arXiv:nucl-th/0412022v1 7 Dec 2004
Nonequilibrium Models of Relativistic Heavy-Ion
H. St¨ ocker1,2, E. L. Bratkovskaya1, M. Bleicher1, S. Soff1, and
1Institut f¨ ur Theoretische Physik, Johann Wolfgang Goethe – Universit¨ at, Robert
Mayer Str. 8-10, 60054 Frankfurt am Main, Germany
2Frankfurt Institute for Advanced Studies (FIAS), Robert Mayer Str. 8-10, 60054
Frankfurt am Main, Germany
3Physics Department, Tsinghua University, Beijing 100084, China
models on the collective flow observables from AGS to RHIC energies. A critical
discussion of the present status of the CERN experiments on hadron collective flow
is given. We emphasize the importance of the flow excitation function from 1 to 50
A·GeV: here the hydrodynamic model has predicted the collapse of the v1-flow and
of the v2-flow at ∼ 10 A·GeV; at 40 A·GeV it has been recently observed by the
NA49 collaboration. Since hadronic rescattering models predict much larger flow than
observed at this energy we interpret this observation as evidence for a first order phase
transition at high baryon density ρB. Moreover, the connection of the elliptic flow v2to
jet suppression is examined. It is proven experimentally that the collective flow is not
faked by minijet fragmentation. Additionally, detailed transport studies show that the
away-side jet suppression can only partially (< 50%) be due to hadronic rescattering.
Furthermore, the change in sign of v1,v2 closer to beam rapidity is related to the
occurence of a high density first order phase transition in the RHIC data at 62.5, 130
and 200 A·GeV.
We review the results from the various hydrodynamical and transport
PACS numbers: 25.75.-q, 25.75.Ld
1. Introduction: Old and new observables for the QGP phase transition
Lattice QCD results [1, 2] show a crossing, but no first order phase transition to the QGP
for vanishing or small chemical potentials µB, i.e. at the conditions accessible at central
rapidities at RHIC full energies. A first order phase transition does occur according to
the QCD lattice calculations [1, 2] only at high baryochemical potentials or densities, i.e.
at SIS-300 and lower SPS energies and in the fragmentation region of RHIC, y ≈ 4 − 5
[3, 4]. The critical baryochemical potential is predicted [1, 2] to be µc
and the critical temperature Tc≈ 150−160 MeV. We do expect a phase transition also at
finite strangeness. Predictions for the phase diagram of strongly interacting matter for
realistic non-vanishing net strangeness are urgently needed to obtain a comprehensive
picture of the QCD phase structure. Multi-Strangeness degrees of freedom are very
B≈ 400±50 MeV
Nonequilibrium Models of Relativistic Heavy-Ion Collisions2
0 200 400600800 10001200
[Karsch et al.]
0 2 3 = y (RHIC)
[Cleymans et al.]
Au+Au, 11 A GeV
Pb+Pb, 40 A GeV
Pb+Pb, 160 A GeV
Au+Au, 21300 A GeV
µ µB [MeV]
with the critical end point at µB
400 MeV,T ≈ 160 MeV as predicted by
Lattice QCD . In addition, the time
evolution in the T−µ-plane of a central cell
in UrQMD calculations  is depicted for
different bombarding energies. Note, that
the calculations indicate that bombarding
probe a first order phase transition.
RHIC (see insert at the µBscale) this point
is accessible in the fragmentation region
only (taken from ).
The new phase diagram
<∼ 40 A·GeV are needed to
promising probes for the properties of the dense and hot matter . The strangeness
distillation process [6, 7] predicts dynamical de-admixture of s and ¯ s quarks, which yields
unique signatures for QGP creation: high multistrange hyperon-/-matter production,
strangelet formation and unusual antibaryon to baryon ratios ect.
A comparison of the thermodynamic parameters T and µB extracted from the
UrQMD-transport model in the central overlap regime of Au+Au collisions  with
the QCD predictions is shown in Fig 1, where the full dots with errorbars denote the
’experimental’ chemical freeze-out parameters – determined from fits to the experimental
yields – taken from Ref. . The triangular and quadratic symbols (time-ordered in
vertical sequence) stand for temperatures T and chemical potentials µBextracted from
UrQMD transport calculations in central Au+Au (Pb+Pb) collisions at RHIC (21.3
A·TeV), 160, 40 and 11 A·GeV  as a function of the reaction time (separated by 1
fm/c steps from top to bottom). The open symbols denote nonequilibrium configurations
and correspond to T parameters extracted from the transverse momentum distributions,
whereas the full symbols denote configurations in approximate pressure equilibrium in
longitudinal and transverse direction.
During the nonequilibrium phase (open symbols) the transport calculations show
much higher temperatures (or energy densities) than the ’experimental’ chemical freeze-
out configurations at all bombarding energies (≥ 11 A·GeV). These numbers are also
higher than the critical point (circle) of (2+1) flavor - Lattice QCD calculations by the
Bielefeld-Swansea-collaboration  (large open circle) and by the Wuppertal-Budapest-
collaboration . The energy density at µc,Tcis in the order of ≈ 1 GeV/fm3(or slightly
below). At RHIC energies a cross-over is expected at midrapidity, when stepping down
in temperature during the expansion phase of the ’hot fireball’. The baryon chemical
potential µBfor different rapidity intervals at RHIC energies has been obtained from a
statistical model analysis by the BRAHMS Collaboration based on measured antihadron
to hadron yield ratios . For midrapidity one finds µB ≃ 0, whereas for forward
rapidities µB increases up to µB ≃ 130 MeV at y = 3. Thus, only extended forward
rapidity measurement (y ≈ 4 − 5) will allow to probe large µBat RHIC. The detectors
Nonequilibrium Models of Relativistic Heavy-Ion Collisions3
at RHIC at present offer only a limited chemical potential range. This situation changes
at lower SPS (and top AGS) as well as at the future GSI SIS-300 energies: sufficiently
large chemical potentials µBshould allow for a first order phase transition  (to the
right of the critical point in the (T,µB) plane). The transport calculations show high
temperatures (high energy densities) in the very early phase of the collisions, only. Here,
hadronic interactions are weak due to formation time effects and yield little pressure.
Diquark, quark and gluon interactions should cure this problem.
2. Directed and elliptic flow
2.1. General considiration
Hydrodynamic flow and shock formation has been proposed early [13, 14] as the
key mechanism for the creation of hot and dense matter during relativistic heavy-
ion collisions. The full three-dimensional hydrodynamical flow problem is much
more complicated than the one-dimensional Landau model : the 3-dimensional
compression and expansion dynamics yields complex triple differential cross-sections,
which provide quite accurate spectroscopic handles on the equation of state.
bounce-off, the squeeze-out and the antiflow [16, 17, 18, 19, 20] (third flow component
[21, 22]) serve as differential barometers for the properties of compressed, dense matter
from SIS to RHIC. Presently, the most employed flow observables are :
Here, pxdenotes the momentum in x-direction, i.e. the transversal momentum within
the reaction plane and pythe transversal momentum out of the reaction plane. The total
transverse momentum is given as pT =
Thus, v1measures the ”bounce-off”, i.e. the strength of the directed flow in the reaction
plane, and v2gives the strength of the second moment of the azimuthal particle emission
distribution, i.e. ”squeeze-out” for v2< 0 [13, 14, 16, 17, 18, 19, 20]. In particular, it
has been shown [14, 16, 17, 18, 19, 20] that the disappearence or ”collapse” of flow is a
direct result of a first order phase transition.
Several hydrodynamic models have been used in the past, starting with the one-
fluid ideal hydrodynamic approach. It is well known that the latter model predicts
far too large flow effects. To obtain a better description of the dynamics, viscous
fluid models have been developed [24, 25, 26]. In parallel, so-called three-fluid models,
which distinguish between projectile, target and the fireball fluids, have been considered
[29, 30]. Here viscosity effects appear only between the different fluids, but not inside
the individual fluids. The aim is to have at our disposal a reliable, three-dimensional,
relativistic three-fluid model including viscosity [25, 26].
Flow can be described very elegantly in hydrodynamics (cf. Refs. [31, 32, 33, 34])
by a proper choice of initial conditions which have very strong influence on the final
results. In this respect, it is important to consider also microscopic multicomponent
y; the z-axis is in the beam direction.
Nonequilibrium Models of Relativistic Heavy-Ion Collisions4
(pre-) hadron transport theory, e.g. models like qMD , IQMD , RQMD ,
UrQMD  or HSD , as control models for viscous hydro and as background models
to subtract interesting non-hadronic effects from data. If Hydro with and without quark
matter EoS, hadronic transport models without quark matter – but with strings – are
compared to data, can we learn whether quark matter has been formed? What degree
of equilibration has been reached? What does the equation of state look like? How are
the particle properties, self energies, cross sections changed?
To estimate systematic model uncertainties, the results of the different microscopic
transport models also have to be carefully compared. The two robust hadron/string
based models, HSD and UrQMD, are considered in the following.
2.2. Review of AGS and SPS results
Microscopic (pre-)hadronic transport models describe the formation and distributions
of many hadronic particles at AGS and SPS rather well . Furthermore, the nuclear
equation of state has been extracted by comparing to flow data which are described
reasonably well up to AGS energies [41, 42, 43, 21, 44, 45]. Ideal hydro calculations, on
the other hand, predict far too much flow at these energies . Thus, viscosity effects
have to be taken into account in hydrodynamics.
In particular, ideal hydro calculations are factors of two higher than the measured
sideward flow at SIS  and AGS, while the directed flow px/m measurement of the
E895 collaboration shows that the p and Λ data are reproduced reasonably well  in
UrQMD, i.e. in a hadronic transport theory with reasonable cross-sections, i.e. realistic
mean-free-path of the constituents.
Only ideal hydro calculations predict, however, the appearance of a so-called ”third
flow component”  or ”antiflow”  in central collisions. We stress that this only
holds if the matter undergoes a first order phase transition to the QGP. The signal is
that around midrapidity the directed flow, px(y), of protons develops a negative slope!
In contrast, a hadronic EoS without QGP phase transition does not yield such an exotic
”antiflow” (negative slope) wiggle in the proton flow v1(y).
The ideal hydrodynamic directed proton flow px (Fig. 2) shows even negative
values between 8 and 20 A·GeV. An increase back to positive flow is predicted with
increasing energy, when the compressed QGP phase is probed.
predicted minimum of the proton flow in the data? The hydro calculations suggest
that this ”softest point collapse” is at ELab≈ 8 A·GeV. This has not been verified by
the AGS data! However, a linear extrapolation of the AGS data indicates a collapse of
the directed proton flow at ELab≈ 30 A·GeV (Fig. 2).
Recently, substantial support for this prediction has been obtained by the low
energy 40 A·GeV SPS data of the NA49 collaboration  (Fig. 3). These data clearly
show the first proton ”antiflow” around mid-rapidity, in contrast to the AGS data as
well as to the UrQMD and HSD calculations involving no phase transition (Fig. 3,
l.h.s.). Thus, at bombarding energies of 30-40 A·GeV, a first order phase transition to
But, where is the
Nonequilibrium Models of Relativistic Heavy-Ion Collisions5
Figure 2. Left: Measured SIS and AGS proton dpx/dy-slope data compared to a
three-fluid hydro calculation . A linear extrapolation of the AGS data indicates a
collapse of flow at ELab≈ 30 A·GeV, i.e. for the lowest SPS- and the upper FAIR-
energies at GSI. Right: Net-baryon density in momentum space for Pb+Pb at 8 A·GeV
for b=3 fm at time 8.4 fm/c calculated in three-fluid hydro  for condition s/ρ < 10.
the baryon rich QGP most likely is already observed; the first order phase transition
line is crossed (cf. Fig. 1). This is the energy region where the new FAIR- facility at
GSI will operate. There are good prospects that the baryon flow collapses and other
first order QGP phase transition signals can be studied at the lowest SPS energies as
well as at the RHIC fragmentation region y > 4 − 5. These experiments will enable a
detailed study of the first order phase transition at high µBand of the properties of the
baryon rich QGP.
3. Proton elliptic flow collapse at 40 A·GeV - evidence for a first order
phase transition at highest net baryon densities
At SIS energies microscopic transport models reproduce the data on the excitation
function of the proton elliptic flow v2quite well: A soft, momentum-dependent equation
of state [50, 51, 52] seems to account for the data. The observed proton flow v2below
∼ 5 A·GeV is smaller than zero, which corresponds to the squeeze-out predicted by
hydrodynamics long ago [13, 14, 16, 17, 18, 19, 20]. The AGS data exhibit a transition
from squeeze-out to in-plane flow in the midrapidity region. The change in sign of the
proton v2 at 4-5 A·GeV is in accord with transport calculations – UrQMD  and
HSD [44, 45]). At higher energies, 10-160 A·GeV, a smooth increase of the flow v2is
predicted from the string-hadronic transport models. In fact, the 158 A·GeV data of
the NA49 Collaboration suggest that this smooth increase proceeds between AGS and
SPS as predicted.
This is in strong contrast to recent NA49 data at 40 A·GeV (cf. Fig. 3, r.h.s.):
A sudden collapse of the proton flow v2is observed for central, midcentral as well as
for peripheral protons. This collapse of v2for protons around midrapidity at 40 A·GeV