bigblob/.style= shape=circle, draw=blue, fill=red 11institutetext: Niels Bohr
International Academy, Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark 22institutetext: School of
Mathematics & Hamilton Mathematics Institute, Trinity College Dublin, College
Green, Dublin 2, Ireland 33institutetext: Higgs Centre for Theoretical
Physics, School of Physics and Astronomy, The University of Edinburgh, EH9
3FD, Scotland 44institutetext: Centre for Theoretical Physics, School of
Physical and Chemical Sciences, Queen Mary University of London, 327 Mile End
Road, London E1 4NS, United Kingdom

# Observables from the Spinning Eikonal

Andres Luna 2,3,4 Nathan Moynihan 3 Donal O’Connell 3 Alasdair Ross

###### Abstract

We study the classical dynamics of spinning particles using scattering
amplitudes and eikonal exponentiation. We show that observables are determined
by a simple algorithm. A wealth of complexity arises in perturbation theory as
positions, momenta and spins must be iteratively corrected at each order. Even
though we restrict ourselves to one-loop computations at quadratic order in
spin, nevertheless we encounter and resolve a number of subtle effects.
Finally, we clarify the links between our work and various other eikonal
approaches to spinning observables.

##  1 Introduction

The direct detection of gravitational waves by the LIGO/Virgo collaboration
LIGOScientific:2016aoc ; LIGOScientific:2017vwq  launched a new era in multi-
messenger astronomy, driving a demand for high-precision predictions for
binary dynamics in general relativity (GR). In this context, the connection
between scattering amplitudes in quantum field theory and the two-body problem
in GR Damour:2016gwp ; Damour:2017zjx ; Bjerrum-Bohr:2018xdl  has seen
increased interest. Indeed, in recent years, new systematic methods have been
developed for extracting potentials and physical observables from scattering
amplitudes Cheung:2018wkq ; Kosower:2018adc ; Cristofoli:2019neg ; Bjerrum-
Bohr:2019kec ; Brandhuber:2021eyq . These aim to leverage modern methods for
calculating amplitudes, including the double copy Bern:2019prr  and advanced
integration techniques Parra-Martinez:2020dzs . By manifestly maintaining
Lorentz invariance, these amplitudes-based approaches fit naturally in the
post-Minkowskian (PM) framework, where observables are expanded in Newton’s
constant $G$ while keeping their exact velocity dependence. This extraction of
classical physics from scattering amplitudes was used to produce the first
conservative two-body Hamiltonian at $\mathcal{O}(G^{3})$ and
$\mathcal{O}(G^{4})$ Bern:2019nnu ; Bern:2019crd ; Bern:2021dqo ; Bern:2021yeh
(see also Refs. Kalin:2020fhe ; Dlapa:2021npj ; Dlapa:2021vgp ; Dlapa:2022lmu
; Bjerrum-Bohr:2022ows ; Jakobsen:2023ndj ; Jakobsen:2023hig  for results at
these orders from alternative formulations).

As gravitational-wave detectors become more sensitive Punturo:2010zz ;
LISA:2017pwj ; Reitze:2019iox , the spin of the black holes or neutron stars
in the binary will play a more important role in the interpretation of
signals. Furthermore, adding spin leads to more complex dynamics, since
angular momentum can be exchanged between the bodies and the orbital motion,
and the system is no longer confined to a plane.

There has also been swift development in the study of the dynamics of spinning
objects interacting gravitationally within the PM approximation, as a result
of the application of a variety of techniques including classical GR
Bini:2017xzy ; Vines:2017hyw ; Bini:2018ywr  (see also Refs. Damgaard:2022jem
; Hoogeveen:2023bqa ). Later, a connection was established between Kerr black
holes and a novel description of scattering amplitudes for massive spinning
fields in terms of the spinor helicity formalism Guevara:2017csg ;
Guevara:2018wpp ; Chung:2018kqs ; Moynihan:2019bor ; Chung:2019yfs ;
Chung:2019duq ; Guevara:2019fsj ; Chung:2020rrz ; Vines:2018gqi . This
fascinating relation has driven a quest to find scattering amplitudes which
correctly describe the Kerr black hole to any order in spin, beyond linearised
level in $G$. This question has been approached using massive higher-spin
quantum field theories in Refs. Chiodaroli:2021eug ; Cangemi:2022abk ;
Cangemi:2022bew ; Ochirov:2022nqz ; Cangemi:2023ysz ; Alessio:2023kgf .
Alternatively, Refs. Bautista:2022wjf ; Bautista:2023sdf  used solutions of
the Teukolsky equation to derive the Compton amplitude (see Ref.
Bautista:2023szu  for a study of two-body dynamics derived from it).

More important for the purposes of this paper is the extraction of classical
effects from scattering amplitudes. This has been achieved using diverse
techniques, including spinning generalisations of effective field theories to
produce Hamiltonians Bern:2020buy ; Bern:2020uwk ; Kosmopoulos:2021zoq ;
Bern:2022kto , and of the observable-based method of Kosower, Maybee and
O’Connell (KMOC) Maybee:2019jus ; Menezes:2022tcs . Another successful
approach has been the use of heavy-particle effective theories
Damgaard:2019lfh ; Aoude:2020onz ; Aoude:2020ygw ; Aoude:2021oqj ;
Chen:2021kxt ; Chen:2022clh ; Aoude:2022trd ; Bjerrum-Bohr:2023jau ;
Haddad:2023ylx ; Aoude:2023vdk ; Bjerrum-Bohr:2023iey , which directly target
classical contributions. Important effects like radiation and absorption have
been studied in Refs. Brandhuber:2023hhl ; DeAngelis:2023lvf  and
Saketh:2022xjb ; Aoude:2023fdm ; Aoude:2023dui ; Chen:2023qzo , respectively.

The electromagnetic case, has been shown to be similar in structure
Saketh:2021sri ; Bern:2021xze , with the post-Lorentzian (PL) expansion being
the analog of the gravitational PM expansions, and so it has been used as a
toy model to study spin effects Arkani-Hamed:2019ymq ; Kim:2023drc ;
Bern:2023ity  (see also Refs. delaCruz:2020bbn ; delaCruz:2021gjp  for non-
abelian generalisations).

In parallel to the program of Amplitudes, worldline-inspired effective field
theories have been used with great success for high precision computations
Liu:2021zxr ; Jakobsen:2021lvp ; Jakobsen:2021zvh ; Jakobsen:2023ndj ;
Jakobsen:2023hig  (see also Refs. Ben-Shahar:2023djm ; Scheopner:2023rzp  for
the derivation of classical amplitudes from the worldline).

In this paper, we focus on the eikonal approximation, which has a long and
successful history of obtaining classical gravitational physics from
scattering amplitudes Ciafaloni:2018uwe ; DiVecchia:2021bdo ;
Cristofoli:2021jas ; Heissenberg:2021tzo ; Emond:2021lfy ; DiVecchia:2022nna ;
Haddad:2021znf ; Adamo:2021rfq ; DiVecchia:2022piu ; Gatica:2023iws ;
Georgoudis:2023eke  — see Ref. DiVecchia:2023frv  for a recent review. In
particular, we follow-up on Ref. Cristofoli:2021jas , refining the techniques
developed there to obtain spin effects in classical observables. In this
approximation, the incoming energy is large compared to momentum transfers,
and the amplitude can be separated into two factors. One of these factors is
the exponent of an eikonal function which we denote as $\chi$, and we refer to
it as the eikonal phase. This part of the amplitude encodes the classical
physics. The remaining terms in the amplitude, which do not exponentiate, are
quantum mechanical. The eikonal phase $\chi$ is analogous to a classical
action, and so it can be used to derive classical observables at definite
orders in Newton’s constant $G$, such as the scattering angle or the linear
impulse.

We are interested in the eikonal phase related to four-point amplitudes in a
quantum theory of massive, spinning fields. The exponentiation properties of
the gravitational impact-parameter space amplitude in this setup have been
investigated, for example in Refs. Adamo:2021rfq ; Haddad:2021znf  (see Refs.
Heissenberg:2023uvo ; Alessio:2022kwv ; Bianchi:2023lrg  for related studies).
Our main result is a simple procedure to derive classical observables,
resulting in compact formulas for the impulse and the spin kick directly from
the Eikonal phase.

We begin in section 2 by introducing a translation operator in momentum space,
and analysing its effect on the relation between the final state in a
scattering process and the eikonal phase. We show that, in conjunction with
the stationary phase conditions, it generates the relevant iterations to
compute observables. In section 3, we detail a prescription for obtaining
observables from the eikonal phase in our setup, and in section 4 we compare
to existing approaches in the literature. Finally, in section 5 we discuss our
results and conclude. We defer the presentation of results up to quadratic
order in spin in the 2PM approximation order to appendix A. We also show
explicitly the Fourier transform and other useful formulas in appendix B.

Note added: As this manuscript was in its final stages, Ref. Gatica:2023iws
appeared on the arXiv, which contains some overlapping results.

##  2 Translation Operators and Eikonal exponentiation

We begin with the assumption that eikonal exponentiation for spinning
particles is of the form Cristofoli:2021jas

$\displaystyle\exp\bigg{(}$ $\displaystyle{i\chi(\Pi(p_{1},p_{2})\cdot
x,p_{1},p_{2})/\hbar}\bigg{)}^{a_{1}^{\prime},a_{2}^{\prime}}_{a_{1},a_{2}}$
(1)
$\displaystyle\qquad=\delta^{a_{1}^{\prime}}_{a_{1}}\delta^{a_{2}^{\prime}}_{a_{2}}+i\int\hat{\mathrm{d}}^{4}q\,\hat{\delta}(2p_{1}\cdot
q)\hat{\delta}(2p_{2}\cdot q)\,e^{-iq\cdot
x/\hbar}\,\mathcal{A}_{4}(p_{1},p_{2},q)^{a_{1}^{\prime},a_{2}^{\prime}}_{a_{1},a_{2}}\,,$

where $\mathcal{A}_{4}(p_{1},p_{2},q)$ is the all-order $2\rightarrow 2$
amplitude with momentum transfer $q$, and we will ignore the quantum
remainder. The eikonal phase $\chi$ is a function of the momentum that
determines the scattering plane, e.g. the plane orthogonal to $p_{1},p_{2}$
above. We note especially that $\chi$ depends on $x$ via its projection onto
this plane $\Pi(p_{1},p_{2})\cdot x$.

Let’s now consider a two-particle initial state with spin

$\displaystyle\ket{\Psi}$
$\displaystyle\equiv\int\mathrm{d}\Phi\left(p_{1},p_{2}\right)\phi_{b}\left(p_{1},p_{2}\right)\xi_{1}^{a_{1}}\xi_{2}^{a_{2}}a^{\dagger}\left(p_{1}\right)_{a_{1}}a^{\dagger}\left(p_{2}\right)_{a_{2}}|0\rangle$
(2)
$\displaystyle\equiv\int\mathrm{d}\Phi\left(p_{1},p_{2}\right)\phi_{b}\left(p_{1},p_{2}\right)\xi_{1}^{a_{1}}\xi_{2}^{a_{2}}\left|p_{1},a_{1};p_{2},a_{2}\right\rangle$
$\displaystyle\equiv\int\mathrm{d}\Phi\left(p_{1},p_{2}\right)\phi_{b}\left(p_{1},p_{2}\right)\left|p_{1},\xi_{1};p_{2},\xi_{2}\right\rangle\,,$

where $a_{1}$ and $a_{2}$ are the spins. We evolve this state with the
$S$-matrix in the usual way to give a final state

$\displaystyle S\ket{\Psi}=\ket{\Psi}+\int$
$\displaystyle\mathrm{d}\Phi\left(p_{1}^{\prime},p_{2}^{\prime}\right)\hat{\mathrm{d}}^{4}q\,\phi_{b}\left(p_{1}^{\prime}-q,p_{2}^{\prime}+q\right)\ket{p_{1}^{\prime},a_{1}^{\prime};p_{2}^{\prime},a_{2}^{\prime}}$
(3) $\displaystyle\times\hat{\delta}\left(2p_{1}^{\prime}\cdot
q-q^{2}\right)\hat{\delta}\left(2p_{2}^{\prime}\cdot
q+q^{2}\right)i\mathcal{A}_{4}(p_{1}^{\prime},p_{2}^{\prime},q)_{a_{1}a_{2}}^{a^{\prime}_{1}a^{\prime}_{2}}\,\xi_{1}^{a_{1}}\xi_{2}^{a_{2}}\,.$

We wish to write this final state in terms of the eikonal given in eq. (1),
and we can do so by noting that the shifted delta functions in the state above
can be written in terms of the _unshifted_ functions via a translation
operator, i.e.

$\displaystyle\hat{\delta}(2p_{1}^{\prime}\cdot
q-q^{2})\hat{\delta}(2p_{2}^{\prime}\cdot q+q^{2})=e^{q\cdot
Y}\hat{\delta}(2p_{1}^{\prime}\cdot q)\hat{\delta}(2p_{2}^{\prime}\cdot
q)e^{-q\cdot Y},$ (4)

where $e^{q\cdot Y}$ acts on everything to its right, and we have defined

$\displaystyle Y_{\mu}=\frac{1}{2}\frac{\partial}{\partial
p_{2}^{\prime\mu}}-\frac{1}{2}\frac{\partial}{\partial p_{1}^{\prime\mu}}.$
(5)

Adopting the notation $\widetilde{X}=e^{q\cdot Y}Xe^{-q\cdot Y}$, and
suppressing the little group indices for the moment, we can include an
amplitude to find

$\displaystyle\hat{\delta}(2p_{1}^{\prime}\cdot
q-q^{2})\hat{\delta}(2p_{2}^{\prime}\cdot q+q^{2})\mathcal{A}_{4}$
$\displaystyle=e^{q\cdot Y}\hat{\delta}(2p_{1}^{\prime}\cdot
q)\hat{\delta}(2p_{2}^{\prime}\cdot q)e^{-q\cdot Y}\mathcal{A}_{4}$ (6)
$\displaystyle=\hat{\delta}(2\tilde{p}_{1}\cdot
q)\hat{\delta}(2\tilde{p}_{2}\cdot q)\left[\tilde{\mathcal{A}}_{4}+e^{q\cdot
Y}[e^{-q\cdot Y},\mathcal{A}_{4}]\right]\,.$

We recognise the first term on the right as the inverse Fourier transform of
the eikonal in eq. (1), i.e.

$\displaystyle i\hat{\delta}(2p_{1}\cdot q)$
$\displaystyle\hat{\delta}(2p_{2}\cdot
q)\mathcal{A}_{4}(s,q^{2})^{a_{1}^{\prime}a_{2}^{\prime}}_{a_{1},a_{2}}$ (7)
$\displaystyle=\frac{1}{\hbar^{4}}\int\mathrm{d}^{4}x\,e^{iq\cdot
x/\hbar}\left\\{\exp\bigg{(}{i\chi(\Pi(p_{1},p_{2})\cdot
x,p_{1},p_{2})/\hbar}\bigg{)}^{a_{1}^{\prime},a_{2}^{\prime}}_{a_{1},a_{2}}-\delta^{a_{1}^{\prime}}_{a_{1}}\delta^{a_{2}^{\prime}}_{a_{2}}\right\\}\,,$

and so we conclude that the final state can be written

$\displaystyle S\ket{\Psi}=$
$\displaystyle\int\mathrm{d}\Phi\left(p_{1}^{\prime},p_{2}^{\prime}\right)\hat{\mathrm{d}}^{4}q\,\phi_{b}\left(p_{1}^{\prime}-q,p_{2}^{\prime}+q\right)\ket{p_{1}^{\prime};p_{2}^{\prime}}$
(8) $\displaystyle\times\left(e^{q\cdot Y}\int\mathrm{d}^{4}xe^{iq\cdot
x}e^{i\chi(\Pi(p_{1}^{\prime},p_{2}^{\prime})\cdot
x,p_{1}^{\prime},p_{2}^{\prime})}e^{-q\cdot Y}+e^{q\cdot
Y}\hat{\delta}(2p_{1}^{\prime}\cdot q)\hat{\delta}(2p_{2}^{\prime}\cdot
q)[e^{-q\cdot Y},\mathcal{A}_{4}]\right),\,$

or, restoring the little group indices and expressing everything in terms of
shifted functions,

$\displaystyle S\ket{\Psi}=$
$\displaystyle\int\mathrm{d}\Phi\left(p_{1}^{\prime},p_{2}^{\prime}\right)\hat{\mathrm{d}}^{4}q\,\phi_{b}\left(p_{1}^{\prime}-q,p_{2}^{\prime}+q\right)\ket{p_{1}^{\prime},a_{1}^{\prime};p_{2}^{\prime},a_{2}^{\prime}}$
(9) $\displaystyle\times\left(\int\mathrm{d}^{4}xe^{iq\cdot
x}\exp\bigg{(}{i\chi(x_{\perp},\tilde{p}_{1},\tilde{p}_{2})}\bigg{)}^{a_{1}^{\prime},a_{2}^{\prime}}_{a_{1},a_{2}}+\hat{\delta}(2\tilde{p}_{1}\cdot
q)\hat{\delta}(2\tilde{p}_{2}\cdot
q)\left[\mathcal{A}_{4}-\tilde{\mathcal{A}}_{4}\right]^{a_{1}^{\prime},a_{2}^{\prime}}_{a_{1},a_{2}}\right)\xi^{a_{1}}\xi^{a_{2}}.\,$

For the final state to be solely described by the eikonal, it’s essential that
the second term becomes negligible in the classical limit. This is achieved if
the amplitudes are shift-invariant up to $\mathcal{O}(\hbar)$ corrections — we
require that

$\displaystyle\hat{\delta}(2\tilde{p}_{1}\cdot
q)\hat{\delta}(2\tilde{p}_{2}\cdot q)e^{-q\cdot Y}[e^{q\cdot
Y},\mathcal{A}(q)]=\mathcal{O}(\hbar).$ (10)

One can check that $Y^{\mu}s^{n}=Y^{\mu}t^{n}=0$, which implies that an
amplitude which is a pure polynomial in $s,t$ is unaffected by the shift.
However, the mass term is an exception, since for example

$\displaystyle\tilde{m}_{1}^{2}=m_{1}^{2}+\tilde{p}_{1}\cdot
q+\mathcal{O}(\hbar).$ (11)

Thankfully, this additional term is taken care of by the delta functions and
we conclude that the amplitude with external scalars is indeed invariant at
all orders. Including massive, spinning external particles complicates the
situation, introducing terms of the form $\epsilon\cdot p$ and
$\epsilon\cdot\epsilon$, where $\epsilon$ is a polarization tensor (or spinor)
for the massive spin-$s$ particle in the initial or final state.

For the moment, we will confine ourselves to studying linear in spin
amplitudes at one loop order, which means we need to show that $e^{-q\cdot
Y}[e^{q\cdot Y},\mathcal{A}(q)]=\mathcal{O}(\hbar)$ at $\mathcal{O}(s_{i})$.
In the classical limit, we can exchange all polarizations for spin-vectors
$s_{i}^{\mu}$, and so we conclude that the scattering amplitude is a Lorentz
scalar function of the vectors $\\{s_{1},p_{1},p_{2},q\\}$. Since we have
established that Mandelstam variables and mass terms are shift-invariant (up
to terms killed by the delta function), we only need to consider the possible
terms that arise involving spin. At linear order in spin, using the vectors at
our disposal, we can only construct the scalar
$\varepsilon(p_{1},p_{2},q,s_{1})$. This is clearly invariant under the
explicit shifts of the momentum $p_{i}\rightarrow p_{i}\pm q/2$, however we
also need to check that it is invariant under the shift of the spin. Since $q$
is of order $\hbar$, we can express the shift as a boost

$\displaystyle
s_{1}(p_{1}+\frac{1}{2}q)=s_{1}(p_{1})+\frac{1}{2m_{1}^{2}}p_{1}^{\mu}(q\cdot
s_{1}),$ (12)

such that the spin contribution also vanishes.

We have shown that $\varepsilon(p_{1},p_{2},q,s_{1})$ is invariant under the
shift, and therefore so is the amplitude:
$\tilde{\mathcal{A}}_{4}-\mathcal{A}_{4}=0$.

With this in mind, let’s consider the change in an observable $\mathbb{O}$
given by

$\displaystyle\Delta\mathbb{O}=\braket{\Psi}{S^{\dagger}\hat{\mathbb{O}}S}{\Psi}-\braket{\Psi}{\hat{\mathbb{O}}}{\Psi}=\braket{\Psi}{S^{\dagger}[\hat{\mathbb{O}},S]}{\Psi}\,.$
(13)

In the limit of large classical spin and converting all commutators into
Poisson brackets, we can express this final state in terms of purely scalar
functions as Cristofoli:2021jas

$\displaystyle\ket{\Psi}$
$\displaystyle=\frac{1}{\hbar^{4}}\int\mathrm{d}\Phi\left(p_{1}^{\prime},p_{2}^{\prime}\right)\int\hat{\mathrm{d}}^{4}q\mathrm{~{}d}^{4}x\phi_{b}\left(p_{1}^{\prime}-q,p_{2}^{\prime}+q\right)e^{iq\cdot
x/\hbar}$ (14)
$\displaystyle\times\left(\left\\{\mathbb{O}_{1}\left(p_{1}^{\prime}\right),e^{i\chi\left(x_{\perp},\tilde{p}_{1},\tilde{p}_{2};\langle\tilde{w}\rangle\right)/\hbar}\right\\}_{\text{B.H.
}}+\mathbb{O}_{1}(q)e^{i\chi\left(x_{\perp},\tilde{p}_{1},\tilde{p}_{2};\langle\tilde{w}\rangle\right)/\hbar}\right)\left|p_{1}^{\prime},\xi_{1};p_{2}^{\prime},\xi_{2}\right\rangle,$

where we have adopted the notation

$\displaystyle\left\\{\mathbb{O}_{1}\left(p_{1}^{\prime}\right),e^{i\chi\left(x_{\perp},s;\langle\tilde{w}\rangle\right)/\hbar}\right\\}_{\text{B.H.
}}\equiv
e^{i\chi/\hbar}\left(-\\{\mathbb{O},\chi\\}-\frac{1}{2}\\{\chi,\\{\mathbb{O},\chi\\}\\}+\cdots\right).$
(15)

Since all our functions are now scalar (the expectation values of the
operators), we can perform the $q,x$ integrals by stationary phase, finding
the following conditions on $q$ and $x$

$\displaystyle Q^{\mu}$ $\displaystyle=-\frac{\partial}{\partial
x_{\mu}}\chi(\Pi(p_{1}^{\prime}+\frac{1}{2}Q,p_{2}^{\prime}-\frac{1}{2}Q)\cdot
x,p_{1}^{\prime}+\frac{1}{2}Q,p_{2}^{\prime}-\frac{1}{2}Q,\tilde{s}_{1},\tilde{s}_{2})$
(16) $\displaystyle=-\frac{\partial}{\partial
x_{\mu}}\chi(x_{\perp},\tilde{p}_{1},\tilde{p}_{2},\tilde{s}_{1},\tilde{s}_{2}),$

and

$\displaystyle X^{\mu}=b^{\mu}-\frac{\partial}{\partial
Q_{\mu}}\chi(x_{\perp},\tilde{p}_{1},\tilde{p}_{2},\tilde{s}_{1},\tilde{s}_{2}).$
(17)

Notice that all functions that appear in the eikonal that are dependent on
$p_{1}^{\prime},p_{2}^{\prime}$ must be evaluated on the shifts determined by
the translation.

The final state can then be expressed as

$\displaystyle\ket{\Psi}$
$\displaystyle=\frac{1}{\hbar^{4}}\int\mathrm{d}\Phi_{b}\left(p_{1}^{\prime},p_{2}^{\prime}\right)\phi\left(p_{1}^{\prime}-Q,p_{2}^{\prime}+Q\right)e^{iQ\cdot
X/\hbar}$ (18)
$\displaystyle\times\left(\left\\{\mathbb{O}_{1}\left(p_{1}^{\prime}\right),e^{i\chi\left(x_{\perp},\tilde{p}_{1},\tilde{p}_{2};\langle\tilde{w}\rangle\right)/\hbar}\right\\}_{\text{B.H.
}}+\mathbb{O}_{1}(Q)e^{i\chi\left(x_{\perp},\tilde{p}_{1},\tilde{p}_{2};\langle\tilde{w}\rangle\right)/\hbar}\right)\left|p_{1}^{\prime},\xi_{1};p_{2}^{\prime},\xi_{2}\right\rangle,$

where now we note that

$\displaystyle\tilde{p}_{1}=p_{1}^{\prime}-\frac{1}{2}Q=p_{1}+\frac{1}{2}Q,~{}~{}~{}~{}~{}\tilde{p}_{2}=p_{2}^{\prime}+\frac{1}{2}Q=p_{2}-\frac{1}{2}Q.$
(19)

We can of course evaluate $\bra{\Psi}S^{\dagger}$ by stationary phase too,
finding

$\displaystyle\bra{\Psi}S^{\dagger}=\int
d\Phi(p_{1},p_{2})\phi_{b}(p_{1}-Q,p_{2}+Q)e^{-iQ\cdot
X/\hbar}e^{-i\chi\left(x_{\perp},\tilde{p}_{1},\tilde{p}_{2};\langle\tilde{w}\rangle\right)/\hbar}\bra{p_{1},\xi_{1};p_{2},\xi_{2}},$
(20)

such that

$\displaystyle\Delta\mathbb{O}_{1}$
$\displaystyle=\braket{\Psi}{S^{\dagger}[\mathbb{O}_{1},S]\Psi}$ (21)
$\displaystyle=\biggl{\langle}\\!\\!\\!\biggl{\langle}e^{-i\chi\left(x_{\perp},\tilde{p}_{1},\tilde{p}_{2};\langle\tilde{w}\rangle\right)/\hbar}\left\\{\mathbb{O}_{1}(p_{1}),e^{i\chi\left(x_{\perp},\tilde{p}_{1},\tilde{p}_{2};\langle\tilde{w}\rangle\right)/\hbar}\right\\}_{B.H.}+\mathbb{O}_{1}(Q)\biggr{\rangle}\\!\\!\\!\biggr{\rangle}$
$\displaystyle=\mathbb{O}_{1}(Q)-\\{\mathbb{O}_{1}(p_{1}),\chi\\}-\frac{1}{2}\\{\chi,\\{\mathbb{O}_{1}(p_{1}),\chi\\}\\}+\cdots$

If we consider the impulse, taking $\mathbb{O}_{1}=\mathbb{P}^{\mu}_{1}$, then
only the first term is non-zero and we find the expected result

$\displaystyle\Delta p_{1}^{\mu}=Q^{\mu}=-\frac{\partial}{\partial
x_{\mu}}\chi(x_{\perp},\tilde{p}_{1},\tilde{p}_{2}).$ (22)

The angular impulse, or spin kick, is determined by choosing
$\mathbb{O}_{1}=\mathbb{W}_{1}^{\mu}$, which now does give rise to an
expansions in Poisson brackets involving the spin vector given by

$\displaystyle
s_{ij}^{\mu}(p)\hat{\delta}(p-p^{\prime})=\frac{1}{m}\braket{p,a_{i}}{\mathbb{W}^{\mu}}{p^{\prime},a_{j}},$
(23)

such that

$\displaystyle\Delta
s_{1}^{\mu}=s_{1}^{\mu}\left(Q\right)-\left\\{s_{1}^{\mu},\chi\right\\}-\frac{1}{2}\left\\{\chi,\left\\{s_{1}^{\mu},\chi\right\\}\right\\}+\cdots,$
(24)

where the spin vector without indices is understood as being an expectation
value. We can think of this expression for the spin-kick in terms of a boost
and a little group rotation, covariantly expressed as

$\displaystyle
W^{\mu}_{~{}\nu}(p,n)=\exp\left(\frac{\theta}{m}\varepsilon^{\mu}_{~{}\nu}(p,n)\right).$
(25)

To see that this generates the terms above, we note that, at leading order, we
have

$\displaystyle\\{s_{1}^{\mu},\chi\\}=\frac{1}{m_{1}}\varepsilon^{\mu}\left(\frac{\partial\chi}{\partial
s_{1}},p_{1},s_{1}\right),~{}~{}~{}~{}~{}\frac{1}{2}\\{\chi,\\{s_{1}^{\mu},\chi\\}\\}=\frac{1}{2}\frac{1}{m_{1}^{2}}\varepsilon^{\mu\nu}\left(\frac{\partial\chi}{\partial
s_{1}},p_{1}\right)\varepsilon_{\nu}\left(\frac{\partial\chi}{\partial
s_{1}},p_{1},s_{1}\right),$ (26)

and similarly at higher order in $\chi$. This tells us that the infinite sum
of Poisson brackets is nothing more than a little group rotation around the
axis $n^{\mu}=\frac{1}{\theta}\frac{\partial\chi}{\partial s_{\mu}}$, i.e.

$\displaystyle
s_{1}^{\mu}-\left\\{s_{1}^{\mu},\chi\right\\}-\frac{1}{2}\left\\{\chi,\left\\{s_{1}^{\mu},\chi\right\\}\right\\}+\cdots=W^{\mu}_{~{}\nu}\left(p_{1},\frac{\partial\chi}{\partial
s_{1}}\right)s_{1}^{\nu}$ (27)

We can represent the final spin as the composition of a boost and rotation
acting on the initial spin

$\displaystyle s_{1}^{\mu}(p_{1})+\Delta s_{1}^{\mu}$
$\displaystyle=W^{\mu}_{~{}\nu}K^{\nu}_{~{}\rho}s_{1}^{\rho}(p_{1})$ (28)
$\displaystyle=\exp\left(-\frac{1}{m_{1}}\varepsilon^{\mu}_{~{}\nu}\left(\frac{\partial\chi}{\partial
s},p_{1}\right)\right)\exp\left(\frac{2}{m_{1}^{2}}Q^{[\nu}p_{1\rho]}\right)s_{1}^{\rho}(p_{1})$
$\displaystyle=s_{1}^{\mu}-(s_{1}(p_{1})\cdot
Q)p_{1}^{\mu}-\left\\{s_{1}^{\mu},\chi\right\\}-\frac{1}{2}\left\\{\chi,\left\\{s_{1}^{\mu},\chi\right\\}\right\\}+\cdots$

where on the last line we have expressed the “direct” term at first order in
$Q$ as

$\displaystyle s_{1}^{\mu}(Q)=\frac{1}{m_{1}^{2}}\left((Q\cdot
p_{1})s_{1}^{\mu}(p_{1})-(s_{1}(p_{1})\cdot Q)p_{1}^{\mu}\right),$ (29)

and dropped the higher order in $Q$ terms. It is useful to point out that the
condition $s_{1}\cdot p_{1}=(s_{1}+\Delta s_{1})(p_{1}+\Delta p)=0$ coupled
with Lorentz invariance requires that $p_{1}^{\mu}+\Delta p_{1}^{\mu}$
transform in the same way.

##  3 Classical spin dynamics from the Eikonal

To compute either the impulse or the spin kick above linear order in $G$, we
need to evaluate the eikonal on the shifted variables — we shift
$p_{i}^{\prime}\rightarrow p_{i}^{\prime}\pm\frac{1}{2}\Delta p$ along with
$s_{i}^{\prime}\rightarrow s_{i}^{\prime}\pm\frac{1}{2}\Delta s$, and we will
need the value of $x_{\perp}^{\mu}$ at the appropriate order in $G$.
Considering only particle one to be spinning, we can expand the stationary
phase condition in eq. (17) as

$\displaystyle X^{\mu}$ $\displaystyle=b^{\mu}-\frac{\partial\chi}{\partial
x_{\perp}^{\nu}}\frac{\partial x_{\perp}^{\nu}}{\partial
Q_{\mu}}-\frac{\partial\tilde{\chi}}{\partial Q_{\mu}}\bigg{|}_{Q=0}$ (30)
$\displaystyle=b^{\mu}+Q_{\nu}\frac{\partial x_{\perp}^{\nu}}{\partial
Q_{\mu}}-\frac{\partial\tilde{\chi}}{\partial Q_{\mu}}\bigg{|}_{Q=0}$

where the second term reflects the implicit $Q$ dependence on the unshifted
eikonal (where all of the dependence is in $x_{\perp}$) and the third the
explicit dependence caused by the shifts. Notice that for amplitudes involving
only scalars, the second term is absent due to the invariance of the
Mandelstam variables under the shift.

To find $x_{\perp}$, we follow the strategy in appendix B of
Cristofoli:2021jas  and write

$\displaystyle x_{\perp}^{\mu}=X^{\mu}-(X\cdot e_{0})e_{0}^{\mu}+(X\cdot
e_{q})e_{q}^{\mu},$ (31)

where $x_{\perp}\cdot e_{0}=x_{\perp}\cdot e_{q}=0$ and

$\displaystyle\left\\{\begin{array}[]{l}e_{0}^{\mu}\equiv
N_{0}\left(\tilde{p}_{1}^{\mu}+\tilde{p}_{2}^{\mu}\right)\\\ e_{q}^{\mu}\equiv
N_{q}\left(\tilde{p}_{1}^{\mu}-\tilde{p}_{2}^{\mu}\right)-N_{0q}\left(\tilde{p}_{1}^{\mu}+\tilde{p}_{2}^{\mu}\right)\end{array}\right.$
(32)

Let’s concern ourselves with finding $x_{\perp}$ to linear order in both $G$
and spin $s_{1}^{\mu}$. At this order, the classical dependence111As noted
earlier, the Mandelstam variables are not explicitly dependent on
$\tilde{p}_{i}$ since they are shift invariant. on spin and $\tilde{p}_{i}$ is
always of the form
$\varepsilon(x_{\perp},\tilde{p}_{1},\tilde{p}_{2},\tilde{s}_{1})$, whose
coefficient doesn’t explicitly depend on $\tilde{p}_{1}$, $\tilde{p}_{2}$ or
$\tilde{s}_{1}$. Ultimately, we wish to express observables in terms of
initial and final spins and momentum $s_{i}$ and $p_{i}$, so we will need to
relate these to their shifted counterparts $\tilde{s}_{i}$ and
$\tilde{p}_{i}$, as these are what will show up in the eikonal. Lorentz
invariance and the spin orthogonality condition $s\cdot p=(s+\Delta
s)\cdot(p+\Delta p)=0$ ensure that both $p_{i}^{\mu}$ and $s_{i}^{\mu}$ must
undergo the same Lorentz transformation relating them to their shifted
counterparts, i.e.

$\displaystyle\tilde{s}_{1}^{\mu}(\tilde{p}_{1})=\Lambda\left(\frac{1}{2}Q,p_{1}^{\prime}\right)^{\mu}_{~{}\nu}s_{1}^{\nu}(p_{1}^{\prime}),~{}~{}~{}~{}~{}\tilde{p}_{1}^{\mu}=\Lambda\left(\frac{1}{2}Q,p_{1}^{\prime}\right)^{\mu}_{~{}\nu}p_{1}^{\prime\nu}.$
(33)

Since by eq. (28) we know how the spin-vector transforms during the course of
a collision, we must assume that after the shift we have performed half of
this transformation, and thus the correct Lorentz transformation is of the
form

$\displaystyle\Lambda\left(\frac{1}{2}Q,p_{1}^{\prime}\right)^{\mu}_{~{}\nu}=\exp\left(-\frac{1}{2m_{1}}\varepsilon^{\mu}_{~{}\rho}\left(\frac{\partial\chi}{\partial
s},p_{1}\right)\right)\exp\left(\frac{1}{m_{1}^{2}}Q^{[\rho}p_{1\nu]}\right),$
(34)

which gives

$\displaystyle\tilde{s}_{1}^{\mu}(\tilde{p}_{1})=s_{1}^{\mu}(p_{1})+\frac{1}{2}\Delta
s_{1}^{\mu},~{}~{}~{}~{}~{}~{}\tilde{p}_{1}^{\mu}=p_{1}^{\mu}+\frac{1}{2}\Delta
p^{\mu}.$ (35)

###  3.1 Shifted Eikonal Functions

In order to compute observables at order $G^{2}$, we will need the tree-level
and one-loop eikonal functions evaluated on the shifted momentum
$\tilde{p}_{i}$ and shifted spin $\tilde{s}_{i}$. We will restrict our
attention to the linear in spin, $\mathcal{O}(G^{2})$ sector for now, and so
we will consider the spinor-scalar case, where the tree-level shifted phase is
given by

$\displaystyle\chi_{1~{}b}^{a}$
$\displaystyle=\int\hat{d}^{4}q~{}\hat{\delta}(2p_{1}^{\prime}\cdot
q)\hat{\delta}(2p_{2}^{\prime}\cdot q)e^{-iq\cdot
x_{\perp}}\mathcal{M}^{tree}_{4}[1,1^{\prime},2,2^{\prime}]^{a}_{~{}b}\Bigg{|}_{p_{1}^{\prime}\rightarrow\tilde{p}_{1},~{}~{}~{}p_{2}^{\prime}\rightarrow\tilde{p}_{2}}.$
(36)

The classical contributions to the $t$-channel unshifted spinor-scalar
amplitude, in terms of $p_{1}^{\prime}$ and $p_{2}^{\prime}$, is given by

$\displaystyle\mathcal{M}_{4~{}b}^{a}$
$\displaystyle=-\frac{\kappa^{2}}{16q^{2}}\bigg{[}\frac{\alpha_{1}^{(0)}}{m_{1}}\bar{u}^{a}(p_{1}^{\prime})u_{b}(p_{1}^{\prime}-q)-\frac{2\alpha_{1}^{(1,1)}}{m_{1}^{2}m_{2}}\varepsilon_{\mu\nu\rho\sigma}q^{\nu}p_{2}^{\prime\mu}p_{1}^{\prime\rho}s^{\sigma}(p_{1}^{\prime})^{a}_{~{}b}\bigg{]}$
(37)

where $a,b$ are little group indices and we have ignored terms set to zero by
the delta functions in the eikonal. The coefficients in the case of gravity
are given by

$\displaystyle\alpha_{1}^{(0)}=4m_{1}^{2}m_{2}^{2}(2\gamma^{2}-1),~{}~{}~{}~{}~{}\alpha_{1}^{(1,1)}=-8im_{1}^{2}m_{2}^{2}\gamma.$
(38)

At this point, it is tempting to express
$\bar{u}^{a}(p_{1}^{\prime})u_{b}(p_{1}^{\prime}-q)=2m_{1}\delta^{a}_{b}+\mathcal{O}(\hbar)$,
however this would be premature: we need to make sure we have found all
possible spin structures before taking the $\hbar\rightarrow 0$ limit. We will
perform this step after evaluating the amplitude on the shifted momenta
$\tilde{p}_{i}$, taking $p_{i}^{\prime}\rightarrow\tilde{p}_{i}$. We find

$\displaystyle\mathcal{M}_{4~{}b}^{a}$
$\displaystyle=-\frac{\kappa^{2}}{16q^{2}}\bigg{[}\frac{\alpha_{1}^{(0)}}{m_{1}}\bar{u}^{a}(\tilde{p}_{1})u_{b}(\tilde{p}_{1}-q)-\frac{2\alpha_{1}^{(1,1)}}{m_{1}^{2}m_{2}}\varepsilon_{\mu\nu\rho\sigma}q^{\nu}\tilde{p}_{2}^{\mu}\tilde{p}_{1}^{\rho}s^{\sigma}(\tilde{p}_{1})^{a}_{~{}b}\bigg{]}$
(39)

In the perturbative regime where $p_{i}^{\prime}\gg Q$, we can write the
remaining spinor contraction as

$\displaystyle\bar{u}^{a}(\tilde{p}_{1})u_{b}(\tilde{p}_{1}-q)$
$\displaystyle=\bar{u}^{a}(p_{1}^{\prime}-\frac{1}{2}Q)e^{\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}}u_{b}(p_{1}^{\prime}-\frac{1}{2}Q)$
(40)
$\displaystyle=\bar{u}^{a}(p_{1}^{\prime})e^{-\frac{i}{2}\bar{\omega}_{\alpha\beta}S^{\alpha\beta}}e^{\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}}e^{\frac{i}{2}\bar{\omega}_{\rho\sigma}S^{\rho\sigma}}u_{b}(p_{1}^{\prime}),$

where222We take $\tilde{p}_{1}^{2}=m_{1}^{2}+\frac{Q^{2}}{4}\simeq m_{1}^{2}$
here.

$\displaystyle\omega_{\mu\nu}=-\frac{1}{m_{1}^{2}}(\tilde{p}_{1\mu}q_{\nu}-\tilde{p}_{1\nu}q_{\mu}),~{}~{}~{}~{}~{}\bar{\omega}_{\mu\nu}=-\frac{1}{2m_{1}^{2}}(p^{\prime}_{1\mu}Q_{\nu}-p^{\prime}_{1\nu}Q_{\mu}).$
(41)

We can use the BCH formula to express the exponentials to first order as

$\displaystyle
e^{-\frac{i}{2}\bar{\omega}_{\alpha\beta}S^{\alpha\beta}}e^{\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}}e^{\frac{i}{2}\bar{\omega}_{\rho\sigma}S^{\rho\sigma}}$
$\displaystyle=e^{\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}}+\frac{i}{2}\bar{\omega}_{\alpha\beta}[e^{\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}},S^{\alpha\beta}]+\mathcal{O}(\bar{\omega}^{2})$
(42)
$\displaystyle=e^{\frac{i}{2}\omega_{\mu\nu}S^{\mu\nu}}-\frac{1}{4}\bar{\omega}_{\mu\nu}\omega_{\alpha\beta}[S^{\mu\nu},S^{\alpha\beta}]+\mathcal{O}(\bar{\omega}^{2}).$

Using the Lorentz algebra, along with
$p_{1\mu}^{\prime}\bar{u}^{a}(p_{1}^{\prime})S^{\mu\nu}u_{b}(p_{1}^{\prime})=0$,
we find

$\displaystyle\bar{u}^{a}(\tilde{p}_{1})u_{b}(\tilde{p}_{1}-q)$
$\displaystyle=2m_{1}\delta^{a}_{~{}b}+\frac{i}{2m_{1}^{2}}Q_{\mu}q_{\nu}\bar{u}^{a}(p_{1}^{\prime})S^{\mu\nu}u_{b}(p_{1}^{\prime})$
(43)
$\displaystyle=2m_{1}\delta^{a}_{~{}b}+\frac{i}{m_{1}^{2}}\varepsilon(Q,q,p_{1}^{\prime},s^{a}_{1~{}b}(p_{1}^{\prime})),$

where we have used

$\displaystyle S^{\mu\nu
a}_{~{}~{}~{}~{}b}=\frac{1}{2m}\bar{u}^{a}(p)S^{\mu\nu}u_{b}(p)=\frac{1}{m}\varepsilon^{\mu\nu}(p,s^{a}_{~{}b}).$
(44)

Plugging this in then gives the tree-level amplitude shifted to
$\mathcal{O}(Q,\hbar)$ as required

$\displaystyle\tilde{\mathcal{M}}_{4~{}b}^{a}$
$\displaystyle=-\frac{\kappa^{2}}{8q^{2}}\bigg{[}\alpha_{1}^{(0)}\left(\delta^{a}_{~{}b}+\frac{i}{m_{1}^{3}}\varepsilon\left(Q,q,p_{1}^{\prime},s^{a}_{1~{}b}(p_{1}^{\prime})\right)\right)-\frac{\alpha_{1}^{(1,1)}}{m_{1}^{2}m_{2}}\varepsilon(q,\tilde{p}_{1},\tilde{p}_{2},s(\tilde{p}_{1})^{a}_{~{}b})\bigg{]}.$
(45)

With this in hand, we can compute the second component of $X^{\mu}$, for which
we will need

$\displaystyle\frac{\partial\tilde{\chi}_{1}}{\partial Q_{\mu}}\Bigg{|}_{Q=0}$
$\displaystyle=\int\hat{d}^{4}q~{}\hat{\delta}(2\tilde{p}_{1}\cdot
q)\hat{\delta}(2\tilde{p}_{2}\cdot q)e^{-iq\cdot
x_{\perp}}\frac{\partial}{\partial
Q_{\mu}}\tilde{\mathcal{M}}_{4}\Bigg{|}_{Q=0}$ (46)
$\displaystyle=\frac{G}{4m_{1}^{3}m_{2}^{2}\sqrt{\gamma^{2}-1}b^{2}}\left(m_{2}\alpha_{1}^{(0)}\varepsilon^{\mu}(s_{1},u_{1},b)+i\alpha_{1}^{1,1}\left(m_{1}\varepsilon^{\mu}(s_{1},u_{1},b)+m_{2}\varepsilon^{\mu}(s_{1},u_{2},b)\right)\right),$

where we have used

$\displaystyle\varepsilon(q,\tilde{p}_{1},\tilde{p}_{2},s_{1}(\tilde{p}_{1}))=\varepsilon(q,\tilde{p}_{1},\tilde{p}_{2},s_{1}(p_{1}^{\prime})+\frac{1}{2}\Delta
s_{1})=\varepsilon(q,\tilde{p}_{1},\tilde{p}_{2},s_{1}(p_{1}^{\prime}))+\mathcal{O}(G).$
(47)

In order to now compute $x_{\perp}^{\mu}$, we project to the plane
perpendicular to $\tilde{p}_{1}$ and $\tilde{p}_{2}$ via

$\displaystyle
x_{\perp}^{\mu}=\Pi^{\mu}_{~{}\nu}(\tilde{p}_{1},\tilde{p}_{2})X^{\nu}=X^{\mu}-(X\cdot
e_{0})e_{0}^{\mu}+(X\cdot e_{q})e_{q}^{\mu}.$ (48)

It is particularly useful to work in a basis

$\displaystyle\left\\{b^{\mu},u_{1}^{\mu},u_{2}^{\mu},\varepsilon^{\mu}(b,u_{1},u_{2})\right\\}=\left\\{b^{\mu},u_{1}^{\mu},u_{2}^{\mu},L^{\mu}\right\\},$
(49)

where $L^{\mu}=\varepsilon^{\mu}(b,u_{1},u_{2})$ is related to the angular
momentum333Note, however, that this doesn’t have dimensions of angular
momentum. It differs by a normalisation factor from, for example, Ref.
Jakobsen:2022zsx . This basis is particularly useful when considering
contributions to observables that are projected into the
$\tilde{p}_{1}$-$\tilde{p}_{2}$ plane, since it allows us to immediately throw
away any terms that happen to be in the direction $u_{1}^{\mu}$ or
$u_{2}^{\mu}$ at the maximal order in $G$ that we care about, since the
projection will always generated higher order terms. With this in mind, we
find that

$\displaystyle\Pi^{\mu}_{~{}\nu}\frac{\partial\tilde{\chi}_{1}}{\partial
Q_{\nu}}\Bigg{|}_{Q=0}=\frac{G(m_{2}\alpha_{1}^{(0)}+i(m_{1}+\gamma
m_{2})\alpha_{1}^{(1,1)})s_{1}\cdot
u_{2}}{4m_{1}^{3}m_{2}^{2}(\gamma^{2}-1)^{3/2}b^{2}}\varepsilon^{\mu}(b,u_{1},u_{2})$
(50)

We can compute the products of $X$ with the projected directions

$\displaystyle X\cdot e_{0}$ $\displaystyle=b\cdot e_{0}=0,$ (51)
$\displaystyle X\cdot e_{q}$ $\displaystyle=b\cdot e_{q}+\mathcal{O}(G^{2})$

At order $G$ then, we find that $x_{\perp}$ is given by

$\displaystyle x_{\perp}^{\mu}$ $\displaystyle=b^{\mu}-(b\cdot
e_{q})e_{q}^{\mu}+\frac{G(m_{2}\alpha_{1}^{(0)}+i(m_{1}+\gamma
m_{2})\alpha_{1}^{(1,1)})s_{1}\cdot
u_{2}}{4m_{1}^{3}m_{2}^{2}(\gamma^{2}-1)^{3/2}b^{2}}\varepsilon^{\mu}(b,u_{1},u_{2})$
(52) $\displaystyle=b^{\mu}-\frac{b\cdot\Delta
p}{2m_{1}m_{2}(\gamma^{2}-1)}K^{\mu}+\frac{G(m_{2}\alpha_{1}^{(0)}+i(m_{1}+\gamma
m_{2})\alpha_{1}^{(1,1)})s_{1}\cdot
u_{2}}{4m_{1}^{3}m_{2}^{2}(\gamma^{2}-1)^{3/2}b^{2}}\varepsilon^{\mu}(b,u_{1},u_{2})$

where we have defined

$\displaystyle K^{\mu}=(m_{1}+\gamma m_{2})u_{2}^{\mu}-(m_{2}+\gamma
m_{1})u_{1}^{\mu}.$ (53)

The leading order, we can write

$\displaystyle b\cdot\Delta p$
$\displaystyle=b_{\mu}\frac{\partial\tilde{\chi}_{1}}{\partial x_{\mu}}$ (54)
$\displaystyle=\frac{G(m_{1}b^{2}\alpha_{1}^{(0)}+i\alpha_{1}^{(1,1)}L\cdot
s_{1})}{2m_{1}^{2}m_{2}\sqrt{\gamma^{2}-1}b^{2}}$

and therefore that the leading order contribution to $x_{\perp}$ is

$\displaystyle x_{\perp}^{\mu}$
$\displaystyle=b^{\mu}+\frac{G(m_{1}b^{2}\alpha_{1}^{(0)}+i\alpha_{1}^{(1,1)}L\cdot
s_{1})}{2m_{1}^{2}m_{2}\sqrt{\gamma^{2}-1}b^{2}}K^{\mu}+\frac{G(m_{2}\alpha_{1}^{(0)}+i(m_{1}+\gamma
m_{2})\alpha_{1}^{(1,1)})s_{1}\cdot
u_{2}}{4m_{1}^{3}m_{2}^{2}(\gamma^{2}-1)^{3/2}b^{2}}L^{\mu}.$ (55)

We have now computed all the ingredients we need to compute observables up to
order $\mathcal{O}(G^{2})$, but before we do it’s useful to summarise the
overall process.

###  3.2 Computing Spinning Observables from the Eikonal

The impulse is obtained from the Eikonal phase using the following formula

$\displaystyle\Delta p^{\mu}$
$\displaystyle=-\Pi^{\mu}_{~{}\nu}\frac{\partial\tilde{\chi}}{\partial
x_{\perp\nu}},$ (56)

where the tilde reminds us that we need to fully evaluate all of the shifted
variables as in the previous subsection. With this in mind, it is useful to
work with mass-rescaled variables

$\displaystyle\tilde{u}_{i}\equiv\frac{\tilde{p}_{i}}{\tilde{m}_{i}},~{}~{}~{}~{}~{}\tilde{a}_{i}\equiv\frac{\tilde{s}_{i}}{\tilde{m}_{i}},$
(57)

and we recall that the dependence is of the form

$\displaystyle\tilde{\chi}=\tilde{\chi}(\tilde{u}_{i},\tilde{a}_{i},x_{\perp}),\quad\Delta
p=\Delta\tilde{p}(\tilde{u}_{i},\tilde{a}_{i},x_{\perp}),$ (58)

As we have explicitly shown, we can express the shifts in terms of the more
familiar variables

$\displaystyle x_{\perp}^{\mu}$ $\displaystyle=b^{\mu}\,+\frac{\Delta
b^{\mu}}{2},$ $\displaystyle\tilde{u}_{i}^{\mu}$
$\displaystyle=u_{i}^{\mu}+\frac{\Delta u_{i}^{\mu}}{2},$ (59)
$\displaystyle\tilde{a}_{i}^{\mu}$ $\displaystyle=a_{i}^{\mu}+\frac{\Delta
a_{i}^{\mu}}{2}.$

Let’s motivate this by computing some observables at $\mathcal{O}(G^{2})$,
considering the eikonal phases of the form

$\displaystyle\tilde{\chi}=\tilde{\chi}_{1}+\tilde{\chi}_{2}+\mathcal{O}(G^{3}).$
(60)

The dependence up to the linear-in-spin order is given by

$\displaystyle\tilde{\chi}_{1}=-\frac{2G}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}}\,\,\big{(}$
$\displaystyle\alpha_{1}^{(0)}\log{|x_{\perp}|}+i\alpha_{1}^{(1,1)}\frac{\varepsilon(\tilde{a}_{1},\tilde{u}_{1},\tilde{u}_{2},x_{\perp})}{|x_{\perp}|^{2}}\big{)}\,,$
(61) $\displaystyle\tilde{\chi}_{2}=\frac{\pi
G^{2}}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}}\,\,\big{(}$
$\displaystyle\alpha_{2}^{(0)}\frac{1}{|x_{\perp}|}+i\alpha_{2}^{(1,1)}\frac{\varepsilon(\tilde{a}_{1},\tilde{u}_{1},\tilde{u}_{2},x_{\perp})}{|x_{\perp}|^{3}}\big{)}\,,$
(62)

where $|x_{\perp}|=\sqrt{-x_{\perp}^{2}}$, and the explicit coefficients in
the case of gravity are

$\displaystyle\alpha_{1}^{(0)}$
$\displaystyle=4m_{1}^{2}m_{2}^{2}(2\gamma^{2}-1)\,,\qquad\
\alpha_{2}^{(0)}=3m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})(5\gamma^{2}-1)\,,$ (63)
$\displaystyle\alpha_{1}^{(1,1)}$
$\displaystyle=-8im_{1}^{2}m_{2}^{2}\gamma\,,\qquad\qquad\alpha_{2}^{(1,1)}=-\frac{im_{1}^{2}m_{2}^{2}(4m_{1}+3m_{2})(5\gamma^{2}-3)\gamma}{(\gamma^{2}-1)}\,.$
(64)

The now-familiar scalar impulse can be expressed as

$\displaystyle\Delta p_{1,\,\textrm{scalar}}^{\mu}$
$\displaystyle=-\frac{G}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}|b|^{2}}\Big{(}2\alpha_{1}^{(0)}+\frac{G\pi}{|b|}\alpha_{2}^{(0)}\Big{)}x_{\perp}^{\mu}$
$\displaystyle=-\frac{G}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}|b|^{2}}\bigg{\\{}\Big{(}2\alpha_{1}^{(0)}+\frac{G\pi}{|b|}\alpha_{2}^{(0)}\Big{)}b^{\mu}+\frac{G}{2m_{1}^{2}m_{2}^{2}(\gamma^{2}-1)^{3/2}}(\alpha_{1}^{(0)})^{2}K^{\mu}\bigg{\\}},$

where the second term in the last line is obtained as an iteration by plugging
in eq. (55).

For the terms which are linear in the spin $a_{1}$, the impulse takes the form

$\displaystyle\Delta p_{1,\,a_{1}}^{\mu}=$
$\displaystyle\frac{G}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}|b|^{4}}\bigg{\\{}i\alpha_{1}^{(1,1)}\Big{(}(a_{1}\cdot
L)b^{\mu}+2(a_{1}\cdot
b)L^{\mu}\Big{)}+\frac{G\pi}{|b|}i\alpha_{2}^{(1,1)}\Big{(}(a_{1}\cdot
L)b^{\mu}+(a_{1}\cdot b)L^{\mu}\Big{)}$
$\displaystyle-\frac{G}{2m_{1}^{2}m_{2}^{2}(\gamma^{2}-1)^{3/2}}\Big{(}c_{\Delta
p,\textrm{iter}}^{(1)}(a_{1}\cdot u_{2})L^{\mu}+c_{\Delta
p,\textrm{iter}}^{(2)}(a_{1}\cdot L)K^{\mu}\Big{)}\bigg{\\}},$ (65)

with iteration coefficients

$\displaystyle c_{\Delta p,\textrm{iter}}^{(1)}=$
$\displaystyle(m_{1}+m_{2}\gamma)(2\alpha_{1}^{(0)}i\alpha_{1}^{(1,1)})+m_{2}\big{(}({\alpha_{1}^{(0)}})^{2}+(\gamma^{2}-1)({i\alpha_{1}^{(1,1)}})^{2}\big{)},$
$\displaystyle c_{\Delta p,\textrm{iter}}^{(2)}=$
$\displaystyle-2\alpha_{1}^{(0)}i\alpha_{1}^{(1,1)}.$ (66)

Plugging in the values for the coefficients, we find the spinning impulse

$\displaystyle\Delta p_{1}^{\mu}=$
$\displaystyle~{}\frac{2Gm_{1}m_{2}}{\sqrt{\gamma^{2}-1}b^{4}}\bigg{[}\left((2\gamma^{2}-1)b^{2}-2\gamma\varepsilon(b,a_{1},u_{1},u_{2})\right)b^{\mu}+2m_{2}\gamma
b\cdot a_{1}L^{\mu}\bigg{]}$ (67)
$\displaystyle+G^{2}m_{1}m_{2}\bigg{[}\left(\frac{(2\gamma^{2}-1)(2(2\gamma^{2}-1)b^{2}+8\gamma\varepsilon(b,a_{1},u_{1},u_{2})}{(\gamma^{2}-1)^{2}b^{4}}\right)K^{\mu}$
$\displaystyle+\left(\frac{\pi\gamma\left(5\gamma^{2}-3+\right)b\cdot
a_{1}\left(4m_{1}+3m_{2}\right)}{4\left(-1+\gamma^{2}\right)^{3/2}\left(-b^{2}\right)^{5/2}}-\frac{2a_{1}\cdot
u_{2}\left(4\gamma
m_{1}-8\gamma^{3}m_{1}+m_{2}-4\gamma^{2}m_{2}\right)}{\left(-1+\gamma^{2}\right)^{2}b^{4}}\right)L^{\mu}$
$\displaystyle+\left(\frac{\pi\left(3\left(1-6\gamma^{2}+5\gamma^{4}\right)b^{2}\left(m_{1}+m_{2}\right)-2\gamma\left(5\gamma^{2}-3\right)\left(4m_{1}+3m_{2}\right)\varepsilon(b,a_{1},u_{1},u_{2})\right)}{4\left(-1+\gamma^{2}\right)^{3/2}\left(-b^{2}\right)^{5/2}}\right)b^{\mu}\bigg{]}$

In rescaled variables, the spin kick can be obtained from the expression

$\displaystyle\Delta a_{i}^{\mu}=a_{i}^{\mu}\left(\Delta
u\right)-\left\\{a_{i}^{\mu},\tilde{\chi}\right\\}-\frac{1}{2}\left\\{\tilde{\chi},\left\\{a_{i}^{\mu},\tilde{\chi}\right\\}\right\\}+\cdots,$
(68)

However, we note that this is an iterative expression in the sense that each
term in the rotation contains a fundamental bracket $\\{a_{1}^{\mu},\chi\\}$.
At order $\mathcal{O}(G^{2})$, we can take advantage of this to write it in
terms of shifted variables as

$\displaystyle\Delta\tilde{a}_{i}^{\mu}(\tilde{u}_{i},\tilde{a}_{i},x_{\perp})$
$\displaystyle=(\tilde{u}_{i}\cdot\Delta
u_{i})\tilde{a}_{i}^{\mu}-(\tilde{a}_{i}\cdot\Delta
u_{i})\tilde{u}_{i}^{\mu}-\varepsilon^{\mu\beta\gamma\delta}\frac{\partial\tilde{\chi}_{n}}{\partial\tilde{a}_{i}^{\beta}}\tilde{u}_{i\gamma}\tilde{a}_{i\delta}+\mathcal{O}(G^{3})\,.$
(69)

An analogous process of evaluating this formula with the Eikonal phases eqs.
(61)-(62), and translating into unshifted variables results in the spin kick
being expressed as

$\displaystyle\Delta
a_{1}^{\mu}=\frac{G}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}|b|^{2}}\bigg{\\{}$
$\displaystyle\Big{(}2i\alpha_{1}^{(1,1)}+\frac{G\pi}{|b|}i\alpha_{2}^{(1,1)}\Big{)}\Big{(}(a_{1}\cdot
u_{2})b^{\mu}-(a_{1}\cdot b)u_{2}^{\mu}\Big{)}$ (70)
$\displaystyle\Big{(}2\gamma
i\alpha_{1}^{(1,1)}+2\alpha_{1}^{(0)}+\frac{G\pi}{|b|}(\gamma
i\alpha_{2}^{(1,1)}+\alpha_{2}^{(0)})\Big{)}(a_{1}\cdot b)u_{1}^{\mu}$
$\displaystyle-\frac{G}{2m_{1}^{2}m_{2}^{2}(\gamma^{2}-1)^{3/2}}$
$\displaystyle\Big{(}c_{\Delta a_{1},\textrm{iter}}^{(1)}b^{-2}(a_{1}\cdot
b)b^{\mu}+c_{\Delta a_{1},\textrm{iter}}^{(2)}(a_{1}\cdot
u_{2})u_{1}^{\mu}+c_{\Delta a_{1},\textrm{iter}}^{(3)}(a_{1}\cdot
u_{2})u_{2}^{\mu}\Big{)}\bigg{\\}},$

with iteration coefficients given by

$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(1)}$
$\displaystyle=m_{2}(-1+\gamma^{2})\big{(}-({\alpha_{1}^{(0)}})^{2}+(-1+\gamma^{2})({i\alpha_{1}^{(1,1)}})^{2}\big{)},$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(2)}$
$\displaystyle=m_{2}(-1+\gamma^{2})\big{(}2\alpha_{1}^{(0)}i\alpha_{1}^{(1,1)}+\gamma({i\alpha_{1}^{(1,1)}})^{2}\big{)}+(m_{1}+m_{2}\gamma)({\alpha_{1}^{(0)}})^{2},$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(3)}$
$\displaystyle=m_{2}(-1+\gamma^{2})\big{(}-({i\alpha_{1}^{(1,1)}})^{2}\big{)}.$

For the explicit form of the Eikonal obtained by plugging the values eqs.
(63)-(64), our results match exactly those of Ref. Liu:2021zxr  after
expressing them in the same basis of vectors.

More generally, we can use eq. (59) to express result in terms of our
preferred past-infinity variables for any observable. This results in

$\displaystyle\Delta\mathbb{O}_{i}^{(1),\mu}=$
$\displaystyle\Delta\tilde{\mathbb{O}}_{i}^{(1),\mu}(u_{i},a_{i},b),$ (71)
$\displaystyle\Delta\mathbb{O}_{i}^{(2),\mu}=$
$\displaystyle\Delta\tilde{\mathbb{O}}_{i}^{(2),\mu}(u_{i},a_{i},b)+\Delta\mathbb{O}^{(2),\mu}_{i,\textrm{iter}}.$
(72)

where the superscript $(i)$ correspond to the order in the post-Minkowskian
approximation, and the observable $\mathbb{O}$ stands for either the momentum
$p$, as in eq. (56) or the spin vector $a$, as in eq. (69), but they are now
evaluated on the un-shifted variables. The iteration term can be expressed up
to the second post-Minkowskian order as

$\Delta\mathbb{O}^{(2),\mu}_{i,\textrm{iter}}=\sum_{i}\frac{\partial\Delta\mathbb{O}_{i}^{(1),\mu}}{\partial
u_{i}^{\nu}}\frac{\Delta
u_{i}^{(1),\nu}}{2}+\sum_{i}\frac{\partial\Delta\mathbb{O}_{i}^{(1),\mu}}{\partial
a_{i}^{\nu}}\frac{\Delta
a_{i}^{(1),\nu}}{2}+\frac{\partial\Delta\mathbb{O}_{i}^{(1),\mu}}{\partial
b^{\nu}}\frac{\Delta b^{(1),\nu}}{2}.$ (73)

Note that since every impulse entering eq. (73) is taken at the first post-
Minkowskian approximation, there is no distinction between $\Delta\mathbb{O}$
and $\Delta\tilde{\mathbb{O}}$. Also, while $\Delta u_{i}^{(1),\mu}$ and
$\Delta a_{i}^{(1),\mu}$ are obtained by means of the formulas eqs. (56) and
(69), the change in impact parameter $\Delta b^{(1),\mu}$ is obtained from the
second stationary phase condition. Using eq. (73), we can write observables at
the second post-Minkowskian approximation compactly as,

$\displaystyle\Delta\mathbb{O}^{(i,j)}=\frac{G}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}|b|^{2(\zeta_{\mathbb{O}}+i+j)}}\bigg{\\{}$
$\displaystyle\sum_{k}\mathcal{O}_{\Delta\mathbb{O},\chi}^{(i,j,k)}\Big{(}c_{\Delta\mathbb{O},\chi_{1}}^{(i,j,k)}+\frac{G\pi}{|b|}c_{\Delta\mathbb{O},\chi_{2}}^{(i,j,k)}\Big{)}$
(74)
$\displaystyle-\frac{G}{2m_{1}^{2}m_{2}^{2}(\gamma^{2}-1)^{3/2}}\sum_{l}\mathcal{O}_{\Delta\mathbb{O},\textrm{iter}}^{(i,j,l)}c_{\Delta\mathbb{O},\textrm{iter}}^{(i,j,l)}\bigg{\\}},$

where $\mathbb{O}$ means either $p$ or $a_{1}$ for the impulse or the spin-
kick, respectively, and the observable-dependent factor $\zeta_{\mathbb{O}}$
in the denominator is defined by

$\displaystyle\zeta_{p}=1,\qquad\qquad\zeta_{a_{1}}=0.$ (75)

The indices $i$ and $j$ label the powers of $a_{1}$ and $a_{2}$, respectively.
The structures $\mathcal{O}_{\Delta p,\chi}$ correspond to contributions which
are linear in the Eikonal phase at either the first or second post-Minkowskian
order, and so the coefficients $c_{\Delta p,\chi_{i}}$ depend on one
$\alpha_{i}$, while the $\mathcal{O}_{\Delta p,\textrm{iter}}$ correspond to
iterations, and their coefficients $c_{\Delta p,\textrm{iter}}$ depend on
products of two $\alpha_{1}$’s. We have obtained results up to quadratic order
in the spin of the particles, which are presented in an appendix.

##  4 Relation to other approaches

By application of this prescription, we have derived covariant observables
from the eikonal phase, namely the classical impulse and the spin kick, up to
the second post-Minkowskian approximation, and up to quadratic order in the
spin of the particles. Results at these order were first obtained in covariant
form in Ref. Liu:2021zxr , by means of the spinning generalisation of the
(worldline-based) post-Minkowskian effective field theory (PMEFT) of Ref.
Kalin:2020mvi . The comparison between both results is straightforward, up to
expressing them in a basis of vectors. We have verified they perfectly agree
with each other.

However, there are other existing derivations of these quantities from the
Eikonal in the literature, with supposedly different formulations, e.g. those
coming from a canonical approach Bern:2020buy  and those derived from
worldline techniques Jakobsen:2021zvh . It is interesting to compare these
approaches, and so we do so in this section.

###  4.1 Relation to covariant (worldline) methods

Let us comment on the relation to another derivation of these observables from
an eikonal phase, which was obtained in Ref. Jakobsen:2021zvh  using a so-
called worldline quantum field theory (WQFT). First, we verified that both
eikonal phases are a perfect match. Notably, their results for observables are
given in terms of the spin tensor, which is related to the (Pauli-Lubanski)
spin vector by means of

$\displaystyle
S^{\mu\nu}=\frac{1}{m}\epsilon^{\mu\nu\sigma\rho}p_{\sigma}S_{\rho},\qquad
S^{\mu}=\frac{1}{2m}\epsilon^{\mu\nu\sigma\rho}p_{\nu}S_{\sigma\rho}.$ (76)

In consequence, we find that to compare their spin-tensor change $\Delta
S^{\mu\nu}$ with our spin kick $\Delta S^{\mu}$ we need to evaluate

$\displaystyle\Delta S^{\mu\nu}=\frac{1}{m}\epsilon^{\mu\nu\sigma\rho}(\Delta
p_{\sigma}S_{\rho}+p_{\sigma}\Delta S_{\rho}+\Delta p_{\sigma}\Delta
S_{\rho}),$ (77)

or alternatively

$\displaystyle\Delta S^{\mu}=\frac{1}{2m}\epsilon^{\mu\nu\sigma\rho}(\Delta
p_{\nu}S_{\sigma\rho}+p_{\nu}\Delta S_{\sigma\rho}+\Delta p_{\nu}\Delta
S_{\sigma\rho}).$ (78)

While our results agree perfectly, it is worth commenting on some similarities
and slight differences in our approaches. First, they obtain their result
through the relation

$\displaystyle\Delta\tilde{S}^{\mu\nu}=\frac{4}{m}\tilde{S}^{\rho[\mu}\frac{\partial\tilde{\chi}}{\partial\tilde{S}_{\nu]}^{\
\ \rho}},$ (79)

where the tilde once again denotes shifted variables. Indeed, these are
naturally chosen by the time-symmetric propagators in their worldline.
Furthermore, they introduce a shift in their eikonal phase of the form

$\displaystyle\tilde{S}_{i}^{\mu\nu}$
$\displaystyle\rightarrow\tilde{S}_{i}^{\mu\nu}+2(u_{i}\cdot
S_{i})^{[\mu]}u_{i}^{\nu]}$ (80) $\displaystyle\tilde{b}^{\mu}$
$\displaystyle\rightarrow\tilde{b}^{\mu}+\xi_{2}\tilde{u}_{2}^{\mu}-\xi_{1}\tilde{u}_{1}^{\mu}+\tilde{S}_{2}^{\mu\nu}u_{2,\nu}-\tilde{S}_{1}^{\mu\nu}u_{1,\nu},$
(81)

where the $\xi_{i}$ depend in general on $b\cdot u_{i}$ and
$S_{i}^{\mu\nu}u_{i,\nu}$. While such shifts vanish on-shell, they do produce
contributions in the directions of $\tilde{u}_{i}$ upon taking derivatives of
the eikonal phase with respect to $\tilde{b}$ and $\tilde{S}^{\mu\nu}$ in a
similar fashion to eq. (56). This shift was demanded to preserve symmetry in
their worldline. In order to ensure we rightly reproduce longitudinal terms in
the impulse, we also perform the following shift in impact parameter in the
Eikonal phase

$\tilde{b}^{\mu}\rightarrow\tilde{b}^{\mu}+\xi_{2}\tilde{u}_{2}^{\mu}-\xi_{1}\tilde{u}_{1}^{\mu},$
(82)

where

$\xi_{1}=\frac{\tilde{b}\cdot(\gamma\tilde{u}_{2}-\tilde{u}_{1})}{\gamma^{2}-1},\qquad\xi_{2}=\frac{\tilde{b}\cdot(\tilde{u}_{2}-\gamma\tilde{u}_{1})}{\gamma^{2}-1}.$
(83)

However, we have found that the spin shift in eqs. (80)-(81) is unsubstantial
for the spin vector, while required for the spin-tensor.

###  4.2 Relation to canonical methods (the $D_{SL}$ formula)

Concurently with the worldline result of Ref. Liu:2021zxr , results at the 2PM
and quadratic in spin orders were obtained using scattering amplitudes in Ref.
Kosmopoulos:2021zoq , building on the formalism introduced in Ref.
Bern:2020buy . The comparison between amplitudes-based results and worldline
approaches is obscured by the fact of them being written in terms of different
variables, namely canonical or covariant. Such a comparison is now well
understood and fairly explored (see, for example, discussions in Ref.
Jakobsen:2022zsx ).

Instead, we will use our newly-built intuition to provide some insight into
the peculiar form of the so-called $D_{SL}$ formulas, which directly obtain
impulses and spin kicks from the eikonal phase. These were reverse-engineered
by first obtaining the observables through $\mathcal{O}(G^{2})$ using
Hamilton’s equations. Let us try to shed some light onto their structure. We
start with their usual form

$\displaystyle\Delta\mathbb{O}^{i}\,$
$\displaystyle=-\\{\mathbb{O}^{i},\,\chi\\}-\frac{1}{2}\,\\{\chi,\\{\mathbb{O}^{i},\,\chi\\}\\}-D_{SL}\left(\chi,\\{\mathbb{O}^{i},\,\chi\\}\right)+\frac{1}{2}\,\\{\mathbb{O}^{i},\,D_{SL}\left(\chi,\chi\right)\\}\,,$
(84)

where $\mathbb{O}$ can stand for either the transverse momentum $p_{\perp}$ or
the spin $S$. Here, we’ll show that the iteration piece

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle-\frac{1}{2}\,\\{\chi_{1},\\{\mathbb{O}^{i},\chi_{1}\\}\\}-D_{SL}\left(\chi_{1},\\{\mathbb{O}^{i},\chi_{1}\\}\right)+\frac{1}{2}\,\\{\mathbb{O}^{i},D_{SL}\left(\chi_{1},\chi_{1}\right)\\}\,.$
(85)

can be instead written as

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta
S^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
L^{j}}\Delta
L^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
p^{j}}\Delta p^{(1),j}\,,$ (86)

in complete analogy with equation (73), thus replacing the mysterious form of
the $D_{SL}$ formula with the more natural evaluation in shifted variables.

To prove this, let us start by reviewing these formulas. The different
brackets appearing in eq. (85) are given by

$\displaystyle\\{p^{i}_{\perp},f\\}\equiv-\frac{\partial f}{\partial
b^{i}}\,,\qquad\\{S^{i},f\\}\equiv\epsilon^{ijk}\,\frac{\partial f}{\partial
S^{j}}S^{k}\,,\qquad
D_{SL}\left(f,g\right)\equiv-\,\epsilon^{ijk}S^{k}\,\frac{\partial f}{\partial
S^{i}}\,\underline{\frac{\partial g}{\partial L^{j}}}\,.$ (87)

The functions $f$ and $g$ in general depend on $S^{i}$, $p^{i}$ and $b^{i}$
(though $p^{i}$ is taken as inert in the brackets). For this reason, the $L$
derivative in the $D_{SL}$ bracket is interpreted as

$\displaystyle\underline{\frac{\partial}{\partial
L^{j}}}\equiv\frac{\partial}{\partial
L^{j}}-\frac{p^{j}p^{k}}{p^{2}}\frac{\partial}{\partial L^{k}}$ (88)

Here, we write the formula considering only one spinning particle, but the
generalisation to two spinning particles is straightforward. We start by
writing separately the first two orders in the post-Minkowskian approximation

$\displaystyle\Delta\mathbb{O}^{(1),i}$
$\displaystyle=-\\{\mathbb{O}^{i},\chi_{1}\\},$ (89)
$\displaystyle\Delta\mathbb{O}^{(2),i}$
$\displaystyle=-\\{\mathbb{O}^{i},\chi_{2}\\}+\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}$
(90)

where $\chi_{n}$ is the nPM Eikonal phase, and we focus on the iteration term,
which is defined in eq. (85). Reinserting back eq. (89) into the iteration, we
note we can write

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle\frac{1}{2}\,\\{\chi_{1},\Delta\mathbb{O}^{(1),i}\\}+D_{SL}\big{(}\chi_{1},\Delta\mathbb{O}^{(1),i}\big{)}+\frac{1}{2}\,\\{\mathbb{O}^{i},D_{SL}\left(\chi_{1},\chi_{1}\right)\\}\,.$
(91)

Now, from the definition of the Poisson bracket, we know

$\displaystyle\\{\chi_{1},\Delta\mathbb{O}^{(1),i}\\}=\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\\{\chi_{1},S^{j}\\}=\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta S^{(1),j},$ (92)

where in the last equality we have used eq. (84) for spin at 1PM

$\displaystyle\Delta S^{(1),i}=-\\{S^{i},\chi_{1}\\}.$ (93)

It is also useful to analyse the expression for the $D_{SL}$ bracket, when one
of its arguments is the eikonal phase. This is

$\displaystyle D_{SL}(\chi_{1},g)$
$\displaystyle=-\epsilon^{ijk}S^{k}\frac{\partial\chi_{1}}{\partial
S^{i}}\frac{\partial g}{\underline{\partial L^{j}}}$ (94)
$\displaystyle=\\{S^{j},\chi_{1}\\}\frac{\partial g}{\underline{\partial
L^{j}}}=-\Delta S^{(1),j}\frac{\partial g}{\underline{\partial L^{j}}}=\Delta
L^{(1),j}\frac{\partial g}{\underline{\partial L^{j}}},$ (95)

where we have used the conservation of total angular momentum

$\displaystyle\Delta J^{i}=\Delta S^{i}+\Delta L^{i}=0.$ (96)

Then, the iteration can be written as

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta
S^{(1),j}+\frac{\partial\Delta\mathbb{O}^{(1),i}}{\underline{\partial
L^{j}}}\Delta L^{j}+\frac{1}{2}\,\\{\mathbb{O}^{i},\Delta
L^{(1),j}\frac{\partial\chi_{1}}{\underline{\partial L^{j}}}\\}\,.$ (97)

Now, since the Poisson bracket acts as a differential on its arguments, we
find it satisfies the Leibniz-like condition

$\displaystyle\\{\mathbb{O}^{i},\Delta
L^{(1),j}\frac{\partial\chi_{1}}{\underline{\partial L^{j}}}\\}$
$\displaystyle=\Delta
L^{(1),j}\\{\mathbb{O}^{i},\frac{\partial\chi_{1}}{\underline{\partial
L^{j}}}\\}+\\{\mathbb{O}^{i},\Delta
L^{(1),j}\\}\frac{\partial\chi_{1}}{\underline{\partial L^{j}}}$ (98)
$\displaystyle=-\frac{\partial\Delta\mathbb{O}^{(1),i}}{\underline{\partial
L^{j}}}\Delta L^{(1),j}+\\{\mathbb{O}^{i},\Delta
L^{(1),j}\\}\frac{\partial\chi_{1}}{\underline{\partial L^{j}}}.$ (99)

where, in the first term of the second line we have used eq. (89), along with
the fact that the action of the Poisson bracket commutes with the derivative
${\partial}/{\underline{\partial L^{j}}}$. Inserting this back we have

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta
S^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\underline{\partial
L^{j}}}\Delta L^{j}-\frac{1}{2}\\{\mathbb{O}^{i},\Delta
L^{(1),j}\\}\frac{\partial\chi_{1}}{\underline{\partial L^{j}}}\,.$ (100)

Recalling eq. (88), the iteration becomes

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta
S^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
L^{j}}\Delta L^{j}$
$\displaystyle-\frac{p^{j}p^{k}}{2p^{2}}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
L^{k}}\Delta L^{j}-\frac{1}{2}\\{\mathbb{O}^{i},\Delta
L^{(1),j}\\}\frac{\partial\chi_{1}}{\underline{\partial L^{j}}}\,.$ (101)

Finally, we have verified that up to the orders in spin of our interest, the
second line in eq. (101) satisfies

$\displaystyle-\frac{p^{j}p^{k}}{2p^{2}}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
L^{k}}\Delta L^{j}-\frac{1}{2}\\{\mathbb{O}^{i},\Delta
L^{(1),j}\\}\frac{\partial\chi_{1}}{\underline{\partial
L^{j}}}=\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
p^{j}}\Delta p^{(1),j}.$ (102)

This allows to write the iteration as

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta
S^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
L^{j}}\Delta
L^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
p^{j}}\Delta p^{(1),j}\,,$

which had already been presented in eq. (86). We have thus shown that the
iteration terms on the $D_{SL}$ formula simply encode the evaluation in
shifted variables of lower order impulses.

This analysis readily extends to the more generic case considered in Ref.
Bern:2023ity , where a relaxation of the spin supplementary condition results
in extra degrees of freedom, which are encoded in the boost vector $K^{i}$. In
that case, the operator $\mathbb{O}$ can stand for $K$ as well, and we need to
generalise the algebra as

$\displaystyle\\{S_{i},S_{j}\\}=\epsilon_{ijk}S_{k}\,,\quad\\{S_{i},K_{j}\\}=\epsilon_{ijk}K_{k}\,,\quad\\{K_{i},K_{j}\\}=-\epsilon_{ijk}S_{k}\,,$
(103)

while promoting the $D_{SL}$ operator to

$\displaystyle D_{L}(f,g)\equiv-\epsilon_{ijk}\bigg{(}S_{k}\frac{\partial
f}{\partial S_{i}}+K_{k}\frac{\partial f}{\partial
K_{i}}\bigg{)}\frac{\partial g}{\underline{\partial L^{j}}}\,.$ (104)

The crucial point is that due to the generalisation of the algebra, we now
have

$\displaystyle\\{\chi_{1},\Delta\mathbb{O}^{(1),i}\\}=\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\\{\chi_{1},S^{j}\\}=\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta S^{(1),j}+\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
K^{j}}\Delta K^{(1),j}.$ (105)

Furthermore, we can again consider

$\displaystyle D_{L}(\chi_{1},g)$
$\displaystyle=\\{S^{j},\chi_{1}\\}\frac{\partial g}{\underline{\partial
L^{j}}}=\Delta L^{(1),j}\frac{\partial g}{\underline{\partial L^{j}}},$ (106)

where the two derivatives in eq. (104) correspond to the two brackets in the
algebra involving $S^{i}$. The rest of the derivation follows in analogy, and
results in

$\displaystyle\Delta\mathbb{O}_{\textrm{iter}}^{(2),i}=$
$\displaystyle\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
S^{j}}\Delta
S^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
K^{j}}\Delta
K^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
L^{j}}\Delta
L^{(1),j}+\frac{1}{2}\frac{\partial\Delta\mathbb{O}^{(1),i}}{\partial
p^{j}}\Delta p^{(1),j}\,,$

again showing the iterations simply come from the evaluation in shifted
variables. We have verified the validity of this formula to second order in
the coupling and linear order in either $S^{i}$ or $K^{i}$, for the impulse,
spin kick and boost kick, as in Ref. Bern:2023ity .

##  5 Conclusions

In this paper, we studied the classical dynamics of spinning particles using
scattering amplitudes and eikonal resummation. A key assumption of our work
was that that the loop expansion of scattering amplitudes can be rewritten as
an exponential. This eikonal exponentiation is by now well understood for
scalar particles (see DiVecchia:2023frv  for a comprehensive review). However
the spinning case is more complicated, since the relevant amplitudes are
matrices in spin space. Consequently, the exponential form of the amplitudes
is also a matrix. It would be particularly useful to deepen our understanding
of eikonal exponentation for spinning amplitudes.

Having assumed eikonal exponentiation, our path to determining observables
followed standard methods. We chose an initial state in the domain of validity
of the classical theory. Time evolution was conveniently captured using the
$S$ matrix. Physical observables were determined as expectations of
appropriate operators n the final state. This is very much in the spirit of
KMOC Kosower:2018adc ; Maybee:2019jus  and other work on observables and the
eikonal DiVecchia:2021bdo ; Cristofoli:2021jas ; DiVecchia:2022nna ;
DiVecchia:2022piu ; Gatica:2023iws ; Georgoudis:2023eke . As emphasised in
Cristofoli:2021jas ; Georgoudis:2023eke , care is required in the use of
eikonal methods to determine the final state in a scattering event because the
delta-function support of the eikonal differs from the actual on-shell
condition for final states. In this paper, we introduced a systematic way of
accounting for this mismatch involving a translation operator in momentum
space. This translation shifts momenta in the eikonal such that the correct,
physical, on-shell condition is satisfied.

The translation led directly to an interesting structure of the eikonal
function for spinning particles. A particularly important point is that, after
translation and at the order we studied, the classical eikonal organises
itself around the “half-way” point of the scattering event. That is, the
translated eikonal depends on $p+\Delta p/2$: the momentum plus half the total
impulse, rather than on a particle’s momentum $p$ alone. Similarly the
classical eikonal function depends on the “half-way” spin rather than initial
or final spins.

A beautiful aspect of the eikonal approach to classical dyanmics is that the
actual values of observables follow from stationary phase conditions. One of
these stationary phase equations picks a specific spatial point $x$, which is
the impact parameter $b$ at lowest order. In the case of spin, we found the
determination of $x$ to be particularly subtle. It was very important to
retain the matrix structure of the amplitude in this step. In many respects,
our approach is similar to the worldline approach of reference
Jakobsen:2021zvh . However, in the worldline approach the position was fixed
by demanding conservation of angular momentum. We note that the position
appearing in the eikonal in our approach is in general different to the
position relevant for conservation of angular momentum. The two points happen
to agree at the accuracy we required in this paper, but in general we expect
these points to differ at higher orders.

Our approach correctly reproduced known expressions for the impulse and spin
kick at one loop, and to quadratic order in spin. These observables are known
to have a complicated structure, but in our approach this complexity arose
from an algorithm which is very simple. Observables become complicated because
as one continues to higher orders, one has to recall that lower order
expressions for the observable involved positions, momenta and spins which
must all be corrected to go to the next order in accuracy. This iterative
structure generates complex expressions from simple rules. Clearly it will be
interesting to study these ideas at higher orders (both in terms of loops and
spins). We expect the same basic ideas to apply, though considerable care will
be required to account for effects which we were able to neglect.

The literature already contains a number of papers examining amplitudes,
observables and classical physics using eikonal methods. We showed how our
approach is consistent with the canonical approach of Ref. Bern:2020buy  and
the worldline formalism of Ref. Jakobsen:2021zvh . Assuming that our
assumption of eikonal exponentiation is valid, our approach opens the way
towards an all-order understanding of spinning observables in the contect of
the eikonal.

It is perhaps worth observing that the KMOC formalism Kosower:2018adc ;
Maybee:2019jus  already provides an amplitudes-based method of computing
classical observables for spinning particles which is expected to hold to all
orders in perturbation theory. However eikonal exponentiation may provide a
way to determine observables at large scattering angle; the KMOC formalism (as
currently understood) is intrinsically perturbative in the scattering angle.

Much work remains to be performed before we can really say that we understand
how spin, amplitudes, and classical physics work together in detail. We
completely neglected radiation, for example. It would be very interesting to
understand how radiation and spin fit together in a more general eikonal
picture. As in the case of scalar particles, radiation requires incorporating
the state of the radiation field, presumably as a coherent-state factor along
the lines of references Ciafaloni:2018uwe ; Cristofoli:2021jas ;
DiVecchia:2022nna ; DiVecchia:2022piu . A more general eikonal along these
lines could be very useful for understanding spin effects in gravitational
waveforms which have very recently been studied using KMOC Kosower:2018adc ;
Cristofoli:2021vyo  in references DeAngelis:2023lvf ; Brandhuber:2023hhl ;
Aoude:2023dui . We look forward to working on this topic in the near future.

### Acknowledgments

We thank Giulia Isabella, Guanda Lin, Jan Steinhoff, Fei Teng, Justin Vines
and Mao Zeng for useful discussions. A. L. is supported by funds from the
European Union’s Horizon 2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement No. 847523 ‘INTERACTIONS’. The work of NM was
supported by the Science and Technology Facilities Council (STFC) Consolidated
Grants ST/T000686/1 “Amplitudes, Strings & Duality” and ST/X00063X/1
“Amplitudes, Strings & Duality”. No new data were generated or analysed during
this study. DOC is supported in part by the STFC grant “Particle Physics at
the Higgs Centre”. We are also grateful to the organisers of the “Amplifying
gravity at all scales” program at Nordita, where parts of this project were
undertaken.

##  Appendix A Observables

We now present results for observables up to the 2PM approximation, and up to
quadratic order in the spin of the particles. We may take as starting point an
eikonal phase of the form

$\displaystyle\tilde{\chi}_{n}=$
$\displaystyle\frac{\zeta_{n}}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}}\,\,\Big{\\{}\alpha_{n}^{(0,1)}\log^{n-2}{|x_{\perp}|^{2}}+i\alpha_{n}^{(1,1)}\frac{\tilde{L}\cdot\tilde{a}_{1}}{|x_{\perp}|^{2}}+i\alpha_{n}^{(1,2)}\frac{\tilde{L}\cdot\tilde{a}_{2}}{|x_{\perp}|^{2}}$
(107) $\displaystyle+$
$\displaystyle{\alpha^{\prime}}_{n}^{(2,1)}\frac{(\tilde{L}\cdot\tilde{a}_{1})(\tilde{L}\cdot\tilde{a}_{2})}{|x_{\perp}|^{4}}+\alpha_{n}^{(2,2)}\frac{(\tilde{u}_{2}\cdot\tilde{a}_{1})(\tilde{u}_{1}\cdot\tilde{a}_{2})}{|x_{\perp}|^{2}}+{\alpha^{\prime}}_{n}^{(2,3)}\frac{(x_{\perp}\cdot\tilde{a}_{1})(x_{\perp}\cdot\tilde{a}_{2})}{|x_{\perp}|^{4}}$
$\displaystyle+$
$\displaystyle{\alpha^{\prime}}_{n}^{(2,4)}\frac{(\tilde{L}\cdot\tilde{a}_{1})^{2}}{|x_{\perp}|^{4}}+\alpha_{n}^{(2,5)}\frac{(\tilde{u}_{2}\cdot\tilde{a}_{1})^{2}}{|x_{\perp}|^{2}}+{\alpha^{\prime}}_{n}^{(2,6)}\frac{(x_{\perp}\cdot\tilde{a}_{1})^{2}}{|x_{\perp}|^{4}}\Big{\\}}\,,$

which can be understood as coming from a Fourier transform of a scattering
amplitude (shown below), and whose kinematic prefactor comes from the
integration over the two directions tansverse to the scattering plane. Here
$|x_{\perp}|=\sqrt{-x_{\perp}^{2}}$, and the normalisation factors in position
space $\zeta$ are given by

$\displaystyle\zeta_{1}=-2G,\qquad\qquad\zeta_{2}=\frac{\pi
G^{2}}{|x_{\perp}|},$ (108)

and we use the vector

$\tilde{L}^{\mu}\equiv\varepsilon^{\mu}(\tilde{u}_{1},\tilde{u}_{2},x_{\perp}).$
(109)

The coefficients $\alpha^{\prime}$ are given by

$\displaystyle{\alpha^{\prime}}_{1}^{(2,1)}=$
$\displaystyle-{\alpha^{\prime}}_{1}^{(2,3)}=\alpha_{1}^{(2,1)}-(\gamma^{2}-1)\alpha_{1}^{(2,3)},$
$\displaystyle{\alpha^{\prime}}_{2}^{(2,1)}=$
$\displaystyle-2\alpha_{2}^{(2,1)}+(\gamma^{2}-1)\alpha_{2}^{(2,3)},\quad{\alpha^{\prime}}_{2}^{(2,3)}=\alpha_{2}^{(2,1)}-2(\gamma^{2}-1)\alpha_{2}^{(2,3)},$
$\displaystyle{\alpha^{\prime}}_{1}^{(2,4)}=$
$\displaystyle-{\alpha^{\prime}}_{1}^{(2,6)}=\alpha_{1}^{(2,4)}-(\gamma^{2}-1)\alpha_{1}^{(2,6)},$
$\displaystyle{\alpha^{\prime}}_{2}^{(2,4)}=$
$\displaystyle-2\alpha_{2}^{(2,4)}+(\gamma^{2}-1)\alpha_{2}^{(2,6)},\quad{\alpha^{\prime}}_{2}^{(2,6)}=\alpha_{2}^{(2,4)}-2(\gamma^{2}-1)\alpha_{2}^{(2,6)}.$

As discussed in the text, we perform a shift in impact parameter in the
Eikonal phase

$x_{\perp}^{\mu}\rightarrow
x_{\perp}^{\mu}+\xi_{2}\tilde{u}_{2}^{\mu}-\xi_{1}\tilde{u}_{1}^{\mu},$ (110)

where

$\xi_{1}=\frac{x_{\perp}\cdot(\gamma\tilde{u}_{2}-\tilde{u}_{1})}{\gamma^{2}-1},\qquad\xi_{2}=\frac{x_{\perp}\cdot(\tilde{u}_{2}-\gamma\tilde{u}_{1})}{\gamma^{2}-1}.$
(111)

For the 2PM results, we split it in general as

$\displaystyle\Delta\mathbb{O}^{(i,j)}=\frac{G}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}|b|^{2(\zeta_{\mathbb{O}}+i+j)}}\bigg{\\{}$
$\displaystyle\sum_{k}\mathcal{O}_{\Delta\mathbb{O},\chi}^{(i,j,k)}\Big{(}c_{\Delta\mathbb{O},\chi_{1}}^{(i,j,k)}+\frac{G\pi}{|b|}c_{\Delta\mathbb{O},\chi_{2}}^{(i,j,k)}\Big{)}$
(112)
$\displaystyle-\frac{G}{2m_{1}^{2}m_{2}^{2}(\gamma^{2}-1)^{3/2}}\sum_{l}\mathcal{O}_{\Delta\mathbb{O},\textrm{iter}}^{(i,j,l)}c_{\Delta\mathbb{O},\textrm{iter}}^{(i,j,l)}\bigg{\\}},$

where $\mathbb{O}$ means either $p$ or $a_{1}$ for the impulse or the spin
kick, respectively, and the observable-dependent factor $\zeta_{\mathbb{O}}$
is given by

$\displaystyle\zeta_{p}=1,\qquad\qquad\zeta_{a_{1}}=0.$ (113)

The indices $i$ and $j$ label the powers of $a_{1}$ and $a_{2}$, respectively.
The structures $\mathcal{O}_{\Delta p,\chi}$ correspond to contributions
linear in the Eikonal phase at either the first or second post Minkowskian
approximation, and so the coefficients $c_{\Delta p,\chi_{i}}$ depend on one
$\alpha_{i}$, while the $\mathcal{O}_{\Delta p,\textrm{iter}}$ correspond to
iterations, and their coefficients $c_{\Delta p,\textrm{iter}}$ depend on
products of two $\alpha_{1}$’s.

The structures for $\Delta p$ and $\Delta a_{1}$ which contribute linearly in
the Eikonal phase are

$\mathcal{O}_{\Delta p,\chi}^{(0,0,1)}=b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,0,1)}=(a_{1}\cdot u_{2})b^{\mu}$
---|---
$\mathcal{O}_{\Delta p,\chi}^{(1,0,1)}=(a_{1}\cdot L)b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,0,2)}=(a_{1}\cdot b)u_{1}^{\mu}$
$\mathcal{O}_{\Delta p,\chi}^{(1,0,2)}=(a_{1}\cdot b)L^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,0,3)}=(a_{1}\cdot b)u_{2}^{\mu}$
$\mathcal{O}_{\Delta p,\chi}^{(1,1,1)}=(\gamma^{2}-1)(a_{1}\cdot b)(a_{2}\cdot b)b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,1,1)}=(a_{2}\cdot L)\big{(}(a_{1}\cdot b)u_{2}^{\mu}-(a_{1}\cdot u_{2})b^{\mu}\big{)}$
$\mathcal{O}_{\Delta p,\chi}^{(1,1,2)}=\big{(}(a_{1}\cdot b)(a_{2}\cdot L)+(a_{1}\cdot L)(a_{2}\cdot b)\big{)}L^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,1,2)}=(a_{2}\cdot b)\big{(}(a_{1}\cdot u_{2})L^{\mu}-(a_{1}\cdot L)u_{2}^{\mu}\big{)}$
$\mathcal{O}_{\Delta p,\chi}^{(1,1,3)}=(a_{1}\cdot L)(a_{2}\cdot L)b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,1,3)}=(a_{2}\cdot u_{1})\big{(}(a_{1}\cdot b)L^{\mu}-(a_{1}\cdot L)b^{\mu}\big{)}$
$\mathcal{O}_{\Delta p,\chi}^{(1,1,4)}=b^{2}(a_{1}\cdot u_{2})(a_{2}\cdot u_{1})b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,1,4)}=(a_{2}\cdot b)(a_{1}\cdot L)u_{1}^{\mu}$
$\mathcal{O}_{\Delta p,\chi}^{(2,0,1)}=(\gamma^{2}-1)(a_{1}\cdot b)^{2}b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(1,1,5)}=(a_{1}\cdot b)(a_{2}\cdot L)u_{1}^{\mu}$
$\mathcal{O}_{\Delta p,\chi}^{(2,0,2)}=(a_{1}\cdot b)(a_{1}\cdot L)L^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(2,0,1)}=(a_{1}\cdot u_{2})(a_{1}\cdot L)b^{\mu}$
$\mathcal{O}_{\Delta p,\chi}^{(2,0,3)}=(a_{1}\cdot L)^{2}b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(2,0,2)}=(a_{1}\cdot u_{2})(a_{1}\cdot b)L^{\mu}$
$\mathcal{O}_{\Delta p,\chi}^{(2,0,4)}=b^{2}(a_{1}\cdot u_{2})^{2}b^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\chi}^{(2,0,3)}=(a_{1}\cdot b)(a_{1}\cdot L)u_{1}^{\mu}$
| $\mathcal{O}_{\Delta a_{1},\chi}^{(2,0,4)}=(a_{1}\cdot b)(a_{1}\cdot
L)u_{2}^{\mu}$

The 1PM coefficients for $\Delta p$ are

$\displaystyle c_{\Delta p,\chi_{1}}^{(0,0,1)}$
$\displaystyle=-2\alpha_{1}^{(0,1)}$ $\displaystyle c_{\Delta
p,\chi_{1}}^{(1,0,1)}$ $\displaystyle=c_{\Delta
p,\chi_{1}}^{(1,0,2)}=2i\alpha_{1}^{(1,1)},$ $\displaystyle-c_{\Delta
p,\chi_{1}}^{(1,1,1)}$ $\displaystyle=c_{\Delta
p,\chi_{1}}^{(1,1,2)}=c_{\Delta
p,\chi_{1}}^{(1,1,3)}=4(\alpha_{1}^{(2,1)}-\alpha_{1}^{(2,3)}),\qquad
c_{\Delta p,\chi_{1}}^{(1,1,4)}=0,$ $\displaystyle-2c_{\Delta
p,\chi_{1}}^{(2,0,1)}$ $\displaystyle=c_{\Delta
p,\chi_{1}}^{(2,0,2)}=2c_{\Delta
p,\chi_{1}}^{(2,0,3)}=8(\alpha_{1}^{(2,4)}-\alpha_{1}^{(2,6)}),\qquad
c_{\Delta p,\chi_{1}}^{(2,0,4)}=0,$

while the 1PM coefficients for $\Delta a_{1}$ are

$\displaystyle c_{\Delta a_{1},\chi_{1}}^{(1,0,1)}$ $\displaystyle=-c_{\Delta
a_{1},\chi_{1}}^{(1,0,3)}=2i\alpha_{1}^{(1,1)},$ $\displaystyle c_{\Delta
a_{1},\chi_{1}}^{(1,0,2)}$ $\displaystyle=\gamma c_{\Delta
a_{1},\chi_{1}}^{(1,0,1)}+2\alpha_{1}^{(0,1)}$ $\displaystyle c_{\Delta
a_{1},\chi_{1}}^{(1,1,1)}$ $\displaystyle=-c_{\Delta
a_{1},\chi_{1}}^{(1,1,2)}=-2\alpha_{1}^{(2,1)}+2\alpha_{1}^{(2,3)}$
$\displaystyle c_{\Delta a_{1},\chi_{1}}^{(1,1,3)}$ $\displaystyle=0$
$\displaystyle c_{\Delta a_{1},\chi_{1}}^{(1,1,4)}$ $\displaystyle=c_{\Delta
a_{1},\chi_{1}}^{(1,1,5)}=-2i\alpha_{1}^{(1,2)}-\gamma c_{\Delta
a_{1},\chi_{1}}^{(1,1,1)}$ $\displaystyle c_{\Delta a_{1},\chi_{1}}^{(2,0,1)}$
$\displaystyle=c_{\Delta a_{1},\chi_{1}}^{(2,0,2)}=-c_{\Delta
a_{1},\chi_{1}}^{(2,0,4)}/2=4\alpha_{1}^{(2,4)}-4\alpha_{1}^{(2,6)}$
$\displaystyle c_{\Delta a_{1},\chi_{1}}^{(2,0,3)}$ $\displaystyle=2\gamma
c_{\Delta a_{1},\chi_{1}}^{(2,0,1)}-4i\alpha_{1}^{(1,1)}.$

The 2PM coefficients for both $\Delta p$ and $\Delta a_{1}$ are

$c_{\Delta p,\chi_{2}}^{(0,0,1)}=-\alpha_{2}^{(0,1)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,0,1)}=i\alpha_{2}^{(1,1)}$
---|---
$c_{\Delta p,\chi_{2}}^{(1,0,1)}=2i\alpha_{2}^{(1,1)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,0,2)}=\alpha_{2}^{(0,1)}+\gamma i\alpha_{2}^{(1,1)}$
$c_{\Delta p,\chi_{2}}^{(1,0,2)}=i\alpha_{2}^{(1,1)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,0,3)}=-i\alpha_{2}^{(1,1)}$
$c_{\Delta p,\chi_{2}}^{(1,1,1)}=-3\alpha_{2}^{(2,1)}+6\alpha_{2}^{(2,3)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,1,1)}=-2\alpha_{2}^{(2,1)}+\alpha_{2}^{(2,3)}$
$c_{\Delta p,\chi_{2}}^{(1,1,2)}=3\alpha_{2}^{(2,1)}-3\alpha_{2}^{(2,3)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,1,2)}=\alpha_{2}^{(2,1)}-2\alpha_{2}^{(2,3)}$
$c_{\Delta p,\chi_{2}}^{(1,1,3)}=6\alpha_{2}^{(2,1)}-3\alpha_{2}^{(2,3)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,1,3)}=-\alpha_{2}^{(2,2)}$
$c_{\Delta p,\chi_{2}}^{(1,1,4)}=3\alpha_{2}^{(2,2)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,1,4)}=-i\alpha_{2}^{(1,2)}+\gamma c_{\Delta a_{1},\chi_{2}}^{(1,1,2)}$
$c_{\Delta p,\chi_{2}}^{(2,0,1)}=-3\alpha_{2}^{(2,4)}+6\alpha_{2}^{(2,6)}$ | $c_{\Delta a_{1},\chi_{2}}^{(1,1,5)}=-2i\alpha_{2}^{(1,2)}-\gamma c_{\Delta a_{1},\chi_{2}}^{(1,1,1)}$
$c_{\Delta p,\chi_{2}}^{(2,0,2)}=6\alpha_{2}^{(2,4)}-6\alpha_{2}^{(2,6)}$ | $c_{\Delta a_{1},\chi_{2}}^{(2,0,1)}=4\alpha_{2}^{(2,4)}+2\alpha_{2}^{(2,5)}-2\alpha_{2}^{(2,6)}$
$c_{\Delta p,\chi_{2}}^{(2,0,3)}=6\alpha_{2}^{(2,4)}-3\alpha_{2}^{(2,6)}$ | $c_{\Delta a_{1},\chi_{2}}^{(2,0,2)}=2\alpha_{2}^{(2,4)}-2\alpha_{2}^{(2,5)}-4\alpha_{2}^{(2,6)}$
$c_{\Delta p,\chi_{2}}^{(2,0,4)}=3\alpha_{2}^{(2,5)}$ | $c_{\Delta a_{1},\chi_{2}}^{(2,0,3)}=-3i\alpha_{2}^{(1,1)}+\gamma c_{\Delta a_{1},\chi_{2}}^{(2,0,4)}$
| $c_{\Delta
a_{1},\chi_{2}}^{(2,0,4)}=-6\alpha_{2}^{(2,4)}+6\alpha_{2}^{(2,6)}$

The iteration structures are

$\mathcal{O}_{\Delta p,\textrm{iter}}^{(0,0,1)}=p_{cm}^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,0,1)}=b^{-2}(a_{1}\cdot b)b^{\mu}$
---|---
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(1,0,1)}=(a_{1}\cdot u_{2})L^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,0,2)}=(a_{1}\cdot u_{2})u_{1}^{\mu}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(1,0,2)}=(a_{1}\cdot L)p_{cm}^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,0,3)}=(a_{1}\cdot u_{2})u_{2}^{\mu}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(1,1,1)}=(a_{2}\cdot u_{1})(a_{1}\cdot t)^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,1,1)}=(a_{2}\cdot u_{1})\big{(}(a_{1}\cdot u_{2})L^{\mu}-(a_{1}\cdot L)u_{2}^{\mu}\big{)}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(1,1,2)}=(a_{1}\cdot u_{2})(a_{2}\cdot t)^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,1,2)}=b^{-2}(a_{2}\cdot b)\big{(}(a_{1}\cdot b)L^{\mu}-(a_{1}\cdot L)b^{\mu}\big{)}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(1,1,3)}=(\gamma^{2}-1)(a_{1}\cdot b)(a_{2}\cdot b)p_{cm}^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,1,3)}=b^{-2}(a_{1}\cdot b)\big{(}(a_{2}\cdot b)L^{\mu}+(a_{2}\cdot L)b^{\mu}\big{)}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(1,1,4)}=(a_{1}\cdot L)(a_{2}\cdot L)p_{cm}^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,1,4)}=(a_{1}\cdot u_{2})(a_{2}\cdot L)u_{2}^{\mu}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(2,0,1)}=(a_{1}\cdot u_{2})(a_{1}\cdot t)^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,1,5)}=(a_{2}\cdot u_{1})(a_{1}\cdot L)u_{1}^{\mu}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(2,0,2)}=(\gamma^{2}-1)(a_{1}\cdot b)^{2}p_{cm}^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(1,1,6)}=(a_{1}\cdot u_{2})(a_{2}\cdot L)u_{1}^{\mu}$
$\mathcal{O}_{\Delta p,\textrm{iter}}^{(2,0,3)}=(a_{1}\cdot L)^{2}p_{cm}^{\mu}$ | $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(2,0,1)}=(a_{1}\cdot u_{2})^{2}L^{\mu}$
| $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(2,0,2)}=b^{-2}(a_{1}\cdot
b)(a_{1}\cdot L)b^{\mu}$
| $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(2,0,3)}=b^{-2}(a_{1}\cdot
b)^{2}L^{\mu}$
| $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(2,0,4)}=(a_{1}\cdot
u_{2})(a_{1}\cdot L)u_{2}^{\mu}$
| $\mathcal{O}_{\Delta a_{1},\textrm{iter}}^{(2,0,5)}=(a_{1}\cdot
u_{2})(a_{1}\cdot L)u_{1}^{\mu}$

where we have used the definitions

$\displaystyle p_{cm}$ $\displaystyle\equiv(m_{2}+\gamma
m_{1})u_{1}^{\mu}-(m_{1}+\gamma m_{2})u_{2}^{\mu}$ (114)
$\displaystyle(a_{i}\cdot t)^{\mu}$
$\displaystyle\equiv{a_{i}}_{\lambda}t^{\lambda\mu},\qquad\qquad
t^{\lambda\mu}\equiv(\gamma^{2}-1)b^{\lambda}b^{\mu}-3L^{\lambda}L^{\mu}.$
(115)

The iteration coefficients for the impulse are

$\displaystyle c_{\Delta p,\textrm{iter}}^{(0,0,1)}$ $\displaystyle=g_{0,0}$
$\displaystyle c_{\Delta p,\textrm{iter}}^{(1,0,1)}$
$\displaystyle=2(m_{1}+m_{2}\gamma)g_{0,1}+m_{2}(g_{0,0}+(-1+\gamma^{2})g_{1,1})$
$\displaystyle c_{\Delta p,\textrm{iter}}^{(1,0,2)}$ $\displaystyle=-2g_{0,1}$
$\displaystyle c_{\Delta p,\textrm{iter}}^{(1,1,1)}$
$\displaystyle=-(m_{2}+m_{1}\gamma)(g_{1,2}+g_{0,3})-m_{1}(g_{0,1}+(-1+\gamma^{2})g_{2,3})$
$\displaystyle c_{\Delta p,\textrm{iter}}^{(1,1,2)}$
$\displaystyle=(m_{1}+m_{2}\gamma)(g_{1,2}+g_{0,3})+m_{2}(g_{0,2}+(-1+\gamma^{2})g_{1,3})$
$\displaystyle c_{\Delta p,\textrm{iter}}^{(1,1,3)}$
$\displaystyle=2(g_{1,2}-2g_{0,3}),\qquad c_{\Delta
p,\textrm{iter}}^{(1,1,4)}=2(g_{1,2}+2g_{0,3})$ $\displaystyle c_{\Delta
p,\textrm{iter}}^{(2,0,1)}$
$\displaystyle=(m_{1}+m_{2}\gamma)(g_{1,1}+2g_{0,4})+m_{2}(g_{0,1}+2(-1+\gamma^{2})g_{1,4})$
$\displaystyle c_{\Delta p,\textrm{iter}}^{(2,0,2)}$
$\displaystyle=(g_{1,1}-4g_{0,4}),\qquad c_{\Delta
p,\textrm{iter}}^{(2,0,3)}=(g_{1,1}+4g_{0,4})$

The iteration coefficients for the spin kick are

$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(1,0,1)}$
$\displaystyle=m_{2}(-1+\gamma^{2})(-g_{0,0}+(-1+\gamma^{2})g_{1,1})$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(1,0,2)}$
$\displaystyle=m_{2}(-1+\gamma^{2})(2g_{0,1}+\gamma
g_{1,1})+(m_{1}+m_{2}\gamma)g_{0,0}$ $\displaystyle c_{\Delta
a_{1},\textrm{iter}}^{(1,0,3)}$ $\displaystyle=m_{2}(-1+\gamma^{2})(-g_{1,1})$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(1,1,1)}$
$\displaystyle=-c_{\Delta p,\textrm{iter}}^{(1,1,1)}$ $\displaystyle c_{\Delta
a_{1},\textrm{iter}}^{(1,1,2)}$
$\displaystyle=m_{2}(-1+\gamma^{2})(-g_{0,2}+(-1+\gamma^{2})g_{1,3})+(m_{1}+m_{2}\gamma)(-1+\gamma^{2})(g_{1,2}-g_{0,3})$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(1,1,3)}$
$\displaystyle=m_{2}(-1+\gamma^{2})(2g_{0,2}-2(-1+\gamma^{2})g_{1,3}),\qquad
c_{\Delta a_{1},\textrm{iter}}^{(1,1,4)}=m_{2}(-1+\gamma^{2})(2g_{1,3})$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(1,1,5)}$
$\displaystyle=m_{1}(g_{0,0}+\gamma g_{0,1}+(-1+\gamma^{2})(g_{2,2}+\gamma
g_{2,3}))+(m_{2}+m_{1}\gamma)(2g_{0,2}+\gamma g_{0,3}+\gamma g_{1,2})$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(1,1,6)}$
$\displaystyle=m_{2}(-1+\gamma^{2})(-2g_{1,2}-2g_{0,3}-2\gamma
g_{1,3})+(m_{1}+m_{2}\gamma)(-2g_{0,2})$ $\displaystyle c_{\Delta
a_{1},\textrm{iter}}^{(2,0,1)}$ $\displaystyle=-c_{\Delta
p,\textrm{iter}}^{(2,0,1)}$ $\displaystyle c_{\Delta
a_{1},\textrm{iter}}^{(2,0,2)}$
$\displaystyle=m_{2}(-1+\gamma^{2})(-6(-1+\gamma^{2})g_{1,4}+3g_{0,1})+(m_{1}+m_{2}\gamma)(-1+\gamma^{2})(2g_{0,4}-g_{1,1})$
$\displaystyle c_{\Delta a_{1},\textrm{iter}}^{(2,0,3)}$
$\displaystyle=m_{2}(-1+\gamma^{2})(4g_{0,1}-8(-1+\gamma^{2})g_{1,4})-c_{\Delta
a_{1},\textrm{iter}}^{(2,0,2)}$ $\displaystyle c_{\Delta
a_{1},\textrm{iter}}^{(2,0,4)}$
$\displaystyle=4m_{2}g_{0,1}+4(m_{1}+m_{2}\gamma)g_{0,4}-(-1+\gamma^{2})^{-1}c_{\Delta
a_{1},\textrm{iter}}^{(2,0,2)}$ $\displaystyle c_{\Delta
a_{1},\textrm{iter}}^{(2,0,5)}$ $\displaystyle=m_{2}(-1+\gamma^{2})((-6\gamma
g_{1,4}-6g_{0,4}-3g_{1,1}))+m_{2}(-\gamma g_{0,1}-g_{0,0})$
$\displaystyle\qquad\qquad\qquad\qquad-(m_{2}+m_{1}\gamma)(2g_{0,4})-(m_{1}+m_{2}\gamma)(4g_{0,1}+\gamma
g_{1,1}),$

where

$\displaystyle g_{i,j}$ $\displaystyle\equiv\beta_{i}\beta_{j},$ (116)
$\displaystyle\beta_{0}$
$\displaystyle=\alpha_{1}^{(0,1)},\qquad\beta_{1}=i\alpha_{1}^{(1,1)},\qquad\beta_{2}=i\alpha_{1}^{(1,2)},$
$\displaystyle\beta_{3}$
$\displaystyle=-\alpha_{1}^{(2,1)}+\alpha_{1}^{(2,3)},\qquad\beta_{4}=-\alpha_{1}^{(2,4)}+\alpha_{1}^{(2,6)}.$
(117)

The $\alpha_{n}^{(i,j)}$ coefficients are those appearing in a scattering
amplitude of the form

$\displaystyle\mathcal{M}^{\textrm{nPM}}=\zeta_{q,n}\Big{\\{}$
$\displaystyle\alpha_{n}^{(0,1)}+\alpha_{n}^{(1,1)}\mathcal{E}_{1}+\alpha_{n}^{(1,2)}\mathcal{E}_{2}$
(118) $\displaystyle+$
$\displaystyle\alpha_{n}^{(2,1)}\mathcal{E}_{1}\mathcal{E}_{2}+\alpha_{n}^{(2,2)}q^{2}(u_{2}\cdot
a_{1})(u_{1}\cdot a_{2})+(-1+\gamma^{2})\alpha_{n}^{(2,3)}(q\cdot
a_{1})(q\cdot a_{2})$ $\displaystyle+$
$\displaystyle\alpha_{n}^{(2,4)}\mathcal{E}_{1}^{2}\ \
\,+\alpha_{n}^{(2,5)}q^{2}(u_{2}\cdot
a_{1})^{2}\qquad\quad\,+(-1+\gamma^{2})\alpha_{n}^{(2,6)}(q\cdot
a_{1})^{2}\Big{\\}}\,,$

where the order-dependent factors $\zeta_{n}$ are given by

$\displaystyle\zeta_{q,1}=-\frac{4\pi
G}{q^{2}},\qquad\zeta_{q,2}=\frac{2\pi^{2}G^{2}}{\sqrt{-q^{2}}},$ (119)

and
$\mathcal{E}_{i}\equiv\epsilon^{\mu\nu\rho\sigma}{u_{1}}_{\mu}{u_{2}}_{\nu}q_{\rho}a_{1}{}_{\sigma}$.
These amplitudes are given in a manifestly shift-symmetric basis. For
uniformity, we have included terms containing $q^{2}$ in the numerator also
for the 1PM, but they produce local terms upon Fourier transform, and can be
discarded. From such an amplitude, we can obtain an Eikonal phase like the one
above through Fourier transform. In the case of gravity, the tree-level
coefficients $\alpha_{1}^{(i,j)}$ take the explicit form

$\displaystyle\alpha_{1}^{(0,1)}$
$\displaystyle=4m_{1}^{2}m_{2}^{2}(2\gamma^{2}-1)\,,\qquad\alpha_{1}^{(1,1)}=-8im_{1}^{2}m_{2}^{2}\gamma\,,\qquad\alpha_{1}^{(1,2)}=-8im_{1}^{2}m_{2}^{2}\gamma,$
$\displaystyle\alpha_{1}^{(2,1)}$
$\displaystyle=\frac{4m_{1}^{2}m_{2}^{2}(1-2\gamma^{2})}{\gamma^{2}-1}\,,\qquad\alpha_{1}^{(2,4)}=\frac{2m_{1}^{2}m_{2}^{2}(2\gamma^{2}-1)(C_{E}-1)}{\gamma^{2}-1}\,,$
(120) $\displaystyle\alpha_{1}^{(2,2)}$
$\displaystyle=0\,,\,\,\qquad\alpha_{1}^{(2,3)}=0,\qquad\alpha_{1}^{(2,5)}=0\,,\qquad\alpha_{1}^{(2,6)}=0\,,$

while for the one-loop amplitude, the coefficients $\alpha_{2}^{(i,j)}$ are
given by

$\displaystyle\alpha_{2}^{(0,1)}$
$\displaystyle=3m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})(5\gamma^{2}-1)\,,$
$\displaystyle\alpha_{2}^{(1,1)}$
$\displaystyle=-\frac{im_{1}^{2}m_{2}^{2}(4m_{1}+3m_{2})(5\gamma^{2}-3)\gamma}{(\gamma^{2}-1)}\,,\qquad\alpha_{2}^{(1,2)}=-\frac{im_{1}^{2}m_{2}^{2}(3m_{1}+4m_{2})(5\gamma^{2}-3)\gamma}{(\gamma^{2}-1)}$
$\displaystyle\alpha_{2}^{(2,1)}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})(20\gamma^{4}-21\gamma^{2}+3)}{(\gamma^{2}-1)^{2}}\,,\qquad\alpha_{2}^{(2,2)}=\frac{m_{1}^{2}m_{2}^{2}(m_{1}+m_{2})(5\gamma^{2}-3)\gamma}{(\gamma^{2}-1)^{2}}\,,$
$\displaystyle\alpha_{2}^{(2,3)}$ $\displaystyle=0\,.$

For the $S_{1}^{2}$ terms of the amplitude, we find it useful to split the
coefficients as

$\displaystyle\alpha_{2}^{(2,i)}=\frac{m_{1}^{2}m_{2}^{2}}{16(\gamma^{2}-1)^{2}}\left(\alpha_{2,\textrm{BH}}^{(2,i)}+C_{E}\alpha_{2,C_{E}}^{(2,i)}\right),\qquad
i=4,5,6.$ (121)

where the Wilson coefficient $C_{E}$ vanishes for the black hole, and so the
$\alpha_{2,\textrm{BH}}$ capture all the information in that limit. The pieces
of the coefficient are given by

$\displaystyle\alpha_{2,\textrm{BH}}^{(2,4)}$
$\displaystyle=-2\left(m_{1}\left(95\gamma^{4}-102\gamma^{2}+15\right)+4m_{2}\left(15\gamma^{4}-15\gamma^{2}+2\right)\right),$
$\displaystyle\alpha_{2,C_{E}}^{(2,4)}$
$\displaystyle=\left(m_{1}\left(95\gamma^{4}-102\gamma^{2}+23\right)+4m_{2}\left(15\gamma^{4}-13\gamma^{2}+2\right)\right)\,,$
$\displaystyle\alpha_{2,\textrm{BH}}^{(2,5)}$
$\displaystyle=-2\left(m_{1}\left(35\gamma^{4}-30\gamma^{2}+3\right)-4m_{2}\left(3\gamma^{2}-1\right)\right),$
$\displaystyle\alpha_{2,C_{E}}^{(2,4)}$
$\displaystyle=\left(m_{1}\left(35\gamma^{4}-30\gamma^{2}+11\right)+4m_{2}\left(3\gamma^{2}+1\right)\right)\,,$
$\displaystyle\alpha_{2,C_{E}}^{(2,6)}$
$\displaystyle=-4\left(3m_{1}(5\gamma^{2}-1)+m_{2}(15\gamma^{2}-1)\right)\,\qquad\alpha_{2,\textrm{BH}}^{(2,6)}=0.$

##  Appendix B Useful Expressions

The Fourier transform is given by

$\displaystyle I^{\mu_{1}\mu_{2}\cdots\mu_{\beta}}\left(D,\alpha\right)$
$\displaystyle=\int\hat{\mathrm{d}}^{D}\hat{\delta}\left(2p_{1}\cdot
q\right)\hat{\delta}\left(2p_{2}\cdot q\right)e^{iq\cdot
x}\frac{q^{\mu_{1}}q^{\mu_{2}}\cdots q^{\mu_{\beta}}}{q^{2\alpha}}$ (122)
$\displaystyle=(-i\frac{\partial}{\partial
x_{\mu_{1}}})(-i\frac{\partial}{\partial
x_{\mu_{2}}})\cdots(-i\frac{\partial}{\partial
x_{\mu_{\beta}}})\frac{1}{4m_{1}m_{2}\sqrt{\gamma^{2}-1}}\frac{\Gamma(D/2-1-\alpha)}{2^{2\alpha}\pi^{D/2-1}\Gamma(\alpha)}\frac{1}{|x|^{d-2-2\alpha}}$
$\displaystyle I\left(4-2\epsilon,1\right)=-\frac{1}{16\pi
m_{1}m_{2}\sqrt{\gamma^{2}-1}}\log
x^{2}+\mathcal{O}(\frac{1}{\epsilon})+const$ (123) $\displaystyle
I^{\mu}\left(4,1\right)=i\frac{1}{8\pi
m_{1}m_{2}\sqrt{\gamma^{2}-1}}\frac{x^{\mu}}{x^{2}}$ (124)

The values for $N_{q},N_{0}$ and $N_{0q}$ are derived from the conditions
$e_{0}^{2}=-e_{q}^{2}=1$ and $e_{q}\cdot e_{0}=0$, which at order $G^{0}$ are
given by

$\displaystyle N_{0}$
$\displaystyle=\frac{1}{\sqrt{(p_{1}+p_{2})^{2}}}=\frac{1}{\sqrt{m_{1}^{2}+m_{2}^{2}+2\gamma
m_{1}m_{2}}},$ (125) $\displaystyle N_{q}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{(p_{1}+p_{2})^{2}}{((p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2})}}=\frac{1}{2}\sqrt{\frac{m_{1}^{2}+m_{2}^{2}+2\gamma
m_{1}m_{2}}{m_{1}^{2}m_{2}^{2}(\gamma^{2}-1)}},$ $\displaystyle N_{0q}$
$\displaystyle=-\frac{1}{2}\frac{m_{1}^{2}-m_{2}^{2}}{\sqrt{(p_{1}+p_{2})^{2}((p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2})}}=-\frac{1}{2}\frac{m_{1}^{2}-m_{2}^{2}}{\sqrt{(m_{1}^{2}+m_{2}^{2}+2\gamma
m_{1}m_{2})(m_{1}^{2}m_{2}^{2}(\gamma^{2}-1))}}$

It is often useful to resolve certain vectors onto a basis
$\\{u_{1}^{\mu},u_{2}^{\mu},b^{\mu},\varepsilon^{\mu}(b,u_{1},u_{2})\\}$. A
few useful expressions are

$\displaystyle\varepsilon^{\mu}(b,s,u_{1})$ $\displaystyle=\frac{s\cdot
u_{2}}{\gamma^{2}-1}\varepsilon^{\mu}\left(b,u_{1},u_{2}\right)-\frac{\gamma\varepsilon\left(b,s,u_{1},u_{2}\right)}{\gamma^{2}-1}u_{1}^{\mu}+\frac{\varepsilon\left(b,s,u_{1},u_{2}\right)}{\gamma^{2}-1}u_{2}^{\mu}$
(126) $\displaystyle\varepsilon^{\mu}(b,s,u_{2})$ $\displaystyle=\frac{\gamma
s\cdot
u_{2}}{\gamma^{2}-1}\varepsilon^{\mu}\left(b,u_{1},u_{2}\right)-\frac{\varepsilon\left(b,s,u_{1},u_{2}\right)}{\gamma^{2}-1}u_{1}^{\mu}+\frac{\gamma\varepsilon\left(b,s,u_{1},u_{2}\right)}{\gamma^{2}-1}u_{2}^{\mu}$
$\displaystyle\varepsilon^{\mu}(s,u_{1},u_{2})$ $\displaystyle=\frac{b\cdot
s}{b^{2}}\varepsilon^{\mu}\left(b,u_{1},u_{2}\right)+\frac{\varepsilon\left(b,s,u_{1},u_{2}\right)}{b^{2}}b^{\mu}$

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