$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{3}\left(N_{c}^{2}+36\right)\frac{N_{c}}{864}\left(-\frac{1}{\epsilon^{3}}+70\hat{\zeta}_{3}+\mathcal{O}(\epsilon^{2})\right){\cal
M}^{{\rm tree},[8_{a}]}_{gg\to gg},$ (D.2b)
$\displaystyle\mathcal{\hat{M}}^{(-,3,1),[8_{a}]}_{qg\to qg}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{3}\left(N_{c}^{2}+36\right)\frac{N_{c}}{864}\left(-\frac{1}{\epsilon^{3}}+70\hat{\zeta}_{3}+\mathcal{O}(\epsilon^{2})\right){\cal
M}^{{\rm tree},[8_{a}]}_{qg\to qg},$ (D.2c)

where $\hat{\zeta}_{3}$ is defined in eq. (4.71).

At four loops we have from eq. (5.36)

$\displaystyle\mathcal{\hat{M}}^{(-,4,2),[8]}_{qq\to qq}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{4}\left[\frac{1}{\epsilon^{4}}\left(\frac{N_{c}^{2}}{384}-\frac{1}{384}\right)-\frac{\hat{\zeta}_{3}}{\epsilon}\left(\frac{13N_{c}^{2}}{24}+\frac{49}{96}\right)+\mathcal{O}(\epsilon)\right]{\cal
M}^{{\rm tree},[8]}_{qq\to qq}$ (D.3a)
$\displaystyle\mathcal{\hat{M}}^{(-,4,2),[8_{a}]}_{gg\to gg}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{4}\left[\frac{1}{\epsilon^{4}}\left(\frac{N_{c}^{2}}{64}+\frac{1}{16}\right)-\frac{\hat{\zeta}_{3}}{\epsilon}\left(\frac{101}{32}N_{c}^{2}+\frac{101}{8}\right)+\mathcal{O}(\epsilon)\right]{\cal
M}^{{\rm tree},[8_{a}]}_{gg\to gg},$ (D.3b)
$\displaystyle\mathcal{\hat{M}}^{(-,4,2),[8_{a}]}_{qg\to qg}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{4}N_{c}^{2}\left(\frac{1}{192\epsilon^{4}}-\frac{101\hat{\zeta}_{3}}{96\epsilon}+\mathcal{O}(\epsilon)\right){\cal
M}^{{\rm tree},[8_{a}]}_{qg\to qg}.$ (D.3c)

As noted below eq. (5.36), $\hat{\mathcal{M}}^{(-,4,2)}$ is non-planar, which
is confirmed by the subleading-$N_{c}$ nature of eqs. (D.3a)-(D.3c).

There are other non-zero components to the odd NNLL reduced amplitudes. The
singlet of $qq$ scattering

$\displaystyle\mathcal{\hat{M}}^{(-,2,0),[1]}_{qq\to qq}=$
$\displaystyle\,-(i\pi)^{2}r_{\Gamma}^{2}\left(N_{c}^{2}-4\right)\sqrt{N_{c}^{2}-1}\frac{1}{2N_{c}^{2}}S^{(2)}(\epsilon){\cal
M}^{{\rm tree},[8]}_{qq\to qq},$ (D.4a)
$\displaystyle\mathcal{\hat{M}}^{(-,3,1),[1]}_{qq\to qq}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{3}\left(N_{c}^{2}-4\right)\sqrt{N_{c}^{2}-1}\frac{1}{48N_{c}}\left(\frac{1}{\epsilon^{3}}+2\hat{\zeta}_{3}+\mathcal{O}(\epsilon^{2})\right){\cal
M}^{{\rm tree},[8]}_{qq\to qq},$ (D.4b)
$\displaystyle\mathcal{\hat{M}}^{(-,4,2),[1]}_{qq\to qq}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{4}\left(N_{c}^{2}-4\right)\sqrt{N_{c}^{2}-1}\frac{1}{256}\left(\frac{1}{\epsilon^{4}}-\frac{2\hat{\zeta}_{3}}{\epsilon}+\mathcal{O}(\epsilon)\right){\cal
M}^{{\rm tree},[8]}_{qq\to qq},$ (D.4c)

and the decuplet of $gg$ scattering

$\displaystyle\mathcal{\hat{M}}^{(-,2,0),[10+\bar{10}]}_{gg\to gg}=$
$\displaystyle\,-(i\pi)^{2}r_{\Gamma}^{2}\frac{\sqrt{N_{c}^{2}-4}}{\sqrt{2}}S^{(2)}(\epsilon){\cal
M}^{{\rm tree},[8_{a}]}_{gg\to gg},$ (D.5a)
$\displaystyle\mathcal{\hat{M}}^{(-,3,1),[10+\bar{10}]}_{gg\to gg}=$
$\displaystyle\,-(i\pi)^{2}r_{\Gamma}^{3}\frac{\sqrt{N_{c}^{2}-4}}{\sqrt{2}}\frac{N_{c}}{12}\left(\frac{1}{\epsilon^{3}}+38\hat{\zeta}_{3}+\mathcal{O}(\epsilon^{2})\right){\cal
M}^{{\rm tree},[8_{a}]}_{gg\to gg},$ (D.5b)
$\displaystyle\mathcal{\hat{M}}^{(-,4,2),[10+\bar{10}]}_{gg\to gg}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{4}\frac{\sqrt{N_{c}^{2}-4}}{\sqrt{2}}\bigg{[}\frac{5}{384\epsilon^{4}}\left(N_{c}^{2}+\frac{12}{5}\right)$
$\displaystyle\hskip
99.58464pt-\frac{91\hat{\zeta}_{3}}{192\epsilon}\left(N_{c}^{2}+\frac{1212}{91}\right)+\mathcal{O}(\epsilon)\bigg{]}{\cal
M}^{{\rm tree},[8_{a}]}_{gg\to gg}.$ (D.5c)

The other components have definite even signature and so do not contribute to
the odd amplitudes.

##  Appendix E The soft anomalous dimension in the high-energy limit

This appendix provides the known values of the corrections $\mathbf{\Delta}$
to the soft anomalous dimension dipole formula in the high energy limit.
$\mathbf{\Delta}$ start at three loops. The soft anomalous dimension in the
high-energy limit is introduced in eq. (2.26), and at three-loop order it is

$\mathbf{\Gamma}^{(3)}_{ij\to
ij}\left(L,\frac{-t}{\lambda^{2}}\right)=\frac{1}{2}\gamma_{K}^{(3)}\left[L\mathbf{T}_{t}^{2}+i\pi\mathbf{T}_{s-u}^{2}\right]+\Gamma_{i}^{(3)}\left(\frac{-t}{\lambda^{2}}\right)+\Gamma_{j}^{(3)}\left(\frac{-t}{\lambda^{2}}\right)+\sum_{m=0}^{2}\mathbf{\Delta}^{(3,m)}L^{m}$
(E.1)

with $\mathbf{\Delta}^{(3,m)}$ being corrections to the dipole formula at
three loops. The even signature part of the soft anomalous dimension at NLL is
two-loop exact Lipatov:1976zz ; Kuraev:1976ge ; Caron-Huot:2017zfo , that is,
the terms $\mathbf{\Delta}^{(+,\ell,\ell-1)}$ vanish at three loops and above.
The corrections at three loops were calculated explicitly in ref.
Almelid:2015jia , after which projections onto definite signatures and a
decomposition in a basis of colour operators suitable for the Regge limit were
found in ref. Caron-Huot:2017fxr . They read

$\displaystyle\mathbf{\Delta}^{(-,3,2)}$
$\displaystyle=\text{Im}\left[\mathbf{\Delta}^{(3,2)}\right]=0$ (E.2a)
$\displaystyle\mathbf{\Delta}^{(+,3,2)}$
$\displaystyle=\text{Re}\left[\mathbf{\Delta}^{(3,2)}\right]=0$ (E.2b)
$\displaystyle\mathbf{\Delta}^{(-,3,1)}$
$\displaystyle=i\pi\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\frac{1}{4}\zeta_{3}$
(E.2c) $\displaystyle\mathbf{\Delta}^{(+,3,1)}$
$\displaystyle=\text{Re}\left[\mathbf{\Delta}^{(3,1)}\right]=0$ (E.2d)
$\displaystyle\mathbf{\Delta}^{(-,3,0)}$
$\displaystyle=i\pi\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\frac{11}{4}\zeta_{4}$
(E.2e) $\displaystyle\mathbf{\Delta}^{(+,3,0)}$
$\displaystyle=\frac{1}{4}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\Big{(}4\zeta_{2}\zeta_{3}-\zeta_{5}\Big{)}-\frac{f^{(3)}}{2}\bigg{\\{}-\frac{5}{8}C_{A}^{2}\mathbf{T}_{t}^{2}+f^{abe}f^{cde}$
$\displaystyle\times\Big{[}\\{{\bf T}_{t}^{a},{\bf T}_{t}^{d}\\}\left(\\{{\bf
T}_{s-u}^{b},{\bf T}_{s-u}^{c}\\}+\\{{\bf T}_{s+u}^{b},{\bf
T}_{s+u}^{c}\\}\right)+\\{{\bf T}_{s-u}^{a},{\bf T}_{s-u}^{d}\\}\\{{\bf
T}^{b}_{s+u},{\bf T}_{s+u}^{c}\\}\Big{]}\bigg{\\}}.$ (E.2f)

where $f^{(3)}=\frac{1}{4}(\zeta_{2}\zeta_{3}+\zeta_{5})$. The generators in
eq. (E.2) are defined as

$\displaystyle{\bf T}_{s-u}^{a}\equiv\frac{1}{\sqrt{2}}\left({\bf
T}_{s}^{a}-{\bf T}_{u}^{a}\right)\hskip 28.45274pt{\bf
T}_{s+u}^{a}\equiv\frac{1}{\sqrt{2}}\left({\bf T}_{s}^{a}+{\bf
T}_{u}^{a}\right).$ (E.3)

These results are completely general: they require no direct action on the
tree amplitude. An alternative expression for the colour factor multiplying
$f^{(3)}$ in eq. (E.2) is derived in appendix G, and given by eq. (G.41). The
latter is only valid upon acting directly upon the octet in colour space,
${\bf T}^{a}_{i}{\bf T}^{a}_{j}$.

In section 6.5 we summarise the state-of-the-art knowledge of the soft
anomalous dimension at four-loop order in the high-energy limit. It may be
written as

$\mathbf{\Gamma}^{(4)}_{ij\to
ij}\left(L,\frac{-t}{\lambda^{2}}\right)=\frac{1}{2}\gamma_{K}^{(4)}\left[L\mathbf{T}_{t}^{2}+i\pi\mathbf{T}_{s-u}^{2}\right]+\Gamma_{i}^{(4)}\left(\frac{-t}{\lambda^{2}}\right)+\Gamma_{j}^{(4)}\left(\frac{-t}{\lambda^{2}}\right)+\sum_{m=0}^{3}\mathbf{\Delta}^{(4,m)}L^{m}.$
(E.4)

At NLL, there is a tower of signature-odd contributions to the soft anomalous
dimension (i.e. signature-even amplitude) emanating from two-Reggeon-exchange
ladder diagrams, which start at four loops Caron-Huot:2013fea  and they have
been determined to all orders Caron-Huot:2017zfo ; Caron-Huot:2020grv . In the
present paper we computed the signature-even component of the four-loop soft
anomalous dimension through NNLL, complementing information from lower
logarithmic accuracy in refs. Caron-Huot:2017zfo ; Caron-Huot:2013fea ;
Falcioni:2020lvv . The results are:

$\displaystyle\mathbf{\Delta}^{(+,4,3)}$
$\displaystyle=\text{Re}\left[\mathbf{\Delta}^{(4,3)}\right]=0$ (E.5a)
$\displaystyle\mathbf{\Delta}^{(-,4,3)}$
$\displaystyle=-i\pi\frac{\zeta_{3}}{24}\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\mathbf{T}_{t}^{2}$
(E.5b) $\displaystyle\mathbf{\Delta}^{(+,4,2)}$
$\displaystyle=\zeta_{2}\zeta_{3}\bigg{(}\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}-\frac{1}{4}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]+\frac{3}{4}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\bigg{)}\,.$
(E.5c)

Note that eqs. (E.2c), (E.2e) and (E.5b) contain a single power of
$i\pi\mathbf{T}_{s-u}^{2}$, consistent with the odd signature of the soft
anomalous dimension. The odd contribution $\mathbf{\Delta}^{(-,4,2)}$ is not
yet known, and neither are the subleading logarithms $\mathbf{\Delta}^{(4,1)}$
and $\mathbf{\Delta}^{(4,0)}$.

##  Appendix F The cusp and the soft anomalous dimensions

In this appendix we study the relation between the soft anomalous dimension
for $n$-parton scattering in general kinematics, ${\bf\Gamma}_{n}$, and the
lightlike cusp anomalous dimension $\Gamma^{\rm{cusp}}_{i}$. This relation is
governed by the collinear anomaly constraint of eq. (7.3) and concerns only
terms that are linear in the logarithm
$l_{ij}=\log\frac{-s_{ij}}{\lambda^{2}}$. All other terms in ${\bf\Gamma}_{n}$
depend on the kinematics only via conformally-invariant cross ratios (CICRs)
and will not be discussed here.

We have seen two particular solutions to the inhomogeneous equations (7.3).
The first, given by eq. (7.10), has a trivial colour structure, while the
second, given by the terms that depend on the scale in eq. (7.13), namely
$\bm{\Gamma}_{n}^{\rm{dip}}$ and $\bm{\Gamma}_{n,\rm{Q4T-2,3L}}$, has a non-
trivial colour structure, and it can be shown to satisfy eq. (7.3) using
colour conservation. We will verify that the difference between these two
particular solutions is a function of conformally-invariant cross ratios
(CICRs). Specialising to four-parton scattering and using colour conservation,
the latter will be expressed in terms of colour operators that involve
generators of just two of the four partons, a representation that is
convenient for deriving the Regge limit of eqs. (7.85) and (7.86), which is
the final goal of this appendix.

##### Relating particular solutions of the collinear anomaly constraints.

The soft anomalous dimension must satisfy the collinear anomaly constraint
Becher:2009cu ; Becher:2009qa ; Gardi:2009qi ; Gardi:2009zv  of eq. (7.3),
namely the inhomogeneous differential equations

$\sum_{j\neq
i}\frac{d{\bf\Gamma}_{n}(\\{s_{kl}\\},\lambda,\alpha_{s})}{d(l_{ij})}=\Gamma^{\rm{cusp}}_{i}(\alpha_{s})\,,$
(F.1)

for all $i$, where the derivatives on the left-hand side are taken with
respect to the logarithms $l_{ij}=\log\frac{-s_{ij}}{\lambda^{2}}$ in eq.
(7.4), while the right-hand side is the lightlike cusp anomalous dimension
Korchemsky:1991zp ; Korchemsky:1987wg  for particle $i$. The latter has a
simple dependence on the representation of the Wilson line. In particular, it
is proportional to the quadratic Casimir through three loops, a property known
as _Casimir scaling_ , and it acquires additional dependence on the
representation through a quartic Casimir starting at four loops Boels:2017ftb
; Boels:2017skl ; Moch:2017uml ; Grozin:2017css , as stated in eq. (2.28),
namely

$\Gamma_{i}^{\rm{cusp}}(\alpha_{s})=\frac{1}{2}\gamma_{K}(\alpha_{s})C_{i}+\sum_{R}g_{R}(\alpha_{s})\frac{d_{RR_{i}}}{N_{R_{i}}}+\mathcal{O}(\alpha_{s}^{5}),$
(F.2)

where $C_{i}$ is the quadratic Casimir and $\frac{d_{RR_{i}}}{N_{R_{i}}}$ the
quartic Casimir from eq. (7.26). Explicit expressions for
$\frac{d_{AA}}{N_{A}},\frac{d_{AF}}{N_{F}}$ are given in eq. (4.42). In
appendix B, $\gamma_{K}$ is given up to three loops in eq. (B) and $g_{F}$ and
$g_{A}$ are provided at four-loop order in eq. (B.3).

In section 7.1 we encountered two different particular solutions to the
inhomogeneous equations (F.1): the first Mark , in eq. (7.10), true for $n\geq
3$, is expressed directly in terms of $\Gamma_{i}^{\rm{cusp}}(\alpha_{s})$

$\displaystyle\begin{split}\Gamma_{n}^{\prime}\left(\left\\{\frac{s_{ij}}{\lambda^{2}}\right\\},\alpha_{s}\right)&\equiv\frac{1}{2(n-1)}\sum_{i=1}^{n}\sum_{j\neq
i}^{n}l_{ij}\left(\Gamma^{\rm{cusp}}_{i}(\alpha_{s})+\Gamma^{\rm{cusp}}_{j}(\alpha_{s})-\frac{1}{(n-2)}\sum_{k\neq
i,j}^{n}\Gamma_{k}^{\text{cusp}}(\alpha_{s})\right),\end{split}$ (F.3)

and thus naturally incorporates all higher Casimirs. The second was presented
in ref. Becher:2019avh , starting with the sum-over-dipoles formula and
including quartic operators coupled to two or three partons, so as to satisfy
(F.1) subject to colour conservation. It reads:

$\displaystyle\begin{split}\bm{\Gamma}^{\rm{lin.}}_{n}(\\{s_{ij}\\},\lambda,\alpha_{s})\equiv&\left(\bm{\Gamma}_{n}^{\rm{dip}}(\\{s_{ij}\\},\lambda,\alpha_{s})-\sum_{i}^{n}\,\gamma_{i}(\alpha_{s})\right)\,+\,\bm{\Gamma}_{n,\rm{Q4T-2,3L}}(\\{s_{ij}\\},\lambda,\alpha_{s})\\\
=&-\frac{1}{4}\gamma_{K}(\alpha_{s})\sum_{(i,j)}\,{\bf T}_{i}\cdot{\bf
T}_{j}\,l_{ij}\\\
&-\frac{1}{2}\sum_{R}g_{R}(\alpha_{s})\Bigg{[}\sum_{(i,j)}\,\big{(}\bm{\bm{{\cal
D}}}_{iijj}^{R}+2\bm{\bm{{\cal
D}}}_{iiij}^{R}\big{)}l_{ij}+\sum_{(i,j,k)}\bm{\bm{{\cal
D}}}_{ijkk}^{R}\,l_{ij}\Bigg{]}\,,\end{split}$ (F.4)

where

$\bm{\bm{\mathcal{D}}}_{ijkl}^{R}=\frac{1}{4!}\sum_{\sigma\in
S_{4}}\operatorname{Tr}_{R}\left[T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right]{\bf
T}_{i}^{a}{\bf T}_{j}^{b}{\bf T}_{k}^{c}{\bf T}_{l}^{d}.$ (F.5)

When all the partonic indices $\\{i,j,k,l\\}$ are the same, it is a quartic
Casimir written in eq. (7.26).

At this point we have two particular solutions to eq. (F.1), as can be checked
by substitution and using colour conservation. Crucially, both these solutions
are strictly linear in the logarithm $l_{ij}$ and we can directly compare them
and define their difference as

$\mathbf{\delta\Gamma}_{n}^{\rm{lin.}}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)\equiv\mathbf{\Gamma}^{\rm{lin.}}_{n}(\\{s_{ij}\\},\lambda,\alpha_{s})-\Gamma_{n}^{\prime}\left(\left\\{\frac{s_{ij}}{\lambda^{2}}\right\\},\alpha_{s}\right),$
(F.6)

which is linear in the logarithms, and at the same time only depends on
combinations that are rescaling invariant, satisfying the homogeneous version
of eq. (F.1). At $n=3$ we have $\mathbf{\delta\Gamma}_{3}^{\rm{lin.}}=0$,
while for four or more partons $\mathbf{\delta\Gamma}_{n}^{\rm{lin.}}$ is
nonzero with its form given below. Note that in this section we consider only
linear terms and ignore homogeneous solutions that could have more complicated
dependence on the velocities.

Computing the difference between eq. (F.4) and eq. (F.3) and using colour
conservation we obtain:

$\mathbf{\delta\Gamma}^{\rm{lin.}}_{n}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)=-\sum_{(i,j,k,l)}\\!\frac{\beta_{ijkl}}{n-2}\Bigg{[}\frac{\gamma_{K}(\alpha_{s})}{4(n-1)}{\bf
T}_{i}\cdot{\bf
T}_{j}+\sum_{R}\frac{g_{R}(\alpha_{s})}{2}\left(\frac{1}{(n-1)}\bm{\mathcal{D}}^{R}_{iiij}+\bm{\mathcal{D}}^{R}_{ijkk}\right)\Bigg{]},$
(F.7)

for $n\geq 4$. The first term in the square brackets is related to the
quadratic Casimir. The second, which is related to the quartic Casimir, will
be denoted by $\mathbf{\delta\Gamma}^{\rm{lin.}}_{n,{\rm Q}}$.

##### Four-parton scattering.

Specialising to four-parton scattering, $n=4$, in eq. (F.7), we find that
further simplification is possible owing to colour conservation. The following
three ways of expressing the quartic part of eq. (F.7) are all equivalent,
each using a combination of just two of the three types of colour operators
$\\{\bm{\mathcal{D}}^{R}_{iijj},\bm{\mathcal{D}}^{R}_{iiij},\bm{\mathcal{D}}^{R}_{ijkk}\\}$:

$\mathbf{\delta\Gamma}^{\rm{lin.}}_{4,{\rm
Q}}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)=\begin{cases}-\sum\limits_{R}g_{R}(\alpha_{s})\sum\limits_{(i,j,k,l)\in
S_{4}}\beta_{ijkl}\left(\frac{1}{12}\bm{\mathcal{D}}^{R}_{iiij}+\frac{1}{4}\bm{\mathcal{D}}^{R}_{ijkk}\right)\\\\[22.76228pt]
-\sum\limits_{R}g_{R}(\alpha_{s})\sum\limits_{(i,j,k,l)\in
S_{4}}\beta_{ijkl}\left(-\frac{1}{12}\bm{\mathcal{D}}^{R}_{iijj}+\frac{1}{3}\bm{\mathcal{D}}^{R}_{ijkk}\right)\\\\[22.76228pt]
-\sum\limits_{R}g_{R}(\alpha_{s})\sum\limits_{(i,j,k,l)\in
S_{4}}\beta_{ijkl}\left(\frac{1}{3}\bm{\mathcal{D}}^{R}_{iiij}+\frac{1}{4}\bm{\mathcal{D}}^{R}_{iijj}\right).\end{cases}$
(F.8)

Of these three we favour using the third, where each term depends on colour
generators of just two of the four legs. With this, we can then express eq.
(F.4) for four partons as

$\displaystyle\mathbf{\Gamma}_{4}^{\rm{lin.}}\left(\left\\{\frac{s_{ij}}{\lambda^{2}}\right\\},\alpha_{s}\right)$
$\displaystyle=\Gamma_{4}^{\prime}\left(\left\\{\frac{s_{ij}}{\lambda^{2}}\right\\},\alpha_{s}\right)+\mathbf{\delta\Gamma}_{4}^{\rm{lin}}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)$
$\displaystyle=\frac{1}{6}\sum_{i=1}^{4}\sum_{j\neq
i}^{4}l_{ij}\left(\Gamma^{\rm{cusp}}_{i}(\alpha_{s})+\Gamma^{\rm{cusp}}_{j}(\alpha_{s})-\frac{1}{2}\sum_{k\neq
i,j}^{4}\Gamma_{k}^{\text{cusp}}(\alpha_{s})\right)$
$\displaystyle\quad\mbox{}-\sum_{(i,j,k,l)\in
S_{4}}\beta_{ijkl}\Bigg{[}\frac{\gamma_{K}(\alpha_{s})}{24}{\bf
T}_{i}\cdot{\bf
T}_{j}+\sum_{R}g_{R}(\alpha_{s})\left(\frac{1}{3}\bm{\mathcal{D}}^{R}_{iiij}+\frac{1}{4}\bm{\mathcal{D}}^{R}_{iijj}\right)\Bigg{]}.$
(F.9)

The first line is proportional to the unit matrix in colour space while the
second line involves non-trivial colour operators. In appendix G, each part of
this expression will be translated to a Regge-limit colour basis and the Regge
limit is then taken.

##### Linear contributions to the soft anomalous dimension in the Regge limit.

Let us now summarise the expressions for the cusp-related contributions in the
soft anomalous dimension in the Regge limit using eq. (F) with results from
appendix G.

Using the result from eq. (G.8), the first line of eq. (F) in the Regge limit
is

$\displaystyle\Gamma_{4}^{\prime}\left(L,\frac{-t}{\lambda^{2}},\alpha_{s}\right)=\left(\frac{2}{3}L+\log\frac{-t}{\lambda^{2}}\right)\left(\Gamma^{\rm{cusp}}_{i}(\alpha_{s})+\Gamma^{\rm{cusp}}_{j}(\alpha_{s})\right).$
(F.10)

The subscript $i$ corresponds to the projectile (partons 1 and 4), while $j$
corresponds to the target (partons 2 and 3). The whole expression is even
under the exchange of $s\leftrightarrow u$.

Next we consider the second line of eq. (F) in the Regge limit. Using results
from eqs. (G.11) and (G.14), and then separating the expression into
signature-even and odd parts, we obtain:

$\displaystyle\begin{split}\delta\mathbf{\Gamma}^{\rm{lin.}}_{4}(L,\alpha_{s})&=L\Bigg{\\{}\frac{\gamma_{K}(\alpha_{s})}{6}\left(3\mathbf{T}_{t}^{2}-2C_{i}-2C_{j}\right)+\sum_{R}\frac{g_{R}(\alpha_{s})}{3}\Big{(}2\bm{\mathcal{D}}_{tttt}^{R}-\bm{\mathcal{D}}_{ssss}^{R}-\bm{\mathcal{D}}_{uuuu}^{R}\Big{)}\Bigg{\\}}\\\
&\quad\mbox{}+\frac{i\pi}{2}\Bigg{\\{}\gamma_{K}(\alpha_{s})\mathbf{T}_{s-u}^{2}+\sum_{R}g_{R}(\alpha_{s})\Big{(}\bm{\mathcal{D}}_{ssss}^{R}-\bm{\mathcal{D}}_{uuuu}^{R}\Big{)}\Bigg{\\}}\,,\end{split}$
(F.11)

where $\mathbf{T}_{s-u}^{2}=\frac{1}{2}({\bf T}_{s}^{2}-{\bf T}_{u}^{2})$ and
where we use the notation in eq. (7.50) for the quartic operators associated
with the three channels. The first line in eq. (F.11) is real and signature-
even, while the second line is imaginary and signature-odd.

To proceed it is useful to define a cusp anomalous dimension operator
associated with each channel in the following way:

$\bm{\Gamma}_{p}^{\rm{cusp}}(\alpha_{s})\equiv\frac{1}{2}\gamma_{K}(\alpha_{s}){\bf
T}_{p}^{2}+\sum_{R}g_{R}(\alpha_{s})\bm{\mathcal{D}}^{R}_{pppp}+{\cal
O}(\alpha_{s}),$ (F.12)

where $p\in\\{s,t,u\\}$. We can then write eq. (F.11) as

$\displaystyle\begin{split}\delta\mathbf{\Gamma}^{\rm{lin.}}_{4}(L,\alpha_{s})&=L\bigg{(}\bm{\Gamma}_{t}^{\rm{cusp}}(\alpha_{s})-\frac{\gamma_{K}(\alpha_{s})}{3}\left(C_{i}+C_{j}\right)-\sum_{R}\frac{g_{R}(\alpha_{s})}{3}\Big{(}\bm{\mathcal{D}}_{ssss}^{R}+\bm{\mathcal{D}}_{uuuu}^{R}+\bm{\mathcal{D}}_{tttt}^{R}\Big{)}\bigg{)}\\\
&\quad\mbox{}+\frac{i\pi}{2}\Big{(}\bm{\Gamma}_{s}^{\rm{cusp}}(\alpha_{s})-\bm{\Gamma}_{u}^{\rm{cusp}}(\alpha_{s})\Big{)}.\end{split}$
(F.13)

Combining eqs. (F.10) and (F.13) we observe that the quadratic Casimir terms
multiplying the high-energy logarithm $L$ cancel. Separating the contributions
by signature
$\mathbf{\Gamma}_{4}^{\rm{lin.}}=\mathbf{\Gamma}_{4}^{\rm{lin.}\,(+)}+\mathbf{\Gamma}_{4}^{\rm{lin.}\,(-)}$,
we find

$\displaystyle\mathbf{\Gamma}_{4}^{\rm{lin.}\,(+)}\left(L,\frac{-t}{\lambda^{2}},\alpha_{s}\right)$
$\displaystyle=L\bm{\Gamma}_{t}^{\rm{cusp}}(\alpha_{s})\,+\,\log\frac{-t}{\lambda^{2}}\left(\Gamma^{\rm{cusp}}_{i}(\alpha_{s})+\Gamma^{\rm{cusp}}_{j}(\alpha_{s})\right)$
(F.14a)
$\displaystyle+L\sum_{R}\frac{g_{R}(\alpha_{s})}{3}\left(-\bm{\mathcal{D}}^{R}_{tttt}-\bm{\mathcal{D}}^{R}_{ssss}-\bm{\mathcal{D}}^{R}_{uuuu}+2\left(\frac{d_{RR_{i}}}{N_{R_{i}}}+\frac{d_{RR_{j}}}{N_{R_{j}}}\right)\right)$
$\displaystyle\mathbf{\Gamma}_{4}^{\rm{lin.}\,(-)}\left(\alpha_{s}\right)$
$\displaystyle=\frac{i\pi}{2}\Big{(}\bm{\Gamma}_{s}^{\rm{cusp}}(\alpha_{s})-\bm{\Gamma}_{u}^{\rm{cusp}}(\alpha_{s})\Big{)}.\,$
(F.14b)

We thus see that quadratic Casimir coefficient $\gamma_{K}(\alpha_{s})$ does
not anymore appear in isolation, but only as part of the complete channel-
related cusp anomalous dimension operator of eq. (F.12). Note that the quartic
coefficient $g_{R}(\alpha_{s})$ does feature on its own. The two expressions
in eq. (F.14) constitute part of the final expressions for the Regge-limit
$\mathbf{\Gamma}_{4}$ in eqs. (7.85) and (7.86), where additional
contributions involving the quartic Casimir arise, which naturally combine
with the second line in eq. (F.14a).

Upon considering $\mathbf{\Gamma}_{4}^{\rm{lin.}}$ to be acting on the tree
amplitude, and specialising to the adjoint representation $R=A$, we may use
results from eqs. (G.8), (G.11), (G.19) and (G.36), to obtain explicit
expressions in terms of quartic Casimirs and commutators:

$\displaystyle\mathbf{\Gamma}_{4,A}^{\rm{lin.}(+)}\left(L,\frac{-t}{\lambda^{2}},\alpha_{s}\right)\mathcal{M}_{\text{tree}}$
$\displaystyle=\Bigg{\\{}L{\bf\Gamma}_{t}^{\rm{cusp}}\,+\,\log\frac{-t}{\lambda^{2}}\left(\Gamma^{\rm{cusp}}_{i}+\Gamma^{\rm{cusp}}_{j}\right)$
(F.15a)
$\displaystyle\qquad-L\frac{g_{A}(\alpha_{s})}{6}\bigg{(}2\left(\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}\right)-\frac{1}{2}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]$
$\displaystyle\qquad+\frac{3}{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{2}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\bigg{)}\Bigg{\\}}\mathcal{M}_{\text{tree}}$

$\displaystyle\mathbf{\Gamma}_{4,A}^{\rm{lin.}(-)}\left(\alpha_{s}\right)\mathcal{M}_{\text{tree}}$
$\displaystyle=$ (F.15b)
$\displaystyle\frac{i\pi}{2}\Bigg{\\{}\bigg{[}\frac{2}{C_{A}}\Gamma_{A}^{\rm{cusp}}(\alpha_{s})\,+\,4g_{A}(\alpha_{s})\left(\frac{d_{AR_{i}}}{C_{i}N_{R_{i}}}+\frac{d_{AR_{j}}}{C_{j}N_{R_{j}}}-\frac{d_{AA}}{2C_{A}N_{A}}-\frac{C_{A}^{3}}{16}\right)\bigg{]}\mathbf{T}_{s-u}^{2}$
$\displaystyle-\frac{g_{A}(\alpha_{s})}{8}\bigg{(}3\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}+\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\mathbf{T}_{t}^{2}-3\mathbf{T}_{t}^{2}[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\mathbf{T}_{t}^{2}\bigg{)}\Bigg{\\}}\mathcal{M}_{\text{tree}}.$

In these expressions we used the complete cusp anomalous dimension, rather
than the coefficient of the quadratic component, $\gamma_{K}(\alpha_{s})$.
Recalling eq. (4.42), we see that the remaining terms proportional to the
quartic component $g_{A}(\alpha_{s})$ are manifestly subleading in the
large-$N_{c}$ limit. Thus, in the planar limit, the leading contributions come
exclusively from the $\Gamma^{\rm{cusp}}$ terms and the collinear anomalous
dimensions.

##  Appendix G Converting colour structures to the Regge-limit basis

In this section, we study contributions to the soft anomalous dimension up to
four loops in eq. (7.13) at $\mathcal{O}(L^{0})$ and $\mathcal{O}(L)$, which
have not been discussed in section 7.2. We will translate the respective
colour structures to the Regge-limit basis and then take the Regge limit. We
will also elaborate on how the $\bm{\Gamma}_{\rm{5T-4L}}$ colour structures in
a Regge-limit basis were derived.

First let us revisit eq. (7.13), following the analysis we conducted in
appendix F of the terms that are linear in the logarithms
$l_{ij}=\log\frac{-s_{ij}}{\mu^{2}}$. These linear terms can be expressed in
terms of the particular solution to eq. (7.3) presented in eq. (7.10), plus
linear homogeneous terms provided in eq. (F.7). We can thus write eq. (7.13)
as

$\displaystyle\begin{split}\mathbf{\Gamma}_{4}\left(\\{s_{ij}\\},\lambda,\alpha_{s}(\lambda^{2})\right)&=\Gamma^{\prime}_{4}\left(\\{s_{ij}\\},\lambda,\alpha_{s}\right)+\delta\mathbf{\Gamma}^{\rm{lin}}_{4}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)+\mathbf{\Gamma}_{4,\rm
4T-3L}\left(\alpha_{s}\right)\\\\[2.84544pt]
&+\sum_{m=1}^{4}\,\gamma_{m}(\alpha_{s})+\mathbf{\Gamma}_{4,\rm
4T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)+\mathbf{\Gamma}_{4,\rm
Q4T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)\\\\[2.84544pt]
&+\mathbf{\Gamma}_{4,\rm
5T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right).\end{split}$ (G.1)

The first term
$\Gamma^{\prime}_{4}\left(\\{s_{ij}\\},\lambda,\alpha_{s}\right)$ is the only
one with scale dependence and it is by itself a particular solution to the
inhomogeneous equations (7.3). It is a unit matrix in colour space (hence it
is not written in boldface notation). It contributes at ${\cal O}(L)$ and
${\cal}(L^{0})$ with a simple expression in the Regge limit derived below in
eq. (G.8). The second term
$\delta\mathbf{\Gamma}^{\rm{lin}}_{4}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)$
consists of non-trivial colour operators; it depends linearly on the logarithm
of CICRs, and hence contributes at ${\cal O}(L)$ and ${\cal O}(L^{0})$. It
comprises two contributions: one is written in terms of colour dipoles, while
the other involves quartic Casimir operators. The former was evaluated in the
Regge limit in ref. DelDuca:2011ae  and it is re-derived here in eq. (G.11),
while the latter is given in the Regge limit in eq. (G.14). The third term on
the first line of eq. (G.1), $\mathbf{\Gamma}_{4,\rm
4T-3L}\left(\alpha_{s}\right)$, is independent of the kinematics, and hence
only contributes at ${\cal O}(L^{0})$. An expression for its colour structures
in terms of operators $\mathbf{T}^{a}_{s-u}$, $\mathbf{T}^{a}_{s+u}$ and
$\mathbf{T}^{a}_{t}$ was given in ref. Caron-Huot:2017fxr . Here we derive an
alternative expression in the Regge-limit basis, eq. (G.41), involving quartic
Casimirs related to the representations of the scattered particles. The
collinear anomalous dimension $\gamma_{m}$ remains unchanged in the Regge
limit and it only contributes at ${\cal O}(L^{0})$. The remaining two terms on
the second line of eq. (G.1) were discussed in detail in section 7.2 and will
not be considered further here. Finally, the term on the last line of eq.
(G.1) was briefly discussed in section 7.2 with further details given in the
final part of this appendix.

### $\Gamma_{4}^{\prime}$ in the Regge limit

This linear expression is a particular inhomogeneous solution to eq. (7.3) and
is given in eq. (F.3). The cusp anomalous dimension is discussed in the
previous appendix and its form is in eq. (7.5). For four partons, the
particular inhomogenous solution is

$\displaystyle\Gamma_{4}^{\prime}\left(\left\\{\frac{s_{ij}}{\lambda^{2}}\right\\},\alpha_{s}\right)$
$\displaystyle=\frac{1}{6}\sum_{i=1}^{4}\sum_{j\neq
i}^{4}l_{ij}\left(\Gamma^{\rm{cusp}}_{i}(\alpha_{s})+\Gamma^{\rm{cusp}}_{j}(\alpha_{s})-\frac{1}{2}\sum_{k\neq
i,j}^{4}\Gamma_{k}^{\text{cusp}}(\alpha_{s})\right).$ (G.2)

Performing the sum and using the fact that for $2\to 2$ scattering the
logarithms are expressed in terms of Mandelstam variables

$\displaystyle l_{34}$
$\displaystyle=l_{12}=\log\frac{se^{-i\pi}}{\lambda^{2}};\qquad
l_{24}=l_{13}=\log\frac{-u}{\lambda^{2}};\qquad
l_{23}=l_{14}=\log\frac{-t}{\lambda^{2}},$ (G.3)

then eq. (G.2) becomes

$\displaystyle\Gamma_{4}^{\prime}\left(\left\\{\underline{s}\right\\},\lambda,\alpha_{s}\right)$
$\displaystyle=\frac{1}{6}\left(\log\frac{se^{-i\pi}}{\lambda^{2}}+\log\frac{-u}{\lambda^{2}}+\log\frac{-t}{\lambda^{2}}\right)\bigg{(}\Gamma^{\rm{cusp}}_{1}(\alpha_{s})+\Gamma^{\rm{cusp}}_{2}(\alpha_{s})$
$\displaystyle\quad\mbox{}+\Gamma^{\rm{cusp}}_{3}(\alpha_{s})+\Gamma^{\rm{cusp}}_{4}(\alpha_{s})\bigg{)},$
(G.4)

where $\left\\{\underline{s}\right\\}=\\{s,t,u\\}$. The subscript of $i$
corresponds to partons 1 and 4 of the projectile, while $j$ corresponds to
partons 2 and 3 of the target. Using target-projectile notation, we therefore
make the following identification

$\Gamma^{\rm{cusp}}_{R_{1}}=\Gamma^{\rm{cusp}}_{R_{4}}=\Gamma^{\rm{cusp}}_{i}\hskip
28.45274pt\text{and}\,\,\hskip
28.45274pt\Gamma^{\rm{cusp}}_{R_{2}}=\Gamma^{\rm{cusp}}_{R_{3}}=\Gamma^{\rm{cusp}}_{j},$
(G.5)

so eq. (G) becomes

$\displaystyle\Gamma_{4}^{\prime}\left(\left\\{\underline{s}\right\\},\lambda,\alpha_{s}\right)=\frac{1}{3}\left(\log\frac{se^{-i\pi}}{\lambda^{2}}+\log\frac{-u}{\lambda^{2}}+\log\frac{-t}{\lambda^{2}}\right)\left(\Gamma^{\rm{cusp}}_{i}(\alpha_{s})+\Gamma^{\rm{cusp}}_{j}(\alpha_{s})\right).$
(G.6)

Eq. (G.6) is even under the exchange of $s\leftrightarrow u$. Upon taking the
Regge limit of

$s\simeq-u\gg-t,$

the logarithms are expressed in terms of the signature-even logarithm $L$ of
eq. (2.9) up to power-suppressed terms, getting

$\displaystyle\begin{split}\log\frac{-u}{\lambda^{2}}\simeq\log\frac{s}{\lambda^{2}}\simeq
L+\log\frac{-t}{\lambda^{2}}+\frac{i\pi}{2}.\end{split}$ (G.7)

In the Regge limit, eq. (G.6) is therefore expressed as

$\displaystyle\Gamma_{4}^{\prime}\left(L,\frac{-t}{\lambda^{2}},\alpha_{s}\right)=\left(\frac{2}{3}L+\log\frac{-t}{\lambda^{2}}\right)\left(\Gamma^{\rm{cusp}}_{i}(\alpha_{s})+\Gamma^{\rm{cusp}}_{j}(\alpha_{s})\right).$
(G.8)

This expression is signature-even and real. It is a unit matrix in colour
space and contains the complete cusp anomalous dimensions for partons $i$ and
$j$.

### $\delta{\bm{\Gamma}}^{\rm{lin.}}$ in the Regge limit

We now translate the colour structures multiplying CICRs in
$\delta\bm{\Gamma}^{\rm{lin.}}$ from eq. (F)

$\displaystyle\delta\bm{\Gamma}_{4}^{\rm{lin.}}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)$
$\displaystyle=\mathbf{\delta\Gamma}_{4,{\rm
2T-2L}}^{\rm{lin.}}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)+\mathbf{\delta\Gamma}_{4,{\rm
Q}}^{\rm{lin.}}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)$
$\displaystyle=-\sum_{(i,j,k,l)\in
S_{4}}\beta_{ijkl}\Bigg{[}\frac{\gamma_{K}(\alpha_{s})}{24}{\bf
T}_{i}\cdot{\bf
T}_{j}+\sum_{R}g_{R}(\alpha_{s})\left(\frac{1}{3}\bm{\mathcal{D}}^{R}_{iiij}+\frac{1}{4}\bm{\mathcal{D}}^{R}_{iijj}\right)\Bigg{]}\,,$
(G.9)

into the Regge-limit basis containing $\mathbf{T}_{t}^{2}$ and
$\mathbf{T}_{s-u}^{2}$ operators. The logarithm of the CICRs, $\beta_{ijkl}$,
is defined in eq. (7.7). The first term in eq. (G) is related to the dipole
term and the second to the quartic Casimir terms. We will analyse each of
these terms in turn.

##### The dipole component in the Regge limit.

The first term in eq. (G) is

$\mathbf{\delta\Gamma}_{4,{\rm
2T-2L}}^{\rm{lin.}}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)\equiv-\frac{\gamma_{K}(\alpha_{s})}{24}\sum_{(i,j,k,l)\in
S_{4}}\beta_{ijkl}\,{\bf T}_{i}\cdot{\bf T}_{j}.$ (G.10)

We expand the sum and we write $\beta_{ijkl}$ in terms of logarithms of the
Mandelstam invariants, using eqs. (7.4), (7.7) and (G.3). The Regge limit is
taken using eq. (G.7). Then we express the dipole operators,
$\mathbf{T}_{i}\cdot\mathbf{T}_{j}$, in terms of the channel operators defined
in eq. (2.17), obtaining

$\displaystyle\begin{split}\mathbf{\delta\Gamma}_{4,{\rm
2T-2L}}^{\rm{lin.}}\left(L,\alpha_{s}\right)&=\frac{\gamma_{K}(\alpha_{s})}{6}\bigg{[}L\Big{(}3\mathbf{T}_{t}^{2}-2C_{i}-2C_{j}\Big{)}+3i\pi\mathbf{T}_{s-u}^{2}\bigg{]},\end{split}$
(G.11)

where $j$ corresponds to partons 2 and 3 and $i$ corresponds to partons 1 and
4 in the target-projectile notation. The first bracket is signature even and
real. The term with $\mathbf{T}_{s-u}^{2}$ is signature odd and imaginary.

##### The quartic component $\delta{\bm{\Gamma}}^{\rm{lin.}}_{4,Q}$ in the
Regge limit.

We now turn to the second term on the last line of eq. (G) given by

$\displaystyle\begin{split}\delta\bm{\Gamma}_{4,\rm{Q}}(\\{\beta_{ijkl}\\},\alpha_{s})&=-\sum_{R}g_{R}(\alpha_{s})\sum_{(i,j,k,l)\in
S_{4}}\beta_{ijkl}\,\bigg{(}\frac{1}{3}\bm{\mathcal{D}}^{R}_{iiij}+\frac{1}{4}\bm{\mathcal{D}}^{R}_{iijj}\bigg{)},\end{split}$
(G.12)

where the general expression $\bm{\mathcal{D}}^{R}_{ijkl}$ is given in eq.
(7.25). The function $g_{R}(\alpha_{s})$ is the quartic component of the cusp
anomalous dimension of eq. (2.28); its value at four-loop order is given in
eq. (B.3).

After expanding the sum in eq. (G.12), then using eq. (7.52), the colour
structures are expressed in terms of the operators
$\bm{\mathcal{D}}^{R}_{ssss}$, $\bm{\mathcal{D}}^{R}_{tttt}$ and
$\bm{\mathcal{D}}^{R}_{uuuu}$ introduced in eq. (7.50). The logarithms are
expressed in terms of $\\{s,t,u\\}$ with eq. (G.3), and then separating into
signature-even and odd parts, we have

$\displaystyle\begin{split}\delta\bm{\Gamma}_{\rm{Q}}(\\{\underline{s}\\},\lambda,\alpha_{s})&=\sum_{R}\frac{g_{R}(\alpha_{s})}{6}\Bigg{[}\Big{(}-i\pi+\log\frac{s}{\lambda^{2}}-\log\frac{-u}{\lambda^{2}}\Big{)}\Big{(}3\bm{\mathcal{D}}^{R}_{uuuu}-3\bm{\mathcal{D}}^{R}_{ssss}\Big{)}\\\
&\quad\mbox{}+\Big{(}-i\pi+\log\frac{s}{\lambda^{2}}+\log\frac{-u}{\lambda^{2}}-2\log\frac{-t}{\lambda^{2}}\Big{)}\bigg{(}-2\bm{\mathcal{D}}^{R}_{ssss}-2\bm{\mathcal{D}}^{R}_{uuuu}+4\bm{\mathcal{D}}^{R}_{tttt}\bigg{)}\Bigg{]}.\end{split}$
(G.13)

The first line is signature odd under $s\leftrightarrow u$ in both the colour
and kinematics, and the last line is even in both. Taking the Regge limit and
using eq. (G.7), the expression becomes

$\displaystyle\begin{split}\delta\bm{\Gamma}_{\rm{Q}}\left(L,\alpha_{s}\right)&=\sum_{R}g_{R}(\alpha_{s})\bigg{\\{}\frac{L}{3}\bigg{[}-\bm{\mathcal{D}}^{R}_{ssss}-\bm{\mathcal{D}}^{R}_{uuuu}+2\bm{\mathcal{D}}^{R}_{tttt}\bigg{]}+\frac{i\pi}{2}\left(\bm{\mathcal{D}}^{R}_{ssss}-\bm{\mathcal{D}}^{R}_{uuuu}\right)\bigg{\\}}.\end{split}$
(G.14)

The first bracket, which contributes at $\rm{N^{3}LL}$ at four loops, is real
and signature even, while the second is odd and imaginary. We can split them
into even and odd terms with

$\delta\bm{\Gamma}_{\rm{Q}}\left(L,\alpha_{s}\right)=\delta\bm{\Gamma}_{\rm{Q}}^{(+)}\left(L,\alpha_{s}\right)+\delta\bm{\Gamma}_{\rm{Q}}^{(-)}\left(\alpha_{s}\right)$
(G.15)

where

$\delta\bm{\Gamma}_{\rm{Q}}^{(+)}\left(L,\alpha_{s}\right)\equiv\sum_{R}g_{R}(\alpha_{s})\frac{L}{3}\bigg{[}-\bm{\mathcal{D}}^{R}_{ssss}-\bm{\mathcal{D}}^{R}_{uuuu}+2\bm{\mathcal{D}}^{R}_{tttt}\bigg{]}$
(G.16)

and

$\delta\bm{\Gamma}_{\rm{Q}}^{(-)}\left(\alpha_{s}\right)\equiv\frac{i\pi}{2}\sum_{R}g_{R}(\alpha_{s})\left(\bm{\mathcal{D}}^{R}_{ssss}-\bm{\mathcal{D}}^{R}_{uuuu}\right),$
(G.17)

which we will now consider in turn in the adjoint representation.

##### The even quartic component in the adjoint representation in the Regge
limit.

Let us consider eq. (G.16) in the adjoint representation:

$\delta\bm{\Gamma}_{\rm{Q,A}}^{(+)}\left(\alpha_{s}\right)\equiv
g_{A}(\alpha_{s})\frac{L}{3}\bigg{[}-\bm{\mathcal{D}}^{A}_{ssss}-\bm{\mathcal{D}}^{A}_{uuuu}+2\bm{\mathcal{D}}^{A}_{tttt}\bigg{]},$
(G.18)

which we wish to express in terms of the Regge-limit basis comprised of
commutators of $\mathbf{T}_{t}^{2}$ and $\mathbf{T}_{s-u}^{2}$ and quartic
Casimirs. The results for the quartic channel operators acting on the tree
amplitude in eqs. (7.63) and (7.64) from section 7.2.2 may be substituted into
eq. (G.18) to give

$\displaystyle\delta\bm{\Gamma}_{\rm{Q,A}}^{(+)}\left(L,\alpha_{s}\right)\mathcal{M}_{\text{tree}}$
$\displaystyle=\frac{L}{3}g_{A}(\alpha_{s})\Bigg{\\{}3\frac{d_{AA}}{N_{A}}-2\left(\frac{d_{AR_{i}}}{N_{R_{i}}}+\frac{d_{AR_{j}}}{N_{R_{j}}}\right)-\bigg{(}\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}\bigg{)}$
(G.19)
$\displaystyle\quad+\frac{1}{4}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]-\frac{3}{4}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{C_{A}}{4}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\Bigg{\\}}\mathcal{M}_{\text{tree}}\,,$

where we note that the last term in the first line is subleading in the
large-$N_{c}$ limit using eq. (4.42), while the second line is readily non-
planar, being written in terms of commutators of $\mathbf{T}_{t}^{2}$ and
$\mathbf{T}_{s-u}^{2}$. $\delta\bm{\Gamma}_{\rm{Q,A}}^{(+)}$ is combined the
other terms that are linear in $L$ in eq. (F.15b). The adjoint quartic
contributions in eq. (G.19) are separated there, because the first term in the
curly brackets, $3\frac{d_{AA}}{N_{A}}$, forms part of
${\mathbf{\Gamma}}_{t}^{\rm{cusp}}$ once combined with the contribution from
$\gamma_{K}$, meanwhile the second forms part of the non-planar terms.

##### The odd quartic component in the adjoint representation in the Regge
limit.

Next we consider eq. (G.17) in the adjoint representation:

$\delta\bm{\Gamma}^{(-)}_{\rm{Q,A}}\left(\alpha_{s}\right)=\frac{i\pi}{2}g_{A}(\alpha_{s})\Big{(}\bm{\mathcal{D}}^{A}_{ssss}-\bm{\mathcal{D}}^{A}_{uuuu}\Big{)}.$
(G.20)

In order to express it in terms of nested commutators, we first use eq. (7.52)
to write

$\displaystyle\begin{split}\Big{(}\bm{\mathcal{D}}^{A}_{ssss}-\bm{\mathcal{D}}^{A}_{uuuu}\Big{)}\mathcal{M}_{\text{tree}}&=\Big{\\{}\bm{\mathcal{D}}^{A}_{1111}+4\bm{\mathcal{D}}^{A}_{1112}+6\bm{\mathcal{D}}^{A}_{1122}+4\bm{\mathcal{D}}^{A}_{1222}+\bm{\mathcal{D}}^{A}_{2222}\\\
&-\left(\bm{\mathcal{D}}^{A}_{1111}+4\bm{\mathcal{D}}^{A}_{1113}+6\bm{\mathcal{D}}^{A}_{1133}+4\bm{\mathcal{D}}^{A}_{1333}+\bm{\mathcal{D}}^{A}_{3333}\right)\Big{\\}}\mathcal{M}_{\text{tree}}.\end{split}$
(G.21)

There are three types of terms to consider. First, identifying

$\bm{\mathcal{D}}^{A}_{1111}=\frac{d_{AR_{i}}}{N_{R_{i}}}\,,\hskip
28.45274pt\bm{\mathcal{D}}^{A}_{2222}=\bm{\mathcal{D}}^{A}_{3333}=\frac{d_{AR_{j}}}{N_{R_{j}}},$
(G.22)

we observe that these projectile and target Casimir terms cancel out in eq.
(G.21). Next we consider operators with one attachment to one line and three
attachments to the opposite line, as shown in figure 4b, for example

$\displaystyle\bm{\bm{\mathcal{D}}}^{R}_{1112}$
$\displaystyle=d^{abcd}_{R}\left({\bf T}^{a,b,c}\right)_{i}{\bf T}^{d}_{j}\,,$
(G.23)

with the fully symmetrised trace defined as

$d^{abcd}_{R}\equiv\frac{1}{4!}\sum_{\sigma\in
S_{4}}\operatorname{Tr}_{R}\left(T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right).$
(G.24)

We notice the colour factor $\mathbf{C}^{(3)}_{13}$, defined in eq. (4.12),
corresponds to a particular case of eq. (G.23), in which the representation
$R$ is the adjoint. Therefore, we compute eq. (G.23) by following the steps in
eqs. (4.38a) and (4.40), getting

$\displaystyle d^{abcd}_{R}\left({\bf T}^{a,b,c}\right)_{i}{\bf T}^{d}_{j}$
$\displaystyle=A{\bf T}_{i}\cdot{\bf
T}_{j}=\frac{d_{RR_{i}}}{N_{R_{i}}C_{R_{i}}}{\bf T}_{i}\cdot{\bf T}_{j}$
(G.25)

where the constant $A$ is computed by taking the trace of the first equation.
This yields

$\bm{\mathcal{D}}^{R}_{1112}=\frac{1}{2}\left(\mathbf{T}_{s-u}^{2}-\frac{\mathbf{T}_{t}^{2}}{2}\right)\frac{d_{RR_{i}}}{N_{R_{i}}C_{i}}.$
(G.26)

We conclude that the terms with three attachments to one line and a single
attachment to the other line in eq. (G.21) take the form

$\displaystyle\bm{\mathcal{D}}^{R}_{klll}=\begin{cases}{\displaystyle\frac{1}{2}\left(\mathbf{T}_{s-u}^{2}-\frac{\mathbf{T}_{t}^{2}}{2}\right)\frac{d_{RR_{i}}}{N_{R_{i}}C_{i}}}&\text{if
$(k,l)\in\\{(2,1),(3,4)\\}$}\vspace{0.2cm}\\\
{\displaystyle\frac{1}{2}\left(\mathbf{T}_{s-u}^{2}-\frac{\mathbf{T}_{t}^{2}}{2}\right)\frac{d_{RR_{j}}}{N_{R_{j}}C_{j}}}&\text{if
$(k,l)\in\\{(1,2),(4,3)\\}$}\vspace{0.2cm}\\\
{\displaystyle-\frac{1}{2}\left(\mathbf{T}_{s-u}^{2}+\frac{\mathbf{T}_{t}^{2}}{2}\right)\frac{d_{RR_{i}}}{N_{R_{i}}C_{i}}}&\text{if
$(k,l)\in\\{(3,1),(2,4)\\}$}\vspace{0.2cm}\\\
{\displaystyle-\frac{1}{2}\left(\mathbf{T}_{s-u}^{2}+\frac{\mathbf{T}_{t}^{2}}{2}\right)\frac{d_{RR_{j}}}{N_{R_{j}}C_{j}}}&\text{if
$(k,l)\in\\{(1,3),(4,2)\\}$}\vspace{0.2cm}\\\ \end{cases}$ (G.27)

The third type of terms in eq. (G.21) involves two attachments to both the
target and projectile, for example

$\displaystyle\begin{split}\bm{\mathcal{D}}^{A}_{1122}\,\left({\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}\right)&=d^{abcd}_{A}\left({\bf
T}^{a,b,x}\right)_{i}\left({\bf T}^{c,d,x}\right)_{j}\\\
&=\bigg{\\{}\frac{1}{2}\left(\text{Tr}\left[F^{a}F^{b}F^{c}F^{d}+F^{d}F^{c}F^{b}F^{a}\right]\right)-\frac{C_{A}}{6}\left(f^{ade}f^{bce}-f^{abe}f^{cde}\right)\bigg{\\}}\\\
&\times\left({\bf T}^{a,b,x}\right)_{i}\left({\bf T}^{c,d,x}\right)_{j}\\\
&\equiv\left(\bm{\mathcal{D}}^{A,\text{Tr}}_{1122}+\bm{\mathcal{D}}^{A,\text{ff}}_{1122}\right)\,\left({\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}\right).\end{split}$ (G.28)

The colour structure relation from eq. (7.55) gives the second line above. The
terms are then separated with superscripts $\rm{Tr}$ for the trace terms and
$\rm{ff}$ for terms with two structure constants. The first term is

$\displaystyle\begin{split}\bm{\mathcal{D}}^{(A,\rm{Tr})}_{1122}\,\left({\bf
T}^{x}_{i}\,{\bf
T}^{x}_{j}\right)&=\frac{1}{2}\left(\text{Tr}\left[F^{a}F^{b}F^{c}F^{d}+F^{d}F^{c}F^{b}F^{a}\right]\right)\left({\bf
T}^{a,b,x}\right)_{i}\left({\bf T}^{c,d,x}\right)_{j}\\\
&=\left(\mathbf{T}^{[a,[b,c]],d,x}\right)_{i}\left(\mathbf{T}^{[b,[a,d]],c,x}\right)_{j}.\end{split}$
(G.29)

Upon using the identities of section 6.4 and appendix C, it yields

$\displaystyle\begin{split}\bm{\mathcal{D}}^{(A,\rm{Tr})}_{1122}\,\left({\bf
T}^{x}_{i}\,{\bf
T}^{x}_{j}\right)&=\Big{\\{}-\frac{1}{64}C_{A}^{4}-\frac{1}{32}\left(\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}-\mathbf{T}_{t}^{2}[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\mathbf{T}_{t}^{2}\right)\\\
&\quad\mbox{}+\frac{1}{4}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}],\mathbf{T}_{t}^{2}+\frac{1}{16}\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\Big{\\}}\,\left({\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}\right).\end{split}$ (G.30)

Focusing on the term involving two structure constants now, we find

$\displaystyle\bm{\mathcal{D}}^{A,\text{ff}}_{1122}\,\left({\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}\right)$
$\displaystyle=-\frac{C_{A}}{6}\left(f^{ade}f^{bce}-f^{abe}f^{cde}\right)\left({\bf
T}^{a,b,x}\right)_{i}\left({\bf T}^{c,d,x}\right)_{j}$
$\displaystyle=\frac{C_{A}}{6}\left({\bf T}^{[d,e],b,x}\right)_{i}\left({\bf
T}^{[e,b],d,x}\right)_{j}-\frac{C_{A}^{3}}{48}\left(\mathbf{T}_{s-u}^{2}-\frac{\mathbf{T}_{t}^{2}}{2}\right)\,\left({\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}\right),$ (G.31)

where the second term is reduced to a dipole using

$f^{abg}{\bf T}^{a}_{i}{\bf T}^{b}_{i}=\frac{iC_{A}}{2}{\bf T}_{i}^{g}$ (G.32)

and then eq. (LABEL:eq:relTsu-Tt/2) is employed. The commutator relation eq.
(4.34) is used to express the first term as commutators, then upon using eqs.
(LABEL:eq:relTsu-Tt/2) and (LABEL:eq:relTsu+Tt/2), it becomes

$\displaystyle\begin{split}\left({\bf T}^{[d,e],b,x}\right)_{i}\left({\bf
T}^{[e,b],d,x}\right)_{j}&=\frac{1}{24}\bigg{\\{}(\mathbf{T}_{s-u}^{2})^{2}\mathbf{T}_{t}^{2}+\frac{1}{2}\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{1}{4}\mathbf{T}_{s-u}^{2}(\mathbf{T}_{t}^{2})^{2}\\\
&\quad\mbox{}+\frac{1}{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}-\frac{1}{4}(\mathbf{T}_{t}^{2})^{2}\mathbf{T}_{s-u}^{2}+\frac{1}{8}(\mathbf{T}_{t}^{2})^{3}\bigg{\\}}\,\left({\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}\right).\end{split}$ (G.33)

Substituting into eq. (G.31), it then reads

$\displaystyle\begin{split}\bm{\mathcal{D}}^{A,\text{ff}}_{1122}\,\left({\bf
T}^{x}_{i}\,{\bf
T}^{x}_{j}\right)&=\bigg{\\{}\frac{C_{A}^{4}}{64}-\frac{C_{A}^{3}}{32}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{48}\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{C_{A}^{2}}{24}(\mathbf{T}_{s-u}^{2})^{2}+\frac{C_{A}^{2}}{48}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\\\
&\quad\mbox{}-\frac{C_{A}}{96}(\mathbf{T}_{t}^{2})^{2}\mathbf{T}_{s-u}^{2}\bigg{\\}}\,\left({\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}\right).\end{split}$ (G.34)

The contributions to $\bm{\mathcal{D}}^{A}_{1133}$ can be found by simply
applying crossing $\mathbf{T}_{s-u}^{2}\to-\mathbf{T}_{s-u}^{2}$ to the
$\bm{\mathcal{D}}^{A}_{1122}$ terms. Substituting the expressions from eqs.
(G.27), (G.30) and (G.34) into eq. (G.21), we have

$\displaystyle\begin{split}\Big{(}\bm{\mathcal{D}}^{A}_{ssss}-\bm{\mathcal{D}}^{A}_{uuuu}\Big{)}\mathcal{M}_{\text{tree}}&=\bigg{\\{}4\mathbf{T}_{s-u}^{2}\left(\frac{d_{AR_{i}}}{N_{R_{i}}C_{i}}+\frac{d_{AR_{j}}}{N_{R_{j}}C_{j}}\right)+\frac{1}{8}\Big{(}-2C_{A}^{3}\mathbf{T}_{s-u}^{2}\\\
&-\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\mathbf{T}_{t}^{2}\Big{]}\Big{)}-\frac{3}{8}\left(\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}-\mathbf{T}_{t}^{2}[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\mathbf{T}_{t}^{2}\right)\bigg{\\}}\mathcal{M}_{\text{tree}}.\end{split}$
(G.35)

The imaginary contribution multiplying the $g_{A}$ is

$\displaystyle\begin{split}\delta\mathbf{\Gamma}_{\rm{Q,A}}^{(-)}\left(\alpha_{s}\right)\,\mathcal{M}_{\text{tree}}=&\frac{i\pi}{2}g_{A}(\alpha_{s})\bigg{\\{}4\mathbf{T}_{s-u}^{2}\left(\frac{d_{AR_{i}}}{N_{R_{i}}C_{i}}+\frac{d_{AR_{j}}}{N_{R_{j}}C_{j}}\right)+\frac{1}{8}\bigg{(}-2C_{A}^{3}\mathbf{T}_{s-u}^{2}\\\
&-3\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}-\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\mathbf{T}_{t}^{2}+3\mathbf{T}_{t}^{2}[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\mathbf{T}_{t}^{2}\bigg{)}\bigg{\\}}\,\mathcal{M}_{\text{tree}},\end{split}$
(G.36)

with $j$ being the target and $i$ the projectile.

### The four-generator three-line term ($\mathbf{4T}$$-$$\mathbf{3L}$) in the
Regge-limit basis

The third term in eq. (G.1) is given by eq. (7.14) and reads

$\mathbf{\Gamma}_{n,\rm
4T-3L}(\\{s_{ij}\\},\lambda,\alpha_{s})=f(\alpha_{s})\sum_{(i,j,k)}\,\bm{{\cal
T}}_{iijk}=-f(\alpha_{s})\sum_{(i,j,k)}f^{abe}f^{cde}\\{{\bf T}_{i}^{a},{\bf
T}_{i}^{d}\\}_{+}{\bf T}_{j}^{b}{\bf T}_{k}^{c},$ (G.37)

with individual colour structures being symmetric under $j\leftrightarrow k$
and the sum being symmetric under the interchange of any two of the three
partons. For these to contribute at four-loop order they must be multiplied by
either $C_{A}$ or $n_{f}T_{F}$ so there is some dependence on the matter
content of the theory.

Considering $2\to 2$ scattering, we expand the sum over four distinct partonic
legs and eq. (G.37) becomes

$\displaystyle\hskip-28.45274pt\mathbf{\Gamma}_{\rm
4T-3L}(\alpha_{s})=-f(\alpha_{s})f^{abe}f^{cde}\Big{\\{}$
$\displaystyle\\!\\!\Big{(}{\bf T}_{1}^{a}{\bf T}_{1}^{d}+{\bf T}_{1}^{d}{\bf
T}_{1}^{a}\Big{)}\Big{(}{\bf T}_{2}^{b}{\bf T}_{3}^{c}+{\bf T}_{2}^{b}{\bf
T}_{4}^{c}+{\bf T}_{3}^{b}{\bf T}_{4}^{c}\Big{)}$ $\displaystyle+\,\Big{(}{\bf
T}_{2}^{a}{\bf T}_{2}^{d}+{\bf T}_{2}^{d}{\bf T}_{2}^{a}\Big{)}\Big{(}{\bf
T}_{1}^{b}{\bf T}_{3}^{c}+{\bf T}_{1}^{b}{\bf T}_{4}^{c}+{\bf T}_{3}^{b}{\bf
T}_{4}^{c}\Big{)}$ $\displaystyle+\,\Big{(}{\bf T}_{3}^{a}{\bf T}_{3}^{d}+{\bf
T}_{3}^{d}{\bf T}_{3}^{a}\Big{)}\Big{(}{\bf T}_{1}^{b}{\bf T}_{2}^{c}+{\bf
T}_{1}^{b}{\bf T}_{4}^{c}+{\bf T}_{2}^{b}{\bf T}_{4}^{c}\Big{)}$
$\displaystyle+\,\Big{(}{\bf T}_{4}^{a}{\bf T}_{4}^{d}+{\bf T}_{4}^{d}{\bf
T}_{4}^{a}\Big{)}\Big{(}{\bf T}_{1}^{b}{\bf T}_{2}^{c}+{\bf T}_{1}^{b}{\bf
T}_{3}^{c}+{\bf T}_{2}^{b}{\bf T}_{3}^{c}\Big{)}\Big{\\}}\,,$

where we drop the number of legs ($4$) in the subscript. By repeatedly using
the commutation relation in eq. (4.34), it is possible to write eq. (G.37) in
terms of the product of three generators associated to the projectile and
three generators associated to the target.

As an example of following these rules, the first line of eq. (G) is

$\displaystyle f^{abe}f^{cde}\Big{(}{\bf T}_{1}^{a}{\bf T}_{1}^{d}+{\bf
T}_{1}^{d}{\bf T}_{1}^{a}\Big{)}\Big{(}{\bf T}_{2}^{b}{\bf T}_{3}^{c}+{\bf
T}_{2}^{b}{\bf T}_{4}^{c}+{\bf T}_{3}^{b}{\bf T}_{4}^{c}\Big{)}$
$\displaystyle=\,-\Big{(}{\bf T}_{1}^{a}[{\bf T}_{1}^{e},{\bf T}_{1}^{c}][{\bf
T}_{2}^{e},{\bf T}_{2}^{a}]{\bf T}_{3}^{c}+{\bf T}_{1}^{a}{\bf
T}_{1}^{d}\Big{[}[{\bf T}_{2}^{c},{\bf T}_{2}^{d}],{\bf T}_{2}^{a}\Big{]}{\bf
T}_{4}^{c}$ $\displaystyle+{\bf T}_{1}^{a}{\bf T}_{1}^{d}\Big{[}[{\bf
T}_{3}^{c},{\bf T}_{3}^{d}],{\bf T}_{3}^{a}\Big{]}{\bf T}_{4}^{c}+[{\bf
T}_{1}^{e},{\bf T}_{1}^{c}]{\bf T}_{1}^{a}[{\bf T}_{2}^{e},{\bf
T}_{2}^{a}]{\bf T}_{3}^{c}$ $\displaystyle+{\bf T}_{1}^{d}{\bf
T}_{1}^{a}\Big{[}[{\bf T}_{2}^{c},{\bf T}_{2}^{d}],{\bf T}_{2}^{a}\Big{]}{\bf
T}_{4}^{c}+{\bf T}_{1}^{d}{\bf T}_{1}^{a}\Big{[}[{\bf T}_{3}^{c},{\bf
T}_{3}^{d}],{\bf T}_{3}^{a}\Big{]}{\bf T}_{4}^{c}\Big{)}.$

Similar expressions can be found for the remaining three lines of eq. (G).
Using these rules we write eq. (G.37) as

$\displaystyle\mathbf{\Gamma}_{\rm 4T-3L}(\alpha_{s})$ $\displaystyle=$
$\displaystyle-f(\alpha_{s})\bigg{(}\sum_{(i,j,k)\in i_{f}}{\bf
T}_{i}^{a}[{\bf T}_{i}^{e},{\bf T}_{i}^{c}][{\bf T}_{j}^{e},{\bf
T}_{j}^{a}]{\bf T}_{k}^{c}+[{\bf T}_{i}^{e},{\bf T}_{i}^{c}]{\bf
T}_{i}^{a}[{\bf T}_{j}^{e},{\bf T}_{j}^{a}]{\bf T}_{k}^{c}$ (G.40)
$\displaystyle+$ $\displaystyle\sum_{(i,j,k)\in j_{f}}{\bf T}_{i}^{a}{\bf
T}_{i}^{d}\Big{[}[{\bf T}_{j}^{c},{\bf T}_{j}^{d}],{\bf T}_{j}^{a}\Big{]}{\bf
T}_{k}^{c}+\Big{[}[{\bf T}_{j}\cdot{\bf T}_{k},{\bf T}_{i}\cdot{\bf
T}_{j}],{\bf T}_{i}\cdot{\bf T}_{j}\Big{]}\bigg{)}\hskip 28.45274pt$

where

$i_{f}=\\{(1,2,3),(2,1,4),(3,1,4),(4,2,3)\\},$
$j_{f}=\\{(2,1,3),(3,1,2),(1,2,4),(4,1,2),(1,3,4),(4,1,3),(2,3,4),(3,2,4)\\}.$

Then when $\mathbf{\Gamma}_{4,\rm 4T-3L}(\alpha_{s})$ acts on the tree-level
amplitude we can rewrite the expression as chains of attachments on the target
and projectile. We then use the identities of eqs. (4.32) and (4.45), to
express eq. (G.37) as

$\displaystyle\bm{\Gamma}_{\rm
4T-3L}(\alpha_{s})\mathcal{M}_{\text{tree}}=f(\alpha_{s})\bigg{(}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+\frac{C_{A}^{3}}{2}-6\frac{d_{AR_{i}}}{N_{R_{i}}C_{i}}-6\frac{d_{AR_{j}}}{N_{R_{j}}C_{j}}\bigg{)}\mathcal{M}_{\text{tree}},$
(G.41)

where, as usual, $j$ is the target and $i$ the projectile. We see that
$\bm{\Gamma}_{\rm 4T-3L}(\alpha_{s})$ is given by even colour structures and
that it is non-planar owing to the property of commutators and eq. (5.10).

### The five-generator four-line term ($\mathbf{5T}$$-$$\mathbf{4L}$) in the
Regge-limit basis

The last term in eq. (G.1) is given by eq. (7.65) which we quote again here

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
5T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)&=\sum_{(i,j,k,l)}\,\bm{\mathcal{T}}_{ijkli}\,{\cal
H}_{1}(\beta_{ijlk},\beta_{iklj};\alpha_{s})\\\
&=\sum_{(i,j,k,l)}if^{adg}f^{bch}f^{egh}\,\\{{\bf T}_{i}^{a},{\bf
T}_{i}^{e}\\}_{+}{\bf T}_{j}^{b}{\bf T}_{k}^{c}{\bf T}_{l}^{d}\,{\cal
H}_{1}(\beta_{ijlk},\beta_{iklj};\alpha_{s}).\end{split}$ (G.42)

We specialise to $2\to 2$ scattering and our aim is to eq. (G.42) in terms of
expressions with definite symmetry under $s\leftrightarrow u$. After
introducing the functions ${\cal H}_{1}^{(+)}$, ${\cal H}_{1}^{(-)}$ and
$\tilde{{\cal H}}_{1}^{(-)}$ in eq. (7.66), we expand the sum in eq. (G.42)
and then use the properties of $\beta_{ijkl}$ in eq. (7.9) and the
antisymmetry of $\bm{\mathcal{T}}_{ijkli}$ and ${\cal H}_{1}$, to obtain

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
5T-4L}^{(4)}(\\{\beta_{ijkl}\\})=&\,2{\cal
H}_{1}^{(+,4)}(\\{\beta_{ijkl}\\})\Big{\\{}\bm{\mathcal{T}}_{13421}+\bm{\mathcal{T}}_{24312}+\bm{\mathcal{T}}_{31243}+\bm{\mathcal{T}}_{42134}\\\
&\hskip
85.35826pt+\big{(}\bm{\mathcal{T}}_{12431}+\bm{\mathcal{T}}_{21342}+\bm{\mathcal{T}}_{34213}+\bm{\mathcal{T}}_{43124}\big{)}\Big{\\}}\\\
&+2{\cal
H}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\Big{\\{}\bm{\mathcal{T}}_{13421}+\bm{\mathcal{T}}_{24312}+\bm{\mathcal{T}}_{31243}+\bm{\mathcal{T}}_{42134}\\\
&\hskip
85.35826pt-\left(\bm{\mathcal{T}}_{12431}+\bm{\mathcal{T}}_{21342}+\bm{\mathcal{T}}_{34213}+\bm{\mathcal{T}}_{43124}\right)\Big{\\}}\\\
&+2\tilde{{\cal
H}}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\Big{\\{}\bm{\mathcal{T}}_{12341}+\bm{\mathcal{T}}_{21432}+\bm{\mathcal{T}}_{34123}+\bm{\mathcal{T}}_{43214}\Big{\\}}.\end{split}$
(G.43)

The colour coefficient of ${\cal H}_{1}^{(+)}$ is even under the exchange of
$s\leftrightarrow u$, while the coefficients of both ${\cal
H}_{1}^{(-)},\tilde{{\cal H}}_{1}^{(-)}$ are odd. We decompose the
$\bm{\mathcal{T}}_{ijkli}$ into two terms as

$\bm{\mathcal{T}}_{ijkli}=if^{adg}f^{bch}f^{egh}{\bf T}_{i}^{a}{\bf
T}_{i}^{e}{\bf T}_{j}^{b}{\bf T}_{k}^{c}{\bf
T}_{l}^{d}-\frac{C_{A}}{4}f^{ade}f^{bce}{\bf T}_{i}^{a}{\bf T}_{j}^{b}{\bf
T}_{k}^{c}{\bf T}_{l}^{d}.$ (G.44)

By repeatedly expressing the structure constants in eq. (G.44) as commutators
according to eq. (4.34), it is possible to write an expression with three
structure constants and five generators, in terms of four generators on the
projectile line $i$ (corresponding to partons $1$ and $4$), and four
generators on the target line $j$ (corresponding to partons $2$ and $3$)
acting on the tree-level amplitude. For the four indices $(i,j,k,l)$, there
are two independent cases to consider. First, we fix $i$ and $k$ on one line
and then require $j$ and $l$ to attach to the opposite line with the following
expression:

$\displaystyle\begin{split}if^{adg}f^{bch}f^{egh}{\bf T}_{i}^{a}{\bf
T}_{i}^{e}{\bf T}_{j}^{b}{\bf T}_{k}^{c}{\bf T}_{l}^{d}&=[{\bf T}_{i}^{g},{\bf
T}_{i}^{d}]{\bf T}_{i}^{e}\Big{[}{\bf T}_{j}^{c},[{\bf T}_{j}^{e},{\bf
T}_{j}^{g}]\Big{]}{\bf T}_{k}^{c}{\bf T}_{l}^{d}.\end{split}$ (G.45)

The second case when rewriting the first term in eq. (G.44) is to fix $i$ and
$l$ on the same line and $j$ and $k$ on the opposite line. This time we apply
the Jacobi identity

$f^{ace}f^{bde}=f^{abe}f^{cde}+f^{ade}f^{bce},$ (G.46)

in such a way to avoid operators with dotted colour generators attached with
both ends on the same line which do not allow for a straightforward
translation to the Regge-limit basis. This ensures that there are four
generators attached to the upper and lower lines with the form

$\displaystyle if^{adg}f^{bch}f^{egh}{\bf T}_{i}^{a}{\bf T}_{i}^{e}{\bf
T}_{j}^{b}{\bf T}_{k}^{c}{\bf T}_{l}^{d}=$ $\displaystyle\,{\bf
T}_{i}^{a}[{\bf T}_{i}^{c},{\bf T}_{i}^{h}\,]\Big{[}{\bf T}_{j}^{h},[{\bf
T}_{j}^{a},{\bf T}_{j}^{d}]\Big{]}{\bf T}_{k}^{c}{\bf T}_{l}^{d}$
$\displaystyle+{\bf T}_{i}^{a}[{\bf T}_{i}^{h},{\bf T}_{i}^{b}]{\bf
T}_{j}^{b}\Big{[}{\bf T}_{k}^{h},[{\bf T}_{k}^{a},{\bf T}_{k}^{d}]\Big{]}{\bf
T}_{l}^{d}.$ (G.47)

Using the above relations we can write eq. (G.43) as a sum over the lists

$l_{1}=\\{(1,3,4,2),(2,4,3,1),(3,1,2,4),(4,2,1,3)\\}$
$l_{2}=\\{(1,2,3,4),(2,1,4,3),(3,4,1,2),(4,3,2,1)\\}$

to give

$\displaystyle\mathbf{\Gamma}_{\rm
5T-4L}^{(4)}(\\{\beta_{ijkl}\\})\mathcal{M}_{\text{tree}}=2\Bigg{\\{}{\cal
H}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\bigg{[}C_{A}f^{ade}f^{bce}{\bf
T}_{1}^{a}{\bf T}_{2}^{b}{\bf T}_{3}^{c}{\bf T}_{4}^{d}$ (G.48)
$\displaystyle\hskip 10.0pt+\\!\\!\sum_{(i,j,k,l)\in l_{1}}\bigg{(}[{\bf
T}_{i}^{d},{\bf T}_{i}^{g}]{\bf T}_{i}^{e}\Big{[}{\bf T}_{j}^{c},[{\bf
T}_{j}^{g},{\bf T}_{j}^{e}]\Big{]}{\bf T}_{k}^{c}{\bf T}_{l}^{d}-[{\bf
T}_{i}^{d},{\bf T}_{i}^{g}]{\bf T}_{i}^{e}\Big{[}{\bf T}_{l}^{c},[{\bf
T}_{l}^{g},{\bf T}_{l}^{e}]\Big{]}{\bf T}_{k}^{c}{\bf
T}_{j}^{d}\bigg{)}\bigg{]}$ $\displaystyle\hskip 10.0pt+{\cal
H}_{1}^{(+,4)}(\\{\beta_{ijkl}\\})\bigg{[}-C_{A}\left(f^{ace}f^{bde}+f^{abe}f^{cde}\right){\bf
T}_{1}^{a}{\bf T}_{2}^{b}{\bf T}_{3}^{c}{\bf T}_{4}^{d}$ $\displaystyle\hskip
10.0pt+\\!\\!\sum_{(i,j,k,l)\in l_{1}}\bigg{(}[{\bf T}_{i}^{d},{\bf
T}_{i}^{g}]{\bf T}_{i}^{e}\Big{[}{\bf T}_{j}^{c},[{\bf T}_{j}^{g},{\bf
T}_{j}^{e}]\Big{]}{\bf T}_{k}^{c}{\bf T}_{l}^{d}+[{\bf T}_{i}^{d},{\bf
T}_{i}^{g}]{\bf T}_{i}^{e}\Big{[}{\bf T}_{l}^{c},[{\bf T}_{l}^{g},{\bf
T}_{l}^{e}]\Big{]}{\bf T}_{k}^{c}{\bf T}_{j}^{d}\bigg{)}\bigg{]}$
$\displaystyle\hskip 10.0pt+\tilde{{\cal
H}}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\bigg{[}-C_{A}f^{ade}f^{bce}{\bf
T}_{1}^{a}{\bf T}_{2}^{b}{\bf T}_{3}^{c}{\bf T}_{4}^{d}$ $\displaystyle\hskip
10.0pt+\\!\\!\sum_{(i,j,k,l)\in l_{2}}\bigg{(}{\bf T}_{i}^{a}[{\bf
T}_{i}^{c},{\bf T}_{i}^{g}\,]\Big{[}{\bf T}_{j}^{g},[{\bf T}_{j}^{a},{\bf
T}_{j}^{d}]\Big{]}{\bf T}_{k}^{c}{\bf T}_{l}^{d}+{\bf T}_{i}^{a}[{\bf
T}_{i}^{g},{\bf T}_{i}^{b}]{\bf T}_{j}^{b}\Big{[}{\bf T}_{k}^{g},[{\bf
T}_{k}^{a},{\bf T}_{k}^{d}]\Big{]}{\bf
T}_{l}^{d}\bigg{)}\bigg{]}\Bigg{\\}}\mathcal{M}_{\text{tree}}.$

The terms without commutators can be written in the Regge-limit basis using
eqs. (7.42)-(7.43b). The commutators can be expanded and divided into two
types of terms. Firstly, those that are just products of four dipoles can
immediately be translated into the Regge limit basis with eq. (2.17). The
second type is those that appear in the entangled combinations of $d_{1},\dots
d_{8}$ in appendix C. They are then written in a similar way to eq. (7.60) to
use the identities in section 6.3.2 and appendix C. Putting everything
together we can write eq. (G.43) as

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
5T-4L}^{(4)}(\\{\beta_{ijkl}\\})\mathcal{M}_{\text{tree}}&=\Bigg{[}{\cal
H}_{1}^{(+,4)}(\\{\beta_{ijkl}\\})\bigg{(}\frac{3C_{A}}{4}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]-\frac{C_{A}}{2}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\\\
&-\frac{1}{4}\mathbf{T}_{t}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{s-u}^{2}\bigg{)}+\frac{1}{4}\tilde{{\cal
H}}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}\\\
&+{\cal
H}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\bigg{(}-\frac{1}{2}\Big{[}\mathbf{T}_{s-u}^{2},\big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\big{]}\Big{]}+\frac{1}{8}\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}\bigg{)}\Bigg{]}\mathcal{M}_{\text{tree}}.\end{split}$
(G.49)

Finally, using eq. (4.48) we can eliminate one of the even-signature
commutator terms in eq. (G.49) by substituting

$\mathbf{T}_{t}^{2}\big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\big{]}\mathbf{T}_{s-u}^{2}\mathcal{M}_{\text{tree}}=\bigg{(}\frac{2}{3}\mathbf{T}_{t}^{2}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{2},\mathbf{T}_{t}^{2}\right]-\mathbf{T}_{s-u}^{2}\big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\big{]}\mathbf{T}_{t}^{2}\bigg{)}\mathcal{M}_{\text{tree}},$
(G.50)

where the two commutators on the right-hand side have already been used in eq.
(5.3) to express the four-loop reduced amplitude and are part of our basis.
This gives

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
5T-4L}^{(4)}(\\{\beta_{ijkl}\\})\mathcal{M}_{\text{tree}}&=\Bigg{[}{\cal
H}_{1}^{(+,4)}(\\{\beta_{ijkl}\\})\bigg{(}-\frac{C_{A}}{2}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+C_{A}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\\\
&-\frac{1}{6}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\bigg{)}+\frac{1}{4}\tilde{{\cal
H}}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}\\\
&+{\cal
H}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\bigg{(}-\frac{1}{2}\Big{[}\mathbf{T}_{s-u}^{2},\big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\big{]}\Big{]}+\frac{1}{8}\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}\bigg{)}\Bigg{]}\mathcal{M}_{\text{tree}},\end{split}$
(G.51)

which is eq. (7.67).

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