#  $\scriptstyle\rm LAPTH-004/24$ The Collinear Limit of the Four-Point Energy
Correlator
in $\mathcal{N}=4$ Super Yang-Mills Theory

Dmitry Chicherin,a Ian Moult,b Emery Sokatchev,a Kai Yan,c,d Yunyue Zhuc a
LAPTh, Universite Savoie Mont Blanc, CNRS, B.P.110, F-74941 Annecy-le-Vieux,
France
b Department of Physics, Yale University, New Haven, CT 06511, USA
c School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China
d Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai
200240, China

###### Abstract

We present a compact formula, expressed in terms of classical polylogarithms
up to weight three, for the leading order four-point energy correlator in
maximally supersymmetric Yang-Mills theory, in the limit where the four
detectors are collinear. This formula is derived by combining a simplified,
manifestly dual conformal invariant form of the $1\to 4$ splitting function
obtained from the square of the tree-level five-particle form factor of
stress-tensor multiplet operators, with a novel integration-by-parts algorithm
operating directly on Feynman parameter integrals. Our results provide
valuable data for exploring the structure of physical observables in
perturbation theory, and for calculations of jet substructure observables in
quantum chromodynamics.

##  .1 Introduction

Explicit calculations of observables in quantum field theory (QFT) have played
a crucial role in uncovering hidden simplicity and structure, and in providing
theoretical data for the development of new computational approaches. While a
large amount of data exists for perturbative scattering amplitudes, less
attention has been focused on physical cross-section level observables
relevant for colliders. Motivated by the remarkable simplicity hidden in
scattering amplitudes, we can hope for more exciting surprises in the study of
physical observables.

An interesting class of physical observables are $N$-point correlation
functions of energy flux Basham  _et al._ (1978a, b, 1979a, 1979b), which we
denote $\rm{E^{N}C}$. The $\rm{E^{N}C}$ are infrared finite Kinoshita (1962);
Lee and Nauenberg (1964), exhibit an operator product expansion Hofman and
Maldacena (2008); Kologlu  _et al._ (2021); Chang  _et al._ (2022), and can be
directly measured in experiment Akrawy  _et al._ (1990); Decamp  _et al._
(1991); Adeva  _et al._ (1991); Abe  _et al._ (1994); Komiske  _et al._
(2022); Chen  _et al._ (2022a); Lu (2023); Fan (2023); Tamis (2023), making
them ideal candidates for explorations in perturbation theory. Although the
$\rm{E^{N}C}$ was computed at strong coupling Hofman and Maldacena (2008) in
$\mathcal{N}=4$ super-Yang-Mills (sYM), little is known about its structure at
weak coupling. The $\rm{E^{2}C}$ was computed at leading-order in the coupling
expansion (LO) in quantum chromodynamics (QCD) in the seminal work of Basham
_et al._ (1978b), and more recently at next-to-leading order (NLO) in QCD
Dixon  _et al._ (2018); Luo  _et al._ (2019) and NNLO in ${\cal N}=4$ sYM
Belitsky  _et al._ (2014); Henn  _et al._ (2019). Explorations of higher point
correlators were initiated with the calculation of the ${\rm{E^{3}C}}$ in the
triple-collinear limit Chen  _et al._ (2020a), and for generic angles Yan and
Zhang (2022); Yang and Zhang (2022). These calculations provided important
data for the light-ray OPE Chen  _et al._ (2021a, b, 2022b); Chang and
Simmons-Duffin (2023), and were crucial for developing new collider physics
observables Komiske  _et al._ (2022); Chen  _et al._ (2022a), motivating the
exploration of higher point correlation functions.

$P_{2}$$P_{1}$$P_{4}$$P_{3}$$P_{5}$$\bigotimes$$z_{2}$$z_{3}$$z_{1}$$z_{4}$$=0$$=z$$=1$$=\frac{z-w}{1-w}$
Figure 1: The collinear limit of the four-point energy correlator parametrized
in terms of momenta, see Eq. (1), and complex parameters $(z,\bar{z})$ and
$(w,\bar{w})$, see Eq. (10).

A simplified limit of the $\rm{E^{N}C}$ is the multi-collinear limit, shown in
Fig. 1. In this limit, the $\rm{E^{N}C}$ becomes independent of the source,
and reduces to a function of $2(N-2)$ independent angles. This limit is also
motivated by jet substructure Larkoski  _et al._ (2017); Asquith  _et al._
(2018), where the $N$ detectors all lie in a single jet Dixon  _et al._
(2019); Chen  _et al._ (2020b); Lee  _et al._ (2022); Chen  _et al._ (2023).
At LO the collinear limit of the ${\rm E^{N}C}$ is given as a phase space
integral of the tree-level universal $1\to{N}$ splitting functions Amati  _et
al._ (1978a, b); Ellis  _et al._ (1978); Catani and Grazzini (1999),
$\mathcal{P}^{(0)}_{1\rightarrow{N}}$, as

$\displaystyle{\rm E^{N}C}\overset{\rm
coll.}{=}\hskip-2.84526pt\int_{0}^{1}\hskip-1.42262ptdx_{1}\cdots
dx_{N}\,\delta(1-\sum_{i}x_{i})\,(x_{1}\cdots
x_{N})^{2}\,\mathcal{P}^{(0)}_{1\rightarrow{N}}\,,$

where $x_{i}$ is the fraction of energy carried by the particle absorbed by
detector $i$. Remarkably, the $N-$point correlators define a class of
manifestly _finite integrals_ in $(N-1)-$dimensional projective space, making
it desirable to develop methods that directly compute in four dimensions,
profiting from the simplifications of _finite_ Feynman loop integrals Caron-
Huot and Henn (2014); Henn  _et al._ (2023); Gambuti  _et al._ (2023). This
structure is similar to the Feynman-parameter representation of loop
integrals, allowing the application of recently developed techniques Artico
and Magnea (2023a, b); Britto (2023); Chen (2020); Lee (2014). Given this
richness in both physical and mathematical contexts, it is imperative to
develop efficient techniques for finite integrals, well-adapted to the
computation of the $\rm E^{N}C$ for arbitrary $\rm{N}$.

In this _Letter_ we obtain a new result for the LO four-point correlator in
$\mathcal{N}=4$ sYM by squaring a $1\to 5$ form factor to derive a simplified
expression for the $1\to 4$ splitting function, which is then integrated using
a novel integration-by-parts technique that operates directly in the energy
parameter $(x_{i})$ space.

##  .2 Super Form Factors and Splitting Functions

$\displaystyle\hskip-8.53581pt\begin{array}[]{c}\includegraphics[width=108.12054pt]{NMHV-
left.eps}\end{array}\begin{array}[]{c}\bm{\times}\end{array}\begin{array}[]{c}\includegraphics[width=108.12054pt]{NMHV-
right.eps}\end{array}$
Figure 2: The product of chiral and antichiral MHV diagrams used to compute
the form factor squared.

We begin by deriving a compact form for the tree-level form factors squared,
$|F_{N+1}({\cal O})|^{2}=|\langle{0}|{\cal
O}(q)|{p_{1},h_{1},\ldots,p_{N+1},h_{N+1}}\rangle|^{2}$, for $N=3,4$, where
${\cal O}$ is a member of the stress-tensor multiplet in planar ${\cal N}=4$
sYM. Here $(p_{i},h_{i})$ denote massless on-shell momenta and helicities. The
particles in the final state (gluons, gluinos and scalars) form on-shell
supermultiplets. The stress-tensor multiplet includes, amongst others, the
protected half-BPS operator ${\cal O}_{20^{\prime}}$, the stress-energy tensor
and the Lagrangian ${\cal L}$ of the theory. ${\cal N}=4$ supersymmetry
relates their form factors squared and summed over all helicities in the on-
shell multiplets, in a simple way, e.g., $|F_{N+1}({\cal
L})|^{2}=(q^{2})^{2}|F_{N+1}({\cal O}_{20^{\prime}})|^{2}$, with
$q=\sum_{i=1}^{N+1}p_{i}$ being the total injected momentum.

To compute the splitting function $\mathcal{P}^{(0)}_{1\rightarrow{N}}$, we
consider the multi-collinear limit of $|F_{N+1}|^{2}$, in which the angles
between the first $N$ particles become small simultaneously (see Fig. 1). The
on-shell particle momenta $p^{\mu}$ have the lightcone components

$\displaystyle p^{+}_{i}=x_{i}\,,\qquad p^{-}_{i}=x_{i}|z_{i}|^{2}\,,\qquad
p^{\perp}_{i}=x_{i}z_{i}\,,$ (1)

where the energy fractions $x_{i}={2p^{0}_{i}}/{q^{0}}$ satisfy
$\sum_{1}^{N+1}x_{i}=2$. In the limit the complex parameters $z_{i}$, for
$i=1,\ldots,N$, go to zero at the same rate. We also have $x_{N+1}=1$ and we
set $z_{N+1}=\infty$ by a Lorentz transformation. The Mandelstam variables
simplify to

$\displaystyle s_{i_{1}i_{2}\ldots
i_{k}N+1}=x_{i_{1}}+x_{i_{2}}+\ldots+x_{i_{k}}\equiv x_{i_{1}i_{2}\ldots
i_{k}}\,,$
$\displaystyle{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{i_{1}i_{2}\ldots
i_{k}}}=\sum_{i<j}x_{i}x_{j}|z_{i}-z_{j}|^{2},\ \,\,i,j=1,\ldots,N.$

For the case $N=4$, ${\cal N}=4$ supersymmetry implies that we need only the
square of the sum of the MHV and NMHV $F_{5}({\cal L})$. Tree-level form
factors in ${\cal N}=4$ sYM are computed efficiently by the so-called MHV
rules Cachazo  _et al._ (2004); Brandhuber  _et al._ (2011). We consider the
planar limit, $N_{c}\to\infty$, and specify a single color orientation. The
MHV matrix element squared is $\mathbb{F}\overline{\mathbb{F}}^{\rm
MHV}_{N+1}=N_{c}^{N+1}{(q^{2})^{4}}/({s_{12}s_{23}\ldots s_{1(N+1)}})$. The
NMHV form factor consists of a sum of diagrams shown on the left panel in Fig.
2 (the right panel corresponds to the conjugate $\overline{\rm NMHV}$ form
factor). They are labeled anti-clockwise by $b^{\prime}>b\in[1,N+1]$ denoting
the set of particles coming out of the interaction vertex. The complementary
set with $a^{\prime}=b^{\prime}+1$ and $a=b-1$ consists of particles connected
to the (non-polynomial) Lagrangian operator, denoted by a cross. The momentum
$P_{bb^{\prime}}=\sum_{i=b}^{b^{\prime}}p_{i}$ flows from the operator to the
interaction vertex. In the multi-collinear limit, if the non-collinear leg
$N+1$ is on the interaction side, the contribution of the graph to the highest
order pole vanishes. Finally, the matrix element squared
$\mathbb{F}\overline{\mathbb{F}}_{N+1}$ is obtained by multiplying the sum of
all diagrams of the same color orientation by its conjugate and summing over
the helicities of the final states. We find that the ratio between the
$(N+1)-$point $\rm{NMHV}\times\overline{\rm{NMHV}}$ and
$\rm{MHV}\times\overline{MHV}$ form factors in the multi-collinear limit
$1||2\dots||N$ is given by

$\displaystyle\frac{\mathbb{F}\overline{\mathbb{F}}^{\rm
NMHV}_{N+1}}{\mathbb{F}\overline{\mathbb{F}}^{\rm MHV}_{N+1}}\overset{\rm
coll.}{=}\sum_{\\{b,b^{\prime}\\};\\{c,c^{\prime}\\}}\Bigg{(}\sum_{i=b}^{b^{\prime}}\sum_{j=c}^{c^{\prime}}\sum_{k=e}^{e^{\prime}}x_{i}x_{j}x_{k}z_{ik}\bar{z}_{kj}\Bigg{)}^{4}\times\frac{z_{b-1\,b}z_{b^{\prime}\,b^{\prime}+1}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{b\ldots
b^{\prime}}}K_{b-1,b^{\prime}}K_{b,b^{\prime}}L_{b,b^{\prime}}L_{b,b^{\prime}+1}}\frac{\bar{z}_{c-1\,c}\bar{z}_{c^{\prime}\,c^{\prime}+1}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{c\ldots
c^{\prime}}}\overline{K}_{c-1,c^{\prime}}\overline{K}_{c,c^{\prime}}\overline{L}_{c,c^{\prime}}\overline{L}_{c,c^{\prime}+1}}\,,$
$\displaystyle z_{ij}=z_{i}-z_{j},\quad
K_{ab}\equiv\sum_{i=a}^{b}z_{ai}x_{i},\quad
L_{ab}\equiv\sum_{i=a}^{b}z_{bi}x_{i},\quad 1\leq b<b^{\prime}\leq N\,,\;1\leq
c<c^{\prime}\leq N,\;[e,e^{\prime}]=[b,b^{\prime}]\cap[c,c^{\prime}]\,.$ (2)

The unphysical poles in Eq. (.2) cancel out in the sum of all terms. ${\cal
N}=4$ supersymmetry implies a helicity decomposition of the matrix elements
squared, e.g.,
$\mathbb{F}\overline{\mathbb{F}}_{5}=2\mathbb{F}\overline{\mathbb{F}}_{5}^{\rm
MHV}+2\mathbb{F}\overline{\mathbb{F}}_{5}^{\rm NMHV}$. The splitting function
is obtained from the multi-collinear limit of
$\mathbb{F}\overline{\mathbb{F}}_{N+1}$ by symmetrizing over all permutations
of the collinear particles, and by dividing by the two-particle matrix element
squared to cancels the dependence on the operator ${\cal L}$,

$\displaystyle{\cal P}^{(0)}_{1\to N}={N_{c}^{N-1}\over(x_{1}\ldots
x_{N})^{2}|z_{12}\ldots z_{N-1N}|^{2}}\,{\cal
G}_{N}+\text{perm}(1,2,\ldots,N)\quad\text{where}\quad{\cal
G}_{N}:=\lim_{1||2\dots||N}{\mathbb{F}\overline{\mathbb{F}}_{N+1}\over\mathbb{F}\overline{\mathbb{F}}^{\rm
MHV}_{N+1}}\ .$ (3)

$-1$$0$$1$$2$$3$$4$$5$$6$ Figure 3: Dual coordinates for the five-particle
form factor. The points lie on an infinite contour with a period equal to the
injected momentum $q$.

Our general result in Eq. (.2) reproduces that for $N=3$ in Chen  _et al._
(2020a), but in a way which makes one of the remarkable features of amplitudes
Drummond  _et al._ (2007, 2008, 2010) and form factors Alday and Maldacena
(2007); Maldacena and Zhiboedov (2010); Bork (2014); Bianchi  _et al._ (2019);
Ben-Israel  _et al._ (2018) in ${\cal N}=4$ sYM manifest, namely the dual
conformal symmetry acting linearly on the particle momenta. To this end we
introduce dual coordinates $y_{i}$ by the rule $p_{i}=y_{i}-y_{i-1}$. In the
amplitude case with vanishing total momentum $q=0$ the dual points form a
closed contour. For the form factor $q\neq 0$, and the dual points lie on an
infinite contour with period $q$, such that $y_{ij}=y_{i+N+1,j+N+1}$ and
$y_{i+N+1,i}\equiv q$, where $y_{i,j}\equiv y_{i}-y_{j}$. This is illustrated
in Fig. 3 for the relevant case of the five-particle form factor. To reproduce
the result of Chen  _et al._ (2020a), we choose the points
$y_{0},\ldots,y_{5}$ on the contour and write the result in terms of dual
conformal cross-ratios
$(a,b,c,d)={y^{2}_{ab}y^{2}_{cd}}/({y^{2}_{ac}y^{2}_{bd}})$,

$\displaystyle\hskip-1.42262pt\frac{\mathbb{F}\overline{\mathbb{F}}^{\rm
NMHV}_{4}}{\mathbb{F}\overline{\mathbb{F}}^{\rm MHV}_{4}}\overset{\rm
coll.}{=}({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,2,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}4})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},3,2,0)+(3,1,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}4})\,.$

A new result reported here is the splitting function for $N=4$ in
$\mathcal{N}=4$ sYM. It is given by a very compact dual conformal function of
the points $y_{i}$ with $i=-1,0,\ldots,5$:

$\displaystyle\frac{\mathbb{F}\overline{\mathbb{F}}^{\rm
NMHV}_{5}}{\mathbb{F}\overline{\mathbb{F}}^{\rm MHV}_{5}}\overset{\rm
coll.}{=}$
$\displaystyle-1+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,2,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},2,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,3,0)+(4,1,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},3,2,0)+(4,2,1,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+(0,4,3,1)$
$\displaystyle+(0,4,3,1)({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,3,1)(3,1,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,2,0)(0,2,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})$
$\displaystyle+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,2,4)(4,2,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},3,2,0)(4,2,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,2,0)(4,2,1,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,3,0)({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,2,4)$
$\displaystyle+(4,1,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})(0,2,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,3,1)({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,2,0)+(3,1,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})(4,2,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})$
$\displaystyle+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},4,3,1)({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,2,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+(3,1,0,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}{5}})({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},2,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,2,4)({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})+(0,2,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}{5}})({\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}-1},1,3,{\color[rgb]{0.5,0.0,0.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.0,0.0}5})\,.$

##  .3 Phase-space Integration from Integration-by-Parts

To compute the $\rm E^{N}C$ in the collinear limit we invent an automated
method exploiting simplifications from integer-dimensional integration-by-
parts (IBPs) identities. Further developing algorithms from Henn  _et al._
(2023), we show how to find a basis of master integrals which is both free
from divergences and avoids higher-degree poles, and illustrate this algorithm
for the specific case of the $\rm E^{4}C$.

1\. Family Definitions: The splitting function exhibits a uniform scaling
weight in the energy fractions, allowing them to be treated as projective
coordinates $[x_{1}:\cdots:x_{N}]\in{\rm P}_{{N}-1}(\mathbb{R}_{+})$,
parametrizing the ${N}-$body collinear phase space. This yields a master
formula for the ${\rm E^{N}C}$ in the collinear limit,

$\displaystyle{\rm E^{N}C}$
$\displaystyle\overset{\text{coll.}}{=}\frac{1}{|z_{12}\cdots
z_{\tiny{{N}-1{N}}}|^{2}}\hskip-1.42262pt\int\hskip-1.42262pt\frac{d^{N}x}{\text{GL}(1)}(x_{1}+\cdots+x_{N})^{-{N}}{\cal
G}_{{N}}$ $\displaystyle\;+\;\text{perm}\,(z_{1},\cdots,z_{N})\,,$ (4)

where ${\cal G}_{N}$ is defined in Eq. (3) and we introduced the short-hand
notation

$\displaystyle\int\frac{d^{N}x}{\text{GL}(1)}\,\equiv\int_{0}^{\infty}dx_{1}\cdots
dx_{N}\,\delta(1-\sum_{i}x_{i})\,.$

Investigating the integrand topologies, we observe that the integrals for the
${\rm E^{4}C}$ belong to a family $A_{1,1,1,1,1,1,1,0,0,0}$ defined as

$\displaystyle
A_{a_{1},\cdots,a_{10}}\equiv\int\frac{d^{4}x}{\text{GL}(1)}\,\frac{1}{\prod_{i}D_{i}^{a_{i}}}\,,$
(5) $\displaystyle
D_{1}={\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}},\quad
D_{2}={\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}},\quad
D_{3}={\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}},\quad$
$\displaystyle D_{4}=x_{1234},\quad D_{5}=x_{234},\quad D_{6}=x_{123},\quad
D_{7}=x_{34},\quad$ $\displaystyle D_{8}=s_{12},\quad D_{9}=s_{23},\quad
D_{10}=s_{34}\,,$

with multi-particle poles only in $D_{1-7}$. The integrals for the ${\rm
E^{3}C}$ are given by the simpler family $B_{1,1,1,1,0,0}$,

$\displaystyle
B_{a_{1},\cdots,a_{6}}\equiv\int\frac{d^{3}x}{\text{GL}(1)}\frac{s_{12}^{-a_{5}}\,s_{23}^{-a_{6}}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}^{a_{1}}\,x_{12}^{a_{2}}\,x_{23}^{a_{3}}\,x_{123}^{a_{4}}}\,.$
(6)

When computing the energy correlators, we wish to work with integrands
satisfying the following conditions:

(a) Homogeneity: The integrand has an overall scaling dimension $(-{N})$ under
the transformation
$(x_{1},\cdots,x_{N})\rightarrow(\kappa\,x_{1},\cdots,\kappa\,x_{N})$.

(b) Finiteness: The integral is free from IR divergences as any subset of
energy variables go to zero. Given condition (a), condition (b) is equivalent
to the UV power-counting behavior $\prod_{i}D_{i}^{-a_{i}}\sim
O(\kappa^{-1-|S|})$ as
$S\rightarrow\kappa\,S\,,\kappa\rightarrow\infty\,,\;\forall\,S\subset\\{x_{1},\cdots,x_{N}\\}\,.$

Such integrals fulfill the same criteria as the ‘admissible integrals’ in Henn
_et al._ (2023), for which four-dimensional IBP methods are particularly well
suited. One can reduce the integrals with poles of degree up to $N$ onto a
basis of master integrals of lower complexity. The latter can be evaluated
with the Maple program HyperInt Panzer (2015).

Our IBP algorithm operates directly on the projective coordinates
$[x_{1}:\cdots:x_{N}]$. Integrals defined without a dimensional regulator
exhibit several distinctive features. First, there exist integrand-level
partial fractioning identities among the integrals with different propagator
indices. It is advantageous to set up partial-fractioning identities among the
(finite) integrands

$\displaystyle
A_{a_{1},\cdots,a_{7};\;q_{1},\cdots,q_{4}}\equiv\int\frac{d^{4}x}{\text{GL}(1)}\,\frac{x_{1}^{-q_{1}}x_{2}^{-q_{2}}x_{3}^{-q_{3}}x_{4}^{-q_{4}}}{D_{1}^{a_{1}}D_{2}^{a_{2}}\cdots
D_{7}^{a_{7}}}\,,$ (7)

where we demand that $a_{i}\geq 0,q_{i}\leq 0,\forall\,i$, and place an upper
bound on the total numerator degree. This allows us to set up a simple basis
of finite integrands with $x_{i}-$monomials in the numerator. Second, boundary
terms are generated in the IBP relations on the surface of the integration
domain: $x_{i}=0,\,i\in\\{1,2,3,4\\}$. These boundary terms are given by
lower-point finite integrals over coordinates on the codimension-1 boundary
$[x_{j}:x_{k}:x_{l}]\equiv[x_{1}:\ldots\hat{x}_{i}\ldots:x_{4}]$,

$\displaystyle
B^{[jkl]}_{a_{1},a_{2},a_{3},a_{4};\;q_{1},q_{2},q_{3}}\equiv\int\frac{d^{3}x}{\text{GL}(1)}\,\frac{x_{j}^{-q_{1}}x_{k}^{-q_{2}}x_{l}^{-q_{3}}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{jkl}}^{a_{1}}\,x_{jk}^{a_{2}}\,x_{kl}^{a_{3}}\,x_{jkl}^{a_{4}}}\,,$
(8)

which are equivalent to the three-point energy integral family in Eq. (6) upon
setting $(j,k,l)=(1,2,3)$. Hence our algorithm must proceed iteratively, by
generating the relevant lower-point integrals and recycling them for the
reduction of the higher-point integral families.

2\. IBP Algorithm: A generic IBP relation in projective space has the form

$\displaystyle\int\frac{d^{N}x}{{\rm GL}(1)}\,O_{i}\circ
f=-\int\frac{d^{N-1}x}{{\rm GL}(1)}\,v\circ f\bigg{|}_{x_{i}=0}\,,$ (9)
$\displaystyle\quad v=\prod_{k}x_{k}^{-q_{k}}\,,\quad
f=\frac{1}{\prod_{j}D_{j}^{a_{j}}}\,,$

where $O_{i}=\frac{\partial}{\partial x_{i}}\,v\,,\,i=1,\cdots,{N}$, are
differential operators acting on projective coordinates, and we demand
$a_{j}\geq 0,\,q_{k}\leq 0,\,\forall\,j,k$. The validity of the IBP relation
requires condition (a) and (b) both fulfilled on the left-hand side of Eq.
(9). Hence we use individual scaling dimensions to impose a well-defined power
counting on the IBP differential operator $O_{i}$ in each integration region.
This means that under the scaling of any subset of integration variables
$S\rightarrow\kappa\,S$ ,

$\displaystyle
O_{i}\rightarrow\kappa^{\beta_{S}}O_{i}\,,\;\forall\,S\subset\\{x_{1},x_{2},x_{3},x_{4}\\}\,,$

where the scaling dimension is given by $\beta_{S}=-\sum_{k\in
S}q_{k}-\delta_{i\in S}\,.$ Given a system of operators graded by individual
scaling dimensions, we derive finite IBPs in the following steps:

(i) _Seeding_ : The seeding process is optimized by an integrand reduction
procedure. In a given sector we pair potential vectors $v$ with an integrand
$f$, and impose condition (a) and (b) on $O_{i}\circ f$. We generate partial
fractioning identities on finite fields for $v\circ f$, organized in _block
triangular form_ Liu and Ma (2019); Guan  _et al._ (2020), and solve for an
integrand basis for IBP generation.

(ii) _IBP generation_ : Applying the differential operators to the integrand,
we generate IBP relations ${\cal M}_{N}^{a}$ in the form of Eq. (9), where the
left-hand side is expanded over integrals in the ${N}-$point family and the
right-hand side defines the boundary integral families. Each integral that
appears in ${\cal M}_{N}^{a}$ is finite by construction.

(iii) _Integrand reduction_ : ${\cal M}_{N}^{a}$ encloses the complete set of
$N-$point finite integrals. Among them we generate all partial fractioning
identities ${\cal M}_{N}^{b}$ following the sector-by-sector procedure
described in step (i).

(iv) _Reducing boundary integrals_ : When possible, we integrate out the
boundary integrals generated on the right-hand-side of ${\cal M}_{N}^{a}$.
Otherwise, we repeat our procedure on the $({N}-1)$-point integral family.

(v) _Integral reduction for the full family_ : We carry out the reduction of
${\cal M}_{{N}}^{a}$ together with ${\cal M}_{{N}}^{b}$ iteratively, starting
from ${N}=2$.

In the case of the $\rm{E^{4}C}$, our algorithm generates $3\times 10^{3}$
finite integrals in the form of Eq. (7) with total numerator degree $p<7$. To
reduce the target integrals in Eq. (.3), the reduction was carried out for
$30^{4}$ iterations with different kinematic values over finite fields,
allowing functional reconstruction powered by Peraro (2016, 2019).

3\. Master Integrals and Symbol Alphabets: For the four-point integral family
in Eq. (5), after performing the integrand reduction, and discarding empty
topologies that contain no finite integrals, the four-point integrals are
mapped onto six sub-topologies, plus their images under a _reflection_
symmetry which flips the detector orientation : $1\leftrightarrow
4,2\leftrightarrow 3$. The six topologies are divided into the following three
categories:

Type-I (one $3-$particle cut) : (2,4,5,6) (2,4,6,7) ,

Type-II ($4-$ and $3-$particle cut) : (1,2,4,5) (1,2,4,7) ,

Type-III (two $3-$particle cuts): (2,3,4,5) (2,3,5,6),

where each topology is labeled by the location of its maximal cuts. After IBP
reduction, we find 28 four-point master integrals $A_{1-28}$, 14 three-point
boundary integrals $B_{1-14}$ and one constant function $C_{0}$. Using the
parametrization (see Fig. 1)

$\displaystyle(z_{1},z_{2},z_{3},z_{4})=(0,\,z,\,1,\,\frac{z-w}{1-w})\,,$ (10)

the master integrals evaluate to polylogarithmic functions up to weight three
in $(z,\bar{z})$ and $(w,\bar{w})$.

The boundary integrals $B_{1-14},C_{0}$ all arise from three-point topologies
$B^{[ijk]}_{1,0,1,1}$ with different detector orientations. For
$(i,j,k)=(1,2,3)$, the master integrals define two dilogarithmic and two
logarithmic functions, whose symbol alphabets read

$\displaystyle\mathcal{A}_{123}=\left\\{z,\bar{z},1-z,1-\bar{z},1-|z|^{2}\right\\}\,.$

The type-I and II topologies contain 18 four-point master integrals $A_{1-18}$
which couple to four distinct boundary integral families:
$B^{[123]}_{1,0,1,1}$, $B^{[124]}_{1,0,1,1}$, $B^{[134]}_{1,0,1,1}$ and
$B^{[234]}_{1,0,1,1}$. Their symbol alphabets are drawn from the set

$\displaystyle\overline{\mathcal{A}}_{123}\cup\overline{\mathcal{A}}_{124}\cup\overline{\mathcal{A}}_{134}\cup\overline{\mathcal{A}}_{234}\cup\mathcal{A}_{\text{Quad}}\,,$

where $\overline{\mathcal{A}}_{123}\equiv\mathcal{A}_{123}\cup\\{z-\bar{z}\\}$
and

$\displaystyle\mathcal{A}_{\text{Quad}}$
$\displaystyle\equiv\\{w\bar{z}-\bar{w}z,\,w\bar{z}-\bar{w},\,z\bar{w}-w,$
$\displaystyle 1-w-\bar{w}+z\bar{w},\,1-w-\bar{w}+w\bar{z}\\}\,.$

$\overline{\mathcal{A}}_{124},\overline{\mathcal{A}}_{134},\overline{\mathcal{A}}_{234}$
can be generated from $\overline{\mathcal{A}}_{123}$ through an $S_{4}$
symmetry transformation.

Figure 4: The intersection between the hyperplane $x_{1234}=0$ and the
hypersurfaces
${\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}=0$
and
${\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}=0$
projected onto the $(x_{2},x_{3})-$plane, gives cubic-root letters in the
${\rm{E^{4}C}}$.

In the type-III topologies we identify 11 four-point master integrals which
couple to two boundary integral families, $B^{[123]}_{1,0,1,1}$ and
$B^{[234]}_{1,1,0,1}$, related by _reflection_. The type-III integrands do not
factorize linearly in terms of $(z,\bar{z},w,\bar{w})$. One must further
introduce roots of a cubic polynomial, $(a,b,c)$, defined by

$\displaystyle a\,b\,c=-|z|^{2}|w|^{2},\quad a+b+c=1-w-\bar{w}-z-\bar{z}\,,$
$\displaystyle
a^{-1}+b^{-1}+c^{-1}=1-z^{-1}-\bar{z}^{-1}-w^{-1}-\bar{w}^{-1}\,.$ (11)

The symbol alphabet of the type-III integrals are drawn from the set

$\displaystyle\mathcal{A}_{123}\cup\mathcal{A}_{\text{Tri}^{2}}\cup\\{1\leftrightarrow
4,2\leftrightarrow 3\\},$

where

$\displaystyle\mathcal{A}_{\text{Tri}^{2}}=$
$\displaystyle\left\\{|z|^{2}-|w|^{2},z-|w|^{2},\bar{z}-|w|^{2}\,,\right.$
(12)
$\displaystyle\;\left.\frac{a}{b},\frac{a+|w|^{2}}{b+|w|^{2}},\frac{a+w}{b+w}\frac{b+\bar{w}}{a+\bar{w}}\right\\}\cup\\{a\rightarrow
b,b\rightarrow c\\}\,,$

and the _reflection_ maps $(z,w)$ to $(1/w,1/z)$ and $(a,b,c)$ to
$(1/a,1/b,1/c)$. The origin of the cubic-root letters can be understood from
the maximal-cut condition
${\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}={\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}=x_{1234}=0$,
whose solutions, when projected onto the $(x_{2},x_{3})-$plane, define a
singular cubic curve $(a\,x_{2}-x_{3})(b\,x_{2}-x_{3})(c\,x_{2}-x_{3})=0$, as
illustrated in Fig. 4. The structure of the master integrals in this sector
are discussed in App. A. Despite the cubic-root dependence, the master
integrals are single-valued functions whose branch cuts cancel on the
Euclidean sheet Brown (2004); Dixon  _et al._ (2012); Bourjaily  _et al._
(2023). They all satisfy the expected first-entry condition Gaiotto  _et al._
(2011), that the first entry of the symbol is $|z_{ij}|^{2}$. In App. B we
define the full set of master integrals $A_{1-28}$, $B_{1-14}$ and $C_{0}$,
and provide their transformation into a basis of uniform transcendental
weight.

##  .4 Results

Integrating the squared form factor, we obtain an analytic result for the
collinear limit of the $\rm E^{4}C$ in $\mathcal{N}=4$ sYM. We find that the
LO $\rm E^{4}C$ function space is comprised of polylogarithms up to weight 3,
whose symbol alphabets are drawn from the set

$\displaystyle\mathcal{A}_{4}(1,2,3,4)$
$\displaystyle=\overline{\mathcal{A}}_{123}\cup\mathcal{A}_{\text{Quad}}\cup\mathcal{A}_{\text{Tri}^{2}}\cup\text{perm}(1,2,3,4)\,.$

Interestingly, the first entry of the symbol only sees five letters:
$\\{|z|^{2},|1-z|^{2},|w|^{2},|1-w|^{2},|z-w|^{2}\\}$. The final answer can be
expressed in terms of a basis of single-valued polylogarithmic functions as
well as their images under $S_{4}$-permutations of the detector orientations,

$\displaystyle\hskip-7.11317pt{\rm{E^{4}C}}^{\rm
LO,coll}_{\mathcal{N}=4}(z_{ij})$
$\displaystyle=\frac{\sum\limits_{I=1}^{51}R_{I}(z,\omega)\,F_{I}(z,\omega)}{|z_{12}|^{2}|z_{23}|^{2}|z_{34}|^{2}}+\text{perm}(1,2,3,4)\,.$

In the ancillary files, we provide the analytic expressions for the basis
$\vec{F}$ and algebraic coefficients $\vec{R}$.

We have checked our result through direct numerical integration over the
energy fractions. An additional check comes from the investigation of
kinematic limits. In particular, we can access the triple-collinear limit by
sending $z_{1},z_{2},z_{3}$ to the origin; and the double-collinear limit by
taking $|z_{12}|^{2},|z_{34}|^{2}$ to zero simultaneously. In these limits the
${\rm E^{4}C}$ factorizes Chen  _et al._ (2020a),

$\displaystyle{\rm{E^{4}C}}\xrightarrow{z_{1},z_{2},z_{3}\sim
0}\,\frac{4}{3|z_{4}|^{2}}{\rm{E^{3}C}}(z_{1},z_{2},z_{3})\,,$
$\displaystyle{\rm{E^{4}C}}\xrightarrow{z_{12},z_{34}\sim
0}\frac{2}{3|z_{23}|^{2}}\,{\rm{E^{2}C}}(z_{1},z_{2})\times{\rm{E^{2}C}}(z_{3},z_{4})\,,$

into lower-point correlators, $\rm{E^{2}C},\rm{E^{3}C}$, defined in Eq. (.3).

##  .5 Outlook

Our result provides a number of directions for further exploring the
perturbative structure of physical observables. First, it is important to
study the singularity structure of the result to understand if its alphabet
can be predicted a priori. There has been recent progress in the study of the
Landau equations Mizera and Telen (2022); Fevola  _et al._ (2023a, b),
intersection theory Mizera (2018); Mizera and Pokraka (2020); De and Pokraka
(2023), as well as the structure of integrals directly in Feynman parameter
space Arkani-Hamed and Yuan (2017); Gong and Yuan (2022); Arkani-Hamed  _et
al._ (2022); Bourjaily  _et al._ (2021); Britto (2023), all of which are
ideally suited for analyzing the integrals appearing in the ${\rm{E^{N}C}}$.
Second, it is interesting to push towards higher point correlators, and to
understand if statements can be made about the function space of the generic
${\rm{E^{N}C}}$. Third, our result suggests to further explore the origin and
the role of dual conformal symmetry for physical observables. Finally, it is
highly desirable to extend our calculation to real world QCD, where our
insight into the $\mathcal{N}=4$ sYM alphabet should prove invaluable.

Acknowledgements: We acknowledge enlightening discussions with Nima Arkani-
Hamed, Hofie Hannesdottir, Gregory Korchemsky, Andrzej Pokraka, Gang Yang,
Xiaoyuan Zhang and Hua Xing Zhu. This work is supported by the National
Natural Science Foundation of China under Grant No. 12357077. KY would like to
thank the sponsorship from Yangyang Development Fund. IM is supported by DOE-
HEP-GR120647.

##  Appendix A Appendix A

In the type-III topologies, the origin of the cubic-root letters is sector
$({\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}2},{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}3},4)$,
where the solutions to maximal-cut condition
${\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}={\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}}=x_{1234}=0$
define three projective lines. Projected onto the $(x_{2},x_{3})-$plane, the
line equations become a singular cubic curve
$(a\,x_{2}-x_{3})(b\,x_{2}-x_{3})(c\,x_{2}-x_{3})=0$, with $a,b,c$ defined in
Eq. (.3). Examining the leading singularity Cachazo (2008); Wasser (2022);
Henn  _et al._ (2020) of the master integrals in sector
$({\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}2},{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}3},4)$,
we observe a subsystem
$\\{f_{1},f_{2}\\}\equiv\frac{|z_{13}z_{34}|^{2}}{|z_{23}|^{4}}\\{A_{22},A_{23}\\}$
whose leading singularities contain odd structures in the cubic roots,

$\displaystyle f_{1}$ $\displaystyle=\frac{1}{(c-a)(a-b)}\int d\ln x_{3}\wedge
d\ln\frac{a\,x_{2}-x_{3}}{b\,x_{2}-x_{3}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{2}}{D_{4}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{3}}{D_{4}}$
$\displaystyle-\frac{1}{(b-c)(c-a)}\int d\ln x_{3}\wedge
d\ln\frac{b\,x_{2}-x_{3}}{c\,x_{2}-x_{3}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{2}}{D_{4}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{3}}{D_{4}}$
$\displaystyle f_{2}$ $\displaystyle=\frac{a}{(c-a)(a-b)}\int d\ln
x_{3}\wedge\ln\frac{a\,x_{2}-x_{3}}{b\,x_{2}-x_{3}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{2}}{D_{4}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{3}}{D_{4}}$
$\displaystyle-\frac{c}{(b-c)(c-a)}\int d\ln
x_{3}\wedge\ln\frac{b\,x_{2}-x_{3}}{c\,x_{2}-x_{3}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{2}}{D_{4}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{3}}{D_{4}}\,.$
(13)

The above expressions have manifest cyclic symmetry. Thus we introduce pure
functions $F_{a,b,c}$ through

$\displaystyle f_{1}$
$\displaystyle\equiv\frac{F_{a}}{(c-a)(a-b)}+\frac{F_{b}}{(a-b)(b-c)}+\frac{F_{c}}{(b-c)(c-a)}\,,\quad$
$\displaystyle f_{2}$
$\displaystyle\equiv\frac{a\,F_{a}}{(c-a)(a-b)}+\frac{b\,F_{b}}{(a-b)(b-c)}+\frac{c\,F_{c}}{(b-c)(c-a)}\,.$

Eq. (A) defines two dlog integrals $\\{F_{a}-F_{b},F_{b}-F_{c}\\}$, whose
symbol alphabets contain the odd cubic-root letters given in the last line of
Eq. (12). More specifically for $F_{a}$, we read off the following cubic-root
letters in its symbol:

$\displaystyle a\,,a+|z|^{2}\,,a+|w|^{2}\,,u_{a}\equiv(a+w)(a+\bar{w})\,,$
$\displaystyle
v_{a}\equiv(a+z)(a+\bar{z})\,,\,\frac{a+w}{a+\bar{w}}\,,\,\frac{a+z}{a+\bar{z}}\,.$

The identities
$u_{a}=\frac{|w|^{2}(a+|w|^{2})(b+|z|^{2})(c+|z|^{2})}{b\,c\,(|w|^{2}-|z|^{2})},v_{a}=\frac{a\,(a+|z|^{2})(b+|w|^{2})(c+|w|^{2})}{|w|^{2}(|w|^{2}-|z|^{2})}$,
together with the shuffle algebra Ree (1958), allow us to show in the first
and second entry of $\mathcal{S}[F_{a}]$, the cubic-root alphabets appear as
cyclic products: $a\,b\,c\,,(a+|z|^{2})(b+|z|^{2})(c+|z|^{2})$ and
$(a+|w|^{2})(b+|w|^{2})(c+|w|^{2})$. Therefore $\mathcal{S}[F_{a}]$ contains
only rational letters in the first and second entry, and in particular the
first-entry condition for single-valuedness is satisfied. Moreover, in the
cyclic combination $\\{F_{a}+F_{b}+F_{c}\\}$, the dependence on cubic roots
fully cancels, and its symbol letters are rational in all entries.

The top sector
$({\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}2},{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}3},4,5)$
contains two additional finite dlog integrals $f_{3},f_{4}$, which are linear
combinations of $A_{23},A_{24},A_{25},A_{26}$. They can be cast into the
following dlog form,

$\displaystyle f_{3}$ $\displaystyle=\int d\ln x_{3}\wedge
d\ln\frac{w\,x_{2}+x_{3}}{\bar{w}\,x_{2}+x_{3}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{2}}{D_{4}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{3}}{D_{5}}\,,$
$\displaystyle f_{4}$ $\displaystyle=\frac{1}{6}\int d\ln x_{3}\wedge
d\ln\frac{[(a\,x_{2}-x_{3})(b\,x_{2}-x_{3})(c\,x_{2}-x_{3})]^{2}}{[(w\,x_{2}+x_{3})(\bar{w}\,x_{2}+x_{3})]^{3}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{2}}{D_{4}}\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{3}}{D_{4}}$
$\displaystyle+\frac{1}{2}\int d\ln x_{3}\wedge
d\ln[(w\,x_{2}+x_{3})(\bar{w}\,x_{2}+x_{3})]\wedge
d\ln\frac{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}D_{2}}{D_{4}}\wedge
d\ln\frac{D_{5}}{D_{4}}\,,$

which suggests that $f_{3},f_{4}$ do not depend on the cubic roots. While the
first piece of $f_{4}$ contains $F_{a}+F_{b}+F_{c}$, its dependence on the
cubic roots is spurious.

##  Appendix B Appendix B

We provide the definition of the master integrals in the Type-I, II and III
subtopologies. The integrands for the four-point master integrals $A_{1-28}$
are

$\displaystyle
A_{1,2}:\quad\frac{x_{3}|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{34}^{2}\,x_{1234}}\,,\quad\frac{|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{34}\,x_{1234}},\quad$
$\displaystyle
A_{3}:\quad\frac{|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{234}\,x_{1234}}\,,$
$\displaystyle
A_{4,5}:\quad\frac{x_{3}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{123}\,x_{34}\,x_{1234}},\quad\frac{x_{2}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{123}\,x_{34}\,x_{1234}}\,,$
$\displaystyle
A_{6,7}:\quad\frac{x_{3}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{123}\,x_{234}\,x_{1234}},\quad\frac{x_{1}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{123}\,x_{234}\,x_{1234}}.$
$\displaystyle
A_{8}:\quad\frac{x_{3}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}^{2}\,x_{34}},\quad
A_{9}:\quad\frac{|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,x_{34}\,x_{1234}}\,,\quad$
$\displaystyle
A_{10}:\quad\frac{|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,x_{234}\,x_{1234}}\,,\quad
A_{11}:\quad\frac{x_{3}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{34}},\quad$
$\displaystyle
A_{12,13}:\quad\frac{x_{2}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{234}}\,,\quad\frac{x_{3}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{234}}\,,$
$\displaystyle A_{14-16}:$
$\displaystyle\qquad\frac{x_{1}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{1234}},\quad\frac{x_{2}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{1234}},\quad\frac{x_{3}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{1234}},$
$\displaystyle
A_{17}:\quad\frac{x_{1}x_{2}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{34}\,x_{1234}},\quad
A_{18}:\quad\frac{x_{2}^{2}|z_{12}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{1234}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{234}\,x_{1234}}.$
$\displaystyle A_{19-23}:\quad$
$\displaystyle\qquad\frac{x_{1}x_{2}|z_{23}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{1234}^{2}},\quad\frac{x_{1}x_{3}|z_{23}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{1234}^{2}}\,,\quad\frac{x_{2}x_{4}|z_{23}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{1234}^{2}}\,,$
$\displaystyle\qquad\frac{x_{2}|z_{23}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{1234}},\quad\frac{x_{3}|z_{23}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{1234}}\,,\quad$
$\displaystyle
A_{24,25}:\quad\frac{x_{2}^{2}|z_{23}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{234}\,x_{1234}},\quad\frac{x_{2}x_{3}|z_{23}|^{4}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{234}\,x_{1234}},$
$\displaystyle A_{26-28}:\quad$
$\displaystyle\frac{x_{2}x_{3}|z_{23}z_{14}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}x_{123}x_{234}},\quad\frac{x_{2}^{2}|z_{23}z_{14}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}x_{123}x_{234}},\quad\frac{x_{3}^{2}|z_{23}z_{14}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}x_{123}x_{234}}\,.$

The integrands for the three-point boundary integrals $B_{1-14}$ are

$\displaystyle
B_{1-3}:\quad\frac{x_{1}|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{123}^{2}},\quad\frac{x_{2}|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{123}^{2}},\quad\frac{|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{123}},\quad$
$\displaystyle
B_{4}:\quad\frac{x_{2}|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{123}}\,x_{23}\,x_{123}},$
$\displaystyle
B_{5,6}:\quad\frac{x_{1}|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{124}}\,x_{124}^{2}},\quad\frac{|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{124}}\,x_{124}},\quad
B_{7}:\quad\frac{x_{2}|z_{12}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{124}}\,x_{24}\,x_{124}},$
$\displaystyle
B_{8}:\quad\frac{|z_{34}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{134}}\,x_{134}},\quad
B_{9}:\quad\frac{x_{3}|z_{34}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{134}}\,x_{24}\,x_{134}},$
$\displaystyle
B_{10-12}:\quad\frac{x_{2}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{234}^{2}},\quad\frac{x_{3}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{234}^{2}},\quad\frac{|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{234}},\quad$
$\displaystyle
B_{13}:\quad\frac{x_{3}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{34}\,x_{234}},\quad$
$\displaystyle
B_{14}:\quad\frac{x_{3}|z_{23}|^{2}}{{\color[rgb]{0.0,0.0,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.0,0.5}s_{234}}\,x_{23}\,x_{234}}\,.$

The two-point boundary integrals reduce to a constant function, which we
define as $C_{0}=1$. The complete basis of master integrals
$\\{\vec{A},\vec{B},C_{0}\\}$ can be brought into a UT basis
$\\{\vec{I}_{A},\vec{I}_{B},C_{0}\\}$ by a linear transformation. We provide
the $43\times 43$ transformation matrix $T$ in the ancillary files.

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