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\begin{document}
%\preprint{JINR}
\title{Instanton Corrections to Quark Form Factor \protect \\ at Large Momentum Transfer}
\author{Alexander E. Dorokhov}
\email{dorokhov@thsun1.jinr.ru}
\author{Igor O. Cherednikov}
\altaffiliation[Also at:]{
Institute for Theoretical Problems of Microphysics,
Moscow State University, 119899 Moscow, Russia} \email{igorch@thsun1.jinr.ru}
% \homepage{}
\affiliation{
Bogoliubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research,
141980 Dubna, Russia}
\date{\today}
\begin{abstract}
Within the Wilson integral formalism,
we discuss the structure of nonperturbative corrections to the quark form factor
at large momentum transfer analyzing the infrared renormalon and instanton effects.
We show that the nonperturbative effects determine the initial value for the
perturbative evolution of the quark form factor and attribute their general structure
to the renormalon ambiguities of the perturbative series.
It is demonstrated that the instanton contributions result in the finite renormalization
of the next-to-leading perturbative result and numerically are characterized by small
factor reflecting the diluteness of the QCD vacuum within the instanton liquid model.
\end{abstract}
\pacs{12.38.Lg, 11.10.Gh}
%\keywords{Suggested keywords}%Use showkeys class option if keyword
                              %display desired
\maketitle
\section{\label{sec:level1}Introduction}

The various aspects of the instanton induced effects in the high energy QCD processes
had been addressed at the very beginning of the instanton era (see, {\it e.g.},
\cc{EL}), and this study has been continued in the later decade
\cc{BAL}. Recently, the interest to them has been revived \cc{Sh, KKL, RW, SRP, DCH},
and the hope of direct detection of the instanton induced effects has
appeared \cc{RH}. One of the important questions in the description of hadronic exclusive
processes is the behaviour of the form factors in various energy
domains (see, {\it e.g.,} \cc{MAG}).
The present paper is devoted to the analysis of the corrections to elastic quark form factor
induced by instanton \cc{REV} and infrared (IR) renormalon \cc{REN} effects to
the color singlet quark form factor at the large momentum transfer treating the nonperturbative effects
within the framework of the instanton liquid model of QCD vacuum.

The color singlet quark form factor is determined via the elastic scattering
amplitude of a quark in electromagnetic field:
\be
{\cal M}_\m =F_q\[(p_1-p_2)^2\]  \bar u(p_1) \g_\m v(p_2) \ \ ,
\ee where $u(p_1), \ v(p_2)$ are the spinors of outgoing and incoming
quarks.
The kinematics of the process is described in terms of the scattering
angle $\c$:
\be \cosh \c = {(p_1 p_2) \over m^2} = 1 + {Q^2 \over 2 m^2} \ \ ,
\ \ Q^2 = - (p_2 - p_1)^2>0 \ \ , \ \ p_1^2=p_2^2 =m^2  \ .\label{m1} \ee

It is known that the leading large-$Q^2$ asymptotics of the quark
form factor is given by the exponentiation of the one-loop term \cc{DL}:
\be
F^{(1)}_q(Q^2) = \exp\(-{\a_s \over 4\pi} C_F \ln^2{Q^2 \over \l^2} \)
+ O\(\a_s^n \ln^{2n-1}{Q^2 \over \l^2}\) \ , \label{dl}
\ee where $\l$ is an IR cutoff parameter.
In general, for a  correct consideration of the non-leading asymptotic contributions one has to
resum all perturbative (such as $O\(\a_s^n \ln^{2n-1}{Q^2}\)$, $O\(\a_s^n \ln^{2n-2}{Q^2}\)$,
{\it etc})
as well as nonperturbative terms. An effective framework for resummation of perturbative and nonperturbative
contributions is
provided by the Wilson integral approach \cc{NACH}.
Within this framework, the resummation of all logarithmic terms
coming from the soft gluon subprocesses
allows us to express the quark form factor (\ref{dl}),
in terms of the vacuum average of the gauge invariant path ordered
Wilson integral \cc{MMP}
\be
W (C_\c)  = {1 \over N_c} \Tr \<0|  \pa \exp \( i g \int_{C_\c} \! d x_{\m} \hat A_{\m}
(x) \)|0\> \ . \label{1a} \ee
In Eq. (\ref{1a}) the integration path corresponding to considering process
goes along the closed contour $C_\c$: the angle (cusp) with infinite sides.
The gauge field
\be
\hat A_{\m} (x) = T^a A^a_{\m}(x)\ , \   \ T^a = {\lambda^a
\over 2} \ \ ,  \ee belongs to the Lie algebra of the gauge group
$SU(N_c)$, while the Wilson loop operator $\pa \ex^{ig\int\! dx A(x)}$ lies in
its fundamental representation.

In our recent paper \cc{DCH}, we applied the Wilson integral formalism to
evaluation of the perturbative and nonperturbative contributions to the
color singlet quark form factor at the low normalization point $\m$ of order
of the inverse instanton size within the instanton liquid model. In the
present work, considering the renormalization group (RG) evolution equation we extend the analysis
to the limit of large momentum transfers focusing on the
asymptotic behaviour. We show that the nonperturbative effects determine the initial
value for the perturbative evolution, find their general structure by
analyzing the renormalon ambiguities of the perturbative series, and establish the correspondence
between them and the instanton induced contribution.

The paper is organized as follows:
In Section II we reproduce the known results of the perturbative one-loop
calculation, and derive the evolution equations taking into account the
nonperturbative contribution as the initial value for perturbative
evolution. In Section III,
we study the consequences of the IR renormalon ambiguities of the perturbative series and
show how the later prescribes the form of the nonperturbative corrections to the asymptotic behavior
of the form factor at large-$Q^2$.
In Section IV, these nonperturbative effects are estimated
in the weak-field approximation within the instanton model of the QCD vacuum.
Finally, the large-$Q^2$ behaviour of the form factor is analyzed taking into account the leading
perturbative, IR renormalon, and instanton induced contributions. The later
are found to be determined by small factor expressed via the
parameters of the instanton liquid model.

\section{Analysis of the perturbative contributions to the Wilson integral}

The Wilson integral (\ref{1a}) can be presented as a series:
$$
W(C_\c)  = 1 + {1 \over N_c}\<0| \sum_{n=2} \ (ig)^n \int_{C_\c}\int_{C_\c} ...\int_{C_\c}
 \! dx_{\m_n}^n \ dx_{\m_{n-1}}^{n-1}... dx_{\m_1}^1 \cdot $$
\be \cdot \ \theta (x^n, x^{n-1}, ... , x^1)
\ \Tr \[\hat A_{\m_n} (x^n) \hat A_{\m_{n-1}} (x^{n-1})... \hat A_{\m_{1}} (x^1) \]|0\>
\ , \label{expan1} \ee where the function $\theta (x)$ orders the color matrices along
the integration contour.
In the present work, we restrict ourselves with the study of the  leading order
(one-loop  --- for the perturbative gauge field and weak-field limit for the
instanton) terms of the expansion (\ref{expan1}) which are given by the
expression:
\be
\w1 (C_\c) =  -{g^2 C_F \over 2}
\ \int_{C_\c}\! dx_\m \int_{C_\c}\! dy_\n \ \prop (x-y)
\ , \label{g1} \ee where the gauge field propagator $\prop(z)$ in
$n$-dimensional space-time $(n = 4 - 2 \ve)$ can be presented in the form:
\be
\prop(z) = g_{\m\n}
\pd_z^2 \D_1(\ve, z^2, \m^2/\l^2) - \pd_\m\pd_\n \D_2(\ve, z^2, \m^2/\l^2) \ .
\label{st1} \ee
The exponentiation theorem for non-abelian path-ordered Wilson integrals
\cc{KR, ET} allows us to express (to one-loop accuracy) the Wilson integral (\ref{1a})
as the exponentiated one-loop term of the series (\ref{expan1}):
\be
W(C_\c) = \exp\[\w1(C_\c) + O(\a_s^2)\] \ . \label{eq:expn1}
\ee

In general, the expression (\ref{g1}) contains ultraviolet (UV) and IR divergences, that
can be multiplicatively renormalized in a consistent way \cc{BRA}.
In contrast to the previous paper \cc{DCH}, we use the dimensional
regularization in order to work with UV singularities, and define the ``gluon mass''
$\l^2$ as the IR regulator and the parameter $\mu^2$ as the UV normalization point.
The dimensionally regularized formula for the leading order (LO) term (\ref{g1})
can be written as \cc{DCH}:
\be
\w1 (C_\c; \ve, \m^2/\l^2, \a_s(\mu))
= 8 \pi \a_s(\mu) C_F h (\c) (1 - \ve)  \D_1(\ve, 0, \m^2/\l^2) \ ,
 \label{pe1} \ee
where $h(\c)$ is the universal cusp factor:
\be
h(\c) = \c \coth \c -1 \  \ee
and in case of the perturbative field
\be
\D_1 (\ve, 0, \m^2/\l^2) = - {1 \over 16\pi^2} \(4\pi {\m^2 \over
\l^2}\)^{\ve} \ {\G(\ve) \over 1 - \ve} \ .  \label{eq:pr01}
\ee
The independence of the expression (\ref{pe1}) of the function $\D_2$ is a
direct consequence of the gauge invariance.
Then, in the one-loop approximation,
\be
W(C_\c; \ve, \m^2/\l^2, \a_s(\mu)) = 1 - {\a_s(\m) \over 2\pi} C_F h(\c) \({1 \over \ve}
- \g_E + \ln 4\pi + \ln{\m^2\over\l^2}
\),
\ee and the cusp dependent renormalization constant \cc{BRA}, within the
$\overline{MS}$-scheme, reads:
\be
Z_{cusp} (C_\c; \ve, \m^2/\l^2, \a_s(\mu)) =
1 + {\a_s(\m) \over 2\pi} C_F h(\c) \({1 \over \ve} - \g_E + \ln 4\pi \) \
.
\ee
The detailed description of the renormalization procedure
within the present approach has been made in \cc{DCH} and will be omitted
here for brevity.

Using the Eq. (\ref{pe1}), one finds the known one-loop result
for the perturbative field, which  contains the dependence
on the UV normalization point $\m^2$ and IR cutoff $\l^2$
({\it e.g.}, \cc{KR}):
\be \w1_{pt} (C_\c)
=  - {\a_s (\m) \over 2 \pi} C_F  h(\c) \ln {\m^2 \over \l^2}
\ . \label{5} \ee
Therefore, in the leading order the kinematic dependence of the expression (\ref{g1}) is
factorized into the function $h(\c)$, which at large-$Q^2$ is approximated by:
\be h(\c) \propto \ln {Q^2 \over m^2} \ . \label{eq:lar1} \ee
In this regime, the dependence of $W$ on the UV normalization scale $\m$ (which can also be treated
as an arbitrary factorization scale dividing the hard and soft subprocesses \cc{KRC1}) is governed by the
RG equation:
\be
\(\m{\pd \over \pd \m} + \b(g) {\pd \over \pd g} \)
{d \  \ln \ W (Q^2) \over d \ \ln Q^2}
 = - \G_{cusp} (\a_s(\m)) \ , \label{2}
\ee where $\G_{cusp}(\a_s)$ is the universal cusp anomalous dimension evaluated
in the perturbation theory. In Eq. (\ref{2}), we take the logarithmic
derivative in $Q^2$ in order to avoid problems with light-cone singularities
at $m^2=0$ \cc{KRC1}.
The solution of the RG equation leads to the evolution equation
\be
{d \ \ln W (Q^2) \over d \ \ln Q^2} = -
\int_{\l^2}^{\m^2} \! {d\x \over 2\x} \ \G_{cusp}(\a_s(\x) ) +
{d \  W_{np} (Q^2) \over d \ \ln Q^2} \ , \label{inc1}
\ee where the function $W_{np}$ gives the initial condition at $\m^2 = \l^2$
and has to be found by the nonperturbative methods \cc{KRREN, TAF}. Solving Eq.
(\ref{inc1}), we take the arbitrary upper bound for the squared momenta of soft gluons
equal to the hard scale: $\m^2 = Q^2$, and find
\be
\ln \ {W (Q^2) \over W(Q_0^2)} = - \int^{Q^2}_{Q_0^2}\! {d x \over x}\[
\ \int_{\l^2}^{x} \! {d \x \over 2
\x}\ \G_{cusp}\(\a_s(\x)\) - {d \  W_{np} (x) \over d \ \ln\  x} \] \ ,
\label{eq:ev1}
\ee what immediately leads to the
conclusion that the leading large-$Q^2$ behaviour of the quark form factor $F_q(Q^2)$
including all logarithmic corrections is controlled by
the universal cusp anomalous dimension (\ref{2}) and can be expressed in
the following form (for comparison, see \cc{KRC1}):
$$
F_q(Q^2) = W(Q^2) = $$ \be = \exp\[-\int_{Q_0^2}^{Q^2}\! {d\x \over 2\x} \  \ln{Q^2 \over \x}
\ \G_{cusp}(\a_s(\x) )  - \ln {Q^2 \over Q_0^2}\ \int_{\l^2}^{Q_0^2} \! {d\x \over
2\x} \ \G_{cusp}(\a_s(\x) ) +  W_{np} (Q^2) \] W_0 \ , \label{1} \ee
where $W_0 = W(Q_0^2)$ contains both perturbative and nonperturbative
contributions. From the one-loop result (\ref{5}),
the cusp anomalous dimension which satisfies the RG equation
(\ref{2}) in one-loop order is given by:
\be {\G_{cusp}^{(1)}} ( \a_s (\m)) = {\a_s (\m) \over \pi} C_F \ .  \label{cp} \ee
Substituting the anomalous dimension (\ref{cp}) in the one-loop approximation for the strong
coupling into the Eq. (\ref{1}), one
finds
$$
F_q^{(1)}(Q^2) = $$ \be = \exp\[- {2C_F \over \b_0} \[\ln {Q^2 \over \L^2}
\ln{\ln(Q^2/\L^2) \over \ln(Q_0^2/\L^2)} - \ln {Q^2 \over Q_0^2}
\(1 - \ln{\ln(Q_0^2/\L^2) \over \ln(\l^2/\L^2)} \) \] +  W_{np}(Q^2) \]
F^{(1)}(Q_0^2)\ , \label{npc0} \ee where $\L$ is the QCD scale. The singularity in Eq. (\ref{npc0})
originates from the region where the IR cutoff approaches $\L$, {\it
i. e., } where the coupling constant $\a_s$ increases, and then may have a
nonperturbative nature.

\section{Effects of the IR renormalons}

In order to determine the structure of the nonperturbative function $W_{np}$ in
Eqs. (\ref{1}, \ref{npc0}), it is instructive to study the
corrections due to IR renormalons \cc{REN}. In the present situation, one can expect
the corrections proportional to the powers of both scales: $\m^2$ and
$\l^2$. However, taking into account the evolution in $\m^2$ to the hard
characteristic scale of the process $Q^2$ (\ref{eq:ev1}), we treat the power $\m^2$-terms
to be strongly suppressed, and focus on the power $\l^2$-corrections.
To find them, let us consider the perturbative function $\D_1(\ve, 0, \m^2/\l^2)$ in the eq. (\ref{pe1}).
The insertion of the fermion bubble 1-chain to the one-loop order expression
(\ref{g1}) is equivalent to replacement of the frozen coupling constant $g^2$ by the
running one $g^2 \to g^2 (k^2) = 4\pi \a_s(k^2)$
\cc{KRREN}:
\be \widetilde\D_1(\ve, 0, \m^2/\l^2) = - 4\pi \m^{2\ve}  \int \! \dk \a_s(k^2){\ex^{ikz} \delta(z^2)
 \over k^2(k^2+\l^2)}\ .
\label{ren1} \ee
For the sake of convenience, we work here in Euclidean space.
Using the integral representation for the one-loop running coupling $\a_s(k^2) = \int_0^\infty \!
d\s (\L^2/k^2)^{\s b }$, $b = \b_0 /4\pi$, we find:
\be
\widetilde\D_1(\ve, 0, \m^2/\l^2) = - {1 \over \b_0(1-\ve)} \(4\pi {\m^2 \over \l^2}\)^\ve \int_0^{\infty} \!
dx\
{\G(1-x -\ve) \G(1+x+\ve) \over (x +\ve ) \G(1 - \ve )} \({\L^2 \over \l^2}\)^x \ . \label{gamma} \ee
To define properly the integral in
r. h. s. of Eq.(\ref{gamma}), one needs to specify a prescription to go around the poles, which
are at the points $\bar x_n = n , \ n \in {\mathbb N}$. Of course, the result
of integration will depend on this prescription giving an ambiguity proportional
to $\(\L^2 / \l^2 \)^n$ for each pole. Then, the IR renormalons produce the power
corrections to the one-loop perturbative result, which we assume to exponentiate with the
later \cc{KRREN, TAF}. Extracting from (\ref{gamma}) the UV singular part in vicinity of the
origin $x =0$, we divide the integration interval $[0, \infty]$ in two parts:
$[0, \d]$ and $[\d, \infty]$, where $\d < 1$. This procedure allows us to
evaluate separately the ultraviolet and the renormalon-induced pieces. For the
ultraviolet piece, we apply the expansion of the integrand in
$\D_1$ in powers of small $x$ and replace the ratio of $\G$-functions by
$\exp(-\g_E \ve)$:
\be \widetilde\D_1^{UV} (\ve, 0, \m^2/\l^2) = -  {1 \over \b_0 (1- \ve)}
\sum_{k, n=0} (-)^n { \(  \ln 4\pi - \g_E + \ln {\m^2  \over \l^2}\)^k
\over k! \ve^{n-k+1}}  \
\int_0^{\d} \! dx \ x^{n} \ \(\L^2 \over \l^2 \)^x \ , \label{rnm1} \ee
which after subtraction of the poles in the $\overline{MS}$-scheme becomes:
\be
\widetilde\D_1^{UV} (0, \m^2/\l^2) =  {1 \over \b_0 ( 1- \ve)}\  \sum_{n=1}\(\ln {\m^2 \over \l^2}
\)^n {(-)^n \over n!} \ \int_0^{\d} \! dx x^{n-1} \(\L^2 \over \l^2 \)^x \ .
\label{eq:iks}\ee
In analogy with results of \cite{Mikh98}, this expression may be rewritten in a closed form  as
\be
\widetilde\D_1^{UV} ( 0, \m^2/\l^2) =
{1 \over \b_0 (1- \ve)}
\int_{0}^{\delta}\frac{dx}{x}
\left\{\ex^{-x\ln\frac{\mu^{2}}{\L^{2}}}
-  \ex^{-x\ln\frac{\l^{2}}{\L^{2}}
}\right\}.
\label{Dex}\ee
Substituting
\be
{d \ \w1 (Q^2) \over d \ \ln Q^2}= 2 C_F (1-\ve) \widetilde\D_1^{UV}(0, \m^2/\l^2)
\label{Dex1}\ee
into Eq. (\ref{2}) one finds
\be
\(\m{\pd \over \pd \m} + \b(g) {\pd \over \pd g} \)
{d \  \ln \ W^{(1)} (Q^2) \over d \ \ln Q^2}
 = - \G_{cusp}^{(1)} (\a_s(\m)) \(1-\exp{\[-\d\frac{4\pi}{\b_0\a_s(\m)}\]}\).
\ee The second exponent in the last equation yields the power suppressed
terms $\(\L^2/Q^2\)^\d$ in large-$Q^2$ regime.
In the leading logarithmic approximation (LLA) Eq. (\ref{Dex1}) is reduced to:
\be
{d \ \w1 (Q^2) \over d \ \ln Q^2} = - {2C_F \over \b_0} \(
\ln{\ln(\m^2/\L^2) \over \ln(\l^2/\L^2)}  \) \ . \label{ptr2}
\ee
The last expression obviously satisfies the perturbative evolution
equation (\ref{2}).

The remaining integral in Eq. (\ref{gamma}) over the interval $[\d, \infty]$
is evaluated at $\ve =0$ since there are no UV singularities.
The resulting expression does not depend on the normalization point $\m$, and thus
it is determined by
the IR region including nonperturbative effects. It contains the renormalon ambiguities
due to different prescriptions in going around the poles $\bar x_n$ in the Borel
plane which yields the power corrections to the quark form factor.

After the substitution $\m^2 = Q^2$ and integration over $ d (\ln \ Q^2)$, we find
in LLA (for comparison, see Eq. (\ref{npc0})):
$$
F_q^{ren}(Q^2) = $$ \be = \exp\[- {2C_F \over \b_0} \[\ln {Q^2 \over \L^2}
\ln{\ln(Q^2/\L^2) \over \ln(Q_0^2/\L^2)} - \ln {Q^2 \over Q_0^2}
\(1 - \ln{\ln(Q_0^2/\L^2) \over \ln(\l^2/\L^2)} \) \] -  \ln {Q^2 \over Q_0^2}
\f_{ren} (\l^2, \L^2) \] F^{ren}(Q_0^2)\ , \label{npc}
\ee
where the function $ \f_{ren}(\l^2, \L^2) = \sum_{k=0} \f_k (\L^2/\l^2)^k$
accumulates the effects of
the IR renormalons, as well as the other nonperturbative information. The coefficients
$\f_k $ cannot be calculated in perturbation theory and can be treated
as the minimal set of nonperturbative parameters.  It is worth noting that
the logarithmic $Q^2$-dependence of the renormalon induced corrections in the large-$Q^2$
regime is factorized, and thus the Eq. (\ref{npc}) reproduces
exactly the structure of nonperturbative contributions found in the
one-loop evolution equation (\ref{npc0}).

\section{Large-$Q^2$ behaviour of the instanton induced contribution}

Let us consider the instanton induced corrections to the perturbative result. The instanton field
is given by
\be \hat A_\m (x; \r) = A^a_{\m} (x; \r) {\sigma^a \over 2} = {1 \over g}
 {\hbox{\bf R}}^{ab} \sigma^a {\eta^{\pm}}^b_{\m\n} (x-z_0)_\n \vf
(x-z_0; \r) , \label{if1}\ee
where ${\hbox{\bf R}}^{ab}$ is the color orientation matrix $(a = {1,..., (N_c^2-1)}, b=1,2,3)$,
$\sigma^a$'s are the Pauli matrices,
and $(\pm)$ corresponds to the instanton, or anti-instanton.
The averaging of the Wilson operator over the nonperturbative vacuum is reduced to the integration
over the coordinate of the instanton center $z_0$, the color orientation and the
instanton size $\r$.  The measure for the averaging over the instanton ensemble
reads $dI = d{\hbox{\bf R}} \ d^4 z_0 \ dn(\r) $, where
$ d{\hbox{\bf R}}$ refers to the averaging over color orientation,
and $dn(\r)$ depends on the choice of the instanton size distribution.
Taking into account (\ref{if1}),
we write the Wilson integral (\ref{1a}), which defines the instanton induced
contribution to the nonperturbative part in (\ref{1}), in the single instanton approximation
in the form:
\be
W_I(C_\c) = {1\over N_c}  \<0| \Tr  \exp \( i \sigma^a \phi^a \)|0\> \ ,
\label{wI1}\ee
where
\be \phi^a(z_0,\rho) =
{\hbox{\bf R}}^{ab} {\eta^{\pm}}^b_{\m\n} \int_{C_\g} \! dx_\m \ (x-z_0)_\n
\vf (x-z_0; \r) \ . \label{iin} \ee We omit the path
ordering operator $\pa$ in (\ref{wI1}) because the instanton field
(\ref{if1}) is a hedgehog in color space, and so it locks the color
orientation by space coordinates. Although
in certain situations, the integrals of this type (Eq. (\ref{iin})) can be evaluated
explicitly \cc{Sh}, the calculation of the
total integral (\ref{iin}) for a given contour requires an additional
work, so we must restrict
ourselves with the weak-field approximation.
In contract to our previous paper \cc{DCH}, we use here the cutoff $\l^2$ to
regularize the IR divergences in the instanton case, while the UV
divergences do not appear at all due to the finite instanton size.
Then, in case of the instanton field, the LO contribution reads
\be
W_I^{(1)}(C_\c) = 2 h(\c) \int\! dn(\r) \ \D_1^I(0, \r^2\l^2) \ ,
\ee where
\be
\D_1^I(0, \r^2\l^2) = -  \int \! {d^4 k \over (2\pi)^4} \ex^{ikz}
\d(z^2) \[2 \tilde \vf'(k^2; \r)\]^2\ . \ee
Here, $\tilde \vf (k^2; \r)$
is the Fourier transform of the instanton profile
function $\vf (z^2; \r)$ and $\tilde \vf'(k^2; \r)$ is it's derivative with respect to
$k^2$.
Note, that for the instanton calculations, it is necessary to map
the scattering angle, $\c$, to the Euclidean space by the analytical continuation \cc{EUC}
$ \c \to i\g \ $, and perform the inverse transition to the Minkowski space-time in the final
expressions in order to restore the $Q^2$-dependence.
In the singular gauge, when the profile function is
\be
\vf (z^2) = {\r^2 \over z^2 (z^2 + \r^2)} \ ,
\ee one gets:
\be
\D_1^I (0, \r^2\l^2) = {\pi^2 \r^4 \over 4} \[\ln (\r^2 \l^2) \ \F_0(\r^2\l^2) + \F_1(\r^2\l^2)
\]\ ,
\ee where
\be
\F_0 (\r^2\l^2) = {1 \over \r^4\l^4} \int_0^1\! {dz \over z (1-z)} \ \[1+ \ex^{\r^2\l^2} -
2 \ex^{z \cdot \r^2\l^2} \]
\ \ , \ \ \lim_{\l^2\to 0}\F_0(\r^2\l^2) = 1 \ ,
\ee and
\be
\F_1(\r^2\l^2) = \sum_{n=1}\int_0^1 \! dxdydz \ {[-\r^2\l^2 (xz + y(1-z))]^n \over n!
n}\ex^{\r^2\l^2 [xz + y(1-z)] } \ \ , \ \ \lim_{\l^2\to 0}\F_1(\r^2\l^2) = 0
\
\ee are the IR-finite expressions. At high energy the instanton induced contribution
is reduced to the form:
\be
{d \ w_I(Q^2) \over d \ \ln\ Q^2} = {\pi^2 \over 2}
\dI \ \r^4 \[\ln (\r^2 \l^2) \ \F_0(\r^2\l^2) + \F_1(\r^2\l^2)
\] \equiv - B_I(\l^2)\ .  \label{II1}\ee
Here we used the exponentiation \cc{DCH} of the single-instanton result
in a dilute instanton ensemble $ W_I = \exp\(w_I\)$ and took only the LO term of the weak-field
expansion (\ref{g1}): $\w1 \to w_I $.

In order to estimate the magnitude of the instanton induced effect
we consider the standard  distribution function \cc{tH} supplied with the
exponential suppressing factor, what has been suggested in
\cc{SH2} (and discussed in \cc{DEMM99} in the framework of constrained instanton
model) in order to describe the lattice data \cc{LAT}:
\be
dn(\r) = {d\r \over \r^5} \ C_{N_c} \(2\pi \over \a_s(\m_r) \)^{2N_c} \exp\(- {2\pi \over \a_s (\m_r)}
\) \(\r\m_r\)^{\b} \exp\(- 2 \pi \s \r^2\) \ , \label{dist1}
\ee
where the constant $C_{N_c}$ is
\be
C_{N_c} = {0.466 \ \ex^{-1.679 N_c} \over (N_c-1)! (N_c-2)!}\approx 0.0015 \ ,
\ee
$\s$ is the string tension,  $\b= \b_0+O(\a_s(\m_r))$, and $\m_r$ is
the normalization point \cc{MOR}. Given the distribution (\ref{dist1})
the main parameters of the instanton liquid model---the mean
instanton size $\bar \r$ and the instanton density $\bar n$---will read:
\be
\bar  \r = {\G(\b/2 - 3/2) \over \G(\b/2 - 2)} {1 \over \sqrt{2 \pi \s} } \ , \ee
\be
\bar n = {C_{N_{c}} \G (\b/2 - 2) \over 2} \(2\pi \over \a_S(\bar \r^{-1}) \)^{2N_c}
\({\L_{QCD} \over \sqrt{2\pi \s}}\)^{\b} (2\pi \s)^2 \ . \label{nbar}
\ee In Eq. (\ref{nbar}) we choose, for convenience, the normalization scale
$\m_r$ of order of the instanton inverse mean size $\bar\r^{-1}$.
Note, that these quantities correspond to the mean size $\r_0$ and
density $n_0$ of instantons used in the model \cc{ILM}, where the size distribution
(\ref{dist1}) is approximated by the delta-function :
$
dn(\r) = n_0 \d(\r-\r_0) d\r \ .$

Thus, we find the leading instanton contribution (\ref{II1}) in the form:
\be
B_I=  K \pi^2 \bar n {\bar \rho}^4 \ln{2\pi \s \over \l^2} \(1 + O\({\l^2\over 2\pi\s}\)\)
, \label{pow1} \ee
where
\be
K = {\G(\b_0/2) [\G (\b_0/2-2)]^3 \over 2 \ [\G(\b_0/2-3/2)]^4} \approx
0.74 \ ,
\ee
and we used the one loop expression for the running coupling constant
\be
\a_s (\bar \r^{-1}) = - {2\pi \over \b_0 \ln \ {\bar \r \L}} \ \ ,
 \ \ \b_0 = {11 N_c -
2 n_f \over 3}\ \ . \ee
The packing
fraction $ \pi^2 \bar n {\bar \rho}^4 $ characterizes diluteness
of the instanton liquid and within the conventional picture its value is estimated to be
$ 0.12 \ ,$
if one takes the model parameters as (see \cc{REV}):
\be  {\bar n} \approx 1 fm^{-4},
\ \ {\bar \r} \approx 1/3 fm \ ,
\ \ \s \approx (0.44 \ GeV)^2. \label{param} \ee
The leading contribution to the quark form factor at asymptotically large $Q^2$
is provided by the (perturbative) evolution governed by
the cusp anomalous dimension (\ref{cp}).
Thus, the instantons yield the sub-leading effects
to the large-$Q^2$ behaviour accompanied by a
numerically small factor \be B_I   \approx 0.02 \ , \ee
as compared to the perturbative term
$\frac{2C_F}{\beta_0}\approx0.24$.

Therefore, from Eqs. (\ref{II1}) and (\ref{npc}),
we find the expression for the quark form factor at large-$Q^2$
with the one-loop perturbative contribution and the nonperturbative contributions
(the function $W_{np}$ in Eq. (\ref{npc0})) which include both the IR renormalon
and the instanton induced terms:
$$
F_q(Q^2) = $$ \be = \exp\[- {2C_F \over \b_0} \(\ln {Q^2 \over \L^2}
\ln{\ln(Q^2/\L^2) \over \ln(Q_0^2/\L^2)} - \ln {Q^2 \over Q_0^2}
\( 1 -\ln{\ln(Q_0^2/\L^2) \over \ln(\l^2/\L^2)} \) \)  - \ln {Q^2 \over Q_0^2}
\(B_I + \f_{ren}  \) \]
\ .  \label{eq:final} \ee
It is clear, that while the asymptotic (double-logarithmic) behaviour is controlled by the
perturbative cusp anomalous dimension,
the leading nonperturbative corrections results in a finite renormalization of the
next-to-leading (logarithmic) perturbative term.

We have to comment that the weak field limit used in the instanton calculations
may deviate from the exact result. Nevertheless, we expect that using of the instanton
solution in the singular gauge, that concentrate the field at small distances,
leads to the reasonable numerical estimate of the full effect.
Thus, the resulting diminishing of the instanton contributions with respect
to the perturbative result appears to be reasonable output.

\section{Conclusion}

We analyzed the structure of the nonperturbative corrections to the quark
form factor at large momentum
transfer. The minimal set of the nonperturbative parameters is found considering infrared renormalon
ambiguities of the perturbative series. In order to model the nonperturbative effects
we studied the quark scattering process in the background of the instanton vacuum.
The instanton induced contribution to the color singlet quark form factor is calculated in
the large momentum transfer regime. It was shown that the instanton induced corrections
correspond to the leading term proportional to $\ln{(Q^2)}$
that was found from the IR renormalon analysis to be permitted for nonperturbative contribution.
The magnitude of these corrections is determined by the small instanton liquid packing fraction parameter,
and they can be treated as finite renormalization of the subleading perturbative
part (\ref{eq:final}).

Let us emphasize that our results are quite sensitive to the prescription how to  make the integration
over instanton sizes finite. For example, if one used the sharp cutoff then the instanton
would produce strongly suppressed power corrections like $\propto (\Lambda/Q)^{\beta_0}$.
However, we think that the distribution function (\ref{dist1}) should be considered
as more realistic, since it reflects more properly the structure of
the instanton ensemble modeling the QCD vacuum.
Indeed, this shape of distribution was recently advocated in \cc{SH2, DEMM99} and supported by the lattice
calculations \cc{LAT}. Finally, we think that
the instanton induced effects may become more important in the hadronic processes with
the squared momentum transfer $-t$ small compared to the hard characteristic
scale, {\it e. g.,} the total center-of-mass energy: $-t << s$. The explicit
evaluation of the instanton effects in such processes will be the subject of
forthcoming study.
\section{ Acknowledgments}
\noindent

We are grateful to A. P. Bakulev, N. I. Kochelev,  O. V. Teryaev
and especially to S. V. Mikhailov
for useful discussions.
This work is partially supported by RFBR (Grant nos. 02-02-81023, 02-02-16194,
01-02-16431), the Heisenberg-Landau
program (Grant no. HL-2002-13),  and INTAS (Grant no. 00-00-366). The work of
I. Ch. is supported
also by the RFBR under Grant no. 00-15-96577. He is also grateful to
the Theory group at DESY
(Hamburg), and especially to F. Schrempp,  for invitation, kind hospitality
and financial
support.
\vspace{.3cm}
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