#  Dynamics for Super-Extremal Kerr Binary Systems at ${\cal O}(G^{2})$

Yilber Fabian Bautista  Institut de Physique Théorique, CEA, Université
Paris–Saclay, F–91191 Gif-sur-Yvette cedex, France

###### Abstract

Using the recently derived higher spin gravitational Compton amplitude from
low-energy analytically continued ($a/Gm\gg 1$) solutions of the Teukolsky
equation for the scattering of a gravitational wave off the Kerr black hole,
observables for non-radiating super-extremal Kerr binary systems at second
post-Minkowskian (PM) order and up to sixth order in spin are computed. The
relevant 2PM amplitude is obtained from the triangle-leading singularity in
conjunction with a generalization of the holomorphic classical limit for
massive particles with spin oriented in generic directions. Explicit results
for the 2PM eikonal phase written for both Covariant and Canonical spin
supplementary conditions – CovSSC and CanSSC respectively – as well as the 2PM
linear impulses and individual spin kicks in the CanSSC are presented. The
observables reported in this letter are expressed in terms of generic contact
deformations of the gravitational Compton amplitude, which can then be
specialized to Teukolsky solutions. In the latter case, the resulting 2PM
observables break the newly proposed spin-shift symmetry of the 2PM amplitude
starting at the fifth order in spin. Aligned spin checks as well as the high
energy behavior of the computed observables are discussed.

Introduction. The application of QFT-inspired methods to compute observables
in classical gravity has seen tremendous advances in the last years Cheung
_et al._ (2018); Kosower  _et al._ (2019); Bern  _et al._ (2022a, b, 2021,
2020, 2019); Dlapa  _et al._ (2023, 2022); Kälin  _et al._ (2020); Mogull  _et
al._ (2021); Jakobsen  _et al._ (2022a); Elkhidir  _et al._ (2023); Brandhuber
_et al._ (2023); Herderschee  _et al._ (2023); Georgoudis  _et al._ (2023); Di
Vecchia  _et al._ (2021) due to their potential relevance for analyzing
measured signals in gravitational wave (GW) detectors Abbott  _et al._ (2016).
The use of these techniques is justified by the separation of scales that
allows to treat physics problems in an effective manner. For instance, an
isolated black hole (BH) seen from far away can be thought of as a point
particle, and its finite size effects such as spin multipole moments can be
modeled by effective operators in (classical) EFT constructions Levi and
Steinhoff (2015); Bern  _et al._ (2022c); Chung  _et al._ (2020a); Jakobsen
and Mogull (2022a); Liu  _et al._ (2021); Arkani-Hamed  _et al._ (2017); Chung
_et al._ (2019); Bautista  _et al._ (2022); Aoude  _et al._ (2022a, b). These
operators are accompanied by free coefficients parametrizing the UV ignorance
of the effective model; they can be fixed by matching computations in the
effective and the full theory. Following this logic, an astonishingly simple
description of an isolated linearized Kerr BH Vines (2018) – effectively a
super-extremal (SE) Kerr BH with spin parameter $a^{\star}=\frac{a}{Gm}\gg 1$,
where $m$ and $a$ are its mass and ring radius respectively – as an elementary
particle of infinite spin minimally coupled to gravity, has recently appeared
in the literature Guevara  _et al._ (2019); Chung  _et al._ (2019); Arkani-
Hamed  _et al._ (2019); Aoude  _et al._ (2020).

Actual Kerr BHs are however neither isolated objects in nature nor linearized
solutions of the field equations as they possess spin parameters satisfying
$a^{\star}\leq 1$. A correct description of these objects requires therefore
resummations of the perturbative computations. Accomplishing these
resummations from a perturbative amplitude approach is a heroic task not yet
addressed along these lines of reasoning, and therefore relying on the
effective one-body formalism Buonanno and Damour (1999, 2000); Damour (2001);
Vines (2018); Buonanno  _et al._ (2006); Khalil  _et al._ (2023), as an
alternative way to study these more realistic Kerr BHs scenarios 111In
principle, in an EFT approach, a spin resummation prescription can be obtained
by allowing the effective coefficients to be functions of $a^{\star}$ Bautista
_et al._ (2022); Saketh and Vines (2022). . Nevertheless, perturbative
approaches are still very useful to gain insights into the physics of actual
Kerr BH systems, and in this work, we follow these lines to study systems made
not of actual Kerr BHs, but of their close relatives, SE Kerr BHs (fastly
rotating objects).

Although SE Kerr BHs are not real objects in nature, they share many features
of actual Kerr BHs, as the two are related via analytic continuation of the
spin parameter from the physical region $a^{\star}\leq 1$, to the SE
$a^{\star}\gg 1$ region. For observables that cannot probe the nature of the
BH horizon – which we shall refer to as  true conservative observables 222For
the systems considered in this work, we discard radiative effects encoded in
the emission of gravitational waves towards future null infinity.  – this
continuation should not possess any subtlety and their values computed for one
kind of objects or the others should coincide in the overlapping
(continuation) region. In this sense, SE Kerr observables readily encapsulate
part of the dynamic for actual Kerr BHs Vines (2018); Bini and Damour (2017);
Chung  _et al._ (2020a); Bern  _et al._ (2020); Kosmopoulos and Luna (2021);
Chen  _et al._ (2022); Jakobsen and Mogull (2022a); Levi and Steinhoff (2021,
2015); Levi and Yin (2022); Kim  _et al._ (2022a); Liu  _et al._ (2021);
Jakobsen and Mogull (2022a); Goldberger  _et al._ (2018); Jakobsen  _et al._
(2022b, c, a); Jakobsen and Mogull (2022b); Blanchet (2014); Porto (2016);
Levi (2018); Levi and Steinhoff (2016, 2021); Levi  _et al._ (2020a);
Antonelli  _et al._ (2020a); Levi  _et al._ (2020b); Antonelli  _et al._
(2020b); Kim  _et al._ (2021); Levi  _et al._ (2021); Levi and Teng (2021);
Maia  _et al._ (2017); Cho  _et al._ (2022, 2021); Kim  _et al._ (2022a, b);
Levi  _et al._ (2022); Levi and Yin (2022). Observables that can probe the
nature of the BH horizon – absorptive observables accounting for fluxes of
energy at the BH horizon – are more subtle since the notion of absorption does
not exist for objects without a horizon; hence, SE Kerr BH observables look
always  conservative. These can however receive contributions from effective
operators that mimic the physical effects happening at the horizon of an
actual Kerr BH, but whose definite identification requires (perhaps) matching
computations to actual dissipative Kerr BH’s observables Tagoshi  _et al._
(1997); Poisson (2004); Chatziioannou  _et al._ (2013, 2016); Isoyama and
Nakano (2018); Goldberger and Rothstein (2006); Goldberger  _et al._ (2020);
Porto (2008); Saketh  _et al._ (2022, 2022).

In this work, we compute conservative observables for the scattering of two SE
Kerr BHs at $\mathcal{O}(G^{2})$, but whose content can potentially encode
true conservative as well as absorptive effects for actual Kerr BHs. For this,
we use the recently extracted higher spin gravitational Compton amplitude from
low-energy solutions of the Teukolsky equation in the SE region Bautista  _et
al._ (2022). The observables of interest in this work are the linear impulse
and the individual spin kicks, for generic spin orientations. We will use the
eikonal phase as an intermediate object, and compute the observables using
formula (14), proposed by the authors of references Bern  _et al._ (2020);
Kosmopoulos and Luna (2021). Since contact deformation of the Compton
amplitude enters the 2PM amplitude only through the triangle diagrams (fig.
1), we expect this formula (14) (showed to be valid for orders $a^{n\leq 4}$
at 2PM Bern  _et al._ (2020); Kosmopoulos and Luna (2021); Chen  _et al._
(2022)) to continue to be valid for higher spin orders. We compute the 2PM
amplitude for generic contact deformation of the Compton amplitude and discuss
the consequences the evaluation of these free coefficients via different
prescriptions has on the computed observables.

A tree-level gravitational Compton amplitude from the Teukolsky equation. In
Bautista  _et al._ (2022), an ansatz for the $\mathcal{O}(G)$ opposite
helicity, higher spin gravitational Compton amplitude of the form (momentum
cons. $p_{4}=p_{1}+k_{2}+k_{3}$)

$A_{4}^{(S)}=A_{4}^{0}\left(e^{(2w+k_{3}-k_{2})\cdot a}{+}P_{\xi}(k_{2}\cdot
a,{-}k_{3}\cdot a,w\cdot a)\right)_{2S}\,,$ (1)

was written invoking physical constraints such as locality, unitarity, and
crossing symmetry, together with a prescription to write contact deformations
that match Teukolsky solutions only in a non-trivial manner 333A more general
ansatz for a gravitational Compton amplitude including neutron stars was
provided recently in Haddad (2023). Other approaches to computing the higher
spin gravitational Compton amplitude have been explored in the literature
Aoude  _et al._ (2022a); Chiodaroli  _et al._ (2022); Bern  _et al._ (2022c);
Aoude  _et al._ (2022a, b); Haddad (2023); Cangemi  _et al._ (2022); Bjerrum-
Bohr  _et al._ (2023); Alessio (2023) .

The scalar contribution, $\langle A_{4}^{0}\rangle$ in (1), encodes the
helicity and physical pole structure of the amplitude, whereas the terms
inside the big parenthesis are only functions of the kinematic invariants and
the spin of the massive legs. Explicitly, the former reads

$\langle A_{4}^{0}\rangle=32\pi Gm^{2}\frac{(\epsilon_{2}\cdot
u)^{2}(\tilde{\epsilon}_{3}\cdot
u)^{2}}{\xi},\,\,\,\xi=\frac{(s-m^{2})^{2}}{m^{2}t}\,,$ (2)

which we choose to evaluate in the gauge

$\epsilon_{2}=\frac{\sqrt{2}|3]\langle
2|}{[32]}\propto\tilde{\epsilon}_{3}=\frac{\sqrt{2}|3]\langle 2|}{\langle
32\rangle}\,,\,\,\,w^{\mu}:=\frac{u\cdot
k_{2}}{u\cdot\epsilon_{2}}\epsilon^{\mu}_{2}\,.$ (3)

Here we have included the massless vector $w^{\mu}$ entering in (1), which is
constructed from the spinors of the massless legs, and $u=p_{1}/m$, is the
incoming massive leg’s four-velocity, with $m$ its respective mass.

The function $P_{\xi}$ contains contact deformations of the BCFW exponential –
some of which at the same time cure the unphysical singularities that appear
starting at $\mathcal{O}(a^{5})$ when the exponential function is expanded –
was written as a Laurent expansion in the optical parameter $\xi$, having the
explicit form given in eq. $(3.47)$ in Bautista  _et al._ (2022), which we
abstain from including explicitly in this work for brevity. This expansion was
further parametrized by three multi-variable polynomials
$p^{(m)}_{|a|},q^{(m)}_{|a|},r^{(m)}_{|a|}$, functions of $(k_{2}\cdot
a,-k_{3}\cdot a,w\cdot a)$, symmetric in their first two arguments, but also
including a linear correction in $\omega|a|$, with
$\omega\approx(s-m^{2})/2m$, the energy of the massless legs. Up to the sixth
order in spin, these polynomials have the explicit form given by eqs.
$(3.48-3.49)$ in Bautista  _et al._ (2022), and contain two type of spin
operators: regular operators, functions of only $\mathcal{R}=\\{k_{2}\cdot
a,k_{3}\cdot a,w\cdot a\\}$, and  exotic operators, functions of
$\mathcal{R}\cup\\{\omega|a|\\}$. The former will encapsulate the real
contributions to the solution to the Teukolsky equation, whereas the latter
accounts for contributions that are imaginary when the BH rotation parameter
satisfies the inequality $a^{\star}\leq 1$.

Each effective operator in $p^{(m)}_{|a|},q^{(m)}_{|a|},r^{(m)}_{|a|}$ is
accompanied by a free coefficient $c_{i}^{(j)},d_{i}^{(j)},f_{i}^{(j)}$
respectively; contact deformations were shown to appear starting at fourth
order in spin as known from the work Chung  _et al._ (2019). These free
coefficients were further fixed by requiring that (1) matches the
$\mathcal{O}(G)$ sector of the low energy limit ($\epsilon=Gm\omega\ll 1$) of
the scattering amplitude for the scattering of a GW off the Kerr BH, computed
with the tools of black hole perturbation theory Sasaki and Tagoshi (2003);
Bautista  _et al._ (2021). Explicit solutions up to six order in spin can be
found in Table 1 in Bautista  _et al._ (2022), which we include in table 5 of
the supplementary material for this work for the reader’s convenience.
Remarkably, up to the fourth order in spin, the Teukolsky solution perfectly
matches the classical limit of the undeformed minimal coupling gravitational
Compton amplitude of Arkani-Hamed, Huang, and Huang Arkani-Hamed  _et al._
(2017), given by the expansion of the exponential in (1), up to
$\mathcal{O}(a^{4})$. Up to this order, Teukolsky solutions are polynomials in
$a^{\star}$, therefore, providing a unique answer for the analytically
continued Kerr results to the SE Kerr approximation $a^{\star}\gg 1$. Starting
at the fifth order in spin, Teukolsky solutions contain complex, non-rational
functions of $a^{\star}$, which are in addition discontinuous at
$a^{\star}=1$. Therefore, a prescription for analytic continuation to the SE
region was needed. The two different prescriptions provided in Bautista  _et
al._ (2022) were labeled by the parameter $\eta$, which takes values $\pm 1$,
with the sign determined by the continuation procedure. Non-trivially,
contributions in the Teukolsky solutions that were real before the
continuation uniquely fix the free coefficients for regular operators in (1),
and are independent of such continuation prescriptions, whereas the pieces
that were imaginary before the continuation, fix the coefficients of exotic
operators, modulo a sign. The exotic contributions are believed to encode only
physical effects happening at the BH horizon for actual Kerr ($a^{\star}\leq
1$) systems, but their precise physical interpretation in a realistic Kerr
context is beyond the scope of this work. We refer the reader to the recent
work Saketh  _et al._ (2022) for a related analysis of horizon dissipation for
Kerr systems (see also Tagoshi  _et al._ (1997); Poisson (2004); Chatziioannou
_et al._ (2013, 2016); Isoyama and Nakano (2018); Goldberger and Rothstein
(2006); Goldberger  _et al._ (2020); Porto (2008); Saketh  _et al._ (2022)).

Some comments from this matching are in order: 1) After analytic continuation,
imaginary contributions become real, therefore providing conservative
information for the SE Kerr system. This is a consequence of erasing the BH
horizon in the continuation procedure, therefore, removing any source of
dissipation. 2) In the matching of (1) to the low energy limit of Teukolsky
solutions, terms of the form $\epsilon^{n}(a^{\star})^{m}f(a^{\star})$ for
$m\neq n$ and $f$ and arbitrary function of the rotation parameter, were
discarded as they do not contribute to the tree-level amplitude. Similarly,
terms of the form $\epsilon^{n}\log\epsilon\,(a^{\star})^{m}f(a^{\star})$ that
do not produce $\mathcal{O}(G)$ contributions were removed. These terms might
however become important for a Compton amplitude that matches the actual Kerr
($a^{\star}\leq 1$) solutions 444 Table 2 in Bautista  _et al._ (2022) contain
coefficients that match the Compton ansatz (1) to low energy Kerr solutions
without analytic continuations, but have however discarded
$\epsilon^{n}\log\epsilon\,(a^{\star})^{m}f(a^{\star})$ terms.  .

For the same helicity sector, the $\mathcal{O}(G)$ Teukolsky solutions were
shown to match spectacularly the analogous exponential $\langle A_{0}\rangle
e^{-(k_{2}+k_{3})\cdot a}$, with checks made up to $\mathcal{O}(a^{6})$ in the
SE region. The results were also shown to be independent of the continuation
prescription.

Leading PM eikonal phase for super-extremal Kerr. In this work we are
interested in computing canonical observables for binary systems at the second
PM order. Binary 2PM observables however will necessarily require 1PM
information entering as iteration terms in the operator formulation Kosower
_et al._ (2019), the 2PM Hamiltonian Bern  _et al._ (2020); Chen  _et al._
(2022), or equivalent as quadratic eikonal contributions in the formula (14)
for the computation of canonical observables directly from the eikonal phase
Bern  _et al._ (2020); Kosmopoulos and Luna (2021). Driven by this, in this
section we revisit the computation of the 1PM eikonal phase for the scattering
of two SE Kerr BHs to all orders in spins. We start by recalling the tree
level, all orders in spin two-body amplitude is obtained from the unitarity
gluing of two SE Kerr 3-point amplitudes Guevara  _et al._ (2019); Chung  _et
al._ (2019); Vines (2018), resulting in the compact expression Chung  _et al._
(2020a):

$M^{(\text{1PM})}_{\text{bare}}{=}{-}\frac{16\pi
G\,(m_{1}m_{2})^{2}}{q^{2}}\cosh\left(2\theta{+}i\frac{\mathcal{E}_{1}/m_{1}{+}\mathcal{E}_{2}/m_{2}}{m_{1}m_{2}\sinh(\theta)}\right)\,.$
(4)

Here we have used the notation
$\mathcal{E}_{i}=\epsilon_{\mu\nu\rho\sigma}q^{\mu}p_{1}^{\nu}p_{2}^{\rho}s_{i}^{\sigma}$,
where $p_{i},s_{i}$ are the respective momenta and spins of the incoming BHs,
and $q$ is the momentum transfer. The Lorentz boost factor was introduced via
the hyperbolic functions
$\cosh(\theta)=\sigma=\frac{p_{1}{\cdot}p_{2}}{m_{1}m_{2}}\,,\,\,\,\sinh(\theta)=\sigma
v\,,$ with $v$ the two bodies’ relative velocity. The “bare” label in (4)
indicates massive spin polarization tensors have been removed, and the
observables computed with this prescription have the rotational gauge freedom
fixed by the Tulczyjew-Dixon covariant spin supplementary condition (CovSSC)
$p_{b}S^{ab}=0$ W. (1957, 1959). Hamiltonian observables however are
customarily computed in the canonical Newton-Wigner SSC (CanSSC)
$S^{ab}(p_{b}+\sqrt{p^{2}}\delta_{0b})$, with $\\{a,b\\}$ local frame indices
Newton and Wigner (1949); Levi and Steinhoff (2015). A way to make the
previous amplitude satisfy the latter constraint is provided by dressing the
bare amplitude with the Thomas-Wigner rotation factors 555More precisely, this
corresponds to a canonical alignment of the incoming and outgoing massive
polarization tensors through a Lorentz transformation Chung  _et al._ (2020b,
a). ,

$M_{\text{dressed}}^{\text{(nPM)}}=M^{(\text{nPM})}_{\text{bare}}U_{1}U_{2}\,,\,\,\,U_{i}=e^{\frac{i\tau\mathcal{E}_{i}}{Em_{i}(E_{i}+m_{i})}}\,,\,\,\,n=1,2.$
(5)

where $E{=}E_{1}{+}E_{2}$, is the sum of the individual bodies’ energies, and
$\tau{=}1$, is a parameter that keeps track of the CanSSC prescription. These
rotation factors are written in the center of mass (CoM) frame, where the
momenta of the BHs are parametrized in the following way (see $e.g.$ Bern  _et
al._ (2020) for details):

$p_{1}{=}{-}(E_{1},\bm{p})\,,\,\,p_{2}{=}{-}(E_{2},\bm{p})\,,\,\,q{=}(0,\bm{q})\,,\,\,\bm{p}{\cdot}\bm{q}{=}\frac{\bm{q}^{2}}{2}\,,$
(6)

Here, $\bm{p}$ is the asymptotic incoming three-momentum – sometimes also
referred to as $\bm{p}_{\infty}$ – and $\bm{q}$ is the three-momentum transfer
in the scattering process. The covariant spin operators can analogously be
mapped to their CoM representations via

$\begin{split}\mathcal{E}_{i}&=E\,\bm{s}_{i}\cdot(\bm{p}\times\bm{q})\,,\,\,\,q\cdot
s_{i}=\bm{q}\cdot\bm{s}_{i}+\mathcal{O}(\bm{q}^{2})\,,\\\
|s_{i}|&=|\bm{s}_{i}|\,,\,\,p_{i}\cdot
s_{j}=\epsilon_{ij}\frac{E}{m_{i}}\bm{p}\cdot\bm{s}_{i}\,,\,\,\epsilon_{12}=-\epsilon_{21}=1\,.\end{split}$
(7)

The eikonal phase for this $2\to 2$ scattering process is then obtained from
the 2-dimensional Fourier transform of the two-body amplitude into impact
parameter space. For the generic 1PM and 2PM cases, we have Amati  _et al._
(1987); Melville  _et al._ (2014); Akhoury  _et al._ (2021); Di Vecchia  _et
al._ (2019)

$\chi^{\text{(nPM)}}_{\tau}{=}\frac{1}{4m_{1}m_{2}\sqrt{\sigma^{2}-1}}\int\frac{d^{2}\bm{q}}{(2\pi)^{2}}e^{-i\bm{q}{\cdot}\bm{b}}M_{\tau}^{\text{(nPM)}}\,,\,n=1,2,$
(8)

with $M_{\tau}^{\text{(2PM)}}$ computed from the triangle leading singularity
(LS) fig. 1. Here we have left explicit the $\tau$ label which for $\tau=0$,
we input the bare (CovSSC) amplitude into (8), whereas for $\tau=1$, the
Thomas-Wigner rotation factors (CanSSC) need to be supplemented.

Going back to the 1PM analysis, the spin operators entering in (4) and (5)
simply become shifts of the impact parameter when the explicit evaluation of
(8) is performed. At the 1PM order, one arrives at the all-spin eikonal phase

$\chi_{\tau}^{\text{(1PM)}}=-\frac{m_{1}m_{2}G}{2\sqrt{\sigma^{2}-1}}\sum_{\pm}(\cosh(2\theta)\pm\sinh(2\theta))\log(\bm{b}_{\tau}^{(\pm)2})\,,$
(9)

where
$\bm{b}_{\tau}^{(\pm)}=\bm{b}+\sum_{i=1,2}\left(\frac{-\tau}{E_{i}+m_{i}}\pm\frac{E}{m_{1}m_{2}\sinh(\theta)}\right)\frac{\bm{p}\times\bm{s}_{i}}{m_{i}}$,
and the $\pm$ sum remembers the helicity sum for the exchanged graviton. Let
us stress this formula for the eikonal is valid for generic spin orientations
and recovers previous lower spin results Bern  _et al._ (2020); Kosmopoulos
and Luna (2021); Chung  _et al._ (2020a).

2PM amplitude from the Triangle LS and the non-aligned HCL. Having the 1PM
eikonal at our disposal, the next ingredient to compute 2PM observables via
(14), is the 2PM eikonal phase for compact objects with generic spin
orientations. The relevant one-loop amplitude entering in (8) is controlled by
the LS (see fig. 1):

$\frac{1}{8m_{2}\sqrt{-t}}\int_{\Gamma_{\rm{LS}}}\frac{dy}{2\pi
y}A_{4}^{(a_{1})}(A)A_{3}^{(a_{2})}(B)A_{3}^{(a_{2})}(C),$ (10)

where the contour $\Gamma_{\text{LS}}$ computes the residue at $y=0$ minus
that at $y=\infty$ Cachazo and Guevara (2020). The momentum labels of the
building tree-level blocks are
$\\{A\\}=\\{p_{1},{-}\tilde{p}_{1},k_{2}^{+},k_{3}^{-}\\}$,
$\\{B\\}=\\{p_{2},{-}l,{-}k_{2}^{-}\\}$ and
$\\{C\\}=\\{-\tilde{p}_{2},l,{-}k_{3}^{+}\\}$; namely, only the opposite
helicity configuration of the Compton amplitude will be relevant for super-
extremal Kerr observables 666It can be easily shown that for the same helicity
sector for the Compton, the integrand in (10) factorizes in the form
$A_{4}^{(0)}A_{3}^{(0)}A_{3}^{(0)}e^{-(k_{2}+k_{3})\cdot(a_{1}+a_{2})}\to
A_{4}^{(0)}A_{3}^{(0)}A_{3}^{(0)}e^{-q{\cdot}(a_{1}+a_{2})}$, where we have
used (11) to rewrite the exponents. That is, the spin-dependent part is
independent of the loop integration variable $y$, and the LS (10) reduces to
that for the scalar case which evaluates to zero as shown in the seminal work
of Guevara Guevara (2019). Amplitudes for neutron stars on the other hand
receive contributions from both helicity configurations of the Compton
amplitude Chen  _et al._ (2022); Aoude  _et al._ (2022b). .

We use the holomorphic classical limit (HCL) parametrization for the non-
aligned spin scenario discussed in the supplementary material for this letter,
as an alternative construction of the 2PM amplitude to the usual – but very
related – unitarity methods Forde (2007); Badger (2009); Bern  _et al._
(2020); Chen  _et al._ (2022); Aoude  _et al._ (2022a). An advantage of our
method however is that the classical limit of the triangle graph can be taken
from the beginning of the computation. For the scalar contributions, the HCL
parametrization is the usual one Guevara (2019); Guevara  _et al._ (2019);
Bautista  _et al._ (2022). In the gauge (3), the scalar Compton amplitude (2)
has the HCL form $A_{4}^{0}(A)=\pi
Gm_{1}^{2}\frac{(2y-v(1+y^{2}))^{4}\sigma^{2}}{v^{2}y^{2}(1-y^{2})^{2}}\,,$
whereas for the product of the two 3-point amplitudes we have
$A_{3}^{0}(B)A_{3}^{0}(C)=8\pi Gm_{2}^{4}\,.$ The non-aligned HCL form for the
spin-contributions to the 2PM amplitude becomes however very different from
their aligned spin construction. The massless momenta hitting the spin vectors
$a_{1}^{\mu}$ in (1), and $a_{2}^{\mu}$ in the 3-point amplitudes
$e^{-k_{2}{\cdot}a_{2}}\times e^{k_{3}{\cdot}a_{2}}$, are now given by

$\begin{split}k_{2}^{\mu}&{=}\frac{|q|\sqrt{z^{2}{-}1}(m_{2}p_{1}^{\mu}{-}m_{1}\sigma
p_{2}^{\mu}){+}m_{1}m_{2}\sigma
vq^{\mu}{+}iz\mathcal{E}^{\mu}}{2m_{1}m_{2}\sigma v}\,,\\\
k_{3}&{=}{-}\frac{|q|\sqrt{z^{2}{-}1}(m_{2}p_{1}^{\mu}{-}m_{1}\sigma
p_{2}^{\mu}){-}m_{1}m_{2}\sigma
vq^{\mu}{+}iz\mathcal{E}^{\mu}}{2m_{1}m_{2}\sigma v},\\\
w^{\mu}&{=}{-}\frac{|q|\sqrt{z^{2}{-}1}\big{(}m_{2}p_{1}^{\mu}z{+}m_{1}\sigma
p_{2}^{\mu}(v-z)\big{)}{+}i(z^{2}-1)\mathcal{E}^{\mu}}{2m_{1}m_{2}(1{-}vz)\sigma}\,,\end{split}$
(11)

with $z=\frac{1+y^{2}}{2y}$ and
$\mathcal{E}^{\mu}=\epsilon^{\alpha\beta\gamma\mu}q_{\alpha}p_{1\beta}p_{2\gamma}$.
These expressions contain the leading in $|q|=\sqrt{-q^{2}}$ contribution to
the classical amplitude, therefore discarding unnecessary quantum information
before loop integration. Notice in particular, the combination
$k_{2}^{\mu}-k_{3}^{\mu}$ is independent of the transfer momentum $q^{\mu}$;
this will be relevant when we discuss below some caveats of the aligned spin
constructions of Guevara  _et al._ (2019); Bautista  _et al._ (2022) for the
computation of the aligned spin scattering angle for the higher spin
$(a^{n>4})$ cases.

Figure 1: Triangle leading-singularity configuration Cachazo and Guevara
(2020).

Having at hand the non-aligned HCL parametrization of the building blocks
entering in (10), together with the HCL form of the optical parameter
$\xi\rightarrow-\sigma^{2}v^{2}\frac{(1-y^{2})^{2}}{4y}$, it is now an easy
task to compute the LS (10) using the Compton amplitude (1), since the problem
has been reduced to a simple residue calculation. We organize the result of
the residue evaluation as follows: Given the set of spin operators
$\mathcal{H}=\\{\mathcal{E}_{1},q{\cdot}s_{2},\sqrt{-q^{2}}p_{1}{\cdot}s_{2},\mathcal{E}_{2},q{\cdot}s_{1},\sqrt{-q^{2}}p_{2}{\cdot}s_{1}\\}$,
for the regular operators, and the spin operator basis
$\mathcal{H}\to\tilde{\mathcal{H}}=\mathcal{H}\cup\\{|q||s_{1}|,|q||s_{2}|\\}$
for the exotic terms appearing starting at the fifth order in spin, we write
the 2PM contribution of the triangle cut in the form

$M_{\text{bare}}^{\text{2PM}}=\frac{\pi
G^{2}}{\sqrt{-q^{2}}}\sum_{i,j}\Big{(}A_{i,j}\mathcal{H}^{\otimes^{i}}_{j}+B_{i,j}\tilde{\mathcal{H}}^{\otimes^{i}}_{j}\Big{)}\,.$
(12)

Then, the main outputs from the LS evaluation are the $\\{A_{i,j},B_{i,j}\\}$
coefficients for a given spin structure in
$\\{\mathcal{H}^{\otimes^{i}}_{j},\tilde{\mathcal{H}}^{\otimes^{i}}_{j}\\}$
respectively.

Up to the fourth order in spin, the LS construction easily recovers the triple
cut coefficients reported in Bern  _et al._ (2020); Kosmopoulos and Luna
(2021); Chen  _et al._ (2022) for SE Kerr BHs with generic spin orientations,
upon evaluating to zero the contact deformations present at spin 4 in (1), as
dictated by the Teukolsky solution (table 5 .) In this work, we extend the 2PM
computation up to the sixth order in spin including both, regular and exotic
contributions to (12). We provide LS results for generic coefficients
parametrizing the Compton ansatz (1), which can then be specialized (if
desired) to Teukolsky solutions (table 5) for the matching prescription
described above.

Let us for brevity include in this letter only the explicit results for the
$s_{1}^{5}\times s_{2}^{0}$ sector of the 2PM triangle ( amplitude results for
the additional spin sectors, and up to the sixth order, are included in the
ancillary files for this work anc ). The spin structures for this sector are
summarized in table 1, and the explicit coefficients for generic contact
deformations are given in table 2. Additionally, we include in table 3 a
dictionary that maps between our generic coefficient amplitude and the one
reported in Bern  _et al._ (2022c), hence providing a connection of the
regular operators in (1) to the Lagrangian construction in Bern  _et al._
(2022c).

| $j$ |  | $j$ |  | $j$ |
---|---|---|---|---|---|---
$\frac{\mathcal{H}_{j}^{\otimes 5}}{\mathcal{E}_{1}}$ | $1$ | $\mathcal{E}_{1}^{4}$ | $2$ | $-q^{2}(p_{2}\cdot s_{1})^{2}\mathcal{E}_{1}^{2}$ | $3$ | $q^{4}(p_{2}\cdot s_{1})^{4}$
$4$ | $(q\cdot s_{1})^{2}\mathcal{E}_{1}^{2}$ | $5$ | $-q^{2}(q\cdot s_{1})^{2}(p_{2}\cdot s_{1})^{2}$ | $6$ | $(q\cdot s_{1})^{4}$
$\frac{\mathcal{\tilde{H}}_{j}^{\otimes 5}}{(p_{2}\cdot s_{1})\mathcal{E}_{1}|s_{1}|}$ | $1$ | $-q^{2}\mathcal{E}_{1}^{2}$ | $2$ | $q^{4}(p_{2}\cdot s_{1})^{2}$ | $3$ | $-q^{2}(q\cdot s_{1})^{2}$
Table 1: The independent spin structures for the $s_{1}^{5}\times s_{2}^{0}$
sector of the 2PM amplitude (12).

| $j$ |  |  | $j$ |
---|---|---|---|---|---
$A_{5,j}$ | $1$ | $\frac{i\sigma\left(7-13\sigma^{2}\right)}{48\left(-1+\sigma^{2}\right)^{3}m_{1}^{7}m_{2}^{3}}-\frac{i\sigma\left(-96-60c_{2,1}+480c_{2,2}+135c_{3,0}+7\sigma^{4}\left(-4+120c_{2,2}+15c_{3,0}-15c_{3,1}\right)-90c_{3,1}+\sigma^{2}\left(304+60c_{2,1}-1320c_{2,2}-240c_{3,0}+195c_{3,1}\right)\right)}{1920\left(-1+\sigma^{2}\right)^{3}m_{1}^{8}m_{2}^{2}}$ |  |
$2$ | $\frac{i\sigma\left(-5+11\sigma^{2}\right)}{24\left(-1+\sigma^{2}\right)^{3}m_{1}^{5}m_{2}^{3}}-\frac{i\sigma\left(4\left(-36+75c_{2,1}+30c_{2,2}-\sigma^{2}\left(82+180c_{2,1}+240c_{2,2}-195c_{3,0}\right)+7\sigma^{4}\left(4+15c_{2,1}+30c_{2,2}-15c_{3,0}\right)-90c_{3,0}\right)+15\left(9-16\sigma^{2}+7\sigma^{4}\right)c_{3,1}\right)}{1920\left(-1+\sigma^{2}\right)^{3}m_{1}^{6}m_{2}^{2}}$ |
$3$ | $\frac{i\sigma\left(-1+3\sigma^{2}\right)}{48\left(-1+\sigma^{2}\right)^{3}m_{1}^{3}m_{2}^{3}}-\frac{i\sigma\left(3\left(-12+40c_{2,1}-40c_{2,2}-55c_{3,0}+25c_{3,1}\right)+\sigma^{2}\left(16+20\left(-13+7\sigma^{2}\right)c_{2,1}+120c_{2,2}+5\left(68-35\sigma^{2}\right)c_{3,0}+5\left(-29+14\sigma^{2}\right)c_{3,1}\right)\right)}{640\left(-1+\sigma^{2}\right)^{3}m_{1}^{4}m_{2}^{2}}$ |
$4$ | $-\frac{i\sigma\left(14\sigma^{2}\left(-4+120c_{2,2}+15c_{3,0}-15c_{3,1}\right)+3\left(16+20c_{2,1}-280c_{2,2}-60c_{3,0}+45c_{3,1}\right)\right)}{1920\left(-1+\sigma^{2}\right)m_{1}^{6}}$ | $\frac{i\left(6c_{4,0}+7\left(-1+\sigma^{2}\right)c_{4,1}+2\left(4-7\sigma^{2}\right)c_{4,2}\right)}{64\left(-1+\sigma^{2}\right)m_{1}^{6}m_{2}}$ | $1$ | $B_{5,j}$
$5$ | $\frac{30i\sigma\left(4+12c_{2,1}-15c_{3,0}+6c_{3,1}\right)-7i\sigma^{3}\left(16+60c_{2,1}+120c_{2,2}-60c_{3,0}+15c_{3,1}\right)}{1920\left(-1+\sigma^{2}\right)m_{1}^{4}}$ | $\frac{i\left(2\left(-4+7\sigma^{2}\right)c_{4,0}-7\left(-1+\sigma^{2}\right)c_{4,1}-6c_{4,2}\right)}{64\left(-1+\sigma^{2}\right)m_{1}^{4}m_{2}}$ | $2$
$6$ | $-\frac{i\sigma\left(-3+7\sigma^{2}\right)m_{2}^{2}\left(-4+120c_{2,2}+15c_{3,0}-15c_{3,1}\right)}{1920m_{1}^{4}}$ | $\frac{i\left(-1+7\sigma^{2}\right)m_{2}\left(c_{4,1}-2c_{4,2}\right)}{64m_{1}^{4}}$ | $3$

Table 2: 2PM amplitude coefficients for the $s_{1}^{5}\times s_{2}^{0}$ sector of the two-body problem (12).  $\boxed{\begin{split}c_{2,2}&\to\frac{374-165E_{1}+513E_{3}-33E_{4}-330E_{5}-495c_{3,0}+495c_{3,1}}{3960}\,,\\\ c_{2,1}&\to\frac{153E_{3}-11\left(4+3E_{4}+15E_{5}\right)+2475c_{3,0}-990c_{3,1}}{1980}\,,\\\ E_{7}&\to\frac{17}{30}+\frac{21E_{3}}{22}-\frac{E_{4}}{2}-\frac{E_{5}}{2}\,,\,\,E_{2}\to-E_{1}+\frac{3}{2}\left(E_{3}+E_{4}\right)\,,\,\,H_{3}\to\frac{3}{2}\,,\\\ C_{i}&\to 1\,,i=2,3,4,5\,,\,\,H_{2}\to 1\,,\,\,E_{6}\to\frac{1}{6}\left(-2-3E_{3}\right)\,,\,\,H_{5}\to\frac{15E_{3}}{11}\,.\end{split}}$ Table 3: Coefficient dictionary relating the regular terms of the $s_{1}^{5}\times s_{2}^{0}$ sector of the 2PM amplitude in this letter and the generic 2PM amplitude included in the ancillary files for ref. Bern  _et al._ (2022c). The fixing $C_{i},H_{2}\to 1$ agrees with the minimal coupling matching of the 3-pt amplitude to the linearized Kerr metric, built in the Compton ansatz (1). Spin-shift symmetric solutions evaluated using the values of the last two columns of table 5 agree with those given in Bern  _et al._ (2022c).  | $j$ |  | $j$ |
---|---|---|---|---
$A_{5,j}$ | $1$ | $\frac{i\sigma\left(14m_{1}-2(13m_{1}+8m_{2})\sigma^{2}+7m_{2}\sigma^{4}\right)}{96m_{1}^{8}m_{2}^{3}(\sigma^{2}-1)^{3}}$ | $4$ | $\frac{7i\sigma^{3}}{48m_{1}^{6}(\sigma^{2}-1)}$
$2$ | $\frac{i\sigma\left(2m_{1}(11\sigma^{2}-5)+m_{2}(24-43\sigma^{2}+28\sigma^{4})\right)}{48m_{1}^{6}m_{2}^{3}(\sigma^{2}-1)^{3}}$ | $5$ | $\frac{7i\sigma(4\sigma^{2}-3)}{48m_{1}^{4}(\sigma^{2}-1)}$
$3$ | $\frac{i\sigma\left(m_{1}(6\sigma^{2}-2)+m_{2}(51-104\sigma^{2}+56\sigma^{4})\right)}{96m_{1}^{4}m_{2}^{3}(\sigma^{2}-1)^{3}}$ | $6$ | $\frac{im_{2}^{2}\sigma(7\sigma^{2}-3)}{96m_{1}^{4}}$
$B_{5,j}$ | $1$ | $\frac{i\eta(2+7\sigma^{2})}{24m_{1}^{6}m_{2}(\sigma^{2}-1)}$ | $3$ | $\frac{i\eta m_{2}(-1+7\sigma^{2})}{24m_{1}^{4}(\sigma^{2}-1)}$
$2$ | $\frac{i\eta(-5+14\sigma^{2})}{24m_{1}^{4}m_{2}(\sigma^{2}-1)}$ |  |
Table 4: 2PM LS coefficients for the $s_{1}^{5}\times s_{2}^{0}$ sector of the
2PM amplitude explicitly evaluated on the Teukolsky solutions table 5 for
$\alpha=1$.

Spin-shift symmetry violation for Teukolsky solutions and the high-energy
limit. Let us discuss in some detail the results of these tables. It is
illustrative to consider explicitly the 2PM LS coefficients evaluated on the
Teukolsky solutions given in table 5; we include them in table 4. Inspection
of these results reveals a breaking of the invariance of the amplitude under
the transformation $a_{i}^{\mu}\to a_{i}^{\mu}+\varsigma_{i}q^{\mu}/q^{2}\,,$
with $\varsigma_{i}$ arbitrary constants, observed for the $a_{i}^{n\leq 4}$
cases Bern  _et al._ (2022c); Aoude  _et al._ (2022a, b). This happens not
only due to the presence of the exotic operators but also because of the non-
zero contributions of regular operators of the form $q\cdot s_{i}$ in table 1.
One can also check shift-symmetric solutions for the Compton amplitude (table
5 ) induce a shift symmetric 2PM amplitude, as first observed in Aoude  _et
al._ (2022b).

Let us now comment on the high energy behavior of the 2PM amplitude for
Teukolsky solutions (table 4). In the high energy limit ($\sigma\to\infty$),
the 1PM amplitude scales as $\mathcal{O}(\sigma^{2})$ (here
$\mathcal{E}_{i}\sim\sigma$). Having a well-defined 2PM amplitude in the high
energy limit means it should grow no faster than $\sigma^{2}$ as
$\sigma\to\infty$ Bern  _et al._ (2022c). This is however not the case for
Teukolsky solutions of table 4. For instance, the term
$A_{5,1}\mathcal{E}_{i}^{5}$ in (12) grows as $\sim\sigma^{4}$ as
$\sigma\to\infty$, and analogously for the remaining contributions. This
singular high-energy behavior propagates to the 2PM observables as we will
discuss below in the context of the aligned spin scattering angle. It is very
intriguing however that, as can be checked explicitly, shift-symmetric
solutions as dictated by the last two columns of table 5 do possess a well-
defined high energy limit, even if some operators remain to be generic 777As
noted in Bern  _et al._ (2020), having a well-defined energy limit does not
fully fix a spin-shift symmetric amplitude, since amplitudes with well-defined
high energy limit can still contain $(q\cdot a)^{2\alpha}$ operators. However,
once the amplitude is shift-symmetric, it also possesses a healthy high energy
limit. .

Eikonal phase and the aligned spin limit. The LS computation provides us with
the ingredients needed to obtain the 2PM eikonal phase (8) for spinning
objects satisfying both CovSSC and CanSSC, the latter obtained from the former
by the shift of the impact parameter
$\bm{b}\to\bm{b}+\sum_{i=1,2}\left(\frac{-\tau}{E_{i}+m_{i}}\right)\frac{\bm{p}\times\bm{s}_{i}}{m_{i}}$,
a consequence of dressing the amplitude with the rotation factors (5). Results
for the eikonal phase in the CanSSC will be needed when evaluating canonical
observables via (14). We evaluate explicitly (8) and present the results for
the eikonal phase in the CovSSC and in CoM coordinates in the ancillary files
anc . The eikonal phase is organized schematically as
$\chi_{\tau=0}^{(\text{2PM})}=\sum_{i,j}\frac{\pi
G^{2}}{b^{2j+1}}\Big{(}L_{i,j}\,b\cdot\mathcal{S}^{j}+Ld_{i,j}\,b\cdot\tilde{\mathcal{S}}^{j}\Big{)}$
with $\\{\mathcal{S}^{j},\tilde{\mathcal{S}}\\}$ operators in the CoM version
of $\\{(\\{b\\}\cup\mathcal{H})^{\otimes
j},(\\{b\\}\cup\tilde{\mathcal{H}})^{\otimes j}\\}$ respectively,
$b^{\mu}=(0,\bm{b},0)$, and the coefficients $\\{L_{i,j},Ld_{i,j}\\}$ are
functions of $m_{i},E_{i},\sigma$, and linear combinations of the amplitude
coefficients $\\{A_{i,j},B_{i,j}\\}$ respectively. From this, we then
specialize the eikonal phase to the case in which the rotating objects have
their spins aligned in the direction of the angular momentum of the system,
$\bm{b}\cdot\bm{s}_{i}=\bm{p}\cdot\bm{s}_{i}=0$ (see $e.g.$ Kosmopoulos and
Luna (2021)) and compute the 2PM aligned spin scattering angle via
$\frac{\partial\chi_{\tau=0}}{\partial|b|}$. In this limit, the exotic
operators do not contribute to the scattering angle as observed in Bautista
_et al._ (2022), therefore for Teukolsky solutions, the angle itself is
independent of the analytic continuation procedure used to match (1) to the GW
scattering process. Results up to the sixth order in spin for the contributing
regular operators are included in the ancillary files for generic contact
deformations in (1) anc .

Let us now comment on the comparison of the aligned spin scattering angle
results in this work to the ones presented in Guevara  _et al._ (2019);
Bautista  _et al._ (2022) for generic, spin-shift-symmetric Bern  _et al._
(2022c); Aoude  _et al._ (2022a, b), and Teukolsky Compton coefficients
Bautista  _et al._ (2022). The HCL prescription taken in Guevara  _et al._
(2019); Bautista  _et al._ (2022) to compute the aligned spin scattering angle
by fixing $\beta=1$, to obtain the spin operators given by eq. (5.15) in
Bautista  _et al._ (2022), in turn hiddenly sets $q{\cdot}s_{j}\to
i\frac{\mathcal{E}_{j}}{m_{1}m_{2}\sigma v}$ as well as $|q|\to
0\Rightarrow|q|p_{i}{\cdot}s_{j}\to 0$ in (11), as can be seen from the last
line of (22). The latter replacement does not have any problem since in the
aligned spin limit the equality $\bm{p}{\cdot}\bm{s}_{i}=0$ is satisfied.
Similarly, the former identification does not possess any subtleties for
combinations of spin operators of the form $(k_{2}-k_{3}){\cdot}s_{i}$, since
these combinations themselves are independent of $q^{\mu}$, as can be checked
by direct inspection of eq. (11). However, for spin operators where
$k_{i}{\cdot}s_{i}$ have individual appearances – as is the case for some of
the contact deformations in (1) – the identification $q.s_{j}\to
i\frac{\mathcal{E}_{j}}{m_{1}m_{2}\sigma v}$ discards $\mathcal{O}(q^{2})$
terms (see (22)) that are important for the aligned spin angle. For quadratic
terms, for instance, this map is equivalent to setting
$(q{\cdot}s_{i})^{2}\to(q{\cdot}s_{i})^{2}-q^{2}s_{i}^{2}\,,$ which in two-
dimensional impact parameter space implies
$3(\bm{b}{\cdot}\bm{s}_{i})^{2}-\bm{b}^{2}\bm{s}_{\perp i}^{2}\to
3(\bm{b}{\cdot}\bm{s}_{i})^{2}-2\bm{b}^{2}\bm{s}_{\perp i}^{2}\,,$ – with
$\bm{s}_{\perp i}$ the components of the spin along $\bm{b}$ – therefore
removing some terms that survive in the aligned spin limit
($\bm{b}{\cdot}\bm{s}_{i}=0$). An analogous analysis follows for higher spin
contributions. Interestingly, this identification removes the terms in the
amplitude that did not have a well-defined high energy limit; this is the
reason reference Bautista  _et al._ (2022) reported scattering angles for
higher spins that do not diverge as $v\to 1$. The spin-shift symmetric result
of Bern  _et al._ (2020); Aoude  _et al._ (2022b) has also a well-defined
high-energy limit. Finally, let us note the angle for the lower spin cases
Guevara  _et al._ (2019) did not face this ambiguity since only the
combination $(2w-k_{2}-k_{3})\cdot a_{i}$ – independent of $q{\cdot}a_{i}$
terms – appeared in the Compton amplitude.

In the following, we incorporate back the aforementioned terms for the aligned
spin scattering angle. We continue the theme of the spin 5 sector for
briefness. At this order we obtain

$\begin{split}\theta^{(5)}=&\,\theta^{(5)}_{\text{BGKV}}-\frac{9\pi
EG^{2}}{256\,b^{7}\,v(1-v^{2})}\Big{[}(a_{1}^{5}m_{2}{+}a_{2}^{5}m_{1})K_{1}\\\
&\hskip
71.13188pt+(a_{1}^{3}m_{2}{+}a_{2}^{3}m_{1})a_{1}a_{2}\,K_{2}\Big{]}\,,\end{split}$
(13)

where
$K_{1}=80(1-v^{2})c_{2,1}+40(8+13v^{2})c_{2,2}+(4-11v^{2})(4-15c_{3,0})-105v^{2}c_{3,1}$
and $K_{2}=-40\big{[}2(1-v^{2})c_{1,1}+(8+13v^{2})c_{1,2}\big{]}$, and with
$\theta^{(5)}_{\text{BGKV}}$ the scattering angle reported in Bautista  _et
al._ (2022). Teukoslky solutions (table 5) give
$K_{1}|_{\text{Teuk.}}=20(12-5v^{2})$ and $K_{2}|_{\text{Teuk.}}=0$, therefore
resulting into a scattering angle divergent as $v\to 1$ as expected from the
discussion above. Let us stress this result is independent of the analytic
continuation procedure for the matching of eq. 1 to the Teukolsky solutions.
Finally, for a shift-symmetric amplitude (see the last two columns of table
5), $K_{1}=K_{2}=0$ recovers the result of Bern  _et al._ (2022c); Aoude  _et
al._ (2022b), and the angle is well behaved in the high-energy limit. Sixth
order in spin corrections follow analogously, explicit results can be found in
the ancillary files anc .

2PM canonical observables. We are finally in ready to compute the
conservative 2PM canonical observables for $2\to 2$ scattering of SE Kerr BHs.
For this, we will follow the prescription provided by the authors of Bern  _et
al._ (2020); Kosmopoulos and Luna (2021). It was noticed in these references
that the conservative observables
$\Delta\mathcal{O}\in\\{\Delta\bm{p}_{\perp},\Delta\bm{s}_{a}\\}$ at 2PM can
be obtained from the eikonal phase in the CanSSC
$\chi\equiv\chi_{\tau=1}=\chi^{\text{1PM}}_{\tau=1}+\chi^{\text{2PM}}_{\tau=1}+\cdots$,
via

$\begin{split}\Delta\mathcal{O}=&-\\{\mathcal{O},\chi\\}-\frac{1}{2}\\{\chi,\\{\mathcal{O},\chi\\}\\}-\mathcal{D}_{SL}(\chi,\\{\mathcal{O},\chi\\})\\\
&+\frac{1}{2}\\{\mathcal{O},\mathcal{D}_{SL}(\chi,\chi)\\}\,,\end{split}$ (14)

with the Poisson bracket given by

$\\{f,g\\}{=}\frac{\partial f}{\partial p_{\perp}^{j}}\frac{\partial
g}{\partial b^{j}}{-}\frac{\partial g}{\partial p_{\perp}^{j}}\frac{\partial
f}{\partial b^{j}}{+}\sum_{a=1,2}\epsilon^{ijk}\frac{\partial f}{\partial
s^{i}_{a}}\frac{\partial g}{\partial s^{j}_{a}}s_{a,k}\,,$ (15)

and the spin derivative operator

$\mathcal{D}_{SL}(f,g){=}\frac{1}{\bm{p}^{2}}\sum_{a,1,2}\left(\frac{\partial
f}{\partial s_{a}^{j}}\frac{\partial g}{\partial
b^{j}}\bm{s}_{a}{\cdot}\bm{p}{-}p^{j}\frac{\partial f}{\partial
s_{a}^{j}}\bm{s}_{a}{\cdot}\nabla_{\bm{b}}g\right)$ (16)

Using this prescription, and the results for the 1 and 2PM eikonal phase
derived above, we have computed the 2PM observables up to the six order in
spin for generic contact deformations in eq. 1, which can then specialize to
Teukolsky solutions. The expressions are however too long to be included in
this letter and we, therefore, provide them in the ancillary material for this
work anc . We include results for the transverse impulse and individual spin
kicks in the CanSSC up to the sixth order in spin for both regular  and exotic
contributions to the 2PM amplitude in the CoM frame. Up to the fourth order in
spin our results completely agree with those reported in Bern  _et al._
(2020); Kosmopoulos and Luna (2021); Chen  _et al._ (2022) upon setting to
zero the contact deformations at this order, and after the authors of Chen
_et al._ (2022) fixed some of the reported observables that had an initial
disagreement with the ones presented in this work.

Conclusions. In this letter, we have computed the canonical observables for
the conservative SE Kerr two-body problem at second order in the PM expansion
and up to sixth order in spin, for generic spin orientation. The results in
this work are presented for generic Compton contact deformations, which can be
specialized to Teukolsky solutions. In the latter case, the 2PM amplitude
breaks the conjecture spin-shift symmetry for Kerr BHs Bern  _et al._ (2022c);
Aoude  _et al._ (2022a, b), however producing observables with a non-smooth
high energy behavior. This leaves as an open problem understanding if the
unhealthy high-energy behavior is a consequence of the analytic continuation
to the SE Kerr region and observables for actual Kerr solutions
($a^{\star}\leq 1$) feature a well-defined high energy limit 888Non-smooth
high energy limit in the two-body context has been recently reported in the
4PM results for scalar BHs Dlapa  _et al._ (2023), which are expected to be
improved by the non-perturbative solutions Gruzinov and Veneziano (2016); Di
Vecchia  _et al._ (2022). , perhaps guiding possible realizations of the spin-
shift symmetry in this scenario. For the generic coefficient case, we have
presented a dictionary that maps the Compton operators in this letter to the
Lagrangian and Hamiltonian constructions in Bern  _et al._ (2022c). Finally,
the identification of regular and exotic contributions with true conservative
and absorptive contributions in the Kerr binary problem is a subject that
requires further scrutiny and we leave for future work.

Acknowledgments. We would like to thank Rafael Aoude, Stefano De Angelis,
Leonardo de la Cruz, Alfredo Guevara, Kays Haddad, Carlo Heissenberg, Andreas
Helset, Chris Kavanagh, Dimitrios Kosmopoulos, David Kosower, Andres Luna and
M. V. S. Saketh for useful discussion. We are also grateful to Jung-Wook Kim
and Ming-Zhi Chung for the continuum discussion and for sharing an updated
version of their 2PM observables Chen  _et al._ (2022) up to spin 4. We would
like to especially thank Justin Vines for sharing his unpublished notes on the
generalization of the HCL for non-aligned spins, and Leonardo de la Cruz for
his comments on the manuscript. This work has been supported by the European
Research Council under Advanced Investigator Grant ERC–AdG–885414.

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## Supplementary Material

##  Appendix A Triangle Leading singularity and the holomorphic classical
limit for generic spin orientation

In this appendix, we present an extension of the Holomorphic Classical Limit
(HCL) Guevara (2019) and the triangle Leading Singularity (LS) Cachazo and
Guevara (2020) constructions for spinning particles for generic spin
directions. This is based on an unpublished note by Justin Vines to whom we
are grateful.

We start by constructing a suitable local 4-dimensional reference frame all’a
Penrose, Rindler, and Chandrasekhar Penrose and Rindler (1984) as follows:
Given a two-dimensional basis of massless spinors
$\mathcal{L}=\\{|\chi\rangle,|\psi\rangle\\}$, together with the co-basis
$\tilde{\mathcal{L}}=\\{[\chi|,[\psi|\\}$, and whose elements satisfy the
normalization conditions
$\langle\chi\psi\rangle=[\chi\psi]=\frac{1}{\sqrt{2}},$ we can adapt the
spinor bases to the vector tetrad
$\\{e^{\mu}_{(a)}\\}=\\{\mathfrak{t}^{\mu},\mathfrak{x}^{\mu},\mathfrak{y}^{\mu},\mathfrak{z}^{\mu}\\}$,
with $\mathfrak{t}^{2}=-1$ and
$\mathfrak{x}^{2}=\mathfrak{y}^{2}=\mathfrak{z}^{2}=-1$, via the usual vector-
spinors map
$V^{\alpha\dot{\alpha}}=(\sigma^{\mu})^{\alpha\dot{\alpha}}v_{\mu}$, as
follows:

$\begin{split}\mathfrak{T}^{\alpha\dot{\alpha}}&=(\sigma^{\mu})^{\alpha\dot{\alpha}}\mathfrak{t}_{\mu}=\sqrt{2}(|\chi\rangle[\psi|+|\psi\rangle[\chi|)\,,\\\
\mathfrak{Z}^{\alpha\dot{\alpha}}&=(\sigma^{\mu})^{\alpha\dot{\alpha}}\mathfrak{z}_{\mu}=\sqrt{2}(|\chi\rangle[\psi|-|\psi\rangle[\chi|)\,,\\\
\mathfrak{X}^{\alpha\dot{\alpha}}&=(\sigma^{\mu})^{\alpha\dot{\alpha}}\mathfrak{x}_{\mu}=\sqrt{2}(|\chi\rangle[\chi|+|\psi\rangle[\psi|)\,,\\\
\mathfrak{Y}^{\alpha\dot{\alpha}}&=(\sigma^{\mu})^{\alpha\dot{\alpha}}\mathfrak{y}_{\mu}=i\sqrt{2}(|\chi\rangle[\chi|-|\psi\rangle[\psi|)\,.\end{split}$
(17)

Here $\sigma^{\mu}$ are the Pauli matrices and shall not be confused with the
Lorentz boost factor $\sigma$ which is a scalar.

This tetrad spans the 4-dimensional flat spacetime and therefore any vector
(including the spin) can be constructed from a linear combination of vectors
in $\\{e^{\mu}_{(a)}\\}$. Notice in general the spinors in $\mathcal{L}$ are
not related to those in $\tilde{\mathcal{L}}$ by complex conjugation and
therefore the tetrad decomposition admits vectors with complex values. Thus, a
generic vector $q^{\mu}$ will be given by the decomposition

$Q^{\alpha\dot{\alpha}}=(\sigma^{\mu})^{\alpha\dot{\alpha}}q_{\mu}=a_{q}|\chi\rangle[\chi|+b_{q}|\chi\rangle[\psi|+c_{q}|\psi\rangle[\chi|+d_{q}|\psi\rangle[\psi|\,,$
(18)

where $a_{q}=\frac{1}{2}\langle\chi|Q|\chi]$, and analogously for the other
components. Lorentz invariant products are obtained in the usual form

$\displaystyle q^{2}$
$\displaystyle=\frac{1}{2}\text{tr}(Q\cdot\bar{Q})=\frac{1}{2}(c_{q}b_{q}-a_{q}d_{q})\,,\,\,\,\,p{\cdot}q=\frac{1}{2}\text{tr}(Q\cdot\bar{P})=\frac{1}{4}(b_{q}c_{p}+b_{p}c_{q}-a_{q}d_{p}-a_{p}d_{q})\,.$
(19)

2PM Triangle kinematics: The next task is to use this reference frame to
parametrize the momenta of the particles in the triangle cut fig. 1. Unlike
for the aligned spin scenario, here we will take the momentum transfer
$q=k_{2}+k_{3}$ so that $q^{2}=2k_{2}{\cdot}k_{3}\neq 0$. We orient the tetrad
$\\{e_{(a)}^{\mu}\\}$ in such a way that the BH 2 is at rest, the BH 1 moves
in the $\mathfrak{z}$-direction with relative velocity $v$, and the component
of $q^{\mu}$ orthogonal to $p_{1}^{\mu}$ and $p_{2}^{\mu}$ lies in the
$\mathfrak{x}$-direction. It follows then

$p_{1}^{\mu}=m_{1}\sigma(\mathfrak{t}^{\mu}+v\mathfrak{z}^{\mu})\,,\,\,\,p_{2}^{\mu}=m_{2}\mathfrak{t}^{\mu}\,,\,\,\,q{\cdot}\mathfrak{y}=0\,.$
(20)

The momentum transfer $q^{\mu}$ can be parametrized using the decomposition
(18). With help of the on-shell conditions
$p_{1}{\cdot}q=q^{2}/2,\,\,p_{2}{\cdot}q=-q^{2}/2$ and
$q{\cdot}\mathfrak{y}=0$, and solving for the coefficients
$a_{q},b_{q},c_{q},d_{2}$ in terms of the kinematic variables
$m_{1},m_{2},\sigma,|q|$, where $|q|=\sqrt{-q^{2}}$, one explicitly gets

$b_{q}=\frac{m_{2}+m_{1}(1+v)\sigma}{\sqrt{2}m_{1}m_{2}\sigma
v}|q|^{2}\,,\,\,\,c_{q}=-\frac{m_{2}+m_{1}(1-v)\sigma}{\sqrt{2}m_{1}m_{2}\sigma
v}|q|^{2}\,,\,\,d_{q}=a_{q}=\pm\sqrt{b_{q}^{2}c_{q}^{2}+2|q|^{2}}\,.$ (21)

HCL parametrization: The final task is to find a suitable parametrization for
the internal massless momenta in such a way the classical limit of the
triangle diagram fig. 1 is easily obtained. For that, let us introduce the new
spinor bases $\\{|\lambda\rangle,|\eta\rangle\\}$ and
$\\{[\lambda|,[\eta|\\}$, following Guevara Guevara (2019). In these new
bases, the external momenta are parametrized as

$\begin{split}P_{1}&=|\hat{\eta}]\bra{\hat{\lambda}}+|\hat{\lambda}]\bra{\hat{\eta}}\,,\,\,\,\,\tilde{P}_{1}=\beta^{\prime}|\hat{\eta}]\bra{\hat{\lambda}}+\frac{1}{\beta^{\prime}}|\hat{\lambda}]\bra{\hat{\eta}}+|\hat{\lambda}]\bra{\hat{\lambda}},\\\
P_{2}&=|\eta]\bra{\lambda}+|\lambda]\bra{\eta}\,,\,\,\,\,\tilde{P}_{2}=\beta|\eta]\bra{\lambda}+\frac{1}{\beta}|\lambda]\bra{\eta}+|\lambda]\bra{\lambda},\\\
Q&=P_{1}-\tilde{P}_{1}=-P_{2}+\tilde{P}_{2}\,,\,\,\,\,\Rightarrow\,\,|q|^{2}=m_{2}^{2}\frac{(\beta-1)^{2}}{\beta}\,,\end{split}$
(22)

where $Q$ is the complex momentum transfer matrix. Thus, as $|q|\to 0$ (as
$\beta\to 1$) one recovers the usual HCL result
$Q\to|\lambda]\langle\lambda|$, but now we keep subleading terms in
$(\beta-1)$. The on-shell conditions $P_{1}^{2}=\tilde{P}_{1}^{2}=m_{1}^{2}$
and $P_{2}^{2}=\tilde{P}_{2}^{2}=m_{2}^{2}$, impose the normalization for the
spinors $\braket{\hat{\lambda}\hat{\eta}}=[\hat{\lambda}\hat{\eta}]=m_{1}$ and
$\braket{\lambda\eta}=[\lambda\eta]=m_{2}$. For the internal gravitons, the
spinor helicity variables are analogously parametrized via

$\begin{split}\ket{k_{2}}&=\frac{1}{\beta+1}\left(\left(\beta^{2}-1\right)\ket{\eta}-\frac{1+\beta
y}{y}\ket{\lambda}\right),\,\,\,|k_{2}]=\frac{1}{\beta+1}\left(\left(\beta^{2}-1\right)y|\eta]+(1+\beta
y)|\lambda]\right),\\\
\ket{k_{3}}&=\frac{1}{\beta+1}\left(\frac{\beta^{2}-1}{\beta}\ket{\eta}+\frac{1-y}{y}\ket{\lambda}\right),\,\,\,|k_{3}]=\frac{1}{\beta+1}\left(-\beta\left(\beta^{2}-1\right)y|\eta]+(1-\beta^{2}y)|\lambda]\right).\end{split}$
(23)

Here $y$ is the loop integration parameter entering in (10). Imposing
consistency between the HCL (22) and the tetrad (18) (20) parametrizations,
allows us to solve for $\\{|\lambda\rangle,|\eta\rangle,[\lambda|,[\eta|\\}$
in terms of $\\{|\chi\rangle,|\psi\rangle,[\chi|,[\psi|\\}$, up to an
irrelevant little group scale $\bar{\omega}$ that cancels from the final
result. Using,
$\beta=1+\frac{|q|}{2m_{2}^{2}}(|q|\mp\sqrt{|q|^{2}+4m_{2}^{2}})$, as well as
the solutions (21) for the on-shell conditions for the external momenta, to
leading order in $|q|$, we find 999Here one is free to choose either of the
solutions for the quadratic equations. This in turn provides a remarkable
simplification in the computation as one does not have to average between the
positive and negative solutions, as it is customarily done when computing the
amplitude coefficients via unitarity methods Forde (2007); Badger (2009); Bern
_et al._ (2020); Chen  _et al._ (2022); Aoude  _et al._ (2022a)

$\begin{split}|\lambda\rangle&=\bar{\omega}\sqrt{|q|}(|\chi\rangle+|\psi\rangle)+\mathcal{O}(q^{3/2})\,,\,\,\,\,[\lambda|=\frac{\sqrt{2}|q|}{\bar{\omega}}([\chi|-[\psi|)+\mathcal{O}(q^{3/2})\,,\\\
|\eta\rangle&=-\frac{m_{2}\bar{\omega}}{\sqrt{|q|}}|\chi\rangle+\mathcal{O}(q^{1/2})\,,\,\,\,\,[\eta|=\frac{m_{2}}{\sqrt{|q|\bar{\omega}}}[\chi|+\mathcal{O}(q^{1/2})\,.\end{split}$
(24)

These solutions can be replaced into (23) to obtain analog expressions for the
internal graviton variables. Explicitly, in the tetrad basis, to leading order
in $|q|$, the internal graviton momenta take the form

$\begin{split}k_{2}^{\mu}&=\frac{|q|}{2\sigma
v}\big{[}\sqrt{z^{2}-1}(u^{\mu}-\sigma\mathfrak{t}^{\mu})+\sigma
v(\mathfrak{x}^{\mu}+iz\mathfrak{y}^{\mu})\big{]}\,,\\\
k_{3}^{\mu}&=-\frac{|q|}{2\sigma
v}\big{[}\sqrt{z^{2}-1}(u^{\mu}-\sigma\mathfrak{t}^{\mu})-\sigma
v(\mathfrak{x}^{\mu}-iz\mathfrak{y}^{\mu})\big{]}\,,\\\
w^{\mu}&=-\frac{|q|}{2\sigma(1-vz)}\Big{[}(u^{\mu}-\sigma\mathfrak{t}^{\mu})z+\sigma
v(\sqrt{z^{2}-1}\mathfrak{t}^{\mu}+i(z^{2}-1)\mathfrak{y}^{\mu})\Big{]}\,,\end{split}$
(25)

where the incoming 4-velocity $u^{\mu}=p_{1}^{\mu}/m_{1}$. Here we have used
(3) as a definition for $w^{\mu}$, and the gauge (3), together with the
Lorentz products $p_{1}{\cdot}k_{2}=-\frac{1}{2}m_{1}|q|\sigma
v\sqrt{z^{2}-1}$ and
$p_{1}{\cdot}\epsilon_{2}=-\frac{1}{2[23]}m_{1}|q|\sigma(1+vz)$ entering in
(3). We can further identify the momentum transfer
$q^{\mu}=|q|\mathfrak{x}^{\mu}$, whereas
$\mathfrak{y}^{\alpha}=\varepsilon^{\delta\alpha\beta\gamma}\mathfrak{t}_{\delta}\mathfrak{x}_{\beta}\mathfrak{z}_{\gamma}$,
since
$\varepsilon_{\alpha\beta\gamma\delta}\mathfrak{t}^{\alpha}\mathfrak{x}^{\beta}\mathfrak{y}^{\gamma}\mathfrak{z}^{\delta}=1$,
with $\varepsilon^{\alpha\beta\gamma\delta}$ the four-dimensional Levi-Civita
symbol. In terms of the external momenta (20), and with
$\mathfrak{y}^{\alpha}=\frac{1}{m_{1}m_{2}\sigma v|q|}\mathcal{E}^{\alpha}$,
where $\mathcal{E}^{\alpha}$ was defined bellow (11), the massless momenta
(25) recover the main text expressions (11).

$k_{3}$

Spin | Spurious-pole | Free Coeffs. | Teukolsky Solutions |  | Spin-Shift-Sym. | Free Coeffs.
---|---|---|---|---|---|---
$a^{4}$ |  | $c_{1}^{(i)},\,\,i=0,1,2$ | $c_{1}^{(i)}=0,\,\,i=0,1,2$ |  | $\begin{aligned} c_{1}^{(i)}&=0,\,i=1,2\end{aligned}$ | $c_{1}^{(0)}$
$a^{5}$ | $c_{3}^{(2)}=4/15-c_{3}^{(0)}+c_{3}^{(1)}$ | $\begin{aligned} c_{2}^{(i)}&,\,\,i=0,1,2\\\ c_{3}^{(i)}&,\,\,i=0,1\\\ c_{4}^{(i)}&,\,\,i=0,1,2\end{aligned}$ | $\begin{aligned} c_{2}^{(i)}&=0,\,\,i=0,1,2\\\ c_{3}^{(0)}&=\alpha\frac{64}{15}\,,\,c_{3}^{(1)}=\alpha\frac{16}{3}\,,\\\ c_{3}^{(2)}&=\frac{4}{15}(1+4\alpha)\,,\\\ c_{4}^{(0)}&=\eta\alpha\frac{64}{15}\,,\\\ c_{4}^{(1)}&=\eta\alpha\frac{16}{5}\,,\,c_{4}^{(2)}=\eta\frac{4}{15}\end{aligned}$ |  | $\begin{aligned} c_{j}^{(i)}&=0,\,\,i=1,2,\,\,j=2,3\\\ c_{3}^{(0)}&=\frac{4}{15},\,\,c_{4}^{(i)}=0\,,i=0,1,2\end{aligned}$ | $c_{2}^{(0)}$
$a^{6}$ | $\begin{aligned} c_{10}^{(2)}&=c_{10}^{(1)}-c_{10}^{(0)}\\\ d_{1}^{(0)}=&-\frac{8}{45}\\\ &+\sum_{j=5}^{7}\sum_{i=0}^{2}(-1)^{i}c_{j}^{(i)}\\\ f_{1}^{(0)}=&\frac{4}{45}+c_{6}^{(0)}-c_{6}^{(1)}\\\ &+\sum_{i=0}^{2}(-1)^{i}c_{8}^{(i)}\end{aligned}$ | $\begin{aligned} c_{5}^{(i)}&,\,\,i=0,1,2\\\ c_{6}^{(i)}&,\,\,i=0,1,3\\\ c_{7}^{(i)}&,\,\,i=0,1,2\\\ c_{8}^{(i)}&,\,\,i=0,1,3\\\ c_{9}^{(i)}&,\,\,i=0,1,2\\\ c_{10}^{i}&,\,\,i=0,1\end{aligned}$ | $\begin{aligned} c_{j}^{(i)}&=0,\,\,i=0,1,2,\,\,j=5,7\\\ c_{6}^{(0)}&=\alpha\frac{128}{45}\,,\,c_{6}^{(1)}=\alpha\frac{32}{9}\,,\\\ c_{6}^{(2)}&=\frac{8}{45}(1{+}4\alpha)\,,c_{8}^{(0)}={-}\alpha\frac{512}{45},\,\\\ c_{8}^{(1)}&={-}\alpha\frac{160}{9},\,c_{8}^{(2)}={-}\frac{16}{45}(1{+}19\alpha),\\\ c_{9}^{(0)}&=-\eta\alpha\frac{128}{45}\,,\,c_{9}^{(1)}=-\eta\alpha\frac{32}{15}\,,\\\ c_{9}^{(2)}&=-\eta\frac{8}{45},\\\ c_{10}^{(0)}&=-\eta\alpha\frac{256}{45},\\\ c_{10}^{(1)}&=-\eta\alpha\frac{352}{45},c_{10}^{(2)}=-\eta\alpha\frac{32}{15}\\\ d_{1}^{(0)}&=0\,,\,f_{1}^{(0)}=-\frac{4}{45}(1+4\alpha)\end{aligned}$ |  | $\begin{aligned} c_{j}^{(i)}&=0,\,\,i=0,1,2,\,\,j=5,9,10\\\ c_{j}^{(i)}&=0,\,\,i=1,2,\,\,j=6,7,8\\\ c_{8}^{(0)}&=-\frac{4}{45}-c_{6}^{(0)}\\\ f_{1}^{(0)}&=0\\\ d_{1}^{(0)}&=-\frac{8}{45}+c_{6}^{(0)}+4c_{7}^{(0)}\end{aligned}$ | $c_{6}^{(0)},c_{7}^{(0)}$

Table 5: Second column:. Spurious pole cancellation constraints on the Compton
ansatz. Third column: Free contact coefficients after imposing the constraint
of the second column. Fourth column: Match to low-energy Teukolsky solutions.
Here $\alpha=1$, is a coefficient tracking non-rational (digamma functions)
contributions, and $\eta=0,\pm 1$, is a parameter that keeps track of the
regular ($\eta=0$), and exotic ($\eta=\pm 1$) pieces of the scattering
amplitude. Fifth column: Spin-shift symmetry constraints on the Compton
ansatz. Sixth column: Free parameters for a spin-shift symmetric Compton
ansatz. Table recreated from Bautista  _et al._ (2022).