\section{Introduction}

The scientific plans of the Large Hadron Collider (LHC) span from
tests of complex dynamics of the Standard Model \citep{Salgado:2011wc}
to 
 searches 
for physics beyond it \citep{Albrow:2008pn}. Quantum Chromodynamics
(QCD) being a part of the Standard Model is the basic theory which
is used to 
set up the calculational background for the collisions at LHC.
 Application of perturbative
QCD relies on 
various
 factorization theorems which allow to decompose
a given process into a long-distance part 
called parton density
and a short
distance hard process.
In particular, for inclusive and jet production processes the appropriate long-distance 
part is realized in terms of various parton distribution functions.
Here we will focus on high energy factorization
\citep{Catani:1990eg} which applies when
the energy scale involved in the scattering process  is high. The evolution
equations of high energy factorization sum up logarithms of energy
accompanied by a coupling constant. Depending on the energy range and
observable one uses: the BFKL \citep{Kuraev:1977fs,Balitsky:1978ic},
BK \citep{Balitsky:1995ub,Kovchegov:1999yj}, CCFM \citep{Ciafaloni:1987ur,Catani:1989sg}
or recently proposed new evolution equations accounting for both saturation
and processes at large momentum transfers \citep{Kutak:2011fu,Kutak:2012qk}.

The important ingredients of high energy factorization are unintegrated
gluon densities, which depend not only on the longitudinal hadron momentum
fraction but on the transversal momentum as well. They have to be
convoluted with the hard process which is calculated with the initiating
gluons being off-shell. In case of processes with non-gluonic final
states, like $q$, $\bar{q}$ production for instance, the corresponding
gauge invariant amplitudes can be calculated by applying so called
high energy projectors, that resolve degrees of freedom relevant at
high energies. The more general framework given  in terms of Lipatov's action \citep{Lipatov:1995pn}
provides a prescription to calculate gauge invariant matrix elements
in a general case for any partonic process initiated by off-shell
gluons \citep{Antonov:2004hh}. For recent studies and applications, we refer to \citep{Chachamis:2012gh,Hentschinski:2011tz,Braun:2011qm,Bartels:2012mq}.

At present, various automatic numerical tools exist for the calculation of multi-gluon amplitudes for ordinary on-shell processes \citep{Mangano:2002ea,Cafarella:2007pc,Gleisberg:2008fv,Alwall:2011uj,Kleiss:2010hy}.
They operate on the amplitude level and use helicity methods, and thus are very efficient and universal.
The present study is a first step aiming at similar methods for off-shell high energy amplitudes.
We develop a new prescription for calculating gauge invariant
high energy factorizable tree-level amplitudes with one of the gluons being off-shell.
Our results turn out to be in agreement with results obtained using
Lipatov's action.

Our prescription can be summarized as follows.
It is known \citep{Leonidov:1999nc} that the ordinary Feynman graphs contributing to an amplitude in an on-shell calculation are in general not sufficient to obtain gauge invariant results if any of the external legs are off-shell.
Thus one usually
needs to consider a larger on-shell process and disentangle the required gauge invariant
off-shell amplitudes. 
 Our approach is however different: we require
the Slavnov-Taylor identities to hold for the off-shell connected
amplitudes, which in turn induces the necessary contributions which
by comparison to \citep{Leonidov:1999nc} can be interpreted as bremsstrahlung
 graphs introduced to restore gauge invariance. Our construction
holds for any number of final state gluons and as an example we give a detailed
derivation and discussion for the helicity amplitudes of the process $g^{*}g\rightarrow gg$. These
amplitudes are the dominant contribution to
forward-central di-jet production \citep{Deak:2010gk,Deak:2011ga,Deak:2009ae} and  are of particular interest for
testing parton densities at low longitudinal momentum fraction of the gluons
with a special focus on saturation effects (for overview see \citep{Kutak:2011rv}
and for recent jet studies of these effects see \citep{Kutak:2012rf}).

An extensive analyzis of tree-level multi-gluon helicity amplitudes in the high-energy limit was done also in Ref. \citep{DelDuca:1999ha}. There, the impact factors for $g^* g\rightarrow Ng$, $N\leq 3$ were disentangled at the amplitude level, however they do not correspond exactly to the kinematic situation considered in the present work.

The paper is organized as follows. Section \ref{sec:GeneralFr} is
introductory and is mainly devoted to set the notation. In particular,
we overview  high energy factorization focusing on the kinematic
setup and we introduce the framework of color
ordered helicity amplitudes which is used later in the paper. In Section
\ref{sec:Off-shell_amps} we identify the problem of gauge non-invariance
of high energy factorization amplitudes and 
 repair it using
the Slavnov-Taylor identities. In Section \ref{sec:Results} we describe
analytical and numerical tests for gauge invariant helicity amplitudes,
in particular we list the helicity amplitudes and their square for
$g^{*}g\rightarrow gg$. We also cross check our result with those
obtained by usage of Lipatov's effective action. In the appendices
we collect the color-ordered Feynman rules we needed in our construction
of gauge invariant amplitudes as well as some technicalities on the
Slavnov-Taylor identities.


\section{Prerequisites and notation}

\label{sec:GeneralFr}


\subsection{High energy factorization}

\label{sec:HighEnFact}

Our considerations of the off-shell amplitudes are embedded in the
formalism of high energy factorization. Let us thus start with a brief
recollection of this topic \citep{Catani:1990eg,Catani:1990xk,Catani:1990ka,Catani:1993ww}.

The basic observation is that the cross section for a hadronic process
can be decomposed at high energies into transversal momentum dependent
parton densities and the hard partonic cross section with off-shell
initial state partons. Thus, the intermediate object that appears
is the amplitude for the process 
\begin{equation}
g^{*}(k_{1})\, g^{*}(k_{2})\rightarrow X,\label{eq:hard_proc}\end{equation}
 where $g^{*}$ denotes an off-shell gluon, and $X$ abbreviates the
on-shell final state particles. In the considered high energy limit,
the incident momenta are para\-metrized as \begin{gather}
k_{1}^{\mu}\simeq\xi_{1}n_{a}^{\mu}+k_{1T}^{\mu},\label{eq:highen_mom1}\\
k_{2}^{\mu}\simeq\xi_{2}n_{b}^{\mu}+k_{2T}^{\mu},\end{gather}
 where $n_{a}$, $n_{b}$ are light-like momenta corresponding to
the two incident hadrons $a$ and $b$. They satisfy $n_{a}\cdottk_{iT}=n_{b}\cdottk_{iT}=0$,
and $k_{i}^{2}=k_{iT}^{2}=-\big|\vec{k}_{iT}\big|^{2}.$

In \citep{Catani:1990eg}, for example, heavy quark production $g^{*}(k_{1})\, g^{*}(k_{2})\to Q\,\overline{Q}$
is considered. The amplitude is calculated by taking into account
the usual tree-level Feynman graphs in the axial gauge with gauge
vector $n^{\mu}=\alpha n_{a}^{\mu}+\beta n_{b}^{\mu}$. Then the gluon
propagator is given by \begin{equation}
D^{\mu\nu}\left(k\right)=S\left(k\right)\, d^{\mu\nu}\left(k,n\right),\end{equation}
 with the scalar part\begin{equation}
S\left(k\right)=\frac{-i}{k^{2}+i\epsilon}\label{eq:scalar_prop}\end{equation}
and the transverse projector \begin{equation}
d^{\mu\nu}\left(k,n\right)=\eta^{\mu\nu}-\frac{k^{\mu}n^{\nu}+k^{\nu}n^{\mu}}{k\cdott n}+n^{2}\frac{k^{\mu}k^{\nu}}{\left(k\cdott n\right)^{2}}.\label{eq:proj_def}\end{equation}
 Above $\eta^{\mu\nu}$ is the metric tensor and we have suppressed
the color indices in $D^{\mu\nu}$. The off-shell legs include propagators,
and are contracted with eikonal couplings defined as 
\begin{gather}
\mathfrak{e}_{1}^{\mu}=\big|\vec{k}_{1T}\big| n_{a}^{\mu},\label{eq:eik1}\\
\mathfrak{e}_{2}^{\mu}=\big|\vec{k}_{2T}\big| n_{b}^{\mu}.\label{eq:eik2}\end{gather}
 It is easy to check, that 
effectively an amputated amplitude is contracted with the following vector
\begin{equation}
-i\,\mathfrak{e}_{1}^{\mu}\, D_{\mu\nu}\left(k_{1}\right)=\frac{k_{T\,1}^{\mu}}{\xi_{1}\big|\vec{k}_{1T}\big|}
\end{equation}
 and similarly for $k_{2}$, which then is shown to lead to the
correct result in the collinear limit. Furthermore, it is shown that
the result is independent of the choice of gauge-vector, proving gauge invariance.

The above might be taken as a prescription to calculate amplitudes
for arbitrary processes, but it will in general not lead to gauge-invariant
results. To achieve the latter, additional contributions are needed. 
They are 
usually obtained by considering a larger, fully on-shell, process
from which an amplitude for the off-shell process is extracted. The
situation is somewhat simpler when one of the gluons, say $k_{2}$,
reaches collinear limit, i.e.\ when $k_{2}^{2}\rightarrow0$. In
the following sections, we shall investigate such amplitudes for the
process \begin{equation}
g^{*}(k_{1})\, g(k_{2})\rightarrow g(k_{3})\ldots g(k_{N})\label{eq:process}\end{equation}
 for an arbitrary number of final state gluons.




\subsection{Color-ordered helicity amplitudes}

\label{sec:helic_color}

Let us denote our purely gluonic tree-level amplitude with $N$ external gluons
(one off-shell and $N-1$ on-shell) as $\mathcal{M}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)$.
For the precise definition see Section \ref{sec:Off-shell_amps}. The polarization vectors of the
on-shell gluons are denoted as $\varepsilon_{i}$. Whenever necessary,
we shall indicate the polarization state explicitly using $\varepsilon_{i}^{(\lambda)}$
with $\lambda=\pm$. Following 
e.g.\ the Kleiss-Stirling construction~\citep{Kleiss:1985yh},
the polarization vectors can be defined in terms of the gluon momentum
$k_{i}$ and an auxiliary light-like momentum $q_{i}$ satisfying
$k_{i}\cdott q_{i}\neq0$. Thus, whenever needed we use the notation
$\varepsilon_{i}^{(\lambda)}(q_{i})$ to indicate explicitly what
reference vector is used. The polarization vectors satisfy several
conditions, of which we would like to highlight two, namely \begin{gather}
\varepsilon_{i}^{(\lambda)}(q_{i})\cdott k_{i}=0,\label{eq:transversality}\\
\varepsilon_{i}^{(\lambda)}(q_{i})\cdott q_{i}=0.\label{eq:eps*q}\end{gather}
 When one changes the reference momentum, the corresponding polarization
vector transforms as
\begin{equation}
\varepsilon_{i}^{\mu}\left(q'_{i}\right)=\varepsilon_{i}^{\mu}\left(q_{i}\right)+\beta_{i}\left(q'_{i},q_{i}\right)k_{i}^{\mu},\label{eq:refmom_change}
\end{equation}
where the function $\beta_{i}$ depends on the precise representation
for the polarization vectors. It is important to note, that as long
as the amplitude $\mathcal{M}$ satisfies the Ward identity 
\begin{equation}
\mathcal{M}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{i-1},k_{i},\varepsilon_{i+1},\ldots,\varepsilon_{N}\right)=0,\quad i=2,\dots ,N
\end{equation}
 the reference momenta can be freely changed, which has proved to
be very useful for obtaining compact expressions for multi-gluon amplitudes
(see e.g.\ \citep{Dixon:1996wi} for a review). In explicit calculations
we use the spinor representation for the polarization vectors
\begin{equation}
\varepsilon_{i}^{\mu\,\pm}\left(q_{i}\right)=\pm\frac{\left\langle q_{i}^{\mp}\right|\gamma^{\mu}\left|k_{i}^{\mp}\right\rangle }{\sqrt{2}\,\left\langle q_{i}^{\mp}|k_{i}^{\pm}\right\rangle },
\end{equation}
where
\begin{gather}
\left\langle k_{i}^{-}|k_{j}^{+}\right\rangle =\overline{u}^{-}\left(k_{i}\right)u^{+}\left(k_{j}\right)\equiv\left\langle ij\right\rangle ,\\
\left\langle k_{i}^{+}|k_{j}^{-}\right\rangle =\overline{u}^{+}\left(k_{i}\right)u^{-}\left(k_{j}\right)\equiv\left[ij\right]
\end{gather}
and the positive-energy spinors with definite helicities are defined
as $u^{\pm}\left(k_{i}\right)=\frac{1}{2}\left(1\pm\gamma_{5}\right)u\left(k_{i}\right)$. The spinor products can be explicitly calculated in the light-cone basis as
\begin{gather}
\left\langle ij\right\rangle = k_{\perp\, i}\,\sqrt{\frac{k^+_j}{k^+_i}} - k_{\perp\, j}\,\sqrt{\frac{k^+_i}{k^+_j}}, \\
\left[ ij\right] = -k^*_{\perp\, i}\sqrt{\frac{k^+_j}{k^+_i}} + k^*_{\perp\, j}\sqrt{\frac{k^+_i}{k^+_j}},
\end{gather}
where $k^{\pm} = k^0\pm k^3$ and $k_{\perp} = k^1+ik^2$.
We also use the following compact notation for spinor products with some $\dsl{\Gamma}$ matrix insertion
\begin{gather}
\left\langle q^{-}\right|\dsl{\Gamma}\left|k^{-}\right\rangle \equiv \left\langle q\right|\dsl{\Gamma}\left|k\right], \\
\left\langle q^{+}\right|\dsl{\Gamma}\left|k^{+}\right\rangle \equiv \left[ q\right|\dsl{\Gamma}\left|k\right\rangle.
\end{gather}

Another important ingredient in this respect is the use of color-ordered
amplitudes \citep{Mangano:1987xk}. They are defined via
\begin{equation}
\mathcal{M}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)=\sum_{\underset{{\scriptstyle \textrm{permutations}}}{\textrm{non-cyclic}}}\mathrm{Tr}\left(t^{a_{1}}\ldots t^{a_{N}}\right)\,\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right),
\label{eq:colordecom}
\end{equation}
 where the summation is over all non-cyclic permutations of the color
indices $\left\{ a_{1},\ldots,a_{N}\right\} $ corresponding to the
external gluons. The matrices $t^{a}$ are the generators of color
$\mathrm{SU}\left(N_{c}\right)$ group normalized as $\mathrm{Tr}\left( t^a t^b\right) = \delta^{ab}$. 
Note, that
the ordering of the arguments in $\mathcal{A}$ does matter and corresponds
to the order of color matrices under the trace. There are several
important properties of color-ordered amplitudes. Here we want to
mention explicitly two of them. First, gauge invariance of $\mathcal{M}$
implies gauge invariance of $\mathcal{A}$. Second, the full amplitude
squared is given by 
\begin{multline}
\sum_{\mathrm{colors}}\left|\mathcal{M}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)\right|^{2}  = N_{c}^{N-2}\left(N_{c}^{2}-1\right)\!\!\!\!\!\sum_{\underset{{\scriptstyle \textrm{permutations}}}{\textrm{non-cyclic}}}\left|\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)\right|^{2}
 \\ +\mathcal{O}\left(N_{c}^{N-2}\right),\label{eq:ampl_squared}
\end{multline}
 where the contribution $\mathcal{O}\left(N_{c}^{N-2}\right)$ vanishes
for $N\leq5$.


\section{Amplitudes with one off-shell leg}

\label{sec:Off-shell_amps}


\begin{figure}
\begin{center}
\psfragscanon {\small \psfrag{1}{$k_{2}$} \psfrag{2}{$k_{1}$}
\psfrag{3}{$k_{3}$} \psfrag{4}{$k_{N}$}\includegraphics[width=3cm]{figure1}\psfragscanoff }
\end{center}
\caption{\small The color-ordered amplitude $\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)$
defined in the main text. The special external vertex corresponds
to eikonal coupling $\mathfrak{e}_{1}^{\mu}=\big|\vec{k}_{T}\big| n_{a}^{\mu}$, while
all the other external lines are amputated and contracted with the
corresponding polarization vectors. The arrows show the choice of
the momenta signs, while the vertical dots stand for the remaining
external gluons.\label{fig:M_def}}
\end{figure}


The subject of our investigations are gluonic amplitudes with one
off-shell leg, \Equation{eq:process}. More precisely, they are obtained
from the corresponding momentum space Green function by means of the
following reduction formula\begin{multline}
\mathcal{M}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)=\lim_{k_{1}\cdot n_{a}\rightarrow0}\,\lim_{k_{2}^{2},\ldots,k_{N}^{2}\rightarrow0}\,\mathfrak{e}_{1}^{\mu_{1}}\varepsilon_{2}^{\mu_{2}}\ldots\varepsilon_{N}^{\mu_{N}}\,\\
S^{-1}\left(k_{2}\right)\ldots S^{-1}\left(k_{N}\right)\tilde{G}_{\mu_{1}\ldots\mu_{N}}\left(k_{1},\ldots,k_{N}\right),\label{eq:reduction}\end{multline}
where 
the
 scalar propagators are defined in Eg.~(\ref{eq:scalar_prop})
and $\tilde{G}$ is momentum space Green function defined as usual,
by Fourier transforming the vacuum matrix element of a time ordered
product of gluon fields. Note, that the first limit is a formal way
of implementing the kinematic condition (\ref{eq:highen_mom1}). The
amplitude defined in \Equation{eq:reduction} is incidentally \emph{not
gauge invariant}, see the discussion below.

Let us remark, that for ordinary, completely on-shell amplitudes the
common practice is to amputate the external gluon legs together with
potential projectors (\ref{eq:proj_def}), if one uses axial gauge
for $\tilde{G}$. In other words, in such case we can choose different
gauges for the internal (off-shell) and the external (on-shell) lines
and the scattering amplitude does not depend on this choice. It
is of course also possible to do so at intermediate stages of a calculation
when dealing with gauge-dependent quantities, however one has to be
consistent, as different gauge-dependent pieces are to eventually compose
a gauge invariant object.

In this spirit, in what follows we work in the axial gauge with $n_{a}$
being the gauge vector for internal off-shell lines (including line
$k_{1}$) while the external on-shell gluons are reduced in the Feynman
gauge, unless otherwise specified. That is, any vertex adjacent to
an external on-shell gluon is contracted with the corresponding polarization
vector directly.

In what follows, we consider a color-ordered amplitude for some specific
color ordering, say $\left(a_{1},a_{2},\ldots,a_{N}\right)$, where
$a_{i}$ is the color quantum number of the gluon with momentum $k_{i}$.
As follows from (\ref{eq:reduction}), the off-shell leg $k_{1}$
contains the full propagator and is contracted with the eikonal vertex
$\mathfrak{e}_{1}$ defined in \Equation{eq:eik1}, which we graphically
denote as a circle with a cross (\Figure{fig:M_def}). According
to Section \ref{sec:helic_color}, the amplitude under consideration
is denoted as $\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)$.
For simplicity, we choose\begin{equation}
k_{1}^{\mu}=n_{a}^{\mu}+k_{T}^{\mu}\label{eq:kinem_k1}\end{equation}
 as the momentum of the off-shell gluon (i.e.\ we absorb $\xi$ into $n_a$ 
in \Equation{eq:highen_mom1}). 

Because of the off-shell external leg, taking into account the usual
tree-level Feynman graphs and using the usual Feynman rules will not
lead to a gauge invariant amplitude. In particular, it will not satisfy
the usual Ward identities, i.e.\ \begin{equation}
\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{i-1},k_{i},\varepsilon_{i+1},\ldots,\varepsilon_{N}\right)\neq0,\quad i=2,\ldots,N,\label{eq:ward_viol}\end{equation}
which are indispensable for the calculation of helicity amplitudes.
Indeed, it is not the Ward identity of the above form that follows
from the local gauge invariance. Rather, one should consider the non-abelian
Slavnov-Taylor (\mbox{S-T}) identites, which relate the amplitude
$\mathcal{A}$ with nonphysical polarization (i.e.\ insertion of
$k_{i}$) and the corresponding diagrams with ghosts. In the next
section, we shall wander this path to construct a new amplitude \begin{equation}
\tilde{\mathcal{A}}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)=\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)+\mathcal{W}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right),\label{eq:Aprime_def}\end{equation}
 such that \begin{equation}
\tilde{\mathcal{A}}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{i-1},k_{i},\varepsilon_{i+1},\ldots,\varepsilon_{N}\right)=0,\quad i=2,\ldots,N.\label{eq:ward_ok}\end{equation}

Since we provide a prescription to calculate $\tilde{\mathcal{A}}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)$ in a specific gauge,
the fact that it satisfies Eq.~(\ref{eq:ward_ok}) is an indication that it is the correct gauge invariant result. Moreover, we show in Appendix 
\ref{sec:App_sing_behav} that (\ref{eq:Aprime_def}) posseses correct collinear and soft behaviour in the corresponding limits. 

\subsection{Restoration of gauge invariance}

\label{sec:Gauge_restor}

In what follows, we skip the first argument (corresponding to the
off-shell leg) in the color-ordered amplitude $\mathcal{A}$, i.e.\ we
write $\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\ldots,\varepsilon_{N}\right)\equiv\mathcal{A}\left(\varepsilon_{2},\ldots,\varepsilon_{N}\right)$
for compactness. In the gauge we specified above, our solution for
the `gauge restoring' amplitude $\mathcal{W}$ reads 
\begin{equation}
\mathcal{W}\left(\varepsilon_{2},\ldots,\varepsilon_{N}\right)=\left(-\frac{g}{\sqrt{2}}\right)^{N-2}\frac{-\big|\vec{k}_{T}\big|\,\varepsilon_{2}\cdott n_{a}\ldots\varepsilon_{N}\cdott n_{a}}{k_{2}\cdott n_{a}\,\left(k_{2}-k_{3}\right)\cdott n_{a}\,\dots \left(k_{2}-\ldots-k_{N-1}\right)\cdott n_{a}}.\label{eq:waste-1}
\end{equation}
The above formula can be interpreted as a straight Wilson line along $n_{a}$ with
the external gluons attached to it, however such an interpretation
is beyond the scope of this paper and shall be analyzed elsewhere.

As already mentioned, according to the S-T identities, the insertion
of $k_{i}$ generates additional gauge terms on the r.h.s of (\ref{eq:ward_viol}).
We shall see that by controlling those terms one can indeed reconstruct
$\mathcal{W}$. The S-T identities in different gauges together with
some less common issues are recollected in Appendix \ref{sub:App_ST}.

Let us consider first the (color-ordered) momentum space Green function,
from which our amplitude is obtained according to the reduction procedure
(\ref{eq:reduction}). Since we have chosen the Feynman gauge for
the external on-shell lines, the relevant S-T identity involves the
space-time derivative of the fields and in turn translates into a
momentum insertion as in (\ref{eq:ward_viol}), c.f.\ (\ref{eq:App_ST}).

\begin{figure}
\begin{raggedright} A) 
\par
\end{raggedright}
\par
\begin{centering}
\psfragscanon \psfrag{1}{$=$} \psfrag{2}{$+$} \psfrag{4}{$+\,\dots$}
\psfrag{3}{$+$} \psfrag{5}{$+\,\dots$}
\includegraphics[width=10cm]{figure2}\psfragscanoff 
\par\end{centering}
\par
\begin{raggedright} B) 
\par
\end{raggedright}
\par
\begin{centering}
\psfragscanon \psfrag{1}{$=$} \psfrag{2}{$+$} \psfrag{3}{$\!\!\! =$}\includegraphics[width=10cm]{figure3}\psfragscanoff 
\par\end{centering}

\caption{{\small A) The Slavnov-Taylor identity for the Green function (with
external propagators) contracted with $k_{2}$. The contraction is
graphically denoted by the arrow transforming to a solid line (see
Appendix \ref{sub:App_color_FR}). The dashed lines with arrows are
standard notation for ghosts. Horizontal dots denote analogous diagrams
for the rest of external legs. B) Application of the identity for the amplitude for a process
with $N-1$ on-shell gluons and one off-shell leg contracted with
an eikonal vertex. Most of the gauge diagrams vanish as described
in the main text, but not all due to the off-shellness of $k_{1}$.\label{fig:ST_Green}}}
\end{figure}


The S-T identities for the Green functions are most conveniently given
in graphical form. We show an example in \Figure{fig:ST_Green}A
for the contraction with $k_{2}$. It generates a set of diagrams
involving ghosts and picks out a ghost-gluon vertex (see (\ref{eq:BRST_K})
in Appendix \ref{sub:App_ST}) without an external propagator. The
ghosts propagating to the shaded blob, which is assumed to be calculated
in the axial gauge, may need some explanation. Let us note that the ghosts
can be introduced also in the axial gauge but they decouple for ordinary
processes, see e.g.\ \citep{Leibbrandt:1987qv} and Appendix \ref{sub:App_ST}
for additional explanations. The ghost-gluon coupling is proportional
to $n_{a}^{\mu}$ and is transverse to the gluon propagator in the
axial gauge $n_{a}^{\mu}d_{\mu\nu}\left(k,n_{a}\right)=0$, thus any
internal gluon cannot be coupled to a ghost. In our case, however,
a ghost can couple to external on-shell gluons, which are taken
in the Feynman gauge and for which the projector $d_{\mu\nu}$ is
absent.

When going from the Green function to the amplitude $\mathcal{A}$
most of the additional gauge diagrams vanish (\Figure{fig:ST_Green}B),
either due to the transversality condition (\ref{eq:transversality}),
or because the residue at the physical pole is zero (due to the lack
of the external propagator, c.f. the reduction formula (\ref{eq:reduction})).
However, since $k_{1}$ is off-shell eventually one term survives,
namely the one with the ghost-gluon vertex with the amputated external
line (the second term of BRST transformation vertex (\ref{eq:BRST_K}))
contracted with the eikonal coupling. Note, that the first term in
the middle in \Figure{fig:ST_Green}B vanishes due to the relation
$k_{1}\cdott n_{a}=0$.

\begin{figure}
\begin{centering}
\psfragscanon {\small \psfrag{1}{$k_{1}$} \psfrag{2}{$k_{2}$}
\psfrag{3}{$k_{3}$} \psfrag{4}{$k_{4}$}\includegraphics[width=6.5cm]{figure4}\psfragscanoff }
\par\end{centering}

\caption{\small The diagrams contributing to the Slavnov-Taylor identity for the color-ordered
amplitude $\mathcal{A}\left(\mathfrak{e}_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)$,
corresponding to the the process $g^{*}g\rightarrow gg$. We show
the momentum flow in the leftmost diagram only. The right diagram
is zero if the axial gauge with $n_{a}$ gauge vector is chosen. \label{fig:gauge_diag_1}}
\end{figure}


In order to illustrate the above statements, let us look at the specific
realization of the gauge diagrams from \Figure{fig:ST_Green}B
for the simplest process $g^{*}g\rightarrow gg$. We display the diagrams
in \Figure{fig:gauge_diag_1} for the color-ordered amplitude.
Since we work in the axial gauge for the internal lines, no gluon
line can attach to a ghost unless it is an external line. There is
only one possible diagram with a triple-gluon coupling in which it
is connected to the eikonal vertex (right diagram in \Figure{fig:gauge_diag_1}).
But this diagram is also zero, since we have chosen $n_{a}$ as the
gauge vector.

The remaining gauge diagrams can easily be calculated and one can
explicitly check the S-T identities (we checked it also for a general
axial gauge with gauge vector $n\neq n_{a}$). For instance, for the
left diagram from \Figure{fig:gauge_diag_1} we obtain using the
Feynman rules collected in Appendix \ref{sub:App_color_FR} \begin{equation}
\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{3}\, n_{a}\cdott\varepsilon_{4}}{\left(k_{2}-k_{3}\right)\cdott n_{a}}\equiv\mathcal{W}_{1}\left(k_{2},\varepsilon_{3},\varepsilon_{4}\right).\label{eq:W1k2}\end{equation}
 The arguments of $\mathcal{W}_{1}$ are chosen in this way for further
convenience and show what leg is contracted with the momentum in the
corresponding S-T relation. %

\begin{figure}
\begin{centering}
\psfragscanon {\small \psfrag{1}{$\mathcal{W}_{2a}$} \psfrag{2}{$\mathcal{W}_{2b}$}
\psfrag{3}{$\mathcal{W}_{3}$}\includegraphics[width=9cm]{figure5}\psfragscanoff }
\par\end{centering}

\caption{\small The color-ordered gauge diagrams for the contractions of the amplitude
$\mathcal{A}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)$
with the external momenta; here $\mathcal{W}_{2a}$ and $\mathcal{W}_{2b}$
correspond to $\mathcal{A}\left(\varepsilon_{2},k_{3},\varepsilon_{4}\right)$,
while $\mathcal{W}_{3}$ corresponds to $\mathcal{A}\left(\varepsilon_{2},\varepsilon_{3},k_{4}\right)$.
The momentum flow is the same as in \Figure{fig:gauge_diag_1}.\label{fig:gauge_diag_2}}
\end{figure}


We can derive analogous identities involving contractions for different
external legs. The corresponding gauge diagrams are depicted in \Figure{fig:gauge_diag_2}.
Using analogous naming convention as in \Equation{eq:W1k2} they
read \begin{equation}
\mathcal{W}_{2a}\left(\varepsilon_{2},k_{3},\varepsilon_{4}\right)=\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{2}\, n_{a}\cdott\varepsilon_{4}}{\left(k_{1}+k_{2}\right)\cdott n_{a}},\end{equation}
 \begin{equation}
\mathcal{W}_{2b}\left(\varepsilon_{2},k_{3},\varepsilon_{4}\right)=\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{2}\, n_{a}\cdott\varepsilon_{4}}{\left(k_{1}-k_{4}\right)\cdott n_{a}},\end{equation}
 \begin{equation}
\mathcal{W}_{3}\left(\varepsilon_{2},\varepsilon_{3},k_{4}\right)=-\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{2}\, n_{a}\cdott\varepsilon_{3}}{\left(k_{1}+k_{2}\right)\cdott n_{a}}.\end{equation}
 An immediate observation is that the gauge diagrams are proportional
to $\big|\vec{k}_{T}\big|$, thus they vanish when all the particles become on-shell;
consequently the l.h.s. of (\ref{eq:ward_viol}) equals to zero
 in that limit.

The `gauge restoring' amplitude $\mathcal{W}$ in (\ref{eq:Aprime_def})
can be obtained using the above results. First, we define $\mathcal{W}_{x}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)$
(note the arguments) for $x=1,2a,2b,3$ by multiplying the external
ghost lines with the longitudinal projections of the corresponding
polarization vectors. More precisely, we multiply the ghost leg $i$ with
$\varepsilon_{i}\cdott n_{a}/k_{i}\cdott n_{a}$. Hence we have \begin{equation}
\mathcal{W}_{1}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)=-\mathcal{W}_{3}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)=\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{2}\, n_{a}\cdott\varepsilon_{3}\, n_{a}\cdott\varepsilon_{4}}{k_{2}\cdott n_{a}\, k_{4}\cdott n_{a}},\end{equation}


\begin{equation}
\mathcal{W}_{2a}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)=\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{2}\, n_{a}\cdott\varepsilon_{3}\, n_{a}\cdott\varepsilon_{4}}{k_{2}\cdott n_{a}\, k_{3}\cdott n_{a}},\end{equation}
 \begin{equation}
\mathcal{W}_{2b}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)=-\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{2}\, n_{a}\cdott\varepsilon_{3}\, n_{a}\cdott\varepsilon_{4}}{k_{3}\cdott n_{a}\, k_{4}\cdott n_{a}},\end{equation}
 Then, we define \begin{multline}
\mathcal{W}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)=\mathcal{W}_{1}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)+\mathcal{W}_{2a}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)\\
+\mathcal{W}_{2b}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)+\mathcal{W}_{3}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right).\end{multline}
 Adding the diagrams, we recover the result anticipated in \Equation{eq:waste-1}
for $N=4$\begin{equation}
\mathcal{W}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)=-\frac{g^{2}}{2}\big|\vec{k}_{T}\big|\,\frac{n_{a}\cdott\varepsilon_{2}\, n_{a}\cdott\varepsilon_{3}\, n_{a}\cdott\varepsilon_{4}}{k_{2}\cdott n_{a}\, k_{4}\cdott n_{a}}.\end{equation}
 It is straightforward to check that indeed (\ref{eq:ward_ok}) is
satisfied with this choice of $\mathcal{W}$. It is a general
property of $\mathcal{W}$ in our choice of gauge, that the sum of
the diagrams with ghosts (and further replaced by longitudinal gluon
polarization) collapses to single terms. The complete proof of \Equation{eq:waste-1}
for any number of gluons is relegated to Appendix~\ref{sub:App_Proof_waste}.

We want to close this section with the following remark. Note that
if we choose the reference momentum for any of the polarization vectors
to be $n_{a}$, the `gauge restoring' amplitude is zero \begin{equation}
\mathcal{W}\left(\varepsilon_{2},\ldots,\varepsilon_{i-1},\varepsilon_{i}\left(n_{a}\right),\varepsilon_{i+1},\ldots,\varepsilon_{N}\right)=0\end{equation}
 for any leg $i$. It follows from the property of polarization vectors
(\ref{eq:eps*q}). Let us now note, that we can always project a polarization
vector with any reference momentum $q_{i}$ to the one with $n_{a}$
using\begin{equation}
\varepsilon_{i}^{\mu}\left(q_{i}\right)d_{\mu}^{\nu}\left(k_{i},n_{a}\right)=\varepsilon_{i}^{\nu}\left(n_{a}\right),\end{equation}
 with $d^{\mu\nu}\left(k_{i},n_{a}\right)$ defined in (\ref{eq:proj_def}).
It follows simply from the transversality condition $n_{a}\cdott d\left(k_{i},n_{a}\right)=0$.
The above remark leads to a conclusion that the gauge invariant amplitude
(\ref{eq:Aprime_def}) can be written as \begin{equation}
\tilde{\mathcal{A}}\left(\varepsilon_{2},\ldots,\varepsilon_{N}\right)=\mathcal{A}\left(\varepsilon_{2},\ldots,\varepsilon_{i-1},d\left(k_{i},n_{a}\right)\cdott\varepsilon_{i},\varepsilon_{i+1},\ldots,\varepsilon_{N}\right),\label{eq:Atil_proj}\end{equation}
 that is one can apply the transverse projector $d^{\mu\nu}\left(k_{i},n_{a}\right)$
to any number (at least one) of external on-shell legs. In particular,
one can apply the projectors to all on-shell legs. Then, (\ref{eq:ward_ok})
is trivially satisfied for any $i$, since $k_{i}\cdott d\left(k_{i},n_{a}\right)=0$
for $k_{i}^{2}=0$. Thus one could have the impression that it is also
possible for any gauge vector $n\neq n_{a}$, as the projector remains
transverse to $k_{i}$ and the `naive' Ward identity is satisfied.
This however does not mean that such amplitude is gauge invariant
-- we relegate this issue to Appendix \ref{sec:App_axial_extproj}.


\section{Analytical and numerical tests}

\label{sec:Results}


\subsection{Numerical tests}

We tested (\ref{eq:ward_ok}) numerically up to $N=12$, calculating
$\mathcal{A}$ with numerical Berends-Giele recursion~\citep{Berends:1987me}.
We also tested (\ref{eq:ward_ok}) for color-dressed amplitudes up
to $N=8$, by calculating $\mathcal{M}$ in the color-flow representation following the method of~\citep{Papadopoulos:2005ky,Duhr:2006iq}.
The color-dressed version of $\mathcal{W}$ was calculated by straightforward
multiplication with the color-dependent factor and summation over
the necessary permutations, {\it i.e.}\ following \Equation{eq:colordecom} with the color-factor in the color-flow representation \citep{Kanaki:2000ms,Maltoni:2002mq}.


\subsection{Analytic expressions for $g^{*}g\rightarrow gg$}

Let us start with the amplitude with full dependence on polarization
vectors. Defining \begin{equation}
\varepsilon_{1}^{\mu}=\frac{k_{T}^{\mu}}{\big|\vec{k}_{T}\big|}\end{equation}
 we get explicitly for  color-ordering $\left(a_{1},a_{2},a_{3},a_{4}\right)$
\begin{multline}
\tilde{\mathcal{A}}\left(\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right)=\tilde{\mathcal{A}}\left(\varepsilon_{4},\varepsilon_{3},\varepsilon_{2}\right)=\frac{g^{2}}{2\, k_{2}\cdott k_{3}k_{3}\cdott k_{4}k_{2}\cdott n_{a}k_{4}\cdott n_{a}}\\
(n_{a}\cdott\varepsilon_{3}(k_{2}\cdott k_{3}k_{3}\cdott\varepsilon_{4}k_{4}\cdott n_{a}\varepsilon_{1}\cdott\varepsilon_{2}+k_{2}\cdott n_{a}k_{3}\cdott k_{4}k_{3}\cdott\varepsilon_{2}\varepsilon_{1}\cdott\varepsilon_{4})k_{1}^{2}\\
+k_{4}\cdott n_{a}(k_{2}\cdott n_{a}((k_{2}\cdott k_{3}k_{4}\cdott\varepsilon_{3}\,\mathfrak{t}-k_{3}\cdott k_{4}(k_{2}\cdott\varepsilon_{3}\,\mathfrak{r}+2k_{2}\cdott k_{3}\varepsilon_{1}\cdott\varepsilon_{3}))\varepsilon_{2}\cdott\varepsilon_{4}\\
+k_{2}\cdott k_{3}(2(k_{1}\cdott\varepsilon_{4}k_{2}\cdott\varepsilon_{3}\varepsilon_{1}\cdott\varepsilon_{2}-k_{1}\cdott\varepsilon_{3}(k_{2}\cdott\varepsilon_{4}\varepsilon_{1}\cdott\varepsilon_{2}+k_{1}\cdott\varepsilon_{2}\varepsilon_{1}\cdott\varepsilon_{4})\\
+k_{1}\cdott\varepsilon_{2}(k_{3}\cdott\varepsilon_{4}\varepsilon_{1}\cdott\varepsilon_{3}-k_{2}\cdott\varepsilon_{3}\varepsilon_{1}\cdott\varepsilon_{4}))-(2\mathfrak{u}_{13}\cdott\varepsilon_{2}-k_{3}\cdott\varepsilon_{2}\,\mathfrak{t})\varepsilon_{3}\cdott\varepsilon_{4})\\
+k_{3}\cdott k_{4}(2k_{2}\cdott\varepsilon_{3}\,\mathfrak{s}_{24}+k_{3}\cdott\varepsilon_{2}(2\mathfrak{s}_{34}-\mathfrak{r}\,\varepsilon_{3}\cdott\varepsilon_{4})))\\
+k_{2}\cdott k_{3}((k_{4}\cdott\varepsilon_{3}n_{a}\cdott\varepsilon_{4}+k_{3}\cdott n_{a}\varepsilon_{3}\cdott\varepsilon_{4})\mathfrak{p}\cdott\varepsilon_{2}-k_{1}\cdott\varepsilon_{1}k_{1}\cdott\varepsilon_{2}k_{3}\cdott\varepsilon_{4}n_{a}\cdott\varepsilon_{3}))\\
+k_{2}\cdott n_{a}(k_{3}\cdott k_{4}(k_{2}\cdott\varepsilon_{3}n_{a}\cdott\varepsilon_{2}\varepsilon_{1}\cdott\varepsilon_{4}k_{1}^{2}-k_{1}\cdott\varepsilon_{1}k_{1}\cdott\varepsilon_{4}(k_{2}\cdott\varepsilon_{3}n_{a}\cdott\varepsilon_{2}+k_{3}\cdott\varepsilon_{2}n_{a}\cdott\varepsilon_{3}))\\
+\varepsilon_{2}\cdott\varepsilon_{3}(k_{4}\cdott n_{a}((\mathfrak{t}\, k_{1}\cdott k_{3}-2k_{3}\cdott k_{4}k_{3}\cdott\varepsilon_{1})k_{3}\cdott\varepsilon_{4}+2k_{3}\cdott k_{4}\mathfrak{u}_{13}\cdott\varepsilon_{4})\\
+k_{3}\cdott k_{4}k_{3}\cdott n_{a}\mathfrak{p}\cdott\varepsilon_{4}))+k_{1}\cdott\varepsilon_{1}\,\varepsilon_{2}\cdott n_{a}\,\varepsilon_{3}\cdott n_{a}\,\varepsilon_{4}\cdott n_{a}\, k_{2}\cdott k_{3}\, k_{3}\cdott k_{4})\end{multline}
 where we have used \begin{gather}
\mathfrak{p}^{\alpha}=k_{1}\cdott\varepsilon_{1}\, k_{1}^{\alpha}-k_{1}^{2}\varepsilon_{1}^{\alpha},\\
\mathfrak{r}=k_{1}\cdott\varepsilon_{1}+2k_{2}\cdott\varepsilon_{1}-2k_{3}\cdott\varepsilon_{1},\\
\mathfrak{s}_{ij}=k_{1}^{\alpha}\varepsilon_{1}^{\beta}\left(\varepsilon_{i\,\alpha}\varepsilon_{j\,\beta}-\varepsilon_{i\,\beta}\varepsilon_{j\,\alpha}\right),\\
\mathfrak{t}=k_{1}\cdott\varepsilon_{1}+2k_{2}\cdott\varepsilon_{1},\\
\mathfrak{u}_{ij}^{\alpha}=k_{i}^{\alpha}k_{j}\cdott\varepsilon_{1}-k_{i}\cdott k_{j}\varepsilon_{1}^{\alpha}.\end{gather}
 We have not used any particular reference momenta for the polarization
vectors -- by applying different choices the above expression can
be simplified. Note, that we are allowed to play with the reference
momenta as the above amplitude is gauge invariant; the $\mathcal{W}$
piece is already included above (the last term). The other color-ordered
amplitudes are obtained by exchanging the relevant polarization vectors
and momenta in the expression above.

Using the spinor helicity formalism one can calculate those amplitudes
explicitly for different helicity combinations. For this purpose it
proves most convenient to choose $n_{a}$ as a reference vector for
each of the on-shell gluons. Labeling spinors associated with the
light-like momenta $n_{a}$, $k_{2}$, $k_{3}$, $k_{4}$ with $a$,
$2$, $3$, $4$ respectively, and further using the notation introduced
in Section \ref{sec:helic_color} (see also~\citep{Mangano:1987xk}),
we find \begin{align}
\tilde{\mathcal{A}}(\varepsilon_{2}^{-}(n_{a}),\varepsilon_{3}^{-}(n_{a}),\varepsilon_{4}^{-}(n_{a})) & 
  = 0,\\
\tilde{\mathcal{A}}(\varepsilon_{2}^{+}(n_{a}),\varepsilon_{3}^{+}(n_{a}),\varepsilon_{4}^{+}(n_{a})) & 
  = 0,\\
\tilde{\mathcal{A}}(\varepsilon_{2}^{-}(n_{a}),\varepsilon_{3}^{-}(n_{a}),\varepsilon_{4}^{+}(n_{a})) & 
  = \frac{g^{2}}{\sqrt{2}}\,\frac{[3|\kTslash|a\rangle}{\big|\vec{k}_{T}\big|}\,\frac{\sqr{4}{a}^{3}}{\sqr{3}{a}\sqr{a}{2}\sqr{2}{3}\sqr{3}{4}},\\
\tilde{\mathcal{A}}(\varepsilon_{2}^{+}(n_{a}),\varepsilon_{3}^{+}(n_{a}),\varepsilon_{4}^{-}(n_{a})) & 
  = \frac{g^{2}}{\sqrt{2}}\,\frac{\langle a|\kTslash|3]}{\big|\vec{k}_{T}\big|}\,\frac{\ang{4}{a}^{3}}{\ang{a}{3}\ang{a}{2}\ang{2}{3}\ang{3}{4}},\\
\tilde{\mathcal{A}}(\varepsilon_{2}^{-}(n_{a}),\varepsilon_{3}^{+}(n_{a}),\varepsilon_{4}^{+}(n_{a})) & 
  = \frac{g^{2}}{\sqrt{2}}\,\frac{\langle a|\kTslash|3]}{\big|\vec{k}_{T}\big|}\,\frac{\ang{a}{2}^{3}}{\ang{a}{3}\ang{2}{3}\ang{3}{4}\ang{4}{a}},\\
\tilde{\mathcal{A}}(\varepsilon_{2}^{+}(n_{a}),\varepsilon_{3}^{-}(n_{a}),\varepsilon_{4}^{-}(n_{a})) & 
  = \frac{g^{2}}{\sqrt{2}}\,\frac{[3|\kTslash|a\rangle}{\big|\vec{k}_{T}\big|}\,\frac{\sqr{a}{2}^{3}}{\sqr{3}{a}\sqr{2}{3}\sqr{3}{4}\sqr{4}{a}},\\
\tilde{\mathcal{A}}(\varepsilon_{2}^{-}(n_{a}),\varepsilon_{3}^{+}(n_{a}),\varepsilon_{4}^{-}(n_{a})) & 
  = \frac{g^{2}}{\sqrt{2}}\,\frac{[3|\kTslash|a\rangle}{\big|\vec{k}_{T}\big|}\,\frac{\sqr{3}{a}^{3}}{\sqr{a}{2}\sqr{2}{3}\sqr{3}{4}\sqr{4}{a}},\\
\tilde{\mathcal{A}}(\varepsilon_{2}^{+}(n_{a}),\varepsilon_{3}^{-}(n_{a}),\varepsilon_{4}^{+}(n_{a})) & 
  = \frac{g^{2}}{\sqrt{2}}\,\frac{\langle a|\kTslash|3]}{\big|\vec{k}_{T}\big|}\,\frac{\ang{a}{3}^{3}}{\ang{a}{2}\ang{2}{3}\ang{3}{4}\ang{4}{a}}.\end{align}
 All momenta are assumed to be incoming in the formulas above. In
what follows we skip the reference momenta indication in the amplitudes.

We may decompose $k_{T}^{\mu}$ in terms of $k_{3}^{\mu}$ and another
light-like momentum $p^{\mu}$ following
\begin{equation}
k_{T}^{\mu}=p^{\mu}+\frac{k_{T}^{2}}{2k_{T}\!\cdot\! k_{3}}\, k_{3}^{\mu}~.
\end{equation}
Then, we have
\begin{equation}
\big|\vec{k}_{T}\big|^2=-k_{T}^{2}=2\frac{(k_{3}\!\cdot\! p)(p\!\cdot\! n_{a})}{k_{3}\!\cdot\! n_{a}}~,
\end{equation}
and
\begin{equation}
\big|[3|\kTslash|a\rangle\big|^{2}=\big|[3|p\hspace{-4.5pt}/|a\rangle\big|^{2}=4|k_{3}\!\cdot\! p||p\!\cdot\! n_{a}|~,\end{equation}
so
\begin{equation}
\left|\frac{[3|\kTslash|a\rangle}{\big|\vec{k}_{T}\big|}\right|^{2}=4|k_{3}\!\cdot\! p||p\!\cdot\! n_{a}|\,\frac{|k_{3}\!\cdot\! n_{a}|}{2|k_{3}\!\cdot\! p||p\!\cdot\! n_{a}|}=2|k_{3}\!\cdot\! n_{a}|~.\end{equation}
 So for the squared helicity amplitudes, we find \begin{align}
|\tilde{\mathcal{A}}(\varepsilon_{2}^{-},\varepsilon_{3}^{-},\varepsilon_{4}^{+})|^{2} & =|\tilde{\mathcal{A}}(\varepsilon_{2}^{+},\varepsilon_{3}^{+},\varepsilon_{4}^{-})|^{2}=\frac{g^{4}}{2}\,\frac{\inp{k_{4}}{n_{a}}^{4}}{\inp{n_{a}}{k_{2}}\inp{k_{2}}{k_{3}}\inp{k_{3}}{k_{4}}\inp{k_{4}}{n_{a}}}\\
|\tilde{\mathcal{A}}(\varepsilon_{2}^{-},\varepsilon_{3}^{+},\varepsilon_{4}^{+})|^{2} & =|\tilde{\mathcal{A}}(\varepsilon_{2}^{+},\varepsilon_{3}^{-},\varepsilon_{4}^{-})|^{2}=\frac{g^{4}}{2}\,\frac{\inp{k_{2}}{n_{a}}^{4}}{\inp{n_{a}}{k_{2}}\inp{k_{2}}{k_{3}}\inp{k_{3}}{k_{4}}\inp{k_{4}}{n_{a}}}\\
|\tilde{\mathcal{A}}(\varepsilon_{2}^{-},\varepsilon_{3}^{+},\varepsilon_{4}^{-})|^{2} & =|\tilde{\mathcal{A}}(\varepsilon_{2}^{+},\varepsilon_{3}^{-},\varepsilon_{4}^{+})|^{2}=\frac{g^{4}}{2}\,\frac{\inp{k_{3}}{n_{a}}^{4}}{\inp{n_{a}}{k_{2}}\inp{k_{2}}{k_{3}}\inp{k_{3}}{k_{4}}\inp{k_{4}}{n_{a}}}~.\end{align}
Applying \Equation{eq:ampl_squared} we find agreement with \citep{Deak:2009xt}. The overall factor $(k_1\cdott k_2/k_1\cdott p_2)^2$ in that publication is due to a difference in the the definition of $p_2$ there and $n_a$ here.


\subsection{Comparison with Lipatov's effective action approach}

\begin{figure}
\epsfig{figure=figure16.eps,width=0.99\linewidth}
\caption{\small \label{Fig:Lipatov}The effective reggeon-gluon-gluon-gluon vertex
as obtained from \LipFig{6}, by expanding all blobs on the r.h.s\ into
graphs using \LipFig{4}.}
\end{figure}


Let us now show that the gluonic amplitudes with one off-shell leg
augmented with the `gauge restoring' amplitude (\ref{eq:waste-1})
are equivalent to the amplitudes obtained using the effective reggeon-gluon
vertices of~\cite{Antonov:2004hh}. In the vertices, no external gluon is necessarily
on-shell, and the reggeon essentially is an off-shell gluon with momentum
$k_{1}^{\mu}=E(n^{+})^{\mu}+k_{T}^{\mu}$, where $(n^{+})^{\mu}=n_{a}^{\mu}/E$.
If all gluons, except the reggeon, are on-shell, the vertices correspond
to our amplitudes, modulo a factor $E/\big|\vec{k}_{T}\big|$.

This can most easily be understood considering \Figure{Fig:Lipatov}.
It graphically depicts the reggeon-gluon-gluon-gluon vertex as presented
in~\cite{Antonov:2004hh}. Straight lines are gluons, and the wavy line attached
to a straight line is the off-shell gluon without a propagator, but
contracted with $\big|\vec{k}_{T}\big|^{2}(n^{+})^{\mu}=(\big|\vec{k}_{T}\big|/E)\mathfrak{e}_{1}^{\mu}$.
Here seems to be a difference with our approach, in which $\mathfrak{e}_{1}^{\mu}$
is contracted to the off-shell leg {\em including the propagator},
and in particular including the projector in the axial gauge. Notice,
however, that the collection of all graphs with a wavy line attached
to a straight line is exactly the set of usual Feynman graphs, calculated
with usual Feynman rules, and before contracting them with $\mathfrak{e}_{1}^{\mu}$
their sum forms an {\em off-shell current} $J_{1}^{\mu}$, satisfying
current conservation
\begin{equation}
k_{1}\cdott J_{1}=0.\label{eq:curcons}\end{equation}

The off-shell current is not gauge-invariant, but \Equation{eq:curcons}
holds in any gauge. Consequently, we have
\begin{equation}
\mathfrak{e}_{1}^{\mu}d_{\mu\nu}(k_{1},n)J_{1}^{\nu}=\mathfrak{e}_{1}\cdott J_{1}-\frac{\mathfrak{e}_{1}\cdott k_{1}}{n\cdott k_{1}}\, n\cdott J_{1}=\mathfrak{e}_{1}\cdott J_{1},\end{equation}

where the second equality follows from the fact that $\mathfrak{e}_{1}\cdott k_{1}=\big|\vec{k}_{1T}\big| n_{a}\cdott k_{1}=0$.
So we see that it does not matter for the off-shell external line
whether the projector is present or not, in any gauge.

The vertices in~\Figure{Fig:Lipatov} indicated by a square represent
the effective vertices of \LipEqn{18}, which have the essential
feature that all gluons attached to it are contracted with $(n^{+})^{\mu}$.
Since the results of~\cite{Antonov:2004hh}\ are gauge-invariant, the graphs can
be calculated in the axial gauge with gauge-vector $(n^{+})^{\mu}$,
with the consequence that the last two graphs vanish, since they contain
a gluon propagator attached to the square vertex. The fourth graph
on the r.h.s.\ stays, and can readily be recognized as our `gauge
restoring' amplitude (\ref{eq:waste-1}).
The other graphs are the usual graphs one would take into account
in a fully on-shell calculation, and are calculated following the
usual Feynman rules. The same argumentation holds for vertices with
arbitrary number of on-shell gluons. All graphs containing square
vertices vanish, except one which corresponds to (\ref{eq:waste-1}),
and the other graphs are the usual ones taken into account in a fully
on-shell calculation.


\section{Summary and outlook}

\label{sec:summary}


We provided a prescription to calculate gauge-invariant multi-gluon
scattering amplitudes within high energy factorization, for which one initial state
gluon is off-shell. Gauge-invariance was ensured by employing Slavnov-Taylor identities,
enabling the construction of the necessary extra contributions to the amplitude besides those coming from the usual Feynman graphs.
In particular, we presented compact expressions for the helicity
amplitudes of the process $g^{*}g\rightarrow gg$, and observed that they are in agreement with existing calculations. Also, we found agreement of our prescription with the effective
action approach of Lipatov.
The generalization to two off-shell legs is currently under study.





\section*{Acknowledgments}

The authors are grateful to F. Hautmann for extensive discussions
and reading the preceding notes. P.K. would like to thank also H. Arodz, L. Motyka, W.~Slominski and L. Szymanowski.
We also acknowledge useful discussions with M. Deak and H.~Jung.

During this research P.K. and K.K have been supported by NCBiR grant
LIDER/02/35/L-2/10/NCBiR/2011.

\nocite{Arodz:2010}
\nocite{Mangano:1990by}

\bibliographystyle{apsrevM}
