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\title*{Power Corrections to the Parton Model in DIS}
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\toctitle{Power Corrections to the Parton Model in DIS}
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\titlerunning{Power corrections in DIS}
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\author{Valentin Zakharov}

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\authorrunning{V.I. Zakharov}
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\institute{Max-Planck Institut f\"ur Physik,
F\"ohringer Ring 6, 80805 M\"unchen, Germany.}
%Princeton University, Princeton NJ 08544, USA
%\and Universit\'{e} de Paris-Sud,
%     Laboratoire d'Analyse Num\'{e}rique,
%     B\^{a}timent 425,\\
%     F-91405 Orsay Cedex, France}

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\begin{abstract}
We review the techniques to evaluate the $1/Q^2$ corrections
to the Deep Inelastic structure functions.
The importance of the
power corrections is that they indicate
the kinematical region beyond which the
leading-twist calculations are no longer valid.
\end{abstract}



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\section{Introduction}
%
%Of course this text has no scientific relevance but should be used
%as a practical example of the Springer layout specifications for
%\emph{physics books}~\cite{fhjl1.1}.
This Workshop is to discuss the HERA physics, which is primarily the
physics of the Deep Inelastic scattering at small $x$.
We approach the DIS at small $x$ from the pure perturbative side and
represent a generic structure function $F_i$ as
\begin{equation}
F_i(x,Q^2)~=~F_i^{p.th}(x,Q^2)+{\Lambda^2_{QCD}\over Q^2}h_i(x)
\end{equation}
where $F_i^{p.th}$ is the leading twist contribution. Note
that the $1/Q^2$ term is considered to be a correction to the
leading twist.
However, if the power correction reaches, say, $10\%$
at some ``moderate'' $Q^2$, it blows up fast at smaller $Q^2$.
Note also that we do not reserve for 
a possible $\ln Q^2$ dependence of the power correction 
but this is just for simplicity.

One of the basic questions addressed by the theory of the DIS at small $x$
is when one is to stop using the leading order perturbative 
calculation which might extend much further than one would
expect naively (for a recent review and further references see
\cite{ball}).
There do exist estimates of the higher twist effects
which show that the $1/Q^2$ corrections are pretty large at small $x$
\cite{stein,smye}:
\begin{equation}
{\delta F_2(x\sim 10^{-3},Q^2)\over F_2(x\sim 10^{-3},Q^2)}~
\approx~-{1~GeV^2\over Q^2}\label{estimate}
\end{equation}
where we picked up $x\sim 10^{-3}$ for the sake of definiteness.
The result has been found for any $x$ \cite{stein,smye}.

If true, the estimate (\ref{estimate}) is an important piece of 
information. There has been not  much discussion yet of
this estimate in the literature. Thus, we are going to provide 
an overview of the assumptions made to derive (\ref{estimate}).
Responding to the request of the organizers, we 
will try to make the presentation understandable to a broader audience.

\section{ General Remarks on the Power Corrections}

We consider QCD and processes determined by physics at short distances.
Which means that there is a generic
large mass scale, $Q\gg \Lambda_{QCD}$ where $\Lambda_{QCD}$ is the
position of
the Landau pole in the coupling:
\begin{equation}
\alpha_s(Q^2)~\approx~{1\over b_0ln{Q^2/\Lambda_{QCD}^2}}\label{coupling}.
\end{equation}
In case of DIS, $Q$ is the 4-momentum of
the virtual photon.

Moreover, we consider power corrections 
$(\Lambda_{QCD}^2/Q^2)^k$ to the parton model.
Note first of all that the power corrections appear to be a pure 
non-perturbative effect. Indeed, on one hand we have 
\begin{equation}
\left({\Lambda_{QCD}\over Q}\right)^k~=~exp(-k/2b_0\alpha_s(Q^2)).
\end{equation}
On the other hand, the function $exp(-const/\alpha)$
with a positive $const$ is a classical example from 
the math courses of a function which has a trivial 
Taylor expansion at $\alpha=0$:
\begin{equation}
exp(-const/\alpha)|_{\alpha=0}~=~
0+0\cdot\alpha+0\cdot\alpha^2+...
\end{equation}
since the function itself and all its derivatives vanish at $\alpha=0$.
Thus, this function, being a non-zero, vanishes identically
as a perturbative expansion, which is the expansion at $\alpha=0$. 

It could be quite well that the power corrections,
unlike pure perturbative contributions, 
are sensitive to the mechanism of confinement
(for a recent review and further references see, e.g., 
\cite{gpz,dokshitzer}).
However, here 
we would not go into this issue since the interest in the estimates
like (\ref{estimate}) is rather pragmatic than theoretical.

The only point we would like to mention is that we are assuming the validity
of the standard picture for the non-perturbative effects.
While
calculating the perturbative expansions is a well defined procedure
in QCD, at least as a matter of principle, the definition of
the non-perturbative terms,
at first sight
is close to saying that these are unknown
terms, the rest of the amplitudes upon subtraction of the perturbative
contributions. In other words, working with the power corrections   
relies to a great extent on intuition and heuristic models.
In particular, we would not like to use any concrete form
of the non-perturbative fluctuations since very little is known
about this.
Nevertheless,
there is one basic assumption underlying all further analysis.
Namely, we will assume here
that the non-perturbative fields are soft. In other words, the
typical size of the non-perturbative fluctuations is of order 
$\Lambda_{QCD}^{-1}$. One could challenge this assumption as well
\cite{gpz} but so far 
there were no alternative models for the power corrections
in DIS developed.

Finally, let us mention that actually working 
with an infinite perturbative expansion 
would be awfully difficult in practice.
Thus, in reality one is always relying on a kind of a {\it truncated}
series, keeping, say, only the leading perturbative plus power corrections.
The assumption behind this truncation is that
the power corrections are somehow {\it enhanced} numerically.
There is no proof of this enhancement but it is an indispensable
ingredient of any phenomenology based on the power corrections.
Historically, this assumption was made first 
in case of the so called
QCD sum rules and worked well. It might fail, however, 
in other cases. 

If there exist high-precision data one may try to
check this assumptions varying the number of terms in
the perturbative expansion kept explicit and 
determining the corresponding variation in the fitted
values of the power corrections. So far, this
procedure was implemented in the most careful way
in case of the DIS at large $x$ \cite{kataev}. 
The analysis certainly indicates that the fitted
value of $h_i$ diminishes quite dramatically with
inclusion of higher terms in the perturbative expansion.
A similar conclusion is reached in a very detailed model study of
power
corrections to the thrust \cite{cardi}.
Although such conclusions might seem quite disturbing, we 
would like to treat them again rather pragmatically. 
Namely, we would understand the estimate (\ref{estimate})
as a correction to the leading log approximation
and leave open the question whether it would be absorbed
into higher order perturbative corrections.

\section{Operator Product Expansion}

The basic theoretical tool which allows to 
exploit the idea that the non-perturbative fields are soft
is the Operator Product expansion (OPE).
In particular, it allows for a systematic treatment of the power corrections
in DIS \cite{vainshtein}. The most important outcome of the
analysis is that the power corrections to the structure functions start
with $Q^{-2}$ terms and we will outline briefly the derivation.

The OPE corresponds to a particular way of preparing 
ordinary Feynman graphs \cite{svz}. The basic ingredient is 
to separate hard and soft lines. By hard lines we understand the lines
which carry virtual momentum of order $Q$ while soft lines carry
momentum of order $\Lambda_{QCD}$. Then, the hard lines are treated
according to the standard Feynman rules. Moreover, in a space-time picture
of the interaction
all the points along these lines are close to each other, $\Delta x\sim 1/Q$
and, in the first approximation can be reduced to a single point.
Note that it is true only in the Euclidean space, while in the Minkowskian
space there could be large cancellations between space- and time- components
of the space-time interval. 
The relation of the results obtained within the OPE
to the physical processes (measurable in the Minkowski space)
is provided by dispersion relations. 

On the other hand, soft lines are not integrated out at all and their
effect is absorbed into matrix elements of the corresponding 
operators. The logic behind this step is that the soft lines 
with $p\sim \Lambda_{QCD}$ are modified drastically 
by confinement effects. 
Note that in the approximation that all the hard lines are
reduced to a single point the soft lines originate and end up at the
same point. Thus, instead of propagators of soft particles 
which are unknown functions
of soft momenta we have now only matrix elements of local operators,
that is numbers, not functions.

In this way, the route of the flow of the large
momentum $Q$ becomes crucial. 
For a given order of perturbation theory all the possibilities
for this flow should be tried.
Examples of the OPE approach to 
some Feynman graphs in the case of DIS is given in Fig. 1. 
\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{dis1.eps}
\end{center}
\caption[]{Examples of applying the OPE to
Feynman graphs. One follows the route of the large momentum $Q$ flow.
All the points along these, hard lines are reduced to a single point
indicates by a black blob. The large momentum $Q$ is brought in 
and taken away by 
the virtual photons, denoted by wavy lines. The dashed line denotes gluon
and solid lines are quarks.}
\label{eps1.1}
\end{figure}

Applying furthermore the standard dispersion relations for
the forward scattering of the virtual photon off the nucleon
one arrives to the well-known predictions for the moments:
\begin{equation}
M_n(Q^2)\equiv\int{F_i(x,Q^2)x^N}dx=M_n(Q_0^2)
\left({\alpha_s(Q^2)
\over\alpha_s(Q_0^2)}\right)^{\gamma_N}
\end{equation}
where $\gamma_N$ are calculable and reflect the effect of the hard lines while
$M_n(Q_0^2)$ are to be borrowed once from experiment
for some $Q_0^2$, as a reflection of our inability to evaluate
the effect of the soft quark lines. Moreover, for each moment we have a new
matrix operator and the corresponding matrix element.

Note that radiative corrections are dominated by virtual momenta
$k\sim Q$, as indicated in Fig. 1. However, there is also a possibility that
the gluon momentum $k$ is small, $k^2\ll Q^2$.
Then the effect 
of the gluon exchange
is to be included into new matrix elements, as indicated
in Fig. 2
\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{dis2.eps}
\end{center}
\caption[]{Emergence of higher twists. If the virtual gluon is
soft, $k\sim \Lambda_{QCD}$ its effect cannot be evaluated
but is rather to be included into matrix elements of operators
containing an extra power of the gluon field strength tensor.}
\label{eps1.2}
\end{figure}

Restricting momenta of virtual gluons to the infrared region costs 
suppression
of the corresponding contributions by powers of $Q^{-2}$,
due to phase space (and gauge invariance).
Applying the OPE allows to immediately find out this power.
Indeed, the leading twist (see Fig. 1) is related to the matrix elements
of operators bilinear in the quark fields, which can generically be
represented as $\bar{q}\Gamma q$ where $\Gamma$ is some matrix in the
spinor and color spaces. If a gluon line is added, see Fig. 2, then 
the corresponding operator becomes $\bar{q}F\Gamma q$ where $F$ is the gluon
field strength tensor (and we suppress all Lorentz and color indices).
The dimension of this extra factor, $F$, is $d=2$ and it is to be compensated
by the corresponding factor $Q^{-2}$:
\begin{equation}
\delta M_n~\sim ~M_n^{(0)}{\beta _n \Lambda^2_{QCD}\over Q^2}.
\end{equation} 

To summarize, the OPE 
allows to parameterize in a systematic way
the infrared sensitive contributions to the ordinary Feynman graphs.
In particular, in case of the DIS
one readily concludes that the leading power
corrections are of order $Q^{-2}$. These corrections are associated with
matrix elements of operators bilinear in quark fields and linear in the
gluon field. Borrowing the matrix elements from experiment 
amounts to introducing new structure functions. Without this input,
the use of the OPE does not allow for any concrete numerical predictions
for the power corrections.

\section{Renormalons or Infrared Sensitive Gluon Mass}

Renormalons (for review see, e.g., \cite{review}) 
represent a particular model for the power corrections which 
avoids the introduction of a new
structure function.
Qualitatively, the basic idea can be explained as follows.
The general formulation of the OPE outlined above
relates the $Q^{-2}$ correction
to an admixture of gluons in the nucleon wave function. 
One could argue, however, that the nucleon wave function is 
build on the valence quarks alone and all the gluon effects can be understood
in terms of the gluon bremsstrahlung from the valence quarks.

In fact this idea, to reduce the gluon structure functions to the quark
structure functions has been tried first in the leading twist case \cite{nsvz}.
Namely, one assumes that at some $Q_0^2\sim\Lambda_{QCD}^2$
there are only quark distributions, with no gluons. Gluons  
evolve at larger $Q^2$, as a result of bremsstrahlung. The model
is not without a success although one cannot go too far 
because of the matching point at low $Q_0^2$.

Now, we need technical means 
to realize the idea that the soft gluons responsible for the power-like
corrections (see Fig. 2 and discussion of it) are produced exclusively 
through bremsstrahlung. Introduce a non-vanishing gluon mass squared 
$\lambda^2$
and try first to expand in $\lambda^2$, see Fig. 3.
The expansion amounts to replacing the gluon propagator by
$(k^2+\lambda^2)^{-1}\to k^{-2}\lambda^2k^{-2}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{dis3.eps}
\end{center}
\caption[]{Introduction of a (fictitious) gluon mass $\lambda^2\to 0$
into the radiative corrections to the parton model. One evaluates contribution
proportional to $\lambda^2\ln\lambda^2$.}
\label{eps3}
\end{figure}
Upon this replacement 
the integral over the momentum of the gluon $k$
becomes logarithmically divergent in the infrared
and as a result we come to terms of
order $\lambda^2\ln\lambda^2$. Indeed,
\begin{equation}
\lambda^2\int_{\lambda}{d^4k\over k^4}~\sim~\lambda^2\ln\lambda^2,
\end{equation} 
where the lower bound of the integration $k\sim \lambda$ is introduced
because at such $k$ the expansion in $\lambda^2$ is no longer valid.

This estimate illustrates a general point
that terms non-analytical in $\lambda^2$,
like $\lambda^2\ln\lambda^2$
can develop only through contributions
of soft gluons with momenta $k\sim \lambda$. Thus, 
non-analyticity in $\lambda^2$ can be used
to mark
the infrared sensitive part of the Feynman integrals
\cite{chetyrkin}. To make connection with QCD
one replaces
\begin{equation}
\lambda^2\ln\lambda^2~\to~\Lambda^2_{QCD},
\end{equation}
with the overall coefficients in this relation left as a fitting parameter.
Indeed, for the sake of parameterization of
the contribution of momenta $k\sim\Lambda_{QCD}$ we can simply
adjust $\lambda\sim\Lambda_{QCD}$. 
In the particular case of the DIS, detailed calculations along these lines
can be found in \cite{marchesini}.

Note that so far we have not mentioned renormalons. 
Actually,
this is another way, consistent with the gauge invariance, 
to isolate contributions of
soft gluons. Namely, one inserts
fermionic bubbles into the gluon line, see Fig. 4.
It can be readily seen that if the number
of fermionic bubbles is large, the virtual 
momentum of the gluon, $k_{eff}$ dominating the Feynman integrals 
is small.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{dis4.eps}
\end{center}
\caption[]{Renormalon chain inserted into a gluon line. 
The number of the fermionic bubbles is assumed to be large
large, $n\gg 1$.}
\label{eps4}
\end{figure}
Indeed, each bubble is proportional to $b_0\ln(k^2/Q^2)$
where $b_0$ is the fermionic contribution
to the first coefficient in the $\beta$-function
and the log factor vanishes at the gluon momentum $k=Q$
because of normalizing the running coupling at $k=Q$.
Moreover, assume that the number of the fermionic bubbles, $n$ is
large, $n\gg 1$. Then the integral over the gluon line
takes the form
\begin{equation}
\int{d^4k\over k^2}\left(b_0\ln(k^2/Q^2)\right)^n~\sim b_0^n n!~~.
\end{equation}
and is large at large $n$. What is even more important, it is dominated
by low momenta $$(k^2)_{eff}~\sim~ Q^2\cdot exp(-n).$$ The type of graphs
represented in Fig. 4 is called renormalon chain. The motivation
to use the renormalon chain is exactly the same as to isolate
the terms of order $\lambda^2\ln\lambda^2$, see above. Namely,
the renormalon chain enhances greatly infrared sensitive 
contributions to the Feynman graphs
and allows for their parameterization. Using
the renormalons one comes to the same results for the power corrections
as by using the infinitesimal gluon mass, see above.

To summarize, renormalons allow to parameterize contribution
of soft gluons to the radiative corrections in DIS. 
The method is equivalent to introducing an infinitesimal gluon mass $\lambda$
and following the terms non-analytical in $\lambda^2$. 
The model has much greater predictive power than the general OPE
since no independent quark-gluon structure function is introduced
(see Fig. 2). Instead, the power correction 
to a structure function is proportional to the structure function itself
(see Figs. 3,4).
 
\section{Reservations, Successes, Predictions}

The great simplification brought by the renormalon model 
comes not without a price tag. The basic new assumption
is that external quark lines are considered hard
with respect to the gluon lines even if both are in fact soft.  
We called this assumption a valence quark approximation,
and it might be valid only for rough estimates.

There are some further caveats as well. 
In particular, the effect of, say, two soft gluon
emission is not suppressed compared to a single soft
gluon considered so far. Indeed, the emission of soft
gluons is governed by $\alpha_s(\Lambda^2)$ which is not small.
Moreover, in case of some observables one can prove that
the multiple gluon emission is important numerically
\cite{zakharov}. 

Thus, the renormalon approach is of openly heuristic and approximate nature
and can be justified only by its success, if any, in the description of the 
data. On the other hand renormalons do fit nicely the $x$ dependence of
the power corrections at large $x$, see, e.g., \cite{forte,liuti}.
This is especially true if one truncates both  
the perturbative expansion and expansion in powers of $Q^{-2}$
to their first respective terms. 

This agreement of the theory and data allows to fix the unknown overall
coefficient in front of the power corrections
and extend the calculation to the region of small $x$ as well. 
Since small $x$ are dominated by the singlet structure functions one
has to include the effect of the infinitesimal gluon mass
in graphs of the type represented in Fig. 5 \cite{stein,smye}.  
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{dis5.eps}
\end{center}
\caption[]{Power corrections to the singlet structure functions.
The black blob indicates insertion of the infinitesimal gluon mass.}
\label{eps5}
\end{figure}   
The prediction for the $x$ dependence of the power corrections appears
to be highly non-trivial. The correction practically dies away at moderate
$x$ but blows up again, with a negative sign at small $x$ \cite{stein,smye}.
I would expect that there should be a simple kinematical reason 
for such a behaviour but I am not in the position to identify it at the moment.

Thus, we come back to the estimate (\ref{estimate}) and may conclude that
despite many potential uncertainties the estimate looks quite conservative
and might well be realistic.

\section{Acknowledgments}

I would like to thank the organizers of the workshop, 
and B. Kniehl in particular for the invitation to the workshop
with its enjoyable atmosphere. I am gratefully acknowledging 
numerous discussions of the power corrections in DIS with R. Akhoury.  

\begin{thebibliography}{8.}
\addcontentsline{toc}{section}{References}

\bibitem{ball}R.D. Ball, S. Forte, hep-ph/9906222.

\bibitem{forte}S. Forte, hep-ph/9812382.

\bibitem{stein}E. Stein, M. Maul, L. Mankiewicz, A. Sch\"afer,
{\it Nucl. Phys.} \textbf{B536}, 318 (1998) (hep-ph/9803342).

\bibitem{smye}G.E. Smye, hep-ph/9812251.

\bibitem{gpz}F.V. Gubarev, M.I. Polikarpov, V.I. Zakharov, hep-ph/9908292.  

\bibitem{dokshitzer}Yu. L. Dokshitzer, hep-ph/9812252.

\bibitem{kataev}A.L. Kataev, G. Parente, A.V. Sidorov, hep-ph/9907310.
 
\bibitem{cardi}E. Gardi, G. Grunberg, hep-ph/9908458.   

\bibitem{vainshtein}E.V. Shuryak, A.I. Vainshtein,
{\it Nucl. Phys.} \textbf{B201}, 141 (1982).

\bibitem{svz}M.A. Shifman, A.I. Vainshtein, V.I. Zakharov,
{\it Nucl. Phys.} \textbf{B147}, 385 (1978);\\
V.A. Novikov, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov,
{\it Fortsch. Phys.} \textbf{32} 585 (1985).    

\bibitem{review} R. Akhoury, V.I. Zakharov, {\it Nucl. Phys. Proc. Suppl.}
\textbf{54A} 217 (1997) (hep-ph/9610492);\\
B.R. Webber,{ \it Nucl. Phys. Proc. Suppl.} \textbf{71} 66 (1999) 
(hep-ph/97122360;\\   
M. Beneke, hep-ph/9810263.  

\bibitem{nsvz}
V.A. Novikov, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov,
{\it Annals Phys.} \textbf{105}, 276 (1977);\\  
M. Gluck, E. Reya, {\it Phys. Lett.} \textbf{B270} 65 (1991).  
 
\bibitem{chetyrkin}
K.G. Chetyrkin, V.P. Spiridonov,
{\it Sov. J. Nucl. Phys.} \textbf{47}, 522 (1988);\\
M. Beneke, V.M. Braun, V.I. Zakharov, {\it Phys. Rev. Lett.}
\textbf{73}, 3058 (1994).
 
\bibitem{marchesini}
Yu.L. Dokshitzer, G. Marchesini, B.R. Webber,
{\it Nucl. Phys.} \textbf{B469}, 93 (1996). 

\bibitem{zakharov}
V.I. Zakharov, {\it Prog. Theor. Phys. Suppl.} \textbf{131}, 107 (1998) 
(hep-ph/9802416). 

\bibitem{liuti}
S. Liuti, {\it Nucl. Phys. Proc. Suppl.} \textbf{74}, 380 (1999) 
(hep-ph/9809248). 
\end{thebibliography}


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