Kerr binary dynamics from minimal coupling and double copy
Francesco Alessioa,b

a NORDITA, KTH Royal Institute of Technology and Stockholm University,
Hannes Alfvéns väg 12, SE-11419 Stockholm, Sweden
b Department of Physics and Astronomy, Uppsala University,
Box 516, SE-75120 Uppsala, Sweden

We construct a new Yang-Mills Lagrangian based on a notion of minimal coupling
that incorporates classical spin effects. The construction relies on the
introduction of a new covariant derivative, which we name “classical spin
covariant derivative”, that is compatible with the three-point interaction of
the $\sqrt{\mathrm{Kerr}}$ solution with the gauge field. The resulting
Lagrangian, besides the correct three-point coupling, predicts a unique choice
for contact terms and therefore it can be used to compute higher-point
amplitudes such as the Compton, unaffected by spurious poles. Using double
copy techniques we use this theory to extract gravity amplitudes and
observables that are relevant to describe Kerr binary dynamics to all orders
in the spin. In particular, we compute the 2PM ($\mathcal{O}(G^{2}_{N})$)
$2\rightarrow 2$ elastic scattering amplitude between two classically spinning
objects to all orders in the spin and use it to extract the 2PM scattering
angle.

##  1 Introduction

The idea that black holes share certain features with elementary particles is
not new and, in fact, it has quite a long history (see e.g.[1, 2, 3]). One of
the reasons that makes these two classes of apparently unrelated objects
similar is that they are both characterized by few charges, namely the mass,
the intrinsic angular momentum i.e. the spin and the electric charge. Although
being exact solutions of the full, non-linear Einstein equations, black holes
seem to be simple objects, at least if compared to other solutions such as
neutron stars.

Beside being interesting in itself, the aforementioned analogy can be
concretely used to perform analytical precision computations relevant in the
context of gravitational waves physics, that has recently dragged a lot of
attention because of the many detections made by LIGO, Virgo and Kagra of
waves emitted by strongly gravitating binary systems [4, 5, 6, 7, 8, 9, 10] .
During the very first phase of the coalescence of two black holes, known as
the inspiral phase, the two compact objects are widely separated and the
gravitational interaction between them is weak. In this regime one can neglect
the internal structure of the bodies, sized by the Schwarzschild radius, and
successfully use perturbation theory to extract the perturbative expansion in
the gravitational coupling constant of the observables one is interested in.
This is called post-Minkowskian (PM) regime and it is usually approached
either by using classical general relativity (GR) methods or scattering
amplitude techniques in quantum field theory (QFT) [11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58]. However, because of on-shell techniques and many
simplifications, state-of-the-art theoretical predictions are made within the
latter approach, where one treats the two gravitating bodies as elementary
point-like particles, computes a scattering amplitude and then extracts its
classical limit, which roughly amounts to sending $\hbar$ to zero, after
having correctly restored it in the theory. Eventually open orbits
observables, obtained by differentiating the classical scattering amplitude,
can be mapped into bounded orbits ones [59, 60, 61] and can be compared to
those derived in the more familiar Post-Newtonian (PN) regime (see e.g. [62]
for a review), where, together with the weak field approximation, one also
considers small velocities.

It is not surprising that the theory used to describe the gravitational
interaction between Schwarzschild black holes, only characterized by their
masses, is in the PM regime simply that of minimally coupled massive scalars
to gravity111In this framework, adding non-minimal terms to the Lagrangian
means considering tidal deformations contributions.. Computations can be
significantly simplified using double copy [63, 64, 65, 66, 67] and, infact,
it has been shown that scalar QCD provides the single copy theory to compute
the relevant building blocks and amplitudes for the the more complicated
theory of gravity [68, 69, 15, 3, 70, 71, 72, 73]. State-of-the-art results
for the scattering between two spinless black holes in both the conservative
and radiative sectors are up to 4PM ($\mathcal{O}(G^{4}_{N})$).

However, more realistic black holes are also characterized by the spin vector
$S^{\mu}$ [74]. Because of this additional scale the problem is much more
complicated and, on top of each PM order, there is a spin multipole expansion
in the classical spin vector that, in general, depends on the nature of the
compact body one is describing [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85,
86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,
104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118,
119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133].
Quantum-mechanically, the spin vector satisfies
$S^{\mu}S_{\mu}=-\hbar^{2}s(s+1)$, where $s$ is the spin quantum number.
Hence, when trying to attack the classical scattering problem using QFT
methods, it seems natural to make use of spinning massive fields, with higher
and higher value of the spin quantum number $s$ , in such a way to keep the
value of $S^{\mu}S_{\mu}$ finite when sending $\hbar$ to zero. This will yield
a spin multipole expansion in the form of a polynomial truncated at order
$(S^{\mu})^{2s}$. The idea of using quantum spinning fields to mimic Kerr
black holes is corroborated by the “minimally coupled”222Here the notion of
“minimal coupling” does not have anything to do with the QFT one, which we
will exploit later on in the paper. three-point amplitudes presented in [75]
that, in the classical limit, match the classical computation of the
linearised stress-energy tensor of a Kerr black hole [78]. The latter resums
all the classical spin dependence of the amplitude into a simple and compact
exponentiated form that can be interpreted, on the support of the three-point
kinematics, as an exponentiated version of the subleading soft graviton
theorem [81, 134]. Using this three-point amplitude, the 1PM classical
amplitude for the scattering between two Kerr black holes has been easily
constructed to all order in spin [81, 86]. However, going to 2PM requires the
knowledge of the tree-level Compton amplitude and, when one constructs such
object using BCFW recursion relations [135], the opposite-helicity sector is
affected, for $s>2$, i.e. for higher spins, by the presence of unphysical
spurious poles. The same problem appears for gauge theories when $s>1$, when
starting from the $\sqrt{\mathrm{Kerr}}$ three-point amplitude. There has been
a lot of effort to understand how to remove these poles. The Lagrangian
approaches of [102, 112] give healthy Compton amplitudes up to $s=5/2$.
Assuming that the Compton amplitude additionally satisfies a “shift-symmetry”
to all orders in spin leads to a result unaffected by the above-mentioned
spurious poles [110]. A microscopical reason for assuming such symmetry is,
however, far from being clear. All these efforts and proposals should
eventually be compared with solutions of the classical Teukolsky equation,
recently found in [125].

In this paper, we follow a different path. We try to answer the question of
whether it is possible, in principle, to construct an effective Lagrangian
field theory describing the interactions of Kerr black holes with gravity
depending on the classical spin $S^{\mu}$ and therefore carrying, as a built-
in feature, an infinite spin multipole expansion. There are several
motivations to ask this question. The first is of purely theoretical nature.
As argued above, black holes are, from a certain perspective, simple and this
simplicity, for the Schwarzschild case, is reflected in the standard QFT
notion of minimal coupling. On the other hand, Kerr black-holes look much more
sophisticated but they are actually not, as clear in Kerr-Schild coordinates
[136]. Indeed, Kerr spacetime can be entirely reconstructed by performing a
Newman-Janis transformation [137] to the radial coordinate characterizing the
spinless solution333A discussion on Kerr-Schild coordinates can be found in
appendix A.. Constructing a new field theory for Kerr by implementing this
transformation directly at the level of the Lagrangian would represent at
least a good candidate to describe Kerr black holes as elementary particles. A
second, but not less important, motivation concerns the frontier of precision
computations. Having a Lagrangian built upon the criterion of simplicity would
make it possible to proceed beyond the first PM order in the perturbative
expansion of gravitational observables. One could argue that, unless there is
a classical GR computation that counterparts the analysis of this paper, these
observables could have little to do with Kerr black holes. However, in any
case, the theory we present here could always be regarded as a “minimal” model
on top of which one is free to add operators of higher order in the curvature,
opportunely multiplied by Wilson coefficients, in order to better and better
approximate true classically spinning solutions.

In practice we do not construct here the theory of gravity, but we limit
ourselves to build its single copy, gauge theory counterpart. Then,
implementing a form of the double copy that adapts to classically spinning
theories, we obtain gravity amplitudes.

The paper is organized as follows. We start in section 2 where, after
discussing the usual notion of minimal coupling and its role in scalar QED, we
construct a new theory by introducing a covariant derivative which is
compatible with the $\sqrt{\mathrm{Kerr}}$ three-point amplitude and that
carries information about its interaction with the gauge field. We show
explicit expressions for the Lagrangian and for the contact terms,
automatically generated by this definition of minimal coupling. The Newman-
Janis shift, discussed in some detail in appendix A, is a constitutive element
of our Lagrangian. In section 3 we consider tree-level amplitudes and we show
that, on-shell, spinning amplitudes involving an arbitrary number of massless
legs entirely factorize in terms of spinless ones and that the full spin
contribution can remarkably be encoded into a simple exponential factor.
Because they come from a local Lagrangian, such amplitudes will be unaffected
by unphysical poles. The difference between the opposite-helicity Compton
amplitude we find within our approach and the “minimally coupled” one is
spelled out in appendix B. Using double copy for classically spinning
theories, in section 4, we obtain gravity amplitudes relevant to describe Kerr
black holes binary dynamics. Section 5 is devoted to the computation of the
2PM amplitude, eikonal and scattering angle using on-shell techniques.
Integrals appearing in the one-loop amplitude and eikonal are computed in
appendix D and E. We conclude in 6 with a discussion. Throughout the paper we
use mostly minus signature.

##  2 Minimal coupling and classical spin

In this section we start by considering scalar QED, review its Feynman rules
and compute the well-known Compton amplitude. Afterwards, by analogy and
requiring compatibility with the three-point coupling of the
$\sqrt{\mathrm{Kerr}}$ solution, we build a new Lagrangian describing
classically spinning scalars coupled to the electromagnetic field.

#### Scalar QED

A scalar $\phi$ with mass $m$ and electric charge $e$ minimally coupled to the
electromagnetic field is described by the scalar QED Lagrangian,

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m^{2}|\phi|^{2},\hskip
14.22636ptD_{\mu}=\partial_{\mu}+ieA_{\mu}.$ (2.1)

Under gauge transformations,

$\displaystyle A_{\mu}\rightarrow
A_{\mu}-\frac{i}{e}U\partial_{\mu}U^{*},\hskip 28.45274pt\phi\rightarrow
U\phi,\hskip 28.45274ptU=e^{ie\hskip 1.42271pt\chi}\in U(1),$ (2.2)

the covariant derivative transforms as $D_{\mu}\phi=UD_{\mu}\phi$. This is the
usual QFT notion of minimal coupling, that consists in providing interactions
between matter and gauge fields with the least number of derivatives in a
gauge invariant way. Clearly other non-minimal interaction terms, containing
more derivatives, may be added to the Lagrangian, but we will not consider
them here. Beside ensuring gauge invariance, the replacement
$\partial_{\mu}\phi\rightarrow D_{\mu}\phi$ in the free Lagrangian
automatically yields three- and four-point couplings,

$\displaystyle\mathcal{L}_{3\mathrm{pt}}=ieA^{\mu}(\phi\partial_{\mu}\phi^{*}-\phi^{*}\partial_{\mu}\phi),\qquad\mathcal{L}_{4\mathrm{pt}}=e^{2}A^{\mu}A_{\mu}|\phi|^{2},$
(2.3)

from which we easily read the Feynman rules,

$\vbox{\hbox{\includegraphics[width=102.43008pt,height=113.81102pt]{3pt.png}}}\begin{aligned}
\hskip 2.84544pt=-ie\hskip 1.42271ptp^{\mu},\end{aligned}\begin{aligned}
\hskip
56.9055pt\vbox{\hbox{\includegraphics[width=102.43008pt,height=113.81102pt]{ct.png}}}\hskip
2.84544pt=2ie^{2}\eta^{\mu\nu},\end{aligned}$ (2.4)

where $\phi$ ($\phi^{*}$) has momentum $p_{1}$ ($p_{2}$), $p\equiv
p_{2}-p_{1}$ and where, by total momentum conservation for the first diagram
$p_{1}+p_{2}+q=0$ and for the second $p_{1}+q_{1}+q_{2}+p_{4}=0$444In our
convention all momenta are outgoing.. The tree-level Compton amplitude is the
sum of the three diagrams below,

$i\mathcal{A}^{\mu\nu}_{4}(p_{i},q_{i})=\vbox{\hbox{\includegraphics[width=82.51282pt,height=110.96556pt]{C.png}}}\begin{aligned}
\hskip
2.84544pt=\end{aligned}\vbox{\hbox{\includegraphics[width=71.13188pt,height=110.96556pt]{g1.png}}}\hskip
2.84544pt\begin{aligned}
+\end{aligned}\vbox{\hbox{\includegraphics[width=71.13188pt,height=110.96556pt]{g2.png}}}\hskip
2.84544pt\begin{aligned}
+\end{aligned}\vbox{\hbox{\includegraphics[width=71.13188pt,height=110.96556pt]{g3.png}}}.$
(2.5)

Using the scalar propagator, we get for $\mathcal{A}_{4}^{\mu\nu}$,

$\displaystyle\mathcal{A}^{\mu\nu}_{4}(p_{i},q_{i})=-2e^{2}\bigg{(}\frac{p_{1}^{\mu}p_{1}^{\nu}+q_{2}^{\mu}p_{1}^{\nu}}{p_{1}\cdot
q_{2}}+\frac{p_{1}^{\mu}p_{1}^{\nu}+p_{1}^{\mu}q_{3}^{\nu}}{p_{1}\cdot
q_{3}}-\eta^{\mu\nu}\bigg{)}.$ (2.6)

Contracting with physical polarization vectors satisfying
$\varepsilon_{i}^{2}=0=\varepsilon_{i}\cdot q_{i}$ and using the on-shell
conditions $q^{2}_{i}=0$ we get,

$\displaystyle\mathcal{A}_{4}(p_{i},q_{i})\equiv\varepsilon_{1\nu}\varepsilon_{2\mu}\mathcal{A}^{\mu\nu}_{4}(p_{i},q_{i})=-\frac{2e^{2}}{p_{1}\cdot
q_{2}p_{1}\cdot q_{3}}p_{1}\cdot F_{2}\cdot F_{3}\cdot p_{1},$ (2.7)

where we introduced the field strength
$F_{i}^{\mu\nu}=2\varepsilon_{i}^{[\mu}q_{i}^{\nu]}$555Symmetrization and
antisymmetrization are defined as
$A^{(\mu}B^{\nu)}=\frac{1}{2}(A^{\mu}B^{\nu}+A^{\nu}B^{\mu})$ and
$A^{[\mu}B^{\nu]}=\frac{1}{2}(A^{\mu}B^{\nu}-A^{\nu}B^{\mu})$. and used that,
on-shell, $p_{1}\cdot(q_{2}+q_{3})=-q_{2}\cdot q_{3}$. Note that
$\mathcal{A}_{4}$ is gauge invariant because it is written purely in terms of
gauge invariant field strengths. If we knew only the three-point coupling in
equation (2.3) i.e. if we summed only the first two diagrams in (2.5), we
could have guessed the four-point contact term, necessary to achieve gauge
invariance, just by requiring that $\delta_{\chi}\mathcal{A}_{4}=0$. This
suggests that, in a minimally coupled gauge theory, contact terms are always
generated as a consequence of the three-point coupling that, in turn, comes
from the notion of a “good” covariant derivative.

As widely discussed in the literature and in the introduction, amplitudes
describing Schwarzschild black holes come from minimally coupled scalars to
gravity. Such minimal coupling is, at the gauge theory level, encoded in the
notion of covariant derivative and indeed, the classical Compton amplitude for
Schwarzschild black holes can be simply obtained by double copying
$\mathcal{A}_{4}$ in (2.7). This shows that the simplicity of black holes is
intimately connected to the notion of minimal coupling. In the following we
will assume, relying on the considerations of the previous section and in
particular on the Newman-Janis algorithm, discussed in appendix A, that Kerr
black holes should also be described by a minimally coupled, simple, theory.
In other words, we will ask whether the classical three-point amplitude for
$\sqrt{\mathrm{Kerr}}$, known to all orders in the spin, yields a good
covariant derivative and can therefore be used to generate contact terms in
the same way they are derived in the spinless case. The answer to this
question, quite surprisingly, is yes.

####  $\sqrt{\mathrm{Kerr}}$ and classically spinning gauge theory

In the worldline picture, the Coulomb field can be thought of as generated by
a charged scalar sourced by a static point particle and hence it is a solution
of the Maxwell equations,

$\displaystyle\partial_{\mu}F^{\mu\nu}(x)=J^{\nu}(x),\qquad
J^{\mu}(x)=e\int\frac{d\tau}{2\pi}u^{\mu}\delta^{(4)}(x-u\tau),$ (2.8)

where $\tau$ is the proper time of the point particle moving on the worldline
$x(\tau)$ and $u^{\mu}(\tau)=\dot{x}(\tau)$ is the unit tangent vector to the
curve satisfying $u_{\mu}u^{\mu}=1$ (see e.g. [70]). Taking the Fourier
transform of $J^{\mu}(x)$ yields,

$\displaystyle\tilde{J}^{\mu}(q)=\int\frac{d^{4}x}{(2\pi)^{4}}e^{iq\cdot
x}J^{\mu}(x)=e\hskip 1.42271ptu^{\mu}\delta(q\cdot u).$ (2.9)

From a Lagrangian field theory point of view, the coupling of the charged
particle to the electromagnetic field is encoded in the term $A_{\mu}J^{\mu}$
and therefore the Feynman rule for the three-point amplitude is the Fourier
transform $\tilde{J}^{\nu}(q)$ of $J^{\nu}(x)$. Indeed, introducing
$p^{\mu}=mu^{\mu}$ the Fourier transform (2.9) is proportional to the first
equation in (2.4). The additional delta function appearing in (2.9) enforces
three-point kinematics $p_{1}\cdot q=p_{2}\cdot q=p\cdot q=0$.

As shown in the context of gravity in [78], the $\sqrt{\mathrm{Kerr}}$
solution, which is the electromagnetic field generated by a rotating disc of
radius $a$, is sourced by the following current,

$\displaystyle
J^{\mu}(x)=e\int\frac{d\tau}{2\pi}\mathrm{exp}\\{\epsilon^{\mu}{}_{\nu}(a,\partial)\\}u^{\nu}\delta^{(4)}(x-u\tau),\qquad\epsilon^{\mu}{}_{\nu}(a,\partial)\equiv\epsilon^{\mu}{}_{\nu\rho\sigma}a^{\rho}\partial^{\sigma},$
(2.10)

where the rescaled spin vector $a^{\mu}$ is related to the classical Pauli-
Lubanski spin vector $S^{\mu}$ by $S^{\mu}=ma^{\mu}$ and satisfies the
Tulczjew-Dixon SSC condition $a\cdot u=0$ [138, 139]. This time, performing
the Fourier transform yields [81],

$\displaystyle\tilde{J}^{\mu}(q)=\int\frac{d^{4}x}{(2\pi)^{4}}e^{iq\cdot
x}J^{\mu}(x)=e\hskip
1.42271pt\mathrm{exp}\\{i\epsilon^{\mu}{}_{\nu}(a,q)\\}u^{\nu}\delta(q\cdot
u),$ (2.11)

which leads to the following three-point amplitude,

$\vbox{\hbox{\includegraphics[width=102.43008pt,height=113.81102pt]{3pt.png}}}\begin{aligned}
\hskip 2.84544pt=-ie\hskip
1.42271pt\mathrm{exp}\\{i\epsilon^{\mu}{}_{\nu}(a,q)\\}p^{\nu}\equiv-ie\hskip
1.42271pt\Lambda^{\mu}{}_{\nu}(a,q)p^{\nu}.\end{aligned}$ (2.12)

In practice, switching from scalar QED to $\sqrt{\mathrm{Kerr}}$ amounts to
multiplying the three-point Feynman rule of the former by a matrix
$\Lambda^{\mu}{}_{\nu}(a,q)$, which is the exponential of the antisymmetric
tensor $\omega_{\mu\nu}=-\omega_{\nu\mu}=\epsilon_{\mu\nu}(a,q)$. Remarkably,
it was shown in [81, 86] that the amplitude (2.12) corresponds to the
classical limit of “minimally coupled” three-point amplitudes constructed in
[75]. Indeed, contracting (2.12) with $\varepsilon_{\mu}$ and expanding the
exponential gives,

$\displaystyle-ie\hskip
1.42271pt\varepsilon^{\mu}\bigg{[}\eta_{\mu\nu}+i\epsilon_{\mu\nu}(a,q)-\frac{1}{2!}\epsilon_{\mu\gamma}(a,q)\epsilon^{\gamma}{}_{\nu}(a,q)+\dots\bigg{]}p^{\nu}$
$\displaystyle=$ $\displaystyle ie\hskip 1.42271pt\varepsilon\cdot
p\bigg{[}1+i\frac{\varepsilon\cdot S\cdot q}{\varepsilon\cdot
p}+\frac{1}{2}(a\cdot q)^{2}+\dots\bigg{]}=ie\hskip 1.42271pt\varepsilon\cdot
p\hskip
1.42271pt\mathrm{exp}\left\\{\frac{i}{2}\frac{S^{\mu\nu}F_{\mu\nu}}{\varepsilon\cdot
p}\right\\},$ (2.13)

where $S^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}p_{\alpha}a_{\beta}$ is the spin
tensor of the massive particle and where we repeatedly used the relations,

$\displaystyle\epsilon^{\mu\nu\rho\sigma}\epsilon_{\alpha\beta\gamma\delta}=-4!\delta^{[\mu}_{\alpha}\delta^{\nu}_{\beta}\delta^{\rho}_{\gamma}\delta^{\sigma]}_{\delta},\qquad\epsilon^{\mu\nu\rho\lambda}\epsilon_{\alpha\beta\gamma\lambda}=-3!\delta^{[\mu}_{\alpha}\delta^{\nu}_{\beta}\delta^{\rho]}_{\gamma},$
(2.14)

together with, crucially, the three-point kinematics $p\cdot q=0$, the on-
shell conditions $\varepsilon^{2}=q^{2}=\varepsilon\cdot q=0$ and the SSC
$a\cdot p=0$ as well. It is suggestive to note [81] that the exponential
factor appearing in (2) is exactly an exponentiated version of the subleading
soft photon theorem [140]. The poles appearing when expanding the exponential
in (2) for $a>1$ are cancelled due to three-point kinematics,

$\displaystyle\frac{i}{2}\frac{S^{\mu\nu}F_{\mu\nu}}{\varepsilon\cdot
p}=i\frac{\varepsilon\cdot S\cdot q}{\varepsilon\cdot
p}=-\frac{a\cdot\varepsilon\hskip 1.42271ptp\cdot q-a\cdot q\hskip
1.42271ptp\cdot\varepsilon}{p\cdot\varepsilon}=a\cdot q,$ (2.15)

where we used $\varepsilon\cdot S\cdot q=\sqrt{(\varepsilon\cdot S\cdot
q)^{2}}$ and the Gram determinants in (2.14) with the assumption of real
polarization vector $\varepsilon$.

Asking that the structure appearing in (2.12) comes from a standard QFT notion
of minimal coupling, i.e. from the existence of a covariant derivative,
implies that the latter should be,

$\displaystyle D_{\mu}^{(a)}=\partial_{\mu}-ie\hskip
1.42271pt\mathrm{exp}\\{\epsilon^{\nu}{}_{\mu}(a,\partial)\\}A_{\nu}.$ (2.16)

Under gauge transformations as in (2.2) we have,

$\displaystyle D_{\mu}^{(a)}\phi\rightarrow$
$\displaystyle\partial_{\mu}(U\phi)-ie\hskip
1.42271pt\mathrm{exp}\\{\epsilon^{\nu}{}_{\mu}(a,\partial)\\}(A_{\nu}+\partial_{\nu}\chi)(U\phi)=UD_{\mu}^{(a)}\phi,$
(2.17)

where we used that the matrix exponential acts trivially on pure gauge
configurations,
$\mathrm{exp}\\{\epsilon^{\nu}{}_{\mu}(a,\partial)\\}\partial_{\nu}\chi=\partial_{\mu}\chi$.
Equation (2.17) shows that the operator $D_{\mu}^{(a)}$ is indeed a good
covariant derivative and it can be used to construct consistent interactions
between matter and gauge fields. We will denote $D_{\mu}^{(a)}$ as classical
spin covariant derivative. We are naturally led to consider the gauge
invariant Lagrangian,

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D^{(a)}_{\mu}\phi)^{*}(D^{\mu}_{(a)}\phi)-m^{2}|\phi|^{2}=\mathcal{L}_{\mathrm{free}}+\mathcal{L}_{3\mathrm{pt}}+\mathcal{L}_{4\mathrm{pt}},$
(2.18)

where,

$\displaystyle\mathcal{L}_{3\mathrm{pt}}=ie\hskip
1.42271pt\mathrm{exp}\\{\epsilon_{\mu\nu}(a,\partial)\\}A^{\mu}(\phi\partial^{\nu}\phi^{*}-\phi^{*}\partial^{\nu}\phi),$
(2.19)
$\displaystyle\mathcal{L}_{4\mathrm{pt}}=e^{2}\mathrm{exp}\\{\epsilon_{\mu\rho}(a,\partial)\\}A^{\mu}\mathrm{exp}\\{\epsilon_{\nu}{}^{\rho}(a,\partial)\\}A^{\nu}|\phi|^{2},$
(2.20)

that automatically yields the three-point diagram in (2.12) and the four-point
contact term,

$\displaystyle\vbox{\hbox{\includegraphics[width=102.43008pt,height=113.81102pt]{ct.png}}}\hskip
2.84544pt=2ie^{2}\Lambda^{\mu}{}_{\rho}(a,q_{3})\Lambda^{\nu\rho}(a,q_{2}).$
(2.21)

Before proceeding further we note here that, in general, the field strength is
defined as the commutator of covariant derivatives,
$F_{\mu\nu}=\frac{i}{e}[D_{\mu},D_{\nu}]$. Using this definition, together
with the classical spin connection in (2.16), would change the photon
propagator. However, in the theory (2.18) we are keeping the free part as it
is in the standard YM Lagrangian, without modifying the field strength as
$F_{\mu\nu}\rightarrow F^{(a)}_{\mu\nu}$. In other words, the photon field
gets dressed with the spin exponential matrix only when it interacts locally
with the massive, classically spinning field $\phi$. Away from it photons are
not affected by the spin. This remark is not important in the abelian version
of the theory we are presenting here, because none of the diagrams appearing
in the Compton (2.5) contains a photon propagator. When extending this theory
to gluons, however, beside the two massive channels and the contact term,
there is also a massless channel involving the gluon propagator.

It comes as a consequence of the above considerations that the spinning
Compton amplitude $\mathcal{A}_{4}^{\mu\nu}(a)$ is simply factorized in terms
of the spinless one as,

$\displaystyle\mathcal{A}_{4}^{\mu\nu}(a;p_{i},q_{i})=\Lambda^{\mu}{}_{\rho}(a,q_{3})\Lambda^{\nu}{}_{\sigma}(a,q_{2})\mathcal{A}_{4}^{\rho\sigma}(p_{i},q_{i}).$
(2.22)

We need to contract with polarization vectors $\varepsilon_{2\nu}$ and
$\varepsilon_{3\mu}$ and hence it is convenient to define new vectors
$\varepsilon^{\prime\mu}_{i}=\Lambda_{\nu}{}^{\mu}(a,q_{i})\varepsilon^{\nu}_{i}$
so that,

$\displaystyle\mathcal{A}_{4}(a;p_{i},q_{i})=\varepsilon_{2\nu}\varepsilon_{3\mu}\mathcal{A}_{4}^{\mu\nu}(a;p_{i},q_{i})=-\frac{2e^{2}}{p_{1}\cdot
q_{2}p_{1}\cdot q_{3}}p_{1}\cdot F^{\prime}_{2}\cdot F^{\prime}_{3}\cdot
p_{1},$ (2.23)

where $F_{i}^{\prime\mu\nu}=2\varepsilon^{\prime[\mu}_{i}q_{i}^{\nu]}$. In
general, all the spinning tree-level amplitudes
$\mathcal{A}_{N}^{\mu_{2}\dots\mu_{N-1}}(a)$ involving two massive and $N-2$
massless lines with momenta $q_{2},\dots,q_{N-1}$ following from the
Lagrangian (2.18) can be written in terms of the spinless ones as,

$\mathcal{A}^{\mu_{2}\dots\mu_{N-1}}_{N}(a;p_{i},q_{i})=\vbox{\hbox{\includegraphics[width=96.73918pt,height=113.81102pt]{nC.png}}}\\\
\begin{aligned} \hskip
2.84544pt=\Lambda^{\mu_{2}}{}_{\nu_{2}}(a,q_{2})\dots&\Lambda^{\mu_{N-1}}{}_{\nu_{N-1}}(a,q_{N-1})\\\
&\mathcal{A}_{N}^{\nu_{2}\dots\nu_{N-1}}(p_{i},q_{i}).\end{aligned}$ (2.24)

##  3 Spin exponentiation

The Lagrangian presented in the previous section predicts tree-level spinning
amplitudes containing various factors of
$\Lambda_{\nu}{}^{\mu}(a,q_{i})\varepsilon^{\nu}_{i}$ for each external
massless line. Here we would like to analyse these factors more in detail and
to extract an alternative expression that possibly does not contain the Levi-
Civita symbol. We introduce the shorthand notation
$\epsilon^{\mu_{1}\dots\mu_{n}}{}_{\nu_{1}\dots\nu_{n}}v^{\nu_{1}}_{1}\dots
v^{\nu_{m}}_{m}\equiv\epsilon^{\mu_{1}\dots\mu_{n}}(v_{1},\dots,v_{m})$ for
$m$ vectors $v_{1},\dots,v_{m}$, that generalizes the one used in (2.10) and
we start by expanding the exponential as,

$\displaystyle\mathrm{exp}\\{i\epsilon_{\nu}{}^{\mu}(a,q)\\}\varepsilon^{\nu}=\hskip
1.42271pt$
$\displaystyle\varepsilon^{\mu}-i\epsilon^{\mu}(\varepsilon,a,q)+\frac{1}{2!}\epsilon^{\alpha}(\varepsilon,a,q)\epsilon_{\alpha}{}^{\mu}(a,q)$
$\displaystyle+$
$\displaystyle\frac{i}{3!}\epsilon^{\alpha}(\varepsilon,a,q)\epsilon_{\alpha}{}^{\beta}(a,q)\epsilon_{\beta}{}^{\mu}(a,q)+\dots$
(3.1)

Defining the dual field strength
$F^{*\mu\nu}_{i}=\frac{i}{2}\epsilon^{\mu\nu}{}_{\rho\sigma}F^{\rho\sigma}=i\epsilon^{\mu\nu}(\varepsilon,q)$
666Usually the dual field strength $F^{*}_{\mu\nu}$ is defined without the $i$
factor. and using the on-shell equality,

$\displaystyle\epsilon^{\alpha}(\varepsilon,a,q)\epsilon_{\alpha}{}^{\beta}(a,q)=a\cdot
q(a\cdot q\varepsilon^{\beta}-a\cdot\varepsilon q^{\beta})=a\cdot
qF^{\beta\rho}a_{\rho},$ (3.2)

the series in (3) can easily be resummed,

$\displaystyle\mathrm{exp}\\{i\epsilon_{\nu}{}^{\mu}(a,q)\\}\varepsilon^{\nu}=\hskip
1.42271pt$
$\displaystyle\varepsilon^{\mu}+F^{*\mu\rho}a_{\rho}+\frac{1}{2!}(a\cdot
q_{i})a_{\rho}F^{\mu\rho}+\frac{1}{3!}(a\cdot
q)^{2}F^{*\mu\rho}a_{\rho}+\dots$ $\displaystyle=\hskip 1.42271pt$
$\displaystyle\frac{a\cdot\varepsilon}{a\cdot
q}q^{\mu}+\frac{F^{*\mu\rho}a_{\rho}}{a\cdot q}\sinh(a\cdot
q)+\frac{F^{\mu\rho}a_{\rho}}{a\cdot q}\cosh(a\cdot q).$ (3.3)

So far, we have not specified the helicity of the massless state
$\varepsilon$. A field strength admits a decomposition into its positive and
negative helicity components $F^{\mu\nu}_{h}$ with $h=\pm$ [73],

$\displaystyle F^{+}_{\mu\nu}=\frac{1}{2}(F_{\mu\nu}-F^{*}_{\mu\nu}),\qquad
F^{-}_{\mu\nu}=\frac{1}{2}(F_{\mu\nu}+F^{*}_{\mu\nu}),$ (3.4)

that, plugged into (3) yields,

$\displaystyle\mathrm{exp}\\{i\epsilon_{\nu}{}^{\mu}(a,q)\\}\varepsilon^{\nu}=\frac{a\cdot\varepsilon}{a\cdot
q}q^{\mu}+\frac{F_{+}^{\mu\rho}a_{\rho}}{a\cdot q}e^{-a\cdot
q}+\frac{F_{-}^{\mu\rho}a_{\rho}}{a\cdot q}e^{a\cdot q}.$ (3.5)

Therefore for fixed helicity $h$ we find,

$\displaystyle\mathrm{exp}\\{i\epsilon_{\nu}{}^{\mu}(a,q)\\}\varepsilon^{\nu}_{h}=e^{-ha\cdot
q}\varepsilon^{\mu}_{h}-\frac{(e^{-ha\cdot q}-1)}{a\cdot
q}a\cdot\varepsilon_{h}q^{\mu},\qquad h=\pm.$ (3.6)

We conclude that, up to a gauge transformation, positive and negative helicity
vectors are eigenvectors of the operator
$\mathrm{exp}\\{i\epsilon_{\nu}{}^{\mu}(a,q)\\}$ with eigenvalues $e^{-ha\cdot
q}$. An immediate consequence is that the spinning three-point amplitude in
(2.12), on-shell ($p\cdot q=0$) and once specified the helicity of the emitted
photon, reads simply $(p\cdot\varepsilon_{h})e^{-ha\cdot q}$. More in general,
any tree-level gauge invariant amplitude of the form (2.24) will contain a
product of spin exponentials,

$\displaystyle\mathcal{A}^{h_{2}\dots h_{N-1}}_{N}(a;p_{i},q_{i})$
$\displaystyle=\epsilon^{h_{2}}_{\mu_{2}}\Lambda^{\mu_{2}}{}_{\nu_{2}}(a,q_{2})\dots\epsilon^{h_{N-1}}_{\mu_{N-1}}\Lambda^{\mu_{N-1}}{}_{\nu_{N-1}}(a,q_{N-1})\mathcal{A}_{N}^{\nu_{2}\dots\nu_{N-1}}(p_{i},q_{i})$
$\displaystyle=e^{-\sum_{i=2}^{N-1}h_{i}\hskip 0.28436pta\cdot
q_{i}}\mathcal{A}^{h_{2}\dots h_{N-1}}_{N}(p_{i},q_{i}).$ (3.7)

In particular, for $N=4$, i.e. for the Compton amplitude, the equal- and
opposite-helicity sectors are,

$\displaystyle\mathcal{A}_{4}^{\pm\pm}(a;p_{i},q_{i})=e^{\mp
a\cdot(q_{2}+q_{3})}\mathcal{A}_{4}^{\pm\pm}(p_{i},q_{i}),\quad\mathcal{A}_{4}^{\pm\mp}(a;p_{i},q_{i})=e^{\mp
a\cdot(q_{2}-q_{3})}\mathcal{A}_{4}^{\pm\mp}(p_{i},q_{i}),$ (3.8)

respectively. Notice that the opposite-helicity Compton amplitude in (3.8) is
“shift symmetric” [110]. Indeed, if we perform
$\delta_{\xi}a=\xi(q_{2}+q_{3})$ we have
$\delta_{\xi}\mathcal{A}_{4}^{\pm\mp}(a)=\mathcal{A}_{4}^{\pm\mp}$ using
$q_{i}^{2}=0$.

So far, we were only concerned about gauge theories. In the next section 4 we
will introduce a notion of double copy for the theory just presented and we
will use it to extract gravity amplitudes that will be the main building
blocks of the 2PM computation we will present in section 5.

However, before turning to gravity, we end this section with some
considerations on the amplitudes in (3.8) and their relation with the ones
present in the literature. Using the kinematics specified in (B.1)-(B.4) we
find that in the classical limit the Compton amplitudes in (3.8) are,

$\displaystyle\mathcal{A}_{4}^{\pm\pm}(a;p_{i},q_{i})=2e^{2}\sin^{2}(\theta/2)e^{\pm\mathcal{G}(\omega,a,\theta)},\quad\mathcal{A}_{4}^{\pm\mp}(a;p_{i},q_{i})=2e^{2}\cos^{2}(\theta/2)e^{\pm\mathcal{F}(\omega,a,\theta)},$
(3.9)

with,

$\displaystyle\mathcal{G}(\omega,a,\theta)=\omega a^{x}\sin\theta-2\omega
a^{z}\sin^{2}(\theta/2),$ (3.10)
$\displaystyle\mathcal{F}(\omega,a,\theta)=-\omega a^{x}\sin\theta-2\omega
a^{z}\cos^{2}(\theta/2).$ (3.11)

Remarkably, the equal-helicity sector we find within our approach coincides
with the one found in [123, 125], once we send $\omega\rightarrow-\omega$.
Equivalently, it is compatible with the “minimally coupled” amplitudes
presented in [75] in massive spinor helicity variables, obtained using BCFW
recursion relations and sewing together three-point amplitudes in (2.12) and
(2.14). However, the spin exponential structure in the second of (3.9) differs
from the one in [75, 83, 125, 129], for $s<2$. A detailed comparison of the
opposite-helicity sectors can be found in appendix B.

##  4 Double copy for classical spin

In this section we will use the gauge theory just discussed to get gravity
amplitudes. In order to do so, we will implement the double copy for
classically spinning particles,

$\displaystyle\mathrm{Kerr}(a)=\sqrt{\mathrm{Kerr}(a)}\otimes\sqrt{\mathrm{Kerr}(0)}.$
(4.1)

The first simple example where it is possible to verify that this version of
the double copy is satisfied is at three-point. The classical gravity
amplitude involving two Kerr black holes and a graviton was found in [78] and
it reads,

$\displaystyle\mathcal{M}_{3}^{\mu\nu}(a;p,q)=i\kappa\hskip
0.28436ptp^{(\mu}\mathrm{exp}\\{i\epsilon^{\nu)}{}_{\rho}(a,q)\\}p^{\rho},\qquad\kappa=\sqrt{32\pi
G_{N}},$ (4.2)

and it can easily be obtained using the prescription in (4.1),

$\displaystyle\mathcal{M}_{3}^{\mu\nu}(a;p,q)=-i\mathcal{A}_{3}^{(\mu}(p,q)\mathcal{A}^{\nu)}_{3}(a;p,q)|_{e\rightarrow\sqrt{\kappa}}.$
(4.3)

Let us now turn to the 1PM elastic scattering amplitude between two Kerr black
holes with rescaled spin vectors $a_{1}$ and $a_{2}$. We can consider the
product,

$\displaystyle
q^{2}\mathcal{A}^{1\mathrm{PM}}(a;p_{i},q)\mathcal{A}^{1\mathrm{PM}}(p_{i},q)=$
$\displaystyle\frac{(e_{1}m_{1})^{2}(e_{2}m_{2})^{2}\sigma^{2}}{4q^{2}}\bigg{[}\sum_{\pm}(1\pm
v)^{2}e^{\pm i\frac{\epsilon(p_{1},p_{2},q,a)}{m_{1}m_{2}\sigma v}}$
$\displaystyle+2(1-v^{2})\cos(\frac{\epsilon(p_{1},p_{2},q,a)}{m_{1}m_{2}\sigma
v})\bigg{]},$ (4.4)

where $\mathcal{A}^{1\mathrm{PM}}(a)$ is the classical 1PM amplitude for the
elastic $2\rightarrow 2$ scattering. It has been calculated in appendix C and
it is explicitly given in (C.8). In (4) $a=a_{1}+a_{2}$, $\sigma=p_{1}\cdot
p_{2}/m_{1}m_{2}$ and $v=\sqrt{\sigma^{2}-1}/\sigma$. We see that the first
term in the above square bracket will reproduce, once reinserting all
numerical factors, the classical amplitude for the one graviton exchange of
two Kerr black holes shown in [86]. However there is another term, whose
origin can be traced back to the fact that when we naively double copy gauge
theory amplitudes having internal massless propagators, the result, together
with the graviton contribution, will contain also that of the dilaton and the
axion 777The axion will only appear when the matter content of the theory has
non-vanishing spin.. Indeed, writing explicitly the product in (4) yields,

$\displaystyle q^{2}$
$\displaystyle\mathcal{A}^{1\mathrm{PM}}(a;p_{i},q)\mathcal{A}^{1\mathrm{PM}}(p_{i},q)=\frac{1}{q^{2}}\mathcal{A}^{\mu}_{3}(a_{1};p_{1},q)\mathcal{A}^{\nu}_{3}(a_{2};p_{2},-q)\mathcal{A}^{\rho}_{3}(p_{1},q)\mathcal{A}_{3}^{\sigma}(p_{2},-q)\eta_{\mu\nu}\eta_{\rho\sigma}$
$\displaystyle=\frac{1}{q^{2}}\mathcal{A}^{\mu}_{3}(a_{1};p_{1},q)\mathcal{A}^{\nu}_{3}(a_{2};p_{2},-q)\mathcal{A}^{\rho}_{3}(p_{1},q)\mathcal{A}_{3}^{\sigma}(p_{2},-q)(\Pi^{(h)}_{\mu\rho;\nu\sigma}+\Pi^{(b)}_{\mu\rho;\nu\sigma}+\Pi^{(\phi)}_{\mu\rho;\nu\sigma}),$
(4.5)

where we defined,

$\displaystyle\Pi^{(h)}_{\mu\rho;\nu\sigma}=\frac{1}{2}(\eta_{\mu\nu}\eta_{\rho\sigma}+\eta_{\mu\sigma}\eta_{\rho\nu}-\eta_{\mu\rho}\eta_{\nu\sigma}),$
(4.6)
$\displaystyle\Pi^{(b)}_{\mu\rho;\nu\sigma}=\frac{1}{2}(\eta_{\mu\nu}\eta_{\rho\sigma}-\eta_{\mu\sigma}\eta_{\rho\nu}),$
(4.7)
$\displaystyle\Pi^{(\phi)}_{\mu\rho;\nu\sigma}=\frac{1}{2}\eta_{\mu\rho}\eta_{\nu\sigma},$
(4.8)

the projectors onto graviton, axion and dilaton, respectively. Therefore, in
order to get the pure gravity result, we have to subtract the last two terms
in (4) to (4). By direct computation, the axion contribution is,

$\displaystyle\frac{1}{q^{2}}\mathcal{A}^{\mu}_{3}(a_{1};p_{1},q)$
$\displaystyle\mathcal{A}^{\nu}_{3}(a_{2};p_{2},-q)\mathcal{A}^{\rho}_{3}(p_{1},q)\mathcal{A}_{3}^{\sigma}(p_{2},-q)\Pi^{(b)}_{\mu\rho;\nu\sigma}$
$\displaystyle=-\frac{(e_{1}m_{1})^{2}(e_{2}m_{2})^{2}}{2q^{2}}\sin(\frac{\epsilon(p_{1},p_{2},q,a_{1})}{m_{1}m_{2}\sigma
v})\sin(\frac{\epsilon(p_{1},p_{2},q,a_{2})}{m_{1}m_{2}\sigma v}),$ (4.9)

whereas that of the dilaton is,

$\displaystyle\frac{1}{q^{2}}\mathcal{A}^{\mu}_{3}(a_{1};p_{1},q)$
$\displaystyle\mathcal{A}^{\nu}_{3}(a_{2};p_{2},-q)\mathcal{A}^{\rho}_{3}(p_{1},q)\mathcal{A}_{3}^{\sigma}(p_{2},-q)\Pi^{(\phi)}_{\mu\rho;\nu\sigma}$
$\displaystyle=\frac{(e_{1}m_{1})^{2}(e_{2}m_{2})^{2}}{2q^{2}}\cos(\frac{\epsilon(p_{1},p_{2},q,a_{1})}{m_{1}m_{2}\sigma
v})\cos(\frac{\epsilon(p_{1},p_{2},q,a_{2})}{m_{1}m_{2}\sigma v}).$ (4.10)

Notice that the axion contribution vanishes as the spin goes to zero.
Subtracting (4) and (4) to (4) yields the full 1PM gravity amplitude,

$\displaystyle\mathcal{M}^{\mathrm{1PM}}(a;p_{i},q)=\frac{\kappa^{2}}{4}\frac{m_{1}^{2}m_{2}^{2}\sigma^{2}}{q^{2}}\sum_{\pm}(1\pm
v)^{2}e^{\pm i\frac{\epsilon(p_{1},p_{2},q,a)}{m_{1}m_{2}\sigma v}},$ (4.11)

where we replaced $e_{i}\rightarrow\sqrt{\kappa}$. Notice that the previous
formula admits an interpretation in terms of the Newman-Janis shift discussed
in detail in appendix A. Indeed, in order to extract observables out of
$\mathcal{M}^{1\mathrm{PM}}(a)$, we need to go to impact parameter space and
compute the eikonal, given by the Fourier transform,

$\displaystyle\tilde{\mathcal{M}}^{1\mathrm{PM}}(\mathbf{a};p_{i},\mathbf{b})$
$\displaystyle=\int\frac{d^{4-2\epsilon}q}{(2\pi)^{4-2\epsilon}}\mathcal{M}^{1\mathrm{PM}}(a;p_{i},q)\hat{\delta}(2p_{1}\cdot
q)\hat{\delta}(2p_{2}\cdot q)e^{iq\cdot b},$
$\displaystyle=\frac{1}{4m_{1}m_{2}\sqrt{\sigma^{2}-1}}\int\frac{d^{2-2\epsilon}\mathbf{q}}{(2\pi)^{2-2\epsilon}}\mathcal{M}^{1\mathrm{PM}}(\mathbf{a};p_{i},\mathbf{q})e^{-i\mathbf{q}\cdot\mathbf{b}},$
(4.12)

where we introduced the infrared regulator $\epsilon=(4-D)/2$,
$\hat{\delta}(x)=2\pi\delta(x)$ and where $\mathbf{q}$ and $\mathbf{a}$ are
the spatial component of the transfer momentum $q$ and of the rescaled spin
vector $a$. Plugging (4.11) into (4) gives,

$\displaystyle\tilde{\mathcal{M}}^{1\mathrm{PM}}(\mathbf{a};p_{i},\mathbf{b})$
$\displaystyle=\frac{\kappa^{2}m_{1}m_{2}\sigma^{2}}{64\sqrt{\sigma^{2}-1}}\frac{\Gamma(-\epsilon)}{\pi^{1-\epsilon}}\sum_{\pm}\frac{(1\pm
v)^{2}}{\absolutevalue{\mathbf{b}\pm i\mathbf{a}}^{-2\epsilon}}$
$\displaystyle=\frac{\kappa^{2}m_{1}m_{2}\sigma^{2}}{64\sqrt{\sigma^{2}-1}}\frac{\Gamma(-\epsilon)}{\pi^{1-\epsilon}}\bigg{[}\frac{(1+v)^{2}}{\absolutevalue{\mathbf{b}}^{-2\epsilon}}+\frac{(1-v)^{2}}{\absolutevalue{\mathbf{b}^{*}}^{-2\epsilon}}\bigg{]}\bigg{|}_{\mathbf{b}\rightarrow\mathbf{b}+i\mathbf{a}},$
(4.13)

which is reminiscent of a Newman-Janis shift (see e.g. (A.7)) of the spinless
result. Therefore, the effect of the classical spin covariant derivative in
(2.12) on 1PM amplitudes in impact parameter space and hence on spinning
observables, is equivalent to an implementation of such shift at the level of
the Lagrangian.

Similarly, we can now consider the gravity Compton amplitude. In general,
there is also a massless channel contributing to the latter and hence we
should double copy a non abelian Yang-Mills Compton instead of (3.8). However,
in the classical limit, gravitons self-interactions are subleading [72] and
therefore we can simply use spinning QED Compton amplitudes constructed
previously. At four-points the double copy reads,

$\displaystyle\mathcal{M}_{4}^{h_{2},h_{3}}(a;p_{i},q_{i})$
$\displaystyle=\frac{\kappa^{2}}{e^{4}}\frac{8p_{1}\cdot q_{2}p_{1}\cdot
q_{3}}{q_{2}\cdot
q_{3}}\mathcal{A}_{4}^{h_{2},h_{3}}(a;p_{i},q_{i})\mathcal{A}_{4}^{h_{2},h_{3}}(p_{i},q_{i})$
$\displaystyle=\frac{\kappa^{2}}{2}\frac{1}{q_{2}\cdot q_{3}p_{1}\cdot
q_{2}p_{1}\cdot q_{3}}e^{-a\cdot(h_{2}q_{2}+h_{3}q_{3})}(p_{1}\cdot
F_{2}^{h_{2}}\cdot F_{3}^{h_{3}}\cdot p_{1})^{2}.$ (4.14)

Again, the equal-helicity sector is reproduced exactly to all orders in the
spin, differently from the opposite-helicity one. Using the complex vectors
$w$ and $w^{*}$ in (B.5), up to $a^{4}$, the Kerr gravity Compton amplitude
are given by,

$\displaystyle\tilde{\mathcal{M}}_{4}^{+-}(a;p_{i},q_{i})=e^{-2a\cdot
w}\mathcal{M}_{4}^{+-}(a;p_{i},q_{i}),\quad\tilde{\mathcal{M}}_{4}^{-+}(a;p_{i},q_{i})=e^{2a\cdot
w^{*}}\mathcal{M}_{4}^{-+}(a;p_{i},q_{i}).$ (4.15)

From $a^{5}$ on these amplitude are affected by spurious poles and therefore
they are not physical.

##  5 Classical 1-loop amplitude and 2PM observables

In order to get the classical 2PM amplitude we consider the following cut,

$i\mathcal{M}^{2\mathrm{PM}}(a;p_{i},q)=\vbox{\hbox{\includegraphics[width=187.78836pt,height=130.88284pt]{2PM.png}}}=\int\frac{d^{4}l}{(2\pi)^{4}}M^{2\mathrm{PM}}(a;p_{i},q_{i}),$
(5.1)

where $l$ is the loop momentum and can be chosen to be either $q_{2}$ or
$q_{3}$. Explicitly, the one-loop integrand is,

$\displaystyle M^{2\mathrm{PM}}(a;p_{i},q_{i})=$
$\displaystyle\mathcal{M}^{\mu_{1}\nu_{1};\rho_{1}\sigma_{1}}_{4}(a;p_{1},-q-q_{3},q_{3})[P(q_{3})]_{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}[P(q_{3}+q)]_{\rho_{1}\sigma_{1};\rho_{2}\sigma_{2}}$
$\displaystyle\mathcal{M}^{\mu_{2}\nu_{2};\rho_{2}\sigma_{2}}_{4}(a;p_{2},-q-q_{3},-q_{3}),$
(5.2)

where $q=-q_{2}-q_{3}$ is the transfer momentum and where
$[P(q)]_{\mu\nu;\rho\sigma}$ is the de Donder propagator,

$\displaystyle[P(q)]_{\mu\nu;\rho\sigma}=-\frac{i}{q^{2}}\Pi^{(h)}_{\mu\nu;\rho\sigma}.$
(5.3)

Instead of computing the product in (5), we find it convenient to switch to
on-shell variables here. In terms of spinor-helicity variables, the Compton
amplitudes in (4.11) are given by,

$\displaystyle\mathcal{M}^{++}_{4}(a;p_{i},q_{i})=\frac{\kappa^{2}}{4}e^{-a\cdot(q_{2}+q_{3})}\frac{m^{4}[23]^{4}}{t\tau_{12}\tau_{13}},\quad$
$\displaystyle\mathcal{M}^{--}_{4}(a;p_{i},q_{i})=\frac{\kappa^{2}}{4}e^{a\cdot(q_{2}+q_{3})}\frac{m^{4}\langle
23\rangle^{4}}{t\tau_{12}\tau_{13}},$ (5.4)
$\displaystyle\mathcal{M}^{+-}_{4}(a;p_{i},q_{i})=\frac{\kappa^{2}}{4}e^{-a\cdot(q_{2}-q_{3})}\frac{[2|p_{1}|3\rangle^{4}}{t\tau_{12}\tau_{13}},\quad$
$\displaystyle\mathcal{M}^{-+}_{4}(a;p_{i},q_{i})=\frac{\kappa^{2}}{4}e^{a\cdot(q_{2}-q_{3})}\frac{[3|p_{1}|2\rangle^{4}}{t\tau_{12}\tau_{13}},$
(5.5)

where $\tau_{ij}=2p_{i}\cdot q_{j}$, $t=s_{23}=2q_{2}\cdot q_{3}=q^{2}$.
Following similar considerations e.g. as in [141, 20], the full integrand of
the classical one-loop amplitude can easily be recast in the following form,

$\displaystyle M^{2\mathrm{PM}}(a;p_{i},q_{i})$
$\displaystyle=\frac{\kappa^{4}}{16}\frac{1}{t\tau_{12}\tau_{13}\tau_{22}\tau_{23}q_{2}^{2}q_{3}^{2}}\bigg{\\{}2t^{4}m_{1}^{4}m_{2}^{4}\cosh(a\cdot(q_{3}+q_{2}))$
$\displaystyle+\cosh(a\cdot(q_{3}-q_{2}))[\mathrm{tr}_{+}(q_{2}p_{2}q_{3}p_{1})^{4}+\mathrm{tr}_{-}(q_{2}p_{2}q_{3}p_{1})^{4}]$
$\displaystyle+\sinh(a\cdot(q_{3}-q_{2}))[\mathrm{tr}_{+}(q_{2}p_{2}q_{3}p_{1})^{4}-\mathrm{tr}_{-}(q_{2}p_{2}q_{3}p_{1})^{4}]\bigg{\\}},$
(5.6)

where the chiral traces are,

$\displaystyle\mathrm{tr}_{\pm}(q_{2}p_{2}q_{3}p_{1})$
$\displaystyle=\frac{1}{2}\tau_{22}\tau_{13}+\frac{1}{2}\tau_{23}\tau_{12}-\frac{t}{2}(s-m_{1}^{2}-m_{2}^{2})\pm
2i\epsilon(q_{2},p_{2},q_{3},p_{1})$
$\displaystyle\equiv\mathcal{E}\pm\mathcal{O},$ (5.7)

where $s=(p_{1}+p_{2})^{2}$ and where we introduced the notation [20],

$\displaystyle\mathcal{E}=\frac{1}{2}\tau_{22}\tau_{13}+\frac{1}{2}\tau_{23}\tau_{12}-\frac{t}{2}(s-m_{1}^{2}-m_{2}^{2}),\qquad\mathcal{O}=2i\epsilon(q_{2},p_{2},q_{3},p_{1}).$
(5.8)

Squaring $\mathcal{O}$ and computing the Gram determinant gives,

$\displaystyle\mathcal{O}^{2}=-4\epsilon(q_{2},p_{2},q_{3},p_{1})^{2}=\mathcal{E}^{2}-(m_{1}^{2}t-\tau_{12}\tau_{13})(m_{2}^{2}t-\tau_{22}\tau_{23}).$
(5.9)

Using $2p_{1}\cdot q=q^{2}$ and $2p_{2}\cdot q=-q^{2}$ we get,

$\displaystyle\frac{1}{\tau_{13}}+\frac{1}{\tau_{12}}=-\frac{t}{\tau_{13}\tau_{12}},\qquad\frac{1}{\tau_{23}}+\frac{1}{\tau_{22}}=\frac{t}{\tau_{23}\tau_{22}},$
(5.10)

that, inserted into (5) yields for $M^{2\mathrm{PM}}(a)$,

$\displaystyle
M^{2\mathrm{PM}}(a;p_{i},q_{i})=-\frac{\kappa^{4}}{16}\frac{1}{q_{2}^{2}q_{3}^{2}}\bigg{[}f_{\mathrm{even}}(\tau_{12},\tau_{13},\tau_{22},\tau_{23})\cosh(a\cdot(q_{3}-q_{2}))$
$\displaystyle+f_{\mathrm{odd}}(\tau_{12},\tau_{13},\tau_{22},\tau_{23})\sinh(a\cdot(q_{3}-q_{2}))\bigg{]}\bigg{(}\frac{1}{\tau_{13}\tau_{23}}+\frac{1}{\tau_{13}\tau_{22}}+\frac{1}{\tau_{12}\tau_{23}}$
$\displaystyle+\frac{1}{\tau_{12}\tau_{22}}\bigg{)},$ (5.11)

where we neglected the first term in (5) because it corresponds to a “box”
diagram whose classical limit, in $D=4$, is compensated by the Born
subtraction of the effective theory (see e.g. [30]) and where we defined,

$\displaystyle
f_{\mathrm{even}}(\tau_{12},\tau_{13},\tau_{22},\tau_{23})\equiv\frac{1}{t^{4}}(2\mathcal{E}^{4}+2\mathcal{O}^{4}+12\mathcal{E}^{2}\mathcal{O}^{2}),$
(5.12) $\displaystyle
f_{\mathrm{odd}}(\tau_{12},\tau_{13},\tau_{22},\tau_{23})\equiv\frac{1}{t^{4}}(8\mathcal{E}^{3}\mathcal{O}+8\mathcal{E}\mathcal{O}^{3}),$
(5.13)

to be the even and odd contributions in the spin multipole expansion,
respectively. We start by analysing the even part. We notice that the
variables $\tau_{ij}$ are not all independent and infact they are related by,

$\displaystyle\tau_{22}=2p_{2}\cdot
q_{2}=t-\tau_{23},\qquad\tau_{12}=2p_{1}\cdot q_{2}=-t-\tau_{13}.$ (5.14)

This enables us to define the functions,

$\displaystyle{}_{(1)}f_{\mathrm{even}}(\tau_{13},\tau_{23})=f_{\mathrm{even}}(-t-\tau_{13},\tau_{13},t-\tau_{23},\tau_{23})=\sum_{i,j=0}^{4}{}_{(1)}f_{\mathrm{even}}^{ij}(\tau_{13})^{i}(\tau_{23})^{j},$
(5.15)
$\displaystyle{}_{(2)}f_{\mathrm{even}}(\tau_{13},\tau_{22})=f_{\mathrm{even}}(-t-\tau_{13},\tau_{13},\tau_{22},t-\tau_{22})=\sum_{i,j=0}^{4}{}_{(2)}f_{\mathrm{even}}^{ij}(\tau_{13})^{i}(\tau_{22})^{j},$
(5.16)
$\displaystyle{}_{(3)}f_{\mathrm{even}}(\tau_{12},\tau_{23})=f_{\mathrm{even}}(\tau_{12},-t-\tau_{12},t-\tau_{23},\tau_{23})=\sum_{i,j=0}^{4}{}_{(3)}f_{\mathrm{even}}^{ij}(\tau_{12})^{i}(\tau_{23})^{j},$
(5.17)
$\displaystyle{}_{(4)}f_{\mathrm{even}}(\tau_{12},\tau_{22})=f_{\mathrm{even}}(\tau_{12},-t-\tau_{12},\tau_{22},t-\tau_{22})=\sum_{i,j=0}^{4}{}_{(4)}f_{\mathrm{even}}^{ij}(\tau_{12})^{i}(\tau_{22})^{j}.$
(5.18)

Inserting the above expansions in equation (5) will give terms having either
two massless propagators, the so-called “bubbles”, or three propagators of
which two massless and one massive, the so called “triangles”. Only the latter
contribute in the classical limit and we now proceed to extract them. We have,

$\displaystyle M^{2\mathrm{PM}}_{\mathrm{even}}(a;p_{i},q_{i})=$
$\displaystyle-\frac{\kappa^{4}}{16}\frac{1}{q_{2}^{2}q_{3}^{2}}\sum_{i=0}^{3}\bigg{[}\frac{{}_{(1)}f_{\mathrm{even}}^{0,i+1}(\tau_{23})^{i}+{}_{(2)}f_{\mathrm{even}}^{0,i+1}(\tau_{22})^{i}}{\tau_{13}}$
$\displaystyle+\frac{{}_{(3)}f_{\mathrm{even}}^{0,i+1}(\tau_{23})^{i}+{}_{(4)}f_{\mathrm{even}}^{0,i+1}(\tau_{22})^{i}}{\tau_{12}}+\frac{{}_{(1)}f_{\mathrm{even}}^{i+1,0}(\tau_{13})^{i}+{}_{(3)}f_{\mathrm{even}}^{i+1,0}(\tau_{12})^{i}}{\tau_{23}}+$
$\displaystyle+\frac{{}_{(2)}f_{\mathrm{even}}^{i+1,0}(\tau_{13})^{i}+{}_{(4)}f_{\mathrm{even}}^{i+1,0}(\tau_{12})^{i}}{\tau_{22}}\bigg{]}\cosh(a\cdot(q_{3}-q_{2}))$
$\displaystyle\equiv{}_{\triangleright}M^{2\mathrm{PM}}_{\mathrm{even}}(a;p_{i},q_{i})+{}_{\triangleleft}M^{2\mathrm{PM}}_{\mathrm{even}}(a;p_{i},q_{i}),$
(5.19)

where we included in the $\triangleright$ and $\triangleleft$ contributions
diagrams having triangular topology with one massive propagator with mass
$m_{1}$ and $m_{2}$, respectively, as shown in Figure 1.

Figure 1: The two diagrams containing classical information for the
$2\rightarrow 2$ elastic scattering at 2PM.

We choose as independent loop momentum $l=q_{3}$ and therefore we substitute
in (5) $q_{2}=-l-q$ and $\tau_{12}=-t-\tau_{13}$ and $\tau_{22}=t-\tau_{23}$.
Taking the classical $t\rightarrow 0$ limit yields,

$\displaystyle\mathcal{M}^{2\mathrm{PM}}_{\mathrm{even}}(a;p_{i},q)=i\frac{\kappa^{4}}{16}\bigg{[}c_{\triangleright}\mathcal{I}_{\triangleright}(a)+(c_{\triangleright})_{\mu\nu}\mathcal{I}^{\mu\nu}_{\triangleright}(a)+(\triangleright\leftrightarrow\triangleleft)\bigg{]},$
(5.20)

where,

$\displaystyle
c_{\triangleright}=4m_{1}^{2}[m_{1}^{4}+m_{1}^{2}(m_{2}^{2}-2s)+(m_{2}^{2}-s)^{2}],\qquad(c_{\triangleright})_{\mu\nu}=-\frac{8m_{1}^{4}p_{2\mu}p_{2\nu}}{t},$
(5.21)

and,

$\displaystyle\mathcal{I}^{\mu_{1}\dots\mu_{n}}_{\triangleright}(a)=\int\frac{d^{4}l}{(2\pi)^{4}}\frac{l^{\mu_{1}}\dots
l^{\mu_{n}}\cosh(2a\cdot l+a\cdot
q)}{l^{2}(l+q)^{2}[(p_{1}+l)^{2}-m_{1}^{2}]},$ (5.22)

and where quantities with $\triangleleft$ can be obtained from
$\triangleright$ by sending $m_{1}\leftrightarrow m_{2}$ and
$p_{1}\leftrightarrow p_{2}$.

Since we are ultimately interested in obtaining the scattering angle, from now
on we consider an aligned spin configuration, where the spins of the black
holes are aligned to the $z$ axis and perpendicular to the scattering plane
$xy$. We also choose to work in the center of mass frame and we fix,

$\displaystyle p_{1}=(-E_{1},-p,0,0),\qquad p_{2}=(-E_{2},p,0,0),\qquad
a_{i}=(0,0,0,a_{i}).$ (5.23)

Using (D.10) and (D.15), we find in the classical limit,

$\displaystyle\mathcal{M}^{2\mathrm{PM}}_{\mathrm{even}}(a;p_{i},q)=$
$\displaystyle\frac{\kappa^{4}}{16}\frac{1}{32\sqrt{-q^{2}}}\bigg{\\{}4m_{1}[m_{1}^{4}+m_{1}^{2}(m_{2}^{2}-2s)+(m_{2}^{2}-s)^{2}]I_{0}(\mathscr{F}(a,q))$
$\displaystyle+2m_{1}\bigg{[}m_{1}^{2}m_{2}^{2}-\frac{(s-m_{1}^{2}-m_{2}^{2})^{2}}{4}\bigg{]}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)}+(m_{1}\leftrightarrow
m_{2})\bigg{\\}},$ (5.24)

where $I_{0}(x)$ and $I_{1}(x)$ are modified Bessel functions of the first
kind and their argument is,

$\displaystyle\mathscr{F}(a,q)=\sqrt{(a\cdot q)^{2}-a^{2}q^{2}},$ (5.25)

which is shift symmetric. We checked that equation (5) reduces to the standard
2PM result for $a=0$, e.g. [30]. Let us now consider the part of the 2PM
amplitude containing odd powers of the classical spin vector. We get,

$\displaystyle\mathcal{M}^{2\mathrm{PM}}_{\mathrm{odd}}(a;p_{i},q)=\frac{\kappa^{4}}{16}\bigg{[}d_{\triangleright}\epsilon_{\mu}(q,p_{2},p_{1})\mathcal{J}^{\mu}_{\triangleright}(a)+(d_{\triangleright})_{\nu}\epsilon_{\mu}(q,p_{2},p_{1})\mathcal{J}^{\mu\nu}_{\triangleright}(a)+(\triangleright\leftrightarrow\triangleleft)\bigg{]},$
(5.26)

where,

$\displaystyle
d_{\triangleright}=\frac{8m_{1}^{2}(m_{1}^{2}+m_{2}^{2}-s)}{t},\qquad(d_{\triangleright})_{\nu}=-\frac{16m_{1}^{2}p_{2\nu}}{t},$
(5.27)

and,

$\displaystyle\mathcal{J}^{\mu_{1}\dots\mu_{n}}(a)=\int\frac{d^{4}l}{(2\pi)^{4}}\frac{l^{\mu_{1}}\dots
l^{\mu_{n}}\sinh(2a\cdot l+a\cdot
q)}{l^{2}(l+q)^{2}[(l+p_{1})^{2}-m_{1}^{2}]}.$ (5.28)

We find that in the classical limit only the vector integral contributes in
(5.26) and we have, using (D.26),

$\displaystyle\mathcal{M}^{2\mathrm{PM}}_{\mathrm{odd}}(a;p_{i},q)=$
$\displaystyle\frac{\kappa^{4}}{16}\bigg{[}\frac{m_{1}(m_{1}^{2}+m_{2}^{2}-s)}{32\sqrt{-q^{2}}}i\epsilon(a,q,p_{2},p_{1})\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)}$
$\displaystyle+(m_{1}\leftrightarrow m_{2})\bigg{]}.$ (5.29)

Computing the Gram determinant gives (see (C.5)-(C.6)),

$\displaystyle\epsilon(a,q,p_{2},p_{1})=i\mathscr{F}(a,q)m_{1}m_{2}\sqrt{\sigma^{2}-1}+\mathcal{O}(t^{2}).$
(5.30)

Substituting into (5) gives the final result for the classical 2PM amplitude
odd in the spin,

$\displaystyle\mathcal{M}^{2\mathrm{PM}}_{\mathrm{odd}}(a;p_{i},q)=$
$\displaystyle-\frac{\kappa^{4}}{16}\frac{\sqrt{\sigma^{2}-1}}{32\sqrt{-q^{2}}}\bigg{[}m_{1}(m_{1}^{2}+m_{2}^{2}-s)I_{1}(\mathscr{F}(a,q))$
$\displaystyle+(m_{1}\leftrightarrow m_{2})\bigg{]},$ (5.31)

where the Lorentz factor is related to $s$ by
$\sigma=\frac{1}{2m_{1}m_{2}}(s-m_{1}^{2}-m_{2}^{2})$.

The 2PM eikonal phase is,

$\displaystyle\tilde{\mathcal{M}}^{2\mathrm{PM}}(\mathbf{a};p_{i},\mathbf{b})$
$\displaystyle=\frac{1}{4m_{1}m_{2}\sqrt{\sigma^{2}-1}}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\mathcal{M}^{2\mathrm{PM}}(\mathbf{a};p_{i},\mathbf{q})e^{-i\mathbf{q}\cdot\mathbf{b}}$
$\displaystyle=\tilde{\mathcal{M}}^{2\mathrm{PM}}_{\mathrm{even}}(\mathbf{a};p_{i},\mathbf{b})+\tilde{\mathcal{M}}_{\mathrm{odd}}^{2\mathrm{PM}}(\mathbf{a};p_{i},\mathbf{b}).$
(5.32)

where,

$\displaystyle\tilde{\mathcal{M}}^{2\mathrm{PM}}_{\mathrm{even}}(\mathbf{a};p_{i},\mathbf{b})=$
$\displaystyle\frac{\kappa^{4}}{16}\frac{1}{128m_{1}m_{2}\sqrt{\sigma^{2}-1}}\bigg{\\{}4m_{1}[m_{1}^{4}+m_{1}^{2}(m_{2}^{2}-2s)+(m_{2}^{2}-s)^{2}]\tilde{I}_{0}[\mathbf{a},\mathbf{b}]$
$\displaystyle+2m_{1}\bigg{[}m_{1}^{2}m_{2}^{2}-\frac{(s-m_{1}^{2}-m_{2}^{2})^{2}}{4}\bigg{]}\tilde{\bigg{(}\frac{I_{1}}{\mathscr{F}}\bigg{)}}[\mathbf{a},\mathbf{b}]+(m_{1}\leftrightarrow
m_{2})\bigg{\\}},$ (5.33)

and,

$\displaystyle\tilde{\mathcal{M}}^{2\mathrm{PM}}_{\mathrm{odd}}(\mathbf{a};p_{i},\mathbf{b})=$
$\displaystyle-\frac{\kappa^{4}}{16}\frac{1}{128m_{1}m_{2}}\bigg{[}m_{1}(m_{1}^{2}+m_{2}^{2}-s)\tilde{I}_{1}[\mathbf{a},\mathbf{b}]$
$\displaystyle+(m_{1}\leftrightarrow m_{2})\bigg{]},$ (5.34)

where the Fourier transforms of the Bessel functions are computed in appendix
E in (E), (E) and (E). The 2PM scattering angle is defined through the usual
relation,

$\displaystyle\Theta^{2\mathrm{PM}}(a;p,b)=-\frac{1}{p}\frac{\partial}{\partial
b}\mathcal{M}^{\mathrm{2PM}}(a;p,b)=\Theta^{2\mathrm{PM}}_{\mathrm{even}}(a;p,b)+\Theta^{2\mathrm{PM}}_{\mathrm{odd}}(a;p,b),$
(5.35)

where,

$\displaystyle\Theta^{2\mathrm{PM}}_{\mathrm{even}}(a;p,b)=-\frac{1}{p}\frac{\partial}{\partial
b}\mathcal{M}^{\mathrm{2PM}}_{\mathrm{even}}(a;p,b),\qquad\Theta^{2\mathrm{PM}}_{\mathrm{odd}}(a;p,b)=-\frac{1}{p}\frac{\partial}{\partial
b}\mathcal{M}^{\mathrm{2PM}}_{\mathrm{odd}}(a;p,b).$ (5.36)

##  6 Conclusions and outlook

We introduced a new Yang-Mills type of Lagrangian based on a notion of minimal
coupling that incorporates classical spin effects. In particular, we found
that the three-point amplitude of $\sqrt{\mathrm{Kerr}}$ can consistently be
described using a “classical spin covariant derivative”, whose implementation
at the level of the Lagrangian automatically yields contact terms. We used
such theory to compute the Compton amplitude to all orders in the spin, first
in gauge theory and then, using classical double copy, in gravity. Such
amplitudes, coming directly from a local Lagrangian, are not affected by
spurious poles and they reproduce the well-known result in the equal-helicity
sector whereas they do not in the opposite-helicity one. In particular, we
showed that the opposite-helicity Compton amplitude coming from our model does
not depend on the complex vector $w$ present e.g. in [92, 123]. We concluded
with a 1-loop (2PM) computation of the scattering amplitude for an elastic
$2\rightarrow 2$ process in an aligned spin configuration, to all orders in
the classical spin vector, from which we extracted the 2PM eikonal and
scattering angle.

As remarked above, the field theory constructed here does not capture all the
features characterizing $\sqrt{\mathrm{Kerr}}$ and Kerr solutions and their
interactions with the gauge fields. However, it has to be regarded as a
minimal model to which one is free to add gauge invariant terms of higher
order in the curvature in order to reproduce correct amplitudes and
observables relevant for the dynamics of true Kerr binaries in the PM regime.
The main advantage of this theory relies in its intrinsic simplicity, which we
believe to be a fundamental criterion for the description of Kerr black holes
as elementary particles. Indeed, the 1PM and the 2PM elastic scattering
amplitudes have a very simple mathematical structure. Concerning future
directions, it would be interesting to go further in the PM expansion and to
explicitly compute the 3PM scattering amplitude and eikonal, both in the
conservative and radiative sectors 888Note that the result contained in [113]
for the radiative sector at 3PM, relying only on the three-point amplitude as
main building block, is fully consistent with the theory presented here. as
well as to construct a worldline version of this theory.

### Acknowledgements

I would like to thank Maor Ben-Shahar, Zvi Bern, Luca Buoninfante, Paolo Di
Vecchia, Alessandro Georgoudis, Kays Haddad, Carlo Heissenberg, Gustav
Jakobsen, Henrik Johansson, Gustav Mogull, Stavros Mougiakakos, Alexander
Ochirov, Paolo Pichini, Jan Plefka and Justin Vines for very useful
discussions. I thank Henrik Johansson and Paolo Pichini for their interest
during initial stages of the project. I am especially grateful to Kays Haddad,
Henrik Johansson and Paolo Pichini for enlightening discussions and to Paolo
Di Vecchia and Paolo Pichini for comments on this draft. The research of FA is
fully supported by the Knut and Alice Wallenberg Foundation under grant KAW
2018.0116. Nordita is partially supported by Nordforsk.

##  Appendix A Kerr-Schild coordinates and Newman-Janis shift

In this appendix we review some features of the Schwarzschild and Kerr
solutions using the Kerr-Schild ansatz [74, 136, 142], that consists in
parametrizing the full spacetime metric $g_{\mu\nu}$ as,

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+\hskip 1.42271pt\phi\hskip
1.42271ptk_{\mu}k_{\nu},\hskip
28.45274pt\bar{g}_{\mu\nu}{k}^{\mu}k^{\nu}=0=g_{\mu\nu}{k}^{\mu}k^{\nu},$
(A.1)

where $\bar{g}_{\mu\nu}$ is a background solution satisfying
$\bar{R}_{\mu\nu}=0$. Because $k^{\mu}$ is null, its indices can be raised and
lowered with both $g_{\mu\nu}$ and $\bar{g}_{\mu\nu}$. Vacuum Einstein
equations $R_{\mu\nu}=0$, together with (A.1) imply [71],

$\displaystyle
R^{(1)}_{\mu\nu}\equiv\bar{\nabla}_{\rho}\bigg{(}\bar{\nabla}_{(\mu}(\phi\hskip
1.42271ptk_{\nu)}k^{\rho})-\frac{1}{2}\bar{\nabla}^{\rho}(\phi\hskip
1.42271ptk_{\mu}k_{\nu})\bigg{)}=0.$ (A.2)

We are interested in describing black holes and hence, from now on we choose
$\phi$ to depend only on the radial coordinate $r$ and on the coordinate $z$,
i.e. we specialize to axisymmetric solutions. Furthermore, we fix
$k^{\mu}=(1,-1,0,0)$ [87] and choose $\bar{g}_{\mu\nu}$ to be the flat
Minkowski metric $\eta_{\mu\nu}$.

If we work in spherical coordinates $(r,\theta,\phi)$ defined by
$x+iy=re^{i\phi}\sin\theta$ and $z=r\cos\theta$, it is immediate to prove that
$R^{(1)}_{\mu\nu}=0$ implies,

$\displaystyle\phi+r\partial_{r}\phi=0,\hskip
28.45274pt\partial_{r}\partial_{\theta}\phi=0,$ (A.3)

whose solution is,

$\displaystyle\phi_{S}=\frac{r_{s}}{r},\hskip 28.45274pt,$ (A.4)

where $r_{s}=2G_{N}m$, that is precisely the Schwarzschild black hole in Kerr-
Schild coordinates.

To obtain the Kerr solution, we perform a change of coordinates and
parametrize $\eta_{\mu\nu}$ with oblate spheroidal coordinates, defined by
$x+iy=(r+ia)e^{i\phi}\sin\theta$ and $z=r\cos\theta$. Notice that so far, the
parameter $a$ is not related to the spin and it appears as mere parameter
characterizing the foliation of the spatial part of the spacetime metric.
Equation (A.2) reduces to,

$\displaystyle\frac{r^{2}-a^{2}\cos^{2}\theta}{r^{2}+a^{2}\cos^{2}\theta}\phi+r\partial_{r}\phi=0,\hskip
28.45274pt\partial_{r}\partial_{\theta}\phi-\partial_{r}\bigg{(}\frac{2a^{2}\phi\sin\theta\cos\theta}{r^{2}+a^{2}\cos^{2}\theta}\bigg{)}=0,$
(A.5)

solved by,

$\displaystyle\phi_{K}=\frac{r_{s}r}{r^{2}+a^{2}\cos^{2}\theta},$ (A.6)

which is the Kerr-Schild form of Kerr spacetime. We note here that performing
a change of coordinates from spherical to oblate spheroidal on the
Schwarzschild solution (A.4) does not yield Kerr. Indeed, the two black holes
are not diffeomorphic. Only performing the diffeomorphism on the ansatz for
the metric, before solving Einstein equations, gives Kerr spacetime. One of
the advantages of using Kerr-Schild coordinates is that they emphasize the
similarities between Schwarzschild and Kerr and that there is a simple and
clear connection between them called Newman-Janis shift [137]. It consists in
temporarily complexifying the radial coordinate of the Schwarzschild solution
and then perform the complex translation $r\rightarrow r+ia\cos\theta$ to
reconstruct Kerr spacetime as follows:

$\displaystyle\phi_{K}=\frac{1}{2}(\phi_{S}+\phi_{S}^{*})|_{r\rightarrow
r+ia\cos\theta}=\frac{r_{s}}{2}\bigg{(}\frac{1}{r+ia\cos\theta}+\frac{1}{r-ia\cos\theta}\bigg{)}.$
(A.7)

This equation displays that all the information about the spinning solution
are actually encoded into the spinless one. All the curvature invariants
constructed with the Kerr metric can be obtained by Newman-Janis shifting the
Schwarzschild ones, e.g.

$\displaystyle
R^{S}_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}_{S}=\frac{48G^{2}m^{2}}{r^{6}},$
(A.8)

and,

$\displaystyle
R^{K}_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}_{K}=24G^{2}m^{2}\frac{2r^{6}-30a^{2}r^{4}\cos^{2}\theta+30a^{4}r^{2}\cos^{4}\theta-2a^{6}\cos^{6}\theta}{(r^{2}+a^{2}\cos^{2}\theta)^{6}},$
(A.9)

are simply related by,

$\displaystyle
R^{K}_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}_{K}=\frac{1}{2}(R^{S}_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}_{S}+(R^{S}_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}_{S})^{*})|_{r\rightarrow
r+ia\cos\theta}.$ (A.10)

The construction of the field theory described in the paper is based on an
implementation of the Newman-Janis shift at the level of the Lagrangian.

We end this section by showing the gauge theory, single copy classical
solutions corresponding to Schwarzschild and Kerr, namely Coulomb and
$\sqrt{\mathrm{Kerr}}$. We parametrize the gauge field as,

$\displaystyle A^{\mu}=\phi k^{\mu},\hskip
28.45274pt\eta_{\mu\nu}k^{\mu}k^{\nu}=0.$ (A.11)

with $k^{\mu}=(1,-1,0,0)$. In spherical and oblate spheroidal coordinates, the
solutions to equations of motion $D_{\mu}F^{\mu\nu}=0$ for $\phi$ turn out to
be,

$\displaystyle A^{\mu}_{S}=\phi_{S}k^{\mu},\hskip
28.45274ptA^{\mu}_{K}=\phi_{K}k^{\mu},$ (A.12)

respectively, where again $\phi_{S}$ and $\phi_{K}$ are given in (A.4) and
(A.6) with the replacement $r_{s}\rightarrow g$. As discussed in [3, 72, 73,
71, 87], the classical double copy structure between the gauge theory and
gravity solutions is evident.

##  Appendix B Comparison with other Compton amplitudes

In this appendix we compare the opposite-helicity Compton amplitudes in (3.8)
following from the Lagrangian (2.18) with other present in the literature. In
order to do so, we specify the kinematics setup as e.g. in [125]999In our
convention all the momenta are outgoing and hence, in order to compare with
[125], we need to send $p_{1}\rightarrow-p_{1}$, $q_{2}\rightarrow-k_{2}$,
$q_{3}\rightarrow k_{3}$ and $p_{4}\rightarrow p_{4}$ in (B.2) and (B.3).,

$\displaystyle p_{1}=-(m,0,0,0),\qquad$ $\displaystyle
q_{2}=-\hbar(\omega,0,0,\omega),$ (B.1) $\displaystyle
q_{3}=\frac{\hbar\omega}{1+\frac{2\hbar\omega}{m}\sin^{2}(\theta/2)}(1,0,\sin\theta,\cos\theta),\qquad$
$\displaystyle p_{4}=-p_{1}-q_{2}-q_{3},$ (B.2)
$\displaystyle\epsilon_{2}^{+}=\frac{1}{\sqrt{2}}(0,1,i,0),\qquad$
$\displaystyle\epsilon_{2}^{-}=-\frac{1}{\sqrt{2}}(0,1,-i,0),$ (B.3)
$\displaystyle\epsilon_{3}^{+}=\frac{1}{\sqrt{2}}(0,\cos\theta,i,\sin\theta),\qquad$
$\displaystyle\epsilon_{3}^{-}=-\frac{1}{\sqrt{2}}(0,\cos\theta,-i,\sin\theta),$
(B.4)

and $a=\frac{1}{\hbar}(0,a^{x},a^{y},a^{z})$ 101010In the remainder of this
appendix we explicitly reinsert the $\hbar$ dependence of the various
quantities.. As explained at the end of section 3, the opposite-helicity
sector differs from the one originally presented in [75]. We can parametrize
the difference by introducing the complex vectors [92, 123],

$\displaystyle
w=-\frac{\omega(q_{2}-q_{3})+\frac{i}{m\hbar}\epsilon(q_{2},q_{3},p_{1})}{2\omega\cos^{2}(\theta/2)},\qquad
w^{*}=-\frac{\omega(q_{2}-q_{3})-\frac{i}{m\hbar}\epsilon(q_{2},q_{3},p_{1})}{2\omega\cos^{2}(\theta/2)},$
(B.5)

and therefore, by denoting by $\tilde{\mathcal{A}}^{\pm\mp}_{4}(a)$ the
opposite-helicity Compton amplitudes shown in [125] for $s<2$, their relation
to the ones in (3.9) is,

$\displaystyle\tilde{\mathcal{A}}^{+-}_{4}(a;p_{i},q_{i})=e^{-2a\cdot
w}\mathcal{A}_{4}^{+-}(a;p_{i},q_{i}),\qquad\tilde{\mathcal{A}}^{-+}_{4}(a;p_{i},q_{i})=e^{2a\cdot
w^{*}}\mathcal{A}_{4}^{+-}(a;p_{i},q_{i}).$ (B.6)

Indeed, the functions
$\tilde{\mathcal{F}}_{+-}(\omega,a,\theta)=\mathcal{F}(\omega,a,\theta)-2a\cdot
w$ and
$\tilde{\mathcal{F}}_{-+}(\omega,a,\theta)=-\mathcal{F}(\omega,a,\theta)+2a\cdot
w^{*}$ are,

$\displaystyle\tilde{\mathcal{F}}_{+-}(\omega,a,\theta)=\omega(a^{x}+a^{z}\sin\theta+2ia^{y}-a^{x}\cos\theta)\tan(\theta/2),$
(B.7)
$\displaystyle\tilde{\mathcal{F}}_{-+}(\omega,a,\theta)=-\omega(a^{x}+a^{z}\sin\theta-2ia^{y}-a^{x}\cos\theta)\tan(\theta/2)=\tilde{\mathcal{F}}^{*}_{+-}(-\omega,a,\theta),$
(B.8)

and they match the ones in [125].

Let us further analyse the mismatch by looking explicitly at the first order
in the spin. Using equation (2.23) and (3) it is simple to show that,

$\displaystyle\mathcal{A}_{4}(a;p_{i},q_{i})=-\frac{e^{2}}{p_{1}\cdot
q_{2}\hskip 1.42271ptp_{1}\cdot
q_{3}}[\omega_{0}+\omega_{1}+\mathcal{O}(a^{2})],$ (B.9)

where,

$\displaystyle\omega_{0}$ $\displaystyle=2p_{1}\cdot F_{2}\cdot F_{3}\cdot
p_{1},$ (B.10) $\displaystyle\omega_{1}$ $\displaystyle=-i(p_{1}\cdot
F_{3}\cdot q_{2}F_{2\mu\nu}+p_{1}\cdot F_{2}\cdot q_{3}F_{3\mu\nu})S^{\mu\nu}$
$\displaystyle\hskip 14.22636pt+2(p_{1}\cdot q_{3}F_{3}\cdot F_{2}^{*}\cdot
a+p_{1}\cdot q_{2}F_{2}\cdot F_{3}^{*}\cdot a)\equiv\omega_{11}+\omega_{12}.$
(B.11)

If we denote by $\tilde{\omega}_{1}$ the coefficient linear in the spin in
(E.3) of [125], the difference with respect to (B.11) is,

$\displaystyle\Delta\omega_{1}=\omega_{1}-\tilde{\omega}_{1}=$
$\displaystyle\hskip 0.28436pt(p_{1}\cdot q_{3}F_{3}\cdot F_{2}^{*}\cdot
a+p_{1}\cdot q_{2}F_{2}\cdot F_{3}^{*}\cdot a)$
$\displaystyle-i\frac{p_{1}\cdot(q_{2}-q_{3})}{2}(F_{2}^{\mu}{}_{\rho}F_{3}^{\rho\nu}-F_{3}^{\mu}{}_{\rho}F_{2}^{\rho\nu})S_{\mu\nu}.$
(B.12)

Quite interestingly, $\Delta\omega_{1}$ is exactly vanishing when plugging
polarization vectors having the same helicity as it should, for the equal-
helicity sector result following from our model is correct. However, when the
two helicities are opposite $\Delta\omega_{1}$ is not zero. In particular, the
term in the last line of (B) vanishes for opposite helicities, whereas the
second term in (B.11) does not. Consequently, using the relations in (B.6), we
get

$\displaystyle\frac{\omega_{11}^{+-}}{p_{1}\cdot F_{2}^{+}\cdot F_{3}^{-}\cdot
p_{1}}=-a\cdot(q_{2}-q_{3})-2a\cdot w,\qquad\frac{\omega_{12}^{+-}}{p_{1}\cdot
F_{2}^{+}\cdot F_{3}^{-}\cdot p_{1}}=2a\cdot w,$ (B.13)
$\displaystyle\frac{\omega_{11}^{-+}}{p_{1}\cdot F_{2}^{-}\cdot F_{3}^{+}\cdot
p_{1}}=a\cdot(q_{2}-q_{3})+2a\cdot
w^{*},\qquad\frac{\omega_{12}^{-+}}{p_{1}\cdot F_{2}^{-}\cdot F_{3}^{+}\cdot
p_{1}}=-2a\cdot w^{*},$ (B.14)

in the classical limit.

##  Appendix C 1PM amplitude for $\sqrt{\mathrm{Kerr}}$

The 1PM amplitude between two $\sqrt{\mathrm{Kerr}}$ particles having charges
$e_{1}$ and $e_{2}$ exchanging one photon with momentum $q$ can be obtained by
gluing together two three-point amplitudes with the Feynman propagator, as
shown in (C.1),

$i\mathcal{A}^{1\mathrm{PM}}(a;p_{i},q)=\vbox{\hbox{\includegraphics[width=128.0374pt,height=125.19194pt]{1PM.png}}},$
(C.1)

because in the classical limit only the $t=q^{2}$ pole contributes. Before
doing so, we use three-point kinematics to rewrite
$\mathcal{A}^{\mu}_{3}(a,p)$ in a more convenient form,

$\displaystyle\mathcal{A}^{\mu}_{3}(a;p,q)$ $\displaystyle=-ie\hskip
0.85355pt\mathrm{exp}\\{i\epsilon^{\mu}{}_{\rho}(a,q)\\}p^{\rho}$
$\displaystyle=-ie\left(p^{\mu}\cosh(a\cdot
q)+i\frac{\epsilon^{\mu}(p,a,q)}{a\cdot q}\sinh(a\cdot q)\right).$ (C.2)

Therefore, the 1PM amplitude is given by,

$\displaystyle i\mathcal{A}^{1\mathrm{PM}}(a;p_{i},q)=$
$\displaystyle-\frac{i}{q^{2}}\eta_{\mu\nu}\mathcal{A}_{3}^{\mu}(a_{1};p_{1},q)\mathcal{A}^{\nu}_{3}(a_{2},p_{2},-q)$
$\displaystyle=$ $\displaystyle\frac{i}{q^{2}}e_{1}e_{2}\bigg{(}p_{1}\cdot
p_{2}\cosh(a_{1}\cdot q)\cosh(a_{2}\cdot q)$
$\displaystyle+\epsilon^{\mu}(p_{1},a_{1},q)\epsilon_{\mu}(p_{2},a_{2},q)\frac{\sinh(a_{1}\cdot
q)\sinh(a_{2}\cdot q)}{a_{1}\cdot qa_{2}\cdot q}$
$\displaystyle+i\epsilon(p_{2},p_{1},a_{1},q)\cosh(a_{2}\cdot
q)\frac{\sinh(a_{1}\cdot q)}{a_{1}\cdot q}$
$\displaystyle-i\epsilon(p_{1},p_{2},a_{2},q)\cosh(a_{1}\cdot
q)\frac{\sinh(a_{2}\cdot q)}{a_{2}\cdot q}\bigg{)}.$ (C.3)

Using $q\cdot p_{1}=0=q\cdot p_{2}$ and $p_{1}\cdot a_{1}=0=p_{2}\cdot a_{2}$,
together with $q^{2}=0$, we compute the following Gram determinants [86],

$\displaystyle\epsilon^{\mu}(p_{1},a_{1},q)\epsilon_{\mu}(p_{2},a_{2},q)=m_{1}m_{2}\sigma
a_{1}\cdot qa_{2}\cdot q,$ (C.4)
$\displaystyle\epsilon(p_{1},p_{2},a_{1},q)=im_{1}m_{2}(a_{1}\cdot
q)\sqrt{\sigma^{2}-1},$ (C.5)
$\displaystyle\epsilon(p_{1},p_{2},a_{2},q)=im_{1}m_{2}(a_{2}\cdot
q)\sqrt{\sigma^{2}-1},$ (C.6)

where $\sigma=\frac{p_{1}\cdot p_{2}}{m_{1}m_{2}}$ is the Lorentz factor.
Substituting the above relations back into (C) we get,

$\displaystyle i\mathcal{A}^{1\mathrm{PM}}(a;p_{i},q)=$
$\displaystyle\frac{ie_{1}e_{2}m_{1}m_{2}}{q^{2}}\bigg{\\{}\sigma\bigg{[}\cosh\left(a_{1}\cdot
q\right)\cosh\left(a_{2}\cdot q\right)+\sinh\left(a_{1}\cdot
q\right)\sinh\left(a_{2}\cdot q\right)\bigg{]}$
$\displaystyle+\sqrt{\sigma^{2}-1}\bigg{[}\cosh\left(a_{2}\cdot
q\right)\sinh\left(a_{1}\cdot q\right)+\cosh\left(a_{1}\cdot
q\right)\sinh\left(a_{2}\cdot q\right)\bigg{]}\bigg{\\}}$ $\displaystyle=$
$\displaystyle\frac{ie_{1}e_{2}m_{1}m_{2}}{q^{2}}\bigg{[}\sigma\cosh(a\cdot
q)+\sqrt{\sigma^{2}-1}\sinh(a\cdot q)\bigg{]},$ (C.7)

where $a=a_{1}+a_{2}$. Introducing the relative velocity of the two bodies
$v=\frac{\sqrt{\sigma^{2}-1}}{\sigma}$ we get for the classical 1PM amplitude,

$\displaystyle\mathcal{A}^{1\mathrm{PM}}(a;p_{i},q)=\frac{e_{1}m_{1}e_{2}m_{2}}{2q^{2}}\sigma\sum_{\pm}(1\pm
v)e^{\pm i\frac{\epsilon(p_{1},p_{2},q,a)}{m_{1}m_{2}\sigma v}},$ (C.8)

compatibly with [87], where we inverted (C.5) and (C.6) to get $a\cdot
q=i\frac{\epsilon(p_{1},p_{2},q,a)}{m_{1}m_{2}\sigma v}$.

It is instructive to show how to get the 1PM amplitude also by summing over
the helicities of intermediate polarizations,

$\displaystyle
i\mathcal{A}^{\mathrm{1PM}}(a;p_{i},q_{i})=-\frac{i}{q^{2}}\sum_{h=\pm}\mathcal{A}_{3}^{\mu}(a_{1};p_{1},q)\epsilon^{h}_{\mu}(q)\epsilon^{-h}_{\nu}(-q)\mathcal{A}_{3}^{\nu}(a_{2};p_{2},-q).$
(C.9)

Using now the definition of $\epsilon^{\prime\mu}$, the previous equation
becomes,

$\displaystyle
i\mathcal{A}^{\mathrm{1PM}}(a;p_{i},q_{i})=-\frac{i}{q^{2}}\sum_{h=\pm}\mathcal{A}_{3}^{\mu}(p_{1},q)\epsilon^{\prime
h}_{\mu}(a_{1},q)\epsilon^{\prime-h}_{\nu}(a_{2},-q)\mathcal{A}_{3}^{\nu}(p_{2},-q).$
(C.10)

We introduce projectors onto positive and negative helicity states,

$\displaystyle\Pi^{+}_{\mu\nu}(q)=\epsilon_{\mu}^{+}(q)\epsilon_{\nu}^{-}(-q)=\frac{1}{2}\eta_{\mu\nu}+\frac{1}{2}\frac{q_{\mu}k_{\nu}+q_{\nu}k_{\mu}}{q\cdot
k}+\frac{i}{2}\frac{\epsilon_{\mu\nu}(q,k)}{q\cdot k},$ (C.11)
$\displaystyle\Pi^{-}_{\mu\nu}(q)=\epsilon_{\mu}^{-}(q)\epsilon_{\nu}^{+}(-q)=\frac{1}{2}\eta_{\mu\nu}+\frac{1}{2}\frac{q_{\mu}k_{\nu}+q_{\nu}k_{\mu}}{q\cdot
k}-\frac{i}{2}\frac{\epsilon_{\mu\nu}(q,k)}{q\cdot k},$ (C.12)

where $k$ is an arbitrary reference null vector. They satisfy,

$\displaystyle\Pi^{\pm}_{\mu}{}^{\rho}(q)\Pi_{\rho}^{\pm}{}^{\nu}(q)=\Pi_{\mu}^{\pm}{}^{\nu}(q),\qquad\Pi^{\pm}_{\mu}{}^{\rho}(q)\Pi_{\rho}^{\mp}{}^{\nu}(q)=0,$
(C.13)
$\displaystyle\Pi^{+}_{\mu\nu}(q)+\Pi^{-}_{\mu\nu}(q)=\eta_{\mu\nu}+\frac{q_{\mu}k_{\nu}+q_{\nu}k_{\mu}}{q\cdot
k}=\Pi_{\mu\nu}(q),\qquad\Pi_{\mu}{}^{\rho}(q)\Pi_{\rho}{}^{\nu}(q)=\Pi_{\mu}{}^{\nu}(q).$
(C.14)

Up to gauge transformations $\epsilon^{\prime\mu}_{h}=e^{-ha\cdot
q}\epsilon^{\mu}_{h}$ and hence the sum appearing in (C.10) reads,

$\displaystyle\sum_{h=\pm}e^{-ha_{1}\cdot
q}\epsilon^{h}_{\mu}(q)e^{ha_{2}\cdot q}\epsilon^{h}_{\nu}(-q)=$
$\displaystyle\bigg{(}\eta_{\mu\nu}-\frac{q_{\mu}k_{\nu}+q_{\nu}k_{\mu}}{q\cdot
k}\bigg{)}\cosh(a\cdot q)$
$\displaystyle-i\frac{\epsilon_{\mu\nu}(q,k)}{q\cdot k}\sinh(a\cdot
q)\equiv\Pi_{\mu\nu}(a;q),$ (C.15)

where $a=a_{1}+a_{2}$. $\Pi_{\mu\nu}(a)$ is a spin-dressed projector and it
satisfies,

$\displaystyle\Pi_{\mu}{}^{\rho}(a;q)\Pi_{\rho}{}^{\nu}(b;q)=\Pi_{\mu}{}^{\nu}(a+b;q).$
(C.16)

The 1PM amplitude in (C.9) can be written in terms of $\Pi_{\mu}{}^{\nu}(a;q)$
as,

$\displaystyle
i\mathcal{A}^{1\mathrm{PM}}(a;p_{i},q_{i})=-\frac{i}{q^{2}}\mathcal{A}_{3}^{\mu}(p_{1},q)\Pi_{\mu\nu}(a;q)\mathcal{A}_{3}^{\nu}(p_{2},-q).$
(C.17)

This equation shows that the full dependence on the classical spin of the 1PM
amplitude is actually encoded into a suitable “spin dressing” of the Feynman
propagator.

Similarly, the gravity 1PM amplitude in (4.11) can be written as,

$\displaystyle i\mathcal{M}^{1\mathrm{PM}}(a;p_{i},q_{i})$
$\displaystyle=-\frac{i}{q^{2}}\mathcal{M}_{3}^{\mu\nu}(a_{1};p_{1},q)\Pi_{\mu\nu;\rho\sigma}(q)\mathcal{M}_{3}^{\rho\sigma}(a_{2};p_{2},-q)$
$\displaystyle=-\frac{i}{q^{2}}\mathcal{M}_{3}^{\mu\nu}(p_{1},q)\Pi_{\mu\nu;\rho\sigma}(a;q)\mathcal{M}_{3}^{\rho\sigma}(p_{2},-q),$
(C.18)

where $\Pi_{\mu\nu;\rho\sigma}(a;q)$ is a spin-dressed de Donder projector,

$\displaystyle\Pi_{\mu\nu;\rho\sigma}(a;q)=$
$\displaystyle\frac{1}{2}\bigg{[}\Pi_{\mu\rho}(a;q)\Pi_{\nu\sigma}(a;q)+\Pi_{\mu\sigma}(a;q)\Pi_{\nu\rho}(a;q)-\Pi_{\mu\nu}(a;q)\Pi_{\rho\sigma}(a;q)\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\Pi_{\mu\rho}(q)\Pi_{\nu\sigma}(q)+\Pi_{\mu\sigma}(q)\Pi_{\nu\rho}(q)-\Pi_{\mu\nu}(q)\Pi_{\rho\sigma}(q)\bigg{]}\cosh(a\cdot
q)$ (C.19) $\displaystyle-\frac{i}{2q\cdot
k}\bigg{[}\epsilon_{\nu\sigma}(q,k)\Pi_{\mu\rho}(q)+\epsilon_{\mu\rho}(q,k)\Pi_{\nu\sigma}(q)\bigg{]}\sinh(a\cdot
q).$

satisfying,

$\displaystyle\Pi_{\mu\nu}{}^{\alpha\beta}(a;q)\Pi_{\alpha\beta}{}^{\rho\sigma}(b;q)=\Pi_{\mu\nu}{}^{\rho\sigma}(a+b;q).$
(C.20)

##  Appendix D One-loop integrals

For the 2PM amplitude, we need to solve the integral
$\mathcal{I}^{\mu_{1}\dots\mu_{n}}_{\triangleright}(a)$ in (5.22), and hence
we start by considering the family of integrals,

$\displaystyle\mathcal{I}^{\mu_{1}\dots\mu_{n}}_{\triangleright}(0)=\int\frac{d^{4}l}{(2\pi)^{4}}\frac{l^{\mu_{1}}\dots
l^{\mu_{n}}}{l^{2}(l+q)^{2}[(p_{1}+l)^{2}-m_{1}^{2}]}.$ (D.1)

We find,

$\displaystyle\mathcal{I}_{\triangleright}(0)=-\frac{i}{32m_{1}\sqrt{-q^{2}}}+\dots,$
(D.2)

and using the Passarino-Veltmano reduction [143, 144, 145, 146],

$\displaystyle\mathcal{I}^{\mu}_{\triangleright}(0)=-\frac{i}{32m_{1}\sqrt{-q^{2}}}\bigg{[}-\frac{1}{2}q^{\mu}+\frac{1}{4}\frac{q^{2}}{m_{1}^{2}}p_{1}^{\mu}+\dots\bigg{]},$
(D.3)
$\displaystyle\mathcal{I}^{\mu\nu}_{\triangleright}(0)=-\frac{i}{32m_{1}\sqrt{-q^{2}}}\bigg{[}\frac{3}{8}q^{\mu}q^{\nu}+\frac{1}{8}\frac{q^{2}}{m_{1}^{2}}p^{\mu}_{1}p_{1}^{\nu}-\frac{3}{8}\frac{q^{2}}{m_{1}^{2}}p_{1}^{(\mu}q^{\nu)}-\frac{1}{8}q^{2}\eta^{\mu\nu}+\dots\bigg{]},$
(D.4)
$\displaystyle\mathcal{I}^{\mu\nu\rho}_{\triangleright}(0)=-\frac{i}{32m_{1}\sqrt{-q^{2}}}\bigg{[}-\frac{5}{16}q^{\mu}q^{\nu}q^{\rho}+\frac{3}{16}\frac{q^{4}}{m_{1}^{4}}p^{\mu}_{1}p_{1}^{\nu}p_{1}^{\rho}-\frac{3}{16}\frac{q^{2}}{m_{1}^{2}}q^{(\mu}p^{\rho}_{1}p^{\nu)}_{1}$
$\displaystyle\hskip
85.35826pt+\frac{15}{32}\frac{q^{2}}{m_{1}^{2}}q^{(\mu}q^{\nu}p_{1}^{\rho)}+\frac{3}{16}q^{2}\eta^{(\mu\nu}q^{\rho)}-\frac{3}{16}\frac{q^{2}}{m_{1}^{2}}\eta^{(\mu\nu}p_{1}^{\rho)}+\dots\bigg{]},$
(D.5)

where $\dots$ denote subleading terms in the classical limit. Expanding in
powers of the total spin vector $a^{\mu}$ the integral
$\mathcal{I}^{\mu_{1}\dots\mu_{n}}_{\triangleright}(a)$, we see that we need
to compute the contractions of the above tensors with $a_{\mu_{1}}\dots
a_{\mu_{n}}$. With the aligned spin and the kinematics specified in (5.23),
all the scalar products $a\cdot p_{i}$ vanish. Hence, we are only interested
in keeping the terms in (D.3)-(D) having products of $q$’s and $\eta$’s. For
the latter there is a general formula [147],

$\displaystyle\mathcal{I}^{\mu_{1}\dots\mu_{n}}(0)=$
$\displaystyle-\frac{i}{32m_{1}\pi^{\frac{3}{2}}\sqrt{-q^{2}}}\sum_{m=0}^{\left[\frac{n}{2}\right]}\frac{(-1)^{n}\Gamma(\frac{1}{2}-m)\Gamma(n-m+\frac{1}{2})\Gamma(m+\frac{1}{2})}{\Gamma(n+1)}\left(\frac{q^{2}}{2}\right)^{m}$
$\displaystyle\\{[\eta]^{m}[q]^{n-2m}\\}^{\mu_{1}\dots\mu_{n}},$ (D.6)

in the classical limit, where the symbol
$\\{[\eta]^{m}[q]^{n-2m}\\}^{\mu_{1}\dots\mu_{n}}$ comprises all the symmetric
structures one can construct with $q$ and $\eta$ having $n$ covariant indices,
e.g.

$\displaystyle\\{[\eta]^{0}[q]^{1}\\}^{\mu_{1}}=q^{\mu_{1}},$
$\displaystyle\\{[\eta]^{1}[q]^{0}\\}^{\mu_{1}\mu_{2}}=\eta^{\mu_{1}\mu_{2}},$
$\displaystyle\\{[\eta]^{1}[q]^{1}\\}^{\mu_{1}\mu_{2}\mu_{3}}=\eta^{\mu_{1}\mu_{2}}q^{\mu_{3}}+\eta^{\mu_{3}\mu_{1}}q^{\mu_{2}}+\eta^{\mu_{2}\mu_{3}}q^{\mu_{1}}.$

We have, expanding $\cosh(x)$,

$\displaystyle\mathcal{I}_{\triangleright}(a)=$
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{(2n)!}\int\frac{d^{4}l}{2\pi)^{4}}\frac{(2a\cdot
l+a\cdot q)^{2n}}{l^{2}(l+q)^{2}[(p_{1}+l)^{2}-m_{1}^{2}]}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{2n}\frac{(a\cdot
q)^{k}2^{2n-k}}{(2n-k)!k!}a_{\mu_{1}}\dots
a_{\mu_{2n-k}}\int\frac{d^{4}l}{(2\pi)^{4}}\frac{l^{\mu_{1}}\dots
l^{\mu_{2n-k}}}{l^{2}(l+q)^{2}[(p_{1}+l)^{2}-m_{1}^{2}]}$ $\displaystyle=$
$\displaystyle-\frac{i}{32m_{1}\pi^{\frac{3}{2}}\sqrt{-q^{2}}}\sum_{n=0}^{\infty}\sum_{k=0}^{2n}\sum_{m=0}^{[n-\frac{k}{2}]}\frac{(-1)^{k}2^{2n-k}}{(2n-k)!k!}\frac{\Gamma(\frac{1}{2}-m)\Gamma(2n-k+\frac{1}{2})\Gamma(m+\frac{1}{2})}{\Gamma(2n-k+1)}$
$\displaystyle\left(\frac{q^{2}a^{2}}{2}\right)^{m}(a\cdot
q)^{2n-2m}\frac{(2n-k)!}{m!(2n-k-2m)!2^{m}},$ (D.7)

where we used that,

$\displaystyle a_{\mu_{1}}\dots
a_{\mu_{n}}\\{[\eta]^{m}[q]^{n-2m}\\}^{\mu_{1}\dots\mu_{n}}=a^{2m}(a\cdot
q)^{n-2m}\frac{n!}{m!(n-2m)!2^{m}}.$ (D.8)

For $n=0,1,2$ the sum in (D) gives,

$\displaystyle-\frac{i}{32m_{1}\sqrt{-q^{2}}},\qquad-\frac{i[(a\cdot
q)^{2}-a^{2}q^{2}]}{128m_{1}\sqrt{-q^{2}}},\qquad-\frac{i[(a\cdot
q)^{2}-a^{2}q^{2}]^{2}}{2048m_{1}\sqrt{-q^{2}}}.$ (D.9)

These quantities are related to the series expansion of the modified Bessel
function of the first kind $I_{0}(\sqrt{(a\cdot q)^{2}-a^{2}q^{2}})$ and, in
particular we have,

$\displaystyle\mathcal{I}_{\triangleright}(a)=\int\frac{d^{4}l}{(2\pi)^{4}}\frac{\cosh(2a\cdot
l+a\cdot
q)}{l^{2}(l+q)^{2}[(p_{1}+l)^{2}-m_{1}^{2}]}=-\frac{i}{32m_{1}\sqrt{-q^{2}}}I_{0}(\mathscr{F}(a,q)),$
(D.10)

where we introduced for convenience,

$\displaystyle\mathscr{F}(a,q)=\sqrt{(a\cdot q)^{2}-a^{2}q^{2}},$ (D.11)

in the aligned spin case, which is shift-symmetric [110]. We could also
derive,

$\displaystyle
a_{\mu}\mathcal{I}^{\mu}_{\triangleright}(a)=\frac{i}{32m_{1}\sqrt{-q^{2}}}\frac{(a\cdot
q)}{2}I_{0}(\mathscr{F}(a,q)),$ (D.12) $\displaystyle
a_{\mu}a_{\nu}\mathcal{I}^{\mu\nu}_{\triangleright}(a)=-\frac{i}{32m_{1}\sqrt{-q^{2}}}\frac{1}{4}\bigg{\\{}[\mathscr{F}^{2}(a,q)+(a\cdot
q)^{2}]I_{0}(\mathscr{F}(a,q))$ $\displaystyle\hskip
66.86414pt-\mathscr{F}(a,q)I_{1}(\mathscr{F}(a,q))\bigg{\\}}.$ (D.13)

Using the above results, it is simple to show that the Passarino-Veltman
decomposition generalizes also to the case of integrals of the form (5.22)
and, in particular, we find,

$\displaystyle\mathcal{I}_{\triangleright}^{\mu}(a)=-\frac{i}{32m_{1}\sqrt{-q^{2}}}\bigg{(}-\frac{1}{2}q^{\mu}+\frac{1}{4}\frac{q^{2}}{m_{1}^{2}}p_{1}^{\mu}-\frac{1}{8}\frac{q^{4}}{m_{1}^{2}}\frac{a\cdot
q}{\mathscr{F}(a,q)^{2}}a^{\mu}\bigg{)}I_{0}(\mathscr{F}(a,q)),$ (D.14)
$\displaystyle\mathcal{I}_{\triangleright}^{\mu\nu}(a)=Aq^{\mu}q^{\nu}+Bq^{(\mu}p_{1}^{\nu)}+Cp_{1}^{\mu}p_{1}^{\nu}+D\eta^{\mu\nu}+Ea^{\mu}a^{\nu}+Fa^{(\mu}q^{\nu)}+Ga^{(\mu}p_{1}^{\nu)},$
(D.15)

where,

$\displaystyle A=$
$\displaystyle-\frac{i}{128m_{1}\sqrt{-q^{2}}}\bigg{[}\frac{2(a\cdot
q)^{2}-a^{2}q^{2}}{\mathscr{F}^{2}(a,q)}I_{0}(\mathscr{F}(a,q))-\frac{(a\cdot
q)^{2}+a^{2}q^{2}}{\mathscr{F}(a,q)^{3}}I_{1}(\mathscr{F}(a,q))\bigg{]},$
(D.16) $\displaystyle B=$
$\displaystyle\frac{i}{128m_{1}\sqrt{-q^{2}}}\frac{q^{2}}{m_{1}^{2}}\bigg{[}\frac{2(a\cdot
q)^{2}-a^{2}q^{2}}{\mathscr{F}(a,q)^{2}}I_{0}(\mathscr{F}(a,q))-\frac{(a\cdot
q)^{2}+a^{2}q^{2}}{\mathscr{F}(a,q)^{3}}I_{1}(\mathscr{F}(a,q)\bigg{]},$
(D.17) $\displaystyle C=$
$\displaystyle-\frac{i}{128m_{1}\sqrt{-q^{2}}}\frac{q^{2}}{m_{1}^{2}}\bigg{[}\frac{q^{2}}{2m_{1}^{2}}\frac{2(a\cdot
q)^{2}-a^{2}q^{2}}{\mathscr{F}(a,q)^{2}}I_{0}(\mathscr{F}(a,q))+\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)}\bigg{]},$
(D.18) $\displaystyle D=$
$\displaystyle\frac{i}{128m_{1}\sqrt{-q^{2}}}q^{2}\bigg{[}\frac{q^{2}}{4m_{1}^{2}}\frac{2(a\cdot
q)^{2}-a^{2}q^{2}}{\mathscr{F}(a,q)^{2}}I_{0}(\mathscr{F}(a,q))+\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)}\bigg{]},$
(D.19) $\displaystyle E=$
$\displaystyle-\frac{i}{64m_{1}\sqrt{-q^{2}}}q^{4}\bigg{[}\frac{I_{0}(\mathscr{F}(a,q))}{2\mathscr{F}(a,q)^{2}}-\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)^{3}}\bigg{]},$
(D.20) $\displaystyle F=$
$\displaystyle\frac{i}{32m_{1}\sqrt{-q^{2}}}q^{2}(a\cdot
q)\bigg{[}\frac{I_{0}(\mathscr{F}(a,q))}{2\mathscr{F}(a,q)^{2}}-\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)^{3}}\bigg{]},$
(D.21) $\displaystyle G=$
$\displaystyle-\frac{i}{64m_{1}\sqrt{-q^{2}}}\frac{q^{4}(a\cdot
q)}{m_{1}^{2}}\bigg{[}\frac{I_{0}(\mathscr{F}(a,q))}{2\mathscr{F}(a,q)^{2}}-\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)^{3}}\bigg{]}.$
(D.22)

Using similar arguments we find,

$\displaystyle\mathcal{J}_{\triangleright}(a)=\int\frac{d^{4}l}{(2\pi)^{4}}\frac{\sinh(2a\cdot
l+a\cdot q)}{l^{2}(l+q)^{2}[(p_{1}+l)^{2}-m_{1}^{2}]}=0,$ (D.23)

and,

$\displaystyle
a_{\mu}\mathcal{J}^{\mu}_{\triangleright}(a)=-\frac{i}{64m_{1}\sqrt{-q^{2}}}\mathscr{F}(a,q)I_{1}(\mathscr{F}(a,q)),$
(D.24) $\displaystyle
a_{\mu}a_{\nu}\mathcal{J}_{\triangleright}^{\mu}(a)=\frac{i}{64m_{1}\sqrt{-q^{2}}}(a\cdot
q)\mathscr{F}(a,q)I_{1}(\mathscr{F}(a,q)).$ (D.25)

The Passarino-Veltman decomposition now is,

$\displaystyle\mathcal{J}^{\mu}_{\triangleright}(a)=-\frac{i}{64m_{1}\sqrt{-q^{2}}}\bigg{[}(a\cdot
q)q^{\mu}-\frac{q^{2}(a\cdot
q)}{2m_{1}^{2}}p_{1}^{\mu}-\frac{q^{2}}{2}a^{\mu}\bigg{]}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)},$
(D.26)
$\displaystyle\mathcal{J}^{\mu\nu}_{\triangleright}(a)=Aq^{\mu}q^{\nu}+Bq^{(\mu}p_{1}^{\nu)}+Cp_{1}^{\mu}p_{1}^{\nu}+D\eta^{\mu\nu}+Ea^{\mu}a^{\nu}+Fa^{(\mu}q^{\nu)}+Ga^{(\mu}p_{1}^{\nu)},$
(D.27)

with,

$\displaystyle A=\frac{i(a\cdot
q)}{64m_{1}\sqrt{-q^{2}}}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)},$
(D.28) $\displaystyle B=-\frac{i(a\cdot
q)}{64m_{1}\sqrt{-q^{2}}}\frac{q^{2}}{m_{1}^{2}}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)},$
(D.29) $\displaystyle C=\frac{i(a\cdot
q)}{64m_{1}\sqrt{-q^{2}}}\frac{2q^{4}}{m_{1}^{4}}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)},$
(D.30) $\displaystyle D=-\frac{i(a\cdot
q)}{64m_{1}\sqrt{-q^{2}}}\frac{q^{4}}{4m_{1}^{2}}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)},$
(D.31) $\displaystyle E=-\frac{i(a\cdot
q)}{64m_{1}\sqrt{-q^{2}}}\frac{q^{6}}{2m_{1}^{2}}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)^{3}},$
(D.32) $\displaystyle
F=-\frac{i}{64m_{1}\sqrt{-q^{2}}}q^{2}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)},$
(D.33) $\displaystyle
G=\frac{i}{64m_{1}\sqrt{-q^{2}}}\frac{q^{4}}{2m_{1}^{2}}\frac{I_{1}(\mathscr{F}(a,q))}{\mathscr{F}(a,q)}.$
(D.34)

We stress again here that the results for the various integrals above are
valid only if $a\cdot p_{1}=0$. Furthermore, it is straightforward to prove
the following relations,

$\displaystyle\frac{\partial\mathcal{I}_{\triangleright}(a)}{\partial
a_{\mu}}=2\mathcal{J}^{\mu}_{\triangleright}(a)+q^{\mu}\mathcal{J}_{\triangleright}(a),$
(D.35)
$\displaystyle\frac{\partial^{2}\mathcal{I}_{\triangleright}(a)}{\partial
a_{\mu}\partial
a_{\nu}}=4\mathcal{I}_{\triangleright}^{\mu\nu}(a)+2q^{\mu}\mathcal{I}_{\triangleright}^{\nu}(a)+2q^{\nu}\mathcal{I}^{\mu}_{\triangleright}(a)+q^{\mu}q^{\nu}\mathcal{I}_{\triangleright}(a),$
(D.36) $\displaystyle\frac{\partial\mathcal{J}_{\triangleright}(a)}{\partial
a_{\mu}}=2\mathcal{I}^{\mu}_{\triangleright}(a)+q^{\mu}\mathcal{I}_{\triangleright}(a),$
(D.37)
$\displaystyle\frac{\partial^{2}\mathcal{J}_{\triangleright}(a)}{\partial
a_{\mu}\partial
a_{\nu}}=4\mathcal{J}_{\triangleright}^{\mu\nu}(a)+2q^{\mu}\mathcal{J}_{\triangleright}^{\nu}(a)+2q^{\nu}\mathcal{J}^{\mu}_{\triangleright}(a)+q^{\mu}q^{\nu}\mathcal{J}_{\triangleright}(a),$
(D.38)

that are compatible with what we have found so far.

##  Appendix E 2PM eikonal phase integrals

In order to compute the 2PM eikonal, we need to evaluate the integral,

$\displaystyle\mathcal{I}_{n}[\mathbf{a};\alpha]\equiv\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{(\mathbf{a}\cdot\mathbf{q})^{n}}{(\mathbf{q}^{2})^{\alpha}}e^{-i\mathbf{q}\cdot\mathbf{b}}=i^{n}a^{i_{1}}\dots
a^{i_{n}}\frac{\Gamma(1-\alpha)}{2^{2\alpha}\pi\Gamma(\alpha)}\frac{\partial^{n}}{\partial
b^{i_{1}}\dots\partial b^{i_{n}}}(\mathbf{b}^{2})^{\alpha-1}.$ (E.1)

Using,

$\displaystyle\frac{\partial^{n}}{\partial b^{i_{1}}\dots\partial
b^{i_{n}}}(\mathbf{b}^{2})^{\alpha-1}=\sum_{m=0}^{[\frac{n}{2}]}\frac{2^{n-m}\Gamma(\alpha)}{\Gamma(\alpha-n+m)}(\mathbf{b}^{2})^{\alpha-1-n+m}\\{[\delta]^{m}[\mathbf{b}]^{n-2m}\\}^{i_{1}\dots
i_{n}},$ (E.2)

and (D.8) we obtain,

$\displaystyle\mathcal{I}_{n}[\mathbf{a};\alpha]=\frac{1}{2^{2\alpha}\pi}\sum_{m=0}^{[\frac{n}{2}]}\frac{\Gamma(1-\alpha)(2i)^{n}n!}{2^{2m}\Gamma(\alpha-n+m)m!(n-2m)!}(\mathbf{b}^{2})^{\alpha-1-n+m}(\mathbf{a}^{2})^{m}(\mathbf{a}\cdot\mathbf{b})^{n-2m}.$
(E.3)

We are interested in computing the Fourier transform of modified Bessel
functions appearing in the 2PM amplitude (5) and (5). We have,

$\displaystyle\tilde{I}_{0}[\mathbf{a},\mathbf{b}]$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{e^{-i\mathbf{b}\cdot\mathbf{q}}}{(\mathbf{q}^{2})^{\frac{1}{2}}}I_{0}(\mathscr{F}(\mathbf{a},\mathbf{q}))=\sum_{n=0}^{\infty}\frac{1}{2^{2n}(n!)^{2}}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{e^{-i\mathbf{b}\cdot\mathbf{q}}}{(\mathbf{q}^{2})^{\frac{1}{2}}}[(\mathbf{a}\cdot\mathbf{q})^{2}-\mathbf{a}^{2}\mathbf{q}^{2}]^{n}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(-1)^{k}(\mathbf{a}^{2})^{k}}{2^{2n}n!k!(n-k)!}a_{i_{1}}\dots
a_{i_{2n-2k}}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{e^{-i\mathbf{b}\cdot\mathbf{q}}}{(\mathbf{q}^{2})^{\frac{1}{2}-k}}q^{i_{1}}\dots
q^{i_{2n-2k}}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(-1)^{k}(\mathbf{a}^{2})^{k}}{2^{2n}n!k!(n-k)!}\mathcal{I}_{2n-2k}[\mathbf{a};1/2-k].$
(E.4)

Similarly,

$\displaystyle\tilde{\bigg{(}\frac{I_{1}}{\mathscr{F}}\bigg{)}}[\mathbf{a},\mathbf{b}]$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{e^{-i\mathbf{q}\cdot\mathbf{b}}}{(\mathbf{q}^{2})^{\frac{1}{2}}}\frac{I_{1}(\mathscr{F}(\mathbf{a},\mathbf{q}))}{\mathscr{F}(\mathbf{a},\mathbf{q})}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(-1)^{k}(\mathbf{a}^{2})^{k}}{2^{2n+1}(n+1)!k!(n-k)!}\mathcal{I}_{2n-2k}[\mathbf{a};1/2-k],$
(E.5)

and,

$\displaystyle\tilde{I}_{1}[\mathbf{a},\mathbf{b}]$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{e^{-i\mathbf{b}\cdot\mathbf{q}}}{(\mathbf{q}^{2})^{\frac{1}{2}}}I_{1}(\mathscr{F}(\mathbf{a},\mathbf{q}))$
$\displaystyle=\sum_{n,l=0}^{\infty}\sum_{k=0}^{n}\frac{(-1)^{k}(2l)!(\mathbf{a}^{2})^{l+k}}{2^{2n+2l+1}(n+1)!(1-2l)(l!)^{2}k!(n-k)!}\mathcal{I}_{2n+1-2k-2l}[\mathbf{a};1/2-k-l].$
(E.6)

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