Mass ordering of differential elliptic flow and its violation for \phi mesons

Mass ordering of differential elliptic flow and its violation for mesons

Tetsufumi Hirano hirano@phys.s.u-tokyo.ac.jp Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan    Ulrich Heinz Department of Physics, The Ohio State University, Columbus, OH 43210, USA CERN, Physics Department, Theory Division, CH-1211 Geneva 23, Switzerland    Dmitri Kharzeev Nuclear Theory Group, Physics Department, Brookhaven National Laboratory, Upton, NY 11973-5000, USA    Roy Lacey Department of Chemistry, SUNY Stony Brook, Stony Brook, NY 11794-3400, USA    Yasushi Nara Akita International University, 193-2 Okutsubakidai, Yuwa-Tsubakigawa, Akita 010-1211, Japan
July 15, 2019
Abstract

We simulate the dynamics of Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC) with a hybrid model that treats the dense early quark-gluon plasma (QGP) stage macroscopically as an ideal fluid, but models the dilute late hadron resonance gas (HG) microscopically using a hadronic cascade. By comparing with a pure hydrodynamic approach we identify effects of hadronic viscosity on the transverse momentum spectra and differential elliptic flow . We investigate the dynamical origins of the observed mass-ordering of  for identified hadrons, focusing on dissipative effects during the late hadronic stage. We find that, at RHIC energies, much of the finally observed mass-splitting is generated during the hadronic stage, due to build-up of additional radial flow. The meson, having a small interaction cross section, does not fully participate in this additional flow. As a result, it violates the mass-ordering pattern for that is observed for other hadron species. We also show that the early decoupling of the meson from the hadronic rescattering dynamics leads to interesting and unambiguous features in the -dependence of the nuclear suppression factor and of the ratio.

pacs:
25.75.-q, 12.38.Mh, 25.75.Ld, 24.10.Nz

I Introduction

A presently hotly debated question is whether the quark-gluon plasma (QGP) created in Au+Au collisions at the Relativistic Heavy Ion Collider (RHIC) experiments () represents a “perfect liquid” reviews (); HG05 (); Romatschke:2007mq (); Song:2007fn (), i.e. a fluid whose shear viscosity to entropy ratio is at or close to the conjectured Son_visc () lower bound . A key observable in this context is the elliptic flow of hadrons emitted anisotropically in non-central collisions Ollitrault (). At the highest RHIC energy of  GeV, the observed values near midrapidity (), for not too large impact parameters ( fm) and transverse momenta ( GeV/), agree well with predictions from ideal fluid dynamics reviews () (i.e. assuming zero viscosity), including STARv2 (); PHENIXv2 () the predicted dependence of on the transverse momentum and hadron masses HKHRV01 (). Evidence for non-zero shear viscosity in the collision fireball is obtained from deviations from ideal fluid dynamical behavior. This is manifested in the experimental data via a gradual break-down of the ideal fluid description when collisions are studied at larger impact parameters and at lower energies NA49v2sys () or when measurements are made away from midrapidity PHOBOSv2eta (); Hiranov2eta (); HT02 (). In previous work HHKLN () we have shown that a large (and possibly the dominant) fraction of these deviations from ideal hydrodynamics is due to “late viscosity”, caused by dissipative effects in the dilute hadronic rescattering stage that stretches between hadronization of the QGP and final kinetic freeze-out. The question whether there is also non-negligible “early viscosity” in the dense QGP phase remains open. An answer to this question requires a proper viscous hydrodynamic treatment of the QGP fluid which is presently being pursued vigorously Romatschke:2007mq (); Song:2007fn (); visc_hydro (). It also depends on still unknown details of the initial conditions in heavy-ion collisions, in particular the initial spatial eccentricity of the fireball HHKLN (); Drescher:2006pi (); Lappi:2006xc (); Drescher:2006ca ().

In this paper we report additional results from the hybrid model study presented in Ref. HHKLN (), focusing our attention on a detailed investigation of dissipative effects during the late hadronic rescattering stage. The early QGP stage, including its hadronization, is described by ideal fluid dynamics. Specifically, we address the questions of (i) how radial and elliptic flow evolve during the hadronic stage when it is described by a realistic hadronic rescattering cascade, rather than by an ideal fluid; (ii) how these differences affect the shapes of the finally observed hadronic -spectra and their differential elliptic flow ; and (iii) whether the differences between ideal fluid and realistic kinetic behavior during the late hadronic stage are similar for all hadronic species, or whether different magnitudes of their scattering cross sections translate into measurably different characteristics of their observed spectra and elliptic flow.

The paper is organized as follows: For completeness, we begin in Section II with a short review of the hybrid model HHKLN () employed in this study. Our results are presented in Section III, in three subsections organized along the questions raised in the preceding paragraph. We close the paper by presenting our conclusions and some perspectives in Section IV.

Ii The model

Our study is based on a hybrid model which combines an ideal fluid dynamical description of the QGP stage with a realistic kinetic simulation of the hadronic stage HHKLN (). Relativistic hydrodynamics is the most relevant formalism to understand the bulk and transport properties of the QGP since it directly connects the collective flow pattern developed during the QGP stage with its equation of state (EOS). It is based on the key assumption of local thermalization. Since this assumption breaks down during both the very anisotropic initial matter formation stage and the dilute late hadronic rescattering stage, the hydrodynamic model can be applied only during the intermediate period, between initial thermalization after about 1 fm/ reviews () and the completion of the quark-hadron transition which, in central Au+Au collisions at RHIC energies, happens after about 10 fm/.

In absence of a non-equilibrium dynamical model for the early pre-equilibrium stage of the collision, its output is replaced by a set of initial conditions for the hydrodynamic evolution which are tuned to experimental measurements of the final state in central () collisions reviews (). To describe the breakdown of the hydrodynamical model during the late hadronic stage, due to expansion and dilution of the matter, one has two options: One can either impose a sudden transition from thermalized matter to non-interacting, free-streaming hadrons through the Cooper-Frye prescription CF (), imposed at a suitable value of the decoupling temperature or decoupling energy density reviews (), or make a transition from the macroscopic hydrodynamic description to a microscopic kinetic description at a suitable value for the switching temperature where both descriptions are simultaneously valid BassDumitru (); TLS01 (); NonakaBass (); HHKLN (); fn1 (), letting the subsequent kinetic decoupling play itself out automatically by following the microscopic evolution until all interactions have ceased. We here use both approaches alternatively, in order to isolate effects that are specifically caused by dissipative effects in the hadron rescattering cascade.

ii.1 Ideal Hydrodynamics

For the space-time evolution of the perfect QGP fluid we solve numerically the equations of motion of ideal fluid dynamics, for a given initial state, in three spatial dimensions and in time ((3+1)-d ideal hydrodynamics) Hiranov2eta ():

(1)
(2)

Here , , and are energy density, pressure, and four-velocity of the fluid, respectively. Due to its smallness at collider energies, we neglect the net baryon density. As an algorithm to solve the above equations we choose the Piecewise Parabolic Method (PPM) PPM (). It is known to be a very robust scheme for solving non-relativistic gas dynamics including shock wave formation and has been employed in many fields. We first applied it in Hirano:2000eu () to solve Eulerian hydrodynamics for relativistic heavy-ion collisions, Eqs. (1) and (2). Use of this algorithm enables us to describe the space-time evolution of relativistic fluids accurately even if the matter passes through a first-order phase transition. The PPM is a higher order extension of the piecewise linear method employed, for example, in the rHLLE algorithm rhlle (). We solve Eq. (1) in coordinates Hiranov2eta () where ) and are longitudinal proper time and space-time rapidity, respectively, adequate for the description of collisions at ultra-relativistic energies. The grid sizes are  fm/,  fm, and . We have checked the grid size dependence of our final results and observed sufficient convergence with the given choice of grid parameters, as long as smooth initial conditions such as those discussed below are used.

ii.2 Equation of State

For the high temperature ( MeV) QGP phase we use the EOS of massless non-interacting parton gas (, , quarks and gluons) with a bag pressure :

(3)

The bag constant is tuned to  MeV to ensure a first order phase transition to a hadron resonance gas at critical temperature  MeV. The hadron resonance gas model at includes all hadrons up to the mass of the resonance. Systematic studies with various models of the EOS including a more realistic cross-over one will be discussed elsewhere.

For a meaningful discrimination between the ideal fluid and hadron cascade descriptions of the hadron phase, and a realistic direct comparison of hydrodynamic results with experimental data, our hadron resonance gas EOS implements chemical freeze-out at  MeV, as observed in RHIC collisions BMRS01 (). This is achieved by introducing appropriate temperature-dependent chemical potentials for all hadronic species in such a way that their numbers including all decay contributions from higher-lying resonances, , are conserved during the evolution HT02 (); Bebie (); spherio (); Teaney:2002aj (); Kolb:2002ve (); Huovinen:2007xh (). [Here is the number of the -th hadron, and is the effective branching ratio (a product of branching ratio and degeneracy) of a decay process .] In this “PCE model” HT02 () only strongly interacting resonances with large decay widths (whose decays do not alter ) remain chemically equilibrated below the chemical freeze-out temperature.

The hadronic chemical composition described by hydrodynamics using the PCE model EOS is roughly consistent with that of the hadronic cascade models, as long as the latter are initialized at with thermal and chemical equilibrium distributions fn2 (). This is crucial for a meaningful comparison between hydrodynamic and kinetic descriptions of hadronic matter since the chemical composition of the hadron resonance gas has a significant influence on the hydrodynamic evolution of the hadronic transverse momentum spectra HG05 (): While the non-equilibrium hadronic chemical potentials do not affect the EOS of the hadronic phase HT02 (), and thus lead to almost identical evolution of radial flow and total momentum anisotropy as for a chemically equilibrated hadron gas, they significantly alter the relationship between energy density and temperature, leading to cooler temperatures and hence to steeper transverse momentum spectra at identical kinetic decoupling energy densities HT02 (). This effect is seen most dramatically in the time-dependence of the mean transverse momentum for pions HG05 (): decreases with proper time after chemical freeze-out whereas with continued hadronic chemical equilibrium it would increase with time. Clear conclusions about hadronic dissipative effects on the shapes of the transverse momentum spectra can therefore only be drawn from a comparison with hydrodynamic models that implement chemical and kinetic freeze-out separately.

ii.3 Initial Conditions

Contrary to Ref. HHKLN () where we studied both Glauber model and Color Glass Condensate (CGC) type initial conditions, for the comparative study presented here we concentrate on the Glauber model, suitably generalized to account for the longitudinal structure of particle multiplicity HHKLN (); AG05 (). We assume an initial entropy distribution of massless partons according to

(4)

where is the position perpendicular to the beam axis, is a normalization factor, the “soft fraction” is explained below, the parameter is the beam rapidity, and is a suitable parametrization of the shape of rapidity distribution in collisions:

(5)

We study Au+Au collisions at  GeV and use , , and , so chosen as to reproduce the charged hadron pseudorapidity distributions measured in these collisions PHOBOS_dNdeta (). and are the number of wounded nucleons in each of the two nuclei and the number of binary nucleon-nucleon collisions, respectively. These are calculated from the Glauber model nuclear thickness function Kolb:2001qz ():

(6)
(7)
(8)

Here  mb is the inelastic nucleon-nucleon cross section, and (where is the impact parameter).

The soft/hard fraction was adjusted to reproduce the measured centrality dependence PHOBOS_Nch () of the charged hadron multiplicity at midrapidity. At , Eq. (4) reduces to where KH05 (); this parameterization is equivalent to the one used in KLN01 (), , with . From Eq. (4), we can compute the entropy density at the initial time  fm/ reviews () of the hydrodynamic evolution, , which provides the initial energy density and pressure distributions through the tabulated EOS described above.

Glauber model initial conditions have a long tradition for hydrodynamic simulations of heavy-ion collisions. In our previous study HHKLN () we showed that with such initial conditions “late viscosity” effects during the dilute hadronic rescattering stage are sufficient to explain all observed deviations of elliptic flow measurements from ideal fluid dynamical predictions. No significant additional viscous effects during the early QGP stage were necessary. We also noted, however, that this conclusion depends crucially on this particular choice of initial conditions, specifically the initial source eccentricity predicted by the Glauber model. The good agreement between theory and experiment disappears when one instead calculates the initial conditions from the KLN model KLN01 (); Hirano:2004rs (); Kuhlman:2006qp (); Drescher:2006pi (); Lappi:2006xc (); Drescher:2006ca (), which is based on CGC ideas and, for the same impact parameter, produces almost 30% larger source eccentricities. If Nature gives preference to such more eccentric initial conditions, additional viscous effects and/or a softer EOS for the QGP stage may be needed to reproduce the experimental data HHKLN (); Drescher:2007cd (). Here, we will not pursue this line of thought any further, but focus on the case of Glauber model initial conditions and the specific modifications of hadron spectra and flow caused by “late hadronic viscosity”.

ii.4 Hadronic Cascade Model

In our hybrid model simulations we switch from ideal hydrodynamics to a hadronic cascade model at the switching temperature  MeV. The subsequent hadronic rescattering cascade is modeled by JAM jam (), initialized with hadrons distributed according to the hydrodynamic model output, calculated with the Cooper-Frye formula CF () along the  MeV hypersurface rejecting inward-going particles. We have checked HHKLN () that switching from an ideal hydrodynamic to a hybrid model description does not entail a major readjustment of initial conditions: Keeping the same initial conditions and hard/soft fraction as previously determined within a purely hydrodynamic approach (see reviews (); HT02 () for a detailed discussion of that procedure) we find HHKLN () that the centrality dependence of at midrapidity remains consistent with the experimental data even if we switch below to the hadronic cascade. Effects on the hadron spectra and elliptic flow are significant, however, and will be discussed in the next section.

As customary in hadronic cascade models jam (); RQMD (); UrQMD (), JAM implements experimental hadronic scattering cross section data where available and uses the additive quark model where data do not exist, assuming the following formula for the total cross section:

(9)

Here is the total nucleon-nucleon cross section, is the number of constituent quarks in a hadron, and is the number of strange quarks in a hadron. For hadrons composed entirely of strange quarks, such as and , the cross sections become very small, due to the suppression factors in brackets in Eq. (9). Only when we calculate spectra for mesons in Sec. III.3, the decay channels for mesons are switched off in the hadronic cascade calculations. Since the life time of mesons ( fm/) is longer than the typical life time of the system (-20 fm/), and the number of mesons is small compared to pions, kaons, and nucleons, this prescription is not expected to affect the bulk space-time evolution during the hadronic stage.

Iii Results

In Ref. HHKLN () we investigated the effect of hadronic dissipation on elliptic flow and found that it significantly suppresses the -integrated at forward and backward rapidity and in peripheral collisions. In the following we explore the origins of this finding in more detail, by investigating hadronic dissipative effects on hadron spectra and differential elliptic flow . We finally explore specifically the spectra and elliptic flow of mesons as an example of a hadron that is only weakly coupled to the rest of the expanding hadronic fireball.

iii.1 Hadronic dissipative effects on spectra and elliptic flow

In this subsection, we compare results from the hybrid model with the ones from ideal hydrodynamics. In ideal hydrodynamic calculations it is assumed that even the late hadron resonance gas phase is characterized by essentially vanishing mean free paths and thus behaves as a perfect fluid, all the way down to kinetic decoupling of the hadron momenta at  MeV. (This value is obtained by a simultaneous fit of the pion and proton spectra in central collisions which allows to separate the effects of radial flow and thermal motion at kinetic freeze-out reviews ().) As discussed, chemical freeze-out is implemented at  MeV by using an EOS with non-equilibrium chemical potentials which hold the stable particle yields constant (and close to the ones in the cascade model approach) during the hydrodynamic evolution of the hadronic phase. The key difference between the hydrodynamic and hybrid model approaches is, thus, the finite mean free path for momentum-changing collisions in the hadronic cascade.

Figure 1: (Color online) spectra with and without hadronic rescattering for (a) protons and (b) pions at midrapidity for Au+Au collisions at  fm, compared with results from ideal hydrodynamics decoupling at  MeV.

In Fig. 1, -spectra for protons and pions are shown for both, the hybrid model and the ideal hydrodynamic approach. For comparison, we also plot the -spectra without hadronic rescattering, obtained by setting all cross sections to zero in the hadron cascade or by setting  MeV in the hydrodynamic approach (both procedures give the same spectra, by construction). Note here that we include contributions from all resonances (except for weak decays unless explicitly noted otherwise) in ideal hydrodynamic and hybrid-model results. One sees that hadronic rescattering in the JAM cascade pushes the protons to higher in exactly the same way as the growing radial flow does in the hydrodynamic approach, if one chooses for the latter a kinetic decoupling temperature of  MeV. The reasonable fit of the measured proton -spectra phenix:pid200 () up to  GeV/ by the hydrodynamic model reviews (); HT02 (); Kolb:2002ve () thus persists in the hybrid model approach (see Fig. 3 in the following subsection).

The lack of visible dissipative effects on the proton spectra is probably an artifact caused by a judicial choice of the kinetic freeze-out temperature  MeV in the hydrodynamic approach, which was driven by the wish to reproduce the measured proton spectra with this model. This accident does not repeat itself for the pions, shown in Fig. 1(b). For pions, the -spectrum becomes slightly steeper when evolved hydrodynamically (the steepening effects due to cooling are not quite compensated by the increasing radial flow) whereas it gets hardened by hadronic rescattering effects in the hybrid approach.

This pattern is consistent with theoretical expectations: In the ideal fluid approach, work in the longitudinal direction reduces the transverse energy per unit rapidity Gyulassy:1983ub (); Ruuskanen:1984wv (). Since pions dominate the medium but their number is fixed after chemical freeze-out, this leads to a decrease of the average per pion HG05 (), explaining the steeper pion spectrum from ideal hydrodynamics. (This argument is not quantitative since it neglects the shifting balance of transverse energy carried by pions and heavier particles such as protons which are more strongly affected by the developing radial flow HG05 (). Also note that it does not remain true if a chemical equilibrium EOS is used in the hadronic phase where the pion number decreases with temperature and the average transverse energy per pion thus increases HG05 ().) In contrast to the ideal fluid, the hadron gas in the JAM cascade is highly viscous. Shear viscosity is known to reduce the longitudinal and increase the transverse pressure visc_hydro (), reducing the loss of transverse energy due to longitudinal work and increasing the transverse flow due to larger transverse pressure gradients visc_hydro (). In addition, there are viscous corrections to the (flow-boosted) thermal equilibrium form of the distribution function at kinetic freeze-out which lead to an additional viscous distortion of the -spectrum which actually increases with Teaney_visc (). For Bjorken expansion of a homogeneous cylinder this distortion can be written analytically as Teaney_visc ()

(10)

where is the spectrum calculated from a boosted thermal equilibrium distribution along the decoupling surface at freeze-out time and temperature , and the expression in brackets preceding it is the -dependent viscous correction, parametrized by the sound attenuation length (where is the shear viscosity).

The viscous flattening of the pion spectrum relative to the pure hydrodynamic approach seen in Fig. 1(b) receives contributions from both factors in Eq. (10): is flattened by the larger transverse flow generated by the viscously increased transverse pressure, and additional flattening comes from the factor in brackets, due to a non-zero value for in a viscous fluid. We don’t know which of the two effects is larger; we only note that the pion spectrum from the hybrid model can be fitted very well by simply multiplying the hydrodynamic model spectrum with the factor in brackets in Eq. (10), taking  MeV and adjusting . How meaningful such a fit is (given that the form (10) makes unrealistic assumptions about the fireball expansion) remains to be seen when realistic viscous hydrodynamic studies become available.

Figure 2: (Color online) for pions and protons in at fm. Results for pions (solid) and protons (dotted) from ideal hydrodynamics with  MeV are compared with the ones for pions (dashed) and protons (dash-dotted) from the hybrid model.
Figure 3: (Color online) Centrality dependences of the spectra for (a) pions, (b) kaons, and (c) protons obtained from our hydro+cascade hybrid model, compared with data from the PHENIX Collaboration phenix:pid200 () for  GeV Au+Au collisions. Impact parameters are (from top to bottom) , , and  fm, corresponding to the 0-5%, 20-30%, and 30-40% centrality ranges, respectively.

While these considerations provide a qualitative explanation for the harder pion -spectrum from the JAM cascade compared to ideal hydrodynamics, the same arguments should also hold for protons where no such effects are seen in Fig. 1(a). As already stated, this is presumably a consequence of an accidental cancellation of delicate thermal and flow effects with viscous corrections for our specific choice of in the hydrodynamic model. Again, a full understanding of these results may require comparison with a viscous hydrodynamic treatment Romatschke:2007mq (); Song:2007fn ().

Figure 2 shows the dependence of for pions and protons in semi-central Au+Au collisions ( fm) at midrapidity (), comparing results from the hybrid model with ideal hydrodynamics. Whereas, after an initial quadratic rise which extends over a larger -range for the heavier protons than the lighter pions HKHRV01 (), the differential elliptic flow  from ideal hydrodynamics increases almost linearly with , this increase is tempered in the results from the hadronic cascade. The differences between the two models is seen to grow with increasing . Again, this is qualitatively just as expected from shear viscous effects Teaney_visc (); Romatschke:2007mq (); Song:2007fn (). Obviously, the different transport properties of the hadronic matter in JAM and in hydrodynamics are seen more clearly in the differential elliptic flow  than in the spectra.

iii.2 Spectra and elliptic flow for , , and

In this subsection, we compare our results from the hybrid model with experimental data for identified hadrons. In Fig. 3, transverse momentum spectra for pions, kaons, and protons from the hybrid model are compared with PHENIX data phenix:pid200 (), for three impact parameters (centrality classes) as shown in the figure. (The impact parameters are adjusted to give the correct average number of participants for each centrality class, as quoted in phenix:pid200 ().) In all cases, the experimental data are reasonably well reproduced by the hybrid model for low transverse momenta to -2.0 GeV/. Additional components (such as thermal quark recombination and jet fragmentation, including energy loss of fast partons in the fireball medium) would be required to reproduce the data above GeV/. It should be emphasized that, unlike in the purely hydrodynamic approach where the slope is controlled by the choice of kinetic freeze-out temperature and the correct hadron yields are ensured by appropriate choice of non-equilibrium hadron chemical potentials (see Sec. II.2), the hybrid model has no adjustable parameters to reproduce both slope and normalization of the transverse momentum spectra. Hadronic cascade processes automatically describe both chemical and kinetic freeze-out.

In Figure 4, we compare the dependence of for pions, kaons, and protons with the STAR data for star:anisotropy (), for four centrality classes. For the 0-5% centrality class we show only pions since the quality of the kaon and proton data at this centrality is insufficient for a meaningful comparison with theory. The hybrid model correctly describes the mass ordering of the differential elliptic flow, , as seen in the data within the low- region covered by the figure. Quantitatively, it provides a reasonable description up to 50% centrality, except for the most central collisions: Our result for pions at  fm is significantly smaller than the data. This can be attributed to the absence of eccentricity fluctuations in our model calculations Drescher:2006ca (); MS03 ().

Figure 4: (Color online) Transverse momentum dependence of the elliptic flow coefficient for pions (dotted blue), kaons (solid red), and protons (dashed green) from the hybrid model, compared with STAR data for from 200  GeV Au+Au collisions, in four centrality classes star:anisotropy ().

To better understand the origin of the mass ordering in , we compare in Fig. 5, for a selected impact parameter of  fm, the above hybrid model result with a calculation where all hadronic rescattering is turned off, allowing only for decay of the unstable hadron resonances. Whereas just after hadronization the differential elliptic flow for pions and protons looks very similar, the mass splitting gets strongly enhanced by hadronic rescattering. The smallness of the pion-proton mass splitting at is partially accidental, because the splitting caused by the radial flow already established during the hydrodynamic QGP phase HKHRV01 () is significantly decreased by the effect of resonance decays which reduces the pion elliptic flow by about 15% Kolb:1999it (); Hirano:2000eu (). Hadronic evolution below steepens the slope of for pions HT02 (), due to the generation of additional (integrated) and the reduction of their mean transverse momentum HG05 (). (Note that for pions the slope of can be simply approximated as HG05 ().)

Figure 5: (Color online) Transverse momentum dependence of the elliptic flow parameter for pions and protons. Solid (dashed) lines are with (without) hadronic rescattering.

For heavy hadrons, on the other hand, radial flow reduces at low HKHRV01 (). Assuming positive elliptic flow, , the stronger transverse flow in the reaction plane pushes heavy particles to larger more efficiently in the reaction plane than perpendicular to it. In extreme cases HKHRV01 () this can, for heavy particles, even lead to a depletion of low- emission into the reaction plane when compared with out-of-plane emission, i.e. to a negative  at low (even though their -integrated total elliptic flow is positive). But even without going to extremes, this mechanism generically reduces  at low for heavy hadrons. So it is the generation of additional radial flow in the hadronic stage which is responsible for (most of) the mass-splitting of  observed in the low region.

This mechanism works even if the (extra) radial flow is not perfectly hydrodynamic, i.e. if (as is the case in the hadron cascade) the system does not remain fully thermalized, with locally isotropic momentum distributions. Any type of anisotropic collective transverse motion will cause such a mass-splitting of  at low , as long as the hadron in question participates in the flow. It is worth mentioning that in hydrodynamic calculations about half of the final radial flow in Au+Au collisions at RHIC is generated during the hadronic stage (see Fig. 7 in Kolb:2000sd () and Fig. 5 in HT02 ()). A similar increase in radial flow generated by the JAM cascade is documented in Fig. 1(a).

From these observations we conclude that the large magnitude of the integrated and the strong mass ordering of the differential  observed at RHIC result from a subtle interplay between perfect fluid dynamics of the early QGP stage and dissipative dynamics of the late hadronic stage: The large magnitude of is due to the large overall momentum anisotropy, generated predominantly in the early QGP stage, whereas the strong mass-splitting between the slopes of  at low reflects the redistribution of this momentum anisotropy among the different hadron species, driven by the continuing radial acceleration and cooling of the matter during the hadronic rescattering phase.

iii.3 Spectra and elliptic flow for mesons

As noted in Sec. II.4, mesons (consisting of strange quarks) have considerably smaller scattering cross sections in JAM than non-strange hadrons Shor:1984ui (). They are therefore expected to show larger dissipative effects in our hybrid model and to not fully participate in the additional radial flow generated during the hadronic rescattering stage. In kinetic theory language, one expects that the mesons decouple from rest of the system earlier than other, non-strange hadrons vanHecke:1998yu (), thereby possibly opening a window to extract direct information on collective phenomena in the partonic stage from -meson spectra Shor:1984ui ().

To study mesons in our hybrid model we stabilize them by turning off their decay channels during the hadronic cascade.

Figure 6: (Color online) Normalized distribution of freezeout times for pions (dashed), protons (dotted), and mesons (solid) for in Au+Au collisions at  fm.

Let us first check how early mesons decouple from the rest of the system. Figure 6 shows the normalized distribution of freeze-out times for pions, protons, and mesons near midrapidity in central collisions ( fm). Clearly, mesons decouple earlier than pions and protons. The freeze-out time distribution for mesons has a prominent peak at  fm/, roughly equal to the time of completion of QGP hadronization in hydrodynamic simulations. This indicates that only very few rescatterings happen for mesons during the hadronic evolution. Similar results were obtained with the RQMD cascade in vanHecke:1998yu () for baryons at SPS energies and in CLLSX () for mesons and baryons at RHIC energies. The freeze-out time distributions for pions and protons are broadened by both elastic scatterings and resonance decays. The long resonance decay tails of the distributions are important for interpreting the pion source function that was recently reconstructed by the PHENIX Collaboration Adler:2006as () using imaging methods.

Figure 7: (Color online) Transverse momentum spectra for mesons reconstructed from decays in central (blue line), semi-central (red line) and peripheral (green line) Au+Au collisions, compared with PHENIX phenix:phi () and STAR star:phi () data. Results from semi-central and peripheral collisions are divided by 10 and 100, respectively. Predictions from ideal hydrodynamics with  MeV are also shown as dashed lines.

In Figure 7, spectra for mesons from the hybrid model are compared with PHENIX phenix:phi () and STAR star:phi () data. Similar to the spectra for pions, kaons, and protons in Fig. 3, we see good agreement with experiment at low ( GeV/). The discrepancy between our results and experiment at larger may indicate the appearance of a quark-antiquark recombination component in the intermediate region Greco:2003xt (); Fries:2003vb (). In the presence of such a component it is questionable to use the -meson spectra over the whole available region to extract the thermal freeze-out temperature and flow for mesons Schweda:2005sy (); such a thermal model fit Schnedermann:1993ws () should be restricted to the region  GeV/ even if data in that region are hard to obtain.

Figure 8: (Color online) The ratio as a function of (left panel) and of transverse kinetic energy (right panel), for different scenarios: central Au+Au collisions in the hybrid model, without hadronic rescattering, and in the hydrodynamic model with  MeV (dotted). The corresponding ratio for proton-proton collisions (extracted from the PYTHIA fit to the experimental data shown in Fig. 10 below) is shown for comparison as the dashed line. See text for more discussion.

In the hydrodynamic model simulations with MeV, shown as dashed lines in Fig. 7, the mesons pick up more additional radial flow during the hadronic stage, resulting in flatter -spectra than in the hybrid model and in the data in the low region. As we will show further below, better data at lower and a simultaneous analysis of the differential elliptic flow in this region should allow to further discriminate between different descriptions of the hadronic rescattering stage.

The effects of radial flow, and the difference in how additional radial flow generated during the hadronic rescattering stage is picked up by protons and mesons (which have rather similar masses), can be enhanced by studying the or transverse kinetic energy dependence of the ratio. A thermalized medium without radial flow features -scaling, i.e. all -spectra have identical slopes, and for such a static fireball the ratio, when plotted as a function of transverse kinetic energy , would be a constant horizontal line. For a thermalized expanding medium, -scaling is broken by radial flow (which couples differently to particles with different masses), resulting in a non-zero slope of the ratio . Perhaps somewhat counterintuitively, this slope of the ratio does not grow monotonically with the radial flow but, after an initial rise, decreases again when the flow becomes so large that the hadron -spectra become very flat; in the limit of “infinite flow” (i.e. ) the hadron -spectra, and thus their ratios, become again perfectly flat.

In Figure 8 we show the ratio, both as a function of transverse kinetic energy (right panel) and of (left panel). It should be noted here that weak decay contribution is not included in proton yields. In the latter case the connection to radial flow is less straightforward, since the kinematics of the transformation from to depends on mass and introduces additional growth with for the ratio. In both representations one sees, however, by comparing the curves for the hydro+cascade model without rescattering (corresponding to ideal hydrodynamics with  MeV) and for the ideal hydrodynamic model with  MeV, that (i) the ratio increases with or due to radial flow effects, and that (ii) the rate of increase drops when the freeze-out temperature is decreased, due to build-up of additional radial flow. Surprisingly, the ratio increases even in collisions, but for entirely different reasons, unrelated to collective flow: The spectrum from collisions shown in Fig. 10 below is considerably flatter than the proton spectrum, leading to the prominent rise of the ratio with . The most interesting feature of Fig. 8 is that the ratio from the hybrid model does not at all increase with or (except at very low  MeV/). Instead, it decreases over almost the entire range of transverse kinetic energy shown in the figure. This decrease is due to the flattening of the proton spectrum by hadronically generated radial flow in which the weakly coupled mesons do not participate. The comparison with collisions and hydrodynamic model simulations in Fig. 8 shows that the observation of such a decreasing ratio would be an unambiguous signature for early decoupling of mesons from the hadronic rescattering dynamics.

Figure 9: (Color online) Transverse momentum dependence of the elliptic flow parameters for pions (dotted blue), protons (dashed green), and mesons (solid red), for Au+Au collisions at  fm. (a) Before hadronic rescattering. (b) After hadronic rescattering. (c) Ideal hydrodynamics with  MeV. The results for pions and protons are the same as shown in Fig. 5.

We now proceed to the discussion of dissipative effects during the hadronic rescattering stage on the differential elliptic flow . Figure 9 shows  from the hybrid model for , , and . We consider semi-central collisions (20-30% centrality), choosing impact parameter  fm. In the absence of hadronic rescattering we observe the hydrodynamically expected mass ordering (Fig. 9(a)), but just as in Fig. 5 (dashed lines) the mass splitting is small. Figure 9(b) shows the effects of hadronic rescattering: while the  curves for pions and protons separate as discussed before (at low the pion curve moves up while the proton curve moves down),  for the meson remains almost unchanged Chen:2006ub (). As a result of rescattering the proton elliptic flow ends up being smaller than that of the meson, for  GeV/, even though . Hadronic dissipative effects are seen to be particle specific, depending on their scattering cross sections which couple them to the medium. The large cross section difference between the protons and mesons in the hadronic rescattering phase leads to a violation of the hydrodynamic mass ordering at low in the final state.

This is the most important new result of our work. Current experimental data Afanasiev:2007tv (); Abelev:2007rw () neither confirm nor contradict this predicted behavior, due to the difficulty of reconstructing low- mesons from their decay products. If it turns out that high precision -meson data at low show violation of mass ordering, it will be evidence for strong momentum anisotropy having developed already during the QGP stage, with the contribution carried by mesons not being redistributed in by late hadronic rescattering. At intermediate , recent data Afanasiev:2007tv (); Abelev:2007rw () confirm the prediction from the quark coalescence model Molnar:2003ff (); Nonaka:2003hx () that there the elliptic flow should scale with the number of constituent quarks: , in spite of the similar and masses which are much larger than those of the pions and mesons. We hope that the present paper motivates an effort to extend these data to lower in order to test our prediction here that, at low , in spite of . While the former observation suggests that at intermediate ( GeV/) quark coalescence during the quark-hadron phase transition controls the finally observed elliptic flow of all hadrons, without measurable distortion by subsequent hadronic reinteractions, confirmation of our prediction would confirm the importance of hadronic rescattering on low- hadrons, with results that depend on the magnitude of the scattering cross sections of the various hadron species.

We close this section with a discussion of the implications of our hybrid model results for the nuclear modification factor

(11)

The observed suppression of pion yields at intermediate to high experiments () provides evidence of jet quenching in relativistic heavy-ion collisions. For baryons, this suppression effect is counteracted in the intermediate region by collective flow effects which, at low , lead to a rise of the (or, more generally, heavy/light) ratio as a function of . Collective flow effects extend into the intermediate region  GeV/ even though the hydrodynamic picture

Figure 10: (Color online) Invariant cross sections as a function of in non-singly diffractive collisions for pions, protons, star:pp_pikp () and mesons. Dotted, dashed, and solid lines are results from PYTHIA for pions , protons, and mesons, respectively.
Figure 11: (Color online) Nuclear modification factors for pions (blue), mesons (red), and protons/antiprotons (green), for Au+Au collisions at fm (corresponding to 0-10% centrality). Shown are predictions from (a) the hybrid model and (b) from ideal hydrodynamics with  MeV.

is known to gradually break down above  GeV/ Heinz:2004ar (). Quark coalescence is one of the key mechanisms by which low- collectivity on the quark-gluon level is transferred to the hadron spectra at intermediate during the hadronization process Greco:2003xt (); Fries:2003vb (); Molnar:2003ff (), leading to (unsuppressed) values of (or of , the ratio of yields per number of binary collisions in central and peripheral collisions) of order unity for baryons at -3 GeV/ experiments (); Abelev:2007rw (). We will show that hadronic rescattering following QGP hadronization affects at low instead.

The PHENIX phenix:phi () and STAR Abelev:2007rw () Collaborations have recently measured for mesons. The PHENIX data show a suppression of mesons by about a factor 2 (with relatively large error bars) in the region  GeV/, consistent with that of pions, while protons and antiprotons are unsuppressed phenix:phi (). This seems to be in contradiction with collective flow arguments which predict since , but consistent with the valence quark scaling predicted by the quark coalescence model Greco:2003xt (); Fries:2003vb (); Molnar:2003ff (). The more recent and precise STAR data Abelev:2007rw (), on the other hand, show an for mesons that follows the one for pions and exceeds the one for protons for  GeV/, but then follows the rise of the proton above the pion one for  GeV/, lagging only slightly behind the protons and reaching a value halfway between pions and protons in the region  GeV/ where peaks at a value of .

Given this somewhat contradictory experimental situation, we offer a prediction from our hybrid model (cautioning beforehand that this model does not include any quark-recombination contributions which are expected to become important above  GeV/) in Figure 11. To construct this Figure, we first fitted the experimentally measured -spectra for pions and protons star:pp_pikp () as well as for mesons star:pp_phi () in non-singly diffractive (NSD) collisions (i.e. inelastic collisions excluding single diffractive events). The fit, shown in Fig. 10, is performed with the help of the event generator PYTHIA 6.403 Sjostrand () which, once properly tuned, yields smooth reference -spectra for collisions. PYTHIA is based on leading order perturbative QCD for semi-hard processes combined with a Lund string fragmentation scheme for soft particles. It works quite well for pions, protons and mesons with default parameters Sjostrand (), except for a necessary readjustment of the factor to . We note that here exceptionally this comparison includes all resonance decays including weak ones since the STAR data show the inclusive spectra. We take the resulting spectra as our reference, after removing weak decay contributions and multiplying them with the ratio to correct for the NSD trigger. For the required cross sections PYTHIA provides the estimates mb and mb.

With these reference spectra the nuclear modification factors can now be calculated from the results shown in Figs. 1, 3 and 7. For pions, protons, and mesons they are shown as functions of in Fig. 11, for Au+Au collisions at impact parameter  fm (i.e. 0-10% centrality). Figure 11(a) shows the predictions for the hybrid model. While for pions is almost flat, , the curves for protons and mesons increase with as expected from radial flow arguments (radial flow hardens the spectra for heavy particles). The rate of increase for the mesons is very similar to that for protons, culminating in a peak value of at  GeV/ for ’s whereas the for protons peaks at a value of near  GeV/. Figure 11(b) shows the corresponding curves for the ideal fluid dynamical simulation with  MeV. For pions and protons, the differences to the hybrid model are minor (at least in the range covered in the Figure), reiterating the observation made in connection with Fig. 1 that the buildup of additional radial flow during the hadronic stage is similar in both models and viscous effects become clearly recognizable only at larger . For mesons one observes a much faster rise of in the hydrodynamic approach, resulting in a larger peak value of at a larger value ( GeV/) than for the hybrid model. The reason for these effects is obviously the larger amount of radial flow picked up during the hadronic stage in the hydrodynamic model and the resulting hardening of the spectrum. The much weaker rise of in the hybrid model can thus be traced directly to the lack of meson rescattering during the hadronic stage.

We note that, even in the hydrodynamic model, the nuclear modification factor doesn’t show a monotonic mass-ordering at low . Naive expectations based on the mass-ordering of the spectral slopes (which reflect radial flow effects) are invalidated by the fact that the -spectra from collisions are flatter than the corresponding proton spectra. Since these spectra enter the denominator of , they distort its dependence differently for protons and mesons.

We also comment that at  GeV/, the characteristics of the observed mass-scaling violation in Fig. 11(a) are qualitatively similar to those expected (and observed) in the quark coalescence picture at intermediate ( GeV/) Nonaka:2003hx (). The differences are quantitative: our prediction for features neither a monotonic mass-ordering at low nor the strict valence quark scaling predicted by the quark-coalescence picture at intermediate .

Iv Conclusions

We have studied effects of hadronic dissipation on the spectra, differential elliptic flow, and nuclear modification factor of pions, kaons, protons, and mesons from Au+Au collisions at RHIC, using a hybrid model which treats the early QGP phase macroscopically as a perfect fluid and the late hadronic phase microscopically with a hadronic cascade. For transverse momenta below  GeV/ and not too peripheral collisions, the hybrid model gives a reasonable description of the measured pion, kaon, proton and meson -spectra. In peripheral collisions ( fm and larger) the model spectra tend to be somewhat steeper than measured. The centrality dependence of the differential elliptic flow  of pions, kaons and protons is better described by the hybrid model than in a purely hydrodynamic approach.

For pions, kaons, and protons, which have relatively large scattering cross sections, hadronic rescattering is seen to generate additional collective transverse flow, but not so for the much more weakly interacting mesons. However, even for pions and protons the extra hadronic transverse flow effects are not “ideal” but exhibit obvious viscous features: Their -spectra are hardened while the growth of their elliptic flow  with increasing is tempered by viscous corrections whose importance is in both cases observed to increase with transverse momentum. The well-known mass-splitting of the differential elliptic flow  observed in hydrodynamic models is seen to be mostly generated during the hadronic rescattering phase and to be largely due to a redistribution of the momentum anisotropy built up during the QGP stage. This redistribution is caused by the mass-dependent flattening of the transverse momentum spectra by additional radial flow generated during the hadronic stage. The much more weakly interacting mesons do not participate in this additional radial flow and thus are not affected by this redistribution of momentum anisotropies: their differential elliptic flow remains almost unaffected by hadronic rescattering. The net result of dissipative hadronic rescattering is therefore that the differential elliptic flow  of protons drops below that of the mesons, in violation of the hydrodynamic mass-ordering. A similar violation of the mass-ordering is seen in the nuclear modification factor at GeV/ where, after hadronic rescattering, the curve for mesons ends up between those for pions and protons even though the is heavier than both of them. For the ratio, the lack of interaction between the mesons and its accelerating hadronic environment should manifest itself in an unexpected but unambiguous decrease with increasing transverse kinetic energy.

The results presented here underscore the conclusion of Ref. HHKLN () that hadronic dissipation may be very important at RHIC and at lower beam energies and should be properly accounted for in attempts to quantitatively account for the experimental data collected from heavy-ion collisions. With  and for low- mesons and the dependence of the ratio on or transverse kinetic energy , we have identified three additional critical observables which should be helpful in sorting out the interplay between hydrodynamic evolution during the early QGP stage and dissipative hadronic expansion during the late stage of the hot and dense fireballs created in these collisions. An accurate extraction of the value for the specific shear viscosity of the QGP created at RHIC requires a proper accounting for effects from late hadronic viscosity. Here, an attempt has been made to do this, by coupling the hydrodynamic model to a hadronic cascade.

Acknowledgements.
This work was supported by the U.S. DOE under contracts DE-FG02-01ER41190 (U.H.), DE-AC02-98CH10886 (D.K.) and DE-FG02-87ER40331.A008 (R.L.). The work of T.H. was partly supported by Grant-in-Aid for Scientific Research No. 19740130.

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