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#  Systematic integral evaluation for spin-resummed binary dynamics

Gang Chen<EMAIL_ADDRESS>Niels Bohr International Academy, Niels Bohr
Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark  Jung-Wook Kim (김정욱)  jung-
<EMAIL_ADDRESS>Max Planck Institute for Gravitational Physics (Albert
Einstein Institute),
Am Mühlenberg 1, D-14476 Potsdam, Germany  Tianheng Wang
<EMAIL_ADDRESS>Center for Theoretical Physics, Seoul National
University,
1 Gwana-ro, Gwanak-gu, 08826, Seoul, South Korea

###### Abstract

Computation of spin-resummed observables in post-Minkowskian dynamics
typically involve evaluation of Feynman integrals deformed by an exponential
factor, where the exponent is a linear sum of the momenta being integrated.
Such integrals can be viewed as tensor integral generating functions, which
provide alternative approaches to tensor reduction of Feynman integrals. We
develop a systematic method to evaluate tensor integral generating functions
using _conventional_ multiloop integration techniques. The spin-resummed
aligned-spin eikonal at second post-Minkowskian order is considered as a
phenomenologically relevant example where evaluation of tensor integral
generating functions is necessary.

Scattering amplitudifferential equations, heavy-mass effective theory,
gravitational waves, classical spin

††preprint:  SNUTP24-003

##  I Introduction

Inspired by quantum field theoretic approaches to gravity Iwasaki (1971);
Donoghue (1994); Bjerrum-Bohr  _et al._ (2003); Holstein and Donoghue (2004);
Holstein and Ross (2008); Neill and Rothstein (2013); Bjerrum-Bohr  _et al._
(2014); Damour (2018), modern methods to compute scattering amplitude have
been widely applied to the study of classical dynamics of binary black hole
scattering, from conservative dynamics Bjerrum-Bohr  _et al._ (2018); Cheung
_et al._ (2018); Cristofoli  _et al._ (2019); Bern  _et al._ (2019a, b);
Parra-Martinez  _et al._ (2020); Di Vecchia  _et al._ (2020, 2021a, 2021b);
Herrmann  _et al._ (2021); Bjerrum-Bohr  _et al._ (2021a, b); Damgaard  _et
al._ (2021); Brandhuber  _et al._ (2021a); Bjerrum-Bohr  _et al._ (2022a, b);
Bern  _et al._ (2022a, b, 2021a); Damgaard  _et al._ (2023); Di Vecchia  _et
al._ (2023); Heissenberg (2023); Adamo and Gonzo (2023) and gravitational
bremsstrahlung Kosower  _et al._ (2019); Brandhuber  _et al._ (2023);
Herderschee  _et al._ (2023); Elkhidir  _et al._ (2023); Georgoudis  _et al._
(2023); Caron-Huot  _et al._ (2023); Bohnenblust  _et al._ (2023); Bini  _et
al._ (2023); Georgoudis  _et al._ (2024a); Adamo  _et al._ (2024); Bini  _et
al._ (2024); Alessio and Di Vecchia (2024); Georgoudis  _et al._ (2024b);
Brunello and De Angelis (2024) to effects of spin Guevara (2019); Arkani-Hamed
_et al._ (2021); Guevara  _et al._ (2019a); Chung  _et al._ (2019); Guevara
_et al._ (2019b); Arkani-Hamed  _et al._ (2020); Aoude  _et al._ (2020); Chung
_et al._ (2020); Guevara  _et al._ (2021); Chen  _et al._ (2022a); Kosmopoulos
and Luna (2021); Chiodaroli  _et al._ (2022); Bautista  _et al._ (2021);
Cangemi  _et al._ (2022); Ochirov and Skvortsov (2022); Damgaard  _et al._
(2019); Bern  _et al._ (2021b); Comberiati and de la Cruz (2022); Maybee  _et
al._ (2019); Haddad (2022); Chen  _et al._ (2022b); Menezes and Sergola
(2022); Febres Cordero  _et al._ (2023); Alessio and Di Vecchia (2022); Bern
_et al._ (2022c); Aoude  _et al._ (2022a, b, 2023); Bautista (2023); De
Angelis  _et al._ (2023); Brandhuber  _et al._ (2024); Aoude  _et al._ (2024);
Riva  _et al._ (2022); Luna  _et al._ (2023) (see also refs. Antonelli  _et
al._ (2019); Damour (2020); Kälin  _et al._ (2020); Dlapa  _et al._ (2022a, b,
c); Jakobsen  _et al._ (2023, 2021a); Mougiakakos  _et al._ (2021); Jakobsen
_et al._ (2021b); Mogull  _et al._ (2021); Jakobsen  _et al._ (2022); Vines
(2018); Vines  _et al._ (2019); Liu  _et al._ (2021); Jakobsen and Mogull
(2022); Damgaard  _et al._ (2022); Bianchi  _et al._ (2023); Gonzo and Shi
(2024) for related approaches), motivated in part by the detection of
gravitational waves by the LIGO-Virgo collaboration Abbott  _et al._ (2016).
The relevant perturbative expansion is the post-Minkowskian (PM) expansion,
where general relativistic corrections are added to the unperturbed special
relativistic kinematics. The study of PM dynamics is more than a theoretical
interest, as they can be used in building waveform models for gravitational
wave detection Buonanno  _et al._ (2024).

The typical integrals encountered in the study of PM dynamics can be
formulated as Feynman integrals, which can be handled efficiently by modern
amplitude techniques such as the integration-by-parts (IBP) reduction Smirnov
(2008); Lee (2015); Larsen and Zhang (2016) and the method of differential
equations Gehrmann and Remiddi (2000); Kotikov (1991); Henn (2013). The
function space of the master integrals and their associated graph topologies
were studied for the conservative scattering dynamics at 4PM and 5PM orders
Bern  _et al._ (2022a); Dlapa  _et al._ (2022a); Damgaard  _et al._ (2023);
Frellesvig  _et al._ (2024a); Driesse  _et al._ (2024); Klemm  _et al._
(2024); Frellesvig  _et al._ (2024b). The function space remains practically
the same when spin effects are incorporated perturbatively into the non-
spinning dynamics.

However, the function space becomes completely different when _exact_ spin
dependence—or spin resummation—is considered. The motivation for studying spin
resummation in binary Kerr dynamics is that gravitational wave physics is
becoming a precision science; one potential obstruction for high-precision
gravitational wave physics is inaccurate modelling of spin effects in
gravitational waveform models Dhani  _et al._ (2024), which needs to be
improved with better understanding of spin effects in gravitational two-body
dynamics.

The spin-induced multipole moments of black holes can be generated by the
Newman-Janis shift Newman and Janis (1965); Arkani-Hamed  _et al._ (2020),
where the position of the black hole is complexified and shifted in the
imaginary spin direction. The translation operator in the imaginary direction
is $\exp(i\alpha\cdot\mathbf{\nabla})=\exp(\alpha\cdot K)$, and the same
factor appears when the Newman-Janis shift is promoted to a dynamical
statement Kim  _et al._ (2024). In particular, the typical Feynman integrals
encountered in the context of PM dynamics are modified by exponential factors
corresponding to translations along the imaginary spin direction.

Such integrals can be considered in a more general setting as _tensor integral
generating functions_ (TIGFs), where a typical Feynman integrand is deformed
by an exponential factor of integration variables. The TIGFs provide an
alternative route to tensor reduction of Feynman integrals Feng (2023). More
importantly, consideration of such integrals allows us to treat the impact
parameter space Fourier transform on an equal footing with loop momenta
integration, and leads us to new insights on PM dynamics presented in impact
parameter space Brunello and De Angelis (2024). For example, the spurious
poles from the intermediate loop integral reduction, and computation of the
terms that contribute to the loop integrals but vanish under the impact
parameter space transform, can be avoided by performing IBP reduction at the
full integrand level.

Unfortunately, known multiloop integration techniques are not directly
applicable to evaluation of such modified Feynman integrals. We introduce a
method to tame the exponential factor and convert it into an extra delta
constraint, rendering the integrand suitable for application of _conventional_
multiloop integration techniques. Interestingly, the extra delta constraint
leads to significantly different function space of the master integrals when
compared to the original master integrals with the same topology.

As a phenomenologically important application, the developed method is used to
compute the 2PM spin-resummed eikonal for binary Kerr scattering. To simplify
the analysis we consider a generalisation of the aligned spin configuration,
where the spin vectors are aligned but allowed to have components along the
impact parameter. The minimal coupling three-point amplitude Arkani-Hamed  _et
al._ (2021) and the spin-resummed heavy-mass effective field theory (HEFT)
Compton amplitude Bjerrum-Bohr  _et al._ (2024, 2023) are used to build the
integrand from unitarity cuts Bern  _et al._ (1994, 1995); Bjerrum-Bohr  _et
al._ (2003) and heavy-mass/velocity cuts Brandhuber  _et al._ (2021a);
Bjerrum-Bohr  _et al._ (2021a, b).

##  II General methods for tensor integral generating functions

TIGFs are obtained from a typical loop integrand as a deformation by an
exponential factor, viz. 111Although generating functions are usually defined
by real exponents, imaginary exponents were used to avoid factors of $i$ when
solving for the integrals. Such an analytic continuation is known in the
probability literature as characteristic functions. This analytic continuation
is allowed since Feynman integrals should be understood as generalised
functions which may not be well-defined by conventional definitions of
integrals.

$\displaystyle\mathcal{I}^{(\alpha)}[\mathbf{y}]:=\int\prod_{j=1}^{L}d^{D}K_{j}\frac{e^{i(\sum_{j=1}^{L}\alpha_{j}\cdot
K_{j})}\Big{(}\prod\limits_{k=r+1}^{n}\delta^{\lambda_{k}{-}1}(\mathcal{D}_{k})\Big{)}}{\mathcal{D}_{1}^{\lambda_{1}}\cdots\mathcal{D}_{r}^{\lambda_{r}}}\,,$
(1)

where $\delta^{\lambda}(x)=\frac{d^{\lambda}}{dx^{\lambda}}\delta(x)$,
$(\alpha)=(\alpha_{1}^{\mu},\cdots,\alpha_{L}^{\mu})$ is the vector of
arguments, $\mathbf{y}$ is the vector of kinematic invariants and
$\mathcal{D}_{i}$ are the inverse propagators; $(K+p)^{2}-m^{2}$ or $2(K\cdot
p)$ when eikonalised 222The delta constraints can be converted to denominators
through the distributional identity $2\pi
i\,\delta(x)=(x-i0^{+})^{-1}-(x+i0^{+})^{-1}$ and its derivatives, therefore
they can be considered as propagator factors.. In the rest of the letter, we
set $D=4-2\epsilon$ for dimensional regularisation. Integrals with non-trivial
numerator factors can be obtained from TIGFs as derivative operators of
$\alpha_{j}^{\mu}$ acting on them, which is a less-explored tensor reduction
method for Feynman integrals Feng (2023). This alternative tensor reduction
approach may provide more efficient methods to reduce irreducible numerators,
which are one of the bottlenecks in evaluating Feynman integrals.

Some specialised techniques were developed for their reduction and evaluation,
including modification of IBP reduction to incorporate the extra exponential
factor Feng (2023); Feng  _et al._ (2024); Li  _et al._ (2024); Brunello  _et
al._ (2024); Brunello and De Angelis (2024). We take a different route and
develop a systematic method to reduce and evaluate TIGFs using _conventional_
multiloop integration techniques. Our key proposal is converting TIGF
integrands (1) into a typical Feynman integrand; we introduce an auxiliary
parameter $t$ and Fourier transform the exponential factor into a delta
constraint,

$\displaystyle\mathcal{I}^{(\alpha)}[\mathbf{y}]$ $\displaystyle=\int
dte^{it}\tilde{\mathcal{I}}^{(\alpha)}[t,\mathbf{y}]\,,$ (2)

where $\tilde{\mathcal{I}}^{(\alpha)}[t,\mathbf{y}]$ is defined as

$\displaystyle\int\prod_{j=1}^{L}d^{D}K_{j}$
$\displaystyle\frac{\delta(\sum_{j=1}^{L}\alpha_{j}\cdot
K_{j}-t)\Big{(}\prod\limits_{k=r+1}^{s}\delta^{\lambda_{k}{-}1}(\mathcal{D}_{k})\Big{)}}{\mathcal{D}_{1}^{\lambda_{1}}\cdots\mathcal{D}_{r}^{\lambda_{r}}}\,.$
(3)

$\tilde{\mathcal{I}}^{(\alpha)}[t,\mathbf{y}]$ is a typical Feynman integrand,
and conventional IBP reduction can be applied to reduce it to a set of master
integrals

$\displaystyle\tilde{\mathcal{I}}_{1}^{(\alpha)}[t,\mathbf{y}]\,,\,\cdots\,,\,\tilde{\mathcal{I}}_{n}^{(\alpha)}[t,\mathbf{y}]\,.$
(4)

The conventional method of evaluating master integrals using differential
equations applies

$\displaystyle\begin{aligned}
\partial_{t}\tilde{\mathcal{I}}_{i}^{(\alpha)}[t,\mathbf{y}]&=\sum_{j=1}^{n}A_{ij}(t,\mathbf{y})\tilde{\mathcal{I}}_{j}^{(\alpha)}[t,\mathbf{y}]\,,\\\
\partial_{y}\tilde{\mathcal{I}}_{i}^{(\alpha)}[t,\mathbf{y}]&=\sum_{j=1}^{n}B_{ij}(t,\mathbf{y})\tilde{\mathcal{I}}_{j}^{(\alpha)}[t,\mathbf{y}]\,.\end{aligned}$
(5)

An important class of this problem is when the $t$-dependence factorises from
the $\mathbf{y}$-dependence so that the differential equations (5) decouple,

$\displaystyle A_{ij}=A_{ij}(t)\,,$ $\displaystyle
B_{ij}=B_{ij}(\mathbf{y})\,,$ (6)

and spin-resummed PM dynamics falls into this class. In such a case, the
$t$-dependence can be factored from the $\mathbf{y}$-dependence,

$\displaystyle\tilde{\mathcal{I}}_{i}^{(\alpha)}[t,\mathbf{y}]=f_{i}(t)\,\widehat{\mathcal{I}}_{i}^{(\alpha)}[\mathbf{y}]\,,$
(7)

and the $t$ Fourier integral of (2) factors from the loop integrals, which can
be absorbed by normalisation of the effective master integrals
$\widehat{\mathcal{I}}_{i}^{(\alpha)}[\mathbf{y}]$. The conventional
differential equation approach can be applied to the second set of equations
in (5) for evaluation of the effective master integrals
$\widehat{\mathcal{I}}_{i}^{(\alpha)}[\mathbf{y}]$. An observable is then
given as a linear combination of the effective master integrals.

In the rest of the letter, we apply the aforementioned techniques to the
evaluation of spin-resummed one-loop eikonal phase as a concrete
phenomenologically relevant example.

##  III Spin-resummed 2PM eikonal

The classical eikonal can be understood as the generator of scattering
observables Gonzo and Shi (2024); Kim  _et al._ (2024), which is computed as
the impact-parameter-space Fourier transform of the classical $2\to 2$ one-
loop amplitude Brandhuber  _et al._ (2021a). The one-loop HEFT amplitude with
classical spin Chen and Wang (2024) is our starting point

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a) {$v_{1}$}; \vertex[right=1.2cm of a] (f2) [HV]{H};
\vertex[right=1.cm of f2] (c){}; \vertex[above=1.6cm of a](ac){$v_{2}$};
\vertex[right=0.8cm of ac] (ad) [dot]{}; \vertex[right=0.7cm of ad] (f2c)
[dot]{}; \vertex[above=1.6cm of c](cc){}; \vertex[above=0.8cm of a] (cutL);
\vertex[right=2.0cm of cutL] (cutR); \vertex[right=0.4cm 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}}\begin{split}&M_{\rm
HEFT}(q,v_{1},v_{2},a_{1},a_{2})={-1\over 2!}\\\
&\int{d^{D}\ell_{1}\over(2\pi)^{D-1}}\Big{(}{\delta(2m_{2}\ell_{1}{\cdot}v_{2})\over\ell_{1}^{2}\ell_{2}^{2}}M_{a_{2}}(\ell_{1},v_{2})\\\
&\phantom{as}\times
M_{a_{2}}(\ell_{2},v_{2})\,M_{a_{1}}(\ell_{1},\ell_{2},v_{2})\Big{)}\,,\end{split}$
(8)

where $\ell_{2}=q-\ell_{1}$. $M_{a_{2}}$ and $M_{a_{1}}$ are the three-point
and four-point HEFT amplitudes provided in the appendix.

The key ingredient is the HEFT Compton amplitude Bjerrum-Bohr  _et al._ (2023,
2024), which—contrary to the minimal coupling Compton amplitude Arkani-Hamed
_et al._ (2021)—is free of unphysical singularities. However, the amplitude
has spurious singularities entering through the entire functions

$\displaystyle G_{1}(x_{1})G_{1}(x_{2})$ $\displaystyle={\sinh(x_{1})\over
x_{1}}{\sinh(x_{2})\over x_{2}}\,,$ (9) $\displaystyle G_{2}(x_{1},x_{2})$
$\displaystyle={1\over x_{2}}\Big{(}{\sinh(x_{12})\over
x_{12}}-\cosh(x_{2}){\sinh(x_{1})\over x_{1}}\Big{)}\,,$

where $x_{12}=x_{1}+x_{2}$ and $x_{i}=a_{1}{\cdot}\ell_{i}$.

Figure 1: $G_{1}(x_{1})G_{1}(x_{2})$ (left) and $G_{2}(x_{1},x_{2})$ (right)

As can be seen from the graph of the two functions in fig. 1, the apparent
singularities of (9) do not correspond to actual singularities. Nevertheless,
the spurious singularities can be problematic in the evaluation of the
integrals, especially when the integrations are performed numerically as the
singularities spoil numerical stability. We therefore introduce their integral
representations

$\displaystyle G_{1}(x_{1})G_{1}(x_{2})$
$\displaystyle=\int_{0}^{1}d\sigma_{1}d\sigma_{2}\cosh(\sigma_{1}x_{1})\cosh(\sigma_{2}x_{2})\,,$
$\displaystyle G_{2}(x_{1},x_{2})$
$\displaystyle=\int_{0}^{1}d\sigma_{1}d\sigma_{2}\Big{(}\sigma_{1}\sinh\left(\sigma_{1}x_{1}+\sigma_{1}\sigma_{2}x_{2}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\sinh\left(\sigma_{2}x_{2}\right)\cosh\left(\sigma_{1}x_{1}\right)\Big{)}\,,$
(10)

which removes spurious denominators at the price of auxiliary integrations
over $\sigma_{i}$. Analogous integral representations are used for $G_{1}$
functions appearing in the three-point amplitudes when constructing the
integrand (8).

The eikonal $\chi$ is obtained by transforming the HEFT amplitude to impact
parameter space

$\displaystyle\begin{aligned}
\chi&=\int{d^{D}q\over(2\pi)^{D-2}}{e^{iq{\cdot}b}\delta(q{\cdot}v_{1})\delta(q{\cdot}v_{2})\over
4m_{1}m_{2}}\\\ &\phantom{=asdfasdf}\times M_{\rm
HEFT}(q,v_{1},v_{2},a_{1},a_{2})\,.\end{aligned}$ (11)

We focus on a slight generalisation of the aligned spin configuration of a
binary Kerr system; $a^{\mu}\equiv a_{1}^{\mu}=\xi a_{2}^{\mu}$ and
$v_{1}{\cdot}a_{2}=v_{2}{\cdot}a_{1}=0$ with $a{\cdot}b\neq 0$.

Exchanging the order of integration to pull out the $\sigma_{i}$ integrals,
the eikonal can be schematically written as

$\displaystyle\chi=\int_{0}^{1}\prod_{j=1}^{4}d\sigma_{j}\sum_{\alpha}\mathcal{I}^{(\alpha)}[\mathbf{y}]\,,$
(12)

where

$\displaystyle y_{1}$ $\displaystyle=v_{1}{\cdot}v_{2}\,,$ $\displaystyle
y_{2}$ $\displaystyle={a{\cdot}a\over-b{\cdot}b}\,,$ $\displaystyle y_{3}$
$\displaystyle=b{\cdot}b\,,$ $\displaystyle y_{4}$
$\displaystyle={b{\cdot}a\over-b{\cdot}b}\,$ (13)

are the Lorentz scalars the eikonal depends on. The integral representation
(III) reorganises the original integrand as a sum over integrands of the form
(1) with additional numerator factors, which we classify into distinct
_sectors_ $(\alpha)$ defined by the exponential factor
$\exp(iq{\cdot}\hat{b}+i\ell_{1}{\cdot}\hat{a})$.

$\displaystyle(\alpha)$ $\displaystyle\equiv(\hat{b}\,,\,\hat{a})$ (14)
$\displaystyle\equiv(b+c^{\prime}_{1}(\sigma)\tilde{a}_{1}+c^{\prime}_{2}(\sigma)\tilde{a}_{2}\,,\,c^{\prime}_{3}(\sigma)\tilde{a}_{1}+c^{\prime}_{4}(\sigma)\tilde{a}_{2})\,,$
$\displaystyle(\mathbf{y})$
$\displaystyle=(y_{1}\,,\,\hat{y}_{2}\,,\,\hat{y}_{3}\,,\,\hat{y}_{4}\,,\,\sigma_{1}\,,\,\sigma_{2}\,,\,\sigma_{3}\,,\,\sigma_{4})\,,$
$\displaystyle(K_{1},K_{2})$ $\displaystyle=(q\,,\,\ell_{1})\,.$

The Lorentz scalars $\hat{y}_{i}$ are defined similar to (13) with hatted
variables $\hat{b},\hat{a}$. The tilded spin vectors
$\tilde{a}_{j}^{\mu}=ia_{j}^{\mu}$ are analytic continuation of the real
spin(-length) vectors of the black holes $a_{j}^{\mu}$. After evaluation, the
master integrals can be analytically continued back to real spin.

After employing the methods developed in the previous section, a typical
integrand for a given sector $(\alpha)$ takes the form

$\displaystyle\mathcal{I}^{(\alpha)}[\mathbf{y}]=\int_{-\infty}^{\infty}dte^{it}\int{d^{D}qd^{D}\ell_{1}\over(2\pi)^{2D-3}}\delta(q{\cdot}\hat{b}+\ell_{1}{\cdot}\hat{a}-t)$
(15)
$\displaystyle\times{\delta(q{\cdot}v_{1})\delta(q{\cdot}v_{2})\delta(\ell_{1}{\cdot}v_{2})\over
8m_{1}m^{2}_{2}\,\ell_{1}^{2}(q-\ell_{1})^{2}}\Big{(}\sum_{r=0}^{2}{N^{(r)}(\mathbf{y})\over(\ell_{1}{\cdot}v_{1})^{r}}+{N^{(3)}(\mathbf{y})\over
q^{2}}\Big{)}\,,$

where denominators of the integrand were kept explicit. The relevant
topologies are given in fig. 2. Each sector can be reduced to master integrals
by applying conventional multiloop integration techniques such as IBP
reduction.

$\lambda_{3}$$\lambda_{5}$$\lambda_{1}$$\lambda_{4}$$\lambda_{6}$$\lambda_{2}$$\lambda_{3}$$\lambda_{5}$$\lambda_{1}$$\lambda_{4}$$\lambda_{6}$$\lambda_{2}$$\lambda_{8}$

Figure 2: The graph topologies of master integrals, edges labelled by
$\lambda_{r}$ defined in (22). The dashed line is the delta constraint
originating from the exponential factor. The thick lines are remaining delta
constraints, while the thin lines are physical propagators.

All master integrals satisfy the factorisation condition (6). Moreover,
solving the differential equations in $t$ yields power-law $t$-dependence of
the master integrals. All terms resulting from reducing (15) have
$t$-dependence of the form

$\displaystyle{t^{j-4\epsilon}\over\epsilon}\,,$ $\displaystyle j\in[0,6]\,.$
(16)

As remarked, the contributions from the $t$ Fourier transform can be treated
as a constant factor. It is remarked that the divergent $\frac{1}{\epsilon}$
term in dimensional regularisation vanishes under the $t$ Fourier transform,
therefore the constant factor is finite.

An exemplary term in the $N^{(3)}$ category is

$\displaystyle\mathcal{I}_{ex}[\mathbf{y}]=\int{d^{D}qd^{D}\ell_{1}\over(2\pi)^{2D-3}}e^{iq{\cdot}b}{\delta(q{\cdot}v_{1})\delta(q{\cdot}v_{2})\delta(\ell_{1}{\cdot}v_{2})\over\ell_{1}^{2}(q-\ell_{1})^{2}q^{2}}$
(17) $\displaystyle\times\frac{(16\pi
G_{N})^{2}m_{1}m_{2}^{2}\left(\ell_{1}{\cdot}v_{1}\right){}^{2}\cosh\left((1-\xi)a_{1}{\cdot}q+2\xi
a_{1}{\cdot}\ell_{1}\right)}{16}\,.$

The $\cosh$ function splits the integrand into two sectors; sector
$(1)=(\hat{a}_{1},\hat{b}_{1})=(2\tilde{a}_{1}\xi\,,\,\tilde{a}_{1}(1-\xi)+b)$
and sector
$(2)=(\hat{a}_{2},\hat{b}_{2})=(-2\tilde{a}_{1}\xi\,,\,\tilde{a}_{1}(\xi-1)+b)$.
Each sector reduces to three master integrals by IBP reduction,

$\displaystyle\mathcal{I}^{(\alpha)}_{ex}[\mathbf{y}]=-(16\pi
G_{N})^{2}{\pi\over
2}\frac{m_{1}m_{2}^{2}(y_{1}^{2}-1)}{32\hat{y}_{2}\hat{y}_{3}}$ (18)
$\displaystyle\phantom{as}\times\Big{(}2\hat{y}_{3}\left(\hat{y}_{2}+2\hat{y}_{4}-1\right)\widehat{\mathcal{I}}_{1}^{(\alpha)}[\mathbf{y}]+2\widehat{\mathcal{I}}_{2}^{(\alpha)}[\mathbf{y}]+\widehat{\mathcal{I}}_{3}^{(\alpha)}[\mathbf{y}]\Big{)}\,,$

where the master integrals are defined by (21) and the superscript $(\alpha)$
denotes the sectors. (17) becomes

$\displaystyle\mathcal{I}_{ex}[\mathbf{y}]=\mathcal{I}^{(1)}_{ex}[\mathbf{y}]+\mathcal{I}^{(2)}_{ex}[\mathbf{y}]\,,$
(19)

where each $\mathcal{I}^{(\alpha)}_{ex}[\mathbf{y}]$ is complex but the sum
$\mathcal{I}_{ex}[\mathbf{y}]$ is real.

The integrals in (12) reduce to 320 sectors. Each sector has five master
integrals. However, only three master integrals per sector appear in the final
result,

$\displaystyle\chi$
$\displaystyle=\int_{0}^{1}\prod_{j=1}^{4}d\sigma_{j}\sum_{\alpha=1}^{320}\Big{(}c_{1,\alpha}(\mathbf{y})\widehat{\cal
I}^{(\alpha)}_{1}[\mathbf{y}]+c_{2,\alpha}(\mathbf{y})\widehat{\cal
I}^{(\alpha)}_{2}[\mathbf{y}]$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+c_{3,\alpha}(\mathbf{y})\widehat{\cal
I}^{(\alpha)}_{3}[\mathbf{y}]\Big{)}\,,$ (20)

where the master integrals are

$\displaystyle\widehat{\cal I}^{(\alpha)}_{1}[\mathbf{y}]$
$\displaystyle={D-4\over t^{2D-8}}J^{(\alpha)}_{1,1,1,1,1,1,0,0}\,,$
$\displaystyle\widehat{\cal I}^{(\alpha)}_{2}[\mathbf{y}]$
$\displaystyle={1\over t^{2D-10}}J^{(\alpha)}_{2,1,1,1,1,1,0,0}\,,$
$\displaystyle\widehat{\cal I}^{(\alpha)}_{3}[\mathbf{y}]$
$\displaystyle={1\over t^{2D-10}}J^{(\alpha)}_{1,1,1,1,1,1,0,1}\,,$ (21)

defined as

$\displaystyle
J^{(\alpha)}_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6},\lambda_{7},\lambda_{8}}\equiv\int{d^{D}qd^{D}\ell_{1}\over(2\pi)^{2D-3}}\delta^{\lambda_{6}-1}(\hat{a}{\cdot}\ell_{1}+\hat{b}{\cdot}q-t)$
$\displaystyle\phantom{asdfasdf}\times{\delta^{\lambda_{3}-1}(\ell_{1}{\cdot}v_{2})\delta^{\lambda_{4}-1}(q{\cdot}v_{1})\delta^{\lambda_{5}-1}(q{\cdot}v_{2})\over(\ell_{1}^{2})^{\lambda_{1}}((q-\ell_{1})^{2})^{\lambda_{2}}(\ell_{1}{\cdot}v_{1})^{\lambda_{7}}(q^{2})^{\lambda_{8}}}\,.$
(22)

The numerators $N^{(0,1,2)}$ of (12) reduce to $\widehat{\cal
I}^{(\alpha)}_{1,2}[\mathbf{y}]$, while the numerators $N^{(3)}$ of (12)
reduce to $\widehat{\cal I}^{(\alpha)}_{1,2,3}[\mathbf{y}]$. The integrals
$\widehat{\cal I}^{(\alpha)}_{1,2}[\mathbf{y}]$ correspond to the left
topology of fig. 2, while the integral $\widehat{\cal
I}^{(\alpha)}_{3}[\mathbf{y}]$ corresponds to the right topology of fig. 2.
The apparent $t$-dependence in (21) makes the master integrals $\widehat{\cal
I}^{(\alpha)}_{1,2,3}[\mathbf{y}]$ independent of $t$. As mentioned
previously, the other two master integrals, $J^{(\alpha)}_{1,1,1,2,1,1,0,0}$
and $J^{(\alpha)}_{1,1,1,1,1,1,1,0}$, never appear and their evaluation is
unnecessary for evaluating the eikonal. It is also remarked that both
$c_{j,\alpha}(\mathbf{y})$ and $\widehat{\cal I}^{(\alpha)}_{j}[\mathbf{y}]$
are finite in dimensional regularisation.

The final result for the eikonal phase is available in the public repository
KerrEikonal2pm Chen  _et al._ (2024). We emphasise that the result contains
all-orders-in-spin contributions for the generalised aligned spin
configuration with $a{\cdot}b\neq 0$.

##  IV Evaluation of the Master integrals

We evaluate the master integrals using the method of differential equations
Gehrmann and Remiddi (2000); Kotikov (1991); Henn (2013). The
$\hat{y}_{3}$-dependence of the master integrals can be determined from
dimensional analysis, since the only dimensionful variable is $\hat{y}_{3}$.
Moreover, the differential operator $\partial_{y_{1}}$ is diagonal in the
system of master integrals, and the $y_{1}$-dependence of the master integrals
can be solved separately.

The nontrivial systems of differential equations are given by $\hat{y}_{2,4}$.
The two systems are degenerate for the master integrals
$\widehat{\mathcal{I}}_{1,2}$ and they only depend on the combination
$\hat{y}_{2}^{\prime}=\hat{y}_{2}+2\hat{y}_{4}$. The differential equations
over $\hat{y}_{2}^{\prime}$ for $\widehat{\mathcal{I}}_{1,2}$ are solved by
complete elliptic integrals 333We adopt Mathematica’s definition for the
elliptic integrals.,

$\displaystyle\widehat{\mathcal{I}}_{1}[\mathbf{y}]$
$\displaystyle=\frac{C}{\sqrt{-\hat{y}_{3}}\sqrt{y_{1}^{2}-1}}\,\frac{K(\hat{y}_{2}^{\prime})}{\pi}\,,$
$\displaystyle\widehat{\mathcal{I}}_{2}[\mathbf{y}]$
$\displaystyle=-\frac{C\sqrt{-\hat{y}_{3}}}{\sqrt{y_{1}^{2}-1}}\,\frac{(\hat{y}_{2}^{\prime}-1)K(\hat{y}_{2}^{\prime})+E(\hat{y}_{2}^{\prime})}{\pi}\,,$
(23)

where the integration constant $C=-\frac{1}{16\pi^{2}}$ is fixed from the
spinless limit $\hat{y}_{2}^{\prime}\to 0$.

The differential equations satisfied by $\widehat{\mathcal{I}}_{3}$ are

$\displaystyle\partial_{\hat{y}^{\prime}_{2}}\widehat{\mathcal{I}}_{3}[\mathbf{y}]={-1\over
2\left(\hat{y}^{\prime}_{2}-2\hat{y}_{4}\right)\left(\hat{y}^{\prime}_{2}+\hat{y}^{2}_{4}-2\hat{y}_{4}\right)}\times$
$\displaystyle\Big{(}\hat{y}_{4}^{2}\widehat{\mathcal{I}}_{3}[\mathbf{y}]-\frac{C}{\pi}\frac{(\sqrt{-\hat{y}_{3}})}{\sqrt{y_{1}^{2}-1}}\left(\hat{y}^{\prime}_{2}+2\hat{y}_{4}\left(\hat{y}_{4}-1\right)\right)E\left(\hat{y}^{\prime}_{2}\right)$
$\displaystyle+\frac{C}{\pi}\frac{(\sqrt{-\hat{y}_{3}})}{\sqrt{y_{1}^{2}-1}}\left(1-\hat{y}_{4}\right)\left(\hat{y}^{\prime}_{2}-2\hat{y}_{4}\right)K\left(\hat{y}^{\prime}_{2}\right)\Big{)}\,,$
(24)
$\displaystyle\partial_{\hat{y}_{4}}\widehat{\mathcal{I}}_{3}[\mathbf{y}]={1\over\left(\hat{y}^{\prime}_{2}-2\hat{y}_{4}\right)\left(\hat{y}^{\prime}_{2}+\hat{y}^{2}_{4}-2\hat{y}_{4}\right)}\Big{(}\hat{y}_{4}\left(\hat{y}^{\prime}_{2}-\hat{y}_{4}\right)\widehat{\mathcal{I}}_{3}[\mathbf{y}]$
$\displaystyle-C\frac{(\sqrt{-\hat{y}_{3}})}{\sqrt{y_{1}^{2}-1}}\left(\hat{y}^{\prime}_{2}-1\right)\left(\hat{y}^{\prime}_{2}-2\hat{y}_{4}\right)K\left(\hat{y}^{\prime}_{2}\right)$
$\displaystyle-C\frac{(\sqrt{-\hat{y}_{3}})}{\sqrt{y_{1}^{2}-1}}\left(\hat{y}_{4}\hat{y}^{\prime}_{2}+\hat{y}^{\prime}_{2}-2\hat{y}_{4}\right)E\left(\hat{y}^{\prime}_{2}\right)\Big{)}\,.$
(25)

The common denominator factor in the two equations implies that the master
integral should be understood as an integral over the elliptic curve

$\displaystyle{\cal
Y}^{2}=(\hat{y}^{\prime}_{2}-2z)(z^{2}-2z+\hat{y}^{\prime}_{2})\,,$ (26)

where $z$ is the integration variable to be identified as $-\hat{y}_{4}$ after
integration. $\widehat{\mathcal{I}}_{3}[\mathbf{y}]$ is computed by solving
(24) at $\hat{y}_{4}=0$, which yields the boundary value for the remaining
integration over $\hat{y}_{4}$. Carrying out the $\hat{y}_{4}$ integration, we
have

$\displaystyle\widehat{\mathcal{I}}_{3}[\mathbf{y}]$
$\displaystyle=\frac{C(\sqrt{-\hat{y}_{3}})}{\sqrt{y_{1}^{2}-1}}\frac{\sqrt{\hat{y}^{\prime}_{2}+\hat{y}_{4}\left(\hat{y}_{4}-2\right)}}{\sqrt{\hat{y}^{\prime}_{2}-2\hat{y}_{4}}}\Bigg{[}\frac{2E(\hat{y}^{\prime}_{2})+\pi}{2\pi}$
$\displaystyle\phantom{=a}+\int_{0}^{\hat{y}_{4}}dz\Big{(}\frac{\left(1-\hat{y}^{\prime}_{2}\right)\left(\hat{y}^{\prime}_{2}-2z\right)K\left(\hat{y}^{\prime}_{2}\right)}{\pi\,{\cal
Y}\,(z^{2}-2z+\hat{y}^{\prime}_{2})}$ (27)
$\displaystyle\phantom{=asdfasdfasdf}-\frac{\left(\hat{y}^{\prime}_{2}z-2z+\hat{y}^{\prime}_{2}\right)E\left(\hat{y}^{\prime}_{2}\right)}{\pi\,{\cal
Y}\,(z^{2}-2z+\hat{y}^{\prime}_{2})}\Big{)}\Bigg{]}\,.$

The $z$-integral can be reduced by the IBP relations following from the exact
differentials

$\displaystyle d\left(\frac{\hat{y}^{\prime}_{2}-2z}{{\cal
Y}}\right)\,,\,d\left(\frac{(\hat{y}^{\prime}_{2}-2z)^{2}}{{\cal
Y}}\right)\,.$ (28)

Eliminating integrals with undesired denominators, we arrive at

$\displaystyle\widehat{\mathcal{I}}_{3}[\mathbf{y}]$
$\displaystyle=\,\frac{C\sqrt{-\hat{y}_{3}}}{\sqrt{y_{1}^{2}-1}}\Bigg{[}\frac{\sqrt{\hat{y}^{\prime}_{2}+\hat{y}_{4}^{2}-2\hat{y}_{4}}}{\pi\sqrt{\hat{y}^{\prime}_{2}-2\hat{y}_{4}}}\Big{(}{\pi-2K(\hat{y}^{\prime}_{2})\over
2}$
$\displaystyle-\left(E(\hat{y}^{\prime}_{2})-K(\hat{y}^{\prime}_{2})\right)\int_{0}^{\hat{y}_{4}}\frac{dz}{{\cal
Y}}-K(\hat{y}^{\prime}_{2})\int_{0}^{\hat{y}_{4}}\frac{zdz}{{\cal Y}}\Big{)}$
$\displaystyle+\,\frac{E(\hat{y}^{\prime}_{2})+\left(1-\hat{y}_{4}\right)K(\hat{y}^{\prime}_{2})}{\pi}\Bigg{]}\,,$
(29)

where the integrals can be converted to incomplete elliptic integrals, which
can be evaluated to arbitrary numerical precision. The integration constants
were fixed by requiring regularity of the spin expansion around
$\hat{a}^{\mu}=0$. Since $\hat{a}^{\mu}\to 0$ is a singular limit of the
underlying curve (26), we expand (IV) around $\hat{y}_{4}$ first and then
expand in $\hat{y}_{2}$ to obtain the series expansion of
$\widehat{\mathcal{I}}_{3}$. The master integrals
$\widehat{\mathcal{I}}_{1,2,3}$ were checked against brute-force calculations
based on the formulas in appendix B of ref. Kim  _et al._ (2024), and found to
agree up to $\mathcal{O}(\hat{a}^{20})$.

##  V Results

The eikonal phase (III) is exact to all orders in spin, and can be expanded in
spin before the $\sigma_{j}$-integration for a check against perturbative-in-
spin calculations. The spin-expanded $\sigma_{j}$ integrand turns out to be
polynomial in $\sigma_{j}$, and the auxiliary $\sigma_{j}$ integrals can be
evaluated exactly. We provide files for evaluating the spin-expanded
$\sigma_{j}$ integrand in the public repository KerrEikonal2pm Chen  _et al._
(2024). We have checked that (III) expanded to $\mathcal{O}(a^{8})$ is
consistent with the results of ref. Chen and Wang (2024).

Figure 3: The eikonal (III) as a function of ${|a|\over|b|}$ and
$\cos(\theta)=\frac{-a\cdot b}{|a||b|}$ with numerical errors $\sim 10^{-15}$.

The merit of the eikonal phase (III) is that we can study its full spin
dependence quantitatively, since the $\sigma_{j}$ integrations can be
numerically evaluated with high precision. As an example of numerically
studying the eikonal (III), we consider the scattering of a test Kerr black
hole on a Schwarzschild background, corresponding to the substitution

$\displaystyle\chi_{a_{2}=0}=\chi|_{\xi\rightarrow 0}\,,$ (30)

which is presented in fig. 3. The parameters $y_{1}={12\over 10}$ and
$y_{3}=-1$ are chosen for the ranges $|\sqrt{-y_{2}}|<0.75$ and
$\cos(\theta)\in[0,0.9]$, where $\theta$ denotes the angle between $a^{\mu}$
and $b^{\mu}$ such that $\sqrt{-y_{2}}\cos(\theta)=y_{4}$.

The numerical calculations can be tested against analytic predictions. For
example, the 2PM aligned-spin ($\cos(\theta)=0$) eikonal is expected to have
the singularity $\propto(|b|^{2}-|a|^{2})^{-3/2}$ based on analytic studies
Damgaard  _et al._ (2022); Aoude  _et al._ (2022a); Kim  _et al._ (2024). Fig.
4 shows numerical evaluation of (III) in the aligned-spin configuration
plotted against the reference curve $\frac{\beta}{(|b|^{2}-|a|^{2})^{3/2}}$,
where the constant $\beta$ was fit to (III) at $|b|=1.02\,|a|$ and other
unspecified parameters are the same as in fig. 3. The two lines on fig. 4 show
a good agreement near the singular point $|b|=|a|$, having $\lesssim 5\,\%$
relative difference for $|b|\leq 1.04\,|a|$. This singularity can be
interpreted as the dynamics detecting the ring singularity from spin
resummation, since the eikonal has a singularity only at $|b|=0$ when spin is
treated perturbatively.

$1$$1.05$$1.1$$1.15$$1.2$$5$$6$$7$$8$$9$$|b|/|a|$$\log(\frac{\chi}{G_{N}^{2}m_{1}m_{2}^{2}})$Aligned-
spin $\chi$ near singular pointeikonalreference Figure 4: Aligned-spin $\chi$
near the singular point $|b|=|a|$. The reference
$\frac{\beta}{(|b|^{2}-|a|^{2})^{3/2}}$ intersects at $|b|=1.02\,|a|$.

##  VI Conclusions and outlook

In this letter, a systematic method to reduce and evaluate TIGFs was
introduced and applied to the scattering dynamics of a binary Kerr system in
the generalised aligned-spin configuration. The corresponding 2PM eikonal
phase (III) was presented in a closed form, which can be studied analytically
by expanding to arbitrary orders in spin, or can be studied numerically
through numerical integrations for exact spin dependence.

An important application of TIGFs is generation of tensor integrals, directly
providing an alternative method of performing tensor reduction in Feynman
integrals Feng (2023). The approach may prove useful for reduction of
irreducible numerators, which in many cases becomes a bottleneck when
evaluation of multiloop Feynman integrals are involved. In this regard,
understanding the criteria for the factorisation of the $t$-dependence (e.g.
from intersection theory Cho and Matsumoto (1995); Mastrolia and Mizera
(2019); Frellesvig  _et al._ (2019); Brunello and De Angelis (2024)) will be
important, as the factorisation played a crucial role in obtaining closed-form
expressions for the TIGFs considered.

Another future direction to explore would be the study of function space
complexity for TIGFs. We have seen that elliptic integrals appear in effective
two-loop (2PM) topologies for TIGFs (see also ref. Kim  _et al._ (2024) for
electromagnetism). It is reasonable to expect that more complex functions
(e.g. elliptic multi-polylogarithms Broedel  _et al._ (2018), integrals over
Calabi-Yau manifolds Bourjaily  _et al._ (2022)) will appear in effective
three-loop (3PM) graph topologies when integrals are deformed into TIGFs.
Moreover, we can attempt to quantify the increase in transcendentality of the
function space when typical Feynman integrals are deformed into TIGFs. This
may even lead to an exploration of new geometries that have not yet been
associated with Feynman integrals.

Special cases of TIGFs appear ubiquitously in quantum-field-theory-inspired
approaches to classical gravitating systems, especially in the calculation of
scattering waveforms Brandhuber  _et al._ (2023); Herderschee  _et al._
(2023); Elkhidir  _et al._ (2023); Georgoudis  _et al._ (2023); Caron-Huot
_et al._ (2023); Bohnenblust  _et al._ (2023); Bini  _et al._ (2023);
Georgoudis  _et al._ (2024a); Adamo  _et al._ (2024); Bini  _et al._ (2024);
Alessio and Di Vecchia (2024); Georgoudis  _et al._ (2024b); Brunello and De
Angelis (2024). It would also be interesting to apply the developed methods to
these problems.

Coming back to the initial motivation for the study of TIGFs, it would be
interesting to apply the developed methods to 3PM all-orders-in-spin dynamics
and study spin resummation. Whether we have the exact three-graviton-Kerr
five-point amplitude—which is necessary for constructing the 3PM integrand—is
less relevant, as long as the conjectured five-point amplitude correctly
captures features of the dynamical Newman-Janis shift; we expect the
singularity structures of the binary Kerr dynamics to be governed by the
dynamical Newman-Janis shift, and correct singularity structures are the most
important when we attempt to resum the perturbative two-body dynamics Kim  _et
al._ (2024). Compared to the 2PM dynamics studied in this letter, the spin-
resummed 3PM dynamics is qualitatively different in that it is the first order
where next-to-leading-order effects in the mass-ratio expansion enters into
the dynamics Vines  _et al._ (2019), thereby including the first beyond
background-probe limit effects. The insights gained from studying singularity
structures of spin-resummed binary Kerr dynamics may motivate new resummation
schemes for spinning binary dynamics, providing a more accurate incorporation
of spin effects in waveform models used by gravitational wave observatories,
potentially having far-reaching consequences for astrophysics and multi-
messenger astronomy.

## Acknowledgements

It is a pleasure to thank Emil Bjerrum-Bohr, Andreas Brandhuber, Graham Brown,
Marcos Skowronek, Zhengwen Liu, Roger Morales, Gabriele Travaglini for
interesting conversations. JWK would like to thank Stefano De Angelis and Fei
Teng for stimulating discussions. The authors would like to thank Bo Feng,
Andres Luna, and Roger Morales for comments on the draft. 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.

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##  Appendix A Appendix: The Compton amplitude

The three point amplitude with one graviton momentum $p_{1}^{\mu}$ and
classical black hole spin $a^{\mu}$ Guevara  _et al._ (2019a); Chung  _et al._
(2019); Arkani-Hamed  _et al._ (2020); Aoude  _et al._ (2020); Johansson and
Ochirov (2019); Bjerrum-Bohr  _et al._ (2023) is

$\displaystyle M_{3}(p_{1}{\cdot}a)=\sqrt{32\pi
G_{N}}\,(\bar{p}{\cdot}\varepsilon_{1})(w_{1}{\cdot}\varepsilon_{1})\,,$ (31)

where $\bar{p}^{\mu}=mv^{\mu}$ denotes massive particle’s momentum,

$\displaystyle\begin{split}w_{1}^{\mu}&\coloneqq\cosh(p_{1}{\cdot}a)\,\bar{p}^{\mu}-i\frac{\sinh(p_{1}{\cdot}a)}{p_{1}{\cdot}a}(p_{1}{\cdot}S)^{\mu}\,,\end{split}$
(32)

and $(p_{1}{\cdot}S)^{\mu}=p_{1\nu}S^{\nu\mu}$. The $S^{\mu\nu}$ denotes the
spin tensor $S^{\mu\nu}=-\epsilon^{\mu\nu\rho\sigma}\bar{p}_{\rho}a_{\sigma}$.

The Compton amplitude with graviton momenta $p_{1,2}$ is constructed from the
double copy Bern  _et al._ (2008, 2010), kinematic Hopf algebra Brandhuber
_et al._ (2021b, 2022a, 2022b), and classical spin bootstrap Bjerrum-Bohr  _et
al._ (2024, 2023),

$\displaystyle{M_{4}\over 32\pi
G_{N}}=-\Big{(}\\!\frac{\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}}{p_{2}{\cdot}\bar{p}}\Big{)}\Bigg{(}\frac{w_{1}{\cdot}F_{1}{\cdot}F_{2}{\cdot}w_{2}}{2(p_{1}{\cdot}p_{2})(p_{1}{\cdot}\bar{p})}-\frac{p_{1}{\cdot}\bar{p}-p_{2}{\cdot}\bar{p}}{4(p_{1}{\cdot}p_{2})(p_{1}{\cdot}\bar{p})}\Big{(}iG_{2}\left(x_{1},x_{2}\right)(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}S{\cdot}p_{2})+iG_{2}(x_{1},x_{2})(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}S{\cdot}p_{1})$
$\displaystyle+iG_{1}\left(x_{12}\right)\text{tr}\left(F_{1}{\cdot}S{\cdot}F_{2}\right)+G_{1}(x_{1})G_{1}(x_{2})\big{(}(a{\cdot}F_{1}{\cdot}\bar{p})(a{\cdot}F_{2}{\cdot}p_{1})\\!-\\!(a{\cdot}F_{1}{\cdot}p_{2})(a{\cdot}F_{2}{\cdot}\bar{p})-\\!\frac{p_{2}{\cdot}\bar{p}\\!-\\!p_{1}{\cdot}\bar{p}}{2}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)\big{)}\Big{)}\Bigg{)}\,$
$\displaystyle+\\!{\Big{(}(\partial_{x_{1}}\\!-\\!\partial_{x_{2}})G_{1}(x_{1})G_{1}(x_{2})\Big{)}\over
4(\bar{p}{\cdot}p_{1})(\bar{p}{\cdot}p_{2})}\Big{(}\bar{p}{\cdot}p_{2}(\bar{p}^{2}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p})+a^{2}(\bar{p}_{4}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}))\\!-\\!(1\leftrightarrow
2)\\!\Big{)}$
$\displaystyle+\Big{(}{i(\partial_{x_{1}}\\!-\\!\partial_{x_{2}})G_{2}(x_{1},x_{2})\over
4(\bar{p}{\cdot}p_{1})(\bar{p}{\cdot}p_{2})}\Big{)}\Big{(}(\bar{p}{\cdot}p_{2})(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p})((a{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}\widetilde{F}_{1}{\cdot}\bar{p}){-}(a{\cdot}F_{1}{\cdot}\bar{p})(a{\cdot}\widetilde{F}_{2}{\cdot}\bar{p}))+(1\leftrightarrow
2)\Big{)}$
$\displaystyle+\\!\Big{(}\\!{(\partial_{x_{1}}\\!-\\!\partial_{x_{2}})^{2}\over
2!}{G_{1}(x_{1})}{G_{1}(x_{2})}\Big{)}\Big{(}(a{\cdot}F_{1}{\cdot}\bar{p})(a{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)\\!-\\!{a^{2}\over
2}((a{\cdot}F_{1}{\cdot}F_{2}{\cdot}p)(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p})-(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)(\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}))\\!\Big{)}$
$\displaystyle\\!\\!\\!+\Big{(}{i(\partial_{x_{1}}-\partial_{x_{2}})^{2}\over
2!}{G_{2}(x_{1},x_{2})}\Big{)}\Big{(}\\!-{1\over
2}\big{(}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)(a{\cdot}F_{2}{\cdot}\bar{p})(a{\cdot}\widetilde{F}_{1}{\cdot}\bar{p})\\!-\\!(1\leftrightarrow
2)\big{)}\Big{)}\,,$ (33)

where $x_{i}=a{\cdot}p_{i}$,
$F_{i}^{\mu\nu}=p^{\mu}_{i}\varepsilon^{\nu}_{i}-\varepsilon^{\mu}_{i}p^{\nu}_{i}$,
and
$\widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$.
The differences between this Compton amplitude and other proposals Cangemi
_et al._ (2023); Bautista  _et al._ (2023) stem from the differences in their
respective formalisms and assumptions Bjerrum-Bohr  _et al._ (2023, 2024),
which are inherited by the one-loop amplitude Chen and Wang (2024). The
recursion relation for the integral representation of the $G$ functions is

$\displaystyle G_{1}(x_{1})$
$\displaystyle{=}\int_{0}^{1}d\sigma_{1}\cosh(\sigma_{1}x_{1})\,,$ (34)
$\displaystyle G_{2}(x_{1},x_{2})$
$\displaystyle{=}\int_{0}^{1}d\sigma_{2}\Big{[}\partial_{x_{1}}G_{1}(x_{1}+\sigma_{2}x_{2})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\sinh(\sigma_{2}x_{2})\cosh(\sigma_{1}x_{1})\Big{]}\,,$ (35) $\displaystyle
G_{r}(x_{1},...,x_{r})$
$\displaystyle{=}{\int_{0}^{1}}d\sigma_{r}\Big{[}\partial_{x_{1}}G_{r-1}(x_{1}{+}\sigma_{r}x_{r},x_{2},...,x_{r-1})$
$\displaystyle-G_{r-1}(x_{1},...,x_{r-1})\sinh(\sigma_{r}x_{r})\Big{]}\,.$
(36)