aainstitutetext: Higgs Centre for Theoretical Physics, School of Physics and
Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, Scotland,
UKbbinstitutetext: Department of Physics and Astronomy, The University of
Manchester, Manchester M13 9PL, UK

# One-loop five-parton amplitudes in the NMRK limit

Emmet P. Byrne<EMAIL_ADDRESS>
###### Abstract

We analyse the real part of one-loop five-parton amplitudes in the next-to-
multi-Regge kinematic (NMRK) limit, to leading power, and to finite order in
the dimensional regularisation parameter. To leading logarithmic (LL)
accuracy, it is known that five-parton amplitudes in this limit are given to
all-orders by a single factorised expression, in which the pair of partons
which are not well-separated in rapidity are described by a two-parton
emission vertex. In this study, we investigate the one-loop amplitudes at
next-to-leading logarithmic (NLL) accuracy, and find that is has a more
complex structure. In particular, it is found that the purely gluonic
amplitudes are compatible with an analogous factorisation of individual colour
structures. From the one-loop amplitudes we extract one-loop two-parton
emission vertices, which are functions of a subset of the momenta of the
amplitude. In the multi-Regge kinematic (MRK) limit, the vertices themselves
factorise into the known one-loop single-parton emission vertices and Lipatov
vertex, with rapidity dependence governed by the one-loop gluon Regge
trajectory, as required by compatibility with the known MRK limit of
amplitudes. The one-loop two-parton emission vertices are necessary
ingredients for the construction of the next-to-next-to leading order (NNLO)
jet impact factors in the BFKL framework.

###### Keywords:

QCD, Factorisation, Regge limit, BFKL

##  1 Introduction

The multi-Regge kinematic (MRK) limit of amplitudes, in which final-state
particles are strongly ordered in rapidity, has long been a source of insight
into the all-orders structure of amplitudes Lipatov:1976zz ; DelDuca:2019tur .
Knowledge of the MRK limit has been used to compute or constrain amplitudes in
general kinematics DelDuca:2009au ; DelDuca:2010zg ; Dixon:2014iba ;
Henn:2016jdu ; Caron-Huot:2016owq ; Almelid:2017qju ; Caron-Huot:2019vjl ;
Falcioni:2021buo ; Caola:2021izf . Historically, the next-to-multi-Regge
kinematic (NMRK) limit (or single-Regge limit), where one of the strong
rapidity-ordering requirements is relaxed, has received far less attention
than the more tractable MRK limit, with some notable exceptions, including
refs. White:1973ola ; Brower:1974yv . The NMRK limit is an intriguing
theoretical arena, where on the one hand the simplified kinematics make
possible a detailed study of high-multiplicity amplitudes, revealing
structures which are obscured in general kinematics Bartels:2008ce ;
Byrne:2022wzk , and on the other hand, the NMRK limit captures more of the
complexity and richness of amplitudes than does the MRK limit. The NMRK limit
and its MRK boundaries provide an ideal test bed for studying which properties
of MRK amplitudes generalise to more generic kinematic spaces. In this work,
we find that the known MRK limit of amplitudes motivates a highly convenient
framework for organising the kinematic structure of the amplitudes in the NMRK
limit. In particular, this organisation makes it trivial to take the further
MRK limit of NMRK-factorised expressions.

The behaviour of amplitudes in Regge limits is important for determining the
asymptotic high-energy limit of cross sections in QCD Lipatov:1976zz ;
Kuraev:1976ge . The leading-logarithmic (LL) behaviour of amplitudes in QCD is
described by the $t$-channel exchange of a so-called Reggeised gluon. In the
language of Regge theory (for pedagogical reviews see refs. Collins:1977jy ;
White:2019ggo ; Mizera:2023tfe ), the amplitude is governed by a simple Regge
pole in the complex angular momentum plane. The Balitsky-Fadin-Kuraev-Lipatov
(BFKL) equation Lipatov:1976zz ; Kuraev:1976ge ; Kuraev:1977fs ;
Balitsky:1978ic  sums the LL contributions to QCD scattering processes to all
orders in the coupling (for pedagogical introductions see refs. DelDuca:1995hf
; Forshaw:1997dc ; Fadin:1998sh ). The leading order (LO) kernel of this
equation consists of the square of the Lipatov vertex Lipatov:1976zz , which
can be obtained by taking the MRK limit of tree-level amplitudes. At next-to-
leading logarithmic (NLL) accuracy, the real part of amplitudes are similarly
described by a simple Regge pole Fadin:2006bj ; Fadin:2015zea , which allows
the BFKL approach to be extended to NLL accuracy Fadin:1998py ;
Ciafaloni:1998gs ; Kotikov:2000pm ; Kotikov:2002ab . This requires the kernel
to be extended to next-to-leading order (NLO). The NLO kernel may be computed
using building blocks which can be obtained by taking the MRK limit of one-
loop amplitudes as well as building blocks which can be obtained by taking the
NMRK limit of tree-level amplitudes. At next-to-next-to leading-logarithmic
(NNLL) accuracy, the real part of the amplitude is described by a Regge cut in
addition to a Regge pole Fadin:2020lam ; Caron-Huot:2017fxr ; DelDuca:2001gu ;
Fadin:2017nka . Recently, schemes have been introduced to disentangle the pole
and cut contributions at fixed order Falcioni:2021dgr ; Fadin:2021csi . In
particular, the three-loop Regge trajectory has recently been determined
DelDuca:2021vjq ; Caola:2021izf ; Caola:2022dfa . It is hoped that the BFKL
approach may yet be extended to describe the evolution of the Regge-pole
contribution.

As reviewed in ref. Byrne:2022wzk , there has been much progress towards
extending the BFKL approach to NNLL accuracy. More recently, knowledge of the
one-loop Lipatov vertex has been extended to ${\mathcal{O}}(\epsilon^{2})$
Fadin:2023roz , as required for the NNLO kernel, and the recent calculation of
two-loop five-parton amplitudes in QCD Agarwal:2023suw ; DeLaurentis:2023nss ;
DeLaurentis:2023izi  has made possible the extraction of the two-loop Lipatov
vertex.

Figure 1: Ingredients of the NNLO jet impact factors: (a)-(d) double-virtual
corrections; (e)-(g) real-virtual corrections; (h)-(j) double-real
corrections. Punctures in blobs denote loop corrections. An asterisk indicates
an off-shell gluon. The grey dashed line indicates the final-state cut.

In order to provide NNLL accurate predictions for physical processes it is
necessary to also know not only the kernel, but also the relevant impact
factors to NNLO accuracy. Let us briefly review the current status of the NNLO
impact factors for the production of a jet, which are depicted in fig. 1. The
double-virtual corrections to the impact factors (figs. 1, 1, 1 and 1) require
knowledge of the two-loop single-parton peripheral-emission111The term _impact
factor_ has been used in the literature to refer to both amplitude-level and
cross-section-level quantities. We will use the term impact factor exclusively
in the latter context, and refer to the amplitude-level factorised expressions
as peripheral-emission vertices, defined in opposition to the central-emission
vertices of ref. Byrne:2022wzk . More often, we will refer to the vertices by
their explicit parton types, taking all momenta outgoing, e.g. “the
${\bar{q}}qgg^{*}$ vertex”. Such vertices are also commonly called parton-
parton-parton-Reggeon (PPPR) vertices, but in this work we do not wish to risk
implying that the factorisation of amplitudes at one loop implies Reggeisation
at higher orders. We use the term _Lipatov vertex_ to refer to the
$g^{*}gg^{*}$ central-emission vertex at any loop order. vertices
DelDuca:2014cya  as well as the interference of two one-loop single-parton
peripheral-emission vertices Fadin:1992rh ; Fadin:1992zt ; Fadin:1995km ;
DelDuca:1998kx . The double-real corrections to the impact factors (figs. 1, 1
and 1) require knowledge of the three-parton emission vertices. These are
known for the pure-gluon case DelDuca:1999iql ; Antonov:2004hh ; Duhr:2009uxa
, but the tree-level ${\bar{q}}qggg^{*}$ vertices are not yet known. In this
paper we obtain some of the missing ingredients for the real-virtual
correction to the impact factors (figs. 1, 1 and 1), namely the one-loop two-
parton peripheral-emission vertices for both the $gggg^{*}$ and
${\bar{q}}qgg^{*}$ case. The one-loop $gggg^{*}$ vertex has previously been
determined for two colour-orderings and helicity-independence of this vertex
was incorrectly assumed Canay:2021yvm . In section 6 we clarify the need for
determining a third distinct colour ordering of this vertex, and in section 7
we show that two distinct helicity configurations of this vertex are required
for the NNLO impact factor.

To regulate the infra-red divergences which will appear when performing the
phase space integration of the two-parton emission vertices it will be
necessary to know the one-loop two-parton peripheral-emission vertices to
order $\epsilon$ in the soft and collinear limits and to order $\epsilon^{2}$
in the soft-collinear limit. In this paper we focus exclusively on finite
(${\mathcal{O}}(\epsilon^{0})$) corrections, which can be inferred from
available one-loop amplitudes Bern:1993mq ; Bern:1994fz . These amplitudes
already reveal some interesting structure of amplitudes in the NMRK limit
beyond tree-level. We find that, in contrast to the tree-level amplitudes, the
one-loop amplitudes are not compatible with an all-orders factorisation of the
amplitude. Rather, they are at best compatible with an all-orders
factorisation of two different colour structures of the amplitude.

The one-loop two-parton emission vertices also contain additional colour
structures compared to the tree-level vertices. For the $gggg^{*}$ vertex,
this is a totally symmetric colour structure, while for the ${\bar{q}}qgg^{*}$
vertex it is a product of delta functions in the adjoint and fundamental
representations. In the latter case, the new colour structure survives
interference with the tree-level vertex, hence contributes to the NNLO kernel.

The factorisation-structure and colour-structure of the ${\mathcal{N}}{=}4$
part of the one-loop $gggg^{*}$ is analogous to the structure of the
${\mathcal{N}}{=}4$ $g^{*}ggg^{*}$ vertex obtained in ref. Byrne:2022wzk . We
expect the one-loop $g^{*}ggg^{*}$ and $g^{*}{\bar{q}}qg^{*}$ vertices in QCD
to have analogous structure to the $gggg^{*}$ and ${\bar{q}}qgg^{*}$ vertices
presented in this work.

The structure of this paper is as follows. In section 2 we review the relevant
properties of QCD amplitudes in the high-energy limit. In section 3 we
introduce a minimal set of kinematic variables to describe $2\to 3$ scattering
which we will use throughout the paper. In section 4 we will review the
properties of tree-level five-parton amplitudes in the NMRK limit and in
section 5 we will review the properties of one-loop four parton amplitudes in
the Regge limit. These two chapters provide the necessary background for the
study of one-loop five-parton amplitudes, which is the topic of the remaining
chapters. In section 6 we study the colour structure of one-loop five-parton
amplitudes at leading power in the NMRK limit. In section 7 and section 8 we
study the respective factorisation properties of colour-ordered and colour-
dressed one-loop five-gluon amplitudes in the NMRK limit. In section 9 and
section 10 we perform the analogous studies of one-loop five-parton amplitudes
with an external quark-antiquark pair. Finaly, we conclude in section 11.

##  2 Review of high-energy limit of amplitudes in QCD

Our conventions for general kinematics are given in section A.1 and our
notation for kinematics in Regge limits is given in section A.2. In this
section we briefly review the all-orders structure of amplitudes in the Regge
limit, given by eq. 292. At tree-level, amplitudes are dominated by the
$t$-channel exchange of a gluon, and to leading power have the factorised form
Lipatov:1976zz ; Kuraev:1976ge

$\displaystyle\begin{split}\mathcal{A}_{4}^{(0)}(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},f_{4}^{\lambda_{3}},f_{1}^{\lambda_{3}})&\underset{\mathrm{Regge}}{\to}2s\
\mathcal{C}^{(0)}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)\
\frac{1}{t}\
\mathcal{C}^{(0)}_{g^{*}f_{4}f_{1}}(-q,p_{4}^{\lambda_{4}},p_{1}^{\lambda_{1}})\,.\end{split}$
(1)

Figure 2: Illustrations of the factorised forms of amplitudes discussed in the
text: (a) factorisation of the $2\to 2$ amplitude in the Regge limit; (b)
factorisation of the $2\to 3$ amplitude in the MRK limit; (c) factorisation of
the $2\to 3$ amplitude in the forward NMRK limit. The green and blue blobs
denote peripheral- and central-emission vertices respectively, to all orders
in the coupling $g_{\mathrm{s}}$. The solid external lines denote the
permitted parton types, as given in the main text. The zig-zag lines denote a
Reggeised gluon.

All the process-dependent information is contained in the vertices, which are
only non-zero when the representation of the external partons permits the
$t$-channel exchange of an anti-symmetric octet:

$\displaystyle\mathcal{C}^{(0)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)$
$\displaystyle=g_{\mathrm{s}}\,(F^{c})_{a_{2}a_{3}}\,C^{(0)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)\,,$
(2a)
$\displaystyle\mathcal{C}^{(0)}_{{\bar{q}}qg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)$
$\displaystyle=g_{\mathrm{s}}\,(T^{c})_{\imath_{3}\bar{\imath}_{2}}\,C^{(0)}_{{\bar{q}}qg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)\,.$
(2b)

Here $T$ are generators in the fundamental representation of $SU(N_{c})$ with
normalisation $\mbox{tr}\left(T^{a}T^{b}\right)=\delta_{ab}$222We acknowledge
the use of the Mathematica package ColorMath Sjodahl:2012nk  in computing
colour factors in this work.. The structure constants, $f$, follow from the
commutatation relation

$[T^{a},T^{b}]_{ij}=if^{abc}\,T^{c}_{ij}.$ (3)

We often make use of the notation $(F^{b})_{ac}=if^{abc}$ for generators in
the adjoint representation, whereby the Jacobi identity is

$[F^{a},F^{b}]_{d_{i}d_{j}}=if^{abc}F^{c}_{d_{i}d_{j}}.$ (4)

Equation 2 introduces several conventions we use throughout this paper. In
order to emphasise the analogy between amplitudes and factorised vertices we
have listed the three outgoing momenta of the vertices. Although this is
redundant, as the vertices conserve momentum, this notation will be helpful
when keeping track of the colour-orderings of higher-multiplicity vertices.
For amplitudes and vertices alike, we use caligraphic type for colour-dressed
objects and roman type for colour-stripped objects. In the case of eq. 2, the
colour-stripped vertices are simply helicity conserving phases. Using the
conventions of section A.1, these phases are DelDuca:1995zy ; DelDuca:1996km

$\displaystyle C^{(0)}_{ggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},q)$
$\displaystyle=1\,,\qquad
C^{(0)}_{g^{*}gg}(-q,p_{4}^{\oplus},p_{1}^{\ominus})=-\frac{q_{\perp}^{*}}{q_{\perp}}\,,$
(5a) $\displaystyle
C^{(0)}_{{\bar{q}}qg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},q)$
$\displaystyle=1\,,\qquad
C^{(0)}_{g^{*}{\bar{q}}q}(-q,p_{4}^{\oplus},p_{1}^{\ominus})=\left(\frac{q_{\perp}^{*}}{q_{\perp}}\right)^{\frac{1}{2}}\,.$
(5b)

The vertices are antisymmetric under $p_{2}^{\lambda_{2}}\leftrightarrow
p_{3}^{\lambda_{3}}$ or $p_{4}^{\lambda_{4}}\leftrightarrow
p_{1}^{\lambda_{1}}$, and parity conjugation of the vertices is given by
complex conjugation. To leading-logarithmic (LL) accuracy in $\frac{s}{t}$,
the amplitudes are given to all-orders by the factorised form Lipatov:1976zz ;
Kuraev:1976ge

$\displaystyle\begin{split}\mathcal{A}_{4}(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},f_{4}^{\lambda_{3}},f_{1}^{\lambda_{3}})&\underset{\mathrm{Regge}}{\to}2s\
\mathcal{C}^{(0)}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)\
\frac{1}{t}\left(\frac{s}{\tau}\right)^{\alpha(t)}\
\mathcal{C}^{(0)}_{g^{*}f_{4}f_{1}}(-q,p_{4}^{\lambda_{4}},p_{1}^{\lambda_{1}})\,,\end{split}$
(6)

which is illustrated in fig. 2. Here $\tau$ is some positive scale of the
order $t$. We consider the amplitude to have the expansion in the bare
coupling $g_{\mathrm{s}}$,

$\mathcal{A}_{4}=\mathcal{A}_{4}^{(0)}+\mathcal{A}_{4}^{(1)}+{\cal
O}(g_{\mathrm{s}}^{6})\,,$ (7)

where the superscript $(l)$ denotes the loop-order and
$\mathcal{A}^{(l)}_{4}\sim{\cal O}(g_{\mathrm{s}}^{2(1+l)})$. In eq. 6, the
virtual corrections are given to all orders by the exponentiation of the one-
loop Regge trajectory Lipatov:1976zz ; Kuraev:1976ge

$\displaystyle\alpha^{(1)}(t)=c_{\Gamma}\ g_{\mathrm{s}}^{2}\
\frac{2N_{c}}{\epsilon}\left(\frac{\mu^{2}}{-t}\right)^{\epsilon}\,,$ (8)

where $c_{\Gamma}$ is the ubiquitous loop factor

$c_{\Gamma}=\frac{1}{(4\pi)^{2-\epsilon}}\frac{\Gamma(1+\epsilon)\Gamma(1-\epsilon)^{2}}{\Gamma(1-2\epsilon)}\,.$
(9)

Beyond LL, it is useful to introduce the concept of signature, or symmetry
under crossing $s\leftrightarrow u$. In the Regge limit this is equivalent to
$s\leftrightarrow-s$, and signature-even ($[+]$) and signature-odd ($[-]$)
ampltiudes are given by

$\displaystyle\begin{split}\mathcal{A}_{4}^{[\pm]}(s,t)=\frac{1}{2}\Big{(}&\mathcal{A}_{4}(s,t)\pm\mathcal{A}_{4}(-s,t)\Big{)}\,.\end{split}$
(10)

At NLL, a factorised form analogous to eq. 6 only holds for the signature-odd
amplitude. The generalisation of eq. 6 to NLL accuracy is Caron-Huot:2017fxr ;
Fadin:1993wh

$\displaystyle\begin{split}{\mathrm{Disp}\left[{\mathcal{A}^{[-]}_{4}(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},f_{4}^{\lambda_{4}},f_{1}^{\lambda_{1}})}\right]}&\underset{\mathrm{Regge}}{\to}2s\
\mathcal{C}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)\
\mathcal{R}_{g^{*}}(q;s)\
\mathcal{C}_{g^{*}f_{4}f_{1}}(-q,p_{4}^{\lambda_{4}},p_{1}^{\lambda_{1}})\,.\end{split}$
(11)

In this paper we only consider the dispersive part of amplitudes, defined by
taking the real part of the transcendental functions. For $2\to 2$ scattering
the Regge cut starts to contribute to the absorptive part of the signature-
even amplitude at one-loop, and (beyond the planar limit) to the dispersive
part of the signature-odd amplitude at two-loops Falcioni:2021dgr . As
outlined in the introduction, our interest is in the Regge-pole contribution
to the amplitude, so we limit our analysis to the dispersive part of the
signature-odd amplitude. In eq. 11 we have introduced the notation

$\displaystyle\mathcal{R}_{g^{*}}(q;s)=\frac{1}{t}\times\frac{1}{2}\
\Bigg{[}\bigg{(}\frac{s}{\tau}\bigg{)}^{\alpha(t)}+\bigg{(}\frac{-s}{\tau}\bigg{)}^{\alpha(t)}\Bigg{]}\,,$
(12)

for a signature-even Reggeisation factor. This factor is relevant for the
signature-odd amplitude due to the overall factor of $s$ in eq. 11. We
consider the factorised terms in eq. 11 to have an expansion in the coupling.
At NLL we need to consider the expansion of the peripheral-emission vertices
to one loop,

$\displaystyle\mathcal{C}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)=\mathcal{C}^{(0)}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)+\mathcal{C}^{(1)}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)+\mathcal{O}\left(g_{\mathrm{s}}^{5}\right)$
(13)

with the (unrenormalised) one-loop correction to eq. 2a given by Fadin:1992rh
; Fadin:1992zt ; DelDuca:1998kx

$\displaystyle\mathcal{C}^{(1)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},q)=g_{\mathrm{s}}^{3}\,c_{\Gamma}\,F^{c}_{a_{2}a_{3}}\,C^{(0)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},q)$
(14a)
$\displaystyle\times\Bigg{\\{}\left(\frac{\mu^{2}}{-t}\right)^{\epsilon}\left[N_{c}\left(-\frac{2}{\epsilon^{2}}-\frac{11}{6\varepsilon}+\frac{1}{\varepsilon}\log\left(\frac{\tau}{-t}\right)-\frac{32}{9}-\frac{\delta_{R}}{6}+\frac{\pi^{2}}{2}\right)+N_{f}\left(\frac{1}{3\varepsilon}+\frac{5}{9}\right)\right]\Bigg{\\}}$
and the (unrenormalised) one-loop correction to eq. 2b given by Fadin:1995km ;
DelDuca:1998kx
$\displaystyle\mathcal{C}^{(1)}_{{\bar{q}}qg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},q)=g_{\mathrm{s}}^{3}\,c_{\Gamma}\,T^{c}_{\imath_{3}\bar{\imath}_{2}}\,C^{(0)}_{{\bar{q}}qg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},q)$
(14b)
$\displaystyle\times\Bigg{\\{}\left(\frac{\mu^{2}}{-t}\right)^{\epsilon}\bigg{[}N_{c}\left(-\frac{1}{\epsilon^{2}}+\frac{1}{3\epsilon}+\frac{1}{\varepsilon}\log\left(\frac{\tau}{-t}\right)+\frac{19}{18}-\frac{\delta_{R}}{3}+\frac{\pi^{2}}{2}\right)-N_{f}\left(\frac{1}{3\epsilon}+\frac{5}{9}\right)$
$\displaystyle\qquad+\frac{1}{N_{c}}\left(\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{7}{2}+\frac{\delta_{R}}{2}\right)\bigg{]}.$

The above vertices are written in terms of the regularisation parameter
$\delta_{R}$, where $\delta_{R}=1$ in CDR/HV schemes and $\delta_{R}=0$ in the
DR/FDH schemes Catani:1996pk . In the following, because we will adopt an
organisation of amplitudes into supersymmetric multiplets, we set
$\delta_{R}=0$. At one-loop the helicity-violating $ggg^{*}$ vertex
Fadin:1993qb ; DelDuca:1998kx ,

$\displaystyle\begin{split}\mathcal{C}^{(1)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},q)&=g_{\mathrm{s}}^{3}F^{c}_{a_{2}a_{3}}(N_{c}-N_{f})\frac{1}{3(4\pi)^{2}}\left(\frac{-q_{\perp}^{*}}{q_{\perp}}\right)\,,\end{split}$
(15a)
$\displaystyle\begin{split}\mathcal{C}^{(1)}_{ggg^{*}}(p_{4}^{\oplus},p_{1}^{\oplus},q)&=g_{\mathrm{s}}^{3}F^{c}_{a_{4}a_{1}}(N_{c}-N_{f})\frac{1}{3(4\pi)^{2}}\,,\end{split}$
(15b)

also contributes at leading-power in the Regge limit. This vertex did not
contribute to the NLO impact factors, but the square of eq. 15 does contribute
to the NNLO impact factors.

In eq. 11 we must also consider the expansion of the Regge trajectory to two
loops,

$\displaystyle\alpha(t)=\alpha^{(1)}(t)+\alpha^{(2)}(t)+\mathcal{O}\left(g_{\mathrm{s}}^{6}\right)\,,$
(16)

but as the subject of this paper is the analysis of one-loop amplitudes, we
will only need eq. 8. The expansion eq. 16 is equivalent to an expansion of
the Reggeised propagator,

$\displaystyle\mathcal{R}_{g^{*}}(q;s)=\mathcal{R}_{g^{*}}^{(0)}(q)+\mathcal{R}_{g^{*}}^{(1)}(q;s)+\mathcal{O}\left(g_{\mathrm{s}}^{3}\right)\,.$
(17)

As for the vertices, eq. 2, it is useful to consider the kinematic part of eq.
17, stripped of colour factors and coupling constants, that is

$\displaystyle\mathcal{R}^{(0)}_{g^{*}}(q)$
$\displaystyle=R^{(0)}_{g^{*}}(q)\,,$ (18)
$\displaystyle\mathcal{R}^{(1)}_{g^{*}}(q;s)$
$\displaystyle=g_{\mathrm{s}}^{2}\,N_{c}\,R^{(1)}_{g^{*}}(q;s)\,,$ (19)

where for clarity, we list the colour-stripped factors at tree-level and one-
loop accuracy:

$\displaystyle R^{(0)}_{g^{*}}(q)$ $\displaystyle=\frac{1}{t}\,,$ (20)
$\displaystyle R^{(1)}_{g^{*}}(q;s)$
$\displaystyle=c_{\Gamma}\frac{1}{t}\frac{2}{\epsilon}\left(\frac{\mu^{2}}{t}\right)^{\epsilon}\log\left(\frac{s}{\tau}\right)\,.$
(21)

The factorised form of eq. 11 generalises beyond the $2\to 2$ case, for
example, the MRK limit of the $2\to 3$ amplitude (eq. 293) is Lipatov:1976zz ;
Kuraev:1976ge

$\displaystyle\begin{split}{\mathrm{Disp}\left[{\mathcal{A}_{5}^{[-,-]}\left(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},g_{4}^{\lambda_{4}},f_{5}^{\lambda_{5}},f_{1}^{\lambda_{1}}\right)}\right]}\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}2s\
&\mathcal{C}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q_{1})\
\mathcal{R}_{g^{*}}(q_{1};s_{34})\\\
\times&\mathcal{V}_{g^{*}gg^{*}}(-q_{1},p_{4}^{\lambda_{4}},q_{2})\\\
\times&\mathcal{R}_{g^{*}}(q_{2};s_{45})\
\mathcal{C}_{g^{*}f_{5}f_{1}}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})\,,\end{split}$
(22)

which is valid to LL accuracy at amplitude level. Here the amplitude consists
of antisymmetric colour representations exchanged in both $t_{1}$ and $t_{2}$
channels. As above, we restrict our study to the dispersive part of the
ampltiude. The phases of amplitudes in the MRK are properly treated with the
analytic representation of amplitudes given in refs. Fadin:1993wh ;
Bartels:1980pe ; Bartels:2008ce  but eq. 22 is sufficient for our present
purposes.

The central-emission vertex has an expansion in the coupling, analogous to eq.
13. In the present work we will need to know this vertex to one-loop accuracy,

$\displaystyle\begin{split}\mathcal{V}_{g^{*}gg^{*}}(-q_{1},p_{4}^{\lambda_{4}},q_{2})=\mathcal{V}_{g^{*}gg^{*}}^{(0)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})+\mathcal{V}_{g^{*}gg^{*}}^{(1)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})+\mathcal{O}\left(g_{\mathrm{s}}^{5}\right)\,.\end{split}$
(23)

The leading-order Lipatov vertex is given by

$\displaystyle\begin{split}\mathcal{V}_{g^{*}gg^{*}}^{(0)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})&=i\,g_{\mathrm{s}}\,f^{c_{1}c_{2}a_{4}}\,V_{g^{*}gg^{*}}^{(0)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})\,,\\\
\end{split}$ (24)

with colour-stripped vertex

$\displaystyle\begin{split}V^{(0)}_{g^{*}gg^{*}}(-q_{1},p_{4}^{\lambda_{4}},q_{2})&=\frac{q_{1\perp}^{\ast}q_{2\perp}}{p_{4\perp}}\,,\\\
\end{split}$ (25)

and the (unrenormalised) one-loop Lipatov vertex is given by Fadin:1993wh ;
Fadin:1994fj ; DelDuca:1998cx

$\displaystyle\begin{split}&\mathcal{V}_{g^{*}gg^{*}}^{(1)}(-q_{1},p_{4}^{\oplus},q_{2})=i\,c_{\Gamma}\,g_{\mathrm{s}}^{3}\,f^{c_{1}c_{2}a_{4}}\,V_{g^{*}gg^{*}}^{(0)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})\\\
&\times\Bigg{\\{}N_{c}\bigg{[}-\frac{1}{\epsilon^{2}}\left(\frac{\mu^{2}}{\left|p_{4\perp}\right|^{2}}\right)^{\epsilon}+\frac{1}{\epsilon}\left(\left(\frac{\mu^{2}}{-t_{1}}\right)^{\epsilon}+\left(\frac{\mu^{2}}{-t_{2}}\right)^{\epsilon}\right)\log\left(\frac{\tau}{\left|p_{4\perp}\right|^{2}}\right)-\frac{1}{2}\log^{2}\left(\frac{t_{1}}{t_{2}}\right)+\frac{\pi^{2}}{3}\bigg{]}\\\
&\qquad-\frac{\beta_{0}}{2}\left(t_{1}+t_{2}+2q_{1\perp}q_{2\perp}^{*}\right)\frac{L_{0}\left(t_{1}/t_{2}\right)}{t_{2}}\\\
&\qquad-\frac{(N_{c}-N_{f})}{3}\left|p_{4\perp}\right|^{2}\left[\left(2t_{1}t_{2}+\left(t_{1}+t_{2}+2\left|p_{2\perp}\right|^{2}\right)q_{1\perp}q_{2\perp}^{*}\right)\frac{L_{2}\left(t_{1}/t_{2}\right)}{t_{2}^{3}}+\frac{q_{1\perp}q_{2\perp}^{*}}{2t_{1}t_{2}}\right]\Bigg{\\}}\,.\end{split}$
(26)

The $L_{i}$ functions are defined in appendix B, and the coefficient of the
one-loop beta function in QCD is

$\beta_{0}=\frac{11N_{c}-2N_{f}}{3}\,.$ (27)

In this paper we are concerned with the factorisation of amplitudes in the
NMRK limit, eq. 294. The factorised form

$\displaystyle\begin{split}{\mathrm{Disp}\left[{\mathcal{A}_{5}^{[-]}\left(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},f_{4}^{\lambda_{3}},f_{5}^{\lambda_{5}},f_{1}^{\lambda_{1}}\right)}\right]}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s\
&\mathcal{C}_{f_{2}f_{3}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})\\\
\times&\mathcal{R}_{g^{*}}(q_{2};s_{45})\\\
\times&\mathcal{C}_{g^{*}f_{5}f_{1}}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})\,,\end{split}$
(28)

is valid to LL accuracy, as detailed in section 4. Here the superscript $[-]$
now denotes the exchange of anti-symmetric colour representations in just the
$t_{2}$ channel. In this paper we investigate whether eq. 28 holds also to NLL
accuracy, which would allow us to define a one-loop correction to the two-
parton emission vertex of the form

$\mathcal{C}^{(1)}_{f_{2}f_{3}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2}).$

We impose the requirement that this vertex factorises in the further MRK and
${\mathrm{MRK}^{\prime}}$ limits according to eq. 22, i.e.

$\displaystyle\mathcal{C}_{f_{2}f_{3}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})$
$\displaystyle\underset{y_{3}\gg
y_{4}}{\to}\mathcal{C}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q_{1})\mathcal{R}_{g^{*}}(q_{1};s_{34})\mathcal{V}_{g^{*}gg^{*}}^{(0)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})\,,$
(29)
$\displaystyle\mathcal{C}_{f_{2}f_{3}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})$
$\displaystyle\underset{y_{4}\gg
y_{3}}{\to}\mathcal{C}_{f_{2}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{4}^{\lambda_{4}},q_{1}^{\prime})\mathcal{R}_{g^{*}}(q_{1}^{\prime};s_{43})\mathcal{V}_{g^{*}gg^{*}}^{(0)}(-q_{1}^{\prime},p_{3}^{\lambda_{3}},q_{2})\,,$
(30)

whenever the partonic identities $f_{i}$ permit an antisymmetric octet
exchange in the $t$\- or $t^{\prime}$\- channel respectively. We will find
that eq. 28 holds for the $\bar{q}qgg^{*}$ case, but it fails in the
$gggg^{*}$ case. The pure-gluon case is the only partonic channel which has
leading-power behaviour in both MRK and ${\mathrm{MRK}^{\prime}}$ limits, and
we find that as a consequence, eq. 28 must be modified to include at least two
such factorised terms, as in eq. 225. In the $\bar{q}qgg^{*}$ case, the
amplitude is power suppressed in one or both of the MRK and
${\mathrm{MRK}^{\prime}}$ limits, depending on whether a fermion or gluon is
physically incoming. This enables us to define a colour dressed vertex, eq.
276, for which the expansion of eq. 28 is correct up to
${\mathcal{O}}(g_{s}^{5})$.

Of course, the all-orders structure of amplitudes cannot be determined from an
analysis of one-loop amplitudes. Vital clues about the all-orders structure of
the five-parton ampliudes in the NMRK limit can be obtained from the two-loop
five parton amplitudes, which have recently been computed Agarwal:2023suw ;
DeLaurentis:2023nss ; DeLaurentis:2023izi . It is hoped that the current paper
will provide a clear account of what we can already learn from the available
one-loop amplitudes, and which will aid these future studies.

##  3 A minimal set of variables

In the next section we will review the structure of $2\to 3$ tree-level
amplitudes in the NMRK limit, eq. 294. In this section, following ref.
Byrne:2022wzk , we introduce a minimal set of dimensionless variables for
$2\to 3$ scattering, which encode transverse momentum conservation and which
make it straightforward to take Regge limits of rational functions. The
longitudinal degrees of freedom are given by the ratios of lightcone
coordinates,

$X=\frac{p_{3}^{+}}{p_{4}^{+}}\,,\qquad\qquad
Y=\frac{p_{4}^{+}}{p_{5}^{+}}\,,\qquad\qquad X,Y\in\mathbb{R}\,,$ (31a) and
the remaining transverse degree of freedom is given by 333After completion of
this work, the author became aware of a very similar treatment of transverse
momenta in ref. Schwartz:2018obd , including a Lorentz invariant
representation of the transverse variables given in appendix D of that
reference.
$z=\frac{q_{1\perp}}{p_{4\perp}}=-\frac{p_{3\perp}}{p_{4\perp}}\,,\qquad\qquad
z\in\mathbb{C}\,.$ (31b)

In practise, we can move to the minimal parameterisation by eliminating all
but one transverse coordinate, e.g.

$p_{4\perp}\to-\frac{1}{1-z}\,q_{2\perp},\qquad
q_{1\perp}\to-\frac{z}{1-z}\,q_{2\perp}\,.$ (32)

We see that the variable $z$ automatically, imposes transverse momentum
conservation, i.e. the relation

$\frac{-q_{1\perp}+p_{4\perp}+q_{2\perp}}{q_{2\perp}}=0\,$ (33)

is manifest. For Lorentz invariant quantities with non-zero mass dimension,
$q_{2\perp}$ will provide the dimensionfull scale.

The parameterisation eq. 31 allows for simple permutation of
$p_{3}\leftrightarrow p_{4}$,

$X\underset{P_{12435}}{\to}\frac{1}{X}\,,\qquad
Y\underset{P_{12435}}{\to}XY\,,\qquad
z\underset{P_{12435}}{\to}\frac{1}{z}\,.$ (34)

Our choice of frame with $p_{1\perp}=p_{2\perp}=0$ might suggest it not
possible to perform the analogous permutation of $p_{2}\leftrightarrow p_{3}$.
However, we can write the minimal variables in a Lorentz invariant form,

$\displaystyle X=\frac{s_{13}}{s_{14}}\,,\qquad
Y=\frac{s_{14}}{s_{15}}\,,\qquad z=-\frac{\langle 2|3|1]}{\langle 2|4|1]}\,.$
(35)

These expressions permit us to obtain maps for arbitrary permutations of
momenta, for example, $p_{2}\leftrightarrow p_{3}$,

$X\underset{P_{13245}}{\to}-\left(\frac{1+Y+XY}{Y}\right)\,,\qquad
Y\underset{P_{13245}}{\to}Y\,,\qquad
z\underset{P_{13245}}{\to}\frac{z(1+Y+XY)}{Y(z+X)}\,.$ (36)

or $p_{2}\leftrightarrow p_{4}$,

$X\underset{P_{14325}}{\to}-\left(\frac{XY}{1+Y+XY}\right)\,,\qquad
Y\underset{P_{14325}}{\to}-(1+Y+XY)\,,\qquad
z\underset{P_{14325}}{\to}\frac{Y(z+X)}{(1+Y+XY)}\,.$ (37)

It can be checked that such maps satisfy the required group structure, e.g.

$\displaystyle P_{13245}^{2}=P_{12435}^{2}=P_{14325}^{2}$
$\displaystyle=\mathrm{id}\,,$ (38) $\displaystyle P_{12435}\circ
P_{13245}\circ P_{14325}\circ P_{13245}$ $\displaystyle=\mathrm{id}\,.$ (39)

A key advantage of the minimal set eq. 31 is that they make it simple to take
generalised Regge limits. For example, the MRK limit is obtained by taking
$X,Y\to\infty$ at the same rate, keeping $z$ finite, while the forward NMRK
limit is given by $Y\to\infty$ with $X$ and $z$ finite.

In this paper, we are particularly interested in the leading power of
amplitudes in the forward NMRK limit, in which we can obtain factorised
vertices that depend only on $X$ and $z$. For dimensionless rational
functions, the maps eqs. 36, 37 and 34 commute with this limit, that is,
within the limit, the permutation of momenta $p_{2}$, $p_{3}$ and $p_{4}$ is
given by the $X\to\infty$ limit of these above maps, which are respectively

$\displaystyle\\{X,z\\}$
$\displaystyle\underset{P_{243}}{\to}\bigg{\\{}\frac{1}{X},\frac{1}{z}\bigg{\\}}\,,$
(40) $\displaystyle\\{X,z\\}$
$\displaystyle\underset{P_{324}}{\to}\bigg{\\{}-(1+X),\frac{z(1+X)}{z+X}\bigg{\\}}\,,$
(41) $\displaystyle\\{X,z\\}$
$\displaystyle\underset{P_{432}}{\to}\bigg{\\{}-\frac{X}{1+X},\frac{z+X}{1+X}\bigg{\\}}\,.$
(42)

##  4 Review of tree-level five-parton amplitudes in the NMRK limit

In this section we review the structure of five-parton tree-level amplitudes
in the forward NMRK limit and their connection to the factorisation formulae
given in section 2. Even though the factorised expressions reviewed in this
section have been known for some time, we review them here in order to
introduce our notation and to collect the prerequisite expressions for
extending the analysis to loop level.

Truncating eq. 28 to leading order, it reads

$\displaystyle\begin{split}\mathcal{A}_{5}^{(0)[-]}\left(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},f_{4}^{\lambda_{3}},f_{5}^{\lambda_{5}},f_{1}^{\lambda_{1}}\right)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s\
&{\mathcal{C}}^{(0)}_{f_{2}f_{3}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})\\\
\times&\mathcal{R}^{(0)}_{g^{*}}(q_{2})\\\
\times&{\mathcal{C}}^{(0)}_{g^{*}f_{5}f_{1}}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})\,.\end{split}$
(43)

The relevant colour-dressed vertex is

$\mathcal{C}^{(0)}_{gggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})=g_{\mathrm{s}}^{2}\sum_{\sigma\in
S_{2}}(F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}})_{a_{2}c_{2}}\
C^{(0)}_{gggg^{*}}(p_{2}^{\lambda_{2}},p_{\sigma_{3}}^{\lambda_{\sigma_{3}}},p_{\sigma_{4}}^{\lambda_{\sigma_{4}}},q_{2})\,,$
(44)

in the pure gluon case, and

$\displaystyle\begin{split}\mathcal{C}^{(0)}_{{\bar{q}}qgg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})=g_{\mathrm{s}}^{2}\Big{[}&(T^{a_{4}}T^{c_{2}})_{\imath_{3}\bar{\imath}_{2}}\
C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})\\\
+&(T^{c_{2}}T^{a_{4}})_{\imath_{3}\bar{\imath}_{2}}\
C^{(0)}_{{\bar{q}}qgg^{*}}(p_{4}^{\lambda_{4}},p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q_{2})\Big{]}\,,\end{split}$
(45)

in the case of an external quark-antiquark pair. A notable feature of these
expressions is that they consist of two colour structures each. In ref.
DelDuca:1995ki ; DelDuca:1996km , they were obtained at fixed helicity from
colour-ordered amplitudes. The colour-ordered amplitudes themselves obeyed a
colour-stripped analogue of the Regge pole factorisation in eq. 28. It is
interesting to consider whether this pattern holds beyond tree level, that is,
whether

$\displaystyle\begin{split}\mathrm{Disp}\left[A_{5}^{[-]}\left(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},f_{4}^{\lambda_{3}},f_{5}^{\lambda_{5}},f_{1}^{\lambda_{1}}\right)\right]\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s\
&C_{f_{2}f_{3}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})\\\
\times&R_{g^{*}}(q_{2};s_{45})\\\
\times&C_{g^{*}f_{5}f_{1}}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})\,,\end{split}$
(46)

holds to LL or NLL accuracy, where for colour-stripped amplitudes we define

$\displaystyle\begin{split}&A_{5}^{[\pm]}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},5^{\lambda_{5}},1^{\lambda_{1}})\\\
&=\frac{1}{2}\left(A_{5}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},5^{\lambda_{5}},1^{\lambda_{1}})\pm
A_{5}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},1^{\lambda_{1}},5^{\lambda_{5}})\right)\,.\end{split}$
(47)

Comparing the truncation of eq. 46 to NLO with the known one-loop amplitudes,
we will find that, with the exception of some subtleties discussed in section
7.1.1, eq. 46 holds to NLL, and proves to be a useful framework to organise
amplitudes, even in general kinematics. The case where the colour-stripped
factorised form, eq. 46, fails to hold in section 7.1.1 is responsible for the
failure of the colour-dressed factorised form, eq. 28, to hold to NLL for the
pure-gluon amplitudes, as discussed in section 8.

The left-hand side of eq. 46 represents the expansion

$\displaystyle A_{n}=g_{\mathrm{s}}^{n-2}\ A_{n}^{(0)}+g_{\mathrm{s}}^{n}\
A_{n}^{(1)}+{\mathcal{O}}(g_{\mathrm{s}}^{n+2})\,,$ (48)

and the factorised terms on the right-hand side represent analogous expansions
of colour-stripped versions of the expressions defined in section 2,

$\displaystyle R_{g^{*}}$
$\displaystyle=R_{g^{*}}^{(0)}\quad\quad\quad\,+g_{\mathrm{s}}^{2}\
R_{g^{*}}^{(1)}\quad\quad+{\mathcal{O}}(g_{\mathrm{s}}^{4})\,,$ (49a)
$\displaystyle C_{g^{*}f_{5}f_{1}}$ $\displaystyle=g_{\mathrm{s}}\
C^{(0)}_{g^{*}f_{5}f_{1}}\ \ \,+g_{\mathrm{s}}^{3}\ C^{(1)}_{g^{*}f_{5}f_{1}}\
\ +{\mathcal{O}}(g_{\mathrm{s}}^{5})\,,$ (49b) $\displaystyle
C_{f_{2}f_{3}f_{4}g^{*}}$ $\displaystyle=g_{\mathrm{s}}^{2}\
C^{(0)}_{f_{2}f_{3}f_{4}g^{*}}+g_{\mathrm{s}}^{4}\
C^{(1)}_{f_{2}f_{3}f_{4}g^{*}}+{\mathcal{O}}(g_{\mathrm{s}}^{6})\,.$ (49c)

The truncation of eq. 46 to leading order has the simple form

$\displaystyle\begin{split}A^{(0)[-]}_{4}\left(f_{2},f_{3},f_{4},f_{5},f_{1}\right)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s\
&C^{(0)}_{f_{2}f_{3}f_{4}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})\\\
\times&R_{g^{*}}^{(0)}(q_{2})\\\
\times&C_{g^{*}f_{5}f_{1}}^{(0)}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})\,,\end{split}$
(50)

which allows the determination of the following colour-ordered peripheral-
emission vertices from tree-level five-gluon amplitudes DelDuca:1995ki :

$\displaystyle
C^{(0)}_{gggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})$
$\displaystyle=0$ (51) $\displaystyle
C^{(0)}_{gggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})$
$\displaystyle=\frac{1}{q_{2\perp}}\frac{(z-1)^{2}X}{z(z+X)}\,,$ (52)
$\displaystyle
C^{(0)}_{gggg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},q_{2})$
$\displaystyle=\frac{1}{q_{2\perp}}\frac{(z-1)^{2}X^{3}}{z(z+X)(1+X)^{2}}\,.$
(53)

These vertices constitute a minimal set from which all others can be obtained
through parity conjugation or the crossing of momenta as in section 3. The
peripheral-emission vertices inherit identities from the parent partial
amplitudes in general kinematics, for example, in the NMRK limit, the
reflection identity for amplitudes

$A^{(0)}_{5}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},5^{\lambda_{5}},1^{\lambda_{1}})=(-1)^{5}A^{(0)}_{5}(4^{\lambda_{4}},3^{\lambda_{3}},2^{\lambda_{2}},1^{\lambda_{1}},5^{\lambda_{5}})\,,$
(54)

together with the lower-point identity

$C^{(0)}_{ggg^{*}}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})=-C^{(0)}_{ggg^{*}}(-q_{2},p_{1}^{\lambda_{1}},p_{5}^{\lambda_{5}})\,,$
(55)

gives rise to an analogous reflection identity for the $gggg^{*}$ vertices,

$\displaystyle
C^{(0)}_{gggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})=C^{(0)}_{gggg^{*}}(p_{4}^{\lambda_{4}},p_{3}^{\lambda_{3}},p_{2}^{\lambda_{2}},q_{2})\,.$
(56)

Similarly, the NMRK limit of the $U(1)$ decoupling identity

$A^{(0)}_{5}(2,3,4,5,1)+A^{(0)}_{5}(2,4,3,5,1)+A^{(0)}_{5}(4,2,3,5,1)+A^{(0)}_{5}(2,3,5,4,1)=0\,,\,$
(57)

leads to the vertex identity

$\displaystyle
C^{(0)}_{gggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})+C^{(0)}_{gggg^{*}}(p_{3}^{\lambda_{3}},p_{2}^{\lambda_{2}},p_{4}^{\lambda_{4}},q_{2})+C^{(0)}_{gggg^{*}}(p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},p_{2}^{\lambda_{2}},q_{2})=0\,,$
(58)

due to the power supression of the final term in eq. 57. Supersymmetric Ward
identities Grisaru:1976vm  are also inherited by the vertices, allowing us to
straightforwardly obtain peripheral-emission vertices with an external quark-
antiquark pair. For example, consider the supersymmetric Ward identity

$A_{5}^{(0)}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},4^{\oplus},5^{\oplus},1^{\ominus})=\frac{\langle
1\ 3\rangle}{\langle 1\
2\rangle}A_{5}^{(0)}(2^{\ominus},3^{\oplus},4^{\oplus},5^{\oplus},1^{\ominus})\,.$
(59)

In the forward NMRK limit this yields the vertex relation

$C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})=-i\sqrt{\frac{X}{(1+X)}}C^{(0)}_{gggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\,.$
(60)

Likewise, the supersymmetric Ward identity

$A_{5}^{(0)}(2^{\oplus},3_{\bar{q}}^{\ominus},4_{q}^{\oplus},5^{\oplus},1^{\ominus})=\frac{\langle
1\ 4\rangle}{\langle 1\
3\rangle}A_{5}^{(0)}(2^{\oplus},3^{\ominus},4^{\oplus},5^{\oplus},1^{\ominus})\,,$
(61)

yields the vertex relation

$C^{(0)}_{g{\bar{q}}qg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},q_{2})=\sqrt{\frac{1}{X}}C^{(0)}_{gggg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},q_{2})\,.$
(62)

We note that the simple ratios of spinor products in eqs. 59 and 61 encode
both the difference in little-group weight of the amplitudes and the correct
power-dependence predicted by Regge theory (that is, in the MRK limit the
vertex eq. 62 is suppressed by a factor $X^{\frac{1}{2}}$ compared to the
pure-gluon vertex because it is governed by an effective $t$-channel exchange
of a spin-$\frac{1}{2}$ particle).

##  5 Review of one-loop four-parton amplitudes in the Regge limit

In this section we review the properties of one-loop four-parton amplitudes in
the Regge limit. In the following sections we will utilise the DDM colour
decompositions DelDuca:1999rs  for one-loop five-point amplitudes. These
useful bases were found after the original determination of the colour-dressed
vertices eqs. 14a and 14b so it will be useful to re-derive these vertices
starting from the four-parton DDM bases. We will need to subtract these lower
point gauge invariant objects in order to define the two-parton peripheral-
emission vertices, so we collect the necessary expressions here. In section
5.1 we review the colour-structure of $n$-parton one-loop amplitudes. In
sections 5.2 and 5.3 we specialise to the four-parton case and review the
simplifications which occur in the Regge limit. In section 5.4 we bring
everything together and define the colour-dressed one-parton peripheral-
emission vertices.

The procedure followed in this section is in one-to-one correspondence with
the analysis we will subsequently perform in the five-parton case. This
section therefore acts as a microcosm of the remainder of the paper,
introducing most of the notation and organisational principles we will need
later, but presented in the simpler and well-understood context of $2\to 2$
scattering.

###  5.1 Colour structure of one-loop amplitudes

Beyond tree level it is useful to make a distinction between partial and
primitive amplitudes. By primitive amplitude we mean a gauge invariant subset
of colour-ordered Feynman diagrams. By partial amplitude we mean a kinematic
coefficient of a colour structure where the colour ordering is not necessarily
fixed.

For the one-loop amplitudes that we analyse in this paper we will use the DDM
colour bases introduced in ref. DelDuca:1999rs . For $n$-gluon amplitudes the
relevant colour decomposition is

$\displaystyle\begin{split}\mathcal{A}_{n}^{(1)}(g_{1},\ldots,g_{n})&=g_{\mathrm{s}}^{n}\sum_{\sigma\in
S_{n-1}/R}\mbox{tr}\left(F^{a_{\sigma_{1}}}\cdots
F^{a_{\sigma_{n}}}\right)A^{(1,\,g)}_{n}\left({\sigma_{1}}^{\lambda_{\sigma_{1}}},\ldots,{\sigma_{n}}^{\lambda_{\sigma_{n}}}\right)\\\
&+N_{f}\ g_{\mathrm{s}}^{n}\sum_{\sigma\in
S_{n}}\mbox{tr}\left(T^{a_{\sigma_{1}}}\cdots
T^{a_{\sigma_{n}}}\right)A^{(1,\,f)}_{n}\left(\sigma_{1}^{\lambda_{\sigma_{1}}},\ldots,{\sigma_{n}}^{\lambda_{\sigma_{n}}}\right)\,,\end{split}$
(63)

where $S_{n-1}=S_{n}/Z_{n}$ is the group of non-cyclic permutations and $R$ is
the reflection operation $R(1,2,\ldots,n-1,n)=R(n,n-1,\ldots,2,1)$. Here,
$A_{n}^{(1,\ m)}$ are the one-loop primitive amplitudes with $n$ external
partons, and particle content $m$ circulating in the loop. We use the notation
$g$ or $f$ to indicate a gluon or fermion is circulating in the loop. We will
later also consider the possibility of a complex scalar, $m{=}s$, circulating
in the loop. We note the $n$-gluon partial amplitudes obey the reflection
identity

$A_{n}^{(1,\,m)}\left(1,2,\ldots,n-1,n\right)=(-1)^{n}A_{n}^{(1,\,m)}\left(n,n-1,\ldots,2,1\right)\,.$
(64)

For amplitudes with $(n-2)$ gluons and a quark-antiquark pair the analogous
colour decomposition is DelDuca:1999rs

$\displaystyle\begin{split}\mathcal{A}_{n}^{(1)}\left(\bar{q}_{1},q_{2},g_{3},\ldots,g_{n}\right)=&g_{\mathrm{s}}^{n}\Bigg{[}\sum_{p=2}^{n}\sum_{\sigma\in
S_{n-2}}\left(T^{c_{2}}T^{a_{\sigma_{3}}}\cdots
T^{a_{\sigma_{p}}}T^{c_{1}}\right)_{\imath_{2}\bar{\imath}_{1}}\left(F^{a_{\sigma_{p+1}}}\cdots
F^{a_{\sigma_{n}}}\right)_{c_{1}c_{2}}\\\ &\times
A^{(1,\,g_{R})}_{n}\left(1_{{\bar{q}}}^{\lambda_{1}},{\sigma_{p+1}}^{\lambda_{\sigma_{p+1}}},\ldots,{\sigma_{n}}^{\lambda_{\sigma_{n}}},2_{q}^{\lambda_{2}},{\sigma_{3}}^{\lambda_{3}},\ldots,{\sigma_{p}}^{\lambda_{p}}\right)\\\
&\mkern-180.0mu+\frac{N_{f}}{N_{c}}\sum_{j=1}^{n-1}\sum_{\sigma\in
S_{n-2}/S_{n;j}}\operatorname{Gr}_{n;j}^{(\bar{q}q)}\left(\sigma_{3},\ldots,\sigma_{n}\right)A_{n;j}^{(1,\,f)}\left(1_{\bar{q}}^{\lambda_{1}},2_{q}^{\lambda_{2}};{\sigma_{3}}^{\lambda_{3}},\ldots,{\sigma_{n}}^{\lambda_{n}}\right)\Bigg{]}\,.\end{split}$
(65)

For primitive amplitudes with an external quark-antiquark pair, diagrams
without closed fermion or scalar loops are assigned to $m{=}g$ because these
diagrams mix under gauge transformations with diagrams with closed gluon
loops. Following ref. Bern:1994fz , we separate primitive amplitudes with an
external quark-antiquark pair into individually gauge invariant terms,

$A^{(1,\,m)}_{n}=A^{(1,\,m_{R})}_{n}+A^{(1,\,m_{L})}_{n}\,.$ (66)

The label $R$ or $L$ denotes the gauge invariant subset of colour-ordered
Feynman diagrams in which the fermion line passes to the right or left of the
closed loop (with the diagrammatic convention that colour ordered partons are
labeled clockwise). The $L$ and $R$ primitive amplitudes are related by a
reflection identity analogous to eq. 64:

$A_{n}^{(1,\,m_{R})}\left(1_{\bar{q}},3,4,\ldots,2_{q},\ldots,n-1,n\right)=(-1)^{n}A_{n}^{(1,\,m_{L})}\left(1_{\bar{q}},n,n-1,\ldots,2_{q},\ldots,4,3\right)\,.$
(67)

Note that for $m{\in}\\{f,s\\}$ the following primitive amplitudes vanish:

$\displaystyle A_{n}^{(1,\,m_{R})}(1_{\bar{q}},2_{q},3,4,\ldots,n)$
$\displaystyle=A_{n}^{(1,\,m_{L})}(1_{\bar{q}},n,\ldots,4,3,2_{q})=0\,,$ (68a)
$\displaystyle A_{n}^{(1,\,m_{R})}(1_{\bar{q}},3,2_{q},4,\ldots,n)$
$\displaystyle=A_{n}^{(1,\,m_{L})}(1_{\bar{q}},n,\ldots,4,2_{q},3)=0\,.$ (68b)

These identities can be understood by considering the Feynman diagrams which
contribute to the respective amplitudes. The amplitudes in eq. 68a and eq. 68b
permit only tadpole and bubble diagrams respectively, which vanish in
dimensional regularisation. We emphasise that eq. 68 does not hold for $m{=}g$
because, as stated previously, the external fermion line may form part of the
gluon loop.

The pure-gluon decomposition, eq. 63, is written solely in terms of primitive
amplitudes. This is not the case for one-loop amplitudes with an external
quark-antiquark pair, eq. 65, where partial amplitudes, not primitive
amplitudes, multiply the colour structures
$\operatorname{Gr}^{(q{\bar{q}})}_{n;j}$. These colour structures are defined
by

$\displaystyle\begin{split}\operatorname{Gr}_{n;1}^{(\bar{q}q)}(3,\ldots,n)&=N_{c}\left(T^{a_{3}}\cdots
T^{a_{n}}\right)_{\imath_{2}\bar{\imath}_{1}}\,,\\\
\operatorname{Gr}_{n;2}^{(\bar{q}q)}(3;4,\ldots,n)&=0\,,\\\
\operatorname{Gr}_{n;j}^{(\bar{q}q)}(3,\ldots,j+1;j+2,\ldots,n)&=\mbox{tr}\left(T^{a_{3}}\cdots
T^{a_{j+1}}\right)\left(T^{a_{j+2}}\cdots
T^{a_{n}}\right)_{\imath_{2}\bar{\imath}_{1}},\quad 3\leq j\leq n-2,\\\
\operatorname{Gr}_{n;n-1}^{(\bar{q}q)}(3,\ldots,n)&=\mbox{tr}\left(T^{a_{3}}\cdots
T^{a_{n}}\right)\delta_{\imath_{2}\bar{\imath}_{1}}\,,\end{split}$ (69)

and $S_{n;j}=Z_{j-1}$ is the subgroup of $S_{n-2}$ that leaves
$\operatorname{Gr}^{(q{\bar{q}})}_{n;j}$ invariant. The partial amplitudes
which multiply these colour structures can be written as a sum of the simpler
primitive amplitudes Bern:1994fz ,

$\displaystyle A_{n;1}^{(1,\,f)}\left(1_{\bar{q}},2_{q},3,\ldots,n\right)$
$\displaystyle=A_{n}^{(1,\,f_{L})}\left(1_{\bar{q}},2_{q},3,\ldots,n\right),$
(70a) $\displaystyle
A_{n;j>1}^{(1,\,f)}\left(1_{\bar{q}},2_{q};3,\ldots,j{+}1;j{+}2,\ldots,n\right)$
$\displaystyle=(-1)^{j}\sum_{\sigma\in\operatorname{COP}\\{\alpha\\}\\{\beta\\}}A_{n}^{(1,\,f_{R})}\left({\sigma_{1}}_{\bar{q}},{\sigma_{2}}_{q},\sigma_{3},\ldots,\sigma_{n}\right),$
(70b)

with
$\\{\alpha\\}=\\{j{+}1,j,\ldots,4,3\\},\\{\beta\\}=\\{1,2,j{+}2,j{+}3,\ldots,n{-}1,n\\}$,
and where $\operatorname{COP}\\{\alpha\\}\\{\beta\\}$ is the set of all
permutations of $\\{1,\ldots,n\\}$ with leg 1 held fixed that preserve the
cyclic ordering of the elements $\alpha_{i}$ within $\\{\alpha\\}$ and the
elements $\beta_{i}$ within $\\{\beta\\}$, while allowing for all possible
relative orderings of the $\alpha_{i}$ with respect to the $\\{\beta\\}$
Bern:1993mq . This concludes our review of the colour structure of one-loop
amplitudes. In the next section we show how the colour structure of one-loop
four-parton amplitudes simplifies in the Regge limit.

###  5.2 Colour structure of one-loop four-parton amplitudes in the Regge
limit

In the previous section, we reviewed the DDM colour decomposition for
$n$-parton amplitudes in general kinematics. In this section we specify to the
four-parton case and show how the colour structure simplifies in the Regge
limit.

Let us first consider the physical scattering of gluons $g_{1}(p_{1})\
g_{2}(p_{2})\to g_{3}(p_{3})\ g_{4}(p_{4})$, with all outgoing momenta eq.
277, in the Regge limit eq. 292. We specify to the $n{=}4$ case of eq. 63,

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)}(g_{1},g_{2},g_{3},g_{4})&=g_{\mathrm{s}}^{4}\sum_{\sigma\in
S_{3}/R}\mbox{tr}\left(F^{a_{\sigma_{1}}}F^{a_{\sigma_{2}}}F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}}\right)A^{(1,\,g)}_{4}\left({\sigma_{1}},{\sigma_{2}},{\sigma_{3}},{\sigma_{4}}\right)\\\
&+N_{f}\ g_{\mathrm{s}}^{4}\sum_{\sigma\in
S_{4}}\mbox{tr}\left(T^{a_{\sigma_{1}}}T^{a_{\sigma_{2}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{4}}}\right)A^{(1,\,f)}_{4}\left({\sigma_{1}},{\sigma_{2}},{\sigma_{3}},{\sigma_{4}}\right)\,.\end{split}$
(71)

We are interested in the leading behaviour of this amplitude in the Regge
limit. We note that all primitive amplitudes with colour-orderings in which
$a_{1}$ and $a_{4}$ are not adjacent are power supressed in this limit. The
structure of the amplitude eq. 71 in the Regge limit therefore simplifies to

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)}(g_{1},g_{2},g_{3},g_{4})\underset{\mathrm{Regge}}{\to}g_{\mathrm{s}}^{4}\sum_{\sigma\in
S_{2}}\mbox{tr}\left(F^{a_{2}}F^{a_{3}}F^{a_{\sigma_{4}}}F^{a_{\sigma_{1}}}\right)&A^{(1,\,g)}_{4}\left(2,3,\sigma_{4},\sigma_{1}\right)\\\
+N_{f}\ g_{\mathrm{s}}^{4}\sum_{\sigma\in
S_{2}}\big{[}\mbox{tr}\left(T^{a_{2}}T^{a_{3}}T^{a_{\sigma_{4}}}T^{a_{\sigma_{1}}}\right)+\mbox{tr}\left(T^{a_{3}}T^{a_{2}}T^{a_{\sigma_{1}}}T^{a_{\sigma_{4}}}\right)\big{]}&A^{(1,\,f)}_{4}\left(2,3,\sigma_{4},\sigma_{1}\right)\,,\end{split}$
(72)

where we have made use of the reflection identity eq. 64. The observation that
the terms in eq. 72 are pairwise related by
$\\{p_{4},\lambda_{4},a_{4}\\}\leftrightarrow\\{p_{1},\lambda_{1},a_{1}\\}$
naturally suggests we consider amplitudes of definite signature in the
$t$-channel, so we define the equivalent of eq. 10 for colour-ordered
amplitudes,

$\displaystyle\begin{split}A_{4}^{(1,\,m)[\pm]}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},1^{\lambda_{1}})=\frac{1}{2}\Big{(}&A_{4}^{(1,\,m)}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},1^{\lambda_{1}})\\\
\pm&A_{4}^{(1,\,m)}(2^{\lambda_{2}},3^{\lambda_{3}},1^{\lambda_{1}},4^{\lambda_{4}})\Big{)}\,.\end{split}$
(73)

The signature-odd amplitude has a particularly simple structure. Using
identities eq. 3, eq. 4 and the trace identities

$\displaystyle\mbox{tr}\left(T^{a}T^{b}T^{c}\right)$
$\displaystyle=\frac{1}{2}\left(d^{abc}+if^{abc}\right)\,,$ (74a)
$\displaystyle\mbox{tr}\left(F^{a}F^{b}F^{c}\right)$ $\displaystyle=iN_{c}\
f^{abc}\,,$ (74b)

we see that the signature-odd amplitude consists of a single colour structure,
namely that of a colour-octet exchange in the $t$-channel:

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)[-]}(g_{1},g_{2},g_{3},g_{4})\to
g_{\mathrm{s}}^{4}F^{d}_{a_{2}a_{3}}F^{d}_{a_{4}a_{1}}\Big{\\{}N_{c}\
&A^{(1,\,g)[-]}_{4}\left(2,3,4,1\right)\\\ +N_{f}\
&A^{(1,\,f)[-]}_{4}\left(2,3,4,1\right)\Big{\\}}\,.\end{split}$ (75)

Let us now perform a similar study of the physical scattering of a gluon and
quark, $g_{1}(p_{1})\ q_{2}(p_{2})\to q_{3}(p_{3})\ g_{4}(p_{4})$. The $n{=}4$
case of eq. 65 reads

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)}\left(\bar{q}_{2},q_{3},g_{4},g_{1}\right)=g_{\mathrm{s}}^{4}\Bigg{\\{}\sum_{\sigma\in
S_{2}}\bigg{[}&\left(T^{c_{2}}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}(F^{a_{\sigma_{4}}}F^{a_{\sigma_{1}}})_{c_{1}c_{2}}A_{4}^{(1,\,g_{R})}\left(2_{\bar{q}},\sigma_{4},\sigma_{1},3_{q}\right)\\\
+&(T^{c_{2}}T^{a_{\sigma_{4}}}T^{c_{1}})_{\imath_{3}\bar{\imath}_{2}}(F^{a_{\sigma_{1}}})_{c_{1}c_{2}}A_{4}^{(1,\,g_{R})}\left(2_{\bar{q}},\sigma_{1},3_{q},\sigma_{4}\right)\\\
+&(T^{c_{2}}T^{a_{\sigma_{4}}}T^{a_{\sigma_{1}}}T^{c_{1}})_{\imath_{3}\bar{\imath}_{2}}\Delta_{c_{1}c_{2}}A_{4}^{(1,\,g_{R})}\left(2_{\bar{q}},3_{q},\sigma_{4},\sigma_{1}\right)\bigg{]}\\\
+\frac{N_{f}}{N_{c}}\bigg{[}\sum_{\sigma\in
S_{2}}&N_{c}(T^{a_{\sigma_{4}}}T^{a_{\sigma_{1}}})_{\imath_{3}\bar{\imath}_{2}}A_{4;1}^{(1,\,f)}\left(2_{\bar{q}},3_{q};\sigma_{4},\sigma_{1}\right)\\\
+\sum_{\sigma\in
S_{2}/Z_{2}}&\mbox{tr}(T^{a_{\sigma_{4}}}T^{a_{\sigma_{1}}})\delta_{\imath_{3}\bar{\imath}_{2}}A_{4;3}^{(1,\,f)}\left(2_{\bar{q}},3_{q};\sigma_{4},\sigma_{1}\right)\bigg{]}\Bigg{\\}}\,.\end{split}$
(76)

Note that we use the symbol $\delta$ and $\Delta$ for the identity in the
fundamental and adjoint representations respectively. As was shown in ref.
Bern:1994fz , the partial amplitude $A_{4;3}^{(1,\,f)}$ is in fact zero. It is
instructive to recall why this is the case. Using eq. 70b, we expand the
partial amplitude in terms of primitive amplitudes,

$\displaystyle\begin{split}A_{4;3}^{(1,\,f)}\left(2_{\bar{q}},3_{q};4,1\right)&=A_{4}^{(1,\,f_{R})}\left(2_{\bar{q}},3_{q},4,1\right)+A_{4}^{(1,\,f_{R})}\left(2_{\bar{q}},3_{q},1,4\right)\\\
&+A_{4}^{(1,\,f_{R})}\left(2_{\bar{q}},4,3_{q},1\right)+A_{4}^{(1,\,f_{R})}\left(2_{\bar{q}},1,3_{q},4\right)\\\
&+A_{4}^{(1,\,f_{R})}\left(2_{\bar{q}},4,1,3_{q}\right)+A_{4}^{(1,\,f_{R})}\left(2_{\bar{q}},1,4,3_{q}\right)\,.\end{split}$
(77)

The first two lines vanish due to eq. 68a and eq. 68b respectively. The final
two primitive amplitudes consist solely of Feynman diagrams with closed
fermion triangles. Therefore, Furry’s theorem implies

$A_{4}^{(1,\,f_{R})}\left(2_{\bar{q}},4,1,3_{q}\right)=-A_{4}^{(1,\
f_{R})}\left(2_{\bar{q}},1,4,3_{q}\right)\,.$ (78)

Thus, we see that eq. 77 is zero, and eq. 76 is simply expressed in terms of
primitive amplitudes.

As for the pure-gluon case above, let us now utilise the fact that primitive
amplitudes that do not have $p_{4}$ and $p_{1}$ cyclically adjacent are power
suppressed in the Regge limit. This observation leads to the simplified colour
structure

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)}\left(\bar{q}_{2},q_{3},g_{4},g_{1}\right)\underset{\mathrm{Regge}}{\to}g_{\mathrm{s}}^{4}\sum_{\sigma\in
S_{2}}\bigg{[}&\left(T^{c_{2}}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}(F^{d_{\sigma_{4}}}F^{d_{\sigma_{1}}})_{c_{1}c_{2}}A_{4}^{(1,\,g_{R})}\left(2_{\bar{q}},\sigma_{4},\sigma_{1},3_{q}\right)\\\
+&(T^{c_{2}}T^{d_{\sigma_{4}}}T^{d_{\sigma_{1}}}T^{c_{1}})_{\imath_{3}\bar{\imath}_{2}}\delta_{c_{1}c_{2}}A_{4}^{(1,\,g_{R})}\left(2_{\bar{q}},3_{q},\sigma_{4},\sigma_{1}\right)\\\
&+N_{f}\
(T^{\sigma_{4}}T^{\sigma_{1}})_{\imath_{3}\bar{\imath}_{2}}A_{4;1}^{(1,\,f_{L})}\left(2_{\bar{q}},3_{q},\sigma_{4},\sigma_{1}\right)\bigg{]}\,.\end{split}$
(79)

Finally, by using the identities

$\displaystyle\left(T^{a}T^{b}\right)_{ij}F^{d}_{ba}$ $\displaystyle=N_{c}\
T^{d}_{ij}\,,$ (80) $\displaystyle\left(T^{a}T^{d}T^{a}\right)_{ij}$
$\displaystyle=-\frac{1}{N_{c}}\ T^{d}_{ij}\,,$ (81)

we find that the signature-odd amplitude consists of only $t$-channel octet
exchange,

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)[-]}\left(\bar{q}_{2},q_{3},g_{4},g_{1}\right)\underset{\mathrm{Regge}}{\to}g_{\mathrm{s}}^{4}T^{d}_{\imath_{3}\bar{\imath}_{2}}F^{d}_{a_{4}a_{1}}\Big{\\{}N_{c}\
&A_{4}^{(1,\,g_{L})[-]}\left(2_{\bar{q}},3_{q},4,1\right)\\\ -\frac{1}{N_{c}}\
&A_{4}^{(1,\,g_{R})[-]}\left(2_{\bar{q}},3_{q},4,1\right)\\\ +N_{f}\
&A_{4}^{(1,\,f_{L})[-]}\left(2_{\bar{q}},3_{q},4,1\right)\Big{\\}}\,,\end{split}$
(82)

as in eq. 75. We have used the reflection identity eq. 67 to write the
primitive amplitudes in the form provided by Bern:1994fz . In the next
section, we analyze these leading-power amplitudes in the Regge limit.

###  5.3 One-loop primitive vertices for the peripheral emission of a gluon or
quark

Having studied the colour structure of amplitudes to leading power in the
Regge limit, we now turn our attention to the structure of the leading-power
primitive amplitudes. As usual, we start with the pure-gluon case, where eq.
71 requires us to obtain the leading behaviour of

$A_{4}^{(1,\,g)[-]}(2,3,4,1)\qquad\qquad\textrm{and}\qquad\qquad
A_{4}^{(1,\,f)[-]}(2,3,4,1)\,.$

It is highly convenient to utilise the _supersymmetric decomposition_ of one-
loop gluon amplitudes Bern:1993mq ; Bern:1994zx ; Bern:1994cg . We can
decompose one-loop pure-gluon primitive amplitudes in supersymmetric theories
into a sum of primitive amplitudes with a spin-1 boson, spin-$\frac{1}{2}$
fermion or complex scalar circulating in the loop. In this paper we consider
the following supersymmetric multiplets:

$\displaystyle A_{n}^{(1,\,\mathcal{N}=4)}$ $\displaystyle=A_{n}^{(1,\,g)}+4\
A_{n}^{(1,\,f)}+3\ A_{n}^{(1,\,s)}\,,$ (83a) $\displaystyle
A_{n}^{(1,\,\mathcal{N}=1_{V})}$
$\displaystyle=A_{n}^{(1,\,g)}+A_{n}^{(1,\,f)}\,,$ (83b) $\displaystyle
A_{n}^{(1,\,\mathcal{N}=1_{\chi})}$
$\displaystyle=A_{n}^{(1,\,f)}+A_{n}^{(1,\,s)}\,,$ (83c)

where the subscripts $V$ and $\chi$ denote vector and chiral multiplets
respectively. We can then write the primitive amplitudes required for QCD in
terms of supersymmetric amplitudes,

$\displaystyle A_{n}^{(1,\ g)}$ $\displaystyle=A_{n}^{(1,\ \mathcal{N}=4)}-4\
A_{n}^{(1,\ \mathcal{N}=1_{\chi})}+A_{n}^{(1,\ s)}$ (84a) $\displaystyle
A_{n}^{(1,\ f)}$ $\displaystyle=A_{n}^{(1,\ \mathcal{N}=1_{\chi})}-A_{n}^{(1,\
s)}\,.$ (84b)

The main benefit of these decompositions is that amplitudes in supersymmetric
theories are cut-constructable, that is, they can be computed entirely from
generalised unitarity cuts Bern:1993mq ; Bern:1994zx ; Bern:1994cg . The
remaining complex scalar term in eq. 84a has a rational term which cannot be
computed from taking discontinuities. It is nevertheless simpler to compute
than the gluon- or quark-loop amplitudes, because the complex scalar does not
propagate spin information around the loop.

An analogous supersymmetric decomposition can be used to organise one-loop
amplitudes with an external quark-antiquark pair. We follow ref. Bern:1994fz
in relating $g$ and $f$ amplitudes to a ${\mathcal{N}}{=}1$ vector multiplet,

$A_{n}^{(1,\
\mathcal{N}=1_{V})}=A_{n}^{(1,\,g_{L})}+A_{n}^{(1,\,g_{R})}+A_{n}^{(1,\,f_{L})}+A_{n}^{(1,\,f_{R})}\,.$
(85)

The left-hand side of this equation can be immediately obtained from the
corresponding pure-gluon amplitude via a supersymmetric Ward identity,
reducing the number of independent amplitudes which we need to analyse.

At one-loop, it is useful to analyse the qualitatively-different MHV and non-
MHV amplitudes separately. The non-MHV amplitudes are zero at tree-level, and
at one-loop they are finite, purely rational functions. We study the Regge
limit of four-parton non-MHV amplitudes in section 5.3.1, where we find they
give rise to helicity-violating peripheral-emission vertices. The one-loop MHV
amplitudes contain divergences in $\epsilon$ and contain products of rational
and transcendental functions. We study the Regge limit of four-parton MHV
amplitudes in section 5.3.1, where we find they give rise to helicity-
conserving peripheral-emission vertices.

####  5.3.1 Helicity-violating vertices

The finite one-loop four-gluon amplitudes are the _all-plus_ amplitudes
Bern:1991aq ; Kunszt:1993sd ,

$\displaystyle
A_{4}^{(1,\,\mathcal{N}=4)}\left(1^{\oplus},2^{\oplus},3^{\oplus},4^{\oplus}\right)$
$\displaystyle=0\,,$ (86a) $\displaystyle
A_{4}^{(1,\,\mathcal{N}=1_{\chi})}\left(1^{\oplus},2^{\oplus},3^{\oplus},4^{\oplus}\right)$
$\displaystyle=0\,,$ (86b) $\displaystyle
A_{4}^{(1,\,s)}\left(1^{\oplus},2^{\oplus},3^{\oplus},4^{\oplus}\right)$
$\displaystyle=\frac{1}{3(4\pi)^{2}}\frac{s_{12}s_{23}}{\langle
12\rangle\langle 23\rangle\langle 34\rangle\langle 41\rangle}\,,$ (86c)

and the _single-minus_ amplitudes,

$\displaystyle
A_{4}^{(1,\,\mathcal{N}=4)}\left(1^{\ominus},2^{\oplus},3^{\oplus},4^{\oplus}\right)$
$\displaystyle=0\,,$ (87a) $\displaystyle
A_{4}^{(1,\,\mathcal{N}=1_{\chi})}\left(1^{\ominus},2^{\oplus},3^{\oplus},4^{\oplus}\right)$
$\displaystyle=0\,,$ (87b) $\displaystyle
A_{4}^{(1,\,s)}\left(1^{\ominus},2^{\oplus},3^{\oplus},4^{\oplus}\right)$
$\displaystyle=\frac{1}{3(4\pi)^{2}}\frac{\langle
24\rangle[24]^{3}}{[12]\langle 23\rangle\langle 34\rangle[41]}\,.$ (87c)

The all-plus amplitudes are power suppressed in the Regge limit, while the
single-minus amplitudes have leading-power contributions, e.g.,

$\displaystyle A_{4}^{(1,\,s)}(2^{\ominus},3^{\oplus},4^{\oplus},1^{\oplus})$
$\displaystyle\underset{\mathrm{Regge}}{\to}\frac{1}{3(4\pi)^{2}}\frac{s}{t}\,,$
(88) $\displaystyle
A_{4}^{(1,\,s)}(2^{\oplus},3^{\oplus},4^{\oplus},1^{\ominus})$
$\displaystyle\underset{\mathrm{Regge}}{\to}\frac{1}{3(4\pi)^{2}}\frac{s}{t}\left(\frac{q_{\perp}^{*}}{q_{\perp}}\right)^{2}\,.$
(89)

Both these sets of observations are compatible with Regge factorisation of
one-loop colour-ordered amplitudes, as discussed in section 4. The four-parton
analogue of eq. 46 is

$\displaystyle\begin{split}\mathrm{Disp}\left[A_{4}\left(f_{2}^{\lambda_{2}},f_{3}^{\lambda_{3}},f_{4}^{\lambda_{3}},f_{1}^{\lambda_{1}}\right)\right]\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s\
&C_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)\\\
\times&R_{g^{*}}(q;s_{34})\\\
\times&C_{g^{*}f_{4}f_{1}}(-q,p_{4}^{\lambda_{4}},p_{1}^{\lambda_{1}})\end{split}$
(90)

with expansion of the colour-ordered vertices,

$\displaystyle
C_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)=g_{\mathrm{s}}\
C^{(0)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)+g_{\mathrm{s}}^{3}\
C^{(1)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)+{\mathcal{O}}(g_{\mathrm{s}}^{5})\,.$
(91)

Expanding eq. 90 to order $g_{\mathrm{s}}^{6}$ we obtain

$\displaystyle\begin{split}\frac{1}{2s}{\mathrm{Disp}\left[{A_{4}^{(1)}(2,3,4,1)}\right]}\underset{\mathrm{Regge}}{\to}&C^{(1)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)R_{g^{*}}^{(0)}(q;s_{34})C_{g^{*}f_{5}f_{1}}^{(0)}(-q,p_{4}^{\lambda_{4}},p_{1}^{\lambda_{1}})\\\
+&C^{(0)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)R_{g^{*}}^{(1)}(q;s_{34})C_{g^{*}f_{5}f_{1}}^{(0)}(-q,p_{4}^{\lambda_{4}},p_{1}^{\lambda_{1}})\\\
+&C^{(0)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q)R_{g^{*}}^{(0)}(q;s_{34})C_{g^{*}f_{5}f_{1}}^{(1)}(-q,p_{4}^{\lambda_{4}},p_{1}^{\lambda_{1}})\,.\end{split}$
(92)

The fact the tree-level all-plus and single-minus amplitudes vanish implies

$C^{(0)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},q)=C^{(0)}_{g^{*}gg}(-q,p_{5}^{\oplus},p_{1}^{\oplus})=0\,.$
(93)

Subsequently, the ansatz eq. 90 correctly predicts the amplitudes eq. 86c
should be power-suppressed in the Regge limit. Likewise, for the single-minus
case, eq. 93 implies only the first or last line of eq. 92 will contribute at
leading power, and we can extract DelDuca:1998cx

$\displaystyle C_{ggg^{*}}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\oplus},q)$
$\displaystyle=\frac{1}{3(4\pi)^{2}}\left(\frac{-q_{\perp}^{*}}{q_{\perp}}\right)\,,$
(94) $\displaystyle C_{ggg^{*}}^{(1,\,s)}(p_{4}^{\oplus},p_{1}^{\oplus},q)$
$\displaystyle=\frac{1}{3(4\pi)^{2}}\,.$ (95)

Just as in eq. 5, the vertices are antisymmetric under $p_{2}\leftrightarrow
p_{3}$ or $p_{4}\leftrightarrow p_{1}$, and the parity conjugate is given by
complex conjugation. The fact that only the scalar term in eq. 87c is non-zero
leads to the neat property

$C_{ggg^{*}}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\oplus},q)=C_{ggg^{*}}^{(1,\,g)}(p_{2}^{\oplus},p_{3}^{\oplus},q)=-C_{ggg^{*}}^{(1,\,f)}(p_{2}^{\oplus},p_{3}^{\oplus},q)\,.$
(96)

####  5.3.2 Helicity-conserving vertices

Let us now proceed with the analysis of MHV amplitudes. It is highly
convenient to normalise one-loop colour-ordered MHV amplitudes by their tree-
level amplitude,

$A^{(1,\,m)}_{n}(1,\ldots,n)=c_{\Gamma}\ A_{n}^{(0)}(1,\ldots,n)\
a_{n}^{(1,\,m)}(1,\ldots,n).$ (97)

Throughout this paper, we will use an analogous notation for factorised
vertices, where lower-case symbols are used for quantities which have been
normalised by the tree-level quantity, and $c_{\Gamma}$. For example, we write
the one-loop vertices for a peripheral and central emission of a gluon as

$\displaystyle
C^{(1,\,m)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q_{1})$
$\displaystyle=c_{\Gamma}\,C_{ggg^{*}}^{(0)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q_{1})\
c_{ggg^{*}}^{(1,\,m)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q_{1})\,,$ (98)
$\displaystyle V^{(1,\,m)}_{g^{*}gg^{*}}(q_{1},p_{4}^{\lambda_{4}},q_{2})$
$\displaystyle=c_{\Gamma}\,V_{g^{*}gg^{*}}^{(0)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})\
v_{g^{*}gg^{*}}^{(1,\,m)}(-q_{1},p_{4}^{\lambda_{4}},q_{2})$ (99)

respectively, and likewise we consider the normalised one-loop Reggeisation
factor

$\displaystyle R^{(1,\,m)}_{g^{*}}(q;s)$
$\displaystyle=c_{\Gamma}\,R^{(0)}_{g^{*}}(q)\ r^{(1,\,m)}_{g^{*}}(q;s)\,.$
(100)

For MHV amplitudes, eq. 92 can then be written

$\displaystyle\begin{split}&\frac{1}{2s}{\mathrm{Disp}\left[{A_{4}^{(1,\,m)}(2^{\lambda_{2}},3^{-\lambda_{2}},4^{-\lambda_{1}},1^{\lambda_{1}})}\right]}\underset{\mathrm{Regge}}{\to}c_{\Gamma}\,C^{(0)}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)\\\
&\hskip 30.00005pt\times
R_{g^{*}}^{(0)}(q;s_{34})\times\Big{[}c^{(1,\,m)}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)+r_{g^{*}}^{(1,\,m)}(q;s_{34})+c_{g^{*}f_{4}f_{1}}^{(1,\,m)}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\Big{]}\\\
&\hskip 30.00005pt\times
C_{g^{*}f_{4}f_{1}}^{(0)}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\,.\end{split}$
(101)

We see that the factorisation ansatz eq. 46 implies the Regge limit of one-
loop coefficients will simply be given by a sum of universal, process
independent functions which depend on a subset of the momenta of the process:

$\displaystyle\begin{split}\mathrm{Re}\left[a_{4}^{(1\,m)}(2,3,4,1)\right]\underset{\mathrm{Regge}}{\to}c^{(1,\,m)}_{f_{2}f_{3}g^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)+r_{g^{*}}^{(1,\,m)}(q;s_{34})+c_{g^{*}f_{4}f_{1}}^{(1,\,m)}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\,.\end{split}$
(102)

The simplest one-loop amplitudes are pure-gluon amplitudes in $\mathcal{N}=4$
sYM, where the one-loop corrections, $a^{(1,\,\mathcal{N}=4)}$, are helicity-
independent and purely transcendental Green:1982sw

$\displaystyle a_{4}^{(1,\
\mathcal{N}=4)}(2^{\lambda_{2}},3^{-\lambda_{2}},4^{-\lambda_{1}},1^{\lambda_{1}})=-\frac{2}{\epsilon^{2}}\left[\left(\frac{\mu^{2}}{-s_{12}}\right)^{\epsilon}+\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}\right]+\ln^{2}\left(\frac{-s_{12}}{-s_{23}}\right)+\pi\,.$
(103)

As observed in ref. Bartels:2008ce , this function admits an _exact_
decomposition into one-loop building blocks. In particular, in the physical
region eq. 277, the real part can be written:

$a_{4}^{(1,\,\mathcal{N}=4)}(2,3,4,1)=c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},q)+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q;s_{12})+c_{g^{*}gg}^{(1,\,\mathcal{N}=4)}(-q,p_{4},p_{1})\,.$
(104)

Here, the real (thus parity invariant), one-loop correction to the peripheral-
emission vertex is

$c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(2^{\lambda_{2}},3^{-\lambda_{2}},q_{1})=\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}\left(-\frac{2}{\epsilon^{2}}+\frac{1}{\epsilon}\log\left(\frac{\tau}{-s_{23}}\right)+\frac{\pi^{2}}{2}\right)\,,$
(105)

and the large logarithmic term

$r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q;s)=\frac{\alpha^{(1)}(t)}{N_{c}\
c_{\Gamma}}\log\left(\frac{s}{\tau}\right)=\frac{2}{\epsilon}\left(\frac{\mu^{2}}{t}\right)^{\epsilon}\log\left(\frac{s}{\tau}\right)$
(106)

is the colour-stripped one-loop truncation of the Reggeised gluon in the $t$
channel.

The primitive amplitudes with a $\mathcal{N}=1_{\chi}$ multiplet or complex
scalar circulating in the loop are not helicity independent. Following ref.
Bern:1994fz ; Bern:1993mq , we write the amplitudes in terms of a divergent
piece, $V$, (which contains purely transcendental functions of the momenta)
and a further, finite term $G$, (which generically contains both rational
functions and products of rational and transcendental functions of the
momenta) 444We use the conventional notation $V$ for the divergent, purely
transcendental part of the one-loop correction. We hope this does not cause
confusion with the central-emission vertices, which will always include
subscript flavour indices.,

$a^{(1,\,m)}=V^{(1,\,m)}+G^{(1,\,m)}\,.$ (107)

For the ‘split’ helicity configuration, the one-loop amplitudes are given by

$\displaystyle
V^{(1,\,\mathcal{N}=1_{\chi})}(1^{\oplus},2^{\oplus},3^{\ominus},4^{\ominus})$
$\displaystyle=\frac{1}{\epsilon}\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}+2\,,$
(108) $\displaystyle
G^{(1,\,\mathcal{N}=1_{\chi})}(1^{\oplus},2^{\oplus},3^{\ominus},4^{\ominus})$
$\displaystyle=0\,,$ (109) $\displaystyle
V^{(1,\,s)}(1^{\oplus},2^{\oplus},3^{\ominus},4^{\ominus})$
$\displaystyle=\frac{1}{3}V^{(1,\ \mathcal{N}=1_{\chi})}+\frac{2}{9}\,,$ (110)
$\displaystyle G^{(1,\,s)}(1^{\oplus},2^{\oplus},3^{\ominus},4^{\ominus})$
$\displaystyle=0\,.$ (111)

These amplitudes have no logarithmic dependence on $s_{12}$, and trivially
admit an exact decomposition into peripheral-emission vertices,

$\displaystyle a_{4}^{(1,\
\mathcal{N}=1_{\chi})}(2^{\oplus},3^{\ominus},4^{\ominus},1^{\oplus})$
$\displaystyle=c_{ggg^{*}}^{(1,\
\mathcal{N}=1_{\chi})}(p_{2},p_{3},q)+c_{ggg^{*}}^{(1,\
\mathcal{N}=1_{\chi})}(p_{4},p_{1},-q)\,,$ (112) $\displaystyle a_{4}^{(1,\
s)}(2^{\oplus},3^{\ominus},4^{\ominus},1^{\oplus})$
$\displaystyle=c_{ggg^{*}}^{(1,\ s)}(p_{2},p_{3},q)+c_{ggg^{*}}^{(1,\
s)}(p_{4},p_{1},-q)\,,$ (113)

where we simply take the vertices to be half the one-loop functions, i.e.,

$\displaystyle c_{ggg^{*}}^{(1,\ \mathcal{N}=1_{\chi})}(p_{2},p_{3},q)$
$\displaystyle=\frac{1}{2\epsilon}\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}+1,$
(114) $\displaystyle c_{ggg^{*}}^{(1,\ s)}(p_{2},p_{3},q)$
$\displaystyle=\frac{1}{3}c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})+\frac{1}{9}\,.$ (115)

Note the lack of any logarithmic dependence on $s$ implies

$r_{g^{*}}^{(1,\,\mathcal{N}=1_{\chi})}=r^{(1,\,s)}_{g^{*}}=r^{(1,\,f)}_{g^{*}}=0\,,$
(116)

and correspondingly, the scale $\tau$ does not enter into eqs. 114 and 115.
This fact also implies the well-known equivalence of gluon Regge trajectories
in QCD and supersymmetric theories with spin-1 particles,

$r_{g^{*}}^{(1,\,\mathcal{N}=4)}=r_{g^{*}}^{(1,\,\mathcal{N}=1_{V})}=r_{g^{*}}^{(1,\,g)}\,.$
(117)

The _alternating-helicity_ amplitudes are considerably more complicated (see
ref. Bern:1994cg ). Of course, in the Regge limit, the split-helicity
amplitudes coincide with the alternating-helicity amplitudes, e.g.,

$\displaystyle
a_{4}^{(1,\,\mathcal{N}=1_{\chi})}(2^{\oplus},3^{\ominus},4^{\oplus},1^{\ominus})$
$\displaystyle\underset{\mathrm{Regge}}{\to}c_{ggg^{*}}^{(1,\,\mathcal{N}=1_{\chi})}(p_{2}^{\oplus},p_{3}^{\ominus},q)+c_{g^{*}gg}^{(1,\,\mathcal{N}=1_{\chi})}(-q,p_{4}^{\oplus},p_{1}^{\ominus})\,,$
(118) $\displaystyle
a_{4}^{(1,\,s)}(2^{\oplus},3^{\ominus},4^{\oplus},1^{\ominus})$
$\displaystyle\underset{\mathrm{Regge}}{\to}c_{ggg^{*}}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\ominus},q)+c_{g^{*}gg}^{(1,\,s)}(-q,p_{4}^{\oplus},p_{1}^{\ominus})\,.$
(119)

Via the decomposition eq. 84, the peripheral-emission vertices eqs. 105, 114
and 115 are sufficient to construct the vertices relevant for QCD,

$\displaystyle c_{ggg^{*}}^{(1,\
g)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)$
$\displaystyle=\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}\left(-\frac{2}{\epsilon^{2}}-\frac{11}{6\epsilon}+\frac{1}{\epsilon}\log\left(\frac{\tau}{-s_{23}}\right)+\frac{\pi^{2}}{2}\right)\,,$
(120) $\displaystyle c_{ggg^{*}}^{(1,\
f)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)$
$\displaystyle=\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}\left(\frac{1}{3\epsilon}+\frac{5}{9}\right)\,.$
(121)

For completeness, we can also write down the impact factor for a
$\mathcal{N}=1_{V}$ multiplet circulating in the loop,

$\displaystyle c_{ggg^{*}}^{(1,\
\mathcal{N}=1_{V})}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)$
$\displaystyle=\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}\left(-\frac{2}{\epsilon^{2}}-\frac{3}{2\epsilon}+\frac{1}{\epsilon}\log\left(\frac{\tau}{-s_{23}}\right)+\frac{\pi^{2}}{2}\right)\,.$
(122)

Let us now consider the factorisation of one-loop primitive amplitudes with an
external quark-antiquark pair, recalling the supersymmetric organisation eq.
85. We can immediately obtain the primitive amplitudes in $\mathcal{N}=1_{V}$
from pure-gluon amplitudes by noting that in supersymmetric theories, the Ward
identities such as eqs. 59 and 61 hold to all orders. For MHV amplitudes, this
in turn implies the equality of one-loop corrections

$a_{5}^{(1,\,\mathcal{N}=1_{V})}(2_{\bar{q}},3_{q},4,1)=a_{5}^{(1,\,\mathcal{N}=1_{V})}(2,3,4,1)\,.$
(123)

Thus, from the Regge factorisation of the amplitude, eq. 102, we see the SUSY
Ward identity eq. 123 immediately leads to the identification of vertices

$c_{{\bar{q}}qg^{*}}^{(1,\
\mathcal{N}=1_{V})}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)=c_{ggg^{*}}^{(1,\
\mathcal{N}=1_{V})}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)\,.$ (124)

Let us now study the non-supersymmetric amplitudes on the right-hand side of
eq. 85, which are given in ref. Bern:1994fz . We find that in the Regge limit,
only the $g_{L}$ term has large-logarithmic dependence. The Regge
decomposition of this amplitude,

$\displaystyle\begin{split}a_{4}^{(1,\,g_{L})}(2_{\bar{q}}^{\lambda_{2}},3_{q}^{-\lambda_{2}},4^{-\lambda_{1}},1^{\lambda_{1}})\underset{\mathrm{Regge}}{\to}&c_{{\bar{q}}qg^{*}}^{(1,\
g_{L})}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)+r_{g^{*}}^{(1,\,g)}(q,s_{34})\\\
+&c_{g^{*}gg}^{(1,\
g)}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\,,\end{split}$ (125)

defines the vertex

$c_{{\bar{q}}qg^{*}}^{(1,\
g_{L})}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)=\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}\left(-\frac{1}{\epsilon^{2}}+\frac{1}{\epsilon}+\frac{1}{\epsilon}\log\left(\frac{\tau}{-s_{23}}\right)+\frac{\pi^{2}}{2}+\frac{19}{18}\right)\,.$
(126)

The fermion- and scalar-loop contributions are more subtle. Indeed, the
amplitudes are zero for all helicity configurations, and for both $L$ and $R$
contributions Bern:1994fz ,

$\displaystyle
a_{4}^{(1,\,{\mathcal{N}=1_{\chi}}_{L})}(2_{\bar{q}},3_{q},4,1)$
$\displaystyle=a_{4}^{(1,\,f_{L})}(2_{\bar{q}},3_{q},4,1)=a_{4}^{(1,\,s_{L})}(2_{\bar{q}},3_{q},4,1)=0\,,$
(127) $\displaystyle
a_{4}^{(1,\,{\mathcal{N}=1_{\chi}}_{R})}(2_{\bar{q}},3_{q},4,1)$
$\displaystyle=a_{4}^{(1,\,f_{R})}(2_{\bar{q}},3_{q},4,1)=a_{4}^{(1,\,s_{R})}(2_{\bar{q}},3_{q},4,1)=0\,.$
(128)

Nevertheless, if these colour-ordered amplitudes obey the factorisation eq.
102, we must take the quark-emission vertices to be equal and opposite that of
the gluon-emission vertices, i.e.

$\displaystyle c_{{\bar{q}}qg^{*}}^{(1,\,f)}(p_{2},p_{3},q)$
$\displaystyle=-c_{ggg^{*}}^{(1,\,f)}(p_{2},p_{3},q)\,,$ (129) $\displaystyle
c_{{\bar{q}}qg^{*}}^{(1,\,s)}(p_{2},p_{3},q)$
$\displaystyle=-c_{ggg^{*}}^{(1,\,s)}(p_{2},p_{3},q)\,.$ (130)

We note the ampltiudes in eq. 128 vanish, due to eq. 68a. We further note that
eq. 68a holds for an arbitrary number of additional gluons. Considering the
MRK limit of such amplitudes informs us the $f_{R}$ and $s_{R}$ amplitudes in
particular cannot contribute to the peripheral emission vertex of a quark,

$\displaystyle
c_{{\bar{q}}qg^{*}}^{(1,\,f_{R})}(p_{2},p_{3},q)=c_{{\bar{q}}qg^{*}}^{(1,\,s_{R})}(p_{2},p_{3},q)$
$\displaystyle=0\,.$ (131)

It follows from eq. 129 and eq. 130 that

$\displaystyle c_{{\bar{q}}qg^{*}}^{(1,\,f_{L})}(p_{2},p_{3},q)$
$\displaystyle=-c_{ggg^{*}}^{(1,\,f)}(p_{2},p_{3},q)\,,$ (132) $\displaystyle
c_{{\bar{q}}qg^{*}}^{(1,\,s_{L})}(p_{2},p_{3},q)$
$\displaystyle=-c_{ggg^{*}}^{(1,\,s)}(p_{2},p_{3},q)\,.$ (133)

Returning to the supersymmetric decomposition eq. 85, we can obtain the Regge
limit of the $g_{R}$ primitive amplitudes from our knowledge of the amplitudes
studied above,

$a_{4}^{(1,\,g_{R})}(2_{\bar{q}},3_{q},4,1)=a_{4}^{(1,\,\mathcal{N}=1_{V})}(2_{\bar{q}},3_{q},4,1)-a_{4}^{(1,\,g_{L})}(2_{\bar{q}},3_{q},4,1)-a_{4}^{(1,\,f_{L})}(2_{\bar{q}},3_{q},4,1)\,.$
(134)

We find that the large logarithmic terms cancel, as do the $ggg^{*}$-vertices.
This means that the $g_{R}$ amplitudes only contribute to the
${\bar{q}}qg^{*}$ vertex,

$\displaystyle\begin{split}a_{4}^{(1,\
g_{R})}(2_{\bar{q}},3_{q},4,1)\underset{\mathrm{Regge}}{\to}&c_{{\bar{q}}qg^{*}}^{(1,\
\mathcal{N}=1_{V})}(p_{2},p_{3},q)-c_{{\bar{q}}qg^{*}}^{(1,\
g_{L})}(p_{2},p_{3},q)+c_{{\bar{q}}qg^{*}}^{(1,\ f_{L})}(p_{2},p_{3},q)\\\
\equiv&c_{{\bar{q}}qg^{*}}^{(1,\ g_{R})}(p_{2},p_{3},q)\,.\end{split}$ (135)

Explicitly, this vertex is

$c_{{\bar{q}}qg^{*}}^{(1,\
g_{R})}(p_{2},p_{3},q)=\left(\frac{\mu^{2}}{-s_{23}}\right)^{\epsilon}\left(-\frac{1}{\epsilon^{2}}-\frac{3}{2\epsilon}-\frac{7}{2}\right)\,.$
(136)

That the $g_{R}$ amplitude only contributes to the quark vertex makes
intuitive sense, when considering the Feynman diagrams that contribute to this
amplitude. The fact that eq. 135 is not compatible with eq. 90 is of no
concern, because the supersymmetric decompositions are only valid at one loop.
Further validation of eq. 135 is provided by considering the Regge limit of
colour-dressed amplitudes, which is the topic of the next section.

###  5.4 Colour-dressed one-loop vertices for the peripheral emission of one
parton

In section 5.2 we reviewed the colour structure of four-parton one-loop
amplitudes in the Regge limit, and in section 5.3 we reviewed the
factorisation of the primitive amplitudes in this limit. In this section we
will combine those results, which will allow us to re-derive the colour-
dressed peripheral-emission vertices eq. 14b, eq. 14a and eq. 15.

Compatibility with the factorisation eq. 11 means the one-loop amplitudes are
given by the truncation to $g_{\mathrm{s}}^{4}$, that is,

$\displaystyle\begin{split}{\mathrm{Disp}\left[{\mathcal{A}^{(1)[-]}_{4}(f_{2},f_{3},f_{4},f_{1})}\right]}\underset{\mathrm{Regge}}{\to}&2s\
\mathcal{C}^{(1)}_{f_{2}f_{3}g^{*}}(p_{2},p_{3},q)\
\mathcal{R}^{(0)}_{g^{*}}(q)\
\mathcal{C}^{(0)}_{g^{*}f_{4}f_{1}}(-q,p_{4},p_{1})\\\ +&2s\
\mathcal{C}^{(0)}_{f_{2}f_{3}g^{*}}(p_{2},p_{3},q)\
\mathcal{R}^{(1)}_{g^{*}}(q;s_{34})\
\mathcal{C}^{(0)}_{g^{*}f_{4}f_{1}}(-q,p_{4},p_{1})\\\ +&2s\
\mathcal{C}^{(0)}_{f_{2}f_{3}g^{*}}(p_{2},p_{3},q)\
\mathcal{R}^{(0)}_{g^{*}}(q)\
\mathcal{C}^{(1)}_{g^{*}f_{4}f_{1}}(-q,p_{4},p_{1})\,.\end{split}$ (137)

Let us consider eq. 71 for specific helicity configurations, and substitute
the Regge limit of partial amplitudes obtained in section 5.3. We begin with
the trivial all-plus case, which we state for completeness:

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)[\pm]}(g_{2}^{\oplus},g_{3}^{\oplus},g_{4}^{\oplus},g_{1}^{\oplus})&\underset{\mathrm{Regge}}{\to}0\,.\end{split}$
(138)

For the single-minus case, the antisymmetry of the vertices lead to only the
signature-odd amplitude contributing in the Regge limit

$\displaystyle\begin{split}\mathcal{A}_{4}^{(1)[-]}(g_{2}^{\oplus},g_{3}^{\oplus},g_{4}^{-\lambda_{1}},g_{1}^{\lambda_{1}})\underset{\mathrm{Regge}}{\to}2s\,&\Big{[}g_{\mathrm{s}}^{3}\,c_{\Gamma}\,(N_{c}-N_{f})\,F^{d}_{a_{2}a_{3}}C^{(1,\,s)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},q)\Big{]}\times
R_{g^{*}}^{(0)}(q)\\\
\times&\Big{[}g_{\mathrm{s}}F^{d}_{a_{4}a_{1}}C_{g^{*}gg}^{(0)}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\Big{]}\,.\end{split}$
(139)

Comparing with eq. 137, we see that eq. 139 is indeed compatible with simple
Regge-pole factorisation and we obtain the vertex

$\mathcal{C}^{(1)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},q)=g_{\mathrm{s}}^{3}\,F^{c}_{a_{2}a_{3}}\,(N_{c}-N_{f})\,C^{(1,\,s)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},q)\,,$
(140)

in agreement with eq. 15a.

For the two-minus case, we have

$\displaystyle\begin{split}&{\mathrm{Disp}\left[{\mathcal{A}_{4}^{(1)[-]}(g_{2}^{\lambda_{2}},g_{3}^{-\lambda_{2}},g_{4}^{-\lambda_{1}},g_{1}^{\lambda_{1}})}\right]}\underset{\mathrm{Regge}}{\to}2s\,\left[g_{\mathrm{s}}\,F^{d}_{a_{2}a_{3}}C^{(0)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)\right]\times
R_{g^{*}}^{(0)}(q)\\\ &\mkern 128.0mu\times
c_{\Gamma}g_{\mathrm{s}}^{2}\Big{\\{}\left(N_{c}\
c_{ggg^{*}}^{(1,\,g)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)+N_{f}\
c_{ggg^{*}}^{(1,\,f)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)\right)\\\
&\mkern 128.0mu\hphantom{c_{\Gamma}g_{\mathrm{s}}^{2}\Bigg{\\{}}\
+N_{c}\,r_{g^{*}}^{(1,\,g)}(q;s_{34})\\\ &\mkern
128.0mu\hphantom{c_{\Gamma}g_{\mathrm{s}}^{2}\Bigg{\\{}}\ +\left(N_{c}\
c_{g^{*}gg}^{(1,\,g)}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})+N_{f}\
c_{g^{*}gg}^{(1,\,f)}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\right)\Big{\\}}\\\
&\mkern
128.0mu\times\left[g_{\mathrm{s}}F^{d}_{a_{4}a_{1}}C^{(0)}_{g^{*}gg}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\right]\end{split}$
(141)

Comparing these one-loop amplitudes with the Reggeisation ansatz eq. 137
allows us to define the one-loop colour dressed vertex

$\displaystyle\begin{split}{\mathcal{C}}^{(1)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)=&c_{\Gamma}\,g_{\mathrm{s}}^{3}F^{d}_{a_{2}a_{3}}\,C^{(0)}_{ggg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)\\\
&\times\left(N_{c}\
c_{ggg^{*}}^{(1,\,g)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)+N_{f}\
c_{ggg^{*}}^{(1,\,f)}(p_{2}^{\lambda_{2}},p_{3}^{-\lambda_{2}},q)\right)\,\end{split}$
(142)

in agreement with eq. 14a. Similarly, eq. 82 becomes

$\displaystyle\begin{split}&{\mathrm{Disp}\left[{\mathcal{A}_{4}^{(1)[-]}\left(\bar{q}_{2}^{\lambda_{2}},q_{3}^{-\lambda_{2}},g_{4}^{-\lambda_{1}},g_{1}^{\lambda_{1}}\right)}\right]}\underset{\mathrm{Regge}}{\to}2s\,\left[g_{\mathrm{s}}\,T^{d}_{\imath_{3}\bar{\imath}_{2}}C^{(0)}_{{\bar{q}}qg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{-^{\lambda_{2}}\lambda_{2}},q)\right]\times
R^{(0)}_{g^{*}}(q)\\\ &\qquad\times c_{\Gamma}\
g_{\mathrm{s}}^{2}\Bigg{\\{}\left(N_{c}\ c^{(1,\
g_{L})}_{{\bar{q}}qg^{*}}(p_{2},p_{3},q)-\frac{1}{N_{c}}c^{(1,\
g_{R})}_{{\bar{q}}qg^{*}}(p_{2},p_{3},q)+N_{f}\,c^{(1,\
f_{L})}_{{\bar{q}}qg^{*}}(p_{2},p_{3},q)\right)\\\
&\qquad\qquad\qquad+N_{c}\,r_{g^{*}}^{(1,\,g)}(q;s_{34})\\\
&\qquad\qquad\qquad+\left(N_{c}\ c^{(1,\,g)}_{g^{*}gg}(-q,p_{4},p_{1})+N_{f}\
c^{(1,\,f)}_{g^{*}gg}(-q,p_{4},p_{1})\right)\Bigg{\\}}\\\
&\qquad\times\left[g_{\mathrm{s}}\,F^{d}_{a_{4}a_{1}}C^{(0)}_{g^{*}gg}(-q,p_{4}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\right]\,,\end{split}$
(143)

from which we obtain

$\displaystyle\mathcal{C}^{(1)}_{{\bar{q}}qg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},q)=g_{\mathrm{s}}^{3}\,c_{\Gamma}\,T^{c}_{\imath_{3}\bar{\imath}_{2}}\,C^{(0)}_{{\bar{q}}qg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},q)$
(144) $\displaystyle\times\left(N_{c}\ c^{(1,\
g_{L})}_{{\bar{q}}qg^{*}}(p_{2},p_{3},q)-\frac{1}{N_{c}}c^{(1,\
g_{R})}_{{\bar{q}}qg^{*}}(p_{2},p_{3},q)+N_{f}\,c^{(1,\
f_{L})}_{{\bar{q}}qg^{*}}(p_{2},p_{3},q)\right)\,,$

in agreement with eq. 14b. In eq. 143 we see that the result eq. 135 is
responsible for producing a $1/N_{c}$ term in the ${\bar{q}}qg^{*}$ vertex and
not the $g^{*}gg$ vertex.

In this extended introduction we have analysed the one-loop four-parton
amplitudes in the new DDM bases, learning how to interpret the leading
behaviour of primitive amplitudes in the Regge limit. In the remaining
sections we extend this reasoning to the case of five-parton one-loop
amplitudes.

##  6 Colour structure of one-loop five-parton amplitudes in a peripheral NMRK
limit

In section 5.2 we analysed the colour structure of one-loop four-parton
amplitudes in the Regge limit. In this section we perform a similar analysis
of one-loop five-parton amplitudes in the forward NMRK limit, eq. 294. Let us
first consider the five-gluon amplitude for the physical process

$g(p_{1})g(p_{2})\to g(p_{3})g(p_{4})g(p_{5}).$

Starting from the $n=5$ case of eq. 63,

$\displaystyle\begin{split}\mathcal{A}_{5}^{(1)}(g_{1},g_{2},g_{3},g_{4},g_{5})&=g_{\mathrm{s}}^{5}\sum_{\sigma\in
S_{4}/\mathcal{R}}\mbox{tr}\left(F^{a_{\sigma_{1}}}\cdots
F^{a_{\sigma_{5}}}\right)A^{(1,\
g)}_{5}\left({\sigma_{1}},\ldots,{\sigma_{5}}\right)\\\ &+N_{f}\
g_{\mathrm{s}}^{5}\sum_{\sigma\in
S_{5}}\mbox{tr}\left(T^{a_{\sigma_{1}}}\cdots T^{a_{\sigma_{5}}}\right)A^{(1,\
f)}_{5}\left(\sigma_{1},\ldots,{\sigma_{5}}\right)\,,\end{split}$ (145)

we repeat the steps of section 5.2. Making use of the fact that the only
colour orderings which contribute to leading power are those that have gluons
1 and 5 cyclically adjacent, we obtain

$\displaystyle\begin{split}\mathcal{A}_{5}^{(1)}(g_{1},g_{2},g_{3},g_{4},g_{5})\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}g_{\mathrm{s}}^{5}\sum_{\sigma\in
Z_{3}}\Big{[}\mbox{tr}\left(F^{a_{\sigma_{2}}}F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}}F^{a_{5}}F^{a_{1}}\right)&A^{(1,\
g)}_{5}\left({\sigma_{2}},{\sigma_{3}},{\sigma_{4}},5,1\right)\\\
+\mbox{tr}\left(F^{a_{\sigma_{2}}}F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}}F^{a_{1}}F^{a_{5}}\right)&A^{(1,\
g)}_{5}\left({\sigma_{2}},{\sigma_{3}},{\sigma_{4}},1,5\right)\\\
+N_{f}\big{(}\mbox{tr}\left(T^{a_{\sigma_{2}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{4}}}T^{a_{5}}T^{a_{1}}\right)-\mbox{tr}\left(T^{a_{\sigma_{4}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{2}}}T^{a_{1}}T^{a_{5}}\right)\big{)}&A^{(1,\
f)}_{5}\left({\sigma_{2}},{\sigma_{3}},{\sigma_{4}},5,1\right)\\\
+N_{f}\big{(}\mbox{tr}\left(T^{a_{\sigma_{2}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{4}}}T^{a_{1}}T^{a_{5}}\right)-\mbox{tr}\left(T^{a_{\sigma_{4}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{2}}}T^{a_{5}}T^{a_{1}}\right)\big{)}&A^{(1,\
f)}_{5}\left({\sigma_{2}},{\sigma_{3}},{\sigma_{4}},1,5\right)\Big{]}\,,\end{split}$
(146)

where the sum is over cyclic permutations of the set $\\{2,3,4\\}$. Equation
146 naturaly suggests that we consider amplitudes of definite signature in the
$t_{2}=s_{51}$ channel. For the signature-odd amplitude, we obtain

$\displaystyle\begin{split}&\mathcal{A}_{5}^{(1)[-]}(g_{1},g_{2},g_{3},g_{4},g_{5})\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}g_{\mathrm{s}}^{5}\left(-F^{d}_{a_{5}a_{1}}\right)\\\
&\times\sum_{\sigma\in
Z_{3}}\Bigg{[}\mbox{tr}\big{(}F^{a_{\sigma_{2}}}F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}}F^{d}\big{)}A^{(1,\
g)[-]}_{5}\left({\sigma_{2}},{\sigma_{3}},{\sigma_{4}},5,1\right)\\\
&\quad+N_{f}\Big{(}\mbox{tr}\big{(}T^{a_{\sigma_{2}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{4}}}T^{d}\big{)}+\mbox{tr}\big{(}T^{a_{\sigma_{4}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{2}}}T^{d}\big{)}\Big{)}A^{(1,\
f)[-]}_{5}\left({\sigma_{2}},{\sigma_{3}},{\sigma_{4}},5,1\right)\Bigg{]}\,,\end{split}$
(147)

where we have used the structure constant definitions and Jacobi identity
(eqs. 4 and 3) to simplify the traces

$\displaystyle\mbox{tr}\big{(}\cdots[F^{a},F^{b}]\cdots\big{)}$
$\displaystyle=\mbox{tr}\big{(}\cdots F^{c}\cdots\big{)}F^{c}_{ba}\,,$ (148a)
$\displaystyle\mbox{tr}\big{(}\cdots[T^{a},T^{b}]\cdots\big{)}$
$\displaystyle=\mbox{tr}\big{(}\cdots T^{c}\cdots\big{)}F^{c}_{ba}\,.$ (148b)

We note that an overall factor $F^{d}_{a_{5}a_{1}}$ factorises from the colour
structures in eq. 147, and furthermore, that the remaining colour structures
are exactly that of the one-loop four-gluon amplitude, eq. 71, with $a_{1}\to
d$.

We now turn our attention to the one-loop amplitude with an external quark-
antiquark pair. We consider the physical process

$q(p_{1})g(p_{2})\to q(p_{3})g(p_{4})g(p_{5})$

as an example, but the following analysis applies to the other physical
processes obtained by crossing particles with momenta in the set
$\\{p_{2},p_{3},p_{4}\\}$, e.g., $gg\to q{\bar{q}}g.$ We start from the $n=5$
case of eq. 65:

$\displaystyle\begin{split}\mathcal{A}_{5}^{(1)}\left(\bar{q}_{2},q_{3},g_{4},g_{5},g_{1}\right)=g^{5}\Bigg{\\{}\sum_{\sigma\in
S_{3}}\bigg{[}&\left(T^{c_{2}}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}\left(F^{a_{\sigma_{4}}}F^{a_{\sigma_{5}}}F^{a_{\sigma_{1}}}\right)_{c_{1}c_{2}}A_{5}^{(1,\
g_{R})}\left(2_{\bar{q}},\sigma_{4},\sigma_{5},\sigma_{1},3_{q}\right)\\\
+&\left(T^{c_{2}}T^{a_{\sigma_{4}}}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}\left(F^{a_{\sigma_{5}}}F^{a_{\sigma_{1}}}\right)_{c_{1}c_{2}}A_{5}^{(1,\
g_{R})}\left(2_{\bar{q}},\sigma_{5},\sigma_{1},3_{q},\sigma_{4}\right)\\\
+&\left(T^{c_{2}}T^{a_{\sigma_{4}}}T^{a_{\sigma_{5}}}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}\left(F^{d_{\sigma_{1}}}\right)_{c_{1}c_{2}}A_{5}^{(1,\
g_{R})}\left(2_{\bar{q}},\sigma_{1},3_{q},\sigma_{4},\sigma_{5}\right)\\\
+&\left(T^{c_{2}}T^{a_{\sigma_{4}}}T^{a_{\sigma_{5}}}T^{a_{\sigma_{1}}}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}\delta_{c_{1}c_{2}}A_{5}^{(1,\
g_{R})}\left(2_{\bar{q}},3_{q},\sigma_{4},\sigma_{5},\sigma_{1}\right)\bigg{]}\\\
+\frac{N_{f}}{N_{c}}\bigg{[}\sum_{\sigma\in
S_{3}}&N_{c}(T^{a_{\sigma_{4}}}T^{a_{\sigma_{5}}}T^{a_{\sigma_{1}}})_{\imath_{3}\bar{\imath}_{2}}A_{5;1}^{(1,\
f)}\left(2_{\bar{q}},3_{q};\sigma_{4},\sigma_{5},\sigma_{1}\right)\\\
+\sum_{\sigma\in
S_{3}/Z_{2}}&\mbox{tr}\left(T^{a_{\sigma_{4}}}T^{a_{\sigma_{5}}}\right)(T^{\sigma_{1}})_{\imath_{3}\bar{\imath}_{2}}A_{5;3}^{(1,\
f)}\left(2_{\bar{q}},3_{q};\sigma_{4},\sigma_{5},\sigma_{1}\right)\\\
+\sum_{\sigma\in
S_{3}/Z_{3}}&\mbox{tr}(T^{a_{\sigma_{4}}}T^{a_{\sigma_{5}}}T^{a_{\sigma_{1}}})\delta_{\imath_{3}\bar{\imath}_{2}}A_{5;4}^{(1,\
f)}\left(2_{\bar{q}},3_{q};\sigma_{4},\sigma_{5},\sigma_{1}\right)\bigg{]}\Bigg{\\}}\end{split}$
(149)

As for the pure-gluon case, the signature-odd amplitude simplifies
considerably in the NMRK limit:

$\displaystyle\begin{split}&\mathcal{A}_{5}^{(1)[-]}\left(\bar{q}_{2},q_{3},g_{4},g_{5},g_{1}\right)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}g^{5}\left(-F^{d}_{a_{5}a_{1}}\right)\\\
&\hskip
40.00006pt\times\Bigg{\\{}\big{(}T^{c_{2}}T^{c_{1}}\big{)}_{\imath_{3}\bar{\imath}_{2}}\big{(}F^{a_{4}}F^{d}\big{)}_{c_{1}c_{2}}\,A_{5}^{(1,\,g_{R})[-]}\left(2_{\bar{q}},4,5,1,3_{q}\right)\\\
&\hskip
50.00008pt+\big{(}T^{c_{2}}T^{c_{1}}\big{)}_{\imath_{3}\bar{\imath}_{2}}\big{(}F^{d}F^{a_{4}}\big{)}_{c_{1}c_{2}}\,A_{5}^{(1,\,g_{R})[-]}\left(2_{\bar{q}},5,1,4,3_{q}\right)\\\
&\hskip
50.00008pt+\big{(}T^{c_{2}}T^{a_{4}}T^{c_{1}}\big{)}_{\imath_{3}\bar{\imath}_{2}}\big{(}F^{d}\big{)}_{c_{1}c_{2}}\,A_{5}^{(1,\,g_{R})[-]}\left(2_{\bar{q}},5,1,3_{q},4\right)\\\
&\hskip
50.00008pt+\big{(}T^{c_{2}}T^{d}T^{c_{1}}\big{)}_{\imath_{3}\bar{\imath}_{2}}\big{(}F^{a_{4}}\big{)}_{c_{1}c_{2}}\,A_{5}^{(1,\,g_{R})[-]}\left(2_{\bar{q}},4,3_{q},5,1\right)\\\
&\hskip
50.00008pt+\left(T^{c_{2}}T^{a_{4}}T^{d}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}\delta_{c_{1}c_{2}}\,A_{5}^{(1,\,g_{R})[-]}\left(2_{\bar{q}},3_{q},4,5,1\right)\\\
&\hskip
50.00008pt+\left(T^{c_{2}}T^{d}T^{a_{4}}T^{c_{1}}\right)_{\imath_{3}\bar{\imath}_{2}}\delta_{c_{1}c_{2}}A_{5}^{(1,\,g_{R})[-]}\left(2_{\bar{q}},3_{q},5,1,4\right)\\\
&\hskip
50.00008pt+\frac{N_{f}}{N_{c}}\bigg{[}N_{c}(T^{a_{4}}T^{d})_{\imath_{3}\bar{\imath}_{2}}A_{5;1}^{(1,\,f)[-]}\left(2_{\bar{q}},3_{q};4,5,1\right)\\\
&\hskip
70.0001pt+N_{c}(T^{d}T^{a_{4}})_{\imath_{3}\bar{\imath}_{2}}A_{5;1}^{(1,\,f)[-]}\left(2_{\bar{q}},3_{q};5,1,4\right)\\\
&\hskip
70.0001pt+\mbox{tr}(T^{a_{4}}T^{d})\delta_{\imath_{3}\bar{\imath}_{2}}A_{5;4}^{(1,\,f)[-]}\left(2_{\bar{q}},3_{q};4,5,1\right)\bigg{]}\Bigg{\\}}\end{split}$
(150)

As for the pure-gluon case, we see that an overall colour factor of
$F^{d}_{a_{5}a_{1}}$ factorises from all terms, that is to say, of the
antisymmetric representations, only $\mathbf{8}_{a}$ contributes to leading
power. The remaining colour structures are equal to the colour structures of
the one-loop four-parton amplitude eq. 76 with the replacement of $a_{1}\to
d$.

We now write the partial amplitudes in the final two lines of eq. 147 in terms
of primitive amplitudes. From eq. 70a we simply obtain

$\displaystyle A_{5;1}^{(1,\,f)[-]}\left(2_{\bar{q}},3_{q};4,5,1\right)=$
$\displaystyle A_{5}^{(1,\,f_{L})[-]}\left(2_{\bar{q}},3_{q},4,5,1\right)\,,$
(151) $\displaystyle
A_{5;1}^{(1,\,f)[-]}\left(2_{\bar{q}},3_{q};5,1,4\right)=$ $\displaystyle
A_{5}^{(1,\,f_{L})[-]}\left(2_{\bar{q}},3_{q},5,1,4\right)\,,$ (152)

even in general kinematics. For the remaining partial amplitude, the leading-
power terms of eq. 70b are

$\displaystyle\begin{split}A_{5;4}^{(1,\,f)[-]}\left(2_{\bar{q}},3_{q};4,5,1\right)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&A_{5}^{(1,\,f_{R})[-]}\left(2_{\bar{q}},1,5,3_{q},4\right)\\\
+&A_{5}^{(1,\,f_{R})[-]}\left(2_{\bar{q}},1,5,4,3_{q}\right)\\\
+&A_{5}^{(1,\,f_{R})[-]}\left(2_{\bar{q}},4,1,5,3_{q}\right)\,.\end{split}$
(153)

This partial amplitude has the same colour structure as the four-parton
partial amplitude eq. 77. Recall that in the four-parton case, this partial
amplitude vanished due to a combination of eq. 68a and eq. 78. If we were to
take seriously the analogy of the antisymmetrised pair of momenta $p_{1}$ and
$p_{5}$ with an effective of-shell gluon in the NMRK limit, we would expect
that the first term in eq. 153 vanishes due to an analogue of eq. 68a, while
the second and third terms cancel due to Furry’s theorem (which in QED is
valid for on-shell and off-shell photons). In the next section we will study
the one-loop five-parton primitive amplitudes in the NRMK limit and we will
see that this is indeed the case.

###  6.1 Simplified colour bases

While the DDM bases are very useful for the gauge-invariant organisation of
the kinematics, they are overcomplete. In this section we rewrite eq. 147 and
eq. 150 in a minimal basis. As in ref. Byrne:2022wzk , we choose a basis where
two elements are the colour-structures of the tree-level vertices eqs. 44 and
45. This choice is particularly useful for comparing the NLO and NNLO
contributions to the BFKL kernel. For the pure-gluon case we obtain

$\displaystyle\begin{split}&\mathcal{A}^{(1)[-]}_{5}(2,3,4,5,1)=g_{\mathrm{s}}^{5}(-F^{d}_{a_{5}a_{1}})\\\
&\times\Bigg{\\{}\sum_{\sigma\in
S_{2}}(F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}})_{a_{2}c_{2}}\\\ &\hskip
20.00003pt\times\
\bigg{[}N_{c}\,\left(\frac{2}{3}A_{5}^{(1,g)}(2,\sigma_{3},\sigma_{4},5,1)-\frac{1}{3}A_{5}^{(1,g)}(\sigma_{3},2,\sigma_{4},5,1)-\frac{1}{3}A_{5}^{(1,g)}(\sigma_{3},\sigma_{4},2,5,1)\right)\\\
&\hskip
30.00005pt+N_{f}\,\left(\frac{2}{3}A_{5}^{(1,f)}(2,\sigma_{3},\sigma_{4},5,1)-\frac{1}{3}A_{5}^{(1,f)}(\sigma_{3},2,\sigma_{4},5,1)-\frac{1}{3}A_{5}^{(1,f)}(\sigma_{3},\sigma_{4},2,5,1)\right)\bigg{]}\\\
&\qquad+d_{A}^{a_{2}a_{3}a_{4}c_{2}}\left(A_{5}^{(1,g)}(2,3,4,5,1)+A_{5}^{(1,g)}(3,2,4,5,1)+A_{5}^{(1,g)}(3,4,2,5,1)\right)\\\
&\qquad+2\,d_{F}^{a_{2}a_{3}a_{4}c_{2}}\left(A_{5}^{(1,f)}(2,3,4,5,1)+A_{5}^{(1,f)}(3,2,4,5,1)+A_{5}^{(1,f)}(3,4,2,5,1)\right)\Bigg{\\}}\end{split}$
(154)

The new colour structures at one-loop are the totally symmetrised traces in
the adjoint and fundamental representations,

$\displaystyle d_{A}^{a_{1}a_{2}a_{3}a_{4}}$
$\displaystyle=\frac{1}{4!}\sum_{\sigma\in
S_{4}}\operatorname{tr}\left(F^{a_{\sigma_{1}}}F^{a_{\sigma_{2}}}F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}}\right)\,,$
(155) $\displaystyle d_{F}^{a_{1}a_{2}a_{3}a_{4}}$
$\displaystyle=\frac{1}{4!}\sum_{\sigma\in
S_{4}}\operatorname{tr}\left(T^{a_{\sigma_{1}}}T^{a_{\sigma_{2}}}T^{a_{\sigma_{3}}}T^{a_{\sigma_{4}}}\right)\,.$
(156)

Noting the correspondence between colour-representations and loop content $m$,
we define the following two linear combinations of primitive amplitudes:

$\displaystyle A_{FF}^{(1,\,m)}(2,3,4,5,1)$
$\displaystyle=\frac{2}{3}A_{5}^{(1,\,m)}(2,3,4,5,1)-\frac{1}{3}A_{5}^{(1,\,m)}(2,4,3,5,1)-\frac{1}{3}A_{5}^{(1,\,m)}(4,2,3,5,1)\,,$
(157) $\displaystyle A_{d}^{(1,\,m)}(2,3,4,5,1)$
$\displaystyle=A_{5}^{(1,\,m)}(2,3,4,5,1)+A_{5}^{(1,\,m)}(2,4,3,5,1)+A_{5}^{(1,\,m)}(4,2,3,5,1)\,.$
(158)

Similarly, for the quark-antiquark case we find

$\displaystyle\begin{split}&\mathcal{A}_{5}^{(1)[-]}\left(\bar{q}_{2},q_{3},g_{4},g_{5},g_{1}\right)\to
g^{5}\left(-F^{d}_{51}\right)\\\
&\times\Bigg{\\{}\big{(}T^{d}T^{a_{4}}\big{)}_{\imath_{3}\bar{\imath}_{2}}\bigg{(}N_{c}A_{5}^{(1,\
g_{L})[-]}\left(2_{\bar{q}},3_{q},5,1,4\right)-\frac{1}{N_{c}}A_{5}^{(1,\
g_{R})[-]}\left(2_{\bar{q}},3_{q},5,1,4\right)\\\ &\hskip
240.00037pt+N_{f}A_{5}^{(1,\
f_{L})[-]}\left(2_{\bar{q}},3_{q},5,1,4\right)\bigg{)}\\\ &\hskip
10.00002pt+\big{(}T^{a_{4}}T^{d}\big{)}_{\imath_{3}\bar{\imath}_{2}}\bigg{(}N_{c}A_{5}^{(1,\
g_{L})[-]}\left(2_{\bar{q}},3_{q},4,5,1\right)-\frac{1}{N_{c}}A_{5}^{(1,\
g_{R})[-]}\left(2_{\bar{q}},3_{q},4,5,1\right)\\\ &\hskip
240.00037pt+N_{f}A_{5}^{(1,\
f_{L})[-]}\left(2_{\bar{q}},3_{q},4,5,1\right)\bigg{)}\\\ &\hskip
10.00002pt+\delta_{\imath_{3}\bar{\imath}_{2}}\Delta_{da_{4}}\bigg{(}A_{5}^{(1,\
g_{L})[-]}\left(2_{\bar{q}},3_{q},5,1,4\right)+A_{5}^{(1,\
g_{R})[-]}\left(2_{\bar{q}},3_{q},5,1,4\right)\\\ &\hskip
55.00008pt+A_{5}^{(1,\
g_{L})[-]}\left(4,2_{\bar{q}},3_{q},5,1\right)+A_{5}^{(1,\
g_{R})[-]}\left(4,2_{\bar{q}},3_{q},5,1\right)\\\ &\hskip
55.00008pt+A_{5}^{(1,\
g_{L})[-]}\left(2_{\bar{q}},4,3_{q},5,1\right)+A_{5}^{(1,\
g_{R})[-]}\left(2_{\bar{q}},4,3_{q},5,1\right)\bigg{)}\Bigg{\\}}\end{split}$
(159)

We see the kinematic coefficient of the $\delta\Delta$ colour structure can be
conveniently written in terms of

$\displaystyle A_{\delta\Delta}^{(1,\,m)}(2_{\bar{q}},3_{q},4,5,1)$
$\displaystyle=A_{5}^{(1,\,m)}(2_{\bar{q}},3_{q},4,5,1)+A_{5}^{(1,\,m)}(2_{\bar{q}},4,3_{q},5,1)+A_{5}^{(1,\,m)}(4,2_{\bar{q}},3_{q},5,1)\,.$
(160)

We note that the totally symmetric colour-structures $d_{A}$ and $d_{F}$ will
not contribute to the NNLO impact factors, as they vanish when interfered with
the tree-level colour structures. This is not the case for the $\delta\Delta$
colour structures.

Our goal is to study the factorisation properties of eqs. 157 and 158 for
$m{\in}\\{g,f\\}$ and eq. 160 for $m{\in}\\{g_{L},g_{R}\\}$. In the next
section we first study the factorisation of the constituent primitive
amplitudes.

##  7 One-loop primitive vertices for the peripheral emission of two gluons

In this chapter we investigate the NMRK limit of one-loop, colour-ordered,
five-gluon amplitudes. Such amplitudes can be organised along the lines of
colour ordering, helicity configuration, and loop content. We have organised
this chapter firstly in terms of loop content, that is, by what particle type,
or particle multiplet, is circulating in the loop. Following the organisation
of ref. Bern:1993mq , we investigate the $\mathcal{N}=4$ amplitudes in section
7.1, the $\mathcal{N}=1_{\chi}$ amplitudes in section 7.2 and the amplitudes
with a complex scalar circulating in the loop in section 7.3. Other multiplets
(not least the physically relevant gluon and fermion contributions) can be
obtained through the supersymmetric rearrangements outlined in section 5.3.

Each section in this chapter is further subdivided by helicity configuration.
Colour-ordered amplitudes with differing helicity configuration typically have
qualitatively different structure. An extreme example of this is provided by a
comparison of the amplitudes in section 5.3.1 with section 5.3.2. For our
purposes, there are two separate points worth emphasising at this stage.
Firstly, a given two-gluon peripheral emission vertex, say

$C_{gggg^{*}}^{(1,\,m)}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\,$

can be obtained by taking the NMRK limit of four distinct five-gluon
amplitudes,

$A_{5}^{(1,\,m)}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus},p_{1}^{\ominus})\,,\qquad
A_{5}^{(1,\,m)}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{1}^{\oplus},p_{5}^{\ominus})\,,$
$A_{5}^{(1,\,m)}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\ominus},p_{1}^{\oplus})\,,\qquad
A_{5}^{(1,\,m)}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{1}^{\ominus},p_{5}^{\oplus})\,.$

The former two amplitudes, being split helicity configurations, have a simpler
analytic structure than the latter two amplitudes. In, the following, we
choose to present the extraction of the peripheral emission vertex from the
simplest starting amplitude (although in practice we have checked each
helicity configuration to verify the universality of the factorisation).
Secondly, in close analogue to amplitudes, the vertices themselves have
qualitatively different structure based on the helicity configuration of the
_vertex_. We need to consider three distinct helicity configurations,

$C_{gggg^{*}}^{(1,\,m)}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\,,\qquad
C_{gggg^{*}}^{(1,\,m)}(p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},q_{2})\,,\qquad
C_{gggg^{*}}^{(1,\,m)}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\,.$

The other helicity configurations can be obtained by complex conjugation and
by the reflection property of the vertices which is inherited from eq. 64. We
note that a supersymmetric ward identity guarantees the all-plus and single-
minus amplitudes vanish to all loop orders in supersymmetric theories. We
therefore only need to consider the all-plus vertex for amplitudes with a
circulating complex scalar.

Our final subdivision of vertices is by colour ordering. For a given helicity
configuration we need to consider the three colour orderings which appear in
eq. 147,

$C_{gggg^{*}}^{(1,\,m)}(p_{2},p_{3},p_{4},q_{2})\,,\qquad
C_{gggg^{*}}^{(1,\,m)}(p_{3},p_{4},p_{2},q_{2})\,,\qquad
C_{gggg^{*}}^{(1,\,m)}(p_{4},p_{3},p_{2},q_{2})$

With a notable exception to be discussed in section section 7.1.1, these
colour orderings can be obtained by application of the kinematic maps in
section 3. Having discussed the organisational structure of this chapter, we
begin with the analysis of amplitudes in $\mathcal{N}=4$.

###  7.1 $\mathcal{N}=4$ contribution

The analysis of one-loop five-gluon amplitudes in $\mathcal{N}=4$ sYM proceeds
similarly to the discussion in section 5.3. As in the four-gluon case, eq.
103, the one-loop corrections are helicity independent, and given by a simple
transcendental function Bern:1993mq

$\displaystyle\begin{split}a_{5}^{(1,\
\mathcal{N}=4)}(1,2,3,4,5)&=-\frac{1}{\epsilon^{2}}\sum_{i=1}^{5}\left(\frac{\mu^{2}}{-s_{i,i+1}}\right)^{\epsilon}+\frac{5}{6}\pi^{2}-\frac{\delta_{R}}{3}\\\
&+\sum_{i=1}^{5}\log\left(\frac{-s_{i,i+1}}{-s_{i+1,i+2}}\right)\log\left(\frac{-s_{i+2,i-2}}{-s_{i-2,i-1}}\right)\,.\end{split}$
(161)

We emphasise that the all-plus and single-minus amplitudes are zero in
supersymmetric theories, so we only need to extract the single-minus vertex.
For consistent organisation of this chapter we treat these identical helicity
configurations under the following subsection.

####  7.1.1  $\\{g^{\ominus},g^{\oplus},g^{\oplus},g^{*}\\}$,
$\\{g^{\oplus},g^{\ominus},g^{\oplus},g^{*}\\}$ and
$\\{g^{\oplus},g^{\oplus},g^{\ominus},g^{*}\\}$

As mentioned in the introduction to this chapter, when no large logarithms are
present in the amplitude, obtaining vertices with different cyclic orderings
is a simple matter of applying the kinematic maps in section 3. This is not
the case for one-loop amplitudes in $\mathcal{N}=4$, and we treat the
following two colour orderings separately.

#### Cyclic ordering $\bm{\\{p_{2},p_{3},p_{4},q_{2}\\}}$

As in the four-gluon case discussed in section 5.3, the function eq. 161
admits an _exact_ factorisation into one-loop MRK building blocks
Bartels:2008ce . We thus write the real part of the one-loop correction in the
physical region of eq. 277 as

$\displaystyle\begin{split}a_{5}^{(1,\,\mathcal{N}=4)}(2,3,4,5,1)=&c_{ggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{2},p_{3},q_{1})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1};s_{34})\\\
+&v^{(1,\,\mathcal{N}=4)}(t_{1},\frac{s_{34}s_{45}}{s_{345}},t_{2})\\\
+&r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{2};s_{45})+c_{g^{*}gg}^{(1,\,\mathcal{N}=4)}(-q_{2},p_{5},p_{1})\,,\end{split}$
(162)

where we have used the special function Byrne:2022wzk

$\displaystyle\begin{split}v^{(1,\
\mathcal{N}=4)}(t_{1},\eta,t_{2})=&-\frac{1}{\epsilon^{2}}\,\left(\frac{\mu^{2}}{\eta}\right)^{\epsilon}+\frac{\pi^{2}}{3}-\frac{1}{2}\log^{2}\left(\frac{t_{1}}{t_{2}}\right)\\\
&+\frac{1}{\epsilon}\,\left[\left(\frac{\mu^{2}}{-t_{1}}\right)^{\epsilon}+\left(\frac{\mu^{2}}{-t_{2}}\right)^{\epsilon}\right]\log\left(\frac{\tau}{\eta}\right)\,.\end{split}$
(163)

As indicated by the lack of subscripts, and the different treatment of the
arguments of this function, we treat this function differently to the other
building blocks hitherto encountered. We will not use this function as a
factorised building block in its own right, but rather to organise the one-
loop corrections to amplitudes and multi-gluon emission vertices such that
their Regge limits are transparent. For example, from eq. 162 the peripheral-
emission vertex for two gluons can be immediately idendified as

$\displaystyle\begin{split}c_{gggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},p_{4},q_{2})=&c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},q_{1})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1};s_{34})\\\
+&v^{(1,\,\mathcal{N}=4)}(t_{1},\frac{s_{34}\,p_{4}^{+}}{(p_{3}^{+}+p_{4}^{+})},t_{2})\,,\end{split}$
(164)

where we needed only to take the NMRK limit of the $\eta$ variable

$\frac{s_{34}s_{45}}{s_{345}}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}\frac{s_{34}\,p_{4}^{+}}{(p_{3}^{+}+p_{4}^{+})}$
(165)

to obtain a function that depends on the momenta local in rapidity (that is,
on $\\{p_{2},p_{3},p_{4}\\}$).

In turn, this vertex has a trivial decomposition in the MRK limit

$\displaystyle\begin{split}c_{gggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},p_{4},q_{2})\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}&c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},q_{1})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1};s_{34})\\\
+&v_{g^{*}gg^{*}}^{(1,\,\mathcal{N}=4)}(-q_{1},p_{4},q_{2})\,,\end{split}$
(166)

where the one-loop Lipatov vertex in $\mathcal{N}=4$ (c.f. the weight-two
terms in eq. 26) is conveniently written in terms of the special function eq.
163,

$\displaystyle
v^{(1,\,\mathcal{N}=4)}_{g^{*}gg^{*}}(-q_{1},p_{4},q_{2})=v^{(1,\,\mathcal{N}=4)}(-|q_{1\perp}|^{2},|p_{4\perp}|^{2},-|q_{2\perp}|^{2})\,.$
(167)

As mentioned in the introduction to this chapter, it is generally just a
matter of applying the kinematic maps in section 3 to obtain the vertices for
other cyclic orderings of the momenta. However, the presence of large
logarithms in $\mathcal{N}=4$ ampltiudes introduces some subtleties which are
discussed in the following section.

#### Cyclic ordering $\bm{\\{p_{3},p_{2},p_{4},q_{2}\\}}$

The exact decomposition eq. 162 holds for any cyclic permutation of the
indices, so it is tempting therefore to write the amplitude with colour
ordering $\\{3,2,4,5,1\\}$ as

$\displaystyle\begin{split}a_{5}^{(1,\,\mathcal{N}=4)}(3,2,4,5,1)=&c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{3},p_{2},q_{1})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1};s_{24})\\\
+&v^{(1,\,\mathcal{N}=4)}(t_{1},\frac{s_{24}s_{45}}{s_{245}},t_{2})\\\
+&r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{2};s_{45})+c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{5},p_{1},-q_{2})\,.\end{split}$
(168)

While this is correct, eq. 168 suggests we should take the peripheral-emission
vertex for this colour ordering to be

$\displaystyle\begin{split}c_{gggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{3},p_{2},p_{4},q_{2})=&c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{3},p_{2},q_{1})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1};s_{24})\\\
+&v^{(1,\,\mathcal{N}=4)}(t_{1},\frac{s_{24}\,p_{4}^{+}}{(p_{2}^{+}+p_{4}^{+})},t_{2})\,,\end{split}$
(169)

that is, the vertex eq. 164, under $P_{324}$. This vertex has the desired MRK
limit,

$\displaystyle\begin{split}c_{gggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{3},p_{2},p_{4},q_{2})\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}&c_{ggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{3},p_{2},q_{1})+r_{g^{*}}^{(1,\
\mathcal{N}=4)}(q_{1};p_{3}^{+}p_{4}^{-})\\\ +&v_{g^{*}gg^{*}}^{(1,\
\mathcal{N}=4)}(q_{1},p_{4},q_{2})\,.\end{split}$ (170)

However, this does not obtain the correct ${\mathrm{MRK}^{\prime}}$ limit.
Indeed, we after somewhat perverse manipulations, we can write the
${\mathrm{MRK}^{\prime}}$ limit of eq. 168 in terms of the expected
${\mathrm{MRK}^{\prime}}$ building blocks plus an unwanted large logarithmic
term:

$\displaystyle\begin{split}c_{gggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{3},p_{2},p_{4},q_{2})\underset{\scriptscriptstyle{\mathrm{MRK}^{\prime}}}{\to}&c_{ggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{2},p_{4},q_{1}^{\prime})+r_{g^{*}}^{(1,\
\mathcal{N}=4)}(q_{1}^{\prime};p_{3}^{+}p_{4}^{-})\\\ +&v_{g^{*}gg^{*}}^{(1,\
\mathcal{N}=4)}(-q_{1}^{\prime},p_{3},q_{2})+\frac{2}{\epsilon}\left(\frac{\mu^{2}}{|q_{2\perp}|^{2}}\right)^{\epsilon}\log\left(\frac{p_{3}^{+}}{p_{4}^{+}}\right)\,.\end{split}$
(171)

We therefore cannot take our vertex to be eq. 169. Rather than starting from
eq. 168, we could instead exploit the reflection identity eq. 64 to write the
_same_ one-loop amplitude in the form 555As in ref. Byrne:2022wzk , the signs
of the arguments of the functions are fixed by the requirement that each
building block is real.

$\displaystyle\begin{split}a_{5}^{(1,\
\mathcal{N}=4)}(4,2,3,1,5)=&c_{ggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{4},p_{2},q_{1}^{\prime})+r_{g^{*}}^{(1,\
\mathcal{N}=4)}(q_{1}^{\prime};s_{43})\\\ +&v^{(1,\
\mathcal{N}=4)}(q_{1}^{\prime},\frac{s_{43}\,s_{35}}{s_{435}},q_{2})\\\
+&r_{g^{*}}^{(1,\ \mathcal{N}=4)}(q_{2};-s_{31})+c_{ggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{5},p_{1},-q_{2})\,.\end{split}$ (172)

This alternative rewriting of the one-loop amplitude suggests we should take
the impact factor to be

$\displaystyle\begin{split}c_{gggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{4},p_{2},p_{2},q_{2})=&c_{ggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{4},p_{2},q_{1}^{\prime})+r_{g^{*}}^{(1,\
\mathcal{N}=4)}(q_{1}^{\prime};s_{43})\\\ +&v^{(1,\
\mathcal{N}=4)}(q_{1}^{\prime},\frac{s_{43}\,p_{3}^{+}}{(p_{3}^{+}+p_{4}^{+})},q_{2})\,,\end{split}$
(173)

which correctly describes the known ${\mathrm{MRK}^{\prime}}$ limit, but not
the MRK limit.

At this point we recall the NMRK limit of the tree-level amplitude of the
colour-ordering in question,

$\displaystyle\begin{split}A^{(0)}_{5}(3,2,4,5,1)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}s\,C^{(0)}_{gggg^{*}}(p_{3},p_{2},p_{4},q_{2})\frac{1}{t_{2}}C^{(0)}_{g^{*}gg}(-q_{2},p_{5},p_{1})\,.\end{split}$
(174)

Recalling eq. 58, we write

$C^{(0)}_{gggg^{*}}(p_{3},p_{2},p_{4},q_{2})=-C^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})-C^{(0)}_{gggg^{*}}(p_{2},p_{4},p_{3},q_{2})\,.$
(175)

The first and second term vanish in the ${\mathrm{MRK}^{\prime}}$ and
${\mathrm{MRK}}$ limits respectively. This observation gives us a way to
utilise the two-fold nature of the one-loop amplitude. We use eq. 57 and eq.
64 to write the full colour-ordered amplitude as

$\displaystyle\begin{split}&\frac{1}{2s}A_{5}^{(1,\
\mathcal{N}=4)}(3,2,4,5,1)\\\
&\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}-c_{\Gamma}C^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})\frac{1}{t_{2}}C^{(0)}_{g^{*}gg}(-q_{2},p_{5},p_{1})a_{5}^{(1,\
\mathcal{N}=4)}(3,2,4,5,1)\\\
&\phantom{\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}}-c_{\Gamma}C^{(0)}_{gggg^{*}}(p_{2},p_{4},p_{3},q_{2})\frac{1}{t_{2}}C^{(0)}_{g^{*}gg}(-q_{2},p_{5},p_{1})a_{5}^{(1,\
\mathcal{N}=4)}(4,2,3,5,1)\,.\end{split}$ (176)

We then use the decomposition eq. 168 for the first line and the decomposition
eq. 172 for the second line. Explicitly,

$\displaystyle\begin{split}&\frac{1}{2s}A_{5}^{(1,\
\mathcal{N}=4)}(3,2,4,5,1)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}-c_{\Gamma}\sum_{\sigma\in
S_{2}}C^{(0)}_{gggg^{*}}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})\\\
&\times\Bigg{\\{}c_{gggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})+r_{g^{*}}^{(1,\
\mathcal{N}=4)}(q_{2};p_{\sigma_{4}}^{+}p_{5}^{-})+c_{ggg^{*}}^{(1,\
\mathcal{N}=4)}(p_{5},p_{1},-q_{2})\Bigg{\\}}\\\
&\frac{1}{t_{2}}C^{(0)}_{g^{*}gg}(-q_{2},p_{5},p_{1})\,.\end{split}$ (177)

In this way, the decomposition of the loop-correction which trivialises the
${\mathrm{MRK}}$ or ${\mathrm{MRK}^{\prime}}$ limits only has support in those
limits. The requirement of yielding the correct ${\mathrm{MRK}}$ and
${\mathrm{MRK}^{\prime}}$ limits has forced us to write eq. 177 as a sum over
two permutations, which will carry over to the colour-dressed amplitude, eq.
224.

###  7.2 $\mathcal{N}=1_{\chi}$ contribution

The analysis of one-loop five-gluon amplitudes in $\mathcal{N}=1_{\chi}$ are
in one respect simpler than the amplitudes in $\mathcal{N}=4$, due to the lack
of any large logarithms in the NMRK or MRK limit. Therefore, the complications
discussed in section 7.1.1 do not arise. However, the amplitudes are not
helicity independent. We need to consider two distinct helicity configurations
of the starting amplitudes, and two distinct helicity configurations of the
vertices. For the convenience of the reader we list the amplitudes here, which
have been taken from refs. Bern:1993mq ; Canay:2021yvm . The amplitudes are
MHV so we use the notation introduced in eq. 97, and we also use the notation
of eq. 107 separating the amplitude into a divergent part

$\displaystyle
V^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus})=\frac{1}{\epsilon}+\frac{1}{2}\left[\log\left(\frac{\mu^{2}}{-s_{23}}\right)+\log\left(\frac{\mu^{2}}{-s_{51}}\right)\right]+2\,,$
(178a) $\displaystyle
V^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\oplus})=\frac{1}{\epsilon}+\frac{1}{2}\left[\log\left(\frac{\mu^{2}}{-s_{34}}\right)+\log\left(\frac{\mu^{2}}{-s_{51}}\right)\right]+2\,,$
(178b)

and a finite part

$\displaystyle
G^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus})$
$\displaystyle=-\frac{I_{1234}+I_{1245}}{2s_{12}s_{51}}L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\,$
(179a)
$\displaystyle\begin{split}G^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\oplus})&=\frac{I_{1325}+I_{1342}}{2s_{13}s_{51}}L_{0}\left(\frac{-s_{34}}{-s_{51}}\right)\\\
&-\frac{I_{1324}I_{1342}}{s_{13}^{2}s_{51}^{2}}Ls_{1}\left(\frac{-s_{23}}{-s_{51}},\frac{-s_{34}}{-s_{51}}\right)-\frac{I_{1325}I_{1352}}{s_{13}^{2}s_{34}^{2}}Ls_{1}\left(\frac{-s_{12}}{-s_{34}},\frac{-s_{51}}{-s_{34}}\right)\,.\end{split}$
(179b)

We will now extract the two distinct helicity configurations for the
$gggg^{*}$ vertex, as outlined in the introduction.

####  7.2.1  $\\{g^{\ominus},g^{\oplus},g^{\oplus},g^{*}\\}$

As discussed in the introduction to this chapter, it is possible to obtain the
vertex

$c^{(1,\,\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})$

from the split-helicity amplitude

$a^{(1,\,\mathcal{N}=1_{\chi})}_{5g}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus},p_{1}^{\ominus}).$

Inspecting eq. 178a, we note that, even in general kinematics, the real part
of this term is nothing other than the sum of two peripheral-emission
vertices, eq. 114, that is,

$\displaystyle\mathrm{Re}\left[V^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus})\right]=c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})+c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{5},p_{1},-q_{2})\,.$ (180)

As an aside, we note this simple separation of the peripheral emission
vertices holds also for the one-loop $n$-gluon split-helicity amplitude
Bern:1994cg , so the one-loop $n{-}4$-gluon all-plus central-emission vertex
is a simple, finite function that can be readily extracted from the $n$-gluon
amplitude.

Returning to the present five-gluon case, we have only to take the forward
NMRK limit of eq. 179a. We first write it in terms of the minimal variables,

$\displaystyle\begin{split}G^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus})&=\frac{1}{2}\left(\frac{X+Y|z|^{2}+XY(z-{\bar{z}}+1)}{X+|z|^{2}+XY|z-1|^{2}}\right)L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\,.\end{split}$
(181)

It is now straightforward to take various Regge limits. In particular, we can
obtain the contribution to the forward-emission vertex by taking the
$Y\to\infty$ (forward NMRK) limit,

$\displaystyle\begin{split}G^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus})&\underset{Y\to\infty}{\to}\frac{1}{2}\left(\frac{|z|^{2}+X(z-{\bar{z}}+1)}{X|z-1|^{2}}\right)L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\,.\end{split}$
(182)

One check of this result is to take the further $X\to\infty$ (MRK) limit,

$\displaystyle\begin{split}G^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus})&\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}\frac{1}{2}\left(\frac{z-{\bar{z}}+1}{|z-1|^{2}}\right)L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\,,\end{split}$
(183)

This is the one-loop correction to the Lipatov vertex in
$\mathcal{N}=1_{\chi}$, which can be inferred from e.g. ref. DelDuca:1998cx ,

$\displaystyle\begin{split}v_{g^{*}gg^{*}}^{(1,\
\mathcal{N}=1_{\chi})}(-q_{1},p_{4},q_{2})&=\frac{1}{2}\left(|q_{1\perp}|^{2}+|q_{2\perp}|^{2}-2q_{1\perp}q_{2\perp}^{*}\right)\frac{L_{0}\left(\frac{|q_{1\perp}|^{2}}{|q_{2\perp}|^{2}}\right)}{|q_{2\perp}|^{2}}\\\
&=\frac{1}{2}\left(\frac{z-{\bar{z}}+1}{|z-1|^{2}}\right)L_{0}\left(\frac{|q_{1\perp}|^{2}}{|q_{2\perp}|^{2}}\right)\,.\end{split}$
(184)

Collecting the two contributions to the peripheral-emission vertex, we obtain

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})=c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})+\frac{1}{2}\left(\frac{|z|^{2}+X(z-{\bar{z}}+1)}{X|z-1|^{2}}\right)L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\,,\end{split}$
(185)

which has the transparent MRK limit

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\underset{X\to\infty}{\to}c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})+v_{g^{*}gg^{*}}^{(1,\
\mathcal{N}=1_{\chi})}(q_{1},p_{4},q_{2})\,.\end{split}$ (186)

The other colour orderings can be obtained using the kinematic maps derived in
section 3. For example, applying the map $P_{243}$ given in eq. 40 we obtain

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{2}^{\ominus},p_{4}^{\oplus},p_{3}^{\oplus},q_{2})=c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{4},q_{1})+\frac{1}{2}\left(\frac{{\bar{z}}+z({\bar{z}}-1)+X}{|z-1|^{2}}\right)L_{0}\left(\frac{-t_{1}^{\prime}}{-t_{2}}\right)\,,\end{split}$
(187)

while applying the map $P_{324}$ given in eq. 41 to the vertex eq. 185 we
obtain

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{3}^{\ominus},p_{2}^{\oplus},p_{4}^{\oplus},q_{2})=&c^{(1,\,\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{3},p_{2},q_{1})\\\
+&\frac{1}{2}\left(\frac{z(2-{\bar{z}})+X(z-{\bar{z}}+1)}{X|z-1|^{2}}\right)L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\,.\end{split}$
(188)

####  7.2.2  $\\{g^{\oplus},g^{\ominus},g^{\oplus},g^{*}\\}$

We cannot obtain
$c^{(1,\,\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},q_{2})$
from a split helicity configuration, but must consider, for example,

$a^{(1,\
\mathcal{N}=1_{\chi})}_{5g}(p_{1}^{\oplus},p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},p_{5}^{\ominus})$

in the forward NMRK limit. Again, we see that $V^{(1,\,\mathcal{N}=1_{\chi})}$
is nothing other than the sum of $ggg^{*}$ vertices

$\displaystyle\mathrm{Re}\left[V^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\oplus},p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},p_{5}^{\ominus})\right]=c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})+c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{5},p_{1},-q_{2})\,.$ (189)

The forward NMRK limit of the remaining term is

$\displaystyle\begin{split}&G^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\oplus},p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},p_{5}^{\ominus})\underset{Y\to\infty}{\to}-\frac{z|z|^{2}+z(z-1)X-(z-{\bar{z}}+1)X^{2}}{2|z-1|^{2}(X+z)X}L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\\\
&\hskip
60.00009pt-\frac{Xz}{(X+z)^{2}}Ls_{-1}\left(\frac{-s_{23}}{-s_{51}},\frac{-s_{24}}{-s_{51}}\right)+\frac{z(X+{\bar{z}})}{|z-1|^{2}(X+z)}L_{0}\left(\frac{-s_{24}}{-s_{51}}\right)\,.\end{split}$
(190)

We note the presence of terms of transcendental weight 2 that appear in this
vertex. This is unlike the factorised building blocks of the LO and NLO BFKL
kernel and impact factors, where the terms of highest transcendental weight
coincide with the $\mathcal{N}=4$ building blocks. Whether such weight-2 terms
survive phase-space integration, and therefore contribute to the NNLO impact
factor, is a question that will be left to a future investigation. In any case
these weight-2 terms are suppressed in the MRK limit, and indeed we find

$\displaystyle\begin{split}G^{(1,\,\mathcal{N}=1_{\chi})}(p_{1}^{\oplus},p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},p_{5}^{\ominus})&\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}v^{(1,\,\mathcal{N}=1_{\chi})}_{g^{*}gg^{*}}(-q_{1},p_{4},q_{2})\end{split}$
(191)

as expected. Adding the contributions from $V^{(1,\,\mathcal{N}=1_{\chi})}$
and $G^{(1,\,\mathcal{N}=1_{\chi})}$ we obtain the vertex

$\displaystyle\begin{split}c^{(1,\
\mathcal{N}=1_{\chi})}(p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},q_{2})&=-\frac{Xz}{(X+z)^{2}}Ls_{-1}\left(\frac{-t_{1}}{-t_{2}},\frac{-t_{1}^{\prime}}{-t_{2}}\right)+\frac{z(X+{\bar{z}})}{|z-1|^{2}(X+z)}L_{0}\left(\frac{-t_{1}^{\prime}}{-t_{2}}\right)\\\
&-\frac{z|z|^{2}+z(z-1)X-(z-{\bar{z}}+1)X^{2}}{2|z-1|^{2}(X+z)X}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+c^{(1,\ \mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})\,,\end{split}$
(192)

which has the MRK limit

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},q_{2})\underset{X\to\infty}{\to}c^{(1,\
\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})+v_{g^{*}gg^{*}}^{(1,\
\mathcal{N}=1_{\chi})}(q_{1},p_{4},q_{2})\,.\end{split}$ (193)

As in section 7.2.1, the other cyclic orderings of momenta can be obtained
using the maps provided in section 3. For example, the map $P_{324}$ (eq. 40)
gives

$\displaystyle\begin{split}c^{(1,\
\mathcal{N}=1_{\chi})}_{gggg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},q_{2})&=\frac{z(X+z)}{X^{2}}Ls_{-1}\left(\frac{-t_{1}}{-t_{2}},\frac{-s_{34}}{-t_{2}}\right)+\frac{z(X+z)(1+X)}{X^{2}|z-1|^{2}}L_{0}\left(\frac{-s_{34}}{-t_{2}}\right)\\\
&+\frac{2z^{2}+z(1+z)X+(z-{\bar{z}}+1)(X+z)X}{2|z-1|^{2}X^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+c^{(1,\ \mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2},p_{3},q_{1})\,,\end{split}$
(194)

In the following, because the expressions for the vertices are lengthier, we
will only display one representative ordering for each helicity configuration.
Note that unlike previous expressions for vertices, we have not written the
transcendental terms in eq. 194 in a manifestly real way. Here and in the
following, we choose to write each Mandelstam invariant with a minus sign, as
in eq. 194, but it is implicit that the dispersive part of all factorised
vertices should be taken, for the physical scattering region in question (here
eq. 277). Due to the simple analytic structure of the vertices, this
convention is of little consequence for the present section, but in section 9,
where several distinct scattering processed are relevant, this convention will
allow for simpler analytic continuation to those regions.

###  7.3 Complex scalar contribution

The analysis of amplitudes with a circulating complex scalar proceeds in close
analogy to the $\mathcal{N}=1_{\chi}$ case. Indeed, the divergent terms are
given by

$\displaystyle
V^{(1,\,s)}=\frac{1}{3}V^{(1,\,\mathcal{N}=1_{\chi})}+\frac{2}{9}\,.$ (195)

Recalling eq. 115, we see that the identification of $V^{(1,\,s)}$ with impact
factors proceeds in exact analogy with the $\mathcal{N}=1_{\chi}$ case. The
characteristic difference in the scalar case is the presence of non-cut-
constructible rational terms. The split-helicity amplitude is still a
relatively simple function,

$\displaystyle\begin{split}G^{(1,\,s)}(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},p_{5}^{\oplus})&=\frac{1}{3}G^{(1,\,\mathcal{N}=1_{\chi})}+\frac{I_{1234}I_{1245}\left(I_{1234}+I_{1245}\right)}{3s_{12}^{3}s_{51}^{3}}L_{2}\left(\frac{-s_{23}}{-s_{51}}\right)\\\
&+\frac{I_{1235}^{2}}{3s_{12}^{2}s_{23}s_{51}}\left(1-\frac{s_{35}}{s_{12}}\right)+\frac{I_{1234}I_{1245}}{6s_{12}^{2}s_{23}s_{51}}\,,\end{split}$
(196)

which, as we will see in section 7.3.2 leads to a relatively simple
$g^{\ominus}g^{\oplus}g^{\oplus}g^{*}$ vertex.

The other helicity configuration is more involved,

$\displaystyle\begin{split}&G^{(1,\,s)}(p_{1}^{\ominus},p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\oplus})=\frac{2}{3}\frac{I_{1324}^{3}I_{1342}}{s_{13}^{4}s_{24}s_{51}^{3}}L_{2}\left(\frac{-s_{23}}{-s_{51}}\right)+\frac{2}{3}\frac{I_{1352}^{3}I_{1325}}{s_{13}^{4}s_{25}s_{34}^{3}}L_{2}\left(\frac{-s_{12}}{-s_{34}}\right)\\\
&\qquad-\frac{I_{1324}^{2}I_{1342}^{2}}{s_{13}^{4}s_{24}^{2}s_{51}^{2}}\left[2Ls_{1}\left(\frac{-s_{23}}{-s_{51}},\frac{-s_{34}}{-s_{51}}\right)+L_{1}\left(\frac{-s_{23}}{-s_{51}}\right)+L_{1}\left(\frac{-s_{34}}{-s_{51}}\right)\right]\\\
&\qquad-\frac{I_{1325}^{2}I_{1352}^{2}}{s_{13}^{4}s_{25}^{2}s_{34}^{2}}\left[2Ls_{1}\left(\frac{-s_{12}}{-s_{34}},\frac{-s_{51}}{-s_{34}}\right)+L_{1}\left(\frac{-s_{12}}{-s_{34}}\right)+L_{1}\left(\frac{-s_{51}}{-s_{34}}\right)\right]\\\
&\qquad+\frac{1}{3s_{51}^{3}}L_{2}\left(\frac{-s_{34}}{-s_{51}}\right)\left[-\frac{I_{1325}I_{1342}\left(I_{1325}+I_{1342}\right)}{s_{13}^{3}}+2\frac{I_{1342}^{3}I_{1324}}{s_{13}^{4}s_{24}}+2\frac{I_{1325}^{3}I_{1352}}{s_{13}^{4}s_{25}}\right]\\\
&\qquad+\frac{I_{1325}+I_{1342}}{6s_{13}s_{51}}L_{0}\left(\frac{-s_{34}}{-s_{51}}\right)+\frac{I_{1325}^{2}I_{1342}^{2}}{3s_{13}^{4}s_{23}s_{51}s_{34}s_{12}}\\\