# Systematics of the Multi-Regge Three-Loop Symbol

Till Bargheer

★

DESY 16-115

Systematics of the Multi-Regge Three-Loop Symbol

Till Bargheer

DESY Theory Group, DESY Hamburg
Notkestraße 85, D-22607 Hamburg, Germany

<EMAIL_ADDRESS>
Abstract

We review the systematics of Mandelstam cut contributions to planar scattering
amplitudes in the multi-Regge limit. Isolating the relevant cut terms, we
explain how the BFKL expansion can be used to construct the perturbative
$n$-point multi-Regge limit amplitude in certain kinematic regions from a
finite number of basic building blocks. At three loops and at leading
logarithmic order, two building blocks are required. Their symbols are
extracted from the known three-loop six-point and seven-point symbols for
general kinematics. The new seven-point building block is constructed in terms
of single-valued multiple polylogarithms to the extent it can be determined
using the symbol as well as further symmetry and consistency constraints.
Beyond the leading logarithmic order, the subleading and sub-subleading terms
require two and one further building block, respectively. The latter could
either be reconstructed from further perturbative data, or from BFKL integrals
involving yet-unknown corrections to the central emission block.

toc

##  1 Introduction

Recent advances in the study of scattering amplitudes have sparked renewed
interest in the multi-Regge limit of high-energy scattering. Besides its
phenomenological significance, it has long been noted that the perturbative
expansion simplifies considerably in this limit: Typically, the perturbative
series has to be (and in fact can be!) resummed due to the appearance of large
logarithms, leading to factorized all-order expressions for scattering
processes. A further enhancement comes about in the case of planar
$\mathcal{N}=4$ super Yang–Mills theory: Here, the multi-Reggeon states that
resum all-order gluon exchanges are governed by the integrable
Balitsky–Fadin–Kuraev–Lipatov (BFKL) [1, 2, 3] and
Bartels–Kwieciǹski–Praszałowicz (BKP) [4, 5] Hamiltonians. This first
appearance of integrability in the planar theory was observed long before the
extensive discoveries and applications of integrable structures that took
place during the past fifteen years [6]. Since the proposal of the
exponentiated Bern–Dixon–Smirnov (BDS) amplitude [7], the systematics of
multi-Regge limit amplitudes in planar $\mathcal{N}=4$ super Yang–Mills theory
have been understood to a remarkable extent. In fact, after a disagreement at
strong coupling had casted doubt on the correctness of the BDS amplitude [8],
it was the absence of the expected Regge pole and cut terms that invalidated
the proposal at weak coupling [9], and that prompted the correction of the BDS
amplitude by the dual conformally invariant remainder function beyond five
points [10, 11].

By now, the remainder function has been constructed to high loop orders by
constraining the possible function space through physical symmetry and
analyticity requirements [12, 13]. This _bootstrap program_ relies on various
input, ranging from the mathematical theory of the relevant functions [14, 15]
to recursion relations [16] and the expansion around collinear limits as
dictated by integrability [17, 18]. In all cases, knowledge about the multi-
Regge limit has provided important boundary data to the bootstrap enterprise.
Conversely, these recent methods admit to compute the BFKL data, and hence the
multi-Regge-limit remainder function, to unprecedented orders [19, 20, 12, 21,
22, 23]. To date, this fruitful interplay has mostly been restricted to the
six-point case. An extension to seven-point functions has been initiated
recently [24]. Going to even higher points will require a better understanding
of the relevant function space. It is conceivable that the Regge limit will
again provide valuable boundary data in this regard.

It has been understood that obtaining the full analytic structure of multi-
Regge limit amplitudes requires to analyze the amplitudes in all possible
kinematic regions [25, 26, 27, 28]. In fact, while the integrable structure at
strong coupling becomes particularly amenable in the multi-Regge limit [29], a
discrepancy with the expectation from weak coupling has been observed in one
of the kinematic regions at seven points [30]. Recently, a systematic study of
the $n$-point two-loop remainder function in all kinematic regions at weak
coupling has been put forward [31]. The ability to study any number of points
relied on the known two-loop _symbol_ of the remainder function for all
multiplicities [32]. Passing from polylogarithmic functions to their symbols
constitutes a major simplification, both for the analysis of the relevant
expressions and for the systematics of the multi-Regge limit.

The goal of the present work is two-fold: One aim is to understand the results
of the previous study [31] from the perspective of Regge cut contributions.
Secondly, we want to lift the analysis to the three-loop level. To this end,
we first isolate the Regge cut contributions that contribute to a given
region, and then expand the relevant contributions to the three-loop order.
Judiciously grouping the resulting terms, we find that the $n$-point three-
loop remainder function, in the simplest class of kinematic regions, reduces
to a linear combination of five building block functions. At the symbol level,
the reduction extends to all kinematic regions. The symbols of the two
building blocks required to reconstruct the $n$-point remainder function at
leading logarithmic order are extracted from the known perturbative data. To
the extent that it is fixed by the symbol as well as symmetries, the new
seven-point building block is constructed in terms of multiple polylogarithms.
The results of this work are assembled in a computer-readable file attached to
this submission.

#### Overview.

Section 2 briefly summarizes the systematics of planar scattering amplitudes
in the multi-Regge limit in a self-contained way. Section 3 highlights the
simplifications and restrictions implied by specializing to certain kinematic
regions, or by passing from functions to symbols. In Section 4, the two-loop
analysis of the multi-Regge limit remainder function is revisited from the
Regge cut point of view. Section 5 extends the analysis to three loops, where
the remainder function can be decomposed into a few basic building blocks. The
latter are discussed in Section 6. We construct a function for the seven-point
building block in Section 7, and Section 8 presents the conclusion.

#### Note added:

The simultaneous paper [33] has some overlap with the present work. In
particular, there is a connection between the “factorization theorem” of [33]
and the application of the reduction identities (4.6) to the expansion of the
$n$-point cut contribution carried out in this work.

##  2 Background

#### Multi-Regge Kinematics.

The $2\to(n-2)$ multi-Regge limit is the $n$-particle generalization of the
simple $s\gg t$ Regge limit for $2\to 2$ scattering. To describe a general
amplitude, we will use the $(3n-10)$ independent Lorentz invariants

$\displaystyle
t_{j}\equiv\boldsymbol{q}_{j}^{2}\,,\qquad\boldsymbol{q}_{j}\equiv\boldsymbol{p}_{2}+\boldsymbol{p}_{3}+\dots+\boldsymbol{p}_{j-1}\,,\qquad
j=4,\dots,n\,,$ (2.1) $\displaystyle s_{j}\equiv
s_{j-1,j}\equiv(\boldsymbol{p}_{j-1}+\boldsymbol{p}_{j})^{2}\,,\qquad
j=4,\dots,n\,,$ (2.2)
$\displaystyle\eta_{j}\equiv\frac{s_{j}s_{j+1}}{(\boldsymbol{p}_{j-1}+\boldsymbol{p}_{j}+\boldsymbol{p}_{j+1})^{2}}\,,\qquad
j=4,\dots,n-1\,.$ (2.3)

Here, $\boldsymbol{p}_{1},\dots,\boldsymbol{p}_{n}$ are the $n$ external
momenta. By convention, they are all incoming, but may have either energy
sign. The $2\to(n-2)$ multi-Regge limit is characterized by a large separation
of rapidities among the produced particles. In terms of the above kinematic
variables, the limit is attained for

$|s|\gg|s_{4}|,\dots,|s_{n}|\gg t_{4},\dots,t_{n}\,,$ (2.4)

where $s=(\boldsymbol{p}_{1}+\boldsymbol{p}_{2})^{2}$ is the total energy. See
Figure 1 for an illustration of the kinematics. Many quantities in the multi-
Regge limit only depend on the kinematics in the transverse space to the
$(\boldsymbol{p}_{1},\boldsymbol{p}_{2})$ plane. We hence define

$\boldsymbol{p}_{j}=\alpha_{j}\boldsymbol{p}_{1}+\beta_{j}\boldsymbol{p}_{2}+\boldsymbol{p}_{j}^{\perp}\,,\qquad\boldsymbol{p}_{1}\cdot\boldsymbol{p}_{j}^{\perp}=\boldsymbol{p}_{2}\cdot\boldsymbol{p}_{j}^{\perp}=0\,,\qquad
j=4,\dots,n-1\,,$ (2.5)

and similarly for $\boldsymbol{q}_{4},\dots,\boldsymbol{q}_{n}$. It is often
convenient to switch to complex variables $p_{j}$, $q_{j}$ whose real and
imaginary parts equal the two components of the transverse momenta
$\boldsymbol{p}_{j}^{\perp}$ and $\boldsymbol{q}_{j}^{\perp}$, respectively:

$\boldsymbol{p}_{j}^{\perp}=\bigl{(}\Re(p_{j}),\Im(p_{j})\bigr{)}\,,\qquad\boldsymbol{q}_{j}^{\perp}=\bigl{(}\Re(q_{j}),\Im(q_{j})\bigr{)}\,.$
(2.6)

Frequently used combinations of the transverse momenta are the complex
anharmonic ratios

$w_{j}=\frac{p_{j-1}q_{j+1}}{q_{j-1}p_{j}}\,,\qquad j=5,\dots,n-1\,.$ (2.7)

Planar $\mathcal{N}=4$ super Yang–Mills theory enjoys dual conformal
invariance. Invariant quantities in this theory can thus only depend on
conformally invariant cross ratios

$U_{ij}\equiv\frac{x_{i+1,j}^{2}x_{i,j+1}^{2}}{x_{ij}^{2}x_{i+1,j+1}^{2}}\,,\qquad
3\leq|i-j|\leq n-2$ (2.8)

of the dual coordinates

$\boldsymbol{p}_{j}\equiv x_{j}-x_{j-1}\,,\qquad x_{ij}=x_{i}-x_{j}\,.$ (2.9)

A basis of kinematically independent invariant cross ratios is provided by

$u_{j,1}=U_{j-3,j}\,,\qquad u_{j,2}=U_{j-2,n}\,,\qquad
u_{j,3}=U_{1,j-1}\,,\qquad j=5,\dots,n-1\,.$ (2.10)

In the multi-Regge limit, these cross ratios converge to $1$ or $0$:

$u_{j,1}\to 1\,,\qquad u_{j,2}\to 0\,,\qquad u_{j,3}\to 0\,.$ (2.11)

The ratios of subleading terms remain finite, and are related to the
anharmonic ratios (2.7) via

$\frac{u_{j,2}}{1-u_{j,1}}\to\frac{1}{|1+w_{j}|^{2}}\,,\qquad\frac{u_{j,3}}{1-u_{j,1}}\to\frac{|w_{j}|^{2}}{|1+w_{j}|^{2}}\,.$
(2.12)

Figure 1: Kinematic variables.

#### Kinematic Regions.

In order to understand the full analytic structure of the multi-Regge limit
amplitude, it is important to analyze it in all physical kinematic regions.
Our starting point will be the physical region in which the energies of all
particles $3,\dots,n$ are negative (which means that those particles are
effectively outgoing, instead of incoming). In this region, all subenergies
$s_{j}$, $j=4,\dots,n$, are negative.111The Minkowski metric is assumed to
have signature $({-}{+}{+}{+})$. In all other physical regions that we will
consider, some of the particles $4,\dots,n-1$ have positive energies (those
particles become incoming), and hence some of the invariants $s_{j}$ become
positive. These other regions are sometimes called “Mandelstam regions”.222One
could also consider regions in which the energies of particles $3$ and/or $n$
are positive, but those regions do not add further analytic structure to the
amplitude, and will thus not be considered in the following. They can be
reached from the all-outgoing region by analytic continuation of the
kinematics. The various regions will be labeled by the subsets
$I\subset\\{4,\dots,n-1\\}$ of particles whose energies have been continued to
positive values. Alternatively, we will often label regions by
$\rho=(\rho_{4},\dots,\rho_{n-1})\in\mathbb{Z}_{2}^{n-4}$, with $\rho_{j}=\pm
1$ (or just $\rho_{j}=\pm$) indicating whether the respective particle has
been flipped (its energy has been continued) $({-})$ or not $({+})$.

Importantly, the various regions become disconnected in the strict multi-Regge
limit. That is to say, in order to continue the kinematics from one region to
the other, one has to complexify the subenergies $s_{k}$ (e.g. by continuing
them along big circles).

#### Multi-Regge Limit Amplitudes.

Scattering in the multi-Regge limit is dominated by the exchange of “Reggeized
gluons” (or “Reggeons”), which are effective particles that resum the
contributions of entire classes of gluonic Feynman diagrams of all loop
orders. The simplest example is the four-point amplitude in _planar_
$\mathcal{N}=4$ super Yang–Mills theory, for which _all_ perturbative
contributions can be resummed and factorized into a single diagram:

$\mathcal{A}_{4}^{\mathrm{MRL}}=\parbox[c]{0.0pt}{\includegraphics{Fig4pt}}=\Gamma(t)s^{\omega(t)}\Gamma(t)\,.$
(2.13)

Here, $\includegraphics{FigGGRvertex}=\Gamma(t)$ is the gluon-gluon-Reggeon
vertex (see e.g. [9]), and  stands for the exchange of a single Reggeon with
propagator $s^{\omega(t)}$, where $\omega(t)$ is the (real-valued) Regge
trajectory. At five points, two different kinematic regions can be considered:
The produced particle $4$ can either be flipped $({-})$ or not $({+})$.
Strikingly, the factorization property of the four-point amplitude extends to
this case: In both regions, the planar five-point amplitude again factorizes
into a single diagram,

$\mathcal{A}_{5}^{\mathrm{MRL}(\pm)}=\parbox[c]{0.0pt}{\includegraphics{Fig5pt}}=\Gamma(t_{4})\,s_{4}^{\omega(t_{4})}\,\Gamma_{45}\,s_{5}^{\omega(t_{5})}\,\Gamma(t_{5})\,,$
(2.14)

where the (complex) gluon production vertex [11]

$\parbox[c]{0.0pt}{\includegraphics{FigRRGvertex}}=\Gamma_{j,j+1}=|\Gamma_{j,j+1}|e^{\pm
i\pi\,\hat{\omega}_{j,j+1}}\,,\qquad\hat{\omega}_{j,j+1}=\hat{\omega}(t_{j},t_{j+1},\eta_{j})\,,$
(2.15)

only depends on the kinematic region through the sign of its phase.

A general $n$-point multi-Regge-limit amplitude in any given kinematic region
$\rho$ receives contributions from Regge pole [34] as well as Mandelstam cut
terms [35, 36],

$\mathcal{A}_{n,\rho}^{\mathrm{MRL}}=\mathcal{A}_{n,\rho}^{\text{Regge
pole}}+\mathcal{A}_{n,\rho}^{\text{Mandelstam cut}}\,.$ (2.16)

Both the pole terms and the cut terms depend on the kinematic region $\rho$.
The origin of the Mandelstam cut terms are non-trivial contributions from
multi-Reggeon bound state exchange in intermediate $t$-channels. For planar
amplitudes of up to five points, such contributions are suppressed by powers
of $1/N_{\text{c}}$, and the amplitudes factorize as indicated above. At six
points, the first cut term appears, in the region $({-}{-})$ where both
intermediate momenta have been flipped [10]. In a generic region $\rho$, the
six-point amplitude therefore reads [11, 27]

$\mathcal{A}_{6,\rho}^{\mathrm{MRL}}=\text{(pole
terms)}^{\rho}+c_{6,1,4}^{\rho}\,\parbox[c]{0.0pt}{\includegraphics{Fig6pt2}}\,,$
(2.17)

where the region-dependent coefficient $c_{6,1,4}^{\rho}$ is non-vanishing for
$\rho=({-}{-})$. Here, the cut diagram stands for all contributions from two-
Reggeon bound state exchange in the $t_{5}$ channel. This picture generalizes
to higher multiplicities: The planar $n$-point multi-Regge limit amplitude is
a sum of region-dependent Regge pole terms as well as Mandelstam cut
contributions with region-dependent coefficients [9, 10, 11, 37, 26, 27, 28,
38]:

$\mathcal{A}_{n,\rho}^{\mathrm{MRL}}=\text{(pole
terms)}^{\rho}+\sum_{j}c_{n,1,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt2}}+\sum_{j}c_{n,2,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt22}}+\dots{\\\
}+\sum_{j}d_{n,1,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt232}}+\sum_{j<k}e_{n,j,k}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt212}}+\dots$
(2.18)

Here, the symbol  stands for the insertion of zero or more complex gluon
production vertices (2.15). For planar amplitudes, the number of exchanged
Reggeons can at most increase or decrease by one when passing from one
$t$-channel to the next.333The number $M_{n}$ of admissible diagrams that can
contribute to the $n$-point amplitude, as a sequence in $n$, equals the
Motzkin sequence, OEIS A001006 [39], with $M_{n}/M_{n-1}\to 3$ for
$n\to\infty$. All other contributions are suppressed by powers of
$1/N_{\text{c}}$. The pole terms as well as the cut-term prefactors can in
principle be obtained from the general quantum field theory principles of
locality & unitarity. The procedure particularly relies on an expansion of the
amplitude into a sum of terms that each have no overlapping energy
discontinuities, following the Steinmann relations [40]. Determining the cut
contributions in this way is a very intricate and tedious procedure that has
to be carried out region by region. This formidable task has been completed
for the seven-point amplitude [28], and a study of the eight-point case is
underway [38], but a generalization to higher multiplicities appears
difficult. Below, we will see that the coefficients $c_{n,b,j}$ are actually
fixed by the two-loop analysis [31].

In fact, the _Mandelstam criterion_ [36] significantly constrains the set of
cut terms that can contribute to any given kinematic region: It asserts that
any cut contribution in which the multi-Reggeon states span the adjacent
$t$-channels $t_{j},\dots,t_{k}$ cannot contribute to regions in which
$s_{j-1}>0$ or $s_{k+1}>0$, that is444The reason is that in such cases, one of
the Feynman loop integrals can be closed trivially, since all singularities
lie on the same side of the integration contour [41].

$c_{n,k-j,j}^{\rho}=0\qquad\text{if}\qquad\rho_{j-1}=\rho_{j}\quad\text{or}\quad\rho_{k}=\rho_{k+1}\,,$
(2.19)

and similarly for the further coefficients in (2.18). Here, the subscripts
$n$, $b$, and $j$ in $c_{n,b,j}^{\rho}$ label the total number of particles,
the number of $t$-channels taking part in the multi-Reggeon state, and the
produced gluon that bounds the multi-Reggeon state on the left. For example,
as indicated above, the six-particle cut term (2.17) is only present in the
$({-}{-})$ region:

$c_{6,1,4}^{({+}{+})}=c_{6,1,4}^{({+}{-})}=c_{6,1,4}^{({-}{+})}=0\,.$ (2.20)

#### BDS and Remainder Function.

The MHV amplitudes of planar $\mathcal{N}=4$ super Yang–Mills theory can be
decomposed into two factors:

$\mathcal{A}_{n}^{\mathrm{MHV}}=\mathcal{A}_{n}^{\text{BDS}}R_{n}$ (2.21)

Here, $\mathcal{A}_{n}^{\text{BDS}}$ is the Bern–Dixon–Smirnov amplitude [7],
wich equals the tree-level amplitude times the exponentiated one-loop
amplitude, and which in fact produces the correct all-loop four-point and
five-point amplitudes. Starting at six points, it however fails to reproduce
the correct Regge pole contributions, and it misses all Regge cut terms
(beyond one loop) [10, 9, 11]. Hence it cannot be the full amplitude, but has
to be corrected by a non-trivial _remainder function_ $R_{n}$. Since the BDS
amplitude correctly captures all infrared singularities and dual conformal
weights, the remainder function is infrared finite and dual conformally
invariant, and thus can only depend on dual conformally invariant cross ratios
(2.8). By definition, it is only non-trivial starting from six points and two
loops.

Passing to the multi-Regge limit, and stripping off the universal absolute
value, the BDS amplitude reduces to a region-dependent phase factor. From the
latter, one can separate off a conformally invariant, infrared finite part
$\exp(i\delta_{n}^{\rho})$, which again is region-dependent, and contains the
finite part of the one-loop Regge cut terms [11, 27]

$\frac{\mathcal{A}_{n}^{\mathrm{BDS,MRL},\rho}}{\Gamma(t_{4})\,|s_{4}^{\omega_{4}}|\,|\Gamma_{45}|\,|s_{5}^{\omega_{5}}|\,|\Gamma_{56}|\,|s_{6}^{\omega_{6}}|\dots|s_{n-1}^{\omega_{n-1}}|\,|\Gamma_{n-1,n}|\,|s_{n}^{\omega_{n}}|\Gamma(t_{n})}=\exp(i\phi_{n}^{\rho})\exp(i\delta_{n}^{\rho})\,.$
(2.22)

The universal denominator is a generalization of the five-point amplitude
(2.14), and it subsumes all dependence on the absolute values of the gluon
production vertices $\Gamma_{k,k+1}$ and Reggeon propagators
$s_{k}^{\omega_{k}}$. The region-dependent phase $\exp(i\phi_{n}^{\rho})$
absorbs the remaining infrared divergences. The finite, conformally invariant
piece $\exp(i\delta_{n}^{\rho})$ combines in a non-trivial way with the
remainder function to a region-dependent linear combination of reduced pole
and cut terms [27, 28]:

$\exp(i\delta_{n}^{\rho})R_{n}^{\rho}=\text{(reduced pole terms)}^{\rho}\\\
+\sum_{j}c_{n,1,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt2r}}+\sum_{j}c_{n,2,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt22r}}+\dots{\\\
}+\sum_{j}d_{n,1,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt232r}}+\sum_{j<k}e_{n,j,k}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt212r}}+\dots$
(2.23)

Here, the grayed-out parts of the cut diagrams have been divided out, and the
(black) cut pieces stand for the remainder after the division.

#### Factorized Cut Integrals.

All reduced cut terms in (2.23) are infrared-finite, conformally invariant
functions of the complex anharmonic ratios $w_{k}$ (2.7). Just like the pole
terms of the four-point and five-point amplitudes, they enjoy the virtue of
Regge factorization, in the following sense: The multi-Reggeon bound states
that propagate in the intermediate $t$-channels are governed by the BFKL [1,
2, 3] and BKP [4, 5] equations. The solutions to these equations are most
naturally expressed in terms of their $\mathrm{SL}(2,\mathbb{C})$
representation labels $(n,\nu)$. Expressing all quantities in terms of these
variables, the cut contribution factorizes into a simple product: Reading a
cut diagram from left to right, each $t$-channel $m$-Reggeon state contributes
one BFKL (or BKP) _Green’s function_ $G_{m}(n_{k},\nu_{k})$, each gluon
emission that increments or decrements the number of exchanged reggeons from
$m$ to $m\pm 1$ contributes an _impact factor_ $\Phi_{m,m\pm
1}(n_{k-1},\nu_{k-1},n_{k},\nu_{k})$, and each intermediate gluon $k$ that
gets emitted from an $m$-Reggeon bound state contributes a _central emission
block_ $C_{m}(n_{k-1},\nu_{k-1},n_{k},\nu_{k})$. Obtaining the full cut
contribution requires completing the state sums in all $t$-channels by summing
and integrating over all $n_{k}$ and $\nu_{k}$. The summation and integration
amounts to a Fourier–Mellin transform from the $(n_{k},\nu_{k})$ variables to
the complex anharmonic ratios $w_{k}$ that provide the kinematic dependence.

The subsequent analysis will focus on the cuts of the type shown in the middle
line of (2.23). For those terms, only the simplest impact factors [10]

$\displaystyle\Phi_{\mathrm{L},k}$
$\displaystyle\equiv\Phi_{0,1}(n_{k},\nu_{k})=\frac{1}{2}\frac{(-1)^{n}}{i\nu_{k}+n_{k}/2}\left(\frac{q_{k-1}}{p_{k-1}}\right)^{-i\nu_{k}-n_{k}/2}\left(\frac{\bar{q}_{k-1}}{\bar{p}_{k-1}}\right)^{-i\nu_{k}+n_{k}/2}+\mathcal{O}(g)\,,$
$\displaystyle\Phi_{\mathrm{R},k}$
$\displaystyle\equiv\Phi_{1,0}(n_{k},\nu_{k})=-\frac{1}{2}\frac{1}{i\nu_{k}-n_{k}/2}\left(\frac{q_{k+1}}{p_{k}}\right)^{i\nu_{k}+n_{k}/2}\left(\frac{\bar{q}_{k+1}}{\bar{p}_{k}}\right)^{i\nu_{k}-n_{k}/2}+\mathcal{O}(g)$
(2.24)

and emission blocks [26]

$C_{k}\equiv
C_{1}(n_{k},\nu_{k},n_{k+1},\nu_{k+1})=-\frac{1}{2}\left(\frac{q_{k+1}}{p_{k}}\right)^{i\nu_{k}+n_{k}/2}\left(\frac{\bar{q}_{k+1}}{\bar{p}_{k}}\right)^{i\nu_{k}-n_{k}/2}\cdot\\\
\cdot\left(\frac{q_{k}}{p_{k}}\right)^{-i\nu_{k+1}-n_{k+1}/2}\left(\frac{\bar{q}_{k}}{\bar{p}_{k}}\right)^{-i\nu_{k+1}+n_{k+1}/2}\tilde{C}(n_{k},\nu_{k},n_{k+1},\nu_{k+1})+\mathcal{O}(g)$
(2.25)

are needed. The required Green’s function stems from the BFKL color-octet
channel and takes the form [10]

$G_{k}\equiv G_{2}(n_{k},\nu_{k})=\varepsilon_{k}^{gE_{n_{k},\nu_{k}}}\,.$
(2.26)

Here, $\varepsilon_{k}\equiv-\sqrt{u_{k,2}u_{k,3}}$ are combinations of
“small” cross ratios (2.10) that approach zero in the multi-Regge limit,
$E_{n,\nu}$ is the BFKL color-octet eigenvalue, and

$g\equiv\frac{g_{\text{YM}}^{2}N_{\text{c}}}{8\pi^{2}}$ (2.27)

is the planar coupling constant. The general two-Reggeon cut term $f_{k}$
spanning $k$ $t$-channels therefore takes the form [26, 30]555The cut
contribution is normalized such that the cut coefficient
$c_{6,1,4}^{\rho=({-}{-})}$ of the six-point remainder function becomes unity.
This choice differs from the normalization used in [28] by a factor of $2i$.

$f_{k}(\varepsilon_{5},\dots,\varepsilon_{k+4};w_{5},\dots,w_{k+4})\equiv\parbox[c]{0.0pt}{\includegraphics{Fignptcut}}=\\\
i\,g\sum_{n_{5},\dots,n_{k+4}}\int\mathrm{d}\nu_{5}\dots\mathrm{d}\nu_{k+4}\,\Phi_{\mathrm{L},5}\,\varepsilon_{5}^{gE_{n_{5},\nu_{5}}}\,C_{5}\,\varepsilon_{6}^{gE_{n_{6},\nu_{6}}}\,C_{6}\dots\,C_{k+3}\,\varepsilon_{k+4}^{gE_{n_{k+4},\nu_{k+4}}}\,\Phi_{\mathrm{R},k+4}\,.$
(2.28)

One can see that the exponentials of kinematic variables in the impact factors
and emission blocks indeed combine into Fourier–Mellin integral transformation
kernels

$w_{k}^{i\nu_{k}+n_{k}/2}\,\bar{w}_{k}^{i\nu_{k}-n_{k}/2}=\rho_{k}^{2i\nu_{k}}\,e^{in_{k}\varphi_{k}}\qquad\text{for}\qquad
w_{k}=\rho_{k}e^{i\varphi_{k}}\,.$ (2.29)

#### Perturbative Expansion.

The expression (2.28) is valid to all orders in the coupling $g$, where all
coupling dependence is contained in the impact factors $\Phi_{\text{L,R}}$,
the emission blocks $C_{k}$, and the BFKL eigenvalues $E_{n_{k},\nu_{k}}$.
Upon a perturbative expansion, the BFKL Green’s functions (2.26) expand in
powers of $g$ and of $\log(\varepsilon_{k})$; the latter are the large
logarithms that are characteristic of the multi-Regge limit. Including
subleading terms of the BFKL eigenvalues, impact factors, and emission blocks,
the cut contribution (2.28) at each order $g^{\ell}$ in the coupling constant
becomes a polynomial of degree $(\ell-1)$ in the large logarithms
$\log\varepsilon_{k}$. Retaining only the leading terms in large logarithms
amounts to the leading logarithmic approximation (LLA), the first subleading
terms constitute the next-to-leading logarithmic approximation (NLLA), and so
on. At order $g^{\ell}$, there are LLA terms of order
$\log(\varepsilon_{k})^{\ell-1}$ all the way to Nℓ-1LLA terms of order
$\log(\varepsilon_{k})^{0}$. At a given loop order, the coefficient of each
monomial in $\log(\varepsilon_{k})$ is a function of the kinematics that
exclusively depends on the complex anharmonic ratios $w_{k}$ (2.7).

##  3 Symbols and Regions

#### Transcendentality and Symbols.

Scattering amplitudes in planar $\mathcal{N}=4$ super Yang–Mills theory
display the property of uniform (or maximal) transcendentality, which means
that every term in the $\ell$-loop amplitude has the same transcendentality
(or transcendental weight) $2\ell$. This concept relies on the assumption that
the amplitude can be expanded in products of multiple polylogarithms (iterated
integrals over $\operatorname{dlog}$ integrands, MPLs for short) [14], $\pi$,
and zeta values.666It is expected that this class of functions is not
sufficient to describe all amplitudes to all orders in general kinematics. For
example, elliptic integrals appear in the ten-point N3MHV amplitude [42].
However, based on the singularity structure of the integrand, it is safe to
assume that all MHV amplitudes can be expressed as rational polynomials of
multiple polylogarithms and zeta values [43, 44], and multi-Regge limit MHV
amplitudes inherit this property. Every $m$-fold iterated integral is assigned
transcendentality $m$. Zeta values can be defined as MPLs evaluated on certain
values, and they inherit the transcendentality of their parent functions. For
example, the polylogarithms $\operatorname{Li}_{m}(x)$ as well as the zeta
values $\zeta_{m}$ have transcendentality $m$, and $\pi$ has transcendental
weight $1$. Under multiplication, transcendentality behaves additively.

Multiple polylogarithms obey many functional identities, which makes them
unwieldy, especially in expressions with many terms. All such functional
relations trivialize when one projects all MPLs to their _symbols_ [45].777For
reviews, see [46, 47]. The latter discard all information contained in the
choice of integration base point. In particular, the symbols are agnostic of
all ambiguities lying in the choice of functional branch. Since all branch
ambiguities of MPLs have subleading _functional_ transcendentality
(transcendentality of functional origin, as opposed to numerical
transcendentality), one typically discards _all_ terms of subleading
functional transcendentality when mapping an expression to its symbol.

When projecting the amplitude to its symbol, the expression (2.23) simplifies
considerably: The reduced pole terms consist of trigonometric functions whose
arguments include factors of $\pi$ [27, 28], hence their perturbative
expansion contains extra powers of $\pi$, which implies that they carry
subleading functional transcendental weight; they therefore get discarded. Cut
terms that involve more than two Reggeons stem from double (or higher)
discontinuities, hence they also have subleading transcendentality and get
projected out. Terms with multiple disconnected multi-Reggeon states (such as
the last term in (2.23)) are products of lower-loop cut terms, hence also
these have subleading transcendental weight and get discarded. On the left
hand side of the equation, the factor $\exp(i\delta_{n}^{\rho})$ can be
truncated to $1$, since all higher terms again include additional factors of
$\pi$. In summary, at the level of the symbol, the remainder function is a
linear combination of two-Reggeon cut terms:

$R_{n}^{\rho}\simeq\sum_{j}c_{n,1,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt2r}}+\sum_{j}c_{n,2,j}^{\rho}\parbox[c]{0.0pt}{\includegraphics{Fignpt22r}}+\dots$
(3.1)

Here, “$\simeq$” denotes equality at the symbol level.888Strictly speaking,
the symbol vanishes, since the right-hand side contains an overall factor of
$2\pi i$. What is meant by “$\simeq$” is that the symbols on both sides agree
after pulling out the overall $2\pi i$ factor. Moreover, here and in the
following, the remaining (black) cut pieces are understood to be one-loop
subtracted, as the one-loop part is (by definition) contained in the BDS
factor that has been divided out. The dots stand for further two-Reggeon cut
terms that span any number of adjacent emitted gluons.

#### Symbols and Regions.

At the symbol level, the discontinuity of an iterated integral along a closed
continuation path only depends on the overall winding numbers of the path
around the singular points of the integrand. From this property alone, it
follows [31] that the symbols $\operatorname{S}[\cdot]$ of the multi-Regge-
limit remainder function in the various kinematic regions obey the relations

$\operatorname{S}[R_{n}^{I}]=\sum_{\\{k,l\\}\subset
I}\operatorname{S}[R_{n}^{\\{k,l\\}}]\,.$ (3.2)

and

$\operatorname{S}[R_{n}^{\\{k,l\\}}]=\operatorname{S}[R_{n}^{[k,l]}]-\operatorname{S}[R_{n}^{[k,l-1]}]-\operatorname{S}[R_{n}^{[k+1,l]}]+\operatorname{S}[R_{n}^{[k+1,l-1]}]\,.$
(3.3)

These relations hold independently of the loop order. The first relation
states that the symbol in any region $I\subset\\{4,\dots,n-1\\}$ is a sum of
symbols in regions $\\{k,l\\}$ where only two momenta $\boldsymbol{p}_{k}$ and
$\boldsymbol{p}_{l}$ are flipped. The second relation in turn expresses the
symbol in those two-flip regions as a linear combination of symbols in regions
where all flipped momenta $k,\dots,l$ are adjacent, labeled by $[k,l]$. It is
therefore sufficient to consider the symbol in those all-adjacent regions.

Note that, since the cut terms can be assumed to be functionally independent,
the relations (3.2,3.3) among symbols imply identical relations for the cut
prefactors $c_{n,b,j}^{\rho}$ in the various regions:

$c_{n,b,j}^{I}=\sum_{\\{k,l\\}\subset I}c_{n,b,j}^{\\{k,l\\}}\,,\qquad
c_{n,b,j}^{\\{k,l\\}}=c_{n,b,j}^{[k,l]}-c_{n,b,j}^{[k,l-1]}-c_{n,b,j}^{[k+1,l]}+c_{n,b,j}^{[k+1,l-1]}\,.$
(3.4)

It is not difficult to see that these relations are consistent with the
Mandelstam criterion (2.19) described above. They completely determine the
coefficients of all two-Reggeon cut contributions of the type shown in (3.1)
to the $n$-point remainder function in any kinematic region $\rho$ in terms of
the coefficients $c_{n,b,j}^{[k,l]}$ of these cut terms in the all-adjacent
regions $\rho=[k,l]$.

In fact, the Mandelstam criterion (2.19) implies that there is only a single
two-Reggeon cut contribution to the $n$-point multi-Regge limit remainder
function in any all-adjacent region $[k,l]$, namely

$R_{n,\mathrm{cut}}^{[k,l]}=c_{n,l-k,k}^{[k,l]}\parbox[c]{0.0pt}{\includegraphics{Fignpt2anyr}}\,,$
(3.5)

where the subscript “cut” indicates that the Regge pole terms are not
included, and the dots stand for the omission of $(l-k-2)$ emission blocks. In
other words,

$c_{n,b,j}^{[k,l]}=0\qquad\text{unless}\qquad j=k\,,\quad\text{and}\quad
b=l-k\,.$ (3.6)

In particular, the cut terms in all such regions equal (up to variable
substitution and the prefactors) the cut terms of the $(l-k+5)$-point
remainder function in the region where _all_ intermediate momenta are flipped:

$R_{n,\mathrm{cut}}^{[k,l]}(\varepsilon_{k+1},\dots,\varepsilon_{l};w_{k+1},\dots,w_{l})=\frac{c_{n,l-k,k}^{[k,l]}}{c_{n^{\prime},n^{\prime}-4,4}^{[4,n^{\prime}-1]}}R_{n^{\prime},\mathrm{cut}}^{[4,n^{\prime}-1]}(\varepsilon_{k+1},\dots,\varepsilon_{l};w_{k+1},\dots,w_{l})\,,$
(3.7)

with $n^{\prime}=l-k+5$. Since the symbol of the remainder funtion is agnostic
of the pole terms, the equations (3.5) and (3.7) hold for the full remainder
function at the symbol level.

##  4 Two-Loop Expansion

We now want to analyze the two-Reggeon contribution (3.5) for any number of
gluons at the perturbative level. The following deconstruction is not
restricted to symbols, but holds at the level of full functions. By
definition, all cut diagrams of the type (3.5) are understood to be one-loop
subtracted. The simplest case involves only two emitted gluons. Perturbatively
expanding the BFKL Green’s function and the impact factors, this simplest
diagram consists of three terms at the two-loop level:

$f_{1}(\varepsilon_{5};w_{5})=\parbox[c]{0.0pt}{\includegraphics{Fig6ptcut}}=\parbox[c]{0.0pt}{\includegraphics{Fig6ptp010}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp100}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp001}}+\mathcal{O}(g^{3})\,.$
(4.1)

Here, a naked line for the impact factor stands for its leading contribution
(2.24), whereas additional dots denote loop corrections. A vertical line in
the $t$-channel two-Reggeon state stands for the one-loop (order $g^{1}$)
piece of the BFKL Green’s function (2.26),

$G_{2}(n_{k},\nu_{k})=\varepsilon_{k}^{gE_{n_{k},\nu_{k}}}=1+g\,E_{n_{k},\nu_{k}}^{(0)}\log(\varepsilon_{k})+\mathcal{O}(g^{2})\,,$
(4.2)

where

$E_{n_{k},\nu_{k}}=\sum_{\ell=0}^{\infty}g^{\ell}E_{n_{k},\nu_{k}}^{(\ell)}$
(4.3)

is the expansion of the BFKL eigenvalue. Due to the factor
$\log(\varepsilon_{k})$ in the one-loop Green’s function, the first term in
(4.1) provides the leading logarithmic approximation (LLA) at this two-loop
order. The subleading NLLA contribution consists of the second and third
diagrams, which have no line insertions, and stem from the trivial piece
$G_{2}(g=0)=1$ of the Green’s function.

Turning to the longer two-Reggeon cut that appears in the $({-}{-}{-})$ region
of the seven-point remainder function, the two-loop expansion yields five
terms,

$f_{2}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})=\parbox[c]{0.0pt}{\includegraphics{Fig7ptcut}}=\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01000}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00010}}\\\
+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10000}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00100}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00001}}+\mathcal{O}(g^{3})\,.$
(4.4)

Here, the emission block makes its first appearance. A plain dotted line
stands for the leading-order emission block (2.25), and additional dots again
denote loop corrections. The LLA piece now consists of two terms, where either
of the Green’s functions in the first or second $t$-channel have been expanded
to one-loop order. Hence the first term is proportional to
$\log(\varepsilon_{5})$, while the second term is proportional to
$\log(\varepsilon_{6})$.999Without loss of generality, it is assumed that the
leftmost particle at the beginning of the cut is particle $4$. The second line
provides the three NLLA terms.

A key fact for the subsequent analysis is the following observation [26]: Any
number of adjacent leading-order emission blocks, not separated by BFKL
eigenvalue insertions, can be absorbed in a neighboring leading-order impact
factor (again not separated by BFKL eigenvalue insertions). The result is the
original impact factor, whose momentum gets replaced by the sum of combined
momenta. Similarly, any number of adjacent leading-order emission blocks can
be combined into a multi-gluon emission block, whose functional form is
identical to the single-gluon block, but whose outgoing momentum is replaced
by the sum of all combined momenta. Diagrammatically, we will denote these
identities as

$\parbox[c]{0.0pt}{\includegraphics{Figifpvid1}}\equiv\parbox[c]{0.0pt}{\includegraphics{Figifpvid2}}\,,\qquad\text{and}\qquad\parbox[c]{0.0pt}{\includegraphics{Figpvid1}}\equiv\parbox[c]{0.0pt}{\includegraphics{Figpvid2}}\,.$
(4.5)

Here, the dots stand for the insertion of any number of leading-order emission
blocks. The identity for impact factors (left) was demonstrated in [26], and
the identity for emission blocks (right) follows straightforwardly. For
completeness, the identities are derived in Appendix A. Using these
identities, one can reduce almost all diagrams in (4.4) to six-point diagrams.
For example,

$\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01000}}=\parbox[c]{0.0pt}{\includegraphics{Fig6ptp010l12}}\,,\qquad\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00010}}=\parbox[c]{0.0pt}{\includegraphics{Fig6ptp010l21}}\,.$
(4.6)

Each term in the two-loop expression (4.4) _a priori_ depends on both complex
anharmonic ratios $w_{5}$ and $w_{6}$ (2.7). But due to the identity (4.6), it
is clear that all dependence of the first term in (4.4) on $w_{5}$ and $w_{6}$
factors into a dependence on the single complex ratio

$v_{5,6;5}\equiv\frac{p_{4}q_{7}}{q_{4}(p_{5}+p_{6})}=\frac{w_{5}}{\bigl{(}1+\frac{1}{w_{6}}\bigr{)}}\,.$
(4.7)

Similarly, the second term in (4.4) only depends on the single complex ratio

$v_{5,6;6}\equiv\frac{(p_{4}+p_{5})q_{7}}{q_{4}p_{6}}=(1+w_{5})w_{6}\,.$ (4.8)

Restricting to the LLA (the first line in (4.4)), and using the identities
(4.6), the three-particle cut therefore reduces to a sum of two copies of the
two-particle cut,

$f_{2,(2)}^{\text{LLA}}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})=f_{1,(2)}^{\text{LLA}}(\varepsilon_{5};v_{5,6;5})+f_{1,(2)}^{\text{LLA}}(\varepsilon_{6};v_{5,6;6})\,.$
(4.9)

Promoting this equation to the full two-loop cut contribution (including the
NLLA piece) requires adding an extra NLLA term to the equation:

$f_{2,(2)}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})=f_{1,(2)}(\varepsilon_{5};v_{5,6;5})+f_{1,(2)}(\varepsilon_{6};v_{5,6;6})+g_{2}(v_{5,6;5},v_{5,6;6})\,,$
(4.10)

where

$g_{2}(v_{5,6;5},v_{5,6;6})=\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00100}}-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp100l21}}-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp001l12}}$
(4.11)

is a finite function of $v_{5,6;5}$ and $v_{5,6;6}$ (or of $w_{4}$ and $w_{5}$
via the relations (4.7,4.8)). Here, the last two terms appear in the two
$f_{1,(2)}$ terms but not in $f_{2,(2)}$ and thus need to be subtracted. They
are, by analogy with (4.5), defined by evaluating the one-loop impact factors
on the sums of momenta $p_{4}+p_{5}$ and $p_{5}+p_{6}$, respectively.

This two-loop analysis straightforwardly generalizes to the cut contribution
for any number of particles. Using the identities (4.5), the LLA part of the
general cut at two loops can be written as

$f_{k,(2)}^{\text{LLA}}(\varepsilon_{5},\dots,\varepsilon_{4+k};w_{5},\dots,w_{4+k})=\sum_{j=5}^{4+k}\,\parbox[c]{0.0pt}{\includegraphics{Fig6ptp010lgen}}=\sum_{j=5}^{4+k}f_{1,(2)}^{\text{LLA}}(\varepsilon_{j};v_{5,4+k;j})\,.$
(4.12)

Here, the variables

$v_{k,l;j}\equiv\frac{q_{k-1}-q_{j}}{q_{k-1}}\frac{q_{l+1}}{q_{j}-q_{l+1}}=\frac{(1+(1+(\dots(1+w_{k})w_{k+1})\dots)w_{j-1})w_{j}}{1+\left(1+\left(\dots\left(1+\frac{1}{w_{l}}\right)\frac{1}{w_{l-1}}\right)\dots\right)\frac{1}{w_{j+1}}}$
(4.13)

for $j=k,\dots,l$ are anharmonic ratios that generalize (4.7,4.8); they are
obtained by grouping the adjacent momenta
$p_{k-1}+\dots+p_{j-1}=q_{k-1}-q_{j}$ and $p_{j}+\dots+p_{l}=q_{j}-q_{l+1}$.
The inversion of this formula is

$w_{j}=\frac{(v_{k,l;j-1}-v_{k,l;j})(1+v_{k,l,j+1})}{(1+v_{k,l;j-1})(v_{k,l;j}-v_{k,l;j+1})}\,,$
(4.14)

assuming the boundary conditions $v_{k,l;k-1}=0$ and $v_{k,l;l+1}=\infty$.
Including the NLLA terms of $f_{k,(2)}$ and $f_{1,(2)}$ on both sides of
equation (4.12), and again applying the reduction identities (4.5), one can
see that all subleading terms combine into a sum of seven-point NLLA pieces
$g_{2}$ (4.11), evaluated with different complex ratios:

$f_{k,(2)}(\varepsilon_{5},\dots,\varepsilon_{4+k};w_{5},\dots,w_{4+k})=\sum_{j=5}^{4+k}f_{1,(2)}(\varepsilon_{j};v_{j})+\sum_{j=5}^{3+k}g_{2}(v_{j},v_{j+1})\,,\qquad
v_{j}\equiv v_{5,4+k;j}\,.$ (4.15)

This concludes the two-loop analysis of the general two-Reggeon cut (3.5). For
any number of emitted particles, the latter can be deconstructed into a sum of
two building blocks, one of them being the simplest two-particle cut $f_{1}$,
the other being the NLLA remainder $g_{2}$ of the three-particle cut $f_{2}$.

Using equation (3.5), the result (4.15) directly implies an analogous relation
for the cut piece of the two-loop remainder function in the region
$({-}{-}\dots-)$ where all momenta have been flipped,

$R_{n,(2),\mathrm{cut}}^{\mathrm{MRL},({-}{-}\dots-)}(\varepsilon_{5},\dots,\varepsilon_{n-1};w_{5},\dots,w_{n-1})=\frac{c_{n}}{c_{6}}\sum_{j=5}^{n-1}R_{6,(2),\mathrm{cut}}^{\mathrm{MRL},({-}{-})}(\varepsilon_{j};v_{j})+c_{n}\sum_{j=5}^{n-2}g_{2}(v_{j},v_{j+1})\,,$
(4.16)

with the abbreviations $c_{n}\equiv c_{n,n-5,4}^{[4,n-1]}$ and $v_{j}\equiv
v_{5,n-1;j}$. With the help of (3.7), very similar relations hold for the
remainder function _symbol_ in any region $[k,l]$ where any number of adjacent
momenta have been flipped.

#### Relation to Previous Work.

At leading logarithmic order, the relation (4.15) together with the variable
map (4.13) has been obtained before [26]. Here, we have generalized it to the
full two-loop level, including the NLLA terms. In fact, an explicit study [31]
of the known two-loop symbol [32] has lead to the slightly stronger
observation

$R_{n,(2),\mathrm{cut}}^{\mathrm{MRL},({-}{-}\dots-)}(\varepsilon_{5},\dots,\varepsilon_{n-1};w_{5},\dots,w_{n-1})\simeq\sum_{j=5}^{n-1}R_{6,(2),\mathrm{cut}}^{\mathrm{MRL},({-}{-})}(\varepsilon_{j};v_{j})+c_{7}\sum_{j=5}^{n-2}g_{2}(v_{j},v_{j+1})\,.$
(4.17)

Also this result had been obtained previously at leading logarithmic order
[48]. Comparing (4.16) with (4.17), one finds that the coefficients of all
simple two-Reggeon cut contributions must be identical,101010Since [31]
analyzed the two-loop symbol for up to ten points, the equality has only been
rigorously established for $n\leq 10$.

$c_{n,l-k,k}^{[k,l]}=c_{6,1,4}^{[4,5]}=1\,,\qquad n\geq 7\,,\quad 4\leq
k<l<n\,.$ (4.18)

Here, the second equality follows from the deliberate choice of normalization
(2.28) for the cut integral.

##  5 Three-Loop Expansion

We are now in a position to extend the previous analysis to the three-loop
order. At three loops, the simplest cut contribution $f_{1}$ expands to

$f_{1,(3)}=\parbox[c]{0.0pt}{\includegraphics{Fig6ptp020}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp110}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp01l0}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp011}}\\\
+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp200}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp101}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp002}}\,.$
(5.1)

Compared to the two-loop case, there are a few new ingredients at three loops:
Two line insertions in the two-Reggeon state (as in the first term) stand for
terms where the BFKL Green’s function (2.26) has been expanded to second order
in the coupling $g$, while the BFKL eigenvalue $E_{n,\nu}$ has been kept at
leading order. A line insertion dressed with a dot stands for one power of the
one-loop correction to the eigenvalue $E_{n,\nu}$. Each line (leading order or
loop corrected) comes with one power of the respective large logarithm
$\log(\varepsilon_{k})$. In other words, expanding

$G_{2}(n_{k},\nu_{k})=\varepsilon_{k}^{gE_{n_{k},\nu_{k}}}=\\\
1+g\,E_{n_{k},\nu_{k}}^{(0)}\log(\varepsilon_{k})+{\textstyle\frac{1}{2}}g^{2}\bigl{(}E_{n_{k},\nu_{k}}^{(0)}\log(\varepsilon_{k})\bigr{)}^{2}+g^{2}E_{n_{k},\nu_{k}}^{(1)}\log(\varepsilon_{k})+\mathcal{O}(g^{3})\,,$
(5.2)

where $E_{n_{k},\nu_{k}}^{(\ell)}$ is the $\ell$-loop BFKL eigenvalue, the
third term in (5.2) produces the first term in (5.1), whereas the fourth term
in (5.2) produces the third term in (5.1). The first term in (5.1) constitutes
the LLA part, the next three terms provide the NLLA contribution, and the
three terms on the second line form the NNLLA piece.

Passing now to the longer cut $f_{2}$, one finds the following terms at three
loops and leading logarithmic order:

$f_{2,(3)}=\parbox[c]{0.0pt}{\includegraphics{Fig7ptp02000}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01010}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00020}}+\mathcal{O}(\mathrm{NLLA})\\\
=\parbox[c]{0.0pt}{\includegraphics{Fig6ptp020l12}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01010}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp020l21}}+\mathcal{O}(\mathrm{NLLA})\,.$
(5.3)

As shown in the second line, two of the LLA diagrams can again be reduced to
six-point diagrams, using (4.5). But, unlike in the two-loop case, one LLA
diagram remains that cannot be reproduced by six-point data. Removing the six-
point pieces by subtracting two instances of $f_{1,(3)}$ functions (5.1), one
finds the remainder (without loss of generality, the emitted gluons are
labeled by $\\{4,5,6\\}$)

$g_{3}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})\equiv
f_{2,(3)}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})-f_{1,(3)}(\varepsilon_{5};v_{5,6;5})-f_{1,(3)}(\varepsilon_{6};v_{5,6;6})\\\
=\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01010}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01100}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00110}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01001}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10010}}\\\
-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp110l21}}-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp011l12}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10001}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10100}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00101}}\\\
+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00200}}-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp101l21}}-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp101l12}}-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp200l21}}-\parbox[c]{0.0pt}{\includegraphics{Fig6ptp002l12}}\,.$
(5.4)

Note that all terms involving the next-to-leading-order BFKL eigenvalue are
captured by the short cut terms $f_{1(3)}$. It is now straightforward to see
that the general $k$-point cut diagram $f_{k,(3)}$, to leading logarithmic
order, becomes a sum of six-point functions $f_{1,(3)}$ and seven-point
functions $g_{3}$:111111Note that, contrary to the two-loop case (4.11), the
three-loop building block $g_{3}$ is defined in terms of the original cross
ratios $w_{5}$, $w_{6}$ rather than the combinations $v_{5,6;5}$, $v_{5,6;6}$.

$f_{k,(3)}^{\text{LLA}}(\varepsilon_{5},\dots,\varepsilon_{4+k};w_{5},\dots,w_{4+k})\\\
=\sum_{j=5}^{4+k}f_{1,(3)}^{\text{LLA}}(\varepsilon_{j};v_{5,4+k;j})+\sum_{i=5}^{3+k}\sum_{j=i+1}^{4+k}g_{3}^{\text{LLA}}(\varepsilon_{i},\varepsilon_{j};v_{5,j-1;i},v_{i+1,4+k;j})\,.$
(5.5)

Including all NLLA and NNLLA diagrams in the functions $f_{k,(3)}$,
$f_{1,(3)}$, and $g_{3}$ on both sides of the above equation, and judiciously
organizing all terms, one finds that the subleading contributions can be
combined into two further NLLA building blocks $g_{\mathrm{L}}$,
$g_{\mathrm{R}}$, and one further NNLLA building block $h$. The full three-
loop cut function $f_{k,(3)}$ can be written as

$\\!f_{k,(3)}(\varepsilon_{5},\dots,\varepsilon_{k+4};w_{5},\dots,w_{k+4})=\sum_{j=5}^{k+4}f_{1,(3)}(\varepsilon_{j};v_{5,4+k;j})+\sum_{i=5}^{k+3}\sum_{j=i+1}^{k+4}g_{3}(\varepsilon_{i},\varepsilon_{j};v_{5,j-1;i},v_{i+1,k+4;j})\\\
+\sum_{i=5}^{k+2}\sum_{j=i+1}^{k+3}g_{\mathrm{L}}(\varepsilon_{i};v_{5,j-1;i},v_{i+1,j;j},v_{j+1,k+4;j+1})+\sum_{i=5}^{k+2}\sum_{j=i+2}^{k+4}g_{\mathrm{R}}(\varepsilon_{j};v_{5,i;i},v_{i+1,j-1;i+1},v_{i+2,k+4;j})\\\
+\sum_{i=5}^{k+1}\sum_{j=i+2}^{k+3}h(v_{5,i;i},v_{i+1,j-1;i+1},v_{i+2,j;j},v_{j+1,k+4;j+1})\,.$
(5.6)

The NLLA building block $g_{\text{L}}$ depends on four intermediate momenta.
It takes the form

$g_{\mathrm{L}}(\varepsilon_{5};w_{5},w_{6},w_{7})=\parbox[c]{0.0pt}{\includegraphics{Fig8ptp0100100}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01001l112}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp01100l121}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp011l13}}\,.$
(5.7)

In the third term of (5.6), this function gets summed over partitions of the
sequence of momenta $(p_{4},\dots,p_{k+4})$ into subsequences

$(p_{4},\dots,p_{i-1})\,,\quad(p_{i},\dots,p_{j-1})\,,\quad(p_{j})\,,\quad\text{and}\quad(p_{j+1},\dots,p_{k+4})\,.$
(5.8)

The building block $g_{\text{R}}$ is a mirror of $g_{\text{L}}$:

$g_{\mathrm{R}}(\varepsilon_{7};w_{5},w_{6},w_{7})=\parbox[c]{0.0pt}{\includegraphics{Fig8ptp0010010}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10010l211}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00110l121}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp110l31}}\,,$
(5.9)

and in the fourth term of (5.6), it gets summed over the partitions

$(p_{4},\dots,p_{i-1})\,,\quad(p_{i})\,,\quad(p_{i+1},\dots,p_{j-1})\,,\quad\text{and}\quad(p_{j},\dots,p_{k+4})\,.$
(5.10)

Finally, the N2LLA building block $h$ reads:

$h(w_{5},w_{6},w_{7},w_{8})=\parbox[c]{0.0pt}{\includegraphics{Fig9ptp001000100}}-\parbox[c]{0.0pt}{\includegraphics{Fig8ptp1000100l2111}}-\parbox[c]{0.0pt}{\includegraphics{Fig8ptp0010001l1112}}\\\
+\parbox[c]{0.0pt}{\includegraphics{Fig8ptp1000100l1211}}+\parbox[c]{0.0pt}{\includegraphics{Fig8ptp0010001l1121}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10001l221}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10001l122}}\\\
-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10100l131}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00101l131}}-\parbox[c]{0.0pt}{\includegraphics{Fig7ptp00200l131}}+\parbox[c]{0.0pt}{\includegraphics{Fig7ptp10001l212}}\\\
+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp200l41}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp101l41}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp101l14}}+\parbox[c]{0.0pt}{\includegraphics{Fig6ptp002l14}}\,.$
(5.11)

The last term in (5.6) sums this function over partitions of the intermediate
momenta into subsequences

$(p_{4},\dots,p_{i-1})\,,\quad(p_{i})\,,\quad(p_{i+1},\dots,p_{j-1})\,,\quad(p_{j})\,,\quad\text{and}\quad(p_{j+1},\dots,p_{k+4})\,.$
(5.12)

For the case $k=3$, which is relevant for the eight-point remainder function
in the $({-}{-}{-}{-})$ region, the last sum in (5.6) has to be replaced by
the single term $\tilde{h}(w_{5},w_{6},w_{7})$, where $\tilde{h}$ is obtained
from $h$ (5.11) by removing the middle particle (and the associated LO
emission block, if applicable).

Using (3.5), the deconstruction (5.6) implies an analogous relation for the
three-loop remainder function in the region $\rho=[4,n-1]=({-}{-}\dots-)$
where all intermediate momenta have been flipped:

$R_{n,(3),\mathrm{cut}}^{({-}{-}\dots-)}(\varepsilon_{4},\dots,\varepsilon_{n-2};w_{4},\dots,w_{n-2})=\sum_{j=5}^{n-1}R_{6,(3),\mathrm{cut}}^{({-}{-})}(\varepsilon_{j};v_{j})+\sum_{\begin{subarray}{c}i,j=5\\\
i<j\end{subarray}}^{n-1}g_{3}(\varepsilon_{i},\varepsilon_{j};v_{5,j-1;i},v_{i+1,n-1;j})\\\
+\sum_{i=5}^{n-3}\sum_{j=i+1}^{n-2}g_{\mathrm{L}}(\varepsilon_{i};v_{5,j-1;i},v_{i+1,j;j},v_{j+1,n-1;j+1})+\sum_{i=5}^{n-3}\sum_{j=i+2}^{n-1}g_{\mathrm{R}}(\varepsilon_{j};v_{5,i;i},v_{i+1,j-1;i+1},v_{i+2,n-1;j})\\\
+\sum_{i=5}^{n-4}\sum_{j=i+2}^{n-2}h(v_{5,i;i},v_{i+1,j-1;i+1},v_{i+2,j;j},v_{j+1,k+4;j+1})\,.$
(5.13)

Here, the identities (4.18) among the cut coefficients have already been taken
into account. Via (3.7), equivalent relations hold for the remainder function
in all regions $\rho=[k,l]$ where any number of adjacent momenta
$\\{k,\dots,l\\}$ has been flipped. In more general regions, the remainder
function receives contributions from further cut terms (of the type shown in
the last line of (2.23)). Passing to the remainder function _symbol_ , these
further cut terms drop out (due to their subleading functional
transcendentality). Thus, by (3.2,3.3), the deconstruction (5.13) implies a
decomposition of the remainder function symbol in _any_ kinematic region in
terms of the symbols of the five building blocks $f_{1,(3)}$, $g_{3}$,
$g_{L}$, $g_{R}$, and $h$.

##  6 Building Blocks

In principle, each term in the perturbative expansion of the Regge cut diagram
(3.5) can be computed from the integral representation (2.28), once the
expressions for the BFKL eigenvalue, impact factor, and central emission block
are known to the desired perturbative order. In the previous sections, we have
shown that, by judiciously organizing all terms in the expansion, the
$n$-point two-loop and three-loop cut contributions can be reconstructed from
a few basic building blocks that are functions of the anharmonic ratios
$w_{j}$. Once these building block functions are known, the Regge cut
contribution to the remainder function can be computed via the formulas
(4.16,5.13).

Here, we will content ourselves with treating the building block functions at
the level of the symbol. The symbol of the two-loop NLLA building block
$g_{2}(v_{1},v_{2})$ has been obtained [31] by taking the multi-Regge limit of
the known two-loop remainder function symbol [32] and using the decomposition
(4.17).

At three loops, both the six-point and seven-point remainder function symbols
are known [49, 24]. By its definition, this data is sufficient to extract the
symbol of the building block $g_{3}$ (5.4), which contains LLA, NLLA, and
NNLLA parts. Applying in turn the first line of the three-loop decomposition
(5.13), this admits a reconstruction of the $n$-point remainder function
symbol at leading logarithmic order.

We compute the multi-Regge limit symbol of the three-loop remainder function
in the same way as for the two-loop symbol. The procedure is detailed in [31],
here we only give a brief summary: Starting with the known six-point and
seven-point symbols for general kinematics, we expand all first entries in
terms of the cross ratios (2.10) via the symbol rule

$(xy)\otimes(z)=(x)\otimes(z)+(y)\otimes(z)\,.$ (6.1)

Next, we collect all terms with the same cross ratio $U_{k,l}$ in the first
entry, strip off the first entry, and multiply by $2\pi i$. The result is the
symbol of the discontinuity under continuation along the path $U_{k,l}\to
e^{2\pi i}U_{k,l}$. In order to obtain the multi-Regge limit symbol of each
discontinuity, we express the kinematic invariants in the symbol entries in
terms of the OPE variables

$\\{T_{j},S_{j},F_{j}\\}=\\{e^{-\tau_{j}},e^{\sigma_{j}},e^{i\phi_{j}}\\}\,,\qquad
j=5,\dots,n-1\,,$ (6.2)

of [18],121212Compared to [18], we cyclically shift the momentum twistors,
such that $Z_{i}^{\text{here}}=Z_{i+1}^{\text{BSV}}$. set
$S_{j}=1/(T_{j}r_{j})$, and take the limit $T_{j}\to 0$, keeping only the
leading term in each entry. For the six-point case,

$r_{5}^{2}={w_{5}\bar{w}_{5}}\,,\qquad F_{5}^{2}={w_{5}/\bar{w}_{5}}\,,\qquad
T_{5}^{2}={\varepsilon_{5}/r_{5}}\,,$ (6.3)

whereas for seven points,

$\displaystyle r_{5}^{2}={w_{6}\bar{w}_{6}}\,,\qquad
F_{5}^{2}={w_{6}/\bar{w}_{6}}\,,\qquad T_{5}^{2}={\varepsilon_{6}/r_{5}}\,,$
$\displaystyle r_{6}^{2}=1/{w_{5}\bar{w}_{5}}\,,\qquad
F_{6}^{2}={\bar{w}_{5}/w_{5}}\,,\qquad T_{6}^{2}={\varepsilon_{5}/r_{6}}\,.$
(6.4)

Finally, again expanding all terms via (6.1), one can extract all large
logarithms via the shuffle relations

$\log(\varepsilon_{j})\bigl{(}x\otimes y\otimes\dots\otimes
z\bigr{)}=\bigl{(}\varepsilon_{j}\otimes x\otimes y\otimes\dots\otimes
z\bigr{)}+\bigl{(}x\otimes\varepsilon_{j}\otimes y\otimes\dots\otimes
z\bigr{)}\\\ +\bigl{(}x\otimes y\otimes\varepsilon_{j}\otimes\dots\otimes
z\bigr{)}+\cdots+\bigl{(}x\otimes y\otimes\dots\otimes\varepsilon_{j}\otimes
z\bigr{)}+\bigl{(}x\otimes y\otimes\dots\otimes
z\otimes\varepsilon_{j}\bigr{)}\,.$ (6.5)

At seven points and three loops, the resulting expression for each
discontinuity is a degree-two polynomial in $\log(\varepsilon_{5})$ and
$\log(\varepsilon_{6})$, whose coefficients are symbols with five entries that
exclusively depend on $w_{5}$, $w_{6}$, and their complex conjugates. Starting
in the kinematic region $({+}{+}{+})$ in which no intermediate momentum is
flipped, each other kinematic region is associated with specific winding
numbers for all cross ratios $U_{k,l}$. Summing the corresponding
discontinuities then yields the remainder function symbol in the respective
kinematic region. In particular, the region $({-}{-}{-})$ that contains the
three-particle cut $f_{2}$, only the cross-ratio $U_{2,6}$ (2.8) winds non-
trivially. Applying the change of variables (4.7,4.8) and subtracting the
respective six-point three-loop symbols (5.4), one finally obtains the symbol
of the building block $g_{3}$.

The NLLA and NNLLA building blocks (5.7,5.9,5.11) first appear in the three-
loop eight-point and nine-point amplitudes, and can thus not (yet) be
extracted from available perturbative data. In principle these functions could
be computed term by term from the integral representation (2.28). While the
BFKL eigenvalue and impact factor are known explicitly to N2LLA and N3LLA [10,
50, 37, 20, 12], and relating the multi-Regge limit to the Wilson loop OPE
[51] led to all-order proposals [52], the missing ingredient is the NLO and
NNLO central emission block (2.25).

In principle, the NLO emission block could be extracted from the building
block $g_{2}$ by subtracting the two reducible terms and inverting the
Fourier–Mellin transform. This however requires knowledge of the full function
$g_{2}$, which at present is only known at leading transcendental weight [31].

The attached Mathematica file MRL3LLA.m contains the symbols for the building
blocks $R_{6,(3)}^{\mathrm{MRL}}$ and $g_{3}$, as well as a function that
reconstructs the three-loop leading-logarithmic-order remainder function
symbol in any kinematic region from these building blocks.

#### Note on the Alphabet.

The three-loop three-particle building block
$g_{3}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})$ has the same alphabet
$\aleph$ (letters appearing in the entries of the symbol) as the two-loop
three-particle building block $g_{2}(w_{5},w_{6})$:

$\aleph_{w}=\\{w_{5},1+w_{5},w_{6},1+w_{6},1+w_{6}+w_{5}w_{6}\\}\cup\\{\text{c.c.}\\}\,,$
(6.6)

where “c.c.” stands for the complex conjugate set of letters. Using the
expansion (5.13), and expanding all variables $v_{k,l;j}$ in terms of
$w_{5},\cdots,w_{n-1}$, the alphabet (of the terms in the first line) becomes
big and complicated for larger $n$. Had one started with the $n$-point symbol,
it would have been difficult to guess the variable transformation (4.13) that
simplifies the alphabet and symbol terms.

Beyond seven points, the full alphabet of the remainder function remains
unknown, even in the multi-Regge limit. At six and seven points, the alphabet
apparently does not change with the loop order, with the full alphabet already
visible at two loops. It appears likely that this pattern breaks at eight
points (beyond the leading logarithmic approximation), since this is the first
instance at which the three-loop building blocks involve more independent legs
than the two-loop building blocks. It would be interesting to work out the
consequences of the deconstruction (5.13) on the higher-point alphabets in
more detail.

##  7 The Function $g_{3}$

Clearly, it is desirable to obtain the building blocks of the multi-Regge
limit amplitude at function level. The function for the three-loop six-point
building block $f_{1,(3)}$ has been derived in [49]. Here, we focus on
constructing the function for the new three-loop seven-point building block
$g_{3}$, which, together with $f_{1,(3)}$, determines the three-particle cut
$f_{2,(3)}$ (5.6). While we will not be able to determine the function $g_{3}$
completely, we can severely constrain it using the knowledge of its symbol as
well as further constraints from symmetry and consistency with the collinear
limit.

#### Structure of the Function $g_{3}$.

MHV amplitudes in multi-Regge kinematics are rational polynomials in multiple
polylogarithms, $\pi$, and (multiple) zeta values, where all occuring
monomials have the same (uniform) transcendental weight [53]. At loop order
$\ell$, the remainder function has weight $2\ell$. The cut terms that make up
the remainder function in the multi-Regge limit are discontinuities of the
full remainder function and thus carry an overall factor $2\pi i$, which is
therefore multiplied by a function of uniform weight $(2\ell-1)$. Collecting
large logarithms, the function $g_{3}$ decomposes as follows:131313We call the
function $g_{3}(\varepsilon_{5},\varepsilon_{6};x,y)\equiv
g_{3}(\varepsilon_{5},\varepsilon_{6};w_{5}(x,y),w_{6}(x,y))$ by the same name
as $g_{3}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})$.

$g_{3}(\varepsilon_{5},\varepsilon_{6};x,y)=2\pi
i\sum_{m,n=0}^{1}\log(\varepsilon_{5})^{i}\log(\varepsilon_{6})^{j}\Bigl{(}g_{3}^{m,n}(x,y)+2\pi
i\,h_{3}^{m,n}(x,y)\Bigr{)}\,.$ (7.1)

At each order in the large logarithms $\log(\varepsilon_{j})$, we have split
the function into real parts $g_{3}^{m,n}$ and imaginary parts $h_{3}^{m,n}$,
each of which is a fixed-weight combination of multiple polylogarithms and
zeta values with real coefficients.

#### Ordinary Multiple Polylogarithms.

In order to construct the function $g_{3}$ by matching a general ansatz to its
symbol, we first need an irreducible basis of multiple polylogarithms of the
right class. Multiple polylogarithms, also called Goncharov polylogarithms
[14], can be defined recursively as iterated integrals

$G(a_{1},\ldots,a_{n};z)\equiv\begin{cases}\displaystyle\frac{1}{n!}\log^{n}z&\text{if
}a_{1}=\ldots=a_{n}=0\,,\\\\[8.61108pt]
\displaystyle\int_{0}^{z}\frac{dt}{t-a_{1}}G(a_{2},\ldots,a_{n};t)&\text{otherwise,}\end{cases}$
(7.2)

with $G(;z)=1$. The sequence of parameters $(a_{1},\dots,a_{n})$ is called the
weight vector, and the length of the weight vector equals the transcendental
weight (or transcendentality) of the function $G(a_{1},\dots,a_{n};z)$.
Multiple zeta values are defined in terms of multiple polylogarithms evaluated
at unity, and inherit their transcendental weight: $\zeta_{k}$ has weight $k$,
$\zeta_{j,k}$ has weight $j+k$, and so forth. $\pi$ has weight $1$.

As noted in [31], using the variables

$x=-v_{6}=-(1+w_{5})w_{6}\,,\qquad
y=-1/v_{5}=-\frac{1+w_{6}}{w_{5}\,w_{6}}\,,$ (7.3)

the alphabet (6.6) of the symbol of $g_{3}$ becomes

$\aleph_{xy}=\\{x,x-1,y,y-1,xy-1\\}\cup\\{\text{c.c.}\\}\,.$ (7.4)

Multiple polylogarithms whose symbols draw their entries from this alphabet
belong to the class of two-dimensional harmonic polylogarithms (2dHPLs) [54].
A generating set for these is given by141414The choice of generating set is
not unique. We used a different basis in [31], but found the choice (7.5) more
suitable for the present analysis.

$\bigl{\\{}G(\vec{a},x)\,|\,a_{i}\in\\{0,1\\}\bigr{\\}}\cup\bigl{\\{}G(\vec{a},1/y)\,|\,a_{i}\in\\{0,1,x\\}\bigr{\\}}\cup\\{\text{c.c.}\\}\,,$
(7.5)

where $\\{\text{c.c.}\\}$ stands for the complex conjugates of the previous
sets. Multiple polylogarithms satisfy shuffle and stuffle algebra relations,
hence the above generating set is overcomplete. An irreducible basis of
generators is provided by the subset whose weight vectors form Lyndon words in
the ordered sets of letters $\\{0,1\\}$ and $\\{0,1,x\\}$, respectively [55].
Including the complex conjugate generators, the resulting irreducible set
consists of $10$, $8$, $20$, $42$, and $108$ basis functions at weights $1$,
$2$, $3$, $4$, and $5$. Including all possible products of lower-weight
functions yields $10$, $63$, $320$, $1433$, and $5190$ linearly independent
terms at weights $1$, $2$, $3$, $4$, and $5$.

#### Single-Valuedness.

Besides the consistency with its known symbol, the function $g_{3}$ has to
satisfy various constraints. One of them is single-valuedness: Due to
unitarity, a physical amplitude can only have branch points where one of the
cross ratios vanishes (or becomes infinite). Since the cross ratios are
expressed in terms of absolute squares of the complex variables $w_{5}$ and
$w_{6}$ (2.12), a rotation $(w_{5}-z,\bar{w}_{5}-\bar{z})\to(e^{+2\pi
i}(w_{5}-z),e^{-2\pi i}(\bar{w}_{5}-\bar{z}))$ around any point $z$ in the
complex plane can never let a cross ratio wind around zero (or infinity). The
same is true for rotations of $w_{6}$, and therefore also for rotations of $x$
and $y$. The conclusion is that the coefficient functions (7.1) of $g_{3}$
must be single-valued functions of the complex variables $x$ and $y$. This
property has been essential for the determination of the six-point multi-Regge
limit to high loop orders [19, 56, 12, 57].

One could in principle implement the single-valuedness constraint by first
constructing a function using the basis (7.5) and then requiring all
monodromies to vanish. However, it turns out that the single-valuedness
constraint can be satisfied directly at the level of the basis: A suitable
basis of single-valued multiple polylogarithms was recently constructed for
any number of points [33].151515See also [23]. We can therefore satisfy the
single-valuedness constraint by employing the single-valued basis, without
losing generality.161616In the first revision of this paper, I had constructed
the function $g_{3}$ using the basis (7.5), which was the state of the art at
the time the preprint of this paper appeared on the arXiv. I thank the JHEP
referee for requesting a construction based on the single-valued basis that
was published at around the same time [33], and which significantly reduces
the number of free parameters that remain after applying all constraints. The
single-valued basis can be constructed purely algebraically from the basis of
ordinary multiple polylogarithms (7.5) using the Hopf algebra structure that
underlies the multiple polylogarithm algebra [14]: Each holomorphic element
$G$ of the ordinary basis (7.5) gets promoted to a single-valued function
$G^{\mathbf{s}}$ by the single-valued map

$\boldsymbol{s}:G(\vec{a},z)\mapsto
G^{\mathbf{s}}(\vec{a},z)\equiv(-1)^{|\vec{a}|}\mu(\bar{S}\otimes\operatorname{id})\Delta
G(\vec{a},z)\,,$ (7.6)

where $\Delta$ is the coproduct, $\operatorname{id}$ is the identity,
$\bar{S}$ is the complex conjugate of the antipode map of the Hopf algebra,
and $\mu$ denotes the multiplication operator $\mu(a\otimes b)=a\cdot b$. The
details are spelled out in Section 3.4.3 of [33], and we will not reproduce
them here. The antiholomorphic elements of (7.5) can equally be promoted to
single-valued functions, which however are not independent from the single-
valued functions generated from the holomorphic elements. A full basis of
single-valued 2dHPLs is therefore provided by the single-valued completions of
the holomorphic elements of the ordinary basis (7.5). Since this halves the
size of the algebra basis, it significantly reduces the number of linearly
independent elements in a general ansatz at any fixed weight. For example,
while a general (real) weight-five ansatz constructed from the ordinary basis
(7.5) as well as zeta values has $6305$ terms (and therefore as many
undetermined coefficients), the corresponding ansatz constructed from the
single-valued basis has only $756$ terms.

To summarize, the single-valued algebra basis that we will employ is

$\bigl{\\{}G^{\mathbf{s}}(\vec{a},x)|\vec{a}\in\operatorname{Lyn}\\{0,1\\}\bigr{\\}}\cup\bigl{\\{}G^{\mathbf{s}}(\vec{a},1/y)|\vec{a}\in\operatorname{Lyn}\\{0,1,x\\}\bigr{\\}}\,,$
(7.7)

where $\operatorname{Lyn}\\{0,1\\}$ and $\operatorname{Lyn}\\{0,1,x\\}$ denote
the sets of Lyndon words formed from the ordered sets of letters $\\{0,1\\}$
and $\\{0,1,x\\}$, respectively. Every single-valued function
$G^{\mathbf{s}}(\vec{a},z)$ is constructed from the ordinary multiple polylog
$G(\vec{a},z)$ according to the algebraic prescription (7.6). In addition, we
assume that $\\{\zeta_{2},\zeta_{3},\zeta_{2,3},\zeta_{5}\\}$ form the
algebraically independent set of (multiple) zeta values up to weight five.

#### The Ansatz and Symbol Constraints.

We start with a general polynomial in single-valued basis functions (7.7) and
zeta values, such that all monomials have identical total weight. Given that
the three-loop amplitude in general kinematics has weight six, and taking into
account the overall factor of $2\pi i$ as well as the large logarithms
$\log(\varepsilon_{5})$, $\log(\varepsilon_{6})$, one finds that the LLA real
part $g_{3}^{1,1}$ has to have weight three, the NLLA real parts $g_{3}^{1,0}$
and $g_{3}^{0,1}$ have weight four, and the NNLLA real part $g_{3}^{0,0}$ has
weight five. The corresponding imaginary parts have weight one less than the
real parts, due to the extra factor $2\pi i$. The sizes of the general ansätze
for all component functions are displayed in Table 1.

Function | $g_{3}^{1,1}$ | $h_{3}^{1,1}$ | $g_{3}^{1,0}$ | $h_{3}^{1,0}$ | $g_{3}^{0,1}$ | $h_{3}^{0,1}$ | $g_{3}^{0,0}$ |  $h_{3}^{0,0}$ | total
---|---|---|---|---|---|---|---|---|---
General ansatz | 71 | 20 | 236 | 71 | 236 | 71 | 756 | 236 | 1697
Match to symbol | 6 | 20 | 25 | 71 | 25 | 71 | 91 | 236 | 545
Parity invariance | 6 | 16 | 21 | 51 | 21 | 51 | 67 | 151 | 384
Target-projectile symmetry | 4 | 10 | 21 | 51 | 0 | 0 | ? | 83 | 236
Vanishing collinear limit I | 2 | 7 | 15 | 44 | 0 | 0 | 55 | 76 | 199
Consistency with the WLOPE I | 1 | 5 | 11 | 37 | 0 | 0 | 49 | 71 | 174
Vanishing collinear limit II | 1 | 5 | 11 | 37 | 0 | 0 | 45 | 65 | 164
Consistency with the WLOPE II | 1 | 5 | 11 | 37 | 0 | 0 | 42 | 59 | 155
Table 1: Numbers of free parameters in the components of the function $g_{3}$
before imposing constraints, after matching to the known symbols, and after
imposing various constraints. The first two components constitute the LLA
part, the next four functions represent the NLLA part, and the last two
functions form the NNLLA part of the function $g_{3}$.

The symbol of the function $g_{3}$ uniquely fixes all terms in the real parts
$g_{3}^{m,n}$ with the highest functional weight, that is all terms that are
free of zeta values. We can perform the match by expanding the single-valued
functions $G^{\mathbf{s}}(\vec{a},z)$ into combinations of ordinary multiple
polylogarithms $G(\cdot,\cdot)$, and by applying the symbol map

$\operatorname{S}[G(a_{1},\ldots,a_{n};z)]=\sum_{i=1}^{n}\Bigl{(}\operatorname{S}[G(a_{1},\ldots,\hat{a}_{i},\ldots,a_{n};z)]\otimes(a_{i}-a_{i-1})\\\
-\operatorname{S}[G(a_{1},\ldots,\hat{a}_{i},\ldots,a_{n};z)]\otimes(a_{i}-a_{i+1})\Bigr{)}\,,$
(7.8)

where $a_{0}=z$, $a_{n+1}=0$, and hatted indices are omitted. The match to the
symbol fixes the majority of terms in the real parts $g_{3}^{m,n}$ (see Table
1), but the symbol is insensitive to all terms of subleading functional
weight, including all terms in the imaginary parts $h_{3}^{m,n}$.

#### Parity Invariance and Target-Projectile Symmetry.

While the terms with subleading functional weight are not seen by the symbol,
they can be constrained by symmetry requirements. Firstly, MHV amplitudes are
invariant under parity (spatial reflection), which is realized by
$w_{i}\leftrightarrow\bar{w}_{i}$ in the multi-Regge limit [48], that is
$x\leftrightarrow\bar{x}$ and $y\leftrightarrow\bar{y}$. Secondly, the multi-
Regge limit amplitude should be invariant under target-projectile symmetry
(exchange of the two ingoing momenta), which amounts to symmetry under
$w_{5}\leftrightarrow 1/w_{6}$ [58], that is $x\leftrightarrow y$ and
$\bar{x}\leftrightarrow\bar{y}$. The sum of six-point terms that is subtracted
in the definition (5.4) of the function $g_{3}$ is separately invariant under
these transformations, and hence we can require parity as well as target-
projectile symmetry for the function $g_{3}$ by itself. These symmetries
significantly reduce the number of free parameters in the components of
$g_{3}$, as summarized in Table 1. In particular, target-projectile symmetry
also swaps $\varepsilon_{5}$ and $\varepsilon_{6}$, such that it fixes
$g_{3}^{0,1}$ and $h_{3}^{0,1}$ uniquely in terms of $g_{3}^{1,0}$ and
$h_{3}^{1,0}$.

Both parity and target projectile symmetry are not trivially implemented: The
parity map replaces all holomorphic weight vectors and arguments of our
single-valued basis functions $G^{\mathbf{s}}$ with their complex conjugates.
Again using the antipode, these conjugate single-valued functions can be re-
expressed in terms of single-valued functions with holomorphic arguments [33],
but those will not necessarily be elements of the basis (7.7). Similarly, the
target-projectile inversion map $x\leftrightarrow y$ produces non-basis
functions. In order to derive constraints for our ansatz coefficients, all
non-basis functions need to be re-expressed in terms of basis functions, which
is possible due to the many relations among multiple polylogarithms, such as
shuffle and stuffle algebra relations. The single-valued map [33] is an
algebra homomorphism, hence every identity among ordinary multiple
polylogarithms lifts to a corresponding identity among single-valued multiple
polylogarithms. In this way, single-valued multiple polylogarithms inherit the
shuffle and stuffle algebra relations from their ordinary counterparts, as
well as the simpler rescaling property

$G^{\mathbf{s}}(a_{1},\dots,a_{n};z)=G^{\mathbf{s}}(ca_{1},\dots,ca_{n};cz)\qquad\text{for
}a_{n}\neq 0\text{ and }c\neq 0\,.$ (7.9)

While one can relate non-basis functions back to basis functions by suitably
combining the right shuffle and stuffle identities, it is often more direct to
just match a non-basis function to a combination of basis functions using
numerics, at least up to the relatively low weight that we consider here. For
example, all multiple polylogarithms up to weight four can be expressed in
terms of classical polylogarithms $\operatorname{Li}_{m}(z)$ as well as
$\operatorname{Li}_{2,2}(z)$ using e.g. the Mathematica package provided by
[59]. Since classical polylogarithms can be readily evalueated numerically, it
is straightforward to match all non-basis functions up to weight four against
combinations of basis functions and zeta values. However, numerics beyond
weight four are not readily available, and thus implementing target-projectile
symmetry for $g_{3,\mathrm{R}}^{0,0}$ would require to compile all function
identities at weight five by algebraic means. We have not attempted to do so,
as it is rather laborious, and looking at Table 1, target-projectile symmetry
for $g_{3,\mathrm{R}}^{0,0}$ would yield around $\sim\\!15$ more constraints,
which would not get us significantly closer to determining the function
$g_{3}$ completely. Parity invariance is less demanding in that regard, as the
only functions that cannot be related back to basis functions by simple
shuffle algebra relations are harmonic polylogarithms of weight four or less.
In Appendix C, we list some of the relations among single-valued
polylogarithms that are needed to evaluate parity and target-projectile
symmetry, and we provide all further relations in an ancillary file.

#### Collinear Limit.

Another set of constraints comes from the expansion around the collinear
limit. Since the BDS amplitude correctly captures the leading behavior in the
collinear limit in the Mandelstam regions that we consider, the remainder
function has to vanish in this limit. In order to take the collinear limit, we
map our variables $(x,\bar{x})$ and $(y,\bar{y})$ back to $F_{5,6}$ and
$r_{5,6}$ via (7.3) and (6.4), which gives

$\displaystyle x$
$\displaystyle=-\frac{F_{5}r_{5}(1+F_{6}r_{6})}{F_{6}r_{6}}\,,$
$\displaystyle\qquad\bar{x}$
$\displaystyle=-{r_{5}(F_{6}+r_{6})}{F_{5}r_{6}}\,,$ (7.10) $\displaystyle y$
$\displaystyle=-\frac{F_{6}(1+F_{5}r_{5})r_{6}}{F_{5}r_{5}}\,,$
$\displaystyle\bar{y}$
$\displaystyle=-\frac{(F_{5}+r_{5})r_{6}}{F_{6}r_{5}}\,.$ (7.11)

While the Regge limit sits at $T_{j}\to 0$, $S_{j}\to\infty$ with
$r_{j}=1/(S_{j}T_{j})$ fixed, the collinear limit is defined by $T_{j}\to 0$
with $S_{j}$ finite. From the Regge limit, the combined Regge-collinear limit
is therefore attained by letting $r_{j}\to\infty$, that is

$x\approx-F_{5}r_{5}\to-\infty\,,\qquad y\approx-F_{6}r_{6}\to-\infty\,.$
(7.12)

In this limit, the harmonic polylogarithm part

$\bigl{\\{}G(\vec{a},x)\,|\,a_{i}\in\\{0,1\\}\bigr{\\}}\cup\\{\text{c.c.}\\}\,,$
(7.13)

of the basis (7.5) expands into logarithms and inverse powers of $x$ (and
$\bar{x}$). For the other part of the basis:

$\bigl{\\{}G(\vec{a},1/y)\,|\,a_{i}\in\\{0,1,x\\}\bigr{\\}}\cup\\{\text{c.c.}\\}\,,$
(7.14)

the expansion is even simpler, since the argument $1/y$ tends to zero, while
the weights $x$ tend to infinity. The basis functions expand to

$\displaystyle G(0,1/y)$ $\displaystyle=\log(1/y)\,,$ (7.15) $\displaystyle
G(0,\dots,0,1,1/y)$ $\displaystyle=-1/y+\mathcal{O}(1/y^{2})\,,$ (7.16)
$\displaystyle G(0,\dots,0,x,1/y)$
$\displaystyle=-1/xy+\mathcal{O}(1/y^{2})\,,$ (7.17)

and all functions $G(\dots,1/y)$ with more than two non-zero weights are of
order $\mathcal{O}(1/y^{2})$. After writing all single-valued functions
$G^{\mathbf{s}}$ in terms of ordinary multiple polylogarithms and applying the
above, we obtain the expansions of the ansätze near the collinear limit
(7.12). In doing so, one has to be careful in picking consistent branches for
all occurring logarithms. Every single-valued multiple polylogarithm expands
to a power series in $\log(r_{i})$ and $1/r_{i}$, where the series
coefficients are rational functions of $F_{5}$ and $F_{6}$ as well as zeta
values.

The first constraint comes from the fact that the remainder function should
vanish in the collinear limit, that is there should be no terms that are free
of $1/r_{i}$ factors. This already implies 37 further constraints on the
ansatz, as can be seen in the fifth line in Table 1. Moreover, we can require
consistency with the general form of the Wilson loop OPE that governs the
remainder function in the collinear limit [17, 18]. The general systematics of
the Wilson loop OPE predicts that the remainder function in the combined
Regge-collinear limit (at three loops and in any kinematic region) takes the
form

$\displaystyle R_{7}^{\mathrm{MRL-coll}}$
$\displaystyle=\frac{\cos(\phi_{5})}{r_{5}}f_{5}\bigl{(}\log(\varepsilon_{6}),\log(r_{5})\bigr{)}+\frac{\cos(\phi_{6})}{r_{6}}f_{6}\bigl{(}\log(\varepsilon_{5}),\log(r_{6})\bigr{)}$
$\displaystyle+\frac{\cos(\phi_{5}+\phi_{6})}{r_{5}r_{6}}h\bigl{(}\log(\varepsilon_{5}),\log(\varepsilon_{6}),\log(r_{5}),\log(r_{6})\bigr{)}$
$\displaystyle+\frac{\cos(\phi_{5}-\phi_{6})}{r_{5}r_{6}}\bar{h}\bigl{(}\log(\varepsilon_{5}),\log(\varepsilon_{6}),\log(r_{5}),\log(r_{6})\bigr{)}+\mathcal{O}(r_{5}^{-2})+\mathcal{O}(r_{6}^{-2})\,,$
(7.18)

where $F_{i}=e^{i\phi_{i}}$, and $f_{5}$, $f_{6}$, $h$, and $\bar{h}$ are
polynomials in the respective logarithms. In particular, the dependence on
$\phi_{5}$ and $\phi_{6}$ is very restricted.171717The form (7.18) is valid in
the Euclidean region as well as the $({+}{+}{+})$ region. During the analytic
continuation into the $({-}{-}{-})$ region, all cross ratios $U_{ij}$ follow
closed loops with identical start and end points. Moreover, in the
Basso–Sever–Vieira expressions for the cross ratios in general kinematics
[60], $\phi_{5}$ and $\phi_{6}$ only appear in the combinations
$\cos(\phi_{5})$, $\cos(\phi_{6})$, and $\cos(\phi_{5}+\phi_{6})$. The cosine
is an entire function, and hence the general form (7.18) is preserved under
the analytic continuation into the $({-}{-}{-})$ region. A general combination
of multiple polylogarithms would also produce sine functions of $\phi_{5}$,
$\phi_{6}$, and $\phi_{5}\pm\phi_{6}$. It turns out that our parity and
target-projectile symmetric ansatz is already free of such sine terms, which
is an important cross-check of our result. Moreover, terms where
$\cos(\phi_{5})$ multiplies $\log(\varepsilon_{5})$ or $\log(r_{6})$ should be
absent, and the same is true for products of $\cos(\phi_{6})$ with
$\log(\varepsilon_{6})$ or with $\log(r_{5})$.181818Note the flipping of the
indices $5$ and $6$ in (6.4). The absence of such terms provides yet more
constraints on the coefficients in our ansatz for $g_{3}$, as can be seen in
the sixth line in Table 1.

When considering the above constraints, one has to keep in mind that the
remainder function in the $({-}{-}{-})$ region consists of the function
$g_{3}$ as well as two copies of the six-point $({-}{-})$ region remainder
function (5.13). The six-point three-loop remainder function in multi-Regge
kinematics has been determined in [49, 20]. In principle, there could be
cross-terms between the six-point functions and the function $g_{3}$, such
that only their sum vanishes and satisfies (7.18) in the collinear limit.
However, we have checked that all coefficients (LLA, NLLA, and NNLLA, real and
imaginary parts) of the six-point function separately vanish and satisfy
(7.18) in the seven-point Regge-collinear limit, for both arguments $v_{5}=-x$
and $v_{6}=-y$. Hence also $g_{3}$ has to satisfy these constraints by itself.

In fact, the Regge-collinear limit is not unique: By cyclically rotating the
tessellation of the heptagon that defines the OPE variables (6.2) and taking
appropriate limits in the variables $S_{i}$, we can probe different limits in
the space of multi-Regge kinematics. Not all collinear limits have an overlap
with the multi-Regge limit: The requirement is that the vanishing of “small”
cross ratios $u_{j,2}$, $u_{j,3}$ is compatible with the collinear limit
$T_{5},T_{6}\to 0$. One further case where this is satisfied is the cyclic
rotation of the Basso–Sever–Vieira variables by $4$ sites, that is we use the
momentum twistors $Z^{\text{here}}_{i}=Z^{\text{BSV}}_{i+4}$, where
$Z^{\text{BSV}}_{i}$ are defined in Appendix A of [18].191919In the case
considered above, we used $Z_{i}=Z^{\text{BSV}}_{i+1}$. In this case, the
collinear-Regge limit is attained by setting $S_{5}=r_{5}T_{5}$,
$S_{6}=1/(r_{6}T_{6})$, and letting $T_{5},T_{6}\to 0$. The multi-Regge
parameters $w_{5}$, $w_{6}$ are then related to the OPE variables by

$r_{5}^{2}=\frac{1}{w_{5}\bar{w}_{5}}\,,\qquad
r_{6}^{2}=\frac{1}{w_{6}\bar{w}_{6}}\,,\qquad
F_{5}^{2}=\frac{w_{5}}{\bar{w}_{5}}\,,\qquad
F_{6}^{2}=\frac{w_{6}}{\bar{w}_{6}}\,,$ (7.19)

which implies

$x=-\frac{F_{6}(F_{5}+r_{5})}{r_{5}r_{6}}\,,\qquad
y=-\frac{r_{5}(F_{6}+r_{6})}{F_{5}F_{6}}\,.$ (7.20)

Conversely, the combined collinear-Regge limit is attained from the multi-
Regge limit by inverting (7.19) for $w_{5}$, $w_{6}$, and letting
$r_{5},r_{6}\to\infty$, which implies

$x\approx-\frac{F_{6}}{r_{6}}\to 0\,,\qquad
y\approx-\frac{r_{5}r_{6}}{F_{5}F_{6}}\to-\infty\,.$ (7.21)

In this case, the expansion of the basis functions (7.5) is even simpler,
since all arguments $x$, $\bar{x}$, $1/y$, and $1/\bar{y}$ tend to zero.
Expanding the ansätze for our component functions, we can again require _(i)_
vanishing of all components in the collinear limit, and _(ii)_ agreement with
the general form (7.18) of the Wilson loop OPE. These constraints further
reduce the ansätze by a few parameters, as shown in the last two lines of
Table 1. It turns out that this second collinear limit does not yield new
further constraints at LLA and NLLA. The NNLLA functions on the other hand do
get constrained further.

#### The Final Answer.

Putting all pieces together, one arrives at the most general combination of
multiple polylogarithms that is parity symmetric, target-projectile symmetric,
agrees with the symbol of $g_{3}$, vanishes in the collinear limit, and
matches the general form of the Wilson loop OPE in the collinear limit. The
resulting function is too bulky for display here, but is attached in the
Mathematica file g3fctn.m. It still contains $155$ undetermined coefficients,
as summarized in Table 1. The space of parameters could perhaps be further
reduced by matching subleading terms in the expansion in $1/r_{i}$ around the
collinear limit to the predictions from the Wilson loop OPE [18], or by
inspecting the double discontinuity of the symbol. We defer a more detailed
analysis of these further constraints to future work.

The functions $g_{3}^{1,1}$ and $h_{3}^{1,1}$ constitute the LLA part of the
function $g_{3}$, they solely stem from the first diagram in (5.4), and are
not affected by the subtraction of the six-point functions $f_{1,(3)}$. We
display the full LLA part of $g_{3}$ in Appendix D. The NLLA parts
$g_{3}^{1,0}$ and $h_{3}^{1,0}$ comprise the diagrams $2$, $4$, and $7$ in
(5.4), and the functions $g_{3}^{0,1}$, and $h_{3}^{0,1}$ consist of the
diagrams $3$, $5$, and $6$. Finally, $g_{3}^{0,0}$ and $h_{3}^{0,0}$
constitute the NNLLA part of the function $g_{3}$, and are composed of the
last eight diagrams in (5.4). Notably, the NLLA parts $g_{3}^{1,0}$ and
$h_{3}^{1,0}$ of transcendental weight four are expressed solely in terms of
products of lower-weight functions $G^{\mathbf{s}}$. That is, they are free of
weight-four functions $G^{\mathbf{s}}$ that cannot be expressed in terms of
lower-weight functions. This is a general feature of single-valued multiple
polylogarithms: Every real even-weight single-valued multiple polylogarithm is
expressible in terms of (products of) lower-weight functions. This property
follows from the defining map (7.6) together with the fact that complex
conjugation acts on single-valued functions through the antipode map.202020I
thank the JHEP referee for pointing out this fact.

##  8 Conclusion

#### Summary.

Exponentiation and factorization are core features of the Regge limit. In the
expansion around large logarithms, they admit a reconstruction of perturbative
amplitudes to any multiplicity, once the BFKL building blocks (eigenvalues,
impact factors, emission blocks) are known. In this work, we have made this
reconstruction explicit, up to the three-loop order.

A central result is the relation (5.13), which expresses the simplest cut
contribution to the $n$-point remainder function at three loops in terms of a
few basic building blocks. It should be emphasized that the identity has a
two-fold meaning: On the one hand, it holds at the level of the complete
remainder function’s _symbol_. On the other hand, it holds at the level of
full _functions_ once one restricts the remainder function to its simplest cut
contribution as in (3.5), neglecting the Regge pole terms as well as higher
Regge cut contributions such as the ones in the last line of (2.23). The
decomposition of the two-Reggeon cut contribution into building blocks is
closely tied to the map (4.13) between conventional multi-Regge limit
variables $w_{i}$ and “building-block variables” $v_{i}$.

The second main result is the determination of the three-loop building block
$g_{3}$ at the level of the symbol from the known seven-point three-loop
symbol for general kinematics. Together with the symbol of the known six-point
building block [49], this permits the reconstruction of the three-loop
remainder function symbol at leading logarithmic order, as implemented in the
attached Mathematica file.

Finally, we have constructed a function representative for the building block
$g_{3}$, based on the knowledge of its symbol as well as the relevant function
space, and by imposing further constraints such as parity invariance, target-
projectile symmetry, and consistency with the Wilson loop OPE.

#### Outlook.

It would be interesting to better understand the general relation between the
BFKL building blocks—impact factors, eigenvalues, and emission blocks—and the
perturbative building blocks that we found for the full cut contributions. Of
course, this relation is in principle provided by the Fourier–Mellin
transform. However, the action of the inverse Fourier–Mellin transform on
general expressions of multiple polylogs is (to the author’s knowledge) not
understood systematically. Especially, it would be interesting to understand
how much can be learnt about the BFKL building blocks when the cut
contributions are only known at the symbol level. A better understanding of
this point would admit to extract the NLO emission block from two-loop data,
from which the three-loop NLO building blocks $g_{L}$ and $g_{R}$ could then
be constructed.

We have only fully determined the three-loop seven-point building block
$g_{3}$ at leading functional transcendental weight. The parts with lower
functional weight (which are multiplied by $\pi$ and zeta values) have been
constrained by symmetry requirements, but still contain considerable
uncertainty in the form of unfixed rational coefficients. It would be
desirable to further constrain the space of parameters, for example by a more
detailed comparison to the Wilson loop OPE [18]. This would require to
explicitly compute the functions $f_{5}$, $f_{6}$, $h$, and $\bar{h}$ by
taking the relevant discontinuity of the three-loop OPE answer. In fact,
Basso, Caron-Huot and Sever could extract the two-particle cut $f_{k,(1)}$ to
all loop orders from the six-point Wilson loop OPE by an ingenuous analytic
continuation in the spectral parameter plane[52]. Of course, reconstructing
the full three-loop cut contribution $f_{k,(3)}$ for any number of points also
requires knowledge of the higher building blocks $g_{\text{L}}$,
$g_{\text{R}}$ (at NLLA), and $h$ (at NNLLA). Beyond that, constructing the
full multi-Regge limit remainder function at subleading functional
transcendentality in all kinematic regions also requires to take more general
multi-Reggeon cut terms into account, such as the ones shown in the last line
of (2.23). While it is possible to project out these more general cut terms by
restricting to kinematic regions where only adjacent momenta have been
flipped, these higher cut terms form an interesting subject on their own, and
it would be very interesting to understand them systematically.

### Acknowledgments

I sincerely wish to thank Jochen Bartels for many very instructive and
enjoyable discussions, as well as for comments on the manuscript. I also want
to thank Johannes Brödel, Vsevolod Chestnov, Georgios Papathanasiou, Volker
Schomerus, and Martin Sprenger for valuable discussions. My work is supported
by a Marie Curie International Outgoing Fellowship within the 7th European
Community Framework Programme under Grant No. PIOF-GA-2011-299865.

##  Appendix A Reduction Identities

Here, we want to derive the reduction identities (4.5). The absorption of
adjacent emission blocks into impact factors was demonstrated in [26], and the
analysis directly implies the reduction identity for emission blocks alone. We
reproduce it here for completeness.

The identities are most easily understood in momentum space. At leading order,
the central emission block simply consists of a single effective Reggeon-
Reggeon-gluon vertex attached to the upper Reggeon line in the two-Reggeon
state, see Figure 2. For a produced gluon with definite helicity, this
effective vertex equals [61]

$\parbox[c]{0.0pt}{\includegraphics{FigRRGvertextree}}=-\sqrt{2}\,\frac{k_{2}\bar{k}_{3}}{\bar{p}_{2}}\,.$
(A.1)

Compared to the full amplitude, the remainder function has the tree amplitude
divided out. We are computing cut contributions to the remainder function,
hence we need to divide by the tree-level expression for gluon emission

$-\sqrt{2}\,\frac{q_{2}\bar{q}_{3}}{\bar{p}_{2}}\,.$ (A.2)

At leading order, the central emission block in momentum space therefore
equals

$C_{2}\equiv
C(q_{2},k_{2},p_{2})=\frac{k_{2}\bar{k}_{3}}{q_{2}\bar{q}_{3}}=\frac{\bar{k}_{2}(k_{2}+p_{2})}{\bar{q}_{2}(q_{2}+p_{2})}\,.$
(A.3)

Combining two leading-order emission blocks requires to include the
intermediate transverse propagator $1/|k_{3}|^{2}$, again divided by the
corresponding tree-level expression $1/|q_{3}|^{2}$:

$C_{2}\cdot\frac{|q_{3}|^{2}}{|k_{3}|^{2}}\cdot
C_{3}=\frac{k_{2}\bar{k}_{3}}{q_{1}\bar{q}_{3}}\cdot\frac{|q_{3}|^{2}}{|k_{3}|^{2}}\cdot\frac{k_{3}\bar{k}_{4}}{q_{3}\bar{q}_{4}}=\frac{k_{2}\bar{k}_{4}}{q_{2}\bar{q}_{4}}=\frac{\bar{k}_{2}(k_{2}+p_{2}+p_{3})}{\bar{q}_{2}(q_{2}+p_{2}+p_{3})}\,.$
(A.4)

This clearly equals the single emission block (A.3) with the emitted momentum
$p_{2}$ replaced by the sum of momenta $p_{2}+p_{3}$. Iterating the procedure
straightforwardly yields the reduction identity for emission blocks, on the
right in (4.5).

The leading-order impact factor consists of a single gluon emission from the
bottom Reggeon line, as in Figure 2. It thus reads [10]

$-\sqrt{2}\,\frac{q_{1}(\bar{q}_{2}-\bar{k}_{2})}{\bar{k}^{\prime}}=-\sqrt{2}\,\frac{q_{1}\bar{q}_{2}}{\bar{p}_{1}}-\sqrt{2}\,\frac{|q_{1}|^{2}\bar{k}_{2}}{\bar{k}^{\prime}\bar{p}_{1}}\,.$
(A.5)

On the right, the emission factor has been split into a “local” piece (first
term) and a “non-local” part (second term). The local piece plays a role for
the one-loop amplitude, but does not affect the remainder function [10, 26].
It therefore can be dropped for the purpose of computing discontinuities of
the remainder function. Thus only the second term remains for the leading-
order impact factor. Dividing by the tree expression
$-\sqrt{2}\,q_{1}\bar{q}_{2}/\bar{p}_{1}$, it becomes

$\Phi_{\mathrm{L},1}\equiv\Phi_{\text{L}}(q_{1},k^{\prime},p_{1})=\frac{\bar{q}_{1}\bar{k}_{2}}{\bar{k}^{\prime}\bar{q}_{2}}=\frac{\bar{q}_{1}(\bar{k}^{\prime}+\bar{p}_{1})}{\bar{k}^{\prime}(\bar{q}_{1}+\bar{p}_{1})}\,.$
(A.6)

Combining this impact factor with an adjacent emission block, one again needs
to include the intermediate propagator factor, which yields

$\Phi_{\mathrm{L},1}\cdot\frac{|q_{2}|^{2}}{|k_{2}|^{2}}\cdot
C_{2}=\frac{\bar{q}_{1}\bar{k}_{2}}{\bar{k}^{\prime}\bar{q}_{2}}\cdot\frac{|q_{2}|^{2}}{|k_{2}|^{2}}\cdot\frac{k_{2}\bar{k}_{3}}{q_{2}\bar{q}_{3}}=\frac{\bar{q}_{1}\bar{k}_{3}}{\bar{k}^{\prime}\bar{q}_{3}}=\frac{\bar{q}_{1}(\bar{k}^{\prime}+\bar{p}_{1}+\bar{p}_{2})}{\bar{k}^{\prime}(\bar{q}_{1}+\bar{p}_{1}+\bar{p}_{2})}\,.$
(A.7)

This equals the original impact factor (A.6) with the emitted momentum $p_{1}$
replaced by the sum of momenta $p_{1}+p_{2}$. Again, iterating the procedure
yields the reduction identity for impact factors, on the left in (4.5).

Figure 2: Leading-order impact factor and emission vertices. The total
$t$-channel momenta are denoted $q_{i}$ (see Figure 1), hence the momentum on
the lower Reggeon line equals
$q_{1}+k^{\prime}=q_{2}-k_{2}=q_{3}-k_{3}=q_{4}-k_{4}$.

##  Appendix B Four-Loop Expansion

At four loops, the two-Reggeon cut at LLA evidently expands to a sum of six-
point, seven-point, and eight-point functions:

$f_{k,(4)}=\sum_{j=5}^{k+4}f_{1,(4)}(\varepsilon_{j};v_{5,k+4;j})+\sum_{i=5}^{k+3}\sum_{j=i+1}^{k+4}g_{2,(4)}(\varepsilon_{i},\varepsilon_{j};v_{5,j-1;i},v_{i+1,k+4;j})\\\
+\sum_{i=5}^{k+2}\sum_{j=i+1}^{k+3}\sum_{m=j+1}^{k+4}g_{3,(4)}(\varepsilon_{i},\varepsilon_{j},\varepsilon_{m};v_{5,j-1;i},v_{i+1,m-1;j},v_{j+1,k+4;m})+\mathcal{O}(\mathrm{NLLA})\,,$
(B.1)

where

$g_{2,(4)}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})\equiv
f_{2,(4)}(\varepsilon_{5},\varepsilon_{6};w_{5},w_{6})-f_{1,(4)}(\varepsilon_{5};v_{5,6;5})-f_{1,(4)}(\varepsilon_{6};v_{5,6;6})\,,$
(B.2)

and

$\displaystyle
g_{3,(4)}(\varepsilon_{5},\varepsilon_{6},\varepsilon_{7};w_{5},w_{6},w_{7})\equiv\mspace{-180.0mu}$
$\displaystyle
f_{3,(4)}(\varepsilon_{5},\varepsilon_{6},\varepsilon_{7};w_{5},w_{6},w_{7})-f_{1,(4)}(\varepsilon_{5};v_{5,7;5})-f_{1,(4)}(\varepsilon_{6};v_{5,7;6})-f_{1,(4)}(\varepsilon_{7};v_{5,7;7})$
$\displaystyle-
g_{2,(4)}(\varepsilon_{5},\varepsilon_{6};w_{5},v_{6,7;6})-g_{2,(4)}(\varepsilon_{5},\varepsilon_{7};v_{5,6;5},v_{6,7;7})-g_{2,(4)}(\varepsilon_{6},\varepsilon_{7};v_{5,6;6},w_{7})\,.$
(B.3)

At NLLA, there are three more building blocks that all stem from nine-point
data:

$\displaystyle
g_{4,(4),1}(\varepsilon_{5},\varepsilon_{6},\varepsilon_{7},\varepsilon_{8};w_{5},w_{6},w_{7},w_{8})$
$\displaystyle=\text{\small$\biggl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&0&1&0&0&1&0&0\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&1&0&1&0&0&1\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&2&0&0&1&0&0\end{matrix}\biggr{]}$}$ (B.4)
$\displaystyle\mspace{-220.0mu}-\text{\small$\biggl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&1&0&1&1&0&0\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&1&0&0&1&1&0\end{matrix}\biggr{]}$}+\text{\small$\biggl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&2&0&0&1\end{matrix}\biggr{]}$}+\text{\small$\biggl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&2&1&0&0\end{matrix}\biggr{]}$}+\text{\small$\biggl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&1&0&1&1\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}1&&4\\\\[-2.0pt]
0&2&1\end{matrix}\biggr{]}$}+\mathcal{O}(\text{NNLLA})\,,$ $\displaystyle
g_{4,(4),2}(\varepsilon_{5},\varepsilon_{6},\varepsilon_{7},\varepsilon_{8};w_{5},w_{6},w_{7},w_{8})$
$\displaystyle=\text{\small$\biggl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&0&0&1&0&0&1&0\end{matrix}\biggr{]}$}+\text{\small$\biggl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&1&1&1&0\end{matrix}\biggr{]}$}+\mathcal{O}(\text{NNLLA})\,,$ (B.5)
$\displaystyle
g_{4,(4),3}(\varepsilon_{5},\varepsilon_{6},\varepsilon_{7},\varepsilon_{8};w_{5},w_{6},w_{7},w_{8})$
$\displaystyle=\text{\small$\biggl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&1&0&1&0\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&0&1&0&1&0\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&0&2&0\end{matrix}\biggr{]}$}$ (B.6)
$\displaystyle\mspace{-220.0mu}-\text{\small$\biggl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&1&0&1&0\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&1&1&0&0&1&0\end{matrix}\biggr{]}$}+\text{\small$\biggl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&0&2&0\end{matrix}\biggr{]}$}+\text{\small$\biggl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&1&2&0\end{matrix}\biggr{]}$}+\text{\small$\biggl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&1&0&1&0\end{matrix}\biggr{]}$}-\text{\small$\biggl{[}\begin{matrix}4&&1\\\\[-2.0pt]
1&2&0\end{matrix}\biggr{]}$}+\mathcal{O}(\text{NNLLA})\,.$

Here, each bracket stands for a BFKL diagram, with the following notation: In
the bottom sequence, the numbers alternatingly stand for impact factors /
emission blocks and BFKL Green’s functions. For the impact factors and
emission blocks, the number specifies the loop order. For the Green’s
functions, it specifies the number of (leading order) BFKL eigenvalues,
accompanied by the respective large logarithm $\log(\varepsilon_{j})$; i.e.
the numbers $1$, $2$, and $3$ stand for the first, second, and third terms in
(5.2). The numbers in the top row specify how many momenta are attached to the
respective impact factor or emission block. The expansion (B.1) of the two-
Reggeon cut extends to

$f_{k,(4)}=f_{1,(4)}^{\\{i_{1},i_{2}\\}}+g_{2,(4)}^{\\{i_{1},i_{2},i_{3}\\}}+g_{3,(4)}^{\\{i_{1},i_{2},i_{3},i_{4}\\}}\\\
+g_{4,(4),1}^{\\{i_{1},i_{2},i_{3},1,i_{4}\\}}+g_{4,(4),2}^{\\{i_{1},i_{2},1,i_{3},i_{4}\\}}+g_{4,(4),3}^{\\{i_{1},1,i_{2},i_{3},i_{4}\\}}+\mathcal{O}(\mathrm{NNLLA})\,.$
(B.7)

Here, each term stands for a sum over partitions $\\{i_{1},i_{2},\dots\\}$,
$\sum_{j}i_{j}=k+1$, of the external momenta $p_{4},\dots,p_{k+4}$ into
subsequences $(p_{4},\dots,p_{3+i_{1}})$,
$(p_{4+i_{1}},\dots,p_{3+i_{1}+i_{2}})$, …, whose sums get attached to the
momentum slots of the respective building block. For example,

$g_{4,(4),1}^{\\{i_{1},i_{2},i_{3},1,i_{4}\\}}\equiv\sum_{i=5}^{k+1}\sum_{j=i+1}^{k+2}\sum_{m=j+2}^{k+4}g_{4,(4),1}(\varepsilon_{i},\varepsilon_{j},\varepsilon_{m-1},\varepsilon_{m};v_{5,j-1;i},v_{i+1,m-2;j},v_{j+1,m-1;m-1},v_{m,k+4;m})\,.$
(B.8)

At NNLLA and NNNLLA, there are many more terms and building blocks. The
complete expansion of the two-Reggeon cut reads

$\displaystyle f_{k,(4)}={}$ $\displaystyle
f_{1,(4)}^{\\{i_{1},i_{2}\\}}+g_{2,(4)}^{\\{i_{1},i_{2},i_{3}\\}}+g_{3,(4)}^{\\{i_{1},i_{2},i_{3},i_{4}\\}}+g_{4,(4),1}^{\\{i_{1},i_{2},i_{3},1,i_{4}\\}}+g_{4,(4),2}^{\\{i_{1},i_{2},1,i_{3},i_{4}\\}}+g_{4,(4),3}^{\\{i_{1},1,i_{2},i_{3},i_{4}\\}}$
$\displaystyle+g_{4,(4),4}^{\\{1,i_{1},1,i_{2},i_{3}\\}}+g_{4,(4),5}^{\\{1,i_{1},i_{2},1,i_{3}\\}}+g_{4,(4),6}^{\\{i_{1},1,1,i_{2},i_{3}\\}}+g_{4,(4),7}^{\\{i_{1},1,i_{2},1,i_{3}\\}}+g_{4,(4),8}^{\\{i_{1},1,i_{2},i_{3},1\\}}$
$\displaystyle+g_{4,(4),9}^{\\{i_{1},i_{2},1,1,i_{3}\\}}+g_{4,(4),10}^{\\{i_{1},i_{2},1,i_{3},1\\}}+g_{5,(4),1}^{\\{i_{1},1,i_{2},1,i_{3},i_{4}\\}}+g_{5,(4),2}^{\\{i_{1},1,i_{2},i_{3},1,i_{4}\\}}+g_{5,(4),3}^{\\{i_{1},i_{2},1,i_{3},1,i_{4}\\}}$
$\displaystyle+g_{4,(4),11}^{\\{i_{1},1,i_{2},1,1\\}}+g_{4,(4),12}^{\\{i_{1},1,1,i_{2},1\\}}+g_{4,(4),13}^{\\{i_{1},1,1,1,i_{2}\\}}+g_{4,(4),14}^{\\{1,i_{1},1,i_{2},1\\}}+g_{4,(4),15}^{\\{1,i_{1},1,1,i_{2}\\}}+g_{4,(4),16}^{\\{1,1,i_{1},1,i_{2}\\}}$
$\displaystyle+g_{5,(4),4}^{\\{i_{1},1,i_{2},1,i_{3},1\\}}+g_{5,(4),5}^{\\{i_{1},1,i_{2},1,1,i_{3}\\}}+g_{5,(4),6}^{\\{i_{1},1,1,i_{2},1,i_{3}\\}}+g_{5,(4),7}^{\\{1,i_{1},1,i_{2},1,i_{3}\\}}+g_{6,(4)}^{\\{i_{1},1,i_{2},1,i_{3},1,i_{4}\\}}\,.$
(B.9)

Terms in the second and third lines start at $\mathcal{O}(\text{NNLLA})$,
terms in the last two lines are of order $\mathcal{O}(\text{NNNLLA})$. The
individual terms are listed explicitly in the following. The completion of the
NLLA terms (B.4)-(B.6) is given by

$\displaystyle g_{4,(4),1}\mspace{-40.0mu}$
$\displaystyle\mspace{40.0mu}=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&0&1&0&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&1&0&1&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&2&0&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&1&0&1&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&1&0&0&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&2&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&2&1&0&0\end{matrix}\Bigr{]}$}$
$\displaystyle+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&1&0&1&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&4\\\\[-2.0pt]
0&2&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&1&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&0&1&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&0&1&0&0&1&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&1&0&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
1&0&0&1&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&0&1&0&1&1&0\end{matrix}\Bigr{]}$}$
$\displaystyle-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&0&1&1&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&1&0&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&1&0&0&2&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&1&1&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
1&0&0&0&1&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
1&0&0&1&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
1&1&0&0&1&0&0\end{matrix}\Bigr{]}$}$
$\displaystyle-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&(1)&0&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
1&0&0&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
1&1&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&0&1&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&1&1&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&1&0&0&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&(1)&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&3\\\\[-2.0pt]
1&1&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&0&1&0&0&0&2\end{matrix}\Bigr{]}$}\mspace{-100.0mu}$
$\displaystyle-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
1&0&0&0&2&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
1&0&0&0&1&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&0&1&0&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
0&0&2&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
1&0&0&0&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
2&0&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&3\\\\[-2.0pt]
1&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&3\\\\[-2.0pt]
1&0&2\end{matrix}\Bigr{]}$}$ (B.10) $\displaystyle
g_{4,(4),2}\mspace{-40.0mu}$
$\displaystyle\mspace{40.0mu}=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&0&0&1&0&0&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&1&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
1&0&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
1&1&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&1&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&1&1&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&2&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&1&2&0&0\end{matrix}\Bigr{]}$}$
$\displaystyle+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&1&(1)&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&(1)&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&4\\\\[-2.0pt]
0&1&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&4\\\\[-2.0pt]
0&(1)&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&4\\\\[-2.0pt]
1&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}4&&1\\\\[-2.0pt]
1&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}4&&1\\\\[-2.0pt]
1&(1)&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}4&&1\\\\[-2.0pt]
2&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&1&0&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
2&0&1&0&0\end{matrix}\Bigr{]}$}$
$\displaystyle+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&2&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
1&0&2&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&3&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
1&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&4\\\\[-2.0pt]
0&0&3\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&4\\\\[-2.0pt]
1&0&2\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&4\\\\[-2.0pt]
2&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}4&&1\\\\[-2.0pt]
1&0&2\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}4&&1\\\\[-2.0pt]
2&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}4&&1\\\\[-2.0pt]
3&0&0\end{matrix}\Bigr{]}$}\mspace{-20.0mu}$ (B.11) $\displaystyle
g_{4,(4),3}\mspace{-40.0mu}$
$\displaystyle\mspace{40.0mu}=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&1&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&0&1&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&0&2&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&1&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&1&1&0&0&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&0&2&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&3&&1\\\\[-2.0pt]
0&0&1&2&0\end{matrix}\Bigr{]}$}$
$\displaystyle+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&1&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}4&&1\\\\[-2.0pt]
1&2&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&1&1&0&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&0&1&1&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&1&0&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&1&0&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&0&1&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&1&1&0&1&0&0\end{matrix}\Bigr{]}$}$
$\displaystyle-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&1&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
1&0&1&0&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&2&0&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&1&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&1&1&0&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&1&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&0&1&1\end{matrix}\Bigr{]}$}$
$\displaystyle-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&0&(1)&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&1&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&0&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&1&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
2&0&0&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&0&(1)&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}3&&2\\\\[-2.0pt]
1&1&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
2&0&0&0&1&0&0\end{matrix}\Bigr{]}$}\mspace{-40.0mu}$
$\displaystyle-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&2&0&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
1&0&1&0&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
2&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&2&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
2&0&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&0&0&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1\\\\[-2.0pt]
1&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}3&&2\\\\[-2.0pt]
2&0&1\end{matrix}\Bigr{]}$}$ (B.12)

Here, a $(1)$ in the place of a Green’s function stands for the one-loop
correction to the BFKL eigenvalue, i.e. for the fourth term in (5.2). The
following terms start at $\mathcal{O}(\text{NNLLA})$:

$\displaystyle g_{4,(4),4}={}$
$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&1&0&0&1&0\end{matrix}\Bigr{]}$ (B.13) $\displaystyle g_{4,(4),5}={}$
$\displaystyle\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
1&0&0&1&0&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
2&0&0&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
1&0&1&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&0&2&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&2&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}3&&2\\\\[-2.0pt]
1&0&2\end{matrix}\Bigr{]}$}$ (B.14) $\displaystyle g_{4,(4),6}={}$
$\displaystyle\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&1&0&0&1&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&0&1&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&1\\\\[-2.0pt]
0&0&1&0&2&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&1&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&1&0&0&0&1\end{matrix}\Bigr{]}$}$ (B.15) $\displaystyle g_{4,(4),7}={}$
$\displaystyle\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&1&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&1&0&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&1&0&0&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&2&&1\\\\[-2.0pt]
1&0&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&2&&1\\\\[-2.0pt]
1&1&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&2\\\\[-2.0pt]
1&0&0&1&1\end{matrix}\Bigr{]}$}\mspace{-20.0mu}$
$\displaystyle+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&2\\\\[-2.0pt]
1&1&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2\\\\[-2.0pt]
0&0&1&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2\\\\[-2.0pt]
0&1&1&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&2&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&2&0&0&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&2&&1\\\\[-2.0pt]
2&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&2&&1\\\\[-2.0pt]
1&0&2&0&0\end{matrix}\Bigr{]}$}$
$\displaystyle+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&2&&1\\\\[-2.0pt]
1&0&1&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&2\\\\[-2.0pt]
1&0&0&0&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&2\\\\[-2.0pt]
2&0&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2\\\\[-2.0pt]
0&0&1&0&2\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2\\\\[-2.0pt]
0&0&2&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2\\\\[-2.0pt]
1&0&1&0&1\end{matrix}\Bigr{]}$}$ (B.16) $\displaystyle g_{4,(4),8}={}$
$\displaystyle\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&1&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&2\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&1&0&2&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&0&2&0&0&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&3\\\\[-2.0pt]
2&0&1\end{matrix}\Bigr{]}$}$ (B.17) $\displaystyle g_{4,(4),9}={}$
$\displaystyle\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&0&0&1&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
1&0&1&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&1\\\\[-2.0pt]
0&0&2&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
0&0&1&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
1&0&0&0&1&0&1\end{matrix}\Bigr{]}$}$ (B.18) $\displaystyle g_{4,(4),10}={}$
$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&0&0&1&0&0&0&1\end{matrix}\Bigr{]}$ (B.19) $\displaystyle g_{5,(4),1}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{11}\begin{matrix}1&&1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&1&0&0&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&2&&1&&1\\\\[-2.0pt]
1&0&1&0&0&1&0\end{matrix}\Bigr{]}$}$ (B.20) $\displaystyle g_{5,(4),2}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{11}\begin{matrix}1&&1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&1&0&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&2\\\\[-2.0pt]
1&0&0&1&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&2\\\\[-2.0pt]
0&0&1&0&0&1&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&1&&1&&1\\\\[-2.0pt]
1&0&0&1&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&3\\\\[-2.0pt]
0&0&1&1&0&0&1\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&3\\\\[-2.0pt]
0&1&1&0&0&0&1\end{matrix}\Bigr{]}$}\mspace{-200.0mu}$
$\displaystyle+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&2&&1\\\\[-2.0pt]
1&1&0&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2&&1\\\\[-2.0pt]
0&0&1&0&1&1&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2&&1\\\\[-2.0pt]
0&0&1&1&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2&&1\\\\[-2.0pt]
0&1&1&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}3&&3\\\\[-2.0pt]
1&1&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&1&&2\\\\[-2.0pt]
0&0&1&0&2&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&2&&1\\\\[-2.0pt]
0&0&2&0&1&0&0\end{matrix}\Bigr{]}$}\mspace{-200.0mu}$ (B.21) $\displaystyle
g_{5,(4),3}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{11}\begin{matrix}1&&1&&1&&1&&1&&1\\\\[-2.0pt]
0&1&0&0&1&0&0&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&2&&2\\\\[-2.0pt]
0&1&0&0&1&0&1\end{matrix}\Bigr{]}$}$ (B.22)

Finally, the following terms only contribute at NNNLLA:

$\displaystyle g_{4,(4),11}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&1&0&1\end{matrix}\Bigr{]}$}$ $\displaystyle g_{4,(4),12}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&1&0&0&0&1\end{matrix}\Bigr{]}$}$ (B.23) $\displaystyle g_{4,(4),13}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&1&0&1&0&0\end{matrix}\Bigr{]}$}+\text{\scriptsize$\Bigl{[}\begin{matrix}2&&1&&2\\\\[-2.0pt]
1&0&1&0&1\end{matrix}\Bigr{]}$}$ $\displaystyle g_{4,(4),14}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&1&0&0&0&1\end{matrix}\Bigr{]}$}$ (B.24) $\displaystyle g_{4,(4),15}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&1&0&1&0&0\end{matrix}\Bigr{]}$}$ $\displaystyle g_{4,(4),16}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&1&&1\\\\[-2.0pt]
1&0&1&0&0&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&1&&1&&2\\\\[-2.0pt]
1&0&1&0&0&0&1\end{matrix}\Bigr{]}$}$ (B.25) $\displaystyle g_{5,(4),4}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{11}\begin{matrix}1&&1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&1&0&0&0&1\end{matrix}\Bigr{]}$}$ $\displaystyle g_{5,(4),5}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{11}\begin{matrix}1&&1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&1&0&1&0&0\end{matrix}\Bigr{]}$}\mspace{150.0mu}$ (B.26)
$\displaystyle g_{5,(4),6}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{11}\begin{matrix}1&&1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&1&0&0&0&1&0&0\end{matrix}\Bigr{]}$}$ $\displaystyle g_{5,(4),7}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{11}\begin{matrix}1&&1&&1&&1&&1&&1\\\\[-2.0pt]
1&0&0&0&1&0&0&0&1&0&0\end{matrix}\Bigr{]}$}$ (B.27) $\displaystyle g_{6,(4)}$
$\displaystyle=\text{\scriptsize$\Bigl{[}\setcounter{MaxMatrixCols}{13}\begin{matrix}1&&1&&1&&1&&1&&1&&1\\\\[-2.0pt]
0&0&1&0&0&0&1&0&0&0&1&0&0\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}1&&2&&2&&2\\\\[-2.0pt]
0&0&1&0&1&0&1\end{matrix}\Bigr{]}$}-\text{\scriptsize$\Bigl{[}\begin{matrix}2&&2&&2&&1\\\\[-2.0pt]
1&0&1&0&1&0&0\end{matrix}\Bigr{]}$}$ (B.28)

The large number of terms at NNLLA and NNNLLA shows that the decomposition
into building blocks is less effective than at two and three loops. The reason
is that the reduction identities (4.5) are only established for the leading
order emission block. It would be interesting to see whether the (thus far
unknown) NLO emission block satisfies similar reduction identities, in which
case many of the above terms could be reduced and absorbed into a smaller
number of building blocks, reducing the complexity of the decomposition.

##  Appendix C Polylogarithm Identities

When applying the target-projectile transformation $x\leftrightarrow y$ or the
parity map $x\to\bar{x}$, $y\to\bar{y}$ to the ansatz functions for the
components of $g_{3}$, some single-valued basis functions are mapped to non-
basis single-valued multiple polylogarithms. In order to impose the required
symmetries on the ansätze, these non-basis functions need to be re-expressed
in terms of basis functions. This can always be achieved with the help of
shuffle and stuffle relations, as well as the rescaling property (7.9).
Shuffle relations take the form

$G(\vec{a};z)G(\vec{b};z)=\sum_{\vec{c}\,\in\,\vec{a}\shuffle\vec{b}}G(\vec{c};z)\,,$
(C.1)

where the sum runs over all shuffles $\vec{a}\shuffle\vec{b}$ of the weight
vectors $\vec{a}$ and $\vec{b}$, that is over all permutations of their
components that preserve the ordering of elements within both $\vec{a}$ and
$\vec{b}$. The shuffle relation follows directly from the iterated integral
definition of multiple polylogarithms (7.2), and they hold for ordinary as
well as single-valued multiple polylogarithms. Stuffle relations on the other
hand are less transparent in the integral representation; they follow directly
from the series representation of $G(\vec{a};z)$ around $z=0$, see for example
[47].

At weights one and two, the following identities are needed to evaluate the
parity and target-projectile invariance conditions:

$\displaystyle G^{\mathbf{s},y}_{0}$
$\displaystyle=-G^{\mathbf{s},\check{y}}_{0}\,,$ $\displaystyle
G^{\mathbf{s},\check{x}}_{0}$ $\displaystyle=-G^{\mathbf{s},x}_{0}\,,$
$\displaystyle G^{\mathbf{s},1}_{x}$
$\displaystyle=-G^{\mathbf{s},x}_{0}+G^{\mathbf{s},x}_{1}\,,$ (C.2)
$\displaystyle G^{\mathbf{s},x}_{x\check{y}}$
$\displaystyle=-G^{\mathbf{s},\check{y}}_{0}+G^{\mathbf{s},\check{y}}_{1}\,,$
$\displaystyle G^{\mathbf{s},\check{y}}_{x\check{y}}$
$\displaystyle=-G^{\mathbf{s},x}_{0}+G^{\mathbf{s},x}_{1}\,,$ $\displaystyle
G^{\mathbf{s},1}_{0,x}$
$\displaystyle={\textstyle\frac{1}{2}}(G^{\mathbf{s},x}_{0})^{2}-G^{\mathbf{s},x}_{0,1}\,,$
(C.3) $\displaystyle G^{\mathbf{s},x}_{0,x\check{y}}$
$\displaystyle={\textstyle\frac{1}{2}}(G^{\mathbf{s},\check{y}}_{0})^{2}-G^{\mathbf{s},\check{y}}_{0,1}\,,$
$\displaystyle G^{\mathbf{s},\check{y}}_{0,x\check{y}}$
$\displaystyle={\textstyle\frac{1}{2}}(G^{\mathbf{s},x}_{0})^{2}-G^{\mathbf{s},x}_{0,1}\,,$
(C.4) $\displaystyle G^{\mathbf{s},\check{y}}_{x\check{y},x}$
$\displaystyle=-G^{\mathbf{s},x}_{0}G^{\mathbf{s},\check{y}}_{1}+G^{\mathbf{s},x}_{1}G^{\mathbf{s},\check{y}}_{1}+G^{\mathbf{s},\check{y}}_{0,x}-G^{\mathbf{s},\check{y}}_{1,x}\,,\mspace{-150.0mu}$
(C.5)

Here and in the following, we use the condensed notation
$G^{\mathbf{s},z}_{a_{1},\dots,a_{n}}\equiv
G^{\mathbf{s}}(a_{1},\dots,a_{n};z)$, and $\check{y}\equiv 1/y$. At weight
three, we have for example the following identities:

$\displaystyle G^{\mathbf{s},x}_{0,0,x\check{y}}$
$\displaystyle=-{\textstyle\frac{1}{6}}(G^{\mathbf{s},\check{y}}_{0})^{3}+G^{\mathbf{s},\check{y}}_{0,0,1}\,,$
(C.6) $\displaystyle G^{\mathbf{s},x}_{0,x\check{y},x\check{y}}$
$\displaystyle=-{\textstyle\frac{1}{6}}(G^{\mathbf{s},\check{y}}_{0})^{3}+G^{\mathbf{s},\check{y}}_{0}G^{\mathbf{s},\check{y}}_{0,1}-G^{\mathbf{s},\check{y}}_{0,0,1}-G^{\mathbf{s},\check{y}}_{0,1,1}+2\zeta_{3}\,,$
(C.7) $\displaystyle G^{\mathbf{s},\check{y}}_{0,0,x\check{y}}$
$\displaystyle=-{\textstyle\frac{1}{6}}(G^{\mathbf{s},x}_{0})^{3}+G^{\mathbf{s},x}_{0,0,1}\,,$
(C.8) $\displaystyle G^{\mathbf{s},\check{y}}_{0,x,x\check{y}}$
$\displaystyle=-{\textstyle\frac{1}{2}}(G^{\mathbf{s},x}_{0})^{2}G^{\mathbf{s},\check{y}}_{1}+G^{\mathbf{s},\check{y}}_{1}G^{\mathbf{s},x}_{0,1}+G^{\mathbf{s},x}_{0}G^{\mathbf{s},\check{y}}_{0,1}-G^{\mathbf{s},x}_{1}G^{\mathbf{s},\check{y}}_{0,1}-G^{\mathbf{s},x}_{0}G^{\mathbf{s},\check{y}}_{0,x}+G^{\mathbf{s},x}_{1}G^{\mathbf{s},\check{y}}_{0,x}$
$\displaystyle\mspace{20.0mu}+G^{\mathbf{s},\check{y}}_{1}G^{\mathbf{s},\check{y}}_{0,x}-2G^{\mathbf{s},\check{y}}_{0,0,x}-G^{\mathbf{s},\check{y}}_{0,x,1}\,,$
(C.9) $\displaystyle G^{\mathbf{s},\check{y}}_{0,x\check{y},x}$
$\displaystyle={\textstyle\frac{1}{2}}(G^{\mathbf{s},x}_{0})^{2}G^{\mathbf{s},\check{y}}_{1}-G^{\mathbf{s},\check{y}}_{1}G^{\mathbf{s},x}_{0,1}-G^{\mathbf{s},\check{y}}_{1}G^{\mathbf{s},\check{y}}_{0,x}+G^{\mathbf{s},\check{y}}_{0,0,x}+G^{\mathbf{s},\check{y}}_{0,1,x}+G^{\mathbf{s},\check{y}}_{0,x,1}\,,$
(C.10) $\displaystyle G^{\mathbf{s},\check{y}}_{0,x\check{y},x\check{y}}$
$\displaystyle=-{\textstyle\frac{1}{6}}(G^{\mathbf{s},x}_{0})^{3}+G^{\mathbf{s},x}_{0}G^{\mathbf{s},x}_{0,1}-G^{\mathbf{s},x}_{0,0,1}-G^{\mathbf{s},x}_{0,1,1}+2\zeta_{3}\,,$
(C.11) $\displaystyle G^{\mathbf{s},\check{y}}_{x\check{y},x,x}$
$\displaystyle=-{\textstyle\frac{1}{2}}G^{\mathbf{s},x}_{0}(G^{\mathbf{s},\check{y}}_{1})^{2}+{\textstyle\frac{1}{2}}G^{\mathbf{s},x}_{1}(G^{\mathbf{s},\check{y}}_{1})^{2}+G^{\mathbf{s},\check{y}}_{1}G^{\mathbf{s},\check{y}}_{0,x}-G^{\mathbf{s},\check{y}}_{0,1,x}-G^{\mathbf{s},\check{y}}_{0,x,1}$
$\displaystyle\mspace{20.0mu}+G^{\mathbf{s},\check{y}}_{0,x,x}-G^{\mathbf{s},\check{y}}_{1,1,x}-G^{\mathbf{s},\check{y}}_{1,x,x}\,,$
(C.12) $\displaystyle G^{\mathbf{s},\check{y}}_{x\check{y},x\check{y},x}$