11institutetext: Centre for Theoretical Physics, Department of Physics and
Astronomy, Queen Mary University of London, E1 4NS, United
Kingdom22institutetext: Higgs Centre for Theoretical Physics, School of
Physics and Astronomy, The University of Edinburgh, EH9 3FD, Scotland

# NS-NS Spacetimes from Amplitudes

Ricardo Monteiro 1 Silvia Nagy 2 Donal O’Connell 1 David Peinador Veiga 2
Matteo Sergola

###### Abstract

Recent work has shown how on-shell three-point amplitudes in gauge theory and
gravity, representing the coupling to massive particles, correspond in the
classical limit to the curvature spinors of linearised solutions. This
connection, made explicit via the KMOC formalism in split metric signature,
turns the double copy of scattering amplitudes into the double copy of
classical solutions. Here, we extend this framework to the universal massless
sector of supergravity, which is the complete double copy of pure gauge
theory. Our extension relies on a Riemann-Cartan curvature incorporating the
dilaton and the B-field. In this setting, we can determine the most general
double copy arising from the product of distinct gauge theory solutions, say a
dyon and $\sqrt{\text{Kerr}}$. This gives a double-copy interpretation to
gravity solutions of the type Kerr-Taub-NUT-dilaton-axion. We also discuss the
extension to heterotic gravity. Finally, we describe how this formalism for
the classical double copy relates to others in the literature, namely (i) why
it is an on-shell momentum space analogue of the convolutional prescription,
and (ii) why a straightforward prescription in position space is possible for
certain vacuum solutions.

QMUL-PH-21-53

##  1 Introduction

At first glance, scattering amplitudes and classical general relativity seem
to have little in common. Amplitudes are quantum-mechanical objects, most
directly relevant to phenomenology at particle colliders. Nevertheless recent
years have seen an explosion of work studying applications of amplitudes to
gravity, and (conversely) the implications of deep results in gravity for
scattering amplitudes, see, for example the very recent references
Herrmann:2021tct ; Cristofoli:2021vyo ; newzvi ; Bern:2021dqo ;
Herrmann:2021lqe ; DiVecchia:2021bdo ; Bautista:2021wfy ; Bern:2020gjj ;
Moynihan:2020gxj ; Cristofoli:2020uzm ; Parra-Martinez:2020dzs ;
Haddad:2020tvs ; AccettulliHuber:2020oou ; Moynihan:2020ejh ; A:2020lub ;
Sahoo:2020ryf ; delaCruz:2020bbn ; Bonocore:2020xuj ; Mogull:2020sak ;
Emond:2020lwi ; Cheung:2020gbf ; Mougiakakos:2020laz ; Carrasco:2020ywq ;
Kim:2020cvf ; Bjerrum-Bohr:2020syg ; Gonzo:2020xza ; delaCruz:2020cpc ;
Cristofoli:2021jas .

Much of this work has been motivated by the advent of gravitational-wave
astrophysics Chung:2018kqs ; Chung:2019duq ; Bern:2020buy ; Neill:2013wsa ;
Cheung:2018wkq ; Bern:2019crd ; Bern:2019nnu ; Bjerrum-Bohr:2013bxa ; Bjerrum-
Bohr:2014zsa ; Luna:2016due ; Damour:2016gwp ; Goldberger:2016iau ;
Cachazo:2017jef ; Guevara:2017csg ; Damour:2017zjx ; Luna:2017dtq ;
Kosower:2018adc ; Maybee:2019jus ; Aoude:2021oqj ; Laddha:2018rle ;
Laddha:2018vbn ; Bjerrum-Bohr:2018xdl ; Cristofoli:2019neg ; Guevara:2019fsj ;
Kalin:2019rwq ; Kalin:2019inp ; Bini:2020hmy ; Aoude:2020onz ; Cheung:2020gyp
; Cheung:2020sdj ; Kalin:2020fhe ; Haddad:2020que ; Kalin:2020lmz ;
DiVecchia:2020ymx ; Bern:2020uwk ; Huber:2020xny . Like collider
phenomenology, gravitational-wave physics will develop into a precision
science — and precision is a core success of the wider amplitudes programme.
Scattering amplitudes have applications to gravitational waves because they
can be used to compute interaction potentials Chung:2018kqs ; Chung:2019duq ;
Bern:2020buy , or more general effective Lagrangians Neill:2013wsa ;
Cheung:2018wkq ; Bern:2019crd ; Bern:2019nnu , which are relevant to compact
binary coalescences. In scattering systems, amplitudes can be used rather
directly to evaluate observables of interest Kosower:2018adc ;
Bautista:2019tdr ; Maybee:2019jus ; Herrmann:2021tct ; Cristofoli:2021vyo ;
Aoude:2021oqj , and at least some of these observables may then be
analytically continued to the bound regime Herrmann:2021lqe ; Kalin:2019rwq ;
Kalin:2019inp ; Bini:2020hmy .

An additional motivation exists, however, for studying scattering amplitudes
in classical gravity. This motivation is founded on the existence of the
double copy in amplitudes.

The double copy, first discovered in string theory by Kawai, Lewellen and Tye
klt  and more recently invigorated and deepened by Bern, Carrasco and
Johansson Bern:2008qj ; Bern:2010ue , is an algorithm for computing scattering
amplitudes in gravity given amplitudes in Yang-Mills (YM) theory Bern:2008qj
.111The double copy was reviewed comprehensively in reference Bern:2019prr .
From the perspective of the double copy, gravity looks to some extent like the
“square” of YM theory. The double copy then provides an entirely new
perspective on classical gravity, very different to the Einstein equation and
the usual geometric approach to general relativity. Since this new perspective
should make some things clear which were previously opaque, it is very
interesting to study what the double copy really means for classical
spacetimes. Indeed, this has been an important topic in the recent literature,
for instance, Monteiro:2014cda ; Ridgway:2015fdl ; White:2016jzc ;
Goldberger:2017frp ; Adamo:2017nia ; Goldberger:2017vcg ; Carrillo-
Gonzalez:2017iyj ; Bahjat-Abbas:2017htu ; Shen:2018ebu ; Ilderton:2018lsf ;
Berman:2018hwd ; Bahjat-Abbas:2018vgo ; Plefka:2018dpa ; Gurses:2018ckx ;
lumonioc ; Cardoso:2016amd ; Andrzejewski:2019hub ; Plefka:2019hmz ;
CarrilloGonzalez:2019gof ; Sabharwal:2019ngs ; Bah:2019sda ; Borsten:2019prq ;
Borsten:2021hua ; Borsten:2021zir ; Bahjat-Abbas:2020cyb ; Elor:2020nqe ;
Adamo:2020qru ; Alfonsi:2020lub ; Keeler:2020rcv ; Easson:2020esh ;
Easson:2021asd ; Pasarin:2020qoa ; Gumus:2020hbb ; Alawadhi:2021uie ;
Lee:2018gxc ; Berman:2020xvs ; Lescano:2021ooe ; Angus:2021zhy ;
Cheung:2021zvb ; Alkac:2021seh ; Ferrero:2020vww ; Alkac:2021bav ;
Gonzo:2021drq ; Saotome:2012vy ; Borsten:2020zgj ; Campiglia:2021srh ;
Godazgar:2021iae ; Shi:2021qsb ; Moynihan:2021rwh .

In this article, our interest is in a systematic exploration of the double
copy at its simplest: the statement that, up to a constant factor, the three
point amplitude in gravity is the square of the three point amplitude in YM
theory. This statement holds (famously) for the three-point amplitude
involving three gravitons, which is the square of the three-gluon amplitude.
It also holds for the three-point amplitude describing the interaction between
a massive particle and a graviton, which is the square of the three-point
amplitude describing the interaction between a charge and a gluon. In fact, we
may equally well say that the gravitational three-point amplitude is the
square of the amplitude for a photon interacting with a point charge, because
at the level of these three-point amplitudes the non-linearity of the YM field
is irrelevant.

The study of amplitudes applied to classical gravity has now given us a rather
direct link between amplitudes and solutions of the Einstein equation. This
link proceeds by viewing the Riemann curvature tensor far from the source as
an operator in the quantum theory, whose expectation value can be computed
perturbatively from amplitudes. This is particularly simple at linearised
level, where the Riemann tensor is gauge invariant. In the correspondence
regime where quantum effects are negligible, this expectation value must be
the classical value of the curvature. This idea is an application of the
“KMOC” formalism Kosower:2018adc ; Maybee:2019jus , which was originally
developed with gravitational waves in mind. From our perspective, however, the
link allows us to directly determine what the double copy means at the level
of classical solutions.

In previous work Monteiro:2020plf , four of the authors studied the simplest
example of this link. The photon/charge three-point amplitude computes the
Coulomb field, and maps under the double copy to the gravitational massive
three-point amplitude, which computes the (linearised) Riemann curvature of
the Schwarzschild solution. Thus, we see that the double copy induces a map
between Coulomb and Schwarzschild. As phrased, the map applies to linearised
gravity; but since Schwarzschild is a Kerr-Schild spacetime, this map extends
from the linearised solution to the exact solution with appropriate
coordinates. This relation between Coulomb and Schwarzschild is an example of
the “classical double copy”, previously proposed by two of the authors and
White in Monteiro:2014cda . Thus, the KMOC formalism allowed us to show (at
least in this example) that the classical double copy _is_ the double copy.

Yet the classical double copy has been applied to solutions far more general
than Coulomb and Schwarzschild. Indeed, it can be phrased more precisely at
the level of curvature spinors Penrose:1960eq ; Newman:1961qr ; Penrose:1983mf
; Penrose:1985bww : the Weyl spinor in gravity and the Maxwell spinor in
electrodynamics (i.e. linearised YM). In that context, the classical double
copy becomes the statement that the Weyl spinor is a symmetrised square of the
Maxwell spinor, up to an overall scalar factor typed . In this Weyl form, the
classical double copy was applied to all vacuum Petrov type D solutions of the
Einstein equation and has been explored also for vacuum type N solutions
Godazgar:2020zbv . A twistorial interpretation of the Weyl double copy is
being developed in White:2020sfn ; Chacon:2021wbr ; Chacon:2021hfe ;
Chacon:2021lox ; Adamo:2021dfg ; Guevara:2021yud ; Farnsworth:2021wvs .

In this article, our main goal is to connect the Weyl double copy to
scattering amplitudes beyond the example in Monteiro:2020plf . To do so, we
must consider more general amplitudes than the simple Coulomb (charge/photon)
amplitude and its double copy to a Schwarzschild (mass/graviton) amplitude.
Luckily, more general amplitudes are indeed available. In gauge theory, two
non-trivial deformations of the Coulomb amplitude exist. The first of these
introduces a complex phase $e^{i\eta\theta}$ in the amplitude, where $\eta$ is
the helicity of the photon Huang:2019cja . This deformation has the
interpretation of an electric/magnetic duality rotation, allowing us to
introduce magnetic charges. The second deformation is slightly more
complicated. For a photon with momentum $k$, it introduces a factor $e^{-\eta
k\cdot a}$ in the amplitude, where $a$ is a four-vector parameter. Rather
remarkably, this deformation leads to an amplitude describing a particle with
large classical spin $a^{\mu}$ interacting with a photon. It may be derived
kerrpart  by studying the large spin limit of the “minimally coupled”
amplitudes of Arkani-Hamed, Huang and Huang ah3 , and is known to be a form of
the Newman-Janis shift nj .

Indeed, in previous work Emond:2020lwi , these amplitudes, and their
gravitational double copies, were explored in the context of the classical
double copy. The double copies were understood as describing the Kerr-Taub-NUT
family of spacetimes. However, in this work, it was not yet understood how to
compute the curvature directly from amplitudes, and so the evidence in favour
of the double copy was gathered by scattering a particle off the source, and
checking that the results agree when the scattering is computed with
amplitudes or the geodesic equation. Now, we are equipped to compute the
curvature directly.

Furthermore, our greater control over the relation between the amplitudes
double copy and the classical double copy now allows us to explore more
general double copies, where we allow dilatons and axions to propagate in
addition to gravitons. These extra degrees of freedom arise in a very simple
manner. Given amplitudes $\mathcal{A}_{\pm}$ for a photon of helicity $\pm$,
we may construct the double copies $\mathcal{A}_{+}\mathcal{A}_{+}$ and
$\mathcal{A}_{-}\mathcal{A}_{-}$. These amplitudes describe gravitons, since
the helicities add up. We can also construct, if we wish, two more amplitudes:
$\mathcal{A}_{+}\mathcal{A}_{-}$ and $\mathcal{A}_{-}\mathcal{A}_{+}$. These
amplitudes source a complex scalar, interpreted as a scalar dilaton and a
pseudoscalar axion. If we include these scalar amplitudes in our definition of
the gravitational theory, then the full non-linear solutions will involve an
interplay between the purely gravitational and scalar degrees of freedom.

The more general double copy at three-point level in fact connects Yang-Mills
theory to the universal massless sector of supergravity, sometimes known as
NS-NS gravity. It is also known as $\mathcal{N}=0$ supergravity, although of
course this is a purely bosonic theory with no supersymmetry in sight.
Nevertheless, its significant role in string theory makes this system
particularly interesting.

There is an immediate obstacle in the way of a direct application of
amplitudes to solutions of NS-NS gravity. The KMOC formalism connects
curvatures to amplitudes. What kind of curvature should we use for NS-NS
gravity? We find that the right object to use is a generalisation of the
Riemann curvature associated with a torsionful, but metric-compatible,
connection. The spinorial form of this curvature decomposes, at linearised
level, into a fully symmetric (therefore spin two) object recovering the Weyl
spinor, and additional objects describing the field strengths of the scalars.
As we will see, the family of solutions we encounter are of type Kerr-Taub-
NUT-axion-dilaton.

It is further possible to generalise the double copy by considering asymmetric
products of gauge theories: that is, to multiply two different gauge theories
in the construction of the gravitational theory. The simplest example of this
idea is to multiply pure YM theory by a Yang-Mills-scalar theory. At the level
of amplitudes, this means that we may also construct amplitudes of the type
$\mathcal{A}_{\pm}\mathcal{A}_{0}$, where $\mathcal{A}_{0}$ describes the
source interacting with a massless scalar particle. The resulting double copy
theory is Einstein-Yang-Mills-axion-dilaton. These heterotic double copies are
considered here for the first time in the context of the Weyl double copy,
although some solutions in this theory have been studied using a double-field-
theory extension of the Kerr-Schild double copy Cho:2019ype  as well as in the
convolutional approach in supersymmetric theories Cardoso:2016ngt ;
Cardoso:2016amd . Interestingly, the Kerr-Newman solution has also been
associated to scattering amplitudes Moynihan:2019bor , but not as a double
copy. The natural double copy theory is heterotic gravity.

Now, we turn to a crucial issue facing any application of three-point
amplitudes. This issue is that, for real kinematics in Lorentzian signature,
the on-shell constraints on the three particles force the energy of the
massless particle (photon, graviton or scalar) to vanish. This fact forces us
to consider an analytic continuation away from Lorentzian signature, as is
familiar from the BCFW recursion relations bcfw . In our case, we could
consider a complex contour of integration, but as in Monteiro:2020plf  we
choose instead to work in “split” metric signature222For related works on
split signature and its application, see also Srednyak:2013ylj ; Mason:2005qu
; Barrett:1993yn ; Atanasov:2021oyu ; Crawley:2021auj ; Guevara:2021yud .
$(+,+,-,-)$. We may analytically continue from this signature back to the
physically-relevant Lorentzian case. However, it may be worth remarking that
split signature spacetimes have their own significance in general relativity,
playing a role for example in the definition of Plebanski coordinates
Chong:2004 .

Finally, we will also consider the relation of our formalism for the classical
double copy, based on scattering amplitudes in momentum space, to position-
space prescriptions. In particular, we will see that it provides an on-shell
momentum space version of the convolutional prescription of Anastasiou:2014qba
; LopesCardoso:2018xes ; Anastasiou:2018rdx ; Borsten:2019prq ;
Borsten:2020xbt ; Borsten:2021zir . Another natural question is why some
solutions admit an equivalent double copy interpretation that is local in
position space (i.e. not written as a convolution), as in the Kerr-Schild
double copy Monteiro:2014cda  and in the original application of the Weyl
double copy typed . For various examples, we will identify the property of the
solutions that allows for this simplicity.

The structure of our paper is as follows. We will begin with a short overview
which aims to give the reader a bird’s eye view of the main ideas in our
paper. In section 3, we explain the geometric construction of the generalised
curvature relevant when axions and dilatons are included in the double copy,
thereby introducing a key tool which we will use in the remainder of the
article. We move on, in section 4, to review point charges in split signature
and the determination of curvatures using amplitudes in that signature. In
section 5, we describe the family of gauge solutions of interest to us. The
double copy of these solutions is the content of section 6. We work in the
context of NS-NS gravity for generality. In section 7, we study the asymmetric
double copy case of heterotic gravity. Section 8 deals with the Fourier
transform from momentum to position space. We conclude with a discussion of
our work and its implications.

##  2 Overview

The basic idea of our paper is that classical solutions can be summarised by
three-point scattering amplitudes which generate them in the classical limit.
These are amplitudes for the emission by a massive particle (the source) of a
massless boson associated to the classical field. The amplitudes have support
in Lorentzian signature with complex momenta or in split signature with real
momenta. Much of the paper consists of the detailed construction of solutions
from the amplitudes, i.e. making the statements in this section explicit.

Let us start with the amplitudes in gauge theory, using Lorentzian signature
for now:

$\displaystyle\text{Gauge theory:}\quad\mathcal{A}_{\eta}\,e^{\eta\,(-k\cdot
a+i\theta)}\,,\;\eta=\pm 1\,.$ (1)

Here, $\eta$ denotes the helicity of the gauge boson, and $k_{\mu}$ denotes
its momentum. The case $\eta=1$ generates a self-dual field, and $\eta=-1$ an
anti-self-dual field. The correspondence is

$\displaystyle|k\rangle_{A}|k\rangle_{B}\;\mathcal{A}_{+}\,e^{-k\cdot
a+i\theta}\;\;\stackrel{{\scriptstyle\text{oFT}}}{{\longrightarrow}}\;\;\phi_{AB}(x)\,,$
(2) $\displaystyle[k|_{\dot{A}}[k|_{\dot{B}}\;\mathcal{A}_{-}\,e^{k\cdot
a-i\theta}\;\;\stackrel{{\scriptstyle\text{oFT}}}{{\longrightarrow}}\;\;\tilde{\phi}_{\dot{A}\dot{B}}(x)\,,$
(3)

where the null momentum of the gauge boson is
$k_{A\dot{B}}=|k\rangle_{A}[k|_{\dot{B}}$ , and oFT stands for on-shell
Fourier transform (from $k$ to $x$). The Maxwell field strength is, as usual,

$\displaystyle
F_{A\dot{A}B\dot{B}}=\phi_{AB}\,\epsilon_{\dot{A}\dot{B}}+\tilde{\phi}_{\dot{A}\dot{B}}\,\epsilon_{AB}\,.$
(4)

Both the self-dual part and the anti-self-dual part obtained from the
amplitudes are solutions to the equations of motion in their own right, but
only their combination (when $\tilde{\phi}_{\dot{A}\dot{B}}$ is the complex
conjugate of $\phi_{AB}$) yields a real Maxwell field in Lorentzian signature.
Meanwhile, the parameters $a^{\mu}$ and $\theta$ in equation (1) characterise
the massive particle. When both vanish, the contributions from $\eta=1$ and
$\eta=-1$ combine to generate the Coulomb solution, as described in
Monteiro:2020plf . The timelike vector $a^{\mu}$ parametrises the spin, and
means that the classical ‘particle’ is actually an extended object of size
$\,a=\sqrt{|a^{\mu}a_{\mu}|}\,$ sourcing a dyonic gauge field. In the case
$\theta=0$, this gauge field has been dubbed $\sqrt{\text{Kerr}}$, as it is
the ‘single copy’ of Kerr Monteiro:2014cda ; Arkani-Hamed:2019ymq . The dyonic
parameter $\theta$ represents an electric-magnetic duality transformation
dcyutin , where the cases $\,\theta=0\\!\\!\mod\pi\,$ and
$\,\theta=\pi/2\\!\\!\mod\pi\,$ describe pure electric charge and pure
magnetic charge, respectively.

We may generate linearised gravity solutions from the amplitudes obtained via
the double copy, using ‘left’ and ‘right’ copies of (1):

$\displaystyle\text{Gravity:}\quad\mathcal{A}_{\eta_{L}}\,e^{\eta_{L}\,(-k\cdot
a_{L}+i\theta_{L})}\times\mathcal{A}_{\eta_{R}}\,e^{\eta_{R}\,(-k\cdot
a_{R}+i\theta_{R})}\,.$ (5)

As the gravitational analogue of the gauge field strength, we will introduce a
generalised curvature tensor that incorporates all the gravity fields. In a
spinorial decomposition, we have

$\displaystyle\mathfrak{R}_{A\dot{A}B\dot{B}C\dot{C}D\dot{D}}$
$\displaystyle=\mathbf{X}_{ABCD}\,\epsilon_{\dot{A}\dot{B}}\,\epsilon_{\dot{C}\dot{D}}+\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}\,\epsilon_{AB}\,\epsilon_{CD}$
(6) $\displaystyle\hskip
28.45274pt+\mathbf{\Phi}_{AB\dot{C}\dot{D}}\,\epsilon_{\smash{\dot{A}\dot{B}}}\,\epsilon_{CD}+\tilde{\mathbf{\Phi}}_{\dot{A}\dot{B}CD}\,\epsilon_{AB}\,\epsilon_{\smash{\dot{C}\dot{D}}}~{}.$

At linearised level, the first line on the right-hand side becomes the Weyl
tensor, while the second line is built from double-derivatives of a complex
scalar whose real components are the dilaton and the axion. In terms of the
amplitudes (5), the cases $(\eta_{L},\eta_{R})=(1,1)$ and
$(\eta_{L},\eta_{R})=(-1,-1)$ correspond to self-dual and anti-self gravitons,
respectively. The correspondence is

$\displaystyle|k\rangle_{A}|k\rangle_{B}|k\rangle_{C}|k\rangle_{D}\;(\mathcal{A}_{+})^{2}\,e^{-k\cdot(a_{L}+a_{R})+i(\theta_{L}+\theta_{R})}\;\;\stackrel{{\scriptstyle\text{oFT}}}{{\longrightarrow}}\;\;{\mathbf{X}}_{ABCD}(x)\,,$
(7)
$\displaystyle[k|_{\dot{A}}[k|_{\dot{B}}[k|_{\dot{C}}[k|_{\dot{D}}\;(\mathcal{A}_{-})^{2}\,e^{k\cdot(a_{L}+a_{R})-i(\theta_{L}+\theta_{R})}\;\;\stackrel{{\scriptstyle\text{oFT}}}{{\longrightarrow}}\;\;\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}(x)\,.$
(8)

These self-dual and anti-self-dual parts combine to form the linearised Weyl
tensor of the Kerr-Taub-NUT solution with rotation parameter $a_{L}+a_{R}$ and
dyonic parameter $\theta_{L}+\theta_{R}$. But we also have the cases of
$(\eta_{L},\eta_{R})=(1,-1)$ and $(\eta_{L},\eta_{R})=(-1,1)$, corresponding
to

$\displaystyle|k\rangle_{A}|k\rangle_{B}[k|_{\dot{C}}[k|_{\dot{D}}\;\mathcal{A}_{+}\mathcal{A}_{-}\,e^{-k\cdot(a_{L}-a_{R})+i(\theta_{L}-\theta_{R})}\;\;\stackrel{{\scriptstyle\text{oFT}}}{{\longrightarrow}}\;\;{\mathbf{\Phi}}_{AB\dot{C}\dot{D}}(x)\,,$
(9)
$\displaystyle[k|_{\dot{A}}[k|_{\dot{B}}|k\rangle_{C}|k\rangle_{D}\;\mathcal{A}_{+}\mathcal{A}_{-}\,e^{k\cdot(a_{L}-a_{R})-i(\theta_{L}-\theta_{R})}\;\;\stackrel{{\scriptstyle\text{oFT}}}{{\longrightarrow}}\;\;\tilde{\mathbf{\Phi}}_{\dot{A}\dot{B}CD}(x)\,.$
(10)

It is important to realise that including these extra objects is a choice: we
have to decide in advance if we want our source to couple to axions and
dilatons when performing the double copy.

Even if we do include the dilaton and the axion, they do not backreact on the
metric at linearised level.333In the usual language of supergravity, we are
defining these fields in the Einstein frame, where the propagator
unambiguously separates the dilaton and axion from the gravitons. In fact,
notice that each of the four choices of $(\eta_{L},\eta_{R})$ generates a
linearised solution to the $\mathcal{N}=0$ supergravity equations of motion in
its own right, because the equations for the four curvature spinors decouple
at linearised level. We obtain a solution from any linear superposition of the
contributions from the four distinct scattering amplitudes. For instance, a
solution where the only non-vanishing curvature component is
${\mathbf{X}}_{ABCD}$ is a self-dual vacuum metric. Any linearised solution
can be corrected order by order in perturbation theory, leading to a fully
non-linear solution. A particular solution is characterised by a particular
coupling of the massive ‘particle’ to the gravity fields. Clearly, only some
linear combinations of the four $(\eta_{L},\eta_{R})$-solutions have a
generalised curvature tensor in Lorentzian signature that is real, namely
those for which ${\mathbf{X}}_{ABCD}$ and
$\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}$ are complex conjugate, and
${\mathbf{\Phi}}_{AB\dot{C}\dot{D}}$ and
$\tilde{\mathbf{\Phi}}_{\dot{A}\dot{B}CD}$ are also complex conjugate.
Equivalently, we can say that the real solutions are those where
$(\eta_{L},\eta_{R})=(\pm 1,\pm 1)$ contribute equally, so that the metric is
real, and $(\eta_{L},\eta_{R})=(\pm 1,\mp 1)$ contribute equally, so that the
dilaton and the axion are real.

The non-uniqueness of the double copy of classical solutions is obvious in
this framework. In previous work Luna:2016hge ; Kim:2019jwm ; Luna:2020adi ,
it was argued that the most general real spacetime that can be interpreted as
a double copy of the Coulomb solution is the JNW solution Janis:1968zz , which
has two parameters: mass (‘graviton parameter’) and dilaton parameter;
Schwarzschild is the case with vanishing dilaton parameter. In the framework
we present here, the two parameters arise from the linear combination of the
real graviton field, such that $(\eta_{L},\eta_{R})=(\pm 1,\pm 1)$, and the
real dilaton field with vanishing axion, $(\eta_{L},\eta_{R})=(\pm 1,\mp 1)$.

Notice that there is an intricate interplay of parameters in the double copy.
Let us define $\bar{a}=a_{L}+a_{R}$ and $\Delta a=a_{L}-a_{R}$, and likewise
$\bar{\theta}=\theta_{L}+\theta_{R}$ and $\Delta\theta=\theta_{L}-\theta_{R}$.
Of these four parameters appearing in the linearised gravity solution, only
the parameters $\bar{a}$ and $\bar{\theta}$ appear in the graviton components,
whereas only the parameters $\Delta a$ and $\Delta\theta$ appear in the
complex scalar and its conjugate. Due to the fact that
$\mathcal{A}_{+}\mathcal{A}_{-}$ is a constant (i.e. $k$-independent), as we
will review later, the complex scalar is generated by $e^{-k\cdot\Delta
a+i\Delta\theta}$, and its conjugate by $e^{k\cdot\Delta a-i\Delta\theta}$.
Focusing on the duality parameters $\theta$, the effect of $\bar{\theta}$ is
to perform a gravitational Ehlers-type ‘electric-magnetic’ duality
transformation of the metric Huang:2019cja ; Alawadhi:2019urr ;
Banerjee:2019saj , whereas the effect of $\Delta\theta$ is to perform an
axion-dilaton duality transformation, well known from supergravity.

In fact, the double copy (5) is not yet fully general. That is because there
is a (bi-adjoint) scalar field that is often implicit.444We will be working at
linearised level, which is the starting point in the perturbative construction
of the solution. Hence, just like we consider an Abelian gauge field, we
consider a bi-Abelian bi-adjoint scalar. It is useful to recall the form of
the standard KLT relations for scattering amplitudes klt : schematically,

$\mathcal{A}_{\text{grav}}\sim\mathcal{A}_{\text{YM}}\times\mathcal{A}_{\text{scalar}}^{-1}\times\mathcal{A}_{\text{YM}}\,.$

Let us focus on three point scattering. In the original context of KLT, the
three-point amplitudes involve massless states: three scalars, or three
gluons, or three gravity states. Moreover, at three points,
$\mathcal{A}_{\text{scalar}}$ is just a constant that is usually implicit.
However, the relevant three-point amplitude for a linearised classical
solution is that for a massive particle to emit one massless particle. So we
have to characterise this massive particle in gravity, gauge theory and scalar
theory (in fact, only in two of them since the double copy fixes the third
one). This is the origin of further non-uniqueness in the classical double
copy, which we will now illustrate. For simplicity, we will consider rotation
but not a dyonic parameter, and we will restrict ourselves to real Lorentzian
vacuum solutions in gravity. Hence, for gravity, we will be interested in the
Kerr solution. In terms of amplitudes, the solution is generated by combining
the self-dual and anti-self-dual parts,

$\displaystyle(\mathcal{A}_{+})^{2}\,e^{-\,k\cdot
a}\oplus(\mathcal{A}_{-})^{2}\,e^{\,k\cdot a}\;:\;\;\text{Kerr($a$)}\,,$ (11)

where $\text{Kerr($a=0$)}=\text{Schwarzschild}$. We can define the gauge
theory ‘single copy’ as

$\displaystyle\mathcal{A}_{+}\,e^{-k\cdot a}\oplus\mathcal{A}_{-}\,e^{k\cdot
a}\;:\;\;\sqrt{\text{Kerr}}(a)\,,$ (12)

where $\sqrt{\text{Kerr}}(a=0)=\text{Coulomb}$. For the (bi-adjoint) scalar,
it is natural to write

$\displaystyle e^{-k\cdot a}\oplus e^{k\cdot a}\;:\;\;\text{scalar($a$)}\,.$
(13)

The case $\text{scalar($0$)}=1/r$ was described in Monteiro:2020plf : it
corresponds to the scalar sourced by a static point particle. As we will see
in this paper, the case with rotation scalar($a$) corresponds to the ‘zeroth
copy’ appearing as $\phi$ in the Kerr-Schild double copy for the Kerr solution
Monteiro:2014cda . The ‘particle’ source for scalar($a$) is not exactly a
point particle, but has size $a=\sqrt{|a^{\mu}a_{\mu}|}$, which is the precise
counterpart to the sources for Kerr($a$) and for $\sqrt{\text{Kerr}}(a)$.555It
may be surprising that, in the case of scalar($a$), the rotating source leads
to chiral amplitudes $e^{\mp k\cdot a}$ for scalar field emission. These
‘self-dual’ and ‘anti-self-dual’ parts of the scalar are precisely, and
respectively, the complex parts appearing as $S$ and $\bar{S}$ in the Weyl
double copy for the self-dual and anti-self-dual parts of the Weyl spinor
typed . So this picture is consistent with both the Kerr-Schild double copy
and the Weyl double copy. In the choice made here, of taking the ‘particle’
source to be naturally related in all three cases (gravity, gauge theory,
scalar), the three-point KLT-like relation associated to (11) is

$\displaystyle\frac{(\mathcal{A}_{+}\,e^{-k\cdot a})^{2}}{e^{-k\cdot
a}}\oplus\frac{(\mathcal{A}_{-}\,e^{k\cdot a})^{2}}{e^{k\cdot
a}}\;:\;\;\text{Kerr($a$)}\,.$ (14)

The point we want to make about the non-uniqueness arising from the bi-adjoint
scalar is that (14) is not the only possible double-copy interpretation of the
Kerr solution. Another obvious possibility is

$\displaystyle\frac{(\mathcal{A}_{+}\,e^{-\,k\cdot
a/2})^{2}}{1}\oplus\frac{(\mathcal{A}_{-}\,e^{\,k\cdot
a/2})^{2}}{1}\;:\;\;\text{Kerr($a$)}\,.$ (15)

Here, the single copy is

$\displaystyle\mathcal{A}_{+}\,e^{-k\cdot a/2}\oplus\mathcal{A}_{-}\,e^{k\cdot
a/2}\;:\;\;\sqrt{\text{Kerr}}(a/2)\,,$ (16)

and the zeroth copy is

$\displaystyle 1\oplus 1\;:\;\;\text{scalar}(0)=\frac{1}{r}\,,$ (17)

which is sourced by a point particle. So the difference between the two
double-copy interpretations of the Kerr solution seen here, (14) versus (15),
is the definition of the scalar solution, that is, of the precise nature of
the massive ‘particle’ in the three-point amplitude for the scalar. There is
clearly a continuum of choices, and we have neglected above the dyonic
parameter, which introduces another degree of freedom. As in the examples
above, however, this freedom in the scalar just leads to different double-copy
interpretations of the gravity solution, not to a new range of gravity
solutions. In order to ensure the finiteness of this paper, we will not
discuss all the possibilities, but hopefully the examples discussed here are
clear.

Above, we wrote down the Lorentzian three-point amplitudes. However, these are
only supported on-shell for complex momenta. This means that the contour of
integration for the on-shell Fourier transform (oFT) is complex.
Alternatively, the amplitudes have support on real momenta in split signature.
We will choose the latter option to define the oFT, and then analytically
continue the solutions to Lorentzian signature. This option not only makes the
oFT more transparent because the integration contour is real, but also gives
us as a bonus the solutions in split signature, where, as already explored in
Monteiro:2020plf , there are causality features of interest in their own
right. Regarding the amplitudes above, the most important feature of the
analytic continuation to split signature is the dependence on the rotation and
dyonic parameters:

$\displaystyle\text{Lorentzian sign.}\;\;e^{\eta\,(-k\cdot
a+i\theta)}\quad\leftrightarrow\quad\text{Split sign.}\;\;e^{\eta\,(ik\cdot
a+\theta)}\,,$ (18)

where we take $a^{\mu}$ and $\theta$ to be real on both sides. We will discuss
the justification in section 5.

In the remainder of the paper, we will present the details of the construction
of solutions that is summarised in this section, illustrated by a variety of
examples. We will, in addition (but not summarised here for brevity), discuss
the case of heterotic gravity as a double copy, and the connection of our
formalism to previous position-space prescriptions for the classical double
copy.

##  3 Generalised curvature and NS-NS fields

As we have seen, the double copy of Yang-Mills theory is not only pure
Einstein gravity, but more generally can be taken to be NS-NS gravity. Besides
the graviton, this theory includes a scalar field $\phi$, the dilaton, and a
two-form field $B_{\mu\nu}$ known as the B-field or the Kalb-Ramond field. A
complete classical double copy map should include all three fields on its
gravitational side. Examples of such maps have been found using double field
theory, both for certain exact solutions Lee:2018gxc ; Kim:2019jwm ;
Lescano:2021ooe ; Berman:2020xvs ; Angus:2021zhy  and for perturbative
solutions Cho:2021nim ; Diaz-Jaramillo:2021wtl . In all these studies, the map
is written in terms of fields, in contrast to the Weyl double copy, where the
map relates curvatures, which are gauge invariant at the linearised level. In
this section, we will address this challenge by defining a generalised
curvature that packages all the NS-NS fields in geometric degrees of freedom.
We will show later that this generalised curvature tensor is the appropriate
object obtained from a double copy of field strengths.

The standard notion of geometry in general relativity, a (pseudo-)Riemanian
manifold $(M,g)$ endowed with the Levi-Civita connection $\nabla$, can be
generalised by relaxing the requirements on the connection. If we allow the
connection to have torsion, while insisting on metric-compatibility, the
result is called Riemann-Cartan geometry.

Consider a $d$-dimensional manifold $M$ equipped with a metric $g_{\mu\nu}$
and an affine connection $\mathfrak{D}$. In a coordinate basis, the covariant
derivative acts on a vector $V$ as

$\mathfrak{D}_{\nu}V^{\mu}=\partial_{\nu}V^{\mu}+\Gamma^{\mu}{}_{\nu\rho}\,V^{\rho}~{}.$
(19)

In general, the affine symbols $\Gamma^{\mu}{}_{\nu\rho}$ do not have to be
symmetric. Their anti-symmetric part is the torsion tensor666See (189) for our
(anti-)symmetrisation conventions, chosen for later convenience.,
$T^{\mu}{}_{\nu\rho}\equiv\frac{1}{2}(\Gamma^{\mu}{}_{\nu\rho}-\Gamma^{\mu}{}_{\rho\nu})=\frac{1}{2}\,\Gamma^{\mu}{}_{[\nu\rho]}$
. We will take $(M,g,\mathfrak{D})$ to be a Riemann-Cartan manifold by
requiring that the connection is metric-compatible,

$\mathfrak{D}_{\lambda}\,g_{\mu\nu}=0\,.$

This condition constrains the affine symbols to take the form

$\Gamma^{\mu}{}_{\nu\rho}=\Big{\\{}{}^{\,\mu}_{\nu\rho}\Big{\\}}+K^{\mu}{}_{\nu\rho}~{},$
(20)

where the first term denotes the standard Christoffel symbols of the Levi-
Civita connection and the second, a tensor called _contorsion_ , must satisfy
$K_{\mu\nu\rho}=-K_{\rho\nu\mu}$ . It can be written uniquely in terms of the
torsion as

$K^{\mu}{}_{\nu\rho}=\frac{1}{2}\,g^{\mu\lambda}\left(g_{\nu\tau}\,T^{\tau}_{\
\lambda\rho}+g_{\rho\tau}\,T^{\tau}_{\
\lambda\nu}+g_{\lambda\tau}\,T^{\tau}_{\ \nu\rho}\right)~{}.$ (21)

This generalised connection defines a generalised Riemann tensor, which in our
conventions we write as

$\mathfrak{R}_{\mu\nu\rho}{}^{\lambda}=\mathfrak{D}_{\nu}\Gamma^{\lambda}{}_{\mu\rho}-\mathfrak{D}_{\mu}\Gamma^{\lambda}{}_{\nu\rho}+\Gamma^{\lambda}{}_{\nu\tau}\Gamma^{\tau}{}_{\mu\rho}-\Gamma^{\lambda}{}_{\mu\tau}\Gamma^{\tau}{}_{\nu\rho}~{}.$
(22)

It is important to note that this tensor does not have the symmetries of the
usual Riemann tensor. It satisfies
$\mathfrak{R}_{\mu\nu\rho\sigma}=\frac{1}{2}\,\mathfrak{R}_{[\mu\nu]\rho\sigma}=\frac{1}{2}\,\mathfrak{R}_{\mu\nu[\rho\sigma]}$,
but $\mathfrak{R}_{\mu\nu\rho\sigma}\neq\mathfrak{R}_{\rho\sigma\mu\nu}$ due
to the lack of symmetry in the last two indices of the contorsion. Using (20),
it can be shown that

$\mathfrak{R}_{\mu\nu\rho}{}^{\lambda}=R_{\mu\nu\rho}{}^{\lambda}+\nabla_{\nu}K^{\lambda}{}_{\mu\rho}-\nabla_{\mu}K^{\lambda}{}_{\nu\rho}+K^{\lambda}{}_{\nu\tau}K^{\tau}{}_{\mu\rho}-K^{\lambda}{}_{\mu\tau}K^{\tau}{}_{\nu\rho}~{},$
(23)

where $\nabla$ denotes the Levi-Civita connection and
$R_{\mu\nu\sigma}{}^{\lambda}$ its Riemann tensor. In general, $\mathfrak{R}$
will denote curvatures with torsion, whereas $R$ is reserved for the standard
Riemannian curvatures of the metric.

Riemann-Cartan manifolds have extra geometrical degrees of freedom in the
contorsion. These degrees of freedom can be used to accommodate the NS-NS
fields, giving them a geometric status similar to the metric. The dilaton is
assigned to the trace of the contorsion while the B-field is related to its
fully antisymmetric component

$\displaystyle
K^{\mu}{}_{\nu\rho}=\frac{\kappa}{2\sqrt{3}}e^{-\frac{4\kappa\phi}{d-2}}\,H^{\mu}{}_{\nu\rho}-\frac{2\,\kappa}{(d-2)\sqrt{d-1}}\left(\,{\delta^{\mu}}_{\nu}\,\partial_{\rho}\phi\,-\,g_{\nu\rho}\,g^{\mu\sigma}\,\partial_{\sigma}\phi\right)~{},$
(24)

where $H=\mathrm{d}B$ is the curvature of the B-field and $\kappa$ is the
gravitational coupling constant. The contorsion (24) was chosen such that the
Ricci scalar is

$\mathfrak{R}=R-\frac{4\kappa^{2}}{d-2}\nabla_{\mu}\phi\,\nabla^{\mu}\phi-\frac{\kappa^{2}}{12}e^{-\frac{8\kappa\phi}{d-2}}\,H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{4\kappa\,\sqrt{d-1}}{d-2}\nabla^{\mu}\nabla_{\mu}\phi~{},$
(25)

the motivation being that $\sqrt{|g|}\,\mathfrak{R}$ is equivalent to the
usual NS-NS Lagrangian density in the Einstein frame, up to a boundary term:

$\displaystyle S$
$\displaystyle=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{d}x\sqrt{|g|}\left(R-\frac{4\kappa^{2}}{d-2}\nabla_{\mu}\phi\,\nabla^{\mu}\phi-\frac{\kappa^{2}}{12}e^{-\frac{8\kappa\phi}{d-2}}\,H_{\mu\nu\rho}H^{\mu\nu\rho}\right)~{},$
(26) $\displaystyle=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{d}x\sqrt{|g|}\
\mathfrak{R}~{}.$ (27)

Similar constructions have been proposed since the discovery of the NS-NS
action Scherk:1974mc . Although the originally proposed connection was not
metric compatible, it assigned connection degrees of freedom to the dilaton
and B-fields. This motivated a series of works trying to recast higher-order
terms of the bosonic string Lagrangian exclusively in terms of generalised
curvature invariants Nepomechie:1985us ; Gross:1986mw ; Bento:1986hx ;
Bern:1987wz . A similar connection is also used in the context of double field
theory Hull:2009mi . A metric-compatible connection was introduced in
Saa:1993mi , wich together with a non-parallel volume element reproduces the
NS-NS Lagrangian in the string frame. Other generalised connections, also
metric-compatible, have been used to endow Einstein-dilaton gravity with a
geometric interpretation Dunn:1974fac ; Fonseca-Neto:2012cvy . A drawback of
these geometric formulations of NS-NS gravity is that, in order to obtain the
correct equations of motion, one needs to impose constraints on the torsion
Dereli:1995zj . For example, the totally antisymmetric component, which we set
proportional to $H_{\mu\nu\rho}$, is not completely free, since $H$ is exact.
Hence, the geometric interpretation of the massless modes is not entirely
clear Hehl:2007bn .

We will be interested in the curvature at linear order in the fields. Starting
from $g_{\mu\nu}=\eta_{\mu\nu}+\kappa\,h_{\mu\nu}$ , and expanding to
linearised order, we obtain

$\mathfrak{R}_{\mu\nu}{}^{\rho\sigma}=-\frac{\kappa}{2}\partial_{[\mu}\partial^{[\rho}h_{\nu]}{}^{\sigma]}+\,\frac{\kappa}{(d-2)\sqrt{d-1}}\,{\delta_{[\mu}}^{[\rho}\partial_{\nu]}\partial^{\sigma]}\phi+\,\frac{\kappa}{2\sqrt{3}}\partial_{[\mu}\partial^{[\rho}B_{\nu]}{}^{\sigma]}~{}.$
(28)

In $d=4$, the field redefinitions

$\displaystyle\phi\to\frac{\sqrt{3}}{2}\phi~{},\qquad\qquad B\to\sqrt{3}B~{},$
(29)

simplify the factors to reduce the linearised Riemann tensor to

$\mathfrak{R}_{\mu\nu}{}^{\rho\sigma}=-\frac{\kappa}{2}\left(\partial_{[\mu}\partial^{[\rho}h_{\nu]}{}^{\sigma]}-\,{\delta_{[\mu}}^{[\rho}\partial_{\nu]}\partial^{\sigma]}\phi-\partial_{[\mu}\partial^{[\rho}B_{\nu]}{}^{\sigma]}\right)~{}.$
(30)

This expression highlights the fact that the generalised Riemann packages all
the NS-NS fields. At this order, the packaging can be taken one step further
by using the ‘fat graviton’ defined in Luna:2016hge  777Some factors differ
from Luna:2016hge  due to different normalisation conventions.

$\displaystyle\mathfrak{H}_{\mu\nu}=\mathfrak{h}_{\mu\nu}-B_{\mu\nu}-P^{q}_{\mu\nu}\,(2\phi+\mathfrak{h})~{},$
(31)

where $\mathfrak{h}_{\mu\nu}$ is the trace-reversed graviton and
$P^{q}_{\mu\nu}$ is a projector

$\displaystyle\mathfrak{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\,h\,\eta_{\mu\nu}~{},\qquad
P^{q}_{\mu\nu}=\frac{1}{2}\left(\eta_{\mu\nu}-\frac{q_{\mu}\partial_{\nu}+q_{\nu}\partial_{\mu}}{q\cdot\partial}\right)~{}.$
(32)

The constant auxiliary null vector $q^{\mu}$ is related to gauge choices. In
fact, the terms involving $q^{\mu}$ drop out of the gauge-invariant curvature,
which can be written as the compact expression

$\displaystyle\mathfrak{R}_{\mu\nu}{}^{\rho\sigma}=-\frac{\kappa}{2}\,\partial_{[\mu}\partial^{[\rho}\mathfrak{H}_{\nu]}{}^{\sigma]}~{}.$
(33)

In this sense, our generalised curvature is the ‘fat Riemann’ associated to
the ‘fat graviton’.

There is yet another way to rewrite (30). In four dimensions, the two-form
$B_{\mu\nu}$ can be traded for a pseudoscalar axion $\sigma$, defined by

$\displaystyle
H_{\mu\nu\rho}=-e^{2\sqrt{3}\,\phi}\,\epsilon_{\mu\nu\rho\sigma}\partial^{\sigma}\sigma~{}.$
(34)

At linearised order, the exponential in the expression above equals 1, and the
fat Riemann is

$\mathfrak{R}_{\mu\nu}{}^{\rho\sigma}=-\frac{\kappa}{2}\left(\partial_{[\mu}\partial^{[\rho}h_{\nu]}{}^{\sigma]}-\,{\delta_{[\mu}}^{[\rho}\partial_{\nu]}\partial^{\sigma]}\phi+\epsilon^{\rho\sigma\lambda}{}_{[\mu}\partial_{\nu]}\partial_{\lambda}\sigma\right)~{}.$
(35)

Later in the paper, we will see how the different products of gauge theory
amplitudes are associated to the different components of the generalised
curvature. We will work in $d=4$, where it is convenient to use the spinor-
helicity formalism for the amplitudes. The relation between the amplitudes and
the generalised curvature is, therefore, much clearer if we also express the
latter spinorially. As described in appendix D, the generalised Riemann tensor
can be decomposed into spinors as

$\displaystyle\mathfrak{R}_{A\dot{A}B\dot{B}C\dot{C}D\dot{D}}$
$\displaystyle=\mathbf{X}_{ABCD}\,\epsilon_{\dot{A}\dot{B}}\,\epsilon_{\dot{C}\dot{D}}+\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}\,\epsilon_{AB}\,\epsilon_{CD}$
(36) $\displaystyle\hskip
28.45274pt+\mathbf{\Phi}_{AB\dot{C}\dot{D}}\,\epsilon_{\smash{\dot{A}\dot{B}}}\,\epsilon_{CD}+\tilde{\mathbf{\Phi}}_{\dot{A}\dot{B}CD}\,\epsilon_{AB}\,\epsilon_{\smash{\dot{C}\dot{D}}}~{},$

where we use the bold typeface in order to distinguish the spinors from those
of $R$. The spinors $\mathbf{X}_{ABCD}$, $\mathbf{\Phi}_{AB\dot{C}\dot{D}}$
and their duals are symmetric in their first and second pairs of indices.
Recall that, generically,
$\mathfrak{R}_{\mu\nu\rho\sigma}\neq\mathfrak{R}_{\rho\sigma\mu\nu}$ . This
asymmetry implies that $\mathbf{X}_{ABCD}\neq\mathbf{X}_{CDAB}$ and
$\mathbf{\Phi}_{AB\dot{C}\dot{D}}\neq\tilde{\mathbf{\Phi}}_{\dot{C}\dot{D}AB}$
. The spinor $\mathbf{X}_{ABCD}$ is not completely symmetric and can be
reduced further as

$\displaystyle\mathbf{X}_{ABCD}=\mathbf{\Psi}_{ABCD}-\big{(}\mathbf{\Sigma}_{A(C}\,\epsilon_{D)B}+\mathbf{\Sigma}_{B(C}\,\epsilon_{D)A}\big{)}+\mathbf{\Lambda}(\epsilon_{AC}\,\epsilon_{BD}+\epsilon_{AD}\,\epsilon_{BC})~{},$
(37)

where $\mathbf{\Psi}_{ABCD}$ and $\mathbf{\Sigma}_{AB}$ are completely
symmetric. A similar decomposition holds for
$\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}$ . Restricting to
linearised level, we can compare the right-hand side of (36) to the right-hand
side of (30): the first line of the former corresponds to the graviton
contribution, whereas the second line corresponds to contributions from
combinations of the dilaton and the axion (which is the single degree of
freedom of the B-field in $d=4$). We will make this more explicit in a later
section.

##  4 Static Point Charge in Split Signature

In this section, we will review how to extract the classical field
configurations sourced by a static particle using scattering amplitudes in
split signature Monteiro:2020plf . The choice of signature is motivated by the
fact that the three-point amplitude does not vanish for real kinematics in
split signature. Alternatively, the same calculations could be carried out in
Lorentzian signature, provided that momenta are complexified. We will address
this point more directly at the end of the section.

###  4.1 Review

Let us consider a static particle source in split signature. We will use
coordinates $(t_{1},t_{2},x,y)$, with signature $(+,+,-,-)$.888We will also
denote two dimensional space-like vectors in bold, eg. $\mathbf{x}=(x,y)$.
Since we have two time directions, we should specify the worldline of the
particle, which we choose to be the $t_{2}$ axis with tangent vector
$u^{\mu}=(0,1,0,0)$. We will model this massive particle as a non-dynamical
scalar wave packet, following the prescription in Kosower:2018adc . The
expectation value of the momentum of the wave packet should be $\langle
p^{\mu}\rangle=m\,u^{\mu}$. We define the state as

$|\psi\rangle=\int\mathrm{d}\Phi(p)\,\varphi(p)\,|p\rangle,\qquad\mathrm{d}\Phi(p)=\hat{\mathrm{d}}^{4}p\,\hat{\delta}(p^{2}-m^{2})\Theta(E_{2})\,,$
(38)

where the wave function $\varphi(p)$ is sharply-peaked around the classical
momentum $m\,u^{\mu}$. The notation $\hat{\mathrm{d}}$ and $\hat{\delta}$
packages factors of $2\pi$, as explained in appendix A. Notice that the theta
function inside $\mathrm{d}\Phi(p)$ enforces positive energy along $t_{2}$,
the worldline direction of the particle. The existence of the other time
direction $t_{1}$ implies that there is another energy, $E_{1}$.999Note that,
in the KMOC formalism, the momentum carried by messengers is of order $\hbar$,
so in the classical limit our massive particle is indeed static.

Now, we want to obtain the electromagnetic field sourced by the particle. The
existence of two time dimensions implies that we need to specify boundary
conditions for the fields. We choose to impose that the “messenger” fields
(photons and gravitons) must be in a vacuum state for $t_{1}\to-\infty$. This
endows the $(t_{1},x,y)$ codimension-1 space with a sense of causality in
which fields are sourced at $t_{1}=0$ by the instantaneous appearance of the
particle. Consequently, we use the mode expansion for the gauge field operator

$\displaystyle
A^{\mu}(x)=\sum_{\eta=\pm}\int\mathrm{d}\Phi(k)\,\hbar^{-\frac{1}{2}}\left(a_{\eta}(k)\varepsilon_{\eta}^{\mu}(k)e^{-i\frac{k\cdot
x}{\hbar}}+a^{\dagger}_{\eta}(k)\varepsilon_{\eta}^{\mu}(k)e^{i\frac{k\cdot
x}{\hbar}}\right).$ (39)

This time the measure is

$\mathrm{d}\Phi(k)=\hat{\mathrm{d}}^{4}k\,\hat{\delta}(k^{2})\Theta(E^{1})\,.$
(40)

Note that the theta function implies that $E_{1}$ must be positive. We will
assume from now on that $\mathrm{d}\Phi(k)$ carries a $\Theta(E_{1})$ for the
gauge field while $\mathrm{d}\Phi(p)$ carries $\Theta(E_{2})$ for the massive
particle. The associated field strength tensor is

$F^{\mu\nu}(x)=-i\sum_{\eta=\pm}\int\mathrm{d}\Phi(k)\hbar^{-\frac{3}{2}}\left(a_{\eta}(k)k^{[\mu}\varepsilon_{\eta}^{\nu]}e^{-i\frac{k\cdot
x}{\hbar}}-a^{\dagger}_{\eta}(k)k^{[\mu}\varepsilon_{\eta}^{\nu]}e^{i\frac{k\cdot
x}{\hbar}}\right)\,.$ (41)

The powers of $\hbar$ will be absent in the classical limit, so from now we
will set $\hbar=1$. We want to obtain the field sourced by our particle when
it is coupled to the electromagnetic field with a charge $Q$. For $t_{1}<0$,
we impose that there must be no messengers, so the field vanishes,

$\displaystyle\langle\psi|F^{\mu\nu}|\psi\rangle=0\,.$ (42)

For positive $t_{1}$, the state evolves with

$|\psi_{\text{out}}\rangle=\lim_{t^{1}\to\infty}U(-t^{1},t^{1})|\psi\rangle=S|\psi\rangle,$
(43)

and the goal is to compute the expectation value of the field

$\displaystyle\langle
F^{\mu\nu}\rangle\equiv\langle\psi|S^{\dagger}F^{\mu\nu}S|\psi\rangle\,.$ (44)

Similarly, we can obtain the spinorial counterpart of the field strength
tensor

$\displaystyle\phi_{AB}(x)=\sigma^{\mu\nu}{}_{AB}F_{\mu\nu}(x)\,.$ (45)

The $\sigma^{\mu\nu}$ matrices are symmetric on their spinor indices $A$ and
$B$. These matrices project two-forms onto their self-dual parts,101010In our
nomenclature, a two-form $F$ is self-dual if $\star F=F$, and anti-self-dual
if $\star F=-F$. and are proportional to the generators of
SL$(2,\mathbbm{R})$.

It was shown in Monteiro:2020plf  that, under the classical limit, the final
state of the electromagnetic field is coherent:

$\displaystyle
S|\psi\rangle=\frac{1}{\mathcal{N}}\int\mathrm{d}\Phi(p)\varphi(p)\exp\left[\sum_{\eta}\int\mathrm{d}\Phi(k)\,\hat{\delta}(2p\cdot
k)\,i\mathcal{A}_{-\eta}(k)\,a^{\dagger}_{\eta}(k)\right]|p\rangle\,,$ (46)

where $\mathcal{N}$ is a normalisation factor ensuring that
$\langle\psi|S^{\dagger}S|\psi\rangle=1$. For our static electric charge, the
amplitude is the three-point scalar QED vertex,

$\begin{split}&\mathcal{A}_{\eta}(k)=-2Q\,p\cdot\varepsilon_{\eta}({k}).\end{split}$
(47)

The expression (46) drastically simplifies the evaluation of expectation
values. This is because the annihilation operator acts as a derivative on the
state,

$\displaystyle a_{\eta}(k)S|\psi\rangle$ $\displaystyle=\hat{\delta}(2p\cdot
k)\,i\mathcal{A}_{-\eta}(k)\,S|\psi\rangle$ (48)
$\displaystyle=\frac{\delta}{\delta a^{\dagger}_{\eta}(k)}S|\psi\rangle\,,$

and annihilation operators can be replaced by amplitudes. The field strength
is therefore

$\displaystyle\langle\psi|S^{\dagger}\,F^{\mu\nu}(x)\,S|\psi\rangle$
$\displaystyle=-2\operatorname{Re}i\sum_{\eta}\int\mathrm{d}\Phi(k)\,\langle\psi|S^{\dagger}\,a_{\eta}(k)\,S|\psi\rangle\,k^{[\mu}\varepsilon_{\eta}^{\nu]}\,e^{-ik\cdot
x}$ (49)
$\displaystyle=\frac{1}{m}\operatorname{Re}\sum_{\eta}\int\mathrm{d}\Phi(k)\,\hat{\delta}(u\cdot
k)\mathcal{A}_{-\eta}(k)\,k^{[\mu}\varepsilon_{\eta}^{\nu]}\,e^{-ik\cdot
x}\,.$

As shown in Monteiro:2020plf , this can be expressed as a differential
operator acting on a scalar potential $S(x)$,

$\displaystyle\langle F^{\mu\nu}(x)\rangle=Q\,u^{[\mu}\partial^{\nu]}S(x)~{},$
(50)

with

$\displaystyle S(x)$
$\displaystyle=2\operatorname{Re}\int\mathrm{d}\Phi(k)\hat{\delta}(k\cdot
u)e^{-ik\cdot x}~{}.$ (51)
$\displaystyle=i\int\hat{\mathrm{d}}^{4}k\,\hat{\delta}(k_{1}^{2}-\mathbf{k}^{2})\Theta(k_{1})\hat{\delta}(k\cdot
u)\left(e^{-ik\cdot x}-e^{ik\cdot x}\right)~{}.$ (52)

Similarly, the Maxwell spinor is

$\displaystyle\langle\psi|S^{\dagger}\,\phi_{AB}(x)\,S|\psi\rangle$
$\displaystyle=-\frac{\sqrt{2}}{m}\operatorname{Re}\int\mathrm{d}\Phi(k)\,\hat{\delta}(u\cdot
k)\,|k\rangle_{A}|k\rangle_{B}\,e^{-ik\cdot x}\mathcal{A}_{+}(k),$ (53)

while the conjugate one reads

$\displaystyle\langle\psi|S^{\dagger}\tilde{\phi}_{\dot{A}\dot{B}}(x)\,S|\psi\rangle=\frac{\sqrt{2}}{m}\,\text{Re}\int\mathrm{d}\Phi({k})\,\hat{\delta}(u\cdot{k})\,[k|_{\dot{A}}[k|_{\dot{B}}\,e^{-i{k}\cdot
x}\,\mathcal{A}_{-}(k)\,.$ (54)

Thus, the two helicity amplitudes correspond directly to the two different
chiralities of Maxwell spinor.

###  4.2 Split signature vs Lorentzian signature

All the classical fields we have obtained are written as integrals of three-
point amplitudes over on-shell momentum space. Therefore, these integrals have
no support in Lorentzian signature for real kinematics. We have avoided this
problem by using split signature. Alternatively, we could have proceeded in
Lorentzian signature provided that we integrate over complex momenta. To
illustrate these two alternatives, consider the scalar potential introduced in
(51). It can be shown that the scalar potential is related to the retarded and
advanced Green’s functions,

$\displaystyle S(x)=G_{\rm ret}(x)-G_{\rm adv}(x)~{},$ (55)

where

$\displaystyle G_{\rm ret}(x)$
$\displaystyle=-\int\hat{\mathrm{d}}^{4}k\frac{e^{-i\,k\cdot
x}\hat{\delta}(k\cdot
u)}{(k_{1}+i\epsilon)^{2}-\mathbf{k}^{2}}=\frac{\Theta(t_{1})\Theta(t_{1}^{2}-r^{2})}{2\pi\sqrt{t_{1}^{2}-r^{2}}}~{},$
(56) $\displaystyle G_{\rm adv}(x)$
$\displaystyle=-\int\hat{\mathrm{d}}^{4}k\frac{e^{-i\,k\cdot
x}\hat{\delta}(k\cdot
u)}{(k_{1}-i\epsilon)^{2}-\mathbf{k}^{2}}=\frac{\Theta(-t_{1})\Theta(t_{1}^{2}-r^{2})}{2\pi\sqrt{t_{1}^{2}-r^{2}}}~{}.$

The existence of different Green’s functions is linked to the freedom to
choose boundary conditions. Our choice is that the field should vanish for
$t_{1}<0$, which selects the retarded propagator. Hence, under these boundary
conditions, we can write

$\displaystyle S(x)=0\quad\text{for}\;t_{1}<0\,,\quad S(x)=G_{\rm
ret}(x)\quad\text{for}\;t_{1}>0\,.$ (57)

In Lorentzian signature, the time coordinate $t_{1}$ is replaced by another
space coordinate, $z$, dual to $k_{1}$. Now, all three coordinates orthogonal
to $t_{2}$ have the same signature and no $i\epsilon$ prescription is needed.
Consequently, the only Green’s function is

$\displaystyle G(x)=-\int\hat{\mathrm{d}}^{4}k\,\hat{\delta}(k\cdot
u)\,\frac{e^{-ik\cdot x}}{k^{2}}=\frac{1}{4\pi\sqrt{r^{2}+z^{2}}}~{}.$ (58)

Analogously to the split signature case, we can recast this Green’s function
into an integral of the form (51). However, the Lorentzian delta function
$\hat{\delta}(k_{1}^{2}+\mathbf{k}^{2})$ has no roots on the real line of
$k_{1}$. As a result, the integration contour on $k_{1}$ must de deformed in
the complex plane. The appropriate contour is

$\displaystyle S(x)$
$\displaystyle=-\Theta(z)\,\operatorname{Re}\,i\int_{-i\infty}^{0}\\!\hat{\mathrm{d}}k_{1}\int\hat{\mathrm{d}}^{2}\mathbf{k}\,\hat{\delta}(k_{1}^{2}+\mathbf{k}^{2})e^{-ik\cdot
x}$ (59) $\displaystyle\hskip
56.9055pt-\Theta(-z)\,\operatorname{Re}\,i\int_{0}^{i\infty}\\!\hat{\mathrm{d}}k_{1}\int\hat{\mathrm{d}}^{2}\mathbf{k}\,\hat{\delta}(k_{1}^{2}+\mathbf{k}^{2})e^{-ik\cdot
x}~{}.$

The prescription to analytically continue the retarded term in (51) to
Lorentzian signature is summarised in figure 1.

Figure 1: Analytical continuation of the split signature contour to Minkowski
signature.

The splitting or doubling of the contour might seem surprising. Ultimately, it
is a reminder that the correct split signature propagator to analytically
continue to Lorentzian signature is the Feynman propagator,

$\displaystyle G_{\rm F}=-\int\hat{\mathrm{d}}^{4}k\frac{e^{-i\,k\cdot
x}\,\hat{\delta}(k\cdot u)}{k_{1}^{2}-\mathbf{k}^{2}+i\epsilon}~{},$ (60)

which is time-symmetric. This is shown graphically in figure 2, where only the
Feynman propagator contour can be deformed into the Lorentzian contour without
crossing the poles.

Figure 2: Contour of the different Green’s functions. From left to right:
retarded, advanced, Feynman and the last one corresponds to Lorentzian
signature.

In position space, the correct analytic continuation of the scalar potential
is 111111This statement is explained more extensively in section 5 of
Monteiro:2020plf .

$\frac{\Theta(t_{1})\Theta(t_{1}^{2}-r^{2})}{2\pi\sqrt{t_{1}^{2}-r^{2}}}\quad\to\quad\frac{1}{4\pi\sqrt{r^{2}+z^{2}}}~{}.$
(61)

Another difference in Lorentzian and split signature appears in the
electromagnetic duality. In (1,3) signature, we define the self-dual and anti-
self-dual electromagnetic tensors as

$\displaystyle F_{\mu\nu}^{+}$
$\displaystyle=\frac{1}{2}\left(F_{\mu\nu}-iF_{\mu\nu}^{*}\right)\,\,,$ (62)
$\displaystyle F_{\mu\nu}^{-}$
$\displaystyle=\frac{1}{2}\left(F_{\mu\nu}+iF_{\mu\nu}^{*}\right)~{},$

such that $F^{\pm*}_{\mu\nu}=\pm iF^{\pm}_{\mu\nu}$. The electromagnetic
stress-energy tensor can be expressed as

$\displaystyle
T_{\mu}{}^{\nu}=F^{+}_{\mu\rho}\,F^{-\rho\nu}+F^{-}_{\mu\rho}\,F^{+\rho\nu}~{}.$
(63)

Under electromagnetic duality with parameter $\theta$,

$\displaystyle F_{\mu\nu}$
$\displaystyle\to\cos\theta\,F_{\mu\nu}+\sin\theta\,F^{*}_{\mu\nu}~{},$ (64)
$\displaystyle F^{*}_{\mu\nu}$
$\displaystyle\to\cos\theta\,F^{*}_{\mu\nu}-\sin\theta\,F_{\mu\nu}\,\,.$

The self-dual and anti-self-dual tensors pick up a phase, $F^{\pm}_{\mu\nu}\to
e^{\pm i\theta}F^{\pm}_{\mu\nu}$, implying that the stress-energy tensor is
preserved.

In split signature however, the self-dual and anti-self-dual field strength
tensors are

$\displaystyle F_{\mu\nu}^{+}$
$\displaystyle=\frac{1}{2}\left(F_{\mu\nu}+F_{\mu\nu}^{*}\right)~{},$ (65)
$\displaystyle F_{\mu\nu}^{-}$
$\displaystyle=\frac{1}{2}\left(F_{\mu\nu}-F_{\mu\nu}^{*}\right)~{},$

such that $F^{\pm}_{\mu\nu}{}^{*}=\pm F^{\pm}_{\mu\nu}$. The stress-energy
tensor is still (63). In this occasion, to keep it invariant we need to have

$\displaystyle F_{\mu\nu}$
$\displaystyle\to\cosh\theta\,F_{\mu\nu}+\sinh\theta\,F^{*}_{\mu\nu}~{},$ (66)
$\displaystyle F^{*}_{\mu\nu}$
$\displaystyle\to\cosh\theta\,F^{*}_{\mu\nu}+\sinh\theta\,F_{\mu\nu}~{},$

such that $F^{\pm}_{\mu\nu}\to e^{\pm\theta}F^{\pm}_{\mu\nu}$. This difference
in duality transformations (64) and (66) can be interpreted as
$\theta\to-i\theta$ under analytic continuation.

##  5 Gauge Fields from Amplitudes

In section 4.1, we only considered the most basic amplitude in QED for a
static point particle. Now, we will generalise this amplitude to allow for
magnetic charge and classical spin. The magnetic charge will be achieved by an
electromagnetic duality rotation, which transforms the amplitudes as
Huang:2019cja ; Emond:2020lwi

$\mathcal{A}_{\eta}(k)\to\mathcal{A}_{\eta}(k)\,e^{\theta\,\eta}~{},$ (67)

Notice that the rotation parameter has been continued from Lorentzian space
$\theta\to-i\theta$, as previously motivated. Angular momentum can be induced
by a Newman-Janis shift Arkani-Hamed:2019ymq ; Guevara:2020xjx . It acts on
the amplitudes as

$\mathcal{A}_{\eta}(k)\to\mathcal{A}_{\eta}(k)\,e^{i\,\eta\,k\cdot a}~{}.$
(68)

The vector $a^{\mu}$ is related to the classical angular momentum. It will be
taken to lie along the Wick rotated coordinate: $a^{\mu}=(a,0,0,0)$.
Consequently, the Lorentzian exponent $-\eta\,k\cdot a$ has been analytically
continued to split signature as $i\,\eta\,k\cdot a$. Neither of these
transformations obstructs the exponentiation leading to the coherent state
reviewed in section 4.1. The same arguments hold, just changing the amplitudes
as in (67) and (68).

The next subsections are devoted to the effects of these deformations on the
field strength tensor and spinors. Later on, these transformations will be
carried over to gravity via the double copy. The duality angle will be
associated to the NUT charge whereas $a$ will be the Kerr angular momentum
parameter. The effect of the transformations is summarised in table 1.

Transformation | Gauge theory | Pure gravity
---|---|---
None | Coulomb | Schwarzschild
EM rotation | dyon | Taub-NUT
Newman-Janis shift | $\sqrt{\text{Kerr}}$ | Kerr
EM rotation $+$ NJ shift | spinning dyon | Kerr-Taub-NUT
Table 1: Effect of the transformations (67) and (68).

###  5.1 General amplitude

Both transformations can be applied simultaneously to the same amplitude,

$\mathcal{A}_{\eta}\to\mathcal{A}_{\eta}e^{\,\eta\,(ik\cdot a+\theta)}~{}.$
(69)

Performing this replacement in (53) yields the transformed Maxwell spinor

$\displaystyle\langle\phi_{AB}(x)\rangle$
$\displaystyle=-\operatorname{Re}\int\mathrm{d}\Phi(k)\,\hat{\delta}(2p\cdot
k)\,2\sqrt{2}\,|k\rangle_{A}|k\rangle_{B}\,\mathcal{A}_{+}(k)\,e^{ik\cdot
a+\theta}\,e^{-i{k}\cdot x}\,\,.$ (70)

It is immediately clear that

$\displaystyle\langle\phi_{AB}(x)\rangle=e^{\theta}\langle\phi_{AB}^{\text{Coul.}}(x-a)\rangle\,,$
(71)

where $\langle\phi_{AB}^{\text{Coul.}}(x)\rangle$ is the Maxwell spinor of the
Coulomb solution. A similar expression can be obtained for the conjugate
spinor,

$\displaystyle\langle\tilde{\phi}_{\dot{A}\dot{B}}(x)\rangle=e^{-\theta}\langle\tilde{\phi}_{\dot{A}\dot{B}}^{\text{Coul.}}(x+a)\rangle$
(72)

The interpretation of this transformation in terms of the Newman-Janis shift
and of electromagnetic duality is manifest. In the following sections, we will
explore these transformations in the tensorial formalism. The advantage of the
spinorial formalism will be obvious.

####  5.1.1 Scalar potential

As a preliminary step, we will study the combined effect of the
transformations on the scalar potential

$\displaystyle S_{a,\theta}(x)$
$\displaystyle:=2\operatorname{Re}i\int\mathrm{d}\Phi(k)\hat{\delta}(k\cdot
u)\,e^{-i\,k\cdot(x-a)}e^{\theta}$ (73)
$\displaystyle=\frac{e^{\theta}}{2\pi}\,\frac{\Theta((t_{1}-a)^{2}-r^{2})}{\sqrt{(t_{1}-a)^{2}-r^{2}}}=e^{\theta}S_{0,0}(x-a)$

where $r$ is defined as the $2d$ radius $\sqrt{x^{2}+y^{2}}$. The first line
implies that the effect of the spin vector $a^{\mu}$ is merely a shift in the
spacetime coordinates $x^{\mu}$ along $t_{1}$. The last line was obtained by
introducing an $i\epsilon$ prescription for convergence, details for which can
be found in appendix B of Monteiro:2020plf .

####  5.1.2 Field strength tensor

The classical field strength tensor can be obtained as the expectation value

$\displaystyle\langle F^{\mu\nu}(x)\rangle$
$\displaystyle\equiv\langle\psi|S^{\dagger}\,F^{\mu\nu}(x)\,S|\psi\rangle$
(74)
$\displaystyle=-2Q\operatorname{Re}\sum_{\eta}\int\mathrm{d}\Phi(k)\,\hat{\delta}(k\cdot
u)\,e^{-ik\cdot(x+\eta\,a)}\,k^{[\mu}\varepsilon_{\eta}^{\nu]}\,e^{-\theta\,\eta}\,\varepsilon_{-\eta}\cdot
u\,.$

In the second line, we have substituted the amplitude into (49). The integrand
can be expanded as

$\langle
F^{\mu\nu}(x)\rangle=-2Q\operatorname{Re}\int\mathrm{d}\Phi(k)\,\hat{\delta}(k\cdot
u)\,e^{-ik\cdot x}\,\\\
\times\left(k^{[\mu}\varepsilon_{+}^{\nu]}\,e^{-\theta-i\,k\cdot
a}\,\varepsilon_{-}\cdot u+k^{[\mu}\varepsilon_{-}^{\nu]}\,e^{\theta+i\,k\cdot
a}\,\varepsilon_{+}\cdot u\right)\,.$ (75)

Using the null tetrad

$\displaystyle\eta^{\mu\nu}=k^{(\mu}n^{\nu)}-\epsilon_{+}^{(\mu}\epsilon_{-}^{\nu)}~{},\qquad
k\cdot n=1~{},$ (76)

the tangent vector can be decomposed into

$\displaystyle u^{\mu}=(u\cdot n)\,k^{\mu}-\varepsilon_{-}\cdot
u\,\varepsilon_{+}^{\mu}-\varepsilon_{+}\cdot u\,\varepsilon_{-}^{\mu}\,\,,$
(77)

which implies that the above expression can be rearranged as

$\displaystyle\langle F^{\mu\nu}(x)\rangle$
$\displaystyle=2Q\operatorname{Re}\int\mathrm{d}\Phi(k)\,\hat{\delta}(k\cdot
u)\,e^{-ik\cdot x}$ (78) $\displaystyle\times\left(\cos(k\cdot
a-i\theta)\,k^{[\mu}u^{\nu]}+{i\,\sin(k\cdot
a-i\theta)}\left(k^{[\mu}\varepsilon_{+}^{\nu]}\,\varepsilon_{-}\cdot
u-k^{[\mu}\varepsilon_{-}^{\nu]}\,\varepsilon_{+}\cdot u\right)\right)\,.$

This can be further simplified to

$\displaystyle\langle F^{\mu\nu}(x)\rangle$
$\displaystyle=2Q\operatorname{Re}\int\mathrm{d}\Phi(k)\,\hat{\delta}(k\cdot
u)\,e^{-ik\cdot x}$ (79) $\displaystyle\times\left(\cos(k\cdot
a-i\theta)\,k^{[\mu}u^{\nu]}-\frac{i\,\sin(k\cdot
a-i\theta)}{2}\epsilon^{\mu\nu\rho\sigma}k_{[\rho}u_{\sigma]}\right),\,$

using the result of the equation (194). Expanding the sine and cosine and
making use of the previously defined $S_{a,\theta}(x)$, we obtain

$\displaystyle\langle F^{\mu\nu}(x)\rangle$
$\displaystyle=Q\,\partial^{[\mu}u^{\nu]}\left[\frac{S_{a,\theta}(x)+S_{-a,-\theta}(x)}{2}\right]-\frac{Q}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{[\rho}u_{\sigma]}\left[\frac{S_{a,\theta}(x)-S_{-a,-\theta}(x)}{2}\right].$
(80)

In this expression, the derivatives act on the terms in brackets, since
$u^{\mu}$ has constant components. We have obtained the fields for generic $a$
and $\theta$. In the next subsections, we will focus on each transformation
individually and interpret their effects.

###  5.2 Newman-Janis shift

A pure Newman-Janis shift has $\theta=0$,

$\displaystyle S_{a,0}(x)$
$\displaystyle=\frac{1}{2\pi}\,\frac{\Theta\left((t_{1}-a)^{2}-r^{2}\right)}{\sqrt{(t_{1}-a)^{2}-r^{2}}}~{}.$
(81)

It is worth remarking on some features of this field. First of all, it encodes
the complex zeroth copy of Kerr. This can be checked by rotating back to (1,3)
signature as $t_{1}\to i\,z$

$\displaystyle(t_{1}-a)^{2}-r^{2}\to(i\,z-a)^{2}-r^{2}=-(z+ia)^{2}-r^{2}=-\left(\tilde{R}+i\,a\,\cos\vartheta\right)^{2}~{}.$
(82)

The “Kerr radius” $\tilde{R}$ and polar angle $\vartheta$ are implicitly
defined by

$\frac{r^{2}}{\tilde{R}^{2}+a^{2}}+\frac{z^{2}}{\tilde{R}^{2}}=1~{},\qquad\cos\vartheta=\frac{z}{\tilde{R}}~{}.$
(83)

Thus,

$\frac{1}{\sqrt{(t_{1}-a)^{2}-r^{2}}}\quad\to\quad\frac{i}{\tilde{R}+i\,a\,\cos\vartheta}\,$
(84)

which is proportional to the scalar (3.32) in Luna:2018dpt . The fact that we
recover the complex scalar supports the idea of associating the double copy
for amplitudes in (2,2) signature to the Weyl double copy. Secondly, notice
that the spin $a$ appears only in the combination $(t_{1}-a)$ in equation
(81). This is the Newman-Janis shift at work: in (2,2) signature, the shift is
a _real_ translation, in the $t_{1}$ direction, at the level of the field
strength. (The shift acts in a more subtle way on the potential. At the level
of the effective action, the shift can be interpreted as replacing the usual
worldline action with a worldsheet structure Guevara:2020xjx .)

Setting $\theta=0$ in the field strength tensor yields

$\displaystyle\langle F^{\mu\nu}(x)\rangle$
$\displaystyle=Q\,\partial^{[\mu}u^{\nu]}\left[\frac{S_{a,0}(x)+S_{-a,0}(x)}{2}\right]-\frac{Q}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{[\rho}u_{\sigma]}\left[\frac{S_{a,0}(x)-S_{-a,0}(x)}{2}\right].$
(85)

Note that in Minkowski signature the first bracket corresponds to the real
part of $S$ while the second corresponds to the imaginary part. In this
formulation, the interpretation of the sign of the Newman-Janis translation
looks somewhat complicated, compared to the Maxwell spinors seen above.

The field (85) is the split signature equivalent of the $\sqrt{\text{Kerr}}$
solution Monteiro:2014cda ; kerrpart . Instead of checking this claim by
direct comparison, which would be tedious due to coordinate transformations,
we can derive (79) with $\theta=0$ in a purely classical way. To do so, we
solve the Maxwell equations in the presence of a $\sqrt{\text{Kerr}}$ source,

$\begin{split}\partial_{\mu}F^{\mu\nu}(x)=Q\int\mathrm{d}\tau\,{\exp(a*\partial)^{\nu}}_{\rho}u^{\rho}\delta^{(4)}(x-u\tau)\equiv
j^{\nu}_{\sqrt{\text{Kerr}}}(x),\,\,\,\,\,(a*b)_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}a^{\rho}b^{\sigma}.\end{split}$
(86)

Note that the source for $\sqrt{\text{Kerr}}$ is formally the one for Coulomb
(see for instance justin2017  or Guevara:2020xjx ) but acted upon by the
differential operator ${\exp(a*\partial)^{\nu}}_{\rho}$. We observe that the
exponential $\exp(a*\partial)$ remains invariant under analytic continuation
to split signature. This is because the derivative picks up a factor of $-i$,
which is cancelled out by the $i$ picked up by the volume form.

We solve (86) by Fourier transform with the boundary conditions outlined in
4.1. We get

$\begin{split}A^{\mu}(x)&=2Q\,\text{Re}\,i\\!\int{\mathrm{d}}\Phi(k)\,\hat{\delta}(k\cdot
u)e^{-ik\cdot x}{\exp(-ia*k)^{\mu}}_{\rho}u^{\rho}.\end{split}$ (87)

The action of the exponential matrix can be further simplified. In fact, on
the support of the on-shell measure, it can be shown that

${\exp(-ia*k)^{\mu}}_{\rho}u^{\rho}=u^{\mu}\cos a\cdot
k-i\epsilon^{\mu}(a,k,u)\frac{\sin a\cdot k}{a\cdot k},$ (88)

where we defined
$\epsilon^{\mu}(a,b,c):=\epsilon^{\mu\alpha\beta\gamma}a_{\alpha}b_{\beta}c_{\gamma}$.
We obtain finally

$A^{\mu}(x)=2Q\,\text{Re}\,i\\!\int{\mathrm{d}}\Phi(k)\,\hat{\delta}(k\cdot
u)e^{-ik\cdot x}\left(u^{\mu}\cos a\cdot k-i\epsilon^{\mu}(a,k,u)\frac{\sin
a\cdot k}{a\cdot k}\right).$ (89)

The Maxwell tensor is then easily computed,

$F^{\mu\nu}(x)=2Q\,\text{Re}\,\\!\int{\mathrm{d}}\Phi(k)\,\hat{\delta}(k\cdot
u)e^{-ik\cdot x}\left(k^{[\mu}u^{\nu]}\cos a\cdot
k-\frac{i}{2}\epsilon^{\mu\nu\rho\sigma}k_{[\rho}u_{\sigma]}\sin a\cdot
k\right),$ (90)

which is equal to the $F_{\mu\nu}$ we had in the purely spinning case with
$\theta=0$.

Furthermore, starting from (90) we can also confirm the expressions (71) and
(72) in the $\theta=0$ case. Projecting on a spinor basis, these are found to
be

$\begin{split}\phi^{\sqrt{\text{Kerr}}}_{AB}(x)&={{\sigma}^{\mu\nu}}_{AB}F_{\mu\nu}(x)\\\
&=2\sqrt{2}Q\,\text{Re}\,\\!\int{\mathrm{d}}\Phi(k)\hat{\delta}(k\cdot
u)e^{-ik\cdot(x-a)}\varepsilon_{+}\cdot u\,|k\rangle_{A}|k\rangle_{B}\\\
&=\phi^{\text{Coul.}}_{AB}(x-a),\end{split}$ (91)

we report the negative-helicity spinor too

$\begin{split}\tilde{\phi}^{\sqrt{\text{Kerr}}}_{\dot{A}\dot{B}}(x)&={\tilde{\sigma}^{\mu\nu}}_{\,\,\,\,\,\,\dot{A}\dot{B}}F_{\mu\nu}(x)\\\
&=-2\sqrt{2}Q\,\text{Re}\,\\!\int{\mathrm{d}}\Phi(k)\hat{\delta}(k\cdot
u)e^{-ik\cdot(x+a)}\varepsilon_{-}\cdot u\,[k|_{\dot{A}}[k|_{\dot{B}}\\\
&=\tilde{\phi}^{{\text{Coul.}}}_{\dot{A}\dot{B}}(x+a),\end{split}$ (92)

matching the expressions first obtained in Guevara:2020xjx .

Notice again that the action of the Newman-Janis translation on the Maxwell
spinors is beautifully simple: $\phi^{\sqrt{\text{Kerr}}}_{AB}$ is a
translation of $\phi^{\text{Coul.}}_{AB}$ in one direction, while
$\tilde{\phi}^{\sqrt{\text{Kerr}}}_{\dot{A}\dot{B}}$ is a translation of
$\tilde{\phi}^{{\text{Coul.}}}_{\dot{A}\dot{B}}$ in the opposite direction.
This is in contrast to the more complicated structure at the level of the
field strength (85). We see that the notion of chirality is intimately related
to the structure of the Newman-Janis shift.

###  5.3 Duality rotation

To investigate the effect of the EM rotation on the fields, we take $a$ to
zero,

$\displaystyle\langle F^{\mu\nu}(x)\rangle$
$\displaystyle=Q\,\partial^{[\mu}u^{\nu]}\left[\frac{S_{0,\theta}(x)+S_{0,-\theta}(x)}{2}\right]-\frac{Q}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{[\rho}u_{\sigma]}\left[\frac{S_{0,\theta}(x)-S_{0,-\theta}(x)}{2}\right]\,.$
(93)

Substituting the value of the scalar integrals,

$\displaystyle\langle F^{\mu\nu}(x)\rangle$
$\displaystyle=\cosh\theta\,Q\,\partial^{[\mu}u^{\nu]}\left[\frac{\Theta(\rho^{2})}{2\pi\,\rho}\right]-\sinh\theta\,\frac{Q}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{[\rho}u_{\sigma]}\left[\frac{\Theta(\rho^{2})}{2\pi\rho}\right]~{},$
(94)

where $\rho^{2}\equiv x^{2}-(x\cdot u)^{2}$. In the region $\rho^{2}>0$, we
can Wick rotate back to Lorentzian signature. The duality angle transforms as
$\theta\to i\theta$ and so $\cosh\theta\to\cos\theta$ and
$\sinh\theta\to-i\sin\theta$. This factor of $i$ is absorbed by the
continuation of the volume form to Lorentzian signature, yielding

$\langle
F^{\mu\nu}\rangle=\cos\theta\,F^{\mu\nu}_{\text{Coul.}}+\sin\theta\,F^{*\mu\nu}_{\text{Coul.}}~{},$
(95)

where the star denotes Hodge conjugation.121212The Hodge dual has been defined
as $F^{*}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}~{}.$
We conclude that the transformation (67) generates a dyon by a duality
rotation of the field.

##  6 NS-NS fields from Amplitudes

Motivated by the double copy of amplitudes, we will consider the map

$\displaystyle\mathcal{M}_{\eta_{L}\eta_{R}}=-\frac{\kappa}{4\,Q^{2}}\,c_{\eta_{L}\eta_{R}}\,\mathcal{A}^{(L)}_{\eta_{L}}\,\mathcal{A}^{(R)}_{\eta_{R}}~{},$
(96)

where there are four choices for $(\eta_{L},\eta_{R})$:

$\displaystyle(+,+)\,,\quad(-,-)\,,\quad(+,-)\,,\quad(-,+)\,.$ (97)

These correspond, respectively, to the gravity field being: positive-helicity
graviton, negative-helicity graviton, complex scalar (dilaton and axion), and
conjugate complex scalar. In general, we allow for four distinct couplings
$c_{\eta_{L}\eta_{R}}$ of our massive particle to these gravity fields. Any
choice of these couplings will lead to a linearised gravity solution. In
practice, we will be most interested in the case where the particle couples
equally to the two chiralities, in which case we take $c_{++}=c_{--}$ and
$c_{+-}=c_{-+}$.

Consider the following mode expansion of the fat Riemann operator

${\mathfrak{R}}^{\mu\nu\rho\sigma}=\kappa\,\operatorname{Re}\int\mathrm{d}\Phi(k)\bigg{[}\sum_{\eta_{L}\eta_{R}}\,a_{\eta_{L}\eta_{R}}\,\varepsilon_{\eta_{L}}^{[\mu}(k)k^{\nu]}\varepsilon_{\eta_{R}}^{[\rho}(k)k^{\sigma]}\bigg{]}e^{-ik\cdot
x}~{}.$ (98)

The operator version of the linearised spinor coefficients are computed by
contracting with the sigma matrices Penrose:1983mf

$\displaystyle\mathbf{X}_{ABCD}$
$\displaystyle={\sigma^{\mu\nu}}_{AB}{\sigma^{\rho\sigma}}_{CD}\,\mathfrak{R}_{\mu\nu\rho\sigma}~{},\qquad\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}={\tilde{\sigma}^{\mu\nu}}_{\,\,\,\,\,\dot{A}\dot{B}}{\tilde{\sigma}^{\rho\sigma}}_{\,\,\,\,\,\dot{C}\dot{D}}\,\mathfrak{R}_{\mu\nu\rho\sigma}~{},$
(99) $\displaystyle\mathbf{\Phi}_{AB\dot{C}\dot{D}}$
$\displaystyle={\sigma^{\mu\nu}}_{AB}{\tilde{\sigma}^{\rho\sigma}}_{\,\,\,\,\,\dot{C}\dot{D}}\,\mathfrak{R}_{\mu\nu\rho\sigma}~{},\qquad\mathbf{\tilde{\Phi}}_{\dot{A}\dot{B}CD}={\tilde{\sigma}^{\mu\nu}}_{\,\,\,\,\,\dot{A}\dot{B}}{{\sigma}^{\rho\sigma}}_{CD}\,\mathfrak{R}_{\mu\nu\rho\sigma}~{}.$
(100)

These contractions are easily computed applying

$\displaystyle{\sigma_{\mu\nu}}^{AB}\,k^{[\mu}\varepsilon_{-}^{\nu]}$
$\displaystyle=-\sqrt{2}\,\langle k|^{A}\langle k|^{B}\,,$ (101)
$\displaystyle{\tilde{\sigma}_{\mu\nu}}^{\,\,\,\,\,\,\dot{A}\dot{B}}\,k^{[\mu}\varepsilon_{-}^{\nu]}$
$\displaystyle=0\,,$
$\displaystyle{\sigma_{\mu\nu}}^{AB}\,k^{[\mu}\varepsilon_{+}^{\nu]}$
$\displaystyle=0\,,$ (102)
$\displaystyle{\tilde{\sigma}_{\mu\nu}}^{\,\,\,\,\,\,\dot{A}\dot{B}}\,k^{[\mu}\varepsilon_{+}^{\nu]}$
$\displaystyle=\sqrt{2}\,|k]^{\dot{A}}|k]^{\dot{B}}\,.$

The resulting spinors are

$\displaystyle{\mathbf{X}}_{ABCD}$
$\displaystyle=\kappa\,\operatorname{Re}\,2\int\mathrm{d}\Phi(k)\;a_{--}|k\rangle_{A}|k\rangle_{B}|k\rangle_{C}|k\rangle_{D}\,e^{-ik\cdot
x}~{},$ (103) $\displaystyle\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}$
$\displaystyle=\kappa\,\operatorname{Re}\,2\int\mathrm{d}\Phi(k)\;a_{++}[k|_{\dot{A}}[k|_{\dot{B}}[k|_{\dot{C}}[k|_{\dot{D}}\,e^{-ik\cdot
x}~{},$ (104) $\displaystyle{\mathbf{\Phi}}_{AB\dot{C}\dot{D}}$
$\displaystyle=-\kappa\,\operatorname{Re}2\int\mathrm{d}\Phi(k)\;a_{-+}\,|k\rangle_{A}|k\rangle_{B}[k|_{\dot{C}}[k|_{\dot{D}}\,e^{-ik\cdot
x}~{},$ (105) $\displaystyle{\tilde{\mathbf{\Phi}}}_{\dot{A}\dot{B}CD}$
$\displaystyle=-\kappa\,\operatorname{Re}2\int\mathrm{d}\Phi(k)\;a_{+-}\,[k|_{\dot{A}}[k|_{\dot{B}}|k\rangle_{C}|k\rangle_{D}\,e^{-ik\cdot
x}~{}.$ (106)

In order to link these objects to the amplitudes (96), we would like to use
the equivalent of (48) for the NS-NS fields. Appendix C shows that the
gravitational final state is also coherent, so

$\displaystyle S|\psi\rangle$
$\displaystyle=\frac{1}{\mathcal{N}}\int\mathrm{d}\Phi(p)\varphi(p)\exp\bigg{[}\int\mathrm{d}\Phi(k)\,i\,\hat{\delta}(2p\cdot
k)\,$ (107) $\displaystyle\hskip
40.0pt\times\left(\sum_{\eta_{L}\eta_{R}}\mathcal{M}_{-\eta_{L},-\eta_{R}}(k)\,a^{\dagger}_{\eta_{L}\eta_{R}}(k)\right)\bigg{]}|p\rangle\,,$

which is analogous to (46). Hence, we conclude that

$\displaystyle a_{\eta_{L}\eta_{R}}(k)S|\psi\rangle$
$\displaystyle=\hat{\delta}(2p\cdot
k)\,i\mathcal{M}_{-\eta_{L},-\eta_{R}}(k)\,S|\psi\rangle$ (108)
$\displaystyle=\frac{\delta}{\delta
a^{\dagger}_{\eta_{L}\eta_{R}}(k)}S|\psi\rangle\,.$

Equation (108) implies that we can easily exchange annihilation operators for
amplitudes inside expectation values, so that we find

$\langle{\mathfrak{R}}^{\mu\nu\rho\sigma}\rangle=\kappa\,\operatorname{Re}\,i\int\mathrm{d}\Phi(k)\hat{\delta}(2k\cdot
p)\bigg{[}\sum_{\eta}\,\mathcal{M}_{-\eta_{L},-\eta_{R}}\,\varepsilon_{\eta_{L}}^{[\mu}(k)k^{\nu]}\varepsilon_{\eta_{R}}^{[\rho}(k)k^{\sigma]}\bigg{]}e^{-ik\cdot
x}~{}.$ (109)

The same can be done in the spinor coefficients. The application of the map
(96) results in

$\displaystyle\langle{\mathbf{X}}_{ABCD}\rangle$
$\displaystyle=-\frac{\kappa^{2}c_{++}}{2Q^{2}}\,\operatorname{Re}\,i\int\mathrm{d}\Phi(k)\hat{\delta}(2p\cdot
k)\,\mathcal{A}_{+}^{(L)}\mathcal{A}_{+}^{(R)}\,|k\rangle_{A}|k\rangle_{B}|k\rangle_{C}|k\rangle_{D}\,e^{-ik\cdot
x}~{},$ (110)
$\displaystyle\langle\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}\rangle$
$\displaystyle=-\frac{\kappa^{2}c_{--}}{2Q^{2}}\,\operatorname{Re}\,i\,\int\mathrm{d}\Phi(k)\hat{\delta}(2p\cdot
k)\,\mathcal{A}_{-}^{(L)}\mathcal{A}_{-}^{(R)}\,[k|_{\dot{A}}[k|_{\dot{B}}[k|_{\dot{C}}[k|_{\dot{D}}\,e^{-ik\cdot
x}~{},$ (111) $\displaystyle\langle{\mathbf{\Phi}}_{AB\dot{C}\dot{D}}\rangle$
$\displaystyle=+\frac{\kappa^{2}c_{+-}}{2\,Q^{2}}\,\operatorname{Re}i\int\mathrm{d}\Phi(k)\hat{\delta}(2p\cdot
k)\,\mathcal{A}_{+}^{(L)}\mathcal{A}_{-}^{(R)}\,\,|k\rangle_{A}|k\rangle_{B}[k|_{\dot{C}}[k|_{\dot{D}}\,e^{-ik\cdot
x}~{},$ (112)
$\displaystyle\langle{\tilde{\mathbf{\Phi}}}_{\dot{A}\dot{B}CD}\rangle$
$\displaystyle=+\frac{\kappa^{2}c_{-+}}{2\,Q^{2}}\,\operatorname{Re}i\int\mathrm{d}\Phi(k)\hat{\delta}(2p\cdot
k)\,\mathcal{A}_{-}^{(L)}\mathcal{A}_{+}^{(R)}\,\,[k|_{\dot{A}}[k|_{\dot{B}}|k\rangle_{C}|k\rangle_{D}\,e^{-ik\cdot
x}~{}.$ (113)

The above expressions make it clear that every quadratic term in the
amplitudes sources a different component of the spinorial curvature. The
double copy structure is remarkably explicit when comparing with the gauge
field spinors (53) and (54). Moreover, we can obtain the equivalent of (71)
and (72) by considering the gauge amplitudes

$\displaystyle\mathcal{A}^{(L)}_{\eta}=-2Q(p\cdot\varepsilon_{\eta})e^{\eta(\theta_{L}+ik\cdot
a_{L})},$ (114)
$\displaystyle\mathcal{A}^{(R)}_{\eta}=-2Q(p\cdot\varepsilon_{\eta})e^{\eta(\theta_{R}+ik\cdot
a_{R})},$

which under the double copy map imply that

$\displaystyle\langle{\mathbf{X}}_{ABCD}(x)\rangle$
$\displaystyle=e^{\bar{\theta}}\langle{\mathbf{X}}_{ABCD}^{\text{JNW}}(x-\bar{a})\rangle~{},$
(115)
$\displaystyle\langle\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}(x)\rangle$
$\displaystyle=e^{-\bar{\theta}}\langle\tilde{\mathbf{X}}_{\dot{A}\dot{B}\dot{C}\dot{D}}^{\text{JNW}}(x+\bar{a})\rangle~{},$
$\displaystyle\langle{\mathbf{\Phi}}_{AB\dot{C}\dot{D}}(x)\rangle$
$\displaystyle=e^{\Delta\theta}\langle{\mathbf{\Phi}}_{AB\dot{C}\dot{D}}^{\text{JNW}}(x-\Delta
a)\rangle~{},$
$\displaystyle\langle{\tilde{\mathbf{\Phi}}}_{\dot{A}\dot{B}CD}(x)\rangle$
$\displaystyle=e^{-\Delta\theta}\langle{\tilde{\mathbf{\Phi}}}_{\dot{A}\dot{B}CD}^{\text{JNW}}(x+\Delta
a)\rangle~{},$

where we have defined

$\bar{\theta}:=\theta_{L}+\theta_{R}~{},\qquad\Delta\theta:=\theta_{L}-\theta_{R}~{},$
(116) $\bar{a}:=a_{L}+a_{R},\qquad\Delta a:=a_{L}-a_{R}\,.$ (117)

The superscript JNW refers to the solution where both single copies are
Coulomb. Notice that, at linearised level, the first two spinors in (115)
match those of the Schwarzschild solution. The various parameters are
elegantly distributed over the different spinors. The parameter $\bar{a}$
corresponds to the spin of Kerr, and appears as expected via the Newman-Janis
shift, while $\bar{\theta}$ corresponds to the split-signature version of the
rotation between the mass and the NUT parameter; together, these two
parameters correspond to the Kerr-Taub-NUT solution. The parameters $\Delta a$
and $\Delta\theta$ correspond, respectively, to a novel type of Newman-Janis
shift for the axion and dilaton, and to the standard axion-dilaton
supergravity duality transformation. While this draft was in preparation, the
Kerr and Kerr-Taub-NUT solutions were discussed, also in split signature, in
Crawley:2021auj ; Guevara:2021yud .

The spinorial language is better fitted for displaying the double copy, but it
is instructive to think about the dilaton and axion. We can map (109) to the
field degrees of freedom using (30), together with the mode expansions of the
fields

$h^{\mu\nu}=2\,\operatorname{Re}\sum_{\eta}\int\mathrm{d}\Phi(k)a_{\eta\eta}(k)\varepsilon^{\mu}_{\eta}(k)\varepsilon^{\nu}_{\eta}(k)\,e^{-ik\cdot
x}~{},$ (118)
$\phi=2\,\operatorname{Re}\int\mathrm{d}\Phi(k)\,a_{\phi}(k)\,e^{-ik\cdot
x}~{},$ (119)
$B^{\mu\nu}=2\,\operatorname{Re}\int\mathrm{d}\Phi(k)\,a_{B}(k)\left(\varepsilon^{\mu}_{+}(k)\varepsilon^{\nu}_{-}(k)-\varepsilon^{\mu}_{-}(k)\varepsilon^{\nu}_{+}(k)\right)\,e^{-ik\cdot
x}~{}.$ (120)

Substituting in (30) implies that (109) can be re-expressed as

${\mathfrak{R}}^{\mu\nu\rho\sigma}=\kappa\,\operatorname{Re}\int\mathrm{d}\Phi(k)\bigg{[}\sum_{\eta}\,a_{\eta}\varepsilon_{\eta}^{[\mu}(k)k^{\nu]}\varepsilon_{\eta}^{[\rho}(k)k^{\sigma]}\\\
+a_{\phi}\,k^{[\mu}\eta^{\nu][\rho}k^{\sigma]}+a_{B}\,k^{[\mu}(\varepsilon_{+}^{\nu]}\varepsilon_{-}^{[\rho}-\varepsilon_{-}^{\nu]}\varepsilon_{+}^{[\rho})k^{\sigma]}\bigg{]}e^{-ik\cdot
x}~{}.$ (121)

The first term in the second line of (121) needs simplification. This is
achieved by expanding the flat metric in terms of the null tetrad

$\displaystyle
k^{[\mu}\eta^{\nu][\rho}k^{\sigma]}=-k^{[\mu}\varepsilon_{+}^{\nu]}\varepsilon_{-}^{[\rho}k^{\sigma]}-k^{[\mu}\varepsilon_{-}^{\nu]}\varepsilon_{+}^{[\rho}k^{\sigma]}\,.$
(122)

Comparison to (98) then implies the following relations between annihilation
operators

$\displaystyle a_{++}=a_{+}~{},\qquad\qquad a_{-+}=a_{\phi}+a_{B}~{},$ (123)
$\displaystyle a_{--}=a_{-}~{},\qquad\qquad a_{-+}=a_{\phi}-a_{B}~{},$

and hence the corresponding relation for amplitudes

$\displaystyle\mathcal{M}_{\eta\eta}$
$\displaystyle=-\frac{\kappa}{4\,Q^{2}}\,c_{\eta\eta}\,\mathcal{A}^{(L)}_{\eta}\,\mathcal{A}^{(R)}_{\eta}~{},$
(124) $\displaystyle\mathcal{M}_{\phi}$
$\displaystyle=-\frac{\kappa}{4\,Q^{2}}\;\frac{1}{2}\left(c_{+-}\mathcal{A}^{(L)}_{+}\mathcal{A}^{(R)}_{-}+c_{-+}\mathcal{A}^{(L)}_{-}\mathcal{A}^{(R)}_{+}\right)~{},$
$\displaystyle\mathcal{M}_{B}$
$\displaystyle=-\frac{\kappa}{4\,Q^{2}}\;\frac{1}{2}\,\left(c_{+-}\mathcal{A}^{(L)}_{+}\mathcal{A}^{(R)}_{-}-c_{-+}\mathcal{A}^{(L)}_{-}\mathcal{A}^{(R)}_{+}\right)~{}.$

In the next sections this prescription will be put into practice to compute
the classical fields obtained by double copying the amplitudes discussed in
section 5. In the following we will restrict to the case $c_{++}=c_{--}$ and
$c_{+-}=c_{-+}$ since these solutions naturally continue to real solutions in
Minkowski signature.

###  6.1 Duality rotation

We will now turn to a concrete example. Consider left and right amplitudes
that differ in their EM duality angle,

$\displaystyle\mathcal{A}^{(L)}_{\eta}=-2Q(p\cdot\varepsilon_{\eta})e^{\theta_{L}\eta},$
(125)
$\displaystyle\mathcal{A}^{(R)}_{\eta}=-2Q(p\cdot\varepsilon_{\eta})e^{\theta_{R}\eta},$

the effect of this difference will be the existence of a rotation between
dilaton and axion. The double copied amplitudes are obtained by applying the
map (124),

$\displaystyle\mathcal{M}_{\eta}$
$\displaystyle=-c_{++}\kappa\,m^{2}(u\cdot\varepsilon_{\eta})^{2}\,e^{\bar{\theta}\eta}~{},$
(126) $\displaystyle\mathcal{M}_{\phi}$
$\displaystyle=\frac{\kappa\,c_{+-}}{2}\,p^{2}\,\cosh\Delta\theta=\frac{\tilde{c}\,m}{2}\,\,\cosh\Delta\theta~{},$
$\displaystyle\mathcal{M}_{B}$
$\displaystyle=\frac{\kappa\,c_{+-}}{2}\,p^{2}\,\sinh\Delta\theta=\frac{\tilde{c}\,m}{2}\,\,\sinh\Delta\theta~{},$

where we have defined $\tilde{c}=\kappa\,c_{+-}\,m$. To test the effect of the
rotation on the metric, let us compute the transformed Weyl tensor,

$\displaystyle\langle
W^{\mu\nu\rho\sigma}(x)\rangle=-\operatorname{Re}i\kappa^{2}\,c_{++}\,m^{2}\int\mathrm{d}\Phi(k)\,\hat{\delta}(2k\cdot
p)\,e^{-ik\cdot x}$ $\displaystyle\left[(\varepsilon_{+}\cdot
u)^{2}k^{[\mu}\varepsilon_{-}^{\nu]}k^{[\rho}\varepsilon_{-}^{\sigma]}\,e^{\bar{\theta}}\right.$
(127) $\displaystyle\left.\quad+(\varepsilon_{-}\cdot
u)^{2}k^{[\mu}\varepsilon_{+}^{\nu]}k^{[\rho}\varepsilon_{+}^{\sigma]}\,e^{-\bar{\theta}}\right]\,.$

A little algebra shows that this can be rewritten as

$\displaystyle\langle
W^{\mu\nu\rho\sigma}(x)\rangle=-\operatorname{Re}i\kappa^{2}\,c_{++}\,m^{2}\int\mathrm{d}\Phi(k)\,\hat{\delta}(2k\cdot
p)\,e^{-ik\cdot x}$ (128)
$\displaystyle\times\left[\cosh\bar{\theta}\left(k^{[\mu}u^{\nu]}k^{[\rho}u^{\sigma]}+\frac{1}{2}k^{[\mu}\eta^{\nu][\rho}k^{\sigma]}\right)\right.\left.-\frac{1}{2}\sinh\bar{\theta}\,\epsilon^{\mu\nu\tau\lambda}\left(k_{[\tau}u_{\lambda]}k^{[\rho}u^{\sigma]}+\frac{1}{2}k_{[\tau}\delta_{\lambda]}^{[\rho}k^{\sigma]}\right)\right]\,.$

The first term, with the hyperbolic cosine, corresponds to the Schwarzschild
solution, expression (3.26) of Monteiro:2020plf , which we will denote
$W^{\mu\nu\rho\sigma}_{\text{Schw.}}$. Making use of this notation leads to
the compact result

$\displaystyle\langle W^{\mu\nu\rho\sigma}\rangle=\cosh\bar{\theta}\
W^{\mu\nu\rho\sigma}_{\text{Schw.}}-\sinh\bar{\theta}\,\frac{1}{2}\epsilon^{\mu\nu\tau\lambda}\
W_{\ \ \tau\lambda}^{\text{Schw.}\ \rho\sigma}.$ (129)

The second term represents the dual of $W_{\text{Schw.}}$, in analogy with the
result of section 5.3. We conclude that the angle $\bar{\theta}$ indeed
rotates the mass and the NUT charge of the solution Huang:2019cja ;
Alawadhi:2019urr .

The Weyl tensor we have computed represents the graviton degrees of freedom in
$\mathfrak{R}^{\mu\nu\rho\sigma}$. The next step is to obtain the classical
expectation value of the dilaton and axion degrees of freedom. Instead of
computing the corresponding components of $\mathfrak{R}^{\mu\nu\rho\sigma}$,
we will obtain the field profiles $\langle\phi\rangle$ and
$\langle\sigma\rangle$ directly.

Let us start with the classical expectation value of the dilaton field,
$\langle\phi\rangle=\langle\psi|S^{\dagger}\phi S|\psi\rangle$. We use the
field operator (119), exponentiation of the coherent state (107) with the
amplitude given in (126). The result is

$\displaystyle\langle\phi(x)\rangle=\tilde{c}\,m\,\cosh\Delta\theta\
\operatorname{Re}i\int\mathrm{d}\Phi(k)\,\hat{\delta}(2p\cdot k)e^{-ik\cdot
x}~{}.$ (130)

Performing the integration as in (73), we obtain

$\displaystyle\langle\phi(x)\rangle=\tilde{c}\,\cosh\Delta\theta\
\frac{\Theta(\rho^{2})}{8\pi}\,\frac{1}{\rho}~{}.$ (131)

We will now tackle the axion field. Recalling (120) and taking a derivative,
we quickly find

$\displaystyle\langle
H_{\mu\nu\rho}(x)\rangle=\langle\frac{1}{2}\partial_{[\mu}B_{\nu\rho]}(x)\rangle=\frac{\tilde{c}}{2}\,\sinh\Delta\theta\operatorname{Re}\int\mathrm{d}\Phi(k)\hat{\delta}(u\cdot
k)k_{[\mu}\varepsilon_{\nu}^{+}\varepsilon^{-}_{\rho]}\,e^{-ik\cdot x}~{}.$
(132)

At this stage, it is very helpful to note that

$\displaystyle\epsilon^{\mu\nu\rho\sigma}k_{[\nu}\varepsilon^{+}_{\rho}\varepsilon^{-}_{\sigma]}=k^{[\mu}n^{\nu}\varepsilon_{+}^{\rho}\varepsilon_{-}^{\sigma]}k_{[\nu}\varepsilon^{+}_{\rho}\varepsilon^{-}_{\sigma]}=-3!\,k^{\mu}~{}.$
(133)

Hence,

$\displaystyle\Rightarrow\langle\epsilon^{\mu\nu\rho\sigma}H_{\nu\rho\sigma}(x)\rangle$
$\displaystyle=-3\,\tilde{c}\,\sinh\Delta\theta\,\operatorname{Re}\int\mathrm{d}\Phi(k)\,\hat{\delta}(k\cdot
u)\,k^{\mu}\,e^{-ik\cdot x}$ (134)
$\displaystyle=-3\,\tilde{c}\,\sinh\Delta\theta\,\partial^{\mu}\left(\frac{\Theta(\rho^{2})}{4\pi}\,\frac{1}{\rho}\right)~{}.$

This expression provides direct information on the axion $\sigma$. To see how,
note from equation (34), expanded to leading order, that the relation between
$H$ and $\sigma$ is simply

$\displaystyle
H_{\mu\nu\rho}=-\epsilon_{\mu\nu\rho\sigma}\,\partial^{\sigma}\sigma\Rightarrow\epsilon^{\mu\nu\rho\sigma}H_{\nu\rho\sigma}=-3!\,\partial^{\mu}\sigma~{}.$
(135)

Comparing with the previous expression, we find

$\displaystyle\langle\sigma(x)\rangle=\tilde{c}\,\sinh\Delta\theta\,\frac{\Theta(\rho^{2})}{8\pi}\,\frac{1}{\rho}~{}.$
(136)

###  6.2 Newman-Janis shift

Now we turn our attention to the spin parameter. Just as we did in the
previous section, we can use the prescription (96) to source an axion and a
dilaton. However, we now consider products of gauge theory amplitudes with
different spins

$\displaystyle\mathcal{A}^{(L)}_{\eta}=-2Q(p\cdot\varepsilon_{\eta})e^{i\,\eta\,a_{L}\cdot
k},$ (137)
$\displaystyle\mathcal{A}^{(R)}_{\eta}=-2Q(p\cdot\varepsilon_{\eta})e^{i\,\eta\,a_{R}\cdot
k}.$

These yield the following gravity amplitudes

$\displaystyle\mathcal{M}_{\eta}$
$\displaystyle=-\kappa\,c_{++}\,m^{2}(u\cdot\varepsilon_{\eta})^{2}\,e^{i\,\eta\,\bar{a}\cdot
k}~{},$ (138) $\displaystyle\mathcal{M}_{\phi}$
$\displaystyle=\frac{\tilde{c}\,m}{2}\,\,\cos(\Delta a\cdot k)~{},$
$\displaystyle\mathcal{M}_{B}$
$\displaystyle=\frac{\tilde{c}\,m}{2}\,\,\sin(\Delta a\cdot k)~{}.$

Once more, the graviton components of the fat curvature tensor found in (109)
reduce to the Weyl tensor

$W^{\mu\nu\rho\sigma}(x)=\kappa\,c_{++}\,\operatorname{Re}\,i\int\mathrm{d}\Phi(k)\hat{\delta}(2k\cdot
p)e^{-ik\cdot
x}\sum_{\eta}\,\mathcal{M}_{\eta}\varepsilon_{-\eta}^{[\mu}k^{\nu]}\varepsilon_{-\eta}^{[\rho}k^{\sigma]}e^{i\eta
k\cdot\bar{a}}.$ (139)

It is not difficult to see that this matches the classical computation with a
spinning source. The linearised EOMs are, writing
${g_{\mu\nu}(x)=\eta_{\mu\nu}+\kappa\,h_{\mu\nu}(x)}$

$\partial^{2}h^{\mu\nu}(x)=-\kappa\,{{\mathsf{P}}^{\mu\nu}}_{\alpha\beta}T^{\alpha\beta}(x),\qquad{{{\mathsf{P}}}^{\mu\nu}}_{\alpha\beta}=\frac{1}{4}{\delta^{(\mu}}_{(\alpha}{\delta^{\nu)}}_{\beta)}-\frac{1}{2}\eta^{\mu\nu}\eta_{\alpha\beta},$
(140)

with the following stress-energy tensor for Kerr Guevara:2020xjx ; justin2017

$T^{\mu\nu}(x)={m}\int\mathrm{d}\tau\,u^{(\mu}{\exp(\bar{a}*\partial)^{\nu)}}_{\rho}u^{\rho}\delta^{(4)}(x-u\tau)\,.$
(141)

Solving (140) with the usual boundary conditions, we find the linearised
metric

$\begin{split}h^{\mu\nu}(x)&=-\kappa
m^{2}\text{Re}\,i\int{\mathrm{d}}\Phi(k)\hat{\delta}(p\cdot k)e^{-ik\cdot
x}\,{{{\mathsf{P}}}^{\mu\nu}}_{\alpha\beta}\,u^{(\alpha}{\exp(-i\bar{a}*k)^{\beta)}}_{\rho}u^{\rho}\\\
&=-\kappa m^{2}\text{Re}\,i\int{\mathrm{d}}\Phi(k)\hat{\delta}(p\cdot
k)e^{-ik\cdot
x}\left[\left(u^{\mu}u^{\nu}-\frac{1}{2}\eta^{\mu\nu}\right)\cos(\bar{a}\cdot
k)\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{i}{2}u^{(\mu}\epsilon^{\nu)}(\bar{a},k,u)\left.\frac{\sin(\bar{a}\cdot
k)}{\bar{a}\cdot k}\right],\end{split}$ (142)

from which the curvature can be computed. After some tedious but
straightforward algebra one finds

$\begin{split}W^{\mu\nu\rho\sigma}(x)&=-\kappa^{2}m^{2}\text{Re}\,i\sum_{\eta}\int{\mathrm{d}}\Phi(k)\hat{\delta}(2p\cdot
k)e^{-ik\cdot x}(\varepsilon_{\eta}\cdot
u)^{2}k^{[\mu}\varepsilon_{-\eta}^{\nu]}k^{[\rho}\varepsilon_{-\eta}^{\sigma]}e^{i\eta
k\cdot\bar{a}}.\end{split}$ (143)

The result matches the one we obtained from amplitudes upon setting
$c_{++}=1$.

For the dilaton and the axion, the calculations are formally analogous to the
ones outlined in 6.1, except that now we have momentum dependent trigonometric
functions which characterise the spin mixing. For this reason we omit the
explicit computations and report the final results. We find for the dilaton

$\displaystyle\langle\phi(x)\rangle$
$\displaystyle=\frac{\tilde{c}}{2}\,\operatorname{Re}i\int\mathrm{d}\Phi(k)\,\hat{\delta}(u\cdot
k)e^{-ik\cdot x}\cos(\Delta a\cdot k)~{}$ (144)
$\displaystyle=\frac{\tilde{c}}{8}\left(S_{\Delta a,0}(x)+S_{-\Delta
a,0}(x)\right),$

referencing to the definition of the scalar potential (73). The axion is
instead given by

$\displaystyle\langle\epsilon^{\mu\nu\rho\sigma}H_{\nu\rho\sigma}(x)\rangle$
$\displaystyle=-3\,\tilde{c}\,\partial^{\mu}\operatorname{Re}i\int\mathrm{d}\Phi(k)\,\hat{\delta}(k\cdot
u)\,e^{-ik\cdot x}\sin(\Delta a\cdot k),$ (145)

telling us that the scalar $\sigma$ is at leading order

$\displaystyle\langle\sigma(x)\rangle$
$\displaystyle=\frac{\tilde{c}}{2}\operatorname{Re}i\int\mathrm{d}\Phi(k)\,\hat{\delta}(u\cdot
k)e^{-ik\cdot x}\sin(\Delta a\cdot k)~{}$ (146)
$\displaystyle=\frac{\tilde{c}}{8}\left(S_{\Delta a,0}(x)-S_{-\Delta
a,0}(x)\right).$

###  6.3 Comparison with known solutions

The linearised solution obtained in section 6.1 corresponds to an axi-dilaton
Taub-NUT black hole. This solution is known exactly; see (17), (19) in
Burgess:1994kq . It is interesting to check that our results agree with the
linearisation of the known solution. There, dilaton and axion are given
as131313Ignoring factors of $\sqrt{3}$ that can be absorbed into $\delta$ at
linear order..

$\displaystyle
e^{-\phi}=(1+\epsilon^{2})\frac{\Lambda^{\delta}}{\epsilon^{2}\Lambda^{2\delta}+1}~{},\qquad\sigma=\frac{\epsilon(\Lambda^{2\delta}-1)}{\epsilon^{2}\Lambda^{2\delta}+1}~{},$
(147)

where

$\Lambda=1-\frac{R_{0}}{R}~{},$

$\delta\,R_{0}$ is the charge of the dilaton and $\epsilon$ is a duality
rotation parameter between dilaton and axion. Notice that $R$ is the (3,1)
signature equivalent of $\rho$. At linearised level, the fields decouple and
the metric is equivalent to Taub-NUT. Expanding at linear order the other
fields and defining $\epsilon=-\tan\frac{\Delta\theta}{2}$, we find

$\displaystyle\phi=\cos\Delta\theta\,\frac{\delta\,R_{0}}{R}~{},\qquad\sigma=\sin\Delta\theta\,\frac{\delta\,R_{0}}{R}~{}.$
(148)

Our solution (131), (136) agrees with this up to an overall constant
($\tilde{c}=16\pi\,\delta\,R_{0}$).141414After rotating back to Minkowski
signature, the hyperbolic trigonometric functions turn into standard
trigonometric functions. Additionally, one has to continue $\sigma\to i\sigma$
due to its pseudo-scalar nature, which cancels the factor of $i$ from the
sine. The factor of $16$ takes into account a factor of $2$ generated by the
analytic continuation of the propagator (61). Then, $\Delta\theta$ is just the
parameter inside $\text{SL}(2,\mathbb{R})$ that generates linear rotations
between dilaton and axion.

In the special case where $\theta_{L}=\theta_{R}$ both single copies are
identical, and we have no mixing: $\Delta\theta=0$. From (126), we see that
this implies that the axion will vanish, leaving a linearised solution that
would be the equivalent to Taub-NUT plus the dilaton. In Burgess:1994kq , this
corresponds to (17) and (18).

On the contrary, if $\theta_{R}=-\theta_{L}$, $\bar{\theta}$ vanishes and the
resulting metric has vanishing NUT charge. The result is a linearised
Schwarzschild metric plus axion plus dilaton, corresponding to the
linearisation of (10) and (13) in Burgess:1994kq .

When both rotation angles are zero, both the NUT charge and axion vanish. We
are left with a linearised JNW solution. This simplified solution will be used
in section 8 to check for the existence of a double copy relation in position
space.

The solutions considered in section 6.2 involving spin are not so well
understood in the literature. There have been attempts to apply a Newman-Janis
shift to the JNW solution, with the prospects of obtaining a spinning
generalisation. However, these claimed generalisations fail to satisfy the
Einstein-dilaton equations of motion Bogush:2020lkp . Although linear, our
solution might help to find a satisfactory generalisation of the JNW metric
with spin.

##  7 Heterotic double copy

It is straightforward to get an extra set of gauge fields in the double copy,
by simply supplementing one of the YM factors with a scalar $X$. For instance,
let us assume that the ‘left gauge theory’ is coupled to a bi-adjoint
scalar151515This should not be confused with the bi-adjoint scalar that
appears in the denominator of various formulations of the double copy. which
transforms under the symmetry groups $G$ and $\hat{G}$. The Lagrangians for
our gauge theory factors are given by:

$\displaystyle\mathcal{L}^{(L)}=$
$\displaystyle-\tfrac{1}{4}\text{Tr}\left[F_{\mu\nu}^{(L)}F^{(L)\mu\nu}\right]+\tfrac{1}{2}\text{Tr}\left[D_{\mu}X^{\hat{a}}D^{\mu}X_{\hat{a}}\right]-\tfrac{g^{2}}{4}\left(f_{abc}X^{b\hat{b}}X^{c\hat{c}}\right)\left(f^{a}_{\
b^{\prime}c^{\prime}}X^{b^{\prime}}_{\ \hat{b}}X^{c^{\prime}}_{\
\hat{c}}\right)$ (149)
$\displaystyle+\tfrac{gg^{\prime}}{3!}if_{abc}F_{\hat{a}\hat{b}\hat{c}}X^{a\hat{a}}X^{b\hat{b}}X^{c\hat{c}}\,\,,$
$\displaystyle\mathcal{L}^{(R)}=$
$\displaystyle-\tfrac{1}{4}\text{Tr}\left[F_{\mu\nu}^{(R)}F^{(R)\mu\nu}\right]\,\,.$

where $f_{abc}$ and $F_{\hat{a}\hat{b}\hat{c}}$ are the structure constants of
$G$ and $\hat{G}$, respectively, and $g^{\prime}$ is a dimensionful arbitrary
constant. In the above, the trace is taken over the group $G$. We can think of
$\mathcal{L}^{(L)}$ as coming from the dimensional reduction of a pure YM
theory, together with the gauging of (a subset of) the global symmetry, which
introduces the cubic interactions Chiodaroli:2014xia . The double copy will
then give an Einstein-Yang-Mills-dilaton-axion theory. Upon linearisation, the
interaction terms in (149) drop out. For our purposes of studying the
linearised solutions of heterotic gravity, it suffices to take our gauge
groups to be $U(1)$, resulting in a single Maxwell field in the gravity
theory.

If we denote the amplitude for the 3-point interaction of the scalar $X$ with
the static particle by $\mathcal{A}_{X}$, we can write schematically:

$\left(\mathcal{A}_{+}^{(L)}+\mathcal{A}_{-}^{(L)}+\mathcal{A}_{X}\right)\
\otimes\
\left(\mathcal{A}_{+}^{(R)}+\mathcal{A}_{-}^{(R)}\right)=\left\\{\begin{matrix}\vspace{7pt}\mathcal{A}_{\pm}^{(L)}\otimes\mathcal{A}_{\pm}^{(R)}\to\mathcal{M}_{\pm}\\\
\vspace{7pt}\mathcal{A}_{(+}^{(L)}\otimes\mathcal{A}_{-)}^{(R)}\to\mathcal{M}_{\phi}\\\
\vspace{7pt}\mathcal{A}_{[+}^{(L)}\otimes\mathcal{A}_{-]}^{(R)}\to\mathcal{M}_{B}\\\
\mathcal{A}_{X}\otimes\mathcal{A}_{\pm}^{(R)}\to\mathcal{A}_{\pm}\end{matrix}\right.$
(150)

where $\mathcal{A}_{\pm}$ is the 3-point amplitude for the gauge field in the
heterotic theory. The amplitude $\mathcal{A}_{X}$ is a constant
$\mathcal{A}_{X}=Q_{X}$. Then, the double copy for the graviton, dilaton and
axion amplitudes, already given in (124), is supplemented by

$\mathcal{A}_{\pm}=\frac{c_{X}}{Q_{X}}\ \mathcal{A}_{X}\
\mathcal{A}_{\pm}^{(R)}\,\,,\ $ (151)

for some arbitrary constant $c_{X}$. The dilaton and axion fields are still
given by (131) and (136), as they are unaffected by the addition of the gauge
field in the linearised approximation. The gauge field will have electric and
magnetic charges given by

$\displaystyle Q_{E}^{d.c.}=$ $\displaystyle
c_{X}\,Q\,\text{cosh}(\theta_{R})\,\,,$ (152) $\displaystyle Q_{M}^{d.c.}=$
$\displaystyle c_{X}\,Q\,\text{sinh}(\theta_{R})\,\,.$

###  7.1 Comparison with known solution

We now want to compare with Burgess:1994kq . Here, the general spherically
symmetric, static and asymptotically flat solution to the heterotic theory is
obtained via a series of transformations. Starting with the graviton-dilaton
solution, characterised by parameters $l$ and $\delta$, one can perform three
$\text{SL}(2,\mathbb{R})$ transformations, with parameters $\omega,\epsilon$
and $\rho$161616For each $\text{SL}(2,\mathbb{R})$, imposing the vanishing of
the dilaton and the axion at infinity reduces the three parameters of
$\text{SL}(2,\mathbb{R})$ to one., interspersed by a discrete T-duality and an
$\text{O}(1,1)$ transformation with parameter $x$ to obtain the full family of
axion-dilaton-Taub-NUT metric coupled to a vector field. The axion and dilaton
charges are given by

$\displaystyle\hat{Q}_{D}$
$\displaystyle=\tfrac{1-\rho^{2}}{1+\rho^{2}}\bar{Q}_{D}-\tfrac{2\rho}{1+\rho^{2}}\bar{Q}_{A}\,\,,$
(153) $\displaystyle\hat{Q}_{A}$
$\displaystyle=\tfrac{1-\rho^{2}}{1+\rho^{2}}\bar{Q}_{A}+\tfrac{2\rho}{1+\rho^{2}}\bar{Q}_{D}\,\,,$

with

$\displaystyle\bar{Q}_{D}$
$\displaystyle=\tfrac{l}{2}\left[(1+x)\tfrac{1-\epsilon^{2}}{1+\epsilon^{2}}\delta+(1-x)\tfrac{1-\omega^{2}}{1+\omega^{2}}\sqrt{1-\delta^{2}}\right]\,\,,$
(154) $\displaystyle\bar{Q}_{A}$
$\displaystyle=\tfrac{l}{2}\left[(1+x)\tfrac{2\epsilon}{1+\epsilon^{2}}\delta+(1-x)\tfrac{2\omega}{1+\omega^{2}}\sqrt{1-\delta^{2}}\right]\,\,.$

Using the notation $\rho=\tan\tfrac{\theta_{\rho}}{2}$,
$\epsilon=\tan\tfrac{\theta_{\epsilon}}{2}$ and restricting to
$\epsilon=\omega$, the charges can be shown to reduce to

$\displaystyle\hat{Q}_{D}$
$\displaystyle=\text{cos}(\theta_{\rho}+\theta_{\epsilon})\tfrac{l}{2}\left[(1+x)\delta+(1-x)\sqrt{1-\delta^{2}}\right]\,\,,$
(155) $\displaystyle\hat{Q}_{A}$
$\displaystyle=\text{sin}(\theta_{\rho}+\theta_{\epsilon})\tfrac{l}{2}\left[(1+x)\delta+(1-x)\sqrt{1-\delta^{2}}\right]\,\,.$

Then, comparison with (131) and (136) gives171717As explained in subsection
6.3, the hyperbolic trigonometric functions will turn into standard
trigonometric functions upon rotating back to Minkowski signature.

$\displaystyle\Delta\theta=$
$\displaystyle\theta_{\rho}+\theta_{\epsilon}\,\,,$ (156)
$\displaystyle\frac{\tilde{c}}{16\pi}=$
$\displaystyle\tfrac{l}{2}\left[(1+x)\delta+(1-x)\sqrt{1-\delta^{2}}\right]\,\,,$

This reduces to the solution without the gauge field for $\theta_{\rho}=0$ and
$x=1$ (note that $x=1$ corresponds to the identity matrix in the
$\text{O}(1,1)$ transformation used in Burgess:1994kq ). The electric and
magnetic charges in Burgess:1994kq , with the restriction $\epsilon=\omega$,
reduce to

$\displaystyle\hat{Q}_{E}=$
$\displaystyle\sqrt{x^{2}-1}l\left(\sqrt{1-\delta^{2}}-\delta\right)\cos\left(\tfrac{\theta_{\rho}}{2}+\theta_{\epsilon}\right)\,\,,$
(157) $\displaystyle\hat{Q}_{M}=$
$\displaystyle\sqrt{x^{2}-1}l\left(\sqrt{1-\delta^{2}}-\delta\right)\sin\left(\tfrac{\theta_{\rho}}{2}+\theta_{\epsilon}\right)\,\,.$

Then, comparing with (152), we get the identifications

$\displaystyle\theta_{R}$
$\displaystyle=\tfrac{\theta_{\rho}}{2}+\theta_{\epsilon}\,\,,$ (158)
$\displaystyle c_{X}\,Q$
$\displaystyle=\sqrt{x^{2}-1}l\left(\sqrt{1-\delta^{2}}-\delta\right)\,\,.$

Finally, we remark that the restriction we imposed on the parameters (i.e.
setting $\epsilon=\omega$) was only necessary in order to obtain a
straightforward map. One can always write a completely general (though more
cumbersome) map via

$\displaystyle\frac{\tilde{c}}{16\pi}$ $\displaystyle=$
$\displaystyle\sqrt{\hat{Q}_{D}^{2}+\hat{Q}_{A}^{2}}\,,\qquad\qquad\Delta\theta$
$\displaystyle=$
$\displaystyle\text{tan}^{-1}\left(\frac{\hat{Q}_{A}}{\hat{Q}_{D}}\right)\,\,,$
(159) $\displaystyle c_{X}Q$ $\displaystyle=$
$\displaystyle\sqrt{\hat{Q}_{E}^{2}+\hat{Q}_{M}^{2}}\,,\qquad\qquad\theta_{R}$
$\displaystyle=$
$\displaystyle\text{tan}^{-1}\left(\frac{\hat{Q}_{M}}{\hat{Q}_{E}}\right)\,\,.$

##  8 Double copy in position space

We have found expressions that exhibit clear double copy relations in on-shell
momentum space. In this section, we will discuss whether these straightforward
double copy relations can be carried over to position space. In particular, we
will see how the position-space Weyl double copy relations Luna:2018dpt
emerge from amplitudes. In most of this section, we will set
$a_{L,R}=\theta_{L,R}=0$, since this simple scenario is enough to illustrate
our points. To simplify the discussion, each term in the expectation value of
(121) will be analysed individually. We will omit the axion term, since it
vanishes when $a_{L,R}=\theta_{L,R}=0$.

First, we shall consider the terms associated to the graviton amplitude, which
we will denote by $\mathfrak{R}^{(h)}{}_{\mu\nu\rho\sigma}$. This is precisely
the Schwarzschild Riemann (and Weyl) tensor considered in Monteiro:2020plf .
After some algebraic manipulations,

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle=\operatorname{Re}\,i\,m^{2}\kappa^{2}\int\mathrm{d}\Phi(k)\hat{\delta}(2k\cdot
p)e^{-ik\cdot
x}\left(k^{[\mu}u^{\nu]}k^{[\rho}u^{\sigma]}-\frac{1}{2}k^{[\mu}\eta^{\nu][\rho}k^{\sigma]}\right)~{}.$
(160)

The second term in the brackets makes the expression traceless. Alternatively,
we can write

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\,\operatorname{Re}\,i\,m^{2}\kappa^{2}\int\mathrm{d}\Phi(k)\hat{\delta}(2k\cdot
p)e^{-ik\cdot x}k^{[\tau}u^{\lambda]}k^{[\eta}u^{\omega]}~{},$ (161)

where $\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}$ projects out
the trace, as in the definition of the Weyl tensor

$\displaystyle W^{\mu\nu\nu\rho}$
$\displaystyle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\,R^{\tau\lambda\eta\omega}~{},$
(162) $\displaystyle\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}$
$\displaystyle=\delta^{\mu}_{\tau}\delta^{\nu}_{\lambda}\delta^{\rho}_{\eta}\delta^{\sigma}_{\omega}+\frac{1}{2}g_{\tau\eta}\delta^{[\mu}_{\lambda}g^{\nu][\rho}\delta^{\sigma]}_{\omega}+\frac{1}{6}\,g_{\tau\eta}\,g_{\lambda\omega}\,g^{\mu[\rho}\,g^{\sigma]\nu}~{}.$

Next, we can take the factors of $k$ outside the integral as derivatives

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle$
$\displaystyle=-\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\,\partial^{[\tau}u^{\lambda]}\partial^{[\eta}u^{\omega]}\,\operatorname{Re}\,i\,m^{2}\kappa^{2}\int\mathrm{d}\Phi(k)\hat{\delta}(2k\cdot
p)e^{-ik\cdot x}$ (163)
$\displaystyle=-\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\,\partial^{[\tau}u^{\lambda]}\partial^{[\eta}u^{\omega]}\,\frac{\,m\kappa^{2}}{2}\,\operatorname{Re}\,i\int\mathrm{d}\Phi(k)\hat{\delta}(k\cdot
u)e^{-ik\cdot x}~{}.$

For positive $t^{1}$, it can be checked that

$\operatorname{Re}\,i\,\int\mathrm{d}\Phi(k)\hat{\delta}(k\cdot u)e^{-ik\cdot
x}=-\frac{1}{2}\int\hat{\mathrm{d}}^{3}k\,\frac{e^{-ik\cdot x}}{k^{2}}~{}.$
(164)

The integral on the right is performed over the three-dimensional subspace of
momenta orthogonal to the worldline $k\cdot u=0$, so
$k_{\mu}=(k_{1},0,k_{3},k_{4})$. This prescription will be used in the rest of
this section. Additionally, the divergence is resolved by the $i\epsilon$
prescription $(k)^{2}=(k^{1}+i\epsilon)^{2}-|\mathbf{k}|^{2}$, which selects
the retarded contour. Substituting in the Riemann tensor and taking the
derivatives inside the integral, we get

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle$
$\displaystyle=-\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{\,m\kappa^{2}}{4}\int\hat{\mathrm{d}}^{3}k\,\frac{e^{-ik\cdot
x}}{k^{2}}k^{[\tau}u^{\lambda]}k^{[\eta}u^{\omega]}~{}.$ (165)

This already looks like a double copy of the field strength tensor

$F^{\mu\nu}(x)=-Q\int\hat{\mathrm{d}}^{3}k\,\frac{e^{-ik\cdot
x}}{k^{2}}k^{[\mu}u^{\nu]}~{}.$ (166)

To obtain a concrete double copy expression, we introduce a delta function

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle$
$\displaystyle=-\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{\,m\kappa^{2}}{4}\int\hat{\mathrm{d}}^{3}k\,\hat{\mathrm{d}}^{3}q\,\hat{\delta}^{3}(k-q)\frac{e^{-ik\cdot
x}}{k^{2}}k^{[\tau}u^{\lambda]}q^{[\eta}u^{\omega]}$ (167)
$\displaystyle=-\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{\,m\kappa^{2}}{4}\int\hat{\mathrm{d}}^{3}k\,\hat{\mathrm{d}}^{3}q\,\mathrm{d}^{3}y\
e^{-iy\cdot(q-k)}\,\frac{e^{-ik\cdot
x}}{k^{2}}k^{[\tau}u^{\lambda]}q^{[\eta}u^{\omega]}$
$\displaystyle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{\,m\kappa^{2}}{4}\int\mathrm{d}^{3}y\
\int\hat{\mathrm{d}}^{3}k\,\,\frac{e^{-ik\cdot(x-y)}}{k^{2}}k^{[\tau}u^{\lambda]}\,\int\hat{\mathrm{d}}^{3}q\,\
e^{-iy\cdot q}q^{[\eta}u^{\omega]}~{}.$

The last line is already a convolution of $F^{\tau\lambda}$ with the integral
in $q$. To complete the calculation we will need the scalar field and its
formal inverse

$S(x)=-\int\hat{\mathrm{d}}^{3}k\,\frac{e^{-ik\cdot x}}{k^{2}}~{},\qquad
S^{-1}(x)=-\int\hat{\mathrm{d}}^{3}k\,e^{-ik\cdot x}\,k^{2}~{},$ (168)

satisfying181818The symbol $\circ$ denotes convolution: $(f\circ
g)(x)=\int\mathrm{d}yf(y)g(x-y)$.

$\displaystyle\left(S\circ
S^{-1}\right)(x)=\delta(t_{1})\delta^{2}(\mathbf{x})\,.$ (169)

We emphasise that the expression for $S^{-1}$ is only formal, due to the
divergence of the integral. $S^{-1}$ is really defined operationally, acting
via the convolution. It will always appear in convolutions where this
divergence is cancelled, yielding a finite result. The calculation can be
carried out following a strategy similar to the previous one. Inserting a
delta function and a factor which equals $1$ on its support, we have

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle$
$\displaystyle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{\,m\kappa^{2}}{4}\
\int\mathrm{d}^{3}y\
\int\hat{\mathrm{d}}^{3}k\,\,\frac{e^{-ik\cdot(x-y)}}{k^{2}}k^{[\tau}u^{\lambda]}$
(170) $\displaystyle\hskip
142.26378pt\int\,\hat{\mathrm{d}}^{3}q\,\hat{\mathrm{d}}^{3}l\,\hat{\delta}(l-q)\,\frac{l^{2}}{q^{2}}\,e^{-iy\cdot
q}q^{[\eta}u^{\omega]}$
$\displaystyle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{\,m\kappa^{2}}{4}\int\mathrm{d}^{3}y\,\,\mathrm{d}^{3}z\,\int\hat{\mathrm{d}}^{3}k\,\,\frac{e^{-ik\cdot(x-y)}}{k^{2}}k^{[\tau}u^{\lambda]}$
$\displaystyle\hskip
142.26378pt\int\hat{\mathrm{d}}^{3}q\,\frac{1}{q^{2}}\,e^{-iq\cdot(y-z)}q^{[\eta}u^{\omega]}\,\int\hat{\mathrm{d}}^{3}l\,l^{2}\,e^{-il\cdot
z}\,~{}.$

Finally, we recognise another convolution,

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}(x)\rangle$
$\displaystyle=\frac{\,m\,\kappa^{2}}{4\,Q^{2}}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\
\left(F^{\tau\lambda}\circ S^{-1}\circ F^{\eta\omega}\right)(x)~{}.$ (171)

We conclude that this contribution of the Riemann tensor is the convolution of
two copies of the field strength tensor with an inverse power of the scalar
field. We remark that the convolutions are performed over a 3-dimensional
subspace of spacetime, reflecting the fact that all our solutions are
independent of $t_{2}$.

The convolution we have obtained is the most natural operation from the point
of view of the double copy Anastasiou:2014qba ; LopesCardoso:2018xes ;
Anastasiou:2018rdx ; Borsten:2019prq ; Borsten:2020xbt ; Borsten:2021zir .
However, we know that for some cases (like Schwarzschild) the relation must
factorise in position space, turning convolutions into ordinary products. In
this factorisation, the projector plays an important role. On its own, $F\circ
S^{-1}\circ F$ does not factorise, but the offending terms are pure traces
that are projected out by $\mathcal{P}$, leaving a neat factorised expression.
In order to gain a better understanding of this, let us go back to (165) and
take the derivatives out:

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle$
$\displaystyle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\partial^{[\tau}u^{\lambda]}\partial^{[\eta}u^{\omega]}\frac{\,m\kappa^{2}}{4}\int\hat{\mathrm{d}}^{3}k\frac{e^{-ik\cdot
x}}{k^{2}}$ (172)
$\displaystyle=-\frac{\,m\kappa^{2}}{4}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\partial^{[\tau}u^{\lambda]}\partial^{[\eta}u^{\omega]}\,S(x)~{}.$

The crucial point now is that the double derivative action on $S(x)$
factorises into single derivatives under the contraction of the projector
$\mathcal{P}$, in the sense to be described below. This happens in the strict
interior of the split signature light-cone, where the curvature is non-
vanishing and type D. It is simpler if we first perform an analytic
continuation to (1,3) signature to prove the factorisation using the
properties of Kerr-Schild vectors recently reviewed in Luna:2016due . First,
we note that the analytic continuation of the scalar $S(x)$ is

$S(x)=\frac{1}{4\pi R}~{},$ (173)

where $R$ can be interpreted as the retarded null distance between a point
$x^{\mu}$ and a static worldline $y^{\mu}(\tau)$ tangent to $u^{\mu}$.
Similarly, we can define a Kerr-Schild vector

$\displaystyle K^{\mu}=\frac{[x^{\mu}-y^{\mu}(\tau)]_{\rm ret}}{R}~{}.$ (174)

It is not hard to prove

$\partial_{\mu}R=u_{\mu}-K_{\mu}~{},\qquad\partial_{\mu}K_{\nu}=\frac{1}{R}\left(\eta_{\mu\nu}+K_{\mu}K_{\nu}-K_{\mu}u_{\nu}-K_{\nu}u_{\mu}\right)~{}.$
(175)

These two identities imply

$\displaystyle\partial_{\mu}S$ $\displaystyle=-4\pi\,S^{2}\,(u_{\mu}-K_{\mu})$
(176) $\displaystyle\partial_{\mu}\partial_{\nu}S$
$\displaystyle=3(4\pi)^{2}\,S^{3}\,(u_{\mu}-K_{\mu})(u_{\nu}-K_{\nu})+(4\pi)^{2}S^{3}\left(\eta_{\mu\nu}-u_{\mu}u_{\nu}\right)~{}.$

The last line can be rewritten as

$\displaystyle\partial_{\mu}\partial_{\nu}S=3\,\frac{\partial_{\mu}S\,\partial_{\nu}S}{S}+(4\pi)^{2}\,S^{3}\left(\eta_{\mu\nu}-u_{\mu}u_{\nu}\right)~{}.$
(177)

Upon substitution in (172), the contribution from the last term on the right-
hand side of the expression above vanishes. Hence,

$\displaystyle\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\partial^{[\tau}u^{\lambda]}\partial^{[\eta}u^{\omega]}\,S(x)=\frac{3}{S(x)}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\left(\partial^{[\tau}u^{\lambda]}S(x)\right)\left(\partial^{[\eta}u^{\omega]}S(x)\right)~{}.$
(178)

This expression completes the argument, because the factors in parenthesis
equal the field strength tensor up to some constants,

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle$
$\displaystyle=-\frac{3\kappa^{2}\,m}{4}\,\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\,\frac{F^{\tau\lambda}F^{\eta\omega}}{S}~{}.$
(179)

This is the Weyl double copy relation for Schwarzschild, which is also valid
in higher dimensions for an appropriate constant factor.

Above, we have presented the proof for the simplest example, where the
deformation parameters $\bar{\theta}$ and $\bar{a}$ are set to zero. Since the
Kerr-Taub-NUT solution satisfies the Weyl double copy, a similar factorisation
must happen for generic $\bar{\theta}$ and $\bar{a}$. The proof is rather
straightforward. The key observation is that (178) holds also for
$S_{\bar{a},\bar{\theta}}$, since the effects of $\bar{a}$ and $\bar{\theta}$
are a translation and a constant scaling respectively. Then, after some
algebra, one can prove that

$\displaystyle\langle\mathfrak{R}^{(h)\,\mu\nu\rho\sigma}\rangle$
$\displaystyle=-\frac{3\kappa^{2}\,m}{4}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\left(\frac{F_{-}^{\tau\lambda}F_{-}^{\eta\omega}}{S_{\bar{a},\bar{\theta}}}+\frac{F_{+}^{\tau\lambda}F_{+}^{\eta\omega}}{S_{-\bar{a},-\bar{\theta}}}\right)~{},$
(180)

where $F_{+}$ and $F_{-}$ are the self-dual and anti-self-dual parts of the
field strength as defined in (65). This result represents the Weyl double copy
expressed in tensorial form, a map that was also studied in Alawadhi:2019urr ;
Alawadhi:2020jrv .

It would be interesting to see whether these simple position-space double copy
relations can be extended to include the dilaton. We start from the linear
dilaton contribution in position space,

$\displaystyle\langle\mathfrak{R}^{(\phi)\,\mu\nu\rho\sigma}(x)\rangle=\frac{\tilde{c}\,m\,\kappa}{2}\,\operatorname{Re}\,i\,\int\mathrm{d}\Phi(k)\hat{\delta}(2k\cdot
p)e^{-ik\cdot x}k^{[\mu}\eta^{\nu][\rho}k^{\sigma]}~{}.$ (181)

On the support of the delta functions, we can write this as

$\displaystyle\langle\mathfrak{R}^{(\phi)\,\mu\nu\rho\sigma}(x)\rangle=-\frac{\tilde{c}\,m\,\kappa}{2}\,\operatorname{Re}\,i\int\mathrm{d}\Phi(k)\hat{\delta}(2k\cdot
p)e^{-ik\cdot x}f^{\lambda[\mu}\eta^{\nu][\rho}f^{\sigma]}{}_{\lambda}~{},$
(182)

where $f^{\mu\nu}=k^{[\mu}u^{\nu]}$. We can follow the same steps as before to
obtain a position space convolutional double copy

$\displaystyle\langle\mathfrak{R}^{(\phi)\,\mu\nu}{}_{\rho\sigma}(x)\rangle=-\frac{\tilde{c}\,\kappa}{8\,Q^{2}}\left(F^{\lambda[\mu}\circ
S^{-1}\circ F_{\lambda[\rho}\right)\\!(x)\ \,{\delta^{\nu]}}_{\sigma]}~{}.$
(183)

In contrast to what happened for the graviton contribution, these convolutions
cannot be turned into products. To see that, we could proceed as for the
graviton and rewrite (182) as a second order differential operator acting on
the Lorentz continuation of $S(x)$. Then, we would like to use (177) to
accomplish the factorisation. The result would be

$\displaystyle\langle\mathfrak{R}^{(\phi)\,\mu\nu}{}_{\rho\sigma}(x)\rangle=\frac{\tilde{c}\,\kappa}{8}\left(3\frac{F^{\lambda[\mu}\,\delta^{\nu]}_{[\rho}\,F_{\sigma]\lambda}}{S}+(4\pi)^{2}S^{3}\left(\delta^{[\mu}_{[\rho}\delta^{\nu]}_{\sigma]}-\delta^{[\mu}_{[\rho}u^{\nu]}\,u_{\sigma]}\right)\right)~{}.$
(184)

While the first term exhibits a local position-space double copy form, the
others do not. Thus, the double copy of the dilatonic contribution is natural
only in terms of convolutions and it is non-local in position space.
Interestingly, the JNW solution admits an exact double copy interpretation in
position space, based on a Kerr-Schild-like construction in double field
theory Kim:2019jwm . However, unlike the Kerr-Schild double copy for
Schwarzschild, the dilatonic deformation makes the relation non-local in
position space.

##  9 Discussion

Our understanding of the link from scattering amplitudes to classical
solutions has matured. It has become straightforward to construct the
linearised solutions associated to massive three-point amplitudes in four
dimensions. The technical developments which have clarified this link are the
KMOC formalism Kosower:2018adc ; Maybee:2019jus ; Cristofoli:2021vyo , which
constructs classical observables from amplitudes, and the analytic
continuation Monteiro:2020plf  to split signature metrics. This second step
allows us to use the methods of KMOC in the context of three-point amplitudes.

In this paper, we took advantage of these developments to perform a
comprehensive survey of the classical fields constructed by the double copy
from gauge theory three-point amplitudes. As anticipated by previous work on
the classical double copy Monteiro:2014cda ; Luna:2015paa ; Luna:2018dpt ;
Huang:2019cja ; Emond:2020lwi , we saw that magnetic charge in gauge theory
indeed double copies to NUT charge in gravity. Furthermore, our methods
confirmed the three-point amplitudes associated by more indirect arguments
Huang:2019cja ; Emond:2020lwi  to Taub-NUT and its spinning generalisation,
Kerr-Taub-NUT. Indeed a split-signature form of Kerr-Taub-NUT and its
scattering amplitude also appeared very recently in independent work
Crawley:2021auj ; Guevara:2021yud .

The most basic example of a classical double copy is that from Coulomb to
Schwarzschild Monteiro:2014cda . However, the literature also contains a
_different_ double copy of Coulomb: namely the JNW solution Goldberger:2016iau
; Luna:2016hge . The origin of this non-uniqueness was discussed in
Kim:2019jwm , where a Kerr-Schild-type exact double copy interpretation of JNW
was also presented, and in Luna:2020adi , where an off-shell convolutional
approach based on the BRST formulation (including Fadeev-Popov ghosts)
Anastasiou:2018rdx  was used. We are also able to understand the origin of the
non-uniqueness using the framework introduced in our paper. It arises directly
from choices inherent in the standard double copy of scattering amplitudes. At
the level of amplitudes, it is always possible to _define_ the gravitational
theory by declaring that its three-point amplitudes are either the double copy
of two same-helicity gluons (resulting in Einstein gravity) or the double copy
of two same-helicity gluons _and_ two opposite-helicity gluons (resulting in
NS-NS gravity). Making the former choice, the double copy of Coulomb is indeed
Schwarzschild. The latter choice, by contrast, leads to the JNW solution. So
we are free to choose the couplings of the massive particle.

It is fascinating that the Kerr solution corresponds to a particularly simple
three-point amplitude Arkani-Hamed:2019ymq . Clearly, this fact is related to
the Newman-Janis shift, which is an all-orders property of Kerr Newman:1965tw
. The “single” copy of the Kerr solution, $\sqrt{\text{Kerr}}$, is a solution
of the Maxwell equations which is also endowed with a simple three-point
amplitude. Turning on a magnetic charge in addition to the spin leads to a
spinning dyonic solution which is, to date, the most general known three-point
amplitude in pure gauge theory. The double copy of this amplitude in pure
gravity is the Kerr-Taub-NUT solution Emond:2020lwi . However, it is also
possible to perform the double copy of these amplitudes in NS-NS gravity
where, as we have seen, the resulting class of solutions is of the type Kerr-
Taub-NUT-dilaton-axion. This generalises the previous discussion of the double
copy from Coulomb to the JNW solution to the more general three-point
amplitudes.

The double copy of gauge theory contains even more general possibilities.
Including scalar matter in the definition of the gauge theory introduces a new
three-point amplitude: this time between the pointlike source and the scalar
field. The resulting “heterotic” double copy is a gravitational theory
including dilatons and axions as well as gauge fields. To date, this is the
most general class of theories in which we can explicitly relate the double
copy for three-point amplitudes to explicit classical fields.

Our work also dealt with another apparent mystery in the classical double
copy. For scattering amplitudes, the double copy is clearly a creature of
momentum space: it is local in that context. Yet the classical double copy is
frequently presented in position space — the question is then how does
locality in position space somehow become locality in momentum space? In
section 8, we showed that this locality arises non-trivially only in specific
cases which include the Kerr-Taub-NUT solution.

In spite of this progress, there is still much to understand. The classical
double copy obviously applied to solutions which are not (yet) connected to
scattering amplitudes. An important example is the (A)dS-Schwarzschild metric,
which is related by the Kerr-Schild double copy to an electromagnetic solution
with a point charge immersed in a background of constant charge density
Luna:2015paa . Given that the classical double copy connects to scattering
amplitudes as well as configurations with a cosmological constant, perhaps it
can lead to some insight on the double copy in the presence of a cosmological
constant.

We focussed throughout on four-dimensional metrics. Of course this case is
particularly important, and (since gravitational wave phenomenology motivates
much of the recent interest in amplitudes and classical gravity) it is also
true that particular attention has been given to the relevant amplitudes in
four dimensions. However, black holes in higher dimensions have a number of
fascinating properties which have received intense scrutiny in the literature.
It would be interesting to generalise our methods to this case. The spinorial
structure of the generalised curvature which was so helpful for us would need
to be understood; we expect that this should link to the higher-dimensional
spinorial decomposition of the curvature described in reference
Monteiro:2018xev .

Finally, let us highlight an outstanding mystery facing the classical double
copy. In this article, we focused on linearised solutions of gravity. But in
terms of the single copy, our solutions are exact. In the case of
Schwarzschild (and indeed Kerr-Taub-NUT), the exact Weyl tensor, in an
appropriate tetrad, is linear in mass (and NUT charge) and so equals its
linearised approximation. However, this is not the case in general. How should
we understand the curvature corrections in these cases from the perspective of
amplitudes? Perhaps we can turn to the beautiful computations of Mougiakakos
and Vanhove Mougiakakos:2020laz  for further insight. A related, but perhaps
more difficult, question is what is the meaning of the Kerr-Schild property of
certain spacetime metrics in terms of scattering amplitudes?

## Acknowledgements

We are grateful to Rashid Alawadhi, Kanghoon Lee, Nathan Moynihan and Chris
White for related discussions. SN is supported by STFC grant ST/T000686/1. RM
and DPV are supported by the Royal Society via a University Research
Fellowship and a Studentship Grant, respectively.

##  Appendix A Conventions

We work in a Minkowski spacetime with signature

$\eta_{\mu\nu}=\text{diag}(+1,-1,-1,-1),$ (185)

which is then continued to the split signature one

$\eta_{\mu\nu}=\text{diag}(+1,+1,-1,-1).$ (186)

Our Fourier transforms conventions are

$\displaystyle f(x)$ $\displaystyle=\int\hat{\mathrm{d}}^{4}k\,e^{-ik\cdot
x}\,\tilde{f}(k)\,,$ (187) $\displaystyle\tilde{f}(k)$
$\displaystyle=\int\mathrm{d}^{4}x\,e^{ik\cdot x}\,f(x)\,,$

where to clean up factors of $2\pi$ we write

$\displaystyle\hat{\mathrm{d}}^{n}k=\frac{\mathrm{d}^{n}k}{(2\pi)^{n}}\,,\qquad\hat{\delta}^{n}(k)=(2\pi)^{n}\delta^{n}(k)\,.$
(188)

We also define (anti-)symmetrized brackets as

$\displaystyle v^{(\mu}w^{\nu)}=v^{\mu}w^{\nu}+v^{\nu}w^{\mu},\qquad
v^{[\mu}w^{\nu]}=v^{\mu}w^{\nu}-v^{\nu}w^{\mu}.$ (189)

##  Appendix B Volume form and null tetrad

This appendix is devoted to relating the null tetrad to the volume form. In
four-dimensional spacetimes, any four-form must be proportional to the volume
form. In particular,

$k_{[\mu}n_{\nu}\varepsilon_{+\,\rho}\varepsilon_{-\,\sigma]}\propto\epsilon_{\mu\nu\rho\sigma}~{}.$
(190)

The proportionality factor can be obtained as follows. First, note that

$\displaystyle\epsilon^{\mu\nu\rho\sigma}\epsilon_{\mu\nu\rho\sigma}=4!~{}.$
(191)

Next, we square the antisymmetrised tetrad,

$\displaystyle
k_{[\mu}n_{\nu}\varepsilon_{+\,\rho}\varepsilon_{-\,\sigma]}k^{[\mu}n^{\nu}\varepsilon_{+}{}^{\rho}\varepsilon_{-}{}^{\sigma]}=4!(k\cdot
n)^{2}(\varepsilon_{+}\cdot\varepsilon_{-})^{2}=4!~{}.$ (192)

Comparing both equalities, we conclude that

$\epsilon_{\mu\nu\rho\sigma}=\pm
k_{[\mu}n_{\nu}\varepsilon_{+\,\rho}\varepsilon_{-\,\sigma]}~{}.$ (193)

The sign on the right is not fixed, as it depends on the particular choice of
tetrad. Right-handed tetrads satisfy the relation with the plus sign. Left-
handed tetrads can be turned into right-handed ones by interchanging
$\varepsilon_{+}$ and $\varepsilon_{-}$. Therefore, without loss of
generality, we can assume that the tetrad is right-handed.

Finally, we can obtain an useful relation by contracting the volume form with
$k_{\rho}$ and $u_{\sigma}$,

$\displaystyle\epsilon^{\mu\nu\rho\sigma}k_{\rho}\,u_{\sigma}$
$\displaystyle=-\varepsilon_{-}\cdot
u\,\epsilon^{\mu\nu\rho\sigma}k_{\rho}\varepsilon_{+\,\sigma}-\varepsilon_{+}\cdot
u\,\epsilon^{\mu\nu\rho\sigma}k_{\rho}\varepsilon_{-\,\sigma}$ (194)
$\displaystyle=-k^{[\mu}n^{\nu}\varepsilon^{+\,\rho}\varepsilon^{-\,\sigma]}\left(\varepsilon_{-}\cdot
u\,k_{\rho}\varepsilon_{+\,\sigma}+\varepsilon_{+}\cdot
u\,k_{\rho}\varepsilon_{-\,\sigma}\right)$
$\displaystyle=-\left(\varepsilon_{-}\cdot
u\,k^{[\mu}\varepsilon_{+}^{\nu]}-\varepsilon_{+}\cdot
u\,k^{[\mu}\varepsilon_{-}^{\nu]}\right)~{}.$

##  Appendix C Gravitational coherent state

This section describes the exponentiation of the coherent state in the
presence of the graviton, dilaton and axion fields. Given that our initial
state contains only one creation operator for the massive particle, the
S-matrix can be expanded in multiplicities as

$\displaystyle
S|\psi\rangle=\frac{1}{\mathcal{N}}(1+iT_{3}+iT_{4}+\cdots)|\psi\rangle\,,$
(195)

where the $T_{n}$ are defined by

$\displaystyle T_{n+2}$
$\displaystyle=\frac{1}{n!}\sum_{\eta_{1},\ldots,\eta_{n}}\int\mathrm{d}\Phi(p^{\prime})\mathrm{d}\Phi(p)\prod_{i=1}^{n}\mathrm{d}\Phi(k_{i})\,\sum_{m=0}^{n}\frac{n!}{l!\,m!\,(n-l-m)!}$
(196)
$\displaystyle\mathcal{M}^{(l,m,n-m-l)}_{-\eta_{1},\ldots,-\eta_{m}}(p\rightarrow
p^{\prime},k_{1},\cdots,k_{n})\times\hat{\delta}^{4}\left(p-p^{\prime}-\sum
k_{i}\right)\,a^{\dagger}(p^{\prime})a(p)$ $\displaystyle\qquad\qquad
a^{\dagger}_{\eta_{1}}(k_{1})\cdots
a^{\dagger}_{\eta_{l}}(k_{m})a^{\dagger}_{\phi}(k_{l+1})\cdots