\documentclass[10pt]{article}

\usepackage[final]{epsfig}
\usepackage{amsmath}
\usepackage{parskip}

\setlength{\oddsidemargin}{0in}
\setlength{\evensidemargin}{0in}
\setlength{\textwidth}{6.25in}
\setlength{\topmargin}{-0.25in}
\setlength{\textheight}{8.5in}
\def\lsim{\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
    \raise1pt\hbox{$<$}}}         %less than or approx. symbol
\def\gsim{\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
    \raise1pt\hbox{$>$}}}         %greater than or approx. symbol
\def\ut#1{$\underline{\smash{\vphantom{y}\hbox{#1}}}$}
\def\overleftrightarrow#1{\vbox{\ialign{##\crcr
    $\leftrightarrow$\crcr
    \noalign{\kern 1pt\nointerlineskip}
    $\hfil\displaystyle{#1}\hfil$\crcr}}}
\renewcommand{\baselinestretch}{1.0}


\renewcommand{\thefootnote}{\fnsymbol{footnote})}

\begin{document}

\hspace{12cm}{\bf TUM/T39-02-18}\\
\hspace*{12cm}{\bf ECT$^\ast$-02-25}\\


\begin{center}
{\bf Statistical Hadronization and the Equation of State of the Quark Gluon
Plasma\footnote{work supported in part by BMBF, GSI and by the European Commission under
contract HPMT-CT-2001-00370}}
\end{center}
%-\vspace{.25 in}
\begin{center}
{Thorsten Renk$^{a}$}

{\small \em $^{a}$ Physik Department, Technische Universit\"{a}t M\"{u}nchen,
D-85747 Garching, GERMANY\\
and ECT$^\ast$, I-38050 Villazzano (Trento), ITALY}

\end{center}
\vspace{0.25 in}

\begin{abstract}
Statistical hadronization models are extremly succesful in describing
measured ratios of hadrons produced in heavy-ion collisions
for a wide range of beam energies
from SIS to RHIC.
Using the idea of statistical hadronization at the phase boundary
within the framework of a recently proposed model for the thermodynamics
of fireballs, we establish a relation between the
equation of state (EoS) in the partonic phase
probed at the phase transition temperature $T_C$ and the measured
hadron ratios. In this way, the ratios can be predicted parameter-free. 
We demonstrate that this framework gives a
consistent description of conditions both at SPS and RHIC.
As the data on dilepton emission from heavy ion collisions 
give evidence for strong in-medium modifications of hadronic
properties, we schematically investigate the sensitivity of our
results to such modifications and qualitatively
sketch a consistent scenario involving such in-medium effects.
\end{abstract}



\vspace {0.25 in}

\section{Introduction}
\label{sec_introduction}

The goal of current investigations in heavy-ion collisions
is to establish the existence of the quark-gluon plasma (QGP)
phase and to study its properties. Several signals have been proposed 
as indicators for
its creation. However, no unambiguous proof has been established so far.
A large part of the problem lies in the difficulty to separate
effects of the evolution of the hot and dense medium from
characteristic changes in the physics of reactions
taking place inside this medium. Therefore, it is mandatory
to aim at a consistent description of as many observables
as possible within a single model of the medium evolution.

In a recent paper \cite{MyDileptons}, we have proposed a model
for the evolution of a fireball assuming local thermal equilibrium
and isentropic expansion. In this approach, the evolution is constrained
by two major pieces of information: first, by 
measured hadronic momentum spectra and Hanbury-Brown Twiss (HBT) interferometry
data which reflect the freeze-out state; secondly, by information on the
Equation of State (EoS) obtained in lattice 
simulations and represented in terms of a 
quasiparticle picture. We have demonstrated that this evolution
scenario is in agreement with data on dilepton emission
measured by the CERES collaboration \cite{CERES}.

No explicit statement about the chemical composition
of the fireball is made. Instead, we use phenomenological
arguments to include the effects of enhanced pion phase space
density into the EoS of hot hadronic matter. On the other
hand, statistical hadronization models are extremely successful
in describing the measured ratios of different hadron
species for a range of collision energies from SIS to RHIC
(see e.g. \cite{PBM1,PBM2, PBM3}).
It is the aim of this paper to show that our fireball
evolution model is consistent with the idea of statistical
hadronization.

The paper is organized as follows: First, we outline the version
of statistical hadronization used in our model. After
comments on technical details, we present results and compare to
data both for SPS and RHIC conditions.
We then investigate the sensitivity of these results
to the model parameters, specifically we study the role
of possible in-medium modifications of particle properties.
We conclude by sketching a possible scenario in which
in-medium modifiactions are possible while statistical
hadronization is still in agreement with the data.

\section{The statistical hadronization model}

As outlined in \cite{PBM1,PBM2,PBM3},
we assume that hadronization at the critical temperature
$T_C$ creates a system of hadrons in chemical equilibrium which can
be described by the grand canonical ensemble, i.e. for each particle
species $i$ we expect the density $n_i$ to be given by
%
\begin{equation}
\label{E-GCE}
n_i = \frac{d_i}{2\pi^2}\int_0^\infty
\frac{p^2 dp}{\exp\{[E_i(p)-\mu_i]/T_C\}\pm 1}.
\end{equation}
%
Here, $d_i$ denotes the degeneracy factor of particle 
species $i$ (spin, isospin, particle / antiparticle), 
the +(-) sign is used for fermions (bosons) and 
$E_i(p)=\sqrt{m_i^2 + p^2}$.
The chemical potential $\mu_i$ takes care of conserved
baryon number $B_i$ and strangeness $S_i$ for each species:
%
\begin{equation}
\mu_i = \mu_B B_i - \mu_S S_i
\end{equation}
%
We neglect a (small) contribution $-\mu_{I_3} I_i^3$
coming from the isospin asymmetry in the colliding nuclei. 

The baryochemical potential $\mu_B$ is then fixed by
the requirement that the net number of baryons inside
the thermalized region is equal to the number of collision
participants $N_{part}$ if the total volume $V$ is known:
%
\begin{equation}
\label{E-Baryons}
V \sum_i n_i B_i = N_{part}.
\end{equation}
%
Similarly, strangeness conservation demands
%
\begin{equation}
\label{E-Strangeness}
V \sum_i n_i S_i = 0.
\end{equation}
%
Thus, the only parameter of the model is the fireball volume $V$.
At the phase transition, however, this volume can be determined
from the EoS of the QGP evaluated at $T=T_C$ using the entropy
density $s(T)$ as
% 
\begin{equation}
\label{E-Volume}
V(T_C) = S_{tot}/s(T_C),
\end{equation}
%
if the total entropy content $S_{tot}$ of the fireball is known.
This quantity, however,  can be
obtained from measuring charged particle multiplicities $N^+$ and $N^-$ 
in suitable rapidity bins and
calculating
%
\begin{equation}
%
D_Q = \frac{N^+ - N^-}{N^+ + N^-} \label{D_Q}.
%
\end{equation}
%
The quantity $D_Q$ stands for the inverse of the specific entropy 
per net baryon $S/B$, and the product $D_Q(S/B)$ roughly measures the entropy 
per pion \cite{ENTROPY-BARYON}. For SPS collisions at 
160 AGeV, we find an entropy per net baryon $S/B = 26$ for
central collisons. For RHIC 6\% central Au-Au collisions 
at 130 AGeV, the specific 
entropy $S/B = 220$ at midrapidity is substantially higher due to the
larger particle multiplicity 
and the smaller net baryon content in the central region. 

Thus, all ingredients entering Eq.~(\ref{E-GCE}) are determined
and we can evaluate the expression for a suitable
choice of hadrons and resonances.

We include all mesons and mesonic resonances up to masses of
1.5 GeV and all baryons and baryonic resonances up to masses
of 2 GeV.
This amounts to 30 (strange and nonstrange) mesonic states
and 36 (nonstrange to multistrange) baryonic states.
In order to compare to experimental results, we calculate
their decay into particles which are long-lived as compared
to the fireball, such as $\pi, K, \eta, N, \Lambda, \Sigma$ and
$\Omega$.

In order to account for interactions between particles, which at
small distances become repulsive, we assume a hard core radius
$R_{C}$ of 0.3 fm for all particles and resonances. The corresponding
excluded volume $V_{ex} = \sum_i N_i V_{ex}^i$,
with $V_i^{ex} = \frac{4\pi}{3} R_C^3$ for all species,  is subtracted 
from the volume obtained in Eq.~(\ref{E-Volume}), which in turn
affects the total number of produced particles. As the excluded
volume itself depends on the total number of particles, we iterate
the correction for a self-consistent result.
The hard core radius of 0.3 fm is determined by comparison  with
p-p collisions \cite{HardCore}.  In the absence of such information
for the other mesons and baryons, we assume its universality.

All data on particles is taken from \cite{ParticleDataBook}.
For many higher-lying states, the properties as well as the
decay channels are poorly known. In these cases, we proceed
as follows: If  a quantity (e.g. masses and widths)
is given only within a certain range, the arithmetic
mean of this range is used in the model. Decay channels
which are reported to be 'seen' are assumed to receive
equal contributions from the branching ratio which is left
after all known channels have been accounted for.
Branching ratios less than 1\% have been neglected. Decay
chains (such as $a_2 \rightarrow \rho \pi \rightarrow \pi \pi\pi$)
have been followed through.  For resonances with large width,
we integrate Eq.~(\ref{E-GCE}) over the mass
range of the resonance using a Breit-Wigner distribution.

\section{Results}

The resulting hadron ratios are shown in Fig.~\ref{F-RatiosStd} for the
case of 158 AGeV central Pb-Pb collisions at SPS, compared with
the experimentally measured values \cite{R1,R2,R3,R4,R5,R6,R7,R8,R9,R10,R10a}. 
One observes
that the overall agreement with data is satisfactory with few exceptions, 
though not quite as good as in e.g. \cite{PBM2}
where a two parameter fit to the data was performed. Note that not
only the ratios of hadron yields agree to experiment in the present approach but also the
absolute numbers, as the baryochemical potential $\mu_B$ is explicitly linked
to the (known) number of participants.

The calculation yields a baryochemical potential $\mu_B = 250$ MeV and a strange chemical
potential $\mu_S = 26.5$ MeV. Note that these quantities depend on the number
of resonances included into the calculation; therefore, a direct comparison with
the values obtained in other types of models is not meaningful.



\begin{figure}[htb]
\begin{center}
\epsfig{file=ratios_sps.eps, width=8cm}
\end{center}
\caption{\label{F-RatiosStd}Hadron ratios in the statistical hadronization
model (dashed bands) as compared to experimental results (filled circles) for SPS,
158 AGeV central Pb-Pb collisions.}
\end{figure}

For central collisions at RHIC at 130 AGeV beam energy   we present the results in Fig.~\ref{F-RatiosRHIC} and
compare to experimental data measured around midrapidity \cite{RHIC1,RHIC2,RHIC3,RHIC4,RHIC5,RHIC6}.
As the distribution of net baryon number in rapidity is very inhomogeneous at RHIC, the
naive application of Eq.~(\ref{E-Baryons}) fails, as we obtain a low
entropy per baryon, $S/B = 75$. At midrapidity, however, this ratio is closer to
$S/B = 220$. Using this value to calculate the ratios in a suitable interval around
midrapidity, we find much better agreement to the data (see Fig.~\ref{F-RatiosRHIC}).

\begin{figure}[htb]
\begin{center}
\epsfig{file=ratios_rhic130.eps, width=8cm}
\end{center}
\caption{\label{F-RatiosRHIC}Hadron ratios in the statistical hadronization
model at midrapidity (dashed bands) as compared to experimental results (filled circles) for RHIC,
130AGeV central Au-Au collisions at midrapidity.}
\end{figure}

\section{In-medium modifications}

In the last section, we have demonstrated that the idea of statistical hadronization
combined with our fireball evolution model is able to achieve good agreement 
in comparison with
data, provided that one uses the vacuum masses and widths of all resonances in
Eq.~(\ref{E-GCE}).  On the other hand, we have used the same fireball evolution
to calculate dilepton emission and here we found that a significant broadening of
the $\rho$ meson was responsible for the observed enhancement in the invariant
mass region below 700 MeV \cite{MyDileptons}.
The underlying thermal field theory calculations of the modifications
of the vector meson properties at finite temperature \cite{Omega,OldDileptons}
and density \cite{FiniteDensity} indicate not only a modified $\rho$
but also broadening and mass shift of the $\omega$ and broadening of the $\phi$
due to the interaction with the medium.

All these calculations (as any other perturbative expansion) become
unreliable in the vicinity of the phase transition, therefore these results
cannot strictly be taken over to the current calculations where we require
these quantities close to $T_C$. However, we may take them as a hint that
two possible modifications  of particle properties in a hot and dense medium may take
place:

\emph{Mass shifts} of  particles in the medium are commonly related to
the restoration of chiral symmetry. In e.g. \cite{BrownRho}, by investigating
scale invariance of an effective Lagrangian, the in-medium scaling laws
%
\begin{equation}
m_\rho^\ast/m_\rho \approx m_\omega^\ast/m_\omega \approx m_N^\ast/m_N \approx
\left(\langle \overline{q}q \rangle^\ast/\langle \overline{q}q \rangle \right)^{1/3}
\end{equation}
%
were established (Brown-Rho scaling). In this equation, asterisks denote
quantities at finite density, $m_\rho, m_\omega$ and $m_N$ the
masses of the $\rho$ and $\omega$ meson, $m_N$ is the nucleon mass
and $\langle \overline{q}q \rangle$ stands for the chiral condensate.

\emph{Decay widths} of particles are in general increased in a medium due to
the presence of new interaction channels. In \cite{Omega} e.g., it was shown that
the $\omega$ meson resonance experiences strong broadening in a hot environment due to
the presence of the scattering process $\omega \pi \rightarrow \pi \pi$.
Even in the absence of such effects, the decay width at finite temperature is
enhanced if the decay products are bosons due to the presence of
bosons of the same type in
the heat bath (Bose-enhancement). As pions are the most abundant species
in a thermal environment and most decays involve one or more pions, this
effect should influence almost all resonances in a hot medium.
The modification of decay widths is only relevant in Eq.~(\ref{E-GCE}) if
the in-medium width is a sizable fraction of the particle mass.
In this case, a large contribution to the particle yield comes from
masses lower than the peak mass in the Breit-Wigner distribution, which are
exponentially enhanced. This enhancement more than counterbalances
the suppression of contributions shifted into the higher mass
region. Therefore, an increase in the decay width of broad
resonances acts similarly as a mass reduction.

There is a third possibility of in-medium particle properties which has
no direct influence on the particle spectral function (and is
therefore not visible in the dilepton data):
The binding potential between the constituents of hadrons
could be partially screened by thermal fluctuations, leading to an
increase of the hadronic core radius.

Such \emph{increased radii} would lead to an enhanced excluded volume
correction. This effect hardly influences the results of
\cite{PBM1,PBM2,PBM3}, as the fireball volume is implicitly determined by
matching the fitted baryochemical potential to the number of participants,
but in the present approach we can expect to observe the influence of increased
hard core radii, as the volume is kept fixed.

In a first step, we examine the effects of in-medium mass shifts in a qualitative
way by tentatively multiplying the vacuum masses of all hadrons with the exception of  the
pion by a constant $c$. The result is shown in Fig.~\ref{F-MVar}.

\begin{figure}[htb]
\begin{center}
\epsfig{file=ratios_mvar.eps, width=8cm}
\end{center}
\caption{\label{F-MVar}Hadron ratios in the statistical hadronization model, for
vacuum particle masses (dashed), assuming a reduction by 10\% (dash-dotted) and 20 \% (solid)
as compared to data (filled circles) for SPS, 158 AGeV central Pb-Pb collisions.}
\end{figure}

The result shows that even a moderate mass reduction of 10\% in the medium is not in line with
the observed hadron ratios. In particular, particle-antiparticle ratios are strongly affected.
In the case of $\overline{p}/p$, one might argue that the relevant inelastic
(annihilation) cross section is not small as compared to the elastic one and therefore
this ratio cannot really be fixed at $T_C$, but must be  adjusted dynamically in the
subsequent evolution. Indeed, it was shown in \cite{Rapp-ppbar} that this
is possible if one takes the statistical hadronization prediction as an initial
condition for rate equations. It is unclear if this is still possible for
different initial conditions, but even if this is the case, this is not an option
for the multistrange particle/antiparticle ratios.


The overall behaviour of the result can be qualitatively
understood as follows: The reduced nucleon mass implies a
lower value of the baryochemical potential in order to produce the observed number of participants,
this in turn affects single and double strange particles and implies changes in $\mu_S$ via
Eq.~(\ref{E-Strangeness}). Therefore, ratios of particles and antiparticles
with 2 or 3 non-strange valence quarks, 
such as $\overline{p}/p$
or $\overline{\Lambda}/\Lambda$ are most affected. Multistrange particle/antiparticle
pairs follow the trend, though in a way less pronounced.

In a second run, we investigate the effect of thermal broadening of resonances,
increasing all widths by a constant multiplicative factor $c_\Gamma$. The resulting
hadron ratios are shown in Fig.~\ref{F-GammaVar}.

\begin{figure}[htb]
\begin{center}
\epsfig{file=ratios_gammavar.eps, width=8cm}
\end{center}
\caption{\label{F-GammaVar}Hadron ratios in the statistical hadronization model, for
vacuum decay widths (dashed), assuming an increase by 20\% (dash-dotted) and 50 \% (solid)
as compared to data (filled circles) for SPS, 158 AGeV central Pb-Pb collisions.}
\end{figure}

One observes the same qualitative behaviour as for a mass reduction, as we have argued before.
The effects of the increased width are, however, less dramatic. An increase by 20\% in the
width of all resonances is still in line with all data except $\overline{p}/p$ and even
an increase of 50\%  is still acceptable for most of the ratios. This is reassuring, as there
is almost certainly thermal broadening of resonances in a hot medium.

As the behaviour for both broadening of resonances and mass reduction (which are
both in line with the dilepton data) is qualitatively the same for the hadron ratios,
the effects of reduced masses cannot be compensated by introducing additional
broadening. So if any of the effects fails in the description of the data, a combination
of both will also fail. 

In the third run, we explore the effect of different choices for the core radius $R_C$
on the ratios. In order to investigate the sensitivity of our results to
the initial choice of $R_C$, we do not only consider thermally increased radii
but also a reduced initial choice. In Fig.~\ref{F-Rvar}, we show the model predictions
for a reduction of $R_C$ by 25\% and for an increase of the same amount.


\begin{figure}[htb]
\begin{center}
\epsfig{file=ratios_rvar.eps, width=8cm}
\end{center}
\caption{\label{F-Rvar}Hadron ratios in the statistical hadronization model, for
the standard choice of $R_C = 0.3$ fm (dashed), assuming $R_C = 0.225$ fm (dash-dotted) and
$R_C = 0.375$ fm (solid)
as compared to data (filled circles) for SPS, 158 AGeV central Pb-Pb collisions.}
\end{figure}

We observe that the overall sensitivity of the resulting hadron ratios
to the hadronic  core radius is rather weak. On the other hand, an enhanced
excluded volume
correction (dotted line in Fig.~\ref{F-Rvar}) acts in a rather peculiar way:
In order to arrive at the same number of participants, the baryochemical
potential has to \emph{increase}. This effect is opposite to the behaviour 
for in-medium mass reductions or increase of decay width. Therefore,
one might expect that the net effect of both thermal broadening of resonances \emph{and}
increased core radius due to a thermally screened binding potential
is moderate and partial compensation may occur.

In order to test this conjecture, we study a scenario in which
the width of resonances has been increased by 50\% and simultaneously the
core radius has been set to 0.45 fm instead of 0.3 fm. The result is
shown in Fig.~\ref{F-Radius}.

\begin{figure}[htb]
\begin{center}
\epsfig{file=ratios_gammaRvar.eps, width=8cm}
\end{center}
\caption{\label{F-Radius}Hadron ratios in the statistical hadronization model assuming
all decay widths increased by 50\%, for
the standard choice of $R_C = 0.3$ fm (dash-dotted) and $R_C = 0.45$ fm (solid)
as compared to data (filled circles) for SPS, 158 AGeV central Pb-Pb collisions.}
\end{figure}

One observes that indeed the expected compensation occurs and the model
prediction approaches the data points.  No effort has been made to
obtain a best description of the data using $c_\Gamma$ and $R_{hc}$ as
fit parameters. 


\section{Conclusions}

Assuming a thermalized system and statistical hadronization at the phase boundary
with subsequent resonance decays,
we have demonstrated that our recently proposed fireball evolution
scenario \cite{MyDileptons} leads to a reasonable description of the data
on hadron ratios. This is essentially a statement about the EoS in the
partonic phase --- once the entropy and baryon content of the relevant
rapidity region is known, the volume at the phase transition follows
uniquely and statistical hadronization can be calculated parameter-free
if one uses vacuum properties of particles and resonances.

On the other hand, in-medium modifications are mandatory
if one tries to explain the dilepton invariant mass spectrum
measured by the CERES collaboration \cite{CERES}. Specifically, the $\rho$ channel
requires strong broadening.
Thermal effective field theory calculations indicate the presence of such
effects for other particles as well.

In a schematic investigation, we have demonstrated that such strong
broadening of resonances or mass reductions are not in line with
the measured data, and no combination of these two effects can be.
However, if one also consideres the effect of a screened binding
potential in a thermal environment, the core radius
of hadrons should grow and this effect can compensate the
effects of both increased decay widths and mass reductions
to some degree. This result is reassuring, as it allows to reconcile the
in-medium modifications observed in the dilepton data with
the statistical model description of measured hadron ratios in a
consistent model framework.



\section*{Acknowledgements}

I would like to thank W.~Weise, A.~Polleri, R.~A.~Schneider, P.~Braun-Munzinger and J.~Stachel
for interesting discussions and helpful comments.


\begin{thebibliography}{99}

\bibitem{MyDileptons}
T.~Renk, R.~A.~Schneider and W.~Weise,
Phys.\ Rev.\ C {\bf 66} (2002) 014902.

\bibitem{CERES}
G.~Agakichiev et al., CERES collaboration, Phys. Rev. Lett. {\bf 75}
(1995) 1272;
G.~Agakichiev et al., CERES collaboration, Phys. Lett. {\bf B422}
(1998) 405.

\bibitem{PBM1}
P.~Braun-Munzinger, J.~Stachel, J.~P.~Wessels and N.~Xu,
Phys.\ Lett.\ B {\bf 344} (1995) 43.

\bibitem{PBM2}
P.~Braun-Munzinger, I.~Heppe and J.~Stachel,
Phys.\ Lett.\ B {\bf 465} (1999) 15.

\bibitem{PBM3}
P.~Braun-Munzinger, D.~Magestro, K.~Redlich and J.~Stachel,
Phys.\ Lett.\ B {\bf 518} (2001) 41.


\bibitem{ENTROPY-BARYON} J. Letessier, A. Tounsi, U. Heinz, J. Sollfrank and J. Rafelski, Phys. Rev. {\bf D51} (1995) 3408. 

\bibitem{ParticleDataBook}
D.E.~Groom et al. (Particle Data Group),
Eur.\ Phys.\ {\bf C 15} (2000) 1.

\bibitem{HardCore}
see, e.g., A.~Bohr and B.~Mottelson, Nucl. Structure (Benjamin, New York 1969), Vol.1, 266.


\bibitem{R1}
G.~Roland (NA49 Collaboration),
Nucl.\ Phys.\ {\bf A 638} (1998) 91c.

\bibitem{R2}
M.~Kaneta, (NA44 Collaboration),
Nucl.\ Phys.\ {\bf A 639} (1998) 419c.

\bibitem{R3}
P.~G.~Jones (NA49 Collaboration),
Nucl.\ Phys.\ {\bf A 610} (1996) 188c.


\bibitem{R4}
E.~Andersen et al. (WA97 Collaboration),
J.\ Phys.\ {\bf G 25} (1999) 171,\\
Phys.\ Lett.\ {\bf B 449} (1999) 401.

\bibitem{R5}
H.~Appelsh\"{a}user et al. (NA49 Collaboration),
Phys.\ Lett.\ {\bf B 444} (1998) 523.

\bibitem{R6}
F.~Gabler (NA49 Collaboration),
J.\ Phys.\ {\bf G 25} (1999) 199.

\bibitem{R7}
S.~Margetis (NA49 Collaboration),
J.\ Phys.\ {\bf G 25} (1999) 189.

\bibitem{R8}
D.~Jouan (NA50 Collaboration),
Nucl.\ Phys.\ {\bf A 638} (1998) 483c,\\
A.~de Falco (NA50 Collaboration),
Nucl.\ Phys.\ {\bf A 638} (1998) 487c.

\bibitem{R9}
F.~P\"{u}hlhofer (NA49 Collaboration),
Nucl.\ Phys.\ {\bf A 638} (1998) 431c.

\bibitem{R10}
H.~Appelsh\"{a}user et al. (NA49 Collaboration),
Eur.\ Phys.\ J.\ {\bf C 2} (1998) 611.

\bibitem{R10a}
A.~Mischke,
%``Energy dependence of Lambda and Antilambda production at CERN-SPS  energies,''
.


\bibitem{RHIC1}
C.~Adler {\it et al.}  (STAR Collaboration),
%``Mid-rapidity Lambda and anti-Lambda production in Au + Au collisions at  s(NN)**(1/2) = 130-GeV,''
Phys.\ Rev.\ Lett.\  {\bf 89} (2002) 092301.

\bibitem{RHIC2}
K.~Adcox {\it et al.}  (PHENIX Collaboration),
%``Measurement of the Lambda and anti-Lambda particles in Au + Au  collisions at s(NN)**(1/2) = 130-GeV,''
Phys.\ Rev.\ Lett.\  {\bf 89} (2002) 092302.

\bibitem{RHIC3}
C.~Adler {\it et al.},
%``Midrapidity phi production in Au+Au collisions at sNN =130 GeV,''
Phys.\ Rev.\ C {\bf 65} (2002) 041901.

\bibitem{RHIC4}
C.~Adler {\it et al.}  (STAR Collaboration),
%``Mid-rapidity anti-proton to proton ratio from Au + Au collisions at  s(N N)**(1/2) = 130-GeV,''
Phys.\ Rev.\ Lett.\  {\bf 86} (2001) 4778.

\bibitem{RHIC5}
C.~Suire (STAR Collaboration), talk given at Quark Matter 2002, Nantes.

\bibitem{RHIC6}
J.~Castillo (STAR Collaboration), talk given at Quark Matter 2002, Nantes.

\bibitem{Omega}
R.~A.~Schneider and W.~Weise,
Phys.\ Lett.\ B {\bf 515} (2001) 89.

\bibitem{OldDileptons}
R.~A.~Schneider and W.~Weise,
Eur.\ Phys.\ J.\ A {\bf 9} (2000) 357.

\bibitem{FiniteDensity}
F.~Klingl, N.~Kaiser and W.~Weise, Nucl. Phys. {\bf A606} (1996) 329.

\bibitem{BrownRho}
G.~E.~Brown and M.~Rho,
Phys.\ Rev.\ Lett.\ {\bf 66} (1991) 2720.

\bibitem{Rapp-ppbar}
R.~Rapp and E.~V.~Shuryak,
Phys.\ Rev.\ Lett.\  {\bf 86} (2001) 2980.


\end{thebibliography}



\end{document}

