The effect of flow on hadronic spectra in an excluded-volume model

Abstract

Recently, we proposed a thermodynamically consistent excluded-volume model for the HG fireball and we noticed that our model gives a suitable description of various properties of multiparticle production and their ratios in the entire range of temperatures and baryon densities. Our aim in this paper is to obtain the variations of freeze-out volume in a slice of unit rapidity, i.e. dV/dy, as well as total volume of the fireball with respect to center-of-mass energy , and to compare our model calculations with the corresponding thermal freeze-out volume obtained from the Hanbury–Brown–Twiss pion interferometry method. We also test the validity of our model in extracting the total multiplicities as well as the central rapidity densities of various hadrons and comparing them with the recent results. We further calculate the rapidity and transverse momentum spectra of various particles produced in different heavy-ion collider experiments in order to examine the role of flow by matching our predictions with the available experimental results. Finally, we extend our analysis for the production of light nuclei, hypernuclei and their antinuclei over a broad energy range from alternating gradient synchrotron to large hadron collider energies.

Full text

arXiv:1202.4852v1 [hep-ph] 22 Feb 2012
Rapidity and Transverse Mass Spectra of Hadrons in a New
Excluded-Volume Model II
S. K. Tiwari∗, P. K. Srivastava, and C. P. Singh
Department of Physics, Banaras Hindu University, Varanasi 221005, INDIA
Abstract
Remarkable success gained by various thermal and statistical approaches in describing the particle
multiplicities and their ratios has emphasized the formation of a fireball consisting of chemically
equilibrated hot and dense hadron gas (HG) produced in the ultrarelativistic heavy-ion collisions. In
an earlier paper referred as I, we proposed a thermodynamically consistent excluded-volume model
for the HG fireball and we noticed that the model gives a suitable description for various properties
of multiparticle production and their ratios in the entire range of temperatures and baryon densities.
Furthermore, a numerical calculation indicates that the model respects causality and the values of
the transport coefficients (such as shear viscosity to entropy (η/s) ratio, and the speed of sound
etc.) suitably match with the predictions of other HG models. The aim in this paper is to obtain
the variations of freeze-out volume in a slice of unit rapidity i.e. dV/dy as well as total volume of
the fireball with respect to center-of-mass energy (√SN N ) and confront our model calculations with
the corresponding thermal freeze-out volume obtained from the Hanbury-Brown-Twiss (HBT) pion
interferometry. We also test the validity of our model in extracting the total multiplicities as well
as the central rapidity densities of various hadrons such as π+,K+,p,φ, Λ, Ξ−, Ω−,¯
Λ etc. and
in getting the rapidity as well as transverse momentum distributions of various particles produced
in different heavy-ion collider experiments in order to examine the role of any hydrodynamic flow
on them by matching our predictions with the experimental results. Furthermore, we extend our
analysis for the production of light nuclei, hypernuclei and their antinuclei over a broad energy
range from Alternating Gradient Synchrotron (AGS) to Large Hadron Collider (LHC) energies, for
which the experimental data have started appearing.
PACS numbers: 12.38.Mh, 12.38.Gc, 25.75.Nq, 24.10.Pa
∗corresponding author: sktiwari4bhu@gmail.com

I. INTRODUCTION
The ultimate goal of ultra-relativistic heavy-ion collisions is to produce a highly excited
and dense matter which possibly involves a phase transition from a hot, dense hadron gas
(HG) to a deconfined quark matter called as quark-gluon plasma (QGP) [1-3]. In this state
the degrees of freedom are those of quarks and gluons only. One hopes that by colliding
heavy nuclei one can create a fireball with an extremely large energy density extending over
a sufficiently large space-time volume so that an equilibrated quark-gluon plasma may be
formed. However, experimental and theoretical investigations made so far reveal that it is
indeed difficult to get an unambiguous evidence for QGP formation. It is very important
to understand the dynamics of the collisions in order to suggest any unique signal for QGP.
Such information can be obtained by analyzing the properties of various particles which are
emitted from various stages of the collisions. The global properties and dynamics of later
stages can be best studied via hadronic observables such as hadron yields, ratios, rapidity
distributions and transverse mass spectra [4].
In our previous paper henceforth referred as I [5], we have proposed a thermodynamically
consistent excluded-volume model and studied different thermodynamical properties of HG
such as number density, energy density, pressure etc.. It is indeed surprising that the predic-
tions of our geometrical model regarding the detailed features of various hadron ratios and
their variations with respect to the center-of-mass energy (√sN N ) are successfully tested
with the available experimental data upto Relativistic Heavy-Ion Collider (RHIC) energies.
We know that if a system passes through a mixed phase in a first order phase transition,
then it can possess a much larger space-time extent than what we expect from a system if it
remains in the hadronic phase only. Thus the freeze-out volume Vof the fireball formed in
heavy-ion collisions, is a significant quantity which can give a hint regarding the occurrence
of mixed phase during a phase transition. Furthermore, the freeze-out volume throws light
on the collective expansion and/or anisotropic flow of the fluid existing in the fireball before
a final thermal freeze-out occurs. In this paper, we calculate the variations of chemical
freeze-out volumes for π+,K+, and K−with respect to √sN N and compare our predictions
with the HBT interferometry data for the thermal freeze-out volume. It gives an indepen-
dent confirmation pertinent to correctness of the geometrical assumptions involved in our
model on the space dynamics of the fireball. We also attempt to show the variations of total

multiplicities as well as the multiplicity in a slice of unit rapidity at midrapidity (y= 0) for
various emitted hadrons, e.g., π+,K+,K−,φ, Λ, Ξ−,Ω−, and ¯
Λ. Further, we extend our
model to deduce the rapidity as well as transverse mass spectra of hadrons and compare
them with the experimental data available in order to illustrate the role of hydrodynamic
flow of the fluid.
In heavy-ion collisions, rapidity densities of produced particles are strongly related to
the energy density and/or entropy density created in the collisions [6]. Further, the depen-
dence of transverse mass spectra of hadrons on √sNN can yield insight into the evolution
of a radial flow present in the dense fluid formed in the collision [7]. Earlier different types
of approaches have been taken into account for studying rapidity distributions and trans-
verse mass spectra [8-13]. Hadronic spectra from purely thermal models usually reveal an
isotropic distribution of particles [8] and hence the rapidity spectra obtained with the ther-
mal models do not reproduce the features of the experimental data satisfactorily. Similarly,
the transverse mass spectra from the thermal models reveal a more steeper curve than that
observed experimentally. The comparisons illustrate that the fireball formed in heavy-ion
collisions does not expand isotropically in nature and there is a prominent input of col-
lective flow in the longitudinal and transverse directions which finally causes anisotropy in
the rapidity and transverse mass distributions of the hadrons after the freeze-out. A lot of
work has been done in recent years on thermal model calculations incorporating the effect
of flow [14-23]. Here we mention some kinds of models of thermal and collective flow used
in the literature. Hydrodynamical properties of the expanding fireball have been initially
discussed by Landau and Bjorken for the stopping and central-rapidity regimes, respectively
[9]. However, collisions even at RHIC energies reveal that they are neither fully stopped,
nor fully transparent. As the collision energy increases, the longitudinal flow grows stronger
and leads to a cylindrical geometry as postulated in Ref. [10-12]. They assume that the
fireballs are distributed uniformally in the longitudinal direction and demonstrate that the
available data can consistently be described in a thermal model with inputs of chemical
equilibrium and flow, although they have used the experimental data for small systems only.
They use two simple parameters : transverse flow velocity (βr) and temperature (T) in their
models. In Ref. [13], non-uniform flow model is used to analyze the spectra specially to
reproduce the dip at midrapidity in the rapidity spectra of baryons by assuming that the
fireballs are distributed non-uniformly in the longitudinal phase space. In Ref. [24-26], the

rapidity-dependent baryon chemical potential has been invoked to study the rapidity spectra
of hadrons. In certain hydrodynamical models [27,28], measured transverse momentum (pT)
distributions in Au −Au collisions at √sN N = 130 GeV [29-31] have been described suc-
cessfully by incorporating a radial flow. In Ref. [32], rapidity spectra of mesons have been
studied using viscous relativistic hydrodynamics in a 1+1 dimension assuming a non-boost
invariant Bjorken flow in the longitudinal direction. They have also analyzed the effect of
the shear viscosity on the longitudinal expansion of the matter. Shear viscosity counteracts
the gradients of the velocity field, as a consequence slows down the longitudinal expansion.
In this paper, we attempt to calculate the rapidity density of various particles at midra-
pidity by using our excluded-volume model of HG and we find a good agreement between
our model calculations and the experimental data. However, when we calculate the rapidity
distributions of various particles, we find that the distribution always takes a narrow shape
in comparison to the experimental data and thus indicating that our thermal model alone is
not capable to describe the experimental data in forward and backward rapidity regions. In
a similar way, the transverse mass spectra of hadrons as obtained in our model again does
not describe the experimental data properly. Our analysis clearly necessitates the presence
of an additional flow factor to be introduced in the model in order to provide a suitable
description of the data.
We plan to study the rapidity distributions and transverse mass spectra of hadrons using
our model in I [5] for a hot, dense HG which is based on an excluded-volume approach [33]
formulated in a thermodynamically consistent way. We have assigned an equal hard-core
size to each type of baryons in the HG while the mesons are treated as pointlike particles.
We use the chemical freeze-out criteria to relate the thermal parameters Tand µBwith the
collision energies [5]. We further emphasize that we have used this hadronic equation of state
(EOS) together with a suitable EOS for QGP in order to obtain the QCD phase diagram and
the associated critical point [34]. The plan of the paper runs as follows : the next section
is dedicated to the formulation of our model for the study of the rapidity distributions
and transverse mass spectra of hadrons using purely thermal source. In the third section,
we modify the formula for the rapidity distributions by incorporating a flow velocity in
the longitudinal direction and similarly the formula for the transverse mass spectra is also
modified by incorporating a collective flow in the longitudinal as well as in the transverse
direction. In section IV, we compare the experimental data with our predictions regarding

rapidity density, transverse mass spectra and transverse momentum spectra of hadrons at
RHIC energies (130 GeV and 200 GeV ) and LHC energy (2.76 T eV ). We have also shown
the rapidity spectra of π−at Super Proton Synchrotron (SPS) energies. As √sN N increases,
the longitudinal as well as the transverse flow also increase which means that the experiments
will observe a larger amount of collective flow at LHC. We have also deduced total mean
multiplicities, mid-rapidity yields of various hadrons and their rapidity densities and studied
their variations with √sNN . Thus we determine the freeze-out volume and/or dV /dy in order
to ascertain whether the emissions of all hadrons occur from the same hypersurface of the
fireball. We also analyze the experimental data on the production of light nuclei, hypernuclei
and their antiparticles over a broad energy going from AGS to RHIC energies. This analysis
thus fully demonstrates the validity of our EOS in describing all the features of a hot, dense
HG. Finally, in the last section we succinctly give the conclusions and summary.
II. HADRONIC SPECTRA WITH THE THERMAL SOURCE
A. Rapidity Distributions
To study the rapidity distributions and transverse mass spectra of the hadrons, we extend
excluded-volume model as proposed in I [5], which was found to describe the hadron ratios
and yields at various √sNN from lower AGS energies upto RHIC energies with remarkable
success. We have incorporated the excluded-volume correction directly in the grand canon-
ical partition function of HG in a thermodynamically consistent manner. We obtain the
number density nex
ifor ith species of baryons after excluded-volume correction as follows [5]:
nex
i= (1 −R)Iiλi−Iiλ2
i
∂R
∂λi
+λ2
i(1 −R)I′
i,(1)
where R=X
i
nex
iV0
iis the fractional occupied volume by the baryons [5,34]. V0
i= 4π r3/3
is the eigen-volume of each baryon having a hard-core radius rand λiis the fugacity of ith
baryon. Further, Iiis the integral of the baryon distribution function over the momentum
space [5].
We can rewrite Eq. (1) in the following manner :

nex
i=giλi
(2π)3h(1 −R)−λi
∂R
∂λiZ∞
0
d3k
hexp Ei
T+λii
−λi(1 −R)Z∞
0
d3k
hexp Ei
T+λii2i.
(2)
This reveals that the invariant distributions are [10,11] :
Ei
d3Ni
dk3=giV λi
(2π)3h(1 −R)−λi
∂R
∂λiEi
hexp Ei
T+λii
−λi(1 −R)Ei
hexp Ei
T+λii2i.
(3)
Using :
Ei
d3Ni
dk3=dNi
dy mTdmTdφ,(4)
we get :
dNi
dy mTdmTdφ =giV λi
(2π)3h(1 −R)−λi
∂R
∂λiEi
hexp Ei
T+λii
−λi(1 −R)Ei
hexp Ei
T+λii2i,
(5)
Here yis the rapidity variable and transverse mass mT=pm2+pT2. Also Eiis the energy
of ith baryon and Vis the total volume of the fireball formed at chemical freeze-out and Ni
is the total number of ith baryons. We assume that the freeze-out volume of the fireball for
all types of hadrons at the time of the homogeneous emissions of hadrons remains the same.
By inserting Ei=mTcoshy in Eq. (5) and integrating the whole expression over trans-
verse component we can get the rapidity distributions of baryons as follows:

dNi
dy th =giV λi
(2π2)h(1 −R)−λi
∂R
∂λiZ∞
0
m2
Tcoshy dmT
hexp mTcoshy
T+λii
−λi(1 −R)Z∞
0
m2
Tcoshy dmT
hexp mTcoshy
T+λii2i.
(6)
Eq.(6) gives the rapidity distributions of baryons arising due to a stationary thermal
source. Here we evaluate the rapidity distributions of hadrons at pT= 0 as the data have
been taken for the pT= 0 case.
Similarly, the rapidity density of mesons can be had by using the following formula :
dNm
dy th =gmV λm
(2π2)Z∞
0
m2
Tcoshy dmT
hexp mTcoshy
T−λmi.(7)
Here gm,λmare the degeneracy factor and fugacity of the meson m, respectively.
B. Transverse Mass Spectra
We use Boltzmann statistics in deriving formula for the transverse mass spectra because
we want to calculate spectra of hadrons only at RHIC energies, where the effect of quantum
statistics is found to be negligible [10]. In the Boltzmann’s approximation, Eq.(5) can be
reduced to a simple form :
dNi
dy mTdmTdφ =giV λi
(2π)3h(1 −R)−λi
∂R
∂λiiEihexp −Ei
Ti.(8)
Putting Ei=mTcoshy in Eq.(8) and integrating over rapidity (y) we get the transverse
mass spectra as follows :
dNi
mTdmT
=giV λi
(2π)3h(1 −R)−λi
∂R
∂λiiZ∞
0
mTcoshy hexp −mTcoshy
Tidydφ, (9)
or, this means :
dNi
mTdmT
=giV λi
(2π2)h(1 −R)−λi
∂R
∂λiimTK1mT
T,(10)

where K1mT
Tis the modified Bessel’s function and is given by the following expression
:
K1mT
T=Z∞
0
coshy hexp −mTcoshy
Tidy. (11)
Similarly transverse mass spectrum of mesons can be evaluated as follows :
dNm
mTdmT
=gmV λm
(2π2)mTK1mT
T.(12)
III. HADRONIC SPECTRA WITH THE EFFECT OF FLOW
In the previous section, we have obtained the expression for rapidity distributions as well
as transverse mass spectra arising from a stationary thermal source only. In this section, we
modify the expression for rapidity spectra i.e. Eq. (6), by incorporating a flow velocity to a
stationary thermal system in the longitudinal direction and thus we attempt to explain the
experimental data in the full rapidity region. The resulting rapidity spectrum of ith hadron,
after the incorporation of the flow velocity in the longitudinal direction is [10,11]:
dNi
dy =Zηmax.
−ηmax. dNi
dy th(y−η)dη, (13)
where dNi
dy th can be calculated by using Eq.(6) for the baryons and by Eq.(7) for the
mesons. The average longitudinal velocity is [13,35]:
hβLi=tanhηmax.
2.(14)
Here ηmax. is an important parameter which provides the upper rapidity limit for the longitu-
dinal flow velocity at particular √sNN and it’s value is determined by the best experimental
fit. The value of ηmax. increases with the increasing √sNN and hence βLalso increases.
In the case of transverse mass spectra, we incorporate flow velocity in both the directions,
longitudinal as well as transverse directions in order to describe the experimental data
satisfactorily. However, we assume radial type of flow velocity in the transverse direction

which imparts a radial velocity boost on top of the thermal distribution. We define the four
velocity field in both the directions as follows [36] :
uµ(ρ, η) = (coshρ coshη, ¯ersinhρ, coshρ sinhη),(15)
where ρis the radial flow velocity in the transverse direction and ηis the flow velocity in
the longitudinal direction. Once we have defined the flow velocity field, we can calculate the
invariant momentum spectrum by using the following formula [10,37] :
Ei
d3Ni
dk3=giV λi
(2π)3h(1 −R)−λi
∂R
∂λiiZexp−kµuµ
Tkλdσλ.(16)
In the derivation of Eq.(16) we assume that an isotropic thermal distribution of hadrons is
boosted by the local fluid velocity uµ. Now the resulting spectrum can be written as follows
[10] :
dNi
mTdmTdy =giV λimT
(2π)3h(1 −R)−λi
∂R
∂λiiZr dr dφ dζ
×exp−mTcosh(y−η)coshρ −pTsinhρ cosφ
T.
(17)
The freeze-out hypersurface dσλin Eq.(16) is parametrized in cylindrical coordinates (r, φ, ζ),
where the radius rcan lie between 0 and Ri.e. the radius of the fireball at freeze-out, the
azimuthal angle φlies between 0 and 2π, and the longitudinal space-time rapidity variable
ζvaries between −ηmax. and ηmax.. Now, integrating Eq.(17) over φas well as ζ, we get the
final expression for the transverse mass spectra [10] :
dNi
mTdmT
=giV λimT
(2π2)h(1 −R)−λi
∂R
∂λiiZR
0
r dr K1mTcoshρ
TI0pTsinhρ
T.(18)
Here I0pTsinhρ
Tis the modified Bessel’s function given by :
I0pTsinhρ
T=1
2πZ2π
0
exppTsinhρ cosφ
Tdφ, (19)
where ρis given by ρ=tanh−1βr, with the velocity profile chosen as βr=βsξn[10,11].
βsis the maximum surface velocity and is treated as a free parameter and ξ=r/R. The
average of the transverse velocity can be evaluated as [31]:

(GeV)
NN
s
1 10 2
10
)
3
dV/dy (fm
1000
2000
3000
4000
5000
)
3
Total freeze−out Volume (fm
0
2000
4000
6000
8000
10000
12000
AGS (Au−Au)
SPS (Pb−Pb)
RHIC (Au−Au)
A’
B’
C’
A
B
C
FIG. 1: Energy dependence of the freeze-out volume for the central nucleus-nucleus collisions.
The symbols are the HBT data for freeze-out volume VH BT for the π+[38]. A′,B′and C′are the
total freeze-out volume and A,Band Cdepict the dV/dy as found in our model for π+,K+and
K−, respectively.
< βT>=Rβsξnξ dξ
Rξ dξ =2
2 + nβs.(20)
In our calculation we use a linear velocity profile, n= 1 and Ris the maximum radius
of the expanding source at freeze-out (0 < ξ < 1) [31]. Similarly following equation can be
used to calculate transverse mass spectra for mesons :
dNm
mTdmT
=gmV λmmT
(2π2)ZR
0
r dr K1mTcoshρ
TI0pTsinhρ
T.(21)
IV. RESULTS AND DISCUSSIONS
We have used freeze-out temperature and baryon chemical potential as determined by
fitting the particle ratios as described in I [5]. Then we use suitable parametrizations for T
and µBin terms of center-of-mass energies. We have used hard-core radius r= 0.8f m for
all types of baryons. We have considered all the particles and the resonances upto mass of

(GeV)
NN
s
1 10 2
10
<N>
−4
10
−3
10
−2
10
−1
10
1
10
2
10
3
10
+
π+
K−
K
φΛ0.02 −
Ξ0.1
+
Ω
+
−
Ω0.2 Λ0.02
A
B
CD
EFG
H
FIG. 2: Variations of total multiplicities of π+,K+,K−,φ, Λ, Ξ−, (Ω−+¯
Ω+), and ¯
Λ with
respect to center-of-mass energy predicted by our model. Experimental data measured in central
Au −Au/P b −P b collisions [39-55] have also been shown for comparison. In this figure, A, B,
C, D, E, F, G, and H represent the multiplicities of π+,K+,K−,φ, Λ, Ξ−, (Ω−+¯
Ω+), and ¯
Λ,
respectively.
2GeV/c2in our calculation. We have used resonances having well defined masses, widths
etc. and branching ratios for sequential decays are also suitably incorporated.
Although the emission of hadrons from a statistical thermal model essentially invokes the
idea of equilibration, it does not reveal any information regarding the existence of a QGP
phase before hadronization. However, if the constituents of the fireball have gone through
a mixed phase, volume Vof the fireball at freeze-out is expected to be much larger than
what we expect from a system if it remains throughout in the hadronic phase only. In
Fig. 1, we have shown Vand dV /dy as obtained in our excluded-volume model and their
variations with the center-of-mass energy. For this purpose, we have used the data for total
multiplicities of π+,K+,K−and after dividing with their corresponding number densities
obtained in our model, we get Vfor π+,K+, and K−, respectively. Similarly for deducing
dV/dy, we use the data for dN/dy and divide by the corresponding number density. We
have compared predictions of our model with the experimental data obtained from the pion

(GeV)
NN
S
1 10 2
10
y=0
(dN/dy)
-4
10
-3
10
-2
10
-1
10
1
10
2
10
+
π+
K
Λ-
K
Λ
Our Model with Boltzmann’s Statistics
Our Model with Quantum Statistics
FIG. 3: Variation of rapidity distributions of various hadrons with respect to √sN N at midrapidity.
Lines show our model calculation. Symbols are the experimental data [7,56,57].
(GeV)
NN
S
1 10 2
10
y=0
(dN/dy)
-4
10
-3
10
-2
10
-1
10
1
10
2
10
p
Our Model
φ
Our Model
p
Our Model
-
Ξ
Our Model
+
Ξ
Our Model
FIG. 4: Energy dependence of rapidity distributions of hadronic species at midrapidity (y=
0) for central nucleus-nucleus collisions. Lines are the model calculations and symbols are the
experimental results [7,56,57] at various √sNN .

interferometry (HBT) [38] which in fact reveal thermal (kinetic) freeze-out volume. Our
results support the finding that the decoupling of strange mesons from the fireball takes
place earlier than the π-mesons. Moreover, a flat minimum occurs in the curves around
energy ≈8GeV , then the volume rapidly increases. At RHIC energy, the volume of the
fireball is around 12000 f m3which is much larger than the volume of gold nucleus used in
the collisions.
In order to calculate total multiplicities of hadrons, we first determine the total freeze-out
volume for K+by dividing the experimentally measured multiplicities of K+at different
center-of-mass energies with it’s number density as calculated in our model at different
center-of-mass energies. We assume that the fireball after expansion, achieves the stage of
chemical equilibrium and the freeze-out volume of the fireball for all types of hadrons at the
time of the homogeneous emissions of hadrons remains same for all particles. This freeze-out
volume thus extracted for K+, has been used to calculate multiplicities of all other hadrons
from corresponding number densities at different √sN N . Figure 2 shows the center-of-mass
energy dependence of multiplicities of hadrons π+,K+,K−,φ, Λ, Ξ−, (Ω−+¯
Ω+), and
¯
Λ as predicted by our model calculation. We also show here corresponding experimental
data measured in central Au −Au/P b −P b collisions [39-55] for comparison. We observe
an excellent agreement between our model calculation and experimental data for the total
multiplicities of all particles except φ, Ξ−, and Ω−, where we see small deviations. Again,
the thermal multiplicities for all these particles are larger than the experimental values. This
analysis again suggests a new and different mechanisms for the production of these particles.
In Fig.3, we compare the experimental midrapidity data [7,56,57] of various hadron species
over a broad energy range from AGS to RHIC energies with the results of our model calcu-
lations. We use the same freeze-out volume of the fireball for each hadron as extracted for
K+in our model calculation. We also show the comparison between the results calculated
using our model with the Boltzmann’s statistics, as used in our previous paper [58], with
the results calculated with full quantum statistics. As expected, both results differ only at
lower energies. However, the results with quantum statistics give much closer agreement
with the experimental data. Figure 4 shows the energy dependence of midrapidity distri-
butions of various hadrons like p,φ, ¯p, Ξ−, and ¯
Ξ+. We observe that the results obtained
in our model show a close agreement with the experimental data [7,56,57]. However, our
model calculation again differs at higher energies for multistrange particles in comparison

Rapidity (y)
-5 -4 -3 -2 -1 0 1 2 3 4 5
/dy
+
π
dN
0
50
100
150
200
250
300
350
400 +
π=0.92)
L
βOur Model+Flow (
=0.88)
L
βOur Model+Flow (
Our Model
FIG. 5: Rapidity distribution of π+at √sN N = 200GeV . Dotted line shows the rapidity distribu-
tion calculated in our thermal model. Solid line and dashed line show the results obtained after
incorporating longitudinal flow in our thermal model. Symbols are the experimental data [59].
to experimental data, which appears to be a common problem of all thermal models.
In Fig.5, we present the rapidity distribution of π+for central Au +Au collisions at
√sNN = 200 GeV over full rapidity range. Dotted line shows the distribution of π+due to
stationary thermal source which describes only the midrapidity data while it fails to describe
the experimental data at other rapidities. Solid line shows the rapidity distributions of π+
after incorporation of longitudinal flow in our thermal model and again in a good agreement
with the experimental data [59]. After fitting the experimental data, we get the value of
ηmax. = 3.20 and hence the longitudinal flow velocity βL= 0.922 at √sN N = 200 GeV .
For comparison and testing the appropriateness of parameter, we also show the rapidity
distributions at different values of the parameter, ηmax. = 2.80 i.e. βL= 0.88, which is
shown by the dashed line in the figure. We find that the results now differ.
In Fig.6, we analyze the experimental data [60] on rapidity distributions of π−at various
√sNN in terms of our model calculation to constrain the allowed distribution of the longitu-
dinal velocities of the fluid elements. We find that our model results are in good agreement
with the experimental data at these energies. Figure 7 demonstrates the variation of longi-
tudinal flow velocities extracted in our model with respect to √sN N and it shows that the
longitudinal flow velocity increases with the increasing √sNN . We compare our results with

Rapidity (y)
-6 -4 -2 0 2 4 6
/dy
-
π
dN
0
50
100
150
200
250
300
350
400 17.3 GeV
Experimental Data
Our Model+Flow
12.3 GeV
Experimental Data
Our Model+Flow
8.76 GeV
Experimental Data
Our Model+Flow
FIG. 6: Rapidity distribution of π−at √sNN = 8.76 GeV, 12.3GeV and 17.3GeV . Lines are our
model calculation and symbols are experimental data [60].
(GeV)
NN
s
1 10 2
10
>
L
β<
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Our Model
Netrakanti [35]
FIG. 7: Variations of the longitudinal flow velocities with respect to √sN N . Open circles are the
results of Ref.[35].
that calculated in Ref.[35] and find that a slight difference exists between these two results.
Figures 8 and 9 demonstrate the rapidity distributions of proton and anti-proton, respec-
tively at √sNN = 200 GeV . Again, our curves describe successfull the experimental data
[61]. Similarly in Figures 10 and 11 we show the rapidity distributions for K+and K−,

Rapidity (y)
-5 -4 -3 -2 -1 0 1 2 3 4 5
/dy
p
dN
0
5
10
15
20
25
30
35
40
45
50
p
Our Model+Flow
Our Model
FIG. 8: Rapidity distribution of proton for central Au −Au collisions at √sNN = 200 GeV .
Dotted line shows the rapidity distribution due to purely thermal source and solid line shows the
result after incorporating the longitudinal flow velocity in our thermal model. Symbols are the
experimental data [61].
Rapidity (y)
-5 -4 -3 -2 -1 0 1 2 3 4 5
/dy
p
dN
0
5
10
15
20
25
30
35
40
p
Our Model+Flow
Our Model
FIG. 9: Rapidity distribution of anti-proton for central Au −Au collisions at √sNN = 200 GeV .
Dotted line shows the rapidity distribution due to purely thermal source and solid line shows the
result after incorporating the longitudinal flow in our thermal model. Symbols are the experimental
data [61].

Rapidity (y)
-5 -4 -3 -2 -1 0 1 2 3 4 5
/dy
+
K
dN
0
10
20
30
40
50
60
70
80
90
100
+
K
Our Model+Flow
Our Model
FIG. 10: Rapidity distribution of K+at √sNN = 200 GeV . Dotted line shows the rapidity distri-
bution due to purely thermal source and solid line shows the result after incorporating longitudinal
flow in the thermal model. Symbols are the experimental data [62].
respectively at √sNN = 200 GeV . We have used the same value of parameter βLand Vfor
each particle.
Figure 12 shows the analysis of the transverse mass spectra of π+and proton for the
most central collisions of Au −Au at √sNN = 130 GeV in terms of our model calculation.
We have neglected the contributions from resonance decays in our calculations here because
the contributions from resonance decays affect the transverse mass spectra only at the lower
transverse mass side i.e. mT<0.3GeV . So, our model calculation show some difference
with the experimental data [31] for mT<0.3GeV but there is a good agreement between
our model calculation and experimental results for mT>0.3GeV . We also show our
model results for π+at different values of βsjust for comparison. Similarly Figure 13 shows
the transverse mass spectra of K+and φfor the most central collisions of Au −Au at
√sNN = 130 GeV . Our calculation is quite successfull in describing the experimental data
[31,64]. However, our model fails again in the case of multistrange particle φ. We need
to devise some different mechanism to explain the experimental results for multistrange
particles. By fitting the experimental results, we get the value of βs= 0.45 and hence
transverse flow velocity βr= 0.30 at √sN N = 130 GeV , and is less than that extracted in
Ref.[31]. It may be due to different freeze-out temperature used in our model.

Rapidity (y)
-5 -4 -3 -2 -1 0 1 2 3 4 5
/dy
-
K
dN
0
10
20
30
40
50
60
70
80
90
100
-
K
Our Model+Flow
Our Model
FIG. 11: Rapidity distribution of K−at √sN N = 200GeV . Dotted line shows the rapidity distri-
bution due to purely thermal source and solid line shows the result after incorporating longitudinal
flow in our thermal model. Symbols are the experimental data [63].
(GeV)
0
−m
T
m
0.5 1 1.5 2 2.5 3 3.5 4
)
2
dy)) (1/(GeV)
T
dm
T
N/(m
2
))(dπ(1/(2
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
1
10
2
10
3
10 +
π=0.55)
s
βOur Model+Flow ( =0.45)
s
βOur Model+Flow (
Our Model
p
Our Model+Flow
Our Model
FIG. 12: Transverse mass spectra for π+and proton for the most central collision at √sN N =
130GeV . Dashed and dotted lines are the results of transverse mass spectra, calculated in our
thermal model, for π+and proton, respectively. Solid and dash-dotted lines are the results for π+
and proton, respectively, obtained after incorporation of flow in our thermal model. Symbols are
the experimental data [31].

(GeV)
0
−m
T
m
0.5 1 1.5 2 2.5 3 3.5 4
)
2
dy)) (1/(GeV)
T
dm
T
N/(m
2
))(dπ(1/(2
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
1
10
2
10
3
10
+
K
Our Model+Flow
Our Model
φ
Our Model+Flow
Our Model
FIG. 13: Transverse mass spectra for K+and φfor the most central collision at √sNN = 130 GeV .
Dashed and dotted lines are the results of transverse mass spectra, calculated in our thermal
model, for K+and φ, respectively. Solid and dash-dotted lines are the results for π+and proton,
respectively, obtained after incorporation of flow in thermal model. Symbols are the experimental
data [31,64].
In Fig. 14, we show the transverse mass spectra of π+and proton for the most central
collisions of Au +Au at √sN N = 200 GeV . From this figure it is clear that the spectra
due to stationary thermal source do not satisfy the experimental data [63] at higher mT
while the thermal source with collective flow can describe the experimental data very well.
For comparison, we also show the transverse mass spectra for π+at different βs. In Fig.
15, we present the transverse-mass spectra for K+and φat √sN N = 200 GeV . Lines are
our model calculations and the symbols represent the experimental data [63,64]. Again, our
model fails to describe the experimental data for φat lower mT. At this √sN N the value
of βs, after best fitting, comes out to be 0.50 and hence transverse flow velocity βr= 0.33.
The transverse flow velocity is able to reproduce the transverse mass spectra of almost all
the hadrons at √sN N = 200 GeV . We also see that the transverse flow velocity increases
with the increasing √sN N .
We also present the pT- spectra of hadrons at various center-of-mass energies. In Fig.
16, we show the (pT) spectra of π+,K+and pfor the most central collisions of Au −Au

(GeV)
0
−m
T
m
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
)
2
dy)) (1/(GeV)
T
dm
T
N/(m
2
))(dπ(1/(2
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
1
10
2
10
3
10
+
π=0.60)
s
βOur Model+Flow (
=0.50)
s
βOur Model+Flow (
Our Model
p
Our Model+Flow
Our Model
FIG. 14: Transverse mass spectra for π+and proton for the most central collisions at √sNN =
200 GeV . Dashed and dotted lines are the transverse mass spectra due to purely thermal source
for π+and proton, respectively. Solid and dash-dotted lines are the results for π+and proton,
respectively obtained after incorporation of flow in thermal model. Symbols are the experimental
data [63].
at √sNN = 200 GeV . Our model calculations are in close agreement with the experimental
data [63]. In Fig. 17, we show the pTspectra of π−,K−and ¯pfor the P b −P b collisions at
√sNN = 2.76 T eV at LHC. Our model calculations give a good fit of the experimental results
[65]. We also compare our results for ¯pspectrum with the hydrodynamical model [66], which
explains successfully the π−, and K−spectra but strongly fails to describe the ¯pspectrum.
We find that our model results are closer with the experimental data than that of Ref.[66].
In Ref. [66], Shen et al. have employed 2 + 1-dimensional viscous hydrodynamics with the
lattice QCD-based EOS. They use Cooper-Frye prescription to implement kinetic freeze-
out by converting the hydrodynamic output to particle spectra. Due to lack of a proper
theoretical and phenomenological knowledge, they use the same parameters for P b −P b
collisions at LHC energy, which was used for Au −Au collisions at √sNN = 200 GeV .
Furthermore, they use the temperature independent η/s ratio in their calculation. After
fitting the experimental data, we get βs= 0.80 and hence βr= 0.53 at this energy which
indicates that the collective flow is stronger at this energy than that observed at RHIC

(GeV)
0
−m
T
m
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
)
2
dy)) (1/(GeV)
T
dm
T
N/(m
2
))(dπ(1/(2
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
1
10
2
10
3
10
+
K
Our Model+Flow
Our Model
φ
Our Model+Flow
Our Model
FIG. 15: Transverse mass spectra for K+and φfor the most central collision at √sN N = 200GeV .
Dashed and dotted lines are the transverse mass spectra due to purely thermal source for K+and
φ, respectively. Solid and dash-dotted lines are the results for K+and φ, respectively, obtained
after incorporation of flow in thermal model. Symbols are the experimental data [63,64].
(GeV)
T
p
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
)
2
dy)) (1/(GeV)
T
dp
T
N/(p
2
)) (dπ(1/(2
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
1
10
2
10
3
10
+
π
Our Model+Flow
Our Model
+
K
Our Model+Flow
Our Model
p
Our Model+Flow
Our Model
FIG. 16: Transverse momentum spectra for π+,p, and K+for the most central Au−Au collision at
√sNN = 200GeV . Lines are the results of our model calculation and symbols are the experimental
results [63].

(GeV)
T
p
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
dy) (1/GeV)
T
N/(dp
2
d
−2
10
−1
10
1
10
2
10
3
10
−
π=0.88)
s
βOur Model+Flow ( =0.80)
s
βOur Model+Flow (
−
K
Our Model+Flow
p
Our Model+Flow
Viscous− Hydrodynamics
in Our Modelφ0.25
FIG. 17: Transverse momentum spectra of various hadrons for the most central collisions of P b −P b
at √sNN = 2.76 T eV from LHC. Lines are the results of model calculations and symbols are the
experimental results [65]. Thick-dashed line is the prediction of viscous-hydrodynamical model [66]
for ¯p. Dashed line is the prediction of our model calculation for φmeson.
(GeV)
NN
s
1 10 2
10
y=0
/dy)
π
/(dN
y=0
/dy)
φ
1000 (dN
2
4
6
8
10
12 Experimental Data
Our Model
HGM
UrQMD Model
FIG. 18: Energy dependence of the φ/π, (π= 1.5(π++π−)) ratio at midrapidity. Dash-dotted line
is the result of statistical hadron gas model (HGM) [67]. Solid line represents our model calculation
and dashed line is the result of UrQMD model [68]. Symbols are the experimental data points [7].

(GeV)
NN
S
1 10 2
10 3
10 4
10
Ratio
−10
10
−8
10
−6
10
−4
10
−2
10
1
Λ/Λ
Our Model
/pp
Our Model
/dd
Our Model
3
/He
3
He
Our Model
4
/He
4
He
FIG. 19: The energy dependence of anti-baryon to baryon yield and antinuclides to nuclides ratios.
Lines are our model calculation and symbols represent the experimental data [74].
energies. We also predict the pTspectra for φmeson at this energy. For comparison, we
also show the spectra at different values of parameter βs= 0.88.
In order to emphasize how our thermal model fails for multistrange particles, we show in
Fig. 18 the variations of the φ/π, (π= 1.5(π++π−)) ratio at midrapidity with √sN N . We
compare our results with the experimental data [7] and statistical hadron gas model (HGM)
[67]. We find that our model provides a good description of the experimental data at lower
energies but fails to describe the data at higher energies. HGM model, which has included
the strangeness saturation factor γsto account for the partial equilibration of strangeness,
describes the data well at the lower and upper energies while it completely disagrees with
the experimental data at the intermediate energies. We also show the results of the hadronic
transport model Ultra-relativistic Quantum Molecular Dynamical (UrQMD) [68] in which
kaon coalescence mechanism is used for the production of φ. The predictions of UrQMD lie
well below the experimental data at all energies which suggests that this mechanism fails
badly in explaining the production mechanism of φ.
We present an analysis of light nuclei, hypernuclei and their anti-particles using the
chemical freeze-out concept within our model calculation. The productions of light nuclei
and hypernuclei at chemical freeze-out points may not be appropriately calculated, because
their binding energies are of the order of few MeV and the chemical freeze-out temperatures

(GeV)
NN
S
1 10 2
10 3
10
Ratio
−3
10
−2
10
−1
10
1
d/p
Our Model
/pΛ
Our Model
3
/He
Λ
3
H
Our Model
3
He
/
Λ
3
H
Our Model
Λ
3
/H
Λ
3
H
Our Model
FIG. 20: The energy dependence of various baryons, antibaryons, nuclei, and antinuclei yield ratios.
Lines are our model calculation and symbols represent the experimental results [74].
are around 100 −165 MeV . But we know that the relative yield of particles composed of
nucleons is mainly determined by the entropy per baryon which is fixed at chemical freeze-
out line in our model [5]. This was first outlined in [69] and was subsequently emphasized
in [70]. This constitutes the basis of thermal analyses for the yields of light nuclei [71,72].
Thus the production yields of light nuclei and hypernuclei is entirely governed by the entropy
conservation. Recently the first measurement of the lightest (anti) hypernuclei was done by
the STAR experiment at RHIC [73]. We want to ask an interesting question whether the
productions of antinuclei can also be explained by our model. In this paper, we predict and
throw light on the production yields of light nuclei, hypernuclei, and heavy baryons (anti-
baryons) within our thermal model approach and we compare our thermal model calculations
with the experimental data. The thermal freeze-out parameters for these yields are also taken
as the same as used for other ratios.
In Fig. 19, we show the energy dependence of ¯p/p, ¯
Λ/Λ,¯
He3/He3,¯
He4/He4and ¯
d/d
yield ratios over a very broad energy range. We take three times of volume and mass of each
nucleon to calculate the volume and mass of He3in our calculation. Similarly, we calculate
densities of other nuclei and hypernuclei using the similar method. We compare our results
with the experimental data [74] and find a good agreement between these two. We stress that
the agreement between the experimental and the calculated curves for the ratio ¯
He3/He3

is a powerful argument that indeed entropy conservation governs the production of light
nuclei. In Fig. 20, we show the variations of Λ/p, d/p with √sN N . We compare our results
with the experimental data points [74]. We also predict the energy dependence of nuclei as
well as antinuclei yield ratio such as H3
Λ/He3and ¯
H3
¯
Λ/¯
He3over a broad energy range going
from lower AGS energies to LHC energy. Some data have started appearing and we find
that our model gives a good fit to the data.
V. CONCLUSIONS
In conclusion, we find that our model provides a good fit to the variations of total multi-
plicities as well as mid-rapidity densities of various particles and we deduce a large freeze-out
volume of the fireball at RHIC energy and this picture supports the idea of a mixed phase
after QGP formation before hadronization because a huge size of a homogeneous fireball
source can only arise if a mixed phase has occurred before the formation of a hot, dense
HG. Further, we present an analysis of rapidity distributions and transverse mass spectra
of hadrons in central nucleus-nucleus collisions at various center-of-mass energies using our
recently proposed equation of state (EOS). We see that the stationary thermal source alone
cannot describe the experimental data fully unless we incorporate flow velocities in the lon-
gitudinal as well as in the transverse direction in our thermal model and as a result our
modified model predictions show a good agreement with the experimental data. Our anal-
ysis shows that a collective flow develops at each √sN N which increases further with the
increasing √sNN . The description of the rapidity distributions and transverse mass spectra
of hadrons at each √sN N matches very well with the experimental data. Although, we are
not able to describe successfully the spectra for multistrange particles which suggests that
a somewhat different type of mechanism is required. We find that the particle yields and
ratios measured in heavy-ion collisions are described well by our thermal model and show an
overwhelming evidence for chemical equilibrium at all beam energies. The rapidity distribu-
tions and transverse mass spectra which essentially are dependent on thermal parameters,
also show a systematic behaviour and their interpretations most clearly involve the pres-
ence of a collective flow in the description of the thermal model. We have also found that
our model together with a quasiparticle EOS for QGP gives the QCD phase boundary and
the location of critical point almost overlapping on the freeze-out curve as predicted earlier

[34]. In conclusion, we find that our model provides an excellent description for the thermal
and hadronic properties of the HG fireball and associated phase transitions existing in the
ultra-relativistic heavy-ion collisions.
In summary, we have formulated a thermodynamically consistent EOS for a hot and
dense HG fireball created in the ultra-relativistic heavy-ion collisions. We have incorporated
a geometrical hard-core size for baryons and antibaryons only. In our description mesons may
possess a size but they can penetrate and overlap on each other. Our prescription has also
made use of full quantum statistics [5]. It is encouraging to note that our results for particle-
ratios and their energy-dependence, total multiplicities, rapidity densities and rapidity and
momentum spectra of various particles consistently match with the experimental data. We
should emphasize that the hadrons emerging from a completely thermalized HG fireball do
not give any information regarding a primordial QGP phase existing before HG. However,
the large freeze-out volume of the fireball obtained in this analysis hints at a mixed phase
occurring before pure HG phase.
VI. ACKNOWLEDGMENTS
S.K.T. and P.K.S. are grateful to the Council of Scientific and Industrial Research (CSIR),
New Delhi, and University Grants Commission (UGC), New Delhi for providing research
grants.
[1] C. P. Singh, Phys. Rept. 236, 147 (1993); Int. J. Mod. Phys. A 7, 7185 (1992).
[2] M. Gazdzicki, M. Gorenstein and P. Seyboth, Acta Physica Polonica B 42, 307 (2011).
[3] E. V. Shuryak, Phys. Rept. 61, 71 (1980).
[4] J. Letessier and J. Rafelski, Hadrons and Quark-Gluon Plasma, Cambridge University Press,
U.K. (2004).
[5] S. K. Tiwari, P. K. Srivastava, and C. P. Singh, Phys. Rev. C 85, 014908 (2012).
[6] W. Busza and R. Leodoux, Ann. Rev. Nucl. Part. Sci. 36, 119 (1988).
[7] C. Blume and C. Markert, Prog. Part. Nucl. Phys. 66, 834 (2011).
[8] P. Huovinen, P. F. Kolb, and W. Heinz, Nucl. Phys. A 5, 698 (2002).

[9] L. D. Landau, Izv. Akad. Nauk. Sec. Fiz. 17, 51 (1953); J. D. Bjorken, Phys. Rev. D 27, 140
(1983).
[10] E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev. C 48, 2462 (1993).
[11] P. Braun-Munzinger et al., Phys.Lett. B 365, 1 (1996); Phys. Lett. B 344, 43 (1995).
[12] E. Schnedermann, and U. Heinz, Phys. Rev. Lett. 69, 2908 (1992).
[13] S. Q. Feng, and Y. Zhong, Phys. Rev. C 83, 034908 (2011); S. Q. Feng and X. B. Yuan, Sci.
China Ser. G 52, 198 (2009).
[14] T. Hirano, K. Morita, S. Muroya, and C. Nonaka, Phys. Rev. C 65, 061902 (2002); K. Morita,
S. Muroya, C. Nonaka, and T. Hirano, ibid. 66, 054904 (2002).
[15] M. Gyulassy, P. Levai, and I. Vitev, Phys. Rev. Lett. 85, 5535 (2000); Nucl. Phys. B 594, 371
(2001).
[16] D. Kharzeev and M. Nardi, Phys. Lett. B 507, 121 (2001).
[17] J. Cleymans and K. Redlich, Phys. Rev. Lett. 81, 5284 (1998); J. Cleymans and H. Satz,
Z. Phys.C 57, 135 (1993); J. Cleymans, H. Oeschler, and K. Redlich, Phys. Lett. B 485, 27
(2000); P. Braun-Munzinger, D. Magestro, K. Redlich, and J. Stachel, Phys. Lett. B 518, 41
(2001).
[18] D. H. Rischke, S. Bernard, and J. A. Maruhn, Nucl. Phys. A 595, 346 (1995); D. H. Rischke,
Y. Pursun, and J. A. Maruhn, ibid.595, 383 (1995); 596, 717(E) (1996).
[19] S. A. Bass et al., Prog. Part. Nucl. Phys. 41, 225 (1998).
[20] Y. M. Sinyukov, S. V. Akkelin, and Y. Hama, Phys. Rev. Lett. 89, 052301 (2002); S. V.
Akkelin, M. S. Borysova, and Yu. M. Sinyukov, Acta Phys. Hung. A 22, 165 (2005).
[21] D. K. Srivastava, J. Alam, and B. Sinha, Phys. Lett. B 296, 11 (1992); D. K. Srivastava, J.
Alam, S. Chakrabarty, B. Sinha, and S. Raha, Ann. Phys. 228, 104 (1993); D. K. Srivastava,
J. Alam, S. Chakrabarty, S. Raha, and B. Sinha, Phys. Lett. B 278, 225 (1992).
[22] P. F. Kolb, U. W. Heinz, P. V. Ruuskanen, and S. A. Voloshin, Phys. Lett. B 503, 58 (2001);
P. F. Kolb, P. Huovinen, U. W. Heinz, and H. Heiselberg, ibid. 500, 232 (2001); P. F. Kolb,
U. W. Heinz, P. Huovinen, K. J. Eskola, and K. Tuominen, Nucl. Phys. A 696, 197 (2001).
[23] U. Mayer and U. Heinz, ibid. 56, 439 (1997); E. Schnedermann and U. Heinz, ibid. 50, 1675
(1994); U. Heinz, Nucl. Phys. A 661, 140c (1999).
[24] F. Becattini and J. Cleymans, J. Phys. G 34, S959 (2007); F. Becattini, J. Cleymans and J.
Strumpfer, Proceeding of Science, (CPOD07), 012 (2007) [arXiv:0709.2599[hep-ph]].

[25] B. Biedron, and W. Broniowski, Phys. Rev. C 75, 054905 (2007); W. Broniowski and B.
Biedron, J. Phys. G 35, 0440189 (2008).
[26] J. Cleymans, J. Strumpfer and L. Turko, Phys. Rev. C 78, 017901 (2008).
[27] W. Broniowski, and W. Florkowski, Phys. Rev. Lett. 87, 272302 (2001); Phys. Rev. C 65,
064905 (2002).
[28] D. Teaney, J. Lauret, and E. V. Shuryak, Phys. Rev. Lett. 86, 4783 (2001).
[29] C. Adler et al., (STAR Collaboration), Phys. Rev. Lett. 87, 262302 (2001).
[30] K. Adcox et al., (PHENIX Collaboration), Phys. Rev. Lett. 88, 242301 (2002).
[31] K. Adcox et al., (PHENIX Collaboration), Phys. Rev. C 69, 024904 (2004).
[32] P. Bozek, Phys. Rev. C 77, 034911 (2008).
[33] C. P. Singh, B. K. Patra and K. K. Singh, Phys. Lett. B 387, 680 (1996).
[34] P. K. Srivastava, S. K. Tiwari, and C. P. Singh, Phys. Rev. D 82, 014023 (2010); Nucl. Phys.
A 862-863 CF, 424 (2011) ; C. P. Singh, P. K. Srivastava and S. K. Tiwari, Phys. Rev. D 80,
114508 (2009); Phys. Rev. D 83, 039904 (E) (2011).
[35] P. K. Netrakanti, and B. Mohanty, Phys. Rev. C 71, 047901 (2005).
[36] P. V. Ruuskanen, Acta Physica Polonica B 18, 551 (1987).
[37] F. Cooper and G. Frye, Phys. Rev. D 10, 186 (1974).
[38] D. Adamova et al., (CERES Collaboration), Phys. Rev. Lett. 90, 022301 (2003).
[39] F. Becattini et al., Phys. Rev. C 64, 024901 (2001).
[40] J. L. Klay et al., (E895 Collaboration), Phys. Rev. C 68, 054905 (2003).
[41] S. V. Afanasiev et al., (NA49 Collaboration), Phys. Rev. C 66, 054902 (2002).
[42] M. Gazdzicki, (NA49 Collaboration), J. Phys. G, 30, S701 (2004).
[43] I. G. Bearden et al., (BRAHMS Collaboration), Phys. Rev. Lett. 94, 162301 (2005).
[44] T. Anticic et al., (NA49 Collaboration), Phys. Rev. Lett. 93, 022302 (2004).
[45] A. Richard et al., (NA49 Collaboration), J. Phys. G, 31, S115 (2005).
[46] S. V. Afanasiev et al., (NA49 Collaboration), Phys. Lett. B 491, 59 (2000).
[47] C. Meurer et al., (NA49 Collaboration), J. Phys. G, 30, S1325 (2004).
[48] S. V. Afanasiev et al., (NA49 Collaboration), Phys. Lett. B 538, 275 (2002).
[49] L. Ahle et al., (E802 Collaboration), Phys. Rev. C 57, 466 (1998).
[50] L. Ahle et al., (E866 and E917 Collaboration), Phys. Lett. B 476, 1 (2000).
[51] L. Ahle et al., (E866 and E917 Collaboration), Phys. Lett. B 490, 53 (2000).

[52] L. Ahle et al., (E802 Collaboration), Phys. Rev. C 58, 3523 (1998).
[53] C. Pinkenburg et al., (E895 Collaboration), Nucl. Phys. A 698, 495c (2002).
[54] S. Albergo et al., (E896 Collaboration), Phys. Rev. Lett. 88, 062301 (2002).
[55] P. Chung et al., (E895 Collaboration), Phys. Rev. Lett. 91, 0202301 (2003).
[56] A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 772, 167 (2006).
[57] M. M. Aggarwal et al., (STAR Collaboration), Phys. Rev. C 83, 024901 (2011).
[58] M. Mishra and C. P. Singh, Phys. Rev. C 76, 024908 (2007); Phys. Rev. C 78, 024910 (2008);
Phys. Lett. B 651, 119 (2007).
[59] I. G. Bearden et al., (BRAHMS Collaboration), Phys. Rev. Lett. 94, 162301 (2005).
[60] S. V. Afanasiev et al., (NA49 Collaboration), Phys. Rev. C 66, 054902 (2002).
[61] I. G. Bearden et al., (BRAHMS Collaboration), Phys. Rev. Lett. 93, 102301 (2004).
[62] H. Yang, J. Phys. G, 35, 104129 (2008).
[63] S. S. Adler et al., (PHENIX Collaboration), Phys. Rev. C 69, 034909 (2004).
[64] B. I. Abelev et al., (STAR Collaboration), Phys. Rev. C 79, 064903 (2009).
[65] M. Floris, (ALICE collaboration), arXiv:1108.3257v1[hep-ex].
[66] C. Shen, U. Heinz, P. Huovinen, and H. Song, Phys. Rev. C 84, 044903 (2011).
[67] A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 772, 167 (2006).
[68] M. Bleicher et al., J. Phys. G 25, 1859 (1999).
[69] P. J. Siemens, and J. I. Kapusta, Phys. Rev. Lett. 43, 1486 (1979).
[70] D. Hahn, and H. St¨ocker, Nucl. Phys. A 476, 718 (1988).
[71] P. Braun-Munzinger, and J. Stachel, J. Phys. G 21, L17 (1995).
[72] P. Braun-Munzinger, and J. Stachel, J. Phys. G 28, 1971 (2002).
[73] B. I. Abelev et al., (STAR Collaboration), Science 328, 58 (2010).
[74] A. Andronic, P. Braun-Munzinger, J. Stachel, and H. St¨ocker, Phys. Lett. B 697, 203 (2011).