fixing the form of the soft anomalous dimension at this order. As was
expected, three loop corrections to the dipole formula (7.12) depend
exclusively on CICRs. The structure of the result is of direct relevance to
the functions we shall encounter at four loops, and therefore we shall review
it below in the context of the general form the anomalous dimension takes at
four loops.

Taking into account the complete set of connected colour structures complying
with the non-Abelian exponentiation theorem Gatheral1983ExponentiationOE ;
Frenkel1984NonabelianEE ; Gardi:2013ita , Becher and Neubert wrote down
Becher:2019avh  a general parametrisation191919Previous work along these lines
has been done e.g. in refs. Gardi:2009qi ; Becher:2009qa ; Dixon:2009ur ;
Ahrens:2012qz ; Almelid:2017qju . – with unknown kinematic functions – which
satisfied the aforementioned collinear anomaly constraints of eq. (7.3) along
with Bose symmetry. The contributions appearing through four loops can be
classified as follows:

$\displaystyle\begin{split}\mathbf{\Gamma}_{n}\left(\\{s_{ij}\\},\lambda,\alpha_{s}(\lambda^{2})\right)&=\mathbf{\Gamma}^{\rm
dip.}_{n}\left(\\{s_{ij}\\},\lambda,\alpha_{s}\right)+\mathbf{\Gamma}_{n,\rm
4T-3L}\left(\alpha_{s}\right)+\mathbf{\Gamma}_{n,\rm
4T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)\\\\[2.84544pt]
&+\mathbf{\Gamma}_{n,\rm
Q4T-2,3L}\left(\\{s_{ij}\\},\lambda,\alpha_{s}\right)+\mathbf{\Gamma}_{n,\rm
Q4T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)\\\\[2.84544pt]
&+\mathbf{\Gamma}_{n,\rm
5T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)+\mathbf{\Gamma}_{n,\rm
5T-5L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)+{\cal
O}(\alpha_{s}^{5})\,,\end{split}$ (7.13)

where the subscript for each term includes, in addition to the number of
partons $n$, the following attributes of each (connected) colour
factor:$\,{\mathrm{Q}}$ for a quartic Casimir related contribution; a number
followed by ${\mathrm{T}}$ to indicate the number generators, and a number
followed by ${\mathrm{L}}$ to indicate the number of distinct lines that
interact.

The first term in eq. (7.13) is the sum-over-dipoles formula, which is the
complete result for $\mathbf{\Gamma}_{n}$ to two loops, while all others start
contributing at three (the second and third terms) and four loops (all
others). Notice that this formula includes terms with explicit dependence on
the scale (the first and the fourth) as well as terms that depend exclusively
on CICRs (all others). The former can be identified with those that constitute
a particular solution satisfying eq. (7.3) (through four loops) and are
strictly linear in $l_{ij}$, while the latter involve higher transcendental
functions of the CICRs. The last two terms, consisting of five generators, are
excluded based on the argument of ref. Vladimirov:2017ksc . We retain them
here to see, independently of the latter argument, what constraints emerge
from the Regge-limit analysis. Let us now introduce explicitly each term of
eq. (7.13) in turn, where we adopt much of the notation from ref.
Becher:2019avh .

The second and third terms in (7.13) start at three loops, where they were
explicitly computed Almelid:2015jia . These terms involve the colour and
kinematic degrees of freedom of subsets of three or four partons, they are
non-planar and depend exclusively on CICRs. They read:

$\displaystyle\mathbf{\Gamma}_{n,\rm 4T-3L}\left(\alpha_{s}\right)$
$\displaystyle=$ $\displaystyle f(\alpha_{s})\sum_{(i,j,k)}\bm{{\cal
T}}_{iijk},$ (7.14) $\displaystyle\mathbf{\Gamma}_{n,\rm
4T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)$ $\displaystyle=$
$\displaystyle\sum_{(i,j,k,l)}\bm{{\cal T}}_{ijkl}\,{\cal
F}(\beta_{ijlk},\beta_{iklj};\alpha_{s}),$ (7.15)

where the summation is over tuples (with no restriction on the relative order
of indices). The colour structure involves four generators:

$\bm{{\cal T}}_{ijkl}\equiv f^{ade}f^{bce}\\{{\bf T}_{i}^{a},{\bf
T}_{j}^{b},{\bf T}_{k}^{c},{\bf T}_{l}^{d}\\}_{+}.$ (7.16)

Notice that the curly brackets represent symmetrisation, defined as

$\\{{\bf T}_{i}^{a_{1}},{\bf T}_{j}^{a_{2}},\dots\bf
T_{l}^{a_{n}}\\}_{+}\equiv\frac{1}{n!}\sum_{\pi}{\bf T}_{i}^{a_{\pi(1)}}{\bf
T}_{j}^{a_{\pi(2)}}\dots\bf T_{l}^{a_{\pi(n)}},$ (7.17)

where the sum is over all permutations of the indices. The symmetrisation only
acts on generators attached to the same line, as those attached to distinct
lines commute. For example,

$\bm{{\cal T}}_{iijk}=f^{ade}f^{bce}\\{{\bf T}_{i}^{a},{\bf
T}_{i}^{b}\\}_{+}{\bf T}_{j}^{c}{\bf
T}_{k}^{d}=\frac{1}{2}f^{ade}f^{bce}\left({\bf T}_{i}^{a}{\bf T}_{i}^{b}+{\bf
T}_{i}^{b}{\bf T}_{i}^{a}\right){\bf T}_{j}^{c}{\bf T}_{k}^{d}.$

The functions $f(\alpha_{s})$ and ${\cal
F}(\beta_{ijlk},\beta_{iklj};\alpha_{s})$ have a perturbative expansion

$f(\alpha_{s})=\left(\frac{\alpha_{s}}{\pi}\right)^{3}f^{(3)}+\left(\frac{\alpha_{s}}{\pi}\right)^{4}\sum_{R}f^{(4)}_{R}+O(\alpha_{s}^{5}),$
(7.18)

and

${\cal
F}(\beta_{ijlk},\beta_{iklj};\alpha_{s})=\left(\frac{\alpha_{s}}{\pi}\right)^{3}{\cal
F}^{(3)}(\beta_{ijlk},\beta_{iklj})+\left(\frac{\alpha_{s}}{\pi}\right)^{4}\sum_{R}{\cal
F}^{(4)}_{R}(\beta_{ijlk},\beta_{iklj})+O(\alpha_{s}^{5})\,,$ (7.19)

where $f^{(\ell)}$ are transcendental constants while ${\cal F}^{(\ell)}$ are
transcendental functions of the CICRs defined in eq. (7.6). At four loops,
these two functions involve a sum over the gauge group representations $R$,
which we write explicitly in eqs. (7.18) and (7.19). This is a general feature
of the colour structures appearing in the anomalous dimension: there is an
implicit sum over the representations, once they are considered at a loop
order higher than when they first appear. This is a manifestation of the fact
that any structure first appears owing to a purely gluonic diagram, and, as
such, it has a universal nature, being entirely independent of the matter
contents of the theory.

The functions $f(\alpha_{s})$ and ${\cal
F}(\beta_{ijlk},\beta_{iklj};\alpha_{s})$ have been calculated at three loops
Almelid:2015jia . Expressing the CICRs in terms of variables $z_{ijkl}$ and
$\bar{z}_{ijkl}$:

$\displaystyle\rho_{ijkl}$ $\displaystyle=z_{ijkl}\bar{z}_{ijkl},\,\,\,\hskip
28.45274pt\rho_{ilkj}=(1-z_{ijkl})(1-\bar{z}_{ijkl}),$ (7.20)

the function ${\cal F}(\alpha_{s})$ reads

$\displaystyle{\cal F}^{(3)}(\beta_{ijlk},\beta_{iklj})$
$\displaystyle=\frac{1}{32}\bigg{(}F(1-z_{ijlk})-F(z_{ijlk})\bigg{)},$ (7.21)

where in turn $F(z)$ is a function of single-valued harmonic polylogarithms
Brown:2004ugm ; Dixon:2012yy ; Brown:2013gia ; Schnetz:2013hqa :

$F(z)\equiv{\cal L}_{10101}(z)+2\zeta_{2}\Big{[}{\cal L}_{001}(z)+{\cal
L}_{100}(z)\Big{]}\,,$ (7.22)

while

$\displaystyle f^{(3)}$
$\displaystyle=\frac{1}{4}\left(\zeta_{5}+2\zeta_{2}\zeta_{3}\right)\,.$
(7.23)

The other terms in eq. (7.13) start at four loops. The quartic term involving
four generators with attachments to two and three legs can be expressed as
Becher:2019avh

$\mathbf{\Gamma}_{n,\rm
Q4T-2,3L}\left(\\{s_{ij}\\},\lambda,\alpha_{s}\right)=-\frac{1}{2}\sum_{R}g_{R}(\alpha_{s})\bigg{[}\sum_{(i,j)}\,\big{(}\bm{{\cal
D}}_{iijj}^{R}+2\bm{{\cal D}}_{iiij}^{R}\big{)}l_{ij}+\sum_{(i,j,k)}\bm{{\cal
D}}_{ijkk}^{R}\,l_{ij}\bigg{]},$ (7.24)

where the colour operator is defined as

$\bm{{\cal D}}_{ijkl}^{R}\equiv\frac{1}{4!}\sum_{\sigma\in
S_{4}}\operatorname{Tr}_{R}\left(T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right)\,{\bf
T}_{i}^{a}{\bf T}_{j}^{b}{\bf T}_{k}^{c}{\bf T}_{l}^{d}\,.$ (7.25)

Similarly to the dipole term, $\mathbf{\Gamma}_{n,\rm Q4T-2,3L}$ is part of
the inhomogeneous solution of eq. (7.3): upon substituting eq. (7.24) into the
left-hand side of eq. (7.3) and using colour conservation, one obtains the
quartic Casimir component of $\Gamma^{\rm{cusp}}_{i}(\alpha_{s})$, where

$\bm{\mathcal{D}}^{R}_{iiii}=\frac{d_{RR_{i}}}{N_{R_{i}}}=\frac{1}{4!}\sum_{\sigma\in\mathcal{S}_{4}}\mathrm{Tr}_{R}\left[T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right]\mathbf{T}^{a}_{i}\mathbf{T}^{b}_{i}\mathbf{T}^{c}_{i}\mathbf{T}^{d}_{i}\,,$
(7.26)

and where $g_{R}(\alpha_{s})$ is the function multiplying
$\frac{d_{RR_{i}}}{N_{R_{i}}}$ in eq. (7.5). As discussed in section 2.3, the
function $g_{R}(\alpha_{s})$ begins at four loops and is known at this order
Henn:2019swt ; Huber:2019fxe ; vonManteuffel:2020vjv . The result is quoted in
eq. (B.3). We provide a more detailed discussion concerning the relation
between $\mathbf{\Gamma}_{n}(\alpha_{s})$ and
$\Gamma^{\rm{cusp}}_{i}(\alpha_{s})$ in appendix F.

The remaining terms in (7.13) are part of the solution to the homogeneous
equation associated to eq. (7.3), therefore, the functions appearing in these
terms depend exclusively on CICRs. The term ${\mathrm{Q4T-4L}}$ involves the
quartic Casimir operator as well, and reads202020Owing to the complete
permutation symmetry of the colour factor $\bm{{\cal D}}_{ijkl}^{R}$ with
respect to $i,j,k$ and $l$, the kinematic function ${\cal G}_{R}$ admits a
similar symmetry. Consequently, ${\cal G}_{R}$ may be factored out of the sum
over the permutations of a given subset of indices.

$\mathbf{\Gamma}_{n,\rm
Q4T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)=\sum_{R}\sum_{(i,j,k,l)}\\!\bm{{\cal
D}}_{ijkl}^{R}\,{\cal G}_{R}(\beta_{ijlk},\beta_{iklj};\alpha_{s}).$ (7.27)

Finally, there are then two terms involving five colour generators: they are
given by

$\displaystyle\mathbf{\Gamma}_{n,\rm
5T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)$
$\displaystyle=\sum_{(i,j,k,l)}\\!\bm{{\cal T}}_{ijkli}\,{\cal
H}_{1}(\beta_{ijlk},\beta_{iklj};\alpha_{s}),$ (7.28a)
$\displaystyle\mathbf{\Gamma}_{n,\rm
5T-5L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)$
$\displaystyle=\sum_{(i,j,k,l,m)}\\!\bm{{\cal T}}_{ijklm}\,{\cal
H}_{2}(\beta_{ijkl},\beta_{ijmk},\beta_{ikmj},\beta_{jiml},\beta_{jlmi};\alpha_{s}),$
(7.28b)

where the colour structure is defined as

$\bm{{\cal T}}_{ijklm}=if^{adf}f^{bcg}f^{efg}\\{{\bf T}_{i}^{a},{\bf
T}_{j}^{b},{\bf T}_{k}^{c},{\bf T}_{l}^{d},{\bf T}_{m}^{e}\\}_{+}.$ (7.29)

The functions ${\cal G}_{R}(\alpha_{s})$, ${\cal H}_{1}(\alpha_{s})$ and
${\cal H}_{2}(\alpha_{s})$ start at four loops, and have not yet been
computed. Similarly, the four-loop contributions to the functions
$f(\alpha_{s})$ and ${\cal F}(\alpha_{s})$ are to date unknown. In all these
cases, the structure of these functions can be constrained by analysing
amplitudes in specific kinematic limits, where additional information can be
obtained. The collinear limit offers one such instance Becher:2009qa ;
Dixon:2009ur ; Almelid:2017qju ; Becher:2019avh , and we briefly summarise
below what constraints it provides based on ref. Becher:2019avh , before
returning to the Regge limit.

It is well known that when two particles in either the initial or final state
become collinear, the amplitude $\mathcal{M}_{n}$ factorises into a splitting
amplitude Sp and the parent amplitude $\mathcal{M}_{n-1}$ Berends:1988zn ;
Mangano_1991 ; Bern_1995 ; Kosower_1999 ; Catani:2011st , with

$\lim_{p_{1}||p_{2}}\mathcal{M}_{n}\left(\\{p_{1},\dots,p_{n}\\},\lambda,\alpha_{s}\right)=\textbf{Sp}\left(\\{p_{1},p_{2}\\},\lambda,\alpha_{s}\right)\mathcal{M}_{n-1}\left(\\{p_{1}+p_{2},p_{3},\dots
p_{n}\\},\lambda,\alpha_{s}\right).$ (7.30)

The splitting amplitude has an anomalous dimension defined as

$\frac{d}{d\log\lambda}\textbf{Sp}\left(\\{p_{1},p_{2}\\},\lambda,\alpha_{s}\right)={\bf\Gamma}_{\text{Sp}}\left(\\{p_{1},p_{2}\\},\lambda,\alpha_{s}\right)\textbf{Sp}\left(\\{p_{1},p_{2}\\},\lambda,\alpha_{s}\right)$
(7.31)

which, just like the function Sp itself, is independent of the momenta and
colour generators of the particles that are not collinear. Performing infrared
factorisation of each of the ingredients in eq. (7.30), one obtains
Becher:2009qa ; Dixon:2009ur

${\bf\Gamma}_{\bf
Sp}\left(\\{p_{1},p_{2}\\},\lambda,\alpha_{s}\right)=\underset{p_{1}||p_{2}}{\text{lim}}{\bf\Gamma}_{n}\left(\\{p_{1},\dots,p_{n}\\},\lambda,\alpha_{s}\right)-{\bf\Gamma}_{n-1}\left(\\{p_{1}+p_{2},p_{3},\dots,p_{n}\\},\lambda,\alpha_{s}\right).$
(7.32)

This provides the non-trivial constraint on ${\bf\Gamma}_{n}$ itself: the
splitting amplitude anomalous dimension on the left-hand side only depends on
the two particles that become collinear, hence so must the right-hand side of
eq. (7.32). This translates into concrete constraints for the functions in eq.
(7.13). The functions $f(\alpha_{s})$ and ${\cal F}(\alpha_{s})$ are related
by the condition Becher:2009qa ; Dixon:2009ur ; Almelid:2017qju ;
Becher:2019avh

$\underset{\beta_{12ij}\to-\infty}{\text{lim}}{\cal
F}(\beta_{12ij},0;\alpha_{s})=\frac{f(\alpha_{s})}{2},$ (7.33)

which, in particular, provide a constraint for the coefficients $f^{(4)}_{R}$
and ${\cal F}^{(4)}_{R}$. Similarly, the functions ${\cal G}_{R}(\alpha_{s})$
and $g_{R}(\alpha_{s})$ are related by Becher:2019avh

$\underset{\beta_{12ij}\to-\infty}{\text{lim}}{\cal
G}_{R}(\beta_{12ij},0;\alpha_{s})=-\frac{g_{R}(\alpha_{s})}{12}\,\beta_{12ij}.$
(7.34)

Furthermore, one has Becher:2019avh

$\underset{\beta_{12ij}\to-\infty}{\text{lim}}{\cal
H}_{1}(\beta_{12ij},0;\alpha_{s})=0\,.$ (7.35)

Last, we have constraints from the high-energy limit, which is of course the
topic of this section. Given our explicit calculation of $2\to 2$ parton
scattering in this limit, we are able to determine the four-loop contribution
to the functions appearing in eq. (7.13) in this limit. In order to proceed we
need first to specialise eq. (7.13) to the case of two-parton scattering, and
then take the high-energy limit. In this kinematic configuration no
constraints can be obtained for ${\cal H}^{(4)}_{2}$, which involves at least
five partons. However, we are able to obtain constraints for ${\cal F}^{(4)}$
and ${\cal G}_{R}^{(4)}$, as well as ${\cal H}^{(4)}_{1}$.

###  7.2 The soft anomalous dimension in the high-energy limit

We now take the general form of the soft anomalous dimension as written in eq.
(7.13), and specialise it to the case of $2\to 2$ particle scattering in the
high-energy limit. In short, the procedure is as follows.

  * •

In eq. (7.13) we drop the contributions which only appear for more than four
external partons, i.e., we do not consider ${\cal H}_{2}$.

  * •

We express the colour operators of eq. (7.13) in what we call _a Regge-limit
basis_ , i.e., in terms of a minimal212121By using the matrix expression of
$\mathbf{T}_{t}^{2}$ and $\mathbf{T}_{s-u}^{2}$, obtained by specialising to
projectile and target states either in the fundamental or in the adjoint
representation Caron-Huot:2017fxr , we verified that there are no linear
relations among the colour structures appearing in the reduced amplitude of
eq. (5.3). set of colour operators made out of ${\bf T}_{t}^{2}$, ${\bf
T}_{s-u}^{2}$, their commutators and quartic Casimir operators, as discussed
in section 4.2. Notice that, in particular, this will naturally split the
terms in eq. (7.13) into even and odd signature contributions.

  * •

We specialise the kinematic functions appearing in eq. (7.13) to the high-
energy limit. First, owing to Bose symmetry, each kinematic function will
acquire a definite signature symmetry, matching the symmetry of the
corresponding colour operator it multiplies. Furthermore, each function will
be implicitly understood as an expansion in the high-energy logarithm $L$
defined in eq. (2.9).

We expand the soft anomalous dimension in powers of the strong coupling,
according to

$\mathbf{\Gamma}_{n}(\\{s_{ij}\\},\lambda,\alpha_{s})=\sum_{\ell}\left(\frac{\alpha_{s}}{\pi}\right)^{\ell}\mathbf{\Gamma}^{(\ell)}_{n}(\\{s_{ij}\\},\lambda)\,,$
(7.36)

where $\ell$ is the loop order. In what follows we are interested to obtain
constraints on the coefficient functions appearing in
$\mathbf{\Gamma}^{(4)}_{4}$ by using the results for the NLL, ${\cal
O}(\alpha_{s}^{4}L^{3})$, and the NNLL, ${\cal O}(\alpha_{s}^{4}L^{2})$, in
the anomalous dimension, summarised in section 6.5. The four-loop order
coefficients $\gamma_{K,R}^{(4)}$, $g_{R}^{(4)}$, $f^{(4)}_{R}$ are associated
exclusively with linear and kinematically-independent contributions, ${\cal
O}(L^{1})$ and ${\cal O}(L^{0})$, and we will not consider them in this
section. Their high-energy limit is considered instead in appendix G, and we
will return to analyse the resulting ${\cal O}(L^{1})$ terms in section 7.4.
This leaves the terms proportional to ${\cal F}$, ${\cal G}_{R}$ and ${\cal
H}_{1}$, i.e. we consider

$\displaystyle\begin{split}\mathbf{\Gamma}_{4}^{(4)}\left(\\{s_{ij}\\},\lambda\right)&=\mathbf{\Gamma}_{\rm
4T-4L}^{(4)}\left(\\{\beta_{ijkl}\\}\right)+\mathbf{\Gamma}_{\rm
Q4T-4L}^{(4)}\left(\\{\beta_{ijkl}\\}\right)+\mathbf{\Gamma}_{\rm
5T-4L}^{(4)}\left(\\{\beta_{ijkl}\\}\right)+\mathcal{O}(L)\,,\end{split}$
(7.37)

where for individual terms we drop the subscript indicating the number of
partons, since we focus exclusively on the $n=4$ case below. The remaining
subscripts are the defining characteristics of the colour operator as in eq.
(7.13).

####  7.2.1 Four-generator four-line term ($\mathbf{4T}$$-$$\mathbf{4L}$)

We start by considering the first term in eq. (7.37), i.e.

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm{4T-4L}}(\\{\beta_{ijkl}\\},\alpha_{s})&\equiv\sum_{(i,j,k,l)}f^{ade}f^{bce}{\bf
T}_{i}^{a}{\bf T}_{j}^{b}{\bf T}_{k}^{c}{\bf T}_{l}^{d}\,{\cal
F}(\beta_{ijlk},\beta_{iklj};\alpha_{s}).\end{split}$ (7.38)

An expression for this term that is specialised to two-parton scattering in
the high-energy limit has been discussed already in refs. Caron-Huot:2017fxr ;
Almelid:2017qju , however, we provide here a short derivation for pedagogical
purposes, in order to introduce useful notation for the elaboration of the
other terms in eq. (7.37).

The colour structure in eq. (7.38) is antisymmetric under the exchange of
$i\leftrightarrow l$ or $j\leftrightarrow k$. Due to Bose symmetry, the
function ${\cal F}(\alpha_{s})$ must be antisymmetric under the exchange of
the same indices. Under this exchange one has ${\cal
F}(\beta_{ijlk},\beta_{iklj};\alpha_{s})=-{\cal
F}(\beta_{iklj},\beta_{ijlk};\alpha_{s})$. Using this property, we write eq.
(7.38) for the case of two-parton scattering as

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
4T-4L}(\\{\beta_{ijkl}\\},\alpha_{s})=&\,8{\bf T}_{1}^{a}{\bf T}_{2}^{b}{\bf
T}_{3}^{c}{\bf T}_{4}^{d}\bigg{[}f^{abe}f^{cde}{\cal
F}(\beta_{1324},\beta_{1423};\alpha_{s})\\\ &\hskip
56.9055pt+f^{ace}f^{bde}{\cal F}(\beta_{1234},\beta_{1432};\alpha_{s})\\\
&\hskip 56.9055pt+f^{ade}f^{bce}{\cal
F}(\beta_{1243},\beta_{1342};\alpha_{s})\bigg{]}.\end{split}$ (7.39)

As we have seen, in the high-energy limit, signature symmetry plays a major
role. In eq. (7.39) it can be implemented by considering symmetric and
antisymmetric combinations under the exchange $2\leftrightarrow 3$. This leads
us to introduce the following two functions:

$\displaystyle\begin{split}{\cal
F}^{(+)}(\\{\beta_{ijkl}\\},\alpha_{s})&\equiv\frac{1}{2}\bigg{\\{}{\cal
F}(\beta_{1324},\beta_{1423};\alpha_{s})+{\cal
F}(\beta_{1234},\beta_{1432};\alpha_{s})\bigg{\\}},\\\ {\cal
F}^{(-)}(\\{\beta_{ijkl}\\},\alpha_{s})&\equiv\frac{1}{2}\bigg{\\{}{\cal
F}(\beta_{1234},\beta_{1432};\alpha_{s})-{\cal
F}(\beta_{1324},\beta_{1423};\alpha_{s})\bigg{\\}}+{\cal
F}(\beta_{1243},\beta_{1342};\alpha_{s}),\end{split}$ (7.40)

such that eq. (7.39) becomes

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
4T-4L}(\\{\beta_{ijkl}\\},\alpha_{s})=&\,8{\bf T}_{1}^{a}{\bf T}_{2}^{b}{\bf
T}_{3}^{c}{\bf
T}_{4}^{d}\bigg{[}\left(f^{abe}f^{cde}+f^{ace}f^{bde}\right){\cal
F}^{(+)}(\\{\beta_{ijkl}\\},\alpha_{s})\\\ &\hskip
85.35826pt+\,f^{ade}f^{bce}{\cal
F}^{(-)}(\\{\beta_{ijkl}\\},\alpha_{s})\bigg{]}.\end{split}$ (7.41)

Due to Bose symmetry, the symmetry of ${\cal F}^{(\pm)}$ must be mirrored into
the colour structure. This becomes evident when expressing the colour
operators in eq. (7.41) in our Regge-limit basis. Using the colour algebra
identity of eq. (4.34), i.e. $f^{abc}{\bf T}^{c}=-i[{\bf T}^{a},{\bf T}^{b}]$,
we have for instance

$\displaystyle f^{abe}f^{cde}{\bf T}_{1}^{a}{\bf T}_{2}^{b}{\bf T}_{3}^{c}{\bf
T}_{4}^{d}$ $\displaystyle=$ $\displaystyle-\Big{[}{\bf T}_{1}\cdot{\bf
T}_{2},[{\bf T}_{3}\cdot{\bf T}_{4},{\bf T}_{1}\cdot{\bf T}_{3}]\Big{]}$
(7.42) $\displaystyle=$ $\displaystyle-\frac{1}{8}\Big{[}{\bf T}_{s}^{2},[{\bf
T}_{s}^{2},{\bf T}_{u}^{2}]\Big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left(\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}+2\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\right),$

where we used the definitions in eqs. (2.17) and (2.19), and any Casimirs
arising vanish in the commutators. With similar steps, the two other colour
operators in eq. (7.41) are written as

$\displaystyle f^{ace}f^{bde}{\bf T}_{1}^{a}{\bf T}_{2}^{b}{\bf T}_{3}^{c}{\bf
T}_{4}^{d}$
$\displaystyle=\frac{1}{16}\left(-\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}+2\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\right)\,,$
(7.43a) $\displaystyle f^{ade}f^{bce}{\bf T}_{1}^{a}{\bf T}_{2}^{b}{\bf
T}_{3}^{c}{\bf T}_{4}^{d}$
$\displaystyle=-\frac{1}{8}\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}.$
(7.43b)

Inserting the expressions in eqs. (7.42) and (7.43) into eq. (7.41) we get

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
4T-4L}(\\{\beta_{ijkl}\\},\alpha_{s})&=2{\cal
F}^{(+)}(\\{\beta_{ijkl}\\},\alpha_{s})\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\\\
&\quad\mbox{}-{\cal
F}^{(-)}(\\{\beta_{ijkl}\\},\alpha_{s})\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\,.\end{split}$
(7.44)

It is easy to see that the symmetry properties of ${\cal F}^{(\pm)}$ are
nicely mirrored into the colour structure: the first nested commutator is
signature-even, containing two $\mathbf{T}_{s-u}^{2}$ operators, while the
second is odd, having a single $\mathbf{T}_{s-u}^{2}$.

At three loops, using the properties of the variables $z_{ijkl}$ introduced in
(7.20):

$z_{ijkl}=\frac{1}{z_{ikjl}}=1-z_{ilkj}=\frac{z_{ijlk}}{z_{ijlk}-1},$ (7.45)

one can write the functions ${\cal F}^{(\pm)}$ as

$\displaystyle\begin{split}{\cal
F}^{(+,3)}\left(\\{\beta_{ijkl}\\}\right)&=\frac{1}{64}F_{1}(z_{1234}),\\\
{\cal
F}^{(-,3)}\left(\\{\beta_{ijkl}\\}\right)&=\frac{1}{64}\Big{(}F_{2}(z_{1234})-F_{3}(z_{1234})\Big{)},\end{split}$
(7.46)

where the functions $F_{1}$, $F_{2}$ and $F_{3}$ have been introduced in ref.
Almelid:2017qju , and read

$\displaystyle\begin{split}F_{1}(z)&\equiv F(1-1/z)-F(1/z)+F(1-z)-F(z),\\\
F_{2}(z)&\equiv F(1/z)-F(1-1/z)+F(1/(1-z))-F(z/(z-1)),\\\ F_{3}(z)&\equiv
F(z)-F(1-z)+F(z/(z-1))-F(1/(1-z))=-F_{1}(z)-F_{2}(z).\end{split}$ (7.47)

Here we consider the four-loop contribution to eq. (7.44). Taking into account
the perturbative expansion introduced in eq. (7.19) one has

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm{4T-4L}}^{(4)}(\\{\beta_{ijkl}\\})&=2\left(\sum_{R}{\cal
F}^{(+,4)}_{R}(\\{\beta_{ijkl}\\})\right)\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\\\
&\hskip 100.0pt-\left(\sum_{R}{\cal
F}^{(-,4)}_{R}(\\{\beta_{ijkl}\\})\right)\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]},\end{split}$
(7.48)

where we recall that the sum over representations starts at this order due to
an additional internal loop, which gives rise to either a factor of $C_{A}$,
or $n_{f}T_{F}$, depending on the particles propagating in the loop222222Here
we have only considered QCD particle types: adjoint gluons and fundamental
quarks. The factor will change depending on the gauge theory considered.. The
first term in eq. (7.48) is signature even, and the second signature odd. It
is worthwhile recalling that, upon expansion, the soft anomalous dimension in
eq. (7.48) will be multiplied by the odd tree-level amplitude in eq. (2.5):
hence, odd signature in the amplitude corresponds to even signature in the
soft anomalous dimension. Taking this into account, we can already make a few
observations. At NLL accuracy there are only gluonic contributions in the even
amplitude, as calculated in ref. Caron-Huot:2017zfo . Therefore, only ${\cal
F}^{(-,4)}_{A}|_{\rm NLL}$ will be non-zero in eq. (7.48), while ${\cal
F}^{(-,4)}_{F}|_{\rm NLL}=0$. Similarly, the NNLL contribution to the odd
amplitude, first presented in ref. Falcioni:2020lvv  and discussed in detail
in section 5 above, is also given in terms of gluonic contributions only.
Following the reasoning above, we expect that ${\cal F}^{(+,4)}_{A}|_{\rm
NNLL}$ may be non-zero and ${\cal F}^{(+,4)}_{F}|_{\rm NNLL}=0$. No
predictions for ${\cal F}^{(-,4)}_{R}|_{\rm NNLL}$ can be made at this stage,
however, given that the even amplitude is still unknown at this logarithmic
accuracy.

####  7.2.2 Quartic Casimir four-generator four-line term
($\mathbf{Q4T}$$-$$\mathbf{4L}$)

The quartic Casimir term only appears starting at four loops. Restricting to
the case of $2\to 2$ scattering and writing explicitly the colour structure,
eq. (7.27) becomes

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
Q4T-4L}(\\{\beta_{1234}\\},\alpha_{s})=&\sum_{R}\,{\cal
G}_{R}(\beta_{1243},\beta_{1342};\alpha_{s})\times\\\ &\hskip
56.9055pt\sum_{\sigma\in
S_{4}}\operatorname{Tr}_{R}\left(T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right){\bf
T}_{1}^{a}{\bf T}_{2}^{b}{\bf T}_{3}^{c}{\bf T}_{4}^{d}\,,\end{split}$ (7.49)

where again there is a sum over the representations $R$ propagating in the
loop. Here we extracted the function ${\cal G}_{R}$ out of the sum over
permutations of the legs $(i,j,k,l)$ using its symmetry: the colour structure
is symmetric under the exchange of any pair of indices due to the symmetrised
trace. Having done that, we performed the sum over permutations $(i,j,k,l)$ on
the colour structure. Because ${\cal
G}_{R}(\beta_{ijlk},\beta_{iklj};\alpha_{s})$ is a completely symmetric
function under permutations, ${\cal G}^{(-)}_{R}=0$, and we can identify
${\cal G}^{(+)}_{R}={\cal G}_{R}$.

In order to conveniently express eq. (7.49), we first introduce some new
colour notation for terms involving a symmetrised trace over four generators
attached to four numbered partonic generators. The colour structures can be
expressed using the colour-flow channels defined in eq. (2.17), with

$\displaystyle\begin{split}\bm{\mathcal{D}}^{R}_{pppp}\equiv&\,\frac{1}{4!}\sum_{\sigma\in
S_{4}}\operatorname{Tr}_{R}\left(T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right){\bf
T}_{p}^{a}{\bf T}_{p}^{b}{\bf T}_{p}^{c}{\bf T}_{p}^{d},\end{split}$ (7.50)

where $p\in\\{s,t,u\\}$, for example

$\displaystyle\begin{split}\bm{\mathcal{D}}^{R}_{ssss}=&\,\frac{1}{4!}\sum_{\sigma\in
S_{4}}\operatorname{Tr}_{R}\left(T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right){\bf
T}_{s}^{a}{\bf T}_{s}^{b}{\bf T}_{s}^{c}{\bf T}_{s}^{d}\\\
=\,&\bm{\mathcal{D}}^{R}_{1111}+4\bm{\mathcal{D}}^{R}_{1112}+6\bm{\mathcal{D}}^{R}_{1122}+4\bm{\mathcal{D}}^{R}_{1222}+\bm{\mathcal{D}}^{R}_{2222}.\end{split}$
(7.51)

A general formula is

$\bm{\mathcal{D}}^{R}_{pppp}=\,\bm{\mathcal{D}}^{R}_{kkkk}+4\bm{\mathcal{D}}^{R}_{kkkl}+6\bm{\mathcal{D}}^{R}_{kkll}+4\bm{\mathcal{D}}^{R}_{klll}+\bm{\mathcal{D}}^{R}_{llll},$
(7.52)

where

$\bm{\mathcal{D}}^{R}_{pppp}=\begin{cases}\bm{\mathcal{D}}^{R}_{ssss}&(k,l)\in\\{(1,2),(3,4)\\}\\\
\bm{\mathcal{D}}^{R}_{uuuu}&(k,l)\in\\{(1,3),(2,4)\\}\\\
\bm{\mathcal{D}}^{R}_{tttt}&(k,l)\in\\{(1,4),(2,3)\\}.\end{cases}$ (7.53)

The expression in eq. (7.52) is symmetric under $k\leftrightarrow l$. For each
of the channels, corresponding to the respective Mandelstam invariants
$p\in\\{s,t,u\\}$, the indices $(k,l)$ can be assigned to be either of the two
combinations shown in eq. (7.53).

Using colour conservation, we can write the colour structure of eq. (7.49) as

$\displaystyle\sum_{\sigma\in
S_{4}}\operatorname{Tr}_{R}\left(T^{\sigma(a)}T^{\sigma(b)}T^{\sigma(c)}T^{\sigma(d)}\right){\bf
T}_{1}^{a}{\bf T}_{2}^{b}{\bf T}_{3}^{c}{\bf
T}_{4}^{d}\mathcal{M}_{\text{tree}}$
$\displaystyle=\bigg{[}2\Big{(}\bm{\mathcal{D}}^{R}_{ssss}+\bm{\mathcal{D}}^{R}_{uuuu}+\bm{\mathcal{D}}^{R}_{tttt}\Big{)}-4\left(\frac{d_{RR_{i}}}{N_{R_{i}}}+\frac{d_{RR_{j}}}{N_{R_{j}}}\right)\bigg{]}\mathcal{M}_{\text{tree}},$
(7.54)

where the quartic Casimirs correspond to the projectile $i$ (partons 1 and 4)
and the target $j$ (partons 2 and 3). The whole expression is signature-even
as expected. This expression is useful as it holds for any representation. We
will see in eqs. (7.85) and (7.86) and in appendix G that the colour
structures multiplying the quartic component of the cusp anomalous dimension
$g_{R}$ can be expressed in a similar way.

##### Adjoint representation.

In the following we restrict our attention to the four-loop coefficient ${\cal
G}^{(+,4)}_{R}$ in the adjoint representation, $R=A$. The reason we focus
specifically on this representation was already explained at the end of the
previous section considering ${\cal F}^{(+,4)}_{R}$, that is: the result for
the signature-odd amplitude at NNLL accuracy, presented in section 5, only
receives a contribution from purely gluonic diagrams. Thus, only ${\cal
G}^{(+,4)}_{A}$ contributes to the sum over $R$ in eq. (7.49) at NNLL
accuracy. It is then possible to use the identity Becher:2019avh

$\displaystyle\begin{split}\sum_{\sigma\in
S_{4}}\operatorname{Tr}\left(F^{\sigma(a)}F^{\sigma(b)}F^{\sigma(c)}F^{\sigma(d)}\right)&=12\left[\mbox{Tr}\big{(}F^{a}F^{b}F^{c}F^{d}\big{)}+\mbox{Tr}\big{(}F^{d}F^{c}F^{b}F^{a}\big{)}\right]\\\
&\hskip
56.9055pt+4C_{A}\left(f^{abe}f^{cde}-f^{ade}f^{bce}\right),\end{split}$ (7.55)

to write

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
Q4T-4L,A}^{(4)}(\\{\beta_{ijkl}\\})&=\big{(}{\bf Q}_{1}^{(4),A}+{\bf
Q}_{2}^{(4),A}\big{)}\,{\cal
G}_{A}^{(+,4)}(\beta_{1234},\beta_{1432}),\end{split}$ (7.56)

where we have defined

$\displaystyle{\bf Q}_{1}^{(4),A}$ $\displaystyle=$ $\displaystyle 12\,{\bf
T}^{a}_{1}{\bf T}^{b}_{2}{\bf T}^{c}_{3}{\bf
T}^{d}_{4}\,\left[\mbox{Tr}\big{(}F^{a}F^{b}F^{c}F^{d}\big{)}+\mbox{Tr}\big{(}F^{d}F^{c}F^{b}F^{a}\big{)}\right],$
(7.57a) $\displaystyle{\bf Q}_{2}^{(4),A}$ $\displaystyle=$ $\displaystyle
4C_{A}\,{\bf T}^{a}_{1}{\bf T}^{b}_{2}{\bf T}^{c}_{3}{\bf
T}^{d}_{4}\left(f^{abe}f^{cde}-f^{ade}f^{bce}\right),$ (7.57b)

and $(F^{x})^{ab}\equiv if^{axb}$. The second term, i.e. ${\bf
Q}_{2}^{(4),A}$, can readily be written in a Regge-limit basis by using the
identities in eqs. (7.42) and (7.43b). We get

${\bf
Q}_{2}^{(4),A}=\frac{C_{A}}{4}\left(3\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}+2\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\right).$
(7.58)

Concerning ${\bf Q}_{1}^{(4),A}$, repeated use of the commutator relation eq.
(4.34) allows us to write it as

${\bf Q}_{1}^{(4),A}=12\bigg{\\{}\Big{[}\big{[}{\bf T}^{b}_{1},{\bf
T}^{e}_{1}\big{]},{\bf T}^{f}_{1}\Big{]}{\bf T}_{2}^{b}\Big{[}\big{[}{\bf
T}^{d}_{3},{\bf T}^{f}_{3}\big{]},{\bf T}^{e}_{3}\Big{]}{\bf T}_{4}^{d}+\,{\bf
T}_{1}^{a}\Big{[}{\bf T}^{e}_{2},\big{[}{\bf T}^{f}_{2},{\bf
T}^{a}_{2}\big{]}\Big{]}{\bf T}_{3}^{c}\Big{[}{\bf T}^{f}_{4},\big{[}{\bf
T}^{e}_{4},{\bf T}^{c}_{4}\big{]}\Big{]}\bigg{\\}}.$ (7.59)

Notice that we apply the commutator relation such as to obtain an expression
with manifest target-projectile symmetry (where, as usual, partons $1$ and $4$
represent the projectile while partons $2$ and $3$ the target). At this point
we recall that the colour operator in the soft anomalous dimension acts on the
tree-level amplitude, to give the part of the four-loop amplitude from which
the single-pole singularities are extracted. Therefore, it is sufficient to
obtain a representation for the colour operator ${\bf Q}_{1}^{(4),A}$ when
acting on the tree-level colour structure ${\bf T}_{i}\cdot{\bf T}_{j}$, as
defined in eq. (2.5). The commutators in eq. (7.59) can then be expressed as
attachments to the projectile ($i$) or target ($j$), as in section 4.2.3, so
eq. (7.59) becomes

${\bf Q}_{1}^{(4),A}({\bf T}_{i}\cdot{\bf T}_{j})=12\left({\bf
T}^{([[b,e],f],x,d)}\right)_{i}\left({\bf
T}^{(b,x,[[d,f],e])}\right)_{j}\,\,+\,\,i\leftrightarrow j.$ (7.60)

It is now in a suitable form to apply the identities in section 4.2 and
appendix C, converting the operator to the Regge-limit basis:

$\displaystyle\begin{split}{\bf Q}_{1}^{(4),A}({\bf T}_{i}\cdot{\bf
T}_{j})&=\bigg{\\{}2\left(\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}\right)-\frac{3C_{A}}{4}\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\\\
&\hskip
14.22636pt-\,\frac{1}{2}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]+\frac{3}{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\bigg{\\}}\,({\bf
T}_{i}\cdot{\bf T}_{j}).\end{split}$ (7.61)

Inserting eqs. (7.58) and (7.61) into eq. (7.56) we have

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
Q4T-4L,A}^{(4)}(\\{\beta_{ijkl}\\})\mathcal{M}_{\text{tree}}&={\cal
G}_{A}^{(+,4)}(\beta_{1234},\beta_{1432})\bigg{\\{}2\left(\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}\right)-\frac{1}{2}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\\\
&\quad\mbox{}+\frac{3}{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{2}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\bigg{\\}}\,\mathcal{M}_{\text{tree}}.\end{split}$
(7.62)

Notice that after a cancellation of the signature-odd commutator term between
${\bf Q}_{1}^{(4),A}$ and ${\bf Q}_{2}^{(4),A}$, the resulting colour operator
in eq. (7.62) is manifestly signature-even, as anticipated at the beginning of
the section. Importantly, we observe that the quartic four-generator four-leg
term $\mathbf{\Gamma}_{\rm Q4T-4L,A}^{(4)}$ is entirely non-planar, given that
the commutators in eq. (7.62) and the combination $d_{AA}/N_{A}-C_{A}^{4}/24$
are separately non-planar (see eq. (5.35)). $\mathbf{\Gamma}_{\rm
Q4T-4L,A}^{(4)}$ now is expressed in the Regge-limit basis, and eq. (7.62)
will be used in section 7.3, along with the other terms, to derive constraints
based on the explicit NNLL results of section 5.

Finally, we can also equate eq. (7.2.2) to eq. (7.62) in the adjoint
representation to express the previously-unknown signature-even combination of
quartic $s$ and $u$ channel operators acting on the tree amplitude, in terms
of nested commutators:

$\displaystyle\begin{split}\Big{(}\bm{\mathcal{D}}^{A}_{ssss}+\bm{\mathcal{D}}^{A}_{uuuu}\Big{)}\mathcal{M}_{\text{tree}}&=\bigg{(}2\left(\frac{d_{AR_{i}}}{N_{R_{i}}}+\frac{d_{AR_{j}}}{N_{R_{j}}}\right)-\frac{C_{A}^{4}}{24}-\frac{1}{4}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\\\
&\quad+\frac{3}{4}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{4}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\bigg{)}\mathcal{M}_{\text{tree}},\end{split}$
(7.63)

while the quartic $t$-channel operator acting on the tree amplitude simply
gives

$\bm{\mathcal{D}}^{A}_{tttt}\mathcal{M}_{\text{tree}}=\frac{d_{AA}}{N_{A}}\mathcal{M}_{\text{tree}}.$
(7.64)

These results will be useful in appendix G, where we analyse the colour
structures multiplying the quartic component of the cusp anomalous dimension
$g_{R}$.

####  7.2.3 Five-generator four-line term ($\mathbf{5T}$$-$$\mathbf{4L}$)

The third term in eq. (7.37) reads

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
5T-4L}\left(\\{\beta_{ijkl}\\},\alpha_{s}\right)&=\sum_{(i,j,k,l)}\,\bm{\mathcal{T}}_{ijkli}\,{\cal
H}_{1}(\beta_{ijlk},\beta_{iklj};\alpha_{s})\\\
&=\sum_{(i,j,k,l)}if^{adg}f^{bch}f^{egh}\,\\{{\bf T}_{i}^{a},{\bf
T}_{i}^{e}\\}_{+}{\bf T}_{j}^{b}{\bf T}_{k}^{c}{\bf T}_{l}^{d}\,{\cal
H}_{1}(\beta_{ijlk},\beta_{iklj};\alpha_{s}).\end{split}$ (7.65)

The colour structure is antisymmetric under $j\leftrightarrow k$ so
$\bm{\mathcal{T}}_{ijkli}=-\bm{\mathcal{T}}_{ikjli}$ and therefore ${\cal
H}_{1}$ is antisymmetric under a swap of its arguments, due to Bose symmetry.
We want to write an expression with manifest symmetry under $s\leftrightarrow
u$, which can be achieved by exploiting the symmetries under swaps of
$2\leftrightarrow 3$ or $1\leftrightarrow 4$ of $\bm{\mathcal{T}}_{ijkli}$ and
${\cal H}_{1}$. Similarly to ${\cal F}(\beta_{ijlk},\beta_{iklj},\alpha_{s})$,
let us introduce symmetric and antisymmetric combinations under
$2\leftrightarrow 3$ of ${\cal H}_{1}$:

$\displaystyle{\cal H}_{1}^{(+)}(\\{\beta_{ijkl}\\},\alpha_{s})$
$\displaystyle\equiv\frac{1}{2}\Big{\\{}{\cal
H}_{1}(\beta_{1324},\beta_{1423};\alpha_{s})+{\cal
H}_{1}(\beta_{1234},\beta_{1432};\alpha_{s})\Big{\\}},$ (7.66a)
$\displaystyle{\cal H}_{1}^{(-)}(\\{\beta_{ijkl}\\},\alpha_{s})$
$\displaystyle\equiv\frac{1}{2}\Big{\\{}{\cal
H}_{1}(\beta_{1324},\beta_{1423};\alpha_{s})-{\cal
H}_{1}(\beta_{1234},\beta_{1432};\alpha_{s})\Big{\\}},$ (7.66b)
$\displaystyle\tilde{{\cal H}}_{1}^{(-)}(\\{\beta_{ijkl}\\},\alpha_{s})$
$\displaystyle\equiv{\cal H}_{1}(\beta_{1243},\beta_{1342};\alpha_{s}).$
(7.66c)

As shown in appendix G, with these definitions we can write eq. (7.65) as

$\displaystyle\begin{split}\mathbf{\Gamma}_{\rm
5T-4L}^{(4)}(\\{\beta_{ijkl}\\})\mathcal{M}_{\text{tree}}&=\Bigg{[}{\cal
H}_{1}^{(+,4)}(\\{\beta_{ijkl}\\})\bigg{(}-\frac{C_{A}}{2}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+C_{A}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\\\
&-\frac{1}{6}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\bigg{)}+\frac{1}{4}\tilde{{\cal
H}}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\\\
&+{\cal
H}_{1}^{(-,4)}(\\{\beta_{ijkl}\\})\bigg{(}-\frac{1}{2}\bigg{[}\mathbf{T}_{s-u}^{2},\Big{[}\mathbf{T}_{s-u}^{2},\big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\big{]}\Big{]}\bigg{]}+\frac{1}{8}\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\bigg{)}\Bigg{]}\mathcal{M}_{\text{tree}},\end{split}$
(7.67)

with all colour operators expressed in terms of nested commutators. Thus, as
expected, all three terms in eq. (7.37), given in eqs. (7.48), (7.62) and
(7.67), are strictly non-planar.

####  7.2.4 The Regge limit of the soft anomalous dimension

We have now specialised eq. (7.37) to the case of two-parton scattering, and
expressed the colour operators involved in these terms in a Regge-limit basis
in eqs. (7.48), (7.62) and (7.67). In order to compare the resulting
expression for the soft anomalous dimension with the high-energy limit
calculation summarised in section 6.5 we formally consider each of the
kinematic functions as an expansion in the high-energy logarithm $L$, for
instance:

${\cal F}^{(-,4)}_{A}(L)={\cal F}^{(-,4,3)}_{A}L^{3}+{\cal
F}^{(-,4,2)}_{A}L^{2}+{\cal F}^{(-,4,1)}_{A}L+{\cal F}^{(-,4,0)}_{A},$ (7.68)

with unknown coefficients ${\cal F}^{(-,\ell,n)}_{A}$ for $\ell=4$ and
$n=3,2,1,0$, which we expect on general grounds to be transcendental numbers
of weight $2\ell-1-n=7-n$ or lower.

Separating the four-loop soft anomalous dimension $\bm{\Gamma}^{(4)}$ into
components with definite signature symmetry, we have at four loops

$\bm{\Gamma}^{(4)}_{\text{Regge}}=\bm{\Gamma}^{(+,4)}_{\text{Regge}}+\bm{\Gamma}^{(-,4)}_{\text{Regge}}\,,$
(7.69)

where we added the subscript Regge, to indicate that the Regge limit has been
taken. Explicitly, using the results in eqs. (7.48), (7.62) and (7.67) we
obtain

$\displaystyle\mathbf{\Gamma}^{(+,4)}_{\text{Regge}}(L)\mathcal{M}_{\text{tree}}=$
$\displaystyle\Bigg{\\{}2{\cal
F}_{A}^{(+,4)}(L)\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}$
(7.70) $\displaystyle\quad\mbox{}+{\cal
G}_{A}^{(+,4)}(L)\bigg{(}2\left(\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}\right)-\frac{1}{2}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]$
$\displaystyle\hskip
71.13188pt+\frac{3}{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{2}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\bigg{)}$
$\displaystyle\quad\mbox{}+{\cal
H}_{1}^{(+,4)}(L)\bigg{(}-\frac{1}{2}C_{A}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+C_{A}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]$
$\displaystyle\hskip
56.9055pt-\frac{1}{6}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\bigg{)}\Bigg{\\}}\mathcal{M}_{\text{tree}}+{\cal
O}(L),$

for the signature-even part, while the odd component reads

$\displaystyle\mathbf{\Gamma}^{(-,4)}_{\text{Regge}}(L)\mathcal{M}_{\text{tree}}$
$\displaystyle=\Bigg{\\{}-\left(\sum_{R}{\cal
F}_{R}^{(-,4)}(L)\right)\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}$
(7.71) $\displaystyle\hskip-4.0pt+{\cal
H}_{1}^{(-,4)}(L)\bigg{(}-\frac{1}{2}\bigg{[}\mathbf{T}_{s-u}^{2}\Big{[}\mathbf{T}_{s-u}^{2},\big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\big{]}\Big{]}\bigg{]}+\frac{1}{8}\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\bigg{)}$
$\displaystyle\hskip-4.0pt+\frac{1}{4}\tilde{{\cal
H}}_{1}^{(-,4)}(L)\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\Bigg{\\}}\mathcal{M}_{\text{tree}}+{\cal
O}(L).$

Functions that do not contribute through NNLL, i.e., only generate ${\cal
O}(L)$ and ${\cal O}(L^{0})$ contributions, are not shown in eqs. (7.70) and
(7.71). We discuss these in section 7.4 and appendix G.

###  7.3 Constraints on the kinematic functions in the soft anomalous
dimension

We are now ready to compare the general parametrisation of the four-loop soft
anomalous dimension to the explicit results of our calculation in the high-
energy limit. Before considering the four-loop case, where the kinematic
functions are unknown, it is useful to conduct a similar exercise at three
loops, where the functions are known Almelid:2015jia  and their high-energy
limit has been previously obtained Caron-Huot:2017fxr ; Almelid:2017qju .

Using eqs. (7.44) and (G.41) we have

$\displaystyle\mathbf{\Delta}^{(+,3)}_{\text{Regge}}(L)\mathcal{M}_{\text{tree}}$
$\displaystyle=\bigg{\\{}2\Big{(}{\cal F}^{(+,3,2)}L^{2}+{\cal
F}^{(+,3,1)}L+{\cal
F}^{(+,3,0)}\Big{)}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}$
(7.72a)
$\displaystyle+f^{(3)}\bigg{(}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+\frac{C_{A}^{3}}{2}-6\frac{d_{AR_{i}}}{N_{R_{i}}C_{i}}-6\frac{d_{AR_{j}}}{N_{R_{j}}C_{j}}\bigg{)}\bigg{\\}}\mathcal{M}_{\text{tree}}$
$\displaystyle\mathbf{\Delta}^{(-,3)}_{\text{Regge}}(L)\mathcal{M}_{\text{tree}}$
$\displaystyle=-\Big{(}{\cal F}^{(-,3,2)}L^{2}+{\cal F}^{(-,3,1)}L+{\cal
F}^{(-,3,0)}\Big{)}\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\mathcal{M}_{\text{tree}}$
(7.72b)

Matching these expressions to $\mathbf{\Delta}$ in appendix E we obtain the
following expansion coefficients:

$\displaystyle\begin{array}[]{lll}{\displaystyle{\cal
F}^{(+,3,2)}=0},&{\displaystyle{\cal F}^{(+,3,1)}=0},&{\displaystyle{\cal
F}^{(+,3,0)}=\frac{1}{8}\left(4\zeta_{3}\zeta_{2}-\zeta_{5}\right)}\\\
{\displaystyle{\cal F}^{(-,3,2)}=0},&{\displaystyle{\cal
F}^{(-,3,1)}=-\frac{i\pi}{4}\zeta_{3}},&{\displaystyle{\cal
F}^{(-,3,0)}=-\frac{11i\pi}{4}\zeta_{4}}\,,\end{array}$ (7.75)

consistently with refs. Caron-Huot:2017fxr ; Almelid:2017qju ; Almelid:2015jia
; Henn:2016jdu . A similar procedure will be followed below at four loops
where there are more functions contributing, all of which are yet unknown. To
this end we consider the expressions in eqs. (7.70) and (7.71) in the Regge-
limit basis, which we comapre with the explicit results we obtained through
NNLL accuracy in eq. (6.48).

##### Constraints at four loops NLL for signature-odd functions.

We start by considering the signature-odd contribution to the soft anomalous
dimension at NLL accuracy. We expand the functions in eq. (7.71) as in eq.
(7.68), and match it to eq. (7.71) order by order in the high-energy logarithm
$L$. At $\mathcal{O}(L^{3})$, equating eq. (7.71) to eq. (6.46), we have

$\displaystyle\begin{split}-i\pi\frac{\zeta_{3}}{24}\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\mathbf{T}_{t}^{2}\,\,\overset{!}{=}\,\,&-\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\,{\cal
F}^{(-,4,3)}_{A}\\\
&-\frac{1}{2}\bigg{[}\mathbf{T}_{s-u}^{2},\Big{[}\mathbf{T}_{s-u}^{2},\big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\big{]}\Big{]}\bigg{]}{\cal
H}^{(-,4,3)}_{1}\\\
&+\frac{1}{8}\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\left(2\tilde{{\cal
H}}^{(-,4,3)}_{1}+{\cal H}^{(-,4,3)}_{1}\right).\end{split}$ (7.76)

Firstly, ${\cal H}^{(-,4,3)}_{1}$ is the only term involving a colour operator
$\propto(\mathbf{T}_{s-u}^{2})^{3}$, which does not appear on the left-hand
side, so we conclude that ${\cal H}^{(-,4,3)}_{1}=0$. Next, $\tilde{{\cal
H}}^{(-,4,3)}_{1}$ multiplies a fully nested commutator, which also cannot be
matched to the colour operators on the left-hand side, so it must vanish as
well. In order to match the single term that remains, we recall that at four
loops the soft anomalous dimension acts directly on the tree-level amplitude,
so we can use
$\mathbf{T}_{t}^{2}\mathcal{M}_{\text{tree}}=C_{A}\mathcal{M}_{\text{tree}}$.
This is consistent with the expectation that ${\cal F}^{(-,4,3)}_{A}$ should
contain a factor of $C_{A}$, while ${\cal F}^{(-,4,3)}_{F}$ does not
contribute at NLL accuracy (see the discussion following eq. (7.48)). We
deduce

$\displaystyle{\cal F}^{(-,4,3)}_{A}$ $\displaystyle=i\pi
C_{A}\frac{\zeta_{3}}{24}\hskip 56.9055pt{\cal F}^{(-,4,3)}_{F}=0$ (7.77a)
$\displaystyle{\cal H}^{(-,4,3)}_{1}$ $\displaystyle=0\hskip
89.626pt\tilde{{\cal H}}^{(-,4,3)}_{1}=0\,.$ (7.77b)

We note that the even amplitude at four loops for NNLL (and beyond) is still
unknown. As a consequence,
$\mathbf{\Gamma}^{(-,4,m)}=\text{Im}[\mathbf{\Gamma}^{(4,m)}]$, with
$m=\\{0,1,2\\}$ remain unconstrained.

##### Constraints at four loops NLL for signature-even functions.

Consider now kinematic functions multiplying signature-even colour structures,
which must be real and symmetric under $s\leftrightarrow u$.

We express eq. (7.70) at $L^{3}$ order and equate it to the relevant $L^{3}$
coefficient in eq. (6.47), which vanishes identically, getting

$\displaystyle 0$
$\displaystyle\,\,\overset{!}{=}\,\,\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\bigg{\\{}2{\cal
F}^{(+,4,3)}_{A}+\frac{C_{A}}{2}\left({\cal G}^{(+,4,3)}_{A}-{\cal
H}^{(+,4,3)}_{1}\right)\bigg{\\}}$ (7.78)
$\displaystyle\quad\mbox{}+\left(2\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{12}-\frac{1}{2}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]+\frac{3}{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\right){\cal
G}^{(+,4,3)}_{A}$
$\displaystyle\quad\mbox{}+\bigg{(}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}-\frac{1}{6}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\bigg{)}{\cal
H}^{(+,4,3)}_{1}.$

The only function that multiplies quartic Casimir terms is ${\cal
G}^{(+,4,3)}_{A}$ so it must be zero. In the last line ${\cal
H}^{(+,4,3)}_{1}$ multiplies linearly independent colour structures so it must
vanish as well. While ${\cal F}^{(+,4,3)}_{A}$ appears in combination with
other functions, those vanish hence so does ${\cal F}^{(+,4,3)}_{A}$. At
$L^{3}$ order we thus obtain the following constraints:

$\displaystyle{\cal F}^{(+,4,3)}_{A}$ $\displaystyle=0\hskip 71.13188pt{\cal
F}^{(+,4,3)}_{F}=0$ (7.79a) $\displaystyle{\cal G}^{(+,4,3)}_{A}$
$\displaystyle=0\hskip 73.97733pt{\cal G}^{(+,4,3)}_{F}=0$ (7.79b)
$\displaystyle{\cal H}^{(+,4,3)}_{1}$ $\displaystyle=0.$ (7.79c)

These results are of course in line with the fact that the signature-even NLL
anomalous dimension is two-loop exact.

##### Constraints at four loops NNLL for signature-even functions.

At $L^{2}$ order, upon equating the relevant terms of eq. (7.70) to eq. (6.39)
we have

$\displaystyle\zeta_{2}\zeta_{3}C_{\mathbf{\Delta}}^{(+,4,2)}$
$\displaystyle\,\,\overset{!}{=}\,\,2C_{\mathbf{\Delta}}^{(+,4,2)}{\cal
G}^{(+,4,2)}_{A}+\bigg{(}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}-\frac{1}{6}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\bigg{)}{\cal
H}^{(+,4,2)}_{1}$ (7.80) $\displaystyle\hskip
8.5359pt+\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\bigg{\\{}2{\cal
F}^{(+,4,2)}_{A}+\frac{C_{A}}{2}\left({\cal G}^{(+,4,2)}_{A}-{\cal
H}^{(+,4,2)}_{1}\right)\bigg{\\}}.$

We immediately see the same colour term $C_{\mathbf{\Delta}}^{(+,4,2)}$,
defined in eq. (6.39), on the left-hand side and multiplying ${\cal
G}^{(+,4,2)}_{A}$ on the right-hand side. This fixes ${\cal G}^{(+,4,2)}_{A}$
and implies that the combination of the other terms must be zero. ${\cal
H}^{(+,4,2)}_{1}$ multiplies colour operators that are linearly independent of
the others, and must vanish. Finally the combination of functions in the
second line of eq. (7.80) multiplying the fully-nested commutator must vanish,
and since ${\cal H}^{(+,4,2)}_{1}=0$, it implies a simple relation between
${\cal F}^{(+,4,2)}_{A}$ and ${\cal G}^{(+,4,2)}_{A}$. The constraints at
$L^{2}$ order for even-signature functions are then

$\displaystyle{\cal F}^{(+,4,2)}_{A}$
$\displaystyle=-C_{A}\frac{\zeta_{2}\zeta_{3}}{8}\hskip 71.13188pt{\cal
F}^{(+,4,2)}_{F}=0$ (7.81a) $\displaystyle{\cal G}^{(+,4,2)}_{A}$
$\displaystyle=\frac{\zeta_{2}\zeta_{3}}{2}\hskip 96.73918pt{\cal
G}^{(+,4,2)}_{F}=0$ (7.81b) $\displaystyle{\cal H}^{(+,4,2)}_{1}$
$\displaystyle=0.$ (7.81c)

The expressions for
$\mathbf{\Delta}^{(+,4,m)}=\text{Re}[\mathbf{\Delta}^{(4,m)}]$, $m=\\{0,1\\}$
are currently not known, so our firm constraints for the even signature part
of the soft anomalous dimension at four loops end at NNLL accuracy. The $m=1$
term, however, has a rather special status due to its connection with the cusp
anomalous dimension, which we discuss in the next section before summarising
the complete set of constraints.

###  7.4 The soft anomalous dimension at four loops

In this section, we present expressions parametrising the four-loop soft
anomalous dimension in the high-energy limit through all powers of the high-
energy logarithm $L$. Although this goes beyond the logarithmic accuracy of
any explicit calculation of the amplitude, we also discuss here the
generalisation of the relation in eq. (2.40) between the cusp anomalous
dimension and the singularities of the gluon Regge trajectory to four loops.
We show that this generalisation is natural despite the presence of quartic
Casimir contributions and it leads to an extra set of constraints on the soft
anomalous dimension.

##### The soft anomalous dimension at four loops.

To begin, it is useful to define an operator representation of the cusp
anomalous dimension

$\bm{\Gamma}_{p}^{\rm{cusp}}\equiv\frac{1}{2}\gamma_{K}(\alpha_{s}){\bf
T}_{p}^{2}+\sum_{R}g_{R}(\alpha_{s})\bm{\mathcal{D}}^{R}_{pppp}+{\cal
O}(\alpha_{s}^{5}),$ (7.82)

associated with a channel $p\in\\{s,t,u\\}$, where we suppress corrections
containing sextic and higher Casimir operators. The quartic operator
$\bm{\mathcal{D}}^{R}_{pppp}$ is defined in eq. (7.52).

When the $t$-channel operators act on the tree amplitude, they reproduce it,
multiplied by the respective adjoint Casimir, namely

$\mathbf{T}_{t}^{2}\mathcal{M}_{\text{tree}}=C_{A}\mathcal{M}_{\text{tree}},\hskip
42.67912pt\bm{\mathcal{D}}^{R}_{tttt}\mathcal{M}_{\text{tree}}=\frac{d_{RA}}{N_{A}}\mathcal{M}_{\text{tree}},$
(7.83)

which yields

$\bm{\Gamma}_{t}^{\rm{cusp}}\mathcal{M}_{\text{tree}}={\Gamma}_{A}^{\rm{cusp}}\mathcal{M}_{\text{tree}}\,,$
(7.84)

where ${\Gamma}_{A}^{\rm{cusp}}$ on the right-hand side is simply the cusp
anomalous dimension defined by an adjoint Wilson line. In contrast, when
$\bm{\mathcal{D}}^{R}_{ssss}$ and $\bm{\mathcal{D}}^{R}_{uuuu}$ act on the
tree amplitude they generate mixing into other colour states, similarly to
their quadratic counterparts ${\bf T}_{s}^{2}$ and ${\bf T}_{u}^{2}$. In
particular, their adjoint signature-even combination is given in eq. (7.63)
and signature-odd combination is given in eq. (G.35).

With these definitions and properties in place we are ready to present the
general form of the soft anomalous dimension for $2\to 2$ scattering in the
high-energy limit. The signature-even part reads

$\displaystyle\bm{\Gamma}^{(+,4)}_{ij\to
ij}\left(L,\frac{-t}{\lambda^{2}}\right)\mathcal{M}_{\text{tree}}$
$\displaystyle=\Bigg{\\{}L\,\bm{\Gamma}_{t}^{\rm{cusp},(4)}+\log\frac{-t}{\lambda^{2}}\left(\Gamma_{i}^{\rm{cusp},(4)}+\Gamma_{j}^{\rm{cusp},(4)}\right)+2\gamma_{i}^{(4)}+2\gamma_{j}^{(4)}$
(7.85)
$\displaystyle+\left(\sum_{R}f^{(4,R)}\right)\bigg{(}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+\frac{C_{A}^{3}}{2}-6\frac{d_{AR_{i}}}{N_{R_{i}}C_{i}}-6\frac{d_{AR_{j}}}{N_{R_{j}}C_{j}}\bigg{)}$
$\displaystyle+2\sum_{R}\left({\cal
G}^{(4)}_{R}(L)-\frac{g^{(4)}_{R}}{6}L\right)\left(\bm{\mathcal{D}}^{R}_{tttt}+\bm{\mathcal{D}}^{R}_{ssss}+\bm{\mathcal{D}}^{R}_{uuuu}-2\left(\frac{d_{RR_{i}}}{N_{R_{i}}}+\frac{d_{RR_{j}}}{N_{R_{j}}}\right)\right)$
$\displaystyle+2\left(\sum_{R}{\cal
F}^{(+,4)}_{R}(L)\right)\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+{\cal
H}_{1}^{(+,4)}(L)\bigg{(}C_{A}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]$
$\displaystyle-\frac{1}{2}C_{A}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}-\frac{1}{6}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\bigg{)}\Bigg{\\}}\mathcal{M}_{\text{tree}},$

while the signature-odd part is

$\displaystyle\bm{\Gamma}^{(-,4)}_{ij\to ij}(L)\mathcal{M}_{\text{tree}}$
$\displaystyle=\Bigg{\\{}\frac{i\pi}{2}\bigg{[}\bm{\Gamma}_{s}^{\rm{cusp},(4)}-\bm{\Gamma}_{u}^{\rm{cusp},(4)}\bigg{]}-\left(\sum_{R}{\cal
F}^{(-,4)}_{R}(L)\right)\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}$
(7.86) $\displaystyle+{\cal
H}^{(-,4)}_{1}(L)\bigg{(}-\frac{1}{2}\bigg{[}\mathbf{T}_{s-u}^{2},\Big{[}\mathbf{T}_{s-u}^{2},\big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\big{]}\Big{]}\bigg{]}+\frac{1}{8}\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\bigg{)}$
$\displaystyle+\tilde{{\cal
H}}^{(-,4)}_{1}(L)\,\frac{1}{4}\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\Bigg{\\}}\mathcal{M}_{\text{tree}}.$

These two expressions generalise eqs. (7.70) and (7.71), respectively, to
include ${\cal O}(L^{1})$ and ${\cal O}(L^{0})$ terms. The derivation of these
contributions is presented in appendices F and G.

In both the signature-even expression of eq. (7.85) and the odd one in eq.
(7.86) we have restored the full $p$-channel ($p\in\\{s,t,u\\}$) cusp
anomalous dimension $\bm{\Gamma}_{p}^{\rm{cusp}}$ of eq. (7.82), consisting of
both the quadratic and quartic components. Specifically, the function
$\gamma_{K}$, which was originally used to express the dipole term in eq.
(7.13), only appears now as part of the full cusp anomalous dimension
$\bm{\Gamma}_{p}^{\rm{cusp}}$, along with its quartic counterpart $g_{R}$. The
way the full $\bm{\Gamma}_{p}^{\rm{cusp}}$ gets restored is explained in
appendix F.

Note that, in line with the general expectation, all the contributions that
survive in the planar limit – the terms in the first line of eq. (7.85) and
the first term in the first line of eq. (7.86) – involve just one or two
partons, while all those involving three or four partons are non-planar. For
most terms the behaviour in the large-$N_{c}$ limit is already manifest in the
above equations owing to the (nested) commutator structure, which is
inherently non-planar, or the behaviour of the quartic Casimir contributions
given in eq. (5.10). There are a couple of terms for which a closer
examination is required: the first of these is the third line in eq. (7.85),
where in the adjoint representation one may use eq. (7.62) to obtain a
manifestly non-planar expression (while the fundamental representation
contribution is automatically subleading in the large-$N_{c}$ limit). The
second is the first term in the first line of eq. (7.86), which contains
planar as well as non-planar contributions, as one may verify using eq.
(F.15b) to express the linear terms.

For pure Yang-Mills, or SYM, where only the adjoint representation is
relevant, one may substitute eq. (7.62) and (F.15b) into the above equations
to obtain more explicit expressions the soft anomalous dimension in the high-
energy limit, separated by signature, including all powers of $L$. The
signature-even part is

$\displaystyle\bm{\Gamma}^{(+,4)}_{ij\rightarrow
ij,\,{\rm(S)YM}}(L)\mathcal{M}_{\text{tree}}$
$\displaystyle=\Bigg{\\{}L\,\bm{\Gamma}_{t}^{\rm{cusp},(4)}+\log\frac{-t}{\lambda^{2}}\left(\Gamma_{i}^{\rm{cusp},(4)}+\Gamma_{j}^{\rm{cusp},(4)}\right)+2\gamma_{i}^{(4)}+2\gamma_{j}^{(4)}$
(7.87)
$\displaystyle+f^{(4,A)}\bigg{(}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+\frac{C_{A}^{3}}{2}-6\frac{d_{AR_{i}}}{N_{R_{i}}C_{i}}-6\frac{d_{AR_{j}}}{N_{R_{j}}C_{j}}\bigg{)}$
$\displaystyle+\left({\cal
G}^{(4)}_{A}(L)-\frac{g^{(4)}_{A}}{6}L\right)\bigg{(}2\left(\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}\right)-\frac{1}{2}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]$
$\displaystyle+\frac{3}{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{2}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\bigg{)}$
$\displaystyle+2{\cal
F}^{(+,4)}_{A}(L)\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+{\cal
H}^{(+,4)}(L)\bigg{(}-\frac{1}{2}C_{A}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}$
$\displaystyle+C_{A}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]-\frac{1}{6}\mathbf{T}_{t}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\mathbf{T}_{s-u}^{2}\bigg{)}\Bigg{\\}}\mathcal{M}_{\text{tree}},$

and the signature-odd part is

$\displaystyle\bm{\Gamma}^{(-,4)}_{ij\rightarrow
ij,\,{\rm(S)YM}}(L)\mathcal{M}_{\text{tree}}$
$\displaystyle=\Bigg{\\{}\frac{i\pi}{C_{A}}\Gamma_{A}^{\rm{cusp},(4)}\mathbf{T}_{s-u}^{2}-{\cal
F}^{(-,4)}_{A}(L)\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}$
(7.88)
$\displaystyle+2\,{i\pi}g_{A}^{(4)}\left(\frac{d_{AR_{i}}}{C_{i}N_{R_{i}}}+\frac{d_{AR_{j}}}{C_{j}N_{R_{j}}}-\frac{2d_{AA}}{C_{A}N_{A}}-\frac{C_{A}^{3}}{16}\right)\mathbf{T}_{s-u}^{2}$
$\displaystyle-\frac{i\pi
g_{A}^{(4)}}{16}\bigg{(}3\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\big{]}\Big{]}+\Big{[}\mathbf{T}_{t}^{2},[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\Big{]}\mathbf{T}_{t}^{2}-3\mathbf{T}_{t}^{2}[\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}]\mathbf{T}_{t}^{2}\bigg{)}$
$\displaystyle+{\cal
H}^{(-,4)}_{1}(L)\bigg{(}-\frac{1}{2}\bigg{[}\mathbf{T}_{s-u}^{2},\Big{[}\mathbf{T}_{s-u}^{2},\big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\big{]}\Big{]}\bigg{]}+\frac{1}{8}\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\bigg{)}$
$\displaystyle+\frac{1}{4}\tilde{{\cal
H}}^{(-,4)}_{1}(L)\bigg{[}\mathbf{T}_{t}^{2},\Big{[}\mathbf{T}_{t}^{2},\big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\big{]}\Big{]}\bigg{]}\Bigg{\\}}\mathcal{M}_{\text{tree}}.$

These expressions make manifest the fact that the planar contributions are all
captured by the $\Gamma^{\rm{cusp}}$ and collinear anomalous dimension terms.

##### The singularities of the Regge trajectory and the cusp anomalous
dimension.

We have seen that the connection Korchemskaya:1994qp ; Korchemskaya:1996je
between the infrared singularities of the gluon Regge trajectory and the
integral of the cusp anomalous dimension (2.40), namely

$\tilde{\alpha}_{g}(t)=K+\mathcal{O}(\epsilon^{0})\,,$ (7.89)

where

$\displaystyle K(\alpha_{s}(\mu^{2}))$
$\displaystyle\equiv-\frac{1}{4}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\gamma_{K}(\alpha_{s}(\lambda^{2}))=\sum_{n=0}^{\infty}\left(\frac{\alpha_{s}(\mu^{2})}{\pi}\right)^{n}K^{(n)}=\frac{1}{2\epsilon}\frac{\alpha_{s}(\mu^{2})}{\pi}+\ldots,$
(7.90)

extends to three loops, despite the presence of a Regge cut contribution at
this order, i.e. ${\cal O}(\alpha_{s}^{3}L^{1})$. For clarity we recall that
this is contingent on defining the trajectory $\tilde{\alpha}_{g}$ in the
Regge-cut scheme, which we defined by absorbing all planar contributions
generated at two and three loops into the Regge-pole term. Specifically, the
trajectory $\tilde{\alpha}_{g}$ was related to the one in the MRS scheme by
eq. (5.60), and the resulting cut-scheme subtracted trajectory is finite (see
eq. (A.10)):

$\displaystyle\hat{\tilde{\alpha}}_{g}(t)\equiv\tilde{\alpha}_{g}(t)-K={\cal
O}(\epsilon^{0})\,.\,$ (7.91)

From the perspective of the soft anomalous dimension this entails a remarkably
simple structure through three loops, namely its even-signature part takes the
form

$\displaystyle\begin{split}\mathbf{\Gamma}^{(+)}_{ij\to
ij}\left(\alpha_{s},L,\frac{-t}{\lambda^{2}}\right)=\frac{1}{2}\gamma_{K}(\alpha_{s})L\mathbf{T}_{t}^{2}&\,+\,\Gamma_{i}\left(\alpha_{s},\frac{-t}{\lambda^{2}}\right)\,+\,\Gamma_{j}\left(\alpha_{s},\frac{-t}{\lambda^{2}}\right)\\\
&\,\,\,+\,\mathbf{\Delta}^{(+,3,0)}\left(\frac{\alpha_{s}}{\pi}\right)^{3}+{\cal
O}(\alpha_{s}^{4})\,,\end{split}$ (7.92)

where, crucially, we used the fact that232323Note that its signature-odd
counterpart, $\mathbf{\Delta}^{(-,3,1)}$ is non-vanishing, see appendix E for
details $\mathbf{\Delta}^{(+,3,1)}=0$ Almelid:2015jia ; Caron-Huot:2017fxr .
We thus see that the only term in the (signature-even) soft anomalous
dimension which is linear in the high-energy logarithms $L$ is the one
proportional to the cusp anomalous dimension. Consequently, the exponentiation
of the singularities via eq. (6.3) directly determines the singularities of
the exponent of $(s/(-t))$, which is precisely the singular part of the gluon
Regge trajectory.

This naturally generalises to four loops, where quartic Casimir contributions
become relevant, as displayed in the definition of the cusp anomalous
dimension eq. (7.5). In order to write an equation such as eq. (7.89) valid
through four loops (and beyond), let us define

$K_{\rm
cusp}(\alpha_{s}(\mu^{2}))\equiv-\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\Gamma^{\rm{cusp}}_{A}(\alpha_{s}(\lambda^{2}))\,.$
(7.93)

Generalising eq. (7.89), we have

$C_{A}\tilde{\alpha}_{g}(t)=K_{\rm cusp}+{\cal O}(\epsilon^{0})$ (7.94)

through four loops, now including the quartic Casimir contributions. The
Regge-pole exponential in eq. (2.39) can then be expressed as $\exp(K_{\rm
cusp}L)$.

##### Linear terms in the soft anomalous dimension at four loops.

Let us now analyse the implications of this relation from the perspective of
the soft anomalous dimension. We do that by directly comparing the two
exponentiation pictures, that of the singularities via eq. (6.3) on the one
hand and that of the high-energy logarithms as a Regge pole, on the other. The
exponents in the two pictures take the form:

$-\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\mathbf{\Gamma}^{(+)}_{ij\to
ij}\left(\alpha_{s},L,\frac{-t}{\lambda^{2}}\right)\quad\longleftrightarrow\quad
C_{A}\tilde{\alpha}_{g}(t)L\,.$ (7.95)

For the two to agree for the terms that are simultaneously ${\cal
O}(1/\epsilon)$ and ${\cal O}(L^{1})$, one requires, just as in eq. (7.92) at
three loops, that the linear term in $L$ within $\mathbf{\Gamma}^{(+)}_{ij\to
ij}$ would be given precisely by $L\bm{\Gamma}_{t}^{\rm{cusp}}$, where
$\bm{\Gamma}_{t}^{\rm{cusp}}$ is defined in eq. (7.82). Upon acting on the
tree amplitude, the $t$-channel operators produce Casimirs in the adjoint
representation according to eq. (7.83), and one recovers the cusp anomalous
dimension in the adjoint representation as in eq. (7.84). In this way the
singularities of the gluon Regge trajectory satisfy eq. (7.94).

We thus conclude that the natural generalisation of the relation of eq. (7.94)
between the gluon Regge trajectory and the cusp anomalous dimension amounts to
the requirement that the terms linear in $L$ within the signature-even part of
the soft anomalous dimension would simply be $L\bm{\Gamma}_{t}^{\rm{cusp}}$.
This conjecture can also be formulated as

$\displaystyle\left.\frac{d}{dL}\mathbf{\Gamma}^{(+)}_{ij\to
ij}\left(\alpha_{s}(\mu^{2}),\frac{-t}{\mu^{2}}\right)\right|_{L=0}{\cal
M}^{\rm tree}_{ij\to ij}\,=\,$
$\displaystyle\,\Gamma^{\rm{cusp}}_{A}(\alpha_{s}(-t))\,{\cal M}^{\rm
tree}_{ij\to ij}\,,$ (7.96)

which of course holds through three loops using eq. (7.92) and (7.82), where
only the quadratic Casimir term ${\bf T}_{t}^{2}$ is present in
$\bm{\Gamma}_{t}^{\rm{cusp}}$. In contrast, at four loops also the
$\bm{\mathcal{D}}^{R}_{tttt}$ becomes relevant.

With this in mind, let us examine the general structure of the signature-even
soft anomalous dimension at four loops in eq. (7.85). The expected
$L\bm{\Gamma}_{t}^{\rm{cusp},(4)}$ term is indeed there. So, for our
conjecture to hold, all other terms which depend on $L$ must not contain any
further linear contribution. Differentiating eq. (7.85) and suppressing higher
logarithms one finds:

$\displaystyle\begin{split}\frac{d\bm{\Gamma}^{(+,4)}_{ij\to
ij,\text{Regge}}}{dL}\bigg{|}_{L=0}\\!\\!\mathcal{M}_{\text{tree}}&=\Bigg{\\{}\bm{\Gamma}_{t}^{\rm{cusp},(4)}\,+\,2\left(\sum_{R}{\cal
F}^{(+,4,1)}_{R}\right)\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}\\\
&\hskip-30.0pt+2\sum_{R}\left({\cal
G}^{(4,1)}_{R}-\frac{g^{(4)}_{R}}{6}\right)\bigg{(}\bm{\mathcal{D}}^{R}_{tttt}+\bm{\mathcal{D}}^{R}_{ssss}+\bm{\mathcal{D}}^{R}_{uuuu}-2\left(\frac{d_{RR_{i}}}{N_{R_{i}}}+\frac{d_{RR_{j}}}{N_{R_{j}}}\right)\bigg{)}\\\
&\hskip-30.0pt+{\cal
H}^{(+,4,1)}\bigg{(}-\frac{1}{2}C_{A}\Big{[}\mathbf{T}_{s-u}^{2},[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\Big{]}+C_{A}\mathbf{T}_{s-u}^{2}[\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}]\\\
&\quad\mbox{}\hskip
142.26378pt-\frac{1}{6}\mathbf{T}_{t}^{2}[(\mathbf{T}_{s-u}^{2})^{2},\mathbf{T}_{t}^{2}]\bigg{)}\Bigg{\\}}\mathcal{M}_{\text{tree}}\,,\end{split}$
(7.97)

which satisfies the conjectured relation in eq. (7.96) subject to the
following constraints

$\displaystyle{\cal F}^{(+,4,1)}_{R}=0,\hskip 28.45274pt{\cal G}^{(4,1)}_{R}$
$\displaystyle=\frac{g^{(4)}_{R}}{6},\hskip 28.45274pt{\cal
H}^{(+,4,1)}_{1}=0\,.$ (7.98)

The coefficients $g_{R}^{(4)}$ are known in QCD Boels:2017ftb ; Boels:2017skl
; Moch:2017uml ; Grozin:2017css ; Henn:2019swt ; Huber:2019fxe ;
vonManteuffel:2020vjv ; Agarwal:2021zft ; see eq. (B.3). We stress that the
constraints in eq. (7.98), which concern the uncharted territory of N3LLs,
have a very different status as compared to those of eqs. (7.77), (7.79) and
(7.81): while the latter are based on explicit calculations in the high-energy
limit, the former relies on a conjectured generalisation of the relation
between the gluon Regge trajectory and the cusp anomalous dimension. One may
note the intriguing similarity between the constraint on ${\cal G}^{(4)}_{R}$
in eq. (7.98) and the collinear constraint Becher:2019avh  on the same
function in eq. (7.34). We stress that these are different kinematic limits.
Whereas in the high-energy limit the function ${\cal G}^{(4)}_{A}$ does have a
double logarithmic contribution – see eq. (7.81a) – in the collinear limit eq.
(7.34) forbids any non-linear dependence on the relevant logarithm.

##### Summary: Regge-limit constraints on $\mathbf{\Gamma}_{n}^{(4)}$.

We derived the soft anomalous dimension in the high-energy limit and used it
to constrain the kinematic functions parametrising this quantity in general
kinematics as proposed in ref. Becher:2019avh . The computed four-loop result,
taking into account NLLs of both even and odd signature, along with the newly-
computed NNLL of even signature, appears in eq. (6.48). In turn, upon taking
the general-kinematics parametrisation and specialising it to $2\to 2$
kinematics in the Regge limit, we obtained eqs. (7.70) and (7.71). Having
chosen a common basis of colour operators for both the computed result and the
parametrised one, the values of the expansion coefficients of the unknown
kinematic functions in powers of $L$ can be directly deduced, and are
summarised in eqs. (7.77), (7.79) and (7.81).

In addition, analysing the connection between the gluon Regge trajectory and
the cusp anomalous dimension originally proposed in ref. Korchemskaya:1994qp ;
Korchemskaya:1996je , we conjectured that the linear term in the soft
anomalous dimension in the Regge limit is given precisely by
$\mathbf{\Gamma}_{t}^{\rm cusp}L$ such that eq. (7.96) holds. This directly
implies an additional set of constraints on the N3LL signature-even
contributions to the soft anomalous dimension according to eq. (7.98).

Signature even | Signature odd
---|---
| $L^{3}$ | $L^{2}$ |  $L^{1}$ (conj.) |  | $L^{3}$ | $L^{2}$ | $L^{1}$
${\cal F}^{(+,4)}_{A}$ | 0 | $-\frac{C_{A}}{8}\zeta_{2}\zeta_{3}$ | 0 | ${\cal F}^{(-,4)}_{A}$ | $i\pi\frac{C_{A}}{24}\zeta_{3}$ | ? | ?
${\cal F}^{(+,4)}_{F}$ | 0 | 0 | 0 | ${\cal F}^{(-,4)}_{F}$ | 0 | ? | ?
${\cal G}^{(+,4)}_{A}$ | 0 | $\frac{1}{2}\zeta_{2}\zeta_{3}$ | $\frac{1}{6}g_{A}^{(4)}$ |  |  |  |
${\cal G}^{(+,4)}_{F}$ | 0 | 0 | $\frac{1}{6}g_{F}^{(4)}$ |  |  |  |
${\cal H}^{(+,4)}_{1}$ | 0 | 0 | 0 | ${\cal H}^{(-,4)}_{1}$ | 0 | ? | ?
|  |  |  | $\tilde{{\cal H}}^{(-,4)}_{1}$ | 0 | ? | ?
Table 1: Constraints on the high-energy limit of the kinematic functions
entering the soft anomalous dimension at four loops, separated by signature.
Note that ${\cal G}$ only has a signature-even component, and ${\cal
H}_{1}^{(4)}$ is purely gluonic. All constraints at order $L^{3}$ and $L^{2}$
in this table are based on explicit computations in the high-energy limit,
while those for order $L^{1}$ are based on the conjectured generalisation of
the relation between cusp singularities and the Regge pole to four loops. The
coefficients $g_{R}^{(4)}$ are known in QCD Boels:2017ftb ; Boels:2017skl ;
Moch:2017uml ; Grozin:2017css ; Henn:2019swt ; Huber:2019fxe ;
vonManteuffel:2020vjv ; Agarwal:2021zft  and are quoted in eq. (B.3).

The full set of constraints on the four-loop kinematic functions is summarised
in Table 1. Here, the left half of the table summarises the constraints on
signature-even (real) functions, while the right half the signature-odd
(imaginary) ones. While the function ${\cal G}_{R}$ multiplying the quartic
four-line term is by construction signature-even, the two other kinematic
functions ${\cal F}_{R}$ and ${\cal H}_{1}$ have both even and odd components
and their decomposition is given in eqs. (7.40) and (7.66), respectively. We
note that our current knowledge of the signature-even contributions is far
greater than that of the odd. In the table we represented unknown expansion
coefficients by question marks.

As a final note we emphasise that the above constraints are fully consistent
with the result by Vladimirov Vladimirov:2017ksc , that only colour operators
consisting of an even number of generators can appear in the soft anomalous
dimension. This implies that the functions multiplying the five generator term
${\cal H}_{1}$ and ${\cal H}_{2}$ in the soft anomalous dimension in eq.
(7.13) vanish identically. This is in line with the last row in Table 1, as
well as the collinear-limit constraints on these functions in ref.
Becher:2019avh .

##  8 Conclusion

In this work we take a step forward in the understanding of $2\to 2$ gauge-
theory amplitudes in the high-energy limit, by studying the tower of NNLL in
the signature-odd (real) amplitude and computing these explicitly through four
loops. This tower of corrections is particularly interesting for the analysis
of the Regge limit, because amplitudes at this logarithmic accuracy develop a
rich structure, featuring both a Regge pole and a Regge cut. Furthermore,
taking the high-energy limit gives us access to properties of four-loop
amplitudes, which are beyond the reach of perturbative calculations with
state-of-the-art techniques. Chief among these is the long-distance
singularity structure in fixed-angle scattering, for which the high-energy
limit is highly constraining.

In order to compute amplitudes in the Regge limit, we employ the method
described in refs. Caron-Huot:2013fea ; Caron-Huot:2017fxr . In essence this
approach allows us to compute transition amplitudes between the projectile and
the target at widely separated rapidities, each described by a state
consisting of a given numbers of Reggeons. The Balitsky-JIMWLK Hamiltonian is
then used to evolve the Reggeon states to the same rapidity. At NNLL accuracy
this involves, beyond the single Reggeon state, also triple-Reggeon states and
mixing amongst these. The sum of all transition amplitudes defines a reduced
amplitude, which is related to the full amplitude by simple multiplicative
factors, eq. (2.44).

We classify the transition amplitudes entering the NNLL of the reduced
amplitude to all loop orders, in eq. (3.60). These fall into two distinct
categories, one being a purely Single Reggeon State (SRS) transition and the
other including four transitions involving Multiple Reggeon States (MRS). The
former features a single Reggeon in both the projectile and the target,
undergoing trivial rapidity evolution as a single Reggeon across the entire
rapidity interval. The latter include all transitions in which a triple-
Reggeon state is generated at any stage during the evolution, be it at the
projectile or the target ends, or during the course of rapidity evolution.
Specifically, the aforementioned four are: $3\to 3$ transitions, $1\to 3$ or
$3\to 1$ and $1\to 1$ which are mediated by a triple-Reggeon state in the
evolution. We show that at NNLL accuracy MRS transition amplitudes can be
computed to any perturbative order by iterating the _leading-order_ Balitsky-
JIMWLK Hamiltonian. Thus, the MRS are universal quantities in any gauge
theory, which do not depend on the matter content.

We computed the reduced amplitudes through four loops, given in eqs. (5.13),
(5.21) and (5.36), providing a detailed derivation of results presented in
ref. Falcioni:2020lvv . In particular, we developed a new method to calculate
the colour factor of the amplitude, when the target and the projectile belong
to general representations of the gauge group. This allowed us to derive new
colour identities and obtain expressions of the reduced amplitudes in an
operator form, which is suitable to investigate universal features of both the
infrared and of the high-energy factorisation. We found that only the $3\to 3$
transitions feature non-trivial colour structure, where different colour
components mix during evolution. All the other MRS transitions are
proportional to the colour octet exchange to all perturbative orders. We
observed that $3\to 1$ and $1\to 3$ transitions at three and at four loops, in
eqs. (5.20) and (5.30) respectively, cancel exactly against corresponding
terms (which involve quartic Casimirs associated with the representations of
the projectile and the target) in the $3\to 3$ exchanges of eqs. (5.18) and
(5.26). We conjecture that such a mechanism is in place to all perturbative
orders and that it completely removes all contributions to the amplitude from
$3\to 1$ and $1\to 3$ transitions. As a result, only the $1\to 1$ transition
generates mixing between states with one and three Reggeons, as we check
explicitly at four loops. There, a further cancellation takes place: the $1\to
1$ contribution in eq. (5.32) cancels the planar terms of $3\to 3$ transitions
in eq. (5.26), to all orders in $\epsilon$. This renders the reduced amplitude
at four loops, eq. (5.3), manifestly non-planar.

The complete cancellation of the contributions emerging from mixing between
single and triple Reggeon states, against corresponding terms associated with
quartic Casimirs in the $3\to 3$ evolution, is highly suggestive of a general
pattern, extending to all orders in this tower of logarithms. As we have seen,
it leads to a partial cancellation of planar contributions in the reduced
amplitude at three loops, and a complete cancellation of such at four loops.
Our expectation is that the reduced amplitude will be non-planar at any order
beyond four loops. The only planar contributions in the reduced amplitude then
occur at two and three loops, before the full set of single-triple Reggeon
transitions opens up.

The non-planar nature of the total contribution to the reduced amplitude from
multiple Reggeon states at four loops (and likely beyond) points to a simple
relation between these quantities and Regge cuts, which are known to arise
only from non-planar diagrams Mandelstam:1963cw ; Eden:1966dnq ;
Collins:1977jy . However, the separation between single-Reggeon state (SRS)
and multiple-Reggeon state (MRS) contributions to the amplitude as defined in
our calculation, is not in one-to-one correspondence with the separation
between the Regge pole and the Regge cut contributions. This is already clear
at two and at three loops, where MRS do contain planar contributions. Hence,
the MRS give rise to both pole and cut contributions, while the SRS
contributes exclusively to the Regge-pole exchange.

In order to elucidate the separation between Regge cut and pole, we rely again
on the structure of the reduced amplitudes in the planar limit. We find that
MRS contributions that are leading in $N_{c}$ appear only in the colour octet
component and are independent of the process, both at two loops, eq. (5.15),
and at three loops, eq. (5.22). Following this analysis of colour factors, we
show that both the SRS contribution and the planar terms of the MRS
contribution may be described by Regge-pole factorisation, while all remaining
non-planar MRS terms define a cut contribution, as done in eq. (5.40). We name
this separation of the amplitude the Regge-cut scheme. It departs from the one
adopted in ref. Caron-Huot:2017fxr , dubbed MRS scheme, where the SRS
contribution alone is factorised as a Regge pole. The change of scheme
modifies the definition of the impact factors and of the Regge trajectory,
which determine the Regge pole contribution, by the planar part of the MRS
contribution. Notably, the two-loop impact factors and the three-loop Regge
trajectory completely characterise the Regge-pole contribution to the NNLL to
all orders. At four loops and beyond there is no parameter which could allow
one to shuffle planar MRS contributions to the Regge pole. Therefore, starting
at four loops the MRS transition amplitudes must contribute exclusively to the
cut and must be entirely non-planar. This is indeed what we find in our four-
loop calculation, eqs. (5.3) and (5.36).

In section 5.4 we construct the complete amplitudes and then we distinguish
pole and cut contributions to the amplitude according to the Regge-cut scheme
(5.40). At two loops, we provide in eq. (5.50) the definition of the Regge cut
coefficient $\mathcal{M}^{(-,2,0),\,\text{cut}}_{ij\to ij}$ in an operator
form, which is valid to all orders in $\epsilon$ for every colour component in
any process. This coincides with the MRS contribution of eq. (5.13), with its
planar limit, eq. (5.15), subtracted. We find that, in the octet component,
$\mathcal{M}^{(-,2,0),\,\text{cut}}_{ij\to ij}$ agrees with the Regge-pole
factorisation breaking term $R^{(2),0,[8]}_{ij}$, defined in refs.
DelDuca:2013ara ; DelDuca:2014cya  on the basis of infrared factorisation. We
determine the corresponding quark and gluon impact factors in this scheme, eq.
(5.48), by giving their relation with the results in the MRS scheme Caron-
Huot:2017fxr . Remarkably, it is possible to move into the impact factors
further terms that appear in $\mathcal{M}^{(-,2,0),\,\text{cut}}_{ij\to ij}$
and are subleading in $N_{c}$, as done in eq. (5.51). This follows from the
structure of the non-planar terms in the reduced amplitude at two loops, given
in eq. (5.12). By following this redefinition, we obtain a new cut,
$\mathcal{M}^{(-,2,0),\,\text{FL-cut}}_{ij\to ij}$, defined in eq. (5.52),
which agrees with the two-loop Regge cut $A_{\text{eik}}\,C_{ij}^{C}$ computed
by Fadin and Lipatov Fadin:2016wso ; Fadin:2017nka .

At three loops, the Regge cut $\mathcal{M}^{(-,3,1),\,\text{cut}}_{ij\to ij}$
takes the form of eq. (5.64). It includes a term proportional to
$\mathcal{M}^{(-,2,0),\,\text{cut}}_{ij\to ij}$ plus the reduced amplitude at
three loops $\hat{\mathcal{M}}^{(-,3,1)}_{ij\to ij}$, eq. (5.21), with its
planar part subtracted. The latter is assigned to the Regge pole and thus it
enters the Regge trajectory at three loops. Eq. (5.60) provides the relation
between the three-loop trajectory in the Regge-cut scheme and in the MRS
scheme of ref. Caron-Huot:2017fxr . In that work, the three-loop trajectory
was determined in the MRS scheme for $\mathcal{N}=4$ SYM. There, it was also
pointed out that the MRS scheme breaks a well-known relation
Korchemskaya:1994qp ; Korchemskaya:1996je  between the infrared singularities
of the gluon Regge trajectory and $K(\alpha_{s})$ of eq. (2.30a), the integral
over the lightlike cusp anomalous dimension. This relation holds for the two-
loop Regge trajectory, but it is violated at three loops in the MRS scheme. In
contrast, we find that the three-loop Regge trajectory in the Regge-cut
scheme, $\tilde{\alpha}_{g}^{(3)}$, features precisely the singularities
predicted by the cusp anomalous dimension, as shown in eq. (5.62).

We compute also the finite contribution to $\tilde{\alpha}_{g}^{(3)}$ in
$\mathcal{N}=4$ SYM in full colour. Notably, we find that the latter agrees
with the known result in the planar theory Drummond:2007aua ; Naculich:2007ub
, without any non-planar correction. In other words, the trajectory features a
maximally non-Abelian colour factor, which is in line with the expected
eikonal origin for this quantity Korchemskaya:1994qp ; Korchemskaya:1996je ;
Falcioni:2019nxk .

Our three-loop analysis suggests that the Regge-cut scheme captures the
analytic structure of high-energy amplitudes. As a confirmation, we find that,
in this scheme, the Regge cut agrees with the function $R^{(3),1,[8]}_{ij\to
ij}$ of refs. DelDuca:2013ara ; DelDuca:2014cya , which contains the
factorisation-breaking singularities in the octet component. However,
different choices are also possible. In particular, as mentioned above, using
eq. (5.12) we identify a specific set of non-planar terms in the reduced two-
loop amplitude that are consistent with Regge-pole factorisation. Absorbing
these into the Regge-pole term at two loops (eq. (5.52)) modifies the
contribution of the Regge cut at three loops of eq. (5.64), only by replacing
$\mathcal{M}^{(-,2,0),\,\text{cut}}_{ij\to ij}$ with the expression of the cut
in the new scheme, $\mathcal{M}^{(-,2,0),\,\text{FL-cut}}_{ij\to ij}$. We
verify that the three-loop cut defined in this way coincides with the cut
contribution, $-A_{\text{eik}}C_{ij}^{C}\,(C_{R}+C_{3})$, in refs.
Fadin:2016wso ; Fadin:2017nka . Notably, the three-loop Regge trajectory is
not affected by colour subleading terms and, even with the FL definition of
the cut, it maintains its relation with the lightlike cusp anomalous
dimension, as well as its maximally non-Abelian colour factor. Therefore, our
new analysis of the colour factors in the reduced amplitudes, allows us to
find the precise relation between the computational scheme introduced in refs.
Caron-Huot:2013fea ; Caron-Huot:2017fxr  and the study of factorisation
breaking and of the Regge cut, performed respectively in refs. DelDuca:2013ara
; DelDuca:2013dsa ; DelDuca:2014cya  and Fadin:2016wso ; Fadin:2017nka ,
finding complete agreement.

Our expression for the Regge cut at four loops is given in eq. (5.71), in
terms of the reduced amplitude at four loops and the cut contributions at two
and three loops. Since the former, eq. (5.36), is non-planar by direct
computation, and the latter two terms are defined in the Regge-cut scheme to
be non-planar by construction, we find the four-loop cut contribution to the
amplitude $\mathcal{M}^{(-,4,2),\text{cut}}_{ij\to ij}$ is non-planar as a
whole. Furthermore, we show in eq. (5.72) that by the same mechanism, in this
scheme the non-planar nature of the reduced amplitude ensures that the cut
remains non-planar to all loop orders.

In sections 6 and 7 we proceed with the investigation of infrared
factorisation at four loops, employing our explicit NNLL calculation as an
input. The comparison between the exponentiation of infrared singularities and
that of high-energy logarithms is useful in several ways. First, it is a
highly non-trivial check of the results. Second, it provides a rich source of
constraints on the yet-unknown soft anomalous dimension at four loops (see
below). Third, it allows us to extract the hard function containing finite
terms in the amplitude through four loops, both in QCD and in $\mathcal{N}=4$
super Yang-Mills (SYM), finding an intriguing relation between the hard
function and the finite parts of the gluon Regge trajectory, eq. (6.40). The
planar terms in the hard function in SYM agree with the predicted
large-$N_{c}$ limit Bern:2005iz ; Drummond:2007aua .

We study the soft anomalous dimension through four loops. In the high-energy
limit, we separate the contributions of the dipole formula Gardi:2009qi ;
Gardi:2009zv ; Becher:2009cu ; Becher:2009qa , from a general remainder
$\bf\Delta$ that starts at three loops, expanding both in powers of the
signature-even logarithm $L$, defined in eq. (2.9). While dipole contributions
are at most linear in $L$, the remainder contains higher powers of the
logarithm, for instance its imaginary part contains terms $L^{3}$ at four
loops Caron-Huot:2013fea ; Caron-Huot:2017zfo . Notably, the real part of the
remainder at three loops ${\bf\Delta}^{(+,3)}$ does not depend on $L$
Almelid:2015jia . In particular, since it lacks linear terms in $L$, it does
not contribute to the tower of NNLLs in the soft anomalous dimension to that
order Caron-Huot:2017fxr . Here we compute the real part of the remainder to
four loops, ${\bf\Delta}^{(+,4)}$, through NNLL, i.e. ${\cal
O}(\alpha_{s}^{4}L^{2})$, finding the first non-vanishing contribution to the
NNLL tower, eq. (6.39). This quantity is manifestly non-planar: it is written
in terms of commutators of the channel operators and the combination of
Casimir invariants $\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}$, which is
subleading in the large-$N_{c}$ limit. This is a strong check on our
calculation, because in the planar limit only diagrams connecting up to two
legs can contribute to the soft anomalous dimension.

We characterise our result for the soft anomalous dimension in eq. (6.39)
further, by comparing it with the general parametrisation for the four-loop
soft anomalous dimension in general kinematics Becher:2019avh , which consists
of all connected Gardi:2013ita  colour structures that may arise at this
order, each multiplying a yet-unknown kinematic function. We compute the high-
energy limit of this parametrisation of the soft anomalous dimension through
NNLLs, both in the real part, eq. (7.70), and in the imaginary part, eq.
(7.71). Focusing on the terms in the real part of the anomalous dimension, we
find only three contributions. Two of them involve colour quadrupole
correlations, featuring four generators, one on each of the four lines. Of
these one colour structure is of the same form that appears at three loops
Almelid:2015jia ; Almelid:2017qju  multiplied by the kinematic function
$\mathcal{F}_{A}^{(+,4)}$, while the second involves a quartic Casimir type
structure (a symmetric trace of four adjoint generators) multiplied by
$\mathcal{G}_{A}^{(+,4)}$. In the Regge-limit both are expressed in terms of
nested commutators of channel colour operators, where the second also features
the combination $\frac{d_{AA}}{N_{A}}-\frac{C_{A}^{4}}{24}$. The final
potential contribution to the soft anomalous dimension, featuring the unknown
function $\mathcal{H}_{1}^{(+,4)}$, generates correlations among four lines
using five colour generators. Matching the general parametrisation with our
result of the anomalous dimension provides non-zero constraints on the high-
energy limit of the functions $\mathcal{F}_{A}^{(+,4)}$, in eq. (7.81a), and
$\mathcal{G}_{A}^{(+,4)}$, in eq. (7.81b). Interestingly, the function
$\mathcal{H}^{(+,4)}_{1}$ must vanish to this logarithmic accuracy. This is
consistent with the result of ref. Vladimirov:2017ksc , which shows that the
correlation of an odd number of colour operators is always prohibited in the
soft anomalous dimension. We determine all constraints on the parametrisation
in ref. Becher:2019avh  that can be derived from the available information on
the Regge limit and we summarise our findings in Table 1.

We expect that the interplay between high-energy and infrared factorisation
will provide further insight into gauge-theory dynamics. This conclusion is
already suggested by arguments about the gluon Regge trajectory. We have
pointed out that this quantity – both its singular and its finite parts – is
expected to be associated to the anomalous dimension of Wilson-line geometries
Korchemskaya:1994qp ; Korchemskaya:1996je ; Falcioni:2019nxk . Here we
verified, up to three loops, the correspondence between the infrared
singularities of the Regge trajectory and the terms proportional to the
quadratic Casimir in the lightlike cusp anomalous dimension. We conjecture
that this relation generalises to four loops and beyond as in eq. (7.94),
where we identify the singularities of the Regge trajectory with the integral
of the complete cusp anomalous dimension, eq. (7.5), including quartic Casimir
(and higher) contributions. This has profound implications on the structure of
the soft anomalous dimension, beyond the accuracy of our calculation.
Specifically, at four loops, the N3LL contribution to the soft anomalous
dimension must be related to the four-loop Regge trajectory. More generally,
if our conjecture holds, linear terms in $L$ in the real part of the soft
anomalous dimension in the Regge limit must be simply proportional to complete
cusp anomalous dimension in the adjoint representation, or equivalently we
expect eq. (7.96) should hold to all orders. At four loops this provides three
new constraints on the soft anomalous dimension, given in eq. (7.98). The
vanishing of $\mathcal{H}^{(+,4)}_{1}$ is of course consistent with the
finding of ref. Vladimirov:2017ksc . The results are included in Table 1,
which provides important input to the bootstrap program to determine the soft
anomalous dimension in general kinematics, which has already been successful
at the three-loop level Almelid:2017qju . Our work paves the way for
bootstrapping this quantity to four loops, where direct calculations are not
yet feasible.

###### Acknowledgements.

We would like to thank Simon Caron-Huot for insightful comments and Claude
Duhr and Andrew McLeod for collaboration on a related project on the soft
anomalous dimension. EG, GF and NM are supported by the STFC Consolidated
Grant ‘Particle Physics at the Higgs Centre’. GF is supported by the ERC
Starting Grant 715049 ‘QCDforfuture’ with Prinipal Investigator Jennifer
Smillie. CM’s work is supported by the Italian Ministry of University and
Research (MIUR), grant PRIN 20172LNEEZ. LV is supported by Fellini, Fellowship
for Innovation at INFN, funded by the European Union’s Horizon 2020 research
programme under the Marie Skłodowska-Curie Cofund Action, grant agreement no.
754496.

##  Appendix A Coefficients of the Regge pole amplitude

In this appendix we collect the coefficients describing the Regge pole part of
the two-parton scattering amplitude, namely, the Regge trajectory and impact
factors. As discussed in section 2.1, this component of the amplitude is
scheme-dependent, starting at NNLL accuracy. Below we begin by compoiling the
coefficients in the MRS scheme of eq. (2.38) and then proceed to discuss the
cut scheme of eq. (2.39).

Following the definition in eq. (6.15), we split the Regge trajectory into a
component proportional to the integral of the cusp anomalous dimension,
$K(\alpha_{s})$ defined in eq. (2.30a), and a remainder, $\hat{\alpha}_{g}$:

$\alpha_{g}(t)=K\left(\alpha_{s}(-t)\right)+\hat{\alpha}_{g}(t).$ (A.1)

Expanding the term in this equation according to eq. (2.6), in QCD one has

$\displaystyle\begin{split}K^{(1)}&=\frac{{\gamma}_{K}^{(1)}}{4\epsilon},\\\
K^{(2)}&=\frac{{\gamma}_{K}^{(2)}}{8\epsilon}-\frac{b_{0}\,{\gamma}_{K}^{(1)}}{32\epsilon^{2}},\\\
K^{(3)}&=\frac{{\gamma}_{K}^{(3)}}{12\epsilon}-\frac{b_{0}\,{\gamma}_{K}^{(2)}+b_{1}\,{\gamma}_{K}^{(1)}}{48\epsilon^{2}}+\frac{b_{0}^{2}\,{\gamma}_{K}^{(1)}}{192\epsilon^{3}},\end{split}$
(A.2a)

where the coefficients $\gamma_{K}^{(i)}$ are given in (B). The remainder,
cusp-subtracted trajectory $\hat{\alpha}_{g}(t)$ is known up to two loops in
QCD. Its coefficients read Lipatov:1976zz ; Kuraev:1976ge ; Fadin:1995xg ;
Fadin:1996tb ; Fadin:1995km ; Blumlein:1998ib

$\displaystyle\hat{\alpha}_{g}^{(1)}$
$\displaystyle=\frac{1}{2\epsilon}(r_{\Gamma}-1)=-\frac{1}{4}\zeta_{2}\,\epsilon-\frac{7}{6}\zeta_{3}\,\epsilon^{2}+{\cal
O}(\epsilon^{3}),$ (A.3a) $\displaystyle\hat{\alpha}_{g}^{(2)}$
$\displaystyle=C_{A}\left(\frac{101}{108}-\frac{\zeta_{3}}{8}\right)-\frac{7n_{f}}{54}+{\cal
O}(\epsilon)\,.$ (A.3b)

The cusp-subtracted trajectory at three loops has been calculated in ${\cal
N}=4$ SYM, Caron-Huot:2017fxr , extracting it from the two-parton scattering
amplitude obtained in ref. Henn:2016jdu . It reads

$\hat{\alpha}_{g}^{(3)}\rvert_{\text{SYM}}=C_{A}^{2}\left(-\frac{\zeta_{2}}{144\epsilon^{3}}+\frac{5\zeta_{4}}{192}\frac{1}{\epsilon}+\frac{107}{144}\zeta_{2}\zeta_{3}+\frac{\zeta_{5}}{4}+{\cal
O}\left(\epsilon\right)\right).$ (A.4)

The description of the Regge-pole component of the amplitude is completed by
the information provided by the quark and gluon impact factors. Following the
definition in eq. (2.41), we split the impact factors into a term
$Z_{i/j}(t)$, defined as the integral of the anomalous dimension
$\Gamma_{i/j}$, see eq. (2.42), and a collinear-subtracted remainder
$D_{i/j}(t)$:

$C_{i/j}(t)=Z_{i/j}(t)\,D_{i/j}(t).$ (A.5)

In terms of the coefficients in eqs. (B) and (B), and setting $\mu^{2}=-t$,
the perturbative expansion of $Z_{i}(t)$ (see eq. (2.43a)) reads

$\displaystyle\begin{split}Z_{i}^{(0)}&=1,\\\
Z_{i}^{(1)}&=-\,C_{i}\,{\gamma}_{K}^{(1)}\frac{1}{4\epsilon^{2}}+\frac{\gamma_{i}^{(1)}}{\epsilon},\\\
Z_{i}^{(2)}&=C^{2}_{i}\left({\gamma}_{K}^{(1)}\right)^{2}\frac{1}{32\epsilon^{4}}\,+C_{i}\,\Bigg{[}\frac{1}{\epsilon^{3}}\frac{{\gamma}_{K}^{(1)}}{4}\left(\frac{3b_{0}}{16}-\gamma_{i}^{(1)}\right)-\,\frac{1}{\epsilon^{2}}\frac{{\gamma}_{K}^{(2)}}{16}\Bigg{]}\\\
&\hskip
14.22636pt+\frac{1}{\epsilon^{2}}\frac{\gamma_{i}^{(1)}}{2}\left(\gamma_{i}^{(1)}-\frac{b_{0}}{4}\right)+\frac{\gamma_{i}^{(2)}}{2\epsilon}.\end{split}$
(A.6)

The one- and two-loop coefficients of the quark and gluon collinear-subtracted
impact factors have been calculated in the MRS scheme of eq. (2.38) in Caron-
Huot:2017fxr . For instance, at one loop one has

$\displaystyle\begin{split}D_{g}^{(1)}&=-N_{c}\left(\frac{67}{72}-\zeta_{2}\right)+\frac{5}{36}n_{f}+\epsilon\bigg{[}N_{c}\left(-\frac{101}{54}+\frac{11}{48}\zeta_{2}+\frac{17}{12}\zeta_{3}\right)+n_{f}\left(\frac{7}{27}-\frac{\zeta_{2}}{24}\right)\bigg{]}\\\
&+\epsilon^{2}\bigg{[}N_{c}\left(-\frac{607}{162}+\frac{67}{144}\zeta_{2}+\frac{77}{72}\zeta_{3}+\frac{41}{32}\zeta_{4}\right)+n_{f}\left(\frac{41}{81}-\frac{5}{72}\zeta_{2}-\frac{7}{36}\zeta_{3}\right)\bigg{]}+{\cal
O}(\epsilon^{3})\,,\end{split}$ (A.7a)
$\displaystyle\begin{split}D_{q}^{(1)}&=N_{c}\left(\frac{13}{72}+\frac{7}{8}\zeta_{2}\right)+\frac{1}{N_{c}}\left(1-\frac{1}{8}\zeta_{2}\right)-\frac{5}{36}n_{f}+\epsilon\bigg{[}N_{c}\left(\frac{10}{27}-\frac{\zeta_{2}}{24}+\frac{5}{6}\zeta_{3}\right)\\\
&+\frac{1}{N_{c}}\left(2-\frac{3}{16}\zeta_{2}-\frac{7}{12}\zeta_{3}\right)+n_{f}\left(-\frac{7}{27}+\frac{\zeta_{2}}{24}\right)\bigg{]}+\epsilon^{2}\bigg{[}N_{c}\left(\frac{121}{162}-\frac{13}{144}\zeta_{2}-\frac{7}{36}\zeta_{3}+\frac{35}{64}\zeta_{4}\right)\\\
&+\frac{1}{N_{c}}\left(4-\frac{\zeta_{2}}{2}-\frac{7}{8}\zeta_{3}-\frac{47}{64}\zeta_{4}\right)+n_{f}\left(-\frac{41}{81}+\frac{5}{72}\zeta_{2}+\frac{7}{36}\zeta_{3}\right)\bigg{]}+{\cal
O}(\epsilon^{3})\,.\end{split}$ (A.7b)

At two loops:

$\displaystyle\begin{split}D_{g}^{(2)}&=-\frac{\zeta_{2}}{32\epsilon^{2}}N_{c}^{2}+N_{c}^{2}\bigg{(}-\frac{26675}{10368}+\frac{335}{288}\zeta_{2}+\frac{11}{18}\zeta_{3}-\frac{\zeta_{4}}{64}\bigg{)}\\\
&+N_{c}n_{f}\bigg{(}\frac{2063}{3456}-\frac{25}{144}\zeta_{2}+\frac{\zeta_{3}}{72}\bigg{)}+\frac{n_{f}}{N_{c}}\bigg{(}-\frac{55}{384}+\frac{\zeta_{3}}{8}\bigg{)}-\frac{25}{2592}n_{f}^{2}+{\cal
O}(\epsilon)\,,\end{split}$ (A.8a)
$\displaystyle\begin{split}D_{q}^{(2)}&=-\frac{\zeta_{2}}{32\epsilon^{2}}N_{c}^{2}+N_{c}^{2}\bigg{(}\frac{22537}{41472}+\frac{87}{64}\zeta_{2}+\frac{41}{144}\zeta_{3}-\frac{15}{256}\zeta_{4}\bigg{)}+\frac{28787}{10368}+\frac{19}{32}\zeta_{2}\\\
&-\frac{205}{288}\zeta_{3}-\frac{47}{128}\zeta_{4}+\frac{1}{N_{c}^{2}}\bigg{(}\frac{255}{512}+\frac{21}{64}\zeta_{2}-\frac{15}{32}\zeta_{3}-\frac{83}{256}\zeta_{4}\bigg{)}\\\
&+N_{c}n_{f}\bigg{(}-\frac{325}{648}-\frac{\zeta_{2}}{4}-\frac{23}{144}\zeta_{3}\bigg{)}+\frac{n_{f}}{N_{c}}\bigg{(}-\frac{505}{1296}-\frac{\zeta_{2}}{16}-\frac{19}{144}\zeta_{3}\bigg{)}+\frac{25}{864}n_{f}^{2}+{\cal
O}(\epsilon)\,.\end{split}$ (A.8b)

The whole impact factors $C_{i}$ can be found inserting the results from eqs.
(A.6)-(A.8b) into eq. (A.5), and expanding order by order in the strong
coupling. At one loop we get

$\displaystyle C_{q}^{(1)}=$
$\displaystyle-\frac{C_{F}}{2\epsilon^{2}}-\frac{3C_{F}}{4\epsilon}+C_{A}\left(\frac{3\zeta_{2}}{4}+\frac{85}{72}\right)+C_{F}\left(\frac{\zeta_{2}}{4}-2\right)-\frac{5n_{f}}{36}+\epsilon\bigg{[}C_{A}\left(\frac{64}{27}-\frac{11\zeta_{2}}{48}+\frac{\zeta_{3}}{4}\right)$
$\displaystyle+C_{F}\left(\frac{3\zeta_{2}}{8}+\frac{7\zeta_{3}}{6}-4\right)+n_{f}\left(\frac{\zeta_{2}}{24}-\frac{7}{27}\right)\bigg{]}+\epsilon^{2}\bigg{[}C_{A}\left(\frac{769}{162}-\frac{85\zeta_{2}}{144}-\frac{77\zeta_{3}}{72}-\frac{3\zeta_{4}}{16}\right)$
$\displaystyle+C_{F}\left(\zeta_{2}+\frac{7\zeta_{3}}{4}+\frac{47\zeta_{4}}{32}-8\right)+n_{f}\left(\frac{5\zeta_{2}}{72}+\frac{7\zeta_{3}}{36}-\frac{41}{81}\right)\bigg{]}+\mathcal{O}(\epsilon^{3}),$
(A.9a) $\displaystyle C_{g}^{(1)}=$
$\displaystyle-\frac{C_{A}}{2\epsilon^{2}}-\frac{b_{0}}{4\epsilon}+C_{A}\left(\zeta_{2}-\frac{67}{72}\right)+\frac{5n_{f}}{36}+\epsilon\bigg{[}C_{A}\left(\frac{11\zeta_{2}}{48}+\frac{17\zeta_{3}}{12}-\frac{101}{54}\right)$
$\displaystyle+n_{f}\left(\frac{7}{27}-\frac{\zeta_{2}}{24}\right)\bigg{]}+\epsilon^{2}\bigg{[}C_{A}\left(\frac{67\zeta_{2}}{144}+\frac{77\zeta_{3}}{72}+\frac{41\zeta_{4}}{32}-\frac{607}{162}\right)$
$\displaystyle+n_{f}\left(\frac{41}{81}-\frac{5\zeta_{2}}{72}-\frac{7\zeta_{3}}{36}\right)\bigg{]}+\mathcal{O}(\epsilon^{3}).$
(A.9b)

In the Regge-cut scheme, the one-loop and two-loop Regge trajectories are
identical to the MRS scheme, since multiple-Reggeon exchanges do not
contribute to the odd amplitude at LL and NLL. Thus, with
$\hat{\tilde{\alpha}}_{g}=\tilde{\alpha}_{g}-K$, and using eq. (5.60) for the
three-loop case, we have

$\displaystyle\hat{\tilde{\alpha}}_{g}^{(1)}=$
$\displaystyle\,{\hat{\alpha}}_{g}^{(1)}=\frac{1}{2\epsilon}(r_{\Gamma}-1),$
(A.10a) $\displaystyle\hat{\tilde{\alpha}}_{g}^{(2)}=$
$\displaystyle\,{\hat{\alpha}}_{g}^{(2)}=C_{A}\left(\frac{101}{108}-\frac{\zeta_{3}}{8}\right)-\frac{7n_{f}}{54}+O(\epsilon),$
(A.10b) $\displaystyle\hat{\tilde{\alpha}}_{g}^{(3)}\rvert_{\text{SYM}}=$
$\displaystyle\,N_{c}^{2}\left(\frac{5}{24}\zeta_{2}\zeta_{3}+\frac{\zeta_{5}}{4}+\mathcal{O}(\epsilon)\right).$
(A.10c)

Similarly, the impact factors at one loop in the Regge-cut scheme are
identical to the MRS scheme: $\tilde{C}_{i}^{(1)}=C_{i}^{(1)}$ for both quarks
and gluons. However, due to eq. (5.48), at two loops we have, in the cut
scheme

$\displaystyle\tilde{C}_{q}^{(2)}=$
$\displaystyle\frac{C_{F}^{2}}{8\epsilon^{4}}+\frac{1}{\epsilon^{3}}\left(\frac{11C_{A}C_{F}}{32}+\frac{3C_{F}^{2}}{8}-\frac{C_{F}n_{f}}{16}\right)+\frac{1}{\epsilon^{2}}\bigg{[}C_{F}^{2}\left(\frac{41}{32}-\frac{\zeta_{2}}{8}\right)-\frac{3C_{A}^{2}\zeta_{2}}{32}+\frac{C_{F}n_{f}}{24}$
$\displaystyle-
C_{A}C_{F}\left(\frac{5\zeta_{2}}{16}+\frac{23}{48}\right)\bigg{]}+\frac{1}{\epsilon}\bigg{[}C_{A}C_{F}\left(-\frac{19\zeta_{2}}{24}+\frac{11\zeta_{3}}{16}-\frac{1513}{576}\right)+C_{F}^{2}\left(\frac{221}{64}-\frac{4\zeta_{3}}{3}\right)$
$\displaystyle+C_{F}n_{f}\left(\frac{\zeta_{2}}{24}+\frac{89}{288}\right)\bigg{]}+C_{A}^{2}\left(\frac{73\zeta_{2}}{32}-\frac{43\zeta_{3}}{48}-\frac{19\zeta_{4}}{32}+\frac{13195}{3456}\right)$
$\displaystyle-
C_{A}C_{F}\left(\frac{1171\zeta_{2}}{576}-\frac{175\zeta_{3}}{48}-\frac{17\zeta_{4}}{8}+\frac{40423}{3456}\right)+C_{F}^{2}\left(\frac{1151}{128}+\frac{17\zeta_{2}}{32}-\frac{29\zeta_{3}}{8}-\frac{65\zeta_{4}}{32}\right)$
$\displaystyle+C_{F}n_{f}\left(\frac{265}{216}+\frac{17\zeta_{2}}{288}+\frac{\zeta_{3}}{6}\right)-C_{A}n_{f}\left(\frac{385}{432}+\frac{5\zeta_{2}}{16}+\frac{7\zeta_{3}}{24}\right)+\frac{25}{864}n_{f}^{2}+\mathcal{O}(\epsilon),$
(A.11a) $\displaystyle\tilde{C}_{g}^{(2)}=$
$\displaystyle\frac{C_{A}^{2}}{8\epsilon^{4}}+\frac{1}{\epsilon^{3}}\left(\frac{77C_{A}^{2}}{96}-\frac{7C_{A}n_{f}}{48}\right)+\frac{1}{\epsilon^{2}}\left[C_{A}^{2}\left(\frac{103}{96}-\frac{17\zeta_{2}}{32}\right)-\frac{49C_{A}n_{f}}{144}+\frac{n_{f}^{2}}{36}\right]$
$\displaystyle+\frac{1}{\epsilon}\left[C_{A}^{2}\left(\frac{853}{864}-\frac{11\zeta_{2}}{12}-\frac{31\zeta_{3}}{48}\right)+C_{A}n_{f}\left(\frac{\zeta_{2}}{6}-\frac{19}{72}\right)+\frac{C_{F}n_{f}}{16}+\frac{5n_{f}^{2}}{216}\right]$
$\displaystyle+C_{A}^{2}\left(\frac{415\zeta_{2}}{576}-\frac{11\zeta_{3}}{9}-\frac{\zeta_{4}}{2}+\frac{10525}{10368}\right)+C_{A}n_{f}\left(-\frac{\zeta_{2}}{16}+\frac{17\zeta_{3}}{36}-\frac{113}{324}\right)$
$\displaystyle+C_{F}n_{f}\left(\frac{55}{192}-\frac{\zeta_{3}}{4}\right)+n_{f}^{2}\left(\frac{29}{864}-\frac{\zeta_{2}}{144}\right)+\mathcal{O}(\epsilon).$
(A.11b)

##  Appendix B Anomalous dimensions

In this appendix we collect the coefficients of the various anomalous
dimensions considered in the main text. All anomalous dimensions are expanded
in powers of the strong coupling according to

$\gamma_{\phi}=\sum_{\ell=1}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{\ell-1}\gamma^{(\ell)}_{\phi}.$
(B.1)

First of all we have the cusp anomalous dimension, defined in eq. (2.28),
which involves quadratic and quartic Casimir terms, recently calculated up to
four loops in QCD Boels:2017ftb ; Boels:2017skl ; Moch:2017uml ;
Grozin:2017css ; Henn:2019swt ; Huber:2019fxe ; vonManteuffel:2020vjv ;
Agarwal:2021zft ; Bruser:2019auj ; Bruser:2020bsh . In QCD, the quadratic
Casimir component, $\gamma_{K}(\alpha_{s})$, has the following expansion
coefficients through three loops242424Three-loop contributions to lightlike
cusp anomalous dimension were first determined in Berger:2002sv ; Moch:2004pa
, by using the connection between this quantity and the large-$x$ limit of
non-singlet splitting functions Korchemsky:1988si . The complete calculation
of the non-singlet three-loop splitting functions has been recently confirmed
in ref. Blumlein:2021enk . Independent calculations of the three-loop cusp
anomalous dimension were also obtained by computing form factors FormFactors ;
Gehrmann:2010ue  and cusped Wilson loop Grozin:2014hna ; Grozin:2015kna  to
this loop order. More recently, such calculations have been completed at four
loops Boels:2017ftb ; Boels:2017skl ; Moch:2017uml ; Grozin:2017css ;
Henn:2019swt ; Huber:2019fxe ; vonManteuffel:2020vjv ; Agarwal:2021zft ;
Bruser:2019auj ; Bruser:2020bsh . Korchemsky:1985xj ; Korchemsky:1987wg ;
Moch:2004pa :

$\displaystyle{\gamma}_{K}^{(1)}$ $\displaystyle=$ $\displaystyle 2,$
$\displaystyle{\gamma}_{K}^{(2)}$ $\displaystyle=$
$\displaystyle\left(\frac{67}{18}-\zeta_{2}\right)C_{A}-\frac{10}{9}T_{R}n_{f},$
$\displaystyle{\gamma}_{K}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{C_{A}^{2}}{96}\left(490-\frac{1072}{3}\zeta_{2}+88\zeta_{3}+264\zeta_{4}\right)+\frac{C_{F}T_{R}n_{f}}{32}\left(-\frac{220}{3}+64\zeta_{3}\right)$
(B.2)
$\displaystyle+\,\frac{C_{A}T_{R}n_{f}}{96}\left(-\frac{1672}{9}+\frac{320}{3}\zeta_{2}-224\zeta_{3}\right)-\frac{2T_{R}^{2}n_{f}^{2}}{27},$

where the fundamental trace is $T_{R}={\rm Tr}(t^{a}t^{a})=\frac{1}{2}$. The
second term, $g_{R}(\alpha_{s})$, multiplying the quartic Casimir, starts at
four loops, and depends on the gauge-group representation $R$. Its
coefficients for $R=A$ (adjoint) and $R=F$ (fundamental) in QCD read

$\displaystyle\begin{split}g_{A}^{(4)}&=\frac{\zeta_{3}}{6}-\frac{3\zeta_{3}^{2}}{2}+\frac{55\zeta_{5}}{12}-\frac{\zeta_{2}}{2}-\frac{31\zeta_{6}}{8},\\\
g_{F}^{(4)}&=n_{f}\left(\zeta_{2}-\frac{\zeta_{3}}{3}-\frac{5\zeta_{5}}{3}\right)\,.\end{split}$
(B.3)

The contribution in ${\cal N}=4$ SYM, for $\gamma_{K}(\alpha_{s})$ and
$g_{A}(\alpha_{s})$, is obtained, according to principle of maximum
trascendentality, by retaining only the terms with highest trascendental
weight at each order.

Next, we have the collinear anomalous dimension $\gamma_{i}$ corresponding to
the parton $i$ FormFactors ; DelDuca:2014cya ; Falcioni:2019nxk ;
Dixon:2017nat , which is part of the anomalous dimension $\Gamma_{i}$ and
$\Gamma_{j}$ defined in eq. (2.29). The collinear anomalous dimension
$\gamma_{i}$ has been recently calculated up to four loops
vonManteuffel:2020vjv ; Agarwal:2021zft . We provide here its coefficients up
to two loops, as needed in the main text, for quarks and gluons. One has
FormFactors ; Gehrmann:2010ue

$\displaystyle\gamma_{q}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{3}{4}\,C_{F},$ $\displaystyle\gamma_{q}^{(2)}$
$\displaystyle=$
$\displaystyle\frac{C_{F}^{2}}{16}\left(-\frac{3}{2}+12\zeta_{2}-24\zeta_{3}\right)$
(B.4)
$\displaystyle+\,\frac{C_{A}C_{F}}{16}\left(-\frac{961}{54}-11\zeta_{2}+26\zeta_{3}\right)+\frac{C_{F}T_{R}n_{f}}{16}\left(\frac{130}{27}+4\zeta_{2}\right),$

for quarks, and

$\displaystyle\gamma_{g}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{b_{0}}{4},$ $\displaystyle\gamma_{g}^{(2)}$
$\displaystyle=$
$\displaystyle\frac{C_{A}^{2}}{16}\left(-\frac{692}{27}+\frac{11}{3}\zeta_{2}+2\zeta_{3}\right)+\frac{C_{A}T_{R}n_{f}}{16}\left(\frac{256}{27}-\frac{4}{3}\zeta_{2}\right)+\frac{C_{F}T_{R}n_{f}}{4},$
(B.5)

for gluons.

##  Appendix C Computing colour factors for arbitrary representations

In this appendix we review computational techniques to evaluate the colour
factors of the transition amplitudes. Universality of the Regge limit implies
that the colour structure of every amplitude is given by the same colour
operators, regardless whether the scattering process involves quarks or gluons
in the initial and final state. In order to determine such operators, we need
to develop techniques to evaluate colour tensors for general representations
of the external particles. Indeed, while it is straightforward to compute
directly colour Feynman rules by specialising the representations of the
scattering particles, such explicit results would completely obscure
universality of the Regge limit. Instead, we would like to express our results
in terms of Casimir operators of the colour channel operators defined in eq.
(2.22)

$\mathbf{T}^{2}_{t},\quad\mathbf{T}^{2}_{s-u}=\frac{\mathbf{T}^{2}_{s}-\mathbf{T}^{2}_{u}}{2},$
(C.1)

which manifest the signature properties under $s\leftrightarrow u$ crossing.
These operators emerge naturally in diagrams that feature connections among
the outermost Reggeon indices, for example

$\displaystyle\begin{split}\includegraphics{Figures/T1T2-crop.pdf}&\raisebox{30.0pt}{$=\left(\mathbf{T}_{1}^{a}\cdot\mathbf{T}_{2}^{a}\right)\,\mathcal{M}_{4},$}\\\
&=\frac{1}{2}\left(\mathbf{T}^{2}_{s-u}-\frac{\mathbf{T}^{2}_{t}}{2}\right)\,\mathcal{M}_{4}.\end{split}$
(C.2)

where, in the second line, we applied colour conservation according to eq.
(2.18). The result above is independent on the four-point matrix element
$\mathcal{M}_{4}$, thus providing a graphical derivation of the relation in
eq. (LABEL:eq:relTsu-Tt/2). We apply a similar procedure whenever a Reggeon is
emitted from an initial-state parton and absorbed by a final-state one,
according to eq. (LABEL:eq:relTsu+Tt/2). Therefore, colour structures up to
two loops are written for general representations by

  * ($i$)

Using the Lie algebra, eq. (4.34), to write three-point vertices
$(F^{a})_{bc}$ in terms of Reggeons connecting the target and the projectile.

  * ($ii$)

Applying repeatedly eqs. (LABEL:eq:relTsu-Tt/2) and (LABEL:eq:relTsu+Tt/2) to
obtain the colour-channel operators of eq. (C.1), acting on the tree-level
amplitude.

The second step may not applicable for diagrams where all Reggeons have one or
more internal attachment, namely they are all either emitted or absorbed
between two other Reggon vertices. We refer to these irreducible
configurations as entangled colour structures.

Indeed, entangled colour structures may occur starting at three loops. At
three loops, we find two such colour tensors252525At three loops there are two
additional irreducible configurations, depicted in figure 12a and 12b. These,
however, can be recast into the form of eq. (C.2) using commutation relations.
corresponding to the graphs in figure 11a and 11b.

(a)  (b)

Figure 11: Diagrammatic representation of the irreducible configurations: (a)
$d_{A}$ in eq. (C.3) and (b) $d_{B}$ in eq. (C.4).

(a)  (b)

Figure 12: Both the double cross diagram (a) and the saltire diagram (b) are
immediately written in terms of colour dipole operators by commuting the pair
of Reggeon emission vertices at the end of either the top or of the bottom
line.

While these diagrams drop out the three-loop amplitude, such entangled
configurations do not cancel in general and we will encounter them in the
four-loop calculation. Hence, we need to extend the techniques summarised
above.

### Permutation diagrams

We begin by introducing a compact notation for colour factors involving $k$
Reggeon attachments on both target and projectile. These configuration are
naturally associated to permutations of $k$ indices. Choosing the top line as
target state $i$, the diagram in figure 11a is written as

$\displaystyle\begin{split}d_{A}&=\Big{(}{\bf T}^{a_{1}}{\bf T}^{a_{2}}{\bf
T}^{a_{3}}{\bf T}^{a_{4}}\Big{)}_{i}\Big{(}{\bf T}^{a_{3}}{\bf T}^{a_{1}}{\bf
T}^{a_{4}}{\bf
T}^{a_{2}}\Big{)}_{j}\equiv\left(\begin{array}[]{cccc}a_{1}&a_{2}&a_{3}&a_{4}\\\
a_{3}&a_{1}&a_{4}&a_{2}\end{array}\right).\end{split}$ (C.3)

Similarly, the diagram in figure 11b is

$\displaystyle\begin{split}d_{B}&=\Big{(}{\bf T}^{a_{1}}{\bf T}^{a_{2}}{\bf
T}^{a_{3}}{\bf T}^{a_{4}}\Big{)}_{i}\Big{(}{\bf T}^{a_{2}}{\bf T}^{a_{4}}{\bf
T}^{a_{1}}{\bf
T}^{a_{3}}\Big{)}_{j}\equiv\left(\begin{array}[]{cccc}a_{1}&a_{2}&a_{3}&a_{4}\\\
a_{2}&a_{4}&a_{1}&a_{3}\end{array}\right).\end{split}$ (C.4)

We do not have an expression of $d_{A}$ and $d_{B}$ separately which is valid
for general representations. However, we are interested only in the
combination $d_{A}+d_{B}$, which manifests the symmetry under the interchanged
of the projectile and the target. For this combination we have the following
identity:

$\displaystyle d_{A}+d_{B}$
$\displaystyle=\left(\begin{array}[]{cccc}a_{1}&a_{2}&a_{3}&a_{4}\\\
a_{4}&a_{1}&a_{3}&a_{2}\end{array}\right)+\left(\begin{array}[]{cccc}a_{1}&a_{2}&a_{3}&a_{4}\\\
a_{1}&a_{4}&a_{2}&a_{3}\end{array}\right)\,,$ (C.9)

which we will prove below. The two terms on right-hand side of eq. (C.9),
depicted in figures 13a and 13b, feature outmost Reggeon interactions
represented by the indices $a_{1}$ and $a_{4}$, respectively. Therefore, these
terms are easily written in terms of colour channel operators by applying eqs.
(LABEL:eq:relTsu-Tt/2) and (LABEL:eq:relTsu+Tt/2), as described in step $(ii)$
above. The resulting two-loop graphs are again reducible, and one obtains:

(a)  (b)

Figure 13: Diagrammatic representation of the terms on the right hand side of
eq. (C.9).

$\displaystyle\begin{split}d_{A}+d_{B}&=\frac{1}{4}\left\\{\left(\mathbf{T}_{s-u}^{2}\right)^{3}-\frac{1}{4}\left(\mathbf{T}_{t}^{2}\right)^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{4}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{C_{A}^{2}}{4}\mathbf{T}_{s-u}^{2}\right\\}{\bf
T}_{i}^{a}\,{\bf T}_{j}^{a},\end{split}$ (C.10)

Eq. (C.10) is a general expression of $d_{A}+d_{B}$ for arbitrary
representations of external particles. This is the only relation needed to
compute three-loop colour structures. The identity (C.9) was crucial to obtain
the result. This identity is conveniently derived by starting from an
auxiliary three-loop configuration

$\displaystyle\begin{split}\tilde{d}^{(3)}&=\Big{(}{\bf T}^{a_{1}}{\bf
T}^{a_{2}}{\bf T}^{a_{3}}\Big{)}_{i}\,\Big{(}{\bf T}^{x}{\bf T}^{a_{1}}{\bf
T}^{a_{3}}{\bf T}^{x}{\bf T}^{a_{2}}\Big{)}_{j}.\end{split}$ (C.11)

The colour factor above is not associated to a permutation of indices, because
it features a boomerang, namely the contraction of a pair of indices on the
same line (in this case the projectile). Using the Lie algebra relations,
there are two ways of moving the $x$ on line $i$: either by commuting $x$ with
$a_{1}$ or $x$ with $a_{3}$. We find respectively

$\displaystyle\tilde{d}^{(3)}_{1}=$
$\displaystyle\,\Big{(}C_{2}(j)-\frac{C_{A}}{2}\Big{)}\left({\bf
T}^{a_{1}}{\bf T}^{a_{2}}{\bf T}^{a_{3}}\right)_{i}\left({\bf T}^{a_{1}}{\bf
T}^{a_{3}}{\bf T}^{a_{2}}\right)_{j}$ (C.12a) $\displaystyle\hskip
128.0374pt+if^{a_{3}xk}\left({\bf T}^{a_{1}}{\bf T}^{a_{2}}{\bf
T}^{a_{3}}\right)_{i}\left({\bf T}^{x}{\bf T}^{a_{1}}{\bf T}^{k}{\bf
T}^{a_{2}}\right)_{j},$ $\displaystyle\tilde{d}^{(3)}_{2}=$
$\displaystyle\,\Big{(}C_{2}(j)-\frac{C_{A}}{2}\Big{)}\left({\bf
T}^{a_{1}}{\bf T}^{a_{2}}{\bf T}^{a_{3}}\right)_{i}\left({\bf T}^{a_{1}}{\bf
T}^{a_{3}}{\bf T}^{a_{2}}\right)_{j}$ (C.12b) $\displaystyle\hskip
128.0374pt+if^{xa_{1}k}\left({\bf T}^{a_{1}}{\bf T}^{a_{2}}{\bf
T}^{a_{3}}\right)_{i}\left({\bf T}^{k}{\bf T}^{a_{3}}{\bf T}^{x}{\bf
T}^{a_{2}}\right)_{j}.$

The two expressions are of course identical, so their difference must vanish

$0=i\left({\bf T}^{a_{1}}{\bf T}^{a_{2}}{\bf
T}^{a_{3}}\right)_{i}\Big{[}f^{a_{3}xk}\left({\bf T}^{x}{\bf T}^{a_{1}}{\bf
T}^{k}{\bf T}^{a_{2}}\right)_{j}-f^{a_{1}xk}\left({\bf T}^{x}{\bf
T}^{a_{3}}{\bf T}^{k}{\bf T}^{a_{2}}\right)_{j}\Big{]}.$ (C.13)

Finally, writing the structure constants in terms of commutators on line $i$,
we obtain eq. (C.9), concluding the proof.

### Four-loop colour factors

All the colour structures appearing at four loops are written in terms of
contractions of five pairs of generators, by applying repeatedly the Lie
algebra. We identify eight independent colour factors that cannot be reduced
in terms of $\mathbf{T}_{s-u}^{2}$ and $\mathbf{T}_{t}^{2}$ by following steps
$(i)$ and $(ii)$ above. We choose to collect them into the following terms:

$\displaystyle\begin{split}d_{1}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{2}&a_{5}&a_{3}&a_{1}&a_{4}\end{array}\right)+\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{4}&a_{1}&a_{3}&a_{5}&a_{2}\end{array}\right),\end{split}$ (C.14a)
$\displaystyle\begin{split}d_{2}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{2}&a_{4}&a_{1}&a_{5}&a_{3}\end{array}\right)+\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{3}&a_{1}&a_{5}&a_{2}&a_{4}\end{array}\right),\end{split}$ (C.14b)
$\displaystyle\begin{split}d_{3}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{2}&a_{4}&a_{5}&a_{1}&a_{3}\end{array}\right)+\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{4}&a_{1}&a_{5}&a_{2}&a_{3}\end{array}\right),\end{split}$ (C.14c)
$\displaystyle\begin{split}d_{4}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{2}&a_{5}&a_{1}&a_{4}&a_{3}\end{array}\right)+\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{3}&a_{1}&a_{5}&a_{4}&a_{2}\end{array}\right),\end{split}$ (C.14d)
$\displaystyle\begin{split}d_{5}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{3}&a_{2}&a_{5}&a_{1}&a_{4}\end{array}\right)+\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{4}&a_{2}&a_{1}&a_{5}&a_{3}\end{array}\right),\end{split}$ (C.14e)
$\displaystyle\begin{split}d_{6}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{3}&a_{4}&a_{1}&a_{5}&a_{2}\end{array}\right)+\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{3}&a_{5}&a_{1}&a_{2}&a_{4}\end{array}\right),\end{split}$ (C.14f)
$\displaystyle\begin{split}d_{7}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{3}&a_{5}&a_{1}&a_{4}&a_{2}\end{array}\right),\end{split}$ (C.14g)
$\displaystyle\begin{split}d_{8}&=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}\\\
a_{4}&a_{2}&a_{5}&a_{1}&a_{3}\end{array}\right),\end{split}$ (C.14h)

where each term $d_{1}\dots d_{8}$ is manifestly symmetric under target-
projectile exchange $i~{}\leftrightarrow~{}j$. In addition, $d_{1}$ is
symmetric under signature symmetry, because the two terms in eq. (C.14a) are
related to each other by reversing the order of indices on one of the lines.
In order to express these colour factors in terms of channel operators, we
consider again the configurations generated by operating with the Lie algebra
on the corresponding boomerang diagrams. In particular, we consider the
following four-loop diagrams

$\displaystyle\begin{split}\tilde{d}^{(4)}_{L}&=\Big{(}{\bf T}^{a_{1}}{\bf
T}^{a_{2}}{\bf T}^{a_{3}}{\bf T}^{a_{4}}\Big{)}_{i}\,\Big{(}{\bf T}^{x}{\bf
T}^{a_{\sigma(1)}}{\bf T}^{a_{\sigma(2)}}{\bf T}^{x}{\bf
T}^{a_{\sigma(3)}}{\bf T}^{a_{\sigma(4)}}\Big{)}_{j},\\\
\tilde{d}^{(4)}_{R}&=\Big{(}{\bf T}^{a_{1}}{\bf T}^{a_{2}}{\bf T}^{a_{3}}{\bf
T}^{a_{4}}\Big{)}_{i}\,\Big{(}{\bf T}^{a_{\sigma(1)}}{\bf
T}^{a_{\sigma(2)}}{\bf T}^{x}{\bf T}^{a_{\sigma(3)}}{\bf
T}^{a_{\sigma(4)}}{\bf T}^{x}\Big{)}_{j},\\\ \tilde{d}^{(4)}_{C}&=\Big{(}{\bf
T}^{a_{1}}{\bf T}^{a_{2}}{\bf T}^{a_{3}}{\bf
T}^{a_{4}}\Big{)}_{i}\,\Big{(}{\bf T}^{a_{\sigma(1)}}{\bf T}^{x}{\bf
T}^{a_{\sigma(2)}}{\bf T}^{a_{\sigma(3)}}{\bf T}^{x}{\bf
T}^{a_{\sigma(4)}}\Big{)}_{j},\end{split}$ (C.15)

which generalise the three-loop boomerang of eq. (C.11) by including one more
pair of indices, and the target-projectile symmetric configurations obtained
from eq. (C.15). Starting from these boomerang configurations, we operate as
in eqs. (C.12) and (C.13) and we get six independent linear relations for
$d_{1}\dots d_{8}$. We derive one more constraint from the colour factor

$\displaystyle\tilde{d}^{(4)}_{P}$
$\displaystyle=\text{Tr}\left[F^{x}F^{a}F^{b}F^{c}\right]\text{Tr}\left[F^{y}F^{a}F^{b}F^{c}\right]\,{\bf
T}^{x}_{i}\,{\bf
T}^{y}_{j}=\left(\frac{d_{AA}}{N_{A}}+\frac{C_{A}^{4}}{12}\right)\,{\bf
T}^{x}_{i}\,{\bf T}^{x}_{j},$ (C.16)

which can be written as a combination of $d_{1}\dots d_{8}$ by using the Lie
algebra to replace traces of generators in the adjoint representation with
commutators of ${\bf T}_{i}$ or ${\bf T}_{j}$. The seven identities obtained
in this way determine the signature-odd contributions to $d_{1}\dots d_{8}$.
In turn, this is sufficient in order to express the real part of the
amplitude, thus allowing us to perform the calculations of section 4.2.
However, we need one more equation in order to determine also contributions of
even signature. In particular, such terms were needed to compute the Regge
limit of the soft anomalous dimension, discussed in section 7. The last
constraint was determined by writing a general ansatz for $d_{3}$ in terms of
products of Casimir operators $\mathbf{T}_{s-u}^{2}$ and $\mathbf{T}_{t}^{2}$
acting on the tree-level amplitude ${\bf T}^{x}_{i}\,{\bf T}^{x}_{j}$. The
unknown coefficients were fitted by comparing the ansatz with explicit results
obtained by specialising the generators ${\bf T}_{i}$ and ${\bf T}_{j}$ in eq.
(C.14c) either in the adjoint or in the fundamental representation. We report
here the final expressions of the colour factors in eqs. (C.14a)-(C.14h),
which all apply for any representation of the external particles. The results
are:

$\displaystyle d_{1}$
$\displaystyle=\Bigg{\\{}\frac{1}{12}\left(\frac{d_{AA}}{N_{A}}+\frac{5}{96}C_{A}^{4}\right)-\frac{3}{32}C_{A}^{2}\,\left(\mathbf{T}_{s-u}^{2}\right)^{2}+\frac{C_{A}}{32}\left[5\,\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{7}{3}\mathbf{T}_{t}^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]$
$\displaystyle+\frac{1}{8}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{4}+\frac{3}{4}\Big{[}\mathbf{T}_{t}^{2},\mathbf{T}_{s-u}^{2}\Big{]}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{5}{12}\left(\mathbf{T}_{t}^{2}\right)^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]\Bigg{\\}}\,{\bf
T}^{x}_{i}\,{\bf T}^{x}_{j},$ (C.17a)
$\displaystyle\begin{split}d_{2}&=\Bigg{\\{}\frac{1}{12}\left(\frac{d_{AA}}{N_{A}}+\frac{5}{96}C_{A}^{4}\right)-\frac{C_{A}^{2}}{8}\left(\mathbf{T}_{s-u}^{2}\right)^{2}+\frac{C_{A}}{16}\left[3\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{7}{6}\mathbf{T}_{t}^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]\\\
&+\frac{1}{8}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{4}-\frac{3}{4}\mathbf{T}_{s-u}^{2}\left(\mathbf{T}_{t}^{2}\right)^{2}\mathbf{T}_{s-u}^{2}+\frac{1}{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{1}{6}\left(\mathbf{T}_{t}^{2}\right)^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]\\\
&-\frac{C_{A}^{3}}{64}\mathbf{T}_{s-u}^{2}-\frac{C_{A}^{2}}{64}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{16}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{3}+\frac{3}{4}\left(\mathbf{T}_{t}^{2}\right)^{2}\mathbf{T}_{s-u}^{2}\right]\\\
&+\frac{1}{16}\Bigg{[}\Big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\Big{]}\left(\mathbf{T}_{s-u}^{2}\right)^{2}-\left(\mathbf{T}_{s-u}^{2}\right)^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{1}{4}\left(\mathbf{T}_{t}^{2}\right)^{3}\mathbf{T}_{s-u}^{2}\Bigg{]}\Bigg{\\}}\,{\bf
T}^{x}_{i}\,{\bf T}^{x}_{j},\end{split}$ (C.17b)
$\displaystyle\begin{split}d_{3}&=d_{6}=\Bigg{\\{}\frac{1}{8}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{4}-\frac{1}{4}\Big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\Big{]}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{1}{4}\left(\mathbf{T}_{t}^{2}\right)^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]\\\
&+\frac{C_{A}^{3}}{64}\mathbf{T}_{s-u}^{2}-\frac{C_{A}^{2}}{16}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{C_{A}}{16}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{3}-\frac{3}{2}\left(\mathbf{T}_{t}^{2}\right)^{2}\mathbf{T}_{s-u}^{2}\right]\\\
&-\frac{1}{16}\left(\mathbf{T}_{s-u}^{2}\Big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\Big{]}\mathbf{T}_{s-u}^{2}+\frac{1}{2}\left(\mathbf{T}_{t}^{2}\right)^{3}\mathbf{T}_{s-u}^{2}\right)\Bigg{\\}}\,{\bf
T}^{x}_{i}\,{\bf T}^{x}_{j},\end{split}$ (C.17c)
$\displaystyle\begin{split}d_{4}&=d_{5}=\Bigg{\\{}\frac{1}{8}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{4}-\frac{1}{4}\Big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\Big{]}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{1}{4}\left(\mathbf{T}_{t}^{2}\right)^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]\\\
&-\frac{C_{A}^{3}}{64}\mathbf{T}_{s-u}^{2}+\frac{C_{A}^{2}}{16}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}+\frac{C_{A}}{16}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{3}-\frac{3}{2}\left(\mathbf{T}_{t}^{2}\right)^{2}\mathbf{T}_{s-u}^{2}\right]\\\
&+\frac{1}{16}\left(\mathbf{T}_{s-u}^{2}\Big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\Big{]}\mathbf{T}_{s-u}^{2}+\frac{1}{2}\left(\mathbf{T}_{t}^{2}\right)^{3}\mathbf{T}_{s-u}^{2}\right)\Bigg{\\}}\,{\bf
T}^{x}_{i}\,{\bf T}^{x}_{j},\end{split}$ (C.17d)
$\displaystyle\begin{split}d_{7}&=d_{8}=\Bigg{\\{}\frac{1}{24}\left(\frac{d_{AA}}{N_{A}}+\frac{5}{96}C_{A}^{4}\right)-\frac{C_{A}^{2}}{16}\left(\mathbf{T}_{s-u}^{2}\right)^{2}+\frac{C_{A}}{32}\left[3\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{7}{6}\mathbf{T}_{t}^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]\\\
&+\frac{1}{16}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{4}-\frac{3}{4}\mathbf{T}_{s-u}^{2}\left(\mathbf{T}_{t}^{2}\right)^{2}\mathbf{T}_{s-u}^{2}+\frac{1}{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{1}{6}\left(\mathbf{T}_{t}^{2}\right)^{2}\left(\mathbf{T}_{s-u}^{2}\right)^{2}\right]\\\
&+\frac{C_{A}^{3}}{128}\mathbf{T}_{s-u}^{2}+\frac{C_{A}^{2}}{128}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{C_{A}}{32}\left[\left(\mathbf{T}_{s-u}^{2}\right)^{3}+\frac{3}{4}\left(\mathbf{T}_{t}^{2}\right)^{2}\mathbf{T}_{s-u}^{2}\right]\\\
&-\frac{1}{32}\Bigg{[}\Big{[}\mathbf{T}_{s-u}^{2},\mathbf{T}_{t}^{2}\Big{]}\left(\mathbf{T}_{s-u}^{2}\right)^{2}-\left(\mathbf{T}_{s-u}^{2}\right)^{2}\mathbf{T}_{t}^{2}\mathbf{T}_{s-u}^{2}-\frac{1}{4}\left(\mathbf{T}_{t}^{2}\right)^{3}\mathbf{T}_{s-u}^{2}\Bigg{]}\Bigg{\\}}\,{\bf
T}^{x}_{i}\,{\bf T}^{x}_{j}.\end{split}$ (C.17e)

##  Appendix D The reduced amplitude in an explicit colour basis

It is interesting to evaluate the NNLL reduced amplitude for different
external partons. We will compute the reduced NNLL odd amplitude at two loops,
three loops and four loops for $qq$, $gg$ and $qg$ scattering. We utilise the
orthornormal $t$-channel basis of ref. DelDuca:2014cya , and use the same
notation as in appendix B of ref. Caron-Huot:2017fxr  with the relevant colour
tensors given in eq. (2.15).

Projecting the two-loop NNLL amplitude in eq. (5.13) in the octet channel for
$qq$, $gg$ and $qg$ scattering we find

$\displaystyle\mathcal{\hat{M}}^{(-,2,0),[8]}_{qq\to qq}=$
$\displaystyle\,\left[2D_{q}^{(2)}+D_{q}^{(1)}D_{q}^{(1)}-(i\pi)^{2}r_{\Gamma}^{2}S^{(2)}(\epsilon)\left(\frac{N_{c}^{2}}{6}-1+\frac{3}{N_{c}^{2}}\right)\right]{\cal
M}^{{\rm tree},[8]}_{qq\to qq},$ (D.1a)
$\displaystyle\mathcal{\hat{M}}^{(-,2,0),[8_{a}]}_{gg\to gg}=$
$\displaystyle\,\left[2D_{g}^{(2)}+D_{g}^{(1)}D_{g}^{(1)}-(i\pi)^{2}r_{\Gamma}^{2}S^{(2)}(\epsilon)\left(\frac{N_{c}^{2}}{6}+6\right)\right]{\cal
M}^{{\rm tree},[8_{a}]}_{gg\to gg},$ (D.1b)
$\displaystyle\mathcal{\hat{M}}^{(-,2,0),[8_{a}]}_{qg\to qg}=$
$\displaystyle\,\left[D_{q}^{(2)}+D_{g}^{(2)}+D_{q}^{(1)}D_{g}^{(1)}-(i\pi)^{2}r_{\Gamma}^{2}S^{(2)}(\epsilon)\left(\frac{N_{c}^{2}}{6}+1\right)\right]{\cal
M}^{{\rm tree},[8_{a}]}_{qg\to qg},$ (D.1c)

where we have normalised by the tree-amplitude octet projection defined in eq.
(2.13) and $S^{(2)}(\epsilon)$ is given in eq. (5.14).

The three-loop reduced amplitude of eq. (5.21) in the $t$-channel octet
representation is

$\displaystyle\mathcal{\hat{M}}^{(-,3,1),[8]}_{qq\to qq}=$
$\displaystyle\,(i\pi)^{2}r_{\Gamma}^{3}\left(N_{c}^{2}+18-\frac{18}{N_{c}^{2}}\right)\frac{N_{c}}{864}\left(-\frac{1}{\epsilon^{3}}+70\hat{\zeta}_{3}+\mathcal{O}(\epsilon^{2})\right){\cal
M}^{{\rm tree},[8]}_{qq\to qq},$ (D.2a)
$\displaystyle\mathcal{\hat{M}}^{(-,3,1),[8_{a}]}_{gg\to gg}=$