A study of QCD coupling constant\\ and power corrections in \\ the fixed target deep inelastic measurements \thanks{Talk presented at X Int. Workshop on Deep Inelastic Scattering (DIS2002), Cracow, 30 April - 4 May 2002. A shorter version will appear in the Proceedings.} % you can use '\\' to break lines
We reanalyze %high statistic deep inelastic scattering data of BCDMS Collaboration by including proper cuts of ranges with large systematic errors. We perform also the %combine fits of high statistic deep inelastic scattering data of BCDMS, SLAC, NM and BFP Collaborations taking the data separately and in combined way and find good agreement between these analyses. We %and extract the values of both the QCD coupling constant $\alpha_s(M^2_Z)$ up to NLO level and of the power corrections to the %proton structure function $F_2$.
dis02L.tex
\documentclass{appolb1}
\usepackage{epsfig}
% epsfig package included for placing EPS figures in the text
%------------------------------------------------------

\newcommand{\z}{&&\hspace*{-1cm}}
\newcommand{\zz}{&&\hspace*{-4cm}}
\newcommand{\ep}{\varepsilon}
\newcommand{\cita} [1] {$^{\hbox{\scriptsize \cite{#1}}}$}
%
\newcommand{\prepr}[1] {\begin{flushright} {\bf #1} \end{flushright}
  \vskip 1.5cm}
\newcommand{\titul}[1] {\begin{center}{\large\bf #1 }
\end{center}\vskip 1.cm}
\newcommand{\autor}[1] {\begin {center}
                {\large \lineskip .5em #1 } \end   {center} }
\newcommand{\lugar}[1] {\begin{center} {\it #1} \end{center}}
\newcommand{\abstr}[1] {{\begin{center} \vskip .5cm {\bf Abstract
                        \vspace{0pt}} \end{center}}\begin{quote} #1
                        \end{quote}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\MSbar}{\overline{\rm MS}}
\newcommand{\as}{\alpha_s}
\newcommand{\asMZ}{\alpha_s(M^2_Z)}
\newcommand{\ar}{\overline a_s}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                %
%    BEGINNING OF TEXT                           %
%                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
% \eqsec  % uncomment this line to get equations numbered by (sec.num)
\title{A study of QCD coupling constant\\ and power 
corrections  in \\
the fixed target deep inelastic measurements
\thanks{Talk presented at X Int. Workshop on Deep Inelastic Scattering 
(DIS2002), Cracow, 30 April - 4 May 2002. 
A shorter version will appear in the Proceedings.}
% you can use '\\' to break lines
}
\author{V.G. Krivokhijine and A.V. Kotikov
%Put here the name(s) of the Author(s)
\address{Joint Institute for Nuclear
Research, 141980 Dubna, Russia}
}
\maketitle
\begin{abstract}
We reanalyze 
%high statistic 
deep inelastic scattering data 
of BCDMS Collaboration by including proper cuts of  ranges
with large systematic errors. 
We perform also the 
%combine 
fits of high statistic deep inelastic scattering data 
of BCDMS, SLAC, NM and BFP Collaborations  
taking the data separately and in combined way and find good agreement
between these analyses. We
%and 
extract the values of both
the QCD coupling constant $\alpha_s(M^2_Z)$ up to
NLO level and of the power corrections to the 
%proton 
structure function $F_2$. 
\end{abstract}
%\PACS{12.38 Bx, 13.60 Hb, 06.20 Jr}

\section{ Introduction }


The deep inelastic scattering (DIS) leptons on hadrons is the basical
 process to study the values of the parton distribution functions (PDF)
which are universal (after choosing of factorization and renormalization 
schemes) and
can be used in other processes.
The accuracy of the present data for deep inelastic
%(DIS) 
structure functions (SF) reached the level at which
the $Q^2$-dependence of logarithmic QCD-motivated terms and power-like ones
%are observed and 
may be studied separately 
%(see, for example, the recent reviews 
(for a review, see 
%the recent papers 
\cite{Beneke} and references 
therein).


In the present paper we sketch the results of our analysis \cite{KriKo}
at the next-to-leading order (NLO)
%large experience in fits of data
of perturbative QCD for
the most known DIS SF 
$F_2(x,Q^2)$ 
taking into account experimental data \cite{SLAC1}-\cite{BFP} of
SLAC, NM,  BCDMS and BFP Collaborations.
We
stress the power-like effects, so-called twist-4 (i.e.
$\sim 1/Q^2$) 
%and twist-6 (i.e. $\sim 1/Q^4$) 
contributions.
To our purposes we represent the SF $F_2(x,Q^2)$ as the contribution
of the leading twist part $F_2^{pQCD}(x,Q^2)$ 
described by perturbative QCD, 
%including 
when the target mass corrections are taken into account,
and the  
nonperturbative part (``dynamical'' twist-four terms):
% $\sim 1/Q^2$):
\vskip -0.5cm
\begin{equation}
F_2(x,Q^2) 
\equiv F_2^{full}(x,Q^2)
=F_2^{pQCD}(x,Q^2)\,
%\left(
\Bigl(
1+\frac{\tilde h_4(x)}{Q^2}
\Bigr),
\label{1}
\end{equation}
%\vskip -0.2cm
where $\tilde h_4(x)$ is magnitude of twist-four terms.

Contrary to standard fits (see, for example, \cite{Al2000}- \cite{fits}) 
when the direct
numerical calculations based on 
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi 
(DGLAP) 
equation \cite{DGLAP} are used to evaluate structure functions, 
we use the exact solution of DGLAP equation
for the Mellin moments $M_n^{tw2}(Q^2)$ of
%$F_2^{full}(x,Q^2)$, $F_2^{pQCD}(x,Q^2)$ and
SF $F_2^{tw2}(x,Q^2)$
and
the subsequent reproduction of 
%$F_2^{k}(x,Q^2)$
$F_2^{full}(x,Q^2)$
%, $F_2^{pQCD}(x,Q^2)$  and/or $F_2^{tw2}(x,Q^2)$ 
at every needed $Q^2$-value with help of the Jacobi 
Polynomial expansion method 
\footnote{We 
%would like to 
note here that there is similar method 
\cite{Ynd}, based on Bernstein polynomials. The method has been used 
in the analyses at the NLO level in \cite{KaKoYaF}
and at the NNLO level in \cite{SaYnd}.}
\cite{Barker,Kri}
(see similar analyses at the NLO level 
\cite{Kri,Vovk}
and at the next-next-to-leading order (NNLO) level and above \cite{PKK}.



In this paper we 
do not present exact formulae of $Q^2$-dependence
of SF $F_2$ which are 
%also 
given in \cite{KriKo}. We note only that
the PDF 
%moments $M_{n}^{tw2}(Q^2)$ 
at some $Q^2_0$ is theoretical input of our analysis and 
the twist-four term $\tilde h_4(x)$
is considered as a set of free parameters (one constant
$\tilde h_4(x_i)$ per $x_i$-bin):
$\tilde h_4^{free}(x)=\sum_{i=1}^{I} \tilde h_4(x_i)$, 
where $I$ is the number of bins.


\vskip -0.3cm
\section{ 
Fits of $F_2$
%: procedure
}
\label{sec:form}



First of all, we choose the cut $Q^2 \geq 1$ GeV$^2$ in all our studies.
For $Q^2 < 1$ GeV$^2$, the applicability of twist expansion is very
questionable. 
Secondly, we choose  $Q^2_0$ = 90 GeV$^2$ ($Q^2_0$ = 20 GeV$^2$) for the 
nonsinglet (combine nonsinglet and singlet) evolution, i.e.
%and $Q^2_0$ = 20 GeV$^2$ for the combine nonsinglet and singlet evolution
quite large values of the normalization point
$Q^2_0$: our
perturbative formulae should be applicable at the value of
$Q^2_0$. Moreover, the higher order corrections $\sim \as^k(Q^2_0)$ 
and $\sim (\as(Q^2)-\as(Q^2_0))^k$
($k \geq 2$) should be
%, coming from normalization conditions of PDF, are 
less important at these
%higher 
$Q^2_0$ values.

%\noindent
We use MINUIT program \cite{MINUIT} for
minimization of 
$\chi^2(F_2)$.
We consider
%use 
free normalizations of data for different experiments. 
For the reference, we use the most stable deuterium BCDMS data
at the value of energy $E_0=200$ GeV 
($E_0$ is the initial energy lepton beam). 
%The usage 
Using other types of data as reference gives
negligible changes in our results. The usage of fixed normalization
for all data leads to fits with a bit worser $\chi^2$.


\vspace{-0.2cm}
\subsection { BCDMS ~~${}^{12}C + H_2 + D_2$ data }
%{\bf 4.1.} {\bf SLAC data}

We start our analysis with the most precise experimental data 
\cite{BCDMS1} obtained  by BCDMS muon
%$\mu h$ 
scattering experiment at the high $Q^2$ values.
The full set of data is 762 
%(607) 
points.


It is well known that the original analyses 
%of 
given by BCDMS Collaboration itself (see
also Ref. \cite{ViMi}) lead to quite small values 
 $\alpha_s(M^2_Z)=0.113$.
Although in some recent papers (see, for example, 
\cite{Al2000,H1})
more higher values of the coupling constant 
$\alpha_s(M^2_Z)$ have been observed, we think that
an additional reanalysis of BCDMS data should be very useful. 

Based on study \cite{Kri2} 
%(see also \cite{H1BCDMS}) 
we proposed in
\cite{KriKo} that 
the reason for small values
of $\alpha_s(M^2_Z)$ coming from BCDMS data was the existence of the subset
of the data having large systematic errors. 
We studied this subject by 
introducing several so-called $Y$-cuts 
\footnote{Hereafter we use the kinematical variable $Y=(E_0-E)/E_0$,
where 
%$E_0$ and 
$E$ is
%are initial and 
scattering energies of lepton.} 
%respectively.}
(see \cite{KriKo}): 
%\vspace{-1.5cm}
%\vskip -0.7cm
\bea
& &y \geq 0.14 \,\,~~~\mbox{ when }~~~ 0.3 < x \leq 0.4, ~~~~~~~ 
%\nonumber \\& &
y \geq 0.16 \,~~~\mbox{ when }~~~ 0.4 < x \leq 0.5 \nonumber \\
& &y \geq Y_{cut3} ~~~\mbox{ when }~~~ 0.5 < x \leq 0.6, ~~~~~~
%\nonumber \\ & &
y \geq Y_{cut4} ~~~\mbox{ when }~~~ 0.6 < x \leq 0.7 \nonumber \\
& &y \geq Y_{cut5} ~~~\mbox{ when }~~~ 0.7 < x \leq 0.8 
\label{cut}
\eea
%\vskip -0.3cm
\noindent
and
%We use 
several $N$ sets  for the cuts at $0.5 < x \leq 0.8$: 
%
%%%  Tab. 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
%& & & & & & & \\
$N$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
%& & & & & & & \\
\hline \hline
$Y_{cut3}$ & 0 & 0.14 & 0.16 & 0.16 & 0.18 & 0.22 & 0.23 \\  
%\hline
$Y_{cut4}$ & 0 & 0.16 & 0.18 & 0.20 & 0.20 & 0.23 & 0.24 \\
$Y_{cut5}$ & 0 & 0.20 & 0.20 & 0.22 & 0.22 & 0.24 & 0.25 \\
\hline
\end{tabular}
\caption{The values of $Y_{cut3}$, $Y_{cut4}$ and $Y_{cut5}$.
}\label{tab2}
\end{center}
\end{table}


%  Fig. 2,   Fig. 3
\begin{figure}[t]
\begin{minipage}[t]{0.48\linewidth}
\begin{center}
%\includegraphics[width=2.37in]{Fig3a.eps} \\
%\includegraphics[width=2.2in]{sistno1.ps} \\
\includegraphics[width=3.1in]{sistno1.ps} \\
\end{center}
\vskip -0.7cm
\caption{
The study of systematics at different $Y_{cut}$ values
in the fits based on nonsinglet evolution (i.e. when $x \geq 0.25$).
The inner (outer) error-bars show statistical (systematic) errors.
}
 \label{fig:3}
\end{minipage}%
\hspace{0.04\textwidth}%
\begin{minipage}[t]{0.48\linewidth}
\begin{center}
%\includegraphics[width=2.5in]{Fig4a.eps} \\
%\includegraphics[width=2.2in]{sistsi1.ps} \\
\includegraphics[width=3.1in]{sistsi1.ps} \\
\end{center}
\vskip -0.7cm
 \caption{
All other notes are as in Fig. 1 with two
 exceptions:  the fits based on combine 
%singlet and nonsinglet 
evolution
%no a $x_{cut}$ and  $Q_0^2=20$ GeV$^2$. Moreover, 
and the points $N_{Ycut}=1,2,3,4,5$ correspond the values
$N=1,2,4,5,6$ in the Table 1.
}\label{fig:4}
\end{minipage}
\end{figure}



\vskip -0.5cm
From 
%the Table 2 and 
the Figs. 1 and 2 we can see that the $\alpha_s$ values are obtained
for $N=1 \div 6$ of $Y_{cut3}$, $Y_{cut4}$ and $Y_{cut5}$ are very stable and
statistically consistent. 

\vskip -0.5cm
\subsection{ SLAC, BCDMS, NM and BFP data }
\label{subsec:revi1}
After these $Y$-cuts have been incorporated (with $N=6$) for BCDMS data, 
the full set of combine data is 1309 
%(797) 
points.
The results of the fits 
%for the coupling constant $\as(M_Z^2)$
are compiled in Summary.
% (see Eqs. (\ref{re2n})-(\ref{re2s})).





\begin{figure}[t]
%\begin{figure}[tb]
%\vskip -0.5cm
%
\begin{center}
\psfig{figure=ht1.ps,width=16cm,height=13cm}
%\psfig{figure=ht1.ps,width=11cm,height=7cm}
\end{center}
\vskip -1cm
\caption{
The values of the twist-four terms. The black and white points correspond 
to the small-$x$ asymptotics $\sim x^{-\omega}$ of sea quark and gluon
distributions
with $\omega =0$ and
$\omega =0.18$, respectively. The statistical errors are displayed only. 
}
\label{HT}
\end{figure}




%\vspace{-0.5cm}
\section{ Summary }


We
have demonstrated several steps of our study \cite{KriKo}
of the $Q^2$-evolution of DIS structure function $F_2$ fitting all
%modern 
fixed target experimental data. 


From the fits we have obtained the value of the normalization 
$\asMZ$
of QCD coupling constant. First of all, we have reanalyzed the BCDMS data 
cutting the range with large systematic errors. As it is possible to see
in 
the Figs. 1 and 2, 
the value of $\asMZ$ rises strongly when
the cuts of systematics were incorporated. In another side, 
the value of $\asMZ$ does not dependent on the concrete type of the
cut within 
%in the range of 
modern statistical errors.

Fitting SLAC, BCDMS, NM and BFP data,
we have found in \cite{KriKo} that at $Q^2 \geq 10 \div 15$ GeV$^2$ 
the formulae of pure perturbative
QCD (i.e. twist-two approximation together with target mass corrections)
are in good agreement with all data
\footnote{We note that at small $x$ values, the perturbative QCD
works well starting with $Q^2 = 1.5 \div 2$ GeV$^2$
and higher twist corrections are important only at very low $Q^2$:
$Q^2 \sim 0.5$ GeV$^2$ (see \cite{Q2evo,HT,Slope} and references therein).
As it is was observed in \cite{DoShi,bfklp} (see also discussions in
\cite{Q2evo,HT,BoAnd}) the good agreement between perturbative QCD and
experiment seems connect with large effective argument of coupling
constant at low $x$ range.}. 
When we have added twist-four corrections, we have very good agreement
between QCD (i.e. first two coefficients of Wilson expansion)
and the data starting already with $Q^2 = 1$ GeV$^2$, where the Wilson
expansion should begin
%start 
to be applicable.
 The 
results for  $\asMZ$ are very similar (see \cite{KriKo}) for the 
both types of analyses and have the following form:
\bea
\as(M_Z^2) &=& 0.1177 \pm 0.0007 ~\mbox{(stat)}
\pm 0.0021 ~\mbox{(syst)} \pm 0.0009 ~\mbox{(norm)}, 
\label{re2s}
\eea
where the symbols ``stat'', ``syst'' and 
``norm'' mark the 
statistical error, systematic one and the 
error of normalization of experimental data.



We would like to note that we have good agreement also with the analysis 
\cite{H1} of
combined H1 and BCDMS data, which has been given by H1 Collaboration very 
recently. 
Our results for $\as(M_Z^2)$ are in good agreement also with 
the average value for coupling constant,
%for $\asMZ$, 
presented in the recent studies (see \cite{Al2000,LEP}
and references therein) and in
famous 
%Altarelli and Bethke 
reviews \cite{Breview}.

%{\bf 3.} 
At the end of our paper we would like to discuss the contributions of 
%shortly two additional subjects: the IRR-model predictions to 
higher twist corrections. 
In our study \cite{KriKo}
%here 
we have reproduced well-known $x$-shape of
the twist-four corrections at the large and intermediate values of 
Bjorken variable $x$ (see the Fig. 3 and \cite{ViMi,KriKo}).
%also, for example, the results of very popular article \cite{ViMi}).

%We would like to note about 
Note that there is a small-$x$ rise of the magnitude of
twist-four corrections, when we use flat parton distributions at $x \to 0$.
As we have discussed in Ref \cite{KriKo},
%already in the Section 3,
there is a strong correlation 
\footnote{ 
This correlation comes because of very limited numbers of 
experimental data used here lie at the low $x$ region. Indeed, only the NMC
experimental data contribute there. 
We hope to incorporate the HERA data \cite{H1,ZEUS}
in our future investigations.}
between the small-$x$ behavior of twist-four
corrections and the type of the corresponding asymptotics of the 
leading-twist parton distributions. The possibility to have  a
singular type of
the asymptotics leads (in our fits)
to the appearance of the rise of sea quark and gluon 
distributions as $\sim x^{-0.18}$ at low $x$ values,
that is in full agreement with low $x$ HERA data
and with theoretical studies \cite{Q2evo,bfklp}.
 At this case
the rise of the magnitude
of twist-four corrections is completely
%fully 
canceled. This cancellation
is in full agreement
with theoretical and phenomenological studies 
%and low $x$ experimental data of H1 and L3 Collaborations 
%(see discussions in \cite{KriKo}). 
(see \cite{Q2evo,Bartels1,HT}).



\vskip 0.2cm
%
{\bf Acknowledgments.}
The study is supported in part by 
%the RFBR grant 02-02-17513 and  by 
the Heisenberg-Landau program.
%~~Authors
A.V.K.  would like to express his sincerely thanks to the Organizing
  Committee of the X Int. Workshop on Deep Inelastic Scattering 
(DIS2002)  for the kind 
invitation
%, the financial support at  such remarkable Conferences, 
and  for fruitful discussions.
%A.V.K. 
He was supported in part by Alexander von Humboldt
fellowship and INTAS  grant N366.



\vspace{-0.5cm}
\begin{thebibliography}{99}

\bibitem{Beneke} M. Beneke, Phys.Report. 317 (1999) 1;
M. Beneke, V.M. Braun, hep-ph/0010208. 
%
\bibitem{KriKo} A.V. Kotikov, V.G. Krivokhijine, Dubna preprint E2-2001-190 
(hep-ph/0108224);
%%CITATION = HEP-PH 0108224;%%
in: 
{\it Proceedings of the XVIth International Workshop 
``High Energy Physics and Quantum Field Theory``} (2001), Moscow 
(hep-ph/0206221);
%%CITATION = HEP-PH 0206221;%%
in: {\it Proc. of the Int.
Workshop ``Renormalization Group 2002''} (2002), High Tatras, Slovakia
(hep-ph/0207222).
%%CITATION = HEP-PH 0207222;%%
%
\bibitem{Stirling} K. Long et al., {\it in} Proc. of the 9th Int. Workshop 
on Deep Inelastic
Scattering, DIS 2001 (2001), Bologna
%, R. Nisius, W.J. Stirling, 
(hep-ph/0109092).
%Summary of Structure Function section on DIS2001.
%
\bibitem{SLAC1}
SLAC Collab., L.W. Whitlow et al., Phys.Lett.  B282 (1992) 475. 
SLAC report
357 (1990).
%
\bibitem{NMC}
NM Collab., M. Arneodo et al.,  Nucl. Phys.  
B483 (1997) 3. 
%
\bibitem{BCDMS1}
BCDMS Collab., A.C.~ Benevenuti et al., Phys.Lett. 
%{\bf B195}(1987) 91, 97; 
B223 (1989) 485; B237 (1990) 592; B195 (1987) 91; Preprints CERN-EP/89-06,
CERN-EP/89-170, Preprint CERN-EP/87-100.
%
\bibitem{BFP} BFP Collab.: P.D. Mayers et al., Phys.Rev. 
D34 (1986) 1265.
%
\bibitem{Al2000}
S.~I.~Alekhin, 
Eur.Phys.J. C10 (1999) 395;
Phys.Rev. D59 (1999) 114016;
D63 (2001) 094022; Phys.Lett. B519 (2001) 57.
%Preprint IHEP 01-30, Protvino, 2000 (hep-ph/0107197).
%
\bibitem{ViMi} M. Virchaux, A. Milsztajn, Phys.Lett. B274 (1992) 221.
%
\bibitem{fits} A.D. Martin et al., 
%R.G. Roberts, W.J. Stirling and R.S. Thorne, 
Eur.Phys.J. C14 (2000) 133;
M. Glueck et al., 
%E. Reya and A. Vogt, 
Eur.Phys.J. C5 (1998) 461;
STEQ Collab., H.Lai et al., 
Eur.Phys.J. C12 (2000) 375.
%
%
\bibitem{DGLAP}  V.N.~Gribov, L.N.~Lipatov, Sov.J.Nucl.Phys.
{\bf 15} (1972) 438;
%%CITATION = YAFIA,15,781;%%
{\bf 15} (1972) 675;
%%CITATION = YAFIA,15,1218;%% 
L.N.~Lipatov, Sov.J.Nucl.Phys. 20 (1975) 94;
%%CITATION = SJNCA,20,94;%%
G.~Altarelli, G. Parisi, Nucl. Phys. B126 (1977) 298;
 Yu.L.~Dokshitzer, JETP 46 (1977) 641.
%
\bibitem{Barker}
G.~Parisi, N.~Sourlas, Nucl. Phys. B151 (1979) 421;
 I.S.~Barker et al., 
%C.B.~Langensiepen and  G.~Shaw,  
Nucl. Phys. B186 (1981) 61;
%\\  I.S.~Barker, B.R.~Martin, and  G.~Shaw,
Z.Phys. C19 (1983) 147;
%\\ I.S.~Barker and  B.R.~Martin, Z. Phys. 
C24 (1984) 255.
%
\bibitem{Kri}
V.G.~Krivokhizhin et al.,
%S.P.~Kurlovich, V.V.~Sanadze, I.A.~Savin, A.V.~Sidorov and
%N.B.~Skachkov, 
Z.Phys. C36 (1987) 51;
%\vspace{-0.2cm}
 Z. Phys.
C48 (1990) 347.
%
\bibitem{Vovk}
V.I.~ Vovk, Z.Phys. C47 (1990) 57;
A.V.~ Kotikov, 
%et al., 
G.~ Parente and J.~ Sanchez Guillen,  
Z.Phys. C58 (1993) 465;
%%CITATION = ZEPYA,C58,465;%%
%
%\bibitem{KKDIS}
A.V. Kotikov and V.G. Krivokhijine, {\it in} Proc. of the 6th Int. 
Workshop on Deep Inelastic
Scattering, DIS 98 (1998), Brussels (hep-ph/9805353).
%%CITATION = HEP-PH 9805353;%%
%
\bibitem{PKK}
G.~Parente et al., 
%A.V.~Kotikov and  V.G.~Krivokhizhin,
Phys.Lett.  B333 (1994) 190;
%%CITATION = HEP-PH 9405290;%%
%
%\bibitem{KKPS1}
A.L.~Kataev et al., 
%A.V.~Kotikov, G.~Parente and A.V.~Sidorov,
Phys.Lett. B388 (1996) 179;
%%CITATION = HEP-PH 9605367;%%
%
%\bibitem{KKPS2}
%A.L.~Kataev, A.V.~Kotikov, G.~Parente and A.V.~Sidorov,
Phys. Lett. 
B417 (1998) 374;
%%CITATION = HEP-PH 9706534;%%
%Nucl.Phys.Proc.Suppl. B64 (1998) 138;
hep-ph/9709509;
%%CITATION = HEP-PH 9709509;%%
%
%\bibitem{KPS}
%A.L.~Kataev, G.~Parente and A.V.~Sidorov,
Nucl.Phys. B573 (2000) 405.
%
%\bibitem{KPS1}
%A.L.~Kataev, G.~Parente and A.V.~Sidorov,
%Preprint CERN-TH/2001-58 (hep-ph/0106221).
%
%\bibitem{Buras}
%A.~Buras,  Rev.Mod.Phys. 52 (1980) 199.

%\bibitem{Yndu}
%F.J. Yndurain, Quantum Chromodynamics (An Introduaction to the Theory
%of Quarks and Gluons).-Berlin, Springer-Verlag (1983).
%
%\bibitem{YF93}
%A.V.~Kotikov,  Phys. Atom. Nucl. {\bf 56} (1993) 1276.
%%%CITATION = PANUE,56,1276;%%
%%[ Yad. Fiz. {\bf 56} (1993) N9, 217].
%
%
\bibitem{Ynd} 
F.J. Yndurain, Phys.Lett. B74 (1978) 68.
%
\bibitem{KaKoYaF} 
B. Escobles et al., 
%M.J. Herrero, C. Lopez, and F.J. Yndurain, 
Nucl.Phys. B242 (1984) 329;
D.I. Kazakov, A.V. Kotikov,
 Yad.\ Fiz.\  46 (1987) 1767 
[Sov.J.Nucl.Phys. 46 (1987) 1057];
%%CITATION = YAFIA,46,1767;%%
%
%\bibitem{KaKo} D.I. Kazakov and A.V. Kotikov,
%Theor.Math.Phys. {\bf 73} (1987) 1264;
Nucl.Phys. B307 (1988) 721;
%%CITATION = NUPHA,B307,721;%%
B345 (1990) 299
%%CITATION = NUPHA,B345,299;%%
%[Yad.\ Fiz.\  {\bf 46} (1987) 1767].
%
\bibitem{SaYnd} 
J. Santiago, F.J. Yndurain, Nucl.Phys. B563 (1999) 45.
Nucl. Phys. {\bf B611} (2001) 447.
%
\bibitem{MINUIT}
F.~James, M.Ross, ``MINUIT'', CERN Computer Center Library, 
D 505, Geneve, 1987.
%
\bibitem{H1} H1 Collab.: C. Adloff et al., 
Eur. Phys. J. C21 (2001) 33.
%Preprint DESY-00-181 (hep-ex/0012053).
%
\bibitem{Kri2}
V. Genchev et al.,  {\it in} Proc. Int. Conference of Problems of High Energy
Physics (1988), Dubna, V.2., p.6;
%V.G.~Krivokhizhin et al., ????\\
%
%\bibitem{Syst}
A. Milsztaijan et al., Z.Phys. C49 (1991) 527.
%
%\bibitem{H1} H1 Collab.: C. Adloff et al.,
%Eur. Phys. J. {\bf C21} (2001) 33.
%\vspace{-0.2cm}
%
%\bibitem{ZEUS} ZEUS Collab.: S. Chekanov et al., 
%Eur. Phys. J. {\bf C21} (2001) 443.
%\vspace{-0.2cm}
%
\bibitem{Q2evo} A.V. Kotikov and G. Parente,
Nucl. Phys. B549 (1999) 242; 
%%CITATION = HEP-PH 9807249;%%
%Nucl. Phys. (Proc. Suppl.) {\bf 99A} (2001) 196 (hep-ph/0010352); 
hep-ph/0010352;
%%CITATION = HEP-PH 0010352;%%
{\it in} Proc. of the Int. Conference PQFT98 (1998), Dubna
(hep-ph/9810223);
%%CITATION = HEP-PH 9810223;%%
{\it in} Proc. of the 8th Int. Workshop on Deep Inelastic
Scattering, DIS 2000 (2000), Liverpool, p. 198 
(hep-ph/0006197).
%%CITATION = HEP-PH 0006197;%%
%
%
%\bibitem{Bartels1}
%J. Bartels, K. Golec-Biernat, K. Peters,
%Eur. Phys. J. {\bf C17} (2000) 121.
%\vspace{-0.2cm}
%
%
\bibitem{HT} A.V. Kotikov and G. Parente,
 {\it in} Proc. Int. Seminar Relativistic Nuclear Physics and Quantum 
Chromodynamics (2000), Dubna
(hep-ph/0012299); 
%%CITATION = HEP-PH 0012299;%%
{\it in} Proc. of the 9th Int. Workshop on Deep Inelastic
Scattering, DIS 2001 (2001), Bologna 
(hep-ph/0106175).
%%CITATION = HEP-PH 0106175;%%
%
\bibitem{Slope} A.V. Kotikov and G. Parente, Preprint US-FT/3-02
(hep-ph/0207276). 
%%CITATION = HEP-PH 0207276;%%
%
\bibitem{DoShi}
Yu.L. Dokshitzer, D.V. Shirkov, 
Z. Phys. C67 (1995) 449;
%
%\bibitem{Rsmallx} 
A.V. Kotikov, JETP Lett. 59 (1994) 1;
Phys. Lett. B338 (1994) 349.
%%CITATION = PHLTA,B338,349;%%
%
%
%\bibitem{Wong}  
W.K.~Wong, 
Phys. Rev.~D54 (1996) 1094
%%
\bibitem{bfklp}  S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov and G.B.
Pivovarov, JETP. Lett. {\bf 70} (1999) 155; 
in: {\it Proc. of the
PHOTON2001}, Ascona, Switzerland, 2001 (CERN-TH/2001-341, SLAC-PUB-9069,
hep-ph/0111390);
%\newline
V.T. Kim, L.N. Lipatov and G.B. Pivovarov, in: {\it Proc. of the
VIIIth Blois Workshop at IHEP}, Protvino, Russia, 1999 (IITAP-99-013,
hep-ph/9911228); %%CITATION = HEP-PH 9911228;%%
in: {\it Proc. of the Symposium on Multiparticle Dynamics (ISMD99)},
Providence, Rhode Island, 1999 (IITAP-99-014, hep-ph/9911242).
%%CITATION = HEP-PH 9911242;%%
%
%\bibitem{bfklp1}  S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov and G.B.
%Pivovarov, 
%in: {\it Proc. of the
%PHOTON2001}, Ascona, Switzerland, 2001 (CERN-TH/2001-341, SLAC-PUB-9069,
hep-ph/0111390.
%
\bibitem{BoAnd}  Bo Andersson et al., hep-ph/0204115.
%%CITATION = HEP-PH 0204115;%%
%%
%\bibitem{NeVo} 
%W.L. van Neerven, A. Vogt, Nucl.Phys. B568 (2000) 263; B603 (2001) 42;
%
%%\bibitem{NeVo1} 
%W.L. van Neerven and A. Vogt, Nucl. Phys. B588 (2000) 345.
%
\bibitem{LEP} G. Dissertori, hep-ex/0105070;
T. Affolder et al., hep-ex/0108034;
%
%\bibitem{Bethke} 
P.A. Movilla Fernandez et al., 
%S. Bethke, O.Biebel and S. Kluth,
Preprint MPI-Ph/2001-005 (hep-ex/0105059).
%
\bibitem{Breview}
G. Altarelli, hep-ph/0204179;
S. Bethke, J.Phys. G26   (2000) R27 (hep-ex/0004021).
%
%
%\bibitem{DELPHI}
%DELPHI Collab., P. Abreu et al., Eur. Phys. J. {\bf C6} (1999) 19.
%
%\bibitem{Al1999}
%S.~I.~Alekhin, 
%Phys. Rev. {\bf D59} (1999) 114016.
%
\bibitem{ZEUS} ZEUS Collab.: S. Chekanov et al., 
Eur. Phys. J. {\bf C21} (2001) 443.
%
\bibitem{Bartels1}
J. Bartels, K. Golec-Biernat, K. Peters,
Eur. Phys. J. {\bf C17} (2000) 121;
J. Blumlein, V. Ravindran, Jian-hong Ruan and Wei Zhu, 
Phys. Lett. {\bf B504} (2001) 235. 
%
\end{thebibliography}


\end{document}



hep-ph0208188.extracted_bib
\begin{thebibliography}{99}

\bibitem{Beneke} M. Beneke, Phys.Report. 317 (1999) 1;
M. Beneke, V.M. Braun, hep-ph/0010208. 
%
\bibitem{KriKo} A.V. Kotikov, V.G. Krivokhijine, Dubna preprint E2-2001-190 
(hep-ph/0108224);
%%CITATION = HEP-PH 0108224;%%
in: 
{\it Proceedings of the XVIth International Workshop 
``High Energy Physics and Quantum Field Theory``} (2001), Moscow 
(hep-ph/0206221);
%%CITATION = HEP-PH 0206221;%%
in: {\it Proc. of the Int.
Workshop ``Renormalization Group 2002''} (2002), High Tatras, Slovakia
(hep-ph/0207222).
%%CITATION = HEP-PH 0207222;%%
%
\bibitem{Stirling} K. Long et al., {\it in} Proc. of the 9th Int. Workshop 
on Deep Inelastic
Scattering, DIS 2001 (2001), Bologna
%, R. Nisius, W.J. Stirling, 
(hep-ph/0109092).
%Summary of Structure Function section on DIS2001.
%
\bibitem{SLAC1}
SLAC Collab., L.W. Whitlow et al., Phys.Lett.  B282 (1992) 475. 
SLAC report
357 (1990).
%
\bibitem{NMC}
NM Collab., M. Arneodo et al.,  Nucl. Phys.  
B483 (1997) 3. 
%
\bibitem{BCDMS1}
BCDMS Collab., A.C.~ Benevenuti et al., Phys.Lett. 
%{\bf B195}(1987) 91, 97; 
B223 (1989) 485; B237 (1990) 592; B195 (1987) 91; Preprints CERN-EP/89-06,
CERN-EP/89-170, Preprint CERN-EP/87-100.
%
\bibitem{BFP} BFP Collab.: P.D. Mayers et al., Phys.Rev. 
D34 (1986) 1265.
%
\bibitem{Al2000}
S.~I.~Alekhin, 
Eur.Phys.J. C10 (1999) 395;
Phys.Rev. D59 (1999) 114016;
D63 (2001) 094022; Phys.Lett. B519 (2001) 57.
%Preprint IHEP 01-30, Protvino, 2000 (hep-ph/0107197).
%
\bibitem{ViMi} M. Virchaux, A. Milsztajn, Phys.Lett. B274 (1992) 221.
%
\bibitem{fits} A.D. Martin et al., 
%R.G. Roberts, W.J. Stirling and R.S. Thorne, 
Eur.Phys.J. C14 (2000) 133;
M. Glueck et al., 
%E. Reya and A. Vogt, 
Eur.Phys.J. C5 (1998) 461;
STEQ Collab., H.Lai et al., 
Eur.Phys.J. C12 (2000) 375.
%
%
\bibitem{DGLAP}  V.N.~Gribov, L.N.~Lipatov, Sov.J.Nucl.Phys.
{\bf 15} (1972) 438;
%%CITATION = YAFIA,15,781;%%
{\bf 15} (1972) 675;
%%CITATION = YAFIA,15,1218;%% 
L.N.~Lipatov, Sov.J.Nucl.Phys. 20 (1975) 94;
%%CITATION = SJNCA,20,94;%%
G.~Altarelli, G. Parisi, Nucl. Phys. B126 (1977) 298;
 Yu.L.~Dokshitzer, JETP 46 (1977) 641.
%
\bibitem{Barker}
G.~Parisi, N.~Sourlas, Nucl. Phys. B151 (1979) 421;
 I.S.~Barker et al., 
%C.B.~Langensiepen and  G.~Shaw,  
Nucl. Phys. B186 (1981) 61;
%\\  I.S.~Barker, B.R.~Martin, and  G.~Shaw,
Z.Phys. C19 (1983) 147;
%\\ I.S.~Barker and  B.R.~Martin, Z. Phys. 
C24 (1984) 255.
%
\bibitem{Kri}
V.G.~Krivokhizhin et al.,
%S.P.~Kurlovich, V.V.~Sanadze, I.A.~Savin, A.V.~Sidorov and
%N.B.~Skachkov, 
Z.Phys. C36 (1987) 51;
%\vspace{-0.2cm}
 Z. Phys.
C48 (1990) 347.
%
\bibitem{Vovk}
V.I.~ Vovk, Z.Phys. C47 (1990) 57;
A.V.~ Kotikov, 
%et al., 
G.~ Parente and J.~ Sanchez Guillen,  
Z.Phys. C58 (1993) 465;
%%CITATION = ZEPYA,C58,465;%%
%
%\bibitem{KKDIS}
A.V. Kotikov and V.G. Krivokhijine, {\it in} Proc. of the 6th Int. 
Workshop on Deep Inelastic
Scattering, DIS 98 (1998), Brussels (hep-ph/9805353).
%%CITATION = HEP-PH 9805353;%%
%
\bibitem{PKK}
G.~Parente et al., 
%A.V.~Kotikov and  V.G.~Krivokhizhin,
Phys.Lett.  B333 (1994) 190;
%%CITATION = HEP-PH 9405290;%%
%
%\bibitem{KKPS1}
A.L.~Kataev et al., 
%A.V.~Kotikov, G.~Parente and A.V.~Sidorov,
Phys.Lett. B388 (1996) 179;
%%CITATION = HEP-PH 9605367;%%
%
%\bibitem{KKPS2}
%A.L.~Kataev, A.V.~Kotikov, G.~Parente and A.V.~Sidorov,
Phys. Lett. 
B417 (1998) 374;
%%CITATION = HEP-PH 9706534;%%
%Nucl.Phys.Proc.Suppl. B64 (1998) 138;
hep-ph/9709509;
%%CITATION = HEP-PH 9709509;%%
%
%\bibitem{KPS}
%A.L.~Kataev, G.~Parente and A.V.~Sidorov,
Nucl.Phys. B573 (2000) 405.
%
%\bibitem{KPS1}
%A.L.~Kataev, G.~Parente and A.V.~Sidorov,
%Preprint CERN-TH/2001-58 (hep-ph/0106221).
%
%\bibitem{Buras}
%A.~Buras,  Rev.Mod.Phys. 52 (1980) 199.

%\bibitem{Yndu}
%F.J. Yndurain, Quantum Chromodynamics (An Introduaction to the Theory
%of Quarks and Gluons).-Berlin, Springer-Verlag (1983).
%
%\bibitem{YF93}
%A.V.~Kotikov,  Phys. Atom. Nucl. {\bf 56} (1993) 1276.
%%%CITATION = PANUE,56,1276;%%
%%[ Yad. Fiz. {\bf 56} (1993) N9, 217].
%
%
\bibitem{Ynd} 
F.J. Yndurain, Phys.Lett. B74 (1978) 68.
%
\bibitem{KaKoYaF} 
B. Escobles et al., 
%M.J. Herrero, C. Lopez, and F.J. Yndurain, 
Nucl.Phys. B242 (1984) 329;
D.I. Kazakov, A.V. Kotikov,
 Yad.\ Fiz.\  46 (1987) 1767 
[Sov.J.Nucl.Phys. 46 (1987) 1057];
%%CITATION = YAFIA,46,1767;%%
%
%\bibitem{KaKo} D.I. Kazakov and A.V. Kotikov,
%Theor.Math.Phys. {\bf 73} (1987) 1264;
Nucl.Phys. B307 (1988) 721;
%%CITATION = NUPHA,B307,721;%%
B345 (1990) 299
%%CITATION = NUPHA,B345,299;%%
%[Yad.\ Fiz.\  {\bf 46} (1987) 1767].
%
\bibitem{SaYnd} 
J. Santiago, F.J. Yndurain, Nucl.Phys. B563 (1999) 45.
Nucl. Phys. {\bf B611} (2001) 447.
%
\bibitem{MINUIT}
F.~James, M.Ross, ``MINUIT'', CERN Computer Center Library, 
D 505, Geneve, 1987.
%
\bibitem{H1} H1 Collab.: C. Adloff et al., 
Eur. Phys. J. C21 (2001) 33.
%Preprint DESY-00-181 (hep-ex/0012053).
%
\bibitem{Kri2}
V. Genchev et al.,  {\it in} Proc. Int. Conference of Problems of High Energy
Physics (1988), Dubna, V.2., p.6;
%V.G.~Krivokhizhin et al., ????\\
%
%\bibitem{Syst}
A. Milsztaijan et al., Z.Phys. C49 (1991) 527.
%
%\bibitem{H1} H1 Collab.: C. Adloff et al.,
%Eur. Phys. J. {\bf C21} (2001) 33.
%\vspace{-0.2cm}
%
%\bibitem{ZEUS} ZEUS Collab.: S. Chekanov et al., 
%Eur. Phys. J. {\bf C21} (2001) 443.
%\vspace{-0.2cm}
%
\bibitem{Q2evo} A.V. Kotikov and G. Parente,
Nucl. Phys. B549 (1999) 242; 
%%CITATION = HEP-PH 9807249;%%
%Nucl. Phys. (Proc. Suppl.) {\bf 99A} (2001) 196 (hep-ph/0010352); 
hep-ph/0010352;
%%CITATION = HEP-PH 0010352;%%
{\it in} Proc. of the Int. Conference PQFT98 (1998), Dubna
(hep-ph/9810223);
%%CITATION = HEP-PH 9810223;%%
{\it in} Proc. of the 8th Int. Workshop on Deep Inelastic
Scattering, DIS 2000 (2000), Liverpool, p. 198 
(hep-ph/0006197).
%%CITATION = HEP-PH 0006197;%%
%
%
%\bibitem{Bartels1}
%J. Bartels, K. Golec-Biernat, K. Peters,
%Eur. Phys. J. {\bf C17} (2000) 121.
%\vspace{-0.2cm}
%
%
\bibitem{HT} A.V. Kotikov and G. Parente,
 {\it in} Proc. Int. Seminar Relativistic Nuclear Physics and Quantum 
Chromodynamics (2000), Dubna
(hep-ph/0012299); 
%%CITATION = HEP-PH 0012299;%%
{\it in} Proc. of the 9th Int. Workshop on Deep Inelastic
Scattering, DIS 2001 (2001), Bologna 
(hep-ph/0106175).
%%CITATION = HEP-PH 0106175;%%
%
\bibitem{Slope} A.V. Kotikov and G. Parente, Preprint US-FT/3-02
(hep-ph/0207276). 
%%CITATION = HEP-PH 0207276;%%
%
\bibitem{DoShi}
Yu.L. Dokshitzer, D.V. Shirkov, 
Z. Phys. C67 (1995) 449;
%
%\bibitem{Rsmallx} 
A.V. Kotikov, JETP Lett. 59 (1994) 1;
Phys. Lett. B338 (1994) 349.
%%CITATION = PHLTA,B338,349;%%
%
%
%\bibitem{Wong}  
W.K.~Wong, 
Phys. Rev.~D54 (1996) 1094
%%
\bibitem{bfklp}  S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov and G.B.
Pivovarov, JETP. Lett. {\bf 70} (1999) 155; 
in: {\it Proc. of the
PHOTON2001}, Ascona, Switzerland, 2001 (CERN-TH/2001-341, SLAC-PUB-9069,
hep-ph/0111390);
%\newline
V.T. Kim, L.N. Lipatov and G.B. Pivovarov, in: {\it Proc. of the
VIIIth Blois Workshop at IHEP}, Protvino, Russia, 1999 (IITAP-99-013,
hep-ph/9911228); %%CITATION = HEP-PH 9911228;%%
in: {\it Proc. of the Symposium on Multiparticle Dynamics (ISMD99)},
Providence, Rhode Island, 1999 (IITAP-99-014, hep-ph/9911242).
%%CITATION = HEP-PH 9911242;%%
%
%\bibitem{bfklp1}  S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov and G.B.
%Pivovarov, 
%in: {\it Proc. of the
%PHOTON2001}, Ascona, Switzerland, 2001 (CERN-TH/2001-341, SLAC-PUB-9069,
hep-ph/0111390.
%
\bibitem{BoAnd}  Bo Andersson et al., hep-ph/0204115.
%%CITATION = HEP-PH 0204115;%%
%%
%\bibitem{NeVo} 
%W.L. van Neerven, A. Vogt, Nucl.Phys. B568 (2000) 263; B603 (2001) 42;
%
%%\bibitem{NeVo1} 
%W.L. van Neerven and A. Vogt, Nucl. Phys. B588 (2000) 345.
%
\bibitem{LEP} G. Dissertori, hep-ex/0105070;
T. Affolder et al., hep-ex/0108034;
%
%\bibitem{Bethke} 
P.A. Movilla Fernandez et al., 
%S. Bethke, O.Biebel and S. Kluth,
Preprint MPI-Ph/2001-005 (hep-ex/0105059).
%
\bibitem{Breview}
G. Altarelli, hep-ph/0204179;
S. Bethke, J.Phys. G26   (2000) R27 (hep-ex/0004021).
%
%
%\bibitem{DELPHI}
%DELPHI Collab., P. Abreu et al., Eur. Phys. J. {\bf C6} (1999) 19.
%
%\bibitem{Al1999}
%S.~I.~Alekhin, 
%Phys. Rev. {\bf D59} (1999) 114016.
%
\bibitem{ZEUS} ZEUS Collab.: S. Chekanov et al., 
Eur. Phys. J. {\bf C21} (2001) 443.
%
\bibitem{Bartels1}
J. Bartels, K. Golec-Biernat, K. Peters,
Eur. Phys. J. {\bf C17} (2000) 121;
J. Blumlein, V. Ravindran, Jian-hong Ruan and Wei Zhu, 
Phys. Lett. {\bf B504} (2001) 235. 
%
\end{thebibliography}