\documentstyle[twocolumn,prl,aps]{revtex}
\input epsf
\begin{document}
%\draft
\title{Charmonium Suppression by Comover Scattering in Pb+Pb
Collisions}
\author{Sean Gavin$^{a}$ and Ramona Vogt$^{b,c}$}
\address{
$^a$Physics Department, Brookhaven National Laboratory, Upton, NY, 
USA\\
$^b$Nuclear Science Division, Lawrence Berkeley National Laboratory, 
Berkeley, CA, USA\\
$^c$Physics Department, University of California, Davis, CA, USA}
\date{\today}
\maketitle
\begin{abstract}
The first reports of $\psi$ and $\psi^\prime$ production from
experiment NA50 at the CERN SPS are compared to calculations based on
a hadronic model of charmonium suppression developed previously.  Data
on centrality dependence and total cross sections are in good accord
with these predictions.
\end{abstract}

\pacs{PACS numbers: 25.75.Dw, 12.38.Mh, 24.85.+p}



\begin{narrowtext}

Experiment NA50 has reported an abrupt decrease in
$\psi$ production in Pb+Pb collisions at 158 GeV per nucleon
\cite{na50}.  Specifically, the collaboration presented a striking
`threshold effect' in the $\psi$--to--continuum ratio by plotting it
as a function of a calculated quantity, the mean path length of the
$\psi$ through the nuclear medium, $L$, as shown in fig.~1a.  This apparent
threshold has sparked considerable excitement as it may signal the
formation of quark--gluon plasma in the heavy Pb+Pb system \cite{bo}.
%
\begin{figure}
\vskip -1.0in
\epsfxsize=3.1in
\leftline{\epsffile{fig1.ps}}
\vskip -1.4in
\caption[]{(a) The NA50 \cite{na50} comparison of $\psi$ production in
Pb+Pb and S+U collisions as a function of the average path length $L$,
see eq.\ (3).  $B$ is the $\psi\rightarrow \mu^+\mu^-$ branching
ratio. (b) Transverse energy dependence of Pb+Pb data.  Curves in (a)
and (b) are computed using eqs.\ (4--6).}
\end{figure}
%

Our aim is to study the Pb results in the context of a hadronic model
of charmonium suppression \cite{gv,gstv}.  We first demonstrate
that the behavior in the NA50 plot, fig.~1a, is not a threshold effect
but, rather, reflects the approach to the geometrical limit of $L$ as
the collisions become increasingly central.  When plotted as a
function of the {\it measured} neutral transverse energy $E_{T}$ as in
fig.~1b, the data varies smoothly as in S+U measurements in fig.~3b
below \cite{na50,na38,na38c,na38d,na38e}.  The difference between S+U and
Pb+Pb data lies strictly in the relative magnitude.  To
assess this magnitude, we compare $\psi$ and $\psi^\prime$ data to
expectations based on the hadronic comover model \cite{gv,gstv}.
The curves in fig.~1 represent our calculations using parameters fixed
earlier in Ref.\ \cite{gstv}.  Our result is essentially the same as
the Pb+Pb prediction in \cite{gv}.


The hadronic contribution to the suppression arises from scattering of
the nascent $\psi$ with produced particles -- the comovers -- and
nucleons \cite{gv,gstv}.  To determine the suppression from nucleon
absorption of the $\psi$, we calculate the probability that a
$c{\overline c}$ pair produced at a point $(b, z)$ in a nucleus
survives scattering with nucleons to form a $\psi$.  The standard
\cite{gstv,gh} result is
%
\begin{equation}
S_{A} = {\rm exp}\{-\int_z^\infty\! dz\, \rho_{A}(b, z) \sigma_{\psi N}\}
\end{equation}
%
where $\rho_{A}$ is the nuclear density, $b$ the impact parameter and
$\sigma_{\psi N}$ the absorption cross section for $\psi$--nucleon
interactions.  One can estimate $S_{A}\sim \exp\{-
\sigma_{\psi N} \rho_0 L_{A}\}$, where $L_{A}$ is the path length
traversed by the $c\overline{c}$ pair.  

Suppression can also be caused by scattering with mesons that happen
to travel along with the $c\overline{c}$ pair (see refs.\ in
\cite{gv}).  The density of such comovers scales roughly as $E_{T}$.
The corresponding survival probability is
%
\begin{equation}
S_{\rm co} = {\rm exp}\{- \int\! d\tau n\, 
\sigma_{\rm co} v_{\rm rel}\},
\end{equation}
%
where $n$ is the comover density and $\tau$ is the time in the $\psi$
rest frame.  We write $S_{\rm co}\sim {\rm exp}\{-\beta E_{T}\}$,
where $\beta$ depends on the scattering frequency,
the formation time of the comovers and the transverse size of the
central region, $R_{T}$, {\it cf.} eq.\ (8).

To understand the saturation of the Pb data with $L$ in fig.~1a, we apply
the schematic approximation of Ref.~\cite{gh} for the moment to write
%
\begin{equation}
{{\sigma^{AB}_\psi(E_{T})}\over{\sigma^{AB}_{\mu^+\mu^-}(E_{T})}}
\propto \langle S_{A}S_{B}S_{\rm co}\rangle
\sim 
{\rm e}^{-\sigma_{\psi N}\rho_{0}L}{\rm e}^{-\beta E_{T}},
\end{equation}
%
where the brackets imply an average over the collision geometry for
fixed $E_{T}$ and $\sigma(E_T) \equiv d\sigma/dE_T$.  The path length
$L\equiv \langle L_{A}+L_{B}\rangle$ and transverse size $R_T$ depend
on the collision geometry.  The path length grows with $E_{T}$,
asymptotically approaching the geometric limit $R_A + R_B$.  Explicit
calculations show that nucleon absorption begins to {\it saturate} for
$b < R_A$, where $R_A$ is the smaller of the two nuclei, see fig.~4
below.  On the other hand, $E_{T}$ continues to
grow for $b < R_A$ due, {\it e.g.}, to fluctuations in the number of
$NN$ collisions.  Equation (2) falls exponentially in this regime
because $\beta$, like $L$, saturates.

In fig.~1b, we compare the Pb data to calculations of the
$\psi$--to--continuum ratio that incorporate nucleon and comover
scattering.  The contribution due to nucleon absorption indeed levels
off for small values of $b$, as expected from eq.\ (3).  Comover
scattering accounts for the remaining suppression.

These results are {\it predictions} obtained using the computer code
of Ref.~\cite{gv} with parameters determined in Ref.~\cite{gstv}.
However, to confront the present NA50 analysis \cite{na50}, we 
account for changes in the experimental coverege as follows:
%
\begin{itemize}
\item Calculate the continuum dimuon yield in the new mass range $2.9
< M < 4.5$~GeV.  
\item Adjust the $E_T$ scale to the pseudorapidity
acceptance of the NA50 calorimeter, $1.1 < \eta < 2.3$.
\end{itemize} 
%
The agreement in fig.~1 depends on these updates.

We now review the details of our calculations, highlighting the
adjustments as we go.  For collisions at a fixed $b$, the
$\psi$--production cross section is
%
\begin{equation}
\sigma_\psi^{AB}(b)
=
\sigma^{NN}_{\psi}\!\int\! d^2s dz dz^\prime\,\rho_A(s,z)
\rho_B(b-s,z^\prime)\, S,
\end{equation}
%
where $S\equiv S_AS_BS_{\rm co}$ is the product of the survival
probabilities in the projectile $A$, target $B$ and comover matter.
The continuum cross section is
%
\begin{equation}
\sigma_{\mu^{+}\mu^{-}}^{AB}(b) = 
\sigma^{NN}_{\mu^+\mu^-}\!\int\! d^2s dz dz^\prime\,\rho_A(s,z)
\rho_B(b-s,z^\prime).
\end{equation}
%
The magnitude of (4,5) and their ratio are fixed by the elementary
cross sections $\sigma^{NN}_{\psi}$ and
$\sigma^{NN}_{\mu^{+}\mu^{-}}$.  We calculate $\sigma^{NN}_{\psi}$
using the phenomenologically--successful color evaporation model
\cite{hpc-psi}.  The continuum in the mass range used by NA50, $2.9 <
M < 4.5$~GeV, is described by the Drell--Yan process.  To confront
NA50 and NA38 data in the appropriate kinematic regime, we compute
these cross sections at leading order following \cite{hpc-psi,hpc-dy}
using GRV LO parton distributions with a charm $K$--factor $K_c= 2.7$
and a color evaporation coefficient $F_\psi =2.54\%$ and a Drell--Yan
$K$--factor $K_{DY}=2.4$.  Observe that these choices were fixed by
fitting $pp$ data at all available energies \cite{hpc-psi}.  Computing
$\sigma^{NN}_{\mu^{+}\mu^{-}}$ for $2.9<M<4.5$~GeV corresponds to the
first update.

To obtain $E_T$ dependent cross sections from eqs.\ (4) and (5), we
write
%
\begin{equation}
\sigma^{AB}(E_{T}) =
\int\! d^2b\, P(E_T,b) \sigma^{AB}(b).
\end{equation}
%
The probability $P(E_T,b)$ that a collision at impact parameter $b$
produces transverse energy $E_T$ is related to the minimum--bias
distribution by
%
\begin{equation}
\sigma_{\rm min}(E_{T}) = \int\! d^{2}b\; P(E_{T}, b).
\end{equation}
%
We parametrize $P(E_{T}, b) = C\exp\{- (E_{T}- {\overline
E}_{T})^2/2\Delta\}$, where ${\overline E}_{T}(b) = \epsilon {\cal
N}(b)$, $\Delta(b) = \omega \epsilon {\overline E}_{T}(b)$,
$C(b)=(2\pi\Delta(b))^{-1}$ and ${\cal N}(b)$ is the number of
participants (see, {\it e.g.}, Ref.~\cite{gv}).
We take $\epsilon$ and $\omega$ to be phenomenological
calorimeter--dependent constants.

We compare the minimum bias distributions for total hadronic $E_T$
calculated using eq.\ (7) for $\epsilon = 1.3$~GeV and $\omega = 2.0$
to NA35 S+S and NA49 Pb+Pb data \cite{na49}.  The agreement in fig.~2a
builds our confidence that eq.\ (7) applies to the heavy Pb+Pb system.
%
\begin{figure}
\vskip -1.1in
\epsfxsize=2.6in
\centerline{\epsffile{fig2.ps}}
\vskip -0.55in
%\vspace{12.5pc}
\caption{Transverse energy distributions from eq.\ (7).
The S--Pb comparison (a) employs the same parameters.}
\end{figure}
%
Figure 2b shows the distribution of neutral transverse energy
calculated using eqs.\ (5) and (6) to simulate the NA50 dimuon
trigger. We take $\epsilon = 0.35$~GeV, $\omega = 3.2$, and
$\sigma^{NN}_{\mu^+\mu^-}\approx 37.2$~pb as appropriate for the
dimuon--mass range $2.9 < M < 4.5$~GeV.  The $E_T$ distribution for
S+U~$\rightarrow \mu^+\mu^- + X$ from NA38 was described \cite{gstv}
using $\epsilon = 0.64$~GeV and $\omega = 3.2$ -- the change in
$\epsilon$ corresponds roughly to the shift in particle production
when the pseudorapidity coverage is changed from $1.7 < \eta < 4.1$
(NA38) to $1.1 < \eta < 2.3$ (NA50).  Taking $\epsilon = 0.35$~GeV for
the NA50 acceptance is the second update listed earlier.

We now apply eqs.\ (1,2,4) and (5) to charmonium suppression in Pb+Pb
collisions.  To determine nucleon absorption, we used $p$A data to fix
$\sigma_{\psi N}\approx 4.8$~mb in Ref.~\cite{gstv}. This choice is in
accord with the latest NA38 and NA51 $pA$ data, see fig.~3a.
To specify comover scattering \cite{gstv}, we assumed that the
dominant contribution to $\psi$ dissociation comes from exothermic
hadronic reactions such as $\rho + \psi \rightarrow D+ \overline{D}$.
We further took the comovers to evolve from a formation time
$\tau_{0}\sim 2$~fm to a freezeout time $\tau_{F}\sim R_{T}/v_{\rm
rel}$ following Bjorken scaling, where $v_{\rm rel}\sim 0.6$ is
roughly the average $\psi-\rho$ relative velocity.  The
survival probability, eq.\ (2), is then
%
\begin{equation}
S_{\rm co} = \exp\{ - \sigma_{\rm co}v_{\rm rel}n_{0}\tau_{0} 
\ln(R_{T}/v_{\rm rel}\tau_{0})\}
\end{equation}
%
where $\sigma_{\rm co} \approx 2\sigma_{\psi N}/3$, $R_{T}\approx
R_{A}$ and $n_{0}$ is the initial density of sufficiently massive
$\rho, \omega$ and $\eta$ mesons.  To account for the variation of
density with $E_{T}$, we take $n_{0} = {\overline
n}_{0}E_{T}/{\overline E}_{T}(0)$ \cite{gv}. A value $\overline{n}_{0}
= 0.8$~fm$^{-3}$ was chosen to fit the central S+U datum.  Since we
fix the density in central collisions, this simple {\it ansatz} for
$S_{\rm co}$ may be inaccurate for peripheral collisions.  [Densities
$\sim 1$~fm$^{-3}$ typically arise in hadronic models of ion
collisions, e.g., refs.~\cite{cascade}.  The internal consistency of
hadronic models at such densities demands further study.]

We expect the comover contribution to the suppression to increase in
Pb+Pb relative to S+U for central collisions because both the
initial density and lifetime of the system can increase.  To be
conservative, we assumed that Pb and S beams achieve the same mean initial
density.  Even so, the lifetime of the system essentially doubles in
Pb+Pb because $R_T \sim R_{A}$ increases to 6.6~fm from 3.6~fm in S+U.
The increase in the comover contribution evident in comparing figs.~1b
and 3b is described by the seemingly innocuous logarithm in eq.\ (8),
which increases by $\approx 60\%$  in the larger Pb system.
%
\begin{figure}
\vskip -1.8in
\epsfxsize=3.0in
\rightline{\epsffile{fig3.ps}}
\vskip -0.1in
%\vspace{12.5pc}
\caption[]{(a) $p$A cross sections \cite{na50} in the NA50 acceptance
and (b) S+U ratios from '91 \cite{na38c} and '92 \cite{na50}
runs. The '92 data are scaled to the '91 continuum.  The dashed line
indicates the suppression from nucleons alone.  The $pp$ cross section
in (a) is constrained by the global fit to $pp$ data in
ref.~\cite{hpc-psi}.}
\end{figure}
%

In Ref.~\cite{gstv}, we pointed out that comovers were necessary to
explain S+U data from the NA38 1991 run \cite{na38}.  Data just
released \cite{na50} from their 1992 run support this conclusion.  The
'91 $\psi$ data were presented as a ratio to the dimuon continuum in
the low mass range $1.7 < M < 2.7$~GeV, where charm decays are an
important source of dileptons.  On the other hand, the '92 $\psi$ data
\cite{na50,na38e} are given as ratios to the Drell--Yan cross section
in the range $1.5< M < 5.0$~GeV.  That cross section is extracted from
the continuum by fixing the $K$--factor in the high mass region
\cite{na38f}.  To compare our result from Ref.~\cite{gstv} to these
data, we scale the '92 data by an empirical factor.  This factor is
$\approx 10\%$ larger than our calculated factor
$\sigma^{NN}_{DY}(92)/\sigma^{NN}_{\rm cont.}(91) \approx 0.4$; these
values agree within the NA38 systematic errors.  [NA50 similarly
scaled the '92 data to the high--mass continuum to produce fig.~1a.]
Because our fit is driven by the highest $E_T$ datum, we see from
fig.~3b that a fit to the '92 data would not appreciably change our
result.  Note that a uniform decrease of the ratio would increase the
comover contribution needed to explain S+U collisions.

To see why saturation occurs in Pb+Pb collisions but not in S+U, we
compare the NA50 $L(E_T)$ \cite{na50} to the average impact parameter
$\langle b\rangle (E_T)$ in fig.~4.  To best understand fig.~1a, we
show the values of $L(E_T)$ computed by NA50 for this figure.  We use
our model to compute $\langle b\rangle = \langle b
T_{AB}\rangle/\langle T_{AB}\rangle$, where $\langle f(b)\rangle
\equiv \int\!d^2b\; P(E_T,b)f(b)$ and $T_{AB} =
\int\!d^{2}sdzdz^\prime \rho_{A}(s,z)\rho_{B}(b-s,z^\prime)$. [Note
that NA50 reports similar values of $\langle b\rangle (E_T)$
\cite{na50}.] In the $E_T$ range covered by the S experiments, we see
that $\langle b\rangle$ is near $\sim R_{\rm S} = 3.6$~fm or larger.
In this range, increasing $b$ dramatically reduces the collision
volume and, consequently, $L$.  In contrast, in Pb+Pb collisions
$\langle b\rangle \ll R_{\rm Pb} =$~6.6~fm for all but the lowest
$E_T$ bin, so that $L$ does not vary appreciably.
%
\begin{figure}
\vskip -1.8in
\epsfxsize=3.0in
\rightline{\epsffile{fig4.ps}}
\vskip -0.1in
%\epsfysize=2.0in
%\rightline{\epsffile{fig1_hb.ps}}
\caption[]{$E_T$ dependence of $L$ (solid) used by NA50 \cite{na50}
(see fig.~1a) and the average impact parameter $\langle b\rangle$
(dot--dashed).  The solid line covers the measured $E_T$ range.}
\end{figure}
%


NA50 and NA38 have also measured the total $\psi$--production cross
section in Pb+Pb \cite{na50} and S+U reactions \cite{na38c}.  To
compare to that data, we integrate eqs.\ (4, 6) to obtain the total
$(\sigma/AB)_{\psi} = 0.95$~nb in S+U at 200~GeV and 0.54~nb for Pb+Pb
at 158~GeV in the NA50 spectrometer acceptance, $0.4 > x_{F}> 0$ and
$-0.5 < \cos\theta < 0.5$ (to correct to the full angular range and $1
> x_{F} > 0$, multiply these cross sections by $\approx 2.07$).  The
experimental results in this range are $1.03 \pm 0.04 \pm 0.10$~nb for
S+U collisions \cite{na38} and $0.44 \pm 0.005 \pm 0.032$ nb for Pb+Pb
reactions \cite{na50}.  Interestingly, in the Pb system we find a
Drell--Yan cross section $(\sigma/AB)_{{}_{DY}} = 37.2$~pb while NA50
finds $(\sigma/AB)_{{}_{DY}} = 32.8\pm 0.9\pm 2.3$~pb.  Both the
$\psi$ and Drell--Yan cross sections in Pb+Pb collisions are somewhat
above the data, suggesting that the calculated rates at the $NN$ level
may be $\sim 20-30\%$ too large at 158~GeV.  This discrepancy is
within ambiguities in current $pp$ data near that low energy
\cite{hpc-psi}.  Moreover, nuclear effects on the parton densities
omitted in eqs.\ (4,5) can affect the total S and Pb cross sections at
this level.

We remark that if one were to neglect comovers and take $\sigma_{\psi
N} = 6.2$~mb, one would find $(\sigma/AB)_{\psi} = 1.03$~nb in S+U at
200~GeV and 0.62~nb for Pb+Pb at 158~GeV.  The agreement with S+U data
is possible because comovers only contribute to the total cross
section at the $\sim 18\%$ level in the light system.  This is
expected, since the impact--parameter integrated cross section is
dominated by large $b$ and the distinction between central and
peripheral interactions is more striking for the asymmetric S+U
system.  As in Ref.~\cite{gstv}, the need for comovers is evident for
the $E_{T}$--dependent ratios, where central collisions are singled
out.
 
%
\begin{figure}
\vskip -1.8in
\epsfxsize=3.0in
\rightline{\epsffile{fig5.ps}}
\vskip -0.1in
\caption[]{Comover suppression of $\psi^\prime$ compared to (a) NA38 and
NA51 $p$A data \cite{na50,na38e} and (b) NA38 S+U data \cite{na38d}
(filled points) and preliminary data \cite{na50}.}
\end{figure}
%
%
\begin{figure}
\vskip -1.2in
\epsfxsize=3.0in
\rightline{\epsffile{fig6.ps}}
\vskip -1.5in
\caption{Comover suppression in Pb+Pb~$\rightarrow \psi^\prime +X$.}
\end{figure}
%
To apply eqs.\ (4-6) to calculate the $\psi^{\prime}$--to--$\psi$
ratio as a function of $E_{T}$, we must specify
$\sigma_{\psi^{\prime}}^{NN}$, $\sigma_{\psi^{\prime} N}$, and
$\sigma_{\psi^{\prime} {\rm co}}$.  Following Ref.~\cite{hpc-psi}, we
use $pp$ data to fix $B\sigma_{\psi^{\prime}}^{NN}/B\sigma_{\psi}^{NN}
= 0.02$ (this determines $F_{\psi^\prime}$).  The value of
$\sigma_{\psi^{\prime} N}$ depends on whether the nascent
$\psi^{\prime}$ is a color singlet hadron or color octet
$c\overline{c}$ as it traverses the nucleus. In the singlet case, one
expects the absorption cross sections to scale with the square of the
charmonium radius.  Taking this ansatz and assuming that the
$\psi^\prime$ forms directly while radiative $\chi$ decays account for
40\% of $\psi$ production, one expects $\sigma_{\psi'}\sim
2.1\sigma_{\psi}$ for interactions with either nucleons or comovers
\cite{gstv}.  For the octet case, we take $\sigma_{\psi^{\prime} N}
\approx \sigma_{\psi N}$ and fix $\sigma_{\psi^{\prime} {\rm
co}}\approx 12$~mb to fit the S+U data.  In fig.~5a, we show that the
singlet and octet extrapolations describe $p$A data equally well.

Our predictions for Pb+Pb collisions are shown in fig.~6.  In the
octet model, the entire suppression of the $\psi^{\prime}$--to--$\psi$
ratio is due to comover interactions.  In view of the schematic nature
of our approximation to $S_{\rm co}$ in eq.\ (8), we regard the
agreement with data of singlet and octet extrapolations as equivalent.

In summary, the Pb data \cite{na50} cannot be described by nucleon
absorption alone.  This is seen in the NA50 plot, fig.~1a, and
confirmed by our results.  The saturation with $L$ but not $E_T$
suggests an additional density--dependent suppression mechanism.
Earlier studies pointed out that additional suppression was already
needed to describe the S+U results \cite{gstv}; recent data
\cite{na50} support that conclusion (see, however, \cite{bo}).
Comover scattering explains the additional suppression.  Nevertheless,
it is unlikely that this explanation is unique.  SPS
inverse--kinematics experiments ($B < A$) and AGS $p$A studies near
the $\psi$ threshold can help pin down model uncertainties.

%We thank C.~Gerschel and M.~Gonin for discussions of the NA50 data, and 
%M.~Gyulassy, R.~Pisarski and M.~Tytgat for insightful comments.
This work was supported in part by US-DOE contracts DE-AC02-76CH00016
and DE-AC03-76SF0098.

{\it Note Added} --- After the completion of this manuscript, we
learned of cascade calculations \cite{cascade} that confirm our
conclusions.  Such calculations do not employ the simplifications
({\it e.g.\ } $n_0\propto E_T$) needed to derive (8).  Some of these
authors took $\sigma_{\psi N} \sim 6$~mb (instead of $\sim 5$~mb)
to fit the NA51 data in fig.~3a somewhat better.
%
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%
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%
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%
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%
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%
%%%%cross sections
\bibitem{na38c}
C.\ Baglin {\it et al.} (NA38) Phys.\ Lett.\ B270 (1991) 105.
%
%%%%psiprime
\bibitem{na38d} C.\ Baglin {\it et al.} (NA38) 
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%
%%%%92 data
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%
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%
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%
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%
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%
% drell yan
\bibitem{na38f}
M.~C. Abreu {\it et al.}, Nucl.\ Phys.\ A566 (1994) 77c.
%
\bibitem{cascade} W.~Cassing and C.~M.~Ko,  (1996);
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%

\end{references}

\end{narrowtext}
\end{document}

