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\draft

\title{Analysis of the freeze-out parameters for RHIC, SPS and AGS based on
${ {dE_{T}} \over {d\eta} } / { {dN_{ch}} \over {d\eta} }$ ratio
measurement }
\author{Dariusz Prorok}
\address{Institute of Theoretical Physics, University of
Wroc{\l}aw,\\ Pl.Maksa Borna 9, 50-204  Wroc{\l}aw, Poland}
\date{December 6, 2002}
\maketitle
\begin{abstract}
In the presented paper ${ {dE_{T}} \over {d\eta} } / { {dN_{ch}}
\over {d\eta} }$ ratio is analyzed in the framework of a
single-freeze-out thermal hadron gas model. Decays of hadron
resonances are taken into account in evaluations of the ratio. The
predictions of the model at the freeze-out parameters previously
established from observed particle yields agree very well with the
ratio measured at RHIC, SPS and AGS.
\end{abstract}

\pacs{PACS: 25.75.Dw, 24.10.Pa, 24.10.Jv}


In this letter, a statistical model is tested in recovering values
of ${ {dE_{T}} \over {d\eta} }_{\mid \eta=0} / { {dN_{ch}} \over
{d\eta} }_{\mid \eta=0}$ ratio measured at RHIC
\cite{Adcox:2001ry}, SPS \cite{Aggarwal:2000bc} and AGS
\cite{Barrette:1994kr}. For RHIC and SPS the ratio equals about
$0.8$ GeV, but for AGS it is $0.72$ GeV. So far, the model has
been applied successfully in explanation of particle ratios and
distributions observed in heavy-ion collisions
\cite{Braun-Munzinger:1994xr,Cleymans:1996cd,Braun-Munzinger:1999qy,Braun-Munzinger:2001ip,Florkowski:2001fp,Michalec:2001um,Broniowski:2001we,Broniowski:2001uk}.
Since the transverse energy measurement is independent and easier
(no particle identification is necessary), it gives an unique
opportunity to verify the concept of the appearance of a thermal
system during a heavy-ion collision.


In the presented paper, the statistical model with single
freeze-out is used
\cite{Florkowski:2001fp,Michalec:2001um,Broniowski:2001we,Broniowski:2001uk}.
The model reproduces very well ratios and $p_{T}$ spectra of
particles observed at RHIC
\cite{Florkowski:2001fp,Broniowski:2001we,Broniowski:2001uk}. The
main assumption of the model is the simultaneous occurrence of
chemical and thermal freeze-outs. The new data on $K^{*}(892)^{0}$
production revealed by the STAR Collaboration \cite{Adler:2002sw}
support strongly this assumption. Since ${ {dE_{T}} \over {d\eta}
}_{\mid \eta=0} / { {dN_{ch}} \over {d\eta} }_{\mid \eta=0}$
measurement has been done at midrapidity, the presented analysis
is valid in the Central Rapidity Region (CRR) of heavy-ion
collisions under consideration.


Therefore, it is assumed that a noninteracting gas of stable
hadrons and resonances at chemical and thermal equilibrium is
present at the CRR. A static fireball is considered, which is in
the spirit of
\cite{Braun-Munzinger:1994xr,Cleymans:1996cd,Braun-Munzinger:1999qy,Braun-Munzinger:2001ip,Florkowski:2001fp,Michalec:2001um},
where freeze-out parameters are found out from particle ratio
measurements also for a static case. Then the distributions of
various species of primordial particles are given by usual
Bose-Einstein and Fermi-Dirac formulae. Only baryon number
$\mu_{B}$ and strangeness $\mu_{S}$ chemical potentials are taken
into account, here. The isospin chemical potential $\mu_{I_{3}}$
has very low value in collision cases considered
\cite{Braun-Munzinger:1999qy,Florkowski:2001fp} and therefore can
be neglected. For given $T$ and $\mu_{B}$, $\mu_{S}$ is determined
from the requirement that the overall strangeness of the gas
equals zero. In this way, the temperature $T$ and the baryon
chemical potential $\mu_{B}$ are the only independent parameters
of the model.


Theoretically, the transverse energy is defined as the sum of
transverse masses of all $L$ interacting and produced particles
\cite{Albrecht:1991fg}

\begin{equation}
E_{T}^{th} = \sum_{i = 1}^{L} \sqrt{m_{i}^{2}+(p_{T}^{i})^{2}} \;,
\label{Thent}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%eq.1

\noindent where $m_{i}$ and $p_{T}^{i}$ are the mass and the
transverse momentum of particle $i$. But experimentally, the
measured quantity is

\begin{equation}
E_{T} = \sum_{i = 1}^{L} E_{i} \cdot \sin{\theta_{i}} \;,
\label{Exent}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%eq.2

\noindent where $E_{i}$ and $\theta_{i}$ are the energy and the
polar angel of particle $i$. Precisely, this what is measured in a
calorimeter is the kinetic energy for nucleons and the total
energy for all other particles \cite{Adcox:2001ry}.


Since the thermal system is dealt with, definitions (\ref{Thent})
and (\ref{Exent}) should be generalized appropriately. At the
temperature $T$ and the baryon chemical potential $\mu_{B}$, the
transverse energy density $\epsilon_{T}^{i}$ of specie $i$ could
be defined as ($\hbar=c=1$ always)

\begin{equation}
\epsilon_{T}^{i}= (2s_{i}+1)
\int_{-\infty}^{\infty}dp_{z}\;\int_{0}^{\infty}dp_{T}\;p_{T}\cdot
E_{T}^{i}(p_{z},p_{T})\; f_{i}(p;T,\mu_{B})\;, \label{energyt}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%eq.3

\noindent where $E_{T}^{i}(p_{z},p_{T})$ is the suitable
expression for the transverse energy of the specie and $m_{i}$,
$s_{i}$ and $f_{i}(p;T,\mu_{B})$ are its mass, spin and the
momentum distribution, respectively.


At the freeze-out the thermal system ceases and there are only
freely escaping particles instead of the fireball. The measured
${{dE_{T}} \over {d\eta} }_{\mid \eta=0}$ is fed from two sources:
(a) stable hadrons which survived until catching in a detector,
(b) secondaries produced by decays and sequential decays of
primordial resonances after the freeze-out. Therefore, if the
contribution to the transverse energy from particles (a) is
described, the distribution $f_{i}$ in (\ref{energyt}) is the
Bose-Einstein or Fermi-Dirac distribution at the freeze-out. But
if the contribution from particles (b) is considered, the
distribution $f_{i}$ is the spectrum of the finally detected
secondary and could be obtained from elementary kinematics of a
many-body decay or from the superposition of two or more such
decays (for details, see
\cite{Florkowski:2001fp,Broniowski:2001uk}; also
\cite{Sollfrank:1990qz,DeWit:it} could be very useful). In fact,
if one considers detected specie $i$, then $f_{i}$ is the sum of
final spectra of $i$ resulting from a single decay (cascade) over
all such decays (cascades) of resonances that at least one of the
final secondaries is of the kind $i$.


Since measured ${ {dN_{ch}} \over {d\eta} }_{\mid \eta=0}$ has
also its origin in the above-mentioned sources (a) and (b), to
properly define the density of charged particles, decays should be
also taken into account. Thus, the density of measured charged
particle $j$ reads

\begin{equation}
n^{j} = n_{primordial}^{j} + \sum_{i} \alpha(j,i)\;
n_{primordial}^{i} \;, \label{nchj}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%eq.4

\noindent where $n_{primordial}^{i}$ is the density of specie $i$
at the freeze-out and $\alpha(j,i)$ is the final number of specie
$j$ which can be received from all possible simple or sequential
decays of particle $i$. The density $n_{primordial}^{i}$ is given
by the usual integral of the Bose-Einstein or Fermi-Dirac
distribution.


Now, in the midrapidity region the theoretical equivalent of ${
{dE_{T}} \over {d\eta} }_{\mid \eta=0} / { {dN_{ch}} \over {d\eta}
}_{\mid \eta=0}$ could be defined as

\begin{equation}
{{{ {dE_{T}} \over {d\eta} }_{\mid \eta=0}} \over {{ {dN_{ch}}
\over {d\eta} }_{\mid \eta=0}}} \equiv { {\epsilon_{T}} \over
{n_{ch}} } \;, \label{thratio}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%eq.5

\noindent where the transverse energy density $\epsilon_{T}$ and
the density of charged particles $n_{ch}$ are given by

\begin{equation}
\epsilon_{T} = \sum_{i \in A} \epsilon_{T}^{i} \;, \label{enertot}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%eq.6

\begin{equation}
n_{ch} = \sum_{j \in B} n^{j} \;. \label{nchtot}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%eq.7

\noindent Note that there are two different sets of final
particles, $A$ and $B$ ($B \subset A$). $B$ denotes final charged
particles and these are $\pi^{+},\; \pi^{-},\; K^{+},\; K^{-},\;
p$ and $\bar{p}\;$, whereas $A$ also includes photons,
$K_{L}^{0},\; n$ and $\bar{n}\;$ \cite{Adcox:2001ry}.


To proceed further, some simplifications are necessary. This is
because the complete treatment of resonance decays in
$\epsilon_{T}$ is complex and consuming a lot of computer working
time in numerical calculations. Therefore the initial set of
resonances should be as small as possible. The lifetime of at
least $10$ fm is chosen as the necessary condition, with the
exception of $K^{*}(892)$ mesons, since one of them is measured at
RHIC \cite{Adler:2002sw}. This reduces constituents of the hadron
gas to $40$ species, but note that all particles listed in
\cite{Braun-Munzinger:1994xr,Cleymans:1996cd,Braun-Munzinger:1999qy,Braun-Munzinger:2001ip,Florkowski:2001fp}
are included. Of course, the condition is arbitrary but makes
sense because most neglected resonances have the lifetime of the
order of a few fm, so they could be thought of as decaying already
at the pre-equilibrium stage. Anyway, on the basis of an analysis
with decays not included in $\epsilon_{T}$, it has been found out
that the ratio (\ref{thratio}) changes slowly with the number of
species of the hadron gas in the region of $T$ and $\mu_{B}$
considered.


In the following, a point-like gas is assumed. It has been checked
that for the excluded volume hadron gas model
\cite{Hagedorn:1978kc,Hagedorn:1980kb,Yen:1997rv} the results are
exactly the same. There are two reasons for that: the first, the
volume corrections placed in denominators of expressions for
various densities cancel each other in a ratio; the second, the
eigenvolume of a hadron and the pressure in the region of $T$ and
$\mu_{B}$ considered are so small that their product correction to
the chemical potential is negligible there.


As it has been already mentioned, the main difficulty in complete
treatment of decays in numerical evaluation of $\epsilon_{T}$ is
their complexity. Therefore some further simplifications should be
done. First of all, some decays and cascades are neglected: (i)
four-body decays, (ii) superpositions of two three-body decays,
(iii) superpositions of two three-body and one two-body decays,
(iv) superpositions of four two-body decays, (v) some decays of
heavy resonances with very small branching ratios. Their maximal
contribution to $\epsilon_{T}$ has been evaluated at $2 \%$. Thus,
the real $\epsilon_{T}$ could be $2 \%$ higher maximally than its
evaluation.



In application to heavy-ion collisions, it is assumed that the
rest frame of the hadron gas is the c.m.s of two colliding ions.
But RHIC is the opposite beam experiment, whereas SPS and AGS are
fixed target ones. So, the laboratory frame is the c.m.s only in
the RHIC case. Since the measurement is done in the laboratory
frame, the momentum distributions in (\ref{energyt}) and in
integrals for $n_{primordial}^{i}$ should be Lorentz transformed
for SPS and AGS. For these cases, it is assumed that there is the
overall uniform flow (of the gas) with constant velocity $v$ equal
to the velocity $v_{c.m.s}$ of the c.m.s relative to the target.
And then, instead of energy E in the distributions, $p \cdot
u=\gamma \cdot (E-p_{z} v)$ is put, where $u$ is the appropriate
four-velocity and $\gamma = (1 - v^{2})^{-1/2}$. The above
expression is simply the energy $E^{*}$ in the frame moving with
the velocity $v$. Note, that since the momentum distributions of
particles are functions only of the magnitude of momentum here,
the change from the momentum dependence to the energy dependence
is trivial. The $v_{c.m.s}$ is calculated for $158 \cdot A$ GeV
Pb-Pb collisions at SPS (this results in $v_{c.m.s} = 0.994$) and
for $11 \cdot A$ GeV Au-Au collisions at AGS ($v_{c.m.s} =
0.918$).

\vskip 1cm

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|} \hline { \hbox to 3truecm{}}& \multicolumn{4}{c|}{} &
 \cr & \multicolumn{4}{c|}{${\epsilon_{T}} / {n_{ch}}$ [GeV]} &
\cr & \multicolumn{4}{c|}{} &
 \cr \cline{2-5} & for $<m_{T}>$ & \multicolumn{3}{c|}{for $<E \cdot {{p_{T}} \over
p}>$} & ${ {dE_{T}} \over {d\eta}} / {{dN_{ch}} \over {d\eta} }$
  \cr \cline{3-5} &  &  &
 & $E \rightarrow E-m_{n}$ &
 \cr & $v_{c.m.s} =
0$ & $v_{c.m.s} = 0$ & $v_{c.m.s} \not= 0$ & for nucleons, & [GeV]
\cr & & & & $v_{c.m.s}$ approp. & \cr \hline $T = 175$ MeV & & & &
& \cr $\mu_{B} = 51$ MeV & 0.96 & 0.88 & & 0.80 &
$0.8^{+0.08}_{-0.06}$ \protect\cite{Adcox:2001ry} \cr RHIC
\protect\cite{Braun-Munzinger:2001ip} & & & & & \cr \hline $T =
165$ MeV & & & & & \cr $\mu_{B} = 41$ MeV & 0.91 & 0.84 &  & 0.77
& $0.8^{+0.08}_{-0.06}$ \protect\cite{Adcox:2001ry} \cr RHIC
\protect\cite{Florkowski:2001fp} & & & & & \cr \hline $T = 168$
MeV & & & & & \cr $\mu_{B} = 266$ MeV & 1.03 & 0.94 & 0.76 & 0.75
& $0.8 \pm 0.2$ \protect\cite{Aggarwal:2000bc} \cr SPS
\protect\cite{Braun-Munzinger:1999qy} & & & & & \cr \hline $T =
164$ MeV & & & & & \cr $\mu_{B} = 234$ MeV & 0.99 & 0.90 & 0.74 &
0.72 & $0.8 \pm 0.2$ \protect\cite{Aggarwal:2000bc} \cr SPS
\protect\cite{Michalec:2001um} & & & & & \cr \hline $T = 130$ MeV
& & & & & \cr $\mu_{B} = 540$ MeV & 1.30 & 1.13 & 0.79 & 0.66 &
$0.72 \pm 0.08$ \protect\cite{Barrette:1994kr} \cr AGS
\protect\cite{Braun-Munzinger:1994xr} & & & & & \cr \hline $T =
110$ MeV & & & & & \cr $\mu_{B} = 540$ MeV & 1.17 & 1.01 & 0.69 &
0.57 & $0.72 \pm 0.08$ \protect\cite{Barrette:1994kr} \cr AGS
\protect\cite{Cleymans:1996cd} & & & & & \cr \hline
\end{tabular}
\end{center}
\begin{table}
\caption{Values of ${{\epsilon_{T}} \over {n_{ch}}}$ ratio
calculated with the averaging of transverse mass or energy times
$\sin{\theta}$ ($\theta$ is the polar angel) in the numerator. In
the first column estimates of freeze-out parameters obtained from
the analysis of particle ratios
\protect\cite{Braun-Munzinger:1994xr,Cleymans:1996cd,Braun-Munzinger:1999qy,Braun-Munzinger:2001ip,Florkowski:2001fp,Michalec:2001um}
are listed. In the last column experimental data are given. The
velocity $v_{c.m.s}$ is the velocity of the center of the mass of
colliding nuclei relatively to the laboratory frame and equals 0
for RHIC, 0.994 for SPS and 0.918 for AGS ($v_{c.m.s}$ appropriate
in the fifth column means that).} \label{Table1}
\end{table}


The final results of calculations of (\ref{thratio}) are presented
in TABLE\,\ref{Table1}. In the first column estimates of
freeze-out parameters obtained from the analysis of particle
ratios
\cite{Braun-Munzinger:1994xr,Cleymans:1996cd,Braun-Munzinger:1999qy,Braun-Munzinger:2001ip,Florkowski:2001fp,Michalec:2001um}
are listed. In the second column corresponding values of the ratio
calculated with the use of the transverse mass as the expression
for the transverse energy in (\ref{energyt}) are placed.
Generally, the predictions do not agree with the data and the
discrepancy increases with lowering the collision energy. In
columns from third to fifth, results of evaluations of the ratio
for $E \cdot \sin{\theta}$ as the expression for the transverse
energy in (\ref{energyt}) are listed. To check how the change from
the transverse mass to the measured quantity itself influences the
results, the values of the ratio calculated in the rest frame of
the gas are put in the third column. Note that in spite of
substantial improvement obtained for SPS (now the results are
within error bars of the data) and AGS, the discrepancy seen for
the latter is still too big. But when calculations are done in the
proper frame (see the fourth column of TABLE\,\ref{Table1}), the
very good agreement with the data is reached. In the fifth column
of TABLE\,\ref{Table1}, the final, most realistic results are
presented, namely the appropriate $v_{c.m.s}$ is put for every
collider and the fact that for nucleons only kinetic energy is
measured is taken into account. For clearness, these results have
been also put together with the data in Fig.\,\ref{Fig.1.}.


\begin{figure}
\begin{center}{
{\epsfig{file=Etncsnn.eps,width=7cm}} }\end{center}
\caption{Values of $\epsilon_{T} / n_{ch}$ ratio from the fifth
column of TABLE\,\ref{Table1}. Black dots and crosses denote
evaluations of the ratio at higher and lower temperature for a
given collider, respectively (see the first column of
TABLE\,\ref{Table1}). Also data points for AGS
\protect\cite{Barrette:1994kr} (circle), SPS
\protect\cite{Aggarwal:2000bc} (triangle) and RHIC
\protect\cite{Adcox:2001ry} (square) are depicted.}
\label{Fig.1.}%1
\end{figure}

One can notice that the accuracy of theoretical predictions rises
with the collision energy. For AGS, $\epsilon_{T} / n_{ch}$ ratio
is within error bars of the experimental point only for the higher
temperature freeze-out, whereas for RHIC both estimates fit very
well. This could mean that AGS energy is the low limit of
applicability of a statistical model. Alternatively, one could
call the application of the same gas for both RHIC and AGS in
question. This is because of a different baryon content of both
cases. AGS gas should be much more baryon rich than RHIC one. If
one considers contributions to primordial $\epsilon_{T}$ (i.e.
decays are not included in calculation of $\epsilon_{T}$) from
constituents of the gas assumed here, nucleons weighted most for
AGS, but pions are the biggest fraction in RHIC case. So, probably
additional baryon resonances should be included in a gas
corresponding to AGS case. In fact, some preliminary estimates of
primordial $\epsilon_{T}$ indicate that adding more species into
the gas causes the increase of $\epsilon_{T} / n_{ch}$ ratio at
AGS freeze-out. Also taking the expansion of the gas into account
could improve the results for AGS case. Roughly speaking, the
expansion produces additional energy, so it could increase
$\epsilon_{T}$. These problems need much more detailed analysis
and will be under further investigations.


To conclude, a statistical model has been used to reproduce ${
{dE_{T}} \over {d\eta} } / { {dN_{ch}} \over {d\eta} }$ ratio
measured at RHIC, SPS and AGS. The importance of presented
analysis lies in the fact that the ratio is an independent
observable, so it can be used as a new tool to verify the
consistence of predictions of a statistical model for all
colliders simultaneously. The point-like non-interacting hadron
gas with $40$ species has been used in final calculations. Decays
and sequential decays of constituents of the gas have been taken
into account. In spite of the simplicity of the model, theoretical
predictions for $\epsilon_{T} / n_{ch}$ ratio agree very well with
the data for the wide range of collision energies starting from
AGS up to RHIC. The predictions have been made at the previous
estimates of freeze-out parameters obtained from the analysis of
measured particle ratios for RHIC
\cite{Braun-Munzinger:2001ip,Florkowski:2001fp}, SPS
\cite{Braun-Munzinger:1999qy,Michalec:2001um} and AGS
\cite{Braun-Munzinger:1994xr,Cleymans:1996cd}. This means that the
applicability of a statistical model to heavy-ion collisions has
been confirmed strongly in an independent way.



The author acknowledges very stimulating discussions with Peter
Braun-Munzinger and Ludwik Turko. This work was supported in part
by the Polish Committee for Scientific Research under Contract No.
KBN - 2 P03B 030 18.


\begin{thebibliography}{99}
\bibitem{Adcox:2001ry}
K.~Adcox {\it et al.}  [PHENIX Collaboration],
%``Measurement of the mid-rapidity transverse energy distribution from  s(N N)**(1/2) = 130-GeV Au + Au collisions at RHIC,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 052301 (2001)
.
%%CITATION = ;%%

\bibitem{Aggarwal:2000bc}
M.~M.~Aggarwal {\it et al.}  [WA98 Collaboration],
%``Scaling of particle and transverse energy production in 208-Pb + 208-Pb  collisions at 158-A-GeV,''
Eur.\ Phys.\ J.\ C {\bf 18}, 651 (2001) .
%%CITATION = ;%%

\bibitem{Barrette:1994kr}
J.~Barrette {\it et al.}  [E877 Collaboration],
%``Charged particle pseudorapidity distributions in Au + Al, Cu, Au, and U collisions at 10.8-A/GeV/c,''
Phys.\ Rev.\ C {\bf 51}, 3309 (1995) .
%%CITATION = ;%%

\bibitem{Braun-Munzinger:1994xr}
P.~Braun-Munzinger, J.~Stachel, J.~P.~Wessels and N.~Xu,
%``Thermal equilibration and expansion in nucleus-nucleus collisions at the AGS,''
Phys.\ Lett.\ B {\bf 344}, 43 (1995) .
%%CITATION = ;%%

\bibitem{Cleymans:1996cd}
J.~Cleymans, D.~Elliott, H.~Satz and R.~L.~Thews,
%``Thermal hadron production in Si-Au collisions,''
Z.\ Phys.\ C {\bf 74}, 319 (1997) .
%%CITATION = ;%%

\bibitem{Braun-Munzinger:1999qy}
P.~Braun-Munzinger, I.~Heppe and J.~Stachel,
%``Chemical equilibration in Pb + Pb collisions at the SPS,''
Phys.\ Lett.\ B {\bf 465}, 15 (1999) .
%%CITATION = ;%%

\bibitem{Braun-Munzinger:2001ip}
P.~Braun-Munzinger, D.~Magestro, K.~Redlich and J.~Stachel,
%``Hadron production in Au Au collisions at RHIC,''
Phys.\ Lett.\ B {\bf 518}, 41 (2001) .
%%CITATION = ;%%

\bibitem{Florkowski:2001fp}
W.~Florkowski, W.~Broniowski and M.~Michalec,
%``Thermal analysis of particle ratios and p(T) spectra at RHIC,''
Acta Phys.\ Polon.\ B {\bf 33}, 761 (2002)
.
%%CITATION = ;%%

\bibitem{Michalec:2001um}
M.~Michalec,
%``Thermal description of particle production in ultra-relativistic  heavy-ion collisions,''
arXiv:.
%%CITATION = ;%%

\bibitem{Broniowski:2001we}
W.~Broniowski and W.~Florkowski,
%``Explanation of the RHIC p(T)-spectra in a thermal model with expansion,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 272302 (2001)
.
%%CITATION = ;%%

\bibitem{Broniowski:2001uk}
W.~Broniowski and W.~Florkowski,
%``Description of strange particle production in Au+Au collisions of sNN =130 GeV in a single-freeze-out model,''
Phys.\ Rev.\ C {\bf 65}, 064905 (2002) .
%%CITATION = ;%%

\bibitem{Adler:2002sw}
C.~Adler {\it et al.}  [STAR Collaboration],
%``K*(892)0 production in relativistic heavy ion collisions at  S(NN)**(1/2) = 130-GeV,''
arXiv:.
%%CITATION = ;%%

\bibitem{Albrecht:1991fg}
R.~Albrecht {\it et al.}  [WA80 Collaboration],
%``Distributions of transverse energy and forward energy in O-16 and S-32 induced heavy ion collisions at 60-A/GeV and 200-A/GeV,''
Phys.\ Rev.\ C {\bf 44}, 2736 (1991).
%%CITATION = PHRVA,C44,2736;%%

\bibitem{Sollfrank:1990qz}
J.~Sollfrank, P.~Koch and U.~W.~Heinz,
%``The Influence Of Resonance Decays On The P(T) Spectra From Heavy Ion Collisions,''
Phys.\ Lett.\ B {\bf 252}, 256 (1990).
%%CITATION = PHLTA,B252,256;%%

\bibitem{DeWit:it}
B.~De Wit and J.~Smith, {\it Field Theory In Particle Physics.
Vol. 1}, (North-Holland Personal Library, Amsterdam, Netherlands,
1986), pp.114-119.

\bibitem{Hagedorn:1978kc}
R.~Hagedorn, I.~Montvay and J.~Rafelski,
%``Thermodynamics Of Nuclear Matter From The Statistical Bootstrap Model,''
in {\it Proceedings of the Workshop on Hadronic Matter at Extreme
Energy Density, Erice, Italy, Oct 13-21, 1978}, edited by N.
Cabibbo and L. Sertorio (Plenum Press, New York, 1980), pp.
49-148.

\bibitem{Hagedorn:1980kb}
R.~Hagedorn and J.~Rafelski,
%``Hot Hadronic Matter And Nuclear Collisions,''
Phys.\ Lett.\ B {\bf 97}, 136 (1980).
%%CITATION = PHLTA,B97,136;%%

\bibitem{Yen:1997rv}
G.~D.~Yen, M.~I.~Gorenstein, W.~Greiner and S.~N.~Yang,
%``Excluded volume hadron gas model for particle number ratios in  A + A collisions,''
Phys.\ Rev.\ C {\bf 56}, 2210 (1997) .
%%CITATION = ;%%
\end{thebibliography}
\end{document}

