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\title{Formation of antideuterons in heavy ion collisions}
\author{{B.L. Ioffe, I.A. Shushpanov and K.N. Zyablyuk}\\
{\it \normalsize Institute of Theoretical and Experimental Physics},\\
{\it \normalsize B.Cheremushkinskaya 25, 117218 Moscow, Russia},\\
{\tt \normalsize ioffe@vitep1.itep.ru, shushpan@heron.itep.ru, zyablyuk@heron.itep.ru}}
\date{}
\begin{document}

\maketitle

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}



\begin{abstract}
The antideuteron production rate at high-energy heavy ions
collisions is calculated basing on the concept of $\bar{d}$
formation by antinucleons which move in the mean field of
freeze-out fireball constituents (mainly pions). The explicit
formula is presented for the coalescence parameter $B_2$ in terms
of deuteron binding energy and fireball volume.
\end{abstract}

\bigskip

\centerline{PACS numbers:  25.75.-q, 25.75.Dw, 12.38.Mh}

\vspace{10mm}

1.{\it Introduction}. The production of antideuterons in heavy ion
collisions was observed at SPS \cite{SPS}, RHIC \cite{RHIC} and AGS \cite{AGS}.
It is desirable to have theoretical understanding of this interesting phenomenon, which
also may shed some light on the mechanism of heavy ion collisions
\cite{4}-\cite{6}. The theoretical description of antinuclei
production in heavy ion collisions was mainly restricted to statistical
models \cite{7,8}, where the small binding energy of antideuteron and
its large size were not accounted. In the model of \cite{Mr} the antideutron
production rate was found in terms of the deutron wave function and
antinucleon distributions, but the interaction of the antideutron with
the fireball enviroment was not considered.
 Because of small binding energy
in comparison to kinetic energies of other particles,
antideuterons can survive only if they are produced at the final
stage of evolution of the fireball, when the fireball may be
considered as a hadronic gas. The dominant mechanism then is the
formation of antideuteron through the reaction $\bar{p} + \bar{n} \to
\bar{d}$. It is clear, that the rate of this reaction vanishes,
when the size of $\bar{d}$ increases, i.e.~its binding energy
$\varepsilon$ is going to zero. Therefore, one may expect an
essential dependence of $\bar{d}$ production rate on
$\varepsilon$. The antideuteron production (unlike $d$-production)
cannot proceed in the collision of peripheral (spectator) nucleons.
So, this process is sensitive to the dynamics of heavy ion collisions.

The formation process $\bar{p} +\bar{n} \to \bar{d}$ is not
realized if all participating particles are on mass shell. In the
fireball, even  at the last stage of its evolution, when $\bar{d}$
is formed, $\bar{p}$, $\bar{n}$ and $\bar{d}$ are not on mass
shell -- they interact with surrounding matter. This fact strongly
enhances the reaction rate being compared with the main process of $d$
production in vacuum $p + n \to d + \pi$.

 According to the dominant coalescence mechanism it is convenient to
 characterize $\bar{d}$ production in heavy ion
 collisions by the coalescence parameter:   %1
\be
B_2 \, =\,  E_{\bar{d}} \frac{d^3 N_{\bar{d}}}{d^3 p_{\bar d}}   \left(
E_{\bar{p}} \frac{d^3 N_{\bar{p}}}{d^3 p_{\bar{p}}} \, E_{\bar{n}} \frac{d^3
N_{\bar{n}}}{d^3 p_{\bar{n}}} \right)^{-1},
\ee
where we can put $d^3N_{\bar{p}}/d^3 p_{\bar{p}}=d^3
N_{\bar{n}}/d^3p_{\bar n},~p_{\bar{p}}=p_{\bar{n}}=p_{\bar{d}}/2$.
 In what follows, we will
consider only the central heavy ion collisions.

 2. {\it Theory.}
Consider the final stage of fireball evolution -- the so called
freeze-out fireball, when the fireball may be treated as a gas of
weakly interacting hadrons. Assume, that particle propagations at
this stage  may be described classically using kinetic equations.
We use the notation $q_i(x,p)$, $i = \bar{p}$, $\bar{n}$, $\bar{d}$,
$\pi$  for the double density in coordinate and momentum
spaces, $n_i(x) = \int~ q_i(x,p) d^3 p$ are the densities.
($q_i(x,p)$ -- are Lorentz invariant.) Let us work in the c.m.~system of
colliding ions. The kinetic equation for $q_{\bar{d}}(p_{\bar{d}},
x)$ may be written in explicitly Lorentz covariant form
($x$-dependence at the right-hand side is omitted):  %2
$$
\frac{m_{\bar{d}}}{E_{\bar{d}}} \frac{\partial q_{\bar{d}}(p_{\bar{d}},x)}{\partial x_{\mu}}
\, u_{\mu}^{\bar{d}} \,= \, \frac{\partial q_{\bar{d}}}{\partial t} \, +\, {\bf v}_{\bar{d}} \nabla
q_{\bar{d}} \, =\, \int d^3 p_{\bar{p}}\, d^3 p_{\bar{n}} \, q_{\bar{p}}
(p_{\bar{p}}) \, q_{\bar{n}} (p_{\bar{n}})\,  \sigma_{\bar{p} \bar{n}
\to \bar{d}} \, v^{rel}_{\bar{p} \bar{n}} \, \times
$$
\be
\times  \delta^3(p_{\bar{p}}  + p_{\bar{n}} - p_{\bar{d}}) \, - \,
q_{\bar{d}}(p_{\bar{d}}) \left[ \, \int d^3 p_{\pi} \, q_{\pi} (p_{\pi}) \,
\sigma_{\pi \bar{d}} \, v^{rel}_{\pi \bar{d}} \, +  \, \ldots  \, \right]
\ee
where
$u_{\mu}^{\bar{d}} = (1, {\bf v}_{\bar{d}}) /
\sqrt{1-v^2_{\bar{d}}}$ is the $\bar{d}$ 4-velocity, the ellipsis
mean similar terms for collisions of $\bar{d}$ with other
constituents of the fireball ($p,n$ etc.) and $v^{rel}_{\bar{p}
\bar{n}}$, $v^{rel}_{\pi \bar{d}}$ are the differences of
$\bar{p}$, $\bar{n}$ and $\pi, \bar{d}$ velocities
$v^{rel}_{\bar{p}\bar{n}}=\mid {\bf v}_{\bar{p}}-{\bf
v}_{\bar{n}}\mid $ etc. The terms, when $\bar{d}$ appears in the
momentum interval $p_{\bar{d}} + \Delta p_{\bar{d}}$ due to
elastic collisions are neglected. Necessary applicability
condition of (2) is $\lambda=p^{-1}_i \ll d$, where $d$ is the mean
distance between fireball constituents.

The cross section  $\sigma_{\bar{p} \bar{n} \to \bar{d}} = \sigma_{pn \to d}$ is equal to: %3
\be
\sigma_{pn \to d} = \frac{3}{4} \cdot \frac{\pi}{4} ~
\frac{g^2}{E_p E_n E_d}~ \frac{1}{v^{rel}_{pn}}~ \delta(E_p + E_n
- E_d),
\ee
where $E_p,E_n,E_d$ are $p,n$ and $d$ total energies,
3/4 is the spin factor and $g$ is the coupling constant of low
energy effective $pnd$ intraction (in the $d$ c.m.~system). The value
of $g^2$ was found by Landau \cite{9} from the requirement of
coincidence (at the deuteron pole) of the $pn$ scattering amplitude in
effective theory with the amplitude in the Bethe-Peierls theory of the
low-energy $pn$-scattering \cite{10}. In the limit of zero range of
nuclear forces $g^2$ is %4
\be
g^2 \,= \,128 \pi\, m_N \sqrt{m_N \varepsilon},
\ee
where $m_N$ is the nucleon mass, $\varepsilon = 2.2\, {\rm MeV}$ is the deuteron binding
energy.  The account of non-zero range $r_0$ increases $g^2$ by a
factor of $(1-\sqrt{m_N \varepsilon} r_0)^{-1} \approx 1.6$ \cite{10,11}.

The mass of the particle moving in medium is shifted being compared with its
vacuum value. Similarly, due to interaction with medium
constutuents, the width $\Gamma$ appears (or width broadening, if the
particle has its proper width). This phenomenon is analogous to  existence
of the refraction and attenuation indices in optics. The mass shift $\Delta
m(E)$ and $\Gamma(E)$ are expressed through the forward scattering amplitude
$f(E)$ of the particle on medium constituent (see \cite{12,13} and references
therein). %5
\be
\Delta \, m(E) \, = \, - \, 2 \pi \, \frac{n}{m} \, Ref (E)
\ee  %6
\be
\Gamma(E)\, =\, 4 \pi \,\frac{n}{m}{\rm Im}\, f(E)\, =\, \frac{np}{m}\, \sigma(E),
\ee
where $E$, $p$ and $m$ are particle energy, momentum and mass,
$n$ is the density of the constutuent in medium. Eqs.~(5),(6) take
place in the system, where constutuents are at rest. In case
of moving constituents the corresponding Lorentz boost must be
done. (By definition $\Delta m$ and $\Gamma$ are Lorentz invariant,
for details see \cite{14}).

Therefore, $\bar{p}$, $\bar{n}$ and $\bar{d}$ in the reaction $\bar{p} + \bar{n}
\to \bar{d}$ can be considered as Breit-Wigner resonances with varying
masses distributed according to the Breit-Wigner formula. The first term
in the r.h.s.~of (2) after substituting (3) is represented by  %7
$$
I\,=\,\frac{3 \pi}{16} g^2\int\frac{d^3p_{\bar{p}} \, d^3p_{\bar{n}}}{E^{\prime}_{\bar{p}}\, E^{\prime}_{\bar{n}}
\, E^{\prime}_{\bar{d}}} \,q_{\bar{p}} (p_{\bar{p}})\, q_{\bar{n}}
(p_{\bar{n}})\, \delta^3 (p_{\bar{p}}+p_{\bar{n}}-p_{\bar{d}})\,
\delta(E^{\prime}_{\bar{p}}+E^{\prime}_{\bar{n}}-
E^{\prime}_{\bar{d}})\, dm^{\prime}_{\bar{p}}
\, dm^{\prime}_{\bar{n}}\, dm^{\prime}_{\bar{d}} \times
$$
\be
\times ~ \frac{\Gamma_{\bar{p}}/2
\pi}{(m^{\prime}_{\bar{p}}-m_{\bar{p}})^2 +
\Gamma^2_{\bar{p}}/4}~\frac{\Gamma_{\bar{n}}/2
\pi}{(m^{\prime}_{\bar{n}} - m_{\bar{n}})^2 +
\Gamma^2_{\bar{n}}/4}~\frac{\Gamma_{\bar{d}}/2
\pi}{(m^{\prime}_{\bar{d}} - m_{\bar{d}})^2 +
\Gamma^2_{\bar{d}}/4}
\ee
where $E^{\prime}_{\bar{p}} = \sqrt{p^2_{\bar{p}}+m^{\prime 2}_{\bar{p}}}$ etc. With a good
accuracy $\Gamma_{\bar{p}} = \Gamma_{\bar{n}} = \Gamma_{\bar{d}}/2
\equiv \Gamma$ and $q_{\bar{p}}(p_{\bar{p}}) = q_{\bar{n}}(p_{\bar{n}}),$
$p_{\bar{p}}=p_{\bar{n}}=p_{\bar{d}}/2$. Assume, that $\Gamma \ll
m$ and is much smaller than characteristic momenta in
$\bar{p}$, $\bar{n}$ distributions. (In fact, $\Gamma \sim 300
\, {\rm MeV}$ -- see below, $\Delta m$ are small, $\Delta m \sim 30 \, {\rm MeV}$).
Then we can omit $\prime$ in $E^{\prime}_i$ in (7), except for
$\delta$-function and integrate over $m^{\prime}_i$ from $-\infty$
to $+\infty$. We get %8
\be
I \, =\, \frac{3 \pi^2}{8 E_{\bar{d}}} g^2 \sqrt{\frac{\Gamma}{m_N}}
q^2_{\bar{p}} (p_{\bar{p}})
\ee
It can be shown, that the term in square brackets in (2) is equal to %9
\be
(m_d/E_{\bar{d}}) \, \Gamma_{\bar{d}}\, =\, 2 \, (m_N/E_{\bar{p}}) \, \Gamma.
\ee
Let us sketch the proof. Consider the pion
contribution to $\Gamma$. Let us perform the boost from the pion rest
system, where (6) takes place, to the $\bar{d}$ rest system. We
have $n_{\pi}(\pi~ \mbox{at rest}) = n_{\pi}(\bar{d}~ \mbox{at rest})(m_{\pi}/E_{\pi})$
and from kinematics $m_d E_{\pi}(\bar{d}~\mbox{at  rest}) = m_{\pi} E_{\bar{d}}(\pi~\mbox{at rest})$.
Since $\sigma$ is the same in both systems, $p_{\bar{d}}=v_{\bar{d}} E_{\bar{d}}
(\pi~\mbox{at rest})$, $v_{\bar{d}}(\pi~\mbox{at rest})=v^{rel}_{\pi
\bar{d}}$ ($\bar{d}~\mbox{at rest})$, %9
\be
\Gamma_{\bar{d}} \, =\,  \int d^3 p_{\pi} \, q_{\pi} (p_{\pi}) \,
v^{rel}_{\pi \bar{d}} \, \sigma_{\pi \bar{d}}(\bar{d}~\mbox{at rest})
\ee
As follows from (2), due to Lorentz invariance, the  second term in
the r.h.s.~of (2) has the form %11
\be
\int d^3 p_{\pi} \, q_{\pi}(p_{\pi}) \, \sigma_{\pi\bar{d}} \,
v^{rel}_{\pi\bar{d}}\,=\, a\, \frac{m_{\bar{d}}}{E_{\bar{d}}} \, .
\ee
Going to the $\bar{d}$ rest system, after comparison  with (10) we get $a=\Gamma_{\bar{d}}$.
 The proof for other constutuents is similar, but their contribution to $\Gamma$ is small
 ($\sim 10-20$\%).

Suppose, that the rate of antideuteron collisions with other constituents of
 the fireball resulting in antideuteron dissociation is much larger than
the rate of fireball expansion. This happens at collisions of
heavy nuclei at high energies, when the size of freeze-out
fireball is large because of large number of produced pions per
nucleon. In this case one may expect the balance: the first term
in the r.h.s. of (2) is equal to the second one and %10
\be
q_{\bar{d}}(p_{\bar{d}}) \, =\, \frac{I}{\Gamma_{\bar{d}}
(m_{\bar d}/E_{\bar{d}})}\, =\, \frac{3 \pi^2}{32\, m_N} \, \frac{1}{\sqrt{\Gamma\,
m_N}} \, g^2\, q^2_{\bar{p}} (p_{\bar{p}})
\ee
$d^3 N_{\bar{d}}/d^3 p_{\bar{d}}$ entering (1) is obtained from (12) by integration
over fireball volume %11
\be
\frac{d^3 N_{\bar{d}}(p_{\bar{d}})}{d^3 p_{\bar{d}}} \, =\, \int d^3x\, q_{\bar{d}}(p_{\bar{d}},x)
\ee

Using (1), (12), (13) and (4) (with $r_0$ correction) we find for
the  coalescence parameter
\be
\label{b20}
B^{th}_2 \, = \, \frac{24 \pi^3}{E_{\bar{p}}} \times 1.6 \,
\sqrt{\frac{\varepsilon}{\Gamma}}  \,
\,{\int d^3x \, q^2_{\bar p}(p_{\bar p}, x)\over \left[ \int d^3x \, q_{\bar p}(p_{\bar p}, x) \right]^2}
\ee
Since the $x$-dependence of $ q_{\bar p}(p_{\bar p}, x)$ is not known, we replace (\ref{b20}) by:
\be
B^{th}_2\, =\, \frac{24 \pi^3}{E_{\bar{p}}} \times 1.6 \,
\sqrt{\frac{\varepsilon}{\Gamma}} \,
\frac{2}{V}\,\frac{\overline{n^2}_{\bar{p}}}{(\bar{n}_{\bar{p}})^2}
\ee
where $V$ is the fireball volume, $\bar{n}_p$ and $\overline{n^2}_p$ are
the mean and mean square $\bar{p}$ densitites in the fireball.
(The coordinate dependence of $\sqrt{\Gamma}$ is neglected).
$B^{th}_2$ (14) is Lorenz invariant, as it should be. The
freeze-out volume $V$ may be understood as a mean value of the
fireball volume at a stage, where, on one side, hadrons are
already formed, i.e., mean distances between them are larger than
the confinement radius $R_c \sim 1/m_{\rho} \sim (1/4)\,{\rm  fm}$, but on
the other side, hadron interactions are still essential. The
antinucleon  distributions $n_{\bar{p}}({\bf r}),n_{\bar{n}}({\bf
r})$ inside the  fireball are essentially nonuniform: at the
pre-freeze-out and freeze-out stages  antinucleons strongly
annihilated in the internal part of the fireball and in much less
extent in its  external layer of the thickness of order
$\bar{p}(\bar{n})$ annihilation length $l_{ann}$
(this effect was considered in \cite{Mr}). For this reason
$\overline{n^2}_p/\bar{n}^2_p$ may be remarkable larger than 1.
For the same reason the antinucleons and antideuterons from
the backside of the fireball (relative to the observer) are
absorbed in the fireball and cannot reach the detector (see Fig.~1).
Therefore, only one half of the fireball volume contributes to the
number of registered ${\bar p}$, ${\bar n}$ and ${\bar d}$. The corresponding factor
approximately equal to 2 is accounted in (15).


\begin{figure}[tb]
\hspace{50mm} \epsfig{file=fire_b.eps, width=60mm}
\caption{Fireball at the freeze-out stage. The effective volume $V_{eff}$ is the half of the
outer shell of thikness $l_{ann}$, from which the antiprotons reach detector}
\label{fig_1}
\end{figure}

$\Gamma$ may be calculated in one or another freeze-out fireball
model. Then by comparison with the data the parameter
$V^{-1}(\overline{n^2}_{\bar{p}}/\bar{n}^2_{\bar{p}})$ can be
found, what would allow to check various models of fireball
evolution.

3. {\it Comparison with the data}. Consider the NA44 experiment at
SPS (CERN): $Pb+Pb$ collisions at $\sqrt{s} = 17 A \, {\rm GeV}$ \cite{SPS}.
Antideuterons were observed at $0.6 < p_{\bar{d}t} < 1.6 \,{\rm GeV}$ and
in the rapidity interval 1.9 to 2.1 in lab.~system, which
corresponds to $\bar{p}_{\bar{p}t} = 0.55\,{\rm GeV}$, $(E_{\bar{p}})_{c.m.} = 1.5\,{\rm GeV}$.
The spectra and particle yields at such collisions are given in \cite{15}. The number
of active nucleons, participating in collision ("wounded"
nucleons) $N_N$ and the number of produced pions are presented in
\cite{16}: $N_N = 362$, $N_{\pi} = 1890$, $Q_{\pi} = N_{\pi}/N_N = 5.2$
(see also \cite{17} for the review of the data on heavy ion
collisions).

We accept the following model for the freeze-out (final) stage of
fireball evolution \cite{13}. (A related model had been suggested long
ago \cite{18,19}: it may be called Fermi--Pomeranhuk model). Neglect
for a moment contributions of all particles except for nucleons
and pions. Assume that at freeze-out stage any participant --
nucleon or pion occupies the volume $v_N$ or $v_{\pi}$,
respectively. Then
\be
n_N \,= \, \frac{N_N}{V}\, = \, \frac{n^0_N}{1 + Q_{\pi} \beta} \; , \qquad
n_{\pi} \, =\, \frac{N_{\pi}}{V}\, =\, \frac{n^0_N Q_{\pi}}{1 + Q_{\pi} \beta}
\ee
where $n^0_N = 1/v_N$, $\beta = v_{\pi}/v_N$. For numerical
estimations we take $n^0_N = 0.26 \, {\rm fm}^{-3}$, 1.5 times standard
nucleus density and $\beta = (r_{\pi}/r_N)^3 \approx 0.55$, where
$r_{\pi} = 0.66\, {\rm  fm}$ and $r_N = 0.81 \, {\rm fm}$ are pion and nucleon
electric radii.

Check first the applicability conditions of our approach. We have:
$n = n_N + n_{\pi} \approx 0.42\, {\rm fm}^{-3}$ and the mean distance
between the fireball constutuents is $d = 1/n^{1/3} = 1.3\,{\rm fm}$.
Evidently, the condition $\lambda_{\bar{p}} = 1/p_{\bar{p}} \ll d$
is well satisfied. Check now the balance condition -- that the
probability of deuteron dissociation  exceeds its escape or
fireball expansion rate. The former is given by $2\Gamma(m_N/E_{\bar{p}})$.
Using (10) and the pion spectrum, presented in \cite{15}, $\Gamma$ was found:
$\Gamma\approx 300\, {\rm MeV}$.
The estimation for the escape rate (or fireball expansion) is $w\sim (1/4)\,{\rm fm}^{-1}$ We have:
$2 \Gamma(m_N/E_p) \approx 2.0\, {\rm fm}^{-1} \gg 0.25\,{\rm fm}^{-1}$. So, this condition is also fulfilled.
Eq.6 is legitimate, if $Im f(E) \ll d$ \cite{12,13}. Since
${\rm Im}\, f \approx 1\,{\rm fm}$, this condition is not well satisfied. For this
reason the value of $\Gamma$, presented above, has a large (may be
50\%) uncertainty. This fact, however, does not influence to much
the value $B^{th}_2$, since $\sqrt{\Gamma}$ enters (14). One may
expect, that because of their larger velocities in comparison with
nucleons, pions form a  halo around  the fireball. So, their
density, and, consequently, the value of $\Gamma$, probably, are overestimated.


At the parameters used above the fireball volume
comes out to be:  $V = 6.2\times 10^3\,{\rm fm}^3$ (15\% correction for
other particles, except for pions and nucleons were accounted). In
the case of sphere its radius is equal to $R = 11.4\,{\rm fm}$. If we
assume, that antiprotons  are mainly concentrated in the outer
shell of the fireball  of the thickness of $l_{ann}\approx 3\,{\rm fm}$,
then $\overline{n^2}/\bar{n}^2\approx 2$ and we get for the coalescence parameter
\be
B^{th}_2\, =\, 3.5 \times 10^{-4}\,{\rm GeV}^2
\ee
Experimentally [1], for the average value of the most central 10\% events it was found:
$B^{exp}_2 = (4.4 \pm 1.3) \times 10^{-4}\,{\rm GeV}^2$. However,
$B^{exp}_2$ strongly depends on centrality: the results for
$0-5\%$ centrality are about 1.5 times lower. Taking in mind all
uncertaintlies -- theoretical and experimental, we believe, that
the NA44 data for coalescence parameter are not in contradiction with
theoretical expectation.


Turn now to the STAR experiment at RHIC: $Au+Au$ collisions at
$\sqrt{s} = 130 A \,{\rm GeV}$ \cite{RHIC}. Antideuterons were measured at $0.5 <
p_t < 0.8\,{\rm GeV}$ and in the rapidity interval $\vert \Delta y_{c.m.}
\vert < 0.3$, $18\%$ of central collisioins were collected.  We
take  $\bar{E}_{\bar{p}, c.m.} = 1.05\,{\rm GeV}$. The number of
"wounded" nucleons in the $18\%$ central $Au+Au$ collisions can be
estimated as  $N_N = 320$ \cite{20}. Multiplicity of negative hadrons
$\bar{h}$ (mainly, pions)  was measured in \cite{21} at pseudorapidity
$\eta = 0$ only  and it was found an increasing of $d h^-/d
\eta\mid _{\eta=0}$ by $52\%$ comparing with the SPS data at
$\sqrt{s} = 17\,{\rm GeV}$. But it is known
that $d h/d \eta/_{\eta=0}$ increase faster with
energy than the total multiplicity. We
estimate $Q_{\pi} = N_{\pi}/N_N \approx 7 \pm 1$. (A value close
to the presented above, can be found from the data compilation
\cite{22}). At $N_N = 320$ with account of $20\%$ correction for
$K$-mesons and hyperons the freeze-out volume is equal to $V =
7.2\times 10^3\,{\rm fm}^3$. The coalescence parameter is equal to
\be
B^{th}_2 = 4.4 \times 10^{-4}\, {\rm GeV}^2
\ee
($\Gamma = 320\,{\rm MeV}$, $\bar{n}^2/(\bar{n})^2$ was put to be 2). Experimentally, STAR
found $B^{exp}_2 = (4.5 \pm 0.3 \pm 1.0) \times 10^{-4}\,{\rm GeV}^2$.
So, the agreement  with experiment is satisfactory.

The main uncertainty of $B_2^{th}$ comes from the fireball volume $V$
which was calculated by (16) with the density $n^0_N$ $1.5$ times larger
than standard nuclear density. However, the width  $\Gamma$ also depends
on the fireball volume, so that $B_2^{th}\sim 1/\sqrt{V}$,
which suppresses this uncertainty twice. We expect the
accuracy of our estimations (17), (18) to be about  $50\%$.


In  E864 experiment \cite{AGS} at AGS the antideuterons  were observed in
$Au+Pt$ collisions at $\sqrt{s}=4.8\,A\,{\rm GeV}$. 10\% of central
collisions we selected. From the data we take: $p_{\bar{p}t}
=0.17\,{\rm GeV}$, $\overline{E}_{\bar{p},c.m.}=0.99\,{\rm GeV}$. The number of
``wounded'' nucleons and $\pi/N$ ratio are $N_N=350$,
$Q_{\pi}=1.6$ (see \cite{17} and references herein).In the same way as
before, we find: $V=2.8\times 10^3\,{\rm fm}^3$, $\Gamma=220\, {\rm MeV}$,
$l_{ann}=1.2\,{\rm fm}$. In this case the validity conditions of our
approach are at the edge of their applicability. So, the
theoretical expectations for $B_2$ are valid only by the order of
magnitude:
\be
B^{th}_2 \sim 1.5 \times 10^{-3}\, {\rm GeV}^2
\ee
in comparison with $B^{exp}_2 = (4.1\pm 2.9\pm 2.3)\times 10^{-3}\,{\rm GeV}^2$.

4.~{\it Summary and Acknowledgements.} The coalescence parameter $B_2$ for
the antideuteron production in heavy ions collisions was calculated.
It was supposed, that the $\bar{d}$ production proceeds at freeze-out
fireball, when the fireball may be treated as a gas of weakly
interacting hadrons. The $\bar{d}$ production is described as
the formation process $\bar{p}+\bar{n}\to \bar{d}$, where
$\bar{p}$, $\bar{n}$, $\bar{d}$ are moving in the mean field of the fireball
constituents (mainly pions). It was shown, that in case of large
$N_{\pi}/N_N$ ratio one may expect a balance: the number of
produced antideuterons is equal to the number of dissotiated
$\bar{d}$ due to collisions with pions. The balance  condition
determines $\bar{d}$ production rate and the value of coalescence
parameter $B_2$. The later is expressed in terms of deuteron
binding energy, and freeze-out fireball volume. The theoretical
values of $B_2$ are in satisfactory agreement with experimental
data at SPS, RHIC and AGS but more data at various nuclei and various
energies of collision and $\bar{d}$ energies would be very
desirable. The comparison of the data with theory would allow
to check various models of fireball evolution.

We are thankful to S.Kiselev, Yu.Kiselev, A.Smirnitsky and N.Rabin
for information about experimental data. This work was supported
in part by CRDF grant RP2-2247 and INTAS grant 2000-587.



\newpage

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