11institutetext: Department of Physics, Kobe University, Kobe 657-8501,
Japan22institutetext: Department of Physics and Astronomy, National Taiwan
University, Taipei 10617, Taiwan33institutetext: Department of Physics and
Center for Theoretical Physics, National Taiwan University, Taipei 10617,
Taiwan44institutetext: Centre for Theoretical Physics, Department of Physics
and Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS, United
Kingdom55institutetext: Kavli Institute for Theoretical Physics, University of
California, Santa Barbara, CA 93106, USA

# Gravitational Faraday effect from on-shell amplitudes

Wei-Ming Chen 2 Ming-Zhi Chung 2,3 Yu-tin Huang 4,5 Jung-Wook Kim
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>jung-
<EMAIL_ADDRESS>
###### Abstract

Effects of massive object’s spin on massive-massless $2\to 2$ classical
scattering is studied. Focus is set on the less-considered dimensionless
expansion parameter $\lambda/b$, where $\lambda$ is the massless particle’s
wavelength and $b$ is the impact parameter. Corrections in $\lambda/b$ start
to appear from $\mathcal{O}(G^{2})$, with leading correction terms tied to the
gravitational Faraday effect, which is a special case of the Lense-Thirring
effect. We compute the eikonal phase up to $\mathcal{O}(G^{2})$ and extract
spin effect on the scattering angle and time delay up to 14th order in spin.
The gravitational Faraday effect at linear order in spin Ishihara:1987dv  is
reproduced by $\lambda/b$ correction terms, which we compute to higher orders
in spin. We find that the equivalence principle, or universality, holds up to
NLO for general spinning bodies, i.e. away from geometric optics limit.
Furthermore, in the black hole limit, we confirm the absence of particular
spin structure observed Bern:2020buy ; Aoude:2020ygw ; Kosmopoulos:2021zoq ;
Chen:2021qkk ; Aoude:2022trd ; Bern:2022kto ; Aoude:2022thd , along with the
associated shift symmetry Bern:2022kto , and argue that it holds to arbitrary
spin order at $\mathcal{O}(G^{2})$ in the massless probe limit.

KOBE-COSMO-22-04 , QMUL-PH-22-16

##  1 Introduction

The fact that the classical limit of scattering amplitudes can correctly
capture black hole dynamics, is on itself a fascinating subject irrespective
of its remarkable success in applications to gravitational wave physics, which
has helped brought into the spotlight since the land mark detection by the
LIGO/Virgo collaboration LIGOScientific:2016aoc ; LIGOScientific:2017vwq .
That point-like sources can faithfully reproduce effects of gravitational
field in the weak field limit was appreciated long ago in the seminal work of
ref.Duff:1973zz , where it was shown that the Schwarzschild metric can be
perturbatively reconstructed order by order in $G$ through Feynman diagrams.
This was later bolstered by the computations of perturbative stess-energy form
factors Holstein:2004dn ; Donoghue:2001qc . Later development focused on the
Post Minkowskian (PM) expansion Bertotti:1956pxu ; Kerr:1959zlt ;
Bertotti:1960wuq ; Portilla:1979xx ; Westpfahl:1979gu ; Portilla:1980uz ;
Bel:1981be ; Westpfahl:1985tsl ; Ledvinka:2008tk ; Damour:2017zjx  for
conservative two-body potential of binary black holes, which can be extracted
from the gravitational scattering amplitude of massive scalars Donoghue:1994dn
; Bjerrum-Bohr:2002gqz ; Bjerrum-Bohr:2013bxa ; Neill:2013wsa . By matching
the field theory amplitude to effective field theory (EFT) description
Cheung:2018wkq , the on-shell simplification led to rapid progress,
culminating in the state of the art 4 PM conservative potential Bern:2019nnu ;
Bern:2019crd ; Bern:2021dqo . As the PM expansion is exact in velocity
($v^{2}$), the results can be directly compared and matched to the post-
Newtonian (PN) expansion Einstein:1938yz ; Ohta:1973je ; Jaranowski:1997ky ;
Damour:1999cr ; Blanchet:2000nv ; Damour:2001bu ; Damour:2014jta ;
Jaranowski:2015lha .

The extension to spinning black holes brings in a new angle to the
correspondence. It was already shown in the early work of refs.Holstein:2008sx
; Vaidya:2014kza  that the spinning Hamiltonian can be captured by the
scattering of elementary spinning particle. However, the absence of elementary
particle beyond spin-2 casts an intriguing conundrum for the program. This was
partially overcome by the kinematic definition of minimal coupling for
arbitrary spin particle Arkani-Hamed:2017jhn  whose classical spin-limit
($\hbar\rightarrow 0$, $s\rightarrow\infty$ and $s\hbar$ held fixed)
successfully reproduced the 1 PM dynamics of Kerr black hole Maybee:2019jus ;
Guevara:2018wpp ; Chung:2018kqs ; Chung:2019duq ; Guevara:2019fsj ; Arkani-
Hamed:2019ymq ; Aoude:2020onz ; Aoude:2021oqj . For non black hole spinning
bodies, in the PN EFT approach one introduces spins as extra degrees of
freedom Porto:2005ac  on the worldline for the point particle effective
action, accompanied by an infinite number of spin-induced multipole operators
whose strengths are parametrised by the corresponding Wilson coefficients
Porto:2008jj ; Levi:2015msa ; see refs.Levi:2019kgk ; Levi:2020uwu ;
Levi:2020lfn ; Kim:2021rfj  for recent works, ref.Liu:2021zxr  in the context
of PM EFT, and refs.Jakobsen:2021lvp ; Jakobsen:2021zvh ; Jakobsen:2022fcj
for worldline QFT. On the amplitude side, this can be incorporated by matching
the Wilson coefficients to those of non-minimal couplings Chung:2019duq ;
Chung:2020rrz , or to introduce a field theory effective lagrangian
Bern:2020buy . The latter approach has been implemented at 2 PM to obtain the
spin-dependent part of the conservative hamiltonian up to fifth order in spin
Kosmopoulos:2021zoq ; Bern:2022kto .

In going beyond the quartic order in spin at 2 PM, from the on-shell approach
one needs the gravitational Compton amplitude for the interaction of higher
spin states which are not unique Arkani-Hamed:2017jhn ; Chung:2018kqs . A
proposal utilizing appropriately conserved off-shell current was given for
spin-5/2 in ref.Chiodaroli:2021eug  (see ref.Falkowski:2020aso  for a compact
expression of the ambiguity). When translated to the effective field theory
Lagrangian approach in ref.Bern:2020buy , the issue becomes the determination
of black hole Wilson coefficients at 2 PM. In a nutshell, the question is can
one identify the underlying principles that determine the Kerr black hole
limit ?

A natural starting point is to examine the 1 PM dynamics and to take note of
any special aspects emerging in the Kerr limit. It was observed in
ref.Aoude:2020mlg  that the change in spin-entanglement entropy is minimized
for minimal couplings, which was later identified as the suppression of spin-
flipping sector in the eikonal phase Chen:2021huj . The absence of spin-
flipping sectors for the 1 PM eikonal amplitude is not surprising as minimal
coupling for massive spinning higher spins are defined such that it has
optimal power counting in the UV, which corresponds to helicity preserving
interactions. Besides optimal UV behaviour, an interesting aspect of the 1 PM
potential is the presence of shift symmetry. For general spinning bodies, the
1 PM potential depends on the spin vector through the combination
$\epsilon(q,p_{2},p_{1},S)$ Chung:2020rrz , which exhibits the independence on
the longitudinal part of the spin vector along the impact parameter plane,
i.e. the potential is invariant under the shift symmetry proposed in
ref.Bern:2022kto ,

$\displaystyle S_{i}^{\mu}\to S_{i}^{\mu}+\xi_{i}\frac{q^{\mu}}{q^{2}}\,.$ (1)

On the other hand, preservation of this symmetry at 2 PM becomes a nontrivial
constraint. Earlier, it was noted that in the spin bilinear sector the spin-
spin interaction $(S_{1}\cdot S_{2})$ only occurs through a special
combination Holstein:2008sx ; Guevara:2017csg ; Damgaard:2019lfh ;
Bern:2020buy

$\displaystyle(q\cdot S_{1})(q\cdot S_{2})-q^{2}(S_{1}\cdot S_{2})\,,$

which is secretly related to shift symmetry, as will be explained. This
combination can be generalised to the case where spin vectors are taken from
the same body, i.e.

$\displaystyle(q\cdot S_{i})(q\cdot S_{j})-q^{2}(S_{i}\cdot S_{j})\,,$ (2)

where $i=j$ is also allowed. It was noted in refs.Aoude:2020ygw ;
Kosmopoulos:2021zoq  that for 2 PM classical amplitude, from which the
conservative Hamiltonian is extracted, the black hole limit (defined by
absence of tidal terms or by correct spin-quadrupole moments) is special in
that the structure eq.(2) appears, at least to _quadratic or bilinear_ order
in spin. A related observation is that when the spin operators are expanded on
the basis of Lorentz invariants

$\displaystyle\\{(q\cdot S_{i})\,,\,(p_{j}\cdot
S_{i})\,,\,\epsilon(q,p_{1},p_{2},S_{i})\\}\,,$ (3)

the black hole limit is special in that $(q\cdot S_{i})$ terms all vanish, at
least to _quartic_ order in spin Chen:2021qkk (appendix B).111The validity of
the exponential form of the Kerr Compton amplitude used in
refs.Guevara:2018wpp ; Guevara:2019fsj ; Bautista:2019tdr ; Aoude:2020onz ;
Chen:2021qkk  to ${\cal O}(S^{4})$ has been confirmed by comparing to black
hole perturbation theory. The authors would like to thank Justin Vines for
sharing conclusions of their upcoming work JustinNew . The two observations
are related since the structure eq.(2) appears when the product
$\epsilon(q,p_{1},p_{2},S_{i})\epsilon(q,p_{1},p_{2},S_{j})$ is reorganised
using Levi-Civita identities. Also, the latter observation is equivalent to
shift symmetry eq.(1); $\epsilon(q,p_{2},p_{1},S)$ is manifestly invariant,
while $(p_{j}\cdot S_{i})$ is invariant classically. Recently, these
equivalent conditions on the spin structures of the Hamiltonian were used with
an additional assumption of favorable high energy behaviour to fix Wilson
coefficients that determine spinning-spinless sectors at 2 PM Aoude:2022trd ;
Bern:2022kto ; Aoude:2022thd .222The authors would like to thank Kays Haddad
for clarification on the relation of the spin structure eq.(2) to the Kerr
limit.

Figure 1: A cartoon of gravitational Faraday effect. The polarisation
direction (black double arrow) of the electromagnetic wave propagating along
the direction of the spinning axis gets rotated (red double arrow) by the
frame-dragging effect of the spinning body.

In this paper, we revisit the classical limit of the gravitational Compton
amplitude with spin effects included. The goal is twofold: on the one hand a
rotating object will induce Lense–Thirring (or gravitomagnetic frame-dragging)
effects, which can rotate the polarization plane of propagating
electromagnetic or gravitational waves as in fig.1. This effect should be
visible through the classical Compton amplitude describing the scattering of
photons or gravitons off a spinning object. At the same time, the role of
shift symmetry or the distinctive spin-spin interaction in determining the
black hole limit can be tested in this context, as the amplitude will be
parameterised by the worldline Wilson coefficients for linear coupling to
gravitons.

Earlier studies of gravitational Compton amplitude focused on the
photon/graviton and massive scalar system, where one is interested in the
corrections to the scattering angle (light bending) of null geodesics Bjerrum-
Bohr:2014zsa ; Bjerrum-Bohr:2016hpa ; Bai:2016ivl ; Chi:2019owc  and Shapiro
time delay Camanho:2014apa ; AccettulliHuber:2020oou . An important tool is
the eikonal approximation, corresponding to large center of mass energy and
small scattering angle ($s\gg t$), where the amplitude exponentiates
Cheng:1969eh ; Abarbanel:1969ek ; Levy:1969cr ; Amati:1987wq ; Amati:1987uf ;
Kabat:1992tb ; Akhoury:2013yua ; KoemansCollado:2019ggb . Importantly, since
the amplitude depends on the external helicity states, instead of the eikonal
phase we have the eikonal phase matrix Camanho:2014apa ;
AccettulliHuber:2020oou

$\left(\begin{array}[]{cc}\delta_{{+},{+}}&\delta_{{-},{+}}\\\
\delta_{{+},{-}}&\delta_{{-},{-}}\end{array}\right)=\bar{\delta}\,\mathbb{I}{+}\alpha_{i}\,\sigma_{i}\,,$
(4)

where we use $2\to 2$ convention for the helicity states, and the matrix is
written in a manifest Hermitian form. The phases $\bar{\delta},\alpha_{i}$,
which are functions of impact parameter $b$ and frequency $\omega$ of the
massless mode, can be perturbatively computed in powers of $G$. Previous
computations focused on
$\bar{\delta}=(\delta_{{+},{+}}{+}\delta_{{-},{-}})/2$, from which one
extracts the scattering angle and time delay, via

$\theta=\frac{1}{\omega}\frac{\partial}{\partial
b}\bar{\delta}(b,\omega),\quad\delta
t=\frac{\partial}{\partial\omega}\bar{\delta}(b,\omega)\,.$ (5)

For scalar Compton $\alpha_{3}=0$ due to the lack of parity violating
interactions, while $\alpha_{1,2}$ was studied for the effects of higher
derivative corrections to Einstein-Hilbert term, which generate non-trivial
same helicity sector at tree and one-loop Camanho:2014apa ;
AccettulliHuber:2020oou . These helicity non-preserving components modify
circular polarisation states to elliptical polarised states.

We will compute the Compton eikonal matrix for general spinning bodies up to
${\cal O}(G^{2})$. The resulting scattering angle and time delay will be a
function of Wilson coefficients. In contrast with the eikonal phase for
massive scattering amplitudes, there is a new dimensionless parameter in the
classical regime: $1/\omega b=\lambda/b$ where $\lambda$ is the wavelength of
the gravitational/EM wave. Thus for spinning bodies we have a double expansion
in $\lambda/b$ and $a/b$, where $a=S/m$ is the spin length. From dimensional
analysis, we can see that $\lambda/b$ corrections in the classical limit only
occurs at ${\cal O}(G^{2})$:

$\displaystyle{\cal O}(G^{2}):\quad\bar{\delta}$ $\displaystyle=$
$\displaystyle\frac{15\pi
G^{2}m^{2}\omega}{4b}\left[1{-}\frac{(\hat{k}_{2}\times\vec{a})\cdot\vec{b}}{3b^{2}}{+}\mathcal{O}\left(\frac{a^{2}}{b^{2}}\right)\right],$
$\displaystyle\alpha:=-\frac{\alpha_{3}}{|h|}$ $\displaystyle=$
$\displaystyle\frac{5\pi
G^{2}m^{2}(\hat{k}_{2}\cdot\vec{a})}{4b^{3}}\left[1{-}(1{+}C_{\rm
S^{2}})\frac{9(\hat{k}_{2}\times\vec{a})\cdot\vec{b}}{8b^{2}}{+}\mathcal{O}\left(\frac{a^{2}}{b^{2}}\right)\right]\,,$
(6)

where $\hat{k}_{2}$ is the spatial direction of the incoming massless plane
wave. Here we are giving the result in terms of an $a/b$ expansion. Note that
LO for $\alpha_{3}$ is subleading for the individual eikonal phase
$\delta_{+,+}$ and $\delta_{-,-}$. We identify $\alpha$ as the rotation angle
of the gravitational Faraday effect Ishihara:1987dv ; Nouri-Zonoz:1999jls ;
Sereno:2004jx ; Brodutch:2011eh ; Farooqui:2013rga ; Shoom:2020zhr ;
Deriglazov:2021gwa ; Chakraborty:2021bsb ; Li:2022izh  whose name is an
analogy to the magneto-optical effect bearing the same name; for linearly
polarised EM waves, the polarisation direction rotates due to the “magnetic”
background. Indeed we see that $\alpha$ is only present when there is a non-
zero spin and the leading linear in spin term matches with that computed using
parallel transport along null geodesics Ishihara:1987dv . While the leading
term for $\alpha$ is universal, the subleading terms depend on the Wilson
coefficients. We provide the result up to 14th order in spin, for both photons
and gravitons.

Note that the spin dependence in the computation of the classical Compton
amplitude stems from the product of spinning three-point amplitudes. In the
black hole limit, where one has minimal coupling, the product of two three-
point amplitudes in the classical limit can be written as

$\left(\prod_{i=2,3}M_{3,i}\right)\exp\left(-\sum_{i=2,3}\frac{\epsilon^{\mu\nu\rho\sigma}(F_{i})_{\mu\nu}p_{1\rho}S_{\sigma}}{2(P\cdot\varepsilon_{i})}\right)$
(7)

where $\prod_{i}M_{3,i}$ is the product of scalar three-point amplitudes,
$(F_{i})_{\mu\nu}=-2ik_{i[\mu}\varepsilon_{i,\nu]}$ is the field strength, and
$\sum_{i}$ sums over the massless legs with $P$ representing the momentum of
the intermediate massive leg ($P=-p_{1}-k_{2}$). The expression can be derived
from eq.(A.6) of ref.Chen:2021qkk  with $z_{i}=1$, neglecting quantum
commutator terms from the BCH formula. The expression is gauge invariant with
on-shell kinematics, and a similar expression for spinning soft factor
described in terms of the field strength $F_{\mu\nu}$ appeared in
ref.Bautista:2019tdr . A remarkable property of this form is that under the
shift operation in eq.(1), it transforms as where $\sum_{i\in{+}}k_{i}$ sums
over the momenta of positive helicity legs and $\sum_{j\in{-}}k_{j}$ sums over
negativity helicity. Thus one immediately sees that for helicity preserving
configuration, the product is shift invariant! For helicity non-conserving
configuration, the shift would not be an invariance, which was already
observed in ref.Aoude:2022trd . As only the former contribute to the 2 PM
classical Compton amplitude, this suggests that it is in fact shift invariant.
We will demonstrate that this is indeed the case.

Finally by considering the normalized quantity $\alpha_{3}/|h|$ along with
$\bar{\delta}$, one can show that equivalence principle holds at leading order
in $\lambda/b$ expansion. At subleading order in $\lambda/b$ both of these
properties begin to deviate. The deviations can be understood as the eikonal
limit being the geometric optics limit, where one ignores the wave nature of
electric-magnetic/gravitational waves and treat it as null rays. This is valid
when the wave length is much smaller than the impact parameter
$\frac{\lambda}{b}\ll 1$. Thus expansion in $\frac{\lambda}{b}$ would
correspond to corrections to the approximation.

This paper is organized as follows: we begin with a brief discussion of
$\lambda/b$ corrections to the eikonal phase and their physical interpretation
in sec.2. Then in sec.3 we compute the $\mathcal{O}(G^{1})$ eikonal phase for
general spinning body, and derive physical observables. In sec.4 , we present
the computation and result for $\mathcal{O}(G^{2})$ eikonal phase.

##  2 General discussion of $\lambda/b$ corrections

The kinematics of interest has four length scales; the Compton wavelength
$\lambda_{C}\sim m^{-1}$ controlling the “quantumness” of the massive body,
the wavelength $\lambda=\omega^{-1}$ of the massless particle,333For massless
particles we consider the wavenumber four-vector
$\bar{k}^{\mu}:=k^{\mu}/\hbar$ as the classical quantity. the spin length
$a\sim|\vec{S}|m^{-1}$ controlling the “size” of the massive body and the
impact parameter ${b}$ controlling the separation. In the (classical)
geometric optics regime the scale hierarchy is444The hierarchy $\lambda\ll a$
is only needed for the conceptual foundation of geometric optics and is
irrelevant for any of the analyses in this paper. In terms of wavepackets,
there is an additional scale hierarchy $\lambda\ll\xi\ll b$ where $\xi$ is the
wavepacket size Cristofoli:2021vyo . JWK would like to thank Donal O’Connell
for discussions on this point.

$\displaystyle\lambda_{C}\ll\lambda\ll a\ll b\,.$ (8)

Treating the internal nature of the massive body as quantum, the separation
between “quantum” and “classical” effects amounts to utilizing the scale
hierarchy between $\lambda_{C}$ and all other scales:

$\displaystyle\frac{\lambda_{C}}{b}\sim\frac{\hbar}{mb}\,,\,\frac{\lambda_{C}}{a}\sim\frac{\hbar}{|\vec{S}|}\,,\,\frac{\lambda_{C}}{\lambda}\sim\frac{\hbar\omega}{m}\,,$
(9)

where $\hbar\omega$ is the energy of the massless particle of momentum.
Considering that the transfer momentum $q\sim b^{-1}$ is the Fourier dual of
the impact parameter $b$, the classical limit corresponds to the following
$\hbar$ scaling for the variables:

$q^{\mu}\rightarrow\hbar q^{\mu},\quad k^{\mu}\rightarrow\hbar k^{\mu},\quad
S^{\mu}\rightarrow\frac{S^{\mu}}{\hbar},\quad m\rightarrow
m,\quad\omega\rightarrow\hbar\omega\,,\quad G\rightarrow\frac{G}{\hbar}\,.$
(10)

Due to the separation of scale in eq.(8), we have two classical expansion
parameters:

$\frac{a}{b}\sim\frac{|\vec{S}|}{mb},\quad\frac{\lambda}{b}=\frac{1}{b\omega}\,.$
(11)

Thus we will only compute the leading term in $\hbar\to 0$ limit, with the
spin and energy expansion ${a}/{|\vec{b}|},\,(\omega b)^{-1}$ being an
additional expansion in the classical regime. From now on we will suppress
$\hbar$ dependence for brevity by setting $\hbar=1$.

$p_{1}$$k_{2}$$k_{3}$$p_{4}$ Figure 2: The diagram shows the configuration of
the Compton amplitudes in our consideration. The wavy line represents the
massless particle and solid line is the massive spinning object. The arrows
show the directions of momentum flow.

Let’s consider the Compton amplitude where the incoming massless state (with
$k_{2}$) scatters off a massive object (with momenta $p_{1}$) as in fig.2. Due
to vanishing/suppressed helicity-flip amplitudes in general relativity, the
outgoing massless state (with $k_{3}$) will be of the same helicity as the
incoming state. The Compton amplitude of interest is $M(\mathbf{1}2^{\pm
h}3^{\pm h}\mathbf{4})$, where $h=1,2$ denotes photon and graviton
respectively. The eikonal approximation is an all-order resummation of $M_{4}$
where the amplitude is reorganised as a phase factor $iM_{4}=e^{i\delta}-1$,
which has been extensively studied in the literature Cheng:1969eh ;
Abarbanel:1969ek ; Levy:1969cr ; Amati:1987wq ; Amati:1987uf ; Kabat:1992tb ;
Akhoury:2013yua ; KoemansCollado:2019ggb . The eikonal phase $\delta(b)$ is
obtained by Fourier transforming the $2\rightarrow 2$ amplitude to impact
parameter space555Different definitions for the impact parameter space are
also used in the literature, which is reviewed in appendix A.

$\displaystyle\begin{aligned}
e^{i\delta_{{\pm},{\pm}}}-1&:=\int\frac{d^{D}q}{(2\pi)^{D}}\hat{\delta}(2p_{1}\cdot
q)\hat{\delta}(2k_{2}\cdot q)\left[e^{-ib\cdot q}iM_{4}(\mathbf{1}2^{\pm
h}3^{\pm h}\mathbf{4})\right]\\\
&=\frac{i}{4m\omega}\int\frac{d^{D-2}q}{(2\pi)^{D-2}}\left[e^{i\vec{b}\cdot\vec{q}}M_{4}(\mathbf{1}2^{\pm
h}3^{\pm h}\mathbf{4})\right]\,,\end{aligned}$ (12)

where $\hat{\delta}(x)=2\pi\delta(x)$ and Mandelstam invariants are
parametrised as:

$\displaystyle
s=(p_{1}{+}k_{2})^{2}=2m\omega{+}m^{2}\,,\,t=(k_{3}{-}k_{2})^{2}={-}|\vec{q}|^{2}\,,\,u=(p_{1}{-}k_{3})^{2}=m^{2}{-}2m\omega+|\vec{q}|^{2}\,.$
(13)

This parametrises $\omega=(p_{1}\cdot k_{2})/m$ as the frequency of the
incoming massless particle in the rest frame of $p_{1}$. The transfer momentum
from the massive to massless state is $q^{\mu}=k_{3}^{\mu}-k_{2}^{\mu}$ and
its spatial part in centre of momentum (COM) frame is the three-vector
$\vec{q}$.

$\displaystyle\begin{aligned}
p_{1}^{\mu}=(\sqrt{m^{2}+k^{2}},-\vec{{k}})\,&,\,p_{4}^{\mu}=(\sqrt{m^{2}+k^{2}},-\vec{{k}}-\vec{q})\,,\\\
k_{2}^{\mu}=(k,\vec{{k}})\,&,\,k_{3}^{\mu}=(k,\vec{{k}}+\vec{q})\,,\end{aligned}$
(14)

where the COM frame variable $k$ is given by $\omega$ through the relation

$\displaystyle(k^{2}/m^{2})=\frac{\sqrt{1+4(\omega^{2}/m^{2})}-1}{2}=(\omega^{2}/m^{2})\times[1+{\cal
O}(\hbar^{2})]\,,$ (15)

and can be considered as $\omega$ in the classical limit.

###  2.1 Dimensional analysis

Before we begin, let’s first perform a dimension analysis to determine the
allowed operators that can accompany various subleading (in $\lambda/b$)
corrections. Each element of the eikonal phase matrix, which we collectively
denote by $\delta$, can be expanded as a perturbative series in the coupling
$G$, the spin $S$, and the dimensionless ratio $\lambda/b$.

$\displaystyle\begin{gathered}\delta=\sum_{L=0}^{\infty}G^{L{+}1}\delta^{(L)}=\sum_{L=0}^{\infty}G^{L{+}1}\left(\sum_{j=0}^{\infty}\delta_{\text{S}^{j}}^{(L)}\frac{a^{j}}{b^{j}}\right)=\sum_{L=0}^{\infty}G^{L{+}1}\left(\sum_{j=0}^{\infty}\frac{a^{j}}{b^{j}}\left(\sum_{i=0}^{\infty}\delta_{\text{S}^{j}}^{(L,i)}\frac{\lambda^{i}}{b^{i}}\right)\right)\,,\end{gathered}$
(17)

where the superscript $(L)$ denotes the $L$-loop (${\cal O}(G^{L+1})$)
contribution and subscript $\text{S}^{n}$ denotes the ${\cal O}(S^{n})$
contribution of the given $L$-loop contribution. The second subscript $i$ of
the paired subscript $(L,i)$ denotes subleading $(\lambda/b)^{i}$ order
contribution to the given $L$-loop contribution.

Let’s define $M_{4}^{(L,0,0)}$ as the terms of the Compton amplitude $M_{4}$
that determine the leading spin-less sector $\delta^{(L,0)}_{\text{S}^{0}}$ of
the $L$-loop eikonal phase $\delta^{(L)}$, and $M_{4}^{(L,i,j)}$ as that which
determines $(\lambda/b)^{i}(a/b)^{j}$ corrections
($\delta^{(L,i)}_{\text{S}^{j}}$). The classical limit of background-test mass
$2\to 2$ scattering has only one master scalar integral (the $L$-loop fan
integral) at each loop order, while other remaining integrals contribute to
the iteration terms required from the exponentiation in the eikonal
approximation Brandhuber:2021eyq . Thus as long as the eikonal approximation
is valid and iteration holds, only one integral is relevant at each loop.
Defining the ratio:

$\displaystyle F_{(L,i,j)}$
$\displaystyle:=\frac{M_{4}^{(L,i,j)}}{M_{4}^{(L,0,0)}}\,,$ (18)

becomes simply the ratios of the integral coefficient of the master scalar
integral, and thus functions of Lorentz invariants (the helicity weight is
cancelled in the ratio). To analyse $F_{(L,i,j)}$ we first write all possible
Lorentz invariants that can be constructed from kinematics. There are
Mandelstam invariants which we can combine into a dimensionless expansion
parameter $\lambda/b$:666We ignore ${\cal O}(1)$ numerical factors in power
counting analysis.

$\displaystyle(s-m^{2})\sim
m\omega\sim\frac{m}{\lambda}\,,\,t=-|\vec{q}|^{2}\sim\frac{1}{b^{2}}$
$\displaystyle\Rightarrow$
$\displaystyle\left[\frac{m\sqrt{-t}}{s-m^{2}}\right]\sim\frac{\lambda}{b}$
(19)

The introduction of the spin vector brings three new invariants due to the
transverse condition $p\cdot S=0$. The three Lorentz invariants can be
combined into the independent dimensionless expansion parameter $a/b$ as:

$\displaystyle\begin{aligned} (q\cdot
S)&\sim\frac{ma}{b}&&\Rightarrow&&&\left[\frac{q\cdot
S}{m}\right]&\sim\frac{a}{b}\\\ (n\cdot S)&\sim\frac{m^{2}a}{\lambda
b}&&\Rightarrow&&&\left[\frac{n\cdot S}{m(s-m^{2})}\right]&\sim\frac{a}{b}\\\
(k_{2}\cdot
S)&\sim\frac{ma}{\lambda}&&\Rightarrow&&&\left[\frac{\sqrt{-t}(k_{2}\cdot
S)}{s-m^{2}}\right]&\sim\frac{a}{b}\,.\end{aligned}$ (20)

The orthogonal vector $n^{\mu}$ is defined as
$n^{\mu}=\epsilon^{\mu\alpha\beta\gamma}p_{1\alpha}k_{2\beta}q_{\gamma}$.
These invariants form a basis for $F_{(L,i,j)}$

$\displaystyle F_{(L,i,j)}=\sum_{\begin{subarray}{c}\alpha+\beta+\gamma=j\\\
\alpha,\beta,\gamma\geq
0\end{subarray}}c_{i,\alpha,\beta,\gamma}\left[\frac{m\sqrt{-t}}{s-m^{2}}\right]^{i}\left[\frac{q\cdot
S}{m}\right]^{\alpha}\left[\frac{n\cdot
S}{m(s-m^{2})}\right]^{\beta}\left[\frac{\sqrt{-t}(k_{2}\cdot
S)}{s-m^{2}}\right]^{\gamma}\,,$ (21)

where $c_{i,\alpha,\beta,\gamma}$ are numerical coefficients. Since
$F_{(L,i,j)}$ is a rational function, the powers of $\sqrt{-t}$ must add up to
an even number and we have the constraint $i+\gamma\in 2\mathbb{Z}$. We list
the consequences of this constraint:

  1. 1.

There are no $\lambda/b$ corrections at ${\cal O}(G^{1})$. The scalar term
$M_{4}^{(0,0,0)}$ contains a $t^{-1}$ pole prefactor. As any expansion in
$\lambda/b$ must be combined to form integer powers of $-t=|\vec{q}|^{2}$, the
corrections will cancel the $t^{-1}$ pole and result in derivatives of delta
functions in impact parameter space, which do not contribute to long range
effects.

  2. 2.

At ${\cal O}(G^{2})$, the scalar term $M_{4}^{(1,0,0)}$ contains a
$1/|\vec{q}|$ prefactor from the scalar triangle integral. The non-analytic
structure of the $1/|\vec{q}|$ prefactor yields long range contributions which
cannot be cancelled by integer powers of $-t$, allowing arbitrary order
$\lambda/b$ corrections. The condition $i+\gamma\in 2\mathbb{Z}$ forces odd
powers of $\lambda/b$ corrections to accompany odd powers of $(k_{2}\cdot S)$,
and the first $\lambda/b$ correction to be at $(\lambda/b)^{2}$ order for
spinless case ($j=0$). The corrections at this order are dependent on
conventions for the eikonal phase as explained in appendix A.

As we will argue in the following, the polarisation plane rotation is an
$\mathcal{O}(\lambda/b)$ correction from the leading eikonal approximation,
thus it is first observed at one-loop order. his is consistent with the
conclusion in ref.Asada:2000vn  that gravitational Faraday effect cannot be
observed at ${\cal O}(G^{1})$ order.

###  2.2 Subleading $\lambda/b$ corrections and rotations of polarization
planes

Previous studies on eikonal phases of gravitational Compton scattering mostly
ignored subleading effects in the $\lambda/b$ expansion. These subleading
corrections are related to the wave-like nature of the massless particle since
they depend on the wavelength $\lambda$ of the particle, but which physical
properties are these terms reflecting?

To answer this question, we consider geometric optics. The propagation of
light (or gravitational wave) is treated as a classical curve where the
tangent vector at each point of the curve is normal to the wavefront. For an
introduction, see e.g. §53 of ref.landau1975classical . We consider the
propagation of waves as a classical trajectory in the limit
$\lambda\rightarrow 0$ or $\lambda/b\ll 1$. Selecting a component of the wave
(e.g. any component of $\vec{E}$ or $\vec{B}$ for a electromagnetic wave) and
denoting it as $f$, we can write an ansatz777The terminology _WKB
approximation_ is also used in the literature for this ansatz, e.g.
ref.Isaacson:1968hbi .

$\displaystyle f=ae^{i\psi}\,,$ (22)

where $a=a(x,t)$ is the amplitude of the particular component and
$\psi=\psi(x,t)$ is the phase of the wave. $\psi$ is called the _eikonal_ and
its derivative $\bar{k}_{\mu}=-\partial_{\mu}\psi$ is regarded as the local
wavenumber four-vector of the ray, which is normal to the wavefront and
tangent to the propagation direction landau1975classical . Since $\psi$
changes by $2\pi$ as one transverses one $\lambda$, the limit $\lambda\to 0$
implies large gradient of $\psi$. Now recall that the wave equation is given
by:

$\displaystyle\frac{\partial^{2}f}{\partial x_{\mu}\partial
x^{\mu}}=\left[\frac{\partial^{2}a}{\partial x_{\mu}\partial
x^{\mu}}e^{i\psi}+2i\frac{\partial a}{\partial
x_{\mu}}\frac{\partial\psi}{\partial
x^{\mu}}e^{i\psi}+\frac{\partial^{2}\psi}{\partial x_{\mu}\partial
x^{\mu}}f\right]-\frac{\partial\psi}{\partial
x_{\mu}}\frac{\partial\psi}{\partial x^{\mu}}f=0\,.$ (23)

Since $\partial\psi\sim\frac{1}{\lambda}$ is large, we expect the first three
terms in square brackets to be subleading compared to the last by a factor of
$\lambda/b$. Amongst the three terms in the square brackets, since
$\frac{\partial a}{\partial x_{\mu}}$ is accompanied by a factor of
$\frac{\partial\psi}{\partial x^{\mu}}$, the subleading corrections at this
order will be dominated by the change in the wave amplitude denoted as $a$,
which also encodes polarisation data. Thus if we can identify $\psi$ with
$\delta(b)$, the $\lambda/b$ corrections reflect polarisation dependence. We
provide an argument in appendix B on why we can identify the eikonal $\psi$ in
eq.(22) as the eikonal phase $\delta(b)$ in eq.(12). Indeed in section 4, our
explicit results show that

$\delta_{{+},{+}}=\delta_{{-},{-}}\times\left[1+{\cal
O}\left(\lambda/b\right)\right]\,.$ (24)

that is, the polarization dependence occurs at subleading order in
$\lambda/b$.

The difference $2\alpha_{3}\equiv(\delta_{+,+}-\delta_{-,-})$, which
characterises the first subleading order in $\lambda/b$, has an interpretation
as _the rotation angle of the polarisation plane_. To see this, consider
positive and negative helicity one-particle states propagating along the $z$
direction. The two states will acquire opposite phases under a rotation of
angle $\alpha$ around the $z$-axes, which is simply a little group rotation.

$|{+}\rangle^{\prime}\rightarrow e^{-i\alpha
J^{12}}|{+}\rangle=e^{-ih\alpha}|{+}\rangle,\quad|{-}\rangle^{\prime}\rightarrow
e^{-i\alpha J^{12}}|{-}\rangle=e^{+ih\alpha}|{-}\rangle\,.$ (25)

The difference of the phase gained by positive and negative helicity photons
($|h|=1$) is tied to the rotation angle of the polarisation plane $\alpha$ by

$\displaystyle\alpha$
$\displaystyle=-\frac{\delta_{\gamma,{+},{+}}-\delta_{\gamma,{-},{-}}}{2}\,.$
(26)

There is an additional factor of 2 for gravitons ($|h|=2$),

$\displaystyle\alpha$
$\displaystyle=-\frac{\delta_{g,{+},{+}}-\delta_{g,{-},{-}}}{4}\,,$ (27)

which compensates for the spin of the graviton. We show that indeed the
difference in eikonal phase reproduces the result of ref.Ishihara:1987dv ,
where the polarisation plan rotation angle was computed by considering a
parallel transport along the light ray’s null geodesic. Finally, we see that
the angle can be identified with the $\sigma_{3}$ component of the eikonal
phase matrix: $\alpha=-\alpha_{3}/|h|$.

##  3 The ${\cal O}(G^{1})$ Compton eikonal phase for spinning objects

We begin with the ${\cal O}(G^{1})$ Compton eikonal phase for general spinning
body. This will allow us to introduce the setup which will carry over to the
${\cal O}(G^{2})$ computation in the next section. Furthermore, the ${\cal
O}(G^{1})$ result will be important in checking whether the box and the cross
box contributions at ${\cal O}(G^{2})$ matches with the square of the former,
as required from exponentiation pattern of the eikonal amplitude.

While helicity non-preserving amplitudes are present for the Einstein-Hilbert
action AccettulliHuber:2020oou , they are suppressed by ${\cal
O}((\lambda/b)^{4})$ compared to their helicity preserving counterparts, both
at tree and one-loop level. As we are not interested in $\lambda/b$ expansion
to such high orders, we will solely focus on helicity preserving processes.

###  3.1 The tree amplitude for eikonal phase

Here we will only consider the helicity conserving configuration
$\widetilde{M}_{4}(\mathbf{1}2^{{+}h}3^{{+}h}\mathbf{4})$ of the classical
amplitude. Here we use $\widetilde{\;}$ to make a distinction between the
classical amplitude from “semi”-classical amplitude. From the previous
discussion we’ve seen that helicity dependence of the eikonal phase can only
be subleading in $\lambda/b$, while dimensional analysis tells us that such
corrections start at ${\cal O}(G^{2})$. Thus this tell us that at ${\cal
O}(G^{1})$, the amplitudes
$\widetilde{M}_{4}(\mathbf{1}2^{{\pm}h}3^{{\pm}h}\mathbf{4})$ must be
identical. Indeed this is verified through our explicit computation. The
gravitational coupling constant is normalised as $\kappa=\sqrt{32\pi G}$,
where $G$ is the Newton’s constant.

The eikonal phase is given by the fourier transform of the Compton amplitude
in the $q^{2}\rightarrow 0$ limit. At ${\cal O}(G^{1})$ this limit is given by
the product of two three-point amplitudes, one corresponding to graviton
coupling to massive spinning and the other massless helicity states. This can
be extracted from the following integral formula, motivated by the
exponentiated representation for amplitudes of Kerr black holes
Guevara:2018wpp ; Guevara:2019fsj ; Bautista:2019tdr ; Aoude:2020onz ;
Chen:2021qkk 888The expression is $2\to 2$ continued from the $4\to 0$
expression. Spinor notations follow that of ref.Chung:2018kqs .

$\displaystyle\begin{gathered}M_{4}^{s}=M_{4}^{s=0}\oint\frac{dz}{2\pi
iz}\left(\sum_{j}C_{\text{S}^{j}}z^{j}\right)\exp\left(-iK^{\mu}L^{\nu}J_{\mu\nu}\right)\,,\\\
K^{\mu}=-\frac{q^{\mu}}{z}\,,\,L^{\mu}=\frac{\langle{3}|\sigma^{\mu}|{2}]}{\langle{3}|p_{1}|{2}]}\,,\end{gathered}$
(30)

where the _scalar factor_ $M_{4}^{s=0}$ is the spin-independent part, i.e. it
represents the Compton amplitude for two massive scalar and two identical
helicity state generated via a graviton exchange. The above can be viewed as a
semi-classical amplitude where the spin is taken to the classical limit, i.e.
$s\rightarrow\infty$ and $\hbar\rightarrow 0$ with $\hbar s$ held fixed.
Indeed taking $s$ finite, and $C_{\text{S}^{j}}=1$, this reduces to the
Compton amplitude for minimally coupled spinning particle Chen:2021qkk . From
$M_{4}^{s}$ we can derive the fully classical amplitude $\widetilde{M}$ by
further expanding the kinematic variables with appropriate $\hbar$ counting.

The _spin factor_ refers to the remaining auxiliary residue integral terms in
eq.(30), where the contour is taken to be encircling the origin counter-
clockwise. The spin factor is motivated to capture the full $t$-channel
(graviton pole) factorization limit, which is modified from the spinless case
by the Wilson coefficients $C_{\text{S}^{n}}$ parametrising the spin-multipole
moments Levi:2015msa ; Chung:2018kqs ; Chung:2019duq ; Chung:2020rrz . In
particular, the $t=0$ limit can be attained by either setting
$|{2}\rangle\propto|{3}\rangle$, or $|2]\propto|3]$. For the former, the
scalar factor $M_{4}^{s=0}$ factorizes to (in all incoming convention)

$\displaystyle-tM_{4}^{s=0}\to M_{3}^{+h,-h,+2}\times M_{3^{-}}^{s=0}\,,$ (31)

where $M_{3}^{+h,-h,+2}$ is the amplitude of a helicity $h$ particle coupling
to a positive helicity graviton and
$M_{3^{-}}^{s=0}=-\kappa(p_{1}\cdot\varepsilon_{3^{-}})^{2}$ is the amplitude
for a massive scalar coupling to a negative helicity graviton. The spin factor
in eq.(30) becomes

$\displaystyle
K^{\mu}=-\frac{q^{\mu}}{z}\,,\,L^{\mu}\to\frac{\varepsilon_{q^{-}}^{\mu}}{\varepsilon_{q^{-}}\cdot
p_{1}}$ $\displaystyle\Rightarrow$
$\displaystyle\exp(-iK^{\mu}L^{\nu}J_{\mu\nu})\to\exp\left(i\frac{q^{\mu}\varepsilon_{q^{-}}^{\nu}J_{\mu\nu}}{(p_{1}\cdot\varepsilon_{q^{-}})z}\right)\,,$
(32)

which reproduces the spin factor of eq.(B.1) in ref.Chen:2021qkk .999The
$|{2}]$ spinors of $L^{\mu}$ are interpreted as the auxiliary spinor of
$\varepsilon_{q^{-}}^{\mu}$. The relative sign difference is due to outgoing
convention for $q$. The factorisation behaviour for the case $|2]\propto|3]$
can be analysed in a similar manner. Although eq.(30) has spurious poles due
to the factor $\langle{3}|p_{1}|{2}]$ in the denominator of the exponent, the
spurious poles can be resolved by terms that do not affect the $t$-channel
pole which determines the ${\cal O}(G^{1})$ eikonal phase, therefore only
considering the terms in eq.(30) is enough to compute the ${\cal O}(G^{1})$
eikonal phase.

The scalar amplitudes for massive (uncharged) scalar interacting with massless
spin-$|h|$ states through graviton exchange are given as

$\displaystyle\frac{M_{4}^{s=0}}{(\kappa/2)^{2}}$
$\displaystyle=\left\\{\begin{aligned} &-\frac{(s-u)^{2}}{4t}&&&h&=0\\\
&-\frac{[{2}|p_{1}|{3}\rangle(s-u)}{2t}&&&h&=1/2\\\
&-\frac{[{2}|p_{1}|{3}\rangle^{2}}{t}&&&h&=1\\\
&\frac{[{2}|p_{1}|{3}\rangle^{3}(s-u)}{2(s-m^{2})(u-m^{2})t}&&&h&=3/2\\\
&\frac{[{2}|p_{1}|{3}\rangle^{4}}{(s-m^{2})(u-m^{2})t}&&&h&=2\end{aligned}\right.\,.$
(33)

Explicit evaluation of relevant spinor contractions yield

$\displaystyle\begin{gathered}[{2}|p_{1}|{3}\rangle=[{3}|p_{1}|{2}\rangle=2m\omega\sqrt{1-\frac{|\vec{q}|^{2}(1+\frac{2\omega}{m})}{4\omega^{2}}}=2m\omega\times\left[1+{\cal
O}(q^{2}/\omega^{2}\,,\,\hbar)\right]\,,\,\\\
[23]=-{\langle}23{\rangle}=|\vec{q}|\,,\end{gathered}$ (36)

up to phase factors for little group scaling, thus all scalar amplitudes in
eq.(33) can be treated as

$\displaystyle M_{4}^{s=0}\simeq\frac{32\pi Gm^{2}\omega^{2}}{\vec{q}^{2}}\,,$
(37)

in the eikonal phase computation, regardless of the helicity. Therefore, the
classical part of the eikonal phase at tree order and the observables derived
from it are independent of the massless particle’s helicity, which can be
understood as a manifestation of the equivalence principle.

Before closing we remark that the $M_{4}^{s}$ in eq.(30) for $h=2$ differs
from the classical gravitational Compton amplitude in eq.(B.12) of
ref.Chen:2021qkk , as the latter was further constrained to have the correct
massive factorization channel residues. However for our purpose, the ${\cal
O}(G^{1})$ eikonal phase, both yield the same result as we demonstrate in
appendix C.

###  3.2 The ${\cal O}(G^{1})$ eikonal phase and observables

$\vec{d}$$\vec{k}_{2}$$\vec{a}_{\perp}$$\vec{a}$ Figure 3: Kinematic set-up of
the vectors describing spin orientation. $\vec{a}$ is the spin-length vector
and $\vec{k}_{2}$ is the momentum of the incoming massless particle.
$\vec{a}_{\perp}$ is the projected component of $\vec{a}$ orthogonal to
$\vec{k}_{2}$. The vectors $\vec{d}$ , $\vec{a}_{\perp}$ and $\vec{k}_{2}$ are
respectively orthogonal with $|\vec{d}|=|\vec{a}_{\perp}|$.

Following conventions of refs.Bern:2020buy ; Kosmopoulos:2021zoq ;
Chen:2021qkk , we identify the Lorentz generator $J_{\mu\nu}$ as the spin
tensor
$S_{\mu\nu}=-\frac{1}{m}\epsilon_{\mu\nu\alpha\beta}p_{1}^{\alpha}S^{\beta}$.
The exponent in the spin factor eq.(30) becomes

$\displaystyle
i\frac{q^{\mu}\langle{3}|\sigma^{\nu}|{2}]S_{\mu\nu}}{z\langle{3}|p_{1}|{2}]}=-\frac{i(n\cdot
S)}{m^{2}\omega z}+{\cal O}(|\vec{q}|^{2},\hbar^{1})\,,$ (38)

where $n^{\mu}=\epsilon^{\mu\alpha\beta\gamma}p_{1\alpha}k_{2\beta}q_{\gamma}$
and only the relevant term for ${\cal O}(G^{1})$ eikonal phase has been kept.
Note that the apparent spurious $\langle{3}|p_{1}|{2}]$ pole simply becomes
$\omega$ in the $q^{2}\rightarrow 0$ limit. Furthermore, the $\omega\to 0$
limit is non-singular since $n^{\mu}$ vanishes. Similar comment applies to our
${\cal O}(G^{2})$ analysis. The classical amplitude is then given by:

$\displaystyle\begin{gathered}\widetilde{M}_{4}|_{\mathcal{O}(G^{1})}=-\frac{32\pi
Gm^{2}\omega^{2}}{q^{2}}\oint\frac{dz}{2\pi
iz}\left(\sum_{j}C_{\text{S}^{j}}z^{j}\right)\exp\left(-\frac{i(n\cdot
S)}{m^{2}\omega z}\right)\,.\end{gathered}$ (40)

To obtain the eikonal phase, we Fourier transform to impact parameter space,

$\displaystyle\begin{aligned}
\delta^{(0)}(b)&=\frac{1}{4m\omega}\int\frac{d^{D-2}q}{(2\pi)^{D-2}}\left[e^{i\vec{b}\cdot\vec{q}}\widetilde{M}_{4}(q)|_{\mathcal{O}(G^{1})}\right]\,,\\\
\end{aligned}$ (41)

which identifies $\nabla_{b}\Leftrightarrow i\vec{q}$. We define the following
impact parameter space variables.

$\displaystyle\begin{aligned}
\vec{a}=\frac{\vec{S}}{m}\,,&&\hat{k}_{2}=\frac{\vec{k}_{2}}{\omega}\,,&&\vec{d}=\hat{k}_{2}\times\vec{a}\,,&&\vec{a}_{\perp}=\vec{a}-(\vec{a}\cdot\hat{k}_{2})\hat{k}_{2}\,.\end{aligned}$
(42)

The unit vector $\hat{k}_{2}$ defines the direction orthogonal to the impact
parameter space ($\hat{k}_{2}\cdot\vec{b}=0$), and $\vec{a}_{\perp}$ is the
projection of the spin-length vector $\vec{a}$ onto the impact parameter plane
so that $\vec{a}\cdot\vec{b}=\vec{a}_{\perp}\cdot\vec{b}$. Since
$\vec{d}\cdot\vec{a}_{\perp}=0$, the two vectors $\vec{a}_{\perp}$ and
$\vec{d}$ span the two-dimensional impact parameter space. These variables are
related to the spin Lorentz invariants by

$\displaystyle\begin{aligned} -\frac{i(n\cdot
S)}{m^{2}\omega}=\left(\hat{k}_{2}\times\vec{a}\right)\cdot\left(i\vec{q}\right)&\Leftrightarrow\left(\hat{k}_{2}\times\vec{a}\right)\cdot\nabla_{b}=\vec{d}\cdot\nabla_{b}\\\
-\frac{i(q\cdot
S)}{m}=\vec{a}\cdot\left(i\vec{q}\right)&\Leftrightarrow\vec{a}\cdot\nabla_{b}=\vec{a}_{\perp}\cdot\nabla_{b}\\\
\frac{k_{2}\cdot
S}{m\omega}&\Leftrightarrow-\hat{k}_{2}\cdot\vec{a}\end{aligned}$ (43)

The tree order eikonal phase $\delta^{(0)}$ can be obtained from the tree
amplitude eq.(30) by expanding eq.(12) to leading coupling order,

$\displaystyle\begin{aligned}
\delta^{(0)}(b)&=\frac{1}{4m\omega}\oint\frac{dz}{2\pi
iz}\left(\sum_{j}C_{\text{S}^{j}}z^{j}\right)\int\frac{d^{D-2}q}{(2\pi)^{D-2}}\left[e^{i\left[\vec{b}+({\vec{d}}/{z})\right]\cdot\vec{q}}\widetilde{M}^{s=0}_{4}(q)\right]\,,\end{aligned}$
(44)

where the order of integration has been changed, and the exponent has been
simplified using eq.(38). Inserting eq.(37) and performing the Fourier
integral in $D=4$ leads to the eikonal phase

$\displaystyle\delta^{(0)}(b)-\delta^{(0)}_{\text{IR}}$
$\displaystyle=-\left(\frac{\kappa}{2}\right)^{2}\frac{m\omega}{2\pi}\oint\frac{dz}{2\pi
iz}\left(\sum_{j}C_{\text{S}^{j}}z^{j}\right)\log\left|\vec{b}+\frac{\vec{d}}{z}\right|$
$\displaystyle=-2Gm\omega\left[2\log
b+\frac{2\vec{d}\cdot\vec{b}}{b^{2}}+C_{\text{S}^{2}}\left(-\frac{2(\vec{d}\cdot\vec{b})^{2}}{b^{4}}+\frac{d^{2}}{b^{2}}\right)\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdfasd}+C_{\text{S}^{3}}\left(\frac{8(\vec{d}\cdot\vec{b})^{3}}{3b^{6}}-\frac{2(\vec{d}\cdot\vec{b})d^{2}}{b^{4}}\right)\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdfasd}+C_{\text{S}^{4}}\left(-\frac{4(\vec{d}\cdot\vec{b})^{4}}{b^{8}}+\frac{4(\vec{d}\cdot\vec{b})^{2}d^{2}}{b^{6}}-\frac{d^{4}}{2b^{4}}\right)+\cdots\right]\,,$
(45) $\displaystyle\delta^{(0)}_{\text{IR}}$
$\displaystyle=-\frac{4Gm\omega}{4-D}$ (46)

where we have separated the IR divergence $\delta^{(0)}_{\text{IR}}$. Setting
all Wilson coefficients to Kerr value $C_{\text{S}^{n}}=1$ the residue
integral localises to $z=1$ and we obtain the eikonal phase eq.(5.32) of
ref.Bautista:2021wfy , reproduced below.

$\displaystyle\delta^{(0)}_{\text{Kerr}}(b)-\delta^{(0)}_{\text{IR}}$
$\displaystyle=-4Gm\omega\log|\vec{b}+\vec{d}|\,.$

The time delay can be computed from eq.(45) using eq.(5), where the IR scale
$b_{0}\gg b$ was introduced as the reference so that the time delay becomes a
finite quantity Camanho:2014apa ; AccettulliHuber:2020oou

$\displaystyle\delta t^{(0)}$
$\displaystyle=-\left(\frac{\kappa}{2}\right)^{2}\frac{m}{2\pi}\oint\frac{dz}{2\pi
iz}\left(\sum_{j}C_{\text{S}^{j}}z^{j}\right)\log\left|\vec{b}+\frac{\vec{d}}{z}\right|$
$\displaystyle=-2Gm\left[2\log\frac{b}{b_{0}}+\frac{2\vec{d}\cdot\vec{b}}{b^{2}}+C_{\text{S}^{2}}\left(-\frac{2(\vec{d}\cdot\vec{b})^{2}}{b^{4}}+\frac{d^{2}}{b^{2}}\right)\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdfasd}+C_{\text{S}^{3}}\left(\frac{8(\vec{d}\cdot\vec{b})^{3}}{3b^{6}}-\frac{2(\vec{d}\cdot\vec{b})d^{2}}{b^{4}}\right)\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdfasd}+C_{\text{S}^{4}}\left(-\frac{4(\vec{d}\cdot\vec{b})^{4}}{b^{8}}+\frac{4(\vec{d}\cdot\vec{b})^{2}d^{2}}{b^{6}}-\frac{d^{4}}{2b^{4}}\right)+\cdots\right]\,.$
(47)

The conventional scattering angle is not well-defined in presence of spin, as
the motion is no longer restricted to lie on the orbital plane. One way to
proceed is to define the scattering angle $\theta$ as a vectorial quantity
$\vec{\theta}$ that is directed along the direction of the impulse Kim:2020cvf
. The impulse for the massless probe $\vec{q}$ can be computed from the
eikonal via

$\displaystyle\vec{q}$ $\displaystyle=\nabla_{b}\delta(b)\,,$ (48)

therefore the vectorial scattering angle $\vec{\theta}$ is given as

$\displaystyle\vec{\theta}:=\frac{\nabla_{b}\bar{\delta}(b,\omega)}{|\vec{k}_{2}|}=\frac{1}{\omega}\nabla_{b}\bar{\delta}(b,\omega)\,,$
(49)

since the size of the incoming momentum $\vec{k}_{2}$ is $\omega$. The
vectorial scattering angle is

$\displaystyle\vec{\theta}^{(0)}$
$\displaystyle=-\left(\frac{\kappa}{2}\right)^{2}\frac{m}{2\pi}\oint\frac{dz}{2\pi
iz}\left(\sum_{j}C_{\text{S}^{j}}z^{j}\right)\frac{\vec{b}+\vec{d}/z}{|\vec{b}+\vec{d}/z|^{2}}$
$\displaystyle=-4Gm\left[\frac{\vec{b}}{b^{2}}+\left(\frac{\vec{d}}{b^{2}}-\frac{2(\vec{d}\cdot\vec{b})}{b^{4}}\vec{b}\right)-C_{\text{S}^{2}}\left(\frac{2(\vec{d}\cdot\vec{b})^{2}}{b^{4}}\vec{d}+\frac{b^{2}d^{2}-4(\vec{d}\cdot\vec{b})^{2}}{b^{6}}\vec{b}\right)\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdf}+C_{\text{S}^{3}}\left(\frac{4(\vec{d}\cdot\vec{b})^{2}-b^{2}d^{2}}{b^{6}}\vec{d}+\frac{4(\vec{d}\cdot\vec{b})(b^{2}d^{2}-2(\vec{d}\cdot\vec{b})^{2})}{b^{8}}\vec{b}\right)\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdf}+C_{\text{S}^{4}}\left(\frac{4(\vec{d}\cdot\vec{b})(b^{2}d^{2}-2(\vec{d}\cdot\vec{b})^{2})}{b^{8}}\vec{d}\right.\right.$
$\displaystyle\phantom{=}\left.\left.\phantom{asdfasdfasdfasdfasdf}+\frac{b^{4}d^{4}-12b^{2}d^{2}(\vec{d}\cdot\vec{b})^{2}+16(\vec{d}\cdot\vec{b})^{4}}{b^{10}}\vec{b}\right)+\cdots\right]\,.$
(50)

Calculation of ${\cal O}(G^{2})$ scattering angle requires a modified equation
suggested in ref.Bern:2020buy  as eq.(7.15), but since the scattering angle is
not the main focus of this paper the calculations will be omitted.

##  4 The ${\cal O}(G^{2})$ eikonal phase and observables

$1$$2$$3$$4$$\ell$(a) triangle$1$$2$$3$$4$$\ell$(b) box$1$$2$$3$$4$$\ell$(c)
crossed box Figure 4: The cut diagrams relevant to classical physics. The
thick sold lines, wavy lines and double wavy lines represent massive spinning
bodies, massless particles and gravitons, respectively. The cut propagators
are labeled by the red dotted lines. $1$ and $2$ are ingoing, $3$ and $4$ are
outgoing, and $\ell$ labels the loop momentum.

In this section, we compute the ${\cal O}(G^{2})$ eikonal phase for massless
helicity states scattering of a general spinning body. Expanding LHS of
eq.(12) to ${\cal O}(G^{2})$ yields two terms.

$\displaystyle\left.e^{i\delta}-1\right|_{{\cal
O}(G^{2})}=i\delta^{(1)}-\frac{\left[\delta^{(0)}\right]^{2}}{2}\,.$ (51)

The matching terms on the RHS of eq.(12) is the one-loop (${\cal O}(G^{2})$)
amplitude, which can be expanded on a basis of master scalar integrals, whose
coefficients can be computed via generalized unitarity methods Bern:1994zx ;
Bern:1994cg ; Bern:1997sc ; Forde:2007mi ; Kilgore:2007qr . Only the scalar
triangle integral with one massive propagator ${\cal I}_{\triangle}$ and the
box integrals ${\cal I}_{\square}$ and ${\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}$ are relevant for the
classical limit Holstein:2004dn

$\displaystyle\left.\widetilde{M}_{4}\right|_{{\cal
O}(G^{2})}=\left[c_{\triangle}\,{\cal I}_{\triangle}+c_{\square}\,{\cal
I}_{\square}+c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}\,{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}\right]\Big{|}_{\hbar\rightarrow
0}\,.$ (52)

The other triangle topology, which requires the Compton amplitude for
evaluation, only contains massless propagators and become irrelevant in the
classical limit. Thus, the kinematical setup allows us to compute higher spin
orders without encountering any spurious singularities. The contributions from
the box integrals, $c_{\square}{\cal
I}_{\square}+c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}$, are known to
reproduce the iteration terms of the eikonal phase, $-[\delta^{(0)}]^{2}/2$
Akhoury:2013yua ; KoemansCollado:2019ggb . Therefore for computing the one-
loop eikonal phase $\delta^{(1)}$ it suffices to compute the triangle
coefficient $c_{\triangle}$. However, to check consistency, we will also
compute the box and cross box coefficients and study exponentiation.
Interestingly, the exponentiation is manifest _only if_ we assume black hole
values for the spin coupling.

We will give the leading $(\lambda/b)$ corrections from the triangle
coefficient as an expansion in the spin $a/b$. This will capture the classical
gravitational Faraday effect, and reproduce the linear in spin term computed
via null geodesic parallel transport computation in ref.Ishihara:1987dv . We
will give the subleading in $a/b$ corrections up to $a^{14}$. The ${\cal
O}(G^{2})$ eikonal phase also provides a consistency test to the proposed
universal spin-tensor structure conjectured in ref.Aoude:2022trd  (and
equivalently shift symmetry in ref.Bern:2022kto ) in the black hole limit for
the scattering phase.

###  4.1 Extraction of scalar integral coefficients

It will be more convenient to use the triple cut of the loop integrals as the
seed to compute the triangle, box and crossed box coefficients. The triple cut
can be used to extract the triangle coefficient and, by imposing one more cut
condition, can lead to quadruple-cuts to generate the box and crossed box
coefficients. In our case, the triple-cut is formed by gluing a massless-
massless-graviton-graviton amplitude to a product of two three-point graviton-
massive-massive amplitudes together.

Here we list the relevant tree amplitudes with the ingoing momenta of $1$ and
$2$ and outgoing momenta $3$ and $4$. The explicit form of the massless-
massless-graviton-graviton amplitude is

$\displaystyle M_{4}^{EH}(1^{+h},2^{+2},3^{+2},4^{+h})$ $\displaystyle=$
$\displaystyle\left(-\frac{[24]}{[12]}\right)^{4-2h}\frac{[12]^{4}\langle
34\rangle^{4}}{stu}\,,$ (53)

where $\pm h$ are the helicities of the massless particles. The product of two
three-point graviton-massive-massive amplitudes can be found in
ref.Chen:2021qkk ,

$\displaystyle C_{4}(1^{s},2^{+},3^{+},4^{s})$ $\displaystyle\equiv$
$\displaystyle
M_{3}(1^{s},2^{+},P)M_{3}(-P,3^{+},4^{s})=-\frac{[2|p_{1}|3\rangle^{4}}{t^{2}}\mathcal{S}_{\\{++\\}}(1,2,3,4)\,,$
(54) $\displaystyle C_{4}(1^{s},2^{-},3^{+},4^{s})$ $\displaystyle\equiv$
$\displaystyle M_{3}(1^{s},2^{-},P)M_{3}(-P,3^{+},4^{s})=-\frac{\langle
23\rangle^{4}}{t^{2}}\mathcal{S}_{\\{-+\\}}(1,2,3,4)\,,$ (55)

where $P=p_{1}+k_{2}$ and

$\displaystyle\mathcal{S}_{\\{++\\}}(1,2,3,4)$ $\displaystyle=$
$\displaystyle\oint\prod_{k=1,2}\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{m}C_{\text{S}^{m}}z_{k}^{m}\right)e^{-i\left(\frac{v_{1}}{z_{1}}+\frac{v_{2}}{z_{2}}\right)}\,,$
(56) $\displaystyle\mathcal{S}_{\\{-+\\}}(1,2,3,4)$ $\displaystyle=$
$\displaystyle\oint\prod_{k=1,2}\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{m}C_{\text{S}^{m}}z_{k}^{m}\right)e^{-i\left(\frac{v_{1}}{z_{1}}-\frac{v_{2}^{*}}{z_{2}}+\frac{v_{3}}{z_{1}z_{2}}\right)}\,,$
(57) $\displaystyle v_{1}=-i(k_{2}\cdot S)\,,$ $\displaystyle
v_{2}=-\frac{i\left[(2m^{2}-t)(k_{2}\cdot S)+2m^{2}(q\cdot S)+2i(n\cdot
S)\right]}{2m^{2}}\,,$ $\displaystyle\hskip
28.45274ptv_{3}=\frac{-it(k_{2}\cdot S)+2(n\cdot S)}{2m^{2}}\,,$

$v_{2}^{*}\,$ is the complex conjugate of $v_{2}$ and the irrelevant quantum
contributions from $f(x)$ factor in ref.Chen:2021qkk  were omitted. The ${\cal
O}(S^{n})$ contribution is obtained by expanding the exponential to ${\cal
O}(S^{n})$, and then performing the auxiliary $z_{k}$ integrals.

According to the triple-cut diagram shown in fig.4, the cut conditions are

$\displaystyle\ell^{2}=\ell\cdot p_{1}=0\,,~{}~{}~{}2\ell\cdot q=-q^{2}\,.$
(58)

Also, the triple-cut involves two types of cuts, depending on whether the two
cut gravitons have equal helicity. For external helicity $h<2$ only the
opposite helicity cut is non-vanishing, while for gravitons, the equal
helicity is non-vanishing for the helicity non-preserving case:

$\displaystyle C_{\\{++\\}}^{\\{h\\}}$ $\displaystyle=$
$\displaystyle\left.\sum_{h^{\prime}=\pm
2}C_{4}(1^{s_{a}},\ell^{+h^{\prime}},-\ell_{1}^{+h^{\prime}},4^{s_{a}})M_{4}^{EH}(2^{+h},\ell_{1}^{-h^{\prime}},-\ell^{-h^{\prime}},3^{+h})\right|_{\mathrm{eq.}\eqref{triplec}}$
(59) $\displaystyle=$
$\displaystyle\left.\frac{\left[-\mathrm{tr}_{-}\left(k_{3}\ell
p_{1}\ell_{1}\right)\right]^{4-2h}}{t^{3}(2\ell_{1}\cdot k_{3})(2\ell\cdot
k_{3})}\left[\frac{\mathrm{tr}_{-}\left(k_{2}\ell
p_{1}\ell_{1}k_{3}p_{1}\right)}{[3|p_{1}|2\rangle}\right]^{2h}\mathcal{S}_{\\{++\\}}(1,\ell,-\ell_{1},4)\right.$
$\displaystyle\hskip
244.69386pt+\left(\ell\,\longleftrightarrow\,\ell_{1}\right)\bigg{|}_{\mathrm{eq.}\eqref{triplec}}\,,~{}~{}~{}~{}~{}~{}$
$\displaystyle C_{\\{-+\\}}^{\\{h=2\\}}$ $\displaystyle=$
$\displaystyle\sum_{h^{\prime}=\pm
2}C_{4}(1^{s_{a}},\ell_{\,}^{-h^{\prime}},-\ell_{1}^{+h^{\prime}},4^{s_{a}})M_{4}^{EH}(2^{-2},\ell_{1}^{-h^{\prime}},-\ell_{\,}^{+h^{\prime}},3^{+2})\bigg{|}_{\mathrm{eq.}\eqref{triplec}}$
(60) $\displaystyle=$ $\displaystyle\frac{m^{4}\langle
23\rangle^{4}t}{(2\ell\cdot k_{3})(2\ell_{1}\cdot
k_{3})}\mathcal{S}_{\\{-+\\}}(1,\ell,-\ell_{1},4)+\left(\,\ell\,\longleftrightarrow\,\ell_{1}\,\right)\bigg{|}_{\mathrm{eq.}\eqref{triplec}}\,,$
$\displaystyle C_{\\{-+\\}}^{\\{h\neq 2\\}}$ $\displaystyle=$ $\displaystyle
0\,,$ (61)

where $\ell_{1}=-\ell-q$,
$\mathrm{tr}_{-}(\dots)=\frac{1}{2}\mathrm{tr}[(1-\gamma_{5})\dots]$ and the
subscripts denote the sign of $\\{h_{2},h_{3}\\}$. As mentioned before, we
will use these triple-cuts to compute all the relevant integral coefficients
in the following subsections.

####  4.1.1 Extraction of box and crossed box coefficients

The box coefficient can be computed from the quadruple cut using unitarity
methods Bern:1994zx ; Bern:1994cg ; Bern:1997sc ; Forde:2007mi ;
Kilgore:2007qr . We can impose one more cut condition (see fig.4) $\ell\cdot
k_{2}=0$ to get the box cut or $\ell\cdot k_{3}=0$ to get the crossed box cut.
For each quadruple cut, there are two solutions to loop momentum which solve
the cut conditions. We use the loop momentum parametrisation of
ref.Bern:2020buy  to compute the quadruple cut, with small modifications to
account for masslessness of $k_{2}^{\mu}$.

$\displaystyle\begin{aligned} \ell^{\mu}&=\alpha p_{1}^{\mu}+\beta
k_{2}^{\mu}+\gamma q^{\mu}+\delta\eta^{\mu}\,,\\\
\eta^{\mu}&=\langle{p_{1}^{\flat}}|\sigma^{\mu}|{k_{2}}]\,,\,p_{1}^{\flat}=p_{1}-\frac{m}{2\omega}k_{2}\,.\end{aligned}$
(62)

$p_{1}^{\flat}$ is a null vector computed using Mandelstam parametrisations
eq.(13). The solutions to the cut conditions,

$\displaystyle\ell^{2}=p_{1}\cdot\ell=k_{2}\cdot\ell=0\,,\,q\cdot\ell=-\frac{q^{2}}{2}\,,$
(63)

are given as

$\displaystyle\ell_{-}^{\mu}=-\frac{q^{2}}{2q\cdot\eta}\eta^{\mu}\,,\,\ell_{+}^{\mu}=\frac{-(2q^{2}\omega)p_{1}^{\mu}+2q^{2}(m+\omega)k_{2}^{\mu}-4m\omega^{2}q^{\mu}}{mq^{2}+2q^{2}\omega+4m\omega^{2}}+\frac{q^{2}}{2q\cdot\eta}\eta^{\mu}\,.$
(64)

The coefficient of the box integral can be computed as an average over the two
loop momentum solutions $\ell_{\pm}$ ,

$\displaystyle c_{\square,+\pm}^{\\{h\\}}$
$\displaystyle=\frac{1}{2}\sum_{\ell=\ell_{\pm}}\Big{[}C_{\square}^{+\pm}(\ell,-\ell_{1})\mathcal{S}_{\\{+\pm\\}}(1,\ell,-\ell_{1},4)+C_{\square}^{+\pm}(\ell_{1},-\ell)\mathcal{S}_{\\{+\pm\\}}(1,\ell_{1},-\ell,4)\Big{]}_{\ell\cdot
k_{2}=0}\,,$ (65)

where $C_{\square}^{\pm+}$ is the part of $[C_{\\{\pm+\\}}(-2\ell\cdot
k_{2})]_{\ell\cdot k_{2}=0}$ without the spin factor
$\mathcal{S}_{\\{\pm+\\}}$. The (helicity factor stripped) crossed box
coefficient can be obtained from that of the box by the substitution
$k_{2}\to-k_{3}$, $k_{3}\to-k_{2}$, $h_{2}\to-h_{3}$, and $h_{3}\to-h_{2}$.
For example,

$\displaystyle\left.\frac{c_{\square,++}^{\\{h\\}}}{(\langle
3|p_{1}|2])^{h}}\right|_{\pm k_{2}\leftrightarrow\mp
k_{3}}=\frac{c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}},--}^{\\{h\\}}}{(\langle
2|p_{1}|3])^{h}}\,.$ (66)

Similar relations hold for other helicity sectors such as
$c_{\square,+-}^{\\{h\\}}$ and
$c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}},++}^{\\{h\\}}$.

####  4.1.2 Extraction of triangle coefficients

For the triple-cut as shown in fig.4, the cut-satisfied loop momentum can be
parametrized as

$\displaystyle\ell^{\mu}_{\pm}=-\left(\frac{t}{t-4m^{2}}\right)p_{1}^{\mu}+\left(\frac{2m^{2}}{t-4m^{2}}\right)q^{\mu}+zP_{\pm}^{\mu}+\left[\frac{m^{2}t}{4\gamma_{\pm}(t-4m^{2})}\right]\frac{Q_{\pm}^{\mu}}{z}\,,$
(67)

where the $\pm$ labels two independent solutions for the cut conditions and

$\displaystyle\gamma_{\pm}=\frac{1}{2}\left(-t\pm\sqrt{t(t-4m^{2})}\right)\,.$
(68)

The explicit forms of $P$ and $Q$ are not important but they satisfy the
following relations

$\displaystyle P_{\pm}^{2}=Q_{\pm}^{2}=P_{\pm}\cdot p_{1}=P_{\pm}\cdot
q=Q_{\pm}\cdot p_{1}=Q_{\pm}\cdot q=0\,,~{}~{}~{}P_{\pm}\cdot
Q_{\pm}=2\gamma_{\pm}\,,$ (69)

which imply a useful formula for the loop computation,

$\displaystyle P_{\pm}^{\mu}Q_{\pm}^{\nu}$ $\displaystyle=$
$\displaystyle\gamma_{\pm}\eta^{\mu\nu}+\frac{4\gamma_{\pm}}{t-4m^{2}}\left[p_{1}^{\mu}p_{1}^{\nu}+\frac{m^{2}q^{\mu}q^{\nu}}{t}+\frac{p_{1}^{\mu}q^{\nu}+q^{\mu}p_{1}^{\nu}}{2}-\frac{i(2m^{2}+\gamma_{\pm})}{2\gamma_{\pm}}\epsilon^{\mu\nu\rho\sigma}(p_{1})_{\rho}q_{\sigma}\right]\,.$

By averaging the residues of the triple-cut with loop momentum solutions
eq.(67) at $z=\infty$, we can obtain the triangle coefficients
$c_{\\{+,\pm\\}}^{\\{h\\}}$, formally as,

$\displaystyle c_{\\{+,\pm\\}}^{\\{h\\}}=\frac{1}{2}{\rm
Res}_{z=\infty}\left[\sum_{\ell=\ell_{\pm}}C_{\\{+\pm\\}}^{\\{h\\}}(\ell)\right]\,.$
(71)

Note that for the helicity preserving case, we can rewrite eq.(59) as

$\displaystyle C_{\\{+,+\\}}^{\\{h\\}}$ $\displaystyle=$
$\displaystyle-\left(\frac{[2\ell]}{[3\ell]}\right)^{2h}\frac{\mathrm{tr}\left[(1-\gamma_{5})k_{3}\ell
p_{1}\ell_{1}\right]^{4}S_{\\{++\\}}(1,\ell,-\ell_{1},4)}{t^{3}(2\ell\cdot
k_{3})(2\ell_{1}\cdot
k_{3})}\bigg{|}_{\mathrm{eq.}\eqref{triplec}}+(\ell\longleftrightarrow\ell_{1})\,.$

In the leading order in $(q/\omega)$-expansion, $[2\ell]\simeq[3\ell]$, so
$C_{\\{+,+\\}}^{\\{h\\}}$ is independent of $|h|$ and this can be understood
as a manifestation of the equivalence principle. However, in the subleading
order $C_{\\{+,+\\}}^{\\{h\\}}$ is no longer $|h|$ independent and the
equivalence principle ceases to apply.

###  4.2 Eikonal phase and its exponentiation

The main contribution to tree order eikonal phase matrix comes from helicity
preserving processes, therefore studying $c_{\square}^{++}$ is enough for
checking exponentiation. The box coefficient for the helicity non-preserving
case $c_{\square}^{+-}$ either vanishes ($|h|<2$) or is suppressed by ${\cal
O}((q/\omega)^{4})$ compared to $c_{\square}^{++}$ ($|h|=2$) which is beyond
the regime where exponentiation holds.

An interesting behaviour of the quadruple cut eq.(65) is that only one
helicity configuration contributes to each loop momentum solution $\ell_{+}$
and $\ell_{-}$, thus the sum over all cut solutions and intermediate massless
states becomes a sum over two terms

$\displaystyle c_{\square,++}^{\\{h\\}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}C_{\square}^{++}(\ell_{+},-\ell_{1+})\mathcal{S}_{\\{++\\}}(1,\ell_{+},-\ell_{1+},4)$
(73) $\displaystyle\hskip
85.35826pt+C_{\square}^{++}(\ell_{1-},-\ell_{-})\mathcal{S}_{\\{++\\}}(1,\ell_{1-},-\ell_{-},4)\Big{]}_{\ell\cdot
k_{2}=0}\,,~{}~{}~{}$

where the spin factors are

$\displaystyle{\cal S}_{\\{++\\}}(\ell_{\pm})$
$\displaystyle=\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]e^{ib_{\pm}^{\mu\nu}S_{\mu\nu}}\,,$
(74)

where ${\cal S}_{\\{++\\}}(\ell_{+})$ and ${\cal S}_{\\{++\\}}(\ell_{-})$
refer to the spin factors appearing in the first and second line of (73)
respectively. Note that the _spin factors ${\cal S}_{\\{++\\}}(\ell_{\pm})$
are distinct_. By using the loop momentum solutions (64) and the explicit form
of spin factor (56) subjected to the cut conditions the exponent of the spin
factor can be reorganised as

$\displaystyle ib_{\pm}^{\mu\nu}S_{\mu\nu}$
$\displaystyle=-\left(\frac{1}{z_{1}}+\frac{1}{z_{2}}\right)\frac{i(n\cdot
S)}{2m^{2}\omega}\mp\left(\frac{1}{z_{1}}-\frac{1}{z_{2}}\right)\frac{(q\cdot
S)}{2m}+{\cal O}(|\vec{q}|^{2},\hbar^{1})\,,$ (75)

where the subleading classical terms of order ${\cal
O}(|\vec{q}|^{2}\hbar^{0})$ have been omitted, since they do not contribute to
exponentiation. However, the distinct spin factors can be reorganised into a
common factor that can be pulled out from the sum eq.(73), due to the symmetry
of the auxiliary integrals in $z_{k}$; $b_{+}^{\mu\nu}$ becomes
$b_{-}^{\mu\nu}$ to the order considered in eq.(75), when exchanging the
auxiliary variables $z_{1}\leftrightarrow z_{2}$ for the cut solution
$\ell_{+}$. The full box coefficient is now a product of the common spin
factor and the scalar factor $c_{\square\,,\,(s=0)}^{++}$, which is the box
integral coefficient for the massive scalar-massless scattering.

$\displaystyle\begin{aligned}
c_{\square,++}^{\\{h\\}}&=\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]e^{ib^{\mu\nu}S_{\mu\nu}}\times
c_{\square\,,\,(s=0)}^{++}\left[1+{\cal
O}(|\vec{q}|^{2},\hbar^{1})\right]\,,\\\
c_{\square\,,\,(s=0)}^{++}&=\frac{C_{\square}^{++}(\ell_{+},-\ell_{1+})+C_{\square}^{++}(\ell_{1-},-\ell_{-})}{2}\,,\end{aligned}$
(76)

where we have omitted the subscript in $b_{-}^{\mu\nu}=b^{\mu\nu}$ for
brevity. Up to the considered order, the spin factor
$e^{ib^{\mu\nu}S_{\mu\nu}}$ is the same spin factor eq.(153) of the
gravitational Compton amplitude eq.(152).

Applying the rules (66) to eq.(76) yields the factorised form of the crossed
box coefficient

$\displaystyle\begin{aligned}
c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}^{++}&=\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]e^{ib^{\mu\nu}S_{\mu\nu}}\times
c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}\,,\,(s=0)}^{++}\left[1+{\cal
O}(|\vec{q}|^{2},\hbar^{1})\right]\,.\end{aligned}$ (77)

The box coefficient and the crossed box coefficient for the massive scalar are
equal in the classical limit, i.e.

$\displaystyle
c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}\,,\,(s=0)}^{++}=c_{\square\,,\,(s=0)}^{++}\left[1+{\cal
O}(\hbar^{1})\right]\,,\,c_{\square\,,\,(s=0)}^{++}=16(\kappa/2)^{4}m^{4}\omega^{4}\left[1+{\cal
O}(|\vec{q}|^{2},\hbar^{1})\right]\,,$ (78)

which means the sum of eq.(76) and eq.(77), the full box topology
contribution, also factorises into the spin factor and the scalar factor,

$\displaystyle\begin{aligned} \left[c_{\square}^{++}{\cal
I}_{\square}+c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}^{++}{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}\right]&=\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]e^{ib^{\mu\nu}S_{\mu\nu}}\times
c_{\square\,,\,(s=0)}^{++}\left[{\cal I}_{\square}+{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}\right]\,,\end{aligned}$
(79)

up to subleading ${\cal O}(|\vec{q}|^{2},\hbar^{1})$ terms, which are
irrelevant for exponentiation. The validity of approximating box integral
contributions by eq.(79) has been explicitly checked up to ${\cal O}(S^{4})$
order; the difference between the classical box and the classical crossed box
coefficient for a specified spin sector appears at quantum subleading order

$\displaystyle c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}^{++}$
$\displaystyle=c_{\square}^{++}\times\left[1+{\cal
O}\left(\hbar^{1}\right)\right]\,.$ (80)

The difference between $c_{\square}^{++}$ and $c_{\square}^{--}$, which
signifies breaking of exponentiation as $\alpha_{3}=0$ at tree order, appears
at ${\cal O}((q/\omega)^{3})$ subleading order.

$\displaystyle c_{\square}^{--}$
$\displaystyle=c_{\square}^{++}\times\left[1+{\cal
O}\left((q/\omega)^{3},\hbar^{1}\right)\right]\,.$ (81)

The difference term should be added as corrections to one-loop eikonal phase,
corresponding to NLO for $\alpha$ (contributions to $\alpha^{(1,1)}$).
Moreover, the ${\cal O}(|\vec{q}|^{2}\hbar^{0})$ subleading classical terms of
the spin factor eq.(75) also results in terms not captured by exponentiation.
These terms yield NLO corrections to $\bar{\delta}$ (contributions to
$\bar{\delta}^{(1,1)}$).

In impact parameter space, eq.(79) scales as $\propto\omega^{2}$ and provides
the iteration term of tree eikonal phase AccettulliHuber:2020oou . In
particular the non-spinning part can be matched to the square of the tree-
eikonal phase:

$\displaystyle c_{\square\,,\,(s=0)}^{++}\left[{\cal I}_{\square}+{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}\right]\Leftrightarrow-\frac{[\delta_{\text{S}^{0}}^{(0)}(b)]^{2}}{2}\,.$
(82)

The full spinning contribution in impact parameter space can be written as

$\displaystyle\left[c_{\square}^{++}{\cal
I}_{\square}+c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}^{++}{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}\right]$
$\displaystyle\Leftrightarrow\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]e^{ib^{\mu\nu}S_{\mu\nu}}\times\left[-\frac{[\delta_{\text{S}^{0}}^{(0)}(b)]^{2}}{2}\right]\,,$
(83)

where the spin factor exponent is a derivative operator

$\displaystyle ib^{\mu\nu}S_{\mu\nu}$
$\displaystyle=\left(\frac{1}{z_{1}}+\frac{1}{z_{2}}\right)\frac{\vec{d}\cdot\nabla_{b}}{2}+\left(\frac{1}{z_{1}}-\frac{1}{z_{2}}\right)\frac{i(\vec{a}_{\perp}\cdot\nabla_{b})}{2}\,,$
(84)

acting on the “square” of non-spinning tree eikonal phase. Computation of
${\cal O}(S^{n})$ terms follow the same procedure of computing the box
coefficients; the exponential is expanded to ${\cal O}(S^{n})$, the $z_{k}$
auxiliary integrals are performed, and then the derivative operators
$\nabla_{b}$ are evaluated.

Note that for general Wilson coefficients the resulting expression will be
different from “squaring” the tree eikonal phase eq.(45). For example, the
finite part is different for the ${\cal O}(S^{2})$ term

$\displaystyle\left[c_{\square}^{++}{\cal
I}_{\square}+c_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}^{++}{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}+\frac{[\delta^{(0)}(b)]^{2}}{2}\right]_{{\cal
O}(S^{2})}$
$\displaystyle=\frac{4G^{2}m^{2}\omega^{2}((\vec{a}_{\perp}\cdot\vec{b})^{2}-(\vec{d}\cdot\vec{b})^{2})}{b^{4}}(C_{\text{S}^{2}}-1)\,.$
(85)

where for the LHS we’ve taken the quadratic in spin part of the difference
between the box contribution and the square of tree eikonal phase.

Such mismatch similarly occurs for higher spin orders. In the Kerr limit
$C_{\text{S}^{j}}=1$ the spin factor simplifies to

$\displaystyle\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]e^{ib^{\mu\nu}S_{\mu\nu}}\stackrel{{\scriptstyle
C_{\text{S}^{j}}\to 1}}{{\rightarrow}}\exp\left(-\frac{i(n\cdot
S)}{m^{2}\omega}\right)\Leftrightarrow\exp\left(\vec{d}\cdot\nabla_{b}\right)\,,$
(86)

which is a constant shift by $\vec{d}=\hat{k}_{2}\times\vec{a}$ in impact
parameter space. This is consistent with the fact that the tree eikonal phase
for Kerr is obtained from the spinless case by a constant shift in impact
parameter space Bautista:2021wfy ,

$\displaystyle\delta^{(0)}_{\text{Kerr}}(\vec{b})$
$\displaystyle=\delta^{(0)}_{\text{Scalar}}(\vec{b}+\vec{d})\,,$ (87)

and exponentiation becomes manifest. The fact that the exponentiation of the
eikonal phase correctly captures the eikonal limit of the amplitude only for
black hole, or equivalently minimal coupling, is reminiscent of similar
statement for $g=2$ in exponentiation of electromagnetic scattering
Cheng:1971gf ; Meng:1972xt ; Weinberg:1971cdi ; Czyz:1975bf .

###  4.3 Results

We are interested in the helicity-preserving parts of the eikonal phase in the
classical limit, which we decompose as

$\displaystyle\bar{\delta}$
$\displaystyle:=\frac{\delta_{+,+}+\delta_{-,-}}{2}\,,\,\alpha:=\frac{\delta_{-,-}-\delta_{+,+}}{2|h|}\,.$
(88)

The superscripts and subscripts for $\delta(b)$ defined in eq.(17) will be
used to denote the same perturbative expansions of $\bar{\delta}$ and
$\alpha$. As explained in section 2.2, $\alpha$ is identified as the
polarisation plane rotation angle. For all spin sectors we find

$\displaystyle\frac{\alpha}{\bar{\delta}}$ $\displaystyle\sim{\cal
O}\left(\frac{1}{\omega b}\right)\,,$ (89)

confirming the expectation eq.(24) based on geometric optics analysis.

The leading order terms of $\bar{\delta}$ and $\alpha$ have the dependence

${\rm
LO}:\quad\bar{\delta}_{\mathrm{LO}}\sim\frac{G^{2}m^{2}\omega}{b},\quad\alpha_{\mathrm{LO}}\sim\frac{G^{2}m^{2}}{b^{2}}$
(90)

where each is multiplied by dimensionless combinations of spin vectors, impact
parameter and momenta. The results are independent of helicity, which can be
understood as the equivalence principle in the geometric optics limit. In the
Kerr limit $C_{\text{S}^{n}}=1$, we find that only particular spin structures
appear, which can be connected to the shift symmetry eq.(1) considered by Bern
et.al. Bern:2022kto . Furthermore, the leading terms
$\bar{\delta}_{\mathrm{LO}}$ were checked to be consistent with eq.(5.38) of
ref.Bautista:2021wfy . We report triangle coefficients $c_{\mathrm{LO}}$
together with $\bar{\delta}_{\mathrm{LO}}$ and $\alpha_{\mathrm{LO}}$ to
${\cal O}(S^{6})$ in the main text, and report higher spin order terms in the
ancillary file.

The NLO terms have the dependence

${\rm
NLO}:\quad\bar{\delta}_{\mathrm{NLO}}\sim\frac{G^{2}m^{2}}{b^{2}},\quad\alpha_{\mathrm{NLO}}\sim\frac{G^{2}m^{2}}{b^{3}\omega}\,,$
(91)

which do not depend on $|h|$; the equivalence principle still holds. However,
the NLO terms are purely imaginary in impact parameter space, and new spin
structures that are inconsistent with the shift symmetry eq.(1) appear in the
Kerr limit. Together with the triangle coefficients $c_{\bar{\delta}/\alpha}$,
we also report residual contributions from box topology
$c_{\square}^{\bar{\delta}/\alpha}$ that contribute to
$\bar{\delta}_{\mathrm{NLO}}$ and $\alpha_{\mathrm{NLO}}$, up to ${\cal
O}(S^{2})$ in the main text and to ${\cal O}(S^{4})$ in appendix D.

####  4.3.1 Leading order

non-vanishing coefficients in BH limit | vanishing coefficients in BH limit
---|---
$\begin{array}[]{c}{}\hfil\\\\[-11.38109pt] Z_{2,1}=1+C_{\mathrm{S}^{2}}\\\ {}\hfil\\\\[-11.38109pt] \end{array}$ | $Z_{2,2}=1-C_{\mathrm{S}^{2}}$
$\begin{array}[]{c}{}\hfil\\\\[-11.38109pt] Z_{3,1}=3C_{\mathrm{S}^{2}}+C_{\mathrm{S}^{3}}\\\ {}\hfil\\\\[-11.38109pt] \end{array}$ | $Z_{3,2}=C_{\mathrm{S}^{2}}-C_{\mathrm{S}^{3}}$
$Z_{4,1}=3C_{\mathrm{S}^{2}}^{2}+4C_{\mathrm{S}^{3}}+C_{\mathrm{S}^{4}}$ | $\begin{array}[]{c}{}\hfil\\\\[-11.38109pt] Z_{4,3}=C_{\mathrm{S}^{2}}^{2}-C_{\mathrm{S}^{4}}\\\ Z_{4,4}=3C_{\mathrm{S}^{2}}^{2}-4C_{\mathrm{S}^{3}}+C_{\mathrm{S}^{4}}\\\ {}\hfil\\\\[-11.38109pt] \end{array}$
$Z_{5,1}=10C_{\mathrm{S}^{2}}C_{\mathrm{S}^{3}}+5C_{\mathrm{S}^{4}}+C_{\mathrm{S}^{5}}$ | $\begin{array}[]{c}{}\hfil\\\\[-11.38109pt] Z_{5,2}=2C_{\mathrm{S}^{2}}C_{\mathrm{S}^{3}}-C_{\mathrm{S}^{4}}-C_{\mathrm{S}^{5}}\\\ Z_{5,3}=2C_{\mathrm{S}^{2}}C_{\mathrm{S}^{3}}-3C_{\mathrm{S}^{4}}+C_{\mathrm{S}^{5}}\\\ {}\hfil\\\\[-11.38109pt] \end{array}$
$Z_{6,1}=10C_{\mathrm{S}^{3}}^{2}+15C_{\mathrm{S}^{2}}C_{\mathrm{S}^{4}}+6C_{\mathrm{S}^{5}}+C_{\mathrm{S}^{6}}$ | $\begin{array}[]{c}{}\hfil\\\\[-11.38109pt] Z_{6,2}=2C_{\mathrm{S}^{3}}^{2}+C_{\mathrm{S}^{2}}C_{\mathrm{S}^{4}}-2C_{\mathrm{S}^{5}}-C_{\mathrm{S}^{6}}\\\ Z_{6,3}=10C_{\mathrm{S}^{3}}^{2}-15C_{\mathrm{S}^{2}}C_{\mathrm{S}^{4}}+6C_{\mathrm{S}^{5}}-C_{\mathrm{S}^{6}}\\\ Z_{6,4}=2C_{\mathrm{S}^{3}}^{2}-C_{\mathrm{S}^{2}}C_{\mathrm{S}^{4}}-2C_{\mathrm{S}^{5}}+C_{\mathrm{S}^{6}}\\\ {}\hfil\\\\[-11.38109pt] \end{array}$
Table 1: The table provides the specific combination of Wilson coefficients
used in the results of $\bar{\delta}$ and $\alpha$. Note that $Z_{4,2}$
appears in Bern:2022kto  which does not appear in our results.

The triangle coefficients are normalised by
$iM_{4}=(\kappa/2)^{4}c_{\triangle}{\cal I}_{\triangle}+\cdots$. Our results
can be summarized into the defined coefficient average
$c_{\bar{\delta}}=\frac{c_{++}+c_{--}}{2}$ and coefficient difference
$c_{\alpha}=\frac{c_{--}-c_{++}}{2|h|}$. Both $c_{\bar{\delta}}$ and
$c_{\alpha}$ are independent of $|h|$ at this order.

To linear order in spin,

$\displaystyle\begin{aligned}
c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{1}}=20im^{2}\omega(n\cdot
S)\,,\,&&c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{1}}=5m^{3}q^{2}(k_{2}\cdot
S)\,,\\\ \bar{\delta}^{\text{S}^{1}}_{\mathrm{LO}}=-\frac{5\pi
G^{2}m^{2}\omega(\vec{d}\cdot\vec{b})}{b^{3}}\,,\,&&\alpha^{\text{S}^{1}}_{\mathrm{LO}}=\frac{5\pi
G^{2}m^{2}(\hat{k}_{2}\cdot\vec{a})}{4b^{3}}\,,\end{aligned}$ (117)

where $\alpha$ matches with the null geodesic parallel transport computation
of ref.Ishihara:1987dv .

At quadratic order in spin

$\displaystyle c_{\bar{\delta},,\mathrm{LO}}^{\mathrm{S}^{2}}$
$\displaystyle=\frac{5}{16}\Big{\\{}Z_{2,1}\left[19(n\cdot
S)^{2}+7m^{2}q^{2}(k_{2}\cdot S)^{2}\right]+12Z_{2,2}m^{2}\omega^{2}(q\cdot
S)^{2}\Big{\\}}\,,$ $\displaystyle\bar{\delta}^{\text{S}^{2}}_{\mathrm{LO}}$
$\displaystyle=\frac{5\pi
G^{2}m^{2}\omega}{64}\left\\{Z_{2,1}\left[19\left(\frac{3(\vec{d}\cdot\vec{b})^{2}-d^{2}b^{2}}{b^{5}}\right)-7\frac{(\hat{k}_{2}\cdot\vec{a})^{2}}{b^{3}}\right]\right.$
$\displaystyle\hskip
170.71652pt\left.+12\,Z_{2,2}\left[\frac{3(\vec{a}\cdot\vec{b})^{2}-a_{\perp}^{2}b^{2}}{b^{5}}\right]\right\\}\,,$
$\displaystyle c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{2}}$
$\displaystyle=-\frac{15iZ_{2,1}mq^{2}(n\cdot S)(k_{2}\cdot
S)}{8\omega}\,,~{}~{}\alpha^{\text{S}^{2}}_{\mathrm{LO}}=-\frac{45Z_{2,1}\pi
G^{2}m^{2}(\hat{k}_{2}\cdot\vec{a})(\vec{d}\cdot\vec{b})}{32b^{5}}\,.$ (118)

At cubic order in spin

$\displaystyle c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{3}}$
$\displaystyle=-\frac{i(n\cdot S)}{8m^{2}\omega}\left[Z_{3,1}\left\\{9(n\cdot
S)^{2}+7m^{2}q^{2}(k_{2}\cdot S)^{2}\right\\}+20Z_{3,2}m^{2}\omega^{2}(q\cdot
S)^{2}\right]\,,$ $\displaystyle\bar{\delta}^{\mathrm{S}^{3}}_{\mathrm{LO}}$
$\displaystyle=-\frac{3\pi
G^{2}m^{2}\omega(\vec{d}\cdot\vec{b})}{32}\left\\{Z_{3,1}\left[9\left(\frac{5(\vec{d}\cdot\vec{b})^{2}-3d^{2}b^{2}}{b^{7}}\right)-7\left(\frac{(\hat{k}_{2}\cdot\vec{a})^{2}}{b^{5}}\right)\right]\right.$
$\displaystyle\hskip
28.45274pt\phantom{=asdfasdf}\left.\phantom{asdfasdfasdf}+20Z_{3,2}\left[\frac{5(\vec{a}\cdot\vec{b})^{2}-a_{\perp}^{2}b^{2}}{b^{7}}\right]\right\\}\,,$
$\displaystyle c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{3}}$
$\displaystyle=-\frac{q^{2}(k_{2}\cdot
S)}{96m\omega^{2}}\left[Z_{3,1}\left\\{39(n\cdot S)^{2}-7m^{2}q^{2}(k_{2}\cdot
S)^{2}\right\\}+60Z_{3,2}m^{2}\omega^{2}(q\cdot S)^{2}\right\\}\,,$
$\displaystyle\bar{\alpha}^{\text{S}^{3}}_{\mathrm{LO}}$
$\displaystyle=\frac{3\pi
G^{2}m^{2}(\hat{k}_{2}\cdot\vec{a})}{128}\left\\{Z_{3,1}\left[13\left(\frac{5(\vec{d}\cdot\vec{b})^{2}-d^{2}b^{2}}{b^{7}}\right)+7\frac{(\hat{k}_{2}\cdot\vec{a})^{2}}{b^{5}}\right]\right.$
$\displaystyle\hskip
56.9055pt\phantom{=asdfasdfasdf}\left.\phantom{asdf}+20Z_{3,2}\left[\frac{5(\vec{a}\cdot\vec{b})^{2}-a_{\perp}^{2}b^{2}}{b^{7}}\right]\right\\}\,.$
(119)

At quartic order in spin

$\displaystyle c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{4}}$
$\displaystyle=\frac{-1}{1536m^{4}\omega^{2}}\Big{\\{}Z_{4,1}\left[294m^{2}q^{2}(n\cdot
S)^{2}(k_{2}\cdot S)^{2}-21m^{4}q^{4}(k_{2}\cdot S)^{4}+239(n\cdot
S)^{4}\right]$ $\displaystyle~{}+60Z_{4,3}m^{2}\omega^{2}(q\cdot
S)^{2}\left[19(n\cdot S)^{2}+7m^{2}q^{2}(k_{2}\cdot
S)^{2}\right]+120\,Z_{4,4}m^{4}\omega^{4}(q\cdot S)^{4}\Big{\\}}\,,$
$\displaystyle\bar{\delta}^{\text{S}^{4}}_{\mathrm{LO}}$
$\displaystyle=\frac{3\pi
G^{2}m^{2}\omega}{256}\left\\{\frac{7Z_{4,1}}{4}\left[-7\,(\hat{k}_{2}\cdot\vec{a})^{2}\left(\frac{5(\vec{d}\cdot\vec{b})^{2}-d^{2}b^{2}}{b^{7}}\right)\right.\right.$
$\displaystyle\phantom{=}\left.\left.\phantom{asdfasdfasdf}-\frac{3(\hat{k}_{2}\cdot\vec{a})^{4}}{2b^{5}}+\frac{239}{42}\left(\frac{35(\vec{d}\cdot\vec{b})^{4}-30(\vec{d}\cdot\vec{b})^{2}d^{2}b^{2}+3d^{4}b^{4}}{b^{9}}\right)\right]\right.$
$\displaystyle\phantom{=}\left.\phantom{asdf}+\frac{5Z_{4,3}}{2}\left[-7\,(\hat{k}_{2}\cdot\vec{a})^{2}\left(\frac{5(\vec{a}\cdot\vec{b})^{2}-a_{\perp}^{2}b^{2}}{b^{7}}\right)\right.\right.$
$\displaystyle\phantom{=}\left.\left.\phantom{asdfasd}+19\left(\frac{35(\vec{a}\cdot\vec{b})^{2}(\vec{d}\cdot\vec{b})^{2}-5(\vec{a}\cdot\vec{b})^{2}d^{2}b^{2}-5(\vec{d}\cdot\vec{b})^{2}a_{\perp}^{2}b^{2}+d^{2}a_{\perp}^{2}b^{4}}{b^{9}}\right)\right]\right.$
$\displaystyle\phantom{=}\left.\left.\phantom{asdf}+5\,Z_{4,4}\left[\frac{35(\vec{a}\cdot\vec{b})^{4}-30(\vec{a}\cdot\vec{b})^{2}a_{\perp}^{2}b^{2}+3a_{\perp}^{4}b^{4}}{b^{9}}\right]\right.\right\\}\,,$
(120) $\displaystyle c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{4}}$
$\displaystyle=\frac{iq^{2}(n\cdot S)(k_{2}\cdot
S)}{384m^{3}\omega^{3}}\Big{\\{}Z_{4,1}\left[23(n\cdot
S)^{2}-7m^{2}q^{2}(k_{2}\cdot S)^{2}\right]+90Z_{4,3}m^{2}\omega^{2}(q\cdot
S)^{2}\Big{\\}}\,,$ $\displaystyle\alpha^{\text{S}^{4}}_{\mathrm{LO}}$
$\displaystyle=-\frac{5\pi
G^{2}m^{2}(\hat{k}_{2}\cdot\vec{a})(\vec{d}\cdot\vec{b})}{512}\left\\{Z_{4,1}\left[23\left(\frac{7(\vec{d}\cdot\vec{b})^{2}-3d^{2}b^{2}}{b^{9}}\right)+21\frac{(\hat{k}_{2}\cdot\vec{a})^{2}}{b^{7}}\right]\right.$
$\displaystyle\left.\hskip
213.39566pt+90Z_{4,3}\left[\frac{7(\vec{a}\cdot\vec{b})^{2}-a_{\perp}^{2}b^{2}}{b^{9}}\right]\right\\}\,.$
(121)

At quintic order in spin

$\displaystyle c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{5}}$
$\displaystyle=\frac{i(n\cdot
S)}{768m^{6}\omega^{3}}\Big{\\{}Z_{5,1}\left[-3m^{4}q^{4}(k_{2}\cdot
S)^{4}+22m^{2}q^{2}(k_{2}\cdot S)^{2}(n\cdot S)^{2}+13(n\cdot S)^{4}\right]$
$\displaystyle+12Z_{5,2}m^{2}\omega^{2}(q\cdot
S)^{2}\left[7m^{2}q^{2}(k_{2}\cdot S)^{2}+9(n\cdot
S)^{2}\right]+40Z_{5,3}m^{4}\omega^{4}(q\cdot S)^{4}\Big{\\}}\,,$
$\displaystyle\bar{\delta}^{\text{S}^{5}}_{\mathrm{LO}}$
$\displaystyle=-\frac{5\pi
G^{2}m^{2}\omega(\vec{d}\cdot\vec{b})}{1024b^{11}}\left\\{Z_{5,1}\left[-9\,b^{4}(\vec{k}_{2}\cdot\vec{a})^{4}+22(\vec{k}_{2}\cdot\vec{a})^{2}\left(3b^{4}d^{2}-7b^{2}(\vec{d}\cdot\vec{b})^{2}\right)\right.\right.$
$\displaystyle\hskip
142.26378pt\left.+13\left(-70b^{2}d^{2}(\vec{d}\cdot\vec{b})^{2}+63(\vec{d}\cdot\vec{b})^{4}+15b^{4}d^{4}\right)\right]$
$\displaystyle+12Z_{5,2}\left[7(\vec{k}_{2}\cdot\vec{a})^{2}\left(a_{\perp}^{2}b^{4}-7b^{2}(\vec{a}\cdot\vec{b})^{2}\right)-189(\vec{a}\cdot\vec{b})^{2}\left(b^{2}d^{2}-3(\vec{d}\cdot\vec{b})^{2}\right)\right.$
$\displaystyle\left.\left.+9a_{\perp}^{2}b^{2}\left(3b^{2}d^{2}-7(\vec{d}\cdot\vec{b})^{2}\right)\right]-120Z_{5,3}\left(14a_{\perp}^{2}b^{2}(\vec{a}\cdot\vec{b})^{2}-21(\vec{a}\cdot\vec{b})^{4}-a_{\perp}^{4}b^{4}\right)\right\\}\,,$
$\displaystyle c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{5}}$
$\displaystyle=\frac{q^{2}(k_{2}\cdot
S)}{15360m^{5}\omega^{4}}\Big{\\{}Z_{5,1}\left[9m^{4}q^{4}(k_{2}\cdot
S)^{4}-50m^{2}q^{2}(k_{2}\cdot S)^{2}(n\cdot S)^{2}+105(n\cdot S)^{4}\right]$
$\displaystyle+20Z_{5,2}m^{2}\omega^{2}(q\cdot S)^{2}\left[39(n\cdot
S)^{2}-7m^{2}q^{2}(k_{2}\cdot S)^{2}\right]+200m^{4}\omega^{4}Z_{5,3}(q\cdot
S)^{4}\Big{\\}}\,,$ $\displaystyle\alpha^{\text{S}^{5}}_{\mathrm{LO}}$
$\displaystyle=\frac{15\pi
G^{2}m^{2}(\vec{k}_{2}\cdot\vec{a})}{4096b^{11}}\left\\{Z_{5,1}\left[9b^{4}(\vec{k}_{2}\cdot\vec{a})^{4}-10(\vec{k}_{2}\cdot\vec{a})^{2}\left(b^{4}d^{2}-7b^{2}(\vec{d}\cdot\vec{b})^{2}\right)\right.\right.$
$\displaystyle\left.\hskip
142.26378pt+21\left(-14b^{2}d^{2}(\vec{d}\cdot\vec{b})^{2}+21(\vec{d}\cdot\vec{b})^{4}+b^{4}d^{4}\right)\right]$
$\displaystyle-4Z_{5,2}\left[7(\vec{k}_{2}\cdot\vec{a})^{2}\left(a_{\perp}^{2}b^{4}-7b^{2}(\vec{a}\cdot\vec{b})^{2}\right)+91(\vec{a}\cdot\vec{b})^{2}\left(b^{2}d^{2}-9(\vec{d}\cdot\vec{b})^{2}\right)\right.$
$\displaystyle\left.\left.-13a_{\perp}^{2}b^{2}\left(b^{2}d^{2}-7(\vec{d}\cdot\vec{b})^{2}\right)\right]+40Z_{5,3}\left(a_{\perp}^{4}b^{4}-14a_{\perp}^{2}b^{2}(\vec{a}\cdot\vec{b})^{2}+21(\vec{a}\cdot\vec{b})^{4}\right)\right\\}\,.$
(122)

At sextic order in spin

$\displaystyle c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{6}}$
$\displaystyle=\frac{1}{491520m^{8}\omega^{4}}\Big{\\{}Z_{6,1}\left[33m^{6}q^{6}(k_{2}\cdot
S)^{6}-345m^{4}q^{4}(k_{2}\cdot S)^{4}(n\cdot S)^{2}\right.$
$\displaystyle\left.\hskip 142.26378pt+1615m^{2}q^{2}(k_{2}\cdot S)^{2}(n\cdot
S)^{4}+745(n\cdot S)^{6}\right]$
$\displaystyle-40Z_{6,2}m^{2}\omega^{2}(q\cdot
S)^{2}\left[21m^{4}q^{4}(k_{2}\cdot S)^{4}-294m^{2}q^{2}(k_{2}\cdot
S)^{2}(n\cdot S)^{2}-239(n\cdot S)^{4}\right]$
$\displaystyle+320Z_{6,3}m^{6}\omega^{6}(q\cdot
S)^{6}+400Z_{6,4}m^{4}\omega^{4}(q\cdot S)^{4}\left[7m^{2}q^{2}(k_{2}\cdot
S)^{2}+19(n\cdot S)^{2}\right]\Big{\\}}\,,$
$\displaystyle\bar{\delta}^{\text{S}^{6}}_{\mathrm{LO}}$
$\displaystyle=-\frac{15\pi
G^{2}m^{2}\omega}{131072b^{13}}\left\\{Z_{6,1}\left[33b^{6}(\vec{k}_{2}\cdot\vec{a})^{6}-69(\vec{k}_{2}\cdot\vec{a})^{4}\left(b^{6}d^{2}-7b^{4}(\vec{d}\cdot\vec{b})^{2}\right)\right.\right.$
$\displaystyle\hskip
85.35826pt+323(\vec{k}_{2}\cdot\vec{a})^{2}\left(-14b^{4}d^{2}(\vec{d}\cdot\vec{b})^{2}+21b^{2}(\vec{d}\cdot\vec{b})^{4}+b^{6}d^{4}\right)$
$\displaystyle\hskip
85.35826pt\left.+149\left(-105b^{4}d^{4}(\vec{d}\cdot\vec{b})^{2}+315b^{2}d^{2}(\vec{d}\cdot\vec{b})^{4}-231(\vec{d}\cdot\vec{b})^{6}+5b^{6}d^{6}\right)\right]$
$\displaystyle+8Z_{6,2}\left[98(\vec{k}_{2}\cdot\vec{a})^{2}\left(a_{\perp}^{2}b^{4}\left(b^{2}d^{2}-7(\vec{d}\cdot\vec{b})^{2}\right)-7(\vec{a}\cdot\vec{b})^{2}\left(b^{4}d^{2}-9b^{2}(\vec{d}\cdot\vec{b})^{2}\right)\right)\right.$
$\displaystyle-21(\vec{k}_{2}\cdot\vec{a})^{4}\left(a_{\perp}^{2}b^{6}-7b^{4}(\vec{a}\cdot\vec{b})^{2}\right)+239\left(a_{\perp}^{2}b^{2}\left(-14b^{2}d^{2}(\vec{d}\cdot\vec{b})^{2}+21(\vec{d}\cdot\vec{b})^{4}+b^{4}d^{4}\right)\right.$
$\displaystyle\left.\left.-7(\vec{a}\cdot\vec{b})^{2}\left(-18b^{2}d^{2}(\vec{d}\cdot\vec{b})^{2}+33(\vec{d}\cdot\vec{b})^{4}+b^{4}d^{4}\right)\right)\right]$
$\displaystyle+64Z_{6,3}\left[-105a_{\perp}^{4}b^{4}(\vec{a}\cdot\vec{b})^{2}+315a_{\perp}^{2}b^{2}(\vec{a}\cdot\vec{b})^{4}-231(\vec{a}\cdot\vec{b})^{6}+5a_{\perp}^{6}b^{6}\right]$
$\displaystyle+80Z_{6,4}\left[19\left(-14a_{\perp}^{2}b^{2}(\vec{a}\cdot\vec{b})^{2}\left(b^{2}d^{2}-9(\vec{d}\cdot\vec{b})^{2}\right)+21(\vec{a}\cdot\vec{b})^{4}\left(b^{2}d^{2}-11(\vec{d}\cdot\vec{b})^{2}\right)\right.\right.$
$\displaystyle\left.\left.\left.+a_{\perp}^{4}b^{4}\left(b^{2}d^{2}-7(\vec{d}\cdot\vec{b})^{2}\right)\right)+7(\vec{k}_{2}\cdot\vec{a})^{2}\left(-14a_{\perp}^{2}b^{4}(\vec{a}\cdot\vec{b})^{2}+21b^{2}(\vec{a}\cdot\vec{b})^{4}+a_{\perp}^{4}b^{6}\right)\right]\right\\}\,,$
$\displaystyle c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{6}}$
$\displaystyle=-\frac{iq^{2}(k_{2}\cdot S)(n\cdot
S)}{737280m^{7}\omega^{5}}\Big{\\{}3600Z_{6,4}m^{4}\omega^{4}(q\cdot S)^{4}$
$\displaystyle\hskip 56.9055pt+Z_{6,1}\left[81m^{4}q^{4}(k_{2}\cdot
S)^{4}-290m^{2}q^{2}(k_{2}\cdot S)^{2}(n\cdot S)^{2}+465(n\cdot S)^{4}\right]$
$\displaystyle\hskip 56.9055pt+240Z_{6,2}m^{2}\omega^{2}(q\cdot
S)^{2}\left[23(n\cdot S)^{2}-7m^{2}q^{2}(k_{2}\cdot S)^{2}\right]\Big{\\}}\,,$
$\displaystyle\alpha^{\text{S}^{6}}_{\mathrm{LO}}$
$\displaystyle=-\frac{105\pi
G^{2}m^{2}(\vec{k}_{2}\cdot\vec{a})(\vec{d}\cdot\vec{b})}{65536b^{13}}\left\\{Z_{6,1}\left[27b^{4}(\vec{k}_{2}\cdot\vec{a})^{4}-58(\vec{k}_{2}\cdot\vec{a})^{2}\left(b^{4}d^{2}-3b^{2}(\vec{d}\cdot\vec{b})^{2}\right)\right.\right.$
$\displaystyle\hskip
170.71652pt\left.+31\left(-30b^{2}d^{2}(\vec{d}\cdot\vec{b})^{2}+33(\vec{d}\cdot\vec{b})^{4}+5b^{4}d^{4}\right)\right]$
$\displaystyle+16Z_{6,2}\left[23\left((\vec{a}\cdot\vec{b})^{2}\left(33(\vec{d}\cdot\vec{b})^{2}-9b^{2}d^{2}\right)+a_{\perp}^{2}b^{2}\left(b^{2}d^{2}-3(\vec{d}\cdot\vec{b})^{2}\right)\right)\right.$
(123)
$\displaystyle\left.\left.-7(\vec{k}_{2}\cdot\vec{a})^{2}\left(a_{\perp}^{2}b^{4}-9b^{2}(\vec{a}\cdot\vec{b})^{2}\right)\right]+240Z_{6,4}\left[a_{\perp}^{4}b^{4}-18a_{\perp}^{2}b^{2}(\vec{a}\cdot\vec{b})^{2}+33(\vec{a}\cdot\vec{b})^{4}\right]\right\\}\,.$

####  4.3.2 Subleading order

The residuals from the box contribution not cancelled by iteration of tree
contribute to subleading $\bar{\delta}$ and $\alpha$. The coefficients are
normalised by $iM_{4}=(\kappa/2)^{4}c_{\square}[{\cal I}_{\square}+{\cal
I}_{\,\rotatebox{90.0}{\scalebox{0.6}[0.9]{$\bowtie$}}}]+\cdots$ with
coefficient average
$c_{\square}^{\bar{\delta}}=\frac{c_{\square}^{++}+c_{\square}^{--}}{2}$ and
coefficient difference
$c_{\square}^{\bar{\alpha}}=\frac{c_{\square}^{--}-c_{\square}^{++}}{2|h|}$.
All integral coefficients ($c_{\bar{\delta}}$, $c_{\alpha}$,
$c_{\square}^{\bar{\delta}}$, and $c_{\square}^{\alpha}$) are independent of
$|h|$ at this order.

Note that $\bar{\delta}$ and $\alpha$ are purely imaginary at this order,
which are written as a sum of two terms. The first part containing $\pi$ is
from the triangle coefficients and the secont part without $\pi$ is from the
box coefficients. The box coefficient contributions were obtained by removing
${\cal O}(\omega^{2})$ terms corresponding to “squaring” of the tree eikonal
phase.

At linear order in spin

$\displaystyle
c^{\,\text{S}^{1}}_{\bar{\delta},\mathrm{NLO}}=0\,,~{}~{}~{}\bar{c}^{\,\text{S}^{1},\,\bar{\delta}}_{\square,\mathrm{NLO}}=4im^{2}q^{2}\omega(n\cdot
S)\,,~{}~{}~{}\bar{\delta}^{\,\text{S}^{1}}_{\mathrm{NLO}}=0+i\left[\frac{8G^{2}m^{2}(\vec{d}\cdot\vec{b})}{b^{4}}\right]\,,$
$\displaystyle
c^{\,\text{S}^{1}}_{\alpha,\mathrm{NLO}}=\frac{5m^{3}q^{2}(q\cdot
S)}{2}\,,~{}~{}~{}c^{\,\text{S}^{1},\,\alpha}_{\square,\mathrm{NLO}}=2m^{3}q^{4}(k_{2}\cdot
S)\,,$ $\displaystyle\alpha^{\,\text{S}^{1}}_{\mathrm{NLO}}=i\left[\frac{15\pi
G^{2}m^{2}(\vec{a}\cdot\vec{b})}{8b^{5}\omega}\right]-i\left[\frac{8G^{2}m^{2}(\hat{k_{2}}\cdot\vec{a})}{b^{4}\omega}\right]\,.$
(124)

At quadratic order in spin

$\displaystyle c^{\,\text{S}^{2}}_{\bar{\delta},\mathrm{NLO}}$
$\displaystyle=\frac{35(1+C_{\text{S}^{2}})m^{2}q^{2}(q\cdot S)(k_{2}\cdot
S)}{16}\,,$
$\displaystyle\bar{c}^{\,\text{S}^{2},\,\bar{\delta}}_{\square,\mathrm{NLO}}$
$\displaystyle=q^{2}\left(C_{\text{S}^{2}}+1\right)\left(m^{2}q^{2}(k_{2}\cdot
S)^{2}+2(n\cdot S)^{2}\right)\,,$
$\displaystyle\frac{\bar{\delta}^{\,\text{S}^{2}}_{\mathrm{NLO}}}{iG^{2}m^{2}}$
$\displaystyle=-\frac{105(1+C_{\text{S}^{2}})\pi(\hat{k}_{2}\cdot\vec{a})(\vec{a}\cdot\vec{b})}{64b^{5}}+\frac{4(C_{\text{S}^{2}}+1)\left[b^{2}a^{2}-4(\vec{d}\cdot\vec{b})^{2}\right]}{b^{6}}\,,$
$\displaystyle c^{\,\text{S}^{2}}_{\alpha,\mathrm{NLO}}$
$\displaystyle=-\frac{15i(1+C_{\text{S}^{2}})mq^{2}(q\cdot S)(n\cdot
S)}{16\omega}\,,$ $\displaystyle
c^{\,\text{S}^{1},\,\alpha}_{\square,\mathrm{NLO}}$
$\displaystyle=-\frac{imq^{4}\left(C_{\text{S}^{2}}+1\right)(k_{2}\cdot
S)(n\cdot S)}{\omega}\,,$
$\displaystyle\frac{\omega\,\alpha^{\,\text{S}^{2}}_{\mathrm{NLO}}}{iG^{2}m^{2}}$
$\displaystyle=-\frac{225(1+C_{\text{S}^{2}})\pi(\vec{a}\cdot\vec{b})(\vec{d}\cdot\vec{b})}{64b^{7}}+\frac{16(C_{\text{S}^{2}}+1)(\vec{d}\cdot\vec{b})(\hat{k_{2}}\cdot\vec{a})}{b^{6}}\,.$
(125)

$a^{2}=d^{2}+(\hat{k_{2}}\cdot\vec{a})^{2}=a_{\perp}^{2}+(\hat{k_{2}}\cdot\vec{a})^{2}$
has been used to simplify $\bar{\delta}^{\,\text{S}^{2}}_{\mathrm{NLO}}$.

###  4.4 Discussions: equivalence principle, spin structures and massless
limits

In the geometric optics limit, the observables considered in this paper—the
observables eq.(5) derived from $\bar{\delta}$ and the rotation angle
$\alpha$—has geometric interpretations in classical general relativity which
do not depend on worldline degrees of freedom. This means the observables do
not depend on the details of the particle, which can be understood as a
manifestation of the equivalence principle. For the computations considered in
this paper, the equivalence principle is realised as the independence of
$\bar{\delta}$ and $\alpha$ from helicity $|h|$ of the massless particle. The
term equivalence principle will be used to denote this helicity independence
when referring to eikonal phase variables.

Since $\lambda/b$ is the classical expansion parameter that parametrises
deviations from the geometric optics limit, we expect the equivalence
principle to be obeyed by $\bar{\delta}$ and $\alpha$ at leading order in
$\lambda/b$. Indeed this is the case. The LO $\bar{\delta}$ and $\alpha$ obeys
the equivalence principle, at least up to one-loop order. The general
expectation is that the equivalence principle for $\bar{\delta}$ and $\alpha$
is broken at sufficiently high orders in $\lambda/b$, since the corrections in
$\lambda/b$ can be interpreted as wave property corrections to the point
particle approximation; the equivalence principle only applies to the strict
point particle limit where tidal effects can be neglected, while a
characteristic property of waves is that they cannot be localised to a point.
Interestingly, the equivalence principle is still obeyed at NLO for both
$\bar{\delta}$ and $\alpha$.

Spin structures

As mentioned in the introduction, the product of the two three point
amplitudes transform nicely under the spin-shift eq.(1) in the black hole
limit. In particular, this suggests that the 2PM classical Compton amplitude
is invariant. Let us verify this explicitly.

All LO triangle coefficients $c_{\bar{\delta},\mathrm{LO}}$ and
$c_{\alpha,\mathrm{LO}}$ have the special property that their dependence on
$(q\cdot S)$ vanish in the Kerr limit $C_{\text{S}^{j}}\to 1$. That is,
projecting the spin-vector on the basis of

$\\{(q\cdot S)\,,\,(k_{2}\cdot S)\,,\,(n\cdot S)\\},$ (126)

only the latter two appears. In the all massive setup relevant for the 2PM
potential, similar phenomenon was observed, which lead to the conclusion that
the result is invariant under the spin-shift. Indeed while $(n\cdot S)$ is
manifestly invariant under the shift, the change in $(p_{2}\cdot S_{1})$ is a
_quantum_ effect: $(p_{2}\cdot S_{1})\sim{\cal O}(\hbar^{-1})$ is shifted by
$\frac{p_{2}\cdot q}{q^{2}}\sim{\cal O}(\hbar^{0})$ where we’ve used momentum
conservation $(p_{j}\cdot q)=\pm q^{2}/2$; $\frac{(p_{j}\cdot
q)/q^{2}}{(p_{j}\cdot S_{i})}={\cal O}(\hbar^{1})$, thus the shift is a
quantum effect and the classical result is invariant. In our massless-massive
setup one naively encounters a violation of this symmetry. This is because the
term $(k_{2}\cdot S)\sim{\cal O}(\hbar^{0})$ shifts as $\frac{k_{2}\cdot
q}{q^{2}}\sim{\cal O}(\hbar^{0})$, i.e. it is shifted by a classical effect.
This observation poses a puzzle as the LO triangle coefficient is not shift
invariant, but the integrand used to compute it was manifestly shift
invariant.

The apparant paradox is resolved when NLO triangle coefficients are also
included in the analysis. The NLO triangle coefficients all contain a factor
of $(q\cdot S)$, which is manifestly not shift invariant. However, the terms
arising from the shift $(q\cdot S)\to(q\cdot S)+\xi$ is _no longer subleading_
in the $\lambda/b$ expansion, and the shift-generated terms of both the LO and
NLO triangle coefficients are of the same order in the $\lambda/b$ expansion.
Explicit calculations show that the two terms cancel, and the shift symmetry
is restored by the interplay of LO and NLO triangle coefficients.

As a concrete example, consider triangle coefficients at linear order in spin.

$\displaystyle
c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{1}}=20im^{2}\omega(n\cdot
S)\,,\,c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{1}}=5m^{3}q^{2}(k_{2}\cdot
S)\,,\,c^{\,\text{S}^{1}}_{\alpha,\mathrm{NLO}}=\frac{5m^{3}q^{2}(q\cdot
S)}{2}\,.$ (127)

Under the shift eq.(1) each coefficient transforms as

$\displaystyle\begin{aligned}
c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{1}}&\to
c_{\bar{\delta},\mathrm{LO}}^{\mathrm{S}^{1}}\,,\\\
c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{1}}&\to
c_{\alpha,\mathrm{LO}}^{\mathrm{S}^{1}}-\frac{5m^{3}q^{2}}{2}\xi_{1}\,,\\\
c^{\,\text{S}^{1}}_{\alpha,\mathrm{NLO}}&\to
c^{\,\text{S}^{1}}_{\alpha,\mathrm{NLO}}+\frac{5m^{3}q^{2}}{2}\xi_{1}\,,\end{aligned}$
(128)

and shift-generated terms cancel due to the interference between LO and NLO
contributions. Similar behaviour can be observed for terms of higher order in
spin or $\lambda/b$ expansion.

Leading order terms as the massless limit

It is known that the leading order $\bar{\delta}_{\mathrm{LO}}$ can be
obtained as the massless limit of the massive case.101010JWK would like to
thank Francesco Comberiati and Leonardo de la Cruz for pointing this out.
Denoting the massive eikonal phase as $\chi$, the explicit map is

$\displaystyle\bar{\delta}_{\mathrm{LO}}$ $\displaystyle=\lim_{m_{2}\to
0}\left.\chi_{S_{1}^{n}S_{2}^{0}}\right|_{m_{2}\sigma=\omega}\,.$ (129)

The subscript denotes that we substitute $\sigma=\omega/m_{2}$ and then take
the $m_{2}\to 0$ limit. We also find that $\alpha_{\mathrm{LO}}$ can be
obtained as the massless limit of the massive case. We first align the spin of
the massive probe particle $m_{2}$ so that the probe particle has a well-
defined helicity. In the rest frame of $m_{1}$, we can write the relevant
four-vectors as

$\displaystyle\begin{aligned}
p_{1}^{\mu}=(m_{1},\vec{0})\,,\,p_{2}^{\mu}=(\omega,k\hat{z})\,,\,S_{2}^{\mu}=(hk/\omega,h\hat{z})\,,\end{aligned}$
(130)

where $\omega^{2}=(k^{2}+m_{2}^{2})$ and $h$ is the helicity defined by
$h=\frac{\vec{p}_{2}\cdot\vec{S}_{2}}{|\vec{p}_{2}|}$. The aligned
configuration sets $n\cdot S_{2}=0$, and “factoring” the helicity $h$ by
setting $p_{1}\cdot S_{2}=-m_{1}\sigma$ we find that $\alpha$ corresponds to
the massless limit of $\chi_{S_{2}^{1}}$.111111This scaling keeps Pauli-
Lubanski vector
$W_{2}^{\mu}=-\frac{1}{2}\epsilon^{\mu\alpha\beta\gamma}k_{2\alpha}J_{\beta\gamma}=m_{2}S_{2}^{\mu}$
finite in the massless limit.

$\displaystyle\alpha_{\mathrm{LO}}$ $\displaystyle=\lim_{m_{2}\to
0}\left.\chi_{S_{1}^{n}S_{2}^{1}}\right|_{m_{2}\sigma=\omega\,,\,n\cdot
S_{2}=0\,,\,p_{1}\cdot S_{2}=-\frac{m_{1}\omega}{m_{2}}}\,.$ (131)

The limit has been found by comparing to the results given in appendix B of
ref.Chen:2021qkk . It would be interesting to link the NLO imaginary terms of
the eikonal phase variables to massive eikonal phase.

Purely imaginary NLO terms

Both $\bar{\delta}_{\mathrm{NLO}}$ and $\alpha_{\mathrm{NLO}}$ are purely
imaginary in impact parameter space. Imaginary terms of the eikonal phase are
usually interpreted as signs of non-conservative effects such as dissipation
or radiation DiVecchia:2020ymx ; DiVecchia:2021ndb ; DiVecchia:2021bdo ;
Bjerrum-Bohr:2021din ; Brandhuber:2021eyq ; Alessio:2022kwv . However, the
purely imaginary NLO terms $\bar{\delta}_{\mathrm{NLO}}$ and
$\alpha_{\mathrm{NLO}}$ cannot be interpreted this way, as $2\to 2$ scattering
at one-loop order is purely elastic. On the other hand, the imaginary term is
NLO in $\lambda/b$, which is the order the wave amplitude $a$ correction
enters the geometric optics eikonal $\psi$ as can be seen in eq.(23).
Therefore a possible interpretation is that the imaginary terms encode
(de)amplification of the waves due to spin of the massive body, which may have
connections with polarising effects discussed in section 3.3 of
ref.Bautista:2021wfy .

##  5 Conclusion

In this paper, we derive the $\mathcal{O}(G^{2})$ eikonal phase of graviton
and photons scattering of general spinning objects. This corresponds to the
classical limit of the one-loop amplitude, which in terms of scalar integral
basis is given by the scalar triangle integral. We extract its coefficient by
computing the triple cut, which is the product of massless four-point
amplitude and two massive spinning three-point amplitudes. From the result we
extract the leading order gravitational Faraday effect,
$\alpha:=-\alpha_{3}/|h|$ for polarized electromagnetic and gravitational
waves, which is given as an expansion in the ratio of spin-length and impact
parameter, $a/b$. While the linear term for Kerr black hole matches with
result in Ishihara:1987dv , we provide the result for general spinning body up
to 14th order in spin.

In computing the 2PM eikonal phase, the contribution from the box and cross-
box nicely reproduces the square of the 1 PM eikonal phase in the Kerr-limit,
indicating that the exponentiation correctly captures the eikonal limit
amplitude. This is no longer true for general Wilson coefficients. Such
observation appears to be consistent with long known result that for
electromagnetic scattering Cheng:1971gf ; Meng:1972xt ; Weinberg:1971cdi ;
Czyz:1975bf , the amplitude only exponentiates for $g=2$. However this appears
to render the extraction of classical observables for neutron stars from
eikonal amplitude challenging. We leave the resolution of this issue to future
work.

In black hole limit, the $\mathcal{O}(G^{2})$ eikonal phase is invariant under
the shift transformation of the spin vector eq.(1) Aoude:2022trd ;
Bern:2022kto ; Aoude:2022thd , whose origin can be traced back to the
exponentiated form of the product of spinning three-point amplitudes. As
discussed in the introduction, for ${\cal O}(G^{2})$ calculation the spin
factor gets multiplied by a factor of

$\exp\left(-\xi\frac{(\sum_{i{\in}{+}}k_{i}\cdot
q){-}(\sum_{j{\in}{-}}k_{j}\cdot q)}{q^{2}}\right)$ (132)

which is in general non-unity. Beyond $\mathcal{O}(G^{2})$, one would have to
consider the product of $n$ three-point amplitudes in the cut, where $n\geq
3$. We expect the simple form of eq.(132) to be generalised for arbitrary
multiple products of three-point amplitudes, when the calculations are
reformulated in the framework such as HEFT Brandhuber:2021kpo ;
Brandhuber:2021eyq /HPET Damgaard:2019lfh ; Aoude:2020onz  where massive-
massless coupling vertices are treated in a homogeneous manner. If the
expectation is indeed the case, then the simplicity of the violation brings
hope that one can write down the explicit form of the “anomaly”, by inserting
this factor into the unitarity cuts. We leave this for future exploration.

Finally we find that at LO in $\lambda/b$ the eikonal phase is universal
between photons and gravitons, while deviation occurs at NNLO. Given that the
expansion in $\lambda/b$ is an expansion around the geometric optics
approximation, the violation can be interpreted as the breakdown of eikonal
approximation. Indeed we see that at this order, the overall box topology
contribution no longer matches the square of the tree eikonal phase as
required from exponentiation. It would be interesting nonetheless to explore
whether sensible observables can be extracted from these terms.

###### Acknowledgements.

We are grateful for Zvi Bern, Andreas Brandhuber, Kays Haddad, Michèle Levi,
Andrés Luna, Donal O’Connell, Gabriele Travaglini, Radu Roiban and Justin
Vines for discussions and valuable feedback on the first draft of the
manuscript. JWK would like to thank Manuel Accettulli Huber and Stefano De
Angelis for sharing computation details of ref.AccettulliHuber:2020oou  and
Andrés Luna for sharing unlisted box coefficients of ref.Kosmopoulos:2021zoq
for comparison. WMC is supported by in part by JSPS KAKENHI Grant Number
21F21317. MZC and YTH is supported by MoST Grant No. 109-2112-M-002 -020 -MY3.
JWK was supported by the Science and Technology Facilities Council (STFC)
Consolidated Grant ST/T000686/1 “Amplitudes, Strings and Duality”. JWK would
like to thank KITP for their hospitality during the stay at the programme
“High-Precision Gravitational Waves”, where part of this work was completed.
This research was supported in part by the National Science Foundation under
Grant No. NSF PHY-1748958.

##  Appendix A Eikonal phase normalisation

We examine the different conventions and resulting differences in the
definition of the eikonal phase. In the $2\to 2$ convention for the $T$-matrix
element $M_{4}$, where $S={\hbox to0.0pt{1\hss}\hskip 1.6pt\hbox{1}}+iT$, the
definitions for the eikonal phase used in the literature generically takes the
form

$\displaystyle\begin{aligned}
e^{i\delta(b)}-1&:=\int\frac{d^{D}q}{(2\pi)^{D}}\,(2\pi)\delta[V\cdot
q]\,(2\pi)\delta[W\cdot q]\,e^{-ib\cdot q}\,iM_{4}\\\
&=\int\frac{d^{D-2}q}{(2\pi)^{D-2}}\frac{i}{\sqrt{\left|V^{2}W^{2}-(V\cdot
W)^{2}\right|}}\left[e^{i\vec{b}\cdot\vec{q}}M_{4}(q)\right]\,,\end{aligned}$
(133)

where $q^{\mu}=p_{3}^{\mu}-p_{2}^{\mu}$ and different definitions differ by
the vectors $V^{\mu}$ and $W^{\mu}$ used to define the impact parameter space.
For strict exponentiation of the classical terms at the diagram level the
vectors are defined as Brandhuber:2021eyq

$\displaystyle\begin{aligned} V=p_{4}+p_{1}\,,\,W=p_{3}+p_{2}\,,\end{aligned}$
(134)

and the independent variable is average three-momentum $\vec{\bar{p}}$. The
geometric interpretation of eikonal variables in Fig.1 of ref.Bern:2020gjj
becomes natural for this choice of $V^{\mu}$ and $W^{\mu}$. In centre of
momentum (COM) frame the momenta are given as

$\displaystyle\begin{aligned}
p_{1}^{\mu}=(\sqrt{m_{1}^{2}+\vec{\bar{p}}^{2}+\vec{q}^{2}/4},-\vec{\bar{p}}+\vec{q}/2)\,&,\,p_{4}^{\mu}=(\sqrt{m_{1}^{2}+\vec{\bar{p}}^{2}+\vec{q}^{2}/4},-\vec{\bar{p}}-\vec{q}/2)\,,\\\
p_{2}^{\mu}=(\sqrt{m_{2}^{2}+\vec{\bar{p}}^{2}+\vec{q}^{2}/4},\vec{\bar{p}}-\vec{q}/2)\,&,\,p_{3}^{\mu}=(\sqrt{m_{2}^{2}+\vec{\bar{p}}^{2}+\vec{q}^{2}/4},\vec{\bar{p}}+\vec{q}/2)\,,\end{aligned}$
(135)

with $\vec{\bar{p}}\cdot\vec{q}=0$. The square of the Jacobian factor is

$\displaystyle(V\cdot
W)^{2}-V^{2}W^{2}=16|\vec{\bar{p}}|^{2}\left[\sqrt{m_{1}^{2}+\vec{\bar{p}}^{2}+\vec{q}^{2}/4}+\sqrt{m_{2}^{2}+\vec{\bar{p}}^{2}+\vec{q}^{2}/4}\right]^{2}\,.$
(136)

For both massive case ($m_{1}\neq 0$ and $m_{2}\neq 0$) we rescale
$\vec{q}\to\hbar\vec{q}$. Only the $q$-dependent terms carry extra powers of
$\hbar^{2}$ and the terms can be dropped in the classical limit.

$\displaystyle(V\cdot
W)^{2}-V^{2}W^{2}=16|\vec{\bar{p}}|^{2}\left[\sqrt{m_{1}^{2}+\vec{\bar{p}}^{2}}+\sqrt{m_{2}^{2}+\vec{\bar{p}}^{2}}\right]^{2}+{\cal
O}(\hbar^{2})\,.$ (137)

For $m_{2}=0$ we rescale $\vec{\bar{p}}\to\hbar\vec{\bar{p}}$ and
$\vec{q}\to\hbar\vec{q}$. The classical limit is given as

$\displaystyle(V\cdot
W)^{2}-V^{2}W^{2}=16\hbar^{2}m_{1}^{2}|\vec{\bar{p}}|^{2}\left[1+{\cal
O}(\hbar^{2})\right]\,.$ (138)

When comparing with the Mandelstam parametrisation using $\omega$, we have the
relation

$\displaystyle\hbar m_{1}\omega=p_{1}\cdot p_{2}$ $\displaystyle\Rightarrow$
$\displaystyle\omega=|\vec{\bar{p}}|\sqrt{1+\frac{\vec{q}^{2}}{4\vec{\bar{p}}^{2}}}\left[1+{\cal
O}(\hbar^{1})\right]\,.$ (139)

Therefore to the first subleading order in $q/\omega$ or $q/\bar{p}$, the two
ratios can be considered equal $q/\omega=q/\bar{p}\times[1+{\cal
O}((q/\bar{p})^{2})]$.

We use a simpler prescription used in the literature KoemansCollado:2019ggb ;
AccettulliHuber:2020oou

$\displaystyle\begin{aligned} V=2p_{1}\,,\,W=2p_{2}\,,\,(V\cdot
W)^{2}-V^{2}W^{2}=16m_{1}^{2}m_{2}^{2}\left(\sigma^{2}-1\right)\,,\end{aligned}$
(140)

where $\sigma:=\frac{p_{1}\cdot p_{2}}{m_{1}m_{2}}$. A slightly different
parametrisation for the COM frame momenta is more useful for this
prescription:

$\displaystyle\begin{aligned}
p_{1}^{\mu}=(\sqrt{m_{1}^{2}+\vec{{p}}^{2}},-\vec{{p}})\,&,\,p_{4}^{\mu}=(\sqrt{m_{1}^{2}+\vec{{p}}^{2}},-\vec{{p}}-\vec{q})\,,\\\
p_{2}^{\mu}=(\sqrt{m_{2}^{2}+\vec{{p}}^{2}},\vec{{p}})\,&,\,p_{3}^{\mu}=(\sqrt{m_{2}^{2}+\vec{{p}}^{2}},\vec{{p}}+\vec{q})\,,\end{aligned}$
(141)

which yields the squared Jacobian factor

$\displaystyle(V\cdot
W)^{2}-V^{2}W^{2}=16m_{1}^{2}m_{2}^{2}\left(\sigma^{2}-1\right)=16|\vec{{p}}|^{2}\left[\sqrt{m_{1}^{2}+\vec{{p}}^{2}}+\sqrt{m_{2}^{2}+\vec{{p}}^{2}}\right]^{2}\,,$
(142)

which is exact. In the classical limit, this factor is equivalent to the
factor eq.(137) up to the loop order considered in this paper if $\bar{p}$ is
identified with $p$. For $m_{2}=0$ the factor simplifies to

$\displaystyle(V\cdot W)^{2}-V^{2}W^{2}=16\hbar^{2}m_{1}^{2}\omega^{2}\,,$
(143)

which gives the normalisation factor used in eq.(12). The difference of
normalisation factors eq.(138) and eq.(143) starts at ${\cal
O}((q/\omega)^{2})$ order, therefore the two prescriptions are
indistinguishable up to the first subleading order in $q/\omega$.

##  Appendix B Linking eikonal phase to geometric optics

We present an argument why the eikonal phase $\delta(b)$ can be identified as
the geometric optics eikonal $\psi$, which can be viewed as a simplified
version of the arguments presented in ref.Cristofoli:2021jas . A related
approach is to identify the eikonal phase as the Hamilton’s principal function
for the massive case Bern:2021dqo ; Damgaard:2021ipf ; Brandhuber:2021eyq ;
Kol:2021jjc . We also comment that Hamiltonian mechanics was developed from
the correspondence between the eikonal and Hamilton’s principal function.

To build the connection, we identify the field component $f$ in eq.(22) as the
following matrix element in a specified gauge, e.g. radiation gauge

$\displaystyle
ae^{i\psi}=f=\langle{\vec{p}^{\prime}}|A^{\mu}(x)|{\vec{p};\vec{k},\varepsilon}\rangle\,,$
(144)

where $\vec{p}$ is the momentum of the incoming massive particle,
$\vec{p}^{\prime}$ is the momentum of the outgoing massive particle, and
$\vec{k}$ is the momentum of the incoming photon with polarisation
$\varepsilon$. This matrix element can be converted to $2\to 2$ scattering
amplitude of a photon scattering off a massive particle by the LSZ reduction
formula

$\displaystyle
M_{4}(p;k,\varepsilon;k^{\prime},\varepsilon^{\prime};p^{\prime})$
$\displaystyle=\int
d^{d}xe^{ik^{\prime}_{\mu}x^{\mu}}[\varepsilon^{\prime}_{\mu}]^{\ast}\partial^{\nu}\partial_{\nu}\langle{\vec{p}^{\prime}}|A^{\mu}(x)|{\vec{p};\vec{k},\varepsilon}\rangle\,,$
(145)

which can be “inverted” to obtain an asymptotic form of the matrix element in
eq.(144)

$\displaystyle\langle{\vec{p}^{\prime}}|A_{\mu}(x)|{\vec{p};\vec{k},\varepsilon}\rangle\simeq\int\frac{d^{d}k^{\prime}}{(2\pi)^{d}}~{}\varepsilon^{\prime}_{\mu}\,M_{4}(p;k,\varepsilon;k^{\prime},\varepsilon^{\prime};p^{\prime})\,e^{-ik^{\prime}_{\mu}x^{\mu}}\,,$
(146)

where $A_{4}$ is the scattering amplitude. The “sum” over $k^{\prime}$ can be
decomposed as a sum over transferred momentum $q=k^{\prime}-k$ and the
“forward momentum” $\bar{k}=(k+k^{\prime})/2$, which we set as
$\bar{k}^{\mu}=(\omega,\bar{k}_{z}\hat{z})$. Combining with the identification
eq.(144), eq.(146) is then recast as

$\displaystyle ae^{i\psi}\simeq\int\frac{d\omega
d\bar{k}_{z}}{(2\pi)^{2}}\int\frac{d^{d-2}q}{(2\pi)^{d-2}}~{}J\,e^{-i(\omega
t-\bar{k}_{z}z)}\left[e^{i\vec{b}\cdot\vec{q}}M_{4}(q)\right]\,.$ (147)

where $\vec{b}$ is the impact parameter space coordinate121212In this case,
the coordinate values of $x^{\mu}$ in eq.(146) except the time $t$ and $z$
component. and $J$ denotes the unimportant terms; the on-shell condition
deltas, the Jacobian, and the polarisation vector. Note that the sign of the
Fourier factor $e^{i\vec{b}\cdot\vec{q}}$ originates from the positive
frequency plane wave factor $e^{-ik^{\prime}_{\mu}x^{\mu}}$ with outgoing
momentum $k^{\prime}=k+q$. Using eq.(12), eq.(147) can be recast as

$\displaystyle e^{i\psi+[\log a]}\simeq\int\frac{d\omega
d\bar{k}_{z}}{(2\pi)^{2}}~{}[\delta_{\text{o-s}}]\,e^{-i(\omega
t-\bar{k}_{z}z)+i\delta(b)+[\log N^{\prime}]}\,,$ (148)

where $N^{\prime}=J/N$ is a slowly varying factor and $\delta_{\text{o-s}}$
gives the on-shell conditions. In other words, we can approximate the eikonal
$\psi$ of geometric optics as

$\displaystyle\psi(t,z,\vec{b})-i[\log a]$ $\displaystyle\simeq-\omega
t+\bar{k}_{z}z+\delta(\vec{b})-i[\log N^{\prime}]\,.$ (149)

where terms in the square brackets are “small” compared to other terms. The
polarisation dependence is given by the ratio $\partial[\log a]/\partial\psi$,
and estimating the amplitude variation length scale as $\partial[\log a]\sim
b^{-1}$ we have the ratio $\partial[\log a]/\partial\psi\sim\lambda/b$.

##  Appendix C Simplifying the spin factor for long-distance physics

The ansatz for the classical amplitude, eq.(30), is tailored to correctly
capture the $t$-channel residue. While the $t$-channel is the only physical
channel for $h\leq 1$, this is no longer true for $h>1$ and $s$\- and
$u$-channels will also become physical. For example, the gravitational Compton
amplitude eq.(B.12) of ref.Chen:2021qkk , reproduced below with $2\to 2$
continuation from $4\to 0$, has a different spin factor from eq.(30).

$\displaystyle\begin{gathered}M_{4}^{s}=M_{4}^{s=0}\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]\exp\left(-i\bar{K}^{\mu}L^{\nu}J_{\mu\nu}\right)\,,\\\
\bar{K}^{\mu}=\frac{k_{2}^{\mu}}{z_{1}}-\frac{k_{3}^{\mu}}{z_{2}}\,,\,L^{\mu}=\frac{\langle{3}|\sigma^{\mu}|{2}]}{\langle{3}|p_{1}|{2}]}\,.\end{gathered}$
(152)

The scalar factor $A_{4}^{s=0}$ is the $h=2$ case of eq.(33), and the
integration contours of the spin factor encircle the origins $z_{1,2}=0$
counter-clockwise. While the spin factor is motivated to capture the $s$\- and
$u$-channel poles, it also correctly captures the $t$-channel pole. This is
because in the $|{2}\rangle\propto|{3}\rangle$ limit ($|{2}]\propto|{3}]$
limit) the vector $L^{\mu}$ becomes proportional to $k_{2}^{\mu}$
($k_{3}^{\mu}$), so the $z_{1}$ ($z_{2}$) dependence of $\bar{K}^{\mu}$ drops
out and the spin factor of eq.(152) effectively reduces to that of eq.(30).

The gravitational Compton amplitude eq.(152) was motivated by the fact that
all physical channel poles are correctly captured, therefore it will only
differ from the “real” Compton amplitude by unknown polynomial terms, at least
up to ${\cal O}(J^{4})$ since there are no spurious poles. Since polynomial
terms become (derivatives of) Dirac delta interactions in impact parameter
space and thus cannot affect long-distance physics, it is concluded that the
amplitude eq.(152) correctly captures the long-distance physics of graviton
scattering from a spinning compact object. The conclusion poses a mystery;
while eq.(30) is _linear_ in Wilson coefficients, eq.(152) is _quadratic_. How
can their long-distance behaviours be equivalent?

In fact, there is a nontrivial cancellation for eq.(152) and the long-distance
behaviour reduces to that of eq.(30). To simplify the analysis, we recast the
spin factor exponent of eq.(152) in a form similar to eq.(38). Substituting
the Lorentz generator $J^{\mu\nu}$ by the spin tensor $S^{\mu\nu}$ and using
the definition $q^{\mu}=k_{3}^{\mu}-k_{2}^{\mu}$,

$\displaystyle-i\bar{K}^{\mu}L^{\nu}S_{\mu\nu}=-\left(\frac{1}{z_{1}}+\frac{1}{z_{2}}\right)\frac{i(n\cdot
S)}{2m^{2}\omega}+\left(\frac{1}{z_{1}}-\frac{1}{z_{2}}\right)\frac{(q\cdot
S)}{2m}+{\cal O}(|\vec{q}|^{2},\hbar^{1})\,.$ (153)

Due to the antisymmetry of $z_{1}$ and $z_{2}$, any terms with odd powers of
$(q\cdot S)$ become irrelevant and only even powers of $(q\cdot S)$ need to be
considered. The even powers of $(q\cdot S)$ can be traded for even powers of
$(n\cdot S)$

$\displaystyle\left[\frac{(q\cdot S)}{m}\right]^{2}=\left[-\frac{i(n\cdot
S)}{m^{2}\omega}\right]^{2}+{\cal O}(q^{2})\,,$ (154)

which is equivalent to the relation
$(\vec{d}\cdot\nabla_{b})^{2}=a_{\perp}^{2}\nabla_{b}^{2}-(\vec{a}_{\perp}\cdot\nabla_{b})^{2}$
in impact parameter space. The Laplacian cannot contribute to the long-
distance behaviour at tree level, therefore it can be ignored in the analysis.

We now focus on the (effective) coefficient of $(n\cdot S)^{N}$ term when the
exponentials in eq.(30) and eq.(152) are expanded to ${\cal O}(S^{N})$ order.
We will ignore the irrelevant $(N!)^{-1}$ factor from the exponential function
in both cases. For eq.(30) we get

$\displaystyle\oint\frac{dz}{2\pi
iz}\left(\sum_{j}C_{\text{S}^{j}}z^{j}\right)\left(-\frac{i(n\cdot
S)}{zm^{2}\omega}\right)^{N}=C_{\text{S}^{N}}\left(-\frac{i(n\cdot
S)}{m^{2}\omega}\right)^{N}\,.$ (155)

For eq.(152) the expansion is rather involved, as all even power contributions
from $(q\cdot S)$ must be included. Using the binomial theorem, collecting
even powers of $(q\cdot S)$, and using eq.(154) to reorganise the expression
as $(n\cdot S)^{N}$, the relevant ${\cal O}(S^{N})$ term from eq.(152) can be
written as

$\displaystyle\begin{gathered}\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]g_{N}(z_{1},z_{2})\left(-\frac{i(n\cdot
S)}{m^{2}\omega}\right)^{N}\,,\\\
g_{N}(z_{1},z_{2}):=\frac{1}{2^{N}}\sum_{M=0}^{\lfloor N/2\rfloor}{N\choose
2M}\left(\frac{1}{z_{1}}+\frac{1}{z_{2}}\right)^{N-2M}\left(\frac{1}{z_{1}}-\frac{1}{z_{2}}\right)^{2M}\,,\end{gathered}$
(158)

where $\lfloor x\rfloor$ is the floor function. The sum in
$g_{N}(z_{1},z_{2})$ can be explicitly evaluated.

$\displaystyle\begin{aligned}
g_{N}(z_{1},z_{2})&=\frac{1}{2^{N}}\sum_{J=0}^{N}\frac{1+(-1)^{J}}{2}{N\choose
J}\left(\frac{1}{z_{1}}+\frac{1}{z_{2}}\right)^{N-J}\left(\frac{1}{z_{1}}-\frac{1}{z_{2}}\right)^{J}\\\
&=\frac{1}{2^{N+1}}\left[\sum_{J=0}^{N}{N\choose
J}\left(\frac{1}{z_{1}}+\frac{1}{z_{2}}\right)^{N-J}\left(\frac{1}{z_{1}}-\frac{1}{z_{2}}\right)^{J}\right.\\\
&\phantom{=asdfasdfasdf}\left.+\sum_{J=0}^{N}{N\choose
J}\left(\frac{1}{z_{1}}+\frac{1}{z_{2}}\right)^{N-J}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}}\right)^{J}\right]\\\
&=\frac{1}{2}\left(\frac{1}{z_{1}^{N}}+\frac{1}{z_{2}^{N}}\right)\,.\end{aligned}$
(159)

Therefore eq.(158) becomes

$\displaystyle\oint\prod_{k=1,2}\left[\frac{dz_{k}}{2\pi
iz_{k}}\left(\sum_{j}C_{\text{S}^{j}}z_{k}^{j}\right)\right]\left(\frac{1}{2z_{1}^{N}}+\frac{1}{2z_{2}^{N}}\right)\left(-\frac{i(n\cdot
S)}{m^{2}\omega}\right)^{N}=C_{\text{S}^{N}}\left(-\frac{i(n\cdot
S)}{m^{2}\omega}\right)^{N}\,,$ (160)

which is the same as eq.(155). In conclusion, the ansatz eq.(30) may be used
in place of the “correct” gravitational Compton amplitude eq.(152) for
evaluating ${\cal O}(G^{1})$ eikonal phase.

##  Appendix D Eikonal phase at NLO to ${\cal O}(S^{4})$

The relation
$a^{2}=d^{2}+(\hat{k_{2}}\cdot\vec{a})^{2}=a_{\perp}^{2}+(\hat{k_{2}}\cdot\vec{a})^{2}$
was used to simplify some expressions.

###  D.1 Cubic order in spin

$\displaystyle c^{\,\text{S}^{3}}_{\bar{\delta},\mathrm{NLO}}$
$\displaystyle=-\frac{7iq^{2}(3C_{\text{S}^{2}}+C_{\text{S}^{3}})(q\cdot
S)(k_{2}\cdot S)(n\cdot S)}{8\omega}\,,$ (161)
$\displaystyle\bar{c}^{\,\text{S}^{3},\,\bar{\delta}}_{\square,\mathrm{NLO}}$
$\displaystyle=-\frac{iq^{2}(n\cdot
S)}{2m^{2}\omega}\left[(C_{\text{S}^{3}}+3C_{\text{S}^{2}})\\{m^{2}q^{2}(k_{2}\cdot
S)^{2}+(n\cdot S)^{2}\\}\right.$
$\displaystyle\phantom{=asdfasdfasdf}\left.+(C_{\text{S}^{2}}-C_{\text{S}^{3}})m^{2}\omega^{2}(q\cdot
S)^{2}\right]\,,$ (162)
$\displaystyle\frac{\bar{\delta}^{\,\text{S}^{3}}_{\mathrm{NLO}}}{iG^{2}m^{2}}$
$\displaystyle=\frac{105\pi(3C_{\text{S}^{2}}+C_{\text{S}^{3}})(\hat{k}_{2}\cdot\vec{a})(\vec{d}\cdot\vec{b})(\vec{a}\cdot\vec{b})}{32b^{7}}$
$\displaystyle\phantom{=as}+\frac{4(\vec{d}\cdot\vec{b})}{b^{8}}\left[(C_{\text{S}^{3}}+3C_{\text{S}^{2}})\left\\{6(\vec{d}\cdot\vec{b})^{2}-b^{2}(d^{2}+2a^{2})\right\\}\right.$
$\displaystyle\phantom{=asdfasdfasdf}\left.-(C_{\text{S}^{2}}-C_{\text{S}^{3}})\left\\{b^{2}a_{\perp}^{2}-6(\vec{a}\cdot\vec{b})^{2}\right\\}\right]\,,$
(163) $\displaystyle c^{\,\text{S}^{3}}_{\alpha,\mathrm{NLO}}$
$\displaystyle=-\frac{q^{2}(q\cdot
S)}{64m\omega^{2}}\left[(3C_{\text{S}^{2}}+C_{\text{S}^{3}})\left\\{13(n\cdot
S)^{2}-7m^{2}q^{2}(k_{2}\cdot S)^{2}\right\\}\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdfasdf}-20\,(C_{\text{S}^{3}}-C_{\text{S}^{2}})m^{2}\omega^{2}(q\cdot
S)^{2}\right]\,,$ (164)
$\displaystyle\bar{c}^{\,\text{S}^{3},\,\alpha}_{\square,\mathrm{NLO}}$
$\displaystyle=-\frac{q^{4}(k_{2}\cdot
S)\left((C_{\text{S}^{2}}-C_{\text{S}^{3}})m^{2}\omega^{2}(q\cdot
S)^{2}+(C_{\text{S}^{3}}+3C_{\text{S}^{2}})(n\cdot
S)^{2}\right)}{4m\omega^{2}}\,,$ (165)
$\displaystyle\frac{\omega\,\alpha^{\,\text{S}^{3}}_{\mathrm{NLO}}}{iG^{2}m^{2}}$
$\displaystyle=-\frac{75\pi(\vec{a}\cdot\vec{b})}{64b^{9}}\left[(C_{\text{S}^{3}}-C_{\text{S}^{2}})\,\\{7(\vec{a}\cdot\vec{b})^{2}-3a_{\perp}^{2}b^{2}\\}\right.$
$\displaystyle\phantom{=}\left.\phantom{asdf}-\frac{(3C_{\text{S}^{2}}+C_{\text{S}^{3}})}{20}\left\\{21(\hat{k}_{2}\cdot\vec{a})^{2}b^{2}+13\left(7(\vec{d}\cdot\vec{b})^{2}-d^{2}b^{2}\right)\right\\}\right]$
$\displaystyle\phantom{=}+\frac{4(\hat{k_{2}}\cdot\vec{a})}{b^{8}}\left[(C_{\text{S}^{3}}+3C_{\text{S}^{2}})\left\\{b^{2}d^{2}-6(\vec{d}\cdot\vec{b})^{2}\right\\}\right.$
$\displaystyle\phantom{=asdfasdfasdfasdf}\left.+(C_{\text{S}^{2}}-C_{\text{S}^{3}})\left\\{b^{2}a_{\perp}^{2}-6(\vec{a}\cdot\vec{b})^{2}\right\\}\right]\,.$
(166)

###  D.2 Quartic order in spin

$\displaystyle c^{\,\text{S}^{4}}_{\bar{\delta},\mathrm{NLO}}$
$\displaystyle=-\frac{7q^{2}(q\cdot S)(k_{2}\cdot
S)}{256m^{2}\omega^{2}}\left[10(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})m^{2}\omega^{2}(q\cdot
S)^{2}\right.$
$\displaystyle\phantom{=asdf}\left.+(3C_{\text{S}^{2}}^{2}+4C_{\text{S}^{3}}+C_{\text{S}^{4}})\left\\{7(n\cdot
S)^{2}-m^{2}q^{2}(k_{2}\cdot S)^{2}\right\\}\right]\,,$ (167)
$\displaystyle\bar{c}^{\,\text{S}^{4},\,\bar{\delta}}_{\square,\mathrm{NLO}}$
$\displaystyle=-\frac{q^{2}}{24m^{4}\omega^{2}}\left[(C_{\text{S}^{4}}+4C_{\text{S}^{3}}+3C_{\text{S}^{2}}^{2})(n\cdot
S)^{2}\left\\{2(n\cdot S)^{2}+3m^{2}q^{2}(k_{2}\cdot S)^{2}\right\\}\right.$
$\displaystyle\phantom{=asdf}\left.\phantom{asdf}+3(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})m^{2}\omega^{2}(q\cdot
S)^{2}\left\\{2(n\cdot S)^{2}+m^{2}q^{2}(k_{2}\cdot S)^{2}\right\\}\right]\,,$
(168)
$\displaystyle\frac{\bar{\delta}^{\,\text{S}^{4}}_{\mathrm{NLO}}}{iG^{2}m^{2}}$
$\displaystyle=-\frac{105\pi(\hat{k}_{2}\cdot\vec{a})(\vec{a}\cdot\vec{b})}{512b^{9}}\left[5\,(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})\,\\{7(\vec{a}\cdot\vec{b})^{2}-3a_{\perp}^{2}b^{2}\\}\right.$
$\displaystyle\phantom{=}\left.\phantom{asdf}+\frac{(3C_{\text{S}^{2}}^{2}+4C_{\text{S}^{3}}+C_{\text{S}^{4}})}{2}\left\\{3(\hat{k}_{2}\cdot\vec{a})^{2}b^{2}+7\left(7(\vec{d}\cdot\vec{b})^{2}-d^{2}b^{2}\right)\right\\}\right]$
$\displaystyle\phantom{=}-\frac{2}{b^{10}}\left[(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})\left\\{6(\vec{a}\cdot\vec{b})^{2}(8(\vec{d}\cdot\vec{b})^{2}-b^{2}a^{2})+b^{2}a_{\perp}^{2}(b^{2}a^{2}-6(\vec{d}\cdot\vec{b})^{2})\right\\}\right.$
$\displaystyle\phantom{=as}\left.+(C_{\text{S}^{4}}+4C_{\text{S}^{3}}+3C_{\text{S}^{2}}^{2})\left\\{b^{4}d^{2}a^{2}-6b^{2}(\vec{d}\cdot\vec{b})^{2}(d^{2}+a^{2})+16(\vec{d}\cdot\vec{b})^{4}\right\\}\right]\,,$
(169) $\displaystyle c^{\,\text{S}^{1}}_{\alpha,\mathrm{NLO}}$
$\displaystyle=\frac{iq^{2}(q\cdot S)(n\cdot
S)}{768m^{3}\omega^{3}}\left[(3C_{\text{S}^{2}}^{2}+4C_{\text{S}^{3}}+C_{\text{S}^{4}})\left\\{23(n\cdot
S)^{2}-21m^{2}q^{2}(k_{2}\cdot S)^{2}\right\\}\right.$
$\displaystyle\phantom{=}\left.\phantom{asdfasdfasdfasdf}+90(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})m^{2}\omega^{2}(q\cdot
S)^{2}\right]\,,$ (170)
$\displaystyle\bar{c}^{\,\text{S}^{4},\,\alpha}_{\square,\mathrm{NLO}}$
$\displaystyle=\frac{iq^{4}(k_{2}\cdot S)(n\cdot
S)}{24m^{3}\omega^{3}}\left[(C_{\text{S}^{4}}+4C_{\text{S}^{3}}+3C_{\text{S}^{2}}^{2})(n\cdot
S)^{2}\right.$
$\displaystyle\phantom{=asdf}\left.\phantom{asdfasdfasdfasdf}+3m^{2}\omega^{2}(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})(q\cdot
S)^{2}\right]\,,$ (171)
$\displaystyle\frac{\omega\,\alpha^{\,\text{S}^{4}}_{\mathrm{NLO}}}{iG^{2}m^{2}}$
$\displaystyle=-\frac{105\pi(\vec{d}\cdot\vec{b})(\vec{a}\cdot\vec{b})}{512b^{11}}\left[45\,(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})\,\\{3(\vec{a}\cdot\vec{b})^{2}-a_{\perp}^{2}b^{2}\\}\right.$
$\displaystyle\phantom{=}\left.\phantom{asdf}+\frac{(3C_{\text{S}^{2}}^{2}+4C_{\text{S}^{3}}+C_{\text{S}^{4}})}{2}\left\\{21(\hat{k}_{2}\cdot\vec{a})^{2}b^{2}+23\left(3(\vec{d}\cdot\vec{b})^{2}-d^{2}b^{2}\right)\right\\}\right]$
$\displaystyle\phantom{=}+\frac{4(\hat{k_{2}}\cdot\vec{a})(\vec{d}\cdot\vec{b})}{b^{10}}\left[(C_{\text{S}^{4}}+4C_{\text{S}^{3}}+3C_{\text{S}^{2}}^{2})\left(8(\vec{d}\cdot\vec{b})^{2}-3b^{2}d^{2}\right)\right.$
$\displaystyle\phantom{=asdf}\left.\phantom{asdf}-3(C_{\text{S}^{2}}^{2}-C_{\text{S}^{4}})\left(b^{2}a_{\perp}^{2}-8(\vec{a}\cdot\vec{b})^{2}\right)\right]\,.$
(172)

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