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\begin{document}

\begin{center}
{\bf\Large On charmonia and charmed mesons photoproduction\\
 at high energy}

%
\vspace{3mm}
%
{\sl V.A.Saleev}\footnote{Email: saleev@ssu.samara.ru} \\
{Samara State University, Samara, 443011, Russia and \\
Samara Municipal Nayanova University, Samara, 443001, Russia}\\ and\\
{\sl D.V.Vasin}\\
{Samara State University, Samara, 443011, Russia}
%
%\authorrunning{V.A.Saleev and D.V.Vasin}
%


%\maketitle              % typesets the title of the contribution


\end{center}

\begin{abstract}
In this report we compare the predictions of the collinear parton
model and the $k_T$-factorization approach in $J/\Psi $ and
$D^\star$ meson photoproduction at HERA energies. It is shown that
obtained $D^\star$ meson spectra over $p_T$ and $\eta $ are very
similar in the parton model and $k_T$-factorization approach and
they underestimate the experimental data. Opposite, the
predictions of the both approaches for $p_T$- and $z$-spectra in
the $J/\Psi $ photoproduction are very different as well as the
prediction obtained for the spin parameter $\alpha(p_T)$.
\end{abstract}

\section{Hard processes in the parton model and $k_T$-factorization
approach.}

Nowadays, there are two approaches which are used in a study of
the charmonia and charmed mesons photoproduction at high energies.
In the conventional collinear parton model \cite{1} it is
suggested that hadronic cross section, for example $\sigma (\gamma
p \to c\bar cX,s)$, and the relevant partonic cross section $\hat
\sigma (\gamma g\to c\bar c,\hat s)$ are connected as follows
%
\begin{equation}
\sigma ^{PM}(\gamma p\to c\bar cX,s)=\int {dx}
G(x,\mu ^2)\hat \sigma (\gamma g\to c\bar c,\hat s)\mbox{,}
\end{equation}
%
where $\hat s=xs$, $G(x,\mu ^2)$ is the collinear gluon
distribution function in a proton, $x$ is the fraction of a proton
momentum, $\mu ^2$ is the typical scale of a hard process. The
$\mu ^2$ evolution of the gluon distribution $G(x,\mu ^2)$ is
described by DGLAP evolution equation \cite{2}. In the so-called
$k_T$-factorization approach hadronic and partonic cross sections
are related by the following condition \cite{3,4,5}:
%
\begin{equation}
\sigma ^{SHA}(\gamma p\to c X,s)=\int {\frac {dx}{x}} \int {d\vec
k_T^2}\int {\frac{d\varphi}{2\pi}}\Phi (x,\vec k_T^2,\mu ^2)\hat
\sigma (\gamma g^\star \to c\bar c,\hat s,\vec k_T^2)\mbox{,}
\end{equation}
%
where $\hat \sigma (\gamma g^\star \to c\bar c,\hat s,\vec k_T^2)$
is the $c\bar c$-pair photoproduction cross section on off
mass-shell gluon, $k^2={k_T}^2=-\vec k_T^2$ is the gluon
virtuality, $\hat s=xs-\vec k_T^2$, $\varphi $ is the azimuthal
angle in the transverse XOY plane between vector $\vec k_T$ and
the fixed OX axis. The unintegrated gluon distribution function
$\Phi (x,\vec k_T^2,\mu ^2)$ can be related to the conventional
gluon distribution by
%
\begin{equation}
xG(x,\mu ^2)=\int\limits_{0}^{\mu ^2} {\Phi (x,\vec k_T^2,\mu
^2)d\vec k_T^2}\mbox{,}
\end{equation}
%
where $\Phi (x,\vec k_T^2,\mu ^2)$ satisfies the BFKL evolution
equation \cite{6}. In formulae~(2) the four-vector of a gluon
momentum is presented as follows:
%
\begin{equation}
k=x p_N+k_T\mbox{,}
\end{equation}
%
where $k_T=(0,\vec k_T,0)$, $p_N=(E_N,0,0,E_N)$ is the four-vector
of a proton momentum. At the $x\ll 1$ the off mass-shell gluon has
dominant longitudinal polarization along the proton momentum.
Taking into account the gauge invariance of a total amplitude
involving virtual gluon we can write the polarization four-vector
in two different forms:
%
\begin{equation}
\varepsilon ^\mu (k)=\frac{k_T^\mu}{|\vec k_T|}
\end{equation}
%
or
%
\begin{equation}
\varepsilon ^\mu (k)=-\frac{x p_N^\mu}{|\vec k_T|}\mbox{.}
\end{equation}
%
As it will be shown above formulae (5) and (6)
give the equal answers in calculating of squared amplitudes under
consideration.

Our calculation in the parton model is down using the GRV LO
\cite{7} parameterization for the collinear gluon distribution
function $G(x,\mu^2)$. In the case of the $k_T$-factorization
approach we use the following parameterizations for an
unintegrated gluon distribution function $\Phi(x,\vec
k_T^2,\mu^2)$: JB by J.~Bluemlein \cite{8}; JS by H.~Jung and
G.~Salam \cite{9}; KMR by M.A.~Kimber, A.D.~Martin and M.G.~Ryskin
\cite{10}. The detail analysis of the evolution equations lied in
a basis of the different parameterizations is over our
consideration. To compare different parameterizations we have
plotted their as a function of $x$ at the fixed $\vec k_T^2$ and
$\mu^2$ in Fig.1 and as a function of $\vec k_T^2$ at the fixed
$x$ and $\mu^2$ in Fig.2.

\begin{figure}[h]
\begin{center}
\includegraphics[width=1.0\textwidth, clip=]{Fig1.eps}
\end{center}
\caption[]{The unintegrated gluon distribution function
$\Phi(x,\vec k_T^2,\mu^2)$ versus $x$ at the fixed values of $\mu$
and $k_T^2$.} \label{eps1}
\end{figure}
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.0\textwidth, clip=]{Fig2.eps}
\end{center}
\caption[]{The unintegrated gluon distribution function
$\Phi(x,\vec k_T^2,\mu^2)$ versus $k_T^2$ at the fixed values of
$\mu$ and $x$.} \label{eps2}
\end{figure}

%The effective collinear gluon distribution functions obtained
%using equation (3) from unintegrated over $\vec k_T^2$
%distributions are show in Fig.3 together with GRV LO
%parameterization.

Note, that all parameterizations of an unintegrated gluon
distribution function describe the data from HERA collider for the
structure function $F_2(x,Q^2)$ well \cite{8,9,10}.

\section{$D^\star$ meson photoproduction in LO QCD.}

The photoproduction of the $D^\star$ meson was studding
experimentally by H1 and ZEUS collaborations at HERA ep-collider
($E_e=27.5$ GeV, $E_N=820$ GeV) \cite{11,12}. Because of the large
mass of a $c$-quark usually it is assumed that $D^\star$ meson
production may be described in the fragmentation approach
\footnote{The another approach based on recombination scenario was
suggested recently in \cite{14}} \cite{13}, where
%
\begin{equation}
\sigma(\gamma p\to D^\star X,p)=\int {D_{c\to D^\star }}(z,\mu ^2)
\sigma(\gamma p\to cX,p_1=p/z)dz
\end{equation}
%
and $D_{c\to D^\star }(z,\mu ^2)$ is the universal fragmentation
function of a $c$-quark into the $D^\star$ meson at the scale $\mu
^2=m_D^2+p_T^2$. The fraction of the $D^\star$ produced by a
$c$-quark as measured by OPAL Collaboration
 \cite{15},
%
$$
\omega _{c\to D^\star }=\int\limits_{0}^{1} {D_{c\to D^\star }(z,\mu
^2)}dz=0.222\pm 0.014\mbox{,}
$$
%
has been used in our LO QCD calculations to normalize the
fragmentation function.

The Peterson \cite{13} fragmentation function was used as a
phenomenological factor:
%
\begin{equation}
D_{c\to D^\star }(z,\mu _0^2)=N \frac{z (1-z)^2}{[(1-z)^2+\epsilon
z]^2}\mbox{.}
\end{equation}
%
In the high energy limit or in the case of a massless quark one
has following relation for the four-vectors $p=z p_1$, however in
the discussed here process the $D^\star$ meson energy is not so
large in compare to $M_{D^\star}$ and the following prescription
was used
%
\begin{equation}
\vec p=z \vec p_1
\end{equation}
%
together with the mass-shell condition for the $c$-quark energy
and momentum $E_1^2=\vec p_1^2+m_c^2$. We have used
$\epsilon=0.06$ as a middle value between two recent fits of
$D^\star$ meson spectra in $e^+e^-$-annihilation, which based on
massive charm ($\epsilon=0.036$)\cite{16} and massless charm
($\epsilon=0.116$)\cite{17} calculations. The squared matrix
element for the subprocess $\gamma g^\star\to c\bar c$ after
summation over a gluon polarization accordingly (6) may be written
as follows \cite{4,5,18}:
%
\begin{equation}
\overline{|M|^2}=16 \pi ^2 e_c^2 \alpha _s \alpha \cdot (\hat
s+\vec k_T^2)^2 \biggl[ \frac{\alpha _1^2+\alpha _2^2}{(\hat
t-m_c^2)(\hat u-m_c^2)} -\frac{2 m_c^2}{\vec k_T^2}\left(
\frac{\alpha _1}{\hat u-m_c^2}-\frac{\alpha _2}{\hat
t-m_c^2}\right) ^2\biggr]
\end{equation}
%
where $\hat s$, $\hat t$ and $\hat u$ are usual Mandelstam
variables,
$$
\alpha _1=\frac{m_c^2+\vec p_{1T}^2}{m_c^2-\hat t}\mbox{, }\alpha
_2=\frac{m_c^2+\vec p_{2T}^2}{m_c^2-\hat u}\mbox{,}
$$
$\vec p_{1T}$ and $\vec p_{2T}$ are the transverse momenta of $c$-
and $\bar c$-quarks, $\vec k_T=\vec p_{1T}+\vec p_{2T}$.

Using formulas (5) for a BFKL gluon polarization four-vector we
can rewrite (10) in the another form:
%
\begin{eqnarray}
\overline{|M|^2}&=&\frac{16 \pi ^2 e_c^2 \alpha _{s} \alpha }{
     (m_c^2 - \hat t)^2(m_c^2 - \hat u)^2}
     \biggl[ m_{c}^2 \biggl( -2m_c^6 - 4m_c^2\vec p_{1T}^2\vec k_T^2 +
m_c^2\vec k_T^4
     + 8\vec p_{1T}^2\vec k_T^4 +\nonumber\\&&+
     3\vec k_T^6 +
     (4m_c^4 + 12\vec p_{1T}^2\vec k_T^2 + 5\vec k_T^4)\hat s -
     (3m_c^2 - 4\vec p_{1T}^2 - 3\vec k_T^2)\hat s^2 + \hat s^3 \biggr)
     +\nonumber\\&&+
     \biggl( 8m_c^6 + 8m_c^2\vec p_{1T}^2\vec k_T^2 - 2m_c^2\vec k_T^4
     - 4\vec p_{1T}^2\vec k_T^4 - \vec k_T^6 -
     12m_c^4\hat s - 4\vec p_{1T}^2\vec k_T^2\hat s -\nonumber\\&&- \vec
k_T^4\hat s + 6m_c^2\hat s^2
      - \vec k_T^2\hat s^2 -
     \hat s^3\biggr) \hat t - \biggl( 4\vec p_{1T}^2\vec k_T^2 -\vec
k_T^4 + 3(-2m_c^2
     +\hat s)^2\biggr) \hat t^2 +\nonumber\\&&+
     4\biggl( 2m_c^2 - \hat s\biggr) \hat t^3 - 2\hat t^4
     -
     4|\vec p_{1T}| \biggl( |\vec k_T| \cos (\varphi) \Bigl( -2m_c^6
     -\nonumber\\&&- \Bigl( \vec k_T^2 - \hat s - 2\hat t \Bigr) \hat
t\Bigl( 2 m_c^2-\hat u\Bigr)
     +
     m_c^4\Bigl( \vec k_T^2 + 3\hat s + 6\hat t\Bigr) +
     m_c^2\Big( 3\vec k_T^4 + \hat s^2 +\nonumber\\&&+ \vec k_T^2(4\hat
s - 2\hat t) - 6\hat s\hat t
      - 6\hat t^2\Bigr)\Bigr)
     + |\vec p_{1T}| \cos (2\varphi )\Bigl( m_c^4\vec k_T^2 + \vec
k_T^2\hat t\Bigl( 2 m_c^2 - \hat u\Bigr)
     -\nonumber\\&&-
     m_c^2\Bigl( 2\vec k_T^4 + \hat s^2 + \vec k_T^2(3\hat s + 2\hat
     t)\Bigr)\Bigr)
     \biggr) \biggr] \mbox{,}
\end{eqnarray}
%
where $\varphi$ is the angle between $\vec p_{1T}$ and $\vec k_T$.

In the last case (11) it is easy to find the parton model limit:
%
\begin{equation}
\lim\limits_{|\vec k_T|\to 0} \int\limits_{0}^{2 \pi} {\frac
{d\varphi}{2 \pi}}
\overline{|M|^2}=\overline{|M_{PM}|^2}\mbox{,}
\end{equation}
%
where
%
$$\vec p_{1T}^2=\vec p_{2T}^2=\frac {(\hat u-m_c^2)(\hat t-m_c^2)}{\hat
s}-m_c^2 \mbox{,}$$
%
and
%
%This is famous formulae, but we can not use it.
%\begin{eqnarray}
%\overline{|M_{PM}|^2}=16 \pi ^2 e_c^2 \alpha _{s} \alpha \cdot
%\biggl[ -4\biggl( \frac{m_c^2}{m_c^2-\hat
%t}+\frac{m_c^2}{m_c^2-\hat u}\biggr) ^2 +4\biggl(
%\frac{m_c^2}{m_c^2-\hat t}+\frac{m_c^2}{m_c^2-\hat u}\biggr)
%\\+\frac{m_c^2-\hat u}{m_c^2-\hat t}+\frac{m_c^2-\hat t}{m_c^2-\hat u}
%\biggr]
%\end{eqnarray}
%
\begin{eqnarray}
\overline{|M_{PM}|^2}&=&-\frac{16 \pi ^2 e_c^2
\alpha_{s}\alpha}{(9(m_c^2 - \hat t)^2 (-m_c^2 + \hat s + \hat
t)^2)} \biggl[-2 m_c^8 + 8 m_c^6 \bigl(\hat s + \hat t\bigr)
-\nonumber\\&&- \hat t \bigl(\hat s^3 + 3 \hat s^2 \hat t + 4\hat
s\hat t^2 + 2\hat t^3\bigr) + m_c^2\bigl(\hat s^3 + 6 \hat s^2\hat
t + 8 \hat t^3 + 4 \hat s\hat t (3 \hat t + \hat u)\bigr)
-\nonumber\\&&- m_c^4 \bigl(7 \hat s^2 + 12 \hat t^2 + 4 \hat s (4
\hat t + \hat u)\bigr)\biggr]
\end{eqnarray}
%

\section{$D^\star$ meson photoproduction at HERA.}

In  this part we will compare our results obtained with leading
order matrix elements for the partonic subprocess $\gamma g\to
c\bar c$ in the conventional parton model as well as in the
$k_T$-factorization approach with data from HERA ep-collider. The
data under consideration taken by the ZEUS Collaboration
\cite{11}. Inclusive photoproduction of the $D^{\star\pm}$ mesons
has been measured for the photon-proton center-of-mass energies in
the range $130 < W < 280$ GeV and the photon virtuality $Q^2 < 1$
GeV$^2$. At low $Q^2$ the cross section for $ep\to eD^\star X$ are
related to $\gamma p$ cross section using the equivalent photon
approximation \cite{19}:
%
$$
d\sigma_{ep}=\int{\sigma_{\gamma p}}\cdot f_{\gamma
/e}(y)dy\mbox{,}
$$
%
where $f_{\gamma /e}(y)$ denotes the photon flux integrated over
$Q^2$ from the kinematic limit of $Q^2_{min}=m_e^2 y^2/(1-y)$ to
the upper limit $Q^2_{max} = 1$ GeV$^2$, $y=W^2/s$, $s=4 E_N E_e$,
$E_N$ and $E_e$ are the proton and electron energies in the
laboratory frame.

The exact formulas for $f_{\gamma /e}(y)$ is taken from \cite{20}:
%
$$
f_{\gamma /e}(y)=\frac{\alpha}{2
\pi}\biggl[\frac{1+(1-y)^2}{y}\log{\frac{Q^2_{max}}{Q^2_{min}}}+2
m_e^2 y
\bigl(\frac{1}{Q^2_{min}}-\frac{1}{Q^2_{max}}\bigr)\biggr]\mbox{.}
$$
%
The limits of integration over $y$ are
$y_{max\choosen{min}}=W^2_{max\choosen{min}}/s$. In our
calculations we used formulas for differential cross section in
the following form:
%
\begin{eqnarray}
\frac{d\sigma(ep \to eD^\star X)}{d\eta dp_{T}}&=&\int dy
f_{\gamma /e}(y)\int{\frac{dz}{z}D_{c\to
D^\star}(z,\mu^2)}\int{\frac{d\varphi}{2 \pi } } \int{d\vec k_T^2}
\frac{\Phi(x,\vec k_T^2,\mu^2)}{x} \times\nonumber\\&&
\times\frac{2 |\vec p_1| |\vec p_{1T}|}{E_1 (2 E_N
(E_1-p_{1z})-W^2)}\cdot\frac{|\bar M|^2}{16 \pi x W^2}
\end{eqnarray}
%
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\textwidth, clip=]{Fig3.eps}
\end{center}
\caption[]{The $\eta$ spectra of the $D^\star$ meson at the
various cut on a transverse momentum ($p_T>2, 4, 6$ GeV,
correspondingly from up to down) and $130 < W < 280$ GeV.}
\label{eps3}
\end{figure}
%
The differential cross section as a function of the $D^\star$
pseudorapidity, which is defined as $\eta=-\ln(\tg
\frac{\theta}{2})$, where the polar angle $\theta$ is measured
with respect to the proton beam direction, is shown in Fig.3 where
the kinematic ranges for the $D^\star$ meson transverse momentum
are $2 < p_T < 12$ GeV, $4 < p_T < 12$ GeV, $6 < p_T < 12$ GeV,
correspondingly from up to down.



We see that the results of calculations performed in the collinear
parton model as well as in the $k_T$-factorization approach with
LO in $\alpha_s$ matrix elements need additional $K$-factor
$(K\approx 2)$ to describe the data. The value of this $K$-factor
is usual for a heavy quark production cross section in the
relevant energy range. Opposite the results obtained in
$k_T$-factorization approach with JB parameterization in
\cite{21}, where the strong enhance for the cross sections at all
$\eta$ in the $k_T$-factorization approach in compare to the
collinear parton model was demonstrated, we see only deformation
of the $\eta$-spectra. We have obtained that at low $p_{Tmin}$ the
maximum value of the cross section even higher in the collinear
parton model and only at the large positive $\eta$ the
$k_T$-factorization approach gives more large values.

The $p_T$ spectrum of $D^\star$ meson in photoproduction at
$|\eta| < 1.5$ and $130 < W < 280$ GeV are shown in Fig.4. All
theoretical curves are under experimental points. As it was
already mentioned typical value of the $K$-factor is equal 2.

%
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth, clip=]{Fig4.eps}
\end{center}
\caption[]{The $p_T$-spectrum of $D^\star$ meson.} \label{eps4}
\end{figure}

Our results show that the introducing of a gluon transverse
momentum $k_T$ in the framework of the $k_T$-factorization
approach does't increase the $D^\star$ meson photoproduction cross
section at the large $p_T$ as it is predicted for the $J/\Psi$
photoproduction \cite{22,23} (see next part of the paper). We see
the small effect in the case of JS parameterization \cite{9} only.

The dependence of the $D^\star$ meson production cross section on
a total photon-proton center-of-mass energy W is shown in Fig.5.
Nowadays the experimental data for $d\sigma /dW$ are absent. We
see that the difference between the results obtained with the
various parameterizations of a unintegrated gluon distribution
function is about 50\%. As well in the case of the $\eta$-spectra.
At the small $p_{Tmin}$ the cross section calculated in the parton
model is larger than predictions obtained in the
$k_T$-factorization approach.


\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\textwidth, clip=]{Fig5.eps}
\end{center}
\caption[]{The theoretical predictions for the $W$-spectra at the
various cut on the $D^\star$ meson transverse momentum ($p_T>2, 4,
6$ GeV, correspondingly from up to down) and $|\eta|<1.5$. }
\label{eps5}
\end{figure}

The main uncertainties of our calculation come from the choice of
a $c$-quark mass in the partonic matrix elements (10),(11) and
(13), and from the choice of a parameter $\epsilon$ in the
Peterson fragmentation function $D_{c \to
D^\star}(z,\mu^2)$.

However, even at the very extremely choice ($m_c\approx 1.3$ GeV
and $\epsilon \approx 0.02$) our theoretical predictions describe
only shapes of the $p_T$- and $\eta$-spectra, but don't describe
absolute values of the measured cross sections. This fact shows
the famous role of the next to leading order corrections in the
$D^\star$ meson photoproduction as in the parton model as in the
$k_T$-factorization approach.

\section{$J/\psi$ photoproduction in LO QCD}

It is well known that in the processes of $J/\psi$ meson
photoproduction on protons at high energies the photon-gluon
fusion partonic subprocess dominates \cite{24}. In the framework
of the general factorization approach of QCD the $J/\psi$
photoproduction cross section depends on the gluon distribution
function in a proton, the hard amplitude of $c\bar c$-pair
production as well as the mechanism of a creation colorless final
state with quantum numbers of the $J/\psi$ meson. In such a way,
we suppose that the soft interactions in the initial state are
described by introducing a gluon distribution function, the hard
partonic amplitude is calculated using perturbative theory of QCD
at order in $\alpha_s(\mu^2)$, where $\mu\sim m_c$, and the soft
process of the $c\bar c$-pair transition into the $J/\psi$ meson
is described in nonrelativistic approximation using series in the
small parameters $\alpha_s$ and $v$ (relative velocity of the
quarks in the $J/\psi$ meson). As is said in nonrelativistic QCD
(NRQCD) \cite{25}, there are color singlet mechanism, in which the
$c\bar c$-pair is hardly produced in the color singlet state, and
color octet mechanism, in which the $c\bar c$-pair is produced in
the color octet state and at a long distance it transforms into a
final color singlet state in the soft process. However, as it was
shown in papers \cite{23,26}, the data from the DESY ep-collider
\cite{27} in the wide region of $p_T$ and $z$ may be described
well in the framework of the color singlet model and the color
octet contribution is not needed. Based on the above mentioned
result we will take into account in our analysis only the color
singlet model contribution in the $J/\psi$ meson photoproduction
\cite{24}. We consider here the role of a proton gluon
distribution function in the $J/\psi$ photoproduction in the
framework of the conventional parton model as well as in the
framework of the $k_T$-factorization approach \cite{3,4,5}.

\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\textwidth, clip=]{Fig6.eps}
\end{center}
\caption[]{Diagrams used for description partonic process
$\gamma+g\to J/\psi+g$. } \label{eps6}
\end{figure}

There are six Feynman diagrams (Fig.6) which describe the partonic
process $\gamma g \to J/\psi g$ at the leading order in $\alpha_s$
and $\alpha$. In the framework of the color singlet model and
nonrelativistic approximation the production of the $J/\psi$ meson
is considered as the production of a quark-antiquark system in the
color singlet state with orbital momentum $L=0$ and spin momentum
$S=1$. The binding energy and relative momentum of quarks in the
$J/\psi$ are neglected. In such a way $M=2 m_c$ and $p_c=p_{\bar
c}=\displaystyle{\frac{p}{2}}$, where $p$ is the 4-momentum of the
$J/\psi$,  $p_c$ and $p_{\bar c}$ are 4-momenta of quark and
antiquark. Taking into account the formalism of the projection
operator \cite{28} the amplitude of the process $\gamma g\to
J/\psi g$ may be obtained from the amplitude of the process
$\gamma g\to \bar c c g$ after replacement:
\begin{equation}
V^i(p_{\bar c})\bar U^j(p_c) \rightarrow
\frac{\Psi(0)}{2\sqrt{M}}\hat \varepsilon (p)(\hat
p+M)\frac{\delta^{ij}}{\sqrt{3}}\mbox{,}
\end{equation}
where $\hat \varepsilon (p)=\varepsilon_{\mu}(p)\gamma^{\mu}$,
$\varepsilon_{\mu}(p)$ is a 4-vector of the $J/\psi$ polarization,
$\displaystyle{\frac{\delta^{ij}}{\sqrt{3}}}$ is the color factor,
$\Psi(0)$ is the nonrelativistic meson wave function at the
origin. The matrix elements of the process $\gamma g^\star\to
J/\psi g$ may be presented as follows:
\begin{equation}
M_i=KC^{ab}\varepsilon_{\alpha}(q_1)\varepsilon^a_{\mu}(q)
\varepsilon^b_{\beta}(q_2)\varepsilon_{\nu}(p)
M_i^{\alpha\beta\mu\nu}\mbox{,}
\end{equation}
\begin{equation}
M_1^{\alpha\beta\mu\nu}= \mbox{Tr}\left[\gamma^{\nu}(\hat
p+M)\gamma^{\alpha}\frac{\hat p_c-\hat
q_1+m_c}{(p_c-q_1)^2-m_c^2}\gamma^{\mu}\frac{-\hat p_{\bar c}-\hat
q_2+m_c}{(p_{\bar c}+q_2)^2-m_c^2}\gamma^{\beta}\right]\mbox{,}
\end{equation}
\begin{equation}
M_2^{\alpha\beta\mu\nu}= \mbox{Tr}\left[\gamma^{\nu}(\hat
p+M)\gamma^{\beta}\frac{\hat p_c+\hat
q_2+m_c}{(p_c+q_2)^2-m_c^2}\gamma^{\alpha}\frac{\hat k -\hat
p_{\bar c}+m_c}{(q-p_{\bar c})^2-m_c^2}\gamma^{\mu}\right]\mbox{,}
\end{equation}
\begin{equation}
M_3^{\alpha\beta\mu\nu}= \mbox{Tr}\left[\gamma^{\nu}(\hat
p+M)\gamma^{\alpha}\frac{\hat p_c-\hat
q_1+m_c}{(p_c-q_1)^2-m_c^2}\gamma^{\beta}\frac{\hat k-\hat p_{\bar
c}+m_c}{(q-p_{\bar c})^2-m_c^2}\gamma^{\mu}\right]\mbox{,}
\end{equation}
\begin{equation}
M_4^{\alpha\beta\mu\nu}= \mbox{Tr}\left[\gamma^{\nu}(\hat
p+M)\gamma^{\mu}\frac{\hat p_c-\hat
k+m_c}{(p_c-q)^2-m_c^2}\gamma^{\alpha}\frac{-\hat p_{\bar c}-\hat
q_2+m_c}{(q_2+p_{\bar c})^2-m_c^2}\gamma^{\beta}\right]\mbox{,}
\end{equation}
\begin{equation}
M_5^{\alpha\beta\mu\nu}= \mbox{Tr}\left[\gamma^{\nu}(\hat
p+M)\gamma^{\beta}\frac{\hat p_c+\hat
q_2+m_c}{(p_c+q_2)^2-m_c^2}\gamma^{\mu}\frac{\hat q_1 -\hat
p_{\bar c}+m_c}{(q_1-p_{\bar
c})^2-m_c^2}\gamma^{\alpha}\right]\mbox{,}
\end{equation}
\begin{equation}
M_6^{\alpha\beta\mu\nu}= \mbox{Tr}\left[\gamma^{\nu}(\hat
p+M)\gamma^{\mu}\frac{\hat p_c-\hat
k+m_c}{(p_c-q)^2-m_c^2}\gamma^{\beta}\frac{\hat q_1-\hat p_{\bar
c}+m_c}{(q_1-p_{\bar c})^2-m_c^2}\gamma^{\alpha}\right]\mbox{,}
\end{equation}
where  $q_1$ is the 4-momentum of the photon,  $q$ is the
4-momentum of the initial gluon, $q_2$ is the  4-momentum of the
final gluon,
%
$$
K=e_ceg_s^2\frac{\Psi(0)}{2\sqrt{M}}\mbox{,}\qquad
C^{ab}=\frac{1}{\sqrt{3}}\mbox{Tr}[T^aT^b]\mbox{,}\quad
e_c=\frac{2}{3}\mbox{,}\quad e=\sqrt{4\pi\alpha}\mbox{,}\quad
g_s=\sqrt{4\pi\alpha_s}\mbox{.}
$$
%
The summation on the photon, the $J/\psi$ meson and final gluon
polarizations is carried out by covariant formulae:
\begin{eqnarray}
&&\sum_{spin}\varepsilon_{\alpha}(q_1)\varepsilon_{\beta}(q_1)
=-g_{\alpha\beta},\\
&&\sum_{spin}\varepsilon_{\alpha}(q_2)\varepsilon_{\beta}(q_2)
=-g_{\alpha\beta},\\
&&\sum_{spin}\varepsilon_{\mu}(p)\varepsilon_{\nu}(p)
=-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{M^2}.
\end{eqnarray}
In case of the initial BFKL gluon we use the prescription (5).
%\begin{equation}
%\sum_{spin}\varepsilon_{\mu}(q)\varepsilon_{\nu}(q)=
%\frac{q_{T\mu}q_{T\nu}}{{\bf k}^2_{T}}.
%\end{equation}
For studing  $J/\psi$ polarized photoproduction we introduce the
4-vector of the longitudinal polarization as follows:
\begin{equation}
\varepsilon_L^{\mu}(p)=\frac{p^{\mu}}{M}-\frac{Mp_N^{\mu}}{(pp_N)}.
\end{equation}
In the high energy limit of $s=2(q_1p_N)>>M^2$ the polarization
4-vector satisfies usual conditions
$(\varepsilon_L\varepsilon_L)=-1$, $(\varepsilon_L p)=0$.

Traditionally for a description of charmonium photoproduction
processes the invariant variable $z=(pp_N)/(q_1p_N)$ is used. In
the rest frame of the proton one has $z=E_{\psi}/E_{\gamma}$. In
the $k_T$-factorization approach the differential on $p_{T}$ and
$z$ cross section of the $J/\psi$ photoproduction may be written
as follows:
\begin{equation}
\frac{d\sigma(\gamma p\to J/\Psi
X)}{dp^2_{T}dz}=\frac{1}{z(1-z)}\int_0^{2\pi}
\frac{d\varphi}{2\pi}\int_0^{\mu^2} d{\bf k}^2_{T}\Phi (x,{\bf
k}^2_{T},{\mu}^2) \frac{\overline{|M|^2}}{16\pi(xs)^2}\mbox{.}
\end{equation}

The analytical calculation of the $\overline{|M|^2}$ is performed
with help of REDUCE package and results are saved in the FORTRAN
codes as a function of $\hat s=(q_1+q)^2$, $\hat t=(p-q_1)^2$,
$\hat u=(p-q)^2$, ${\bf p}_T^2$, ${\bf k}^2_{T}$ and  $\cos
(\varphi)$. We directly have tested that
\begin{equation}
\lim_{{{\bf k}^2_{T}\to 0}}\int_0^{2\pi}\frac{d\varphi}{2\pi}
\overline{|M|^2}= \overline{|M_{PM}|^2}\mbox{,}
\end{equation}
where ${\bf p}_T^2= \displaystyle{\frac{\hat t \hat u}{\hat s}}$
in the $\overline{|M|^2}$ and $\overline{|M_{PM}|^2}$ is the
square of the amplitude in the conventional parton model
\cite{24}. In the limit of ${\bf k}^2_{T}=0$ from formula (27) it
is easy to find the differential cross section in the parton
model, too:
\begin{equation}
\frac{d\sigma^{PM}(\gamma p\to J/\Psi X)}{dp^2_{T}dz}=
\frac{\overline{|M_{PM}|^2}xG(x,\mu^2)}{16\pi(xs)^2z(1-z)}\mbox{.}
\end{equation}
However, making calculations in the parton model we use formula
(27), where integration over ${\bf k}^2_{T}$ and $\varphi$ is
performed numerically, instead of (29). This method fixes the
common normalization factor for both approaches and gives a direct
opportunity to study effects connected with virtuality of the
initial BFKL gluon in the partonic amplitude.

\section{$J/\psi$ photoproduction at HERA}

After we fixed the selection of the gluon distribution functions
$G(x,\mu^2)$ or $\Phi (x,{\bf k}^2_{T},{\mu}^2)$ there are two
parameters only, which values determine the common normalization
factor of the cross section under consideration: $\Psi(0)$ and
$m_c$. The value of the $J/\psi$ meson wave function at the origin
may be calculated in a potential model or obtained from
experimental well known decay width $\Gamma(J/\psi\to
\mu^+\mu^-)$. In our calculation we used the following choice
$|\Psi(0)|^2=0.064$ GeV$^3$ which corresponds to NRQCD coefficient
$<O^{J/\psi},{\bf 1}^3S_1>=1.12$ GeV$^3$ as the same as in Ref.
\cite{26}. Note, that this value is a little smaller (30 \%) than
the value which was used in our paper \cite{23}. Concerning a
charmed quark mass, the situation is not clear up to the end. From
one hand, in the nonrelativistic approximation one has
$m_c=\displaystyle{\frac{M}{2}}$, but there are many examples of
taking smaller value of a $c$-quark mass in the amplitude of a
hard process, for example $m_c=1.4$ GeV. Taking into consideration
above mentioned we perform calculations at $m_c=1.5$  GeV. The
cinematic region under consideration is determined by the
following conditions: $Q^2<1$ GeV$^2$, $60<W<240$ GeV, $0.3<z<0.9$
and $p_T>1$ GeV, which correspond to the H1 Collaboration data
\cite{29}. We assume that the contribution of the color octet
mechanism is large at the $z>0.9$ only. In the region of the small
values of the  $z<0.2$ the contribution of the resolved photon
processes \cite{30} as well as the charm excitation processes
\cite{31} may be large, too. All of these contributions are not in
our consideration.


Figures 7--10 show our results which were obtained as in the
conventional parton model as well as in the $k_T$-factorization
approach with the different parameterizations of the unintegrated
gluon distribution function. The dependence of the results on
selection of a hard scale parameter $\mu$ is much less than the
dependence on selection of a $c$-quark mass and selection of a
parameterization. We put $\mu^2=M^2+ \bf p_T^2$ in a gluon
distribution function and in a running constant $\alpha_s(\mu^2)$.

\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\textwidth, clip=]{Fig7.eps}
\end{center}
\caption[]{The  $J/\psi$ spectrum on $p^2_{T}$ at the $60<W<240$
GeV and $0.3<z<0.9$. } \label{eps7}
\end{figure}

%Figure 10 shows the dependence of the total $J/\psi$
%photoproduction cross section on $\sqrt s$. It is visible that the
%difference between the parton model prediction and the result of
%$k_T$-factorization approach is much less than between the
%results obtained at different values of $c$-quark mass in the both
%models. At the $m_c=1.55$ GeV the obtained cross sections in 1.5
%--2 times are less than experimental data \cite{4}, but at the
%$m_c=1.4$ GeV the theoretical curves lie even a shade higher of
%the experimental points.

%There are not contradictions between the theoretical predictions
%and data for the $z$-spectrum of the $J/\psi$ mesons. Figure 5
%shows that the experimental points lie inside the theoretical
%corridor as in the parton model as in the $k_T$-factorization
%approach.

 The count of a transverse momentum of the BFKL gluons in the
$k_T$-factorization approach results in a flattening of the
$p_T$-spectrum of the $J/\psi$ as contrasted by predictions of the
parton model. For the first time this effect was indicated in the
Ref. \cite{22}, and later in the Ref. \cite{23}. Figure 7 shows
the result of our calculation for the $p_T$ spectrum of the
$J/\psi$ mesons. Using the $k_T$-factorization approach we have
obtained the harder $p_T$-spectrum of the  $J/\psi$ than has been
predicted in the LO parton model. It is visible that at large
values of $p_T$ only the $k_T$-factorization approach gives
correct description of the data \cite{29}. However, it is
impossible to consider this visible effect as a direct indication
on nontrivial developments of the small-x physics. In the article
\cite{26} was shown that the calculation in the NLO approximation
gives a harder $p_T$ spectrum of the $J/\psi$ meson, too, which
will agree with the data at the large $p_T$.

In the $k_T$-factorization approach JB parameterization \cite{8}
gives $p_T$-spectrum, which is very close to experimental data.
From the another hand in the case of JS parameterization \cite{9}
the additional  $K$-factor approximately equal 2 is needed.

\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\textwidth, clip=]{Fig8.eps}
\end{center}
\caption[]{The
 $J/\psi$ spectrum on $z$ at the $60<W<240$ GeV (
$p_{T}>1,2,3$ GeV, correspondingly from up to down).  }
\label{eps8}
\end{figure}

%\vspace{50mm}

\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\textwidth, clip=]{Fig9.eps}
\end{center}
\caption[]{The total $J/\psi$ photoproduction cross section versus
$W$ at the $0.3<z<0.8$ and $p_T>1$ GeV.} \label{eps9}
\end{figure}

The $z$ spectra are shown in Fig.8 at the various choice of the
$p_T$ cut: $p_T > 2$, $4$ and $6$ GeV, correspondingly. The
relation between the theoretical predictions and experimental data
is the same as in Fig.7. The $k_T$ factorization approach give
more correct description of the data especially at large value of
$z$ where the curve obtained in the collinear parton model tends
to zero.


Fig.9 shows the dependence of the total $J/\psi$ photoproduction
cross section on $W$ at $0.3<z<0.8$ and $p_T>1$ GeV. The shape of
this dependence agrees well with the result obtained using JS
parameterization \cite{9} or KMR \cite{10} parameterization.
However, the predicted absolute value of the cross section
$\sigma_{\gamma p}$ is smaller by factor 2 than obtained data
\cite{29}. The results of calculation using JB \cite{8} or GRV
\cite{7} parameterizations are larger and coincide with the data
\cite{29} better.





As it was mentioned above, the main difference between the
$k_T$-factorization approach and the conventional parton model is
nontrivial polarization of the BFKL gluon. It is obvious, that
such a spin condition of the initial gluon should result in
observed spin effects during the birth of the polarized $J/\psi$
meson. We have performed calculations for the spin parameter
$\alpha$ as a function $z$ or $p_T$ in the conventional parton
model and in the $k_T$-factorization approach :
\begin{equation}
\alpha (z)=\displaystyle{\frac {\frac{d\sigma_{tot}}{dz}
-3\frac{d\sigma_L}{dz}}
{\frac{d\sigma_{tot}}{dz}+\frac{d\sigma_L}{dz}}},\qquad \alpha
(p_T)=\displaystyle{\frac { \frac{d\sigma_{tot}}{dp_{T}}
-3\frac{d\sigma_L}{dp_T}}
{\frac{d\sigma_{tot}}{dp_T}+\frac{d\sigma_L}{dp_T}}}
\end{equation}
Here  $\sigma_{tot}=\sigma_L+\sigma_T$ is the total $J/\psi$
production cross section, $\sigma_L$ is the production cross
section for the longitudinal polarized $J/\psi$ mesons, $\sigma_T$
is the production cross section for the transverse polarized
$J/\psi$ mesons. The parameter $\alpha$ controls the angle
distribution for leptons in the decay $J/\psi \to l^+l^-$ in the
$J/\psi$ meson rest frame:
\begin{equation}
\frac{d\Gamma}{d \cos(\theta)}\sim 1+\alpha\cos^2(\theta).
\end{equation}


\begin{figure}[h]
\begin{center}
\includegraphics[width=.6\textwidth, clip=]{Fig10.eps}
\end{center}
\caption[]{Parameter $\alpha$ as a function of $p_T$ at the $0.3 <
z < 0.9$, $60 < W < 240$ GeV.} \label{eps10}
\end{figure}
%Figure 7 shows the parameter $\alpha (z)$, which is calculated in
%the parton model (curve 2) and in the $k_T$-factorization approach
%(curve 1). We see that both curves lie near zero at $z<0.8$ and
%increase at $z>0.8$. The large difference between predictions is
%visible only at $z>0.9$ where our consideration is not adequate.
%Let's remark, that parameter is gentle depends on mass of a
%charmed quark and we demonstrate here only outcomes obtained at
%the $m_c=1.55$.
The theoretical results for the parameter $\alpha(z)$ are very
close to each other irrespective of the choice of an approach or a
gluon parameterization \cite{23}.


For the parameter $\alpha (p_T)$ we have found strongly opposite
predictions in the parton model and in the $k_T$-factorization
approach, as it is visible in Fig.10. The parton model predicts
that  $J/\psi$ mesons should have transverse polarizations at the
large $p_T$ ($\alpha(p_T)=0.6$ at the $p_T=6$ GeV), but
$k_T$-factorization approach predicts that $J/\psi$ mesons should
be longitudinally polarized ( $\alpha(p_T)=-0.4$ at the $p_T=6$
GeV). The experimental points lie in the range $0 < p_T < 5$ GeV
and they have the large errors. However, it is visible that
$\alpha(p_T)$ decrease as $p_T$ changes from 1 to 5 GeV. This fact
coincide with theoretical prediction obtained in the
$k_T$-factorization approach. Nowadays, a result of the NLO parton
model calculation in the case of the polarized $J/\psi$ meson
photoproduction is unknown. It should be an interesting subject of
future investigations. If the count of the NLO corrections will
not change predictions of the LO parton model for $\alpha (p_T)$,
the experimental measurement of this spin effect will be a direct
signal about BFKL gluon dynamics.

Nowadays, the experimental data on $J/\Psi$ polarization in
photoproduction at large $p_T$ are absent. However there are
similar data from CDF Collaboration \cite{32}, where $J/\psi$ and
$\psi'$ $p_T$-spectra and polarizations have been measured.
Opposite the case of $J/\psi$ photoproduction, the hadroproduction
dada needs to take into account the large color-octet contribution
in order to explain $J/\psi$ and $\psi'$ production at Tevatron in
the conventional collinear parton model. The relative weight of
color-octet contribution may be smaller if we use
$k_T$-factorization approach, as was shown recently in
\cite{33,34,35,36}. The predicted using collinear parton model
transverse polarization of $J/\psi$ at large $p_T$ is not
supported by the CDF data, which can be roughly explained by the
$k_T$-factorization approach \cite{34}. In conclusion, the number
of theoretical uncertainties in the case of $J/\psi$ meson
hadroproduction is much more than in the case of photoproduction
and they need more complicated investigation, which is why the
future experimental analysis of $J/\psi$ photoproduction at THERA
will be clean check of the collinear parton model and the
$k_T$-factorization approach.


The authors would like to thank M.~Ivanov and S.~Nedelko for kind
hospitality during workshop "Heavy quark - 2002" in Dubna,
S.~Baranov, A.~Lipatov and O.~Teryaev for discussions on the
$k_T$-factorization approach of QCD and H.~Jung for the valuable
information on unintegrated gluon distribution functions. This
work has been supported in part by the Russian Foundation for
Basic Research under Grant 02-02-16253.

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\end{document}


