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QMUL-PH-24-09

# Spinning binary dynamics in cubic effective field theories of gravity

Andreas Brandhuber  Graham R. Brown  Paolo Pichini,
Gabriele Travaglini  and Pablo Vives Matasan

###### Abstract

We study the binary dynamics of two Kerr black holes with arbitrary spin
vectors in the presence of parity-even and parity-odd cubic deformations of
gravity. We first derive the tree-level Compton amplitudes for a Kerr black
hole in cubic gravity, which we then use to compute the two-to-two amplitudes
of the massive bodies to leading order in the deformation and the post-
Minkowskian expansion. The required one-loop computations are performed using
the leading singularity approach as well as the heavy-mass effective field
theory (HEFT) approach. These amplitudes are then used to compute the leading-
order momentum and spin kick in cubic gravity in the KMOC formalism. Our
results are valid for generic masses and spin vectors, and include all the
independent parity-even and parity-odd cubic deformations of Einstein-Hilbert
gravity. We also present spin-expanded expressions for the momentum and spin
kicks, and the all-order in spin deflection angle in the case of aligned
spins.

###### Contents

  1. 1 Introduction
  2. 2 Classical Compton amplitudes from cubic deformations
  3. 3 One-loop spinning amplitudes from leading singularities
    1. 3.1 Kinematics
    2. 3.2 Introducing the triple cut
    3. 3.3 Leading singularities with spinor parameterisation
      1. 3.3.1 Parameterisation of the momenta and integration measure
      2. 3.3.2 Taking the classical limit
      3. 3.3.3 Leading singularities for $I_{1}$
      4. 3.3.4 Leading singularities for $I_{2}$ and $G_{3}$
    4. 3.4 Leading singularities without spinor variables
  4. 4 One-loop spinning amplitudes from unitarity and the heavy-mass limit
    1. 4.1 Review of the heavy-mass limit for Einstein-Hilbert
    2. 4.2 Introducing $R^{3}$ corrections: the scalar case
    3. 4.3 Introducing $R^{3}$ corrections: the spinning case
  5. 5 Parity-odd cubic deformations
  6. 6 Observables from KMOC
    1. 6.1 The momentum kick
    2. 6.2 The spin kick
    3. 6.3 Fourier transforms to impact parameter space
    4. 6.4 Explicit results and discussion
      1. 6.4.1 Results for the momentum kick
      2. 6.4.2 Expanded in spin momentum and spin kicks
      3. 6.4.3 Aligned spin case
  7. 7 Conclusions

##  1 Introduction

The detection of gravitational waves is entering a high-precision era, with
large numbers of gravitational-wave observations by the LIGO-Virgo-KAGRA
collaboration and the prospect of increased accuracy in forthcoming missions
such as LISA. As a consequence there is a need for ever more precise
theoretical predictions for the dynamics of binary black holes and neutron
stars. Available techniques to achieve this goal now include scattering
amplitude methods, developed over the years both in theoretical models and
with a view to concrete applications to high-precision measurements at the
LHC.111For a recent review see [1]. By modelling encounters of two celestial
objects as a scattering of two pointlike particles, modern amplitude methods
can be imported to study these processes in perturbation theory. Crucial in
these derivations is the observation [2] that only non-analytic terms in the
amplitudes, arising from long-distance propagation of gravitons, are relevant
to obtain the leading low-energy corrections, making unitarity-based methods
perfectly suited for these calculations.

Two natural avenues are available to explore. In the first, the focus is
solely on the Einstein-Hilbert (EH) action, and specifically on computing
perturbative corrections to various quantities of interest to high orders in
$GM/b$, where $G$ is Newton’s constant, $b$ is the impact parameter and $M$ is
the characteristic mass scale of the process. Progress in this direction
includes deriving Newton’s potential at second Post-Minkowskian (PM) order [3,
4, 5, 6], the potential at 3PM [7, 8, 9, 10, 11, 12, 13] and at 4PM [14, 15,
16], processes with radiation emission and waveforms [17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28], spin effects [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, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75], and
applications of the worldline formalism [76, 77, 78, 79, 80, 81, 82, 83, 84,
85, 86, 87, 88, 89, 90, 91, 92, 93], which was recently used to obtain the
first 5PM results [94, 95]. Another direction entertains the possibility that
general relativity is only the leading-order approximation of a yet to be
discovered (more) complete theory. An efficient way to study this idea is
through the framework of effective field theories (EFTs), initially developed
in the context of gravity in [2, 96]. In this language, higher-derivative
operators are added to the EH action, and their effect on binary dynamics is
computed as a perturbative series in inverse powers of a scale, on which one
can then set bounds, see for example [97, 98, 99].222In particular, in [98] a
bound of $\ell_{\rm EFT}\leq 38.2$ km was found for the fundamental length
scale of cubic theories, for the choice $\beta_{1}{=}\tilde{\beta}_{1}$,
$\beta_{2}{=}\tilde{\beta}_{2}{=}0$ for the couplings in the effective action
(1.3). One is then tasked to write down independent, gauge-invariant
interactions of increasing mass dimension, which can augment the EH action.

This approach was followed recently in [100] for the case of dimension-eight
operators. The first new couplings in fact arise at dimension four, that is
quadratic in the curvatures, but they are known to not affect scattering
amplitudes [101, 102, 103] (see [104] for a recent derivation with modern
amplitude methods). At dimensions six we encounter two independent parity-even
couplings together with their two parity-odd cousins, which are cubic in the
Riemann tensor. These cubic interactions will be the focus of the present
paper. They have been investigated already in a number of works, both using
traditional relativity approaches333See also [105, 106, 107, 108, 109] for
related work in other modified theories of gravity. [110, 111, 98], as well as
amplitude techniques applied to the case of spinless binary systems [112, 113,
114, 115], or systems where only one black hole is spinning and of much larger
mass than the other [116]. In particular, classical and quantum corrections to
Newton’s potential were computed at leading order in the perturbations [112,
113], as well as the deflection angles for massless probes [112, 114].
However, in a generic encounter we expect both celestial bodies to have non-
zero spin vectors $a_{1}$ and $a_{2}$ and arbitrary masses $m_{1}$ and
$m_{2}$, and this is the situation we wish to address in this work.

Cubic interactions are known to give rise to time advance and hence causality
violations already at tree level [117], but only at scales beyond the regime
of validity of the effective field theory approximation [118], specifically
for $b\lesssim\alpha^{1/4}$, where $\alpha$ is the typical coupling of a cubic
interaction [114].444A detailed discussion of such violations in the amplitude
framework for all parity-even cubic couplings can be found in [114].
Therefore, cubic interactions are the first nontrivial corrections to general
relativity to be included in an EFT description of gravitational phenomena.
Specifically, the two independent parity-even ones can be chosen as

$\displaystyle I_{1}$
$\displaystyle\coloneq{R^{\alpha\beta}}_{\mu\nu}{R^{\mu\nu}}_{\rho\sigma}{R^{\rho\sigma}}_{\alpha\beta}\
,\qquad
I_{2}\coloneq{R^{\mu\nu\alpha}}_{\beta}{R^{\beta\gamma}}_{\nu\sigma}{R^{\sigma}}_{\mu\gamma\alpha}\
,$ (1.1)

though, alternatively to $I_{2}$, the combination

$\displaystyle G_{3}\coloneq I_{1}-2I_{2}\ $ (1.2)

is often considered. Therefore, in the following we will consider the
gravitational effective action555In our normalisations, Newton’s constant is
defined as $G=\kappa^{2}/(32\pi)$.

$\begin{split}S=\int\\!d^{4}x\sqrt{-g}\,\bigg{[}&-\frac{2}{\kappa^{2}}R\,+\,\beta_{1}I_{1}+\beta_{2}G_{3}+\tilde{\beta}_{1}\tilde{I}_{1}+\tilde{\beta}_{2}\tilde{G}_{3}\bigg{]}\,,\end{split}$
(1.3)

where $\tilde{I}_{1}$ and $\tilde{G}_{3}$ are defined as $I_{1}$ and $G_{3}$
except for replacing one Riemann tensor by its dual
$\tilde{R}^{\mu\nu\alpha\beta}{=}(1/2)\,\epsilon^{\mu\nu\rho\sigma}{R_{\rho\sigma}}^{\alpha\beta}$
in both terms. Notably, the interactions $I_{1}$ and $G_{3}$ also appear in
the low-energy effective action of bosonic string theory, with the particular
values

$\beta_{1}=-\frac{2}{\kappa^{2}}\frac{\,\alpha^{\prime\,2}}{48}\,,\qquad\beta_{2}=-\frac{2}{\kappa^{2}}\frac{\,\alpha^{\prime\,2}}{24}\,,\qquad\tilde{\beta}_{1,2}=0,$
(1.4)

for the coefficients $\beta_{1,2}$ and $\tilde{\beta}_{1,2}$. An interesting
feature of the $G_{3}$ operator is that it is topological in six dimensions,
and has vanishing four-dimensional graviton amplitudes [119, 115]. It does
however lead to modifications of the classical potential of a two-body system
[112, 113], although, unlike $I_{1}$, it does not cause deflection of massless
particles [112, 114]. Furthermore, it can be reinterpreted as a tidal
deformation after performing an appropriate field redefinition [115]. We also
note recent interest in parity violation, following tentative indications in
the Cosmic Microwave Background and the large-scale structure of galaxies, see
e.g. [120, 121, 122, 123, 124, 125, 126, 127].

After describing what deformations we are interested in, we are left to
discuss how to investigate their effect on observables. We will do so using
the KMOC formalism [128], further developed for spinning objects in [34], and
soon after applied in [35] to tree-level scattering of black holes in EH for
arbitrary spin directions and to all orders in the black-hole multipole
expansions. In particular we will compute, to leading order in the cubic
couplings, the momentum and spin kicks – the variations in the momentum and
spin of one of the two spinning bodies after the interaction has occurred.
These quantities vanish at tree level, or first Post-Minkowskian order (1PM),
and receive a non-vanishing classical contribution at one loop (2PM) which we
derive. We obtain these quantities as functions of the masses, the spin
vectors, the impact parameter and the relative velocity of the two objects,
and work in the post-Minkowskian expansion, with $Gm_{1,2}{\ll}b$ and
$|\vec{a}_{1,2}|{\lesssim}Gm_{1,2}$, and hence $|\vec{a}_{1,2}|{\ll}b$.

It turns out that at leading order in the deformations, the momentum and spin
kicks can be extracted entirely from the Fourier transform to impact parameter
space of the classical one-loop amplitude of four heavy spinning objects. The
latter quantity is therefore what we have to compute. We will do so by
exploiting recent progress on classical amplitudes of spinning particles [129,
31, 130, 32, 33, 131, 132, 81], together with an application of one-loop
techniques [133, 30, 134] based on triple cuts [135] or the Heavy-mass
Effective Field Theory (HEFT) approach developed in [136, 13, 137, 138, 22].

The first ingredients needed for our calculation are the spinning graviton
Compton amplitudes in cubic gravity, which we derive making use of the energy-
momentum tensor for a Kerr black hole [29, 33, 32, 139, 41, 43] and the cubic
vertices arising from $I_{1}$ and $G_{3}$. The resulting Compton amplitudes
are non-vanishing only for equal-helicity gravitons (for outgoing momenta),
and are given in (2.13) and (2.23) for $I_{1}$ and $G_{3}$, respectively. One
notices the particularly simple form of the latter amplitude, which is
reflected in a correspondingly simpler result for the one-loop amplitude
(3.40), in comparison to the $I_{1}$ amplitude (3.35). The amplitudes from
parity-odd interactions can be derived without a new calculation, and are
closely related to those from the parity-even ones, see (5.7). The results for
the momentum and spin kicks, valid to all orders in the masses and spins of
the heavy objects, can be found in our Cubic Corrections to Spinning
Observables from Amplitudes GitHub repository. For the sake of presentation,
we limit ourselves to quoting the explicit expressions of the all-order in
spin momentum kick, the linear in spin correction to the momentum and spin
kicks, as well as the scattering angle for the case where the two black holes
have aligned spins. An interesting observation in the case of aligned spins is
that, for parity-odd interactions, the scattering is non-planar unlike for the
parity-even deformations.

The rest of the paper is organised as follows. In Section 2 we derive the
classical gravitational Compton amplitudes for Kerr black holes using the
corresponding classical energy-momentum tensor, together with the cubic
vertices induced by the new interactions. In Section 3, we use these
gravitational Compton amplitudes to derive the classical one-loop amplitudes
for the two-to-two scattering of Kerr black holes with different masses and
spins using triple cuts. In Section 4 we perform the same computation, this
time using the HEFT formalism. The derivation for the parity-odd interactions
$\tilde{I}_{1}$ and $\tilde{G}_{3}$ is presented in Section 5. In Section 6 we
use the KMOC approach and the classical one-loop amplitudes derived earlier to
compute the momentum and spin kicks to all orders in spin, and later examine
the aligned spin case in detail.

Throughout this paper we will suppress the coefficients of the cubic
interactions $\beta_{1},\beta_{2},\tilde{\beta}_{1},\tilde{\beta_{2}}$, which
can always be reinstated at the end.

##  2 Classical Compton amplitudes from cubic deformations

We begin by quoting the expressions of the new three-point graviton vertices
arising from $I_{1}$ and $I_{2}$, for gravitons of momenta $k_{i}$, $i=1,2,3$,
where gravitons $1$ and $2$ are on shell, and $3$ is off shell with momentum
$k_{3}=-k_{1}-k_{2}$. For convenience we will write them as if they appear
with coefficient $1$ in the action, i.e. for $\beta_{1}=\beta_{2}=1$; at the
end of the calculation these coupling constants can be set to any value of
interest, in particular to the choices (1.4) arising from bosonic string
theory.666We also mention that $R^{3}$ graviton amplitudes were first
considered in [140]. Throughout this section we focus on the parity-even
interactions, and will discuss the Compton amplitudes arising from the parity-
odd ones in Section 5.

We assume that the polarisation tensors have the factorised form
$\varepsilon_{k_{i}}^{\mu\nu}=\varepsilon_{k_{i}}^{\mu}\varepsilon_{k_{i}}^{\nu}$
and obey the usual transversality conditions
$\varepsilon_{k_{i}}{\cdot}k_{i}=0$. The linearised Riemann tensor of the on-
shell particles then has the very simple form

$\displaystyle
R_{i}^{\mu\nu\alpha\beta}={-}\left(\frac{\kappa}{2}\right)F_{i}^{\mu\nu}F_{i}^{\alpha\beta}\
\ \text{with}\ \
F_{i}^{\mu\nu}=k_{i}^{\mu}\varepsilon_{k_{i}}^{\nu}-k_{i}^{\nu}\varepsilon_{k_{i}}^{\mu}\
,\quad i=1,2\,,$ (2.1)

and, for the off-shell Riemann tensor,

$\displaystyle
R_{3}^{\mu\nu\alpha\beta}=-\left(\frac{\kappa}{2}\right)\Big{(}k_{3}^{\mu}k_{3}^{\beta}\,h_{3}^{\nu\alpha}+k_{3}^{\nu}k_{3}^{\alpha}\,h_{3}^{\mu\beta}-k_{3}^{\nu}k_{3}^{\beta}\,h_{3}^{\mu\alpha}-k_{3}^{\mu}k_{3}^{\alpha}\,h_{3}^{\nu\beta}\Big{)}\,,$
(2.2)

where $h_{3}^{\mu\nu}$ is an off-shell graviton with momentum $k_{3}$. The two
vertices arising from an insertion of $I_{1}$ or $I_{2}$ can then be written
in manifestly gauge invariant form as777Our convention for symmetrisation is
$A_{(\mu\nu)}=\frac{1}{2}(A_{\mu\nu}+A_{\nu\mu})$, and all momenta are assumed
to be outgoing in this section.

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}=-i\,3!\left(\frac{\kappa}{2}\right)^{3}\,4\,\text{Tr}(F_{1}{\cdot}F_{2})(k_{2}{\cdot}F_{1})^{(\mu}(k_{1}{\cdot}F_{2})^{\nu)}\
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}=i\,3!\left(\frac{\kappa}{2}\right)^{3}\,(k_{3}{\cdot}[F_{1},F_{2}])^{\mu}(k_{3}{\cdot}[F_{1},F_{2}])^{\nu}\
.$ (2.4)

In the rest of the paper we will pick the gauge where
$k_{1}{\cdot}\varepsilon_{k_{2}}=k_{2}{\cdot}\varepsilon_{k_{1}}=0$. In this
gauge, the two vertices are

$\displaystyle V_{I_{1}}^{\mu\nu}(k_{1},k_{2})$
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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=i\,3!\left(\frac{\kappa}{2}\right)^{3}\,8(k_{1}{\cdot}k_{2})^{3}\,(\varepsilon_{k_{1}}{\cdot}\varepsilon_{k_{2}})\varepsilon^{(\mu}_{k_{1}}\varepsilon_{k_{2}}^{\
\nu)}\ ,$ (2.5) $\displaystyle V_{I_{2}}^{\mu\nu}(k_{1},k_{2})$
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=i\,3!\left(\frac{\kappa}{2}\right)^{3}\,(\varepsilon_{k_{1}}{\cdot}\varepsilon_{k_{2}})^{2}(k_{1}{\cdot}k_{2})^{2}(k_{1}-k_{2})^{\mu}(k_{1}-k_{2})^{\nu}\
\,.$ (2.6)

In spinor-helicity variables the vertices become, for the $1^{++}2^{++}$
helicity configuration,

$\displaystyle V_{I_{1}}^{\mu\nu}(k_{1}^{++},k_{2}^{++})$
$\displaystyle=-i\frac{3!}{4}\left(\frac{\kappa}{2}\right)^{3}\,[1\,2]^{4}\Big{(}\langle
2|\mu|1]\langle 1|\nu|2]+\mu\leftrightarrow\nu\Big{)}\,\ ,$ (2.7)
$\displaystyle V_{I_{2}}^{\mu\nu}(k_{1}^{++},k_{2}^{++})$
$\displaystyle=i\frac{3!}{4}\left(\frac{\kappa}{2}\right)^{3}\,[1\,2]^{4}(k_{1}{\cdot}k_{2})^{2}(k_{1}-k_{2})^{\mu}(k_{1}-k_{2})^{\nu}\
,$ (2.8)

while
$V_{I_{1}}(k_{1}^{\pm\pm},k_{2}^{\mp\mp})=V_{I_{2}}(k_{1}^{\pm\pm},k_{2}^{\mp\mp})=0$.

In order to build the classical Compton amplitudes induced by the cubic
interactions, we contract the three-point vertices above with888The factor of
$-i$ is chosen in such a way that the spinless ($a^{\mu}=0$) limit of the
corresponding amplitude matches the amplitude obtained from the three-point
scalar-scalar-graviton vertex  $\displaystyle
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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i\left(\frac{\kappa}{2}\right)\Big{[}-\eta^{\mu\nu}(p_{1}{\cdot}p_{2}+m^{2})+p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\nu}p_{1}^{\mu}\Big{]}\
,$ (2.9) where in the classical limit $p_{1}\to p$ and $p_{2}\to-p$.
$-iT^{\mu\nu}$, where $T^{\mu\nu}$ is the linearised energy-momentum tensor
for a classical Kerr black hole of mass $m$, momentum $p^{\mu}$ and spin
$S^{\mu}\coloneq m\,a^{\mu}$ [29, 32, 139, 41, 43],

$\displaystyle
T^{\mu\nu}=\left(\frac{\kappa}{2}\right)2\left[\cosh(a{\cdot}q)p^{\mu}p^{\nu}-i\frac{\sinh(a{\cdot}q)}{a{\cdot}q}p^{(\mu}\epsilon^{\nu)}_{\
\alpha\beta\gamma}p^{\alpha}a^{\beta}q^{\gamma}\right]\,,$ (2.10)

where $q=k_{1}+k_{2}$. We refer to $a^{\mu}$ as the ring radius vector [55].
For a Kerr black hole in the rest frame and spinning around the $z$-axis we
have $a^{\mu}=(0,0,0,a)$, where $a$ is the radius of the ring singularity; the
spin vector $S^{\mu}$ is related to the spin tensor as

$\displaystyle
S^{\mu}=\frac{1}{2m}\epsilon^{\mu\nu\alpha\beta}p_{\nu}S_{\alpha\beta}\ .$
(2.11)

Note that the ring radius satisfies

$\displaystyle p{\cdot}a(p)=0\,.$ (2.12)

Using (2.10) and the standard de Donder propagator, one obtains the following
spinning Compton amplitudes:

$\displaystyle\begin{split}M_{I_{1}}(p,k_{1},k_{2})&=i\,\left(\frac{\kappa}{2}\right)^{4}24\,(k_{1}{\cdot}k_{2})^{2}(\varepsilon_{k_{1}}{\cdot}\varepsilon_{k_{2}})\bigg{\\{}\cosh(a{\cdot}q)\bigg{[}2(p{\cdot}\varepsilon_{k_{1}})(p{\cdot}\varepsilon_{k_{2}})-m^{2}(\varepsilon_{k_{1}}{\cdot}\varepsilon_{k_{2}})\Big{]}\\\
&-i\frac{\sinh(a{\cdot}q)}{a{\cdot}q}\Big{[}(p{\cdot}\varepsilon_{k_{1}})\epsilon(\varepsilon_{k_{2}}paq)+(p{\cdot}\varepsilon_{k_{2}})\epsilon(\varepsilon_{k_{1}}paq)\Big{]}\bigg{\\}}\,,\end{split}$
(2.13)

and

$\displaystyle\begin{split}M_{I_{2}}(p,k_{1},k_{2})&=i\,\left(\frac{\kappa}{2}\right)^{4}6\,(k_{1}{\cdot}k_{2})\,(\varepsilon_{k_{1}}{\cdot}\varepsilon_{k_{2}})^{2}\bigg{\\{}\cosh(a{\cdot}q)\Big{[}\big{(}p{\cdot}(k_{1}-k_{2})\big{)}^{2}+m^{2}(k_{1}{\cdot}k_{2})\Big{]}\\\
&-2i\,\frac{\sinh(a{\cdot}q)}{a{\cdot}q}p{\cdot}(k_{1}-k_{2})\
\epsilon(k_{1}k_{2}pa)\Big{]}\bigg{\\}}\,,\end{split}$ (2.14)

and we note that

$\displaystyle
M_{I_{1}}(p,k_{1}^{\pm\pm},k_{2}^{\mp\mp})=M_{I_{2}}(p,k_{1}^{\pm\pm},k_{2}^{\mp\mp})=0\
.$ (2.15)

The second line of (2.13) can be rewritten without Levi-Civita symbols using
the relation

$\displaystyle(p{\cdot}\varepsilon_{k_{1}}^{\pm})\epsilon(\varepsilon_{k_{2}}^{\pm}paq)+(p{\cdot}\varepsilon_{k_{2}}^{\pm})\epsilon(\varepsilon_{k_{1}}^{\pm}paq)=\mp
ip{\cdot}(k_{1}-k_{2})\Big{[}(p{\cdot}\varepsilon_{k_{1}}^{\pm})(a{\cdot}\varepsilon_{k_{2}}^{\pm})-(p{\cdot}\varepsilon_{k_{2}}^{\pm})(a{\cdot}\varepsilon_{k_{1}}^{\pm})\Big{]}\,,$
(2.16)

which can be proven via the identities

$\displaystyle\begin{split}\epsilon(\varepsilon^{\pm}_{k_{1}}paq)&=\mp
i\Big{[}a{\cdot}(k_{1}-k_{2})(p{\cdot}\varepsilon^{\pm}_{k_{1}})-p{\cdot}(k_{1}-k_{2})(a{\cdot}\varepsilon^{\pm}_{k_{1}})\Big{]}\
,\\\ \epsilon(\varepsilon^{\pm}_{k_{2}}paq)&=\pm
i\Big{[}a{\cdot}(k_{1}-k_{2})(p{\cdot}\varepsilon^{\pm}_{k_{2}})-p{\cdot}(k_{1}-k_{2})(a{\cdot}\varepsilon^{\pm}_{k_{2}})\Big{]}\
.\end{split}$ (2.17)

In order to arrive at (2.17) we have used

$\displaystyle\varepsilon_{k_{1}}^{+}$
$\displaystyle=-\sqrt{2}\,\frac{\lambda_{2}\tilde{\lambda}_{1}}{\langle
1\,2\rangle}\,,\qquad\varepsilon_{k_{2}}^{+}=\sqrt{2}\,\frac{\lambda_{1}\tilde{\lambda}_{2}}{\langle
1\,2\rangle}\,,$ (2.18) $\displaystyle\varepsilon_{k_{1}}^{-}$
$\displaystyle=\sqrt{2}\,\frac{\lambda_{1}\tilde{\lambda}_{2}}{[1\,2]}\,,\qquad\
\
\,\varepsilon_{k_{2}}^{-}=-\sqrt{2}\,\frac{\lambda_{2}\tilde{\lambda}_{1}}{[1\,2]}\,,$
(2.19)

and the identity

$\displaystyle\epsilon(abcd)=\frac{i}{4}\langle a|bcd-dcb|a]\,,$ (2.20)

valid if $a$ is a null vector.999We follow the spinor conventions of [141]. We
can then further rewrite (2.13) and (2.14) in spinor-helicity form:

$\displaystyle\begin{split}M_{I_{1}}(p,k_{1}^{++},k_{2}^{++})&=i\left(\frac{\kappa}{2}\right)^{4}3!\frac{[1\,2]^{4}}{q^{2}}\bigg{\\{}-4\cosh(a{\cdot}q)\
(p{\cdot}k_{1})(p{\cdot}k_{2})\,\\\
&+\frac{1}{2}p{\cdot}(k_{1}-k_{2})\frac{\sinh(a{\cdot}q)}{a{\cdot}q}\Big{(}[1|p|2\rangle[2|a|1\rangle-[2|p|1\rangle[1|a|2\rangle\Big{)}\bigg{\\}}\,,\end{split}$
(2.21)

and

$\displaystyle\begin{split}M_{I_{2}}(p,k_{1}^{++},k_{2}^{++})&=\frac{i}{2}\left(\frac{\kappa}{2}\right)^{4}\,3!\,\frac{[1\,2]^{4}}{q^{2}}\bigg{\\{}\cosh(a{\cdot}q)\Big{[}\big{(}p{\cdot}(k_{1}-k_{2})\big{)}^{2}+m^{2}(k_{1}{\cdot}k_{2})\Big{]}\\\
&+\frac{1}{2}p{\cdot}(k_{1}-k_{2})\frac{\sinh(a{\cdot}q)}{a{\cdot}q}\Big{(}[1|p|2\rangle[2|a|1\rangle-[2|p|1\rangle[1|a|2\rangle\Big{)}\bigg{\\}}\
,\end{split}$ (2.22)

and similarly for the amplitudes with two negative-helicity gravitons.

Two comments are in order here. First, we can now write the $G_{3}$ amplitude
$M_{G_{3}}=M_{I_{1}}-2M_{I_{2}}$, getting

$\displaystyle
M_{G_{3}}(p,k_{1}^{++},k_{2}^{++})=-3\,i\,\left(\frac{\kappa}{2}\right)^{4}\cosh(a{\cdot}q)[1\,2]^{4}\left(m^{2}+\frac{q^{2}}{2}\right)\,,$
(2.23)

and observe the remarkable feature that it does not contain a term
proportional to $\sinh(a{\cdot}q)/a{\cdot}q$. Also note that the $q^{2}/2$
term can be dropped in the classical limit. Second, in the spinless limit
these expressions agree with those derived in [112], up to a factor of
$(-2/\kappa^{2})(\alpha^{\prime}/4)^{2}$ which was introduced there to
normalise the $I_{1}$ and $G_{3}$ couplings as they appear in the low-energy
effective action of the bosonic string. In the spinless case, one has

$\displaystyle\begin{split}\left.M_{I_{1}}(p,k_{1}^{++},k_{2}^{++})\right|_{a=0}&=-4i\left(\frac{\kappa}{2}\right)^{4}3!\
\frac{[1\,2]^{4}}{q^{2}}(p{\cdot}k_{1})(p{\cdot}k_{2})\,,\\\
\left.M_{I_{2}}(p,k_{1}^{++},k_{2}^{++})\right|_{a=0}&=\frac{i}{2}3!\left(\frac{\kappa}{2}\right)^{4}\frac{[1\,2]^{4}}{q^{2}}\Big{[}\big{(}p{\cdot}(k_{1}-k_{2})\big{)}^{2}+m^{2}(k_{1}{\cdot}k_{2})\Big{]}\
,\\\
\left.M_{I_{1}}(p,k_{1}^{\pm\pm},k_{2}^{\mp\mp})\right|_{a=0}&=\left.M_{I_{2}}(p,k_{1}^{\pm\pm},k_{2}^{\mp\mp})\right|_{a=0}=0\
.\end{split}$ (2.24)

##  3 One-loop spinning amplitudes from leading singularities

In this section we study the two-to-two scattering process of Kerr black holes
in the presence of cubic deformations of gravity. There is no tree-level
contribution to this process, hence the leading term requires a one-loop
computation. Here we will perform this calculation using leading
singularities, in two ways: first, using the approach of [133, 30], slightly
adapted here to the spinning case;101010See also [116] for the case of two
spinning/two non-spinning particles. we will then revisit the computation
using the parameterisation of [134], which offers certain advantages compared
to the previous one.

###  3.1 Kinematics

Our spinning particles have masses $m_{1}$ and $m_{2}$ and spin vectors
$a_{1}$ and $a_{2}$,

$p_{1}=\bar{p}_{1}-\dfrac{q}{2}$$p_{2}=\bar{p}_{2}+\dfrac{q}{2}$$p_{2}^{\prime}=\bar{p}_{2}-\dfrac{q}{2}$$\
\ p_{1}^{\prime}=\bar{p}_{1}+\dfrac{q}{2}$ (3.1)

with

$\displaystyle p_{1}^{2}=(p_{1}^{\prime})^{2}=m_{1}^{2}\,,\qquad
p_{2}^{2}=(p_{2}^{\prime})^{2}=m_{2}^{2}\,.$ (3.2)

It is often convenient to introduce the barred variables [142, 9]

$\displaystyle\begin{split}p_{1}&=\bar{p}_{1}-\frac{q}{2}\,,\qquad
p_{1}^{\prime}=\bar{p}_{1}+\frac{q}{2}\,,\\\
p_{2}&=\bar{p}_{2}+\frac{q}{2}\,,\qquad
p_{2}^{\prime}=\bar{p}_{2}-\frac{q}{2}\,,\end{split}$ (3.3)

which satisfy

$\displaystyle\bar{p}_{1}{\cdot}q=\bar{p}_{2}{\cdot}q=0\,,$ (3.4)

and $\bar{p}_{i}^{2}=\bar{m}_{i}^{2}$, with

$\bar{m}_{i}=\sqrt{m_{i}^{2}-\frac{q^{2}}{4}}=m_{i}-\frac{q^{2}}{4m_{i}}+\mathcal{O}(m_{i}^{-2})\,.$
(3.5)

We also introduce four-velocities $u_{1}$ and $u_{2}$ as

$\displaystyle p_{1}\coloneq m_{1}u_{1}\,,\qquad p_{2}\coloneq m_{2}u_{2}\,,$
(3.6)

with $u_{1}^{2}=u_{2}^{2}=1$, as well as the Lorentz factor

$\displaystyle\sigma\coloneqq
u_{1}{\cdot}u_{2}\coloneq\frac{1}{\sqrt{1-v^{2}}}\ ,$ (3.7)

where $v$ is the velocity of one particle in the rest frame of the other. In
the rest frame of particle $p_{1}$ the four-velocities have the form
$u_{1}^{\mu}=(1,0,0,0)$ and $u_{2}^{\mu}=(\sigma,0,0,v\,\sigma)$.

###  3.2 Introducing the triple cut

To begin, we note that the tree-level three-point amplitudes with two spinning
particles and one graviton remains unchanged, hence the first $R^{3}$
corrections can only enter the four-point spinning amplitude at one loop
through the $R^{3}$ Compton amplitudes. The relevant one-loop triple cut
(leading singularity) is then shown in Figure 1; there is an additional
diagram where the two types of particle are swapped. The top amplitude is one
of the Compton amplitudes induced by the cubic interactions, as derived in
Section 2.

$(L)+(180:1)$$p_{1}$$(R)+(0:1)$$p_{1}^{\prime}=p_{1}+q$$(top)+(180:1.5)$$p_{2}$$(top)+(0:1.5)$$p_{2}^{\prime}=p_{2}-q$$\ell_{1}$$\ell_{2}$$L=p_{1}+\ell_{1}$$M_{I}$$M_{3}$$M_{3}$
Figure 1: The leading singularity triangle diagram
$\mathcal{M}^{\bigtriangleup}$, where the bottom (top) particles have mass
$m_{1}$ ($m_{2}$). $M_{I}$ represents the Compton amplitude in the presence of
a deformation $I$ with $I\in(I_{1},I_{2},G_{3})$.

The massive-line cut in the figure implies a sum over the polarisation states
of the exchanged massive particle. This is usually realised by inserting a
physical-state projector in between the two three-point amplitudes. For
instance, consider the finite-spin three-point amplitudes given below, with
two massive spin-$s$ states and one positive-helicity graviton,

$\displaystyle\begin{split}&M_{3}(-p_{1},L,-\ell_{1}^{++})=-i\kappa\frac{\langle\bm{p_{1}L}\rangle^{2s}}{m_{1}^{2s}}(\varepsilon_{-\ell_{1}}^{+}{\cdot}p_{1})^{2}\,,\qquad\\\
&M_{3}(-L,p_{1}^{\prime},-\ell_{2}^{++})=-i\kappa\frac{\langle\bm{Lp_{1}^{\prime}}\rangle^{2s}}{m_{1}^{2s}}(\varepsilon_{-\ell_{2}}^{+}{\cdot}p_{1}^{\prime})^{2}\,,\end{split}$
(3.8)

where the bold letters denote massive spinors defined as

$\displaystyle|\bm{p}\rangle\coloneq z_{a}^{(p)}|p^{a}\rangle\,,$ (3.9)

where $a=1,2$ is the $SU(2)$ little-group index and $z_{a}^{(p)}$ is a
polarisation variable. The massive factorisation cut in terms of these
amplitudes is

$-\kappa^{2}\frac{(\langle\bm{p_{1}}L_{a}\rangle\langle
L^{a}\bm{p_{1}^{\prime}}\rangle)^{2s}}{m_{1}^{4s}}(\varepsilon_{-\ell_{1}}^{+}\cdot
p_{1})^{2}(\varepsilon_{-\ell_{2}}^{+}\cdot
p_{1}^{\prime})^{2}=-\kappa^{2}\frac{\langle\bm{p_{1}}\bm{p_{1}^{\prime}}\rangle^{2s}}{m_{1}^{2s}}(\varepsilon_{-\ell_{1}}^{+}\cdot
p_{1})^{2}(\varepsilon_{-\ell_{2}}^{+}\cdot p_{1}^{\prime})^{2},$ (3.10)

where $|L_{a}\rangle\langle L^{a}|=\epsilon_{ab}|L^{b}\rangle\langle L^{a}|$
is the state sum for the exchanged particle $L$, and $\epsilon_{ab}$ is the
two-dimensional Levi-Civita tensor. Note that in arriving at the right-hand
side of (3.10) we have used $\langle\bm{p_{1}}L_{a}\rangle\langle
L^{a}\bm{p_{1}^{\prime}}\rangle=m_{1}\langle\bm{p_{1}}\bm{p_{1}^{\prime}}\rangle$.
To compute the classical limit of (3.10), we use the methods outlined in [67]
to obtain

$-\kappa^{2}e^{\mp(\ell_{1}+\ell_{2}){\cdot}a_{1}}(\varepsilon_{-\ell_{1}}^{\pm}{\cdot}p_{1})^{2}(\varepsilon_{-\ell_{2}}^{\pm}{\cdot}p_{1}^{\prime})^{2},$
(3.11)

where $a_{1}$ is the spin vector of particle 1. Note that we have included the
case of two negative-helicity gravitons, obtained from (3.10) by conjugation.
Rewriting
$e^{\mp(\ell_{1}+\ell_{2}){\cdot}a_{1}}=e^{\mp\ell_{1}{\cdot}a_{1}}e^{\mp\ell_{2}{\cdot}a_{1}}$,
we see that (3.11) is just the product of two classical three-point
amplitudes; hence, the state sum trivialises in the classical limit.
Summarising, we can equivalently make use of the classical three-point
amplitudes with one graviton of momentum $\ell$ and two massive particles with
momenta $p$ and $-(p+\ell)$ and spin vector $a$, which are [32]

$\displaystyle
M_{3}(p,\ell^{++})=-i\left(\frac{\kappa}{2}\right)e^{a{\cdot}\ell}\left(\frac{\langle\xi|p|\ell]}{\langle\xi\ell\rangle}\right)^{2}\,,\qquad
M_{3}(p,\ell^{--})=-i\left(\frac{\kappa}{2}\right)e^{-a{\cdot}\ell}\left(\frac{\langle\ell|p|\xi]}{[\ell\xi]}\right)^{2}\,,\qquad$
(3.12)

where $\xi$ and $\tilde{\xi}$ are reference spinors.

###  3.3 Leading singularities with spinor parameterisation

####  3.3.1 Parameterisation of the momenta and integration measure

In this approach, one introduces the following parameterisation of the massive
momenta $p_{1}$ and $p_{1}^{\prime}$ in Figure 1 [30],

$\displaystyle p_{1}$
$\displaystyle=\lambda\tilde{\eta}+\eta\tilde{\lambda}\,,\qquad
p_{1}^{\prime}=\lambda\tilde{\lambda}+\beta\,\lambda\tilde{\eta}+\beta^{-1}\,\eta\tilde{\lambda}\
.$ (3.13)

Then

$\displaystyle
p_{1}^{2}=(p_{1}^{\prime})^{2}=\langle\lambda\eta\rangle[\tilde{\lambda}\tilde{\eta}]=m_{1}^{2}\,,$
(3.14)

which one can solve by setting
$\langle\lambda\eta\rangle=[\tilde{\lambda}\tilde{\eta}]=m_{1}$. The momentum
transfer is

$\displaystyle
q\coloneq\ell_{1}+\ell_{2}=p_{1}^{\prime}-p_{1}=\lambda\tilde{\lambda}+(\beta-1)\lambda\tilde{\eta}-\frac{\beta-1}{\beta}\eta\tilde{\lambda}\
,$ (3.15)

with

$\displaystyle q^{2}=-m_{1}^{2}\frac{(\beta-1)^{2}}{\beta}\ ,$ (3.16)

hence the limit $q^{2}\to 0$ can be taken by sending $\beta\to 1$. The loop
momentum $L$ of Figure 1 can be parameterised as

$\displaystyle L=z\ell+\omega\lambda\tilde{\lambda}\,,$ (3.17)

where the null momentum $\ell$ can be expanded as

$\displaystyle\ell=A\,\lambda\tilde{\eta}+B\,\eta\tilde{\lambda}+AB\,\lambda\tilde{\lambda}+\eta\tilde{\eta}\
.$ (3.18)

After imposing the triple-cut conditions

$\displaystyle\ell_{1}^{2}=\ell_{2}^{2}=L^{2}-m_{1}^{2}=0\,,$ (3.19)

there is only one integration variable left, which can be taken to be
$y=-z/(\beta-1)^{2}$. The cut loop momenta $\ell_{1}$ and $\ell_{2}$ can then
be solved as [30]

$\displaystyle\begin{split}\ell_{1}=&\frac{1}{\beta+1}\Big{[}\frac{1+\beta
y}{y}\lambda-(\beta^{2}-1)\eta\Big{]}\times\frac{1}{\beta+1}\Big{[}(1+\beta
y)\tilde{\lambda}+y(\beta^{2}-1)\tilde{\eta}\Big{]}\,,\\\
\ell_{2}=&\frac{i}{\beta+1}\Big{[}\frac{1-y}{y}\lambda+\frac{\beta^{2}-1}{\beta}\eta\Big{]}\times\frac{i}{\beta+1}\Big{[}(1-\beta^{2}y)\tilde{\lambda}-y\beta(\beta^{2}-1)\tilde{\eta}\Big{]}\,.\end{split}$
(3.20)

The one-loop integration measure,

$\displaystyle\frac{d^{4}L}{(2\pi)^{4}}\frac{1}{\ell_{1}^{2}}\frac{1}{\ell_{2}^{2}}\frac{1}{L^{2}-m_{1}^{2}}\
,$ (3.21)

then becomes, after integrating out $\omega$, $A$ and $B$,

$\displaystyle d\mathfrak{m}_{\rm eff}=\frac{1}{32\pi
m_{1}^{2}}\frac{\beta}{\beta^{2}-1}\ \frac{dy}{y}\ .$ (3.22)

We have normalised this effective measure in such a way that by integrating it
and picking the residue at $y=\infty$ one gets the expected result for the
classical part of a scalar triangle, that is111111Here we have exploited the
nice fact that the classical part of the triangle integral is simply $-1/4$
times the result of its leading singularity [133, 32].

$\displaystyle\ointctrclockwise_{y=\infty}\\!d\mathfrak{m}_{\rm eff}=2\pi
i\,{\rm Res}\left[\frac{1}{32\pi m_{1}^{2}}\frac{\beta}{\beta^{2}-1}\
\frac{1}{y}\right]_{y=\infty}=-\frac{i}{32\,m_{1}\sqrt{-q^{2}}}+\mathcal{O}(1)\
,$ (3.23)

after using (3.16).

####  3.3.2 Taking the classical limit

In the absence of spin, the classical limit can simply be taken by expanding
around $q^{2}=0$, or equivalently $\beta=1$ (this is know as the holomorphic
classical limit, or HCL [30]). In the presence of spin, the correct classical
limit can be obtained by recalling that $q$ and $a$ scale with $\hbar$ as

$\displaystyle(q,a)\to(\hbar q,\hbar^{-1}a)\ ,$ (3.24)

and then expanding in $\hbar$. The result of the leading singularity
calculation is conveniently expressed in terms of the following variables:

$\displaystyle\begin{split}u&\coloneqq\langle\eta|p_{2}|\lambda]\,,\qquad
v\coloneqq\langle\lambda|p_{2}|\eta]\,,\\\
w&\coloneqq\langle\eta|a|\lambda]\,,\qquad\
z\coloneqq\langle\lambda|a|\eta]\,,\end{split}$ (3.25)

where

$\displaystyle u+v=2p_{1}{\cdot}p_{2}\coloneq 2m_{1}m_{2}\,\sigma\,,\qquad
u\,v=m_{1}^{2}m_{2}^{2}+\langle\eta|p_{1}|\eta]\langle\lambda|p_{1}|\lambda]\,,$
(3.26)

where

$\displaystyle\langle\lambda|p_{2}|\lambda]=2(q{\cdot}p_{2})+(\beta-1)\Big{(}\frac{u}{\beta}-v\Big{)}\,,\qquad\langle\lambda|a|\lambda]=2(q{\cdot}a)+(\beta-1)\Big{(}\frac{w}{\beta}-z\Big{)}\,,$
(3.27)

and $\sigma$ was introduced in (3.7). We can then implement the rescaling
(3.24) on these variables as

$\displaystyle\begin{split}(\lambda,\tilde{\lambda})&\to\hbar^{\frac{1}{2}}(\lambda,\tilde{\lambda})\,,\quad(\eta,\tilde{\eta})\to\hbar^{-\frac{1}{2}}(\eta,\tilde{\eta})\,,\quad
p{\cdot}q\to\hbar^{2}p{\cdot}q\,,\\\
(u,v)&\to(u,v)\,,\qquad(w,z)\to\hbar^{-1}(w,z)\,,\quad\beta-1\to\hbar(\beta-1)\,.\end{split}$
(3.28)

An important comment is in order here. Simply counting powers of $\beta{-}1$
would discard terms such as
$(\beta{-}1)\langle\lambda|p_{2}|\lambda]\langle\eta|a|\eta]$ compared to
terms such as $u\langle\lambda|a|\lambda]$, though they scale identically in
$\hbar$. Expanding in $\hbar$ (and not in $\beta{-}1$) we can then capture
fully the classical contribution. Related issues of the $\beta\to 1$
prescription in the presence of spin are discussed in [134].

####  3.3.3 Leading singularities for $I_{1}$

There are two triple cuts to consider, where the helicities of the internal
gravitons are $(++,++)$ or $(--,--)$. There is no $(++,--)$ internal helicity
assignment as the corresponding Compton amplitude vanishes. The leading
singularities for these two internal helicity assignments are given by

$\displaystyle\begin{split}\mathcal{M}_{I_{1}}^{\bigtriangleup,(I)}&=\ointctrclockwise_{y=\infty}\\!d\mathfrak{m}_{\rm
eff}\
M_{I_{1}}(p_{2},\ell_{1}^{++},\ell_{2}^{++})M_{3}(p_{1}^{\prime},-\ell_{2}^{--})M_{3}(-p_{1},-\ell_{1}^{--})\,,\\\
\mathcal{M}_{I_{1}}^{\bigtriangleup,(II)}&=\ointctrclockwise_{y=\infty}\\!d\mathfrak{m}_{\rm
eff}\
M_{I_{1}}(p_{2},\ell_{1}^{--},\ell_{2}^{--})M_{3}(p_{1}^{\prime},-\ell_{2}^{++})M_{3}(-p_{1},-\ell_{1}^{++})\,,\end{split}$
(3.29)

which requires the cubic gravity Compton amplitude given in (2.21), its parity
conjugate expression, and the three-point amplitudes in (3.12). Note that the
exponential factors contained in these three-point amplitudes combine to
$e^{\pm a_{1}{\cdot}(\ell_{1}+\ell_{2})}=e^{\pm a_{1}{\cdot}q}$ in the
first/second triple cut in (3.29).

The triple cuts in (3.29) will now be evaluated on the cut using the
parameterisation described in Section 3.3.1. Any instance of the internal
graviton momenta $\ell_{1}$ and $\ell_{2}$ can be written in terms of the
$\lambda$ and $\eta$ spinors, meaning everything can be expressed using the
spinor traces (3.25). For instance,

$\langle\ell_{1}\ell_{2}\rangle=i\frac{\beta-1}{\beta}\frac{m_{1}}{y}\,,\qquad[\ell_{1}\ell_{2}]=-i(\beta-1)y\,m_{1}\,.$
(3.30)

This highlights the fact that angle and square brackets are not simply related
by complex conjugation, since we are working with complex momenta.

It turns out that the integrands (and hence leading singularities) for the two
internal helicity assignments are identical, hence the two exponential factors
combine as $e^{a_{1}{\cdot}q}+e^{-a_{1}{\cdot}q}=2\cosh{(a_{1}{\cdot}q)}$. We
then take the residue at $y=\infty$, and take the leading-in-$\hbar$
contribution using the scalings (3.28). The first contribution is at
$\mathcal{O}(\hbar^{3})$, for which we get121212We used the SpinorHelicity4D
package [143] for some spinor manipulations.

$\displaystyle\mathcal{M}_{I_{1}}^{\bigtriangleup}=\cosh(a_{1}{\cdot}q)\Big{(}\mathcal{M}_{I_{1}}^{\bigtriangleup}|_{\cosh{}}+\mathcal{M}_{I_{1}}^{\bigtriangleup}|_{\sinh{}}\Big{)}\,,$
(3.31)

with

$\displaystyle\mathcal{M}_{I_{1}}^{\bigtriangleup}|_{\cosh{}}={}$
$\displaystyle
i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{64}m_{1}^{4}(\beta-1)^{3}\left[(u+v)^{2}-4m_{1}^{2}m_{2}^{2}\right]\cosh{(a_{2}{\cdot}q)}\,,$
(3.32) $\displaystyle\mathcal{M}_{I_{1}}^{\bigtriangleup}|_{\sinh{}}={}$
$\displaystyle-i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{128}m_{1}^{4}(\beta-1)^{3}\frac{\sinh{(a_{2}{\cdot}q)}}{a_{2}{\cdot}q}(u+v)\Big{[}(u-v)\langle\lambda|a_{2}|\lambda]$
$\displaystyle+$
$\displaystyle(z-w)\langle\lambda|p_{2}|\lambda]+(\beta-1)\Big{(}\langle\eta|p_{2}|\eta]\langle\lambda|a_{2}|\lambda]-\langle\eta|a_{2}|\eta]\langle\lambda|p_{2}|\lambda]\Big{)}\Big{]}\,.$
(3.33)

Notice that here it is apparent that the classical $\hbar$ expansion does not
match the $(\beta-1)$ expansion when spin is included, as the $\sinh$ term
contains both a term proportional to $(\beta-1)^{3}$ and $(\beta-1)^{4}$,
which however are of the same order in $\hbar$.

The $\sinh$ term can be simplified using the identity

$\displaystyle-4i\epsilon(p_{1}p_{2}a_{2}q)={}$
$\displaystyle(u-v)\langle\lambda|a_{2}|\lambda]+(z-w)\langle\lambda|p_{2}|\lambda]$
$\displaystyle+2(\beta-1)\Big{(}\langle\eta|p_{2}|\eta]\langle\lambda|a_{2}|\lambda]-\langle\eta|a_{2}|\eta]\langle\lambda|p_{2}|\lambda]\Big{)}+\mathcal{O}(\hbar)\,,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
(3.34)

which can be derived using (2.20). Using then (3.26) to rewrite the remaining
spinor traces in terms of familiar kinematic variables, we can recast the
$\mathcal{M}_{I_{1}}^{\bigtriangleup}$ leading singularity in the compact form

$\mathcal{M}_{I_{1}}^{\bigtriangleup}=i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{16}m_{1}^{3}m_{2}^{2}\,|q|^{3}\cosh(a_{1}{\cdot}q)\left[\left(\sigma^{2}-1\right)\cosh(a_{2}{\cdot}q)-i\sigma\epsilon(u_{1}u_{2}a_{2}q)\frac{\sinh(a_{2}{\cdot}q)}{a_{2}{\cdot}q}\right]\,,$
(3.35)

where we have defined

$\displaystyle|q|\coloneq\sqrt{-q^{2}}\,,$ (3.36)

with $q^{2}{<}0$. Finally, to obtain the complete one-loop result we need to
add a term $\mathcal{M}_{I_{1}}^{\bigtriangledown}$ obtained from
$\mathcal{M}_{I_{1}}^{\bigtriangleup}$ by performing the the simultaneous
replacements131313One should also swap $u_{1}\leftrightarrow u_{2}$ and
$q\leftrightarrow-q$ but these two changes compensate each other.
$m_{1}\leftrightarrow m_{2}$ and $a_{1}\leftrightarrow a_{2}$,

$\displaystyle\mathcal{M}_{I_{1}}^{\bigtriangledown}\coloneq\left.\mathcal{M}_{I_{1}}^{\bigtriangleup}\right|_{m_{1}\leftrightarrow
m_{2},a_{1}\leftrightarrow a_{2}}\,.$ (3.37)

Doing so we arrive at the final answer

$\displaystyle\mathcal{M}_{I_{1}}=\mathcal{M}_{I_{1}}^{\bigtriangleup}+\mathcal{M}_{I_{1}}^{\bigtriangledown}\,,$
(3.38)

for the one-loop classical amplitude. This quantity will be the main
ingredient to compute the momentum and spin kicks in Section 6.

####  3.3.4 Leading singularities for $I_{2}$ and $G_{3}$

The calculation for the $I_{2}$ and $G_{3}$ leading singularities is very
similar to the case of $I_{1}$, with the relevant cubic gravity Compton
amplitudes for the $I_{2}$ and $G_{3}$ cases now being given in (2.22) and
(2.23), respectively. Proceeding as before, we arrive at the following
results:

$\displaystyle\mathcal{M}_{I_{2}}^{\bigtriangleup}$
$\displaystyle=i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{32}m_{1}^{3}m_{2}^{2}\,|q|^{3}\cosh(a_{1}{\cdot}q)\left[\sigma^{2}\,\cosh(a_{2}{\cdot}q)-i\sigma\epsilon(u_{1}u_{2}a_{2}q)\frac{\sinh(a_{2}{\cdot}q)}{a_{2}{\cdot}q}\right]\,,$
(3.39) $\displaystyle\mathcal{M}_{G_{3}}^{\bigtriangleup}$
$\displaystyle=-i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{16}m_{1}^{3}m_{2}^{2}\,|q|^{3}\cosh(a_{1}{\cdot}q)\,\cosh(a_{2}{\cdot}q)\,.$
(3.40)

Note the interesting feature of the classical one-loop amplitude for the
$G_{3}$ case, which is free of a $\sinh$ term (as the Compton amplitude
(2.23)). Furthermore, we can obtain the $G_{3}$ amplitude from the $I_{1}$
amplitude by (formally) setting ${\sigma}\rightarrow 0$.

Finally, as in the $I_{1}$ case, to obtain the complete one-loop result we
need to add terms $\mathcal{M}_{I_{2}}^{\bigtriangledown}$,
$\mathcal{M}_{G_{3}}^{\bigtriangledown}$ obtained by simultaneously replacing
$m_{1}\leftrightarrow m_{2}$, $a_{1}\leftrightarrow a_{2}$ in
$\mathcal{M}_{I_{2}}^{\bigtriangleup}$,
$\mathcal{M}_{G_{3}}^{\bigtriangleup}$, so that the complete classical one-
loop amplitudes are

$\displaystyle\mathcal{M}_{I_{2}}=\mathcal{M}_{I_{2}}^{\bigtriangleup}+\mathcal{M}_{I_{2}}^{\bigtriangledown}\,,\qquad\mathcal{M}_{G_{3}}=\mathcal{M}_{G_{3}}^{\bigtriangleup}+\mathcal{M}_{G_{3}}^{\bigtriangledown}\,.$
(3.41)

The results for $I_{1}$ in (3.35) and $I_{2}$ in (3.39) in the spinless limit
agree with [112, 113]. In addition, in the limit where $a_{2}\rightarrow 0$
and $m_{1}\gg m_{2}$ (i.e. in the probe limit) we match the results of [116].

###  3.4 Leading singularities without spinor variables

In this section we revisit the leading singularity computation of Section 3.3
using an efficient, alternative parameterisation of [134] that avoids the use
of auxiliary spinor variables to solve the triple-cut on-shell conditions. We
begin by introducing a convenient linear combination $\tilde{p}_{2}$ of
$p_{1}$ and $p_{2}$,

$\displaystyle\begin{split}\tilde{p}_{2}^{\mu}&=p_{2}^{\mu}-\frac{m_{2}}{m_{1}}\sigma\,p_{1}^{\mu}=m_{2}(u_{2}^{\mu}-\sigma\,u_{1}^{\mu})=m_{2}\tilde{u}_{2}^{\mu}\
,\\\ \end{split}$ (3.42)

which is orthogonal to $p_{1}$, i.e. $\tilde{p}_{2}{\cdot}p_{1}=0$. It is
convenient to parameterise the internal graviton momentum $\ell_{1}$ as

$\displaystyle\ell_{1}^{\mu}=\alpha|q|\tilde{u}_{2}^{\mu}+\beta|q|u_{1}^{\mu}+\gamma
q^{\mu}+i\delta E^{\mu}\ ,$ (3.43)

where
$E^{\mu}=\epsilon^{\mu\alpha\beta\gamma}\tilde{u}_{2\alpha}u_{1\beta}q_{\gamma}$,
with $|q|$ being defined in (3.36). In the first step we perform three residue
integrals corresponding to the triple cut loci conditions (3.19), which leads
to the following one-parameter family of solutions:

$\displaystyle\begin{split}\ell_{1}^{\mu}&=\frac{q^{\mu}}{2}+\frac{\sqrt{z^{2}-1}}{2v\sigma}|q|\,\tilde{u}_{2}^{\mu}+\frac{iz}{2v\sigma}E^{\mu}\
,\\\
\ell_{2}^{\mu}&=\frac{q^{\mu}}{2}-\frac{\sqrt{z^{2}-1}}{2v\sigma}|q|\,\tilde{u}_{2}^{\mu}-\frac{iz}{2v\sigma}E^{\mu}\
,\end{split}$ (3.44)

where the parameter $z$ is related to the parameter $y$ in Section 3.3 via
$z=\frac{1+y^{2}}{2y}$. The remaining integration measure is proportional to
$\frac{1}{\sqrt{z^{2}-1}}$ where the precise normalisation comes from Jacobian
factors arising at the various steps of the computation. For the evaluation of
the contour integrals it is useful to quote here its residue in the
parameterisation given above:

$\displaystyle{\underset{z=\infty}{\mathrm{Res}}\,}\frac{1}{\sqrt{z^{2}-1}}=-1\,.$
(3.45)

The rest of the computation of the leading singularities is very similar to
that of (3.29) in the previous section. In the present case the effective
measure is

$\displaystyle d\mathfrak{m}_{\mathrm{eff}}=\frac{1}{64\pi
m_{1}|q|}\frac{dz}{\sqrt{z^{2}-1}}\,,$ (3.46)

where the contour has to be taken around $z=+\infty$ and, as before, the
normalisation is chosen to give directly the classical contribution from the
triangle integral. The two triple cuts are then

$\displaystyle\begin{split}\mathcal{M}_{I}^{\bigtriangleup,(I)}&=\ointctrclockwise_{z=+\infty}\\!d\mathfrak{m}_{\mathrm{eff}}\
M_{I}(p_{2},\ell_{1}^{++},\ell_{2}^{++})M_{3}(p_{1}^{\prime},-\ell_{2}^{--})M_{3}(-p_{1},-\ell_{1}^{--})\,,\\\
\mathcal{M}_{I}^{\bigtriangleup,(II)}&=\ointctrclockwise_{z=+\infty}\\!d\mathfrak{m}_{\mathrm{eff}}\
M_{I}(p_{2},\ell_{1}^{--},\ell_{2}^{--})M_{3}(p_{1}^{\prime},-\ell_{2}^{++})M_{3}(-p_{1},-\ell_{1}^{++})\,,\end{split}$
(3.47)

with $I{\in}(I_{1},I_{2},G_{3})$, and $d\mathfrak{m}_{\mathrm{eff}}$ given in
(3.46). Also note that, as in (3.29), the first/second triple cut contains a
factor of $e^{\pm a_{1}{\cdot}q}$ from the two three-point amplitudes.

We will not present full details of the evaluation of these residues, but make
a number of comments. First, the first term, $q^{\mu}/2$, in (3.44) never
contributes as it either contracts to zero or gives terms like
$p_{1,2}{\cdot}q\propto q^{2}$ which can be dropped. Similarly, terms
involving $E^{\mu}$ contract to zero or give rise to terms with vanishing
residue. So effectively we can set

$\displaystyle\ell_{1,2}\rightarrow\pm\frac{\sqrt{z^{2}-1}}{2v\sigma}|q|\,\tilde{u}_{2}^{\mu}\
,$ (3.48)

and it is easy to see that only terms quadratic in $\ell_{1}$ can contribute.
The corresponding residue is

$\displaystyle{\underset{z=\infty}{\mathrm{Res}}\,}\sqrt{z^{2}-1}=+\frac{1}{2}\
.$ (3.49)

Plugging the tree amplitudes from previous sections into the residue integral
and performing the state sum we arrive at the following results for the
classical amplitudes (setting the couplings of $I_{1}$ and
$G_{3}=I_{1}-2I_{2}$ to unity),

$\displaystyle\begin{split}\mathcal{M}_{I_{1}}^{\bigtriangleup}&=i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{16}m_{1}^{3}m_{2}^{2}|q|^{3}\cosh(a_{1}{\cdot}q)\left[(\sigma^{2}-1)\cosh(a_{2}{\cdot}q)-i\sigma\frac{\sinh(a_{2}{\cdot}q)}{a_{2}{\cdot}q}\epsilon(u_{1}u_{2}a_{2}q)\right],\\\
\mathcal{M}_{G_{3}}^{\bigtriangleup}&=-i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{16}m_{1}^{3}m_{2}^{2}|q|^{3}\cosh(a_{1}{\cdot}q)\cosh(a_{2}{\cdot}q),\end{split}$
(3.50)

in agreement with (3.35) and (3.40). For the bosonic string one
obtains141414Recall that the particular values of $\beta_{1}$ and $\beta_{2}$
in bosonic string theory are given in (1.4).

$\displaystyle\begin{split}\mathcal{M}_{I_{1},{\rm
BS}}^{\bigtriangleup}&=-i\frac{(\alpha^{\prime})^{2}\kappa^{4}m_{1}^{3}m_{2}^{2}}{2^{13}}|q|^{3}\cosh(a_{1}{\cdot}q)\left[(\sigma^{2}-1)\cosh(a_{2}{\cdot}q)-i\sigma\frac{\sinh(a_{2}{\cdot}q)}{a_{2}{\cdot}q}\epsilon(u_{1}u_{2}a_{2}q)\right],\\\
\mathcal{M}_{G_{3},{\rm
BS}}^{\bigtriangleup}&=i\frac{(\alpha^{\prime})^{2}\kappa^{4}m_{1}^{3}m_{2}^{2}}{2^{12}}|q|^{3}\cosh(a_{1}{\cdot}q)\cosh(a_{2}{\cdot}q)\
.\end{split}$ (3.51)

##  4 One-loop spinning amplitudes from unitarity and the heavy-mass limit

###  4.1 Review of the heavy-mass limit for Einstein-Hilbert

In this section we will cover an alternative method to construct the classical
amplitudes above based on unitarity cuts and the heavy-mass limit. The
discussion here follows the approach of Heavy-mass Effective Field Theory
(HEFT) [144, 145, 146, 147, 37, 39, 148, 136, 13, 137], where the classical
limit is taken by expanding in the $m{\rightarrow}\infty$ limit. This is
equivalent to the usual $\hbar{\rightarrow}0$ expansion for spinless particles
since the typical graviton momentum $q$ is much less than the mass $m$ of the
particle.

To impose this hierarchy of scales one can either work with classical
wavenumbers $\hat{q}=q/\hbar$ and then take $\hbar$ small with fixed
$\hat{q}$, or simply take $m$ large compared to the exchanged graviton
momentum. The same logic extends to the case of spinning particles, except
that one has to simultaneously expand in the limit of $m{\rightarrow}\infty$
and $s{\rightarrow}\infty$ (where $s$ is the spin quantum number of the
massive particle) while keeping the ratio $s/m=|a|$ finite. In practice, it is
more convenient to work with the barred variables of (3.3) (since they satisfy
$\bar{p}_{i}{\cdot}q=0$ exactly) and perform the expansion as
$\bar{m}{\rightarrow}\infty$. Once we have obtained the one-loop four-point
classical amplitudes, we can safely remove the bars from our expressions as
discussed later in this section.

We begin with a review of the heavy-mass expansion in the case of pure general
relativity coupled to scalars, before including $R^{3}$ corrections and spin.
The one-loop four-point amplitude can be constructed by performing a unitarity
cut in the $q^{2}$-channel and gluing together two Compton amplitudes. This
extracts the piece of the amplitude containing discontinuities in $q^{2}$,
which contribute to long-range effects after performing a Fourier transform in
$q$. The analytic terms in $q$ would only give rise to local contributions in
impact parameter space. To obtain the classical piece of this cut we must
expand in the classical limit (or equivalently the heavy-mass limit) and
extract the piece with the right scaling. Since we are already working at the
level of the $q^{2}$ cut we can perform the classical expansion on each
Compton amplitude separately.

To warm up, we will first consider the case of the usual scalar Compton
amplitude without $R^{3}$ corrections

$\displaystyle
M_{4}=M_{4,\bar{m}^{3}}+M_{4,\bar{m}^{2}}+\mathcal{O}(\bar{m}^{1}),$ (4.1)

where $M_{4,\bar{m}^{3}}$ and $M_{4,\bar{m}^{2}}$ are the hyperclassical and
classical pieces of the Compton amplitude respectively. These are given
explicitly by

$\displaystyle M_{4,\bar{m}^{3}}$
$\displaystyle\coloneqq\mathcal{A}_{3}(\bar{p},k_{1})\,\pi\delta(\bar{p}{\cdot}k_{1})\,\mathcal{A}_{3}(\bar{p},k_{2})\,,\quad\mathcal{A}_{3}(\bar{p},k_{i})\coloneqq-i\kappa(\bar{p}{\cdot}\varepsilon_{k_{i}})^{2}$
(4.2) $\displaystyle M_{4,\bar{m}^{2}}$
$\displaystyle\coloneqq\mathcal{A}_{4}(\bar{p},k_{1},k_{2})=-i\frac{\kappa^{2}}{q^{2}}\left(\frac{\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}}{\bar{p}{\cdot}k_{1}}\right)^{2}\,.$
(4.4)

Note that the “hyperclassical” piece of the Compton amplitude is given by an
iteration of three-point amplitudes. To compute the classical piece of the
one-loop amplitude we will only need the first two terms in the heavy-mass
expansion of the Compton amplitude. However, other classical observables may
require further subleading terms, see for example [28].

We will write the terms in the expansion above diagrammatically as follows

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) {$p$};
\vertex[right=1.8cm of a] (f2)[HV]{$M_{4}$}; \vertex[right=1.8cm of f2]
(c){$p^{\prime}$}; \vertex[above=1.8cm of f2] (gm){}; \vertex[left=0.8cm of
gm] (g2){$k_{1}$}; \vertex[right=0.8cm of gm] (g20){$k_{2}$}; \diagram* { (g2)
-- [rmomentum'={}](f2), (g20) -- [rmomentum={}](f2), (a) -- [fermion,thick]
(f2) -- [fermion,thick] (c), (g2)--[double, vector, line width =
.75](f2),(g20)--[double, vector, line width = .75](f2) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}{=}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) {$p$};
\vertex[right=1.8cm of a] (f2)[HV]{H}; \vertex[right=1.8cm of f2] (c)[HV]{H};
\vertex[right=1.8cm of c] (d){$p^{\prime}$}; \vertex[above=1.8cm of f2]
(g1){$k_{1}$}; \vertex[above=1.8cm of c] (g2){$k_{2}$}; \vertex[right=0.9 cm
of f2] (cut); \vertex[above=0.3cm of cut] (cutu); \vertex[below=0.3cm of cut]
(cutb); \diagram* { (g1) -- [rmomentum'={}] (f2), (g2) -- [rmomentum={}] (c),
(a) -- [fermion,thick] (f2) -- [thick] (c)-- [fermion,thick] (d),
(g1)--[double, vector, line width = .75](f2), (g2)--[double, vector, line
width = .75](c), (cutu)--[red,thick] (cutb) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}{+}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) {$p$};
\vertex[right=1.8cm of a] (f2)[HV]{H}; \vertex[right=1.8cm of f2]
(c){$p^{\prime}$}; \vertex[above=1.8cm of f2] (gm){}; \vertex[left=0.8cm of
gm] (g2){$k_{1}$}; \vertex[right=0.8cm of gm] (g20){$k_{2}$}; \diagram* { (g2)
-- [rmomentum'={}](f2), (g20) -- [rmomentum={}](f2), (a) -- [fermion,thick]
(f2) -- [fermion,thick] (c), (g2)--[double, vector, line width =
.75](f2),(g20)--[double, vector, line width = .75](f2) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}{+\cdots}$ (4.5)

where the red cut propagator denotes the delta function
$\pi\delta(\bar{p}{\cdot}k_{1})$.

To compute the classical part of the scalar four-point amplitude we need the
terms in the heavy-mass expansion which are
$\mathcal{O}(\bar{m}_{1}^{3}\bar{m}_{2}^{2})$ and
$\mathcal{O}(\bar{m}_{1}^{2}\bar{m}_{2}^{3})$. To obtain these we simply glue
one copy of (4.2) and (4.4) together in a unitary cut

$\displaystyle=\frac{1}{2}\sum_{h_{1},h_{2}}\int\frac{d^{D}\ell_{1}}{(2\pi)^{D}}\delta(\ell_{1}^{2})\delta(\ell_{2}^{2})\mathcal{A}_{3}^{h_{1}}(\bar{p}_{1},\ell_{1})\pi\delta(\bar{p}_{1}{\cdot}\ell_{1})\mathcal{A}_{3}^{h_{2}}(\bar{p}_{1},\ell_{2})\mathcal{A}_{4}^{h_{1},h_{2}}(\bar{p}_{2},\ell_{1},\ell_{2})\,,$
(4.6)

with $\ell_{2}=-\ell_{1}+q$. The diagram above is
$\mathcal{O}(\bar{m}_{1}^{3}\bar{m}_{2}^{2})$ and the
$\mathcal{O}(\bar{m}_{1}^{2}\bar{m}_{2}^{3})$ contribution is found by
swapping the particle labels $1$ and $2$. In principle, there are also
hyperclassical contributions to the one-loop amplitude which come from gluing
two copies of (4.2) together. However, these
$\mathcal{O}(\bar{m}_{1}^{3}\bar{m}_{2}^{3})$ terms will cancel out in the
observables we are interested in here.

To compute the cut above we perform the graviton propagator sum using

$\displaystyle\sum_{h}\varepsilon^{\mu}_{-k}{\varepsilon}^{\nu}_{-k}{\varepsilon}^{\alpha}_{k}\varepsilon^{\beta}_{k}=\frac{1}{2}\Big{(}P^{\mu\alpha}P^{\nu\beta}+P^{\mu\beta}P^{\nu\alpha}\Big{)}-\frac{1}{D-2}P^{\mu\nu}P^{\alpha\beta}\
,$ (4.7)

where

$P^{\mu\nu}\coloneqq\eta^{\mu\nu}-\frac{k^{\mu}t^{\nu}+k^{\nu}t^{\mu}}{k{\cdot}t}\,,\quad
k{\cdot}\varepsilon_{k}=t{\cdot}\varepsilon_{k}=0\,,$ (4.8)

$k^{\mu}$ is the graviton momentum and $t^{\mu}$ is a null reference momentum
transverse to $\varepsilon_{k}$. In practice, we can replace
$P^{\mu\nu}\rightarrow\eta^{\mu\nu}$ at the beginning since the dependence on
the reference momentum $t^{\mu}$ will drop out of the gauge invariant state
sum.151515For the manipulation of expressions in this section, we used the
Mathematica package DDimPackage [149]. However, the reference momentum
dependence can be more subtle in the spinning case, as we will see later.

After performing the state sum and simplifying the result, we lift the
propagators $\ell_{1},\ell_{2}$ off the cuts (i.e. we replace
$\delta(\ell_{i}^{2})\rightarrow i/\ell_{i}^{2}$), and use integration by
parts (IBP) relations to reduce the integrand.161616We perform the IBP
reductions in this paper using LiteRed [150, 151]. We are then left with an
amplitude that is proportional to the scalar cut triangle. In $D=4$ we have,

$\mathcal{M}_{4}=i\frac{3\pi}{32}\kappa^{4}\bar{m}_{1}^{3}\bar{m}_{2}^{2}(5\bar{\sigma}^{2}-1)I_{3}(\bar{u}_{1},q^{2})+(1\leftrightarrow
2)\,,$ (4.9)

where

$I_{3}(\bar{u}_{i},q^{2})\coloneqq\int\\!\frac{d^{4}\ell_{1}}{(2\pi)^{4}}\frac{\delta(\bar{u}_{i}{\cdot}\ell_{1})}{\ell_{1}^{2}(\ell_{1}-q)^{2}}=\frac{1}{16\pi}\frac{1}{|q|}\,.$
(4.10)

Thus the final result for the one-loop classical four-point amplitude is

$\mathcal{M}_{4}=i\left(\frac{\kappa}{2}\right)^{4}\frac{3}{32}\bar{m}_{1}^{2}\bar{m}_{2}^{2}(\bar{m}_{1}+\bar{m}_{2})\frac{5\bar{\sigma}^{2}-1}{|q|}\,,$
(4.11)

where we have defined $\bar{\sigma}\coloneq\bar{u}_{1}{\cdot}\bar{u}_{2}$.

An important comment on barred vs. unbarred variables is in order here. Since
hyperclassical terms exist for the four-point amplitude above, the distinction
between $\bar{m}_{i}$ and $m_{i}$ is in principle important, since upon re-
expanding in $m_{i}$ these hyperclassical terms could feed down into the
classical amplitude above. However, in the elastic one-loop case we are
considering, these feed-down terms only contribute to quantum corrections and
do not affect the classical amplitude. This can be seen by noting that
$\bar{m}_{i}$ and $m_{i}$ differ by a term proportional to $q^{2}$ (see
(3.5)); furthermore the only other non-zero kinematic invariant that we can
build out of the massive momenta is
$\bar{\sigma}=\bar{u}_{1}{\cdot}\bar{u}_{2}$, however due to four-point
kinematics this also differs from $\sigma$ by a term proportional to $q^{2}$,
as can be seen from (3.3). Thus we are free to remove the bars in (4.11), and
a similar conclusion applies to the amplitudes in the presence of cubic
corrections.

###  4.2 Introducing $R^{3}$ corrections: the scalar case

In this paper we are interested in the corrections to the classical scattering
of two spinning black holes from the operators $I_{1}$ and $G_{3}$ (or
equivalently $I_{1}$ and $I_{2}$). At leading order in their Wilson
coefficients, these operators only enter through their corrections to the
Compton amplitudes described in Section 2. In particular, the three-point
amplitudes involving two heavy particles and one graviton are left unchanged.
In the heavy-mass expansion, the $R^{3}$ corrections to the Compton amplitudes
begin at $\mathcal{O}(\bar{m}^{2})$, not $\mathcal{O}(\bar{m}^{3})$ as for the
full Compton amplitudes Section 2. This is due to a lack of massive
propagators in the $R^{3}$ Compton amplitudes, which are responsible for the
$\mathcal{O}(\bar{m}^{3})$ terms in (4.2).

For the reasons above, the first corrections to the classical four-point
massive amplitude due to $R^{3}$ interactions begin at one loop, not tree
level. To compute these we will again first focus on the scalar case. The
procedure is identical to the one detailed in the previous section: perform a
cut in the $q^{2}$-channel, then glue heavy-mass expanded amplitudes together
to obtain the classical piece, and finally, IBP reduce the result. The only
difference is that now we use the $R^{3}$ corrections to the Compton amplitude
at $\mathcal{O}(\bar{m}^{2})$. Explicitly we have the
$\mathcal{O}(\bar{m}_{1}^{2}\bar{m}_{2}^{2})$ cut

$\displaystyle=\frac{1}{2}\sum_{h_{1},h_{2}}\int\frac{d^{D}\ell_{1}}{(2\pi)^{D}}\delta(\ell_{1}^{2})\delta(\ell_{2}^{2})\mathcal{A}_{3}^{h_{1}}(\bar{p}_{1},\ell_{1})\pi\delta(\bar{p}_{1}{\cdot}\ell_{1})\mathcal{A}_{3}^{h_{2}}(\bar{p}_{1},\ell_{2})M_{4,R^{3}}^{h_{1},h_{2}}(\bar{p}_{2},\ell_{1},\ell_{2})$
(4.12)

where the $R^{3}$ correction to the classical Compton amplitude,
$M_{4,R^{3}}$, is either (2.13) or (2.14) in the spinless limit. As before
there is another $\mathcal{O}(\bar{m}_{1}^{2}\bar{m}_{2}^{3})$ cut obtained by
swapping the particle labels 1 and 2. After performing the state sum using
(4.7) and then IBP reducing the result, we obtain an integrand proportional to
the same scalar triangle as before. For $I_{1}$ and $G_{3}$ corrections we
have

$\displaystyle\begin{split}\mathcal{M}_{4,I_{1}}&=3i\pi\left(\frac{\kappa}{2}\right)^{6}\bar{m}_{1}^{3}\bar{m}_{2}^{2}(\bar{\sigma}^{2}-1)(q^{2})^{2}I_{3}(\bar{u}_{1},q^{2})+(1\leftrightarrow
2)\\\
&=i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{16}\bar{m}_{1}^{2}\bar{m}_{2}^{2}(\bar{m}_{1}+\bar{m}_{2})|q|^{3}(\bar{\sigma}^{2}-1)\,,\end{split}$
(4.13)

and

$\displaystyle\begin{split}\mathcal{M}_{4,G_{3}}&=-3i\pi\left(\frac{\kappa}{2}\right)^{6}\bar{m}_{1}^{3}\bar{m}_{2}^{2}(q^{2})^{2}I_{3}(\bar{u}_{1},q^{2})+(1\leftrightarrow
2)\\\
&=-i\left(\frac{\kappa}{2}\right)^{6}\frac{3}{16}\bar{m}_{1}^{2}\bar{m}_{2}^{2}(\bar{m}_{1}+\bar{m}_{2})|q|^{3}\,.\end{split}$
(4.14)

As previously mentioned we can replace $\bar{m}_{i}$ with $m_{i}$ and
$\bar{u}_{i}$ with $u_{i}$. In particular, in this case there are no
hyperclassical $R^{3}$ corrections which could feed down to the classical
terms above. After performing such a replacement, the above expressions match
the results of [112].

###  4.3 Introducing $R^{3}$ corrections: the spinning case

To add spin to the one-loop classical four-point $R^{3}$ amplitude we must add
spin dependence to the two Compton amplitudes that appear on either side of
the $q^{2}$-cut. First, let us examine the expansion of spinning Compton
amplitudes in the classical limit. Here we focus on the case of the Compton
amplitude with two positive (or two negative) helicity gravitons, as this is
the only contribution we will need for the $R^{3}$ computation.

Ideally, we wish to start from an unexpanded finite-spin Compton amplitude and
then take the classical limit $s/m{\rightarrow}|a|$, $s{\rightarrow}\infty$.
The spinning Compton amplitudes in EH gravity with two positive (or two
negative) helicity gravitons can be found by demanding appropriate
factorisation into three-point amplitudes [39, 130, 152],

$\displaystyle
M_{4}(-\bm{p}^{s},\bm{p^{\prime}}^{s},k_{1}^{++},k_{2}^{++})=\frac{\langle\bm{p}\,\bm{p^{\prime}}\rangle^{2s}}{m^{2s}}M_{4}(-p,p^{\prime},k_{1}^{++},k_{2}^{++})\,,$
(4.15) $\displaystyle
M_{4}(-\bm{p}^{s},\bm{p^{\prime}}^{s},k_{1}^{--},k_{2}^{--})=\frac{[\bm{p}\,\bm{p^{\prime}}]^{2s}}{m^{2s}}M_{4}(-p,p^{\prime},k_{1}^{--},k_{2}^{--})\,,$
(4.16)

where the massive spinors $|\bm{p}^{s}\rangle$ and
$|\bm{p^{\prime}}^{s}\rangle$ are defined in (3.9) (with similar definitions
for $|\bm{p}^{s}]$ and $|\bm{p^{\prime}}^{s}]$), and
$M_{4}(p,p^{\prime},k_{1}^{\pm\pm},k_{2}^{\pm\pm})$ is the scalar Compton
amplitude. We will only need the leading piece in the classical expansion of
the above amplitude for our purposes. This will be the
$\mathcal{O}(\bar{m}^{3})$ piece of the scalar Compton amplitude multiplied by
the large spin (and large mass) limit of the spinning prefactor above.
Explicitly, in an analogous manner to (3.11) with $l_{i}=-k_{i}$, we have

$\displaystyle\begin{split}M_{4,\bar{m}^{3}}(\bar{p},a,k_{1}^{\pm\pm},k_{2}^{\pm\pm})=e^{\pm(k_{1}+k_{2}){\cdot}a}\mathcal{A}_{3}(\bar{p},k_{1}^{\pm\pm})\,\pi\delta(\bar{p}{\cdot}k_{1})\,\mathcal{A}_{3}(\bar{p},k_{2}^{\pm\pm})\,,\end{split}$
(4.17)

with

$\displaystyle\mathcal{A}_{3}(\bar{p},k_{i}^{\pm\pm})\coloneqq-i\kappa(\bar{p}{\cdot}\varepsilon_{k_{i}}^{\pm})^{2}\,.$
(4.18)

The spinning $R^{3}$ Compton amplitudes are $\mathcal{O}(\bar{m}^{2})$ to the
leading order in the classical limit. Additionally, as we mentioned in Section
2, they vanish for $(+,-)$ and $(-,+)$ helicity configurations. As a result,
to obtain a classical contribution to the four-point spinning one-loop
amplitude we need to glue the $\mathcal{O}(\bar{m}^{2})$ $R^{3}$ Compton
amplitudes to the $\mathcal{O}(\bar{m}^{3})$ equal-helicity amplitude in
(4.17)

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