# Toward double copy on arbitrary backgrounds

Anton Ilderton  and William Lindved

###### Abstract

Double copy relates scattering amplitudes in a web of gravitational and gauge
theories. Although it has seen great success when applied to amplitudes in
vacuum, far less is known about double copy in arbitrary gravitational and
gauge backgrounds. Focussing on the simplest pair production amplitudes of
scalar QCD in a background gauge field, we construct, at next-to-leading order
in perturbation theory, a double copy map to particle production in general
metrics (and associated axio-dilatons) constructed from the gauge background.
We connect our results to convolutional and classical double copy and, turning
to examples, identify a class of gauge fields which generate FRW spacetimes
via double copy. For this case we are able to conjecture the all-orders form
of the double copy map.

##  1 Introduction

Double copy relates scattering in gauge theory to scattering in gravity, by
replacing the colour structure of amplitudes with a copy of the kinematic
structure [1, 2, 3, 4, 5, 6]. As well as offering a practical method to
streamline gravitational scattering calculations by mapping them to simpler
gauge theory calculations, it suggests a deep connection between gauge and
gravitational theories. For reviews see [7, 8, 9, 10].

Double copy has been extremely successful when applied to amplitudes in
vacuum. If it is a fundamental relationship between gauge theory and gravity
then double copy should presumably hold in some form on any background, and
not just for perturbative scattering on trivial backgrounds, i.e. vacuum.
There are many physical situations, in both gravity and gauge theory, which
motivate the study of background fields, the presence of which often leads to
explicitly non-linear and non-perturbative phenomena. Examples include
gravitational memory [11], QCD in strong magnetic fields [12, 13], magnetar
environments [14, 15, 16], lab-astrophysics [17, 18] and intense laser physics
[19].

In asking how double copy looks on arbitrary backgrounds there are, very
broadly, two directions to consider. First, one can ask whether gauge theory
amplitudes in a given spacetime map to gravity amplitudes in the same
spacetime; this is (implicity) what is done, and works, for perturbation
theory in vacuum, where the background is Minkowski. Here quite some progress
has been made in the case of AdS space and related CFT correlators [20, 21,
22, 23, 24, 25, 26, 27].

Second, one can ask whether scattering on a given gauge background maps to
scattering in some double-copy-related spacetime. Again progress can be made
by turning to specific examples; a double copy map has been found which
relates scattering on gauge and gravitational plane waves [28, 29, 30], which
holds _non_ -perturbatively in the coupling to the background. The study of
conserved quantities in highly symmetric backgrounds has also revealed double
copy maps for geodesics and related observables between e.g.
$\sqrt{\text{Schw}}$ and Schwarzschild, $\sqrt{\text{Kerr}}$ and Kerr [31],
and between electromagnetic vortices and gravitational waves [32]. For
_classical_ double copy in curved spacetime see e.g. [33, 34, 35, 36, 37].

Here we adopt a simple but direct approach to this topic; focussing on
fundamental matter coupled to Yang-Mills, i.e. scalar QCD, and its double copy
[38, 39, 40], we consider _arbitrary_ gauge and gravitational backgrounds, and
look at the simplest scattering processes on them, namely two-point amplitudes
at tree level. Unlike in vacuum, such amplitudes are highly non-trivial on
backgrounds, as they encode geodesic motion of test particles, associated
memory effects, and also particle production. Further, these amplitudes can be
arbitrarily complex even at tree level because of the in-principle arbitrary
_functional_ degrees of freedom in the background (i.e. tree amplitudes will
not typically be rational functions of kinematic data).

While one aim of this investigation is to, eventually, identify double-copy
structures in the _non_ -perturbative parts of scattering amplitudes on
backgrounds, we will begin in this paper by treating all interactions with the
background perturbatively. Calculating order by order in perturbation theory
we will find non-trivial relations even at leading order in the coupling,
since three-point tree amplitudes are kinematically allowed in a background
(and not just for zero momentum bosons).

This paper is organised as follows. In Sec. 2 we introduce the theories of
interest, namely (fundamental) scalars coupled to Yang-Mills, and to axio-
dilaton gravity. We calculate the leading order contribution to particle
production in arbitrary backgrounds, and identify a double copy map between
the gauge and gravitational theories, with the respective backgrounds related
by the ‘convolutional double copy’ [41, 42, 43, 44]. In Sec. 3 we extend this
calculation, and the double copy map, to next-to-leading order (NLO), i.e.
beyond the linearised theory. Here the nonlinearity of Einstein’s equations
makes itself known, imposing constraints on the relation between the gauge and
gravitational backgrounds if the double copy is to hold order by order in
perturbation theory.

In Sec. 4 we conjecture an all-orders form for the double copy. We then
analyse the dilaton contribution to our 2-point amplitudes, and show non-
perturbatively that it is equivalent to a non-conformal matter coupling, which
greatly simplifies computations. Turning to examples, we focus on pair
creation in FRW spacetimes, and in time-dependent (chromo) electric fields.
The simplifications offered by going to specific choices of background allow
us to extend our calculations to third perturbative order. This reveals some
potential subtleties which may arise beyond NLO. It also allows us to
conjecture an explicit all-orders form for the double copy map to FRW
spacetimes, and to revisit and reinterpret the constraints imposed for general
backgrounds at NLO. We conclude in Sec. 5.

### Notation and conventions

$\hbar=1$ throughout. Mostly minus metric. Factors of $2\pi$ in delta
functions and integration measures are absorbed into hats, so
$\hat{\delta}=(2\pi)\delta$ and $\hat{\mathrm{d}}=\mathrm{d}/(2\pi)$.

##  2 Leading order

###  2.1 Gauge theory

We consider a massive scalar field $\varphi(x)$, in the fundamental
representation, minimally coupled to a YM field $A_{\mu}(x)\equiv
A_{\mu}^{a}(x)\mathsf{T}^{a}$ with $\mathsf{T}^{a}$ the group generators. The
Lagrangian is simply

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}+(D_{\mu}\varphi)^{\dagger}D^{\mu}\varphi-m^{2}\varphi^{\dagger}\varphi\,,$
(2.1)

in which the covariant derivative is $D_{\mu}=\partial_{\mu}-igA_{\mu}$. We
introduce a background by separating $A_{\mu}\to
A^{\text{b.g.}}_{\mu}+A_{\mu}$, in which $A_{\mu}$ is now to be quantised,
along with $\varphi$, on the background $A_{\mu}^{\text{b.g.}}$. This
background may be sourced or not, and we make no assumptions on its spacetime
dependence, except for the tacit assumption that the potential vanishes
asymptotically.

We have in mind physical situations where the background is strong, i.e. the
coupling of the matter field to the background $\propto
gA^{\text{b.g.}}/m\gg~{}g$. This is well described by a truncation of the
theory where we neglect fluctuations of the gauge field and consider a scalar
in the background with Lagrangian

$\mathcal{L}\to(D_{\mu}\varphi)^{\dagger}D^{\mu}\varphi-m^{2}\varphi^{\dagger}\varphi\,,$
(2.2)

where now $D_{\mu}=\partial_{\mu}-igA^{\text{b.g.}}_{\mu}$. We drop ‘b.g.’
from the gauge field from here on.

As the only dynamical field in the theory is now $\varphi$, there are only two
interesting amplitudes; the $1\to 1$ amplitude for a massive particle
scattering on the background, and the $0\to 2$ pair production amplitude.
Formally these are related by a change in sign for one of the particle
momenta, but they obviously differ physically; the $0\to 2$ amplitude is
purely quantum, for example, and for the sake of concreteness we focus on this
throughout.

As discussed in the introduction, the eventual aim is to make non-perturbative
statements about double copy in backgrounds, but we will begin here by working
perturbatively in the coupling to the background111To be strict, this
corresponds to assuming the hierarchy $1\gg gA/m\gg g$.. As such, the Feynman
rules of our truncated theory (2.2) are given in Fig. 1; note that at every
vertex we conserve momentum and integrate ${\hat{\mathrm{d}}}^{4}k$ for each
background field momentum. The leading order contribution to the pair
production amplitude, call it $\mathcal{A}_{(1)}$, is simply the three-point
diagram with the massive legs taken on-shell. One immediately finds

${i\mathcal{A}_{(1)}=-(2g)}\int\\!\hat{\mathrm{d}}^{4}k\,\hat{\delta}^{4}(p+q-k)A^{a\mu}(k){\mathsf{T}}^{a}_{ij}{\big{(}p_{\mu}-\tfrac{1}{2}k_{\mu}\big{)}}\;,$
(2.3)

in which $p_{\mu}$ and $q_{\mu}$ are the momenta of the produced pair, while
${\mathsf{T}}^{a}_{ij}\equiv c_{i}{\mathsf{T}}^{a}c_{j}$ with $c$ the colour
vectors of the pair. The amplitude is easy to understand; it is a three-point
amplitude in vacuum, with an off-shell gluon, integrated against the
background field profile. This remaining integral is trivially evaluated,
setting the argument of the gauge field (the momentum influx from the field)
equal to the produced particle momentum, as expected. We leave the integral
unevaluated as this will later help expose the double copy structure, just as
in vacuum.

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\times$}; \diagram* { (o) -- [fermion, momentum = $p$]
(s1), (o) -- [anti fermion, momentum' = $q$] (s2), (b) -- [gluon] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=ig{\mathsf{T}}^{a}A^{a}(k)\cdot(p-q)\qquad\quad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[above left=of o] (b1){$\times$}; \vertex[below left=of o]
(b2){$\times$}; \diagram* { (o) -- [fermion, momentum = $p$] (s1), (o) --
[anti fermion, momentum' = $q$] (s2), (b1) -- [gluon] (o), (b2) -- [gluon]
(o), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=ig^{2}{\mathsf{T}}^{a}{\mathsf{T}}^{b}A^{a}(k_{1})\cdot
A^{b}(k_{2}),$
Figure 1:  The Feynman rules for a massive (fundamental) scalar particle
coupled to the background YM field $A_{\mu}$, indicated by a cross. Massive
particle momenta are outgoing.

No gauge has been assumed. It can be checked without difficulty that (2.3) is
invariant, on the support of the momentum-conserving delta function, under
linearised gauge transformations or, equivalently, invariant up to terms of
order $g^{2}$ under the full gauge transformation, here parameterised by
$\omega(x)$,

$A_{\mu}(x)\to A_{\mu}(x)+\partial_{\mu}\omega(x)-ig[A_{\mu}(x),\omega(x)]\;.$
(2.4)

###  2.2 Gravity

Turning to gravity, we begin with the conjectured all-order action for the
double copy of a (single flavour of) fundamental scalar coupled to Yang-Mills,
as given in [40]; this is

$S=\int\\!\mathrm{d}^{4}x\,\sqrt{-g}\bigg{(}{\frac{1}{\kappa^{2}}\bigg{[}-2R+\frac{g^{\mu\nu}\partial_{\mu}{\bar{Z}}\partial_{\nu}Z}{\big{(}1-\tfrac{1}{4}{\bar{Z}}Z\big{)}^{2}}\bigg{]}}+\frac{g^{\mu\nu}\partial_{\mu}\varphi^{\dagger}\partial_{\nu}\varphi-m^{2}e^{-\phi}\varphi^{\dagger}\varphi}{\big{(}1-\frac{\kappa^{2}}{32}\varphi^{\dagger}\varphi\big{)}^{2}}\bigg{)}\;,$
(2.5)

in which the matter field $\varphi$ is a complex scalar, while $Z$ encodes the
dilaton $\phi$ and axion $\chi$ as

$Z\equiv 2\frac{\chi+i(e^{-\phi}-1)}{\chi+i(e^{-\phi}+1)}\;.$ (2.6)

We have chosen alternative scalings of the latter fields, compared to the
literature, which will make clearer the separation into background and
dynamical pieces.

The action (2.5) was derived by perturbatively matching gauge theory and
gravity amplitudes, and then conjecturing the all-orders form shown based on
observed patterns. We note the striking similarity of the contact terms in the
matter sector to the structures in the axio-dilaton gravity sector [40]. We
will later see similar structures appearing through double copy in the purely
gravitational sector.

We perform a field redefinition $\bar{g}_{\mu\nu}=e^{-\phi}g_{\mu\nu}$ (hence
$\bar{g}^{\mu\nu}=e^{\phi}g^{\mu\nu}$ and $\bar{g}=ge^{-4\phi}$), such that
the dilaton couples to all matter terms in the same way, which allows for a
particularly clear double copy prescription. Inserting this into (2.5) and
transforming the Ricci scalar accordingly [45, App. G], the action (2.5)
becomes, immediately dropping the bar on the new metric for clarity and
writing out the dilaton and axion terms explicitly [46],

$S=\int\\!\mathrm{d}^{4}x\,\sqrt{-g}e^{\phi}\bigg{[}\frac{-2R-2g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+e^{\phi}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi}{\kappa^{2}}+\frac{g^{\mu\nu}\partial_{\mu}\varphi^{\dagger}\partial_{\nu}\varphi-m^{2}\varphi^{\dagger}\varphi}{\big{(}1-\tfrac{\kappa^{2}}{32}\varphi^{\dagger}\varphi\big{)}^{2}}\bigg{]}.$
(2.7)

To make the equivalent truncation as in gauge theory we separate the metric,
dilaton and axion into backgrounds and dynamical fluctuations, subsequently
dropping the latter as we assume a regime where the background dominates. We
then expand the background around flat space, writing

$g^{\mu\nu}=\eta^{\mu\nu}-\kappa
h^{\mu\nu}\;,\qquad\phi=\kappa\phi_{(1)}\;,\qquad\chi=\kappa\chi_{(1)}\;.$
(2.8)

Since $\kappa$ is negligible compared to the stronger coupling to the
background, $\kappa h$, we can also drop the contact terms. The surviving part
of the action is then (2.7) without the contact terms and where all axio-
dilaton gravity fields are interpreted as backgrounds. Only one term in the
action contains dynamical fields, namely

$\int\\!\mathrm{d}^{4}x\,\sqrt{-g}\,{e^{\phi}}\big{(}g^{\mu\nu}\partial_{\mu}\varphi^{\dagger}\partial_{\nu}\varphi-m^{2}\varphi^{\dagger}\varphi\big{)}\;,$
(2.9)

and we note that the axion does not couple directly to the matter field.

$\begin{split}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\times$};\diagram* { (o) -- [fermion, momentum = $p$]
(s1), (o) -- [anti fermion, momentum' = $q$] (s2), (b) -- [photon] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=i\kappa
h^{\mu\nu}(k)\left[p_{\mu}q_{\nu}-\frac{1}{2}\left(p\cdot
q+m^{2}\right)\eta_{\mu\nu}\right]\\\ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\times$};\diagram* { (o) -- [fermion, momentum = $p$]
(s1), (o) -- [anti fermion, momentum' = $q$] (s2), (b) -- [scalar] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=-i\kappa\phi(k)\left(p\cdot
q+m^{2}\right),\end{split}$ Figure 2:  The Feynman rules for a massive
(complex) scalar particle coupled to a background metric perturbation
$h^{\mu\nu}$, and dilaton $\phi$, in the linearised approximation.

We assume that the metric, like the gauge field, is asymptotically flat [47],
and moreover that $h\to 0$ asymptotically – relaxing this is not difficult, as
we comment on later, but it is a natural starting point when working
perturbatively. The Feynman rules are given in Fig. 2 and the leading order
amplitude is

$i\mathcal{M}_{(1)}=-\kappa\int\\!\hat{\mathrm{d}}^{4}k\,\hat{\delta}^{4}(p+q-k)\bigg{[}h^{\mu\nu}(k)\Big{(}p_{\mu}q_{\nu}-\frac{1}{2}(p\cdot
q+m^{2})\eta_{\mu\nu}\Big{)}-\phi_{(1)}(k)(p\cdot q+m^{2})\bigg{]}\;.$ (2.10)

The metric contribution can be checked against the literature; the pair
creation amplitude and probability were discussed extensively in222The non-
minimal coupling included in [48] seems to change sign between the Lagrangian
and the Feynman rules. We mention this only because a non-minimal coupling
will appear below, and the sign is important. [48]. Again, no gauge has been
chosen for the metric. Invariance of (2.10) under linearised diffeomorphisms
is easily verified.

###  2.3 Double copy

In this section we identify a double copy relation between the gauge and
gravitational amplitudes (2.3) and (2.10). There are essentially two questions
to ask; first, how can the functional degrees of freedom in the gauge field be
encoded in a metric perturbation and/or axio-dilaton and, second, how are the
remaining kinematic structures in (2.3) and (2.10) related? We take these in
reverse order, since the answer to the second question will inform the first.

#### Kinematics

The double copy form of the kinematic structure is revealed simply by
rearranging terms in the gravity amplitude (2.10):

${i\mathcal{M}_{(1)}=\kappa}\int\\!\hat{\mathrm{d}}^{4}k\,\hat{\delta}^{4}(p+q-k)\bigg{[}h^{\mu\nu}(k)(p-\tfrac{1}{2}k)_{\mu}(p-\tfrac{1}{2}k)_{\nu}+\tfrac{1}{4}R_{(1)}(k)+\tfrac{1}{2}k^{2}\phi_{(1)}(k)\bigg{]}\;.$
(2.11)

in which

$R_{(1)}(k)=k^{2}{h^{\sigma}}_{\sigma}(k)-k_{\mu}k_{\nu}h^{\mu\nu}(k)\;,$
(2.12)

is (the Fourier transform of) the linearised Ricci scalar, $R=\kappa R_{(1)}$.
The first term in the square brackets of (2.11) is explicitly what we want –
two copies of the kinematic terms in the gauge theory amplitude. What remains
should therefore vanish, which constrains the dilaton to be a function of the
metric variables according to

$\displaystyle\phi_{(1)}(k)$ $\displaystyle=-\frac{1}{2k^{2}}R_{(1)}(k)$
(2.13)
$\displaystyle=-\tfrac{1}{2}{h^{\sigma}}_{\sigma}(k)+\frac{k_{\mu}h^{\mu\nu}(k)k_{\nu}}{2k^{2}}\;.$
(2.14)

To understand this, recall that for the double copy of perturbative scattering
amplitudes in vacuum, graviton polarisations $\epsilon^{\mu\nu}$ emerge as the
symmetric traceless product of gluon polarisations, and the dilaton emerges
through the trace. Here the dilaton is again essentially a trace, as seen in
the first term of (2.14), and we will see immediately below that the metric
$h^{\mu\nu}$ appears as the product of gauge fields. The second term in
(2.14), going like $khk$ would not arise for scattering of on-shell states in
vacuum, as then $k\epsilon k=0$. Here it may be nonzero, though, since our
background may be sourced.

The origin of (2.13) is readily identified – the equations of motion for the
dilaton which follow from the _gravitational_ part of the original action are,
assuming a vanishing axion,

${\sqrt{-g}e^{\phi}\bigg{[}-2R-2g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\bigg{]}+4\partial_{\mu}\big{(}\sqrt{-g}e^{\phi}g^{\mu\nu}\partial_{\nu}\phi\big{)}=0\;.}$
(2.15)

Linearising this equation and going to Fourier space, one immediately recovers
(2.13); hence the dilaton is constrained to obey the equations of motion in a
prescribed metric and with no matter fields. We similarly take the axion to
obey its equation of motion, for which $\chi=0$ is always a solution, and we
adopt this from here on. The potential for cancellations between the dilaton
and metric contributions is made _explicit_ by our choice of field variables
in (2.7), and is a benefit of that choice. Had we used the original variables
of (2.5) we would have had to allow the metric to depend on the dilaton, and
then separate out a trace-like term to find the same cancellation. Our change
of variables seems to do this for us.

#### Functional degrees of freedom

We expect, from the form of double copy in flat space, that the product of
gauge fields _in Fourier space_ should yield the Fourier transform of the
metric perturbation, $h(k)\sim A(k)A(k)$. This is in the spirit of the
convolutional double copy (i.e. the metric should be a convolution of gauge
fields in position space), which has been applied in Minkowski [41] and in
(A)dS [44], see also [42, 43]. (For special cases where the double copy is
local in position space, suggesting a convolution in Fourier space, see [49].)

Dimensional analysis requires, however, the presence of some other structure
which cancels the dimensionality of one copy of the gauge field. This suggests
the introduction of a biadjoint scalar (BAS) field $\Phi$, see e.g. [50, 44],
such that

$h^{\mu\nu}(k)=A^{\mu a}(k)\Phi^{-1}_{ab}(k)A^{\nu b}(k).$ (2.16)

There is a great deal of flexibility in choosing $\Phi$, and one could also
take the product of different gauge fields $A^{\mu}$ and ${\tilde{A}}^{\mu}$
as $h\sim A\Phi^{-1}{\tilde{A}}$, a freedom analogous to choosing two
different gauge groups when taking the zeroth copy. Exploiting these freedoms
may be useful for simplifying calculations but does not affect which metrics
can be obtained through (2.16) – any choice of $\Phi$ can ultimately be
absorbed into a transformation of the gauge field.

With this relation between the metric perturbation and gauge field, we can
obtain the correct gravity amplitude from (2.3) by replacing $g$ with
$\kappa/2$, familiar from the literature, and by replacing the colour factor
with a copy of the remainder of the amplitude,

${\mathsf{T}}^{a}_{ij}\to-\Phi^{-1}_{ab}(k)A^{b\nu}(k)\big{(}p_{\nu}-\tfrac{1}{2}k_{\nu}\big{)}\;,$
(2.17)

tied to the original amplitude by the BAS. Much like in vacuum the coupling,
delta functions and momentum integrals are not copied – here and below, our
double copy prescription will work at the level of _integrands_. Note that we
have let $g\to\kappa/2$ before copying the numerator; as such there is a
factor of 2 in the amplitude that is effectively not copied, but this is
equivalent to instead defining the colour factor in terms of generators
rescaled by $\sqrt{2}$, as is typical [7], and copying all numerical factors.

We comment briefly on the sources of our gauge field and metric (2.16), since
less is known about sourced solutions, compared to vacuum solutions, in
classical double copy, see though [51]. If the gauge background is sourced by
$J_{\mu}$ we may write, at the linearised level,
$A_{\mu}^{a}(k)=\big{(}k_{\mu}k_{\nu}A^{a\nu}(k)-J_{\mu}^{a}(k)\big{)}/k^{2}$.
Substituting into (2.16), and writing e.g. $h=A\circ A$ to compactify notation
for products with the BAS [44], the Einstein equations for the metric (2.16)
become

$\begin{split}G_{\mu\nu}(k)&=\frac{1}{2k^{2}}\bigg{[}J_{\mu}(k)\circ{J}_{\nu}(k)-\bigg{(}\eta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\bigg{)}J_{\alpha}(k)\circ{J}^{\alpha}(k)\bigg{]}\;.\end{split}$
(2.18)

The term in round brackets is the $k$-transverse propagator, which appears
because $k\cdot J=0$ in the gauge theory, and guarantees that
$k_{\mu}G^{\mu\nu}=0$. (For discussion of how gravitational symmetries emerge
from gauge theory via double copy see [52] and [53].)

#### Summary

Beginning with a background gauge field, we can form a background metric from
the product of gauge fields in Fourier space, in the spirit of the
convolutional double copy. The metric thus inherits all functional degrees of
freedom originally present, and we have seen that the dilaton is then fixed
via its equations of motion. Scattering massive fields on these gauge and
gravitational backgrounds, we find that the leading order amplitudes indeed
obey a double copy relation. The gravitational amplitude is, finally,

${i\mathcal{M}_{(1)}=\kappa}\int\\!\hat{\mathrm{d}}^{4}k\,\hat{\delta}^{4}(p+q-k)h^{\mu\nu}(k)\Big{[}(p-\tfrac{1}{2}k)_{\mu}(p-\tfrac{1}{2}k)_{\nu}\Big{]}\;,$
(2.19)

which is to be compared against the gauge theory amplitude (2.3). Identifying
$h$ as in (2.16), the double copy structure is evident.

It is interesting to note how the choice of colour structure in $A_{\mu}$
affects which metrics can be obtained through (2.16). If the gauge field
contains only a single generator, for example, the resulting metric
perturbation is necessarily a bivector. Constructing the most general metric
would seem to require as many generators as there are spacetime dimensions.
There are then many different choices of $A_{\mu}$ which will yield the same
metric.

The question to ask next is whether these double copy results persist beyond
the linearised approximation and, if so, what new structures arise? In
particular we note that the convolutional double copy is only defined at the
linearised level, which makes the question of how functional degrees of
freedom double copy, beyond leading order, especially interesting. These are
the topics of the next section.

##  3 Next to leading order

Here we generalise the calculations above to next to leading order (NLO) in
perturbation theory. As before we present the individual gauge and gravity
calculations first, then discuss how to relate them via double copy.

###  3.1 Gauge theory

It is straightforward to compute the NLO gauge theory amplitude using the
Feynman rules in Fig. 1. Recall that momentum is conserved at each vertex, but
that there is an accompanying integral over the background four-momentum. We
continue to leave all such integrals unevaluated and therefore introduce for
them the compact notation

$\int_{n}\equiv\int\\!\prod_{i=1}^{n}\hat{\mathrm{d}}^{4}k_{i}\,\,\hat{\delta}^{4}\bigg{(}p+q-\sum_{j=1}^{n}k_{j}\bigg{)}\;.$
(3.1)

This also allows us to focus on the _integrands_. With this, the NLO gauge
theory amplitude is easily found to be

${i\mathcal{A}_{(2)}}={-(2g)^{2}}\int_{2}\frac{{({\mathsf{T}}^{a}{\mathsf{T}}^{b})_{ij}}}{\rho}A^{\mu
a}(k_{1})A^{\nu
b}(k_{2})\left[\left(p-\tfrac{1}{2}k_{1}\right)_{\mu}\left(q-\tfrac{1}{2}k_{2}\right)_{\nu}+\frac{\rho}{4}\eta_{\mu\nu}\right]\;,$
(3.2)

where $\rho\equiv(p-k_{1})^{2}-m^{2}$ is the propagator denominator. The first
term in (3.2) comes from the single diagram with two three-point vertices – as
each interaction is with the background, there is no distinction between
different orderings of gluon legs. The second term comes from the seagull
diagram. The amplitude (3.2) exhibits a particularly simple colour structure,
which generalises directly to higher orders, see [39] and [54] for more
comments on this ‘single quark line’ type colour factor. Gauge invariance is
readily checked. The $\mathcal{O}(g^{2})$ amplitude, that is the sum
$\mathcal{A}_{(1)}+\mathcal{A}_{(2)}$, is invariant up to terms of order
$g^{3}$ under (2.4); the nonlinear part of the transformation of (2.3) cancels
against the linearised transformation of (3.2).

Diagrams which may be neglected, or contribute irrelevant phases, in vacuum
(e.g. bubbles, tadpoles, disconnected contributions), can contain nontrivial
physical information in backgrounds (on e.g. vacuum decay and self focussing).
Disconnected loop diagrams can contribute to our two-point amplitude already
at order $\kappa^{2}$ but we expect that these pieces can be double-copied
separately; here we will focus on the connected tree-level piece to pair
production.

###  3.2 Gravity

The analogous NLO calculation in gravity is only complicated by the fact that,
as we move beyond the linearised approximation, the amplitude will pick up
contributions from higher order corrections to the metric (and dilaton). This
requires a slightly more careful counting of corrections than in gauge theory.

From here on we work with a general perturbation to Minkowski spacetime, again
without choosing a particular gauge, which we specify through the inverse
metric

$g^{\mu\nu}(x)=\eta^{\mu\nu}-\mathfrak{H}^{\mu\nu}(x),$ (3.3)

in which the perturbation may now contain (in preparation for both the NLO
calculation and higher-order calculations below) terms of all orders $\kappa$,

$\mathfrak{H}^{\mu\nu}(x)\equiv\sum_{n=1}^{\infty}\kappa^{n}h^{\mu\nu}_{(n)}(x)\;.$
(3.4)

(The perturbations $h^{\mu\nu}_{(n)}(x)$ can readily be related order by order
to a general perturbation of the metric $g_{\mu\nu}$.) Note that to avoid
clutter we will when possible omit the subscript on the linear piece, writing
$h_{(1)}^{\mu\nu}\equiv h^{\mu\nu}$, which also matches the notation of Sec.
2. Given the relations (2.14) and (2.15) seen above, we similarly expand the
dilaton as

$\phi(x)=\sum_{n=1}^{\infty}\kappa^{n}\phi_{(n)}(x)\;.$ (3.5)

We expand the Lagrangian in (2.9) up to second order in $\mathfrak{H}$ and
$\phi$, which is enough to capture all terms which can contribute to NLO:

$\begin{split}\mathcal{L}=\,&\mathcal{L}_{0}\\\
&+\Big{(}\frac{1}{2}\mathfrak{H}+\phi\Big{)}\mathcal{L}_{0}-\mathfrak{H}^{\mu\nu}\partial_{\mu}\varphi^{\dagger}\partial_{\nu}\varphi\\\
&\quad+\bigg{(}\frac{1}{4}{\mathfrak{H}^{\mu\nu}\mathfrak{H}_{\mu\nu}}+\frac{1}{8}\mathfrak{H}^{2}+\frac{1}{2}{\mathfrak{H}\phi}+\frac{1}{2}{\phi^{2}}\bigg{)}\mathcal{L}_{0}-\Big{(}\frac{1}{2}{\mathfrak{H}}+\phi\Big{)}\mathfrak{H}^{\mu\nu}\partial_{\mu}\varphi^{\dagger}\partial_{\nu}\varphi{\,+\ldots}\;,\end{split}$
(3.6)

in which $\mathcal{L}_{0}$ is the free scalar Lagrangian in flat space, and
indices are raised and lowered using the Minkowski metric as usual.

$\begin{split}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\circ$}; \diagram* { (o) -- [fermion, momentum = $p$]
(s1), (o) -- [anti fermion, momentum' = $q$] (s2), (b) -- [photon] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=i\mathfrak{H}^{\mu\nu}(k)\left[p_{\mu}q_{\nu}-\frac{1}{2}\left(p\cdot
q+m^{2}\right)\eta_{\mu\nu}\right]\\\ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\circ$}; \diagram* { (o) -- [fermion, momentum = $p$]
(s1), (o) -- [anti fermion, momentum' = $q$] (s2), (b) -- [scalar] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=-i\phi(k)\left(p\cdot
q+m^{2}\right),\end{split}$ Figure 3:  Feynman rules for a complex scalar
interacting with an all orders background graviton and dilaton.

The Feynman rules generated by the _second_ line of (3.6) are formally the
same as those in Fig. 2 except that the background $\mathfrak{H}$ now contains
all orders in $\kappa$, which we represent graphically with a circle ‘$\circ$’
on the external leg in Fig. 3. So, this diagram evaluated on-shell, for
example, gives contributions at every order in $\kappa$.

The leading order Feynman rules of Fig. 2 are recovered by extracting the
linear piece $\kappa h^{\mu\nu}_{(1)}$ of the full perturbation, denoted by
replacing the circle with a cross in the diagram. More generally we have e.g.

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\circ$}; \diagram* { (o) -- [fermion] (s1), (o) --
[anti fermion] (s2), (b) -- [photon] (o), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\times$}; \diagram* { (o) -- [fermion] (s1), (o) --
[anti fermion] (s2), (b) -- [photon] (o), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\times\times$}; \diagram* { (o) -- [fermion] (s1), (o)
-- [anti fermion] (s2), (b) -- [photon] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\times\\!\times\\!\times$}; \diagram* { (o) --
[fermion] (s1), (o) -- [anti fermion] (s2), (b) -- [photon] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\;\cdots,$ (3.7)

with each term on the right hand side picking out the contribution at a
specific order in $\kappa$, concretely

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[left=of o] (b){$\underbrace{\times\cdots\times}_{n}$}; \diagram* { (o)
-- [fermion, momentum = $p$] (s1), (o) -- [anti fermion, momentum' = $q$]
(s2), (b) -- [photon] (o), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=i\kappa^{n}h_{(n)}^{\mu\nu}(k)\bigg{[}p_{\mu}q_{\nu}-\frac{1}{2}\left(p\cdot
q+m^{2}\right)\eta_{\mu\nu}\bigg{]}\;.$ (3.8)

This notation allows us to visually keep track of the perturbation parameter
through the number of crosses in a diagram.

$\begin{split}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[above left=of o] (b1){$\circ$}; \vertex[below left=of o]
(b2){$\circ$}; \diagram* { (o) -- [fermion, momentum = $p$] (s1), (o) -- [anti
fermion, momentum' = $q$] (s2), (b1) -- [photon] (o), (b2) -- [photon] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=i\mathfrak{H}^{\mu\nu}(k_{1})\mathfrak{H}^{\alpha\beta}(k_{2})\left[\frac{\eta_{\alpha\beta}}{2}p_{\mu}q_{\nu}-\left(\frac{\eta_{\alpha\mu}\eta_{\beta\nu}}{4}+\frac{\eta_{\mu\nu}\eta_{\alpha\beta}}{8}\right)\left(p\cdot
q+m^{2}\right)\right]\\\ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[above left=of o] (b1){$\circ$}; \vertex[below left=of o]
(b2){$\circ$}; \diagram* { (o) -- [fermion, momentum = $p$] (s1), (o) -- [anti
fermion, momentum' = $q$] (s2), (b1) -- [photon] (o), (b2) -- [scalar] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=i\mathfrak{H}^{\mu\nu}(k_{1})\phi(k_{2})\left[p_{\mu}q_{\nu}-\frac{\eta_{\mu\nu}}{2}\left(p\cdot
q+m^{2}\right)\right]\\\ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[small] \vertex(o) at
(0,0); \vertex[above right=of o] (s1); \vertex[below right=of o] (s2);
\vertex[above left=of o] (b1){$\circ$}; \vertex[below left=of o]
(b2){$\circ$}; \diagram* { (o) -- [fermion, momentum = $p$] (s1), (o) -- [anti
fermion, momentum' = $q$] (s2), (b1) -- [scalar] (o), (b2) -- [scalar] (o), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&=-\frac{i}{2}\phi(k_{1})\phi(k_{2})\left(p\cdot
q+m^{2}\right),\end{split}$ Figure 4:  Feynman rules that begin contributing
to the pair creation amplitude at second order in the coupling. As before,
replacing a circle on an external leg with $n$ crosses indicates taking the
order $n$ contribution of the corresponding background (metric or dilaton).

At each perturbative order there is a new set of diagrams to take into
account, built from a larger set of Feynman rules – at NLO the four-point
vertices in Fig. 4 are introduced, for example. They contribute along with the
order $\kappa^{2}$ parts of the diagrams in Fig. 3. The remaining four tree-
level diagrams are those with two three-point vertices.

Following the results found at leading order we take the dilaton to obey its
equations of motion – the leading piece $\phi_{(1)}$ is again replaced by the
linearised Ricci scalar through (2.13), and expanding (2.15) to second order
we have

$\begin{split}4\partial^{2}\phi_{(2)}=2R_{(2)}+2\phi_{(1)}R_{(1)}&+hR_{(1)}+2\partial\phi_{(1)}\cdot\partial\phi_{(1)}\\\
&-4\partial_{\mu}\bigg{[}\bigg{(}\frac{h}{2}\eta^{\mu\nu}+\phi_{(1)}\eta^{\mu\nu}-h^{\mu\nu}\bigg{)}\partial_{\nu}\phi_{(1)}\bigg{]},\end{split}$
(3.9)

which allows us to express $\phi_{(2)}$ in terms of the metric perturbations.

###  3.3 Double copy

Putting together all the diagrams and frequently utilising symmetries related
to interchanges of Lorentz indices and momenta $k_{i}$ under the integral, we
find that the NLO amplitude is

$\begin{split}&{i}\mathcal{M}_{(2)}={\frac{1}{8}}\kappa^{2}\int_{2}h^{\mu\nu}(k_{1})h^{\alpha\beta}(k_{2})\eta_{\nu\beta}\left(p-q\right)_{\mu}\left(p-q\right)_{\alpha}{+\frac{1}{4}}\kappa^{2}\int_{1}h_{(2)}^{\mu\nu}(k_{1})\left(p-q\right)_{\mu}\left(p-q\right)_{\nu}\\\
&{+}\kappa^{2}\int_{2}\frac{h^{\mu\nu}(k_{1})h^{\alpha\beta}(k_{2})}{\rho}\left[\left(p-\tfrac{1}{2}k_{1}\right)_{\mu}\left(q-\tfrac{1}{2}k_{2}\right)_{\alpha}+\frac{\rho}{4}\eta_{\mu\alpha}\right]\left[\left(p-\tfrac{1}{2}k_{1}\right)_{\nu}\left(q-\tfrac{1}{2}k_{2}\right)_{\beta}+\frac{\rho}{4}\eta_{\nu\beta}\right].\end{split}$
(3.10)

Compare this to (3.2); the second line of the gravity amplitude displays a
clear double copy structure. Informed by the replacements at leading order we
can obtain the second line from the gauge amplitude as follows: leave the
propagator, integral and factor of $(2g)^{2}$ untouched, and then replace the
generators with a second copy of the remainder of the integrand,

$({\mathsf{T}}^{a}{\mathsf{T}}^{b})_{ij}\to-\Phi_{aa^{\prime}}^{-1}(k_{1})\Phi_{bb^{\prime}}^{-1}(k_{2})A^{\alpha
a^{\prime}}(k_{1})A^{\beta
b^{\prime}}(k_{2})\left[\left(p-\tfrac{1}{2}k_{1}\right)_{\alpha}\left(q-\tfrac{1}{2}k_{2}\right)_{\beta}+\frac{\rho}{4}\eta_{\alpha\beta}\right],$
(3.11)

along with BAS fields. With the identification (2.16), the indices work out
precisely so that we recover the second line of (3.10).

We are then left with the question of how to deal with the terms on line one
of (3.10), in order to have a working double copy prescription. With our
current setup the gravity amplitude matches our double copy prediction only if
those terms vanish, which in turn requires

$h_{(2)}^{\mu\nu}(x)=-\tfrac{1}{2}h^{\mu\sigma}(x)\eta_{\sigma\rho}h^{\rho\nu}(x).$
(3.12)

Higher order corrections to the metric are thus _determined_ from the linear
perturbation by enforcing a double copy structure. As such, while the choice
of gauge field and BAS still determines the metric, the story is not as
straightforward as (2.16) suggests – we will comment further on this result,
and the resummation of the perturbative series, below.

A notable special case is Kerr-Schild spacetimes [55, 56, 57], where there are
no higher order corrections to the metric and
$h^{\mu\sigma}{h_{\sigma}}^{\nu}=0$ by construction, rendering (3.12) trivial.
Since in addition $h^{\sigma}_{\sigma}=0$ for Kerr-Schild, it seems likely
that no further corrections are generated at higher orders in perturbation
theory. It would then follow that if $\kappa h$ is in Kerr-Schild form, no
resummation is necessary. (This class includes plane waves, for which an all-
orders double copy map has already found [28].)

We note that, instead of putting restrictions on the metric, the first line of
(3.10) could be generated through the double copy if the gauge field and BAS
have series expansions in $g$. We will not pursue this option further here,
though.

##  4 Examples and all-order results

In this section we will use the above results to conjecture a double copy map
at all orders. We will then identify a non-perturbative way to greatly
simplify the dilaton contribution to our two-point amplitudes, which we will
make use of when we turn to particular choices of background fields. This will
allow us to explicitly extend the LO and NLO results above to higher orders in
perturbation theory, and provide further all-orders conjectures.

###  4.1 All-orders conjecture

As is apparent from the Feynman rules of Fig. 1, the tree-level amplitude at
order $g^{m}$ of a single quark line interacting with a background can always
be written as

$i\mathcal{A}_{(m)}=g^{m}\int_{m}\frac{\prod_{s=1}^{m}A^{\mu_{s}a_{s}}(k_{s})}{D_{m}}N_{\mu_{1}\dots\mu_{m}}\Big{(}\textstyle{\prod}_{s=1}^{m}t^{a_{s}}\Big{)}_{ij}\;,$
(4.1)

where $D_{m}$ is the product of propagator denominators from the diagram with
$m$ three-point vertices, $t^{a}=\sqrt{2}{\mathsf{T}}^{a}$ are generators
normalised as in the vacuum double copy formalism, and
$N_{\mu_{1}\dots\mu_{m}}$ is some tensor built from kinematic variables.
Following the procedure we have seen at leading and next to leading order, we
make the replacements

$g\to\frac{\kappa}{2}\;,\qquad{\Big{(}{\textstyle{\prod}}_{s=1}^{m}\,t^{a_{s}}\Big{)}_{ij}}\to{\Big{(}\textstyle{\prod}_{s=1}^{m}\Phi_{a_{s}b_{s}}^{-1}(k_{s})\mathcal{A}^{\nu_{s}b_{s}}(k_{s})\Big{)}N_{\nu_{1}\dots\nu_{m}}}\;,$
(4.2)

and, identifying the metric perturbation through (2.16), we arrive at a
conjectured double copy amplitude

$i\mathcal{M}_{(m)}=\left(\frac{\kappa}{2}\right)^{m}\int_{m}\frac{\prod_{s=1}^{m}h^{\mu_{s}\nu_{s}}(k_{s})}{D_{m}}N_{\mu_{1}\dots\mu_{m}}N_{\nu_{1}\dots\nu_{m}}\;.$
(4.3)

This double copy map can be put into a particularly familiar form by
introducing the multi-indices $\mathbf{a}=(a_{1},\dots,a_{s})$ and
$\mathbf{b}=(b_{1},\dots,b_{s})$, through which we define colour factors,
kinematic numerators, and an object built from BAS fields as

$c_{\mathbf{a}}\equiv\Big{(}{\textstyle{\prod}}_{s=1}^{m}\,t^{a_{s}}\Big{)}_{ij}\;,\qquad
n_{\mathbf{a}}\equiv\Big{(}\textstyle{\prod}_{s=1}^{m}A^{\mu_{s}a_{s}}(k_{s})\Big{)}N_{\mu_{1}\dots\mu_{m}}\;,\qquad\Phi_{\mathbf{ab}}^{-1}\equiv\prod_{s=1}^{m}\Phi^{-1}_{a_{s}b_{s}}(k_{s})\;.$
(4.4)

In terms of these quantities the gauge theory amplitude and its proposed
double copy are compactly expressed as

$i\mathcal{A}_{(m)}=g^{m}\sum_{\mathbf{a}}\int_{m}\frac{n_{\mathbf{a}}c_{\mathbf{a}}}{D_{m}}\;,\qquad
i\mathcal{M}_{(m)}=\left(\frac{\kappa}{2}\right)^{m}\sum_{\mathbf{a,b}}\int_{m}\frac{n_{\mathbf{a}}\Phi^{-1}_{\mathbf{ab}}n_{\mathbf{b}}}{D_{m}}\;.$
(4.5)

In this form the amplitudes mirror the structure of double copy in vacuum, and
our replacement rules (4.2) now effectively substitute colour factors for
kinematic numerators through
$c_{\mathbf{a}}\to\Phi^{-1}_{\mathbf{ab}}n_{\mathbf{b}}$, along with some
extra structure in the form of BAS fields.

The sum runs over all colour structures $\mathbf{a}$, meaning every
combination of colour indices $(a_{1},\dots a_{n})$ is double copied
separately. Note that in general the colour factors $c_{\mathbf{a}}$ are not
unique due to commutation relations between the generators. Consequently there
is some ambiguity in identifying the kinematic numerators $n_{\mathbf{a}}$
which, as we will see, can affect the double copy amplitude.

On the gravity side the expansion parameter $\kappa h$, which is generated by
the choice of gauge and BAS fields, is seemingly related to the full metric
perturbation $g-\eta$ in a non-trivial way. We will investigate this in the
next section and see, in an analysis of examples, that it is still possible to
choose the linear perturbation so as to produce a specific metric of interest.

###  4.2 Simplifying the dilaton contribution

Define the field

$w(x)\equiv\exp\bigg{(}\frac{\phi(x)}{2}\bigg{)}\;.$ (4.6)

In terms of $w(x)$ the dilaton equation of motion (2.15) simplifies
considerably; it becomes the _linear_ equation

$\square
w(x)-\frac{1}{4}R(x)w(x)=0\;,\qquad\text{where}\qquad\square\equiv\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\;.$
(4.7)

This is the equation of motion for a massless scalar _non-minimally coupled_
to the metric with $\xi=-1/4$. We will now show that the dilaton contribution
to two-point amplitudes can be traded entirely for the same non-minimal
coupling of the matter field $\varphi$ to the metric – this offers a
considerable simplification when extending the previous calculations to higher
orders, as we will do below.

The pair production amplitude we consider may also be computed using the
perturbiner approach [58, 59, 60, 61, 62, 63, 64], which states that tree-
level two-point amplitudes for a field $\varphi$ on a background are computed
by the quadratic part of the classical action evaluated on solutions of the
equations of motion (chosen to implement the appropriate scattering boundary
conditions). This result has recently been used in eikonal scattering [65],
scattering on various backgrounds [28, 66, 67, 65], celestial holography [68],
pair creation [69], and cosmology [70, 71, 72].

For us, the equation of motion following from the action (2.9) is just the
wave equation

$(\square+m^{2})\,\varphi=0\;.$ (4.8)

Evaluating (2.9) on any solution to (4.8) reduces, via integration by parts,
to a boundary term. If $\varphi_{p}$ represents the solution of (4.8)
corresponding to a free particle of momentum $p_{\mu}$ in the asymptotic
future, then the boundary term to calculate for pair production is

$\int_{S}\\!\mathrm{d}S^{\mu}\,e^{iq\cdot
x}{\overset{\leftrightarrow}{\partial}}_{\mu}\varphi_{p}(x)\;,$ (4.9)

which is just the standard Klein-Gordon inner product on an asymptotic surface
$S$ with timelike normal. Another way arrive at this result is to observe that
overlaps like (4.9) simply compute the various Bogoliubov coefficients
relating asymptotic particle wavefunctions for in/out states, in other words
amplitudes, see e.g. [73].

If we define $\pi(x)=w(x)\varphi_{p}(x)$, and choose boundary conditions such
that $w(x)$ and $\mathrm{d}S^{\mu}\partial_{\mu}w(x)$ vanish on the boundary
$S$, it follows trivially that (4.9) may be computed replacing $\varphi_{p}$
with $\pi$. Since we take $w$ to obey its equation of motion (4.7), it is
easily checked that the new field $\pi$ obeys the equations of motion of a
massive scalar with non-minimal coupling $\xi=-1/4$;

$(\square+m^{2})\,\pi(x)-\frac{1}{4}R(x)\pi(x)=0\;.$ (4.10)

The important point is that we can compute the pair production amplitude
simply by solving (4.10) and calculating (4.9) – at no point do we have to
_explicitly_ solve the dilaton equations of motion. To compute our amplitudes
we can thus drop the dilaton entirely and work instead with the action

$\int\\!\mathrm{d}^{4}x\,\sqrt{-g}\big{(}g^{\mu\nu}\partial_{\mu}\pi^{\dagger}\partial_{\nu}\pi-m^{2}\pi^{\dagger}\pi+\frac{1}{4}R\pi^{\dagger}\pi\big{)}\;.$
(4.11)

This makes calculations significantly simpler, as we will exploit below.

Interestingly, the numerical value of the non-minimal coupling here is the
_same_ as arises in the worldline approach (to scalar fields interacting with
gravity) as a finite counterterm which remains after cancellation of
divergences associated with the path integral measure. See [74] for a
discussion and history. It might be interesting to see if the worldline
coupling could be reinterpreted in terms of the dilaton.

###  4.3 Example: pair creation in FRW spacetimes

A natural gravitational analogue of particle creation in a time-dependent
field is particle creation in a time-dependent metric [75, 76, 77, 78]. We
focus here on the example of FRW spacetime, beginning by asking how this fits
into our proposed double copy scheme at the linearised level.

Taking the FRW line element in the standard form

$\mathrm{d}\tau^{2}=\mathrm{d}t^{2}-a^{2}(t)\mathrm{d}\bm{x}^{2},$ (4.12)

for $a(t)$ the scale factor, the inverse metric can be written

$g^{\mu\nu}=\eta^{\mu\nu}-\bigg{(}\frac{1}{a^{2}(t)}-1\bigg{)}{X_{3}^{\mu\nu}},$
(4.13)

in which we have introduced

${X_{3}^{\mu\nu}}\equiv e^{\mu}_{i}e^{\nu}_{i},$ (4.14)

for $\\{e_{1}^{\mu},e_{2}^{\mu},e_{3}^{\mu}\\}$ an orthonormal basis of
spacelike vectors. Note that the tensor $X_{3}^{\mu\nu}$ is constant and obeys
$X_{3}^{\mu\sigma}\eta_{\sigma\rho}X_{3}^{\rho\nu}=-X_{3}^{\mu\nu}$. In order
to obtain this structure via the proposed double copy map (2.16) we consider a
gauge field of the form

$A_{a}^{\mu}=\zeta^{i}_{a}(t)e_{i}^{\mu}\;,$ (4.15)

in which the functions $\zeta^{i}_{a}(t)$ are naturally assumed to be only
$t$-dependent given the metric of interest. With this, we identify a metric
perturbation

$\begin{split}h^{\mu\nu}&=A_{a}^{\mu}\Phi_{ab}^{-1}A_{b}^{\nu}\\\
&=\big{(}\zeta^{i}_{a}\Phi_{ab}^{-1}\zeta^{j}_{b}\big{)}e_{i}^{\mu}e_{j}^{\nu}\;.\end{split}$
(4.16)

We now choose $\zeta$ and $\Phi$ such that
$\zeta^{i}_{a}\Phi_{ab}^{-1}\zeta^{j}_{b}\propto\delta_{ij}$ in order to
generate the required tensor structure $X_{3}^{\mu\nu}$. One interesting
choice, given that most examples of classical double copy are essentially
abelian, see though [79], is to consider a gauge group with at least the same
number of colours as spatial dimensions, for example SU(2) in $3+1$
dimensions; one may then take $\zeta^{i}_{a}(t)=\zeta(t)\delta_{ia}$ for
$\zeta$ a scalar. The gauge field is hence

$A^{\mu}=\zeta(t){\mathsf{T}}^{i}e_{i}^{\mu}\;,$ (4.17)

and if we additionally choose $\Phi_{ab}^{-1}=\Phi^{-1}\delta_{ab}$ diagonal,
then the metric perturbation is

$h^{\mu\nu}=\Phi^{-1}\zeta^{2}(t)X_{3}^{\mu\nu}\;,$ (4.18)

which has the desired tensor form.

We now choose the scalar functions such that $\Phi^{-1}\zeta^{2}(t)=f(t)$; our
general results then immediately relate the gauge and gravity amplitudes up to
NLO, and (3.12) gives the FRW metric

$g^{\mu\nu}=\eta^{\mu\nu}-X_{3}^{\mu\nu}(\kappa
f(t)+\frac{1}{2}\kappa^{2}f^{2}(t)+\ldots)\;.$ (4.19)

For a given scale factor $a(t)$, we would like to choose $\kappa f(t)$ such
that the resummed metric matches (4.13). This requires knowledge of how higher
order corrections to the metric relate to the linear perturbation, which we
investigate below.

### Higher orders

Since we have restricted to a specific metric, we are able to push our
amplitude calculations to the next order in perturbation theory. This
illustrates some of the subtleties which can appear beyond NLO.

The $\mathcal{O}(g^{3})$ contribution to the gauge theory amplitude in the
background (4.17) is

$i\mathcal{A}_{(3)}=-(2g)^{3}{\mathsf{T}}^{i}{\mathsf{T}}^{j}{\mathsf{T}}^{k}\int_{3}\frac{\zeta(k_{1})\zeta(k_{2})\zeta(k_{3})}{\rho_{1}\rho_{3}}\bigg{[}(e_{i}\cdot
p)(e_{j}\cdot p)(e_{k}\cdot p)+{(e_{i}\cdot
p)}\frac{\rho_{3}}{4}\delta_{jk}+{(e_{k}\cdot
p)}\frac{\rho_{1}}{4}\delta_{ij}\bigg{]}\;,$ (4.20)

as follows directly from the Feynman rules in Fig. 1. The first term in (4.20)
comes from the diagram with three 3-point vertices, while the second and third
terms comes from diagrams containing a four-point vertex.

According to the proposal (4.2) the expected double copy amplitude is

$i\mathcal{M}_{(3)}=\kappa^{3}\int_{3}\frac{f(k_{1})f(k_{2})f(k_{3})}{\rho_{1}\rho_{3}}{\bf
p}^{2}\bigg{[}{\bf p}^{4}+\frac{1}{2}(\rho_{1}+\rho_{3}){\bf
p}^{2}+\frac{3}{16}(\rho_{1}^{2}+\rho_{3}^{2})+\frac{1}{8}\rho_{1}\rho_{3}\bigg{]}\;,$
(4.21)

in which we have used $(e_{i}\cdot p)(e_{i}\cdot
p)=p_{\mu}X_{3}^{\mu\nu}p_{\nu}={\bf p}^{2}$.

We turn to the gravitational calculation. The only tensor structure which will
be produced by the double copy map is $X_{3}^{\mu\nu}$, we generalise (4.19)
to

$g^{\mu\nu}=\eta^{\mu\nu}-X_{3}^{\mu\nu}\sum\limits_{n=1}^{\infty}c_{n}\kappa^{n}f^{n}(t)\;,$
(4.22)

with $c_{1}=1$, $c_{2}=\tfrac{1}{2}$ already fixed by (3.12), and the
remaining constants $c_{n}$ to be determined. The corresponding on-shell
dilaton is easily found order-by-order, and is a function only of $t$. (We
give the explicit form as part of a later example, see (4.31).)

A direct calculation in gravity recovers precisely the amplitude (4.21) if the
metric has

$c_{3}=\frac{5}{16}\;.$ (4.23)

So, as at second order, the metric perturbations are determined at third order
in terms of the linearised perturbation, if the double copy is to hold. We
will, below, make an all-orders conjecture for the $c_{n}$. First, though, we
highlight a subtle point.

Observe that we could have used the fact that
${\mathsf{T}}^{j}{\mathsf{T}}^{j}$ is a quadratic Casimir to write (4.20) as

$i\mathcal{A}_{(3)}\overset{!}{=}-(2g)^{3}{\mathsf{T}}^{i}{\mathsf{T}}^{j}{\mathsf{T}}^{k}\int_{3}\frac{\zeta(k_{1})\zeta(k_{2})\zeta(k_{3})}{\rho_{1}\rho_{3}}\bigg{[}(e_{i}\cdot
p)(e_{j}\cdot p)(e_{k}\cdot p)+(e_{i}\cdot
p)\frac{\rho_{1}+\rho_{3}}{4}\delta_{jk}\bigg{]}\;,$ (4.24)

in which only the terms containing a 4-point vertex are affected. This would
suggest that the double copied amplitude is

$i\mathcal{M}_{(3)}\overset{!}{=}\kappa^{3}\int_{3}\frac{f(k_{1})f(k_{2})f(k_{3})}{\rho_{1}\rho_{3}}{\bf
p}^{2}\bigg{[}{\bf p}^{4}+\frac{1}{2}(\rho_{1}+\rho_{3}){\bf
p}^{2}+\frac{3}{16}(\rho_{1}+\rho_{3})^{2}\bigg{]}\;,$ (4.25)

which is _not_ the same as (4.21). While this amplitude can still be recovered
from an FRW metric, but with a different coefficient $c_{3}=9/16$ in the
third-order metric perturbation, we will below continue to use our initial
approach, in which we do not use any group identities. One reason for this is
that, as we are about to see, it will allow us to conjecture all-orders
results.

###  4.4 Example: double copy of electric fields and FRW-like spacetimes

In this section we work with a family of three backgrounds $A_{{d}}^{\mu}$,
for ${d}\in\\{1,2,3\\}$, where

$A_{d}^{\mu}=\zeta(t)\sum_{i=1}^{d}{\mathsf{T}}^{i}e_{i}^{\mu}\;.$ (4.26)

(Note that we continue to work in $3+1$ dimensions in each case.) For ${d}=3$
we recover the FRW case already considered. The case ${d}=1$ is effectively
abelian, since there is then only a single generator ${\mathsf{T}}^{1}$ in the
game and $A_{1}^{\mu}$ represents a time-dependent (chromo) electric field
with profile $\zeta(t)$ and polarisation $e_{1}^{\mu}$. (For formal
equivalences between particle creation in electric fields and FRW spacetimes
see e.g. [80].) The case ${d}=2$ interpolates between the abelian and FRW
cases. We again assume there are at least as many generators as spacetime
dimensions. Through (2.16) and (3.12) the double copy map will generate, for
each ${d}$, metric perturbations proportional to

$X_{d}^{\mu\nu}\equiv\sum_{i=1}^{d}e^{\mu}_{i}e^{\nu}_{i}\;,$ (4.27)

i.e. the metric perturbation is diagonal and nontrivial only in directions $1$
to ${d}\leq 3$. $X_{d}$ has the properties
$X_{d}^{\mu\sigma}\eta_{\sigma\rho}X_{d}^{\rho\nu}=-X_{d}^{\mu\nu}$ and
$X_{d}^{\mu\nu}\eta_{\mu\nu}=-{d}$.

Beginning with the gauge theory, the $\mathcal{O}(g^{3})$ amplitudes are given
by taking (4.20) and restricting the sums over $i,j,k$ to run from $1$ to
${d}$ only. As in (4.20) we take the result generated directly from the
Feynman rules, without exploiting any properties of the gauge group. It
follows that the expected double copied gravitational amplitude is, at
$\mathcal{O}(\kappa^{3})$,

$i\mathcal{M}_{(3)}=\kappa^{3}\int_{3}\frac{f(k_{1})f(k_{2})f(k_{3})}{\rho_{1}\rho_{3}}{\bf
p}_{d}^{2}\bigg{[}{\bf p}_{d}^{4}+\frac{1}{2}(\rho_{1}+\rho_{3}){\bf
p}_{d}^{2}+\frac{{d}}{16}(\rho_{1}^{2}+\rho_{3}^{2})+\frac{1}{8}\rho_{1}\rho_{3}\bigg{]}\;,$
(4.28)

in which

${\bf p}_{d}^{2}\equiv p\cdot X_{d}\cdot p=\sum_{i=1}^{d}p_{i}p_{i}\;.$ (4.29)

We turn to the direct calculation of the $\mathcal{O}(\kappa^{3})$ amplitude
in the gravitational theory, with the metric ansatz

$g^{\mu\nu}_{d}\equiv\eta^{\mu\nu}-X^{\mu\nu}_{d}\sum\limits_{j=1}^{\infty}c_{j}\kappa^{j}f^{j}(t)\;,$
(4.30)

where, as before, $c_{1}=1$, $c_{2}=1/2$ fixed by our general results, and the
remaining $c_{j}$ to be determined. The corresponding dilaton solution to
order $\kappa^{3}$ is

$\phi(t)=\frac{{d}}{2}\kappa
f(t)-\frac{{d}}{8}\int_{\infty}^{t}\\!\mathrm{d}s\int_{\infty}^{s}\\!\mathrm{d}u\,\kappa^{2}f^{\prime}(u)^{2}+\frac{{d}}{12}(6c_{3}-1)\kappa^{3}f(t)^{3}+\ldots\;,$
(4.31)

with boundary conditions chosen such that
$\\{\phi(t),\partial_{t}\phi(t)\\}\to 0$ as $t\to\infty$. Evaluating the
$\mathcal{O}(\kappa^{3})$ gravitational amplitude via the Feynman rules, one
recovers exactly (4.28) for

$c_{3}=\frac{{d}+2}{16}\;.$ (4.32)

Now that we have several related examples, we are prompted to speculate on the
behaviour of higher-order terms in the series (4.22) (and the general series
(3.4)). We expect that there will be a similar constraint to (3.12) at
$\mathcal{O}\left(\kappa^{3}\right)$, i.e. that $h_{(3)}\sim hhh$, with some
linear combination of terms having differently contracted indices on the right
hand side. The explicit $d$-dependence in $c_{3}$ indicates that trace terms
begin contributing to $c_{j}$ for $j>2$, whereas there is no trace term in the
general result at $\mathcal{O}(\kappa^{2})$. At higher orders, different
powers of traces could give various dependencies on powers of ${d}$, and it is
not obvious what their coefficients are. However, motivated by the structure
of the contact terms which appear in both the matter and axio-dilaton sectors
of the original Lagrangian (2.5) we make the following conjecture for the
resummed series (4.30):

$g^{\mu\nu}_{d}=\eta^{\mu\nu}-\frac{\kappa f(t)}{\Big{(}1-\frac{{d}}{4}\kappa
f(t)\Big{)}^{2/{d}}}X_{d}^{\mu\nu}\;.$ (4.33)

It is easily checked that the expansion of (4.33) recovers $c_{1}=1$,
$c_{2}=1/2$ and (4.32).

There are of course infinitely many functions which could reproduce the
patterns in (4.28), (4.31) and (4.32). To provide some small additional
evidence for the conjecture (4.33), we have checked that it continues to hold
at the next order in perturbation theory, $\mathcal{O}(\kappa^{4})$, for
${d}=1$, i.e. the double copy map holds provided the metric perturbation has
$c_{4}=1/16$, which agrees with the fourth-order expansion of (4.33). (For
this we used the ‘Ricci form’ of the action (4.11) which simplifies
calculations.)

Given this, let us comment on the structures in the resummed metric (4.33).
For ${d}=2$ we have a resummed geometric series, while for the abelian case
${d}=1$ the denominator is a square, which is when the form of the metric most
strongly resembles that of the contact terms in the original Lagrangian (2.5).
We comment that in the same case the kinematic parts of the amplitudes
$\mathcal{M}_{(n)}$ also simplify to perfect squares, with (writing out only
the relevant structures) $\mathcal{M}_{(1)}\sim{\bf p_{1}}^{2}$,
$\mathcal{M}_{(2)}\sim({\bf p}_{1}^{2}+\tfrac{1}{4}\rho_{1})^{2}$ and, from
(4.28), $\mathcal{M}_{(3)}\sim{\bf p}_{1}^{2}({\bf
p}_{1}^{2}+\frac{1}{4}(\rho_{1}+\rho_{3}))^{2}$.

It would seem natural for well-behaved gauge backgrounds to map to well-
behaved gravitational backgrounds, or perhaps that singularities should map to
corresponding singularities, as for e.g. the classical double copy of the
Coulomb field to the Schwarzchild metric [55]. This draws attention to the
fact that, for all ${d}$, the resummed perturbations (4.33) appear to be
singular for sufficiently large perturbations. However, observe that (4.33)
corresponds to a scale factor

$\frac{1}{a^{2}(t)}=1+\frac{\kappa f(t)}{\Big{(}1-\frac{{d}}{4}\kappa
f(t)\Big{)}^{2/{d}}}\;,$ (4.34)

and it is easily verified that this relation is _invertible_ in the sense that
there is a finite $\kappa f(t)$ for _any_ given finite $a^{2}(t)$. The
apparent divergence in our inverse metric simply corresponds to taking the
scale factor $a(t)\to 0$, which is obviously singular (while the zero on the
right hand side of (4.34), i.e. the zero of the inverse metric, corresponds to
taking $a(t)\to\infty$). Thus, if (4.33) holds, we are able to recover any FRW
spacetime from the double copy map. This is reassuring since it suggests that
the conditions, like (3.12), generated by the double copy map are not really
constraints, but are rather delivering a metric in a particular
parameterisation. Furthermore, our conjecture potentially offers an
amplitudes-based definition of _classical double copy_ for FRW, a type O
spacetime, whereas the majority of classical double copy results hold for
Kerr-Schild or type D spacetimes [81]. It would therefore, in future work, be
extremely interesting to further test, or verify, (4.33).

##  5 Conclusions

We have considered non-trivial two-point amplitudes of massive particles on
arbitrary gauge and gravitational background fields. Working perturbatively in
the coupling to the backgrounds, we have established a double copy map between
the gauge and gravitational amplitudes. In this map the gauge background
defines a linearised metric perturbation, from which higher order metric
perturbations (and an associated dilaton) are constructed.

While working perturbatively with the lowest multiplicity amplitudes available
is perhaps the simplest approach one could take to the question of if or how
double copy applies in general backgrounds, it is nevertheless direct and
exposes a surprising amount of structure, even at leading and next-to-leading
order in perturbation theory.

Our results have given insight into classical and convolutional double copy.
While the latter has only been formulated at the level of linearised
perurbations, we have seen that there is a sense in which this is all that is
needed – demanding that the double copy holds order by order in perturbation
theory constructs a metric from the linearised perturbation. Notably, we have
for specific examples been able to conjecture all-orders results which
generate e.g. all FRW spacetimes through double copy. We also found a non-
perturbative way to simplify the dilaton contribution to our amplitudes,
replacing it by a non-minimal coupling. While we do not expect this
replacement to hold at higher points or loops it is useful for the computation
of two-point tree amplitudes on backgrounds, which are of broad interest [82,
67, 65, 66, 68, 69, 72].

As we worked perturbatively, we made the assumption that all our backgrounds
were asymptotically flat. This can be relaxed, though. For potentials/metric
perturbations which are only gauge-equivalent to zero, we can deal with the
resulting memory terms as in e.g. [83, 84]. This may give some insight into
the issues with static terms in the double copy, seen in [85]. For backgrounds
which are genuinely non-flat asymptotically, we may appeal to the perturbiner
approach [58, 59, 60, 61, 62] in order to define analogues of scattering
amplitudes (e.g. boundary to boundary correlators in AdS [82]).

There are many possible directions for future research, including extensions
of our results to higher orders, loops, and other numbers of dimensions [86,
87], as well as other gauge theories. The focus could initially be on specific
backgrounds where calculations are simpler (the class of fields in [88, 89,
69] are one candidate) or which have features of interest, e.g. horizons [56,
90]. It would also be interesting to investigate the self-dual sector [91, 92,
93, 79, 94, 95]. Another obvious target is higher-point amplitudes of the
matter and gauge fields, i.e. scattering of particles _with_ radiation; the
more complex colour structures of such amplitudes would allow us to start
exploring colour-kinematic duality in non-flat backgrounds [96, 97, 98, 99].
We hope that having general results, even to NLO, may help to better expose
how gauge transformations and diffeomorphisms are related through double copy.
Finally, an open question is ‘what is the single copy of amplitudes in a
_given_ gravitational background?’ For the family of FRW-like spacetimes
considered here, our results suggest that the answer is the gauge field
(4.26), although, recall e.g. (4.34), it is necessary to go to all orders to
see this explicitly.

We hope to explore some of these questions in future work.

We thank T. Adamo, D. Akpinar, R. Gonzo, S. Klisch, C. Schubert and C. Shi for
useful discussions. The authors are supported by the STFC consolidator grant
ST/X000494/1, “Particle Theory at the Higgs Centre” (A.I.) and an STFC
studentship (W.L.).

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