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\title{Kaon and Proton Ratios from Central Pb+Pb Collisions 
       at the CERN SPS}

\author{M. Kaneta\address{ Department of Physics, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima 739-0046, Japan}
        {\it for the NA44 Collaboration}\address{ NBI - LANL - Columbia - Nantes - Hiroshima - CERN - Rusjer Boskovic - OSU - SUNY - Lund - Texas A\&M - BNL} \\
        \vspace{1mm}
        {\small 
         I.G.Bearden, H.B{\o}ggild, J.Boissevain, J.R.Dodd, B.Erazmus, S.Esumi, C.W.Fabjan, D.Ferenc, \\
         D.E.Fields, A.Franz, J.J.Gaardh{\o}je, M.Hamelin, A.G.Hansen, O.Hansen, D.Hardtke, \\
         H.van Hecke, E.B.Holzer, T.Humanic, P.Hummel, B.V.Jacak, R.Jayanti, K.Kaimi, M.Kaneta, \\
         M.Kopytine, M.Leltchouk, A.Ljubicic Jr., B.L{\o}rstad, N.Maeda, R.Malina, M.Murray, \\
         H.Ohnishi, G.Paic, S.U.Pandey, F.Piuz, J.Pluta, V.Polychronakos, M.Potekhin, G.Poulard, \\
         D.Reichhold, A.Sakaguchi,  J.Schmidt-S{\o}rensen, J.Simon-Gillo, W.Sondheim, M.Spegel, \\
         T.Sugitate, J.P. Sullivan, Y.Sumi, W.J.Willis, K.L.Wolf, N.Xu  and D.S.Zachary}
       }

\begin{document}

\maketitle

\begin{abstract}
  $K^-/K^+$ and $\bar{p}/p$ ratios measured in 158 A$\cdot$GeV Pb+Pb 
collisions are shown as a function of transverse momentum $P_T$ and centrality
in top 8.5\% central region.
  Little centrality dependence of the $K^-/K^+$ and $\bar{p}/p$ ratios is 
observed.
  The transverse mass $m_{T}$ distribution and $dN/dy$ of $K^+$, $K^-$,
$p$ and $\bar{p}$ around mid-rapidity are obtained.
  The temperature $T_{ch}$ and the chemical potentials for both light and 
strange quarks ($\mu_q, \mu_s$) at chemical freeze-out are determined by 
applying simple thermodynamical model to the present data.
  The resultant  $\mu_q, \mu_s$ and $T_{ch}$ are compared with those
obtained from similar analysis of SPS S+A and AGS Si+A data.
  The chemical freeze-out temperature $T_{ch}$ at CERN energies
is higher than thermal freeze-out temperature $T_{f\!o}$ which is
extracted from $m_{T}$ distribution of charged hadrons. 
  At AGS energies $T_{ch}$ is close to $T_{f\!o}$.
\end{abstract}


\section{INTRODUCTION}
  Studies of particle ratios address the nature of high
energy/baryon density states formed in ultra relativistic heavy ion 
collisions.
  In the collisions, the energetic beam makes the hot and dense
collision complex.
  Lots of hadrons are created and a huge number of collisions occurs
many of these produce strangeness.
  Therefore, thermo-chemical model seems adequate to describe the system.
  After the high density era, the system expands and it becomes out
of equilibrium.
  In this situation, we can discuss thermodynamical quantities at the
time of chemical and thermal freeze-out from the particle ratios
and transverse momentum spectra.
 In SPS and AGS experiments, the transverse mass distributions of
hadrons suggest that there are local thermal equilibrium state and some
indications of transverse flow.

\section{NA44 EXPERIMENT}

  The NA44 focusing spectrometer is designed to measure one and two
particle momentum distribution of charged hadrons around mid
rapidity.
  The TOF resolution is 100ps.
  Particle identification is done by TOF and two different threshold 
Cherenkov counters.
  Momentum resolution $\delta p/p$ is about 0.5\% and the centrality is
measured by multiplicity counters behind the target.
  Details about the spectrometer can be found in 
Ref.~\cite{AS01}.

\section{PARTICLE RATIOS AND THERMODYNAMICAL ANALYSES}
\subsection{Kaon and proton distributions around mid-rapidity}
  Fig.~\ref{fig:ratio_pt} shows $K^{-}/K^{+}$ and $\bar{p}/p$ ratios as
a function of centrality and $P_T$.
  The centrality of event is defined by pulse height of the
multiplicity counter which is proportional to number of relativistic
particles in forward direction.
  We select three groups of highly central events; the most central
3.5\%, 3.5-6.0\% and 6.0-8.5\%.
  The particle ratios both $K^{-}/K^{+}$ and $\bar{p}/p$ vary weakly 
in the three groups.
  The $m_{T}$ distributions of $K^{+}$, $K^{-}$, $p$ and $\bar{p}$ in the 
top 8.5\% central region are shown in Fig.~\ref{fig:mtdist}.
  The line in the figures is fit of a single exponential function: $A\exp(m_{T}/T)$,
where $T$ is the slope parameter.
The values of $T$ and $dN/dy$ derived from integration of the fit
function are in Table \ref{tab:mtdndy}.  

\vspace{-1mm}
\begin{figure}[h]
  \begin{minipage}[t]{78mm}
    \epsfile{file=ratios_pt4.eps,scale=0.38}
    \vspace{-10mm}
    \caption{$K^{-}/K^{+}$ and $\bar{p}/p$ ratio as a function of $P_{T}$ and centrality.}
    \label{fig:ratio_pt}
  \end{minipage}
  \hspace{\fill}
  \begin{minipage}[t]{79mm}
    \epsfile{file=slope_kp_047.eps,scale=0.605}
    \vspace{-10mm}
    \caption{$m_{T}$ distribution of $K^{+}, K^{-}, p$ and $\bar{p}$. 
             The vertical axis is absolute cross section.}
    \label{fig:mtdist}
  \end{minipage}
  \vspace{-1cm}
\end{figure}


\begin{table}[h]
  \caption{The slope parameter and $dN/dy$. 
           The systematic errors are less than 4\% for slope parameter 
           and less than 20\% for $dN/dy$. 
           The values with correction of $\Lambda$ and $\Sigma^{+}$
           decay effect for proton yield are shown in the bracket. 
           The systematic errors in the bracket are from the correction factor only.}
  \label{tab:mtdndy}
  \begin{tabular}{clclcl}
    \hline
              & slope parameter$[$MeV$]$ & & $dN/dy$ & & y \\ 
    \hline
    \multicolumn{1}{c}{%
      $\begin{array}{c}
         \makebox[9mm]{$K^{+}$}  \\
         \makebox[9mm]{$K^{-}$}  
       \end{array}$
    }
              & \multicolumn{1}{@{}l@{}}{
                  $\begin{array}{l}
                     237\pm3  \\
                     \hline 
                     238\pm3  
                   \end{array}$
                }
              & \makebox[2mm]
              & \multicolumn{1}{@{}l@{}}{
                  $\begin{array}{l}
                     21.3\pm0.8 \\
                     \hline
                     11.5\pm0.4 
                   \end{array}$
                }
              & \makebox[2mm]
              &  2.4-3.5 \\
    \hline
    \multicolumn{1}{c}{%
      $\begin{array}{c}
         \makebox[9mm]{$p$}        \\
         \makebox[9mm]{$\bar{p}$}  
       \end{array}$
    }
              & \multicolumn{1}{@{}l@{}}{
                  $\begin{array}{ll}
                     303\pm6 & (331\pm10\pm11) \\
                     \hline 
                     281\pm8 & (316\pm13\pm15)   
                   \end{array}$
                }
              &
              & \multicolumn{1}{@{}l@{}}{
                  $\begin{array}{ll}
                     25.3\pm1.0  & (20.1\pm1.2\pm1.3)   \\
                     \hline
                     1.64\pm0.09 & (1.11\pm0.19\pm0.11)
                   \end{array}$
                }
              &
              &    2.3-2.9 \\
    \hline
  \end{tabular}
\end{table}

  It is noted that particle identification is made about 20 m
downstream from target.
  Thus, observed particle yield should be corrected to take into
account decays of unstable particles.
  We estimate the effect of $\Lambda$ and $\Sigma^{+}$ decays to
proton yield by a Monte Carlo simulation.
  The inputs of simulation are parameterised $dN/dy$ and $m_{T}$ slopes
distributions by RQMD (V1.08) and experimental results from Ref.~\cite{PJ01}.
  The values with correction of $\Lambda$ and $\Sigma^{+}$ decay effect
are shown in the bracket on the Table \ref{tab:mtdndy}.
  To estimate the systematic error, the ratio ($\Lambda$+$\Sigma^{+}$)/$p$ 
were increasied and decreased by a factor 1.5.

\subsection{Strangeness neutral hadron gas model}
  A system in chemical equilibrium is described by temperature and
chemical potentials.
  Here, we analyse the present data by a thermo-chemical model of
Ref.~\cite{JL02}(Strangeness neutral model). 
  It considers chemical equilibrium between strange and antistrange
hadrons under a condition of zero net strange charge.
  The model assumes full chemical equilibrium for $u$ and $d$ quarks.
  On the other hand, there is strangeness saturation factor $\gamma_{s}$;
  $\gamma_{s}=1$ means full chemical equilibrium for $s$ quark.
  In this model, we assume hadron gas including well-established higher mass 
resonances up to 2.0 GeV.

\subsection{$\mu_{q}$, $\mu_{s}$ and $T_{ch}$ at the chemical freeze-out}
  We determined chemical potentials, temperature and $\gamma_{s}$ 
at the chemical freeze-out in the following way.
  $\mu_{s}$ is given as a function of $\mu_{q}$ and $T_{ch}$
for different $\gamma_{s}$ from the model.
%   The calculation is done by numerical method with 5 MeV step
% for the potentials and temperature and 0.05 step for $\gamma_{s}$.  
  The ratios from $K^{+}$, $K^{-}$, $p$ and $\bar{p}$ are 
re-described by $\mu_{q}$, $\mu_{s}$, $T_{ch}$ and $\gamma_{s}$.
  Since net strange charge is zero, three of the four parameters
$\mu_{q}$, $\mu_{s}$, $T_{ch}$ and $\gamma_{s}$ are independent.
  For the ratios, $\chi^{2}$ map for various 
$\mu_{q}$, $T_{ch}$ and $\gamma_{s}$ is made.
  As a region of $\chi^{2}$ minimum + 1, $T_{ch}$ is obtained as $155\sim170$ MeV,
$\mu_{q} = 75\sim85$ MeV, $\mu_{s} = 22\sim31$ MeV and $\gamma_{s} = 0.75\sim0.95$.

\begin{figure}[b]
  \vspace{-7mm}
  \begin{center}
    \epsfile{file=phasediagram.eps,scale=0.6685}
    \vspace{-10mm}
    \caption{Various $\mu_{q}$, $\mu_{s}$ and $T_{ch}$ in heavy ion collisions at the SPS energies and the AGS energies.
             The points for S+A and for Si+A are referred from Ref.s~$[$6-9$]$.}
    \label{fig:sum_pha}
  \end{center}
\end{figure}

  Fig.~\ref{fig:sum_pha}~(a) shows a compilation of results for $\mu_q$ and
$T_{ch}$ at SPS and AGS energies including those of different
combinations of projectile and target.
   As seen in the  Fig.~\ref{fig:sum_pha}~(a), the $T_{ch}$ at SPS
energies is higher than at AGS energies, whereas $\mu_{q}$ at SPS
energies is lower than at AGS energies.
  This is consistent with higher baryon stopping at AGS energies.
  Fig.~\ref{fig:sum_pha}~(b) show $\mu_{s}$. It is 3$\sim$40 MeV at
SPS energies, while 60$\sim$120 MeV at AGS energies.
  In this compilation, dependence of $\mu_{q}$, $\mu_{s}$ and $T_{ch}$ 
on collision system is less clear than $\sqrt{s}$ dependence, 
since variation between different analysis is not negligible.

  The thermal equilibrium temperature $T_{f\!o}$ is independent of the 
collision system at AGS and SPS energy region and seems to saturate 
form AGS energies.
  The saturation temperature is about 145 MeV\cite{NX01}.
  The $T_{ch}$ is higher than $T_{f\!o}$ at SPS energies.
  On the other hand, $T_{f\!o}$ at AGS energies $T_{ch}$ is close to $T_{f\!o}$. 

\section{SUMMARY}
  We report $K^{-}/K^{+}$ and $\bar{p}/p$ ratios of Pb+Pb central
collisions at SPS energies as a function of $P_{T}$ and centrality.
  We observe little centrality dependence in either kaons and protons 
ratios in the top 8.5\% central region.
  The $m_{T}$ distribution and $dN/dy$ of $K^{+}, K^{-}$, $p$ and 
$\bar{p}$ around mid-rapidity are shown.
  The inverse slope parameters of $K^{+}$ and $K^{-}$ are the same within errors.
On the other hand, the inverse slope parameter of $\bar{p}$ is smaller than one of $p$.
  Assuming the system around mid-rapidity is in chemical equilibrium,
from ratios of $K^{+}, K^{-}$, $p$ and $\bar{p}$,
the chemical freeze-out temperature is found
to be $T_{ch} = 155\sim170$ MeV, the chemical potential of light and
strange quark was calculated to be: $\mu_{q} = 75\sim85$ MeV and
$\mu_{s} = 22\sim31$ MeV, $\gamma_{s} = 0.75\sim0.95$, respectively.
  The chemical freeze-out temperature is larger than the freeze-out
temperature.
  Those results are quite natural with the view point of thermodynamics.

  We compile the value of $\mu_{q}$, $\mu_{s}$ and $T_{ch}$ in heavy ion
collisions at SPS and AGS energies.
  The general trend is that $\mu_q$ and $\mu_s$ ($T_{ch}$) at SPS energies
is lower (higher) than at AGS energies.
  The $T_{ch}$ is larger than $T_{f\!o}$ in S+A and Pb+Pb collisions at
SPS energies.
  At AGS energies, both temperatures are close to each other.

\section*{ACKNOWLEDGEMENT}
NA44 collaboration thanks the staff of the CERN PS-SPS accelerator complex
for their excellent work. 
We are grateful to the technical staff at CERN and the collaborating
institutes for their valuable contribution.
We also wish to thank for the support given by the Austrian Fonds zur
F\"orderung der Wissenschaftlichen Forschung; the Science Research
Council of Denmark; the Japanese Society for the Promotion of Science;
the Ministry of Education, Science and Culture, Japan; the Science
Research Council of Sweden; the National Science Foundation, the US
Department of Energy, and the W.B. Keck Foundation.

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\bibitem{NX01}
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% referred in the figure. 3
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\end{thebibliography}

\end{document}
