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\begin{document}
\title{
%\vspace{-3cm}
%\vskip 1cm
Power corrections in the deep inelastic scattering
%\footnote{
%\uppercase{P}resented at the \uppercase{R}ingberg
%\uppercase{W}orkshop: {\it \uppercase{N}ew \uppercase{T}rends in
%\uppercase{HERA}  \uppercase{P}hysics 2003}, \uppercase{S}ep. 28
%-- \uppercase{O}ct. 3, 2003,  
%\uppercase{R}ingberg \uppercase{C}astle,
%\uppercase{R}ottach-\uppercase{E}gern, \uppercase{G}ermany.}
}

\author{S. ALEKHIN} 

\address{Institute for High Energy Physics (IHEP), \\
Pobeda, 1, Protvino 142281 Russia \\
E-mail: alekhin@sirius.ihep.su}

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

\abstracts{We outline the impact of the power corrections on the study of 
deep inelastic scattering and describe the determination of the power corrections 
from the existing deep inelastic scattering data.}

One of the main ingredients of the high energy phenomenology of the hadrons 
interaction is factorization. It allows to greatly simplify theoretical 
calculations using the connection of different processes through convolution
of the coefficient functions 
(i.e. parton cross sections) with the parton distributions functions
(PDFs). However factorization is valid only for the hard interactions, i.e.
in the limit of large masses, transverse momenta, etc. For the soft
interactions it is broken due to parton correlations being non-negligible.
The formalism for the description of 
the factorization breakdown is given by the operator-product
expansion (OPE)\cite{Wilson:zs}. In terms of this expansion the factorized 
part of the cross section corresponds to the contribution of the 
leading-twist (LT) operators and the factorization-breaking 
part -- to the higher-twist (HT) ones. 
The coefficient functions for the LT terms and the dependence of the 
PDFs on the transverse momentum $Q$ can be calculated in the 
pQCD-improved parton model. In accordance to these calculations
the LT (twist-2) structure functions of deep inelastic scattering (DIS) 
demonstrate logarithmic $Q$-dependence.
The $Q$-dependence of the HT terms is mainly defined by the factor of
$1/Q^{\tau-2}$, where $\tau$ is the corresponding value of twist. 
Due to the $1/Q$ factors the HT terms rapidly fall with $Q$ and in the 
limit of large $Q$ the LT terms give the dominant contribution.
However the coefficients at the power-like factors cannot be 
calculated using existing approaches and therefore the on-set 
scale of this asymptotic regime is a priori unclear.
The importance of this problem was recognized just after first attempts to 
confront the pQCD predictions to the earliest available
data since it was understood that the measured $Q$-dependence of 
the DIS structure functions can be equally well described both by the 
log-like and the power-like behavior\cite{Abbott:1980as}. 
With more precise measurements in the wide region of $Q$, which appeared later,
we can better separate the LT and the HT terms, 
however the impact of the HT terms on different aspects of the DIS 
phenomenology is not understood completely. In this report we describe  
the modern status of this problem and outline possible ways to resolve it.

\begin{figure}[ht]
\centerline{\epsfig{file=ht0.ps,width=10cm,height=8cm}}
\caption{The $x$-dependence of twist-4 term in the proton structure function
$F_2$ determined from the fit of Ref.{\protect \cite{Alekhin:2002fv}} 
to the combined SLAC-BCDMS-NMC-H1-ZEUS
data. The value of $\alpha_{\rm s}$ given in the panel was obtained from the
same fit.}
\label{fig:ht2}
\end{figure}

For  DIS the OPE up to twist-4 terms reads
\begin{equation}
F_{i}(x,Q^2) = F_{i}^{\rm LT,TMC}(x,Q^2) +
\frac{H_{i}(x,Q^2)}{Q^2} + {O}\left(1/Q^4 \right), 
\label{eqn:add}
\end{equation}
where $i$ marks different structure functions and
$F^{\rm LT,TMC}$ are the LT terms corrected for the target mass 
effects\cite{Nachtmann:1973mr}. 
These target mass effects give contribution to the 
power-like terms of the OPE, but are regularly considered separately  
since they are just of kinematical origin and can be expressed as 
a convolution of the LT terms with the  
coefficient functions proportional to negative powers of 
$Q$ (see\cite{Georgi:1976ve}).
The functions $H$ give the dynamical HT contributions
and correspond to the operators, which include correlations
of the quark and gluon fields\cite{Shuryak:1981kj}. 
Calculation of matrix elements of these operators in field theory 
is very difficult\cite{Ellis:1982cd} and only very limited results have been 
obtained so far\cite{Braun:1986ty,Dressler:1999zi}. The predictions of 
most of the models for the HT terms contain essential free parameters 
and cannot be used without a fit to the data.
For this reason the main source of information about the HT 
comes from  phenomenological studies. These studies are regularly based 
on the QCD extrapolation from the high-$Q$ region to the low-$Q$ one, where 
a discrepancy between the extrapolation and data is considered as 
a manifestation of the HT terms. This approach requires accurate 
treatment of all sources of theoretical uncertainties, which can have an impact 
on the extrapolation, and therefore to bias the results or even generate 
 fake HT terms. Besides, since the  
the existing DIS data from any one single experiment 
do not cover the whole kinematic region 
necessary for the study of such kind, the combination of all available data 
is desirable. In this case the fake HT terms can be also generated 
due to incomparability of the separate data sets, 
especially if the kinematics of different experiments do not overlap.

\begin{figure}[ht]
\centerline{\epsfig{file=irr.ps,width=10cm,height=10cm}}
\caption{Comparison of the isospin asymmetry in the HT terms of 
the structure functions $F_2$ and $F_{\rm T}=F_2+F_{\rm L}$ extracted from the 
global fit of Ref.{\protect \cite{Alekhin:2003qq}} (solid curves)
and calculated from the IRR model (dashes).}
\label{fig:irr}
\end{figure}

The first firm evidence of the HT dynamical terms was found 
in\cite{Virchaux:1991jc}. 
In this analysis the twist-4 terms in the structure functions
$F_2$ for proton and deuteron were determined from the next-to-leading order
(NLO) QCD fit to the combined SLAC-BCDMS data with account of 
the HT terms parameterized in the factorized form:
\begin{equation}
F_{2}(x,Q^2) = F_{2}^{LT,TMC}(x,Q^2)\left[1+\frac{h_2(x)}{Q^2}\right]. 
\label{eqn:fac}
\end{equation}
This form is equivalent to (\ref{eqn:add}) modulo the impact of the anomalous 
dimensions of LT and HT operators. The latter must exist as follows 
from the theoretical expectations\cite{Bukhvostov:te} and were indeed found in 
the phenomenological analysis of\cite{Ricco:1998yr}.
However at $x\le 0.75$, which are relevant 
for the analysis of\cite{Virchaux:1991jc}, 
the results of fits based on (\ref{eqn:add}) and (\ref{eqn:fac})
are comparable within the experimental errors\cite{Alekhin:1999iq}.
This indicates that the logarithmic corrections to the $Q$-dependence of the 
HT terms, which are generated by the anomalous dimensions, are unimportant 
in this region of $x$. The interpretation of the  
HT terms observed in the NLO analysis of\cite{Virchaux:1991jc}
for a long time 
was questionable due to the concern that they can be simulated by the missing 
higher-order QCD corrections. This concern is based on 
the observation that the $Q$-dependence of the terms proportional 
to the large powers of the strong coupling constant $\alpha_{\rm s}$ can simulate 
power-like behavior in the limited range of $Q$
(see\cite{Dokshitzer:1999ai}).
However such a simulation is possible only for the specific choice of powers 
of $\alpha_{\rm s}$ and $Q$ and/or insufficiently precise data.
In DIS the next-to-next-to-leading (NNLO) QCD corrections do not simulate 
the power-like behavior within the precision of existing data and as a result
the change of the dynamical twist-4 terms from the NLO to the NNLO fit
does not exceed the experimental 
errors\cite{Alekhin:2002nv} (similar stability was observed earlier 
in\cite{Martin:2000gq}, although conclusion about the non-vanishing of the 
dynamical HTs was not drawn 
due to the fact that they were not separated from the target-mass effects). 
The same stability should be valid also for the higher-order corrections
since the N$^3$LO correction is important at  
very high and very low $x$ only as follows from 
the analyses based on the resummation 
technique\cite{Altarelli:2000mh,Schaefer:2001uh,Gardi:2002xm,Bianchi:2003hi}.

The $x$-dependence of the twist-4 term in the proton structure function $F_2$ 
extracted from the data with the assumption, that functions $H(x,Q)$ do not depend
on $Q$, is given in Fig.\ref{fig:ht2}. This term is maximal at $x\sim 0.6$
and is determined in this region with the accuracy of $O(10\%)$ that 
allows to put a constraint on some matrix elements of the twist-4 
operators\cite{Choi:cu}.

\begin{figure}[ht]
\centerline{\epsfig{file=hln.ps,width=10cm,height=8cm}}
\caption{The HT terms in the nucleon structure function $F_{\rm L}$
determined from the global fit of Ref.{\protect \cite{Alekhin:2002fv}}.}
\label{fig:hln}
\end{figure}

The impact of the HT terms is important up to relatively large values of $Q$.
E.g. at $x=0.6$ the experimental error in $F_2$ averaged over global data set
is less than the HT terms at $Q^2\gtrsim 20~{\rm GeV}^2$ only.
In practice this means that inaccount of the HT terms can lead to bias 
of different quantities extracted from the DIS data. It is of 
particular importance for the extraction of $\alpha_{\rm s}$. 
Indeed, in the global fit with 
constraints $H=0$ the value of $\alpha_{\rm s}(M_{\rm Z})=0.1215+-0.0003$
(c.f. $\alpha_{\rm s}(M_{\rm Z})=0.1143+-0.0014$ with the HT
terms fitted)\cite{Alekhin:2002rk}. For the fit without account of the 
HT terms we observe the shift of the 
central value and an essential decrease of the error. Both effects
are due to the value of $\alpha_{\rm s}$ and are  strongly anti-correlated with $H_2$.
Essentially this means that for the global fit of existing data 
the error in $\alpha_{\rm s}$ is determined by the uncertainty in the HT 
terms and if these terms could be reliably estimated (e.g. from  
lattice calculations) the error in $\alpha_{\rm s}$ would be suppressed 
several times. The cut on data with $Q^2 \lesssim 20~{\rm GeV}^2$ 
aimed to decrease the
uncertainty in $\alpha_{\rm s}$ due to the HT terms is ineffective since 
in this case
the error in $\alpha_{\rm s}$ rises due to loss of data corresponding to the 
fastest rate of the QCD evolution. The promising way in this direction is to 
collect new statistically significant samples of data at HERA.

\begin{figure}[ht]
\centerline{\epsfig{file=fit.ps,width=10cm,height=8cm}}
\caption{The 1$\sigma$ band of the global fit (curves) compared 
to the data on $F_2$ used in the fit.} 
\label{fig:fit}
\end{figure}

A good precision in the determination of $H_2$
allows a conclusive comparison to the theoretical models.
One of the most predictive model of the HTs is based on the 
infrared renormalon (IRR)
approach\cite{Beneke:1998ui}. This model does not fix the general 
normalization of the HT terms, but puts a stringent constraint on their  
$x$-dependence. For the non-singlet terms the IRR model gives 
an especially simple predictions\cite{Stein:1996wk,Dasgupta:1996hh}
\begin{equation}
H_i^{\rm NS} = A \cdot F_{\rm 2}^{\rm LT,NS}\otimes C_i^{\rm IRR}, 
\label{eqn:irr}
\end{equation}
where $\otimes$ denotes the Mellin convolution, $C_i^{\rm IRR}$ are the 
corresponding coefficient functions and $A$ is the normalization factor.
These predictions are in qualitative agreement to the data, however 
the subtle details are not reproduced. For example a study of 
the isospin asymmetry, which can be used for a test of the basic 
features of the IRR model, since $C_i^{\rm IRR}$ are 
flavor-independent\cite{Brodsky:2000zu}, demonstrates
significant disagreement 
between the HT terms extracted from the data and the IRR predictions
(see Fig.\ref{fig:irr}).

\begin{figure}[ht]
\centerline{\epsfig{file=rslac.ps,width=10cm,height=8cm}}
\caption{The JLAB and SLAC measurements of $R=\sigma_{\rm L}/\sigma_{\rm T}$
compared to 1$\sigma$ band for $R^{\rm LT,TMC}$
extracted from the NNLO global fit of Ref.{\protect \cite{Alekhin:2002fv}}.}  
\label{fig:jlab}
\end{figure}

The HT term in $F_L$ extracted from the global fit to the DIS data 
is given in Fig.\ref{fig:hln}. 
The observed $x$-dependence of $H_{\rm L}$
is less regular than the 
one of $H_2$. It might be that due to $F_{\rm L}$ extracted 
from the global fit it is rather sensitive to disagreements in the data.
This point is illustrated in Fig.\ref{fig:fit}, which contains the 
comparison of the results of the global fit to the data on $F_2$ 
at $x\approx 0.012$.
It is evident that due to the large gap between separate data sets 
any disagreement 
between the NMC data at low $Q$ and/or the HERA ones at high $Q$ can 
be compensated by the corresponding HT term in $F_{\rm L}$, if one
performs the fit to the data of the cross section in order to extract 
$H_{\rm L}$. Therefore the 
spike at small $x$ might be driven by such a disagreement and more data 
at small $x$ and $Q^2=2\div 30~{\rm GeV}^2$ will be extremely useful for 
clarification of this point.
An alternative explanation of this spike is the effect of missing higher-order 
QCD corrections that is partially supported by the relative 
perturbative instability of $H_{\rm L}$, at least at $x\sim 0.2$.
Indeed, the higher-order corrections at small $x$
obtained using the $k_{\rm T}$ resummation technique are very 
large\cite{Catani:1994sq} and with the use of
the approximate 3-loop coefficient function for 
$F_{\rm L}^{\rm LT}$ from\cite{Martin:2000gq}
based on the calculations of\cite{Catani:1994sq,Larin:1993vu}
one probably should observe a decrease of $H_{\rm L}$ extracted from the fit.
However, one has to take into account that the uncertainty in the 
3-loop coefficient function of\cite{Martin:2000gq} due to missing 
terms in the 3-loop corrections is very large and 
the impact of the exact 3-loop correction might turn 
insufficient to explain the observed spike.
The spike might be generated by a 
new phenomena such as the parton saturation\cite{Gribov:tu} and, 
eventually, the OPE up to twist-4 might be insufficient\cite{Liuti:2001qk}.
In agreement with the latter the non-vanishing twist-6 contribution 
to $F_{\rm L}$ was found in the analysis of\cite{Ricco:1998yr}. 
This analysis was based on the data from the resonance region and therefore 
in order to obtain this result 
the duality hypothesis\cite{Bloom:1971ye} was heavily used. However, 
the recent JLAB-E94-110 data\cite{Liang} beyond the resonance
region confirm this observation. In Fig.\ref{fig:jlab} the JLAB 
measurements of $R=\sigma_{\rm L}/\sigma_{\rm T}$
are compared to the earlier SLAC data\cite{Whitlow:1990gk}
and the LT term extracted 
from the global fit. Excess of the SLAC data over the LT term with account 
of the target mass correction at $Q^2 \sim 2~{\rm GeV}^2$ is interpreted as 
a manifestation of the HT term in $R$ and, correspondingly, 
in $F_{\rm L}$\cite{Miramontes:ni}.
At small $Q$ the relation between data and the LT term is 
opposite and the most natural interpretation within OPE is that 
the effect of the positive twist-4 contribution
is overwhelmed by the negative twist-6 contribution.
If so, the structure of the HT terms is very 
non-trivial, at least for $F_{\rm L}$.
In summary, despite the predictions based on the HT terms given in 
Fig.\ref{fig:hln} are in good agreement with the independent 
measurements of $F_{\rm L}$ at small $x$ and $Q$ by the H1 collaboration 
(see\cite{Alekhin:2003yh} for the details),
the status of the determination of $H_{\rm L}$ is rather unclear 
and for a better understanding 
we need efforts both from the theoretical and the experimental side. 

The dynamical 
HT terms in the polarized structure functions are even less known 
than for the unpolarized ones, although the potential impact of 
the HT terms is very important in this case due to the typical relatively 
low $Q$ of the data used in the analysis. The recent indication  
on the the HT terms in $g_1$\cite{Leader:2002ni} was not entirely confirmed 
in the latter study\cite{Bartelski:2003fq} and, evidently, a more 
careful examination is necessary.

The HT terms in the DIS of neutrinos are also poorly known because of the low 
precision of the available data (see recent review\cite{Kataev:2001fv}
for more details). The essential progress in this area can be achieved 
only with the very intense neutrino beams 
(neutrino factories)\cite{Mangano:2001mj}. 
These facilities would allow to constraint the dynamical HT terms
in the neutrino-nucleon structure function at the level of precision 
comparable to the case of the charged leptons' DIS. 
However, since the neutrino experiments provide a much wider 
set of constraints, this additional input would allow to 
significantly improve disentangling the matrix element of 
different operators, which describe the partons' correlation.

In conclusion, our understanding of the power corrections 
in the DIS structure functions is rather limited. The kinematical 
component of the power correction (target mass effects) can be 
calculated with a reasonable accuracy, but the dynamical HT terms  
are essentially unknown for the most of structure functions.
Meanwhile  further study of HT terms is necessary in order to outline the 
kinematical region, where the QCD factorization can be applied and 
the universality of PDFs can be proven. From a practical point of view
it is necessary to provide unbiased extrapolations of PDFs 
to the high-energy region for the searches of the effects of new physics.
Eventually, study of the dynamical HT terms give information about 
the regime intermediate between asymptotic freedom and confinement.
All this makes the precise determination of the HT terms 
an important and promising topic.

The research was supported in part by the RFBR grant 
03-02-17177 and by the National Science Foundation 
under Grant No. PHY99-07949.

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\end{document}


