compat=1.1.0 graviton/.style=circle, draw=green!60, fill=green!5, very thick,
minimum size=7mm codot/.style=/tikz/shape=circle, /tikz/fill=red,/tikz/minimum
size=0.1cm,/tikz/inner sep=1.8pt sb/.style= /tikz/shape=circle,
/tikz/fill=black, /tikz/minimum size=0.3cm,/tikz/inner sep=1.pt myblob/.style=
/tikz/shape=rectangle, /tikz/fill=red, /tikz/minimum size=0.2cm,

#  Kinematic Hopf Algebra for BCJ Numerators in
Heavy-Mass Effective Field Theory and Yang–Mills Theory

Andreas Brandhuber1<EMAIL_ADDRESS>Gang Chen1<EMAIL_ADDRESS>Henrik Johansson2,3<EMAIL_ADDRESS>Gabriele Travaglini1
<EMAIL_ADDRESS>Congkao Wen1<EMAIL_ADDRESS>$\mbox{}^{1}$Centre for
Theoretical Physics, Department of Physics and Astronomy, Queen Mary
University of London,
Mile End Road, London E1 4NS, United Kingdom  $\mbox{}^{2}$Department of
Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
$\mbox{}^{3}$Nordita, Stockholm University and KTH Royal Institute of
Technology, Hannes Alfvéns väg 12, 10691 Stockholm, Sweden

###### Abstract

We present a closed formula for all Bern-Carrasco-Johansson (BCJ) numerators
describing $D$-dimensional tree-level scattering amplitudes in a heavy-mass
effective field theory with two massive particles and an arbitrary number of
gluons. The corresponding gravitational amplitudes obtained via the double
copy directly enter the computation of black-hole scattering and
gravitational-wave emission. Our construction is based on finding a kinematic
algebra for the numerators, which we relate to a quasi-shuffle Hopf algebra.
The BCJ numerators thus obtained have a compact form and intriguing features:
gauge invariance is manifest, locality is respected for massless exchange, and
they contain poles corresponding to massive exchange. Counting the number of
terms in a BCJ numerator for $n{-}2$ gluons gives the Fubini numbers
$\mathsf{F}_{n-3}$, reflecting the underlying quasi-shuffle Hopf algebra
structure. Finally, by considering an appropriate factorisation limit, the
massive particles decouple, and we thus obtain a kinematic algebra and all
tree-level BCJ numerators for $D$-dimensional pure Yang-Mills theory.

Scattering amplitudes, heavy-mass effective theory, Hopf algebra, quasi-
shuffle product

††preprint:  NORDITA 2021-091, QMUL-PH-21-45, SAGEX-21-34, UUITP-60/21

##  I Introduction

Quantum field theory holds many surprising discoveries, one of which is the
Bern-Carrasco-Johansson (BCJ) duality between colour and kinematics Bern  _et
al._ (2008, 2010). In addition to providing a field-theory underpinning of the
Kawai-Lewellen-Tye (KLT) open-closed string relations Kawai  _et al._ (1986),
the duality hints at a hidden algebraic structure in a variety of gauge
theories. Scattering amplitudes in these theories can be written as a sum of
cubic diagrams, each one expressed as the product of a colour and a kinematic
factor. The colour factors satisfy Jacobi relations inherited from the gauge-
group Lie algebra, and the kinematic numerators satisfy corresponding
kinematic Jacobi relations Bern  _et al._ (2008). Through the double-copy
construction, gravitational amplitudes can be obtained from the kinematic
numerators.

A central question is to identify the hidden algebra behind the kinematic
relations. In this Letter we provide an explicit construction, in two related
contexts. First we will study the amplitudes in an effective theory of heavy
particles coupled to gluons, or gravitons Georgi (1990); Luke and Manohar
(1992); Neubert (1994); Manohar and Wise (2000); Damgaard  _et al._ (2019).
These theories, which we will refer to as HEFT (heavy-mass effective field
theory), are obtained from a Yang-Mills (YM) theory, or general relativity, by
restricting to the leading-order term in an inverse mass expansion. This is an
appropriate approximation for the dynamics of particles with momentum exchange
much smaller than their masses. Astrophysical black-hole scattering in general
relativity satisfies this, and the relevant gravitational amplitudes were
recently studied through a gauge-invariant double copy Brandhuber  _et al._
(2021a). The underlying gauge-theory factors are the central objects, and we
will here unravel their algebraic structure, including that of pure YM theory
after factorising out the heavy particles.

The understanding of the kinematic algebra has so far only progressed in small
steps. The first successful construction of the algebra was limited to the
self-dual sector of YM theory Monteiro and O’Connell (2011). In that case the
algebra corresponds to area-preserving diffeomorphisms, and explicit
representations of the generators were found. Self-dual YM is far from a
complete theory, having vanishing tree amplitudes and a non-CPT invariant
spectrum, yet it is the first confirmation of BCJ duality with explicit
generators and cubic Feynman rules. Another example of the duality was found
in the nonlinear sigma model Chen and Du (2014), as realised in Cheung and
Shen (2017) using a cubic Lagrangian. The corresponding kinematic algebra was
later Cheung  _et al._ (2018a) tied to that of higher-dimensional Poincaré
symmetry Cheung  _et al._ (2018b).

Efforts to identify the kinematic algebra have recently been renewed for YM
theory Chen  _et al._ (2019, 2021), and for HEFT Brandhuber  _et al._ (2021a).
The common idea is to realise the algebra with abstract vector and tensor
currents, multiplied through a fusion product. A consistent fusion product was
worked out for the maximally-helicity-violating (MHV) and next-to-MHV sectors
of YM theory Chen  _et al._ (2019, 2021). The approach was then applied to
HEFT in Brandhuber  _et al._ (2021a), giving explicit expressions for two
heavy particles coupled to gluons or gravitons, and the fusion products were
presented up to six particles. This approach is well-adapted for gravitational
physics, and was used to compute the black hole scattering angle in a post-
Minkowskian (PM) expansion at 3PM order Brandhuber  _et al._ (2021b) (see also
Bern  _et al._ (2019); Damour (2020); Kälin  _et al._ (2020); Di Vecchia  _et
al._ (2021); Herrmann  _et al._ (2021); Bjerrum-Bohr  _et al._ (2021); Bern
_et al._ (2021)).

In this Letter, we construct a kinematic algebra for HEFT, and by
factorisation, infer that the same algebra also works for pure YM theory. In
particular, we give a representation of all the generators, and all fusion
products needed for computing tree-level HEFT amplitudes with two heavy
particles and an arbitrary number of gluons/gravitons. Interestingly, the
obtained fusion product has the same structure as the quasi-shuffle product,
known from the mathematical literature, specifically in the context of
combinatorial Hopf algebras of shuffles and quasi-shuffles Hoffman (2000);
Aguiar and Mahajan (2010); Fauvet  _et al._ (2017). The quasi-shuffle Hopf
algebra generates all ordered partitions for a given set Hoffman (2000) (often
called $\mathbb{S}\mathbb{C}$ – the linear species of set compositions, or
ordered partitions). Mapping the generators to gauge-invariant expressions, we
obtain a closed formula for all tree-level BCJ numerators relevant to the
HEFT. The numerators are gauge invariant, manifestly crossing symmetric and
factorise into lower-point numerators on the massive poles. The underlying
quasi-shuffle Hopf algebra implies that the counting of the number of terms in
a numerator with $n{-}2$ gluons gives the Fubini numbers $\mathsf{F}_{n-3}$,
which counts the number of ordered partitions of $n{-}3$ elements.

Finally, all the considerations in HEFT directly translate to pure YM theory.
The pure-gluon BCJ numerators, and the corresponding expressions for the
generators, are obtained from the natural on-shell factorisation limit
Brandhuber  _et al._ (2021a), which removes the two heavy particles and
replaces them with an additional gluon (with label $n{-}1$). This is
straightforward: replace the heavy-particle velocity $v$ with the last
polarisation vector, $v\rightarrow\epsilon_{n-1}$, and impose the last on-
shell condition $p^{2}_{1\ldots n{-}2}\rightarrow 0$. This operation does not
modify the generator fusion rules, and hence YM theory admits the same
kinematic algebra. The heavy-mass poles become spurious in this limit, and
cancel out once the amplitude is assembled.

##  II The HEFT kinematic algebra

A novel colour-kinematic duality and double copy for HEFT was obtained in
Brandhuber  _et al._ (2021a), by four of the present authors. Ignoring
couplings, the YM and gravity tree amplitudes with two heavy particles and
$n{-}2$ gluons/gravitons are

$\displaystyle\begin{split}A(12\ldots
n{-}2,v)&\,=\,\sum_{\Gamma\in\rho}{\mathcal{N}(\Gamma,v)\over
d_{\Gamma}}\,,\\\ M(12\ldots
n{-}2,v)&\,=\,\sum_{\Gamma\in\tilde{\rho}}{\big{[}\mathcal{N}(\Gamma,v)\big{]}^{2}\over
d_{\Gamma}}\,,\end{split}$ (1)

where $\rho$ ($\tilde{\rho}$) denotes all (un)ordered nested commutators of
the particle labels $\\{1,\ldots\,,n{-}2\\}$, where the leftmost label is
fixed to 1. The ordering is important since here we work with colour-ordered
YM amplitudes. Considering the set $\\{1,2,3\\}$, we have
$\rho=\\{[[1,2],3],[1,[2,3]]\\}$ and
$\tilde{\rho}=\\{[[1,2],3],[[1,3],2],[1,[2,3]]\\}$. In general, labels
$1,\ldots,n{-}2$ are reserved for the gluons/gravitons and the heavy particles
are assigned $n{-}1$ and $n$, and $v$ is the velocity that characterises the
heavy particles.

The nested commutators are in one-to-one correspondence with cubic graphs (and
hence BCJ numerators), and the corresponding massless scalar-like propagator
denominators are denoted as $d_{\Gamma}$. For instance, the nested commutator
$[[1,2],3]$ corresponds to the following cubic graph, associated BCJ
numerator, and propagator denominator:

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\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=0.13cm 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}}\\!\\!\\!\\!\leftrightarrow\,\mathcal{N}({[[1,2],3]},v)\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
d_{[[1,2],3]}=p^{2}_{12}p^{2}_{123}\,,$ (2)

where $p_{i_{1}\ldots i_{r}}:=p_{i_{1}}+\cdots+p_{i_{r}}$ and the red square
denotes the heavy-particle source.

The BCJ numerator $\mathcal{N}(\Gamma,v)$ is a function of a nested set of
labels $\Gamma$, and it has an expansion which parallels that of the
commutator, e.g.

$\displaystyle\mathcal{N}([1,[2,3]],v)$
$\displaystyle=\,\mathcal{N}(123,v)-\mathcal{N}(132,v)$
$\displaystyle\,\,-\mathcal{N}(231,v)+\mathcal{N}(321,v)\ ,$ (3)

and we refer to the object $\mathcal{N}(1\ldots n{-}2,v)$ as the pre-
numerator. In analogy with a Lie algebra, this quantity should be obtained by
multiplying generators through an associative fusion product. Thanks to the
nested commutator structure, the BCJ numerators will automatically satisfy
kinematic Jacobi identities.

Explicit pre-numerators can be obtained from the constraint imposed by
requiring that they lead to correct amplitudes, and in Brandhuber  _et al._
(2021a) this was done up to six points. In the following, it will be crucial
to find representations of the pre-numerators where any non-locality will
correspond to a massive physical pole $\sim{1\over v\cdot P}$, where $P$ is a
sum of gluon momenta 111Note that $\sum_{i=1}^{n-2}v{\cdot}p_{i}=0$. . This
linearised propagator arises because of the large-mass expansion. Our results
will be an improvement compared to Brandhuber  _et al._ (2021a), since in that
work additional spurious poles were present in the pre-numerators. We find the
following explicit new results up to five points:

$\displaystyle\mathcal{N}(1,v)$ $\displaystyle={v{\cdot}\varepsilon_{1}}\,,$
$\displaystyle\mathcal{N}(12,v)$
$\displaystyle=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}v}{2v{\cdot}p_{1}}\,,$
$\displaystyle\mathcal{N}(123,v)$
$\displaystyle=-\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}V_{12}{\cdot}F_{3}{\cdot}v}{3v{\cdot}p_{1}v{\cdot}p_{12}}-\frac{v{\cdot}F_{1}{\cdot}F_{3}{\cdot}V_{1}{\cdot}F_{2}{\cdot}v}{3v{\cdot}p_{1}v{\cdot}p_{13}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\,+\frac{v{\cdot}F_{1}{\cdot}F_{2}{\cdot}F_{3}{\cdot}v}{3v{\cdot}p_{1}}\,,$
(4)

where
$F^{\mu\nu}_{i}\\!\\!:=p^{\mu}_{i}\varepsilon^{\nu}_{i}{-}\varepsilon^{\mu}_{i}p^{\nu}_{i}$,
and
$V_{\tau}^{\mu\nu}\\!\\!:=v^{\mu}\sum_{j\in\tau}p_{j}^{\nu}=v^{\mu}p^{\nu}_{\tau}$.

Following Chen  _et al._ (2019, 2021); Brandhuber  _et al._ (2021a), the pre-
numerators are presumed to be constructible in an algebraic fashion, by
multiplying abstract generators of the kinematic algebra via a fusion product,

$\displaystyle\mathcal{N}(12\ldots n{-}2,v):=\langle T_{(1)}\star
T_{(2)}\star\cdots\star T_{(n{-}2)}\rangle\,,$ (5)

where the $T_{(i)}$s are generators carrying the gluon label $i$, and $\star$
denotes the bilinear and associative fusion product 222The $\star$-product is
bilinear if $(aT_{\sigma}+bT_{\gamma})\star T_{\rho}=aT_{\sigma}\star
T_{\rho}+bT_{\gamma}\star T_{\rho}$,
$T_{\rho}\star(aT_{\sigma}+bT_{\gamma})=aT_{\rho}\star
T_{\sigma}+bT_{\rho}\star T_{\gamma}$, where $a$, $b$ are numbers.. The angle
bracket represents a linear map from the abstract generators to gauge- and
Lorentz-invariant functions. It preserves the multi-linearity with respect to
the polarisation vectors and the linear scaling in the velocity $v$ of the
heavy particles.

The starting point of the construction is $\langle
T_{(i)}\rangle=v{\cdot}\varepsilon_{i}$, which is the unique choice that
respects all the properties listed above, and furthermore generates the
correct three-point amplitude. We can then combine two generators to obtain

$\displaystyle\mathcal{N}(12,v):=\langle T_{(1)}\star T_{(2)}\rangle=-\langle
T_{(12)}\rangle\,,$ (6)

where we choose $\langle
T_{(12)}\rangle={v{\cdot}F_{1}{\cdot}F_{2}{\cdot}v\over 2v{\cdot}p_{1}}$ to
reproduce Eq. (II) 333Combining two three-point numerators together to form
the four-point one is a natural operation, indeed $\langle T_{(12)}\rangle$
factorises on the massive pole as $\langle
T_{(12)}\rangle\big{|}_{v{\cdot}p_{1}\rightarrow 0}=(-{p_{1}{\cdot}p_{2}\over
2v{\cdot}p_{1}})\langle T_{(1)}\rangle\langle T_{(2)}\rangle$.. Similarly, at
five points one finds

$\displaystyle T_{(12)}\star T_{(3)}$
$\displaystyle=-T_{(123)}+T_{(12),(3)}+T_{(13),(2)}\,,$ (7)

with

$\displaystyle\langle
T_{(123)}\rangle={v{\cdot}F_{1}{\cdot}F_{2}{\cdot}F_{3}{\cdot}v\over
3v{\cdot}p_{1}}\,,\,\,\,\langle
T_{(12),(3)}\rangle={v{\cdot}F_{1}{\cdot}F_{2}{\cdot}V_{12}{\cdot}F_{3}{\cdot}v\over
3v{\cdot}p_{1}v{\cdot}p_{12}}\,,$ $\displaystyle\langle
T_{(13),(2)}\rangle={v{\cdot}F_{1}{\cdot}F_{3}{\cdot}V_{1}{\cdot}F_{2}{\cdot}v\over
3v{\cdot}p_{1}v{\cdot}p_{13}}\,\,.$ (8)

The particular index assignments in the obtained generators are consistent
with a general formula, which we find to work to any number of points,

$\displaystyle\langle
T_{(1\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})}\rangle:=\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[myblob]{}; \vertex[left=0.8cm of a] (a0)[myblob]{};
\vertex[right=0.8cm of a] (a2)[myblob]{}; \vertex[right=0.8cm of a2]
(a3)[myblob]{}; \vertex[right=0.8cm of a3] (a4)[myblob]{}; \vertex[above=0.8cm
of a] (b1){$\tau_{1}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $}; \vertex[above=0.8cm of a2]
(b2){$\tau_{2}$}; \vertex[above=0.8cm of a3] (b3){$\cdots$};
\vertex[above=0.8cm of a4] (b4){$\tau_{r}$}; \vertex[above=0.8cm of a0]
(b0){$1\leavevmode\nobreak\ \leavevmode\nobreak\ $}; \vertex[above=0.8cm of
a0](j1){$$}; \vertex[right=0.2cm of j1](j2){$$}; \vertex[right=0.6cm of
j2](j3){$$}; \vertex[right=0.4cm of j3](j4){$$}; \vertex[right=0.8cm of
j4](j5){$$}; \vertex[right=0.2cm of j5](j6){$$}; \vertex[right=0.6cm of
j6](j7){$$}; \vertex[right=0.0cm of j7](j8){$$}; \vertex[right=0.8cm of
j8](j9){$$}; \diagram*{(a)--[very thick](a0),(a)--[very thick](a2),(a2)--[very
thick](a3), (a3)--[very thick](a4),(a0) -- [thick] (j1),(a) -- [thick]
(j2),(a)--[thick](j3),(a2) -- [thick] (j4),(a2)--[thick](j5),(a4) -- [thick]
(j8),(a4)--[thick](j9)}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle={v{\cdot}F_{1\tau_{1}}{\cdot}V_{\Theta(\tau_{2})}{\cdot}F_{\tau_{2}}\cdots
V_{\Theta(\tau_{r})}{\cdot}F_{\tau_{r}}{\cdot}v\over(n{-}2)v{\cdot}p_{1}v{\cdot}p_{1\tau_{1}}\cdots
v{\cdot}p_{1\tau_{1}\tau_{2}\cdots\tau_{r-1}}}\,.$ (9)

The $\tau_{i}$s are ordered non-empty sets such that
$\tau_{1}\cup\tau_{2}\cup\cdots\cup\tau_{r}=\\{2,3,\ldots,n{-}2\\}$ and
$\tau_{i}\cap\tau_{j}=\emptyset$, i.e. they constitute a partition. The set
$\Theta(\tau_{i})$ consists of all indices to the left of $\tau_{i}$ and
smaller than the first index in $\tau_{i}$; that is
$\Theta(\tau_{i})=(\\{1\\}\cup\tau_{1}\cup\cdots\cup\tau_{i-1})\cap\\{1,\ldots,\tau_{i[1]}\\}$.
Note that the denominators in Eq. (II) are the advertised massive propagators.
For convenience, we also define $F_{\tau_{i}}$ as the ordered contraction of
several linearised field strengths $F^{\mu\nu}_{j}$ with indices in
$\tau_{i}$, e.g. $F^{\mu\nu}_{12}=F^{\mu\alpha}_{1}F_{2\alpha}^{\ \nu}$.

To clarify the formula, consider a non-trivial example,
$T_{(1458),(26),(37)}$, that is mapped to

$\displaystyle\langle
T_{(1458),(26),(37)}\rangle={v{\cdot}F_{1458}{\cdot}V_{1}{\cdot}F_{26}{\cdot}V_{12}{\cdot}F_{37}{\cdot}v\over
v{\cdot}p_{1}\,v{\cdot}p_{1458}\,v{\cdot}p_{124568}}\ .$ (10)

We may further clarify the $\Theta(\tau_{i})$s by drawing a “musical diagram”,
where the gluon labels (notes) are filled in progressively from left to right
and each horizontal line indicates which set in the partition they belong to:
A given $\Theta(\tau_{i})$ is associated with the first gluon on the
horizontal line $\tau_{i}$, and the set includes all labels “southwest” of
this gluon. Specifically, in this example, the relevant sets used in Eq. (10)
are $\Theta(26)=\\{1\\}$, and $\Theta(37)=\\{1,2\\}$. Furthermore, the
contraction of field strengths can be read out by following each horizontal
$\tau_{i}$-line in this musical diagram. A horizontal line can be thought of
as the fundamental representation of the Lorentz group, and the linearised
field strengths as Lorentz generators acting in this space.

Let us return to the algebra of the abstract generators. The pre-numerators
can be recursively constructed from only knowing the following fusion product:

$\displaystyle T_{(1\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})}\star T_{(j)}.$
(12)

We assume that the possible outcome of this fusion product maintains the
relative order of the labels in the left and right generator. Then by assuming
we have a complete set of generators, we can only produce the terms

$\displaystyle T_{(1j),(\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})},\quad
T_{(1\tau_{1}),(\tau_{2}),\ldots,(\tau_{i}),(j),(\tau_{i+1}),\ldots,(\tau_{r})},$
$\displaystyle
T_{(1\tau_{1}),(\tau_{2}),\ldots,(\tau_{i}j),\ldots,(\tau_{r})},\quad\text{where}\quad
i\in\\{1,\cdots,r\\}.$ (13)

By writing up a general ansatz, and fixing the free coefficients by comparing
to the correct amplitudes via the map in Eq. (II), we find a simple all-
multiplicity solution. The fusion product is captured by the general formula

$\displaystyle T_{(1\tau_{1}),\ldots,(\tau_{r})}\star
T_{(j)}=\hskip-14.22636pt\sum_{\sigma\in\\{(\tau_{1}),\ldots,(\tau_{r})\\}\shuffle\\{(j)\\}}T_{(1\sigma_{1}),\ldots,(\sigma_{r+1})}$
$\displaystyle\hskip
56.9055pt-\sum_{i=1}^{r}T_{(1\tau_{1}),\ldots,(\tau_{i-1}),(\tau_{i}j),(\tau_{i+1}),\ldots,(\tau_{r})}\,,$
(14)

where $\shuffle$ denotes the shuffle product between two sets, e.g.
$\\{A,B\\}\shuffle\\{C\\}=\\{ABC,ACB,CAB\\}.$ A proof for Eq. (II) will be
given in the next section; here we will study examples. For $n=4,5$, Eqs. (6)
and (7) are recovered, and at six points, the fusion products are

$\displaystyle T_{(123)}\star T_{(4)}$
$\displaystyle=-T_{(1234)}+T_{(123),(4)}+T_{(14),(23)}$ $\displaystyle
T_{(12),(3)}\star T_{(4)}$ $\displaystyle=-T_{(12),(34)}-T_{(124),(3)}$
$\displaystyle+T_{(12),(3),(4)}+T_{(12),(4),(3)}+T_{(14),(2),(3)}$
$\displaystyle T_{(13),(2)}\star T_{(4)}$
$\displaystyle=-T_{(13),(24)}-T_{(134),(2)}+T_{(13),(2),(4)}$
$\displaystyle+T_{(13),(4),(2)}+T_{(14),(3),(2)}\,,$ (15)

leading to the six-point pre-numerator

$\displaystyle\mathcal{N}(1234,v)$ $\displaystyle=\langle-
T_{{(12)},{(3)},{(4)}}-T_{{(12)},{(4)},{(3)}}-T_{{(14)},{(2)},{(3)}}$
$\displaystyle-
T_{{(14)},{(3)},{(2)}}-T_{{(13)},{(2)},{(4)}}-T_{{(13)},{(4)},{(2)}}$
$\displaystyle+T_{{(123)},{(4)}}+T_{{(124)},{(3)}}+T_{{(134)},{(2)}}$
$\displaystyle+T_{{(12)},{(34)}}+T_{{(13)},{(24)}}+T_{{(14)},{(23)}}-T_{{(1234)}}\rangle\,.$
(16)

As already advertised, the algebra defined by the fusion product in Eq. (II)
is known in the context of combinatorial Hopf algebras of shuffles and quasi-
shuffles Hoffman (2000); Aguiar and Mahajan (2010); Fauvet  _et al._ (2017).
Specifically, our fusion product defines a quasi-shuffle Hopf algebra that
generates all ordered partitions for a given set Hoffman (2000). Indeed, the
subscripts of the $T$s are precisely all possible ordered partitions of
$\\{2,3,\ldots,n{-}2\\}$. This is also interpreted in Fauvet  _et al._ (2017)
as a Hopf monoid in the category of coalgebra species. These Hopf algebras are
endowed with a product that is commutative and associative Hoffman (2000);
Cartier (2002); Ihara  _et al._ (2006), with a coproduct, counit and antipode
Hoffman (2000) (see the Appendix for more details).

We have thus found a realisation of the kinematic algebra for HEFT by mapping
it to a quasi-shuffle Hopf algebra. Note that the associativity of the fusion
product is a natural property – for example, we can construct a BCJ numerator
either as $((T_{(1)}\star T_{(2)})\star T_{(3)})\star\cdots$ or
$\cdots\star(T_{(n-4)}\star(T_{(n{-}3)}\star T_{(n{-}2)}))$. To complete the
story, we must also give the fusion product for the most general generators.
Assuming the fusion product is associative and preserves the relative order
for the left and right generators, one obtains a unique result Aguiar and
Mahajan (2010),

$\displaystyle T_{(1\tau_{1}),\cdots,(\tau_{r})}\star
T_{(\omega_{1}),\cdots,(\omega_{s})}=$
$\displaystyle\sum_{\begin{subarray}{c}\sigma|_{\\{\tau\\}}=\\{(\tau_{1}),\cdots,(\tau_{r})\\}\\\
\sigma|_{\\{\omega\\}}=\\{(\omega_{1}),\cdots,(\omega_{s})\\}\end{subarray}}(-1)^{t-r-s}T_{(1\sigma_{1}),(\sigma_{2}),\cdots,(\sigma_{t})}\,,$
(17)

where $\tau_{i}$ and $\omega_{j}$ do not contain the label $1$, as this index
is always fixed to be the leftmost index of any expression, and thus it is
inert to the algebra. The fusion product of two generators, neither containing
label 1, is also given by Eq. (II) after dropping the 1. We use $\\{\tau\\}$
and $\\{\omega\\}$ to denote the total set of labels in the $\tau_{i}$ and
$\omega_{i}$, respectively. By $\sigma|_{\\{\tau\\}}$ we mean a restriction to
the elements in $\\{\tau\\}$, e.g.
$\\{(235),(4),(67)\\}|_{\\{2,3,4\\}}=\\{(23),(4)\\}$.

The number of ordered partitions of $\\{2,3,\ldots,n{-}2\\}$ are known as the
Fubini numbers 444The $n^{\rm th}$ Fubini number, also known as ordered Bell
number, also gives the number of possible outcomes of a horse race with $n$
horses, including dead heats.  Mező (2019)

$\displaystyle\mathsf{F}_{n{-}3}=\sum_{r=1}^{n{-}3}r!\ {n{-}3\brace r}\,,$
(18)

which therefore also counts the number of terms in the pre-numerator of an
$n$-point HEFT amplitude 555For instance, the pre-numerators
$\mathcal{N}(1,v)$, $\mathcal{N}(12,v)$, $\mathcal{N}(123,v)$ and
$\mathcal{N}(1234,v)$ contain 1, 1, 3 and 13 terms, respectively.. Here
${n\brace k}$ denotes the number of $k$-partitions on $n$ objects (also known
as Stirling partition number of the second kind). The Fubini numbers also give
the Hilbert series of $\mathbb{S}\mathbb{C}$ Fauvet  _et al._ (2017).

From the kinematic algebra, the closed form of the pre-numerator is directly
obtained as

$\displaystyle\mathcal{N}(1\ldots
n{-}2,v)=\sum_{r=1}^{n{-}3}\sum_{\tau\in\mathbf{P}_{\\{2,\cdots,n{-}2\\}}^{(r)}}({-}1)^{n+r}\langle
T_{(1\tau_{1}),\ldots,(\tau_{r})}\rangle\,,$ (19)

where $T_{(1\tau_{1}),\ldots,(\tau_{r})}$ is defined in Eq. (II) and
$\mathbf{P}_{\\{2,\cdots,n{-}2\\}}^{(r)}$ denotes all the ordered partitions
of $\\{2,3,\ldots,n{-}2\\}$ into $r$ subsets. This closed expression
automatically induces a recursion relation for the pre-numerator:

$\displaystyle\mathcal{N}(12\ldots n{-}2,v)=\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[myblob]{}; \vertex[right=0.8cm of a](v2)[myblob]{};
\vertex[above=0.9cm of v2](jr1){$\leavevmode\nobreak\ \leavevmode\nobreak\
\cdots n{-}2$}; \vertex[left=0.5cm of jr1](jr2){$2$}; \vertex[above=0.9cm of
a](j1){$1$}; \diagram*{(a) -- [very thick] (v2),(v2) -- [thick] (jr1),(v2) --
[thick] (jr2),(a) -- [thick] (j1)}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\sum_{\tau_{R}}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[myblob]{}; \vertex[right=0.9cm of a](v2)[myblob]{};
\vertex[above=0.9cm of v2](jr1){$\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $}; \vertex[right=0.3cm of
jr1](jr2){\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{ }$0$}; \vertex[left=0.3cm of
jr1](jr3){$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \tau_{R}$};
\vertex[above=0.3cm of a](b)[dot]{}; \vertex[above=0.63cm of
b](d){$1\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \tau_{L}$}; \vertex[left=0.6cm of b](c);
\vertex[left=0.22cm of b](c23); \vertex[above=0.13cm of c23](v23)[dot]{};
\vertex[above=.4cm of c](j1){$\leavevmode\nobreak\ $}; \vertex[right=0.7cm of
j1](j2){$\leavevmode\nobreak\ \leavevmode\nobreak\ $}; \vertex[right=0.5cm of
j2](j3){$\leavevmode\nobreak\ \leavevmode\nobreak\ $}; \diagram*{(a) --
[thick] (b),(a) -- [very thick] (v2),(v2) -- [thick] (jr3),(v2) -- [thick]
(jr2),(b) -- [thick] (j1),(v23) -- [thick] (j2),(b)--[thick](j3)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=(-1)^{n}{v{\cdot}F_{12\ldots
n{-}2}{\cdot}v\over(n{-}2)v{\cdot}p_{1}}$ (20)
$\displaystyle-\sum_{\tau_{R}\subset\\{2,\cdots,n{-}2\\}}(-1)^{n_{R}}\mathcal{N}(1\tau_{L},v){(n{-}2{-}n_{R})H_{\Theta_{\tau_{R}},\tau_{R}}\over(n{-}2)v{\cdot}p_{1\tau_{L}}},$

where $\tau_{L}\cup\tau_{R}=\\{2,3,\cdots,n{-}2\\}$,
$\tau_{L},\tau_{R}\neq\emptyset$, and we have defined

$\displaystyle H_{\sigma,\tau}:=p_{\sigma}{\cdot}F_{\tau}{\cdot}v\,.$ (21)

Here $n_{R}$ denotes the number of indices in $\tau_{R}$. From Eq. (II), we
can see that the number of terms satisfies the recursion relation

$\displaystyle\mathsf{F}_{n{-}3}=\sum_{i=0}^{n-4}\binom{n{-}3}{i}\mathsf{F}_{i}\,,$
(22)

where $\mathsf{F}_{0}=1$. This is the well-known recursion relation for the
Fubini numbers Gross (1962). To illustrate the simplicity of the pre-
numerator, we quote the fairly modest number of terms up to ten points:

$n$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---
$\mathsf{F}_{n{-}3}$ |  1 |  1 |  3 |  13 |  75 |  541 |  4683 |  47293

In the next section we present a general proof of our construction of the BCJ
numerators.

##  III Proof of the form of the pre-numerator

We here give the proof of the BCJ numerators. Using
$\mathcal{N}([12{\ldots}n{-}2],v)\\!:=\\!\mathcal{N}([{\ldots}[[1,2],3]{\ldots},n{-}2],v)$,
we show that they give correct amplitudes, as obtained from the pre-numerator
in Eq. (19). In addition, we will show that the following simple relation
holds, valid in the HEFT:

$\displaystyle\mathcal{N}([12\ldots n{-}2],v)=(n{-}2)\,\mathcal{N}(12\ldots
n{-}2,v)\,.$ (23)

The outline of the proof is as follows. Starting from the factorisation
properties on massive poles of our HEFT numerators as derived from the KLT
formula Bjerrum-Bohr  _et al._ (2011), we prove that $\mathcal{N}([12\ldots
n{-}2],v)$ has the same factorisation. We will then consider the difference
between $\mathcal{N}([12\ldots n{-}2],v)$ and the correct BCJ numerator (as
derived from KLT), which is free of poles. Using arguments similar to those of
Arkani-Hamed  _et al._ (2018); Rodina (2019), we will then show that gauge
invariance ensures that this difference vanishes.

The starting point is the factorisation on massive poles of BCJ numerators in
HEFT. Using KLT relations, one can easily show that

(24) $\displaystyle\mathcal{N}_{\rm KLT}([1\ldots n{-}2],v)$
$\displaystyle{\overset{v{\cdot}p_{1\tau_{L}}}{\longrightarrow}}p_{{}_{\Theta(\tau_{R})}}{\cdot}p_{\tau_{R[1]}}\,\times$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathcal{N}_{\rm
KLT}([1\tau_{L}],v)\mathcal{N}_{\rm KLT}([\tau_{R}],v)\,,$

where $\tau_{L}\cup\tau_{R}=\\{2,3,\ldots,n{-}2\\}$ and $\tau_{R[1]}$ denote
the first index in $\tau_{R}$. The red cross denotes the cut on the physical
pole. The derivation of this formula is given in the Appendix.

The next step is to prove that the closed form Eq. (19) has the same
factorisation as Eq. (24), which we will now do inductively. The seed of the
induction is the factorisation where the right-hand side of Eq. (24) only
contains one gluon, that is, $n_{R}=1$, and we focus on the massive pole
$1\over v{\cdot}p_{1\tau_{L}}$. The factorisation is then immediately read off
from Eq. (II): only one term in the second diagram in that equation
contributes, with the residue given by

$\displaystyle\mathcal{N}([1i_{2}\ldots i_{n{-}3}],v)\,p_{1i_{2}\ldots
i_{n{-}3}}{\cdot}p_{i_{n{-}2}}\,\mathcal{N}(i_{n{-}2},v)\,.$ (25)

In the next step, we assume that factorisation for $n_{R}=j{-}1$ at the
massive pole $1\over v{\cdot}p_{1\tau_{L}}$ has the same form as Eq. (24), and
we then derive that for $n_{R}=j$. According to Eq. (II), in this channel the
residue of $(n{-}2)\,\mathcal{N}$ at the massive pole is

$\displaystyle\mathcal{N}([1\tau_{L}],v)\,\sum_{r=1}^{n_{R}}\sum_{\sigma\in\mathbf{P}_{\tau_{R}}^{(r)}}{H_{{}_{\Theta(\sigma_{1})},\sigma_{1}}\cdots
H_{{}_{\Theta(\sigma_{r})},\sigma_{r}}\over v{\cdot}p_{\sigma_{1}}\cdots
v{\cdot}p_{\sigma_{1}\sigma_{2}\cdots\sigma_{r-1}}}\,.$ (26)

As we show in the Appendix, in the limit $v{\cdot}p_{1\tau_{L}}\rightarrow 0$,
the sum in Eq. (26) becomes precisely a BCJ numerator,

$\displaystyle\sum_{r=1}^{n_{R}}\sum_{\sigma\in\mathbf{P}_{\tau_{R}}^{(r)}}{H_{{}_{\Theta(\sigma_{1})},\sigma_{1}}\cdots
H_{{}_{\Theta(\sigma_{r})},\sigma_{r}}\over v{\cdot}p_{\sigma_{1}}\cdots
v{\cdot}p_{\sigma_{1}\sigma_{2}\cdots\sigma_{r-1}}}=p_{{}_{\Theta(\tau_{R})}}{\cdot}p_{\tau_{R[1]}}\mathcal{N}([\tau_{R}],v)\,.$
(27)

This establishes the required factorisation property of Eq. (24) for the
proposed formula in Eq. (19).

Next, we consider the difference

$\displaystyle f=(n{-}2)\,\mathcal{N}(1\ldots n{-}2,v)-\mathcal{N}_{\rm
KLT}([1\ldots n{-}2],v).$ (28)

As the factorisation on the heavy-mass poles is the same, and both contain
only such poles, $f$ must be a polynomial. An adaptation of the argument of
Arkani-Hamed  _et al._ (2018); Rodina (2019) allows us to show that $f=0$. To
this end, we note that the velocity $v$ can appear in two possible ways.
First, through the combination $p{\cdot}v$ with $p$ being any of the momenta.
This multiplies a polynomial function of dimension $n{-}4$ built from $n{-}2$
gluon momenta and $n{-}2$ polarisations. As is well known, and pointed out
recently in Arkani-Hamed  _et al._ (2018); Rodina (2019), no such gauge-
invariant function exists and hence it must vanish. Second, $v$ can appear in
the combination $v{\cdot}F$, which now multiplies a function of dimensions
$n{-}4$, constructed from $n{-}2$ gluon momenta and $n{-}3$ polarisations. As
before, such a function must vanish. Hence we conclude that $f=0$, and
therefore

$\displaystyle(n{-}2)\,\mathcal{N}(1\ldots n{-}2,v)=\mathcal{N}_{\rm
KLT}([1\ldots n{-}2],v)\,.$ (29)

This completes the derivation of our BCJ numerator.

##  IV From HEFT to Yang-Mills

It is straightforward to obtain the kinematic algebra, and the BCJ numerators,
of pure YM theory from the HEFT construction, by exploiting the factorisation
property of the HEFT amplitude on a gluon pole Brandhuber  _et al._ (2021a).
The massive particles decouple on the pole, and we obtain the BCJ numerator
for YM amplitudes:

$\displaystyle\mathcal{N}^{\rm YM}([1\ldots n{-}1])$
$\displaystyle=\mathcal{N}([1\ldots
n{-}2],v)\big{|}^{v\rightarrow\epsilon_{n{-}1}}_{p_{1\ldots
n{-}2}^{2}\rightarrow 0}\,.$ (30)

The same replacement should be performed on the expressions of the generators
of the algebra given in Eq. (II), with no modification to the fusion rules We
have also explicitly verified Eq. (IV) for $D$-dimensional YM amplitudes up to
nine points.

The BCJ numerators thus obtained are manifestly gauge invariant and crossing
symmetric for all gluons except the last one, $n{-}1$. The price to pay is
that they also contain spurious poles of the form
$\frac{1}{\varepsilon_{n-1}{\cdot}P}$, which however can be eliminated in the
complete amplitudes – in practice, one can use only independent variables
(after imposing on-shell conditions and momentum conservation) in the
amplitudes, then terms with spurious poles should cancel out, or we can simply
drop them by hand.

As for the BCJ numerators, the spurious poles can also be eliminated
explicitly. We take the MHV sector of the five-point case as an example to
illustrate this idea. According to Eqs. (19) and (IV), the corresponding
numerator is

$\displaystyle\Big{(}v{\cdot}\varepsilon_{1}{p_{1}{\cdot}\varepsilon_{2}p_{12}{\cdot}\varepsilon_{3}p_{2}{\cdot}vv{\cdot}p_{3}\over
v{\cdot}p_{1}v{\cdot}p_{12}}+v{\cdot}\varepsilon_{1}{p_{1}{\cdot}\varepsilon_{3}p_{1}{\cdot}\varepsilon_{2}p_{3}{\cdot}vv{\cdot}p_{2}\over
v{\cdot}p_{1}v{\cdot}p_{13}}$
$\displaystyle-v{\cdot}\varepsilon_{1}{p_{1}{\cdot}\varepsilon_{2}p_{2}{\cdot}\varepsilon_{3}p_{3}{\cdot}v\over
v{\cdot}p_{1}}\Big{)}\Big{|}_{v\rightarrow\varepsilon_{4}}=\varepsilon_{1}{\cdot}\varepsilon_{4}p_{1}{\cdot}\varepsilon_{2}p_{12}{\cdot}\varepsilon_{3}\,,$
(31)

which is in agreement with Chen  _et al._ (2019).

Finally, we remark that our BCJ numerators for $n$-point pure YM amplitudes
have some similarity with those given in Cheung and Mangan (2021). Both
constructions maintain gauge invariance and crossing symmetry of $n{-}1$
external legs, and contain linear spurious poles. Intriguingly, we also
observe that the numerator of Cheung and Mangan (2021) has
$2\,\mathsf{F}_{n-3}$ terms, while ours has $\mathsf{F}_{n-3}$ terms. It would
be interesting to further unravel the connections between these two
constructions.

##  V Conclusion

In this Letter, we constructed a kinematic algebra that manifests BCJ colour-
kinematics duality in tree-level HEFT and YM theory, and showed that it can be
mapped to a quasi-shuffle Hopf algebra. It is intriguing to note that Hopf
algebras have already appeared in several different contexts in quantum field
theory and string theory, e.g. renormalisation theory Connes and Kreimer
(2000), symbols and co-actions of loop integrals, Goncharov  _et al._ (2010);
Duhr (2012); Abreu  _et al._ (2014); Brown (2017) and string
$\alpha^{\prime}$-expansion Broedel  _et al._ (2013). The obtained kinematic
algebra is very simple in terms of the abstract generators, and a non-trivial
aspect of the construction is the map between these generators and the
kinematic variables (momenta and polarisations), for which we find a simple
closed formula that exhibits manifest gauge invariance (see e.g. Bjerrum-Bohr
_et al._ (2011); Mafra  _et al._ (2011); Du and Teng (2017); Edison and Teng
(2020); Cheung and Mangan (2021) for other all-multiplicity BCJ
constructions).

Several questions remain open. First, it would be interesting to derive our
BCJ numerators from a Lagrangian description, which may expose hidden
symmetries/structures of the theory. The non-localities of the numerators are
both mild and physical in HEFT, thus a Lagrangian approach seems feasible. It
may also prove fruitful to try to find representations of the generators in
the form of differential operators in kinematic variables, thus re-introducing
kinematics in the fusion rules. On the mathematical side one may note that a
Hopf algebra should have a coproduct and counit: what do these operations
imply for the numerator and amplitude? The extension of our construction to
loop amplitudes, as well as other theories, is an important avenue. Finally,
it would be interesting to find a more direct construction of the pure YM
kinematic algebra, without passing through the HEFT. We leave these questions
for future investigation.

### Acknowledgements

We thank Kays Haddad, Sanjaye Ramgoolam, Bill Spence and Mao Zeng for useful
discussions, and Maor Ben-Shahar, Lucia Garozzo, Fei Teng and Tianheng Wang
for collaborations on related topics. This work was supported by the Science
and Technology Facilities Council (STFC) Consolidated Grants ST/P000754/1
“String theory, gauge theory & duality” and ST/T000686/1 “Amplitudes, strings
& duality” and by the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 764850 “SAGEX”.
HJ is supported by the Knut and Alice Wallenberg Foundation under grants KAW
2018.0116 (From Scattering Amplitudes to Gravitational Waves) and KAW
2018.0162, the Swedish Research Council under grant 621-2014-5722, and the
Ragnar Söderberg Foundation (Swedish Foundations’ Starting Grant). CW is
supported by a Royal Society University Research Fellowship No. UF160350.

##  Appendix A Appendix: Factorisation of BCJ numerators from KLT relations

In this section we derive the factorisation properties of the BCJ numerators
from KLT. We begin by observing that one can define a DDM Del Duca  _et al._
(2000) basis of amplitudes $A(1\beta,v)$ also for the HEFT theory, where
$\beta$ contains all permutations of the gluons $\\{2,\ldots,n{-}2\\}$. This
is possible because of a generalisation of the Kleiss-Kuijf (KK) relations
Kleiss and Kuijf (1989) which the HEFT amplitudes satisfy,

$\displaystyle\sum_{\sigma\in\beta_{L}\shuffle\beta_{R}}A(\sigma,v)=0\,,$ (32)

where $\beta_{L}\cup\beta_{R}=\\{1,2,\ldots,n{-}2\\}$ and
$\beta_{L},\beta_{R}\neq\emptyset$. One can then extend the results of Vaman
and Yao (2010), and write all colour-ordered HEFT amplitudes in terms of
numerators in the DDM basis:

$\displaystyle A(1\beta,v)=\sum_{\alpha\in
S_{n{-}3}}\mathsf{m}(1\beta|1\alpha)\,\mathcal{N}_{\rm KLT}([1\alpha],v)\,.$
(33)

Thanks to the off-shell condition $p_{12\ldots n{-}2}^{2}\neq 0$, the inverse
of $\mathsf{m}(1\beta|1\alpha)$ exists and is equal to the standard KLT matrix
$\mathcal{S}$ Bjerrum-Bohr  _et al._ (2010). We then get the KLT
representation of the BCJ numerator:

$\displaystyle\mathcal{N}_{\rm KLT}([1\alpha],v)=\sum_{\beta\in
S_{n{-}3}}\mathcal{S}(1\alpha|1\beta)A(1\beta,v)\,.$ (34)

Using the KK relation and the crossing symmetry of KLT matrix in HEFT, one can
show that $\mathcal{N}_{\rm KLT}$ is fully crossing symmetric, which leads to

$\displaystyle\mathcal{N}_{\rm KLT}(12\ldots n{-}2,v)$ $\displaystyle={1\over
n{-}2}\,\mathcal{N}_{\rm KLT}([12\ldots n{-}2],v).$ (35)

Then to prove the closed formula for the pre-numerator in Eq. (19) we only
need to show that

$\displaystyle(n{-}2)\,\mathcal{N}(12\ldots n{-}2,v)=\mathcal{N}_{\rm
KLT}([12\ldots n{-}2],v)\,.$ (36)

The above equation implies that its left-hand side inherits the crossing
symmetry from $\mathcal{N}_{\rm KLT}$, and hence it can be identified as the
BCJ numerator $\mathcal{N}([12\ldots n{-}2],v)$.

Using Eq. (34), the known factorisation of HEFT amplitudes, and the recursive
relation for the KLT matrix Bern  _et al._ (1999); Bjerrum-Bohr  _et al._
(2010); Carrasco  _et al._ (2017); Frost  _et al._ (2020)

$\displaystyle\mathcal{S}(1\ldots
j|1\beta_{L}j\beta_{R})=p_{\Theta(j)}{\cdot}p_{j}\,\mathcal{S}(1\ldots
j{-}1|1\beta_{L}\beta_{R})\,,$ (37)

one finally arrives at Eq. (24).

As an illustration, we consider the five-point case. According to Eq. (34), we
have

$\displaystyle\mathcal{N}_{\rm KLT}([123],v)$
$\displaystyle=p_{1}{\cdot}p_{2}(p_{12}{\cdot}p_{3}\,A(123,v)+p_{1}{\cdot}p_{3}A(132,v))\,.$

There are three different poles, and the corresponding factorisations are

pole | residue of $\mathcal{N}_{\rm KLT}([123],v)$
---|---
${1\over v{\cdot}p_{12}}$ | $p_{12}{\cdot}p_{3}\,\mathcal{N}_{\rm KLT}([12],v)\,\mathcal{N}_{\rm KLT}(3,v)$
${1\over v{\cdot}p_{13}}$ | $p_{1}{\cdot}p_{2}\,\mathcal{N}_{\rm KLT}([13],v)\,\mathcal{N}_{\rm KLT}(2,v)$
${1\over v{\cdot}p_{1}}$ | $p_{1}{\cdot}p_{2}\,\mathcal{N}_{\rm KLT}(1,v)\,\mathcal{N}_{\rm KLT}([23],v)$

in agreement with Eq. (24).

##  Appendix B Proof of factorisation of BCJ numerators

In this Appendix we prove Eq. (27). We begin with the factorisation property
in Eq. (26), which we quote here for convenience,

$\displaystyle\mathcal{N}([1\tau_{L}],v)\,\sum_{r=1}^{n_{R}}\sum_{\sigma\in\mathbf{P}_{\tau_{R}}^{(r)}}{H_{{}_{\Theta(\sigma_{1})},\sigma_{1}}\cdots
H_{{}_{\Theta(\sigma_{r})},\sigma_{r}}\over v{\cdot}p_{\sigma_{1}}\cdots
v{\cdot}p_{\sigma_{1}\sigma_{2}\cdots\sigma_{r-1}}}\,.$ (38)

It is convenient to factor out the dependence on the field strength of the
added particle, $F_{\tau_{R[1]}}$, which according to Eq. (II) always appears
dotted with $p_{{}_{\Theta(\tau_{R})}}$. This allows us to rewrite Eq. (38) in
the form

$\displaystyle\mathcal{N}([1\tau_{L}],v)\Big{[}p_{{}_{\Theta(\tau_{R})}}{\cdot}p_{\tau_{R[1]}}\,\varepsilon_{\tau_{R[1]}}{\cdot}K_{R}-p_{{}_{\Theta(\tau_{R})}}{\cdot}\varepsilon_{\tau_{R[1]}}\,p_{\tau_{R[1]}}{\cdot}K_{R}\Big{]}\,.$
(39)

The resulting $K^{\mu}_{R}$ has a rather complex form which will not be needed
explicitly in the following.

The next step is to prove that $p_{\tau_{R[1]}}{\cdot}K_{R}=0$. To do so, we
note that $p_{\tau_{R[1]}}{\cdot}K_{R}$ has the form, schematically,

$\displaystyle
f_{1}(p{\cdot}v)H_{i_{1},\tau_{j}}+f_{2}(p{\cdot}v)H_{i_{1},\tau_{j_{1}}}H_{i_{2},\tau_{j_{2}}}+\cdots$
$\displaystyle+f_{n{-}3}(p{\cdot}v)H_{i_{1},\tau_{j_{1}}}\cdots
H_{i_{n{-}3},\tau_{j_{n-4}}}\,.$ (40)

This expression must have degree one in $v$ (or equivalently in the mass $m$)
hence all the $f_{i}$ with $i>1$ contain massive poles of the form $p{\cdot}v$
for some $p$. We will now prove that $f_{i}=0$ for all $i>1$ by showing that
Eq. (B) cannot have any massive poles; we will then separately show that terms
of the form $f_{1}H$ also vanish.

Consider an arbitrary massive pole $p_{\tau_{R_{b}}}{\cdot}v\rightarrow 0$ in
the expression in Eq. (39), as shown in the diagram below, with
$\tau_{R_{a}}\cup\tau_{R_{b}}=\tau_{R}$. This amounts to taking a double
residue on the numerator: first $p_{\tau_{R_{b}}}{\cdot}v\rightarrow 0$, then
$p_{\tau_{R_{a}}}{\cdot}v\rightarrow 0$. According to the induction assumption
in the derivation in the main text, the result of this operation is

$\displaystyle\mathcal{N}([1\tau_{L}],v)\,p_{{}_{\Theta(\tau_{R_{a}})}}{\cdot}p_{\tau_{R_{a}[1]}}$
$\displaystyle\times\,$
$\displaystyle\mathcal{N}([\tau_{R_{a}}],v)p_{{}_{\Theta(\tau_{R_{b}})}}{\cdot}p_{\tau_{R_{b}[1]}}\,\mathcal{N}([\tau_{R_{b}}],v)\,.$
(42)

Note that the factor
$p_{{}_{\Theta(\tau_{R_{a}})}}{\cdot}p_{\tau_{R_{a}[1]}}$, which is arbitrary,
is identical to $p_{{}_{\Theta(\tau_{R})}}{\cdot}p_{\tau_{R[1]}}$ and appears
in the first term of Eq. (39). Hence we conclude that the second term in that
equation cannot contribute in the limit $p_{\tau_{R_{b}}}{\cdot}v\rightarrow
0$, or equivalently $f_{i}=0$ for all $i>1$.

The next step is to show that the terms of the form $f_{1}\,H$ also vanish
after performing the sums in Eq. (38). This can be verified directly.
Collecting all the relevant terms $T_{\cdots,(\tau_{R})}$,
$T_{\cdots,(\tau_{R[1]}),(\tau_{R[2]}\cdots\tau_{R[n_{R}]})}$,
$T_{\cdots,(\tau_{R[2]}\cdots\tau_{R[n_{R}]}),(\tau_{R[1]})}$, we find

$\displaystyle-
H_{\tau_{R[1]},\tau_{R[2]}\ldots\tau_{R[n_{R}]}}+H_{\Theta(\tau_{R})\tau_{R[1]},\tau_{R[2]}\ldots\tau_{R[n_{R}]}}$
$\displaystyle+{v{\cdot}p_{\tau_{R[1]}}\over
v{\cdot}p_{1\tau_{L}}+v{\cdot}p_{\tau_{R[2]}\cdots\tau_{R[n_{R}]}}}H_{\Theta(\tau_{R}),\tau_{R[2]}\cdots\tau_{R[n_{R}]}}=0\,,$
(43)

where $\tau_{R[i]}$ denotes the $i^{\rm th}$ index in $\tau_{R}$ and we have
used the on-shell condition
${-}v{\cdot}p_{\tau_{R[1]}}=v{\cdot}p_{1\tau_{L}}+v{\cdot}p_{\tau_{R[2]}\cdots\tau_{R[n_{R}]}}$.
This finishes the proof of $p_{\tau_{R[1]}}{\cdot}K_{R}=0$. Summarising, we
have shown that Eq. (39) reduces to

$\displaystyle\mathcal{N}([1\tau_{L}],v)\Big{(}p_{{}_{\Theta(\tau_{R})}}{\cdot}p_{\tau_{R[1]}}\varepsilon_{\tau_{R[1]}}{\cdot}K_{R}\Big{)}\,.$
(44)

Furthermore, by comparing with Eq. (B), we see that
$\varepsilon_{\tau_{R[1]}}{\cdot}K_{R}$ and $\mathcal{N}([\tau_{R}],v)$ have
the same factorisation behaviour, and hence their difference is a polynomial

$\displaystyle
g=\varepsilon_{\tau_{R[1]}}{\cdot}K_{R}-\mathcal{N}([\tau_{R}],v)\,.$ (45)

An adaptation of the argument of Arkani-Hamed  _et al._ (2018); Rodina (2019)
will now allow us to show that $g=0$. To this end, we note that the velocity
$v$ can appear in two possible ways. First, through the combination
$p{\cdot}v$ with $p$ being any of the momenta. This multiplies a polynomial
function of dimension $n_{R}{-}2$ constructed from $n_{R}$ gluon momenta and
$n_{R}$ polarisations. According to Arkani-Hamed  _et al._ (2018); Rodina
(2019), no such gauge-invariant function exists and hence it must vanish.
Second, $v$ can appear in the combination $v{\cdot}F$, which now multiplies a
function of dimensions $n_{R}{-}2$, constructed from $n_{R}$ gluon momenta and
$n_{R}{-}1$ polarisations. As before, such a function must vanish. From this
we deduce that $g$ vanishes, ending our proof.

##  Appendix C Appendix: Seven-point pre-numerator

For completeness, we provide one more example of a pre-numerator, namely at
seven points:

$\displaystyle\mathcal{N}(12345,v)=\langle-
T_{{(12)},{(3)},{(4)},{(5)}}-T_{{(12)},{(3)},{(5)},{(4)}}$ $\displaystyle-
T_{{(12)},{(4)},{(3)},{(5)}}-T_{{(12)},{(4)},{(5)},{(3)}}-T_{{(12)},{(5)},{(3)},{(4)}}$
$\displaystyle-
T_{{(12)},{(5)},{(4)},{(3)}}+T_{{(12)},{(3)},{(45)}}+T_{{(12)},{(45)},{(3)}}$
$\displaystyle+T_{{(12)},{(34)},{(5)}}+T_{{(12)},{(5)},{(34)}}-T_{{(12)},{(345)}}$
$\displaystyle+T_{{(12)},{(35)},{(4)}}+T_{{(12)},{(4)},{(35)}}+T_{{(123)},{(4)},{(5)}}$
$\displaystyle+T_{{(123)},{(5)},{(4)}}-T_{{(123)},{(45)}}-T_{{(1234)},{(5)}}-T_{{(1235)},{(4)}}$
$\displaystyle+T_{{(124)},{(3)},{(5)}}+T_{{(124)},{(5)},{(3)}}-T_{{(124)},{(35)}}-T_{{(1245)},{(3)}}$
$\displaystyle+T_{{(125)},{(3)},{(4)}}+T_{{(125)},{(4)},{(3)}}-T_{{(125)},{(34)}}-T_{{(13)},{(2)},{(4)},{(5)}}$
$\displaystyle-
T_{{(13)},{(2)},{(5)},{(4)}}-T_{{(13)},{(4)},{(2)},{(5)}}-T_{{(13)},{(4)},{(5)},{(2)}}$
$\displaystyle-
T_{{(13)},{(5)},{(2)},{(4)}}-T_{{(13)},{(5)},{(4)},{(2)}}+T_{{(13)},{(2)},{(45)}}$
$\displaystyle+T_{{(13)},{(45)},{(2)}}+T_{{(13)},{(24)},{(5)}}+T_{{(13)},{(5)},{(24)}}-T_{{(13)},{(245)}}$
$\displaystyle+T_{{(13)},{(25)},{(4)}}+T_{{(13)},{(4)},{(25)}}+T_{{(134)},{(2)},{(5)}}+T_{{(134)},{(5)},{(2)}}$
$\displaystyle-
T_{{(134)},{(25)}}-T_{{(1345)},{(2)}}+T_{{(135)},{(2)},{(4)}}+T_{{(135)},{(4)},{(2)}}$
$\displaystyle-
T_{{(135)},{(24)}}-T_{{(14)},{(2)},{(3)},{(5)}}-T_{{(14)},{(2)},{(5)},{(3)}}$
$\displaystyle-
T_{{(14)},{(3)},{(2)},{(5)}}-T_{{(14)},{(3)},{(5)},{(2)}}-T_{{(14)},{(5)},{(2)},{(3)}}$
$\displaystyle-
T_{{(14)},{(5)},{(3)},{(2)}}+T_{{(14)},{(2)},{(35)}}+T_{{(14)},{(35)},{(2)}}$
$\displaystyle+T_{{(14)},{(23)},{(5)}}+T_{{(14)},{(5)},{(23)}}-T_{{(14)},{(235)}}+T_{{(14)},{(25)},{(3)}}$
$\displaystyle+T_{{(14)},{(3)},{(25)}}+T_{{(145)},{(2)},{(3)}}+T_{{(145)},{(3)},{(2)}}-T_{{(145)},{(23)}}$
$\displaystyle-
T_{{(15)},{(2)},{(3)},{(4)}}-T_{{(15)},{(2)},{(4)},{(3)}}-T_{{(15)},{(3)},{(2)},{(4)}}$
$\displaystyle-
T_{{(15)},{(3)},{(4)},{(2)}}-T_{{(15)},{(4)},{(2)},{(3)}}-T_{{(15)},{(4)},{(3)},{(2)}}$
$\displaystyle+T_{{(15)},{(2)},{(34)}}+T_{{(15)},{(34)},{(2)}}+T_{{(15)},{(23)},{(4)}}+T_{{(15)},{(4)},{(23)}}$
$\displaystyle-
T_{{(15)},{(234)}}+T_{{(15)},{(24)},{(3)}}+T_{{(15)},{(3)},{(24)}}+T_{{(12345)}}\rangle.$
(46)

As anticipated, it contains 75 terms, which is the Fubini number
$\mathsf{F}_{4}$.

##  Appendix D Appendix: More on quasi-shuffle Hopf algebras

As we remarked in the main text, in a Hopf algebra additional structures are
required besides the fusion product. Here we discuss the notions of coproduct
and counit (which make the algebra a bialgebra), and antipode (which make the
bialgebra a Hopf algebra). We begin by recalling that the fusion product is
commutative,

$\displaystyle T_{(1\tau_{1}),\cdots,(\tau_{r})}\star
T_{(\omega_{1}),\cdots,(\omega_{s})}=T_{(\omega_{1}),\cdots,(\omega_{s})}\star
T_{(1\tau_{1}),\cdots,(\tau_{r})}\,,$ (47)

which makes it an associative and commutative quasi-shuffle product. The
commutativity is a natural property because the Hopf algebra is associated
with ordered partitions of a given set. This is also consistent with the
application to HEFT amplitudes, where we consider the pre-numerator of a
particular ordering of gluons, for example the canonical one
$\mathcal{N}(12\ldots n{-}2,v)$. Once $\mathcal{N}(12\ldots n{-}2,v)$ is
obtained using the fusion products, one can then commute the particle labels
and obtain the BCJ numerator $\mathcal{N}([12\ldots n{-}2],v)$.

One can further introduce a number of operations on the generators Hoffman
(2000). The first one is the coproduct $\Delta$, which satisfies
$\Delta(T_{\sigma})\star\Delta(T_{\gamma})=\Delta(T_{\sigma}\star T_{\gamma})$
and

$\displaystyle\Delta(T_{(1\tau_{1}),\cdots,(\tau_{r})})$
$\displaystyle=T_{(1\tau_{1}),\cdots,(\tau_{r})}\otimes\mathbb{I}-T_{(1)}\otimes
T_{(\tau_{1}),\cdots,(\tau_{r})}$
$\displaystyle+\sum_{i=1}^{r-1}T_{(1\tau_{1}),\cdots,(\tau_{i})}\otimes
T_{(\tau_{i+1}),\cdots,(\tau_{r})}\,,$
$\displaystyle\Delta(T_{(\tau_{1}),\cdots,(\tau_{r})})$
$\displaystyle=T_{(\tau_{1}),\cdots,(\tau_{r})}\otimes\mathbb{I}+\mathbb{I}\otimes
T_{(\tau_{1}),\cdots,(\tau_{r})}$
$\displaystyle+\sum_{i=1}^{r-1}T_{(\tau_{1}),\cdots,(\tau_{i})}\otimes
T_{(\tau_{i+1}),\cdots,(\tau_{r})}\,,$ (48)

where $\mathbb{I}$ is the identity element in the algebra with the property
$\mathbb{I}\star T_{\sigma}=T_{\sigma}\star\mathbb{I}=T_{\sigma}$. The second
one is the counit $\epsilon$, which satisfies
$\epsilon(T_{\sigma})\star\epsilon(T_{\gamma})=\epsilon(T_{\sigma}\star
T_{\gamma})$ and

$\displaystyle\epsilon(T_{(\tau_{1}),\cdots,(\tau_{r})})$
$\displaystyle=0,\quad\epsilon(\mathbb{I})=1\,.$ (49)

Finally we introduce the antipode $S$ , with the requirement
$\star(\mathbb{I}\otimes
S)\Delta(T_{\sigma})=\star(S\otimes\mathbb{I})\Delta(T_{\sigma})=\epsilon(T_{\sigma})\mathbb{I}$,
and satisfying

$\displaystyle S(T_{(\tau_{1}),\cdots,(\tau_{r})})$
$\displaystyle=-T_{(\tau_{1}),\cdots,(\tau_{r})}$ (50)
$\displaystyle-\sum_{i=1}^{r-1}S(T_{(\tau_{1}),\cdots,(\tau_{i})})\star
T_{(\tau_{i+1}),\cdots,(\tau_{r})}\,.$

##  Appendix E An equivalent kinematic algebra in HEFT and Yang-Mills

In this section we consider the situation when the gluons are not in canonical
ordering $1,2,\ldots,n{-}2$. To do this, we define the kinematic algebra by
extending the generators to contain a superscript accounting for the ordering,
and define the fusion product as

$\displaystyle T^{(\alpha)}_{(\tau_{1}),\cdots,(\tau_{r})}\star
T^{(\beta)}_{(\omega_{1}),\cdots,(\omega_{s})}=$
$\displaystyle\sum_{\begin{subarray}{c}\sigma|_{\\{\tau\\}}=\\{(\tau_{1}),\cdots,(\tau_{r})\\}\\\
\sigma|_{\\{\omega\\}}=\\{(\omega_{1}),\cdots,(\omega_{s})\\}\end{subarray}}(-1)^{t-r-s}T^{(\alpha\beta)}_{(\sigma_{1}),(\sigma_{2}),\cdots,(\sigma_{t})}\,,$
(51)

where the superscripts $\alpha$ and $\beta$ represent the possibly non-
canonical orderings of the gluons, $\alpha$ ($\beta$) is a permutation of
$\tau_{1}\cup\cdots\cup\tau_{r}$ ($\omega_{1}\cup\cdots\cup\omega_{s}$) and
$\alpha\cap\beta=\emptyset$. Furthermore $\tau_{i},\omega_{j},\sigma_{k}$ are
subsets which preserve the ordering of $\alpha,\beta,\alpha\beta$
respectively. We call $T^{(\alpha)}_{(\tau_{1}),\cdots,(\tau_{r})}$ the
extended generators.

The fusion product of these extended generators is associative but non-
commutative, and can be considered as a non-abelian extension of the quasi-
shuffle product. When $\alpha,\beta$ are in canonical ordering it reduces to
the one in the main text, in which case we omit the generators’ superscripts.
The extended generators can again be mapped to gauge- and Lorentz-invariant
functions, which are defined as follows

$\displaystyle\langle
T^{(\alpha)}_{(\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})}\rangle:=\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[myblob]{}; \vertex[left=0.8cm of a] (a0)[myblob]{};
\vertex[right=1.0cm of a] (a2)[myblob]{}; \vertex[right=0.8cm of a2]
(a3)[myblob]{}; \vertex[right=0.8cm of a3] (a4)[myblob]{}; \vertex[above=0.8cm
of a] (b1){$\overbrace{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\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{}{0}\leavevmode\nobreak\
\leavevmode\nobreak\ }^{\tau_{1}}$\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ }; \vertex[above=0.8cm of a2] (b2){$\tau_{2}$};
\vertex[above=0.8cm of a3] (b3){$\cdots$}; \vertex[above=0.8cm of a4]
(b4){$\tau_{r}$}; \vertex[above=0.8cm of a0] (b0){$\leavevmode\nobreak\ $};
\vertex[above=0.8cm of a0](j1){$$}; \vertex[right=0.2cm of j1](j2){$$};
\vertex[right=0.6cm of j2](j3){$$}; \vertex[right=0.6cm of j3](j4){$$};
\vertex[right=0.8cm of j4](j5){$$}; \vertex[right=0.2cm of j5](j6){$$};
\vertex[right=0.6cm of j6](j7){$$}; \vertex[right=0.0cm of j7](j8){$$};
\vertex[right=0.8cm of j8](j9){$$}; \diagram*{(a)--[very thick](a0),(a)--[very
thick](a2),(a2)--[very thick](a3), (a3)--[very thick](a4),(a0) -- [thick]
(j1),(a) -- [thick] (j2),(a)--[thick](j3),(a2) -- [thick]
(j4),(a2)--[thick](j5),(a4) -- [thick] (j8),(a4)--[thick](j9)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle={v{\cdot}F_{\tau_{1}}{\cdot}V_{\Theta^{\alpha}(\tau_{2})}{\cdot}F_{\tau_{2}}\cdots
V_{\Theta^{\alpha}(\tau_{r})}{\cdot}F_{\tau_{r}}{\cdot}v\over
v{\cdot}p_{\tau_{1[1]}}v{\cdot}p_{\tau_{1}}\cdots
v{\cdot}p_{\tau_{1}\tau_{2}\cdots\tau_{r-1}}}\,,$ (52)

where $\Theta^{\alpha}(\tau_{i})$ denotes the set of all the indices that are
to the left of the first index of $\tau_{i}$ in the ordered set $(\alpha)$ and
are contained in the subsets $\tau_{j}$ with $j<i$. The mapped function
vanishes if $\Theta^{\alpha}(\tau_{i})$ is the empty set.

The pre-numerator is then generated from the fusion product of the extended
generators through

$\displaystyle\mathcal{N}^{\prime}(i_{1}i_{2}\ldots i_{n-2},v)=\langle
T^{(i_{1})}_{(i_{1})}\star T^{(i_{2})}_{(i_{2})}\star\cdots\star
T^{(i_{n{-}2})}_{(i_{n{-}2})}\rangle\,.$ (53)

Importantly, this pre-numerator $\mathcal{N}^{\prime}(i_{1}i_{2}\ldots
i_{n-2},v)$ is identical to the pre-numerator $\mathcal{N}(i_{1}i_{2}\ldots
i_{n-2},v)$ obtained from an appropriate permutation of the labels in
$\mathcal{N}(12\ldots{n{-}2},v)$ from the main text.

We can further define commutation relations using the new fusion product:

$\displaystyle[T^{(\alpha)}_{\sigma},T^{(\beta)}_{\gamma}]=T^{(\alpha)}_{\sigma}\star
T^{(\beta)}_{\gamma}-T^{(\beta)}_{\gamma}\star T^{(\alpha)}_{\sigma}\,,$ (54)

and the extended generators form an infinite-dimensional Lie algebra. Then the
BCJ numerator for any cubic graph can be expressed as the nested commutator of
the corresponding $T^{(j)}_{(j)}$ associated with that graph.

As an example consider the five-point case, where

$\displaystyle\mathcal{N}^{\prime}([[1,2],3],v)$
$\displaystyle:=\langle[[T^{(1)}_{(1)},T^{(2)}_{(2)}],T^{(3)}_{(3)}]\rangle$
$\displaystyle=\langle[T^{(1)}_{(1)}\star T^{(2)}_{(2)}-T^{(2)}_{(2)}\star
T^{(1)}_{(1)},T^{(3)}_{(3)}]\rangle\,.$ (55)

We now compute each contribution, starting with

$\displaystyle[T^{(1)}_{(1)}\star
T^{(2)}_{(2)},T^{(3)}_{(3)}]=[T^{(12)}_{(1),(2)}{+}T^{(12)}_{(2),(1)}{-}T^{(12)}_{(12)},T^{(3)}_{(3)}]$
$\displaystyle=T^{(123)}_{(1),(2),(3)}{+}T^{(123)}_{(1),(3),(2)}{+}T^{(123)}_{(3),(1),(2)}{-}T^{(123)}_{(13),(2)}{-}T^{(123)}_{(1),(23)}$
$\displaystyle-(T^{(312)}_{(1),(2),(3)}{+}T^{(312)}_{(1),(3),(2)}{+}T^{(312)}_{(3),(1),(2)}{-}T^{(312)}_{(31),(2)}{-}T^{(312)}_{(1),(32)})$
$\displaystyle+T^{(123)}_{(2),(1),(3)}{+}T^{(123)}_{(2),(3),(1)}{+}T^{(123)}_{(3),(2),(1)}{-}T^{(123)}_{(23),(1)}{-}T^{(123)}_{(2),(13)}$
$\displaystyle-(T^{(312)}_{(2),(1),(3)}{+}T^{(312)}_{(2),(3),(1)}{+}T^{(312)}_{(3),(2),(1)}{-}T^{(312)}_{(32),(1)}{-}T^{(312)}_{(2),(31)})$
$\displaystyle-(T^{(123)}_{(12),(3)}+T^{(123)}_{(3),(12)}-T^{(123)}_{(123)})$
$\displaystyle+(T^{(312)}_{(12),(3)}+T^{(312)}_{(3),(12)}-T^{(312)}_{(312)}).$
(56)

According to the definition in Eq. (E), all generators with a single index in
the first component (i.e. $T_{(i),\sigma}$) are mapped to zero since a single
field strength $F_{i}$ is antisymmetric. Furthermore,
$T^{(123)}_{(23),(1)},T^{(312)}_{(12),(3)}$ also vanish due to the fact that
the sets $\Theta(1)$ and $\Theta^{(312)}(3)$ for these two generators are
empty. Finally, we arrive at

$\displaystyle\langle[T^{(1)}_{(1)}\star T^{(2)}_{(2)},T^{(3)}_{(3)}]\rangle$
$\displaystyle=\langle T^{(123)}_{(123)}\rangle-\langle
T^{(123)}_{(13),(2)}\rangle-\langle T^{(123)}_{(12),(3)}\rangle$
$\displaystyle-(\langle T^{(312)}_{(312)}\rangle-\langle
T^{(312)}_{(31),(2)}\rangle-\langle T^{(312)}_{(32),(1)}\rangle)$
$\displaystyle=\mathcal{N}(123,v)-\mathcal{N}(312,v).$ (57)

The other term in Eq. (E) is obtained similarly, and

$\displaystyle\langle[T^{(2)}_{(2)}\star T^{(1)}_{(1)},T^{(3)}_{(3)}]\rangle$
$\displaystyle=\mathcal{N}(213,v)-\mathcal{N}(321,v)\,.$ (58)

As expected we find

$\displaystyle\mathcal{N}^{\prime}([[1,2],3],v)=\mathcal{N}([[1,2],3],v)\,.$
(59)

Therefore, we are able to obtain the BCJ numerators directly from the non-
abelian extended quasi-shuffle Hopf algebra. It can be shown for an arbitrary
number of gluons that the BCJ numerator defined from the nested commutator of
the extended generators is equal to the BCJ numerator obtained from the nested
commutator of the indices of the pre-numerator in the kinematic Hopf algebra.

Summarising, we can arrive at the BCJ generators in two ways: either via the
construction presented in the main text, or with the extended generators
introduced here. The diagram below represents these two possibilities, where
one can either proceed right and down, or down and right:

$\displaystyle T_{(1)}\star\cdots\star
T_{(n{-}2)}\xrightarrow[\begin{subarray}{c}\text{extended}\\\
\text{generators}\end{subarray}]{\text{commutator}}[\ldots[T^{(1)}_{(1)},T^{(2)}_{(2)}],\ldots,T^{(n-2)}_{(n{-}2)}]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Bigg{\downarrow}\langle\leavevmode\nobreak\
\leavevmode\nobreak\ \rangle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Bigg{\downarrow}\langle\leavevmode\nobreak\
\leavevmode\nobreak\ \rangle$ $\displaystyle\mathcal{N}(12\ldots
n{-}2,v)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\xrightarrow[\text{gluon labels}]{\text{commutator}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathcal{N}([12\ldots n{-}2],v)\,.$
(60)

## References

  * Bern  _et al._ (2008) Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. D78, 085011 (2008), arXiv:0805.3993 [hep-ph] .
  * Bern  _et al._ (2010) Z. Bern, J. J. M. Carrasco, and H. Johansson, Phys. Rev. Lett. 105, 061602 (2010), arXiv:1004.0476 [hep-th] .
  * Kawai  _et al._ (1986) H. Kawai, D. C. Lewellen, and S. H. H. Tye, Nucl. Phys. B269, 1 (1986).
  * Georgi (1990) H. Georgi, Phys. Lett. B 240, 447 (1990).
  * Luke and Manohar (1992) M. E. Luke and A. V. Manohar, Phys. Lett. B 286, 348 (1992), arXiv:hep-ph/9205228 .
  * Neubert (1994) M. Neubert, Phys. Rept. 245, 259 (1994), arXiv:hep-ph/9306320 .
  * Manohar and Wise (2000) A. V. Manohar and M. B. Wise,  _Heavy quark physics_ , Vol. 10 (Cambridge University Press, 2000).
  * Damgaard  _et al._ (2019) P. H. Damgaard, K. Haddad, and A. Helset, JHEP 11, 070 (2019), arXiv:1908.10308 [hep-ph] .
  * Brandhuber  _et al._ (2021a) A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, JHEP 07, 047 (2021a), arXiv:2104.11206 [hep-th] .
  * Monteiro and O’Connell (2011) R. Monteiro and D. O’Connell, JHEP 07, 007 (2011), arXiv:1105.2565 [hep-th] .
  * Chen and Du (2014) G. Chen and Y.-J. Du, JHEP 01, 061 (2014), arXiv:1311.1133 [hep-th] .
  * Cheung and Shen (2017) C. Cheung and C.-H. Shen, Phys. Rev. Lett. 118, 121601 (2017), arXiv:1612.00868 [hep-th] .
  * Cheung  _et al._ (2018a) C. Cheung, G. N. Remmen, C.-H. Shen, and C. Wen, JHEP 04, 129 (2018a), arXiv:1709.04932 [hep-th] .
  * Cheung  _et al._ (2018b) C. Cheung, C.-H. Shen, and C. Wen, JHEP 02, 095 (2018b), arXiv:1705.03025 [hep-th] .
  * Chen  _et al._ (2019) G. Chen, H. Johansson, F. Teng, and T. Wang, JHEP 11, 055 (2019), arXiv:1906.10683 [hep-th] .
  * Chen  _et al._ (2021) G. Chen, H. Johansson, F. Teng, and T. Wang, JHEP 10, 042 (2021), arXiv:2104.12726 [hep-th] .
  * Brandhuber  _et al._ (2021b) A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, JHEP 10, 118 (2021b), arXiv:2108.04216 [hep-th] .
  * Bern  _et al._ (2019) Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019), arXiv:1901.04424 [hep-th] .
  * Damour (2020) T. Damour, Phys. Rev. D 102, 024060 (2020), arXiv:1912.02139 [gr-qc] .
  * Kälin  _et al._ (2020) G. Kälin, Z. Liu, and R. A. Porto, Phys. Rev. Lett. 125, 261103 (2020), arXiv:2007.04977 [hep-th] .
  * Di Vecchia  _et al._ (2021) P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, (2021), arXiv:2104.03256 [hep-th] .
  * Herrmann  _et al._ (2021) E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, (2021), arXiv:2104.03957 [hep-th] .
  * Bjerrum-Bohr  _et al._ (2021) N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Planté, and P. Vanhove, (2021), arXiv:2105.05218 [hep-th] .
  * Bern  _et al._ (2021) Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 126, 171601 (2021), arXiv:2101.07254 [hep-th] .
  * Hoffman (2000) M. E. Hoffman, Journal of Algebraic Combinatorics 11, 49 (2000).
  * Aguiar and Mahajan (2010) M. Aguiar and S. A. Mahajan,  _Monoidal functors, species and Hopf algebras_ , Vol. 29 (American Mathematical Society Providence, RI, 2010).
  * Fauvet  _et al._ (2017) F. Fauvet, L. Foissy, and D. Manchon, in  _Annales de l’Institut Fourier_ , Vol. 67 (2017) pp. 911–945.
  * Note (1) Note that $\sum@\slimits@_{i=1}^{n-2}v{\cdot}p_{i}=0$.
  * Note (2) The $\star$-product is bilinear if $(aT_{\sigma}+bT_{\gamma})\star T_{\rho}=aT_{\sigma}\star T_{\rho}+bT_{\gamma}\star T_{\rho}$, $T_{\rho}\star(aT_{\sigma}+bT_{\gamma})=aT_{\rho}\star T_{\sigma}+bT_{\rho}\star T_{\gamma}$, where $a$, $b$ are numbers.
  * Note (3) Combining two three-point numerators together to form the four-point one is a natural operation, indeed $\langle T_{(12)}\rangle$ factorises on the massive pole as $\langle T_{(12)}\rangle\leavevmode@ifvmode{\hbox{$\left|\vbox to1.0pt{}\right.$}}_{v{\cdot}p_{1}\rightarrow 0}=(-{p_{1}{\cdot}p_{2}\over 2v{\cdot}p_{1}})\langle T_{(1)}\rangle\langle T_{(2)}\rangle$.
  * Cartier (2002) P. Cartier, Asterisque-Société Mathématique de France 282, 137 (2002).
  * Ihara  _et al._ (2006) K. Ihara, M. Kaneko, and D. Zagier, Compositio Mathematica 142, 307 (2006).
  * Note (4) The $n^{\rm th}$ Fubini number, also known as ordered Bell number, also gives the number of possible outcomes of a horse race with $n$ horses, including dead heats.
  * Mező (2019) I. Mező,  _Combinatorics and number theory of counting sequences_ (Chapman and Hall/CRC, 2019).
  * Note (5) For instance, the pre-numerators $\mathcal{N}(1,v)$, $\mathcal{N}(12,v)$, $\mathcal{N}(123,v)$ and $\mathcal{N}(1234,v)$ contain 1, 1, 3 and 13 terms, respectively.
  * Gross (1962) O. A. Gross, The American Mathematical Monthly 69, 4 (1962).
  * Bjerrum-Bohr  _et al._ (2011) N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard, and P. Vanhove, JHEP 01, 001 (2011), arXiv:1010.3933 [hep-th] .
  * Arkani-Hamed  _et al._ (2018) N. Arkani-Hamed, L. Rodina, and J. Trnka, Phys. Rev. Lett. 120, 231602 (2018), arXiv:1612.02797 [hep-th] .
  * Rodina (2019) L. Rodina, JHEP 09, 084 (2019), arXiv:1612.06342 [hep-th] .
  * Cheung and Mangan (2021) C. Cheung and J. Mangan, JHEP 11, 069 (2021), arXiv:2108.02276 [hep-th] .
  * Connes and Kreimer (2000) A. Connes and D. Kreimer, Commun. Math. Phys. 210, 249 (2000), arXiv:hep-th/9912092 .
  * Goncharov  _et al._ (2010) A. B. Goncharov, M. Spradlin, C. Vergu, and A. Volovich, Phys.Rev.Lett. 105, 151605 (2010), arXiv:1006.5703 [hep-th] .
  * Duhr (2012) C. Duhr, JHEP 08, 043 (2012), arXiv:1203.0454 [hep-ph] .
  * Abreu  _et al._ (2014) S. Abreu, R. Britto, C. Duhr, and E. Gardi, JHEP 10, 125 (2014), arXiv:1401.3546 [hep-th] .
  * Brown (2017) F. Brown, Commun. Num. Theor. Phys. 11, 453 (2017), arXiv:1512.06409 [math-ph] .
  * Broedel  _et al._ (2013) J. Broedel, O. Schlotterer, and S. Stieberger, Fortsch. Phys. 61, 812 (2013), arXiv:1304.7267 [hep-th] .
  * Mafra  _et al._ (2011) C. R. Mafra, O. Schlotterer, and S. Stieberger, JHEP 07, 092 (2011), arXiv:1104.5224 [hep-th] .
  * Du and Teng (2017) Y.-J. Du and F. Teng, JHEP 04, 033 (2017), arXiv:1703.05717 [hep-th] .
  * Edison and Teng (2020) A. Edison and F. Teng, JHEP 12, 138 (2020), arXiv:2005.03638 [hep-th] .
  * Del Duca  _et al._ (2000) V. Del Duca, L. J. Dixon, and F. Maltoni, Nucl. Phys. B571, 51 (2000), arXiv:hep-ph/9910563 [hep-ph] .
  * Kleiss and Kuijf (1989) R. Kleiss and H. Kuijf, Nucl. Phys. B312, 616 (1989).
  * Vaman and Yao (2010) D. Vaman and Y.-P. Yao, JHEP 11, 028 (2010), arXiv:1007.3475 [hep-th] .
  * Bjerrum-Bohr  _et al._ (2010) N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng, and T. Sondergaard, Phys. Rev. D 82, 107702 (2010), arXiv:1005.4367 [hep-th] .
  * Bern  _et al._ (1999) Z. Bern, L. J. Dixon, M. Perelstein, and J. S. Rozowsky, Nucl. Phys. B546, 423 (1999), arXiv:hep-th/9811140 [hep-th] .
  * Carrasco  _et al._ (2017) J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, JHEP 06, 093 (2017), arXiv:1608.02569 [hep-th] .
  * Frost  _et al._ (2020) H. Frost, C. R. Mafra, and L. Mason, (2020), arXiv:2012.00519 [hep-th] .