Transverse-mass spectra in Au+Au and Pb+Pb collisions in incident energy
range from 2A to 160A GeV are analyzed within the model of 3-fluid dynamics. It
is shown that dynamical description of freeze-out, accepted in this model,
naturally explains the incident energy behavior of inverse-slope parameters of
these spectra observed in experiment. Simultaneous reproduction of the
inverse-slopes of all considered particles (protons, pions and kaons) suggests
that these particles belong to the same hydrodynamic flow at the instant of
their freeze-out.Comment: 4 pages, 3 figure

Ivanov, Yu. B.

Russkikh, V. N.

English

arXiv

Yu. B. Ivanova

V. N. Russkikh

Springer - Publisher Connector

DOI 10.1140/epja/i2008-10634-2
Letter
Eur. Phys. J. A 37, 139–142 (2008) THE EUROPEAN
PHYSICAL JOURNAL A
Letter
Transverse-mass eﬀective temperature in heavy-ion collisions
from AGS to SPS
Yu.B. Ivanova and V.N. Russkikh
Gesellschaft fu¨r Schwerionenforschung mbH, Planckstr. 1, D-64291 Darmstadt, Germany and
Kurchatov Institute, Kurchatov sq. 1, Moscow 123182, Russia
Received: 14 April 2008 / Revised: 16 June 2008
Published online: 5 August 2008 – c© Societa` Italiana di Fisica / Springer-Verlag 2008
Communicated by J. Wambach
Abstract. Transverse-mass spectra in Au + Au and Pb + Pb collisions in the incident energy range from
2A to 160A GeV are analyzed within the model of 3-ﬂuid dynamics (3FD). It is shown that the dynamical
description of freeze-out, accepted in this model, naturally explains the incident-energy behavior of the
inverse-slope parameters of these spectra observed in experiment. The simultaneous reproduction of the
inverse slopes of all considered particles (p, π and K) suggests that these particles belong to the same
hydrodynamic ﬂow at the instant of their freeze-out.
PACS. 24.10.Nz Hydrodynamic models – 25.75.-q Relativistic heavy-ion collisions
Experimental data on transverse-mass spectra of kaons
produced in central Au + Au [1] or Pb + Pb [2] collisions
reveal peculiar dependence on the incident energy. The
inverse-slope parameter (the so-called eﬀective tempera-
ture T ) of these spectra at mid rapidity increases with inci-
dent energy in the energy domain of BNL Alternating Gra-
dient Synchrotron (AGS) and then saturates at the ener-
gies of CERN Super Proton Synchrotron (SPS). In refs. [3,
4] it was assumed that this saturation is associated with
the deconﬁnement phase transition. This assumption was
indirectly conﬁrmed by the fact that microscopic trans-
port models, based on hadronic degrees of freedom, failed
to reproduce the observed behavior of the kaon inverse
slope [5,6]. Hydrodynamic simulations of ref. [7] succeeded
to describe this behavior. However, in order to reproduce it
these hydrodynamic simulations required incident-energy
dependence of the freeze-out temperature which almost
repeated the shape of the corresponding kaon eﬀective
temperature. This happened even in spite of using the
equation of state (EoS) involving the phase transition into
quark-gluon plasma (QGP). This way, the puzzle of kaon
eﬀective temperatures was just translated into a puzzle of
freeze-out temperatures. Moreover, the results of ref. [7]
imply that the peculiar incident-energy dependence of the
kaon eﬀective temperature may be associated with the dy-
namics of freeze-out.
a e-mail: Y.Ivanov@gsi.de
In this paper we would like to present calculations of
eﬀective temperatures within the 3FD model [8–11] which
is suitable for simulating heavy-ion collisions in the range
from AGS to SPS energies. We perform our simulations [8,
10,11] with a simple, hadronic EoS [12], which involves
only a density-dependent mean ﬁeld providing saturation
of cold nuclear matter at normal nuclear density and with
the proper binding energy. The 3FD model with the in-
termediate EoS turned out to be able to reasonably re-
produce a great body of experimental data [8] in a wide
energy range from AGS to SPS. In particular, transverse-
mass spectra of protons were reproduced. This was done
with the unique set of model parameters summarized in
ref. [8]. Problems were met only in the description of the
transverse ﬂow [10]. The directed ﬂow requires a softer
EoS at top AGS and SPS energies (in particular, this
desired softening may signal the occurrence of the phase
transition into QGP). A similar softening is needed for the
reproduction of recent data on the rapidity distributions
of the net-baryon number in central Pb + Pb collisions at
energies 20A–80A GeV [11].
The transverse-mass spectra are most sensitive to the
freeze-out parameters of the model. In fact, the inverse
slopes of these spectra represent a combined eﬀect of the
temperature and collective transverse ﬂow of expansion.
Figure 1 demonstrates this important interplay. Had it
been only the eﬀect of thermal excitation, inverse slopes
for diﬀerent hadronic species would approximately equal.
140 The European Physical Journal A
0.1
0.15
0.2
0.25
0.3
T
 [
G
eV
]
K+ K-
0 4 8 12 16
s1/2 [GeV]
0.1
0.15
0.2
0.25
0.3
p
0 4 8 12 16 20
E866, E917 (AGS)
NA49 (SPS)
p
Fig. 1. Inverse-slope parameters of transverse-mass spectra of
kaons, pions and protons at mid rapidity produced in central
Au + Au and Pb + Pb collisions as a function of the invariant
incident energy. The solid lines correspond to the purely expo-
nential ﬁt (λ = 0, see eq. (1)), while the dashed lines present
results with λ = −1 for pions and with λ = 1 for protons.
Experimental data are from refs. [1,2,13,14].
The collective transverse ﬂow makes them diﬀerent. These
two eﬀects partially compensate each other: the later
the freeze-out occurs, the lower the temperature and the
stronger the collective ﬂow are. Nevertheless, transverse-
mass spectra turn out to be sensitive to the instant of the
freeze-out.
3FD results for inverse-slope parameters of transverse-
mass spectra of kaons, pions and protons produced in cen-
tral Au + Au and Pb + Pb collisions are presented in ﬁg. 1.
The inverse slopes T were deduced by ﬁtting the calcu-
lated spectra by the formula
d2N
mT dmT dy
∝ (mT )
λ
exp
(
−
mT
T
)
, (1)
where mT and y are the transverse mass and rapidity, re-
spectively. Though the purely exponential ﬁt with λ = 0
does not always provide the best ﬁt of the spectra, it al-
lows a systematic way of comparing spectra at diﬀerent
incident energies1. In order to comply with experimental
ﬁts at AGS energies (and hence with displayed experi-
mental points), we also present results with λ = −1 for
pions2 [1] and with λ = 1 for protons [14]. These results
1 Accordingly to the experimental ﬁt of proton spectra at
SPS energies [13], we ﬁtted the calculated spectra by func-
tion (1) with λ = 0 only at mT − mN > 0.2 GeV/c
2, where
mN is the nucleon mass, and did not care of low-mT parts of
these spectra.
2 In fact, λ was treated as a ﬁt parameter in ref. [1]. However,
since the resulting values of the λ parameter turned out to be
are obtained with precisely the same EoS and set of pa-
rameters (friction, freeze-out and formation time) as those
used in ref. [8], which was found to be the best for other ob-
servables. No special tuning was done to reproduce these
eﬀective temperatures.
Numerical problems, discussed in ref. [8], prevented us
from simulations at RHIC energies. Already for the central
Pb + Pb collision at the top SPS energy the code requires
7.5 GB of (RAM) memory. At the top RHIC energy, the
required memory is three order of magnitude higher, which
is unavailable in modern computers.
As seen from ﬁg. 1, the reproduction of eﬀective tem-
peratures is quite reasonable. Moreover, the pion and pro-
ton eﬀective temperatures also reveal saturation at SPS
energies, if they are deduced from the purely exponential
ﬁt with λ = 0. It is important that it is achieved with
a single freeze-out parameter εfrz = 0.4 GeV/fm
3, the
critical freeze-out energy density, which is the same for
all considered incident energies above 2A GeV, both for
chemical and thermal freeze-out. Only for smaller ener-
gies we used smaller values: εfrz(2A GeV) = 0.3 GeV/fm
3
and εfrz(1A GeV) = 0.2 GeV/fm
3. In order to clarify why
this happens, let us turn to the 3FD freeze-out procedure,
which is analyzed in ref. [15] in more detail.
The freeze-out criterion we use is
ε < εfrz, (2)
where ε = uμT
μνuν is the total energy density of all three
ﬂuids in the proper reference frame, where the composed
matter is at rest. This total energy density is deﬁned in
terms of the total energy momentum tensor Tμν ≡ Tμνp +
Tμνt +T
μν
f being the sum of energy momentum tensors T
μν
α
of separate ﬂuids (projectile-like, target-like and ﬁreball
ones) and the total collective 4-velocity of the matter uμ =
uνT
μν/uλT
λκuκ. Note that the latter deﬁnition is, in fact,
an equation determining uμ. A very important feature of
our freeze-out procedure is an anti-bubble prescription.
The matter is allowed to be frozen out only if a) either the
matter is located near the boarder with vacuum (this piece
of matter gets locally frozen out), b) or the maximal value
of the total energy density in the system is less than εfrz:
max ε < εfrz (3)
(the whole system gets instantly frozen out).
In the 3FD model this freeze-out simultaneously ter-
minates both chemical and kinetic processes.
Before the instant of the global freeze-out, cf. (3), the
freeze-out removes matter from the surface of the hydro-
dynamically expanding system. This removed matter gives
rise to observable spectra of hadrons. This kind of freeze-
out is similar to the model of “continuous emission” pro-
posed in ref. [16]. There the particle emission occurs from
a surface layer of the mean-free-path width. In our case
the physical pattern is the same, only the mean free path
is shrunk to zero.
compatible with −1 within the error bars, we performed our
ﬁt with λ = −1.
Yu.B. Ivanov and V.N. Russkikh: Transverse-mass eﬀective temperature in heavy-ion collisions . . . 141
0 4 8 12 16 20
s1/2 [GeV]
0
0.1
0.2
0.3
<
û
ou
t>
 [
G
eV
/f
m
3 ]
Fig. 2. Actual average freeze-out energy density in central
(zero impact parameter) Pb + Pb collisions as a function of
invariant incident energy.
Condition (2) ensures only that the actual freeze-out
energy density (let us call it εout), at which the freeze-out
actually occurs, is less than εfrz. Therefore, εfrz can be
called a “trigger” value of the freeze-out energy density.
As explained in ref. [15], a natural value of this actual
freeze-out energy density is εout ≈ εs/2, i.e. at that the
middle of the fall from the near-surface value of the energy
density, εs, to zero. To ﬁnd out the actual value of εout,
we have to analyze the results of a particular simulation.
In our previous paper [8] we have performed only a rough
analysis of this kind. This is why in the main text of ref. [8]
we mentioned the value of approximately 0.2 GeV/fm3
for εout and in the appendix explained how the freeze-
out actually proceeded. (In terms of ref. [8] (εfrz[1] and
εcodefrz[1]) our present quantities are εfrz = ε
code
frz[1] and εout =
εfrz[1].) The results of a more comprehensive analysis for
central (b = 0) Pb + Pb collisions are presented in ﬁg. 2,
which shows the εout value averaged over the space-time
evolution of the collision: 〈εout〉. As seen, 〈εout〉 reveals
saturation at the SPS energies, very similar to that in
eﬀective temperatures in ﬁg. 1. This happens in spite of
the fact that our freeze-out condition involves only a single
constant parameter εfrz.
The “step-like” behavior of 〈εout〉 is a consequence
of the freeze-out dynamics, as was demonstrated in
ref. [15]. At low (AGS) incident energies, the energy den-
sity achieved at the border with vacuum, εs, is lower than
εfrz. Therefore, the surface freeze-out starts at lower en-
ergy densities. It further proceeds at lower densities up to
the global freeze-out because the freeze-out front moves
not faster than with the speed of sound, like any pertur-
bation in the hydrodynamics. Hence it cannot overcome
the supersonic barrier and reach dense regions inside the
expanding system. With the incident-energy rise the en-
ergy density achieved at the border with vacuum gradu-
ally reaches the value of εfrz and then even overshoots it. If
the overshoot happens, the system ﬁrst expands without
0 4 8 12 16 20
s1/2 [GeV]
0.2
0.4
0.6
0.8
1
Tfrz/mp
nfrz/n0
b
mfrz/mN
Fig. 3. Average temperature Tfrz/mπ (over the pion mass),
transverse velocity β = vT /c, baryon density nfrz/n0 (over
the normal nuclear density) and baryon chemical potential
μfrz/mN (over the nucleon mass) at the freeze-out in central
Au + Au (at AGS energies, b = 2 fm) and Pb + Pb (at SPS
energies, b = 2.5 fm) collisions as a function of the invariant
incident energy.
freeze-out. The freeze-out starts only when εs drops to the
value of εfrz. Then the surface freeze-out occurs really at
the value εs ≈ εfrz and thus the actual freeze-out energy
density saturates at the value 〈εout〉 ≈ εfrz/2. This freeze-
out dynamics is quite stable with respect to numerics [15].
Figure 3 presents average temperatures, transverse ve-
locities, baryon densities and chemical potentials achieved
at the freeze-out in central collisions. Hadronic gas EoS
was used to determine these quantities. The freeze-out
temperature Tfrz has a similar “step-like” behavior. How-
ever, its absolute values are essentially lower than the ef-
fective temperatures in ﬁg. 1. This fact once again illus-
trates that inverse slopes represent a combined eﬀect of
the temperature and collective transverse ﬂow of expan-
sion associated with the collective transverse velocity β.
Note that β even slightly decreases with incident energy.
At SPS energies the freeze-out temperatures in ﬁg. 3 are
noticeably lower than those deduced from hadron multi-
plicities in the statistical model [17,18]. The reason for
this is as follows. Whereas the statistical model assumes a
single uniform ﬁreball, in the 3FD simulations at the late
stage of the evolution the system eﬀectively consists of sev-
eral “ﬁreballs”: two (one baryon-rich and one baryon-free)
ﬁreballs at lower SPS energies and three (two baryon-rich
and one baryon-free) ﬁreballs at top SPS energies [15].
Therefore, whereas high multiplicities of mesons and an-
tibaryons are achieved by means of high temperatures in
the statistical model, the 3FD model explains them by
142 The European Physical Journal A
an additional contribution of the baryon-free ﬁreball at a
lower temperature. In particular, this is the reason why
two diﬀerent freeze-out points (chemical and kinetic ones)
are not needed in the 3FD model. The freeze-out baryon
density nfrz exhibits a maximum at incident energies of
Elab = 10A–30A GeV which are well within range of the
planned FAIR in GSI. This observation agrees with that
deduced from the statistical model [19], even baryon den-
sity values in the maximum are similar to those presented
in ref. [19].
Returning to the question if the considered “step-like”
behavior of eﬀective temperatures is a signal of phase
transition into QGP, we should admit that this is not quite
clear as yet. It depends on the nature of the freeze-out
parameter εfrz = 0.4 GeV/fm
3 which should be further
clariﬁed. EoS is not of prime importance for this behavior.
The only constrain on the EoS is that it should be in some
way reasonable. Moreover, our preliminary results indi-
cate that a completely diﬀerent EoS [20] with 1st-order
phase transition to QGP still reasonably reproduces this
“step-like” behavior even in spite of the fact that it fails
to describe a large body of other data. This happens
because the same freeze-out pattern is accepted there.
In fact, EoS is just the pressure as a function of baryon
and energy densities: P (nB , ε). In this calculation we used
hadronic EoS [8,12]. However, this is just an interpreta-
tion of the function P (nB , ε), which we use, in hadronic
terms. Moreover, our EoS is too soft at high densities to
be matched with even heavy-quark bag-model EoS [20] or
quasiparticle ﬁts to lattice QCD data [21] in order to con-
struct the 1st-order phase transition. (The hadronic and
quark pressures as functions of the baryon chemical po-
tential should have a crossing point in order to construct
the 1st-order phase transition.) Therefore, it would not
be surprising if the same EoS can be reinterpreted also
in terms of a very smooth cross-over phase transition to
quark-gluon matter.
This hydrodynamic explanation of the considered
“step-like” behavior of eﬀective temperatures together
with the failure of kinetic approaches implies that a heavy
nuclear system really reveals a hydrodynamic motion dur-
ing its expansion. The simultaneous reproduction of the
inverse slopes of all considered particles (p, π and K) im-
plies that these particles belong to the same hydrodynamic
ﬂow at the instant of their freeze-out. An indirect sup-
port of this conjecture is the recent success of the GiBUU
model [22] in the reproduction of kaon inverse slopes. That
was achieved by taking into account three-body interac-
tions, which essentially increased the equilibration rate.
We are grateful to I.N. Mishustin, L.M. Satarov, V.V. Skokov,
V.D. Toneev, and D.N. Voskresensky for fruitful discussions.
This work was supported by the Deutsche Forschungsgemein-
schaft (DFG project 436 RUS 113/558/0-3), the Russian Foun-
dation for Basic Research (RFBR grant 06-02-04001 NNIO a),
the Russian Federal Agency for Science and Innovations (grant
NSh-3004.2008.2).
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Transverse-mass effective temperature in heavy-ion collisions from AGS to SPS

MUCC (Crossref)

Ivanova, Yu. B.

GSI Repository

Transverse-mass spectra in Au + Au and Pb + Pb collisions in the incident energy range from 2A to 160A GeV are analyzed within the model of 3-fluid dynamics (3FD). It is shown that the dynamical description of freeze-out, accepted in this model, naturally explains the incident-energy behavior of the inverse-slope parameters of these spectra observed in experiment. The simultaneous reproduction of the inverse slopes of all considered particles (ρ, π and K) suggests that these particles belong to the same hydrodynamic flow at the instant of their freeze-out

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Transverse-Mass Effective Temperature in Heavy-Ion Collisions from AGS to SPS

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