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\vspace*{3mm}
\begin{center}
{\Large \bf  
The constraints on the non-singlet polarized parton densities 
from the infrared-renormalon model}\\
\vspace{0.1cm}
{\bf A.L. Kataev}\\
\vspace{0.1cm}
Institute for Nuclear Research of the Academy of Sciences of 
Russia,\\ 117312, Moscow, Russia\\
\end{center}
%\vspace*{1cm}
\begin{center}
{\bf ABSTRACT}\\
\end{center}

We argue   that the infrared-renormalon approach 
gives the constraints on the next-to-leading order non-singlet 
polarized parton densities. The advocated feature follows from the 
consideration  of the effect revealed in the process 
of the next-to-leading order fits to the  data for the assymetry 
of  polarized
lepton-nucleon scattering   
which result in the 
approximate nullification 
of the   $1/Q^2$-correction to $A_1^N(x,Q^2)$.


%\vspace*{5mm}
\noindent

The study of the QCD predictions for the photon-nucleon asymmetry 
$A_1^N=\frac{\sigma_{1/2}-\sigma_{3/2}}{\sigma_{1/2}+\sigma_{3/2}}$,
where subscripts denote the total angular momentum of the 
photon-nucleon pair along the incoming lepton's direction,
plays the essential role in the analysis of polarized deep inelastic scattering 
(DIS) (see e.g. Ref. \cite{Altarelli:1998gn}). It is related to the well-known 
structure function $g_1^N$ of polarized DIS by the following way 
\begin{equation}
A_1^N(x,Q^2)=(1+\gamma^2)\frac{g_1^N(x,Q^2)}{F_1^N(x,Q^2)}
\end{equation}
where the kinematic factor $\gamma$ is defined as $\gamma=(4M_N^2x^2/Q^2)$ 
and $g_1^N(x,Q^2)$ is the structure function (SF) of 
polarized DIS, while $F_1^N(x,Q^2)$ SF 
enters into the cross-section of unpolarizeds charged lepton-hadron DIS 
(see e.g. Ref. \cite{Leader:2002ni}). Quite recently several procedures of the 
study of  the     
 $Q^2$-behavior of $A_1^N$ were discussed in the literature (see Refs. 
\cite{Kotikov:1996vr}-\cite{Leader:2001kh}). 
Moreover, in Refs. \cite{Leader:1999qp,Leader:2001kh,Leader:2002ni} 
by fitting existing   CERN,  DESY and SLAC  data for polarized DIS 
the  $1/Q^2$ dynamical power correction to $A_1^N$
was  extracted. 
In general it   gives additional contribution to the perturbation theory 
part of $(A_1^N)_{PT}$ and can  be parameterized as 
\begin{equation} 
A_1^N(x,Q^2)=(A_1^N(x,Q^2))_{PT}+\frac{h^{A_1}(x)}{Q^2}~~~~.
\end{equation}
It is interesting, that in the process of the fits of 
Refs. \cite{Leader:1999qp,Leader:2001kh,Leader:2002ni} 
it was found that the $x$-shape of $h^{A_1}(x)$ is   consistent 
with zero (see e.g.  Fig.1 from Ref.\cite{Leader:2002ni}).

%\end{equation}
\begin{figure}
\includegraphics[width=8cm]{Fig1.ps}
%\includegraphics[width=8cm]{0212085.Fig1.ps}
\caption{The results of extraction of $h^{A_1}(x)$ from the next-to-leading 
order  fits of 
Ref. \cite{Leader:2002ni} in the JET scheme \cite{Carlitz:ab} .}   
\end{figure}


In this note we are describing the possible consequences 
of this effect in the 
non-singlet (NS) approximation, which is valid for the  $x$-cut $x\gtrsim 0.25$.
Our consideration will be based on  the infrared-renormalon (IRR) approach, 
developed  in QCD in Ref. \cite{Zakharov:1992bx} and reviewed in detail 
in Ref. \cite{Beneke:1998ui}.    
This 
approach was used in Ref. \cite{Dasgupta:1996hh} to 
study   the behavior of the  $1/Q^2$ corrections 
to the NS contributions to  $F_2$ and $F_1$
SFs of unpolarized  DIS of charged leptons on nucleons 
and the pure NS  $xF_3$ SF of $\nu N$ DIS using 
$\overline{MS}$-scheme 
calculations\footnote{Note that we avoid considerations of the 
IRR renormalon free   expansions in QCD copling constants with  
the ``freezing-type'' behaviour at small $Q^2$ (see 
Refs. \cite{Krasnikov:1996jq}).} .
It is interesting that  the predicted in Ref. \cite{Dasgupta:1996hh}
$x$-shape of the IRR induced power corrections to $xF_3$ was supported 
in Refs. \cite{Kataev:1997nc,Alekhin:1998df}
by the  leading order (LO) and next-to-leading order (NLO) 
fits to CCFR'97 data ( 
the detailed description and refinements of the fits of 
Ref. \cite{Kataev:1997nc} is given in 
Refs. \cite{Kataev:1999bp,Kataev:2001kk}). Therefore, it is 
worth to consider  the consequences  of calculations of the    
IRR 
contributions to the NS part of  $g_1^N$ SF  of polarized 
deep-inelastic scattering, which was also performed in 
Ref. \cite{Dasgupta:1996hh}.

It should be stressed that the calculations  of Ref. \cite{Dasgupta:1996hh}
predict that in the NS approximation (or in the valence-quarks approximation) 
the contributions 
of  the $1/Q^2$ corrections  to $F_1$ and $xF_3$
SFs are the same.
Indeed, the corresponding results of  Ref. \cite{Dasgupta:1996hh} can be 
re-written in the following way:
\begin{equation}
\label{F1}
h^{F_1}(x,\mu^2)=h^{F_3}(x,\mu^2)= A_2^{'}\int_x^1\frac{dz}{z}C_1(z)q^{NS}(x/z,\mu^2)dz
\end{equation} 
where 
\begin{equation}
\label{C1}
C_1(z)= -\frac{4}{(1-x)_{+}}+2(2+x+2x^2)-5\delta(1-x)-\delta^{'}(1-x)~~~,
\end{equation}
the $'+'$-prescription, for any test function, is defined as 
\begin{equation}
\int_0^1F(x)_{+}f(x)dx=\int_0^1F(x)[f(x)-f(1)]dx~~~,
\end{equation} 
and 
\begin{equation}
q^{NS}(x,\mu^2)=\sum_{i=1}^{n_f}\bigg(e_i^2-\frac{1}{n_f}\sum_{k=1}^{n_f}e_k^2
\bigg)
\bigg(q_i(x,\mu^2)+\overline{q}_i(x,\mu^2)\bigg)
\end{equation} 
are the  NS parton densities, 
$\mu^2$ is the normalization point of order 1 GeV$^2$ and $A_2^{'}$ is  
is the IRR model parameter, to be extracted from the fits to the 
concrete data. Its most precise value was extracted from the  low-energy 
$xF_3$ data, collected by the IHEP-JINR Neutrino detector at the 
IHEP 70 GeV proton synchrotron (see Ref. \cite{Alekhin:2001zj}).
As was discussed  in Ref. \cite{Kataev:2001fv}, the result of 
Ref. \cite{Alekhin:2001zj} $A_2^{'}$=$-0.130\pm 0.056$ (exp) GeV$^2$ is 
in agreement with 
the value extracted from the NLO analysis of the CCFR'97 $xF_3$ 
data \cite{Seligman:mc},
namely with  $A_2^{'}$=$-0.125\pm0.053$ (stat) GeV$^2$ \cite{Kataev:2001kk}. 
It should be noted, that the identity of Eq. (\ref{F1}) does not contradict 
point of view, expressed in Refs. \cite{Kotikov:1996vr,Kotikov:1998ew}, 
that to study the $Q^2$ behavior of $A_1(Q^2)$ in the NS approximation 
it might be convenient to use   instead of theoretical expression for  $F_1$
the concrete $xF_3$ data.


Consider  now the case of $g_1$ SF of polarized DIS. In general, the 
IRR contributions to $g_1$ were studied in Ref. \cite{Stein:1998wr}.
In the NS approximation the IRR contributions 
to $g_1$ were calculated in  Ref. \cite{Dasgupta:1996hh}, where the following 
result was obtained 
\begin{equation}
\label{g1}
h^{g_1}(x,\mu^2)= A_2\int_x^1\frac{dz}{z}C_1(z)\Delta^{NS}(x/z,\mu^2)dz~~~~.
\end{equation}
Here  $ \Delta^{NS}(x,\mu^2)$ are the NS polarized parton densities 
and   the 
IRR model coefficient function $C_1(z)$ is the same,      
as in the case of the 
IRR model contributions to the  $1/Q^2$ corrections 
for $F_1$ and $xF_3$ SFs of unpolarized deep-inelastic scattering
(see Eq. (\ref{C1})). As to the IRR model parameter $A_2$, in general 
one should not expect that it has the same value as the parameter 
$A_2^{'}$ in Eq. (\ref{F1}). In principle, it should be extracted 
from the separate  fits to $g_1$ data in the NS approximation.
However, it is worth to stress, that in the NS approximation 
the IRR contributions to $g_1^N$ and $F_1^N$ are closely related.
(the similar feature was revealed while 
comparing IRR model contributions to the Bjorken  sum rule for $g_1^N$
SF \cite{Broadhurst:1993ru} and still unmeasured  Bjorken sum rule for 
$F_1^{\nu N}$ SF \cite{Broadhurst:2002bi}). 

Using now the similarity of the IRR model predictions of Eqs. (3), (7) 
and the property of approximate absence of the $1/Q^2$ contribution to the 
assymetry $A_1^N$, revealed in the process of  
fits of Refs. \cite{Leader:2002ni,Leader:1999qp,Leader:2001kh}, we find 
the following constraint
\begin{equation}
A_2 \Delta^{NS}(x,\mu^2) \backsimeq A_2^{'} q^{NS}(x,\mu^2)
\end{equation} 
which is valid both at the LO and NLO.
This constraint is the main result of our note.

It is interesting that the well-known LO bound of Ref. \cite{Altarelli:1998gn}
\begin{equation}
|\Delta(x,Q^2)|\leq q(x,Q^2)
\end{equation}
is now transformed to the LO relation between  the IRR model parameters
of Eq. (3) and Eq. (7), namely 
\begin{equation}
A_2^{'}\leq A_2~~.
\end{equation}
We hope that it will be possible to check this constraint and the 
relation of Eq. (8)  using 
the fits of the concrete  data for $g_1$ SF.

This work is done within the program of the RFBR Grant N 02-01-00601.
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\end{thebibliography}
\end{document}
