Gauge Theory Webs and Surfaces
Ozan Erdo\u{g}an
George Sterman
We analyze the perturbative cusp for massless gauge theories in coordinate space, and express it as the exponential of a two-dimensional integral. The exponent has a geometric interpretation, which links the renormalization scale with invariant distances.
Short_websurface_v2.tex
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\begin{document}

\preprint{YITP-SB-11-33}

\renewcommand{\thefigure}{\arabic{figure}}

\title{Gauge Theory Webs and Surfaces}

\author{Ozan Erdo\u{g}an, George Sterman}

\affiliation{C.N.\ Yang Institute for Theoretical Physics, 
Stony Brook University, Stony Brook, New York, 11794--3840, USA}

\date{\today}

\begin{abstract}
We analyze the perturbative cusp for
massless gauge theories in coordinate space, and express it
as the exponential of a two-dimensional integral.  
The exponent has a geometric
interpretation, which links the renormalization scale with invariant
distances.
\end{abstract}

\maketitle

{\it Introduction}.
Wilson lines, or ordered exponentials \cite{Bialynicki-Birula,Wilson:1974sk}
 represent the interaction of energetic partons with relatively softer radiation
in gauge theories.    
For constant velocities, ordered exponentials of semi-infinite length
correspond to the eikonal approximation for energetic partons.
Classic phenomenological applications of ordered exponentials include soft radiation limits in
electron-positron annihilation and deeply inelastic scattering \cite{Korchemsky:1993uz}.  In these
cases, the electroweak current is represented by a color singlet vertex at which
lines with different velocities are coupled.   This vertex is often referred to as a cusp.
The set of all virtual corrections for the cusp is formally identical to a
vacuum expectation value, and can be
written as
\bea
\Gamma^{(f)}(\beta_1,\beta_2)
=
\bigg \langle T\bigg(  \Phi^{(f)}_{\beta _2}(\infty,0)\, \Phi^{(f)}_{\beta _1}(0,-\infty) \bigg)  \bigg \rangle_0\, ,
\label{eq:Gammadef}
\eea
in terms of constant-velocity ordered exponentials,
\bea
\Phi^{(f)}_{\beta _i}(x+\lambda\beta_i,x)
&\ &
\nn\\
&\ & \hspace{-20mm}
=
{\cal P}\exp \left (-ig\int_0^\lambda d\lambda'\beta _i\cdot A^{(f)}(x+\lambda'\beta_i)
\right )\, .
\label{eq:Phidef}
\eea
Here  $f$ labels a representation of the gauge group 
(which we suppress below) and $\beta_i$ is a four-velocity,
taken light-like in the following.   We will
explore all-orders properties of single- and multiple-cusp
products of ordered exponentials, computed perturbatively in coordinate space.

Perturbative corrections to the cusp, Eq.\ (\ref{eq:Gammadef})
are scaleless, and hence vanish in dimensional regularization.
The ultraviolet poles of (\ref{eq:Gammadef}) determine the anomalous dimension of the cusp,
but for an asymptotically free theory neither its ultraviolet nor its infrared behavior can be considered as truly physical.
At very short distances, dynamics is perturbative and recoil cannot be neglected.
At very long distances, it is nonperturbative and dominated by the hadronic
spectrum.   
In this discussion,
we will regard the cusp as an interpolation between these asymptotic
regimes, and we will
concentrate on the structure of the integrals in the intermediate
region.  

We will argue that in any gauge theory the cusp matrix element can be expressed
as the exponential of an integral over a two-dimensional surface.
The corresponding integrand is an expansion in the gauge theory coupling, evaluated at a scale
given by the invariant distance from a point on the surface to the cusp vertex.
 We will go on to apply this result to
multi-cusp polygonal Wilson loops \cite{Korchemskaya:1992je,Drummond:2007aua}.


{\it Exponentiation and Webs}.
The cusp has long been known \cite{Gatheral:1983cz} to be 
 the exponential of a sum of special diagrams called webs, which are  irreducible by cutting two
eikonal lines.
 We represent this result as 
\bea
\Gamma(\beta_1,\beta_2,\vep)
=
\exp E(\beta_1,\beta_2,\vep)\, ,
\eea
in $D=4-2\vep$ dimensions.
The exponent $E$ equals a sum over web diagrams, $d$,
each given by a   group factor
multiplied by a diagrammatic integral, 
\bea
E(\beta_1,\beta_2,\vep) = \sum_{webs\ d} \bar C_d\; {\cal F}_d(\beta_1,\beta_2,\vep)\, ,
\eea
where ${\cal F}_d$ represents the momentum- or coordinate-
space integral for diagram $d$.
 The coefficients of these integrals,
$\bar C_d$ are modified color factors.  Two-loop examples are shown in Fig.~\ref{webfig}\@.
\begin{figure}[b]
\begin{center}
\includegraphics[width=6.5cm]{2loopdiagrams2.pdf}
\end{center}
\caption{Two-loop web diagrams, all with the same
modified color factor~$\bar C_d = C_aC_A$\@. Notice that only for the rightmost
diagram does the modified color factor differ from its original color
factor.}
\label{webfig}
\end{figure}
In momentum space we
can write the exponent $E$ as an integral over 
a single, overall loop momentum integral that
connects the web with the cusp vertex, assuming that
all loop momentum integrals internal to the web 
have been carried out, including as well the necessary
counterterms of the theory \cite{Laenen:2000ij,Berger:2003zh}.   Taking into account the
boost invariance of the cusp, and the invariance
of the ordered exponentials under rescalings of
the velocities $\beta_i$, we have for the
exponent the form,
\bea
E = 
 \int \frac{d^D k}{(2\pi)^D}\, \bar w \left(k^2,{k\cdot \beta_1\, k\cdot \beta_2\over
\beta_1\cdot\beta_2},\mu^2,\alpha_s(\mu^2,\vep),\varepsilon\right)\, .
\label{eq:webexpk}
\eea
Multiplicative renormalization
of the cusp implies additive renormalization
for the exponent $E$.   
In addition, the webs themselves are renormalization-scale independent,
$
\mu{d\over d\mu}\; \bar w \left(k^2,{k\cdot \beta_1 k\cdot \beta_2\over
\beta\cdot\beta'},\mu^2,\alpha_s(\mu^2,\vep),\varepsilon\right)
=0\, .
$
A further property of webs is
the absence of collinear and soft subdivergences.    
That is, in Eq.\ (\ref{eq:webexpk}), collinear poles are generated only when $k^2$ and either $k\cdot \beta_1$
or $k\cdot \beta_2$ vanish, 
infrared poles only when all three vanish and the overall
ultraviolet poles only when all components of $k$
diverge. Eq.~(\ref{eq:webexpk}) thus organizes the same double poles
found in the corresponding partonic form factors
\cite{Berger:2003zh,Magnea:1990zb}. A proof of these properties in
momentum space is given in 
Ref.~\cite{Berger:2003zh}, based on the factorization of soft 
gluons from fast-moving collinear partons.    These considerations suggest that 
when embedded in an on-shell amplitude, the web acts as a unit, almost
like a single gluon, dressed by arbitrary orders in the coupling.
In the following, we observe that this analogy can be extended
to coordinate space.

The coordinate space analog of Eq.\ (\ref{eq:webexpk})
is a double integral over two parameters $\lambda$ and $\sigma$ that measure distances along 
the Wilson lines $\beta_1$ and $\beta_2$, 
respectively,
with a new web function, $w$, which
depends on these variables through the only
available dimensionless combination $\beta_1\cdot
\beta_2\,\lambda\sigma\mu^2$,
\bea
E
&=&
\int_0^\infty \frac{d\lambda}{\lambda}\, 
\int_0^\infty \frac{d\sigma}{\sigma}\ w (\alpha_s(\mu^2,\vep),\lambda\sigma\mu^2,\varepsilon)\, ,
\label{eq:web2}
\eea
where here and below, we set $\beta_1\cdot\beta_2=1$ and choose
timelike kinematics. 
We emphasize that we are interested primarily in the form and symmetries of the
integrand, rather than its convergence
properties.    Nevertheless,
to separate infrared and ultraviolet poles
in the integration, it is necessary that the integrand, $w$ in Eq.\
(\ref{eq:web2})
be free of both infrared and ultraviolet divergences at $\varepsilon=0$ in 
renormalized perturbation theory (aside from the renormalization of the cusp itself).
For finite values of $\lambda$ and $\sigma$, 
there are no infrared divergences from integrations
over the internal vertices of webs \cite{Erdogan:2011}.    We will 
see below that for nonzero $\lambda$ and $\sigma$,
$w$ also has no collinear subdivergences. All $\vep$ poles of the exponent, and therefore the cusp, are then 
 associated with the integrals over $\lambda$ and $\sigma$ in
 (\ref{eq:web2}). 

To derive Eq.\ (\ref{eq:web2}) along with the finiteness of $w$, we follow Ref.\ \cite{Mitov:2010rp}
and write the exponent as a sum over the numbers, $e_a$, of gluons attached 
to the two Wilson lines, 
of velocity
$\beta_a$, $a=1,2$.
For a given choice of $e_a$, the web diagrams are integrals over the
positions $\tau_{j_a}^{(a)}$ of
these ordered vertices of  a
function ${\cal W}_{e_1,e_2}(\tau_{j_a}^{(a)})$ 
found by integrating over all the internal vertices of the corresponding web diagrams.
In the notation of Ref.\ \cite{Mitov:2010rp} we then have
 at $i$th order $(i\ge e_1+e_2)$,
\be
E^{(i)} =
 \sum_{e_1,e_2=1}^{i-1} \prod_{a=1}^2\, 
\prod_{{j_a}=1}^{e_a}\int_{\tau^{(a)}_{{j_a}-1}}^\infty\! d\tau^{(a)}_{j_a}\, 
{\cal W}_{e_1,e_2}^{(i)}\left(\{\tau^{(a)}_{j_a}\}\right)\, ,
\label{eq:calFcalW}
\ee
where $\tau^{(a)}_0\equiv 0$.   Here and below we expand functions as $E=\sum (\as/\pi)^iE^{(i)}$.  
Next, we fix the values of the vertices in Eq.\ (\ref{eq:calFcalW}) that are furthest from
the origin, and denote them as $\tau^{(1)}_{e_1}=\lambda$,
$\tau^{(2)}_{e_2}=\sigma$.   The function
$w$ in Eq.\ (\ref{eq:web2}) is then defined by
\bea
w^{(i)}(\lambda\sigma) &=& \sum_{e_1,e_2}
\prod_{{j_1}=1}^{e_1-1}\int_{\tau^{(1)}_{{j_1}-1}}^{\lambda}d\tau^{(1)}_{j_1}\ \prod_{j_2=1}^{e_2-1}\int_{\tau^{(2)}_{j_2-1}}^{\sigma}\ d\tau^{(2)}_{j_2}\, 
\nonumber\\
&\ & \hspace{5mm} \times\ {\cal W}_{e_1,e_2}^{(i)}\left(\{\tau^{(a)}_{j_a}\}\right)\, ,
\label{eq:calFcalW2}
\eea
where we integrate over all $\tau_{j_a}^{(a)}$, $j_a<e_a$.   We want to show that
these integrals, along with the integrals over internal vertices of the web diagrams, 
do not give rise to singular behavior in the function $w(\sigma\lambda)$, once
the full set of web diagrams is combined at a given order.

Diagram by diagram, one may use an analysis of the analytic 
structure of the coordinate
integrations
\cite{Date:1982un}
combined with a coordinate space
power counting technique  to identify the most general singular subregions
in coordinate space 
\cite{Erdogan:2011}.   These occur whenever a subset of internal vertices of a web diagram becomes ``collinear" by
approaching either Wilson line, while other, ``soft" vertices
remain at finite distances.  


In coordinate space, as in momentum space, however, 
soft and collinear vertices are always attached 
to each other
through lines that carry unphysical polarizations.
Region by region, soft and collinear subdiagrams then factorize \cite{Bodwin:1984hc,Collinsbook}.   
Once in factorized form,
divergent integrals  cancel when all web diagrams are combined at a given order 
\cite{Erdogan:2011} by a method related to the momentum space
arguments given in \cite{Berger:2003zh}.  
It is necessary to implement this cancellation at fixed $\lambda$ and $\sigma$.
Once this is done and subdivergences thereby eliminated,
the integrals over all vertices of the web diagrams converge on scales
set by $\sigma$ and $\lambda$  in (\ref{eq:web2}), and the web acts as a unit.  
Singular behavior of the cusp arises as 
$\sigma$ and $\lambda$ vanish, and in these limits all web
vertices approach the directions of $\beta_1$ or $\beta_2$ together,
as in Fig.\ \ref{fig:2loopwebs}.    This is the 
perturbative
realization
of the web as a geometrical 
object.
\begin{figure}[b]
\begin{center}
\includegraphics[width=6cm]{2loopwebs.pdf}
\end{center}
\caption{Representation of singular regions for a two-loop web diagram.}
\label{fig:2loopwebs}
\end{figure}

The web function $w$ constructed this way is scale invariant, so that
in (\ref{eq:web2}), we may
shift the renormalization scale to the product $(\sigma\lambda)^{-1}$, which
results in an expression with the coupling running as
the leading vertices move up and down the Wilson lines,
\bea
E
&=&
\int_0^\infty \frac{d\lambda}{\lambda}\, 
\int_0^\infty \frac{d\sigma}{\sigma}\, w\left(\alpha_s\left(1/\lambda\sigma,\vep\right),\varepsilon\right)\, .
\label{eq:web3}
\eea
In this all-orders form, dependence on the product $\lambda\sigma$ is entirely
through the running coupling, aside from the overall dimensional factor.
For conformal field theories, Eq.\ (\ref{eq:web3}) for the cusp holds as well 
at strong coupling \cite{Kruczenski:2002fb,Alday:2008yw,Alday:2007hr},
where the coordinates $\lambda$ and $\sigma$  also parameterize a surface.
The generality of these results
can be traced to the symmetries of the problem \cite{Kruczenski:2002fb}.
It is interesting to note, however, that in the strong coupling analysis, the product of
internal coordinates $\lambda\sigma$, which serves as the renormalization scale
in Eq.\ (\ref{eq:web3}),  relates the plane of the Wilson lines to a
minimal surface in five dimensions.    

The lowest order expression for Eq.\ (\ref{eq:web2}) is found directly from
the coordinate space gluon propagator in Feynman gauge,
$
D^{\mu\nu} = \frac{\Gamma(1-\vep)}{4\pi^{2-\vep}}\, \frac{-g^{\mu\nu}}{\left(-x^2+i\ep\right)^{1-\vep}}\, .
$
The resulting expression already illustrates the nontrivial
relationship between the renormalization scale and the positions of the 
vertices, 
\be
E^{\rm (LO)}\,=\, -\ 
C_a\, \Gamma(1-\vep) \frac{\as(\mu)}{2\pi}
\int_0^\infty \frac{d\lambda}{\lambda}\, 
 \frac{d\sigma}{\sigma}\, (2\pi \lambda\sigma \mu^2)^{\vep}\, ,
\label{eq:webLO}
\ee
with $C_a=4/3,3$ for $a=q,g$ in QCD.   
More generally, consistency with momentum space pole structure\ \cite{Korchemsky:1993uz,Berger:2003zh} requires 
\be
w\left(\alpha_s\left(1/[\lambda\sigma]\right),\varepsilon\right)\ =\ 
-\frac{1}{2}\; \Gamma_{\rm cusp} \big(\as\left(1/\lambda\sigma,\vep\right)\big)
 +{\cal O}(\vep) \  ,
\label{eq:web4}
\ee
where $\Gamma_{\rm cusp}(\as)$ is the cusp anomalous dimension, defined for flavor $a$ as in
\cite{Korchemskaya:1992je,Korchemsky:1987wg}, $\Gamma_{\rm cusp}^{(a)}= (\as/\pi) C_a[1+(\as/\pi)K]$, 
plus higher orders, with
$K=[C_A(67/36-\pi^2/12)-(5n_f)/18]$.
We have verified this relation 
for
the coordinate web constructed as above 
at two loops \cite{Erdogan:2011}\@.    Interpreted as a fully scaleless integral,
Eq.\ (\ref{eq:web4}) can be defined by its ultraviolet poles,
and is of course gauge invariant.
If the integrals are cut off in the infrared, 
however,
gauge-dependent corrections at ${\cal O}(\vep)$ in 
Eq.\ (\ref{eq:web4}) remain in general.


{\it Polygon Loops}.
The above reasoning leads to a number of interesting results for 
polygonal closed Wilson loops \cite{Alday:2007hr,Alday:2008yw,Drummond:2007aua}.
These amplitudes also exponentiate in perturbation theory in terms of webs
\cite{Drummond:2007aua}.   To this observation we may apply once again the lack
of subdivergences for webs.

\begin{figure}[b]
\begin{center}
\includegraphics[width=3cm]{cusp_web2.pdf}
\put(-96,-4){$X_a$}
\put(-105,33){$\sigma_a\beta'_a$}
\put(-52,-7){$\lambda_a\beta_a$}
\put(-66,23){$W_a$}
\put(-65,7){$\dots$}
\put(-79,20){$\vdots$}
\end{center}
\caption{A single-cusp web $W_a$, in the sum
of Eq.\ (\ref{eq:Lfsum1}).}
\label{fig:cuspweb}
\end{figure}


Generic diagrams for polygonal loops are shown in
Figs.~\ref{fig:cuspweb} and \ref{fig:sidewebs}. 
In Fig.\ \ref{fig:cuspweb}, for example, the $a$th vertex of the polygon
represents a cusp vertex  that connects two Wilson 
lines, of velocity $\beta_a$ and $\beta'_a$.   

Exponentiation in coordinate space implies that the logarithm of the
polygon~$P$ is a sum of the web configurations represented by the figures,
\bea
\ln P &=& \sum_{{\rm cusps}\ a} W_a \ 
+ \sum_{{\rm sides}\ ab}W_{ab}\ 
+\ W_{\rm plane}\, .
\label{eq:Lfsum1}
\eea
The  first terms organize webs associated entirely with one of the cusps of the
polygon, constructed in terms of the
coordinate webs identified above.   Because each edge is of finite length, there are now additional
terms associated with the end-point contributions, 
which may be combined with webs connecting three sides.   
The cancellation of subdivergences in webs  implies that after a sum over diagrams,
 only  the cusp poles and
 a single, overall
collinear singularity survives  \cite{Drummond:2007aua,Erdogan:2011}.  
There remains a finite contribution from webs that connect
all four (or in general more) of the Wilson lines, and these are represented by the final term in (\ref{eq:Lfsum1}).

\begin{figure}[t]
\begin{center}

\subfigure[]{\includegraphics[width=3cm]{side_web2.pdf}}
\put(-96,-4){$X_a$}
\put(0,-4){$X_b$}
\put(-51,40){$W_{ab}$}
\put(-74,37){$\vdots$}
\put(-14,37){$\vdots$}
\put(-48,14){$\dots$}
\quad\quad\quad
\subfigure[]{\includegraphics[width=3cm]{finiteweb2.pdf}}
\put(-51,40){$W_{\rm fin}$}
\put(-77,40){$\vdots$}
\put(-11,40){$\vdots$}
\put(-48,12){$\dots$}
\put(-48,73){$\dots$}


\end{center}

\caption{(a) A `side' web $W_{ab}$ in 
of Eq.\ (\ref{eq:Lfsum1}), in this case associated
with the light-like side $X_a-X_b$\@. (b) A web
that contributes to $W_{\rm plane}$ in Eq.\ (\ref{eq:Lfsum1}).}

\label{fig:sidewebs}

\end{figure}
Evidently, 
$W_a(\beta_a,\beta'_a)$ is the same as for the finite Wilson lines
in Eq.\ (\ref{eq:web3}), in terms of the lengths of the sides of the polygon,
which can be scaled to some constant, $X$,
\bea
W_a(\beta_a,\beta'_a,X)
&\!=\!&
-\int_{0}^{X}\! \frac{d\lambda_a}{\lambda_a}\, 
\int^0_{-X}\! \frac{d\sigma_a}{\sigma_a}\, w(\alpha_s(1/\lambda_a\sigma_a,\vep),\varepsilon)\, .
\nn
\\
\label{eq:web_sides}
\eea
The web function can depend only on the scalar products
of the velocities, and we may assume for simplicity that these are all the same.

Polygons of this sort have been studied
in the context of a duality to scattering amplitudes~\cite{Alday:2007hr,Drummond:2007aua}.  
Here,
we consider a four-sided polygon 
that projects to a square in the $x_1/x_2$ plane, with side $X$,
as in Figs.\ \ref{fig:cuspweb}--\ref{fig:sidewebs}.   In four dimensions, the loop starts at the origin, travels along the
plus-$x^1$ direction for a `time' $X^0=X$, then changes direction
to $x^2$ for time $X$, and then moves backwards in time and space,
first in the $x^1$ direction, then $x^2$, back to the origin.
We can now use the coordinates $x^1$ and $x^2$ to 
define parameters $\lambda_a$ and $\sigma_a$ for
each of the cusp integrals $W_a$ in Eq.\ (\ref{eq:web_sides}),
\be\begin{array}{lcl}
\sigma_1 = -x_2\, ,& \qquad &\lambda_1 = x_1\, , \\
\sigma_2 = x_1-X\, ,& \qquad & \lambda_2 = x_2\, , \\
\sigma_3 = x_2-X\, , & \quad & \lambda_3 = X-x_1\, , \\
\sigma_4 = -x_1\, , & \quad & \lambda_4 = X-x_2\, .
\end{array}\ee
In 
this notation,
we can add the four cusp
web integrals of Eq.\ (\ref{eq:web_sides}), 
to get
a single
integral over $x_1$ and $x_2$.   The web functions, of course,
depend on the particular forms of $\lambda$ and $\sigma$
above. We find
\bea
\sum_{a=1}^4 W_a(\beta_a,\beta'_a)
&=& \ \int_0^X dx_1\, \int_0^X dx_2
\label{eq:sum_webs}
\\
&\ & \hspace{-29mm}
\frac{(X-x_2)[(X-x_1) w_1 + x_1 w_2]
+
x_2[x_1 w_3 + (X-x_1) w_4]}
{x_1(X-x_1)x_2(X-x_2)}
\, ,
\nn
\eea
where $w_a\equiv w(\alpha_s(\lambda_a(x_1,x_2)\sigma_a(x_1,x_2)))$.
For a conformal theory, all dependence on the $\sigma_a$ and $\lambda_a$
is in the denominators and we can sum over $a$ to get a result in terms
of a constant web function $w_0$. Changing variables to $y_a=1-2x_a/X$, we
derive the unregularized form found from the analysis of extremal
two-dimensional surfaces embedded in a five-dimensional background in \cite{Alday:2007hr}, 
\bea
\sum_{a=1}^4 W_a(\beta_a,\beta'_a)
&=&  \int_{-1}^1 dy_1\, \int_{-1}^1 dy_2\,
\frac{4 w_0}
{(1-y_1^2)(1-y_2^2)}
\, ,
\nn\\
\label{eq:sum_conformal_webs}
\eea
to which we should add
the collinear and finite multi-cusp contributions of Fig.\ \ref{fig:sidewebs}.

{\it Conclusions}.
We have found that when the massless cusp is analyzed in coordinate space,
it is naturally written as the exponential of a two-dimensional integral.   The integrand, 
a web function,
depends on the single invariant scale
through the running of the coupling, which for a theory that is conformal
in four dimensions agrees with 
strong coupling results \cite{Alday:2007hr,Alday:2008yw,Kruczenski:2002fb}.
This agreement extends to aspects of closed, polygonal Wilson loops.
These results do not rely on a planar limit \cite{'tHooft:1973jz},
but it is natural to conjecture that for large $N_c$ the 
integral may take on an even more direct interpretation in terms of
surfaces for non-conformal theories.

In QCD, of course, our 
explicit
knowledge of the web function is limited
to the first few terms in the perturbative series, which run out of predictive power
as the invariant distance increases. The functional form, however,
holds to all orders in perturbation theory, and may point to an
interpolation
between short and long distances.

\acknowledgments
We thank  B. van Rees for discussions. This work was supported by the
National Science Foundation, 
grants PHY-0653342 and PHY-0969739.  

\begin{thebibliography}{}

\bibitem{Bialynicki-Birula}
I.\ Bialynicki-Birula,
%``Gauge invariant variables in the Yang-Mills theory,"
Bull.\ Acad.\ Polon.\ Sci.\ {\bf 11}, 135 (1963);\\
%\cite{Mandelstam:1968hz}
%\bibitem{Mandelstam:1968hz}
  S.~Mandelstam,
  %``Feynman rules for electromagnetic and Yang-Mills fields from the gauge
  %independent field theoretic formalism,''
  Phys.\ Rev.\  {\bf 175}, 1580 (1968).
  %%CITATION = PHRVA,175,1580;%%

  %\cite{Wilson:1974sk}
\bibitem{Wilson:1974sk}
  K.~G.~Wilson,
  %``CONFINEMENT OF QUARKS,''
  Phys.\ Rev.\  D {\bf 10}, 2445 (1974);\\
  %%CITATION = PHRVA,D10,2445;%%
%\cite{Yang:1974kj}
%\bibitem{Yang:1974kj}
  C.~N.~Yang,
  %``Integral formalism for gauge fields,''
  Phys.\ Rev.\ Lett.\  {\bf 33}, 445 (1974);\\
  %%CITATION = PRLTA,33,445;%%
   %\cite{Polyakov:1978vu}
%\bibitem{Polyakov:1978vu}
  A.~M.~Polyakov,
  %``Thermal Properties Of Gauge Fields And Quark Liberation,''
  Phys.\ Lett.\  B {\bf 72}, 477 (1978);\\
  %%CITATION = PHLTA,B72,477;%%
%  \bibitem{Susskind:1979up}
  L.~Susskind,
  %``Lattice Models Of Quark Confinement At High Temperature,''
  Phys.\ Rev.\  D {\bf 20}, 2610 (1979).
  %%CITATION = PHRVA,D20,2610;%%
  
 %\cite{Korchemsky:1993uz}
\bibitem{Korchemsky:1993uz}
  G.~P.~Korchemsky, G.~Marchesini,
  %``Resummation of large infrared corrections using Wilson loops,''
  Phys.\ Lett.\  {\bf B313}, 433-440 (1993).



 %\cite{Korchemskaya:1992je}
\bibitem{Korchemskaya:1992je}
  I.~A.~Korchemskaya, G.~P.~Korchemsky,
  %``On lightlike Wilson loops,''
  Phys.\ Lett.\  {\bf B287}, 169-175 (1992).
 

  %\cite{Drummond:2007aua}
\bibitem{Drummond:2007aua}
 J.~M.~Drummond, G.~P.~Korchemsky and E.~Sokatchev,
 %``Conformal properties of four-gluon planar amplitudes and Wilson loops,''
 Nucl.\ Phys.\  B {\bf 795}, 385 (2008)
 [arXiv:0707.0243 [hep-th]];\\
%\cite{Drummond:2007cf}
%\bibitem{Drummond:2007cf}
 J.~M.~Drummond, J.~Henn, G.~P.~Korchemsky and E.~Sokatchev,
 %``On planar gluon amplitudes/Wilson loops duality,''
 Nucl.\ Phys.\  B {\bf 795}, 52 (2008)
 [arXiv:0709.2368 [hep-th]].


   %\cite{Gatheral:1983cz}
\bibitem{Gatheral:1983cz}
  J.~G.~M.~Gatheral,
  %``Exponentiation Of Eikonal Cross-Sections In Nonabelian Gauge Theories,''
  Phys.\ Lett.\  B {\bf 133}, 90 (1983);\\
  %%CITATION = PHLTA,B133,90;%%
%\cite{Frenkel:1984pz}
%\bibitem{Frenkel:1984pz}
  J.~Frenkel and J.~C.~Taylor,
  %``Nonabelian Eikonal Exponentiation,''
  Nucl.\ Phys.\  B {\bf 246}, 231 (1984);\\
  %%CITATION = NUPHA,B246,231;%%
    %\cite{Sterman:1981jc}
%\bibitem{Sterman:1981jc}
  G.~Sterman, in ``Perturbative Quantum Chromodynamics", D.\ W.\ Duke and J.\ F.\ Owens ed.,
  %``Infrared Divergences In Perturbative QCD. (Talk),''
  AIP Conf.\ Proc.\  {\bf 74}, 22 (American Inst.\ of Phys., 1981).
  %%CITATION = APCPC,74,22;%%

%\cite{Laenen:2000ij}
\bibitem{Laenen:2000ij}
  E.~Laenen, G.~Sterman, W.~Vogelsang,
  %``Recoil and threshold corrections in short distance cross-sections,''
  Phys.\ Rev.\  {\bf D63}, 114018 (2001).
  [hep-ph/0010080].

\bibitem{Berger:2003zh}
  C.~F.~Berger,
  %``Soft gluon exponentiation and resummation,''
  arXiv:hep-ph/0305076;\\
%\cite{Berger:2002sv}
%\bibitem{Berger:2002sv}
  C.~F.~Berger,
  %``Higher orders in A(alpha(s))/(1-x)+ of non-singlet partonic splitting
  %functions,''
  Phys.\ Rev.\  D {\bf 66}, 116002 (2002)
  [arXiv:hep-ph/0209107].

  %\cite{Magnea:1990zb}
\bibitem{Magnea:1990zb}
  L.~Magnea and G.~Sterman,
  %``Analytic continuation of the Sudakov form-factor in QCD,''
  Phys.\ Rev.\  D {\bf 42}, 4222 (1990);\\
%\cite{Catani:1998bh}
%\bibitem{Catani:1998bh} 
  S.~Catani,
  %``The Singular behavior of QCD amplitudes at two loop order,''
  Phys.\ Lett.\ B {\bf 427}, 161 (1998)
  [hep-ph/9802439];\\
  %%CITATION = HEP-PH/9802439;%%
%\cite{Sterman:2002qn}
%\bibitem{Sterman:2002qn}
  G.~Sterman and M.~E.~Tejeda-Yeomans,
  %``Multi-loop amplitudes and resummation,''
  Phys.\ Lett.\  B {\bf 552}, 48 (2003)
  [arXiv:hep-ph/0210130];\\
  %%CITATION = PHLTA,B552,48;%% 
%\cite{Bern:2005iz}
%\bibitem{Bern:2005iz}
 Z.~Bern, L.~J.~Dixon and V.~A.~Smirnov,
 %``Iteration of planar amplitudes in maximally supersymmetric Yang-Mills
 %theory at three loops and beyond,''
 Phys.\ Rev.\  D {\bf 72}, 085001 (2005)
 [arXiv:hep-th/0505205];\\
 %%CITATION = PHRVA,D72,085001;%%
%\cite{Dixon:2008gr}
%\bibitem{Dixon:2008gr}
 L.~J.~Dixon, L.~Magnea and G.~Sterman,
 %``Universal structure of subleading infrared poles in gauge theory
 %amplitudes,''
 JHEP {\bf 0808}, 022 (2008)
 [arXiv:0805.3515 [hep-ph]].
 
\bibitem{Erdogan:2011}  O.\ Erdo\u{g}an and G.\ Sterman, in
  preparation.

%\cite{Mitov:2010rp}
\bibitem{Mitov:2010rp}
  A.~Mitov, G.~Sterman, I.~Sung,
  %``Diagrammatic Exponentiation for Products of Wilson Lines,''
  Phys.\ Rev.\  {\bf D82}, 096010 (2010).
  [arXiv:1008.0099 [hep-ph]].

%\cite{Date:1982un}
\bibitem{Date:1982un} 
  G.~Date, doctoral thesis,
  %``Factorization Theorems In Perturbative Quantum Field Theory,''
  UMI-83-07385.
%%CITATION = UMI-83-07385;%% 


%\cite{Bodwin:1984hc}
\bibitem{Bodwin:1984hc}
 G.~T.~Bodwin,
 %``Factorization of the Drell-Yan Cross-Section in Perturbation Theory,''
 Phys.\ Rev.\ D {\bf 31}, 2616 (1985)
 [Erratum-ibid.\ D {\bf 34}, 3932 (1986)];\\
%\cite{Collins:1985ue}
%\bibitem{Collins:1985ue}
 J.~C.~Collins, D.~E.~Soper and G.~Sterman,
 %``Factorization for Short Distance Hadron - Hadron Scattering,''
 Nucl.\ Phys.\ B {\bf 261}, 104 (1985);
 %%CITATION = NUPHA,B261,104;%%
 %%CITATION = PHRVA,D31,2616;%%
%\cite{Collins:1988ig}
%\bibitem{Collins:1988ig}
% J.~C.~Collins, D.~E.~Soper and G.~Sterman,
 %``Soft Gluons and Factorization,''
 %Nucl.\ Phys.\ B 
 {\bf 308}, 833 (1988).
 %%CITATION = NUPHA,B308,833;%%
  

\bibitem{Collinsbook}
%\cite{Collins:1989gx}
%\bibitem{Collins:1989gx}
  J.~C.~Collins, D.~E.~Soper and G.~Sterman,
  in ``Perturbative Quantum Chromodynamics", A.H.\ Mueller, ed.,
  Adv.\ Ser.\ Direct.\ High Energy Phys.\  {\bf 5}, 1 (World Scientific, 1988)
  [arXiv:hep-ph/0409313];\\
  %%CITATION = 00319,5,1;%%
J.\ Collins, ``Foundations of Perturbative QCD"  (Cambridge Univ. Pr., 2011).

\bibitem{Alday:2007hr}
 L.~F.~Alday and J.~M.~Maldacena,
 %``Gluon scattering amplitudes at strong coupling,''
 JHEP {\bf 0706}, 064 (2007)
 [arXiv:0705.0303 [hep-th]];\\
 %%CITATION = JHEPA,0706,064;%%
%\cite{Alday:2007he}
%\bibitem{Alday:2007he}
 L.~F.~Alday and J.~Maldacena,
 %``Comments on gluon scattering amplitudes via AdS/CFT,''
 JHEP {\bf 0711}, 068 (2007)
 [arXiv:0710.1060 [hep-th]].
 
%\cite{Alday:2008yw}
\bibitem{Alday:2008yw}
 L.~F.~Alday and R.~Roiban,
 %``Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
 %Correspondence,''
 Phys.\ Rept.\  {\bf 468}, 153 (2008)
 [arXiv:0807.1889 [hep-th]].
 %%CITATION = PRPLC,468,153;%%
 
 %\cite{Kruczenski:2002fb}
\bibitem{Kruczenski:2002fb} 
  M.~Kruczenski,
  %``A Note on twist two operators in N=4 SYM and Wilson loops in Minkowski signature,''
��JHEP {\bf 0212}, 024 (2002)
��[hep-th/0210115].
��%%CITATION = HEP-TH/0210115;%%
  


 %\cite{Korchemsky:1987wg}
\bibitem{Korchemsky:1987wg}
  G.~P.~Korchemsky, A.~V.~Radyushkin,
  %``Renormalization of the Wilson Loops Beyond the Leading Order,''
  Nucl.\ Phys.\  {\bf B283}, 342-364 (1987).


%\cite{'tHooft:1973jz}
\bibitem{'tHooft:1973jz}
  G.~'t Hooft,
  %``A Planar Diagram Theory for Strong Interactions,''
  Nucl.\ Phys.\  B {\bf 72}, 461 (1974).
  %%CITATION = NUPHA,B72,461;%% 

 
\end{thebibliography}

\end{document}


1112.4564.extracted_bib
\begin{thebibliography}{}

\bibitem{Bialynicki-Birula}
I.\ Bialynicki-Birula,
%``Gauge invariant variables in the Yang-Mills theory,"
Bull.\ Acad.\ Polon.\ Sci.\ {\bf 11}, 135 (1963);\\
%\cite{Mandelstam:1968hz}
%\bibitem{Mandelstam:1968hz}
  S.~Mandelstam,
  %``Feynman rules for electromagnetic and Yang-Mills fields from the gauge
  %independent field theoretic formalism,''
  Phys.\ Rev.\  {\bf 175}, 1580 (1968).
  %%CITATION = PHRVA,175,1580;%%

  %\cite{Wilson:1974sk}
\bibitem{Wilson:1974sk}
  K.~G.~Wilson,
  %``CONFINEMENT OF QUARKS,''
  Phys.\ Rev.\  D {\bf 10}, 2445 (1974);\\
  %%CITATION = PHRVA,D10,2445;%%
%\cite{Yang:1974kj}
%\bibitem{Yang:1974kj}
  C.~N.~Yang,
  %``Integral formalism for gauge fields,''
  Phys.\ Rev.\ Lett.\  {\bf 33}, 445 (1974);\\
  %%CITATION = PRLTA,33,445;%%
   %\cite{Polyakov:1978vu}
%\bibitem{Polyakov:1978vu}
  A.~M.~Polyakov,
  %``Thermal Properties Of Gauge Fields And Quark Liberation,''
  Phys.\ Lett.\  B {\bf 72}, 477 (1978);\\
  %%CITATION = PHLTA,B72,477;%%
%  \bibitem{Susskind:1979up}
  L.~Susskind,
  %``Lattice Models Of Quark Confinement At High Temperature,''
  Phys.\ Rev.\  D {\bf 20}, 2610 (1979).
  %%CITATION = PHRVA,D20,2610;%%
  
 %\cite{Korchemsky:1993uz}
\bibitem{Korchemsky:1993uz}
  G.~P.~Korchemsky, G.~Marchesini,
  %``Resummation of large infrared corrections using Wilson loops,''
  Phys.\ Lett.\  {\bf B313}, 433-440 (1993).



 %\cite{Korchemskaya:1992je}
\bibitem{Korchemskaya:1992je}
  I.~A.~Korchemskaya, G.~P.~Korchemsky,
  %``On lightlike Wilson loops,''
  Phys.\ Lett.\  {\bf B287}, 169-175 (1992).
 

  %\cite{Drummond:2007aua}
\bibitem{Drummond:2007aua}
 J.~M.~Drummond, G.~P.~Korchemsky and E.~Sokatchev,
 %``Conformal properties of four-gluon planar amplitudes and Wilson loops,''
 Nucl.\ Phys.\  B {\bf 795}, 385 (2008)
 [arXiv:0707.0243 [hep-th]];\\
%\cite{Drummond:2007cf}
%\bibitem{Drummond:2007cf}
 J.~M.~Drummond, J.~Henn, G.~P.~Korchemsky and E.~Sokatchev,
 %``On planar gluon amplitudes/Wilson loops duality,''
 Nucl.\ Phys.\  B {\bf 795}, 52 (2008)
 [arXiv:0709.2368 [hep-th]].


   %\cite{Gatheral:1983cz}
\bibitem{Gatheral:1983cz}
  J.~G.~M.~Gatheral,
  %``Exponentiation Of Eikonal Cross-Sections In Nonabelian Gauge Theories,''
  Phys.\ Lett.\  B {\bf 133}, 90 (1983);\\
  %%CITATION = PHLTA,B133,90;%%
%\cite{Frenkel:1984pz}
%\bibitem{Frenkel:1984pz}
  J.~Frenkel and J.~C.~Taylor,
  %``Nonabelian Eikonal Exponentiation,''
  Nucl.\ Phys.\  B {\bf 246}, 231 (1984);\\
  %%CITATION = NUPHA,B246,231;%%
    %\cite{Sterman:1981jc}
%\bibitem{Sterman:1981jc}
  G.~Sterman, in ``Perturbative Quantum Chromodynamics", D.\ W.\ Duke and J.\ F.\ Owens ed.,
  %``Infrared Divergences In Perturbative QCD. (Talk),''
  AIP Conf.\ Proc.\  {\bf 74}, 22 (American Inst.\ of Phys., 1981).
  %%CITATION = APCPC,74,22;%%

%\cite{Laenen:2000ij}
\bibitem{Laenen:2000ij}
  E.~Laenen, G.~Sterman, W.~Vogelsang,
  %``Recoil and threshold corrections in short distance cross-sections,''
  Phys.\ Rev.\  {\bf D63}, 114018 (2001).
  [hep-ph/0010080].

\bibitem{Berger:2003zh}
  C.~F.~Berger,
  %``Soft gluon exponentiation and resummation,''
  arXiv:hep-ph/0305076;\\
%\cite{Berger:2002sv}
%\bibitem{Berger:2002sv}
  C.~F.~Berger,
  %``Higher orders in A(alpha(s))/(1-x)+ of non-singlet partonic splitting
  %functions,''
  Phys.\ Rev.\  D {\bf 66}, 116002 (2002)
  [arXiv:hep-ph/0209107].

  %\cite{Magnea:1990zb}
\bibitem{Magnea:1990zb}
  L.~Magnea and G.~Sterman,
  %``Analytic continuation of the Sudakov form-factor in QCD,''
  Phys.\ Rev.\  D {\bf 42}, 4222 (1990);\\
%\cite{Catani:1998bh}
%\bibitem{Catani:1998bh} 
  S.~Catani,
  %``The Singular behavior of QCD amplitudes at two loop order,''
  Phys.\ Lett.\ B {\bf 427}, 161 (1998)
  [hep-ph/9802439];\\
  %%CITATION = HEP-PH/9802439;%%
%\cite{Sterman:2002qn}
%\bibitem{Sterman:2002qn}
  G.~Sterman and M.~E.~Tejeda-Yeomans,
  %``Multi-loop amplitudes and resummation,''
  Phys.\ Lett.\  B {\bf 552}, 48 (2003)
  [arXiv:hep-ph/0210130];\\
  %%CITATION = PHLTA,B552,48;%% 
%\cite{Bern:2005iz}
%\bibitem{Bern:2005iz}
 Z.~Bern, L.~J.~Dixon and V.~A.~Smirnov,
 %``Iteration of planar amplitudes in maximally supersymmetric Yang-Mills
 %theory at three loops and beyond,''
 Phys.\ Rev.\  D {\bf 72}, 085001 (2005)
 [arXiv:hep-th/0505205];\\
 %%CITATION = PHRVA,D72,085001;%%
%\cite{Dixon:2008gr}
%\bibitem{Dixon:2008gr}
 L.~J.~Dixon, L.~Magnea and G.~Sterman,
 %``Universal structure of subleading infrared poles in gauge theory
 %amplitudes,''
 JHEP {\bf 0808}, 022 (2008)
 [arXiv:0805.3515 [hep-ph]].
 
\bibitem{Erdogan:2011}  O.\ Erdo\u{g}an and G.\ Sterman, in
  preparation.

%\cite{Mitov:2010rp}
\bibitem{Mitov:2010rp}
  A.~Mitov, G.~Sterman, I.~Sung,
  %``Diagrammatic Exponentiation for Products of Wilson Lines,''
  Phys.\ Rev.\  {\bf D82}, 096010 (2010).
  [arXiv:1008.0099 [hep-ph]].

%\cite{Date:1982un}
\bibitem{Date:1982un} 
  G.~Date, doctoral thesis,
  %``Factorization Theorems In Perturbative Quantum Field Theory,''
  UMI-83-07385.
%%CITATION = UMI-83-07385;%% 


%\cite{Bodwin:1984hc}
\bibitem{Bodwin:1984hc}
 G.~T.~Bodwin,
 %``Factorization of the Drell-Yan Cross-Section in Perturbation Theory,''
 Phys.\ Rev.\ D {\bf 31}, 2616 (1985)
 [Erratum-ibid.\ D {\bf 34}, 3932 (1986)];\\
%\cite{Collins:1985ue}
%\bibitem{Collins:1985ue}
 J.~C.~Collins, D.~E.~Soper and G.~Sterman,
 %``Factorization for Short Distance Hadron - Hadron Scattering,''
 Nucl.\ Phys.\ B {\bf 261}, 104 (1985);
 %%CITATION = NUPHA,B261,104;%%
 %%CITATION = PHRVA,D31,2616;%%
%\cite{Collins:1988ig}
%\bibitem{Collins:1988ig}
% J.~C.~Collins, D.~E.~Soper and G.~Sterman,
 %``Soft Gluons and Factorization,''
 %Nucl.\ Phys.\ B 
 {\bf 308}, 833 (1988).
 %%CITATION = NUPHA,B308,833;%%
  

\bibitem{Collinsbook}
%\cite{Collins:1989gx}
%\bibitem{Collins:1989gx}
  J.~C.~Collins, D.~E.~Soper and G.~Sterman,
  in ``Perturbative Quantum Chromodynamics", A.H.\ Mueller, ed.,
  Adv.\ Ser.\ Direct.\ High Energy Phys.\  {\bf 5}, 1 (World Scientific, 1988)
  [arXiv:hep-ph/0409313];\\
  %%CITATION = 00319,5,1;%%
J.\ Collins, ``Foundations of Perturbative QCD"  (Cambridge Univ. Pr., 2011).

\bibitem{Alday:2007hr}
 L.~F.~Alday and J.~M.~Maldacena,
 %``Gluon scattering amplitudes at strong coupling,''
 JHEP {\bf 0706}, 064 (2007)
 [arXiv:0705.0303 [hep-th]];\\
 %%CITATION = JHEPA,0706,064;%%
%\cite{Alday:2007he}
%\bibitem{Alday:2007he}
 L.~F.~Alday and J.~Maldacena,
 %``Comments on gluon scattering amplitudes via AdS/CFT,''
 JHEP {\bf 0711}, 068 (2007)
 [arXiv:0710.1060 [hep-th]].
 
%\cite{Alday:2008yw}
\bibitem{Alday:2008yw}
 L.~F.~Alday and R.~Roiban,
 %``Scattering Amplitudes, Wilson Loops and the String/Gauge Theory
 %Correspondence,''
 Phys.\ Rept.\  {\bf 468}, 153 (2008)
 [arXiv:0807.1889 [hep-th]].
 %%CITATION = PRPLC,468,153;%%
 
 %\cite{Kruczenski:2002fb}
\bibitem{Kruczenski:2002fb} 
  M.~Kruczenski,
  %``A Note on twist two operators in N=4 SYM and Wilson loops in Minkowski signature,''
��JHEP {\bf 0212}, 024 (2002)
��[hep-th/0210115].
��%%CITATION = HEP-TH/0210115;%%
  


 %\cite{Korchemsky:1987wg}
\bibitem{Korchemsky:1987wg}
  G.~P.~Korchemsky, A.~V.~Radyushkin,
  %``Renormalization of the Wilson Loops Beyond the Leading Order,''
  Nucl.\ Phys.\  {\bf B283}, 342-364 (1987).


%\cite{'tHooft:1973jz}
\bibitem{'tHooft:1973jz}
  G.~'t Hooft,
  %``A Planar Diagram Theory for Strong Interactions,''
  Nucl.\ Phys.\  B {\bf 72}, 461 (1974).
  %%CITATION = NUPHA,B72,461;%% 

 
\end{thebibliography}