$P_{w}^{B}=0$ we see that the transcendental terms in eq. (136) become equal,
allowing us to write

$\displaystyle\begin{split}S^{B}_{vw}(W_{2}(\sigma_{B})-W_{1}(\sigma_{B}))+S^{B}_{wu}(W_{3}(\sigma_{B})-W_{1}(\sigma_{B}))\xrightarrow[P^{B}_{w}\to
0]{}(S^{B}_{vw}+S^{B}_{wu})\log^{2}\left(\frac{1}{1+|z|^{2}X}\right).\end{split}$
(138)

The sum of rational terms $S^{B}_{vw}+S^{B}_{wu}$ is finite at $P_{w}^{B}=0$,
so we have shown that the dispersive part of the amplitude eq. (132) is free
from unphysical singularities, as expected.

| $P_{u}^{B}=0$ | $P_{v}^{B}=0$ | $P_{w}^{B}=0$
---|---|---|---
$u_{B}$ | 1 | $\frac{1}{1+|1+\frac{w}{z}|^{2}\frac{1}{X}}$ | $\frac{1}{1+|z|^{2}X}$
$v_{B}$ | $\frac{1}{1+|z|^{2}X}$ | 1 | $\frac{1}{1+|z|^{2}X}$
$w_{B}$ | $\frac{1}{1+|z|^{2}X}$ | $\frac{1}{1+|1+\frac{w}{z}|^{2}\frac{1}{X}}$ | 1
Table 2: Behaviour of the cross ratios in the NMRK at the three unphysical-
singularity surfaces of the rational terms of the ordering $\sigma_{B}$,
listed in eq. (62).

As in section 4.1, we can write the real part of the transcendental functions
exactly in terms of the one-loop trajectory, impact factors and Lipatov
vertex, plus a NMRK remainder term. We first separate the transcendental
function into a ‘central’ and ‘non-central’ pieces,

$\displaystyle\begin{split}2N_{c}\kappa_{\Gamma}{\mathrm{Re}\left[{W_{2}(\sigma_{A})}\right]}=&E(t_{1},s_{34};t_{3},s_{56})+2W_{2}^{A}\,,\end{split}$
(139)

with $E$ given in eq. (96), and we further separate the central function into

$2W_{2}^{A}=U_{A}+\Delta W_{A}(u_{A},v_{A},w_{A})\,.$ (140)

The function $U_{A}$ is given by eq. (100). In other words, this is a
helicity-independent term. In contrast, for the opposite-helicity case the
remainder function is given by

$\Delta
W_{A}(u_{A},v_{A},w_{A})=N_{c}\kappa_{\Gamma}\left(-\frac{1}{4}\log^{2}\left(u_{A}\right)+\frac{1}{2}\log(u_{A})\log(v_{A}w_{A})-\log(v_{A})\log(w_{A})\right)\,.$
(141)

Note that, as for the same-helicity case, eq. (101), the variable assignment
in eq. (100) leads to a NMRK remainder function which is IR-finite and
symmetric under $v_{A}\leftrightarrow w_{A}$. However, we note that unlike eq.
(101), eq. (141) does not vanish in the MRK limit. We will return to this
point later. When working in NMRK, in order to exploit the equality of the
real parts of the transcendental functions under an exchange of
$p_{1}\leftrightarrow p_{6}$ or $p_{2}\leftrightarrow p_{3}$, we make the
approximation eq. (102), and we make the NMRK approximation of the cross
ratios in eq. (141). For the $A^{\prime}$ ordering we similarly write

$\displaystyle\begin{split}2N_{c}\kappa_{\Gamma}{\mathrm{Re}\left[{W_{2}(\sigma_{A^{\prime}})}\right]}&=E(t_{1},s_{35};t_{3},s_{46})+2W_{2}^{A^{\prime}}\,,\\\
2W_{2}^{A^{\prime}}&=U_{A^{\prime}}+\Delta
W_{A}(u_{A^{\prime}},w_{A^{\prime}},v_{A^{\prime}}).\end{split}$ (142)

where $U_{A^{\prime}}$ is related to $U_{A}$ by exchange of
$p_{4}\leftrightarrow p_{5}$.

As in section 4.1, we can write the transcendental function of the
$\sigma_{B}$ ordering in terms of the one-loop factorised pieces in two
different ways,

$\displaystyle\begin{split}2N_{c}\kappa_{\Gamma}{\mathrm{Re}\left[{W_{1}(\sigma_{B})}\right]}=E(t_{1},-s_{24};t_{3},s_{56})+2W_{1}^{B}=E(t_{1},s_{35};t_{3},-s_{14})+2W_{1}^{B^{\prime}}\,,\end{split}$
(143)

with

$\displaystyle\begin{split}2W_{1}^{B}&=U_{B}+\Delta
W_{B}(u_{B},v_{B},w_{B}),\qquad\qquad
2W_{1}^{B^{\prime}}=U_{B^{\prime}}+\Delta
W_{B}(v_{B},u_{B},w_{B})\,.\end{split}$ (144)

The helicity-independent functions $U_{B}$ and $U_{B^{\prime}}$ are given in
eq. (108), and the NMRK remainder is

$\displaystyle\Delta
W_{B}(u_{B},v_{B},w_{B})=N_{c}\kappa_{\Gamma}\left(\log(v_{B})\log\left(\frac{w_{B}}{u_{B}}\right)+\log(u_{B})\log\left(\frac{w_{B}}{v_{B}}\right)\right)\,.$
(145)

We note that, similar to eq. (4.1), eq. (145) is IR finite, symmetric under
$u_{B}\leftrightarrow v_{B}$, but does not vanish in either MRK or
${\mathrm{MRK}^{\prime}}$ limits. We emphasise that eq. (143) is exact, but in
the following we make the NMRK approximation (110) for $U_{B}$ or
$U_{B^{\prime}}$, after which the functions $U_{B}$ and $U_{B^{\prime}}$ are
related by $p_{4}\leftrightarrow p_{5}$. We also make the NMRK approximation
to the cross ratios in eq. (145), after which this function is symmetric under
$p_{4}\leftrightarrow p_{5}$.

Following the procedure of section 4.1, we now wish to express the opposite-
helicity amplitude using the colour-structure basis of eq. (117). To this end,
we apply the colour rotation of eq. (113) to the sum inside the curly brackets
in eq. (132). We find that the structure of the kinematic coefficients is
similar to the same-helicity case, motivating the following expression for
general helicities, generalising eq. (117),

$\displaystyle{\mathrm{Disp}\left[{\mathcal{M}^{(1)}_{\mathcal{N}=4}(p_{4}^{\nu_{4}},p_{5}^{\nu_{5}})}\right]}\xrightarrow[\rm
NMRK]{}$ $\displaystyle\frac{g_{\scriptscriptstyle
S}^{6}}{(4\pi)^{2}}\,s_{12}F^{a_{3}a_{2}c_{1}}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\,\times$
$\displaystyle\sum_{\sigma\in
S_{2}}\Bigg{\\{}(F^{a_{\sigma_{4}}}F^{a_{\sigma_{5}}})_{c_{1}c_{3}}\left(A^{gg(1)}_{\vDash}(q_{1},p^{\nu_{\sigma_{4}}}_{\sigma_{4}},p^{\nu_{\sigma_{5}}}_{\sigma_{5}},q_{3})+A^{gg(0)}(q_{1},p^{\nu_{\sigma_{4}}}_{\sigma_{4}},p^{\nu_{\sigma_{5}}}_{\sigma_{5}},q_{3})E(t_{1},s_{3\sigma_{4}};t_{3},s_{\sigma_{5}6})\right)$
$\displaystyle\qquad\qquad\qquad+\frac{1}{N_{c}}d_{A}^{c_{1}a_{\sigma_{4}}a_{\sigma_{5}}c_{3}}A^{gg(1)}_{d}(q_{1},p^{\nu_{\sigma_{4}}}_{\sigma_{4}},p^{\nu_{\sigma_{5}}}_{\sigma_{5}},q_{3})\Bigg{\\}}\times\frac{1}{t_{3}}F^{a_{6}a_{1}c_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\,.$
(146)

For the opposite-helicity case we find that the kinematic coefficient of the
tree-level colour structure is

$\displaystyle A^{gg(1)}_{\vDash}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})$
(147) $\displaystyle=$
$\displaystyle\frac{1}{3}\Bigg{[}2A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\Big{(}2W^{A}_{2}+W^{B}_{1}\Big{)}-2A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\Big{(}W^{A^{\prime}}_{2}-W^{B^{\prime}}_{1}\Big{)}$
$\displaystyle\quad+2\Big{(}S^{A}_{uv}(W^{A}_{1}-W^{A}_{2})+S^{A}_{wu}(W^{A}_{3}-W^{A}_{2})\Big{)}-\Big{(}S^{A^{\prime}}_{uv}(W^{A^{\prime}}_{1}-W^{A^{\prime}}_{2})+S^{A^{\prime}}_{wu}(W^{A^{\prime}}_{3}-W^{A^{\prime}}_{2})\Big{)}$
$\displaystyle\quad-\Big{(}S^{B}_{vw}(W^{B}_{2}-W^{B}_{1})+S^{B}_{wu}(W^{B}_{3}-W^{B}_{1})\Big{)}\Bigg{]}\,.$

There is some freedom in how to partition the kinematic coefficient of $d_{A}$
as a sum over colour orderings. Before discussing our choice of how we
organise this term, let us investigate the MRK limit of eq. (147). We first
drop the rational terms that are power suppressed in this limit,

$\displaystyle A^{gg(1)}_{\vDash}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})$
(148) $\displaystyle\xrightarrow[\mathrm{MRK}]{}$
$\displaystyle\frac{1}{3}\Bigg{[}A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\Big{(}3U_{\mathrm{MRK}}+2\Delta
W_{A}(u_{A},v_{A},w_{A})+\Delta W_{B}(u_{B},v_{B},w_{B})\Big{)}$
$\displaystyle\quad+2\Big{(}S^{A}_{uv}(W^{A}_{1}-W^{A}_{2})+S^{A}_{wu}(W^{A}_{3}-W^{A}_{2})\Big{)}-S^{B}_{vw}\big{(}W^{B}_{2}-W^{B}_{1}\big{)}\Bigg{]}\,,$

where $U_{\mathrm{MRK}}$ is the helicity-independent MRK limit of the central
transcendental functions, introduced in eq. (123). We now note that the
nonvanishing terms of $\Delta W_{A}$ and $\Delta W_{B}$ cancel against the MRK
limits of the terms proportional to $S^{A}$ and $S^{B}$ respectively, i.e.

$\displaystyle\begin{split}A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\Delta
W_{A}(u_{A},v_{A},w_{A})&\underset{\mathrm{MRK}}{=}-S^{A}_{uv}(W^{A}_{1}-W^{A}_{2})-S^{A}_{wu}(W^{A}_{3}-W^{A}_{2})\,,\\\
A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\Delta
W_{B}(u_{B},v_{B},w_{B})&\underset{\mathrm{MRK}}{=}S^{B}_{vw}(W^{B}_{2}-W^{B}_{1})\,.\end{split}$
(149)

With this we see that the MRK limit of eq. (148),

$\displaystyle\begin{split}A^{gg(1)}_{\vDash}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})&\xrightarrow[\mathrm{MRK}]{}V^{g(0)}(q_{1},p_{4}^{\oplus},q_{2})\frac{1}{t_{2}}V^{g(0)}(q_{2},p_{5}^{\ominus},q_{3})\,U_{\mathrm{MRK}}\,,\end{split}$
(150)

coincides with the MRK limit of the same-helicity amplitude, eq. (125),
manifesting the helicity independence of the MRK limit. A similar analysis
shows that the ${\mathrm{MRK}^{\prime}}$ limit of eq. (148),

$\displaystyle\begin{split}A^{gg(1)}_{\vDash}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})&\xrightarrow[\mathrm{MRK}^{\prime}]{}V^{g(0)}(q_{1},p_{5}^{\ominus},q_{2}^{\prime})\frac{1}{t_{2}^{\prime}}V^{g(0)}(q_{2}^{\prime},p_{4}^{\oplus},q_{3})\,U_{\mathrm{MRK^{\prime}}}\,,\end{split}$
(151)

coincides with the ${\mathrm{MRK}^{\prime}}$ limit of the same-helicity
vertex, eq. (127).

Equation (149) also provides our motivation for writing the kinematic
coefficient of the totally symmetric colour structure as

$\displaystyle\begin{split}A^{gg(1)}_{d}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})&=2A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\left(W_{2}^{A}-W_{1}^{B}\right)\\\
&+S^{A}_{uv}(W^{A}_{1}-W^{A}_{2})+S^{A}_{wu}(W^{A}_{3}-W^{A}_{2})+S^{B}_{vw}(W^{B}_{2}-W^{B}_{1}),\end{split}$
(152)

with the aforementioned property eq. (137) leading to

$\displaystyle\begin{split}A^{gg(1)}_{d}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})&=2A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\left(W^{A^{\prime}}_{2}-W_{1}^{B^{\prime}}\right)\\\
&+S^{A^{\prime}}_{uv}(W^{A^{\prime}}_{1}-W^{A^{\prime}}_{2})+S^{A^{\prime}}_{wu}(W^{A^{\prime}}_{3}-W^{A^{\prime}}_{2})+S^{B}_{wu}(W^{B}_{3}-W^{B}_{1}).\end{split}$
(153)

This choice gives us the simple MRK and MRK′ limits,

$\displaystyle\begin{split}A^{gg(1)}_{d}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})&\xrightarrow[\mathrm{MRK}]{}0\,,\\\
A^{gg(1)}_{d}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})&\xrightarrow[\mathrm{MRK}^{\prime}]{}0\,.\end{split}$
(154)

similar to eqs. (126) and (128).

###  4.3 NLO $gg\to gggg$ squared matrix element in the central NMRK limit

Having obtained the central NMRK limit of the six-gluon one-loop amplitude in
${{\cal N}=4}$ SYM for all leading helicity configurations, we now consider
the contribution of this amplitude to the squared matrix element for $gg\to
gggg$ at NLO, i.e. the interference of this amplitude with the tree-level
amplitude given in eqs. (25) and (26), summed over all colour indices:

$2{\mathrm{Re}\left[{{\cal M}^{(0)*}_{6g}{\cal
M}^{(1)}_{6g}}\right]}=2{\mathrm{Re}\left[{{\cal
M}^{(0)*}_{6g}\mathrm{Disp}\left[{\cal
M}^{(1)}_{6g}\right]}\right]}-2{\mathrm{Im}\left[{{\cal
M}^{(0)*}_{6g}\mathrm{Absorp}\left[{\cal M}^{(1)}_{6g}\right]}\right]}\,.$
(155)

We consider these two contributions in turn.

#### Dispersive part

We first consider the contribution of the dispersive part of the one-loop
amplitude, eq. (146), to eq. (155). The colour structures contributing to
${\cal M}^{(0)*}_{6g}\mathrm{Disp}\big{[}{\cal M}^{(1)}_{6g}\big{]}$ are
depicted as interference diagrams in figure 8. For each of them, after summing
over the colour indices $a_{2}$, $a_{3}$ and $a_{1}$, $a_{6}$, the factors
associated with the impact factors are proportional to the identity in the
adjoint representation, i.e.

$\mbox{tr\,}(F^{c_{1}}F^{\tilde{c}_{1}})=2N_{c}\delta^{c_{1}\tilde{c}_{1}}\,,\qquad\qquad\mbox{tr\,}(F^{c_{3}}F^{\tilde{c}_{3}})=2N_{c}\delta^{c_{3}\tilde{c}_{3}}\,.$
(156)

Figure 8: Three of the colour structures contributing to ${\cal
M}^{(0)*}_{6g}\mathrm{Disp}\big{[}{\cal M}^{(1)}_{6g}\big{]}$ using the basis
of one-loop colour structures in eq. (146). The remaining three colour
structures can be obtained by interchanging $a_{4}\leftrightarrow a_{5}$. A
three-gluon vertex represents the adjoint generator $F$, while the $\otimes$
symbol represents the fully symmetric trace $d_{A}$ of eq. (115).

We then see that the colour structures involving the totally symmetric tensor
$d_{A}$ vanish after contraction with the partially antisymmetric tree-level
colour-structure:

$\displaystyle
4N_{c}^{2}\delta^{c_{1}\tilde{c}_{1}}\delta^{c_{3}\tilde{c}_{3}}d_{A}^{c_{1}a_{4}a_{5}c_{3}}(F^{a_{4}}F^{a_{5}})_{\tilde{c}_{1}\tilde{c}_{3}}=-8N_{c}^{2}d_{A}^{c_{1}a_{4}a_{5}c_{3}}f^{c_{1}a_{4}\tilde{c}_{2}}f^{\tilde{c}_{2}a_{5}c_{3}}=0.$
(157)

The coefficient $A_{d}^{gg(1)}$ of the two-gluon CEV in eq. (146) therefore
does not contribute to the NLO $gg\to 4\,\text{jet}$ cross section, or by
extension, to the N3LO dijet cross section after phase-space integration. We
emphasise that this conclusion applies to any helicity configuration of the
one-loop two-gluon CEV.

By a similar argument, the symmetric colour structure of the two-gluon CEV
does not contribute to the NNLL BFKL kernel. To see this we note that at this
logarithmic accuracy, the relevant interference, or ‘ladder’ diagrams
describing an arbitrary number of real emissions along the ladder, including
precisely one instance of the one-loop two-gluon CEV, have all other ‘rungs’
generated by the interference of tree-level impact factors and one-gluon CEVs
(those appearing in fig. 1 (b)). The colour structure containing $d_{A}$ of
such a ladder diagram is depicted in figure 9. Given that the colour structure
of the tree-level impact factors are either a fundamental generator or adjoint
structure constant, and that the colour-structure of any one-gluon CEV is
simply a structure constant, it follows that upon summing over the colour
indices, the entire ladder of rungs above and below the one-loop two-gluon CEV
collapse with the result proportional the identity in the adjoint
representation (with each rung yielding an extra factor of $2N_{c}$). At the
final stage, the contraction between $d_{A}$ and the structure constant yields
a vanishing result, just as in eq. (157).

Figure 9: A colour structure of an interference, or ‘ladder’ diagram at NNLL
accuracy, containing one instance of the one-loop two-gluon CEV, and all other
real emissions given by tree-level one-gluon CEVs. The $\otimes$ symbol
represents the fully symmetric trace $d_{A}$ of the two-gluon CEV. This colour
structure vanishes upon summing over all colour indices, as discussed in the
main text.

We further note that the symmetric colour structure of the two-gluon CEV is
expected to contribute to the cross section and to the BFKL kernel at N3LL
accuracy: at this order one must consider the square of the one-loop two-gluon
CEV, where the interference involving two $d_{A}$ terms survives, and yields a
quartic Casimir invariant.

Having seen that the symmetric trace of four generators, $d_{A}$, which
appears in the two-gluon CEV, does not contribute to the cross section or the
BFKL kernel at NNLL, it is interesting to ask whether such symmetric colour
structures contribute at this logarithmic accuracy to _any_ of the
interference diagrams making up the BFKL kernel in fig. 3. The answer is
negative: upon summing over colour indices, the impact factors in any of these
interference diagrams would be proportional to the identity in the adjoint
representation. This argument holds for both gluon or quark impact factors.
Having contracted the impact factors, the virtual loops and the central gluons
only involve six colour vertices in total, which cannot form a quartic Casimir
in any of the contributions in fig. 3. This may be contrasted with
contributions to the $2\to 2$ amplitude from triple Reggeon exchange, which do
involve quartic Casimirs at NNLL accuracy Falcioni:2020lvv ; Falcioni:2021buo
; Falcioni:2021dgr .

#### Absorptive part

We now consider the contribution of the absorptive part of the one-loop
amplitude to eq. (155). As mentioned above, for the absorptive part of the
one-loop amplitude, all signature components contribute at leading power in
the NMRK limit, although they do not have logarithmic dependence on the large
invariants $s_{34}$ or $s_{56}$. The four signature components are listed in
eqs. (239)–(242) for $\nu_{4}=\nu_{5}$ and in eqs. (243)–(246) for
$\nu_{4}=-\nu_{5}$. However, the leading-power behaviour of the tree-level
amplitude is given by the $(-,-)$ component alone, and this projects out the
$(-,-)$ component of the absorptive one-loop amplitude in ${\cal
M}^{(0)*}_{6g}\mathrm{Absorp}\big{[}{\cal M}^{(1)}_{6g}\big{]}$ upon summing
over the external colour indices. For the $(-,-)$ component of the absorptive
part of the amplitude (eq. (239) for the $\nu_{4}=\nu_{5}$ case and eq. (243)
for the $\nu_{4}=-\nu_{5}$ case), we can factor out an adjoint generator
associated to the tree-level impact factors. Upon considering the interference
between these absorptive parts and the tree-level amplitude, we therefore find
that the colour factors associated with the impact factors are identical to
the dispersive case above, eq. (156).

To proceed and evaluate the imaginary part of ${\cal
M}^{(0)*}_{6g}\mathrm{Absorp}\big{[}{\cal M}^{(1)}_{6g}\big{]}$ we now
distinguish between the same-helicity case, where the relevant absorptive part
is given in eq. (239), and the opposite-helicity case, where the relevant
absorptive part is given in eq. (243). For the former case, the colour factors
which contain $\mbox{tr\,}(F^{c_{1}}F^{a_{4}}F^{a_{5}}F^{c_{3}})$ or
$\mbox{tr\,}(F^{c_{1}}F^{a_{5}}F^{a_{4}}F^{c_{3}})$ do not contribute to the
absorptive interference term, due to either the vanishing of the colour factor
or the reality of its kinematic coefficient. The remaining contribution is
proportional to

$\displaystyle\begin{split}&\mbox{tr\,}(F^{c_{1}}F^{a_{5}}F^{c_{3}}F^{a_{4}})(F^{a_{4}}F^{a_{5}})_{c_{1}c_{3}}\mathrm{Im}\left[B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})A^{gg(0)*}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\right]\\\
+\,&\mbox{tr\,}(F^{c_{1}}F^{a_{5}}F^{c_{3}}F^{a_{4}})(F^{a_{5}}F^{a_{4}})_{c_{1}c_{3}}\mathrm{Im}\left[B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})A^{gg(0)*}(q_{1},p^{\oplus}_{5},p^{\oplus}_{4},q_{3})\right]=0.\end{split}$
(158)

To show that this quantity is zero we first note that the two colour factors
are equal, which can be shown using the reversal-invariance property of the
trace of four adjoint generators, along with a relabelling of the indices
$a_{4}$ and $a_{5}$. After factoring out the common colour factor one is left
with the imaginary part of a kinematic factor which can be shown to be real
using eq. (54). We conclude that the absorptive part of the amplitude does not
contribute to the squared matrix element for $gg\to gggg$ at NLO in the
central NMRK limit for the helicity configurations with $\nu_{4}=\nu_{5}$.

Repeating the above analysis for the opposite-helicity case we find a non-
vanishing result. Specifically, for $\nu_{4}=\oplus$ and $\nu_{5}=\ominus$ we
find that

$\displaystyle\begin{split}2\mathrm{Im}\Big{[}{\cal
M}^{(0)*}_{6g}(p^{\oplus}_{4},p^{\ominus}_{5})&\,\mathrm{Absorp}\left[{\cal
M}^{(1)}_{6g}(p^{\oplus}_{4},p^{\ominus}_{5})\right]\Big{]}\xrightarrow[\mathrm{NMRK}]{}2\,g_{\scriptscriptstyle
S}^{8}\frac{\alpha_{\scriptscriptstyle
S}}{4\pi}\frac{\pi}{2}\frac{s^{2}}{t_{1}^{2}t_{3}^{2}}\left|C^{g(0)}(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}})\right|^{2}\\\
\times
8\,N_{c}^{5}(N_{c}^{2}-1)\bigg{\\{}&\mathrm{Im}\left[\Big{(}S_{uv}^{A}\log(v_{A})+S_{wu}^{A}\log(w_{A})\Big{)}A^{gg(0)*}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right]\\\
+&\mathrm{Im}\left[\left(S_{uv}^{A^{\prime}}\log(v_{A^{\prime}})+S_{wu}^{A^{\prime}}\log(w_{A^{\prime}})\right)A^{gg(0)*}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right]\\\
-&\mathrm{Im}\left[S_{uv}^{B}B^{gg(0)*}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right]\log\left(\frac{w_{B}}{u_{B}v_{B}}\right)\bigg{\\}}\times\left|C^{g(0)}(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}})\right|^{2}\,,\end{split}$
(159)

which is generically non-zero.

Next we observe that the absorptive interference term for $\nu_{4}=\ominus$
and $\nu_{5}=\oplus$ is equal to the complex conjugate of eq. (159).
Therefore, the absorptive part of the one-loop amplitude in NMRK does not
contribute to the helicity-summed squared matrix element for $gg\to gggg$ at
NLO. Thus, the contribution of the one-loop six-gluon amplitude in the central
NMRK limit to the cross section at this logarithmic accuracy is entirely
driven by the dispersive part.

##  5 All-orders conjectures for amplitudes in a central NMRK limit

In section 4 we discussed the six-gluon ${\cal N}=4$ SYM amplitude to one-loop
accuracy. However, the results of section 4.1 and 4.2 are suggestive of the
factorised form that amplitudes may take in the central NMRK limit at higher
orders, and potentially at higher multiplicity. In this section we therefore
make conjectures for the form that the dispersive part of amplitudes may take
beyond one loop and beyond the six-gluon case.

The separation of the non-central $E$ functions in eq. (146), multiplying the
tree-level two-gluon CEV, strongly suggests a factorised picture, analogous to
eq. (2.2), for the production of two gluons in the central NMRK limit. The
one-loop two-gluon CEV extracted this way is a component of BFKL at NNLL
accuracy. As noted in ref. Canay:2021yvm , one can place limits on the pattern
of Reggeization by requiring that the MRK limit of said CEV agrees with the
known behaviour of amplitudes in MRK. The following Reggeization ansatz for
the production of two gluons in the central NMRK limit is compatible with both
the known MRK limit and with the structure of the dispersive part of the one-
loop amplitude eq. (146),

$\displaystyle\begin{split}&{\mathrm{Disp}\left[{{\cal
M}^{(-,-)}_{6g}}\right]}\xrightarrow[\mathrm{NMRK}]{}s\left[g_{\scriptscriptstyle
S}(F^{a_{3}})_{a_{2}c_{1}}\,C^{g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}})\right]\\\
&\hskip 28.45274pt\times\sum_{\sigma\in
S_{2}}\Bigg{\\{}\frac{1}{t_{1}}\,\frac{1}{2}\left[\left(\frac{s_{3\sigma_{4}}}{\tau}\right)^{\alpha(t_{1})}+\left(\frac{-s_{3\sigma_{4}}}{\tau}\right)^{\alpha(t_{1})}\right]\\\
&\hskip 28.45274pt\times\left[g_{\scriptscriptstyle
S}^{2}(F^{a_{\sigma_{4}}}F^{a_{\sigma_{5}}})_{c_{1}c_{3}}\,A^{gg}_{\vDash}(q_{1},p^{\nu_{\sigma_{4}}}_{\sigma_{4}},p^{\nu_{\sigma_{5}}}_{\sigma_{5}},q_{3})+\frac{1}{N_{c}}d^{c_{1}a_{\sigma_{4}}a_{\sigma_{5}}c_{3}}_{A}A^{gg}_{d}(q_{1},p^{\nu_{\sigma_{4}}}_{\sigma_{4}},p^{\nu_{\sigma_{5}}}_{\sigma_{5}},q_{3})\right]\,\\\
&\hskip
28.45274pt\times\frac{1}{t_{3}}\,\frac{1}{2}\left[\left(\frac{s_{{\sigma_{5}}6}}{\tau}\right)^{\alpha(t_{3})}+\left(\frac{-s_{{\sigma_{5}}6}}{\tau}\right)^{\alpha(t_{3})}\right]\Bigg{\\}}\\\
&\hskip 28.45274pt\times\left[g_{\scriptscriptstyle
S}(F^{a_{6}})_{a_{1}c_{3}}\,C^{g}(p_{1}^{\nu_{1}},p_{6}^{\nu_{6}})\right]\,.\end{split}$
(160)

This factorization is illustrated schematically in figure 10 (a). The first
thing to note is that, unlike amplitudes at LL, or signature-odd amplitudes at
NLL, the amplitude is not fully factorised. Rather, eq. (160) is rather a sum
over two colour orderings, each of which is factorised in the familiar form of
impact factors, Reggeized gluons and a two-gluon CEV.

Further to this point are the assumptions we have made regarding the colour
structure $d_{A}$ beyond one-loop accuracy. In eq. (160) we have assumed that
this totally-symmetric colour factor couples to Reggeized gluons at higher
orders in $\alpha_{\scriptscriptstyle S}$. Specifically, eq. (160) assumes a
particular form for this Reggeization, based on the partitioning of the
kinematic coefficient of $d_{A}$, sections 4.1 and 4.2. While this
partitioning seems natural from the point of view of requiring compatibility
with the known MRK limit, the nature of Reggeization of this term of course
cannot be determined from the one-loop amplitude studied in this paper. It may
be constrained by taking the central NMRK limit of the two-loop six-gluon
amplitude. A similar structure is accessible through the non-central NMRK
limit of the two-loop five-gluon amplitude.

Figure 10: (a) The expected factorised picture of the dispersive part of a
six-gluon amplitude in the central NMRK limit. (b) The expected factorization
of an $n$-gluon amplitude in a central NMRK limit. The coloured blobs
represent impact factors, and one- and two gluon CEVs, defined to all orders
in perturbation theory, similarly to figure 4.

With these assumptions in mind, we may consider how eq. (160) may be
generalised to the case of $2\to n$ scattering in any of the central NMRK
limits where the rapidity separation between a pair of consecutive central
gluons denoted by $m$ and $m+1$ (with $4\leq m\leq n-2$) is not necessarily
large, while all other rapidity separations are large. We propose the
following ansatz for an arbitrary number of legs:

$\displaystyle\begin{split}&{\mathrm{Disp}\left[{{\cal
M}^{[\mathbf{8}_{a}]}_{ng}}\right]}\xrightarrow[\mathrm{NMRK}]{}s\left[g_{\scriptscriptstyle
S}(F^{a_{3}})_{a_{2}c_{1}}\,C^{g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}})\right]\\\
&\times\prod_{i=4}^{m-1}\left(\frac{1}{t_{i-3}}\,\frac{1}{2}\left[\left(\frac{s_{i-1,i}}{\tau}\right)^{\alpha(t_{i-3})}+\left(\frac{-s_{i-1,i}}{\tau}\right)^{\alpha(t_{i-3})}\right]\left[g_{\scriptscriptstyle
S}(F^{a_{i}})_{c_{i-3}c_{i-2}}V^{g}(q_{i-3},p_{i}^{\nu_{i}},q_{i-2})\right]\right)\\\
&\times\sum_{\sigma\in
S_{2}}\Bigg{\\{}\frac{1}{t_{m-3}}\,\frac{1}{2}\left[\left(\frac{s_{m-1,\sigma_{m}}}{\tau}\right)^{\alpha(t_{m-3})}+\left(\frac{-s_{m-1,\sigma_{m}}}{\tau}\right)^{\alpha(t_{m-3})}\right]\\\
&\times\bigg{[}g_{\scriptscriptstyle
S}^{2}(F^{a_{\sigma_{m}}}F^{a_{\sigma_{m+1}}})_{c_{m-3}c_{m-1}}\,A^{gg}_{\vDash}(q_{m-3},p^{\nu_{\sigma_{m}}}_{\sigma_{m}},p^{\nu_{\sigma_{m+1}}}_{\sigma_{m+1}},q_{m-1})\\\
&\hskip
150.0pt+\frac{1}{N_{c}}d^{c_{m-3}a_{\sigma_{m}}a_{\sigma_{m+1}}c_{m-1}}_{A}A^{gg}_{d}(q_{m-3},p^{\nu_{\sigma_{m}}}_{\sigma_{m}},p^{\nu_{\sigma_{m+1}}}_{\sigma_{m+1}},q_{m-1})\bigg{]}\,\\\
&\times\frac{1}{t_{m-1}}\,\frac{1}{2}\left[\left(\frac{s_{\sigma_{m+1},m+2}}{\tau}\right)^{\alpha(t_{m-1})}+\left(\frac{-s_{\sigma_{m+1},m+2}}{\tau}\right)^{\alpha(t_{m-1})}\right]\Bigg{\\}}\\\
&\times\prod_{i=m+2}^{n-1}\left(\left[g_{\scriptscriptstyle
S}(F^{a_{i}})_{c_{i-3}c_{i-2}}V^{g}(q_{i-3},p_{i}^{\nu_{i}},q_{i-2})\right]\frac{1}{t_{i-2}}\,\frac{1}{2}\left[\left(\frac{s_{i,i+1}}{\tau}\right)^{\alpha(t_{i-2})}+\left(\frac{-s_{i,i+1}}{\tau}\right)^{\alpha(t_{i-2})}\right]\right)\\\
&\times\left[g_{\scriptscriptstyle
S}(F^{a_{n}})_{a_{1}c_{n-3}}\,C^{g}(p_{1}^{\nu_{1}},p_{n}^{\nu_{n}})\right]\,,\end{split}$
(161)

which is written entirely in terms of the components extracted from $n\leq 6$
amplitudes. This factorised structure is depicted schematically in figure 10
(b), where real emissions that are strongly ordered in rapidity are described
in a similar way to the MRK structure depicted in figure 4.

##  6 Conclusions

Taking the one-loop six-gluon amplitudes of $\mathcal{N}=4$ SYM Bern:1994zx ;
Bern:1994cg  into the central NMRK limit, in this paper we have obtained the
one-loop corrections to the two-gluon central-emission vertex (CEV) in this
theory, in both the same-helicity and opposite-helicity configurations. This
is a first step toward the determination of the same quantity in QCD.

In contrast with the one-gluon CEV, we find that the two-gluon CEV, eq. (84),
receives one-loop corrections involving a new colour structure consisting of a
fully symmetric trace of four adjoint generators, which is not present at tree
level. We find, however, that this new structure does not contribute to the
squared matrix element for $gg\to gggg$ at NLO, so it does not enter the NNLL
BFKL kernel. We note, however that this symmetric colour structure does
contribute to the squared matrix element at N3LL accuracy, where the
interference between two one-loop two-gluon CEVs gives rise to a quartic
Casimir invariant.

As discussed in section 4.3, none of the NNLL contributions to the BFKL
kernel, which are described by the interference diagrams in fig. 3, gives rise
to a quartic Casimir. This may be contrasted with contributions to the $2\to
2$ amplitude from triple-Reggeon exchange, which do involve quartic Casimirs
at NNLL accuracy Falcioni:2020lvv ; Falcioni:2021buo ; Falcioni:2021dgr .

In addition to the new colour structure, also the kinematic dependence of the
two-gluon CEV is rather complex. In the same-helicity two-gluon CEV, which is
extracted from an MHV amplitude, dilogarithms survive in the NMRK limit, eqs.
(101) and (4.1). The dilogarithms depend on conformally-invariant cross ratios
(42) and (56), the same variables which characterize six-gluon amplitudes in
planar $\mathcal{N}=4$ SYM Drummond:2007au .

Yet another layer of complexity arises in the case of the one-loop NMHV
amplitude, from which the opposite-helicity two-gluon CEV is extracted.
Considering this amplitude in central NMRK, eq. (146), eq. (147) and eq.
(152), we find rational coefficients which are not proportional to the tree
amplitude, accompanied by different transcendental functions, eq. (131). This
structure involves an interplay between spurious poles of these rational
coefficients and zeroes of the transcendental functions multiplying them, see
eqs. (132)–(136) and tables 1 and 2. These features propagate into the one-
loop opposite-helicity CEV.

Another consequence of the more complicated structure of the one-loop NMHV
amplitude is that, in contrast with the MHV case, an absorptive part survives
in the squared matrix element for $gg\to gggg$ at NLO, after summing over
colour indices, as shown in section 4.3. However, upon summing over gluon
helicities, this absorptive contribution vanishes, and thus only the
dispersive part of the one-loop six-gluon amplitude contributes to unpolarised
cross sections at NNLL.

In ref. Bartels:2008ce , it was noted that the transcendental functions of the
one-loop six-gluon MHV amplitude can be written exactly in terms of the known
one-loop Regge components, eq. (97), consisting of impact factors, Reggeized
gluons and Lipatov vertices, plus a remainder term (101) written in terms of
the conformally-invariant cross ratios (42). In eqs. (139) and (141), we have
shown that this holds also for the NMHV amplitude, and thus for the whole one-
loop six-gluon amplitude in $\mathcal{N}=4$ SYM. This parallels the set-up of
the two-loop six-gluon MHV amplitude in planar $\mathcal{N}=4$ SYM, which can
be expressed in terms of two-loop Regge components DelDuca:2008jg  and
remainder functions depending on conformally-invariant cross ratios (42), and
which can be evaluated in the central NMRK limit without any loss of
generality DelDuca:2009au ; DelDuca:2010zg . It would be interesting to extend
this analysis to the two-loop six-gluon amplitude in full $\mathcal{N}=4$ SYM.

It follows in particular that the transcendental functions making up the one-
loop two-gluon CEV can be expressed in terms of one-loop MRK components,
evaluated at NMRK kinematics, plus a remainder function. The former is
independent of the helicity configuration, eq. (95), while the latter has a
distinct functional form for each of the two helicity configurations.

Our one-loop analysis of the two-gluon CEV provides first hints with regards
to the factorization and exponentiation properties of the six-gluon amplitude.
In eq. (160) we have proposed an exponentiation pattern of the six-gluon
amplitude in the central NMRK limit in which rapidity ordering is partially
linked with colour ordering. The guiding principles have been the locality of
the two-gluon CEV in rapidity, and its factorisation in the MRK limit into two
one-gluon CEVs connected by a Regge trajectory. However, the link between
rapidity and colour ordering in (160) is only partial, because of the presence
of a fully-symmetric colour structure, which occurs first at the one-loop
level, eq. (146): an evaluation of the two-loop amplitude in the NMRK limit is
necessary to confirm or disprove this factorized structure. We leave the
investigation of two-loop amplitudes in the NMRK limit to future work.

In the NLL corrections to the BFKL kernel (fig. 2 b), the one-loop one-gluon
CEV is multiplied by the corresponding tree-level CEV, with the gluon $p_{4}$
emitted centrally along the ladder which is integrated over its phase space.
No collinear divergences are allowed in MRK, eq. (7), however in the phase-
space integration, the gluon $p_{4}$ may become soft and yield a pole of
${\cal O}(1/\epsilon)$. In order to generate all the finite terms of the
squared amplitude, the one-loop one-gluon CEV must be evaluated to ${\cal
O}(\epsilon)$ in the limit that the gluon $p_{4}$ is soft Fadin:1996yv ;
DelDuca:1998cx , which requires the use of one-loop soft functions to ${\cal
O}(\epsilon)$ Bern:1998sc . In the NNLL corrections to the BFKL kernel,
similar features are present: the one-loop two-gluon CEV is multiplied by the
corresponding tree-level CEV with the gluons $p_{4}$ and $p_{5}$ emitted along
the ladder, which are integrated over their phase space. In the phase-space
integration, both gluons may become soft, as well as collinear to each other,
yielding poles of up to ${\cal O}(1/\epsilon^{3})$. Accordingly, the one-loop
two-gluon CEV must be evaluated to ${\cal O}(\epsilon^{3})$ in the limit that
gluons $p_{4}$ and $p_{5}$ are soft, and/or collinear to each other. To higher
order in $\epsilon$, the one-loop splitting functions are known Kosower:1999rx
; Bern:1999ry , but the one-loop double-soft functions are yet to be
determined. We leave the analysis of the higher-order terms of the one-loop
two-gluon CEV to future work.

For generic gluon momenta, in order to obtain the two-gluon CEV in QCD using
the supersymmetric decomposition of one-loop amplitudes, the next step is to
perform a similar NMRK analysis of the contributions of a $\mathcal{N}=1$
multiplet and of a scalar circulating in the loop. We also leave this analysis
for the future.

## Acknowledgements

We thank Claude Duhr for useful discussions. EG is supported by the STFC
Consolidated Grant ‘Particle Physics at the Higgs Centre’. EB and JMS are
supported by the ERC Starting Grant 715049 ‘QCDforfuture’. JMS is also
supported by a Royal Society University Research Fellowship. This research was
supported by the US Department of Energy under contract DE–AC02–76SF00515, and
in part by the National Science Foundation under Grant No. NSF PHY–1748958. LD
thanks the Kavli Institute for Theoretical Physics for hospitality during the
completion of this work. For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising from this submission.

##  Appendix A Kinematic regions

###  A.1 General multi-parton kinematics

We consider the production of four partons of momenta $p_{i}$, $i=3,4,5,6$, in
the scattering between two partons of momenta $p_{1}$ and $p_{2}$, as depicted
in figure 11. By convention, we consider the scattering in the unphysical
region where all momenta are taken as outgoing, and then we analytically
continue to the physical region where $p_{1}^{0}<0$ and $p_{2}^{0}<0$. Thus
partons are incoming or outgoing depending on the sign of their energy. Since
the helicity of a positive-energy (negative-energy) massless spinor has the
same (opposite) sign as its chirality, the helicities assigned to the partons
depend on whether they are incoming or outgoing. We label outgoing (positive-
energy) particles with their helicity; if they are incoming the actual
helicity and charge are reversed, e.g. an incoming left-handed parton is
labelled as an outgoing right-handed anti-parton.

Figure 11: Depiction of the production of four gluons of momenta $p_{i}$,
$i=3,4,5,6$, in the scattering between two partons of momenta $p_{1}$ and
$p_{2}$. We take all momenta to be outgoing.

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

$2p\cdot
q=p^{+}q^{-}+p^{-}q^{+}-p_{\perp}q^{*}_{\perp}-p^{*}_{\perp}q_{\perp}\,,$
(162)

the four-momenta are,

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{p_{1}^{-}}{2},0,0,\frac{-p_{1}^{-}}{2}\right)\equiv\left(0,p_{1}^{-};0,0\right)\,,$
$\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle\left(\frac{p_{2}^{+}}{2},0,0,\frac{p_{2}^{+}}{2}\right)\equiv\left(p_{2}^{+},0;0,0\right)\,,$
(163) $\displaystyle p_{i}$ $\displaystyle=$
$\displaystyle\left(\frac{p_{i}^{+}+p_{i}^{-}}{2},{\rm Re}[p_{i\perp}],{\rm
Im}[p_{i\perp}],\frac{p_{i}^{+}-p_{i}^{-}}{2}\right)\equiv|p_{i\perp}|\left(e^{y_{i}},e^{-y_{i}};\cos{\phi_{i}},\sin{\phi_{i}}\right)\,\,,$

where $y$ is the rapidity, and $3\leq i\leq 6$. The first notation in eq.
(163) is the standard representation $p^{\mu}=(p^{0},p^{x},p^{y},p^{z})$,
while in the second we have the $+$ and $-$ light-cone components on the left
of the semicolon, and the transverse components on the right.

From momentum conservation,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\sum_{i=3}^{6}p_{i\perp}\,,$
$\displaystyle p_{2}^{+}$ $\displaystyle=$
$\displaystyle-\sum_{i=3}^{6}p_{i}^{+}\,,$ (164) $\displaystyle p_{1}^{-}$
$\displaystyle=$ $\displaystyle-\sum_{i=3}^{6}p_{i}^{-}\,,$

and using the scalar product eq. (162), the Mandelstam invariants may be
written as,

$\displaystyle s$ $\displaystyle=$ $\displaystyle 2p_{1}\cdot
p_{2}=\sum_{i,j=3}^{6}p_{i}^{+}p_{j}^{-}$ $\displaystyle s_{2i}$
$\displaystyle=$ $\displaystyle 2p_{2}\cdot
p_{i}=-\sum_{j=3}^{6}p_{i}^{-}p_{j}^{+}$ (165) $\displaystyle s_{1i}$
$\displaystyle=$ $\displaystyle 2p_{1}\cdot
p_{i}=-\sum_{j=3}^{6}p_{i}^{+}p_{j}^{-}$ $\displaystyle s_{ik}$
$\displaystyle=$ $\displaystyle 2p_{i}\cdot
p_{k}=p_{i}^{+}p_{k}^{-}+p_{i}^{-}p_{k}^{+}-p_{i\perp}p^{\ast}_{k\perp}-p^{\ast}_{i\perp}p_{k\perp}\,,$

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

For the momenta in eq. (163), we find the following right-handed spinor
products, $\langle k\,p\rangle=\overline{u}_{-}(k)u_{+}(p)$, in the notation
of Ref. DelDuca:1999iql ,

$\displaystyle\langle p_{i}\,p_{j}\rangle$ $\displaystyle=$ $\displaystyle
p_{i\perp}\sqrt{\frac{p_{j}^{+}}{p_{i}^{+}}}-p_{j\perp}\sqrt{\frac{p_{i}^{+}}{p_{j}^{+}}}\,,$
$\displaystyle\langle p_{2}\,p_{i}\rangle$ $\displaystyle=$
$\displaystyle-i\sqrt{\frac{-p_{2}^{+}}{p_{i}^{+}}}\,p_{i\perp}\,,$ (166)
$\displaystyle\langle p_{i}\,p_{1}\rangle$ $\displaystyle=$ $\displaystyle
i\sqrt{-p_{1}^{-}p_{i}^{+}}\,,$ $\displaystyle\langle p_{2}\,p_{1}\rangle$
$\displaystyle=$ $\displaystyle-\sqrt{p_{2}^{+}p_{1}^{-}}\,,$

where we have used the mass-shell condition,

$|p_{i\perp}|^{2}=p_{i}^{+}p_{i}^{-}\,.$ (167)

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

$[k\,p]={\rm sign}(k^{0}p^{0})\langle p\,k\rangle^{\ast}\,.$ (168)

Spinor products are antisymmetric,

$\langle k\,p\rangle=-\langle p\,k\rangle\,,\qquad[k\,p]=-[p\,k]\,.$ (169)

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

$[p|\gamma^{\mu}|k\rangle=\langle k|\gamma^{\mu}|p]\,,$ (170)

and complex conjugation,

$[p|\gamma^{\mu}|k\rangle^{*}={\rm
sign}(k^{0}p^{0})[k|\gamma^{\mu}|p\rangle\,.$ (171)

Through the Fierz rearrangement,

$\langle k|\gamma^{\mu}|p]\langle v|\gamma^{\mu}|q]=2\langle
k\,v\rangle[q\,p]\,,$ (172)

and the Gordon identity,

$[p|\gamma^{\mu}|p\rangle=\langle p|\gamma^{\mu}|p]=2p^{\mu}\,,$ (173)

we obtain that

$\displaystyle\langle k|\not{q}|p]=\langle k\,q\rangle[q\,p]\,,$
$\displaystyle[p|\not{q}|k\rangle=[p\,q]\langle q\,k\rangle\,.$ (174)

###  A.2 Multi-Regge kinematics

In multi-Regge kinematics, we require that the light-cone momenta of the
gluons are strongly ordered and have comparable transverse momentum. In the
six-gluon case, we take

$\displaystyle\begin{split}&p^{+}_{3}\gg p^{+}_{4}\gg p^{+}_{5}\gg
p^{+}_{6}\,,\quad\qquad p^{-}_{3}\ll p^{-}_{4}\ll p^{-}_{5}\ll p^{-}_{6}\,,\\\
&|p_{3\perp}|\simeq|p_{4\perp}|\simeq|p_{5\perp}|\simeq|p_{6\perp}|\,.\end{split}$
(175)

The strong ordering of light-cone momenta in eq. (175) is equivalent to
requiring a strong-ordering on the rapidities,

$y_{3}\gg y_{4}\gg y_{5}\gg y_{6}\,.$ (176)

Momentum conservation in eq. (164) then becomes

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\sum_{i=3}^{6}p_{i\perp}\,,$
$\displaystyle p_{2}^{+}$ $\displaystyle\simeq$ $\displaystyle-p_{3}^{+}\,,$
(177) $\displaystyle p_{1}^{-}$ $\displaystyle\simeq$ $\displaystyle-
p_{6}^{-}\,.$

To leading accuracy, the Mandelstam invariants in eq. (165) are reduced to

$\displaystyle s$ $\displaystyle=$ $\displaystyle 2p_{1}\cdot p_{2}\simeq
p_{3}^{+}p_{6}^{-}\,,$ $\displaystyle s_{2i}$ $\displaystyle=$ $\displaystyle
2p_{2}\cdot p_{i}\simeq-p_{3}^{+}p_{i}^{-}\,,$ (178) $\displaystyle s_{1i}$
$\displaystyle=$ $\displaystyle 2p_{1}\cdot p_{i}\simeq-p_{i}^{+}p_{6}^{-}\,,$
$\displaystyle s_{ij}$ $\displaystyle=$ $\displaystyle 2p_{i}\cdot p_{j}\simeq
p_{i}^{+}p_{j}^{-}\,,$

with $3\leq i,j\leq 6$, and with $p^{+}_{i}>p^{+}_{j}$, for $i<j$. Note that
the mass-shell condition in eq. (167) implies that

$s\,|p_{4\perp}|^{2}|p_{5\perp}|^{2}\simeq s_{34}s_{45}s_{56}\,,$ (179)

which is an example of the general multi-Regge constraint,

$s_{ij}\,\prod_{k=i+1}^{j-1}|p_{k\perp}|^{2}=\prod_{k=i}^{j-1}s_{k,k+1}\,.$
(180)

The spinor products in eq. (166) become,

$\displaystyle\langle p_{i}\,p_{j}\rangle$ $\displaystyle\simeq$
$\displaystyle-\sqrt{\frac{p_{i}^{+}}{p_{j}^{+}}}\,p_{j\perp}\,\qquad{\rm
for}\,y_{i}>y_{j}\;,$ $\displaystyle\langle p_{2}\,p_{i}\rangle$
$\displaystyle\simeq$
$\displaystyle-i\sqrt{\frac{p_{3}^{+}}{p_{i}^{+}}}\,p_{i\perp}\,,$ (181)
$\displaystyle\langle p_{i}\,p_{1}\rangle$ $\displaystyle\simeq$
$\displaystyle i\sqrt{p_{i}^{+}p_{6}^{-}}\,,$ $\displaystyle\langle
p_{2}\,p_{1}\rangle$ $\displaystyle\simeq$
$\displaystyle-\sqrt{p_{3}^{+}p_{6}^{-}}\,,$

with $3\leq i\leq 6$. In eqs. (178) and (181) only the leading terms are
displayed. Here and henceforth, it is understood that, if so required by the
computation, sub-leading terms may need to be retained.

In the Euclidean region, where the Mandelstam invariants are taken as all
negative, the ordering in eq. (175) becomes

$-s_{12}\gg-s_{123},-s_{345}\gg-s_{34},-s_{45},-s_{56}\gg-
s_{23},-s_{61},-s_{234}\,.$ (182)

Introducing a parameter $\lambda\ll 1$, the hierarchy in eq. (176) above is
equivalent to the rescaling,

$\\{s_{123},s_{345}\\}={\cal
O}(\lambda)\,,\qquad\\{s_{34},s_{45},s_{56}\\}={\cal
O}(\lambda^{2})\,,\qquad\\{s_{23},s_{61},s_{234}\\}={\cal O}(\lambda^{3})\,.$
(183)

###  A.3 Central next-to-multi-Regge kinematics

We now consider the production of four partons of momenta $p_{i}$, with $3\leq
i\leq 6$, with the new constraint that partons 4 and 5 are in the central
region along the gluon ladder,

$\displaystyle\begin{split}&p^{+}_{3}\gg p^{+}_{4}\simeq p^{+}_{5}\gg
p^{+}_{6}\,,\quad\qquad p^{-}_{3}\ll p^{-}_{4}\simeq p^{-}_{5}\ll
p^{-}_{6}\,\,,\\\
&|p_{3\perp}|\simeq|p_{4\perp}|\simeq|p_{5\perp}|\simeq|p_{6\perp}|\,,\end{split}$
(184)

and hence,

$y_{3}\gg y_{4}\simeq y_{5}\gg y_{6}.$ (185)

The leading contributions to momentum conservation are the same as in eq.
(177). The Mandelstam invariants of eq. (165) become

$\displaystyle s$ $\displaystyle\simeq$ $\displaystyle p_{3}^{+}p_{6}^{-}\,,$
$\displaystyle s_{2i}$ $\displaystyle\simeq$ $\displaystyle-
p_{3}^{+}p_{i}^{-}\,,$ $\displaystyle s_{1i}$ $\displaystyle\simeq$
$\displaystyle-p_{i}^{+}p_{6}^{-}\,,$ (186) $\displaystyle s_{jk}$
$\displaystyle\simeq$ $\displaystyle p_{j}^{+}p_{k}^{-}\,,\qquad j\neq
4\,\,\,{\rm and}\,\,\,k\neq 5\,,$ $\displaystyle s_{45}$ $\displaystyle=$
$\displaystyle
p_{4}^{+}p_{5}^{-}+p_{4}^{-}p_{5}^{+}-p_{4\perp}^{\ast}p_{5\perp}-p_{5\perp}^{\ast}p_{4\perp}\,,$

with $3\leq i,j,k\leq 6$, with $p^{+}_{j}>p^{+}_{k}$ for $j<k$. Note that the
mass-shell condition in eq. (167) and the Mandelstam invariants in eq. (186)
imply that

$\displaystyle\begin{split}s&\simeq\frac{s_{34}\,s_{56}}{p_{4}^{-}p_{5}^{+}}\\\
&=\frac{s_{34}\,s_{56}\,p_{4}^{+}p_{5}^{-}}{|p_{4\perp}|^{2}|p_{5\perp}|^{2}}\,.\\\
\end{split}$ (187)

All three-particle Mandelstam invariants either simplify to a two-particle
Mandelstam invariant, or are equal to one of the following, up to an overall
sign,

$\displaystyle\begin{split}s_{123}&\simeq(p_{4}^{+}+p_{5}^{+})p_{6}^{-}\,,\\\
s_{234}&\simeq-|p_{3\perp}+p_{4\perp}|^{2}-p_{4}^{-}p_{5}^{+}\,,\\\
s_{235}&\simeq-|p_{3\perp}+p_{5\perp}|^{2}-p_{4}^{+}p_{5}^{-}\,,\\\
s_{345}&\simeq p_{3}^{+}(p_{4}^{-}+p_{5}^{-})\,.\end{split}$ (188)

In the Euclidean region, where the Mandelstam invariants are taken as all
negative, the ordering in eq. (184) becomes

$-s_{12}\gg-s_{34},-s_{56},-s_{123},-s_{345}\gg-
s_{23},-s_{45},-s_{61},-s_{234},-s_{235}\,.$ (189)

Introducing a parameter $\lambda\ll 1$, the hierarchy in eq. (184) above is
equivalent to the rescaling,

$\\{s_{34},s_{56},s_{123},s_{345}\\}={\cal
O}(\lambda)\,,\qquad\\{s_{23},s_{45},s_{61},s_{234},s_{235}\\}={\cal
O}(\lambda^{2})\,.$ (190)

The spinor products in eq. (166) become

$\displaystyle\langle p_{2}\,p_{1}\rangle$ $\displaystyle\simeq$
$\displaystyle-\sqrt{p_{3}^{+}p_{6}^{-}}\,,$ $\displaystyle\langle
p_{2}\,p_{k}\rangle$ $\displaystyle=$
$\displaystyle-i\sqrt{\frac{-p_{2}^{+}}{p_{k}^{+}}}\,p_{k\perp}\simeq-i\sqrt{\frac{p_{3}^{+}}{p_{k}^{+}}}p_{k\perp}\,,$
$\displaystyle\langle p_{k}\,p_{1}\rangle$ $\displaystyle=$ $\displaystyle
i\sqrt{-p_{1}^{-}p_{k}^{+}}\,\simeq i\sqrt{p_{k}^{+}p_{6}^{-}}\,,$ (191)
$\displaystyle\langle p_{j}\,p_{k}\rangle$ $\displaystyle=$ $\displaystyle
p_{j\perp}\sqrt{\frac{p_{k}^{+}}{p_{j}^{+}}}-p_{k\perp}\sqrt{\frac{p_{j}^{+}}{p_{k}^{+}}}\simeq-
p_{k\perp}\,\sqrt{\frac{p_{j}^{+}}{p_{k}^{+}}}\,,\qquad j\neq 4\,\,\,{\rm
and}\,\,\,k\neq 5$ $\displaystyle\langle p_{4}\,p_{5}\rangle$ $\displaystyle=$
$\displaystyle
p_{4\perp}\sqrt{\frac{p_{5}^{+}}{p_{4}^{+}}}-p_{5\perp}\sqrt{\frac{p_{4}^{+}}{p_{5}^{+}}}\,,$

with $3\leq j,k\leq 6$, with $p^{+}_{j}>p^{+}_{k}$.

The following spinor strings occur in the BCFW representation of NMHV tree-
level amplitudes. We list the spinor strings for two representative colour
orderings. For the colour ordering $\\{1,2,3,4,5,6\\}$ the relevant quantities
are

$\displaystyle\begin{split}\langle
p_{1}|\not{p}_{2}+\not{p}_{3}|p_{4}]&\simeq-i|q_{3\perp}|\sqrt{\frac{p_{4}^{+}}{p_{6}^{+}}}(q_{2\perp}^{*}+p_{4\perp}^{*}\frac{p_{5}^{+}}{p_{4}^{+}})\,,\\\
\langle p_{3}|\not{p}_{4}+\not{p}_{5}|p_{6}]&\simeq-
q_{3\perp}^{*}\sqrt{\frac{p_{3}^{+}}{p_{6}^{+}}}\left(p_{4\perp}+p_{5\perp}\right)\,,\\\
\langle p_{5}|\not{p}_{6}+\not{p}_{1}|p_{2}]&\simeq-
ip_{5\perp}\sqrt{\frac{p_{3}^{+}}{p_{5}^{+}}}(q_{2\perp}^{*}+p_{5\perp}^{*}\frac{p_{4}^{-}}{p_{5}^{-}})\,,\end{split}$
(192)

while for the colour ordering $\\{1,4,2,3,5,6\\}$ the relevant quantities are

$\displaystyle\begin{split}\langle
p_{1}|\not{p}_{4}+\not{p}_{2}|p_{3}]&\simeq-i|q_{3\perp}|\sqrt{\frac{p_{3}^{+}}{p_{6}^{+}}}(p_{5\perp}^{*}+q_{3\perp}^{*})\,,\\\
\langle p_{2}|\not{p}_{3}+\not{p}_{5}|p_{6}]&\simeq
iq_{3\perp}^{*}\sqrt{\frac{p_{3}^{+}}{p_{6}^{+}}}(p_{4\perp}+q_{3\perp})\,,\\\
\langle p_{5}|\not{p}_{6}+\not{p}_{1}|p_{4}]&\simeq-
p_{5\perp}q_{2\perp}^{*}\sqrt{\frac{p_{4}^{+}}{p_{5}^{+}}}-p_{4\perp}^{*}q^{\prime}_{2\perp}\sqrt{\frac{p_{5}^{+}}{p_{4}^{+}}}\,.\\\
\end{split}$ (193)

##  Appendix B Lorentz invariant quantities in minimal coordinates

In this section we collect useful expressions for Lorentz-invariant
quantities, in the central NMRK region of appendix A.3, written in terms of
the minimal set of coordinates, $\\{w,z,X\\}$, introduced in eqs. (33) and
(34). We use the shorthand notation for spinors,

$\langle i\,j\rangle=\langle
p_{i}\,p_{j}\rangle,\qquad[i\,j]=[p_{i}\,p_{j}],\qquad\langle i|k+l|j]=\langle
p_{i}|\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p_{l}+\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p_{k}|p_{j}].$ (194)

Of particular importance to the central NMRK limit is the spinor product,

$\displaystyle\langle 4\,5\rangle=-q_{1\perp}\,\frac{w+Xz}{w\sqrt{X}(z-1)}\,,$
(195)

which is treated exactly in this limit. The three-particle invariants of eq.
(188) become

$\displaystyle\begin{split}s_{123}&=|q_{1\perp}|^{2}\frac{p_{4}^{+}}{p_{6}^{+}}\frac{(1+X)|w-1|^{2}|z|^{2}}{X|z-1|^{2}|w|^{2}}\,,\\\
s_{234}&=-|q_{1\perp}|^{2}\,\frac{(1+X|z|^{2})}{X|z-1|^{2}}\,,\\\
s_{235}&=-|q_{1\perp}|^{2}\frac{(|w+z-wz|^{2}+X|z|^{2})}{|z-1|^{2}|w|^{2}}\,,\\\
s_{345}&=|q_{1\perp}|^{2}\frac{p_{3}^{+}}{p_{4}^{+}}\frac{(|w|^{2}+X|z|^{2})}{|z-1|^{2}|w|^{2}}\,.\\\
\end{split}$ (196)

The lightcone-momentum dependence cancels when considering physical quantities
such as amplitudes. The spinor strings of eq. (192) become

$\displaystyle\begin{split}\langle
1|2+3|4]&=-iq_{1\perp}^{*}|q_{1\perp}|\sqrt{\frac{p_{4}^{+}}{p_{6}^{+}}}\frac{|w-1||z|}{|z-1||w|}\frac{(1+X\bar{z})}{X(\bar{z}-1)}\,,\\\
\langle
3|4+5|6]&=|q_{1\perp}|^{2}\sqrt{\frac{p_{3}^{+}}{p_{6}^{+}}}\,\frac{\bar{z}(\bar{w}-1)(w-z)}{|w|^{2}|z-1|^{2}}\,,\\\
\langle
5|6+1|2]&=-i|q_{1\perp}|^{2}\sqrt{\frac{p_{3}^{+}}{p_{5}^{+}}}\frac{(w+X|z|^{2})}{wX|z-1|^{2}}\,,\end{split}$
(197)

while the spinor strings of eq. (193) become

$\displaystyle\begin{split}\langle
1|4+2|3]&=-iq_{1\perp}^{*}|q_{1\perp}|\sqrt{\frac{p_{3}^{+}}{p_{6}^{+}}}\frac{|w-1||z|}{|z-1||w|}\frac{\bar{z}}{(\bar{z}-1)}\,,\\\
\langle
2|3+5|6]&=-i|q_{1\perp}|^{2}\sqrt{\frac{p_{3}^{+}}{p_{6}^{+}}}\frac{\bar{z}(\bar{w}-1)(w+z-wz)}{|w|^{2}|z-1|^{2}}\,,\\\
\langle
5|6+1|4]&=-|q_{1\perp}|^{2}\frac{(w+z-wz+X|z|^{2})}{w\sqrt{X}|z-1|^{2}}\,.\end{split}$
(198)

The spurious surfaces listed in eqs. (44) and (62) can be recognised from eqs.
(197) and (198) respectively.

###  B.1 Cross ratios in the NMRK limit

In general kinematics, the finite, dual conformally invariant part of the six-
gluon amplitude in planar ${\cal N}=4$ SYM can be written in terms of multiple
polylogarithms with the following 9-letter alphabet,

$\\{u,v,w,1-u,1-v,1-w,y_{u},y_{v},y_{w}\\}.$ (199)

The cross ratios $u$, $v$ and $w$ for the canonical ordering are given in eq.
(42), although we appended a further $A$ subscript to distinguish this from
the other colour orderings that are relevant away from the planar limit. The
remaining letters can be constructed from the cross ratios. For
$x\in\\{u,v,w\\}$, $y_{x}$ is defined by

$\displaystyle\begin{split}y_{x}\equiv\frac{x-z_{+}}{x-z_{-}}\,,\end{split}$
(200)

with

$z_{\pm}\equiv\frac{1}{2}(u+v+w-1\pm\sqrt{\Delta}),\quad\qquad\Delta\equiv(1-u-v-w)^{2}-4uvw.$
(201)

In this section we investigate these quantities for the two representative
colour orderings used throughout this paper, namely $\sigma_{A}$ and
$\sigma_{B}$ of eq. (81). The corresponding quantities for
$\sigma_{A^{\prime}}$ can be obtained using the relations given in eq. (50).

### Permutation $\sigma_{A}$

The cross ratios for this colour ordering, $u_{A}$, $v_{A}$ and $w_{A}$, are
given in eq. (42). In NMRK they lie in the range $[0,1]$. In the central NMRK
region we list the other six letters,

$\displaystyle\begin{split}1-u_{A}&=\frac{X|w-z|^{2}}{(1+X)(|w|^{2}+X|z|^{2})}\,,\\\
1-v_{A}&=\frac{|1+Xz|^{2}}{(1+X|z|^{2})(1+X)}\,,\\\
1-w_{A}&=\frac{|w+X|z|^{2}|^{2}}{(|w|^{2}+X|z|^{2})(1+X|z|^{2})}\,,\end{split}$
(202)
$\displaystyle\begin{split}y_{u_{A}}&=\frac{(w+Xz)(1+X\bar{z})(\bar{w}+X|z|^{2})}{(\bar{w}+X\bar{z})(1+Xz)(w+X|z|^{2})}\,,\\\
y_{v_{A}}&=\frac{(\bar{w}-\bar{z})(z-1)(w+X|z|^{2})}{(w-z)(\bar{z}-1)(\bar{w}+X|z|^{2})}\,,\\\
y_{w_{A}}&=\frac{\bar{z}(\bar{w}-1)(w-z)(1+Xz)}{z(w-1)(\bar{w}-\bar{z})(1+X\bar{z})}\,.\end{split}$
(203)

We note that eq. (203) are complex phases, which results from the quantity,

$\sqrt{\Delta_{A}}=\pm\frac{X\Big{(}(|w|^{2}-X|z|^{2})(z-\bar{z})-|z|^{2}(1-X)(w-\bar{w})+(1-X|z|^{2})(w\bar{z}-\bar{w}z)\Big{)}}{(1+X)(1+X|z|^{2})(|w|^{2}+X|z|^{2})}\,,$
(204)

being purely imaginary. Thus its square is negative, $\Delta_{A}<0$. One can
get from any $2\to 4$ scattering kinematics to NMRK by a dual conformal
transformation DelDuca:2009au ; DelDuca:2010zg . Since $\Delta_{A}$ is dual
conformally invariant, this means that $\Delta_{A}<0$ for any nonsingular
$2\to 4$ scattering configuration. We further note that the minimal variables,
introduced in section 3.1, rationalise $\sqrt{\Delta_{A}}$.

### Permutation $\sigma_{B}$

The cross ratios for this colour ordering, $u_{B}$, $v_{B}$ and $w_{B}$, are
given in eq. (56), which in NMRK lie in the range $[0,1]$. In the central NMRK
region we list the other six letters,

$\displaystyle\begin{split}1-u_{B}&=\frac{|w+z-wz|^{2}}{(|w+z-wz|^{2}+X|z|^{2})}\,,\\\
1-v_{B}&=\frac{X|z|^{2}}{(1+X|z|^{2})}\,,\\\
1-w_{B}&=\frac{|w+z-wz+X|z|^{2}|^{2}}{(1+X|z|^{2})(|w+z-wz|^{2}+X|z|^{2})}\,,\end{split}$
(205)
$\displaystyle\begin{split}y_{u_{B}}&=\frac{w+z-wz+X|z|^{2}}{\bar{w}+\bar{z}-\bar{w}\bar{z}+X|z|^{2}}\,,\\\
y_{v_{B}}&=\frac{(w+z-wz)(\bar{w}+\bar{z}-\bar{w}\bar{z}+X|z|^{2})}{(\bar{w}+\bar{z}-\bar{w}\bar{z})(w+z-wz+X|z|^{2})}\,,\\\
y_{w_{B}}&=\frac{(w-1)(z-1)(\bar{w}+\bar{z}-\bar{w}\bar{z})}{(\bar{w}-1)(\bar{z}-1)(w+z-wz)}\,.\end{split}$
(206)

Again we find that eq. (206) are phases which is a result of

$\sqrt{\Delta_{B}}=\pm\frac{X|z|^{2}\Big{(}(\bar{w}+\bar{z}-\bar{w}\bar{z})-(w+z-wz)\Big{)}}{(1+X|z|^{2})(|w+z-wz|^{2}+X|z|^{2})}\,,$
(207)

being purely imaginary. We note that as in eq. (204), the minimal variables
rationalise this quantity.

##  Appendix C The power-suppressed helicity configurations

The tree-level six gluon amplitude (25) describes a total of 50 helicity
configurations. However, the helicity-flip impact factors
$C^{g(0)}(p_{2}^{\pm},p_{3}^{\pm})$ and $C^{g(0)}(p_{1}^{\pm},p_{6}^{\pm})$
are power suppressed in $t_{1}/s$ and $t_{3}/s$, respectively. Of the 20 NMHV
configurations described by eq. (25), 12 are power suppressed in this manner,
owing to one or two helicity-flip impact factors, as depicted in figure 12.
Similarly, of the 30 MHV configurations described by eq. (25), 22 are power
suppressed due to one or two helicity-flip impact factors and are depicted in
figure 13.

Figure 12: Six of the NMHV helicity configurations that are power suppressed
in the central NMRK limit. The remaining six power-suppressed NMHV
configurations can be obtained by flipping all helicities. The labelling of
the gluons follows figure 11.

The remaining 16 helicity configurations of eq. (25) are all associated to
amplitudes which do not have a helicity-flip impact factor, and these are the
only configurations that we consider in this paper.

Figure 13: Eleven of the MHV helicity configurations that are power suppressed
in the central NMRK limit. The remaining eleven power suppressed MHV
configurations can be obtained by flipping all helicities. The labelling of
the gluons follows figure 11.

##  Appendix D NMHV amplitudes in general kinematics

In this appendix we list the tree-level and one-loop six-gluon amplitudes in
${\cal N}=4$ SYM which are used in the main text to obtain the two-gluon
opposite-helicity CEV. For these amplitudes we use the shorthand notation for
spinors listed in eq. (194).

###  D.1 Tree-level six-gluon NMHV amplitudes

Here we list a representation of the tree-level six-gluon NMHV colour-ordered
amplitudes obtained through the BCFW on-shell recursion relations
Britto:2004ap ; Britto:2005fq ,

$\displaystyle\begin{split}M^{(0)}\left(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\oplus},p_{6}^{\oplus}\right)=&\
\frac{\langle 1|2+3|4]^{4}}{D_{2}^{*}}+\frac{\langle
3|4+5|6]^{4}}{D_{3}},\end{split}$ (208)
$\displaystyle\begin{split}M^{(0)}\left(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\ominus},p_{5}^{\oplus},p_{6}^{\oplus}\right)=&\
\frac{\langle 4|1+2|3]^{4}}{D_{1}^{*}}+\frac{\langle
2\,4\rangle^{4}[5\,6]^{4}}{D_{2}}+\frac{\langle
1\,2\rangle^{4}[3\,5]^{4}}{D_{3}^{*}},\end{split}$ (209)
$\displaystyle\begin{split}M^{(0)}\left(p_{1}^{\ominus},p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\ominus},p_{6}^{\oplus}\right)=&\
\frac{\langle 1\,3\rangle^{4}[4\,6]^{4}}{D_{1}}+\frac{[2\,4]^{4}\langle
1\,5\rangle^{4}}{D_{2}^{*}}+\frac{[2\,6]^{4}\langle
3\,5\rangle^{4}}{D_{3}}.\end{split}$ (210)

The functions in the denominators are

$D_{1}(1,2,3,4,5,6)=-\langle 1\,2\rangle\langle
2\,3\rangle\left[4\,5\right]\left[5\,6\right]s_{123}\langle 1|2+3|4]\langle
3|4+5|6]\,,$ (211)

and cyclic permutations thereof. We use the shorthand notation

$D_{i}(\sigma)=D_{i}(\sigma_{i},\sigma_{i+1},\sigma_{i+2},\sigma_{i+3},\sigma_{i+4},\sigma_{i+5}),$
(212)

where for the amplitudes in this section it is assumed – and left implicit –
that the ordering $\sigma$ is always taken to be that of the parent $M^{(0)}$,
i.e. $\left\\{1,2,3,4,5,6\right\\}$. $D_{i+1}(\sigma)$ is obtained from
$D_{i}(\sigma)$ by replacing $\sigma_{i}\to\sigma_{i+1}$ in the latter, with
all indices taken modulo 6. However, there are only three independent cyclic
orderings due to the relation $D_{i+3}(\sigma)=D_{i}^{*}(\sigma)$.

We will see that the individual rational terms that occur in eqs. (208–210)
will appear also in the one-loop amplitudes listed in the following
subsection. The coefficient of the $\epsilon^{-2}$ pole of these one-loop
amplitudes must be proportional to the tree-level amplitude, thus, using the
one-loop amplitudes, we obtain an alternative set of representations for the
tree-level amplitudes:

$\displaystyle
M^{(0)}\left(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\oplus},p_{6}^{\oplus}\right)=$
$\displaystyle\ \frac{(s_{123})^{4}}{D_{1}^{*}}+\frac{\langle
2\,3\rangle^{4}[5\,6]^{4}}{D_{2}}+\frac{\langle
1\,2\rangle^{4}[4\,5]^{4}}{D_{3}^{*}},$ (213) $\displaystyle
M^{(0)}\left(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\ominus},p_{5}^{\oplus},p_{6}^{\oplus}\right)=$
$\displaystyle\ \frac{\langle 1\,2\rangle^{4}[5\,6]^{4}}{D_{1}}+\frac{\langle
1|2+4|3]^{4}}{D_{2}^{*}}+\frac{\langle 4|1+2|6]^{4}}{D_{3}},$ (214)
$\displaystyle
M^{(0)}\left(p_{1}^{\ominus},p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\ominus},p_{6}^{\oplus}\right)=$
$\displaystyle\ \frac{\langle 5|1+3|2]^{4}}{D_{1}^{*}}+\frac{\langle
3|2+4|6]^{4}}{D_{2}}+\frac{\langle 1|2+6|4]^{4}}{D_{3}^{*}}.$ (215)

The representations eqs. (208–210) are complementary to the representations
eqs. (213–215) in the sense that where the NMRK limit of the rational terms of
the former give rise directly to $R_{xy}^{I}$ (defined in eqs. (46) and (58)),
the rational terms of the latter give rise to ${R}_{\bar{x}\bar{y}}^{I}$
(defined in eqs. (48) and (60)), and vice versa.

###  D.2 One-loop six-gluon NMHV amplitudes in ${\cal N}=4$ SYM

We will need to study the NMRK limit of the colour-ordered amplitudes derived
in ref. Bern:1994cg , namely:

$\displaystyle\begin{split}\frac{(4\pi)^{2}}{\kappa_{\Gamma}}M^{(1)}_{\mathcal{N}=4}\left(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\oplus},p_{6}^{\oplus}\right)=&\left(\frac{(s_{123})^{4}}{D_{1}^{*}}\right)W_{1}\\\
+&\left(\frac{\langle 1|2+3|4]^{4}}{D_{2}^{*}}+\frac{\langle
2\,3\rangle^{4}[5\,6]^{4}}{D_{2}}\right)W_{2}\\\ +&\left(\frac{\langle
3|4+5|6]^{4}}{D_{3}}+\frac{\langle
1\,2\rangle^{4}[4\,5]^{4}}{D_{3}^{*}}\right)W_{3}\,,\end{split}$ (216)
$\displaystyle\begin{split}\frac{(4\pi)^{2}}{\kappa_{\Gamma}}M^{(1)}_{\mathcal{N}=4}\left(p_{1}^{\ominus},p_{2}^{\ominus},p_{3}^{\oplus},p_{4}^{\ominus},p_{5}^{\oplus},p_{6}^{\oplus}\right)=&\left(\frac{\langle
4|1+2|3]^{4}}{D_{1}^{*}}+\frac{\langle
1\,2\rangle^{4}[5\,6]^{4}}{D_{1}}\right)W_{1}\\\ +&\left(\frac{\langle
1|2+4|3]^{4}}{D_{2}^{*}}+\frac{\langle
2\,4\rangle^{4}[5\,6]^{4}}{D_{2}}\right)W_{2}\\\ +&\left(\frac{\langle
4|1+2|6]^{4}}{D_{3}}+\frac{\langle
1\,2\rangle^{4}[3\,5]^{4}}{D_{3}^{*}}\right)W_{3}\,,\end{split}$ (217)
$\displaystyle\begin{split}\frac{(4\pi)^{2}}{\kappa_{\Gamma}}M^{(1)}_{\mathcal{N}=4}\left(p_{1}^{\ominus},p_{2}^{\oplus},p_{3}^{\ominus},p_{4}^{\oplus},p_{5}^{\ominus},p_{6}^{\oplus}\right)=&\left(\frac{\langle
1\,3\rangle^{4}[4\,6]^{4}}{D_{1}}+\frac{\langle
5|1+3|2]^{4}}{D_{1}^{*}}\right)W_{1}\\\ +&\left(\frac{[2\,4]^{4}\langle
1\,5\rangle^{4}}{D_{2}^{*}}+\frac{\langle 3|2+4|6]^{4}}{D_{2}}\right)W_{2}\\\
+&\left(\frac{[2\,6]^{4}\langle 3\,5\rangle^{4}}{D_{3}}+\frac{\langle
1|2+6|4]^{4}}{D_{3}^{*}}\right)W_{3}\,.\end{split}$ (218)

The transcendental functions are given by

$\displaystyle\begin{split}W_{i}(\sigma)&=-\frac{1}{2\epsilon^{2}}\sum_{j=1}^{6}\left(\frac{\mu^{2}}{-t^{[2]}_{j}}\right)^{\epsilon}-\log\left(\frac{-t^{[3]}_{i}}{-t^{[2]}_{i}}\right)\log\left(\frac{-t^{[3]}_{i}}{-t^{[2]}_{i+1}}\right)-\log\left(\frac{-t^{[3]}_{i}}{-t^{[2]}_{i+3}}\right)\log\left(\frac{-t^{[3]}_{i}}{-t^{[2]}_{i+4}}\right)\\\
&\phantom{\frac{1}{2}}+\log\left(\frac{-t^{[3]}_{i}}{-t^{[2]}_{i+2}}\right)\log\left(\frac{-t^{[3]}_{i}}{-t^{[2]}_{i+5}}\right)+\frac{1}{2}\log\left(\frac{-t^{[2]}_{i}}{-t^{[2]}_{i+3}}\right)\log\left(\frac{-t^{[2]}_{i+1}}{-t^{[2]}_{i+4}}\right)\\\
&+\frac{1}{2}\log\left(\frac{-t^{[2]}_{i+5}}{-t^{[2]}_{i}}\right)\log\left(\frac{-t^{[2]}_{i+1}}{-t^{[2]}_{i+2}}\right)+\frac{1}{2}\log\left(\frac{-t^{[2]}_{i+2}}{-t^{[2]}_{i+3}}\right)\log\left(\frac{-t^{[2]}_{i+4}}{-t^{[2]}_{i+5}}\right)+\frac{\pi^{2}}{3},\end{split}$
(219)

where we use the same notation for the arguments and cyclic permutations as
for the function $D_{i}$, discussed below eq. (211). While in eqs. (216–218)
the ordering argument is left implicit, it is useful to display the ordering
explicitly when the transcendental functions of different colour orderings
appear in the same expression, e.g. in eq. (131). We note that the function
$W_{i}$ has cyclic symmetry in the form $W_{i+3}=W_{i}$ so there are only
three independent cyclic permutations. In eq. (219) we use the compact
notation for Mandelstam invariants introduced in eq. (90), and we use the
prescription of eq. (91). We emphasise that the function $W_{i}$ does not
depend on the individual helicities of the particles.

##  Appendix E Additional forms of the opposite-helicity central-emission
vertex

In ref. Duhr:2009uxa  an alternative form of the opposite-helicity two-gluon
central-emission vertex was found by using the CSW rules Cachazo:2004kj ,

$\displaystyle\begin{split}\left.A^{gg(0)}(q_{1},p_{4}^{\oplus},p_{5}^{\ominus},q_{3})\right|_{4}=&\sqrt{\frac{x_{4}}{x_{5}}}\frac{\left|q_{1\perp}\right|^{2}x_{4}p_{5\perp}^{3}}{\langle
4\,5\rangle\left(x_{5}\left|p_{4\perp}\right|^{2}+x_{4}\left|p_{5\perp}\right|^{2}\right)p_{4\perp}\left(p_{4\perp}+p_{5\perp}\right)}\\\
&+\sqrt{\frac{x_{4}}{x_{5}}}\frac{q_{1\perp}^{*}x_{4}\left(q_{3\perp}+p_{5\perp}\right)^{3}}{\left(q_{3\perp}\sqrt{\frac{x_{4}}{x_{5}}}-\langle
4\,5\rangle\right)\left(x_{5}\left|p_{4\perp}\right|^{2}+x_{4}\left|q_{2\perp}\right|^{2}\right)p_{4\perp}}\\\
&+\frac{q_{1\perp}q_{3\perp}\sqrt{x_{4}x_{5}}x_{5}}{\langle
4\,5\rangle\left(x_{5}p_{4\perp}-x_{4}q_{2\perp}\right)}-\sqrt{\frac{x_{4}}{x_{5}}}\frac{q_{1\perp}^{*}q_{3\perp}x_{4}}{\left(p_{4\perp}+p_{5\perp}\right)[4\,5]}\,.\end{split}$
(220)

In the minimal variables of section 3.1 this is

$\displaystyle\begin{split}\left.A^{gg(0)}(q_{1},p_{4}^{\oplus},p_{5}^{\ominus},q_{3})\right|_{4}=&-\frac{\bar{w}z^{3}|1-z|^{2}X^{2}}{(w-z)(w+Xz)(|w|^{2}+X|z|^{2})}+\frac{X^{2}z^{3}(1-\bar{z})}{(1+Xz)(1+X|z|^{2})}\\\
&+\frac{(1-w)(1-z)zX}{(1+X)(1+Xz)(w+Xz)}+\frac{(1-w)(1-\bar{z})\bar{w}zX^{2}}{(1+X)(w-z)(\bar{w}+X\bar{z})}\,,\end{split}$
(221)

which has the expected physical singularities of eqs. (37) and (38), and also
spurious poles at $P_{u_{A}}=0$ and $\bar{P}_{v_{A}}=0$.

We also list a compact representation we have found

$\displaystyle\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{5}$
$\displaystyle=$
$\displaystyle\frac{Xz(\bar{w}-1)(Xz+w+z-1)}{(X+1)(w+Xz)}-\frac{X\bar{w}(w-1)(\bar{w}+X)z}{(\bar{w}+X\bar{z})(w-z)(X+1)}$
(222)
$\displaystyle+\frac{X\bar{w}(z-1)z^{2}(Xz+|w|^{2})}{(|w|^{2}+X|z|^{2})(w-z)(w+Xz)}+\frac{Xz^{2}}{X|z|^{2}+1}\,,$

which has only a single spurious pole at $P_{u_{A}}=0$. We finally list a
representation which is compact when expressed in terms of light-cone
coordinates, which was obtained by partial fractioning the third term of eq.
(47),

$\displaystyle\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{6}$
(223)
$\displaystyle\qquad=x_{5}-y_{5}\,\frac{|q_{1\perp}|^{2}}{p_{4\perp}p_{5\perp}^{*}}-\frac{q_{1\perp}^{*}(q_{1\perp}-p_{4\perp})^{2}}{p_{4\perp}s_{234}}-\frac{x_{5}p_{4\perp}^{*}(q_{1\perp}-p_{4\perp})}{x_{4}s_{234}}$
$\displaystyle\qquad\quad+\frac{q_{1\perp}^{*}}{p_{4\perp}\langle
4\,5\rangle^{*}}x_{4}\sqrt{\frac{x_{4}}{x_{5}}}(y_{4}q_{1\perp}-p_{4\perp})+\frac{q_{1\perp}}{p_{5\perp}^{*}\langle
4\,5\rangle}x_{5}\sqrt{\frac{x_{5}}{x_{4}}}(y_{5}q_{1\perp}^{*}-p_{5\perp}^{*})\,.$

###  E.1 BCFW representation for the ordering $\sigma_{A^{\prime}}$

In this appendix we present the BCFW representations of the opposite-helicity
vertex for the ordering $\sigma_{A^{\prime}}$. These expressions can be
obtained from the results for the ordering $\sigma_{A}$ in section 3.3 by
applying the rules listed in eq. (50).

For the $\sigma_{A^{\prime}}=\\{1,2,3,5,4,6\\}$ ordering we list the NMRK
limits of the cross ratios,

$\begin{array}[]{ccccc}u_{A^{\prime}}&=&\\!\\!\\!\\!u_{A},\\\
v_{A^{\prime}}&=&{\displaystyle\frac{s_{23}s_{46}}{s_{235}s_{123}}}&\xrightarrow[\mathrm{NMRK}]{}&{\displaystyle\frac{|w|^{2}|z-1|^{2}X}{(1+X)(|w+z-wz|^{2}+X|z|^{2})}}\,,\\\
w_{A^{\prime}}&=&{\displaystyle\frac{s_{35}s_{61}}{s_{345}s_{235}}}&\xrightarrow[\mathrm{NMRK}]{}&{\displaystyle\frac{|z|^{4}|w-1|^{2}X}{(|w|^{2}+X|z|^{2})(|w+z-wz|^{2}+X|z|^{2})}}\,.\end{array}$
(224)

From the BCFW representations of the NMHV amplitudes of eqs. (208, 210, 214)
we obtain the opposite-helicity central-emission vertex,

$\displaystyle\begin{split}A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})=R^{A^{\prime}}_{uv}+R^{A^{\prime}}_{vw}+R^{A^{\prime}}_{wu}\,,\end{split}$
(225)

with

$\displaystyle\begin{split}R^{A^{\prime}}_{uv}&=\frac{|w-1|^{2}\bar{w}X^{2}|z|^{2}(\bar{z}-1)}{(\bar{w}+X\bar{z})(X+1)(w-z)(\bar{w}+\bar{z}-\bar{w}\bar{z}+X\bar{z})}\,,\\\
R^{A^{\prime}}_{vw}&=\frac{wz(w-1)(z-1)(\bar{w}+\bar{z}-\bar{w}\bar{z})^{4}}{\bar{z}(|w+z-wz|^{2}+X|z|^{2})(\bar{w}+\bar{z}-\bar{w}\bar{z}+X\bar{z})(w(\bar{w}+\bar{z}-\bar{w}\bar{z})+X|z|^{2})}\,,\\\
R^{A^{\prime}}_{wu}&=-\frac{(\bar{w}-1)|w|^{2}X^{2}|z-1|^{2}z^{2}|z|^{2}}{(w+Xz)(|w|^{2}+X|z|^{2})(w(\bar{w}+\bar{z}-\bar{w}\bar{z})+X|z|^{2})(w-z)}\,.\end{split}$
(226)

Instead starting from the BCFW representations of eqs. (209, 213, 215), we
obtain the representation,

$\displaystyle\begin{split}A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})=R^{A^{\prime}}_{\bar{u}\bar{v}}+R^{A^{\prime}}_{\bar{v}\bar{w}}+R^{A^{\prime}}_{\bar{w}\bar{u}}\,,\end{split}$
(227)

with

$\displaystyle\begin{split}R^{A^{\prime}}_{\bar{u}\bar{v}}&=\frac{|w-1|^{2}w(z-1)|z|^{2}}{(X+1)(\bar{w}-\bar{z})(w+Xz)(w+z-wz+Xz)}\,,\\\
R^{A^{\prime}}_{\bar{v}\bar{w}}&=\frac{\bar{w}(\bar{w}-1)(\bar{z}-1)z^{2}|z|^{2}X^{2}}{(|w+z-wz|^{2}+X|z|^{2})(w+z-wz+Xz)(\bar{w}(w+z-wz)+X|z|^{2})}\,,\\\
R^{A^{\prime}}_{\bar{w}\bar{u}}&=-\frac{(w-1)|w|^{2}\bar{w}^{4}|z-1|^{2}z}{\bar{z}(|w|^{2}+X|z|^{2})(\bar{w}+X\bar{z})(\bar{w}(w+z-wz)+X|z|^{2})(\bar{w}-\bar{z})}\,.\end{split}$
(228)

The physical-singularity surfaces of this ordering are

$\begin{array}[]{ccccccc}\langle
4\,5\rangle=0&\leftrightarrow&p_{4\perp}\sqrt{\displaystyle\frac{p_{4}^{+}}{p_{5}^{+}}}-p_{5\perp}\sqrt{\displaystyle\frac{p_{5}^{+}}{p_{4}^{+}}}=0&\leftrightarrow&w+Xz=0\,,\\\
s_{123}=0&\leftrightarrow&p_{4}^{+}+p_{5}^{+}=0&\leftrightarrow&1+X=0\,,\\\
s_{235}=0&\leftrightarrow&|q_{2\perp}^{\prime}|^{2}+p_{4}^{+}p_{5}^{-}=0&\leftrightarrow&|w+z-wz|^{2}+X|z|^{2}=0\,,\\\
s_{354}=0&\leftrightarrow&p_{4}^{-}+p_{5}^{-}=0&\leftrightarrow&|w|^{2}+X|z|^{2}=0\,,\end{array}$
(229)

while the unphysical-singularity surfaces are

$\begin{array}[]{ccccccc}\langle
1|2+3|5]=0&\leftrightarrow&P^{A^{\prime}}_{v}=0&\leftrightarrow&q^{\prime\ast}_{2\perp}-p_{5\perp}^{*}{\displaystyle\frac{p_{4}^{+}}{p_{5}^{+}}}=0&\leftrightarrow&\bar{w}+\bar{z}-\bar{w}\bar{z}+X\bar{z}=0\,,\\\
\langle
3|5+4|6]=0&\leftrightarrow&P^{A^{\prime}}_{u}=0&\leftrightarrow&p_{4\perp}+p_{5\perp}=0&\leftrightarrow&w-z=0\,,\\\
\langle
4|6+1|2]=0&\leftrightarrow&P^{A^{\prime}}_{w}=0&\leftrightarrow&q^{\prime\ast}_{2\perp}+p_{5\perp}^{*}{\displaystyle\frac{p_{5\perp}}{p_{4\perp}}\frac{p_{4}^{+}}{p_{5}^{+}}}=0&\leftrightarrow&w(\bar{w}+\bar{z}-\bar{w}\bar{z})+X|z|^{2}=0\,.\\\
\end{array}$ (230)

We note that

$\displaystyle\begin{split}R^{A}_{xy}\ \ {\underset{p_{4}\leftrightarrow
p_{5}}{\leftrightarrow}}\ \
\left(R^{A^{\prime}}_{\bar{x}\bar{y}}\right)^{*}\,.\end{split}$ (231)

##  Appendix F Absorptive part of the one-loop six-gluon amplitude in NMRK

In eq. (83) we decomposed the ${\cal N}=4$ SYM one-loop six-gluon amplitude in
NMRK into a basis of symmetric $(+)$ or antisymmetric $(-)$ representations
exchanged in the $t_{1}$ and $t_{3}$ channels. For one-loop amplitudes where
an adjoint representation is circulating in the loop, the $(-,-)$ component is
given by eq. (84). We list here analogous expressions for the remaining
components:

$\displaystyle\begin{split}&\mathcal{M}^{(1)(+,+)}=\frac{g_{\scriptscriptstyle
S}^{6}}{4}\\\
&\times\bigg{\\{}\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}F^{a_{5}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\left(M^{(1)}(\sigma_{A})+M^{(1)}(\overset{\leftrightarrow}{\sigma_{A}})+M^{(1)}(\underset{\leftrightarrow}{\sigma_{A}})+M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})\right)\\\
&+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{5}}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\left(M^{(1)}(\sigma_{A^{\prime}})+M^{(1)}(\overset{\leftrightarrow}{\sigma_{A^{\prime}}})+M^{(1)}(\underset{\leftrightarrow}{\sigma_{A^{\prime}}})+M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A^{\prime}}}})\right)\\\
&+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}F^{a_{5}}\right)\left(M^{(1)}(\sigma_{B})+M^{(1)}(\overset{\leftrightarrow}{\sigma_{B}})+M^{(1)}(\underset{\leftrightarrow}{\sigma_{B}})+M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})\right)\bigg{\\}}.\end{split}$
(232)

$\displaystyle\begin{split}&\mathcal{M}^{(1)(+,-)}=\frac{g_{\scriptscriptstyle
S}^{6}}{4}\\\
&\times\bigg{\\{}\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}F^{a_{5}}\left[F^{a_{6}},F^{a_{1}}\right]\right)\left(M^{(1)}(\sigma_{A})+M^{(1)}(\overset{\leftrightarrow}{\sigma_{A}})-M^{(1)}(\underset{\leftrightarrow}{\sigma_{A}})-M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})\right)\\\
&+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{5}}F^{a_{4}}\left[F^{a_{6}},F^{a_{1}}\right]\right)\left(M^{(1)}(\sigma_{A^{\prime}})+M^{(1)}(\overset{\leftrightarrow}{\sigma_{A^{\prime}}})-M^{(1)}(\underset{\leftrightarrow}{\sigma_{A^{\prime}}})-M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A^{\prime}}}})\right)\\\
&+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}\left[F^{a_{6}},F^{a_{1}}\right]F^{a_{5}}\right)\left(M^{(1)}(\sigma_{B})+M^{(1)}(\overset{\leftrightarrow}{\sigma_{B}})-M^{(1)}(\underset{\leftrightarrow}{\sigma_{B}})-M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})\right)\bigg{\\}}.\end{split}$
(233)
$\displaystyle\begin{split}&\mathcal{M}^{(1)(-,+)}=\frac{g_{\scriptscriptstyle
S}^{6}}{4}\\\
&\times\bigg{\\{}\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{4}}F^{a_{5}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\left(M^{(1)}(\sigma_{A})-M^{(1)}(\overset{\leftrightarrow}{\sigma_{A}})+M^{(1)}(\underset{\leftrightarrow}{\sigma_{A}})-M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})\right)\\\
&+\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{5}}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\left(M^{(1)}(\sigma_{A^{\prime}})-M^{(1)}(\overset{\leftrightarrow}{\sigma_{A^{\prime}}})+M^{(1)}(\underset{\leftrightarrow}{\sigma_{A^{\prime}}})-M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A^{\prime}}}})\right)\\\
&+\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}F^{a_{5}}\right)\left(M^{(1)}(\sigma_{B})-M^{(1)}(\overset{\leftrightarrow}{\sigma_{B}})+M^{(1)}(\underset{\leftrightarrow}{\sigma_{B}})-M^{(1)}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})\right)\bigg{\\}}.\end{split}$
(234)

In section 4 we found that in the central NMRK, the dispersive amplitude
received a contribution from the $(-,-)$ component alone. The same is not true
for the absorptive part of the amplitude, which is the object of study of this
appendix. To this end we list the NMRK limit of the imaginary parts of the
transcendental functions eq. (89) and eq. (219), for the 12 colour orderings
that appear in eqs. (232)–(234). We begin by giving the results for the
$\sigma_{A}$ orderings:

$\displaystyle\begin{split}{\rm Im}\Big{[}V_{6}(\sigma_{A})\Big{]}=2{\rm
Im}\Big{[}W_{2}(\sigma_{A})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{4\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{-}s_{45}p_{5}^{+}t_{3}}{(\mu^{2})^{4}}\right)\,,\\\
{\rm Im}\Big{[}V_{6}(\overset{\leftrightarrow}{\sigma_{A}})\Big{]}=2{\rm
Im}\Big{[}W_{2}(\overset{\leftrightarrow}{\sigma_{A}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{p_{4}^{-}s_{45}p_{5}^{+}t_{3}}{t_{1}(\mu^{2})^{2}}\right)\,,\\\
{\rm Im}\Big{[}V_{6}(\underset{\leftrightarrow}{\sigma_{A}})\Big{]}=2{\rm
Im}\Big{[}W_{2}(\underset{\leftrightarrow}{\sigma_{A}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{-}s_{45}p_{5}^{+}}{t_{3}(\mu^{2})^{2}}\right)\,,\\\
{\rm
Im}\Big{[}V_{6}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{-}p_{5}^{+}t_{3}}{s_{45}(\mu^{2})^{2}}\right)+2\pi\log\left(\frac{u_{A}}{1-u_{A}}\right)\,,\\\
2{\rm
Im}\Big{[}W_{2}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{-}p_{5}^{+}t_{3}}{s_{45}(\mu^{2})^{2}}\right)-2\pi\log\left(\frac{v_{A}w_{A}}{u_{A}}\right)\,.\end{split}$
(235)

We provide the results for $W_{1}$ and $W_{3}$ as differences from those for
$W_{2}$. Interestingly,
$\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}}$ is the
only ordering which leads to differing imaginary parts for different cyclic
permutations of $W_{i}$:

$\begin{array}[]{cclcccl}{\rm
Im}\Big{[}W_{1}(\sigma_{A})-W_{2}(\sigma_{A})\Big{]}&=&0\,,&&{\rm
Im}\Big{[}W_{3}(\sigma_{A})-W_{2}(\sigma_{A})\Big{]}&=&0\,,\\\ {\rm
Im}\Big{[}W_{1}(\overset{\leftrightarrow}{\sigma_{A}})-W_{2}(\overset{\leftrightarrow}{\sigma_{A}})\Big{]}&=&0\,,&&{\rm
Im}\Big{[}W_{3}(\overset{\leftrightarrow}{\sigma_{A}})-W_{2}(\overset{\leftrightarrow}{\sigma_{A}})\Big{]}&=&0\,,\\\
{\rm
Im}\Big{[}W_{1}(\underset{\leftrightarrow}{\sigma_{A}})-W_{2}(\underset{\leftrightarrow}{\sigma_{A}})\Big{]}&=&0\,,&&{\rm
Im}\Big{[}W_{3}(\underset{\leftrightarrow}{\sigma_{A}})-W_{2}(\underset{\leftrightarrow}{\sigma_{A}})\Big{]}&=&0\,,\\\
{\rm
Im}\Big{[}W_{1}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})-W_{2}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})\Big{]}&=&2\pi\log(v_{A})\,,&&{\rm
Im}\Big{[}W_{3}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})-W_{2}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}})\Big{]}&=&2\pi\log(w_{A})\,.\\\
\end{array}$ (236)

We further note that only for the ordering
$\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{A}}}$ does the
analytic continuation of the dilogarithms in eq. (89) to our physical region
produce an imaginary part. For this colour ordering, the cross ratio

$\frac{s_{63}s_{45}}{s_{632}s_{245}},$

which is related to $u_{A}$ by interchange of $p_{1}\leftrightarrow p_{6}$ and
$p_{2}\leftrightarrow p_{3}$, has two positive invariants in the numerator and
two negative invariants in the denominator. The analytic continuation of the
dilogarithm that depends on this cross ratio leads to factor of
$i\pi\log(1-u_{A})$ which is related to the factorisation breaking term
discovered in ref. Bartels:2008ce . The interpretation of this quantity is
different in our paper; ref. Bartels:2008ce  considers the single planar
colour ordering of the amplitude, analytically continued to the physical
region where $s_{12},s_{45}>0$,
$s_{23},s_{34},s_{56},s_{61},s_{234},s_{345}<0$, while we consider a certain
colour-ordered amplitude in the physical region where
$s_{12},s_{34},s_{45},s_{56},s_{345}>0$, $s_{23},s_{61},s_{234}<0$. However,
the consequence is the same, namely that this term is not compatible with
Regge-pole factorisation of the amplitude, which only applies to its
dispersive part. Instead it contributes to a Regge cut.

Expressions for the $\sigma_{A^{\prime}}$ orderings are related by an
interchange of $p_{4}\leftrightarrow p_{5}$.

For the $\sigma_{B}$ orderings we obtain

$\displaystyle\begin{split}{\rm Im}\Big{[}V_{6}(\sigma_{B})\Big{]}=2{\rm
Im}\Big{[}W_{1}(\sigma_{B})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{5}^{+}p_{5}^{-}t_{3}}{p_{4}^{+}p_{4}^{-}(\mu^{2})^{2}}\right)\,,\\\
{\rm
Im}\Big{[}V_{6}(\overset{\leftrightarrow}{\sigma_{B}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{+}p_{5}^{-}t_{3}}{p_{4}^{-}p_{5}^{+}(\mu^{2})^{2}}\right)+2\pi\log\left(\frac{v_{B}}{1-v_{B}}\right)\,,\\\
2{\rm
Im}\Big{[}W_{1}(\overset{\leftrightarrow}{\sigma_{B}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{+}p_{5}^{-}t_{3}}{p_{4}^{-}p_{5}^{+}(\mu^{2})^{2}}\right)+2\pi\log\left(\frac{u_{B}v_{B}}{w_{B}}\right)\,,\\\
{\rm
Im}\Big{[}V_{6}(\underset{\leftrightarrow}{\sigma_{B}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{-}p_{5}^{+}t_{3}}{p_{4}^{+}p_{5}^{-}(\mu^{2})^{2}}\right)+2\pi\log\left(\frac{u_{B}}{1-u_{B}}\right)\,,\\\
2{\rm
Im}\Big{[}W_{1}(\underset{\leftrightarrow}{\sigma_{B}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{-}p_{5}^{+}t_{3}}{p_{4}^{+}p_{5}^{-}(\mu^{2})^{2}}\right)+2\pi\log\left(\frac{u_{B}v_{B}}{w_{B}}\right)\,,\\\
{\rm
Im}\Big{[}V_{6}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})\Big{]}=2{\rm
Im}\Big{[}W_{1}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})\Big{]}\xrightarrow[\mathrm{NMRK}]{}&-\frac{2\pi}{\epsilon}+\pi\log\left(\frac{t_{1}p_{4}^{+}p_{4}^{-}t_{3}}{p_{5}^{+}p_{5}^{-}(\mu^{2})^{2}}\right)\,.\end{split}$
(237)

Again the orderings that lead to imaginary contributions from the dilogarithms
(namely $\overset{\leftrightarrow}{\sigma_{B}}$,
$\underset{\leftrightarrow}{\sigma_{B}}$) are the same orderings for which the
imaginary parts of the distinct cyclic permutations of $W_{i}$ differ:

$\begin{array}[]{cclcccl}{\rm
Im}\Big{[}W_{2}(\sigma_{B})-W_{1}(\sigma_{B})\Big{]}&=&0\,,&&{\rm
Im}\Big{[}W_{3}(\sigma_{B})-W_{1}(\sigma_{B})\Big{]}&=&0\,,\\\ {\rm
Im}\Big{[}W_{2}(\overset{\leftrightarrow}{\sigma_{B}})-W_{1}(\overset{\leftrightarrow}{\sigma_{B}})\Big{]}&=&2\pi\log\left(\frac{w_{B}}{u_{B}}\right)\,,&&{\rm
Im}\Big{[}W_{3}(\overset{\leftrightarrow}{\sigma_{B}})-W_{1}(\overset{\leftrightarrow}{\sigma_{B}})\Big{]}&=&-2\pi\log\left(u_{B}\right)\,,\\\
{\rm
Im}\Big{[}W_{2}(\underset{\leftrightarrow}{\sigma_{B}})-W_{1}(\underset{\leftrightarrow}{\sigma_{B}})\Big{]}&=&-2\pi\log\left(v_{B}\right)\,,&&{\rm
Im}\Big{[}W_{3}(\underset{\leftrightarrow}{\sigma_{B}})-W_{1}(\underset{\leftrightarrow}{\sigma_{B}})\Big{]}&=&2\pi\log\left(\frac{w_{B}}{v_{B}}\right)\,,\\\
{\rm
Im}\Big{[}W_{2}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})-W_{1}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})\Big{]}&=&0\,,&&{\rm
Im}\Big{[}W_{3}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})-W_{1}(\overset{\leftrightarrow}{\underset{\leftrightarrow}{\sigma_{B}}})\Big{]}&=&0\,.\\\
\end{array}$ (238)

### Absorptive part of the same-helicity amplitude

From the above lists of imaginary parts we can now assemble the absorptive
parts of the colour-dressed amplitude for the case where
$\nu_{4}=\nu_{5}=\oplus$:

$\displaystyle\begin{split}{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(-,-)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\oplus})}\right]}&\xrightarrow[\mathrm{NMRK}]{}g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}F^{a_{3}a_{2}c_{1}}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\\\
\times\frac{\pi}{2}\Bigg{\\{}\mbox{tr\,}\left(F^{c_{1}}F^{a_{4}}F^{a_{5}}F^{c_{3}}\right)&A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\left(\frac{-5}{\epsilon}+\log\left(\frac{t_{1}s_{45}(p_{4}^{-}p_{5}^{+})^{2}t_{3}}{(\mu^{2})^{5}}\frac{u_{A}}{1-u_{A}}\right)\right)\\\
+\mbox{tr\,}\left(F^{c_{1}}F^{a_{5}}F^{a_{4}}F^{c_{3}}\right)&A^{gg(0)}(q_{1},p^{\oplus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-5}{\epsilon}+\log\left(\frac{t_{1}s_{45}(p_{4}^{+}p_{5}^{-})^{2}t_{3}}{(\mu^{2})^{5}}\frac{u_{A^{\prime}}}{1-u_{A^{\prime}}}\right)\right)\\\
+\mbox{tr\,}\left(F^{c_{1}}F^{a_{5}}F^{c_{3}}F^{a_{4}}\right)&B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\left(\frac{-4}{\epsilon}+\log\left(\frac{t_{1}^{2}t_{3}^{2}}{(\mu^{2})^{4}}\frac{v_{B}}{1-v_{B}}\frac{u_{B}}{1-u_{B}}\right)\right)\Bigg{\\}}\\\
&\times\frac{1}{t_{3}}F^{a_{6}a_{1}c_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\,,\end{split}$
(239)
$\displaystyle\begin{split}{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(+,-)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\oplus})}\right]}\xrightarrow[\mathrm{NMRK}]{}&\frac{\pi}{2}g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\frac{1}{t_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\\\
\times\Bigg{\\{}\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}F^{a_{5}}\left[F^{a_{6}},F^{a_{1}}\right]\right)&A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{3}s_{45}}{t_{1}\mu^{2}}\frac{1-u_{A}}{u_{A}}\right)\right)\\\
+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{5}}F^{a_{4}}\left[F^{a_{6}},F^{a_{1}}\right]\right)&A^{gg(0)}(q_{1},p^{\oplus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{3}s_{45}}{t_{1}\mu^{2}}\frac{1-u_{A^{\prime}}}{u_{A^{\prime}}}\right)\right)\\\
+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}\left[F^{a_{6}},F^{a_{1}}\right]F^{a_{5}}\right)&B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\log\left(\left(\frac{p_{5}^{-}}{p_{4}^{+}}\right)^{2}\frac{1-u_{B}}{u_{B}}\frac{v_{B}}{1-v_{B}}\right)\Bigg{\\}}\,,\end{split}$
(240)
$\displaystyle\begin{split}{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(-,+)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\oplus})}\right]}\xrightarrow[\mathrm{NMRK}]{}&\frac{\pi}{2}g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\frac{1}{t_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\\\
\times\Bigg{\\{}\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{4}}F^{a_{5}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)&A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}s_{45}}{t_{3}\mu^{2}}\frac{1-u_{A}}{u_{A}}\right)\right)\\\
+\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{5}}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)&A^{gg(0)}(q_{1},p^{\oplus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}s_{45}}{t_{3}\mu^{2}}\frac{1-u_{A^{\prime}}}{u_{A^{\prime}}}\right)\right)\\\
+\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}F^{a_{5}}\right)&B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\log\left(\left(\frac{p_{5}^{+}}{p_{4}^{-}}\right)^{2}\frac{u_{B}}{1-u_{B}}\frac{1-v_{B}}{v_{B}}\right)\Bigg{\\}}\,,\end{split}$
(241)
$\displaystyle\begin{split}{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(+,+)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\oplus})}\right]}\xrightarrow[\mathrm{NMRK}]{}\frac{\pi}{2}&g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\frac{1}{t_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\\\
\times\Bigg{\\{}\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}F^{a_{5}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)&A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}t_{3}}{s_{45}\mu^{2}}\frac{u_{A}}{1-u_{A}}\right)\right)\\\
+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{5}}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)&A^{gg(0)}(q_{1},p^{\oplus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}t_{3}}{s_{45}\mu^{2}}\frac{u_{A^{\prime}}}{1-u_{A^{\prime}}}\right)\right)\\\
+\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}F^{a_{5}}\right)&B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\oplus}_{5},q_{3})\log\left(\frac{1-u_{B}}{u_{B}}\frac{1-v_{B}}{v_{B}}\right)\Bigg{\\}}\,.\end{split}$
(242)

In section 4.3 we show that the absorptive parts listed in eqs. (239–242) do
not contribute to the NLO $gg\to gggg$ squared matrix element. These terms
will only contribute to the squared matrix element at higher orders in the
coupling.

### Absorptive part of the opposite-helicity amplitude

Here we similarly list the absorptive parts of the colour-dressed amplitude
for the case where $\nu_{4}=-\nu_{5}=\oplus$:

$\displaystyle\begin{split}&{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(-,-)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\ominus})}\right]}\xrightarrow[\mathrm{NMRK}]{}g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}F^{a_{3}a_{2}c_{1}}C^{g(0)}\left(p_{2}^{\ominus},p_{3}^{\oplus}\right)\frac{1}{t_{1}}\\\
\times\frac{\pi}{2}\Bigg{\\{}&\mbox{tr\,}\left(F^{c_{1}}F^{a_{4}}F^{a_{5}}F^{c_{3}}\right)\bigg{[}A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\left(\frac{-5}{\epsilon}+\log\left(\frac{t_{1}s_{45}(p_{4}^{-}p_{5}^{+})^{2}t_{3}}{(\mu^{2})^{5}}\frac{u_{A}}{v_{A}w_{A}}\right)\right)\\\
&\hskip 220.00034pt+S^{A}_{uv}\log(v_{A})+S^{A}_{wu}\log(w_{A})\bigg{]}\\\
+&\mbox{tr\,}\left(F^{c_{1}}F^{a_{5}}F^{a_{4}}F^{c_{3}}\right)\bigg{[}A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-5}{\epsilon}+\log\left(\frac{t_{1}s_{45}(p_{4}^{+}p_{5}^{-})^{2}t_{3}}{(\mu^{2})^{5}}\frac{u_{A^{\prime}}}{v_{A^{\prime}}w_{A^{\prime}}}\right)\right)\\\
&\hskip
220.00034pt+S^{A^{\prime}}_{uv}\log(v_{A^{\prime}})+S^{A^{\prime}}_{wu}\log(w_{A^{\prime}})\bigg{]}\\\
+&\mbox{tr\,}\left(F^{c_{1}}F^{a_{5}}F^{c_{3}}F^{a_{4}}\right)\bigg{[}B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\left(\frac{-4}{\epsilon}+\log\left(\frac{t_{1}^{2}t_{3}^{2}}{(\mu^{2})^{4}}\right)\right)\\\
&\hskip
100.00015pt-S^{B}_{uv}\log\left(\frac{w_{B}}{u_{B}v_{B}}\right)\bigg{]}\Bigg{\\}}\times\frac{1}{t_{3}}F^{a_{6}a_{1}c_{3}}C^{g(0)}\left(p_{6}^{\oplus},p_{1}^{\ominus}\right)\,,\end{split}$
(243)
$\displaystyle\begin{split}&{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(+,-)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\ominus})}\right]}\xrightarrow[\mathrm{NMRK}]{}\frac{\pi}{2}g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\frac{1}{t_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\\\
\times\Bigg{\\{}&\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}F^{a_{5}}\left[F^{a_{6}},F^{a_{1}}\right]\right)\\\
&\times\bigg{[}A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{3}s_{45}}{t_{1}\mu^{2}}\frac{v_{A}w_{A}}{u_{A}}\right)\right)-S^{A}_{uv}\log(v_{A})-S^{A}_{wu}\log(w_{A})\bigg{]}\\\
+&\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{5}}F^{a_{4}}\left[F^{a_{6}},F^{a_{1}}\right]\right)\\\
&\times\bigg{[}A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{3}s_{45}}{t_{1}\mu^{2}}\frac{v_{A^{\prime}}w_{A^{\prime}}}{u_{A^{\prime}}}\right)\right)-S^{A^{\prime}}_{uv}\log(v_{A^{\prime}})-S^{A^{\prime}}_{wu}\log(w_{A^{\prime}})\bigg{]}\\\
+&\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}\left[F^{a_{6}},F^{a_{1}}\right]F^{a_{5}}\right)\\\
&\times\bigg{[}2B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\log\left(\frac{p_{5}^{-}}{p_{4}^{+}}\right)+S^{B}_{vw}\log\left(\frac{u_{B}}{v_{B}w_{B}}\right)-S^{B}_{wu}\log\left(\frac{v_{B}}{w_{B}u_{B}}\right)\bigg{]}\Bigg{\\}}\,,\end{split}$
(244)
$\displaystyle\begin{split}&{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(-,+)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\ominus})}\right]}\xrightarrow[\mathrm{NMRK}]{}\frac{\pi}{2}g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\frac{1}{t_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\\\
\times\Bigg{\\{}&\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{4}}F^{a_{5}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\\\
&\times\bigg{[}A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}s_{45}}{t_{3}\mu^{2}}\frac{v_{A}w_{A}}{u_{A}}\right)\right)-S^{A}_{uv}\log(v_{A})-S^{A}_{wu}\log(w_{A})\bigg{]}\\\
+&\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{5}}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\\\
&\times\bigg{[}A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}s_{45}}{t_{3}\mu^{2}}\frac{v_{A^{\prime}}w_{A^{\prime}}}{u_{A^{\prime}}}\right)\right)-S^{A^{\prime}}_{uv}\log(v_{A^{\prime}})-S^{A^{\prime}}_{wu}\log(w_{A^{\prime}})\bigg{]}\\\
+&\mbox{tr\,}\left(\left[F^{a_{2}},F^{a_{3}}\right]F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}F^{a_{5}}\right)\\\
&\times\bigg{[}2B^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\log\left(\frac{p_{5}^{-}}{p_{4}^{+}}\right)-S^{B}_{vw}\log\left(\frac{u_{B}}{v_{B}w_{B}}\right)+S^{B}_{wu}\log\left(\frac{v_{B}}{w_{B}u_{B}}\right)\bigg{]}\Bigg{\\}}\,,\end{split}$
(245)
$\displaystyle\begin{split}&{\mathrm{Absorp}\left[{\mathcal{M}^{(1)(+,+)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\ominus})}\right]}\xrightarrow[\mathrm{NMRK}]{}\frac{\pi}{2}g_{\scriptscriptstyle
S}^{6}\frac{\kappa_{\Gamma}}{(4\pi)^{2}}s_{12}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}\frac{1}{t_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\\\
\times\Bigg{\\{}&\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}F^{a_{5}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\\\
&\times\bigg{[}A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}t_{3}}{s_{45}\mu^{2}}\frac{u_{A}}{v_{A}w_{A}}\right)\right)+S^{A}_{uv}\log(v_{A})+S^{A}_{wu}\log(w_{A})\bigg{]}\\\
+&\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{5}}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}\right)\\\
&\times\bigg{[}A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\left(\frac{-1}{\epsilon}+\log\left(\frac{t_{1}t_{3}}{s_{45}\mu^{2}}\frac{u_{A^{\prime}}}{v_{A^{\prime}}w_{A^{\prime}}}\right)\right)+S^{A^{\prime}}_{uv}\log(v_{A^{\prime}})+S^{A^{\prime}}_{wu}\log(w_{A^{\prime}})\bigg{]}\\\
+&\mbox{tr\,}\left(\left\\{F^{a_{2}},F^{a_{3}}\right\\}F^{a_{4}}\left\\{F^{a_{6}},F^{a_{1}}\right\\}F^{a_{5}}\right)\times\bigg{[}S^{B}_{uv}\log\left(\frac{w_{B}}{u_{B}v_{B}}\right)\bigg{]}\Bigg{\\}}\,.\end{split}$
(246)

In section 4.3 we show that at the level of the NLO $gg\to gggg$ squared
matrix element, eq. (243) is the only absorptive component of the amplitude
which can contribute for a specific helicity configuration. This contribution
vanishes in the sum over helicities $\nu_{4}$ and $\nu_{5}$. All components
eq. (239)–(246) will contribute at higher orders in the coupling.

##  Appendix G Kinematic limits of the two-gluon central-emission vertex

Starting from the central NMRK region, it is possible to take further
kinematic limits to regions which are contained within NMRK. In these more
restricted kinematic regions further simplifications occur. A key example are
the regions where we additionally require $p_{4}^{+}\gg p_{5}^{+}$ or
$p_{5}^{+}\gg p_{4}^{+}$ which we refer to as the MRK and
${\mathrm{MRK}^{\prime}}$ limits respectively. These limits of the two-gluon
CEV are studied in appendix G.1. From the central NMRK region we may
alternatively take the limit where one of the central gluons becomes soft,
which is the subject of appendix G.2, or the limit where the two central
gluons become collinear to each other, which is the subject of appendix G.3.
These further limits act as checks on the results presented in this paper, and
also provide intuition for the kinematic dependence of the cross ratios and
the two-gluon CEV. One may also investigate the limits where both of the
central gluons become soft, but this is not pursued in this appendix.

###  G.1 MRK limit

Recall that the MRK (MRK′) limit is obtained by taking $X\to\infty$ ($X\to
0$). In the MRK or MRK′ limit, the tree-level two-gluon CEV factorises into a
product of Lipatov vertices times a $t$-channel pole. This factorization
property applies for any helicity configuration, for example, for opposite
helicities

$\displaystyle\begin{split}\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{\rm
MRK}&=V^{g(0)}(q_{1},p^{\oplus}_{4},q_{2})\frac{1}{t_{2}}V^{g(0)}(q_{2},p^{\ominus}_{5},q_{3})\\\
&=\frac{z}{\bar{z}}(\bar{w}-1)(\bar{z}-1),\\\ \end{split}$ (247)
$\displaystyle\begin{split}\left.A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right|_{{\mathrm{MRK}^{\prime}}}&=V^{g(0)}(q_{1},p^{\ominus}_{5},q^{\prime}_{2})\frac{1}{t^{\prime}_{2}}V^{g(0)}(q^{\prime}_{2},p^{\oplus}_{4},q_{3})\\\
&=\frac{z}{\bar{z}}\frac{(\bar{w}(\bar{z}-1)-\bar{z})}{(w(z-1)-z)}(w-1)(z-1)\,,\end{split}$
(248)

with Lipatov vertex $V^{g(0)}$ in eq. (9). Considering the opposite helicity
configuration at one loop, we saw in section 4.2 that different transcendental
terms multiply rational terms which are not simply the tree amplitudes. This
prompts an analysis of these individual rational terms in the MRK and
${\mathrm{MRK}^{\prime}}$ limits. As a consistency check on the MRK and
${\mathrm{MRK}^{\prime}}$ limits of these individual rational terms, the
relevant sums of these rational terms given in eq. (45), eq. (47), eq. (225)
and eq. (227) should reproduce the limits eq. (247) and eq. (248).

For the $\sigma_{A}$ ordering in the MRK, from eqs. (46) and (48) we obtain

$\displaystyle\begin{split}R^{A}_{uv}&\xrightarrow[\mathrm{MRK}]{}\frac{w-1}{w-z}\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{\rm
MRK}\\\
R_{\bar{v}\bar{w}}^{A}&\xrightarrow[\mathrm{MRK}]{}\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{\rm
MRK}\\\
R_{wu}^{A}&\xrightarrow[\mathrm{MRK}]{}-\frac{z-1}{w-z}\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{\rm
MRK}\\\ \end{split}$ (249)

while $R_{\bar{u}\bar{v}}^{A}$, $R_{vw}^{A}$ and $R_{\bar{w}\bar{u}}^{A}$
vanish in this limit. All six of these rational terms vanish in the
${\mathrm{MRK}^{\prime}}$ limit. The limits listed in eq. (249) can be used to
show eq. (247), starting from either of the representations eq. (45) or eq.
(47). On the other hand, for the $\sigma_{A^{\prime}}$ permutation, all
rational terms vanish in MRK. In ${\mathrm{MRK}^{\prime}}$,
$R^{A^{\prime}}_{uv}$, $R_{\bar{v}\bar{w}}^{A^{\prime}}$ and
$R_{wu}^{A^{\prime}}$ vanish, while the remaining terms tend to

$\displaystyle\begin{split}R_{\bar{u}\bar{v}}^{A^{\prime}}&\xrightarrow[\mathrm{MRK}^{\prime}]{}-\frac{(\bar{w}-1)\bar{z}^{2}}{(\bar{w}-\bar{z})(\bar{w}(\bar{z}-1)-\bar{z})}\left.A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right|_{{\mathrm{MRK}^{\prime}}}\\\
R_{vw}^{A^{\prime}}&\xrightarrow[\mathrm{MRK}^{\prime}]{}\left.A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right|_{{\mathrm{MRK}^{\prime}}}\\\
R_{\bar{w}\bar{u}}^{A^{\prime}}&\xrightarrow[\mathrm{MRK}^{\prime}]{}\frac{\bar{w}^{2}(\bar{z}-1)}{(\bar{w}-\bar{z})(\bar{w}(\bar{z}-1)-\bar{z})}\left.A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right|_{{\mathrm{MRK}^{\prime}}}.\end{split}$
(250)

Using the representation eq. (225) or eq. (227) we see that these limits are
consistent with the limit of the tree-level amplitude eq. (248). Finally for
the $B$ configurations, we get terms that survive in the MRK limit,

$\displaystyle\begin{split}R^{B}_{uv}&\xrightarrow[\mathrm{MRK}]{}-\frac{(w-1)(z-1)}{w(z-1)-z}\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{\rm
MRK}\\\ R_{\bar{u}\bar{v}}^{B}&\xrightarrow[\mathrm{MRK}]{}0\\\
R_{vw}^{B}&\xrightarrow[\mathrm{MRK}]{}0\\\
R_{\bar{v}\bar{w}}^{B}&\xrightarrow[\mathrm{MRK}]{}\left.-A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{\rm
MRK}\\\
R_{wu}^{B}&\xrightarrow[\mathrm{MRK}]{}\frac{1}{w(z-1)-z}\left.A^{gg(0)}(q_{1},p^{\oplus}_{4},p^{\ominus}_{5},q_{3})\right|_{\rm
MRK}\\\ R_{\bar{w}\bar{u}}^{B}&\xrightarrow[\mathrm{MRK}]{}0,\end{split}$
(251)

and also terms that survive in the ${\mathrm{MRK}^{\prime}}$ limit,

$\displaystyle\begin{split}R^{B}_{uv}&\xrightarrow[\mathrm{MRK}^{\prime}]{}-\frac{(\bar{w}-1)(\bar{z}-1)}{\bar{w}(\bar{z}-1)-\bar{z}}\left.A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right|_{{\mathrm{MRK}^{\prime}}}\\\
R_{\bar{u}\bar{v}}^{B}&\xrightarrow[\mathrm{MRK}^{\prime}]{}0\\\
R_{vw}^{B}&\xrightarrow[\mathrm{MRK}^{\prime}]{}\frac{1}{\bar{w}(\bar{z}-1)-\bar{z}}\left.A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right|_{{\mathrm{MRK}^{\prime}}}\\\
R_{\bar{v}\bar{w}}^{B}&\xrightarrow[\mathrm{MRK}^{\prime}]{}0\\\
R_{wu}^{B}&\xrightarrow[\mathrm{MRK}^{\prime}]{}0\\\
R_{\bar{w}\bar{u}}^{B}&\xrightarrow[\mathrm{MRK}^{\prime}]{}-\left.A^{gg(0)}(q_{1},p^{\ominus}_{5},p^{\oplus}_{4},q_{3})\right|_{{\mathrm{MRK}^{\prime}}}.\end{split}$
(252)

The MRK and ${\mathrm{MRK}^{\prime}}$ limits of the one-loop two gluon CEV in
${\cal N}=4$ SYM are discussed in section 4. The notation introduced in that
section makes it trivial to obtain the MRK and ${\mathrm{MRK}^{\prime}}$
limits of the transcendental terms, with the exception of terms that can be
compactly described in terms of the cross ratios of eqs. (42), (224) and (56).
To facilitate the investigation of these latter transcendental functions, we
list the cross ratios in the MRK and ${\mathrm{MRK}^{\prime}}$ limits:

$\begin{array}[]{ccc}u_{A}\xrightarrow[\mathrm{MRK}]{}1,&v_{A}\xrightarrow[\mathrm{MRK}]{}\frac{(-t_{1})p_{5}^{+}}{(-t_{2})p_{4}^{+}},&w_{A}\xrightarrow[\mathrm{MRK}]{}\frac{(-t_{3})p_{4}^{-}}{(-t_{2})p_{5}^{-}},\\\
u_{A^{\prime}}\xrightarrow[\mathrm{MRK}^{\prime}]{}1,&v_{A^{\prime}}\xrightarrow[\mathrm{MRK}^{\prime}]{}\frac{(-t_{1})p_{4}^{+}}{(-t_{2}^{\prime})p_{5}^{+}},&w_{A^{\prime}}\xrightarrow[\mathrm{MRK}^{\prime}]{}\frac{(-t_{1})p_{5}^{-}}{(-t_{2}^{\prime})p_{4}^{-}},\\\
u_{B}\xrightarrow[\mathrm{MRK}]{}1,&v_{B}\xrightarrow[\mathrm{MRK}]{}\frac{p_{4}^{-}p_{5}^{+}}{(-t_{2})},&w_{B}\xrightarrow[\mathrm{MRK}]{}\frac{(-t_{1})(-t_{3})}{(-t_{2})p_{4}^{+}p_{5}^{-}},\\\
u_{B}\xrightarrow[\mathrm{MRK}^{\prime}]{}\frac{p_{4}^{+}p_{5}^{-}}{(-t_{2}^{\prime})},&v_{B}\xrightarrow[\mathrm{MRK}^{\prime}]{}1,&w_{B}\xrightarrow[\mathrm{MRK}^{\prime}]{}\frac{(-t_{1})(-t_{3})}{(-t_{2}^{\prime})p_{4}^{-}p_{5}^{+}}.\\\
\end{array}$ (253)

###  G.2 Soft limits

In the limit that either of the gluons of the tree two-gluon CEV become soft,
$\displaystyle p_{4}\to 0$ or $\displaystyle p_{5}\to 0$, the amplitude given
in eq. (26) is singular. It factorises into an eikonal factor, which behaves
like $\displaystyle 1/p_{4}$ or $\displaystyle 1/p_{5}$, and a five-gluon
amplitude in MRK. For example, taking the limit where all components of
$\displaystyle p_{4}$ tend to zero at the same rate, eq. (26) factorises as

$\displaystyle\displaystyle\left.M^{(0)}_{6g}(p_{1}^{\nu_{1}},p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},p_{6}^{\nu_{6}})\right|_{\mathrm{NMRK}}$
(254) $\displaystyle\displaystyle\xrightarrow[p_{4}\to 0]{}$
$\displaystyle\displaystyle\left[\mathrm{Soft}^{(0)}(p_{3},p^{\nu_{4}}_{4},p_{5})\right]_{\mathrm{NMRK}}\,\left[M^{(0)}_{5g}(p_{1}^{\nu_{1}},p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},p^{\nu_{5}}_{5},p_{6}^{\nu_{6}})\right]_{\mathrm{MRK}}\,,$

where the five-gluon amplitude in MRK is

$\left.M^{(0)}_{5g}(p_{1}^{\nu_{1}},p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},p^{\nu_{5}}_{5},p_{6}^{\nu_{6}})\right|_{\mathrm{MRK}}=s\,C^{g(0)}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}})\,\frac{1}{t_{1}}\,V^{g(0)}(q_{1},p^{\nu_{5}}_{5},q_{3})\,\frac{1}{t_{3}}\,C^{g(0)}(p_{1}^{\nu_{1}},p_{6}^{\nu_{6}})\,,$
(255)

with $\displaystyle q_{3}=q_{1}-p_{5}$ and $\displaystyle V^{g(0)}$ given in
eq. (9). In eq. (254), the positive-helicity soft factor, or eikonal factor is

$\mathrm{Soft}^{(0)}(p_{3},p^{\oplus}_{4},p_{5})=\frac{\langle
3\,5\rangle}{\langle 3\,4\rangle\langle 4\,5\rangle}\,.$ (256)

Note that this eikonal factor does not depend on the helicities of particles 3
and 5, and in fact not even on their parton flavour. The negative-helicity
eikonal factor is obtained by complex conjugation. In the NMRK limit the
positive-helicity eikonal factor becomes

$\left.\mathrm{Soft}^{(0)}(p_{3},p^{\oplus}_{4},p_{5})\right|_{\mathrm{NMRK}}=\mathrm{Soft}^{(0)}_{A}(p_{4}^{\oplus},p_{5})=\frac{p_{5\perp}}{p_{4\perp}}\sqrt{\frac{p_{4}^{+}}{p_{5}^{+}}}\,\frac{1}{\langle
4\,5\rangle}\,,$ (257)

where in the first equality we have introduced a notation which makes it clear
that this function no longer depends on $\displaystyle p_{3}$ once we have
taken the NMRK limit. Using this notation, eq. (254) implies that the two-
gluon CEV factorises as

$A^{gg(0)}(q_{1},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},q_{3})\xrightarrow[p_{4}\to
0]{}\mathrm{Soft}^{(0)}_{A}(p_{4}^{\nu_{4}},p_{5})\,V^{g(0)}(q_{1},p^{\nu_{5}}_{5},q_{3})\,.$
(258)

For the equal-helicity vertex eq. (27), which comes from a MHV amplitude, this
limit is straightforward to verify. In fact, for such an MHV amplitude, the
relation at the amplitude level, eq. (254), holds exactly in general
kinematics without even requiring that $\displaystyle p_{4}\to 0$. For the
opposite-helicity vertex, in order to check eq. (258) we have found it most
convenient to use the representation in eq. (45).

As a step towards obtaining the soft limit of the one-loop amplitude, we first
consider the rational coefficients. Specifically, for the limit where all
components of $\displaystyle p_{4}$ tend to zero at the same rate, the minimal
variables of eq. (33) and eq. (34) satisfy

$\displaystyle\displaystyle X\to 0,\qquad
z\to-\frac{q_{1\perp}}{p_{4\perp}},\qquad
w\to\frac{q_{1\perp}}{p_{5\perp}},\qquad Xz\to\ {\rm finite}.$ (259)

In this limit $\displaystyle R^{A}_{uv}$, $\displaystyle R^{A}_{vw}$ and
$\displaystyle R^{A}_{\bar{w}\bar{u}}$ tend to zero. The others are given by

$\displaystyle\displaystyle\begin{split}&R^{A}_{wu}\to\frac{Xz^{2}(\bar{w}-1)}{w+Xz},\\\
&R^{A}_{\bar{u}\bar{v}}\to-\frac{Xz^{2}|w-1|^{2}}{(1+Xz)(w+Xz)},\\\
\qquad&R^{A}_{\bar{v}\bar{w}}\to\frac{Xz^{2}(\bar{w}-1)}{1+Xz}.\end{split}$
(260)

Therefore $\displaystyle R^{A}_{uv}+R^{A}_{vw}+R^{A}_{wu}$ and $\displaystyle
R^{A}_{\bar{u}\bar{v}}+R^{A}_{\bar{v}\bar{w}}+R^{A}_{\bar{w}\bar{u}}$ tend to
the same limit (that of eq. (258)). The results for the $\displaystyle
A^{\prime}$ ordering are no longer related to these by $\displaystyle
p_{4}\leftrightarrow p_{5}$ because the specific choice of taking
$\displaystyle p_{4}\to 0$ breaks the symmetry. We therefore list them here.
The rational coefficients $\displaystyle R^{A^{\prime}}_{uv}$, $\displaystyle
R^{A^{\prime}}_{\bar{v}\bar{w}}$ and $\displaystyle
R^{A^{\prime}}_{\bar{w}\bar{u}}$ tend to zero. The remaining non-zero
coefficients are

$\displaystyle\displaystyle\begin{split}&R^{A^{\prime}}_{\bar{u}\bar{v}}\to-\frac{wz(1-\bar{w})}{w+Xz},\\\
&R^{A^{\prime}}_{vw}\to\frac{wz(1-\bar{w})^{2}}{w\bar{w}-w-Xz},\\\
&R^{A^{\prime}}_{wu}\to\frac{w\bar{w}Xz^{2}(1-\bar{w})}{(w+Xz)(w\bar{w}-w-Xz)},\end{split}$
(261)

such that again $\displaystyle
R^{A^{\prime}}_{uv}+R^{A^{\prime}}_{vw}+R^{A^{\prime}}_{wu}$ and
$\displaystyle
R^{A^{\prime}}_{\bar{u}\bar{v}}+R^{A^{\prime}}_{\bar{v}\bar{w}}+R^{A^{\prime}}_{\bar{w}\bar{u}}$
give the expected tree-level limit:

$\displaystyle\displaystyle-\frac{wz(1-\bar{w})}{w+Xz}=\
\mathrm{Soft}^{(0)}_{A^{\prime}}(p_{4}^{\nu_{4}},p_{5})\,V^{g(0)}(q_{1},p_{5}^{\ominus},q_{3}),$
(262)

where, analogous to eq. (257), we have introduced the notation

$\left.\mathrm{Soft}^{(0)}(p_{5},p_{4}^{\oplus},p_{6})\right|_{\mathrm{NMRK}}=\mathrm{Soft}^{(0)}_{A^{\prime}}(p_{4}^{\nu_{4}},p_{5})=-\sqrt{\frac{p_{5}^{+}}{p_{4}^{+}}}\,\frac{1}{\langle
4\,5\rangle}\,.$ (263)

Finally, for the $\displaystyle B$ colour ordering $\displaystyle
R^{B}_{\bar{u}\bar{v}}$, $\displaystyle R^{B}_{vw}$ and $\displaystyle
R^{B}_{wu}$ tend to zero, leaving

$\displaystyle\displaystyle\begin{split}&R^{B}_{uv}\to-(\bar{w}-1)z,\\\
&R^{B}_{\bar{v}\bar{w}}\to-\frac{Xz^{2}(\bar{w}-1)}{1+Xz-\bar{w}},\\\
&R^{B}_{\bar{w}\bar{u}}\to-\frac{z(\bar{w}-1)^{2}}{\bar{w}-Xz-1}.\end{split}$
(264)

These then lead to the expected tree-level result where $\displaystyle
R^{B}_{uv}+R^{B}_{vw}+R^{B}_{wu}$ and $\displaystyle
R^{B}_{\bar{u}\bar{v}}+R^{B}_{\bar{v}\bar{w}}+R^{B}_{\bar{w}\bar{u}}$ both sum
to

$\displaystyle\displaystyle-(\bar{w}-1)z=\mathrm{Soft}^{(0)}_{B}(p_{4})\,V^{g(0)}(q_{1},p_{5}^{\ominus},q_{3}),$
(265)

where as for the other colour orderings we have introduced

$\left.\mathrm{Soft}^{(0)}(p_{1},p_{4}^{\oplus},p_{2})\right|_{\mathrm{NMRK}}=\mathrm{Soft}^{(0)}_{B}(p_{4})=\frac{1}{p_{4\perp}},$
(266)

which does not even depend on the helicity of the soft particle. The three
relevant eikonal functions in the NMRK obey

$\mathrm{Soft}^{(0)}_{A}(p_{4}^{\nu_{4}},p_{5})+\mathrm{Soft}^{(0)}_{A^{\prime}}(p_{4}^{\nu_{4}},p_{5})+\mathrm{Soft}^{(0)}_{B}(p_{4})=0,$
(267)

consistent with the more general eq. (54), and the eikonal identity.

We now turn to the colour-ordered amplitudes at one-loop. Beginning with the
same-helicity vertex, taking the limit $\displaystyle p_{4}\to 0$ in eqs.
(97), (103) and (106), we find

$\displaystyle\displaystyle\begin{split}N_{c}\kappa_{\Gamma}{\mathrm{Re}\left[{\left.V_{6}(\sigma_{A})\right|_{\rm
NMRK}}\right]}\xrightarrow[p_{4}\to
0]{}\,\,&E(t_{1},s_{35};t_{3},s_{56})+v^{g(1)}(t_{1},|p_{5\perp}|^{2},t_{3})+N_{c}\,s^{(1)}_{A}(p_{4}^{\nu_{4}},p_{5})\\\
N_{c}\kappa_{\Gamma}{\mathrm{Re}\left[{\left.V_{6}(\sigma_{A^{\prime}})\right|_{\rm
NMRK}}\right]}\xrightarrow[p_{4}\to
0]{}\,\,&E(t_{1},s_{35};t_{3},s_{56})+v^{g(1)}(t_{1},|p_{5\perp}|^{2},t_{3})+\,N_{c}\,s^{(1)}_{A^{\prime}}(p_{4},p_{5}),\\\
N_{c}\kappa_{\Gamma}{\mathrm{Re}\left[{\left.V_{6}(\sigma_{B})\right|_{\rm
NMRK}}\right]}\xrightarrow[p_{4}\to
0]{}\,\,&E(t_{1},s_{35};t_{3},s_{56})+v^{g(1)}(t_{1},|p_{5\perp}|^{2},t_{3})+N_{c}\,s^{(1)}_{B}(p_{4}),\end{split}$
(268)

where $\displaystyle E(t_{1},s_{35};t_{3},s_{56})$ is the natural collection
of impact factors and Regge trajectories in the absence of $\displaystyle
p_{4}$ and $\displaystyle v^{g(1)}$ is the one-loop Lipatov vertex correction
given in eq. (86). The one-loop soft factor Bern:1998sc  is

$\displaystyle\displaystyle
s^{(1)}(p_{3},p_{4}^{\nu_{4}},p_{5})=-\kappa_{\Gamma}\left(\frac{\mu^{2}(-s_{35})}{(-s_{34})(-s_{45})}\right)^{\epsilon}\left(\frac{1}{\epsilon^{2}}+\frac{\pi^{2}}{6}\right)+\mathcal{O}(\epsilon^{2}),$
(269)

which must first be analytically continued to our physical region, discussed
in section A.1, and then taken in the NMRK limit. In eq. (268) we have used
the notation

$\displaystyle\displaystyle s^{(1)}_{A}(p_{4},p_{5})$
$\displaystyle\displaystyle={\mathrm{Re}\left[{s^{(1)}(p_{3},p_{4},p_{5})\Big{|}_{\mathrm{NMRK}}}\right]}\,,$
(270) $\displaystyle\displaystyle s^{(1)}_{A^{\prime}}(p_{4},p_{5})$
$\displaystyle\displaystyle={\mathrm{Re}\left[{s^{(1)}(p_{4},p_{5},p_{6})\Big{|}_{\mathrm{NMRK}}}\right]}\,,$
(271) $\displaystyle\displaystyle s^{(1)}_{B}(p_{4})$
$\displaystyle\displaystyle={\mathrm{Re}\left[{s^{(1)}(p_{1},p_{4},p_{2})\Big{|}_{\mathrm{NMRK}}}\right]}\,,$
(272)

which, as for the analogous tree-level functions, more clearly indicate the
simpler kinematic dependence of these one-loop soft functions in the NMRK
limit.

Eq. (268) exactly matches the expected sum of the one-loop corrections to the
non-central pieces, the Lipatov vertex for the single emission of
$\displaystyle p_{5}$ and the one-loop soft function. From eq. (94), these are
multiplied by the soft limit of the corresponding tree-level amplitude for
that ordering.

Turning now to the opposite-helicity case in eq. (132), and starting with the
$\displaystyle A$ colour ordering, we find

$\displaystyle\displaystyle\begin{split}S_{wu}^{A}&\xrightarrow[p_{4}\to
0]{}\mathrm{Soft}^{(0)}_{A}(p_{4}^{\nu_{4}},p_{5})V^{g(0)}(q_{1},p_{5}^{\ominus},q_{3}),\\\
{\mathrm{Re}\left[{W_{1}(\sigma_{A})-W_{2}(\sigma_{A})}\right]}&\xrightarrow[p_{4}\to
0]{}0,\\\
{\mathrm{Re}\left[{W_{3}(\sigma_{A})-W_{2}(\sigma_{A})}\right]}&\xrightarrow[p_{4}\to
0]{}\log\left(\frac{s_{34}t_{3}}{s_{35}t_{1}}\right)\log\left(\frac{|p_{5\perp}|^{2}}{s_{45}}\right).\end{split}$
(273)

This allows us to extract a common factor of the tree amplitude multiplying

$\displaystyle{\mathrm{Re}\left[{2W_{2}(\sigma_{A})+(W_{3}(\sigma_{A})-W_{2}(\sigma_{A}))}\right]}.$

Using eqs. (139) and (140), we find

$\displaystyle\displaystyle\begin{split}&\left.{\mathrm{Re}\left[{2W_{2}(\sigma_{A})+(W_{3}(\sigma_{A})-W_{2}(\sigma_{A}))}\right]}\right|_{\rm
NMRK}\\\ =&\left.{\mathrm{Re}\left[{V_{6}(\sigma_{A})-\Delta
V_{A}(u_{A},v_{A},w_{A})+\Delta
W_{A}(u_{A},v_{A},w_{A})+(W_{3}(\sigma_{A})-W_{2}(\sigma_{A}))}\right]}\right|_{\rm
NMRK}\\\ \xrightarrow[p_{4}\to
0]{}&\left.{\mathrm{Re}\left[{V_{6}(\sigma_{A})}\right]}\right|_{{\rm
NMRK},\,p_{4}\to 0},\end{split}$ (274)

where the cancellation in the last line arises from the soft limit of eqs.
(101), (141) and (273). Eq. (268) then immediately gives the expected one-loop
structure which equals that found for the same-helicity vertex. The helicity-
dependence in the soft limit is contained only in the tree-level factors.

Similarly for the $\displaystyle A^{\prime}$-ordering we find

$\displaystyle\displaystyle\begin{split}S_{uv}^{A^{\prime}}&\xrightarrow[p_{4}\to
0]{}\mathrm{Soft}^{(0)}_{A^{\prime}}(p_{4}^{\nu_{4}},p_{5})V^{g(0)}(q_{1},p_{5}^{\ominus},q_{3}),\\\
\left.{\mathrm{Re}\left[{W_{1}(\sigma_{A^{\prime}})-W_{2}(\sigma_{A^{\prime}})}\right]}\right|_{\rm
NMRK}&\xrightarrow[p_{4}\to
0]{}\log\left(\frac{s_{46}t_{1}}{s_{56}t_{3}}\right)\log\left(\frac{|p_{5\perp}|^{2}}{s_{45}}\right),\\\
\left.{\mathrm{Re}\left[{W_{3}(\sigma_{A^{\prime}})-W_{2}(\sigma_{A^{\prime}})}\right]}\right|_{\rm
NMRK}&\xrightarrow[p_{4}\to 0]{}0.\end{split}$ (275)

Extracting a common coefficient of the tree amplitude leaves

$\displaystyle\displaystyle\left.{\mathrm{Re}\left[{2W_{2}(\sigma_{A^{\prime}})+(W_{1}(\sigma_{A^{\prime}})-W_{2}(\sigma_{A^{\prime}}))}\right]}\right|_{\rm
NMRK}\xrightarrow[p_{4}\to
0]{}\left.{\mathrm{Re}\left[{V_{6}(\sigma_{A^{\prime}})}\right]}\right|_{{\rm
NMRK},\,p_{4}\to 0},$ (276)

as expected. Finally for the $\displaystyle B$ ordering,

$\displaystyle\displaystyle\begin{split}S_{vw}^{B}+S_{wu}^{B}&\xrightarrow[p_{4}\to
0]{}\mathrm{Soft}^{(0)}_{B}(p_{4})V^{g(0)}(q_{1},p_{5}^{\ominus},q_{3}),\\\
\left.{\mathrm{Re}\left[{W_{2}(\sigma_{B})-W_{1}(\sigma_{B})}\right]}\right|_{\rm
NMRK}&\xrightarrow[p_{4}\to
0]{}\log\left(\frac{s_{24}s_{56}}{s_{12}t_{1}}\right)\log\left(\frac{s_{14}s_{35}}{s_{12}t_{3}}\right),\\\
\left.{\mathrm{Re}\left[{W_{3}(\sigma_{B})-W_{1}(\sigma_{B})}\right]}\right|_{\rm
NMRK}&\xrightarrow[p_{4}\to
0]{}\log\left(\frac{s_{24}s_{56}}{s_{12}t_{1}}\right)\log\left(\frac{s_{14}s_{35}}{s_{12}t_{3}}\right).\end{split}$
(277)

As the final two limits are equal, one again factors out the tree amplitude
leaving

$\displaystyle\displaystyle\left.{\mathrm{Re}\left[{2W_{1}(\sigma_{B})+(W_{2}(\sigma_{B})-W_{1}(\sigma_{B}))}\right]}\right|_{\rm
NMRK}\xrightarrow[p_{4}\to
0]{}\left.{\mathrm{Re}\left[{V_{6}(\sigma_{B})}\right]}\right|_{\mathrm{NMRK},\,p_{4}\to
0}.$ (278)

Finally, performing the change of basis of colour structures in eq. (113), we
collect the results of this appendix in the following expression for general
helicities,

$\displaystyle\displaystyle\begin{split}&\mathrm{Disp}\left[\mathcal{M}^{(1)}_{{\cal
N}=4}(p_{4}^{\nu_{4}},p_{5}^{\nu_{5}})\right]\xrightarrow[\mathrm{NMRK,}\
p_{4}\to 0]{}\\\ &\frac{g_{\scriptscriptstyle
S}^{6}}{(4\pi)^{2}}\,s_{12}F^{a_{3}a_{2}c_{1}}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\frac{1}{t_{1}}V^{g(0)}(q_{1},p_{5}^{\nu_{5}},q_{3})\times\\\
&\Bigg{\\{}(F^{a_{4}}F^{a_{5}})_{c_{1}c_{3}}\bigg{[}\mathrm{Soft}_{A}(p_{4}^{\nu_{4}},p_{5})\left(3\,v^{g(1)}(t_{1},|p_{5\perp}|^{2},t_{3})+N_{c}\left(2s^{(1)}_{A}(p_{4},p_{5})+s^{(1)}_{B}(p_{4})\right)\right.\\\
&\left.+3\,E(t_{1},s_{35};t_{3},s_{56})\right)+N_{c}\mathrm{Soft}_{A^{\prime}}(p_{4}^{\nu_{4}},p_{5})\left(s^{(1)}_{A^{\prime}}(p_{4},p_{5})-s^{(1)}_{B}(p_{4})\right)\bigg{]}\\\
&+(F^{a_{5}}F^{a_{4}})_{c_{1}c_{3}}\bigg{[}\mathrm{Soft}_{A^{\prime}}(p_{4}^{\nu_{4}},p_{5})\left(3\,v^{g(1)}(t_{1},|p_{5\perp}|^{2},t_{3})+N_{c}\left(2s^{(1)}_{A^{\prime}}(p_{4},p_{5})+s^{(1)}_{B}(p_{4})\right)\right.\\\
&\left.+3\,E(t_{1},s_{35};t_{3},s_{56})\right)+N_{c}\mathrm{Soft}_{A}(p_{4}^{\nu_{4}},p_{5})\left(s^{(1)}_{A}(p_{4},p_{5})-s^{(1)}_{B}(p_{4})\right)\bigg{]}\\\
&+d_{A}^{c_{1}a_{\sigma_{4}}a_{\sigma_{5}}c_{3}}\bigg{[}\mathrm{Soft}_{A}(p_{4}^{\nu_{4}},p_{5})\left(s^{(1)}_{A}(p_{4},p_{5})-s^{(1)}_{B}(p_{4})\right)\\\
&+\mathrm{Soft}_{A^{\prime}}(p_{4}^{\nu_{4}},p_{5})\left(s^{(1)}_{A^{\prime}}(p_{4},p_{5})-s^{(1)}_{B}(p_{4})\right)\bigg{]}\Bigg{\\}}\,\times\frac{1}{t_{3}}F^{a_{6}a_{1}c_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\,.\end{split}$
(279)

###  G.3 Collinear limits

In this appendix we investigate the behaviour of the one-loop amplitude eq.
(146) in the collinear limit,

$p_{4}\to z_{4}P\,,\qquad p_{5}\to z_{5}P\,,\qquad z_{4}+z_{5}=1\,.$ (280)

We begin by recalling the collinear limit of the colour ordered six-gluon
tree-level amplitude in general kinematics. It factorises into a divergent
splitting amplitude, which behaves like $\displaystyle 1/\langle 45\rangle$ or
$\displaystyle 1/[45]$, and a five-gluon amplitude,

$M^{(0)}_{6g}(p_{1}^{\nu_{1}},p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},p_{6}^{\nu_{6}})\xrightarrow[]{p_{4}||p_{5}}\sum_{\nu=\pm}\mathrm{Split}^{(0)}_{-\nu}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,M^{(0)}_{5g}(p_{1}^{\nu_{1}},p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},P^{\nu},p_{6}^{\nu_{6}}),$
(281)

where we sum over the helicities of the intermediate state $\displaystyle
P^{\nu}$. For an equal-helicity pair, the tree-level splitting amplitudes are

${\rm{Split}}^{(0)}_{\ominus}(p^{\oplus}_{4},p^{\oplus}_{5})=\frac{1}{\sqrt{z_{4}z_{5}}\langle
4\,5\rangle}\,,\qquad{\rm{Split}}^{(0)}_{\oplus}(p^{\oplus}_{4},p^{\oplus}_{5})=0\,,$
(282)

while for an opposite-helicity pair, they are

${\rm{Split}}^{(0)}_{\oplus}(p^{\oplus}_{4},p^{\ominus}_{5})=\frac{z^{2}_{5}}{\sqrt{z_{4}z_{5}}\langle
4\,5\rangle}\,,\qquad{\rm{Split}}^{(0)}_{\ominus}(p^{\oplus}_{4},p^{\ominus}_{5})=-\frac{z^{2}_{4}}{\sqrt{z_{4}z_{5}}[4\,5]}\,.$
(283)

In the central NMRK limit, eq. (281) becomes

$\displaystyle\displaystyle\left.M^{(0)}_{6g}(p_{1}^{\nu_{1}},p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},p_{6}^{\nu_{6}})\right|_{\mathrm{NMRK}}$
$\displaystyle\displaystyle\xrightarrow[]{p_{4}||p_{5}}\sum_{\nu=\pm}\mathrm{Split}^{(0)}_{-\nu}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,\left.M^{(0)}_{5g}(p_{1}^{\nu_{1}},p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},P^{\nu},p_{6}^{\nu_{6}})\right|_{\mathrm{MRK}},$
(284)

where the five-gluon amplitude in MRK is given in eq. (255), with
$\displaystyle q_{3}=q_{1}-P$. In particular, the two-gluon CEV factorises as

$A^{gg(0)}(q_{1},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},q_{3})\stackrel{{\scriptstyle
p_{4}||p_{5}}}{{=}}\sum_{\nu=\pm}\mathrm{Split}^{(0)}_{-\nu}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,V^{g(0)}(q_{1},P^{\nu},q_{3})\,.$
(285)

For the equal-helicity vertex eq. (27), it is straightforward to verify eq.
(285). For the opposite-helicity vertex, we have found it more convenient to
check eq. (285) by starting from eq. (45) or eq. (47) rather than eq. (28),
where as we have mentioned in section 3.2, the BCFW representations make clear
the collinear singularity structure of the amplitudes. Working in the minimal
variables, the collinear limit can be expressed as

$\displaystyle\displaystyle\begin{split}w&\xrightarrow[p_{4}||p_{5}]{}-\frac{z_{4}}{z_{5}}z\,,\\\
X&\xrightarrow[p_{4}||p_{5}]{}\frac{z_{4}}{z_{5}}\,.\end{split}$ (286)

For the $\displaystyle\sigma_{A}$ ordering we find that four of the six
rational terms factorise into a splitting function and Lipatov vertex, while
two of the rational terms do not diverge in this limit (unless we move to the
further soft-collinear limits of $\displaystyle z_{4}\to 0$ or $\displaystyle
z_{4}\to 1$),

$\displaystyle\displaystyle\begin{split}R^{A}_{uv}&\xrightarrow[p_{4}||p_{5}]{}V^{g(0)}(q_{1},P^{\oplus},q_{3})\mathrm{Split}^{(0)}_{\ominus}(p_{4}^{\oplus},p_{5}^{\ominus}),\\\
R_{\bar{u}\bar{v}}^{A}&\xrightarrow[p_{4}||p_{5}]{}V^{g(0)}(q_{1},P^{\ominus},q_{3})\mathrm{Split}^{(0)}_{\oplus}(p_{4}^{\oplus},p_{5}^{\ominus}),\\\
R_{vw}^{A}&\xrightarrow[p_{4}||p_{5}]{}-\frac{(z-1)(zz_{4}+z_{5})z_{5}}{(\bar{z}-1)(\bar{z}z_{4}+z_{5})(|z|^{2}z_{4}+z_{5})},\\\
R_{\bar{v}\bar{w}}^{A}&\xrightarrow[p_{4}||p_{5}]{}-\frac{z^{4}z_{4}^{2}(\bar{z}-1)(\bar{z}z_{4}+z_{5})}{z_{5}(z-1)(zz_{4}+z_{5})(|z|^{2}z_{4}+z_{5})},\\\
R_{wu}^{A}&\xrightarrow[p_{4}||p_{5}]{}V^{g(0)}(q_{1},P^{\ominus},q_{3})\mathrm{Split}^{(0)}_{\oplus}(p_{4}^{\oplus},p_{5}^{\ominus}),\\\
R_{\bar{w}\bar{u}}^{A}&\xrightarrow[p_{4}||p_{5}]{}V^{g(0)}(q_{1},P^{\oplus},q_{3})\mathrm{Split}^{(0)}_{\ominus}(p_{4}^{\oplus},p_{5}^{\ominus}).\end{split}$
(287)

Analogous results for the $\displaystyle\sigma_{A^{\prime}}$ ordering can be
found by exchanging $\displaystyle p_{4}\leftrightarrow p_{5}$ and taking the
complex conjugate, as in eq. (231). For the $\displaystyle\sigma_{B}$
ordering, as expected, none of the rational terms diverge in this limit,

$\displaystyle\displaystyle\begin{split}R^{B}_{uv}&\xrightarrow[p_{4}||p_{5}]{}\frac{|z-1|^{2}|zz_{4}+z_{5}|^{2}}{z_{5}((z-1)z_{4}+z_{5})\bar{z}},\\\
R_{\bar{u}\bar{v}}^{B}&\xrightarrow[p_{4}||p_{5}]{}0,\\\
R_{vw}^{B}&\xrightarrow[p_{4}||p_{5}]{}\frac{z_{5}(z-1)(zz_{4}+z_{5})}{\bar{z}(|z|^{2}z_{4}+z_{5})((z+\bar{z}-1)z_{4}+z_{5})},\\\
R_{\bar{v}\bar{w}}^{B}&\xrightarrow[p_{4}||p_{5}]{}\frac{z^{3}z_{4}^{2}(\bar{z}-1)(\bar{z}z_{4}+z_{5})}{z_{5}(|z|^{2}z_{4}+z_{5})((z+\bar{z}-1)z_{4}+z_{5})},\\\
R_{wu}^{B}&\xrightarrow[p_{4}||p_{5}]{}\frac{z_{5}z_{4}^{2}(\bar{z}-1)(\bar{z}z_{4}+z_{5})}{((z-1)z_{4}+z_{5})((z+\bar{z}-1)z_{4}+z_{5})\left(|z_{5}+(z-1)z_{4}|^{2}+z_{4}z_{5}\right)},\\\
R_{\bar{w}\bar{u}}^{B}&\xrightarrow[p_{4}||p_{5}]{}\frac{(z-1)(zz_{4}+z_{5})((\bar{z}-1)z_{4}+z_{5})^{3}}{z_{5}((z+\bar{z}-1)z_{4}+z_{5})\left(|z_{5}+(z-1)z_{4}|^{2}+z_{4}z_{5}\right)}.\end{split}$
(288)

From these lists of rational terms we can immediately verify that the
divergent pieces satisfy the condition in eq. (285) from the representations
eq. (45) or eq. (47). We note that in this limit the divergent pieces also
satisfy

$\left.A^{gg(0)}(q_{1},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},q_{3})\right|_{p_{4}||p_{5}}=-\left.A^{gg(0)}(q_{1},p^{\nu_{5}}_{5},p^{\nu_{4}}_{4},q_{3})\right|_{p_{4}||p_{5}}.$
(289)

In addition to eq. (280), the amplitude in eq. (26) is divergent in the
collinear regions $\displaystyle p_{3\perp}=-q_{1\perp}\rightarrow 0$ or
$\displaystyle p_{6\perp}=q_{3\perp}\rightarrow 0$. In those regions, eq. (26)
must not diverge more rapidly than $\displaystyle 1/|q_{i\perp}|$, with
$\displaystyle i=1,3$, in order for the related cross section not to diverge
more than logarithmically. Since eq. (26) displays poles as
$\displaystyle|q_{1\perp}|^{2}$ and $\displaystyle|q_{3\perp}|^{2}$ go to
zero, the central-emission vertex must be at least linear in
$\displaystyle|q_{i\perp}|$ as $\displaystyle|q_{i\perp}|\rightarrow 0$,

$A^{gg(0)}(q_{1},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},q_{3})\stackrel{{\scriptstyle|q_{i\perp}|\rightarrow
0}}{{=}}{\cal O}(|q_{i\perp}|)\,,$ (290)

with $\displaystyle i=1,3$, which is indeed fulfilled by eq. (28).

Let us now consider the collinear limit of the one-loop amplitudes in
$\displaystyle{\cal N}=4$ SYM in the NMRK limit. It is most convenient if we
work at the level of colour-ordered amplitudes, where in general kinematics
the colour-ordered one-loop six gluon amplitude has the collinear limit
Bern:1994zx ,

$\displaystyle\displaystyle\begin{split}M^{(1)}_{6g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},p_{6}^{\nu_{6}},p_{1}^{\nu_{1}})\xrightarrow[]{p_{4}||p_{5}}\sum_{\nu=\pm}\Big{(}&\mathrm{Split}_{-\nu}^{(0)}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,M^{(1)}_{5g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},P^{\nu},p_{6}^{\nu_{6}},p_{1}^{\nu_{1}})\\\
+&\mathrm{Split}_{-\nu}^{(1)}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,M^{(0)}_{5g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},P^{\nu},p_{6}^{\nu_{6}},p_{1}^{\nu_{1}})\Big{)}\,,\end{split}$
(291)

where the one-loop splitting functions were obtained in Bern:1994zx . In
$\displaystyle\mathcal{N}=4$ SYM, these splitting functions are proportional
to the tree-level splitting function,

$\mathrm{Split}_{-\nu}^{(1)}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})=\kappa_{\Gamma}\,\mathrm{Split}_{-\nu}^{(0)}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,r_{S}^{\mathrm{SUSY}}(z_{4},s_{45}),$
(292)

where the loop factor,

$r_{S}^{\mathrm{SUSY}}(z_{4},s_{45})=-\frac{1}{\epsilon^{2}}\left(\frac{\mu^{2}}{z_{4}(1-z_{4})(-s_{45})}\right)^{\epsilon}+2\log(z_{4})\log(1-z_{4})-\frac{\pi^{2}}{6},$
(293)

is independent of the helicities. In the central NMRK limit we expect to
obtain

$\displaystyle\displaystyle\begin{split}\left.M^{(1)}_{6g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},p^{\nu_{4}}_{4},p^{\nu_{5}}_{5},p_{6}^{\nu_{6}},p_{1}^{\nu_{1}})\right|&{}_{\mathrm{NMRK}}\xrightarrow[]{p_{4}||p_{5}}\\\
\sum_{\nu=\pm}\bigg{(}&\mathrm{Split}_{-\nu}^{(0)}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,\left.M^{(1)}_{5g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},P^{\nu},p_{6}^{\nu_{6}},p_{1}^{\nu_{1}})\right|_{\mathrm{MRK}}\\\
+&\mathrm{Split}_{-\nu}^{(1)}(p^{\nu_{4}}_{4},p^{\nu_{5}}_{5})\,\left.M^{(0)}_{5g}(p_{2}^{\nu_{2}},p_{3}^{\nu_{3}},P^{\nu},p_{6}^{\nu_{6}},p_{1}^{\nu_{1}})\right|_{\mathrm{MRK}}\bigg{)}.\end{split}$
(294)

Let us check this first for the same-helicity case. As we are working at the
level of colour-ordered amplitudes, we consider the three representative
orderings of eq. (94). For the $\displaystyle\sigma_{A}$ colour ordering, we
find

$\displaystyle\displaystyle\begin{split}s_{234}&\xrightarrow[p_{4}||p_{5}]{}z_{5}t_{1}+z_{4}t_{3}\,,\\\
\end{split}$ (295)

so that in the collinear limit,

$v_{A}+w_{A}\xrightarrow[p_{4}||p_{5}]{}1\,.$ (296)

In particular, this allows us to use the identity,

${\rm Li_{2}}\left(1-x\right)+{\rm
Li_{2}}\left(x\right)=\frac{\pi^{2}}{6}-\log\left(1-x\right)\log\left(x\right).$
(297)

to write

$\displaystyle\displaystyle\begin{split}\Delta
V_{A}(u_{A},v_{A},w_{A})\xrightarrow[p_{4}||p_{5}]{}N_{c}\kappa_{\Gamma}\bigg{(}&-\frac{1}{4}\log^{2}\left(\frac{s_{45}}{|P_{\perp}|^{2}}\right)-\frac{1}{2}\log\left(\frac{s_{45}}{|P_{\perp}|^{2}}\right)\log\left(\frac{t_{1}t_{3}z_{4}z_{5}}{s_{234}^{2}}\right)\\\
&+\log\left(\frac{z_{4}t_{3}}{s_{234}}\right)\log\left(\frac{z_{5}t_{1}}{s_{234}}\right)\bigg{)}\,.\end{split}$
(298)

For the full transcendental function of this colour ordering we find

$\displaystyle\displaystyle\begin{split}N_{c}\kappa_{\Gamma}{\mathrm{Re}\left[{\left.V_{6}(\sigma_{A})\right|_{\mathrm{NMRK}}}\right]}\xrightarrow[p_{4}||p_{5}]{}&E(t_{1},p_{3}^{+}P^{-};t_{3},P^{+}p_{6}^{-})+v^{g(1)}_{\mathcal{N}=4}\left(t_{1},|P_{\perp}|^{2},t_{3}\right)\\\
+&N_{c}{\mathrm{Re}\left[{r^{\mathrm{SUSY}}(z_{4},s_{45})}\right]}\,,\end{split}$
(299)

which is invariant under $\displaystyle p_{4}\leftrightarrow p_{5}$. After
manipulating the colour factors to obtain the natural association of one
adjoint generator with each three-point tree-level factor, we obtain

$\displaystyle\displaystyle\begin{split}{\mathrm{Re}\left[{\mathcal{M}^{(1)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\oplus})}\right]}&\xrightarrow[\mathrm{NMRK},\,p_{4}||p_{5}]{}\\\
\frac{g_{\scriptscriptstyle
S}^{6}}{(4\pi)^{2}}\kappa_{\Gamma}s_{12}&\left[F^{a_{3}a_{2}c_{1}}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\right]\frac{1}{t_{1}}\\\
\times&\left[F^{c_{1}c_{3}c_{2}}V^{g(0)}(q_{1},P^{\oplus},q_{3})\right]\left[F^{c_{2}a_{5}a_{4}}\mathrm{Split}^{(0)}_{\ominus}(p_{4}^{\oplus},p_{5}^{\oplus})\right]\\\
\times&\frac{1}{t_{3}}\left[F^{a_{6}a_{1}c_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\right]\times\bigg{\\{}N_{c}\left.{\mathrm{Re}\left[{V_{6}(\sigma_{A})}\right]}\right|_{\mathrm{NMRK},\,p_{4}||p_{5}}\bigg{\\}}\,.\end{split}$
(300)

The opposite-helicity case proceeds similarly. From eq. (287) we see that in
the collinear limit, both $\displaystyle S^{A}_{uv}$ and $\displaystyle
S^{A}_{wu}$ are equal to the tree-level amplitude, and these terms combine
with the $\displaystyle\Delta W_{A}$ term to play the same role as
$\displaystyle\Delta V_{A}$ in this limit,

$\displaystyle\displaystyle
A^{gg(0)}(q_{1},p_{4}^{\oplus},p_{4}^{\ominus},q_{3})\Delta
W_{A}(u_{A},v_{A},w_{A})+S^{A}_{uv}\log(v_{A})\left(\frac{w_{A}}{u_{A}}\right)+S^{A}_{wu}\log(w_{A})\left(\frac{v_{A}}{u_{A}}\right)$
$\displaystyle\displaystyle\xrightarrow[p_{4}||p_{5}]{}\left(\sum_{\nu=\pm}V^{g(0)}(q_{1},P^{\nu},q_{3})\mathrm{Split}^{(0)}_{-\nu}(p_{4}^{\oplus},p_{5}^{\ominus})\right)\left.\Delta
V_{A}(u_{A},v_{A},w_{A})\right|_{p_{4}||p_{5}}$ (301)

This means we can straightforwardly write

$\displaystyle\displaystyle\begin{split}{\mathrm{Re}\left[{\mathcal{M}^{(1)}_{\mathcal{N}=4}(p_{4}^{\oplus},p_{5}^{\ominus})}\right]}&\xrightarrow[\mathrm{NMRK},p_{4}||p_{5}]{}\\\
\frac{g_{\scriptscriptstyle
S}^{6}}{(4\pi)^{2}}\kappa_{\Gamma}s_{12}&\left[F^{a_{3}a_{2}c_{1}}C^{g(0)}\left(p_{2}^{\nu_{2}},p_{3}^{-\nu_{2}}\right)\right]\frac{1}{t_{1}}\\\
\times&\sum_{\nu=\pm}\left(\left[F^{c_{1}c_{3}c_{2}}V^{g(0)}(q_{1},P^{\nu},q_{3})\right]\left[F^{c_{2}a_{5}a_{4}}\mathrm{Split}^{(0)}_{-\nu}(p_{4}^{\oplus},p_{5}^{\ominus})\right]\right)\\\
\times&\frac{1}{t_{3}}\left[F^{a_{6}a_{1}c_{3}}C^{g(0)}\left(p_{1}^{\nu_{1}},p_{6}^{-\nu_{1}}\right)\right]\times\bigg{\\{}N_{c}\left.{\mathrm{Re}\left[{V_{6}(\sigma_{A})}\right]}\right|_{\mathrm{NMRK},\,p_{4}||p_{5}}\bigg{\\}}.\end{split}$
(302)

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