text:
\section{Introduction}
\label{intro}

The scattering of hadrons at high energy and small
momentum transfer is very interesting.
On the one hand, hadronic scattering in this Regge limit
has been very successfully described \cite{softpom}
by Regge theory \cite{Collins},
based on analyticity and unitarity of the $S$-matrix.
The behaviour of hadronic scattering amplitudes is
encoded in the positions of so--called Regge poles and
cuts in the plane of complex angular momentum.
With Gribov's reggeon
calculus \cite{Gribov} a consistent field theory of interacting
reggeons was found and extensively studied \cite{BakerTerMar}.
To the present day, Regge theory remains one of the deep truths of
particle physics.
The positions of the Regge singularities, however, cannot be calculated
in the framework of Regge theory.

On the other hand, QCD has been firmly established as the
quantum field theory describing the physics of strong
interactions. It should therefore -- at least
in principle -- be possible to derive Regge theory from QCD
and to understand it in terms of quark and gluon
degrees of freedom. This important and fundamental problem
has not yet been solved.
The main difficulty lies in the fact that the Regge limit
is characterized by high parton densities. In addition,
many hadronic scattering processes in the Regge kinematics
are dominated by small momentum scales. Therefore
non--perturbative effects are expected to become important.

Fortunately, there are a few scattering processes that can
be treated perturbatively even in the Regge limit. These processes
involve the scattering of small color dipoles. An example
is the scattering of highly virtual photons,
in which the virtuality of the photons provides a hard scale.
This process is of phenomenological interest \cite{gamgam,Brodsky},
as it can be measured at LEP \cite{L3} and, even better,
at a future Linear Collider. For our present investigation we will
look at it from a more theoretical perspective.
The basic idea is to approach the difficult dynamics of QCD in
the Regge limit from a perturbative starting point, namely highly
virtual $\gamma^* \gamma^*$ scattering. Our hope is that the
results obtained in the study of this special process are of more
general relevance to the general dynamics of QCD in the Regge limit.
In a later step, one should then of course try to understand the effects
of non--perturbative dynamics on the emerging picture.
One such non--perturbative effect would be the diffusion of
momenta into the infra--red \cite{HansZigarre,Muellergeneral},
which can be controlled to a certain extent
in $\gamma^* \gamma^*$ scattering
by demanding sufficiently large photon virtualities.

The first step towards a QCD-based description of hadronic
scattering in the high energy limit was done when the
BFKL (or perturbative) Pomeron was derived \cite{FKL,BL}.
It describes a $t$-channel exchange carrying vacuum
quantum numbers and resums the leading logarithms
of the squared energy $s$ which compensate the smallness of the
strong coupling constant. The BFKL Pomeron can be understood
as the $t$-channel exchange of two (reggeized) gluons. It is
expected to apply to processes which are governed by a single
hard momentum scale.

The BFKL Pomeron results in a power--like growth
$\sigma \sim s^{0.5}$
of the total cross section at high energy. This eventually
leads to a violation of unitarity, since according to the
Froissart bound \cite{Froissartcite,Martin}
the growth of total cross--sections
can at most be logarithmic with the energy.
Even though the actual violation of that bound would
occur only well above the Planck scale it makes the
leading logarithmic approximation inconsistent as far
as our main goal is concerned. Namely, we cannot
expect to derive a consistent picture of the
Regge limit from an approximation
that does not respect unitarity.

Unitarity can be restored by including exchanges with more than
two gluons in the $t$-channel.
A set of so--called unitarity
corrections can be identified that results in a unitary amplitude.
This leads to the generalized leading logarithmic approximation
\cite{Bartelsnuclphys,Bartelskernels,Bartelsinteq}.
In order to satisfy unitarity
also in all sub--channels, the number of gluons in the $t$-channel
should not be fixed, and arbitrary numbers of gluons should be
taken into account. The natural objects to consider are therefore
amplitudes describing the production of $n$ gluons in the $t$-channel.
These $n$-gluon amplitudes obey a tower of coupled integral
equations, the first of which ($n=2$) coincides with the BFKL equation.
In the Regge limit the dynamics of the interaction is effectively
reduced to the two transverse dimension. Consequently, the integral
equations are equations in two--dimensional
tranverse momentum space. The complex angular momentum
acquires the meaning of an energy--like variable, whereas its
conjugate variable, i.\,e.\ rapidity, can be interpreted as a
time--like variable. This is in agreement with the fact that
real gluon emissions in the (generalized) leading logarithmic
approximation are strongly ordered in rapidity.

So far, the amplitudes have been investigated for up to $n=4$ gluons.
Although a complete analytic solution for the four--gluon amplitude
is still missing, a series of remarkable properties
of the amplitudes has been found.
The three--gluon amplitude and
a part of the four--gluon amplitude can be written as superpositions of
two--gluon amplitudes. This is called the reggeization of that parts,
generalizing the well--known reggeization of the gluon
in non--abelian gauge theories \cite{Lipatovreggeierung}.
A consequence of this phenomenon is that the amplitudes exhibit
a very interesting field theory structure. They consist of only very
few building blocks: states of two and four interacting $t$-channel
gluons and a vertex coupling the two-- to the four--gluon state
\cite{BPLB,BW}.
The latter is a number--changing element and turns the
quantum--mechanical problem of the $n$-gluon states
\cite{Bartelskernels,BKP} into a field
theory of unitarity corrections. A further striking feature is
observed after a Fourier transformation to two--dimensional
impact parameter space. The two--gluon state \cite{Lipatov86},
the four--gluon states as well
as the two--to--four vertex are invariant under conformal
transformations of the gluon coordinates \cite{BLW}.
These properties indicate that the whole set of unitarity corrections
in the generalized leading logarithmic approximation
can be described by an effective conformal field theory in
two--dimensional impact parameter space with
rapidity as an additional parameter. It would be a great step
forward to identify this conformal field theory.
That would allow one to apply the extremely powerful
methods of conformally invariant field theory in two dimensions
\cite{BPZconf} (for a review see \cite{Ginsparg}).

These interesting properties of the unitarity corrections have
until now only been observed in the amplitudes with up to four gluons,
and only the most basic elements of the potential field theory have
been calculated explicitly.
More generally, the existence of such a field theory structure
has not yet been proved beyond four gluons.
To find further evidence for the conjectured field theory structure
and to extract its general properties requires the investigation
of higher $n$-gluon amplitudes.
A series of questions arise naturally in this program.
An important goal is of course to find out about the existence of
new elements of the effective field theory, like a possible
two--to--six vertex, and to determine
their properties. A further natural question is
whether the two--to--four vertex can be generalized to the
case in which the two incoming gluons are not in a color singlet
state, and how this generalization looks like. Closely related is
the question of how a repeated two--to--four transition
takes place. The study of the
six--gluon amplitude will provide an answer to the
question whether a Pomeron can split into two Odderons,
the Odderon being the $C=-1$ partner of the Pomeron.
An important issue will also be to test the conformal invariance
of the higher $n$-gluon amplitudes in impact parameter space.

In the present paper we study the amplitudes with five
and six gluons in the $t$-channel.
We show that the five--gluon amplitude reggeizes completely, i.\,e.\
is a superposition of two-- and four--gluon amplitudes.
The corresponding mechanism appears to apply to each $n$-gluon
amplitude with odd $n$. We extract a reggeizing part from the
six--gluon amplitude and derive an integral equation for the
remaining part. This equation contains a term which gives rise to
at least one new element in the effective field theory. The other terms
in the equation can be expressed through the known two--to--four
vertex, indicating that further reggeization takes place in the
six--gluon amplitude. We discuss our findings in the light of
the potential effective field theory.
The six--gluon amplitude allows us for the
first time to determine the perturbative Pomeron--Odderon--Odderon
vertex.
Its precise form and properties, as well as those of the generalization
of the two--to--four vertex to the color non--singlet will be
discussed in a separate publication.
Also the conformal invariance of the five-- and
six--gluon amplitudes will be investigated in a separate publication.

The problem of unitarity in high energy QCD has been addressed
in many different ways, among them the following.
The first attempt was made in ref.\ \cite{GLR},
but in that approach
a set of diagrams has been left out which is included in our
approach. These diagrams turn out to be essential for
the emergence of higher $n$-gluon states.
Considerable progress has been made in investigating these
$n$-gluon states in the $t$-channel.
Much interest was attracted by the symmetric three--gluon state
(the Odderon) \cite{Lipatovodderon,Odderonpapers}.
More generally, the large $N_c$ limit of the $n$-gluon states
has been shown to
be equivalent to a completely integrable model \cite{LipatovXXX},
namely the XXX Heisenberg model with non--compact
$\mathrm{SL}(2,\C)$ spin zero \cite{GregoryFaddeev}. This
remarkable result was subsequently used to obtain further interesting
results \cite{Gregory}, including the Odderon as a special case.
These results are relevant for our present approach since the
$n$-gluon states -- although with finite $N_c$ --
are elements of the potential effective field theory.
The two--to--four transition vertex and its relation to the
triple--Pomeron vertex has been studied in \cite{BraunVacca,Vacca}.
The latter can be obtained from the two--to--four vertex
after projection on BFKL Pomerons. It has been
shown to have the structure of a conformal
three--point function in \cite{Hans}. That reference also
discusses the analytic structure of the four--gluon state
for finite $N_c$.
An alternative approach to QCD at high energy is Mueller's
dipole picture \cite{dipole}, which has been proved to be equivalent
to the BFKL formalism \cite{WallonNavelet,Peschequiv}.
The dipole picture has been used to study unitarity effects as well
\cite{Muellerunitarity,Salam,Peschunitarity}.
The value of the triple--Pomeron coupling obtained in the dipole
picture \cite{Peschanski3p}
agrees with the one calculated from the two--to--four
vertex arising in our present approach \cite{Gregory3p}.
Other related approaches are the formulation of an effective
action for the Regge limit \cite{effaction,Lipatovphysrep},
a similar approach also aiming at a simplified effective
theory for high energy scattering \cite{Verlinde}, and
the method of operator expansion \cite{Balitsky}.
Renormalization group methods have been used to study
unitarization in \cite{Weigert}. In \cite{Whitepreprint}
a supercritical phase of reggeon field theory is studied
in which the Pomeron consists of a single gluon and a
'wee parton'.
Approaches of more non--perturbative nature are
for example the eikonal
approximation in a soft gluon background \cite{Nachtmann},
the model of the stochastic vacuum \cite{Dosch}, and the
semiclassical approach \cite{Buchmueller}.
Also the recently conjectured AdS/CFT correspondence
\cite{Maldacena} has been applied to the Regge limit
\cite{RhoAdS,PeschAdS}.
This short overview is of course not complete.
All the different approaches should be regarded as complementary.
As far as this has been tested, different approaches often lead to
equivalent or similar results. The most difficult problem will
eventually be the combination of the perturbative and non--perturbative
approaches.

The paper is organized as follows. In section \ref{BFKLchap}
we recall important facts about the BFKL equation and
briefly describe the violation of unitarity and the reggeization
of the gluon. In section \ref{ngluon} we motivate the use of
$n$-gluon amplitudes and outline their formal definition.
The integral equations for the amplitudes are described in
detail, including color algebra and the inhomogeneous terms
consisting of quark loops with $n$ gluons attached. We introduce
reggeon momentum diagrams and use them to classify the
occurring momentum space integrals in terms of a small set
of standard integrals. Section \ref{34V} contains a review
of the known results about the three-- and four--gluon
amplitudes as well as a discussion of the field theory structure
found in these amplitudes. In section \ref{5gluons} we
solve the equation for the five--gluon amplitude and discuss
the general mechanism that is expected to cause complete
reggeization in all amplitudes with an odd number of gluons.
Section \ref{d6} is devoted to the study of the six--gluon amplitude.
We extract a reggeizing part and derive the equation for
the remaining part which is then discussed in detail. From that
equation we also derive the Pomeron--Odderon--Odderon vertex.
We conclude with a summary and an outlook.
The two appendices contain algorithms for
computing contractions of color tensors (appendix \ref{colorapp})
and for bringing the momentum space integrals to their
standard forms (appendix \ref{momentumapp}).

\section{The BFKL equation and violation of unitarity}
\label{BFKLchap}

In this section we give a very brief account of the
basic properties of the BFKL equation.
For more detailed accounts of the BFKL theory we
refer the reader to the reviews
\cite{Lipatovphysrep,Jeff,LipatovinMueller}.

\subsection{The BFKL equation}
\label{BFKLsection}

The total hadronic cross section can be related to the
elastic forward scattering amplitude via the
optical theorem,
\begin{equation}
  \sigma_{\mbox{\scriptsize tot}} =
  \frac{1}{s}\, \mbox{Im}\, A_{\mbox{\scriptsize el}}(s,t=0)
\,,
\label{opttheorem}
\end{equation}
where $t$ is the momentum transfer.
It is convenient to use partial wave amplitudes, which
in the high energy limit amounts to performing
a Mellin transformation
\begin{equation}
  A(s,t) = i s \int_{\delta-i \infty}^{\delta+i \infty}
           \frac{d\omega}{2 \pi i} \left( \frac{s}{M^2} \right)^\omega
           A(\omega,t)
\,,
\label{Mellin}
\end{equation}
thereby changing from squared energy $s$ to complex angular
momentum $\omega$.
The high energy behavior of the total cross section is then
determined by the singularities of $A(\omega,t)$ in
the $\omega$-plane, i.\,e.\ Regge poles and
Regge cuts. The rightmost singularity gives the leading
contribution and is identified with the Pomeron.
As it describes an elastic scattering process
(see (\ref{opttheorem})) it carries vacuum quantum numbers.

If there is a hard momentum scale in the process the use
of perturbation theory is justified. The smallness of the
strong coupling constant $\alpha_s$ at large momentum
scales can at high energy be compensated by large
logarithms of the energy.
The resummation of Feynman diagrams contributing
to the leading logarithmic approximation (LLA)
--- $\alpha_s \ll 1$ and $\alpha_s \log(s) \sim 1$ ---
results in the BFKL equation \cite{FKL,BL}.
The longitudinal degrees of freedom decouple in
high energy scattering, and the dynamics takes place
in transverse space only.
The full amplitude can be written in factorized form
\begin{equation}
A(\omega,t) = \int \frac{d^2{\bf k}}{(2\pi)^3}
\frac{d^2{\bf k}'}{(2\pi)^3}
\phi_\omega({\bf k},{\bf k}';{\bf q}) \phi_1({\bf k},{\bf q})
\phi_2({\bf k}',{\bf q})
\,,
\end{equation}
$\phi_1$, $\phi_2$ being the impact factors
of the scattered colorless states. The color neutrality
implies
\begin{equation}
\phi_{1,2}({\bf k}=0,{\bf q}) = \phi_{1,2}({\bf k}={\bf q},{\bf q}) = 0
\,,
\end{equation}
which is important for the infrared finiteness of the amplitude.
The function $\phi_\omega$ can be interpreted as the
partial wave amplitude for the scattering of virtual
gluons with virtualities $-{\bf k}^2$, $-({\bf q}-{\bf k})^2$,
$-{\bf k}'^2$, and $-({\bf q}-{\bf k}')^2$ respectively.
It is described by the BFKL equation, which is an integral equation in
the two--dimensional space of transverse momenta
and of Bethe--Salpeter type. In detail it has the form
\begin{equation}
  \omega \phi_\omega({\bf k},{\bf k}';{\bf q}) =
 \phi^0({\bf k},{\bf k}';{\bf q})
 + \int \frac{d^2{\bf l}}{(2\pi)^3} \,
 \frac{1}{{\bf l}^2 ({\bf q}-{\bf l})^2}
 K_{\mbox{\scriptsize BFKL}}({\bf l},{\bf q}-{\bf l};{\bf k},{\bf q}-{\bf k})
\,\phi_\omega({\bf l},{\bf k}';{\bf q})
\,.
\label{BFKLeq}
\end{equation}
$\phi^0$ is an inhomogeneous term,
depending on the process under consideration.
The integral kernel, the so-called
BFKL or Lipatov kernel, is given by
\begin{eqnarray}
K_{\mbox{\scriptsize BFKL}}({\bf l},{\bf q}-{\bf l};{\bf k},{\bf q}-{\bf k}) &=&
 -  N_c g^2 \left[ {\bf q}^2 - \frac{{\bf k}^2({\bf q}-{\bf l})^2}{({\bf k}-{\bf l})^2}
 - \frac{({\bf q}-{\bf k})^2 {\bf l}^2}{({\bf k}-{\bf l})^2} \right] \nonumber \\
 & & +(2\pi)^3 {\bf k}^2 ({\bf q}-{\bf k})^2
 \left[ \,\beta({\bf k}) + \beta({\bf q} - {\bf k}) \right]
 \delta^{(2)}({\bf k}-{\bf l}).
\label{Lipatovkernel}
\end{eqnarray}
The strong coupling constant is normalized to
$\alpha_s= \frac{g^2}{4\pi}$ and is kept fixed in LLA.
The function $\beta$ in
the kernel is defined as
\begin{equation}
  \beta({\bf k}^2) = \frac{N_c}{2} g^2  \int \frac{d^2{\bf l}}{(2 \pi)^3}
          \frac{{\bf k}^2}{{\bf l}^2 ({\bf l} -{\bf k})^2}
\,.
\label{traject}
\end{equation}
The function $\alpha({\bf k}^2) = 1 + \beta({\bf k}^2)$
is known as the gluon trajectory function. It passes through
the physical
spin $1$ of the gluon at vanishing argument ${\bf k}^2=0$.
Since it is the function $\beta$ that will frequently occur
throughout this paper we call it (in obvious abuse of language)
the trajectory function of the gluon as well.

The factor $(-N_c)$ in the BFKL kernel is a color factor. If the
two gluons entering the amplitude $\phi_\omega$ are not in a
color singlet state the color factor $C_I$ will be
different.
(In that case the amplitude is not infrared finite and has to be
regularized.)
In general, $C_I$ depends on the
irreducible representation $I$ of the two gluons. If
$N_c=3$, the factor $C_I$ equals $-3$, $-\frac{3}{2}$,
$-\frac{3}{2}$, $0$, $1$
for the irreducible representations ${\bf 1}$, ${\bf 8_A}$, ${\bf 8_S}$,
${\bf 10}+{\bf \overline{10}}$, ${\bf 27}$, respectively.
We will use the symbol $K_{\mbox{\scriptsize BFKL}}$
for the BFKL kernel only if the two gluons are in a color singlet
state.

The complex angular momentum $\omega$ acts as an energy
variable in the BFKL equation. It can be shown to be conjugate
to rapidity which thus acquires the meaning of a time variable
in the BFKL equation.

The general form of the solution of the BFKL equation
can be derived from the integral equation by iteration.
Accordingly, the elastic scattering amplitude at high energies
has in LLA the structure of a gluon ladder in the $t$-channel,
\begin{equation}
 s \rightarrow \infty \quad \picbox{limit2to2.pstex_t}
= \sum_{\mbox{\tiny number} \atop \mbox{\tiny of rungs}}
 \,\,\picbox{ladder.pstex_t}
\,,
\label{ladder}
\end{equation}
and the ladder rungs represent BFKL kernels.
The BFKL equation can be solved analytically, but
we will not make use of the explicit form of the solution.
In the present paper we will also not need
the conformal invariance \cite{Lipatov86}
of the BFKL equation in impact parameter space.

Finally, we mention the $t$-channel
reggeon unitarity relation for the BFKL amplitude.
Let $C(\omega;{\bf k},{\bf k}';{\bf q})$ be the amputated
BFKL amplitude, i.\,e.\ the amplitude
$\phi_\omega$ without the reggeon propagator
$(\omega-\beta({\bf k}^2) - \beta(({\bf q}-{\bf k})^2))^{-1}$,
\begin{equation}
 C(\omega;{\bf k},{\bf k}';{\bf q})=
\left(
\omega-\beta({\bf k}^2) - \beta(({\bf q}-{\bf k})^2)
\right)
\phi_\omega({\bf k},{\bf k}';{\bf q})
\end{equation}
with $t=-{\bf q}^2$. After a continuation to the physical
region of the $t$-channel it is possible to show \cite{FKL}
with the help of the BFKL equation that
\begin{equation}
\mbox{disc}_\omega C =
\int d^2{\bf l} \,(2\pi)^3
\frac{1}{{\bf l}^2 ({\bf q}-{\bf l})^2}
\delta \left(\omega-\beta({\bf l}^2) - \beta(({\bf q}-{\bf l})^2)\right)
 C(\omega;{\bf k},{\bf l};{\bf q}) C^*(\omega;{\bf l},{\bf k}';{\bf q})
\,.
\label{tunitaryBFKL}
\end{equation}
The right hand side can be understood as a unitarity
integral for the two--gluon amplitude $C$.

\subsection{Violation of unitarity}
\label{violation}

The leading singularity in the $\omega$-plane
can be determined analytically from the BFKL
equation, and leads a
power--like growth of the amplitude
$A \sim s^{(1+\omega_{\mbox{\tiny BFKL}})}$
with the exponent
$\omega_{{\mbox{\scriptsize BFKL}}}
= \alpha_s N_c \,4 \ln 2/\pi \simeq 0.5 $.
Consequently, the total cross section in the leading logarithmic
approximation grows like
$\sigma_{\mbox{\scriptsize tot}}\sim s^{\omega_{\mbox {\tiny BFKL}}}$.
This result is in conflict with the
Froissart--Martin theorem \cite{Froissartcite,Martin}
which derives a bound on the total hadronic
cross section from unitarity.
In detail, the Froissart--Martin bound is
$\sigma_{\mbox{\scriptsize tot}} \le \mbox{const.} \,\log^2(s)$.
A power--like growth will eventually violate
this bound --- and thus unitarity ---
at asymptotically large energies.
This observation is the starting point for the considerations
in this paper.

\subsection{Reggeization of the gluon}
\label{reggeization}

The phenomenon of reggeization
in non--abelian gauge theories \cite{Lipatovreggeierung} is the following.
The $t$-channel exchange in the BFKL equation carrying
the quantum numbers of a gluon, i.\,e.\ a color octet\footnote{We
speak of 'octet' to mean the adjoint representation
also for general $N_c$.} exchange, gives rise to a special solution.
For antisymmetric
color octet exchange the color factor in the BFKL kernel
is $N_c/2$ instead of $N_c$.
(In this color representation the amplitude is not infrared finite
and a regularization has to be applied.)
Let us further assume that the inhomogeneous term $\phi_0$
is a function of $({\bf k}_1+{\bf k}_2)$.
Then the equation exhibits the solution
\begin{equation}
\phi^{\bf 8_A} ({\bf k}_1+{\bf k}_2) =
\frac{\phi^{\bf 8_A}_0 ({\bf k}_1+{\bf k}_2)}{\omega -\beta({\bf k}_1+{\bf k}_2)}
\,.
\label{Pole}
\end{equation}
This solution has a pole and can thus be interpreted as describing
the propagation of a single particle with momentum $({\bf k}_1+{\bf k}_2)$
and the quantum numbers of a gluon.
In a sense the gluon turns out to be a bound state of two gluons here.
The fact that the gluon is a composite state of gluons is
often termed 'bootstrap'.
It indicates that the correct degrees
of freedom in high energy QCD are not elementary gluons but
so--called reggeized gluons. The reggeized gluon can be understood as a
collective excitation of the gauge field.

When we interchange the two gluons in the color octet amplitude
above we find that its sign changes. This fact gives rise to the notion
of signature. It characterizes the behavior under the exchange
of two gluons, that is the simultaneous interchange of color and
momentum labels. The reggeized gluons obviously carries negative
signature.

\boldmath
\section{Integral equations for $n$-gluon amplitudes}
\unboldmath
\label{ngluon}

\boldmath
\subsection{The $n$-gluon amplitudes}
\unboldmath
\label{amplitudes}

The method suited to restore unitarity in the perturbative approach
is known as the generalized leading logarithmic approximation (GLLA).
It constitutes an approximation scheme in which
a minimal set of non--leading corrections is identified that
leads to a unitary amplitude. The minimal set of
contributions required here comprises
subleading corrections with a larger number $n$ of
reggeized gluons in the $t$-channel. These are what
we call unitarity corrections to the BFKL Pomeron.
It is necessary to include all possible $n$ to eventually fulfill the
requirement of unitarity. Quark exchanges in the $t$-channel
are always suppressed by powers of the energy $s$
with respect to the corresponding gluon exchanges and are
not taken into account in GLLA.

The most complete approach to a systematic treatment
of unitarity corrections in a perturbative framework
was formulated
in \cite{Bartelsnuclphys,Bartelskernels,Bartelsinteq}.
Its aim is to arrive at an effective description of QCD in the Regge
limit in the spirit of a reggeon field theory
\cite{Gribov,BakerTerMar}, the requirement of unitarity
being built in from the very beginning.
Of course, the BFKL amplitude with its two $t$-channel gluons
should be incorporated into the whole approach as the
lowest order contribution.
It appears natural to define partial wave amplitudes
similar to the BFKL amplitude but now with $n$ reggeized gluons
in the $t$-channel. In eq.\ (\ref{tunitaryBFKL}) we have seen the
$t$-channel reggeon unitarity equation for the BFKL amplitude.
It can be summarized symbolically as
\begin{equation}
 \mbox{disc}_\omega A(\omega,t) \sim C_2 C_2^*
\,,
\label{tunitarity2}
\end{equation}
where $C_2$ is an amputated amplitude. We include a
reggeon propagator to arrive at
$D_2=C_2 [\omega-\beta({\bf k}_1)-\beta({\bf k}_2)]^{-1}$.
We have chosen a new symbol $D_2$ for the BFKL amplitude here
since we now want to consider the special physical process
of  $\gamma^* \gamma^*$-scattering.
The amplitude $D_2$ obeys the BFKL
equation (\ref{BFKLeq}) with a special choice of
the inhomogeneous term, namely the coupling of the two gluons
to the photons via a quark loop.
The unitarity relation (\ref{tunitarity2}) can
be generalized to include $n$-gluon intermediate states.
Symbolically, the generalization has the form
\begin{equation}
 \mbox{disc}_\omega A(\omega,t) \sim \sum_{n=2}^\infty C_n C_n^*
\,.
\label{tunitarityn}
\end{equation}
We include a reggeon propagator to find
$D_n=C_n (\omega -\sum_{i=1}^n \beta({\bf k}_i))^{-1}$.
Here the $D_n$ describe the production of $n$ on--shell gluons
in the $t$-channel. They are non--amputated amplitudes,
i.\,e.\ have propagators on the external gluon lines.
The correct treatment of $t$-channel unitarity relations
including multi--particle amplitudes is highly non--trivial.
To our knowledge the most complete survey of this
extensive technology is \cite{Abarbanel},
and the reader is referred to
that reference for the details. We will content ourselves here with
having motivated the physical meaning of the $n$-gluon
amplitudes $D_n$ we are going to study.
Further below we will briefly outline the formal definition of
the amplitudes $D_n$.

Once we take into account subleading
corrections with more reggeized gluons in the $t$-channel
and consider multi--particle amplitudes like the
$n$-gluon amplitudes $D_n$
there exist of course subchannels of the scattering
amplitude and we have to care about unitary also
in the subchannels. The approach initiated in
\cite{Bartelsnuclphys,Bartelskernels,Bartelsinteq}
is designed to ensure unitarity not
only in the direct channel but also in all subchannels.
This implies that the number of gluons in the $t$-channel
gluons is not conserved. Due to that the set of integral
equations for the $n$-gluon amplitudes is turned
into a tower of coupled equations including number--changing
integral kernels. (A detailed
description of the integral equations will follow in
section \ref{inteqsection}.)

The non--conservation of the number of reggeized
gluons in the $t$-channel evolution contrasts
sharply with the situation in the
Bartels--Kwieci\'nski--Prasza{\l}owicz
(BKP) equations \cite{Bartelskernels,BKP}.
The latter describe the $t$-channel evolution of a compound
state of a fixed number of reggeized gluons in the Regge limit.
Their large-$N_c$ limit turned out to be
equivalent to a completely integrable model
\cite{LipatovXXX},
namely the XXX--Heisenberg model with conformal
$\mbox{SL}(2,\C)$ spin $s=0$ \cite{GregoryFaddeev}.
Although the BKP equations do not apply directly
to our $n$-gluon amplitudes $D_n$ they will play an
important role in the effective field theory of
unitarity corrections that we are heading for.
As we will explain in more detail in section \ref{4ftconf}
there will be different elements in the effective field
theory.
First we will have $n$-particle Green functions,
which are number--conserving elements.
Their behavior will be governed by the
BKP equations. In addition, there will be
number--changing elements which we will call vertices.
They arise as a unique feature of the approach pursued here,
and turn the quantum mechanical problem described by
the BKP equations into a quantum field theory.

We will now outline the formal definition
of the $n$-gluon amplitudes $D_n$.
The way the amplitudes $D_n$ are defined
is inspired by Regge theory. A condensed but still rather
extensive description of the methods that have to be used
here can be found in
\cite{Whitepreprint}, more complete
reviews are contained in \cite{BrowerDeTar} and
\cite{WhiteIntJ}.
The procedure starts from a physical $2+n$ multi--particle
scattering process. One identifies certain
kinematical variables with the use of so--called Toller
diagrams and hexagraphs, and after defining partial waves
one can eventually get the desired amplitude by taking
an appropriate mixed Regge and helicity--pole limit.
Although it is in principle possible, we will not carry out
this program for the amplitudes under consideration
in this paper.
The procedure becomes technically complicated very
quickly with the increasing number of gluons in the
amplitude. Moreover, the physical processes we would
have to start with for larger $n$ are rather artifical.
However, the method outlined here appears to be very
natural for a special phenomenological application
of the four--gluon amplitude. In \cite{BW} a part of
the four--gluon amplitude $D_4$ in which the
four gluons form two pairs of color singlets was
used for the description of the process of
high-mass diffractive dissociation in deep
inelastic electron--proton scattering.
The rapidity gap between the proton and the
diffractively produced system is caused by a
colorless exchange between the proton and the
photon. The latter is modelled by a two--gluon
exchange. In the amplitude for this process
the initial state is therefore indeed a three--particle
state, and the cross--section takes the form of
a three--to--three scattering process.
In \cite{BW} the appropriate limit for this process
was identified
as the well--known triple Regge limit in which
$s \gg M^2 \gg Q^2 \gg \Lambda^2_{\mbox{\scriptsize QCD}}$,
where $M^2$ is the invariant mass of the diffractively produced
particles.

The $n$-gluon amplitudes are defined to have
as external lines two photons and $n$ gluons.
The photons are coupled to the gluons via their
splitting into a quark--antiquark pair to
which the gluons are attached.
Being $n$-gluon amplitudes the $D_n$ carry as
arguments $n$ color labels $a_i$ in addition to the transverse
momenta ${\bf k}_i$ of the gluons. The color labels correspond
to generators $t^{a_i}$ of the gauge group $\mbox{SU}(N_c)$
in the adjoint representation.
As partial waves the $D_n$ have also as an argument
the complex angular momentum $\omega$.
Since all $D_n$ will carry the same argument $\omega$ we
will suppress it in our notation.
In our notation we will suppress the photon momenta as well.
The $n$-gluon amplitudes are thus characterized as
\begin{equation}
 D_n^{a_1\dots a_n}({\bf k}_1, \dots, {\bf k}_n)
\,.
\end{equation}
The transverse gluon momenta ${\bf k}_i$ in the amplitude
are all chosen to be outgoing.
The $D_n$ are non--amputated amplitudes, i.\,e.\ they have
propagators for the outgoing reggeized gluons.
Further they are multiply--cut amplitudes. We take
discontinuities in the $n-1$ energy variables defined from
one photon and the $i$ first gluons ($1\le i \le n-1$),
\begin{equation}
 s_i = \left( p_{\gamma^*;1} +
\sum_{j=1}^i p_j \right)^2
\,.
\label{energiesforcuts}
\end{equation}
Here $p_{\gamma^*;1}$ and $p_j$ are the
four--momenta of the left photon and the
gluons, respectively.
The amplitudes $D_n$ can be defined for the non--forward
direction
\begin{equation}
  \sum_{i=1}^n {\bf k}_i \neq 0
\end{equation}
as well. All results in this paper will hold for the non--forward
direction, and we will not mention this in each case separately.

The simplest of the $n$-gluon amplitudes is $D_2$.
It is identical with the well--known BFKL amplitude
discussed in section \ref{BFKLsection}.
There the inhomogeneous term in the BFKL equation was not
specified. In $D_2$ it is given by the
lowest order coupling of the two $t$-channel gluons to the virtual
photons through a quark loop.
The two outgoing gluons in the BFKL amplitude $D_2^{a_1a_2}$
are in a color singlet such that we can factorize the color
structure and define the momentum part $D_2$ by
\begin{equation}
  D_2^{a_1a_2} ({\bf k}_1,{\bf k}_2) = \delta_{a_1a_2} D_2 ({\bf k}_1,{\bf k}_2)
\,.
\label{splitd2}
\end{equation}
We recall two simple but very important properties of
the momentum part $D_2$ of the BFKL amplitude.
The first is that it vanishes when one of its momentum
argument vanishes,
\begin{equation}
  D_2 ({\bf k}_1,{\bf k}_2) |_{{\bf k}_1=0} = D_2 ({\bf k}_1,{\bf k}_2) |_{{\bf k}_2=0} = 0
\label{nullstelld2}
\,.
\end{equation}
The second is the symmetry in its two momentum arguments,
\begin{equation}
  D_2 ({\bf k}_1,{\bf k}_2) = D_2 ({\bf k}_2,{\bf k}_1)
\label{symmetryd2}
\,.
\end{equation}

Concluding this section, we introduce a shorthand notation
for the arguments of $D_2$. Later we will use it for
the arguments of other functions as well.
In the case that an argument of a function, say $D_2$,
is a sum of two or more transverse momenta
we will only give the indices of these momenta, and
a string of indices stands for the sum of the corresponding
momenta.
So we have for example
\begin{equation}
  D_2(12,3) = D_2({\bf k}_1+{\bf k}_2,{\bf k}_3) \,.
\label{123notation}
\end{equation}

\subsection{Integral equations}
\label{inteqsection}
The $n$-gluon amplitudes $D_n$ obey a tower of
coupled integral equations. These have been derived in
\cite{Bartelsinteq} by means of $s$-channel
dispersion relations.
Like the BFKL equation they are equations in two--dimensional
transverse momentum space describing the $t$-channel evolution of
the amplitudes under investigation. In this evolution, the complex
angular momentum $\omega$ again plays the role of an
energy variable. Its conjugate, i.\,e.\ rapidity, acquires the meaning
of the time--like variable in the evolution.

The integral equation for the two--gluon amplitude $D_2^{a_1a_2}$
is of course identical to the BFKL equation,
\begin{equation}
   \left( \omega - \sum_{i=1}^2\beta({\bf k}_i)\right) D_2^{a_1a_2} =
D_{(2;0)}^{a_1a_2}
+ K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2} \otimes D_2^{b_1b_2}
\,.
\label{inteq2}
\end{equation}
We have moved the trajectory functions to the left hand side of
the equation to make the generalization to larger $n$ more transparent.
The inhomogeneous term $D_{(2;0)}$ denotes the lowest order
coupling of the two gluons to the photons via the quark loop.
The quark loop will  be the subject of section \ref{quarkloop}.
The integral kernel $K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2}$ is,
roughly speaking, the BFKL kernel (\ref{Lipatovkernel}) without
the gluon trajectory functions $\beta$. An exact definition of the
integral kernels will be given in section \ref{kernels}.
The superscript $\{b\} \rightarrow \{a\}$ corresponds to the color
labels of the in-- and outgoing gluons.
The convolution symbol $\otimes$ stands for
an integral $\int \frac{d^2 {\bf l}}{(2 \pi)^3}$ over the loop momentum
and a propagator $\frac{1}{{\bf l}^2}$ for each of the two
gluons entering the kernel from above.

The integral equation for the three--gluon amplitude $D_3^{a_1a_2a_3}$
has the form
\begin{equation}
  \left( \omega - \sum_{i=1}^3 \beta({\bf k}_i) \right) D_3^{a_1a_2a_3} =
D_{(3;0)}^{a_1a_2a_3}
+ K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 3} \otimes D_2^{b_1b_2}
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2} \otimes D_3^{b_1b_2b_3}
\,.
\label{inteq3}
\end{equation}
The inhomogeneous term $D_{(3;0)}$ is now the quark loop with
three gluons attached to it.
In (\ref{inteq3}) we find for the first time
a new kernel in the equation.
$K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 3}$ is a transition
kernel from two to three reggeized gluons.
The second term on the right hand side of the equation therefore
tells us that at some point in the $t$-channel evolution we can
have a transition from two to three gluons.
The last term describes the evolution of a system of three gluons,
and the sum extends over all pairwise interactions of the three
reggeized gluons via the kernel $K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2}$.
Let us look at the term in which the first and second gluon
interact via a kernel. In this term the momentum and the color
label of the third gluon are not affected. The kernel should
thus be understood to contain a factor $\delta_{a_3b_3}$.
The symbol $\otimes$ denotes again the integration over
the loop momentum in the first two gluons and propagator
factors for each of them.
The other terms are obtained in analogy.

The equations for higher $n$ are built in a very similar way.
They contain as the respective inhomogeneous term the lowest
order coupling of $n$ gluons to the quark loop. We denote this
lowest order term as $D_{(n;0)}$. A detailed
discussion of the quark loop and explicit formulae for
$n \le 6$ will follow in section \ref{quarkloop}.
In addition, the higher equations contain also higher transition
kernels $K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow m}$
from two to $m$ gluons. A general formula for arbitrary $m$ as well
as the explicit formulae for $m\le 6$ are contained in
section \ref{kernels}.

Since we will  make use of the integral equations for up to $n=6$
in this paper, we now state them explicitly. The general rule
should then be obvious.
For $n=4$ we have
\begin{eqnarray}
  \left( \omega - \sum_{i=1}^4 \beta({\bf k}_i) \right) D_4^{a_1a_2a_3a_4} &=&
D_{(4;0)}^{a_1a_2a_3a_4}
+ K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 4} \otimes D_2^{b_1b_2}
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 3} \otimes D_3^{b_1b_2b_3} \nonumber \\
&&+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2} \otimes D_4^{b_1b_2b_3b_4}
\,,
\label{inteq4}
\end{eqnarray}
for $n=5$ the equation is
\begin{eqnarray}
 \left(\omega - \sum_{i=1}^5 \beta({\bf k}_i) \right)
 D_5^{a_1a_2a_3a_4a_5} &=&
D_{(5;0)}^{a_1a_2a_3a_4a_5}
+ K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 5} \otimes D_2^{b_1b_2} \nonumber \\
&&+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 4} \otimes D_3^{b_1b_2b_3}
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 3} \otimes D_4^{b_1b_2b_3b_4} \nonumber \\
&&+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2} \otimes D_5^{b_1b_2b_3b_4b_5}
\,,
\label{inteq5}
\end{eqnarray}
and finally for $n=6$ the integral equation reads
\begin{eqnarray}
 \left(\omega - \sum_{i=1}^6 \beta({\bf k}_i) \right)
 D_6^{a_1a_2a_3a_4a_5a_6} &=&
D_{(6;0)}^{a_1a_2a_3a_4a_5a_6}
+ K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 6} \otimes D_2^{b_1b_2} \nonumber \\
&&+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 5} \otimes D_3^{b_1b_2b_3}
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 4} \otimes D_4^{b_1b_2b_3b_4} \nonumber \\
&&+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 3} \otimes D_5^{b_1b_2b_3b_4b_5}
\nonumber \\
&&+ \sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2} \otimes D_6^{b_1b_2b_3b_4b_5b_6}
\,.
\label{inteq6}
\end{eqnarray}
Here we again have to explain the meaning of the convolutions
and the summation symbols. In short, the sums contain all
combinations of the respective amplitudes and kernels in which
the $t$-channel gluons do not cross. Before we give an example
of the combinatorics we write the integral equations in
pictorial language,  which makes them easier to understand.
\begin{eqnarray}
\left( \omega - \sum_{i=1}^2 \beta({\bf k}_i)\right)
\pichallo{gld21at85.pstex_t}{3} &=& \pichallo{gld22at85.pstex_t}{3}
 + \pichallo{gld23at85.pstex_t}{3}
\label{inteq2diag}
\\
 \left( \omega - \sum_{i=1}^3 \beta({\bf k}_i)\right)
\pichallo{gld31at85.pstex_t}{3} &=& \pichallo{gld32at85.pstex_t}{3}
+ \pichallo{gld33at85.pstex_t}{3} + \sum \pichallo{gld34at85.pstex_t}{3}
\label{inteq3diag}
\\
 \left( \omega - \sum_{i=1}^4 \beta({\bf k}_i)\right)
\pichallo{gld41at85.pstex_t}{3} &=& \pichallo{gld42at85.pstex_t}{3}
 + \pichallo{gld43at85.pstex_t}{3} + \sum \pichallo{gld44at85.pstex_t}{3}
\nonumber \\
&&+ \sum \pichallo{gld45at85.pstex_t}{3}
\label{inteq4diag}
\\
\left( \omega - \sum_{i=1}^5 \beta({\bf k}_i)\right)
\pichallo{gld51at85.pstex_t}{3} &=&
\pichallo{gld52at85.pstex_t}{3} + \pichallo{gld53at85.pstex_t}{3}
 +\sum \pichallo{gld54at85.pstex_t}{3} +\nonumber \\
&& +\sum \pichallo{gld55at85.pstex_t}{3}
+\sum \pichallo{gld56at85.pstex_t}{3}
\label{inteq5diag}
\end{eqnarray}

\begin{eqnarray}
\left( \omega - \sum_{i=1}^6 \beta({\bf k}_i)\right)
\pichallo{gld61at85.pstex_t}{3} &=&
\pichallo{gld62at85.pstex_t}{3} + \pichallo{gld63at85.pstex_t}{3}
+ \sum \pichallo{gld64at85.pstex_t}{3}
\nonumber \\
&& + \sum \pichallo{gld65at85.pstex_t}{3}
  + \sum \pichallo{gld66at85.pstex_t}{3} \nonumber \\
&&+ \sum \pichallo{gld67at85.pstex_t}{3}
\label{inteq6diag}
\end{eqnarray}
In each diagram only two gluon lines from the amplitudes
enter a kernel.
An integration $\int \frac{d^2 {\bf l}}{(2 \pi)^3}$ over the
loop momentum and a propagator $\frac{1}{{\bf l}^2}$ for
each of the two gluons entering the kernel from above
are implied again.
The momenta and color labels of the other
gluons are not changed.
With the help of the pictorial notation it is also very easy
to understand which combinations of amplitudes and
kernels have to be convoluted such that $t$-channel
gluons do not cross. For example, the
sum in the last but one term in the
equation (\ref{inteq4diag}) for the four--gluon amplitude
extends over the four convolutions
\begin{equation}
\sum \pichallo{gld44at85.pstex_t}{3}
= \pichallo{gld44at85.pstex_t}{3}
+ \pichallo{expsum2at85.pstex_t}{3}
+ \pichallo{expsum3at85.pstex_t}{3}
+ \pichallo{expsum4at85.pstex_t}{3}
\,.
\end{equation}

We will now in turn discuss the elements occurring
in the integral equations:
the inhomogeneous terms $D_{(n;0)}$ representing
the coupling of $n$ gluons to a quark loop and
the integral kernels $K_{2 \rightarrow n}$.
But before doing so, we first have to consider some
color algebra.

\subsection{Color structure}
\label{color}

In this section we collect some essential facts about color algebra.
While doing so we also introduce the so--called birdtrack
notation\footnote{A more complete account of this notation can
be found in \protect\cite{Cvitanovic} where it is also applied to
other Lie groups. Our normalization convention
slightly deviates from \protect\cite{Cvitanovic}.} for
structure constants. This diagrammatic
notation is very useful for the problem of contracting indices of
arbitrary color tensors.
Such a powerful tool is needed here
since tensor contractions constitute an essential part of the
computations in the study of the integral equations.
The diagrammatic method that serves this purpose is described in detail
in appendix \ref{colorapp}.

We are interested in the structure of the gauge group $G=\mbox{SU}(N_c)$
with generators $t^a$ ($a=1,\dots,N_c^2-1$) in the Lie algebra
$\mbox{su}(N_c)$.
The algebra is
\begin{equation}
  [ t^a, t^b ] = i f_{abc} t^c  \,.  \label{algebra}
\end{equation}
For the case of $\mbox{su}(3)$ the $t^a$ are given by the well--known
Gell--Mann matrices $\lambda^a$, $t^a= \lambda^a / 2$.
The antisymmetric structure constants $f_{abc}$ can be expressed
in terms of generators as
\begin{equation}
  f_{abc} = - f_{acb} =
             - 2 i \, [ \mbox{tr} ( t^a t^b t^c )
                          - \mbox{tr} ( t^c t^b t^a ) ] \,,
\label{fabcdef}
\end{equation}
diagrammatically
\begin{equation}
  f_{abc} =
     \picbox{f_abc.pstex_t} =  \,- \hspace{-.2cm} \picbox{f_acb.pstex_t}
     =  - 2 i \left[ \, \picbox{trabc.pstex_t} -
                  \,\picbox{tracb.pstex_t} \, \right] \,.
\end{equation}
The thicker oriented lines stand for  quark color representations,
the unorientated lines correspond to gluon color lines.
The $f_{abc}$ are obviously invariant under cyclic permutations of
the indices. Normalization of generators is such that for the
Killing form
\begin{equation}
\mbox{tr} ( t^a t^b ) = \frac{1}{2} \delta_{ab} \hspace{.7cm}
  \mbox{or} \hspace{.7cm} \picbox{trab.pstex_t}
              = \frac{1}{2} \,\, \picbox{delab.pstex_t} \,.
\label{deltanorm}
\end{equation}
Using birdtrack notation the algebra (\ref{algebra}) becomes
\begin{equation}
   \picbox{lambdaab.pstex_t} \,-\, \picbox{lambdaba.pstex_t} =
   i \, \picbox{lambdacf_abc.pstex_t} \,.
\label{algebrabird}
\end{equation}
The anticommutator of two generators is
\begin{equation}
  \{ t^a, t^b \} = \frac{1}{N_c} \delta_{ab} + d_{abc} t^c \,,
\label{anticommab}
\end{equation}
and the symmetric structure constants $d_{abc}$ are expressed
in terms of generators as
\begin{equation}
  d_{abc}  = d_{acb} = 2 \, [ \mbox{tr} ( t^a t^b t^c )
                          + \mbox{tr} ( t^c t^b t^a ) ] \,,
\label{dabcdef}
\end{equation}
in diagrams
\begin{equation}
  d_{abc}  = \picbox{d_abc.pstex_t} = \picbox{d_acb.pstex_t}
           = 2 \left[ \, \picbox{trabc.pstex_t} +
                  \,\picbox{tracb.pstex_t} \, \right] \,.
\end{equation}
With this we have collected all basic elements of the
birdtrack notation.
We will slightly extend the birdtrack notation by the
definition
\begin{equation}
  \left[ \picbox{3an2f12.pstex_t} \right] \star \Theta^{\{b\}}
   = \picbox{3an2f12mab.pstex_t} \star \Theta^{\{b\}}
   = f_{a_1a_2b_1} \delta_{a_3b_2} \Theta^{b_1b_2}
\end{equation}
for the contraction of the set of color labels $\{b\}$
of an arbitrary color tensor $\Theta^{\{b\}}$.
The symbol $\star$ stands for the contraction of the set $\{b\}$
in color space. The extension to more than two elements in the
set $\{b\}$ is straightforward.

Generalizing (\ref{fabcdef}) and (\ref{dabcdef}) we
define further tensors that are built from traces of
generators in the following way\footnote{In appendix
\ref{colorapp} we will for brevity refer to tensors built from
traces over generators in this way as 'standard
tensors'. To the best of our knowledge this term
does not carry a fixed meaning in the literature on
Lie algebras.}
\begin{eqnarray}
  d^{b_1 b_2 \dots b_n} &=&
           \mbox{tr} (t^{b_1} t^{b_2} \dots t^{b_n} )
         + \mbox{tr} (t^{b_n} \dots t^{b_2} t^{b_1} ) \label{dallgdef} \\
  f^{b_1 b_2 \dots b_n} &=&
    \frac{1}{i} \,  [ \mbox{tr} (t^{b_1} t^{b_2}\dots t^{b_n})
         - \mbox{tr} (t^{b_n} \dots t^{b_2} t^{b_1} ) ]
    \label{fallgdef}
\,.
\end{eqnarray}
The definitions are valid for any $n \in {\rm I\kern-.18em N}$.
For $n=2,3$, however, the tensors arising from (\ref{dallgdef}),
(\ref{fallgdef}) are proportional\footnote{
In detail we have according to (\protect\ref{deltanorm}),
(\protect\ref{dabcdef}), and (\protect\ref{fabcdef})
\begin{equation}
 d^{ab} = \delta_{ab}\,;
\quad \quad
\protect\label{differentnorm1}
 d^{abc} = \frac{1}{2} d_{abc}\,;
\protect\label{differentnorm2}
\quad \quad
 f^{abc} = \frac{1}{2} f_{abc}
\protect\label{differentnorm3}
\,.
\end{equation}
}
to $\delta_{b_1b_2}$, $f_{b_1b_2b_3}$,
and $d_{b_1b_2b_3}$ respectively.
For the cases $n=2,3$ we will stick to the conventional definitions of
structure constants given earlier in (\ref{fabcdef}), (\ref{dabcdef}).

The following three tensors are special cases
of (\ref{dallgdef}), (\ref{fallgdef}). We will make extensive use of
them throughout this paper.
The tensor
\begin{equation}
  d^{a b c d} = \picbox{d1234.pstex_t} =
           \mbox{tr} (t^{a} t^{b} t^{c} t^{d} )
         + \mbox{tr} (t^{d} t^{c} t^{b} t^{a} )
    = \picbox{tr1234.pstex_t} +\, \picbox{tr4321.pstex_t}
\label{d4def}
\end{equation}
was used already in \cite{BW} in the investigation
of the four--gluon amplitude. We now add to this the two tensors
\begin{equation}
  f^{a b c d e} = \picbox{f12345.pstex_t} =
    \frac{1}{i} \,  [ \mbox{tr} (t^{a} t^{b} t^{c} t^{d} t^{e})
         - \mbox{tr} (t^{e} t^{d} t^{c} t^{b} t^{a} ) ]
    \label{f5def}
    = \frac{1}{i} \,
   \left[ \,\picbox{tr12345.pstex_t}
      - \,\picbox{tr54321.pstex_t} \right]
\end{equation}
and
\begin{equation}
d^{abcdef} = \mbox{tr} (t^{a} t^{b} t^{c} t^{d} t^{e}t^{f})
 +  \mbox{tr} (t^{f}t^{e} t^{d} t^{c} t^{b} t^{a} )
\,.
\end{equation}
All three are invariant under cyclic permutations
of the color labels (as are obviously all tensors defined according
to (\ref{dallgdef}), (\ref{fallgdef})).
It will turn out that these color tensors are very well suited
for the whole problem of solving the integral equations.
When interpreting the results in terms of reggeon color
representations, the decomposition of these tensors into
the lower order tensors $f_{abc}$, $d_{abc}$ and $\delta_{ab}$
is also useful:
\begin{eqnarray}
  d^{a b c d} &=&  \frac{1}{2 N_c} \delta_{ab} \delta_{cd}
                   + \frac{1}{4} (d_{abk}d_{kcd} - f_{abk}f_{kcd} )
                   \label{d4decomp} \\
              &=&  \frac{1}{2 N_c} \,\picbox{d12d34.pstex_t}
                   + \frac{1}{4} \left( \, \picbox{d12cdc34.pstex_t}
                     - \picbox{f12cfc34.pstex_t} \right)
                   \label{d4decompdiag} \,,
\end{eqnarray}
as is easily derived using (\ref{algebra}) and (\ref{anticommab}).
From this we get by cyclic permutation
\begin{equation}
 d^{a b c d} =  \frac{1}{2 N_c} \delta_{ad} \delta_{bc}
                   + \frac{1}{4} (d_{adk}d_{kbc} + f_{adk}f_{kbc} ) \,.
\end{equation}
We further mention the property $d^{bacd} = d^{abdc}$,
which turns out to be useful for calculational purposes.
For $f^{a b c d e}$ we have
\begin{eqnarray}
  f^{a b c d e} &=&  \frac{1}{4N_c} ( \delta_{ab} f_{cde}
                   + f_{abc} \delta_{de} ) \nonumber \\
             & &   + \frac{1}{8} ( f_{abk} d_{kcl} d_{lde}
                   + d_{abk} f_{kcl} d_{lde}
                   + d_{abk} d_{kcl} f_{lde}
                   - f_{abk} f_{kcl} f_{lde})
                   \label{f5decomp} \\
             &=& \frac{1}{4N_c}
                         \left( \, \picbox{del_abf_cde.pstex_t}
                                    + \picbox{f_abcdel_de.pstex_t} \right)
                         \nonumber \\
             & & + \frac{1}{8}
                         \left(\, \picbox{fdd.pstex_t}
                                    + \picbox{dfd.pstex_t}
                                    + \picbox{ddf.pstex_t}
                                    - \picbox{fff.pstex_t} \right)
                   \label{f5decompdiag}
\end{eqnarray}
and further identities can be obtained
from (\ref{f5decomp}) by making use of the invariance
under cyclic permutations.
The tensor $d^{abcdef}$ can be decomposed in a similar way,
\begin{eqnarray}
 d^{a b c d e f} &=&
 \frac{1}{4 N_c^2}\, \delta_{ab} \delta_{cd} \delta_{ef} \nonumber \\
& &\,+ \frac{1}{8 N_c} \,
( \delta_{ab} d_{cdk} d_{kef}
  - \delta_{ab} f_{cdk} f_{kef}
  + d_{abk} d_{kcd} \delta_{ef}
  - f_{abk} f_{kcd} \delta_{ef}) \nonumber \\
& &\,+ \frac{1}{16}\,
 ( d_{abk} d_{kcl} d_{ldm} d_{mef}
- d_{abk} d_{kcl} f_{ldm} f_{mef}
- d_{abk} f_{kcl} d_{ldm} f_{mef} \nonumber \\
& & \hspace{.9cm}- d_{abk} f_{kcl} f_{ldm} d_{mef}
- f_{abk} d_{kcl} d_{ldm} f_{mef}
- f_{abk} d_{kcl} f_{ldm} d_{mef} \nonumber \\
& &\hspace{.9cm}- f_{abk} f_{kcl} d_{ldm} d_{mef}
+ f_{abk} f_{kcl} f_{ldm} f_{mef} )
\,.
\label{d6decomp}
\end{eqnarray}
In birdtracks it becomes
\begin{eqnarray}
 d^{a b c d e f}
&=& \frac{1}{4 N_c^2}\, \picbox{deldeldel.pstex_t} \nonumber \\
& &\,+ \frac{1}{8 N_c} \,
 \left( \, \picbox{deldd.pstex_t}
- \picbox{delff.pstex_t}
+ \picbox{dddel.pstex_t}
- \picbox{ffdel.pstex_t} \right) \nonumber \\
& &\,+ \frac{1}{16}\,
 \left( \picbox{dddd.pstex_t}
- \picbox{ddff.pstex_t}
- \picbox{dfdf.pstex_t}
- \picbox{dffd.pstex_t} \right. \nonumber \\
& & \hspace{.9cm} \left.
- \picbox{fddf.pstex_t}
- \picbox{fdfd.pstex_t}
- \picbox{ffdd.pstex_t}
+ \picbox{ffff.pstex_t} \right)
\,.
\label{d6decompdiag}
\end{eqnarray}
and again other possible decompositions of $d^{abcdef}$ are
obtained from this by cyclic permutations.

To conclude this section about color tensors, we mention the
well--known Jacobi identity
\begin{equation}
 f_{abk}f_{kcd} + f_{ack}f_{bkd} + f_{adk}f_{kbc} = 0
\,.
\label{Jacobi}
\end{equation}

\subsection{The quark loop}
\label{quarkloop}
Let us now consider the inhomogeneous terms in the integral
equations (\ref{inteq2})--(\ref{inteq6}).
The terms $D_{(n;0)}^{a_1\dots a_n}$ describe
the lowest order coupling of $n$ gluons to the quark loop.
$D_{(2;0)}^{a_1a_2}$ is the sum of four cut diagrams
\begin{equation}
  D_{(2;0)}^{a_1a_2}({\bf k}_1,{\bf k}_2) = \picbox{d20diag1.pstex_t} +
      \picbox{d20diag2.pstex_t} + \picbox{d20diag3.pstex_t} +
      \picbox{d20diag4.pstex_t}
  \,.
\end{equation}
The two gluons are attached to the quark loop in all
possible ways to preserve gauge--invariance.
The $s$-channel cut (indicated in each diagram by the
dotted vertical line)
implies that the cut quark lines are set on--shell.
$D_{(2;0)}^{a_1a_2}$ depends on the transverse momenta of the two gluons
and on their color indices. The latter dependence is of course
trivial and we can define the momentum part $D_{(2;0)}({\bf k}_1,{\bf k}_2)$
of this amplitude by
\begin{equation}
  D_{(2;0)}^{a_1a_2}({\bf k}_1,{\bf k}_2) =
              \delta_{a_1a_2} D_{(2;0)}({\bf k}_1,{\bf k}_2) \,.
\label{splitD20}
\end{equation}
An analytic expression for $D_{(2;0)}({\bf k}_1,{\bf k}_2)$
was first found in \cite{LevinRyskin}.
The explicit formula for  $D_{(2;0)}$ will not be needed in the following.
We will only make use of  the fact that $D_{(2;0)}$ is symmetric
under the exchange of its transverse momentum arguments
\begin{equation}
 D_{(2;0)}({\bf k}_1,{\bf k}_2) = D_{(2;0)}({\bf k}_2,{\bf k}_1)
\end{equation}
and vanishes if one of its two arguments vanishes,
\begin{equation}
\left. D_{(2;0)}({\bf k}_1,{\bf k}_2) \right|_{{\bf k}_1=0} =
\left. D_{(2;0)}({\bf k}_1,{\bf k}_2) \right|_{{\bf k}_2=0} = 0
\,.
\end{equation}
In addition, it will be important for the consistency of the
integral equations that
\begin{equation}
D_{(2;0)}({\bf k}_1,{\bf k}_2) < \mbox{const.}\, \log {\bf k}_i^2
\end{equation}
in the ultraviolet region, i.\,e.\ the growth with the momenta
is not stronger than logarithmic.

For the amplitudes $D_{(n;0)}$ with $n>2$ we again have to attach
the $n$ gluons to the quark loop in all possible ways to preserve gauge
invariance. But the fact that we are dealing with multiply--cut
amplitudes reduces the number of diagrams to consider.
The cuts forbid the crossing of $t$-channel gluons as indicated
in Fig.~\ref{fig:nocrossing}.
\FIGURE{
\input{nocross.pstex_t}
\caption{Cut amplitude contributing to the coupling of $n=4$
gluons to a quark loop}
\label{fig:nocrossing}
}
The ordering of the gluons along the loop is therefore fixed up to
the possibility of coupling the gluons to the quark or the
antiquark. We thus have $2^n$ cut diagrams
for the inhomogeneous term $D_{(n;0)}$.

It turns out that all the amplitudes $D_{(n;0)}^{a_1 \dots a_n}$
can be expressed in terms of $D_{(2;0)}$, the momentum part of
$D_{(2;0)}^{a_1a_2}$ as defined in (\ref{splitD20}).
To see how this reduction mechanism works let us have a look at
two neighbouring gluons along the quark  loop.
(They are not necessarily neighbouring as arguments
of the amplitude, c.\,f.\ gluons 1 and 3 in Fig.~\ref{fig:nocrossing}.)
In the high energy limit
the quark--gluon vertices have to be contracted with a
longitudinal momentum $p$.
The Dirac trace over the quark loop then contains
\begin{equation}
  \picbox{2to1gluon1.pstex_t} \sim\,
     \mbox{tr}(\dots \slash p \slash k \slash p \dots)
    \,\delta(k^2)  \,.
\end{equation}
The $\delta(k^2)$ comes in since the quark has to be set on--shell.
Using a Sudakov decomposition $k=\alpha q^\prime + \beta p + k_t$
with $q^\prime= q +xp$, $q^{\prime\,2}=p^2=0$, $2 p \cdot q = s$, and
$k_t^2=-{\bf k}^2$ one finds for this expression
\begin{equation}
  \mbox{tr}(\dots \slash p \dots) \,
  \delta(\beta - {\bf k}^2/(\alpha s))
  \, \simeq \picbox{2to1gluon2.pstex_t}\,.
\end{equation}
This means that due to energy--momentum conservation
only the sum ${\bf k}_i + {\bf k}_j$ of the two momenta enters.
We can apply this to all gluons along the quark loop and
thereby reduce the momentum part of each
diagram to one corresponding to a
diagram in which only two gluons are coupled to the quark loop.
The color structure is not affected by this reduction.
A remark is in order concerning the contribution
in which all gluons are coupled to the
quark line or the antiquark line. This term
acts as a regularization term. As we will see below
it can be added and subtracted in such a way that the
full $D_{(n;0)}^{a_1 \dots a_n}$ can be
expressed in terms of $D_{(2;0)}$.

Let us now see how the color structure of the
quark loop amplitudes builds up.
Each diagram contributing to $D_{(n;0)}$
contains a trace over $n$ $\mbox{su}(N_c)$ generators.
The $2^n$ diagrams come in pairs in the following sense.
Consider a diagram with $k$ gluons coupled to the
quark and $n-k$ gluons coupled to the antiquark.
Then there is also a diagram with the $k$ gluons
coupled to the antiquark instead of the quark and the
other $n-k$ gluons now coupled to the quark.
The momentum structure of the two diagrams is the
same up to a factor $(-1)^n$.
(This is because the coupling of a gluon to a quark
effectively differs from that to an antiquark by a sign.)
The color part of the second diagram is again
a trace over generators $t^{a_i}$, but in the trace they now appear
in reversed order compared to the first diagram.
Adding the two diagrams one thus finds a color tensor of the
kind $d^{a_1\dots d_n}$ for an even number $n$ of gluons
and a tensor of the kind $f^{a_1\dots d_n}$ for an odd
number $n$ of gluons
(cf.\ (\ref{dallgdef}), (\ref{fallgdef}) for the definition of
the $d$- and $f$-tensors).
There are $2^{n-1}$ pairs of such diagrams.
Having in mind that due to the photons at the two ends of the loop
the color tensor is not altered if the first
or $n$th gluon is coupled to the quark instead of the antiquark
and vice versa,
we conclude that the number of different color tensors
contributing to the coupling of $n$ gluons to the quark loop is
in general $2^{n-3}$ if $n \ge 3$.
The color structures for two and three gluons attached to the
quark loop are more or less trivial: in both cases there is only one
color tensor ($\delta_{a_1a_2}$ and $f_{a_1a_2a_3}$, respectively).

We have seen that the diagrams contributing to the quark loop
come in $2^{n-1}$ pairs. Among them is a special pair,
namely the one consisting of the two regularization terms
mentioned earlier, in which all gluons couple to the quark line
or the antiquark line. This pair can
be added and subtracted with different color structures
in such a way that the remaining $2^{n-1}-1$ pairs are
regularized to give a $D_{(2;0)}$ each. Therefore the quark loop
$D_{(n;0)}^{a_1 \dots a_n}$ can be expressed as a sum of
$2^{n-1}-1$ amplitudes $D_{(2;0)}$ .

In this paper, we will need the expressions
for the quark loop with up to six gluons attached.
For three gluons coupled to the quark loop we find
\begin{equation}
  D_{(3;0)}^{a_1a_2a_3}({\bf k}_1,{\bf k}_2,{\bf k}_3)
= \frac{1}{2} g f_{a_1a_2a_3} \,
[ D_{(2;0)}(12,3) - D_{(2;0)}(13,2) + D_{(2;0)}(1,23) ]  \,,
\label{d30}
\end{equation}
where we use the notation introduced in (\ref{123notation}).
In the case of four gluons the amplitude contains
two different color structures,
\begin{eqnarray}
D_{(4;0)}^{a_1a_2a_3a_4}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4)
 &=& \! - g^2 d^{a_1a_2a_3a_4} \, [ D_{(2;0)}(123,4) + D_{(2;0)}(1,234)
                                  - D_{(2;0)}(14,23) ]
   \nonumber \\
 & & \! - g^2 d^{a_2a_1a_3a_4}  \, [ D_{(2;0)}(134,2) + D_{(2;0)}(124,3)
                                 - D_{(2;0)}(12,34) \nonumber \\
 && \hspace{2.4cm} - D_{(2;0)}(13,24) ]
\,.
\label{d40}
\end{eqnarray}
When five gluons are coupled to the quark loop
there appear four different color structures in the corresponding
amplitude,
\begin{eqnarray}
\lefteqn{D_{(5;0)}^{a_1a_2a_3a_4a_5}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5)= }
\\
  &=&  - g^3 \{ f^{a_1a_2a_3a_4a_5} \, [
                  D_{(2;0)}(1234,5) + D_{(2;0)}(1,2345) - D_{(2;0)}(15,234)]
   \nonumber \\
  & &  \,
              + f^{a_2a_1a_3a_4a_5} \, [
                  D_{(2;0)}(1345,2) - D_{(2;0)}(12,345)
                  + D_{(2;0)}(125,34) - D_{(2;0)}(134,25) ]
   \nonumber \\
  & &  \,
              + f^{a_1a_2a_3a_5a_4} \, [
                  D_{(2;0)}(1235,4) - D_{(2;0)}(14,235)
                  + D_{(2;0)}(145,23) - D_{(2;0)}(123,45) ]
   \nonumber \\
  & &  \,
              + f^{a_1a_2a_4a_5a_3} \, [
                  D_{(2;0)}(1245,3) - D_{(2;0)}(13,245)
                  + D_{(2;0)}(135,24)
- D_{(2;0)}(124,35) ] \}. \nonumber
\label{d50}
\end{eqnarray}
For six gluons attached to the quark loop we
find the following result. Now eight different
color structures contribute,
\begin{eqnarray}
\lefteqn{D_{(6;0)}^{a_1a_2a_3a_4a_5a_6}
   ({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5,{\bf k}_6)= } \nonumber \\
&=&  g^4 \{ d^{a_1a_2a_3a_4a_5a_6} \,[
  D_{(2;0)}(12345,6) + D_{(2;0)}(1,23456) - D_{(2;0)}(16,2345) ] \nonumber \\
& &\,+\,d^{a_2a_1a_3a_4a_5a_6} \,[
  D_{(2;0)}(13456,2) - D_{(2;0)}(1345,26)
+ D_{(2;0)}(126,345) \nonumber \\
&&\hspace{3cm} -  D_{(2;0)}(12,3456) ] \nonumber \\
& &\, +\,d^{a_1a_2a_3a_4a_6a_5} \,[
 D_{(2;0)}(12346,5) - D_{(2;0)}(1234,56)
+ D_{(2;0)}(156,234) \nonumber \\
&&\hspace{3cm} - D_{(2;0)}(15,2346) ] \nonumber \\
& &\, +\,d^{a_2a_1a_3a_4a_6a_5} \,[
 - D_{(2;0)}(1256,34) - D_{(2;0)}(1346,25)
 + D_{(2;0)}(125,346) \nonumber \\
&& \hspace{3cm} + D_{(2;0)}(134,256) ] \nonumber \\
& &\, +\,d^{a_3a_1a_2a_4a_5a_6} \,[
 D_{(2;0)}(12456,3) - D_{(2;0)}(1245,36)
 + D_{(2;0)}(136,245) +\nonumber \\
&&\hspace{3cm} - D_{(2;0)}(13,2456) ] \nonumber \\
& &\,+\,d^{a_1a_2a_3a_5a_6a_4} \,[
 D_{(2;0)}(12356,4) - D_{(2;0)}(1235,46)
+ D_{(2;0)}(146,235) \nonumber \\
&&\hspace{3cm} - D_{(2;0)}(14,2356) ] \nonumber \\
& &\, +\,d^{a_2a_1a_3a_5a_6a_4} \,[
- D_{(2;0)}(1246,35) - D_{(2;0)}(1356,24)
+ D_{(2;0)}(124,356) \nonumber \\
&&\hspace{3cm} + D_{(2;0)}(135,246) ] \nonumber \\
& &\, +\,d^{a_1a_2a_3a_6a_5a_4} \,[
- D_{(2;0)}(1236,45) - D_{(2;0)}(1456,23)
+ D_{(2;0)}(123,456) \nonumber \\
& & \hspace{3cm} + D_{(2;0)}(145,236) ]
\}
\,.
\label{d60}
\end{eqnarray}

\subsection{Reggeon momentum diagrams}
\label{momentumdiags}
We now introduce a further diagrammatic notation for the momentum
space integrals occurring in our analysis of the integral equations.
It will be applied in the next section where we will present
the integral kernels $K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow m}$.
With the help of so--called
reggeon momentum diagrams we hope to make our results more
transparent and easier to read.
A reggeon momentum integral looks like the following
example:
\begin{equation}
  \picbox{diagexample.pstex_t} \hspace{2cm}
\label{diagex}
\end{equation}
We now state the rules for translating these
diagrams back to explicit integrals. At the same
time we apply the rules step by step to the above
example.
\begin{itemize}
\item[(i)] Assign momentum variables to all lines according
            to momentum conservation (see diagram).
\item[(ii)] Write down an integral $\int \frac{d^2{\bf l}}{(2 \pi)^3}$
            over the loop momentum. (There is always a loop since
            some amplitude, for instance $D_2$, has to be attached to
            the upper two lines and thus be written under the integral.)
\item[(iii)] Find in the diagram the vertex which has two lines
            attached from above (vertex A in (\ref{diagex})).
\item[(iv)] Write down the square of the sum of the momenta
           attached to this vertex from below.
           (In our example $({\bf k}_2+{\bf k}_3)^2$.)
\item[(v)] Write down propagators for the two lines attached
           to this vertex from above.
           (In our example
            $({\bf l}-{\bf k}_1)^{-2} ({\bf l}-{\bf k}_1-{\bf k}_2-{\bf k}_3)^{-2}$.)

\end{itemize}
For the diagram (\ref{diagex}) this results in
\begin{equation}
   \picbox{4a23-1.pstex_t} =
    \int \frac{d^2{\bf l}}{(2 \pi)^3}
    \frac{({\bf k}_2+{\bf k}_3)^2}{({\bf l}-{\bf k}_1)^2 ({\bf l}-{\bf k}_1-{\bf k}_2-{\bf k}_3)^2} \,.
\end{equation}
These rules can easily be inverted in order to construct the reggeon
momentum diagram from a given momentum space integral.
The reggeon momentum diagrams have to be understood
as integral  operators. The integration has to be carried
out with a function of the two upper momenta.
We emphasize that our notation implies
only two propagators for a given reggeon momentum diagram.

A few more examples of the diagrammatic notation for
momentum space integrals are contained in section \ref{standardint}.

\subsection{Integral kernels}
\label{kernels}
The integral kernels $K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow m}$
were calculated in \cite{Bartelskernels} by means of
$s$-channel dispersion relations.
As explained in section \ref{inteqsection} the kernels are
convoluted with different amplitudes in the integral equations.
Only two of the gluons in the respective amplitude
actually interact with each other. The kernel acts trivially
on the momenta and color labels of the other gluons. We
will therefore discuss only the non--trivial action of the
kernel here.

The kernel for the transition
from two gluons with transverse momenta ${\bf q}_1,{\bf q}_2$
and color labels $\{b\} = \{b_1,b_2\}$
to $m$ gluons with transverse momenta ${\bf k}_1,\dots,{\bf k}_m$ and
color labels $\{a\} = \{ a_1,\dots ,a_m\} $ is given
by
\begin{eqnarray}
\lefteqn{K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow m}
              ({\bf q}_1,{\bf q}_2;{\bf k}_1,\dots,{\bf k}_m) = \picbox{kernel2n.pstex_t}
        } \nonumber \\
     &=&   f_{b_1a_1c_1} f_{c_1a_2c_2} \dots f_{c_{m-1}a_mb_2} \;
    g^m  \left[ ({\bf k}_1 + \dots + {\bf k}_m)^2
               - \frac{{\bf q}_2^2({\bf k}_1+\dots+{\bf k}_{m-1})^2}{({\bf k}_m-{\bf q}_2)^2}
      \right. \nonumber \\
     & & \hspace{3.5cm} \left.
               - \frac{{\bf q}_1^2({\bf k}_2+\dots+{\bf k}_m)^2}{({\bf k}_1-{\bf q}_1)^2}
               + \frac{{\bf q}_1^2{\bf q}_2^2
                ({\bf k}_2+\dots+{\bf k}_{m-1})^2}{({\bf k}_1-{\bf q}_1)^2({\bf k}_m-{\bf q}_2)^2}
        \right]
\,.
\label{Kn}
\end{eqnarray}
For the kernels that are needed for up to six gluons in the $t$-channel
this means in our diagrammatic notation for color tensors
and momentum integral kernels
\begin{eqnarray}
   K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2}
              ({\bf q}_1,{\bf q}_2;\{ {\bf k}_i\} )   &=&
       \picbox{color2to2.pstex_t}  g^2  {\bf q}_1^2 {\bf q}_2^2
      \left[ \; \picbox{c12.pstex_t} - \picbox{2b1.pstex_t}
             - \picbox{2b2.pstex_t}  \;  \right]
   \\
   K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 3}
              ({\bf q}_1,{\bf q}_2;\{ {\bf k}_i\} )   &=&
       \picbox{color2to3.pstex_t}  g^3 {\bf q}_1^2 {\bf q}_2^2
      \left[ \; \picbox{c123.pstex_t} - \picbox{3b12.pstex_t}
             -  \picbox{3b23.pstex_t} +  \picbox{3a2-1.pstex_t} \; \right]
\label{k23}
   \\
   K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 4}
              ({\bf q}_1,{\bf q}_2;\{ {\bf k}_i\} )   &=&
       \picbox{color2to4.pstex_t} g^4 {\bf q}_1^2 {\bf q}_2^2
      \left[ \; \picbox{c1234.pstex_t} - \picbox{4b123.pstex_t}
             - \picbox{4b234.pstex_t} + \picbox{4a23-1.pstex_t} \; \right]
   \\
   K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 5}
              ({\bf q}_1,{\bf q}_2;\{ {\bf k}_i\} )   &=&
       \picbox{color2to5.pstex_t} g^5 {\bf q}_1^2 {\bf q}_2^2
      \left[ \;\picbox{c12345.pstex_t} - \picbox{5b1234.pstex_t}
      -  \picbox{5b2345.pstex_t} + \picbox{5a234-1.pstex_t}\; \right]
\\
 K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 6}
              ({\bf q}_1,{\bf q}_2;\{ {\bf k}_i\} )   &=&
       \picbox{color2to6.pstex_t} g^6 {\bf q}_1^2 {\bf q}_2^2
      \left[ \;\picbox{c123456.pstex_t} - \picbox{6b12345.pstex_t}
      -  \picbox{6b23456.pstex_t} + \picbox{6a2345-1.pstex_t}
\; \right] \hspace*{.4cm}
\end{eqnarray}
Here the inverse propagators ${\bf q}_1^2 {\bf q}_2^2$ are required in order to
cancel the propagators implied by our graphical
notation. This is necessary because the kernels
$K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow m}$ in the integral equations are
defined without propagators; there the propagators come in via
the convolution denoted by the symbol $\otimes$.

We emphasize that the kernel
$K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2}$ is not identical to the full
BFKL kernel (\ref{Lipatovkernel}) as it does not contain the
trajectory functions $\beta$. In equation (\ref{inteq2})
they have been moved to the left--hand side to make the
generalization of this equation to $n>2$ more transparent.
Also the kernels $K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow n}$ for $n>2$
do not contain any trajectories and are not infrared finite.
It is only in the full integral equations that the infrared
divergences cancel.

\subsection{Standard integrals}
\label{standardint}

When convoluting the amplitudes $D_n$ with the kernels
$K_{2\rightarrow m}$ according to the integral equations, we have
to deal with a large number of momentum space integrals.
It turns out that all these integrals can be classified by
a small number of standard integrals. These five
standard types
of integrals are even sufficient for addressing the case
of $n$-gluon amplitudes for any value of $n$.
(Strictly speaking this applies to the so--called reggeizing
pieces of the amplitudes as will be explained in the next section.)
We will therefore give a complete list of the
five different standard integrals (or diagrams) which
at the same time serve as a few more examples of
our notation.

All five standard integrals occur already when dealing
with three outgoing gluons. We therefore give the list
for $n=3$ here, the generalization to larger $n$ being obvious.
First we have
\begin{equation}
  a(2,1) = \picbox{3a2-1.pstex_t}
    = \int \frac{d^2{\bf l}}{(2 \pi)^3}
       \frac{{\bf k}_2^2}{({\bf l}-{\bf k}_1)^2 [{\bf l}-({\bf k}_1+{\bf k}_2)]^2}
      \,D_2\!\left({\bf l}, \sum_{j=1}^{3}{\bf k}_j-{\bf l} \right) \,.
\label{adiag}
\end{equation}
The first argument of the function $a$ is the index of the momentum
attached to the vertex A from below. (As in section \ref{momentumdiags}
vertex A is the vertex
in the diagram with two lines attached from above.)
The momentum structure of the diagram is then fixed
by giving one of the other two outgoing momenta
since the diagram has to be folded with a symmetric
function of the upper two momenta,
namely with the BFKL amplitude $D_2$.
We choose the outgoing momentum with
the lowest index as the second argument of the function $a$
thereby completely fixing the corresponding momentum integral
since the momentum carried by the third outgoing gluon line
can easily be inferred from the total number of gluons.
Therefore our notation for $a$ has to be supplied with the
total number of $t$-channel gluons it describes.
(In the case of more than three gluons we here choose the
group of momenta containing the lowest index as the second
argument of $a$.)
Applying this notation,
our earlier example given in (\ref{diagex})
would be assigned the expression $a(23,1)$ when applied
to $D_2$. In section \ref{momentumdiags} the reggeon
momentum diagrams stood for integral operators. In this
and in the following sections we use the same diagrams also
for the convolution with BFKL amplitudes. The meaning
should be clear from the context. The function $a$ and the
functions $b$, $c$, $s$, $t$ to be defined below always mean
the integrals with $D_2$ included.
The second type of diagram is
\begin{equation}
  b(12) =  \picbox{3b12.pstex_t}
      = \int \frac{d^2{\bf l}}{(2 \pi)^3}
         \frac{({\bf k}_1+{\bf k}_2)^2}{{\bf l}^2 [{\bf l}-({\bf k}_1+{\bf k}_2)]^2}
    \,D_2\!\left({\bf l}, \sum_{j=1}^{3}{\bf k}_j-{\bf l} \right)\,.
\label{bdiag}
\end{equation}
The third one is the contact term
\begin{equation}
  c(123) = \picbox{c123.pstex_t} = \int \frac{d^2{\bf l}}{(2 \pi)^3}
 \frac{({\bf k}_1+{\bf k}_2+{\bf k}_3)^2}{{\bf l}^2\left({\bf l}-\sum_{j=1}^3{\bf k}_j\right)^2}
 \,D_2\!\left({\bf l}, \sum_{j=1}^{3}{\bf k}_j-{\bf l} \right)\,,
\label{cdiag}
\end{equation}
which is local in impact parameter space.
The integrals $a$, $b$, and $c$ correspond to real
corrections, that is to $s$-channel gluon production.

Further we have two integral types corresponding to
virtual corrections. These are factorizing and
are connected with what we already know as
the trajectory function $\beta$ of the reggeized gluon:
\begin{eqnarray}
  t(12) =  \picbox{3t12.pstex_t}
      &=& \int \frac{d^2{\bf l}}{(2 \pi)^3}
       \frac{({\bf k}_1+{\bf k}_2)^2}{{\bf l}^2 [{\bf l}-({\bf k}_1+{\bf k}_2)]^2}
     D_2({\bf k}_1+{\bf k}_2,{\bf k}_3)
\nonumber \\
    &=&   \frac{2}{N_c g^2} \beta({\bf k}_1+{\bf k}_2) D_2({\bf k}_1+{\bf k}_2,{\bf k}_3)
   \label{tdiag}
\end{eqnarray}
and
\begin{eqnarray}
  s(1,2) =  \picbox{3s1-2.pstex_t}
   &=& \int \frac{d^2{\bf l}}{(2 \pi)^3}
    \frac{{\bf k}_1^2}{{\bf l}^2({\bf l}-{\bf k}_1)^2}
     D_2({\bf k}_1+{\bf k}_2,{\bf k}_3)
\nonumber \\
    &=& \frac{2}{N_c g^2} \beta({\bf k}_1)  D_2({\bf k}_1+{\bf k}_2,{\bf k}_3) \,.
\label{sdiag}
\end{eqnarray}
This means that the bubble diagram corresponds to a gluon
trajectory function $\beta$ for the line it is drawn on.

It suggests itself here to write the BFKL kernel
using the standard integrals introduced above.
When the kernel is applied to the BFKL amplitude $D_2$
we find
\begin{eqnarray}
K_{\mbox{\scriptsize BFKL}} \otimes D_2 &=&
N_c g^2 \left[ \,c(12) - b(1) - b(2) + \frac{1}{2} t(1)
+\frac{1}{2} t(2) \right]  \\
&=& N_c g^2 \left[ \,\picbox{c12.pstex_t}
- \picbox{2b1.pstex_t} - \picbox{2b2.pstex_t}
+ \frac{1}{2} \,\picbox{2t1.pstex_t} +\frac{1}{2}\,\picbox{2t2.pstex_t}
\right]
\,.
\end{eqnarray}

The notation introduced here makes implicit use of the
fact that the BFKL amplitude $D_2$ is symmetric in
its two momentum arguments. Without this assumption
it would, for example, be necessary to give explicitly a third
argument to fully specify the integral $a$ in (\ref{adiag}).
We have chosen to restrict the short--hand notation to
the case of a symmetric function since
we will only apply it to the case of $D_2$.

As an example of how the convolutions of $n$-gluon
amplitudes with kernels can be reduced to the standard
integrals may serve
\begin{equation}
  D_2({\bf l}_1+{\bf l}_2,{\bf l}_3)
  \otimes K_{2\rightarrow 3}({\bf l}_2,{\bf l}_3;{\bf k}_2,{\bf k}_3,{\bf k}_4)=
  g^3[ b(234) - a(23,1) - b(34) + a(3,12) ] \,.
\end{equation}
This can be easily checked using the explicit definition (\ref{k23})
or (\ref{Kn}) of the kernel.
When studying the $n$-gluon amplitudes $D_n$ we
encounter a problem connected with these convolutions
in momentum space. For each convolution it is relatively easy
to find a representation in terms of the standard integrals,
as the above example shows. The actual problem is the
rapidly increasing number of convolutions we have to deal with
when coming to larger $n$. We will explain this problem in more
detail in the following sections. For $n=5$ gluons the problem is
at the edge of being tractable by hand. For $n=6$ gluons the
problem has to be attacked with the help of a computer.
It is exactly for this reason that we introduce the
classification of momentum space integrals in this section.
In appendix \ref{momentumapp} we give an algorithm
suited for implementation on a computer, for instance
in the PERL script language.
The example above is intended to illustrate  that the notation
indeed allows us to handle the rather complicated integrals in
compact form.

Closing this section, we remark that even the five standard types
of integrals mentioned above are not completely independent if we take
our definitions (\ref{adiag})--(\ref{sdiag}) literally.
Relations between them occur in the case that one of the outgoing
legs of the diagrams has zero momentum.
One finds, for example, that the function $b$ emerges
from $a$ when the two arguments
of $a$ exhaust the outgoing gluons, that is for three gluons we would
get
\begin{equation}
  a(1,23) = b(1)\,.
\end{equation}
Similarly, for three outgoing gluons
\begin{equation}
  b(123) = c(123) \,.
\end{equation}
In addition, we find the function $t$ from $s$ when the second
argument of $s$ vanishes, for example
\begin{equation}
  s(1,-) = t(1) \,.
\end{equation}
In spite of these relations we prefer to treat the five integral
types as fundamentally different because they correspond
to very different locality properties in impact parameter space.
The function $c$, for instance, corresponds to a contact
interaction whereas the function $a$ contains a non--locality.

This concludes our discussion of the elements of the integral
equations and we can now proceed to solving them (at least partially).

\boldmath
\section{Three and four gluons,
         the transition vertex $V_{2 \rightarrow 4}$}
\unboldmath
\label{34V}

\subsection{The three-gluon amplitude}
\label{threegluon}

The amplitudes with three and four gluons, $D_3$ and $D_4$,
were first investigated in \cite{BPLB,BW}.
It was found that the integral equation (\ref{inteq3}) for the
three gluon amplitude can be solved, the solution being
\begin{equation}
D_3^{a_1a_2a_3}({\bf k}_1,{\bf k}_2,{\bf k}_3)
= \frac{1}{2} g f_{a_1a_2a_3} \,
[ D_2(12,3) - D_2(13,2) + D_2(1,23) ]
\,,
\label{d3}
\end{equation}
which can be shown by direct computation. In addition
to performing the color algebra contractions and the convolutions
in momentum space one has to make use of the integral equation
(\ref{inteq2}) for the two--gluon amplitude $D_2$.

The result (\ref{d3}) means that an actual
three--gluon state does not appear.\footnote{This
does not affect the existence of the Odderon, since in
our case the three--gluon system has even $C$-parity.}
In the contrary, the amplitude turns out to be a superposition of
two--gluon states. We call this phenomenon
the reggeization of the amplitude. It generalizes the notion
of reggeization previously attributed to the fact that the BFKL
equation in the color octet channel can be solved by a pole--ansatz
and thus describes a one--reggeon state. In the case of the three--gluon
amplitude reggeization again occurs in a channel corresponding to the
adjoint representation, i.\,e.\ in the octet channel.
We should emphasize that the reggeization of $D_3$ is a property of the
momentum space part of the amplitude. The analytic
properties correspond to those of a superposition of two--gluon
compound states. Nevertheless, $D_3$ remains a three--gluon
amplitude carrying three color labels, i.\,e.\ the color part of the
amplitude is not affected.

It is worth noting that the solution (\ref{d3})
is obtained from the lowest order term (\ref{d30}) by replacing
the quark loop amplitudes $D_{(2;0)}$ by the full BFKL amplitudes
$D_2$ while keeping the color and momentum structure.

We do not give a proof of (\ref{d3}) here. In section \ref{5gluons}
we will discover that the reggeization of the three--gluon
amplitude $D_3$ is actually a special case of a more general
identity that we will discuss in detail.

\boldmath
\subsection{The four-gluon amplitude and the two-to-four transition
vertex $V_{2 \rightarrow 4}$}
\unboldmath
\label{fourgluon}

Our method for analyzing the structure of the $n$-gluon
amplitudes is the following.
One starts with an educated guess for the solution or at least a part
of it, assuming the full solution
to be a sum of the part we have guessed and a remaining
term. That ansatz is inserted into the integral equation and
a new integral equation for the unknown part is derived.
If the guess was good the new integral equation is
in a certain sense simple, and allows one to gain further
information about the unknown part.
If the guess was not optimal, on the other hand,
the new integral equation
will be complicated and not allow us to extract
further information.
Of course, this procedure is not uniquely determined.
As we will see, the quark loop contains very useful
information that can be used for choosing a promising
ansatz.
Let us now see how this method works in practice.

In \cite{BPLB,BW} the four--gluon amplitude was split into two parts,
\begin{equation}
  D_4 = D_4^R + D_4^I \,,
\label{d4ri}
\end{equation}
a reggeizing part $D_4^R$ and a part $D_4^I$ that for
reasons to be explained below is called the irreducible part of the
four--gluon amplitude.
The reggeizing part is --- in analogy with the three--gluon case ---
chosen as the superposition of two--gluon amplitudes,
\begin{eqnarray}
\lefteqn{D_4^{R\,a_1a_2a_3a_4}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4)= } \nonumber \\
  &=&  - g^2 d^{a_1a_2a_3a_4} \, [ D_2(123,4) + D_2(1,234)
                                  - D_2(14,23) ]
   \nonumber \\
 & & - g^2 d^{a_2a_1a_3a_4}  \, [ D_2(134,2) + D_2(124,3)
                                 - D_2(12,34) - D_2(13,24) ] \,.
\label{d4r}
\end{eqnarray}
Again, this is obtained from the lowest order term
$D_{(4;0)}$, see (\ref{d40}),
by the replacement $D_{(2;0)} \to D_2$.
The ansatz for the reggeizing part and thus the
decomposition (\ref{d4ri}) is to some extend arbitrary.
Recently, a different ansatz for the reggeizing part was
investigated in \cite{BraunVacca,Vacca}. That ansatz, as also
discussed in \cite{BW}, is more convenient for the
analysis of high mass diffractive processes.
We will not further discuss other possible choices for the
splitting (\ref{d4ri}) of the amplitude here.
All choices will, of course, lead to
equivalent results for the complete amplitude.
The choice given above is singled out because it leads to a
simple picture for the remaining part $D_4^I$, especially
when interpreted in view of a field theory structure
of unitarity corrections.

The next step is to derive a new integral equation for
the unknown irreducible part $D_4^I$. To this end we
insert the ansatz (\ref{d4ri}),(\ref{d4r}) for the full
amplitude into the original integral equation (\ref{inteq4}).
The known result (\ref{d3}) for the three gluon amplitude
is inserted as well.
Due to the choice of $D_4^R$ we can then apply
the equation (\ref{inteq2}) for the two--gluon amplitude
to the expression $\omega  D_4^R$
on the left hand side.
That exactly eliminates the lowest order term $D_{(4;0)}$
on the right hand side and produces additional terms
containing only the convolution of $D_2$ amplitudes with
the kernel $K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2}$ or
products of $D_2$ with trajectory functions $\beta$.
Besides the terms containing $D_4^I$ all other contributions to
the right hand side consist of convolutions of two--gluon
amplitudes $D_2$ with the integral kernels.
Our new equation thus takes the form
\begin{equation}
  \left(\omega - \sum_{i=1}^4 \beta({\bf k}_i) \right)
  D_4^{I\,a_1a_2a_3a_4} ({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4) =
   V_{2 \rightarrow 4}^{a_1a_2a_3a_4} D_2
  +\sum K^{ \{b\} \rightarrow \{a\} }_{2\rightarrow 2} \otimes D_4^{I\,b_1b_2b_3b_4}
   \,.
\label{inteq4i}
\end{equation}
The sum on the right--hand side of this new equation extends over
all pairwise interactions of the four gluons.
In the inhomogeneous term $V_{2 \rightarrow 4} D_2$
we collect all the terms containing $D_2$,
hence the notation. $V_{2 \rightarrow 4}$ should be
understood as an integral operator acting on the two--gluon
amplitude. As we will explain in more detail below,
it describes the transition from the two--gluon
state to a fully interacting four--gluon system in the $t$-channel.
The explicit expression for the two--to--four transition vertex
was computed in \cite{BPLB,BW}. To arrive at this explicit result the
following steps have to be done. First we have to contract
the color tensors of the amplitudes with those of the kernels.
This is done along the lines described in appendix \ref{colorapp}.
The second step is to bring the momentum space integrals to
their standard form, that is to classify them according to
the scheme explained in section \ref{standardint}.
Both steps result in lengthy calculations because of
the large number of contractions involved. The results of
the $\mbox{su}(N_c)$ tensor contractions are then multiplied with
the corresponding momentum space integrals.
Finally, all terms can be collected to give the vertex
$V_{2 \rightarrow 4}$. Due to cancellations
the resulting expression is comparatively compact.
Remarkably, all terms belonging to the color tensors $d^{a_1a_2a_3a_4}$
and $d^{a_2a_1a_3a_4}$ drop out. One finds
the following color and momentum structure for the vertex:
\begin{eqnarray}
 V_{2 \rightarrow 4}^{a_1a_2a_3a_4}(\{ {\bf q}_j\},{\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4)
 &=&
  \delta_{a_1a_2} \delta_{a_3a_4}
V(\{ {\bf q}_j\},{\bf k}_1,{\bf k}_2;{\bf k}_3,{\bf k}_4)
 \nonumber \\
 & &
 +\, \delta_{a_1a_3} \delta_{a_2a_4}
V(\{ {\bf q}_j\},{\bf k}_1,{\bf k}_3;{\bf k}_2,{\bf k}_4)
 \nonumber \\
 & &
 +\,\delta_{a_1a_4} \delta_{a_2a_3}
V(\{ {\bf q}_j\},{\bf k}_1,{\bf k}_4;{\bf k}_2,{\bf k}_3)
\,.
\label{colV}
\end{eqnarray}
The function $V$ is the same in all three terms on the right hand side.
The ${\bf q}_j$ are the two momenta entering from above. Since
throughout this paper
$V_{2 \rightarrow 4}$ is almost always contracted with a BFKL
amplitude $D_2$ from above, we will omit these arguments
in the following and consider the quantity $V_{2 \rightarrow 4}D_2$.
The vertex function $V({\bf k}_1,{\bf k}_2;{\bf k}_3,{\bf k}_4)$
has the explicit form
\begin{eqnarray}
(V D_2)  ({\bf k}_1,{\bf k}_2;{\bf k}_3,{\bf k}_4) &=& \frac{g^4}{4} \times
\{\,2  \,[\, c(1234)   \nonumber \\
&& \hspace{1.5cm}
-\, b(123) - b(124) - b(134) - b(234) + b(12) + b(34) \nonumber \\
&& \hspace{1.5cm}+\, a(13,2) + a(14,2) + a(23,1) + a(24,1) \nonumber \\
&& \hspace{1.5cm}-\, a(1,2) - a(2,1) - a(3,12) - a(4,12) ]  \nonumber \\
&& \hspace{1.2cm}+ [\, t(123) + t(124) + t(134) + t(234) -t(12) -t(34)
\nonumber \\
&& \hspace{1.5cm}-\, s(13,2) - s(13,4) - s(14,2) - s(14,3) \nonumber \\
&& \hspace{1.5cm}-\, s(23,1) -s(23,4) - s(24,1) - s(24,3) \nonumber \\
&&  \hspace{1.5cm}+\, s(1,2) + s(1,34) + s(2,1) + s(2,34) \nonumber \\
&& \hspace{1.5cm}+ \,s(3,12) + s(3,4)+ s(4,12) + s(4,3) ]  \}
\label{v24}
\end{eqnarray}
where we have made use of the notation introduced in
section \ref{standardint}.

Let us now describe the known properties of the vertex function $V$
and of the full transition vertex $V_{2 \rightarrow 4}$. The first
observation is that
$V({\bf k}_1,{\bf k}_2;{\bf k}_3,{\bf k}_4)$ is symmetric in its first two and in
its last two arguments
\begin{eqnarray}
V({\bf k}_1,{\bf k}_2;{\bf k}_3,{\bf k}_4) &=& V({\bf k}_2,{\bf k}_1;{\bf k}_3,{\bf k}_4) \nonumber \\
                             &=& V({\bf k}_1,{\bf k}_2;{\bf k}_4,{\bf k}_3) \,,
\label{vsymm12}
\end{eqnarray}
and symmetric under the exchange of the first pair of arguments
and the second pair of arguments (that is why our notation separates these
pairs by a semicolon)
\begin{equation}
V({\bf k}_1,{\bf k}_2;{\bf k}_3,{\bf k}_4) = V({\bf k}_3,{\bf k}_4;{\bf k}_1,{\bf k}_2) \,.
\label{vsymm12,34}
\end{equation}
Therefore, according to (\ref{colV}) the full vertex
$V_{2 \rightarrow 4}$ is completely symmetric under the simultaneous
exchange of color and momentum of the gluons.

The combination of integrals in $VD_2$ vanishes when one of the
outgoing momenta vanishes,
\begin{equation}
\left.
(V D_2) ({\bf k}_1,{\bf k}_2;{\bf k}_3,{\bf k}_4)
\right|_{{\bf k}_i = 0} = 0  \;\;\;\;\;\;  (i\in\{1,\dots,4\})
\,.
\end{equation}
This result can be proven easily using
identities of the kind mentioned at the end of section \ref{standardint}
and the fact that the BFKL amplitude $D_2$
vanishes at zero--momentum argument.
This property of $V D_2$ is carried over to the full vertex,
\begin{equation}
\left.
(V_{2 \rightarrow 4} D_2)^{a_1a_2a_3a_4}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4)
\right|_{{\bf k}_i = 0} = 0  \;\;\;\;\;\;  (i\in\{1,\dots,4\})
\,.
\label{nullstellvertex}
\end{equation}
Further, the function $VD_2$ is infrared finite, i.\,e.\ the
infrared divergences in the different integrals contributing to
(\ref{v24}) cancel in the sum.
This can be easily seen after noticing that already certain
combinations of very few standard integrals are infrared
finite, nicely showing the cancellation of divergences between
real and virtual corrections. The combination
\begin{equation}
  b({\bf l}) - \frac{1}{2} \,t({\bf l})
\label{inffinite1}
\end{equation}
is infrared finite for any sum of momenta that is
substituted for ${\bf l}$, as is clear from the corresponding
integrals (see section \ref{standardint}).
The factor $1/2$ comes about because the integrand of the
trajectory function $\beta$ (see (\ref{tdiag}) and
(\ref{traject})) exhibits two divergences of the same form.
Similarly, one can show that the combination
\begin{equation}
 a({\bf l}_2,{\bf l}_1)
-\frac{1}{2} \,s({\bf l}_2,{\bf l}_1) - \frac{1}{2}\,s({\bf l}_2,{\bf l}_3)
\label{inffinite2}
\end{equation}
is infrared finite separately for any partition of the momenta
$\{ {\bf k}_i \}$ into three sums ${\bf l}_1,{\bf l}_2,{\bf l}_3$.
Finally, the term $c({\bf l})$ is infrared finite separately
since in this term (see (\ref{cdiag})) the poles of the
propagators are cancelled by the zeros of the BFKL
amplitude $D_2$ in the integral. The finiteness of these
three groups is independent of the total
number of momenta ${\bf k}_i$ that are split into the groups
denoted by ${\bf l}_j$.
It should be obvious from equation (\ref{v24}) that
all integrals in the vertex function come in exactly
these infrared finite combinations.

We now come back to
the main problem of understanding the full
four--gluon amplitude $D_4$. When the ansatz
(\ref{d4ri}) was made the goal was to arrive at a
simple equation for the yet unknown part $D_4^I$.
The new integral equation (\ref{inteq4i})
is in fact simple: It contains only the vertex
$V_{2 \rightarrow 4}$ and a homogeneous part,
and can therefore now be iterated.
The structure arising from this is
\begin{equation}
  D_4^I = G_4 \cdot V_{2 \rightarrow 4} \cdot D_2 \,,
\label{d4istruct}
\end{equation}
$G_4$ being the Green function of the four--gluon state.
The Green function obeys the BKP equation with
four $t$-channel gluons, which is a four--particle
Schr\"odinger equation.
Its Hamiltonian is given by the homogeneous part of the
integral equation (\ref{inteq4i}),
i.\,e.\ by the sum of all pairwise interactions
$K_{2 \to 2}$ of the four gluons.
Unfortunately, the eigenvalues and eigenfunctions
of the Hamiltonian are not known and (\ref{d4istruct})
remains a formal solution only. Though, some
properties of the four--gluon state have been worked
out in \cite{Hans}.
We will return to the interpretation of the structure
inherent in (\ref{d4istruct}) momentarily.

Even without knowing an analytic formula for $D_4^I$
we can deduce two important properties. Like the
two--gluon amplitude, the irreducible part of the
four--gluon amplitude vanishes (modulo logarithms)
when one of the outgoing
gluon momenta is set to zero,
\begin{equation}
\left.
  D_4^{I\,a_1a_2a_3a_4}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4)
\right|_{{\bf k}_i = 0} = 0  \;\;\;  (i\in\{1,\dots,4\})
\,.
\label{nullstelld4i}
\end{equation}
To prove this we proceed order by order in the iteration of the
Hamiltonian $\sum K_{2 \to 2}$. The identity holds
in lowest order since the vertex itself has this property
(see (\ref{nullstellvertex})).
In the next order, (\ref{nullstelld4i}) holds because
$K_{2 \to 2}$ also vanishes if one of the outgoing
momenta becomes zero, etc.
Similarly, we can show that the irreducible part $D_4^I$ is completely
symmetric in the four gluons, that is under the simultaneous exchange
of color and momentum,
\begin{eqnarray}
D_4^{I\,a_1a_2a_3a_4}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4)
&=& D_4^{I\,a_2a_1a_3a_4}({\bf k}_2,{\bf k}_1,{\bf k}_3,{\bf k}_4) \nonumber \\
&=& D_4^{I\,a_3a_2a_1a_4}({\bf k}_3,{\bf k}_2,{\bf k}_1,{\bf k}_4) \nonumber \\
&=& D_4^{I\,a_4a_2a_3a_1}({\bf k}_4,{\bf k}_2,{\bf k}_3,{\bf k}_1)
\,.
\label{complsymmd4i}
\end{eqnarray}

\subsection{Field theory structure}
\label{4ftconf}

Although we do not have an analytic expression for the
irreducible part, we have
gathered by now quite some knowledge about the structure of the
four--gluon amplitude $D_4$. Neglecting for a moment
color and normalization factors, this structure is
illustrated in the following diagram
\begin{equation}
  D_4({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4)
 = \sum \picbox{solutiond41.pstex_t} + \picbox{solutiond42.pstex_t}
\,.
\label{solutiond4diag}
\end{equation}
The first part is the superposition of two--gluon states
coupled to the quark loop (the reggeizing part $D_4^R$).
In the second (irreducible) part $D_4^I$ a two--gluon system couples
to the quark loop and then undergoes a transition to a
four--gluon compound state via the vertex $V_{2 \rightarrow 4}$.
From this we learn that the complete amplitude consists of only a few
basic building blocks: a quark loop allowing the coupling of the
$t$-channel gluons to external particles,
the two--gluon Green function (BFKL
amplitude), the four--gluon Green function, and the two--to--four
transition vertex. The Green functions describe the quantum mechanical
propagation of bound states of $t$-channel gluons.
With the transition vertex we have in addition a number--changing
element connecting different $n$-reggeon states.
This turns the quantum mechanical problem of $n$-gluon
states into that of a quantum field theory of reggeized gluons.
All this takes place in the $2$-dimensional space of transverse momenta.
The complex angular momentum $\omega$ plays the role
of an energy variable. Its conjugate variable, i.\,e.\ rapidity, plays
the role of the time variable.

We would like to emphasize that phenomenon of reggeization
is crucial for the emergence of a field theory structure.
Without reggeization in the three--gluon amplitude and
in the part $D_4^R$ of the four--gluon
amplitude one would not have been able to arrive at the
simple structure (\ref{solutiond4diag}).

So far only the simplest elements of a potential effective
field theory have been identified: the two--gluon compound state, the
two--to--four transition vertex $V_{2 \rightarrow 4}$
and the four--gluon compound state. An analytic formula
for the latter is still missing. The concept of an effective field
theory has not been derived from first principles. It has to be
tested and further elements should be derived. To achieve this
is seems natural to proceed to higher $n$-gluon amplitudes.
This constitutes the main goal of this paper, namely
to deepen our understanding of the field theory structure
of unitarity corrections by studying the five-- and six--gluon
amplitudes.

\section{Five gluons}
\label{5gluons}

\subsection{A reggeizing part and the integral equation for the
remaining part}
\label{splitd5}

In the first step, our analysis of the five--gluon amplitude $D_5$
follows the same lines as
the study of the three-- and four--gluon amplitudes.
To get started we identify a reggeizing part $D_5^R$ of
the amplitude and split the amplitude accordingly,
\begin{equation}
  D_5 = D_5^R + D_5^I
\,.
\label{d5ri}
\end{equation}
With a well--chosen $D_5^R$ we will come to a new integral
equation for the yet unknown quantity $D_5^I$. Again, this
decomposition is not unique. Our ansatz will lead to
an equation for $D_5^I$ that can even be solved.
This situation is the best we can hope for and
further justification for the ansatz is certainly not needed.
The natural choice for $D_5^R$ is once more suggested by the
inhomogeneous term $D_{(5;0)}$. This means that our ansatz
has exactly the
same color and momentum structure as $D_{(5;0)}$
in (\ref{d50}), but we replace $D_{(2;0)}$ by $D_2$
resulting in
\begin{eqnarray}
\lefteqn{D_5^{R\,a_1a_2a_3a_4a_5}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5)= }
\\
  &=&  - g^3 \{ f^{a_1a_2a_3a_4a_5} \, [
                  D_2(1234,5) + D_2(1,2345) - D_2(15,234)]
   \nonumber \\
  & &  \phantom{ - g^3 }
              + f^{a_2a_1a_3a_4a_5} \, [
                  D_2(1345,2) - D_2(12,345)
                  + D_2(125,34) - D_2(134,25) ]
   \nonumber \\
  & &  \phantom{ - g^3 }
              + f^{a_1a_2a_3a_5a_4} \, [
                  D_2(1235,4) - D_2(14,235)
                  + D_2(145,23) - D_2(123,45) ]
   \nonumber \\
  & &  \phantom{ - g^3 }
              + f^{a_1a_2a_4a_5a_3} \, [
                  D_2(1245,3) - D_2(13,245)
                  + D_2(135,24)
\nonumber
- D_2(124,35) ] \} \,.
\nonumber
\label{d5r}
\end{eqnarray}
This is inserted into the integral equation (\ref{inteq5}).
We insert into that equation
the expressions (\ref{d3}), (\ref{d4ri}), (\ref{d4r}) for $D_3$ and $D_4$
as well. In order to find the new integral equation for $D_5^I$ we
have to simplify and collect all terms not involving $D_5^I$ and
$D_4^I$. These terms will contribute to the inhomogeneous term
of the new equation for $D_5^I$ and we will now discuss them.
From the left--hand side of (\ref{inteq5}) we get
$\omega D_5^R$, which can be treated using the
BFKL equation (\ref{inteq2}). Due to this, the inhomogeneous term
$D_{(5;0)}$ in (\ref{inteq5}) is exactly cancelled and we get
further terms involving only convolutions of $D_2$ functions
with kernels $K_{2 \rightarrow 2}$ or trajectories $\beta$.
From the right--hand side of
(\ref{inteq5}) we get contributions of the type
$K_{2 \rightarrow 5} \otimes D_2$,
$\sum K_{2 \rightarrow 4} \otimes D_3$, $\sum K_{2 \rightarrow 3} \otimes D_4^R$,
and $\sum K_{2 \rightarrow 2} \otimes D_5^R$. All of these can be written
as sums of convolutions of $D_2$ with the kernels $K_{2 \rightarrow m}$.
The corresponding contractions of
color tensors are performed using the diagrammatic method described
in appendix \ref{colorapp}. In that appendix we also give the
explicit formulae for some of the contractions required.
The total number
of contractions needed here is close to one hundred, and they can be
easily obtained from those in the appendix.
The momentum integrals are brought
to their standard forms as classified in section \ref{standardint}.
The respective momentum integrals and color contractions
are then multiplied and can be collected.
This last step amounts to collecting several thousand terms
and sorting them according to the different color tensors,
and we do this with the help of a computer algebra program.

In the derivation of the new integral equation
the terms involving $D_4^I$ remain unchanged.
They will be treated at a later stage of the analysis.
The same is true for the homogeneous term containing $D_5^I$.
The combinations of the $D_4^I$ and $D_5^I$ amplitudes with
the kernels are therefore the same as in the
original equation (\ref{inteq5}).
We thus find the following equation for the unknown part $D_5^I$
of the five--gluon amplitude:
\begin{eqnarray}
 \left(\omega - \sum_{i=1}^5 \beta({\bf k}_i)\right) D_5^I &=&
 \sum   f_{a_1a_2a_3} \delta_{a_4a_5} H(1,2,3;4,5)
 + \sum K^{ \{b\} \rightarrow \{a\} }_{2 \rightarrow 3} \otimes D_4^{I\,b_1b_2b_3b_4}
\nonumber \\
&&
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2 \rightarrow 2} \otimes D_5^{I\,b_1b_2b_3b_4b_5}
\,.
\label{inteq5i}
\end{eqnarray}
The first term on the right hand side is the result of the computation
described above. We will now treat it in more detail.

The first interesting observation concerns its color structure.
All terms proportional to
$f^{a_1a_2a_3a_4a_5}$ (and the other three permutations of this
occurring in (\ref{d5r}))
are cancelled between the different contributions to this inhomogeneous
term and drop out. Something similar happened in the case of $D_4$
where there is no term proportional to $d^{a_1a_2a_3a_4}$
in the vertex $V_{2 \rightarrow 4}$ and only lower tensors (i.\,e.\ products
of $\delta$-tensors) contribute.

Secondly, we observe the following symmetry of the new inhomogeneous
term that we have calculated.
The sum extends over all (ten)
possibilities to have a pair of gluons in a color singlet.
For each of these permutations of the gluons color and momentum
labels are exchanged simultaneously, i.\ e.\ the sum in
(\ref{inteq5i}) stands for
\begin{eqnarray}
  \sum  f_{a_1a_2a_3} \delta_{a_4a_5} H(1,2,3;4,5)
 &=& f_{a_1a_2a_3} \delta_{a_4a_5} H(1,2,3;4,5) +
 f_{a_1a_2a_4} \delta_{a_3a_5} H(1,2,4;3,5) \nonumber \\
&& + \dots
 +  \delta_{a_1a_2} f_{a_3a_4a_5} H(3,4,5;1,2)
\,.
\label{explsumoverH}
\end{eqnarray}
The function $H$ is the same in all ten permutations.
This symmetry is an outcome of our computation, and it has not
been used to derive (\ref{inteq5i}). On the other hand,
it is not an unexpected property
of the inhomogeneous term. Already in the corresponding
equation (\ref{inteq4i}) in the four--gluon case the
inhomogeneous term, i.\,e.\ the vertex $V_{2 \rightarrow 4}$,
had a similar symmetry.

A closer inspection of the function $H$ reveals that it is actually
a superposition of vertex functions $V$ which we encountered in the
discussion of the two--to--four vertex $V_{2 \rightarrow 4}$.
Namely,
\begin{equation}
  H(1,2,3;4,5) = \frac{g}{2}
 [(VD_2)(12,3;4,5) - (VD_2)(13,2;4,5) + (VD_2)(1,23;4,5)]
\,.
\label{h=sumv}
\end{equation}
To obtain this striking result is was necessary to go through
the full calculation of all convolutions of amplitudes with kernels
as described. It is only afterwards that we are able to discover the
simple structure in terms of $V$.
Unfortunately, we do not know a way leading
to (\ref{inteq5i}), (\ref{h=sumv}) that avoids this tedious
calculation.

\subsection{Solving the equation for the remaining part}
\label{solving5i}
Up to this point, our analysis of the five--gluon amplitude
followed essentially the same lines as in the case of four
gluons. Whereas there the new integral equation could
simply be iterated, this is not possible here.
To find the solution for $D_5^I$ we now have to go beyond the
procedure applied for $n=3$ and $4$ gluons.

Taking a close look at the integral equation (\ref{inteq5i})
for $D_5^I$ we discover that its structure bears a strong
resemblance to the equation (\ref{inteq3}) for the three--gluon
amplitude $D_3$.
In the second term on the right hand side
of (\ref{inteq5i}) a pair of gluons of the amplitude $D_4^I$ is
convoluted with a
two--to--three kernel. In the corresponding term in
(\ref{inteq3}) it was the two--gluon (BFKL) amplitude $D_2$ that
was convoluted with the same kernel. There the first
term on the right hand side, i.\,e.\ $D_{(3;0)}$,
was the superposition of quark loop amplitudes $D_{(2;0)}$
that are the lowest order terms in the ladder
expansion of $D_2$. In (\ref{inteq5i}) the corresponding
term (\ref{explsumoverH}) is, according to (\ref{h=sumv}),
the superposition of functions $VD_2$. (In fact it is even
the superposition of full two--to--four reggeon vertices
$V_{2 \rightarrow 4}D_2$ as we will see below.)
These vertex functions, in turn, constitute the lowest
order terms in the ladder expansion\footnote{This statement
has to be taken with some care, since the terms $VD_2$ are
of course not of lowest order in the coupling constant $g$.
In the contrary, $D_2$ already contains an infinite series
of ladder diagrams. What is meant here is that each diagram
in the ladder expansion of $D_4^I$ starts with a full two--gluon ladder
and a vertex attached to this.}
of the irreducible
amplitude $D_4^I$, cf.\ (\ref{d4istruct}).
More specifically, it is for each of the ten terms in (\ref{explsumoverH})
that we find in a three--gluon subsystem exactly the
momentum structure that also determines $D_{(3;0)}$.
The similarity is also evident for the color structure,
namely the three--gluon subsystem comes with
a tensor $f_{abc}$.

Clearly, this suggests to construct a solution $D_5^I$
in analogy with the three--gluon amplitude.
While $D_3$ is
a superposition of BFKL amplitudes $D_2$
we now should choose  $D_5^I$ as a similar superposition of irreducible
four--gluon amplitudes $D_4^I$.
The following combination of $D_4^I$ amplitudes is of this kind
and in fact is a solution to equation (\ref{inteq5i}).
We will outline the proof of this fact momentarily.
\begin{eqnarray}
\lefteqn{D_5^{I\,a_1a_2a_3a_4a_5}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5) =
 \frac{g}{2} \times} \nonumber \\
&&\times \left\{ f_{a_1a_2c} D_4^{I\,ca_3a_4a_5}(12,3,4,5)
+ f_{a_1a_3c} D_4^{I\,ca_2a_4a_5}(13,2,4,5) \right. \nonumber \\
&& \hspace{.5cm}
+ \,f_{a_1a_4c} D_4^{I\,ca_2a_3a_5}(14,2,3,5)
+ f_{a_1a_5c} D_4^{I\,ca_2a_3a_4}(15,2,3,4) \nonumber \\
&& \hspace{.5cm}
+ \,f_{a_2a_3c} D_4^{I\,a_1ca_4a_5}(1,23,4,5)
+ f_{a_2a_4c} D_4^{I\,a_1ca_3a_5}(1,24,3,5) \nonumber \\
&& \hspace{.5cm}
+\, f_{a_2a_5c} D_4^{I\,a_1ca_3a_4}(1,25,3,4)
+ f_{a_3a_4c} D_4^{I\,a_1a_2ca_5}(1,2,34,5) \nonumber \\
&& \hspace{.5cm} \left.
+ \,f_{a_3a_5c} D_4^{I\,a_1a_2ca_4}(1,2,35,4)
+ f_{a_4a_5c} D_4^{I\,a_1a_2a_3c}(1,2,3,45)
\right\}
\label{d5isolution}
\end{eqnarray}
In each of the terms one pair $(i,j)$ of gluons is
merged\footnote{Depending on the context one would like to use different
words for the formula (\protect\ref{d5isolution}).
From the point of view of constructing the  solution
it is clearly a 'merging' of two gluons, with the concept of a
$t$-channel evolution in mind one would prefer to speak of a 'splitting'
of gluons.}
into one gluon which then enters the irreducible four--gluon
amplitude from below.
This gluon in $D_4^I$ has momentum $({\bf k}_i + {\bf k}_j)$ and color label $c$.
The merging of the two gluons in color space happens via
a $f_{a_ia_jc}$ tensor ($i<j$).
The position in the amplitude $D_4^I$ at which the 'composite'
gluon with color $c$
and momentum $({\bf k}_i + {\bf k}_j)$ enters does not matter since $D_4^I$ is
completely symmetric in the four gluons, cf.\ (\ref{complsymmd4i}).
All possible pairs of gluons are treated in the same way.
The way pairs of gluons are merged (or arise from splittings)
becomes more transparent when (\ref{d5isolution}) is written
using birdtrack notation,
\begin{eqnarray}
\lefteqn{D_5^{I\,a_1a_2a_3a_4a_5}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5) =
 \frac{g}{2} \times} \nonumber \\
&&\times \Bigg\{ \left[ \, \picbox{5an4f12.pstex_t} \right]
 \star D_4^{I\,b_1b_2b_3b_4}(12,3,4,5) +
\left[ \, \picbox{5an4f13.pstex_t} \right]
 \star D_4^{I\,b_1b_2b_3b_4}(13,2,4,5) \nonumber \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f14l.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(14,2,3,5)
+ \left[ \, \picbox{5an4f15l.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(15,2,3,4) \nonumber \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f23.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(1,23,4,5)
+ \left[ \, \picbox{5an4f24l.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(1,24,3,5) \nonumber \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f25l.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(1,25,3,4)
+ \left[ \, \picbox{5an4f34.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(1,2,34,5) \nonumber \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f35l.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(1,2,35,4)
+ \left[ \, \picbox{5an4f45.pstex_t} \right]
\star D_4^{I\,b_1b_2b_3b_4}(1,2,3,45) \Bigg\}
\,.
\label{d5isolutiondiag}
\end{eqnarray}
We will come to the interpretation of this structure
in section \ref{interpret5i}.

Now we explain how (\ref{d5isolution}) can be
shown to solve the integral equation (\ref{inteq5i}).
The only pieces of information about the irreducible four--gluon
amplitude $D_4^I$ we need for the purpose of this proof are its complete
symmetry in the four outgoing gluons, cf.\ (\ref{complsymmd4i}),
and the integral equation (\ref{inteq4i}) it fulfills. Fortunately,
an analytic solution of the latter is not required.

We start from the integral equation (\ref{inteq5i}) derived previously
and insert the conjectured solution (\ref{d5isolution}). On the left
hand side we then find ten terms of the kind $\omega D_4^I$. To these
we apply (\ref{inteq4i}). Thereby we produce different terms and
we first concentrate on the terms involving the
vertex function $V$. For example, applying (\ref{inteq4i}) to
\begin{equation}
\omega f_{a_1a_2c} D_4^{I\,ca_3a_4a_5}(12,3,4,5)
\label{onetermind5ilhs}
\end{equation}
produces, due to (\ref{colV}), the expression
\begin{eqnarray}
f_{a_1a_2c} (V_ {2 \rightarrow 4}D_2)^{ca_2a_3a_4} (12,3,4,5) &=&
 f_{a_1a_2a_3} \delta_{a_4a_5} (VD_2)(12,3;4,5) \nonumber \\
&&+ f_{a_1a_2a_4} \delta_{a_3a_5} (VD_2)(12,4;3,5) \nonumber \\
&&+ f_{a_1a_2a_5} \delta_{a_3a_4} (VD_2)(12,5;3,4)
\end{eqnarray}
containing three different vertex functions $V$.
Similar expressions are obtained from the other $\omega D_4^I$
terms on the left hand side. In some cases a minus sign arises due to
the antisymmetry of the structure constant $f_{abc}$. For instance,
from the second term in (\ref{d5isolution}),
\begin{equation}
 f_{a_1a_3c} D_4^{I\,ca_2a_4a_5}(13,2,4,5)
\,,
\end{equation}
in which the pair (1,3)
of gluons is merged we get
\begin{eqnarray}
&&- f_{a_1a_2a_3} \delta_{a_4a_5} (VD_2)(13,2;4,5)
+ f_{a_1a_3a_4} \delta_{a_2a_5} (VD_2)(13,4;2,5) \nonumber \\
&& \hspace{4mm}
+ f_{a_1a_3a_5} \delta_{a_2a_4} (VD_2)(13,5;2,4)
\,.
\end{eqnarray}
Therefore we find exactly the same
thirty vertex functions that occur also on the
right hand side of (\ref{inteq5i}) according
to (\ref{explsumoverH}),(\ref{h=sumv}).
We have thus confirmed that the conjectured solution (\ref{d5isolution})
indeed reproduces the correct lowest order term in the integral equation,
namely the combination of vertex functions $VD_2$ given above.
Moreover, we see that the first term on the right hand side of
the integral equation (\ref{inteq5i})
is not only a superposition of vertex functions $VD_2$ but of full
transition vertices $V_ {2 \rightarrow 4}$ (applied to $D_2$ as usual).

Let us now consider further terms in the integral equation (\ref{inteq5i})
that we have not treated yet, namely those
involving the irreducible four--gluon amplitude $D_4^I$.
Having applied (\ref{inteq4i}) to the $\omega D_4^I$ terms
on the left hand side the homogeneous term of that equation
produces convolutions of $D_4^I$ amplitudes with kernels
$K_{2 \rightarrow 2}$. In these, a kernel acts on $D_4^I$ first
and then the splitting of one gluon into a pair happens according
to the combinations in (\ref{d5isolution}).
On the right hand side of the integral equation (\ref{inteq5i}) the last
term also gives us convolutions of $D_4^I$ amplitudes with kernels
$K_{2 \rightarrow 2}$, but here the order of the convolution and
the splitting of gluons is interchanged: first one gluons splits into
two and then two of the now five gluons interact via a kernel
$K_{2 \rightarrow 2}$.
Among the terms just mentioned a subclass cancels immediately.
Consider the case that the two--to--two kernel acts between two gluons
none of which undergoes a splitting (LHS) or has emerged from a
splitting (RHS). Then the order of interaction and splitting along
the $t$-channel evolution is irrelevant
and these terms are in fact the same on both sides.

The next terms in the integral equation that we look at are the products
of $D_4^I$'s with trajectories $\beta$. These arise on the
left hand side either from $\omega D_4^I$ via (\ref{inteq4i}) or from
the original $\left[ \sum_{i=i}^5 \beta({\bf k}_i)\right] D_5^I$
after (\ref{d5isolution})
is inserted. Those in which the argument of the trajectory function
does not correspond to a gluon undergoing or arising from a splitting
cancel directly between these contributions. It can be easily checked that
the others are exactly cancelled by the terms from the right hand side
in which the two gluons emerging from a splitting interact with each
other via a kernel $K_{2 \rightarrow 2}$.

It is a bit more complicated to study the expressions still left in the
integral equation after the cancellations discussed so far.
These are $D_4^I$'s undergoing a two--to--three transition
via the kernel $K_{2 \rightarrow 3}$ on the RHS and
convolutions of $D_4^I$ functions
with kernels $K_{2 \rightarrow 2}$
in which one of the gluons undergoing or emerging from a
splitting is involved in the interaction (both sides). The latter
do no longer include such convolutions in which the
interaction is between the two gluons emerging from the
splitting. In the terms under consideration three of the five
outgoing gluons participate in the splitting or in the interaction.
The other two gluons do not interact and can be in an arbitrary
color state. Among the five gluons there can be a total of ten
different three--gluon subgroups, and we will argue that the cancellation
takes place in each of these subgroups separately. To this end let us
concentrate on one of these subgroups, say the one with the first three
of the outgoing gluons affected.

The mechanism that makes these contributions cancel between the
two sides of the integral equation is the same that already caused
reggeization in the three--gluon amplitude $D_3$.
This does not come as a surprise since
it was just the similarity of the corresponding integral equations that
lead us to the ansatz (\ref{d5isolution}).
The identity actually bringing about the reggeization of the
three--gluon subsystem is in pictorial language
\begin{eqnarray}
 \picbox{reggel1.pstex_t} + \picbox{reggel2.pstex_t}
+ \picbox{reggel3.pstex_t} &=&
 \frac{2}{g}\,\picbox{regger1.pstex_t} + \picbox{regger2.pstex_t}
+ \picbox{regger3.pstex_t}
\nonumber \\
&& + \picbox{regger4.pstex_t}+ \picbox{regger5.pstex_t}
+ \picbox{regger6.pstex_t}+ \picbox{regger7.pstex_t}
\,.
\label{reggeizebilder}
\end{eqnarray}
Only the three--gluon subsystem is shown, and the horizontal lines
at the top are meant to suggest the irreducible amplitude $D_4^I$
that the gluons enter. The arrows indicate the symmetry of this
amplitude under the simultaneous exchange of color and momentum
of the two gluons. The kernels are the ones defined in
section \ref{kernels}.

The splitting of a gluon is depicted here by the corresponding
color diagram and is meant to indicate the behavior in
momentum space\footnote{Strictly speaking, this
is done in abuse of our notation that usually separates momentum
and color space. Confusion should hardly be possible here as
all terms have been described in detail before.} as well.
The terms on the left (right) hand side of (\ref{reggeizebilder})
are exactly the ones that occur on the left (right) hand side of
the integral equation (\ref{inteq5i}).
To prove (\ref{reggeizebilder}), the convolutions
are evaluated as described in the preceding sections.
However, here the situation is slightly complicated by the fact that
the two gluons entering from above can be in an arbitrary color state.
In the case of $D_3$ these two gluons were in a color singlet state,
effectively reducing all color tensors to an overall $f_{a_1a_2a_3}$.
Here we have to be more careful and treat three independent color classes
separately. (Of course, this could have been done already for $D_3$
but there it was not necessary.) The three classes are
\begin{equation}
 \picbox{color2to3.pstex_t}\,,
 \qquad\qquad \picbox{color2to3ex12.pstex_t}\,,
 \qquad \qquad \picbox{color2to3ex23.pstex_t}
\,.
\label{threecolors}
\end{equation}
For some of the terms in (\ref{reggeizebilder}) it is necessary to
use the symmetry in the upper two gluons which
is a property of $D_4^I$.
Each of the three terms on the left hand side of
(\ref{reggeizebilder}) contributes to two of the three color classes
in (\ref{threecolors}) via the Jacobi identity (\ref{Jacobi}).
Having dissected the integral equation this far, it is finally a
comparatively short calculation to check that (\ref{reggeizebilder})
holds. Thereby we have finished
the proof that $D_5^I$, as given in (\ref{d5isolution}), in fact is
the solution of the integral equation.

\subsection{Interpretation of the result}
\label{interpret5i}

In the preceding section we have been able to solve the
equation for the five--gluon amplitude.
Now we want to interpret our findings in view of a possible
field theory of unitarity corrections.
Let us first summarize the essential results we have obtained.
We have split the five--gluon amplitude into
two parts. The first part was the reggeizing part $D_5^R$
that is the superposition of two--gluon amplitudes $D_2$.
We have found an integral equation for the remaining
part $D_5^I$ and have solved it. It turned
out that the remaining part is the superposition of
irreducible four--gluon amplitudes $D_4^I$.
Neglecting all normalization factors and color tensors,
this situation can be sketched in the following way:
\begin{equation}
  D_5({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5) =
\sum
 \picbox{solutiond51.pstex_t} + \sum \picbox{solutiond52.pstex_t}
\,.
\label{solutiond5diag}
\end{equation}
The first term on the right hand side is the reggeizing
part $D_5^R$ of the amplitude (see (\ref{d5r})). The
sum extends over all partitions of the five gluons into
two groups.
The second term is the one found in the
preceding section, see (\ref{d5isolution}).
Here the sum includes all possible
pairs of gluons that then merge into one.

In the five--gluon amplitude,
we only find elements that are already known:
the amplitudes $D_2$ and $D_4^I$, and the
two--to--four transition vertex $V_ {2 \rightarrow 4}$.
There is no new transition vertex and no new irreducible amplitude
that would include a five--gluon compound state.
The absence of new elements is an intriguing result.
It shows that the five--gluon amplitude reggeizes
completely. This result (\ref{solutiond5diag})
clearly constitutes a generalization of the concept
of reggeization and proves that reggeization also takes place in
more complicated amplitudes.
Especially interesting is the reggeization
in the second part that turns out to be a superposition
of irreducible four--gluon amplitudes.
The mechanism at work here is exactly
the same as in the three--gluon amplitude.

Reggeization was an important prerequisite
for the emergence of the two--to--four
vertex and thus of the field theory structure
in the four--gluon amplitude. That the phenomenon
of reggeization also occurs in the five--gluon amplitude
gives us confidence that
the idea of a field theory structure will be a good
guiding line also for the investigation of the six--gluon
amplitude.

Given that the three-- and five--gluon
amplitudes exhibit complete reggeization caused by the
same mechanism, one is
naturally lead to the question whether the same
is true for each odd number $n$ of gluons. Indeed, as we have seen,
the mechanism leading to reggeization in a three--gluon
subsystem is very general.  It is completely independent of the
structure of the quark loop that we started with
in the analysis of the integral equations.
In deriving (\ref{reggeizebilder})
we only made use of the fact that the amplitude to which the
upper two gluons are attached --- in this case $D_4^I$ ---
is symmetric in the two gluons. Therefore we can conclude
that for a given odd $n$ one important condition for
the reggeization of the $n$-gluon amplitude is
fulfilled as soon as the irreducible part of the $(n-1)$-gluon
amplitude is symmetric.
However, we have to keep in mind that this is only one
of the two conditions leading to complete reggeization.
The second condition necessary for the reggeization
of $D_5^I$ was that the inhomogeneous term in
the integral equation (\ref{inteq5i}) had the
specific form (\ref{h=sumv}), i.\,e.\ could be written
as a special superposition of transition vertices
$V_{2 \rightarrow 4}$.
For a part of an arbitrary $n$-gluon
amplitude with odd $n$ to reggeize it is obviously
necessary that the respective
inhomogeneous term has a very specific form.
With our present knowledge, we are not able to derive this
specific form of the inhomogeneous term for general $n$.
The complete reggeization of
amplitudes with an odd number of gluons
is therefore at present only a (plausible) conjecture.

We would like to add a remark that concerns our choice of notation,
but actually goes beyond a pure issue of notation.
In the splitting of the four--gluon amplitude into two parts
in (\ref{d4ri}) the superscript $I$ in $D_4^I$ was meant to
indicate that this part of the amplitude is irreducible ---
in contrast to the other part. For four gluons this was
a good choice of notation since that part in fact contains
a new irreducible compound state of four reggeized gluons.
In the case of five gluons, we again split the amplitude
into two parts, cf.\ (\ref{d5ri}). The first part $D_5^R$
is a reggeizing one as the superscript $R$ indicates.
But now we have discovered that the remaining part
$D_5^I$ reggeizes as well. (For this reason we have
avoided to call this part 'irreducible'.)
Nevertheless our notation makes perfect sense when
extended in an appropriate way.
The first superscript $I$ or $R$ should be understood
as specifying the (non--)reggeization of the respective
part with respect to the two--gluon state.
We can then introduce a second superscript to indicate
the (non--)reggeization with respect to the four--gluon
state. (We will actually be forced to do so when considering
the six--gluon amplitude in the next section.)
For a proper notation we should thus identify
$D_5^I = D_5^{I,\,R}$,
the second superscript now indicating that this part
reggeizes with respect to the four--gluon state.
The notation can easily be extended to accommodate
reggeization with respect to a potential six--gluon state
or even higher compound states.

\section{Six gluons}
\label{d6}

\subsection{A reggeizing part}
\label{regge6}

Encouraged by the success we have had so far with
that procedure we again
use the quark loop amplitude $D_{(6;0)}$ to construct
from it a reggeizing part $D_6^R$ as a superposition
of two--gluon (BFKL) amplitudes. The full six--gluon
amplitude is then split into two parts,
\begin{equation}
 D_6 = D_6^R + D_6^I
\,,
\label{d6split}
\end{equation}
and it will be our first task to find a new integral
equation for the remaining part $D_6^I$. In detail,
the reggeizing part $D_6^R$ is
\begin{eqnarray}
\lefteqn{D_6^{R\,a_1a_2a_3a_4a_5a_6}
   ({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5,{\bf k}_6)= }
\\
&=&  g^4 \{ d^{a_1a_2a_3a_4a_5a_6} \,[
  D_2(12345,6) + D_2(1,23456) - D_2(16,2345) ] \nonumber \\
& &\,+\,d^{a_2a_1a_3a_4a_5a_6} \,[
  D_2(13456,2) - D_2(1345,26)
+ D_2(126,345) -  D_2(12,3456) ] \nonumber \\
& &\, +\,d^{a_1a_2a_3a_4a_6a_5} \,[
 D_2(12346,5) - D_2(1234,56)
+ D_2(156,234) - D_2(15,2346) ] \nonumber \\
& &\, +\,d^{a_2a_1a_3a_4a_6a_5} \,[
 - D_2(1256,34) - D_2(1346,25)
 + D_2(125,346) + D_2(134,256) ] \nonumber \\
& &\, +\,d^{a_3a_1a_2a_4a_5a_6} \,[
 D_2(12456,3) - D_2(1245,36)
 + D_2(136,245) - D_2(13,2456) ] \nonumber \\
& &\,+\,d^{a_1a_2a_3a_5a_6a_4} \,[
 D_2(12356,4) - D_2(1235,46)
+ D_2(146,235) - D_2(14,2356) ] \nonumber \\
& &\, +\,d^{a_2a_1a_3a_5a_6a_4} \,[
- D_2(1246,35) - D_2(1356,24)
+ D_2(124,356) + D_2(135,246) ] \nonumber \\
& &\, +\,d^{a_1a_2a_3a_6a_5a_4} \,[
- D_2(1236,45) - D_2(1456,23)
+ D_2(123,456)
+ D_2(145,236) ]
\}
\nonumber
\label{d6r}
\end{eqnarray}
as obtained from (\ref{d60}) by the replacement
$D_{(2;0)} \rightarrow D_2$ while keeping the color
and momentum structure.
This expression already indicates one of the major
difficulties we have to overcome during the treatment of
the six--gluon amplitude: the large number of
terms we have to take care of.

\subsection{The integral equation for the remaining part}
\label{inteq6isection}

The original integral equation (\ref{inteq6}) for the
six--gluon amplitude is now used to
derive a new integral equation for the unknown part $D_6^I$.
The method in this step is exactly the same as for the
four-- and five--gluon amplitudes. We
insert into the integral equation our complete knowledge
about the reggeizing parts $D_n^R$
of the amplitudes $D_n$ with up to
$n=6$ gluons, including the ansatz (\ref{d6split}), (\ref{d6r})
for the six--gluon amplitude.
The corresponding formulae for $n\le5$ can be found in the
preceding sections.
Then we apply the BFKL equation
(\ref{inteq2}) to the expression $\omega D_6^R$
on the left hand side. This is possible because
$D_6^R$ was chosen as a superposition of BFKL amplitudes,
cf.\ (\ref{d6r}).
We thereby produce convolutions of $D_2$
amplitudes with two--to--two
kernels and products of $D_2$ amplitudes
with trajectory functions $\beta$.
The insertion of the reggeizing parts $D_n^R$ of the
amplitudes on the right hand side leads to convolutions
of $D_2$ amplitudes with the integral kernels.
We have to perform the corresponding contractions
of color tensors and have to bring the integrals
to their standard forms as classified in section \ref{standardint}.
The main problem consists in the huge
number of combinations of amplitudes with kernels.
We have to perform close to $250$ contractions
of color tensors, and we have to find the standard form of
more than $3500$ integrals. Whereas the color tensors
can still be calculated by hand this is no longer possible
for the huge number of momentum space integrals.
We have therefore developed an algorithm for this purpose
that is suited for the implementation on a computer.
The algorithm is explained in detail in appendix
\ref{momentumapp}. We have written a PERL script
based on this algorithm that produces an output which
can directly be used as an input for a computer algebra
program like MAPLE. The tensor contractions are calculated
with the help of the method described in appendix \ref{colorapp}.
Some of the contractions are given explicitly in that
appendix. Many other contractions are obtained from
these by permutations of the gluon color labels.
The computer algebra program is then used to multiply
the resulting sums of elementary tensors with the
corresponding integrals, and to finally collect all terms.
In the final step more than $2 \cdot 10^4$ integrals
have to be sorted according to their color tensor coefficients.
(This shows that our method of dealing with the integral
equations will in its practical applicability be limited
to relatively small numbers $n$ of gluons.)
Having collected all terms in the equation which contain
the amplitude $D_2$  we have found the inhomogeneous term
of the new integral equation for $D_6^I$.

In the derivation of the new integral equation the terms containing
the irreducible four--gluon amplitude $D_4^I$ and the second part
$D_5^I$ of the five--gluon amplitude remain unchanged.
Their combinations with the kernels are the same as in the
original equation (\ref{inteq6}).
The resulting integral equation for $D_6^I$
is then found to have the form
\begin{eqnarray}
\lefteqn{
\left(\omega - \sum_{i=1}^6 \beta({\bf k}_i)\right)
D_6^{I\,a_1a_2a_3a_4a_5a_6}({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5{\bf k}_6)
= } \nonumber \\
&=&
 ( W^{a_1a_2a_3a_4a_5a_6}D_2)({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5{\bf k}_6)
\nonumber \\
&&
 + \sum f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6) \nonumber \\
&&
 + \sum d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6)
\nonumber\\
&&
 + \sum d^{a_2a_1a_3a_4} \delta_{a_5a_6} J(1,2,3,4;5,6) \nonumber \\
&&
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2 \rightarrow 4} \otimes D_4^{I\,b_1b_2b_3b_4}
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2 \rightarrow 3}\otimes D_5^{I\,b_1b_2b_3b_4b_5}
\nonumber \\
&&
+ \sum K^{ \{b\} \rightarrow \{a\} }_{2 \rightarrow 2}
\otimes D_6^{I\,b_1b_2b_3b_4b_5b_6}
\,.
\label{inteq6i}
\end{eqnarray}
The first four terms on the right hand side are the result of the
computation outlined above. We will now describe them in detail.

The first observation we make is again that certain color structures
are completely cancelled in the equation. All terms proportional
to $d^{a_1a_2a_3a_4a_5a_6}$ (and the other seven permutations
of this occurring in (\ref{d6r})) are cancelled between the different
contributions to the inhomogeneous term and drop out. The
same was observed
for the tensors $d^{a_1a_2a_3a_4}$ and $f^{a_1a_2a_3a_4a_5}$
in the equations for the parts
$D_4^I$ and $D_5^I$ of the four-- and five--gluon amplitudes,
respectively.

As in the integral equations for $D_4^I$ and $D_5^I$ the
inhomogeneous term has a high degree of symmetry which
we will explain for each of the terms below.
This symmetry is not only nice by itself, but it is also a
possibility to check our calculation.

The first term on the right hand side of (\ref{inteq6i})
differs in its structure from the other terms and will be treated
separately in section \ref{v26}. Here we mention already that
it is symmetric in the sense that it is the sum of terms that
are obtained from each other by permutations of the gluons.

The same is true for the second term on the right hand side
of the new integral equation. The sum extends over all
partitions of the six gluons into two groups each of which contains
three gluons,
\begin{eqnarray}
\sum f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6) &=&
f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6)
\nonumber \\
&& + \, f_{a_1a_2a_4} f_{a_3a_5a_6} L(1,2,4;3,5,6)
\nonumber \\
&& + \dots +  f_{a_1a_5a_6} f_{a_2a_3a_4} L(1,5,6;2,3,4)
\,.
\label{explsumoverL}
\end{eqnarray}
The function $L$ is the same in all terms in the sum
and only its arguments are exchanged in the different
terms.
A closer inspection reveals that the function $L$
permits a decomposition into
vertex functions $V$ known from the two--to--four
transition vertex (see section \ref{34V}),
\begin{eqnarray}
\lefteqn{
L(1,2,3;4,5,6) = \frac{g^2}{4} \times } \nonumber \\
&&\times [(VD_2)(12,3;45,6) - (VD_2)(12,3;46,5)
+ (VD_2)(12,3;4,56) \nonumber \\
&&\hspace{.5cm}
- (VD_2)(13,2;45,6) + (VD_2)(13,2;46,5) - (VD_2)(13,2;4,56) \nonumber \\
&&\hspace{.5cm}
+ (VD_2)(1,23;45,6) - (VD_2)(1,23;46,5)
+ (VD_2)(1,23;4,56) ]
\,.
\label{l=sumv}
\end{eqnarray}

The sum in the third term on the right hand side of the
integral equation extends over all partitions of the six gluons
into one group containing four and one group containing
two gluons,
\begin{eqnarray}
\sum d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6) &=&
d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6)
\nonumber \\
&&+ \,d^{a_1a_2a_3a_5}\delta_{a_4a_6} I(1,2,3,5;4,6)
\nonumber \\
&&+ \dots + \delta_{a_1a_2} d^{a_3a_4a_5a_6} I(3,4,5,6;1,2)
\,.
\label{explsumoverI}
\end{eqnarray}
Also in this case we find that the function $I$ is the
same in all terms in the sum. Remarkably, also this function
can be written in terms of the vertex function $V$,
\begin{equation}
I(1,2,3,4;5,6) = - g^2 [
(VD_2)(1,234;5,6) + (VD_2)(123,4;5,6) - (VD_2)(14,23;5,6) ]
\,.
\label{i=sumv}
\end{equation}
The sum in the fourth term on the right hand side of
the new integral equation (\ref{inteq6i}),
\begin{equation}
\sum d^{a_2a_1a_3a_4} \delta_{a_5a_6} J(1,2,3,4;5,6)
\,,
\label{explsumoverJ}
\end{equation}
extends over the same permutations of gluons as the
term discussed before (see (\ref{explsumoverI})).
Again the function $J$ is the same in all terms in the sum,
and it can be written as a superposition
of vertex functions $V$ as
\begin{eqnarray}
J(1,2,3,4;5,6) &=& - g^2 [
(VD_2)(134,2;5,6) + (VD_2)(124,3;5,6) \nonumber \\
&&
\hspace{.8cm} - (VD_2)(12,34;5,6) - (VD_2)(13,24;5,6)]
\,.
\label{j=sumv}
\end{eqnarray}

We would like to emphasize that the symmetry of the
sums contributing to the inhomogeneous term of the
new integral equation is an outcome of our calculation.
We have not used it to derive the new equation. That
we find the symmetry in the resulting equation gives
us confidence that we did not make any errors in the
long and tedious calculation leading to (\ref{inteq6i}).
We also would like to stress that the representation
of a part of the inhomogeneous term as a superposition
of well--known
vertex function $V$ is an outcome of our calculation and
was not used to derive the new equation. Unfortunately, we
do not know a way that directly leads to the comparatively
simple structure arising in the terms discussed above.

\subsection{A new piece in the field theory}
\label{v26}

The first term on the right hand side of
the integral equation (\ref{inteq6i}) differs in its
structure from the other terms.  We will therefore
discuss it separately in this section. The study of the other terms
will be resumed in section \ref{d6ir}.
We start by giving an explicit representation of the term
under consideration and proceed by listing its properties.
After that we will
elaborate on the question how the new term
has to be interpreted in the context of the effective
field theory structure of unitarity corrections.

\subsubsection{Explicit representation and properties}
\label{expl26}

The first term on the right hand side in (\ref{inteq6i})
has the following color and momentum structure:
\begin{equation}
 (W^{a_1a_2a_3a_4a_5a_6}D_2)({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5,{\bf k}_6) =
\sum   d_{a_1a_2a_3} d_{a_4a_5a_6} (W D_2)(1,2,3;4,5,6)
\,.
\label{newpiece}
\end{equation}
The sum extends over all (ten) partitions of the six gluons
into two groups containing three gluons each,
\begin{eqnarray}
\sum   d_{a_1a_2a_3} d_{a_4a_5a_6} (W D_2)(1,2,3;4,5,6)
&=& d_{a_1a_2a_3} d_{a_4a_5a_6} (W D_2)(1,2,3;4,5,6)
\nonumber \\
&&+ \, d_{a_1a_2a_4} d_{a_3a_5a_6} (W D_2)(1,2,4;3,5,6)
+ \dots
\nonumber \\
&&
+ \,d_{a_1a_5a_6} d_{a_2a_3a_4} (W D_2)(1,5,6;2,3,4)
\,.
\label{explsumoverw}
\end{eqnarray}
The function $W D_2$ is the same in all permutations.
(Again this is an outcome of our calculation and was not
assumed at any stage when the equation (\ref{inteq6i})
was derived.)
The sum thus contains the same permutations of the
six gluons as the second
term (\ref{explsumoverL}) on the right hand
side of (\ref{inteq6i}).
For the notation to be consistent
the function $W^{a_1a_2a_3a_4a_5a_6}$ should be
understood as an integral operator acting on a BFKL
amplitude $D_2$. It thus carries two more momentum
arguments ${\bf q}_j$ for the momenta entering from
above. We will again suppress these two momenta in
the following.

In contrast to the other
terms in (\ref{inteq6i})
discussed so far the function $W$ does not permit
a decomposition into the vertex functions $V$ known from the
two--to--four transition vertex.
We therefore give its full momentum space representation
as we have obtained it as a result of our calculation. We again use the
standard integrals defined in section \ref{standardint}.
Then $WD_2$ has the explicit representation
\begin{eqnarray}
\lefteqn{(WD_2)({\bf k}_1,{\bf k}_2,{\bf k}_3;{\bf k}_4,{\bf k}_5,{\bf k}_6) =
  \frac{g^6}{16} \,\times } \nonumber \\
&& \{  2 \,[ \,  c(123456) \nonumber \\
&&\quad\, -\,\, b(12345) - b(12346) - b(12356)- b(12456)- b(13456)
- b(23456) \nonumber \\
&&\quad\, +\, b(1234) + b(1235) + b(1236) + b(1456) + b(2456) + b(3456)
\nonumber \\
&&\quad\, +\, a(1245,3) + a(1246,3) + a(1256,3) + a(1345,2) + a(1346,2)
+ a(1356,2) \nonumber \\
&&\quad\, +\, a(2345,1) + a(2346,1) + a(2356,1) \nonumber \\
&&\quad\, -\,\, b(123) - b(456) \nonumber \\
&&\quad\, -\, a(124,3) - a(125,3) - a(126,3) - a(134,2) - a(135,2)
- a(136,2) +\nonumber \\
&&\quad\, -\, a(234,1) - a(235,1) - a(236,1) \nonumber \\
&&\quad\, -\, a(145,23) - a(146,23) - a(156,23) - a(245,13) - a(246,13)
- a(256,13) \nonumber \\
&&\quad\, -\, a(345,12) - a(346,12) - a(356,12)\nonumber \\
&&\quad\, +\, a(12,3) + a(13,2) + a(23,1) + a(45,123) + a(46,123)
+ a(56,123) \nonumber \\
&&\quad\, +\, a(14,23) + a(15,23) + a(16,23) + a(24,13) + a(25,13)
+ a(26,13) \nonumber \\
&&\quad\, +\, a(34,12) + a(35,12) + a(36,12) \nonumber \\
&&\quad\, -\, a(1,23) - a(2,13) - a(3,12) - a(4,123) - a(5,123) - a(6,123) ]
\nonumber \\
&& +[\, t(12345) + t(12346) + t(12356) + t(12456) + t(13456) + t(23456)
\nonumber \\
&& \quad\, -\, t(1234) - t(1235) - t(1236) - t(1456) - t(2456) - t(3456) \nonumber \\
&& \quad\, -\, s(1245,3) - s(1245,6) - s(1246,3)
- s(1246,5) - s(1256,3) - s(1256,4)
\nonumber \\
&& \quad\, -\, s(1345,2) - s(1345,6) - s(1346,2)
- s(1346,5) - s(1356,2) - s(1356,4)
\nonumber \\
&& \quad\, -\, s(2345,1) - s(2345,6) - s(2346,1)
- s(2346,5) - s(2356,1) - s(2356,4)
\nonumber \\
&& \quad\, +\, t(123) + t(456) \nonumber \\
&& \quad\, +\, s(124,3) + s(124,56) + s(125,3)
+ s(125,46) + s(126,3) + s(126,45)
\nonumber \\
&& \quad\, +\, s(134,2) + s(134,56) + s(135,2)
+ s(135,46) + s(136,2) + s(136,45)
\nonumber \\
&& \quad\, +\, s(234,1) + s(234,56) + s(235,1)
+ s(235,46) + s(236,1) + s(236,45)
\nonumber \\
&& \quad\, +\, s(145,23) + s(145,6) + s(146,23)
+ s(146,5) + s(156,23) + s(156,4)
\nonumber \\
&& \quad\, +\, s(245,13) + s(245,6) + s(246,13)
+ s(246,5) + s(256,13) + s(256,4)
\nonumber \\
&& \quad\, +\, s(345,12) + s(345,6) + s(346,12)
+ s(346,5) + s(356,12) + s(356,4)
\nonumber \\
&& \quad\, -\, s(12,3) - s(12,456) - s(13,2)
- s(13,456) - s(23,1) - s(23,456) \nonumber \\
&& \quad\, -\, s(45,123) - s(45,6) - s(46,123)
- s(46,5) - s(56,123) - s(56,4) \nonumber \\
&& \quad\, -\, s(14,23) - s(14,56) - s(15,23)
- s(15,46) - s(16,23) - s(16,45) \nonumber \\
&& \quad\, -\, s(24,13) - s(24,56) - s(25,13)
- s(25,46) - s(26,13) - s(26,45) \nonumber \\
&& \quad\, -\, s(34,12) - s(34,56) - s(35,12)
- s(35,46) - s(36,12) - s(36,45) \nonumber \\
&& \quad\, +\, s(1,23) + s(1,456) + s(2,13)
+ s(2,456) + s(3,12) + s(3,456) \nonumber \\
&& \quad\, +\, s(4,123) + s(4,56) + s(5,123)
+ s(5,46) + s(6,123) + s(6,45)
] \}
\,.
\label{w123456}
\end{eqnarray}
On first sight this expression appears to be very complicated.
Closer inspection reveals that it has a series of very
interesting properties. Some of them very much resemble
those of the function $V$ we have described in section
\ref{fourgluon}.

Let us first have a look at the symmetry properties of the
function $W$. We find that
$W$ is fully symmetric in its first three arguments
\begin{eqnarray}
W({\bf k}_1,{\bf k}_2, {\bf k}_3;{\bf k}_4,{\bf k}_5,{\bf k}_6)
&=& W({\bf k}_2,{\bf k}_1, {\bf k}_3;{\bf k}_4,{\bf k}_5,{\bf k}_6) \nonumber \\
&=& W({\bf k}_3,{\bf k}_2, {\bf k}_1;{\bf k}_4,{\bf k}_5,{\bf k}_6)
\label{wsymmetry123}
\end{eqnarray}
as well as in its last three arguments,
\begin{eqnarray}
W({\bf k}_1,{\bf k}_2, {\bf k}_3;{\bf k}_4,{\bf k}_5,{\bf k}_6)
&=& W({\bf k}_1,{\bf k}_2, {\bf k}_3;{\bf k}_5,{\bf k}_4,{\bf k}_6) \nonumber \\
&=& W({\bf k}_1,{\bf k}_2, {\bf k}_3;{\bf k}_6,{\bf k}_5,{\bf k}_4)
\,.
\label{wsymmetry456}
\end{eqnarray}
Notably, the color structure corresponding to
that permutation of momenta, i.\,e.\ the tensor
$d_{a_1a_2a_3}d_{a_4a_5a_6}$, has exactly the
same symmetry properties.
Further, $W$ is symmetric under the exchange
of the first three and last three arguments,
\begin{equation}
W({\bf k}_1,{\bf k}_2, {\bf k}_3;{\bf k}_4,{\bf k}_5,{\bf k}_6) =
W({\bf k}_4,{\bf k}_5, {\bf k}_6;{\bf k}_1,{\bf k}_2,{\bf k}_3)
\,.
\end{equation}
From these symmetries of the function $W$ and
from the permutations that enter
the sum in (\ref{newpiece}) we can conclude that the full
expression $W^{a_1a_2a_3a_4a_5a_6}(1,2,3,4,5,6)$
is completely symmetric in the six outgoing gluons, i.\,e.\
under the simultaneous exchange of color labels and
momentum arguments.

Next we look at the behavior of $W$ when
one of its momentum arguments vanishes.
Not unexpectedly, we find that
$W$ vanishes whenever one of the six gluons
carries zero transverse momentum,
\begin{equation}
\left.
W({\bf k}_1,{\bf k}_2, {\bf k}_3;{\bf k}_4,{\bf k}_5,{\bf k}_6)
\right|_{{\bf k}_i=0} = 0
\;\;\;\;\;\;  (i\in\{1,\dots,6\})
\,.
\label{nullstellw}
\end{equation}
Starting from the explicit representation (\ref{w123456})
the proof is straightforward.
Of course, we again have to use the relations
between the different standard integrals
mentioned at the end of section \ref{standardint} and
the fact that the gluon trajectory function $\beta({\bf k})$
vanishes for ${\bf k}=0$.

Further we find that the function $W$ is infrared finite.
The proof requires just a close inspection of the
standard integrals occurring in (\ref{w123456}).
It is easily seen that the integrals come in the
infrared finite combinations (\ref{inffinite1}) and
(\ref{inffinite2}) discussed already in section
\ref{fourgluon}. (Again, the integrals
have been arranged in (\ref{w123456})
in a way that hopefully makes
this transparent.)

We expect the function $W$ to be conformally
invariant in impact parameter space,
but we will not discuss this issue further here.

\subsubsection{Interpretation in view of an effective field theory}
\label{interpret26}

It is now natural to ask where the new piece
$(W D_2)^{a_1a_2a_3a_4a_5a_6}$ finds its place
in the effective field theory of unitarity corrections.
At present, we are not yet able to give a conclusive
answer to this question. However, the following two
possible answers naturally arise.

As we have seen in the preceding section the new piece
has properties
that very much resemble those of the two--to--four
transition vertex $V_{2 \rightarrow 4}$.
It is fully symmetric in the six
gluons, it is infrared finite, and it vanishes when one of the
gluon momenta vanishes.
It is well possible that the new piece
$W^{a_1a_2a_3a_4a_5a_6}$ is a new two--to--six
gluon transition vertex $V_{2 \rightarrow 6}$, and
that it constitutes a new element
of the effective field theory.

The second possibility is the following.
A coupling scheme can be
chosen in the first three gluons and in the remaining
three gluons. Then the function $W$ is split into
several parts according to the symmetry or antisymmetry
under the exchange of the gluons $1$ and $2$, say, and under the
exchange of the pair $(12)$ of gluons with gluon $3$, and analogously
for the other three gluons. Based on the result of this procedure
it is possible to define new two--to--four vertices
with symmetry properties different from those of the
well--known two--to--four vertex $V_{2 \rightarrow 4}$.
Based on these one can in turn define new four--gluon amplitudes
that then become basic elements of the effective field theory.
Due to this a direct transition from two to six reggeized gluons
is avoided, and the new piece becomes a superposition of different
two--to--four vertices. However,
this second possibility has not yet been fully investigated.

The two possibilities have considerably different implications
for the emerging field theory of unitarity corrections, especially
in view of the necessary extension to amplitudes with more
gluons in the $t$-channel. The first possibility corresponds to a
picture in which an infinite number of new transition vertices
$V_{2 \rightarrow 2n}$ occurs
in the field theory, one for each even number of gluons.
The second possibility, in contrast, leads to a picture in which
there are only four two--to--four vertices with different
symmetry properties, and no other vertices of the type
$V_{2 \rightarrow 2n}$. Both possibilities should be
investigated in more detail.
We expect that especially a better understanding of the
conformal invariance of the expected effective field theory
will help to clarify the status of the new piece.

\subsection{Further reggeization}
\label{d6ir}

Now we come to discuss the other terms that are present in
the new integral equation (\ref{inteq6i})
for the part $D_6^I$ of the six--gluon amplitude.
In section \ref{inteq6isection} we have already shown
that these terms can be written as superpositions of
well--known vertex functions $V$. We will in this
section disregard the new piece discussed in the
previous section.

Already in section \ref{5gluons} we encountered
a situation similar to the one which we find here in
the integral equation (\ref{inteq6i}).
Also there the inhomogeneous term of the integral
equation for $D_5^I$ could be written as a superposition
of vertex functions $V$. It was a characteristic
indication for the occurrence of a further reggeization
of the amplitude $D_5^I$ with respect to the
irreducible four--gluon amplitude $D_4^I$.
This idea even allowed
us to find the exact solution of the equation.
We cannot expect that the remaining part $D_6^I$
of the six--gluon amplitude reggeizes completely.
But the occurrence of the vertex functions in the
inhomogeneous term of its equation strongly
suggests that a part of $D_6^I$ will reggeize.
To gain further insight we should therefore
construct an ansatz for the remaining part in order
to simplify the integral equation (\ref{inteq6i}).
The remaining part $D_6^I$ should thus be split
into a reggeizing part and an irreducible part,
\begin{equation}
D_6^I = D_6^{I,\,R} + D_6^{I,\,I}
\,,
\label{d6iirii}
\end{equation}
where this time the term 'reggeizing' refers to
the reggeization with respect to the four--gluon
compound state, cf.\ the discussion at the end of
section \ref{interpret5i}.
The reggeizing part should be a superposition of
irreducible four--gluon amplitudes, symbolically
\begin{equation}
  D_6^{I,\,R} = \sum D_4^I
\label{d6ird4i}
\,.
\end{equation}
The problem is now to find the correct color and momentum
structure for the right hand side of this symbolic equation.

We should have in mind that the inhomogeneous term in the
integral equations for $D_n$, i.\,e.\ the quark loop,
always suggests the best choice
of a reggeizing part $D_n^R$.
To make a good guess for the reggeizing part $D_6^{I,\,R}$
we should therefore have a close look at the inhomogeneous
term of the new integral equation (\ref{inteq6i}).

Let us first look at the terms (\ref{i=sumv}) and
(\ref{j=sumv}) containing the functions $I$ and $J$.
We will pick one permutation in the sums (\ref{explsumoverI})
and (\ref{explsumoverJ}) only, the other permutations
can then be treated in analogy. We see immediately that
the color and momentum structure in the first four gluons
in the terms
\begin{equation}
 d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6)
\end{equation}
and
\begin{equation}
d^{a_2a_1a_3a_4} \delta_{a_5a_6} J(1,2,3,4;5,6)
\end{equation}
is exactly the same as in $D_4^R$.

A second observation is a certain mismatch
between $f_{abc}$ and $d_{abc}$ tensors.
While there are terms of the kind
\begin{equation}
\sum f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6)
\end{equation}
present in the equation which can be written
in terms of $V$, the
corresponding terms with $d_{abc}$ tensors
(the new piece, see section \ref{v26}) cannot be
written in a similar way. This already indicates that
the $f$- and $d$-tensors have to be treated differently.

In order to come from an ansatz of the form (\ref{d6ird4i})
to the cancellation of the inhomogeneous term in
the integral equation we have to use the integral
equation (\ref{inteq4i}) for the irreducible part $D_4^I$
of the four--gluon amplitude. We want to write
the arguments of the $D_4^I$'s in the ansatz
in such a way that they have exactly the same
momentum structure which
we find in the vertex functions $V$ in
(\ref{inteq6i}). This is completely analogous to
the parts $D_n^R$ in which the momentum
structure was taken from the quark loop.
In the case of $D_n^R$ we could also
keep the color structure. This was possible because
the color structure of the two--gluon amplitude $D_2$ was
trivial, i.\,e.\ the two gluons were always in a color singlet
state. That allowed us to factorize the two--gluon
amplitude into a color part ($\delta_{a_1a_2}$) and
a momentum part.
Now the situation is more complicated since such a factorization
is not possible for the irreducible four--gluon amplitude.
We have to use the full amplitude $D_4^{I\,b_1b_2b_3b_4}$.
Since we certainly need the six color labels
$a_1,\dots,a_6$ in the ansatz (\ref{d6ird4i}),
the technical procedure we have to use is the
contraction with a tensor $\Theta$,
\begin{equation}
 \Theta^{a_1a_2a_3a_4a_5a_6;b_1b_2b_3b_4}
 D_4^{I\,b_1b_2b_3b_4}
\,.
\end{equation}
The tensor $\Theta$ is an invariant tensor in the ten--fold
tensor product $\bigotimes_{i=1}^{10} [\mbox{su}(N_c)]$
of the Lie algebra.
It will obviously be very difficult to find the correct
tensors for the contractions in this huge tensor space
without having additional information.
We have to hope that the situation is in a certain sense
more simple. It will be necessary to find restrictions on the
tensors from the inhomogeneous term in the new integral
equation.
Now a problem arises. In the term
\begin{equation}
d^{a_1a_2a_3a_4} \delta_{a_5a_6} V(123,4;5,6)
\end{equation}
for example the first four gluons are in an overall
color singlet state, as are the last two gluons.
Obviously, the color tensor necessary for the contraction
with $D_4^{I\,b_1b_2b_3b_4}$ is fixed by the inhomogeneous
term only in the case in which the gluons with labels $b_1$ and
$b_2$ are in a color singlet state. For the other irreducible
representations we have no hint from the inhomogeneous
term which would restrict the tensor $\Theta$.
This problem seems to be a conceptual one in our approach.
In the first step, that is for identifying a reggeizing part $D_n^R$
in the $n$-gluon amplitude, the quark loop was sufficient
to fix the reggeizing part. In the present situation
it is not completely excluded that the correct
color tensor for the non--singlet states cannot be fixed
unambiguously. Possibly the solution of this problem
requires a better knowledge of the irreducible four--gluon
amplitude $D_4^I$.
In spite of this conceptual problem we expect that
one can find a simple ansatz that leads to further insight.

Unfortunately, we have not yet been able to find a
satisfying and unique solution. It appears that a better
understanding of the phenomenon of reggeization,
especially in the irreducible parts of the amplitudes,
will help to resolve this problem. First steps in this
direction have been done in \cite{Carlothesis}.

In view of the discussion above and in section
\ref{interpret26} it is certainly to early to
draw final conclusions concerning the field theory
structure of the six--gluon amplitude.
The most important result of this section is contained
in the integral equation (\ref{inteq6i})
for the remaining part $D_6^I$
of the six--gluon amplitude. The occurrence of the vertex
function $V$ in this equation is an extremely strong
indication for the fact that further reggeization with
respect to the four--gluon compound state takes place
in the six--gluon amplitude. Exactly this is
the necessary condition for the emergence of the
field theory structure in the unitarity corrections.
We regard this as strong evidence for the existence of
an effective field theory of unitarity corrections.

\subsection{The Pomeron-Odderon-Odderon vertex}
\label{POO}

We now turn to the question which place the Odderon
finds in the effective field theory.
The Odderon is the $C=-1$ partner of the Pomeron, i.\,e.\
it carries negative charge parity. In perturbative QCD
it consists of a compound state of three reggeized gluons
described by the three--particle BKP equation
\cite{Bartelskernels,BKP}.
The three gluons are in a completely symmetric state, and the
color part of the wavefunction is a $d_{abc}$ tensor.
Given the quantum numbers of these states, it is a natural question
whether a BFKL Pomeron can be coupled to two Odderons.
The six--gluon amplitude is the obvious place to look
for such a Pomeron--Odderon--Odderon vertex.
The triple Pomeron vertex is obtained from the
two--to--four
gluon transition vertex $V_{2 \rightarrow 4}$
by projecting it onto three
BFKL eigenfunctions \cite{Hans,Gregory3p}.
We can therefore in analogy try to project the inhomogeneous
term of the new integral equation (\ref{inteq6i}) onto
two Odderon wavefunctions from below.
The inhomogeneous term in that equation consists of
several contributions, and we will concentrate on the
piece discussed in section \ref{v26}. We therefore ask
whether the integral
\begin{eqnarray}
V_{POO}
&=&\int \left( \prod_{i=1}^6 d^2{\bf k}_i \right)
  (W^{a_1a_2a_3a_4a_5a_6}D_2)({\bf k}_1,{\bf k}_2,{\bf k}_3,{\bf k}_4,{\bf k}_5,{\bf k}_6)
\nonumber \\
&&\hspace{.9cm}
\times d_{a_1a_2a_3} d_{a_4a_5a_6}
\Psi_1({\bf k}_1,{\bf k}_2,{\bf k}_3) \Psi_2({\bf k}_4,{\bf k}_5,{\bf k}_6)
\label{poointegral}
\end{eqnarray}
is different from zero. Unfortunately, the wavefunction
$\Psi$ of the Odderon is not known explicitly, but
conformal invariance places strong constraints on it.
In \cite{Lipatovodderon} the wavefunction of the Odderon
in impact parameter space was found to have the general form
\begin{equation}
 \Psi(\rho_1, \rho_2, \rho_3) =
\left( \frac{\rho_{12}\rho_{13}\rho_{23}}{\rho_{10}^2
\rho_{20}^2\rho_{30}^2} \right)^{h/3}
\psi(x)
\label{odderonwavefct}
\,.
\end{equation}
$h$ is the conformal weight of the Odderon state
and $x$ is  the anharmonic ratio
\begin{equation}
  x =\frac{\rho_{12}\rho_{30}}{\rho_{13}\rho_{20}}
\end{equation}
with
$\rho_{ij} = \rho_i - \rho_j$, the $\rho_i$ being
the two--dimensional coordinates in impact
parameter space.
The wavefunction $\Psi$ vanishes when two of the
coordinates of the three gluons in the Odderon
coincide. This property of the Odderon drastically reduces
the number of terms in
$(W^{a_1a_2a_3a_4a_5a_6}D_2)$ that can give
a non--vanishing contribution
to the integral (\ref{poointegral}).
If a term in $(W D_2)$ depends only on the
sum of two momenta, say ${\bf k}_1$ and ${\bf k}_2$, then this
term is after Fourier transformation to impact parameter
space proportional to a delta--function of the two
corresponding coordinates, i.\,e.\ proportional to
$\delta(\rho_1-\rho_2)$.
This implies a zero in the Odderon wavefunction and the
corresponding term does not contribute to (\ref{poointegral}).
Only very few standard integrals in $(W D_2)$ can actually give
non--vanishing contributions. Some of them give
identical results in the integral (\ref{poointegral}) due
to the symmetry of the Odderon wavefunction.
(They are in this sense equivalent to each other
as far as their contribution to (\ref{poointegral}) is concerned.)
The possible contributions to the above integral can
in this way be reduced to the following infrared finite
combination of standard integrals:
\begin{equation}
2 a(14,25) - s(14,25) - s(14,36)
\,.
\label{nonvanishterms}
\end{equation}
We now have to look for these terms in the new piece
$(W^{a_1a_2a_3a_4a_5a_6}D_2)$.
Interestingly, the first of the permutations in
(\ref{explsumoverw}) --- the color tensor of which
exactly matches the color structure of the two
Odderons --- does not contain terms
equivalent to the above combination.
But the other 9 permutations do contain such terms.
The term
\begin{equation}
  d_{a_1a_2a_5} d_{a_3a_4a_6} (W D_2)(1,2,5;3,4,6) \,
\end{equation}
for example,
contains exactly the terms (\ref{nonvanishterms}).
As can be easily shown, each of those other
permutations contains exactly four combinations
equivalent to (\ref{nonvanishterms}).
The corresponding color contraction gives a factor
\begin{equation}
d_{a_1a_2a_5} d_{a_3a_4a_6}
d_{a_1a_2a_3} d_{a_4a_5a_6} =
\left( \frac{N_c^2-4}{N_c} \right) ^2 (N_c^2 -1)
\,.
\end{equation}
We can therefore conclude that a perturbative
Pomeron--Odderon--Odderon vertex exists, and collecting
all terms it becomes
\begin{eqnarray}
V_{POO}&=&
9 \,\cdot \,4 \cdot \,\frac{g^6}{16}
\left( \frac{N_c^2-4}{N_c} \right) ^2 (N_c^2 -1)
\int  \!\left( \prod_{i=1}^6 d^2{\bf k}_i \right) \times \\
&&\times [ 2 a(14,25) - s(14,25) - s(14,36) ]
\Psi_1({\bf k}_1,{\bf k}_2,{\bf k}_3) \Psi_2({\bf k}_4,{\bf k}_5,{\bf k}_6)
\,.
\nonumber
\label{POOvertex}
\end{eqnarray}
This vertex certainly deserves further study.
For example, it should be possible to write
it in the form of a conformal three--point
function as it was possible for the triple--Pomeron vertex.
A more detailed knowledge of the Odderon wavefunction
will eventually allow one to determine the numerical
value of the integral (\ref{POOvertex}).
Another interesting question is whether the
Pomeron--Odderon--Odderon vertex can also be
calculated in the dipole picture of high energy QCD,
as it was possible for the coupling of three Pomerons.
This immediately raises the more fundamental
question of how the Odderon arises in the dipole
picture at all.

\section{Summary and outlook}

We have studied unitarity corrections in high energy QCD
in the generalized leading logarithmic approximation.
The objects of interest in this framework are
amplitudes describing the production of $n$ gluons in the
$t$-channel. These $n$-gluon amplitudes obey a tower of
coupled integral equations. The equation for the two--gluon
amplitudes coincides with the BFKL equation, and
each integral equation involves all amplitudes with a lower
number of gluons. The equation for a given $n$-gluon
amplitude can therefore only be approached after the
equations with less gluons have been solved.

A systematic approach to solving the equations
successively has been presented.
The first step is the identification
of a reggeizing part $D_n^R$ of the amplitude under
consideration. It is a superposition of two--gluon (BFKL)
amplitudes and can be obtained from the lowest order term
in the corresponding integral equation, i.\,e.\ the quark
loop with $n$ gluons attached to it. We have given the
corresponding expressions explicitly for up to six gluons.
In the case of the three--gluon amplitude this part
already solves the integral equation, whereas for $n \ge 4$
a new integral equation for the remaining part has to
be derived. That derivation is technically involved
since the number of convolutions of amplitudes with
integral kernels increases rapidly with the
number of gluons. We have identified a small set of
standard integrals that can be used to classify all
momentum space integrals occurring in the derivation.
A combinatorial method suitable for implementation
on a computer has been developed to bring all integrals
to their standard form. Birdtrack notation is used for performing
the contractions in color space. The resulting equation can then
be studied in a second step.

We have reviewed
the known results about the three-- and four--gluon amplitudes,
and discussed the field theory structure discovered in these
amplitudes. The three--gluon amplitude reggeizes completely
and is a superposition of BFKL amplitudes. The four--gluon
amplitude consists of a reggeizing part (a superposition
of BFKL amplitudes) and an irreducible part. The $t$-channel
evolution of the latter starts with a two--gluon state that couples
to a four--gluon state via the two--to--four vertex
$V_{2 \rightarrow 4}$. The emerging picture is that of an
effective field theory with $n$-gluon states coupled to each other
via number--changing vertices.

Using the methods described above we have then investigated
the amplitudes with five and six gluons in the $t$-channel.
After extracting a reggeizing part, i.\,e.\ a superposition
of BFKL amplitudes, from the five--gluon amplitude we
have derived the equation for the remaining part. The solution
of this equation can be found as a superposition of irreducible
four--gluon amplitudes. Thus the five--gluon amplitude
again reggeizes completely. The mechanism causing this
is the same as in the case of the three--gluon amplitude.
It appears therefore very likely that every amplitude with
an odd number of gluons exhibits complete reggeization.
This would imply that only $t$-channel states with even
numbers of gluons occur in the effective field theory.

A part of the six--gluon amplitude reggeizes again
and is a superposition of BFKL amplitudes. As one of our
main results, the integral equation for the remaining part
has been derived, see eq.~(\ref{inteq6i}).
It contains a new piece with very
interesting symmetry properties. However, we have not
yet been able to clarify whether it should be interpreted as
a new two--to--six vertex or as a superposition of new
two--to--four vertices.
The other terms in the equation have been shown to be
superpositions of two--to--four vertices $V_{2 \rightarrow 4}$.
We have calculated the perturbative Pomeron--Odderon--Odderon
vertex from the new piece.

The emergence of the vertex
$V_{2 \rightarrow 4}$ in some of the terms in the
equation strongly suggests
that a further part of the six--gluon amplitude is a superposition
of irreducible four--gluon states, i.\,e.\ reggeizes with
respect to the four--gluon state.
We have not yet been able to find the correct color structure
in this superposition.
Here we encounter for the first time a situation in which
reggeization occurs in a subsystem of two gluons which do not
form a color singlet. In contrary to the first stage of reggeization
(with respect to the two--gluon state) our equation does not
fix that color structure uniquely. Knowing it would
immediately allow one to generalize also the two--to--four
vertex to the non--singlet case.
The solution of this problem will probably require a better
and more general understanding of the phenomenon
of reggeization. Also information gained in other
approaches could be very helpful here, especially the
the dipole picture seems to be promising in this respect.

The most important property of the five-- and six--gluon
amplitudes found in the present paper is that they exhibit
reggeization. On the other hand, we have seen that exactly
reggeization is the necessary condition for the emergence
of the field theory structure in the unitarity corrections.
Therefore the five-- and six--gluon amplitudes
fit nicely into the picture of a potential effective field theory.
We regard this as  strong evidence for the conjecture
that the whole set of unitarity
corrections can be formulated as an effective field theory
in $2+1$ dimensions, with rapidity acting as the time--like
variable.

An important step will be to prove the conformal invariance
of that effective field theory in impact parameter space also
for the amplitudes with more than four gluons. The
proof should obviously start with the five-- and six--gluon
amplitudes, but can hopefully be extended to all possible
elements of the theory. We hope that the conformal
symmetry can also help to answer the open questions about
reggeization, about the two--to--four vertex in the color
non--singlet, and about
the meaning of the new piece found in the six--gluon amplitude.
In summary, we expect that the unitarity corrections can be cast into
the form of a conformal field theory. This opens the
fascinating possibility of applying the powerful methods
of conformal field theory and --- once the effective
conformal theory is identified --- to derive the general
properties of high energy QCD, now bypassing the
laborious explicit calculation of higher $n$-gluon amplitudes.
These general features should certainly not depend on the
process under consideration. An effective field theory found
in the amplitudes describing virtual photon--photon scattering
is expected to be relevant also to other scattering processes
in the high energy limit.

The NLO corrections to the BFKL equation have recently become
available \cite{FLNLO,Ciafaloni}, and the understanding
of these corrections has been rapidly improved, see for example
\cite{GavinNLO}.
On a long--term basis it would be interesting to compute
the NLO order corrections to all elements of the
effective field theory of unitarity corrections, although
this appears very difficult.

In the present paper we have concentrated on more
theoretical aspects of the unitarity corrections. We would
like to point out that unitarity corrections are also
interesting from a phenomenological point of view.
Unitarity is violated only at asymptotically
high energies in any measurable quantity calculated in
the BFKL formalism. However, this does by not means
imply that the unitarity corrections, for example the four--gluon
state, give a negligible contribution to observable quantities
at presently accessible energies. The
presence of the four--gluon state might well
have a sizable effect on many observables, among them
the total cross--section in virtual photon--photon scattering.
A study of these effects would be very valuable.

\acknowledgments
We would like to thank Gregory Korchemsky, Lev Lipatov,
Alan White, and especially Hans Lotter and Mark W\"usthoff
for most helpful discussions. We have also benefitted from
very enjoyable and encouraging
discussions with the late Vladimir Gribov.

redpajama_set_name: RedPajamaArXiv