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\documentclass{ws-p8-50x6-00}

\begin{document}

\title{Heavy quark production in the semihard QCD approach
at HERA and beyond}

\author{S.P. Baranov}

\address{P.N. Lebedev Physics Institute, Leninsky prosp. 53,  
Moscow 117924, Russia\\ 
E-mail: baranov@sci.lebedev.ru}

\author{A.V. Lipatov}

\address{Department of Physics, M.V. Lomonosov Moscow State
University, Moscow 119899, Russia\\
E-mail: artem\underline{\hphantom{3}}lipatov@mail.ru}

\author{N.P. Zotov}

\address{D.V. Skobeltsyn Institute of Nuclear Physics,
 M.V. Lomonosov Moscow State University,
 Moscow 119899, Russia\\  
E-mail: zotov@theory.sinp.msu.ru}

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% You may repeat \author \address as often as necessary      %
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\maketitle

\abstracts{
Processes of heavy quark production at HERA, TEVATRON and THERA
energies are considered using the semihard ($k_T$ factorization)
QCD approach with emphasis on the BFKL dynamics of gluon
distributions.}

The experimental results on $b\bar b -$pair production cross sections
obtained by the H1 and ZEUS Collaborations at HERA\cite{H}
 and D0 and CDF Collaborations at TEVATRON\cite{T}
 provide a strong impetus for further
theoretical studies.  Comparisons of these results with NLO
pQCD calculations show that they underestimate the cross sections
at HERA and TEVATRON energies. Therefore, it looks certainly
reasonable to try a different approach.

 In this work we focus on the description of $b\bar b-$pair cross sections
at HERA and TEVATRON in the so called semihard ($k_T$ factorization) QCD
approach (SHA)\cite{GLR,CCH},
 which we have applied earlier to open charm\cite{SZ} and
 $J/\Psi$ photoproduction at HERA (see in ref.\cite{SZ}). We also discuss 
the sensitivity of our theoretical results\cite{BZ3} to the BFKL
 type dynamics\cite{BFKL} which may be investigated in the photoproduction
of $D^{*}$ and $J/\Psi$ mesons at THERA energies. 

 In SHA, the unintegrated gluon distribution $\varphi_G(x, k_{T}^2)$
is connected with the conventional 
gluon density $xG(x, Q^2)$ by the following relation
\begin{equation}   
 xG(x, Q^2) =  xG(x, Q_0^2) + \int_{Q_0^2}^{Q^2} dk_T^2 
\varphi_G(x, k_{T}^2), 
\end{equation}
where $Q_0^2$ is the collinear cutoff parameter.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=0.45\linewidth]{gpbb.eps}
\includegraphics[width=0.48\linewidth]{ppbbSLsp1.eps}
\end{center}
\vspace*{-5mm}
\caption{The cross sections of $b\bar b$ production $\sigma(p_T > p_T^{min})$
 at HERA (left panel) and  TEVATRON (right panel): curves 1, 2, 3, 4 and 5 
correspond to the
MT, GRV, RS, LRSS and BFKL parametrizations of gluon distribition.}
\end{figure*}
  We used the results of ref.\cite{CCH}  for the off mass shell parton 
cross sections, and we used several different parameterizations for the
unintegrated gluon distribution (see ref.\cite{SZ} for details),
namely: the LRSS\cite{GLR},
RS\cite{RS} and the so called BFKL~\cite{Blum} parameterizations.
We used the following set of SHA parameters: $Q_0^2 =$4, 2 and 1 GeV$^2$ in (1)
for the RS, LRSS  and BFKL parameterizations; in the case of the BFKL
parameterization the parameter $\Delta = 0.35$\cite{SZ}; everywhere the
charm and beauty  quark masses are $m_c =$1.5 GeV and $m_b =$4.75 GeV.

 The results of our calculations for the total cross section of inelastic
$b\bar b$ photoproduction at HERA as compared to H1\cite{H} data are
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=0.45\linewidth]{xgamthera.eps}
\includegraphics[width=0.45\linewidth]{xgamccfm_thera.eps}
\end{center}
\vspace*{-5mm}
\caption{The differential cross section $d \sigma /dx_\gamma$ (nb) for 
$Q^2<1$ GeV$^2$ with BFKL (left panel) and  CCFM (right panel)
 unintegrated gluon distributions
at THERA.}
 \label{xgamcc}
\end{figure*}
%
\begin{figure}[!t]
\begin{center}
\epsfig{figure=jrthera.eps,width=7cm,height=5cm}
\end{center}
\caption{The fraction of $J/\Psi$ mesons in helicity
zero state (degree of spin alignment).}
\label{fig3}
\end{figure}
published in the paper by Lipatov, Saleev, Zotov\cite{SZ}. We have shown
there that the H1 data are well described by the LRSS parametrization
and by the the BFKL parametrization but only with as small $m_b$ as 
$m_b =$4.25 GeV in the latter case. In Fig. 1a we show our
results for the total cross
section of inelastic $b\bar b$ photoproduction at HERA compared to
 ZEUS data\cite{H}.
We see that only the LRSS parametrization describes the ZEUS data 
(at $m_b =$4.75 GeV). In contrast with this, the cross
section for $b\bar b$
production at TEVATRON\cite{T} is described by the BFKL and RS
parametrizations very well (Fig. 1b). The LRSS parametrization
(at the same values\cite{SZ} of parameters and normalization)
overshoots the D0 (and CDF) data.

 In the ref.\cite{BZ2} the calculations of the associated charm
and dijet production cross section have been made within the SHA
with BFKL and CCFM\cite{HJ} unintegrated gluon distributions at HERA
energies. 
The attention was focused there on the variable $x_\gamma$, which is the
fraction of the photon momentum contributed to a pair of jets with
largest $p_T$. The results of the similar calculations made for THERA
conditions  are shown in Fig. 2  as a futher test of the underlying dynamics.
 The existence of the wide plateau at $x_\gamma < 0.9$ seen in the fugure
comes from the noncollinear gluon evolution, which generates gluons with
non-negligeable transverse momentum. In a significant fraction of events 
the gluon emitted close to the quark box appears to be even harder than
one or even both of the quarks produced in hard interaction.

The effects of initial gluon off-shellness may be, best of all, 
seen in the transverse momentum spectra of $J/\Psi$ mesons\cite{B}.
 In contrast with the conventional (massless) parton model, the SHA shows
that the fraction of $J/\Psi$  mesons in the helicity zero state increases
with their transverse momentum $p_T$. A deviation from the parton model
behaviour becomes well pronounced already from $p_T > 3$ GeV at HERA  
energies\cite{B}, and at $p_T > 6$ GeV the helicity zero polarization
tends to be dominant. The same effect is seen in Fig. 3, where
we show the results of the calculations\cite{BZ3} of the ratio 
$\sigma_{h= 0}/\sigma$ for $J/\Psi$ photoproduction at THERA conditions
made with the BFKL unintegrated gluon distribution. 

The examples considered in this paper demonstrate the effects of the BFKL
gluon evolution on the important and experimentally measurable quantities,
such as the event topology or vector meson spin alignement. At present, 
the theoretical predictions made for HERA and TEVATRON 
conditions have found their
experimental confirmation. A further investigation of the relevant effects
at THERA collider can put our understanding of the hadron structure on
even more solid grounds.

One of us (N.Z.) is gratefull  to Russian Academy of Science and
 the Organizing Committee of  DIS2001 for financial support.
This work has been supported by the Royal Swedish Academy of Sciences.


%
\begin{thebibliography}{99}
%\bibitem{ja}C Jarlskog in {\em CP Violation}, ed. C Jarlskog
%(World Scientific, Singapore, 1988).
%\bibitem{ma}L. Maiani, \Journal{\PLB}{62}{183}{1976}.
%\bibitem{bu}J.D. Bjorken and I. Dunietz, \Journal{\PRD}{36}{2109}{1987}.
%\bibitem{bd}C.D. Buchanan {\it et al}, \Journal{\PRD}{45}{4088}{1992}.
\bibitem{H}
 ZEUS Collab., J. Breitweg {\it et al}, {\em Eur. Phys. J.}
 C {\bf 18}, 625 (2001).\\
H1 Collab., C. Adlof {\it et al}, {\em Phys. Lett.} B {\bf 467}, 156
(1999).
\bibitem{T}
 D0 Collab., B. Abbott {\it et al}, {\em Phys. Lett.} B {\bf 487}, 264
(2000).\\
CDF Collab., F. Abe {\it et al}, {\em Phys. Rev. Lett.}  {\bf 71}, 2396 (1993).
\bibitem{GLR}
 L.V.~Gribov, E.M.~Levin, M.G.~Ryskin, {\em Phys. Rep.} {\bf
100}, 1 (1983). \\  
 E.M.~Levin, M.G.~Ryskin, Yu.M.~Shabelski, A.G.~Shuvaev,
{\em Sov. J. Nucl. Phys.} {\bf 53}, 657 (1991).
\bibitem{CCH}
S. Catani, M. Ciafoloni, F. Hautmann, {\em Nucl. Phys.} B {\bf
366}, 135 (1991). \\
 J.C. Collins and R.K. Ellis, {\em Nucl. Phys.} B {\bf 360}, 3
(1991).
\bibitem{SZ}
 V.A. Saleev, N.P. Zotov N.P., {\em Mod. Phys. Lett.} A {\bf
11}, 25 (1996); S.P. Baranov, N.P. Zotov, {\em Phys. Lett.} B {\bf
458}, 389 (1999); A.V. Lipatov,  V.A. Saleev, N.P. Zotov N.P.,
 {\em Mod. Phys. Lett.} A {\bf15}, 25 (2000).
%\bibitem{LZ}
% V.A. Saleev, N.P. Zotov, {\em Mod. Phys. Lett.} A {\bf
%9}, 151 (1994); A.V. Lipatov, N.P. Zotov, {\em Mod. Phys. Lett.} A {\bf
%15}, 695 (2000).
\bibitem{BZ3}
 S.P. Baranov, N.P. Zotov, .
\bibitem{BFKL} 
E.~Kuraev, L.~Lipatov, V.~Fadin, {\em Sov. Phys. JETP} {\bf
44}, 443 (1976); {\it ibid.} {\bf 45}, 199 (1977);
     Y.~Balitskii, L.~Lipatov, {\em Sov. J. Nucl. Phys.} {\bf
28}, 822 (1978).
\bibitem{RS}
 A.G. Ryskin, Yu.M. Shabelski, {\it Z. Phys.} C {\bf 66}\, 
151 (1995).
\bibitem{Blum}
J. Blumlein, Preprint DESY 95-121.
\bibitem{BZ2}
S.P. Baranov, N.P. Zotov {\em Phys. Lett.} B {\bf 491}, 111 (2000).
\bibitem{HJ}
H. Jung, in {\it Proc. Workshop on MC Generators}, DESY, 
1999, p. 75.
\bibitem{B}
S.P. Baranov,  {\em Phys. Lett.} B {\bf
428}, 377 (1998).
\end{thebibliography}   
\end{document}

\section{Guidelines}
\subsection{Producing the Hard Copy}\label{subsec:prod}
The hard copy may be printed using the advice given in the file
{\em readme$\_$2e.txt}, which is repeated in this section. You should
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from our WWW pages at:

\noindent
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\noindent
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\noindent {\em readme$\_$2e.txt} --- the preliminary guide.

\noindent {\em ws-p8-50x6-00.cls} --- the class file that provides the higher
level latex commands for the proceedings. Don't change these parameters.

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If you wish to use some other form of word-processor, some
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\subsection{Using Other Word-Processing Packages}\label{subsec:wpp}
If you want to use some other form of word-processor to
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It's also important to reproduce the spacing of the text and
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Paragraphs should have a first line indented by about 0.25in
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\subsection{Headings and Text and Equations}
Please preserve the style of the headings, text fonts and line
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Equations should be centered and numbered consecutively, as in
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where only one referencing equation number is wanted.

In latex, it is simplest to give the equation a label, as in
Eq.~(\ref{eq:murnf}) where we have used 
{\em $\backslash$label\{eq:murnf\}} to identify the equation.
You can then use the reference {\em $\backslash$ref\{eq:murnf\}}
when citing the equation in the text which will avoid the need
to manually renumber equations due to later changes. (Look at
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The same method can be used for referring to sections 
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\subsection{Tables}
The tables are designed to have a uniform style throughout the
proceedings volume. It doesn't matter how you choose to place
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The top and bottom horizontal lines should be single 
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minimum. We've chosen a more complicated example purely as an 
illustration of what is possible.

The caption heading for a table should be placed at the top of
the table.

\begin{table}[t]
\caption{Experimental Data bearing on $\Gamma(K \rightarrow \pi \pi \gamma)$
for the $K^0_S$, $K^0_L$ and $K^-$ mesons.\label{tab:exp}}
\begin{center}
\footnotesize
\begin{tabular}{|c|c|c|l|}
\hline
{} &\raisebox{0pt}[13pt][7pt]{$\Gamma(\pi^- \pi^0)\; s^{-1}$} &
\raisebox{0pt}[13pt][7pt]{$\Gamma(\pi^-\pi^0\gamma)\; s^{-1}$} &{}\\
\hline
\multicolumn{2}{|c|}{\raisebox{0pt}[12pt][6pt]{Process 
for Decay}} & &\\
\cline{1-2}
$K^-$   &$1.711 \times 10^7$ 
&\begin{minipage}{1in}
\begin{center}
$2.22 \times 10^4$ \\ (DE $ 1.46 \times 10^3)$
\end{center}\end{minipage} 
&\begin{minipage}{1.5in}
\phantom{xxx}
No (IB)-E1 interference seen but data shows excess events relative to IB over
$E^{\ast}_{\gamma} = 80$ to $100$~MeV
\end{minipage} \\[22pt]
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Figures}
If you wish to `embed' a postscript figure in the file, 
then remove the \% mark
from the declaration of the postscript figure {\em epsfbox} within the figure
description and change the filename to an appropriate one. Also
remove the comment \% mark from the {\em epsfxsize} command and specify the required width of the figure.
System will automatically enlarge or reduce the figure based on the {\em x-size} provided with {\em epsfxsize} command. 
You may need to play around with this as different computer systems
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command {\em figurebox}, which has three arguments. The thrid argument
is for actual figure name.

\begin{figure}[t]
\figurebox{20pc}{15pc}{} % to have a box alone
%\epsfxsize=10pc % will enlarge or reduce the postscript figures based on the xsize
%\epsfbox{xxx.eps} % postscript image file name
\caption{A generalized cactus tree: the confluent
transfer-matrix $S$ transforms the state function $f(x)$ and
$f(z)$ into $f(x)$.  \label{fig:radish}}
\end{figure}

Next adjust the scaling of the figure until it's correctly
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If you prefer to use some other method then it's very important
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The caption heading for a figure should be placed below the
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\subsection{Limitations on the Placement of Tables,
Equations and Figures}\label{sec:plac}
Very large figures and tables should be placed on a page by themselves. One
can use the instruction {\em $\backslash$begin\{figure\}$[$p$]$} or
{\em $\backslash$begin\{table\}$[$p$]$}
to position these, and they will appear on a separate page devoted to
figures and tables. We would recommend making any necessary
adjustments to the layout of the figures and tables
only in the final draft. It is also simplest to sort out line and
page breaks in the last stages.

\subsection{Acknowledgments, Appendices, Footnotes and the Bibliography}
If you wish to have acknowledgments to funding bodies etc.,
these may be placed in a separate section at the end of the
text, before the Appendices. This should not be numbered so use
{\em $\backslash$section$\ast$\{Acknowledgments\}}.

It's preferable to have no appendices in a brief article, but if more
than one is necessary then simply copy the
{\em $\backslash$section$\ast$\{Appendix\}}
heading and type in Appendix A, Appendix B etc. between the brackets.

Footnotes are denoted by a letter superscript
in the text,\footnote{Just like this one.} and references
are denoted by a number superscript.
We have used {\em $\backslash$bibitem} to produce the bibliography.
Citations in the text use the labels defined in the bibitem declaration,
for example, the first paper by Jarlskog~\cite{ja} is cited using the command
{\em $\backslash$cite\{ja\}}.

If you more commonly use the method of square brackets in the line of text
for citation than the superscript method,
please note that you need  to adjust the punctuation
so that the citation command appears after the punctuation mark.

\subsection{Final Manuscript}
The final hard copy that you send must be absolutely clean and unfolded.
It will be printed directly without any further editing. Use a printer
that has a good resolution (300 dots per inch or higher). There should
not be any corrections made on the printed pages, nor should adhesive
tape cover any lettering. Photocopies are not acceptable.

The manuscript will not be reduced or enlarged when filmed so please ensure
that indices and other small pieces of text are legible.

\section{Sample Text}
The following may be (and has been) described as `dangerously irrelevant'
physics. The Lorentz-invariant phase space integral for
a general n-body decay from a particle with momentum $P$
and mass $M$ is given by:
\begin{equation}
I((P - k_i)^2, m^2_i, M) = \frac{1}{(2 \pi)^5}\!
\int\!\frac{d^3 k_i}{2 \omega_i} \! \delta^4(P - k_i).
\label{eq:murnf}
\end{equation}
The only experiment on $K^{\pm} \ra \pi^{\pm} \pi^0 \gamma$ since 1976
is that of Bolotov {\it et al}.~\cite{bu} 
There are two necessary conditions required for any acceptable 
parametrization of the quark mixing matrix. The first is that
the matrix must be unitary, and the second is that it should
contain a CP violating phase $\delta$. In Sec.~\ref{subsec:wpp}
the connection between invariants (of form similar to J) and
unitarity relations will be examined further for the more
general $ n \times n $ case. The reason is that such a matrix is
not a faithful representation of the group, i.e.~it does not
cover all of the parameter space available. 
\begin{equation}
\begin{array}{rcl}
\bf{K} & = &  Im[V_{j, \alpha} {V_{j,\alpha + 1}}^*
{V_{j + 1,\alpha}}^* V_{j + 1, \alpha + 1} ] \\[4pt]
&&{}+ Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2}}^* V_{k + 1, \alpha + 3} ]  \\[4pt]
&&{}+ Im[V_{j + 2, \beta} {V_{j + 2,\beta + 1}}^*
{V_{j + 3,\beta}}^* V_{j + 3, \beta + 1} ]  \\[4pt]
&&{}+ Im[V_{k + 2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2}}^* V_{k + 3, \beta + 3}] \\[4pt]
& & \\
\bf{M} & = &  Im[{V_{j, \alpha}}^* V_{j,\alpha + 1}
V_{j + 1,\alpha} {V_{j + 1, \alpha + 1}}^* ]  \\[4pt]
&&{}+ Im[V_{k, \alpha + 2} {V_{k,\alpha + 3}}^*
{V_{k + 1,\alpha + 2}}^* V_{k + 1, \alpha + 3} ]  \\[4pt]
&&{}+ Im[{V_{j + 2, \beta}}^* V_{j + 2,\beta + 1}
V_{j + 3,\beta} {V_{j + 3, \beta + 1}}^* ]  \\[4pt]
&&{}+ Im[V_{k + 2, \beta + 2} {V_{k + 2,\beta + 3}}^*
{V_{k + 3,\beta + 2}}^* V_{k + 3, \beta + 3}],
\\ & &
\end{array}\label{eq:spa}
\end{equation}
where $ k = j$ or $j+1$ and $\beta = \alpha$ or $\alpha+1$, but if
$k = j + 1$, then $\beta \neq \alpha + 1$ and similarly, if
$\beta = \alpha + 1$ then $ k \neq j + 1$.\footnote{An example of a
matrix which has elements
containing the phase variable $e^{i \delta}$ to second order, 
i.e.~elements with a phase variable $e^{2i \delta}$ is given at
the end of this section.} There are only 162 quark mixing
matrices using these parameters which are 
to first order in the phase variable $e^{i \delta}$ as is the case for
the Jarlskog parametrizations, and for which J is not identically
zero.
It should be noted that these are physically identical and
form just one true parametrization.
\bea
T & = & Im[V_{11} {V_{12}}^* {V_{21}}^* V_{22}]  \nonumber \\[4pt]
&&{}+ Im[V_{12} {V_{13}}^* {V_{22}}^* V_{23}]   \nonumber \\[4pt]
&&{}- Im[V_{33} {V_{31}}^* {V_{13}}^* V_{11}].
\label{eq:sp}
\eea


\section*{Acknowledgments}
This is where one places acknowledgments for funding bodies etc.
Note that there are no section numbers for the Acknowledgments, Appendix
or References. Furthermore, the system will automatically generate the heading for 
the reference section.


\section*{Appendix}
We can insert an appendix here and place equations so that they
are given numbers such as Eq.~(\ref{eq:app}).
\be
x = y.
\label{eq:app}
\ee

\begin{thebibliography}{99}
\bibitem{ja}C Jarlskog in {\em CP Violation}, ed. C Jarlskog
(World Scientific, Singapore, 1988).

\bibitem{ma}L. Maiani, \Journal{\PLB}{62}{183}{1976}.

\bibitem{bu}J.D. Bjorken and I. Dunietz, \Journal{\PRD}{36}{2109}{1987}.

\bibitem{bd}C.D. Buchanan {\it et al}, \Journal{\PRD}{45}{4088}{1992}.

\end{thebibliography}

\end{document}

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