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\begin{center}
{\large \bf
Is there a unique thermal source of dileptons in \\
Pb(158 A$\cdot$GeV) + Au, Pb reactions?
} \\[6mm]
{\sc
K. Gallmeister$^a$, B. K\"ampfer$^a$, O.P. Pavlenko$^{a,b}$} \\[6mm]

$^a$Forschungszentrum Rossendorf, PF 510119, 01314 Dresden, Germany \\[1mm]
$^b$Institute for Theoretical Physics, 252143 Kiev - 143, Ukraine
\end{center}

\centerline{Abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
An analysis of
the dilepton measurements in the reactions Pb(158 A$\cdot$GeV) + Au, Pb
%measured by the CERES and NA50 collaborations 
points to a unique
thermal source contributing to the invariant mass and transverse 
momentum spectra. Effects of the flow pattern are discussed.
\\[2cm]

{\bf Introduction:} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Dileptons are penetrating probes which carry nearly undisturbed
information about early, hot and dense matter stages in relativistic
heavy-ion collisions. Some effort, however, is needed for
disentangling the various sources contributing to the observed
yields and for identifying the messengers from primordial states of
strongly interacting matter.
 
The dielectron spectra for the reaction Pb(158 A$\cdot$GeV) + Au measured by
the CERES collaboration \cite{CERES_1} cannot be described by a superposition
of $e^+ e^-$ decay channels of final hadrons, i.e.\ the hadronic cocktail.
A significant additional source of dielectrons must be there. Since the
data \cite{CERES_1} cover mainly the invariant mass range $M <$ 1.5 GeV the
downward extrapolation of the Drell-Yan process is not 
an appropriate explanation.
Also correlated semileptonic decays of open charm mesons have been excluded
\cite{PBM}. As a widely accepted explanation, a thermal source is found to
account for the data (cf. \cite{CERES_exp,Rapp_Wambach} and further
references therein, in particular with respect to in-medium effects and
chiral symmetry restoration).

Very similar, the NA50 collaboration has found, for the reaction
Pb(158 A$\cdot$GeV) + Pb, that the superposition of
Drell-Yan dimuons and open charm decays does not explain the data
in the invariant mass range 1.5 GeV $< M <$ 2.5 GeV \cite{NA50_1}.
Final state interactions \cite{Lin} or abnormal charm enhancement
\cite{NA50_1,NA50_2} have been proposed as possible explanations. Here we
try to explain the NA50 measurements by a more apparent idea \cite{Shuryak,Rapp},
namely a thermal source. We present a schematic model which at the same time
describes the CERES and NA50 data.

{\bf The model:} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since we include the corresponding detector acceptances a good starting
point for Monte Carlo simulations is the differential dilepton spectrum
\begin{equation}
\frac{dN}{p_{\perp \,1} \, d p_{\perp \,1} \, 
p_{\perp \,2} \, d p_{\perp \,2} dy_1 \, dy_2 \,
d \phi_1 \, d \phi_2}
=
\int d^4 Q \, d^4 x \, \frac{dR}{d^4 Q \, d^4 x}
\delta^{(4)} (Q - p_1 - p_2),
\label{rate_1}
\end{equation}
where $Q = p_1 + p_2$ is the pair four-momentum,
$p_{1,2}$ are the individual lepton four-momenta composed of transverse momenta
$p_{\perp \,1,2}$ and rapidities $y_{1,2}$ and azimuthal angles
$\phi_{1,2}$. Here we extensively employ the quark - hadron duality
\cite{Rapp_Wambach,Rapp} and base the rate $R$ on the lowest-order
quark - antiquark ($q \bar q$) annihilation rate 
(cf. \cite{KKMcLR,Ruuskanen})
\begin{equation}
\frac{dR}{d^4 Q \, d^4 x} = \frac{5 \alpha^2}{36 \pi^4} 
\exp \left\{ - \frac{u \cdot Q}{T} \right\},
\label{rate_2}
\end{equation}
where $u(x)$ is the four-velocity of the medium depending on the space-time
as also the temperature $T(x)$ does.
Note that, due to Lorentz invariance, $u$ necessarily enters this expression.
The above rate is in Boltzmann approximation, and a term related to the
chemical potential is suppressed. As shown in \cite{Rapp_Wambach} the
$q \bar q$ rate deviates from the hadronic one at $M <$ 300 MeV, but in this
range the Dalitz decays dominate anyhow; in addition, in this range the
thermal yield is strongly suppressed by the CERES acceptance.
In the kinematical regions we consider below, the lepton masses can be
neglected.

Performing the space-time and momentum integrations in 
eqs.~(\ref{rate_1}, \ref{rate_2})  one gets
\begin{equation}
\frac{dN}{d p_{\perp \,1} \, d p_{\perp \,2} 
dy_1 \, dy_2 \, d \phi_1 \, d \phi_2}
= 
\frac{5 \alpha^2}{72 \pi^5} p_{\perp \, 1} p_{\perp \, 2}
\int_{t_i}^{t_f} dt \, V(t) \, E,
\label{rate_3}
\end{equation}
\begin{eqnarray}
E &=& \left\{
\begin{array}{ll}   %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exp \left\{ -
\frac{M_\perp \cosh Y \cosh \rho (r,t)}{T(r,t)} \right\}
\frac{\sinh \xi}{\xi}
& \mbox{for}
\quad 3D,\\
K_0 \left( \frac{M_\perp \cosh \rho (r,t)}{T(r,t)} \right)
I_0 \left( \frac{Q_\perp \sinh \rho (r,t)}{T(r,t)} \right)
& \mbox{for}
\quad 2D,\\
\end{array}   %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\right.  \\
%\end{equation}
%\begin{eqnarray}
V(t) &=& \left\{
\begin{array}{ll}  %--------------------------------
4 \pi \int dr \, r^2
& \mbox{for}
\quad 3D,  \\
t \int dr \, r
& \mbox{for}
\quad 2D,  \\
\end{array}   %---------------------------------
\right.
\end{eqnarray}
where $V(t)$ acts on $E$, and $3D$ means spherical symmetric expansion,
while $2D$ denotes the case of longitudinal boost-invariant and
cylinder-symmetrical transverse expansion; the quantity $\xi$ is defined as
$\xi = T^{-1} \sinh \rho \sqrt{M_\perp^2 \cosh Y^2 - M^2}$,
and $\rho(r,t)$ is the radial or transverse expansion rapidity;
$K_0$ and $I_0$ are Bessel functions \cite{KKMcLR}.
The components of the lepton pair four-momentum
$Q = (M_\perp \cosh Y, M_\perp \sinh Y, \vec Q_\perp)$
are related to the individual lepton momenta via
\begin{eqnarray}
M_\perp^2 
&=& p_{\perp \, 1}^2 + p_{\perp \, 2}^2 +
2 p_{\perp \, 1} p_{\perp \, 2} \cosh (y_1 -y_2), \\
\vec Q_\perp &=& \vec p_{\perp \, 1} + \vec p_{\perp \, 2},\\
M^2 &=& M_\perp^2 - Q_\perp^2,\\
\tanh  Y 
&=& 
\frac{p_{\perp 1} \, \sinh y_1 + p_{\perp 2} \, \sinh y_2 }
{p_{\perp 1} \, \cosh y_1 + p_{\perp 2} \, \cosh y_2 }. 
\end{eqnarray}
It turns out that the shape of the invariant mass spectrum
$d N / (dM \, dY \vert_{|Y| < 0.5} \, dt \, d V(t))$, which is determined only
by the emissivity function $E$, does not depend on the flow rapidity 
$\rho$ in the 2D case \cite{KKMcLR}, and in the 3D case 
for $T$ = 120 $\cdots$ 220 MeV and $\rho < 0.6$ there is also no
effect of the flow. The analysis of transverse momentum spectra of
various hadrons species point to an average transverse expansion velocity
$\bar v_\perp \approx$ 0.43 \cite{BK_2} at kinetic freeze-out,
while a combined analysis of hadron spectra
including HBT data yields $\bar v_\perp \approx$ 0.55 \cite{Heinz}.
Therefore, $\rho < 0.6$ is the relevant range for the considered reactions.
  
We note further that the invariant mass spectra 
$d N / (dM \, dY \vert_{|Y| < 0.5} \, dt \, d V(t))$
for the 3D and 2D cases differ only marginally.
Relying on these findings one can approximate the emissivity function
$E$ by that of a ''static'' source at midrapidity, as appropriate only for
symmetric systems,
\begin{equation}
E = \exp \left\{ - \frac{M_\perp \cosh Y}{T(t)} \right\},
\label{rate_4}
\end{equation}
thus getting rid of the peculiarities of the flow pattern.
In contrast to the invariant mass spectra, 
the transverse momentum or transverse mass spectra are
sensitive to the flow pattern \cite{KKMcLR,Asakawa_Ko_Levai,BK_1}, in general.
A value of $\rho =$ 0.4, for example, causes already a sizeable change of the shape 
of the spectra $dN / (dQ_\perp dY \vert_{|y| < 0.5} \, dt \, d V(t))$ 
compared to $\rho = 0$, in particular in the large-$Q_\perp$ region. 
The differences between the 2D and 3D cases are not
larger than a factor of 2 and,
in a restricted $Q_\perp$ interval,
can be absorbed in a renormalization.
The most striking difference of the 2D and 3D scenarios is seen in the
rapidity spectrum: for 2D it is flat, while in the 3D case it is
localized at midrapidity (values of $\rho <$ 0.6 also do not change
the latter rapidity distribution). Below we shall discuss which space is left
to extract from the dilepton data in restricted acceptance regions hints for
the flow pattern when the other dilepton sources are also taken into account.

{\bf Comparison with data:} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In line with the above arguments we base our rate calculations on
eqs.~(\ref{rate_3}, \ref{rate_4}) and use the parameterizations
\cite{Klingel_Weise}
\begin{eqnarray}
T &=& (T_i - T_\infty) \exp \left\{ - \frac{t}{t_2} \right\} + T_\infty,\\
V &=& N \exp \left\{ \frac{t}{t_1} \right\}.
\label{history}
\end{eqnarray}
with $T_i =$ 210 MeV, $T_\infty =$ 110 MeV, $t_1 =$ 5 fm/c, $t_2 =$ 8 fm/c,
$N = \frac{A + B}{2.5 n_0}$ with $A, B$ as mass numbers of the
colliding systems and $n_0 =$ 0.17 fm${}^{-3}$.
We stop the time evolution at $T_f =$ 130 MeV.

In fig.~1 we show the comparison with the preliminary CERES data applying the
appropriate acceptance \cite{CERES_1}. One observes a satisfactory overall
agreement of the sum of the hadronic cocktail and the thermal
contribution with the data.
It is the thermal contribution which fills the hole of
the cocktail around $M = 500$ MeV in the invariant mass distribution
in fig.~1a. In the mass bin $M =$ 0.25 $\cdots$ 0.68 GeV the thermal
yield is seen (fig.~1b) to dominate at small values of the transverse 
momentum $Q_\perp$. In this region of $Q_\perp$ transverse flow
effects are not important. The large-$Q_\perp$ spectrum is dominated
by the cocktail. For higher-mass bins the thermal
yield in the region of the first two data points is nearly 
as strong as the cocktail
and rapidly falls then at larger values of $Q_\perp$ below the cocktail. 
Therefore, the flow effects turn out to be of minor
importance for the present analysis, since within our framework the
transverse flow shows up at larger values of $Q_\perp$.

The question now is whether the same thermal source model 
accounts also for the NA50 data \cite{NA50_1}.
The available NA50 data are not yet efficiency corrected and the
efficiency matrix is not accessible. To have a reference we proceed as follows.
According to the detailed analysis \cite{NA50_1} of the most central
collisions, the shapes of the $M$ and $Q_\perp$ spectra for invariant masses 
below the $J/\psi$ peak are described by the Drell-Yan yield
$+$ 3 $\times$ the yield from correlated semileptonic decays of 
open charm mesons, both ones generated with PYTHIA \cite{PYTHIA}.
We have checked that our PYTHIA simulations coincide
with results of the NA50 collaboration when applying the acceptance cuts. 
Our $K$ factors are adjusted by
a comparison with Drell-Yan data \cite{DY_data} and identified open charm
data (cf. \cite{PBM} for the data compilation). 
Some confidence in our procedure is gained by
correctly reproducing the NA50 interpretation \cite{NA50_1} of the NA38 data
\cite{NA38}, which are efficiency corrected 
and which we can directly analyze: 
the NA38 data are described by the Drell-Yan yield 
$+$ 1.45 $\times$ the open charm contribution;
the absolute normalization is obtained from a fit of the Drell-Yan
region beyond the $J/\psi$.

To get the yield for Pb + Pb collisions
from PYTHIA the overlap function $T_{AA} =$ 31 mb${}^{-1}$ is used.
The resulting spectra (within the kinematical cuts for the NA50
acceptance) are displayed in figs.~2a and 2b by open squares. 
One observes that both the continuum part of the invariant mass spectrum 
(without the $J/\psi$ contribution)
and the transverse momentum spectrum for the mass bin 
$M =$ 1.5 $\cdots$ 2.5 GeV
are nicely reproduced by the sum of Drell-Yan, open charm and thermal
contributions. The thermal yield dominates again at not too large values of
$Q_\perp$ where transverse flow effects can be neglected.
Therefore, it seems that from present dilepton measurements the
transverse flow can hardly be inferred.

{\bf Summary and discussion:} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In summary we have shown that a very simplified, schematic model for thermal
dilepton emission, with parameters adjusted to the CERES data, also
accounts for the measurements of the NA50 collaboration. A direct
comparison with the NA50 data is not possible as long as the efficiency
matrix is not at disposal. Nevertheless, our study points to a common thermal
source seen in different phase space regions in the dielectron and
dimuon channels. This unifying interpretation of different measurements
has to be contrasted with other proposals  of explaining the dimuon
excess in the invariant mass region 1.5 $\cdots$ 2.5 GeV either by
final state hadronic interactions  or an abnormally large
open charm production. The latter one, in principle,
can be checked experimentally \cite{NA50_2},
thus improving our understanding of dilepton sources.

Due to the convolution of the local matter emissivity and the space-time
history of the whole matter and the general dependence on the flow pattern,
it is difficult to decide which type of matter (deconfined or hadron matter)
emits really the dileptons. 
Our model is not aimed at answering this question. Instead,
with respect to chiral symmetry restoration, we 
apply the quark - hadron duality as a convenient way to roughly
describe the dilepton emissivity of matter by a $q \bar q$ rate, 
being aware that higher-order QCD processes change this rate 
\cite{BK_1,Aurenche} (to some extent this might be
included in a changed normalization $N$). In further investigations a more
reliable rate together with a more detailed space-time evolution must
be attempted and chemical potentials controlling the baryon and pion densities
must be included. In this line the model has to be improved before
using it for analyzing 
the dilepton yields in the asymmetric reactions S(200 A$\cdot$GeV) + X
measured by the NA38, HELIOS-3, and CERES collaborations,
where also details of the rapidity distribution become important.

{\bf Acknowledgments:} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Stimulating discussions with 
P. Braun-Munzinger,
W. Cassing, 
O. Drapier,
Z. Lin, 
U. Mosel,
E. Scomparin,
J. Rafelski,
J. Stachel, and
G. Zinovjev are gratefully
acknowledged.
O.P.P. thanks for the warm hospitality of the nuclear theory group
in the Research Center Rossendorf.
The work is supported by BMBF grant 06DR829/1.
\newpage

{\small
\begin{thebibliography}{99}

\vspace*{-1mm}

\bibitem{CERES_1} 
S. Esumi (for the CERES collaboration),
Proc. 15th Winter Workshop on Nuclear Dynamics, Jan. 1999, Park
City,\\
B. Lenkeit, Proc. Int. Nucl. Phys. Conf., Aug. 1998, Paris;
Ph. D. thesis, Heidelberg 1998

\vspace*{-1mm}

\bibitem{PBM} 
P. Braun-Munzinger, D. Miskowiec, A. Drees, C. Lorenco,
Eur. Phys. J. C 1 (1998) 123

\vspace*{-1mm}

\bibitem{CERES_exp} 
P. Huovinen, M. Prakash, 
Phys. Lett. B 450 (1999) 15,\\
C.H. Lee, J. Wirstam, I. Zahed, T.H. Hansson, Phys. Lett. B 448 (1999) 168

\vspace*{-1mm}

\bibitem{Rapp_Wambach} 
R. Rapp, J. Wambach, 

\vspace*{-1mm}

\bibitem{NA50_1} 
E. Scomparin (for the NA50 Collaboration), 
Nucl. Phys. A 610 (1996) 331c,
J. Phys. G 25 (1999) 235c,\\
P. Bordalo, talk at Quark Matter '99, May 1999, Torino

\vspace*{-1mm}

\bibitem{Lin} 
Z. Lin, X.N. Wang, Phys. Lett. B 444 (1998) 245, \\
Z. Lin, C.M. Ko, B. Zang, 

\vspace*{-1mm}

\bibitem{NA50_2} NA50 letter of intent:
http://www.cern.ch/NA50/pixel/loi

\vspace*{-1mm}

\bibitem{Shuryak} E.V. Shuryak, 

\vspace*{-1mm}

\bibitem{Rapp} R. Rapp, 

\vspace*{-1mm}

\bibitem{KKMcLR} K. Kajantie, M. Kataja, L. McLerran, P.V. Ruuskanen,
Phys. Rev. D 34 (1986) 811

\vspace*{-1mm}

\bibitem{Ruuskanen} 
P.V. Ruuskanen, in {\it Quark-Gluon Plasma}, (Ed.)
R. Hwa, World Scientific 1990

\vspace*{-1mm}

\bibitem{BK_2} 
B. K\"ampfer, O.P Pavlenko, A. Peshier, M. Hentschel,
G. Soff, J. Phys. G 23 (1997) 2001c

\vspace*{-1mm}

\vspace*{-1mm}\bibitem{Heinz}
B. Tomasik, U.A. Wiedemann, U. Heinz, 

\vspace*{-1mm}

\bibitem{Asakawa_Ko_Levai} 
M. Asakawa, C.M. Ko, P. Levai,
Phys. Rev. Lett. 70 (1993) 398

\vspace*{-1mm}

\bibitem{BK_1}
B. K\"ampfer, O.P. Pavlenko, A. Peshier, G. Soff,
Phys. Rev. C 52 (1995) 2704

\vspace*{-1mm}

\bibitem{Klingel_Weise} F. Klingl, W. Weise, 

\vspace*{-1mm}

\bibitem{PYTHIA} T. Sj\"ostrand, Comp. Phys. Commun. 82 (1994) 74

\vspace*{-1mm}

\bibitem{NA38} M.C. Abreu et al. (NA38), Phys. Lett. B 423 (1998) 207

\vspace*{-1mm}

\bibitem{DY_data} D.M. Kaplan et al., Phys. Rev. Lett. 40 (1978) 435,\\
A.S. Ito et al., Phys. Rev. D 23 (1981) 604,\\
C.N. Brown et al., Phys. Rev. Lett. 63 (1989) 2637  

\vspace*{-1mm}

\bibitem{Aurenche} P. Aurenche, F. Gelis, R. Kobes, H. Zaraket,
\\
%\bibitem{Thoma} 
M. Thoma, A. Sch\"afer, Phys. Lett. B 451 (1999) 195

\end{thebibliography} }

\newpage
%\centerline{{\bf Figure captions}}

\begin{figure}[t]
\centering
~\\[-1cm]
%\centerline{\epsfxsize=.7 \hsize \epsffile{lhcpt.eps}}
\psfig{file=Ceres.M.eps,width=6.5cm,angle=-90}
\hfill
\psfig{file=Ceres.QT2.eps,width=6.5cm,angle=-90}
~\\[.5cm]
\caption{
The preliminary CERES data \protect\cite{CERES_1}
and the hadronic cocktail \protect\cite{CERES_1} (dashed lines)
and the thermal yield (full curves). The sum
of the cocktail and the thermal yield is shown by the gray curves.
Left panel (a): the invariant mass spectrum,
right panel (b): the transverse momentum spectrum for the mass bin
0.25 $\cdots$ 0.68 GeV.
}
\label{fig.1}
\end{figure}

%\newpage
\begin{figure}[b]
\centering
~\\[-.1cm]
%\centerline{\epsfxsize=.7 \hsize \epsffile{cut10.eps}}
\psfig{file=NA50.M.eps,width=6.5cm,angle=-90}
\hfill
\psfig{file=NA50.QT.eps,width=6.5cm,angle=-90}
~\\[.5cm]
\caption{
The reconstructed measurements of the NA50 collaboration (see text for
details) indicated by open squares (not data!) in comparison with the thermal yield
(full curves), the Drell-Yan contribution (dashed curves) and the
contribution of open charm decays (dash-dotted curves). The sum of
these contributions is displayed by the gray curves.
Left panel (a): the continuum invariant mass spectrum,
right panel (b): the transverse momentum spectrum for the mass bin
1.5 $\cdots$ 2.5 GeV.
}
\label{fig.2}
\end{figure}

\end{document}


