# Supermassive black holes from runaway mergers and accretion in nuclear star
clusters

Konstantinos Kritos,1 Emanuele Berti,1 and Joseph Silk1,2,3
1William H. Miller III Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, Maryland 21218, USA
2Insitut d’Astrophysique de Paris, UMR 7095 CNRS and UPMC, Sorbonne
Université, F-75014 Paris, France
3University of Oxford, Oxford OX1 3RH, United Kingdom E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Rapid formation of supermassive black holes occurs in dense nuclear star
clusters that are initially gas-dominated. Stellar-mass black hole remnants of
the most massive cluster stars sink into the core, where a massive runaway
black hole forms as a consequence of combined effects of repeated mergers and
Eddington-limited gas accretion. The associated gravitational-wave signals of
high-redshift extreme mass-ratio inspirals are a unique signature of the
nuclear star cluster scenario.

###### keywords:

(transients:) black hole mergers, galaxies: nuclei, (galaxies:) quasars:
supermassive black holes

††pubyear: 2024††pagerange: Supermassive black holes from runaway mergers and
accretion in nuclear star clusters–References

Introduction. The emergence of supermassive black holes (SMBHs) in the early
Universe is an outstanding problem in modern cosmology and astrophysics. Many
active galactic nuclei (AGN) powered by SMBHs have been identified at redshift
$z\gtrsim 6$, with spectroscopically inferred black hole (BH) masses in the
range $10^{6}$–$10^{8}\,M_{\odot}$ (Greene et al., 2023; Juodžbalis et al.,
2024; Maiolino et al., 2024). As suggested by recent spectroscopic surveys
with the James Webb Space Telescope (JWST), the abundance of red-selected AGN
is a hundred times that expected in this redshift range (Matthee et al., 2023;
Maiolino et al., 2023). The consensus is that SMBHs have grown out of massive
seed black holes (Volonteri, 2012), and the problem is then reduced to
explaining the formation of those intermediate-mass BH seeds (Greene et al.,
2020). One possible formation channel is the direct collapse of pristine gas
clouds at $z>10$ (Latif et al., 2022; Begelman et al., 2006). Alternative
seeding mechanisms involve repeated BH mergers in dense star clusters
(Antonini et al., 2019; Fragione & Silk, 2020; Atallah et al., 2023; Kritos et
al., 2023) and feeding by debris of successive tidal disruption events (TDEs;
Stone et al., 2017; Rizzuto et al., 2023; Inayoshi et al., 2020). Moreover,
several works have studied the rapid formation of supermassive stellar objects
with masses in the range $10^{3}$–$10^{5}M_{\odot}$ within metal-poor massive
star clusters through runaway collisions of stars and gas accretion (Tagawa et
al., 2020; Vergara et al., 2023; Reinoso et al., 2023; Escala, 2021;
Schleicher et al., 2022; Das et al., 2021; Boekholt et al., 2018). These
supermassive stellar objects can then collapse into intermediate-mass BH
seeds.

Nuclear star clusters (NSCs) are extremely dense and massive stellar systems
that have been observed to reside in the centers of most galaxies in the local
Universe (Neumayer et al., 2020). In the nearby Universe, NSCs span the mass
range from $10^{5}M_{\odot}$ to $10^{9}M_{\odot}$, with evidence for in situ
star formation in the more massive systems (Fahrion et al., 2022). At high
redshift, a number of lensed star clusters with effective radii as small as a
parsec ($\rm pc$) and with masses of order $10^{6}M_{\odot}$ have been
identified (Vanzella et al., 2023; Adamo et al., 2024). There is some
theoretical support for the fragmentation of the central region of
protogalaxies into millions of metal-poor stars, forming clusters with half-
mass radii $\lesssim 0.5\,\rm pc$ (Devecchi et al., 2010).

In this paper, we use a semianalytic NSC model and demonstrate the possibility
of rapidly forming SMBHs through the combined processes of repeated BH mergers
and Eddington-limited gas accretion, assuming that a substantial amount of gas
can be retained within the cluster for a few tens of millions of years ($\rm
Myr$).

Nuclear star cluster model. Our numerical model for the NSC implements a two-
mass system consisting of $N_{\star}$ stars, each of mass $m_{\star}$, and
$N_{\rm BH}$ stellar BHs, each of mass $m_{\rm BH}$ and nonspinning, with an
initial mass $M_{\rm g}$ of residual gas not turned into stars. We find that
typically one of the centrally located and initially stellar mass BHs grows
dramatically via gas, star, and BH accretion, and we denote its mass and spin
by $M_{\rm BH}$ and $\chi_{\rm BH}$, respectively. The total cluster mass is
therefore $M_{\rm cl}=m_{\star}N_{\star}+m_{\rm BH}N_{\rm BH}+M_{\rm g}+M_{\rm
BH}$. Since the growing BH is one of the stellar BHs, initially $M_{\rm
BH,0}=m_{\rm BH}$ and $\chi_{\rm BH,0}=0$. This “oligarchic” growth scenario
can occur in dynamical environments (see e.g. Matsubayashi et al., 2004;
Kovetz et al., 2018), where the formation of multiple massive BHs within the
same system is significantly suppressed by their preferential accretion by the
most massive object (Kritos et al., 2023).

The gas, composed primarily of ionized hydrogen with sound speed $c_{\rm
s}\simeq 10\,\rm km\,s^{-1}$ (Krumholz & Matzner, 2009), is removed
exponentially over a characteristic gas expulsion timescale $\tau_{\rm ge}$
(Baumgardt & Kroupa, 2007). The growing BH may accrete some of this gas. We
assume an Eddington-limited Bondi accretion rate onto the BH seed
$\dot{M}_{\rm acc}$ with Salpeter timescale of $50\,\rm Myr$, and a time-
evolving Plummer gas density $\rho_{\rm g}=3M_{\rm g}/[4\pi(r_{\rm
h}/1.3)^{3}]$, where $r_{\rm h}$ is the half-mass radius. As a consequence of
gradual gas removal, $r_{\rm h}$ expands adiabatically (Hills, 1980).

The half-mass relaxation time $\tau_{\rm rh}$, modified for two-mass systems
following Eq. (10) from Antonini & Gieles (2020), is the characteristic time
over which a fraction $\zeta\simeq 0.1$ of the total cluster energy $E_{\rm
cl}=-\kappa GM_{\rm cl}^{2}/(2r_{\rm h})$ (with $\kappa\simeq 0.4$, see
Spitzer & Hart, 1971) can be transferred throughout the cluster via two-body
relaxation (Breen & Heggie, 2013). The core is dominated by BHs, which
partially decouple from stars due to the impossibility of energy equipartition
(“Spitzer instability”). The core collapses on a time-scale $\tau_{\rm cc}$
given by Eq. (5) from Portegies Zwart & McMillan (2002).

At this point, three-body hard binaries (3bb) can form and heat the system.
The rate of energy production $\dot{E}_{\rm cl}=-\zeta E_{\rm cl}/\tau_{\rm
rh}$ depends on $r_{\rm h}$, and is controlled by the efficiency of two-body
relaxation (“Hénon’s principle”). We compute the energy generated by a single
3bb $Q_{\rm 3bb}$, and use this condition to calculate the 3bb rate $R_{\rm
3bb}=\dot{E}_{\rm cl}/Q_{\rm 3bb}$ (as in Heggie & Hut, 2003, page 244). If
all BHs but one have been ejected from the system, we assume that energy
generation proceeds via tidal disruption of stars accreting onto the runaway
BH. The TDE rate is $R_{\rm TDE}=\dot{E}_{\rm cl}/Q_{\rm TDE}$, where $Q_{\rm
TDE}=GM_{\rm BH}\overline{m}_{\star}/(2r_{\rm T})$ is the heat released from a
single TDE and $r_{\rm T}$ is the tidal radius (Rees, 1988). The main
advantage of our method, as in Antonini & Gieles (2020), is that it is not
necessary to calculate the properties of the core, while we can incorporate
the effect of a cusp around the SMBH.

The differential equations that govern the evolution of the system are given
by

$\displaystyle{d\overline{m}_{\star}\over dt}$
$\displaystyle=-\nu{\overline{m}_{\star}\over t}\Theta(t-t_{\rm se})\,,$ (1a)
$\displaystyle{dN_{\star}\over dt}$ $\displaystyle=-\xi_{\rm
e}{N_{\star}\over\tau_{\rm rh}}-R_{\rm TDE}\,,$ (1b) $\displaystyle{dN_{\rm
BH}\over dt}$ $\displaystyle=-k_{\rm ejs}R_{\rm 3bb}\,,$ (1c)
$\displaystyle{dM_{\rm g}\over dt}$ $\displaystyle=-{M_{\rm g}\over\tau_{\rm
ge}}-\dot{M}_{\rm acc}\,,$ (1d) $\displaystyle{dM_{\rm BH}\over dt}$
$\displaystyle=\dot{M}_{\rm acc}+f_{\rm TDE}\overline{m}_{\star}R_{\rm
TDE}\,,$ (1e) $\displaystyle{dr_{\rm h}\over dt}$ $\displaystyle=\left({\zeta
r_{\rm h}\over\tau_{\rm rh}}+{2r_{\rm h}\over M_{\rm cl}}{dM_{\rm cl}\over
dt}\right)\Theta(t-\tau_{\rm cc})$ $\displaystyle-{r_{\rm h}\over M_{\rm
cl}}\left({N_{\star}}{d\overline{m}_{\star}\over dt}-{M_{\rm g}\over\tau_{\rm
ge}}\right)\,.$ (1f)

Equations (1a) and (1b) can be compared with Alexander et al. (2014). The main
difference is a loss term due to TDEs, $R_{\rm TDE}$, on the right-hand side
of Eq. (1b). The first three terms on the right-hand side of Eq. (1f) follow
Antonini & Gieles (2020), but we also include an adiabatic expansion term due
to gas expulsion. The quantity $k_{\rm ejs}$ in Eq. (1c) is the number of
single BHs ejected per unit binary. We set $\nu=0.07$, $t_{\rm se}=2\,\rm Myr$
(Antonini & Gieles, 2020), and $\xi_{\rm e}=0.0074$ (Alexander et al., 2014).

At each iteration, we sample $k_{\rm 3bb}$, the number of 3bbs formed, from a
Poisson distribution with parameter $\lambda=R_{\rm 3bb}dt$, where $dt$ is the
time-step of the simulation. Every 3bb is formed with components $M_{\rm BH}$
and $m_{\rm BH}$, with its initial semimajor axis $a_{0}$ determined by
sampling the initial hardness parameter $\eta_{0}\geq 1$ (Morscher et al.,
2015). Binaries harden until they reach their ejection ($a_{\rm ejb}$) or
gravitational-wave (GW) radius ($a_{\rm GW}$), assuming thermal eccentricity.
To compute the merger remnant mass $m_{\rm r}$, spin $\chi_{\rm r}$, and
relativistic recoil—or GW kick $v_{\rm k}$—we implement the equations in Sec.
V.A-C of Gerosa & Kesden (2016) and references therein, which are based on
numerical relativity fits and assume isotropic spin directions. If $a_{\rm
GW}>a_{\rm ejb}$ the binary merges within the cluster through the emission of
GWs, and if $v_{\rm k}<v_{\rm esc},$ the merger remnant is retained in the
cluster. In this case, we discretely update the runaway BH mass, spin, and
number by the substitutions $M_{\rm BH}\to m_{\rm r}$, $\chi_{\rm
BH}\to\chi_{\rm r}$, and $N_{\rm BH}\to N_{\rm BH}-1$, while in any case of
ejection, $M_{\rm BH}\to m_{\rm BH}$, $\chi_{\rm BH}\to 0$, and $N_{\rm BH}\to
N_{\rm BH}-2$.

Black hole growth. We implement a fourth-order Runge-Kutta scheme to
numerically solve the system of equations (1) using a time step of
$dt=0.01\,\rm Myr$ (chosen to be much smaller than $\tau_{\rm rh}$). We set
$N_{\star,0}=10^{7}$, $r_{\rm h,0}=0.1\,\rm pc$ (where the “0” refers to
initial values) and $\tau_{\rm ge}=100\,\rm Myr$. Using a Kroupa initial mass
function in the range $[0.08,150]M_{\odot}$, the average initial stellar mass
is $\overline{m}_{\star,0}=0.59M_{\odot}$, while the fraction of BH
progenitors with mass $>20M_{\odot}$ is $\simeq 0.18\%$. We assume $m_{\rm
BH}=10M_{\odot}$, but to quantify uncertainties, we also repeat the analysis
with $m_{\rm BH}=50M_{\odot}$. In Fig. 1, we show the growth of $M_{\rm BH}$
for four selected values of $M_{\rm
g,0}=0,\,10^{6},\,10^{7},\,10^{8}M_{\odot}$.

Figure 1: Time evolution of the growing BH mass $M_{\rm BH}$ for different
initial amounts of gas in the system. The gas expulsion time-scale is set to
$\tau_{\rm ge}=100\,\rm Myr$, the initial stellar number to
$N_{\star,0}=10^{7}$, and the initial half-mass radius to $r_{\rm
h,0}=0.1\,\rm pc$. Thick (thin) lines correspond to simulations with $m_{\rm
BH}=10M_{\odot}$ ($m_{\rm BH}=50M_{\odot}$). For comparison, we also show how
a $10M_{\odot}$ seed (thick gray) and a $50M_{\odot}$ seed (thin gray) would
grow under the sole effect of Eddington accretion.

Figure 2: Time evolution of the cumulative number of BH mergers $N_{\rm me}$
(left panel), merger rate $R_{\rm me}$ (left panel, inset), half-mass radius
$r_{\rm h}$ (right panel), and cluster mass $M_{\rm cl}$ (right panel, inset)
under the same initial conditions used in Fig. 1. Figure 3: Characteristic GW
strain amplitude $h_{\rm c}$ (computed as in Bonetti & Sesana, 2020) as a
function of detector-frame GW frequency for five extreme mass-ratio inspirals
(EMRIs) at redshift $z=10$ (colored lines). We use cosmological parameters
from Planck 18 (Aghanim et al., 2020). The initial orbital frequency and
eccentricity are indicated in the legend; labels on the colored curves refer
to the mass of the primary. The bands indicate the $h_{\rm c}$ range as the
secondary mass is varied between $10M_{\odot}$ (lower bound) and $50M_{\odot}$
(upper bound). The black solid curves correspond to the noise amplitudes of
present and planned GW observatories: aLIGO
(https://dcc.ligo.org/LIGO-T1800044/public), aVIRGO
(https://git.ligo.org/sensitivity-curves/observing-scenario-
paper-2019-2020-update), ET (https://www.et-gw.eu/index.php/etsensitivities),
CE (https://cosmicexplorer.org/sensitivity.html), DECIGO (Yagi & Seto, 2011),
LISA (Robson et al., 2019), and $\mu$Ares (Sesana et al., 2021). Finally, the
shaded regions correspond to lower redshift ($z=1$–$2$) binary sources from
stellar-mass compact astrophysical binaries (CAB), EMRIs, and SMBH binaries.

All simulated systems have $v_{\rm esc,0}>300\,\rm km\,s^{-1}$, allowing
repeated mergers to form a runaway BH (Antonini et al., 2019). The BH rapidly
grows by hundreds of repeated BH mergers (Fig. 2, left panel) with a merger
rate that peaks at $\sim 100\,{\rm Myr}^{-1}$ right after core collapse (see
inset in the left panel of Fig. 2) and steadily drops as the system expands.

Significant amounts (20%–30%) of gas can be accreted within $200\,\rm Myr$,
rapidly bringing the BH mass into the SMBH regime. The residual gas does not
fragment because of heating by UV radiation from cluster stars. Our accretion
channel is similar to the direct collapse scenario but occurs on the Salpeter
timescale, rather than on the dynamical timescale. The combination of mergers
and accretion helps accelerate the growth because Eddington-limited accretion
cannot itself grow a stellar-mass BH seed above $1000M_{\odot}$ on a similar
timescale (compare the gray lines in Fig. 1, which ignore BH mergers).

Starting from the initial radius $r_{\rm h,0}=0.1\,\rm pc$, the cluster
inflates by a factor of 10–100 (depending on $M_{\rm g,0}$), so that $r_{\rm
h,0}$ is on the parsec scale after a Hubble time (see Fig. 2, right panel).
When $m_{\rm BH}=50M_{\odot}$, a larger amount of energy is produced, causing
$r_{\rm h}$ to expand to above $100\,\rm pc$ (a feature that resembles the
bulges of present-day dwarf galaxies). The rapid mass loss in the first few
hundred Myr is due to gas expulsion (see the inset of the right panel of Fig.
2). After core collapse, the gas-richest system (dotted red line) undergoes a
phase of contraction due to balanced evolution that starts at $\simeq 22\,\rm
Myr$. Once the gas has been expelled, the energy production term in Eq. (1f)
dominates, and the system starts to expand again at $t\simeq 200\,\rm Myr$.

We have demonstrated with an example that a combination of gas accretion and
BH mergers is capable of growing stellar-mass BH seeds into the SMBH regime
within only $200\,\rm Myr$. Most notably, if such a cluster forms at $z=15$,
then (depending on the initial gas content) an SMBH can assemble by $z\simeq
10$. These SMBH seeds can then further grow by subsequent gas accretion
episodes during gas inflows into the NSC following a galaxy merger. This
hierarchical scenario can be studied by combining our formalism with galaxy
merger trees.

We recall that very massive NSCs are observed locally in many galaxies. Many
of them are inferred to have gas because we observe young stars. We find that
NSC evolution inevitably results in expansion by an order of magnitude or more
(see the right panel of Fig. 2). In contrast, at formation, many NSCs were
massive and compact. This supports our contention that NSCs are a promising
environment for generating SMBHs via an enhanced merger rate density in the
high-redshift Universe.

The ultimate test of our model would be the observation of individual GW
signals from SMBH–BH mergers in NSC cores, which may be possible with future
GW observatories (Fig. 3). The GW background produced by these sources is a
distinctive signature of the NSC scenario that deserves further study. Our
predictions motivate the development of the science case for next-generation
GW observatories spanning the $\mu{\rm Hz}$ frequency range between LISA and
Pulsar Timing Arrays. Our proposed GW background should be taken into account
when estimating astrophysical confusion noise sources that can affect the
sensitivity of proposed $\mu$Hz detectors, such as $\mu$Ares. For comparison,
the black-dotted line in Fig. 3 shows an estimate of the confusion noise from
SMBH and white dwarf binaries in the $\mu$Ares band (Sesana et al., 2021).

Acknowledgements. We thank Fabio Antonini, Mark Ho-Yeuk Cheung, and Giada
Caneva Santoro for discussions. This paper was supported by the Onassis
Foundation - Scholarship ID: F ZT 041-1/2023-2024. K.K. and E.B. are supported
by NSF Grants No. AST-2006538, PHY-2207502, PHY-090003 and PHY-20043, by NASA
Grants No. 20-LPS20-0011 and 21-ATP21-0010, by the John Templeton Foundation
Grant 62840, by the Simons Foundation, and by the Italian Ministry of Foreign
Affairs and International Cooperation Grant No. PGR01167. This work was
carried out at the Advanced Research Computing at Hopkins (ARCH) core facility
(rockfish.jhu.edu), which is supported by the NSF Grant No. OAC-1920103.

Data availability. The output of the simulations is available upon a
reasonable request to the corresponding author. A Python implementation of our
model through the code Nuce (for “NUclear Cluster Evolution”) is publicly
available on GitHub at the URL https://github.com/Kkritos/Nuce.

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&\qquad+\frac{I_{1324}^{2}I_{1342}^{2}}{3s_{13}^{4}s_{23}s_{24}s_{34}s_{51}}+\frac{I_{1325}^{2}I_{1352}^{2}}{3s_{13}^{4}s_{25}s_{12}s_{34}s_{51}}-\frac{I_{1342}I_{1325}}{6s_{13}^{2}s_{34}s_{51}}\,.\end{split}$
(197)

and, as we will see in section 7.3.3, this leads to a relatively complicated
$g^{\oplus}g^{\ominus}g^{\oplus}g^{*}$ vertex.

Unlike the supersymmetric multiplets studied in section 7.1 and section 7.2,
the all-plus amplitude does not vanish for a circulating complex scalar
Bern:1993mq :

$\displaystyle
A_{5}^{(1,\,s)}\left(1^{\oplus},2^{\oplus},3^{\oplus},4^{\oplus},5^{\oplus}\right)=\frac{1}{6(4\pi)^{2}}\frac{\left(\sum_{i=1}^{5}s_{i-1,\,i}\,s_{i,\,i+1}\right)+\epsilon(1,2,3,4)}{\langle
12\rangle\langle 23\rangle\langle 34\rangle\langle 45\rangle\langle
51\rangle}\,,$ (198)

where the contracted Levi-Civita symbol can be written

$\epsilon(i,j,k,l)=[i\,j]\langle j\,k\rangle[k\,l]\langle l\,i\rangle-\langle
i\,j\rangle[j\,k]\langle k\,l\rangle[l\,i]\,.$ (199)

Similarly, the single-minus amplitude is non-vanishing,

$\displaystyle\begin{split}&A^{(1,\,s)}\left(1^{\ominus},2^{\oplus},3^{\oplus},4^{\oplus},5^{\oplus}\right)=\frac{1}{3(4\pi)^{2}}\frac{1}{[12]\langle
23\rangle\langle 34\rangle\langle 45\rangle[51]}\\\ &\hskip
30.00005pt\times\bigg{[}\left(s_{23}+s_{34}+s_{45}\right)[25]^{2}-[24]\langle
43\rangle[35][25]\\\ &\hskip 60.00009pt-\frac{[12][15]}{\langle
12\rangle\langle 15\rangle}\left(\langle 12\rangle^{2}\langle
13\rangle^{2}\frac{[23]}{\langle 23\rangle}+\langle 13\rangle^{2}\langle
14\rangle^{2}\frac{[34]}{\langle 34\rangle}+\langle 14\rangle^{2}\langle
15\rangle^{2}\frac{[45]}{\langle 45\rangle}\right)\bigg{]}\end{split}$ (200)

The NMRK limits of these amplitudes have already been investigated in ref.
Canay:2021yvm , and the $g^{\oplus}g^{\oplus}g^{\oplus}g^{*}$ vertex has been
extracted. For completeness, in the following section we briefly repeat this
analysis using the minimal set of variables of eq. 31.

####  7.3.1  $\\{g^{\oplus},g^{\oplus},g^{\oplus},g^{*}\\}$

We find the amplitude factorises according to eq. 28 as expected, e.g.

$\displaystyle
A_{5}^{(1,\,s)}(2^{\oplus},3^{\oplus},4^{\oplus},5^{\oplus},1^{\ominus})$
$\displaystyle\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s\,C_{ggg^{*}}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})R^{(0)}_{g^{*}}(q_{2})C^{(0)}_{g^{*}gg}(-q_{2},p_{5}^{\oplus},p_{1}^{\ominus})\,.$
(201)

This allows us to identify the vertex

$\displaystyle\begin{split}&C^{(1,\,s)}_{gggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})=\\\
&\frac{1}{q_{2\perp}}\left(\frac{q_{2\perp}^{*}}{q_{2\perp}}\right)\frac{1}{X+1}\left(\frac{X^{2}(z-1)^{3}{\bar{z}}}{z^{2}({\bar{z}}-1)(X+z)}-\frac{X^{2}(z-1)^{3}\left(X+{\bar{z}}\right)}{(X+1)z({\bar{z}}-1)(X+z)^{2}}+\frac{X(z-1)^{2}}{z(X+z)}\right)\,.\end{split}$
(202)

in agreement with Canay:2021yvm . This vertex in turn factorises in the
further MRK limit:

$\displaystyle\begin{split}C^{(1,\,s)}_{gggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}&\frac{1}{q_{2\perp}}\left(\frac{q_{2\perp}^{*}}{q_{2\perp}}\right)\frac{{\bar{z}}(z-1)^{3}}{z^{2}({\bar{z}}-1)}\\\
&=C^{(1,\,\mathcal{N}=1_{\chi})}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},q_{1})R^{(0)}_{g^{*}}(q_{1})V^{(0)}_{g^{*}gg^{*}}(-q_{1},p_{4},q_{2})\,.\end{split}$
(203)

Let us now consider the all-plus case, where eq. 28 reads

$\displaystyle{\mathrm{Disp}\left[{A(2^{\oplus},3^{\oplus},4^{\oplus},5^{\oplus},1^{\oplus})}\right]}=2s\,C_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})R_{g^{*}}(q_{2};s_{45})C_{g^{*}gg}(-q_{2},p_{5}^{\oplus},p_{1}^{\oplus})\,.$
(204)

Given that both gluon-emission vertices in this expression begin at one loop,
consistency with this factorised form would require this process to first
contribute at leading power at order $g_{\mathrm{s}}^{7}$,

$\displaystyle{\mathrm{Disp}\left[{A^{(2)}(2^{\oplus},3^{\oplus},4^{\oplus},5^{\oplus},1^{\oplus})}\right]}=2s\,C^{(1,\,s)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})R^{(0)}_{g^{*}}(q_{2})C^{(1,\,s)}_{g^{*}gg}(-q_{2},p_{5}^{\oplus},p_{1}^{\oplus})\,.$
(205)

Indeed, we find the one-loop amplitudes eq. 198 are power-supressed in the
NMRK limit, consistent with eq. 204.

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

Following the procedure of section 7.2.1, we obtain

$\displaystyle\begin{split}c^{(1,\,s)}_{gggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})&=c^{(1,\
s)}_{ggg^{*}}(p_{2},p_{3},q_{1})+\frac{1}{6}\left(\frac{|z|^{2}+X(z-{\bar{z}}+1)}{X|z-1|^{2}}\right)L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&-\frac{z({\bar{z}}-1)(X+{\bar{z}})(|z|^{2}+X(z-{\bar{z}}+1))}{3X^{2}(|z-1|^{2})^{3}}L_{2}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&-\frac{1}{3}\frac{z({\bar{z}}-1)X}{{\bar{z}}(z-1)(1+X)^{2}}+\frac{1}{6}\frac{X+{\bar{z}}}{{\bar{z}}(z-1)(1+X)}\,.\end{split}$
(206)

In the MRK limit, we find

$\displaystyle\begin{split}c^{(1,\,s)}_{gggg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})&\underset{X\to\infty}{\to}c^{(1,\
s)}_{ggg^{*}}(p_{2},p_{3},q_{1})+v^{(1,\,s)}_{g^{*}gg^{*}}(-q_{1},p_{4},q_{2})\,,\end{split}$
(207)

as desired, where the Lipatov vertex with a complex scalar circulating in the
loop is

$\displaystyle\begin{split}v^{(1,\,s)}_{g^{*}gg^{*}}(-q_{1},p_{4},q_{2})=&\frac{1}{3}v_{g^{*}gg^{*}}^{(1,\
\mathcal{N}=1_{\chi})}(q_{1},p_{4},q_{2})-\frac{1}{6}\frac{|p_{4\perp}|^{2}}{q_{1\perp}^{*}q_{2\perp}}\\\
-&\frac{1}{3}|p_{4\perp}|^{2}q_{1\perp}q_{2\perp}^{*}(|q_{1\perp}|^{2}+|q_{2\perp}|^{2}-2q_{1\perp}q_{2\perp}^{*})\frac{L_{2}\left(\frac{|q_{1\perp}|^{2}}{|q_{2\perp}|^{2}}\right)}{(-|q_{2\perp}|^{2})^{3}}\\\
=&\frac{1}{3}v_{g^{*}gg^{*}}^{(1,\
\mathcal{N}=1_{\chi})}(q_{1},p_{4},q_{2})+\frac{1}{6}\frac{1}{{\bar{z}}(z-1)}+\frac{1}{3}\frac{z(z-{\bar{z}}+1)}{(|z-1|^{2})^{3}}L_{2}\left(\frac{|q_{1\perp}|^{2}}{|q_{2\perp}|^{2}}\right)\,,\end{split}$
(208)

which can be inferred from e.g. ref. DelDuca:1998cx .

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

Following the procedure of section 7.2.2, we obtain the vertex

$\displaystyle\begin{split}c^{(1,\,s)}_{gggg^{*}}&(p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},q_{2})=c^{(1,\
s)}_{ggg^{*}}(p_{2},p_{3},q_{1})-\frac{X+{\bar{z}}}{6(1+X)(z-1){\bar{z}}}\\\
&+\frac{X(1+z-{\bar{z}})+|z|^{2}}{6X|z-1|^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)+\frac{X(X+{\bar{z}})(X(1+z-{\bar{z}})+|z|^{2})}{3X^{2}(z-1)^{3}({\bar{z}}-1)^{2}}L_{2}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+2\frac{Xz^{2}(X+{\bar{z}})}{|z-1|^{2}(X+z)^{3}}\left[L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)+L_{0}\left(\frac{-t_{1}^{\prime}}{-t_{2}}\right)\right]-\frac{2(X(z-{\bar{z}}+1)+|z|^{2})}{6X|z-1|^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&-\left(\frac{z(X+{\bar{z}})}{|z-1|^{2}(X+z)}\right)^{2}\left[L_{1}\left(\frac{-t_{1}}{-t_{2}}\right)+L_{1}\left(\frac{-t_{1}^{\prime}}{-t_{2}}\right)\right]+\frac{1}{(z-1)^{2}}L_{1}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&-\frac{2}{3(z-1)^{3}}\left(\frac{z(X+{\bar{z}})(X(1+z-{\bar{z}})+|z|^{2})}{X^{2}({\bar{z}}-1)^{2}}+\frac{z^{3}(X+{\bar{z}})^{3}}{X^{2}({\bar{z}}-1)^{3}(X+z)}-1\right)L_{2}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+\frac{2z(X+{\bar{z}})^{3}}{3(|z-1|^{2})^{3}(X+z)}L_{2}\left(\frac{-t_{1}^{\prime}}{-t_{2}}\right)-\frac{2z^{2}X^{2}}{(X+z)^{4}}Ls_{-1}\left(\frac{-t_{1}^{\prime}}{-t_{2}},\frac{-t_{1}}{-t_{2}}\right)\\\
&-\frac{z(X+{\bar{z}})^{3}}{3(X+1)^{2}|z-1|^{2}{\bar{z}}(X+z)}-\frac{X({\bar{z}}-1)+|z|^{2}(1+X)}{3(X+1)(z-1)|z|^{2}}\,.\end{split}$
(209)

We have written this vertex in such a way to make the MRK limit transparent.
In the $X\to\infty$ limit we note that only the first two lines of eq. 209
contribute to leading power, and we can readily verify

$\displaystyle\begin{split}c^{(1,\,s)}_{gggg^{*}}(p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},q_{2})&\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}c^{(1,\
s)}_{ggg^{*}}(p_{2},p_{3},q_{1})+v^{(1,\
s)}_{g^{*}gg^{*}}(-q_{1},p_{4},q_{2})\,.\end{split}$ (210)

As discussed in section 7.1, the tree-level amplitude for this colour ordering
is leading in both MRK and ${\mathrm{MRK}^{\prime}}$ limits. To obtain the
${\mathrm{MRK}^{\prime}}$ limit of eq. 209, we simply note

$c^{(1,\,s)}_{gggg^{*}}(p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},q_{2})=c^{(1,\,s)}_{gggg^{*}}(p_{4}^{\oplus},p_{2}^{\ominus},p_{3}^{\oplus},q_{2})\,,$
(211)

which follows from eq. 64 and the invariance of eq. 209 under
$p_{5}\leftrightarrow p_{1}$. From this, it follows that

$\displaystyle\begin{split}c^{(1,\,s)}_{gggg^{*}}(p_{3}^{\oplus},p_{2}^{\ominus},p_{4}^{\oplus},q_{2})&\underset{\scriptscriptstyle{\mathrm{MRK}^{\prime}}}{\to}c^{(1,\
s)}_{ggg^{*}}(p_{2},p_{4},q_{1}^{\prime})+v^{(1,\
s)}_{g^{*}gg^{*}}(-q_{1}^{\prime},p_{3},q_{2})\,.\end{split}$ (212)

This concludes our discussion of primitive five-gluon amplitudes in the NMRK
limit. The NMRK limit of the relevant partial amplitudes discussed in section
6.1 can be obtained from the results presented in this section, via the
kinematic maps discussed in section 3, or the supersymmetric rearrangements
discussed in section 5.3.

##  8 Colour-dressed one-loop vertex for the peripheral emission of two gluons

In this section we combine the colour analysis of section 6 and the kinematic
analysis of section 7. Having analysed the NMRK limit of the necessary partial
amplitudes, we insert them into eq. 154 to find the NMRK limit of the colour-
dressed amplitude.

Let us first consider eq. 154 for MHV amplitudes. We find that the kinematic
coefficients of the totally symmetric colour structures have the following
form, for $m{\in}\\{g,f\\}$:

$\displaystyle\begin{split}A^{(1,\,m)}_{d}&(2,3,4,5,1)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s_{12}\,C^{(1,\,m)}_{d}(p_{2},p_{3},p_{4},q_{2})\,R^{(0)}_{g^{*}}(q_{2})\,C^{(0)}_{g^{*}gg}(-q_{2},p_{5}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\end{split}$
(213)

Here we have introduced the partial vertex

$\displaystyle\begin{split}C^{(1,\,m)}_{d}(p_{2},p_{3},p_{4},q_{2})=&C^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})\left(c^{(1,\,m)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})-c^{(1,\,m)}_{gggg^{*}}(p_{3},p_{2},p_{4},q_{2})\right)\\\
+&C^{(0)}_{gggg^{*}}(p_{2},p_{4},p_{3},q_{2})\left(c^{(1,\,m)}_{gggg^{*}}(p_{2},p_{4},p_{3},q_{2})-c^{(1,\,m)}_{gggg^{*}}(p_{4},p_{2},p_{3},q_{2})\right)\,.\end{split}$
(214)

We emphasise the fact that the one-loop corrections to
$C_{g^{*}gg}(-q_{2},p_{5},p_{1})$ have canceled in eq. 213, and in the $m{=}g$
case, all large logarithmic corrections have also canceled.

Performing a similar analysis of the kinematic coefficient of the tree-level
colour structures we find,

$\displaystyle\begin{split}&A^{(1,\,m)}_{FF}(2,3,4,5,1)\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}2s_{12}\,R^{(0)}_{g^{*}}(q_{2})\,C^{(0)}_{g^{*}gg}(-q_{2},p_{5}^{-\lambda_{1}},p_{1}^{\lambda_{1}}s)\\\
&\times\Bigg{\\{}C^{(1\,,m)}_{FF}(p_{2},p_{3},p_{4},q_{2})+C^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})\bigg{(}r_{g^{*}}^{(1,\,m)}(q_{2};p_{4}^{+}p_{5}^{-})+c^{(1,\,m)}_{g^{*}gg}(-q_{2},p_{5},p_{1})\bigg{)}\end{split}$
(215)

which again is valid for both $m{\in}\\{g,f\\}$. We have collected the
dependence on the subset of momenta $\\{p_{2},p_{3},p_{4}\\}$ with the
definition

$\displaystyle\begin{split}C^{(1\,,m)}_{FF}(p_{2},p_{3},p_{4},q_{2})=&C^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})\bigg{(}\frac{2}{3}c^{(1,\,m)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})+\frac{1}{3}c^{(1,\,m)}_{gggg^{*}}(p_{3},p_{2},p_{4},q_{2})\bigg{)}\\\
-&C^{(0)}_{gggg^{*}}(p_{2},p_{4},p_{3},q_{2})\left(\frac{1}{3}c^{(1,\,m)}_{gggg^{*}}(p_{2},p_{4},p_{3},q_{2})-\frac{1}{3}c^{(1\,m)}_{gggg^{*}}(p_{4},p_{2},p_{3},q_{2})\right)\,.\Bigg{\\}}\,,\end{split}$
(216)

In contrast to eq. 213, eq. 215 has a large logarithmic term
$r^{(1,m)}_{g^{*}}$, and importantly, the large scale of this logarithm
depends on the colour ordering. The definitions of the partial vertices allow
for easier discussion of the full one-loop amplitude, which becomes

$\displaystyle\begin{split}&{\mathrm{Disp}\left[{\mathcal{A}^{(1)[-]}_{5}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},5^{-\lambda_{1}},1^{\lambda_{1}})}\right]}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}g_{s}^{5}\,2s_{12}\,R^{(0)}_{g^{*}}(q_{1})\,(-F^{d}_{a_{5}a_{1}})C^{(0)}_{g^{*}gg}(-q_{2},p_{5}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\\\
&\times\Bigg{\\{}\sum_{\sigma\in
S_{2}}(F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}})_{a_{2}c_{2}}\bigg{[}N_{c}\,C_{FF}^{(1,\,g)}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})+N_{f}\,C_{FF}^{(1,\,f)}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})\\\
&\hskip
20.00003pt+C^{(0)}_{gggg^{*}}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})\bigg{(}N_{c}\,r_{g^{*}}^{(1,\,g)}(q_{2};p_{\sigma_{4}}^{+}p_{5}^{-})\\\
&\hskip
120.00018pt+N_{c}\,c_{g^{*}gg}^{(1,\,g)}({-q_{2}},p_{5},p_{1})+N_{f}\,c_{g^{*}gg}^{(1,\,f)}({-q_{2}},p_{5},p_{1})\bigg{)}\bigg{]}\\\
&\hskip
20.00003pt+d_{A}^{a_{2}a_{3}a_{4}c_{2}}C_{d}^{(1,\,g)}(p_{2},p_{3},p_{4},q_{2})+2\,d_{F}^{a_{2}a_{3}a_{4}c_{2}}C_{d}^{(1,\,f)}(p_{2},p_{3},p_{4},q_{2})\Bigg{\\}}\,.\end{split}$
(217)

The form of eq. 217 is not compatible with the straightforward factorisation
given in eq. 28. Due to fact that different large logarithms multiply the two
different colour orderings of $C^{(0)}_{gggg^{*}}$, it is not possible to
simply define a colour-dressed one-loop vertex of the form

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

Rather, we define colour-dressed building blocks for each colour structure in
eq. 217. For an individual tree-level colour structure we define

${\mathcal{C}}_{FF}(p_{2},p_{3},p_{4},q_{2})={\mathcal{C}}^{(0)}_{FF}(p_{2},p_{3},p_{4},q_{2})+{\mathcal{C}}_{FF}^{(1)}(p_{2},p_{3},p_{4},q_{2})+\mathcal{O}(g_{\mathrm{s}}^{6})\,,$
(218)

with tree-level and one-loop coefficients

$\displaystyle{\mathcal{C}}^{(0)}_{FF}(p_{2},p_{3},p_{4},q_{2})$
$\displaystyle=g_{\mathrm{s}}^{2}\,(F^{a_{3}}F^{a_{4}})_{a_{2}c_{2}}{\mathcal{C}}^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})\,,$
(219)
$\displaystyle\begin{split}{\mathcal{C}}_{FF}^{(1)}(p_{2},p_{3},p_{4},q_{2})&=c_{\Gamma}g_{\mathrm{s}}^{4}(F^{a_{3}}F^{a_{4}})_{a_{2}c_{2}}\Big{(}N_{c}\,C_{FF}^{(1,\,g)}(p_{2},p_{3},p_{4},q_{2})\\\
&\qquad\qquad\qquad\qquad\,+N_{f}\,C_{FF}^{(1,\,f)}(p_{2},p_{3},p_{4},q_{2})\Big{)}\,,\end{split}$
(220)

respectively. In terms of this notation, we note the colour-dressed tree-level
vertex, eq. 44, can be written

${\mathcal{C}}^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})=\sum_{\sigma\in
S_{2}}{\mathcal{C}}^{(0)}_{FF}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})\,.$
(221)

Similarly, for the symmetric colour structures which first appear at one loop,
we define

${\mathcal{C}}_{d}(p_{2},p_{3},p_{4},q_{2})={\mathcal{C}}_{d}^{(1)}(p_{2},p_{3},p_{4},q_{2})+\mathcal{O}(g_{\mathrm{s}}^{6})\,,$
(222)

which starts at order $g_{\mathrm{s}}^{4}$,

$\displaystyle{\mathcal{C}}_{d}^{(1)}(p_{2},p_{3},p_{4},q_{2})$
$\displaystyle=c_{\Gamma}\,g_{\mathrm{s}}^{4}\left(d_{A}^{a_{2}a_{3}a_{4}c_{2}}C_{d}^{(1,\,g)}(p_{2},p_{3},p_{4},q_{2})+2\,d_{F}^{a_{2}a_{3}a_{4}c_{2}}C_{d}^{(1,\,f)}(p_{2},p_{3},p_{4},q_{2})\right)\,.$
(223)

With these colour-dressed definitions eq. 217 can be written in the more
compact from

$\displaystyle\begin{split}\frac{1}{2s}{\mathrm{Disp}\left[{\mathcal{A}^{(1)[-]}_{5}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},5^{-\lambda_{1}},1^{\lambda_{1}})}\right]}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&\mathcal{C}_{d}^{(1)}(p_{2},p_{3},p_{4},q_{2})\mathcal{R}_{g^{*}}^{(0)}(q_{2})\mathcal{C}_{g^{*}gg}^{(0)}(-q_{2},p_{5},p_{1})\\\
+&\mathcal{C}^{(0)}_{FF}(p_{2},p_{3},p_{4},q_{2})\mathcal{R}_{g^{*}}^{(1)}(q_{2};s_{45})\mathcal{C}_{g^{*}gg}^{(0)}(-q_{2},p_{5},p_{1})\\\
+&\mathcal{C}^{(0)}_{FF}(p_{2},p_{4},p_{3},q_{2})\mathcal{R}_{g^{*}}^{(1)}(q_{2};s_{35})\mathcal{C}_{g^{*}gg}^{(0)}(-q_{2},p_{5},p_{1})\\\
+&\mathcal{C}^{(0)}_{gggg^{*}}(p_{2},p_{3},p_{4},q_{2})\mathcal{R}_{g^{*}}^{(0)}(q_{2})\mathcal{C}_{g^{*}gg}^{(1)}(-q_{2},p_{5},p_{1})\,.\end{split}$
(224)

This is as far as the analysis of the MHV one-loop pure-gluon amplitudes can
carry us. We conclude this section with a brief discussion on how the one-loop
result eq. 224 may fit into an all-orders expansion describing $2\to 3$
amplitudes in the NMRK to NLL accuracy. While, as we have discussed, a single-
term factorisation formula such as eq. 28 is not compatible with the
requirement of a single vertex which itself factorises into the known
${\mathrm{MRK}}$ and ${\mathrm{MRK}^{\prime}}$ limits, one possibility which
_is_ permitted by the one-loop result eq. 224 is

$\displaystyle\begin{split}{\mathrm{Disp}\left[{\mathcal{A}^{[-]}_{5}(2^{\lambda_{2}},3^{\lambda_{3}},4^{\lambda_{4}},5^{-\lambda_{1}},1^{\lambda_{1}})}\right]}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&2s\,\sum_{\sigma\in
S_{2}}\left({\mathcal{C}}_{FF}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})+\frac{1}{2}{\mathcal{C}}_{d}(p_{2},p_{\sigma_{3}},p_{\sigma_{4}},q_{2})\right)\\\
&\times\mathcal{R}_{g^{*}}(q_{2};p_{\sigma_{4}}^{+}p_{5}^{-})\,\mathcal{C}_{g^{*}gg}(-q_{2},p_{5},p_{1})\,,\end{split}$
(225)

which is depicted in fig. 3.

Figure 3: Illustration of eq. 225, which is one possible form of all-orders
factorisation of the dispersive part of five-parton amplitudes which (unlike
the factorised form in fig. 2) is compatible with the one-loop result eq. 224.

Of course, whether the term ${\mathcal{C}}_{d}$ recieves logarithmic
corrections at higher orders, let alone the form they may take, cannot be
informed by this one-loop analysis. Much can be learned, therefore, from
investigating the NMRK limit two-loop five-gluon amplitudes.

Finally, we turn to the simpler case of a single negative helicity gluon. We
obtain the compact results

$\displaystyle\begin{split}C_{FF}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})&=C_{FF}^{(1,\,g)}(p_{2},p_{3},p_{4},q_{2})=-C_{FF}^{(1,\,f)}(p_{2},p_{3},p_{4},q_{2})\\\
&=\frac{1}{q_{2\perp}}\left(\frac{q_{2\perp}^{*}}{q_{2\perp}}\right)\frac{X(z-1)^{3}\left(X^{3}{\bar{z}}+X^{2}(z+1){\bar{z}}+Xz(z+1)+z^{2}\right)}{(X+1)^{2}z^{2}({\bar{z}}-1)(X+z)^{2}}\,,\end{split}$
(226)
$\displaystyle\begin{split}C_{d}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})&=C_{d}^{(1,\,g)}(p_{2},p_{3},p_{4},q_{2})=-C_{d}^{(1,\,f)}(p_{2},p_{3},p_{4},q_{2})\\\
&=-\frac{1}{q_{2\perp}}\left(\frac{q_{2\perp}^{*}}{q_{2\perp}}\right)\frac{3X(z-1)^{2}(z-{\bar{z}})}{(X+1)z({\bar{z}}-1)(X+z)}\,.\end{split}$
(227)

The one-loop amplitude in the NMRK is then

$\displaystyle\begin{split}&\mathcal{A}^{(1)[-]}_{5}(2^{\oplus},3^{\oplus},4^{\oplus},5^{-\lambda_{1}},1^{\lambda_{1}})\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}g_{\mathrm{s}}^{5}\,2s_{12}\,R^{(0)}_{g^{*}}(q_{1})(-F^{d}_{a_{5}a_{1}})C^{(0)}_{g^{*}gg}(-q_{2},p_{5}^{-\lambda_{1}},p_{1}^{\lambda_{1}})\\\
&\times\Bigg{\\{}\sum_{\sigma\in
S_{2}}(F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}})_{a_{2}c_{2}}\
(N_{c}-N_{f})C_{FF}^{(1,\,s)}(p_{2}^{\oplus},p_{\sigma_{3}}^{\oplus},p_{\sigma_{4}}^{\oplus},q_{2})\,\\\
&\quad+\left(d_{A}-2\,d_{F}\right)^{a_{2}a_{3}a_{4}c_{2}}C_{d}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\Bigg{\\}}\,.\end{split}$
(228)

This amplitude is compatible with eq. 28, and we can define the one-loop
colour-dressed vertex

$\displaystyle\begin{split}\mathcal{C}^{(1)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})&=\sum_{\sigma\in
S_{2}}(F^{a_{\sigma_{3}}}F^{a_{\sigma_{4}}})_{a_{2}c_{2}}\
(N_{c}-N_{f})C_{FF}^{(1,\,s)}(p_{2}^{\oplus},p_{\sigma_{3}}^{\oplus},p_{\sigma_{4}}^{\oplus},q_{2})\,\\\
&\quad+\left(d_{A}-2\,d_{F}\right)^{a_{2}a_{3}a_{4}c_{2}}C_{d}^{(1,\,s)}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\,.\end{split}$
(229)

This colour-dressed vertex has the MRK limit

$\mathcal{C}^{(1)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\underset{\scriptscriptstyle{\mathrm{MRK}}}{\to}\mathcal{C}^{(1)}_{ggg^{*}}(p_{2}^{\oplus},p_{3}^{\oplus},q_{1})\mathcal{R}_{g^{*}}^{(0)}(q_{1})\mathcal{V}_{g^{*}gg^{*}}^{(0)}(-q_{1},p_{4}^{\oplus},q_{2}),$
(230)

as required by the known compatibility of the amplitude with eq. 22.

In the remainder of this paper, we perform an analogous investigation of the
one-loop amplitudes with an external quark-antiquark pair.

##  9 One-loop primitive vertices for the peripheral emission of a quark and
gluon

In this section we take the NMRK limit of one-loop primitive amplitudes of
three gluons and a quark-antiquark pair. We use the amplitudes provided by
ref. Bern:1994fz . Expressions are given in that paper for the following
N-1MHV helicity configurations

$\\{{\bar{q}}^{\ominus},q^{\oplus},g^{\oplus},g^{\oplus},g^{\oplus}\\},\quad\\{{\bar{q}}^{\ominus},g^{\oplus},q^{\oplus},g^{\oplus},g^{\oplus}\\},$

which in the NMRK limit only give rise to the one-loop
$\\{g^{\oplus},g^{\oplus},g^{\oplus},g^{*}\\}$ vertex discussed in section
7.3.1, and the MHV configurations

$\displaystyle\\{{\bar{q}}^{\ominus},q^{\oplus},g^{\ominus},g^{\oplus},g^{\oplus}\\},\quad\\{{\bar{q}}^{\ominus},q^{\oplus},g^{\oplus},g^{\ominus},g^{\oplus}\\},\quad$
$\displaystyle\\{{\bar{q}}^{\ominus},q^{\oplus},g^{\oplus},g^{\oplus},g^{\ominus}\\},\quad\\{{\bar{q}}^{\ominus},g^{\ominus},q^{\oplus},g^{\oplus},g^{\oplus}\\}$
$\displaystyle\\{{\bar{q}}^{\ominus},g^{\oplus},q^{\oplus},g^{\ominus},g^{\oplus}\\},\quad$
$\displaystyle\\{{\bar{q}}^{\ominus},g^{\oplus},q^{\oplus},g^{\oplus},g^{\ominus}\\}.$

Let us briefly review how any amplitude can be related to one of the above by
discrete transformations. We note that parity conjugation, via complex
conjugation, can be used to map to a single-minus or two-minus amplitude, and
charge conjugation,

$A_{n}^{(1,\,m_{L})}(\cdots,i_{q}^{\lambda},\cdots,j_{{\bar{q}}}^{(-\lambda)},\cdots)=A_{n}^{(1,\,m_{R})}(\cdots,i_{{\bar{q}}}^{\lambda},\cdots,j_{q}^{(-\lambda)},\cdots)\,,$
(231)

can be used to map to an amplitude where the negative-helicity fermion is the
antiquark. Cyclicity of the amplitude means the first argument can always be
set to be the antiquark. Finally, the reflection identity, eq. 67, can be used
to bring the quark into either the second or third entry.

For each configuration, explicit amplitudes are provided in ref. Bern:1994fz
for particle content $\mathcal{N}=1_{V}$, $g_{L}$,
${\mathcal{N}=1_{\chi}}_{L}$ and $s_{L}$. We remind the reader that the
${\mathcal{N}=1_{\chi}}_{R}$ and $s_{R}$ amplitudes vanish for these
configurations according to eq. 68a and eq. 68b. The fermion contribution can
be obtained via the decomposition

$A_{n}^{(1,\,f_{L})}=A_{n}^{(1,\,{\mathcal{N}=1_{V}}_{L})}-A_{n}^{(1,\,{s}_{L})}\,,$
(232)

and the $g_{R}$ contribution can be subsequently obtained by means of the
supersymmetric decomposition eq. 85, which we duplicate here for the
convenience of the reader:

$A_{n}^{(1,\
\mathcal{N}=1_{V})}=A_{n}^{(1,\,g_{L})}+A_{n}^{(1,\,g_{R})}+A_{n}^{(1,f_{L})}\,.$
(233)

Following the procedure of section 5.3.2, the left hand side of this equation
can be found through the supersymmetric Ward identity

$a_{5}^{(1,\,\mathcal{N}=1_{V})}(2_{\bar{q}},3_{q},4,5,1)=a_{5}^{(1,\,\mathcal{N}=1_{V})}(2,3,4,5,1)\,.$
(234)

It follows that

$c_{{\bar{q}}qgg^{*}}^{(1,\,{\mathcal{N}}=1V)}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})=c_{gggg^{*}}^{(1,\,{\mathcal{N}}=1V)}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})\,,$
(235)

with analogous relations for different orderings of the external particle
species. The organisation of this section is similar to section 7, where we
first categorise the peripheral-emission vertices by internal particle
content, then by helicity configuration, where a minimal set of primitive
vertices is given by the three cyclic orderings

$\\{{\bar{q}}^{\ominus},q^{\oplus},g^{\oplus},g^{*}\\}\,,\qquad\\{{\bar{q}}^{\ominus},g^{\oplus},q^{\oplus},g^{*}\\}\,,\qquad\\{g^{\oplus},{\bar{q}}^{\ominus},q^{\oplus},g^{*}\\}\,.$

###  9.1 $\mathcal{N}=4$ contribution

Although in practice we use a $\mathcal{N}=1_{V}$ multiplet decomposition, eq.
233, to organise the amplitudes and vertices, it is worthwhile to analyse the
amplitudes with a circulating $\mathcal{N}=4$ multiplet for two reasons.
Firstly, ref. Bern:1994fz  writes the $g_{L}$ and $\mathcal{N}=1_{V}$
amplitudes in terms of the function eq. 161, and this proves convenient also
for the factorised expressions. Secondly, it provides us with the simplest
context to study the key difference between the pure-gluon case and the case
with an external quark-antiquark pair.

Recall the discussion of eq. 161 in section 7.1.1. To be concrete, let us
study the physical scattering

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

The treatment of the ordering $\\{p_{2},p_{3},p_{4},q_{2}\\}$ proceeds
similarly to the pure gluon case, and by a SUSY Ward identity we obtain

$\displaystyle\begin{split}\mathrm{Re}\left[a_{5}^{(1,\,\mathcal{N}=4)}(2_{\bar{q}},3_{g},4,5,1)\right]\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&c_{{\bar{q}}qgg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},p_{4},q_{2})\\\
+&r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{2};p_{4}^{+}p_{5}^{-})+c_{g^{*}gg}^{(1,\,\mathcal{N}=4)}(-q_{2},p_{5},p_{1})\end{split}$
(236)

with primitive vertex

$\displaystyle\begin{split}c_{{\bar{q}}qgg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},p_{4},q_{2})=&c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{3},q_{1})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1};s_{34})\\\
+&v^{(1,\,\mathcal{N}=4)}(t_{1},\frac{s_{34}\,p_{4}^{+}}{(p_{3}^{+}+p_{4}^{+})},t_{2})\,.\end{split}$
(237)

Likewise, for the second colour-ordering in eq. 159 we write

$\displaystyle\begin{split}\mathrm{Re}\left[a_{5}^{(1,\,\mathcal{N}=4)}(4,2_{\bar{q}},3_{q},5,1)\right]\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&c_{g{\bar{q}}qg^{*}}^{(1,\,\mathcal{N}=4)}(p_{4},p_{2},p_{3},q_{2})\\\
+&r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{2};p_{4}^{+}p_{5}^{-})+c_{g^{*}gg}^{(1,\,\mathcal{N}=4)}(-q_{2},p_{5},p_{1})\,,\end{split}$
(238)

with primitive vertex

$\displaystyle\begin{split}c_{g{\bar{q}}qg^{*}}^{(1,\,\mathcal{N}=4)}(p_{4},p_{2},p_{3},q_{2})=&c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{3},p_{2},q_{1})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1};s_{24})\\\
+&v^{(1,\,\mathcal{N}=4)}(t_{1},\frac{s_{24}\,p_{4}^{+}}{(p_{2}^{+}+p_{4}^{+})},t_{2})\,.\end{split}$
(239)

In this form, it is transparent that the correct large logarithms are
generated by the ${\bar{q}}qgg^{*}$ and $g{\bar{q}}qg^{*}$ vertices in the MRK
limit. This follows from the $U(1)$-decoupling identities, e.g.

$C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},q_{2})+C^{(0)}_{{\bar{q}}gqg^{*}}(p_{2}^{\oplus},p_{4}^{\oplus},p_{3}^{\ominus},q_{2})+C^{(0)}_{g{\bar{q}}qg^{*}}(p_{4}^{\oplus},p_{2}^{\oplus},p_{3}^{\ominus},q_{2})=0\,,$
(240)

which in the MRK limit becomes

$C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},q_{2})=-C^{(0)}_{g{\bar{q}}qg^{*}}(p_{4}^{\oplus},p_{2}^{\oplus},p_{3}^{\ominus},q_{2})\,,$
(241)

due to the power-suppression of the second term in eq. 240. The one-loop
corrections, eqs. 237 and 239 on the other hand become equal in the MRK limit,
due to the limiting behaviour

$\frac{s_{24}\,p_{4}^{+}}{(p_{2}^{+}+p_{4}^{+})}\underset{{\mathrm{MRK}}}{=}\frac{s_{34}\,p_{4}^{+}}{(p_{3}^{+}+p_{4}^{+})}\,.$
(242)

Thus the kinematic coefficients of the first two colour-structures in eq. 159
are equal and opposite, as required by compatibility with eq. 22.

We also need to consider the $\delta\Delta$ coefficient, eq. 160. This must be
free from large logarithms in the NMRK and must vanish in the MRK limit. To
verify the former condition, we can write, (merely for convenience to separate
out a large logarithmic term and a backward $g^{*}gg$ vertex)

$\displaystyle\begin{split}\mathrm{Re}\left[a_{5}^{(1,\,\mathcal{N}=4)}(2_{\bar{q}},4,3_{q},5,1)\right]\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&c_{{\bar{q}}gqg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{4},p_{3},q_{2})\\\
+&r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{2};p_{3}^{+}p_{5}^{-})+c_{g^{*}gg}^{(1,\,\mathcal{N}=4)}(-q_{2},p_{5},p_{1})\,,\end{split}$
(243)

with vertex

$\displaystyle\begin{split}c_{{\bar{q}}gqg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{4},p_{3},q_{2})=&c_{ggg^{*}}^{(1,\,\mathcal{N}=4)}(p_{2},p_{4},q_{1}^{\prime})+r_{g^{*}}^{(1,\,\mathcal{N}=4)}(q_{1}^{\prime};s_{43})\\\
+&v^{(1,\,\mathcal{N}=4)}(t_{1}^{\prime},\frac{s_{43}\,p_{3}^{+}}{(p_{4}^{+}+p_{3}^{+})},t_{2})\,.\end{split}$
(244)

All large logarithmic behaviour in the $m{=}g$ amplitudes can be conveniently
written in terms of the $\mathcal{N}=4$ function eq. 161, and only the
$m{=}g_{L}$ amplitudes have large logarithms in the NMRK limit. Therefore the
LL behaviour of eq. 160 in the NMRK limit coincides with

$\displaystyle\begin{split}{\mathrm{Disp}\left[{A^{(1,\,\mathcal{N}=4)}_{\delta\Delta}(p_{2},p_{3},p_{4},p_{5},p_{1})}\right]}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})\bigg{(}c^{(1,\,\mathcal{N}=4)}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})\\\
&-c^{(1,\,\mathcal{N}=4)}_{{\bar{q}}gqg^{*}}(p_{2},p_{4},p_{3},q_{2})+r^{(1,\,g)}_{g^{*}}(q_{2},p_{4}^{+}p_{5}^{-})-r^{(1,\,g)}_{g^{*}}(q_{2},p_{3}^{+}p_{5}^{-})\bigg{)}\\\
+&C^{(0)}_{g{\bar{q}}qg^{*}}(p_{4},p_{2},p_{3},q_{2})\bigg{(}c^{(1,\,\mathcal{N}=4)}_{g{\bar{q}}qg^{*}}(p_{4},p_{2},p_{3},q_{2})\\\
&-c^{(1,\,\mathcal{N}=4)}_{{\bar{q}}gqg^{*}}(p_{2},p_{4},p_{3},q_{2})+r^{(1,\,g)}_{g^{*}}(q_{2},p_{4}^{+}p_{5}^{-})-r^{(1,\,g)}_{g^{*}}(q_{2},p_{3}^{+}p_{5}^{-})\bigg{)}\,.\end{split}$
(245)

We note that all dependence on the backward $g^{*}ggg$ vertex cancels, so that
this partial amplitude only contributes to the forward physics, from which we
define the partial vertex

$C^{(1,\,\mathcal{N}=4)}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})\equiv\left.{\mathrm{Disp}\left[{A^{(1,\,\mathcal{N}=4)}_{\delta\Delta}(p_{2},p_{3},p_{4},p_{5},p_{1})}\right]}\right|_{\mathrm{NMRK}}\,.$

We note that this partial vertex is free from large logarithms, because the
difference of Reggeisation factors

$r^{(1,\,g)}_{g^{*}}(q_{2},p_{4}^{+}p_{5}^{-})-r^{(1,\,g)}_{g^{*}}(q_{2},p_{3}^{+}p_{5}^{-})\,,$

depends only on the ratio $X=p_{3}^{+}/p_{4}^{+}$ which is finite in the NMRK
limit. Finally, we note that due to eq. 241 and eq. 242, the entire partial
vertex vanishes in the MRK limit as expected.

###  9.2 $\mathcal{N}=1_{\chi}$ contribution

In this section, we repeat the analysis of section 7.2 but for amplitudes with
an external quark-antiquark pair.

####  9.2.1 $\\{{\bar{q}}^{\ominus},q^{\oplus},g^{\oplus},g^{*}\\}$

The relevant amplitudes have a very simple NMRK limit, e.g.

$\displaystyle\begin{split}a_{5}^{(1,\
{\mathcal{N}=1_{\chi}}_{L})}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},4^{\oplus},5^{\ominus},1^{\oplus})\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&-\frac{(X+z)}{X(z-1)}L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\,.\end{split}$
(246)

To obtain the ${\bar{q}}qgg^{*}$ vertex we use eq. 132 to add and subtract

$c^{(1,\
\mathcal{N}=1_{\chi})}_{{\bar{q}}qg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},q_{1})+c_{g^{*}gg}^{(1,\
\mathcal{N}=1_{\chi})}(-q_{2},p_{5}^{\ominus},p_{1}^{\oplus})=\frac{1}{2}\log\left(\frac{-s_{23}}{-s_{51}}\right)\,,$
(247)

leading to

$\displaystyle\begin{split}a_{5}^{(1,\,{\mathcal{N}=1_{\chi}}_{L})}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},4^{\oplus},5^{\ominus},1^{\oplus})\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&c^{(1,\,\mathcal{N}=1_{\chi})}_{{\bar{q}}qg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},q_{1})\\\
+&\frac{X(1+z-{\bar{z}})-z({\bar{z}}-2)}{2X|z-1|^{2}}L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\\\
+&c_{g^{*}gg}^{(1,\,\mathcal{N}=1_{\chi})}(-q_{2},p_{5}^{\ominus},p_{1}^{\oplus})\,.\end{split}$
(248)

We thus define the 2-parton peripheral-emission vertex to be

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{{\bar{q}}qgg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})=&c^{(1,\,\mathcal{N}=1_{\chi})}_{{\bar{q}}qg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},q_{1})+\frac{X(1+z-{\bar{z}})-z({\bar{z}}-2)}{2X|z-1|^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\,,\end{split}$
(249)

which has the trivial MRK limit

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{{\bar{q}}qgg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})\underset{X\to\infty}{\to}&c^{(1,\,\mathcal{N}=1_{\chi})}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},q_{1})+v_{g^{*}gg^{*}}^{(1,\,\mathcal{N}=1_{\chi})}(-q_{1},p_{4},q_{2})\,.\end{split}$
(250)

####  9.2.2 $\\{{\bar{q}}^{\ominus},g^{\oplus},q^{\oplus},g^{*}\\}$

The amplitudes which would give rise to this vertex do not contribute to eq.
159. Furthermore, these amplitudes are zero:

$A_{5}^{(1,\,{\mathcal{N}=1_{\chi}}_{L})}({\bar{q}}^{\ominus},g^{\oplus},q^{\oplus},g^{\oplus},g^{\ominus})=A_{5}^{(1,\,{\mathcal{N}=1_{\chi}}_{L})}({\bar{q}}^{\ominus},g^{\oplus},q^{\oplus},g^{\ominus},g^{\oplus})=0\,.$
(251)

Following the discussion in section 5.3.2, we can nevertheless define

$c_{{\bar{q}}qgg^{*}}^{(1,\,{\mathcal{N}=1_{\chi}}_{L})}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})=-c_{g^{*}gg}^{(1,\,{\mathcal{N}=1_{\chi}})}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})\,,$
(252)

which only depends on the invariant $s_{51}=s_{234}$.

####  9.2.3 $\\{g^{\oplus},{\bar{q}}^{\ominus},q^{\oplus},g^{*}\\}$

As in section 9.2.1, the NMRK limits of the relevant amplitudes are again very
simple, for example

$\displaystyle\begin{split}a_{5L}^{(1,\
\mathcal{N}=1_{\chi})}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},5^{\ominus},1^{\oplus},4^{\oplus})\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&\frac{1}{(z-1)}L_{0}\left(\frac{-s_{23}}{-s_{51}}\right)\,.\end{split}$
(253)

We remark that this is simply $X/(X+z)$ times the amplitude in eq. 246.
Recalling the NMRK behaviour of the tree level amplitudes,

$\displaystyle\frac{A^{(0)}(4^{\oplus},2_{\bar{q}}^{\ominus},3_{q}^{\oplus},5^{\ominus},1^{\oplus})}{A^{(0)}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},4^{\oplus},5^{\ominus},1^{\oplus})}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}\frac{X}{X+z}$
(254)

we see that

$\displaystyle A^{(1,\
\mathcal{N}=1_{\chi})}(4^{\oplus},2_{\bar{q}}^{\ominus},3_{q}^{\oplus},5^{\ominus},1^{\oplus})+A^{(1,\
\mathcal{N}=1_{\chi})}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},4^{\oplus},5^{\ominus},1^{\oplus})\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}0\,,$
(255)

which is reminiscent of the identity eq. 78, which followed from Furry’s
theorem, but for an off-shell gluon with momentum $p_{5}+p_{1}$.

The vertex resulting from eq. 253 is

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{{\bar{q}}qgg^{*}}(p_{4}^{\oplus},p_{2}^{\ominus},p_{3}^{\oplus},q_{2})=&c^{(1,\,\mathcal{N}=1_{\chi})}_{{\bar{q}}qg^{*}}(p_{2}^{\ominus},p_{3}^{\oplus},q_{1})+\frac{X(1+z-{\bar{z}})+|z|^{2}}{2X|z-1|^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\,,\end{split}$
(256)

which has the transparent MRK limit

$\displaystyle\begin{split}c^{(1,\,\mathcal{N}=1_{\chi})}_{{\bar{q}}qgg^{*}}(p_{4}^{\oplus},p_{2}^{\ominus},p_{3}^{\oplus},q_{2})\underset{X\to\infty}{\to}&c^{(1,\,\mathcal{N}=1_{\chi})}(2_{\bar{q}}^{\ominus},3_{q}^{\oplus},q_{1})+v_{g^{*}gg^{*}}^{(1,\,\mathcal{N}=1_{\chi})}(-q_{1},p_{4},q_{2})\,.\end{split}$
(257)

###  9.3 Complex scalar contribution

The procedure to obtain these vertices proceeds similarly to section 9.2 and
we simply collect the primitive vertices here. The
${\bar{q}}^{\ominus}q^{\ominus}g^{\oplus}g^{*}$ vertex is

$\displaystyle\begin{split}c_{{\bar{q}}qgg^{*}}^{(1,\
s_{L})}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})&=c_{{\bar{q}}qg^{*}}^{(1,\
s_{L})}(p_{2},p_{3},q_{2})+\frac{1}{6}\frac{X(1+z-{\bar{z}})-z({\bar{z}}-2)}{X|z-1|^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+\frac{1}{3}\frac{z|X+z|^{2}(X(1+z-{\bar{z}})+|z|^{2})}{X^{3}(z-1)^{3}({\bar{z}}-1)^{2}}L_{2}\left(\frac{-t_{1}}{-t_{2}}\right)-\frac{|X+z|^{2}}{6X(1+X)(z-1){\bar{z}}}\,.\end{split}$
(258)

The ${\bar{q}}^{\ominus}g^{\oplus}q^{\ominus}g^{*}$ vertex is

$c_{{\bar{q}}gqg^{*}}^{(1,\,s_{L})}(p_{2}^{\ominus},p_{4}^{\oplus},p_{3}^{\oplus},q_{2})=-c_{g^{*}gg}^{(1,\,{s})}(-q_{2},p_{5}^{\lambda_{5}},p_{1}^{\lambda_{1}})\,.$
(259)

Finally, the $g^{\oplus}{\bar{q}}^{\ominus}q^{\ominus}g^{*}$ vertex is

$\displaystyle\begin{split}c_{g{\bar{q}}qg^{*}}^{(1,\
s_{L})}(p_{4}^{\oplus},p_{2}^{\ominus},p_{3}^{\oplus},q_{2})&=c_{{\bar{q}}qg^{*}}^{(1,\
s_{L})}(p_{2},p_{3},q_{2})+\frac{1}{6}\frac{X(1+z-{\bar{z}})+|z|^{2}}{X|z-1|^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+\frac{1}{3}\frac{z(X+{\bar{z}})^{2}(X(1+z-{\bar{z}})+|z|^{2})}{X^{3}(z-1)^{3}({\bar{z}}-1)^{2}}L_{2}\left(\frac{-t_{1}}{-t_{2}}\right)-\frac{(X+{\bar{z}})^{2}}{6X(1+X)(z-1){\bar{z}}}\,.\end{split}$
(260)

###  9.4 Gluon contribution

Again, the analysis proceeds similarly to section 9.2 and we simply collect
the $g_{L}$ primitive vertices here. The $g_{R}$ vertices can be obtained via
the supersymmetric decomposition eq. 85. The
${\bar{q}}^{\ominus}q^{\oplus}g^{\oplus}g^{*}$ vertex is

$\displaystyle\begin{split}c_{{\bar{q}}qgg^{*}}^{(1,\
g_{L})}(p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\oplus},q_{2})&=c_{{\bar{q}}qg^{*}}^{(1,\
g_{L})}(p_{2},p_{3},q_{1})+r_{g^{*}}^{(1,\
g)}(t_{1};s_{34})+v^{(1,\,\mathcal{N}=4)}\left(t_{1},\frac{s_{34}\,p_{4}^{+}}{(p_{3}^{+}+p_{4}^{+})},t_{2}\right)\\\
&+\left(c_{{\bar{q}}qgg^{*}}^{(1,\,s_{L})}(p_{2},p_{3},p_{4},q_{2})-c_{{\bar{q}}qg^{*}}^{(1,\,s_{L})}(p_{2},p_{3},q_{1})\right)\\\
&-4\frac{X(1+z-{\bar{z}})+z}{2X|z-1|^{2}}L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&-\frac{z}{X}L_{s_{-1}}\left(\frac{-t_{1}}{-t_{2}},\frac{-t_{1}^{\prime}}{-t_{2}}\right)-\frac{z}{2X(z-1)^{2}}L_{1}\left(\frac{-t_{1}}{-t_{2}}\right)\,.\end{split}$
(261)

The ${\bar{q}}^{\ominus}g^{\oplus}q^{\oplus}g^{*}$ vertex is

$\displaystyle\begin{split}c_{{\bar{q}}gqg^{*}}^{(1,\
g_{L})}(p_{2}^{\ominus},p_{4}^{\oplus},p_{3}^{\oplus},q_{2})=&c_{{\bar{q}}gqg^{*}}^{(1,\
\mathcal{N}=4)}(p_{2},p_{4},p_{3},q_{1})+\frac{1}{\epsilon^{2}}\left(\frac{\mu^{2}}{-s_{24}}\right)^{\epsilon}+\frac{1}{\epsilon^{2}}\left(\frac{\mu^{2}}{-s_{34}}\right)^{\epsilon}-\frac{3}{2}\left(\frac{\mu^{2}}{-s_{24}}\right)\\\
&+L_{s_{-1}}\left(\frac{-s_{24}}{-t_{2}},\frac{-s_{34}}{-t_{2}}\right)-\frac{2z}{z-1}L_{0}\left(\frac{-s_{24}}{-t_{2}}\right)+\frac{z^{2}}{2(z-1)^{2}}L_{1}\left(\frac{-s_{24}}{-t_{2}}\right)-3\,,\end{split}$
(262)

with $\mathcal{N}=4$ vertex as in eq. 244. Recalling the comment beneath eq.
194, we emphasise that only the dispersive part of eq. 262 for the physical
scattering $q(p_{2})\to q(p_{3})g(p_{4})$ should be taken. Finally, the
$g^{\oplus}{\bar{q}}^{\ominus}q^{\oplus}g^{*}$ vertex is

$\displaystyle\begin{split}&c_{g{\bar{q}}qg^{*}}^{(1,\
g_{L})}(p_{4}^{\oplus},p_{2}^{\ominus},p_{3}^{\oplus},q_{2})=c_{{\bar{q}}qg^{*}}^{(1,\
g_{L})}(p_{2},p_{3},q_{1})+r_{g^{*}}^{(1,\
g)}(t_{1};s_{24})+v^{(1,\,\mathcal{N}=4)}\left(t_{1},\frac{-s_{24}\,p_{4}^{+}}{(p_{3}^{+}+p_{4}^{+})},t_{2}\right)\\\
&+(c_{g{\bar{q}}qg^{*}}^{(1,\
s_{L})}(p_{4},p_{2},p_{3},q_{2})-c_{{\bar{q}}qg^{*}}^{(1,\
s_{L})}(p_{2},p_{3},q_{1}))\\\
&+\left(\frac{X(z+{\bar{z}}-1)-2z(X+{\bar{z}})+4X({\bar{z}}-1)+|z|^{2}}{2X|z-1|^{2}}-\frac{z(2X+z)(X+{\bar{z}})}{|z-1|^{2}(X+z)^{2}}\right)L_{0}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+\left(\frac{X^{2}\left(({\bar{z}}-1)^{2}-z^{2}\right)+Xz\left(-2(z+1){\bar{z}}+{\bar{z}}^{2}+1\right)-z^{2}{\bar{z}}^{2}}{2X(z-1)^{2}({\bar{z}}-1)^{2}(X+z)}\right)L_{1}\left(\frac{-t_{1}}{-t_{2}}\right)\\\
&+\left(-\frac{Xz^{2}}{(X+z)^{3}}+\frac{2Xz}{(X+z)^{2}}+\frac{z}{X+z}\right)L_{s_{-1}}\left(\frac{-t_{1}}{-t_{2}},\frac{-t_{1}^{\prime}}{-t_{2}}\right)\\\
&-\left(\frac{z(X+{\bar{z}})\left(-5X^{2}+Xz(4{\bar{z}}-7)-5X{\bar{z}}+2z^{2}({\bar{z}}-1)-3z{\bar{z}}\right)}{2(z-1)^{2}({\bar{z}}-1)^{2}(X+z)^{2}}\right)L_{1}\left(\frac{-t_{1}^{\prime}}{-t_{2}}\right)\\\
&+\frac{X^{2}(4z+{\bar{z}}-1)+2Xz(z+3{\bar{z}}-1)+z^{2}(3{\bar{z}}-1)}{2(z-1)({\bar{z}}-1)(X+z)^{2}}\,.\end{split}$
(263)

##  10 Colour-dressed one-loop vertex for the peripheral emission of a quark
and gluon

In section 6 we analysed the colour structure of the one-loop ${\bar{q}}qggg$
amplitudes at leading power in the NMRK limit. In the previous section we
analysed the factorisation properties of the primitive amplitudes in the NMRK
limit. Following the procedure of section 8, we now assemble the NMRK limit of
the colour-dressed amplitude.

As in section 8, it is useful to consider colour-dressed vertices for each of
the colour structures in eq. 159. For the tree-level colour structures we
define

${\mathcal{C}}_{TT}(p_{2},p_{3},p_{4},q_{2})={\mathcal{C}}^{(0)}_{TT}(p_{2},p_{3},p_{4},q_{2})+{\mathcal{C}}_{TT}^{(1)}(p_{2},p_{3},p_{4},q_{2})+\mathcal{O}(g_{\mathrm{s}}^{6})\,,$
(264)

where the tree-level terms are taken from eq. 45,

$\displaystyle{\mathcal{C}}^{(0)}_{TT}(p_{2},p_{3},p_{4},q_{2})$
$\displaystyle=g_{\mathrm{s}}^{2}(T^{a_{4}}T^{c_{2}})_{\imath_{3}\bar{\imath}_{2}}\
C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})$
(265) $\displaystyle{\mathcal{C}}^{(0)}_{TT}(p_{4},p_{2},p_{3},q_{2})$
$\displaystyle=g_{\mathrm{s}}^{2}(T^{c_{2}}T^{a_{4}})_{\imath_{3}\bar{\imath}_{2}}\
C^{(0)}_{{\bar{q}}qgg^{*}}(p_{4}^{\lambda_{4}},p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},q_{2})$
(266)

such that

$\displaystyle\begin{split}\mathcal{C}^{(0)}_{{\bar{q}}qgg^{*}}(p_{2}^{\lambda_{2}},p_{3}^{\lambda_{3}},p_{4}^{\lambda_{4}},q_{2})={\mathcal{C}}^{(0)}_{TT}(p_{2},p_{3},p_{4},q_{2})+{\mathcal{C}}^{(0)}_{TT}(p_{4},p_{2},p_{3},q_{2})\,.\end{split}$
(267)

The NMRK limit of the one-loop amplitudes lead us to define the one-loop
correction to these colour-dressed vertices,

$\displaystyle\begin{split}&\mathcal{C}^{(1)}_{TT}(p_{2},p_{3},p_{4},q_{2})=c_{\Gamma}g_{s}^{4}\big{(}T^{a_{4}}T^{d}\big{)}_{\imath_{3}\bar{\imath}_{2}}C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})\\\
&\
\times\left(N_{c}\,c^{(1,\,g_{L})}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})-\frac{1}{N_{c}}\,c^{(1,\,g_{R})}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})+N_{f}\,c^{(1,\,f_{L})}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})\right)\,,\end{split}$
(268)
$\displaystyle\begin{split}&\mathcal{C}^{(1)}_{TT}(p_{4},p_{2},p_{3},q_{2})=c_{\Gamma}g_{s}^{4}\big{(}T^{d}T^{a_{4}}\big{)}_{\imath_{3}\bar{\imath}_{2}}C^{(0)}_{{\bar{q}}qgg^{*}}(p_{4},p_{2},p_{3},q_{2})\\\
&\
\times\left(N_{c}\,c^{(1,\,g_{L})}_{{\bar{q}}qgg^{*}}(p_{4},p_{2},p_{3},q_{2})-\frac{1}{N_{c}}\,c^{(1,\,g_{R})}_{{\bar{q}}qgg^{*}}(p_{4},p_{2},p_{3},q_{2})+N_{f}\,c^{(1,\,f_{L})}_{{\bar{q}}qgg^{*}}(p_{4},p_{2},p_{3},q_{2})\right)\,.\end{split}$
(269)

Similarly we define a colour-dressed vertex for the new colour structure which
first appears at one loop,

$\mathcal{C}^{(1)}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})=\mathcal{C}^{(1)}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})+{\mathcal{O}}(g_{\mathrm{s}}^{4})\,.$
(270)

As in section 9.1, the partial amplitude eq. 160 for $m{\in}\\{g_{L},g_{R}\\}$
only contributes to the forward vertex and we find

$\displaystyle\begin{split}\mathcal{C}^{(1)}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})=c_{\Gamma}g_{s}^{4}\,\delta_{\imath_{3}\bar{\imath}_{2}}\Delta_{da_{4}}\,\Big{(}C^{(1,\,g_{L})}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})+C^{(1,\,g_{R})}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})\Big{)}\,,\end{split}$
(271)

with

$\displaystyle\begin{split}C^{(1,\,g_{L})}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})=&C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})\bigg{(}c^{L(1,\,g)}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})-c^{L(1,\,g)}_{{\bar{q}}gqg^{*}}(p_{2},p_{4},p_{3},q_{2})\\\
&\hskip
120.00018pt+r^{(1,\,g)}_{g^{*}}(q_{2},p_{4}^{+}p_{5}^{-})-r^{(1,\,g)}_{g^{*}}(q_{2},p_{3}^{+}p_{5}^{-})\bigg{)}\\\
+&C^{(0)}_{g{\bar{q}}qg^{*}}(p_{4},p_{2},p_{3},q_{2})\bigg{(}c^{L(1,\,g)}_{g{\bar{q}}qg^{*}}(p_{4},p_{2},p_{3},q_{2})-c^{L(1,\,g)}_{{\bar{q}}gqg^{*}}(p_{2},p_{4},p_{3},q_{2})\\\
&\hskip
120.00018pt+r^{(1,\,g)}_{g^{*}}(q_{2},p_{4}^{+}p_{5}^{-})-r^{(1,\,g)}_{g^{*}}(q_{2},p_{3}^{+}p_{5}^{-})\bigg{)}\,,\end{split}$
(272)

and

$\displaystyle\begin{split}C^{(1,\,g_{R})}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})=&C^{(0)}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})\bigg{(}c^{(1,\,g_{R})}_{{\bar{q}}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})-c^{(1,\,g_{R})}_{{\bar{q}}gqg^{*}}(p_{2},p_{4},p_{3},q_{2})\bigg{)}\\\
+&C^{(0)}_{g{\bar{q}}qg^{*}}(p_{4},p_{2},p_{3},q_{2})\bigg{(}c^{(1,\,g_{R})}_{g{\bar{q}}qg^{*}}(p_{4},p_{2},p_{3},q_{2})-c^{(1,\,g_{R})}_{{\bar{q}}gqg^{*}}(p_{2},p_{4},p_{3},q_{2})\bigg{)}\,.\end{split}$
(273)

As in eq. 245, eq. 272 is free from large logarithms. With these definitions
we can write the NMRK limit of the one-loop amplitudes in the compact form

$\displaystyle\begin{split}&\frac{1}{2s}{\mathrm{Disp}\left[{\mathcal{A}_{5}^{(1)[-]}\left(\bar{q}_{2},q_{3},g_{4},g_{5},g_{1}\right)}\right]}\\\
&\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}\bigg{(}\mathcal{C}^{(1)}_{TT}(p_{2},p_{3},p_{4},q_{2})+\mathcal{C}^{(1)}_{TT}(p_{4},p_{2},p_{3},q_{2})+\mathcal{C}^{(1)}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})\bigg{)}\mathcal{R}^{(0)}_{g^{*}}(q_{2})\mathcal{C}^{(0)}_{g^{*}gg}(-q_{2},p_{5},p_{1})\\\
&\quad\
+\bigg{(}\mathcal{C}^{(0)}_{TT}(p_{2},p_{3},p_{4},q_{2})+\mathcal{C}^{(0)}_{TT}(p_{4},p_{2},p_{3},q_{2})\bigg{)}\mathcal{R}^{(1)}_{g^{*}}(q_{2};p_{4}^{+}p_{5}^{-})\mathcal{C}^{(0)}_{g^{*}gg}(-q_{2},p_{5},p_{1})\\\
&\quad\
+\bigg{(}\mathcal{C}^{(0)}_{TT}(p_{2},p_{3},p_{4},q_{2})+\mathcal{C}^{(0)}_{TT}(p_{4},p_{2},p_{3},q_{2})\bigg{)}\mathcal{R}^{(0)}_{g^{*}}(q_{2})\mathcal{C}^{(1)}_{g^{*}gg}(-q_{2},p_{5},p_{1})\,.\end{split}$
(274)

Unlike the pure-gluon case, eq. 217, eq. 274 is compatible with an all-orders
factorisation of the form

$\displaystyle\begin{split}{\mathrm{Disp}\left[{\mathcal{A}_{5}^{(1)[-]}\left(\bar{q}_{2},q_{3},g_{4},f_{5},f_{1}\right)}\right]}\underset{\scriptscriptstyle{\mathrm{NMRK}}}{\to}&2s\,\mathcal{C}_{\bar{q}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})\mathcal{R}_{g^{*}}(q_{2};p_{4}^{+}p_{5}^{-})\mathcal{C}_{g^{*}f_{5}f_{1}}(-q_{2},p_{5},p_{1})\,,\end{split}$
(275)

where the one-loop correction to eq. 45 is simply given by the sum of the
three colour-dressed vertices

$\displaystyle\begin{split}C^{(1)}_{\bar{q}qgg^{*}}(p_{2},p_{3},p_{4},q_{2})=\mathcal{C}^{(1)}_{TT}(p_{2},p_{3},p_{4},q_{2})+\mathcal{C}^{(1)}_{TT}(p_{4},p_{2},p_{3},q_{2})+\mathcal{C}^{(1)}_{\delta\Delta}(p_{2},p_{3},p_{4},q_{2})\,.\end{split}$
(276)

##  11 Conclusions

In this paper we have analysed the dispersive parts of the one-loop $ggggg$
and ${\bar{q}}qggg$ ampltiudes in the NMRK limit eq. 294. The final results
are presented in a compact notation in eq. 224 and eq. 274 respectively. In
both cases, to leading power, the amplitudes comprise solely of the exchange
of a colour octet in the $t_{2}$ channel. Equation 274 is compatible with an
all-orders factorisation of the form eq. 28, which defines the colour dressed
one-loop ${\bar{q}}qgg^{*}$ vertex eq. 276. The pure-gluon case is not quite
as simple eq. 224, but the one-loop amplitude is compatible with an analogous
factorisation of each colour structure, one such possibility being given in
eq. 225. We have defined a colour-dressed one-loop $gggg^{*}$ vertex for each
colour structure in eq. 220 and eq. 223.

As reviewed in the introduction, there has been much progress towards
extending the BFKL approach to NNLL accuracy. In order to make predictions for
jet cross sections at this accuracy, it is necessary to compute the jet impact
factors to NNLO. The vertices extracted in this paper are necessary
ingredients for the real-virtual contribution to the NNLO impact factors. The
new totally symmetric colour structures appearing in the one-loop $gggg^{*}$
vertex do not survive interference with the tree-level amplitude and therefore
are not relevant for the impact factors. The same is not true for the
${\bar{q}}qgg^{*}$ vertex, where the new one-loop colour structure in eq. 271
has non-zero interference with the tree-level colour structures.

While this analysis of one-loop amplitudes allows for the extraction of one-
loop vertices which themselves factorise into known quantities in the further
MRK limit, a one-loop analysis cannot provide information on the logarithmic
corrections these vertices may receive at higher orders. The results of this
paper therefore provide strong motivation to perform a similar investigation
of the NMRK limit of the recently computed two-loop five parton amplitudes.

##  12 Acknowledgements

The author acknowledges useful discussions with Vittorio del Duca, Einan Gardi
and Jenni Smillie. The author thanks Einan Gardi and Jenni Smillie for a
careful reading of the manuscript. The author thanks Maxim Nefedov and Andreas
van Hameren for helpful comments and discussions of this topic at the Low-x
2023 workshop. This work was funded by the ERC Starting Grant 715049
‘QCDforfuture’. For the purpose of open access, the author has applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising from this submission.

##  Appendix A Kinematics

###  A.1 General kinematics

In this paper we consider the scattering processes of $2\to 2$ and $2\to 3$
massless partons. The amplitudes in refs. Bern:1993mq ; Bern:1994fz  are given
in the unphysical region where all momenta are taken as out-going. We
analytically continue to the physical region where $p_{1}^{0}<0$ and
$p_{2}^{0}<0$. In summary, the two scattering regions we consider are the
$n{=}4$ and $n{=}5$ case of

$p_{i}^{2}=0\ \ \forall\
i\in\\{1,\cdots,n\\},\qquad\sum_{i=1}^{n}p_{i}=0\,,\qquad
p_{1}^{0},p_{2}^{0}<0\,,\qquad p_{i}^{0}>0\ \ \forall i\in\\{3,\cdots,n\\},$
(277)

We parameterise the momenta with light-cone coordinates $p^{\pm}=p^{0}\pm
p^{z}$, and complex transverse coordinates $p_{\perp}=p^{x}+ip^{y}$, with
scalar product,

$2\,(p\cdot
q)=p^{+}q^{-}+p^{-}q^{+}-p_{\perp}q^{*}_{\perp}-p^{*}_{\perp}q_{\perp}\,.$
(278)

We take momenta $p_{1}$ and $p_{2}$ to be along the $-$ and $+$ lightcone
directions respectively. Explicitly, the four-momenta are

$\displaystyle\begin{split}p_{1}&=\left(\tfrac{p_{1}^{-}}{2},0,0,\tfrac{-p_{1}^{-}}{2}\right)\equiv\left(0,p_{1}^{-};0\right)\,,\\\
p_{2}&=\left(\tfrac{p_{2}^{+}}{2},0,0,\tfrac{p_{2}^{+}}{2}\right)\ \
\equiv\left(p_{2}^{+},0;0\right)\,,\\\
p_{i}&=\left(\tfrac{p_{i}^{+}+p_{i}^{-}}{2},{\rm Re}[p_{i\perp}],{\rm
Im}[p_{i\perp}],\tfrac{p_{i}^{+}-p_{i}^{-}}{2}\right)\equiv\left(p_{i}^{+},p_{i}^{-};p_{i\perp}\right)\quad\forall
i\in\\{3,\cdots,n\\},\end{split}$ (279)

in terms of cartesian and lightcone coordinates respectively. We also make use
of rapidity, which in terms of lightcone coordinates is

$y_{i}=\frac{1}{2}\log\left(\frac{p_{i}^{+}}{p_{i}^{-}}\right).$ (280)

In terms of the lightcone components the Mandelstam invariants may be written

$\displaystyle s$ $\displaystyle=2\,p_{1}\cdot p_{2}=p_{2}^{+}p_{1}^{-}$
$\displaystyle s_{2i}$ $\displaystyle=2\,p_{2}\cdot
p_{i}=\sum_{i=3}^{5}p_{2}^{+}p_{i}^{-}$ (281) $\displaystyle s_{1i}$
$\displaystyle=2\,p_{1}\cdot p_{i}=\sum_{i=3}^{5}p_{i}^{+}p_{1}^{-}$
$\displaystyle s_{ij}$ $\displaystyle=2\,p_{i}\cdot
p_{j}=p_{i}^{+}p_{j}^{-}+p_{i}^{-}p_{j}^{+}-p_{i\perp}p^{\ast}_{j\perp}-p^{\ast}_{i\perp}p_{j\perp}\,,$

with $3\leq i,j\leq 6$.

For the momenta in eq. 279, we find the following right-handed spinor
products, $\langle k\,p\rangle=\overline{u}^{\ominus}(k)u^{\oplus}(p)$, in the
notation of ref. DelDuca:1999iql ,

$\displaystyle\begin{split}\langle
p_{1}\,p_{2}\rangle&=\sqrt{p_{2}^{+}p_{1}^{-}}\,,\\\ \langle
p_{2}\,p_{i}\rangle&=-i\sqrt{\frac{-p_{2}^{+}}{p_{i}^{+}}}\,p_{i\perp}\,,\\\
\langle p_{i}\,p_{1}\rangle&=i\sqrt{-p_{1}^{-}p_{i}^{+}}\,,\\\ \langle
p_{i}\,p_{j}\rangle&=p_{i\perp}\sqrt{\frac{p_{j}^{+}}{p_{i}^{+}}}-p_{j\perp}\sqrt{\frac{p_{i}^{+}}{p_{j}^{+}}}\,,\\\
\end{split}$ (282)

with $3\leq i,j\leq 6$, where we have used the mass-shell condition,

$|p_{i\perp}|^{2}=p_{i}^{+}p_{i}^{-}\,.$ (283)

Left-handed spinor products, $[k\,p]=\overline{u}^{\oplus}(k)u^{\ominus}(p)$,
are given by complex conjugation,

$[k\,p]={\rm sign}(k^{0}p^{0})\langle p\,k\rangle^{\ast}\,.$ (284)

Spinor products are antisymmetric,

$\langle k\,p\rangle=-\langle p\,k\rangle\,,\qquad[k\,p]=-[p\,k]\,.$ (285)

We also use the currents, $[p|\gamma^{\mu}|k\rangle$ and $\langle
p|\gamma^{\mu}|k]$, which are related by charge conjugation,

$[p|\gamma^{\mu}|k\rangle=\langle k|\gamma^{\mu}|p]\,,$ (286)

and complex conjugation,

$[p|\gamma^{\mu}|k\rangle^{*}={\rm
sign}(k^{0}p^{0})[k|\gamma^{\mu}|p\rangle\,.$ (287)

Through the Fierz rearrangement,

$\langle k|\gamma^{\mu}|p]\langle v|\gamma^{\mu}|q]=2\langle
k\,v\rangle[q\,p]\,,$ (288)

and the Gordon identity,

$[p|\gamma^{\mu}|p\rangle=\langle p|\gamma^{\mu}|p]=2p^{\mu}\,,$ (289)

we obtain that

$\displaystyle\langle k|\not{q}|p]=\langle k\,q\rangle[q\,p]\,,$
$\displaystyle[p|\not{q}|k\rangle=[p\,q]\langle q\,k\rangle\,.$ (290)

The phase conventions of the amplitudes studied in this paper follow from the
choice of polarisation vectors,

$\epsilon_{\mu}^{\oplus}(p,r)=-\frac{[p|\gamma_{\mu}|r\rangle}{\sqrt{2}\langle
p\,r\rangle}\,,\qquad\qquad\epsilon_{\mu}^{\ominus}(p,r)=\frac{\langle
p|\gamma_{\mu}|r]}{\sqrt{2}[p\,r]}\,,$ (291)

where $r$ is an arbitrary lightlike reference momentum.

###  A.2 Kinematic regions

In this section we introduce the kinematic regions relevant for this study,
namely the Regge limit of $2\to 2$ scattering and the MRK and NMRK limits of
$2\to 3$ scattering. The high-energy limit, or Regge limit of $2\to 2$
scattering is given by the limit $s\gg-t$, where $s=s_{12}$ and $t=s_{23}$. In
terms of the rapidity and transverse momenta of the final-state partons, this
limit can be expressed

$y_{3}\gg y_{4}\,,\qquad\qquad|p_{3\perp}|\sim|p_{4\perp}|\,,$ (292)

where the transverse momenta are finite. The momentum flowing in the
$t$-channel is denoted $q=-p_{2}-p_{3}=p_{4}+p_{1}$, as depicted in fig. 4.

Figure 4: Illustration of the Regge-kinematic region (right), obtained from
general kinematics (left) by taking the limit eq. 292. The grey blobs group
final-state momenta where no strict rapidity ordering is assumed.

The MRK limit is the straightforward generalisation of this limit to the
multi-parton case. In this paper we only consider the MRK limit of $2\to 3$
scattering,

$y_{3}\gg y_{4}\gg
y_{5}\,,\qquad\qquad|p_{3\perp}|\sim|p_{4\perp}|\sim|p_{5\perp}|\,.$ (293)

The momenta flowing in the $t$-channels are $q_{1}=-p_{2}-p_{3}$ and
$q_{2}=p_{5}+p_{1}$, that is, $t_{1}=q_{1}^{2}$ and $t_{2}=q_{2}^{2}$.

As stated in the introduction, the NMRK limit (also known as a quasi-multi-
Regge kinematic or QMRK limit) is given by relaxing one of these strict
rapidity orderings. The focus of this paper is on $2\to 3$ scattering, which
has a rich structure of nested NMRK and MRK limits, illustrated in fig. 5. We
restrict to the canonical ordering of particles, and, without loss of
generality, only consider the forward NMRK limit, that is,

$y_{3}\sim y_{4}\gg
y_{5}\,,\qquad\qquad|p_{3\perp}|\sim|p_{4\perp}|\sim|p_{5\perp}|\,.$ (294)

We note that two different MRK limits can be reached from eq. 294. In addition
to the canonical MRK limit eq. 293, the limit

$y_{4}\gg y_{3}\gg
y_{5}\,,\qquad\qquad|p_{3\perp}|\sim|p_{4\perp}|\sim|p_{5\perp}|\,,$ (295)

is also a boundary of the forward NMRK limit. We will refer to the region eq.
295 as the ${\mathrm{MRK}^{\prime}}$ limit, and likewise denote the relevant
$t$-channel invariant by a prime, that is,
$t_{1}^{\prime}=(q_{1}^{\prime})^{2}$ with $q_{1}^{\prime}=-p_{2}-p_{4}$. The
relationship between the MRK, ${\mathrm{MRK}}^{\prime}$ and forward NMRK
limits is depicted in fig. 6.

In practice, to obtain the (N)MRK limit of rational functions, we first
implement the explicit representation of spinor products given in section A.1,
and then map to the minimal set of variables described in section 3.

Figure 5: Illustration of the 3 forward NMRK, 3 backward NMRK and 3! MRK
limits which can be obtained from a $2\to 3$ scattering process in general
kinematics.  Figure 6: Illustration of the heirarchy of limits relevant for
this paper. Starting from general kinematics (left) the forward NMRK region
(centre) is reached by taking the limit eq. 294. The further MRK and
${\mathrm{MRK}}^{\prime}$ regions can be reached by taking the limits eq. 293
or eq. 295 respectively, which are the rapidity boundaries of the NMRK region.

##  Appendix B Special Functions

In this section we list the special functions used in this paper. The only
divergent special function we us is the massless bubble integral,

$K_{0}(x)=\frac{1}{\epsilon}+2-\log(-x)+\mathcal{O}(\epsilon)\,.$ (296)

We use the following weight-1 functions, which are non-singular as $x\to 1$
Bern:1993mq

$\displaystyle L_{0}(x)$ $\displaystyle=\frac{\log(x)}{1-x}\,,$ (297)
$\displaystyle L_{1}(x)$ $\displaystyle=\frac{\log(x)+1-x}{(1-x)^{2}}\,,$
(298) $\displaystyle L_{2}(x)$
$\displaystyle=\frac{1}{(1-x)^{3}}\left[\log(x)-\frac{x}{2}+\frac{1}{2x}\right]\,,$
(299)

and we use the following weight-2 functions, which are non-singular as $x+y\to
1$

$\displaystyle Ls_{-1}(x,y)$ $\displaystyle={\rm Li_{2}}(1-x)+{\rm
Li_{2}}(1-y)+\log(x)\log(y)-\frac{\pi^{2}}{6}\,,$ (300) $\displaystyle
Ls_{1}(x,y)$
$\displaystyle=\frac{1}{(1-x-y)^{2}}\bigg{[}Ls_{-1}(x,y)+(1-x-y)\left[L_{0}(x)+L_{0}(y)\right]\bigg{]}\,.$
(301)

For $x>0$, the argument of the $L_{i}$ functions can be inverted using

$\displaystyle L_{0}\left(\frac{1}{x}\right)$ $\displaystyle=x\,L_{0}(x)\,,$
(302) $\displaystyle L_{1}\left(\frac{1}{x}\right)$
$\displaystyle=-x^{2}\,L_{1}(x)-x\,,$ (303) $\displaystyle
L_{2}\left(\frac{1}{x}\right)$ $\displaystyle=x^{3}\,L_{2}(x)\,.$ (304)

It is often necessary to write the $L_{i}$ functions in terms of functions
with a higher or lower index so that the resulting coefficients are free from
unphysical singularities. To this end we list explicitly the identities

$\displaystyle L_{0}\left(x\right)$
$\displaystyle=(1-x)\,L_{1}(x)-1=(1-x)^{2}\,L_{2}\left(x\right)-\left(\frac{1+x}{2x}\right)\,,$
(305) $\displaystyle L_{1}\left(x\right)$
$\displaystyle=(1-x)\,L_{2}(x)-\frac{1}{2x}\,.$ (306)

We analytically continue these functions by giving all Mandelstam invariants a
small positive imaginary part, $s\to s+i0$. For logarithms of ratios of
invariants, this prescription yields

$\log\left(\frac{-s_{1}}{-s_{2}}\right)=\log\left(\left|\frac{s_{1}}{s_{2}}\right|\right)-i\pi\left(\Theta(s_{1})-\Theta(s_{2})\right)$
(307)

The analytic continuation of the dilogarithm is given in ref.
vanHameren:2005ed .

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