# Intersections of Twisted Forms:
New theories and Double copies

Pouria Mazloumi and Stephan Stieberger  Max-Planck Institut für Physik
Werner–Heisenberg–Institut, 80805 Munich, Germany  Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Tree–level scattering amplitudes of particles have a geometrical description
in terms of intersection numbers of pairs of twisted differential forms on the
moduli space of Riemann spheres with punctures. We customize a catalog of
twisted differential forms containing both already known and new differential
forms. By pairing elements from this list intersection numbers of various
theories can be furnished to compute their scattering amplitudes. Some of the
latter are familiar through their CHY description, but others are unknown.
Likewise, certain pairings give rise to various known and novel double–copy
constructions of spin–two theories. This way we find double copy constructions
for many theories, including higher derivative gravity, (partial massless)
bimetric gravity and some more exotic theories. Furthermore, we present a
derivation of amplitude relations in intersection theory.

###### keywords:

Amplitudes, gauge theory, gravity, intersection theory, twisted forms, double
copy, color kinematics duality

††journal: A

###### Contents

  1. 1 Introduction
  2. 2 CHY integrands for twisted differentials
  3. 3 Catalog of theories and corresponding twisted forms
    1. 3.1 Known theories: their twisted forms and string theory relations
    2. 3.2 More theories from the Einstein Yang–Mills form $\tilde{\varphi}^{EYM}_{\pm,n;r}$
    3. 3.3 Theories involving the Einstein–Maxwell form $\tilde{\varphi}^{EM}_{\pm,n;r}$
  4. 4 New and old theories from a novel twisted form
    1. 4.1 Novel twisted form from disk embedding
    2. 4.2 New theories constructed from the twisted form $\widetilde{\varphi}^{Bosonic}$
  5. 5 New and old double copies
    1. 5.1 Collections of some known double copies
    2. 5.2 Further double copies
    3. 5.3 Double copies of spin–two theories
    4. 5.4 Double copy of $YM+(DF)^{2}$ and higher derivative gravity
    5. 5.5 Bimetric theory as a double copy
  6. 6 Amplitude relations in intersection theory
    1. 6.1 Amplitude relations from intersection theory
    2. 6.2 Expansion Coefficient for EYM amplitude
  7. 7 Concluding remarks
  8. A EYM basis expansion

##  1 Introduction

The geometry of the moduli space of Riemann spheres with punctures is
ubiquitous for a geometrical description of tree–level scattering amplitudes
of particles. This geometry appears for computing intersection numbers of
twisted differential forms, for formulating amplitudes in terms of
Cachazo–He–Yuan (CHY) description and as world–sheet for tree–level closed
string amplitudes. In fact, the latter are formulated as a multi–dimensional
complex integral over the sphere, which by applying complex analysis can be
written in terms of pairs of real integrals subject to some monodromy phases.
This procedure gives rise to the (Kawai–Lewellen–Tye) KLT relations which in
turn give rise to an underlying double copy property intertwining two open
string sectors. This double copy structure can be also be inferred at the
level of twisted intersection forms as twisted period relations. This way by
pairing known and new twisted differential forms we may encounter new double
copy structures from twisted intersection numbers. Intersection numbers
provide a geometric description of tree–level $n$–point amplitudes in various
quantum field theories and likewise a geometric interpretation of the
kinematic Jacobi identity [1]. In this work we want to further investigate the
interrelation between quantum field theory, string scattering amplitudes and
intersection theory. We elaborate both on the computation of the amplitudes
and on the construction of possible double copies which arise from those
theories.

The computation of scattering amplitudes involving spin–two particles is a
tedious challenge in conventional Feynman diagram approach due to the
proliferation of diagrams involved. However, since the inception of
Kawai–Lewellen–Tye relations [2] at the tree–level this task can be relegated
to the computation of the spin–one sector only by means of doubling the
result. In modern language this procedure has been formulated as double copy
structure of gravity based on color kinematics duality [3, 4]. This
manipulation replaces the color structure of one gauge theory with the
kinematical structure of a second gauge theory. This doubling procedure is not
restricted to constructing spin–two interactions from essentially squaring
spin–one interactions but can be applied in principle for any pairs of
theories which allow for color kinematics duality. For a detailed account on
this see [5]. In fact, a peculiar feature of twisted intersection theory is
that it exhibits the double copy structure inherited from string theory in a
more natural way than QFT amplitudes. In intersection theory constructing
double copies corresponds to paring different twisted intersection forms.

For the construction of differential forms of physical relevance in
intersection theory both world–sheet integrands of string amplitudes and the
framework of CHY constructions provide fruitful ingredients. Furthermore, an
extension of open superstring interactions through an embedding of the disk
onto the sphere allows to accommodate couplings between open and closed
strings relevant for twisted forms describing EYM amplitudes [6]. In this work
we shall apply this embedding for the bosonic string theory.

This work is organized as follows. In Section $2$ we introduce some basics of
twisted intersection theory. We refer the reader to [6] for a more detailed
introduction to this subject. Here, we primarily elaborate on how to extract
twisted differentials from CHY integrands and vice versa. In Section 3 we
customize a catalog of twisted differential forms containing both already
known and new differential forms. By pairing various differential forms we
shall obtain intersection numbers describing the amplitudes of the
corresponding theories. For the latter we shall discuss both their Lagrangians
and possible CHY representations. In Section 4 by applying the embedding from
the disk onto the sphere to bosonic open string fields we construct a novel
twisted form. The latter allows to construct amplitudes by intersection
numbers for a variety of theories including Weyl–YM or $(DF)^{2}+\phi^{3}$
theories. In Section 5 we elaborate on the generic underlying structure of
double copy constructions in intersection theory. We shall present old and new
double copies. For the latter an amplitude description has been unknown
previously. This way we find a double copy construction for many theories,
including higher derivative gravity, (partial massless) bimetric gravity and
some more exotic theories. In Section 6 we present a derivation of
Kleiss–Kuijf (KK) and Bern–Carrasco–Johansson (BCJ) amplitude relations in
intersection theory. Finally, the appendices contain some supplementary
material.

##  2 CHY integrands for twisted differentials

Intersection numbers provide a geometric description of tree–level $n$–point
amplitudes in various quantum field theories. Selecting a theory requires
specifying two twisted differential forms $\varphi_{-}$ and $\varphi_{+}$ on
the moduli space ${\cal M}_{0,n}$ of Riemann spheres with $n$ punctures

$\mathcal{M}_{0,n}=\\{(z_{1},\ldots,z_{i},\ldots,z_{n})\in({\bf C}{\bf
P})^{n-3}\ ,\ i\neq j,k,l\ |\ \mathop{{\forall}}_{m\neq n}z_{m}\neq z_{n}\\}\
,$ (2.1)

subject to the three points $z_{j},z_{k},z_{l}$ fixed. The space (2.1) gives
rise to the moduli space of Riemann spheres with $n$ punctures. Due to the
$SL(2,{\bf C})$ invariance we consider $(n-3)$–forms $\varphi$ as elements of
cohomology equivalence classes

$\varphi\simeq\varphi+\nabla_{\pm\omega}\xi\ ,$ (2.2)

with a rational $n-4$ form $\xi$ and the Gauss–Manin connection
$\nabla_{\pm\omega}=d\pm\omega\wedge$ with $d$ the exterior derivative and
some closed one–form (twist) $\omega$. In order to make contact to physical
amplitudes for the twist $\omega$ one can select a potential $W$ with
$\omega=dW$ as generating function such that

$\omega=\alpha^{\prime}\sum_{1\leq i,j\leq n}2p_{i}p_{j}\ d\ln(z_{i}-z_{j})\
,$ (2.3)

with $n$ on–shell momenta $p_{i}$. The factor $\alpha^{\prime}$ is chosen such
that $\omega$ is dimensionless. Differential forms $\varphi$ with the property
(2.2) give rise to classes $[\varphi]$ of twisted forms, cf. Section 2 of Ref.
[6] for further details. The intersection of the two twisted forms is defined
as integral over (2.1) [7]

$\langle\varphi_{+},\varphi_{-}\rangle_{\omega}:=\left(-\frac{\alpha^{\prime}}{2\pi
i}\right)^{n-3}\
\int\limits_{\mathcal{M}_{0,n}}\iota_{\omega}(\varphi_{+})\wedge\varphi_{-}\
.$ (2.4)

The map $\iota_{\omega}(\varphi_{+})\in H^{n-3}_{+\omega}$ is the restriction
of the twisted form over the compact support
$H^{n-3}_{\omega,c}(\mathcal{M}_{0,n},\nabla_{\omega})$ of
$H^{n-3}_{\omega}(\mathcal{M}_{0,n},\nabla_{\omega})$. Otherwise, the integral
over the moduli space $\mathcal{M}_{0,n}$ would not be well–defined since the
latter is non–compact. The one–form (2.3) is related to the scattering
equations

$f_{k}:=\sum_{j\neq k}\frac{p_{k}p_{j}}{z_{j}-z_{k}}=0\ ,\ \ 1\leq k\leq n\ ,$
(2.5)

which in the limit $\alpha^{\prime}\rightarrow\infty$ relate the twisted
intersection numbers (2.4) to the CHY description. For this, the intersection
form (2.4) can be expanded w.r.t. to (twisted) period integrals

$\langle{C_{a}\atop\tilde{C}_{b}}\otimes
KN^{\pm}|\varphi_{\pm}\rangle=\int\limits_{C_{a},\tilde{C}_{b}}KN^{\pm}\;\varphi_{\pm}\
$ (2.6)

relating the twisted cohomology elements $\varphi_{\pm}\in
H_{\pm\omega}^{n-3}$ and their Poincare dual twisted homology cycles
$C_{a},\tilde{C}_{b}\in H^{\pm\omega}_{n-3}(\mathcal{M}_{0,n},KN^{\pm})$
associated with the multivalued function $KN^{\pm}$, respectively. Then, this
expansion allows the saddle point approximation of (2.4) for
$\alpha^{\prime}\rightarrow\infty$:

$\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi_{+},\varphi_{-}\rangle_{\omega}=\alpha^{\prime\frac{n-3}{2}}\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\delta(f_{k})\
\lim_{\alpha^{\prime}\rightarrow\infty}\hat{\varphi}_{+}\;\hat{\varphi}_{-}\
.$ (2.7)

On the other hand, a generic CHY integral describing an $n$–point amplitude of
a wide variety of massless QFTs in $d$ the space–time dimensions assumes the
following form

${\cal A}(n)=\delta^{(d)}\left(p_{1}+\ldots p_{n}\right)\
\int_{\mathcal{M}_{0,n}}d\mu_{n}\ \widetilde{{\cal
I}}(\\{p_{i},\tilde{\epsilon}_{i},z_{i}\\})\ {\cal
I}(\\{p_{l},\epsilon_{l},z_{l}\\})\
\sideset{}{{}^{\prime}}{\prod}_{l=1}^{n}\delta(f_{l})\ ,$ (2.8)

with the measure

$d\mu_{n}=z_{jk}z_{jl}z_{kl}\prod_{i=1\atop i\notin\\{j,k,l\\}}^{n}d^{i}z\ ,$
(2.9)

which is a degree $n-3$ holomorphic form on $\mathcal{M}_{0,n}$ with
$z_{j},z_{k},z_{l}$ being three arbitrary marked points fixed by $SL(2,{\bf
C})$ invariance. Above, the localization of the amplitude at the scattering
equation is achieved through the delta–functions $\delta(f_{i})$ with support
on (2.5) There are $(n-3)!$ solutions $\sigma^{(a)}_{k}$ of the equations
(2.5) and in terms of which the amplitude (2.8) becomes

${\cal A}(n)=\delta^{(d)}\left(p_{1}+\ldots p_{n}\right)\
\sum_{a=1}^{(n-3)!}\left.\Phi^{-1}\ \widetilde{{\cal I}}\ {\cal
I}\right|_{z_{k}=\sigma^{(a)}_{k}}\ ,$ (2.10)

with $\Phi=\det(\partial_{j}f_{a})$ the Jacobian referring to the $n-3$
independent $\delta$–functions. The numerators $\widetilde{{\cal I}}$ and
${\cal I}$ are rational functions of the external data
$\\{p_{l},\epsilon_{l},z_{l}\\}$ denoting momenta, polarizations
$\epsilon_{l}$ and points $z_{i}$ on $\mathcal{M}_{0,n}$. Specifying a certain
theory means selecting a pair of numerators $\widetilde{{\cal I}}$ and ${\cal
I}$.

We may use the numerators $\widetilde{{\cal I}}$ and ${\cal I}$ to define the
pair of $n$–forms

$\varphi_{+}={\cal I}\ d\mu_{n}\ \ \ ,\ \ \ \varphi_{-}=\widetilde{{\cal I}}\
d\mu_{n}$ (2.11)

of the twisted cohomology groups $\varphi_{+}\in H_{+\omega}^{n-3}$ and
$\varphi_{-}\in H^{n-3}_{-\omega}$, respectively. Then the bilinear pairing
through the intersection number (2.4) computes the amplitude (2.10) as

${\cal
A}(n)=\langle\varphi_{+},\varphi_{-}\rangle_{\omega}=\alpha^{\prime\frac{n-3}{2}}\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\delta(f_{k})\
\lim_{\alpha^{\prime}\rightarrow\infty}\hat{\varphi}_{+}\;\hat{\varphi}_{-}\equiv{\cal
A}_{CHY}(n)\ ,$ (2.12)

which in turn by (2.7) is related to the CHY amplitude ${\cal A}_{CHY}(n)$.
Conversely, we may look for pairs of twisted forms to derive previously
unknown CHY constructions from (2.7) like for Weyl–YM and $(DF)^{2}+\phi^{3}$.

Color kinematics (CK) duality [3] and subsequently the double copy structure
of amplitudes imposes constraints on the detailed form of the amplitudes. CK
duality is realized in many field–theory amplitudes including non–linear–sigma
model, (Dirac-)Born–Infeld, Volkov–Akulov and special Galileon theory to be
discussed in the next Section. CK duality of gauge theory and the double copy
structure of gravity or in general gauge/gravity relations are proven by the
geometry of the corresponding amplitudes in string theory by means of
monodromy relations on the string world–sheet [2, 8, 9]. Similar results are
derived at one–loop [10, 11], cf. also [12, 13]. Therefore, it is beneficial
to discuss these properties within the framework of intersection theory. One
of the important features of describing amplitudes by intersection theory is
that many involved characteristics of field–theory amplitudes are understood
in terms of the pole structure of their underlying pairs of twisted forms. In
particular, KLT relations simply follow from linear algebra within the
bilinear combination of twisted forms [14]. Note, that twisted forms $\varphi$
are assumed to be elements of cohomology classes, i.e. defined subject to
(2.2). As a consequence of the latter equivalence one can understand and prove
the KK and BCJ relations as cohomology invariant deformations corresponding to
the total derivative of $\varphi^{color}$, i.e. as a consequence of [1]:

$(d\pm dW\wedge)\,\varphi^{color}_{\pm;n}\simeq 0\ .$ (2.13)

In Subsection 6.1 we shall explicitly demonstrate how to derive KK and BCJ
relations from (2.13). Furthermore, there we shall evidence, that theories
with CK duality generically have a common color form in their pairs of twisted
forms entering (2.4). Conversely, amplitudes of a theory satisfying BCJ–KK
relations imply the existence of $\varphi^{color}$ in the underlying
description by intersection theory.

In twisted intersection theory the realization of double copies can be easily
achieved if the underlying pairs of theories are constructed from pairs of
twisted forms each containing a color factor. Then by inserting an orthogonal
basis into each of the intersection numbers (2.4) by means of expanding each
twisted form $\varphi_{\pm}$ w.r.t. an orthonormal basis of $n$–forms and its
dual basis, respectively these two theories are linked by the KLT kernel, cf.
Section 5 for more details. This procedure has been applied in [7] to express
EYM amplitudes as double copies of YM and gen.YMS, cf. also Section 5.
Equipped with this recipe as building block we will find previously unknown
double copies for spin–two theories in Section 5.

##  3 Catalog of theories and corresponding twisted forms

In this section we shall introduce new twisted forms to describe amplitudes of
different theories. The latter may have a field content of spin–zero, spin–one
and/or spin–two. By properly pairing these newly constructed forms with other
forms we then build double copies for spin–zero, spin–one or spin–two theories
with the latter invariant under linearized diffeomorphisms. The structure of
string scattering amplitudes proves to be indispensable to construct our
twisted forms. The world–sheet monodromy of the punctures of vertex operators
in string amplitudes naturally gives rise to twisted cohomology allowing to
specify string based twisted forms [1, 6]. Nevertheless, the proposed twisted
forms correspond to amplitudes of the underlying quantum field theory. In
fact, the CHY formulation of the latter if existent links the two descriptions
in the $\alpha^{\prime}\rightarrow\infty$ limit (2.12).

###  3.1 Known theories: their twisted forms and string theory relations

We start by giving an overview of theories which can be described by twisted
intersection forms. In fact, there is a whole catalogue of the latter which
can be paired for intersection numbers to represent a variety of different
theories [1]. Many of them also allow for a CHY description. In the following
table we list the known theories, their pair of intersection forms and their
CHY representation if there is one.

Theory | $\varphi_{-,n}$ | $\varphi_{+,n}$ | CHY representation | Amplitude
---|---|---|---|---
bi-adjoint scalar | $\varphi^{color}_{-,n}$ | $\varphi^{color}_{+,n}$ | $\mathcal{C}_{n}\mathcal{C}_{n}$ |  $n$ color scalar
Einstein | $\varphi^{gauge}_{-,n}$ | $\varphi^{gauge}_{+,n}$ | $\mathrm{Pf}^{\prime}\psi_{n}\mathrm{Pf}^{\prime}\psi_{n}$ |  $n$ gravitons
Yang-Mills | $\varphi^{color}_{-,n}$ | $\varphi^{gauge}_{+,n}$ | $\mathcal{C}_{n}\mathrm{Pf}^{\prime}\psi_{n}$ |  $n$ gluons
YM+$(DF)^{2}$ | $\varphi^{color}_{-,n}$ | $\varphi^{bosonic}_{+,n}$ | ?? |  $n$ higher derivative gluons
Einstein–Weyl | $\varphi^{gauge}_{-,n}$ | $\varphi^{bosonic}_{+,n}$ | ?? |  $n$ spin 2
special Galilean (sGal) | $\varphi^{scalar}_{-,n}$ | $\varphi^{scalar}_{+,n}$ | $(\mathrm{Pf}^{\prime}A_{n})^{4}$ |  $n$ higher derivative scalars
NLSM | $\varphi^{color}_{-,n}$ | $\varphi^{scalar}_{+,n}$ | $\mathcal{C}_{n}(\mathrm{Pf}^{\prime}A_{n})^{2}$ |  $n$ scalars
Born–Infeld (BI) | $\varphi^{scalar}_{-,n}$ | $\varphi^{gauge}_{+,n}$ | $(\mathrm{Pf}^{\prime}A_{n})^{2}\,\mathrm{Pf}^{\prime}\psi_{n}$ |  $n$ spin 1
Table 1a: Known theories, their pairs of twisted forms and their CHY
representations.

The twisted forms in Table 1a assume the following form:

$\displaystyle\varphi^{color}_{\pm,n}$ $\displaystyle=d\mu_{n}\
\frac{\mathop{\rm Tr}(T^{c_{1}}T^{c_{2}}\ldots
T^{c_{n}})}{(z_{1}-z_{2})(z_{2}-z_{3})\ldots(z_{n}-z_{1})}\equiv\mathop{\rm
Tr}(T^{c_{1}}T^{c_{2}}\ldots T^{c_{n}})\ PT(1,2,\ldots,n)$
$\displaystyle\varphi^{gauge}_{\pm,n}$
$\displaystyle=d\mu_{n}\int\prod\limits_{i=1}^{n}d\theta_{i}d\bar{\theta_{i}}\frac{\theta_{k}\theta_{l}}{z_{k}-z_{l}}\exp\Bigg{\\{}-\alpha^{\prime
2}\sum\limits_{i\neq j}\frac{\theta_{i}\theta_{j}p_{i}\cdot
p_{j}+\bar{\theta}_{i}\bar{\theta}_{j}\varepsilon_{i}\cdot\varepsilon_{j}+2(\theta_{i}-\theta_{j})\bar{\theta_{i}}\varepsilon_{i}\cdot
p_{j}}{z_{i}-z_{j}\mp\alpha^{\prime-1}\theta_{i}\theta_{j}}\Bigg{\\}}\ ,$
$\displaystyle\varphi^{scalar}_{\pm,n}$
$\displaystyle=d\mu_{n}\;(\mathrm{Pf}^{\prime}A_{n})^{2}=d\mu_{n}\;\frac{\det{A}_{[kl]}}{(z_{k}-z_{l})^{2}}$
(3.1) $\displaystyle\varphi^{bosonic}_{\pm,n}$
$\displaystyle=d\mu_{n}\left(\pm\frac{1}{\alpha^{\prime}}\right)^{\lfloor\frac{n-2}{2}\rfloor}\int\prod\limits_{i=1}^{n}d\theta_{i}d\bar{\theta_{i}}\exp\Bigg{\\{}\sum\limits^{n}_{i\neq
j}\Bigg{(}\pm
2\alpha^{\prime}\frac{\theta_{i}\bar{\theta}_{i}\varepsilon_{i}\cdot
p_{j}}{z_{i}-z_{j}}+\frac{\theta_{i}\bar{\theta}_{i}\theta_{j}\bar{\theta}_{j}\varepsilon_{i}\cdot\varepsilon_{j}}{(z_{i}-z_{j})^{2}}\Bigg{)}\Bigg{\\}}$
$\displaystyle\stackrel{{\scriptstyle\alpha^{\prime}\rightarrow\infty}}{{\longrightarrow}}\
W_{\underbrace{11...1}_{n}}\ d\mu_{n}\ .$

The matrices in the CHY representation are defined in the same way as [15]:

$\psi_{n}=\begin{pmatrix}A&-C^{T}\\\ C&B\end{pmatrix}\ ,$ (3.2)

with the three matrices:

$\displaystyle A_{n}=\begin{cases}0\ ,\,\,\,\,\,\,\,\,i=j\\\
\frac{p_{i}p_{j}}{z_{i}-z_{j}}\ ,\,\,\,\,\,\,\,\,i\neq j\end{cases},\ \,\,$
$\displaystyle C_{n}=\begin{cases}-\sum\limits_{k\neq
i}^{n}\frac{\varepsilon_{i}p_{k}}{z_{i}-z_{k}}\ ,\,\,\,\,\,\,\,\,i=j\\\
\frac{\varepsilon_{i}p_{j}}{z_{i}-z_{j}}\ ,\,\,\,\,\,\,\,\,i\neq
j\end{cases},\,\,$ $\displaystyle B_{n}=\begin{cases}0\ ,\,\,\,\,\,\,\,\,i=j\
,\\\ \frac{\varepsilon_{i}\varepsilon_{j}}{z_{i}-z_{j}}\
,\,\,\,\,\,\,\,\,i\neq j\ .\end{cases}$ (3.3)

Furthermore, in the definition of $\varphi^{scalar}_{\pm;n}$ there appears the
reduced Pfaffian for the matrix $A_{n}$

$\mathrm{Pf}^{\prime}A_{n}\equiv\frac{(-1)^{k+l}}{z_{k}-z_{l}}\;\mathrm{Pf}A_{[kl]}\
,$ (3.4)

with the index $k,l$ denoting removals of rows $k,l$ and columns $k,l$.
Furthermore, by using $\mathrm{Pf}M^{2}=\det M$ we have the following
relation:

$\frac{\det A_{[kl]}}{z^{2}_{kl}}=(\mathrm{Pf}^{\prime}A_{n})^{2}\ .$ (3.5)

In the following we review some basics about the special Galilean, non–linear
sigma model and Born–Infeld theories displayed in Table 1a. The Einstein–Weyl
theory will be discussed in Section 5.

  * 1.

Special Galilean (sGal)

Galilean theories in $d$ dimension are defined as scalar effective field
theories involving higher derivative interactions in the potential [16, 17]
namely

$\displaystyle\mathcal{L}_{Gal}=\frac{1}{2}\;\partial^{\mu}\phi\partial_{\mu}\phi+\sum\limits_{n=3}^{d-1}g_{n}\,(\partial\phi)^{2}(\partial_{\mu}\partial^{\mu}\phi)^{n-2}\
,$ (3.6)

or variants thereof [18]. The Galilean theory is called special if it features
a ${\bf Z}_{2}$ symmetry. The amplitude of $n$ scalars in this theory [18] can
be written in terms of the intersection number (2.4) leading to the correct
CHY formulation (2.12) in the following way:

${\cal
A}_{sGal}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi_{-,n}^{scalar},\varphi_{+,n}^{scalar}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\delta(f_{k})\,\,(\mathrm{Pf}^{\prime}A_{n})^{4}\ .$ (3.7)

  * 2.

Non–linear sigma model (NLSM)

The non–linear sigma model (NLSM) is defined as a theory with scalars $\Phi$
together with an embedding onto a manifold $M$ with a non–linear interaction
potential $V(\Phi)$. The theory for $N$ scalars fields $\Phi$ satisfying an
$U(N)$ symmetry can be written in terms of Cayley parametrization [18]:

$\mathcal{L}_{NLSM}=\frac{1}{8\lambda^{2}}\mathop{\rm
Tr}\left\\{\;\partial_{\mu}U(\Phi)\partial^{\mu}U(\Phi)\;\right\\}\ ,$ (3.8)

with:

$U(\Phi):=(\bm{1}+\lambda\Phi)\;(\bm{1}-\lambda\Phi)^{-1}\ .$ (3.9)

The field $\Phi$ may be written in the adjoint representation as
$\Phi=\phi^{I}T^{I}$ with $T^{I}$ the generators of $U(N)$. Upon expanding
$U(\Phi)$ in terms of $\Phi$ yields the usual scalar kinetic term. The
scattering amplitude of this theory involving $n$ scalars [18] can be given in
terms of the intersection number (2.4) leading to the correct CHY formulation
(2.12) as:

${\cal
A}_{NLSM}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi_{-,n}^{color},\varphi_{+,n}^{scalar}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\delta(f_{k})\,\,\mathcal{C}_{n}\,(\mathrm{Pf}^{\prime}A_{n})^{2}\
.$ (3.10)

  * 3.

Born–Infeld theory (BI)

The Born–Infeld theory is the non–linear generalization of Maxwell theory
[19]. The Lagrangian for this theory in $d=4$ is given by the non–linear
interaction

$\displaystyle\mathcal{L}_{BI}=\ell^{-2}\Bigg{(}\sqrt{-{\det}_{4}(\eta_{\mu\nu}+\ell
F_{\mu\nu})}-1\Bigg{)}\ ,$ (3.11)

where the scale $\ell$ can be related to the inverse string tension
$\alpha^{\prime}$ as $\ell=2\pi\alpha^{\prime}$. The $n$–point amplitude of
this theory [18] can be expressed in terms of the intersection number (2.4)
leading to the CHY representation of this theory (2.12) in terms of the
following localization integral:

${\cal
A}_{BI}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi_{-,n}^{scalar},\varphi_{+,n}^{gauge}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\delta(f_{k})\,\,\mathrm{Pf}^{\prime}\psi_{n}\;(\mathrm{Pf}^{\prime}A_{n})^{2}\
.$ (3.12)

Actually, all twisted forms in Table 1a are derived from string theory, which
in turn also motivated the corresponding CHY descriptions. In our previous
work [6] we were mainly concerned with the two twisted forms $\varphi^{color}$
and $\varphi^{gauge}$. The latter refers to superstring theory and is relevant
to describe both SYM and GR amplitudes. Therefore, it is natural to discuss
the twisted form $\varphi^{bosonic}$ stemming from bosonic string theory.
While the partial gluon subamplitudes $A_{YM}(1,\ldots,n)$ are the building
blocks for the open superstring amplitudes [20, 21, 22], i.e.

${\cal A}_{open\atop
superstring}(1,\pi(2,\ldots,n-2),n-1,n)=\sum_{\rho,\tau\in
S_{n-3}}Z(\pi|\tau)\ S[\rho|\tau]_{1}\ A_{YM}(1,\rho(2,\ldots,n-2),n-1,n)\ ,$
(3.13)

with the iterated disk integrals
$Z(\pi|\tau)=Z(1,\pi(2,\ldots,n-2),n-1,n|1,\tau(2,\ldots,n-2),n,n-1)$, the
same role is played by the subamplitudes $B(1,\ldots,n)$ for $(DF)^{2}+YM$
gauge theory for the open string [23], i.e.:

${\cal A}_{open\atop bosonic\
string}(1,\pi(2,\ldots,n-2),n-1,n)=\sum_{\rho,\tau\in S_{n-3}}Z(\pi|\tau)\
S[\rho|\tau]_{1}\ B(1,\rho(2,\ldots,n-2),n-1,n)\ .$ (3.14)

Above the KLT kernel $S[\rho|\tau]_{1}$ is given by the $k!\times k!$–matrix
[24, 25]

$S[\sigma|\rho]_{\ell}:=S^{(0)}[\,\sigma(1,\ldots,k)\,|\,\rho(1,\ldots,k)\,]_{1}=\prod_{t=1}^{k}\left(p_{1}p_{t_{\sigma}}+\sum_{r<t}p_{r_{\sigma}}p_{t_{\sigma}}\theta(r_{\sigma},t_{\sigma})\right)\
,$ (3.15)

with $j_{\sigma}=\sigma(j)$ and $\theta(r_{\sigma},t_{\sigma})=1$ if the
ordering of the legs $r_{\sigma},t_{\sigma}$ is the same in both orderings
$\sigma(1,\ldots,k)$ and $\rho(1,\ldots,k)$, and zero otherwise. By comparing
the two expressions (3.13) and (3.14) it is natural to construct from the
latter the twisted form $\varphi^{bosonic}$ relevant to $(DF)^{2}+YM$ theory.
The latter involves couplings of YM and $(DF)^{2}$ theory. The bosonic field
theory of the latter is defined by the following Lagrangian [26]

$\displaystyle\mathcal{L}_{YM+(DF)^{2}}$
$\displaystyle=\frac{1}{2}\;(D_{\mu}F^{a\,\mu\nu})^{2}-\frac{g}{3}\;F^{3}+\frac{1}{2}\
(D_{\mu}\varphi^{a})^{2}+\frac{g}{2}\;C^{\alpha\,ab}\varphi^{\alpha}F^{a}_{\mu\nu}F^{b\,\mu\nu}$
(3.16)
$\displaystyle+\frac{g}{3!}\;d^{\alpha\beta\gamma}\varphi^{\alpha}\varphi^{\beta}\varphi^{\gamma}-\frac{1}{2}\;m^{2}(\varphi^{\alpha})^{2}-\frac{1}{4}\;m^{2}F^{2}\
,$

with the field content comprising a massless gluon, a massive gluon and a
massive scalar. The Lagrangian of $(DF)^{2}$–theory only will be displayed in
Eq. (4.10). To the latter the YM part is added as a mass deformation, which in
total gives rise to the Lagrangian (3.16). As a consequence there are mass
terms proportional to $m^{2}$ for the fields $\varphi^{\alpha}$ and
$F_{\mu\nu}^{a}$. Within bosonic string theory this mass is related to the
string tension as

$m^{2}=-\alpha^{\prime-1}\ ,$ (3.17)

accounting for the string tachyonic modes representing both the massive gluon
and the massive scalar. In fact, for $\alpha^{\prime}\rightarrow 0$ we recover
pure YM theory (after multiplying the Lagrangian by an overall factor of
$m^{-2}=\alpha^{\prime}$), while for $\alpha^{\prime}\rightarrow\infty$ we
obtain $(DF)^{2}$ theory. Hence, after taking the limit
$\alpha^{\prime}\rightarrow\infty$ the mass (3.17) goes to zero and the
Lagrangian boils down to $(DF)^{2}$–theory, i.e.

$\lim_{\alpha^{\prime}\rightarrow\infty}\mathcal{L}_{YM+(DF)^{2}}(\alpha^{\prime})\simeq\mathcal{L}_{DF^{2}}\
,$ (3.18)

with the latter referring to (4.10). This is consistent with the CHY
construction, which works only for massless theories. Likewise, the
$\alpha^{\prime}\rightarrow\infty$ limit of the intersection number (2.4)
discards the mass term and reproduces the massless $(DF)^{2}$ theory.

In (3.1) for $\varphi^{bosonic}$ we have introduced the function $W$, which
arises in its $\alpha^{\prime}\rightarrow\infty$ limit. More concretely, the
latter is used e.g. in [27] and the set $L$ of indices of the function $W_{L}$
refer to a product of Lam–Yao cycles:

$W_{\underbrace{11...1}_{n}}=\prod_{i=1}^{n}C_{ii}=\prod\limits_{i=1}^{n}\Bigg{(}\sum\limits_{j=1\atop
j\neq i}^{n}\frac{\varepsilon_{i}\cdot p_{j}}{z_{ij}}\Bigg{)}\ .$ (3.19)

Notice, that $W_{11...1}$ is produced from the twisted form
$\varphi^{bosonic}$ in the limit $\alpha^{\prime}\rightarrow\infty$. In this
limit the twisted form $\varphi^{bosonic}_{\pm,n}$ can be used for describing
$(DF)^{2}$ theory. The limit $\alpha^{\prime}\rightarrow\infty$ removes in
$\varphi^{bosonic}_{\pm,n}$ all the contractions $\epsilon_{i}\epsilon_{j}$ in
agreement with the absence of those terms in $(DF)^{2}$ theory. Therefore, we
define the following form:

$\varphi^{Bosonic}_{\pm,n}=d\mu_{n}\;\left(\pm\frac{1}{\alpha^{\prime}}\right)^{\lfloor\frac{n-2}{2}\rfloor}\int\prod\limits_{i=1}^{n}d\theta_{i}d\bar{\theta_{i}}\;\exp\Bigg{\\{}\sum\limits^{n}_{i\neq
j}\Bigg{(}\pm
2\alpha^{\prime}\frac{\theta_{i}\bar{\theta}_{i}\varepsilon_{i}\cdot
p_{j}}{z_{i}-z_{j}}\Bigg{)}\Bigg{\\}}\ =W_{\underbrace{11...1}_{n}}\ d\mu_{n}\
.$ (3.20)

The last equality is up to an overall sign. On the other hand, for the CHY
representation of $YM+(DF)^{2}$ theory we cannot use the limit
$\alpha^{\prime}\rightarrow\infty$ since this theory is massive with the mass
given by (3.17). As a consequence taking the limit
$\alpha^{\prime}\rightarrow\infty$ removes the mass term in (3.16) referring
to the YM part of this theory. As discussed before, in this limit only the CHY
representation of $(DF)^{2}$–theory is reproduced.

A similar situation applies to Einstein–Weyl theory, cf. Subsection 5.3.2. In
this case the mass term refers to the Einstein term and one obtains only the
conformal gravity (i.e. Weyl) amplitude in the limit
$\alpha^{\prime}\rightarrow\infty$. Additionally, this also relates to the
notion of massive CHY amplitudes. To conclude, while we do have
representations in twisted intersection theory for amplitudes of both
$YM+(DF)^{2}$ and $(DF)^{2}$ theories we only have CHY representations for the
latter. The same is true for conformal gravity and Einstein–Weyl theories, cf.
Section 5.

In this work one of our goals is to both complete and extend the Table 1a.
Furthermore, we want to understand how these constructions relate to string
theory, color–kinematics duality and double copies.

###  3.2 More theories from the Einstein Yang–Mills form
$\tilde{\varphi}^{EYM}_{\pm,n;r}$

The first extension of Table 1a has been accomplished in our publication [6].
There, we have shown, that one can extend the twisted intersection description
(2.4) to Einstein Yang–Mills (EYM) amplitudes by introducing an embedding
formalism. Concretely, we introduced the twisted form:

$\displaystyle\widetilde{\varphi}^{EYM}_{\pm,n;r}=$ $\displaystyle
d\mu_{n+r}\,\,\mathcal{C}(1,2,3,...,n)\int\prod\limits_{i\in{\cal
S}_{r}}d\theta_{i}d\bar{\theta_{i}}\ \exp\Bigg{\\{}\frac{1}{2}\alpha^{\prime
2}\sum\limits_{i,j\in{\cal S}_{r}}\begin{pmatrix}\theta_{i}\\\
\bar{\theta}_{i}\end{pmatrix}^{t}\Psi_{r}\,\,\begin{pmatrix}\theta_{j}\\\
\bar{\theta}_{j}\end{pmatrix}\Bigg{\\}}\Bigg{|}_{\overline{\zeta_{l}}\rightarrow\zeta_{l}}$
(3.21)
$\displaystyle\times\exp\Bigg{\\{}\pm\frac{1}{2}\alpha^{\prime}\sum\limits_{i,j\in{\cal
S}_{r}}\frac{\theta_{i}\theta_{j}\overline{\theta}_{i}\overline{\theta}_{j}(\xi_{i}\cdot\xi_{j})}{(\overline{\zeta}_{i}-\overline{\zeta}_{j})^{2}}\Bigg{\\}}\
.$

With (3.21) we have the following table

Theory | $\varphi_{-}$ | $\varphi_{+}$ | CHY representation | Amplitude
---|---|---|---|---
EYM | $\tilde{\varphi}^{EYM}_{-,n;r}$ | $\varphi^{gauge}_{+,n+r}$ | $\mathcal{C}_{n}\;\mathrm{Pf}\Psi_{S_{r}}\;\mathrm{Pf}^{\prime}\psi_{n+r}$ |  $r$ gravitons, $n$ gluons
Table 1b: Twisted forms for EYM and its CHY representation.

to construct EYM amplitudes involving $n$ gluons and $r$ gravitons as:

$\displaystyle{\cal A}_{EYM}(n;r)$
$\displaystyle=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\tilde{\varphi}^{EYM}_{-,n;r},{\varphi}^{gauge}_{+,n+r}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\ \delta(f_{k})\
\lim_{\alpha^{\prime}\rightarrow\infty}\hat{\tilde{\varphi}}^{EYM}_{-,n;r}\,\,\hat{\varphi}^{gauge}_{+,n+r}$
(3.22) $\displaystyle=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\delta(f_{k})\
\mathcal{C}_{n}\,\mathrm{Pf}\Psi_{S_{r}}\,\mathrm{Pf}^{\prime}\psi_{n+r}\ .$

Actually, we can go further by looking at other theories, which have similar
CHY structures than EYM. The first extension originates from paring
$\tilde{\varphi}^{EYM}_{-,n;r}$ with other twisted forms from (3.1) and
comparing (2.7) with the corresponding CHY representation [18]. This way it is
straightforward to construct amplitudes for generalized Yang–Mills Scalar
(gen.YMS) and extended Dirac Born–Infeld (ext.DBI) supplementing our Table 1a
by the following content:

Theory | $\varphi_{-}$ | $\varphi_{+}$ | CHY representation | Amplitude
---|---|---|---|---
Generalized Yang-Mills scalar (gen.YMS) | $\tilde{\varphi}^{EYM}_{-,n;r}$ | $\varphi^{color}_{+,n+r}$ | $\mathcal{C}_{n}\mathrm{Pf}\Psi_{S_{r}}\mathcal{C}_{n+r}$ |  $r$ gluons $n$ color scalars
Extended Dirac Born-Infeld (ext.DBI) | $\tilde{\varphi}^{EYM}_{-,n;r}$ | $\varphi^{scalar}_{+,n+r}$ | $\mathcal{C}_{n}\mathrm{Pf}\Psi_{S_{r}}(\mathrm{Pf}^{\prime}A_{n+r})^{2}$ |  $r$ gluons $n$ higher derivative scalars

Table 2: Additional theories that can be described through new twisted form
$\widetilde{\varphi}^{EYM}$.

In order to verify our Table 2 we compute the intersection numbers (2.4) in
the limit $\alpha^{\prime}\rightarrow\infty$ given by (2.7) involving the
pairs of twisted forms for gen.YMS and ext.DBI and compare the results with
their corresponding CHY representations.

In the following, we shall give some details about the two theories, which
have appeared in the previous Table 2.

  * 1.

Generalized Yang–Mills scalar (gen.YMS)

Generalized Yang–Mills Scalar (gen.YMS) is described by YM gauge theory
coupled to scalars $\phi^{a\tilde{a}}$ with both one color $a$ and one flavour
index $\tilde{a}$ [18] i.e.:

$\displaystyle\mathcal{L}_{gen.YMS}=-\frac{1}{4}F_{a}^{\mu\nu}F^{a\mu\nu}-\frac{1}{2}(D_{\mu}\phi^{a\tilde{a}})^{2}+\lambda\
f^{abc}f^{\tilde{a}\tilde{b}\tilde{c}}\
\phi^{a\tilde{a}}\phi^{b\tilde{b}}\phi^{c\tilde{c}}\ .$ (3.23)

Actually, if the second set of labels $\tilde{a}$ also represents color
indices we have two color groups and obtain pure YM plus cubic biadjoint
scalar theory $YM+\phi^{3}$. Furthermore, without the three–point interaction
$\lambda\rightarrow 0$ in the Lagrangian (3.23) we are obtaining Yang Mills
Scalar (YMS) theory. As in the definition of (3.23) the group structure of the
theory at hand is extended by a flavour group for multiple scalars
$\phi^{a\tilde{a}}$. However, here we shall consider only one scalar in the
adjoint representation of the gauge group with a single trace for the flavour
group. As proposed above, with (2.7) we can construct the CHY representation
of the gen.YMS amplitude involving $r$ gluons and $n$ scalars:

$\displaystyle{\cal A}_{gen.YMS}(n;r)$
$\displaystyle=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{color}_{-,n+r},\tilde{\varphi}^{EYM}_{+,n;r}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\delta(f_{k})\
\lim_{\alpha^{\prime}\rightarrow\infty}\hat{\varphi}^{color}_{-,n+r}\,\,\hat{\tilde{\varphi}}^{EYM}_{+,n;r}$
(3.24) $\displaystyle=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\delta(f_{k})\ \mathcal{C}_{n}\ \mathrm{Pf}\Psi_{S_{r}}\
\mathcal{C}_{n+r}\ .$

  * 2.

Extended Dirac Born–Infeld

There are two extensions of the the Born–Infeld (BI) theory, which we will
discuss throughout out this work. Firstly, there is the Dirac Born–Infeld
(DBI) theory. The latter is defined as an scalar extension of the BI theory
(3.11) with $n$ scalars and the Lagrangian is given by [18]:

$\displaystyle\mathcal{L}_{DBI}=\ell^{-2}\
\Bigg{(}\sqrt{-{\det}_{4}(\eta_{\mu\nu}-\ell^{2}\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{I}+\ell
F_{\mu\nu})}-1\Bigg{)}\ .$ (3.25)

Furthermore, one can extend this theory by generalizing the scalar kinetic
term by the function $U(\Phi)$ given in (3.9) and leading to extended DBI
theory (ext.DBI) with the corresponding Lagrangian

$\displaystyle\mathcal{L}_{ext.DBI}=\ell^{-2}\
\Bigg{(}\sqrt{-{\det}_{4}(\eta_{\mu\nu}-\frac{\ell^{2}}{4\lambda^{2}}\mathop{\rm
Tr}(\partial_{\mu}U^{\dagger}\partial_{\nu}U)-\ell^{2}W_{\mu\nu}+\ell
F_{\mu\nu})}-1\Bigg{)}\ ,$ (3.26)

and:

$\displaystyle W_{\mu\nu}$
$\displaystyle=\sum\limits_{m=1}^{\infty}\sum\limits_{k=0}^{m-1}\frac{2(m-k)}{2m-1}\;\lambda^{2m+1}\;\mathop{\rm
Tr}(\partial_{[\mu}\Phi\,\,\Phi^{2k}\partial_{\nu]}\Phi\,\,\Phi^{2(m-k)-1})\
.$ (3.27)

For $\lambda\rightarrow 0$ the expression (3.26) yields the DBI action (3.25),
while we recover NSLM for $\ell\rightarrow 0$, respectively. With (2.7) the
amplitude for ext.DBI involving $r$ gluons and $n$ scalars can be constructed
as

$\displaystyle{\cal A}_{ext.DBI}(n;r)$
$\displaystyle=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{scalar}_{-,n+r},\tilde{\varphi}^{EYM}_{+,n;r}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\delta(f_{k})\
\lim_{\alpha^{\prime}\rightarrow\infty}\hat{\varphi}^{scalar}_{-,n+r}\,\,\hat{\tilde{\varphi}}^{EYM}_{+,n;r}\
,$ (3.28) $\displaystyle=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\delta(f_{k})\;\mathcal{C}_{n}\;\mathrm{Pf}\Psi_{S_{r}}\;(\mathrm{Pf}^{\prime}A_{n+r})^{2}\
,$

which is the CHY amplitude given in [18]. The double copy structure of BI, DBI
and ext.DBI theories will be discussed in Section 5 resulting in the
representations (5.20), (5.14) and (5.23), respectively.

###  3.3 Theories involving the Einstein–Maxwell form
$\tilde{\varphi}^{EM}_{\pm,n;r}$

Above we have used the twisted form (3.21) of EYM to compute the intersection
number (2.7) for the amplitudes of EYM, gen.YMS and ext.DBI theories given in
(3.22), (3.24) and (3.28), respectively. While for the latter a known CHY
representation exists [18], their corresponding twisted forms had not been
constructed so far. Similarly, the twisted form for Einstein–Maxwell (EM),
i.e. Einstein gravity with an $U(1)^{m}$ gauge group, as it arises from
compactification of $m$ dimensions, can be used to construct other amplitudes
for theories which interact with Maxwell theory. One would expect these
theories to be defined in some limit of the Yang–Mills theory. We organize
them in the following Table 3 based on [18].

Theory | CHY representation | Amplitude
---|---|---
Einstein–Maxwell (EM) | $\mathrm{Pf}X_{n}\,\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;\mathrm{Pf}^{\prime}\psi_{n+r}$ |  $r$ graviton $n$ photon
Dirac Born–Infeld (DBI) | $\mathrm{Pf}X_{n}\,\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;(\mathrm{Pf}^{\prime}A_{n+r})^{2}$ |  $r$ gluons $n$ color scalars
Yang-Mills scalar (YMS) | $\mathrm{Pf}X_{n}\,\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;\mathcal{C}_{n+r}$ |  $r$ gluons $n$ color scalars
Table 3: Known theories yet without twisted from description.

The matrix $\Psi_{S_{r:n}}$ is similar to the object $\Psi_{S_{r}}$ introduced
in [6]. The latter is a $(2r)\times(2r)$–matrix with only those indices
included, which refer to the highest spin particles of the theory under
consideration. On the other hand, the object $\Psi_{S_{r:n}}$ is a
$(2r+n)\times(2r+n)$–matrix with an additional (half) sector contributing for
the lower spin particles. More concretely, for EM we have [18]

$\Psi_{S_{r:n}}=\begin{pmatrix}A_{ab}&A_{aj}&(-C^{t})_{ab}\\\
A_{ib}&A_{ij}&(-C^{t})_{aj}\\\ C_{ab}&C_{aj}&B_{ab}\end{pmatrix}$ (3.29)

described by the sets of graviton indices $a,b\in\\{1,\ldots,r\\}$ and the
photon (gluon) indices $i,j\in\\{1,\ldots,n\\}$. In cases of DBI and YMS we
have the sets of gluon indices $a,b\in\\{1,\ldots,r\\}$ and the scalar indices
$i,j\in\\{1,\ldots,n\\}$.

The only common part in the CHY representation of all three theories is
comprised by the Pfaffian $\mathrm{Pf}X_{n}$ of the $n\times n$ matrix $X$
with matrix elements:

$\displaystyle X_{ab}=\begin{cases}&\frac{1}{z_{a}-z_{b}}\hskip 2.84544pt\quad
a\neq b\ ,\\\ &0\qquad\quad a=b\ .\end{cases}$ (3.30)

This Pfaffian describes a correlator involving an even number of $n$ fermions,
i.e.:

$\langle\psi(z_{1})\ldots\psi(z_{n})\rangle=\mathrm{Pf}X_{n}\ .$ (3.31)

Therefore, the number $n$ of photons must be even. The twisted form of EM can
be represented by the following $(n+r)$–form:

$\displaystyle\tilde{\varphi}^{EM}_{\pm,n;r}$
$\displaystyle=d\mu_{n+r}\,\,\int\prod\limits_{i\in{\cal
S}_{r}}d\theta_{i}d\bar{\theta_{i}}\,\frac{\theta_{k}\theta_{l}}{z_{k}-z_{l}}\,\exp\Bigg{\\{}\frac{1}{2}\alpha^{\prime
2}\sum\limits_{i,j\in\\{r,n\\}}\begin{pmatrix}\theta_{i}\\\
\bar{\theta}_{i}\end{pmatrix}^{t}\Psi_{S_{r:n}}\,\,\begin{pmatrix}\theta_{j}\\\
\bar{\theta}_{j}\end{pmatrix}\Bigg{\\}}\;\times\mathrm{Pf}X_{n}\ ,$ (3.32)

where we have used the Pfaffian in equation (3.31). Together with the twisted
form for YM $\varphi^{gauge}_{-,n+r}$ we can construct the Einstein-Maxwell
amplitude

$\displaystyle{\cal A}_{EM}(n;r)$
$\displaystyle=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{gauge}_{-,n+r},\tilde{\varphi}^{EM}_{+,n;r}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\ \delta(f_{k})\
\lim_{\alpha^{\prime}\rightarrow\infty}\hat{\varphi}^{gauge}_{-,n+r}\,\,\hat{\tilde{\varphi}}^{EM}_{+,n;r}\
,$ (3.33) $\displaystyle=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\ \delta(f_{k})\
\mathrm{Pf}X_{n}\,\,\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;\,\,\mathrm{Pf}^{\prime}\psi_{n+r}\
,$

involving $r$ gravitons and $n$ photons. Similar constructions can be
established for the two other theories displayed in Table 3. In total, we have
the following pairings of twisted forms:

Theory | $\varphi_{-}$ | $\varphi_{+}$ | CHY representation
---|---|---|---
EM | $\varphi^{gauge}_{-,n+r}$ | $\tilde{\varphi}^{EM}_{+,n;r}$ | $\mathrm{Pf}X_{n}\;\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;\mathrm{Pf}^{\prime}\psi_{n+r}$
DBI | $\varphi^{scalar}_{-,n+r}$ | $\tilde{\varphi}^{EM}_{+,n;r}$ | $\mathrm{Pf}X_{n}\;\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;(\mathrm{Pf}^{\prime}A_{n+r})^{2}$
YMS | $\varphi^{color}_{-,n+r}$ | $\tilde{\varphi}^{EM}_{+,n;r}$ | $\mathrm{Pf}X_{n}\;\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;\mathcal{C}_{n+r}$
Table 4: Theories which can be described by $\tilde{\varphi}^{EM}_{\pm,n;r}$.

##  4 New and old theories from a novel twisted form

So far we have only considered those extensions of Tables 1a and 1b, which
refer to known theories. This has been achieved by paring known twisted forms
with either the EYM form $\tilde{\varphi}^{EYM}_{\pm,n;r}$ or the EM form
$\tilde{\varphi}^{EM}_{\pm,n;r}$ given in (3.21) and (3.32), respectively.
However, we want to extend and generalize our considerations by constructing
new twisted forms relevant for different types of spin–two theories, e.g.
Weyl, conformal or $R^{3}$ gravity. These theories allow for CHY
representations related to constructions in string theory [23] or ambitwistor
string theory [28]. The theories and their CHY representations we shall be
concerned with are displayed in Table 5.

###  4.1 Novel twisted form from disk embedding

The function

$\widetilde{W}_{\underbrace{11\ldots 1}_{r}}=\prod\limits_{i\in{\cal
R}}\Bigg{(}\sum\limits_{j\in{\cal N}}\frac{\varepsilon_{i}\cdot
p_{j}}{z_{ij}}+\sum\limits_{j\in{\cal R}\atop j\neq
i}\frac{\varepsilon_{i}\cdot p_{j}}{z_{ij}}\Bigg{)}\ ,$ (4.1)

which has been introduced in [23], generalizes (3.19) and describes the
Weyl–YM theory. The set ${\cal N}$ encompasses all gluon labels while the set
${\cal R}$ comprises all graviton labels. In the object (4.1) the diagonal
elements of the $C$–block (3.3) entering the matrix $\psi_{n}$ in (3.2)
appear. A special limit of the function (4.1) is exhibited in (3.1), which
gives rise to (3.19) and stems from the twisted form
$\varphi^{bosonic}_{\pm,n}$ originating from the open bosonic string theory.
Here we want to construct the new twisted form
$\widetilde{\varphi}^{Bosonic}_{\pm,n;r}$ associated to (4.1).

While the twisted form $\varphi^{bosonic}_{\pm,n}$ may directly be derived
from the bosonic string disk amplitude for the construction of the form
$\widetilde{\varphi}^{Bosonic}_{\pm,n;r}$ we apply the embedding of disk
fields onto the sphere following [6]. In the latter work we have introduced a
map which embeds the field content of the open superstring sector onto a
specific closed superstring sector. This map has then successively been
applied to construct the twisted form $\widetilde{\varphi}^{EYM}_{\pm;n;r}$.
In similar manner we want to apply this map to embed the field content of a
bosonic open string sector onto a closed string sector on the sphere and then
extract the bosonic twisted form $\widetilde{\varphi}^{Bosonic}_{\pm,n;r}$. We
begin with the embedding of the open string fields on the disk onto the sphere
as:

$\displaystyle X^{\mu}(x)$ $\displaystyle\longmapsto
X^{\mu}(z)+\widetilde{X}^{\mu}(\overline{z})\ \Rightarrow\
\begin{cases}&\partial X\mapsto\partial X\\\ &k\cdot X\mapsto
k\cdot(X+\widetilde{X})\ .\end{cases}$ (4.2)

Then, the bosonic open string vertex operators on the sphere are mapped
according to [6]:

$\displaystyle V_{o}(\varepsilon_{i},k_{i},x_{i})$ $\displaystyle\longmapsto
V_{o}(\varepsilon_{i},k_{i},z_{i})\ e^{ik_{i}\widetilde{X}(\overline{z}_{i})}\
\tilde{J}^{c_{i}}(\bar{z}_{i})\ ,\ \ $ $\displaystyle i=1,\ldots,n\ ,$ (4.3)
$\displaystyle V_{c}(\varepsilon_{s},q_{s},z_{n+s},\bar{z}_{n+s})$
$\displaystyle\longmapsto
V_{c}=V_{o}(\varepsilon_{s},q_{s},z_{n+s})V_{o}(\tilde{\varepsilon}_{s},\tilde{q}_{s},\bar{z}_{n+s})$
$\displaystyle s=1,\ldots,r\ .$

Applying the map (4.3) on a generic disk correlator involving $n$ open and $r$
closed bosonic strings yields the following sphere correlator:

$\displaystyle\Big{\langle}\prod\limits_{i=1}^{n}V_{o}(\varepsilon_{i},k_{i},z_{i})$
$\displaystyle\prod\limits_{s=1}^{r}V_{c}(\varepsilon_{s},q_{s},z_{n+s},\bar{z}_{n+s})\Big{\rangle}_{D_{2}}\mapsto\langle\tilde{J}^{c_{1}}(\bar{z}_{1})\ldots\tilde{J}^{c_{n}}(\bar{z}_{n})\rangle_{S^{2}}$
(4.4)
$\displaystyle\times\Big{\langle}\prod\limits_{i=1}^{n}V_{o}(\varepsilon_{i},k_{i},z_{i})e^{ik_{i}\tilde{X}(\bar{z}_{i})}\prod\limits_{s=1}^{r}V_{c}(\varepsilon_{s},q_{s},z_{n+s},\bar{z}_{n+s})\Big{\rangle}_{S_{2}}\
.$

The factorization property on the sphere allows to write the correlator (4.4)
as the product of holomorphic and anti–holomorphic contractions

$\displaystyle{\cal A}_{embededd}(n;r)$
$\displaystyle=\underbrace{\Big{\langle}\prod\limits_{i=1}^{n}V_{o}(\varepsilon_{i},k_{i},z_{i})\prod\limits_{s=1}^{r}V_{o}(\varepsilon_{s},q_{s},z_{n+s})\Big{\rangle}_{S_{2}}}_{holomorphic}$
(4.5)
$\displaystyle\times\underbrace{\mathcal{C}_{n}\\{\overline{z}_{i}\\}\;\Big{\langle}\prod\limits_{s=1}^{r}V_{o}(\tilde{\varepsilon}_{s},\tilde{q}_{s},\bar{z}_{n+s})\prod\limits_{i=1}^{n}e^{ik_{i}\tilde{X}(\bar{z}_{i})}\Big{\rangle}_{S_{2}}}_{anti-
holomorphic},$

with the gauge current part

$\mathcal{C}_{n}(\\{\overline{z}_{i}\\})=\frac{1}{(\bar{z}_{1}-\bar{z}_{2})(\bar{z}_{2}-\bar{z}_{3})\cdot\ldots\cdot(\bar{z}_{n}-\bar{z}_{1})}\
,$ (4.6)

referring to the single trace term $\mathop{\rm Tr}(T^{c_{1}}\ldots
T^{c_{n}})$. The latter is extracted from the full gauge correlator
$\langle\tilde{J}^{c_{1}}(\bar{z}_{1})\ldots\tilde{J}^{c_{n}}(\bar{z}_{n})\rangle_{S^{2}}$
by projection [6]. As customary in (4.5) the holomorphic part comprises the
contractions of $n+r$ bosonic open strings which give rise to the twisted form
$\varphi^{bosonic}_{\pm,n+r}$ given in (3.1) and [1]. On the other hand, the
anti–holomorphic part of (4.5) can be borrowed to define
$\widetilde{\varphi}^{Bosonic}_{\pm,n;r}$ as follows:

$\mathcal{C}_{n}\\{\overline{z}_{i}\\}\
\Big{\langle}\prod\limits_{s=1}^{r}V_{o}(\tilde{\varepsilon}_{s},\tilde{q}_{s},\bar{z}_{n+s})\prod\limits_{i=1}^{n}e^{ik_{i}\tilde{X}(\bar{z}_{i})}\Big{\rangle}_{S_{2}}\times\overline{KN}^{-1}\;\Bigg{|}_{\bar{z_{i}}\mapsto
z_{i}}=:\widetilde{\varphi}^{Bosonic}_{\pm,n;r}\ .$

Note, that the above step involves a map from anti–holomorphic coordinates
$\bar{z}_{i}$ to holomorphic ones $z_{i}$ together with the removal of the
(anti–holomorphic) Koba–Nielsen factor $\overline{KN}$ [6]. To this end, we
find:

$\displaystyle\widetilde{\varphi}^{Bosonic}_{\pm,n;r}$
$\displaystyle=d\mu_{n+r}\;\left(\pm\frac{1}{\alpha^{\prime}}\right)^{\lfloor\frac{n+r-2}{2}\rfloor}\mathcal{C}_{n}\\{z_{i}\\}\int\prod\limits_{i=1}^{r}d\theta_{i}d\bar{\theta_{i}}\;$
$\displaystyle\times\exp\Bigg{\\{}\pm 2\alpha^{\prime}\,\sum\limits_{i\in{\cal
R}}\theta_{i}\bar{\theta}_{i}\,\Bigg{(}\sum\limits_{j\in{\cal
N}}\frac{\varepsilon_{i}\cdot p_{j}}{z_{i}-z_{j}}+\sum\limits_{j\in{\cal
R}\atop j\neq i}\frac{\varepsilon_{i}\cdot
p_{j}}{z_{i}-z_{j}}\Bigg{)}+\sum\limits_{j\neq i\atop j,i\in{\cal
R}}\frac{\theta_{i}\bar{\theta}_{i}\theta_{j}\bar{\theta}_{j}\varepsilon_{i}\cdot\varepsilon_{j}}{(z_{i}-z_{j})^{2}}\Bigg{\\}}\
.$ (4.7)

Finally, we take the $\alpha^{\prime}\rightarrow\infty$ limit as

$\displaystyle\lim\limits_{\alpha^{\prime}\rightarrow\infty}\widetilde{\varphi}^{Bosonic}_{\pm,n;r}=\mathcal{C}_{n}\
\prod\limits_{i\in{\cal R}}\Bigg{(}\sum\limits_{j\in{\cal
N}}\frac{\varepsilon_{i}\cdot p_{j}}{z_{ij}}+\sum\limits_{j\in{\cal R}\atop
j\neq i}\frac{\varepsilon_{i}\cdot p_{j}}{z_{ij}}\Bigg{)}\
d\mu_{n+r}\equiv\mathcal{C}_{n}\ \widetilde{W}_{\underbrace{11...1}_{r}}\
d\mu_{n+r}\ ,$ (4.8)

which comprises the color form $\mathcal{C}_{n}$ (over the set of legs ${\cal
N}$) and the function $\widetilde{W}_{\underbrace{11...1}_{r}}$ referring to
the set ${\cal R}$ of legs and introduced in (4.1) .

###  4.2 New theories constructed from the twisted form
$\widetilde{\varphi}^{Bosonic}$

In this subsection we display a set of theories whose amplitudes can be
constructed from the intersection number (2.4) involving the twisted form
(3.20) or our newly constructed form (4.7). In fact, the amplitudes are
inferred from (2.7) in the $\alpha^{\prime}\rightarrow\infty$ limit where
(3.20) and (4.8) become relevant. The following three twisted forms stemming
from bosonic string theory are available:

  * 1.

$\varphi^{bosonic}_{\pm,n}$ arises from the full bosonic string amplitude and
includes different orders in $\alpha^{\prime}$, cf. (3.1) and Table 1a. It
does not directly correspond to a CHY representation.

  * 2.

$\varphi^{Bosonic}_{\pm,n}$ is the $\alpha^{\prime}\rightarrow\infty$ limit of
$\varphi^{bosonic}_{\pm,n}$, cf. (3.20). It corresponds to the function
$W_{\underbrace{11...1}_{n}}$ given in (3.19) in the CHY formulation.

  * 3.

$\widetilde{\varphi}^{Bosonic}_{\pm,n;r}$ is the embedded twisted $n+r$–form
(4.7) related to $\varphi^{bosonic}_{\pm,n}$. It corresponds to the function
$\widetilde{W}_{\underbrace{11...1}_{r}}$ given in (4.1) in the CHY
formulation times a color $n$–form $\mathcal{C}_{n}$.

Building up on the previous results we have the following table of theories
and their different representations.

Theory | $\varphi_{-}$ | $\varphi_{+}$ | CHY representation | Amplitude
---|---|---|---|---
$(DF)^{2}$ | $\varphi^{color}_{-,n}$ | $\varphi^{Bosonic}_{+,n}$ | $\mathcal{C}_{n}\;W_{\underbrace{11...1}_{n}}$ |  $n$ gluon (higher derivative)
Conformal Gravity (CG) | $\varphi^{gauge}_{-,n}$ | $\varphi^{Bosonic}_{+,n}$ | $\mathrm{Pf}^{\prime}\psi_{n}\;W_{\underbrace{11...1}_{n}}$ |  $n$ graviton (Weyl)
$(Weyl)^{3}\,\,or\,\,R^{3}$ | $\varphi^{Bosonic}_{-,n}$ | $\varphi^{Bosonic}_{+,n}$ | $\Bigg{(}W_{\underbrace{11...1}_{n}}\Bigg{)}^{2}$ |  $n$ graviton (higher derivative)
$Weyl$–$YM$ | $\widetilde{\varphi}^{Bosonic}_{-,n;r}$ | ${\varphi}^{gauge}_{+,n+r}$ | $\mathcal{C}_{n}\;\widetilde{W}_{\underbrace{11...1}_{r}}\;\mathrm{Pf}^{\prime}\psi_{n+r}$ |  $r$ graviton (Weyl) $n$ gluons
$(DF)^{2}$–Photon | $\varphi^{scalar}_{-,n}$ | $\varphi^{Bosonic}_{+,n}$ | $(\mathrm{Pf}^{\prime}A_{n})^{2}\;W_{\underbrace{11...1}_{n}}$ |  $n$ higher derivative photon
$(DF)^{2}+\phi^{3}$ | $\tilde{\varphi}^{Bosonic}_{-,n;r}$ | $\varphi^{color}_{+,n+r}$ | $\mathcal{C}_{n}\;\widetilde{W}_{\underbrace{11...1}_{r}}\;\mathcal{C}_{n+r}$ |  $r$ gluons (higher derivative) $n$ scalars
Table 5: Theories described by twisted intersection forms
${\varphi}^{Bosonic}_{\pm,n}$ and $\tilde{\varphi}^{Bosonic}_{\pm,n;r}$.

Notice, that the construction of Weyl–YM is similar to the EYM amplitude
(3.22). For the latter we used the open and closed superstring together with
the underlying embedding formalism to construct
$\tilde{\varphi}^{EYM}_{-,n;r}$, while $\tilde{\varphi}^{Bosonic}_{-,n;r}$ is
derived from (4.3) involving the open and closed bosonic string. In the above
table we have added the paring of our newly constructed twisted form (4.7) and
the color form $\varphi^{color}_{\pm,n+r}$. In the
$\alpha^{\prime}\rightarrow\infty$ limit we obtain the corresponding CHY
integrand from the twisted intersection (2.4):

$\lim_{\alpha^{\prime}\rightarrow\infty}\langle\tilde{\varphi}^{Bosonic}_{-,n;r},\varphi^{color}_{+,n+r}\rangle_{\omega}\simeq\mathcal{C}_{n}\;\widetilde{W}_{\underbrace{11...1}_{r}}\;\mathcal{C}_{n+r}\
.$ (4.9)

We can see that the resulting CHY representation in (4.9) corresponds to
$(DF)^{2}+\phi^{3}$ theory. Finally, let us review some of the basics of the
vector theories appearing in Table 5.

  * 1.

Higher derivative gauge theory $(DF)^{2}$

In the literature there is a variety of higher derivative gauge theories
denoted by $(DF)^{2}$. Here we use the following definition from [26]:

$\mathcal{L}_{(DF)^{2}}=\frac{1}{2}(D_{\mu}F^{a\,\mu\nu})^{2}-\frac{g}{3}\;F^{3}+\frac{1}{2}(D_{\mu}\varphi^{a})^{2}+\frac{g}{2}\;C^{\alpha\,ab}\varphi^{\alpha}F^{a}_{\mu\nu}F^{b\,\mu\nu}+\frac{g}{3!}\;d^{\alpha\beta\gamma}\varphi^{\alpha}\varphi^{\beta}\varphi^{\gamma}\
.$ (4.10)

In particular, we have the cubic higher order gauge interaction and gauge
covariant derivative

$\displaystyle F^{3}$
$\displaystyle=f^{abc}F^{a\,\mu}_{\nu}F^{b\,\nu}_{\gamma}F^{c\,\gamma}_{\mu}\
,$ (4.11) $\displaystyle D_{\mu}\varphi^{\alpha}$
$\displaystyle=\partial_{\mu}\varphi^{\alpha}-ig(T^{a}_{R})^{\alpha\beta}A_{\mu}^{a}\,\varphi^{\beta}$

respectively, with $\varphi$ in a real representation $R$ of the gauge group
and real generators of the gauge group $(T^{a}_{R})^{\alpha\beta}$. This
theory has six propagating degrees of freedom. One accounting for the
”auxiliary” scalar field $\varphi$, which is not considered in the scattering
amplitude under consideration. In addition, we have five degrees of freedom
from the higher derivative gauge theory which includes a negative norm state
(i.e. ghost). The amplitude of this theory, which describes the scattering of
$n$ higher derivative spin–one vectors, is given by the intersection number
(2.4) as

$\displaystyle{\cal
A}_{(DF)^{2}}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi_{-,n}^{color},\varphi_{+,n}^{Bosonic}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\
\delta(f_{k})\,\,\mathcal{C}_{n}\,\,W_{\underbrace{11...1}_{n}}\ ,$ (4.12)

leading to the correct CHY formulation.

  * 2.

$(DF)^{2}-{Photon}$

Next, the $(DF)^{2}-{\rm Photon}$ theory involves the higher derivative $U(1)$
gauge theory and the Einstein gravity background. The relevant Lagrangian is
given by:

$\displaystyle\frac{1}{\sqrt{-g}}\mathcal{L}_{(DF)^{2}_{Photon}}=$
$\displaystyle\frac{1}{2\kappa^{2}}R+\frac{1}{4}(D_{\mu}F^{\alpha\beta})^{2}+\frac{1}{8}RF^{2}-\frac{1}{6}\kappa^{2}D_{\mu}F_{\alpha\beta}D^{\alpha}F^{\mu\gamma}F_{\gamma\delta}F^{\delta\beta}$
(4.13)
$\displaystyle+\frac{1}{48}\kappa^{2}(D_{\mu}F^{\alpha\beta})^{2}F^{2}+\mathcal{O}(\kappa^{4})\
.$

The amplitude ${\cal A}_{(DF)^{2}_{Photon}}(n)$ involving $n$ photons can be
computed by the intersection number (2.4). The
$\alpha^{\prime}\rightarrow\infty$ limit (2.7) of the latter yields

${\cal
A}_{(DF)^{2}_{Photon}}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{scalar}_{-,n},\varphi^{Bosonic}_{+,n}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\delta(f_{k})\,\,W_{\underbrace{11...1}_{n}}\;(\mathrm{Pf}^{\prime}A_{n})^{2}\
,$ (4.14)

and reproduces the CHY representation of the amplitude ${\cal
A}_{(DF)^{2}_{Photon}}(n)$ [28]. In (4.14) we have the submatrix $A_{n}$
defined in (3.3) and the function $W_{\underbrace{11...1}_{n}}$ is given in
(3.20).

##  5 New and old double copies

One of the most intriguing features of double copy relations is the connection
between spin–one and spin–two theories and their symmetries. Likewise, the
relation between gauge invariance for spin–one theories and diffeomorphism
invariance associated to spin–two theories. It has been argued in [26] that
taking two spin–one theories which satisfy color–kinematics duality and double
copy them results in a spin–two theory which is invariant under linearized
diffeomorphism. This important feature requires further investigation and
understanding for the massive case which is notable challenging [29]. In that
case one needs to clarify the role of CK duality and the corresponding KK–BCJ
amplitude relations. Then, with these informations one may construct massive
spin–one theories which satisfy this duality and check the corresponding
double copied theory against diffeomorphism invariance for massive spin–two
theories, e.g. dRGT gravity.

In intersection theory double copies can simply be constructed by pairing two
theories whose description in terms of intersection numbers (2.4) comprise a
color–form $\varphi^{color}\equiv PT(a)$. The latter constitutes an
orthonormal basis, i.e.

$\langle\Phi_{+,a},\Phi_{-,b}^{\vee}\rangle_{\omega}=\delta_{ab}\ \ \ ,\ \ \
\Phi_{+,a}=PT(a)\ ,$ (5.1)

and this fact allows us to simply ”glue” two different theories both
containing such a color–form $\varphi^{color}$. More concretely, for two
theories $T_{1}$ and $T_{2}$ given by the intersection numbers

$T_{1}=\langle\varphi^{1}_{+},\varphi^{color}\rangle\qquad\qquad
T_{2}=\langle\varphi^{2}_{-},\varphi^{color}\rangle$ (5.2)

respectively, one schematically obtains for their double copy:

$T_{1}\otimes T_{2}=\langle\varphi^{1}_{+},\varphi^{2}_{-}\rangle\ .$ (5.3)

More precisely, with the orthonormal basis (5.1) we have

$\displaystyle\varphi_{+}^{1}$ $\displaystyle=\sum_{a=1}^{(m-3)!}\langle
PT^{\vee}(a),\varphi_{+}^{1}\rangle_{\omega}\ PT(a)\ ,$
$\displaystyle\varphi_{-}^{2}$
$\displaystyle=\sum_{b=1}^{(m-3)!}\langle\varphi_{-}^{2},PT(b)\rangle_{\omega}\
PT^{\vee}(b)\ ,$ (5.4)

and with

$\langle
PT^{\vee}(a),\varphi_{+}^{1}\rangle_{\omega}=\sum_{b=1}^{(m-3)!}S[a|b]\
\langle PT(b),\varphi_{+}^{1}\rangle_{\omega}\ ,$

we consider the following manipulations leading to a double copy expression

$\displaystyle\langle\varphi_{+}^{1},\varphi^{2}_{-}\rangle$
$\displaystyle=\sum_{a=1}^{(m-3)!}\langle
PT^{\vee}(a),\varphi_{+}^{1}\rangle_{\omega}\
\langle\varphi_{-}^{2},PT(a)\rangle_{\omega}$
$\displaystyle=\sum_{a,b=1}^{(m-3)!}\langle\varphi_{-}^{2},PT(a)\rangle_{\omega}\
S[a|b]\ \langle PT(b),\varphi_{+}^{1}\rangle_{\omega}$
$\displaystyle=\sum_{a,b=1}^{(m-3)!}T_{1}(a)\ S[a|b]\ T_{2}(b)\ ,$ (5.5)

with the intersection form or KLT kernel $S[a|b]$ given in (3.15). Therefore,
the two theories (5.2) involving a color form in their twisted intersection
forms give rise to the double copy (5.5) denoted by $T_{1}\otimes T_{2}$.

###  5.1 Collections of some known double copies

Here we compile a list of different double copy constructions from pairs of
theories discussed before. The latter exhibit a color form $\varphi^{color}$
in their twisted intersection form (2.4). In particular, we shall discuss
double copies for Born–Infeld and $(DF)^{2}_{Photon}$ theories.

  1. (i)

Special Galilean

We start with the special Galilean theory described by gluing two identical
NSLM theories

$\displaystyle T_{1}(a)$ $\displaystyle={\cal
A}_{NLSM}(a)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle
PT(a),\varphi^{scalar}_{+,n}\rangle_{\omega}\ ,$ $\displaystyle T_{2}(b)$
$\displaystyle={\cal
A}_{NLSM}(b)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{scalar}_{-,n},PT(b)\rangle_{\omega}\
.$ (5.6)

With (5.5) we construct the double copy $T_{1}\otimes T_{2}$ of Galilean
theory

${\cal
A}_{sGal}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{scalar}_{-,n},\varphi^{scalar}_{+,n}\rangle_{\omega}$
(5.7)

describing the scattering of $n$ scalars with higher derivative interaction
(3.6), i.e.:

${\cal A}_{sGal}(n)=\sum_{a,b=1}^{(n-3)!}{\cal A}_{NLSM}(a)\ S[a|b]\ {\cal
A}_{NLSM}(b)\ .$ (5.8)

  2. (ii)

Einstein–Yang–Mills (EYM)

Secondly, we consider a double copy from gen.YMS and YM theories

$\displaystyle T_{1}(a)$ $\displaystyle={\cal
A}_{gen.YMS}(a)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle
PT(a),\tilde{\varphi}^{EYM}_{+,n;r}\rangle_{\omega}\ ,$ $\displaystyle
T_{2}(b)$ $\displaystyle={\cal
A}_{YM}(b)=\langle\varphi^{gauge}_{-,n+r},PT(b)\rangle_{\omega}$ (5.9)

to construct the double copy $T_{1}\otimes T_{2}$ of EYM amplitudes [6]

${\cal
A}_{EYM}(n;r)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{gauge}_{-,n+r},\tilde{\varphi}^{EYM}_{+,n;r}\rangle_{\omega}$
(5.10)

describing the scattering of $n$ gluons and $r$ gravitons [18]

${\cal A}_{EYM}(n;r)=\sum_{a,b=1}^{(m-3)!}{\cal A}_{gen.YMS}(a)\ S[a|b]\ {\cal
A}_{YM}(b)\ ,$ (5.11)

with $m=n+r$.

  3. (iii)

Dirac–Born–Infeld (DBI)

Thirdly, we glue together YMS and NSLM theories

$\displaystyle T_{1}(a)$ $\displaystyle={\cal
A}_{YMS}(a)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle
PT(a),\tilde{\varphi}^{EM}_{+,n;r}\rangle_{\omega}\ ,$ $\displaystyle
T_{2}(b)$ $\displaystyle={\cal
A}_{NSLM}(b)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{scalar}_{-,n+r},PT(b)\rangle_{\omega}$
(5.12)

to construct the double copy $T_{1}\otimes T_{2}$ of DBI amplitudes

${\cal
A}_{DBI}(n;r)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{scalar}_{-,n+r},\tilde{\varphi}^{EM}_{+,n;r}\rangle_{\omega}$
(5.13)

describing the scattering of $r$ gluons and $n$ scalars [18]:

${\cal A}_{DBI}(n;r)=\sum_{a,b=1}^{(m-3)!}{\cal A}_{YMS}(a)\ S[a|b]\ {\cal
A}_{NLSM}(b)$ (5.14)

  4. (iv)

Einstein–Maxwell (EM)

Furthermore, EM can be written as double copy of the following two theories

$\displaystyle T_{1}(a)$ $\displaystyle={\cal
A}_{YMS}(a)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle
PT(a),\tilde{\varphi}^{EM}_{+,n;r}\rangle_{\omega}\ ,$ $\displaystyle
T_{2}(b)$ $\displaystyle={\cal
A}_{YM}(b)=\langle\varphi^{gauge}_{-,n+r},PT(b)\rangle_{\omega}\ .$ (5.15)

giving rise to the double copy $T_{1}\otimes T_{2}$ of EM amplitudes

${\cal
A}_{EM}(n;r)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{gauge}_{-,n+r},\tilde{\varphi}^{EM}_{+,n;r}\rangle_{\omega}$
(5.16)

involving of $r$ gravitons and $n$ photons [18]:

${\cal A}_{EM}(n;r)=\sum_{a,b=1}^{(m-3)!}{\cal A}_{YMS}(a)\ S[a|b]\ {\cal
A}_{YM}(b)\ .$ (5.17)

  5. (v)

Born–Infeld (BI) theory

The amplitudes of Born–Infeld theory can be written as double copy of the
following two theories:

$\displaystyle T_{1}(a)$ $\displaystyle={\cal
A}_{NLSM}(a)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle
PT(a),\varphi^{scalar}_{+,n}\rangle_{\omega}\ ,$ $\displaystyle T_{2}(b)$
$\displaystyle={\cal
A}_{YM}(b)=\langle\varphi^{gauge}_{-,n},PT(b)\rangle_{\omega}\ .$ (5.18)

Then, the double copy $T_{1}\otimes T_{2}$ yields the BI amplitudes

${\cal
A}_{BI}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{gauge}_{-,n},\varphi^{scalar}_{+,n}\rangle_{\omega}$
(5.19)

accounting for the scattering of $n$ gluons:

${\cal A}_{BI}(n)=\sum_{a,b=1}^{(n-3)!}{\cal A}_{NLSM}(a)\ S[a|b]\ {\cal
A}_{YM}(b)\ .$ (5.20)

  6. (vi)

Extended Dirac Born–Infeld (ext.DBI) theory

In addition to BI and DBI the amplitudes of ext.DBI theory can be written as
double copy of the following two theories:

$\displaystyle T_{1}(a)$ $\displaystyle={\cal
A}_{NLSM}(a)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle
PT(a),\varphi^{scalar}_{+,n+r}\rangle_{\omega}\ ,$ $\displaystyle T_{2}(b)$
$\displaystyle={\cal
A}_{gen.YMS}(b)=\langle\tilde{\varphi}^{EYM}_{-,n;r},PT(b)\rangle_{\omega}\ .$
(5.21)

Then, the double copy $T_{1}\otimes T_{2}$ yields the ext.DBI amplitudes

${\cal
A}_{ext.DBI}(n,r)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\tilde{\varphi}^{EYM}_{-,n;r},\varphi^{scalar}_{+,n+r}\rangle_{\omega}$
(5.22)

accounting for the scattering of $r$ gluons and $n$ higher derivative scalars
as:

${\cal A}_{ext.DBI}(n,r)=\sum_{a,b=1}^{(m-3)!}{\cal A}_{NLSM}(a)\ S[a|b]\
{\cal A}_{gen.YMS}(b)\ .$ (5.23)

  7. (vii)

$(DF)^{2}_{Photon}$ theory

Finally, the amplitudes (4.14) of $(DF)^{2}_{Photon}$ theory can be written by
double copying the following two theories:

$\displaystyle T_{1}(a)$ $\displaystyle={\cal
A}_{NLSM}(a)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle
PT(a),\varphi^{scalar}_{+,n}\rangle_{\omega}\ ,$ $\displaystyle T_{2}(b)$
$\displaystyle={\cal
A}_{(DF)^{2}}(b)=\langle\varphi^{Bosonic}_{-,n},PT(b)\rangle_{\omega}\ .$
(5.24)

The double copy $T_{1}\otimes T_{2}$ for the amplitudes

${\cal
A}_{(DF)^{2}_{Photon}}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{Bosonic}_{-,n},\varphi^{scalar}_{+,n}\rangle_{\omega}$
(5.25)

involving $n$ higher derivative photons gives:

${\cal A}_{(DF)^{2}_{Photon}}(n)=\sum_{a,b=1}^{(n-3)!}{\cal A}_{NLSM}(a)\
S[a|b]\ {\cal A}_{(DF)^{2}}(b)\ .$ (5.26)

Note, that this double copy involves Einstein gravity and higher–derivative
photon terms, cf. Eq. (4.13).

Finally, let us collect all seven double copies $(i)$–$(vii)$ in the following
table.

Theory | $T_{1}$ | $T_{2}$ | CHY representation
---|---|---|---
sGal | NSLM | NSLM | $(\mathrm{Pf}^{\prime}A_{n})^{4}$
EYM | gen.YMS | YM | ${\cal C}_{n}\mathrm{Pf}\Psi_{S_{r}}\,\mathrm{Pf}^{\prime}\psi_{n+r}$
DBI | YMS | NSLM | $\mathrm{Pf}X_{n}\,\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;(\mathrm{Pf}^{\prime}A_{n+r})^{2}$
EM | YMS | YM | $\mathrm{Pf}X_{n}\,\mathrm{Pf}^{\prime}\Psi_{S_{r:n}}\;\mathrm{Pf}^{\prime}\psi_{n+r}$
BI | NSLM | YM | $(\mathrm{Pf}^{\prime}A_{n})^{2}\;\mathrm{Pf}^{\prime}\psi_{n}$
ext.DBI | NSLM | YM | $\mathcal{C}_{n}\;\mathrm{Pf}\Psi_{S_{r}}\;(\mathrm{Pf}^{\prime}A_{n+r})^{2}$
$DF^{2}$-photon | NSLM | $(DF)^{2}$ | $(\mathrm{Pf}^{\prime}A_{n})^{2}\;W_{\underbrace{11...1}_{n}}$
Table 6: Double copies $T_{1}\otimes T_{2}$ through twisted form description
(5.2).

###  5.2 Further double copies

As discussed in [23, 30] double coping spin–one theories, which satisfy
color–kinematics duality, gives rise to the existence of a diffeomorphism
invariant spin–two theory in the resulting product. This is apparent both at
the level of the amplitudes and the Lagrangian. Our amplitudes are computed by
intersection numbers (2.4) supplemented by a pair of twisted forms. If the
latter comprise a color form $\varphi^{color}$, i.e. we have the
representations (5.2), then generically CK duality is furnished for this
theory, cf. Subsection 6.1. Therefore, those theories (5.2) are candidates for
building blocks of double copies (5.5) or likewise for new spin–two theories.
In the following Table 7a we compile a list of all previously known and newly
constructed double copies from theories containing a color form
$\varphi^{color}$.

Theory $\otimes$ | $(DF)^{2}$ | YM | NLSM |  YM+$(DF)^{2}$
---|---|---|---|---
$(DF)^{2}$ |  Weyl3 or $R^{3}$ Gravity | Conformal Gravity |  $(DF)^{2}$-Photon |  $R^{3}$-Weyl
YM | Conformal Gravity | GR | Born Infeld | Einstein–Weyl
NLSM |  $(DF)^{2}$-Photon | Born-Infeld | Special Galilean |  BI-$(DF)^{2}$ Photon
YM+$(DF)^{2}$ |  $R^{3}$-Weyl | Einstein–Weyl |  BI-$(DF)^{2}$ Photon | HD gravity
Table 7a: Table of different double copies

E.g. the double copy of $(DF)^{2}$ with itself gives rise to the amplitudes of
six–derivative gravity $R^{3}$ originating from the bosonic ambitwistor string
[28]. Note, in contrast double copying $F^{3}$ amplitudes leads to the
amplitudes produced by Einstein gravity coupled to a dilaton field $\phi$, and
deformed by operators of the form $\phi R^{2}$ and $R^{3}$ [31]. On the other
hand, the conformal supergravity (CSG) amplitudes (of the non–minimal
Berkovits–Witten type theory [32]) from Table 5 follow from double copying
$(DF)^{2}$ and SYM theory [26]. From Table 7a it can be evidenced, that there
are new double copies of theories which can be constructed only by using the
property that the color form provides CK duality and can be used as integral
part to double copy two different theories. Besides, there is a close relation
between string theory and the content of Table 7a by means of looking at the
inherent double copy of the closed string in terms of open strings cf. [2, 23,
33]. Moreover, we find a double copy for higher derivative (HD) gravity and
bigravity to be discussed in Subsections 5.4 and 5.5, respectively.

Furthermore, in the next Table 7b the four double copies (5.11), (5.14),
(5.17) and (5.23) are tabulated among some new constructions like EM squared.

Theory $\otimes$ | YMS | gen.YMS | NLSM | YM
---|---|---|---|---
YMS |  EM2 |  EM $+$ YMS | DBI | EM
gen.YMS |  EM $+$ YMS | EYMS | ext.DBI | EYM
Table 7b: Table of different double copies

We use the property, that cubic bi–adjoint scalar theory $\phi^{3}$ acts as
identity element if double copied with another theory or itself [34]. For
example the double copy construction (5.10) of EYM amounts to:

$\text{gen.YMS}\otimes\text{YM}=[YM+\phi^{3}]\otimes YM=(YM\otimes
YM)+YM=GR+YM=\text{EYM}\ .$ (5.27)

Furthermore, in Table 7b we have:

$\displaystyle\text{gen.YMS}\otimes\text{YMS}$
$\displaystyle=[YM+\phi^{3}]\otimes YMS=(YM\otimes
YMS)+YMS=\text{EM}+\text{YMS}\ .$ (5.28)

On the other hand, double copying two copies of gen.YMS amplitudes has been
argued to produce amplitudes in Einstein–Yang–Mills–Scalar (EYMS) theory. The
latter simply follows from compactification of EYM theory [15, 35, 18]. This
result can also been anticipated from the decomposition:

$\displaystyle\text{gen.YMS}\otimes\text{gen.YMS}$
$\displaystyle=\text{gen.YMS}\otimes[YM+\phi^{3}]=EYM+\text{gen.YMS}\ .$
(5.29)

###  5.3 Double copies of spin–two theories

In Table 7a we have gathered various spin–two theories including GR. While the
latter is well–known the other theories may not be familiar. Therefore, for
those we shall discuss their Lagrangians and relevant amplitudes. Actually, we
need to emphasize that for some of these theories the exact form of their
Lagrangian is not known or different names for them are used by various
authors.

####  5.3.1 Conformal Gravity

We start with conformal gravity (CG). This name is associated to different
theories. The simplest definition corresponds to the $({\rm Weyl})^{2}$ action
(pure $(\text{Weyl})^{2}$ conformal gravity), i.e.

$\mathcal{L}_{CG}=\kappa_{W}^{-2}\;\sqrt{-g}\ (W_{\mu\nu\alpha\beta})^{2}\ ,$
(5.30)

with the coupling constant $\kappa_{W}$ and $W_{\mu\nu\alpha\beta}$ the Weyl
tensor

$W_{\alpha\beta\gamma\delta}=R_{\alpha\beta\gamma\delta}-\frac{1}{2}\;(R_{\alpha\gamma}g_{\beta\delta}-R_{\alpha\delta}g_{\beta\gamma}+R_{\beta\delta}g_{\alpha\gamma}-R_{\beta\gamma}g_{\alpha\delta})+\frac{1}{6}\;(g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma})\;R\
,$ (5.31)

which has the same symmetry properties (w.r.t. its indices) than the Riemann
tensor $R_{\alpha\beta\gamma\delta}$. In addition to the square of the Riemann
tensor the square of the Weyl tensor is the second independent quadratic
curvature invariant. Actually, we have the following relation:

$W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}=R_{GB}+2\;\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}\;R^{2}\right)\
,$ (5.32)

with the Gauss–Bonnet term

$R_{GB}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4\;R_{\mu\nu}R^{\mu\nu}+R^{2}\
,$ (5.33)

which for $d=4$ reduces to a topological surface term. The latter can be added
to the action without changing the (classical) theory in a spacetime which is
asymptotically Minkowski. Therefore, in $d=4$ the Lagrangian (5.30) can be
written in terms of the Riemann scalar $R$ and Ricci tensor $R_{\mu\nu}$ as:

$\displaystyle\mathcal{L}_{CG}=\kappa_{W}^{-2}\;\sqrt{-g}\
\left(\frac{1}{3}R^{2}-R^{\mu\nu}R_{\mu\nu}\right)\ .$ (5.34)

According to Table 7a the scattering amplitude ${\cal A}_{CG}(n)$ of this
theory involving $n$ spin–two particles can be represented as a double copy in
agreement with the CHY representation [28]

${\cal
A}_{CG}(n)\sim\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{gauge}_{-,n},\varphi^{Bosonic}_{+,n}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\prod_{k=2}^{n-2}\delta(f_{k})\,\,\mathrm{Pf}^{\prime}\psi_{n}\,\,W_{\underbrace{11...1}_{n}}\
,$ (5.35)

with the object111We should emphasize that the function
$W_{\underbrace{11...1}_{r}}$ in the CHY representation (5.35) does not refer
to the Weyl tensor. (3.19)

$W_{\underbrace{11...1}_{r}}=\widetilde{W}_{\underbrace{11...1}_{r}}\Big{|}_{{\cal
N}=0}=\prod\limits_{i=1}^{r}\Bigg{(}\sum\limits_{j=1\atop j\neq
i}^{r}\frac{\varepsilon_{i}\cdot p_{j}}{z_{ij}}\Bigg{)}\ ,$ (5.36)

derived from (4.1) for $n=0$. CG theory propagates six degrees of freedom
(packaged into the metric $g_{\mu\nu}$) which contain a massless spin–two (two
degrees of freedom), a massless spin–one (two degrees) and a massless ghost
spin–two field (two degrees) [36].

In (5.35) we have exhibited the double copy structure of CG [26]:

${\rm Conformal\ Gravity}=(DF)^{2}\otimes YM$ (5.37)

in terms of intersection numbers (2.4). Note that both factors $(DF)^{2}$ and
$YM$ have vectors $\tilde{A}_{\mu}$ and $A_{\nu}$, respectively, which tensor
according to

$\tilde{A}_{\mu}\otimes A_{\nu}=g_{\mu\nu}\oplus B_{\mu\nu}\otimes\phi$ (5.38)

giving rise to a graviton $g_{\mu\nu}$, an antisymmetric two–form $B_{\mu\nu}$
and a dilaton field $\phi$. Looking at the double copy (5.35) we evidence the
origin of the massless spin–one degrees of freedom as descending from the YM
theory and the massless spin–two and ghost spin–two fields are stemming from
$(DF)^{2}$.

####  5.3.2 Einstein–Weyl gravity

Einstein–Weyl gravity is a modification of GR by adding the square of the Weyl
tensor [36, 37]:

$\mathcal{L}_{EW}=\sqrt{-g}\;\left(m^{2}\;R+\kappa_{W}^{-2}\;W_{\mu\nu\alpha\beta}^{2}\right)\
.$ (5.39)

Above, the parameter $m$ relates to the string scale
$m^{2}\sim\alpha^{\prime-1}$ (likewise to the Planck scale $m\sim M_{\rm
Planck}$). In fact, the mass parameter $m$ interpolates between two–derivative
and four–derivative theories. This way by taking the limit
$\alpha^{\prime}\rightarrow\infty$ we reproduce pure $(\text{Weyl})^{2}$
conformal gravity. Similar to equation (3.18) we have:

$\lim_{\alpha^{\prime}\rightarrow\infty}\mathcal{L}_{EW}(\alpha^{\prime})\simeq\mathcal{L}_{W}\
,$ (5.40)

On the other hand, for $\alpha^{\prime}\rightarrow 0$ we end up at (non–pure)
Einstein gravity. It can be shown, that the EW action (5.39) describes both
massless and massive spin–two up to total derivatives. EW gravity has seven
degrees of freedom accounting for two degrees from the standard massless
spin–two graviton and an additional five (ghost) degrees for the massive
spin–two field all packaged inside $g_{\mu\nu}$. In fact, starting from an
action for a particular bimetric gravity with the two spin–two fields
$g_{\mu\nu}$ and $f_{\mu\nu}$ with a mass term for the latter and eliminating
$f_{\mu\nu}$ through its equations of motion yields (5.39) [38]. We shall
return to this properties in Subsection 5.5.

For EW gravity scattering amplitudes ${\cal A}_{EW}(n)$ involving $n$
gravitons can be constructed through bosonic string amplitudes [23]. According
to (5.40) in the limit $\alpha^{\prime}\rightarrow\infty$ the Einstein–Hilbert
part of the EW theory decouples and therefore the CHY representation of this
theory cannot be constructed by applying this limit at the intersection number
(2.7) of twisted forms. On the other hand, by using the bosonic string content
one can write the amplitudes of the full theory as [1] (cf. also Table 1a):

${\cal
A}_{EW}(n)=\langle\varphi^{gauge}_{+,n},\varphi^{bosonic}_{-,n}\rangle_{\omega}\
.$ (5.41)

The double copy structure of this theory is given by:

$\textit{\rm Einstein--Weyl}=[YM+(DF)^{2}]\otimes
YM\sim\langle\varphi^{gauge}_{-,n},\varphi^{bosonic}_{+,n}\rangle_{\omega}\ .$
(5.42)

Note, that according to (3.18) and (5.40) in the limit
$\alpha^{\prime}\rightarrow\infty$ the double copy structure (5.42) will
reduce to the double copy of conformal gravity (5.37). On the other hand, for
$\alpha^{\prime}\rightarrow 0$ we obtain the double copy for non–pure Einstein
gravity including an anti–symmetric tensor and a dilaton scalar, cf. also
(5.38).

Actually, the double copy structure (5.42) resembles the heterotic string. The
amplitudes for the closed heterotic string also referred to as GR+$R^{2}$ can
be calculated by using the KLT relations implementing the open bosonic string
amplitude (3.14) together with the open superstring amplitude (3.13). The
action up to order $\alpha^{\prime 3}$ was derived in [39]

$\displaystyle S_{\rm{heterotic\ string}}$
$\displaystyle=-\frac{2}{\kappa^{2}}\int{\mathrm{d}}^{d}x\,\sqrt{g}\;\left\\{R-\tfrac{4}{d-2}(\partial_{\mu}\phi)^{2}-\frac{1}{12}H^{2}\right.$
(5.43)
$\displaystyle\qquad\qquad+\frac{\alpha^{\prime}}{8}e^{-2\phi}\;\big{(}R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\big{)}+{\cal
O}(\alpha^{\prime 3})\Big{\\}},$

where $\phi$ represents the dilaton and
$H_{\mu\nu\rho}=3\left(\partial_{[\mu}B_{\nu\rho]}+\tfrac{\alpha^{\prime}}{4}\omega^{ab}_{[\mu}R^{ab}{}_{\nu\rho}\right)$
is the field strength of the anti–symmetric $B$–field and the Chern–Simons
form $\omega$. Note, that in (5.43) the linear order in $\alpha^{\prime}$
corresponds to the Gauss–Bonnet term (5.33). In the limit
$\alpha^{\prime}\rightarrow 0$ we recover standard GR including an
anti–symmetric tensor and a dilaton scalar, cf. (5.38). Likewise, in this
limit $\alpha^{\prime}\rightarrow 0$ the double copy structure (5.42) boils
down to $YM\otimes YM$.

####  5.3.3 $Weyl^{3}$ or $R^{3}$ gravity

Another spin-$2$ higher derivative theory can be constructed through the six
derivative terms like $Weyl^{3}$ or $R^{3}$. Some speculation about the
existence of $Weyl^{3}$ is made in [40], where arguments on dimensional
analysis are presented. The amplitude structure can be reproduced by the
intersection number (2.4) involving a pair of the twisted forms (3.20)

${\cal
A}_{Weyl^{3}}(n)=\lim_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{Bosonic}_{-,n},\varphi^{Bosonic}_{+,n}\rangle_{\omega}\
,$ (5.44)

leading to the following CHY representation:

${\cal A}_{Weyl^{3}}(n)=\int\limits_{\mathcal{M}_{0,n}}d\mu_{n}\
\prod_{k=2}^{n-2}\delta(f_{k})\,\,(W_{\underbrace{11...1}_{n}})^{2}\ .$ (5.45)

Looking at equation (4.12) it is evident that this integrand is built from two
sectors accounting for $(DF)^{2}$. Therefore, the twisted form (3.20) gives
rise to the double copy structure of the amplitude (5.45), i.e.:

$Weyl^{3}=(DF)^{2}\otimes(DF)^{2}\ ,$ (5.46)

or likewise [26]:

$(DF)^{2}\otimes(DF)^{2}\sim(\nabla R)^{2}+R^{3}\ .$ (5.47)

####  5.3.4 $Weyl^{3}$–$(DF)^{2}$

Here we shall discuss a double copy of $(DF)^{2}$ with $(DF)^{2}+\phi^{3}$.
More specifically, we shall discuss how the amplitude of $Weyl^{3}$–$(DF)^{2}$
theory may be described by the two twisted forms $\varphi^{Bosonic}_{\pm,n+r}$
and $\tilde{\varphi}^{Bosonic}_{\pm,n;r}$. The $Weyl^{3}$–$(DF)^{2}$ theory
can be considered as an exotic cousin of EYM theory. Its CHY representation is
assumed to take the following form:

$\displaystyle{\cal A}_{Weyl^{3}-(DF)^{2}}(n;r)$
$\displaystyle=\lim\limits_{\alpha^{\prime}\rightarrow\infty}\langle\varphi^{Bosonic}_{-,n+r},\tilde{\varphi}^{Bosonic}_{+,n;r}\rangle_{\omega}=\int\limits_{\mathcal{M}_{0,n+r}}d\mu_{n+r}\
\prod_{k=2}^{n+r-2}\delta(f_{k})\,\,\mathcal{C}_{n}\widetilde{W}_{\underbrace{11...1}_{r}}W_{\underbrace{11...1}_{n+r}}\
.$ (5.48)

The formula (5.48) is a conjecture, which presumably may be proven by
ambitwistor construction [40]. However, this goes beyond the scope of this
work and thus we write the $Weyl^{3}$-$(DF)^{2}$ theory in terms of the
following double copy of theories:

$Weyl^{3}-(DF)^{2}=(DF)^{2}\otimes[(DF)^{2}+\phi^{3}]=(DF)^{2}\otimes(DF)^{2}+(DF)^{2}\
.$ (5.49)

####  5.3.5 Summary

In the following Table 8 we summarize old and newly constructed double copies
and their pairs of twisted forms leading to spin–two theories.

Theory | $\varphi_{-}$ | $\varphi_{+}$ | CHY representation
---|---|---|---
Einstein–Weyl | $\varphi^{gauge}_{-,n}$ | $\varphi^{bosonic}_{+,n}$ | ??
Conformal Gravity (CG) | $\varphi^{gauge}_{-,n}$ | $\varphi^{Bosonic}_{+,n}$ | $\mathrm{Pf}^{\prime}\psi_{n}W_{\underbrace{11...1}_{n}}$
$Weyl^{3}$ | $\varphi^{Bosonic}_{-,n}$ | $\varphi^{Bosonic}_{+,n}$ | $\Bigg{(}W_{\underbrace{11...1}_{n}}\Bigg{)}^{2}$
$Weyl^{3}-DF^{2}$ | $\widetilde{\varphi}^{Bosonic}_{-,n;r}$ | $\varphi^{Bosonic}_{+,n+r}$ | $\mathcal{C}_{n}\widetilde{W}_{\underbrace{11...1}_{r}}W_{\underbrace{11...1}_{n+r}}$
HD gravity | $\varphi^{bosonic}_{-,n}$ | $\varphi^{bosonic}_{+,n}$ | $??$
Table 8: Old and new double copies for spin–two theories.

###  5.4 Double copy of $YM+(DF)^{2}$ and higher derivative gravity

In this section we look at the double copy of $YM+(DF)^{2}$ theory and show
that it can be related to higher derivative (HD) gravity. Looking at the Table
7a we can write the double copy of the $YM+(DF)^{2}$ in terms of known
spin–two theories in the following way:

$\displaystyle\otimes[(DF)^{2}+YM]$ (5.50)
$\displaystyle\sim\underbrace{(YM\otimes YM)}_{GR}+\underbrace{(DF^{2}\otimes
DF^{2})}_{R^{3}}+\underbrace{(DF^{2}\otimes YM)}_{CG}+\underbrace{(YM\otimes
DF^{2})}_{CG}$ $\displaystyle\sim GR+CG_{1}+CG_{2}+R^{3}\sim
GR+\widetilde{CG}+\mathcal{O}(R^{3})\ .$

Since the amplitude of the $YM+(DF)^{2}$ is described by the intersection
number (2.4) of $\varphi^{bosonic}_{\pm,n}$ and $\varphi^{color}_{\pm,n}$ we
can write the amplitude of the resulting double copied theory as:

${{\cal A}}(n)=\langle\varphi^{bosonic}_{-,n},\varphi^{bosonic}_{+,n}\rangle\
.$ (5.51)

The twisted form $\varphi^{bosonic}_{\pm,n}$ stems from the bosonic string and
is given in (3.1). The amplitude (5.51) only involves the metric $g_{\mu\nu}$
described by the higher derivative Lagrangian, which comprises the conformal
term+GR+$R^{3}$ corrections (5.50). Therefore, looking at the structure of the
double copy in (5.50) we see that this amplitude corresponds to the
interaction of $n$ spin–two fields (i.e.$\,g_{\mu\nu}$) with higher derivative
interactions. Hence, it is worth looking at the theories known as HD gravity.
The latter can be defined by the following Lagrangian [41]:

$\displaystyle\mathcal{L}=$ $\displaystyle
m_{g}^{2}\;\sqrt{g}\;\Bigg{[}\lambda_{1}R(g)+\lambda_{3}\;\left(\frac{1}{3}R^{2}-R^{\mu\nu}R_{\mu\nu}\right)\Bigg{]}+\mathcal{O}(R^{3})\
,$ (5.52)

with the following equation of motion

$B_{\mu\nu}+\frac{\lambda_{1}}{\lambda_{3}}\ G_{\mu\nu}+\mathcal{O}(R^{3})=0\
,$ (5.53)

where $B_{\mu\nu}$ and $G_{\mu\nu}$ are the Bach and Einstein tensors,
respectively. Some possible candidates for the $\mathcal{O}(R^{3})$ operators
in gravity can be found in [42]. The Lagrangian in (5.52) describes GR plus
Weyl (conformal) interactions together with higher order curvature corrections
$\mathcal{O}(R^{3})$. Comparing this Lagrangian to the double copy structure
in (5.50) we can see that the content and the type of the interactions are the
same.

Actually, the amplitudes for the closed bosonic string also referred to as
GR+$R^{2}$+$R^{3}$ can be calculated by using the KLT relations and the open
bosonic string amplitudes (3.14). The action up to order $\alpha^{\prime 3}$
was derived in [43]

$\displaystyle S_{\begin{subarray}{c}\rm{closed}\\\ \rm{bosonic\
string}\end{subarray}}$
$\displaystyle=-\frac{2}{\kappa^{2}}\int{\mathrm{d}}^{d}x\,\sqrt{g}\left\\{R-\tfrac{4}{d-2}(\partial_{\mu}\phi)^{2}-\frac{1}{12}H^{2}\right.$
(5.54)
$\displaystyle\qquad\qquad+\frac{\alpha^{\prime}}{4}e^{-2\phi}\;\big{(}R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\big{)}$
$\displaystyle\qquad\qquad+\alpha^{\prime
2}e^{-4\phi}\;\bigg{(}\frac{1}{16}R^{\mu\nu}{}_{\alpha\beta}R^{\alpha\beta}{}_{\lambda\rho}R^{\lambda\rho}{}_{\mu\nu}-\frac{1}{12}R^{\mu\nu}{}_{\alpha\beta}R^{\nu\lambda}{}_{\beta\rho}R^{\lambda\mu}{}_{\rho\alpha}\bigg{)}+{\cal
O}(\alpha^{\prime 3})\Big{\\}},$

where $\phi$ represents the dilaton and
$H_{\mu\nu\rho}=3\partial_{[\mu}B_{\nu\rho]}$ is the field strength of the
anti–symmetric $B$–field. Note, that in (5.54) the linear order in
$\alpha^{\prime}$ corresponds to the Gauss–Bonnet term which for $d=4$ reduces
to a topological surface term. In (5.50) there is also the higher order
correction $R^{3}$, which in the bosonic closed string (5.54) originates at
quadratic order in $\alpha^{\prime 2}R^{3}$. It should be emphasized that in
higher–point gravitational amplitudes the order $\alpha^{\prime 2}$ cannot be
reproduced from the double copy of the $\alpha^{\prime}$ order of the
corresponding open bosonic string amplitudes (3.14) due to additional
$\alpha^{\prime 2}\zeta_{2}F^{4}$ contributions from a single open string
sector [31, 44, 45].

It is worth mentioning that HD gravity may be related to bimetric gravity. The
latter is constructed through interactions of two spin two fields $g_{\mu\nu}$
and $f_{\mu\nu}$ with a nonlinear interacting potential [46]. It has been
shown [47] that by expanding the potential of bimetric gravity around a
specific solution of $f_{\mu\nu}$ in terms of $g_{\mu\nu}$ and integrating out
$f_{\mu\nu}$ one can match the Lagrangian of bimetric gravity to that of HD
gravity. Therefore, the intersection number associated with the amplitude in
(5.51) can also be related to some particular parameters of the integrated
bimetric gravity known as partially massless bimetric gravity, cf. Subsection
5.5.

###  5.5 Bimetric theory as a double copy

The construction of the double copy (5.50) may be related to some limit of
bimetric gravity. The starting point of this map is the connection between
conformal gravity and bimetric gravity. Looking at the bimetric potential and
its parameters one can find a region in the parameter space of bimetric theory
known as partially massless (PM) [47]. In this parameter configuration, the
bimetric theory in quadratic order in the curvature $R_{\mu\nu}$ matches
conformal gravity.

Bimetric gravity is a consistent and ghost–free theory that consists of two
spin–two fields, i.e. two metric tensors. The action for ghost free bimetric
gravity was written by Hassan and Rosen in $d=4$ dimension as [46]:

$S=m_{g}^{2}\Bigg{(}\int d^{4}x\;\sqrt{g}\;R(g)+\alpha^{2}\int
d^{4}x\sqrt{f}\;R(f)-2\,m^{2}\int d^{4}x\;\sqrt{g}\;V(S;\beta_{n})\Bigg{)}\ .$
(5.55)

Above, we have two independent Einstein–Hilbert kinetic terms for the two
spin–two fields $g_{\mu\nu}$ and $f_{\mu\nu}$ with two different Planck masses
$m_{g}$ and $m_{f}$ (with $\alpha=\frac{m_{f}}{m_{g}}$) and $m$ is the mass
scale of the massive spin–two field after diagonalization of the two metrics.
These two spin–two fields interact via the potential $V(S;\beta_{n})$. This
potential couples the two metrics and creates non–trivial self and mixed
interactions and is given by

$V(S;\beta_{n})=\sum\limits_{n=0}^{4}\beta_{n}\;e_{n}(S)\ ,$ (5.56)

where $\beta_{n}$ are five free dimensionless parameters and the symmetric
polynomials $e_{n}(S)$ depend on the matrix $S^{\mu}_{\hphantom{\mu}\nu}$.
Specifically, we have:

$\displaystyle e_{n}(S)$
$\displaystyle=S^{\mu_{1}}_{\hphantom{\mu_{1}}[\mu_{1}}\dots
S^{\mu_{n}}_{\hphantom{\mu_{n}}\mu_{n}]}\ ,$ (5.57) $\displaystyle
S^{\mu}_{\hphantom{\mu}\nu}S^{\nu}_{\hphantom{\mu}\rho}$
$\displaystyle=g^{\mu\sigma}f_{\sigma\rho}\quad\textrm{namely:}\quad
S^{\mu}_{\hphantom{\mu}\nu}=(\sqrt{g^{-1}f})^{\mu}_{\hphantom{\mu}\nu}\ .$

In the action (5.55) we have two independent Ricci tensors associated with the
two metrics $g_{\mu\nu}$ and $f_{\mu\nu}$, respectively. Therefore, we have
two independent diffeomorphism transformations which keep each of the two
Ricci tensors invariant. There is a special limit of this theory known as
partially massless (PM) bimetric theory, which is defined over the following
background solution in order to eliminate the $f_{\mu\nu}$ [47]

$\displaystyle
f_{\mu\nu}=a^{2}g_{\mu\nu}+\frac{2\gamma}{m^{2}}P_{\mu\nu}+\mathcal{O}(m^{-4})\
,$ (5.58)

where $P_{\mu\nu}$ is the Schouten tensor. Plugging these relations back into
(5.56), in [47] it is shown that the kinetic term of the second metric
$f_{\mu\nu}$ and the bimetric potential assume the following form (in
arbitrary dimension):

$\displaystyle m^{2}\sqrt{g}V(S;\beta_{n})$
$\displaystyle=m^{2}\sqrt{g}\sum\limits_{n=0}^{4}\beta_{n}e_{n}(S)\sim\Lambda_{0}\;\sqrt{g}[1+\kappa
R(g)]$ (5.59)
$\displaystyle+\sqrt{g}\frac{\Lambda_{1}}{m^{2}}\;\left(\frac{d}{4(d-1)}R^{2}-R^{\mu\nu}R_{\mu\nu}\right)+\mathcal{O}(R^{3})\
,$ $\displaystyle\sqrt{f}R(f)$
$\displaystyle\sim(\Lambda_{2})^{d-2}\sqrt{g}R(g)-\frac{\Lambda_{3}}{m^{2}}\sqrt{g}\;\left(\frac{d}{4(d-1)}R^{2}-R^{\mu\nu}R_{\mu\nu}\right)+\mathcal{O}(R^{3})\
.$

The parameters $\Lambda_{i}(\beta_{n})$ are dimensionless. For simplicity we
absorbed other dimensionless parameters, e.g. $\gamma$ in (5.58), in the
parameters $\Lambda_{i}$, for details of the calculation we refer to [47]. We
fix the parameters $\beta_{n}$ in such a way that the term including the
cosmological constant vanishes i.e. $\Lambda_{0}=0$. For the full fledged PM
bimetric one has to fix all remaining $\beta_{n}$ so that also
$\Lambda_{2}(\beta_{n})=0$ [47]. The last two terms of (5.59) correspond to
the CG Lagrangian (5.34). More concretely, one has the following identity, cf.
(5.32)

$(W_{\mu\nu\alpha\beta})^{2}=R_{GB}+\Big{(}\frac{d}{4(d-1)}R^{2}-R^{\mu\nu}R_{\mu\nu}\Big{)}\
,$ (5.60)

with $R_{GB}$ the Gauss-Bonnet term, which is a total derivative in four
dimension and can be discarded in the Lagrangian. After putting for $d=4$ the
relations (5.59) back into (5.55) we evidence that there are two copies of CG
together with the kinetic term for the first metric $g_{\mu\nu}$ described by
GR. Concretely, we obtain:

$\displaystyle\mathcal{L}$
$\displaystyle=m_{g}^{2}\;\sqrt{g}\;\Bigg{[}R(g)+\alpha^{2}\Lambda_{2}^{2}R(g)-\alpha^{2}\frac{\Lambda_{3}}{m^{2}}\;\left(\frac{1}{3}R^{2}-R^{\mu\nu}R_{\mu\nu}\right)$
(5.61)
$\displaystyle-2\,\frac{\Lambda_{1}}{m^{2}}\;\left(\frac{1}{3}R^{2}-R^{\mu\nu}R_{\mu\nu}\right)\
\Bigg{]}+\mathcal{O}(R^{3})\ .$

Combining the terms (5.61) into the action we reproduce (5.52) with:

$\displaystyle\lambda_{1}=(1+\alpha^{2}\Lambda_{2}^{2})\ ,$ (5.62)
$\displaystyle\lambda_{3}=-\frac{1}{m^{2}}\;(2\Lambda_{1}+\alpha^{2}\Lambda_{3})\
.$

Then, the resulting Lagrangian (5.52) describes a HD gravity with the higher
order corrections due to the bimetric theory. Calculating the equations of
motions for the action (5.52) in the leading order222Note, that $R^{3}$ is of
order $m^{4}$. of $m^{2}$ one ends up with the same constraints (5.53) as in
EW, cf. e.g. [48] and the comment below Eq. (5.40).

Recall, that we have eliminated $f_{\mu\nu}$ through (5.58) and obtained the
Lagrangian (5.52). Similar to EW the PM bimetric gravity propagates seven
degrees of freedom (packaged into the metric $g_{\mu\nu}$), which contain a
massless spin–two and a massive spin–two. In PM bimetric gravity the
scattering amplitude ${\cal A}(n)$ describes the interaction of $n$ spin–two
fields $g_{\mu\nu}$. The latter is described by the Lagrangian (5.52)
comprising the conformal term+GR+$R^{3}$ corrections. Eventually, the
amplitude ${\cal A}(n)$ may be computed by using (2.4) and is given by the
intersection number (5.51). Looking at Table 7a we may anticipate for PM
bimetric gravity the double copy structure (5.50). In addition to GR in (5.50)
there are the two CG theories originating in (5.61). The latter can be
combined in one single subject to (LABEL:lambdas), while the $R^{3}$ term is
of higher order $m^{4}$.

A final comment is, that choosing the potential (5.56) as

$V(\beta;S)\approx\ m^{2}\;[(f^{\mu}_{\mu})^{2}-f^{\mu\nu}f_{\mu\nu}]$ (5.63)

leads to Einstein–Weyl gravity. This has been pointed out in [37] and also
commented below Eq. (5.40). Finally, a detailed discussion on string
scattering amplitudes for full bimetric gravity with comments on their double
copy structure can be found in [49, 50].

##  6 Amplitude relations in intersection theory

In this section we describe amplitude relations in the language of twisted
forms. Furthermore, we shall look at the EYM amplitude and expand the latter
in terms of an orthonormal basis of twisted forms describing YM amplitudes.

###  6.1 Amplitude relations from intersection theory

It has been emphasized in [1] that the fundamental KK and BCJ relations in the
language of twisted intersection theory may be understood as the total
derivative (exact twisted form) associated to the color form
$\varphi^{color}_{\pm,n-1}$ i.e.:

$(d\pm d\omega\wedge)\,\varphi^{color}_{\pm;n-1}\sim 0\in H^{n-3}_{\omega}\ .$
(6.1)

Recall, that we are using both $\varphi^{color}$ and $PT(1,2,\ldots,n)$ for
the color form. To derive amplitude relations in intersection theory we first
note that $d\omega$ corresponds to the scattering equations (2.5), i.e.:

$d\omega=\sum\limits_{i=2}^{n-2}S_{i}\;dz_{i}\ ,\qquad\textit{with:}\qquad
S_{i}=\sum\limits^{n-2}_{j\neq i}\frac{s_{ij}}{z_{ij}}\ .$ (6.2)

Secondly, to simplify the calculation we introduce the insertion function
$\mathrm{Ins}(i)_{jk}$s:

$\mathrm{Ins}(i)_{j,k}:=\frac{z_{jk}}{z_{ji}z_{ik}}\ ,$ (6.3)

which acts on the color form $PT(1,\ldots,n)$ as operator:

$\mathrm{Ins}(i)_{j,k}\;PT(1,2,3,\ldots,j,k,...,n)=PT(1,2,3,\ldots,j,i,k,...,n)\
.$ (6.4)

Therefore, by using momentum conservation and performing some rearrangements
we can write $S_{i}$ in terms of $\mathrm{Ins}(i)_{j,k}$ as

$S_{i}=\sum\limits^{n-2}_{j\neq i}x(i)_{j}\;\mathrm{Ins}(i)_{j,j+1}\ ,$ (6.5)

where $x(i)_{j}$ is defined as:

$x(i)_{j}=p_{i}\cdot\sum_{k=2}^{j}p_{k}\ .$ (6.6)

Now, by using $\varphi^{color}_{\pm;n-1}=PT(1,2,\ldots,n-1)$ and
$d\varphi^{color}_{\pm;n-1}=0$ we rewrite (6.1) in the following way:

$\displaystyle\Phi$ $\displaystyle:=(d\pm
d\omega\wedge)\;\varphi^{color}_{\pm;n-1}=\pm(d\omega\wedge)\;\varphi^{color}_{\pm;n-1}$
(6.7)
$\displaystyle=(d\omega\big{|}_{dz_{n}}\wedge)\varphi^{color}_{\pm;n-1}=(S_{n}dz_{n}\wedge)\varphi^{color}_{\pm;n-1}\simeq
0\,\in H^{n-3}_{\omega}\ .$

Above $d\omega|_{dz_{n}}$ projects onto the one–form part $dz_{n}$ of
$d\omega$. Eventually, from (6.7) we deduce the following relation

$\sum\limits^{n-2}_{j\neq
n}x(n)_{j}\;\mathrm{Ins}(n)_{j,j+1}PT(1,2,3,...,n-1)=\sum\limits^{n-2}_{j\neq
n}x(n)_{j}\;PT(1,2,3,..,j,n,j+1...,n-1)=0,$ (6.8)

which assumes the form of a BCJ relation. To produce the BCJ relations for YM
amplitudes we shall calculate the fowling intersection number

$\displaystyle\langle\varphi^{gauge}_{-,n},\Phi\rangle_{\omega}=\langle\varphi^{gauge}_{-,n},(S_{n}dz_{n}\wedge)\varphi^{color}_{\pm;n-1}\rangle_{\omega}$
(6.9) $\displaystyle=\langle\varphi^{gauge}_{-,n},\sum\limits^{n-2}_{j\neq
n}x(n)_{j}PT(1,2,3,..,j,n,j+1...,n-1)\rangle_{\omega}$
$\displaystyle=\sum\limits^{n-2}_{j\neq
n}x(n)_{j}\;\underbrace{\langle\varphi^{gauge}_{-,n},PT(1,2,3,..,j,n,j+1...,n-1)\rangle_{\omega}}_{\mathcal{A}_{YM}(1,2,3,..,j,n,j+1...,n-1)}=0\
,$

from which the BCJ relation for YM amplitudes follows:

$\sum\limits^{n-2}_{j\neq
n}x(n)_{j}\;\mathcal{A}_{YM}(1,2,3,..,j,n,j+1...,n-1)=0\ .$ (6.10)

On the other hand, for the KK relations we start at the $n+1$–form

$\varphi^{color}_{\pm,n+1}(\sigma_{l}):=PT(1,2,...,l,p,l+1,....,n)\ ,$

with $n+1$ legs. One additional leg denoted by $p\equiv n+1$ is appended to
$n$ legs such that there are in total $n+1$ legs. Furthermore $\sigma_{l}$
denotes the particular ordering of those $n+1$ legs as:
$\sigma_{l}\leftrightarrow(1,2,...,l,p,l+1,....,n)$. As a consequence of KK
relations for PT factors we have the identity:

$\displaystyle\sum\limits_{l=1}^{n}\varphi^{color}_{\pm,n+1}(\sigma_{l})=\sum\limits_{l=1}^{n}PT(1,2,\ldots,l,p,l+1,....,n)=0\
.$ (6.11)

Inserting (6.11) into the intersection number (2.4) appended by
$\varphi^{gauge}_{-,n+1}$ yields

$\langle\varphi^{gauge}_{-,n+1},\sum\limits_{l=1}^{n}\varphi^{color}_{+,n+1}(\sigma_{l})\rangle_{\omega}=\sum\limits_{l=1}^{n}\langle\varphi^{gauge}_{-,n+1},\varphi^{color}_{+,n+1}(\sigma_{l})\rangle_{\omega}=0\
,$ (6.12)

from which the KK relation for YM amplitudes of generic helicity
configurations follows:

$\sum\limits_{l=1}^{n}\mathcal{A}_{YM}(1,2,\ldots,l,p,l+1,....,n)=0\ .$ (6.13)

###  6.2 Expansion Coefficient for EYM amplitude

Any intersection number (2.4) may be expanded w.r.t. a orthonormal basis of
$n$–forms $\bigcup\limits_{a=1}^{(n-3)!}\\{\Phi_{+,a}\\}\in
H_{+\omega}^{n-3}$. Generally, together with the dual basis
$\bigcup\limits_{b=1}^{(n-3)!}\\{\Phi^{\vee}_{-,b}\\}\in H_{-\omega}^{n-3}$
with the intersection matrix

$\langle\Phi_{+,a},\Phi^{\vee}_{-,b}\rangle_{\omega}=\delta_{ab}\ ,$ (6.14)

we have the following expansion of twisted $n$–forms
$\varphi^{1}_{+},\varphi^{2}_{-}$

$\displaystyle\varphi^{1}_{+}$
$\displaystyle=\sum_{a=1}^{(n-3)!}\langle\Phi^{\vee}_{-,a},\varphi_{+}\rangle_{\omega}\
\Phi_{+,a}\ ,$ (6.15) $\displaystyle\varphi^{2}_{-}$
$\displaystyle=\sum_{b=1}^{(n-3)!}\langle\varphi_{-},\Phi_{+,b}\rangle_{\omega}\
\Phi^{\vee}_{-,b}\ ,$ (6.16)

respectively, cf. (5.4) for the case of $\Phi_{+,a}=PT(a)$ leading to (5.5):

$\langle\varphi_{+}^{1},\varphi^{2}_{-}\rangle=\sum_{a=1}^{(m-3)!}\langle
PT^{\vee}(a),\varphi_{+}^{1}\rangle_{\omega}\
\langle\varphi_{-}^{2},PT(a)\rangle_{\omega}\ .$ (6.17)

With
$\varphi^{1}_{+}=\tilde{\varphi}^{EYM}_{+,n;r},\varphi^{2}_{-}=\varphi^{EYM}_{-,n;r}=\varphi^{gauge}_{-,n+r}$
the orthogonal decomposition (6.17) can be used to express EYM amplitudes in
terms of a linear combination of a basis of $m:=n+r$–point YM subamplitudes
[6]:

$\mathcal{A}_{EYM}(n;r)=\lim_{\alpha^{\prime}\rightarrow\infty}\
\sum_{a=1}^{(m-3)!}\langle
PT^{\vee}(a),\widetilde{\varphi}^{EYM}_{+,n;r}\rangle_{\omega}\ \ {\cal
A}_{YM}(a)\ .$ (6.18)

In addition, we may use BCJ–KK relations for further simplifications.

To determine in (6.18) the coefficients $\langle
PT^{\vee}(a),\widetilde{\varphi}^{EYM}_{+,n;r}\rangle_{\omega}$ we exemplify
the one graviton case $r=1$ and we label the momentum of this one graviton by
$p\equiv n+1$. From [6] we have for the EYM twisted form
$\tilde{\varphi}^{EYM}_{n;1}$

$\displaystyle\lim\limits_{\alpha^{\prime}\rightarrow\infty}\tilde{\varphi}^{EYM}_{n;1}$
$\displaystyle=d\mu_{n+1}\;PT(1,\ldots,n)\;\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
x_{l})\frac{z_{l,l+1}}{z_{l,p}z_{p,l+1}}$ (6.19)
$\displaystyle=d\mu_{n+1}\;\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
x_{l})\;PT(1,2,...,l,p,l+1,...,n)\ ,$

with the definition $x^{\mu}_{l}=\sum\limits_{j=1}^{l}k^{\mu}_{j}$. Inserting
(6.19) into (6.18) yields:

$\displaystyle\mathcal{A}_{EYM}(n;1)$
$\displaystyle=\lim\limits_{\alpha^{\prime}\rightarrow\infty}\sum\limits_{\alpha\in
S_{n-3}}\langle
PT^{\vee}(\alpha),\tilde{\varphi}^{EYM}_{n;1}\rangle\;\mathcal{A}_{YM}(\alpha)$
(6.20) $\displaystyle=\sum\limits_{\alpha\in
S_{n-3}}\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot x_{l})\;\langle
PT^{\vee}(\alpha),PT(1,2,...,l,p,l+1,...,n)\rangle_{\omega}\;\mathcal{A}_{YM}(\alpha)\
.$

Notice, that above the $\alpha^{\prime}$–dependence has dropped. Now we can
use the orthonormality condition (6.14), i.e.

$\langle PT^{\vee}(\alpha),PT(\beta)\rangle_{\omega}=\delta_{\alpha,\beta}\ ,$
(6.21)

label the ordering of $PT(1,2,...,l,p,l+1,...,n)$ by $\sigma_{l}$ and (6.20)
becomes

$\displaystyle\mathcal{A}_{EYM}(n;1)$ $\displaystyle=\sum\limits_{\alpha\in
S_{n-3}}\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot x_{l})\;\langle
PT^{\vee}(\alpha),PT(\sigma_{l})\rangle_{\omega}\;\mathcal{A}_{YM}(\alpha)\ .$
(6.22) $\displaystyle=\sum\limits_{\alpha\in
S_{n-3}}\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
x_{l})\;\delta_{\alpha,\sigma_{l}}\;\mathcal{A}_{YM}(\alpha)$
$\displaystyle=\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
x_{l})\;\mathcal{A}_{YM}(1,2,...,l,p,l+1,...,n)\ ,$

which is the relation from [51].

Next, let us also evaluate the coefficients $\langle
PT^{\vee}(a),\widetilde{\varphi}^{EYM}_{+,n;r}\rangle_{\omega}$ in (6.18) for
the two graviton case $r=2$. Again, the starting point is the twisted form
$\tilde{\varphi}^{EYM}_{n;2}$ for EYM given in [6]

$\displaystyle\lim_{\alpha^{\prime}\rightarrow\infty}\widetilde{\varphi}^{EYM}_{\pm,n;2}$
$\displaystyle=d\mu_{n+2}\,\,PT(1,2,3,...,n)\int\prod\limits_{i\in{\cal
S}_{2}}d\theta_{i}d\bar{\theta_{i}}\ \exp\Bigg{\\{}\frac{1}{2}\alpha^{\prime
2}\sum\limits_{i,j\in{\cal S}_{2}}\begin{pmatrix}\theta_{i}\\\
\bar{\theta}_{i}\end{pmatrix}^{t}\Psi_{{\cal
S}_{2}}\,\,\begin{pmatrix}\theta_{j}\\\
\bar{\theta}_{j}\end{pmatrix}\Bigg{\\}}\Bigg{|}_{\overline{\zeta_{l}}\rightarrow\zeta_{l}}$
$\displaystyle=d\mu_{n+2}\,\,PT(1,2,3,...,n)\;\mathrm{Pf}^{\prime}\Psi_{2}\ ,$
(6.23)

with ${\cal S}_{2}=\\{n+2,n+4\\}$. To compute the coefficient $\langle
PT_{a}^{\vee},\tilde{\varphi}^{EYM}_{n;2}\rangle$ of the decomposition (6.18)
we expand the twisted form (6.23) in terms of a PT basis (for a derivation we
refer to A) we have:

$\displaystyle\lim\limits_{\alpha^{\prime}\rightarrow\infty}\tilde{\varphi}^{EYM}_{n;2}=d\mu_{n+2}$
$\displaystyle\,\Bigg{\\{}\sum\limits_{k\geq
l=1\atop}^{n-1}(\varepsilon_{q}\cdot x_{l})(\varepsilon_{p}\cdot
x_{k})PT(1,2,...,l,p,l+1,...k,q,k+1,...,n)+(p\leftrightarrow q)$ (6.24)
$\displaystyle\quad-\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
q)(\varepsilon_{q}\cdot
x_{l})\sum\limits_{j=2}^{l-1}PT(1,j-1,q,j,...l-1,p,l,...,n)+(p\leftrightarrow
q)$
$\displaystyle\qquad-\frac{1}{2}\sum_{l=1}^{n-1}(\varepsilon_{p}\cdot\varepsilon_{q})s_{p,l}\sum\limits_{j=2}^{l}PT(1,j-1,q,p,j,...,l,...,n)+(p\leftrightarrow
q)\Bigg{\\}}\ .$

Now we proceed similar to the one graviton case and insert (6.24) into (6.18)
and apply the orthonomality condition (6.21) to get:

$\displaystyle{\cal A}_{EYM}(n;2)=$
$\displaystyle\lim\limits_{\alpha^{\prime}\rightarrow\infty}\sum\limits_{a=1}^{(n-3)!}\langle
PT^{\vee}(a),\tilde{\varphi}^{EYM}_{n;2}\rangle\;{\cal A}_{YM}(a)$ (6.25)
$\displaystyle=$ $\displaystyle\sum\limits_{k\geq
l=1\atop}^{n-1}(\varepsilon_{q}\cdot x_{l})(\varepsilon_{p}\cdot x_{k}){\cal
A}_{YM}(1,2,...,l,p,l+1,...k,q,k+1,...,n)+(p\leftrightarrow q)$
$\displaystyle-\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
q)(\varepsilon_{q}\cdot x_{l})\sum\limits_{j=2}^{l-1}{\cal
A}_{YM}(1,j-1,q,j,...l-1,p,l,...,n)+(p\leftrightarrow q)$
$\displaystyle\qquad-\frac{1}{2}\sum_{l=1}^{n-1}(\varepsilon_{p}\cdot\varepsilon_{q})s_{p,l}\sum\limits_{j=2}^{l}{\cal
A}_{YM}(1,j-1,q,p,j,...,l,...,n)+(p\leftrightarrow q)\ .$

This result provides the expansion of the EYM amplitude involving $n$ gluons
and two gravitons in terms of pure YM $n+2$–point subamplitudes. The
expression (6.25) has been already given in [52], while here we had applied
twisted intersection theory to derive it.

##  7 Concluding remarks

While in [6] we have focused on EYM amplitudes and their description in terms
of twisted intersection theory, in this work we have extended and generalized
these results into several directions. By pairing various differential forms
we have computed intersection numbers (2.4) describing the amplitudes of
various field–theories, cf. Tables 1a–4. First, we have reviewed known
constructions of field–theory amplitudes through their pairs of twisted forms
entering the intersection number (2.4), cf. Table 1a. In particular, theories
which are described by the twisted form of EYM (3.21) are tabulated in Tables
1b and 2. Secondly, for known field–theories (e.g. EM, DBI and YMS) yet
without any twisted from description their pairs of twisted forms has been
given. In fact, one of the twisted forms of these theories is the twisted form
(3.32) for EM, cf. Table 4. On the other hand, a new twisted form (3.20) has
been extracted from the twisted form $\varphi^{bosonic}$ related to the
bosonic string. This form is relevant for describing amplitudes (4.14) of
$(DF)^{2}$ theory. Furthermore, we have applied the embedding from the disk
onto the sphere to bosonic open string fields to construct a novel twisted
form (4.7). The latter allows to construct amplitudes by intersection numbers
for a variety of theories including Weyl–YM or $(DF)^{2}+\phi^{3}$ theories,
cf. Table 5. In Section 5 we have elaborated on the generic underlying
structure of double copy construction in intersection theory. In particular,
we have been concerned with theories whose pairs of twisted differentials
entering (2.4) contain a color form $\varphi^{color}$. For those theories CK
duality is easily exhibited and hence they serve as natural candidates for
double copies, cf. also Table 6. Building up on this construction in Tables 7a
and 7b we have presented old and new double copies. For the latter an
amplitude description has been unknown previously. In particular, in Table 8
we present a collection of previously unknown spin–two double copy
constructions. We find a double copy construction for HD gravity, (partial
massless) bimetric gravity and some more exotic theories. Our findings are not
complete and it would be very interesting to build up on those and extending
the list of possible new double copies. In Section 6 we present a derivation
of Kleiss–Kuijf (KK) and Bern–Carrasco–Johansson amplitude relations in
intersection theory. Finally, the appendices contain some supplementary
material.

While throughout this work we have only discussed tree–level amplitudes and
their underlying twisted intersection description a natural further step is to
extend our results to higher–loops. A subsequent question is finding
appropriate twisted forms on the elliptic curve which generalize (3.1) and
give rise to one–loop integrands of gauge and gravity theories. The one–loop
monodromy results from [10, 12, 13] are of importance for constructing such
differential forms. The KLT relations [2] are the foundation of any double
copy relation at tree–level both in the CHY formalism and twisted intersection
theory. Consequently, one–loop KLT relations on the Riemann torus [53] should
trigger one–loop double copies in twisted intersection theory.

## Acknowledgment

One of the authors (PM) would like to thank Chrysoula Markou for useful
discussions and suggestions and Hrólfur Ásmundsson for very useful
discussions.

##  Appendix A EYM basis expansion

Here we expand the twisted from $\hat{\tilde{\varphi}}^{EYM}_{n;2}$

$\displaystyle\lim_{\alpha^{\prime}\rightarrow\infty}\widetilde{\varphi}^{EYM}_{\pm,n;2}=d\mu_{n+2}\,\,PT(1,2,3,...,n)\;\mathrm{Pf}^{\prime}\Psi_{2}$
(A.1)

w.r.t. a Park–Taylor basis. We start with the matrix $\Psi_{{\cal S}_{2}}$:

$\displaystyle\Psi_{{\cal S}_{2}}=\begin{pmatrix}0&\frac{p\cdot
q}{\overline{z}_{3,4}}&-\Psi^{4}_{1}&\frac{\widetilde{\varepsilon}_{q}\cdot
p}{\overline{z}_{3,4}}\\\ -\frac{p\cdot
q}{\overline{z}_{3,4}}&0&\frac{-\widetilde{\varepsilon}_{p}\cdot
q}{\overline{z}_{3,4}}&-\Psi^{6}_{1}\\\
\Psi^{4}_{1}&\frac{\widetilde{\varepsilon}_{p}\cdot
q}{\overline{z}_{3,4}}&0&\frac{\widetilde{\varepsilon}_{p}\cdot\widetilde{\varepsilon}_{q}}{\overline{z}_{3,4}}\\\
-\frac{\widetilde{\varepsilon}_{q}\cdot
p}{\overline{z}_{3,4}}&\Psi^{6}_{1}&-\frac{\widetilde{\varepsilon}_{p}\cdot\widetilde{\varepsilon}_{q}}{\overline{z}_{3,4}}&0\end{pmatrix}\
.$ (A.2)

The Pfaffian of this matrix is given by

$\displaystyle\mathrm{Pf}\Psi_{{\cal S}_{2}}$
$\displaystyle=\Psi^{p}_{1}\Psi^{q}_{1}-\frac{s_{pq}(\varepsilon_{p}\cdot\varepsilon_{q})}{z_{pq}^{2}}+\frac{(\varepsilon_{p}\cdot
q)(\varepsilon_{q}\cdot p)}{z_{pq}^{2}}\ ,$ (A.3)

with the following definitions:

$\displaystyle\Psi^{p}_{1}=\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
x_{l})\
\frac{\sigma_{l,l+1}}{\sigma_{l,p}\sigma_{l+1,p}}+(\varepsilon_{p}\cdot q)\
\frac{\sigma_{q,n}}{\sigma_{p,q}\sigma_{n,p}}=\tilde{\Psi}^{p}_{1}+(\varepsilon_{p}\cdot
q)\ \frac{\sigma_{q,n}}{\sigma_{p,q}\sigma_{n,p}}\ ,$ (A.4)
$\displaystyle\Psi^{q}_{1}=\sum\limits_{l=1}^{n-1}(\varepsilon_{q}\cdot
x_{l})\
\frac{\sigma_{l,l+1}}{\sigma_{l,q}\sigma_{l+1,q}}+(\varepsilon_{q}\cdot p)\
\frac{\sigma_{p,n}}{\sigma_{q,p}\sigma_{n,q}}=\tilde{\Psi}^{q}_{1}+(\varepsilon_{q}\cdot
p)\ \frac{\sigma_{p,n}}{\sigma_{q,p}\sigma_{n,q}}\ ,$ $\displaystyle
x_{l}=\sum\limits_{m=1}^{l}k^{\mu}_{m}\ ,$

where we denote the first summands in the above terms by
$\tilde{\Psi}^{p}_{1}$ and $\tilde{\Psi}^{q}_{1}$, respectively. Using these
and plugging them back into (A.3) we can cancel the the last term in (A.3),
which is a double pole, and we obtain:

$\displaystyle\mathrm{Pf}\Psi_{{\cal S}_{2}}$
$\displaystyle=\tilde{\Psi}^{p}_{1}\tilde{\Psi}^{q}_{1}+\tilde{\Psi}^{p}_{1}(\varepsilon_{q}\cdot
p)\frac{z_{pn}}{z_{pq}z_{qn}}+\tilde{\Psi}^{q}_{1}(\varepsilon_{p}\cdot
q)\frac{z_{qn}}{z_{pq}z_{pn}}-\frac{s_{pq}\varepsilon_{p}\cdot\varepsilon_{q}}{z_{pq}^{2}}\
.$ (A.5)

Now we can write the twisted from $\hat{\tilde{\varphi}}^{EYM}_{n;2}$ in the
following way:

$\displaystyle\lim\limits_{\alpha^{\prime}\rightarrow\infty}\hat{\tilde{\varphi}}^{EYM}_{n;2}=PT(1,2,...,n)\mathrm{Pf}\Psi_{{\cal
S}_{2}}$ (A.6)
$\displaystyle=PT(1,2,...,n)\Bigg{[}\tilde{\Psi}^{p}_{1}\tilde{\Psi}^{q}_{1}+\tilde{\Psi}^{p}_{1}(\varepsilon_{q}\cdot
p)\frac{z_{pn}}{z_{pq}z_{qn}}+\tilde{\Psi}^{q}_{1}(\varepsilon_{p}\cdot
q)\frac{z_{qn}}{z_{pq}z_{pn}}-\frac{s_{pq}\varepsilon_{p}\cdot\varepsilon_{q}}{z_{pq}^{2}}\Bigg{]}\
.$

At this point we are able to expand this expression in terms of a Park–Taylor
basis and use the orthogonality condition of that basis to obtain amplitude
relations. In order to expand we use the following relations (cf. for the
proof [52]):

$\displaystyle\tilde{\Psi}^{p}_{1}\tilde{\Psi}^{q}_{1}$
$\displaystyle=\sum\limits_{l=1}^{n-1}\sum\limits_{k=1}^{n-1}(\varepsilon_{p}\cdot
x_{l})\,(\varepsilon_{q}\cdot
x_{k})\frac{\sigma_{l,l+1}}{\sigma_{l,p}\sigma_{l+1,p}}\ \
\frac{\sigma_{k,k+1}}{\sigma_{k,q}\sigma_{k+1,q}}\ ,$ (A.7)
$\displaystyle\tilde{\Psi}_{1}^{p}(\varepsilon_{q}\cdot
p)\frac{z_{pn}}{z_{pq}z_{qn}}PT(1,2,3,...,n)$
$\displaystyle=-(\varepsilon_{q}\cdot
p)\sum\limits_{l=1}^{n-1}\varepsilon_{p}\cdot
x_{l}\sum\limits_{\sigma=q\cup\\{1,2,...,l\\}}PT(\sigma,p,l+1,...,n)\ ,$
$\displaystyle\frac{s_{pq}}{z_{p,q}^{2}}$
$\displaystyle=\sum\limits_{i=l}^{n-1}s_{pl}\frac{1}{z_{lp}z_{pq}}\ .$

The last two relations are proven through implantation of multiple BCJ-KK
relations and scattering equations. Using the first relation in (A.7) we
derive:

$\displaystyle\lim\limits_{\alpha^{\prime}\rightarrow\infty}\hat{\tilde{\varphi}}^{EYM}_{n;2}=$
$\displaystyle\sum\limits_{k\geq l=1\atop}^{n-1}(\varepsilon_{q}\cdot
x_{l})(\varepsilon_{p}\cdot
x_{k})PT(1,2,...,l,p,l+1,...k,q,k+1,...,n)+(p\leftrightarrow q)$ (A.8)
$\displaystyle+\tilde{\Psi}_{1}^{p}(\varepsilon_{q}\cdot
p)\frac{z_{pn}}{z_{pq}z_{qn}}PT(1,2,3,...,n)+(p\leftrightarrow q)$
$\displaystyle-\frac{(\varepsilon_{p}\cdot\varepsilon_{q})s_{p,q}}{z_{p,q}^{2}}\
PT(1,2,3,...,n)\ .$

we can simplify this equations further using the second and third relations in
(A.7) after multiply terms with the Park-Taylor we obtain:

$\displaystyle\lim\limits_{\alpha^{\prime}\rightarrow\infty}\tilde{\varphi}^{EYM}_{n;2}=d\mu_{n+2}$
$\displaystyle\,\Bigg{\\{}\sum\limits_{k\geq
l=1\atop}^{n-1}(\varepsilon_{q}\cdot x_{l})(\varepsilon_{p}\cdot
x_{k})PT(1,2,...,l,p,l+1,...k,q,k+1,...,n)+(p\leftrightarrow q)$ (A.9)
$\displaystyle-\sum\limits_{l=1}^{n-1}(\varepsilon_{p}\cdot
q)(\varepsilon_{q}\cdot
x_{l})\sum\limits_{j=2}^{l-1}PT(1,j-1,q,j,...l-1,p,l,...,n)+(p\leftrightarrow
q)$
$\displaystyle-\frac{1}{2}\sum_{l=1}^{n-1}(\varepsilon_{p}\cdot\varepsilon_{q})s_{p,l}\sum\limits_{j=2}^{l}PT(1,j-1,q,p,j,...,l,...,n)+(p\leftrightarrow
q)\Bigg{\\}}$

This is the expansion which we use in (6.24).

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