# Scattering on self-dual Taub-NUT

Tim Adamo  Giuseppe Bogna  Lionel Mason  Atul Sharma

###### Abstract

We derive exact solutions of massless free field equations and tree-level two-
point amplitudes up to spin 2 on self-dual Taub-NUT space-time, as well as on
its single copy, the self-dual dyon. We use Killing spinors to build analogues
of momentum eigenstates, finding that, in the spirit of color-kinematics
duality, those for the self-dual dyon lift directly to provide states on the
self-dual Taub-NUT background if one replaces charge with energy. We discover
that they are forced to have faster growth at infinity than in flat space due
to the topological non-triviality of these backgrounds. The amplitudes for
massless scalars and spinning particles in the $(+\,+)$ and $(+\,-)$ helicity
configurations vanish for generic kinematics as a consequence of the
integrability of the self-dual sector. The $(-\,-)$ amplitudes are non-
vanishing and we compute them exactly in the backgrounds, which are treated
non-perturbatively. It is explained how spin is easily introduced via a
Newman-Janis imaginary shift along the spin-vector leading directly to the
additional well-known exponential factor in the dot product of the spin with
the momenta. We also observe a double copy relation between the gluon
amplitude on a self-dual dyon and graviton amplitude on a self-dual Taub-NUT
space-time.

##  1 Introduction

Physics in black hole backgrounds remains a major avenue for explorations in
classical and quantum gravity. One concrete setting which captures both of
these regimes is that of _scattering_ off a black hole background. In the
classical context, the computation of probe or wave scattering off a black
hole encodes classical observables including scattering angle, waveform and
power emitted (cf., [1, 2, 3, 4, 5, 6, 7, 8]). Indeed, the computation of
scattering amplitudes with emission in curved background space-times provides
the on-shell building blocks of the self-force expansion [9, 10, 11, 12, 13],
which systematizes corrections to geodesic motion due to radiation reaction
(cf., [14, 15]). In the quantum context, scattering near black hole event
horizons is a playground where emergent features of quantum gravity in
strongly curved space-times can potentially be studied [16, 17, 18, 19, 20,
21, 22, 23]. More generally, in proposals for holography in asymptotically
flat space-times, such as celestial holography (cf., [24, 25]), the S-matrix
is the natural boundary observable. Scattering amplitudes on black hole space-
times thus provide key data for asymptotically flat holography.

However, the analytic computation of scattering on black hole space-times is
rife with difficulties. One obvious complication is the fact that the S-matrix
does not exist as a unitary operator in the background QFT, due to the
presence of the event horizon. This problem can be circumvented, at least
perturbatively, by providing a purely variational definition of tree-level
‘scattering amplitudes’ as multi-linear pieces of the classical background
field action, evaluated on recursively constructed solutions to the
background-coupled equations of motion. This is sometimes called the
‘perturbiner’ approach [26, 27, 28, 29], and scattering amplitudes defined in
this way contain all of the expected dynamical information [30, 31, 32].

This does nothing to ameliorate the underlying technical complication of
determining the ingredients necessary to compute a scattering amplitude in a
black hole background. This is apparent already at the level of solving the
linearised, background-coupled equations of motion that determine the
asymptotic scattering states. Even in the toy model of scattering scalar
charges on a Coulomb potential, the solutions to the scalar wave equation are
given by confluent hypergeometric functions, while on a Schwarzschild metric,
solutions of the scalar wave equation are confluent Heun functions (cf.,
[33]). Linearised gravitational waves (i.e., gravitons) are determined by
solving the Regge-Wheeler-Zerilli equations on Schwarzschild [34, 35], or the
Teukolsky equations on Kerr [36, 37]. While these equations are amenable to
numerical methods, constructing general solutions analytically is hard.

One possibility for circumventing these (and other) difficulties is to first
compute scattering amplitudes on _self-dual_ analogues of black hole space-
times. The underlying integrability of the self-dual sector could – in
principle, at least – enable all-multiplicity amplitude computations around
the self-dual background, which could in turn be leveraged to perturbatively
reconstruct the anti-self-dual part of the physical, non-chiral black hole. Of
course, this requires first being able to compute all-multiplicity information
on a self-dual background in the first place, which is far from obvious.
Furthermore, scattering on self-dual backgrounds is of interest in its own
right, particularly in the context of celestial holography, where the only
known top-down construction of a holographic dual involves an asymptotically
flat self-dual (scalar-flat Kähler) bulk geometry [38, 39].

In [40, 41, 42, 43] a framework was developed to compute all-multiplicity
graviton scattering on self-dual _radiative_ space-times, as well as all-
multiplicity gluon scattering on self-dual radiative gauge fields. These
backgrounds are determined by their free radiative data at null infinity. As
they are fully non-linear backgrounds, the resulting scattering amplitudes
resum infinitely many contributions that would arise in conventional
perturbation theory around a trivial vacuum. While this realizes the goal of
obtaining all-multiplicity results around the self-dual sector, black hole
space-times are stationary with no radiation, so do not fall into the
radiative class of metrics.

In this paper, we begin the task of extending the work of [40, 41, 42, 43] to
self-dual black hole space-times. The first order of business in this regard
is to determine the massless free fields, the analogues of momentum
eigenstates, on such backgrounds. These will serve as the external
wavefunctions for massless scattering processes on self-dual black hole
backgrounds. Furthermore, knowing them immediately allows us to determine the
2-point amplitudes on the background, by simply evaluating the quadratic part
of the background field action on two such free fields.

The paradigmatic self-dual black hole is the _self-dual Taub-NUT_ (SDTN)
solution [44]. This is the unique complexified metric in the Taub-NUT family
[45, 46] that is self-dual, with the mass $M$ and NUT parameter $N$ related by
$N=-\mathrm{i}M$. The SDTN metric is also the self-dual part of the Kerr
metric, as can be seen by analytically continuing through an imaginary shift
in the direction of the spin vector. This is the basis of the Newman-Janis
correspondence [47] and is particularly easy to see in terms of the expression
for the ASD Weyl spinor in terms of the valence-two Killing spinor
$\chi_{\alpha\beta}$ [48, 49, 50, 51] and §6 of [52]. We have

$\psi_{\alpha\beta\gamma\delta}=M\frac{\chi_{(\alpha\beta}\chi_{\gamma\delta)}}{\chi^{5}}\,,\qquad\chi^{2}:=\chi_{\alpha\beta}\chi^{\alpha\beta}\,.$
(1.1)

Here $M$ is the mass, and in the rectangular coordinates $\vec{x}$ for the
original Kerr metric [53],
$\chi^{\alpha\beta}=(\vec{x}-\mathrm{i}\vec{a})\cdot\vec{\sigma}_{\alpha\beta}$,
where $\vec{\sigma}_{\alpha\beta}$ is the 3-vector of Pauli matrices and
$\vec{a}$ the spin vector aligned along the $z$-axis. The self-dual Weyl
spinor is obtained from the complex conjugate
$\bar{\chi}_{\dot{\alpha}\dot{\beta}}$ and both are obtained from the
corresponding expression for Schwarzschild by opposite imaginary shifts along
$\pm\vec{a}$. For the full Schwarzschild and Kerr metrics, this requires a
somewhat mysterious nonlinear superposition of shifts in both
$\pm\mathrm{i}\vec{a}$. However, in the self-dual sector, the spinning SD
Taub-NUT metric is simply the complex metric obtained by translating the
original SDTN along $\mathrm{i}\vec{a}$. We will use this fact later to obtain
spinning analogues of our formulae.

Our focus in this paper will be on constructing solutions to the massless free
field equations of spin 0, 1 and 2 on the SDTN background, and computing the
resulting two-point scattering amplitudes. In the spirit of the double copy
[54], we also perform the same computations for the _self-dual dyon_ (SDD),
the self-dual combination of the Coulomb solution and a magnetic monopole.
Indeed we see that our momentum eigenstates on the SDD background lift
directly to provide those on the SDTN background.

We obtain explicit expressions for linear, massless background-coupled fields
on SDD and SDTN backgrounds. In the study of self-dual radiative backgrounds,
linear fields were determined using twistor methods [41, 43], but on the SDD
and SDTN backgrounds more elementary techniques can be used.111Indeed, in
contrast to the radiative case, SDD and SDTN do not have functional degrees of
freedom. Our main tool for constructing our candidate momentum eigenstates is
to introduce novel _charged Killing spinors_. There is then an elementary lift
from charged massless fields on SD dyons to massless fields whose energy is
related to the charge on the SD Taub-NUT. These are special structures on
these backgrounds and result in remarkable, elementary expressions that are
far simpler than the confluent hypergeometric functions expected in the
Coulomb case, or the Heun functions for Schwarzschild or Kerr. This simplicity
allows us to compute the 2-point functions of these massless linear fields by
direct integration.

Another novelty associated to these computations, which can be viewed in both
a positive and negative light, is that both the SDD and SDTN are
_topologically non-trivial_. The SDD is the self-dual superposition of the
Coulomb and Dirac monopole fields; the latter has a non-trivial Chern class on
the spheres of constant radius and the monopole charge is necessarily
quantized [55, 56, 57, 58]. Charged fields therefore have spin weight on the
sphere, and consequently momentum eigenstates have non-trivial spherical
harmonics, leading to higher radial growth at infinity than expected for un-
charged fields. Changes in asymptotic boundary conditions are an expected
feature due to the underlying monopole topology (cf., [59, 60, 61, 62, 63,
64]).

A similar phenomenon arises on SDTN, where the topology enforces a periodic
time variable leading to a circle bundle with non-trivial topology on spheres
of constant radius [46, 65]. This leads to quantization of the energy
$\omega_{i}$ of the particles $4M\omega_{i}\in\mathbb{Z}$, and fields with
non-vanishing energy have non-trivial spin weight on the sphere. Again, this
means that momentum eigenstates on the SDTN space-time have enhanced growth at
infinity.

One could worry that this enhanced growth at infinity renders the scattering
amplitudes of these background-coupled fields ill-defined. Indeed, the usual
way of computing 2-point amplitudes in point-like potentials is to extract the
leading $r^{-1}$ behaviour of the free fields as $r\to\infty$ (cf., [66]), or
evaluate the quadratic action as a large-$r$ boundary term [11], and this
perspective is complicated somewhat by the non-trivial topology reflected in
our solutions. We find it more natural to evaluate the 2-point amplitude by
direct integration of the quadratic action, evaluated on a superposition of
two of the free field solutions. Remarkably, the enhanced polynomial growth in
$r$ is compensated by the plane wave phase in such a way that we obtain
simple, finite results. Our final answer for gravity on the spinning SD Taub-
NUT background is

$\mathcal{A}(1,2)=4\pi^{2}\,M\,(4M\omega_{2})!\,\mathrm{e}^{-(\vec{k}_{1}+\vec{k_{2}})\cdot\vec{a}}\,\frac{(2\omega_{1})^{4M\omega_{2}-2}\,\langle
1\,2\rangle^{4+4M\omega_{2}}}{|\vec{k}_{1}+\vec{k}_{2}|^{8M\omega_{2}+2}}\,\delta_{-\omega_{1},\omega_{2}}\,,$
(1.2)

where the particles have momenta $(\omega_{i},\vec{k}_{i})$ that are null with
energies $\omega_{i}=\pm|\vec{k}_{i}|$, quantized as above and $\langle
12\rangle$ is the usual spinor-helicity contraction of polarization spinors
for the two particles. The factor of
$\mathrm{e}^{-(\vec{k}_{1}+\vec{k_{2}})\cdot\vec{a}}$ follows from (4.16) by
translating the contour used in (4.13) by $\mathrm{i}$ times the spin-vector
$\vec{a}$ as described above. Analogous formulae for the SD dyon follow from
(4.10). Here the anomalous little-group weight arises from the topological
non-triviality above, and can be removed by a suitable normalization of the
states as described later.

#### Summary of the paper:

In section 2, we provide brief reviews of the SD dyon and SD Taub-NUT
backgrounds. Section 3 starts by introducing the main tool, charged solutions
to the Killing spinor equation coupled to the SD dyon. These allow us to make
general solutions on flat space to charged solutions in the self-dual dyon
background. This allows us to introduce general momentum eigenstates and
solutions to the Hamilton-Jacobi equations on these backgrounds. We then show
how these solutions can be lifted directly to the SD Taub-NUT background.
Various other solutions of different helicity and in potential form can then
be found. Section 4 is concerned with deriving 2-point tree-level around the
backgrounds under consideration. Section 5 contains a discussion of the
interpretation, the extension to a spinning SDTN background, the connection to
flat space amplitudes, as well as discussion of possible directions for future
research. Appendix A contains details of the integrals involved in the
evaluation of the 2-point amplitudes.

##  2 The classical backgrounds

We begin by reviewing our classical backgrounds of interest, starting with the
self-dual dyon (SDD) background in Euclidean signature. This starts life as a
background in electrodynamics, but can also be embedded as a Cartan-valued
background in non-abelian gauge theories. We then describe the metric of self-
dual Taub-NUT (SDTN) space in Lorentzian, Euclidean and split signatures. In
the following sections, we will focus on Euclidean signature.

###  2.1 Self-dual dyon

The self-dual dyon (SDD) arises, in the first instance, as a classical
solution of abelian gauge theory. A general dyon profile solving Maxwell’s
equations on Euclidean $\mathbb{R}^{4}$ can be constructed through a
superposition of electric and magnetic charges. The self-dual (SD) profile has
equal electric and magnetic charges and is captured by the gauge field

$A=\frac{\mathrm{d}t}{r}+a\,,\qquad a=(1-\cos\theta)\,\mathrm{d}\phi\,.$ (2.1)

Here $(r,\theta,\phi)$ are standard spherical polar coordinates on space
$\mathbb{R}^{3}-0=\mathbb{R}_{+}\times S^{2}$, and $t$ is the Wick-rotated
time coordinate from the Lorentzian slice.

The first term in (2.1) is the Coulomb part of the potential sourced by the
electric charge of the dyon, whereas the second term $a$ is the Dirac monopole
sourced by the magnetic charge. This has a wire singularity at $\theta=\pi$
which can be moved to $\theta=0$ by the gauge transformation
$\mathrm{e}^{2\mathrm{i}\phi}$. The associated field strength

$F=\mathrm{d}A=\frac{\mathrm{d}t\wedge\mathrm{d}r}{r^{2}}+\sin\theta\,\mathrm{d}\theta\wedge\mathrm{d}\phi$
(2.2)

is smooth on the sphere and is easily seen to be self-dual with respect to the
Hodge star operator associated to the flat Euclidean metric on
$\mathbb{R}^{4}$. As such, it satisfies Maxwell’s equations and the electric
and magnetic charge are equal.

In order to construct non-trivial amplitudes on this background, we can embed
the SDD into the Cartan subalgebra of the gauge algebra of a generic non-
abelian Yang-Mills theory. Let $\mathsf{c}$ be an element of the Cartan
subalgebra and simply set

$A=\mathsf{c}\left(\frac{\mathrm{d}t}{r}+a\right)\,.$ (2.3)

For example, in the simplest case of $\mathrm{SU}(2)$ the element $\mathsf{c}$
can be taken to be the Pauli matrix $\sigma_{3}$. Since the gauge field lies
in an abelian subalgebra of the gauge algebra, the corresponding field
strength reduces to

$F=\mathrm{d}A+\frac{1}{2}\,[A,A]=\mathsf{c}\left(\frac{\mathrm{d}t\wedge\mathrm{d}r}{r^{2}}+\sin\theta\,\mathrm{d}\theta\wedge\mathrm{d}\phi\right)\,.$
(2.4)

This is essentially the same as the abelian field strength, and continues to
satisfy the (non-abelian) self-duality equations.

###  2.2 Self-dual Taub-NUT

#### Lorentzian signature:

Taub-NUT space-time takes the form

$\mathrm{d}s^{2}=f(r)\left(\mathrm{d}t-2N\,a\right)^{2}-\frac{\mathrm{d}r^{2}}{f(r)}-\left(r^{2}+N^{2}\right)\mathrm{d}\Omega^{2}\,,$
(2.5)

where $a$ is the monopole field from (2.1) and

$f(r)=\frac{r^{2}-2Mr-N^{2}}{r^{2}+N^{2}}=1-\frac{M+\mathrm{i}N}{r+\mathrm{i}N}-\frac{M-\mathrm{i}N}{r-\mathrm{i}N}\,.$
(2.6)

Here $M$ is ADM mass and $N$ is the NUT charge; $\mathrm{d}\Omega^{2}$ is the
round sphere metric on $S^{2}$, whereas $\mathrm{d}a$ is the volume form on
$S^{2}$.

In a choice of complex stereographic coordinate
$\zeta=\mathrm{e}^{\mathrm{i}\phi}\tan(\theta/2)$ one has

$\mathrm{d}\Omega^{2}=m\,\bar{m}\,,\qquad
m\coloneqq\frac{2\,\mathrm{d}\zeta}{1+|\zeta|^{2}}\,.$ (2.7)

Gauge transformations of $a\rightarrow a+\mathrm{d}g$, for some function $g$,
are accompanied by coordinate transformations $t\rightarrow t+2Ng$. Globally,
this requires the identification $t\sim t+8\pi\mathrm{i}N$. Setting
$N=-\mathrm{i}M$ produces the complex metric

$\mathrm{d}s^{2}=f(r)\,(\mathrm{d}t+2\mathrm{i}\,Ma)^{2}-\frac{\mathrm{d}r^{2}}{f(r)}-\left(r^{2}-M^{2}\right)\mathrm{d}\Omega^{2}\,,\qquad
f(r)=\frac{r-M}{r+M}\,,$ (2.8)

which is vacuum and self-dual (in the holomorphic, complexified category).

#### Euclidean signature:

The metric (2.8) acquires a Euclidean real slice with the replacement of $t$
by $\mathrm{i}t$, and $r-M$ by $r$, taking $M$ to be real [44]. Setting

$\vec{x}=(r\sin\theta\cos\phi,\,r\sin\theta\sin\phi,\,r\cos\theta)\,,\qquad(\theta,\phi)\in
S^{2}\,,$ (2.9)

the SDTN metric takes the form of a Gibbons-Hawking gravitational instanton
[67, 68]:

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=\left(1+\frac{2M}{r}\right)^{-1}\left(\mathrm{d}t-2Ma\right)^{2}+\left(1+\frac{2M}{r}\right)\mathrm{d}\vec{x}^{2}$
(2.10) $\displaystyle\equiv
V^{-1}(\mathrm{d}t-2Ma)^{2}+V\,\mathrm{d}\vec{x}^{2}\,,$

where
$\mathrm{d}\vec{x}\,^{2}=(\mathrm{d}x^{1})^{2}+(\mathrm{d}x^{2})^{2}+(\mathrm{d}x^{3})^{2}$,
and $V$, $a$ are given by

$V\coloneqq 1+\frac{2M}{r}\,,\qquad a\coloneqq(1-\cos\theta)\mathrm{d}\phi\,.$
(2.11)

More generally, this ansatz gives a self-dual Ricci-flat metric when the
scalar and vector potentials $(V,a)$ solve the Bogomolny equation on
$\mathbb{R}^{3}-0$,

$\partial_{i}V+2M\epsilon_{i}{}^{jk}\partial_{j}a_{k}=0\,,$ (2.12)

where $\partial_{i}\equiv\partial/\partial x^{i}$, $i=1,2,3$. With $(V,a)$
given by (2.11), they describe the Euclidean SDTN metric.

Taking $r\geq 0$ and identifying $t\sim t+8\pi M$, this gives a complete, non-
trivial, self-dual Ricci-flat metric on $\mathbb{R}^{4}$. To see this, first
combine the spherical coordinates $(\theta,\phi)\in S^{2}$ with the periodic
coordinate $t\in S^{1}$ to find the total space of the Hopf fibration
$S^{3}\rightarrow S^{2}$ with fiber $S^{1}$. Together with
$r\in\mathbb{R}_{+}$ these make up coordinates on
$\mathbb{R}^{4}-0=\mathbb{R}_{+}\times S^{3}$. One then completes the space-
time into $\mathbb{R}^{4}$ by adding back the origin to $\mathbb{R}^{4}-0$.
The $S^{3}$s at constant $r$ approach spheres of radius $\sqrt{2Mr}$ as
$r\rightarrow 0$, and shrink to a point at the origin [69].

The locus $r=0$ is a Euclidean analogue of the black hole horizon for SDTN.
However, in Euclidean signature this is not an actual horizon, but only a
single point. The metric has an apparent singularity at $r=0$, but can be
extended across it by a standard change of coordinates. Computing curvature
invariants like the Kretschmann scalar, one finds a true curvature singularity
at $r=-2M$, but for $M>0$, this point is outside of Euclidean
$\mathbb{R}^{4}$. Equivalently, it can be thought of as being hidden behind
the ‘horizon’. This intuition becomes precise in split signature, as we
briefly mention below.

The metric (2.10) is easily verified to be Ricci-flat. For the orientation in
which the volume form is
$\sqrt{|g|}\,\mathrm{d}t\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}$,
(2.10) is also self-dual. That is, the Weyl tensor $C_{\mu\nu\rho\sigma}$,
viewed as a $2$-form on either its first two or last two indices, is self-
dual: $C=*C$, where $*$ is the Hodge star of (2.10) in the chosen orientation.
This means that the metric is hyperkähler: there exist a 2-sphere’s worth of
complex structures on $\mathbb{R}^{4}$ that are compatible as well as
covariantly constant with respect to this metric.

#### Split signature:

The global structure of the SDTN space-time in split (i.e., $(+-\,-\,+)$)
signature was studied in [70, 71]. In this signature, the authors of [70]
interpret it as a genuine black hole geometry.

The Wick rotation that takes (2.10) from Euclidean to split signature while
preserving the sign of the volume form
$\sqrt{|g|}\,\mathrm{d}t\wedge\mathrm{d}x^{1}\wedge\mathrm{d}x^{2}\wedge\mathrm{d}x^{3}$
is given by $(x^{1},x^{2})\mapsto(\mathrm{i}x^{1},-\mathrm{i}x^{2})$, or
equivalently $\theta\mapsto\mathrm{i}\theta,\phi\mapsto-\phi$. The resulting
metric reads222The metric employed in [70] is found by a further change of
coordinates $t\mapsto t-2M$, $r\mapsto r-M$.

$\mathrm{d}s^{2}=\left(1+\frac{2M}{r}\right)^{-1}\big{[}\mathrm{d}t+2M\,(1-\cosh\theta)\,\mathrm{d}\phi\big{]}^{2}+\left(1+\frac{2M}{r}\right)\mathrm{d}\vec{x}^{2}\,,$
(2.13)

where now
$\mathrm{d}\vec{x}^{2}=(\mathrm{d}x^{3})^{2}-(\mathrm{d}x^{1})^{2}-(\mathrm{d}x^{2})^{2}$.
In this signature, the coordinates range over $r,\theta\geq 0$,
$\phi\sim\phi+2\pi$, $t\sim t+8\pi M$.

Once again, the space-time has a coordinate singularity at $r=0$. This happens
to be a genuine horizon instead of a single point as in Euclidean signature,
and the metric can be extended beyond this horizon into the interior of this
black hole by continuing $r$ to the negative reals. The maximal extension
encounters a curvature singularity at $r=-2M$, where the space-time ends. This
justifies the use of the terminology _self-dual black hole_.

Another important distinction between the geometries is that in Euclidean
signature, surfaces of constant $t,r$ are spatial 2-spheres parametrized by
$\theta,\phi$. In split signature, a surface of constant $t$ and constant
$r^{2}=(x^{3})^{2}-(x^{1})^{2}-(x^{2})^{2}$ is a two-dimensional hyperbolic
space $\mathbb{H}_{2}$, also known as the Poincaré disk. So, for $r>0$, the
metric can be thought of as a metric on a circle bundle over
$\mathbb{R}_{+}\times\mathbb{H}_{2}$.

##  3 Massless free fields from Killing spinors

The first step in understanding quantum field theory on curved space-times is
to construct the Hilbert space of states. To study perturbation theory around
the free limit, this is taken to be the space of solutions of the classical
free field equations. In Euclidean signature, these are the Laplace equation
and its higher spin counterparts. In this section, we show that it is possible
to find exact expressions for classical solutions of massless free field
equations on both the SD dyon and SD Taub-NUT backgrounds. This is in stark
contrast to the corresponding situation in real Coulomb potentials or physical
black holes like Schwarzschild or Kerr, where such solutions can only be found
as spherical harmonic expansions with radial wavefunctions given by
complicated special functions like the Heun function (cf., [33]).

Our solutions will instead be plane waves dressed with simple power laws,
reducing to familiar momentum eigenstates in the flat limit $M\to 0$ or
$\mathsf{c}\to 0$. For this reason, we refer to these solutions as _quasi_
-momentum eigenstates, as there is no actual translation invariance in the SDD
or SDTN backgrounds. We first construct scalar and negative helicity states,
and then describe a spin-raising procedure to construct positive helicity
states. We mainly focus on spin 1 states around the SD dyon and spins 1 and 2
around SD Taub-NUT. Our main technique will be to introduce charge-raising and
lowering Weyl spinors that solve certain Killing spinor equations.333There is
considerable literature on Dirac Killing spinors on Taub-NUT and related
geometries, see for example [72, 73, 74, 75, 76]. It would be interesting to
study their relation, if any, to our Weyl Killing spinors. These allow us to
turn standard flat space momentum eigenstates into states on SDD and SDTN
backgrounds.

#### Spinor-helicity conventions:

In what follows, it will be useful to work with spinor-helicity variables
$\kappa^{\alpha}$ and $\tilde{\kappa}^{\dot{\alpha}}$ parametrizing null
momenta $k^{\mu}$ via the decomposition

$k^{\alpha{\dot{\alpha}}}=\frac{1}{\sqrt{2}}\begin{pmatrix}\omega+\mathrm{i}k^{3}&&k^{2}+\mathrm{i}k^{1}\\\
-k^{2}+\mathrm{i}k^{1}&&\omega-\mathrm{i}k^{3}\end{pmatrix}=\kappa^{\alpha}\tilde{\kappa}^{{\dot{\alpha}}}\,.$
(3.1)

Here, $\alpha=0,1$ and $\dot{\alpha}=\dot{0},\dot{1}$ are left- and right-
handed 2-component spinor indices, respectively. One can express
$\kappa^{\alpha}$ and $\tilde{\kappa}^{\dot{\alpha}}$ in terms of
stereographic coordinates
$(z,\tilde{z})\in\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ on the complexified
celestial sphere,

$z:=\frac{k^{1\dot{0}}}{k^{0\dot{0}}}=\frac{-k^{2}+\mathrm{i}k^{1}}{\omega+\mathrm{i}k^{3}}\,,\qquad\tilde{z}:=\frac{k^{0\dot{1}}}{k^{0\dot{0}}}=\frac{k^{2}+\mathrm{i}k^{1}}{\omega+\mathrm{i}k^{3}}\,.$
(3.2)

Up to little group scalings
$(\kappa,\tilde{\kappa})\sim(s\kappa,s^{-1}\tilde{\kappa})$,
$s\in\mathbb{C}^{*}$, we may solve for the spinor-helicity variables to find

$\kappa^{\alpha}=(1,z)\,,\qquad\tilde{\kappa}^{{\dot{\alpha}}}=\frac{\sqrt{2}\,\omega}{1+z\tilde{z}}\,(1,\tilde{z})\,.$
(3.3)

Fixing $\kappa^{0}=1$ as done here happens to be a convenient choice in self-
dual backgrounds. The Levi-Civita invariants
$\varepsilon^{\alpha\beta},\varepsilon^{{\dot{\alpha}}\dot{\beta}}$ of
$\mathrm{SL}(2,\mathbb{C})$ are used to raise and lower spinor indices and to
construct spinor brackets in the standard way:
$\langle\lambda\,\kappa\rangle\coloneqq\varepsilon^{\alpha\beta}\lambda_{\beta}\kappa_{\alpha}=\lambda^{\alpha}\kappa_{\alpha}$
and
$[\tilde{\lambda}\,\tilde{\kappa}]\coloneqq\varepsilon^{\dot{\alpha}\dot{\beta}}\tilde{\lambda}_{\dot{\beta}}\tilde{\kappa}_{\dot{\alpha}}=\tilde{\lambda}^{\dot{\alpha}}\tilde{\kappa}_{\dot{\alpha}}$
(using the conventions of [43]).

Similarly, we can introduce the spinor equivalent $x^{\alpha{\dot{\alpha}}}$
of the space-time coordinates $x^{\mu}\in\mathbb{R}^{4}$. In Euclidean
signature, this is given by

$x^{\alpha{\dot{\alpha}}}=\frac{1}{\sqrt{2}}\begin{pmatrix}x^{0}+\mathrm{i}x^{3}&&x^{2}+\mathrm{i}x^{1}\\\
-x^{2}+\mathrm{i}x^{1}&&x^{0}-\mathrm{i}x^{3}\end{pmatrix}\,.$ (3.4)

In our applications, we will take $x^{0}=t$ to be the periodic coordinate, and
$x^{i}$ to be the spatial coordinates. Similarly, spinor equivalents of
tensors on a 4-manifold with Riemannian metric $g$ can be locally defined by
working in a tetrad basis $\theta^{a}$. The spinor equivalent of $\theta^{a}$
is

$\theta^{\alpha{\dot{\alpha}}}=\frac{1}{\sqrt{2}}\begin{pmatrix}\theta^{0}+\mathrm{i}\theta^{3}&&\theta^{2}+\mathrm{i}\theta^{1}\\\
-\theta^{2}+\mathrm{i}\theta^{1}&&\theta^{0}-\mathrm{i}\theta^{3}\end{pmatrix}\,.$
(3.5)

The spinor components $A_{\alpha{\dot{\alpha}}}$ of any 1-form $A$ are the
components in its expansion
$A=A_{\alpha{\dot{\alpha}}}\theta^{\alpha{\dot{\alpha}}}$ in this co-frame,
and so on for other tensors; see [77, 52] for an extensive review of this
notation. In flat space, one simply sets
$\theta^{\alpha{\dot{\alpha}}}=\mathrm{d}x^{\alpha{\dot{\alpha}}}$.

###  3.1 Killing spinors and raising/lowering charge and helicity

A valence-$p$ anti-self-dual (ASD) Killing spinor on a general curved space-
time is a solution $\chi^{\alpha_{1}\ldots\alpha_{p}}$ to the equation

$\nabla^{(\alpha}{}_{\dot{\alpha}}\chi^{\beta_{1}\ldots\beta_{p})}=0\,.$ (3.6)

We will be interested in ASD Killing spinors of valence $p=1,2$ on SD
backgrounds and their helicity raising and lowering properties.

As shown in §6.4 of [52], a Killing spinor can be used to raise the helicity
of a solution to the helicity $-(n+p)/2$ massless free field equations

$\nabla^{\alpha{\dot{\alpha}}}\psi_{\alpha\alpha_{2}\ldots\alpha_{n+p}}=0$
(3.7)

to generate a solution

$\varphi_{\alpha_{1}\ldots\alpha_{n}}=\psi_{\alpha_{1}\ldots\alpha_{n+p}}\,\chi^{\alpha_{n+1}\ldots\alpha_{n+p}}$
(3.8)

of the helicity $-n/2$ field equations. That this satisfies the massless field
equations follows directly from the Leibnitz rule and (3.6). Observe that
constant spinors give trivial examples of this, but our focus will be on non-
trivial valence 1 and 2 Killing spinors, with those of valence one also
raising and lowering charge.

The valence-2 ASD Killing spinor appropriate to the SD dyon is given on flat
space by choosing a unit timelike vector
$T_{\alpha\dot{\alpha}}=\frac{1}{\sqrt{2}}\,\mathrm{diag}\,(1,1)$ and writing

$\chi^{\alpha\beta}=\sqrt{2}\,T^{(\alpha}{}_{\dot{\alpha}}x^{\beta)\dot{\alpha}}\,.$
(3.9)

Our charge raising and lowering operators will be spinors
$\chi^{\alpha}_{\pm}$ subject to the relations

$\chi^{\alpha\beta}=\chi_{+}^{(\alpha}\chi_{-}^{\beta)}\,,\qquad\chi_{+}^{\alpha}=\sqrt{2}\,\bar{\chi}_{-}^{\dot{\alpha}}\,T^{\alpha}{}_{\dot{\alpha}}\,,$
(3.10)

(or, equivalently, to be complex conjugates in Euclidean signature) which only
defines them up to a phase. This phase cannot be chosen to be global on the
sphere as the line subbundles they span inside the 4d spin bundle are non-
trivial on the sphere. These line bundles are the spin bundle of the 2-sphere
and its dual: $\mathscr{O}(\mp 1)\to S^{2}$, having Chern class $\mp 1$,
respectively. These are the monopole line bundles of minimal degree. However,
as we will see below, when understood as sections of line bundles of the
appropriate Chern class $\pm 1$, the spinors $\chi^{\alpha}_{\pm}$ are indeed
global on the sphere.

We can make explicit choices by first defining stereographic coordinates on
the 2-spheres at constant $t$ and $r$

$\zeta\coloneqq\frac{x^{1}+\mathrm{i}x^{2}}{r+x^{3}}=\mathrm{e}^{\mathrm{i}\phi}\tan\frac{\theta}{2}\,,\qquad\qquad\bar{\zeta}=\frac{x^{1}-\mathrm{i}x^{2}}{r+x^{3}}\coloneqq
e^{-\mathrm{i}\phi}\tan\frac{\theta}{2}\,.$ (3.11)

With these coordinates, we can take

$\chi^{\alpha}_{+}=\sqrt{\frac{r}{1+|\zeta|^{2}}}\;(\bar{\zeta},-1)\,,\qquad\chi^{\alpha}_{-}=\sqrt{\frac{r}{1+|\zeta|^{2}}}\;(1,\zeta)$
(3.12)

Our main result in this subsection is then:

###### Proposition 3.1

For the self-dual dyon in the gauge

$A=\frac{\mathrm{d}t}{r}+a\,,\qquad
a=\frac{\mathrm{i}\,(\zeta\mathrm{d}\bar{\zeta}-\bar{\zeta}\mathrm{d}\zeta)}{1+|\zeta|^{2}}=(1-\cos\theta)\mathrm{d}\phi\,,$
(3.13)

satisfying $F_{A}=*F_{A}$ with respect to the flat metric
$\mathrm{d}s^{2}=\mathrm{d}t^{2}+\mathrm{d}\vec{x}^{2}$, the spinors
$\chi_{\pm}^{\alpha}$ satisfy the charged Killing spinor equations

$\partial^{(\alpha}{}_{\dot{\alpha}}\chi_{\pm}^{\beta)}\pm\frac{\mathrm{i}}{2}\,A^{(\alpha}{}_{\dot{\alpha}}\chi_{\pm}^{\beta)}=0\,,$
(3.14)

with the factors of $\pm\mathrm{i}/2$ reflecting the minimal charges
associated to the spin bundles of the sphere. Thus, $\chi^{\alpha}_{\pm}$ are
charge $\pm 1/2$ Killing spinors.

This is proved by direct calculation. It also follows directly also from the
twistor treatment of the SD dyon, but we will give that argument elsewhere.

The SDD potential $A$ and spinors $\chi_{\pm}^{\alpha}$ as defined have wire
singularities along $\theta=\pi$: the coordinate $\zeta^{\prime}=1/\zeta$ is
regular at $\theta=\pi$ where $\zeta^{\prime}=0$, and we find
$\chi^{\alpha}_{\pm}$ singular there

$\chi^{\alpha}_{+}=\sqrt{\frac{r}{1+|\zeta^{\prime}|^{2}}}\;\left(\frac{|\zeta^{\prime}|}{\bar{\zeta}^{\prime}},-|\zeta^{\prime}|\right)\,,\quad\chi^{\alpha}_{-}=\sqrt{\frac{r}{1+|\zeta^{\prime}|^{2}}}\;\left(|\zeta^{\prime}|,\frac{|\zeta^{\prime}|}{\zeta^{\prime}}\right)\,.$
(3.15)

However, the respective gauge transformation $\mathrm{e}^{\pm
i\phi}=(\zeta/|\zeta|)^{\pm 1}=(|\zeta^{\prime}|/\zeta^{\prime})^{\pm 1}$
removes these singularities

$\chi^{\alpha}_{+}=\sqrt{\frac{r}{1+|\zeta^{\prime}|^{2}}}\;\left(1,-\bar{\zeta}^{\prime}\right)\,,\quad\chi^{\alpha}_{-}=\sqrt{\frac{r}{1+|\zeta^{\prime}|^{2}}}\;\left(\zeta^{\prime},1\right)\,,$
(3.16)

at the price of introducing a singularity at $\theta=0$. Similarly,

$\mathrm{i}\,a=\mathrm{i}\,(1-\cos\theta)\,\mathrm{d}\phi\rightarrow-\mathrm{i}\,(1+\cos\theta)\,\mathrm{d}\phi\,,$
(3.17)

under this transformation, which is now regular at $\theta=\pi$. Thus, as
connections and sections of line bundles, everything is globally defined.

The main consequence that we will use is the following:

###### Corollary 3.1

Let the totally symmetric ASD spinor $\psi_{\alpha_{1}\cdots\alpha_{n+2e}}$
with $2e\in\mathbb{Z}$ satisfy the flat space wave equation,
$\partial^{\alpha_{1}\dot{\alpha}}\psi_{\alpha_{1}\cdots\alpha_{n+2e}}=0$.
Then

$\varphi_{\alpha_{1}\cdots\alpha_{n}}:=\psi_{\alpha_{1}\cdots\alpha_{n+2e}}\,\chi_{+}^{\alpha_{n+1}}\cdots\chi_{+}^{\alpha_{n+2e}}\,,$
(3.18)

satisfies the helicity $-n/2$ zero-rest-mass equations of charge $e$:

$\left(\partial^{\alpha_{1}\dot{\alpha}}+\mathrm{i}e\,A^{\alpha_{1}\dot{\alpha}}\right)\varphi_{\alpha_{1}\cdots\alpha_{n}}=0\,.$
(3.19)

When $n=0$ we similarly have
$\left(\partial^{\alpha\dot{\alpha}}+\mathrm{i}e\,A^{\alpha\dot{\alpha}}\right)\left(\partial_{\alpha\dot{\alpha}}+\mathrm{i}e\,A_{\alpha\dot{\alpha}}\right)\varphi=0$.

Proof: This follows for $n>0$ just as in the flat case, because, following an
application of the Leibnitz rule, each derivative of $\chi_{+}^{\alpha}$ is
automatically symmetrized over its undotted indices and comes with one copy of
$\frac{\mathrm{i}}{2}\,A$ by (3.14). So that $2e$ contractions with
$\chi_{+}^{\alpha}$ generate a solution of charge $\frac{1}{2}\cdot 2e=e$.

The case $n=0$ requires a little more work. Write
$\mathrm{D}_{\alpha\dot{\alpha}}=\partial_{\alpha\dot{\alpha}}+\mathrm{i}e\,A_{\alpha\dot{\alpha}}$
when acting on a field of charge $e$. Taking the case $n=1$, we find

$0=\mathrm{D}_{\beta\dot{\beta}}\mathrm{D}_{\alpha}{}^{\dot{\alpha}}\varphi^{\alpha}=\mathrm{D}^{2}\varphi_{\beta}\,,$
(3.20)

where the first equality follows from the massless equations on
$\varphi_{\alpha}$ and the second from the fact that the term symmetric in
$\alpha\beta$ is proportional to the ASD curvature of $A$, which is zero since
$A$ is SD. Contracting with a further constant spinor then yields a charged
scalar solution to the background-coupled scalar wave equation, and our
original $\psi_{\alpha_{1}\cdots\alpha_{2e}}$ can be taken to arise from a
$\psi_{\alpha_{1}\cdots\alpha_{2e+1}}$ in this way. $\Box$

Of course, we also have the analogous result for $\chi_{-}^{\alpha}$ lowering
charge. Note that although the initial valence-2 Killing spinor
$\chi^{\alpha\beta}$ can be used to lower helicity, it does not alter the
charge, so contracting with $\chi^{\alpha\beta}$ does not generate any more
charged solutions.

###  3.2 Scalar and negative helicity states on SD dyon

Charged quasi-momentum eigenstates of non-positive helicity on the SDD
background can now be generated by beginning with uncharged negative helicity
momentum eigenstates on flat space:

$\psi_{\alpha\cdots\beta}=\kappa_{\alpha}\cdots\kappa_{\beta}\,\mathrm{e}^{\mathrm{i}k\cdot
x}\,,$ (3.21)

where the null momentum has spinor decomposition
$k_{\alpha\dot{\alpha}}=\kappa_{\alpha}\tilde{\kappa}_{\dot{\alpha}}$. Using
Corollary 3.1, we obtain solutions

$\varphi_{\alpha_{1}\cdots\alpha_{n}}=\kappa_{\alpha_{1}}\cdots\kappa_{\alpha_{n}}\,\langle\kappa\,\chi_{+}\rangle^{p}\,\langle\kappa\,\chi_{-}\rangle^{q}\,\mathrm{e}^{\mathrm{i}k\cdot
x}\,,$ (3.22)

of charge $\frac{1}{2}(p-q)$. It is straightforward to check that the charged
zero-rest-mass equations still hold for (3.22) even when one of $p$ or $q$ is
negative, although in this case there will be radial wire singularities where
$\langle\kappa\,\chi_{\pm}\rangle$ appearing in the denominator vanishes.

With the parametrization (3.12), the general quasi-momentum eigenstate of
helicity $-n/2$ is given explicitly by

$\varphi^{(p,q)}_{\alpha_{1}\ldots\alpha_{n}}=\kappa_{\alpha_{1}}\ldots\kappa_{\alpha_{n}}\left(\frac{r}{1+|\zeta|^{2}}\right)^{\frac{p+q}{2}}(\bar{\zeta}z+1)^{p}\,(\zeta-z)^{q}\,\mathrm{e}^{\mathrm{i}k\cdot
x}\,,\qquad z\coloneqq\frac{\kappa^{1}}{\kappa^{0}}\,,$ (3.23)

where constant factors of $\kappa^{0}$ have been removed to give correct
little-group weight. A particular feature here is that for smooth solutions on
the sphere and non-trivial charge, we see that the
$\langle\kappa\chi_{+}\rangle$ factors generate spin-weighted spherical
harmonics as described for example in §4.15 of [77]. These are, in turn,
accompanied by powers of $r$ that, for global solutions on the sphere, are
higher than those expected for uncharged fields on a flat background.

In particular, there are distinguished solutions that are nowhere singular on
the celestial sphere and grow with the lowest possible power of $r$ as
$r\to\infty$. For $n\geq 0$, these are given by

$\boxed{\hskip
10.00002pt\varphi^{+}_{\alpha_{1}\ldots\alpha_{n}}\coloneqq\varphi_{\alpha_{1}\ldots\alpha_{n}}^{(2e,0)}=\kappa_{\alpha_{1}}\ldots\kappa_{\alpha_{n}}\bigg{(}\frac{r}{1+|\zeta|^{2}}\bigg{)}^{e}(\bar{\zeta}z+1)^{2e}\,\mathrm{e}^{\mathrm{i}k\cdot
x}\,,\hskip 10.00002pt}$ (3.24)

for positive quantized charge $2e\in\mathbb{Z}_{\geq 0}$, and

$\hskip 28.45274pt\boxed{\hskip
10.00002pt\varphi^{-}_{\alpha_{1}\ldots\alpha_{n}}\coloneqq\varphi_{\alpha_{1}\ldots\alpha_{n}}^{(0,-2e)}=\kappa_{\alpha_{1}}\ldots\kappa_{\alpha_{n}}\bigg{(}\frac{r}{1+|\zeta|^{2}}\bigg{)}^{-e}(\zeta-z)^{-2e}\,\mathrm{e}^{\mathrm{i}k\cdot
x}\,,\hskip 10.00002pt}$ (3.25)

for negative quantized charge $2e\in\mathbb{Z}_{\leq 0}$. The $n=0$ case
corresponds to scalar solutions. We refer to these solutions as _minimal
fields_ , or _minimal states_.

The other, non-singular solutions can be obtained as derivatives of the
minimal states with respect to the external data like momentum. To see this,
write

$k\cdot
x=\omega\,t+\mathrm{i}\omega\,r\,\frac{(\zeta-z)(\bar{\zeta}-\tilde{z})-(1+\zeta\tilde{z})(1+\bar{\zeta}z)}{(1+|\zeta|^{2})\,(1+z\tilde{z})}\,,$
(3.26)

which implies that for $p,q\geq 0$ we either have

$\varphi^{(p,q)}_{\alpha_{1}\ldots\alpha_{n}}=\left(\frac{1+z\tilde{z}}{2\omega}\,\partial_{\tilde{z}}\right)^{q}\varphi^{+}_{\alpha_{1}\ldots\alpha_{n}}\qquad\text{or}\qquad\varphi^{(p,q)}_{\alpha_{1}\ldots\alpha_{n}}=\left(\frac{1+z\tilde{z}}{2\omega}\,\partial_{\tilde{z}}\right)^{p}\varphi^{-}_{\alpha_{1}\ldots\alpha_{n}}\,,$
(3.27)

depending on the sign of the charge $e$. The corresponding amplitudes for
these more general states can thus be computed as suitable derivatives of the
amplitudes for the minimal states, on which we focus from now on.

Observe that for the minimal scalar states $\varphi^{\pm}$, the gradients
$D_{\alpha\dot{\alpha}}\varphi^{\pm}$ define background-dressed null 4-momenta
$K^{\pm}_{\alpha\dot{\alpha}}(x)=\kappa_{\alpha}\tilde{K}^{\pm}_{\dot{\alpha}}(x)$
via
$\nabla_{\alpha\dot{\alpha}}\varphi^{\pm}=\mathrm{i}K^{\pm}_{\alpha\dot{\alpha}}\varphi^{\pm}$,
where

$\displaystyle\tilde{K}^{\pm}_{\dot{\alpha}}(x)$
$\displaystyle=\tilde{\kappa}_{\dot{\alpha}}-\frac{2\mathrm{i}e}{r}\,\frac{\chi_{\pm}^{\alpha}\,T_{\alpha\dot{\alpha}}}{\langle\kappa\,\chi_{\pm}\rangle}\,.$
(3.28)

Thus, $\varphi^{\pm}$ satisfy the charged Hamilton-Jacobi equation on the SDD
background.

###  3.3 Scalar and negative helicity states on SD Taub-NUT

In this subsection we lift our massless fields that are charged with respect
to the SD dyon background to provide massless fields background coupled to SD
Taub-NUT. The main idea in fact works for general Gibbons-Hawling metrics.

Here we focus on the ASD zero-rest-mass equations

$\nabla^{\alpha\dot{\alpha}}\psi_{\alpha\alpha_{2}\ldots\alpha_{n}}=0\,,$
(3.29)

on the curved SDTN metric. Self-dual Ricci-flatness of the background gives
the simplification that the spin connection on the undotted spinor bundle is
flat, and can be taken to be identically zero in appropriate frames. This is
the case for the standard co-frame $\theta^{a}$ of SDTN given by

$\theta^{0}=\frac{\mathrm{d}t-2Ma}{\sqrt{V}}\,,\qquad\theta^{i}=\sqrt{V}\,\mathrm{d}x^{i}\,,$
(3.30)

for which it can be checked that the ASD 2-forms

$\Sigma^{i}=\theta^{0}\wedge\theta^{i}-\varepsilon^{i}_{jk}\theta^{j}\wedge\theta^{k}\,,$
(3.31)

are closed, leading to the absence of ASD spin coefficients in this frame [78,
79].

Let $e_{a}=(e_{0},e_{i})$ be the corresponding dual vector fields. After
multiplying through by $\sqrt{V}$, these are

$\begin{split}\sqrt{V}(e_{0},e_{i})=(V\partial_{t}\,,\partial_{i}+2Ma_{i}\partial_{t})\qquad\implies\qquad\sqrt{V}\,e_{a}=\delta_{a}^{\mu}(\partial_{\mu}+2MA_{\mu}\partial_{t})\,,\end{split}$
(3.32)

where $A_{\mu}$ are the space-time components of the SD dyon
$A=\mathrm{d}t/r+a$ in the Cartesian coordinates $x^{\mu}=(t,x^{i})$. The
second equality follows by substituting $V\partial_{t}=(1+2M/r)\partial_{t}$.

Now, take the time dependence of $\psi_{\alpha\cdots\beta}$ to be a factor of
$\mathrm{e}^{\mathrm{i}\omega t}$. We can then replace the $\partial_{t}$ by
$\mathrm{i}\omega$ where desired. Thus, when acting on
$\psi_{\alpha\cdots\beta}$ we can write

$\sqrt{V}\,\nabla_{a}\psi_{\alpha\cdots\beta}=\sqrt{V}\,e_{a}\psi_{\alpha\cdots\beta}=\delta_{a}^{\mu}(\partial_{\mu}+2MA_{\mu}\partial_{t})\psi_{\alpha\cdots\beta}\,.$
(3.33)

After multiplying through (3.30) by $\sqrt{V}$, the zero-rest-mass equation
reduces to:

$\left(\partial^{\alpha{\dot{\alpha}}}+2\mathrm{i}M\omega
A^{\alpha{\dot{\alpha}}}\right)\psi_{\alpha\alpha_{2}\ldots\alpha_{n}}=0\ \,.$
(3.34)

Here, $A_{\alpha{\dot{\alpha}}}$ are the components of the SD dyon in the flat
space co-frame $\mathrm{d}x^{\alpha{\dot{\alpha}}}$. We see that (3.30) is
reduced to the background-coupled ASD wave equation on a SDD for fields of
charge $e=2M\omega$.

A similar procedure works for the scalar case $n=0$. We start with the
Laplacian of a scalar $\varphi$,

$\Box\varphi=\frac{1}{|g|^{1/2}}\,\partial_{\mu}(|g|^{1/2}g^{\mu\nu}\partial_{\nu}\varphi)\,,$
(3.35)

with $|g|$ the metric determinant, and try to express it in terms of the
tetrad. Take the indices $\mu,\nu$ to again refer to the coordinates
$x^{\mu}=(t,x^{i})$. Using $g^{\mu\nu}=\delta^{ab}e_{a}^{\mu}e_{b}^{\nu}$,
this becomes

$\begin{split}\Box\varphi&=\frac{1}{|g|^{1/2}}\,\partial_{\mu}(|g|^{1/4}e_{a}^{\mu}\cdot|g|^{1/4}e^{a}\varphi)\\\
&=\frac{1}{|g|^{1/4}}\,e_{a}(|g|^{1/4}e^{a}\varphi)+\frac{1}{|g|^{1/4}}\,\partial_{\mu}(|g|^{1/4}e_{a}^{\mu})\,e^{a}\varphi\,,\end{split}$
(3.36)

where we have abbreviated $e^{a\nu}\partial_{\nu}\varphi=e^{a}\varphi$, etc.
Observe that the second term in this expression vanishes: using $|g|=V^{2}$,
it follows that the vectors $|g|^{1/4}e_{a}$ are divergence-free

$\begin{split}\partial_{\mu}(|g|^{1/4}e_{a}^{\mu})&=\partial_{\mu}\left(\delta_{a}^{\mu}+2M\delta^{\nu}_{a}A_{\nu}\delta^{\mu}_{t}\right)\\\
&=2M\,\delta^{\nu}_{a}\,\partial_{t}A_{\nu}=0\,,\end{split}$ (3.37)

as a consequence of the time-independence of the SDD $A_{\mu}$.

This means that the Laplacian on SDTN is simply

$\begin{split}\Box\varphi&=\frac{1}{\sqrt{V}}\,e_{a}\big{(}\sqrt{V}e^{a}\varphi\big{)}\\\
&=\frac{1}{V}\,\delta^{\mu\nu}(\partial_{\mu}+2MA_{\mu}\partial_{t})(\partial_{\nu}+2MA_{\nu}\partial_{t})\varphi\,.\end{split}$
(3.38)

If $\varphi$ is a momentum eigenstate carrying energy $\omega$, this reduces
to

$\Box\varphi=\frac{1}{V}\,\delta^{\mu\nu}(\partial_{\mu}+2M\mathrm{i}\omega
A_{\mu})(\partial_{\nu}+2M\mathrm{i}\omega A_{\nu})\varphi\,,$ (3.39)

having again used the time-independence of $A_{\nu}$ to commute the
$\partial_{t}$ in the first bracket past the $A_{\nu}$ in the second. Up to
the prefactor of $1/V$, this is precisely the gauge covariant Laplacian in the
SD dyon background!

We can therefore lift our solutions from the previous subsection. Recall that
we must have that $2e\in\mathbb{Z}$ to make $\psi_{\alpha\cdots\beta}$ (or the
scalar $\varphi$) single-valued, so the energy $\omega$ is necessarily
quantized to satisfy $4M\omega\in\mathbb{Z}$. Of course, this also follows on
SDTN from the periodicity in $t$.

We reiterate that in this section we did not use the special form of $(V,a)$
for the SD Taub-NUT. Thus this technique can be applied to any Gibbons-Hawking
metric to obtain background-coupled massless fields from charged fields with
respect to the static SD Maxwell field $A_{\mu}=(V,a)$.

#### Potentials:

In the special case of a negative-helicity graviton, we take $n=4$. In order
to construct a 2-point function, we will need not only the ASD part of the
linearized Weyl tensor $\psi_{\alpha\beta\gamma\delta}$, but also the
corresponding linear perturbation $\gamma_{\alpha\beta}$ to the ASD spin
connection. Introducing the spinor equivalent $\theta^{\alpha\dot{\alpha}}$ of
the coframe and the ASD 2-forms
$\Sigma^{\alpha\beta}\coloneqq\theta^{\alpha}{}_{\dot{\alpha}}\wedge\theta^{\beta\dot{\alpha}}$,
the linearized spin connection is a 1-form satisfying

$\mathrm{d}\gamma_{\alpha\beta}=\psi_{\alpha\beta\gamma\delta}\,\Sigma^{\gamma\delta}\,.$
(3.40)

Its components in the chosen coframe,
$\gamma_{\alpha\beta}=\gamma_{\alpha\beta\delta\dot{\delta}}\theta^{\delta\dot{\delta}}$
can be expressed by introducing a pair of _dressing matrices_ whose form is
motivated by the dressed momentum (3.28)

$\displaystyle H^{\pm\,{\dot{\alpha}}}{}_{\dot{\beta}}(x,\lambda)$
$\displaystyle=\frac{1}{\sqrt{V}}\left(\delta^{{\dot{\alpha}}}_{\dot{\beta}}-\frac{4M}{r}\frac{T^{\alpha\dot{\alpha}}\kappa_{\alpha}\,\chi_{\pm}^{\beta}\,T_{\beta\dot{\beta}}}{\langle\kappa\,\chi^{\pm}\rangle}\right)\,,$
(3.41)

These matrices are unimodular and satisfy the relations

$K^{\pm}_{{\dot{\alpha}}}=\tilde{\kappa}_{\dot{\beta}}H^{\pm\,\dot{\beta}}{}_{{\dot{\alpha}}}\,,\qquad\mathrm{d}(\kappa_{\alpha}H^{\pm\,\dot{\beta}}{}_{{\dot{\alpha}}}\theta^{\alpha{\dot{\alpha}}})=0\,.$
(3.42)

This directly implies that the potential

$\displaystyle\hskip 10.00002pt\gamma^{\pm}_{\alpha\beta\gamma\dot{\gamma}}$
$\displaystyle=\frac{1}{[\eta\tilde{\kappa}]}\,\kappa_{\alpha}\kappa_{\beta}\kappa_{\gamma}\eta_{\dot{\delta}}H^{\pm\,\dot{\delta}}{}_{\dot{\gamma}}\varphi^{\pm}\,,\hskip
10.00002pt$ (3.43)

built from positive and negative energy scalar solutions $\varphi^{\pm}$
satisfies the linearized equations of motion for the ASD spin connection. Here
$\eta_{\dot{\alpha}}$ is an arbitrary constant spinor which will drop out of
gauge invariant observables.

###  3.4 Positive helicity states

In the remainder of this section, we describe positive helicity gluons on the
SD dyon and positive helicity photons and gravitons on SDTN. These can be
obtained by raising the helicity of the scalar solutions following §6.4 of
[52] or Plebanski-like formulations of the self-duality equations [80]. This
is most direct at the level of gauge potentials for the field strengths or
curvature tensors.

For gluons, we seek linearized gauge fields with vanishing ASD linearized
field strength. We can realize these states by linearizing the $K$-matrix
formalism of self-dual Yang-Mills [81, 82]: introduce a scalar potential
$\varphi$ for the gauge potential

$a_{\alpha\dot{\alpha}}=\xi_{\alpha}\,\xi^{\beta}\,\mathrm{D}_{\beta\dot{\alpha}}\varphi\,,$
(3.44)

where $\xi_{\alpha}$ is an arbitrary reference spinor that will drop out from
gauge invariant quantities, including scattering amplitudes. It follows from
the self-duality condition of the background covariant derivative that the
linearized ASD field strength is

$f_{\alpha\beta}=\frac{1}{2}\,\xi_{\alpha}\xi_{\beta}\,\mathrm{D}_{\gamma\dot{\gamma}}\mathrm{D}^{\gamma\dot{\gamma}}\varphi\,,$
(3.45)

so that the state (3.44) represents a positive-helicity gluon whenever
$\varphi$ solves the charged scalar wave equation. Thus the operator
$\xi_{\alpha}\xi^{\beta}\mathrm{D}_{\beta\dot{\alpha}}$ acts a helicity-
raising operator. In terms of the dressed momenta, we can take the positive-
helicity states to have the linearized gauge potential

$\boxed{\hskip
10.00002pta^{\pm}_{\alpha\dot{\alpha}}=\frac{1}{\langle\xi\kappa\rangle}\,\xi_{\alpha}\tilde{K}^{\pm}_{\dot{\alpha}}\varphi^{\pm}\hskip
10.00002pt}$ (3.46)

where factors of $\langle\xi\kappa\rangle$ have been introduced for
normalization.

Next, we come to the SDTN case. Since the SDTN metric is self-dual as well as
Ricci-flat, the spin connection acting on the bundle of left-handed Weyl
spinors is flat. In the coframe (3.30), a component-wise calculation shows
that this left-handed spin connection vanishes identically. Consequently, in
such a coframe, any constant spinor $\xi_{\alpha}$ on $\mathbb{R}^{4}$ is
automatically covariantly constant with respect to the SD Taub-NUT metric.
Hence, it can again be used to raise spin following the rules of [83].

On SDTN, the operator $\xi_{\alpha}\xi^{\beta}\nabla_{\beta\dot{\alpha}}$
still acts as a helicity-raising operator, where $\nabla_{\alpha\dot{\alpha}}$
are now the components of the covariant derivative in the basis
$\theta^{\alpha\dot{\alpha}}$. This means that a positive-helicity photon is
again described by the linearized gauge potential

$a_{\alpha{\dot{\alpha}}}=\xi_{\alpha}\,\xi^{\beta}\,\nabla_{\beta{\dot{\alpha}}}\varphi\,,$
(3.47)

and that the linearized field strength of such a state is purely SD if
$\varphi$ satisfies the wave equation. A positive-helicity spin-2 wavefunction
can be constructed by applying the helicity-raising operator twice to the
scalar solution:

$h_{\alpha{\dot{\alpha}}\beta\dot{\beta}}=\xi_{\alpha}\xi_{\beta}\xi^{\gamma}\xi^{\delta}\nabla_{\gamma{\dot{\alpha}}}\nabla_{\delta\dot{\beta}}\varphi\,.$
(3.48)

A calculation similar to the one leading to (3.45) shows that the ASD part of
its linearized Weyl tensor vanishes. This is a linearization of the
formulation of self-dual Ricci-flat metrics in terms of Plebanski’s second
heavenly potential [80].

##  4 Two-point scattering amplitudes

In the perturbiner formalism, the tree-level two-point amplitude is captured
by the on-shell kinetic term evaluated on a linear combination of external
states in the background of choice. For scalar scattering around both
backgrounds and for photon scattering around SD Taub-NUT, the usual non-chiral
actions suffice. Conversely, it is natural to adopt chiral formulations of
Yang-Mills [84] and general relativity [78] when computing gluon amplitudes
around SD dyon and graviton amplitudes around SD Taub-NUT backgrounds. They
express gauge theory and gravity as deformations around their self-dual
sectors.

These chiral formulations are built perturbatively around $BF$ type theories,
where the linear field $B$ encodes the negative-helicity particles. The $BF$
theories capture the integrable self-dual sectors, so their scattering
amplitudes vanish identically. As an important consequence, the two-point
amplitudes we are interested in are zero for the $(+\,+)$ and $(+\,-)$
helicity configurations. This is because such amplitudes do not receive
contributions from non-self-dual interactions.

On the other hand, since the deformation away from self-duality is quadratic
in the negative-helicity field, there is scope for non-vanishing $(-\,-)$
2-point amplitudes. Calculating these is our main focus. We denote these
amplitudes by
$\mathcal{A}(1_{e/\omega=\pm}^{h_{1}},2^{h_{2}}_{e/\omega=\mp})$, where
$h_{1,2}$ denotes the helicity of each particle and the subscripts denote
whether each particle has positive or negative charge/energy. Other selection
rules also arise from the symmetries of the problem. The translation symmetry
in $t$ imposes that 2-point amplitudes can only be non-vanishing if one of the
states has positive energy and the other state has negative energy. Similarly,
in the SD dyon case, the two states will be required to have equal and
opposite charges due to global charge conservation. In what follows, we
briefly touch upon each of the requisite chiral formulations, before using
them to compute the desired amplitudes.

###  4.1 Scalar amplitudes

The free scalar action is

$S_{\text{scalar}}=\int_{M}\mathrm{d}^{4}x\,\sqrt{|g|}\,g^{\mu\nu}\nabla_{\mu}\varphi\,\nabla_{\nu}\varphi\,.$
(4.1)

As advertised, we consider two cases: one in which $g$ is the flat metric and
$\nabla$ is the covariant derivative $\mathrm{D}$ coupled to a Cartan-valued
SDD embedded in some gauge algebra, and another in which $g$ is the SDTN
metric and $\nabla$ the associated covariant derivative.

Using (3.28), the scalar 2-point amplitude around both backgrounds is given by

$\mathcal{A}(1^{0}_{e=-},2^{0}_{e=+})=2\pi\langle
1\,2\rangle\,\tilde{\delta}\\\
\times\int\mathrm{d}^{3}\vec{x}\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{n}\left(\tilde{\kappa}_{1\,\dot{\alpha}}+\frac{n}{r}\frac{\chi^{-}_{\alpha}\tensor{T}{{}^{\alpha}_{\dot{\alpha}}}}{\langle
1\chi^{-}\rangle}\right)\left(\tilde{\kappa}_{2}^{\dot{\alpha}}+\frac{n}{r}\frac{\chi^{+}_{\beta}T^{\beta\dot{\alpha}}}{\langle
2\chi^{+}\rangle}\right)\mathrm{e}^{\mathrm{i}(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{x}}\,.$
(4.2)

The value of $n$ is $2e_{2}$ on the SDD and $4M\omega_{2}$ on SDTN, while
$\tilde{\delta}$ equals $\delta_{-e_{1},e_{2}}\delta(\omega_{1}+\omega_{2})$
around SDD and equals $\delta_{-\omega_{1},\omega_{2}}$ around SDTN.

As we show in appendix A, the integral can be evaluated explicitly and
vanishes on the support of charge or energy conservation in the SD dyon and SD
Taub-NUT cases respectively. That is,

$\boxed{\hskip 10.00002pt\mathcal{A}(1^{0}_{e=-},2^{0}_{e=+})=0\,.\hskip
10.00002pt}$ (4.3)

This is consistent with a variety of observations. Its vanishing turns out to
be closely related to gauge invariance of the 2-graviton $(-\,-)$ amplitude,
see below. It is also expected from the fact that the free scalar action
uplifts to a local holomorphic action on twistor space, and the space-time
states lift to twistor space states that do not scatter. The latter is
essentially a consequence of the geometric fact that twistor space contains
multiple copies of space-time, and the two scalar states live on disjoint
copies which do not interact [85].

###  4.2 Photon scattering on SD Taub-NUT

The standard QED action on a general curved space-time can be used to compute
the 2-point $(-\,-)$ photon amplitude on SDTN. In terms of the states
previously introduced, the on-shell QED action reads

$\displaystyle S_{\text{QED}}$
$\displaystyle=\frac{1}{4\mathtt{g}^{2}}\int_{M}F_{1^{-}}\wedge F_{2^{+}}$
(4.4)
$\displaystyle=\int_{M}\mathrm{d}^{4}x\,V\,b^{-}_{1\,\alpha\beta}\,b^{+\,\alpha\beta}_{2}\,,$

where $\mathtt{g}$ is the Maxwell coupling, $b^{-}_{1\,\alpha\beta}$ is given
by (3.25) with $n=2$ and $e=2M\omega_{1}<0$, and $b^{+}_{2\,\alpha\beta}$ is
given by (3.24) with $n=2$ and $e=2M\omega_{2}>0$. In particular, we have
taken photon 1 to have negative energy and photon 2 to have positive energy,
and used the ASD condition on the field strength. Upon inserting the
expressions for the quasi-momentum eigenstates and performing the integral
along the time direction, the 2-point amplitude reads

$\mathcal{A}(1^{-1}_{\omega=-},2^{-1}_{\omega=+})=2\pi\,\langle
1\,2\rangle^{2}\,\delta_{-\omega_{1},\omega_{2}}\\\
\times\int\mathrm{d}^{3}\vec{x}\,\left(1+\frac{2M}{r}\right)\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{4M\omega_{2}}\mathrm{e}^{\mathrm{i}(\vec{k}_{1}+\vec{k}_{2})\cdot
x}\,.$ (4.5)

This integral can be evaluated by means of the techniques developed in
appendix A and it vanishes:

$\boxed{\hskip
10.00002pt\mathcal{A}(1^{-1}_{\omega=-},2^{-1}_{\omega=+})=0\,.\hskip
10.00002pt}$ (4.6)

###  4.3 Gluon scattering on SD dyon

For sake of simplicity, let us embed the SD dyon in an $\mathrm{SU}(2)$ gauge
theory by taking the Cartan-valued color factor in (2.3) to be
$\mathsf{c}=\sigma_{3}$, the third Pauli matrix. Then the off-diagonal gluons
have positive and negative charge.

The relevant chiral action for gauge theory is due to Chalmers and Siegel
[84],

$S[B,A]=\int\mathrm{tr}\,B\wedge
F-\frac{\mathtt{g}^{2}}{2}\int\mathrm{tr}\,B\wedge B\,.$ (4.7)

Here, $B$ is an adjoint-valued ASD 2-form, $F$ is the field strength of the
gauge field $A$ and $\mathtt{g}$ is the Yang-Mills coupling. The equation of
motion for $B$ ensures that the ASD component of $F$ is proportional to $B$
itself, so that the on-shell action reduces to

$S_{\text{on-shell}}[B,A]=\frac{\mathtt{g}^{2}}{2}\int\mathrm{tr}\,B\wedge
B\,.$ (4.8)

By virtue of being ASD, $B$ can be expanded as
$B=B_{\alpha\beta}\Sigma^{\alpha\beta}$ on a basis of ASD 2-forms
$\Sigma^{\alpha\beta}$.444Recall that in flat space, we are working with
$\Sigma^{\alpha\beta}=\mathrm{d}x^{\alpha}{}_{{\dot{\alpha}}}\wedge\mathrm{d}x^{\beta\dot{\alpha}}$.
The linearized equation of motion of $F$ becomes the helicity $-1$ free field
equation for $B_{\alpha\beta}$. Therefore $B$ gets identified with negative
helicity gluons.

As previously discussed, the two-point $(-\,-)$ tree-level amplitude is
constructed by inserting the field $B=\mathtt{g}^{-1}(b_{1}+b_{2})$, where the
$b_{i}$ are negative helicity quasi-momentum eigenstates. The trace over the
colour indices implies that the amplitude can be non-vanishing only if the
charge is conserved, $e_{1}+e_{2}=0$. And as usual we have energy conservation
$\omega_{1}+\omega_{2}=0$ as a consequence of the time translation symmetry.

We will assume without loss of generality that $b_{1}$ has a negative charge
$e_{1}<0$. After performing the integral over $t$, the two-point amplitude
reduces to

$\mathcal{A}(1^{-1}_{e=-},2^{-1}_{e=+})=-\pi\,\langle
1\,2\rangle^{2}\,\delta(\omega_{1}+\omega_{2})\,\delta_{-e_{1},e_{2}}\\\
\times\int\mathrm{d}^{3}\vec{x}\,\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{2e_{2}}\mathrm{e}^{\mathrm{i}(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{x}}\,.$
(4.9)

Since the background has a time-like, as well as a spatial $\mathrm{SO}(3)$,
symmetry, we expect the 2-point amplitude to possess these symmetries as well.
The time translation symmetry is manifest from the energy-conserving delta
function, while the $\mathrm{SO}(3)$ invariance can be made manifest by
explicitly computing the integral in (4.9). We defer the detailed computation
of such integrals to appendix A, stating the final expression here:

$\boxed{\hskip
10.00002pt\mathcal{A}(1^{-1}_{e=-},2^{-1}_{e=+})=-8\pi^{2}\,e_{2}\,(2e_{2})!\,\frac{(2\omega_{1})^{2e_{2}-1}\,\langle
1\,2\rangle^{2+2e_{2}}}{|\vec{k}_{1}+\vec{k}_{2}|^{4e_{2}+2}\,(\kappa_{1}^{0}\,\kappa_{2}^{0})^{2e_{2}}}\,\delta(\omega_{1}+\omega_{2})\,\delta_{-e_{1},e_{2}}\,.\hskip
10.00002pt}$ (4.10)

The factor of $(\kappa_{1}^{0}\,\kappa_{2}^{0})^{-2e_{2}}$ ensure that the
wave functions, and hence the amplitude, has the appropriate little group
scaling: as discussed, this needs to be corrected as a consequence of the non-
trivial topology of the SDD background. Rotational invariance can be
manifested by dropping the factor of
$(\kappa_{1}^{0}\,\kappa_{2}^{0})^{-2e_{2}}$ from the wave functions at the
price of introducing an anomalous little-group weight.

###  4.4 Graviton scattering on SD Taub-NUT

General relativity admits a first-order chiral formulation, first developed by
Plebanski [78]; see [86] for a review. The field variables are the tetrad
$\theta^{\alpha\dot{\alpha}}$ and an ASD spin connection
$\Gamma_{\alpha\beta}=\Gamma_{(\alpha\beta)}$; the classical action is

$S[\theta,\Gamma]=\int_{M}\Sigma^{\alpha\beta}\wedge\mathrm{d}\Gamma_{\alpha\beta}+\kappa^{2}\int_{M}\Sigma^{\alpha\beta}\wedge\tensor{\Gamma}{{}_{\alpha}^{\gamma}}\wedge\tensor{\Gamma}{{}_{\beta\gamma}}\,.$
(4.11)

Here
$\Sigma^{\alpha\beta}=\epsilon_{\dot{\alpha}\dot{\beta}}\theta^{\alpha\dot{\alpha}}\wedge\theta^{\beta\dot{\beta}}$,
while $\kappa^{2}=16\pi G$ is the gravitational coupling. The classical
equation of motion for $\Gamma_{\alpha\beta}$ sets
$\kappa^{2}\Gamma_{\alpha\beta}$ equal to the ASD spin connection of the
metric
$g=\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\theta^{\alpha\dot{\alpha}}\theta^{\beta\dot{\beta}}$
on $M$. On the support of this equation of motion, the remaining equation of
motion for $\Sigma^{\alpha\beta}$ reproduces Einstein’s vacuum equations, so
that the on-shell action is vanishing.

However, if we consider a linear perturbation to the spin connection of the
form
$\delta\Gamma^{\alpha\beta}=\kappa^{-1}(\gamma_{1}^{\alpha\beta}+\gamma_{2}^{\alpha\beta})$
and identify the perturbations with negative-helicity quasi-momentum graviton
eigenstates, the on-shell action is non-vanishing and computes the two-point
$(-\,-)$ amplitude around the space-time $M$ with background, non-dynamical
metric $g$. Thus, the two-point amplitude is compactly given by

$\mathcal{A}(1^{-2},2^{-2})=\int_{M}\Sigma^{\alpha\beta}\wedge\tensor{\gamma}{{}_{1\,\alpha}^{\gamma}}\wedge\gamma_{2\,\beta\gamma}\,.$
(4.12)

Upon inserting the on-shell states (3.43) and assuming without loss of
generality that graviton 1 has negative energy, the 2-point amplitude reads

$\mathcal{A}(1^{-2}_{\omega=-},2^{-2}_{\omega=+})=-\frac{2\pi\,\langle
1\,2\rangle^{3}\,\eta_{1\,{\dot{\alpha}}}\,\eta_{2\,\dot{\beta}}}{[\eta_{1}\,1][\eta_{2}\,2]}\>\delta_{-\omega_{1},\omega_{2}}\\\
\times\int\mathrm{d}^{3}\vec{x}\;V\,H_{1}^{-\,{\dot{\alpha}}}{}_{\dot{\gamma}}H_{2}^{+\,\dot{\beta}\dot{\gamma}}\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{4M\omega_{2}}\mathrm{e}^{\mathrm{i}(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{x}}\,,$
(4.13)

where $\eta_{1,2}$ are reference spinors parametrizing the residual gauge
symmetry arising from linearized spin-frame variations and diffeomorphisms.
The 2-point amplitude should be independent of the choice of these spinors.

As previously anticipated, this gauge invariance is also closely related to
the vanishing of the scalar amplitude around SDTN. Recalling the first
equation (3.42) relating the background-dressed momenta and the dressing
matrices, we can express the scalar and graviton amplitudes as

$\displaystyle\mathcal{A}(1^{0}_{\omega=-},2^{0}_{\omega=+})$
$\displaystyle\propto\tilde{\kappa}_{1\,\dot{\alpha}}\tilde{\kappa}_{2\,\dot{\beta}}\int\mathrm{d}^{3}\vec{x}\,VH_{1}^{-\,{\dot{\alpha}}}{}_{\dot{\gamma}}H_{2}^{+\,\dot{\beta}\dot{\gamma}}\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{4M\omega_{2}}\mathrm{e}^{\mathrm{i}(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{x}}\,,$
(4.14) $\displaystyle\mathcal{A}(1^{-2}_{\omega=-},2^{-2}_{\omega=+})$
$\displaystyle\propto\frac{\eta_{1\,\dot{\alpha}}\eta_{2\,\dot{\beta}}}{[\eta_{1}\,1][\eta_{2}\,2]}\int\mathrm{d}^{3}\vec{x}\,VH_{1}^{-\,{\dot{\alpha}}}{}_{\dot{\gamma}}H_{2}^{+\,\dot{\beta}\dot{\gamma}}\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{4M\omega_{2}}\mathrm{e}^{\mathrm{i}(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{x}}$
(4.15)

up to kinematical prefactors. Gauge invariance of the graviton 2-point
amplitude requires the integral to be proportional to
$\tilde{\lambda}_{1}^{\dot{\alpha}}\tilde{\lambda}_{2}^{\dot{\beta}}$, and
this in turns implies the vanishing of the scalar 2-point amplitude.

Finally, the evaluation of the integral in (4.13) leads to the following
expression for the 2-point amplitude

$\boxed{\hskip
10.00002pt\mathcal{A}(1^{-2}_{\omega=-},2^{-2}_{\omega=+})=4\pi^{2}\,M\,(4M\omega_{2})!\,\frac{(2\omega_{1})^{4M\omega_{2}-2}\,\langle
1\,2\rangle^{4+4M\omega_{2}}}{|\vec{k}_{1}+\vec{k}_{2}|^{8M\omega_{2}+2}\,(\kappa_{1}^{0}\,\kappa_{2}^{0})^{4M\omega_{2}}}\,\delta_{-\omega_{1},\omega_{2}}\,,\hskip
10.00002pt}$ (4.16)

Again the factor of $(\kappa_{1}^{0}\,\kappa_{2}^{0})^{-4M\omega_{2}}$ comes
from the original wave functions and can be dropped to manifest rotational
invariance at the price of introducing an anomalous little-group weight. To
obtain the corresponding formula (1.2) for the spinning SD Taub-NUT, we must
perform the imaginary shift along $i\vec{a}$ to reduce the centre of the
spinning SD Taub-NUT to $\vec{x}=0$ as discussed following (3.3). This leads
to the integral (4.13) above except that the factors of
$\mathrm{e}^{i\vec{k}_{i}\cdot x}$ generate the extra exponential factor of
$\mathrm{e}^{-(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{a}}$ appearing in (1.2) under
this shift.

###  4.5 Double copy structure

Double copy refers to a broad set of methods for writing perturbative
scattering amplitudes of gravitational theories as the ‘square’ of
perturbative amplitudes of some non-gravitational theories (cf., [87, 88, 89]
for reviews). In its most developed form, this relates multiparticle graviton
scattering amplitudes of general relativity to gluon scattering amplitudes of
Yang-Mills theory, and there are hints that some of that correspondence
continues to hold in curved backgrounds [30, 90, 91]. Double copy has also
been extended to algebraically special classes of exact solutions to give a
notion of ‘classical double copy’ [92]. The SD Taub-NUT metric is the
classical double copy of a SD dyon in this language [54], and we have seen the
double copy at work in the construction of background coupled fields. In
particular, squaring the non-exponential part of a spin-1 background coupled
field on the SDD gives a spin-2 background coupled field on SDTN under $e\to
M\omega$. This only respects the quantization condition for $e$ even because
conventionally we do not double the $\exp(\mathrm{i}k\cdot x)$ factor.

One does not usually speak of double copy for 2-point amplitudes: in flat
backgrounds these amplitudes are vanishing, and because they are determined by
only the kinetic part of the theory they are insensitive to colour-kinematics
duality. Nevertheless, as we have non-vanishing 2-point amplitudes in the
$(-\,-)$ helicity configuration, it is natural to investigate the existence of
any double copy relationship between (4.10) and (4.16).

Clearly, upon squaring the 2-point gluon amplitude on the SDD a replacement
rule is needed to remove the charges $e_{1,2}$. Since the 2-point amplitude is
entirely governed by the inner product between free fields, via the kinetic
term in the classical action, it seems most natural to implement the same
replacement rule which takes charged zero-rest-mass fields on SDD to zero-
rest-mass fields with the doubling described above, namely $e\to M\omega$.
This leads to a simple double copy-like relation between the amplitudes:

$\mathcal{A}(1^{-2}_{\omega=-},2^{-2}_{\omega=+})\sim\,|\vec{k}_{1}+\vec{k}_{2}|^{2}\,\mathcal{A}^{2}(1^{-1}_{e=-},1^{-1}_{e=+})\Big{|}_{e_{1,2}\to
M\omega_{1,2}}\,,$ (4.17)

where the amplitudes are implicitly understood to be stripped of
charge/momentum conserving delta functions or Kronecker deltas, combinatorial
factors etc.. At present, we lack any sort of first-principles understanding
of this relationship.

##  5 Interpretations and future directions

We have shown that massless free field equations on self-dual dyon and Taub-
NUT backgrounds can be solved with particularly simple quasi-momentum
eigenstates. These are much simpler than solutions on the corresponding non-
chiral Coulomb and Schwarzschild backgrounds, which are only expressible in
harmonic expansions and in terms of special functions. This reflects the
complete integrability underlying the SDD and SDTN backgrounds. This
simplicity allows us to perform the integrations explicitly to yield two-point
functions. In the weak background field limit, these should correspond to a
four point amplitude in a probe limit, i.e., with an incoming and outgoing
massive leg whose momentum is unchanged, interacting with an incoming and
subsequently outgoing massless probe particle. However, the formuale in, for
example, [93, 94], assume momentum conservation, whereas our integration
procedure has focused on the part of the two-point function that is not
supported on conserved momentum. Thus it is difficult to compare our results
to those in the literature. Our results are closer to describing an amplitude
for a negative helicity massless particle of momentum $\vec{k_{1}}$ having its
helicity flipped by the SD background into one of positive helicity with
momentum $\vec{k_{2}}$ as described for more general MHV amplitudes in [95].
It would be interesting to see if some correspondence can be established with
the integrable classical particle scattering results of [96].

Our backgrounds do have the feature of non-trivial topology, leading to a non-
trivial spin weight for charged probe particles on the SDD, and particles with
(quantized) non-zero energy on SDTN. This in turn leads to higher spherical
harmonics and corresponding higher growth in $r$ than expected in flat space.
It would be interesting to recover these results from previous investigations
of dyon scattering, based on harmonic expansions, by restricting to the SD
sector and demanding regularity on the sphere [63, 64].

We introduced spin into our two-point functions in (1.2) following the Newman-
Janis correspondence [47, 97, 98], which in the self-dual case becomes a
simple imaginary shift; with this we obtain spinning SD Taub-NUT as a complex
metric providing the self-dual part of the Kerr metric. This imaginary shift
in the direction of the spin vector $\vec{a}$ can be used to move the contour
of integration so that the integrand in the spinning case is the same as given
above as in for example (4.9) or (4.13), except that the plane wave factors in
the wave function must be shifted to give the additional factor of
$\mathrm{e}^{-(\vec{k_{1}}+\vec{k_{2}})\cdot\vec{a}}$. Once this factor is
taken outside the integrals, the rest reduces to those considered above
leading to the same integrated answer (4.10) or (4.16), but now with the
overall additional factor of
$\mathrm{e}^{-(\vec{k_{1}}+\vec{k_{2}})\cdot\vec{a}}$. This is a general
phenomenon and explains the simple exponential dependence on the spin found in
[99, 94] and implicitly in [93]. These relate to formulae with just ASD
particles on the full background because the two ASD particles on the
background will not interact with the ASD part of the background to the orders
considered.

For Coulomb solutions or physical black holes (Schwarzschild or Kerr) in
Lorentzian signature, the topological issues that we have confronted here, of
course, do not arise. This makes our results connected more directly to
backgrounds with a non-vanishing magnetic charge or NUT parameter. However, it
also makes it clear that if one wishes to pursue the strategy of
perturbatively reconstructing the non-chiral background by expanding around
the SD sector, the topological issues that we have encountered will arise at
the first step. According to the Newman-Janis construction, the Kerr metric is
expressed as a nonlinear superposition of SD Taub-NUT and its conjugate ASD
Taub-NUT, displaced along the imaginary spin vector $\pm\mathrm{i}\,\vec{a}$.
In this way, that the topology of the SDTN is canceled by its conjugate. The
wire singularity then needs only connect the two complex conjugate centres,
reminiscent of the Kerr worldsheet of [100].

Our results relied on two novel techniques. The first was the introduction of
charged Killing spinors that convert flat space massless fields to those that
are charged with respect to the dyon background but with a different spin. The
second was the lifting of charged massless fields on the SD dyon background to
SD Taub-NUT. It will be interesting to explore the full domain of
applicability of these techniques. We have already remarked that the lifting
described in §3.3 of massless fields that are background coupled to a self-
dual static Maxwell field to those that are coupled to the corresponding
Gibbons-Hawking metric. Thus this will apply also to the ALE or ALF $A_{k}$
gravitational instantons. One can show via twistor methods that many self-dual
Maxwell fields that admit charged Killing spinors can be constructed from
twistor space, not just the SD dyon. It would be interesting to see whether
they can be extended beyond these classes, for example the metrics of [101]
looking promising.

Non-trivial SD solutions are never real in Lorentzian signature, but can be in
Euclidean or split signature and indeed the original SDTN was conceived as a
Euclidean signature instanton [44]. When expanding around the SD sector to
obtain results relevant to Lorentzian signature, one envisions representing
the SD background as a complex solution defined on what will eventually be the
physical, Lorentzian-real slice. However, it may then be reasonable to hope
that the contours for the various integrations be deformable to other
signatures, particularly when Feynman propagators are used as these naturally
continue to the complex. On the Euclidean signature background, the full
spheres of constant radius $r$ are the same as those for Lorentzian signature,
and so the consequences of non-trivial topology cannot be avoided. However,
for split (or neutral) signature, these spheres are replaced by hyperbolic
discs, as in [70], (or $1+1$ de Sitter) and so the topological issues are no
longer present. It is conceivable that such choices better reflect an
effective strategy for perturbing around the SD sector. As a possible
application, the states (3.23) are non-singular even when either $p$ or $q$
are negative, because the on-shell condition in split signature implies that
$z_{i}$ lies on the boundary of the unit disc in the complex plane, while
$|\zeta|\leq 1$. In particular, this means that we can scatter states with
$n=0$, that is states with the standard
$O(r^{0})\,\mathrm{e}^{\mathrm{i}k\cdot x}$ behaviour at infinity.
Furthermore, if Lorentzian momenta are chosen, the poles below can be taken to
be in the complement of the disc over which the integration takes place. For
example, the gluon amplitude on the SDD background in split signature takes
the form

$\mathcal{A}(1^{-1}_{e=-},2^{-1}_{e=+})=-\pi\,\delta(\omega_{1}+\omega_{2})\,\delta_{-e_{1},e_{2}}\,\langle
1\,2\rangle^{2}\\\
\times\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{H}_{2}}\frac{\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}}{(1-\zeta\bar{\zeta})^{2}}\left[\frac{(\zeta-
z_{1})(\bar{\zeta}z_{2}-1)}{(\zeta-
z_{2})(\bar{\zeta}z_{1}-1)}\right]^{e_{2}}\mathrm{e}^{\mathrm{i}(\vec{k}_{1}+\vec{k}_{2})\cdot\vec{x}}\,.$
(5.1)

Here $(\zeta,\bar{\zeta})$ are Poincaré disc coordinates on the hyperbolic
space $\mathbb{H}_{2}$. If we take the momenta in split signature, $|z_{i}|=1$
whereas in Lorentz signature we can take $|z_{1}|<1$ and $|z_{2}|>1$. We leave
a systematic evaluation of the integrals in these amplitudes to future work.

## Acknowledgments

We are grateful to Roland Bittleston, Erin Crawley, Alfredo Guevara, Elizabeth
Himwich, Adam Kmec, Uri Kol, Alex Ochirov, Donal O’Connell, Natalie Paquette,
Martin Roček, Andrew Strominger and Chiara Toldo for helpful discussions. TA
is supported by a Royal Society University Research Fellowship, the Leverhulme
Trust grant RPG-2020-386 and the Simons Collaboration on Celestial Holography
MP-SCMPS-00001550-11. GB is supported by a joint Clarendon Fund and Merton
College Mathematics Scholarship. LJM would like to thank the Institutes des
Haut Études Sciéntifique, Bures Sur Yvette, and the Laboratoire Physique at
the ENS, Paris for hospitality while this was being written up and the STFC
for financial support from grant number ST/T000864/1. AS is supported by a
Black Hole Initiative fellowship, funded by the Gordon and Betty Moore
Foundation and the John Templeton Foundation.

##  Appendix A Integration of the two-point amplitudes

Here we give some details on the integration of the two-point amplitudes.
These will be made up from the six basis integrals

$\displaystyle\mathscr{H}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=\int\mathrm{d}^{3}\vec{x}\,\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{n}\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.1) $\displaystyle\mathscr{J}_{n}^{0}(\vec{k},z_{1},z_{2})$
$\displaystyle=\int\mathrm{d}^{3}\vec{x}\,\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{n-1}\frac{1}{1+|\zeta|^{2}}\,\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.2) $\displaystyle\mathscr{J}_{n}^{\zeta}(\vec{k},z_{1},z_{2})$
$\displaystyle=\int\mathrm{d}^{3}\vec{x}\,\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{n-1}\frac{\zeta}{1+|\zeta|^{2}}\,\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.3) $\displaystyle\mathscr{J}_{n}^{\bar{\zeta}}(\vec{k},z_{1},z_{2})$
$\displaystyle=\int\mathrm{d}^{3}\vec{x}\,\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{n-1}\frac{\bar{\zeta}}{1+|\zeta|^{2}}\,\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.4) $\displaystyle\mathscr{J}_{n}^{\zeta\bar{\zeta}}(\vec{k},z_{1},z_{2})$
$\displaystyle=\int\mathrm{d}^{3}\vec{x}\,\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{n-1}\frac{|\zeta|^{2}}{1+|\zeta|^{2}}\,\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.5) $\displaystyle\mathscr{K}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=\int\mathrm{d}^{3}\vec{x}\,\left(\frac{r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}{1+|\zeta|^{2}}\right)^{n-1}\frac{1}{r}\,\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.6)

where $n\in\mathbb{Z}_{\geq 0}$ is a non-negative integer. Here
$\vec{k}=\vec{k}_{1}+\vec{k}_{2}$ with $\vec{k}_{1},\vec{k}_{2}$ the momenta
of of the two wave functions. We will compute such integrals assuming
$\vec{k}$ to be real and analytically continue the final results to complex
$\vec{k}$. This is equivalent to Wick rotating $t\mapsto\mathrm{i}t$ to turn
the SDTN metric into a complex metric on a real slice on which the scattering
particles have Lorentzian momenta. This follows the computation of scattering
amplitudes on instantons via Wick rotation, first developed in [102, 103].

These are oscillatory integrals so we need to use an $\mathrm{i}\epsilon$
prescription to make the integrals convergent. Thus we introduce a Gaussian
factor of $\mathrm{e}^{-\epsilon(|\zeta|^{2}+r)}$ into the integrand, and take
the limit $\epsilon\to 0$. We will ignore the distributional terms supported
on $|\vec{k}|=0$ that will also be present which correspond to forward
scattering contributions. We discard these taking merely the _naïve_
$\epsilon\to 0$ limit.

###  A.1 Decomposition of the amplitudes

In terms of this basis of integrals, the scalar amplitude can be expressed as

$\displaystyle\mathcal{A}(1^{0}_{e=-},2^{0}_{e=+})$
$\displaystyle=2\pi\delta_{-\omega_{1},\omega_{2}}\langle
1\,2\rangle\left[-[12]\mathscr{H}_{n}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})-\frac{1}{2}n^{2}\mathscr{K}_{n}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})\right.$
$\displaystyle\quad-n(\iota_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{2\,\dot{\alpha}}+z_{1}o_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{1\,\dot{\alpha}})\mathscr{J}_{n}^{0}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})$
$\displaystyle\quad+(o_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{1\,\dot{\alpha}}+o_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{2\,\dot{\alpha}})\mathscr{J}_{n}^{\zeta}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})$
$\displaystyle\quad-(z_{1}\iota_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{1\,\dot{\alpha}}+z_{2}\iota_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{2\,\dot{\alpha}})\mathscr{J}_{n}^{\bar{\zeta}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})$
$\displaystyle\left.\quad+(z_{2}o_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{2\,\dot{\alpha}}+\iota_{\alpha}T^{\alpha\dot{\alpha}}\tilde{\kappa}_{1\,\dot{\alpha}})\mathscr{J}_{n}^{\zeta\bar{\zeta}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})\right]\,,$
(A.7)

with $n=2e_{2}$ for scattering around SD dyon and $n=4M\omega_{2}$ for
scattering around SD Taub-NUT, respectively. We have also introduced the
constant spinors $\iota^{\alpha}\coloneqq(1,0)$ and
$o^{\alpha}\coloneqq(0,-1)$, forming a dyad normalized as $\langle\iota
o\rangle=1$. Similarly, the photon amplitude around SD Taub-NUT is

$\mathcal{A}(1^{-1}_{\omega=-},2^{-1}_{\omega=+})=\frac{2\pi}{\mathtt{g}^{2}}\langle
1\,2\rangle^{2}\delta_{-\omega_{1},\omega_{2}}(\mathscr{H}_{4M\omega_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})+2M\mathscr{K}_{1+4M\omega_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2}))\,.$
(A.8)

The 2-point gluon amplitude can be expressed as

$\mathcal{A}(1^{-1}_{e=-},2^{-1}_{e=+})=-\pi\mathtt{g}^{2}\delta(\omega_{1}+\omega_{2})\delta_{-e_{1},e_{2}}\langle
1\,2\rangle^{2}\mathscr{H}_{2e_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})\,,$
(A.9)

with $n=2e_{2}$. Finally, upon expanding the product of the dressing matrices
in (4.9) using (3.41), the 2-point graviton amplitude can be expressed in
terms of these integrals as

$\displaystyle\mathcal{A}(1^{-2}_{\omega=-},2^{-2}_{\omega=+})$
$\displaystyle=-\frac{\kappa^{2}\langle
1\,2\rangle^{3}\eta_{1\,\dot{\alpha}}\eta_{2\,\dot{\beta}}}{[\eta_{1}\,1][\eta_{2}\,2]}2\pi\delta_{-\omega_{1},\omega_{2}}\left(\varepsilon^{\dot{\alpha}\dot{\beta}}\mathscr{H}_{4M\omega_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})\right.$
$\displaystyle\quad+4MT^{\alpha\dot{\alpha}}T^{\beta\dot{\beta}}(\kappa_{1\,\alpha}\iota_{\beta}-z_{1}o_{\alpha}\kappa_{2\,\beta})\mathscr{J}^{0}_{4M\omega_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})$
$\displaystyle\quad+4MT^{\alpha\dot{\alpha}}T^{\beta\dot{\beta}}(-\kappa_{1\,\alpha}o_{\beta}+o_{\alpha}\kappa_{2\,\beta})\mathscr{J}^{\zeta}_{4M\omega_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})$
$\displaystyle\quad+4MT^{\alpha\dot{\alpha}}T^{\beta\dot{\beta}}(z_{2}\kappa_{1\,\alpha}\iota_{\beta}-z_{1}\iota_{\alpha}\kappa_{2\,\beta})\mathscr{J}^{\bar{\zeta}}_{4M\omega_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})$
$\displaystyle\quad+4MT^{\alpha\dot{\alpha}}T^{\beta\dot{\beta}}(-z_{2}\kappa_{1\,\alpha}o_{\beta}+\iota_{\alpha}\kappa_{2\,\beta})\mathscr{J}_{4M\omega_{2}}^{\zeta\bar{\zeta}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})$
$\displaystyle\quad\left.+8M^{2}T^{\alpha\dot{\alpha}}\kappa_{1\,\alpha}T^{\beta\dot{\beta}}\kappa_{2,\beta}\mathscr{K}_{4M\omega_{2}}(\vec{k}_{1}+\vec{k}_{2},z_{1},z_{2})\right)\,,$
(A.10)

with again $n=4M\omega_{2}$. Thus the computation of all the four amplitudes
reduces to the evaluation of (A.1)-(A.6): to do so, we introduce auxiliary
“master integrals” that can be evaluated explicitly and used to recover
(A.1)-(A.6).

###  A.2 $\mathscr{H}_{n}$

Let us start with the computation of (A.1): we first introduce the complex
parameters

$w=\frac{k^{1}+\mathrm{i}k^{2}}{k^{3}}\,,\qquad\bar{w}=\frac{k^{1}-\mathrm{i}k^{2}}{k^{3}}$
(A.11)

so that we can parametrize the 3-vectors $\vec{x}$ and $\vec{k}$ as

$\vec{x}=\frac{r}{1+|\zeta|^{2}}(\zeta+\bar{\zeta},\mathrm{i}(\bar{\zeta}-\zeta),1-|\zeta|^{2})\,,\qquad\vec{k}=\frac{1}{2}k^{3}(w+\bar{w},\mathrm{i}(\bar{w}-w),2)\,.$
(A.12)

Some useful identities are

$\vec{k}\cdot\vec{x}=\frac{k^{3}r}{1+|\zeta|^{2}}(\zeta\bar{w}+\bar{\zeta}w+1-|\zeta|^{2})\,,\qquad\vec{k}\,^{2}=(k^{3})^{2}(1+|w|^{2})\,.$
(A.13)

Upon performing the rescaling $r\to r(1+|\zeta|^{2})$, the integral
$\mathscr{H}_{n}(\vec{k},z_{1},z_{2})$ reduces to

$\mathscr{H}_{n}(\vec{k},z_{1},z_{2})=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,(1+|\zeta|^{2})[r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)]^{n}\mathrm{e}^{\mathrm{i}k^{3}r(\zeta\bar{w}+\bar{\zeta}w+1-|\zeta|^{2})}\,.$
(A.14)

We can now introduce our first master integral as

$\hat{\mathscr{H}}(\vec{k},z_{1},z_{2};t)=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,(1+|\zeta|^{2})\mathrm{e}^{\mathrm{i}k^{3}r(\zeta\bar{w}+\bar{\zeta}w+1-|\zeta|)^{2}+\mathrm{i}tr(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)}\,.$ (A.15)

$\mathscr{H}_{n}(\vec{k},z_{1},z_{2})$ can now be recovered as

$\mathscr{H}_{n}(\vec{k},z_{1},z_{2})=\left.(-\mathrm{i}\partial_{t})^{n}\hat{\mathscr{H}}(\vec{k},z_{1},z_{2};t)\right|_{t=0}\,.$
(A.16)

The advantage in computing $\hat{\mathscr{H}}$ instead of evaluating
$\mathscr{H}_{n}$ directly is manifest, as the former is a simple Gaussian
integral over the stereographic coordinates: introducing the parameters

$A=tz_{2}-k^{3}\,,\quad B=k^{3}\bar{w}+t\,,\quad C=k^{3}w-tz_{1}z_{2}\,,\quad
D=k^{3}-tz_{1}\,,$ (A.17)

we can express the master integral as

$\hat{\mathscr{H}}(\vec{k},z_{1},z_{2};t)=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,(1+|\zeta|^{2})\mathrm{e}^{\mathrm{i}r(A|\zeta|^{2}+B\zeta+C\bar{\zeta}+D)}\,.$
(A.18)

As we will set $t=0$ at the end of the calculation, we are free to think of
$t$ to be small enough so that the imaginary part of $tz_{2}$ does not
dominate our $\mathrm{i}\epsilon$ regulator.

Performing the Gaussian integral leaves us with a Mellin integral over the
radial variable

$\hat{\mathscr{H}}(\vec{k},z_{1},z_{2};t)=\frac{4\pi\mathrm{i}}{A^{3}}\int_{0}^{\infty}\mathrm{d}r\,(r(A^{2}+BC)+\mathrm{i}A)\mathrm{e}^{\mathrm{i}r(AD-
BC)/A}\,.$ (A.19)

Since we are interested in taking derivatives of the master integral at $t=0$
and since the function $(AD-BC)/A$ has an imaginary part at least linear in
$t$, the implied $\mathrm{i}\epsilon$ prescription allows us to make this
residual Mellin integral convergent as well. In this way, we finally find

$\displaystyle\hat{\mathscr{H}}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=-\frac{4\pi\mathrm{i}(A+D)}{(AD-BC)^{2}}$ (A.20)
$\displaystyle=\frac{4\pi\mathrm{i}tz_{12}}{(\vec{k}\,^{2}+tQ)^{2}}\,,$

where in the second line we plugged in the values (A.17) and defined the
quantity

$Q=k^{3}(w-z_{1}-z_{2}-\bar{w}z_{1}z_{2})\,.$ (A.21)

Upon taking $n$ derivatives, we conclude

$\mathscr{H}_{n}(\vec{k},z_{1},z_{2})=4\pi
n\,n!\frac{z_{12}(\mathrm{i}Q)^{n-1}}{(\vec{k}\,^{2})^{n+1}}\,.$ (A.22)

Notice in particular that $\mathscr{H}_{n}$ has the expected mass dimension:
rescaling the radial variable in (A.1) to set $|\vec{k}|$ to 1, we expect
$\mathscr{H}_{n}\sim|\vec{k}|^{-n-3}$, which is precisely the scaling of
(A.22) once we notice that $Q\sim|\vec{k}|$.

###  A.3 $\mathscr{J}_{n}$ and $\mathscr{K}_{n}$

The computation of (A.2)-(A.6) proceeds along the same lines, so we can be
brief. After the rescaling $r\to r(1+|\zeta|^{2})$, the integrals read

$\displaystyle\mathscr{J}^{0}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,[r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)]^{n-1}\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.23) $\displaystyle\mathscr{J}^{\zeta}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,\zeta\,[r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)]^{n-1}\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.24) $\displaystyle\mathscr{J}^{\bar{\zeta}}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,\bar{\zeta}\,[r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)]^{n-1}\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.25) $\displaystyle\mathscr{J}^{\zeta\bar{\zeta}}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,|\zeta|^{2}\,[r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)]^{n-1}\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.26) $\displaystyle\mathscr{K}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,[r(\zeta-
z_{1})(\bar{\zeta}z_{2}+1)]^{n-1}\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{x}}\,,$
(A.27)

The relevant master integrals are thus

$\displaystyle\hat{\mathscr{J}}^{0}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,\mathrm{e}^{\mathrm{i}r(A|\zeta|^{2}+B\zeta+C\bar{\zeta}+D)}\,,$
(A.28) $\displaystyle\hat{\mathscr{J}}^{\zeta}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,\zeta\mathrm{e}^{\mathrm{i}r(A|\zeta|^{2}+B\zeta+C\bar{\zeta}+D)}\,,$
(A.29) $\displaystyle\hat{\mathscr{J}}^{\bar{\zeta}}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,\bar{\zeta}\mathrm{e}^{\mathrm{i}r(A|\zeta|^{2}+B\zeta+C\bar{\zeta}+D)}\,,$
(A.30)
$\displaystyle\hat{\mathscr{J}}^{\zeta\bar{\zeta}}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=\,,2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r^{2}\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,|\zeta|^{2}\mathrm{e}^{\mathrm{i}r(A|\zeta|^{2}+B\zeta+C\bar{\zeta}+D)}$
(A.31) $\displaystyle\hat{\mathscr{K}}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=2\mathrm{i}\int_{0}^{\infty}\mathrm{d}r\,r\int_{\mathbb{C}}\mathrm{d}\zeta\,\mathrm{d}\bar{\zeta}\,\mathrm{e}^{\mathrm{i}r(A|\zeta|^{2}+B\zeta+C\bar{\zeta}+D)}\,,$
(A.32)

and now the desired integrals can be recovered by taking $n-1$ derivatives
only, e.g.

$\mathscr{J}^{0}_{n}(\vec{k},z_{1},z_{2})=\left.(-\mathrm{i}\partial_{t})^{n-1}\hat{\mathscr{J}}^{0}(\vec{k},z_{1},z_{2};t)\right|_{t=0}\,.$
(A.33)

After performing the Gaussian integrals over $\zeta$, $\bar{\zeta}$ and the
Mellin integral over $r$, we find

$\displaystyle\hat{\mathscr{J}}^{0}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=\frac{4\pi\mathrm{i}(k^{3}-tz_{2})}{(\vec{k}\,^{2}+tQ)^{2}}\,,$
(A.34) $\displaystyle\hat{\mathscr{J}}^{\zeta}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=\frac{4\pi\mathrm{i}(k^{3}w-tz_{1}z_{2})}{(\vec{k}\,^{2}+tQ)^{2}}\,,$
(A.35) $\displaystyle\hat{\mathscr{J}}^{\bar{\zeta}}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=\frac{4\pi\mathrm{i}(k^{3}\bar{w}+t)}{(\vec{k}\,^{2}+tQ)^{2}}\,,$
(A.36)
$\displaystyle\hat{\mathscr{J}}^{\zeta\bar{\zeta}}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=-\frac{4\pi\mathrm{i}(k^{3}-tz_{1})}{(\vec{k}\,^{2}+tQ)^{2}}\,,$
(A.37) $\displaystyle\hat{\mathscr{K}}(\vec{k},z_{1},z_{2};t)$
$\displaystyle=\frac{4\pi}{\vec{k}\,^{2}+tQ}\,,$ (A.38)

so that

$\displaystyle\mathscr{J}^{0}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=-4\pi(n-1)!\frac{(\mathrm{i}Q)^{n-2}}{(\vec{k}\,^{2})^{n}}\left(\frac{nk^{3}Q}{\vec{k}\,^{2}}+(n-1)z_{2}\right)\,,$
(A.39) $\displaystyle\mathscr{J}^{\zeta}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=-4\pi(n-1)!\frac{(\mathrm{i}Q)^{n-2}}{(\vec{k}\,^{2})^{n}}\left(\frac{nk^{3}wQ}{\vec{k}\,^{2}}+(n-1)z_{1}z_{2}\right)\,,$
(A.40) $\displaystyle\mathscr{J}^{\bar{\zeta}}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=-4\pi(n-1)!\frac{(\mathrm{i}Q)^{n-2}}{(\vec{k}\,^{2})^{n}}\left(\frac{nk^{3}\bar{w}Q}{\vec{k}\,^{2}}-(n-1)\right)\,,$
(A.41) $\displaystyle\mathscr{J}^{\zeta\bar{\zeta}}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=4\pi(n-1)!\frac{(\mathrm{i}Q)^{n-2}}{(\vec{k}\,^{2})^{n}}\left(\frac{nk^{3}Q}{\vec{k}\,^{2}}+(n-1)z_{1}\right)\,,$
(A.42) $\displaystyle\mathscr{K}_{n}(\vec{k},z_{1},z_{2})$
$\displaystyle=4\pi(n-1)!\frac{(\mathrm{i}Q)^{n-1}}{(\vec{k}\,^{2})^{n}}\,.$
(A.43)

To get the desired expressions for the amplitudes, we can simplify the
formulae for the integrals further using on-shell kinematics, i.e. using the
fact that $\vec{k}=\vec{k}_{1}+\vec{k}_{2}$ and expressing $Q$, $k^{3}$,
$\vec{k}\,^{2}$, $w$, and $\bar{w}$ in terms of $z_{1}$, $z_{2}$,
$\omega_{1}$, and $\omega_{2}$. On the support of $\omega_{1}+\omega_{2}=0$,
the relevant expressions are

$\displaystyle\vec{k}\,^{2}$
$\displaystyle=-\frac{4z_{12}\tilde{z}_{12}\omega_{2}^{2}}{(1+z_{1}\tilde{z}_{1})(1+z_{2}\tilde{z}_{2})}\,,$
(A.44) $\displaystyle k^{3}$
$\displaystyle=-2\mathrm{i}\omega_{2}\frac{z_{1}\tilde{z}_{1}-z_{2}\tilde{z}_{2}}{(1+z_{1}\tilde{z}_{1})(1+z_{2}\tilde{z}_{2})}\,,$
(A.45) $\displaystyle k^{3}w$
$\displaystyle=-2\mathrm{i}\omega_{2}\frac{-z_{12}+\tilde{z}_{12}z_{1}z_{2}}{(1+z_{1}\tilde{z}_{1})(1+z_{2}\tilde{z}_{2})}\,,$
(A.46) $\displaystyle k^{3}\bar{w}$
$\displaystyle=-2\mathrm{i}\omega_{2}\frac{-\tilde{z}_{12}+z_{12}\tilde{z}_{1}\tilde{z}_{2}}{(1+z_{1}\tilde{z}_{1})(1+z_{2}\tilde{z}_{2})}\,,$
(A.47) $\displaystyle Q$ $\displaystyle=-2\mathrm{i}\omega_{1}z_{12}\,.$
(A.48)

With these, we immediately obtain the vanishing of the scalar and photon
amplitudes: for example, in the photon case the amplitude is

$\mathcal{A}(1^{-1}_{\omega=-},2^{-1}_{\omega=+})=\frac{8\pi^{2}}{\mathtt{g}^{2}}\delta_{-\omega_{1},\omega_{2}}\langle
1\,2\rangle^{2}(4M\omega_{2})!\frac{z_{12}^{4M\omega_{2}}}{|\vec{k}_{1}+\vec{k}_{2}|^{8M\omega_{2}+2}}\\\
\times\left(4M\omega_{2}(2\omega_{1})^{4M\omega_{2}-1}+2M(2\omega_{1})^{4M\omega_{2}}\right)\,,$
(A.49)

and the bracket vanishes on the support of $\omega_{1}+\omega_{2}=0$. An
analogous cancellation, although slightly more involved, occurs for
$\mathcal{A}(1^{0}_{e=-},2^{0}_{e=+})$. For the 2-point gluon amplitude, we
find

$\mathcal{A}(1^{-1}_{e=-},2^{-1}_{e=+})=-8\pi^{2}\mathtt{g}^{2}\delta(\omega_{1}+\omega_{2})\delta_{-e_{1},e_{2}}\langle
1\,2\rangle^{2}e_{2}\,(e_{2})!\frac{(2\omega_{1})^{2e_{2}-1}z_{12}^{2e_{2}}}{|\vec{k}_{1}+\vec{k}_{2}|^{4e_{2}+2}}\,,$
(A.50)

and upon expressing $z_{12}=\langle 1\,2\rangle$ in terms of the spinor-
helicity variables, we recover (4.10). The computation for the 2-point
graviton amplitude proceeds along the same lines, except that the technical
details are more involved. Upon inserting (A.1)-(A.6) into (A.10), we find

$\mathcal{A}(1^{-2}_{\omega=-},2^{-2}_{\omega=+})=-8\pi\kappa^{2}\frac{\langle
1\,2\rangle^{3}\eta_{1\,\dot{\alpha}}\eta_{2\,\dot{\beta}}}{[\eta_{1}\,1][\eta_{2}\,2]}\delta_{-\omega_{1},\omega_{2}}(4M\omega_{2}-1)!\frac{(2\omega_{1})^{4M\omega_{2}-1}z_{12}^{4M\omega_{2}+2}}{|\vec{k}_{1}+\vec{k}_{2}|^{8M\omega_{2}+2}}M^{2}\times\\\
\times\frac{-2\omega_{2}^{2}}{(1+z_{1}\tilde{z}_{1})(1+z_{2}\tilde{z}_{2})}\left(\begin{array}[]{cc}1&\tilde{z}_{2}\\\
\tilde{z}_{1}&\tilde{z}_{1}\tilde{z}_{2}\end{array}\right)^{\dot{\alpha}\dot{\beta}}\,.$
(A.51)

We can recognize the tensor structure in the second line as the tensor product
$\tilde{\kappa}_{1}^{\dot{\alpha}}\tilde{\kappa}_{2}^{\dot{\beta}}$, as we
expect it to be because the amplitude ought to be gauge invariant, i.e.,
independent of the choice of the reference spinors $\eta_{1\,\dot{\alpha}}$
and $\eta_{2\,\dot{\alpha}}$. With this simplification and expressing again
$z_{12}$ as $\langle 1\,2\rangle$, we finally recover (4.16).

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