\section{The background field gluon propagator}

The propagator $ G^{\mu\nu}_{ab}(x,y)$
of the semi-fast gluons in the background
of the tree-level field (\ref{Atilde}) and in the LC-gauge
($a^+_c={\cal A}^+_c=0$) has been obtained
in Sect. 6 of Paper I (see also Refs. \cite{AJMV95,HW98})
via the following three steps:
{\it i)} First one has constructed the background field 
propagator $G(x,y)$ of a {\it scalar} field. {\it ii)} Then, 
one has computed the gluon propagator $\acute G^{\mu\nu}_{ab}(x,y)$
for a quantum gluon in the {\it temporal} gauge $\acute a^-_c=0$ 
(but with the background field ${\cal A}^\mu_c=\delta^{\mu i}
{\cal A}^i_c$ still in the LC-gauge). {\it iii)} Finally,
the  LC-gauge propagator $ G^{\mu\nu}_{ab}$
has been obtained from $\acute G^{\mu\nu}_{ab}$ via
a gauge rotation. 

Below, we shall present the final results of these calculations.
For convenience, we start with the free propagator.


\subsection{The free LC-gauge propagator}

Consider first the free propagator (no background field)
$G^{\mu\nu}_{0\,ab}=\delta_{ab}G^{\mu\nu}_0$.
With the retarded prescription
advocated in Sect. 1.3, this reads, in momentum space:
\be\label{LCPROP}
G_0^{i-}(p)&=&{p^i\over p^++i\epsilon}\,G_0(p),\quad
G_0^{-i}(p)={p^i\over p^+-i\epsilon}\,G_0(p),\nn
G_0^{ij}(p)&=&\delta^{ij}G_0(p),\qquad
G_0^{--}(p)=\,{\rm PV}\, {2p^-\over p^+}\,G_0(p)\,,\ee
where $G_0(p)=1/(2p^+p^--p_\perp^2+i\epsilon)$ [the free
propagator of a massless scalar field], and PV denotes
the principal value prescription:\be
{\rm PV}\ {1 \over p^+} \equiv {1 \over 2}
\left ( 
{1 \over p^+\ -i \varepsilon} + {1 \over p^+\ +i \varepsilon}
\right ) \,.
%=\, \lim_{\varepsilon\to 0}\, \frac{ p^+}{(p^+)^2 
%+\varepsilon^2}\,.
\labe{PV}\ee
Note that, strictly speaking, the gauge-fixing prescription 
in eq.~(\ref{LCPROP}) is ``retarded''
only as far as the component $G^{i-}_0$ is concerned; by hermiticity,
the corresponding prescription in $G_0^{-i}$ is
advanced, while in $G_0^{--}$ it is a principal value.

Other prescriptions that are referred to in the main text are 
the ``advanced'' prescription (see, e.g., Refs. \cite{MQ,K96,KM98}) 
which is obtained by changing the sign of $i\epsilon$ for the axial 
poles in eq.~(\ref{LCPROP}) [that is, this is advanced for the
component $G^{i-}_0$], and the
PV-prescription, for which $1/p^+ \equiv {\rm PV}(1/p^+)$
in all the components of the propagator.

For further reference, we note also the expression of the scalar
propagator in the $x^-$ representation:
\be\label{scprop0}
         G_0(x^-,p^-,{ p}_\perp)&=&
\int {dp^+ \over 2 \pi}\  e^{-ip^+x^-}\,
{1 \over {2p^+p^--p_\perp^2 + i\epsilon}}\,\nn
&=& - {i\over {2p^-}}\, \left\{ \theta (x^-) \theta (p^-)
- \theta (-x^-) \theta (-p^-) \right\}
{\rm e}^{ -i\frac{p_{\perp}^2}{2p^-}x^-}\,.\ee

\subsection{ The background field propagator of a scalar field}

Since the background fields are static, and thus we have
homogeneity in time, it is convenient to work in the
$p^-$--representation, that is, to construct the propagator
$G(\vec x,\vec y, p^-)$ for a given $p^-$. This also makes it
easy to implement the strip restriction (\ref{strip-}).
The propagator is known only for the
discontinuous background field
${\cal A}^i(\vec x)=\theta(x^-){i\over g}V\del^i V^\dagger$
in eq.~(\ref{APM}),
which is how the actual field of eq.~(\ref{Atilde}) is ``seen''
by the semi-fast gluons\footnote{With $p^-$ constrained 
as in eq.~(\ref{strip-}), the typical momentum scale
$p^+\sim p^2_\perp/p^-$ for longitudinal dynamics is
relatively small, $p^+\ll \Lambda^+$, so that the 
semi-fast gluons are unable to discriminate the internal 
structure of the ``$\theta$''-function in eq.~(\ref{APM}).}.
The final result for $G(\vec x,\vec y, p^-)$ can be written as:
\be\label{GSCALAR}
G(\vec x,\vec y, p^-)&=&G_0(\vec x-\vec y, p^-)
\Bigl\{\theta (x^-)\theta (y^-)V(x_\perp)V^\dagger (y_\perp)
 + \theta (-x^-)\theta (-y^-)\Bigr\}\nn
&+& 2ip^-\int d^3\vec z\,\, G_0(\vec x-\vec z, p^-)
\, \delta (z^-)\,G_0(\vec z-\vec y, p^-) \nn
&{}&\times\,\left\{ \theta (x^-)\theta (-y^-)
V(x_\perp)V^\dagger(z_\perp) - \theta (-x^-) \theta(y^-)
V(z_\perp)V^\dagger (y_\perp)\right\}.\ee
It can be easily verified that this function is
continuous at both $x^- = 0$ and $y^- = 0$.

On eq.~(\ref{GSCALAR}), we distinguish two type of contributions, 
``crossing'' and ``noncrossing'', corresponding to trajectories which
cross, or do not cross, the plane at $x^-=0$ where is
located the singularity of the background field (or
its source). Consider ``noncrossing'' trajectories first, for
which $x^-$ and $y^-$ are of the same sign:
When they are both negative, the propagation takes place
 in a domain where the field vanishes; this is therefore
free propagation. When $x^-$ and $y^-$ are both positive,
the gluon propagates in a background field which is just a gauge
rotation, ${\cal A}^i={i\over g}V\del^i V^\dagger\,$;
thus, the net effect comes from the difference
between the gauge rotations at the end points. If the trajectory
crosses the discontinuity at  $x^-=0$, say in going from $y^-<0$
to $x^->0$, then there is also a gauge factor $V^\dagger(z_\perp)$
associated with this crossing (at some arbitrary $z_\perp$).

We thus write, with obvious notations, 
$G = G^{(n)} + G^{(c)}$. In the crossing piece $G^{(c)}$, it is
further possible to replace:
\be\label{xyp}
\theta (x^-)\theta (-y^-)\longrightarrow \theta (p^-),\qquad
\theta (-x^-) \theta(y^-)\longrightarrow \theta (-p^-),\ee
because of the correlation between the sign of 
$p^-$ and the direction of propagation in $x^-$, as manifest
on eq.~(\ref{scprop0}).

\subsection{ The gluon propagator in the temporal gauge}

This has the following non-zero components ($G\,=\,$ the scalar
propagator in  eq.~(\ref{GSCALAR})):
\be\label{acuteG}
\acute G^{ij}(\vec x,\vec y, p^-)&=&\delta^{ij} G(\vec x,\vec y, p^-),\nn
\acute G^{+i}(\vec x,\vec y, p^-)&=&{i \over p^-}\, {\cal D}_x^i 
G(\vec x,\vec y, p^-),\qquad
%\,=\,{i \over p^-}\,U(\vec x)\Bigl(\partial^i_x 
%\tilde G(\vec x,\vec y, p^-)\Bigr) U^{\dagger}(\vec y)\,,\nn
\acute G^{i+} (\vec x,\vec y, p^-)\,=\,-\,{i \over p^-}\,
G(\vec x,\vec y, p^-)\,{\cal D}^{\dagger \,j}_y\,,
%\,=\,-\,{i \over p^-}\,
%U(\vec x)\Bigl(\partial^i_y \tilde G(\vec x,\vec y, p^-)\Bigr)
%U^{\dagger}(\vec y),\ee
\nonumber\\
\acute G^{++}(\vec x,\vec y, p^-)&=&{1 \over (p^-)^2}\,\left\{{\cal D}_x^i
G(\vec x,\vec y, p^-){\cal D}^{\dagger \,i}_y\,+\,\delta^{(3)}(\vec x-\vec y)
\right\}\quad.
\ee
Note that, because of the strip restriction (\ref{strip-}) on $p^-$,
the operator $1/p^-$ is never singular.

In the above equations, ${\cal D}^i\equiv\partial^i -ig{\cal A}^i$ and
${\cal D}^{\dagger j}=\partial^{\dagger j} +ig{\cal A}^j$
with the derivative $\partial^{\dagger j}$ acting on the function on its left.
By using the Wilson lines in eq.~(\ref{GSCALAR}), it is possible to
convert these covariant derivatives into ordinary derivatives.
This relies on the following identities, valid
for any function $O(x)$ (cf. eq.~(\ref{Atilde})) :
\be
{\cal D}^i \Bigl[U(\vec x)O(x)\Bigr]\,=\,U(\vec x)\,\partial^i O(x),
\quad \Bigl[O(x)U^{\dagger}(\vec x)\Bigr]{\cal D}^{\dagger \,i}\,=\,
\Bigl(\partial^i O(x)\Bigr)U^{\dagger}(\vec x).\ee
This gives the following expressions for the crossing and 
non-crossing pieces of, e.g., $\acute G^{++}$:
\be\label{G++c}
{\acute G}^{++(c)}(\vec x,\vec y; p^-)&=&
{2i \over p^-} \int d^2z_{\perp}\
\partial^i_x G_0(x^-,x_{\perp}-z_{\perp})\,
\partial^i_y G_0(-y^-,z_{\perp}-y_{\perp})\nonumber\\
&{}&\,\times\,\left \{
\theta(p^-) V(x_\perp)V^\dagger(z_\perp) -
\theta(-p^-) V(z_\perp)V^\dagger (y_\perp)
\right \},
\ee
and
\be\label{G++n}
{\acute G}^{++(n)}(\vec x,\vec y; p^-)&=&
\Bigl\{\theta(x^-)\theta(y^-) V(x_\perp)V^\dagger (y_\perp)
+\theta(-x^-)\theta(-y^-)\Bigr\}
\nonumber\\&{}&\,\,\times\,{1 \over (p^-)^2}\,
\partial^i_x\partial^i_y G_0(\vec x-\vec y; p^-)\,
+\,\delta^{(3)}(\vec x-\vec y){1 \over (p^-)^2}\, .
\ee
In the crossing piece we have also performed the replacement 
(\ref{xyp}). 

In the zero field limit, the above equations yield the free
temporal-gauge propagator in the expected form:
\be\label{TGPROP}
{\acute G}_0^{ij}(p)=\delta^{ij}G_0(p),\quad
{\acute G}_0^{i+}(p)={\acute G}_0^{+i}(p)={p^i\over p^-}\,G_0(p),
\quad{\acute G}_0^{++}(p)=\,{2p^+\over p^-}\,G_0(p)\,,\ee
vwhere $G_0(p)=1/(2p^+p^--p_\perp^2+i\epsilon)$.

\subsection{ The gluon propagator in the LC gauge}

This is related to the temporal-gauge propagator presented
above via the relation:
\be\label{GLC}
iG^{\mu\nu}_{bc}(x,y)\,\equiv\,
\langle {\rm T}\,a_b^\mu(x) a_c^\nu(y)\rangle\,=\,
\left\langle {\rm T}\Bigl(
\acute a^\mu-{\cal D}^\mu{1 \over {\partial^+}} \,\acute a^+ \Bigr)_x^b
\Bigl(\acute a^\nu-{\cal D}^\nu{1 \over {\partial^+}} \,\acute a^+
\Bigr)_y^c\right\rangle,\ee
or, more explicitly,
\begin{mathletters}
\be\label{Gij}
G^{ij} & = &
\acute G^{ij} - {\cal D}^i {1 \over \partial^+}\, \acute G^{+j}
+ \acute G^{i+} {1 \over \partial^+} {\cal D}^{\dagger j}-
{\cal D}^i {1 \over \partial^+}\,{\acute G}^{++} 
{1 \over \partial^+} {\cal D}^{\dagger j}\,,\\\label{G-i}
G^{-i} &= &- \,{\partial^- \over \partial^+}\, \acute G^{+i} -
{\partial^- \over \partial^+} {\acute G}^{++}
 {1 \over \partial^+} {\cal D}^{\dagger i}\,,\\
\label{Gi-}
G^{i-} & =&   \,\acute G^{i+}\, {\partial^- \over \partial^+} - \,
{\cal D}^i \,{1 \over \partial^+}\, {\acute G}^{++} 
{\partial^- \over \partial^+ }\,,\\
\label{G--}
G^{--} & = &- \,{\partial^- \over \partial^+}
{\acute G}^{++} {\partial^- \over \partial^+}\,,
\ee
\label{LCG}
\end{mathletters}
where we use the convention that the 
derivatives written on the right act on the functions on their left;
e.g., $\partial^- F \,\partial^-\equiv \partial^-_x\partial^-_y F(x,y)$.


The axial poles at $p^+=0$ in the expressions above are
regularized according to the ``retarded'' prescription 
in eq.~(\ref{LCPROP}). This implies that the operators $1/\partial^+$
written on the left (right) of $\acute G^{\mu\nu}$ in eq.~(\ref{LCG})
are regularized with a retarded (advanced) prescription.
For instance,
\be \label{delRA}
{1 \over \partial^+}\,{\acute G}^{++} 
{1 \over \partial^+}\,\equiv \,{1 \over \partial^+_R}\,{\acute G}^{++} 
{1 \over \partial^+_A}\,,\ee
where
\begin{mathletters}
\be\label{delret}
\langle x^-|{1 \over {i\partial^+_R}}|y^-\rangle&\equiv&
\int {dp^+\over 2\pi} \,\frac{{\rm e}^{ip^+(x^--y^-)}}
{p^++i\epsilon}\,=\,-i\theta(x^--y^-)\,,\\
\langle x^-|{1 \over {i\partial^+_A}}|y^-\rangle&\equiv&\,
\int {dp^+\over 2\pi} \,\frac{{\rm e}^{ip^+(x^--y^-)}}
{p^+-i\epsilon}\,=\,i\theta(y^--x^-)\,.\label{deladv}
\ee\label{+RA}
\end{mathletters}
In going from eq.~(\ref{GLC}) to eq.~(\ref{LCG}),
we have also used :
\be \langle x^-|{1 \over {i\partial^+_A}}|y^-\rangle\,=\,-
\langle y^-|{1 \over {i\partial^+_R}}|x^-\rangle\,.\ee





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\begin{document}
\tighten
\preprint{Saclay-T01/085, BNL-NT-01/21, 

\title{ Nonlinear Gluon Evolution in the Color Glass Condensate: II}
\author{Elena Ferreiro}
\address{Departamento de F\'{\i}sica de Part\'{\i}culas,
%Universidad de Santiago de Compostela,
15706 Santiago de Compostela, Spain}
\author{Edmond Iancu}
\address{Service de Physique Th\'eorique, CE Saclay,
        F-91191 Gif-sur-Yvette, France}
\author{Andrei Leonidov}
\address{P. N. Lebedev Physical Institute, Moscow, Russia}
\author{Larry McLerran}
\address{ Physics Department, Brookhaven National Laboratory,
                 Upton, NY 11979, USA}
\date{\today }
\maketitle
\begin{abstract}
We complete the construction of the renormalization group equation 
(RGE) for the Color Glass Condenstate begun in Paper I. This is 
the equation which governs the evolution with rapidity
of the statistical weight function for the color glass field.
The coefficients in this equation --- one-loop real and virtual
contributions --- are computed explicitly, to all orders in the
color glass field. The resulting RGE can be interpreted as the
imaginary-time evolution equation, with rapidity as the 
``imaginary time'', for a quantum field theory in two spatial dimensions.
In the weak field limit it reduces to the BFKL equation.
In the general non-linear case, it is equivalent to an
equation by Weigert which summarizes in functional form
the evolution equations for Wilson line operators previously derived 
by Balitsky and Kovchegov.

\end{abstract}

\newpage

\input{S1.tex}
\input{S2.tex}
\input{S3.tex}
\input{S4.tex}
\input{S5.tex}
\input{S6.tex}

\appendix
\setcounter{equation}{0}
\input{A1.tex}
\input{A2.tex}
\input{A3.tex}
\input{A4.tex}
\begin{thebibliography} {99}


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\end{thebibliography}


\end{document}
\section{Introduction}

Hadronic scattering at high energy, or small Bjorken's $x$, uncovers
a novel regime of QCD where the coupling is small ($\alpha_s\ll 1$) but 
the parton densities are so large that conventional perturbation theory 
breaks down, via strong non-linear effects 
\cite{GLR,MQ,BM87,AM0,MV94,AGL,AM2,AM1,Larry01,Levin}. 
In a previous paper \cite{PI}, to be referred to as ``Paper I'' 
throughout this text, we have outlined the construction of an 
effective theory which is well suited to study the non-linear phenomena
at small $x$ and has a transparent physical interpretation: It portrays 
the gluon component of the hadron wavefunction (the relevant
component at small $x$) as a {\it Color Glass Condensate} (CGC).
This is a multiparticle quantum state with 
high occupation numbers, but to the accuracy of interest it can be 
represented as a classical stochastic color field 
with a probability law determined by a functional Fokker-Planck equation.

The latter is a renormalization group equation (RGE) which shows how to
construct the effective theory by integrating out quantum fluctuations
in the background of a strong color field (the CGC).
The non-linear effects included in the RGE via this background field
describe interactions among the gluons 
produced in the quantum evolution towards small $x$. 
This leads to a non-linear 
generalization of the BFKL equation \cite{BFKL} whose 
general structure has been originally identified in Refs. \cite{JKLW97}, 
in an effort to give a field-theoretical justification to
the McLerran-Venugopalan model for the gluon distribution
of a large nucleus \cite{MV94,K96,JKMW97,KM98}. But previous attempts 
\cite{JKW99} to compute the coefficients in this equation 
(beyond the linear, or BFKL, approximation \cite{JKLW97})
have suffered from technical complications and, moreover, appear 
to be inconsistent 
with the results obtained from other perturbative approaches 
\cite{B,K,W}.


In this paper, we continue the 
analysis in Paper I and compute explicitly the coefficients 
in the RGE alluded to above. As recently reported in Ref. \cite{SAT}, 
the non-linear effects encoded in these coefficients lead to 
{\it gluon saturation}, that is, to a limitation on the maximum
gluon density in the hadron wavefunction as $x\to 0$. 
In contrast to the linear BFKL
equation, which predicts the exponential growth of the gluon distribution 
function with $\tau\equiv \ln(1/x)$, our corresponding result 
grows only linearly, and thus respects the unitarity bounds.
This is consistent with a previous result by Mueller \cite{AM2},
and also with some recent analyses \cite{LT99,B00,LL,Levin} 
of the Balitsky--Kovchegov equations \cite{B,K}, 
which are encoded too in our RGE \cite{RGE} 
(see also Sect. \ref{sec:BK} below).

This paper is quite technical and relies heavily on the results of 
Paper I. To facilitate its reading and prepare the calculations to follow,
it is convenient to summarize, in the remaining part of this Introduction, 
the general structure of the effective theory and its quantum evolution. 
In this introductory discussion, we shall follow the presentation 
in Paper I, to which the reader may refer for more details, but we shall also 
anticipate some of the results to be obtained later in this paper.
The remaining sections are organized as follows: 
In Section II we present the explicit calculation of the
``real correction', i.e., the charge-charge correlator $\chi$
which plays the role of the diffusion kernel in the functional RGE.
In Section III we do the same for the ``virtual correction'', i.e.,
the one-point function $\sigma$ which plays the role of a force term. 
In Section IV, we present the final
result for the RGE, and discuss its general structure and 
some of its remarkable properties.
We emphasize, in particular, the Hamiltonian structure of this
equation, and its relation with a similar equation by Weigert \cite{W}.
In Section V, we derive evolution
equations for observables from the general RGE. In the weak field
limit, we thus recover the BFKL equation for the unintegrated gluon
distribution. In the general non-linear case, we obtain the
coupled evolution equations for Wilson-line correlators originally
derived by Balitsky and (in the large $N_c$ limit) also by Kovchegov.
In Section VI we present conclusions and perspectives.
Some explicit calculations, as well as the presentation of
the background field gluon propagator, are deferred to the four 
Appendices.


\subsection{The effective theory for the CGC}
\label{sec:EFT}

The effective theory applies to gluon correlations in the hadron 
 wavefunction as measured in deep inelastic 
scattering at small Bjorken's $x$. It is formulated in the hadron
infinite momentum frame, where small $x$ corresponds to soft
longitudinal momenta\footnote{Throughout, we
use light-cone vector notations, that is,
$v^\mu=(v^+,v^-,{\bf v}_\perp)$, with
$v^+\equiv (1/\sqrt 2)(v^0+v^3)$,
$v^-\equiv (1/\sqrt 2)(v^0-v^3)$, and ${\bf v}_\perp
\equiv (v^1,v^2)$. The dot product reads $k\cdot x = k^- x^+ + k^+ x^-
- {\bf k}_\perp \cdot {\bf x}_\perp$. The hadron four-momentum
reads $P^\mu=(P^+,0,0_\perp)$, with $P^+\to \infty$.}
$k^+=xP^+$, with $P^+$ the hadron momentum, and $x\ll 1$.
The main physical assumption, which is a posteriori verified in
the construction of the effective theory, is that the ``fast''
partons in the hadron wavefunction (i.e., the excitations with
relatively large longitudinal momenta $p^+\gg k^+$) can be replaced,
as far as their effects on the soft correlation functions are 
concerned, by a classical random {\it color source} $\rho^a(x)$, 
whose gross properties are determined by the kinematics.

The separation of scales in longitudinal momenta ($p^+\gg k^+$) 
implies a corresponding separation in (light-cone) energies
($p^-\sim p_\perp^2/2p^+\,\ll\,k^- \sim k_\perp^2/2k^+$),
and therefore also in (light-cone) time scales: The lifetime
$\Delta x^+ \sim 1/k^- \propto k^+$ of the
soft gluons is much shorter than the characteristic time scale
$\sim 1/p^- \propto p^+$ for the dynamics of the fast
partons. Thus, the latter appear to the soft gluons
as nearly on-shell colored particles which propagate on 
the light-cone ($z\simeq t$, or $x^-\simeq 0$)
with large $p^+$ momenta. The associated color charge density 
$\rho^a(x)$ is therefore
{\it static} (i.e., independent of $x^+$), {\it localized} near the 
light-cone (within a small distance $\Delta x^- \sim 1/p^+ \ll 1/k^+$), 
and {\it random} (since this is the instantaneous color charge in the
hadron ``seen'' by the soft gluons at the arbitrary time of their
emission). The correlations of $\rho$ are encoded in a
gauge-invariant {\it weight function} $W_\tau[\rho]$. This is the 
probability density for having a color charge distribution with 
density $\rho_a(x^-,{\bf x}_{\perp})$, normalized as:
\be\label{norm}
\int {\cal D}\rho\, \,W_\tau[\rho]\,=\,1\,.\ee
Note that we use the momentum-space rapidity 
$\tau\equiv\ln(P^+/k^+) = \ln(1/x)$ to indicate the
dependence of the weight function upon the soft scale $k^+$.

Thus, in this effective theory, the
equal-time\footnote{Only equal-time correlators are needed, since these
are the only ones to be measured by a small-$x$ external probe, which
is absorbed almost instantaneously by the hadron. In eq.~(\ref{clascorr}),
it is understood that the fields $A^i_a(x^+,\vec x)$ involve only Fourier
modes with longitudinal momenta $k^+=xP^+$.}
gluon correlation functions at the scale $k^+ =
xP^+$ are obtained as (with ${\vec x}\equiv (x^-,{\bf x}_{\perp})$):
\be\label{clascorr}
\langle A^i_a(x^+,\vec x)A^j_b(x^+,\vec y)
\cdots\rangle_\tau\,=\,
\int {\cal D}\rho\,\,W_\tau[\rho]\,{\cal A}_a^i({\vec x})
{\cal A}_b^j({\vec y})\cdots\,,\ee
where ${\cal A}_a^i\equiv {\cal A}_a^i[\rho]$ is the 
solution to the classical Yang-Mills equations with source
$\rho_a$ :
\be
(D_{\nu} F^{\nu \mu})_a(x)\, =\, \delta^{\mu +} \rho_a(x)\,,
\label{cleq0}
\ee
in the light-cone (LC) 
gauge $A^+_a=0$, which is the gauge which allows for the most
direct contact with the gauge-invariant physical quantities 
\cite{AM1,PI}. For instance, 
the gluon distribution function
($\equiv$ the total number of gluons per unit rapidity
with longitudinal momentum $k^+=xP^+$ and transverse momentum
$k_\perp^2 \le Q^2$) is obtained as \cite{AM1,PI}
\be\label{GCL}
x G(x,Q^2)&=&\frac{1}{\pi}
\int {d^2k_\perp \over (2 \pi)^2}\,\Theta(Q^2-
k_\perp^2)\,\Bigl\langle\,
|{\cal F}^{+i}_a(\vec k)|^2\Bigr\rangle_\tau\,.\ee
where ${\cal F}^{+i}_a=\partial^+ {\cal A}^i_a$ is the 
electric field associated to the classical solution
${\cal A}_a^i[\rho]$, and $\vec k \equiv (k^+,{\bf k}_\perp)$
with $k^+=xP^+=P^+{\rm e}^{-\tau}$.

To deduce explicit expressions for these classical fields,
it is preferable to express the LC-gauge solution
${\cal A}_a^i$ in terms of color source $\tilde\rho_a$
in the {\it covariant} gauge $\partial^\mu \tilde A_\mu =0$
(COV-gauge). This is possible since both the measure and the
weight function in the functional integral (\ref{clascorr})
are gauge-invariant, so that the classical average can be done 
equally well by integrating over the COV-gauge $\tilde\rho$.
In terms of this latter, the classical solution 
${\cal A}_a^\mu[\tilde\rho]$
is known explicitly \cite{PI}, and reads: 
${\cal A}_a^+=0$ (the gauge condition), ${\cal A}^-_a=0$, and 
(in matrix notations appropriate for the adjoint representation:
$\tilde\rho\equiv \tilde\rho_a T^a$, etc.)
\begin{mathletters}\be\label{Aclass}\,
{\cal A}^i\,
(\vec x) &=&{i \over g}\, U(\vec x) \,\partial^i  U^\dagger(\vec x),\\
U^{\dagger}(x^-,x_{\perp})&=&
 {\rm P} \exp
 \left \{
ig \int_{-\infty}^{x^-} dz^-\,{\alpha}(z^-,x_{\perp})
 \right \},\label{Udef}\\
- \nabla^2_\perp \alpha({\vec x})&=&\tilde\rho(\vec x).
\label{alpharho}\ee\label{Atilde}\end{mathletters}
This is the gauge-transform, with gauge function $U(\vec x)$,
of the corresponding solution in the COV-gauge, which has only
one non-trivial component: $\tilde{\cal A}^\mu_a 
=\delta^{\mu +}\alpha_a({\vec x})$.
All the fields above are static, i.e., independent of $x^+$.
Since $\tilde\rho$, and therefore $\alpha$, are localized at
$\Delta x^- < 1/k^+$, the associated field ${\cal A}^i$ appears 
effectively as a $\theta$--function :
\be\labe{APM}
{\cal A}^i(x^-,x_\perp)\,\approx\,\theta(x^-)\,
\frac{i}{g}\,V(\del^i V^\dagger)
\,\equiv\,\theta(x^-){\cal A}^i_\infty(x_\perp),\ee
to any probe with a much lower longitudinal resolution
(i.e., with momenta $q^+\ll k^+$). On the same resolution scale:
\be\labe{UTAF}
U^{\dagger}(x^-,x_{\perp})\,\approx \,
\theta(x^-)\,V^\dagger(x_{\perp}) + \theta(-x^-),\qquad\,
{\cal F}^{+i}(\vec x) \,\approx\,\delta(x^-)\,
{\cal A}^{i}_\infty(x_\perp).\ee
In the equations above, $V$ and $V^\dagger$ are
the asymptotic values of the respective
gauge rotations as $x^-\to\infty$ :
\be\labe{v}
V^\dagger(x_{\perp})\,\equiv\,{\rm P} \exp
 \left \{
ig \int_{-\infty}^{\infty} dz^-\,\alpha (z^-,x_{\perp})
 \right \}.\ee
In practice, $U(x^-,x_{\perp})=V(x_{\perp})$ for any
$x^-\gg 1/k^+$.


Note that the solution to the classical non-linear equation
is known exactly, which makes this approach particularly convenient
to study the non-linear physics at small $x$. The classical 
approximation should be indeed appropriate in this regime, which
is characterized by weak coupling and large occupation numbers.
On the other hand, the weight function $W_\tau[\rho]$ 
is obtained via a quantum calculation in which the quantum 
fluctuations with momenta $k^+\simle p^+ \simle P^+$ are
integrated out in layers of $p^+$, and in the background of
the classical fields ${\cal A}^i$ \cite{JKMW97,JKLW97,PI}. 
This calculation captures the basic 
mechanism leading to large gluon densities --- namely, the BFKL-type 
of evolution towards small $x$ ---, while also including non-linear 
effects (i.e., rescatterings among the produced gluons) 
via the background fields. In the saturation regime, where one
expects color fields as strong as ${\cal A}^i\sim 1/g$
(corresponding, via eq.~(\ref{GCL}),
to gluon densities of order $1/\alpha_s$
\cite{AM1,JKMW97,AM2,SAT}), the background field calculation
must be carried out {\it exactly}, i.e., to all orders in ${\cal A}^i$. 

\subsection{The quantum evolution of the effective theory}
\label{QEVOL}


To describe the quantum evolution, it is convenient to introduce
an arbitrary separation scale $\Lambda^+$ such as 
$k^+ \simle \Lambda^+ \ll P^+$, and assume that the
``fast'' quantum modes with momenta $p^+ \gg\Lambda^+$ have been 
already integrated out and replaced, to the accuracy of interest, by
a classical color source $\rho_a(\vec x)$ with weight function
$W_\Lambda[\rho]$. On the other hand, the ``soft'' gluons with momenta
$p^+ <\Lambda^+$ are still explicitly present in the theory.
The correlation functions at the soft scale
$k^+$ are then obtained from the following generating functional :
\be\labe{PART}
{Z}[J]\,=\,\int {\cal D}\rho\,\,W_\Lambda[\rho]
\,\,\left\{\frac{\int^\Lambda {\cal D}A_a^\mu\,
\delta(A^+_a)\,\,{\rm e}^{\,iS[A,\,\rho]-i\int J\cdot A}}
{\int^\Lambda {\cal D}A_a^\mu\,\delta(A^+_a)\,\,{\rm e}^{\,iS[A,\,\rho]}}
\right\}.\ee
where the ``external current'' $J^\mu_a$ is just a device to generate
Green's functions via functional differentiations,
and should not be confused with the physical source $\rho$.
Eq.~(\ref{PART}) is written fully in the LC gauge (in particular, $\rho$
is the LC-gauge color source), and involves
two functional integrals: ({\it a}) a quantum path integral
over the soft gluon fields $A^\mu$, which generates ($\rho$-dependent)
quantum expectation values at fixed $\rho$, e.g.,
\be\labe{2point} 
\langle {\rm T}\,A^\mu(x)A^\nu(y)\rangle [\rho]
\,=\,\frac{\int^\Lambda {\cal D}A\,\delta(A^+)
\,\,A^\mu(x)A^\nu(y)\,\,{\rm e}^{\,iS[A,\,\rho]}}
{\int^\Lambda {\cal D}A\,\delta(A^+)\,\,{\rm e}^{\,iS[A,\,\rho]}}
,\ee
and ({\it b}) a classical average over $\rho$.
As discussed in Paper I, eq.~(\ref{PART}) has the typical structure
to describe correlations in a glass.


The action $S[A,\,\rho]$ in eqs.~(\ref{PART}) and (\ref{2point})
reads as follows \cite{JKLW97} :
\be\label{ACTION}
S[A,\rho]\,=\,- \int d^4x \,{1 \over 4} \,F_{\mu\nu}^a F^{\mu\nu}_a
\,+\,{i \over {gN_c}} \int d^3 \vec x\, {\rm Tr}\,\Bigl\{ \rho(\vec x)
\,W[A^-](\vec x)\Bigr\}\,\equiv\,S_{YM}\,+\,S_W,\,\,
\ee
where $W[A^-]$ is a Wilson line in the temporal direction:
\be\label{WLINE}
W[A^-](\vec x)\, =\,{\rm P}\, \exp\left[\,ig\int dx^+ A^-(x) \right].
\ee
In the classical, or saddle point, approximation $\delta S/\delta A^\mu
=0$, the action (\ref{ACTION}) generates the desired equations
of motion, that is, eqs.~(\ref{cleq0}) with $A^-=0$. This shows that
the classical field in eqs.~(\ref{Atilde}) is the tree-level background
field in which propagate the quantum fluctuations. 
The general non-linear structure of $S_W$ in eq.~(\ref{WLINE}),
which plays a role only for the quantum corrections (in that it
generates new interaction vertices; cf. Sect. \ref{FRULES} 
below), reflects our eikonal approximation for the interaction 
between fast particles moving in the plus direction (here represented
by $\rho$) and the comparatively slow gluon fields.


In the above formulae, the intermediate scale $\Lambda^+$
enters at two levels: as an upper cutoff on the
longitudinal momenta $p^+$ of the quantum gluons, and in
the weight function $W_\Lambda[\rho]$ for the classical source.
Of course, the
final results for correlation functions at the scale $k^+$
must be independent of this arbitrary scale :
\be
\Lambda^+ \frac{\del {Z}[J]}{\del \Lambda^+}\,=\,0.\ee
%\equiv\,\frac{\del {Z}[J]}{\del \tau}\,=\,0,\ee
This constraint can be formulated as a
renormalization group equation (RGE) for $W_\Lambda[\rho]$ which 
governs its evolution with decreasing $\Lambda^+$.

The initial conditions for this evolution are determined by the
properties of the hadronic matter at $\Lambda^+ \sim P^+$.
These are not really under control in perturbation theory,
but one can try to rely on some non-perturbative model, 
like the valence quark model.  The initial conditions might be under better 
control in the high density environment of a very large nucleus,
since we expect the coupling constant to be weaker at high density.
The indeterminancy of evolution equations due to initial conditions is hardly a new problem in QCD, since the DGLAP and BFKL equations are both
limited by such uncertainty.  In these cases, and we also hope
 here\footnote{The approximate solutions to the RGE recently
found in Ref. \cite{SAT} appear to confirm this expectation.}, 
the solution of the evolution equation for arbitrarily high energies 
is universal and its generic properties are largerly independent of the initial conditions.


Starting with these initial conditions, one then proceeds with the
quantum evolution down to the soft scale of interest
 $\Lambda^+ \sim xP^+\ll P^+$.
In this process, the original source at the scale 
$P^+$ gets dressed by the quantum fluctuations with momenta 
$\Lambda^+ < |p^+| < P^+$. We treat this process in perturbation theory, in the ``leading logarithmic approximation'' (LLA)\footnote{Indeed, it is only 
to this accuracy that the assumed separation of scales in the problem is
maintained by quantum corrections.}
--- i.e., by retaining only the terms 
enhanced by the large logarithm $\ln(1/x)$, to all orders in
$(\alpha_s\ln(1/x))^n$) ---,
which, by itself, is also the accuracy of the BFKL equation, 
but we go beyond BFKL in that we resum also finite density effects, 
which are expected to become increasingly
important as we go to smaller and smaller $\Lambda^+$ (or Bjorken's $x$). 
We do that by performing a background field calculation, that is,
by integrating out the quantum fluctuations at one step 
in the background of the classical field generated by the color
source at the previous step, with the background field simulating
(via its correlations) the finite density effects in the system.

To be more explicit, 
let us describe one step in this renormalization group procedure
in some detail. Assume that we know the effective theory at
the scale $\Lambda^+$ --- as specified by the
corresponding weight function $W_\Lambda[\rho]\equiv W_\tau[\rho]$,
with $\tau=\ln(P^+/\Lambda^+)$ ---, and we are interested
in correlations at the softer scale
$k^+ \sim b\Lambda^+$ with $b\ll 1$, but such as $\alpha_s\ln(1/b)< 1$, for
perturbation theory in powers of $\alpha_s\ln(1/b)$ to make sense.
Our purpose is to construct the new weight function 
$W_{b\Lambda}[\rho]\equiv  W_{\tau+\Delta\tau}[\rho]$,
with $\Delta\tau\equiv \ln(1/b)$, which would determine the 
gluon correlations at this softer scale. As compared to 
$W_\tau[\rho]$, this new weight function must include also the 
quantum effects induced by the ``semi-fast'' gluons 
with longitudinal momenta in the strip
\be\labe{strip}\,\,
 b\Lambda^+ \,\,\ll\,\, |p^+|\,\, \ll\,\,\Lambda^+\,.\ee
To compute these effects, it is convenient to decompose the gluon field 
in eq.~(\ref{PART}) as follows 
\be
A^\mu_c\,=\,{\cal A}^\mu_c[\rho]+a^\mu_c+\delta A^\mu_c,\ee
where ${\cal A}^\mu_c$ is the tree-level field,
$a^\mu_c$ represents the semi-fast fluctuations, and
$\delta A^\mu_c$ refers to the remaining modes with
$|p^+| \le b\Lambda^+$.
By integrating out the fields $a^\mu$, some
new correlations are induced at the soft scale $b\Lambda^+$ 
via the coupling $\delta A^-_c\delta\rho_c$,
where $\delta\rho_c$ 
is the color charge of the semi-fast gluons. 
These correlations have to be computed to lowest order
in $\alpha_s\ln(1/b)$, but to all orders in the 
background fields ${\cal A}^i$. 
This is essentially an one-loop calculation, but with the exact
background field propagator of the semi-fast gluons.
The new correlations are eventually absorbed 
into the functional change $\Delta W \equiv W_{\tau+\Delta\tau} - W_\tau$
in the weight function. Since $\Delta W\propto \Delta \tau$,
this evolution is most conveniently formulated as a (functional) 
renormalization group equation for $W_\tau[\rho]$, which reads
\cite{JKLW97,PI}
\be\label{RGE}
{\del W_\tau[\rho] \over {\del \tau}}\,=\,\alpha_s
\left\{ {1 \over 2} {\delta^2 \over {\delta
\rho_\tau^a(x_\perp) \delta \rho_\tau^b(y_\perp)}} [W_\tau\chi_{xy}^{ab}] - 
{\delta \over {\delta \rho_\tau^a(x_\perp)}}
[W_\tau\sigma_{x}^a] \right\}\,,
\ee 
in compact notation where repeated color indices 
(and coordinates) are understood to be summed (integrated) over. The
 coefficients $\sigma_{x}^a\equiv \sigma_a(x_\perp)$ and
$\chi_{xy}^{ab}\equiv\chi_{ab}(x_\perp,y_\perp)$ in the above
equation are related to the 1-point and 2-point functions
of the color charge $\delta\rho_a(x)$
of the semi-fast gluons via the following relations:
\be\label{sigperp}
\alpha_s\ln{1\over b}\,\sigma_a ({x}_\perp)&\equiv &
\int dx^- \,\langle\delta \rho_a(x)\rangle\,,\nn
\alpha_s\ln{1\over b}\,\chi_{ab}(x_\perp, y_\perp)&\equiv &
\int dx^- \int dy^-\,
\langle\delta \rho_a(x^+,\vec x)\,
\delta \rho_b(x^+,\vec y)\rangle\,,\ee
where the brackets denote the average over quantum fluctuations
in the background of the tree-level color fields ${\cal A}^i$,
as shown in eq.~(\ref{2point}).

In writing eq.~(\ref{RGE}), we have also anticipated the longitudinal
structure of the quantum evolution, which will become manifest only
after performing the quantum calculations in the next sections. Specifically,
we shall see that the {\it induced source} $\langle\delta \rho_a(x)\rangle$
($\equiv$ the correction to $\rho$ generated by the gluons 
with $p^+$ in the strip (\ref{strip})) has support at
% $x^-$ in the strip
\be\label{stripx-}
1/\Lambda^+\,\,\simle\,\,x^-\,\,\simle\,\,1/(b\Lambda^+)\,.\ee
By induction, we shall deduce that $\rho_a(\vec x)$ ($\equiv$ the color
source generated by the quantum evolution down to 
$\Lambda^+=P^+{\rm e}^{-\tau}$) has support 
at\footnote{The fact that the source has support at {\it
positive} $x^-$, rather than at generic  $x^-$ with $|x^-|\simle x^-_\tau$
will be seen to be related to our specific gauge-fixing prescription;
cf. Sect. \ref{sec:sigmaA}.} 
$0\le x^-\simle x^-_\tau$ with 
\be\label{xtau}
 x^-_\tau\,\equiv\, 1/\Lambda^+\,=\,x^-_0{\rm e}^{\tau},\qquad
x^-_0\,\equiv\, 1/P^+\,.\ee
Thus, the color source is built in layers of $x^-$, with a one-to-one
correspondence between the $x^-$--coordinate of a given layer and the 
$p^+$--momenta of the modes that have been integrated to generate that layer. 
%This correspondence is most conveniently expressed
%in terms of the corresponding {\sl rapidities}. We have already
%introduced the {\sl momentum} rapidity 
%$\tau\equiv\ln(P^+/\Lambda^+) = \ln(1/x)$ to specify
%the longitudinal momentum. Let us similarly introduce
%a {\sl space-time} rapidity $\eta$ to indicate the
%longitudinal coordinate $x^-\,$; this is defined as (for positive $x^-$):
%\be\label{etarapdef} \eta
When some new quantum modes, with rapidities
$\tau <\tau' <\tau+\Delta\tau$, are integrated out, 
the preexisting color source at
$0 < x^-< x^-_\tau$ % (or {\sl space-time} rapidities $\eta <\tau$)
remains unchanged, but some new source is added in
the interval (\ref{stripx-}). 
%(i.e., $x^-_\tau < x^-< x^-_{\tau+\Delta\tau}$). 
Because of that, %the corresponding change in the weight function
$\Delta W \equiv W_{\tau+\Delta\tau} - W_\tau$ involves only the
change in $\rho_a$ within that last interval. In the
continuum limit, this generates the functional derivatives of
$W_\tau$ with respect to $\rho_a(\vec x)$ at $x^-=x^-_\tau$
{\it only}, that is, the derivatives with respect to
\be\label{rhotau}
\rho_{\tau}^a(x_\perp)
\,\equiv\, \rho^a(x^-=x^-_\tau,x_\perp),\ee
as shown in eq.~(\ref{RGE}).

On the other hand, given the separation of scales in the problem,
the detailed longitudinal structure of the quantum effects should 
not be too important, and this too has been anticipated in writing
eqs.~(\ref{RGE})--(\ref{sigperp}): In the same way as the original
source with support at $x^-\simle 1/\Lambda^+$ appears effectively
as a $\delta$-function to the semi-fast gluons (with 
wavelenghts $\Delta x^-\sim 1/p^+ \gg 1/\Lambda^+$), 
the induced source $\langle\delta \rho\rangle$,
although relatively delocalized as compared
to $\rho$ (cf. eq.~(\ref{stripx-})), appears to the
soft gluons with $k^+\simle b\Lambda^+$ as a rather
sharp color distribution around $x^-\simeq x^-_\tau$. 
Thus, the soft gluons can probe only
%(and also the
%small-$x$ external probe in DIS) are sensitive only to the
the {\it two-dimensional} color charge distribution 
in the transverse plane, as obtained after integrating out $x^-$, 
cf. eq.~(\ref{sigperp}). 
This explains the 2-dimensional structure of the coefficients
$\sigma$ and $\chi$ in the RGE (\ref{RGE}).


Eq.~(\ref{RGE}) has the structure of the diffusion equation: It
is a second-order (functional) differential equation whose r.h.s.
is a total derivative (as necessary to conserve the total
probability; cf. Sect. \ref{sect:PROP}). Thus, this equation describes
the quantum evolution as a diffusion (with diffusion ``time'' $\tau$)
of the probability density $W_\tau[\rho]$
in the functional space spanned by $\rho_a(x^-,x_\perp)$.
For this equation to be useful, its coefficients $\sigma$
and $\chi$ must be known explicitly as functionals of $\rho$.
It is therefore more convenient to use the COV-gauge source 
$\tilde\rho_a$, or the associated Coulomb field $\alpha_a$,
as the functional variable to be averaged over. Indeed,
$\sigma$ and $\chi$ depend upon the color source
via the classical field ${\cal A}^i$, which is most directly
related to $\alpha_a$, cf. eqs.~(\ref{Atilde}).

As explained in Ref. \cite{PI}, when going from the (background)
LC-gauge to the COV-gauge, the induced color charge $\sigma$ acquires 
a new contribution, in addition to the color rotation with matrix
$U^\dagger$. This is so because the transformation between
the two gauges depends upon the charge content in the problem: this
was $\rho$ at the initial scale $\Lambda^+$, thus giving a gauge
transformation $U^\dagger[\rho]$, but it becomes $\rho+\delta\rho$,
with fluctuating $\delta\rho$, at the new scale $b\Lambda^+$, 
thus inducing a fluctuating component in the corresponding 
gauge function $U^\dagger[\rho+\delta\rho]$.
After averaging out the quantum fluctuations, one is left with a RGE for
$W_\tau[\tilde\rho]$ which is formally similar to eq.~(\ref{RGE}), but
with $\rho \to \tilde\rho$,
$\chi\to \tilde\chi$, and $\sigma\to \tilde\sigma$, where:
\begin{mathletters}
\label{tildesc}
\be\label{tildechi}
\tilde \chi_{ab}(x_\perp,y_\perp)&\equiv&
V^{\dagger}_{ac}(x_\perp)\,
\chi_{cd}(x_\perp,y_\perp)\,V_{d b}(y_\perp),\\
\tilde\sigma_a (x_\perp)&\equiv&
 V^{\dagger}_{ab}(x_\perp)\,\sigma_b(x_\perp)- \delta\sigma_a(x_\perp),
\label{tildesig} \\ 
\delta\sigma_a(x_\perp)&\equiv&
{g\over 2}\,f^{abc}\int d^2y_\perp\,\,
\tilde\chi_{cb}(x_\perp,y_\perp)\,
\langle y_\perp|\,\frac{1}{-\grad^2_\perp}\,|
x_\perp\rangle\,,\label{sigclas}
\ee\end{mathletters}
with $V$ and $V^\dagger$ as defined in eq.~(\ref{v}).
The correction $-\delta\sigma$, to be referred to as the
``classical polarization'', is the result of the quantum evolution
of the gauge transformation itself.

It turns out that the RGE is most conveniently written as
an equation for $W_\tau[\alpha]\equiv
W_\tau[\tilde\rho= - \nabla^2_\perp \alpha]$, in which case it reads:
\be\labe{RGEA0}
{\del W_\tau[\alpha] \over {\del \tau}}\,=\,\alpha_s
\left\{ {1 \over 2} {\delta^2 \over {\delta
\alpha_\tau^a(x_\perp) \delta \alpha_\tau^b(y_\perp)}} 
[W_\tau\eta_{xy}^{ab}] - 
{\delta \over {\delta \alpha_\tau^a(x_\perp)}}
[W_\tau\nu_{x}^a] \right\}\,,
\ee
where $\alpha^a_\tau(x_\perp)\equiv
\alpha^a(x^- = x^-_\tau,x_\perp)$ (cf. eq.~(\ref{rhotau})), and
\begin{mathletters}
\be\label{nudef}
\nu^a (x_\perp)&\equiv&\int d^2z_\perp\,
\langle x_\perp|\,\frac{1}{-\grad^2_\perp}\,|z_\perp\rangle\,
\tilde\sigma^a (z_\perp),\\
\eta^{ab}(x_\perp,y_\perp)&\equiv&\int d^2z_\perp \int d^2u_\perp\,
\langle x_\perp|\,\frac{1}{-\grad^2_\perp}\,|z_\perp\rangle\,
\tilde\chi^{ab}(z_\perp,u_\perp)\,
\langle u_\perp|\,\frac{1}{-\grad^2_\perp}\,|y_\perp\rangle\,.
\label{etadef}\ee
\label{nueta}\end{mathletters}
This form has, in particular, the advantage to eliminate any explicit
reference to the arbitrary infrared cutoff $\mu$ which enters
the solution to eq.~(\ref{alpharho}) for $\alpha$ :
\be\labe{alpha}
\alpha ({\vec x})
%\,=\,\int d^2y_\perp\,
%\langle x_\perp|\,\frac{1}{-\grad^2_\perp}\,|y_\perp\rangle\,
%\tilde\rho (x^-,{\bf y}_\perp)
\,=\,-\int \frac{d^2y_\perp}{2\pi}\,
\ln\Bigl(|{\bf x}_\perp - {\bf y}_\perp|\mu\Bigr)\,
\tilde\rho (x^-,{\bf y}_\perp).\ee
It is our ultimate goal in this paper to compute explicitly the
quantities $\eta$ and $\nu$ as functionals of the field
$\alpha$, and thus completely specify the RGE (\ref{RGEA0}).



\subsection{Feynman rules for $\chi$ and $\sigma$}
\label{FRULES}

The basic quantum calculation is that of the quantities
$\chi$ and $\sigma$ in the LC gauge.
The necessary Feynman rules can be
readily derived from the action (\ref{ACTION}) \cite{JKLW97,PI}, and are
summarized here for later reference. Eqs.~(\ref{sigperp}) involve the color
charge $\delta\rho_a$ of the semi-fast fields $a^\mu_c$. To
the order of interest, this reads
\begin{equation}\label{delta12}
\delta \rho_a (x) =\delta \rho_a^{(1)} (x)+\delta \rho_a^{(2)} (x),
\end{equation}
where $\delta \rho^{(1)}$ is linear in the fluctuations 
$a^\mu$, while $\delta \rho^{(2)}$ is quadratic\footnote{To compare
with the corresponding formulae in Paper I (see, e.g., eqs. (4.41)--(4.42)
in Ref. \cite{PI}), please note a change in our
normalization conventions: Here, the various fields preserve their 
natural normalization fixed by the action (\ref{ACTION}). 
By contrast, in Paper I we have rescaled the classical fields 
and sources by a factor $1/g$.}: 
\be\label{rho10}
 \delta \rho_a^{(1)} (x) & = & 
-2i g{\cal F}^{+i}_{ac} (\vec x) a^{ic} (x) + \nonumber \\
& & +g\rho^{ac} (\vec x)
\int dy^+ \langle x^+ |{\rm PV}\,{1 \over i\partial^-} |y^+ \rangle
 a^{c-}(y^+,\vec x), \\
\delta \rho_a^{(2)}(x)& = & 
 g f^{abc} [\partial^+ a^{b}_{i}(x)
  ]a^{c}_{i}(x) 
 \nonumber\\ &{-}& (g^2/N_c) \,\rho^{b}({\vec x})
  \int dy^+ a^{-c}(y^+,{\vec x}) \int dz^+ a^{-d}(z^+,{\vec x})
\nonumber\\ &{}& \nonumber\,\,\,\, \times\,
  \biggl\{\theta (z^+ -y^+)
  \theta (y^+ -x^+) {\rm Tr} \,(T^a T^c T^d T^b)
\\ &{}&\nonumber 
  \qquad+\ \theta (x^+ -z^+) \theta (z^+ -y^+) {\rm Tr} \,(T^a T^b T^c T^d)
\\ &{}&\qquad
+\ \theta (z^+ -x^+) \theta (x^+ -y^+) {\rm Tr}\,(T^a T^d T^b T^c) \biggr\}.
\label{rho2}
\ee
In the right hand sides of these equations, the terms involving
$a^i_c$ come from the three-gluon vertex in
$S_{YM}$, while the terms involving $a^-_c$
come from the two- and three-point vertices in $S_W$.
It is understood here that only the soft modes with
$k^+\simle b\Lambda^+$ are to be kept in the products of fields.

Consider first the following charge-charge 
correlator\footnote{Our notations are such that
$\hat\chi$ and $\chi$ or $\hat\sigma$ and $\sigma$ differ
just by a factor of $\alpha_s\ln(1/b)$.} which enters
eq.~(\ref{sigperp}) for $\chi$ :
\be\label{chi00}
{\hat \chi}_{ab}({\vec x},{\vec y}) \, \equiv \,
\langle\delta \rho_a(x^+,\vec x)\,
\delta \rho_b(x^+,\vec y)\rangle\,.\ee
By time homogeneity (recall that the background fields
${\cal A}^i$ are static), this equal-time 2-point function 
is actually independent of time. When evaluating $\hat\chi$ 
within our present accuracy,
it is sufficient to keep only the terms of first order in fluctuations:
$\delta \rho \sim \delta \rho^{(1)}$. These are given by 
eq.~(\ref{rho10}) which implies (in condensed notations,
where the PV prescription in $1/p^-$ and the condition 
$x^+=y^+$ are implicit):
\be\label{chi0}
\hat\chi_{ab}(\vec x,\vec y)\,=\,g^2
\left\langle \left(-2i{\cal F}^{+i}
a^i + \,\rho {1 \over i\partial^-} a^- \right)_x^a
\left(
2i a^i {\cal F}^{+i} +a^- {1 \over 
i\partial^-} \rho \right)_y^b \right\rangle.
\ee
This involves the propagator
\be\label{delAcorr}
iG^{\mu\nu}_{ab}(x,y)[{\cal A}]&\equiv&
\langle {\rm T}\,a_a^\mu(x) a_b^\nu(y)\rangle\ee
of the semi-fast gluons in the background of the tree-level fields 
${\cal A}^i$, and in the LC-gauge ($a^+=0$).
Given the specific structure of the background field (\ref{Atilde}),
it has been possible to construct this
propagator {\it exactly}, i.e., to all orders in ${\cal A}^i$
\cite{AJMV95,HW98,PI}. The resulting expression, to be
extensively used in what follows, is presented in Appendix A.

The construction of the propagator in Refs.
\cite{AJMV95,HW98,PI} has relied in an 
essential way on the separation of scales in the problem: 
Because of their low $p^+$ momenta ($p^+\ll \Lambda^+$),
the semi-fast gluons $a^\mu$ are unable to discriminate the
longitudinal structure of the source, but rather ``see'' this 
as a $\delta$-function at $x^-=0$. One is thus reduced to the
problem of the scattering off a $\delta$-type potential, 
whose solution is known exactly. But this also means that,
strictly speaking, the propagator (\ref{delAcorr}) is known 
only far away from the support of the source
(for $|x^-| \gg 1/\Lambda^+$), where ${\cal A}^i$ 
takes the approximate form in eq.~(\ref{APM}).
It is thus an important self-consistency check on the 
calculations to follow to verify that $\chi$ and $\sigma$
are indeed insensitive to the internal structure of the source.

Still for the construction of the propagator, it has been
more convenient to impose the strip restriction (\ref{strip})
on the LC-energy $p^-$, rather than on the longitudinal
momentum $p^+$ \cite{PI}. Indeed, to LLA, the quantum effects are 
due to nearly on-shell gluons, for which the constraint (\ref{strip})
is equivalent to the following constraint on $p^-$ :
\be\labe{strip-}\,
\Lambda^- \,\ll\, |p^-| \,\ll\, \Lambda^-/b\,,\ee
where $\Lambda^-\equiv Q_\perp^2/2\Lambda^+$, and 
$Q_\perp$ is some typical transverse momentum.
(Recall that we assume all the transverse momenta
to be comparable; thus, to LLA, it makes no difference 
what is the precise value of $Q_\perp$.)
But the condition (\ref{strip-}) is better adapted to the
present background field problem, 
where we have homogeneity in time (so that we can work in
the $p^-$--representation), but strong
inhomogeneity in $x^-$ (cf. eqs.~(\ref{APM}) and (\ref{UTAF})).

A final subtlety refers to
the choice of a gauge condition in the LC-gauge propagator:
Recall that, even after imposing the condition $A^+=0$, 
one has still the freedom to perform $x^-$--independent 
gauge transformations. This ambiguity shows up as an unphysical 
pole at $p^+=0$ in the gluon propagator. Since in our calculations
with strip restriction on $p^-$, $p^+$ is not restricted anylonger,
an $i\epsilon$ prescription is needed to regulate this pole. 
This is important since, as our calculations show,
the final results for $\chi$ and $\sigma$ are generally dependent
upon this $i\epsilon$ prescription.
In fact, the discrepancy between our results below and those
reported in Refs. \cite{JKW99} may be partially
attributed to using different gauge-fixing prescriptions.

For consistency with the retarded boundary conditions 
(${\cal A}^i\to 0$ for $x^- \to -\infty$)
imposed on the classical solution (\ref{Atilde}),
we shall use a retarded $i\epsilon$ prescription also
in the LC-gauge propagator of the quantum fluctuations (see Appendix A).
With this choice, the induced source $\langle\delta \rho\rangle$
appears to have support only at {\it positive} $x^-$, as shown
eq.~(\ref{stripx-}). We have also verified that, by using advanced
prescriptions (both in the classical and the quantum calculations),
the support of $\langle\delta \rho\rangle$ would be rather at {\it negative}
$x^-$ (but with $|x^-|$ still constrained as in eq.~(\ref{stripx-})).
In both cases, however, the same results are finally
obtained for $\chi$ and $\sigma$ after integrating out the 
(prescription-dependent) longitudinal structure.
Thus, the RGE comes up the same with both retarded and advanced
prescriptions. On the other hand, 
we have not been able to give a sense to calculations with 
other prescriptions, like principal value or Leibbrandt-Mandelstam. 
(For instance, when computed with a principal value prescription,
the coefficients $\sigma$ and $\chi$ appear to be sensitive to the
internal structure of the source in $x^-$, which contradicts the
assumed separation of scales.)

We conclude this introductory section by giving
explicit expressions for $\chi$ and $\sigma$ 
in terms of the background field
propagator $G^{\mu\nu}_{ab}(x,y)$ of the semi-fast gluons.
For $\chi$, we use eqs.~(\ref{sigperp}) and (\ref{chi0}) to deduce that
\be\label{chi1}
\alpha_s\ln{1\over b}\,
\chi_{ab}({\bf x}_\perp,{\bf y}_\perp)\,=\,
\int dx^- \int dy^-\,\hat\chi_{ab}(\vec x,\vec y),\ee
with (in matrix notations) 
\be\label{chi2}
\frac{1}{g^2}\,\hat \chi(\vec x, \vec y) &=&i\,
2{\cal F}^{+i}_x\, \langle x|G^{ij}|y\rangle \,2{\cal F}^{+j}_y \, +\, 
2{\cal F}^{+i}_x\,\langle x|G^{i-}\,{1 \over i\partial^-}|y\rangle \, \rho_y 
\nonumber\\&{}&\,\,\,\,-\,
\rho_x \, \langle x|{1 \over i\partial^-}\, G^{-i}|y\rangle \, 
2{\cal F}^{+i}_y\,+\,i
\rho_x \langle x|{1 \over i\partial^-} G^{--} {1 \over i\partial^-}
|y\rangle\,\rho_y.\ee
The equal-time limit is implicit here; it is achieved
by taking $y^+=x^++\epsilon$ within 
the time-ordered propagator (\ref{delAcorr})
\cite{PI}. Diagramatically, all the above contributions to $\chi$ 
are represented by {\it tree-like} Feynman graphs (no loops),
with vertices proportional to $\rho$ 
or ${\cal F}^{+i}$ (see Fig. \ref{CHIFIG}).
Since, physically, these are quantum corrections associated 
with the emission of a real (semi-fast) gluon, we shall sometimes
refer to $\chi$ or $\hat\chi$ as the {\it real correction}.
(In the weak field approximation, $\chi$ is responsible
for the real piece of the BFKL kernel \cite{JKLW97,PI};
see also Sect. \ref{sec:BFKL} below.)
\begin{figure}
\protect\epsfxsize=12.cm{\centerline{\epsfbox{CHIFIG.eps}}}
         \caption{Feynman diagrams for the four contributions
to $\chi$ given in eq.~(\ref{chi2}).}
\label{CHIFIG}
\end{figure} 



To the same accuracy, the induced source $\langle\delta \rho\rangle$
involves only the terms in $\delta\rho$ of second order
in the fluctuations :
\be\labe{JIND}\,
\hat\sigma_a(\vec x)\,\equiv\,\langle\delta \rho_a(x)\rangle\,
\,=\,\langle\delta \rho_a^{(2)}(x)\rangle,\ee
with $\delta \rho_a^{(2)}$ given by eq.~(\ref{rho2}).
This yields 
\be\label{sigma0}
\alpha_s\ln{1\over b}\,\sigma_a ({\bf x}_\perp)\,=\,
\int dx^- \, {\rm Tr } \,(T^a \hat\sigma(\vec x)),\ee
where:
\be\label{sigma}
\hat\sigma(\vec x)&\equiv&-g \partial^+_y G^{ii}(x,y)\Big |_{x=y}\,
+\,ig^2\rho({\vec x})\Bigl\langle x\bigg|
 {1 \over i \partial^-} \,G^{--}  {1 \over i\partial^-}\bigg|x 
\Bigr\rangle\nn
&\equiv&\hat\sigma_1(\vec x)\,+\,\hat\sigma_2(\vec x).
\ee
In writing $\hat\sigma_2$ as above, we have used compact but
formal notations for the second contribution
to $\delta \rho_a^{(2)}$ in eq.~(\ref{rho2}), which is non-local
in time. Diagramatically, the two terms in eq.~(\ref{sigma})
are represented by the one-loop diagrams displayed 
in Fig. \ref{SIGFIG}. These are vertex and self-energy corrections
which in the weak field limit (i.e., to linear order in $\rho$)
generate the virtual piece of the BFKL kernel \cite{JKLW97,PI}.
Accordingly, we shall refer to $\sigma$ or $\hat\sigma$ as the
{\it virtual correction}.


\begin{figure}
\protect\epsfxsize=14.cm{\centerline{\epsfbox{SIGFIG.eps}}}
         \caption{Feynman diagrams for $\hat\sigma_1$ (a) and
 $\hat\sigma_2$ (b,c,d). The wavy line with a blob denotes the
background field propagator of the semi-fast gluons;
the continuous line represents the source $\rho$; the precise
vertices can be read off eq.~(\ref{rho2}).}
\label{SIGFIG}
\end{figure} \section{Recovering some known equations}
\label{sec:LIMITS}

If $\langle O[\alpha] \,\rangle_\tau$ is any observable which
can be computed as an average over $\alpha$, as in eq.~(\ref{OBSERV}),
then it satisfies an evolution equation given by (\ref{evolO}), namely,
\be\labe{evolOBS}
{\del \over {\del \tau}}\langle O[\alpha] \,\rangle_\tau\,=\,
\left\langle {1 \over 2}\,{\delta \over {\delta
\alpha_\tau^a(x_{\perp})} }\,\eta_{xy}^{ab}\,
{\delta \over {\delta \alpha_\tau^b(y_{\perp})}}\,O[\alpha]\right
\rangle_\tau\,,\ee
where we have also used the RGE in Hamiltonian form, eq.~(\ref{RGEH}),
and we have integrated twice by parts within the
functional integral over $\alpha$. In what follows we shall apply
this equation for two choices of the operator $O[\alpha]$ : 

\noindent
--- The unintegrated gluon distribution function\footnote{This is
 independent of $k^+$ since the electric field
$ {\cal F}^{i+}(\vec x)$ is almost a $\delta$-function in $x^-$;
cf. eq.~(\ref{UTAF}).} (i.e., the integrand of eq.~(\ref{GCL})):
\be\label{varphi}
\varphi_\tau (k^2_{\perp})\,\equiv\,
k^2_{\perp}\,
\Bigl\langle\,|{\cal F}^{i+}_a(k^+,k_\perp)|^2\Bigr\rangle_\tau\,\ee
in the weak field approximation; we shall thus recover the BFKL
equation, as expected.

\noindent
--- The following two-point function of the Wilson lines
(see below for its interpretation):
\be\label{Stau}
S_\tau(x_{\perp},y_{\perp})\,\equiv\,
\Big\langle {\rm tr}\big(V^\dagger(x_{\perp}) V(y_{\perp})\big)
\Big\rangle_\tau,\ee
for which we shall recover the Balitsky--Kovchegov equation
\cite{B,K}.

\subsection{The BFKL limit}
\label{sec:BFKL}

The BFKL equation is obtained in the weak field (and source)
approximation, in which one can expand the Wilson lines in the
Hamiltonian to lowest non-trivial order. For instance, the BFKL
Hamiltonian in the present formalism is obtained by replacing,
in eqs.~(\ref{H}),
\be
1 - V^\dagger_zV_x\,\longrightarrow\,ig\Big(\alpha^a(x_\perp)
- \alpha^a(z_\perp)\Big)T^a,\qquad \alpha^a(x_\perp)
\equiv \int dx^-\, \alpha^a(x^-,x_\perp)\,.\ee
Clearly, this is formally the same as the 
perturbative expansion (i.e., the expansion in powers of $g$) 
of the Hamiltonian to lowest order. After this expansion, the
Hamiltonian takes the generic form
\be\label{HBFKL} H_{BFKL}\,\sim\,\alpha\,{i \delta \over {\delta
\alpha} }\,\alpha\,{i \delta \over {\delta\alpha} }\,,\ee
which is like the second-quantized Hamiltonian for a
non-relativistic many-body problem: In this weak field limit,
the quantum evolution is diagonal in the number of fields,
i.e., it couples only correlators $\langle
\alpha(1) \alpha(2)\cdots\alpha(n)\rangle_\tau$ with the same number $n$
of fields. The BFKL equation is the corresponding evolution equation
for the 2-point function (\ref{varphi}).

To write down this equation, we first observe that, for
such weak fields, the solution to the Yang-Mills equations
(\ref{cleq0}) can be linearized as well:
\be {\cal F}^{+i}_a(k)\,\approx \,
i(k^i/k^2_\perp)\,\rho_a(k),\ee
which implies an approximate expression for the
unintegrated gluon distribution (\ref{varphi}):
\be\label{varphiWF}
\varphi_\tau (k^2_{\perp})\,\approx\,
\Big\langle\rho_a(k_\perp)\rho_a(-k_\perp)\Big\rangle_\tau,\ee
where
\be \rho_a(k_\perp)\,=\,\int d^2x_\perp\,
{\rm e}^{-ik_\perp\cdot x_\perp}\,\rho_a(x_\perp),\qquad
\rho_a(x_\perp)\,\equiv\,\int dx^-\, \rho_a(x^-,x_\perp).\ee
We thus need the equation satisfied by the charge-charge
correlator $\langle\rho\rho\rangle_\tau$, which follows quite
generally from the RGE (\ref{RGE}) in the $\rho$-representation:
\be\labe{RGE2pLIN}
{d\over {d\tau}}\,
\Big\langle\rho_a(x_\perp)\rho_b(y_\perp)\Big\rangle_\tau\,=\,\alpha_s
\Big\langle\sigma_a(x_\perp)\rho_b(y_\perp)
\,+\,\rho_a(x_\perp)\sigma_b(y_\perp)\,+\,
\chi_{ab}(x_\perp,y_\perp)\Bigr\rangle_\tau\,.\ee
For generic, strong, fields and
sources, the r.h.s. of this equation involves $n$-point correlators 
$\langle\rho(1) \rho(2)\cdots\rho(n)\rangle_\tau$ of arbitrarily high
order $n$. But in the weak field limit, where $\sigma$ is linear in
$\rho$ and $\chi$ is quadratic, this becomes a closed equation for
the 2-point function, in agreement with the general argument after
eq.~(\ref{HBFKL}). The coefficients $\chi$ and $\sigma$ in this
limit will be now obtained by expanding the general formulae
(\ref{CHIFINALDEVT}) and (\ref{sigx}) to lowest order in $\rho$.

Consider $\chi^{(0)}$ first. To the order of interest in $\rho$, one
can replace, in eq.~(\ref{CHIFINALDEVT}),
\be
{\cal A}^i_{\infty}(x_{\perp})\,\approx\,-\partial^i\alpha(x_{\perp})
\,\approx\,(\partial^i/\grad_\perp^2)\rho(x_{\perp}),\qquad
%\longrightarrow\,i(k^i/k^2_\perp)\,\rho_a(k_{\perp}),
{\cal D}^i_{\infty}\,\approx\,\partial^i,\ee
which after simple algebra yields (in matrix notations:
$(\rho_x\rho_y)^{ab}=\rho_x^{ac}\rho_y^{cb}$, etc.) :
\be\label{CHIBFKL}
\chi^{(0)}_{ab}(x_\perp,y_\perp)&=&4
\int {d^2p_\perp \over (2 \pi)^2}\,
\frac{{\rm e}^{ip_{\perp}\cdot(x_{\perp}-y_{\perp})}}
{p_{\perp}^2}\nonumber\\
&{}&\,\,\,\,\times\left\{\rho_x\rho_y-i\rho_x(p^i\partial^i\alpha)_y
+i(p^i\partial^i\alpha)_x\rho_y+ p^2_\perp (\partial^i\alpha)_x
(\partial^i\alpha)_y\right\}^{ab},
\ee
or, after a Fourier transform,
\be\label{chiBFKL}
\chi^{(0)}_{aa}(k_\perp,-k_\perp)\,=\,4N_ck_\perp^2
\int \frac{d^2 p_\perp}{(2\pi)^2}\,\frac{\rho_a(p_\perp)\rho_a(-p_\perp)}
{p^2_{\perp} (k_{\perp}-p_{\perp})^2}\,.\ee
Consider similarly $\sigma^{(0)}$: to linear order in $\rho$,
this can be extracted from any of the expressions (\ref{sigx}),
(\ref{tsigx}), or (\ref{tsigma}), so let us choose the latter
expression, for convenience. To this aim, it is enough to replace,
in eq.~(\ref{tsigma1}),
\be
{\rm Tr}\Bigl(T^a V^\dagger_x V_z\Bigr)\,\approx\,igN_c[\alpha^a(x_\perp)
-\alpha^a(z_\perp)],\ee
as appropriate for weak fields, and thus get, after simple algebra,
\be\label{sigmaq}
\sigma^{(0)}_a (k_\perp)
\,=\,-N_c \,\rho_a(k_\perp)
\int \frac{d^2 p_\perp}{(2\pi)^2}\,{k^2_{\perp} \over p_{\perp}^2 
(p_{\perp}- k_{\perp})^2}\,.\ee

By inserting eqs.~(\ref{chiBFKL}) and (\ref{sigmaq}) into 
the evolution equation (\ref{RGE2pLIN}), and using (\ref{varphiWF}),
one finally obtains:
\begin{eqnarray}
 {\partial \varphi_\tau (k^2_{\perp}) \over \partial \tau} & = & \,\,\,
{\alpha_s N_c \over \pi^2}\,
\int d^2 p_{\perp}
 {k^2_{\perp} \over p^2_{\perp} (k_{\perp}-p_{\perp})^2}\,
 \varphi_\tau(p^2_{\perp}) \nonumber \\
& & -\,
{\alpha_s N_c \over2 \pi^2}\,
\int d^2 p_{\perp}
 {k^2_{\perp} \over p^2_{\perp} (k_{\perp}-p_{\perp})^2}\,
 \varphi_\tau(k^2_{\perp})\,,
\label{BFKL}
\end{eqnarray}
which coincides, as anticipated, with the BFKL equation \cite{BFKL}.
The first term in the r.h.s., which here is generated by 
$\chi^{(0)}$, is the {\it real} BFKL kernel, 
while the second term, coming from $\sigma^{(0)}$,
is the corresponding {\it virtual} kernel.

\subsection{The Balitsky--Kovchegov equation}
\label{sec:BK}

Note first that, although written in a specific gauge --- namely,
the covariant gauge where the color field of the hadron reads
$A^\mu_a=\delta^{\mu+}\alpha_a(\vec x)$ ---, the quantity (\ref{Stau})
has a gauge invariant meaning, as the COV-gauge expression of the
gauge-invariant operator
\be\label{StauP}
S_\tau(x_{\perp},y_{\perp})\,\equiv\,
\Big\langle {\rm tr} \,L (x_{\perp},y_{\perp})\Big\rangle_\tau,
\ee
where $L (x_{\perp},y_{\perp})$ is the Wilson loop obtained by
closing the two Wilson lines along $x^-$ (at $x_{\perp}$ and
$y_{\perp}$, respectively) with arbitrary paths joining
$x_{\perp}$ and $y_{\perp}$ in the transverse planes at $x^-=\infty$
and $x^-=-\infty$.

Physically, $S_\tau(x_{\perp},y_{\perp})$
is the S-matrix element for the scattering of a ``color
dipole'' off the hadron in the eikonal approximation. To clarify this, 
consider DIS in a special frame (the "dipole frame") where most of the
energy is put in the hadron which moves along the positive $z$ direction
---  in this respect, this is like the infinite momentum frame, 
so one can use the CGC effective theory for the hadron wavefunction ---,
but the virtual photon is itself energetic enough for the scattering to
proceed as follows: 
The photon first fluctuates into an energetic 
quark--antiquark pair (a ``color dipole") which then propagates in the
negative $z$ (or positive $x^-$) direction, i.e., towards the hadron,
against which it scatters eventually, with a cross section
(see, e.g., \cite{K,W,Levin} for more details)
\be\label{sigmadipole}
\sigma_{dipole}(\tau,r_\perp)\,=\,2\int d^2b_\perp\,\frac{1}{N_c}
\Big\langle {\rm tr}\Big(1- V^\dagger(x_{\perp}) V(y_{\perp})\Big)
\Big\rangle_\tau.\ee
Here $x_{\perp}$ and $y_{\perp}$ are the transverse coordinates
of the quark and antiquark in the pair,
$r_\perp=x_{\perp}-y_{\perp}$ is the size of the dipole, and
$b_\perp=(x_{\perp}+y_{\perp})/2$ is the impact parameter.
$V^\dagger_x$ and $V_y$ are Wilson lines in the $x^-$ direction,
defined as in eq.~(\ref{v}) but in the fundamental representation.
They describe the eikonal coupling of 
the quark and antiquark to the color field $A^+_a=\alpha_a$ of the hadron. 
The average over the hadronic target in eq.~(\ref{sigmadipole}) 
is here understood as
an average over $\alpha$ in the sense of eq.~(\ref{OBSERV}).
When the scattering energy increases (i.e., $\tau$ increases),
the weight function $W_\tau[\alpha]$ for this average changes due to quantum corrections, and so does the scattering cross section. 
This change is governed by eq.~(\ref{evolOBS}) with 
$O[\alpha]=S_\tau(x_{\perp},y_{\perp})$, that we shall construct
now explicitly.

The final outcome of this calculation, to be detailed in Appendix D,
is an evolution equation for $S_\tau(x_{\perp},y_{\perp})$
that we write down here for a color dipole in some arbitrary 
representation $R$, since this is not more difficult:
\be\labe{evolVR}
{\del \over {\del \tau}}\Big\langle {\rm tr}_R (V^\dagger_x V_y)
\Big\rangle_\tau=-{\alpha_s\over \pi^2}\int d^2z_\perp
\frac{(x_\perp-y_\perp)^2}{(x_\perp-z_\perp)^2(y_\perp-z_\perp)^2 }
\left\langle C_R{\rm tr}_R (V^\dagger_x V_y)-
%\tilde V^{\dagger\,ab}_z
%\,{\rm tr}_R (t^aV^\dagger_xt^b V_y)
{\rm tr}_R (V^\dagger_zt^a V_z V^\dagger_xt^aV_y)
\right\rangle_\tau.\nn\ee
In this equation, all the Wilson lines 
$V^\dagger$, $V$, and the color group generators $t^a$, $t^b$
belong to the generic representation $R$, and $C_R=t^a t^a$.

In particular, for the fundamental representation, the
following Fierz identity:
\be\label{Fierz}
t_a^{ij}\,t_a^{kl}\,=\,{1 \over 2}\,\delta^{il}\delta^{jk}\,-\,
{1 \over 2N_c}\, \delta^{ij}\delta^{kl}\,\ee
allows one to simplify the color trace in the last term in 
eq.~(\ref{evolVR}). By also using $C_R=(N^2_c-1)/2N_c$, one
finally obtains (with $R=F$ kept implicit)
\be\labe{evolV}
{\del \over {\del \tau}}\langle {\rm tr}(V^\dagger_x V_y)
\rangle_\tau=-{\alpha_s\over 2 \pi^2}\int d^2z_\perp
\frac{(x_\perp-y_\perp)^2}{(x_\perp-z_\perp)^2(y_\perp-z_\perp)^2 }
\left\langle N_c {\rm tr}(V^\dagger_x V_y)
- {\rm tr}(V^\dagger_x V_z){\rm tr}(V^\dagger_z V_y)\right\rangle_\tau,\ee
which coincides with the equation derived by Balitsky
within a rather different formalism \cite{B} : via the operator
expansion of high-energy scattering in the target rest frame.

The first observation is that the above equations are
not closed. They relate a 2-Wilson-line correlation 
function to a 4-line function, for which one can derive an
evolution equation too \cite{B}, but one may already
guess that this would be not the end of the story: 
The 4-line function will be in turn coupled, via its evolution 
equation, to a 6-line function, and so on, so that one is really
dealing with an infinite hierarchy of coupled equations 
of which eq.~(\ref{evolV}) is just the first equation.

But a closed equation can still be obtained in
the large $N_c$ limit, since in this limit the 4-line correlation function
in eq.~(\ref{evolV}) factorizes:
\be
\left\langle {\rm tr}(V^\dagger_x V_z)\,
{\rm tr}(V^\dagger_z V_y)\right\rangle_\tau
\longrightarrow 
\left\langle {\rm tr}(V^\dagger_x V_z)\right\rangle_\tau\,
\left\langle{\rm tr}
(V^\dagger_z V_y)\right\rangle_\tau\quad {\rm for}\,\,N_c\to\infty,
\ee
and eq.~(\ref{evolV}) reduces to a closed 
equation\footnote{A different closed equation has been recently
obtained in Ref. \cite{KW01} by evaluating the 4-line correlator
in eq.~(\ref{evolV}) with a Gaussian weight function for finite
$N_c$. Note however that the general solution to the RGE
(\ref{RGEH}) is {\it not} a Gaussian \cite{BIW} and, moreover,
eq.~(\ref{evolV}) is not even consistent with a Gaussian, or
``mean field'', approximation to the general RGE \cite{SAT,SATL}.}
for the quantity ${\cal N}(x_\perp,y_\perp)\equiv 
\langle{\rm tr}(1-V^\dagger_x V_y)\rangle_\tau$ ($=$
the forward scattering amplitude of the color dipole 
with the hadron) : 
\be\labe{evolN}
{\del \over {\del \tau}} {\cal N}_{xy}
={\alpha_s\over 2 \pi^2}\int d^2z_\perp
\frac{(x_\perp-y_\perp)^2}{(x_\perp-z_\perp)^2(y_\perp-z_\perp)^2 } 
\left\{ N_c\Bigl({\cal N}_{xz} + {\cal N}_{zy} - {\cal N}_{xy}\Bigr)
- {\cal N}_{xz}{\cal N}_{zy}\right\}.\,\,\,\,\,\ee
This coincides with the evolution equation obtained by Kovchegov \cite{K}
within still a different formalism\footnote{A similar non-linear equation
describing the multiplication of pomerons was suggested in Ref. \cite{GLR}
and proved in \cite{MQ} in the double-log approximation. More recently,
Braun has reobtained this 
equation by resumming ``fan'' diagrams \cite{B00}.}, 
namely the dipole model by  Mueller \cite{AM3}.

The second observation is that the Wilson lines in the adjoint 
representation which were originally present in the Hamiltonian
(\ref{H}) have melted together with the Wilson lines in the generic
representation of the external dipole to generate the color traces
in the r.h.s. of eq.~(\ref{evolVR}). This melting has been made possible
by relations like:
\be\label{melt}
\tilde V^{\dagger}_{ab}t^b\,=\,Vt^aV^{\dagger},\ee
which allows one to trade a group element $\tilde V^{\dagger}$ in
the adjoint representation (that we denote here with a tilde, for more
clarity) for a pair of group elements $V$ and $V^{\dagger}$ in the
generic representation with generators $t^a$, $t^b$.
Note also the simple correspondence between the two terms
within the braces in the r.h.s. of eq.~(\ref{evolVR}) and the four terms
in eq.~(\ref{eta}) for $\eta$ :
The first two terms, $1$ and $\tilde V^{\dagger}_x \tilde V_y$, in eq.~(\ref{eta}) have contributed equally to the first term, involving
$\langle{\rm tr}_R (V^\dagger_x V_y)\rangle$, in eq.~(\ref{evolVR}),
while the other two terms, $\tilde V^{\dagger}_x \tilde V_z$ and
$\tilde V^{\dagger}_z \tilde V_y$, have given identical contribution to 
the remaining piece, involving the 4-line function, in eq.~(\ref{evolVR}).

The third important observation refers to the transverse kernel
in eq.~(\ref{evolVR}), or (\ref{evolV}). A brief comparison shows that
this is not the same as the original kernel
${\cal K}(x_\perp,y_\perp,z_\perp)$, eq.~(\ref{Kxyz}), of the RGE.
Rather, this has been generated as (see the explicit calculation
in Appendix C)
\be\label{decay}
{\cal K}(x_\perp,x_\perp,z_\perp)+
{\cal K}(y_\perp,y_\perp,z_\perp)- 2
{\cal K}(x_\perp,y_\perp,z_\perp)\,=\,
\frac{(x_\perp-y_\perp)^2}{(x_\perp-z_\perp)^2(y_\perp-z_\perp)^2 }\,.\ee
The final result, also known as ``the dipole kernel'' (since
it is proportional to the probability for the decay of a dipole of size
$x_\perp-y_\perp$ into two dipoles of sizes $x_\perp-z_\perp$
and $y_\perp-z_\perp$, respectively), has the remarkable feature to 
show a better IR behaviour than the original kernel (\ref{Kxyz}).
When $z_\perp \gg x_\perp,\,y_\perp$, the r.h.s. of eq.~(\ref{decay})
decreases like $(x_\perp-y_\perp)^2/z_\perp^4$, so its integral
$\int d^2z_\perp$ is actually finite. This guarantees, in particular,
that the evolution equation (\ref{evolN}) in the large $N_c$ limit
is free of IR problems, and it strongly suggest a similar property for
the exact equation (\ref{evolV}), although a rigorous proof
in this latter case would still require a knowledge of 
the large--$z_\perp$ behaviour of the 4-line function
$\langle
{\rm tr}(V^\dagger_x V_z){\rm tr}(V^\dagger_z V_y)\rangle$.
(All of the kernels occuring in the evolution equation for this 4-line 
correlation function are of the dipole type too \cite{B}.)

The dipole kernel (\ref{decay}) has ultraviolet poles 
at $z_\perp=x_\perp$ and $z_\perp=y_\perp$, but these are inocuous,
since the accompanying Wilson-line operators
in eqs.~(\ref{evolVR}) or (\ref{evolV}) cancel each other
at these points.



