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{\Large \bf 
What information can we obtain from the  yield ratio $\pi^-/\pi^+$
in heavy-ion collisions ?}\\
\vspace*{5mm}
T.Osada and ~M.Biyajima\\
{\small Department of Physics, Faculty of Science, 
        Shinshu University, Matsumoto 390, Japan}\\
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{\bf Introduction:}~~~
Significant enhancement of the pionic yield ratio $\pi^-/\pi^+$ 
in small $m_T$-$m_{\pi}$ region in heavy-ion collisions is 
observed in $[1,2]$. 
It may be caused by the Coulomb interaction between a charged pion and 
the central charged fragments (CCF) with effective charge 
$Z_{\mbox{\scriptsize eff}}$ which consists of nearly stopping nucleons. 
Several authors$[3,4]$ also have argued this phenomenon 
due to the Coulomb interaction of $\pi^{\pm}$-CCF system. 
Using the zero-range Gamow factor, the yield ratio 
is analyzed by Shuryak$[4]$; 
$N^{\pi^{-}}/N^{\pi^{+}}=G(\eta_-)/G(\eta_+)$ where 
$G(\eta_{\pm})=2\pi\eta_{\pm}/[\exp(2\pi\eta_{\pm})-1]$ 
and $\eta_+=Z_{\mbox{\scriptsize eff}}\alpha~m_{\pi}/p_r=-\eta_{-}$. 
To treat Coulomb interaction of $\pi^{\pm}$-CCF system rigorously, 
the Coulomb wave function $\psi({\bf p}_r,{\bf r})$ of the two 
body system should be used. A pion source function $\rho_{\beta}(r)$  
describing the finite interaction region also needs to introduce.\\
{\bf Theoretical formula of the yield ratio:}~~~The single particle spectra 
$N^{\pi^+}$ in terms of $\psi({\bf p}_r,{\bf r})$ and $\rho_{\beta}(r)$ is 
given in the following 
\begin{eqnarray}
N^{\pi^+}({\bf p}_r,\beta) &=& \!\!\int\!d^3{\bf r} ~~\rho_{\beta}(r) 
           \mid   \psi({\bf p}_r,{\bf r})   \mid^2 \nonumber \\
&&\hspace*{-2cm} 
=G(\eta_+) \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{(-i)^n(i)^m}{n+m+1}
A_n(\eta_+)A_m(\eta_+) 
~\big[\int_0^{\infty}\!\!dr r^{n+m+2}~\rho_{\beta}(r)\big]~
(2p_r)^{n+m}, \label{single}
\end{eqnarray}
where $A_n(\eta_+)=\Gamma(i\eta_++n)/\{\Gamma(i\eta_+)(n!)^2\}$ and 
${\bf p}_r$ is a relative momentum of pion 
in the rest frame of CCF. For concrete analysis two types of the 
source function are assumed: $\rho(r)={\cal N}\exp(-r^2/2\beta)$ 
and $\rho(r)={\cal N}\exp(-r/\beta)$. 
Finally the pionic yield ratio $\pi^-/\pi^+$ with fixed 
pions rapidity is given by
\begin{eqnarray}
\pi^-/\pi^+=
\frac{N^{\pi^-}(m_T-m_{\pi};\eta_-,\beta)}
     {N^{\pi^+}(m_T-m_{\pi};\eta_+,\beta)}. \label{ratio}
\end{eqnarray}
{\bf Analysis of experimental data and quasi-scaling in $N^{\pi^{\pm}}$:}~~~
Using eq.(\ref{ratio}) we analyze data of Pb+Pb at 158A GeV and Au+Au 
at 11A GeV in $[1,2]$. 
In the present analysis of those data we find the following 
quasi-scaling behavior 
\begin{eqnarray}
N^{\pi^{\pm}}(m_T-m_{\pi};\lambda~\eta_{\pm},\lambda~\beta) \approx 
N^{\pi^{\pm}}(m_T-m_{\pi};\eta_{\pm},\beta), \label{quasi-scaling}
\end{eqnarray}
where $\lambda>0$. This quasi-scaling behavior makes separated estimation 
of $Z_{\mbox{\scriptsize eff}}$ and $\beta$ difficult. 
Figure 1 demonstrates that two sets of parameters 
give almost the same results. 
Here we define the following function, 
\begin{eqnarray}
S(\lambda~\eta_{\pm}, \lambda~\beta) &\equiv&
\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{(-i)^n(i)^m}{n+m+1}
A_n(\eta_+)A_m(\eta_+) 
~\big[\int_0^{\infty}\!\!dr r^{n+m+2}~\rho_{\beta}(r)\big]~
(2p_r)^{n+m}.
\end{eqnarray} 
Comparing coefficients of the Taylor expansion of the above 
function $S(\lambda~\eta_{\pm}, \lambda~\beta)$ with $1/G(\eta_{\pm})$, 
we find that $S(\lambda~\eta_{\pm}, \lambda~\beta) \approx 
\mbox{const.}\times 1/G(\eta_{\pm})$. 
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\begin{figure}[t]
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\vspace*{-12mm}
{\small {\bf Fig.1}~~~The quasi-scaling property of eq.(\ref{ratio}).}\\
{\small {\bf Fig.2}~~~Estimation of $Z_{\mbox{\scriptsize eff}}$ 
comparing the present results and those of HBT effect.} 
\end{figure}
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This means that the $\lambda$ dependence of l.h.s. of 
eq.(\ref{quasi-scaling}) cancels out in the product of 
$G(\eta_{\pm})$ and $S(\lambda~\eta_{\pm})$. 
The cancellation of the $\lambda$ dependence is cause of 
the quasi-scaling behavior seen in $N^{\pi^{\pm}}$. \\
{\bf Estimation of the baryon number density of the CCF:}~~~
Combining results by means of HBT effect$[6,7]$ with that of 
yield ratio $\pi^-/\pi^+$, we can estimate a possible definite value of 
$Z_{\mbox{\scriptsize eff}}$. Providing that the size of the CCF 
equals to that of the pion interaction region, the root mean 
square radius of the CCF is given by $\langle r_{\pi}^2 \rangle^{1/2}
=\sqrt{3}\beta$ for the Gaussian source. 
From the studies of the HBT analysis for Pb+Pb collisions, 
an inequality 12.7~fm $\leq \langle r_{\pi}^2 \rangle^{1/2} \leq$ 15.1~fm 
is obtained. 
This leads to 125 $\leq Z_{\mbox{\scriptsize eff}} \leq$ 150. 
See Fig.2. 
Then the baryon number density $n_B$ is given by as follows: 
\begin{eqnarray}
\displaystyle{n_B=\frac{2.5\times Z_{\mbox{\scriptsize eff}}}{4\pi/3~
    \langle r_{\pi}^2 \rangle^{3/2}}}. 
\end{eqnarray}
This result gives $0.024 \leq n_B \leq 0.036$ fm$^{-3}$ 
as the baryon number density of the CCF.\\
{\bf Concluding remarks:}~~~We have derived eq.(\ref{single}) for the 
yield ratio $\pi^-/\pi^+$ in heavy-ion collisions. Using eqs. 
(\ref{single}) and (\ref{ratio}) we analyze data in $[1,2]$ and 
find the quasi-scaling. Combining several results estimated by 
the HBT effect$[6,7]$ with our present analysis 
of the yield ratio, we are able to estimate the baryon number density 
of the CCF, $0.024 \leq n_B \leq 0.036$ fm$^{-3}$.\\
{\bf Acknowledgments:}~~~This work is partially supported by 
Japanese Grant-in-Aid for Science Research from the Ministry of Education, 
Science and Culture (\#.06640383) and (\#.08304024).\\
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{\bf References}\\
$[1]$ H. Hamagaki (E802/E866 Collabs.) Genshikaku Kenkyu {\bf 40} 
(1995) 39. \\
$[2]$ H. Boggild et al. (NA44 Collab.),Phys. Lett.{\bf B372}
(1996) 339.\\
$[3]$ M. Gyulassy and S.K. Kauffmann, Nucl. Phys.
{\bf A362} (1981) 503.~See~also~refs.in~[5]. 
$[4]$ E.V. Shuryak, Nucl. Phys.{\bf A533}(1991)761. \\ 
$[5]$ T. Osada, S. Sano, M. Biyajima and G. Wilk, 
Phys. Rev. {\bf C54} (1996) 2167R. \\
$[6]$ A. Franz (NA44 Collab.), talk presented at
Quark Matter '96.\\
$[7]$ T. Osada, M. Biyajima and G. Wilk, submitted to Phys. Rev.{\bf C}. 
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