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SNUTP24-002

# Dynamics of Spinning Binary at 2PM

Gang Chen$\mbox{}^{a}$ Tianheng Wang$\mbox{}^{b,c}$

###### Abstract

We consider the covariant proposal for the gravitational Compton amplitude for
a Kerr black hole. Employing the covariant three- and four-point Compton
amplitudes, we assemble the classical one-loop integrand on the maximal cut at
all orders in spin, utilizing the method of unitarity. Expanding in powers of
spin, we evaluate the one-loop amplitude up to $\mathcal{O}(G^{2}a^{8})$.
Supplemented with extra contact contributions derived from the far-zone data
of the Teukolsky solutions, the one-loop amplitude is in agreement with
results available in the literature. We showcase the classical eikonal in the
aligned-spin case at $\mathcal{O}(G^{2}a^{7})$.

###### Contents

  1. 1 Introduction
  2. 2 Classical Compton amplitudes
    1. 2.1 Covariant Compton amplitude to arbitrary spin order
    2. 2.2 Spin expansion and analysis of contact contributions
  3. 3 Eikonal with spin at 2PM
    1. 3.1 Contribution from the covariant Compton amplitude
    2. 3.2 Contribution from extra contact terms
    3. 3.3 2PM eikonal in the spin expansion
  4. 4 Conclusions and outlook
  5. A Extra contact terms
  6. B UV behaviours
  7. C Decomposition of tensor integrals

##  1 Introduction

The successful detection of gravitational waves [1, 2, 3, 4, 5] has inspired
an explosion of developments in the studies of black hole mergers. Besides
numerical relativity [6, 7, 8] and the formal approaches focusing on the post-
Newtonian expansion [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28], various amplitude-based approaches [29, 30, 31, 32,
33, 34, 35, 36, 37] have proven highly effective in computing higher-order
corrections in the Post-Minkowskian (PM) expansion and revealing the
underlying structures of gravitational interactions.

These amplitude-based approaches view the dynamics of heavy bodies interacting
via weak gravity as that of scattering processes in the classical regime where
the gravitons are soft. A multitude of methods [38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] have been devised to single out
the classical order from the full (quantum) amplitudes. Techniques for
computing loop amplitudes efficiently facilitate computations at high PM
orders. At the point of writing, black hole scattering observables have been
computed up to 4PM [56, 57, 58, 59, 60, 36, 61] for both non-spinning and
spinning binaries and partial results are obtained at 5PM in the non-spinning
case [62].

Incorporating spin degrees of freedom is non-trivial, both conceptually and
computationally. Challenges appear even at tree level. Although the three-
point amplitudes of two massive particles and one graviton are classified in
[63], complications enter at four points, namely the gravitational Compton
scattering. On the one hand, fundamental questions in quantum field theories
with higher-spin particles need to be addressed [64, 65, 66, 67, 68, 69],
towards a full understanding of these amplitudes. On the other hand, the
concept of double copy [70, 71, 72] and general considerations on symmetry and
locality requirements allow us to bootstrap the Compton amplitudes of
arbitrary spins [73, 74], despite incomplete knowledge of the underlying
theory. These results can then be tested against approaches developed from
other perspectives such as the Teukolsky equation [75, 76, 77, 78, 79, 80, 81,
82, 83].

Based on somewhat different assumptions, the Compton amplitude for a Kerr
black hole has been computed both as an expansion in powers of spin [84, 85,
86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97] and as a resummed function
[98, 99, 73, 74]. They all agree up to the quartic order in the spin
expansion, where the spin-shift symmetry is expected to hold. Going to higher
orders in spin, they deviate from one another, as the nuances in their
respective formalism and assumptions become important. These results provide a
pool of data for future investigations to analyze and clarify, which in turn
will help to reveal the structures of the underlying theory. The Compton
amplitude for a Kerr black hole is also an important ingredient in the
calculation of loop amplitudes, which describe the dynamics of Kerr binaries
at higher PM orders [100, 101, 86, 102, 103, 104, 105, 106, 87, 107, 108, 109,
110].

A complementary line of research [111, 112, 113, 114, 115, 116, 117, 118]
based on the worldline quantum field theory (WQFT) formalism [42] provides a
fast-track to high PM calculations for physical observables and generating
functions, up to quadratic orders in spin. Recent development in the worldline
formalism focuses on the radial action, which produces scattering observables
at 6PM beyond the quadratic order in certain kinematic regions [119]. The
gravitational waveform in the spinning case is discussed in [96, 107, 120] up
to the quartic order in spin. Besides Einstein gravity, related works on the
spinning binary systems in gauge [121] and other gravitational backgrounds
[122, 123] have been carried out. Beyond conservative dynamics, tidal and
radiation effects have been explored from both amplitude approaches and
worldline-based formalisms [124, 125, 126, 127, 128, 129]. Recent developments
on the self-force expansion [130, 131, 132, 133, 134] may also shed light on
the dynamics in curved backgrounds.

In this paper, we consider one particular proposal for the Compton amplitude
[73, 74], which is obtained using the double-copy and bootstrap. In section 2,
we review its structures and make comparisons with other proposals at the
level of their spin expansions. In section 3, we compute the one-loop
amplitude using this Compton amplitude from the leading singularity and the
resulting 2PM eikonal phase.

##  2 Classical Compton amplitudes

We begin this section with a review of the classical three-point [135, 35,
136, 73] and four-point Compton amplitudes [73, 74] of two massive particles
with spins minimally coupled to gravity at tree level. The Compton amplitude
follows from the double-copy and factorization requirements and includes
contact terms. The contact terms match with physical data at low orders in
spin and display certain empirical properties, which allows for an
extrapolation. This Compton amplitude is conjectured to hold at all orders in
spin. In the second part of this section, we discuss the comparison with the
Compton amplitudes computed from other approaches [99, 83]. The differences in
the contact terms can be interpreted as being related to the internal
structures of the Kerr black hole. Moreover, we demonstrate a simple procedure
to obtain expressions in covariant variables for these extra contact terms.

###  2.1 Covariant Compton amplitude to arbitrary spin order

Here we review the structures of the classical amplitudes of a heavy particle
emitting gravitons. Throughout this paper, we restrict our discussions to the
kinematics in the heavy-mass limit [41]. The incoming and outgoing massive
momenta typically are parameterised as $\bar{p}=mv$ and
$\bar{p}^{\prime}=(mv-q)$ where $q^{\mu}$ denotes the total momenta of the
emitted graviton(s). The on-shell conditions $\bar{p}^{2}=\bar{p}^{\prime
2}=(mv-q)^{2}=m^{2}$ yield $v^{2}=1$ and $v{\cdot}q=0$.

The three-point amplitude of two massive particles of arbitrary masses and
spins and a graviton is first given in [135] by considering general symmetry
constraints. Restricting to the minimal coupling and taking the heavy-mass
limit, the amplitude corresponds to the Kerr black hole of an arbitrary
classical spin. A compact, covariant form of this three-point amplitude is
found in [35, 136, 73]

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)
{$\overline{p}$}; \vertex[right=1.5cm of a] (f2)[dot]{}; \vertex[right=1.5cm
of f2] (c){$\overline{p}^{\prime}$}; \vertex[above=1.3cm of f2]
(gm){$\varepsilon_{1},p_{1}$}; \diagram* { (a) -- [thick] (f2) -- [thick] (c),
(gm)--[photon,ultra thick](f2) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\,\,\,\mathcal{M}_{3}(1,\overline{p}^{\prime},\overline{p})=-i\kappa(\bar{p}{\cdot}\varepsilon_{1})({\mathsf{w}}{\cdot}\varepsilon_{1})\,,$
(2.1)

where
${\mathsf{w}}^{\mu}\coloneqq\cosh(x_{1})\bar{p}^{\mu}-iG_{1}(x_{1})(p_{1}{\cdot}S)^{\mu}$,
with $G_{1}(x_{1})=\frac{\sinh(x_{1})}{x_{1}}$, $x_{1}=a{\cdot}p_{1}$ and
$(p_{1}{\cdot}S)^{\mu}=p_{1\nu}S^{\nu\mu}$. Here $S^{\mu\nu}$ denotes the spin
tensor which describes the classical spin of the black hole and we have
introduced the spin-length vector $a^{\mu}$, with
$S^{\mu\nu}=-\epsilon^{\mu\nu\rho\sigma}\bar{p}_{\rho}a_{\sigma}$. As
$G_{1}(x_{1})$ is an entire function, (2.1) is local in the sense that it is
free of unphysical poles involving $x_{1}$ in the denominator once expanded in
powers of the spin vector.

At four points, we have the classical gravitational Compton scattering of two
gravitons and the Kerr black hole, depicted below.

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)
{$\overline{p}$}; \vertex[right=1.9cm of a]
(f2)[GR]{$~{}~{}\mathbf{S}~{}~{}$}; \vertex[right=1.9cm of f2]
(c){$\overline{p}^{\prime}$}; \vertex[above=1.3cm of f2] (gm){};
\vertex[left=0.8cm of gm] (g2){$\varepsilon_{1},p_{1}$}; \vertex[right=0.8cm
of gm] (g20){$\varepsilon_{2},p_{2}$}; \diagram* { (a) -- [thick] (f2) --
[thick] (c), (g2)--[photon,ultra thick](f2),(g20)--[photon,ultra thick](f2) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (2.2)

A covariant form of the Compton amplitude for a Kerr black hole is obtained in
[74, 73]. Schematically, this amplitude receives three types of contributions
as follows,

$\displaystyle\mathcal{M}_{4}(1,2,\overline{p}^{\prime},\overline{p})$
$\displaystyle=-{\mathcal{N}_{a}(1,2,\overline{p}^{\prime},\overline{p})\,\mathcal{N}_{0}(1,2,\overline{p},\overline{p}^{\prime})\over
2(p_{1}{\cdot}p_{2})}+{\mathcal{N}_{\rm
r}(1,2,\overline{p}^{\prime},\overline{p})\over
4(\overline{p}{\cdot}p_{1})(\overline{p}{\cdot}p_{2})}+\mathcal{N}_{\rm
c}(1,2,\overline{p}^{\prime},\overline{p}).$ (2.3)

Here the first term, as its form suggests, is assembled from the double-copy
procedure. The second term is designated to take care of the spin-flip
effects. The last denotes the contact contributions which can not be
determined from factorizations.

The double-copy term, constructed in [73], is given by $\mathcal{N}_{a}$
denoting the kinematic numerator of the Yang-Mills Compton amplitude in the
heavy-mass limit and its spinless counterpart $\mathcal{N}_{0}$. The numerator
$\mathcal{N}_{a}$ takes the following form,

$\displaystyle\mathcal{N}_{a}(1,2,\overline{3},\overline{4})=-\frac{{\mathsf{w}}_{1}{\cdot}F_{1}{\cdot}F_{2}{\cdot}{\mathsf{w}}_{2}}{(p_{1}{\cdot}\bar{p})}+\frac{p_{1}{\cdot}\bar{p}-p_{2}{\cdot}\bar{p}}{2(p_{1}{\cdot}\bar{p})}\times$
(2.4)
$\displaystyle\Big{(}iG_{2}\left(x_{1},x_{2}\right)(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}S{\cdot}p_{2})+iG_{2}(x_{1},x_{2})(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}S{\cdot}p_{1})+iG_{1}\left(x_{12}\right)\text{tr}\left(F_{1}{\cdot}S{\cdot}F_{2}\right)$
$\displaystyle+G_{1}(x_{1})G_{1}(x_{2})\big{(}(a{\cdot}F_{1}{\cdot}\bar{p})(a{\cdot}F_{2}{\cdot}p_{1})\\!-\\!(a{\cdot}F_{1}{\cdot}p_{2})(a{\cdot}F_{2}{\cdot}\bar{p})-\\!\frac{p_{2}{\cdot}\bar{p}\\!-\\!p_{1}{\cdot}\bar{p}}{2}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)\big{)}\Big{)}\,,$

where $x_{12}=x_{1}+x_{2}$ and

$\displaystyle G_{2}(x_{1},x_{2})$ $\displaystyle\equiv{1\over
x_{2}}\Big{(}{\sinh(x_{12})\over x_{12}}-\cosh(x_{2}){\sinh(x_{1})\over
x_{1}}\Big{)}.$ (2.5)

$G_{2}(x_{1},x_{2})$ is also an entire function and hence renders
$\mathcal{N}_{a}$ free of unphysical poles in $x_{1}$, $x_{2}$ or $x_{12}$.
The scalar numerator $\mathcal{N}_{0}$ follows from the kinematic Hopf algebra
[137, 138, 139, 140, 141, 142]

$\displaystyle\mathcal{N}_{0}(1,2,\overline{p},\overline{p}^{\prime})=-\Big{(}\\!\frac{\bar{p}^{\prime}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}^{\prime}}{p_{1}{\cdot}\bar{p}^{\prime}}\Big{)}.$
(2.6)

We have $\mathcal{N}_{0}=\mathcal{N}_{a}|_{a\rightarrow 0}$. These numerators
manifestly satisfy the crossing symmetries described by group algebra in
actions [143, 144] as demanded by the color-kinematic duality. The double-copy
term is consistent with all possible massless factorizations.

The second term in eq. (2.3) complements the double-copy contribution to
produce the correct spin-flip effects, starting from the cubic order in spin.
It is determined by the factorization requirement on the physical massive
poles. The numerator $\mathcal{N}_{r}$ reads

$\displaystyle\mathcal{N}_{\rm r}(1,2,\overline{p}^{\prime},\overline{p})=$
$\displaystyle\\!{\Big{(}(\partial_{x_{1}}\\!-\\!\partial_{x_{2}})G_{1}(x_{1})G_{1}(x_{2})\Big{)}\over
4(\bar{p}{\cdot}p_{1})(\bar{p}{\cdot}p_{2})}\Big{(}\bar{p}{\cdot}p_{2}(\bar{p}^{2}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p})$
$\displaystyle+a^{2}(\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}))\\!-\\!(1\leftrightarrow
2)\\!\Big{)}$
$\displaystyle+\Big{(}{i(\partial_{x_{1}}\\!-\\!\partial_{x_{2}})G_{2}(x_{1},x_{2})\over
4(\bar{p}{\cdot}p_{1})(\bar{p}{\cdot}p_{2})}\Big{)}\Big{(}(\bar{p}{\cdot}p_{2})(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p})((a{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}\widetilde{F}_{1}{\cdot}\bar{p})$
$\displaystyle-(a{\cdot}F_{1}{\cdot}\bar{p})(a{\cdot}\widetilde{F}_{2}{\cdot}\bar{p}))+(1\leftrightarrow
2)\Big{)}.$ (2.7)

The consistency conditions resulted from all the physical factorizations [87]
are satisfied.

It is noted in [74, 73] that the three-point amplitude, the double-copy and
the spin-flip contributions to the Compton amplitude share the property that
their _degrees_ are zero. Here the degree is defined as the maximal power of
$1/\chi$, when $\chi$ parameterizes the scaling of the spin variable, namely
$a^{\mu}\rightarrow\chi a^{\mu}$ and $\chi\rightarrow i\infty$.

The contact contribution is not accessible via factorizations and instead is
obtained by matching with physical data [81, 145] at $\mathcal{O}(a^{4})$ and
$\mathcal{O}(a^{5})$. Such expressions also exhibit the characteristic above
that their degrees are zero at low orders in spin. Imposing this as a general
constraint, the following contact contribution is extrapolated to all orders
in spin,

$\displaystyle\mathcal{N}_{\rm c}(1,2,\overline{p}^{\prime},\overline{p})$
$\displaystyle=\\!\Big{(}\\!{(\partial_{x_{1}}\\!-\\!\partial_{x_{2}})^{2}\over
2!}{G_{1}(x_{1})}{G_{1}(x_{2})}\Big{)}\Big{(}(a{\cdot}F_{1}{\cdot}\bar{p})(a{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)\\!$
$\displaystyle-\\!{a^{2}\over
2}((a{\cdot}F_{1}{\cdot}F_{2}{\cdot}p)(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p})+(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)(\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}))\\!\Big{)}$
(2.8)
$\displaystyle\\!\\!\\!+\Big{(}{i(\partial_{x_{1}}-\partial_{x_{2}})^{2}\over
2!}{G_{2}(x_{1},x_{2})}\Big{)}\Big{(}\\!-{1\over
2}\big{(}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)(a{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}\widetilde{F}_{1}{\cdot}\bar{p})\\!-\\!(1\leftrightarrow
2)\big{)}\Big{)}.$

Putting together all three types of contributions, the covariant expression
for the Compton amplitude (2.3) is manifestly gauge invariant and free of
unphysical poles.

###  2.2 Spin expansion and analysis of contact contributions

Here we consider the expansion of the Compton amplitude reviewed above in
powers of the spin vector and discuss the comparison between this Compton
amplitude and known results in the literature computed from two different
approaches, in particular, the higher-spin theory [99] and the Teukolsky
equation [83]. To avoid confusion, in the remaining part of this section, we
refer to eq. (2.3) as $\mathcal{M}_{\rm SP}$, where the subcript $\rm SP$
indicates that this expression is obtained from bootstrapping with only two
parameters $x_{1}$ and $x_{2}$ and is speculated to describe the single-
particle contributions only in [74]. Likewise, we denote the Compton amplitude
computed from the higher-spin theory in [99] as $\mathcal{M}_{\rm HS}$ and the
one extracted from the solution to the Teukolsky equation in [83] as
$\mathcal{M}_{\rm TS-FZ}$, where the “FZ” in the subscript indicates that we
only consider the contributions obtained from imposing the so-called far-zone
asymptotic behaviours.111We refer to the Teukolsky result with
$\alpha\rightarrow 1$ as the “far-zone data” throught this manuscript, as
indicated in [83]. However, we acknowledge ambiguities in this way of
separating the far-zone and near-zone contributions.

The Compton amplitude derived from the higher-spin theory $\mathcal{M}_{\rm
HS}$ contains two entire functions which depend on three parameters [99],
among which is the spheroidicity parameter defined as
$z=2\sqrt{-a{\cdot}a}(\overline{p}{\cdot}p_{1})/m$ [146] .222We use a
different normalization for $z$ from that in [99]. The $z$-dependence in
$\mathcal{M}_{\rm HS}$ follows from the higher-spin framework and shows up as
contact terms only in the classical limit. Such terms are speculated to be
related to the internal structures of the Kerr black hole [147] or the near-
zone physics, which is beyond the scope of $\mathcal{M}_{\rm SP}$ given in eq.
(2.3). On top of the discussions on the comparison at the level of the entire
functions in [74], we have found further agreement with $\mathcal{M}_{\rm HS}$
when taking $z\rightarrow 0$, namely

$\displaystyle\mathcal{M}_{\rm
SP}(1^{\pm},2^{\mp},\overline{p}^{\prime},\overline{p})=\mathcal{M}_{\rm
HS}(1^{\pm},2^{\mp},\overline{p}^{\prime},\overline{p})|_{z\rightarrow 0}.$
(2.9)

This relation is checked order by order up to $\mathcal{O}(a^{20})$.

As for the comparison with [83], it is observed in [74] that the extra contact
terms needed for matching with the far-zone data at $\mathcal{O}(a^{5})$ can
be rewritten as a polynomial in $z$, although the $z$-dependence is traded for
other parameters in the original expression of $\mathcal{M}_{\rm TS-FZ}$.
These extra contact terms vanish as $z\rightarrow 0$. Here we see that this
observation holds up to $\mathcal{O}(a^{8})$. In other words, at a given order
in spin, we can separate $\mathcal{M}_{\rm TS-FZ}$ into two parts, the
$z$-independent part which agrees with $\mathcal{M}_{\rm SP}$ in 4 dimensions
and the $z$-dependent part which vanishes as $z\rightarrow 0$,

$\displaystyle\mathcal{M}_{\rm
SP}(1^{\pm},2^{\mp},\overline{p}^{\prime},\overline{p})=\mathcal{M}_{\rm TS-
FZ}(1^{\pm},2^{\mp},\overline{p}^{\prime},\overline{p})-\mathcal{M}^{(c)}_{\rm
TS-FZ}(z),\quad\text{with }\mathcal{M}^{(c)}_{\rm TS-FZ}(0)=0.$ (2.10)

$\mathcal{M}_{\rm TS-FZ}$ is expressed in terms of the spinor-helicity
variables. $\mathcal{M}_{\rm SP}$ can be readily rewritten and it is
straightforward to find $\mathcal{M}_{\rm TS-FZ}^{(c)}$ in these variables.
But to see that $\mathcal{M}_{\rm TS-FZ}^{(c)}$ indeed vanishes at a given
order in spin as $z\rightarrow 0$, it is more convenient to use an alternative
expression in terms of covariant variables. To this end, we adopt a simple
procedure as follows.

We begin with a general ansatz in terms of the covariant variables. That is,
the ansatz is a linear combination of monomials comprised of powers of the
parameter $z$ and factors of the forms $\mathsf{v}{\cdot}\mathsf{v}^{\prime}$,
$\mathsf{v}{\cdot}F_{i}{\cdot}F_{j}{\cdot}\mathsf{v}^{\prime}$ and
$\mathsf{v}{\cdot}F_{i}{\cdot}\tilde{F}_{j}{\cdot}\mathsf{v}^{\prime}$ where
the vectors $\mathsf{v}$ and $\mathsf{v}^{\prime}$ can be either $a^{\mu}$ or
a momentum. The ansatz must have the correct counting in $a^{\mu}$ and
$\varepsilon_{i}^{\mu}$ and satisfy the dimension and parity requirements.
Imposing that it should agree with the spinor-helicity expression of
$\mathcal{M}^{(c)}_{\rm TS-FZ}(z)$ for all possible helicity configurations,
when restricted to 4 dimensions, we fix the coefficients in the ansatz. Recall
that the same-helicity gravitational Compton amplitude is known to all orders
in spin, which is captured already in $\mathcal{M}_{\rm SP}$. Hence
$\mathcal{M}_{\rm TS-FZ}^{(c)}(z)$ must vanish in the same-helicity
configurations and receive non-zero contributions only in the opposite-
helicity ones.

We note that the monomials satisfying the power counting and parity
requirements form an over-complete basis in 4 dimensions, due to the
identities involving the Levi-Civita symbols. Such relations only hold in 4
dimensions and may contain coefficients that are rational in inner products of
momenta. Hence it is difficult to remove such redundancies in the ansatz
completely. Consequently, there is a multitude of covariant expressions for
$\mathcal{M}_{\rm TS-FZ}^{(c)}$ at a given order in spin. They all evaluate to
the same in 4 dimensions, but may be different in general dimensions. This
leads to certain subtleties when we construct their contribution to one-loop
amplitude, which will be discussed in the next section.

We give the explicit expressions for $\mathcal{M}^{(c)}_{\rm TS-FZ}(z)$ at
$\mathcal{O}(a^{5})$ and $\mathcal{O}(a^{6})$ and higher-order terms are given
in Appendix A. These expressions are manifestly gauge invariant and vanishing
as $z\rightarrow 0$. We emphasize that they should be viewed as valid in 4
dimensions only, although they are expressed in terms of covariant quantities.

$\displaystyle\mathcal{M}^{(c,5)}_{\rm TS-FZ}(z)$
$\displaystyle=2ip_{1}{\cdot}\bar{p}\,a{\cdot}a\left(a{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}\bar{p}+a{\cdot}\widetilde{F}_{1}{\cdot}F_{2}{\cdot}\bar{p}\right)\Bigg{(}\frac{\left(a{\cdot}a\right)(\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p})}{12m^{2}}-\frac{11}{60}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)\Bigg{)}$
$\displaystyle\mathcal{M}^{(c,6)}_{\rm TS-FZ}(z)$
$\displaystyle=\frac{(a{\cdot}a)^{3}\left(p_{1}{\cdot}\overline{p}\right){}^{2}\left(\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\right){}^{2}}{9m^{4}}+\frac{13(a{\cdot}a)^{3}p_{1}{\cdot}\overline{p}p_{1}{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}}{90m^{2}}$
$\displaystyle-\frac{13(a{\cdot}a)^{3}p_{1}{\cdot}\overline{p}p_{2}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}}{90m^{2}}+\frac{2(a{\cdot}a)^{2}p_{1}{\cdot}\overline{p}a{\cdot}p_{1}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}}{15m^{2}}$
$\displaystyle+\frac{16(a{\cdot}a)^{2}p_{1}{\cdot}\overline{p}a{\cdot}p_{2}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}}{45m^{2}}-\frac{(a{\cdot}a)^{2}\left(p_{1}{\cdot}\overline{p}\right){}^{2}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}}{m^{2}}$
$\displaystyle-\frac{2(a{\cdot}a)^{2}p_{1}{\cdot}\overline{p}a{\cdot}p_{2}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}}{15m^{2}}-\frac{16(a{\cdot}a)^{2}p_{1}{\cdot}\overline{p}a{\cdot}p_{1}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}}{45m^{2}}$
$\displaystyle+\frac{14}{45}a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\left(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a\right){}^{2}+\frac{22}{45}a{\cdot}ap_{1}{\cdot}\overline{p}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}p_{1}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}$
$\displaystyle-\frac{22}{45}a{\cdot}ap_{1}{\cdot}\overline{p}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}p_{2}a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}\,.$
(2.11)

Before closing this section, we make several remarks on the various attempts
at obtaining the gravitational Compton in the literature. The Teukolsky
solution yields the scattering phase of the graviton scattered off the Kerr
black hole, by imposing the asymptotic behaviours of the Teukolsky equations
in the near-zone (near the horizon) and the far-zone (at infinity) [81, 145,
83]. The scattering amplitude is then determined by matching at the level of
the scattering phase. In principle, not only the leading but also higher-order
contributions in the PM expansion can enter the non-perturbative scattering
phase. Hence for a meaningful comparison with the tree Compton amplitudes
obtained from field-theory approaches, it is crucial to isolate the leading PM
contribution in the scattering phase. For a complete comparison between these
two types of approaches, loop-level Compton amplitudes are also necessary.

On the other hand, the scattering phase obtained from the Teukolsky equation
contains both near-zone and far-zone contributions. The far-zone contribution
is rational, whereas the near-zone contains transcendental functions. The
Compton amplitudes computed from field-theory approaches are postulated to
capture the rational part. At the level of the scattering phase, there
naturally is another type of ambiguities arising from rewriting the
transcendental functions using various identities, which changes the rational
terms. Therefore, it is also important to determine the rational terms in a
way that is physically meaningful. In this paper, we consider and match with
$\mathcal{M}_{\rm TS-FZ}$ which is extracted from the far-zone contribution
only. In [99], another procedure is discussed which leads to the matching
between $\mathcal{M}_{\rm HS}$ and $\mathcal{M}_{\rm TS}$ at a different
point. It would be interesting to see if there exists a way of rewriting the
transcendental functions such that the remaining rational terms are
independent of $z$.

##  3 Eikonal with spin at 2PM

In this section, we compute the eikonal phase [102] for the spinning Kerr
binary at 2PM. The classical eikonal [35] is an important generating that
computes physical observables. In the non-spinning case, several variations,
such as the radial action [50, 51, 52], the HEFT phase [41] and the WQFT
eikonal [42, 45, 148], have also been investigated. At 2PM, they are
practically the equivalent, given by Fourier transforming the classical one-
loop $2\rightarrow 2$ scattering amplitude to the impact parameter space. As
seen in related works [102, 149], these generating functions generally extend
to the spinning case with spin-related subtleties.

The eikonal and the resulting observables receive all types of contributions
from the Compton amplitudes, as discussed in section 2. Our main focus is the
classical one-loop amplitude is constructed from the covariant Compton
amplitude, using unitarity-based methods [150, 151, 41]. We compute the
general form of the one-loop integrand and perform the loop integration in the
spin expansion explicitly up to $\mathcal{O}(a^{8})$. For the sake of
comparing with physical data, we include the contributions from the extra
$z$-dependent contact terms at $\mathcal{O}(a^{5})$ and $\mathcal{O}(a^{6})$.
Altogether, we find perfect agreement with the far-zone data.

###  3.1 Contribution from the covariant Compton amplitude

The classical one-loop amplitude is computed from the leading singularity as
depicted below (with the mirror diagram implied), where the triple cut is
given by two massless (graviton) cuts and one massive one,

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)
{$\overline{p}_{1}$}; \vertex[right=1.5cm of a] (f2)
[GR]{$~{}~{}\mathbf{S}~{}~{}$}; \vertex[right=1.5cm of f2]
(c){$\overline{p}^{\prime}_{1}$}; \vertex[above=2.0cm of
a](ac){$\overline{p}_{2}$}; \vertex[right=1.0cm of ac] (ad) [dot]{};
\vertex[right=1.0cm of ad] (f2c) [dot]{}; \vertex[above=2.0cm of
c](cc){$\overline{p}^{\prime}_{2}$}; \vertex[above=1.0cm of a] (cutL);
\vertex[right=3.0cm of cutL] (cutR); \vertex[right=0.5cm of ad] (att);
\vertex[above=0.3cm of att] (cut20){$$}; \vertex[below=0.3cm of att] (cut21);
\diagram* { (a) -- [fermion,thick] (f2)-- [fermion,thick] (c),
(f2)--[photon,ultra thick,momentum=$\ell_{1}$](ad), (f2)-- [photon,ultra
thick,momentum'=$\ell_{2}$] (f2c),(ac) -- [fermion,thick] (ad)--
[fermion,thick] (f2c)-- [fermion,thick] (cc), (cutL)--[dashed, red,thick]
(cutR), (cut20)--[ red,thick] (cut21) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad\quad\begin{split}&\mathcal{M}^{(1)}_{a_{1}a_{2}}={1\over
2}\sum_{h_{i}=\pm}{(32\pi
G)^{2}\over(4\pi)^{D/2}}\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta(m_{2}\,v_{2}{\cdot}\ell_{1})\over\ell_{1}^{2}\ell_{2}^{2}}\\\
&\quad\;\times\Big{(}\mathcal{M}_{3}^{-h_{1}}(-\ell_{1},v_{2})\mathcal{M}_{3}^{-h_{2}}(-\ell_{2},v_{2})\mathcal{M}_{4}^{h_{1}h_{2}}(\ell_{1},\ell_{2},v_{1})\Big{)}.\end{split}$
(3.1)

Here the massive cut labelled by a solid red line gives the $\delta$-function
and we have formally lifted the massless cuts $\delta(\ell_{j}^{2})\rightarrow
i/\ell_{j}^{2}$, while keeping in mind that bubble or ultra-local terms, with
zero or negative propagator powers for the massless propagators, should
vanish. The velocities of the heavy bodies are parameterized such that
$v_{1}^{2}=v_{2}^{2}=1$ and $v_{1}{\cdot}v_{2}=\gamma$ as usual. The tree-
level amplitudes $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$ are given by eq.
(2.1) and eq. (2.3). We use dimensional regularization in the loop
integration, namely $D=4-2\epsilon$. The summation is taken over the
helicities of the gravitons and the completeness relation reads

$\displaystyle\sum_{h=\pm}\varepsilon^{\mu}_{k}\varepsilon^{\nu}_{k}\varepsilon^{*\rho}_{-k}\varepsilon^{*\sigma}_{-k}={1\over
2}\left(\mathcal{P}^{\mu\rho}\mathcal{P}^{\nu\sigma}+\mathcal{P}^{\mu\sigma}\mathcal{P}^{\nu\rho}\right)-{1\over
D-2}\mathcal{P}^{\mu\nu}\mathcal{P}^{\rho\sigma},$ (3.2)

where

$\displaystyle\mathcal{P}^{\mu\nu}=\eta^{\mu\nu}-{k^{\mu}n^{\nu}+n^{\mu}k^{\nu}\over
k{\cdot}n}\,$ (3.3)

with the reference null vector $n^{\mu}$ transverse to the polarization
$k{\cdot}\varepsilon=n{\cdot}\varepsilon=0$. The covariant three- and four-
point tree amplitudes are expected to hold in general dimensions and in
practice we can simply set $\mathcal{P}^{\mu\nu}=\eta^{\mu\nu}$. The reference
vector always drops out.

The dependence on the graviton polarization vectors in the tree amplitudes eq.
(2.1) and eq. (2.3) always enters through the (dual) field strengths
$F_{i}^{\mu\nu}$ and $\tilde{F}_{i}^{\mu\nu}$ and is completely manifest.
Since the function $G_{1}(x)$ depends only on momenta and spins, the above
gluing can be performed without expanding in powers of spin. The resulting
contribution to the one-loop integrand at all orders in spin is given in the
Ancillary Files.

The presence of $e^{\ell_{1}{\cdot}a_{j}}$ factors in eq. (3.1) may seem
worrisome. Before expanding in powers of spin, potential divergences due to
such factors can be easily remedies by a shift of the spin vector
$a_{j}\rightarrow i\tilde{a}_{j}$. We illustrate with a simple example that
the integral in eq. (3.1) does not behave worse than the non-spinning case
does in the ultraviolet under this shift in Appendix B. Having evaluated the
integral, we can analytically continue back to real values of $a_{j}$. As for
the spin expansion of this integral, it can be defined as the Taylor expansion
of the well-defined integral at each order. Similar integrals are studied and
the resummed expressions are given in special kinematic regions in [121]. Our
preliminary analysis of simple examples in such kinematic regions show that
the naive evaluation of the spin-expanded integral agrees with the expansion
of the resummed one, order by order.

We now proceed to consider the expansion of eq. (3.1) in powers of spin.
Expanding out the spin tensor $S^{\mu\nu}$ and using the identity333We use the
package KiHA for manipulating the expressions [152].

$\displaystyle\epsilon^{\mu_{1}\cdots\mu_{n}\alpha_{1}\cdots\alpha_{m}}\epsilon_{\mu_{1}\cdots\mu_{n}\beta_{1}\cdots\beta_{m}}=(-1)^{n}n!\delta^{\alpha_{1}\cdots\alpha_{m}}_{\beta_{1}\cdots\beta_{m}},\quad\quad\delta^{\alpha_{1}\cdots\alpha_{m}}_{\beta_{1}\cdots\beta_{m}}=m!\delta^{\left[\alpha_{1}\right.}_{\;\beta_{1}}\cdots\delta^{\left.\alpha_{m}\right]}_{\;\beta_{m}},$
(3.4)

we can always write the integrand in terms of scalar products and at most one
factor of
$\epsilon(A,B,C,D)=\epsilon_{\mu\nu\rho\sigma}A^{\mu}B^{\nu}C^{\rho}D^{\sigma}$.
In fact, we find that at even orders in spin $\mathcal{O}(a^{2n})$, the
integrand contains only scalar products and at odd orders in spin
$\mathcal{O}(a^{2n+1})$, the integrand contains one $\epsilon(A,B,C,D)$
factor. Removing the bubble/ultra-local terms, the spin-expanded expression
has two types of structures,

$\displaystyle\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta(\ell_{1}{\cdot}v_{2})(\ell_{1}{\cdot}a_{1})^{n_{1}}(\ell_{1}{\cdot}a_{2})^{n_{2}}\over\ell_{1}^{2}\ell_{2}^{2}(\ell_{1}{\cdot}v_{1})^{m}},\quad\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta(\ell_{1}{\cdot}v_{2})(\ell_{1}{\cdot}a_{1})^{n_{1}}(\ell_{1}{\cdot}a_{2})^{n_{2}}\ell_{1}^{\mu}\over\ell_{1}^{2}\ell_{2}^{2}(\ell_{1}{\cdot}v_{1})^{m}},$
(3.5)

where $n$, $n_{1}$, $n_{2}$ and $m$ are integers. The vector $\ell_{1}^{\mu}$
in the second term above typically gets contracted with
$\epsilon(A,B,C,\cdot)_{\mu}$ where $A,B,C\in\\{q,v_{1},v_{2},a_{1},a_{2}\\}$.

Such one-loop tensor integrals can be readily decomposed into scalar
integrals. These scalar integrals are dressed by rank-$n$ tensor structures.
As shown in [44], it is convenient to write down such tensor structures in
terms of the following variables

$\displaystyle\theta^{\mu}_{1}:={\gamma
v_{2}^{\mu}-v_{1}^{\mu}\over\gamma^{2}-1},\quad\theta^{\mu}_{2}:={\gamma
v_{1}^{\mu}-v_{2}^{\mu}\over\gamma^{2}-1},\quad\Pi^{\mu\nu}:=\eta^{\mu\nu}-\theta_{1}^{\mu}v_{1}^{\nu}-\theta_{2}^{\mu}v_{2}^{\nu}-{q^{\mu}q^{\nu}\over
q^{2}},$ (3.6)

where $\Pi^{\mu\nu}$ is the $(D-3)$-dimensional metric. It is easy to verify
that a tensor integral of the form

$\displaystyle\mathcal{I}_{1,1,m,1}[\ell_{1}^{\mu_{1}}\cdots\ell_{1}^{\mu_{n}}]=\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta(\ell_{1}{\cdot}v_{2})\ell_{1}^{\mu_{1}}\cdots\ell_{1}^{\mu_{n}}\over\ell_{1}^{2}\ell_{2}^{2}(\ell_{1}{\cdot}v_{1})^{m}}$
(3.7)

can be decomposed into scalar integrals dressed by tensor structures built
from $q^{\mu}$, $\theta_{1}$ and $\Pi^{\mu\nu}$ only. Since all but at most
one of these indices are to be contracted with $a_{1}$ or $a_{2}$, symmetries
of the two structures in eq. (3.5) lead to closed-form expressions for the
decomposition. We present the explicit decomposition of the first structure in
eq. (3.5) here and postpone the derivation to Appendix C,

$\displaystyle\mathcal{I}_{1,1,\alpha,1}[(\ell_{1}{\cdot}a_{1})^{M}(\ell_{1}{\cdot}a_{2})^{N-M}]$
$\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}n_{1}+n_{2}+2n_{3}=N,\\\ n_{i}\geqslant
0,n_{i}\in\mathbb{Z}\end{subarray}}{1\over
N_{n_{3}}}\left(\sum_{k=0}^{n_{3}}\left(\begin{array}[]{c}n_{3}\\\
k\end{array}\right){1\over(\gamma^{2}-1)^{k}}\left({q^{2}\over
2}\right)^{n_{2}+n_{3}-k}\mathcal{I}_{1,1,\alpha-n_{1}-2k,1}[1]\right)$ (3.10)
$\displaystyle\quad\quad\times\Bigg{(}\sum_{\text{cond.}}C_{N,M,n_{1},n_{2},m_{1},m_{2},m_{3}}~{}(a_{1}{\cdot}\theta_{1})^{m_{1}}(a_{2}{\cdot}\theta_{1})^{n_{1}-m_{1}}(a_{1}{\cdot}q)^{m_{2}}(a_{2}{\cdot}q)^{n_{2}-m_{2}}$
$\displaystyle\quad\quad\quad\quad\quad\quad(a_{1}{\cdot}\Pi{\cdot}a_{1})^{m_{3}}(a_{1}{\cdot}\Pi{\cdot}a_{2})^{m_{4}}(a_{2}{\cdot}\Pi{\cdot}a_{2})^{n_{3}-m_{3}-m_{4}}\Bigg{)},$
(3.11)

where $N_{n_{3}}=(D-3)(D-1)\cdots(D-3+2(n_{3}-1))$ for $n_{3}>0$ and
$N_{0}=1$. The summation in the last bracket is taken over all the solutions
to the conditions below

$\displaystyle m_{1}+m_{2}+2m_{3}+m_{4}=M,\quad 0\leqslant m_{i}\leqslant
n_{i},\quad 0\leqslant m_{4}\leqslant n_{3},\quad m_{i}\in\mathbb{Z}.$ (3.12)

The coefficients read

$\displaystyle C_{N,M,n_{1},n_{2},m_{1},m_{2},m_{3}}={M!\over
m_{1}!m_{2}!m_{3}!m_{4}!}{(N-M)!\over(n_{1}-m_{1})!(n_{2}-m_{2})!(n_{3}-m_{3})!}{1\over
2^{n_{3}-m_{4}}}.$ (3.13)

The second structure in eq. (3.5) admits a similar expression with one free
index, which we also give in Appendix C.

After decomposing the tensor integrals, the resulting scalar integrals are
easily cast in a basis of master integrals using the IBP relations.444We use
the package LiteRed for the IBP reduction [153, 154]. Up to
$\mathcal{O}(a^{8})$, we see that only the triangle integral below contributes
in 4 dimensions,555The box integral $\mathcal{I}_{1,1,1,1}$ appears after the
IBP reduction. But its coefficient vanishes when we restrict to 4 dimensions.

$\displaystyle\mathcal{I}_{1,1,0,1}=\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta(\ell_{1}{\cdot}v_{2})\over\ell_{1}^{2}\ell_{2}^{2}}={2^{5-D}\pi^{2}(-q^{2})^{(D-5)/2}~{}\text{sec}\left(\pi
D/2\right)\over\Gamma\left({D\over 2}-1\right)}.$ (3.14)

Evaluating the integral, we arrive at the spin-expanded one-loop amplitude
computed from the covariant Compton amplitude in eq. (2.3). It is independent
of the $z$-parameter. We find perfect agreements with the literature up to
$\mathcal{O}(G^{2}a^{4})$ [145]. Since the terms at low orders in spin are
well established, we choose not to display the relatively bulky expressions.
Starting from $\mathcal{O}(G^{2}a^{5})$, to match with the far-zone data,
$z$-dependent contact terms $\mathcal{M}_{\rm TS-FZ}^{(c)}(z)$ need to be
included. The gluing process involving eq. (2.2) is slightly more subtle,
which we are to discuss shortly. We include the spin-expanded one-loop
amplitude from $\mathcal{O}(a^{5})$ to $\mathcal{O}(a^{8})$ in the Ancillary
Files.

###  3.2 Contribution from extra contact terms

Here we discuss the one-loop contribution from the $z$-dependent contact terms
$\mathcal{M}_{\rm TS-FZ}^{(c)}$ at $\mathcal{O}(G^{2}a^{5})$ and
$\mathcal{O}(G^{2}a^{6})$. It is given by the diagram depicted in eq. (3.1)
with the four-point Compton amplitude substituted by eq. (2.2). Since the
covariant Compton amplitude in eq. (2.3) suffices to match with the far-zone
data up to $\mathcal{O}(a^{4})$, the $\mathcal{O}(G^{2}a^{5})$ $z$-dependence
only enters through the $\mathcal{O}(G^{2}a_{1}^{5}a_{2}^{0})$ or
$\mathcal{O}(G^{2}a_{1}^{0}a_{2}^{5})$ terms. Hence the three-point amplitude
can be taken as the non-spinning one. At $\mathcal{O}(G^{2}a^{6})$, we need
both $\mathcal{O}(G^{2}a^{5}_{1}a_{2}^{1})$ and
$\mathcal{O}(G^{2}a_{1}^{6}a_{2}^{0})$ (and their mirrors), which come from
gluing up the two expressions in eq. (2.2) with appropriate orders in the spin
expansion of the three-point amplitude respectively.

As demonstrated in the previous section, the $z$-dependent contact terms in
eq. (2.2) are constructed such that they agree with those in [83] in 4
dimensions. This allows for ambiguities and the covariant expressions, which
are equivalent in 4 dimensions may have subtle differences, if naively
extrapolated to general dimensions. Hence, their contribution to the one-loop
amplitude needs to be constructed strictly in 4 dimensions as well.

In 4 dimensions, the three-point amplitude in eq. (2.1) simplifies to

$\displaystyle\mathcal{M}_{3}(1^{+},\bar{p}^{\prime},\bar{p})=-i\kappa
e^{+p_{1}{\cdot}a}(\bar{p}{\cdot}\varepsilon_{1}^{+})^{2},\quad\quad\mathcal{M}_{3}(1^{-},\bar{p}^{\prime},\bar{p})=-i\kappa
e^{-p_{1}{\cdot}a}(\bar{p}{\cdot}\varepsilon_{1}^{-})^{2}.$ (3.15)

Recall that the extra contact terms $\mathcal{M}_{\rm TS-FZ}^{(c)}$ also
vanish in the same-helicity configurations, namely, $\mathcal{M}_{\rm TS-
FZ}^{(c,5)}(1^{\pm},2^{\pm},\bar{p}^{\prime},\bar{p})=\mathcal{M}_{\rm TS-
FZ}^{(c,6)}(1^{\pm},2^{\pm},\bar{p}^{\prime},\bar{p})=0$. Consequently, their
one-loop contribution is given by

$\displaystyle\\!\\!\\!\\!\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)
{$\overline{p}_{1}$}; \vertex[right=1.5cm of a] (f2) [GR]{$~{}~{}~{}$};
\vertex[right=1.5cm of f2] (c){$\overline{p}^{\prime}_{1}$};
\vertex[above=2.0cm of a](ac){$\overline{p}_{2}$}; \vertex[right=1.0cm of ac]
(ad) [dot]{}; \vertex[right=1.0cm of ad] (f2c) [dot]{}; \vertex[above=2.0cm of
c](cc){$\overline{p}^{\prime}_{2}$}; \vertex[above=1.0cm of a] (cutL);
\vertex[right=3.0cm of cutL] (cutR); \vertex[right=0.5cm of ad] (att);
\vertex[above=0.3cm of att] (cut20){$$}; \vertex[below=0.3cm of att] (cut21);
\vertex[above right=0.2cm and 0.45cm of cutL] (hpL) {$+$}; \vertex[below
right=0.2cm and 0.6cm of cutL] (hmL) {$-$}; \vertex[above left=0.2cm and
0.45cm of cutR] (hmR) {$-$}; \vertex[below left=0.2cm and 0.6cm of cutR] (hpR)
{$+$}; \diagram* { (a) -- [fermion,thick] (f2)-- [fermion,thick] (c),
(f2)--[photon,ultra thick](ad), (f2)-- [photon,ultra thick] (f2c),(ac) --
[fermion,thick] (ad)-- [fermion,thick] (f2c)-- [fermion,thick] (cc),
(cutL)--[dashed, red,thick] (cutR), (cut20)--[ red,thick] (cut21) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a)
{$\overline{p}_{1}$}; \vertex[right=1.5cm of a] (f2) [GR]{$~{}~{}~{}$};
\vertex[right=1.5cm of f2] (c){$\overline{p}^{\prime}_{1}$};
\vertex[above=2.0cm of a](ac){$\overline{p}_{2}$}; \vertex[right=1.0cm of ac]
(ad) [dot]{}; \vertex[right=1.0cm of ad] (f2c) [dot]{}; \vertex[above=2.0cm of
c](cc){$\overline{p}^{\prime}_{2}$}; \vertex[above=1.0cm of a] (cutL);
\vertex[right=3.0cm of cutL] (cutR); \vertex[right=0.5cm of ad] (att);
\vertex[above=0.3cm of att] (cut20){$$}; \vertex[below=0.3cm of att] (cut21);
\vertex[above right=0.2cm and 0.45cm of cutL] (hpL) {$-$}; \vertex[below
right=0.2cm and 0.6cm of cutL] (hmL) {$+$}; \vertex[above left=0.2cm and
0.45cm of cutR] (hmR) {$+$}; \vertex[below left=0.2cm and 0.6cm of cutR] (hpR)
{$-$}; \diagram* { (a) -- [fermion,thick] (f2)-- [fermion,thick] (c),
(f2)--[photon,ultra thick](ad), (f2)-- [photon,ultra thick] (f2c),(ac) --
[fermion,thick] (ad)-- [fermion,thick] (f2c)-- [fermion,thick] (cc),
(cutL)--[dashed, red,thick] (cutR), (cut20)--[ red,thick] (cut21) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\begin{split}&{1\over
2}{(32\pi
G)^{2}\over(4\pi)^{D/2}}\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta(m_{2}\,v_{2}{\cdot}\ell_{1})\over\ell_{1}^{2}\ell_{2}^{2}}\\\
&\Big{(}e^{(\ell_{1}-\ell_{2}){\cdot}a_{2}}\mathcal{C}_{+-}+e^{-(\ell_{1}-\ell_{2}){\cdot}a_{2}}\mathcal{C}_{-+}\Big{)},\end{split}$
(3.16)

where

$\displaystyle\mathcal{C}_{\pm\mp}=\left.(\bar{p}_{2}{\cdot}\varepsilon_{1}^{\pm})^{2}(\bar{p}_{2}{\cdot}\varepsilon_{2}^{\mp})^{2}\mathcal{M}_{\rm
TS-
FZ}^{(c)}(\ell_{1}^{\mp},\ell_{2}^{\pm},\bar{p}^{\prime}_{1},\bar{p}_{1})\right|_{\text{
completeness relation}}$ (3.17)

In this case, we have to use the “chiral” version of the completeness
relation. Equivalently, we can separate the even and odd terms in spin, as
discussed in [122]. At even orders in spin, we have

$\displaystyle\cosh\left(\ell_{12}{\cdot}a_{2}\right)(\mathcal{C}_{+-}+\mathcal{C}_{-+})=\cosh\left(\ell_{12}{\cdot}a_{2}\right)(\mathcal{C}_{+-}+\mathcal{C}_{++}+\mathcal{C}_{--}+\mathcal{C}_{-+}),$
(3.18)

where $\ell_{12}=\ell_{1}-\ell_{2}$ and $\mathcal{C}_{++}+\mathcal{C}_{--}=0$.
Here we can use the same completeness relation in eq. (3.2), since we have

$\displaystyle\varepsilon^{+\mu}_{k}\varepsilon^{+\nu}_{k}\varepsilon^{-\rho}_{-k}\varepsilon^{-\sigma}_{-k}+\varepsilon^{-\mu}_{k}\varepsilon^{-\nu}_{k}\varepsilon^{+\rho}_{-k}\varepsilon^{+\sigma}_{-k}$
(3.19)

for both gravitons. At odd powers in spin, we have instead

$\displaystyle\sinh\left(\ell_{12}{\cdot}a_{2}\right)(\mathcal{C}_{+-}-\mathcal{C}_{-+})=\sinh\left(\ell_{12}{\cdot}a_{2}\right)(\mathcal{C}_{+-}-\mathcal{C}_{++}+\mathcal{C}_{--}-\mathcal{C}_{-+}).$
(3.20)

Hence we have

$\displaystyle\varepsilon^{+\mu}_{k}\varepsilon^{+\nu}_{k}\varepsilon^{-\rho}_{-k}\varepsilon^{-\sigma}_{-k}-\varepsilon^{-\mu}_{k}\varepsilon^{-\nu}_{k}\varepsilon^{+\rho}_{-k}\varepsilon^{+\sigma}_{-k}={1\over
4}\left(\mathcal{P}^{\mu\rho}\tilde{\mathcal{P}}^{\nu\sigma}+\mathcal{P}^{\mu\sigma}\tilde{\mathcal{P}}^{\nu\rho}+\mathcal{P}^{\nu\rho}\tilde{\mathcal{P}}^{\mu\sigma}+\mathcal{P}^{\nu\sigma}\tilde{\mathcal{P}}^{\mu\rho}\right),$
(3.21)

for one of the two gravitons, where

$\displaystyle\tilde{\mathcal{P}}^{\mu\nu}={i\epsilon^{\mu\nu\rho\sigma}k_{\rho}n_{\sigma}\over
k{\cdot}n}.$ (3.22)

This way we guarantee that the remnant ambiguities introduced in the process
of lifting the spinor-helicity expression for $\mathcal{M}_{\rm TS-FZ}^{(c)}$
to the covariant one have no impact at one loop.

This concludes our discussion on the one-loop contribution from the extra
contact terms. Together with the contribution from the covariant Compton
amplitude, we find agreement with the far-zone data in [145] up to
$\mathcal{O}(G^{2}a^{6})$, which is the highest-order result to our best
knowledge. Comparing with the 2PM calculation from the higher-spin amplitudes
[155] may be helpful for finding an all-order-in-spin remedy for the
$z$-dependence.

###  3.3 2PM eikonal in the spin expansion

Generally speaking, the eikonal exponentiation can be identified with the
classical amplitude Fourier transformed to the impact parameter space. The
classical eikonal phase is then extracted order by order. Recent works [102,
121] show that the classical eikonal can be viewed as a generator for
canonical transformations.666Another work [119] shows that the radial action
can be understood in a similar fashion.

Up to 2PM, the relation between the amplitude and the eikonal remains
straightforward, namely

$\displaystyle\chi^{\rm(nPM)}_{\tau}={1\over
4m_{1}m_{2}\sqrt{\gamma^{2}-1}}\int{d^{D-2}\mathbf{q}\over(2\pi)^{D-2}}e^{-i\mathbf{b}{\cdot}\mathbf{q}}\mathcal{M}^{(n-1)}_{a_{1}a_{2}}U_{1}U_{2},$
(3.23)

where $\mathbf{b}$ is the impact parameter and $\tau$ labels the Thomas-Wigner
rotation factors [85, 86]

$\displaystyle U_{i}=e^{i\tau\mathcal{E}_{i}\over
Em_{i}(E_{i}+m_{i})},\quad\quad\mathcal{E}_{i}:=E\,\mathbf{s}_{i}{\cdot}(\mathbf{p}\times\mathbf{q})$
(3.24)

with $E_{i}$ denoting the energy of the two black holes and $E=E_{1}+E_{2}$.
The bold-faced letters denote the three-vectors. The three-momenta are given
in the center-of-mass (COM) frame, namely
$p_{1}=(E_{1},\mathbf{p}),~{}p_{2}=(E_{2},-\mathbf{p}),~{}q=(0,\mathbf{q})$,
and $\mathbf{s}_{i}$ is the spatial part of $s_{i}^{\mu}=m_{i}a_{i}^{\mu}$.
Different values of $\tau$ correspond to different choices of the spin
supplementary conditions (SSC): $\tau=0$ corresponds to the covariant SSC
$\bar{p}_{\mu}S^{\mu\nu}=0$ whereas $\tau=1$ the canonical SSC. (For more
details, see also [84, 145].) Dressing the amplitude with the Thomas-Wigner
rotation factors is effectively equivalent to shifting the impact parameter
$\mathbf{b}\rightarrow\mathbf{b}+\sum_{i=1,2}\left({-\tau\over
E_{i}+m_{i}}\right)\mathbf{p}\times\mathbf{s}_{i}$. The canonical observables
follow from the the canonical SSC.

Here we showcase the explicit expression for the $z$-independent
$\mathcal{O}(G^{2}a^{7})$ contributions to $\chi^{\rm(2PM)}_{\tau=0}$ in the
aligned-spin case, which is given by the covariant Compton amplitude. To
compare with the far-zone data, it needs to be supplemented by the
contributions from the $z$-dependent contact terms $\mathcal{M}_{\rm TS-
FZ}^{(c,n=5,6,7)}$, which are given in eq. (2.2) and Appendix A. We
parameterize the aliged spins as $a_{2}^{\mu}=\xi a_{1}^{\mu}$ and hence the
powers of $a_{1}$ and $a_{2}$ are tracked by the power of $\xi$.

$\displaystyle\left.\chi^{\rm(2PM)}_{\tau=0}\right|_{\xi^{4},z=0}$
$\displaystyle=-\frac{35\pi
G^{2}\xi^{4}\gamma\left(24\left(9\gamma^{2}-4\right)m_{1}+\left(114-239\gamma^{2}\right)m_{2}\right)}{512\left(\gamma^{2}-1\right)^{3/2}|\mathbf{b}|^{8}}~{}\hat{b}{\cdot}S_{1}{\cdot}p_{2}$
$\displaystyle\quad\quad\left({429(\hat{b}{\cdot}a_{1})^{6}}-{594a_{1}^{2}(\hat{b}{\cdot}a_{1})^{4}}+{216a_{1}^{4}(\hat{b}{\cdot}a_{1})^{2}}-16a_{1}^{6}\right),$
(3.25) $\displaystyle\left.\chi^{\rm(2PM)}_{\tau=0}\right|_{\xi^{5},z=0}$
$\displaystyle=-\frac{21\pi
G^{2}\xi^{5}\gamma\,\hat{b}{\cdot}S_{1}{\cdot}p_{2}}{512\left(\gamma^{2}-1\right)^{3/2}|\mathbf{b}|^{8}}\Big{(}2145\left(\left(38\gamma^{2}-15\right)m_{1}+26\left(1-2\gamma^{2}\right)m_{2}\right)(\hat{b}{\cdot}a_{1})^{6}$
$\displaystyle\quad\quad\quad-33a_{1}^{2}\big{(}\left(3421\gamma^{2}-1351\right)m_{1}+2340\left(1-2\gamma^{2}\right)m_{2}\big{)}(\hat{b}{\cdot}a_{1})^{4}$
$\displaystyle\quad\quad\quad+36a_{1}^{4}\big{(}\left(1141\gamma^{2}-451\right)m_{1}+780\left(1-2\gamma^{2}\right)m_{2}\big{)}(\hat{b}{\cdot}a_{1})^{2}$
$\displaystyle\quad\quad\quad-8a_{1}^{6}\big{(}\left(381\gamma^{2}-151\right)m_{1}+260\left(1-2\gamma^{2}\right)m_{2}\big{)}\Big{)},$
(3.26) $\displaystyle\left.\chi^{\rm(2PM)}_{\tau=0}\right|_{\xi^{6},z=0}$
$\displaystyle=-\frac{7\pi G^{2}\xi^{6}\gamma\,\hat{b}\cdot S_{1}\cdot
p_{2}}{4096\left(\gamma^{2}-1\right)^{3/2}|\mathbf{b}|^{8}}\Big{(}\\!\\!-\\!\\!16a_{1}^{6}\big{(}16\left(81\gamma^{2}-26\right)m_{1}+15\left(77-149\gamma^{2}\right)m_{2}\big{)}$
$\displaystyle\quad\quad\quad-66a_{1}^{2}\big{(}16\left(721\gamma^{2}-226\right)m_{1}+135\left(77-149\gamma^{2}\right)m_{2}\big{)}(\hat{b}{\cdot}a_{1})^{4}$
$\displaystyle\quad\quad\quad+72a_{1}^{4}\big{(}16\left(241\gamma^{2}-76\right)m_{1}+45\left(77-149\gamma^{2}\right)m_{2}\big{)}(\hat{b}{\cdot}a_{1})^{2}$
$\displaystyle\quad\quad\quad+2145\left(16\left(16\gamma^{2}-5\right)m_{1}+3\left(77-149\gamma^{2}\right)m_{2}\right)(\hat{b}{\cdot}a_{1})^{6}\Big{)},$
(3.27) $\displaystyle\left.\chi^{\rm(2PM)}_{\tau=0}\right|_{\xi^{7},z=0}$
$\displaystyle=-\frac{\pi
G^{2}\xi^{7}\gamma\,\hat{b}{\cdot}S_{1}{\cdot}p_{2}}{1024\left(\gamma^{2}-1\right)^{3/2}|\mathbf{b}|^{8}}\Big{(}2145\left(6\left(8\gamma^{2}-1\right)m_{1}+7\left(9-17\gamma^{2}\right)m_{2}\right)(\hat{b}{\cdot}a_{1})^{6}$
$\displaystyle\quad\quad\quad-198a_{1}^{2}\big{(}\left(727\gamma^{2}-97\right)m_{1}+105\left(9-17\gamma^{2}\right)m_{2}\big{)}(\hat{b}{\cdot}a_{1})^{4}$
$\displaystyle\quad\quad\quad+108a_{1}^{4}\big{(}\left(493\gamma^{2}-73\right)m_{1}+70\left(9-17\gamma^{2}\right)m_{2}\big{)}(\hat{b}{\cdot}a_{1})^{2}$
$\displaystyle\quad\quad\quad-8a_{1}^{6}\big{(}\left(513\gamma^{2}-93\right)m_{1}+70\left(9-17\gamma^{2}\right)m_{2}\big{)}\Big{)},$
(3.28)

where we define $\hat{b}^{\mu}=b^{\mu}/|b|$. The terms that are not listed
above can be obtained by swapping the two particles
$\chi^{\rm(2PM)}|_{\xi^{n}}=\left(\chi^{\rm(2PM)}|_{\xi^{7-n}}\right)_{1\leftrightarrow
2}$.

Following the prescription in [102], the impulse $\Delta\mathbf{p}_{\perp}$
and the spin kick $\Delta\mathbf{s}_{1}$ can be generated from the eikonal
phase
$\chi_{\tau=0}=\chi^{\rm(1PM)}_{\tau=0}+\chi^{\rm(2PM)}_{\tau=0}+\mathcal{O}(G^{3})$
via777It is noted in [121] that the operator $\mathcal{D}_{SL}$ is not
necessary once all the constraints are properly imposed.

$\displaystyle\Delta\mathcal{O}=-\\{\mathcal{O},\chi\\}-{1\over
2}\\{\chi,\\{\mathcal{O},\chi\\}\\}-\mathcal{D}_{SL}\left(\chi,\\{\mathcal{O},\chi\\}\right)+{1\over
2}\\{\mathcal{O},\mathcal{D}_{SL}(\chi,\chi)\\},$ (3.29)

where $\Delta\mathcal{O}=\\{\Delta\mathbf{p}_{\perp},\Delta\mathbf{s}_{1}\\}$
and the full impulse is given by $\Delta\mathbf{p}=\Delta
p_{\parallel}\,\mathbf{p}/|\mathbf{p}|+\Delta\mathbf{p}_{\perp}$, which
satisfies the on-shell condition
$(\mathbf{p}+\Delta\mathbf{p})^{2}=\mathbf{p}^{2}$. The Poisson bracket and
the spin derivative operators are defined as

$\displaystyle\\{\mathbf{p}_{\perp},f\\}:=-{\partial
f\over\partial\mathbf{b}},\quad\quad\\{\mathbf{s}_{1},f\\}:={\partial
f\over\partial\mathbf{s}_{1}}\times\mathbf{s}_{i},\quad\quad\mathcal{D}_{SL}(f,g):=-\mathbf{s}_{1}{\cdot}\left({\partial
f\over\partial\mathbf{s}_{1}}{\partial g\over\partial\mathbf{L}}\right),$
(3.30)

with $\mathbf{L}:=\mathbf{b}\times\mathbf{p}$.

##  4 Conclusions and outlook

We consider a particular proposal for the classical gravitational Compton
amplitude, which follows from bootstrapping techniques and is expressed in
covariant variables. This covariant expression is consistent with all possible
massless and massive factorization requirements at all orders in spin and
contains a contact contribution that is obtained by imposing the same
empirical patterns as those observed in other contributions.

In this paper, we further the analysis in [74] regarding the contact terms. In
particular, we verify the matching between the covariant Compton amplitude and
the one derived from the higher-spin theory [99] up to $\mathcal{O}(a^{20})$,
when taking the spheroidicity parameter $z\rightarrow 0$. We also devise a
simple procedure to find a covariant form which accounts for the missing
contact contributions compared to those extracted from the far-zone data. We
confirm that such contributions can be written as a polynomial in $z$ and
vanish as $z\rightarrow 0$ up to $\mathcal{O}(a^{8})$. We believe such
patterns extend to higher orders in spin. These observations corroborate
somewhat the folklore that the $z$-dependent contact terms are associated with
the internal structures of the Kerr black hole. Further studies on multiple
fronts are needed to uncover the precise nature of such contact contributions.

From the covariant Compton amplitude, the calculation of the classical one-
loop amplitude on the triple cut using unitarity-based methods is then
streamlined. We obtain an all-order-in-spin integrand and evaluate the
integral up to $\mathcal{O}(a^{8})$ in the spin expansion. We have also
included the contributions from $z$-dependent contact terms, for which the
computation involves an extra subtlety. The Fourier transform of the one-loop
amplitude to the impact parameter space gives the eikonal phase. We find
perfect agreement with the 2PM far-zone data up to $\mathcal{O}(a^{6})$. It is
certainly interesting to consider the one-loop integral without expanding in
spin.

Two directions are natural to look into, the binary dynamics and the waveform
at higher PM orders. The binary dynamics at $\mathcal{O}(G^{3})$ is given by
the two-loop $2\rightarrow 2$ amplitude and the waveform at
$\mathcal{O}(G^{5/2})$ is given by the one-loop amplitude with a graviton
radiation. The five-point tree-level gravitational Compton amplitude is needed
in both cases. Computing the five-point amplitude to an arbitrary order in
spin is by itself a challenge, although it is conceivable that the
bootstrapping techniques should carry over. Even without the five-point
amplitude at hand, one may consider mass sectors, for instance the “zig-zag”
diagrams at two loops in [41], which only involve the three- and four-point
amplitudes in the construction of loop amplitudes. The behaviours of the loop
amplitudes in these sectors may in turn shed new light on additional
constraints for the structures of the gravitational Compton amplitudes.

## Acknowledgement

We thank F. Y. Baustista, E. Bjerrum-Bohr, A. Brandhuber, G. R. Brown, J.
Gowdy, H. Johansson, J.-W. Kim, S. Lee, P. Pichini, M. Skowronek, G.
Travaglini for insightful discussions. TW is grateful to H. Johansson, J.-W.
Kim and S. Lee for discussions on their respective works and comments on the
manuscript. GC has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No. 847523 “INTERACTIONS”. TW is supported by the NRF grant
2021R1A2C2012350 and the Fellowship of China Postdoctoral Science Foundation
(No. 2022M713228).

##  Appendix A Extra contact terms

Here we present the explicit expressions for the extra contact terms that give
the $z$-dependence of the Compton amplitude extracted from the Teukolsky
solution at $\mathcal{O}(a^{7})$ and $\mathcal{O}(a^{8})$. We choose only the
contribution obtained from fitting the far-zone asymptotic.

$\displaystyle\mathcal{M}^{(c,7)}_{\rm TS-
FZ}=\frac{\left(284a{\cdot}p_{1}+211a{\cdot}p_{2}\right)\left(p_{1}{\cdot}\overline{p}\right){}^{2}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}\widetilde{F}_{2}{\cdot}\overline{p}(a{\cdot}a)^{3}}{945m^{4}}$
$\displaystyle+\frac{8p_{1}{\cdot}\overline{p}\left(a{\cdot}F_{1}{\cdot}p_{2}p_{1}{\cdot}F_{2}{\cdot}\overline{p}+a{\cdot}F_{2}{\cdot}p_{1}p_{2}{\cdot}F_{1}{\cdot}\overline{p}\right)\left(\overline{p}{\cdot}F_{1}{\cdot}\widetilde{F}_{2}{\cdot}\overline{p}-\overline{p}{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}\overline{p}\right)(a{\cdot}a)^{3}}{945m^{2}}$
$\displaystyle+\frac{\left(211a{\cdot}p_{1}+284a{\cdot}p_{2}\right)\left(p_{1}{\cdot}\overline{p}\right){}^{2}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}\overline{p}(a{\cdot}a)^{3}}{945m^{4}}$
$\displaystyle+\frac{p_{1}{\cdot}\overline{p}\left(99\left(a{\cdot}p_{1}\right){}^{2}m^{2}-101a{\cdot}p_{1}a{\cdot}p_{2}m^{2}-78a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}\widetilde{F}_{2}{\cdot}\overline{p}(a{\cdot}a)^{2}}{1890m^{4}}$
$\displaystyle+\frac{4p_{1}{\cdot}\overline{p}\left(-33\left(a{\cdot}p_{1}\right){}^{2}m^{2}+23a{\cdot}p_{1}a{\cdot}p_{2}m^{2}+60a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}\widetilde{F}_{2}{\cdot}\overline{p}(a{\cdot}a)^{2}}{945m^{4}}$
$\displaystyle+\frac{p_{1}{\cdot}\overline{p}\left(-99\left(a{\cdot}p_{2}\right){}^{2}m^{2}+101a{\cdot}p_{1}a{\cdot}p_{2}m^{2}+78a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}\overline{p}{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}\overline{p}(a{\cdot}a)^{2}}{1890m^{4}}$
$\displaystyle-\frac{101a{\cdot}p_{1}\left(p_{1}{\cdot}\overline{p}\right){}^{2}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a\overline{p}{\cdot}F_{1}{\cdot}\widetilde{F}_{2}{\cdot}\overline{p}(a{\cdot}a)^{2}}{135m^{2}}-\frac{101a{\cdot}p_{2}\left(p_{1}{\cdot}\overline{p}\right){}^{2}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a\overline{p}{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}\overline{p}(a{\cdot}a)^{2}}{135m^{2}}$
$\displaystyle-\frac{4p_{1}{\cdot}\overline{p}\left(-33\left(a{\cdot}p_{2}\right){}^{2}m^{2}+23a{\cdot}p_{1}a{\cdot}p_{2}m^{2}+60a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}\overline{p}(a{\cdot}a)^{2}}{945m^{4}}$
$\displaystyle+\frac{212}{945}a{\cdot}p_{1}\left(p_{1}{\cdot}\overline{p}\right){}^{2}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}F_{1}{\cdot}\widetilde{F}_{2}{\cdot}aa{\cdot}a+\frac{212}{945}a{\cdot}p_{2}\left(p_{1}{\cdot}\overline{p}\right){}^{2}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}aa{\cdot}a$
$\displaystyle+\frac{p_{1}{\cdot}\overline{p}\left(-121\left(a{\cdot}p_{1}\right){}^{2}m^{2}-121\left(a{\cdot}p_{2}\right){}^{2}m^{2}+190a{\cdot}p_{1}a{\cdot}p_{2}m^{2}+372a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)}{630m^{2}}$
$\displaystyle~{}~{}~{}\times(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}\widetilde{F}_{1}{\cdot}F_{2}{\cdot}\overline{p}a{\cdot}a)$
(A.1) $\displaystyle\mathcal{M}^{(c,8)}_{\rm TS-
FZ}=\frac{64\left(p_{1}{\cdot}\overline{p}\right){}^{2}p_{1}{\cdot}F_{2}{\cdot}\overline{p}p_{2}{\cdot}F_{1}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}(a{\cdot}a)^{4}}{2835m^{4}}+\frac{32\left(p_{1}{\cdot}\overline{p}\right){}^{2}p_{1}{\cdot}F_{2}{\cdot}F_{1}{\cdot}p_{2}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}(a{\cdot}a)^{4}}{2835m^{2}}$
$\displaystyle+\frac{\left(p_{1}{\cdot}\overline{p}\right){}^{2}\left(m^{2}\left(174\left(a{\cdot}p_{1}\right){}^{2}-55a{\cdot}p_{2}a{\cdot}p_{1}+174\left(a{\cdot}p_{2}\right){}^{2}\right)-278a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)\left(\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\right){}^{2}(a{\cdot}a)^{3}}{5670m^{6}}$
$\displaystyle+\frac{8\left(a{\cdot}p_{1}-a{\cdot}p_{2}\right)}{945m^{2}}\Big{(}p_{1}{\cdot}\overline{p}a{\cdot}F_{1}{\cdot}p_{2}p_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}(a{\cdot}a)^{3}+p_{1}{\cdot}\overline{p}a{\cdot}F_{2}{\cdot}p_{1}p_{2}{\cdot}F_{1}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}(a{\cdot}a)^{3}\Big{)}$
$\displaystyle+\frac{p_{1}{\cdot}\overline{p}\left(m^{2}a{\cdot}p_{1}\left(936a{\cdot}p_{1}-883a{\cdot}p_{2}\right)-483a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)p_{1}{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}(a{\cdot}a)^{3}}{11340m^{4}}$
$\displaystyle+\frac{p_{1}{\cdot}\overline{p}\left(a{\cdot}p_{2}\left(883a{\cdot}p_{1}-936a{\cdot}p_{2}\right)m^{2}+483a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)p_{2}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}(a{\cdot}a)^{3}}{11340m^{4}}$
$\displaystyle+\frac{\left(p_{1}{\cdot}\overline{p}\right){}^{2}}{5670m^{4}}\Big{(}-13m^{2}\left(132\left(a{\cdot}p_{1}\right){}^{2}-223a{\cdot}p_{2}a{\cdot}p_{1}+132\left(a{\cdot}p_{2}\right){}^{2}\right)$
$\displaystyle+1987a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\Big{)}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}(a{\cdot}a)^{2}$
$\displaystyle+\frac{\left(p_{1}{\cdot}\overline{p}\right){}^{2}\left(2m^{2}\left(255\left(a{\cdot}p_{1}\right){}^{2}-443a{\cdot}p_{2}a{\cdot}p_{1}+255\left(a{\cdot}p_{2}\right){}^{2}\right)-487a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)\left(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a\right){}^{2}a{\cdot}a}{5670m^{2}}$
$\displaystyle+\frac{a{\cdot}p_{1}p_{1}{\cdot}\overline{p}\left(3m^{2}a{\cdot}p_{1}\left(33a{\cdot}p_{1}-73a{\cdot}p_{2}\right)-859a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}a{\cdot}a}{1890m^{2}}$
$\displaystyle+\frac{a{\cdot}p_{2}p_{1}{\cdot}\overline{p}\left(3a{\cdot}p_{2}\left(73a{\cdot}p_{1}-33a{\cdot}p_{2}\right)m^{2}+859a{\cdot}a\left(p_{1}{\cdot}\overline{p}\right){}^{2}\right)a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}a{\cdot}a}{1890m^{2}}$
$\displaystyle+\frac{(a{\cdot}a)^{2}p_{1}{\cdot}\overline{p}}{5670m^{4}}\Big{(}3\left(-72\left(a{\cdot}p_{1}\right){}^{3}+136a{\cdot}p_{2}\left(a{\cdot}p_{1}\right){}^{2}+83\left(a{\cdot}p_{2}\right){}^{2}a{\cdot}p_{1}-27\left(a{\cdot}p_{2}\right){}^{3}\right)m^{2}$
$\displaystyle+a{\cdot}a\left(2288a{\cdot}p_{1}+289a{\cdot}p_{2}\right)\left(p_{1}{\cdot}\overline{p}\right){}^{2}\Big{)}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}$
$\displaystyle+\frac{(a{\cdot}a)^{2}p_{1}{\cdot}\overline{p}}{5670m^{4}}\Big{(}3m^{2}\left(27\left(a{\cdot}p_{1}\right){}^{3}-83a{\cdot}p_{2}\left(a{\cdot}p_{1}\right){}^{2}-136\left(a{\cdot}p_{2}\right){}^{2}a{\cdot}p_{1}+72\left(a{\cdot}p_{2}\right){}^{3}\right)$
$\displaystyle-a{\cdot}a\left(289a{\cdot}p_{1}+2288a{\cdot}p_{2}\right)\left(p_{1}{\cdot}\overline{p}\right){}^{2}\Big{)}a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\overline{p}\overline{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\overline{p}$
(A.2)

##  Appendix B UV behaviours

Here we discuss the UV behaviour of the one-loop integral (3.1). Before
expanding in spin, the potential divergences can be avoided, when we shift the
spin vector to the imaginary axis $a_{j}\rightarrow i\tilde{a}_{j}$. We first
illustrate this process with a simple example. Consider the double-copy
contribution to the Compton amplitude eq. (2.3), of which the spin-dependence
is given in eq. (2.4). Naively, the worst UV behaviour would have come from
the term with $G_{1}(x_{1})G_{2}(x_{2})$, yielding the highest UV scaling at
one loop as follows,

$\displaystyle\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta(m_{2}v_{2}{\cdot}\ell_{1})\over\ell_{1}^{2}\ell_{2}^{2}}G_{1}(\ell_{1}{\cdot}a_{1})G_{1}(\ell_{2}{\cdot}a_{1})\cosh(\ell_{1}{\cdot}a_{2})\cosh(\ell_{2}{\cdot}a_{2})\ell_{1}^{\mu}\ell_{1}^{\nu},$
(B.1)

where we have implicitly $\ell_{2}=q-\ell_{1}$ as always. Performing the shift
$a_{j}^{\mu}=i\tilde{a}_{j}^{\mu}$, the exponential of $\ell_{1}{\cdot}a_{j}$
becomes finite. That is, we have the following scaling as
$\ell_{1}\rightarrow\Lambda\ell_{1}$ and $\Lambda\rightarrow\infty$,

$\displaystyle\cosh(\ell_{1}{\cdot}a_{j})\sim\mathcal{O}(\Lambda^{0}),\quad\quad
G_{1}(\ell_{1}{\cdot}a_{j})\sim\mathcal{O}(\Lambda^{-1}),\quad\quad
G_{2}(\ell_{1}{\cdot}a_{j},\ell_{2}{\cdot}a_{j})\sim\mathcal{O}(\Lambda^{-1}).$
(B.2)

This way, we see that the one-loop integral above is well regulated.

We note that the particular derivatives of the entire functions appearing in
eq. (2.3) are more suppressed, namely

$\displaystyle(\partial_{a_{1}{\cdot}\ell_{1}}-\partial_{a_{1}{\cdot}\ell_{2}})G_{2}(a_{1}{\cdot}\ell_{1},a_{1}{\cdot}\ell_{2})\sim\mathcal{O}(\Lambda^{-2}),$
(B.3)
$\displaystyle(\partial_{a_{1}{\cdot}\ell_{1}}-\partial_{a_{1}{\cdot}\ell_{2}})\Big{(}G_{1}(a_{1}{\cdot}\ell_{1})G_{1}(a_{1}{\cdot}\ell_{2})\Big{)}\sim\mathcal{O}(\Lambda^{-2}).$
(B.4)

Since all the entire functions involved in eq. (2.3) are all finite under the
shift of the spin vector, the UV scaling boils down to the remaining factors
given in section 2.1, which can be straightforwardly examined. We find that
all the terms in eq. (3.1) are indeed well-defined under the shift in a
similar fashion, which allows us to evaluate it. In the end, we can
analytically continue back to the original real values of the spin vectors.

##  Appendix C Decomposition of tensor integrals

Here we derive the closed-form expression for the decompositions of the tensor
integrals (2.2). The variables given in eq. (3.6) satisfy the orthogonality
relations below,

$\displaystyle q{\cdot}\theta_{i}=\theta_{1}{\cdot}\theta_{2}=0,\quad
v_{i}{\cdot}\theta_{j}=\delta_{ij},\quad
q_{\mu}\Pi^{\mu\nu}=\theta_{i\mu}\Pi^{\mu\nu}=v_{i\mu}\Pi^{\mu\nu}=0.$ (C.1)

This yields the orthogonality of the tensor basis constructed from these
building blocks.

We denote the (tensor) integrals as

$\displaystyle\mathcal{I}_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}[f(\ell_{1})]\equiv\int{d^{D}\ell_{1}\over\pi^{D/2}}{\delta^{(\alpha_{4}-1)}(\ell_{1}{\cdot}v_{2})f(\ell_{1})\over\ell_{1}^{2\alpha_{1}}(q-\ell_{1})^{2\alpha_{2}}(\ell_{1}{\cdot}v_{1})^{\alpha_{3}}}.$
(C.2)

We consider the special case where we only have non-vanishing integrals with
$\alpha_{1}=\alpha_{2}=\alpha_{4}=1$, since the integrand in eq. (3.1) can not
have higher propagator powers in $\alpha_{1}$ or $\alpha_{2}$ and ultra-local
terms (negative values of $\alpha_{1}$ or $\alpha_{2}$) are excluded. The
scalar integral family is simply $\mathcal{I}_{1,1,\alpha_{3},1}[1]$. The two
types of integrals in (3.5) are
$\mathcal{I}_{1,1,\alpha_{3},1}[(\ell_{1}{\cdot}a_{1})^{M}(\ell_{1}{\cdot}a_{2})^{N-M}]$
and
$\mathcal{I}_{1,1,\alpha_{3},1}[(\ell_{1}{\cdot}a_{1})^{M}(\ell_{1}{\cdot}a_{2})^{N-M}\ell^{\mu}]$.

We consider the tensor numerator with free indices first. The most general
ansatz for the decomposition of
$\mathcal{I}_{1,1,\alpha_{3},1}[\ell_{1}^{\mu_{1}}\cdots\ell_{1}^{\mu_{N}}]$
reads

$\displaystyle\mathcal{I}_{1,1,\alpha_{3},1}[\ell_{1}^{\mu_{1}}\cdots\ell_{1}^{\mu_{N}}]=\sum_{\begin{subarray}{c}n_{1}+n_{2}+2n_{3}=N,\\\
n_{i}\geqslant 0,n_{i}\in\mathbb{Z}\end{subarray}}$
$\displaystyle\mathbb{C}_{n_{1}n_{2}n_{3}}\,\text{sym}\left[\theta_{1}^{\mu_{1}}\cdots\theta_{1}^{\mu_{n_{1}}}q^{\mu_{n_{1}+1}}\cdots
q^{\mu_{n_{1}+n_{2}}}\right.$
$\displaystyle\quad\quad\left.\Pi^{\mu_{n_{1}+n_{2}+1}\mu_{n_{1}+n_{2}+2}}\cdots\Pi^{\mu_{N-1}\mu_{N}}\right],$
(C.3)

where $\text{sym}[\cdots]$ denotes the sum of distinct tensor structures
obtained from index permutations. Generally speaking, the coefficients
$\mathbb{C}_{n_{1}n_{2}n_{3}}$ can be obtained by contracting both sides of
the ansatz with all possible rank-$n$ tensor structures built from
$\\{v_{i}^{\mu},q_{i}^{\mu},\Pi^{\mu\nu}\\}$ and solving the linear equations.
Thanks to the orthogonality of the tensor structures, these equations are
already diagnolized and we can simply read off the coefficients
$\mathbb{C}_{n_{1}n_{2}n_{3}}$ as follows,

$\displaystyle\mathbb{C}_{n_{1}n_{2}n_{3}}={1\over
N_{n_{3}}}\mathcal{I}_{1,1,\alpha_{3},1}\left[(\ell_{1}{\cdot}v_{1})^{n_{1}}(\ell_{1}{\cdot}q)^{n_{2}}(\ell_{1}{\cdot}\Pi{\cdot}\ell_{1})^{n_{3}}\right],$
(C.4)

where we have $N_{n_{3}}=(D-3)(D-1)\cdots(D-3+2(n_{3}-1))$ for $n_{3}>0$ and
$N_{0}=1$. The remaining integral can be rewritten in terms of scalar
integrals,

$\displaystyle\mathcal{I}_{1,1,\alpha_{3},1}\left[(\ell_{1}{\cdot}v_{1})^{n_{1}}(\ell_{1}{\cdot}q)^{n_{2}}(\ell_{1}{\cdot}\Pi{\cdot}\ell_{1})^{n_{3}}\right]=\sum_{k=0}^{n_{3}}\left(\begin{array}[]{c}n_{3}\\\
k\end{array}\right){1\over(\gamma^{2}-1)^{k}}\left({q^{2}\over
2}\right)^{n_{2}+n_{3}-k}\mathcal{I}_{1,1,\alpha_{3}-n_{1}-2k,1},$ (C.7)

where we have repeatedly used $\ell_{1}{\cdot}q={1\over
2}\left(\ell_{1}^{2}+q^{2}-(q-\ell_{1})^{2}\right)$ and that scaleless
integrals are vanishing.

The contraction between the symmetrized tensor structures in (C) and the spin
vectors can be worked out. We have

$\displaystyle\text{Cont}(N,M,n_{1},n_{2})$
$\displaystyle\equiv\text{sym}[\,\overbrace{\theta_{1}\cdots\theta_{1}}^{n_{1}}\,\overbrace{q\cdots
q}^{n_{2}}\,\overbrace{\Pi\cdots\Pi}^{n_{3}}\,]\underbrace{a_{1}\cdots
a_{1}}_{M}\underbrace{a_{2}\cdots a_{2}}_{N-M}$
$\displaystyle=\sum_{\text{cond.}}C_{N,M,n_{1},n_{2},m_{1},m_{2},m_{3}}~{}(a_{1}{\cdot}\theta_{1})^{m_{1}}(a_{2}{\cdot}\theta_{1})^{n_{1}-m_{1}}(a_{1}{\cdot}q)^{m_{2}}(a_{2}{\cdot}q)^{n_{2}-m_{2}}$
$\displaystyle\quad\quad\quad\quad\quad(a_{1}{\cdot}\Pi{\cdot}a_{1})^{m_{3}}(a_{1}{\cdot}\Pi{\cdot}a_{2})^{m_{4}}(a_{2}{\cdot}\Pi{\cdot}a_{2})^{n_{3}-m_{3}-m_{4}},$
(C.8)

where the sums are taken over the solutions to the conditions below

$\displaystyle m_{1}+m_{2}+2m_{3}+m_{4}=M,\quad 0\leqslant m_{i}\leqslant
n_{i},\quad 0\leqslant m_{4}\leqslant n_{3},\quad m_{i}\in\mathbb{Z}.$ (C.9)

The coefficients read

$\displaystyle C_{N,M,n_{1},n_{2},m_{1},m_{2},m_{3}}={M!\over
m_{1}!m_{2}!m_{3}!m_{4}!}{(N-M)!\over(n_{1}-m_{1})!(n_{2}-m_{2})!(n_{3}-m_{3})!}{1\over
2^{n_{3}-m_{4}}}.$ (C.10)

Hence we have arrived at the decomposition of
$\mathcal{I}_{1,1,\alpha_{3},1}[(\ell_{1}{\cdot}a_{1})^{M}(\ell_{1}{\cdot}a_{2})^{N-M}]$.

The remaining integral
$\mathcal{I}_{1,1,\alpha_{3},1}[(\ell_{1}{\cdot}a_{1})^{M}(\ell_{1}{\cdot}a_{2})^{N-M}\ell_{1}^{\mu}]$
is nothing but the above with one free index and can be dealt with similarly.
We first rewrite the symmeterized sum of the tensor structures as follows,

$\displaystyle\text{sym}[\,\overbrace{\theta_{1}\cdots\theta_{1}}^{n_{1}}\,\overbrace{q\cdots
q}^{n_{2}}\,\overbrace{\Pi\cdots\Pi}^{n_{3}}\,]=$
$\displaystyle\quad\theta_{1}^{\mu_{N}}\text{sym}[\,\overbrace{\theta_{1}\cdots\theta_{1}}^{n_{1}-1}\,\overbrace{q\cdots
q}^{n_{2}}\,\overbrace{\Pi\cdots\Pi}^{n_{3}}\,]$
$\displaystyle+q^{\mu_{N}}\text{sym}[\,\overbrace{\theta_{1}\cdots\theta_{1}}^{n_{1}}\,\overbrace{q\cdots
q}^{n_{2}-1}\,\overbrace{\Pi\cdots\Pi}^{n_{3}}\,]$
$\displaystyle+\sum_{i=1}^{N-1}\Pi^{\mu_{i}\mu_{N}}\text{sym}[\,\overbrace{\theta_{1}\cdots\theta_{1}}^{n_{1}}\,\overbrace{q\cdots
q}^{n_{2}}\,\overbrace{\Pi\cdots\Pi}^{n_{3}-1}\,].$ (C.11)

Contracting with $a_{1}^{\mu_{1}}\cdots a_{1}^{\mu_{M}}a_{2}^{\mu_{M+1}}\cdots
a_{2}^{\mu_{N-M-1}}$, we have

$\displaystyle\quad\text{sym}[\,\overbrace{\theta_{1}\cdots\theta_{1}}^{n_{1}}\,\overbrace{q\cdots
q}^{n_{2}}\,\overbrace{\Pi\cdots\Pi}^{n_{3}}\,]\underbrace{a_{1}\cdots
a_{1}}_{M}\underbrace{a_{2}\cdots a_{2}}_{N-1-M}$ $\displaystyle=$
$\displaystyle\quad\theta^{\mu_{N}}_{1}\text{Cont}(N-1,M,n_{1}-1,n_{2})+q^{\mu_{N}}\text{Cont}(N-1,M,n_{1},n_{2}-1)$
(C.12)
$\displaystyle+M(a_{1}{\cdot}\Pi)^{\mu_{N}}\text{Cont}(N-2,M-1,n_{1},n_{2})+(N-1-M)(a_{2}{\cdot}\Pi)^{\mu_{N}}\text{Cont}(N-2,M,n_{1},n_{2}).$

## References

  * [1] LIGO Scientific, Virgo collaboration, _Binary Black Hole Mergers in the first Advanced LIGO Observing Run_ , _Phys. Rev. X_ 6 (2016) 041015 [1606.04856].
  * [2] LIGO Scientific, Virgo collaboration, _Observation of Gravitational Waves from a Binary Black Hole Merger_ , _Phys. Rev. Lett._ 116 (2016) 061102 [1602.03837].
  * [3] LIGO Scientific, Virgo collaboration, _GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence_ , _Phys. Rev. Lett._ 116 (2016) 241103 [1606.04855].
  * [4] LIGO Scientific, VIRGO collaboration, _GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2_ , _Phys. Rev. Lett._ 118 (2017) 221101 [1706.01812].
  * [5] LIGO Scientific, Virgo collaboration, _GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral_ , _Phys. Rev. Lett._ 119 (2017) 161101 [1710.05832].
  * [6] F. Pretorius, _Evolution of binary black hole spacetimes_ , _Phys. Rev. Lett._ 95 (2005) 121101 [gr-qc/0507014].
  * [7] M. Campanelli, C.O. Lousto, P. Marronetti and Y. Zlochower, _Accurate evolutions of orbiting black-hole binaries without excision_ , _Phys. Rev. Lett._ 96 (2006) 111101 [gr-qc/0511048].
  * [8] J.G. Baker, J. Centrella, D.-I. Choi, M. Koppitz and J. van Meter, _Gravitational wave extraction from an inspiraling configuration of merging black holes_ , _Phys. Rev. Lett._ 96 (2006) 111102 [gr-qc/0511103].
  * [9] A. Buonanno and T. Damour, _Effective one-body approach to general relativistic two-body dynamics_ , _Phys. Rev._ D59 (1999) 084006 [gr-qc/9811091].
  * [10] W.D. Goldberger and I.Z. Rothstein, _An Effective field theory of gravity for extended objects_ , _Phys. Rev. D_ 73 (2006) 104029 [hep-th/0409156].
  * [11] B. Kol and M. Smolkin, _Classical Effective Field Theory and Caged Black Holes_ , _Phys. Rev. D_ 77 (2008) 064033 [0712.2822].
  * [12] J.B. Gilmore and A. Ross, _Effective field theory calculation of second post-Newtonian binary dynamics_ , _Phys. Rev. D_ 78 (2008) 124021 [0810.1328].
  * [13] S. Foffa and R. Sturani, _Effective field theory calculation of conservative binary dynamics at third post-Newtonian order_ , _Phys. Rev. D_ 84 (2011) 044031 [1104.1122].
  * [14] S. Foffa, P. Mastrolia, R. Sturani and C. Sturm, _Effective field theory approach to the gravitational two-body dynamics, at fourth post-Newtonian order and quintic in the Newton constant_ , _Phys. Rev. D_ 95 (2017) 104009 [1612.00482].
  * [15] R.A. Porto and I.Z. Rothstein, _Apparent ambiguities in the post-Newtonian expansion for binary systems_ , _Phys. Rev. D_ 96 (2017) 024062 [1703.06433].
  * [16] J. Blümlein, A. Maier and P. Marquard, _Five-Loop Static Contribution to the Gravitational Interaction Potential of Two Point Masses_ , _Phys. Lett. B_ 800 (2020) 135100 [1902.11180].
  * [17] S. Foffa and R. Sturani, _Conservative dynamics of binary systems to fourth Post-Newtonian order in the EFT approach I: Regularized Lagrangian_ , _Phys. Rev. D_ 100 (2019) 024047 [1903.05113].
  * [18] S. Foffa, R.A. Porto, I. Rothstein and R. Sturani, _Conservative dynamics of binary systems to fourth Post-Newtonian order in the EFT approach II: Renormalized Lagrangian_ , _Phys. Rev. D_ 100 (2019) 024048 [1903.05118].
  * [19] J. Blümlein, A. Maier, P. Marquard and G. Schäfer, _Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach_ , _Nucl. Phys. B_ 955 (2020) 115041 [2003.01692].
  * [20] J. Bluemlein, A. Maier, P. Marquard and G. Schaefer, _Testing binary dynamics in gravity at the sixth post-Newtonian level_ , 2003.07145.
  * [21] D. Bini, T. Damour and A. Geralico, _Sixth post-Newtonian local-in-time dynamics of binary systems_ , _Phys. Rev. D_ 102 (2020) 024061 [2004.05407].
  * [22] D. Bini, T. Damour and A. Geralico, _Sixth post-Newtonian nonlocal-in-time dynamics of binary systems_ , _Phys. Rev. D_ 102 (2020) 084047 [2007.11239].
  * [23] J. Blümlein, A. Maier, P. Marquard and G. Schäfer, _The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach_ , _Nucl. Phys. B_ 983 (2022) 115900 [2110.13822].
  * [24] J. Blümlein, A. Maier, P. Marquard and G. Schäfer, _The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach: potential contributions_ , 2010.13672.
  * [25] S. Foffa, R. Sturani and W.J. Torres Bobadilla, _Efficient resummation of high post-Newtonian contributions to the binding energy_ , _JHEP_ 02 (2021) 165 [2010.13730].
  * [26] J. Blümlein, A. Maier, P. Marquard and G. Schäfer, _The 6th post-Newtonian potential terms at $O(G_{N}^{4})$_, _Phys. Lett. B_ 816 (2021) 136260 [2101.08630].
  * [27] J.-W. Kim, M. Levi and Z. Yin, _Quadratic-in-spin interactions at fifth post-Newtonian order probe new physics_ , _Phys. Lett. B_ 834 (2022) 137410 [2112.01509].
  * [28] G. Cho, R.A. Porto and Z. Yang, _Gravitational radiation from inspiralling compact objects: Spin effects to fourth Post-Newtonian order_ , 2201.05138.
  * [29] N.E.J. Bjerrum-Bohr, P.H. Damgaard, G. Festuccia, L. Plante and P. Vanhove, _General Relativity from Scattering Amplitudes_ , _Phys. Rev. Lett._ 121 (2018) 171601 [1806.04920].
  * [30] C. Cheung, I.Z. Rothstein and M.P. Solon, _From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion_ , _Phys. Rev. Lett._ 121 (2018) 251101 [1808.02489].
  * [31] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, _Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order_ , _Phys. Rev. Lett._ 122 (2019) 201603 [1901.04424].
  * [32] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, _Black Hole Binary Dynamics from the Double Copy and Effective Theory_ , _JHEP_ 10 (2019) 206 [1908.01493].
  * [33] D. Neill and I.Z. Rothstein, _Classical Space-Times from the S Matrix_ , _Nucl. Phys._ B877 (2013) 177 [1304.7263].
  * [34] A. Cristofoli, R. Gonzo, D.A. Kosower and D. O’Connell, _Waveforms from Amplitudes_ , 2107.10193.
  * [35] Z. Bern, A. Luna, R. Roiban, C.-H. Shen and M. Zeng, _Spinning black hole binary dynamics, scattering amplitudes, and effective field theory_ , _Phys. Rev. D_ 104 (2021) 065014 [2005.03071].
  * [36] Z. Bern, J. Parra-Martinez, R. Roiban, M.S. Ruf, C.-H. Shen, M.P. Solon et al., _Scattering Amplitudes and Conservative Binary Dynamics at ${\cal O}(G^{4})$_, _Phys. Rev. Lett._ 126 (2021) 171601 [2101.07254].
  * [37] N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, _Quantum gravitational corrections to the nonrelativistic scattering potential of two masses_ , _Phys. Rev. D_ 67 (2003) 084033 [hep-th/0211072].
  * [38] D.A. Kosower, B. Maybee and D. O’Connell, _Amplitudes, Observables, and Classical Scattering_ , _JHEP_ 02 (2019) 137 [1811.10950].
  * [39] B. Maybee, D. O’Connell and J. Vines, _Observables and amplitudes for spinning particles and black holes_ , _JHEP_ 12 (2019) 156 [1906.09260].
  * [40] A. Brandhuber, G. Chen, G. Travaglini and C. Wen, _A new gauge-invariant double copy for heavy-mass effective theory_ , _JHEP_ 07 (2021) 047 [2104.11206].
  * [41] A. Brandhuber, G. Chen, G. Travaglini and C. Wen, _Classical gravitational scattering from a gauge-invariant double copy_ , _JHEP_ 10 (2021) 118 [2108.04216].
  * [42] G. Mogull, J. Plefka and J. Steinhoff, _Classical black hole scattering from a worldline quantum field theory_ , _JHEP_ 02 (2021) 048 [2010.02865].
  * [43] G.U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, _Classical Gravitational Bremsstrahlung from a Worldline Quantum Field Theory_ , _Phys. Rev. Lett._ 126 (2021) 201103 [2101.12688].
  * [44] G.U. Jakobsen and G. Mogull, _Conservative and Radiative Dynamics of Spinning Bodies at Third Post-Minkowskian Order Using Worldline Quantum Field Theory_ , _Phys. Rev. Lett._ 128 (2022) 141102 [2201.07778].
  * [45] T. Wang, _Binary dynamics from worldline QFT for scalar QED_ , _Phys. Rev. D_ 107 (2023) 085011 [2205.15753].
  * [46] J. Parra-Martinez, M.S. Ruf and M. Zeng, _Extremal black hole scattering at $\mathcal{O}(G^{3})$: graviton dominance, eikonal exponentiation, and differential equations_, _JHEP_ 11 (2020) 023 [2005.04236].
  * [47] P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, _The eikonal approach to gravitational scattering and radiation at $\mathcal{O}$(G3)_, _JHEP_ 07 (2021) 169 [2104.03256].
  * [48] C. Heissenberg, _Infrared divergences and the eikonal exponentiation_ , _Phys. Rev. D_ 104 (2021) 046016 [2105.04594].
  * [49] P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, _The eikonal operator at arbitrary velocities I: the soft-radiation limit_ , 2204.02378.
  * [50] P.H. Damgaard, L. Plante and P. Vanhove, _On an exponential representation of the gravitational S-matrix_ , _JHEP_ 11 (2021) 213 [2107.12891].
  * [51] U. Kol, D. O’connell and O. Telem, _The radial action from probe amplitudes to all orders_ , _JHEP_ 03 (2022) 141 [2109.12092].
  * [52] N.E.J. Bjerrum-Bohr, L. Planté and P. Vanhove, _Post-Minkowskian radial action from soft limits and velocity cuts_ , _JHEP_ 03 (2022) 071 [2111.02976].
  * [53] Z. Bern, J.P. Gatica, E. Herrmann, A. Luna and M. Zeng, _Scalar QED as a toy model for higher-order effects in classical gravitational scattering_ , 2112.12243.
  * [54] H. Lee, S. Lee and S. Mazumdar, _Classical observables from partial wave amplitudes_ , _JHEP_ 06 (2023) 096 [2303.07638].
  * [55] H. Lee, K. Lee and S. Lee, _Poincaré generators at second post-Minkowskian order_ , _JHEP_ 10 (2023) 044 [2307.05626].
  * [56] C. Dlapa, G. Kälin, Z. Liu and R.A. Porto, _Dynamics of binary systems to fourth Post-Minkowskian order from the effective field theory approach_ , _Phys. Lett. B_ 831 (2022) 137203 [2106.08276].
  * [57] C. Dlapa, G. Kälin, Z. Liu, J. Neef and R.A. Porto, _Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order_ , 2210.05541.
  * [58] C. Dlapa, G. Kälin, Z. Liu and R.A. Porto, _Bootstrapping the relativistic two-body problem_ , _JHEP_ 08 (2023) 109 [2304.01275].
  * [59] G.U. Jakobsen, G. Mogull, J. Plefka and B. Sauer, _Tidal effects and renormalization at fourth post-Minkowskian order_ , _Phys. Rev. D_ 109 (2024) L041504 [2312.00719].
  * [60] Z. Bern, J. Parra-Martinez, R. Roiban, M.S. Ruf, C.-H. Shen, M.P. Solon et al., _Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at O(G4)_ , _Phys. Rev. Lett._ 128 (2022) 161103 [2112.10750].
  * [61] P.H. Damgaard, E.R. Hansen, L. Planté and P. Vanhove, _Classical Observables from the Exponential Representation of the Gravitational S-Matrix_ , 2307.04746.
  * [62] M. Driesse, G.U. Jakobsen, G. Mogull, J. Plefka, B. Sauer and J. Usovitsch, _Conservative Black Hole Scattering at Fifth Post-Minkowskian and First Self-Force Order_ , 2403.07781.
  * [63] N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, _Scattering amplitudes for all masses and spins_ , _JHEP_ 11 (2021) 070 [1709.04891].
  * [64] Y.M. Zinoviev, _On massive high spin particles in AdS_ , hep-th/0108192.
  * [65] Y.M. Zinoviev, _On massive spin 2 interactions_ , _Nucl. Phys. B_ 770 (2007) 83 [hep-th/0609170].
  * [66] Y.M. Zinoviev, _On massive spin 2 electromagnetic interactions_ , _Nucl. Phys. B_ 821 (2009) 431 [0901.3462].
  * [67] Y.M. Zinoviev, _On spin 3 interacting with gravity_ , _Class. Quant. Grav._ 26 (2009) 035022 [0805.2226].
  * [68] A. Ochirov and E. Skvortsov, _Chiral Approach to Massive Higher Spins_ , _Phys. Rev. Lett._ 129 (2022) 241601 [2207.14597].
  * [69] L. Cangemi and P. Pichini, _Classical limit of higher-spin string amplitudes_ , _JHEP_ 06 (2023) 167 [2207.03947].
  * [70] Z. Bern, J.J.M. Carrasco and H. Johansson, _New Relations for Gauge-Theory Amplitudes_ , _Phys. Rev._ D78 (2008) 085011 [0805.3993].
  * [71] Z. Bern, J.J.M. Carrasco and H. Johansson, _Perturbative Quantum Gravity as a Double Copy of Gauge Theory_ , _Phys. Rev. Lett._ 105 (2010) 061602 [1004.0476].
  * [72] Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, _The Duality Between Color and Kinematics and its Applications_ , 1909.01358.
  * [73] N.E.J. Bjerrum-Bohr, G. Chen and M. Skowronek, _Classical spin gravitational Compton scattering_ , _JHEP_ 06 (2023) 170 [2302.00498].
  * [74] N.E.J. Bjerrum-Bohr, G. Chen and M. Skowronek, _Covariant Compton Amplitudes in Gravity with Classical Spin_ , _Phys. Rev. Lett._ 132 (2024) 191603 [2309.11249].
  * [75] S.A. Teukolsky, _Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations_ , _Astrophys. J._ 185 (1973) 635.
  * [76] W.H. Press and S.A. Teukolsky, _Perturbations of a Rotating Black Hole. II. Dynamical Stability of the Kerr Metric_ , _Astrophys. J._ 185 (1973) 649.
  * [77] S.A. Teukolsky and W.H. Press, _Perturbations of a rotating black hole. III - Interaction of the hole with gravitational and electromagnet ic radiation_ , _Astrophys. J._ 193 (1974) 443.
  * [78] H.S. Chia, _Tidal deformation and dissipation of rotating black holes_ , _Phys. Rev. D_ 104 (2021) 024013 [2010.07300].
  * [79] Y.F. Bautista, A. Guevara, C. Kavanagh and J. Vines, _From Scattering in Black Hole Backgrounds to Higher-Spin Amplitudes: Part I_ , 2107.10179.
  * [80] M.M. Ivanov and Z. Zhou, _Vanishing of Black Hole Tidal Love Numbers from Scattering Amplitudes_ , _Phys. Rev. Lett._ 130 (2023) 091403 [2209.14324].
  * [81] Y.F. Bautista, A. Guevara, C. Kavanagh and J. Vines, _Scattering in black hole backgrounds and higher-spin amplitudes. Part II_ , _JHEP_ 05 (2023) 211 [2212.07965].
  * [82] M.V.S. Saketh, Z. Zhou and M.M. Ivanov, _Dynamical tidal response of Kerr black holes from scattering amplitudes_ , _Phys. Rev. D_ 109 (2024) 064058 [2307.10391].
  * [83] Y.F. Bautista, G. Bonelli, C. Iossa, A. Tanzini and Z. Zhou, _Black Hole Perturbation Theory Meets CFT 2: Kerr Compton Amplitudes from Nekrasov-Shatashvili Functions_, 2312.05965.
  * [84] M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, _The simplest massive S-matrix: from minimal coupling to Black Holes_ , _JHEP_ 04 (2019) 156 [1812.08752].
  * [85] M.-Z. Chung, Y.-T. Huang and J.-W. Kim, _Classical potential for general spinning bodies_ , _JHEP_ 09 (2020) 074 [1908.08463].
  * [86] M.-Z. Chung, Y.-t. Huang, J.-W. Kim and S. Lee, _Complete Hamiltonian for spinning binary systems at first post-Minkowskian order_ , _JHEP_ 05 (2020) 105 [2003.06600].
  * [87] W.-M. Chen, M.-Z. Chung, Y.-t. Huang and J.-W. Kim, _The 2PM Hamiltonian for binary Kerr to quartic in spin_ , _JHEP_ 08 (2022) 148 [2111.13639].
  * [88] R. Aoude, K. Haddad and A. Helset, _Searching for Kerr in the 2PM amplitude_ , _JHEP_ 07 (2022) 072 [2203.06197].
  * [89] R. Aoude, K. Haddad and A. Helset, _Classical Gravitational Spinning-Spinless Scattering at O(G2S $\infty$)_, _Phys. Rev. Lett._ 129 (2022) 141102 [2205.02809].
  * [90] Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban and F. Teng, _Binary Dynamics Through the Fifth Power of Spin at $\mathcal{O}(G^{2})$_, 2203.06202.
  * [91] Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, T. Scheopner, F. Teng et al., _Quantum Field Theory, Worldline Theory, and Spin Magnitude Change in Orbital Evolution_ , 2308.14176.
  * [92] G. Menezes and M. Sergola, _NLO deflections for spinning particles and Kerr black holes_ , _JHEP_ 10 (2022) 105 [2205.11701].
  * [93] F. Febres Cordero, M. Kraus, G. Lin, M.S. Ruf and M. Zeng, _Conservative Binary Dynamics with a Spinning Black Hole at O(G3) from Scattering Amplitudes_ , _Phys. Rev. Lett._ 130 (2023) 021601 [2205.07357].
  * [94] L. Cangemi, M. Chiodaroli, H. Johansson, A. Ochirov, P. Pichini and E. Skvortsov, _Kerr Black Holes Enjoy Massive Higher-Spin Gauge Symmetry_ , 2212.06120.
  * [95] K. Haddad, _Recursion in the classical limit and the neutron-star Compton amplitude_ , _JHEP_ 05 (2023) 177 [2303.02624].
  * [96] S. De Angelis, R. Gonzo and P.P. Novichkov, _Spinning waveforms from KMOC at leading order_ , 2309.17429.
  * [97] Y.-J. Chen, T. Hsieh, Y.-T. Huang and J.-W. Kim, _On-shell approach to (spinning) gravitational absorption processes_ , 2312.04513.
  * [98] L. Cangemi, M. Chiodaroli, H. Johansson, A. Ochirov, P. Pichini and E. Skvortsov, _From higher-spin gauge interactions to Compton amplitudes for root-Kerr_ , 2311.14668.
  * [99] L. Cangemi, M. Chiodaroli, H. Johansson, A. Ochirov, P. Pichini and E. Skvortsov, _Compton Amplitude for Rotating Black Hole from QFT_ , 2312.14913.
  * [100] J. Vines, _Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings_ , _Class. Quant. Grav._ 35 (2018) 084002 [1709.06016].
  * [101] A. Guevara, A. Ochirov and J. Vines, _Scattering of Spinning Black Holes from Exponentiated Soft Factors_ , _JHEP_ 09 (2019) 056 [1812.06895].
  * [102] D. Kosmopoulos and A. Luna, _Quadratic-in-spin Hamiltonian at $\mathcal{O}$(G2) from scattering amplitudes_, _JHEP_ 07 (2021) 037 [2102.10137].
  * [103] T. Scheopner and J. Vines, _Dynamical Implications of the Kerr Multipole Moments for Spinning Black Holes_ , 2311.18421.
  * [104] F. Alessio and P. Di Vecchia, _Radiation reaction for spinning black-hole scattering_ , _Phys. Lett. B_ 832 (2022) 137258 [2203.13272].
  * [105] A. Elkhidir, D. O’Connell, M. Sergola and I.A. Vazquez-Holm, _Radiation and Reaction at One Loop_ , 2303.06211.
  * [106] D. Bini, T. Damour and A. Geralico, _Comparing one-loop gravitational bremsstrahlung amplitudes to the multipolar-post-Minkowskian waveform_ , _Phys. Rev. D_ 108 (2023) 124052 [2309.14925].
  * [107] A. Brandhuber, G.R. Brown, G. Chen, J. Gowdy and G. Travaglini, _Resummed spinning waveforms from five-point amplitudes_ , _JHEP_ 02 (2024) 026 [2310.04405].
  * [108] A. Luna, N. Moynihan, D. O’Connell and A. Ross, _Observables from the Spinning Eikonal_ , 2312.09960.
  * [109] H. Lee and S. Lee, _Poincaré invariance of spinning binary dynamics in the post-Minkowskian Hamiltonian approach_ , _Class. Quant. Grav._ 40 (2023) 245004 [2305.10739].
  * [110] A. Buonanno, G.U. Jakobsen and G. Mogull, _Post-Minkowskian Theory Meets the Spinning Effective-One-Body Approach for Two-Body Scattering_ , 2402.12342.
  * [111] G.U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, _Gravitational Bremsstrahlung and Hidden Supersymmetry of Spinning Bodies_ , 2106.10256.
  * [112] G.U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, _SUSY in the sky with gravitons_ , _JHEP_ 01 (2022) 027 [2109.04465].
  * [113] G.U. Jakobsen and G. Mogull, _Linear response, Hamiltonian, and radiative spinning two-body dynamics_ , _Phys. Rev. D_ 107 (2023) 044033 [2210.06451].
  * [114] G.U. Jakobsen, G. Mogull, J. Plefka, B. Sauer and Y. Xu, _Conservative scattering of spinning black holes at fourth post-Minkowskian order_ , 2306.01714.
  * [115] G.U. Jakobsen, G. Mogull, J. Plefka and B. Sauer, _Dissipative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order_ , _Phys. Rev. Lett._ 131 (2023) 241402 [2308.11514].
  * [116] C. Shi and J. Plefka, _Classical double copy of worldline quantum field theory_ , _Phys. Rev. D_ 105 (2022) 026007 [2109.10345].
  * [117] F. Comberiati and C. Shi, _Classical Double Copy of Spinning Worldline Quantum Field Theory_ , _JHEP_ 04 (2023) 008 [2212.13855].
  * [118] R. Gonzo and C. Shi, _Boundary to bound dictionary for generic Kerr orbits_ , _Phys. Rev. D_ 108 (2023) 084065 [2304.06066].
  * [119] R. Gonzo and C. Shi, _Scattering and bound observables for spinning particles in Kerr spacetime with generic spin orientations_ , 2405.09687.
  * [120] R. Aoude, K. Haddad, C. Heissenberg and A. Helset, _Leading-order gravitational radiation to all spin orders_ , _Phys. Rev. D_ 109 (2024) 036007 [2310.05832].
  * [121] J.-H. Kim, J.-W. Kim and S. Lee, _Massive twistor worldline in electromagnetic fields_ , 2405.17056.
  * [122] A. Brandhuber, G.R. Brown, P. Pichini, G. Travaglini and P.V. Matasan, _Spinning binary dynamics in cubic effective field theories of gravity_ , 2405.13826.
  * [123] Z. Bern, E. Herrmann, R. Roiban, M.S. Ruf, A.V. Smirnov, V.A. Smirnov et al., _Amplitudes, Supersymmetric Black Hole Scattering at $\mathcal{O}(G^{5})$, and Loop Integration_, 2406.01554.
  * [124] G. Kälin, Z. Liu and R.A. Porto, _Conservative Dynamics of Binary Systems to Third Post-Minkowskian Order from the Effective Field Theory Approach_ , _Phys. Rev. Lett._ 125 (2020) 261103 [2007.04977].
  * [125] G. Kälin, Z. Liu and R.A. Porto, _Conservative Tidal Effects in Compact Binary Systems to Next-to-Leading Post-Minkowskian Order_ , _Phys. Rev. D_ 102 (2020) 124025 [2008.06047].
  * [126] M.M. Riva and F. Vernizzi, _Radiated momentum in the post-Minkowskian worldline approach via reverse unitarity_ , _JHEP_ 11 (2021) 228 [2110.10140].
  * [127] C. Dlapa, G. Kälin, Z. Liu and R.A. Porto, _Conservative Dynamics of Binary Systems at Fourth Post-Minkowskian Order in the Large-Eccentricity Expansion_ , _Phys. Rev. Lett._ 128 (2022) 161104 [2112.11296].
  * [128] W.D. Goldberger, J. Li and I.Z. Rothstein, _Non-conservative effects on spinning black holes from world-line effective field theory_ , _JHEP_ 06 (2021) 053 [2012.14869].
  * [129] G. Kälin and R.A. Porto, _Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics_ , 2006.01184.
  * [130] D. Kosmopoulos and M.P. Solon, _Gravitational self force from scattering amplitudes in curved space_ , _JHEP_ 03 (2024) 125 [2308.15304].
  * [131] C. Cheung, J. Parra-Martinez, I.Z. Rothstein, N. Shah and J. Wilson-Gerow, _Effective Field Theory for Extreme Mass Ratios_ , 2308.14832.
  * [132] J. Wilson-Gerow, _Conservative scattering of Reissner-Nordström black holes at third post-Minkowskian order_ , _JHEP_ 05 (2024) 265 [2310.17731].
  * [133] G.U. Jakobsen, _Spin and Susceptibility Effects of Electromagnetic Self-Force in Effective Field Theory_ , _Phys. Rev. Lett._ 132 (2024) 151601 [2311.04151].
  * [134] A. Klemm, C. Nega, B. Sauer and J. Plefka, _CY in the Sky_ , 2401.07899.
  * [135] N. Arkani-Hamed, Y.-t. Huang and D. O’Connell, _Kerr black holes as elementary particles_ , _JHEP_ 01 (2020) 046 [1906.10100].
  * [136] H. Johansson and A. Ochirov, _Double copy for massive quantum particles with spin_ , _JHEP_ 09 (2019) 040 [1906.12292].
  * [137] A. Brandhuber, G. Chen, H. Johansson, G. Travaglini and C. Wen, _Kinematic Hopf Algebra for Bern-Carrasco-Johansson Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory_ , _Phys. Rev. Lett._ 128 (2022) 121601 [2111.15649].
  * [138] A. Brandhuber, G.R. Brown, G. Chen, J. Gowdy, G. Travaglini and C. Wen, _Amplitudes, Hopf algebras and the colour-kinematics duality_ , _JHEP_ 12 (2022) 101 [2208.05886].
  * [139] G. Chen, G. Lin and C. Wen, _Kinematic Hopf algebra for amplitudes and form factors_ , _Phys. Rev. D_ 107 (2023) L081701 [2208.05519].
  * [140] G. Chen, L. Rodina and C. Wen, _Kinematic Hopf algebra for amplitudes from higher-derivative operators_ , _JHEP_ 02 (2024) 096 [2310.11943].
  * [141] G. Chen, L. Rodina and C. Wen, _Kinematic Hopf algebra and BCJ numerators at finite $\alpha^{\prime}$_, 2403.04614.
  * [142] N.E.J. Bjerrum-Bohr, G. Chen, Y. Miao and M. Skowronek, _Color-Kinematic Numerators for Fermion Compton Amplitudes_ , 2404.15265.
  * [143] G. Chen, H. Johansson, F. Teng and T. Wang, _On the kinematic algebra for BCJ numerators beyond the MHV sector_ , _JHEP_ 11 (2019) 055 [1906.10683].
  * [144] G. Chen, H. Johansson, F. Teng and T. Wang, _Next-to-MHV Yang-Mills kinematic algebra_ , _JHEP_ 10 (2021) 042 [2104.12726].
  * [145] Y.F. Bautista, _Dynamics for Super-Extremal Kerr Binary Systems at ${\cal O}(G^{2})$_, 2304.04287.
  * [146] S.R. Dolan, _Scattering and Absorption of Gravitational Plane Waves by Rotating Black Holes_ , _Class. Quant. Grav._ 25 (2008) 235002 [0801.3805].
  * [147] J.-W. Kim and J. Steinhoff, _Spin supplementary condition in quantum field theory: covariant SSC and physical state projection_ , _JHEP_ 07 (2023) 042 [2302.01944].
  * [148] G.U. Jakobsen, G. Mogull, J. Plefka and B. Sauer, _All things retarded: radiation-reaction in worldline quantum field theory_ , _JHEP_ 10 (2022) 128 [2207.00569].
  * [149] V. Witzany, _Hamilton-Jacobi equation for spinning particles near black holes_ , _Phys. Rev. D_ 100 (2019) 104030 [1903.03651].
  * [150] N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Planté and P. Vanhove, _Classical gravity from loop amplitudes_ , _Phys. Rev. D_ 104 (2021) 026009 [2104.04510].
  * [151] N.E.J. Bjerrum-Bohr, P.H. Damgaard, L. Planté and P. Vanhove, _The Amplitude for Classical Gravitational Scattering at Third Post-Minkowskian Order_ , 2105.05218.
  * [152] “Kinematic hopf algebra.”
  * [153] R.N. Lee, _Presenting LiteRed: a tool for the Loop InTEgrals REDuction_ , 1212.2685.
  * [154] R.N. Lee, _LiteRed 1.4: a powerful tool for reduction of multiloop integrals_ , _J. Phys. Conf. Ser._ 523 (2014) 012059 [1310.1145].
  * [155] L. Bohnenblust, L. Cangemi, H. Johansson and P. Pichini, _to appear_ , .