# Manifest colour-kinematics duality and double-copy in the string-based
formalism

Naser Ahmadiniaz<EMAIL_ADDRESS>Helmholtz-Zentrum Dresden-Rossendorf,
Bautzner Landstraße 400, 01328 Dresden, Germany  Filippo Maria Balli
<EMAIL_ADDRESS>Dipartimento di Scienze Fisiche, Informatiche e
Matematiche, Università degli Studi di Modena e Reggio Emilia, Via Campi
213/A, I-41125 Modena, Italy
and INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy
Department of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden
Olindo Corradini<EMAIL_ADDRESS>Cristhiam Lopez-Arcos
<EMAIL_ADDRESS>Alexander Quintero Vélez<EMAIL_ADDRESS>Escuela de
Matemáticas, Universidad Nacional de Colombia Sede Medellín, Carrera 65 $\\#$
59A–110, Medellín, Colombia  Christian Schubert<EMAIL_ADDRESS>Instituto de Física y Matemáticas Universidad Michoacana de San Nicolás de
Hidalgo Edificio C-3, Apdo. Postal 2-82 C.P. 58040, Morelia, Michoacán, México

###### Abstract

The relation for the gravity polarisation tensor as the tensor product of two
gluon polarisation vectors has been well-known for a long time, but a version
of this relation for multi-particle fields is presently still not known. Here
we show that in order for this to happen we first have to ensure that the
multi-particle polarisations satisfy colour-kinematics duality. In previous
work it has been show that this arises naturally from the Bern-Kosower
formalism for one-loop gluon amplitudes, and here we show that the tensor
product for multi-particle fields arise naturally in the Bern-Dunbar-Shimada
formalism for one-loop gravity amplitudes. This allows us to formulate a new
prescription for double-copy gravity Berends-Giele currents, and to obtain
both the colour-dressed Yang-Mills Berends-Giele currents in the Bern-
Carrasco-Johansson gauge and the gravitational Berends-Giele currents
explicitly. An attractive feature of our formalism is that it never becomes
necessary to determine gauge transformation terms. Our double-copy
prescription can also be applied to other cases, and to make this point we
derive the double-copy perturbiners for $\alpha^{\prime}$-deformed gravity and
the bi-adjoint scalar model.

††journal: .

UUITP-47/21

###### Contents

  1. 1 Introduction
  2. 2 The string-based rules in field theory
    1. 2.1 The one-loop $n$-gluon amplitudes
    2. 2.2 The one-loop $n$-graviton amplitudes
    3. 2.3 Symmetric partial integration
  3. 3 Colour-kinematics duality
    1. 3.1 Colour-stripped Berends-Giele currents from the Bern-Kosower formalism
    2. 3.2 Colour-dressed Berends-Giele currents
  4. 4 Double-copy relations for perturbiner expansions
    1. 4.1 Multi-particle polarisation tensors from the Bern-Dunbar-Shimada formalism
    2. 4.2 Double-copy perturbiner expansion
  5. 5 Some other examples
    1. 5.1 $\alpha^{\prime}$-deformations
    2. 5.2 Zeroth-copy
  6. 6 Conclusions
  7. A Berends-Giele currents of multiplicity five

##  1 Introduction

The perturbative expansion of Yang-Mills theory has remarkable aspects and
properties which are extremely difficult to be seen and extracted from the
Feynman rules. Parke and Taylor [1] made a conjucture about the tree-level
scattering amplitudes that are maximally helicity violating (MHV) to be
expressed in terms of simple holomorphic functions. Their observation was
based on a computation for the first few cases but later was proved by Berends
and Giele [2] for the general case. Recent years have seen a tremendous
development in the area of the on-shell matrix elements calculation in quantum
field theory, especially for gauge theory and gravity, such as unitary-based
methods [3, 4], twistors [5], Britto-Cachazo-Feng-Witten (BCFW) recursion [6,
7] and Grassmannians [8, 9], see [10, 11] for recent reviews.

In an independent development, the perturbiner expansion method was introduced
by Rosly and Selivanov [12, 13, 14, 15] as an efficient method to obtain the
tree-level scattering amplitudes for a generic massless quantum field theory
and, in the Yang-Mills case, can as well be used as a tool to compute multi-
particle trees with one particle off-shell, i.e. the so-called Berends-Giele
currents [2]. In fact, it turns out that these currents can be efficiently
packed if a particular gauge is chosen, namely the Bern-Carrasco-Johansson
(BCJ) gauge [16], which displays the very helpful feature known as “colour-
kinematics duality”, where colour-ordered amplitudes obey the same relations
as their associated colour factors. In the Berends-Giele currents, colour-
kinematics duality was made explicit by identifying multi-particle
polarisations which satisfy the so-called generalized Jacobi identities (GJI)
[17].

In order to extend such constructions to gravity, a possible general strategy
which has been widely used is to rely upon perturbative gauge-gravity duality,
which is a consequence of open-closed duality of string theory and at the
perturbative string level gives rise to the so-called Kawai-Lewellen-Tye (KLT)
relations between open string amplitudes and closed string amplitudes [18]. In
the particle limit of string theory ($\alpha^{\prime}\rightarrow 0$), it leads
to relations between tree-level graviton amplitudes and tree-level gluon
amplitudes in Yang-Mills theories, which are often summarized as “${\rm
gravity}=({\rm gauge~{}theory})^{2}$”. Such duality holds even though the
structures of the non-abelian Yang-Mills and the Einstein-Hilbert lagrangians
are rather different: the former contains only up to four-point interactions
while the latter contains infinitely many vertices. Therefore the validity of
the above duality in the field theory limit has been a major puzzle for many
years. Finally, however powerful, these rules are limited to relations between
scattering amplitudes, i.e. they are on-shell expressions.

The original derivation of the KLT relations was done in string theory [18].
Later on they were applied to $n$-point amplitudes in field theories in [19,
20], and more recently they were revisited in a more algebro-topological
framework in [21]. The connection between gravity and gauge theory starts
already at three points. The three-point amplitudes vanish for real on-shell
momenta, but this can be avoided by going to complex momenta which leads to
the following compact result

$\displaystyle\mathscr{M}_{3}(1,2,3)=\mathscr{A}_{3}(1,2,3)\tilde{\mathscr{A}}_{3}(1,2,3)$
(1.1)

where $\mathscr{M}_{3}$ and $\mathscr{A}_{3}(\tilde{\mathscr{A}}_{3})$ are the
three-point gravity and gauge theory amplitudes accordingly [22]. For the
four- and five-point amplitudes the relations look slightly different but
still very simple

$\displaystyle\mathscr{M}_{4}(1,2,3,4)$ $\displaystyle=$ $\displaystyle-
s_{12}\mathscr{A}_{4}(1,2,3,4)\tilde{\mathscr{A}}_{4}(1,2,4,3)$
$\displaystyle\mathscr{M}_{5}(1,2,3,4,5)$ $\displaystyle=$ $\displaystyle
s_{12}s_{34}\mathscr{A}_{5}(1,2,3,4,5)\tilde{\mathscr{A}}_{5}(2,1,4,3,5)+s_{13}s_{24}\mathscr{A}_{5}(1,3,2,4,5)\tilde{\mathscr{A}}_{5}(3,1,4,2,5)$
(1.2)

where $s_{ij}=(k_{i}+k_{j})^{2}=2k_{i}\cdot k_{j}$.

Bern, Carrasco and Johansson [16] discovered a direct way of constructing
gravity amplitudes from gauge theory amplitudes after organizing the latter in
a specific manner, so that the amplitude numerator respects a certain colour-
kinematics duality, which involves colour factors and kinematic numerators;
then by a suitable replacement which is called the “double-copy construction”,
they obtained tree-level gravity amplitudes. Specifically, at tree-level,
colour dressed scattering amplitudes in Yang-Mills theories can be written in
the following form

$\mathscr{A}_{n}\sim\sum_{i}\frac{c_{i}n_{i}}{D_{i}}$ (1.3)

where the $c_{i}$ are the colour factors or numerators consisting of
contractions of the gauge Lie algebra structure constants, the $n_{i}$ are the
kinematic numerators which are sums of Lorentz-invariant contractions of
external momenta and polarisations, and the $D_{i}$ are the propagators which
can be written in term of Mandelstam variables. One particularity is that the
sum is only over cubic or trivalent graphs with the quartic vertices absorbed
by the cubic ones. Colour-kinematics duality means that the relations
satisfied by the colour factors due to Jacobi identities are mirrored by their
respective kinematic numerators, i.e.

$\displaystyle c_{i}+c_{j}+c_{k}=0\ \rightarrow\ n_{i}+n_{j}+n_{k}=0.$ (1.4)

The great advantage of having the amplitude represented like (1.3) is that the
calculation of the associated gravity amplitude is automatic, as soon as the
numerators satisfy colour-kinematics. The gravity amplitudes are obtained in
terms of the gauge theory information simply by replacing the colour factors
by another copy of the kinematic numerators and summing over the same cubic
diagrams. The $n$-point gravity amplitude is given by

$\displaystyle\mathscr{M}_{n}\sim\sum_{i}\frac{n_{i}\tilde{n}_{i}}{D_{i}}.$
(1.5)

Interestingly, the double-copy construction has also been used in more
phenomenological studies, such as in gravitational wave physics (see for
expample Refs. [23, 24, 25]) which since the direct detection of gravitational
waves in the LIGO and VIRGO experiments [26, 27] has become a very active
field of research. Very recently, a classical double-copy relation was
developed in the context of a worldline QFT description of the classical
gravitational scattering of massive bodies [28].

In general, in the literature there are several proofs for the BCJ amplitude
relations from different approaches such as the string-theory monodromy
relations [29, 30, 31] and the BCFW recursion relations [6, 7]. In the field
theory limit, in contrast to the KLT relations, the double-copy construction
has been conjectured to hold for integrands of loop-level amplitudes [32]
which has been tested in various publications [33, 34, 35, 36, 37, 38, 39].
There is a lot of evidence in supporting the conjucture that colour-kinematic
dual representations exist for a variety of gauge theory amplitudes. This fact
has been shown explicitly for tree-level amplitudes with up to eight external
legs [32], and for integrands at several loop-levels in maximally
Supersymmetric-Yang-Mills theory [16, 33, 34, 39]. Less supersymmetric
theories are also expected to respect such duality at both tree and loop
orders, see [32] for a two-loop QCD example. In Ref. [40] the duality between
colour and kinematics has been generalized to three-algebras in three-
dimensional supersymmetric Chern-Simons theories. Double-copy also found
applicability in anyonic models [41].

At loop level, starting from string amplitudes, Bern and Kosower unveiled a
very powerful master formula for the one-loop gluon scattering amplitudes [42,
43, 44], later re-derived by Strassler directly from the particle approach,
i.e. from the worldline formalism [45, 46], which does not require on-shell
conditions nor masslessness. Bern, Dunbar and Shimada then extended the Bern-
Kosower string-based rules to gravity [47, 48]. Their approach made essential
use of the aforementioned open-closed string duality, treating graviton
amplitudes as a double-copy of gluon amplitudes. Such approach is reviewed
below, in Section 2.

A common difficulty in both scenarios, i.e. gluon and gravity amplitudes, is
the explicit realization of Ward identities, which in the off-shell case
relate $n$-particle amplitudes to $(n-1)$-particle amplitudes. In the on-shell
case they are encoded in the transversality condition which involves both one-
particle irreducible and reducible diagrams, which arise when trees are sewn
to the loop. Hence, it is of fundamental importance to have a method which
allows one to efficiently compute these trees. As already mentioned in the
gluon case such multi-particle trees, with one particle off-shell, are dubbed
Berends-Giele currents [2].

Recently, some of the present authors investigated the construction of
Berends-Giele currents for gluons, inspired by the Bern-Kosower replacement
rules [49]. The most relevant aspect of the Bern-Kosower rules to such purpose
is a procedure known as “pinching procedure” that allows one to construct the
reducible parts of the amplitudes from the irreducible parts of the one-loop
amplitudes at the level of the Feynman-Schwinger integrands. First one employs
suitable integrations by parts which effectively remove quartic vertices and
yield integrands which are expressed in terms of “Lorentz cycles”, i.e. traces
of products of the linearized parts of gluon field strenghts in momentum
space. Then, one suitably removes linearly appearing Green’s functions, which
correspond to the external lines to be pinched. In [49] this pinching
procedure was implemented with the introduction of a differential operator,
so-called “pinch operator”, and also the parts of the integrand identified
that have to be pinched in order to extract the multi-particle polarisations
that will conform the Berends-Giele currents. Such polarisations come
naturally in the BCJ gauge. The results of the above paper are summarized in
Section 3.

In the present manuscript, in Section 4, we extend such a construction to
gravity amplitudes, by considering the Bern-Dunbar-Shimada formalism and the
double-copy procedure at the level of the Berends-Giele currents. In other
words, we find the multi-particle polarisation expansion for gravity by means
of the “double pinch operator” which is a double-copy of the one used in the
Yang-Mills case. These multi-particle polarisation currents are thus used to
construct the perturbiner expansion. Unlike other recent approaches [50, 51],
in our method the off-shell double-copy polarisation currents emerge directly
from the Bern-Dunbar-Shimada formalism, rather than from KLT relations.

Our new prescription for the double-copy Berends-Giele currents can be applied
to other models. In Section 5 we present the perturbiners for the cases of the
$\alpha^{\prime}$-deformed gravity and the bi-adjoint scalar model. Finally,
in Section 6, we conclude.

#### Notation

Latin indices $a,b$, etc. from the beginning of the alphabet run over the
$N^{2}-1$ generators of $\mathfrak{su}(N)$. We let these generators be denoted
by $T_{a}$, with structure constants $f_{ab}^{\phantom{ab}c}$ satisfying

$[T_{a},T_{b}]=\mathrm{i}\mkern 1.0mu\sqrt{2}f_{ab}^{\phantom{ab}c}T_{c}.$

For reasons of formal convenience, we set
$\tilde{f}_{ab}^{\phantom{ab}c}=\mathrm{i}\mkern
1.0mu\sqrt{2}f_{ab}^{\phantom{ab}c}$.

By a word we mean a finite string $P=i_{1}i_{2}\cdots i_{n}$ of positive
integers $i_{1},i_{2},\dots,i_{n}\geq 1$. The word consisting of no symbols is
called the empty word, written $\varnothing$. Given a word $P=i_{1}i_{2}\cdots
i_{n}$, we denote by $\lvert P\rvert$ its length $n$. Also, the multi-particle
momentum for such word and its associated Mandelstam invariant are given by
$k_{P}=k_{i_{1}}+k_{i_{2}}+\cdots+k_{i_{n}}$ and $s_{P}=k_{P}^{2}$.

We shall implicitly work with the free Lie algebra generated by all words with
letters in $1,2,\dots,n$. There one can consider the left-to-right bracketing
$\ell$, which is defined recursively by

$\displaystyle\ell(\varnothing)$ $\displaystyle=0,$ $\displaystyle\ell(i)$
$\displaystyle=i,$ $\displaystyle\ell(i_{1}i_{2}\cdots i_{n})$
$\displaystyle=\ell(i_{1}i_{2}\cdots i_{n-1})i_{n}-i_{n}\ell(i_{1}i_{2}\cdots
i_{n-1}).$

Using this notation, for objects labelled by words, the generalised Jacobi
identities of order $k$ can be characterised as

$U_{P\ell(Q)}+U_{Q\ell(P)}=0$

for every pair of non-empty words $P$ and $Q$ such that $\lvert P\rvert+\lvert
Q\rvert=k$. For example,

$\displaystyle\begin{split}U_{12}+U_{21}&=0,\\\ U_{123}+U_{312}+U_{231}&=0,\\\
U_{1234}+U_{2143}+U_{3412}+U_{4321}&=0.\end{split}$ (1.6)

If $U_{P}$ satisfies the generalised Jacobi identities we will use the
notation $U_{\ell(P)}$ instead of $U_{P}$. In particular, this implies that
$U_{[P,Q]}=U_{P\ell(Q)}$.

##  2 The string-based rules in field theory

Bern and Kosower [43, 44] were the first to systematically investigate the
usefulness of the fact that many amplitudes in field theory can be represented
as the infinite string tension limit of appropriately chosen string
amplitudes. In this way they obtained rules for the construction of Feynman-
Schwinger type parameter integral representations of the one-loop on-shell
$n$-gluon amplitudes with a scalar, spinor or gluon loop. This work was then
generalized to the one-loop $n$-graviton amplitudes by Bern, Dunbar and
Shimada [47] with some details filled in later by Dunbar and Norridge [48].

###  2.1 The one-loop $n$-gluon amplitudes

The central object in the Bern-Kosower formalism is the colour-ordered one-
loop $n$-gluon correlator with a massless scalar loop. This amplitude in
general has a one-particle irreducible and a one-particle reducible
contribution. The irreducible one is encoded in the following master formula,

$\displaystyle\Gamma(k_{1},\varepsilon_{1};\ldots;k_{n},\varepsilon_{n})$
$\displaystyle=$ $\displaystyle{(-ig)}^{n}{\rm tr}\,(T^{a_{1}}\cdots
T^{a_{n}}){\int_{0}^{\infty}}dT{(4\pi
T)}^{-\frac{D}{2}}\int_{0}^{T}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\cdots\int_{0}^{\tau_{n-2}}d\tau_{n-1}$
(2.1) $\displaystyle\times\exp\biggl{\\{}\sum_{i,j=1}^{n}\Bigl{(}{1\over
2}G_{ij}k_{i}\cdot k_{j}-i\dot{G}_{ij}\varepsilon_{i}\cdot k_{j}+{1\over
2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}\Bigr{)}\biggr{\\}}\Bigl{|}_{\varepsilon_{1}\ldots\varepsilon_{n}}$

Here

$\displaystyle G_{ij}\equiv
G(\tau_{i},\tau_{j})=|\tau_{i}-\tau_{j}|-\frac{(\tau_{i}-\tau_{j})^{2}}{T}$
(2.2)

so that

$\displaystyle{\dot{G}}_{ij}\equiv{\partial\over\partial\tau_{i}}G_{ij}={\rm
sign}(\tau_{i}-\tau_{j})-2\frac{(\tau_{i}-\tau_{j})}{T}$ (2.3)

(the explicit form of
$\ddot{G}_{ij}\equiv\frac{\partial^{2}}{\partial\tau_{i}^{2}}G_{ij}$ is not
needed). The notation $\bigl{|}_{\varepsilon_{1}\ldots\varepsilon_{n}}$ means
that the exponential has to be expanded keeping only the terms linear in each
of the polarisation vectors $\varepsilon_{1},\ldots,\varepsilon_{n}$. The
resulting integrand is of the form

$\displaystyle\exp\biggl{\\{}\cdot\biggr{\\}}\Bigl{|}_{\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}}\equiv{(-i)}^{n}P_{n}(\dot{G}_{ij},\ddot{G}_{ij}){\rm
e}^{{1\over 2}\sum_{i,j=1}^{n}G_{ij}k_{i}\cdot k_{j}}$ (2.4)

with certain polynomials $P_{n}$.

The reducible contributions can be included by the following “pinching
procedure”: (i) Remove the second derivatives $\ddot{G}_{ij}$ contained in
$P_{n}$ through suitable partial integrations. This step will lead to the
replacement $P_{n}(\dot{G}_{ij},\ddot{G}_{ij})\ \to Q_{n}(\dot{G}_{ij})$. (ii)
Draw all possible $\phi^{3}$ one-loop diagrams $D_{i}$ with $n$ legs, labelled
$1,\ldots,n$ and following the ordering of the colour trace. (iii) The
pinching rule amounts to the replacement

$\displaystyle\dot{G}_{ij}\longrightarrow\frac{2}{s_{ij}}=\frac{2}{(k_{i}+k_{j})^{2}}$
(2.5)

removing the vertex and transferring the label $i$ to the ingoing leg (see
Fig. 1).

Figure 1:  Pinching of a vertex according to the Bern-Kosower rules.

The $\tau_{j}$ \- integration is omitted and the index $j$ replaced by $i$ in
the remaining $G_{kl}$ and $\dot{G}_{kl}$. (iv) The previous replacement can
only occur on a vertex with labels $i<j$, iff $Q_{n}$ contains $\dot{G}_{ij}$
linearly. Moreover, a diagram will contribute iff each vertex except the ones
attached directly to the loop corresponds to a possible pinch. The pinching
procedure starts with the outermost vertices and recursively removes the trees
attached to the loop.

As a further benefit of the integration-by-parts procedure, at this stage the
contributions of the spinor and gluon loop to the $N$-gluon amplitudes can be
constructed at the integrand level using a set of “loop replacement rules”
[43, 44].

The pinching procedure was streamlined in [49] by the introduction of a _pinch
operator_ , that we will present here again in the next section, and the
implementation of the multi-particle techniques that helped to have a better
understanding of the structure of the trees attached to the loop.

###  2.2 The one-loop $n$-graviton amplitudes

The gluon master formula (2.1) was generalized in [47, 48] to a master formula
for the irreducible one-loop $n$-graviton amplitudes with a massless scalar
loop:

$\displaystyle\Gamma[k_{1},h_{1};\cdots;k_{n},h_{n}]$ $\displaystyle=$
$\displaystyle-(-{\kappa\over 4})^{n}\int_{0}^{\infty}{dT\over T}(4\pi
T)^{-{D\over 2}}\int_{0}^{T}d\tau_{1}\cdots\int_{0}^{T}d\tau_{n}$ (2.6)
$\displaystyle\times\,\exp\Biggl{\\{}\sum_{i,j=1}^{n}\biggl{[}{1\over
2}G_{ij}k_{i}\cdot
k_{j}-i(\dot{G}_{ij}\varepsilon_{i}+\dot{\bar{G}}_{ij}\bar{\varepsilon}_{i})\cdot
k_{j}+{1\over 2}\ddot{G}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}$
$\displaystyle+{1\over
2}\ddot{\bar{G}}_{ij}\bar{\varepsilon}_{i}\cdot\bar{\varepsilon}_{j}+{1\over
2}H_{ij}(\varepsilon_{i}\cdot\bar{\varepsilon}_{j}+\varepsilon_{j}\cdot\bar{\varepsilon}_{i})\biggr{]}\Biggr{\\}}\Bigl{|}_{\varepsilon_{1}\ldots\varepsilon_{n}\bar{\varepsilon}_{1}\ldots\bar{\varepsilon}_{n}}$

Here we have used that on-shell the graviton polarisations can be chosen so as
to factorize,
$h_{i}^{\mu\nu}=\varepsilon_{i}^{\mu}\bar{\varepsilon}_{i}^{\nu}$. In the
absence of the terms with $H_{ij}$ this would, after the expansion of the
exponent, lead to a prefactor polynomial that simply factorizes into two
copies of the one of the gluonic case in (2.4),

$\displaystyle\exp\biggl{\\{}\cdot\biggr{\\}}\Bigl{|}_{\varepsilon_{1}\ldots\varepsilon_{n}\bar{\varepsilon}_{1}\ldots\bar{\varepsilon}_{n}}=P_{n}(\dot{\bar{G}}_{ij},\ddot{\bar{G}}_{ij})P_{n}(\dot{G}_{ij},\ddot{G}_{ij}){\rm
e}^{{1\over 2}\sum_{i,j=1}^{n}G_{ij}k_{i}\cdot k_{j}}$ (2.7)

At the string level, this comes from the factorisation of the closed string
modes into left-movers and right-movers. The additional terms involving
$H_{ij}$ stem from the fact that the left- and right-movers are coupled
through the zero mode of the string. In order to study the structure of the
now gravitational trees attached to the loop, (2.7) is all we need, but for
the calculation of the whole one-loop irreducible part we cannot neglect the
contributions that come from these terms.

Differently from the gluon case, it is now generally not possible to remove
all of the $\dot{G}_{ij},\dot{\bar{G}}_{ij}$ using partial integrations in the
variables $\tau_{i}$ alone. Instead, one has to return to the string level and
appeal to the fact that, before taking the infinite string tension limit, the
left-and right movers depended on independent variables $\tau_{i}$ and
$\bar{\tau}_{i}$. This allows one to write
$\dot{G}_{ij}={\partial\over\partial\tau_{i}}G_{ij}$,
$\dot{\bar{G}}_{ij}={\partial\over\partial\bar{\tau}_{i}}{\bar{G}}_{ij}$ and
to treat $\dot{G}_{ij},\ddot{G}_{ij}$ as independent of
$\dot{\bar{G}}_{ij},\ddot{\bar{G}}_{ij}$ in the partial integration procedure.
Additionally, the following rules must be used for derivatives hitting the
universal exponent,

$\displaystyle\frac{\partial}{\partial{\bar{\tau}_{k}}}\dot{G}_{ij}$
$\displaystyle=$ $\displaystyle{1\over
2}(\delta_{ki}H_{ij}-\delta_{kj}H_{ij})$ (2.8)
$\displaystyle{\partial\over\partial{\tau_{k}}}{{\dot{\bar{G}}}}_{ij}$
$\displaystyle=$ $\displaystyle{1\over
2}(\delta_{ki}H_{ij}-\delta_{kj}H_{ij})$ (2.9)
$\displaystyle{\partial\over\partial{\bar{\tau}_{k}}}\ddot{G}_{ij}$
$\displaystyle=$ $\displaystyle 0$ (2.10)
$\displaystyle{\partial\over\partial{\tau_{k}}}{{\ddot{\bar{G}}}}_{ij}$
$\displaystyle=$ $\displaystyle 0$ (2.11)

The $H_{ij}$ are to be treated as constants in the integration-by-parts.

After the removal of the $\ddot{G}_{ij},\ddot{\bar{G}}_{ij}$, the inclusion of
the reducible contributions can be achieved by a pinching procedure that
parallels the one for the gluon case above, except that the condition for the
pinching of a vertex with labels $i<j$ now is that the integrand should
contain both $\dot{G}_{ij}$ and $\dot{\bar{G}}_{ij}$ linearly, and that the
replacement (2.5) has to be modified to

$\displaystyle\dot{G}_{ij}{{\dot{\bar{G}}}}_{ij}\rightarrow\frac{4}{s_{ij}}$
(2.12)

After the recursive removal of all trees attached to the loop one has at hand
a parameter integral representation for the full on-shell $n$-graviton matrix
element with a scalar loop. Representations for other spins in the loop (Weyl
fermion, vector, gravitino, graviton) can again be obtained from this by
certain loop replacement rules that are essentially independent applications
of the above-mentioned QCD rules to the left- and right-mover parts, with an
additional substitution rule $H_{ij}\longrightarrow 2/T$ for the cross terms.

###  2.3 Symmetric partial integration

Returning to the gluon case, the integrand resulting from the integration-by-
parts procedure is unique for the two- and three-gluon cases, but starting
from $n=4$ ambiguities appear [46]. Different algorithms lead to different
equivalent integrands that are all free of $\ddot{G}_{ij}$s and suitable for
an application of the pinching and loop replacement rules. Considering that
the master formula (2.1) possesses (apart from the colour ordering) manifest
permutation (or Bose) symmetry between the $N$ gluons, in [52] the following
_symmetric partial integration_ algorithm was proposed that preserves this
symmetry at each step.

  1. 1.

In every step, partially integrate away all $\ddot{G}_{ij}$s appearing in the
term under inspection simultaneously. This is possible since different
$\ddot{G}_{ij}$s never share variables.

  2. 2.

In the first step, for every $\ddot{G}_{ij}$ partially integrate both over
$\tau_{i}$ and $\tau_{j}$, and take the mean of the results.

  3. 3.

At every following step, any $\ddot{G}_{ij}$ appearing must have been created
in the previous step. Therefore either both $i$ and $j$ were used in the
previous step, or just one of them. If both, the rule is to again use both
variables in the actual step for partial integration, and take the mean of the
results. If only one of them was used in the previous step, then the other one
should be used in the actual step.

This algorithm transforms the polynomial $P_{n}(\dot{G}_{ij},\ddot{G}_{ij})$
into a polynomial $Q_{n}(\dot{G}_{ij})$ that, unlike $P_{n}$, is homogeneous
not only in the polarisations, but also in the momenta. Together with the
manifest permutation invariance, this makes it possible to write $Q_{n}$
extremely compactly using a decomposition into bicycles and tails. A bicycle
of length $k$ is defined by

$\displaystyle\dot{G}(i_{1},i_{2},\cdots,i_{k})\equiv\dot{G}_{i_{1}i_{2}}\dot{G}_{i_{2}i_{3}}\cdots\dot{G}_{i_{k}i_{1}}Z_{k}(i_{1},i_{2},\dots,i_{k})$
(2.13)

where

$\displaystyle
Z_{k}(i_{1},i_{2},\dots,i_{k})\equiv\Bigl{(}\frac{1}{2}\Bigr{)}^{\delta_{k2}}{\rm
tr}(f_{i_{1}}\cdots f_{i_{k}})$ (2.14)

and
$f_{i}^{\mu\nu}=k_{i}^{\mu}\varepsilon_{i}^{\nu}-\varepsilon_{i}^{\mu}k_{i}^{\nu}$
is the gluon field strength tensor. The tails are the left-overs after
factorizing out all possible bicycles. The $k$-tail
$T(i_{1},i_{2},\cdots,i_{k})$ involves $k$ polarisation vectors that have not
yet been absorbed into field strength tensors (it is possible to do so using
further partial integrations with non-local coefficients [53], but here we
will not follow this route).

For example, the cycle decompositions of $Q_{3}$ and $Q_{4}$ read

$\displaystyle\begin{split}Q_{3}&=Q_{3}^{3}+Q_{3}^{2}\\\
Q_{3}^{3}&=\dot{G}(1,2,3)\\\
Q_{3}^{2}&=\dot{G}(1,2)T(3)+\dot{G}(2,3)T(1)+\dot{G}(3,1)T(2)\end{split}$
(2.15)
$\displaystyle\begin{split}Q_{4}&=Q_{4}^{4}+Q_{4}^{3}+Q_{4}^{2}+Q_{4}^{22}\\\
Q_{4}^{4}&=\dot{G}(1,2,3,4)+\dot{G}(1,2,4,3)+\dot{G}(1,3,2,4)\\\
Q_{4}^{3}&=\dot{G}(1,2,3)T(4)+\dot{G}(2,3,4)T(1)+\dot{G}(3,4,1)T(2)+\dot{G}(4,1,2)T(3)\\\
Q_{4}^{2}&=\dot{G}(1,2)T(3,4)+\dot{G}(1,3)T(2,4)+\dot{G}(1,4)T(2,3)\\\ &\hskip
8.5359pt+\dot{G}(2,3)T(1,4)+\dot{G}(2,4)T(1,3)+\dot{G}(3,4)T(1,2)\\\
Q_{4}^{22}&=\dot{G}(1,2)\dot{G}(3,4)+\dot{G}(1,3)\dot{G}(2,4)+\dot{G}(1,4)\dot{G}(2,3)\end{split}$
(2.16)

the superscripts on the left-hand side indicating the cycle-content of a term.
Here the one- and two-tails appear,

$\displaystyle T(a)$ $\displaystyle\equiv$
$\displaystyle\sum_{r}\dot{G}_{ar}\varepsilon_{a}\cdot k_{r}$ (2.17)
$\displaystyle T(a,b)$ $\displaystyle\equiv$
$\displaystyle\sum_{{r,s}\atop{(r,s)\neq(b,a)}}\dot{G}_{ar}\varepsilon_{a}\cdot
k_{r}\dot{G}_{s}\varepsilon_{b}\cdot k_{s}+{1\over
2}\dot{G}_{ab}\varepsilon_{a}\cdot\varepsilon_{b}\Bigl{[}\sum_{r\neq
b}\dot{G}_{ar}k_{a}\cdot k_{r}-\sum_{s\neq a}\dot{G}_{s}k_{b}\cdot
k_{s}\Bigr{]}$ (2.18)

Note that the cycle decomposition of $Q_{N}$ involves the tails of length up
to $N-2$. Up to length 4 the tails are given in [54]; the five-tail was
computed in [49].

For the graviton amplitudes, the above partial integration rules imply that
the symmetric partial integration algorithm can be applied separately in the
variables $\tau_{i}$ to remove the $\ddot{G}_{ij}$ and in the $\bar{\tau}_{i}$
to remove the $\ddot{\bar{G}}_{ij}$, with additional terms involving $H_{ij}$
generated by the first two rules in (2.11). The integrand can thus be ordered
according to the powers of $H_{ij}$, where the terms in the prefactor
polynomial not containing any $H_{ij}$ can be factorised into
$Q_{n}(\dot{\bar{G}})Q_{n}(\dot{G})$ and terms with $m$ factors of $H$
containing $(n-m)$ factors of $\dot{G}$ and $\dot{\bar{G}}$ each.

In this factorised term $Q_{n}(\dot{\bar{G}})Q_{n}(\dot{G})$ we can apply all
that we learned in the gluonic case in [49]. One of the lessons there was that
we have to pinch only the tails in order to extract the multi-particle
polarisations, that happen to obey colour-kinematics duality. Now we will have
“squared” structures in the integrands like

$\displaystyle\bar{T}(1,2,\dots,n-2)T(1,2,\dots,n-2)$ (2.19)

where we use the shorthand notation $\bar{T}(1,2,\dots,n-2)$ to imply that it
only depends on $\dot{\bar{G}}$’s.

The idea now is to apply the procedure we implemented in [49] to calculate the
Berends-Giele currents from the pinching of the squared tails to the
gravitational case with this new structure in (2.19). In the next section we
will review this procedure for gluons before going back to the gravity
version.

##  3 Colour-kinematics duality

In the present section we will investigate how the colour-kinematics duality
can be made manifest at the level of Yang-Mills Berends-Giele currents. We
begin with a short summary of the results obtained in [49], where a systematic
procedure to construct such currents in the BCJ gauge from the Bern-Kosower
formalism for one-loop gluon amplitudes is proposed.

###  3.1 Colour-stripped Berends-Giele currents from the Bern-Kosower
formalism

The main ingredients for the approach advocated in [49] are the symmetric
partial integration algorithm and the Bern-Kosower rules reviewed in the
preceding section. These two combine to produce the permutation invariant
integrand $Q_{n}(\dot{G})$ and, for two adjacent legs $i$ and $j$ with $i<j$,
the “pinch operator” acting on $Q_{n}(\dot{G})$ as

$\mathscr{D}_{ij}Q_{n}(\dot{G})=\frac{\partial}{\partial\dot{G}_{ij}}Q_{n}(\dot{G})\bigg{|}_{\begin{subarray}{c}\dot{G}_{ij}=0\phantom{iiii}\\\
\dot{G}_{jk}\rightarrow\dot{G}_{ik}\end{subarray}}.$ (3.1)

The latter is what diagrammatically corresponds to pinching the two adjacent
legs $i$ and $j$. Thus the complete effect of the pinching procedure in the
Bern-Kosower formalism may be implemented by the iterated action of pinch
operators.

With the help of the foregoing we can build the colour-stripped Yang-Mills
Berends-Giele currents directly in the BCJ gauge for both, the field strength
and the polarisation. For the purposes of the present paper, however, we can
restrict our attention to the latter. In doing so, we note –as pointed out in
at the end of Section IV of [49]– that the part of the polynomial
$Q_{n}(\dot{G})$ responsible for the extraction of the multi-particle fields
associated with the colour-stripped Berends-Giele polarisation current, after
applying the pinch operator consecutively $n-2$ times for each pair of labels
following the cyclic order, is the $(n-2)$-tail $T(1,2,\dots,n-2)$. More
explicitly, the exact meaning of this expression is

$\displaystyle\begin{split}\mathscr{D}_{1(n-1)}\mathscr{D}_{1(n-2)}\cdots\mathscr{D}_{13}\mathscr{D}_{12}T(1,2,\dots,n-2)=\varepsilon_{12\cdots(n-2)}\cdot
k_{n-1}.\end{split}$ (3.2)

The multi-particle polarisation field $\varepsilon^{\mu}_{12\cdots(n-2)}$
obtained this way satisfies the GJI of order $n-2$ in $1,2,\dots,n-2$. This
property for these kind of polynomials has been verified up to degree $n=9$.
For this reason, it is more accurate to write
$\varepsilon^{\mu}_{\ell(12\cdots(n-2))}$ instead of
$\varepsilon^{\mu}_{12\cdots(n-2)}$, or, more generally,
$\varepsilon^{\mu}_{\ell(P)}$ instead of $\varepsilon^{\mu}_{P}$ for any word
$P$.

Armed with these results, it is possible to find the explicit expression for
the colour-stripped Berends-Giele polarisation current in terms of the multi-
particle polarisation fields $\varepsilon^{\mu}_{P}$. To that end, it is very
convenient to introduce a combinatorial artifact that helps us to keep track
of the correspondence between nested Lie brackets and planar binary trees.
This is termed the “binary tree map” in [55], but we will refer to it as the
colour-stripped Berends-Giele map. It is defined as the map $b_{\mathsf{cs}}$
acting on all words and determined recursively by

$\displaystyle\begin{split}b_{\mathsf{cs}}(i)&=i,\\\
b_{\mathsf{cs}}(P)&=\frac{1}{s_{P}}\sum_{P=QR}[b_{\mathsf{cs}}(Q),b_{\mathsf{cs}}(R)],\end{split}$
(3.3)

where $s_{P}$ is the Mandelstam invariant, and where $\sum_{P=QR}$ denotes the
sum over all possible deconcatenations of the word $P$ into $Q$ and $R$. Also
as a matter of notation, for an arbitrary labelled object $U_{P}$, such as the
multi-particle polarisation fields $\varepsilon^{\mu}_{P}$, we bring the
definition from [55] for the replacement of words by such object as

$\llbracket U\rrbracket\circ P=U_{P}.$ (3.4)

With this background in mind, the colour-stripped Berends-Giele polarisation
current associated with the multi-particle polarisation fields
$\varepsilon^{\mu}_{P}$ is simply

$\mathcal{A}^{\mu}_{P}=\llbracket\varepsilon^{\mu}\rrbracket\circ
b_{\mathsf{cs}}(P).$ (3.5)

We note, moreover, that the GJI satisfied by the multi-particle polarisation
fields $\varepsilon^{\mu}_{P}$ translate directly into the shuffle symmetry
$\mathcal{A}^{\mu}_{P\shuffle Q}=0$. As an example, the colour-stripped
Berends-Giele polarisation currents up to multiplicity four would read

$\displaystyle\begin{split}\mathcal{A}^{\mu}_{1}&=\varepsilon^{\mu}_{1},\\\
\mathcal{A}^{\mu}_{12}&=\frac{\varepsilon^{\mu}_{[1,2]}}{s_{12}},\\\
\mathcal{A}^{\mu}_{123}&=\frac{\varepsilon^{\mu}_{[[1,2],3]}}{s_{12}s_{123}}+\frac{\varepsilon^{\mu}_{[1,[2,3]]}}{s_{23}s_{123}},\\\
\mathcal{A}^{\mu}_{1234}&=\frac{\varepsilon^{\mu}_{[[[1,2],3],4]}}{s_{12}s_{123}s_{1234}}+\frac{\varepsilon^{\mu}_{[[1,[2,3]],4]}}{s_{123}s_{1234}s_{23}}+\frac{\varepsilon^{\mu}_{[[1,2],[3,4]]}}{s_{12}s_{1234}s_{34}}+\frac{\varepsilon^{\mu}_{[1,[[2,3],4]]}}{s_{1234}s_{23}s_{234}}+\frac{\varepsilon^{\mu}_{[1,[2,[3,4]]]}}{s_{1234}s_{234}s_{34}}.\end{split}$
(3.6)

In these expressions, the multi-particle polarisation fields
$\varepsilon^{\mu}_{[1,2]}$, $\varepsilon^{\mu}_{[[1,2],3]}$ and
$\varepsilon^{\mu}_{[[[1,2],3],4]}$ can be written in a compact way as

$\displaystyle\begin{split}\varepsilon^{\mu}_{[1,2]}&=\tfrac{1}{2}\big{\\{}(k_{1}\cdot\varepsilon_{2})\varepsilon_{1}^{\mu}-(k_{2}\cdot\varepsilon_{1})\varepsilon_{2}^{\mu}+\varepsilon_{1\nu}f_{2}^{\mu\nu}-\varepsilon_{2\nu}f_{1}^{\mu\nu}\big{\\}},\\\
\varepsilon^{\mu}_{[[1,2],3]}&=\tfrac{1}{2}\big{\\{}(k_{3}\cdot\varepsilon_{[1,2]})\varepsilon_{3}^{\mu}-(k_{12}\cdot\varepsilon_{3})\varepsilon_{[1,2]}^{\mu}+\varepsilon_{3\nu}f_{[1,2]}^{\mu\nu}-\varepsilon_{[1,2]\nu}f_{3}^{\mu\nu}\big{\\}}-k^{\mu}_{123}h_{123},\\\
\varepsilon^{\mu}_{[[[1,2],3],4]}&=\tfrac{1}{2}\big{\\{}(k_{4}\cdot\varepsilon_{[[1,2],3]})\varepsilon_{4}^{\mu}-(k_{123}\cdot\varepsilon_{4})\varepsilon_{[[1,2],3]}^{\mu}+\varepsilon_{4\nu}f_{[[1,2],3]}^{\mu\nu}-\varepsilon_{[[1,2],3]\nu}f_{4}^{\mu\nu}\big{\\}}+(k_{12}\cdot
k_{3})\varepsilon_{3}^{\mu}h_{124}\\\ &\quad\quad\,+(k_{1}\cdot
k_{2})(\varepsilon^{\mu}_{2}h_{134}-\varepsilon^{\mu}_{1}h_{234})-k^{\mu}_{1234}h_{1234},\end{split}$
(3.7)

where we have set

$\displaystyle\begin{split}f^{\mu\nu}_{i}&=k^{\mu}_{i}\varepsilon^{\nu}_{i}-k^{\nu}_{i}\varepsilon^{\mu}_{i},\\\
f^{\mu\nu}_{[1,2]}&=k^{\mu}_{12}\varepsilon^{\nu}_{[1,2]}-k^{\nu}_{12}\varepsilon^{\mu}_{[1,2]}-(k_{1}\cdot
k_{2})(\varepsilon^{\mu}_{1}\varepsilon^{\nu}_{2}-\varepsilon^{\nu}_{1}\varepsilon^{\mu}_{2}),\\\
f^{\mu\nu}_{[[1,2],3]}&=k^{\mu}_{123}\varepsilon^{\nu}_{[[1,2],3]}-k^{\nu}_{123}\varepsilon^{\mu}_{[[1,2],3]}-(k_{12}\cdot
k_{3})(\varepsilon^{\mu}_{[1,2]}\varepsilon^{\nu}_{3}-\varepsilon^{\nu}_{[1,2]}\varepsilon^{\mu}_{3})-(k_{1}\cdot
k_{2})(\varepsilon^{\mu}_{1}\varepsilon^{\nu}_{[2,3]}+\varepsilon^{\mu}_{[1,3]}\varepsilon^{\nu}_{2})\\\
&\quad\,+(k_{1}\cdot
k_{2})(\varepsilon^{\nu}_{1}\varepsilon^{\mu}_{[2,3]}-\varepsilon^{\nu}_{[1,3]}\varepsilon^{\mu}_{2})\\\
h_{123}&=\tfrac{1}{4}(\varepsilon_{1}\cdot\varepsilon_{2})\varepsilon_{3}\cdot(k_{2}-k_{1}),\\\
h_{1234}&=\tfrac{1}{4}\big{[}\varepsilon_{1}\cdot\varepsilon_{2}\,\varepsilon_{3}\cdot
k_{2}\varepsilon_{4}\cdot\,\left(k_{1}-k_{23}\right)+\tfrac{1}{2}\left(\varepsilon_{1}\cdot\varepsilon_{2}\,\varepsilon_{3}\cdot\varepsilon_{4}\,k_{2}\cdot
k_{3}\right)-(123\rightarrow 312)\big{]}\,-(1\leftrightarrow 2).\end{split}$
(3.8)

The bracketed notation in the words tell us about the GJI satisfied by the
given object. Let us remind the GJI up to rank four, from (1.6), for the
polarisations

$\displaystyle\begin{split}&\varepsilon^{\mu}_{[1,2]}+\varepsilon^{\mu}_{[2,1]}=0,\\\
&\varepsilon^{\mu}_{[[1,2],3]}+\varepsilon^{\mu}_{[[1,2],3]}=0,\quad\varepsilon^{\mu}_{[[1,2],3]}+\varepsilon^{\mu}_{[[3,1],2]}+\varepsilon^{\mu}_{[[2,3],1]}=0,\\\
&\varepsilon^{\mu}_{[[1,2],3],4]}+\varepsilon^{\mu}_{[[1,2],3],4]}=0,\quad\varepsilon^{\mu}_{[[1,2],3],4]}+\varepsilon^{\mu}_{[[3,1],2],4]}+\varepsilon^{\mu}_{[[2,3],1],4]}=0,\\\
&\varepsilon^{\mu}_{[[1,2],3],4]}-\varepsilon^{\mu}_{[[1,2],4],3]}+\varepsilon^{\mu}_{[[3,4],1],2]}-\varepsilon^{\mu}_{[[3,4],2],1]}=0.\end{split}$
(3.9)

In addition, using the the symmetry properties of the bracket,
$\varepsilon^{\mu}_{[[1,2],3]}=-\varepsilon^{\mu}_{[3,[1,2]]}$,
$\varepsilon^{\mu}_{[[1,[2,3]],4]}=-\varepsilon^{\mu}_{[[[2,3],1],4]}$,
$\varepsilon^{\mu}_{[1,[[2,3],4]]}=-\varepsilon^{\mu}_{[[[2,3],4],1]}$,
$\varepsilon^{\mu}_{[1,[2,[3,4]]]}=\varepsilon^{\mu}_{[[[3,4],2],1]}$ and
$\varepsilon^{\mu}_{[[1,2],[3,4]]}=\varepsilon^{\mu}_{[[[1,2],3],4]}-\varepsilon^{\mu}_{[[[1,2],4],3]}$,
so that these multi-particle polarisation fields are obtained from the
formulas in (3.7) by a simple relabelling. It is also worth pointing out that
$f^{\mu\nu}_{i}$, $f^{\mu\nu}_{[1,2]}$ and $f^{\mu\nu}_{[[1,2],3]}$ in (3.8)
respectively correspond to the single-, two- and three-particle field strength
produced following the procedure of [49].

Now that we have an explicit form of the the colour-stripped Berends-Giele
polarisation currents, the next task is to write down the colour-stripped
perturbiner expansion. This is a simple matter: we just set it to be
generating series

$A^{\mu}(x)=\sum_{n\geq
1}\sum_{P\in\mathscr{W}_{n}}\mathcal{A}^{\mu}_{P}\operatorname{e}^{\mathrm{i}\mkern
1.0muk_{P}\cdot x}T^{a_{P}},$ (3.10)

where $\mathscr{W}_{n}$ denotes the set of words of length $n$. It is
important to note that the shuffle symmetry satisfied by the constituent
currents $\mathcal{A}^{\mu}_{P}$ guarantees that the generating series (3.10)
is a Lie algebra-valued field. This expansion does not come directly from the
Yang-Mills action, since in our case we have only trivalent vertices with no
use of auxiliary fields.

To complete our discussion we must also mention how the colour-stripped
Berends-Giele polarisation currents $\mathcal{A}^{\mu}_{P}$ are related to the
scattering amplitudes in Yang-Mills theory. At tree level, the colour-ordered
partial amplitude of $n$ gluons is determined through the Berends-Giele
formula

$\mathscr{A}^{\mathrm{tree}}(1,2,\dots,n)=s_{12\cdots(n-1)}\mathcal{A}^{\mu}_{12\cdots(n-1)}\mathcal{A}_{n\mu}.$
(3.11)

The factor $s_{12\cdots(n-1)}$ is inserted to cancel the off-shell propagator
inside $\mathcal{A}_{12\cdots(n-1)}$. Now that we are assuming momentum
conservation and have on-shell external legs. There are other off-shell terms
that cancel out, the ones of the form $k^{\mu}_{P}h_{P}$ at the end of each
polarisation. Finally it may be remarked that, by virtue of the shuffle
symmetry, the partial amplitudes in the form of (3.11) satisfy the Kleiss-
Kuijf relations [56].

###  3.2 Colour-dressed Berends-Giele currents

Now we turn our attention to obtaining the colour-dressed Berends-Giele
polarisation currents from the multi-particle polarisation fields
$\varepsilon_{P}^{\mu}$. These type of currents were obtained first for Yang-
Mills in the Lorenz gauge in [50] using perturbiner methods. Back to our case,
we need to make some small, but important, changes in the notation introduced
thus far. In the first place, we need to modify the colour-stripped Berends-
Giele map (3.3) by a colour-dressed version of it, which we write as
$b_{\mathsf{cd}}$. Here we borrow the prescription already encountered in
[57]. Namely, we define $b_{\mathsf{cd}}$ as the map acting on all ordered
words and determined recursively by

$\displaystyle\begin{split}b_{\mathsf{cd}}(i)&=i,\\\
b_{\mathsf{cd}}(P)&=\frac{1}{2s_{P}}\sum_{P=Q\cup
R}[b_{\mathsf{cd}}(Q),b_{\mathsf{cd}}(R)],\end{split}$ (3.12)

where $\sum_{P=Q\cup R}$ denotes the sum over all possible ways of
distributing the letters of the ordered word $P$ into non-empty ordered words
$Q$ and $R$. We remark that the factor of $2$ in the denominator can be
dropped if we impose the condition that $\lvert Q\rvert\geq\lvert R\rvert$. In
the second place, for each ordered word $P=i_{1}i_{2}\cdots i_{n}$ of length
$n$, we employ the notation $c_{P}^{a}$ to indicate the product of colour
factors determined by

$c_{P}^{a}=\tilde{f}_{a_{i_{1}}a_{i_{2}}}^{\phantom{a_{i_{1}}a_{i_{2}}}b}\tilde{f}_{ba_{i_{3}}}^{\phantom{ba_{i_{3}}}c}\cdots\tilde{f}_{da_{i_{n-1}}}^{\phantom{da_{i_{n-1}}}e}\tilde{f}_{ea_{i_{n}}}^{\phantom{ea_{i_{n}}}a},$
(3.13)

with the understanding that $c_{i}^{a}=\delta^{a}_{\phantom{a}a_{i}}$. We
further put

$c_{[P,Q]}^{a}=\tilde{f}_{bc}^{\phantom{bc}a}c_{P}^{b}c_{Q}^{c}$ (3.14)

for any pair of ordered words $P$ and $Q$. In the third place, given two
arbitrary labelled objects $U_{P}$ and $V_{P}$, we define the replacement of
ordered words by the product of such objects as

$\llbracket U\otimes V\rrbracket\circ P=U_{P}V_{P}.$ (3.15)

By making use of the foregoing, one can show that we can write the colour-
dressed Berends-Giele polarisation currents in the form

$\mathcal{A}_{P}^{a\mu}=\llbracket
c^{a}\otimes\varepsilon^{\mu}\rrbracket\circ b_{\mathsf{cd}}(P).$ (3.16)

At this point, however, we should perhaps emphasise that this way of
representing the colour-dressed Berends-Giele polarisation currents is always
possible regardless of whether or not the multi-particle polarisation fields
$\varepsilon^{\mu}_{P}$ satisfy the GJI. When they do, as it is the case in
the present discussion, we see that such identities mirror the GJI satisfied
by the colour factor $c_{P}^{a}$. Hence, we are led to the conclusion that the
“factorisation” of the colour-dressed Berends-Giele polarisation currents
given in (3.16) is a realisation of the colour-kinematics duality. This will
be pointed out, in a somewhat simplified context, and from a more algebraic
perspective, in [57]. In the next section, we will see that in terms of this
factorisation, the double-copy prescription is straightforward to phrase.

We shall now proceed to write down explicitly the colour-dressed Berends-Giele
polarisation currents up to multiplicity four, in order to familiarise
ourselves with formula (3.16). We first consider the single-particle case in
which $P=1$. Then we at once obtain

$\mathcal{A}_{1}^{a\mu}=\delta^{a}_{\phantom{a}a_{1}}\varepsilon^{\mu}_{1}.$
(3.17)

Next we consider the two-particle case in which $P=12$. In this case, the only
possible way of distributing the letters is $(Q,R)=(1,2)$, and thus we find
that colour-dressed Berends-Giele polarisation current
$\mathcal{A}_{12}^{a\mu}$ acquires the form

$\mathcal{A}_{12}^{a\mu}=\frac{c^{a}_{[1,2]}\varepsilon^{\mu}_{[1,2]}}{s_{12}},$
(3.18)

with colour factor
$c^{a}_{[1,2]}=\tilde{f}_{a_{1}a_{2}}^{\phantom{a_{1}a_{2}}a}$ and two-
particle polarisation field $\varepsilon^{\mu}_{[1,2]}$ given by (3.7). Let us
next take up the three-particle case in which $P=123$. In this case, the
possible ways of distributing the letters that contribute to the sum are
$(Q,R)=(12,3),(13,2),(23,1)$. Therefore, after a straightforward calculation
making use of the recursion (3.12) we obtain for the colour-dressed Berends-
Giele polarisation current $\mathcal{A}_{123}^{a\mu}$ the formula

$\mathcal{A}_{123}^{a\mu}=\frac{c^{a}_{[[1,2],3]}\varepsilon^{\mu}_{[[1,2],3]}}{s_{12}s_{123}}+\frac{c^{a}_{[[1,3],2]}\varepsilon^{\mu}_{[[1,3],2]}}{s_{13}s_{123}}+\frac{c^{a}_{[[2,3],1]}\varepsilon^{\mu}_{[[2,3],1]}}{s_{23}s_{123}},$
(3.19)

with colour factors
$c^{a}_{[[1,2],3]}=\tilde{f}_{a_{1}a_{2}}^{\phantom{a_{1}a_{2}}b}\tilde{f}_{ba_{3}}^{\phantom{ba_{3}}a}$,
$c^{a}_{[[1,3],2]}=\tilde{f}_{a_{1}a_{3}}^{\phantom{a_{1}a_{3}}b}\tilde{f}_{ba_{2}}^{\phantom{ba_{2}}a}$,
$c^{a}_{[[2,3],1]}=\tilde{f}_{a_{2}a_{3}}^{\phantom{a_{2}a_{3}}b}\tilde{f}_{ba_{1}}^{\phantom{ba_{1}}a}$
and three-particle polarisation fields $\varepsilon^{\mu}_{[[1,2],3]}$,
$\varepsilon^{\mu}_{[[1,3],2]}$, $\varepsilon^{\mu}_{[[2,3],1]}$ given by
(3.7) after relabelling. Finally, we consider the four-particle case in which
$P=1234$. In this case, the possible ways of distributing the letters that
contribute to the sum are
$(Q,R)=(123,4),(124,3),(134,2),(234,1),(12,34),(13,24),(23,14)$. By analogy
with the calculation leading to (3.19), we find that the colour-dressed
Berends-Giele polarisation current $\mathcal{A}_{1234}^{a\mu}$ may be
represented in the form

$\displaystyle\begin{split}\mathcal{A}_{1234}^{a\mu}&=\frac{c^{a}_{[[[1,2],3],4]}\varepsilon^{\mu}_{[[[1,2],3],4]}}{s_{12}s_{123}s_{1234}}+\frac{c^{a}_{[[[1,2],4],3]}\varepsilon^{\mu}_{[[[1,2],4],3]}}{s_{12}s_{124}s_{1234}}+\frac{c^{a}_{[[[1,3],4],2]}\varepsilon^{\mu}_{[[[1,3],4],2]}}{s_{13}s_{134}s_{1234}}+\frac{c^{a}_{[[[2,3],4],1]}\varepsilon^{\mu}_{[[[2,3],4],1]}}{s_{23}s_{234}s_{1234}}\\\
&\quad\,+\frac{c^{a}_{[[[1,3],2],4]}\varepsilon^{\mu}_{[[[1,3],2],4]}}{s_{13}s_{123}s_{1234}}+\frac{c^{a}_{[[[1,4],2],3]}\varepsilon^{\mu}_{[[[1,4],2],3]}}{s_{14}s_{124}s_{1234}}+\frac{c^{a}_{[[[1,4],3],2]}\varepsilon^{\mu}_{[[[1,4],3],2]}}{s_{14}s_{134}s_{1234}}+\frac{c^{a}_{[[[2,3],1],4]}\varepsilon^{\mu}_{[[[2,3],1],4]}}{s_{23}s_{123}s_{1234}}\\\
&\quad\,+\frac{c^{a}_{[[[2,4],1],3]}\varepsilon^{\mu}_{[[[2,4],1],3]}}{s_{24}s_{124}s_{1234}}+\frac{c^{a}_{[[[2,4],3],1]}\varepsilon^{\mu}_{[[[2,4],3],1]}}{s_{24}s_{234}s_{1234}}+\frac{c^{a}_{[[[3,4],1],2]}\varepsilon^{\mu}_{[[[3,4],1],2]}}{s_{34}s_{134}s_{1234}}+\frac{c^{a}_{[[[3,4],2],1]}\varepsilon^{\mu}_{[[[3,4],2],1]}}{s_{34}s_{234}s_{1234}}\\\
&\quad\,+\frac{c^{a}_{[[1,2],[3,4]]}\varepsilon^{\mu}_{[[1,2],[3,4]]}}{s_{12}s_{34}s_{1234}}+\frac{c^{a}_{[[1,3],[2,4]]}\varepsilon^{\mu}_{[[1,3],[2,4]]}}{s_{13}s_{24}s_{1234}}+\frac{c^{a}_{[[1,4],[2,3]]}\varepsilon^{\mu}_{[[1,4],[2,3]]}}{s_{14}s_{23}s_{1234}}.\end{split}$
(3.20)

Here the colour factors are easily determined from (3.13) and (3.14) as

$\displaystyle c^{a}_{[[[1,2],3],4]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{2}}b}_{a_{1}a_{2}}\tilde{f}^{\phantom{ba_{3}}c}_{ba_{3}}\tilde{f}^{\phantom{ca_{4}}a}_{ca_{4}},$
$\displaystyle\qquad c^{a}_{[[[1,2],4],3]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{2}}b}_{a_{1}a_{2}}\tilde{f}^{\phantom{ba_{4}}c}_{ba_{4}}\tilde{f}^{\phantom{ca_{3}}a}_{ca_{3}},$
$\displaystyle\qquad c^{a}_{[[[1,3],4],2]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{3}}b}_{a_{1}a_{3}}\tilde{f}^{\phantom{ba_{4}}c}_{ba_{4}}\tilde{f}^{\phantom{ca_{2}}a}_{ca_{2}},$
$\displaystyle c^{a}_{[[[2,3],4],1]}$
$\displaystyle=\tilde{f}^{\phantom{a_{2}a_{3}}b}_{a_{2}a_{3}}\tilde{f}^{\phantom{ba_{4}}c}_{ba_{4}}\tilde{f}^{\phantom{ca_{1}}a}_{ca_{1}},$
$\displaystyle c^{a}_{[[[1,3],2],4]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{3}}b}_{a_{1}a_{3}}\tilde{f}^{\phantom{ba_{2}}c}_{ba_{2}}\tilde{f}^{\phantom{ca_{4}}a}_{ca_{4}},$
$\displaystyle c^{a}_{[[[1,4],2],3]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{4}}b}_{a_{1}a_{4}}\tilde{f}^{\phantom{ba_{2}}c}_{ba_{2}}\tilde{f}^{\phantom{ca_{3}}a}_{ca_{3}},$
$\displaystyle c^{a}_{[[[1,4],3],2]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{4}}b}_{a_{1}a_{4}}\tilde{f}^{\phantom{ba_{3}}c}_{ba_{3}}\tilde{f}^{\phantom{ca_{2}}a}_{ca_{2}},$
$\displaystyle c^{a}_{[[[2,3],1],4]}$
$\displaystyle=\tilde{f}^{\phantom{a_{2}a_{3}}b}_{a_{2}a_{3}}\tilde{f}^{\phantom{ba_{1}}c}_{ba_{1}}\tilde{f}^{\phantom{ca_{4}}a}_{ca_{4}},$
$\displaystyle c^{a}_{[[[2,4],1],3]}$
$\displaystyle=\tilde{f}^{\phantom{a_{2}a_{4}}b}_{a_{2}a_{4}}\tilde{f}^{\phantom{ba_{1}}c}_{ba_{1}}\tilde{f}^{\phantom{ca_{3}}a}_{ca_{3}},$
$\displaystyle c^{a}_{[[[2,4],3],1]}$
$\displaystyle=\tilde{f}^{\phantom{a_{2}a_{4}}b}_{a_{2}a_{4}}\tilde{f}^{\phantom{ba_{3}}c}_{ba_{3}}\tilde{f}^{\phantom{ca_{1}}a}_{ca_{1}},$
$\displaystyle c^{a}_{[[[3,4],1],2]}$
$\displaystyle=\tilde{f}^{\phantom{a_{3}a_{4}}b}_{a_{3}a_{4}}\tilde{f}^{\phantom{ba_{1}}c}_{ba_{1}}\tilde{f}^{\phantom{ca_{2}}a}_{ca_{2}},$
$\displaystyle c^{a}_{[[[3,4],2],1]}$
$\displaystyle=\tilde{f}^{\phantom{a_{3}a_{4}}b}_{a_{3}a_{4}}\tilde{f}^{\phantom{ba_{2}}c}_{ba_{2}}\tilde{f}^{\phantom{ca_{1}}a}_{ca_{1}},$
$\displaystyle c^{a}_{[[1,2],[3,4]]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{2}}b}_{a_{1}a_{2}}\tilde{f}^{\phantom{a_{3}a_{4}}c}_{a_{3}a_{4}}\tilde{f}^{\phantom{bc}a}_{bc},$
$\displaystyle c^{a}_{[[1,3],[2,4]]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{3}}b}_{a_{1}a_{3}}\tilde{f}^{\phantom{a_{2}a_{4}}c}_{a_{2}a_{4}}\tilde{f}^{\phantom{bc}a}_{bc},$
$\displaystyle c^{a}_{[[1,4],[2,3]]}$
$\displaystyle=\tilde{f}^{\phantom{a_{1}a_{4}}b}_{a_{1}a_{4}}\tilde{f}^{\phantom{a_{2}a_{3}}c}_{a_{2}a_{3}}\tilde{f}^{\phantom{bc}a}_{bc}.$

As for the four-particle polarisation fields, keeping in mind the identities
$\varepsilon^{\mu}_{[[1,2],[3,4]]}=\varepsilon^{\mu}_{[[[1,2],3],4]}-\varepsilon^{\mu}_{[[[1,2],4],3]}$,
$\varepsilon^{\mu}_{[[1,3],[2,4]]}=\varepsilon^{\mu}_{[[[1,3],2],4]}-\varepsilon^{\mu}_{[[[1,3],4],2]}$
and
$\varepsilon^{\mu}_{[[1,4],[2,3]]}=\varepsilon^{\mu}_{[[[1,4],2],3]}-\varepsilon^{\mu}_{[[[1,4],3],2]}$,
they are all determined by (3.7) with the necessary relabellings.

Having obtained the expression (3.16) for the colour-dressed Berends-Giele
polarisation currents, we can of course then obtain the colour-dressed
perturbiner expansion. This is simply given as the generating series

$A^{a\mu}(x)=\sum_{n\geq
1}\sum_{P\in\mathscr{OW}_{n}}\mathcal{A}^{a\mu}_{P}\operatorname{e}^{\mathrm{i}\mkern
1.0muk_{P}\cdot x},$ (3.22)

where $\mathscr{OW}_{n}$ denotes the set of ordered words of length $n$. It
should also be noted that the link between the colour-stripped and colour-
dressed perturbiner expansions (3.10) and (3.22) is provided by
$A^{\mu}(x)=A^{\mu}_{a}(x)T^{a}$. Finally we remark that a colour-dressed
perturbiner expansion analogous to (3.22) for the field strength can be
obtained if we instead take the colour-dressed Berends-Giele field strength
current associated with the multi-particle field strength.

Before leaving this section, let us comment on the role the colour-dressed
Berends-Giele polarisation currents $\mathcal{A}^{a\mu}_{P}$ play in the
determination of the scattering amplitudes for Yang-Mills theory. Employing
again the Berends-Giele formula we obtain the colour-dressed $n$-point
amplitude

$\mathscr{A}^{\mathrm{tree}}_{n}=s_{12\cdots(n-1)}\mathcal{A}^{a\mu}_{12\cdots(n-1)}\mathcal{A}_{na\mu}$
(3.23)

where again we assume momentum conservation. It is also interesting to note
that we may rewrite the amplitude (3.23) as

$\mathscr{A}^{\mathrm{tree}}_{n}=\sum_{\Gamma}\frac{c_{\Gamma}n_{\Gamma}}{\prod_{e\in\Gamma}s_{e}},$
(3.24)

where the sum goes over all $(2n-5)!!$ trivalent trees $\Gamma$ with
propagators $s_{e}$ associated to each internal edge $e$ of $\Gamma$. Here
$c_{\Gamma}$ denotes the colour structure attached to each diagram, while
$n_{\Gamma}$ is the remaining part of the numerator involving kinematic
information such as contractions of momenta and polarisation vectors.

##  4 Double-copy relations for perturbiner expansions

In this section we will show that the double-copy prescription to construct
gravity theories as the “square” of Yang-Mills theory finds a natural
interpretation in terms of perturbiner expansions. To accomplish this, we
first briefly discuss a systematic procedure, exactly analogous to the one
found in [49], to obtain the multi-particle polarisation fields on the gravity
side from the Bern-Dunbar-Shimada formalism for one-loop graviton amplitudes.

###  4.1 Multi-particle polarisation tensors from the Bern-Dunbar-Shimada
formalism

We take as point of departure the symmetric partial integration algorithm and
the Bern-Dunbar-Shimada rules explained in Section 2. From these we identify
the permutation invariant integrand
$\bar{Q}_{n}(\dot{\bar{G}})Q_{n}(\dot{G})$. In addition, just as in Section 3
we have associated for two adjacent legs $i$ and $j$ with $i<j$ a pinch
operator, we may likewise define a “double pinch operator” acting on
$\bar{Q}_{n}(\dot{\bar{G}})Q_{n}(\dot{G})$ as

$\bar{\mathscr{D}}_{ij}\mathscr{D}_{ij}\bar{Q}_{n}(\dot{\bar{G}})Q_{n}(\dot{G})=\Bigg{(}\frac{\partial}{\partial\dot{\bar{G}}_{ij}}\bar{Q}_{n}(\dot{\bar{G}})\bigg{|}_{\begin{subarray}{c}\dot{\bar{G}}_{ij}=0\phantom{iiii}\\\
\dot{\bar{G}}_{jk}\rightarrow\dot{\bar{G}}_{ik}\end{subarray}}\Bigg{)}\Bigg{(}\frac{\partial}{\partial\dot{G}_{ij}}Q_{n}(\dot{G})\bigg{|}_{\begin{subarray}{c}\dot{G}_{ij}=0\phantom{iiii}\\\
\dot{G}_{jk}\rightarrow\dot{G}_{ik}\end{subarray}}\Bigg{)}.$ (4.1)

This double pinch operator is thus identical with the one for Yang-Mills
applied independently to both the left- and right-mover parts of the integrand
expression.

Our object is to find the multi-particle polarisation tensors by iterated
action of double pinch operators. Here we may borrow from the analysis carried
out in the Yang-Mills case, where we learned that the part of the polynomial
$Q_{n}(\dot{G})$ relevant to the multi-particle polarisations is the
$(n-2)$-tail. This makes it feasible in the present situation to also consider
the $(n-2)$-tail $\bar{T}(1,2,\dots,n-2)T(1,2,\dots,n-2)$. Applying the double
pinch operator consecutively $n-2$ times for each pair labels to the latter,
one finds

$\displaystyle\begin{split}&\bar{\mathscr{D}}_{1(n-1)}\mathscr{D}_{1(n-1)}\bar{\mathscr{D}}_{1(n-2)}\mathscr{D}_{1(n-2)}\cdots\bar{\mathscr{D}}_{13}\mathscr{D}_{13}\bar{\mathscr{D}}_{12}\mathscr{D}_{12}\bar{T}(1,2,\dots,n-2)T(1,2,\dots,n-2)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\bar{\varepsilon}^{\mu}_{12\cdots(n-2)}\varepsilon^{\nu}_{12\cdots(n-2)}k_{(n-1)\mu}k_{(n-1)\nu}.\end{split}$
(4.2)

This relation ensures that the multi-particle polarisation tensor is given by
$\bar{\varepsilon}^{\mu}_{12\cdots(n-2)}\varepsilon^{\nu}_{12\cdots(n-2)}$. We
may also remark that, by construction, each of the individual factors
$\bar{\varepsilon}^{\mu}_{12\cdots(n-2)}$ and
$\varepsilon^{\nu}_{12\cdots(n-2)}$ satisfies the generalised Jacobi identity
of order $n-2$ in $1,2,\dots,n-2$. Thus this is precisely the “square” of the
Yang-Mills multi-particle polarisation fields derived upon using (3.2).

One further thing to be noted is this. In our preliminary discussion of the
Bern-Dunbar-Shimada formalism, we indicated that when bringing into play the
pinching rules we no longer have an ordering of the tree legs. This means that
the tree attached to the loop is obtained by taking all possible pinches,
which is an exceedingly tedious and onerous task. The main point to be
stressed in connection with (4.2) is that we may infer directly the existence
of a “double-copy” version of the Berends-Giele polarisation currents,
circumventing the need to determine them indirectly using the pinching
procedure.

###  4.2 Double-copy perturbiner expansion

The foregoing discussion contains all the underlying principles that are
necessary for treating the double-copy polarisation currents and the
corresponding perturbiner expansion. Indeed, examining the expression for the
colour-dressed Berends-Giele polarisation current (3.16) and taking note of
(4.2) it is readily verified that the double-copy polarisation currents may be
obtained by replacing the colour factor $\tilde{f}^{a}$ with another copy of
the multi-particle polarisation field $\bar{\varepsilon}^{\mu}$. To be more
precise, the double-copy polarisation current, which we denote by
$\mathcal{G}^{\mu\nu}_{P}$, is expressible in the form

$\mathcal{G}^{\mu\nu}_{P}=\llbracket\bar{\varepsilon}^{\mu}\otimes\varepsilon^{\nu}\rrbracket\circ
b_{\mathsf{cd}}(P).$ (4.3)

This provides a realisation of the off-shell double-copy that arises naturally
in the string-based formalism, as an alternative to previous approaches [50,
58] that mimic the KLT relations adapting them to Berends-Giele currents.

As some examples, bringing to mind (3.17), (3.18), (3.19) and (3.20), the
first instances of the double-copy polarisation current up to multiplicity
four are given by

$\displaystyle\begin{split}\mathcal{G}^{\mu\nu}_{1}&=\bar{\varepsilon}^{\mu}_{1}\varepsilon^{\nu}_{1},\\\
\mathcal{G}_{12}^{\mu\nu}&=\frac{\bar{\varepsilon}^{\mu}_{[1,2]}\varepsilon^{\nu}_{[1,2]}}{s_{12}},\\\
\mathcal{G}_{123}^{\mu\nu}&=\frac{\bar{\varepsilon}^{\mu}_{[[1,2],3]}\varepsilon^{\nu}_{[[1,2],3]}}{s_{12}s_{123}}+\frac{\bar{\varepsilon}^{\mu}_{[[1,3],2]}\varepsilon^{\nu}_{[[1,3],2]}}{s_{13}s_{123}}+\frac{\bar{\varepsilon}^{\mu}_{[[2,3],1]}\varepsilon^{\nu}_{[[2,3],1]}}{s_{23}s_{123}},\\\
\mathcal{G}_{1234}^{\mu\nu}&=\frac{\bar{\varepsilon}^{\mu}_{[[[1,2],3],4]}\varepsilon^{\nu}_{[[[1,2],3],4]}}{s_{12}s_{123}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[1,2],4],3]}\varepsilon^{\nu}_{[[[1,2],4],3]}}{s_{12}s_{124}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[1,3],4],2]}\varepsilon^{\nu}_{[[[1,3],4],2]}}{s_{13}s_{134}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[2,3],4],1]}\varepsilon^{\nu}_{[[[2,3],4],1]}}{s_{23}s_{234}s_{1234}}\\\
&\quad\,+\frac{\bar{\varepsilon}^{\mu}_{[[[1,3],2],4]}\varepsilon^{\nu}_{[[[1,3],2],4]}}{s_{13}s_{123}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[1,4],2],3]}\varepsilon^{\nu}_{[[[1,4],2],3]}}{s_{14}s_{124}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[1,4],3],2]}\varepsilon^{\nu}_{[[[1,4],3],2]}}{s_{14}s_{134}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[2,3],1],4]}\varepsilon^{\nu}_{[[[2,3],1],4]}}{s_{23}s_{123}s_{1234}}\\\
&\quad\,+\frac{\bar{\varepsilon}^{\mu}_{[[[2,4],1],3]}\varepsilon^{\nu}_{[[[2,4],1],3]}}{s_{24}s_{124}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[2,4],3],1]}\varepsilon^{\nu}_{[[[2,4],3],1]}}{s_{24}s_{234}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[3,4],1],2]}\varepsilon^{\nu}_{[[[3,4],1],2]}}{s_{34}s_{134}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[[3,4],2],1]}\varepsilon^{\nu}_{[[[3,4],2],1]}}{s_{34}s_{234}s_{1234}}\\\
&\quad\,+\frac{\bar{\varepsilon}^{\mu}_{[[1,2],[3,4]]}\varepsilon^{\nu}_{[[1,2],[3,4]]}}{s_{12}s_{34}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[1,3],[2,4]]}\varepsilon^{\nu}_{[[1,3],[2,4]]}}{s_{13}s_{24}s_{1234}}+\frac{\bar{\varepsilon}^{\mu}_{[[1,4],[2,3]]}\varepsilon^{\nu}_{[[1,4],[2,3]]}}{s_{14}s_{23}s_{1234}}.\end{split}$
(4.4)

We reiterate that the crucial step in the double-copy procedure we have just
argued is the construction of the multi-particle polarisation fields
$\varepsilon^{\mu}_{P}$ satisfying the GJI as suggested by colour-kinematics
duality.

Since we have already obtained the double-copy polarisation currents we can
now readily obtain the double-copy perturbiner expansion, which is nothing but
the generating series

$G^{\mu\nu}(x)=\sum_{n\geq
1}\sum_{P\in\mathscr{OW}_{n}}\mathcal{G}^{\mu\nu}_{P}\operatorname{e}^{\mathrm{i}\mkern
1.0muk_{P}\cdot x}.$ (4.5)

Like in the Yang-Mills case, (4.5) is not a solution of the Einstein field
equations, for it has been “strictified” to include exclusively cubic
interactions.

Going on-shell now, it remains to say a word about the scattering amplitudes
in the double-copy theory. Recalling the colour-dressed amplitude (3.23), the
Berends-Giele formula for the $n$-point gravity amplitude reads

$\mathscr{M}_{n}^{\mathrm{tree}}=s_{12\cdots(n-1)}\mathcal{G}^{\mu\nu}_{12\cdots(n-1)}\mathcal{G}_{n\mu\nu}.$
(4.6)

Not surprisingly, the previous expression takes the well-known form for
gravity amplitudes in its double-copy version

$\mathscr{M}^{\mathrm{tree}}_{n}=\sum_{\Gamma}\frac{\bar{n}_{\Gamma}n_{\Gamma}}{\prod_{e\in\Gamma}s_{e}},$
(4.7)

which is equivalent to the KLT formula as can be seen in [32]. We also checked
our result up to degree $n=5$ for particular polarisations. At any rate, the
outcome of this approach is that we can calculate the amplitudes for the
double-copy theory in a relatively straightforward manner, without the need
for separately finding local BCJ numerators. This attribute was not apparent
in previous approaches using the perturbiner method, since the generating
series of Berends-Giele currents is usually presented in its colour-stripped
version for the BCJ gauge.

##  5 Some other examples

Now that we found a prescription for the double-copy perturbiners, let us
apply it to other theories beyond Yang-Mills and gravity. In principle it can
be applied to any theory as soon as we guarantee multi-particle fields in the
BCJ gauge. One first example should be the case where the BCJ gauge originally
appeared, ten-dimensional $\mathscr{N}=1$ super Yang-Mills in [17] (more
recently from a new approach in [59]), but for now we will restrict our
presentation only to cases without supersymmetry.

###  5.1 $\alpha^{\prime}$-deformations

For the first example we will calculate the currents and amplitudes for the
deformations of general relativity that come from the $\alpha^{\prime}$
corrections of the closed bosonic string, also referred to as
GR+$R^{2}$+$R^{3}$. The amplitudes for this theory were calculated using the
KLT relations for string theory [18, 60]. The action was found in [61] and it
reads

$\displaystyle\begin{split}S_{\begin{subarray}{c}\mathrm{closed}\\\
\mathrm{bosonic}\end{subarray}}&\sim\int{\mathrm{d}}^{D}x\,\sqrt{g}\left\\{R-2(\partial_{\mu}\varphi)^{2}-\frac{1}{12}H^{2}+\frac{\alpha^{\prime}}{4}e^{-2\varphi}\big{(}R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\big{)}\right.\\\
&\qquad\qquad\qquad\>+\alpha^{\prime
2}e^{-4\varphi}\bigg{(}\frac{1}{16}R^{\mu\nu}{}_{\alpha\beta}R^{\alpha\beta}{}_{\lambda\rho}R^{\lambda\rho}{}_{\mu\nu}-\frac{1}{12}R^{\mu\nu}{}_{\alpha\beta}R^{\nu\lambda}{}_{\beta\rho}R^{\lambda\mu}{}_{\rho\alpha}\bigg{)}+{\cal
O}(\alpha^{\prime 3})\Big{\\}},\end{split}$ (5.1)

where here $\varphi$ represents the dilaton and $H=dB$ represents the field
strength of the $B$-field. The gauge field theory for the double-copy is the
deformed Yang-Mills theory that comes from the low energy limit of the open
string. The action, compatible with colour-kinematics duality [62], is the
following

$\displaystyle S_{\mathrm{YM}+F^{3}+F^{4}}=\int{\mathrm{d}}^{D}x\
\mathrm{tr}\left\\{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{2\alpha^{\prime}}{3}F_{\mu}{}^{\nu}F_{\nu}{}^{\lambda}F_{\lambda}{}^{\mu}+\frac{\alpha^{\prime
2}}{4}[F_{\mu\nu},F_{\lambda\rho}][F^{\mu\nu},F^{\lambda\rho}]\right\\},$
(5.2)

which has the following equations of motion in the Lorenz gauge,
$\partial_{\mu}A^{\mu}=0$,

$\displaystyle\Box A^{\lambda}$ $\displaystyle=$
$\displaystyle[A^{\mu},\partial_{\mu}A^{\lambda}]+[A_{\mu},F^{\mu\lambda}]+2\alpha^{\prime}\big{\\{}[\nabla_{\mu}F^{\mu\nu},F_{\nu}{}^{\lambda}]+[F^{\mu\nu},\nabla_{\mu}F_{\nu}{}^{\lambda}]\big{\\}}$
(5.3) $\displaystyle+2\alpha^{\prime
2}\Big{\\{}\big{[}[\nabla_{\mu}F^{\mu\lambda},F_{\rho\sigma}],F^{\rho\sigma}\big{]}+\big{[}[F^{\mu\lambda},\nabla_{\mu}F_{\rho\sigma}],F^{\rho\sigma}\big{]}+\big{[}[F^{\mu\lambda},F_{\rho\sigma}],\nabla_{\mu}F^{\rho\sigma}\big{]}\Big{\\}}.$

In [63], the authors conducted a detailed analysis for the calculation of the
currents in this gauge using the perturbiner approach [64, 50, 65]. Then, they
applied the non-linear gauge transformation studied in [66] in order to obtain
currents in the BCJ gauge. In general the expressions for the
$\alpha^{\prime}$-deformed multi-particle polarisations have the following
structure

$\displaystyle
a^{\mu}_{P}=\varepsilon^{\mu}_{P}+\alpha^{\prime}\varepsilon^{(1)\mu}_{P}+\alpha^{\prime
2}\varepsilon^{(2)\mu}_{P}.$ (5.4)

We invite the reader to have a look at the explicit expressions in [63].

Our double-copy perturbiner for the $\alpha^{\prime}$-deformation of general
relativity comes out to be

$\displaystyle G^{(\alpha^{\prime})\mu\nu}(x)=\sum_{n\geq
1}\sum_{P\in\mathcal{OW}_{n}}\mathcal{G}^{(\alpha^{\prime})\mu\nu}_{P}\operatorname{e}^{\mathrm{i}k_{P}\cdot
x},$ (5.5)

where the Berends-Giele currents is given by

$\displaystyle\mathcal{G}^{(\alpha^{\prime})\mu\nu}_{P}$ $\displaystyle=$
$\displaystyle\llbracket a^{\mu}\otimes\bar{a}^{\nu}\rrbracket\circ
b_{cd}(P).$ (5.6)

Naturally, in complete analogy with (4.6), the corresponding amplitude reads

$\displaystyle\mathscr{M}_{n}^{(\alpha^{\prime})\mathrm{tree}}$
$\displaystyle=$ $\displaystyle s_{1\dots n-1}\mathcal{G}_{1\dots
n-1}^{(\alpha^{\prime})\mu\nu}\mathcal{G}^{(\alpha^{\prime})}_{n\mu\nu}.$
(5.7)

This we have also checked using the explicit expressions for $a^{\mu}_{P}$
from [63].

###  5.2 Zeroth-copy

Another example whose perturbiner can be obtained in a very straightforward
manner with our approach is the one for the bi-adjoint scalar model. For this
model, originally found in [67], we have a scalar field that takes values in
the tensor product $\mathfrak{su}(N)\otimes\mathfrak{su}(N^{\prime})$, and is
expressible in terms of the generators as $\Phi=\Phi_{aa^{\prime}}T^{a}\otimes
T^{\prime a^{\prime}}$. The corresponding action takes the form

$\displaystyle S_{\text{bi-
adjoint}}=\int{\mathrm{d}}^{D}x\left\\{-\tfrac{1}{2}\Phi^{aa^{\prime}}\Box\Phi_{aa^{\prime}}+\frac{1}{3!}\tilde{f}^{abc}\tilde{f^{\prime}}^{a^{\prime}b^{\prime}c^{\prime}}\Phi_{aa^{\prime}}\Phi_{bb^{\prime}}\Phi_{cc^{\prime}}\right\\}.$
(5.8)

Its Berends-Giele currents were found for the first time in [64] in the
colour-stripped version and the colour-dressed version in [50], both cases
using the perturbiner approach. Here we can obtain it simply by applying the
zeroth-copy [68], now in its analogue perturbiner version. Therefore, for the
bi-adjoint perturbiner we have

$\displaystyle\Phi^{aa^{\prime}}(x)=\sum_{n\geq
1}\sum_{P\in\mathcal{OW}_{n}}\phi^{aa^{\prime}}_{P}\operatorname{e}^{\mathrm{i}k_{P}\cdot
x},$ (5.9)

where for the Berends-Giele currents read

$\displaystyle\phi^{aa^{\prime}}_{P}=\llbracket c^{a}\otimes c^{\prime
a^{\prime}}\rrbracket\circ b_{cd}(P)$ (5.10)

The expressions for the currents are exactly like the ones in (4.4) but
replacing the polarisations by the colour factors presented in Section 3. The
colour-dressed amplitudes can also be calculated directly using the Berends-
Giele formula in (4.6).

##  6 Conclusions

We have seen that colour-kinematics duality and double-copy arise quite
naturally in the string-based formalism in the form of multi-particle fields.
The combination of the string-based rules and the technology developed in the
study of such fields allowed us to reduce the calculation of tree-level
amplitudes in Yang-Mills and gravity to a single basic calculation, namely the
full pinching of the tail that gives the multi-particle polarisation in the
BCJ gauge. Both the colour-dressed Yang-Mills Berends-Giele currents in the
BJC gauge and the gravitational Berends-Giele currents are given explicitly up
to multiplicity five. The most attractive feature of our formalism is that it
never becomes necessary to determine gauge transformation terms. We presented
a new prescription for the off-shell double-copy that has applications to
theories beyond the ones that we can represent by the infinite string tension
limit, as we have demonstrated with the examples of $\alpha^{\prime}$-deformed
gravity and the bi-adjoint scalar model.

In the Yang-Mills case, we have shown in [49] how to feed the obtained multi-
particle tensors back into the Bern-Kosower formalism so as to make the whole
pinching procedure unnecessary. It is not obvious whether this aspect of our
approach can be generalized to the gravity case, since here the existence of
the cross terms seems to start making a real difference. We leave this to
further study.

One application in progress is the calculation of Berends-Giele currents for
gravity coupled to matter fields along the lines of [69] and [70], that could
be compared with [71]. Another application for the near future is to some
cases of supergravity, where Berends-Giele currents have been found for
$\mathscr{N}=1$ Super-Yang-Mills in the BCJ gauge in [17, 55].

## Acknowledgements

We thank Lucia Garozzo for providing us with the file with their
$\alpha^{\prime}$-deformed polarisations. CLA thanks Oliver Schlotterer for
the very fruitful discussions. CS thanks Lance Dixon and Piotr Tourkine for
various informations and discussions. FMB is supported in part by the European
Research Council under ERC-STG-804286 UNISCAMP.

##  A Berends-Giele currents of multiplicity five

In this appendix, we exploit the procedure showed above for the computation of
colour-dressed Berends-Giele polarisation currents in the BCJ gauge and we
show the complete expressions for the current at multiplicity five. The
computation follows from (3.16), where in the five-particle case $P=12345$. In
this case, the word decomposition reads as $(Q,R)=(1234)(5)$, $(1235)(4)$,
$(1245)(3)$, $(1345)(2)$, $(2345)(1)$, $(123)(45)$, $(124)(35)$, $(125)(34)$,
$(134)(25)$, $(145)(23)$, $(135)(24)$, $(234)(15)$, $(235)(14)$, $(245)(13)$,
$(345)(12)$. Therefore, making also use of the recursion in (3.11), we obtain
for the colour-dressed Berends-Giele polarisation current
$\mathcal{A}_{12345}^{a\mu}$ the formula

$\displaystyle\begin{split}\mathcal{A}_{12345}^{a\mu}&=\frac{c^{a}_{[[[[1,2],3],4],5]}\varepsilon^{\mu}_{[[[[1,2],3],4],5]}}{s_{12}s_{123}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[1,3],2],4],5]}\varepsilon^{\mu}_{[[[[1,3],2],4],5]}}{s_{13}s_{123}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[2,3],1],4],5]}\varepsilon^{\mu}_{[[[[2,3],1],4],5]}}{s_{23}s_{123}s_{1234}s_{12345}}\\\
&+\frac{c^{a}_{[[[[1,2],4],3],5]}\varepsilon^{\mu}_{[[[[1,2],4],3],5]}}{s_{12}s_{124}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[1,4],2],3],5]}\varepsilon^{\mu}_{[[[[1,4],2],3],5]}}{s_{14}s_{124}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[2,4],1],3],5]}\varepsilon^{\mu}_{[[[[2,4],1],3],5]}}{s_{24}s_{124}s_{1234}s_{12345}}\\\
&+\frac{c^{a}_{[[[[1,3],4],2],5]}\varepsilon^{\mu}_{[[[[1,3],4],2],5]}}{s_{13}s_{134}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[1,4],3],2],5]}\varepsilon^{\mu}_{[[[[1,4],3],2],5]}}{s_{14}s_{134}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[3,4],1],2],5]}\varepsilon^{\mu}_{[[[[3,4],1],2],5]}}{s_{34}s_{134}s_{1234}s_{12345}}\\\
&+\frac{c^{a}_{[[[[2,3],4],1],5]}\varepsilon^{\mu}_{[[[[2,3],4],1],5]}}{s_{23}s_{234}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[2,4],3],1],5]}\varepsilon^{\mu}_{[[[[2,4],3],1],5]}}{s_{24}s_{234}s_{1234}s_{12345}}+\frac{c^{a}_{[[[[3,4],2],1],5]}\varepsilon^{\mu}_{[[[[3,4],2],1],5]}}{s_{34}s_{234}s_{1234}s_{12345}}\\\
&+\frac{c^{a}_{[[[1,2],[3,4]],5]}\varepsilon^{\mu}_{[[[1,2],[3,4]],5]}}{s_{12}s_{34}s_{1234}s_{12345}}+\frac{c^{a}_{[[[1,3],[2,4]],5]}\varepsilon^{\mu}_{[[[1,3],[2,4]],5]}}{s_{13}s_{24}s_{1234}s_{12345}}+\frac{c^{a}_{[[[1,4],[2,3]],5]}\varepsilon^{\mu}_{[[[1,4],[2,3]],5]}}{s_{14}s_{23}s_{1234}s_{12345}}\\\
&+\Big{(}(1234)(5)\leftrightarrow(1235)(4)\Big{)}+\Big{(}(1234)(5)\leftrightarrow(1245)(3)\Big{)}\\\
&+\Big{(}(1234)(5)\leftrightarrow(1345)(2)\Big{)}+\Big{(}(1234)(5)\leftrightarrow(2345)(1)\Big{)}\\\
&+\frac{c^{a}_{[[[1,2],3],[4,5]]}\varepsilon^{\mu}_{[[[1,2],3],[4,5]]}}{s_{12}s_{123}s_{45}s_{12345}}+\frac{c^{a}_{[[[2,3],1],[4,5]]}\varepsilon^{\mu}_{[[[2,3],1],[4,5]]}}{s_{23}s_{123}s_{45}s_{12345}}+\frac{c^{a}_{[[[1,3],2],[4,5]]}\varepsilon^{\mu}_{[[[1,3],2],[4,5]]}}{s_{13}s_{123}s_{45}s_{12345}}\\\
&+\Big{(}(123)(45)\leftrightarrow(124)(35)\Big{)}+\Big{(}(123)(45)\leftrightarrow(125)(34)\Big{)}+\Big{(}(123)(45)\leftrightarrow(134)(25)\Big{)}\\\
&+\Big{(}(123)(45)\leftrightarrow(145)(23)\Big{)}+\Big{(}(123)(45)\leftrightarrow(135)(24)\Big{)}+\Big{(}(123)(45)\leftrightarrow(234)(15)\Big{)}\\\
&+\Big{(}(123)(45)\leftrightarrow(235)(14)\Big{)}+\Big{(}(123)(45)\leftrightarrow(245)(13)\Big{)}+\Big{(}(123)(45)\leftrightarrow(345)(12)\Big{)},\end{split}$
(A1)

where the colour factors have structures of type

$\displaystyle\begin{split}c^{a}_{[[[[1,2],3],4],5]}=\tilde{f}^{\phantom{a_{1}a_{2}}b}_{a_{1}a_{2}}&\tilde{f}^{\phantom{ba_{3}}c}_{ba_{3}}\tilde{f}^{\phantom{ca_{4}}d}_{ca_{4}}\tilde{f}^{\phantom{da_{5}}a}_{da_{5}},\qquad
c^{a}_{[[[1,2],[3,4]],5]}=\tilde{f}^{\phantom{a_{1}a_{2}}b}_{a_{1}a_{2}}\tilde{f}^{\phantom{a_{3}a_{4}}c}_{a_{3}a_{4}}\tilde{f}^{\phantom{bc}d}_{bc}\tilde{f}^{\phantom{da_{5}}a}_{da_{5}},\\\
&c^{a}_{[[[1,2],3],[4,5]]}=\tilde{f}^{\phantom{a_{1}a_{2}}b}_{a_{1}a_{2}}\tilde{f}^{\phantom{ba_{3}}c}_{ba_{3}}\tilde{f}^{\phantom{a_{4}a_{5}}d}_{a_{4}a_{5}}\tilde{f}^{\phantom{cd}a}_{cd}.\end{split}$
(A2)

Also, in (A1) we made use of the multi-particle polarisation field
$\varepsilon_{[[[[1,2],3],4],5]}^{\mu}$ defined as

$\displaystyle\begin{split}\varepsilon_{[[[[1,2],3],4],5]}^{\mu}&=\frac{1}{2}\big{[}\varepsilon_{5}^{\mu}\left(\varepsilon_{[[[1,2],3],4]}\cdot
k_{5}\right)-\varepsilon_{[[[1,2],3],4]}^{\mu}\left(\varepsilon_{5}\cdot
k_{1234}\right)+\varepsilon_{[[[1,2],3],4]\nu}f_{5}^{\nu\mu}-\varepsilon_{5\nu}f_{[[[1,2],3],4]}^{\nu\mu}\big{]}\\\
&+(k_{123}\cdot k_{4})\varepsilon_{4}^{\mu}h_{1235}+(k_{12}\cdot
k_{3})\big{(}\varepsilon_{3}^{\mu}h_{1245}+\varepsilon_{[3,4]}^{\mu}h_{125}-\varepsilon_{[1,2]}^{\mu}h_{345}\big{)}\\\
&+(k_{1}\cdot
k_{2})\big{(}\varepsilon_{2}^{\mu}h_{1345}+\varepsilon_{[2,3]}^{\mu}h_{145}+\varepsilon_{[2,4]}^{\mu}h_{135}-\varepsilon_{1}^{\mu}h_{2345}-\varepsilon_{[1,3]}^{\mu}h_{245}-\varepsilon_{[1,4]}^{\mu}h_{235}\big{)}-k_{12345}^{\mu}h_{12345}\end{split}$
(A3)

where we have set

$\displaystyle\begin{split}f^{\mu\nu}_{[[[1,2],3],4]}&=k^{\mu}_{1234}\varepsilon^{\nu}_{[[[1,2],3],4]}-(k_{123}\cdot
k_{4})\varepsilon^{\mu}_{[[1,2],3]}\varepsilon^{\nu}_{4}-(k_{12}\cdot
k_{3})(\varepsilon^{\mu}_{[1,2]}\varepsilon^{\nu}_{[3,4]}+\varepsilon^{\mu}_{[[1,2],4]}\varepsilon^{\nu}_{3})\\\
&-(k_{1}\cdot
k_{2})(\varepsilon^{\mu}_{[1,3]}\varepsilon^{\nu}_{[2,4]}+\varepsilon^{\mu}_{[1,4]}\varepsilon^{\nu}_{[2,3]}+\varepsilon^{\mu}_{[[1,3],4]}\varepsilon^{\nu}_{2}-\varepsilon^{\mu}_{[[2,3],4]}\varepsilon^{\nu}_{1})-(\mu\leftrightarrow\nu)\end{split}$
(A4)

and

$\displaystyle\begin{split}h_{12345}&=\frac{1}{4}\bigg{[}\Big{\\{}\varepsilon_{3}\cdot\varepsilon_{4}\varepsilon_{1}\cdot
k_{2}\varepsilon_{2}\cdot k_{3}\varepsilon_{5}\cdot
k_{1}-\varepsilon_{2}\cdot\varepsilon_{3}\varepsilon_{1}\cdot
k_{2}\varepsilon_{4}\cdot k_{3}\varepsilon_{5}\cdot
k_{1}+\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{3}\cdot
k_{2}\varepsilon_{4}\cdot k_{3}\varepsilon_{5}\cdot k_{1}\\\
&+\varepsilon_{3}\cdot\varepsilon_{4}\varepsilon_{1}\cdot
k_{2}\varepsilon_{2}\cdot k_{3}\varepsilon_{5}\cdot
k_{2}-\varepsilon_{2}\cdot\varepsilon_{3}\varepsilon_{1}\cdot
k_{2}\varepsilon_{4}\cdot k_{3}\varepsilon_{5}\cdot
k_{2}-\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{3}\cdot
k_{2}\varepsilon_{4}\cdot k_{3}\varepsilon_{5}\cdot k_{2}\\\
&-\varepsilon_{1}\cdot\varepsilon_{5}\varepsilon_{3}\cdot\varepsilon_{4}\varepsilon_{2}\cdot
k_{3}k_{1}\cdot
k_{2}+\frac{1}{2}\varepsilon_{1}\cdot\varepsilon_{5}\varepsilon_{2}\cdot\varepsilon_{3}\varepsilon_{4}\cdot
k_{3}k_{1}\cdot
k_{2}-\frac{1}{2}\varepsilon_{2}\cdot\varepsilon_{5}\varepsilon_{3}\cdot\varepsilon_{4}\varepsilon_{1}\cdot
k_{2}k_{2}\cdot k_{3}\\\
&-\frac{3}{4}\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{3}\cdot\varepsilon_{4}\varepsilon_{5}\cdot
k_{1}k_{2}\cdot
k_{3}+\frac{3}{4}\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{3}\cdot\varepsilon_{4}\varepsilon_{5}\cdot
k_{2}k_{2}\cdot k_{3}-(123\rightarrow 312)-(1234\rightarrow 4123)\Big{\\}}\\\
&+\Big{\\{}\varepsilon_{2}\cdot\varepsilon_{3}\varepsilon_{1}\cdot
k_{2}\varepsilon_{4}\cdot k_{2}\varepsilon_{5}\cdot
k_{3}-\varepsilon_{2}\cdot\varepsilon_{4}\varepsilon_{1}\cdot
k_{2}\varepsilon_{3}\cdot k_{2}\varepsilon_{5}\cdot
k_{1}-\varepsilon_{2}\cdot\varepsilon_{3}\varepsilon_{1}\cdot
k_{2}\varepsilon_{4}\cdot k_{2}\varepsilon_{5}\cdot k_{1}\\\
&-\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{3}\cdot
k_{2}\varepsilon_{4}\cdot k_{2}\varepsilon_{5}\cdot
k_{2}-\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{3}\cdot
k_{2}\varepsilon_{4}\cdot k_{2}\varepsilon_{5}\cdot
k_{4}+\frac{1}{2}\varepsilon_{2}\cdot\varepsilon_{4}\varepsilon_{3}\cdot\varepsilon_{5}\varepsilon_{1}\cdot
k_{2}k_{2}\cdot k_{3}\\\
&+\frac{1}{2}\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{3}\cdot\varepsilon_{5}\varepsilon_{4}\cdot
k_{2}k_{2}\cdot
k_{3}+\frac{1}{2}\varepsilon_{1}\cdot\varepsilon_{2}\varepsilon_{4}\cdot\varepsilon_{5}\varepsilon_{3}\cdot
k_{2}k_{2}\cdot k_{4}-(123\rightarrow 312)\Big{\\}}\\\
&+\varepsilon_{1}\cdot\varepsilon_{4}\varepsilon_{2}\cdot
k_{1}\varepsilon_{3}\cdot k_{1}\varepsilon_{5}\cdot
k_{1}+\varepsilon_{2}\cdot\varepsilon_{4}\varepsilon_{1}\cdot
k_{2}\varepsilon_{3}\cdot k_{2}\varepsilon_{5}\cdot
k_{4}\bigg{]}-(1\rightarrow 2).\end{split}$ (A5)

Note that, as for the lower-point polarisation fields,
$\varepsilon_{[[[[1,2],3],4],5]}^{\mu}$ is the only five-particle polarisation
field needed in (A1). Indeed, using the identities
$\varepsilon^{\mu}_{[[[1,2],[3,4]],5]}=\varepsilon^{\mu}_{[[[[1,2],3],4],5]}-\varepsilon^{\mu}_{[[[[1,2],4],3],5]}$
and
$\varepsilon^{\mu}_{[[[1,2],3],[4,5]]}=\varepsilon^{\mu}_{[[[[1,2],3],4],5]}-\varepsilon^{\mu}_{[[[[1,2],3],5],4]}$,
all the five-particle polarisation fields appearing in (A1) are obtained from
(A3) by a simple relabelling.

Finally, we report here the formula for the double-copy polarisation current
at five point $\mathcal{G}_{12345}^{\mu\nu}$. As pointed out in Section 4,
this is simply obtained by substituting the colour factors $c^{a}$ in (A1)
with another copy of the multi-particle polarisation field
$\bar{\varepsilon}^{\mu}$. The resulting expression for
$\mathcal{G}_{12345}^{\mu\nu}$ is thus given by

$\displaystyle\begin{split}\mathcal{G}_{12345}^{\mu\nu}&=\frac{\varepsilon^{\mu}_{[[[[1,2],3],4],5]}\bar{\varepsilon}^{\nu}_{[[[[1,2],3],4],5]}}{s_{12}s_{123}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[1,3],2],4],5]}\bar{\varepsilon}^{\nu}_{[[[[1,3],2],4],5]}}{s_{13}s_{123}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[2,3],1],4],5]}\bar{\varepsilon}^{\nu}_{[[[[2,3],1],4],5]}}{s_{23}s_{123}s_{1234}s_{12345}}\\\
&+\frac{\varepsilon^{\mu}_{[[[[1,2],4],3],5]}\bar{\varepsilon}^{\nu}_{[[[[1,2],4],3],5]}}{s_{12}s_{124}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[1,4],2],3],5]}\bar{\varepsilon}^{\nu}_{[[[[1,4],2],3],5]}}{s_{14}s_{124}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[2,4],1],3],5]}\bar{\varepsilon}^{\nu}_{[[[[2,4],1],3],5]}}{s_{24}s_{124}s_{1234}s_{12345}}\\\
&+\frac{\varepsilon^{\mu}_{[[[[1,3],4],2],5]}\bar{\varepsilon}^{\nu}_{[[[[1,3],4],2],5]}}{s_{13}s_{134}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[1,4],3],2],5]}\bar{\varepsilon}^{\nu}_{[[[[1,4],3],2],5]}}{s_{14}s_{134}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[3,4],1],2],5]}\bar{\varepsilon}^{\nu}_{[[[[3,4],1],2],5]}}{s_{34}s_{134}s_{1234}s_{12345}}\\\
&+\frac{\varepsilon^{\mu}_{[[[[2,3],4],1],5]}\bar{\varepsilon}^{\nu}_{[[[[2,3],4],1],5]}}{s_{23}s_{234}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[2,4],3],1],5]}\bar{\varepsilon}^{\nu}_{[[[[2,4],3],1],5]}}{s_{24}s_{234}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[[3,4],2],1],5]}\bar{\varepsilon}^{\nu}_{[[[[3,4],2],1],5]}}{s_{34}s_{234}s_{1234}s_{12345}}\\\
&+\frac{\varepsilon^{\mu}_{[[[1,2],[3,4]],5]}\bar{\varepsilon}^{\nu}_{[[[1,2],[3,4]],5]}}{s_{12}s_{34}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[1,3],[2,4]],5]}\bar{\varepsilon}^{\nu}_{[[[1,3],[2,4]],5]}}{s_{13}s_{24}s_{1234}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[1,4],[2,3]],5]}\bar{\varepsilon}^{\nu}_{[[[1,4],[2,3]],5]}}{s_{14}s_{23}s_{1234}s_{12345}}\\\
&+\Big{(}(1234)(5)\leftrightarrow(1235)(4)\Big{)}+\Big{(}(1234)(5)\leftrightarrow(1245)(3)\Big{)}\\\
&+\Big{(}(1234)(5)\leftrightarrow(1345)(2)\Big{)}+\Big{(}(1234)(5)\leftrightarrow(2345)(1)\Big{)}\\\
&+\frac{\varepsilon^{\mu}_{[[[1,2],3],[4,5]]}\bar{\varepsilon}^{\nu}_{[[[1,2],3],[4,5]]}}{s_{12}s_{123}s_{45}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[2,3],1],[4,5]]}\bar{\varepsilon}^{\nu}_{[[[2,3],1],[4,5]]}}{s_{23}s_{123}s_{45}s_{12345}}+\frac{\varepsilon^{\mu}_{[[[1,3],2],[4,5]]}\bar{\varepsilon}^{\nu}_{[[[1,3],2],[4,5]]}}{s_{13}s_{123}s_{45}s_{12345}}\\\
&+\Big{(}(123)(45)\leftrightarrow(124)(35)\Big{)}+\Big{(}(123)(45)\leftrightarrow(125)(34)\Big{)}+\Big{(}(123)(45)\leftrightarrow(134)(25)\Big{)}\\\
&+\Big{(}(123)(45)\leftrightarrow(145)(23)\Big{)}+\Big{(}(123)(45)\leftrightarrow(135)(24)\Big{)}+\Big{(}(123)(45)\leftrightarrow(234)(15)\Big{)}\\\
&+\Big{(}(123)(45)\leftrightarrow(235)(14)\Big{)}+\Big{(}(123)(45)\leftrightarrow(245)(13)\Big{)}+\Big{(}(123)(45)\leftrightarrow(345)(12)\Big{)}.\end{split}$
(A6)

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