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\newcommand{\JP}{{J/\psi}}
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\begin{document}

\vspace*{1.0cm}
\centerline{\Large{M.V.Lomonosov Moscow State University}}
\centerline{\Large{D.V.Skobeltsyn Institute of Nuclear Physics}}

\vspace*{2.0cm}
\rightline{\large{INP MSU Preprint 96--30/437}}

\vspace*{3.0cm}

\bigskip
\centerline{\Large{The Modified Intrinsic Charm Model}}
\centerline{\Large{And $J/\psi$ Production in Hadronic Collisions}}

\vspace*{1.0cm}
\centerline{\large{Yu.A.Golubkov}}
\centerline{\small{e--mail: golubkov@elma01.npi.msu.su,
~golubkov@vxdesy.desy.de}}

\vspace*{9.0cm}
\centerline{\large{Moscow 1996}}

\vfill\eject
\vspace*{5.0cm}
\centerline{Abstract}
\bigskip
Using the statistical approach additional light
partons have been introduced in the Intrinsic Charm model.
Explicit expressions are obtained for charmed and light quark
distributions. A comparison is made with standard IC model predictions.
It is shown that such a modification of the IC model leads
to better agreement with the $pN$ and $\pi N$ data on $\JP$ production
at low energies and allows to avoid a contradiction to experimental
data at high energy.

\vspace*{3.0cm}
\leftline{\small{\copyright Moscow State University, 
Institute of Nuclear Physics}}

\vfill\eject

\bigskip
\section{Preface}

\npar
Evidence for an excess of charm production at high $x_F$
observed in a number of experiments on
proton--proton \cite{Drij79} and muon--proton \cite{EMC82} collisions
stimulated the development of the model with an assumption that
the hadron wave function decomposition may contain
the \hbox{$\vert\,uud\ac c>$} Fock state component.
The first theoretical estimation for the charmed particles production
due to nonpertubative charm component inside the proton has been made 
by S.Brodsky et al. in \cite{Brodsky80} using old--fashioned
pertubation theory.
The standard Intrinsic Charm (IC) model predicts very hard distributions
for charmed particles due to the fact that the charmed quarks in the
proton carries a large fraction of the proton longitudinal momenta.
But some last experimental data \cite{Kowit94} on the $\JP$ production 
in $pN$ interactions at $P_0\,=\,800\ GeV/c$ show
that the distribution predicted by the model \cite{Brodsky80}
strictly contradicts to the experiment at large $x_F$.

\npar
If the above assumption about excitation of the higher Fock states
with heavy quarks is valid there must be
much larger probability to create additional light partons.
Therefore we have to consider also the Fock
states with arbitrary number of light quarks and gluons.
With natural restriction that this parton system as whole 
has to carry the hadron quantum numbers and sum of the parton momenta
is to be equal to the hadron momentum.
In this case the additional light partons
restrict the available phase space and lead to effective
decrease of the average longitudinal momentum of heavy quarks.
Despite the recreated interest to IC problem no attempts
have been made so far to consider this effect.
This step has obvious physical basis and, as we show,
leads to visible experimental consequences due to 
softer momentum distributions of the partons.
The increase of the number of light quarks leads also to larger
absolute value of total electric charge of the partonic system
and to the larger cross section in the electroproduction processes.

\npar
In present paper we exploit the statistical approach \cite{Kuti71},
derive expressions for distributions of heavy and light partons 
within framework of the non--covariant pertubation theory
and compare the $x_F$ distributions of the $\JP$ particle
to experimental data in $\pi N$ and $pN$ collisions.

\npar
We shall speak mostly about the proton. For meson all the calculations
are completely similar. The only difference is that in the meson case
the lowest Fock state contains two light quarks instead of
three ones and, therefore, number of integrations in the following
expressions will be lesser by unit in comparison to the proton case.

%----------------------------------------------

\section{Description of the model}

\subsection{The original IC model}

\npar
Further in present paper we shall denote as $F^{(n)}$ a Fock state
{\hbox{$|uudc\ac+n>$}}
containing one {\cc} pair, three (two for meson) "valence" quarks,
carring the hadron quantum numbers and n light partons, gluons and
quark--antiquark pairs, with zero summary quantum numbers. 
In papers \cite{Brodsky80} it was proposed that the probability
to create a Fock--state $F^{(n)}$ in the proton is being described 
by the expression:

\begin{equation}
\label{postulate}
dW^{(n)}\,(x_1,x_2,\ldots,x_{n+5})\ \sim
\left (M^2\,-\,\sum_{i=1}^{n+5}\frac{\trm{i}^2}{x_i}\right )^{-2}
\ \delta\left (1-\,\sum_{i=1}^{n+5}\,x_i\right )\,\prod_{i=1}^{n+5}
\,dx_i,
\end{equation}

\noindent where the $\trm{i}$ are the effective transverse
masses of the partons {\hbox{$\trm{i}^2\,=\,m_i^2\,+\,<\trp{i}^2>$}}, 
the $<\trp{}^2>$
is the average transverse momentum squared, M is the proton mass 
and n denotes the number of additional light partons. 

\npar
If to regard valence quarks only, i.e., the Fock state 
\hbox{$F^{(0)}\,=\,|uudc\ac>$}
and to neglect mass of the proton $M$ and all the $\trm{} ^2$ 
for light quarks one easily obtains the one--particle distributions,
normalized on unit:

\begin{equation}
\label{brodist}
\begin{array}{lll}
c_1^{(0)}(x) & = & 1800\,x^2\left [\frac{1}{3}\,(1-x)\,(1+10\,x+x^2)
+2x\,(1-x)\,\ln\,x\right ] \\
q_1^{(0)}(x) & = & 6\,(1-x)^5 \\
\end{array}
\end{equation}

\noindent for charmed and light quarks, respectively.

\npar
Because we are interested only in the hadron Fock states
containing two charmed quarks we can put in the folowing calculations
the overall normalization $N_{IC}$ equal to unit for the sake of simplicity. 
To recalculate final results it is
enough to multiply them by factor $N_{IC}^{exp}$, extracted either from an
experiment or from a theoretical model. This normalization is order of 
$N_{IC}^{exp}\ \sim\ (0.5 - 1)\%$ 
(see \cite{EMC82,Hoffmann83,Golub94,Harris96}).

\subsection{Arbitrary number of light partons}
\label{arbin}

\npar
In this section we use the statistical approach
\cite{Kuti71} to introduce arbitrary number of additional light partons.
Remind that we consider all the partons are being on the mass shell
and use the non--covariant pertubation theory \cite{Brodsky80}.
It is also necessary to keep in mind that the results obtained in present
paper are, generally speaking, valid only for partons with sufficiently
large momentum fractions ("valence" partons). The wee partons
are being created due to bremsstrahlung and have different
distribution. In present paper we don't consider their contribution
and we use the condition $\sum_i\,\int\,dx\,f_i(x)\,=\,1$.
If we introduce additional sea partons following Eq.(\ref{postulate})
and negelect transverse masses of light partons
all the light quarks and gluons will have the same momentum distribution.
Thus we can not distinguish the "valence" and the "sea" light quarks. 
The validity of the approximation (\ref{postulate})
for many partons we consider later in this paper.


\npar
The probability to create a final state is proportional to 
$d^3p/(2\pi \hbar)^3$ --- the number of cells in the momentum space.
Because we don't consider transverse movement we can use 
the longitudinal phase space only.
We define \hbox{$p_k\,=\,P_{max}\,x_k$}, where $P_{max}$ is 
the maximal longitudinal momentum of the parton.
Thus we can write for the probability to create
either one gluon $(g)$ or one quark $(q)$ (remind, that $\hbar$\,=\,1):

\begin{equation}
\label{avalue}
dW\ \sim\ \frac{b}{2\,\pi}\,w_{g,q}\,P_{max}\,dx_g\ \equiv\ a_{g,q}\,dx_{g,q},
\end{equation}

\npar
where, $w_k$ is the statistical weight of the parton depending on 
its colour and spin.
To simplify further formulae we introduced the entity
\hbox{$a_{g,q}\ =\ (b/2\pi)\,w_{g,q}\,P_{max}$}.

\npar
For the on--mass--shell gluons we have eight colours
and two polarization states. Similarly for each quark we have
three colours and two polarization states. Thus we have:

\begin{equation}
\label{weights}
w_g\ =\ 16,\hspace{1cm}w_q\ =\ 6.
\end{equation}

\npar
If we neglect any dynamical effects we can put 
\hbox{$b_g\,\approx\,b_q\,=\,b\,\approx\,const$}.
The parameter $b$ is unknown and we at present
have no indication on its exact value. But we can expect that
it is order of unit.
This parameter has to be adjusted either from experiment or
from a theoretical model. E.g., in the bag model we are able, 
in principle,  to calculate the necessary probabilities and 
distributions but it is completely
unclear how will these values be transformed under the Lorentz boost
to the infinite momentum system.
Therefore the most direct way is a comparison to experimental data.

\npar
We don't distingush the flavours of the \qq pairs, i.e., we consider the
$SU(3)$--symmetric sea.
This suggestion does not play any r\^ole in our case.
So we can write the expression for the probability to create
a state $F^{(n)}$ with one \cc pair, three quarks $uud$ 
(two quarks for a meson),
$n_g$ gluons and $n_q$ quark--antiquark pairs:

\begin{equation}
\label{nprob}
dW^{(n)}_p(x_1,\ldots,x_{n+5})\ \sim\ \sum_{n_g+2n_q=n}\frac{a_g^{n_g}}{n_g!}
\,\frac{a_q^{2n_q}}{(2n_q)!}
\,\frac{x_1^2\,x_2^2}
{(x_1+x_2)^2}\,
\prod_{i=1}^{n+5}\,dx_i\,\delta(1-\sum_{j=0}^{n+5}\,x_j).
\end{equation}

\npar
Here $n\ =\ n_g+2\,n_q$, indecies "1" and "2" denote the $c$ and $\ac$
quarks, respectively. 
The factors $1/n_g!$ and $1/(2n_q)!$ 
take into account indistingushibility of the gluons and the quarks.

\npar
To obtain the total probability for all the Fock states 
containing two charmed quarks
one needs to perform summation over $n$,
$0\,<\,n\,<\,\infty$, and the integration over all $x_i$:

\begin{equation}
\label{totprob}
1\ =\ Z^{-1}_p(a_g,a_q)\,\sum_{n=0}^{\infty}\,
\int\,dW^{(n)}_p(x_1,\ldots ,x_{n+5}),
\end{equation}

\noindent where, $Z^{-1}_p(a_g,a_q)$ is the normalization factor
(we put the overall normalization equal to unit).
If we perform the integration over all $x_i$ in Eq.(\ref{nprob})
we obtain:

\begin{equation}
\label{ntot}
W^{(n)}_p\ =\ \int\,dW^{(n)}_p(x_1,\ldots ,x_{n+5})
\ =\ \frac{1}{5}\,\frac{1}{(n+6)!}
\,\sum_{n_g+2n_q=n}\frac{a_g^{n_g}}{n_g!}\,\frac{a_q^{2n_q}}{(2n_q)!}
\end{equation}

\npar
For the normalization factor we have:

\begin{equation}
Z_p(a_g,a_q)\ =\ \sum_{n=0}^{\infty}\
\,W^{(n)}_p\ =\ \frac{1}{5}\,\sum_{n=0}^{\infty}\,\frac{1}{(n+6)!}
\,\sum_{n_g+2n_q=n}\frac{a_g^{n_g}}{n_g!}
\,\frac{a_q^{2n_q}}{(2n_q)!}
\end{equation}

\npar
The summation over $n\,=\,n_g+2n_q$
can be carried out using properties of the binomial sums. 
As result we obtain for the proton:

\begin{equation}
\label{znorm}
Z_p(a_g,a_q)\ =\ \frac{1}{10}
\,\sum_{n=0}^{\infty}\,\frac{a_+^n\,+\,a_-^n}{n!(n+6)!}
\ \equiv\ \frac{1}{5}\,S_6(a_+,a_-);
\ \ \ \ a_{\pm}\ =\ a_g\,\pm\,a_q.
\end{equation}

\npar
Similarly for meson we have

\begin{equation}
%\label{znorm}
Z_M(a_g,a_q)\ =\ \frac{1}{10}
\,\sum_{n=0}^{\infty}\,\frac{a_+^n\,+\,a_-^n}{n!(n+5)!}
\ \equiv\ \frac{1}{5}\,S_5(a_+,a_-),
\end{equation}

\noindent where we denoted

\begin{eqnarray*}
S_m(a_g,a_q) & = & \frac{1}{2}
\,\sum_{n=0}^{\infty}\,\frac{a_+^n\,+\,a_-^n}{n!(n+m)!}\\
\\
             &  = &  \frac{1}{2}
\,\left \{ a_+^{-m/2}\,I_m(2\sqrt{a_+})+
\,a_-^{-m/2}\,I_m(2\sqrt{a_-}) \right \}.\\
\end{eqnarray*}

\npar
Here, $I_m(z)$ is the modified Bessel function of the m--th order.

\npar
In the considered model in difference to usual parton picture
the distributions of the additional light partons does not look 
like bremsstrahlung spectrum and its value is finite
at $x\,\rightarrow\,0$. 
It follows from Eq.(\ref{znorm}) that 
the coefficients in the sum have a maximum at some number 
$n\,=\,n_{eff}$. Thus we can speak
about effective or average number of light partons in the hadron.
Fig.\ref{avers} presents the average number and average momentum 
fraction of light partons versus the $P_{max}$ in the proton
at the above choice of the relative weights $w_g,\,w_q$.

\begin{figure}[htb] %                            Fig.1. Average x & N
\centerline{\hbox{
\psfig{figure=avers.ps,bbllx=0.5cm,bblly=5.0cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=6.0cm}
}}
\caption{\label{avers}
The average momentum fraction (a) and
the average number (b) of light partons
versus maximal momentum $P_{max}$ for the proton.
}
\end{figure}

\npar
For light parton we obtain the following 1--particle distribution
(for the proton case):

\begin{equation}
\label{qrk1}
q_1(x;a)\ =\ \frac{1}{5}\,Z^{-1}_p(a_g,a_q)\,(1-x)^5
\,S_5\left [a_+(1-x),a_-(1-x) \right ].
\end{equation}

\npar
And for charmed quarks we have the  2--particle distribution:

\begin{equation}
\label{charm2}
c_2(x_c,x_{\ac};a_g,a_q)\ =\ Z^{-1}_p(a_g,a_q)
\,\frac{x_c^2x_{\ac}^2}{(x_c+x_{\ac})^2}
\,(1-x_c-x_{\ac})^2
\,S_2\left [a_+(1-x_c-x_{\ac}),a_-(1-x_c-x_{\ac}) \right ],
\end{equation}

One--particle distribution for charmed quark looks as

\begin{equation}
\label{charm1}
c_1(x_c;a_g,a_q)\ =\ Z^{-1}(a_g,a_q)
\,x_c^2(1-x_c)^5
\,\int_0^1\,\frac{dy\,y^2\,(1-y)^2}{[x_c+(1-x_c)y]^2}
\,S_2\left [a_+(1-x_c)(1-y),a_-(1-x_c)(1-y) \right ].
\end{equation}

\begin{figure}[htb]                %        Fig.
\par
\centerline{\hbox{%
\psfig{figure=quarks.ps,bbllx=1.5cm,bblly=5.5cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=8.0cm}%
}}
\par
\caption{\label{cadist}
Distributions for light (a) and charmed (b)
quarks over $x$ in the proton from Eqs.(11), (13) at 
$P_{max}\ =\ 10,\,and\,100\ GeV$.
The solid line is the distributions predicted by the IC model
without additional sea partons.
}
\par
\end{figure}

\npar
Additional light partons carry zero quantum numbers 
and its number $n$ is not invariant, i.e., depends on the reference frame.
From the definition (\ref{avalue}) and Fig.\ref{avers}
we see that the average number of the light partons 
depends on the reaction energy.
The increase in the energy of the reaction
leads to the increase in the effective number of sea partons
and to softer longitudinal distribution.

\npar
We have to note that the growth of the $<n>$ is faster
than usual logarithmic law due to exponential growth of the number
of cells in the momentum space. 
We assumed the constant distribution over the momentum for
the unconstrained probabilities to produce a light parton.
In general, it is not so and the parameter "b" can
depend on the parton momentum.


%--------------------------------------------------------

\subsection{Self--consistency of the model}

\npar
Now we have to make a rough check of the self--consistency of the
Eq.(\ref{postulate}).
Let us consider one Fock state with fixed number $n$ of additional
light partons.
Perfoming integration in  Eq.(\ref{postulate}) over necessary parton momenta
it is easy to obtain the 2--particle distribution for $c\ac$ pair:

\begin{equation}
\label{twocdist}
c_2^{(n)}(x_c,x_{\ac})\ \sim\ \frac{x_c^2\,x_{\ac}^2}{(x_c\,
+\,x_{\ac})^2}\,(1-x_c-x_{\ac})^{n+2},
\end{equation}

\noindent and respectively $(n+3)$ particle distribution for light partons:

\begin{equation}
\label{nqdist}
q_{n+3}^{(n)}(x_1,x_2,\ldots,x_{n+3})\ \sim
\ \left (1-\sum_{i=1}^{n+3}\,x_i\right )^3.
\end{equation}

\npar
These two expressions lead to 1--particle distributions:

\begin{equation}
\label{onecdist}
c^{(n)}_1(x_c)\ \sim\ x_c^2\,(1-x_c)^{n+5}\,\int_0^1\,dy\,
\frac{y^2(1-y)^{n+2}}{\left [x_c+(1-x_c)\,y\right ]^2}
\end{equation}

\noindent for charmed quarks and

\begin{equation}
\label{qdist}
q_1^{(n)}(x)\ \sim\ \left ( 1-x \right )^{n+5}
\end{equation}

\noindent for light quarks.

\npar
Average fractions of the proton momentum carried by a quark is:

\begin{equation}
\label{avermom}
\begin{array}{lll}
<x_c> & = & \frac{2}{n+7}, \ \ \ \ \ \ {\mbox{for c quark}}\\
\\
<x_q> & = & \frac{1}{n+7}, \ \ \ \ \ \ {\mbox{for light parton}}.\\
\end{array}
\end{equation}

\npar
All the above formulae have been derived in suggestion
that effective transverse masses of heavy quarks are much larger
than for light ones. It is really so but for validity of the
approach we need to satisfy more restrictive condition
for each Fock state $F^{(n)}$:

$$
\frac{2\,<\trm{c}^2>}{<x_c>}\ \gg\ \left | M^2\,-\,
\sum_{i=1}^{n+3}\, \frac{<\trm{i}^2>}{<x_i>} \right |
$$

\npar
We take:
$$
<\trm{c}^2>\,\approx\,2m_c^2\,\approx\,4.5\ GeV^2;
\ \ \ \ <x_c>\,\approx\,2\,<x_q>
$$

\npar
For $<\trm{q}^2>\,\approx\,<\trp{q}^2>\,\approx\,0.1\ GeV^2$ and
for $<x_q>\,\approx\,1/(n+7)$ one can obtain that for
$n\,<\,12$ the above approximation is valid within 30\%
accuracy. We hope that general formulae from previous subsection 
presenting the sum over all states work due to double factorial 
in the denominators of the expansion coefficients, which
suppresses the contributions of the higher states. 
Of course, at small $x_q$ where the distribution 
$q_1(x)\,\gg\,c_1(x)$ the validity of the approximation 
will be definetely brocken but this model considers large values of x.

\npar
So we conclude that our expressions are self--consistent
for sufficiently large $P_{max}\ \leq\ 20$ (see Fig.\ref{avers}(b)).

%------------------------------------------

\section{Application to $\JP$ Production}

\subsection{The recombination model}

\npar
To estimate the longitudinal distributions of the $\JP$ particles we use
the recombination model \cite{Hwa77}.
In this model the differential cross section of final particle 
can be written as:


\begin{equation}
\label{sigrec}
\frac{d\sigma}{dx_F}\ \sim\ \int_0^x\,dx_1\,dx_2\,F(x_1,x_2)\,R(x_1,x_2;x_F),
\end{equation}

\npar
where, \(F(x_1,x_2)\) is the two--particle distribution
for quarks "1" and "2" with momentum fractions $x_1$ and $x_2$,
respectively; $R(x_1,x_2;x_F)$ is the recombination function
describing the probability for two quarks to coalesce in the final
meson with momentum fraction $x_F$. In the simplest case:

$$
R(x_1,x_2;x_F)\ =\ \delta(x_F\,-\,x_1\,-\,x_2),
$$

\noindent ensuring the longitudinal momentum conservation.
In principle for charmed particles the primordial transverse momenta 
of the initial $c$ quarks can reach large values ($\geq \, 1\ GeV$)
and have to be taken into account by obvious way (see, e.g., \cite{Golrec}).
But for the aim of present paper it is not important.
As a result from Eqs.(\ref{sigrec}) and (\ref{charm2})
one obtains for the $x_F$ distribution of $\JP$ particles
the following expressions:

\begin{equation}
\label{psipp}
\begin{array}{lll}
\frac{d\sigma(\JP)(pp)}{dx_F}\ & =\  &\frac{1}{30} Z^{-1}_p(a_g,a_q)
\,x_F^3\,(1-x_F)^2\,S_2\left [a_+(1-x_F),a_-(1-x_F) \right ],\\
\\
\\
\frac{d\sigma(\JP)(\pi p)}{dx_F}\ & =\  &\frac{1}{30} {Z^{-1}_M(a_g,a_q)}
\,x_F^3\,(1-x_F)\,S_2\left [a_+(1-x_F),a_-(1-x_F) \right ],\\
\\
\end{array}
\end{equation}


\npar
for $pp$ and $\pi p$ collisions, respectively.
For hadronic interactions we have in the center--of--mass system
\hbox{$P_{max}\ =\ \sqrt{s}/2$}.
If we normalize our cross
section on specific value we have the only free parameter ---
the constant $b$ from Eq.(\ref{avalue})
At the moment we have no other choice as
to put it equal to unit, $b\ =\ 1$. Note here that at larger values
of $b$ we have softer distributions for $c$ quarks and for $\JP$ particle.

\npar
In paper \cite{Kowit94} the $\JP$ production in the proton--nucleus
interactions has been measured at $\sqrt{s}\ =\ 38.7\ GeV$
and $0.30\,<\,x_F\,<\,0.95$. It is a region where 
we expect the contribution from the intrinsic charm.
Based on estimations from \cite{Vogt91} and private communication
to R.Vogt the authors of the paper \cite{Kowit94} made a conclusion 
that the IC model conradicts to their experimental data.
They used a value for total cross section for IC $\JP$ production 
$\sigma_{tot}(\JP)\ \approx\ 1.8\ nb$.
The present paper is not right place to discuss the absolute value
of the $\JP$ total cross section. It is very complicated subject
and needs special consideration. So we use the value $0.9\ nb$
for $\sigma(\JP)$ in the forward hemisphere at FNAL energy. 
For lower energy data \cite{Badier83} we normilized
our expressions for measured total cross section in the forward direction
for the so called "diffractive" component (see \cite{Badier83}) 
of the $\JP$ yield.

\npar
Results of this comparison are shown in Figs.\ref{ppcomp}
and \ref{pipcomp} for $b\,=\,1$.
In these figures we also plotted the predictions of the standard
IC model based on the same recombination model and with the same normalization.
As one observes the predictions of the usual IC model can really be rejected 
for above value of the total cross section at FNAL energy. 
On another hand predictions followed from our calculations 
don't contradict to the experimental data.
At low energies both models agrees with the experimental data.

\npar
This fact allows us to suggest that the contribution from intrinsic
charm can be observed at sufficiently low initial energies and
is practically negligible at high enrgy even at large $x_F$.
This comparison also confirms that parameter $b\ \sim\ 1$.

\bigskip
\vbox{
\centerline{\bf{Acknowledgement}}

\npar
I want to thank Dr.~G.Wolf very much for useful discussions.
I would like to express my gratitude to DESY for partial support 
during this work.

\npar
I am also thankful to Prof.~P.F.Ermolov for his stimulating interest
to this subject.
}
% ----------------------------------------------------------

%\newpage
%  ------------- References -----------------------

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\end{thebibliography}

\vfill\eject

% ============================================================

% ---------------------------------------------
\begin{figure}[htb] %                 Fig.Kowit& Badier, pp, 200 & 800
\centerline{\hbox{
\psfig{figure=psi_pp200.ps,bbllx=0.5cm,bblly=5.0cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=6.0cm}
\psfig{figure=psi_cu800.ps,bbllx=0.5cm,bblly=5.0cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=6.0cm}
}}
\caption{\label{ppcomp}
$J/\psi$ distributions over $x_F$ in (a) $pp$ at $P_0\,=\,200\ GeV/c$
and (b) in $pCu$ collisions at $P_0\,=\,800\ GeV/c$.}
\end{figure}

% ---------------------------------------------
\begin{figure}[htb] %                  Fig.Badier pi-p, 150 - 280
\centerline{\hbox{
\psfig{figure=psi_pi-p150.ps,bbllx=0.5cm,bblly=5.0cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=6.0cm}
\psfig{figure=psi_pi-p200.ps,bbllx=0.5cm,bblly=5.0cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=6.0cm}
}}
\centerline{\hbox{
\psfig{figure=psi_pi-p280.ps,bbllx=0.5cm,bblly=5.0cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=6.0cm}
\psfig{figure=psi_pi+p200.ps,bbllx=0.5cm,bblly=5.0cm,%
bburx=19.0cm,bbury=23.0cm,clip=t,height=6.0cm}
}}
\caption{\label{pipcomp}
$J/\psi$ distributions over $x_F$ in $\pi^-p$ collisions
at (a) $P_0\,=\,150\ GeV/c$; (b) $P_0\,=\,200\ GeV/c$;
(c) $P_0\,=\,280\ GeV/c$ and (d) in $\pi^+p$ collisions
at $P_0\,=\,200\ GeV/c$.}
\end{figure}

\end{document}


