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minimum size=7mm myblob/.style= /tikz/shape=rectangle, /tikz/fill=red,
/tikz/minimum size=0.3cm, HV/.style= /tikz/shape=circle,
/tikz/fill=rgb:black,1;white,2, /tikz/minimum size=0.01cm,/tikz/inner
sep=1.8pt codot/.style=/tikz/shape=circle, /tikz/fill=red,/tikz/minimum
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/tikz/shape=ellipse, /tikz/fill=rgb:black,1;white,2, /tikz/minimum size=0.3cm,

QMUL-PH-22-18

SAGEX-22-27

# Amplitudes, Hopf algebras and the colour-kinematics duality

Andreas Brandhubera,b Graham R. Browna,b Gang Chena Joshua Gowdya,b
Gabriele Travaglinia,b and Congkao Wena

###### Abstract

It was recently proposed that the kinematic algebra featuring in the colour-
kinematics duality for scattering amplitudes in heavy-mass effective field
theory (HEFT) and Yang-Mills theory is a quasi-shuffle Hopf algebra. The
associated fusion product determines the structure of the Bern-Carrasco-
Johansson (BCJ) numerators, which are manifestly gauge invariant and with
poles corresponding to heavy-particle exchange. In this work we explore the
deep connections between the quasi-shuffle algebra and general physical
properties of the scattering amplitudes. First, after proving the double-copy
form for gravitational HEFT amplitudes, we show that the coproducts of the
kinematic algebra are in correspondence with factorisations of BCJ numerators
on massive poles. We then study an extension of the standard quasi-shuffle
Hopf algebra to a non-abelian version describing BCJ numerators with all
possible gluon orderings. This is achieved by tensoring the original algebra
with a particular Hopf algebra of orderings. In this extended version, a
specific choice of the coproduct in the algebra of orderings leads to an
antipode in the resulting Hopf algebra that has the interpretation of
reversing the gluons’ order within each BCJ numerator.

##  1 Introduction

The colour-kinematics duality and double copy [1, 2, 3] play an important role
in the modern approach to scattering amplitudes and gravitational physics.
Inspired by Kawai-Lewellen-Tye (KLT) relations [4] for string amplitudes,
these remarkable properties were originally proposed for tree-level amplitudes
in pure Yang-Mills theory and gravity, and extended to amplitudes in a variety
of other theories [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24], to loop level [25, 26, 27, 28, 29, 30, 31, 32, 33, 34] and
even to exact classical solutions [35, 36, 37]. Recently the underlying
algebra and geometry of the colour-kinematics duality and double copy have
been studied from several different viewpoints [38, 39, 40, 17, 19, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57], leading to an in-
depth exploration of gravitational physics, gauge field theory and their
relationships.

In this paper we mainly focus on the algebraic aspects of the colour-
kinematics duality. The first critical milestone is to reveal the kinematic
algebra and construct all the duality-satisfying numerators, known as Bern-
Carrasco-Johansson (BCJ) numerators, from an algebraic product. In [57], and
building on [54, 55, 56], four of the present authors and Johansson proposed a
kinematic algebra that underlies the colour-kinematics duality in a heavy-mass
effective theory (HEFT) and, using a decoupling limit, in Yang-Mills (YM)
theory. Remarkably, it was found that the BCJ numerators for tree-level
amplitudes of two heavy particles and any number of gluons can be constructed
from a quasi-shuffle Hopf algebra, which is well studied in the context of
combinatorial Hopf algebras of shuffles and quasi-shuffles [58, 59, 60, 61,
62].

The HEFT BCJ numerators obtained in our approach enjoy several important
properties:

1. They are manifestly gauge invariant.

2. They are local for massless gluons, but contain poles corresponding to the propagation of the heavy particles.

3. Higher-point numerators factorise into products of lower-point ones at the singularities of the massive poles.

4. By decoupling the two heavy particles through a factorisation limit, one obtains BCJ numerators for tree-level amplitudes in pure YM theory, which therefore are also constructed using the same quasi-shuffle Hopf algebra.

This paper sharpens our understanding of the kinematic algebra and its
applications to the understanding of the colour-kinematics duality in HEFT and
YM theory. Our goal is twofold: first, we wish to link various mathematical
aspects of the newly discovered quasi-shuffle Hopf algebra [57] back to the
physical properties of scattering amplitudes; and second, we introduce new
mathematical structures and extend further the quasi-shuffle Hopf algebra to
incorporate certain general properties of scattering amplitudes, which are
otherwise difficult to describe using the standard quasi-shuffle algebra.
These two directions go hand-in-hand.

Specifically, for better describing the ordering of particles in the
scattering amplitudes we are led to a non-abelian extension of the standard
quasi-shuffle Hopf algebra, and important structures of the algebra such as
the coproduct and the antipode act non-trivially on the BCJ numerators. As we
will show, the action of the coproduct is interpreted as a factorisation limit
of the BCJ numerators on the massive poles, whereas the antipode reverses the
ordering of the particles entering the numerators.

There are several reasons to pursue these goals. One stems from the importance
of exploring the kinematic algebra from a mathematical viewpoint. In addition,
one may recall that the gravitational amplitudes obtained after a double copy
describe black hole scattering and gravitational wave emission. The resulting
expressions in the approach described in this paper are extremely compact, and
have led to one of the most efficient ways of studying black hole scattering
[63].

The rest of the paper is organised as follows. In Section 2, we review the
novel colour-kinematics duality and general properties of tree-level
amplitudes in HEFT, including the decoupling limit leading to YM amplitudes.
We also prove the double copy formula for the gravitational amplitudes
involving heavy particles using KLT relations. In Section 3, we introduce the
concept of the pre-numerators in HEFT and their construction from abstract
algebraic generators. These generators obey a quasi-shuffle Hopf algebra, and
the pre-numerators are obtained using a map from the generators to functions
of momenta and polarisation vectors. The BCJ numerators are directly related
to the pre-numerators in a simple manner through the so-called nested
commutators. Section 4 aims to study the kinematic Hopf algebra for the BCJ
numerators in HEFT and to understand the physical interpretation of various
mathematical operations in the Hopf algebra. In particular, the coproduct is
related to the factorisation of the pre-numerators. In Section 5, we introduce
the non-abelian version of the quasi-shuffle Hopf algebra. This is realised by
extending the algebraic generators to include the ordering of external
particles. We show that it is this version of the kinematic algebra that
directly describes the BCJ numerators in an algebraic fashion in the HEFT.
Section 6 is concerned with the factorisation behaviour of BCJ numerators and
tree-level amplitudes in HEFT and their connections with coproducts, using the
non-abelian version of the quasi-shuffle Hopf algebra. Finally, we conclude in
Section 7 and comment on several directions for further research. Three
appendices complete the paper. In Appendix A we provide a recursive definition
of the quasi-shuffle product, while in Appendix B we discuss a connection
between shuffle and quasi-shuffle algebras. Finally, in Appendix C we give an
alternative definition of the pre-numerator from a different mapping rule
which ensures that the pre-numerators themselves are symmetric under the
action of the antipode.

##  2 Colour-kinematics duality and double copy in HEFT

###  2.1 General setup

We will be interested in amplitudes involving two massive scalars with mass
$m$ and $n{-}2$ gluons/gravitons, in the heavy-mass limit [64, 65, 66, 67,
68]. More precisely, we will define the HEFT amplitude to be the piece of the
gluon or graviton amplitude which is of order $m$ or $m^{2}$,
respectively111Strictly speaking, gravity amplitudes contain additional
contact terms of order $m^{3}$ and higher [63] in the heavy-mass limit. These
are given by products of lower-point HEFT amplitudes and delta functions, and
vanish on generic kinematic configurations. In the following we will ignore
such contributions, which is equivalent to dropping all Feynman
$i\epsilon$’s.. For Yang-Mills-scalar amplitudes the HEFT amplitudes are
simply obtained by taking the leading term in the heavy-mass limit. For
instance, the colour-ordered two-scalar/two-gluon amplitude is given by222We
have omitted the overall coupling dependence.

$\mathcal{A}(1,2,3,4)=\frac{2p_{3}{\cdot}F_{1}{\cdot}F_{2}{\cdot}p_{3}}{s_{12}\big{(}s_{13}-m^{2}\big{)}}\,,$
(2.1)

where particles $1,2$ are gluons, $3,4$ are adjoint massive scalars.
$F_{i}^{\mu\nu}{:=}p_{i}^{\mu}\varepsilon_{i}^{\nu}-p_{i}^{\nu}\varepsilon_{i}^{\mu}$
denotes a linearised field strength, and $s_{ij}{:=}(p_{i}+p_{j})^{2}$. To
take the heavy-mass limit we write the momentum of the first and second scalar
(in a convention where all particles are outgoing) as $p_{3}^{\mu}=-mv^{\mu}$
and $p_{4}^{\mu}=mv^{\mu}+q^{\mu}$, respectively, where $v^{\mu}$ is the
velocity of the heavy particle and $q^{\mu}=p_{1}^{\mu}+p_{2}^{\mu}$ is the
sum of the gluon momenta. Then we can take the leading term in the large-$m$
expansion, to yield

$A(12,v)=-\frac{m}{s_{12}}\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}v}{v{\cdot}p_{1}}\,.$
(2.2)

In the above we have replaced the two scalar labels $3,4$ with a single label
for the velocity $v$. For $n$-point HEFT amplitudes we will also make the same
replacement, with legs $n{-}1$ and $n$ always labelling the heavy scalars,

$\displaystyle\mathcal{A}(1,2,\ldots,n{-}1,n)\xrightarrow[]{\rm
HEFT}A(12\ldots n{-}2,v)\,.$ (2.3)

In the same manner as above, we can take the heavy-mass limit of a
gravitational amplitude $\mathcal{M}(1,2,\ldots,n{-}1,n)$ involving two
massive scalars and extract the term which scales like $m^{2}$ giving the
gravitational HEFT amplitude $M(12\ldots n{-}2,v)$

$\displaystyle\mathcal{M}(1,2,\ldots,n{-}1,n)\xrightarrow[]{\rm
HEFT}M(12\ldots n{-}2,v)\,.$ (2.4)

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)[]{$n{-}1$};
\vertex[left=1.9cm of a] (a0)[GR]{$~{}~{}\rm\mathcal{A}~{}~{}$};
\vertex[left=1.9cm of a0] (am1){$n$}; \vertex[above=1.9cm of a0]
(b0){$\cdots$}; \vertex[left=1.6cm of b0] (bm1){$1$}; \vertex[right=1.6cm of
b0] (b1){${n{-}2}$}; \diagram*{(a)--[very thick](a0),(am1)--[very thick](a0),
(a0) -- [thick,gluon] (b0),(a0) -- [thick,gluon] (bm1),(a0) -- [thick,gluon]
(b1)}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\xrightarrow[]{\rm
double~{}copy}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)[]{$n{-}1$};
\vertex[left=1.9cm of a] (a0)[GR]{$~{}~{}\rm\mathcal{M}~{}~{}$};
\vertex[left=1.9cm of a0] (am1){$n$}; \vertex[above=1.9cm of a0]
(b0){$\cdots$}; \vertex[left=1.6cm of b0] (bm1){$1$}; \vertex[right=1.6cm of
b0] (b1){${n{-}2}$}; \diagram*{(a)--[very thick](a0),(am1)--[very thick](a0),
(a0) -- [photon,line width=0.8mm] (b0),(a0) -- [line width=0.8mm,photon]
(bm1),(a0) -- [line width=0.8mm,photon] (b1)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Note that gluons (gravitons) in YM (gravity) amplitudes $\mathcal{A}$
($\mathcal{M}$) are labelled as $1,\ldots,n{-}2$ as in the figure above. We
also quote the useful relation

$v{\cdot}p_{12\ldots n-2}=0+\mathcal{O}(m^{-1})\,,$ (2.5)

which follows from momentum conservation and on-shell conditions, where we
have defined $p_{12\ldots m}{:=}\sum_{k=1}^{m}p_{k}$. From now on we will drop
explicit factors of $m$ and $m^{2}$ since HEFT amplitudes always scale
homogeneously in the mass.

A novel colour-kinematic duality and double copy for HEFT amplitudes were
recently proposed in [56, 57]. The colour-ordered YM tree amplitudes with two
heavy particles and $n{-}2$ gluons are found to take the following form

$\displaystyle A(12\ldots n{-}2,v)$
$\displaystyle\,=\,\sum_{\Gamma\in\rho}{\mathcal{N}(\Gamma,v)\over
d_{\Gamma}}\,,$ (2.6)

where $\rho$ denotes all ordered nested commutators of the labels
$\\{1,\ldots\,,n{-}2\\}$. Each term in the sum is in one-to-one correspondence
with a cubic graph with gluon labels following the colour ordering of the
amplitude, and $d_{\Gamma}$ is the product of inverse scalar propagators of
massless particles associated with that cubic graph. $\mathcal{N}(\Gamma,v)$
represents the HEFT BCJ numerator for the cubic graph $\Gamma$, whose
properties and explicit construction will be studied in detail in this paper.
Note that the number of terms in the sum in (2.6) is given by the Catalan
number $C_{n-3}$ (for $n$-gluon amplitudes in YM, this number becomes
$C_{n-2}$).

As explained in e.g. [55], every nested commutator structure has an associated
cubic graph, where the vertices of the graph correspond to commutators and the
external legs correspond to gluons or gravitons. For gluons we only consider
graphs where the external legs follow the colour ordering of the amplitude. As
an example, for the set $\\{1,2,3\\}$, we have
$\rho{=}\\{[[1,2],3],\,[1,[2,3]]\\}$, and for the set $\\{1,2,3,4\\}$ we have
$\rho{=}\\{[[1,2],[3,4]],\,[[1,2],3],4],\,[1,[2,[3,4]]],\,[[1,[2,3]],4],\,[1,[[2,3],4]]\\}$.
The first nested commutator in each list can be represented pictorially as

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)[myblob]{};
\vertex[above=0.3cm of a](b)[dot]{}; \vertex[left=0.6cm of b](c);
\vertex[left=0.22cm of b](c23); \vertex[above=0.14cm of c23](v23)[dot]{};
\vertex[above=.4cm of c](j1){$1$}; \vertex[right=.7cm of j1](j2){$2$};
\vertex[right=0.5cm of j2](j3){$3$}; \diagram*{(a) -- [thick] (b),(b) --
[thick] (j1),(v23) -- [thick] (j2),(b)--[thick](j3)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\!\\!\quad\longrightarrow\quad\mathcal{N}([[1,2],3],v)\,,\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)[myblob]{};
\vertex[above=0.3cm of a](b)[dot]{}; \vertex[left=0.8cm of b](c);
\vertex[left=0.3cm of b](c12); \vertex[right=0.3cm of b](c34);
\vertex[above=0.17cm of c12](v12)[dot]{}; \vertex[above=0.17cm of
c34](v34)[dot]{}; \vertex[above=.4cm of c](j1){$1$}; \vertex[right=.7cm of
j1](j2){}; \vertex[right=0.5cm of j2](j3){}; \vertex[right=0.35cm of
j3](j4){}; \vertex[below=0.08cm of j4](k4){}; \vertex[below=0.05cm of
j2](k2){}; \vertex[below=0.05cm of j3](w3){}; \vertex[left=0.36cm of
w3](k3){}; \vertex[right=0.6cm of j1](l2){$2$}; \vertex[right=1.0cm of
j1](l3){$3$}; \vertex[right=1.5cm of j1](l4){$4$}; \diagram*{(a) -- [thick]
(b),(b) -- [thick] (j1),(v12) -- [thick]
(k2),(b)--[thick](k4),(v34)--[thick](k3)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\!\\!\quad\longrightarrow\quad\mathcal{N}([[1,2],[3,4]],v)\,,$
(2.7)

where the red box represents the two heavy scalars. For these particular
graphs, we also have $d_{\Gamma}=s_{12}s_{123}$ and
$d_{\Gamma}=s_{12}s_{34}s_{1234}$, respectively.

We will show in the next section that once the gluon amplitudes are recast in
colour-kinematics form (2.6), then the double-copy procedure will allow us to
directly write down the tree-level amplitude of $n{-}2$ gravitons coupled to
two heavy particles as

$\displaystyle M(12\ldots n{-}2,v)$
$\displaystyle\,=\,\sum_{\Gamma\in\widetilde{\rho}}{\big{[}\mathcal{N}(\Gamma,v)\big{]}^{2}\over
d_{\Gamma}}\,.$ (2.8)

There is no colour ordering for gravitons, therefore $\widetilde{\rho}$
denotes all un-ordered nested commutators of $\\{1,\ldots\,,n{-}2\\}$, which
are labels for the external gravitons. For example, in the case of three
gravitons we have $\tilde{\rho}=\\{[[1,2],3],[[1,3],2],[1,[2,3]]\\}$. In
general, the total number of terms of un-ordered nested commutators in (2.8)
is $(2n-7)!!$. Each term in the sum is in one-to-one correspondence with a
cubic graph.

The BCJ numerators are manifestly gauge invariant and contain only physical
heavy-mass propagators, which are linear. Thus, by construction, gluon HEFT
amplitudes contain only single massive poles, while graviton HEFT amplitudes
contain only double massive poles. The latter propery is entirely non-obvious
upon simply expanding graviton amplitudes in the full theory in the large-mass
limit. Moreover, the antisymmetric property for a single graph:

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\, \feynman
\vertex(a)[HV]{$~{}~{}~{}~{}$}; \vertex[above=0.6cm of a](b)[dot]{};
\vertex[left=1.6cm of b](c); \vertex[left=0.82cm of b](cL);
\vertex[above=0.8cm of cL](vL)[HV]{$L$}; \vertex[right=1.64cm of
vL](vR)[HV]{$R$}; \vertex[above=0.6cm of vL](t1){}; \vertex[above=0.6cm of
vR](t2){}; \vertex[above=1.8cm of c](j1){}; \vertex[right=1.2cm of j1](j2){};
\vertex[right=1.1cm of j2](j3){}; \vertex[right=0.8cm of j3](j4){};
\diagram*{(a) -- [thick] (b),(b)--[thick](vL) -- [thick] (j1),(vL) -- [thick]
(j2),(vR)--[thick](j3),(b)--[thick](vR)--[thick](j4)}; \, ,
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\, \feynman
\vertex(a)[HV]{$~{}~{}~{}~{}$}; \vertex[above=0.6cm of a](b)[dot]{};
\vertex[left=1.6cm of b](c); \vertex[left=0.82cm of b](cL);
\vertex[above=0.8cm of cL](vL)[HV]{$R$}; \vertex[right=1.64cm of
vL](vR)[HV]{$L$}; \vertex[above=0.6cm of vL](t1){}; \vertex[above=0.6cm of
vR](t2){}; \vertex[above=1.8cm of c](j1){}; \vertex[right=1.2cm of j1](j2){};
\vertex[right=1.1cm of j2](j3){}; \vertex[right=0.8cm of j3](j4){};
\diagram*{(a) -- [thick] (b),(b)--[thick](vL) -- [thick] (j1),(vL) -- [thick]
(j2),(vR)--[thick](j3),(b)--[thick](vR)--[thick](j4)}; \, ,
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2.9)

and the Jacobi identity for a triplet of graphs:

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\, \feynman
\vertex(a)[HV]{$~{}~{}~{}~{}$}; \vertex[above=0.6cm of a](b)[dot]{};
\vertex[left=0.45cm of b](c23); \vertex[above=0.5cm of c23](v23)[dot]{};
\vertex[left=1.4cm of b](c); \vertex[left=1.02cm of b](cL);
\vertex[above=1.0cm of cL](vL)[HV]{$A$}; \vertex[right=2.04cm of
vL](vR)[HV]{$C$}; \vertex[right=1.1cm of vL](vm)[HV]{$B$}; \vertex[above=2.0cm
of c](j1){}; \vertex[right=0.8cm of j1](j2){}; \vertex[right=0.4cm of
j2](j5){}; \vertex[right=0.8cm of j5](j6){}; \vertex[right=1.4cm of j2](j3){};
\vertex[right=0.7cm of j3](j4){}; \diagram*{(a) --[thick] (b),(b)--[thick](vL)
-- [thick] (j1),(vL) -- [thick]
(j2),(vR)--[thick](j3),(b)--[thick](vR)--[thick](j4),(vm)--[thick](j6),(vm)--[thick](j5),(v23)--[thick](vm)};
\, , \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\, \feynman
\vertex(a)[HV]{$~{}~{}~{}~{}$}; \vertex[above=0.6cm of a](b)[dot]{};
\vertex[left=0.45cm of b](c23); \vertex[above=0.5cm of c23](v23)[dot]{};
\vertex[left=1.4cm of b](c); \vertex[left=1.02cm of b](cL);
\vertex[above=1.0cm of cL](vL)[HV]{$A$}; \vertex[right=2.04cm of
vL](vR)[HV]{$B$}; \vertex[right=1.1cm of vL](vm)[HV]{$C$}; \vertex[above=2.0cm
of c](j1){}; \vertex[right=0.8cm of j1](j2){}; \vertex[right=0.4cm of
j2](j5){}; \vertex[right=0.8cm of j5](j6){}; \vertex[right=1.4cm of j2](j3){};
\vertex[right=0.7cm of j3](j4){}; \diagram*{(a) --[thick] (b),(b)--[thick](vL)
-- [thick] (j1),(vL) -- [thick]
(j2),(vR)--[thick](j3),(b)--[thick](vR)--[thick](j4),(vm)--[thick](j6),(vm)--[thick](j5),(v23)--[thick](vm)};
\, , \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\, \feynman
\vertex(a)[HV]{$~{}~{}~{}~{}$}; \vertex[above=0.6cm of a](b)[dot]{};
\vertex[right=0.45cm of b](c23); \vertex[above=0.5cm of c23](v23)[dot]{};
\vertex[left=1.4cm of b](c); \vertex[left=1.02cm of b](cL);
\vertex[above=1.0cm of cL](vL)[HV]{$A$}; \vertex[right=2.04cm of
vL](vR)[HV]{$C$}; \vertex[right=1.1cm of vL](vm)[HV]{$B$}; \vertex[above=2.0cm
of c](j1){}; \vertex[right=0.8cm of j1](j2){}; \vertex[right=0.4cm of
j2](j5){}; \vertex[right=0.8cm of j5](j6){}; \vertex[right=1.4cm of j2](j3){};
\vertex[right=0.7cm of j3](j4){}; \diagram*{(a) --[thick] (b),(b)--[thick](vL)
-- [thick] (j1),(vL) -- [thick]
(j2),(vR)--[thick](j3),(b)--[thick](vR)--[thick](j4),(vm)--[thick](j6),(vm)--[thick](j5),(v23)--[thick](vm)};
\, , \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2.10)

hold automatically. We also note the following two important relations for
colour-ordered gluon amplitudes:

$\displaystyle\textit{U(1) decoupling relation:}\sum_{\sigma\in
i\shuffle\\{1,\ldots,i-1,i+1,\ldots,n-2\\}}A(\sigma,v)=0\,,$ (2.11)
$\displaystyle\textit{Reflection relation:}\ \ \ \ \ \ \ \ \ A(12\ldots
n{-}2,v)=(-1)^{n{-}1}A(n{-}2\ldots 21,v)\,.$ (2.12)

For example, we have

$\displaystyle\begin{split}A(123,v)+A(213,v)+A(231,v)&=0\,,\\\
A(123,v)-A(321,v)&=0\,.\end{split}$ (2.13)

Finally, it is straightforward to obtain pure Yang-Mills amplitudes or BCJ
numerators from the HEFT numerators by taking the decoupling limit [57]

$\displaystyle v\to\varepsilon_{n-1}\,,\qquad(p_{1}+\cdots+p_{n-2})^{2}\to
0\,,$ (2.14)

This limit maps an $n$-point HEFT amplitude (with two scalars and $n{-}2$
gluons) into an $(n{-}1)$-point YM amplitude, with the two scalars replaced by
gluon $n{-}1$.

###  2.2 Proof of the HEFT double copy

The particular representation of HEFT amplitudes given in (2.6) was proved in
[57] by understanding the factorisation behaviour of the BCJ numerators. The
main goal of this section is to prove the double-copy formula (2.8). To do so,
we start from the KLT relation for the full theory [69]

$\displaystyle{\cal M}(1,2,\ldots,n{-}1,n)=-\sum_{\alpha,\beta\in
S_{n-3}}\mathcal{S}(1\alpha|1\beta){\cal A}(1,\alpha,n{-}1,n){\cal
A}(1,\beta,n,n{-}1)\,\,,$ (2.15)

where $\mathcal{S}(1\alpha|1\beta)$ is the field theory KLT kernel which is an
$(n{-}3)!\times(n{-}3)!$ matrix, and the sum is over the $(n{-}3)!$
permutations of $(2,\ldots,n{-}2)$. In the heavy-mass limit for the particles
$n{-}1$ and $n$, we see that swapping $n{-}1$ and $n$ amounts to sending
$v{\to}-v$, and the HEFT amplitudes are odd under this transformation,

$\displaystyle\begin{split}{\cal
A}(1,\alpha,n{-}1,n)&\stackrel{{\scriptstyle\rm
HEFT}}{{\longrightarrow}}A(1\alpha,v)\ ,\\\ {\cal
A}(1,\alpha,n,n{-}1)&\stackrel{{\scriptstyle\rm
HEFT}}{{\longrightarrow}}A(1\alpha,-v)\,=\,-A(1\alpha,v)\,.\end{split}$ (2.16)

Using this relation we obtain the KLT double copy relation for the HEFT
amplitudes

$\displaystyle M(12\ldots n{-}2,v)$ $\displaystyle=\sum_{\alpha,\beta\in
S_{n-3}}\mathcal{S}(1\alpha|1\beta)A(1\alpha,v)A(1\beta,v)\,.$ (2.17)

We then write the colour-ordered HEFT amplitudes with two massive scalars and
$n{-}2$ gluons as [1, 70, 71, 39, 69]

$\displaystyle A(1\alpha,v)=\sum_{\beta\in
S_{n-3}}\mathsf{m}(1\alpha|1\beta)\mathcal{N}([1\beta],v)\,,$ (2.18)

where $\mathcal{N}([1\beta],v)$ are KLT numerators in the Del Duca-Dixon-
Maltoni (DDM) basis [72]. This basis is associated with fully left-nested
commutators where the first index is fixed to be 1, which we denote as

$[1\beta]\coloneqq[1,\beta_{2},\beta_{3},\ldots,\beta_{n{-}2}]\coloneqq[\ldots[[[1,\beta_{2}],\beta_{3}],\beta_{4}],\ldots,\beta_{n{-}2}]\,,$
(2.19)

where $\beta$ is a permutation of the remaining gluon legs $2,\ldots,n{-}2$.
The matrix $\mathsf{m}$ appearing in (2.18) is the propagator matrix [70], or
inverse of the KLT matrix, satisfying333Note that in YM theory the propagator
matrix $\mathsf{m}$ is singular, while in our case it is invertible because
the sum of the gluon momenta is not lightlike.

$\displaystyle\sum_{\beta^{\prime}\in
S_{n-3}}\mathcal{S}(1\alpha|1\beta^{\prime})\mathsf{m}(1\beta^{\prime}|1\beta)=\delta_{\alpha\beta}\,.$
(2.20)

Plugging (2.18) into (2.17) we get

$\displaystyle\begin{split}M(12\ldots
n{-}2,v)&=\sum_{\alpha,\beta,\beta^{\prime}\in
S_{n-3}}A(1\alpha,v)\mathcal{S}(1\alpha|1\beta)\mathsf{m}(1\beta|1\beta^{\prime})\mathcal{N}([1\beta^{\prime}],v)\\\
&=\sum_{\alpha,\alpha^{\prime}\in
S_{n-3}}\mathsf{m}(1\alpha|1\alpha^{\prime})\mathcal{N}([1\alpha^{\prime}],v)\mathcal{N}([1\alpha],v)\,.\end{split}$
(2.21)

Now, the last line of (2.21) can be recast in the form of (2.8). This can be
seen from the definition of the propagator matrix [70] in the bi-adjoint
scalar theory [71, 39, 73, 74, 75],

$\displaystyle\mathcal{A}_{\text{bi-
adj}}=\sum_{\Gamma\in\widetilde{\rho}}\frac{c_{\Gamma}\,\widetilde{c}_{\Gamma}}{d_{\Gamma}}=\sum_{\alpha,\alpha^{\prime}\in
S_{n-2}}c([1\alpha],n)\mathsf{m}(1\alpha|1\alpha^{\prime})\widetilde{c}([1\alpha^{\prime}],n)\,,$
(2.22)

and then using the colour-kinematics duality to replace the factors of $c$ in
those formulae by factors of $\mathcal{N}$, thus transforming (2.21) into
(2.8). Here $c$ and $\widetilde{c}$ are nothing but the trace of the colour
group generators for the external scalars, that is

$\displaystyle
c([1\alpha],n)=f^{a_{1}a_{\alpha(2)}x_{1}}f^{x_{1}a_{\alpha(3)}x_{2}}\cdots
f^{x_{n-3}a_{\alpha(n-1)}a_{n}}\,,$ (2.23)

with a similar formula for $\widetilde{c}$.

Alternatively, we use the BCJ form (2.6) to rewrite the gauge theory amplitude
appearing in (2.21), giving

$\displaystyle\begin{split}M(12\ldots n{-}2,v)&=\sum_{\alpha\in
S_{n-3}}\sum_{\Gamma\in\rho_{\alpha}}\frac{\mathcal{N}(\Gamma,v)}{d_{\Gamma}}\mathcal{N}([1\alpha],v)\,.\end{split}$
(2.24)

One can then show that (2.24) can be cast in the form of (2.8). For instance,
at five points (2.24) reads

$\displaystyle\begin{split}M(123,v)&=\,\left(\frac{\mathcal{N}([[1,2],3],v)}{s_{12}s_{123}}+\frac{\mathcal{N}([1,[2,3]],v)}{s_{23}s_{123}}\right)\mathcal{N}([[1,2],3],v)\\\
&+\left(\frac{\mathcal{N}([[1,3],2],v)}{s_{13}s_{123}}+\frac{\mathcal{N}([1,[3,2]],v)}{s_{23}s_{123}}\right)\mathcal{N}([[1,3],2],v)\\\
&=\,\frac{\mathcal{N}([[1,2],3],v)^{2}}{s_{12}s_{123}}+\frac{\mathcal{N}([1,[2,3]],v)^{2}}{s_{23}s_{123}}+\frac{\mathcal{N}([[1,3],2],v)^{2}}{s_{13}s_{123}}\,,\end{split}$
(2.25)

where we have used anti-symmetry in $2\leftrightarrow 3$ to combine the terms
with propagator $\frac{1}{s_{23}}$ using the Jacobi identity

$\displaystyle\mathcal{N}([[1,2],3],v)-\mathcal{N}([[1,3],2],v)=\mathcal{N}([1,[2,3]],v)\,.$
(2.26)

The final line of (2.25) is precisely in the form of (2.8). In general, at
higher points, we can always use Jacobi identities to combine terms in this
manner.

For completeness we also derive the propagator matrix
$\mathsf{m}(1\alpha|1\alpha^{\prime})$ in the five-point example. From (2.6)
we have

$\displaystyle\begin{split}A(123,v)&=\frac{\mathcal{N}([[1,2],3],v)}{s_{12}s_{123}}+\frac{\mathcal{N}([1,[2,3]],v)}{s_{23}s_{123}}\
,\\\
A(132,v)&=\frac{\mathcal{N}([[1,3],2],v)}{s_{13}s_{123}}+\frac{\mathcal{N}([1,[3,2]],v)}{s_{23}s_{123}}\
.\end{split}$ (2.27)

Not all the numerators above are in the DDM basis (or fully left-nested), and
thus we rewrite them using the kinematic Jacobi identities as

$\displaystyle\begin{split}\mathcal{N}([1,[2,3]],v)&=\mathcal{N}([1,2,3],v)-\mathcal{N}([1,3,2],v)\
,\\\
\mathcal{N}([1,[3,2]],v)&=-\mathcal{N}([1,2,3],v)+\mathcal{N}([1,3,2],v)\,.\end{split}$
(2.28)

Plugging these into (2.27) we have

$\displaystyle\begin{pmatrix}A(123,v)\\\
A(132,v)\end{pmatrix}=\mathsf{m}(1\\{23\\}|1\\{23\\})\begin{pmatrix}\mathcal{N}([1,2,3],v)\\\
\mathcal{N}([1,3,2],v)\end{pmatrix}\ ,$ (2.29)

where

$\displaystyle\mathsf{m}(1\\{23\\}|1\\{23\\})=\frac{1}{s_{123}}\begin{pmatrix}\frac{1}{s_{12}}+\frac{1}{s_{23}}&-\frac{1}{s_{23}}\\\
-\frac{1}{s_{23}}&\frac{1}{s_{13}}+\frac{1}{s_{23}}\end{pmatrix}\,.$ (2.30)

It is immediate to check that $\mathsf{m}(1\\{23\\}|1\\{23\\})$ is non-
singular since $s_{123}{=}(p_{1}{+}p_{2}{+}p_{3})^{2}{\neq}0$, as mentioned
earlier.

###  2.3 Comparison to Yang-Mills theory

It is interesting to compare (2.18) to the corresponding formula for an
$n$-point gluon amplitude in Yang-Mills theory

$\displaystyle{\cal A}(1,\alpha,n)=\sum_{\beta\in
S_{n-2}}\mathsf{m}(1\alpha|1\beta)\mathcal{N}([1,\beta],n)\,.$ (2.31)

Here the expansion is over the $(n{-}2)!$ fully left-nested DDM numerators
$\mathcal{N}([1,\beta],n)$ with particles $1$ and $n$ fixed, which are widely
studied in the literature [76, 77, 78, 79, 80, 81, 82]. Each term in the sum
corresponds to a half-ladder, or multi-peripheral graph, and in this case the
matrix $\mathsf{m}(1\alpha|1\beta)$ is singular. When we take the decoupling
limit (2.14), mapping our HEFT amplitudes (with two scalars and $n{-}2$
gluons) into $(n{-}1)$-point YM amplitudes, the YM amplitudes thus obtained
are in the form (2.31) (with $n$ replaced by $n{-}1$).

We also note the alternative KLT representation of YM amplitudes with three
legs being fixed [69, 83],

$\displaystyle{\cal A}(1,2,\alpha,n)=\sum_{\beta\in
S_{n-3}}\mathsf{m}(12\alpha|12\beta)\widetilde{\mathcal{N}}([1,2,\beta],n)\,,$
(2.32)

where the matrix $\mathsf{m}(12\alpha|12\beta)$ is now invertible, which
implies that the $\widetilde{\mathcal{N}}$ are gauge invariant and unique, but
may contain poles [84]. The $\widetilde{\mathcal{N}}$ represent a basis of
numerators which are equivalent to the $\mathcal{N}$ up to the kernel of the
non-invertible matrix $\mathsf{m}(1\alpha|1\beta)$ with $\alpha,\beta\in
S_{n-2}$. However, we have used generalised gauge transformations to
explicitly set linearly dependent numerators to zero as per [83].

As an example, we review the four-point case. Here we have, from (2.27),

$\displaystyle\begin{pmatrix}{\cal A}(1,2,3,4)\\\ {\cal
A}(1,3,2,4)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\frac{1}{s_{12}}+\frac{1}{s_{23}}&-\frac{1}{s_{23}}\\\
-\frac{1}{s_{23}}&\frac{1}{s_{13}}+\frac{1}{s_{23}}\end{pmatrix}\begin{pmatrix}\mathcal{N}([1,2,3],4)\\\
\mathcal{N}([1,3,2],4)\end{pmatrix}\,$
$\displaystyle=\begin{pmatrix}\frac{1}{s_{12}}+\frac{1}{s_{23}}&-\frac{1}{s_{23}}\\\
-\frac{1}{s_{23}}&\frac{1}{s_{13}}+\frac{1}{s_{23}}\end{pmatrix}\Bigg{[}\begin{pmatrix}\mathcal{N}([1,2,3],4)\\\
\mathcal{N}([1,3,2],4)\end{pmatrix}+\begin{pmatrix}{s_{12}\over
s_{13}}\mathcal{N}([1,3,2],4)\\\ -\mathcal{N}([1,3,2],4)\end{pmatrix}\Bigg{]}$
$\displaystyle=\begin{pmatrix}\frac{1}{s_{12}}+\frac{1}{s_{23}}&-\frac{1}{s_{23}}\\\
-\frac{1}{s_{23}}&\frac{1}{s_{13}}+\frac{1}{s_{23}}\end{pmatrix}\begin{pmatrix}\widetilde{\mathcal{N}}([1,2,3],4)\\\
0\end{pmatrix}\,,$ (2.33)

so that
$\widetilde{\mathcal{N}}([1,2,3],4)\\!=\\!\mathcal{N}([1,2,3],4)+{s_{12}\over
s_{13}}\mathcal{N}([1,3,2],4)$, and
$\widetilde{\mathcal{N}}([1,3,2],4)\\!=\\!0$. Note that, in contrast to the
numerators $\mathcal{N}$, the numerators $\widetilde{\mathcal{N}}$ are not
crossing symmetric.

##  3 Pre-numerators and quasi-shuffle products

The BCJ numerators introduced in the previous section can be generated from an
object known as the pre-numerator [54, 55, 57], denoted in the same way as the
BCJ numerators but without the commutator structure:

$\mathcal{N}(123\ldots n\\!-\\!2,v)\,.$ (3.1)

We can construct the BCJ numerators by permuting the labels of the pre-
numerator according to the commutator structure, for example

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)[myblob]{};
\vertex[above=0.3cm of a](b)[dot]{}; \vertex[left=0.6cm of b](c);
\vertex[left=0.22cm of b](c23); \vertex[above=0.14cm of c23](v23)[dot]{};
\vertex[above=.4cm of c](j1){$1$}; \vertex[right=.7cm of j1](j2){$2$};
\vertex[right=0.5cm of j2](j3){$3$}; \diagram*{(a) -- [thick] (b),(b) --
[thick] (j1),(v23) -- [thick] (j2),(b)--[thick](j3)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\!\\!:\quad\mathcal{N}([[1,2],3],v):=\mathcal{N}(123,v)-\mathcal{N}(213,v)+\mathcal{N}(321,v)-\mathcal{N}(312,v)\,,$
(3.2)

and similarly for more complicated BCJ numerators. Written in this way the BCJ
numerators manifestly satisfy the Jacobi identities.

The BCJ numerators are unique, however the pre-numerators used to build them
are not uniquely defined: there is still freedom in choosing their exact form,
which we can use to make certain properties of the pre-numerators manifest.
Remarkably, it was shown in [57] that for tree amplitudes in HEFT there exists
a closed-form expression for _crossing symmetric_ pre-numerators in terms of
gauge-invariant quantities for arbitrary $n$. Moreover, the pre-numerators in
the canonical gluon ordering $\mathcal{N}(12\ldots n{-}2,v)$ can be
constructed from a quasi-shuffle algebra using a two-step procedure which we
now outline, following [57].444One can then define the pre-numerators with
other orderings via the simple relabelling $\mathcal{N}(12\ldots j\ldots
i\ldots n{-}2,v):=\mathcal{N}(12\ldots i\ldots j\ldots
n{-}2,v)|_{i\leftrightarrow j}$. This definition also has the benefit of
making crossing symmetry manifest.

First, we build an intermediate object which we call an “algebraic” pre-
numerator [54, 55, 63, 57],

$\displaystyle\widehat{\mathcal{N}}(123\ldots n{-}2):=T_{(1)}\star
T_{(2)}\star T_{(3)}\star\cdots\star T_{(n{-}2)}\,,$ (3.3)

obtained by fusing abstract generators $T_{(i)}$, one for each gluon leg,
using a quasi-shuffle product denoted by $\star$. Then we define a linear map
(denoted by angle brackets $\langle\bullet\rangle$) from these abstract
generators to kinematic quantities

$\mathcal{N}(123\ldots n{-}2,v):=\big{\langle}\widehat{\mathcal{N}}(123\ldots
n{-}2)\big{\rangle}\,.$ (3.4)

The linear map relates each single-index generator, e.g. $\langle
T_{(i)}\rangle$, to a vector current with a heavy source. Multi-index
generators are mapped to multi-rank tensor currents of the same heavy source.
The fusion product of these currents can be understood as an algebraic fusion
rule from lower-rank to higher-rank currents once we perform the map (3.4). In
this sense, the algebra generators are identified as the operators that
generate the tensor currents [54, 55].

In the next two sections we review the quasi-shuffle product, before
describing the map $\langle\bullet\rangle$ to tensor currents and finally pre-
numerators.

###  3.1 The quasi-shuffle product

The quasi-shuffle algebra consists of a vector space of generators
$\mathfrak{A}$, with generators written as

$T_{(\tau_{1}),\ldots(\tau_{r})}\,,$ (3.5)

labelled by subsets of the gluon indices
$\tau_{i}\subset\\{1,2,\ldots,n{-}2\\}$, for example $T_{(12),(45),(3)}$. As
sets, the $\tau_{i}$ are unordered and therefore the labels contained within
each $\tau_{i}$ can be written in any order by definition; we choose for
convenience to write them in the order $\\{1,2,3,\ldots\\}$. However, the
subsets themselves $(\tau_{1}),\ldots,(\tau_{r})$ are ordered.

To begin, we introduce some standard nomenclature for generators: we will
refer to generators with a single subset $T_{(\tau_{1})}$ as _“letters”_ ,
those with multiple subsets $T_{(\tau_{1}),\ldots,(\tau_{r})}$ as _“words”_ ,
and the length of a word is the number of subsets, or letters, it contains.

Next we introduce the quasi-shuffle product $\star$ on the space of generators
$\mathfrak{A}$, first in some examples and then in full generality. The
simplest case to consider is the product of two letters, say, $T_{(1)}$ and
$T_{(2)}$, which gives the four-point algebraic pre-numerator:

$\displaystyle T_{(1)}\star T_{(2)}=T_{(1),(2)}+T_{(2),(1)}-T_{(12)}\,.$ (3.6)

The first two terms above correspond to the “shuffle” part of the quasi-
shuffle product, while the last term is a letter made from “stuffing” $(1)$
and $(2)$ together (hence the quasi-shuffle nature of the algebra). The
“stuffing” terms can be understood as giving rise to “contact term”
corrections to the lower-point rules such that the numerators constructed in
this way lead to the correct amplitudes. For instance, at four points, naively
attaching two three-point numerators together by shuffling would lead to
$T_{(1),(2)}$ and $T_{(2),(1)}$, and the required correction to obtain the
correct numerator (or equivalently amplitude) is precisely $T_{(12)}$.555We
are grateful to Oliver Schlotterer for a discussion on this point.

The product of a letter with a word is also fairly straightforward and follows
the same splitting into shuffle and stuffing terms. As an example, consider

$\displaystyle\begin{split}T_{(1)}\star
T_{(2),(3),(4)}&=T_{(1),(2),(3),(4)}+T_{(2),(1),(3),(4)}+T_{(2),(3),(1),(4)}+T_{(2),(3),(4),(1)}\\\
&-T_{(12),(3),(4)}-T_{(2),(13),(4)}-T_{(2),(3),(14)}\,.\end{split}$ (3.7)

Note that this product preserves the ordering of the letters in the word,
$T_{(2),(3),(4)}$. This is true in general: the quasi-shuffle of two words
$T_{(\tau_{1}),(\tau_{2})\ldots(\tau_{r})}$ and
$T_{(\rho_{1}),(\rho_{2})\ldots(\rho_{r})}$ preserves the ordering of the
$\tau_{i}$ and the $\rho_{j}$. The general formula for the product of two
arbitrary words is given by [57]

$\displaystyle T_{(\tau_{1}),\ldots,(\tau_{r})}\star
T_{(\rho_{1}),\ldots,(\rho_{s})}=\sum_{\genfrac{}{}{0.0pt}{}{\sigma\lvert_{\\{\tau\\}}=\\{(\tau_{1}),\ldots,(\tau_{r})\\}}{\sigma\lvert_{\\{\rho\\}}=\\{(\rho_{1}),\ldots,(\rho_{s})\\}}}(-1)^{t-r-s}T_{(\sigma_{1}),\ldots,(\sigma_{t})}\,,$
(3.8)

where the $\tau_{i}$ or $\rho_{i}$ are now any subsets of
$\\{1,\ldots,n{-}2\\}$. The notation $\sigma\lvert_{\\{\tau\\}}$ means that we
restrict the partition $\sigma$ onto the subset
$\tau\\!=\\!\tau_{1}\cup\tau_{2}\cup\cdots\cup\tau_{r}$, for example
$\\{(234),(56),(78)\\}\lvert_{\\{2,4,6\\}}=\\{(24),(6)\\}$. There is also a
recursive definition of the quasi-shuffle product [58, 61] which we give for
completeness in Appendix A. The quasi-shuffle product defined in this way is
associative and commutative, hence we can perform the products in any order we
choose.

###  3.2 Mapping to the pre-numerator

After constructing the algebraic pre-numerator $\widehat{\mathcal{N}}(12\ldots
n{-}2)$ using (3.8), we then use the angle-bracket map to obtain the pre-
numerator (from which we finally obtain BCJ numerators as e.g. in (3.2)). The
key step is clearly the $\langle\bullet\rangle$ map, which we now discuss.

The angle-bracket map for the special case of a single label generator is
simply

$\displaystyle\langle T_{(i)}\rangle=v{\cdot}\varepsilon_{i}\,,$ (3.9)

while for a generic generator, it is given by [57]

$\displaystyle\langle
T_{(\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})}\rangle=\frac{v{\cdot}F_{\tau_{1}}{\cdot}V_{\Theta(\tau_{2})}{\cdot}F_{\tau_{2}}\ldots{\cdot}V_{\Theta(\tau_{r})}{\cdot}F_{\tau_{r}}{\cdot}v}{(n-2)v{\cdot}p_{\tau_{1}[1]}v\cdot
p_{\tau_{1}}\ldots v\cdot p_{\tau_{r-1}}}\,,$ (3.10)

where the length of $\tau_{1}\cup\cdots\cup\tau_{r}$ is $n{-}2$.
$\Theta(\tau_{i})$ is the subset of labels in
$\tau_{1}\cup\cdots\cup\tau_{i-1}$ which are smaller than the first label in
$\tau_{i}$ for the canonical ordering $1,2,\ldots,n{-}2$, and $\tau_{1}[1]$ is
the first label in $\tau_{1}$. As an example, if $T_{(13),(25),(46)}$ then
$\Theta(25)=\\{1\\},\Theta(46)=\\{1,2,3\\}$ which can also be seen by locating
all the labels to the south-west of the labels $2$ and $4$ in the following
“musical diagram” [57] Here we have also defined
$V_{\tau}^{\mu\nu}=v^{\mu}p_{\tau}^{\nu}$, and $F_{\tau_{i}}$ is the product
of linearised field strengths $F_{j}$ for all $j\in\tau_{i}$ where $\tau_{j}$
is ordered with respect to the canonical ordering $1,2,\ldots,n{-}2$. For
example, $F_{123}=F_{1}{\cdot}F_{2}{\cdot}F_{3}$. Finally, if any of the sets
$\Theta(\tau_{i})$ happens to be empty, then we set that generator to zero.

Note that here we always consider the canonical gluon ordering
$1,2,\ldots,n{-}2$ when we take the map $\langle\bullet\rangle$. This is what
makes the algebraic pre-numerator $\widehat{\mathcal{N}}(12\ldots n{-}2)$
symmetric in the gluon labels, while the kinematic pre-numerator
$\mathcal{N}(12\ldots n{-}2)$ is not. In Section 5 we will consider an
extension to the algebra which includes arbitrary gluon orderings, however for
now we will use the canonical ordering.

For the first few lower-point cases, the pre-numerators are given explicitly
by

$\displaystyle\mathcal{N}(1,v)$ $\displaystyle=\langle
T_{(1)}\rangle=v{\cdot}\varepsilon_{1}\,,$ (3.12)
$\displaystyle\begin{split}\mathcal{N}(12,v)&=\langle T_{(1)}\star
T_{(2)}\rangle=\cancelto{0}{\langle T_{(1),(2)}\rangle}+\cancelto{0}{\langle
T_{(2),(1)}\rangle}-\langle T_{(12)}\rangle\\\
&=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}v}{2v{\cdot}p_{1}}\,,\end{split}$
(3.13)
$\displaystyle\begin{split}\mathcal{N}(123,v)&=\langle\cancelto{0}{T_{\text{(1),(2),(3)}}}\rangle-\langle\cancelto{0}{T_{\text{(1),(23)}}}\rangle-\langle
T_{\text{(12),(3)}}\rangle+\langle\cancelto{0}{T_{\text{(1),(3),(2)}}}\rangle\\\
&-\langle
T_{\text{(13),(2)}}\rangle+\langle\cancelto{0}{T_{\text{(2),(1),(3)}}}\rangle-\langle\cancelto{0}{T_{\text{(2),(13)}}}\rangle+\langle\cancelto{0}{T_{\text{(2),(3),(1)}}}\rangle\\\
&-\langle\cancelto{0}{T_{\text{(23),(1)}}}\rangle+\langle\cancelto{0}{T_{\text{(3),(1),(2)}}}\rangle-\langle\cancelto{0}{T_{\text{(3),(12)}}}\rangle+\langle\cancelto{0}{T_{\text{(3),(2),(1)}}}\rangle+\langle
T_{\text{(123)}}\rangle\\\
&=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}V_{12}{\cdot}F_{3}{\cdot}v}{3v{\cdot}p_{1}v{\cdot}p_{12}}-\frac{v{\cdot}F_{1}{\cdot}F_{3}{\cdot}V_{1}{\cdot}F_{2}{\cdot}v}{3v{\cdot}p_{1}v{\cdot}p_{13}}+\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}F_{3}{\cdot}v}{3v{\cdot}p_{1}}\,.\end{split}$
(3.14)

In the above expressions, we have made explicit the terms that are mapped to
zero according to the rules (3.10). From these examples it seems as though
there is a large redundancy in the pre-numerator, since many terms are set to
zero by the map. Despite this redundancy, we shall see in the next section
that these terms are essential for linking the coproduct to the factorisation
behaviour of the pre-numerator.666There is an alternative construction where
we treat gluon $1$ as special, and in that construction the vanishing terms do
not appear [57].

##  4 Kinematic Hopf algebra for the pre-numerator

The quasi-shuffle product described in the last section gives us an
algorithmic way to produce $n$-point pre-numerators and hence all BCJ
numerators in the HEFT. The quasi-shuffle algebra can be extended to a
bialgebra with the introduction of two new operations: a coproduct: $\Delta$,
and a counit: $\epsilon$, which must satisfy specific compatibility conditions
we will detail later in this section. Additionally, there exists another
operation, the antipode: $S$, which further extends this bialgebra to a Hopf
algebra, and is also subject to various compatibility conditions. In this
section we will describe each one of these new operations in turn, starting
with the coproduct and its relation to (multiple) factorisation of the pre-
numerators.

###  4.1 The coproduct and factorisation

The coproduct is a linear map from the space $\mathfrak{A}$ of generators
$T_{\omega}$, introduced in Section 3.1, to the tensor product space
$\mathfrak{A}\otimes\mathfrak{A}$. For the quasi-shuffle product, the
coproduct is well known [61, 58] and can be derived from two key relations.
First, we define the coproduct of the simplest object, a letter

$\Delta(T_{(\tau)})=\mathbbm{I}\otimes
T_{(\tau)}+T_{(\tau)}\otimes\mathbbm{I}\,,$ (4.1)

where $\mathbbm{I}$ is the identity element of $\star$, the “empty” word. Then
we extend the definition to products of letters using a defining property of
the coproduct: compatibility with the quasi-shuffle product,

$\Delta(T_{(\tau_{1})\ldots(\tau_{r})}\star
T_{(\rho_{1})\ldots(\rho_{s})})=\Delta(T_{(\tau_{1})\ldots(\tau_{r})})\star\Delta(T_{(\rho_{1})\ldots(\rho_{s})})\,,$
(4.2)

where the extension of the product to tensors is defined naturally as
$(A\otimes B)\star(C\otimes D)=(A\star C)\otimes(B\star D)$. We can represent
the above equation (and similar equations later on) diagrammatically as
follows:

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[right=0.8cm of i1](ic){}; \vertex[right=1.6cm of i1](i2){};
\vertex[below=.9cm of ic](v1)[dot]{}; \vertex[below=1.6cm of v1](v2)[codot]{};
\vertex[below=3.2cm of i1](o1); \vertex[below=3.2cm of i2](o2); \diagram*{(i1)
-- [thick, bend right] (v1),(i2) -- [thick, bend left] (v1),(v2) -- [thick,
bend right] (o1),(v2) -- [thick,bend left] (o2),(v1)--[thick](v2)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}~{}=~{}~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[right=1.6cm of i1](i2){}; \vertex[below=.8cm of i1](v1)[codot]{};
\vertex[below=.8cm of i2](v12)[codot]{}; \vertex[below=.8cm of v1](vc);
\vertex[below=.8cm of v12](vc2); \vertex[below=.8cm of vc](v2)[dot]{};
\vertex[below=.8cm of vc2](v22)[dot]{}; \vertex[below=.8cm of v2](o1);
\vertex[below=.8cm of v22](o2); \vertex[left=.7cm of vc](vm1);
\vertex[right=.8cm of vc](vm2); \vertex[right=.7cm of vc2](vm3);
\vertex[left=0.65cm of vc2](vm4){}; \diagram*{(i1) -- [thick] (v1),(v2) --
[thick] (o1),(i2) -- [thick] (v12),(v22) -- [thick] (o2),(v1) -- [thick,bend
right] (vm1),(vm1) -- [thick,bend right] (v2),(v1) -- [thick,bend left]
(vm2),(vm2) -- [thick,bend right] (v22),(v12) -- [thick,bend right]
(vm4),(vm4) -- [thick,bend left] (v2),(v12) -- [thick,bend left] (vm3),(vm3)
-- [thick,bend left] (v22)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (4.3)

where the red dot
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
denotes the coproduct and the black dot ${\bullet}$ the quasi-shuffle product.
Note that such diagrams should be read from top to bottom.

Using (4.1) and (4.2) we can deduce the coproduct of the algebraic pre-
numerator as

$\Delta(T_{(1)}\star T_{(2)}\star\cdots\star
T_{(n{-}2)})=\Delta(T_{(1)})\star\Delta(T_{(2)})\star\cdots\star\Delta(T_{(n-2)})\,.$
(4.4)

Additionally, we can consider the coproduct of an arbitrary word (which will
in general not be a product of other words)

$\displaystyle\Delta(T_{(\tau_{1}),\ldots,(\tau_{r})}):=\sum_{i=0}^{r}T_{(\tau_{1}),\ldots,(\tau_{i})}\otimes
T_{(\tau_{i+1}),\ldots,(\tau_{r})}\,.$ (4.5)

Equation (4.4) is usually the fastest way to compute the coproduct, but first
it is worth using the explicit definition (4.5) in an example. Consider the
coproduct of the four-point algebraic pre-numerator,

$\begin{split}\Delta(\widehat{\mathcal{N}}(12))&=\Delta(T_{(1)}\star
T_{(2)})=\Delta(T_{(1)(2)})+\Delta(T_{(2)(1)})-\Delta(T_{(12)})\\\
&=\mathbbm{I}\otimes T_{(1)(2)}+T_{(1)}\otimes
T_{(2)}+T_{(1)(2)}\otimes\mathbbm{I}\\\ &+\mathbbm{I}\otimes
T_{(2)(1)}+T_{(2)}\otimes T_{(1)}+\mathbbm{I}\otimes T_{(2)(1)}\\\
&-\mathbbm{I}\otimes T_{(12)}-T_{(12)}\otimes\mathbbm{I}\,.\end{split}$ (4.6)

We can also introduce $\Delta^{\prime}$, the reduced coproduct, which simply
removes all trivial terms involving the identity $\mathbbm{I}$. For the four-
point pre-numerator this leaves us with

$\displaystyle\Delta^{\prime}(\widehat{\mathcal{N}}(12))=$ $\displaystyle
T_{(1)}\otimes T_{(2)}+T_{(2)}\otimes
T_{(1)}=\widehat{\mathcal{N}}(1)\otimes\widehat{\mathcal{N}}(2)+\widehat{\mathcal{N}}(2)\otimes\widehat{\mathcal{N}}(1)\,.$
(4.7)

In this example there are two important points to emphasise:

  * 1\.

The coproduct has the general property of splitting up words (and algebraic
pre-numerators) which, we will see, is reminiscent of factorisation on the
massive propagators $v{\cdot}p_{\tau}$.

  * 2.

The non-trivial terms in the coproduct comes precisely from the terms
$T_{(1),(2)}$ and $T_{(2),(1)}$ which did not contribute to the pre-numerator
upon taking the map $\langle\bullet\rangle$ in (3.12).

This is true in general: we can derive an expression for the coproduct of the
$n$-point algebraic pre-numerator in terms of tensor products of lower-point
ones,

$\displaystyle\Delta\widehat{\mathcal{N}}(123\ldots n{-}2)$
$\displaystyle=\sum_{\sigma_{L}\cup\sigma_{R}=\\{1,\ldots,n{-}2\\}}\widehat{\mathcal{N}}(\sigma_{L})\otimes\widehat{\mathcal{N}}(\sigma_{R})\,,$
(4.8)

where here we allow $\sigma_{L}$ and $\sigma_{R}$ to be empty and define
$\widehat{\mathcal{N}}(\emptyset):=\mathbbm{I}$. Thus if we demand
$\sigma_{L}$, $\sigma_{R}$ to be non-empty in the above equation we get
$\Delta^{\prime}(\widehat{\mathcal{N}}(123\ldots n{-}2))$ instead. The proof
of the above formula comes directly from the relation (4.4). Indeed, assuming
that the $(n{-}1)$-point pre-numerator takes the form above, then by induction
for the $n$-point pre-numerator we have

$\displaystyle\begin{split}\Delta\widehat{\mathcal{N}}(123\ldots
n{-}2)&=\Delta(\widehat{\mathcal{N}}(123\ldots n{-}3)\star
T_{(n-2)})=\Delta(\widehat{\mathcal{N}}(123\ldots
n{-}3))\star\Delta(T_{(n-2)})\\\
&=\sum_{\sigma_{L}\cup\sigma_{R}=\\{1,2,\ldots,n{-}3\\}}(\widehat{\mathcal{N}}(\sigma_{L})\otimes\widehat{\mathcal{N}}(\sigma_{R}))\star(\mathbbm{I}\otimes
T_{(n-2)}+T_{(n-2)}\otimes\mathbbm{I})\\\
&=\sum_{\sigma_{L}\cup\sigma_{R}=\\{1,2,\ldots,n{-}2\\}}(\widehat{\mathcal{N}}(\sigma_{L})\otimes\widehat{\mathcal{N}}(\sigma_{R}))\,.\end{split}$
(4.9)

To make the connection between the coproduct and factorisation precise we need
to extend the map $\langle\bullet\rangle$ to include tensor structures. To do
this let us first consider the factorisation property of the four-point pre-
numerator

$\mathcal{N}(12,v)=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}v}{2v{\cdot}p_{1}}\,.$
(4.10)

The factorisation of the above pre-numerator as
$v{\cdot}p_{1}\\!\rightarrow\\!0$ is characterised by the corresponding
residue:

$\begin{split}\text{Res}_{(v{\cdot}p_{1})}\mathcal{N}(12,v)&=v{\cdot}\varepsilon_{1}\frac{p_{1}{\cdot}p_{2}}{2}v{\cdot}\varepsilon_{2}=\mathcal{N}(1,v)\frac{p_{1}{\cdot}p_{2}}{2}\mathcal{N}(2,v)\,.\end{split}$
(4.11)

More generally, the pre-numerator has the following factorisation behaviour,
proven in [57], obtained from the residues at the poles
$v{\cdot}p_{1\tau}{=}0$:

$\text{Res}_{(v{\cdot}p_{1\tau})}\mathcal{N}(12\ldots
n{-}2,v)=\frac{\|\tau\|\|\omega\|}{n-2}\mathcal{N}(\tau,v)(p_{\Theta(\omega)}{\cdot}p_{\omega[1]})\mathcal{N}(\omega,v)\,,$
(4.12)

where $\tau$ and $\omega$ are subsets of $\\{2,3,\ldots,n{-}2\\}$ such that
$\tau\cup\omega=\\{2,3,\ldots,{n-2}\\}$, and $\|\tau\|$, $\|\omega\|$ are
their respective lengths. Again, $\Theta(\omega)$ is the set of labels in
$\\{1,2,\ldots,n{-}2\\}$ which are less that $\omega[1]$. Note that in the
HEFT due to momentum conservation we have $v{\cdot}p_{1\ldots n{-}2}=0$; for
example at five points we could write $v{\cdot}p_{12}=-v{\cdot}p_{3}$. Thus,
in the above we have included leg $1$ in $v{\cdot}p_{1\tau}$ to fix such
ambiguities.

Now we come to a crucial point: each one of these massive poles can be matched
to a term in the coproduct if we define a replacement rule $\mathcal{C}$ for
tensor products as

$\displaystyle\mathcal{C}(T_{\omega_{1}}\otimes T_{\omega_{2}})$
$\displaystyle=\frac{\|\omega_{1}\|\|\omega_{2}\|}{\|\omega_{1}\|+\|\omega_{2}\|}\left({1\over
v{\cdot}p_{\omega_{1}}}\right)T_{\omega_{1}}p_{\Theta(\omega_{2})}{\cdot}p_{\omega_{2}[1]}T_{\omega_{2}}\,,$
(4.13)

where $\Theta$ is defined in the same way as before, and if
$\Theta(\omega)=\emptyset$ then we set $p_{\emptyset}=0$. Note that the
algebraic generators have not yet been evaluated, and should be treated as
commuting objects. Finally, to map to physical quantities we must apply the
angle bracket $\langle\bullet\rangle$ to the generators, as in

$\langle T_{\omega_{1}}T_{\omega_{2}}\rangle=\langle
T_{\omega_{1}}\rangle\langle T_{\omega_{2}}\rangle\,.$ (4.14)

To make the connection between factorisation and the coproduct, we must take
the residue at the relevant poles before we perform the angle bracket map to
kinematics. Explicitly, we have the following relation for any subset
$\tau\subset\\{2,\ldots,n-2\\}$:

$\displaystyle\Big{\langle}\text{Res}_{(v{\cdot}p_{1\tau})}\mathcal{C}\Delta^{\prime}\widehat{\mathcal{N}}(12\ldots
n{-}2)\Big{\rangle}=\text{Res}_{(v{\cdot}p_{1\tau})}\mathcal{N}(12\ldots
n{-}2,v)\,.$ (4.15)

We also note that the pre-numerator has poles which are not colour ordered
e.g. when $v{\cdot}p_{13}=0$, and therefore the residues at these poles must
cancel when we combine these pre-numerators into colour-ordered amplitudes.

In summary, the coproduct decomposes a pre-numerator into a sum of terms, each
of which corresponds to a factorisation channel on a single massive pole. It
is useful to illustrate the connection between coproducts and factorisation in
some examples.

$\bm{n{=}4}$. In this case, we calculated the coproduct of the algebraic pre-
numerator in (4.6). Mapping this result to physical quantities using (4.13) we
obtain

$\displaystyle\begin{split}\operatorname{\mathcal{C}}\Delta^{\prime}\widehat{\mathcal{N}}(12)&=\operatorname{\mathcal{C}}(T_{{(1)}}\otimes
T_{{(2)}})+\cancelto{0}{\operatorname{\mathcal{C}}(T_{{(2)}}\otimes
T_{{(1)}}})\\\ &=\frac{1}{2}\left({1\over
v{\cdot}p_{1}}\right)T_{(1)}p_{1}{\cdot}p_{2}T_{(2)}\,.\end{split}$ (4.16)

Taking the residue at the pole $v{\cdot}p_{1}=0$, and mapping to physical
quantities we get

$\langle\text{Res}_{v{\cdot}p_{1}}\operatorname{\mathcal{C}}\Delta^{\prime}\widehat{\mathcal{N}}(12)\rangle=\frac{1}{2}\langle
T_{(1)}p_{1}{\cdot}p_{2}T_{(2)}\rangle=\frac{1}{2}\langle T_{(1)}\rangle
p_{1}{\cdot}p_{2}\langle
T_{(2)}\rangle=\frac{1}{2}v{\cdot}\varepsilon_{1}p_{1}{\cdot}p_{2}v{\cdot}\varepsilon_{2}\,,$
(4.17)

where the right-hand side is nothing but the factorisation behaviour of the
four-point pre-numerator in (4.10).

$\bm{n{=}5}$. At five points the algebraic pre-numerator is given in (3.14),
and its coproduct is

$\displaystyle\begin{split}\Delta\widehat{\mathcal{N}}(123)&=\mathbbm{I}\otimes\widehat{\mathcal{N}}(123)+\widehat{\mathcal{N}}(123)\otimes\mathbbm{I}\\\
&+\widehat{\mathcal{N}}(1)\otimes\widehat{\mathcal{N}}(23)+\widehat{\mathcal{N}}(2)\otimes\widehat{\mathcal{N}}(13)+\widehat{\mathcal{N}}(3)\otimes\widehat{\mathcal{N}}(12)\\\
&+\widehat{\mathcal{N}}(23)\otimes\widehat{\mathcal{N}}(1)+\widehat{\mathcal{N}}(13)\otimes\widehat{\mathcal{N}}(2)+\widehat{\mathcal{N}}(12)\otimes\widehat{\mathcal{N}}(3)\,.\end{split}$
(4.18)

Proceeding as before, we act on the above equation with the replacement rules
$\operatorname{\mathcal{C}}$ as given in (4.13), to find that only
$\widehat{\mathcal{N}}(1)\otimes\widehat{\mathcal{N}}(23)$,
$\widehat{\mathcal{N}}(13)\otimes\widehat{\mathcal{N}}(2)$, and
$\widehat{\mathcal{N}}(12)\otimes\widehat{\mathcal{N}}(3)$ are non-vanishing.
Thus we get

$\displaystyle\begin{split}\operatorname{\mathcal{C}}\Delta^{\prime}(\widehat{\mathcal{N}}(123))&=\operatorname{\mathcal{C}}\Big{(}\widehat{\mathcal{N}}(1)\otimes\widehat{\mathcal{N}}(23)+\widehat{\mathcal{N}}(13)\otimes\widehat{\mathcal{N}}(2)+\widehat{\mathcal{N}}(12)\otimes\widehat{\mathcal{N}}(3)\Big{)}\\\
&=\frac{2}{3}\left({1\over
v{\cdot}p_{1}}\right)\widehat{\mathcal{N}}(1)p_{1}{\cdot}p_{2}\widehat{\mathcal{N}}(23)+\frac{2}{3}\left({1\over
v{\cdot}p_{13}}\right)\widehat{\mathcal{N}}(13)p_{1}{\cdot}p_{2}\widehat{\mathcal{N}}(2)\\\
&+\frac{2}{3}\left({1\over
v{\cdot}p_{12}}\right)\widehat{\mathcal{N}}(12)p_{12}{\cdot}p_{3}\widehat{\mathcal{N}}(3)\,.\end{split}$
(4.19)

Now if we take the residue at the pole, say, $v\cdot p_{13}=0$, and apply the
map $\langle\bullet\rangle$ we obtain

$\langle\text{Res}_{v{\cdot}p_{13}}\,\mathcal{C}\Delta^{\prime}\widehat{\mathcal{N}}(123)\rangle={v{\cdot}F_{1}{\cdot}F_{3}{\cdot}v\over
3\,v{\cdot}p_{1}}p_{1}{\cdot}p_{2}v{\cdot}\varepsilon_{2}\,,$ (4.20)

which is the expected factorisation of the pre-numerator in the limit
$v{\cdot}p_{13}\to 0$.

We now move on to discuss iterated coproducts and their connection to
generalised factorisation.

###  4.2 Generalised factorisation and iterated coproducts

We now examine the factorisation of the pre-numerators when we take multiple
massive propagators on shell – we call this generalised factorisation.
Interestingly, the result will depend on the order in which we take the
limits, in contrast to amplitudes, where the order of limits is irrelevant. As
an example consider the five-point pre-numerator

$\begin{split}\mathcal{N}(123,v)&=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}V_{12}{\cdot}F_{3}{\cdot}v}{3v{\cdot}p_{1}v{\cdot}p_{12}}-\frac{v{\cdot}F_{1}{\cdot}F_{3}{\cdot}V_{1}{\cdot}F_{2}{\cdot}v}{3v{\cdot}p_{1}v{\cdot}p_{13}}+\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}F_{3}{\cdot}v}{3v{\cdot}p_{1}}\,.\end{split}$
(4.21)

There are three massive poles we can consider here which we write as
$v{\cdot}p_{1}\\!\rightarrow\\!0$, $v{\cdot}p_{12}\\!\rightarrow\\!0$ and
$v{\cdot}p_{13}\\!\rightarrow\\!0$ (recall that $v{\cdot}p_{123}=0$ here so
that $v{\cdot}p_{2}=-v{\cdot}p_{13}$). If we take $v{\cdot}p_{1}\rightarrow 0$
we find, using (4.12)

$\text{Res}_{(v{\cdot}p_{1})}\mathcal{N}(123,v)=\frac{2}{3}\mathcal{N}(1,v)p_{1}{\cdot}p_{2}\mathcal{N}(23,v)\,.$
(4.22)

Then taking $v{\cdot}p_{2}\rightarrow 0$, which is now equivalent to
$v{\cdot}p_{12}\rightarrow 0$ we obtain

$\text{Res}_{(v{\cdot}p_{2},\,v{\cdot}p_{1})}\mathcal{N}(123,v)=\frac{1}{3}\mathcal{N}(1,v)p_{1}{\cdot}p_{2}\mathcal{N}(2,v)p_{2}{\cdot}p_{3}\mathcal{N}(3,v)\,.$
(4.23)

Note that the labels appearing in
$\text{Res}_{(v{\cdot}p_{2},\,v{\cdot}p_{1})}$ are ordered. Alternatively, we
could first take the limit $v{\cdot}p_{12}\rightarrow 0$ then
$v{\cdot}p_{1}\rightarrow 0$ which first gives

$\text{Res}_{(v{\cdot}p_{12})}\mathcal{N}(123,v)=\frac{2}{3}\mathcal{N}(12,v)p_{12}{\cdot}p_{3}\mathcal{N}(3,v)\,,$
(4.24)

and then

$\text{Res}_{(v{\cdot}p_{1},\,v{\cdot}p_{12})}\mathcal{N}(123,v)=\frac{1}{3}\mathcal{N}(1,v)p_{1}{\cdot}p_{2}\mathcal{N}(2,v)p_{12}{\cdot}p_{3}\mathcal{N}(3,v)\,.$
(4.25)

Comparing the above with (4.23) we see that although the pre-numerator has
been split into the same pre-numerators ($\mathcal{N}(1,v)$,
$\mathcal{N}(2,v)$ $\mathcal{N}(3,v)$), the momentum products differ. All of
the different choices of multiple factorisations of $\mathcal{N}(123,v)$ can
be summarised in the following diagram:

${\mathcal{N}(123,v)}$${\frac{2}{3}\mathcal{N}(1,v)p_{1}{\cdot}p_{2}\mathcal{N}(23,v)}$${\frac{2}{3}\mathcal{N}(12,v)p_{12}{\cdot}p_{3}\mathcal{N}(3,v)}$${\frac{2}{3}\mathcal{N}(13,v)p_{1}{\cdot}p_{2}\mathcal{N}(2,v)}$${\frac{1}{3}p_{1}{\cdot}p_{2}p_{2}{\cdot}p_{3}\prod_{i=1}^{3}\mathcal{N}(i,v)}$${\frac{1}{3}p_{1}{\cdot}p_{2}p_{12}{\cdot}p_{3}\prod_{i=1}^{3}\mathcal{N}(i,v)}$${\frac{1}{3}p_{1}{\cdot}p_{3}p_{1}{\cdot}p_{2}\prod_{i=1}^{3}\mathcal{N}(i,v)}$$\scriptstyle{v{\cdot}p_{1}\rightarrow
0}$$\scriptstyle{v{\cdot}p_{12}\rightarrow
0}$$\scriptstyle{v{\cdot}p_{13}\rightarrow
0}$$\scriptstyle{v{\cdot}p_{2}\rightarrow
0}$$\scriptstyle{v{\cdot}p_{1}\rightarrow
0}$$\scriptstyle{v{\cdot}p_{1}\rightarrow 0}$

This process continues for higher-point pre-numerators, and in general the
order in which the limits are taken is important. By applying the coproduct to
a pre-numerator several times we aim to build an object which can recreate
this factorisation structure by taking multiple residues, as we now discuss.

If we apply $k$ coproducts to a generator we land in the $(k+1)$-tensor
product space of $\mathfrak{A}$, which we write as
$\mathfrak{A}^{\otimes(k+1)}$. A $k$-iterated coproduct is denoted as
$\Delta^{(k)}$ and defined recursively as

$\displaystyle\Delta^{(k)}=(\Delta^{(k-1)}\otimes\mathbbm{I})\Delta\,.$ (4.26)

For example $\Delta^{(2)}\,T_{(1),(2),(3)}$ is given by

$\displaystyle\begin{split}\Delta^{(2)}T_{(1),(2),(3)}&=(\Delta\otimes\mathbbm{I})\Delta(T_{(1),(2),(3)})\\\
&=(\Delta\otimes\mathbbm{I})\Big{[}\mathbbm{I}\otimes
T_{(1),(2),(3)}+T_{(1)}\otimes T_{(2)(3)}+T_{(1),(2)}\otimes
T_{(3)}+T_{(1)(2)(3)}\otimes\mathbbm{I}\Big{]}\\\
&=\mathbbm{I}\otimes\mathbbm{I}\otimes T_{(1),(2),(3)}+\mathbbm{I}\otimes
T_{(1)}\otimes T_{(2),(3)}+T_{(1)}\otimes\mathbbm{I}\otimes T_{(2),(3)}\\\
&+\mathbbm{I}\otimes T_{(1),(2)}\otimes T_{(3)}+T_{(1)}\otimes T_{(2)}\otimes
T_{(3)}+T_{(1),(2)}\otimes\mathbbm{I}\otimes T_{(3)}\\\ &+\mathbbm{I}\otimes
T_{(1),(2),(3)}\otimes\mathbbm{I}+T_{(1)}\otimes
T_{(2),(3)}\otimes\mathbbm{I}+T_{(1),(2)}\otimes T_{(3)}\otimes\mathbbm{I}\\\
&+T_{(1),(2),(3)}\otimes\mathbbm{I}\otimes\mathbbm{I}\,,\end{split}$ (4.27)

where in the second equality we have used
$\Delta(\mathbbm{I})=\mathbbm{I}\otimes\mathbbm{I}$. As before, we can
introduce a reduced iterated coproduct $\Delta^{\prime(k)}$, which simply
removes the trivial terms involving the identity. For the example above this
gives

$\Delta^{\prime(2)}T_{(1),(2),(3)}=T_{(1)}\otimes T_{(2)}\otimes T_{(3)}\,.$
(4.28)

Note that the choice of $(\Delta\otimes\mathbbm{I})$ instead of
$(\mathbbm{I}\otimes\Delta)$ in (4.26) is arbitrary, due to the co-
associativity of the coproduct:

$(\mathbbm{I}\otimes\Delta)\Delta(T_{\omega})=(\Delta\otimes\mathbbm{I})\Delta(T_{\omega})\,.$
(4.29)

This property can also be illustrated diagrammatically as

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[below=0.8cm of i1](c1)[codot]{}; \vertex[below left=1.3cm of c1](v1);
\vertex[below right=1.3cm of c1](c2)[codot]{}; \vertex[below =0.95cm of
v1](o1){}; \vertex[below left =1.3cm of c2](o2); \vertex[below right=1.3cm of
c2](o3); \diagram*{(i1) -- [thick] (c1),(c1) -- [thick, bend right] (v1)[] --
[thick] (o1),(c1) -- [thick, bend left] (c2),(c2)--[thick, bend
right](o2),(c2)--[thick, bend left](o3)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[below=0.8cm of i1](c1)[codot]{}; \vertex[below right=1.3cm of c1](v1);
\vertex[below left=1.3cm of c1](c2)[codot]{}; \vertex[below =0.95cm of
v1](o1){}; \vertex[below right =1.3cm of c2](o2); \vertex[below left=1.3cm of
c2](o3); \diagram*{(i1) -- [thick] (c1),(c1) -- [thick, bend left] (v1)[] --
[thick] (o1),(c1) -- [thick, bend right] (c2),(c2)--[thick, bend
left](o2),(c2)--[thick, bend right](o3)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (4.30)

Using the general form of a single coproduct of the algebraic pre-numerators
(4.8), and the recursive definition for the iterated coproduct $\Delta^{(k)}$
in (4.26), we can deduce the action of $\Delta^{(k)}$ on a generic algebraic
pre-numerator:

$\displaystyle\Delta^{(k)}(\widehat{\mathcal{N}}(12\ldots
n{-}2))=\sum_{\tau_{1}\cup\cdots\cup\tau_{k}\cup\tau_{k+1}=\\{1,\ldots,n-2\\}}\widehat{\mathcal{N}}(\tau_{1})\otimes\widehat{\mathcal{N}}(\tau_{2})\otimes\cdots\otimes\widehat{\mathcal{N}}(\tau_{k+1})\,,$
(4.31)

where again we allow any of the $\tau_{i}$ to be empty, and we can obtain the
expression for $\Delta^{\prime(k)}$ by demanding that the $\tau_{i}$ be non-
empty.

We can now naturally extend the replacement rule $\operatorname{\mathcal{C}}$
defined in (4.13) to include an arbitrary number of tensor products,

$\displaystyle\begin{split}\mathcal{C}\Big{(}T_{\omega_{1}}\otimes
T_{\omega_{2}}\otimes\cdots\otimes
T_{\omega_{i}}\Big{)}&:=\frac{\|\omega_{1}\|\|\omega_{2}\|\ldots\|\omega_{i}\|}{\|\omega_{1}\|+\|\omega_{2}\|+\ldots\|\omega_{i}\|}\left({1\over
v{\cdot}p_{\omega_{1}}v{\cdot}p_{\omega_{1}\omega_{2}}\ldots
v{\cdot}p_{\omega_{1}\cdots\omega_{i-1}}}\right)\\\
&\times\Big{(}T_{\omega_{1}}p_{\Theta(\omega_{2})}{\cdot}p_{\omega_{2}(1)}T_{\omega_{2}}\cdots
p_{\Theta(\omega_{i})}{\cdot}p_{\omega_{i}(1)}T_{\omega_{i}}\Big{)}\,,\end{split}$
(4.32)

where $\Theta(\omega_{i})$ is the set of elements in
$\omega_{1}\cup\cdots\cup\omega_{i-1}$ which are less than the first element
of $\omega_{i}$ in the canonical ordering $(1,2\ldots,n{-}2)$. Now we can
explicitly obtain $k$ factorisations of a pre-numerator from the $k$ coproduct
by taking the residues on the appropriate poles and applying the map
$\langle\bullet\rangle$,

$\displaystyle\Big{\langle}\text{Res}_{(v{\cdot}p_{\omega_{1}},\ldots
v{\cdot}p_{\omega_{k}})}\operatorname{\mathcal{C}}\Delta^{\prime(k)}\widehat{\mathcal{N}}(12\ldots
n{-}2)\Big{\rangle}=\text{Res}_{(v{\cdot}p_{\omega_{1}},\ldots
v{\cdot}p_{\omega_{k}})}\Big{(}\mathcal{N}(12\ldots n{-}2,v)\Big{)}\,,$ (4.33)

where once again both of these residues are ordered and taken right to left.

As a nontrivial example, consider the five-point case. After applying the
first coproduct one obtains

$\displaystyle\begin{split}&\Delta^{(2)}(\widehat{\mathcal{N}}(123))=(\Delta\otimes\mathbbm{I})\Delta(\widehat{\mathcal{N}}(123))\\\
&=\Delta(\mathbbm{I})\otimes\widehat{\mathcal{N}}(123)+\Delta(\widehat{\mathcal{N}}(123))\otimes\mathbbm{I}\\\
&+\Delta(T_{(1)})\otimes\widehat{\mathcal{N}}(23)+\Delta(T_{(2)})\otimes\widehat{\mathcal{N}}(13)+\Delta(T_{(3)})\otimes\widehat{\mathcal{N}}(12)\\\
&+\Delta(\widehat{\mathcal{N}}(23))\otimes
T_{(1)}+\Delta(\widehat{\mathcal{N}}(13))\otimes
T_{(2)}+\Delta(\widehat{\mathcal{N}}(12))\otimes T_{(3)}\,,\end{split}$ (4.34)

from which we can identify which pieces contribute to factorisation by using
the replacement rule $\mathcal{C}$ defined in (4.13). Doing so, we see that
the terms which contribute are

$\displaystyle\begin{split}\mathcal{C}\Delta^{\prime(2)}(\widehat{\mathcal{N}}(123))&=\mathcal{C}\Big{(}\Delta(\widehat{\mathcal{N}}(12))\otimes\widehat{\mathcal{N}}(3)\Big{)}+\mathcal{C}\Big{(}\Delta(\widehat{\mathcal{N}}(13))\otimes\widehat{\mathcal{N}}(2)\Big{)}\\\
&=\mathcal{C}\Big{(}T_{(1)}\otimes T_{(2)}\otimes
T_{(3)}\Big{)}+\mathcal{C}\Big{(}T_{(1)}\otimes T_{(3)}\otimes
T_{(2)}\Big{)}\\\ &=\frac{1}{3}\left({1\over
v{\cdot}p_{1}v{\cdot}p_{12}}\right)T_{(1)}p_{1}{\cdot}p_{2}T_{(2)}p_{12}{\cdot}p_{3}T_{(3)}\\\
&+\frac{1}{3}\left({1\over
v{\cdot}p_{1}v{\cdot}p_{13}}\right)T_{(1)}p_{1}{\cdot}p_{3}T_{(3)}p_{1}{\cdot}p_{2}T_{(2)}\,.\end{split}$
(4.35)

Extracting the residue corresponding to $v{\cdot}p_{1}\rightarrow 0$ and then
$v{\cdot}p_{2}\rightarrow 0$, we get

$\displaystyle\begin{split}&\text{Res}_{(v{\cdot}p_{2},v{\cdot}p_{1})}\operatorname{\mathcal{C}}\Delta^{\prime(2)}\widehat{\mathcal{N}}(123)\\\
&=\oint_{v{\cdot}p_{2}=0}{d(v{\cdot}p_{2})}\oint_{v{\cdot}p_{1}=0}{d(v{\cdot}p_{1})}\frac{1}{3}\bigg{(}\frac{T_{(1)}p_{1}{\cdot}p_{2}T_{(2)}p_{12}{\cdot}p_{3}T_{(3)}}{v{\cdot}p_{1}v{\cdot}p_{12}}+\frac{T_{(1)}p_{1}{\cdot}p_{3}T_{(3)}p_{1}{\cdot}p_{2}T_{(2)}}{v{\cdot}p_{1}v{\cdot}p_{13}}\bigg{)}\\\
&=\frac{1}{3}\,T_{(1)}\,p_{1}{\cdot}p_{2}\,T_{(2)}\,p_{2}{\cdot}p_{3}\,T_{(3)}\,,\end{split}$
(4.36)

on which we can act with the map $\langle\bullet\rangle$ to obtain the
factorisation

$\displaystyle\begin{split}\big{\langle}\text{Res}_{(v{\cdot}p_{2},v{\cdot}p_{1})}\operatorname{\mathcal{C}}\Delta^{\prime(2)}\widehat{\mathcal{N}}(123,v)\big{\rangle}&=\frac{1}{3}\,v{\cdot}\varepsilon_{1}\,p_{1}{\cdot}p_{2}\,v{\cdot}\varepsilon_{2}\,p_{2}{\cdot}p_{3}\,v{\cdot}\varepsilon_{3}\,\\\
&=\text{Res}_{(v{\cdot}p_{2},v{\cdot}p_{1})}\,\mathcal{N}(123,v)\,.\end{split}$
(4.37)

The other two factorisations can be obtained similarly:

$\displaystyle\begin{split}\big{\langle}\text{Res}_{(v{\cdot}p_{1},v{\cdot}p_{12})}\operatorname{\mathcal{C}}\Delta^{\prime(2)}\widehat{\mathcal{N}}(123)&\big{\rangle}=\frac{1}{3}v{\cdot}\varepsilon_{1}\,p_{1}{\cdot}p_{2}\,v{\cdot}\varepsilon_{2}\,p_{12}{\cdot}p_{3}\,v{\cdot}\varepsilon_{3}\,,\\\
\big{\langle}\text{Res}_{(v{\cdot}p_{1},v{\cdot}p_{2})}\operatorname{\mathcal{C}}\Delta^{\prime(2)}\widehat{\mathcal{N}}(123)\big{\rangle}&=-\frac{1}{3}v{\cdot}\varepsilon_{1}\,p_{1}{\cdot}p_{3}\,v{\cdot}\varepsilon_{2}\,p_{1}{\cdot}p_{2}\,v{\cdot}\varepsilon_{3}\,.\end{split}$
(4.38)

It is easy to check that the three residues are not linearly independent:

$\displaystyle\begin{split}&\langle\text{Res}_{(v{\cdot}p_{2},v{\cdot}p_{1})}\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}(123))\rangle\\\
=&\,\langle\text{Res}_{(v{\cdot}p_{1},v{\cdot}p_{12})}\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}(123))\rangle+\langle\text{Res}_{(v{\cdot}p_{1},v{\cdot}p_{2})}\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}(123))\rangle\,,\end{split}$
(4.39)

which is a consequence of the global residue theorem of multi-dimensional
integrals [85].

As a final comment, we note that for an $n$-point amplitude we can have non-
trivial iterated coproducts $\Delta^{(k)}$ with $k=2,\ldots,n{-}3$. In the
context of multiple polylogarithms, $\Delta^{(n-3)}$ corresponds to what is
usually known as the symbol [86]. Note however that in the case of iterated
integrals, the relevant algebra is a shuffle algebra [87].777Interestingly,
quasi-shuffle algebras also make an appearance in the context of multiple
polylogarithms when written in terms of nested sums rather than iterated
integrals, see Section 3.4 of [88] for a review. We also discuss the
connection between quasi-shuffle and shuffle algebras in Appendix B.

###  4.3 The counit

To obtain the quasi-shuffle bialgebra we introduce the counit $\epsilon$,
defined by its action on the generators $T_{\omega}$ and the identity
$\mathbbm{I}$:

$\displaystyle\epsilon(\mathbbm{I}):=\mathbbm{I},\qquad\epsilon(T_{\omega}):=0\,.$
(4.40)

The counit can be used to “undo” the action of the coproduct as follows:

$\star(\mathbbm{I}\otimes\epsilon)\Delta(T_{\omega})=\star(\epsilon\otimes\mathbbm{I})\Delta(T_{\omega})=T_{\omega}\,,$
(4.41)

where by $\star$ we mean $\star(T_{\omega_{1}}\otimes
T_{\omega_{2}})=T_{\omega_{1}}\star T_{\omega_{2}}$. Again, the content of
this equation can be shown using diagrams:

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(in){};
\vertex[below=.8cm of in](v1)[codot]{}; \vertex[below=.8cm of v1](vc);
\vertex[right=0.4cm of
vc](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[left=0.6cm of vc](v3);
\vertex[below=.8cm of vc](v4)[dot]{}; \vertex[below=.8cm of v4](v5){};
\diagram*{(in) -- [thick] (v1),(v1) -- [thick,bend left] (v2),(v1) --
[thick,bend right] (v3),(v2) -- [thick,bend left] (v4),(v3) -- [thick,bend
right] (v4), (v4) --[thick](v5)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(in){};
\vertex[below=.8cm of in](v1)[codot]{}; \vertex[below=.8cm of v1](vc);
\vertex[left=0.4cm of
vc](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[right=0.6cm of vc](v3);
\vertex[below=.8cm of vc](v4)[dot]{}; \vertex[below=.8cm of v4](v5){};
\diagram*{(in) -- [thick] (v1),(v1) -- [thick,bend right] (v2),(v1) --
[thick,bend left] (v3),(v2) -- [thick,bend right] (v4),(v3) -- [thick,bend
left] (v4), (v4) --[thick](v5)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[below=3.2cm of i1](v5){}; \vertex[above=1.2cm of i1](v6){};
\diagram*{(i1) -- [thick] (v5)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (4.42)

One can also check directly that the counit is compatible with the quasi-
shuffle product,

$\displaystyle\epsilon(T_{\omega_{1}}\star
T_{\omega_{2}})=\epsilon(T_{\omega_{1}})\star\epsilon(T_{\omega_{2}})\,,$
(4.43)

which can also be summarised in a diagram

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[right=0.6cm of i1](ic){}; \vertex[right=1.2cm of i1](i2){};
\vertex[below=.9cm of ic](v1)[dot]{}; \vertex[below=1.cm of
v1](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[below=0.9cm of v2](o1);
\diagram*{(i1) -- [thick, bend right] (v1),(i2) -- [thick, bend left]
(v1),(v1)--[thick](v2), (v2)--[thick](o1)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[right=0.6cm of i1](ic){}; \vertex[right=1.2cm of i1](i2){};
\vertex[below=.6cm of ic](v1u); \vertex[below=0.9cm of
i1](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[below=0.9cm of
i2](v3)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[below=1.2cm of v1](v4)[dot]{};
\vertex[below=1.0cm of v4](v5)[]{}; \diagram*{(i1) -- [thick] (v2),(i2) --
[thick] (v3),(v3) -- [thick,bend left] (v4),(v2) -- [thick,bend right]
(v4),(v4)--[thick](v5)}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,.$ (4.44)

Unlike the coproduct, the counit does not have a particularly useful
interpretation in terms of a property of the pre-numerator, since it simply
maps the algebraic pre-numerator to zero,

$\epsilon(\widehat{\mathcal{N}}(12\ldots n{-}2,v))=0\,,$ (4.45)

which is easily seen from the definition (4.40).

###  4.4 The antipode

The final operation we need to introduce to form a Hopf algebra is the
antipode $S$, which is a linear map from the space of generators
$\mathfrak{A}$ to itself. Its defining property is

$\displaystyle\star(\mathbbm{I}\otimes
S)\Delta(T_{\omega})=\star(S\otimes\mathbbm{I})\Delta(T_{\omega})=\epsilon(T_{\omega})\,,$
(4.46)

which can also be described diagrammatically as

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(in){};
\vertex[below=.8cm of in](v1)[codot]{}; \vertex[below=.8cm of v1](vc);
\vertex[right=0.4cm of
vc](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{\nullfont\small}$\mathbf{s}$};
\vertex[left=0.6cm of vc](v3); \vertex[below=.8cm of vc](v4)[dot]{};
\vertex[below=.8cm of v4](v5){}; \diagram*{(in) -- [thick] (v1),(v1) --
[thick,bend left] (v2),(v1) -- [thick,bend right] (v3),(v2) -- [thick,bend
left] (v4),(v3) -- [thick,bend right] (v4), (v4) --[thick](v5)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(in){};
\vertex[below=.8cm of in](v1)[codot]{}; \vertex[below=.8cm of v1](vc);
\vertex[left=0.4cm of
vc](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{\nullfont\small}$\mathbf{s}$};
\vertex[right=0.6cm of vc](v3); \vertex[below=.8cm of vc](v4)[dot]{};
\vertex[below=.8cm of v4](v5){}; \diagram*{(in) -- [thick] (v1),(v1) --
[thick,bend right] (v2),(v1) -- [thick,bend left] (v3),(v2) -- [thick,bend
right] (v4),(v3) -- [thick,bend left] (v4), (v4) --[thick](v5)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[below=1.6cm of
i1](v1)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[below=3.2cm of i1](v5){};
\diagram*{(i1) -- [thick] (v1)-- [thick] (v5)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (4.47)

In simple terms, we can take a generator and first split it up via the
coproduct, then act on one of the tensor product factors with $S$, and then
recombine with the product. The result of this operation is that all
generators except for the identity are mapped to zero as per (4.40).888The
antipode of a generator gets its name by analogy with antipodal points on a
sphere whose vectors sum to zero.

The explicit action of the antipode on a generator is defined recursively,

$\displaystyle\begin{split}S(T_{(\tau_{1}),\ldots,(\tau_{r})})&:=-\sum_{i=0}^{r-1}S(T_{(\tau_{1}),\ldots,(\tau_{i})})\star
T_{(\tau_{i+1}),\ldots,(\tau_{r})}\,,\\\
S(\mathbbm{I})&:=\mathbbm{I}\,.\end{split}$ (4.48)

Using this, one can check the following required compatibility properties:

$\displaystyle\begin{split}S(T_{\omega_{1}}\star
T_{\omega_{2}})&=S(T_{\omega_{2}})\star S(T_{\omega_{1}})\,,\\\ (S\otimes
S)\Delta(T_{\omega})&=R_{12}\Delta S(T_{\omega})\,,\\\ \epsilon
S(T_{\omega})&=\epsilon(T_{\omega})\,,\end{split}$ (4.49)

where $R_{12}$ denotes the exchange of the first and second element in the
tensor product. The diagrammatic representation of the above relations are as
follows,

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[right=0.6cm of i1](ic){}; \vertex[right=1.2cm of i1](i2){};
\vertex[below=.9cm of ic](v1)[dot]{}; \vertex[below=1.cm of
v1](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{\nullfont\small}$\mathbf{s}$};
\vertex[below=1.cm of v2](o1); \diagram*{(i1) -- [thick, bend right] (v1),(i2)
-- [thick, bend left] (v1),(v1)--[thick](v2), (v2)--[thick](o1)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[right=0.6cm of i1](ic){}; \vertex[right=1.2cm of i1](i2){};
\vertex[below=.6cm of ic](v1){}; \vertex[below=.6cm of ic](v1u);
\vertex[below=1.3cm of
i1](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{\nullfont\small}$\mathbf{s}$};
\vertex[below=1.3cm of
i2](v3)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{\nullfont\small}$\mathbf{s}$};
\vertex[below=1.5cm of v1](v4)[dot]{}; \vertex[below=1.0cm of v4](v5)[]{};
\diagram*{(i1) -- [thick,bend right](v1)-- [thick,bend left] (v3),(i2) --
[thick,bend left](v1u)-- [thick,bend right] (v2),(v3) -- [thick,bend left]
(v4),(v2) -- [thick,bend right] (v4),(v4)--[thick](v5)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,,$
$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[below=0.9cm of i1](v0)[codot]{}; \vertex[below=0.9cm of v0](vc1);
\vertex[left=0.4cm of
vc1](v1)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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\vertex[right=0.4cm of
vc1](v2)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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\vertex[below=0.9cm of v1](o1); \vertex[below=0.9cm of v2](o2); \diagram*{(i1)
-- [thick](v0),(v0) -- [thick, bend right](v1)--[thick](o1),(v0) -- [thick,
bend left](v2)--[thick](o2)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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i1](vm1)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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\vertex[below=0.6cm of vm1](v0)[codot]{}; \vertex[below=0.5cm of v0](vc1);
\vertex[left=0.5cm of vc1](v1); \vertex[right=0.5cm of vc1](v2);
\vertex[below=0.65cm of vc1](vc2); \vertex[below=0.5cm of vc1](vc2u){};
\vertex[below=1.2cm of v1](o1); \vertex[below=1.2cm of v2](o2); \diagram*{(i1)
-- [thick](vm1) -- [thick](v0),(v0)--[thick,bend right](v1),(v0) --
[thick,bend left](v2),(v1) -- [thick,bend right](vc2)-- [thick,bend
left](o2),(v2) -- [thick,bend left](vc2u)-- [thick,bend right](o1)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,,$
$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[below=1.0cm of
i1](v0)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{\nullfont\small}$\mathbf{s}$};
\vertex[below=1.2cm of
v0](v1)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[below=3.2cm of i1](v5){};
\diagram*{(i1) -- [thick] (v0)-- [thick] (v1)-- [thick] (v5)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}=~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(i1){};
\vertex[below=1.6cm of
i1](v1)[sb]{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{\nullfont\small}$\mathbf{\epsilon}$}; \vertex[below=3.2cm of i1](v5){};
\diagram*{(i1) -- [thick] (v1)-- [thick] (v5)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,.$ (4.50)

To find the action of the antipode on the algebraic pre-numerator we first use
(4.48) to find the antipode of a single letter,

$S(T_{(i)})=-T_{(i)}\,,$ (4.51)

then we use the first relation of (4.49) to find

$\begin{split}S\widehat{\mathcal{N}}(12\ldots n{-}2)&=S(T_{(1)}\star
T_{(2)}\star T_{(3)}\cdots T_{(n{-}2)})\\\ &=S(T_{(n-2)})\star
S(T_{(n-3)})\star S(T_{(n-4)})\star\cdots\star S(T_{(1)})\\\
&=(-1)^{n-2}T_{(1)}\star T_{(2)}\star T_{(3)}\star\cdots\star T_{(n{-}2)}\\\
&=(-1)^{n-2}\widehat{\mathcal{N}}(12\ldots n{-}2)\,,\end{split}$ (4.52)

where in the second-to-last line we have use the fact that the quasi-shuffle
product is commutative. Thus the antipode only changes the pre-numerator by at
most an overall sign. This seems to suggest that, like the counit, there is no
useful interpretation of the antipode. However, we shall see in the next
section that we can extend the quasi-shuffle algebra to a non-abelian version
in which case the antipode becomes non-trivial.

##  5 The non-abelian quasi-shuffle Hopf algebra

In Section 3 we laid out properties of the quasi-shuffle product used to build
pre-numerators and, in turn, BCJ numerators. As an explicit example consider
the four-point BCJ numerator

$\mathcal{N}([1,2],v)=\mathcal{N}(12,v)-\mathcal{N}(21,v)\,.$ (5.1)

$\mathcal{N}(12,v)$ was given in (3.13), and to find $\mathcal{N}(21,v)$ we
simply swap the labels $1$ and $2$ of the pre-numerator $\mathcal{N}(12,v)$.
Thus the four-point BCJ numerator is given by

$\mathcal{N}([1,2],v)=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}v}{2v{\cdot}p_{1}}+\frac{v{\cdot}F_{2}{\cdot}F_{1}{\cdot}v}{2v{\cdot}p_{2}}=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}v}{v{\cdot}p_{1}}\,,$
(5.2)

where in the last equality we have used $(p_{1}+p_{2}){\cdot}v=0$.

Note that in this example the pre-numerator $\mathcal{N}(12,v)$ is not
symmetric in the labels $1$ and $2$, whereas the algebraic pre-numerator
$\widehat{\mathcal{N}}(12)=T_{(1)}\star T_{(2)}$ is symmetric since the quasi-
shuffle product is commutative. The ordering of labels $(1,2)$ is only imposed
on the pre-numerator once we take the angle bracket map
$\langle\bullet\rangle$ given in (3.10). This raises the interesting question
of whether we can incorporate arbitrary orderings of legs, beyond the
canonical ordering $1,2,\ldots,n{-}2$, into the algebra itself while
preserving the Hopf algebra structure.

As we now demonstrate it is possible to extend the quasi-shuffle product to
become non-abelian by endowing our generators with additional dependence on
the overall ordering of the massless particles denoted by $\alpha\in S_{n-2}$.
First steps in this direction were taken in the appendix of the prequel paper
[57], but here we will present the complete construction, the non-abelian
kinematic Hopf algebra, and find, in particular, an amplitude interpretation
for its coproduct and the antipode. The important result will be the ability
to compute BCJ numerators directly at the level of the non-abelian quasi-
shuffle algebra.

###  5.1 Extended generators

The first important step is to introduce extended generators with ordering
$\alpha\in S_{n-2}$ of the form999 Remarkably, an extension of the generators
analogous to this one has led to the kinematic algebra construction for the
BCJ numerators of the scattering amplitudes as well as form factors in Yang-
Mills-scalar theory coupled with a bi-adjoint $\phi^{3}$ interaction with a
general mass [89].

$\displaystyle
T^{(\alpha)}_{(\tau_{1}),(\tau_{2})\ldots(\tau_{r})}:=r^{(\alpha)}\otimes
T_{(\tau_{1}),(\tau_{2})\ldots(\tau_{r})}\,,$ (5.3)

where the $\tau_{i}$ are subsets of $\\{1,2,\ldots,n{-}2\\}$. In practice the
actual labels appearing in the subscript $\tau=\tau_{1}\cup\cdots\cup\tau_{r}$
will be a subset of $\alpha$, for example $T^{(12435)}_{(12),(34)}$. These new
generators correspond to a tensor product extension of the underlying vector
space of the quasi-shuffle algebra. As implied by (5.3), we extend the
generators involving indices $\\{1,2,\ldots,n{-}2\\}$ by tensoring with the
free algebra over the elements of $\\{1,2,\ldots,n{-}2\\}$. This algebra is
generated by taking products and linear combinations of the $n{-}2$ elements
$r^{(i)}$ for $i\in\\{1,2,\ldots,n{-}2\\}$.

As a free algebra, this product simply consists of concatenating indices

$\displaystyle r^{(\alpha)}\cdot r^{(\beta)}:=r^{(\alpha\beta)}\,,$ (5.4)

and in general builds objects with indices consisting of words taken from the
alphabet $\\{1,2,\ldots,n{-}2\\}$. It is clear from its definition that this
product is non-abelian. For now this is the only structure we will impose on
the orderings, however we shall add slightly more in the next section. The
fusion product of the extended generators is defined similarly and is induced
through their definition (5.3) in terms of a tensor structure and through the
previously defined products (3.6) and (5.4). It is given by

$\displaystyle\begin{split}T^{(\alpha)}_{(\tau_{1}),\ldots,(\tau_{r})}\star
T^{(\beta)}_{(\rho_{1}),\ldots,(\rho_{s})}&:=(r^{(\alpha)}\otimes
T_{(\tau_{1}),(\tau_{2})\ldots(\tau_{r})})\star(r^{(\beta)}\otimes
T_{(\rho_{1}),(\rho_{2})\ldots(\rho_{s})})\\\ &=(r^{(\alpha)}\cdot
r^{(\beta)})\otimes(T_{(\tau_{1}),(\tau_{2})\ldots(\tau_{r})}\star
T_{(\rho_{1}),(\rho_{2})\ldots(\rho_{s})})\\\
&=r^{(\alpha\beta)}\otimes\sum_{\genfrac{}{}{0.0pt}{}{\sigma\lvert_{\\{\tau\\}}=\\{(\tau_{1}),\ldots,(\tau_{r})\\}}{\sigma\lvert_{\\{\rho\\}}=\\{(\rho_{1}),\ldots,(\rho_{s})\\}}}(-1)^{t-r-s}T_{(\sigma_{1}),\ldots,(\sigma_{t})}\,.\end{split}$
(5.5)

This new product corresponds to the tensor extension of the original product
via $\star\mapsto\cdot\otimes\star$, where for convenience we use the same
symbol for both.

The actual fusion products we will perform in order to build a BCJ numerator
will involve orderings $\alpha$ and $\beta$ of disjoint sets, hence the
product $\alpha\beta$ will also be an ordering of their union.

This extension of the quasi-shuffle algebra is non-abelian since the product
on the superscripts is non-abelian:

$\displaystyle\begin{split}[T^{(\alpha)}_{(\tau_{1}),\ldots,(\tau_{r})}\,,\,T^{(\beta)}_{(\rho_{1}),\ldots,(\rho_{s})}]&:=T^{(\alpha)}_{(\tau_{1}),\ldots,(\tau_{r})}\star
T^{(\beta)}_{(\rho_{1}),\ldots,(\rho_{s})}-T^{(\beta)}_{(\rho_{1}),\ldots,(\rho_{s})}\star
T^{(\alpha)}_{(\tau_{1}),\ldots,(\tau_{r})}\\\
&=(r^{(\alpha\beta)}-r^{(\beta\alpha)})\otimes(T_{(\tau_{1}),(\tau_{2})\ldots(\tau_{r})}\star
T_{(\rho_{1}),(\rho_{2})\ldots(\rho_{s})})\\\ &\neq 0\,.\end{split}$ (5.6)

We can now build algebraic pre-numerators with any ordering $\sigma$ of labels
$\\{1,2,\ldots,n{-}2\\}$ by taking successive products of extended generators

$\widehat{\mathcal{N}}(\sigma_{1}\sigma_{2}\ldots\sigma_{n{-}2})\coloneqq
T_{(\sigma_{1})}^{(\sigma_{1})}\star
T_{(\sigma_{2})}^{(\sigma_{2})}\star\cdots\star
T_{(\sigma_{n{-}2)}}^{(\sigma_{n{-}2})}\,,$ (5.7)

and further we can define algebraic BCJ numerators by simply expanding in
terms of algebraic pre-numerators. For example, at five points one has

$\widehat{\mathcal{N}}([[1,2],3])=\widehat{\mathcal{N}}(123)-\widehat{\mathcal{N}}(213)+\widehat{\mathcal{N}}(321)-\widehat{\mathcal{N}}(312)\,.$
(5.8)

These algebraic pre-numerators can then be mapped to physical pre-numerators
using a map which we again denote by $\langle\bullet\rangle$. If we recall the
original definition (3.10) of $\langle\bullet\rangle$, we can extend it to
arbitrary orderings as follows,

$\displaystyle\langle
T^{(\alpha)}_{(\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})}\rangle=\frac{v{\cdot}F_{\tau_{1}}{\cdot}V_{\Theta^{\alpha}(\tau_{2})}{\cdot}F_{\tau_{2}}\ldots{\cdot}V_{\Theta^{\alpha}(\tau_{r})}{\cdot}F_{\tau_{r}}{\cdot}v}{(n-2)v{\cdot}p_{\tau_{1}[1]}v{\cdot}p_{\tau_{1}}\ldots
v{\cdot}p_{\tau_{1}\cdots\tau_{r-1}}}\,,$ (5.9)

where now $\Theta^{(\alpha)}(\tau_{i})$ is the subset of
$\tau_{1}\cup\cdots\cup\tau_{i-1}$ which are less than the first leg in
$\tau_{i}$ with respect to the ordering $\alpha$, for example if
$T^{(254163)}_{(13),(25),(46)}$ then $\Theta^{(254163)}(46)=\\{2,5\\}$. If any
of the sets $\Theta^{(\alpha)}(\tau_{i})$ happens to be empty then we map that
generator to zero. The expression $F_{\tau_{j}}$ denotes an ordered product of
$F_{i}$ for $i\in\tau_{j}$ where the $\tau_{j}$ are also ordered with respect
to the ordering $\alpha$.

###  5.2 The Hopf algebra of orderings

The extension of the quasi-shuffle algebra described above can also be turned
into a Hopf algebra by defining a new coproduct, counit and antipode. This can
be achieved by simply tensoring the original Hopf algebra of quasi-shuffles
with another Hopf algebra of orderings which acts on the $r^{(\alpha)}$, as we
now discuss.

The coproduct: To begin building such a Hopf algebra of orderings, we define a
coproduct to the concatenation product (5.4). As in the quasi-shuffle case,
the coproduct is a map from the space of orderings $\mathcal{O}$ to the tensor
product space $\mathcal{O}\otimes\mathcal{O}$. It turns out that there are
many different choices for it. As before, we can define the coproduct by its
action on the simplest object, $r^{(i)}$, and then extend to all other
elements using linearity and compatibility with the product:
$\Delta(r^{(\alpha)}{\cdot}r^{(\beta)})=\Delta(r^{(\alpha)}){\cdot}\Delta(r^{(\beta)})$.
We will also abuse notation here and use the same symbol for the coproduct, as
in the quasi-shuffle Hopf algebra.

To explain the possible choices of coproduct we start with the following
simple ansatz

$\Delta_{a,b}(r^{(i)})\coloneqq a(\mathbbm{I}\otimes
r^{(i)}+r^{(i)}\otimes\mathbbm{I})+b\,r^{(i)}\otimes r^{(i)}\,,$ (5.10)

where $a$ and $b$ are arbitrary constants at least one of which is non-zero.
In order to satisfy co-associativity, and compatibility with a non-trivial
counit and antipode (see (4.42) and (4.47)), we require the coproduct to be
symmetric between the left and right parts of the tensor product. If we
consider the case were $b=0$ and $a=1$ the coproduct will not preserve the
ordering of legs in the pre-numerator, for example:101010Interestingly, this
form of the coproduct for the concatenation product (5.4) appears in [58],
where one considers the (quasi-)shuffle algebra as dual to the free algebra.
However, we could not find any physical interpretation for it.

$\Delta_{a=1,b=0}(r^{(12)})=\mathbbm{I}\otimes r^{(12)}+r^{(1)}\otimes
r^{(2)}+r^{(2)}\otimes r^{(1)}+r^{(12)}\otimes\mathbbm{I}\,.$ (5.11)

The whole point of introducing the extended algebra was to make the ordering
of particles manifest, so instead we will choose $a=0$ and $b=1$, which gives
for a general ordering:

$\Delta(r^{(\alpha)})\coloneqq\Delta_{a=0,b=1}(r^{(\alpha)})=r^{(\alpha)}\otimes
r^{(\alpha)}\,.$ (5.12)

It is straightforward to show that this definition also satisfies co-
associativity as per diagram (4.30).

The counit: The counit can also be simply defined, to yield a bialgebra of the
$r^{(\alpha)}$,

$\epsilon(r^{(\alpha)})=\mathbbm{I}\,,\qquad\epsilon(\mathbbm{I})=\mathbbm{I}\,,$
(5.13)

where again we are abusing notation and using $\mathbbm{I}$ to label the
identity element of the product ${\cdot}$ in (5.4) and the quasi-shuffle
product (3.8). This definition immediately satisfies the defining properties
of the counit, the diagrams (4.42) and (4.44).

The antipode: Finally, to construct a Hopf algebra for the orderings
$r^{(\alpha)}$ we need to define an antipode $S$, which by definition must
satisfy (see diagram (4.47))

$\cdot(S\otimes\mathbbm{I})\Delta(r^{(\alpha)})=\cdot(\mathbbm{I}\otimes
S)\Delta(r^{(\alpha)})=\epsilon(r^{\alpha})\,.$ (5.14)

Using our choices for the coproduct and the counit, this becomes

$S(r^{(\alpha)}){\cdot}r^{(\alpha)}=r^{(\alpha)}{\cdot}S(r^{(\alpha)})=\mathbbm{I}\,.$
(5.15)

It is clear from the above relation that $S(r^{(\alpha)})$ should correspond
to the inverse of $r^{(\alpha)}$, however in the free algebra of the
$r^{(\alpha)}$ no such inverses exist. Additionally, the antipode must be an
antihomomorphism from the space of $r^{(\alpha)}$ to itself, in particular it
must satisfy

$S(r^{(ij)})=S(r^{(j)}){\cdot}S(r^{(i)})\,.$ (5.16)

To solve both of the above relations we must include inverses and instead
consider a group algebra of orderings. The antipode will then be identified
with the map which sends a group element to its inverse – this is a very
general construction applicable to any group known as a group Hopf algebra.

We began with a free algebra of orderings $r^{(\alpha)}$ and in order to
introduce the antipode we require inverse elements. The most general
construction we could employ is a group algebra of the free group with $n{-}2$
generators. This group algebra is constructed in a manner very similar to the
free algebra, and consists of linear combinations of products of generators
$g$ in the group and inverse elements $g^{-1}$, while assuming only the group
axioms. However, to define the group algebra which corresponds to orderings we
require that the antipode of a particular ordering $r^{(\alpha)}$ is also an
ordering of the same indices contained in $\alpha$. As such we define the
inverse of an ordering $\alpha$ to be the reversed ordering
$\overline{\alpha}$:

$r^{(\alpha)}{\cdot}r^{(\beta)}=r^{(\alpha\beta)},\qquad
r^{(\alpha)}{\cdot}r^{(\overline{\alpha})}=\mathbbm{I}\,,$ (5.17)

for example $r^{(\overline{123})}=r^{(321)}$, and as before we can take linear
combinations of orderings. Note that the above conditions imply
$r^{(i)}{\cdot}r^{(i)}=\mathbbm{I}$. Instead of a free group this is the group
freely generated by $n{-}2$ self-inverse elements, where “freely” here means
assuming no other group relations beyond the axioms. This group is known as
the universal Coxeter group with $n{-}2$ generators.

Now that we have inverses in place, it is clear that we should define the
antipode of an ordering as

$S(r^{(\alpha)})=r^{(\overline{\alpha})}\,.$ (5.18)

This definition also satisfies the other compatibility conditions required by
the diagrams (4.50).

To summarise, we have defined a coproduct, counit and antipode for the algebra
of orderings with product given by (5.4). The resulting Hopf algebra has the
interesting property of requiring an underlying group algebra structure of the
universal Coxeter Group such that the antipode acting on an element is
identified with the inverse element given by the reversed ordering. This is
known as a group Hopf algebra and is a general construction that can be made
for any group.

###  5.3 Combining the Hopf algebras

Equipped with a Hopf algebra of orderings, we can now describe the Hopf
algebra structure of the non-abelian quasi-shuffle algebra by tensoring the
ordering algebra $\mathcal{O}$ with the quasi-shuffle algebra $\mathfrak{A}$.
As advertised at the beginning of this section, and much like the fusion
product (5.5) for the extended generators, the coproduct, counit and antipode
of the extended algebra are simply given by a product structure.

The coproduct on the extended generators is

$\displaystyle\begin{split}\Delta(T_{(\tau_{1}),\ldots,(\tau_{r})}^{(\alpha)}):=&(\mathbbm{I}\otimes\mathcal{T}\otimes\mathbbm{I})\Delta(r^{(\alpha)})\otimes\Delta(T_{(\tau_{1}),\ldots,(\tau_{r})})\\\
=&\sum_{i=0}^{r}\left(T_{(\tau_{1}),\ldots,(\tau_{i})}^{(\alpha)}\otimes
T_{(\tau_{i+1}),\ldots,(\tau_{r})}^{(\alpha)}\right)\,,\end{split}$ (5.19)

where
$\mathcal{T}{:}\,\mathcal{O}\otimes\mathfrak{A}\rightarrow\mathfrak{A}\otimes\mathcal{O}$
is a transposition operator which swaps the entries in the tensor product. It
is then an immediate consequence that the coproduct (5.19) is compatible with
the product defined in (5.5) and therefore satisfies the diagram (4.3). We
also again define the reduced coproduct $\Delta^{\prime}$ which removes all
trivial generators without subscripts, i.e. $T^{(\alpha)}$.

Like the coproduct, the counit is extended such that it acts on both the
generators $T_{\omega}$ and the ordering $r^{(\alpha)}$. That is,

$\displaystyle\epsilon(T^{(\alpha)}_{\omega}):=\epsilon(r^{(\alpha)})\otimes\epsilon(T_{\omega})=\mathbbm{I}\otimes\epsilon(T_{\omega})=0\,,$
(5.20)

except for when the element $T_{\omega}=\mathbbm{I}$, in which case we have

$\displaystyle\epsilon(T^{(\alpha)}):=\epsilon(r^{(\alpha)})\otimes\epsilon(\mathbbm{I})=\mathbbm{I}\otimes\mathbbm{I}=\mathbbm{I}\,.$
(5.21)

Again it is trivial to check that this counit is compatible with the product
(5.5) as required by the diagram (4.44).

Having defined the coproduct and counit we have fully specified our bialgebra,
however to be a Hopf algebra we also need an antipode. In our case we can
define the antipode acting on extended generators as

$\displaystyle\begin{split}S(T^{(\alpha)}_{(\tau_{1}),\ldots,(\tau_{r})})&:=S(r^{(\alpha)})\otimes
S(T_{(\tau_{1}),\ldots,(\tau_{r})})=r^{(\overline{\alpha})}\otimes\bigg{(}-\sum_{i=0}^{r-1}S(T_{(\tau_{1}),\ldots,(\tau_{i})})\star
T_{(\tau_{i+1}),\ldots,(\tau_{r})}\bigg{)}\,\\\
&=\sum_{h=0}^{r-1}(-1)^{h+1}\sum_{r>i_{1}>i_{2}\cdots>i_{h}>1}T^{(\overline{\alpha})}_{(\tau_{r}\ldots\tau_{i_{1}+1}),(\tau_{i_{1}}\ldots\tau_{i_{2}+1}),\ldots,(\tau_{i_{h}}\ldots\tau_{1})}\,,\end{split}$
(5.22)

where $(\tau_{i}\ldots\tau_{j})$ denotes the union of the $\tau_{k}$ from
indices $i$ to $j$. It is again a simple exercise to check the required
properties given by diagrams (4.47) and (4.50). Acting on an algebraic pre-
numerator the antipode reverses the gluon labels up to a factor of
$(-1)^{n{-}2}$

$\displaystyle\begin{split}S\left(T^{(1)}_{(1)}\star\cdots\star
T^{(n{-}2)}_{(n{-}2)}\right)&=S(r^{(12\ldots n{-}2)})\otimes
S(T_{(1)}\star\cdots\star T_{(n{-}2)})\\\ &=r^{(n{-}2\ldots
21)}\otimes(-1)^{n{-}2}\,T_{(1)}\star\cdots\star T_{(n{-}2)}\\\
&=(-1)^{n{-}2}\widehat{\mathcal{N}}(n{-}2\ldots 21)\,.\end{split}$ (5.23)

It is interesting to note that the antipode acting on an algebraic BCJ
numerator simply mirrors the associated cubic graph up to an overall minus
sign. For example, the antipode acting on a left-nested BCJ numerator gives

$\displaystyle\begin{split}S\,\widehat{\mathcal{N}}([1,2,3,\ldots,n{-}2])&=(-1)^{n{-}2}\widehat{\mathcal{N}}([n{-}2,n{-}3,n{-}4,\ldots,1])\\\
&=-\widehat{\mathcal{N}}([1,\ldots,[n{-}4,[n{-}3,n{-}2]]\ldots])\,,\end{split}$
(5.24)

where the final commutator in (5.24) is then written in the original order but
has gone from left-nested to right-nested, which gives an additional
$(-1)^{n-3}$ factor and the mirror of the original cubic graph.

As an example, consider the case of four gluons, where we have five cubic
graphs: a pair of graphs and their mirrors with commutators
$\\{[[1,2],3],4],\,[1,[2,[3,4]]]\\}$ and
$\\{[[1,[2,3]],4],\,[1,[[2,3],4]]\\}$, and the mirror-symmetric graph
corresponding to the commutator $[[1,2],[3,4]]$. Under the antipode map we
have

$\displaystyle\begin{split}S\,\widehat{\mathcal{N}}([[1,2],[3,4]])=-\widehat{\mathcal{N}}([[1,2],[3,4]])\,,\\\
S\,\widehat{\mathcal{N}}([[1,2],3],4])=-\widehat{\mathcal{N}}([1,[2,[3,4]]])\,,\\\
S\,\widehat{\mathcal{N}}([[1,[2,3]],4])=-\widehat{\mathcal{N}}([1,[[2,3],4]])\,.\end{split}$
(5.25)

We can also construct an algebraic amplitude by replacing the kinematic BCJ
numerator with the algebraic one,

$\displaystyle\widehat{A}(12\ldots n{-}2)$
$\displaystyle\,\coloneqq\,\sum_{\Gamma\in\rho}{\widehat{\mathcal{N}}(\Gamma)\over
d_{\Gamma}}\,.$ (5.26)

The HEFT amplitude (2.6) satisfies the reflection identity (2.12), which is
inherited from the well-known reflection identity of gluon-scalar amplitudes
before taking the heavy-mass limit. We expect the algebraic amplitude to
satisfy its own reflection identity, generated by the action of the antipode,
which we now describe.

First, we define an operator $\bar{S}$ which consists of the antipode $S$ and
a reversal of the labels in the massless propagators, hence giving the
propagators $d_{\bar{\Gamma}}$ of the mirrored graph $\bar{\Gamma}$. Then,
under the action of $\bar{S}$, the algebraic amplitude picks up an overall
sign

$\displaystyle\begin{split}\bar{S}\,\widehat{A}(12\ldots
n{-}2)&\coloneqq\sum_{\Gamma\in\rho}{S\widehat{\mathcal{N}}(\Gamma)\over
d_{\overline{\Gamma}}}\\\
&=\sum_{\Gamma\in\rho}{-\widehat{\mathcal{N}}(\bar{\Gamma})\over
d_{\overline{\Gamma}}}\\\ &=-\widehat{A}(12\ldots n{-}2)\,,\end{split}$ (5.27)

where we have first used the second line of (5.24), and then the fact that the
HEFT amplitude is a sum over all cubic graphs, and hence every BCJ numerator
appears together with its mirror. The action of $\bar{S}$ simply multiplies by
a factor of $-1$.

The action of $\bar{S}$ on algebraic HEFT amplitudes can also be written as

$\displaystyle\begin{split}\bar{S}\,\widehat{A}(12\ldots
n{-}2)=(-1)^{n{-}2}\widehat{A}(n{-}2\ldots 21)\,,\end{split}$ (5.28)

using the first line of (5.24). Finally, combining (5.27) and (5.28) we have

$\widehat{A}(12\ldots n{-}2)=(-1)^{n{-}1}\widehat{A}(n{-}2\ldots 21)\,,$
(5.29)

which reproduces the reflection identity (2.12) for the algebraic HEFT
amplitude and relates it to the action of the antipode.

In summary, in this section we have constructed a non-abelian quasi-shuffle
Hopf algebra which allows us to treat all possible colour orderings on the
same footing, unlike its abelian counterpart that only applied to the
canonical ordering $(1,2,\ldots,n{-}2)$. As a bonus we were able to give
physical interpretations to the various operations of the Hopf algebra at the
level of the numerators, rather than the pre-numerator. We will discuss in the
next section in detail how the co-product of the new algebra is related to the
factorisation of BCJ numerators.

##  6 Factorisation behaviour of BCJ numerators and amplitudes

In this section we discuss the factorisation properties of the BCJ numerators
and HEFT amplitudes with gluons and gravitons. Just like the pre-numerators,
the BCJ numerators and the HEFT amplitudes are formal polynomials in the
$T_{\omega}$, and we can use the coproduct to characterise the factorisation
behaviour. To do this we must define an extension of the replacement rule
$\operatorname{\mathcal{C}}$ for the coproduct defined earlier in (4.13). This
is given by

$\displaystyle\begin{split}\operatorname{\mathcal{C}}\Big{(}T_{\omega_{1}}^{(\alpha)}\otimes
T_{\omega_{2}}^{(\alpha)}\otimes\cdots\otimes
T_{\omega_{i}}^{(\alpha)}\Big{)}&=\frac{\|\omega_{1}\|\|\omega_{2}\|\ldots\|\omega_{i}\|}{\|\omega_{1}\|+\|\omega_{2}\|+\ldots\|\omega_{i}\|}\frac{1}{v{\cdot}p_{\omega_{1}}v{\cdot}p_{\omega_{1}\omega_{2}}\cdots
v{\cdot}p_{\omega_{1}\cdots\omega_{i-1}}}\\\
&\times\Big{(}T^{(\alpha|_{\omega_{1}})}_{\omega_{1}}p_{\Theta^{(\alpha)}(\omega_{2})}{\cdot}p_{\omega_{2}(1)}T^{(\alpha|_{\omega_{2}})}_{\omega_{2}}\cdots
p_{\Theta^{(\alpha)}(\omega_{i})}{\cdot}p_{\omega_{i}(1)}T^{(\alpha|_{\omega_{i}})}_{\omega_{i}}\Big{)}\,,\end{split}$
(6.1)

where we chose to restrict the ordering in the superscript to include only
those indices appearing in the subscript. For example, we have

$\displaystyle\begin{split}&\langle T_{(13)}^{(524163)}\otimes
T_{(52)(6)}^{(524163)}\otimes T_{(4)}^{(524163)}\rangle=0\,,\\\ &\langle
T_{(52)}^{(524163)}\otimes T_{(46)}^{(524163)}\otimes
T_{(13)}^{(524163)}\rangle=\frac{1}{v{\cdot}p_{52}v{\cdot}p_{5246}}T^{(52)}_{(52)}p_{52}{\cdot}p_{4}T^{(46)}_{(46)}p_{524}{\cdot}p_{1}T^{(13)}_{(13)}\,.\end{split}$
(6.2)

With this definition, for each cubic graph or nested commutator structure
$\Gamma$, we have

$\displaystyle\Big{\langle}\text{Res}_{(v{\cdot}p_{\omega_{1}},\ldots
v{\cdot}p_{\omega_{k}})}\,\mathcal{C}\,\Delta^{\prime(k)}(\widehat{\mathcal{N}}(\Gamma))\Big{\rangle}=\text{Res}_{(v{\cdot}p_{\omega_{1}},\ldots
v{\cdot}p_{\omega_{k}})}\,\mathcal{N}(\Gamma,v)\,.$ (6.3)

Finally, before moving on to some examples we note an additional useful
relation between left-nested BCJ numerators and the pre-numerator, proven in
[57]

$\displaystyle\mathcal{N}([12\ldots n{-}2])=(n-2)\mathcal{N}(12\ldots
n{-}2)\,.$ (6.4)

Note that this relation is only valid at the level of the kinematic pre-
numerator and not at the level of the algebraic pre-numerator.111111We have
attempted to implement this relation at the level of the algebra, for example
by requiring that $\widehat{\mathcal{N}}([1,2])=2\widehat{\mathcal{N}}(12)$.
However this spoils the definition of the coproduct such that it is no longer
well defined.

###  6.1 Five-point example

To illustrate the factorisation of BCJ numerators we consider some examples,
beginning at five points. In this case there are three cubic graphs in the
HEFT, and the corresponding BCJ numerators are
$\widehat{\mathcal{N}}([[1,2],3])$, $\widehat{\mathcal{N}}([1,[2,3]])$, and
$\widehat{\mathcal{N}}([[1,3],2])$. Single-cut factorisations are in one-to-
one correspondence with terms in the single coproduct
$\Delta(\widehat{\mathcal{N}}(\Gamma))$. Using (6.1) one finds

$\displaystyle\begin{split}\operatorname{\mathcal{C}}\Delta^{\prime}(\widehat{\mathcal{N}}([[1,2],3]))&={1\over
v{\cdot}p_{1}}T^{(1)}_{(1)}p_{1}{\cdot}p_{2}\Big{(}{4\over
3}\widehat{\mathcal{N}}(23)-{2\over 3}\widehat{\mathcal{N}}(32)\Big{)}\\\
&-{1\over v{\cdot}p_{2}}T^{(2)}_{(2)}p_{2}{\cdot}p_{1}\Big{(}{4\over
3}\widehat{\mathcal{N}}(13)-{2\over 3}\widehat{\mathcal{N}}(31)\Big{)}\\\
&-{2\over
3v{\cdot}p_{12}}T^{(3)}_{(3)}\Big{(}(p_{1}{\cdot}p_{3}+p_{12}{\cdot}p_{3})\widehat{\mathcal{N}}(12)-(p_{12}{\cdot}p_{3}+p_{2}{\cdot}p_{3})\widehat{\mathcal{N}}(21)\Big{)}\,.\end{split}$
(6.5)

The single coproduct of the other two cubic graphs are related to the above by
swapping the indices and the Jacobi relation

$\displaystyle\begin{split}\operatorname{\mathcal{C}}\Delta^{\prime}(\widehat{\mathcal{N}}([[1,3],2]))&=\operatorname{\mathcal{C}}\Delta^{\prime}(\widehat{\mathcal{N}}([[1,2],3]))\Big{|}_{2\longleftrightarrow
3}\,,\\\
\operatorname{\mathcal{C}}\Delta^{\prime}(\widehat{\mathcal{N}}([1,[2,3]]))&=\operatorname{\mathcal{C}}\Delta^{\prime}(\widehat{\mathcal{N}}([[1,2],3]))-\operatorname{\mathcal{C}}\Delta^{\prime}(\widehat{\mathcal{N}}([[1,3],2]))\,.\end{split}$
(6.6)

Beyond the single-cut factorisation, the (multiple) coproduct gives terms in
correspondence with multi-factorisation behaviour. The double-cut
factorisation behaviour can be found from the double coproduct as follows:

$\displaystyle\begin{split}\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}([[1,2],3]))&=\Big{(}{p_{1}{\cdot}p_{2}p_{12}{\cdot}p_{3}\over
v{\cdot}p_{1}v{\cdot}p_{12}}+{p_{1}{\cdot}p_{3}p_{1}{\cdot}p_{2}\over
v{\cdot}p_{1}v{\cdot}p_{13}}\Big{)}T^{(1)}_{(1)}T^{(2)}_{(2)}T^{(3)}_{(3)}\,,\\\
\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}([[1,3],2]))&=\Big{(}{p_{1}{\cdot}p_{3}p_{13}{\cdot}p_{2}\over
v{\cdot}p_{1}v{\cdot}p_{13}}+{p_{1}{\cdot}p_{2}p_{1}{\cdot}p_{3}\over
v{\cdot}p_{1}v{\cdot}p_{12}}\Big{)}T^{(1)}_{(1)}T^{(2)}_{(2)}T^{(3)}_{(3)}\,,\\\
\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}([1,[2,3]]))&=\Big{(}{-p_{1}{\cdot}p_{3}p_{2}{\cdot}p_{3}\over
v{\cdot}p_{1}v{\cdot}p_{13}}+{p_{1}{\cdot}p_{2}p_{2}{\cdot}p_{3}\over
v{\cdot}p_{1}v{\cdot}p_{12}}\Big{)}T^{(1)}_{(1)}T^{(2)}_{(2)}T^{(3)}_{(3)}\,.\end{split}$
(6.7)

It is also possible but more complicated to check the factorisation behaviour
from the BCJ numerator directly, and doing so we confirm the relation (6.3)
for the five-point BCJ numerators. For example, the double-cut factorisation
behaviour is

$\displaystyle\begin{split}\Big{\langle}\text{Res}_{v{\cdot}p_{2},v{\cdot}p_{1}}\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}(\Gamma))\Big{\rangle}&=\text{Res}_{v{\cdot}p_{2},v{\cdot}p_{1}}\mathcal{N}(\Gamma,v)\,,\\\
\Big{\langle}\text{Res}_{v{\cdot}p_{1},v{\cdot}p_{12}}\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}(\Gamma))\Big{\rangle}&=\text{Res}_{v{\cdot}p_{1},v{\cdot}p_{12}}\mathcal{N}(\Gamma,v)\,,\\\
\Big{\langle}\text{Res}_{v{\cdot}p_{1},v{\cdot}p_{13}}\operatorname{\mathcal{C}}\Delta^{\prime(2)}(\widehat{\mathcal{N}}(\Gamma))\Big{\rangle}&=\text{Res}_{v{\cdot}p_{1},v{\cdot}p_{13}}\mathcal{N}(\Gamma,v)\,,\end{split}$
(6.8)

where the $\Gamma\in\\{[[1,2],3],[[1,3],2],[1,[2,3]]\\}$. Similarly to the
pre-numerator in Section 4.2, only two of the double cuts are independent.

Following the factorisation behaviour for BCJ numerators, the factorisation
behaviour of amplitudes can also be characterised by the coproduct. For
example, consider the amplitude $A(123,v)$, whose algebraic form is

$\displaystyle\widehat{A}(123)={\widehat{\mathcal{N}}([[1,2],3])\over
p_{12}^{2}p_{123}^{2}}+{\widehat{\mathcal{N}}([1,[2,3]])\over
p_{23}^{2}p_{123}^{2}}\,.$ (6.9)

Then the single-cut factorisation on $v{\cdot}p_{1}=0$ is

$\displaystyle\langle\text{Res}_{v{\cdot}p_{1}}\operatorname{\mathcal{C}}\Delta^{\prime}\widehat{A}(123)\rangle={v{\cdot}\varepsilon_{1}p_{1}{\cdot}p_{2}\mathcal{N}([2,3],v)\over
p_{12}^{2}p_{123}^{2}}+{v{\cdot}\varepsilon_{1}p_{1}{\cdot}p_{23}\mathcal{N}([2,3],v)\over
p_{23}^{2}p_{123}^{2}}={1\over 2}v{\cdot}\varepsilon_{1}A(23,v)\,,$ (6.10)

which is expected from the factorisation behaviour on the heavy-mass
propagator. On the other hand, the colour-ordered amplitude has no pole in
$v{\cdot}p_{13}$. Hence, the single cut on $v{\cdot}p_{2}=0$ vanishes,

$\displaystyle\langle\text{Res}_{v{\cdot}p_{2}}\operatorname{\mathcal{C}}\Delta^{\prime}\widehat{A}(123)\rangle={v{\cdot}\varepsilon_{2}p_{1}{\cdot}p_{2}\mathcal{N}([1,3],v)\over
p_{12}^{2}p_{123}^{2}}+{v{\cdot}\varepsilon_{2}(p_{1}{\cdot}p_{2}-p_{13}{\cdot}p_{2})\mathcal{N}([1,3],v)\over
p_{23}^{2}p_{123}^{2}}=0\,.$ (6.11)

The double cut can be calculated similarly; we have only one independent
double cut due to the missing pole at $v{\cdot}p_{2}\\!=\\!0$. This agrees
with the fact that the factorisation of an amplitude does not depend on the
ordering in which we take the cuts, in contrast to the pre-numerator discussed
in Section 4.2.

###  6.2 Six-point example

We now consider the six-point case, and mainly focus on the left-nested
commutator $[1,2,3,4]$. Other commutators in the DDM basis are directly
obtained by relabelling the indices of the gluons while the algebraic
numerators beyond the DDM basis are then derived from the Jacobi relations.

The BCJ numerator for the left-nested commutator $[1,2,3,4]$ has the following
coproduct structure,

$\displaystyle\begin{split}\operatorname{\mathcal{C}}\Delta^{\prime}\widehat{\mathcal{N}}([1,2,3,4])&=\frac{2\widehat{\mathcal{N}}(13)\widehat{\mathcal{N}}(24)p_{1}{\cdot}p_{2}}{v{\cdot}p_{13}}-\frac{\widehat{\mathcal{N}}(31)\widehat{\mathcal{N}}(24)p_{1}{\cdot}p_{2}}{v{\cdot}p_{13}}-\frac{\widehat{\mathcal{N}}(12)\widehat{\mathcal{N}}(43)p_{1}{\cdot}p_{3}}{v{\cdot}p_{12}}\\\
&+\frac{\widehat{\mathcal{N}}(12)\widehat{\mathcal{N}}(34)\left(2p_{1}{\cdot}p_{3}+p_{2}{\cdot}p_{3}\right)}{v{\cdot}p_{12}}-\frac{\widehat{\mathcal{N}}(13)\widehat{\mathcal{N}}(42)p_{1}{\cdot}p_{2}}{v{\cdot}p_{13}}\\\
&+{3\over
4}\Big{(}\frac{\widehat{\mathcal{N}}(4)\widehat{\mathcal{N}}(123)\left(p_{23}{\cdot}p_{4}+2p_{1}{\cdot}p_{4}\right)}{v{\cdot}p_{123}}+\frac{\widehat{\mathcal{N}}(3)\widehat{\mathcal{N}}(412)p_{12}{\cdot}p_{3}}{v{\cdot}p_{3}}\\\
&+\frac{\widehat{\mathcal{N}}(1)(2\widehat{\mathcal{N}}(234)+\widehat{\mathcal{N}}(432)-\widehat{\mathcal{N}}(324)-\widehat{\mathcal{N}}(423))p_{1}{\cdot}p_{2}}{v{\cdot}p_{1}}\\\
&-\frac{\widehat{\mathcal{N}}(3)\widehat{\mathcal{N}}(124)\left(2p_{1}{\cdot}p_{3}+p_{2}{\cdot}p_{3}\right)}{v{\cdot}p_{3}}-\frac{\widehat{\mathcal{N}}(4)\widehat{\mathcal{N}}(312)\left(p_{12}{\cdot}p_{4}+2p_{3}{\cdot}p_{4}\right)}{v{\cdot}p_{123}}\Big{)}\\\
&-(1\leftrightarrow 2)\,.\end{split}$ (6.12)

After mapping to kinematics we can find all the factorisation channels from
$\langle\text{Res}_{v{\cdot}P}\operatorname{\mathcal{C}}\Delta\widehat{\mathcal{N}}([1,2,3,4])\rangle$,
as tabulated below:

$\displaystyle\begin{array}[]{|c||c||c|}\hline\cr
v{\cdot}p_{1}&v{\cdot}p_{134}&v{\cdot}p_{124}\\\
\hline\cr\hline\cr\mathcal{N}(1,v)\mathcal{N}([234],v)p_{1}{\cdot}p_{2}&\mathcal{N}(2,v)\mathcal{N}([134],v)p_{1}{\cdot}p_{2}&\mathcal{N}(3,v)\mathcal{N}([124],v)p_{12}{\cdot}p_{3}\\\
\hline\cr v{\cdot}p_{14}&v{\cdot}p_{13}&v{\cdot}p_{12}\\\
\hline\cr\hline\cr\mathcal{N}([14],v)\mathcal{N}([23],v)p_{1}{\cdot}p_{2}&\mathcal{N}([13],v)\mathcal{N}([24],v)p_{1}{\cdot}p_{2}&\mathcal{N}([12],v)\mathcal{N}([34],v)p_{12}{\cdot}p_{3}\\\
\hline\cr&v{\cdot}p_{123}&\\\
\hline\cr\hline\cr&\mathcal{N}(4,v)\mathcal{N}([123],v)p_{123}{\cdot}p_{4}&\\\
\hline\cr\end{array}$ (6.19)

For the colour-ordered amplitude, we have

$\displaystyle\begin{split}\widehat{A}(1234,v)&={\widehat{\mathcal{N}}([1,2,3,4],v)\over
p_{12}^{2}p_{123}^{2}p_{1234}^{2}}+{\widehat{\mathcal{N}}([[1,2],[3,4]],v)\over
p_{12}^{2}p_{34}^{2}p_{1234}^{2}}-{\widehat{\mathcal{N}}([2,3,4,1],v)\over
p_{23}^{2}p_{234}^{2}p_{1234}^{2}}\\\
&+{\widehat{\mathcal{N}}([3,4,2,1],v)\over
p_{34}^{2}p_{234}^{2}p_{1234}^{2}}-{\widehat{\mathcal{N}}([2,3,1,4],v)\over
p_{23}^{2}p_{123}^{2}p_{1234}^{2}}\,.\end{split}$ (6.20)

The factorisation behaviour of the amplitude can also be obtained from the
coproduct by using the factorisation behaviour of the BCJ numerators.
Considering the pole at $v{\cdot}p_{1}=0$ which is present in the colour-
ordered amplitude, we have

$\displaystyle\begin{split}\langle\text{Res}_{v{\cdot}p_{1}}\operatorname{\mathcal{C}}\Delta\widehat{A}(1234,v)\rangle&=\Big{(}{p_{1}{\cdot}p_{2}\over
p_{12}^{2}p_{123}^{2}p_{1234}^{2}}+{p_{1}{\cdot}p_{234}\over
p_{23}^{2}p_{234}^{2}p_{1234}^{2}}+{p_{1}{\cdot}p_{23}\over
p_{23}^{2}p_{123}^{2}p_{1234}^{2}}\Big{)}\mathcal{N}(1,v)\mathcal{N}([234],v)\\\
&+\Big{(}{p_{1}{\cdot}p_{2}\over
p_{12}^{2}p_{34}^{2}p_{1234}^{2}}+{p_{1}{\cdot}p_{234}\over
p_{34}^{2}p_{234}^{2}p_{1234}^{2}}\Big{)}\mathcal{N}(1,v)\mathcal{N}([2,[3,4]]\\\