%Paper: 
%From: KORCHEMSKY@VAXPR.CINECA.IT
%Date: Fri, 30 OCT 92 16:55 GMT


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%           "Structure function for large $x$             %
%           and renormalization of Wilson loop"           %
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%            G.P.Korchemsky and G.Marchesini              %
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%             26 pages, 6 figures (included)              %
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\begin{document}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\thispagestyle{empty}

\hfill\parbox{35mm}{{\sc UPRF--92--354} \par
                          \par
                     September, 1992}


\vspace*{15mm}

\begin{center}
{\LARGE Structure function for large $x$
\\[4mm]
and renormalization of Wilson loop
\footnote{Research supported in part by Ministero dell'Universt\`a
e della Ricerca Scientifica e Tecnologica.}
}
\vspace{22mm}

{\large G.~P.~Korchemsky}%
\footnote{On leave from the Laboratory of Theoretical Physics,
          JINR, Dubna, Russia}
\footnote{INFN Fellow}

\medskip

and

\medskip

{\large G.~Marchesini}

\bigskip

\medskip

{\em Dipartimento di Fisica, Universit\`a di Parma and \par
INFN, Gruppo Collegato di Parma, I--43100 Parma, Italy} \par
{\tt e-mail: korchemsky@parma.infn.it; marchesini@parma.infn.it}

\end{center}

\vspace*{20mm}

\begin{abstract}
We discuss the relation between hard distributions near the phase
space boundary, such as structure function for large $x$, and
Wilson loop expectation values calculated along paths partially
lying on the light-cone. Due to additional light cone singularities,
multiplicative renormalizability for these expectation values is lost.
Nevertheless we establish the renormalization group equation for
the light like Wilson loops and show that it is equivalent to
the evolution equation for the physical distributions. By performing
a two loop calculation we verify these properties and show that the
universal form of the splitting function near the phase space
boundary originates from the cusp anomalous dimension of Wilson loops.
\end{abstract}

\newpage

\sect{Introduction}

It is well known \ci{IR} that for hard distributions such as deep inelastic
structure functions, fragmentation functions, Drell-Yan pair cross section,
jet production etc.\ near the phase space boundary the perturbative expansion
involves large corrections which needed to be summed. This region is
characterized by the presence of two large scales $Q^2$ and $M^2$ with $Q^2
\gg M^2 \gg \Lambda_{QCD}^2$ and at any order in perturbation theory the
leading contributions are given by double logarithmic terms
$(\as^n\log^{2n}Q^2/M^2).$ The leading contributions can be summed and the
result is given by the exponentiation of the one-loop contribution. This
exponentiated form holds also after the summation of the next-to-leading
contributions $(\as^n \log^{2n-1} Q^2/M^2)$. The exponent is simply modified
by two loop corrections in which the term $\as^2 \log^4 Q^2/M^2$ is absent
and the coefficient of $\as^2 \log^3 Q^2/M^2$ is proportional to the one-loop
beta-function. This fact suggests us to apply renormalization group approach
for summing large perturbative contributions to all orders.

The universal properties of hard processes near the phase space
boundary originate from the universality of soft emission, which is
responsible of the double logarithmic contributions. Soft gluon
emission from fast quark line can be treated by eikonal approximation for
both incoming and outgoing quarks. In this approximation quarks behave as
classical charged particles and their interaction with soft gluons can be
described by path ordered Wilson lines along their classical trajectories.
The corresponding hard distribution is then given by the vacuum averaged
product of the time-ordered Wilson line, corresponding to the amplitude,
and the anti-time ordered Wilson line,  corresponding to the complex
conjugate amplitude. The combination of these Wilson lines forms a path
ordered exponential $W(C)$ along a closed path $C$ which lies partially
on the light-cone and depends on the kinematic of the hard process
\be
\lab{def}
W(C) \equiv \vev{0|\CP\exp\left(ig\oint_{C}dz_\mu A^\mu(z)\right)|0}\,,
\qquad
A_\mu(z)=A_\mu^a(z)\lambda^{a}
\ee
where $\lambda^{a}$ are the gauge group generator. Here the gluon operators
are ordered along the integration path $C$ and not according to time.
Notice that on different parts of the path $C$ the gluon fields are
time or anti-time ordered. This expression is different from the usual
Wilson loop expectation value in which one has ordering along the path for
the colour indices of $\lambda^{a}$ and ordering in time for the gluons fields
$A_\mu^{a}(z)$. In eq.~\re{def} we have ordering both of colour matrices
and gauge fields along the path.

In this paper we study the renormalization properties of $W(C)$ for a path
$C$ partially lying on the light-cone and show that renormalization group
(RG) equation for $W(C)$ corresponds to the evolution equation \ci{EQ} for
the structure function or the fragmentation function near the phase space
boundary. In particular, we discuss the relation between the ``cusp
anomalous dimension'' of Wilson loop and the splitting function $P(z)$ for
$z \to 1$. %We explicitly perform the two-loop calculation of $W(C)$ with the
%path $C$ fixed by the kinematics of hard process.

In sect.~2 we establish the relation between $W(C)$ and the structure
function or fragmentation function near the phase space boundary.
In sect.~3 we perform the two-loop calculation in the Feynman gauge
and in the $\MS-$regularization scheme of $W(C)$ with the path $C$
fixed by the kinematics of hard process. In sect.~4 we deduce the RG
equation for $W(C)$ and show in sect.~5 that this equation gives rise to
the evolution equation for the structure function and fragmentation
function near the phase space boundary. Sect.~6 contains some concluding
remarks.

\sect{Wilson loop and structure function in the soft limit}

Consider the deep inelastic process for an incoming hadron of energy $E$,
longitudinal momentum $P_\ell$ and mass $M$ probed by a hard photon
with momentum $q.$ In the infinite momentum frame we have $P_+ \gg P_-$,
where the light-cone variables for the hadron momentum
$P_\mu=(P_+,\bom{P}_T,P_-)$ are defined by
$$
P_+=(E+P_\ell)/\sqrt 2,  \qqquad
P_-=(E-P_\ell)/\sqrt 2,  \qqquad
\bom{P}_T=(P_1,P_2)=0 \,.
$$
In this frame $xP_+$ is the ``+'' component of the momentum of the quark
probed by the hard photon, where $x=-{q^2}/{2 (P\cdot q)}$ is the Bjorken
variable. For $x\to1$ the hard photon probes at short distances  the quark
which carries nearly all hadron momentum. In this region the hard process
is characterized by two scales: the virtuality of the photon $Q^2=-q^2$,
which gives the short distance scale, and the large distance scale
$(1-x)Q^2$.

In this Section we show that, after factorization of leading twist
contributions,  the structure function for $x\to1$ is given in terms
of a generalized Wilson loop in \re{def}. To show this we first make the
leading twist approximation and then we perform the $x\to1$ limit.


\subsection{Summing collinear gluons}

Due to the factorization theorem \ci{FT} the differential cross-section
of this process is given in terms of the structure function and the
partonic cross section. The structure function is a universal distribution
measuring the probability to find in the hadron a parton with $x$
fraction of the $P_+$ momentum. To leading twist ($Q^2 \gg M^2$) the
structure function has the following representation \ci{Col}
\be
F(x,\mu/M)=\int_{-\infty}^{+\infty} \frac{dy_-}{2\pi}
               \e^{-iP_+y_-x} \vev{P|\, \bar\Psi[y;A]
               \; \CP\exp\left(ig\int_0^y dz_\mu A^{\mu}(z)\right)
               \; \gamma_+\, \Psi[0;A]\, |P}
\lab{str}
\,,
\ee
where the vector $y_\mu$ lies on the light-cone with components
$$
y_\mu=(y_+,\bom y_T,y_-)=(0,\bom 0, y_-) \, .
$$
The state $\bra{P}$ describes the hadron with momentum $P$ and mass
$M.$ Integration over $y_-$ fixes the value of the ``$+$'' component
of the total momentum of the emitted radiation to be $(1-x)P_+.$ The
corresponding transverse and ``$-$'' components are integrated over
by fixing $\bom y_T=0$ and $y_+=0.$ Because of these unrestricted
integrations the structure function contains ultraviolet (UV) divergences
which are subtracted at the reference point $\mu.$ The quark field
operators $\bar\Psi[y;A]$ and $\Psi[0;A]$ are defined in the Heisenberg
representation and depend on the gauge fields $A_\mu(x).$ The path
ordered exponential is evaluated along the line between points $0$ and
$y_\mu$ on the light-cone and ensures the gauge invariance of the
nonlocal composite quark--antiquark operator.

To obtain \re{str} one uses the general form of the structure function
\be
F(x,\mu/M)=\int_{-\infty}^{+\infty} \frac{dy_-}{2\pi}
               \e^{-iP_+y_-x} \vev{P|j_+(y)j_+(0)|P}
\lab{gen} \,,
\ee
where
$$
j_\mu(y)=\bar\Psi[y;A]\gamma_\mu\Psi[y;A]
$$
is the electromagnetic current.

Consider for example the one-loop contribution of fig.~1. In the infinite
momentum frame the recoiling momentum $P'$ has component $P'_- \gg P'_+$.
The vertices for the absorption and emission of the real gluon $k$ are
given by
$$
\frac{\ds{P}\gamma_\mu(\ds{P}-\ds{k})}{-2Pk +i0}
\qquad
\mbox{and}
\qquad
\frac{\ds{P'}\gamma_\mu(\ds{P'}+\ds{k})}{2P'k -i0}
$$
respectively. For zero quark mass the two vertices contain the following
singularities:
\bit
\item $P-$collinear:
     $ k_+ \sim P_+, \quad k_-\sim P_-$
\item $P'-$collinear:
     $ k_+\sim P'_+, \quad k_-\sim  P'_-$
\item soft:
     $\qqqquad k_-\sim k_+ \to 0 \,, $
\eit
with $\bom{k}_T^2 = 2k_+k_-$. According to the LNK theorem \ci{LNK},
by summing all diagrams for the structure function both the $P'-$collinear
and the soft singularities cancel. We then focus our attention only into
the $P-$collinear singularities.

Consider the vertex for the absorption of gluon $k$ with gauge
potential $a_\mu(k)$ in the momentum representation
\be
g\frac{\ds{P'}\ds{a}(k)(\ds{P'}+\ds{k})}{2P'k -i0}
\lab{ver}
\,,
\ee
where the gluon field carries longitudinal and transverse polarizations.
It can be easily shown that when the momentum $k$ is collinear to $P$, the
components of $a_\mu(k)$ transverse to $k$ contribute to higher twist and
can be neglected. Thus, only longitudinal polarization survives and the
gauge potential becomes pure gauge field
$$
a_\mu(k)=k_\mu\left(\frac{ya(k)}{yk-i0}\right)\,.
$$
This property allows us to sum to all orders the contribution of
collinear gluons (leading twist approximation) using Ward identities.
Inserting this gauge field in \re{ver} we obtain that the gluon
emission is factorized into
$$
\ds{P'}\cdot g\int \frac{d^4k}{(2\pi)^4}\left(\frac{ya(k)}{yk-i0}\right)
=\ds{P'}\cdot  ig\int d^4x \ J_\mu(x) A_\mu(x)
=\ds{P'}\cdot  ig\int_0^\infty d\tau\ y_\mu A_\mu(y\tau)
\,,
$$
where
$$
J_\mu(x)=y_\mu \int_0^\infty d\tau \ \delta^4(x-y\tau)
$$
is the classical non-abelian eikonal current  of the scattered quark $P'$
in which we neglect higher twists contributions of order $P'_+/P'_-$.
The field operator $\bar\Psi[0;A]$ for the scattered quark $P'$ which
absorbs collinear gluons with potentials $A_\mu(x)$ to this order is given
by
\be
\lab{one}
\bar\Psi[0;A]=\bar\Psi[0;A=0]
\left(1+ig\int_0^\infty dz_\mu A_\mu(z) + \CO(g^2) \right)
\ee
with integration path along $y_-$ axis.

The generalization of the one loop result \re{one} can be obtained
by using Ward identities. Since the absorbed gluons are pure gauge fields
the dependence on $A_\mu(x)$ of the quark operator $\bar\Psi[0;A]$ is
factorized into a pure phase factor
\be
\lab{col1}
\bar\Psi[0;A]=\bar\Psi[0;A=0]\;\Phi_y[0,\infty;A]
\,,
\ee
where $\Phi_y(z_1,z_2)$ is the path ordered exponential
calculated from point $z_1$ to $z_2$ along the direction $y$
$$
\Phi_y[z_1,z_2;A]\equiv \CP\exp\left(ig\int_{z_1}^{z_2} dz_\mu
A_\mu(z)\right)\,.
$$
For the quark field operator $\Psi[y;A]$, which describes absorption of
collinear gluons in the final state, one obtains
\be
\Psi[y;A]=\Phi_{-y}[\infty,y;A]\;\Psi[y;A=0]
\lab{col2}
\,.
\ee
Eqs.\re{col1} and \re{col2} imply that, to leading twist, the product of
electromagnetic currents in \re{gen} can be replaced by
\ba
j_+(y)j_+(0) &\propto& \bar\Psi[y;A]\Phi_{-y}[\infty,y;A]
                       \Phi_y[0,\infty;A]\gamma_+
                       \Psi[0,A]
\nonumber
\\
&=&\bar\Psi[y;A]\CP\exp\left(ig\int_0^y dz_\mu A_\mu(z)\right)\gamma_+
                 \Psi[0;A]
\lab{apr}
\ea
leading to the expression \re{str} for the structure function.
Here, we have used the following properties of the Wilson lines:
\ba
&\bullet&\  \mbox{hermicity:}\qqqquad
\Phi_y^\dagger[0,\infty;A]=\Phi_{-y}[\infty,0;A]
\nonumber \\
&\bullet&\  \mbox{causality:}\qqqquad
\Phi_y[b,c;A]\Phi_y[a,b;A]=\Phi_y[a,c;A]
\lab{cau}
\\
&\bullet&\  \mbox{unitarity:}\qqqquad
\Phi_y^\dagger[a,b;A]\Phi_y[a,b;A]=1
\nonumber
\ea
Notice, that the approximation \re{apr} is valid in the leading twist
limit for an arbitrary values of the scaling variable $x.$

\subsection{Soft approximation for $x\to 1$}

For $x\to 1$ the total ``+'' momentum of the emitted radiation
$(1-x)P_+$ vanishes and the expression for the product of the two
currents in eq.~\re{apr} can be further simplified. Since in this
region the ``+'' component of the momentum of any emitted real
gluon $q_{i}$ is positive and vanishing, the emitted gluons are
either soft ($q_{i+}\sim q_{i-}\to 0$) or collinear to $P'$
($q_{i-}\gg q_{i+} \to 0$). As already recalled, due to the LNK
theorem, in the latter case real and virtual contributions cancel
after resummation of all diagrams. There are also contributions with
hard virtual gluons. They can be factorized and absorbed into the
coefficient function $H(\mu/M)$. Furthermore we ignore quark emission
which does not contribute to the soft limit.

The soft gluons interact with the incoming quark $P$ via eikonal
vertices. In this approximation the quark behaves as a classical
particle with mass $M.$ Therefore, in the $x\to 1$ limit the $A-$dependence
of the quark fields $\bar\Psi[y;A]$ and $\Psi[0;A]$ is factorized
into phase factors similar to path-ordered exponentials found in the
collinear approximation. The path is along a trajectory of a massive
classical particle with momentum $P$
\be
\bar\Psi[y;A]=\bar\Psi[y;A=0]\Phi_{-P}[y,\infty;A]\,,
\qqqquad
\Psi[0;A]=\Phi_P[\infty,0;A]\Psi[0;A=0]
\lab{sof}
\,.
\ee
It should be noticed that similar phase factors \re{col1} and \re{col2}
describe interaction of quark with collinear and soft gluons although
the two phase space regions are different. This is due to the fact
that in both regions gluons do not change the quark state: the collinear
gluons are pure gauge fields with vanishing strength; the soft gluons
interact with quarks via eikonal vertices which preserve the momentum
and polarization of the quarks.

Combining the phase factors \re{apr} and \re{sof} from collinear and soft
approximations we obtain the following representation for the structure
function in the $x\to 1$ limit:
\be
F(x,\mu/M) = H(\mu/M) \int_{-\infty}^{+\infty} P_+\frac{dy_-}{2\pi}
               \e^{iP_+y_-(1-x)} W(C_S)
\lab{main}
\ee
with $H(\mu/M)$ the coefficient function and
\be
W(C_S)
      =\vev{0|\Phi_{-P}[y,\infty;A]\Phi_y[0,y;A]\Phi_P[\infty,0;A]|0}
      \equiv \vev{0|\CP\exp\left(ig\oint_{C_S}dz_\mu A^\mu(z)\right)|0}
\lab{wl}
\;,
\ee
where the index $S$ indicates that the photon probe has space-like momentum.
The integration path $C_S=\ell_1\cup \ell_2\cup \ell_3$ is shown in fig.~2.
The rays $\ell_1$ and $\ell_3$ correspond to two classical trajectories of
a massive particle going from infinity to $0$ with momentum $P$ and from $y$
to infinity with momentum $-P,$ respectively. The segment $\ell_2=[0,y]$
lies on the light-cone. As explained before, since we are dealing with a
cross-section rather than with an amplitude, the gauge fields are ordered
along the path rather than the time. Path- and time-orderings are related
as follow
$$
\CP=\CT      \ \mbox{for $\ell_1$}\;, \quad
\CP=\bar \CT \  \mbox{for $\ell_3$}\;, \quad
\CP=\CT      \  \mbox{for $\ell_2$ and $y_->0$}\;, \quad
\CP=\bar \CT \  \mbox{for $\ell_2$ and $y_-<0$}\;, \quad
$$
where $T$ and $\bar T$ denote time- and anti-time ordering, respectively.
This path has two cusps at the points $0$ and $y$ where the direction of
the particle is changed by the hard photon probe.

In agreement with the factorization theorems \ci{FT} the expression
\re{main} is given by the product of two functions: the Fourier transform
of the Wilson loop expectation value, which takes into account all
collinear singularities as $x\to 1,$ and the coefficient function, which
takes into account hard contributions from the interaction of quarks and
gluons at short distances. The coefficient function does not depend on
$P_+(1-x)$ or, equivalently, on $y_-.$

\subsection{Fragmentation function in the $x\to 1$ limit}

The Wilson loop representation for the structure function in \re{main}
can be easily generalized for the fragmentation function $D(x,\mu/M)$
in the limit $x\to 1.$ This function gives the probability of finding a
hadron in a quark jet. To the leading twist approximation the fragmentation
function is given by \ci{Col}
\be
D(x,\mu/M)=\int_{-\infty}^{+\infty}\frac{dy_-}{2\pi}
               \e^{-iP_+y_-/x}
             \sum_f \vev{P,f|\bar\Psi[y;A]\Phi_y[\infty,y;A]|0}
                    \gamma_+\vev{0|\Phi_{-y}[0,\infty;A]\Psi[0;A]|P,f}
\lab{fra}
\ee
where $P_\mu=(P_+,\bom 0,P_-)$ is momentum of the observed hadron,
$y=(0,\bom 0, y_-)$ is a vector in coordinate space which lies on
the light-cone, and $f$ denotes the associated final state radiation.
In the infinite momentum frame with $P_+\gg P_-$ integration over
$y_-$ fixes the ``+'' component of the total associated radiation
momentum to be $(1/x-1)P_+$. As in Subsection~2.1, the phase factors
in eq.\re{fra} are obtained from the resummation of gluons interacting
with the recoiling quark $P'$ and collinear to the observed hadron $P$.

In the limit $x\to 1$ the associated radiation becomes soft and we can
perform the eikonal approximation as in the previous section. Namely,
the dependence of gauge fields in the quark and anti-quark field operators
in \re{fra} factorizes into two phase factors as in eq.\re{sof}. In
the soft ($x\to 1$) limit we obtain the following representation for
the fragmentation function
$$
D(x,\mu/M) = H(\mu/M)\int_{-\infty}^{+\infty}P_+ \frac{dy_-}{2\pi}
               \e^{iP_+y_-(1-1/x)} W(C_T)
\,.
$$
The generalized vacuum averaged Wilson loop operator $W(C_T)$ is given by
\baa
W(C_T)
       &=&\sum_f\vev{f|\Phi_P[y,\infty;A]\Phi_y[\infty,y;A]|0\rangle
               \langle 0|\Phi_{-y}[0,\infty;A]\Phi_{-P}[\infty,0;A]|f}
\\
   &=&\vev{0|\Phi_{P}[y,\infty;A]\Phi_y[0,y;A]\Phi_{-P}[\infty,0;A]|0}
\\
    &\equiv& \vev{0|\CP\exp\left(ig\oint_{C_T}dz_\mu A^\mu(z)\right)|0}
\,,
\eaa
where we have used completeness condition for the associated radiation
$1=\sum_f |f\rangle\langle f|$ and causality \re{cau} for the phase factors.
The index $T$ indicates that the photon probe has time-like momentum.

The integration path $C_T=\ell_1\cup \ell_2\cup \ell_3$ is shown in fig.~3
and three parts $\ell_i$ have meanings similar to that described before
for the space-like process. In this case path- and time-orderings are
related as follows
$$
\CP= \CT    \    \mbox{for $\ell_1$}\;,
\quad
\CP= \bar \CT        \   \mbox{for $\ell_3$}\;,
\quad
\CP=\CT         \    \mbox{for $\ell_2$ and $y_->0$}\;,
\quad
\CP=\bar \CT    \   \mbox{for $\ell_2$ and $y_-<0$}
$$
This path has two cusps at the points $0$ and $y$ where the direction of
the particle is changed by the hard photon probe.


\subsection{One-loop calculation}

We perform the one-loop calculation of $W(C_S)$ in order to explain
in this formulation the nature and origin of double logarithms and to
discuss the analytical properties of $W(C_S)$ in the $y_-$ variable.

We parameterize the integration path $C_S=\ell_1\cup \ell_2\cup \ell_3
=\{z_\mu(t); t\in (-\infty,+\infty)\}$ as follows
$$
z_\mu(t)=\left\{\begin{array}{ll}
                \n_\mu t,& \qquad -\infty < t < 0 \\
                 y_\mu t,&  \qquad 0 < t < 1       \\
                 y_\mu-{\n}_\mu(t-1),&  \qquad 1 < t < \infty
               \end{array}
         \right.
\qqqquad n_\mu \equiv \frac {P_\mu}{M}.
$$
The relevant diagrams are given in fig.~4 plus the symmetric ones. We
perform the calculation in the Feynman gauge and in coordinate
representation by using the dimensional regularization. As we shall
see in the next section this representation is very convenient for
the calculation of $W(C_S)$ to two loops. Feynman rules in coordinate
representation are summarized in the Appendix.

The path from point $0$ to $y$ consists actually of the product of
paths from $0$ to $\infty$ and from $\infty$ to $y$ as shown in fig.~5
for the diagram of fig.~4a. The gluon is either real (fig.~5a) or
virtual (fig.~5b). In the both cases the gluon is emitted at point
$z_1=\n t_1$ with $t_1 < 0.$ For real gluon we have $z_2=yt_2$ with
$1<t_2<\infty,$ while for virtual gluon we have $z_2=yt_2$ with
$0<t_2<\infty.$ The two contributions are given by
\be
W^{(1)}_a=(ig)^2C_F (\n y) \int_{-\infty}^0 dt_1
\left\{\int_0^\infty dt_2\ D(z_2-z_1)+\int_\infty^1 dt_2\ D_+(z_2-z_1)
\right\}
\lab{l1a}
\ee
where $C_F$ is the quadratic Casimir operator in the quark representation.
For $y_-$ positive the vector $z=(z_2-z_1)$ is time-like
($z^2>0$ and $z_0>0$) so that cutted and full propagators coincide:
$D_+(z)=D(z).$ Then, the sum of the two diagrams of fig.~5 is
\be
W^{(1)}_a=-g^2 C_F (\n y) \int_{-\infty}^0 dt_1 \int_0^1 dt_2\ D(yt_2-nt_1)
\lab{sum}
\ee
The same result is obtained for $y_-$ negative by deforming the
integration path $\ell_1$. To show this observe first that the
singularities of the integrand in the $t_1-$complex plane lie in the
lower half plane. Therefore we can deform the integration path from
$-\infty <t_1 < 0$ to $0 < t_1 < \infty.$ After this deformation the
vector $yt_2-nt_1$ becomes time-like and we have the same situation as
for $y_-$ positive.

Both the real and virtual contributions in \re{l1a} have infrared (IR)
and UV singularities. The real and virtual IR singularities, coming from
$t_2\to \infty$, cancel in \re{sum}, giving a bonded range for $t_2$.
This cancellation between the two paths from point $y$ to $\infty$ is
general since it is the result of causality of Wilson lines in \re{cau}.
For this reason in fig.~4 we neglect these parts of the integration paths.

Integral in eq.\re{sum} is ultraviolet divergent and we use dimensional
regularization. Using the $D-$dimensional propagator given in the
Appendix we obtain
$$
W_a^{(1)}  =
\gdo \mu^{4-D}\Gamma(D/2-1)C_F(ny)\int_{-\infty}^0 dt_1\int_0^1 dt_2
\left(-t_1^2+2(ny)t_1t_2+i0\right)^{1-D/2}
$$
which is divergent for $D=4.$ To see the origin of these divergences
we change the integration variable to $t_1=-2(ny)t_2t_1'$ and obtain
$$
W_a^{(1)}
=-\frac{g^2}{8\pi^{D/2}}C_F\Gamma(D/2-1)(2\mu(ny)-i0)^{4-D}
\int_0^\infty dt_1' (t_1'(1+t_1'))^{1-D/2}
\int_0^1 dt_2 t_2^{3-D}
$$
For $D\to 4$ we find two  singularities: for $t_2\to 0$ and $t_1' \to 0.$
In the first case we have $z_2-z_1\to 0$, which corresponds to the
integration near the cusp at point $0$. This singularity is called
``cusp singularity'' \ci{away}. For $t_1'\to 0$ we have $z_1\to 0$
and $(z_2-z_1)^2 \to z_2^2=0$ which corresponds to the light-cone
(collinear) singularity. The final unrenormalized result for this diagram
is
\be
W_a^{(1)} =-\gdo C_F(2\mu(ny)-i0)^{4-D}
\frac{\Gamma(3-D/2)\Gamma(D-3)}{(4-D)^2}
\lab{res:a}
\ee
For the diagram of fig.~4b we obtain
\baa
W_b^{(1)}
&=&-(ig)^2 C_F \int_{-\infty}^0 dt_1 \int_0^\infty dt_2\, D_+(y-\n(t_2+t_1))
 \\
   &=&-\gdo \mu^{4-D} \Gamma(D/2-1)C_F\int_{-\infty}^0 dt_1
   \int_0^\infty dt_2
\\ &\times &
   \left[2(y_- -\n_-(t_2+t_1)-i0)(\n_+(t_2+t_1)+i0)\right]^{1-D/2}
\eaa
This integral for $D\to 4$ has an infrared singularity for $z_2-z_1
\to\infty.$ Taking $D > 4$ one gets
\be
W_b^{(1)}
=\gdo C_F(2\mu(ny)-i0)^{4-D}\frac{\Gamma(3-D/2)\Gamma(D-3)}{4-D}
\lab{res:b}
\ee
For the diagram of fig.~4c we have
\ba
W_c^{(1)} &=&\gdo\mu^{4-D}\Gamma(D/2-1)C_F \int_{-\infty}^0
dt_1\int_{t_1}^0dt_2\,(-(t_1-t_2)^2+i0)^{1-D/2}
\nonumber
\\
   &=&-\gdo C_F(\mu^2\n^2)^{2-D/2}\frac{\Gamma(D/2-1)}{3-D}
     \int_{-\infty}^0 dt_1 t_1^{3-D}
\lab{res:c}
\,.
\ea
In this case we have an IR singularity for $D < 4$ as $t_1-t_2\to\infty$
and an UV divergence for $D > 4$  as $t_1\to 0$ and $t_1-t_2\to 0.$
The IR divergence of $W_c^{(1)} $ is canceled by the IR divergence of
$W_b^{(1)} .$ Thus, the sum of the diagrams $W_b^{(1)} +W_c^{(1)}
$ contains only UV divergence. This IR cancellation does not depend on
the scheme we use to regularize IR divergences, \eg by putting a cutoff
in the $t_i-$integration or by giving a fictitious mass to the gluon.

Notice, that the IR and UV poles in $W_c^{(1)}$ have opposite coefficients,
thus one can formally set $W_c^{(1)} =0.$ In this case however the pole
of $W_b^{(1)} $ for $D=4$ has to be interpreted as an UV singularity.

Summing eqs.\re{res:a}, \re{res:b} and the symmetric contributions
we obtain the one-loop expression for the unrenormalized Wilson loop
\re{wl}
\be
W^{(1)}=\gdo C_F(2\mu(\n y)-i0)^{4-D}\Gamma(3-D/2)\Gamma(D-3)
\left(-\frac2{(4-D)^2}+\frac1{4-D}\right).
\lab{res:un}
\ee
By subtracting the poles in the $\MS-$scheme we obtain
\be
W^{(1)}_{ren.}=\alpi C_F\left(-L^2 + L - \frac5{24}\pi^2\right),
\qquad L=\log(\rho - i0)
\lab{res:re}
\ee
where
\be
\rho=(\n y)\mu =(Py)\frac\mu{M},
\qqqquad
y=(0,\bom 0,y_-),
\qqqquad
n=P/M\,.
\lab{res:re'}
\ee
{}   From eqs.\re{res:un} and \re{res:re} we can directly see that
$W(C_S)$ to one loop depends only on the variable $\rho$ and the possible
singularities are in the upper half plane
\be
W(C_S)=W(\rho-i0)\,.
\lab{ana}
\ee
This is due to the fact that $\rho$ is the only scalar dimensionless
variable formed by $n$, $y$ and $\mu$. The ``$-i0$'' prescription
comes from the position of the pole in the free gluon propagator in the
coordinate representation.

\sect{Two-loop calculation}

In this section we perform the two-loop calculation of $W(C_S)$.
In order to simplify the calculation and reduce the number of diagrams
to compute we use the nonabelian exponentiation theorem \ci{ET}.
According to this theorem we can write
$$
W \equiv 1+\sum_{n=1}^\infty \left(\alpi\right)^n W^{(n)}
      =  \exp \sum_{n=1}^\infty \left(\alpi\right)^n w^{(n)}
$$
where $w^{(n)}$ is given by the contributions of $W^{(n)}$ with the
``maximal nonabelian color and fermion factors'' to the $n-$th order
of perturbation theory. At one loop we have only the color factor $C_F.$
At two-loops the maximal nonabelian color factor is $C_AC_F$ and the
fermionic factor is $C_FN_f.$ Here, $C_A$ is the quadratic Casimir
operator in the gluon representation and $N_f$ the number of light
quarks. From this theorem we have $W^{(1)}=w^{(1)}$ and
$$
W^{(2)}=\half(w^{(1)})^2 + w^{(2)}
$$
and the contributions with the abelian color factor $C_F^2$ are
contained only in the first term.

In fig.~6 we list all nonvanishing the two-loop diagrams which contain
color factors $C_FC_A$ and $C_FN_f.$ As for the one loop case, the paths
in these diagrams do not contain the two rays from $y$ to $\infty$ because
their contributions cancel due to causality of Wilson lines in \re{cau}.

The color factor of abelian-like diagrams of figs.~6.1-6.6 is
$C_F(C_F-\half C_A).$ These diagrams contribute to $w^{(2)}$ with
the color factor $-\half C_FC_A.$ Diagrams with gluon self-energy of
figs.~6.7 and 6.8 will contain contributions with both color factors
$C_FC_A$ and $C_FN_f.$ The latter one comes from the quark loop
contribution. Finally, diagrams of figs.~6.9-6.11 are of nonabelian
nature involving three-gluon coupling and their color factor is
proportional to $C_FC_A.$ We have omitted diagrams with abelian color
factor $C_F^2$, diagrams vanishing due to the antisymmetry of three-gluon
vertex, diagrams proportional to $y^2=0$ and self-energy like diagrams
obtained by iterating the one-loop diagram in fig.~4c. The reason for
neglecting of the latter type of diagrams is the following. As in the
one-loop case the self-energy diagrams have both IR and UV poles which
cancel each other. On the other hand in the sum of all two loop diagrams
the IR singularities cancel completely. Therefore, as we did for one-loop
case, we neglect the self-energy diagrams and interpret the IR poles in
the diagrams of fig.~6 as UV singularities.

For the diagrams of figs.~6.2--6.5 and figs.~6.8 and 6.10 we have to
consider contributions involving cutted propagators. As we have observed
before, there is no difference between real and virtual gluon propagating
in the time-like direction ($D(z)=D_+(z)$ for time-like $z$). This gives
rise to a partial cancellation between real and virtual contributions.
This cancellation has been already used to simplify the analysis in the
one-loop case \re{res:c} and we will further exploit it in this section.
It turns out that to compute singular contributions for $D=4$ we can
replace the cutted propagators in fig.~6 by full ones. This is due to
the fact that one can always deform the integration path in such a way
that gluons propagate in the time-like direction.


\subsection{Abelian like diagrams}

The general form of the contribution of the abelian like diagrams of
figs.~6.1--6.6 is ($a=1,\ldots,6$)
\be
W_a=\left(\gdo\right)^2 C_F(C_F-\half C_A) (\rho -i0)^{8-2D} I_a(D)
\lab{abe}
\ee
where $I_a(D)$ is a function of $D$ and the index $a$ refers to the
diagrams in fig.~6 and $\rho$ is given in \re{ana}.

Diagram of fig.~6.1 gives
$$
W_1=g^4 C_F(C_F-\half C_A) (\n y)
\lot{-\infty}0{t_1}0{t_2}001 \, D(z_3-z_1)D(z_4-z_2)
$$
with
$$
D(z_3-z_1)D(z_4-z_2)=
\coef
\left[ (t_1-t_3)^2t_2(t_2-\frac{y_-}{\n_-}t_4)\right]^{1-D/2}
$$
where $z_i=\n t_i,$ $i=1,2,3$ and $z_4=yt_4.$
This diagram contain UV divergence for $z_1-z_3\to 0$ corresponding to
vertex renormalization. The rest of the diagram has the same divergences
as one loop diagram of fig.~4a. Namely, it has a cusp singularity for
$z_4-z_2\to 0$ and light-cone collinear singularity for $z_2\to 0.$
Performing the integration we obtain
$$
I_1=-\frac{\Gamma(D/2-1)\Gamma(7-3D/2)\Gamma(2D-7)}{6(3-D)(4-D)^3}
$$
Diagram of fig.~6.2 gives
$$
W_2=-g^4 C_F(C_F-\half C_A)
\lot{-\infty}0{t_1}0{t_2}00\infty \,D(z_3-z_1)D_+(z_4-z_2)
$$
with
$$
D(z_3-z_1)D_+(z_4-z_2)=\coef
\left[(t_1-t_3)^{2}(t_2+t_4+i0)
(t_2+t_4-\frac{y_-}{\n_-}+i0)\right]^{1-D/2}
$$
where $z_i=\n t_i,$ $i=1,2,3$ and $z_4=y-\n t_4.$ The analysis of
singularities is similar to the previous case. We have a UV singularity
for $z_1-z_3\to 0$ and a IR pole originated from one-loop diagram of
fig.~4b. The result of the integration is
$$
I_2=\frac{\Gamma(D/2-1)\Gamma(7-3D/2)\Gamma(2D-7)}{2(3-D)(4-D)^2(5-D)}
$$
Diagram of fig.~6.3 gives
$$
W_3=-g^4 C_F(C_F-\half C_A) (\n y)^2
\lot{-\infty}001{t_2}10\infty \,D(z_3-z_1)D_+(z_4-z_2)
\,,
$$
where $z_1=\n t_1,$ $z_2=yt_2,$ $z_3=yt_3$ and $z_4=y-\n t_4.$ Since
all singularity in $t_4-$plane lie at the lower half-plane we deform
the $t_4-$integration path from $[0,\infty)$ to $(-\infty,0].$ After
this transformation $z_4$ is replaced by $z_4'=y+\n t_4$ and $z_4'-z_2$
becomes time-like vector for which cutted and full propagators are the
same. We can than replace
$$
D(z_3-z_1)D_+(z_4-z_2) \Rightarrow D(z_3-z_1)D_+(z_4'-z_2)
= D(z_3-z_1)D(z_4'-z_2)
$$
which gives
$$
D(z_3-z_1)D(z_4'-z_2) = \coef
\left[t_1 t_2(t_1-\frac{y_-}{n_-}t_3)
(t_4+\frac{y_-}{n_-}(1-t_2))\right]^{1-D/2}
\,.
$$
This diagram has two light-cone collinear singularities for $z_1\to 0$ and
$z_4\to y.$ Evaluating the integral we obtain
$$
I_3=\frac{\Gamma^2(3-D/2)\Gamma^2(D-3)}{(4-D)^4}
\left(1-\frac{\Gamma^2(5-D)}{\Gamma(9-2D)}\right)
\,.
$$
Diagram of fig.~6.4 gives
$$
W_4= g^4 C_F(C_F-\half C_A)
\lot{-\infty}0{t_1}00\infty{t_3}\infty \,D_+(z_3-z_1)D_+(z_4-z_2)
$$
where $z_i=\n t_i$ for $i=1,2$ and $z_j=y-\n t_j$ for $j=3,4.$ As in the
previous diagram we can deform $z_3$ and $z_4$ integration paths by
replacing $z_j=y-\n t_j$ with $z_j'=y+\n t_j$ for $j=3,4$ corresponding to
the replacement
$$
D_+(z_3-z_1)D_+(z_4-z_2) \Rightarrow D_+(z_3'-z_1)D_+(z_4'-z_2)
= D(z_3'-z_1)D(z_4'-z_2)
$$
and we have
$$
D(z_3'-z_1)D(z_4'-z_2)=\coef\left[
(t_4-t_2+\frac{y_-}{n_-})(t_4-t_2)
(t_3-t_1+\frac{y_-}{n_-})(t_3-t_1)
\right]^{1-D/2}.
$$
%\baa
%D(z_3'-z_1)D(z_4'-z_2) &=&
%\\
%& & \hspace*{-40mm}
%\coef\left[
%(t_4-t_2+\frac{y_-}{n_-})(t_4-t_2)
%(t_3-t_1+\frac{y_-}{n_-})(t_3-t_1)
%\right]^{1-D/2}
%\,.
%\eaa
This diagram has a single IR pole for $z_4-z_2\to\infty$ and
$z_3-z_1\to\infty$ simultaneously. Evaluating the integral we obtain
$$
I_4=-\frac1{4-D}\frac{\Gamma(2D-7)}{2\Gamma^2(D/2-1)}+\CO((4-D)^0)
\,.
$$
Diagram of fig.~6.5 gives
$$
W_5=-g^4 C_F(C_F-\half C_A)
\lot{-\infty}0{t_1}0010\infty \,D(z_3-z_1)D_+(z_4-z_2)
\,,
$$
where $z_i=\n t_i$ for $i=1,2,$ $z_3=yt_3$ and $z_4=y-\n t_4.$
Deforming the $z_4$ integration path we replace $z_4$ with $z_4'=y+\n
t_4$
$$
D(z_3-z_1)D_+(z_4-z_2) \Rightarrow D(z_3-z_1)D_+(z_4'-z_2)
= D(z_3-z_1)D(z_4'-z_2)
$$
and we have
$$
D(z_3-z_1)D(z_4'-z_2)=\coef
\left[t_1(t_1-\frac{y_-}{n_-}t_3)(t_4-t_2+\frac{y_-}{n_-})(t_4-t_2)
\right]^{1-D/2}
\,.
$$
It turns out that this diagram has no singularities for $D=4.$
To confirm this we evaluate the integral and obtain
\baa
I_5&=&-\Gamma^2(D/2-1)\left\{-\frac{\pi^2}3 \right.
    +\frac1{(4-D)^3(3-D)}
     \left(
       \frac{3\Gamma(3-D/2)\Gamma(2D-7)}{\Gamma(3D/2-5)}
\right.
\\
& &- \left.\left.
       \frac{\Gamma(5-D)\Gamma(2D-7)}{\Gamma(D-3)}
      -\frac{2\Gamma(3-D/2)\Gamma(D-3)}{\Gamma(D/2-1)}
    \right)
\right\} = \CO((4-D)^0)
\,.
\eaa
Diagram of fig.~6.6 gives
$$
W_6=g^4 C_F(C_F-\half C_A)  (ny)^2
\lot{-\infty}0{t_1}001{t_3}1 \, D(z_3-z_1) D(z_4-z_2)
$$
where $z_i=\n t_i$ for $i=1,2,$ $z_j=yt_j$ for $j=3,4,$
and we have
$$
D(z_3-z_1)D(z_4-z_2)=\coef
\left[t_1t_2(\frac{y_-}{n_-}t_3-t_1)(\frac{y_-}{n_-}t_4-t_2)
\right]^{1-D/2}
\,.
$$
This diagram has two cusps singularities for $z_3-z_1\to 0$ and
$z_4-z_2\to 0$ and two light-cone collinear singularities for $z_1\to 0$
and $z_2\to 0.$ Evaluating the integral we obtain
\baa
I_6&=&\Gamma^2(D/2-1)
   \left\{\frac1{(4-D)^4}
    \left(
       \frac{\Gamma(5-D)\Gamma(2D-7)}{2\Gamma(D-3)}
      -\frac{\Gamma(7-3D/2)\Gamma(2D-7)}{3\Gamma(D/2-1)}
\right.\right.
\\
& &\hspace{-10mm}-\left.\left.
          \frac14(2-D)(3-D/2)\Gamma(D-3)
          \left(\Gamma(5-D)-\frac{\Gamma(3-D/2)}{\Gamma(D/2-1)}
          \right)
    \right)
    -\frac1{4-D}\frac{\zeta(3)}2
%   + const.
\right\}
+\CO((4-D)^0)
\,,
\eaa
where $\zeta(3)$ is the Reimann function.

\subsection{Self-energy diagrams}

The general form of the contribution of the diagrams with gluon
self-energy in figs.~6.7 and 6.8 is ($a=7,8$)
$$
W_a=\left(\gdo\right)^2
C_F\left((3D-2)C_A-2(D-2)N_f\right) (\rho -i0)^{8-2D} I_a(D)
\,.
$$
The one-loop correction to the gluon propagator in the Feynman gauge
in the coordinate representation is given  by
\be
D^{(1)}(z)=\frac{g^2}{64\pi^D}\left((3D-2)C_A-2(D-2)N_f\right)
\frac{\Gamma^2(D/2-1)}{(D-4)(D-3)(D-1)}
(-z^2+i0)^{3-D}
\lab{pro}
\ee
which differs from the free propagator in the power of $z^2-i0$. For
time-like vector $z$ the cutted one-loop propagator coincides with
the full propagator in \re{pro}. As in the one loop case this allows
us to treat the sum of these diagrams with all possible cuts by using
the full propagator. We obtain
$$
W_7 = -g^2C_F(\n y)\int_{-\infty}^0 dt_1\int_0^1 dt_2\ D^{(1)}(z_2-z_1)
$$
where $z_1=\n t_1$ and $z_2=y t_2$ and
$$
W_8 = g^2C_F(\n y)\int_{-\infty}^0 dt_1\int_0^\infty dt_2\
       D^{(1)}_+(z_2-z_1)
$$
where $z_1=\n t_1$ and $z_2=y-\n t_2.$ By performing the integration we get
$$
I_7=\frac{\Gamma^2(D/2-1)\Gamma(5-D)\Gamma(2D-7)}{16(4-D)^3(1-D)\Gamma(D-2)}
\,,
\qqquad
I_8=-\frac{\Gamma^2(D/2-1)\Gamma(5-D)\Gamma(2D-7)}{8(4-D)^2(1-D)\Gamma(D-2)}
$$

\subsection{Diagrams with three-gluon vertices}

For the diagrams of figs.~6.9--6.11 containing three-gluon vertex we have the
following general expression
$$
W_a=\frac12 C_AC_F\left(\gdo\right)^2 (\rho-i0)^{8-2D} I_a(D)
$$
where $a$ refers to the various diagrams in fig.~6 with three-gluon vertex.
We simplify the analysis of the diagrams with various cuts by computing only
the contributions which are singular for $D\to 4$. In this case we can treat
all gluons as virtual. To show this observe that for the cutted diagrams at
$D=4$ we have only infrared and light-cone collinear singularities. The
infrared singularities cancel. The collinear singularities appear when
gluons propagate along the light-cone, \ie when the intermediate point $z_4$
lies on the segment $[0,y].$ In this case cutted propagators coincide
with the full propagators. Notice that cusp singularities appear when all
gluons interact at small distances. They are present only for diagrams of
figs.~6.9 and 6.11 for $z_i\to 0.$ In this case all gluons are virtual.
Therefore, in the following we study only the contributions to $W_a$ in
which all gluons as virtual.

For the diagram of fig.6.9 we have
$$
W_9=\half g^4 C_AC_F \int_{-\infty}^0 dt_1\int_0^1 dt_2\int_{t_2}^1 dt_3
    \int d^D z_4 \,
    \n^{\mu_1}y^{\mu_2}y^{\mu_3}\Gamma_{\mu_1\mu_2\mu_3}(z_1,z_2,z_3)
    \prod_{i=1}^3 D(z_i-z_4)
$$
where $z_1=\n t_1,$ $z_2=y t_2$ and $z_3=y t_3.$ By using the expression
for the three-gluon vertex in the Appendix we find
\be
\n^{\mu_1}y^{\mu_2}y^{\mu_3}\Gamma_{\mu_1\mu_2\mu_3}(z_1,z_2,z_3)
=i(\n y)\left(y\frac{\partial}{\partial z_2}
            -y\frac{\partial}{\partial z_3}
       \right)
=i(\n y)\left(\frac{\partial}{\partial t_2}
            -\frac{\partial}{\partial t_3}
       \right)
\lab{glu1}
\,.
\ee
Due to this particularly simple form of the three-gluon vertex the
integration over $t_2$ or $t_3$ becomes trivial. For the first term the
integration over $t_2$ gives the contributions from the end points $z_2=0$
and $z_2=z_3.$ For the second term the integration over $t_3$ gives the
contributions from $z_3=z_2$ and $z_3=1.$ In all contributions, the
integral over the intermediate point $z_4$ is factorized into the following
expression
\ba
J(z_1,z_2,z_3)&=&\int d^D z_4 \ \ \prod_{i=1}^3 D(z_i-z_4)
\nonumber
\\
& &\hspace{-16mm}= -\frac{i^{D-1}}{32\pi^D}\frac{\Gamma(D-3)}{4-D}
            \int_0^1 ds (s(1-s))^{D/2-2}
            \left((-z_1+sz_2+(1-s)z_3)^2-i0\right)^{3-D}
\lab{id}
\ea
valid for $z_2$ and $z_3$ on the light-cone ($z_2^2=z_3^2=0$).
The pole at $D=4$ corresponds to a light-cone singularity as  $z_4$
approaches the segment $[0,y].$ The meaning of the integral over the
parameter $s$ is the following. For $D=4$ one integrates over the free
gluon propagator between the points $z_1$ and $z=s z_2+(1-s)z_3$.
Since the $z$ lies on the light-cone between $z_2$ and $z_3$ the vector
$z-z_1$ is time-like. This confirms the expectation expressed at the
beginning of this subsection that all collinear gluons propagate in the
time-like direction.

Performing the remaining integration we obtain
$$
I_{9}=-\frac{\Gamma(5-D)\Gamma(2D-7)}{4(4-D)^3}
       \left(\frac{\Gamma^2(D/2-1)}{(4-D)\Gamma(D-3)}
            -\frac{4\Gamma(7-3D/2)\Gamma(D/2-1)}{3(4-D)\Gamma(5-D)}
            +\frac{\Gamma^2(D/2-1)}{\Gamma(D-2)}
       \right)
\,.
$$
For the diagram of fig.~6.10 one deforms the integration path over $z_3=y-\n
t_3$ into $z_3=y+\n t_3$, replaces cutted propagators by full ones and
obtains
$$
W_{10}=\half g^4 C_AC_F \int_{-\infty}^0 dt_1\int_0^1 dt_2\int_0^\infty dt_3
    \int d^Dz_4 \
    \n^{\mu_1}y^{\mu_2}n^{\mu_3}\Gamma_{\mu_1\mu_2\mu_3}(z_1,z_2,z_3)
    \prod_{i=1}^3 D(z_i-z_4)
\,,
$$
where $z_1=\n t_1,$ $z_2=y t_2$ and $z_3=y+\n t_3.$
By using the three-gluon vertex we find
\baa
\n^{\mu_1}y^{\mu_2}n^{\mu_3}\Gamma_{\mu_1\mu_2\mu_3}(z_1,z_2,z_3)
&=&i\left(y\frac{\partial}{\partial z_1}
            -y\frac{\partial}{\partial z_3}
       \right)
 -i(\n y)\left(\n\frac{\partial}{\partial z_1}
            -\n\frac{\partial}{\partial z_3}
       \right)
\\
&=&i\left(y\frac{\partial}{\partial z_1}
            -y\frac{\partial}{\partial z_3}
       \right)
 -i(\n y)\left(\frac{\partial}{\partial t_1}
             -\frac{\partial}{\partial t_3}
       \right)
\eaa
The first term leads to a contribution which is regular for $D=4.$
To see this notice that the singularities arise when the gluons are
propagating along the light-like vector $y.$
However, this configuration is suppressed by applying the operator
$y\frac{\partial}{\partial z_i}$ to the gluon propagator.
The expression $y\frac{\partial}{\partial z_i}D(z_i-z_4)$ is proportional
to $y(z_i-z_4)$ and vanishes for $z_i-z_4$ parallel to $y.$
The second term is similar to the one of the previous diagram.
We have contributions from the end-points $z_1=0$ and $z_3=y.$
For $z_1=0$ ($z_3=y$) the two vectors $z_1$ and $z_2$ ($z_2$ and $z_3$)
lie on the light-cone and we can apply the identity \re{id}.
Performing the remaining integrations we obtain
$$
I_{10}=-\frac{\Gamma(5-D)\Gamma(2D-7)\Gamma(D/2-1)}{2(4-D)^4}
     \left(\frac{\Gamma(D/2-1)}{\Gamma(D-3)}-
           \frac{\Gamma(7-3D/2)}{\Gamma(5-D)}\right)
$$
For the last diagram of fig.~6.11 we have
$$
W_{11}=\half g^4 C_AC_F
       \int_{-\infty}^0 dt_1\int_{t_1}^0 dt_2\int_0^\infty dt_3
       \int d^D z_4\,\n^{\mu_1}\n^{\mu_2}y^{\mu_3}
       \Gamma_{\mu_1\mu_2\mu_3}(z_1,z_2,z_3)
       \prod_{i=1}^3 D(z_i-z_4)
$$
where $z_1=\n t_1,$ $z_2=\n t_2$ and $z_3=yt_3.$ By using three-gluon
vertex we obtain
\ba
\n^{\mu_1}\n^{\mu_2}y^{\mu_3}\Gamma_{\mu_1\mu_2\mu_3}(z_1,z_2,z_3)
&=&-i\left(y\frac{\partial}{\partial z_1}
            -y\frac{\partial}{\partial z_2}
       \right)
 +i(\n y)\left(\n\frac{\partial}{\partial z_1}
            -\n\frac{\partial}{\partial z_2}
       \right)
\nonumber
\\
&=&-i\left(y\frac{\partial}{\partial z_1}
            -y\frac{\partial}{\partial z_2}
       \right)
 +i(\n y)\left(\frac{\partial}{\partial t_1}
            -\frac{\partial}{\partial t_2}
       \right)
\lab{glu2}
\,.
\ea
For this diagram we have the following singularities: a cusp singularity
for $z_i\to 0$ with $i=1,\ldots, 4$; two independent light-cone collinear
singularities for $z_1,z_2\to 0$ and $z_4$ approaching the segment $[0,y]$;
an ultraviolet singularity from $z_2,z_4\to z_1.$

The operator $y\frac{\partial}{\partial z_i}$ in the first term
suppresses propagation along light-like vector $y$ of gluon from
point $z_4$ to $z_1$ or to $z_2.$ This implies that the contribution
of this part of three-gluon vertex contains only a triple pole in $4-D.$

The second term in \re{glu2} is similar to the one in \re{glu1} for diagram
of fig.~6.9. We have two end-point contributions with $z_2=0$ and $z_1=z_2.$
For $z_2=0$ the two vectors $z_2$ and $z_3$ lie on the light-cone and we can
apply the identity \re{id}. For $z_1=z_2$ the integral is similar to the
one of the gluon self-energy correction.

We finally obtain
\baa
I_{11}&=&\frac14\left(
 \frac{\Gamma(7-3D/2)\Gamma(D/2-1)\Gamma(2D-7)}{3(4-D)^4}
-\frac{\Gamma(5-D)\Gamma(2D-7)\Gamma(3-D/2)\Gamma(D/2-1)}{3(4-D)^3}
\right.
\\
& &-\left.
\frac{\Gamma(5-D)\Gamma(2D-7)}{3(4-D)^2}
-\frac{\pi^2}{36}\frac{\Gamma(5-D)\Gamma(2D-7)}{4-D}
             \right)
\,.
\eaa
Recall that all expressions in this subsection are valid up to terms
which are regular for $D=4$.


\subsection{Renormalization at two-loop order}

Since the diagrams of fig.~6 have nested ultraviolet divergences
corresponding to the renormalization of the vertices and propagators
one has to include additional counter-terms. In the $\MS-$scheme their
contributions are given to two loops by
$$
w_{c.t.}^{(2)}=\left(\alpi\right)^2 C_F\left(\frac{11}3C_A-\frac23 N_f\right)
(\rho-i0)^{4-D}\frac{\Gamma(3-D/2)\Gamma(D-3)}{(4-D)^2}
\left(\frac1{4-D}-\frac12\right)\,.
$$
To obtain the final expression for the renormalized $w$ to two-loops
we add the counter-terms, subtract the poles in the $\MS-$scheme and
take into account combinatorial factors. By using the non-abelian
exponentiation theorem $w^{(2)}$ is obtained by omitting the colour
factor $C_F^2$ in \re{abe}. The final contributions to $w^{(2)}$ from
the various diagrams $(a=1,\ldots,11)$ have the following form
$$
w_a^{(2)}=\left(\alpi\right)^2C_F\left[
C_A( A_a L^4 + B_a L^3 + C_a L^2 + D_a L )
+N_f( E_a L^3 + F_a L^2 + G_a L )
+ \CO(L^0)\right]
$$
where $L$ is given in \re{res:re} and for the various diagrams
the nonvanishing coefficients are given by
$$
       A_6=-\fracs19,
\qquad A_9=\fracs1{18},
\qquad A_{11}=\fracs1{18},
$$
$$
       B_1=-\fracs29,
\qquad B_7=-\fracs59,
\qquad B_9=-\fracs13,
\qquad B_{11}=-\fracs19,
\qquad B_{c.t.}=\fracs{11}{18},
$$
$$
       C_1=-\fracs13,
\qquad C_2=1,
\qquad C_3=-\fracs{\pi^2}6,
\qquad C_6=-\fracs7{72}\pi^2,
\qquad C_7=-\fracs{31}{36},
$$
$$
       C_8=\fracs56,
\qquad C_9=\fracs{25}{144}\pi^2-\fracs12,
\qquad C_{10}=\fracs1{12}\pi^2,
\qquad C_{11}=\fracs{13}{144}\pi^2-\fracs16,
\qquad C_{c.t}=-\fracs{11}{12},
$$
$$
       D_1=-\fracs{13}{72}\pi^2-\fracs13,
\quad D_3=2\zeta(3),
\quad D_4=\fracs12,
\quad D_6=\fracs29\zeta(3),
\quad D_7=-\fracs{47}{54}-\fracs5{16}\pi^2,
\quad D_8=\fracs{31}{36},
$$
$$
       D_9=\fracs1{72}\zeta(3)-\fracs12-\fracs3{16}\pi^2,
\qquad D_{10}=-\fracs14\zeta(3),
\qquad D_{11}=-\fracs{13}{144}+\fracs{19}{72}\zeta(3),
\qquad D_{c.t.}=\fracs{55}{144}\pi^2,
$$
$$
       E_7=\fracs29,
\qquad E_{c.t.}=-\fracs19,
$$
$$
       F_7=\fracs5{18},
\qquad F_8=-\fracs13,
\qquad F_{c.t.}=\fracs16,
$$
$$
       G_7=\fracs18\pi^2+\fracs7{27},
\qquad G_8=-\fracs5{18},
\qquad G_{c.t.}=-\fracs5{72}\pi^2.
$$
Summing all contributions we finally obtain
\be
w^{(2)}= \left(\alpi\right)^2C_F
\left( B L^3 + C L^2 + D L + \CO(L^0) \right)
\lab{res:main}
\ee
where
\baa
B&=&-\fracs{11}{18}C_A+\fracs19 N_f,
\\
C&=&\left(\fracs1{12}\pi^2-\fracs{17}{18}\right)C_A
       +\fracs19 N_f,
\\
D&=&\left(\fracs94\zeta(3)-\fracs7{18}\pi^2-\fracs{37}{108}\right)C_A
 +\left(\fracs1{18}\pi^2-\fracs1{54}\right) N_f
\,.
\eaa
This expression has the following properties:
\bit
\item the coefficient of $L^4$ vanishes;
\item the coefficient of $L^3$ is proportional to the one-loop
beta-function;
\item the interpretation of the remaining coefficients in front of $L^2$
and $L$ will be clear from the RG equation for light-like Wilson loop
we are going to discuss in the next section.
\eit

The two-loop calculation confirms the analytical dependence in \re{ana}
which means that all singularities of $W(C_S)$ in the complex $\rho-$plane
lie in the upper half plane. After integration over $y_-$ in \re{main}
this leads to the spectral property
\be
F(x,\mu/M)=0 \qquad \mbox{for $x > 1.$}
\lab{spe}
\ee
It has been observed \ci{break} that the presence in \re{res:main} of the
$L^3$ term seems to be in contradiction with the renormalization
properties of Wilson loops. Assuming that $W(C_S)$ renormalizes
multiplicatively, the $L^3$ term should vanish. However one should notice
that we are dealing with a light-like Wilson loop which has additional
light-cone singularities. Moreover, the fact that the coefficient of $L^3$
is just the one-loop beta-function suggests that light-like Wilson loop
obeys a RG equation. This equation is discussed in the next section.

\sect{Renormalization group equation for Wilson loop on the light-cone}

To evaluate the structure function for $x\to 1$ given in eq.\re{main}
one needs to compute $W(C_S)$ in \re{wl} to all order in perturbation theory.
A powerful method to resum the expansion is the use of renormalization
group equation. In this section, following Ref. \ci{WL}, we deduce the RG
equation of Wilson loops with path partially lying on the light-cone.

Recall that for $x$ away from 1 the operator product expansion on the
light-cone allows us to relate the $\mu-$dependence of the structure
function with the ultraviolet properties of local composite twist-2
operators obtained by expanding in powers of $y_-$ the matrix element in
\re{str}. This dependence is described by the evolution equations \ci{EQ}
\be
\left(\mu\partder{\mu}+\beta(g)\partder{g}\right)F(x,\mu/M)
=\int_x^1 dz\ P(x/z) F(z,\mu/M)
\lab{eq1}
\ee
where the splitting function $P(z)$ is singular for $z\to1$. For $x\to 1$
the structure function \re{main} is given in terms of $W(C_S)$, which
is a nonlocal operator. This suggests that it is convenient to treat
the Wilson loop as a nonlocal functional of gauge field rather than to
expand it into sum of infinitely many local composite operators. This can
be done by using RG equation of $W(C_S)$, which gives the dependence on
the renormalization point $\mu$. Since $W(C_S)$ is a function of the
single parameter $\rho \sim \mu y_-$, from its $\mu$ dependence we
directly obtain the dependence on $y_-$ and, through Fourier transform in
\re{main}, the dependence on $x$ for the structure function.

\subsection{Wilson loop renormalization away from the light-cone}

Renormalization group equation for the Wilson loop away from the light-cone
is well known \ci{away} and depends on the explicit form of the path.
The integration path for $W(C_S)$ is shown in fig.~2. It has two cusps at
the points $0$ and $y$ where the quark is probed by the photon. The
important property of this path is that the segment $\ell_2$ lies on
the light-cone. As we discussed in the two loop calculation, the presence
of the $L^3$ term entails that the renormalization properties of $W(C_S)$ are
different from the ones of Wilson loops with path away from the light cone.

Suppose for a moment that $\ell_2$ lies away from the light-cone, \ie
$y^2\neq 0.$ Then, the dependence on $\mu$ of the renormalized Wilson loop
expectation value is described by the RG equation
\be
\left(\mu\frac{\partial}{\partial\mu}
+\beta(g)\frac{\partial}{\partial g}\right)
\log W_{(y^2\neq 0)}(C_S)=-\Gamma_{cusp}(\gamma_+,g)-\Gamma_{cusp}(\gamma_-,g)
\lab{rg1}
\ee
where $\gamma_\pm$ are the angles in Minkowski space between vectors
$\pm y_\mu$ and $n_\mu=p_\mu/M$ with $n^2=1$
$$
\cosh \gamma_\pm=\pm\frac{(ny)}{\sqrt{y^2}}
$$
The cusp anomalous dimension $\Gamma_{cusp}(\gamma,g)$ is gauge invariant
function of the cusp angle \ci{away}. In the limit of large angles
$\gamma$ we have \ci{cusp}
$$
\Gamma_{cusp}(\gamma,g)=\gamma \Gamma_{cusp}(g)+\CO(\gamma^0)
$$
where the coefficient $\Gamma_{cusp}(g)$ is known to two loop order and is
given by
\be
\Gamma_{cusp}(g)=\alpi C_F+\left(\alpi\right)^2
C_F\left(C_A\left(\frac{67}{36}-\frac{\pi^2}{12}\right)-N_f\frac5{18}\right)
\lab{cusp}
\ee
For the integration path of fig.~2 we have $y^2=0$ leading to an infinite
$\gamma_\pm,$ thus eq.\re{rg1} becomes meaningless. One can directly check
that the two-loop result of previous section does not satisfy equation
\re{rg1}. However, as shown in the next subsection, there is a simple way
to find the generalization of the renormalization group equation for the
light-cone Wilson loop.

\subsection{Wilson loop renormalization on the light-cone}

Renormalization group equation for Wilson loops on the light-cone has been
proposed in \ci{WL} and can be obtained as follows. First, one slightly
shifts the integration path away from the light-cone by setting
$y^2\neq 0$ and keeping the cusp angle $\gamma_\pm=\half\log(4(ny)^2/y^2)$
large. Since $\gamma_\pm$ is a logarithmic function of $(ny)$ and $y^2$ we
need to define its analytical continuation away from positive $(ny)$ and
$y^2.$ The proper expressions for $\gamma_\pm$ can be deduced from one-loop
calculation of Wilson loop \ci{WL} and are given by
$$
\gamma_+=\half\log\frac{4((ny)-i0)^2}{y^2-i0}\,,
\qqquad
\gamma_-=\half\log\frac{4((ny)+i0)^2}{y^2-i0}
$$
where the ``$-i0$'' prescription comes from the position of the
singularity of the free gluon propagator in the coordinate representation.
By using these expressions we now differentiate the renormalization group
equation \re{rg1} with respect to the variable $(\n y)$
\be
\left(\mu\frac{\partial}{\partial\mu}+\beta(g)\frac{\partial}{\partial g}
\right)\frac{\partial}{\partial (\n y)} \log W(C_S)
=-\Gamma_{cusp}(g)\left(\frac{1}{(\n y)-i0}
+\left(\frac{1}{(\n y)+i0}\right)^\dagger\right)
=-\frac{2\Gamma_{cusp}(g)}{(\n y)-i0}
\lab{rg2}
\,.
\ee
The two terms originate from two cusps of the path of fig.~2 which lie
on the opposite sides of the cut. This is the reason for the appearance
of complex conjugation in the second term.

The variable $y^2$ disappeared from this equation and one can formally
set $y^2=0.$ This is the proposed renormalization group equation for
the Wilson loop on the light-cone \ci{WL}. One easily check that two-loop
expression \re{res:main} of the previous section does satisfy eq.\re{rg2}.

Notice, that Wilson loop on the light-cone depends only on a single
variable $\rho.$ Thus equation \re{rg2} becomes very powerful since one can
integrate it and obtain RG equation for the light-like Wilson loop
\be
\left(\mu\frac{\partial}{\partial\mu}+\beta(g)\frac{\partial}{\partial g}
\right) W(C_S)
=\left(-2\Gamma_{cusp}(g)\log(\rho-i0) - \Gamma(g)\right) W(C_S)
\lab{rg3}
\ee
where $\Gamma(g)$ is the integration constant. From this equation we can
see the origin of the various terms in the two-loop calculations in
\re{res:main}. The appearance of the one-loop beta function in front of
the $L^3$ term is obvious from \re{rg3}. The coefficient of $L^2$ is
proportional to a sum of $\Gamma_{cusp}(g)$ and one-loop beta-function.
The coefficient of $L$ is given by
$$
\Gamma(g)=-\alpi C_F
-\left(\alpi\right)^2C_F\left[
\left(\frac7{18}\pi^2+\frac{37}{108}-\frac94\zeta(3)\right)C_A
 -\left(\frac1{54}-\frac1{18}\pi^2\right) N_f
\right]
$$
While $\Gamma_{cusp}$ is a universal number, $\Gamma(g)$ depends on the
path under consideration. The unusual feature of the RG equation \re{rg3}
is that the anomalous dimension given by the coefficient of $W(C_S)$ in
the r.h.s. depends on $\rho$, \ie on the renormalization point $\mu$ and
$y_-.$ Actually this property leads to the evolution equation for the
structure function as we shall discuss in the next section.

\sect{Evolution equations}

The RG equation for the structure function near the phase space boundary
is obtained by using the representation \re{main}. Introducing the
parameter $\rho$ defined in \re{res:re'} we can write
$$
F(x,\mu/M)=H(\mu/M)\int_{-\infty}^{\infty}
\frac{d\rho}{2\pi}\frac{M}{\mu}
\ \e^{i\frac{M}{\mu}(1-x)\rho} W(\rho-i0)
$$
with the inverse transformation
$$
W(\rho-i0)\sim \int_{-\infty}^1 dx\
\e^{-i\frac{M}{\mu}(1-x)\rho}F(x,\mu/M)
$$
where the range of $x-$integration takes into account the spectral
property of the structure function in \re{spe}. The renormalization group
equation in \re{rg3} gives
\be
{\cal D} \ F(x,\mu/M)\equiv\left(\mu\partder{\mu}+\beta(g)\partder{g}\right)
F(x,\mu/M)
=\int_{x}^1 dz\ P(1+x-z) F(z,\mu/M)
\lab{eq2}
\ee
where
$$
P(z)=\frac{M}{\mu}\int_{-\infty}^{\infty} \frac{d\rho}{2\pi}
\ \e^{i\frac{M}{\mu}(1-z)\rho}\left(-2\Gamma_{cusp}(g)\log(\rho-i0)
-\Gamma(g)+{\cal D} \ \log H(\mu/M)-1\right)\,.
$$
{}  From the analytical property of the integrand we have
$$
P(z)=0 \qquad \mbox{for $z > 1$}\,,
$$
this is the reason for setting to $x$ the lower limit for $z$
in \re{eq2}. To compute $P(z)$ we use the representation
$$
\log(\rho-i0)=\int_0^\infty\frac{d\alpha}{\alpha}
\left(\e^{-i\alpha}-\e^{-i\alpha\rho}\right)\,.
$$
After a carefull treatment of the $\alpha \to 0$ singularities,
we  obtain
$$
P(z)=2\Gamma_{cusp}(g)\left(\frac{\theta(1-z)}{1-z}\right)_+
     +\delta(1-z) h(g)
$$
where
$$
h(g)=-\Gamma(g)+2\Gamma_{cusp}(g)\log\frac{M\e^{\gamma_E}}{\mu}
+{\cal D}\; \log H(\mu/M)-1\,,
$$
with $\gamma_E$ the Euler constant. Due to the factorization theorem
\ci{FT}, $P(z)$ should not depend on $\mu$ and the same is then true
for the function $h(g)$. This means that the $\mu$ dependence in the
term involving the coefficient function should be compensated by a
contribution from $W(C_S)$. The function $h(g)$ is not fixed by the
RG equation of $W(C_S)$ and should be directly computed from Feynman
diagrams. While $\Gamma_{cusp}(g)$ gets contribution only from soft
gluons, both soft and hard gluons contribute to $h(g).$

The evolution equation \re{eq2} in the soft limit $x\sim 1$ and $z\sim 1$
can be written in the standard form \re{eq1} where $P(z)$ is the two-loop
quark splitting function for $z$ near $1.$ The general form of $P(z)$ is
given by
\be\lab{pz}
(1-z) P(z)=2\Gamma_{cusp}(g)\; \theta(1-z) , \qquad \mbox{as $z\to 1$}
\ee
Using the expression \re{cusp} for $\Gamma_{cusp}(g)$ one easily
verifies that the result of two-loop calculations \ci{2L} obeys this
relation.

The evolution equation \re{eq2} for $x\sim 1$ is obtained from the RG
equation for the light-like Wilson loop. Because of the universal
structure of RG equation, one finds that any distribution, which can
be represented in terms of the light-like Wilson loops, satisfies equation
\re{eq2} with the kernel $P(z)$ defined by \re{pz}. This is the case for
the quark and gluon structure and fragmentation functions (see subsect.~2.3)
at large $x$. For the gluon distributions one should replace the colour
factor $C_F$ by $C_A$ in \re{cusp}.

\sect{Concluding remarks}

To conclude we would like to mention a close relation between the analysis
here presented and the heavy quark effective field theory \ci{Geo} which
seems to be a powerful tool for analyzing of heavy meson phenomenology.
The emission of gluons from heavy quarks can be treated by the eikonal
approximation. This implies that the propagation of the heavy quarks through
the cloud of light particles can be described by Wilson lines and that
the effective heavy quark field theory can actually be formulated \ci{KR}
in terms of the Wilson lines. Therefore, the renormalization properties of
Wilson lines discussed in this paper are related to the ones for the
effective theory. For example one finds that the ``velocity dependent
anomalous dimension'' is the cusp anomalous dimension \re{cusp}.

The central point of our analysis was the RG equation \re{rg3} for the
generalized Wilson loop expectation value, $W(C)$, with path partially
lying on the light-cone. The cusp anomalous dimension $\Gamma_{cusp}(g)$
entering into this equation is a new universal quantity of perturbative
QCD which controls the behaviour near the phase space boundary of hard
distributions. The evolution equation for these distributions corresponds
to the RG group equation for $W(C)$, moreover the splitting function near
the phase space boundary is related to the cusp anomalous dimension. This
equation does not imply that $W(C)$ is renormalized multiplicatively. In
particular, at two loop order one finds a non vanishing $L^3$ term with
a coefficient given by the one-loop $\beta-$function. This follows
directly from the RG equation of $W(C)$. Moreover $\Gamma_{cusp}(g)$ is
contained in the coefficient of $L^2$ (see term $C$ in eq.~\re{res:main}).

\bigskip\bigskip

\noindent{\Large{\bf Appendix:\ \ Feynman rules in the coordinate
representation}}

\bigskip

\setcounter{equation}0

\renewcommand{\theequation}{A.\arabic{equation}}

\noindent In this appendix we recall the Feynman rules for
calculation the generalized Wilson loop expectation value
in the coordinate representation using dimensional regularization.
The $D-$dimensional free gluon propagator in the Feynman gauge
is given by
$D^{\mu\nu}(x)=-g^{\mu\nu}D(x)$ where
$$
D(x)=i\int \frac{d^Dk}{(2\pi)^D}\e^{-ikx}
             \frac{1}{k^2+i0}
= \frac{\Gamma(D/2-1)}{4\pi^{D/2}}(-x^2+i0)^{1-D/2}
$$
The ``cutted'' propagator $D_+^{\mu\nu}(x)=-g^{\mu\nu}D_+(x)$
associated to a real gluon is defined as
$$
D_+(x)=\int \frac{d^Dk}{(2\pi)^D}\e^{-ikx}2\pi\theta(k_0)\delta(k^2)
= \frac{\Gamma(D/2-1)}{4\pi^{D/2}}[-2(x_+-i0)(x_--i0)]^{1-D/2}
$$
where the last equality holds only for $\bom{x}_T=0.$
We note that for gluon propagating in the time-like
direction ($x_+>0,$ $x_->0$ and $\bom{x}_T=0$) cutted and full propagator
coincide. For the three-gluon vertex with three gluon propagators
attached we have

\bigskip
$
\Gamma_{\mu_1\mu_2\mu_3}(z_1,z_2,z_3)\int d^D z_4\
\prod_{i=1}^3 D(z_4-z_i)
$ \hfill

\bigskip
\hfill $=-i\left(
  g^{\mu_1\mu_2}\left(\partial_1^{\mu_3}-\partial_2^{\mu_3}\right)
 +g^{\mu_2\mu_3}\left(\partial_2^{\mu_1}-\partial_3^{\mu_1}\right)
 +g^{\mu_1\mu_3}\left(\partial_3^{\mu_2}-\partial_1^{\mu_2}\right)
  \right) \int d^D z_4 \ \ \prod_{i=1}^3 D(z_4-z_i)
$

\bigskip
\noindent For the gluon attached to the point on the ray $\ell_1$
of the integration path of fig.~2 and propagating to $z$ we have
$
ign_\mu \int_{-\infty}^0 dt_1 D(n t_1 -z)
\,.
$
The analogous expressions for a gluon attached to the segment $\ell_2$ and
the ray $\ell_3$ are
$
igy_\mu \int_0^1 dt_2 D(y t_2 -z)
$
and
$
-ign_\mu \int_0^\infty dt_3 D(y-n t_3 -z)
$
respectively.

The scalar products and integration measure in terms of light-cone
variables are
$$
a_\pm=\frac1{\sqrt{2}}(a_0\pm a_3),
\qquad
\bom{a}=(a_1,a_2),
\qquad
(a b)= a_+b_-+a_+b_- -\bom{a}\cdot\bom{b},
\qquad
d^Da=da_+\ da_-\ d^{D-2}\bom{a}
$$
for arbitrary $D-$dimensional vectors $a_\mu$ and $b_\mu$.

\newpage

\bb{99}
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        \rmp{60}{373}{88};  \\
        Yu.L.\ Dokshitzer, V.A.\ Khoze,  A.H.\ Mueller  and S.I.\
        Troyan,  {\em Basics of Perturbative QCD}\/, Editions Fronti\`eres,
        Paris, 1991
\bi{EQ}   V.N.Gribov and L.N.Lipatov, Yad. Fiz. 15 (1972) 781; 1218;
\\        L.N.Lipatov, Yad. Fiz. 20 (1974) 181;
\\        G.Altarelli and G.Parisi, Nucl.Phys. 126B (1977) 298.
\bi{FT} J.C.Collins, D.E.Soper and G.Sterman,
        ``Factorization of Hard Processes in QCD,''
        in ``Perturbative Quantum Chromodynamics'', ed. by A.H.Mueller (World
        Scientific, Singapore, 1989) p.1.
\bi{Col} J.C.Collins and D.E.Soper, Nucl. Phys. B194 (1982) 445.
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\\       T.D.Lee and M.Nauenberg, Phys.Rev. 133 (1964) 1549.
\bi{ET} J.G.M.Gatheral, Phys. Lett. 113B (1984) 90;
\\      J.Frenkel and J.C.Taylor, Nucl. Phys. B246 (1984) 231.
\bi{break} A.Andrasi and J.C.Taylor, Nucl. Phys. B350 (1991) 73.
\bi{WL} I.A.Korchemskaya and G.P.Korchemsky,
        Phys. Lett. 287B (1992) 169.
\bi{away} A.M.Polyakov, Nucl. Phys. B164 (1980) 171;
\\        I.Ya.Aref'eva, Phys. Lett. B93 (1980) 347;
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\\       J. Kalinowski, K. Konishi, P.N. Scharbach and T.R. Taylor,
       \np{181}{253}{81};
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\\       I. Antoniadis and E.G. Floratos, \np{191}{217}{81}.
\bi{Geo} For a review see: H.Georgi, ``Heavy Quark Effective
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\bi{KR}  G.P.Korchemsky and A.V.Radyushkin, \pl{279}{359}{92}.
\eb

\newpage

\begin{center}
{\Large {\bf Figure captions:}}
\end{center}

\bigskip

\bigskip


{\bf Fig.~1}: One-loop Feynman diagram contributing to the structure function
        of deep inelastic scattering. The gluon with momentum $k$ is
        emitted by the quark $P$ and absorbed by the quark $P'$ in the final
        state. We use solid line for quarks, dotted lines for gluons,
        dashed lines for photons and dot-dashed line for the unitary cut.

\bigskip

{\bf Fig.~2}: Integration path $C_S=\ell_1\cup\ell_2\cup\ell_3$
        for the Wilson loop $W(C_S)$ corresponding to the structure function
        for large $x.$ The ray $\ell_1$ is along the time-like vector
        $n_\mu$ from $-\infty$ to $0;$ the  segment $\ell_2$ is from
        point $0$ to $y$ along the light-cone; the ray $\ell_3$ is from
        the point $y$ to $-\infty$ along the vector $-n_\mu$. This path
        has two cusps at points $0$ and $y$ where the quark undergoes hard
        scattering.

\bigskip

{\bf Fig.~3}: Integration path $C_T$ corresponding to the fragmentation
function
        for large $x.$

\bigskip

{\bf Fig.~4}: One-loop diagrams contributing to $W(C_S).$ Here, the double
        line represents the integration path in the Minkowski space as in
        fig.~2. The Feynman rules for these diagrams are given in
        the Appendix.

\bigskip

{\bf Fig.~5}: Diagrams corresponding to fig.~4a for virtual and real gluon
        (see \re{l1a}). Due to a partial cancellation of Wilson lines,
        the sum of these two diagram gives the single contribution of
        fig.~4a. Similar cancellations hold for all diagrams.

\bigskip

{\bf Fig.~6}: Nonvanishing two-loop diagrams containing the ``maximally
        nonabelian'' color $C_AC_F$ and the fermionic $C_FN_f$ factors.
        Due to the exponentiation theorem, these are the only diagram we
        need to evaluate to compute the two-loop contribution of $W(C_S)$.
        The blob denotes the sum of gluon, quark and ghost loops.

\newpage

\begin{center}
{\Large {\bf Figures:}}

\bigskip

\bigskip

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