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\begin{center}
\begin{Large}
\begin{bf}
 CHARMED QUARK AND $J/\Psi$ PHOTOPRODUCTION
 IN THE SEMIHARD APPROACH OF QCD
  AT HERA ENERGIES
\end{bf}
\end{Large}
\vspace*{1cm}

    V.A.Saleev\\    
{\it Samara State University, Samara 443011, Russia\\}

\vspace*{0.5cm}
    N.P.Zotov\\ 
{\it D.V. Skobeltsyn Institute of Nuclear Physics,Moscow State University,\\ 
      Moscow 119899, Russia\\}
\end{center}      
\vspace*{1.5cm}
\begin{abstracts}
{\small We compare our theoretical results for $c\bar c-$quark and
$J/\Psi-$photoproduction cross sections obtained in the semihard approach
 of QCD~\cite{r1,r2} with H1 and ZEUS experimental data~\cite{r3,r4,r5}} 
\end{abstracts}

\vspace*{1.5cm} 
\section{Introduction}
The observation of heavy quark and quakonium photoproduction offers a
unique opportunity to probe the gluon distribution in proton at small $x$ 
by measurements of the total cross sections of these processes or of their
differential distributions.
 
 It is known that at HERA energies and beyond heavy quark and
quarkonium photoproduction processes
 are of the so-called semihard type~\cite{r6}.
In such 
processes by definition a hard scattering scale $Q$ (or heavy quark
mass $M$) is large as compared to the $\Lambda_{QCD}$ parameter,
but $Q$ is much smaller than the total center of mass energies:
$\Lambda_{QCD} \ll Q \ll \sqrt s$. The last condition implies that 
the processes occur in small $x$
region: $x\simeq M^2/s\ll 1$. In such a case the perturbative
QCD expansion has large coefficients $O(\ln^{n}(s/M^2))
\sim O(\ln
^{n}(1/x))$ besides the usual renormalization group ones
which are $O(\ln^{n}(Q^2/\Lambda^2))$.
It means that in the small $x$ region $(x\simeq M^2/s \ll 1)$ we should 
first of all sum up all the terms of the type $(\alpha_s\ln s/M^2)^n$ and 
also the terms $(\alpha_s\ln Q^2/Q^2_0\ln s/M^2)^n$ (moreover $\alpha_s\ln Q^2/Q^2_0$
in DIS). The second problem that appears at small $x$ is that of the violation of
unitarity due to the increase of the gluon density at $x\to 0$. The latter one
is reduced to taking into account the absorption (screening) corrections which should 
stop
the growth of the cross section at $x\to 0$ (in accordance with the unitarity
condition).
Both these problems were resolved in the so called semihard approach (SHA) by 
L. Gribov, E. Levin, M. Ryskin~\cite{r6}.
Therefore we used this SHA for the calculations
of the cross sections of $J/\Psi -$ and $c\bar c -$photoproduction~\cite{r1,r2}.
In present paper the result of these calculations will be compared with new H1
and ZEUS experimental data at HERA enargies~\cite{r3,r4,r5}.

\newpage
\section{Heavy quark photoproduction cross section in the semihard
approach}
In the SHA heavy quark photoproduction is determined by the contribution
of two photon-gluon fusion diagrams
% the sum of the ladder diagrams in double logarithm approximation
%$(\sim (\alpha_s\ln Q^2/Q^2_0\ln 1/x)^n)$ is reduced to the two diagrams (Fig. 1)
with "unusual" properties of the gluons in proton. These gluons are the off mass shell
ones, their virtuality $q^2 = -\vec{q}_T ^2$, and they possess the propeperty of
 the alignment of their polarization vector along their transverse momentum
such as $\epsilon_{\mu} = q_{T\mu}/|\vec{q}_{T}|$~\cite{r6,r7}.
Their distributions in $x$ and $q_T$ in proton is given by
the unintegrated gluon structure function $\varphi (x,q^2_T)$,
 wich is connected with the usual gluon 
distribution function $G(x,Q^2)$ by the following relation
\begin {eqnarray}
\int\limits_0^{Q^2}\varphi
(x,q_{T}^2)dq_{T}^2=xG(x,Q^2).
\end{eqnarray}
The exact expression for the function $\varphi (x,q^2)$ can be obtained as a 
solution of the evolution equation, which, contrary to the parton model case,
is nonlinear due to interactions between the gluons in small $x$ region.

In our calculations we used the following phenomenological parametrization~\cite{r7}:
\begin{eqnarray}
\varphi(x,q_{T}^2)=C\frac{0.05}{x+0.05}(1-x)^3f(x,q_{T}^2),
\end{eqnarray}
where
%The function $f(x,q^2_T)$ and another SHA parameters may be found in ~\cite{r1,r2}
\begin{eqnarray}
f(x,q^2_T)& =&1 ,\qquad q_{T}^2\le q_0^2(x)\nonumber\\
f(x,q^2_T)& =&(\frac{q_0^2(x)}{q_{T}^2})^2, \qquad q_{T}^2>q_0^2(x),
\end{eqnarray}
and $q^2_{0}(x) = Q^2_{0} + \Lambda^2\exp( 3.56 \sqrt{ \ln(x_0/x)})$,
 $Q_{0}^2 = 2 GeV^2$, $\Lambda = 56$ MeV, $x_{0}$ = 1/3.
The parameter $q_0^2(x)$ can be treated as a new infrared-cutoff, which plays
the role of a typical transverse momentum of partons in the parton cascade
of the proton in semihard processes. The behaviour of $q_0(x)$ was 
discussed in~\cite{r6}. It increases with $\ln (1/x)$ and at $x =0.01 - 0.001$
the values of $q_0(x)$ are about $2 - 4 GeV$.

For relatively small virtuality $Q^2 \le q^2_0(x)$ the gluon function
behaves as $xG(x,Q^2) \sim CQ^2$. So we have the saturation of the gluon density
 at
these values $Q^2$.


 The normalization factor $C\simeq 0.97$ mb
 in (2) was obtained in~\cite{r7} where
$b\bar b$-pair
production at Tevatron energy was described.

Thus, in the case of real transverse polarized photon the heavy quark photoproduction
cross section at $x\ll 1$ can be expressed as ~\cite{r2}
\begin{eqnarray}
\frac{d\sigma}{d^2p_{1T}}(\gamma p\to Q\bar Q X)=
\int dy_1^{\ast}\frac{d^2q_{T}}{\pi}
\frac{\varphi (x,q_{T}^2)|\bar M|^2}{16\pi^2(sx)^2\alpha},
\end{eqnarray}
where $p_T, y^{\ast}_1$ are transverse momentum and rapidity (in the center of mass
frame of colliding particles) of heavy quark and $\alpha = 1 - \alpha _1$ with
$\alpha _1 = M_1\exp (y^{\ast}_1)/\sqrt s$.

The matrix element $\bar M$ for a subprocess $\gamma g^{\ast} \to
c\bar c$ depends on the virtuality of the gluon and differs from the one
of the usual parton model. For the square of this matrix element we used the
following form:
%\begin{eqnarray}
%|\bar M|^2=16\pi^2e_Q^2\alpha_s\alpha(x_1x_2s)^2[
% \frac{1}{(\hat u-M^2)(\hat t-M^2)}-
%\frac{1}{q_{1\bot}^2q_{2\bot}^2}
%(1+\frac{\alpha_2\beta_1s}{\hat t-M^2}+
% \frac{\alpha_1\beta_2s}{\hat u-M^2})^2]
%\end{eqnarray}
%For real photon and off-shell gluon it reads:
\begin{eqnarray}
|\bar M|^2=16\pi^2e_Q^2\alpha_s\alpha_{em}(xs)^2[\frac{\alpha_1^2+\alpha^2}
 {(\hat t-M^2)(\hat u-M^2)}+\frac{2M^2}{q_{T}^2}
(\frac{\alpha_1}{\hat u-M^2}-
 \frac{\alpha}{\hat t-M^2})^2],
\end{eqnarray}
where $\hat s,~\hat t,~\hat u$ are usual
Mandelstam variables of partonic subprocess $\gamma g^{\ast} \to c\bar c$.

\section{Quarkonium photoproduction cross section in SHA of QCD and in
colour singlet model}
 We also used similar formulas for heavy quarkonium photoproduction~\cite{r1}.
In the case of real transverse polarized photon, the heavy quarkonium
photoproduction cross section at $x\ll 1$ in SHA is expressed in the form
(we consider here $J/\Psi-$ photoproduction):
\begin {eqnarray}
\sigma (\gamma p\to J/\Psi X)=\int \frac{dz}{z(1-z)}\int dp_T^2\int
\frac{d\phi}
{2\pi}\int d\vec q_T\,^2 \frac{\varphi (x,q_T^2)}
{16\pi (s+q_T^2)^2}
 \sum|\bar M(\gamma g^*\to J/\Psi g)|^2,
\end{eqnarray}
 where (in the lab. frame) $z=E/E_{\gamma}$, $s=2m_pE_{\gamma}$ and
 $p=(E,\vec p_T,p_z)$ is 4--momentum of the quarkonium, $\phi$ is
  the angle between initial gluon and quarkonium transverse
  momenta $\vec q_T$ and $\vec p_T$.
  In (6) $\sum$ indicates an average over two photon polarizations
  and colours of initial gluon
  as well as a sum over polarizations of final particles.

If we take the limit of zero $\vec q_T$, and if we average over the
transverse directions of $\vec q_T$, we obtain the formula of standard
parton model (SPM):
\begin {eqnarray}
\sigma (\gamma p\to J/\Psi X)=\int \frac{dz}{z(1-z)}\int dp_T^2
 \frac{xG(x,Q^2)}
{16\pi (sx)^2}
 \sum|M_{part}(\gamma g\to J/\Psi g)|^2,
\end{eqnarray}
where $\sum$ now indicates an average over colours and
transverse polarizations of real initial gluon and photon
  as well as a sum over polarizations of final particles. We averaged
  over the transverse directions of $\vec q_T$ using expression:
  \begin{equation}
 \int d\vec q_T\,^2\int \frac{d\phi}{2\pi}\varphi(x,\vec q_T)
  \sum |\bar M|^2=xG(x,Q^2)\sum|M_{part}|^2,
   \end{equation}
 where
 \begin{equation}
  \int_{0}^{2\pi}\frac{d\phi}{2\pi}\frac{q_{T\mu}q_{T\nu}}{\vec
q_T\,^2}
   =\frac{1}{2}g_{\mu\nu}.
 \end{equation}

Within the framework of perturbative QCD the photoproduction of $J/\Psi$
 particles is described by subprocess $\gamma g\to
  J/\Psi g$~\cite{r8}. In this approach, so called
  "colour singlet model" (CSM), the quarkonium is represented by a define
 wave function so that the final $c\bar c$ system are colour
 singlet, $J^p=1^-$ state of specified mass. At not very large $p_T$ the
 contribution of these subprocesses are significantly greater than others,
 such as: $\gamma g\to b\bar b$ with $b\to J/\Psi X$.
 That is why we took into account only this.

The matrix element $\bar M$ of the process $\gamma g\ast \to J/\Psi g$
 was calculated using the sum of six diagrams according to the
 CSM. We makes summation over spins and colours of final gluon, $J/\Psi$
 and photon as in SPM. In the case of initial off shell gluon with
 transverse momentum $\vec q_T$ we takes polarization tenzor in 
the form~\cite{r6,r7}:
 \begin{equation}
  d_{\mu\nu}(q)=\varepsilon_{\mu}(q)\varepsilon_{\nu}(q)=
  \frac{q_{T\mu}q_{T\nu}}{\vec q_T\,^2}.
   \end{equation}
\newpage
 The calculation of $\sum|\bar M|^2$ was made analytically by "REDUCE"
 system and result can be expressed in the following form:
 \begin{equation}
 \sum|\bar M|^2=\frac{Bx^2}{\vec q_T\,^2}\sum_{i=1}^{6}F_i(z,\vec q_T\,^2,
  \hat t,\hat u),
\end{equation}
 where
 \begin{equation}
  B=\frac{32\pi^2}{3\alpha}\alpha_s(Q^2)\alpha_s(q^2)\Gamma_{ee}M,
 \end{equation}
  $Q^2=M^2+\vec p_T\,^2$, M is the mass of quarkonium, $\Gamma_{ee}$ is
   the leptonic quarkonium width, $\hat t$ and $\hat u$ are ordinary
   Mandelstam variables, $\hat t=M^2-(M^2+\vec p_T\,^2)/z$,
   $ \hat u=M^2-\vec q_T\,^2-zxs+2p_Tq_T\cos \phi$ and $x=(M^2-\hat u
      -\hat t)/s$. The explicit form of functions $F_i$ are given in~\cite{r9}.
  

 We would like to note the limits of applicability of CSM:
$z \leq 0.8$ and $p^2_T \geq 0.1M^2$~\cite{r10}, where $z =
E_{J/\Psi}/E_{\gamma}$ (in lab. frame) and $p_T$ is the $J/\Psi-$ transverse
momentum. These limits correspond to the region of H1 experimental data~\cite{r4}.


\section{Results}
The results of our calculations for cross sections of
$\gamma p \to c\bar c X$ and $ \gamma p \rightarrow J/\Psi X$  processes~\cite{r1,r2}
 are compared (in Figs. 1, 2) with H1 and ZEUS experimental data~\cite{r3,r4,r5}.
(The curves in Fig.1 correspond to different values of $c-$quark mass:
solid - $m_c$ = 1.3 $GeV$, dashed - 1.4 $GeV$ and dash-dotted - 1.5 $GeV$.
)

\unitlength=1cm

\begin{figure}[ht]
%\begin{minipage}[b]{16cm}
\begin{picture}(16,10)
\epsfysize =12cm
\epsfxsize =14cm
\put(+1,-2){\epsfbox{pic.eps}}
\end{picture}
%\caption{figure}
%\end{minipage}
\end{figure}

 In Fig. 3 the z-distribution of $J/\Psi-$mesons~\cite{r1}
is shown (solid curve) in comparison  with H1 experimental data \cite{r4}
and colour octet
model (COM) result (dash curve)~\cite{r11}. 

\newpage
\section{Conclusions}
We see that our theoretical curves obtained in~\cite{r1,r2} describe very
well the new H1 and ZEUS  data, and the strong increase of z-distribution
obtained in  COM is not confirmed by the experimental data~\cite{r4} and our
CSM results~\cite{r1}.

Thus it means that the results obtained in the semihard approach of QCD
can be used for  extraction of the effective gluon distribution in proton from
the H1 and ZEUS experimental data for heavy quark and quarkonium photoproduction
at HERA energies.

\section{References}

\begin{thebibliography}{99}
\bibitem{r1}
V.A. Saleev, N.P. Zotov, Mod. Phys. Lett. A9 (1994) 151, 1517(E)
\bibitem{r2}
V.A. Saleev, N.P. Zotov, Mod. Phys. Lett. A11 (1996) 25; 
% 
\bibitem{r3}
ZEUS Collab., M. Derrick et al., Phys. Lett. B349 (1995) 225
\bibitem{r4}
H1 Collab., S. Aid et al., DESY 96-037, 1996
\bibitem{r5}
H1 Collab., S. Aid et al., DESY 96-055, 1996
\bibitem{r6}
L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rep. C100 (1983) 1
\bibitem{r7}
E.M. Levin et al. Yad. Fiz. 53 (1991) 1059
\bibitem{r8}
E.L. Berger, D. Jones, Phys. Rev. D23 (1981) 1521
\bibitem{r9}
V.A. Saleev, N.P. Zotov, Mod. Phys. Lett. A9 (1994) 1517
\bibitem{r10}
R. Baier, R. Ruckl, Nucl. Phys. B218 (1983) 289
\bibitem{r11}
M. Cacciari, M. Kramer, DESY 96-005, 1996
\end{thebibliography}

\end{document}

{\bf Figure captions}
\begin{enumerate}
\item
The total charm photoproduction cross section
\item
The $J/\Psi$ photoproduction cross section
\item
The $z-$distribution for $J/\Psi$ photoproduction
\end{enumerate}




 which takes into account
$\alpha_s$ corrections due to soft gluon radiation. The typical value
  of $K$--factor in hard hadron-hadron interactions is $K=2-2.5$. As it
  was noted in Ref. \cite{r11} the choice of $C=0.97~$mb corresponds to
  the upper limit of heavy quark hadroproduction cross sections. For the 
normalitation factor $C_1$ in (8)
 we shall use value of $C=0.65$ mb, which gives better description
  of existing data on charm quark photoproduction at fixed target
  energies $\sqrt s_{\gamma p}=10-30$ GeV.

 In (4) $\Phi_e(x_1,q_{1\bot}^2)$ is the well known virtual photon spectrum
in Weizsacker-Williams approximation ~\cite{r26} before the integration
over  $ q_{1\bot}^2$:
\begin{equation}
 \Phi(x_1, q_{1\bot}^2)=\frac{\alpha}{2\pi}[\frac{1+(1-x_1)^2}
  {x_1\vec q_{1\bot}^2}-\frac{2m_e^2x_1}{\vec q_{1\bot}^4}].
\end{equation}
In this paper we cosider only heavy quark photoproduction processes.

 The effective gluon distributions $xG(x,Q^2)$ obtained from (7), (8)
increase as $x^{-\omega_0}$  at not very small $x$ $(0.01 < x < 
0.15)$,
where $\omega_0 = 0.5$ corresponds to the BFKL Pomeron singularity
~\cite{r10}. This rise goes continuously up to $x = x_0$, where
$x_0$ being a solution of the equation $q^2_0(x_0) = Q^2$. In the region
$x < x_0$ there is the saturation of the gluon distribution:
$xG(x,Q^2)\simeq CQ^2$ or $C_1Q^2$.

The square of matrix element of partonic subprocess $\gamma^{\ast} g^{\ast}
\to Q\bar Q$
can be written as follows:
\begin{equation}
|M|^2=16\pi^2e_Q^2\alpha_s\alpha(x_1x_2s)^2[
 \frac{1}{(\hat u-M^2)(\hat t-M^2)}-
\frac{1}{q_{1\bot}^2q_{2\bot}^2}
(1+\frac{\alpha_2\beta_1s}{\hat t-M^2}+
 \frac{\alpha_1\beta_2s}{\hat u-M^2})^2]
\end{equation}
For real photon and off-shell gluon it reads:
\begin{equation}
|M|^2=16\pi^2e_Q^2\alpha_s\alpha(x_2s)^2[\frac{\alpha_1^2+\alpha_2^2}
 {(\hat t-M^2)(\hat u-M^2)}+\frac{2M^2}{q_{2\bot}^2}
(\frac{\alpha_1}{\hat u-M^2}-
 \frac{\alpha_2}{\hat t-M^2})^2],
\end{equation}
where $\alpha_2=1-\alpha_1$ and $\hat s,~\hat t,~~\hat u$ are usual
Mandelstam variables of partonic subprocess
\begin{eqnarray}
 \hat s&=&(p_1+p_2)^2=(q_1+q_2)^2,\qquad
 \hat t=(p_1-q_1)^2=(p_2-q_2)^2,\nonumber\\
 \hat u&=&(p_1-q_2)^2=(p_2-q_1)^2,\qquad
 \hat s+\hat t+\hat u=2M^2+q_{1\bot}^2+q_{2\bot}^2.
\end{eqnarray}

\section{Discussion of the Results}

\par
The results of our calculations for the total cross sections of
$c$- and $b$-quark photoproduction are shown in Figs.~3 and 4.
Solid curves correspond to the semihard approach predictions and
dashed curves correspond to the SPM results with
the GRV LO parametrization of the gluon distribution~\cite{r27}.
We used in our calculations $m_c=1.5$ GeV and $m_b=4.75$ GeV.
Fig.~3a shows us that the solid curve for 
charm quark cross section obtained in semihard approach with
 parametrization 
(7) describes new ZEUS data ~\cite{r5}
 as well as data from earlier fixed
  target experiments in the range $15 < \sqrt s < 30$ GeV.
The curve of
 the parametrization (8) (Fig.~3b) describes better the 
experimental data at low energies ($6 < \sqrt s < 30$ GeV ) but 
gives worse description ZEUS data ~\cite{r5}. There are some
 differences between
the theoretical curves and ZEUS experimental data (Figs.~3). If these
ones will
remain valid for new data, then its will show to essential resolved photon
contribution
~\cite{r5,r28} and a sensitivity of charm photoproduction cross section
to the photon structure function ~\cite{r29}.

 We would like to note that our results for the parametrization (7)
coincide with the results from Ref. ~\cite{r28}
 for pointlike component of the
charm photoproduction cross section for $MRSD'_{-}$ parton density 
at $\mu = m_{c}$ and ones for parametrization (8) correspond to the 
charm photoproduction cross section in Ref. ~\cite{r28} for
$MRSA$ parton density.
  
   At the energy range $\sqrt s_{\gamma p}\ge 500$
GeV the saturation effects in the gluon distribution function
are important for charm quark photoproduction in semihard
approach (Fig.4): the c-quark cross section grows more slowly
at these energies (solid curve). The beauty quark
photoproduction cross section predicted by semihard approach
is larger than the one predicted by SPM at all energies (Fig.~4).
Our conclusion for b-quark photoproduction is the same as in
Ref.~\cite{r11}, where b-quark hadroproduction was described
using semihard approach.

 The $p_{\bot}$ distributions for $c$- and $b$-quark photoproduction
in the semihard approach (solid curves) and in the SPM
(dashed curves) at the energy $\sqrt s_{\gamma p}=200$ GeV are
 shown in Fig.~5.
 The curves are obtained in the semihard
approach for charm quark photoproduction show the saturation effects
in low $p_{\bot}$ region ($p_{\bot} < 2$ Gev/c). At high
$p_{\bot}$ region ($3 < p_{\bot} < 20$ GeV/c the
heavy quark photoproduction $p_{\bot}$ distributions obtained in
the semihard approach are higher than the ones of the SPM (with GRV LO
parametrization of gluon distribution).
This behaviour of c-quark $p_{\bot}$ distributions in the $k_{\bot}$
factorization approach results from the off mass shell
subprocess cross section as well as the saturation
effects of the gluon structure function, because of $m_c^2\approx q_0(x)^2$
at HERA energies. In the case of beauty quarks $m_b^2>>q_0(x)^2$ and
similar effects are absent.


 Fig.6 show the comparison of heavy quark rapidity
distribution (in the photon-proton center of mass frame) obtained
in the different models:
solid curve shows the $y$ distributio
semihard approach,  dashed curve shows the one in SPM. The 
effects discussed
above are sufficiently large near the kinematic boundaries,
i.e. at big value of $|y^{\ast}|$. We see that the difference between solid
and dashed curvers can't be degrade at all $y^{\ast}$ via change of
normalization of both models.

\section{Conclusions}

\par
 We showed that the semihard approach describes experimental data
for the charm photoproduction cross section at low and HERA energies,
leads to the saturation effects
for the total cross section of charm quark photoproduction
at $\sqrt s_{\gamma p}\ge 500$ GeV as well as to exceeding over the SPM prediction
for beauty quark total cross section at the energy range of
$\sqrt s_{\gamma p}=100-500$ GeV, and
predicts a marked difference for rapidity and transverse
momentum distributions of charm and beauty quark photoproduction,
which can be studied already at HERA $ep-$collider.
\vspace{5mm}

{\Large\bf Acknowledgements}

\vspace{4mm}
 dis96.tex   This  research was supported by the Russian
  Foundation of Basic Research (Grant 93-02-3545).
Authors would like to thank J.Bartels, S.Catani,
G.Ingelman, H.Jung,
J.Lim, M.G.Ryskin and A.P.Martynenko for fruitfull discussions
of the obtained results.

 One of us (N.Z.) would like to thank E.M.Levin
 for discussions of the small $x$
physics and the semihard approach in begin stage of this paper,
L.K.Gladilin and I.A.Korzhavina for discussions of ZEUS 
experimental data,
P.F.Ermolov for interest and support
 and also
  W.Buchmuller, G.Ingelman, R.Klanner,
P.Zerwas and the DESY directorate for hospitality and
support at DESY.


\begin{thebibliography}{1}
\bibitem{r1}
A.Ali et al. Proceedings of the HERA Workshop, Hamburg,
ed. R.D.Peccei, 1988, vol.1, p.395
\bibitem{r2}
A.Ali, D.Wyler. Proceedings of the Workshop"Physics at HERA",
eds W.Buchmuller and G.Ingelman, Hamburg, 1992, vol.2, p.669;
A.Ali. Preprint DESY 93 - 105 
\bibitem{r3}
M.A.G.Aivazis, J.C.Collins, F.I.Olness, W.-K.Tung.
Phys. Rev. D50 (1994) 3102 
\bibitem{r4}
M.Derrick et al. (ZEUS). Phys. Lett. B297 (1992) 404; B315 (1993) 481;
B322 (1994) 287; B332 (1994) 228.  \\
T.Ahmed et al. (H1). Phys. Lett. B297 (1992) 205; A.De Roeck.
Preprint DESY  94 - 005 
\bibitem{r5}
M.Derrick et al. (ZEUS). Preprint DESY 95 - 013 
\bibitem{r6}
L.V.Gribov, E.M.Levin, M.G.Ryskin. Rhys. Rep. C100 (1983) 1
\bibitem{r7}
G.Martinnelli. Univ. of Roma preprint N842 (1991)
\bibitem{r8}
R.K.Ellis, P.Nason. Nucl. Phys. B312 (1989) 551
\bibitem{r9}
V.N.Gribov, L.N.Lipatov. Sov. J. Nucl.Phys. 15 (1972) 438;
L.N.Lipatov. Yad. Fiz. 20 (1974);
G.Altarelli, G.Parisi. Nucl. Phys. B126 (1977) 298;
Yu.L.Dokshitzer. Sov. Phys. JETP 46 (1977) 641
\bibitem{r10}
L.N.Lipatov. Sov. J. Nucl. Phys. 23 (1976) 338;
E.A.Kuraev, L.N.Lipatov, V.S.Fadin. Sov. Phys. JETP 45 (1977) 199;
Y.Y.Balitskii, L.N.Lipatov. Sov. J. Nucl. Phys. 28 (1978) 822;
L.N.Lipatov. Sov. Phys. JETP 63 (1986) 904
\bibitem{r11}
E.M.Levin, M.G.Ryskin, Yu.M.Shabelski, A.G.Shuvaev. Yad. Fiz. 53 (1991)
1059; 54 (1991) 1420
\bibitem{r12}
S.Catani, M.Chiafaloni, F.Hautmann. Nucl. Phys. B366 (1991) 135 .
\bibitem{r13}
J.C.Collins, R.K.Ellis. Nucl.Phys. B360 (1991) 3.
\bibitem{r14}
E.M.Levin, M.G.Ryskin. Phys. Rep. 189 (1990) 276;
E.M.Levin. Review talk at Intern. Workshop on DIS and
Related Subjects (Eilat, Israel, 1994), Preprint
FERMILAB-Conf-94/068-T
\bibitem{r15}
E.M.Levin, M.G.Ryskin, Yu.M.Shabelski, A.G.Shuvaev.
Preprint DESY 91 - 110 
\bibitem{r16}
G.Marchesini, B.R.Webber. Preprint CERN TH-6495/92 (1992);
B.R.Webber. Proceedings of the Workshop"Physics at HERA",
eds W.Buchmuller and G.Ingelman, Hamburg, 1992, vol.1, p.285
\bibitem{r17}
P.Nason, S.Frixione, M.L.Mangano, G.Ridolfi. Preprint CERN-TH.7134/94
\bibitem{r18}
J.Smith, W.L. van Neerven. Nucl. Phys. B374 (1992) 36
\bibitem{r19}
E.Laenen, S.Rienersma, J.Smith, W.L. van Neerven. Nucl. Phys.
B392 (1993) 162 and 229.
\bibitem{r20}
S.Frixione, M.L.Mangano, P.Nason, G.Ridolfi. Phys. Lett. B308 (1993) 137;
Preprint 
 CERN-TH.6921/93
\bibitem{r21}
G.Ridolfi, S.Frixione, M.L.Mangano, P.Nason. Preprint CERN-TH.7377/94
\bibitem{r22}
S.Catani, M.Ciafaloni, F.Hautmann. Proceedings of the Workshop
"Physics at HERA", eds W.Buchmuller and G.Ingelman, Hamburg,
1992, vol.2, p.690; Preprint CERN-TH.6398/92
\bibitem{r23}
V.A.Saleev, N.P.Zotov. Proceedings of the Workshop "Physics at
HERA", Hamburg, 1992, vol.1, p.637.
\bibitem{r24}
V.A.Saleev, N.P.Zotov. Modern Phys. Lett. A9 (1994) 151; E A9
(1994) 1517
\bibitem{r25}
V.A.Saleev, N.P.Zotov. In "Relativistic Nuclear Physics and Quantum 
Chromodynamics", Proceedings of the XIIth Intern. Seminar on H.E.
Physics Problems, Dubna, Russia, 1994; 
\bibitem{r26}
C.F.Weizsacker. Z. Phys. 88 (1934) 612;
E.J.Williams. Phys. Rev. 45 (1934) 729
\bibitem{r27}
M.Gluck, E.Reya, A.Vogt. Z. Phys. C53 (1992) 127; Phys. Lett.
B306 (1993) 391
\bibitem{r28}
S.Frixione, M.L.Mongano, P.Nason, G.Ridolfi. Preprint CERN-TH.7527/94
\bibitem{r29}
M.Drees, F.Halzen. Phys. Rev. Lett. 61 (1988) 275; M.Drees, R.M.
Godbole. Phys. Rev. D39 (1989) 169.

\end{thebibliography}
                %=======================================================
{\bf Figure captions}
\begin{enumerate}
\item
QCD diagrams for open heavy quark photoproduction subprocesses
\item
Diagram for heavy quark elecrtoproduction
\item
a) The total cross section for open charm quark photoproduction:
solid curve - the semihard approach for the parametrization (7),
dashed curve - the SPM.
The solid dots are the ZEUS measurements \cite{r5} and the open dots
are earlier measurements from fixed target experiments. 

b) The solid curve - the semihard approach for the parametrization (8).
  The experimental data as in Fig.~3a 
\item
The total cross section for open charm and beauty quark photoproduction:
curves as in Fig.3a

\item
The $p_{\bot}$ distribution for charm and beauty quark photoproduction
at $\sqrt{s_{\gamma p}} = 200$ GeV: curves as in Fig.3a
\item
The $y^{*}$ distribution for charm and beauty quark photoproduction:
at $\sqrt{s_{\gamma p}} = 200$ GeV: curves as in Fig.3a
\end{enumerate}
\newpage

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\begin{center}
Fig.~1
\end{center}
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\begin{center}
Fig.~2
\end{center}

\end{document}




