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% -----------------------------simplifications----------------------
\def\itx{{\it x~}}             % x in Italian
\def\qq2{$Q^2$}               % Q2
\def\aa1{$A_1(x,Q^2)$}        % A1
\def\ff1{$F_1(x,Q^2)$}        % F1
\def\gg1{$g_1(x,Q^2)$}        % g1
\def\fmv{first moment~}    %
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\begin{document}
\vskip 3cm
\begin{center}
{\Large{\bf
%On
The $Q^2$ dependence of the measured asymmetry $A_1$ \\
from the similarity of $g_1(x,Q^2)$ and  $F_3(x,Q^2)$
structure functions 
}}
\end{center}
\vskip 1.7cm
\begin{center}
{\bf A.V.Kotikov}
\footnote{
~E-mail: kotikov@sunse.jinr.ru; Anatoli.Kotikov@cern.ch}
%,
%{\bf A.P.Nagaitsev}
%\footnote{~E-mail:
%nagajcev@hermes.desy.de;
%nagajcev@sunse.jinr.ru} 
and {\bf D.V.Peshekhonov}
\footnote{
%~Present address: ???, Milano, Italy;
~E-mail: Dimitri.Pechekhonov@infn.trieste.it;
Dmitri.Pechekhonov@cern.ch\\
~presently at INFN sezione di Trieste, Italia}
\\
\vskip 0.5cm

{\it Particle Physics Laboratory\\ Joint Institute for Nuclear Research\\ 
141980 Dubna, Russia.}
\end{center}
\vskip 3cm
{\large{\bf Abstract}}\\
We propose a new approach for taking into account the $Q^2$ dependence of
measured asymmetry $A_1$.
This approach is based on the similarity of the $Q^2$ behaviour 
and the shape of 
the spin-dependent structure function $g_1(x,Q^2)$ and spin averaged 
structure function $F_3(x,Q^2)$. 
%at at kinematical range $x > 0.01$.\\
The analysis is applied on 
%of the SMC 
available experimental data.
\vskip .5cm \hskip -.56cm
PACS number(s): 13.60.Hb, 11.55.Hx, 13.88.+e \\
%\end{flushleft}
\newpage
\section{Introduction}
%\hskip -.56cm

In a recent years there has been a significant progress in the study 
%The main aim of the experiments \cite{SMCp}-\cite{Q2E154} is the measurement 
of the spin-dependent structure function (SF)
 $g_1(x,Q^2)$ (see \cite{SMCp}-\cite{Q2E154}).
% versus $x$~ and $Q^2$ 
%and test of the Bjorken \cite{Bj} and Ellis-Jaffe \cite{EJ} sum rules.
The direct measurement of SF $g_1$ is very elaborate procedure (see, however,
\cite{Gagu})
%for this type of the analysis)
%\\ The value of $g_1(x,Q^2)$  
and ordinary its value is extracted from the  spin dependent 
asymmetry $A_1$ (see, for example \cite{AEL, Ram}) in agreement with the 
following formula:
\begin{eqnarray}
g_1(x,Q^2)~=~A_1(x,Q^2) \cdot F_1(x,Q^2),
%\cdot 
%(1+ \frac{4M^2x^2}{Q^2}),
\label{a}
\end{eqnarray}
where $F_1(x,Q^2)$~is the spin average SF.\\
%and $M$ is the proton mass. \\
\hskip -.56cm
The asymmetry $A_1(x,Q^2)$ is
%corresponds
closely connected with the ratio of
polarized and unpolarized cross-sections and may be "easy"
measured due to cancelation of many experimental uncertantities.
Experimentally asymmetry is extracting only at  few points 
$Q^2_{1i}, ...,Q^2_{ni}$ for each $x_i$ bin. To study the properties of
$g_1(x,Q^2)$ and to calculate the values of  spin dependent
sum rules \cite{Bj,EJ} we have
%leads to the necessity
to know the $A_1$ as a function of $Q^2$.\\ 
%values at $Q^2$ {\it independent} of $x_i$.\\
\hskip -.56cm
The most popular
%conventional
assumption applied on $A_1$ \cite{EK93} is 
\begin{eqnarray}
A_1(x,Q^2) = A_1(x), \label{a1}
\end{eqnarray}
It means that SF $g_1$ and  $F_1$ have the 
same $Q^2$ dependences 
%The assumption (\ref{a1}) is in agreement with the modern experimental
%data but it
what does not  following from the theory. 
On the contrary, the 
behaviour of $F_1$ and $g_1$ as a functions of $Q^2$ is expected to be 
different due to the difference between polarized and unpolarized splitting 
functions\footnote{except the leading order of the quark-quark interaction.}.\\
%
There are several approaches \cite{Q2A}-\cite{NST} how to take into account 
the $Q^2$ dependence of $A_1$.
%pdv% mostly 
They are 
% pdv end %
based on the different approximate solutions of the DGLAP equations.
Some of them have been used already by Spin Muon Collaboration (SMC)
and E154 Collaboration in the last analyses of experimental data
(see \cite{SMC} and \cite{Q2E154}, respectively). These
approaches \cite{Q2A}-\cite{NST} lead to similar results in 
$g_1(x,Q^2)$\footnote{ The form of the $Q^2$-dependence
for $A_1$ is different in approaches \cite{Q2A}-\cite{NST}. However, all of
them are in agreement with the weak $Q^2$-dependence at moderate $x$ 
and quite strong at small $x$ region.
%(and quite small $Q^2$ values,where experimental data are avaluable).
} which are contrast with the calculations
%results
based on Eq.(\ref{a1}).\\
%For example, the difference between new E154
%neutron data of $g_1$, obtained in \cite{Q2E254} with help of (\ref{1a})
%and DGLAP evolution is less then but comparable with experimental 
%uncertanties. \\

In this article we suggest another method how to take into account $Q^2$
dependence of $A_1$
which is based on the 
observation that the splitting functions of  the DGLAP equations for
the SF $g_1$ and  $F_3$ and  the shapes of these SF 
are similar in a wide $x$ range.
%and, thus, the $Q^2$ 
%dependence of them has to be close as a consequence. 
Our approach allows to set  $Q^2$-dependence of $A_1$ in a very simple way 
(see Eq.(\ref{5})) and leads to the results
(some of them have been recently presented \cite{DIS98} on the Workshop
DIS98), which are similar 
with ones based on the  DGLAP evolution.\\


\section{The $Q^2$ dependence of 
%$F_3(x,Q^2)$ and $g_1(x,Q^2)$
structure functions}

%\hskip -.56cm
Lets consider the $Q^2$ evolution of 
nonsinglet (NS) and singlet (SI) parts of the SF separately.\\

\hskip -.56cm
For the SF $F_3$ and
the nonsinglet (NS) parts of $g_1$ and $F_1$
the corresponding DGLAP equations can be presented
as\footnote{We use $\alpha(Q^2)= \alpha_s(Q^2)/{4 \pi}$.} as:\\
\bea
{dg_1^{NS}(x,Q^2) \over dlnQ^2} &=& -{1 \over 2} \gamma_{NS}^-(x, \alpha)
\otimes g_1^{NS}(x,Q^2),
\nonumber \\
%
{dF_1^{NS}(x,Q^2) \over dlnQ^2} &=& -{1 \over 2} \gamma_{NS}^+(x, \alpha)
\otimes F_1^{NS}(x,Q^2),
\label{1} \\
%
{dF_3(x,Q^2) \over dlnQ^2} &=& -{1 \over 2} \gamma_{NS}^-(x, \alpha)
\otimes F_3(x,Q^2),
\nonumber
\eea
where the symbol $\otimes$ means the Mellin convolution:
\bea
f_1(x) \otimes f_2(x) \equiv \int^1_x \frac{dz}{z} f_1(z) f_2(\frac{x}{z})
\nonumber
\eea
%\hskip -.56cm
The splitting functions $\gamma^{\pm}_{NS}(x,\alpha)$ are the reverse
Mellin transforms of the anomalous dimensions $\gamma^{\pm}_{NS}(n, \alpha)= \alpha \gamma^{(0)}_{NS}(n) + \alpha^2 \gamma^{\pm (1)}_{NS}(n) + O(\alpha^3)$ and the Wilson coefficients\footnote{We consider here structure functions 
but not the parton distributions. Note also that $b^{+}_{NS}(n)$ and
$b^{-}_{NS}(n)$ can be defined as $b_{1,NS}(n) = b_{2,NS}(n) - b_{L,NS}(n)$ and
$b_{3,NS}(n)$.}
$\alpha b^{\pm}(n) + O(\alpha^2)$:
 \begin{eqnarray}
\gamma^{ \pm}_{NS}(x,\alpha) ~=~ \alpha
  \gamma^{(0)}_{NS}(x) + \alpha^2 \biggl(
  \gamma^{\pm (1)}_{NS}(x) +
  2\beta_0 b^{\pm}(x)  \biggr) + O(\alpha ^3),
\label{2}  \end{eqnarray}
where $\beta (\alpha)= - \alpha^2 \beta_0 -
 \alpha^3 \beta_1 + O(\alpha^4)$ is QCD $\beta$-function.\\
%
Eqs. (\ref{1}) show that the 
%$Q^2$ dependence of 
DGLAP equations for $F_3$ and for the NS part
of $g_1$  are the same
(it was obtained exactly
%at least 
in first two orders of the perturbative QCD\footnote{It is easy
to demonstrate 
%this equality 
in any order of perturbation theory. The
SF $g_1$ and $F_3$ are the results of the $\gamma_5$ matrix
contribution to the lepton
and hadron parts of deep-inelastic
cross-sections, respectively. In the NS case 
%of $F_3$ and the NS part of $g_1$ 
there is only one $\gamma$-matrix trace,
connecting the lepton and hadron parts. Its contribution $\sim ~
tr(\gamma_5 \gamma_{\mu} \gamma_{\nu} \gamma_{\alpha} \gamma_{\beta} ....)$
is the same in both cases above. For the SI part of $g_1$ there are
diagrams with several traces, which arise in the second order of 
perturbation QCD and lead to the difference between the splitting
functions of SF $F_3$ and the SI part of SF $g_1$.}  
 \cite{Kodaira})
and differ from the one for $F_1$ already in the first subleading order
($\gamma^{+(1)}_{NS} \neq \gamma^{-(1)}_{NS}$ \cite{RoSa} and
$b^+_{NS} - b^-_{NS} = (8/3)x(1-x)$).\\

\hskip -.56cm
For the SI parts of $g_1$ and $F_1$ evolution equations are :\\
\bea
{dg_1^S(x,Q^2) \over dlnQ^2} &=& -{1 \over 2} \biggl[
\gamma_{SS}^{*}(x, \alpha) \otimes g_1^S(x,Q^2) +
\gamma_{SG}^{*}(x, \alpha) \otimes \Delta G(x,Q^2) \biggr],
\nonumber
\\
%
{dF_1^S(x,Q^2) \over dlnQ^2} &=& -{1 \over 2} \biggl[
\gamma_{SS}(x, \alpha) \otimes F_1^S(x,Q^2) +
\gamma_{SG}(x, \alpha) \otimes G(x,Q^2) \biggr],
\label{4}
\eea
where the SI splitting functions $\gamma_{Si}(x, \alpha)$
($i=\{S,G\}$) are represented as
 \bea
\gamma_{SS}(x,\alpha) &=& \alpha
  \gamma^{(0)}_{SS}(x) + \alpha^2 \biggl(
  \gamma^{(1)}_{SS}(x) +  b_G(x) \otimes \gamma^{(0)}_{GS}(x) +
  2\beta_0 b_S(x)  \biggr)
%\nonumber \\
+ O(\alpha ^3),
\nonumber  \\
\gamma_{SG}(x,\alpha) &=& \frac{e}{f} \biggl[ \alpha
  \gamma^{(0)}_{SG}(x) + \alpha^2 \biggl(
  \gamma^{(1)}_{SG}(x) + b_G(x) \otimes \bigl(
  \gamma^{(0)}_{GG}(x) - \gamma^{(0)}_{SS}(x) \bigr)
+ 2\beta_0 b_G(x)
            \nonumber \\
&+&  b_S(x) \otimes \gamma^{(0)}_{SG}(x)
\biggr)
  \biggl] + O(\alpha ^3)  \label{4.1} \eea
%
and $e = \sum_i^f e^2_i$ is
the sum of charge squares of $f$ active quarks.
Equations for the polarized singlet splitting functions
$\gamma_{SS}^{*}(x, \alpha)$ and $\gamma_{SG}^{*}(x, \alpha)$
are similar. They can be obtained from (\ref{4.1}) by replacing
$\gamma^{(0)}_{SG}(x) \to \gamma^{*(0)}_{SG}(x)$,
$\gamma^{(1)}_{Si}(x) \to \gamma^{*(1)}_{Si}(x)$ and $b_i(x) \to
b^*_i(x)$ ($i=\{S,G \}$).\\

\hskip -.56cm
Careful consideration of  the quark part of (\ref{4}) and (\ref{4.1})
shows that
the value $b^*_s(x)$ ($b_s(x)$) coincides with $b^-(x)$ ($b^+(x)$).
The difference between
$\gamma_{NS}^{-(1)}(x)$ and $\gamma_{SS}^{* (1)}(x) +  b^*_G(x) \otimes
\gamma^{(0)}_{GS}(x)$  is negligible because it does not contain
a power singularity at $x \to 0$ (i.e. a singularity
at $n \to 1$ in momentum space).
Moreover, this difference decreases as $O(1-x)$ at $x \to 1$
\cite{MeNe}
(contrary to this, the difference between
$\gamma_{SS}^{(1)}(x) +  b_G(x) \otimes \gamma^{(0)}_{GS}(x)$ and
$\gamma_{SS}^{* (1)}(x) +  b^*_G(x) \otimes \gamma^{(0)}_{GS}(x)$
contains the power singularity at $x \to 0$ (see for example
\cite{Kodaira})).
Thus, the DGLAP equations for $F_3$ and the SI part of $g_1$ have
a close splitting functions, which are essentially different from the
splitting functions of the SI part of $F_1$. \\
The quark part of SI SF $g_1$ itself 
%Note here that the SI part of SF  $g_1$ 
contains two 
%quarks 
components:
valence and sea. Valence part does not connect with the
gluon and obeys the DGLAP equation similar to first equation of (\ref{1}).
The sea part obeys the first equation of (\ref{4}). This contribution
%The r.h.s. of DGLAP of equation for sea quark part contains the gluon
%contribution but it 
was not observed  experimentally yet.\\
%and, seems, it is small numerically 
%(at least at 
%in the range of modern experimental data.\\
The contribution of the gluon distribution in $g_1$
is not so important for the modern data 
(see \cite{GRSV,GS96, Ram, NST, GRam})
(in a contrary to  unpolarized case):  
%also for $F_1$ it
%the gluon term 
%is not negligible at $x \leq 0.3$ 
%but for $g_1$ it is not so important
%%can be neglected 
%(see \cite{GRSV,GS96, Ram, NST, GRam}) 
%%and the discussion in next section) 
%for the modern data: 
data are described well for extremally 
different values of $\Delta G(x,Q^2)$ and even for different  sign.
%(see \cite{Rams}). 
%Because data are sensitive slightly to the size of $\Delta G $ (or
%indicate its non-large values \cite{E581})
Hence we will neglet this term  in our analysis.\\
%already at $x\geq 0.01$.
Thus, 
the valence component seems to dominate in SI part of $g_1$
at the range of the present experimental data\footnote{To support
of this point of view, one can see also the recent analysis of
E154 Collaboration \cite{Q2E154}, where the contribution of sea $+$
gluon and valence parts are divided, studied and presented in Fig. 2
of \cite{Q2E154}.}
and it allows us to expect
%. Thus, 
%one would expect
a similarity of SI part of $g_1$ and SF $F_3$.\\
 
\hskip -.56cm
As we saw from above, 
the shapes and 
DGLAP equations for $g_1$ and $F_3$
are very close in NS and SI analyses\footnote{The similar
shapes of SF $F_3$ and  $g_1$ in the range of measured values of $x$ 
one can see also in the analysis  \cite{BoSo}.}
and
both of them differ from the
corresponding equations for $F_1$.
This
similarity leads to close $Q^2$ dependence of SF $g_1(x,Q^2)$ and
$F_3(x,Q^2)$ .\\
%if these structure functions have a similar $x$-dependence at least at some 
%value of $Q^2$:  the r.h.s. of DGLAP equations (\ref{1}) and (\ref{4})
%contain SF integrals. It is the case: , for example.\\

\hskip -.56cm
%This above observation about
The similarity of $Q^2$ dependence 
of $F_3(x,Q^2)$ and  $g_1(x,Q^2)$
may be also supported by some arguments following from analysis at
$x \to 0$ \footnote{At $x \to 1$ the behaviour of $g_1$ and 
$F_3$ (and $F_1$, too) should be similar because it is governed by valence 
quark distributions.}. 
Although all existed data in polarized DIS (exclude  two first SMC points) 
are outside from the region $x \leq 10^{-2}$, the study
of  small $x$ asymptotics is 
important as for
the future data, as for an extrapolation of the present data to the small $x$
egion.\\
The similarity of the splitting functions of  SF $F_3$ and 
%NS part of
$g_1$
%and, also, the similarity for  $F_3$ and the SI part of $g_1$
have been already demonstrated and, thus,
 we have to discuss now the shapes of  $F_3$ and 
 $g_1$ at small $x$.
It is well known that SF $F_3$ is governed at small $x$ by $\rho$-meson 
trajectory and, thus,
\bea
F_3(x) \sim x^{-1/2}
\label{d.1}
\eea
The $Q^2$-evolution does not change this behaviour. For SF $g_1$ the 
situation is not so clear. From Regge analysis \cite{Hei} the NS part of
$g_1$ is governed by $a_1$ trajectory, i.e.
$
g_1(x) \sim x^{-\alpha_P(a_1)}
$
with the intersept $\alpha_P(a_1)$ having values
$0 \geq \alpha_P(a_1) \geq -1/2$ (see \cite{ElKa1}). However,  
the BFKL inspired approach \cite{BERns} leads to
% and the modern data prefer 
more singular behaviour for NS 
part: 
\bea
g_1^{NS}(x) \sim x^{-0.45},
\label{d.3}
\eea
 which is close to (\ref{d.1}). 
%The data for the proton and neutron polarized
%SF seems increase and decrease at $x \to 0$, respectively.\\
For SI
part of $g_1$ an information is very poor. The BFKL inspeared approach
\cite{BERs}
leads to the small $x$ behaviour $g_1^{SI}(x) \sim x^{-1}$, but really SI part
 of $g_1$\footnote{Correctly,  sea component of SI part of  $g_1$,
which dominates here if it has nonzero magnitude.}
was not observed at small $x$ yet. It is also connected  with the
fact that the deutron SF $g_1^{d}(x)$, which is close to the SI component, 
is comparable with zero at small $x$.\\ 
%There is only one candidat to the 
%behaviour more singular then (\ref{d.3})(see \cite{E154n,Q2E154,SoTe}): 
%it is last E154 
%neutron data. But they may be fitted \cite{Rat} 
%also successfully with more flat
%asymptotics $g_1^{SI}(x) \sim x^{-0.5}(1-4x)$.\\
As a consequences, the shapes of  SF $F_3$ and the
NS part of SF $g_1$ seems to be close also at small $x$ (if BFKL 
approach is correct at least). 
%The shape of the SI part of $g_1$ may be another but its contribution 
%is not detected yet really in experiment.\\
The SI part of $g_1$ may has another shape but modern experimental
data do not allow to study it.\\

\hskip -.56cm
%This observation
The analysis discussed above allows us to conclude that
the function $A_1^*$ defined as:
\bea
A_1^*(x) = {g_1(x,Q^2) \over F_3(x,Q^2)}
\label{4.2}
\eea
has to be practically $Q^2$ independent in the hole region of modern
experimental data \cite{SMCp}-\cite{Q2E154}.
%at $x>0.01$.
%Note, however, that the r.h.s. of DGLAP equations (\ref{1}) and (\ref{4}) 
%are contained integrals on structure functions and, hence, equation 
%(\ref{4.2}) is valid only if  $g_1(x,Q^2)$ and $F_3(x,Q^2)$ have similar 
%$x$-dependence\footnote{It is so (see \cite{BoSo} at $Q^2 = 3 GeV^2$, 
%for example).}. 
\\

\hskip -.56cm
In agreement with Eq.(\ref{4.2}) measured asymmetry $A_1(x_i,Q^2_i)$ can be 
found at some value of $Q^2$ as:
\bea
A_1(x_i,Q^2) =  {F_3(x_i,Q^2) \over F_3(x_i,Q^2_i)} \cdot
{F_1(x_i,Q^2_i) \over F_1(x_i,Q^2)} \cdot A_1(x_i,Q^2_i)
%\cdot \frac{(1+ 4M^2x_i^2/Q^2)}{(1+ 4M^2x_i^2/Q_i^2)}.
\label{5}
\eea

\section{Calculation of $A_1(x,Q^2)$ and $\Gamma_1(Q^2)$.}
%

To apply this approach we used\footnote{Similar analysis of SMC data of
\cite{SMCp} has been done in  \cite{KP} using
%Eq.(\ref{5}) and
old CCFR data \cite{CCFR}.} the 
%Spin Muon Collaboration (SMC) 
SMC \cite{SMC}, E143 \cite{E143} and  E154 \cite{E154n,Q2E154}
collaboration data.
To use relation (\ref{5}) we have parametrized CCFR data on 
$F_2(x,Q^2)$ and $xF_3(x,Q^2)$ 
\cite{CCFRN} in the form the same with NMC fit of the structure function 
$F_2(x,Q^2)$ \cite{NMC} (see Appendix). 
To obtain structure function $F_1(x,Q^2)$ we take the parametrization of 
the CCFR data on $F_2(x,Q^2)$ \cite{CCFRN} and SLAC parametrization of 
$R(x,Q^2)$ \cite{SLAC} 
and use relation :
\begin{equation}
F_1(x,Q^2)= \frac{F_2(x,Q^2)}{2x(1+R(x,Q^2))} \cdot 
(1+ \frac{4M^2x^2}{Q^2}) ,
\label{5.1}
\end{equation}
where  $M$ is the proton mass. \\
The fact that we use in Eq.(\ref{5}) parametrizations of CCFR data
\cite{CCFRN} for both 
SF $xF_3(x,Q^2)$ and $F_2(x,Q^2)$ allows to avoid systematical uncertanties and
nucleon correlation in nuclei.\\
%\footnote{
%In the previous version \cite{KP} with old CCFR data \cite{CCFR} we worked 
%with the NMC parametrization of $F_2(x,Q^2)$ \cite{NMC} as for evaluation 
%of $A_1(x,Q^2)$  (Eq.(\ref{5})) as for the calculation of $g_1(x,Q^2)$   
%(Eq.(\ref{a})). Here the parametrization  of $F_2(x,Q^2)$ of  
%CCFR data \cite{CCFRN} is used for evaluation of $A_1(x,Q^2)$ 
%to avoid systematical uncertanties.
%For the calculation of $g_1(x,Q^2)$ we apply the NMC parametrization
% \cite{NMC}.}.\\
Fig. 1 shows the ratio $A_1(Q^2)/A_1(5GeV^2)$ obtained with Eq.(\ref{5}).
Comparison of Fig. 1 with the results of E154 Collaboration (Fig. 4 in 
\cite{Q2E154}) shows quite good agreement.\\
% for the neutron case. 
%For the proton
%our $Q^2$-evolution is steeper then it was found by E154.\\
 \hskip -.56cm
The SF $g_1(x,Q^2)$ was evaluated
%calculated 
using Eq.(\ref{a}) 
where spin average SF $F_1$ has been calculated using NMC 
parametrization of $F_2(x,Q^2)$ \cite{NMC}.  
Results are presented in Fig. 2 and Fig. 3 for E154 and SMC data,
respectively. Our results are in excelent agreement with the
SMC and E154 Collaboration analyses  based 
on direct DGLAP evolution
(see \cite{SMC} and \cite{Q2E154}, respectively).\\


%
%
%\subsection{ $Q^2$ dependence of $A_1$}
%\subsection {Results on the sum rules}

\hskip -.56cm
To make a comparison with the theory predictions on the sum rules
we have calculated also the 
first moment
value of the structure function $g_1$ at different $Q^2$
%The theoretical predictions on the spin-dependent structure function
%$g_1$ are done for its first moment $\Gamma_1$ :
\begin{equation}
\Gamma_1= \tilde \Gamma_1 + \Delta \tilde \Gamma_1,  \nonumber
\end{equation} 
where 
\begin{equation}
\tilde \Gamma_1(Q^2)=\int_{x_{min}}^{x_{max}} g_1(x,Q^2)dx
~~\mbox{ and }~~
\Delta\tilde \Gamma_1(Q^2)=\int^{x_{min}}_{0} g_1(x,Q^2)dx +
\int^{1}_{x_{max}} g_1(x,Q^2)dx
\nonumber \end{equation} 
are
the integral through  the measured kinematical $x$ region plus an
 estimation for unmeasured ranges, respectively.\\ 
%
The value of $\Delta\tilde \Gamma_1$ coming from the unmeasured 
$x$-regions was estimated using original
methodics by %Spin Muon, E143 and E154
``owner-collaborations''. 
%Obtained values of $\tilde \Gamma_1(Q^2)$, 
%%$\Delta \tilde \Gamma_1(Q^2)$ 
%and $\Gamma_1(Q^2)$ for the 
%proton and deutron at $Q^2$: $3, 5, 10, 30$ and $100$ GeV$^2$,
%together with the theoretical predictions \cite{LRV1}
%are presented in the Table 1.
%The errors for the estimation of $\Delta \tilde \Gamma_1(Q^2)$
%care the same as in \cite{SMCp, SMCd}. 
%Moreover they do not contribute to the errors of
%$\Gamma_1(Q^2)$ in agreement with SMC method and hence
%the errors of $\tilde \Gamma_1(Q^2)$ and $\Gamma_1(Q^2)$ are
%the same.
%
We have to note here that the methodic 
of $\Delta\tilde \Gamma_1$ estimation 
may leads to some underestimation of 
$g_1^{p,d,n}(x,Q^2)$ at small $x$ and of $ \Gamma ^{p,d,n}(Q^2)$, 
as a consequense (see the careful analysis in \cite{GRSV}). 
To clear up this situation it is necessary to have
more precise data at small $x$.\\
%
Values of $\Gamma_1(Q^2)$ which are obtained from the exact solution of the 
DGLAP evolution equation \cite{SMC,Q2E154} of $g_1(x,Q^2)$ and $A_1$
\cite{SMCp}-\cite{Q2E154} and in our approach on the scaling of $A_1^*$ are
quite close each other for all cases discussed here. 
Thus all approaches lead to the similar conclusions on $ \Gamma ^{p,d,n}(Q^2)$
and the results are in a strong disagreement with  the theoretical predictions
\cite{LRV1}, we will consider the effect of $A_1^*$ scaling only for the 
Bjorken Sum rule $\Gamma_1^p -\Gamma_1^n$.\\
%
%
%Obtained values of  $\Gamma_1(Q^2)$ are quite close in all cases discussed
%here, i.e. for exact DGLAP evolution \cite{SMC,Q2E154} of 
%$g_1(x,Q^2)$ and for $A_1$ \cite{SMCp}-\cite{Q2E154} and $A_1^*$
%scalings. The similar conclusion has been observed also in the
%recent articles \cite{SMC,Q2E154} for first two approaches.
%It follows from opposite in sign effects of 
%DGLAP evolution and $A_1^*$
%scaling at small and large $x$-values. As all approaches lead to similar
%conclusions for $ \Gamma ^{p,d,n}(Q^2)$ and because these results are in
%strong disagreement with  the theoretical predictions \cite{LRV1}, we
%will consider carefully the effect of $A_1^*$
%scaling only for the Bjorken Sum rule $\Gamma_1^p -\Gamma_1^n$.\\
%
%are presented in the Table 1.
Deuteron SMC and E143 data allow us to extract the value of $\Gamma_1^n$: 
%(which directly measured also by E154 Collaboration):
\begin{equation}
\Gamma_1^p + \Gamma_1^n=\frac{2\Gamma_1^d}{1-1.5w_d},  \nonumber
\end{equation}
where $w_d$=0.05 is the the probability of the deutron 
to be in a D-state.
Knowledge of proton and neuteron first momenta $\Gamma_1^{p,n}$ allows 
to test the Bjorken sum rule:
\begin{equation}
\Gamma_1^{p-n} \equiv  \int \limits_{0}^{1}(g_1^p(x,Q^2)-g_1^n(x,Q^2))dx=
\Gamma_1^p -\Gamma_1^n   \nonumber
\end{equation}
Results
%of our analysis
are presented in the Table 1 in comparison with 
values published by SMC, E143 and E154 Colaborations and with the theoretical
predictions computed in the third order in the QCD $\alpha_s$ \cite{LV}.\\
%as first moments of $g_1(x,Q^2)$ for proton-deutron target.\\ 
%Note that only the statistical errors are quoted here.
%To the considered accuracy they coincide with the errors
%cited in (\cite{SMCp, SMCd}).
%%-\cite{E143d}). 
%The presented theoretical
%predictions for the Bjorken and Ellis-Jaffe sum rules have been computed in
%\cite{LV} to the third order in the QCD $\alpha_s$.\\
%Fig. 2 shows $Q^2$ dependence of the Bjorken sum rule $\Gamma _1^{p-n} (Q^2)$
% predicted by the theory and obtained with our method.\\
%Note here that the $Q^2$ behaviour of $\Gamma _1^{p-n} (Q^2)$ is strongly 
%depended from the used parametrization of $F_2(x,Q^2)$. 
Let us now describe the main results, which follow from the Table 1 and
 the  Figures.
%Looking carefully at the results presented in the Tables 1, 2 and on the 
%figures we can make the following conclusions
\begin{itemize}
\item Our description of the $Q^2$ evolution of the asymmetry \aa1 
has very simple form (\ref{5}) but gives results which
are in good agreement with 
a powerfull analyses \cite{GRSV,GS96,Q2E154}.
%
\item
Results on $g_1(x,Q^2)$ are in excelent agreement with  SMC and
E154 Collaborations analyses, based 
on direct DGLAP evolution.
\item Our method allows to 
%make a more correct 
test 
%the Bjorken 
sum rules in a simple way with a good accuracy.
Obtained results on the $\Gamma_1^p - \Gamma_1^n$ show that used experimental 
data 
%(essentially proton one of SMC \cite{SMC} and neutron one of E154 
%\cite{Q2E154})
well confirm the Bjorken sum rule  prediction.
%For example, in the case of SMC data our average value coincides with 
%theoretical predictions. 
%\item  
%The values of $\Gamma_1^p$ and $ \Gamma_1^n$ themselves
%obtained here do not change essentially.
%The improvement for the Bjorken
%sum rule is the result of the opposite changes of the $\Gamma_1^p$
%and $ \Gamma_1^n$ values, when Eq.(\ref{4.1}) is used.
\end{itemize}
\hskip -.56cm


\section{Conclusion} 
We have considered the $Q^2$ evolution of the asymmetry \aa1
based on the similarity of $Q^2$ dependence of the SF $g_1(x,Q^2)$ and 
$F_3(x,Q^2)$\footnote{The usefull parametrizations of SF
$F_2(x,Q^2)$ and $xF_3(x,Q^2)$ are obtained for new CCFR data and
presented in Appendix.}.
%We have shown the sourse of this similarity and recalculated the
%measured values of the asymmetry using this similarity
%to the  $Q^2$ values independent from numbers of experimental bins. 
%Obtained $Q^2$ dependence of the asymmetry is in a good agreement with the 
%results obtained by \cite{GRSV,GS96,Q2E154}. 
%Our $Q^2$ dependence
%of the asymmetry is similar to one from \cite{ANR}.\\
Obtained results on  $g_1(x,Q^2)$
are in excelent agreement with the corresponding results of SMC and
E154 Collaborations, based 
on  direct DGLAP evolution.
Our test of the Ellis-Jaffe sum rules for the proton, deuteron and neutron 
are very close to the values published by Spin Muon, E143 and E154 
Collaborations. 
%It is so because 
However, the 
corrections coming due to $Q^2$ evolution of asymmetry \aa1 have an opposite 
signes for the proton and deutron. It leads to the 
%essential 
improvement in agreement between the experiment and the 
theoretical prediction on the Bjorken sum.\\
% (essentially in the case of SMC
%proton data \cite{SMC} and E154 neutron one \cite{Q2E154}).\\
%
%Thus, we have shown that our suggestion
%about the similarity of the $Q^2$ behaviour 
%and the shapes of the spin-dependent structure function $g_1(x,Q^2)$
%and spin averaged structure function $F_3(x,Q^2)$ leads to effects,
%%the phenomenons, 
%which are
%very close to ones obtained by direct DGLAP evolution.
%%On the other hand,
%%the suggestion involves very simple analysis (the application of 
%%eq.(\ref{5})).\\
We believe that future precise data will illuminate a violation of 
our hypothese (probably, at very small $x$ values: $x \leq 10^{-3}$).
This violation will indicate clearly the appearence of nonzero 
contributions from
sea quark and gluon components of SF $g_1(x,Q^2)$ having quite singular 
 shapes at small $x$\footnote{In the case of a similar shapes of SI and NS 
components the separation and study of them will be
quite elaborate procedure.} (see the carefull analysis in \cite{HNS}). 
Thus, check of the 
$Q^2$-dependence of the ratio $A_1^* = g_1/F_3$  for a future
precise data leads to make a qualitative estimations
shapes and 
%the 
$Q^2$-dependences of gluon and sea quark
distributions.\\
%and them contributions to the difference
%between experimental data
%%measurements 
%and theoretical predictions for Ellis-Jaffe sum rule.\\



%\hskip -.56cm
%A small $x$ consideration of $Q^2$ dependence of the asymmetry
%\aa1 will be also important to add but it is the subject of
%future large publication \cite{KNP}.
%, where recalculated CCFR data will be used too.
\vskip 0.5 cm

{\large \bf Acknowledgements}\\

\hskip -.56cm
We are grateful to W.G.~Seligman for
providing us the available CCFR data of Refs.\cite{CCFRN, CCFR},
to A.V.~Efremov and O.V. Teryaev for interest to this work and discussions
%A.L.~Kataev, V.G. Krivokhijine, I.A. Savin,
%G.I. Smirnov and O.V. Teryaev for interest to this work and discussions.
%We are grateful also 
and to A.P. Nagaitsev for a collaboration in the beginning
of this study.\\
% and to anonimous Referee for
%critical notices lead to essential improvement of the article.\\
%We are also grateful to G.~Ridolfi for the possibility to present
%the short version \cite{KP} of this letter on polarized SF section
%of the Workshop DIS96 and to A.L.~Kataev for the information about
%the reanalysis of CCFR data.\\

\hskip -.56cm
%This work is supported partially by the Russian Fund for Fundamental Research,
%Grant N 95-02-04314-a.\\

\hskip -.56cm
{\large \bf Appendix}\\

\hskip -.56cm
The parametrizations are used for CCFR data \cite{CCFR} :
\bea
xF_3(x,Q^2) = F_3^a  \cdot { \biggl( {
log(Q^2/\Lambda ^2) \over log(Q^2_0/\Lambda ^2) } \biggl) }^
{F_3^b}
~\mbox{ and }~
F_2(x,Q^2) = F_2^a  \cdot { \biggl( {
log(Q^2/\Lambda ^2) \over log(Q^2_0/\Lambda ^2) } \biggl) }^
{F_2^b}
%\cdot \biggl( 1+\frac{F_2^c}{Q^2} \biggr)
,
\nonumber
\eea
where
%
\bea
F_3^a &=& x^{C_1} \cdot (1-x)^{C_2} \cdot
%\nonumber \\
\biggl( C_3+C_4 \cdot (1-x) +C_5 \cdot
(1-x)^2
\nonumber \\ & &~+~
C_6 \cdot (1-x)^3 +C_7 \cdot (1-x)^4 \biggr) 
%\cdot
%\biggl[ C_8+C_9 \cdot x+C_{10} \cdot x^2 + C_{11} \cdot x^3 \biggr]
\nonumber \\ & & \nonumber \\
F_2^a &=& x^{B_1} \cdot (1-x)^{B_2} \cdot
%\nonumber \\
\biggl( B_3+B_4 \cdot (1-x) +B_5 \cdot
(1-x)^2
\nonumber \\ & &~+~
B_6 \cdot (1-x)^3 +B_7 \cdot (1-x)^4 \biggr) 
\nonumber \\ & & \nonumber \\
%\eea
%\bea
F_3^b &=& C_{8}+C_{9} \cdot x+{C_{10} \over x+C_{11}}
~\mbox{ and }~
F_2^b = B_{8}+B_{9} \cdot x+{B_{10} \over x+B_{11}}
\nonumber 
%& & \nonumber \\
%F_2^c &=& B_{12}+B_{13} \cdot x+B_{14} \cdot x^2 + B_{15} \cdot x^3
%\nonumber
\eea
and $Q^2_0 = 20~ {\rm GeV}^2$, $\Lambda = 337~{\rm MeV}$. The values of
$Q^2_0$ and $\Lambda $ are fixed in agreement with CCFR analysis 
\cite{CCFRN}.
The values of the coefficients $C_i$ ($i=1,...,11$) 
and $B_i$ ($i=1,...,15$) are 
given in Table 2.\\
%The addition of the higher twist term to $xF_3$ case leads to
%worser fits and this term is neglected at the case.

%\newpage

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\vskip 2cm
\newpage
{\Large {\bf Figure Captions}}\\
\vskip 1cm
\begin{description}
\item [Figure 1.]  $Q^2$ dependence of the ratio
 $A_1(x,Q^2)/A_1(x,5GeV^2)$.~ 
%
\item [Figure 2.]
Structure function $xg^n_1(x,Q^2)$ evolved to $Q^2 = 5 GeV^2$
using our eq.(\ref{5}); DGLAP NLO evolution; the assumption that
$g^n_1/F^n_1$ is independent of $Q^2$. Last two sets are taken
from \cite{Q2E154}.
%%
\item [Figure 3.]
Structure function $xg^p_1(x,Q^2)$ evolved to $Q^2 = 10 GeV^2$
using our eq.(\ref{5}); the assumption that
$g^n_1/F^n_1$ is independent of $Q^2$; DGLAP NLO evolution according
to the analyses of \cite{Q2A,GRSV}. Last two sets are taken from \cite{SMC}.



\end{description}

\vskip 3cm

{\Large {\bf Table Captions}}\\
\vskip 1cm
\begin{description}
\item [Table 1.]  The values of $\Gamma_1^p - \Gamma_1^n$.
The errors are shown only for several points for each set of data.
Uncertanties of our analysis are comparable with ones of 
\cite{SMC,SMCd,E143,Q2E154}.
%
\item [Table 2.]
The values of the coefficients of CCFR data parametrization.



\end{description}


\newpage
%
%



%
%\vskip 2cm
%\hskip -.56cm
%{\bf Table 1.}~~ 
%The first moment values of $g_1$.\\
%of the proton and deuteron.\\ 

%\hskip -.56cm
%{\bf Table 2.}~~ The values of $\Gamma_1^p - \Gamma_1^n$.\\

%\vskip .2cm \hskip -.56cm
%{\bf Table 3.}~~ The values of the coefficients of CCFR data parametrization.\\


%
%\newpage
%
\hskip -.56cm
{\bf Table 1.}~~ The values of $\Gamma_1^p - \Gamma_1^n$.
The errors are shown only for several points for each set of data.
Uncertanties of our analysis are comparable with ones of 
\cite{SMC,SMCd,E143,Q2E154}.\\
%Original value were obtained by SMC at $10~{\rm GeV}^2$ and E143 Collaboration
%at $3~{\rm GeV}^2$.\\
%\vskip -.9cm
\begin{table}[t]
\begin{center}
\begin{tabular}{|l|c|c|l|l|l|}
\hline
$Q^2$ (GeV$^2$) & 100 & 30 & 10 & 5 & 3 \\
\hline
\multicolumn{6}{|c|}{SMC proton \cite{SMC} and deutron data \cite{SMCd}}\\
\hline
$A_1$-scaling & 0.247 & 0.226 & 0.202 & 0.186 & 0.170 \\
$A_1^*$-scaling & 0.210 & 0.201 & 0.191 & 0.184 & 0.176 \\
Analysis of \cite{SMC} &  &  & 0.183   & 0.181 $\pm$ 0.035 &  \\
\hline
\multicolumn{6}{|c|}{SMC proton \cite{SMC} and E154 neutron data
\cite{Q2E154}}\\
\hline
$A_1$-scaling & 0.221 & 0.209 & 0.194 & 0.183 & 0.171 \\ 
$A_1^*$-scaling & 0.194 & 0.190 & 0.185 & 0.181 & 0.176 \\
\hline
\multicolumn{6}{|c|}{E143 proton and deutron data \cite{E143}}\\
\hline
$A_1$-scaling & 0.170 & 0.169 & 0.165 & 0.160 & 0.154 \\
$A_1^*$-scaling & 0.163 & 0.162 & 0.160 & 0.157 & 0.154 \\
Analysis of \cite{E143} &  &  &   & 0.164 $\pm$ 0.021 & 0.164  \\
\hline
\multicolumn{6}{|c|}{E143 proton \cite{E143} and E154 neutron data
\cite{E154n,Q2E154}}\\
\hline
$A_1$-scaling & 0.189 & 0.186 & 0.179 & 0.174 & 0.169 \\ 
$A_1^*$-scaling & 0.172 & 0.172 & 0.171 & 0.169 & 0.166 \\
Analysis of \cite{Q2E154} &  &  &   & 0.171 $\pm$ 0.013 &    \\
%&  &  &   &  $\pm$ 0.006 &    \\
Analysis of \cite{E143} &  &  &   & 0.170 $\pm$ 0.012 &    \\
\hline
Theory & 0.194 & 0.191 & 0.186 & 0.181 $\pm$ 0.002 & 0.177 \\
\hline
\end{tabular}
\end{center}
\end{table}



%
%\newpage
%
\vskip .2cm \hskip -.56cm
{\bf Table 2.}~~The values of the coefficients of CCFR data parametrization.\\
\vskip -.9cm
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
%\hline
\hline
 $ C_1$ &  $ C_2$ &  $ C_3$ &   $ C_4$ &  $ C_5$ & $ C_6$ \\
\hline
 0.33092 & 3.5000  & 6.5739 & -7.1015 & 2.5388 & 7.6944  \\
\hline
%\hline
$ C_7$  &  $ C_8$ &  $ C_9$ &   $ C_{10}$ &   $C_{11}$ & \\
\hline
 -8.4285 &  4.9135 &  -4.9857 &  -8.1629 &  1.8193 & \\
\hline
%\hline \hline 
 $ B_1$ &  $ B_2$ &  $ B_3$ &   $ B_4$ &  $ B_5$ & $ B_6$ \\
\hline
 -0.06101 & 3.5000  & 4.9728 & -3.1309 & -1.3361 &  0.94242 \\
\hline
%\hline
$ B_7$  &  $ B_8$ &  $ B_9$ &   $ B_{10}$ &   $B_{11}$ &
%$B_{12}$
\\
\hline
 0.11729 &  -0.92024 & -1.6489 &  0.61776 &  0.38910 &
%-4.7142
\\
\hline
%\hline
% $B_{13}$ &   $B_{14}$ &   $B_{15}$ &  &  & \\
%\hline
%29.203 &  -99.220 & 115.69 &  & & \\
%\hline \hline
\end{tabular} \end{center} \end{table}


\end{document}



\hskip -.56cm
{\bf Table 1.}~~ 
The first moment values of $g_1$.
The errors are shown only for several points for each set of data.
Uncertanties of our analisis are comparable with one of 
\cite{SMC}-\cite{Q2E154}.
%\newpage
%and the Bjorken sum rule values $\Gamma_1^{p-n}$.\\
\vskip -.9cm
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|l|c|l|l|}
%\hline
\hline
$Q^2$ & \multicolumn{2}{|c|}{$A_1^*$ scaling} 
& \multicolumn{2}{|c|}{$A_1$ scaling (our analysis)} & Theory \\
(GeV$^2$) & \multicolumn{2}{|c|}{(Evolution)}
& \multicolumn{2}{|c|}{(original values)} & predictions \\
%  & \multicolumn{2}{|c|}{} & \multicolumn{2}{|c|}{} &  \\
\cline{2-5}
& & & & & \\
&  $\tilde \Gamma_1 $
%& $\Delta\tilde \Gamma_1$
&$\Gamma_1$  
&  $\tilde \Gamma_1 $
%& $\Delta\tilde \Gamma_1$
&$\Gamma_1$ & \\
\hline %\hline
%\multicolumn{6}{|c|}{}\\
\multicolumn{6}{|c|}{SMC proton data \cite{SMC}}\\
\hline
3 & 0.1253 & 0.131 & 0.121 & 0.127 & 0.160 \\ 
5 & 0.1291    & 0.135~(0.132) & ?0.129? & ?0.135? & 0.167  \\ 
%& & (0.132) & & & \\
10 & 0.1323  & 0.138 & 0.136 & 0.142 & 0.170 \\
%$\pm$ 0.004 \\  
& (0.130) & (0.136 $\pm$ 0.016)
%$\pm$ 0.008)
& (0.133) & (0.139 $\pm$ 0.017) & $\pm$ 0.004 \\
%& &$\pm$ 0.008) & & & \\
30 & 0.1352   & 0.141 & 0.146   & 0.152 & 0.18?  \\
100 & 0.1376  & 0.144 & 0.154   & 0.160& 0.19?  \\
\hline
%\multicolumn{6}{|c|}{}\\
\multicolumn{6}{|c|}{SMC deutron data \cite{SMCd}}\\
\hline
3 & 0.0394 & 0.040 & 0.038 & 0.039 & 0.069 \\ 
5 & 0.0393    & 0.040 & ?0.038? & ?0.039? & 0.070  \\ 
%& & (0.132) & & & \\
10 & 0.0385  & 0.039 & 0.037 & 0.038 & 0.071 \\
%$\pm$ 0.004 \\  
& (0.0407) & (0.041 $\pm$ 0.008)
%$\pm$ 0.003)
& (0.0372) & (0.037 $\pm$ 0.007) & $\pm$ 0.004 \\
%& &$\pm$ 0.003) & & & \\
30 & 0.0367   & 0.037 & 0.035   & 0.036 & 0.07?  \\
100 & 0.0347  & 0.035 & 0.033   & 0.034 & 0.07?  \\
\hline
%\multicolumn{6}{|c|}{}\\
\multicolumn{6}{|c|}{e143 proton data \cite{E143}}\\
\hline
3 & 0.1087 & 0.122 & 0.111 & 0.125 & 0.160 \\ 
& & &(0.116)  & (0.128 $\pm$ 0.003) & \\
5 & 0.1096    & 0.123 & ?0.114?  & ?0.128?  & 0.167  \\ 
& & &(0.116) & (0.128) & $\pm$ 0.004 \\
10 & 0.1095  & 0.124 & 0.115 & 0.132 & 0.170 \\
%$\pm$ 0.004 \\  
30 & 0.1075  & 0.123 & 0.117   & 0.137 & 0.18?  \\
100 & 0.1051  & 0.122 & 0.116   & 0.139 & 0.19?  \\
\hline
%\multicolumn{6}{|c|}{}\\
\multicolumn{6}{|c|}{E143 deutron data \cite{E143}}\\
\hline
3 & 0.0403 & 0.041  & 0.0410 & 0.042 & 0.069 \\
& & &(0.044)  & (0.045 $\pm$ 0.003) & \\
5 & 0.0401    & 0.041  & ?0.0409? &?0.042?  & 0.070  \\ 
& & &(0.043)  & (0.044 $\pm$ 0.003) & $\pm$ 0.004 \\
10 & 0.0393  & 0.040 & 0.0408 & 0.042 & 0.071 \\
%$\pm$ 0.004 \\  
30 & 0.0376   & 0.039  & 0.0398   & 0.041 & 0.07?  \\
100 & 0.0359  & 0.037 & 0.0386   & 0.040 & 0.07?  \\
\hline
%\multicolumn{6}{|c|}{}\\
\multicolumn{6}{|c|}{E154 neutron data \cite{E154n,Q2E154}}\\
\hline
3 & -0.0399 & ?-0.063?  & -0.039 & ?-0.062?  & -0.012 \\ 
5 & -0.0413    & ?-0.064? & ??-0.043?? &??-0.065?? & -0.014  \\ 
 &(-0.035) & (-0.058 $\pm$ 0.011)
%$\pm$ 0.007)
&(-0.036) 
%&$\pm$ 0.004)
& &$\pm$ 0.004 \\
%& &$\pm$ 0.007) & & & \\
10 & -0.0428  & ?-0.066? & -0.047 & ?-0.070? & -0.016 \\
%$\pm$ 0.004 \\  
30 & -0.0446   &?-0.068? & -0.052 & ?-0.075? & -0.01?  \\
100 & -0.0463  & ?-0.070? & -0.057   & ?-0.080? & -0.01?  \\
\hline
%\hline
\end{tabular}
\end{center}
\end{table}





