# Hexagon Bootstrap in the Double Scaling Limit

Vsevolod Chestnov,a Georgios Papathanasioub

DESY 20-221

Hexagon Bootstrap in the Double Scaling Limit

Vsevolod Chestnov,a Georgios Papathanasioub

aII. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany

bDESY Theory Group, DESY Hamburg,
Notkestraße 85, D-22603 Hamburg, Germany

<EMAIL_ADDRESS>
Abstract

We study the six-particle amplitude in planar $\mathcal{N}=4$ super Yang-Mills
theory in the double scaling (DS) limit, the only nontrivial codimension-one
boundary of its positive kinematic region. We construct the relevant function
space, which is significantly constrained due to the extended Steinmann
relations, up to weight 13 in coproduct form, and up to weight 12 as an
explicit polylogarithmic representation. Expanding the latter in the collinear
boundary of the DS limit, and using the Pentagon Operator Product Expansion,
we compute the non-divergent coefficient of a certain component of the Next-
to-Maximally-Helicity-Violating amplitude through weight 12 and eight loops.
We also specialize our results to the overlapping origin limit, observing a
general pattern for its leading divergence.

###### Contents

  1. 1 Introduction
  2. 2 The six-gluon amplitude in the double-scaling limit
    1. 2.1 Analytic structure of the normalized amplitude
    2. 2.2 The double-scaling (DS) limit
    3. 2.3 Integrability and Extended Steinmann conditions
    4. 2.4 Branch cut conditions and transcendental constants
  3. 3 Bootstrapping the DS Functions
    1. 3.1 Solving the integrability and extended Steinmann constraints
    2. 3.2 Promoting coproducts to functions
    3. 3.3 Promoting coproducts to expansions
  4. 4 The Wilson loop OPE and the collinear limit
    1. 4.1 Wilson loop normalization, variables and $R$-invariants in the DS limit
    2. 4.2 NMHV Wilson loop OPE
  5. 5 Results
    1. 5.1 The NMHV amplitude up to 8 loops and weight 12
    2. 5.2 The origin limit
    3. 5.3 Further refinements of the $\mathcal{H}^{\mathrm{DS}}$ space
  6. A The Multiple Zeta Value basis
  7. B Tensor perspective on the coproduct bootstrap

##  1 Introduction

The exact description of quantum interactions is one of the outstanding
questions in theoretical physics. Our best hope for answering this question is
in the realm of the simplest interacting quantum field theory in four
dimensions, ${\mathcal{N}}=4$ super Yang-Mills theory (SYM) [1, 2]. Indeed, in
the large-color or planar limit [3] the integrability of the theory has
enabled the determination of the scaling dimension spectrum of all its single-
trace operators beyond perturbation theory, see for example the reviews [4,
5].

For the quantities actually encoding the outcome of particle interactions,
namely the scattering amplitudes, which in this theory also happen to be
equivalent to null polygonal Wilson loops [6, 7, 8], integrable structures are
currently known to emerge only in certain corners of the space of kinematics.
The best-understood such corner is a certain collinear limit [9], whereby
every term in the series expansion of the Wilson loop or amplitude with
respect to the kinematic variables that become small in this limit, can be
mapped to an exactly solvable flux tube [10, 11, 12, 13, 14, 15, 16, 17, 18,
19]. The flux tube description, known as the Wilson loop (or Pentagon)
Operator Product Expansion (OPE), is complete for the first nontrivial
amplitude of the theory, which as a consequence of dual conformal symmetry
(see [20] for a review) has multiplicity $n=6$111At higher multiplicity, one
final building block known as the “matrix part” is still missing..

In order to obtain closed expressions for this six-particle or ‘hexagon’
amplitude, which will be the focus of this article, both at finite coupling
and in general kinematics, one would thus have to resum the aforementioned
kinematic expansion. A strategy to achieve this ambitious goal, would be to
divide it into two simpler steps: First resum the kinematics order by order in
perturbation theory with respect to the planar coupling $g$, and then resum
the perturbative series222Alternatively, one could start from the strong-
instead of the weak-coupling regime, see [21, 22, 23, 24, 25] for work in this
direction.. Indeed, a great deal is known about the class of polylogarithmic
functions the first step evaluates to, thus greatly facilitating its
realization. And it is not unreasonable to expect that these polylogarithms
can be in turn resummed to more complicated functions of hypergeometric type,
as has been the case with certain integrals contributing to the amplitude
[26], thanks to the existence of differential equations relating different
perturbative orders.

The task of resumming the perturbative OPE series was initiated in [27]333For
more recent work on the weak-coupling OPE resummation, see also [28, 29,
Bork:2019aud, Bork:2020aut]., also building on the earlier work [30, 31],
under one additional simplification: Starting from the one-dimensional
collinear limit, the kinematics was resummed to the “double-scaling” (DS)
limit [32, 13], instead of the full three-dimensional space of general
kinematics of the hexagon. The double-scaling limit is distinguished by the
fact that it is the only codimension-one boundary of the region of positive
kinematics, where the (appropriately normalized [33]) amplitude is
nonvanishing, modulo its discrete symmetries. The positive region, first
considered in the context of amplitude integrands [34] and then adapted to the
space of external kinematics in [35], is part of the Euclidean region, where
amplitudes are free of branch points. Indeed, as is reviewed in e.g. [36], the
only other codimension-one boundary is the soft or equivalently multi-Regge
limit, where the amplitude is nonvanishing only after analytically continuing
away from the Euclidean region.

From the point of view of the flux tube description, the double scaling limit
is advantageous because only a simpler subset, of so-called same-helicity
gluon excitations contribute. These are not charged under the internal
symmetries of the theory, and are labeled by a particle number $N$, which also
corresponds to the dimensionality of their all-loop integral representation.
In [27], in was in particular the $N=1$ excitations which were considered, and
it was realized that existing nested summation algorithms [37] allow their
explicit evaluation in terms of two-dimensional harmonic polylogarithms [38],
or more precisely their subset associated to the $A_{2}$ cluster algebra [35],
in principle at any loop order.

In this work, we take the next step and study the $N=2$ gluon OPE excitations.
These give rise to significantly more complicated integrals where, to the best
of our knowledge, no direct method for their evaluation is available to date
(see [39] for the current state of the art with respect to nested summation
technology), despite the fact that they are expected to lie in the same space
of functions as the $N=1$ excitations. Instead, we will rely on the bootstrap
philosophy, where one first constructs the expected space of functions, and
then locates the physical quantity in question within this space. This
approach has been very successful for determining perturbative six- and seven-
particle amplitudes in planar ${\mathcal{N}}=4$ SYM in general kinematics [40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. In particular, the former are
known through six and seven loops in the Next-to-Maximally Helicity-Violating
(NMHV) and Maximally-Helicity-Violating (MHV) configuration, respectively,
whereas the latter are known through four loops, see also the recent review
[53].

First, we will thus develop the hexagon bootstrap in the simplified setting of
the double scaling limit. We will see that while the limit breaks some of the
symmetries of the amplitude, it still preserves important analytic properties
that tame the growth of the space of relevant functions with respect to the
weight, namely the number of iterated integrations defining them. Among these
properties, a special role will be played by the extended Steinmann relations
[26, 51], which generalize the ordinary Steinmann relations [54, 55, 56] so as
to forbid not only double, but also multiple discontinuities in overlapping
channels. While these channels are normally associated to Mandelstam
invariants, for functions with physical branch cuts the extended Steinmann
relations imply the absence of discontinuities also with respect to more
general kinematic variables, as also predicted by the principle of cluster
adjacency [57, 36]. Specifically in the double-scaling limit, we find that two
of such generalized discontinuities are forbidden. With their aid, we will be
able to construct the corresponding space of ‘Extended Steinmann Double-
Scaling’ (DS for short) functions to weight 12 explicitly, and to weight 13
when the functions are specified iteratively in terms of their first
derivatives (or coproducts, see [58] for a review).

Then, we will proceed to uniquely identify the contribution of $N=1,2$ gluon
OPE excitations inside the DS space. On the one hand, we will work out the
expansion of our functions in the collinear limit, and on the other hand we
will compare that to the sum representation of the OPE predictions, obtained
from their original integral form with the help of Cauchy’s residue theorem,
and organized into finite coefficients multiplying divergent logarithms in the
limit. So as to be able to provide useful boundary data and checks to the
amplitude bootstrap in general kinematics, here we will be focusing on the
contributions to the NMHV (super)amplitude, which carries both rational and
transcendental dependence on the kinematics. Given that the $N>3$ OPE
contrinutions only start contributing at higher loop orders, in this manner we
will be able to determine the finite coefficients in the DS limit of the NMHV
hexagon for a particular, so-called (1111) component of its rational
depenendence through weight 12 and eight loops.

With these results at hand, it is possible to study further interesting
subspaces of the DS limit. As an example, we will indeed also specialize them
to a DS boundary point known as the origin limit [50], where a similar OPE
resummation strategy as the one employed here, has led to finite-coupling
conjectures for the form of the MHV amplitude [59], which exhibits a Sudakov-
like exponentiation. As was observed in the latter paper, and we confirm here,
the NMHV amplitude no longer exhibits this exponentiation. Nevertheless, we
observe a general pattern for its leading divergence at the origin, which may
be valid to all loops.

Finally, we briefly address the question of how ‘minimal’ the DS function
space is, namely whether it also contains redundant functions which are not
present in the amplitude or its derivatives. By comparing with the space
spanned by the latter, we notice that non-constant redundant functions already
start appearing at weight three. Understanding the reason for this redundancy,
and further refining our space by eliminating it, are interesting questions
that we leave for the future. Perhaps more importantly, in follow-up work we
look forward to using the host of explicit results we have obtained, in order
to develop new direct evaluation algorithmms for the two-gluon OPE
contributions, which may also be more broadly applicable to perturbative
quantum field theory. The presence of a similar successful paradigm, where the
knowledge of certain double pentagon ladder integrals [60] to high loop order
[26] subsequently led to new computational methods [61], is very encouraging
in this respect.

This paper is organized as follows. In Section 2, we start by reviewing the
essential analytic properties of the six-particle amplitude in general
kinematics, and then move on to work out their implications in the DS limit.
Relying on these properties, in Section 3 we describe the construction of our
DS function space and its expansion in the collinear limit. The latter is to
be compared with the predictions of the Wilson loop OPE, discussed in Section
4. The thus obtained new results on the on the NMHV amplitude in the DS limit
are presented in Section 5. Finally, we have included two appendices with
further details on our DS space construction. Our results are also attached as
computer-readable files accessible at [62].

##  2 The six-gluon amplitude in the double-scaling limit

In this section we deduce the analytic properties of the six-particle
amplitude in the DS limit. In subsection 2.1 we first review some background
information on the normalization of the amplitude, its analytic structure in
general kinematics, and the class of multiple polylogarithms encoding it. Then
in subsection 2.2 we define the DS limit, and describe the potential amplitude
singularities in the limit. Subsection 2.3 continues with the analysis of the
property of integrability and of the extended Steinmann relations, especially
illustrating the constraining power of the latter. Finally in subsection 2.4
we derive additional conditions obeyed by the amplitude and its derivatives in
the DS limit, stemming from the absence of unphysical branch cuts.

###  2.1 Analytic structure of the normalized amplitude

The infrared divergence structure of the six-particle amplitude is well-
understood [33] and can be factored out, giving rise to a finite normalized
amplitude. Different conventions on this normalization reflect the freedom to
also absorb finite terms in this infrared-divergent factor, which here we will
choose as the BDS-like ansatz $A^{\text{BDS-like}}_{6}$ [63]. Its precise form
will not be important for our purposes, and instead we will be focusing on the
BDS-like normalized amplitudes of the two inequivalent helicity
configurations,

$\mathcal{E}:=\frac{A_{6,\textrm{MHV}}}{\,A^{(0)}_{6,\textrm{MHV}}\,A^{\text{BDS-
like}}_{6}}\,,\quad\mathrm{E}:=\frac{A_{6,\textrm{NMHV}}}{\,A^{(0)}_{6,\textrm{NMHV}}\,A^{\text{BDS-
like}}_{6}}\,,$ (2.1)

where we have also divided out by their known tree-level contribution. While
$\mathcal{E}$ is given just by a single bosonic function of the kinematics,
$\mathrm{E}$ may in turn be further decomposed as

$\displaystyle\mathrm{E}=\tfrac{1}{2}\Bigl{[}\bigl{(}(1)+(4)\bigr{)}E(u,v,w)+\bigl{(}(2)+(5)\bigr{)}E(v,w,u)+\bigl{(}(3)+(6)\bigr{)}E(w,u,v)$
$\displaystyle+\bigl{(}(1)-(4)\bigr{)}\tilde{E}({u},{v},{w})+\bigl{(}(2)-(5)\bigr{)}\tilde{E}({v},{w},{u})+\bigl{(}(3)-(6)\bigr{)}\tilde{E}({w},{u},{v})\Bigr{]}.$
(2.2)

where $(1):=[23456]$ and its cyclic permutations denote the so-called
$R$-invariants [64, 65, 60], namely rational terms of the kinematics and the
Grassmann variables, encoding the superconformal and dual superconformal
symmetry of the amplitude.

The two functions $E$ and $\tilde{E}$ introduced in eq. (2.2), as well as the
entire MHV amplitude $\mathcal{E}$, are bosonic functions of the kinematical
data, which can be conveniently parametrised by a set of three cross-ratios:

$\\{u,v,w\\}:=\Bigl{\\{}\frac{s_{12}s_{45}}{s_{123}s_{345}},\frac{s_{23}s_{56}}{s_{234}s_{123}},\frac{s_{34}s_{61}}{s_{345}s_{234}}\Bigr{\\}},$
(2.3)

which are invariant under the parity transformation. Evidence from all
explicit results to date, as well as from the analysis of the integrand [66]
(note however the caveats pointed out in [67]) that the order $g^{2L}$
($L$-loop) contribution in the weak coupling expansion of
$\mathcal{E},E,\tilde{E}$ can be expressed in terms of _multiple
polylogarithms_ (MPLs) [68, 69, 70, 71] (see also the review [58]) of
transcendental weight $p=2L$. A function $F^{(p)}$ is said to be an MPL of
weight $p$ if its total differential obeys

$dF^{(p)}=\sum_{\mafx@beta\in{\mathcal{A}}}\bigl{(}F^{(p-1)}\bigr{)}^{\mafx@beta}\>d\operatorname{log}(\mafx@beta)\,,$
(2.4)

such that that $\bigl{(}F^{(p-1)}\bigr{)}^{\mafx@alpha}$ is an MPL of weight
$(p-1)$ and so on, where the recursive definition terminates with the usual
logarithms ($p=1$) on the left-hand side, and rational numbers ($p=0$) as
coefficients of the total differentials on the right-hand side. The set
$\mathcal{A}$ of arguments of dlog forms is called the _symbol alphabet_ , and
it encodes positions of possible branch points of the $F^{(p)}$ function.

This recursive nature of the differential of MPLs is a part of a deeper
structure revealed by the _coproduct_ (more precisely, _coaction_)
$\mafx@Delta$ [72, 73, 74, 75], which, very roughly, decomposes an MPL of
weight $p$ into a sum of tensor products of MPLs of lower weight. In
particular, the total differential (2.4) is essentially equivalent to the
$\\{p-1,1\\}$ component of $\mafx@Delta$,

$\mafx@Delta_{p-1,1}F^{(p)}=\sum_{\mafx@beta\in\mathcal{A}}\bigl{(}F^{(p-1)}\bigr{)}^{\mafx@beta}\otimes\bigl{[}\log(\mafx@beta)\mod\,(\mathrm{i}\mafx@pi)\bigr{]}\,.$
(2.5)

Further considering the total differential of $F^{(p-1)}$, or equivalently the
analogue of eq. (2.5) for the latter, then yields the $\\{p-2,1,1\\}$
component of the coproduct,

$\displaystyle\mafx@Delta_{p-2,1,1}F^{(p)}=\sum_{\mafx@alpha,\mafx@beta\in{\mathcal{A}}}\bigl{(}F^{(p-2)}\bigr{)}^{\mafx@alpha,\mafx@beta}\otimes\log\mafx@alpha\otimes\log\mafx@beta\,,$
(2.6)

where we will also refer to the leftmost factor on the right-hand side as the
‘double coproduct’. In the above formula, and in what follows, identification
of logarithmic factors up to $\mathrm{i}\mafx@pi$ is implied in all but the
first slot of the tensor product. We can also continue the decomposition of
the leftmost coproduct factor, until we reach the maximal, $p$ times iterated
coproduct $\\{1,\ldots,1\\}$, which is also known as the _symbol_.

Another very usefull point of view on the MPLs stems from their integral
representation: an MPL is defined to be a $\mathbb{Q}$-linear combination of
the following iterated integrals (sometimes refered to as “hyperlogarithms”):

$\displaystyle
G(a_{1},\ldots,a_{p};z):=\int_{0}^{z}G(a_{2},\ldots,a_{p};t)\;\frac{\mathrm{d}t}{t-a_{1}},\quad
G(;z):=1,$ (2.7)

where the special case of only zero arguments is covered by this rule:

$\displaystyle
G(\underbrace{0,\ldots,0}_{p};z):=\frac{\bigl{(}\log{z}\bigr{)}^{p}}{p!}.$
(2.8)

The space of so-called _hexagon functions_ containing
$\\{\mathcal{E},E,\tilde{E}\\}$ and their coproducts is in fact a much smaller
subspace of all MPLs, due to additional physical and mathematical constraints
that the amplitude satisfies. The idea of the hexagon function bootstrap [40,
41, 42, 43, 44, 46, 47, 50, 51] is to first construct this space, from its
basis then form an ansatz for the amplitude, and finally find a unique
solution for the latter by comparing it to the behavior of the amplitude in
various kinematic limits, known by other means. Below we will briefly review
the additional analytic properties of hexagon functions, which we will then
specialize to the double-scaling limit.

_Symbol alphabet_. The space of hexagon functions containing
$\\{\mathcal{E},E,\tilde{E}\\}$ are MPLs as defined in eq. (2.4), whose
letters are drawn from the following list [40]:

$\cal A^{\text{hex}}=\\{u,v,w,1-u,1-v,1-w,y_{u},y_{v},y_{w}\\}\,.$ (2.9)

Apart from the parity-even cross ratios of eq. (2.3), we have also introduced
parity odd letters that are expressed in terms of the latter
($u_{1}=u,u_{2}=v,u_{3}=w$ and similarly for $y_{i}$) as

$y_{i}:=\frac{u_{i}-z_{+}}{u_{i}-z_{-}}\,,\quad
z_{\pm}:=\frac{1}{2}(-1+u+v+w\pm\sqrt{\mafx@delta})\,,\quad\mafx@delta:=(1-u-v-w)^{2}-4uvw\,.$
(2.10)

Zeros of these expressions label the possible locations of the branch cut
singularities of MPLs in the ansatz.

_First entry condition_. Locality dictates that in the Euclidean region,
amplitudes can only develop singularities at its boundary, where any of the
cross ratios tend to zero. This implies that only the first three letters in
eq. (2.9) are allowed to appear in the first entry of the symbol [32], or
equivalently that the weight-one space of hexagon functions consists of

$F^{(1)}\in\\{\log u,\log v,\log w\\}\,.$ (2.11)

_Integrability conditions_. Any well-defined hexagon function $F$ must satisfy

$\frac{\partial^{2}F}{\partial x_{i}\partial x_{j}}\ =\
\frac{\partial^{2}F}{\partial x_{j}\partial x_{i}}\,,\qquad i\neq j,$ (2.12)

for any choice of variables $x_{1},x_{2},x_{3}$ parametrizing the kinematics,
such as the cross ratios (2.3). Given the relation between total differentials
and coproduct components (2.4, 2.5), these translate to linear relations for
the left factors of the double coproduct of eq. (2.6), known as the
integrability conditions.

_Extended Steinmann relations_. Basic principles of quantum field theory
prohibit virtual particles in any physical process to simultaneously become
on-shell in two overlapping channels [54, 55, 56]. This can be translated into
the vanishing of certain double discontinuities of the associated physical
quantity, or equivalently into restrictions on the first two entries of its
symbol (if it is described by MPLs). In [26, 51] it was realized that these
relations in fact hold not on just the first two, but on any adjacent letters
in the symbol. These ‘extended Steinmann relations’ may be equivalently be
stated in terms of the double coproducts of any hexagon function $F$ as

$\displaystyle F^{v,u}+F^{v,w}+F^{w,u}+F^{w,w}=0,$ (2.13)

together with two other cyclic permutations $u\rightarrow v\rightarrow
w\rightarrow u$. When combined with the other analytic properties mentioned
thus far, they also automatically imply the following double coproduct
relations,

$\displaystyle F^{{1-u,v}}+F^{{1-u,w}}+F^{{1-u,1-v}}+F^{{1-u,1-w}}$
$\displaystyle=0$ (2.14) $\displaystyle
F^{{w,y_{w}}}+F^{{1-v,y_{w}}}+F^{{1-w,y_{w}}}-F^{{v,y_{v}}}-F^{{1-v,y_{v}}}-F^{{1-w,y_{v}}}+F^{{v,y_{w}}}-F^{{w,y_{v}}}$
$\displaystyle=0$ (2.15) $\displaystyle
F^{{1-u,y_{v}}}-F^{{1-u,y_{u}}}+F^{{y_{v},1-w}}-F^{{y_{w},1-w}}$
$\displaystyle=0$ (2.16) $\displaystyle
F^{{1-v,1-u}}+F^{{y_{u},y_{v}}}+F^{{y_{w},y_{w}}}-F^{{y_{u},y_{w}}}-F^{{y_{w},y_{v}}}$
$\displaystyle=0$ (2.17)

plus cyclic permutations. The above relations, eqs. (2.13)-(2.17), on
integrable functions are also predicted by the principle of cluster adjacency
[57, 36], which relies on the fact that the letters (2.9) are isomorphic to
the variables of a mathematical object known as $A_{3}$ cluster algebra [35].
Part of the structure of this object is the arrangement of the cluster
variables in overlapping sets, which may be naturally translated to double
coproduct relations.

_Branch cut conditions and transcendental constants_. All of the above double
coproduct relations in fact define hexagon functions only up to constants
times weight-one logarithms. Not every value of these constants is allowed
however, since it may lead to beyond-the-symbol terms with unphysical branch
cuts. As was discussed in [42, 46], one way to eliminate this possibility is
to additionally impose that hexagon functions are well-behaved in certain
kinematic limits.

Here we will adopt the choice of the aforementioned papers, and consider the
soft limit

$u,w\rightarrow 0,v\rightarrow
1\,\,\,\text{with}\,\,\,\frac{u}{1-v},\frac{w}{1-v}\,\,\,\text{fixed}\,\,\,\Leftrightarrow\,\,\,y_{v}\rightarrow
1\,\,\,\text{with}\,\,\,y_{u},y_{w}\,\,\,\text{fixed}\,,$ (2.18)

where the $\\{p-1,1\\}$ coproduct component of hexagon functions should obey

$F^{1-v}\big{|}_{y_{v}\rightarrow 1}=F^{y_{u}}\big{|}_{y_{v}\rightarrow
1}=F^{y_{w}}\big{|}_{y_{v}\rightarrow 1}=0\,.$ (2.19)

These conditions need to also be supplemented with the set of transcendental
constants that we include as independent functions in our basis, and then they
determine the precise rational coefficients these multiply weight-1 logarithms
with. As is also reviewed in [53], on general grounds these constants should
be drawn from multiple zeta values (MZV), and in [50, 51] it was furthermore
conjectured that only their subset of even ordinary zeta values of weight at
least four,

$\displaystyle\\{\mafx@zeta_{4},\mafx@zeta_{6},\mafx@zeta_{8},\mafx@zeta_{10},\mafx@zeta_{12},\ldots\\}$
(2.20)

is necessary for the six-particle amplitude and its derivatives at any loop
order.

Let us now proceed to work out the consequences of these analytic properties
of the amplitude in the double-scaling limit, which will be the focus of this
article.

###  2.2 The double-scaling (DS) limit

Figure 1:  The $A_{3}$ Stasheff associahedron, relevant for six-particle
scattering. The positive region in the space of kinematics amounts to the
interior of the polytope, whereas pentagonal faces correspond to double-
scaling limits, and square faces to soft (or equivalently multi-Regge) limits.
The two  red pentagonal faces represent the two possible double scaling limits
DS1 and DS2 that are considered in this work.

The double-scaling limit [32, 13, 27] of the six-particle amplitude in the
planar $\mathcal{N}=4$ SYM is generally defined as the limit where one of the
three cross ratios (2.3) goes to zero, whereas the remaining two are held
fixed. In this work we will choose the particular orientation

$\text{DS limit:}\quad w\rightarrow
0\,\,\,\text{with}\,\,\,u,v\,\,\,\text{fixed},$ (2.21)

where it is easy to show that the discriminant of eq. (2.10) reduces to

$\sqrt{\mafx@delta}\xrightarrow[\text{DS}]{}\pm(1-u-v)\left[1-\frac{(1-u-v+2uv)w}{(1-u-v)^{2}}\right]+{\mathcal{O}}(w^{2})\,,$
(2.22)

with different signs denoting the two possible choices of the square root
branch, such that the odd letters become

$y_{u}\rightarrow\left[\frac{u}{1-v}\right]^{\pm 1}\,,\quad
y_{v}\rightarrow\left[\frac{v}{1-u}\right]^{\pm 1}\,,\quad
y_{w}\rightarrow\left[\frac{(1-u)(1-v)w}{(1-u-v)^{2}}\right]^{\pm 1}\,.\quad$
(2.23)

Comparing with eq. (2.9), we thus see that in this limit hexagon functions $F$
reduce to divergent logarithms in $w$, times ‘DS functions’ $f$ with alphabet

$\\{u,v,1-u,1-v,1-u-v\\}\,.$ (2.24)

By virtue of eq. (2.23), the $\\{p-1,1\\}$ coproducts of the DS functions are
related to the hexagon functions $F$ as follows:

$\displaystyle f^{u}$ $\displaystyle=F^{u}\pm F^{y_{u}}\,,$ (2.25)
$\displaystyle f^{v}$ $\displaystyle=F^{v}\pm F^{y_{v}}\,,$ $\displaystyle
f^{1-u}$ $\displaystyle=F^{1-u}\pm(F^{y_{w}}-F^{y_{v}})\,,$ $\displaystyle
f^{1-v}$ $\displaystyle=F^{1-v}\pm(F^{y_{w}}-F^{y_{u}})\,,$ $\displaystyle
f^{1-u-v}$ $\displaystyle=\mp 2F^{y_{w}}\,.$

(plus $f^{\prime w}=F^{w}\pm F^{y_{w}}$ if $f^{\prime}$ denotes the products
of $f$ with $\log w$). As it was with eq. (2.22), the $\pm$ sign here is due
to the fact that hexagon functions are in fact well-defined only in the double
cover of the cross-ratio parametrization of the kinematics. In other words,
there really exist two $w\rightarrow 0$ double-scaling limits DS1 and DS2,
that are related by a parity transformation, $y_{i}\rightarrow 1/y_{i}$. With
the appropriate choice of independent kinematic variables, where the region of
positive kinematics is the particular blowup of the unit cube shown in Figure
1, the two DS limits are the boundaries depicted in  red. As is reviewed in
e.g. [36], the six-particle positive region has the topology of the Stasheff
associahedron, closely related to the $A_{3}$ cluster algebra, whereas
pentagonal faces represent its $A_{2}$ subalgebras.

In what follows, we will use an alphabet equivalent to (2.24), which arises
naturally when evaluating the Wilson loop OPE contributions to the DS limit .
In particular, we define

$\displaystyle x:=-\frac{1-u-v}{v}\,,\quad y:=\frac{1-v}{1-u-v}\,,$ (2.26)

or conversely

$u=\frac{x(1-y)}{1-xy}\,,\quad v=\frac{1}{1-xy}\,,$ (2.27)

such the alphabet (2.24) takes the equivalent form

$\mathcal{A}:=\\{x,y,1-x,1-y,1-xy\\}\,,$ (2.28)

which we will adopt from this point on. For completeness, we also quote how
the $\\{p-1,1\\}$ coproducts of the DS functions in the two equivalent
alphabets are related by virtue of eq. (2.26),

$\displaystyle f^{u}$ $\displaystyle=f^{1-y}\,,$ $\displaystyle f^{v}$
$\displaystyle=-f^{x}-f^{1-x}-f^{1-xy}\,,$ $\displaystyle f^{1-u}$
$\displaystyle=f^{1-x}\,,$ (2.29) $\displaystyle f^{1-v}$
$\displaystyle=f^{y}\,,$ $\displaystyle f^{1-u-v}$
$\displaystyle=f^{x}-f^{y}-f^{1-y}\,.$

Finally, in the above choice of alphabet, it is evident that for DS functions,
the first entry condition (2.11) becomes

$f^{(1)}\in\\{\log(1-xy),\log(x(1-y))\\}\,.$ (2.30)

###  2.3 Integrability and Extended Steinmann conditions

After defining the DS limit, in the previous subsection we also analyzed the
symbol letters and first entry condition of the space of functions expected to
capture six-particle scattering in this limit. Here, we continue by deriving
constraints on the double coproducts of these functions.

First, the integrability conditions are very simple to derive for the alphabet
(2.28) we will be using from a single equation of the form (2.12), with
$x_{1}=x$ and $x_{2}=y$. It gives rise to the following set of six equations,

$\displaystyle f^{x,y}-f^{y,x}=f^{1-x,y}-f^{y,1-x}=f^{x,1-y}-f^{1-y,x}=0,$
(2.31) $\displaystyle f^{1-x,1-xy}+f^{1-x,1-y}-f^{1-xy,1-x}-f^{1-y,1-x}=0,$
$\displaystyle f^{1-x,1-y}+f^{1-xy,1-y}-f^{1-y,1-x}-f^{1-y,1-xy}=0,$
$\displaystyle
f^{x,1-xy}-f^{1-x,1-y}-f^{1-xy,x}+f^{1-xy,y}-f^{y,1-xy}+f^{1-y,1-x}=0.$

Next, we examine the extended Steinmann relations, as well as the additional
relations they imply on integrable functions with the alphabet (2.9) and first
entries (2.11), eqs. (2.13)-(2.17). We find that only two linear combinations
of these 15 equations survive in the DS limit, in particular

$\displaystyle
f^{1-x,y}=f^{x,1-xy}+f^{1-x,1-xy}+f^{1-xy,1-x}+f^{1-xy,1-xy}+f^{1-xy,1-y}-f^{y,1-xy}=0.$
(2.32)

Owing to their origin in general kinematics, we will denote eqs. (2.32) as the
‘DS Extended Steinmann relations’.

One may be tempted to think that the DS Extended Steinmann relations may only
marginally reduce the size of the relevant function space, due to their
relatively small number. As we see in Table 1 however, their effect is in fact
very significant: Focusing momentarily on the space of symbols, i.e.
polylogarithmic functions modulo transcendental constants (these will be
reinstated in the next subsection) with the alphabet (2.28), in the first line
we quote their number as a function of their weight $p$, when the weight-one
or first entry space is constrained as dictated by eq. (2.30). We notice that
this number grows by roughly a factor of three at each weight (more precisely,
it is equal to $3^{p-1}+2^{p-1}$).

On the second line of Table 1, we display how many of the functions of the
first line, additionally obey the DS Extended Steinmann relations (2.32).
Evidently, their number now grows by roughly a factor of two instead of three.
Thus the DS Extended Steinmann relations are responsible for a massive
reduction in the size of the relevant function space, for example by more than
98% for weight 13. They will thus be pivotal for constructing this function
space to high weight, as detailed in the next section, and for bootstrapping
new results for the six-particle NMHV amplitude in the DS limit, as presented
in Section 5.

weight $p$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---|---|---
First entry | 2 | 5 | 13 | 35 | 97 | 275 | 793 | 2315 | 6817 | 20195 | 60073 | 179195 | 535537
Ext. Stein. | 2 | 4 | 9 | 19 | 39 | 78 | 154 | 302 | 591 | 1157 | 2269 | 4460 | 8788

Table 1: The dimension of the space of symbols with physical branch cuts in
the DS limit (first line), that additionally obey the extended Steinmann
relations (second line).

This concludes the analysis of all symbol-level constraints on the DS
functions space. In the next subsection, we discuss the additional constraints
necessary to promote them to functions.

###  2.4 Branch cut conditions and transcendental constants

As briefly reviewed in subsections 2.1 and 2.2, hexagon or DS functions are
allowed to have branch points only when a cross ratio approaches zero (or
infinity). Focusing on DS functions, one should thus require that they are
free of branch point singularities when $u\rightarrow 1$, $v\rightarrow 1$ or
$u\rightarrow 1-v$, or equivalently that their derivatives are free of poles
there. Given that

$\frac{\partial f}{\partial
u}=\frac{f^{u}}{u}-\frac{f^{1-u}}{1-u}-\frac{f^{1-u-v}}{1-u-v}\,,$ (2.33)

the physical branch cut conditions thus translate to

$f^{1-u}\big{|}_{u\rightarrow 1}=f^{1-v}\big{|}_{v\rightarrow
1}=f^{1-u-v}\big{|}_{u\rightarrow 1-v}=0\,.$ (2.34)

Let us now translate the above branch cut limits and conditions in our choice
of independent variables (2.26) and symbol letters (2.28). As far as the
limits are concerned, we have

$\displaystyle u$ $\displaystyle\rightarrow 1:x\rightarrow 1\,\,\text{with $y$
fixed}\,,$ $\displaystyle v$ $\displaystyle\rightarrow 1:y\rightarrow
0\,\,\text{with $x$ fixed}\,,$ (2.35) $\displaystyle u$
$\displaystyle\rightarrow 1-v:x\rightarrow 0\,\,\text{with $xy$ fixed}\,,$

whereas by virtue of (2.29) the branch cut conditions become

$f^{y}\big{|}_{y\rightarrow 0}=f^{1-x}\big{|}_{x\rightarrow
1}=f^{x}-f^{y}-f^{1-y}\big{|}_{\begin{subarray}{c}x\rightarrow 0\hfill\\\
xy\,\,\text{fixed}\end{subarray}}=0\,.$ (2.36)

A potential subtlety with the branch cut conditions (2.34) or (2.36), is that
the limits they describe are not just the edges of the pentagonal faces of the
Stasheff polytope depicted in Figure 1, which are usually simpler to impose.
This is especially the case with the third limit/condition in (2.36), which
requires taking $y\rightarrow\infty$ simultaneously with $x\rightarrow 0$, and
thus lies outside of the $0<x,y<1$ square, where any function with the
alphabet $\mathcal{A}$ given in eq. (2.28) is free of branch cuts.

To avoid this complication, we will instead rely on the fact that the DS limit
overlaps with the soft limit on the edge between a  red pentagonal and a
green square face of the Stasheff polytope in Figure 1, where branch cut
conditions for the general hexagon function space have been derived. From
(2.27), it is clear that the DS limit intersects with the soft limit (2.18)
for $x\rightarrow 0$ with $y$ fixed. Then, using (2.25) and (2.29) to relate
the DS coproducts in the $\mathcal{A}$ alphabet to the hexagon function
coproducts, and specializing to the double-scaling/soft overlap, we can show
that by virtue of eq. (2.19),

$f^{x}-f^{1-y}\big{|}_{x\rightarrow
0}\,=F^{1-v}\mp(F^{y_{w}}+F^{y_{u}})\big{|}_{y_{v}\rightarrow 1}=0\,.$ (2.37)

Instead of eq. (2.36), we will thus choose to impose the following set of
simpler branch cut conditions,

$\displaystyle f^{y}\big{|}_{y\rightarrow 0}=f^{1-x}\big{|}_{x\rightarrow
1}=f^{x}-f^{1-y}\big{|}_{x\rightarrow 0}=0\,.$ (2.38)

Finally, let us come to the question of which transcendental constants we
should include in our DS function space as independent basis elements. While
only the subset (2.20) of MZVs have been found to be necessary in general
kinematics, it is easy to show that in the DS limit we will certainly also
need $\mafx@zeta_{2}$. In particular, this is what the weight-two hexagon
function $\text{Li}_{2}(1-1/w)=-G(0,1;1-1/w)$, see e.g. [51], reduces to in
the limit. A similar analysis at weight three indicates that we also need to
include $\mafx@zeta_{3}$ as an independent constant. Given that these low-
weight constants will be responsible for the bulk of non-trivial functions
with vanishing symbols in our space, we will choose to be agnostic and include
all MZVs, a basis of which is contained in appendix A through weight twelve,
as independent functions in our space. We will then come back and examine the
possible redundancy of our space in the closing subsection 5.3.

Summarizing, we define our space of Extended Steinmann DS functions,
$\mathcal{H}^{\mathrm{DS}}$, to consist of MPLs with the alphabet of eq.
(2.28), whose first entry space is restricted to (2.30), and obeys the
integrability (2.31) DS extended Steinmann (2.32) and branch cut conditions
(2.38), also containing all MZVs as independent functions. In the next
section, we describe the construction of this space in detail.

##  3 Bootstrapping the DS Functions

The goal of this section is to construct a linear space of functions
$\mathcal{H}^{\mathrm{DS}}$, which we refer to as the (Extended Steinmann) DS
space, encoding six-particle scattering in the DS limit, defined in the
previous section. The DS space has a natural grading by transcendental weight,

$\mathcal{H}^{\mathrm{DS}}=\bigoplus_{p\geq
1}\mathcal{H}^{\mathrm{DS}}_{p}\,,$ (3.1)

where the $\mathcal{H}^{\mathrm{DS}}_{p}$ components consist of linear
combinations of polylogarithms and Multiple Zeta Values (MZVs) of weight $p$.
MZVs form a graded subalgebra of their own,

$\mathcal{Z}=\bigoplus_{p\geq 0}\mathcal{Z}_{p}\,,$ (3.2)

with $\mathcal{Z}_{0}=\mathbb{Q}$ and each component at weight $p$ has
dimensionality $|\mathcal{Z}_{p}|:=\dim_{\mathbb{Q}}\mathcal{Z}_{p}$ with
corresponding basis collected in the $p^{\text{th}}$ element of eq. (A.1).

In subsection 3.1, we first explain how to build a coproduct representarion of
$\mathcal{H}^{\mathrm{DS}}$, relying on the analytic properties worked out in
Section 2. Then, in subsection 3.2 we promote this representation to explicit
expressions in terms of MPLs. Finally, in subsection 3.3 carry out the
$x\rightarrow 0$ series expansion of our functions, which will allow us to
match them against predictions for the amplitude in the collinear limit, as we
will discuss in the next section.

###  3.1 Solving the integrability and extended Steinmann constraints

weight | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---
number | $5$ | $12$ | $26$ | $56$ | $116$ | $236$
non-zero | $12$ | $31$ | $96$ | $263$ | $901$ | $2.6\times 10^{3}$
density | $0.24$ | $0.10$ | $0.06$ | $0.04$ | $0.03$ | $0.019$
max | $1$ | $2$ | $2$ | $20$ | $40$ | $560$
weight | $8$ | $9$ | $10$ | $11$ | $12$ | $13$
number | $474$ | $943$ | $1867$ | $3686$ | $7270$ | $14295$
non-zero | $9.9\times 10^{3}$ | $39\times 10^{3}$ | $.2\times 10^{6}$ | $.8\times 10^{6}$ | $3.6\times 10^{6}$ | $15\times 10^{6}$
density | $0.018$ | $0.018$ | $0.024$ | $0.025$ | $0.026$ | $0.029$
max | $1120$ | $5.6\times 10^{4}$ | $6.0\times 10^{8}$ | $1.9\times 10^{12}$ | $5.7\times 10^{17}$ | $2.0\times 10^{25}$
Table 2:  Features of the $\mathcal{H}^{\mathrm{DS}}_{p}$ basis we have
constructed, in its coproduct tensor representation. The first row represents
the weight $p$, the second row gives the dimension of the space (including
MZVs), the third, fourth and fifth rows show the amount of non-zero entries,
densities and maximum entry values of the coproduct tensors, respectively.

Now let us turn to the construction of a particular basis
$\\{f^{(p)}_{j}\\}_{j\in J_{p}}$ of the $\mathcal{H}^{\mathrm{DS}}_{p}$ space
of the DS functions of weight $p\geq 1$, where the elements of the basis are
labeled by a list $J_{p}:=\\{1,2,\ldots,|J_{p}|\\}$. In this section we
construct the $\mathcal{H}^{\mathrm{DS}}_{p}$ in the so-called coproduct form,
then further refine it in Section 2.4, and finally promote it to the full
functional form in Section 3.2. But before we do this, we first need to
properly introduce the initial conditions and our notation for coproducts.

The starting point of our recursive construction is the lowest weight space
$\mathcal{H}^{\mathrm{DS}}_{1}$, which is two dimensional:
$\dim_{\mathbb{Q}}(\mathcal{H}^{\mathrm{DS}}_{1})\equiv|J_{1}|=2$, and is
spanned by the following two logarithms:

$\displaystyle f^{(1)}_{1}:=G(\tfrac{1}{y};x)=\log(1-xy)=-\sum_{n\geq
1}\tfrac{1}{n}(xy)^{n},$ (3.3) $\displaystyle
f^{(1)}_{2}:=G(1;y)+G(0;x)=\log\bigl{(}x(1-y)\bigr{)}.$ (3.4)

Note, that around $x=0$ the first function $f^{(1)}_{1}$ has a well-defined
Taylor expansion, while the second function $f^{(1)}_{2}$ is logarithmically
divergent. In Section 4 such power-and-log expansions of the elements of
$\mathcal{H}^{\mathrm{DS}}$ will become our main tool for calculating
amplitudes in the DS limit.

Next, as was outlined in Section 2.1, the $\\{p-1,1\\}$ coproduct component of
a weight $p$ DS function $f^{(p)}_{i}\in\mathcal{H}^{\mathrm{DS}}_{p}$ can be
expressed as a $\mathbb{Q}$-linear combination of certain $\otimes$-products
with weight $(p-1)$ DS functions
$f^{(p-1)}_{j}\in\mathcal{H}^{\mathrm{DS}}_{p-1}$ in their left entries and
logarithms of the $\mathcal{A}$ alphabet elements in the right. As was first
understood in [48], see also [51], it is more efficient to encode this
component using a single object
$c^{(p)}_{ij\mafx@alpha}\in\mathbb{Q}^{|J_{p}|\times|J_{p-1}|\times|\mathcal{A}|}$,
which we refer to as the coproduct tensor, namely

$\displaystyle\mafx@Delta_{p-1,1}\bigl{(}f_{i}^{(p)}\bigr{)}=\sum_{j,\mafx@alpha}c^{(p)}_{ij\mafx@alpha}\,f^{(p-1)}_{j}\otimes\log(\mafx@alpha),$
(3.5)

where $i\in J_{p}$ and we have omitted the summation ranges $j\in J_{p-1}$ and
$\mafx@alpha\in\mathcal{A}$ for simplicity. It follows from the
coassociativity of the coproduct $\mafx@Delta$, that its $\\{p-2,1,1\\}$
component is then given by an inner product of two coproduct tensors at weight
$p$ and $p-1$:

$\displaystyle\mafx@Delta_{p-2,1,1}\bigl{(}f_{i}^{(p)}\bigr{)}=\sum_{\begin{subarray}{c}j,k\\\
\mafx@alpha,\mafx@beta\end{subarray}}c^{(p)}_{ij\mafx@alpha}\,c^{(p-1)}_{jk\mafx@beta}\,f^{(p-2)}_{k}\otimes\log(\mafx@beta)\otimes\log(\mafx@alpha).$
(3.6)

Now we are ready to discuss the recursive step, i.e. the construction of the
$\mathcal{H}^{\mathrm{DS}}_{p}$ out of the already known
$\mathcal{H}^{\mathrm{DS}}_{p-1}$ space at previous weight. Following the
coproduct bootstrap method, we formulate a set of linear constraints on the
coproducts, whose nullspace defines the $\mathcal{H}^{\mathrm{DS}}_{p}$ space.
To do that, we start with a tensor product
$\mafx@Delta_{p-2,1}(\mathcal{H}^{\mathrm{DS}}_{p-1})\otimes\mathcal{A}$ of
the $\\{p-2,1\\}$ coproduct component of the whole
$\mathcal{H}^{\mathrm{DS}}_{p-1}$ space, encoded in the
$c^{(p-1)}_{jk\mafx@beta}$ tensor, and another copy of the $\mathcal{A}$
alphabet. Using the natural inclusion
$\mafx@Delta_{p-2,1}(\mathcal{H}^{\mathrm{DS}}_{p-1})\otimes\mathcal{A}\subset\mathcal{H}^{\mathrm{DS}}_{p-2}\otimes\mathcal{A}\otimes\mathcal{A}$,
we impose a set $\mathcal{S}:=\\{1,\ldots,|\mathcal{S}|\\}$ of homogeneous
linear conditions on the last two entries of the coproduct via a map
$\mathcal{I}:\mathcal{A}\otimes\mathcal{A}\rightarrow\mathbb{Q}^{|\mathcal{S}|}$
that acts on the tensor space as follows:

$\displaystyle\mathcal{H}^{\mathrm{DS}}_{p-1}\otimes\mathcal{A}\xrightarrow[]{\mafx@Delta_{p-2,1}\otimes\>\operatorname{id}}\mafx@Delta_{p-2,1}(\mathcal{H}^{\mathrm{DS}}_{p-1})\otimes\mathcal{A}\xrightarrow[]{\operatorname{id}\otimes\>\mathcal{I}}\mathcal{H}^{\mathrm{DS}}_{p-2}\otimes\mathbb{Q}^{|\mathcal{S}|},$
(3.7)

and look for its kernel. In our implementation $\mathcal{I}$ contains the
integrability as well as the extended Steinmann conditions shown in eqs.
(2.31) and (2.32), so that $|\mathcal{S}|=8$ in our case. In practice, the map
$\mathcal{I}$ is represented as a tensor $\mathcal{I}_{\mafx@alpha\mafx@beta
s}\in\mathbb{Q}^{|\mathcal{A}|\times|\mathcal{A}|\times|\mathcal{S}|}$, whose
explicit values are stated in eq. (B.2). Contraction of the
$\mathcal{I}_{\mafx@alpha\mafx@beta s}$ tensor together with the coproduct
tensor $c^{(p-1)}_{jk\mafx@beta}$ gives a realization of the
$M:=[\operatorname{id}\otimes\>\mathcal{I}]\circ[\mafx@Delta_{p-2,1}\otimes\operatorname{id}]$
map from eq. (3.7), whose tensorial representation (after some transpositions)
can easily be spelled out:

$\displaystyle
M_{ksj\mafx@alpha}:=\sum_{\mafx@beta}c^{(p-1)}_{jk\mafx@beta}\,\mathcal{I}_{\mafx@alpha\mafx@beta
s}.$ (3.8)

A more explicit derivation of this definition is given in Appendix B. This map
$M$ encodes the basic linear constraints needed for determining the
$\mathcal{H}^{\mathrm{DS}}_{p}$ space.

Now, let
$\tilde{c}^{(p)}_{\tilde{i}(j\mafx@alpha)}\in\mathbb{Q}^{|\tilde{J}_{p}|\times(|J_{p-1}|\cdot|\mathcal{A}|)}$
denote a basis of the $\operatorname{Ker}(M_{(ks)(j\mafx@alpha)})$ nullspace:

$\displaystyle\sum_{(j\mafx@alpha)}\,M_{(ks)(j\mafx@alpha)}\,\tilde{c}^{(p)}_{\tilde{i}(j\mafx@alpha)}=0,\quad\text{for
each }k\in J_{p-1},\>s\in\mathcal{S},\>\tilde{i}\in\tilde{J}_{p},$ (3.9)

where the list $\tilde{J}_{p}:=\\{1,\ldots,|\tilde{J}_{p}|\\}$, for the
moment, labels the basis elements, while the brackets $(j\mafx@alpha)$ and
$(ks)$ denote vectorization of the corresponding tensor indices, which label
bases of the leftmost and the rightmost spaces in eq. (3.7) respectively. To
further reduce the nullspace $\operatorname{Ker}(M_{(\mafx@alpha j)(sk)})$ and
obtain the entire $\mathcal{H}^{\mathrm{DS}}_{p}$ space we exploit the
additional analyticity constraints shown in eq. (2.38). Each of these 3 branch
cut conditions always evaluate to just MZVs, hence they produce
$3\times|\mathcal{Z}_{p-1}|$ additional linear equations at weight $p$.
Resolving these conditions further reduces the list of basis elements
$\tilde{J}_{p}\rightarrow J_{p}$, and leaves us with the final form of the
$c^{(p)}_{ij\mafx@alpha}$ coproduct tensors. Their properties with respect to
the weight $p$ are summarized in Table 2. Note, in particular, that the
produced coproduct tensors tend to be very sparse and their densities never
exceed a few percent. We present the coproduct tensors for weight $p\leq 13$
in the ancillary file coproducts-w2-13.m.

In practice, the natural $\operatorname{GL}(|\tilde{J}_{p}|)$ freedom of
choosing a particular version of the coproduct tensors (prior to imposing the
branch cut conditions) $\tilde{c}^{(p)}_{\tilde{i}(j\mafx@alpha)}$ can be used
to improve the efficiency of the computer implementation. There are, of
course, many possible metrics for optimization: we can look for the most
sparse tensors in the output, the fastest overall execution time, the lowest
values of the $\lVert\cdot\rVert_{\infty}$ norm, or some heuristic mix of
those. The main computational challenge is the Gaussian elimination in the
field of rationals $\mathbb{Q}$ required for the nullspace determination. As
the dimensions of the equation matrix $M_{(ks)(j\mafx@alpha)}$ grow
exponentially with increasing weight $p$, so do the denominators and
numerators in its entries. A very special care is therefore needed in order to
perform the Gaussian elimination at higher weights. Building on the ideas of
[45], we tested several possible optimization strategies, the most promising
of which are presented in Table 3. Our best method of computation consists of
reversing the order of vectors of the NullSpace’s output and application of
the Lenstra–Lenstra–Lovasz (LLL) reduction via Mathematica’s LatticeReduce
command. This way we were able to significantly reduce the sizes of the
integer entries in the produced coproduct tensors $c^{(p)}_{jk\mafx@beta}$, as
reflected in Table 3. At higher weights $p\geq 11$ we also made use of the
Spasm library [76] for performing the row reduction over finite fields and
subsequent rational reconstruction.

weight | A | B | C | D
---|---|---|---|---
2 | 1 | 1 | 1 | 1
3 | 2 | 2 | 1 | 1
4 | 2 | 2 | 2 | 2
5 | 20 | 20 | 10 | 7
6 | 40 | 20 | 60 | 105
7 | 560 | 420 | 2100 | 18366
8 | 1120 | 14850 | 5950 | 227673167
Table 3:  The sizes of maximal entries of the coproduct tensors
$c^{(p)}_{ij\mafx@alpha}$ at weight $p$ in the four computational setups that
we used. The version A, our best choice so far, reorders the rows of the
tensors and uses the LLL reduction; the version B in addition to that also
rescales the rows of the tensors in order to make them integer-valued and only
then applies the LLL; the version C only uses the LLL without any reordering
of the rows; version D does not modify the tensors in any way and serves as a
baseline for numeric optimizations.

###  3.2 Promoting coproducts to functions

Now everything is ready for construction of the full function space
$\mathcal{H}^{\mathrm{DS}}$ that is used to determine the MHV and NMHV six-
gluon amplitudes in the DS limit. We provide two equivalent representations of
the elements of $\mathcal{H}^{\mathrm{DS}}$: one in terms of MPLs and the
other one in terms of power-and-log series expansions. The latter is less
computationally demanding and instrumental for the Wilson loop OPE resummation
as explained in Section 4.2.

Our procedure essentially boils down to iterative integration of the
differentials 2.4 defined by the coproducts from eq. (3.5) along a fixed path
$\mafx@gamma$ shown in Figure 2. The path
$\mafx@gamma:=\mafx@gamma_{1}\mafx@gamma_{2}$ connects the base point of
integration $\\{x,y\\}=\\{0,0\\}$, at which we set the integration constants
to $0$, with a general kinematical point $\\{x,y\\}$ inside the unit square
$0<x,y<1$. Our choice of the integration path $\mafx@gamma$ dictates a special
representation of the DS functions
$f^{(p)}_{j}\in\mathcal{H}^{\mathrm{DS}}_{p}$ in terms of MPLs, see for
example [77] or [78],

$\displaystyle
f^{(p)}_{j}=\sum\limits_{\vec{X},\vec{Y}}g_{j\vec{X}\vec{Y}}\>G(\vec{X};x)\,G(\vec{Y};y),\quad
g_{j\vec{X}\vec{Y}}\in\mathcal{Z}^{|J_{p}|\times 3^{p}\times
2^{p}},\quad\bigl{|}g_{j\vec{X}\vec{Y}}\bigr{|}+\bigl{|}\vec{X}\bigr{|}+\bigl{|}\vec{Y}\bigr{|}=p,$
(3.10)

where the lists (which are also referred to as ‘words’) of arguments $\vec{Y}$
and $\vec{X}$ are drawn from the
$\vec{Y}\in\\{0,1\\}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}$
and
$\vec{X}\in\\{0,1,\tfrac{1}{y}\\}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}$
sets. Here by
<EMAIL_ADDRESS>we mean the set of all words made out of some finite set $\mafx@Sigma$, in
other words it is a disjoint union
$\mafx@Sigma^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}:=\mafx@epsilon\mathbin{\dot{\cup}}\mafx@Sigma\mathbin{\dot{\cup}}\mafx@Sigma^{2}\mathbin{\dot{\cup}}\ldots$,
where $\mafx@epsilon$ is an empty word, while any other word $\vec{X}$ of
length $\bigl{|}\vec{X}\bigr{|}$ lies in the $|\vec{X}|^{\text{th}}$
component: $\vec{X}\in\mafx@Sigma^{|\vec{X}|}$. The lengths of these two lists
and the transcendental weight of the MZV-valued coefficients, denoted as
$\bigl{|}\vec{X}\bigr{|}$, $\bigl{|}\vec{Y}\bigr{|}$, and
$\bigl{|}g_{j\vec{X}\vec{Y}}\bigr{|}$ respectively, add up to the
transcendental weight $p$ of the function for each term in the sum.

Figure 2:  Integration path $\mafx@gamma:=\mafx@gamma_{1}\mafx@gamma_{2}$ connecting the base point of integration $\\{0,0\\}$ and a general kinematical point $\\{x,y\\}$ inside the unit square.  alphabet $\mathcal{A}$ | $x$ | $y$ | $1-x$ | $1-y$ | $1-xy$
---|---|---|---|---|---
$\mafx@gamma_{1}$ | $0$ | $d\operatorname{log}(y)$ | $0$ | $d\operatorname{log}(y-1)$ | $0$
$\mafx@gamma_{2}$ | $d\operatorname{log}(x)$ | $0$ | $d\operatorname{log}(x-1)$ | $0$ | $d\operatorname{log}(x-\tfrac{1}{y})$
Table 4:  Pull-backs of the $d\operatorname{log}$-forms along the two
components of the integration path shown in Figure 2. These
$d\operatorname{log}$-forms act as the integration kernels in eqs. (3.11,
3.12, 3.13). Note, that along $\mafx@gamma_{1}$ the integration variable is
$y$, and along $\mafx@gamma_{2}$ the integration variable is $x$, while $y$
remains constant.

The integral along $\mafx@gamma$ of the $\\{p-1,1\\}$ coproduct component
(3.5) or, equivalently, of the differential of a given DS function splits into
two terms corresponding to the two parts of the integration path. The pull-
backs of the $d\operatorname{log}$-forms shown in Table 4 make sure that both
of these integrations preserve the general form of the (3.10) representation:
the integral along the $\mafx@gamma_{1}$ only modifies the
$g_{j\mafx@epsilon\vec{Y}}G(\vec{Y};y)$ terms in that sum, while the
$\mafx@gamma_{2}$ integration changes only the $G(\vec{X};x)$ factor. To
quantitatively describe the second integral along the $\mafx@gamma_{2}$ path
we introduce the following three linear operators:

$\displaystyle\mathcal{O}_{1}(G(\vec{X};x))$
$\displaystyle:=\int_{0}^{x}G(\vec{X};t)\>d\operatorname{log}{(t)}$
$\displaystyle=G(0,\vec{X};x),$ (3.11)
$\displaystyle\mathcal{O}_{2}(G(\vec{X};x))$
$\displaystyle:=\int_{0}^{x}G(\vec{X};t)\>d\operatorname{log}{(t-1)}$
$\displaystyle=G(1,\vec{X};x),$ (3.12)
$\displaystyle\mathcal{O}_{3}(G(\vec{X};x))$
$\displaystyle:=\int_{0}^{x}G(\vec{X};t)\>d\operatorname{log}{(t-\tfrac{1}{y})}$
$\displaystyle=G(\tfrac{1}{y},\vec{X};x).$ (3.13)

It is evident that these operators just prepend the corresponding letter to
the list of arguments. Later in eqs. (3.19, 3.20, 3.21) we will show how the
same integral operators act on the series expansions of MPLs, which turn out
to be more useful objects in practice.

In this manner, we have succeeded in promoting the coproduct representation of
our DS functions to explicit expressions in terms of MPLs through weight 12.
Due to size restrictions, in the ancillary file ds-w1-10.m we provide these
MPL representations through weight 10 and list the weight $p\leq 3$ functions
explicitly below.444Readers interested in the $p=11,12$ expressions are
welcome to contact the authors. In terms of the following compact notation,

$\displaystyle l_{x}:=\log(x),\qquad\mathcal{G}_{\vec{X}}:=G(\vec{X};x).$
(3.14)

which we will adopt from this point on, the weight $p=1$ functionsof eqs.
(3.3) and (3.4) read

$\displaystyle f^{(1)}_{1}$ $\displaystyle=l_{1-xy},\>f^{(1)}_{2}=l_{x(1-y)}.$
(3.15)

Then, there are five functions at weight $p=2$,

$\displaystyle f^{(2)}_{1}$
$\displaystyle=\mathcal{G}_{0\check{y}},\>f^{(2)}_{2}=\mathcal{G}_{0\check{y}}-\mathcal{G}_{1\check{y}}-\mathcal{G}_{01}+l_{1-x}l_{x(1-y)},\>$
$\displaystyle f^{(2)}_{3}$
$\displaystyle=\tfrac{1}{2}l_{x(1-y)}^{2},\>f^{(2)}_{4}=l_{x(1-y)}l_{1-xy}-l_{1-xy}^{2},\>f^{(2)}_{5}=\mafx@zeta_{2}\,,\>$
(3.16)

and 12 functions at $p=3$,

$\displaystyle f^{(3)}_{1}$
$\displaystyle=\mathcal{G}_{\check{y}0\check{y}},\>f^{(3)}_{2}=\mafx@zeta_{2}l_{1-xy},\>f^{(3)}_{3}=\mafx@zeta_{2}l_{x(1-y)},\>f^{(3)}_{4}=\mathcal{G}_{00\check{y}},\>$
$\displaystyle f^{(3)}_{5}$
$\displaystyle=\mathcal{G}_{01}l_{x(1-y)}+\mathcal{G}_{00\check{y}}-\mathcal{G}_{01\check{y}}-2\mathcal{G}_{001},\>f^{(3)}_{6}=\tfrac{1}{6}l_{x(1-y)}^{3},\>$
$\displaystyle f^{(3)}_{7}$
$\displaystyle=\tfrac{1}{2}l_{1-xy}l_{x(1-y)}^{2}+l_{x(1-y)}[-\mathcal{G}_{0\check{y}}-\mathcal{G}_{\check{y}1}]+\mathcal{G}_{00\check{y}}+\mathcal{G}_{0\check{y}1}+\mathcal{G}_{\check{y}01}-\mathcal{G}_{\check{y}0\check{y}}+\mathcal{G}_{\check{y}1\check{y}},\>$
$\displaystyle f^{(3)}_{8}$
$\displaystyle=\mathcal{G}_{0\check{y}}l_{x(1-y)}-2\mathcal{G}_{0\check{y}\check{y}},\>$
$\displaystyle f^{(3)}_{9}$
$\displaystyle=\tfrac{1}{2}l_{1-x}l_{x(1-y)}^{2}+l_{x(1-y)}[-\mathcal{G}_{01}-\tfrac{1}{2}l_{1-x}^{2}]-\mathcal{G}_{10\check{y}}+\mathcal{G}_{11\check{y}}+\mathcal{G}_{001}+\mathcal{G}_{011}+\mathcal{G}_{101}+\mafx@zeta_{2}l_{1-x},\>$
$\displaystyle f^{(3)}_{10}$
$\displaystyle=l_{x(1-y)}[\mathcal{G}_{0\check{y}}-\mathcal{G}_{1\check{y}}]-\mathcal{G}_{00\check{y}}+\mathcal{G}_{01\check{y}}-2\mathcal{G}_{0\check{y}\check{y}}+2\mathcal{G}_{1\check{y}\check{y}},\>$
$\displaystyle f^{(3)}_{11}$
$\displaystyle=l_{x(1-y)}[-\mathcal{G}_{1\check{y}}-\mathcal{G}_{\check{y}1}+l_{1-xy}^{2}]+\mathcal{G}_{01\check{y}}+\mathcal{G}_{0\check{y}1}-2\mathcal{G}_{0\check{y}\check{y}}+2\mathcal{G}_{1\check{y}\check{y}}+\mathcal{G}_{\check{y}01}-\mathcal{G}_{\check{y}0\check{y}}+\mathcal{G}_{\check{y}1\check{y}}-\tfrac{2}{3}l_{1-xy}^{3},\>$
$\displaystyle f^{(3)}_{12}$<EMAIL_ADDRESS>(3.17)

As was anticipated at the end of Section 2.2, we see the low-weight MZV
constants $\mafx@zeta_{2}$ and $\mafx@zeta_{3}$ are independent elements of
the $\mathcal{H}^{\mathrm{DS}}$ space, the same holds for other MZVs from eq.
(A.1) at higher weights.

Before concluding this subsection, let us also make a technical remark
regarding the efficient representation of MPLs for computer algebra
manipulations [79]: It proves useful to store the $\vec{Y}$ lists of arguments
of MPLs from eq. (3.10) as base 2 (or binary) numbers, and encode the
$\vec{X}$ lists as base 3 (or ternary) numbers via a $\tfrac{1}{y}\rightarrow
2$ replacement, e.g. $G(1,0,\tfrac{1}{y};x)\mapsto G(102_{3};x)$. It allows us
to represent the action of the
$\\{\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}\\}$ integration operators
arithmetically as addition of certain (weight-dependent) numbers, e.g.
$\mathcal{O}_{3}(G(1,0,\tfrac{1}{y};x))=G(\tfrac{1}{y},1,0,\tfrac{1}{y};x)\mapsto
G(102_{3}+2000_{3};x)$. Also note that as a consequence of the branch cut
condition of eq. (2.38), $f^{y}=0$ as $y\rightarrow 0$, the first integral
along the $\mafx@gamma_{1}$ effectively only increases the powers of
$l_{1-y}:=\log(1-y)$, since the $\vec{Y}$ lists in eq. (3.10) are forced to be
free of any zeros inside of them. Therefore we only need to keep track of its
overall length $|\vec{Y}|$. We have found that these tweaks greatly to reduce
the memory and storage usage of our computer implementation.

###  3.3 Promoting coproducts to expansions

For every function inside our $\mathcal{H}^{\mathrm{DS}}$ space, apart from
its MPL representation $f$ we also construct its $x\rightarrow 0$ expansion,
which we shall denote as $\check{f}$ in order to avoid confusion (note that we
have also dropped the weight index to avoid clutter). In the next section we
will use these expansions $\check{f}$ to resum the Wilson loop OPE state sum
and obtain predictions for the NMHV six gluon amplitude at high loop orders,
so before that we need to discuss the relevant features of our construction.

In fact, direct integration of the $x\rightarrow 0$ expansions $\check{f}$
turns out to be easier than expanding the MPL representations we obtained in
the previous subsection, especially at higher weights $p\geq 10$. This way at
each step of the algorithm we only have to deal with $x\rightarrow 0$ power-
and-log expansions of MPLs, which are directly applied to fix the coefficient
of the ansatz in the NMHV amplitude bootstrap problem. We will consider the
$x\rightarrow 0$ expansions $\check{f}$ of the following mixed form:

$\displaystyle\check{f}=\sum_{k=0}^{N}\sum_{l=0}^{k-1}\sum_{m,n}\>q_{klmn}x^{k}y^{l}\bigl{(}\log(x)\bigr{)}^{m}\bigl{(}\log(1-y)\bigr{)}^{n}+\sum\limits_{\tilde{X}}g_{\tilde{X}}\>G(\tilde{X};x)$
(3.18)

where the list of arguments of MPLs $\tilde{X}$ is drawn from a reduced set
$\tilde{X}\in\\{0,\tfrac{1}{y}\\}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}$,
and the coefficients $q_{klmn}\in\mathcal{Z}^{N\times(N-1)\times p\times p}$
and $g_{\tilde{X}}\in\mathcal{Z}^{2^{p}}$ are MZV-valued, so that the overall
weight of the RHS in eq. (3.18) does not exceed $p$, while
$\bigl{|}g_{\tilde{X}}\bigr{|}+\bigl{|}\tilde{X}\bigr{|}=p$. The summation
range $0\leq l<k\leq N$ in the first sum of eq. (3.18) indicates that the MPLs
that contribute to it have at least one “$1$” among their arguments. In
contrast, MPLs from the second sum of eq. (3.18), when expanded, only contain
equal powers of both $x$ and $y$ (apart from simple logarithms). Therefore, we
refer to the first sum in eq. (3.18) as the “non-diagonal part” of the
expansion and denote the corresponding linear space of monomials by
$\mathcal{N}$, while the second sum will be called the “diagonal part”, whose
linear space of MPLs (or $x\rightarrow 0$ expansions) is denoted by
$\mathcal{D}$. Hence an expansion $\check{f}$ belongs to a direct sum of these
two parts: $\check{f}=\mathcal{N}\oplus\mathcal{D}$ and so does the NMHV
amplitude that we are after.

Alongside the $\\{p-1,1\\}$ coproduct components discussed in Section 3.1, our
main computational tool are the integration operators introduced in eqs.
(3.11, 3.12, 3.13). Their action on the non-diagonal $\mathcal{N}$ part of the
expansions $\check{f}$ is given by the following formulas:

$\displaystyle\left.\mathcal{O}_{1}(\check{f})\right|_{\mathcal{N}}$
$\displaystyle:=\sum_{k,l,m,n}q_{klmn}\>x^{k}y^{l}\bigl{(}\log(1-y)\bigr{)}^{n}\,\sum_{i=0}^{m}(-1)^{m-i}\>\frac{m!}{i!}\,k^{-(m-i+1)}\>\bigl{(}\log(x)\bigr{)}^{i}$
(3.19)
$\displaystyle\left.\mathcal{O}^{\prime}_{2}(\check{f})\right|_{\mathcal{N}}$
$\displaystyle:=-\sum_{k,l,m,n}q_{klmn}\>y^{l}\bigl{(}\log(x)\bigr{)}^{m}\bigl{(}\log(1-y)\bigr{)}^{n}\,\sum_{i=k+1}^{N}x^{i}$
(3.20)
$\displaystyle\left.\mathcal{O}^{\prime}_{3}(\check{f})\right|_{\mathcal{N}}$
$\displaystyle:=-\sum_{k,l,m,n}q_{klmn}\>y^{l-k}\bigl{(}\log(x)\bigr{)}^{m}\bigl{(}\log(1-y)\bigr{)}^{n}\,\sum_{i=k+1}^{N}(xy)^{i}$
(3.21)

where the partial operators
$\\{\mathcal{O}^{\prime}_{2},\mathcal{O}^{\prime}_{3}\\}$ are related to the
full integration operators $\\{\mathcal{O}_{2},\mathcal{O}_{3}\\}$ as
$\mathcal{O}_{2}=\mathcal{O}_{1}\circ\mathcal{O}^{\prime}_{2}$ and
$\mathcal{O}_{3}=\mathcal{O}_{1}\circ\mathcal{O}^{\prime}_{3}$. Note, that the
rule (3.19) needs an additional definition in case of a pure $\log(x)$
monomial:

$\displaystyle\mathcal{O}_{1}(\bigl{(}\log(x)\bigr{)}^{m}):=\frac{1}{m+1}\bigl{(}\log(x)\bigr{)}^{m+1},$
(3.22)

in accordance with eq. (2.8).

| $\mathcal{N}$ | $\mathcal{D}$
---|---|---
$\mathcal{O}_{1}$ | $\mathcal{N}$ | $\mathcal{D}$
$\mathcal{O}^{\prime}_{2}$ | $\mathcal{N}$ | $\mathcal{N}$
$\mathcal{O}^{\prime}_{3}$ | $\mathcal{N}$ | $\mathcal{D}$
Table 5:  Action of the integration operators
$\\{\mathcal{O}_{1},\mathcal{O}^{\prime}_{2},\mathcal{O}^{\prime}_{3}\\}$ on
the non-diagonal $\mathcal{N}$ and diagonal $\mathcal{D}$ parts of the
expansions. None of the operators maps the non-diagonal part $\mathcal{N}$
into the diagonal $\mathcal{D}$, and the $\mathcal{O}_{1}$ and
$\mathcal{O}^{\prime}_{3}$ operators even preserve $\mathcal{D}$. These
properties allow us to keep the diagonal $\mathcal{D}$ part of the expansion
in eq. (3.18) in the full MPL form and only expand the non-diagonal
$\mathcal{N}$ part.

The action of
$\\{\mathcal{O}_{1},\mathcal{O}^{\prime}_{2},\mathcal{O}^{\prime}_{3}\\}$ is
summarized in Table 5 in terms of the non-diagonal and diagonal parts
$\mathcal{N}$ and $\mathcal{D}$. A crucial observation about these operators
is that they do not increase the power of $y^{l}$ independently from $x^{k}$
or, in other words, none of them act as $\mathcal{N}\rightarrow\mathcal{D}$,
which is, of course, just a consequence of the eqs. (3.11, 3.12, 3.13). This
feature allows us to work with the diagonal part $\mathcal{D}$ of any
expansion $\check{f}$ separately keeping it in the full unexpanded MPL form.

weight $p$ | $\leq 6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$
---|---|---|---|---|---|---|---
order $N_{\text{opt}}$ | $\leq 4$ | $5$ | $7$ | $9$ | $13$ | $18$ | $25$
Table 6:  Optimal expansion order $N_{\text{opt}}$, corresponding to the
minimal value of $N$ in eq. (3.18), such that the series expansions
$\check{f}$ of all DS functions at a given weight $p$ become linearly
independent.

Finally, let us note an important technical detail of this approach: the
maximal expansion order $N$ needed for the high-weight functions has to be set
in advance for the weight $p=1$ functions, and cannot be altered in between.
We define the optimal expansion order $N(p)=N_{\text{opt}}(p)$ to be the
minimal value of $N$, for which the expansions of the weight $p$ functions we
obtain in this way, are linearly independent. Our empirical findings of the
value of $N_{\text{opt}}(p)$ are summarized in Table 6. We emphasize that had
we chosen to also expand the diagonal part $\mathcal{D}$, the optimal
expansion order would drastically increase. The computational gain obtained
thus justifies our hybrid strategy of only expanding the non-diagonal
$\mathcal{N}$ part in eq. (3.18).

##  4 The Wilson loop OPE and the collinear limit

In the previous sections, we essentially described how to construct an ansatz
for the six-gluon amplitude in the DS limit. Here, we will show how to find a
unique solution for this ansatz, by exploiting independent information for the
amplitude in the collinear boundary of the DS limit, furnished by the Wilson
Loop OPE approach [10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

While our space of functions contains both the MHV and NMHV amplitudes, in
what follows we will be focusing on the latter, which in the literature is
known to lower loop order [50]. After reviewing another amplitude
normalization and set of kinematic variables that will be convenient for our
purposes, in subsection 4.1 we analyze the behavior of the rational,
$R$-invariant part of the NMHV amplitude in the two parity-conjugate DS
limits. Choosing one of the two limits, and the so-called (1111) rational
component of the amplitude, subsection 4.2 then discusses in detail how to
obtain predictions from the Wilson loop OPE, and match them against our
ansatz. The reader interested in the final result, may jump to Section 5.

###  4.1 Wilson loop normalization, variables and $R$-invariants in the DS
limit

In order to describe the Wilson loop dual to the amplitude, it will be
convenient to change the BDS-like normalization (2.1), redistributing a known
factor between the finite and infrared-divergent part of the latter. That is,
in this section we will be considering the NMHV framed Wilson loop,
$\mathcal{W}$, which is related to $\mathrm{E}$ by

$\displaystyle\frac{\mathrm{E}}{W}$
$\displaystyle=\>\exp\bigl{(}-\tfrac{1}{4}\mafx@Gamma_{\text{cusp}}\mathcal{X}\bigr{)}.$
(4.1)

In the rightmost factor in this equation
$\tfrac{1}{4}\mafx@Gamma_{\text{cusp}}=g^{2}-2\mafx@zeta_{2}g^{4}+\mathcal{O}(g^{6})$
is the cusp anomalous dimension, a quantity known to all loops [80], and

$\displaystyle\mathcal{X}$ $\displaystyle:=X-\mathcal{E}^{(1)}$ $\displaystyle
X$ $\displaystyle=-{\rm Li}_{2}(1-u)-{\rm Li}_{2}(1-v)+{\rm Li}_{2}(w)$
$\displaystyle\quad+\log(1-w)^{2}-\log(1-w)\log(\tfrac{v}{u})-\log(u)\log(v)+\tfrac{\mafx@pi^{2}}{6},$
(4.2) $\displaystyle\mathcal{E}^{(1)}$ $\displaystyle={\rm
Li}_{2}(1-\tfrac{1}{u})+{\rm Li}_{2}(1-\tfrac{1}{v})+{\rm
Li}_{2}(1-\tfrac{1}{w}).$ (4.3)

As we mentioned in Section 2, at each loop order $L$, $\mathcal{E}^{(L)}$ is
believed to be a pure transcendental function, and particularly in the DS
limit it should lie in the $\mathcal{H}^{\mathrm{DS}}$ space. On the other
hand, as shown in eq.(2.2) $\mathrm{E}$ is a combination of pure functions and
rational $R$-invariants. The latter are polynomials in the Grassmann variables
encoding the supersymmetry of the theory, and in this work we will
particularly focus on their $(1111)$ component, first studied from the point
of view of the Wilson loop OPE in [11].

Following the latter reference, we will also switch to another choice of
kinematic variables that is convenient for describing both the collinear and
DS limits of the hexagonal Wilson loop. It consists of a triple $\\{S,F,T\\}$,
whose relation to the $\\{u,v,w\\}$ cross-ratios reads:

$\displaystyle u=\frac{F}{F+FS^{2}+ST+F^{2}ST+FT^{2}},\quad
v=\frac{S^{2}}{T^{2}}\,u\,w,\quad w=\frac{T^{2}}{1+T^{2}}.$ (4.4)

In these variables, for the NMHV $(1111)$ component of interest the
$R$-invariants reduce to [50]

$\displaystyle(1)$ $\displaystyle\rightarrow 0,$ $\displaystyle(2)$
$\displaystyle\rightarrow\frac{F^{3}T}{(S+FT)(F+ST+FT^{2})(F+FS^{2}+ST+F^{2}ST+FT^{2})},$
$\displaystyle(3)$ $\displaystyle\rightarrow\frac{1}{(1+T^{2})(1+FST+T^{2})},$
(4.5) $\displaystyle(4)$ $\displaystyle\rightarrow\frac{S}{S+FT},$
$\displaystyle(5)$
$\displaystyle\rightarrow\frac{T(FS+T)^{3}}{F(1+FST+T^{2})(F+FS^{2}+ST+F^{2}ST+FT^{2})},$
$\displaystyle(6)$
$\displaystyle\rightarrow\frac{T^{4}}{F(1+T^{2})(F+ST+FT^{2})}.$

Now let’s see what are the simplifications to the above formulas that follow
from the double scaling kinematics. As was shown in Section 2.2, there are
really two DS limits, which in the OPE variables (4.4) correspond to:

double scaling limit 1 (DS1): $\displaystyle\\{T,F^{-1}\\}\rightarrow 0,\quad$
$\displaystyle TF\;\text{fixed,}$ (4.6) double scaling limit 2 (DS2):
$\displaystyle\\{T,F\\}\rightarrow 0,\quad$ $\displaystyle
TF^{-1}\;\text{fixed.}$ (4.7)

They are related by the parity transformation, in these variables translates
to a simple $F\rightarrow F^{-1}$ replacement. In these limits, we may also
relate the $\\{x,y\\}$ variables of eq. (2.26) with respect to the surviving
OPE variables as
$\\{x,y\\}\big{|}_{\text{DS1}}=\\{-\frac{TF}{S},1+\frac{1}{TFS}\\}$ and

$\displaystyle\\{x,y\\}\big{|}_{\text{DS2}}=\Bigl{\\{}-\frac{T}{SF},1+\frac{F}{TS}\Bigr{\\}}.$
(4.8)

From now on will will thus switch back to our familiar $\\{x,y\\}$ variables.

Also note that the cross-ratios eq. (4.4) are parity-even functions, so they
reduce in both versions of the DS limit to the same values,
$\\{u,v,w\\}\rightarrow\Bigl{\\{}\frac{x(1-y)}{1-xy},\frac{1}{1-xy},0\Bigr{\\}}$.
Therefore, the function $\mathcal{X}$ responsible for the BDS-like
normalization and defined in eq. (4.1) in both of the DS limits reduces to

$\displaystyle\mathcal{X}$
$\displaystyle\xrightarrow[\text{DS}]{}\tfrac{\mafx@pi^{2}}{3}+\tfrac{1}{2}\log(x(1-y))+\tfrac{1}{2}\log(w)^{2}.$
(4.9)

On the other hand, the $(1111)$ components of the $R$-invariants in the two
versions of the double scaling limit differ significantly. Using the
definitions from eqs. (4.5) and the limits from eqs. (4.6) and (4.7) we see
that

$\displaystyle\\{(1),\ldots,(6)\\}\rightarrow\begin{cases}\\{0,-\frac{x^{2}(1-y)}{(1-x)(1-xy)},-\frac{1-y}{y},\frac{1}{1-x},\frac{1}{y(1-xy)},0\\}&\text{for
DS1,}\\\ \\{0,0,1,1,0,0\\}&\text{for DS2.}\end{cases}$ (4.10)

Very interestingly, the DS2 limit completely trivializes the $R$-invariants
and makes the ratio function $\mathrm{E}^{(1111)}$ pure (meaning it becomes a
$\mathbb{Q}$-linear sum of MPLs). From this point on, we will focus on the DS2
limit as it greatly facilitates the search for the NMHV six-particle amplitude
within the DS function space $\mathcal{H}^{\mathrm{DS}}$. Specifically, it
allows us to use the same mixed form of eq. (3.18) for the collinear limit
expansion of the entire amplitude, without having to take additional
contributions from the rational factors into account.

###  4.2 NMHV Wilson loop OPE

Let us now proceed to discuss in detail how to determine the (1111) component
of the six-particle NMHV amplitude in the DS2 limit with the help of the
Wilson loop OPE. We start by briefly reviewing the latter approach, and
especially the predictions it provides for the weak-coupling expansion of the
amplitude, in integral form. We then proceed to trade them with infinite sum
representations relying on Cauchy’s residue theorem. Finally, we match them
against the series representation of our ansatz, carried our in subsection
3.3.

The Wilson loop OPE provides a non-perturbative description of the amplitude
as a state sum expansion around the collinear limit. The DS and collinear
limits are closely related as can be seen from eqs. (4.6) and (4.7). The DS
limit benefits from having only gluonic states with positive (for DS1) or
negative (for DS2) helicities contributing to it. In the DS2 limit we will be
focusing on, discussed in the previous subsection, the $(1111)$ component of
the framed NMHV Wilson loop $W$ may be written as

$\displaystyle W\xrightarrow[\text{DS2}]{}\sum_{N_{-}\geq
0}\>W_{N_{-}}=\sum_{N_{-}\geq 0}\sum_{L\geq 0}\>g^{2L}W^{(L)}_{N_{-}},$ (4.11)

where $W_{N_{-}}$ are contributions from multi-particle excitations consisting
of $N_{-}$ negative helicity gluon bound states, further organized into
$L$-loop contributions to their weak-coupling expansion. As we’ll see below,
$W_{N_{-}}$ starts contributing at $L=N_{-}^{2}+N_{-}$ loops, and so for our
purposes it will be sufficient to restrict to $N\leq 2$, which provides an
accurate desciption up to and including $L=11$ loops.

The one gluon bound state particle contribution $W_{1_{-}}$ is given by

$\displaystyle W_{1_{-}}=\sum_{l_{1}\geq
1}(TF^{-1})^{l_{1}}\int\frac{\mathrm{d}u_{1}}{2\mafx@pi}\>\>T^{\mafx@gamma_{l_{1}}\\!(u_{1})}S^{\,\mathrm{i}\,p_{l_{1}}\\!(u_{1})}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\mafx@mu_{l_{1}}\\!(u_{1})\,h_{-l_{1}}\\!(u_{1}),$
(4.12)

whereas the superposition of two gluon bound states $W_{2_{-}}$ by

$\displaystyle W_{2_{-}}=\sum_{l_{1}\geq l_{2}\geq
1}\frac{1}{1+\mafx@delta_{l_{1},l_{2}}}(TF^{-1})^{l_{1}+l_{2}}\int$
$\displaystyle\frac{\mathrm{d}u_{1}\>\mathrm{d}u_{2}}{(2\mafx@pi)^{2}}\>\>T^{\mafx@gamma_{l_{1}}\\!(u_{1})+\mafx@gamma_{l_{2}}\\!(u_{2})}S^{\,\mathrm{i}\,p_{l_{1}}\\!(u_{1})+\mathrm{i}\,p_{l_{2}}\\!(u_{2})}$
$\displaystyle\qquad\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\mafx@mu_{l_{1}}\\!(u_{1})\,\mafx@mu_{l_{2}}\\!(u_{2})\,h_{-l_{1}}\\!(u_{1})\,h_{-l_{2}}\\!(u_{2})}{P_{l_{1}|\,l_{2}}(u_{1}|u_{2})\,P_{l_{2}|\,l_{1}}(u_{2}|u_{1})}.$
(4.13)

In the above expressions, $p$ and $\mafx@gamma$ denote the momentum and
quantum energy correction of the gluon excitations, whereas $\mafx@mu$, $P$
and $h$ are further physical quantities describing them which are known as the
measure, pentagon transition and NMHV form factor. They have been derived at
finite coupling in [13], and to leading quantum order, their perturbative
expansions read

$\displaystyle p_{l}(u)$
$\displaystyle=2u+2\mathrm{i}g^{2}\bigl{(}\mafx@psi(\mathrm{i}u^{-})-\mafx@psi(-\mathrm{i}u^{+})\bigr{)}+\mathcal{O}(g^{4}),$
(4.14) $\displaystyle\mafx@gamma_{l}(u)$
$\displaystyle=2g^{2}\bigl{(}\mafx@psi(1+\mathrm{i}u^{-})+\mafx@psi(1-\mathrm{i}u^{+})-2\mafx@psi(1)\bigr{)}+\mathcal{O}(g^{4}),$
(4.15) $\displaystyle\mafx@mu_{l}(u)$
$\displaystyle=(-1)^{l}g^{2}\>\bigl{(}u^{+}u^{-}\bigr{)}^{-2}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\operatorname{\mafx@Gamma}(1+\mathrm{i}u^{-})\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+})}{\operatorname{\mafx@Gamma}(l)}+\mathcal{O}(g^{4}),$
(4.16) $\displaystyle P_{l_{1}|\,l_{2}}(u_{1}|u_{2})$
$\displaystyle=\frac{(-1)^{l_{2}}}{g^{2}}\>u^{+}_{1}u^{-}_{1}u^{+}_{2}u^{-}_{2}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\operatorname{\mafx@Gamma}(-\mathrm{i}u^{+}_{1}+\mathrm{i}u^{-}_{2})}{\operatorname{\mafx@Gamma}(1+\mathrm{i}u^{-}_{1})\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+}_{2})}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\operatorname{\mafx@Gamma}(\mathrm{i}u^{-}_{1}-\mathrm{i}u^{-}_{2})}{\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+}_{1}+\mathrm{i}u^{+}_{2})}+\mathcal{O}(g^{4}),$
(4.17) $\displaystyle h_{l}(u)$
$\displaystyle:=\Bigl{(}\tfrac{1}{g^{2}}\,x(u^{+})\,x(u^{-})\Bigr{)}^{\mathrm{sign}(l)},$
(4.18)

expressed in terms of the shifted rapidities

$u^{\pm}_{k}:=u_{k}\pm\tfrac{\mathrm{i}\,l_{k}}{2}\,.$ (4.19)

and Zhukowski variables

$x(u):=\tfrac{1}{2}\bigl{(}u+\sqrt{u^{2}-4g^{2}}\bigr{)}=u+\mathcal{O}(g^{2})\,.$
(4.20)

As a consequence, the $L$-loop contribution to the OPE integrals defined in
eqs. (4.12, 4.13) has the general structure

$\displaystyle W^{(L)}_{1_{-}}=g^{2L}\sum_{l_{1}\geq
1}\Bigl{(}-\tfrac{T}{SF}\Bigr{)}^{l_{1}}\int\frac{\mathrm{d}u_{1}}{2\mafx@pi}\>\frac{S^{2\mathrm{i}u^{-}_{1}}\>\mathcal{P}_{1}^{(L)}}{\bigl{(}u^{+}_{1}u^{-}_{1}\bigr{)}^{2L-1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+}_{1})\operatorname{\mafx@Gamma}(1+\mathrm{i}u^{-}_{1})}{\operatorname{\mafx@Gamma}(l_{1})}$
(4.21)

for one gluon bound state, and

$\displaystyle W^{(L)}_{2_{-}}=g^{2L}\sum_{l_{1}\geq l_{2}\geq
1}\frac{1}{1+\mafx@delta_{l_{1},l_{2}}}\Bigl{(}-\tfrac{T}{SF}\Bigr{)}^{l_{1}+l_{2}}\int\frac{\mathrm{d}u_{1}\>\mathrm{d}u_{2}}{(2\mafx@pi)^{2}}\>\frac{S^{2\mathrm{i}(u^{-}_{1}+u^{-}_{2})}\>\mathcal{P}_{2}^{(L)}}{\bigl{(}u^{+}_{1}u^{-}_{1}u^{+}_{2}u^{-}_{2}\bigr{)}^{2L-7}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}$
(4.22)
$\displaystyle\>\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+}_{1})^{2}\operatorname{\mafx@Gamma}(1+\mathrm{i}u^{-}_{1})^{2}\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+}_{2})^{2}\operatorname{\mafx@Gamma}(1+\mathrm{i}u^{-}_{2})^{2}}{\operatorname{\mafx@Gamma}(l_{1})\operatorname{\mafx@Gamma}(l_{2})\operatorname{\mafx@Gamma}(-\mathrm{i}u^{+}_{1}+\mathrm{i}u^{-}_{2})\operatorname{\mafx@Gamma}(\mathrm{i}u^{-}_{1}-\mathrm{i}u^{+}_{2})}\cdot\frac{\operatorname{\mafx@Gamma}(1+\mathrm{i}u^{+}_{1}-\mathrm{i}u^{+}_{2})\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+}_{1}+\mathrm{i}u^{+}_{2})}{\operatorname{\mafx@Gamma}(\mathrm{i}u^{-}_{1}-\mathrm{i}u^{-}_{2})\operatorname{\mafx@Gamma}(-\mathrm{i}u^{-}_{1}+\mathrm{i}u^{-}_{2})}$

for two excitations, which becomes non-zero starting from the $L=6$ loop
order. Here the $\mathcal{P}_{k}^{(L)}$ symbols collect factors with
polynomial dependence on the shifted rapidities $u^{\pm}_{k}$, the logarithms
of kinematical variables $\\{\log(T),\log(S)\\}$, and on polygamma functions
$\\{\mafx@psi^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}(\mp
u^{\pm}_{k}),\mafx@psi^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}(1\mp
u^{\pm}_{k})\\}$, which appear at higher order in the perturbative expansions
(4.14)-(4.18). For example, the first two non-zero values for the
$\mathcal{P}^{(L)}_{1}$ read:

$\displaystyle\mathcal{P}_{1}^{(2)}$ $\displaystyle=1,$ (4.23)
$\displaystyle\mathcal{P}_{1}^{(3)}$
$\displaystyle=\tfrac{7}{2}(u^{-}_{1})^{2}+u^{-}_{1}u^{+}_{1}+\tfrac{7}{2}(u^{+}_{1})^{2}-(u^{-}_{1}u^{+}_{1})^{2}\Bigl{(}-\tfrac{\mafx@pi^{2}}{3}+\bigl{(}\mafx@psi^{(1)}(1+\mathrm{i}u^{-}_{1})+\mafx@psi^{(1)}(1-\mathrm{i}u^{+}_{1})\bigr{)}$
$\displaystyle\qquad+\tfrac{1}{2}\bigl{(}\mafx@psi(1+\mathrm{i}u^{-}_{1})+\mafx@psi(1-\mathrm{i}u^{+}_{1})\bigr{)}^{2}+\tfrac{1}{2}\bigl{(}\mafx@psi(\mathrm{i}u^{-}_{1})-\mafx@psi(-\mathrm{i}u^{+}_{1})\bigr{)}^{2}$
$\displaystyle\qquad+2\bigl{(}\mafx@psi(1+\mathrm{i}u^{-}_{1})+\mafx@psi(1-\mathrm{i}u^{+}_{1})\bigr{)}\log(T)+2\bigl{(}\mafx@psi(\mathrm{i}u^{-}_{1})-\mafx@psi(-\mathrm{i}u^{+}_{1})\bigr{)}\log(S)\Bigr{)}$
$\displaystyle\quad+\mafx@gamma_{\mathrm{E}}\>(\ldots),$ (4.24)

and similarly for $\mathcal{P}^{(L)}_{2}$:

$\displaystyle\mathcal{P}_{2}^{(6)}$ $\displaystyle=1,$ (4.25)
$\displaystyle\mathcal{P}_{2}^{(7)}$
$\displaystyle=\tfrac{11}{2}\Bigl{[}(u^{-}_{1}u^{+}_{1}u^{-}_{2})^{2}+(u^{-}_{1}u^{+}_{1}u^{+}_{2})^{2}+(1\leftrightarrow
2)\Bigr{]}$
$\displaystyle\quad+u^{-}_{1}u^{+}_{1}u^{-}_{2}u^{+}_{2}\bigl{(}u^{-}_{1}u^{+}_{1}+u^{-}_{1}u^{-}_{2}+u^{-}_{1}u^{+}_{2}+u^{+}_{1}u^{-}_{2}+u^{+}_{1}u^{+}_{2}+u^{-}_{2}u^{+}_{2}\bigr{)}$
$\displaystyle\quad-(u^{-}_{1}u^{+}_{1}u^{-}_{2}u^{+}_{2})^{2}\Bigl{(}-\tfrac{4\mafx@pi^{2}}{3}+\Bigl{[}\tfrac{1}{2}\bigl{(}\mafx@psi(1+\mathrm{i}u^{-}_{1})+\mafx@psi(1-\mathrm{i}u^{+}_{1})\bigr{)}^{2}+\tfrac{1}{2}\bigl{(}\mafx@psi(\mathrm{i}u^{-}_{1})-\mafx@psi(-\mathrm{i}u^{+}_{1})\bigr{)}^{2}$
$\displaystyle\qquad+2\bigl{(}\mafx@psi^{(1)}(1+\mathrm{i}u^{-}_{1})+\mafx@psi^{(1)}(1-\mathrm{i}u^{+}_{1})\bigr{)}-2\mafx@psi(\mathrm{i}u^{-}_{1})\mafx@psi(-\mathrm{i}u^{+}_{1})$
$\displaystyle\qquad+2\bigl{(}\mafx@psi(1+\mathrm{i}u^{-}_{1})+\mafx@psi(1-\mathrm{i}u^{+}_{1})\bigr{)}\log(T)+2\bigl{(}\mafx@psi(\mathrm{i}u^{-}_{1})-\mafx@psi(-\mathrm{i}u^{+}_{1})\bigr{)}\log(S)+(1\leftrightarrow
2)\Bigr{]}$
$\displaystyle\qquad-\bigl{(}\mafx@psi(1+\mathrm{i}u^{-}_{1})+\mafx@psi(1-\mathrm{i}u^{+}_{1})\bigr{)}\bigl{(}\mafx@psi(1+\mathrm{i}u^{-}_{2})+\mafx@psi(1-\mathrm{i}u^{+}_{2})\bigr{)}$
$\displaystyle\qquad-\bigl{(}\mafx@psi(\mathrm{i}u^{-}_{1})-\mafx@psi(-\mathrm{i}u^{+}_{1})\bigr{)}\bigl{(}\mafx@psi(\mathrm{i}u^{-}_{2})+\mafx@psi(-\mathrm{i}u^{+}_{2})\bigr{)}\Bigr{)}$
$\displaystyle\quad+\mafx@gamma_{\mathrm{E}}\>(\ldots).$ (4.26)

Here the ellipsis $\>(\ldots)$ hide terms proportional to the Euler-Mascheroni
constant $\mafx@gamma_{\mathrm{E}}$, which always drop out from the final
answer. Note that the last fraction in eq. (4.22) can actually be simplified
to just a polynomial [27]:

$\displaystyle\frac{\operatorname{\mafx@Gamma}(1+\mathrm{i}u^{+}_{1}-\mathrm{i}u^{+}_{2})\operatorname{\mafx@Gamma}(1-\mathrm{i}u^{+}_{1}+\mathrm{i}u^{+}_{2})}{\operatorname{\mafx@Gamma}(\mathrm{i}u^{-}_{1}-\mathrm{i}u^{-}_{2})\operatorname{\mafx@Gamma}(-\mathrm{i}u^{-}_{1}+\mathrm{i}u^{-}_{2})}=(-1)^{l_{1}-l_{2}+1}(\mathrm{i}u^{+}_{1}-\mathrm{i}u^{+}_{2})(\mathrm{i}u^{-}_{1}-\mathrm{i}u^{-}_{2}),$
(4.27)

which means that it does not contribute any additional poles to the integrand.

After the perturbative expansion of eqs. (4.21) and (4.22) is achieved to the
desired loop order $L$, our next step is application of the Cauchy’s residue
theorem: for $S>1$ the contour of $u_{k}$-integration is closed in the upper
half-plane and the residues at poles
$u_{k}=\mathrm{i}(j_{k}+\tfrac{l_{k}}{2})$ are collected. Following the ideas
of [30, 27], we first change the variables in the integrands according to:

$\displaystyle
u_{k}=\mafx@varepsilon_{k}+\mathrm{i}(j_{k}+\tfrac{l_{k}}{2}),\quad
u^{\pm}_{k}=\mafx@varepsilon_{k}+\mathrm{i}j_{k}+\begin{cases}\mathrm{i}\,l_{k}&\text{for
$+$}\\\ 0&\text{for $-$}\end{cases}\quad\text{for}\quad
j_{k}\in\mathbb{Z}_{\geq 0}.$ (4.28)

Next we expand the integrands in the $\mafx@varepsilon_{k}\rightarrow 0$ limit
and pick the coefficient of the $\mafx@varepsilon_{k}^{-1}$ term, which gives
an analytic expression for the residues as functions on an integer lattice of
the summation variables $\\{j_{1},l_{1}\\}$ in the one excitation case, and
$\\{j_{1},j_{2},l_{1},l_{2}\\}$ in the case of two excitations with $j_{k}\geq
0$ and $l_{1}\geq l_{2}\geq 1$.

Special care is needed for the $j_{k}=0$ boundary, where the corresponding
shifted rapidity $u^{-}_{k}$ in the denominators of the integrands of eqs.
(4.21) and (4.22) develop additional $\mafx@varepsilon_{k}$ singularities. The
lattice of summation variables $\\{j_{1},l_{1}\\}$ gets split into 2 regions
with respect to the $j_{1}>0$ condition: one region is $\\{j_{1}=0\\}$ and the
other is $\\{j_{1}>0\\}$. The first two $u$-dependent
$\operatorname{\mafx@Gamma}$-functions in the denominator of eq. (4.22) also
require separate treatment, as they reduce the overall
$\mafx@varepsilon_{k}$-divergence when
$-\mathrm{i}u^{+}_{1}+\mathrm{i}u^{-}_{2}\in\mathbb{Z}_{\geq 0}$ or
$\mathrm{i}u^{-}_{1}-\mathrm{i}u^{+}_{2}\in\mathbb{Z}_{\geq 0}$. When
combined, these conditions split the lattice of summation variables
$\\{j_{1},j_{2},l_{1},l_{2}\\}$ into 8 regions
$\\{\mathrm{I},\ldots,\mathrm{VIII}\\}$, as illustrated in Figure 3, which we
distinguish by the values of the following 4-tuple of predicates:

$\displaystyle\\{j_{1}>0,j_{2}>0,j_{2}<j_{1}+l_{1},j_{2}>j_{1}-l_{2}\\}.$
(4.29)

In Table 7 we list the eight possible combinations of values of these
predicates as well as the total number of the lattice points that satisfy them
truncated at $j_{1}+j_{2}+l_{1}+l_{2}\leq N_{\text{opt}}$, where we took the
optimal expansion order $N_{\text{opt}}=25$ from Table 6.

Figure 3:  A graphical depiction of the eight regions of the $W_{2_{-}}$ summation variables presented on the $\\{j_{1},j_{2}\\}$-plane, parametrized by the helicities $\\{l_{1},l_{2}\\}$ and listed in Table 7.  region | $j_{1}>0$ | $j_{2}>0$ | $j_{2}<j_{1}+l_{1}$ | $j_{2}>j_{1}-l_{2}$ | number of points
---|---|---|---|---|---
I | ✓ | ✓ | ✓ | ✓ | $3360$
II | ✓ | ✓ | ✓ | ✗ | $2272$
III | ✓ | ✓ | ✗ | ✓ | $1232$
IV | ✓ | ✗ | ✓ | ✓ | $358$
V | ✓ | ✗ | ✓ | ✗ | $864$
VI | ✗ | ✓ | ✓ | ✓ | $764$
VII | ✗ | ✓ | ✗ | ✓ | $458$
VIII | ✗ | ✗ | ✓ | ✓ | $156$
Table 7:  The eight regions of the summation variables
$\\{j_{1},j_{2},l_{1},l_{2}\\}$ and the values of the predicates introduced in
eq. (4.29) that separate them. Here we denote $\\{\text{\char 51},\text{\char
55}\\}=\\{\text{true},\text{false}\\}$. The total number of points (or terms
in the sums) in each region is evaluated for the expansion order $x^{25}$,
which corresponds to $j_{1}+j_{2}+l_{1}+l_{2}\leq N_{\text{opt}}(12)=25$. In
Figure 3 we also illustrate these regions on the $\\{j_{1},j_{2}\\}$-plane.

Finally we arrive at the sum over residues for the one gluon OPE integral
shown in eq. (4.21) containing the following two terms:

$\displaystyle W^{(L)}_{1_{-}}=\sum_{j_{1},l_{1}\geq
1}\>x^{j_{1}+l_{1}}(1-y)^{j_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\mathcal{Q}^{(L)}_{1}}{\bigl{(}j_{1}(j_{1}+l_{1})\bigr{)}^{2L-1}}\cdot\frac{\operatorname{\mafx@Gamma}(j_{1}+l_{1})}{\operatorname{\mafx@Gamma}(l_{1})\operatorname{\mafx@Gamma}(1+j_{1})}+\sum_{l_{1}\geq
1}\>x^{l_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\mathcal{Q}^{(L)}_{2}}{l_{1}^{2L+2}}$
(4.30)

where we collected all the dependence on $\\{j_{1},l_{1}\\}$ parameters, as
well as
$\\{\mafx@psi^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}(j_{1}),\mafx@psi^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}(j_{1}+l_{1})\\}$
and the $\\{\log(T),\log(S)\\}$ logarithms in two factors
$\mathcal{Q}^{(L)}_{1}$ for the $j_{1}>0$ region and $\mathcal{Q}^{(L)}_{2}$
for the $j_{1}=0$ region. Similarly, the residue sum for the two gluon OPE
integral of eq. (4.22) involves eight terms, one for each region shown in
Table 7:

$\displaystyle
W^{(L)}_{2_{-}}=\sum_{\\{j_{1},j_{2},l_{1},l_{2}\\}\in\mathrm{I}}\>x^{j_{1}+j_{2}+l_{1}+l_{2}}(1-y)^{j_{1}+j_{2}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\displaystyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\textstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptstyle\times$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.7}{$\scriptscriptstyle\times$}}}}}\frac{\mathcal{R}^{(L)}_{\mathrm{I}}}{\bigl{(}j_{1}j_{2}(j_{1}+l_{1})(j_{2}+l_{2})\bigr{)}^{2L-6}}\cdot$
(4.31)
$\displaystyle\>\cdot\frac{\operatorname{\mafx@Gamma}(j_{1}+l_{1})^{2}\operatorname{\mafx@Gamma}(j_{2}+l_{2})^{2}}{\operatorname{\mafx@Gamma}(l_{1})\operatorname{\mafx@Gamma}(l_{2})\operatorname{\mafx@Gamma}(1+j_{1})^{2}\operatorname{\mafx@Gamma}(1+j_{2})^{2}\operatorname{\mafx@Gamma}(j_{1}-j_{2}+l_{1})\operatorname{\mafx@Gamma}(-j_{1}+j_{2}+l_{2})}+\text{other
regions $\\{\mathrm{II},\ldots,\mathrm{VIII}\\}$},$

where we hide all the complicated dependence on the summation variables
$\\{j_{1},j_{2},l_{1},l_{2}\\}$ and the polygamma functions
$\\{\mafx@psi^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}(j_{k}),\mafx@psi^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}(j_{k}+l_{k})\\}$
in symbols
$\\{\mathcal{R}^{(L)}_{\mathrm{I}},\ldots,\mathcal{R}^{(L)}_{\mathrm{VIII}}\\}$.

While in [30, 27] it was shown that the sum representation of the one-gluon
contribution (4.30) can be evaluated in terms of MPLs with the help of the
algorithms of ref.[37], to our knowledge there exists no direct method for
similarly evaluating the two-gluon contribution, eq. (4.13). It is for this
reason that we will instead choose to resort to the bootstrap method.

In summary, by virtue of eq. (4.11) the above residue sum representations of
the OPE gluonic contributions $W^{(L)}_{1}$ and $W^{(L)}_{2}$ yield the (1111)
component of the NMHV Wilson loop $W$ in the DS2 limit as a series expansion
around the collinear limit. Converting the latter to the BDS-like
normalization according to eq. (4.1), and recalling that it is a pure function
as a consequence of eq. (4.10), we then equate it to an ansatz built out of
the DS functions we constructed in Section 3, in their series expansion
representation (3.18). This fixes all coefficients of our ansatz, and using
the MPL representation of our DS functions, we thus determine the (1111)
component of $\mathrm{E}^{(L)}$ in this form.

Before presenting our results in the next section, let us also make some
further comments on our computational setup. The form of the sums of eqs.
(4.30) and (4.31) indicates that in every term the power of the $x$-variable
is bigger than the power of the $y$-variable. In our ansatz, they will thus
correspond to the $\mathcal{N}$ “non-diagonal” part of the expansions of eq.
(3.18), implying that its “diagonal” part is zero. This makes our ansatz
expansion of the form (3.18) well-suited for matching onto eqs. (4.30) and
(4.31), and further implies that we may bound the summation ranges of the
latter according to $j_{1}+l_{1}\leq N_{\text{opt}}$ and
$j_{1}+j_{2}+l_{1}+l_{2}\leq N_{\text{opt}}$, where $N_{\text{opt}}$ is the
optimal expansion order discussed in subsection 3.3 and shown in Table 6.

##  5 Results

In this work we constructed a space of functions $\mathcal{H}^{\mathrm{DS}}$
relevant for the DS limit of the six gluon scattering amplitude in
$\mathcal{N}=4$ super-Yang-Mills theory through transcendental weight $p=12$.
We applied the $\mathcal{H}^{\mathrm{DS}}$ space to (partial) calculation of
the NMHV component of the six gluon amplitude up to $L\leq 8$ loops.

In Section 3.1 we employed extended Steinmann and analyticity conditions in
order to construct the coproduct representation of the
$\mathcal{H}^{\mathrm{DS}}$ space. In Section 3.2 and Section 3.3 we present
two closely related realizations of the $\mathcal{H}^{\mathrm{DS}}$: one is in
terms of multiple polylogarithms, and one in terms of power-and-log expansions
around the $\\{u,v,w\\}\rightarrow\\{0,1,0\\}$ kinematical point. The latter
is then combined with the residue sum representation of the one and two gluon
contributions to the Wilson Loop Operator Product Expansion, which is
explained in some detail in Section 4. Combination of these two techniques
allows us to determine the six gluon NMHV amplitude in the DS limit to high
loop order, find new (potentially all-loop) patterns in the analytical form of
the amplitude in the $\\{u,v,w\\}\rightarrow 0$ origin limit, and pinpoint the
excessive elements of the $\mathcal{H}^{\mathrm{DS}}$ that should be
irrelevant for amplitude calculations as will be explained next.

###  5.1 The NMHV amplitude up to 8 loops and weight 12

As the main application of our DS space of functions
$\mathcal{H}^{\mathrm{DS}}$, we have computed the (1111) component of the NMHV
superamplitude $\mathrm{E}^{(L)}$ in the double scaling limit $w\rightarrow
0$, more precisely one of its two parity images defined in eq. (4.7). This
quantity has the following natural decomposition,

$\displaystyle\mathrm{E}^{(L)}(u,v,w)\xrightarrow[]{(1111)}\sum_{p=0}^{2L}\mathrm{E}^{(L)}_{p}(u,v)\cdot\bigl{(}\log(w)\bigr{)}^{2L-p},$
(5.1)

where each coefficient $\mathrm{E}^{(L)}_{p}(u,v)$ of a large logarithm
$\log(w)^{2L-p}$ is a pure transcendental function of weight $p$. In
particular, we have determined $\mathrm{E}^{(L)}_{p}$ for any $L\leq 8$ and
$p\leq 12$, namely the full component for $L\leq 6$, as well as all terms with
two (four) or more powers of $\log(w)$ at $L=7$ ($L=8$). The $p\leq 12$
restriction is due to the maximal weight we have explicit MPL representations
for $\mathcal{H}^{\mathrm{DS}}$ so far, whereas the $L\leq 8$ restriction is
due to the size of the expressions predicted by the Wilson loop OPE.
Nevertheless, we are hopeful that the new results we have obtained will offer
valuable boundary data and consistency checks for NMHV hexagon in general
kinematics at seven [81] and eight loops. We provide expressions
$\sum_{L=0}^{6}\sum_{p=0}^{2L}g^{2L}\mathrm{E}^{(L)}_{p}\cdot\bigl{(}\log(w)\bigr{)}^{2L-p}$
and
$\sum_{L=7}^{8}\sum_{p=0}^{12}g^{2L}\mathrm{E}^{(L)}_{p}\cdot\bigl{(}\log(w)\bigr{)}^{2L-p}$
for the full $(1111)$ component of the NMHV amplitude in the DS limit up to 6
loops and partial results at 7 and 8 loops in the files EDS-L1-8-w0-12.m.

Let us also describe the checks we have performed on our answers. First off,
we confirm BDS-normalized component vanishes in the  soft limit boundary
$x\rightarrow 0$ limit or, equivalently,

$\displaystyle\mathrm{E}^{(L)}\xrightarrow[\text{{\color[rgb]{0.34375,0.68359375,0.23828125}
soft} \&
DS2}]{}\Bigl{(}e^{\tfrac{1}{4}\mafx@Gamma_{\text{cusp}}\mathcal{E}^{(1)}}\Bigr{)}^{(L)},$
(5.2)

as expected. Furthermore, we find perfect agreement with existing results on
the collinear limit expansion of the NMHV amplitude, namely the
$\\{T,T^{2},T^{3}\\}$ terms for $L\leq 4$ from eEEtT3.m of [47], as well as
the first two $\\{T,T^{2}\\}$ terms for $L\leq 6$ from W1111L0-6.m of [50]
(this time for the individual OPE contributions $W^{(L)}_{1_{-}}$ and
$W^{(L)}_{2-}$). Additional checks were also performed in an interesting
boundary point of the DS limit, that we discuss next.

###  5.2 The origin limit

The origin limit $\\{u,v,w\\}\rightarrow 0$ was first analyzed in [50], where
it was observed that at weak coupling the MHV six-particle amplitude takes a
very simple (Sudakov-like) form of exponentiated double logarithms,

$\displaystyle\log(\mathcal{E})=-\tfrac{\mafx@Gamma_{\text{oct}}}{24}l^{2}_{uvw}-\tfrac{\mafx@Gamma_{\text{hex}}}{24}\bigl{(}l^{2}_{u/v}+l^{2}_{v/w}+l^{2}_{w/u}\bigr{)}+C_{0}+\mathcal{O}(u),$
(5.3)

in terms of certain quantities
$\\{\mafx@Gamma_{\text{oct}},\mafx@Gamma_{\text{hex}},C_{0}\\}$ that only
depend on the coupling. More recently these quantities have been conjectured
at finite coupling [59], based on a Wilson loop OPE resummation procedure
morally similar to the one considered in this paper, and backed by further
strong coupling analysis via the gauge/string duality.

Although in the latter paper it was also pointed out that the NMHV amplitude
no longer displays this exponentiation of double logarithms, it may still be
interesting to look for patterns that may hint to an all-loop description.
Clearly, the origin limit is a boundary of the DS limit, corresponding to
$y\rightarrow 1^{+}$, $x\rightarrow-\infty$ in the variables (2.26) underlying
the construction of our DS function space. We have indeed taken this limit of
the general result discussed in the previous section, and through four loops
it reads

$\displaystyle\mathrm{E}\xrightarrow[\text{DS2 \&
origin}]{(1111)}1+g^{2}\Bigl{[}-2\mafx@zeta_{2}+l_{u}l_{v}-\frac{l_{u}^{2}}{2}-\frac{l_{v}^{2}}{2}-\frac{l_{w}^{2}}{2}\Bigr{]}$
$\displaystyle+g^{4}\Bigl{[}\frac{l_{w}^{4}}{8}+l_{w}^{2}(2\mafx@zeta_{2}-\frac{l_{u}l_{v}}{2}+\frac{l_{u}^{2}}{4}+\frac{l_{v}^{2}}{4})$
$\displaystyle\quad-\frac{1}{2}l_{u}^{3}l_{v}+\frac{1}{2}l_{u}^{2}l_{v}^{2}-\frac{1}{2}l_{u}l_{v}^{3}+\frac{l_{u}^{4}}{8}+\frac{3}{2}\mafx@zeta_{2}l_{v}^{2}+\mafx@zeta_{3}l_{v}+\frac{l_{v}^{4}}{8}+\frac{39\mafx@zeta_{2}^{2}}{10}+\frac{3}{2}\mafx@zeta_{2}l_{u}^{2}+\mafx@zeta_{3}l_{u}-5\mafx@zeta_{2}l_{u}l_{v}\Bigr{]}$
$\displaystyle+g^{6}\Bigl{[}-\frac{l_{w}^{6}}{48}+l_{w}^{4}(-\frac{3\mafx@zeta_{2}}{4}+\frac{l_{u}l_{v}}{8}-\frac{l_{u}^{2}}{16}-\frac{l_{v}^{2}}{16})$
$\displaystyle\quad+l_{w}^{2}(-\frac{167\mafx@zeta_{2}^{2}}{20}-\frac{5}{4}\mafx@zeta_{2}l_{u}^{2}-\frac{\mafx@zeta_{3}l_{u}}{2}+\frac{7}{2}\mafx@zeta_{2}l_{u}l_{v}+\frac{1}{4}l_{u}^{3}l_{v}-\frac{1}{4}l_{u}^{2}l_{v}^{2}+\frac{1}{4}l_{u}l_{v}^{3}-\frac{l_{u}^{4}}{16}-\frac{5}{4}\mafx@zeta_{2}l_{v}^{2}-\frac{\mafx@zeta_{3}l_{v}}{2}-\frac{l_{v}^{4}}{16})$
$\displaystyle\quad+l_{w}(\frac{1}{2}\mafx@zeta_{3}l_{u}^{2}+2\mafx@zeta_{2}^{2}l_{u}+\frac{1}{2}\mafx@zeta_{2}l_{u}^{2}l_{v}+\frac{1}{2}\mafx@zeta_{2}l_{u}l_{v}^{2}+2\mafx@zeta_{2}^{2}l_{v}+\frac{1}{2}\mafx@zeta_{3}l_{v}^{2})$
$\displaystyle\quad+\frac{1}{8}l_{u}^{5}l_{v}-\frac{3}{16}l_{u}^{4}l_{v}^{2}+\frac{5}{18}l_{u}^{3}l_{v}^{3}-\frac{3}{16}l_{u}^{2}l_{v}^{4}+\frac{1}{8}l_{u}l_{v}^{5}-\frac{l_{u}^{6}}{48}-\frac{69}{20}\mafx@zeta_{2}^{2}l_{v}^{2}-\frac{1}{2}\mafx@zeta_{2}l_{v}^{4}-\mafx@zeta_{3}l_{v}^{3}-6\mafx@zeta_{2}\mafx@zeta_{3}l_{v}-8\mafx@zeta_{5}l_{v}-\frac{l_{v}^{6}}{48}$
$\displaystyle\quad-\frac{527\mafx@zeta_{2}^{3}}{105}-\frac{1}{2}\mafx@zeta_{2}l_{u}^{4}-\mafx@zeta_{3}l_{u}^{3}-\frac{69}{20}\mafx@zeta_{2}^{2}l_{u}^{2}-6\mafx@zeta_{2}\mafx@zeta_{3}l_{u}-8\mafx@zeta_{5}l_{u}+\frac{19}{6}\mafx@zeta_{2}l_{u}^{3}l_{v}-\frac{5}{2}\mafx@zeta_{2}l_{u}^{2}l_{v}^{2}+\frac{271}{10}\mafx@zeta_{2}^{2}l_{u}l_{v}+\frac{19}{6}\mafx@zeta_{2}l_{u}l_{v}^{3}\Bigr{]}$
$\displaystyle+\mathcal{O}(g^{8}),$ (5.4)

where the symbol $l_{x}$ represents a simple logarithm: $l_{x}:=\log(x)$ as
defined in eq. (3.14). Higher loop order $L\leq 8$ corrections to eq. (5.4)
bounded by transcendental weight $p\leq 12$ can be found in the ancillary file
EDS-origin-L1-8-w0-12.m.

In this manner, we confirm that no additional cancellations occur for the
(1111) component, that would lead to a Sudakov-like form similar to the MHV
amplitude at the origin. We have also further vetted our result by comparing
it with the origin limit of the NMHV amplitude in general kinematics through
six loops [50], finding perfect agreement.

While it may be worthwhile to also look at other components for additional
cancellations, already from the current data we observe an interesting general
pattern for the three highest powers of divergent logarithms for the NMHV/MHV
ratio function. In particular, up to $L\leq 6$ loops we find

$\displaystyle\mathrm{E}/\mathcal{E}\xrightarrow[\text{DS2 \&
origin}]{}\sum_{L=0}^{6}g^{2L}\Bigl{[}\frac{(l_{u}l_{v})^{L}}{(L!)^{2}}$
$\displaystyle\quad+\mafx@zeta_{2}\Bigl{(}-\frac{(l_{u}l_{v})^{L-2}(l_{u}+l_{v})l_{w}}{(L-2)!(L-1)!}-\frac{(l_{u}l_{v})^{L-2}((L-1)^{2}l_{u}^{2}+(4(L-1)^{2}+3(L-1)-1)l_{u}l_{v}+(L-1)^{2}l_{v}^{2})}{(L-1)!L!}\Bigr{)}$
$\displaystyle\quad+\mafx@zeta_{3}\Bigl{(}\frac{(l_{u}l_{v})^{L-3}(l_{u}^{2}+l_{v}^{2})l_{w}}{(L-2)!L!}-\frac{(l_{u}l_{v})^{L-3}(l_{u}+l_{v})((L-2)l_{u}^{2}-(L-1)l_{u}l_{v}+(L-2)l_{v}^{2})}{(L-2)!(L-1)!}\Bigr{)}$
$\displaystyle\quad+\mathcal{O}((l_{u}l_{v})^{2L-4})\Bigr{]}$
$\displaystyle\>+\mathcal{O}(g^{14}),$ (5.5)

It may thus well be that this pattern persists at higher loops, especially
given that the two-gluon OPE excitation already contributes at $L=6$ loops.

###  5.3 Further refinements of the $\mathcal{H}^{\mathrm{DS}}$ space

In this final subsection, let us also discuss how it may be possible to
further reduce the size of the $\mathcal{H}^{\mathrm{DS}}$ functional space,
so as to make bootstrapping at higher loops more tractable. Our construction
of $\mathcal{H}^{\mathrm{DS}}$ includes the entire basis of independent MZVs
from eq. (A.1), while we notice only the ordinary zeta values appear in the
NMHV remainder $\mathrm{E}$ function. This is an indication that perhaps only
the latter are needed as independent constants in our space.

To answer such questions about the potential redundancy of our space, we
studied the “nested derivatives” or the $\\{n,1,\ldots,1\\}$ coproduct
components of the NMHV amplitude expressed in terms of the elements of the
$\mathcal{H}^{\mathrm{DS}}$ space, following the blueprint of [51]. In Table 8
we report our findings on the dimensions of the minimal subspaces, needed to
match the $\mathrm{E}^{(6)}_{p}$ and its coproducts. Note how these dimensions
saturate for increasing weight $p$ and fixed $n^{\text{th}}$ coproduct
component for

$\displaystyle p=2n+2$ (5.6)

at the latest, and the saturated value is smaller than the dimension of the
corresponding DS subspace $\dim_{\mathbb{Q}}\mathcal{H}^{\mathrm{DS}}_{n}$,
which implies existance of unnecessary elements in the
$\mathcal{H}^{\mathrm{DS}}$ space. We find that the first such “extra”
function $f^{(3),\text{extra}}_{1}\in\mathcal{H}^{\mathrm{DS}}_{3}$ at weight
$p=3$ that does not appear in the NMHV amplitude reads:

$\displaystyle f^{(3),\text{extra}}_{1}=f^{(3)}_{2}\equiv
f^{(1)}_{1}\mafx@zeta_{2}=l_{1-xy}\mafx@zeta_{2},$ (5.7)

where $\check{y}:=\tfrac{1}{y}$, $\mathcal{G}_{\vec{X}}:=G(\vec{X};x)$, and
$l_{x}:=\log(x)$. Similarly, at weight $p=4$ the space of “extra” functions is
spanned by the following four elements:

$\displaystyle f^{(4),\text{extra}}_{1}$
$\displaystyle=f^{(1)}_{1}\mafx@zeta_{3}=l_{1-xy}\>\mafx@zeta_{3},$
$\displaystyle f^{(4),\text{extra}}_{2}$
$\displaystyle=f^{(2)}_{1}\mafx@zeta_{2}=\mathcal{G}_{0\check{y}}\>\mafx@zeta_{2},$
$\displaystyle f^{(4),\text{extra}}_{3}$
$\displaystyle=f^{(2)}_{2}\mafx@zeta_{2}=\bigl{(}\mathcal{G}_{0\check{y}}-\mathcal{G}_{1\check{y}}-\mathcal{G}_{01}+l_{1-x}l_{x(1-y)}\bigr{)}\>\mafx@zeta_{2},$
$\displaystyle f^{(4),\text{extra}}_{4}$
$\displaystyle=f^{(2)}_{4}\mafx@zeta_{2}=\bigl{(}l_{x(1-y)}l_{1-xy}-l_{1-xy}^{2}\bigr{)}\>\mafx@zeta_{2},$
(5.8)

and at weight $p=5$ we found the following eleven functions:

$\displaystyle\\{f^{(5),\text{extra}}_{1},\ldots,f^{(5),\text{extra}}_{11}\\}=\\{f^{(3)}_{1}\mafx@zeta_{2},f^{(2)}_{1}\mafx@zeta_{3},f^{(3)}_{4}\mafx@zeta_{2},f^{(3)}_{5}\mafx@zeta_{2},f^{(2)}_{2}\mafx@zeta_{3},f^{(3)}_{9}\mafx@zeta_{2},$
$\displaystyle
f^{(3)}_{7}\mafx@zeta_{2},f^{(3)}_{8}\mafx@zeta_{2},f^{(3)}_{10}\mafx@zeta_{2}f^{(2)}_{4}\mafx@zeta_{3},f^{(3)}_{11}\mafx@zeta_{2}\\}$
(5.9)

Interestingly, we see that while $\mafx@zeta_{2}$ and $\mafx@zeta_{3}$ are
needed as independent constants in our space, not all of their products with
non-constant DS functions appear at higher weight. From these findings we
conclude that our DS functional space $\mathcal{H}^{\mathrm{DS}}$ is indeed
overcomplete, at least for the problem of bootstrapping the NMHV remainder
function in $\mathcal{N}=4$ SYM. It would be interesting to perform a more
careful study of the intricate interplay between the integrability and
extended Steinmann relations of eqs. (2.31) and (2.32), the branch cut
conditions of eq. (2.38), and the coaction principle [51] in order to further
perfect the DS bootstrap. We hope that it will provide valuable insights
useful for the general hexagon bootstrap program as well as for other
computational problems in high energy physics.

$n^{\text{th}}$ comp. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---|---|---|---
$p=2$ | 1 |  |  |  |  |  |  |  |  |  |
$p=3$ | 1 | 1 |  |  |  |  |  |  |  |  |
$p=4$ | 2 | 4 | 2 |  |  |  |  |  |  |  |
$p=5$ | 2 | 4 | 5 | 2 |  |  |  |  |  |  |
$p=6$ | 2 | 5 | 9 | 5 | 2 |  |  |  |  |  |
$p=7$ | 2 | 5 | 10 | 11 | 5 | 2 |  |  |  |  |
$p=8$ | 2 | 5 | 11 | 19 | 12 | 5 | 2 |  |  |  |
$p=9$ | 2 | 5 | 11 | 22 | 28 | 12 | 5 | 2 |  |  |
$p=10$ | 2 | 5 | 11 | 22 | 40 | 28 | 12 | 5 | 2 |  |
$p=11$ | 2 | 5 | 11 | 22 | 44 | 60 | 28 | 12 | 5 | 2 |
$p=12$ | 2 | 5 | 11 | 22 | 45 | 79 | 60 | 28 | 12 | 5 | 2
functions | 2 | 5 | 12 | 26 | 56 | 116 | 236 | 474 | 943 | 1867 | 3686
Table 8:  The number of independent $\\{n,1,\ldots,1\\}$ coproduct components
of all $\log{w}$ coefficients $\mathrm{E}^{(6)}_{p}$ of the $L=6$ loop NMHV
amplitude from eq. (5.2). For the reader’s convenience we copied the total
number of DS functions $f^{(p)}$ at each weight $p$ from Table 2.

## Acknowledgements

GP would like to thank Benjamin Basso, Simon Caron-Huot, Lance Dixon, Falko
Dulat, Matt von Hippel and Andrew McLeod for collaboration on closely related
topics, as well as Lance Dixon and Omer Gurdogan for discussions on related
$A_{2}$ function counts. VC thanks Sven-Olaf Moch for helpfull discussions and
encouragement. The authors acknowledge support from the Deutsche
Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2121
“Quantum Universe” 390833306. The work of VC has also been supported by the
DFG under grant number MO 1801/2-1.

##  Appendix A The Multiple Zeta Value basis

In this work we used the basis and reduction rules for MZVs from the MZV
datamine project [79]. The basis reads as follows:

$\displaystyle\bigl{\\{}\\{\mafx@zeta_{2}\\},\\{\mafx@zeta_{3}\\},\\{\mafx@zeta_{2}^{2}\\},\\{\mafx@zeta_{2}\mafx@zeta_{3},\mafx@zeta_{5}\\},\\{\mafx@zeta_{2}^{3},\mafx@zeta_{3}^{2}\\},\\{\mafx@zeta_{2}^{2}\mafx@zeta_{3},\mafx@zeta_{2}\mafx@zeta_{5},\mafx@zeta_{7}\\},\\{\mafx@zeta_{2}^{4},\mafx@zeta_{2}\mafx@zeta_{3}^{2},\mafx@zeta_{3}\mafx@zeta_{5},\mafx@zeta_{5,3}\\},$
$\displaystyle\\{\mafx@zeta_{2}^{3}\mafx@zeta_{3},\mafx@zeta_{3}^{3},\mafx@zeta_{2}^{2}\mafx@zeta_{5},\mafx@zeta_{2}\mafx@zeta_{7},\mafx@zeta_{9}\\},\\{\mafx@zeta_{2}^{5},\mafx@zeta_{2}^{2}\mafx@zeta_{3}^{2},\mafx@zeta_{2}\mafx@zeta_{3}\mafx@zeta_{5},\mafx@zeta_{5}^{2},\mafx@zeta_{3}\mafx@zeta_{7},\mafx@zeta_{2}\mafx@zeta_{5,3},\mafx@zeta_{7,3}\\},$
$\displaystyle\\{\mafx@zeta_{11},\mafx@zeta_{2}^{4}\mafx@zeta_{3},\mafx@zeta_{2}\mafx@zeta_{3}^{3},\mafx@zeta_{2}^{3}\mafx@zeta_{5},\mafx@zeta_{3}^{2}\mafx@zeta_{5},\mafx@zeta_{3}\mafx@zeta_{5,3},\mafx@zeta_{5,3,3},\mafx@zeta_{2}^{2}\mafx@zeta_{7},\mafx@zeta_{2}\mafx@zeta_{9}\\},$
$\displaystyle\\{\mafx@zeta_{2}^{6},\mafx@zeta_{2}^{3}\mafx@zeta_{3}^{2},\mafx@zeta_{3}^{4},\mafx@zeta_{2}^{2}\mafx@zeta_{3}\mafx@zeta_{5},\mafx@zeta_{2}\mafx@zeta_{5}^{2},\mafx@zeta_{2}^{2}\mafx@zeta_{5,3},\mafx@zeta_{6,4,1,1},\mafx@zeta_{2}\mafx@zeta_{3}\mafx@zeta_{7},\mafx@zeta_{5}\mafx@zeta_{7},\mafx@zeta_{2}\mafx@zeta_{7,3},\mafx@zeta_{3}\mafx@zeta_{9},\mafx@zeta_{9,3}\\}\bigr{\\}}.$
(A.1)

##  Appendix B Tensor perspective on the coproduct bootstrap

Here we give another point of view on the integrability and Steinmann
constraints discussed in Section 3.1 using tensorial notation. To generate a
DS function $f^{(p)}\in\mathcal{H}^{\mathrm{DS}}_{p}$ at weight $p$ lets us
form an ansatz for the $\\{p-1,1\\}$ coproduct component:

$\displaystyle\mafx@Delta_{p-1,1}(f^{(p)})=\sum_{j,\mafx@alpha}c_{j\mafx@alpha}\,f^{(p-1)}_{j}\otimes\log(\mafx@alpha)$
(B.1)

with unknown rational coefficients
$c_{j\mafx@alpha}\in\mathbb{Q}^{|J_{p-1}|\times|\mathcal{A}|}$. Next we apply
eq. (3.5) to functions $f^{(p-1)}_{j}$ of the previous weight, and introduce a
set of integrability and extended Steinmann conditions $\mathcal{S}$ on the
$\\{p-2,1,1\\}$ component of the produced coproduct shown in eqs. (2.31) and
(2.32) (see also the discussion around eq. (3.7)). For practical purposes, we
collect these constraints in a single tensor
$\mathcal{I}_{\mafx@alpha\mafx@beta
s}\in\mathbb{Q}^{|\mathcal{A}|\times|\mathcal{A}|\times|\mathcal{S}|}$, whose
non-zero elements are summarized below:

$\displaystyle\mathcal{I}_{1,1,5}=-\mathcal{I}_{1,2,5}=\tfrac{1}{2}\mathcal{I}_{1,3,5}=\mathcal{I}_{1,4,5}=\mathcal{I}_{1,5,5}=1,$
$\displaystyle\mathcal{I}_{2,3,4}=-\mathcal{I}_{2,4,3}=-\mathcal{I}_{2,4,5}=\mathcal{I}_{2,5,4}=1,$
$\displaystyle-\mathcal{I}_{3,3,4}=-\mathcal{I}_{3,3,5}=\mathcal{I}_{3,4,3}=\mathcal{I}_{3,5,3}=1,$
(B.2)
$\displaystyle\mathcal{I}_{4,1,5}=-\mathcal{I}_{4,2,5}=-\mathcal{I}_{4,3,4}=\mathcal{I}_{4,4,3}=-\mathcal{I}_{4,5,1}=\mathcal{I}_{4,5,2}=1,$
$\displaystyle-\mathcal{I}_{5,1,4}=\mathcal{I}_{5,4,1}=\mathcal{I}_{6,3,2}=\mathcal{I}_{7,2,3}=-\mathcal{I}_{8,1,2}=\mathcal{I}_{8,2,1}=1.$

The action of these conditions on the $\\{p-2,1,1\\}$ coproduct component of
the unknown function $f^{(p)}$ follow from eq. (B.1) and read:

$\displaystyle\sum_{\begin{subarray}{c}j,k\\\
\mafx@alpha,\mafx@beta\end{subarray}}c_{j\mafx@alpha}c^{(p-1)}_{jk\mafx@beta}\,f^{(p-2)}_{k}\,\mathcal{I}_{s\mafx@alpha\mafx@beta}=\sum_{\begin{subarray}{c}j,k\\\
\mafx@alpha\end{subarray}}f^{(p-2)}_{k}\,M_{ksj\mafx@alpha}\,c_{j\mafx@alpha}=0,\quad\text{for
each }s\in\mathcal{S},$ (B.3)

where
$M_{ksj\mafx@alpha}\in\mathbb{Q}^{|J_{p-2}|\times|\mathcal{S}|\times|J_{p-1}|\times|\mathcal{A}|}$
(implying the necessary transposition of tensor indices here) was already
defined in eq. (3.8) above. Now note, that the DS functions of lower weight
$f^{(p-2)}_{k}$ are linearly independent from one another by construction,
which means that eq. (B.3) contains $|J_{p-2}|\cdot|\mathcal{S}|$ constraints
for $|J_{p-1}|\cdot|\mathcal{A}|$ unknowns. The space of DS functions at
weight $p$ lies inside the nullspace of the
$M_{(ks)(j\mafx@alpha)}\in\mathbb{Q}^{(|J_{p-2}|\cdot|\mathcal{S}|)\times(|J_{p-1}|\cdot|\mathcal{A}|)}$
matrix: schematically we write
$\mathcal{H}^{\mathrm{DS}}_{p}\subset\operatorname{Ker}(M_{(ks)(j\mafx@alpha)})$,
where the brackets $(ks)$ and $(j\mafx@alpha)$ denote vectorization of
indices. In Mathematica such a vectorization $M_{ksj\mafx@alpha}\rightarrow
M_{(ks)(j\mafx@alpha)}$ of a rank 4 tensor into a rank 2 one is achieved
simply via application of Flatten[#, {{1, 2}, {3, 4}}]&. The further
processing of this nullspace is described in the main text in Section 3.1.

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