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QMUL-PH-23-18

# Resummed spinning waveforms from five-point amplitudes

Andreas Brandhuber$\mbox{}^{a}$ Graham R. Brown$\mbox{}^{a}$ Gang
Chen$\mbox{}^{b}$,
Joshua Gowdy$\mbox{}^{a}$ and Gabriele Travaglini$\mbox{}^{a}$

###### Abstract

We compute the classical tree-level five-point amplitude for the two-to-two
scattering of spinning celestial objects with the emission of a graviton.
Using this five-point amplitude, we then turn to the computation of the
leading-order time-domain gravitational waveform. The method we describe is
suitable for arbitrary values of classical spin of Kerr black holes and does
not require any expansion in powers of the spin. In this paper we illustrate
it in the simpler case of the scattering of one Kerr and one Schwarzschild
black hole. An important ingredient of our calculation is a novel form of the
Compton amplitude with spinning particles including contact terms derived from
matching to black-hole perturbation theory calculations. This ensures that our
waveform is valid up to at least fourth order in the spin. Our method can be
applied immediately to generate improved waveforms once higher-order contact
terms in the Compton amplitude become available. Finally, we show the formula
for the gravitational memory to all orders in the spin, which is in agreement
with our results.

###### Contents

  1. 1 Introduction
  2. 2 Kinematics of the scattering and spin variables
  3. 3 Classical gravitational Compton amplitude with spin
    1. 3.1 Three-point amplitude
    2. 3.2 The Compton amplitude
  4. 4 Spinning five-point amplitude
  5. 5 The time-domain waveform
    1. 5.1 Waveforms from amplitudes
    2. 5.2 A scalar warm-up
    3. 5.3 General expression of the time-domain waveform for arbitrary spins
  6. 6 The waveform from the scattering of a Schwarzschild and a Kerr black hole
    1. 6.1 The $q_{1}^{2}$-channel
    2. 6.2 The $q_{2}^{2}$-channel
    3. 6.3 Discussion of the resummed spin waveform
  7. 7 Comparison with the spin-expanded integrand
  8. 8 Gravitational memory
    1. 8.1 General strategy
    2. 8.2 Four-point two-to-two spinning amplitude
    3. 8.3 Fourier transform to impact parameter space
    4. 8.4 Result for the gravitational memory
  9. A Simplifying the four-point amplitude
  10. B More on the integrand

##  1 Introduction

Since the first direct observation of gravitational waves [1, 2, 3, 4, 5], a
flurry of observations and theoretical predictions have greatly advanced the
fields of black-hole physics and general relativity. Important questions
regarding the intrinsic properties of black holes, the dynamics of binary
black-hole processes, and more, can all be investigated in depth through high-
precision gravitational-wave observations and theoretical calculations.

One widely used and highly successful analytical tool for the study of binary
black-hole systems is the Post-Newtonian (PN) expansion [6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39] and the effective one-body formulation
[40, 41, 42, 43, 44, 45]. Recently, several varieties of modern methods, e.g.
the double copy [46, 47, 48, 49, 50, 51, 52, 53], the Kosower-Maybee-O’Connell
(KMOC) formalism [54], heavy-mass effective theories [55, 56, 57, 58, 59, 60,
61, 62], the eikonal approach [63, 64, 65, 66], velocity cuts and the
exponential representation of the $S$-matrix [67, 68, 69], worldline effective
theory [70, 71, 72, 73, 74, 75] and worldline quantum field theory [76, 77,
78], have emerged as powerful theoretical frameworks for studying binary
black-hole physics to high Post-Minkowskian (PM) order from different points
of view. In particular they have been successfully applied to compute the
conservative part of the binary dynamics of gravitationally interacting
systems [79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 67, 89, 90, 91, 92, 93, 94,
95, 96, 97, 98, 99, 100, 69, 101, 102, 103, 104, 105, 106, 107] to high orders
in the PM expansion.

Research directly focused on the gravitational waveforms of binary black-hole
systems in the PM expansion is evolving rapidly. The tree-level waveforms for
spinless objects were computed in [108, 109, 110] and reproduced in [77, 111]
in the worldline picture. The tree-level waveform was studied in [112, 113]
using the scattering-amplitude based KMOC formalism [54, 112] and investigated
using the eikonal approach in [64, 63]. At one loop, the study of the
gravitational waveform was initiated recently in [62, 114, 115, 116] where the
principal value contribution was obtained and shown to be consistent between
KMOC and a heavy-mass effective field theory (HEFT) framework. The remaining
terms beyond this principal value part were pointed out in [117] and shown to
give an additional contribution to the waveform. The existence of such terms
was also suggested by comparing with the Multipolar-Post-Minkowskian waveform
in [118].

Gravitational waveforms are influenced by various intrinsic properties of
black holes. One of the most significant factors among them is their spin. An
important building block for including spin effects in waveforms is the
minimal coupling between a classical spinning black hole and a graviton
obtained using the massive spinor-helicity formalism [119]. Further important
developments made use of spinor helicity [120, 121, 122, 123, 124, 125, 56,
126, 127, 128, 129, 130, 131, 132, 133, 134, 135], the covariant amplitude
form [136, 137, 138, 55, 139, 43, 44, 140, 141, 142, 143], gravitational
solutions [144, 145, 146, 147, 148], and the worldline picture [106, 149, 76,
77, 103, 150]. At tree level, the spin contribution to the waveform up to
quadratic order was obtained in [151, 152] using a worldline effective theory.

In this paper, by employing the definition of waveforms in terms of five-point
amplitudes [112], we compute gravitational waveforms involving spinning black
holes, crucially without the need to expand in their spin. The building blocks
entering the recursive BCFW construction [153, 154] of the five-point
amplitude, adapted to the classical amplitude [62], are the three-point and
four-point Compton amplitudes with massive particles of arbitrary classical
spin, which were constructed in [155, 156] using a bootstrap technique which
makes use of entire functions. After expanding in spin, this form of this
Compton amplitude agrees with results obtained from black-hole perturbation
theory [145, 142, 134] for Kerr black holes up to at least fourth order in
spin.

In this work we mainly focus on the time-domain waveform. First, we perform
the Fourier transform over the frequency; the exponential factors in the
spinning amplitude then produce a simple extra delta function when
transforming to impact parameter space [151, 77, 114]. This additional delta
function localises the integral further and simplifies the tree-level
amplitude greatly. Finally, thanks to Cauchy’s theorem, the remaining one-
dimensional integral localises to contour integrals around physical poles
only. We stress here that our approach does not require any expansion in the
spin parameters. Importantly, this allows us to preserve the (partially)
resummed form of the Compton amplitude, and thus enables us to obtain a first
glimpse at large-spin effects in gravitational waveforms.

The rest of the paper is organised as follows. In the next section we
introduce the kinematics of the process, together with the definition of the
spin variables we employ. In Section 3 we introduce the three-point amplitude
and the Compton amplitude with spinning particles. These are then used in
Section 4 to construct the five-point amplitude of four massive spinning
particles with the emission of a gravitational wave, using a particular form
of BCFW recursion relation introduced in [62] for classical amplitudes. In
Section 5 we introduce the general method to compute the time-domain waveforms
and illustrate how this computation reduces to a sum of residues on physical
factorisation poles only, in the simpler case of spinless particles. We then
present the general expression of the waveform for arbitrary spins of the two
black holes. In Section 6 we specialise to the case of a Schwarzschild and a
Kerr black hole, and also present several plots of the waveforms for
increasing values of the spin of the Kerr black hole. In Section 7 we make
some interesting observations by comparing the waveforms obtained using the
resummed Compton amplitudes to those derived from the Compton amplitudes
expanded in the spin parameter. Section 8 presents a short derivation of the
memory of the gravitational wave in the spinning case, to all orders in the
spins of the celestial objects, which we have then used to test our analytic
results. Finally, two appendices complete the paper. In Appendix A we perform
some useful simplification of the expression of the four-point Compton
amplitude, which are convenient in the derivation of the memory; and in
Appendix B we list the coefficients appearing in the $q_{1}^{2}$\- and
$q_{2}^{2}$-channels of the classical, tree-level five-point amplitude derived
in Section 5.

The interested reader can find Mathematica notebooks with expressions for the
spinning HEFT amplitudes with one emitted graviton, and explicit time-domain
waveform results in the system of a Schwarzschild and a Kerr black hole
SpinningWaveform GitHub repository.

Note added: While preparing this manuscript we became aware of the nice work
[157], with which our paper has some overlap. We have checked that our results
agree with theirs.

##  2 Kinematics of the scattering and spin variables

Here we review the kinematics of the scattering of two heavy spinning
particles of masses $m_{1}$ and $m_{2}$ and spin vectors $a_{1}$ and $a_{2}$,
with the emission of a graviton of momentum $k$:

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}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{22.5pt}{0.0pt}\pgfsys@curveto{22.5pt}{12.42656pt}{12.42656pt}{22.5pt}{0.0pt}{22.5pt}\pgfsys@curveto{-12.42656pt}{22.5pt}{-22.5pt}{12.42656pt}{-22.5pt}{0.0pt}\pgfsys@curveto{-22.5pt}{-12.42656pt}{-12.42656pt}{-22.5pt}{0.0pt}{-22.5pt}\pgfsys@curveto{12.42656pt}{-22.5pt}{22.5pt}{-12.42656pt}{22.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@transformcm{0.89676}{0.0}{0.0}{0.89676}{0.0pt}{0.0pt}\pgfsys@invoke{
}\definecolor[named]{pgf@tempcol}{rgb}{.5,.5,.5}\definecolor[named]{pgf@tempcol}{rgb}{1,1,1}\definecolor[named]{pgf@tempcol}{rgb}{1,1,1}\definecolor{pgfshadetemp}{rgb}{0.5,0.5,0.5}\definecolor{pgfshadetemp}{rgb}{1,1,1}\definecolor{pgfshadetemp}{rgb}{1,1,1}\definecolor{pgfshadetemp}{rgb}{1,1,1}{}\lxSVG@sh@defs{}\lxSVG@sh@insert{-69.44pt}{-69.44pt}{\lxSVG@sh}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\end{array}$ (2.1)

As usual we have introduced barred variables, defined as [158, 87]

$\displaystyle\begin{split}p_{1}&=\bar{p}_{1}+\frac{q_{1}}{2}\,,\qquad
p_{1}^{\prime}=\bar{p}_{1}-\frac{q_{1}}{2}\,,\\\
p_{2}&=\bar{p}_{2}+\frac{q_{2}}{2}\,,\qquad
p_{2}^{\prime}=\bar{p}_{2}-\frac{q_{2}}{2}\,,\end{split}$ (2.2)

which satisfy

$\displaystyle\bar{p}_{1}{\cdot}q_{1}=\bar{p}_{2}{\cdot}q_{2}=0\,.$ (2.3)

We also introduce barred masses,

$\displaystyle\bar{m}_{i}^{2}\coloneqq\bar{p}_{i}^{2}=m_{i}^{2}-\frac{q_{i}^{2}}{4}\,,\qquad
i=1,2\,,$ (2.4)

with the HEFT expansion being organised in powers of the $\bar{m}_{i}$.

To parameterise the scattering process we choose five independent Lorentz-
invariant quantities as in [62],

$\begin{split}y\coloneqq v_{1}{\cdot}v_{2}\geq 1\ ,\qquad q_{i}^{2}\leq 0\
,\qquad w_{i}\coloneqq v_{i}{\cdot}k\geq 0\,,\qquad i=1,2,\end{split}$ (2.5)

where the four-velocities are defined from $p_{i}{=}m_{i}v_{i}$, with
$v_{i}^{2}{=}1$. We also note that $y$ is the relativistic factor
$\frac{1}{\sqrt{1-v_{\rm rel}^{2}}}$, where $v_{\rm rel}$ is the relative
velocity of one of the two heavy objects in the rest frame of the other. We
will also be using the barred versions
$\bar{w}_{i}\coloneqq\bar{v}_{i}{\cdot}k$ and
$\bar{y}\coloneqq\bar{v}_{1}{\cdot}\bar{v}_{2}$ of the above quantities, with
$\bar{p}_{i}\coloneqq\bar{m}_{i}\bar{v}_{i}$ and $\bar{v_{i}}^{2}=1$.

The spin tensors for incoming and outgoing massive particles in terms of the
spin vectors $s_{i}$ are given by:

$\displaystyle\begin{split}&S_{i}^{\sigma\nu}(p_{i})=-\frac{1}{m_{i}}\epsilon^{\rho\nu\alpha\beta}p_{i\,\alpha}s_{i\,\beta}(p_{i}),\qquad
S_{i}^{\sigma\nu}(p_{i}^{\prime})=-\frac{1}{m_{i}}\epsilon^{\sigma\nu\alpha\beta}p^{\prime}_{i\,\alpha}s_{i\,\beta}(p_{i}^{\prime})\,\end{split}$
(2.6)

To expand this in the heavy-mass limit we change variables from
$p_{i},p_{i}^{\prime}$ to $\bar{p}_{i}$ and $q_{i}$ as in (2.2). We follow the
method of [125] and use an infinitesimal Lorentz transformation from
$\bar{p}_{i}$ to $\bar{p}_{i}\pm\dfrac{q}{2}$ to write

$\displaystyle s_{i}^{\mu}\big{(}\bar{p}_{i}\pm\frac{q}{2}\big{)}$
$\displaystyle=(\delta_{\nu}^{\mu}\pm{\omega^{\mu}}_{\nu})s_{i}(\bar{p}_{i})^{\nu}$
$\displaystyle=\Big{[}\delta_{\nu}^{\mu}\mp\frac{1}{2\bar{m}^{2}}(\bar{p}_{i}^{\mu}q_{\nu}-q^{\mu}\bar{p}_{i\,\nu})\Big{]}s_{i}(\bar{p}_{i})^{\nu}$
(2.7)
$\displaystyle=s_{i}(\bar{p}_{i})^{\mu}\mp\frac{\bar{p}_{i}^{\mu}}{2\bar{m}^{2}}q{\cdot}s_{i}(\bar{p}_{i})+\mathcal{O}(\bar{m}^{-2})\,.$

This is valid since $\bar{m}_{i}$ (which will eventually be the classical
mass) is much larger than the typical value of $q$. This lets us expand the
spin tensors as

$\displaystyle\begin{split}S_{i}^{\sigma\nu}(p)&=-\frac{1}{\bar{m}_{i}}\epsilon^{\sigma\nu\alpha\beta}\bar{p}_{i\,\alpha}s_{i\,\beta}(\bar{p})-\frac{1}{2\bar{m}_{i}}\epsilon^{\sigma\nu\alpha\beta}q_{\alpha}s_{i\,\beta}(\bar{p}_{i})+\cdots\,,\\\
S_{i}^{\sigma\nu}(p^{\prime})&=-\frac{1}{\bar{m}_{i}}\epsilon^{\sigma\nu\alpha\beta}\bar{p}_{i\,\alpha}s_{i\,\beta}(\bar{p}_{i})+\frac{1}{2\bar{m}_{i}}\epsilon^{\sigma\nu\alpha\beta}q_{\alpha}s_{i\,\beta}(\bar{p}_{i})+\cdots\,,\end{split}$
(2.8)

where, remarkably, the shifts in $s_{i}^{\mu}(p_{i}^{(\prime)})$ drop out to
this order in the $\bar{m}$ expansion, due to the antisymmetry of the Levi-
Civita. We can also define the classical spin parameter as

$\displaystyle
a_{i}^{\mu}\coloneqq\frac{s_{i}^{\mu}(\bar{p}_{i})}{\bar{m}_{i}}\,,$ (2.9)

to write

$\displaystyle\begin{split}S_{i}^{\sigma\nu}(p_{i}^{\prime})&=-\epsilon^{\sigma\nu\alpha\beta}\Big{(}\bar{p}_{i\,\alpha}-\frac{q_{\alpha}}{2}\Big{)}a_{i\,\beta}+\mathcal{O}(\bar{m}_{i}^{-1})\,,\\\
S_{i}^{\sigma\nu}(p_{i})&=-\epsilon^{\sigma\nu\alpha\beta}\Big{(}\bar{p}_{i\,\alpha}+\frac{q_{\alpha}}{2}\Big{)}a_{i\,\beta}+\mathcal{O}(\bar{m}^{-1})\,.\end{split}$
(2.10)

Finally, in the large $\bar{m}_{i}$ limit the two spin tensors in (2.10)
become the same, and we define our classical spin tensors as

$S_{i}^{\mu\nu}\coloneqq-\epsilon^{\mu\nu\rho\sigma}\bar{p}_{i\,\rho}a_{i\,\sigma}\,,$
(2.11)

which satisfies $S^{\mu\nu}_{i}p_{i\nu}{=}0$, known as the spin supplementary
condition [159, 160], while $a_{i}$ satisfies $\bar{p}_{i}{\cdot}a_{i}=0$. We
can also invert this relation

$\displaystyle
a^{\mu}_{i}=-\frac{1}{\bar{m}^{2}_{i}}\epsilon^{\mu\nu\alpha\beta}\bar{p}_{i\,\nu}S_{i\,\alpha\beta}\,.$
(2.12)

Note that $a^{\mu}_{i}$ has mass dimension $-1$ so that $S^{\mu\nu}_{i}$ is
dimensionless. The spin vector of a heavy particle is then
$s^{\mu}_{i}\coloneqq\bar{m}_{i}a^{\mu}_{i}$. Much like $\bar{p}^{\mu}$ and
$\bar{v}^{\mu}$, both $s^{\mu}$ and $a^{\mu}$ are well defined in the
classical/large-$m$ limit.

Finally, the gravitational coupling we use is $\kappa=\sqrt{32\pi G}$.

##  3 Classical gravitational Compton amplitude with spin

###  3.1 Three-point amplitude

The three-point amplitude for two classical massive spinning particles is
given by [119, 121, 122, 161]

$\mathcal{M}_{3}=-i\kappa\,(\bar{p}{\cdot}\varepsilon_{1})^{2}\exp\left(-i\frac{k_{1}{\cdot}S\cdot\varepsilon_{1}}{\bar{p}{\cdot}\varepsilon_{1}}\right)\,,$
(3.1)

where $p$ is the momentum of the massive particle, $k_{1}$ is the momentum of
the graviton with polarisation $\varepsilon_{1}$ and $S$ is the spin tensor of
the massive particle introduced in (2.11). The amplitude (3.1) can also be
written as [139, 162, 155]

$\mathcal{M}_{3}=-i\kappa(\bar{p}{\cdot}\varepsilon_{1})({\mathsf{w}}_{1}{\cdot}\varepsilon_{1})\,,$
(3.2)

where

$\begin{split}{\mathsf{w}}_{1}^{\mu}&\coloneqq\cosh(k_{1}{\cdot}a)\bar{p}^{\mu}-i\frac{\sinh(k_{1}{\cdot}a)}{k_{1}{\cdot}a}(k_{1}{\cdot}S)^{\mu}\,,\end{split}$
(3.3)

and we have used the notation $(k_{1}{\cdot}S)^{\mu}=k_{1\nu}S^{\nu\mu}$.

###  3.2 The Compton amplitude

We now move on to discuss the four-point amplitude. For convenience, in this
section we will call the momenta $p,k_{1},k_{2},p^{\prime}$ where
$p,p^{\prime}$ are the momenta of the massive particles,
$p^{2}{=}(p^{\prime})^{2}{=}m^{2}$ and $k_{1,2}$ are the momenta of the
gravitons, with $k_{1,2}^{2}{=}0$.

(3.4)

The four-point classical Compton amplitude can be divided into three pieces
[156],

$\displaystyle\mathcal{M}_{4}=\frac{-i\mathcal{N}_{\rm
dc}}{2k_{1}{\cdot}k_{2}}+\frac{-i\mathcal{N}_{\rm
r}}{4\bar{p}{\cdot}k_{1}\bar{p}{\cdot}k_{2}}-i\mathcal{N}_{\rm c}.$ (3.5)

The first term is computed ignoring the “spin flip effect” and can be obtained
using the double copy [155],

$\displaystyle\mathcal{N}_{\rm dc}$
$\displaystyle=-\Bigg{[}\frac{{\mathsf{w}}_{1}{\cdot}F_{1}{\cdot}F_{2}{\cdot}{\mathsf{w}}_{2}}{k_{1}{\cdot}\bar{p}}-\Big{(}iG_{2}\left(x_{1},x_{2}\right)(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}S{\cdot}k_{2}+a{\cdot}F_{2}{\cdot}F_{1}{\cdot}S{\cdot}k_{1})$
$\displaystyle+iG_{1}\left(x_{12}\right)\text{tr}\left(F_{1}{\cdot}S{\cdot}F_{2}\right)+G_{1}(x_{1})G_{1}(x_{2})(a{\cdot}F_{1}{\cdot}\bar{p}a{\cdot}F_{2}{\cdot}k_{1}-a{\cdot}F_{1}{\cdot}k_{2}a{\cdot}F_{2}{\cdot}\bar{p})$
$\displaystyle+k_{1}{\cdot}\bar{p}\,G_{1}(x_{1})G_{1}(x_{2})a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a\Big{)}\Bigg{]}\Big{(}\frac{\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}}{k_{2}{\cdot}\bar{p}}\Big{)}\,.$
(3.6)

with

$\displaystyle x_{i}\coloneqq k_{i}{\cdot}a\,,\quad x_{i\ldots
j}\coloneqq(k_{i}+\cdots+k_{j}){\cdot}a\ ,\quad
F^{\mu\nu}_{i}=k_{i}^{\mu}\varepsilon_{i}^{\nu}-\varepsilon_{i}^{\mu}k_{i}^{\nu}\,,$
(3.7)

Note that it contains both massless and massive poles and we already take the
HEFT expansion. This term is the minimal amplitude to fit the test particle
scattering angle in the Kerr metric.

The second term is due to the spin-flip effect, and only gives rise to massive
poles,

$\displaystyle\mathcal{N}_{\rm r}$
$\displaystyle=\Big{(}(\partial_{x_{1}}-\partial_{x_{2}})G_{1}(x_{1})G_{1}(x_{2})\Big{)}$
$\displaystyle\times\Big{(}\bar{p}{\cdot}k_{2}(\bar{p}^{2}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}aa{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p}+a^{2}\bar{p}{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p}a{\cdot}F_{1}{\cdot}F_{2}{\cdot}\bar{p})-(1\leftrightarrow
2)\Big{)}$
$\displaystyle+i\Big{(}(\partial_{x_{1}}-\partial_{x_{2}})G_{2}(x_{1},x_{2})\Big{)}$
$\displaystyle\times\Big{(}\bar{p}{\cdot}k_{2}a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p}(a{\cdot}F_{2}{\cdot}\bar{p}a{\cdot}\widetilde{F}_{1}{\cdot}\bar{p}{-}a{\cdot}F_{1}{\cdot}\bar{p}a{\cdot}\widetilde{F}_{2}{\cdot}\bar{p})+(1\leftrightarrow
2)\Big{)}\,,$ (3.8)

where
$\widetilde{F}^{\mu\nu}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$
denotes the Hodge dual of the linearised field strength. The last contribution
consists of contact terms,

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

The $G$-functions appearing in the expressions above can be defined in terms
of hyperbolic functions as [155]

$\displaystyle G_{1}(x)$ $\displaystyle:=\frac{\sinh(x)}{x}\,,$ $\displaystyle
G_{2}(x_{1},x_{2})$ $\displaystyle:={1\over
x_{2}}\Big{(}{\sinh(x_{{12}})\over{x_{12}}}-\cosh(x_{2})\,{\sinh(x_{1})\over{x_{1}}}\Big{)},$
(3.10)

and are entire functions, free of singularities. Note that
$G_{2}(x_{2},x_{1})=-G_{2}(x_{1},x_{2})$.

The above contact terms in the first two lines of (3.9) only begin
contributing at quartic order in the spin and their numerical coefficients
have been fixed against results at quartic order in [145, 142, 134] arising
from black-hole perturbation theory. The last line in (3.9) contributes from
quintic order in the spin and its numerical coefficient $e_{1}$ is fixed by
the conjectured “spin-shift symmetry” applied at this order [133, 142] to be
$e_{1}=-3/4$.

Following the pattern from the entire functions point of view, we could also
have more contact terms as, for example

$\displaystyle\begin{split}&e_{2}\Big{(}{i(\partial_{x_{1}}-\partial_{x_{2}})^{2}\over
2!}{G_{2}(x_{1},x_{2})}\Big{)}\Big{(}\bar{p}^{2}(a{\cdot}F_{1}{\cdot}F_{2}{\cdot}a)(a{\cdot}F_{2}{\cdot}\widetilde{F}_{1}{\cdot}a)-(1\leftrightarrow
2)\Big{)}\,,\\\ &e_{3}\Big{(}{i(\partial_{x_{1}}-\partial_{x_{2}})^{2}\over
2!}{G_{2}(x_{1},x_{2})}\Big{)}\Big{(}a^{2}(a{\cdot}F_{2}{\cdot}F_{1}{\cdot}\bar{p})(a{\cdot}\widetilde{F}_{1}{\cdot}F_{2}{\cdot}\bar{p})-(1\leftrightarrow
2)\Big{)}\,.\end{split}$ (3.11)

These terms would start contributing at $\mathcal{O}(a^{5})$ but they are
ruled out by spin-shift symmetry. In the future one may want to add these
contact terms and relax the spin-shift symmetry constraint, however we do not
consider this in this paper. Therefore, the results derived here are
applicable to Kerr black holes at least to quartic order in the spin. Thus, we
will set $e_{1}=-3/4,\,e_{2}=0,\,e_{3}=0$, although our method makes it easy
to deal with any values of the $e_{i}$’s and also further contact terms
starting at $\mathcal{O}(a^{6})$ and beyond.

##  4 Spinning five-point amplitude

The crucial ingredient to compute the waveforms is the classical part of the
five-point amplitude of two spinning celestial objects with one radiated
graviton.111In the next section we will see that actually only the residues on
the physical factorisation channels are needed for computing the waveform.
However, since the computation of the five-point amplitude is so simple we
cannot resist to present it here. It can be derived using the HEFT BCFW
recursion relation introduced in [62] and is obtained from the following two
recursive diagrams,

(4.1)

corresponding to the $q_{1}^{2}$ and $q_{2}^{2}$ channels, respectively. In
the scalar case, these BCFW diagrams capture all of the ‘contact terms’ in the
classical amplitude (that is terms without poles in $q_{1}^{2}$ or $q_{2}^{2}$
but possibly with massive poles). In the spinning case we will follow the same
procedure and, although we have no general proof that these contact terms are
captured fully, we have checked that the contributions from the two BCFW
diagrams satisfy the correct soft behaviour. Regardless, such contact terms
without poles in $q_{1}^{2}$ or $q_{2}^{2}$ do not contribute to the tree-
level waveform as we will see in Sections 5 and 7.

The contribution of each of the two diagrams is obtained by gluing a three-
point amplitude with a four-point Compton amplitude, given in (3.1) and (3.5),
respectively. In doing so one has to sum over the intermediate states of the
exchanged graviton, using

$\displaystyle\sum_{h}\varepsilon^{\mu_{a}}_{-\hat{q}}{\varepsilon}^{\nu_{a}}_{-\hat{q}}{\varepsilon}^{\mu_{b}}_{\hat{q}}\varepsilon^{\nu_{b}}_{\hat{q}}=\frac{1}{2}\Big{[}\eta^{\mu_{a}\mu_{b}}\eta^{\nu_{a}\nu_{b}}+\eta^{\mu_{a}\nu_{b}}\eta^{\nu_{a}\mu_{b}}-\frac{2}{D-2}\eta^{\mu_{a}\nu_{a}}\eta^{\mu_{b}\nu_{b}}\Big{]}\
.$ (4.2)

For convenience, we introduce a tensor current by extracting the polarisation
vector from the Compton amplitude:

$\displaystyle\mathcal{J}^{\mu\nu}_{i}\varepsilon_{\mu\nu}=\mathcal{M}_{4}(-q_{i},k,\bar{v}_{i},a_{i}),$
$\displaystyle\mathcal{J}^{\mu\nu}_{i,0}\varepsilon_{\mu\nu}=\mathcal{M}_{4}(-q_{i},k,\bar{v}_{i},a_{i}=0)\,.$
(4.3)

Then, the amplitude in each channel is of the form

$\displaystyle\begin{split}\mathcal{M}_{q_{1}^{2}}&={1\over
q_{1}^{2}}\sum_{h}\Big{(}\cosh(a_{1}{\cdot}q_{1})(\bar{v}_{1}{\cdot}\varepsilon)^{2}\varepsilon_{\mu\nu}\mathcal{J}^{\mu\nu}_{1}-iG(a_{1}{\cdot}q_{1})\bar{v}_{1}{\cdot}\varepsilon
q_{1}{\cdot}S_{1}{\cdot}\varepsilon\varepsilon_{\mu\nu}\mathcal{J}^{\mu\nu}_{1}\Big{)}\\\
&={1\over
q_{1}^{2}}\Big{(}\cosh(a_{1}{\cdot}q_{1})\big{(}\bar{v}_{1}{\cdot}\mathcal{J}_{1}{\cdot}\bar{v}_{1}-{1\over
2}\text{tr}(\mathcal{J}_{1})\big{)}-{i\over
2}G(a_{1}{\cdot}q_{1})\big{(}q_{1}{\cdot}S_{1}{\cdot}\mathcal{J}_{1}{\cdot}\bar{v}_{1}-\bar{v}_{1}{\cdot}\mathcal{J}_{1}{\cdot}S_{1}{\cdot}q_{1}\big{)}\Big{)}\,,\end{split}$
(4.4)

and

$\displaystyle\begin{split}\mathcal{M}_{q_{2}^{2}}&={1\over
q_{2}^{2}}\sum_{h}\Big{(}\cosh(a_{2}{\cdot}q_{2})(\bar{v}_{2}{\cdot}\varepsilon)^{2}\varepsilon_{\mu\nu}\mathcal{J}^{\mu\nu}_{2}-iG(a_{2}{\cdot}q_{2})\bar{v}_{2}{\cdot}\varepsilon
q_{2}{\cdot}S_{2}{\cdot}\varepsilon\varepsilon_{\mu\nu}\mathcal{J}^{\mu\nu}_{2}\Big{)}\\\
&={1\over
q_{2}^{2}}\Big{(}\cosh(a_{2}{\cdot}q_{2})\big{(}\bar{v}_{2}{\cdot}\mathcal{J}_{2}{\cdot}\bar{v}_{2}-{1\over
2}\text{tr}(\mathcal{J}_{2})\big{)}-{i\over
2}G(a_{2}{\cdot}q_{2})\big{(}q_{2}{\cdot}S_{2}{\cdot}\mathcal{J}_{2}{\cdot}\bar{v}_{2}-\bar{v}_{2}{\cdot}\mathcal{J}_{2}{\cdot}S_{2}{\cdot}q_{2}\big{)}\Big{)}\,.\end{split}$
(4.5)

The full amplitude can be obtained directly adding (4.4) and (4.5),

$\displaystyle\mathcal{M}_{5,\rm
HEFT}=\mathcal{M}_{q_{1}^{2}}+\mathcal{M}_{q_{2}^{2}}\,.$ (4.6)

Both channels have the spurious pole $1\over k{\cdot}q_{1}$, which cancels
after summing the two contributions. To see this, we must use the Bianchi
identity in $D$-dimensional momentum space [163]

$\displaystyle\begin{split}A{\cdot}F_{k}{\cdot}B\
k{\cdot}C+B{\cdot}F_{k}{\cdot}C\ A{\cdot}k+C{\cdot}F_{k}{\cdot}A\
k{\cdot}&B=0\,,\end{split}$ (4.7)

where $A,B,C$ can be any vector. For example, a particular application is

$\displaystyle\bar{v}_{1}{\cdot}S_{2}{\cdot}F_{k}{\cdot}q_{2}=\frac{k{\cdot}q_{2}\bar{v}_{1}{\cdot}F_{k}{\cdot}S_{2}{\cdot}\bar{v}_{1}-k{\cdot}S_{2}{\cdot}\bar{v}_{1}\bar{v}_{1}{\cdot}F_{k}{\cdot}q_{2}}{q_{2}{\cdot}\bar{v}_{1}}.$
(4.8)

The resulting expression for the amplitude only contains the following four
field-strength products:

$\displaystyle a_{1}{\cdot}F_{k}{\cdot}\bar{v}_{1},$ $\displaystyle
a_{2}{\cdot}F_{k}{\cdot}\bar{v}_{1},$ $\displaystyle
q_{1}{\cdot}F_{k}{\cdot}\bar{v}_{1},$ $\displaystyle
v_{1}{\cdot}F_{k}{\cdot}\bar{v}_{2},$ $\displaystyle
q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}\bar{v}_{1},$ $\displaystyle
q_{1}{\cdot}S_{2}{\cdot}F_{k}{\cdot}\bar{v}_{1},$ $\displaystyle
v_{1}{\cdot}F_{k}{\cdot}S_{1}{\cdot}\bar{v}_{2},$
$\displaystyle\bar{v}_{1}{\cdot}F_{k}{\cdot}S_{2}{\cdot}\bar{v}_{1},$
$\displaystyle\text{tr}\left(F_{k}{\cdot}S_{1}\right),$
$\displaystyle\text{tr}\left(F_{k}{\cdot}S_{2}\right).$ (4.9)

The complete expression for the five-point amplitude of two spinning black
holes is included in the GitHub repository associated to this paper.

In this paper we will present waveforms in the simpler situation of the
scattering of a Schwarzschild and a Kerr black hole, deferring the study of
the waveform produced by two Kerr black holes to [164]. Without loss of
generality, we will therefore set $a_{2}{=}0$, which dramatically simplifies
the contribution from the $q_{1}^{2}$-channel. Then the amplitude in each
channel has a very compact form

$\displaystyle\begin{split}\mathcal{M}_{q_{1}^{2}}&={1\over
q_{1}^{2}}\Big{[}\cosh(a_{1}{\cdot}q_{1})\big{(}\bar{v}_{1}{\cdot}\mathcal{J}_{2,0}{\cdot}\bar{v}_{1}-\frac{1}{2}\text{tr}(\mathcal{J}_{2,0})\big{)}\\\
&-{i\over
2}G(a_{1}{\cdot}q_{1})\big{(}q_{1}{\cdot}S_{1}{\cdot}\mathcal{J}_{2,0}{\cdot}\bar{v}_{1}-\bar{v}_{1}{\cdot}\mathcal{J}_{2,0}{\cdot}S_{1}{\cdot}q_{1}\big{)}\Big{]}\,,\end{split}$
(4.10)

and

$\displaystyle\mathcal{M}_{q_{2}^{2}}$ $\displaystyle={1\over
q_{2}^{2}}\Big{[}\bar{v}_{2}{\cdot}\mathcal{J}_{1}{\cdot}\bar{v}_{2}-\frac{1}{2}\text{tr}(\mathcal{J}_{1})\Big{]}\,.$
(4.11)

##  5 The time-domain waveform

###  5.1 Waveforms from amplitudes

We begin by briefly reviewing the emergence of waveforms in black-hole
scattering. We consider the classical gravitational field produced by the
scattering of two black holes which are modelled by two massive spinning
particles using the KMOC approach [54, 112]. The corresponding initial two-
particle state has the form

$\displaystyle|\psi\rangle_{\rm
in}\coloneqq\int\\!\prod_{j=1}^{2}d\Phi(p_{j})e^{ip_{1}{\cdot}b}\phi(p_{1})\phi(p_{2})|p_{1},a_{1},p_{2},a_{2}\rangle_{\rm
in}\ .$ (5.1)

Following [54, 112, 62, 114, 115], one finds that

$\displaystyle\begin{split}\langle h^{\rm
out}_{\mu\nu}(x)\rangle_{\rm\psi}\\!=\\!\kappa\,\int\\!\prod_{j=1}^{2}d\Phi(\bar{p}_{j})\,|\phi(\bar{p}_{1})|^{2}|\phi(\bar{p}_{2})|^{2}\Big{[}\sum_{h}\int\\!d\Phi(k)e^{-ik{\cdot}x}\,\varepsilon^{(h)\ast}_{\mu\nu}(\hat{k})\,\big{[}i\,{W}\big{]}\,+\,{\rm
h.c.}\Big{]},\end{split}$ (5.2)

where $k{\coloneqq}\omega\hat{k}$. Here ${W}{=}{W}(\vec{b},k;h)$ is the
spectral waveform for the emission of a graviton of momentum $k$ and helicity
$h$, which at leading order in the PM expansion is222The factor of $-i$
cancels the $i$ from our definition of amplitudes as matrix elements of
$i\,T$.

$\displaystyle{W}(b,k^{h})\coloneqq-i\int\\!d\mu^{(D)}\ e^{iq_{1}{\cdot}b}\
\mathcal{M}_{5,\rm HEFT}(q_{1},q_{2},a_{1},a_{2};h)\,,$ (5.3)

where we have introduced the $D$-dimensional measure (for regularisation
purposes)

$\displaystyle
d\mu^{(D)}\coloneqq\frac{d^{D}q_{1}}{(2\pi)^{D{-}1}}\frac{d^{D}q_{2}}{(2\pi)^{D{-}1}}\,(2\pi)^{D}\delta^{(D)}(q_{1}+q_{2}-k)\delta(2{\bar{p}}_{1}{\cdot}q_{1})\delta(2{\bar{p}}_{2}{\cdot}q_{2})\,,$
(5.4)

with $q_{1,2}{=}p_{1,2}{-}p_{1,2}^{\prime}$ being the momentum transfers, and
$D{=}4{-}2\epsilon$. Here we are ignoring zero-modes in the amplitude which
only have support when the graviton energy $\omega$ is zero.

In the far-field limit, corresponding to large observer distance
$r{\coloneqq}|\vec{x}|$ and time $t$ with fixed retarded time
$u{\coloneqq}t{-}r$, (5.2) can be simplified to333Henceforth, we omit an
overall factor of
$\int\\!\prod_{j=1}^{2}d\Phi(p_{j})\,|\phi(p_{1})|^{2}|\phi(p_{2})|^{2}$.

$\displaystyle\begin{split}\langle h^{\rm
out}_{\mu\nu}(x)\rangle_{\rm\psi}\\!=\\!\frac{\kappa}{4\pi
r}\,\Big{[}\sum_{h}\varepsilon^{(h)\ast}_{\mu\nu}(\hat{k})\int_{0}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\,{W}(b,k^{h})\,+\,{\rm
h.c.}\Big{]}_{k=\omega(1,\mathbf{\hat{x}})}.\end{split}$ (5.5)

Alternatively, extending the $\omega$ integration from $-\infty$ to $+\infty$,

$\displaystyle\begin{split}&\langle h^{\rm
out}_{\mu\nu}(x)\rangle_{\rm\psi}=\frac{\kappa}{4\pi
r}\,\Big{[}\sum_{h}\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\\\
&\Big{[}\varepsilon^{(h)\ast}_{\mu\nu}(\hat{k})\,\theta(\omega)\,\left.{W}\big{(}b,k^{h}\big{)}\right|_{k=\omega(1,\mathbf{\hat{x}})}+\,\varepsilon^{(h)}_{\mu\nu}(\hat{k})\theta(-\omega)\,\left.{W}^{\ast}\big{(}b,k^{h}\big{)}\right|_{k=-\omega(1,\mathbf{\hat{x}})}\Big{]}\,.\end{split}$
(5.6)

We now define444We comment that in our normalisations, the combination
$\langle h_{+}{-}ih_{\times}\rangle$ is proportional to the strain $h(x)$,
specifically $h(x){=}-(1/2)\langle h_{+}{-}ih_{\times}\rangle$, where the
strain is related to the Newman-Penrose scalar $\Psi_{4}$ as
$\Psi_{4}{=}d^{2}h/du^{2}$.

$\displaystyle\begin{split}\langle h_{+}\pm
ih_{\times}\rangle&\coloneqq\langle h_{\mu\nu}^{\rm
out}\rangle\varepsilon^{\mu\nu}_{(\pm\pm)}\coloneqq\frac{1}{4\pi
r}(h_{+}^{\infty}\pm ih_{\times}^{\infty})\,.\end{split}$ (5.7)

Using the properties of the positive/negative helicity polarisation vectors
$\varepsilon_{\mu}^{(\pm)\ast}{=}\varepsilon^{\mu(\mp)}$,
$\varepsilon_{\mu}^{(\pm)\ast}\varepsilon^{\mu(\pm)}=-1$,
$\varepsilon_{\mu}^{(\pm)\ast}\varepsilon^{\mu(\mp)}=0$ we get

$\displaystyle\begin{split}&h^{\infty}_{+}\pm
ih^{\infty}_{\times}=\kappa\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\Big{[}\theta(\omega)\,\left.{W}\big{(}b,k^{\pm}\big{)}\right|_{k=\omega(1,\mathbf{\hat{x}})}+\,\theta(-\omega)\,\left.{W}^{\ast}\big{(}b,k^{\mp}\big{)}\right|_{k=-\omega(1,\mathbf{\hat{x}})}\Big{]}.\end{split}$
(5.8)

We can now combine the two terms in (5.8). In order to do so, we first note
that the five-point spinning amplitude has the form

$\displaystyle-i\mathcal{M}_{5,\rm
HEFT}=\varepsilon_{\mu\nu}(k)m^{\mu\nu}\,,\qquad\text{with}\qquad
m^{\mu\nu}=m^{\mu\nu}_{\rm even}+im^{\mu\nu}_{\rm odd}\,,$ (5.9)

where $m^{\mu\nu}_{\rm even}$ and $m^{\mu\nu}_{\rm odd}$ are real, and contain
even and odd powers of the spin, respectively. Then we observe that we can
separate out the $\omega$ dependence of the amplitude: we perform a rescaling
of $q_{1,2}$ and define

$q_{1,2}=\omega\hat{q}_{1,2},\qquad k=\omega\hat{k}.$ (5.10)

Then we have

$\displaystyle\left.\mathcal{M}_{5,\rm
HEFT}(q_{1},q_{2},k^{h},a_{1},a_{2})\right|_{S^{n}}=\frac{\omega^{n}}{\omega^{2}}\left.\mathcal{M}_{5,\rm
HEFT}(\hat{q}_{1},\hat{q}_{2},\hat{k}^{h},a_{1},a_{2})\right|_{S^{n}}\,,$
(5.11)

where $\left.\right|_{S^{n}}$ denotes the term containing $n$ powers of the
spin in the HEFT amplitude. Note that $\mathcal{M}_{5,\rm
HEFT}(\hat{q}_{1},\hat{q}_{2},\hat{k}^{h},a)$ is $\omega$-independent.
Combining (5.9) and (5.11) we find that

$\displaystyle\left.W^{\ast}(b,k^{h})\right|_{k=-\omega(1,\hat{\mathbf{x}})}=\left.W(b,k^{-h})\right|_{k=\omega(1,\hat{\mathbf{x}})}\,,$
(5.12)

and we can thus rewrite

$\displaystyle h^{\infty}_{+}\pm
ih^{\infty}_{\times}=\kappa\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\left.W(b,k^{\pm})\right|_{k=\omega(1,\hat{\mathbf{x}})}\,.$ (5.13)

For convenience, in the following we will call this quantity

$\displaystyle\begin{split}h^{\infty}(u)&\coloneqq\kappa\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\left.W(b,k)\right|_{k=\omega(1,\hat{\mathbf{x}})}\\\
&=-i\kappa\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\int\\!\frac{d^{4}q_{1}}{(2\pi)^{2}}\delta(2{\bar{p}}_{1}{\cdot}q_{1})\delta(2{\bar{p}}_{2}{\cdot}(k-q_{1}))\
e^{iq_{1}{\cdot}b}\ \mathcal{M}_{5,\rm HEFT}\,,\end{split}$ (5.14)

leaving the dependence on the helicity understood, and where in all formulae
$k{=}\omega(1,\hat{\mathbf{x}})$.

The above no longer appears manifestly real but in fact it is (when expressed
in a basis of real polarisation tensors) thanks to the properties of
$-i\mathcal{M}_{5,\rm HEFT}$ in (5.9) and (5.11). That is, a real term in the
amplitude has an even power of the spin and hence after the re-scaling (5.10)
is an even function of $\omega$; its Fourier transform is thus real. On the
other hand, terms containing a factor of $i$ will feature an odd power of the
spin and so are odd functions of $\omega$; their Fourier transform is thus
imaginary and this cancels the additional factor of $i$, with the final result
being real.

###  5.2 A scalar warm-up

Here we detail the computation of the scalar tree-level waveform, as a warm-up
to the spinning case. Many of the simplifications we discuss here apply to the
spinning waveform as well, in particular the intriguing fact that the
computation boils down to a simple application of Cauchy’s theorem. We begin
with the expression for the waveform (5.14) derived in the previous section
(here and for the rest of the paper we will drop the explicit bars on all of
the variables to reduce clutter)

$\displaystyle\begin{split}h^{\infty}(u)=-i\kappa\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\int\\!\frac{d^{4}q_{1}}{(2\pi)^{2}}\delta(2{p}_{1}{\cdot}q_{1})\delta(2{p}_{2}{\cdot}(k-q_{1}))\
e^{iq_{1}{\cdot}b}\ \mathcal{M}_{5,\rm HEFT}\,,\end{split}$ (5.15)

First, we rescale the momentum transfers by $\omega$, as discussed above,
using $q_{1,2}{=}\omega\hat{q}_{1,2}$ and $k{=}\omega\hat{k}$. The classical
scalar amplitude then scales universally like $\omega^{-2}$, which cancels the
power of $\omega^{2}$ coming from the change of variables, to give

$\displaystyle\begin{split}h^{\infty}(u){=}{-i\kappa}\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\int\\!\frac{d^{4}\hat{q}_{1}}{(2\pi)^{2}}\delta(2{p}_{1}{\cdot}\hat{q}_{1})\delta(2{p}_{2}{\cdot}(\hat{k}-\hat{q}_{1}))\
e^{i\omega\hat{q}_{1}{\cdot}b}\ \mathcal{M}_{5,\rm
HEFT}(\hat{q}_{1},\hat{q}_{2},\hat{k}^{h})\,.\end{split}$ (5.16)

In addition, it is useful to rescale the energy and retarded time by
$\sqrt{-b^{2}}$, as $\omega\rightarrow\omega/\sqrt{-b^{2}}$ and
$u\rightarrow\sqrt{-b^{2}}u$. Effectively this means we are measuring the
retarded time $u$ in units of $\sqrt{-b^{2}}$. With this choice, the tree-
level waveform becomes

$h^{\infty}(u){=}\frac{-i\kappa}{\sqrt{-b^{2}}}\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\int\\!\frac{d^{4}\hat{q}_{1}}{(2\pi)^{2}}\delta(2{p}_{1}{\cdot}\hat{q}_{1})\delta(2{p}_{2}{\cdot}(\hat{k}-\hat{q}_{1}))\
e^{i\omega\frac{\hat{q}_{1}{\cdot}b}{\sqrt{-b^{2}}}}\ \mathcal{M}_{5,\rm
HEFT}(\hat{q}_{1},\hat{q}_{2},\hat{k}^{h})\,.$ (5.17)

In fact, we are free to set $\sqrt{-b^{2}}=1$ in the expression above (and in
all subsequent expressions) since $b^{\mu}$ only appears in the exponent
through $b^{\mu}/\sqrt{-b^{2}}$. To restore $\sqrt{-b^{2}}$ we simply count
the mass dimension of the expression, obtaining the $1/\sqrt{-b^{2}}$ factor
above. Similarly, one can recover the original definition of the retarded time
$u$ by counting mass dimension.

Next, it is useful to split the amplitude into the two terms coming from the
BCFW diagrams (4.1). This gives us two contributions to the waveform, which we
call $h_{1}^{\infty}(u)$ and $h_{2}^{\infty}(u)$

$h_{q_{1}^{2},q_{2}^{2}}^{\infty}(u)={-i\kappa}\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\int\\!\frac{d^{4}\hat{q}_{1}}{(2\pi)^{2}}\delta(2{p}_{1}{\cdot}\hat{q}_{1})\delta(2{p}_{2}{\cdot}(\hat{k}-\hat{q}_{1}))\
e^{i\omega\hat{q}_{1}{\cdot}b}\ \mathcal{M}_{q_{1}^{2},q_{2}^{2}}\,.$ (5.18)

The two contributions $\mathcal{M}_{q_{1}^{2}}$ and $\mathcal{M}_{q_{2}^{2}}$
are related by the replacements
$v_{1}{\leftrightarrow}v_{2},q_{1}{\leftrightarrow}q_{2}$, which allows us to
obtain the waveform contribution in the $q_{2}^{2}$-channel from the
$q_{1}^{2}$-channel. To do this we perform the following replacements

$h_{q_{1}^{2}}^{\infty}(b{\cdot}k-u)\xrightarrow[]{v_{1}\leftrightarrow
v_{2}}h_{q_{2}^{2}}^{\infty}(u)\,,$ (5.19)

which can be seen immediately using the definition (5.18). The asymmetric
shift in the proper time $u$ is due to our asymmetric choice of impact
parameter in (5.1).

To compute the first cut, we decompose $\hat{q}_{1}$ onto a basis of four-
vectors [112]

$\displaystyle\hat{q}_{1}^{\mu}=z_{1}v_{1}^{\mu}+z_{2}v_{2}^{\mu}+z_{v}v_{\perp}^{\mu}+z_{b}b^{\mu}\,,$
(5.20)

where

$v_{\perp}\coloneqq\epsilon(v_{1},v_{2},b,\bullet)\,,$ (5.21)

and then change integration variables from $q_{1}$ to
$z_{1},z_{2},z_{v},z_{b}$. In this parameterisation, we can use the two delta
functions in (5.18) to localise the variables $z_{1}$ and $z_{2}$ to

$z_{1}=\frac{(v_{2}{\cdot}k)(v_{1}{\cdot}v_{2})}{(v_{1}{\cdot}v_{2})^{2}-1}=\frac{w_{2}y}{y^{2}-1},\quad
z_{2}=-\frac{v_{2}{\cdot}k}{(v_{1}{\cdot}v_{2})^{2}-1}=-\frac{w_{2}}{y^{2}-1}\,.$
(5.22)

The remaining integrals are then over $z_{v},z_{b}$ and $\omega$,

$h_{q_{1}^{2}}^{\infty}(u)=\frac{-i\kappa}{(4\pi)^{2}m_{1}m_{2}}\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}dz_{v}dz_{b}e^{-i\omega(u+z_{b})}\mathcal{M}_{q_{1}^{2}}\Big{|}_{z_{1}=\frac{w_{2}y}{y^{2}-1},\,z_{2}=-\frac{w_{2}}{y^{2}-1}}\,.$
(5.23)

The integral over $\omega$ also gives a delta function which we can
immediately use to localise the $z_{b}$ integral,

$h_{q_{1}^{2}}^{\infty}(u)=\frac{-i\kappa}{(4\pi)^{2}m_{1}m_{2}}\int_{-\infty}^{+\infty}\\!dz_{v}\mathcal{M}_{q_{1}^{2}}\Big{|}_{z_{1}=\frac{w_{2}y}{y^{2}-1},\,z_{2}=-\frac{w_{2}}{y^{2}-1},\,z_{b}=-u}\,.$
(5.24)

To compute the final integral in $z_{v}$ we use Cauchy’s residue theorem,
hence we need to examine the pole structure of the $q_{1}^{2}$-cut. The
integrand contains three types of poles in $z_{v}$ which arise from certain
denominator structures in the tree-level amplitude. These are

$\displaystyle\text{Physical
pole:}\quad\frac{1}{q_{1}^{2}}\sim\frac{1}{(z_{v}-iA)(z_{v}+iA)},$ (5.25)
$\displaystyle\text{Spurious
pole:}\quad\frac{1}{q_{1}^{2}\,q_{1}{\cdot}k}\sim\frac{1}{(z_{v}-iA)(z_{v}+iA)(z_{v}-B)}\,,$
(5.26) $\displaystyle\text{Pole at
infinity:}\,\begin{cases}\frac{z_{v}}{q_{1}^{2}}\sim\frac{z_{v}}{(z_{v}-iA)(z_{v}+iA)}\xrightarrow[z_{v}\rightarrow\infty]{}\frac{1}{z_{v}}\,,\\\
\frac{z_{v}^{2}}{q_{1}^{2}\,q_{1}{\cdot}k}\sim\frac{z_{v}^{2}}{(z_{v}-iA)(z_{v}+iA)(z_{v}-B)}\xrightarrow[z_{v}\rightarrow\infty]{}\frac{1}{z_{v}}\,,\end{cases}$
(5.27)

where $A$ and $B$ are real functions of the external kinematics. To compute
the $z_{v}$ integral we will close the integration contour in the upper half
plane to capture the pole at $z_{v}=iA$ and regulate the pole at infinity with
a principal value prescription. This is equivalent to taking the integration
limits $z_{v}\rightarrow-\infty$ and $z_{v}\rightarrow+\infty$ in a symmetric
fashion, and implies that the pole at infinity receives an extra factor of
$\frac{1}{2}$. The spurious pole at $z_{v}=B$ (coming from the factor
$q_{1}{\cdot}k$) lies on the integration contour, however we know that this
pole cancels when we combine the two cuts in $q_{1}^{2}$ and $q_{2}^{2}$.
Hence we are free to ignore the residue on this spurious pole since it would
cancel at the end of the computation (as we have checked explicitly).

In fact, we can further simplify the integral (5.24) using the following
observations. First, the integral of one of the terms with a pole at infinity
in (5.27) is actually zero,

$\int_{-\infty}^{+\infty}\\!dz_{v}\frac{z_{v}}{(z_{v}-iA)(z_{v}+iA)}=0\,.$
(5.28)

This can be seen from the fact that the integrand is odd in $z_{v}$, or that
the residue at $z_{v}=iA$ cancels with half the residue at infinity (recalling
the principal value prescription mentioned earlier). The second term with a
pole at infinity in (5.27) can also be simplified as

$\displaystyle\frac{z_{v}^{2}}{(z_{v}{-}iA)(z_{v}{+}iA)(z_{v}{-}B)}$
$\displaystyle=\frac{((z_{v}{-}B){+}B)^{2}}{(z_{v}-iA)(z_{v}+iA)(z_{v}-B)}$
$\displaystyle=\frac{B}{(z_{v}{-}iA)(z_{v}{+}iA)}+\frac{B^{2}}{(z_{v}{-}iA)(z_{v}{+}iA)(z_{v}{-}B)}+\cdots$
(5.29)

where $+\cdots$ are terms which vanish after integration due to (5.28). The
remaining terms above are in the form of (5.25) and (5.26). Thus, after
simplifications the only terms relevant to the waveform integral (5.24) are

$\frac{1}{q_{1}^{2}}\sim\frac{1}{(z_{v}-iA)(z_{v}+iA)}\,,\qquad\frac{1}{q_{1}^{2}q_{1}{\cdot}k}\sim\frac{1}{(z_{v}-iA)(z_{v}+iA)(z_{v}-B)}\,,$
(5.30)

for which we only compute the residue on the physical pole $z_{v}=iA$. The
computation for the second cut $\mathcal{M}_{q_{2}^{2}}$ proceeds in an
identical way, or alternatively we can obtain the second cut using the
replacements (5.19).

We have thus learned that the computation of the waveform can be efficiently
reduced to the evaluation of residues on physical poles. The same general
principle will be used in the spinning case. The final expression for the
scalar waveform is simply the sum of $h_{q_{1}^{2}}^{\infty}$ and
$h_{q_{2}^{2}}^{\infty}$, and is included in the GitHub repository.

We can choose a frame such that the kinematics are given by

$\displaystyle v_{1}$
$\displaystyle=(1,0,0,0),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}v_{2}=(y,\sqrt{y^{2}-1},0,0)$
$\displaystyle k$
$\displaystyle=(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,v_{\perp}=(0,0,\sqrt{y^{2}-1},0)$
$\displaystyle\varepsilon$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}0,\cos\theta\cos\phi-i\sin\phi,\cos\theta\sin\phi+i\cos\phi,-\sin\theta\big{)},~{}~{}~{}b=(0,0,0,1)\,,$
(5.31)

and then in Figure 1 we present the scalar waveform at fixed angles
$\theta={\pi\over 4}$ and $\phi={\pi\over 4}$ for various values of $y$.

$-4$$-3$$-2$$-1$$0$$1$$2$$3$$4$$-1$$-0.5$$0$$0.5$$1$$\cdot
10^{-2}$$u$${h}_{+}^{\infty}\,\,$Scalar Waveforms$y$=9/8$y$=5/4$y$=3/2 Figure
1: Scalar waveforms $h^{\infty}_{+}$ at various values of $y$.

###  5.3 General expression of the time-domain waveform for arbitrary spins

We now turn to the spinning case. The first observation to make is that, in
principle, the Fourier transform to impact parameter space in (5.14) is ill-
defined due to the large-$q_{1}$ behaviour of the integrand giving rise to an
ultraviolet (UV) divergence. An elegant way to regularise this is to leave the
hyperbolic and exponential functions in the Compton amplitudes unexpanded (in
the spin vectors), introduce a new spin parameter as

$\displaystyle\tilde{a}_{1,2}\coloneqq ia_{1,2}\,,$ (5.32)

and temporarily take $\tilde{a}_{1,2}$ to be real. Assuming that the final
spinning waveform has an expansion around $a_{1,2}\rightarrow 0$, this
analytic continuation should not change the expansion coefficients. In support
of this approach we mention that the $a_{1}\rightarrow 0$ limit of our
waveform gives the correct scalar result, and for $a_{1}\neq 0$ has the
correct gravitational memory (computed in Section 8); and finally, our results
also agree with the recently derived waveform of [157], obtained by expanding
in spin and then integrating, up to and including $\mathcal{O}(a_{1}^{4})$.
Indeed, one can expand the amplitude in the spin parameters before
integration, and the amplitude’s degree of divergence would grow with each
additional order in the spin. However, as we see in Section 7, these
divergences can be ignored since they only contribute to contact terms in
$q_{1}^{2}$ and $q_{2}^{2}$, and both methods (the analytic continuation and
expanding in spin before integration) give the same result.

Proceeding now with the analytic continuation in the spin (5.32), we observe
that in the large-$q_{1}$ limit, i.e. $q_{1}{\rightarrow}\lambda\,q_{1}$ with
$\lambda{\rightarrow}\infty$, the scaling behaviour of the amplitude is now
$\mathcal{O}(\lambda^{-1})$ as $\lambda\to\infty$. Pleasingly, this is
precisely the same behaviour as that of the scalar amplitude. This logarithmic
divergence will appear, identically to the scalar case, as a pole at infinity
which we can again regulate with a principal value prescription. The waveform
is therefore well-defined once we tame this logarithmic divergence,

$\displaystyle\begin{split}h^{\infty}(u)&=-i\kappa\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\int\frac{d^{4}\hat{q}_{1}}{(2\pi)^{2}}\delta(2{p}_{1}{\cdot}\hat{q}_{1})\delta(2{p}_{2}{\cdot}(\hat{k}-\hat{q}_{1}))e^{i\omega\hat{q}_{1}{\cdot}b}\
\omega^{2}(\mathcal{M}_{q_{1}^{2}}+\mathcal{M}_{q_{2}^{2}})\,,\end{split}$
(5.33)

where we have re-scaled $q_{1}$ to $\omega\,\hat{q}_{1}$ as discussed in the
last section. The factor of $\omega^{2}$ comes from the re-scaled measure
while the amplitude itself depends on $\omega$ in a manner which we now
describe.

Writing the hyperbolic functions within the expression of the Compton
amplitude (4.1) in terms of exponential functions, we find that the tree-level
amplitude can be rewritten as a linear combination of at most eight
exponential factors, with a very simple frequency dependence. Specifically, we
find that only three different powers of the frequency $\omega$ can appear for
arbitrary classical spins,

$\displaystyle\begin{split}\mathcal{M}_{q_{1}^{2}}&=\sum_{\varrho_{1},\varrho_{2},\varrho_{3}}^{\pm,\pm,\pm}{e^{\varrho_{1}i\omega\tilde{a}_{1}{\cdot}\hat{q}_{1}+\varrho_{2}i\omega\tilde{a}_{2}{\cdot}\hat{q}_{1}+\varrho_{3}i\omega\tilde{a}_{2}{\cdot}\hat{k}}}{1\over\omega^{2}}\Big{(}\sum_{i=0}^{2}\mathcal{M}^{(i)}_{q_{1}^{2}}(\varrho_{1},\varrho_{2},\varrho_{3})\omega^{i}\Big{)}\,,\\\
\mathcal{M}_{q_{2}^{2}}&=\sum_{\varrho_{1},\varrho_{2},\varrho_{3}}^{\pm,\pm,\pm}e^{\varrho_{1}i\omega\tilde{a}_{2}{\cdot}\hat{q}_{2}+\varrho_{2}i\omega\tilde{a}_{1}{\cdot}\hat{q}_{2}+\varrho_{3}i\omega\tilde{a}_{1}{\cdot}\hat{k}}{1\over\omega^{2}}\Big{(}\sum_{i=0}^{2}\mathcal{M}^{(i)}_{q_{2}^{2}}(\varrho_{1},\varrho_{2},\varrho_{3})\omega^{i}\Big{)}.\end{split}$
(5.34)

Hence, the waveform integral has a simple general structure. In the remainder
of this section we will focus on $\mathcal{M}^{(i)}_{q_{1}^{2}}$, and the case
of $\mathcal{M}^{(i)}_{q_{2}^{2}}$ is similar. Similarly to the scalar case,
the four-dimensional integration is immediately reduced to a two-dimensional
one using the $\delta$-functions in (5.33). Furthermore, for each exponential
factor, the Fourier transform to the time domain generates a third delta
function, which constrains the integration over $q_{1}$ to the hyperplane
defined by555Henceforth, we relabel the integration variables dropping the
hats on the $q_{i}$ variables, and on $k$.

$\displaystyle
b{\cdot}q_{1}+\varrho_{1}\tilde{a}_{1}{\cdot}q_{1}+\varrho_{2}\tilde{a}_{2}{\cdot}q_{1}+\varrho_{3}\tilde{a}_{2}{\cdot}k-u=0\,.$
(5.35)

Following similar manipulations to (5.2) in the scalar case, the master
integrands are of the form

$\displaystyle{1\over(q_{1}{\cdot}X+Y{\cdot}Z)q_{1}^{2}},$
$\displaystyle{1\over q_{1}^{2}},$ $\displaystyle{q_{1}{\cdot}W\over
q_{1}^{2}},$ (5.36)

where $W$ can be chosen to be orthogonal to the localising hyperplane and
$q_{1}{\cdot}X+Y{\cdot}Z$ denotes a generic spurious pole linear in $q_{1}$
and featuring external vectors $X,Y$ and $Z$ which may be the spins $a_{i}$ or
$k$. The first two master integrals are UV convergent, while the last one is
logarithmically divergent. However, the last master integral is an odd
function of $q_{1}$, and hence vanishes when integrated on a symmetric domain,
identically to (5.28) in the scalar case. This corresponds to a principal
value (PV) regularisation of the divergent integral, or equivalently a PV
regularisation of the pole at infinity. With this regularisation, the residue
of the pole at $q_{1}^{2}=0$ of the third term in (5.36) cancels the residue
of the pole at infinity. Therefore, we can drop the last master integral
altogether.

Now that the pole at infinity has been removed, we can perform the integration
of the remaining terms using Cauchy’s theorem on the finite poles. There is
only one physical pole in this channel, namely $q_{1}^{2}{=}0$. The residues
on the spurious poles $k{\cdot}q_{1}$ in the integrand can be discarded since
they cancel when combining with the $q_{2}^{2}$-channel, a fact we have
confirmed by explicit calculations. The residues of the spin-dependent
spurious poles in the three-point and Compton amplitudes (coming from the
entire functions $G_{i}$) cancel when performing an expansion in the spins
$a_{1}$ and $a_{2}$, and so they can also be ignored. A similar statement
holds for these poles in the final integrated waveform.

In summary, the closed-form expression of the time-domain waveform with
arbitrary spin is then

$\displaystyle\begin{split}h^{\infty}(u)=&-i\kappa\sum_{\varrho_{1},\varrho_{2},\varrho_{3}}^{\pm,\pm,\pm}\sum_{j=0}^{2}(i\partial_{u})^{j}\Big{[}\ointctrclockwise_{(q_{1}^{2})^{+}=0}\frac{d^{4}q_{1}}{(2\pi)^{2}}\delta(2{v}_{1}{\cdot}q_{1})\delta(2{v}_{2}{\cdot}(k-q_{1}))\\\
&\delta(b{\cdot}q_{1}+\varrho_{1}\tilde{a}_{1}{\cdot}q_{1}+\varrho_{2}\tilde{a}_{2}{\cdot}q_{1}+\varrho_{3}\tilde{a}_{2}{\cdot}k-u)\,\mathcal{M}^{(i)}_{q_{1}^{2},{\rm
fin}}(\varrho_{1},\varrho_{2},\varrho_{3})\\\
+&\ointctrclockwise_{(q_{2}^{2})^{+}=0}\frac{d^{4}q_{1}}{(2\pi)^{2}}\delta(2{v}_{1}{\cdot}q_{1})\delta(2{v}_{2}{\cdot}(k-q_{1}))\\\
&\delta(b{\cdot}q_{1}+\varrho_{1}\tilde{a}_{2}{\cdot}q_{2}+\varrho_{2}\tilde{a}_{1}{\cdot}q_{2}+\varrho_{3}\tilde{a}_{1}{\cdot}k-u)\,\mathcal{M}^{(i)}_{q_{2}^{2},{\rm
fin}}(\varrho_{1},\varrho_{2},\varrho_{3})\Big{]}\,,\end{split}$ (5.37)

where $\mathcal{M}^{(i)}_{q_{j}^{2},{\rm fin}}$ denotes the UV-convergent part
of the amplitude coming from the first two master integrals in (5.36). We
denote as $(q_{1,2}^{2})^{+}=0$ the physical poles in the upper half plane.

##  6 The waveform from the scattering of a Schwarzschild and a Kerr black
hole

In this paper, we will focus on the case where the first black hole is
spinning while the second is spinless, that is $a_{2}{=}0$.

###  6.1 The $q_{1}^{2}$-channel

For the contribution to the amplitude in the $q_{1}^{2}$-channel, the waveform
integrand is obtained from gluing a three-point spinning amplitude with a
four-point spinless amplitude. The amplitude in this channel is very simple
thanks to our restriction $a_{2}=0$, and from (4.10) we obtain, up to overall
constant pre-factors

$\displaystyle\begin{split}\omega^{2}e^{i\omega b\cdot q_{1}-i\omega
u}\mathcal{M}_{q_{1}^{2}}&=\omega^{2}e^{i\omega b\cdot q_{1}-i\omega
u}\Big{(}\frac{c_{1}\cosh\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega^{2}q_{1}{\cdot}q_{1}k{\cdot}q_{1}}+\frac{c_{2}\left(q_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}\cosh\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega^{2}q_{1}{\cdot}q_{1}k{\cdot}q_{1}}\\\
&+\frac{c_{3}\left(q_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}q_{1}{\cdot}S_{1}{\cdot}v_{2}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega
q_{1}{\cdot}q_{1}k{\cdot}q_{1}}+\frac{c_{4}q_{1}{\cdot}F_{k}{\cdot}v_{1}k{\cdot}S_{1}{\cdot}q_{1}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega q_{1}{\cdot}q_{1}k{\cdot}q_{1}}\\\
&+\frac{c_{5}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}}{\omega
q_{1}{\cdot}q_{1}}+\frac{c_{6}q_{1}{\cdot}F_{k}{\cdot}v_{1}\cosh\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega^{2}q_{1}{\cdot}q_{1}k{\cdot}q_{1}}\\\
&+\frac{c_{7}q_{1}{\cdot}F_{k}{\cdot}v_{1}q_{1}{\cdot}S_{1}{\cdot}v_{2}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega
q_{1}{\cdot}q_{1}k{\cdot}q_{1}}+\frac{c_{8}k{\cdot}S_{1}{\cdot}q_{1}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega q_{1}{\cdot}q_{1}k{\cdot}q_{1}}\\\
&+\frac{c_{9}k{\cdot}q_{1}\cosh\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega^{2}q_{1}{\cdot}q_{1}}+\frac{c_{10}q_{1}{\cdot}F_{k}{\cdot}v_{1}\cosh\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega^{2}q_{1}{\cdot}q_{1}}\\\
&+\frac{c_{11}k{\cdot}q_{1}q_{1}{\cdot}S_{1}{\cdot}v_{2}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega
q_{1}{\cdot}q_{1}}+\frac{c_{12}q_{1}{\cdot}F_{k}{\cdot}v_{1}q_{1}{\cdot}S_{1}{\cdot}v_{2}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega q_{1}{\cdot}q_{1}}\\\
&+\frac{c_{13}k{\cdot}S_{1}{\cdot}q_{1}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega
q_{1}{\cdot}q_{1}}+\frac{c_{14}\cosh\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega^{2}q_{1}{\cdot}q_{1}}\\\
&+\frac{c_{15}q_{1}{\cdot}S_{1}{\cdot}v_{2}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)}{\omega
q_{1}{\cdot}q_{1}}+\frac{c_{16}q_{1}{\cdot}F_{k}{\cdot}v_{1}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}}{\omega
q_{1}{\cdot}q_{1}k{\cdot}q_{1}}\\\ &+\frac{c_{17}G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}}{\omega
q_{1}{\cdot}q_{1}k{\cdot}q_{1}}\Big{)}\,,\end{split}$ (6.1)

where the coefficients $c_{i}$ are independent of $q_{1}$ and $\omega$,and
hence can be factored out in the waveform integration, there precise form is
given in Appendix B. For this channel the amplitude scales with $\omega$ as
$\omega^{0}$ with the remaining $\omega$ dependence exponentiating. In this
channel, there are only two sectors from the exponential factors:

$\displaystyle{\rm(I)}:\quad
e^{-i\omega\left(-\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}q_{1}+u\right)},$
$\displaystyle{\rm(II)}:\quad
e^{-i\omega\left(\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}q_{1}+u\right)}.$ (6.2)

Again we have the parameterisation of $q_{1}$ on the four-dimensional vector
basis given by the vectors

$\displaystyle v_{1},v_{2},b,v_{\perp}.$ (6.3)

As in the scalar case, we temporarily set $b^{2}=-1$ which means regarding the
spins $a_{i}$ and retarded time $u$ as dimensionless and measured in units of
$\sqrt{{-}b^{2}}$. The overall dependence on $b$ can then be reinstated by
counting of mass dimension and gives simply a prefactor of
$\frac{1}{\sqrt{{-}b^{2}}}$.

However, the parameterisation (6.3) is not well suited to the particular
sectors and does not cleanly identify the UV-divergent term in (5.36). It is
more convenient to introduce a sector-dependent basis as

$\displaystyle v_{1},v_{2},\tilde{b}_{j},\tilde{v}_{j},$ (6.4)

where in each sector we introduce an effective impact parameter

$\displaystyle\begin{cases}\tilde{b}_{(\rm I)}\coloneqq-b-\tilde{a}_{1}&\\\
\tilde{b}_{(\rm II)}\coloneqq-b+\tilde{a}_{1}&\end{cases},$ (6.5)

and correspondingly

$\displaystyle\tilde{v}_{j}\coloneqq\epsilon(v_{1},v_{2},\tilde{b}_{j},\bullet),\,\quad
j=\rm I,II\,.$ (6.6)

We then parameterise $q_{1}$ as

$\displaystyle
q_{1}=z_{1}v_{1}+z_{2}v_{2}+z_{v}\tilde{v}_{j}+z_{b}\tilde{b}_{j},\quad j=\rm
I,II\,,$ (6.7)

in terms of the basis vectors defined above. The divergent part in (5.36) is
then of the form

$\displaystyle{z_{v}\over c-z_{v}^{2}},$ (6.8)

which vanishes once we perform the integration as in the scalar case; hence we
drop such terms.

Examples with constrained spin:  In this paper we present results for the case
where the Kerr black hole spin $a_{1}$ satisfies the additional constraint

$\displaystyle\tilde{a}_{1}{\cdot}v_{2}=0\,.$ (6.9)

In this case $b,\tilde{a}_{1}$ are both constrained to the hyperplane
orthogonal to $v_{1}$ and $v_{2}$ as
$\tilde{a}_{1}{{\cdot}}v_{i}{=}b{{\cdot}}v_{i}{=}0$. The $q_{1}$ variable is
also constrained to another parallel hyperplane defined by
$q_{1}{\cdot}v_{1}{=}0,q_{1}{\cdot}v_{2}{=}k{{\cdot}}v_{2}{=}w_{2}$. Then the
extra $\delta$-function after the time-domain Fourier transform is, in the two
sectors,

$\displaystyle\quad\delta\left(-\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}q_{1}+u\right)=\delta(\tilde{b}_{(\rm
I)}{\cdot}q_{1}+u),$
$\displaystyle\quad\delta\left(\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}q_{1}+u\right)=\delta(\tilde{b}_{(\rm
II)}{\cdot}q_{1}+u)$ (6.10)

and the $q_{1}$ integral localises to the line as shown in the following
figure,

$v_{\perp}$$-b$$-b-\tilde{a}$$-b+\tilde{a}$$~{}~{}~{}~{}~{}{-}u$$~{}~{}~{}~{}~{}{-}u$
$\displaystyle\xrightarrow[\text{vector basis}]{\text{changing}}$ (6.11)

The plane depicted here is the one orthogonal to $v_{1}$ and $v_{2}$ which
corresponds to the integrals over $z_{v}$ and $z_{b}$ in each sector (6.7).
The variable $\tilde{b}{\cdot}q_{1}=\tilde{b}^{2}z_{b}$ is localised to $-u$
using (6.10) and the integral over $z_{v}$ is taken along the red line
orthogonal to the basis vector $\tilde{b}$. In the following we use
$\tilde{b}_{\rm(I)},\tilde{b}_{\rm(II)}$ to denote the shifted impact
parameters in the two sectors, and $\tilde{v}_{(\rm I)},\tilde{v}_{(\rm II)}$
to denote the corresponding orthogonal directions. We also note that when we
replace back $\tilde{a}_{1}=ia_{1}$ in terms of the physical spin the
quantities $\tilde{b}_{\rm(I)},\tilde{b}_{\rm(II)}$ and $\tilde{v}_{(\rm
I)},\tilde{v}_{(\rm II)}$ are complex conjugates of each other.

We now go into some explicit examples. First, consider the term

$\frac{c_{14}\cosh\left(\omega a_{1}{\cdot}q_{1}\right)e^{i\omega
b{\cdot}q_{1}-i\omega u}}{q_{1}{\cdot}q_{1}}.$ (6.12)

Then according to (5.37), we need to sum over the two sectors and get

$\displaystyle\frac{c_{14}}{4\sqrt{w_{2}^{2}\tilde{b}_{\rm(I)}{\cdot}\tilde{b}_{\rm(I)}-u^{2}\left(y^{2}-1\right)}}+\frac{c_{14}}{4\sqrt{w_{2}^{2}\tilde{b}_{\rm(II)}{\cdot}\tilde{b}_{\rm(II)}-u^{2}\left(y^{2}-1\right)}},$
(6.13)

which is a real result since $\tilde{b}_{\rm(I)},\tilde{b}_{\rm(II)}$ are a
complex conjugate pair. This is a general feature of the integrals encountered
in the following calculation, namely when replacing $\tilde{a}_{1}=ia_{1}$ the
sector variables are complex but appear in combinations such that the
resulting waveform is real (for a basis of real polarisations). A second
example is

$\displaystyle\frac{c_{13}\omega k{\cdot}S_{1}{\cdot}q_{1}e^{i\omega
b{\cdot}q_{1}-i\omega
u}G_{1}\left(-i\omega\tilde{a}_{1}{\cdot}q_{1}\right)}{q_{1}{\cdot}q_{1}}.$
(6.14)

The contribution to the waveform is

$\displaystyle{ic_{13}\over
4}\Bigg{(}\frac{-k{\cdot}S_{1}{\cdot}\tilde{v}_{\rm(I)}\sqrt{w_{2}^{2}\tilde{b}_{\rm(I)}{\cdot}\tilde{b}_{\rm(I)}+u^{2}\left(1-y^{2}\right)}+u\left(y^{2}-1\right)\tilde{b}_{\rm(I)}{\cdot}S_{1}{\cdot}k-w_{2}\tilde{b}_{\rm(I)}{\cdot}\tilde{b}_{\rm(I)}k{\cdot}S_{1}{\cdot}v_{2}}{u\left(y^{2}-1\right)\tilde{a}_{1}{\cdot}\tilde{b}_{\rm(I)}\sqrt{w_{2}^{2}\tilde{b}_{\rm(I)}{\cdot}\tilde{b}_{\rm(I)}-u^{2}\left(y^{2}-1\right)}-\tilde{a}_{1}{\cdot}\tilde{v}_{\rm(I)}\left(u^{2}\left(y^{2}-1\right)-w_{2}^{2}\tilde{b}_{\rm(I)}{\cdot}\tilde{b}_{\rm(I)}\right)}$
$\displaystyle-\frac{-k{\cdot}S_{1}{\cdot}\tilde{v}_{\rm(II)}\sqrt{w_{2}^{2}\tilde{b}_{\rm(II)}{\cdot}\tilde{b}_{\rm(II)}+u^{2}\left(1-y^{2}\right)}+u\left(y^{2}-1\right)\tilde{b}_{\rm(II)}{\cdot}S_{1}{\cdot}k-w_{2}\tilde{b}_{\rm(II)}{\cdot}\tilde{b}_{\rm(II)}k{\cdot}S_{1}{\cdot}v_{2}}{u\left(y^{2}-1\right)\tilde{a}_{1}{\cdot}\tilde{b}_{\rm(II)}\sqrt{w_{2}^{2}\tilde{b}_{\rm(II)}{\cdot}\tilde{b}_{\rm(II)}-u^{2}\left(y^{2}-1\right)}-\tilde{a}_{1}{\cdot}\tilde{v}_{\rm(II)}\left(u^{2}\left(y^{2}-1\right)-w_{2}^{2}\tilde{b}_{\rm(II)}{\cdot}\tilde{b}_{\rm(II)}\right)}\Bigg{)}.$
(6.15)

In this form, the poles which depend on the spin vector $a_{1}$ are due to the
spurious pole in the $G_{1}$ function. As with the $G_{1}$ function itself,
this pole explicitly cancels once we expand for $|a_{1}|\ll 1$ giving

$\displaystyle-\frac{c_{13}w_{2}\left(uw_{0}\left(y^{2}-1\right)\tilde{a}_{1}{\cdot}b+\left(-uw_{3}y^{2}+uw_{3}+w_{1}w_{2}y-w_{2}^{2}\right)\tilde{a}_{1}{\cdot}v_{\perp}\right)}{2\left(y^{2}-1\right)\left(-u^{2}\left(y^{2}-1\right)-w_{2}^{2}\right){}^{3/2}}$
$\displaystyle+\frac{c_{13}w_{2}^{3}\left(uw_{0}\left(y^{2}-1\right)\tilde{a}_{1}{\cdot}b+\left(-uw_{3}y^{2}+uw_{3}+w_{1}w_{2}y-w_{2}^{2}\right)\tilde{a}_{1}{\cdot}v_{\perp}\right)}{4\left(y^{2}-1\right)^{2}\left(-u^{2}\left(y^{2}-1\right)-w_{2}^{2}\right){}^{7/2}}$
$\displaystyle\times\left(\left(y^{2}-1\right)\left(u^{2}\left(y^{2}-1\right)-4w_{2}^{2}\right)\left(\tilde{a}_{1}{\cdot}b\right){}^{2}+\left(u^{2}\left(y^{2}-1\right)+w_{2}^{2}\right)\left(\tilde{a}_{1}{\cdot}v_{\perp}\right)^{2}\right)+\cdots,$
(6.16)

where
$w_{1}:=k{\cdot}v_{1},w_{2}:=k{\cdot}v_{2},w_{3}:=k{\cdot}b,w_{0}:=k{\cdot}v_{\perp}$.

A third example is

$\displaystyle\frac{c_{3}\omega\left(q_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}q_{1}{\cdot}S_{1}{\cdot}v_{2}e^{i\omega
b{\cdot}q_{1}-iu\omega}G_{1}\left(-i\omega\tilde{a}_{1}{\cdot}q_{1}\right)}{q_{1}{\cdot}q_{1}k{\cdot}q_{1}}.$
(6.17)

In this case, there is a trivial log-divergent term which we remove using the
method described in Section 5.3. Thus the integral gives

$\displaystyle\sum_{j=\rm I}^{\rm II}{(-1)^{j-1}\over
4(y^{2}-1)}\Bigg{[}\frac{ic_{3}\left(\tilde{v}_{j}{\cdot}S_{1}{\cdot}v_{2}\sqrt{w_{2}^{2}\tilde{b}_{j}{\cdot}\tilde{b}_{j}-u^{2}\left(y^{2}-1\right)}+u\left(y^{2}-1\right)\tilde{b}_{j}{\cdot}S_{1}{\cdot}v_{2}\right)}{k{\cdot}\tilde{v}_{j}\sqrt{w_{2}^{2}\tilde{b}_{j}{\cdot}\tilde{b}_{j}+u^{2}\left(1-y^{2}\right)}+u\left(y^{2}-1\right)\tilde{b}_{j}{\cdot}k+w_{2}\left(w_{2}-w_{1}y\right)\tilde{b}_{j}{\cdot}\tilde{b}_{j}}$
$\displaystyle{\left(\tilde{v}_{j}{\cdot}F_{k}{\cdot}v_{1}\sqrt{w_{2}^{2}\tilde{b}_{j}{\cdot}\tilde{b}_{j}+u^{2}\left(1-y^{2}\right)}+u\left(y^{2}-1\right)\tilde{b}_{j}{\cdot}F_{k}{\cdot}v_{1}-w_{2}\tilde{b}_{j}{\cdot}\tilde{b}_{j}v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)^{2}\over\tilde{b}_{j}{\cdot}\tilde{b}_{j}\sqrt{w_{2}^{2}\tilde{b}_{j}{\cdot}\tilde{b}_{j}-u^{2}\left(y^{2}-1\right)}\left(\tilde{a}_{1}{\cdot}\tilde{v}_{j}\sqrt{w_{2}^{2}\tilde{b}_{j}{\cdot}\tilde{b}_{j}-u^{2}\left(y^{2}-1\right)}+u\left(y^{2}-1\right)\tilde{a}_{1}{\cdot}\tilde{b}_{j}\right)}$
$\displaystyle-\frac{ic_{3}\left(\tilde{v}_{j}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}\tilde{v}_{j}{\cdot}S_{1}{\cdot}v_{2}}{\tilde{b}_{j}{\cdot}\tilde{b}_{j}\tilde{a}_{1}{\cdot}\tilde{v}_{j}k{\cdot}\tilde{v}_{j}}\Bigg{]}.$
(6.18)

The last term is the removed log-divergent term that can be removed trivially.
One can also directly check that the spurious poles
$1\over\tilde{a}_{1}{\cdot}\tilde{v}_{j}$ and
$1\over\left(\tilde{a}_{1}{\cdot}\tilde{v}_{j}\sqrt{w_{2}^{2}\tilde{b}_{j}{\cdot}\tilde{b}_{j}-u^{2}\left(y^{2}-1\right)}+u\left(y^{2}-1\right)\tilde{a}_{1}{\cdot}\tilde{b}_{j}\right)$
cancel among the sectors. Again, by expanding for $|{a}_{1}|\ll 1$ we see that
the spurious poles cancel,

$\displaystyle\frac{-ic_{3}w_{2}^{2}\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}b{\cdot}S_{1}{\cdot}v_{2}}{2\left(y^{2}-1\right)\left(u^{2}\left(1-y^{2}\right)-w_{2}^{2}\right){}^{3/2}}\frac{1}{\big{(}w_{1}w_{2}y-w_{2}^{2}-w_{0}\sqrt{u^{2}\left(1-y^{2}\right)-w_{2}^{2}}-uw_{3}\left(y^{2}-1\right)\big{)}^{2}}$
$\displaystyle\times\Big{(}{-w_{0}(u^{2}(y^{2}-1)-w_{2}^{2})\sqrt{u^{2}\left(1-y^{2}\right)-w_{2}^{2}}-u^{3}\left(y^{2}-1\right)^{2}w_{3}-w_{1}w_{2}^{3}y+w_{2}^{4}}\Big{)}$
$\displaystyle+\cdots\,.$ (6.19)

The spin-independent spurious pole is still present and will only cancel after
summing with the corresponding terms in the $q_{2}^{2}$-channel.

###  6.2 The $q_{2}^{2}$-channel

For the second graph in (4.1), the physical propagator is

$\displaystyle{1\over q_{2}^{2}}={1\over(k-q_{1})^{2}}.$ (6.20)

It is convenient to shift the integration variable as $q_{1}\rightarrow
q_{1}+k$ and the physical propagator becomes simply ${1\over q_{1}^{2}}$, the
same as in the $q_{1}^{2}$-channel. The spurious pole $1\over k{\cdot}q_{1}$
is invariant under the shift due to the on-shell condition of the external
graviton, while the spin dependent spurious poles become

$\displaystyle{1\over a{\cdot}q_{1}},$ $\displaystyle{1\over a{\cdot}k}.$
(6.21)

The delta functions coming from the definition of the waveform (5.14) are
shifted correspondingly as

$\displaystyle\delta(2v_{1}{\cdot}q_{1}+2w_{1}),$
$\displaystyle\delta(2v_{2}{\cdot}q_{1}).$ (6.22)

Applying the residue theorem to evaluate the integrals is then the exact same
process as the $q_{1}^{2}$-channel with the following integrand

$\displaystyle(\omega^{2}e^{i\omega\left(b{\cdot}k+b{\cdot}q_{1}\right)-i\omega
u})\mathcal{M}_{q_{2}^{2}}=(\omega^{2}e^{i\omega\left(b{\cdot}k+b{\cdot}q_{1}\right)-i\omega
u})\times$ $\displaystyle\Bigg{[}\frac{\cosh\left(\omega
a_{1}{\cdot}(k+q_{1})\right)\left(2w_{1}^{2}\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)^{2}-4w_{1}yv_{1}{\cdot}F_{k}{\cdot}v_{2}q_{1}{\cdot}F_{k}{\cdot}v_{1}+\left(2y^{2}-1\right)\left(q_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}\right)}{4w_{1}^{2}\omega^{2}q_{1}{\cdot}q_{1}k{\cdot}q_{1}}$
$\displaystyle+\frac{iG_{1}\left(\omega
a_{1}{\cdot}(k+q_{1})\right)}{2w_{1}^{2}\omega
q_{1}{\cdot}q_{1}k{\cdot}q_{1}}\Big{(}w_{1}v_{1}{\cdot}F_{k}{\cdot}v_{2}-yq_{1}{\cdot}F_{k}{\cdot}v_{1}\Big{)}\Big{(}-w_{2}q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}$
$\displaystyle+k{\cdot}S_{1}{\cdot}v_{2}q_{1}{\cdot}F_{k}{\cdot}v_{1}+q_{1}{\cdot}S_{1}{\cdot}v_{2}q_{1}{\cdot}F_{k}{\cdot}v_{1}+v_{1}{\cdot}F_{k}{\cdot}v_{2}k{\cdot}S_{1}{\cdot}q_{1}-w_{2}k{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}\Big{)}$
$\displaystyle+\frac{G_{2}\left(\omega a_{1}{\cdot}q_{1},\omega
a_{1}{\cdot}k\right)\left(c_{33}q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}+\left(c_{25}q_{1}{\cdot}S_{1}{\cdot}v_{2}+c_{48}\right)q_{1}{\cdot}F_{k}{\cdot}v_{1}+c_{41}q_{1}{\cdot}S_{1}{\cdot}v_{2}+c_{19}\right)}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{G_{1}\left(\omega
a_{1}{\cdot}(k+q_{1})\right)\left(c_{29}q_{1}{\cdot}F_{k}{\cdot}v_{1}+c_{43}\right)}{\omega
q_{1}{\cdot}q_{1}}+{G_{1}\left(\omega a_{1}{\cdot}k\right)G_{1}\left(\omega
a_{1}{\cdot}q_{1}\right)\over
q_{1}{\cdot}q_{1}}\Big{(}c_{2}+c_{68}q_{1}{\cdot}F_{k}{\cdot}v_{1}$
$\displaystyle+a_{1}{\cdot}q_{1}\left(c_{63}q_{1}{\cdot}F_{k}{\cdot}v_{1}+c_{70}\right)+c_{62}\left(q_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}+k{\cdot}q_{1}\left(c_{7}q_{1}{\cdot}F_{k}{\cdot}v_{1}+c_{11}\right)\Big{)}$
$\displaystyle+{G_{1}\left(\omega a_{1}{\cdot}k\right)\cosh\left(\omega
a_{1}{\cdot}q_{1}\right)\over\omega
q_{1}{\cdot}q_{1}}\left(c_{12}q_{1}{\cdot}F_{k}{\cdot}v_{1}+c_{51}\right)+G^{\prime\prime}_{e}\left(\omega
a_{1}{\cdot}q_{1},\omega
a_{1}{\cdot}k\right)\omega^{2}\Big{(}\frac{c_{4}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{c_{16}a_{1}{\cdot}q_{1}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{58}a_{1}{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{74}\left(a_{1}{\cdot}q_{1}\right){}^{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{57}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{72}a_{1}{\cdot}q_{1}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}\Big{)}$
$\displaystyle+G^{\prime}_{e}\left(\omega a_{1}{\cdot}q_{1},\omega
a_{1}{\cdot}k\right)\omega\Big{(}\frac{c_{66}a_{1}{\cdot}q_{1}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{64}a_{1}{\cdot}q_{1}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{69}\left(a_{1}{\cdot}q_{1}\right)^{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{82}a_{1}{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{c_{10}k{\cdot}q_{1}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{14}\left(q_{1}{\cdot}F_{k}{\cdot}v_{1}\right)^{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{71}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{22}\left(k{\cdot}q_{1}\right)^{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{67}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{59}}{q_{1}{\cdot}q_{1}}\Big{)}$
$\displaystyle+G^{{}^{\prime\prime}}_{o}\left(\omega a_{1}{\cdot}q_{1},\omega
a_{1}{\cdot}k\right)\omega^{2}\Big{(}\frac{c_{44}a_{1}{\cdot}q_{1}q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{52}a_{1}{\cdot}q_{1}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{46}a_{1}{\cdot}q_{1}k{\cdot}S_{1}{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{c_{53}a_{1}{\cdot}q_{1}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{76}a_{1}{\cdot}q_{1}q_{1}{\cdot}S_{1}{\cdot}v_{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{61}a_{1}{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{81}\left(a_{1}{\cdot}q_{1}\right){}^{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{45}q_{1}{\cdot}S_{1}{\cdot}v_{2}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{c_{5}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{47}k{\cdot}q_{1}q_{1}{\cdot}S_{1}{\cdot}v_{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{6}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}\Big{)}+G^{{}^{\prime}}_{o}\left(\omega
a_{1}{\cdot}q_{1},\omega
a_{1}{\cdot}k\right)\omega\Big{(}\frac{c_{32}a_{1}{\cdot}q_{1}q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{c_{20}a_{1}{\cdot}q_{1}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{13}a_{1}{\cdot}q_{1}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{40}a_{1}{\cdot}q_{1}q_{1}{\cdot}S_{1}{\cdot}v_{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{50}\left(a_{1}{\cdot}q_{1}\right){}^{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{80}a_{1}{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{c_{15}q_{1}{\cdot}S_{1}{\cdot}v_{2}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{34}q_{1}{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{21}q_{1}{\cdot}F_{k}{\cdot}v_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{26}k{\cdot}q_{1}q_{1}{\cdot}S_{1}{\cdot}v_{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{35}k{\cdot}S_{1}{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}$
$\displaystyle+\frac{c_{49}k{\cdot}q_{1}}{q_{1}{\cdot}q_{1}}+\frac{c_{73}q_{1}{\cdot}S_{1}{\cdot}v_{2}}{q_{1}{\cdot}q_{1}}+\frac{c_{60}}{q_{1}{\cdot}q_{1}}\Big{)}\Bigg{]},$
(6.23)

where

$\displaystyle G_{o}^{{}^{\prime}}(x_{1},x_{2})$
$\displaystyle:=(\partial_{x_{1}}-\partial_{x_{2}})G_{2}(x_{1},x_{2}),$
$\displaystyle G_{e}^{{}^{\prime}}(x_{1},x_{2})$
$\displaystyle:=(\partial_{x_{1}}-\partial_{x_{2}})G_{1}(x_{1})G_{1}(x_{2})$
$\displaystyle G_{o}^{{}^{\prime\prime}}(x_{1},x_{2})$
$\displaystyle:={(\partial_{x_{1}}-\partial_{x_{2}})^{2}\over
2}G_{2}(x_{1},x_{2}),$ $\displaystyle G_{e}^{{}^{\prime\prime}}(x_{1},x_{2})$
$\displaystyle:={(\partial_{x_{1}}-\partial_{x_{2}})^{2}\over
2}G_{1}(x_{1})G_{1}(x_{2})\,.$ (6.24)

The coefficients are listed in Appendix B. The integrand is composed of four
parts:

  * •

terms including the functions $G_{1}$ and $\cosh$ and with spurious pole
$1\over k{\cdot}q_{1}$: In this part, the entire function are $G_{1}(\omega
a_{1}{\cdot}(k+q_{1})),\cosh(\omega a_{1}{\cdot}(k+q_{1}))$. All the terms are
of $\mathcal{O}(\omega^{0})$. It is easy to see that the spurious pole is
cancelled when adding the corresponding terms in the $q_{1}^{2}$-channel.

  * •

terms with the functions $\cosh,G_{1},G_{2}$ and without the spurious pole
$1\over k{\cdot}q_{1}$: They are all of $\mathcal{O}(\omega^{0})$.

  * •

terms with the functions $G^{\prime}_{o},G^{\prime}_{e}$: They are of
$\mathcal{O}(\omega^{0})$ and $\mathcal{O}(\omega^{1})$. All of them do not
contain the spurious pole $1\over k{\cdot}q_{1}$

  * •

terms with the functions
$G^{{}^{\prime\prime}}_{o},G^{{}^{\prime\prime}}_{e}$: They are of
$\mathcal{O}(\omega^{0})$, $\mathcal{O}(\omega^{1})$ and
$\mathcal{O}(\omega^{2})$. All of them do not contain the spurious pole
$1\over k{\cdot}q_{1}$. They also do not contain the physical massive pole
${1\over k{\cdot}v_{1}}={1\over w_{1}}$.

Unlike in the $q_{1}^{2}$-channel, here we have more general entire functions
coming from the compton amplitude for the particle with spin $a_{1}$ and
consequently we now have four sectors with different exponential factors

$\displaystyle{\rm(I)}:\quad$
$\displaystyle\exp\left(-i\omega\left(-\tilde{a}_{1}{\cdot}k-\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}k-b{\cdot}q_{1}+u\right)\right),$
$\displaystyle{\rm(II)}:\quad$
$\displaystyle\exp\left(-i\omega\left(-\tilde{a}_{1}{\cdot}k+\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}k-b{\cdot}q_{1}+u\right)\right),$
$\displaystyle{\rm(III)}:\quad$
$\displaystyle\exp\left(-i\omega\left(\tilde{a}_{1}{\cdot}k-\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}k-b{\cdot}q_{1}+u\right)\right),$
$\displaystyle{\rm(IV)}:\quad$
$\displaystyle\exp\left(-i\omega\left(\tilde{a}_{1}{\cdot}k+\tilde{a}_{1}{\cdot}q_{1}-b{\cdot}k-b{\cdot}q_{1}+u\right)\right).$
(6.25)

In each sector, we still use the sector-dependent basis in (6.4) and
parameterise the $q_{1}$ variable of (6.7) with

$\displaystyle\begin{cases}\tilde{b}_{\rm(I)}=\tilde{b}_{\rm(III)}=-b-\tilde{a}_{1}\\\
\tilde{b}_{\rm(II)}=\tilde{b}_{\rm(IV)}=-b+\tilde{a}_{1}\\\ \end{cases},$
$\displaystyle\tilde{v}_{j}=\epsilon(v_{1},v_{2},\tilde{b}_{j},\bullet)\,,$
$\displaystyle j={\rm I,II,III,IV}.$ (6.26)

Using this, the extra $\delta$-functions in each sector are

$\displaystyle{\rm(I)}:\quad$
$\displaystyle\delta\left(-\tilde{a}_{1}{\cdot}k-b{\cdot}k+\tilde{b}_{(\rm
I)}{\cdot}q_{1}+u\right),$ $\displaystyle{(\rm III)}:\quad$
$\displaystyle\delta\left(\tilde{a}_{1}{\cdot}k-b{\cdot}k+\tilde{b}_{(\rm
III)}{\cdot}q_{1}+u\right),$ $\displaystyle{\rm(II)}:\quad$
$\displaystyle\delta\left(-\tilde{a}_{1}{\cdot}k-b{\cdot}k+\tilde{b}_{(\rm
II)}{\cdot}q_{1}+u\right),$ $\displaystyle{\rm(IV)}:\quad$
$\displaystyle\delta\left(\tilde{a}_{1}{\cdot}k-b{\cdot}k+\tilde{b}_{(\rm
IV)}{\cdot}q_{1}+u\right)\,.$ (6.27)

Then the integration localises onto a hyperplane for each sector and the
method is exactly the same as in the last section. The new feature is the
appearance of the entire functions
$G^{\prime}_{o},G^{\prime}_{e},G^{{}^{\prime\prime}}_{o},G^{{}^{\prime\prime}}_{e}$.
The derivatives will lead to entire functions that are not homogeneous with
respect to $\omega$ even while ignoring the exponential factors. Hence the
integrand has three different powers of $\omega$, schematically

$\displaystyle
1\times(\bullet)e^{\bullet}+\omega\times(\bullet)e^{\bullet}+\omega^{2}\times(\bullet)e^{\bullet}.$
(6.28)

This leads to an integral result with the structure

$\displaystyle
1\times(\bullet\bullet)+i\partial_{u}(\bullet\bullet)-\partial^{2}_{u}(\bullet\bullet)\,.$
(6.29)

In practice, our result is obtained from evaluating the $\delta$-functions as
usual and replacing $\omega$ by $i\partial_{u}$ at the end, as shown in
(5.37).

We now perform a numerical check that the result is free of spin-dependent
spurious poles. After a random numerical replacement, the spin-dependent
spurious pole is located at

$\displaystyle\tilde{a}_{1}\cdot v_{\perp}-\frac{42\tilde{a}_{1}\cdot
b}{5}=\xi\rightarrow 0\,.$ (6.30)

We extract the singular terms at the spurious pole, finding

$\displaystyle-\frac{1323\sqrt{3}u}{640\sqrt{-25u^{2}-700u-17444}\xi^{3}}+\frac{5245317u}{1100800\sqrt{-75u^{2}-2100u-58732}\xi^{3}}$
$\displaystyle-\frac{9261\sqrt{3}}{320\sqrt{-25u^{2}-700u-17444}\xi^{3}}+\frac{36717219}{550400\sqrt{-75u^{2}-2100u-58732}\xi^{3}}-\frac{277641}{1100800\xi^{3}}$
$\displaystyle+\cdots\text{1345 more terms}\cdots$ (6.31)

After applying the derivative operators and setting $u{=}0$ we get

$\displaystyle\frac{0.0123034\,+0.140702i}{\xi^{2}}+\frac{0.0205579\,-0.00156661i}{\xi}$
$\displaystyle-\frac{0.0123034\,+0.140702i}{\xi^{2}}-\frac{0.0222046\,-0.0033312i}{\xi}$
$\displaystyle+\frac{0.00164663\,-0.00176459i}{\xi}=0.$ (6.32)

We have also tested that for several other values of $u$ and find the singular
term is always vanishing. This indicates that the final result is free of
spurious poles to any spin order.

###  6.3 Discussion of the resummed spin waveform

The final result of the waveform has three contributions coming from terms
each with up to two $u$-derivatives acting on them

$\displaystyle h^{\infty}(u)$ $\displaystyle=\Big{(}h^{\infty}_{0}(u)+\omega
h^{\infty}_{1}(u)+\omega^{2}h^{\infty}_{2}(u)\Big{)}\Big{|}_{\omega\rightarrow
i\partial_{u}}\,,$ (6.33)

and in terms of $\tilde{a},b,v_{\perp},k,v_{1},v_{2}$ takes the following
schematic form

$\displaystyle\frac{\left(2y^{2}-1\right)v_{1}{\cdot}F_{2}{\cdot}v_{2}\left(v_{\perp}{\cdot}F_{2}{\cdot}v_{1}\left(\tilde{a}_{1}{\cdot}b-1\right)-\tilde{a}_{1}{\cdot}v_{\perp}b{\cdot}F_{2}{\cdot}v_{1}\right)}{8w_{1}^{2}w_{2}\left(\left(y^{2}-1\right)\left(\tilde{a}_{1}{\cdot}b-1\right){}^{2}+\left(\tilde{a}_{1}{\cdot}v_{\perp}\right){}^{2}\right)}$
$\displaystyle+\frac{\left(2y^{2}-1\right)v_{1}{\cdot}F_{2}{\cdot}v_{2}\left(\tilde{a}_{1}{\cdot}v_{\perp}b{\cdot}F_{2}{\cdot}v_{1}-v_{\perp}{\cdot}F_{2}{\cdot}v_{1}\left(\tilde{a}_{1}{\cdot}b+1\right)\right)}{8w_{1}^{2}w_{2}\left(\left(y^{2}-1\right)\left(\tilde{a}_{1}{\cdot}b+1\right){}^{2}+\left(\tilde{a}_{1}{\cdot}v_{\perp}\right){}^{2}\right)}$
$\displaystyle+\cdots\text{ 336 more terms }\cdots$ (6.34)

We note that the poles in $w_{1},w_{2}$ correspond to the physical massive
poles $1\over v_{1}{\cdot}k_{2}$ and $1\over v_{2}{\cdot}k_{2}$. The singular
behaviour on these poles does not depend on the contact terms present in the
Compton amplitude, which by definition are free of such poles, and so this
behaviour is exact up to any spin order. The explicit result in the case of
$a_{1}{\cdot}v_{2}{=}0$ can be found in the GitHub repository. In the
remainder of this subsection, we focus on the properties of the waveform by
plotting its numerical values as a function of the retarded time $u$ and the
spin parameter. As in the scalar case, we can choose a frame such that the
kinematics are given by

$\displaystyle v_{1}$
$\displaystyle=(1,0,0,0),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}v_{2}=(y,\sqrt{y^{2}-1},0,0)$
$\displaystyle k$
$\displaystyle=(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}v_{\perp}=(0,0,\sqrt{y^{2}-1},0)$
$\displaystyle\varepsilon$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}0,\cos\theta\cos\phi-i\sin\phi,\cos\theta\sin\phi+i\cos\phi,-\sin\theta\big{)}\,,~{}~{}b=(0,0,0,1).$
(6.35)

Then we can further parameterise the constrained spin $a_{1}$ such that
$a_{1}{\cdot}v_{2}=0$ as

$\displaystyle a_{1}$ $\displaystyle=(0,0,|a|\cos\psi,|a|\sin\psi),$ (6.36)

where $|a|$ is the magnitude of the spin and $\psi$ the angle of the spin’s
direction in the plane orthogonal to $v_{1}$ and $v_{2}$.

In Figures 2 and 3 we show the time-domain waveform $h_{+}$ at $y={3\over
2},\theta={\pi\over 4},\phi={\pi\over 4}$. In all of our graphs, we set
$\kappa{=}m_{1}{=}m_{2}{=}1$, so each graph is missing a factor of
$\kappa^{4}m_{1}m_{2}$. Figure 2 shows the waveform dependence on the retarded
time $u$ and angle $\psi$. When the magnitude of the spin is equal to $0.2$, a
small spin parameter compared to the magnitude of the impact parameter $|b|$,
the time-domain waveform is similar to the scalar case. The spin effect on the
waveform can then be taken as a perturbation on top of the spinless cases.
However, for a larger magnitudes, for example $0.65$, the time-domain waveform
is modified greatly due to the effects of spin. To highlight the effect of
changing the magnitude of the spin, in the Figure 3 we plot the various
spinning waveforms at fixed spin angle $\psi=\frac{\pi}{4}$.

Figure 2: Waveform of $h^{\infty}_{+}$ at $|a|={0.0}$, $|a|={0.2}$,
$|a|={0.65}$. Note that here $|a|,u$ are in units of the impact parameter
$|b|$ as we have set $|b|=1$. For general impact parameter, one can simply
replace $|a|,u$ with $|a|/|b|,u/|b|$. Similarly in Figure 3.
$-2$$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$2$$-1.5$$-1$$-0.5$$0$$0.5$$1$$\cdot
10^{-2}$$u$${h}_{+}^{\infty}$$|a|$=0.65$|a|$=0.4$|a|$=0.2$|a|$=0.0 Figure 3:
Waveform of $h^{\infty}_{+}\,\,$ at $|a|={0.0}$, $|a|={0.2}$, $|a|={0.65}$
with spin angle $\psi={\pi\over 4}$.

From the waveform, we can extract the gravitational memory effect using

$\displaystyle\Delta h^{\infty}=h^{\infty}(+\infty)-h^{\infty}(-\infty).$
(6.37)

We first study the Taylor expansion around $u\rightarrow\infty$ of the
individual pieces $h^{\infty}_{i}(u)$ which contribute to the waveform in
(6.33) and find they all have similar behaviour

$\displaystyle h^{\infty}_{i}(u)\sim c_{i}+\mathcal{O}\Big{(}{1\over
u}\Big{)}.$ (6.38)

The contributions $h_{1}^{\infty}$ and $h_{2}^{\infty}$ have the derivative
$i\partial_{u}$ acting on them as such their behaviour in the large $u$ limit
is sub-leading and they do not contribute to the memory. The memory can then
by computed from the contribution $h_{0}^{\infty}$ and we find

$\displaystyle\Delta h^{\infty}$
$\displaystyle=\frac{\kappa^{4}m_{1}m_{2}}{8\pi}\frac{v_{1}{\cdot}F_{k}{\cdot}v_{2}}{4w_{1}^{2}w_{2}^{2}\sqrt{y^{2}-1}\left(4\left(a_{1}{\cdot}b\right){}^{2}+\left(a_{1}{\cdot}a_{1}+1\right){}^{2}\right)}$
$\displaystyle\times\Big{(}2a_{1}{\cdot}bv_{1}{\cdot}F_{k}{\cdot}v_{2}\left(\left(1-2y^{2}\right)a_{1}{\cdot}k+w_{0}y\left(a_{1}{\cdot}a_{1}-1\right)\right)$
$\displaystyle+4w_{2}a_{1}{\cdot}b\left(\left(1-2y^{2}\right)a_{1}{\cdot}F_{k}{\cdot}v_{1}+y\left(a_{1}{\cdot}a_{1}-1\right)v_{\perp}{\cdot}F_{k}{\cdot}v_{1}\right)$
$\displaystyle-\left(a_{1}{\cdot}a_{1}+1\right)\left(-2w_{2}b{\cdot}F_{k}{\cdot}v_{1}+w_{3}v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)\left(2yv_{2}{\cdot}S_{1}{\cdot}b+2y^{2}-1\right)\Big{)}.$
(6.39)

In this compact formula, we notice that all the terms contain at least one
pole of $w_{1}$ and $w_{2}$. This indicates that contact terms in the Compton
amplitude do not contribute to the memory at any order in spin. As such we
should expect that the waveform we have computed fully captures all orders in
the spin memory. In addition, we compute a formula for the tree-level
gravitational memory at all orders in spin (8.36) in Section 8 below using a
classical soft factor. The two formulae are indeed in agreement. We also
mention again that we have compared our results to those of [157], finding
agreement (see also [165]).

A graph of the memory, for the same kinematics as before and various values of
the magnitude of the spin and direction, is presented in Figure 4. When $|a|$
tends to 1, there are two singular points at $\psi=0,\pi$ corresponding to
when the spin vector and impact parameter are orthogonal.

Figure 4: Gravitational memory: the top graph (blue) is the imaginary part,
corresponding to $h^{\infty}_{\times}$, and the bottom graph (orange) is the
real part, corresponding to $h^{\infty}_{+}$.

##  7 Comparison with the spin-expanded integrand

If the spin parameter is small with respect to the impact parameter
$|a|\ll|b|$ then we can evaluate the waveform integration order by order in a
spin expansion. When we perform such an expansion the tree-level five-point
amplitude is free of the spin-dependent spurious poles. One can still work in
the $q_{1}^{2}$ and $q_{2}^{2}$ channels separately, which only contain one
spurious pole $\frac{1}{q_{1}{\cdot}k}$. After the usual re-scaling
$q_{i}=\omega\hat{q}_{i}$, the waveform integrand is given by

$\displaystyle\mathcal{M}_{q_{1}^{2}}={1\over\omega^{2}}\Big{(}\sum_{i=0}^{\infty}\mathcal{M}^{(i)}_{q_{1}^{2}}\omega^{i}\Big{)},$
$\displaystyle\mathcal{M}_{q_{2}^{2}}={1\over\omega^{2}}\Big{(}\sum_{i=0}^{\infty}\mathcal{M}^{(i)}_{q_{2}^{2}}\omega^{i}\Big{)}.$
(7.1)

We still integrate over the frequency first but now after expanding in the
spin parameter there is only one sector per cut. Thus the integrand contains
the same delta functions as in the scalar case

$\displaystyle\begin{cases}\quad\delta\left(-b{\cdot}q_{1}+u\right)&q_{1}^{2}\text{-channel}\\\
\quad\delta\left(-b{\cdot}q_{2}+u\right)&q_{2}^{2}\text{-channel}.\end{cases}$
(7.2)

The extra powers of $\omega$ become derivatives in the time, $i\partial_{u}$,
as before. Now using the original parameterisation (5.20), after we localise
$z_{b}$ each term in the integrand belongs to one of the following general
expressions

$\displaystyle{c_{0}+c_{1}z_{v}+c_{2}z_{v}^{2}+c_{3}z_{v}^{3}\cdots\over(z_{v}+\tilde{c})(z_{v}^{2}+\bar{c})},$
$\displaystyle{\bar{c}_{0}+\bar{c}_{1}z_{v}+\bar{c}_{2}z_{v}^{2}+\bar{c}_{3}z_{v}^{3}\cdots\over(z_{v}^{2}+\bar{c})}\,,$
(7.3)

where the $c$’s are functions of the external kinematics. The ${1\over
z_{v}^{2}+c^{\prime}_{2}}$ is the physical $q_{1}^{2}$ (or $q_{2}^{2}$) pole
and ${1\over z_{v}+c^{\prime}_{1}}$ is the spurious pole at $q_{1}{\cdot}k$.
Since the waveform only receives contributions from the physical pole, we can
use polynomial division to reduce the numerators. Explicitly, we perform
polynomial division over the physical pole, and obtain

$\displaystyle{c^{\prime}_{0}+c^{\prime}_{1}z_{v}\over(z_{v}+\tilde{c})(z_{v}^{2}+\bar{c})}+(\text{terms
without physical poles}),$
$\displaystyle{\bar{c}^{\prime}_{0}+\bar{c}^{\prime}_{1}z_{v}\over(z_{v}^{2}+\bar{c})}+(\text{terms
without physical poles}).$ (7.4)

Terms without physical poles correspond to contributions that are proportional
to delta functions in $b$ (and derivatives thereof) and hence do not
contribute to the long-range waveform. Thus we only have the following two
types of master integrals after performing partial fractions over the spurious
pole

$\displaystyle{c^{{}^{\prime\prime}}_{0}\over(z_{v}+\tilde{c})(z_{v}^{2}+\bar{c})}\,,$
$\displaystyle{\bar{c}^{{}^{\prime\prime}}_{0}+\bar{c}^{{}^{\prime\prime}}_{1}z_{v}\over(z_{v}^{2}+\bar{c})}={\bar{c}^{{}^{\prime\prime}}_{0}\over(z_{v}^{2}+\bar{c})}+\text{(terms
that integrate to zero)}.$ (7.5)

The two master integrals can then be evaluated by calculating the residue on
the physical pole. The final result of the $q_{1}$ integral is of the form

$\displaystyle h^{\infty}_{\rm expanded}(u)$
$\displaystyle=\Big{(}\sum_{i=0}^{\infty}\omega^{i}h^{\infty}_{i}(u)\Big{)}|_{\omega\rightarrow
i\partial_{u}}$
$\displaystyle\sim\Bigg{[}\frac{\omega^{4}a_{1}{\cdot}a_{1}a_{1}{\cdot}kv_{\perp}{\cdot}F_{2}{\cdot}v_{1}a_{1}{\cdot}F_{2}{\cdot}v_{1}}{24\left(y^{2}-1\right)}-\frac{i\omega^{3}a_{1}{\cdot}a_{1}v_{\perp}{\cdot}F_{2}{\cdot}v_{1}\text{tr}\left(F_{2},S_{1}\right)}{48\left(y^{2}-1\right)}$
$\displaystyle-\frac{w_{0}\left(2y^{2}-1\right)\left(v_{1}{\cdot}F_{2}{\cdot}v_{2}\right){}^{2}}{8w_{1}^{2}w_{2}^{2}\left(y^{2}-1\right)}-\frac{\left(2y^{2}-1\right)\left(v_{1}{\cdot}F_{2}{\cdot}v_{2}\right){}^{2}}{8w_{1}^{2}w_{2}^{2}\left(y^{2}-1\right)\sqrt{-u^{2}\left(y^{2}-1\right)-w_{2}^{2}}}$
$\displaystyle\times\left(-w_{0}\sqrt{u^{2}\left(-y^{2}\right)+u^{2}-w_{2}^{2}}+uw_{3}y^{2}-uw_{3}+w_{1}w_{2}y-w_{2}^{2}\right)$
$\displaystyle+\cdots\text{more
terms}\cdots\Bigg{]}\Bigg{|}_{\omega\rightarrow i\partial_{u}}.$ (7.6)

The full waveform result expanded in the spin parameter up to $a^{4}$ order is
included in the GitHub repository. Our result contains contributions at orders
beyond $a^{4}$ but these will in general be incomplete until possible
additional contact terms are included in the Compton amplitude.

We now comment on the difference between the resummed spinning waveform versus
the spin-expanded waveform truncated at $\mathcal{O}(a^{4})$. To do so, we
illustrate the spin-expanded waveform at $|a|=0.2$ and $|a|=0.65$ in Figure 5.
Comparing with the resummed result shown for the same values in Figure 2, we
see that for $|a|=0.2$ the spin expanded result at $\mathcal{O}(a^{4})$ is
accurate. However, at $|a|=0.65$ the spin expansion breaks down and the
perturbative result is no longer valid.

Figure 5: waveform of $h^{+}$ at $|a|={0.65}$, $|a|={0.2}$

To see more clearly the difference between the resummed spin result and the
perturbative spin result truncated at $\mathcal{O}(a^{4})$, we also fix
$\psi={\pi\over 4}$. For lower values of spin, for example $|a|={0.2}$, the
expanded and resummed waveforms are nearly identical, as shown in the right-
hand side of Figure 6. Conversely, for large values of the spin, for example
$|a|={0.65}$, the expanded and resummed results are markedly different,
although their limiting values as $u\rightarrow\pm\infty$ are similar.

$-2$$-1$$0$$1$$-1.5$$-1$$-0.5$$0$$0.5$$1$$\cdot
10^{-2}$$u$${h}_{+}^{\infty}\,\,$$y=3/2,|a|=0.65$resummedexpanded

$-2$$-1$$0$$1$$-1$$-0.5$$0$$0.5$$1$$\cdot
10^{-2}$$u$${h}_{+}^{\infty\,\,}$$y=3/2,|a|=0.2$resummedexpanded

Figure 6: Comparison of the expanded and resummed waveforms $h_{+}$ at
$|a|=0.65$ and $|a|=0.2$ with spin angle $\psi={\pi\over 4}$. For $|a|=0.2$
the graphs are indistinguishable.

##  8 Gravitational memory

###  8.1 General strategy

An elegant way to compute the memory was discussed in [114] for the spinless
case, and we adapt it to the case of spinning heavy particles. Given a
function

$\displaystyle
f(u)\coloneqq\int_{-\infty}^{+\infty}\\!\frac{d\omega}{2\pi}e^{-i\omega
u}\tilde{f}(\omega)$ (8.1)

of the retarded time $u$, the memory is defined as

$\displaystyle\begin{split}\Delta f&\coloneqq
f(u\to+\infty)-f(u\to-\infty)=\int_{-\infty}^{+\infty}\\!du\,\frac{d}{du}f(u)\\\
&=-i\int_{-\infty}^{+\infty}\\!{d\omega}\,\delta({\omega})\big{[}\omega\tilde{f}(\omega)\big{]}\,,\end{split}$
(8.2)

showing that it is determined by the pole at $\omega{=}0$, i.e. its soft
limit, as observed by [166].

We now apply (8.2) to (5.13) to compute the gravitational memory, getting

$\displaystyle\Delta(h^{\infty}_{+}\pm
ih^{\infty}_{\times})=-i\frac{\kappa}{2}\left[\lim_{\omega\to
0^{+}}\big{[}\omega
W\big{(}b,k^{\pm}\big{)}\big{]}_{k=\omega(1,\mathbf{\hat{x}})}+\lim_{\omega\to
0^{-}}\big{[}\omega
W^{\ast}\big{(}b,k^{\mp}\big{)}\big{]}_{k=-\omega(1,\mathbf{\hat{x}})}\right]\,.$
(8.3)

From this relation we see that the memory effect arises from the leading soft
behaviour of the five-point amplitude, which factorises into a soft factor
times a four-point amplitude, schematically

$\displaystyle\mathcal{M}_{5}\rightarrow{\rm Soft}\times\mathcal{M}_{4}.$
(8.4)

Correspondingly, as $\omega\to 0$ the waveform tends to its leading soft
limit,

$\displaystyle W_{\rm
soft}\big{(}b,k^{h}\big{)}=-i\int\\!d\mu^{(D)}e^{iq{\cdot}b}S_{\rm W}^{\rm
HEFT}(k,q;h)\mathcal{M}_{4}^{\rm HEFT}(q)\,,$ (8.5)

where [62]

$\displaystyle\begin{split}S_{\rm W}^{\rm
HEFT}&=-\frac{\kappa}{2}\,\varepsilon_{\mu\nu}^{(h)}(k)\left[\frac{p_{1}^{\mu}q^{\nu}+p_{1}^{\nu}q^{\mu}}{p_{1}{\cdot}k}-p_{1}^{\mu}p_{1}^{\nu}\frac{q{\cdot}k}{(p_{1}{\cdot}k)^{2}}\,-\,1\leftrightarrow
2\right]\\\
&=-\frac{\kappa}{2}\,\frac{1}{\omega}\varepsilon_{\mu\nu}^{(h)}(k)\left[\frac{p_{1}^{\mu}q^{\nu}+p_{1}^{\nu}q^{\mu}}{p_{1}{\cdot}\hat{k}}-p_{1}^{\mu}p_{1}^{\nu}\frac{q{\cdot}\hat{k}}{(p_{1}{\cdot}\hat{k})^{2}}\,-\,1\leftrightarrow
2\right]\,,\,\end{split}$ (8.6)

is the classical Weinberg soft factor for the emission of a graviton with
momentum $k{=}\omega\hat{k}$ and helicity $h$, with $q{=}q_{1}{=}-q_{2}$ in
the soft limit and $\hat{k}{=}(1,\mathbf{\hat{x}})$ (see [62] for a derivation
of the classical soft factor and a discussion of classical limits in the HEFT
context).

We then change integration variables $q{\to}-q$, and use

$\displaystyle S_{\rm W}^{\rm HEFT}(-k,-q;h)$ $\displaystyle=S_{\rm W}^{\rm
HEFT}(k,q;h)\,,$ (8.7) $\displaystyle\big{[}S_{\rm W}^{\rm
HEFT}(k,q,-h)\big{]}^{\ast}$ $\displaystyle=S_{\rm W}^{\rm HEFT}(k,q,h)\,,$
(8.9)

also noting that, at tree level in the spinning (and spinless) case,666In the
spinless case we further have $\mathcal{M}_{4}^{\rm
HEFT}(-q)=\mathcal{M}_{4}^{\rm HEFT}(q)$. This is no longer true in the
presence of spin.

$\displaystyle-i\mathcal{M}_{4}^{\rm HEFT}(q)=\big{[}-i\mathcal{M}_{4}^{\rm
HEFT}(-q)\big{]}^{\ast}\,,$ (8.10)

which can be checked from the explicit expression derived later in (8.26).
With these observations, we get

$\displaystyle\begin{split}\left.W_{\rm
soft}^{\ast}\big{(}b,k^{-h}\big{)}\right|_{k=-\omega(1,\hat{\mathbf{x}})}&=\int\\!d\mu^{(D)}e^{-iq{\cdot}b}\big{[}S_{\rm
W}^{\rm HEFT}(-k,q;-h)\big{]}^{\ast}(-i\mathcal{M}_{4}^{\rm
HEFT})^{\ast}(q)\\\ &=\int\\!d\mu^{(D)}e^{iq{\cdot}b}S_{\rm W}^{\rm
HEFT}(-k,-q;h)(-i\mathcal{M}_{4}^{\rm HEFT})^{\ast}(-q)\\\
&=\int\\!d\mu^{(D)}e^{iq{\cdot}b}S_{\rm W}^{\rm
HEFT}(k,q;h)(-i\mathcal{M}_{4}^{\rm HEFT})(q)\,.\end{split}$ (8.11)

Hence we can write

$\displaystyle\begin{split}&\lim_{\omega\to 0^{+}}\big{[}\omega
W\big{(}b,k^{h}\big{)}\big{]}_{k=\omega(1,\mathbf{\hat{x}})}+\lim_{\omega\to
0^{-}}\big{[}\omega
W^{\ast}\big{(}b,k^{-h}\big{)}\big{]}_{k=-\omega(1,\mathbf{\hat{x}})}\\\
&=\int\\!d\mu^{(D)}e^{iq_{\cdot}b}S_{\rm W}^{\rm
HEFT}(\hat{k},q;h)(-i\mathcal{M}_{4}^{\rm HEFT})\,.\end{split}$ (8.12)

In conclusion

$\displaystyle\begin{split}\Delta(h^{\infty}_{+}\pm
ih^{\infty}_{\times})&=-i\,\kappa\int\\!d\mu^{(D)}e^{iq_{\cdot}b}S_{\rm
W}^{\rm HEFT}(\hat{k},q;\pm)\,\big{(}-i\mathcal{M}_{4}^{\rm HEFT}\big{)}(q)\\\
&=-i\,\kappa S_{\rm W}^{\rm HEFT}\Big{(}\hat{k},-i\frac{\partial}{\partial
b};\pm\Big{)}\int\\!d\mu^{(D)}e^{iq{\cdot}b}\,\,\big{(}-i\mathcal{M}_{4}^{\rm
HEFT}\big{)}(q)\\\ &=-i\,\kappa S_{\rm W}^{\rm
HEFT}\Big{(}\hat{k},-i\frac{\partial}{\partial b};\pm\Big{)}\,\delta_{\rm
HEFT}\,,\end{split}$ (8.13)

or

$\displaystyle\begin{split}\Delta(h^{\infty}_{+}\pm
ih^{\infty}_{\times})&=-i\,\kappa S_{\rm W}^{\rm
HEFT}\Big{(}\hat{k},-i\frac{\partial}{\partial b};\pm\Big{)}\,\delta_{\rm
HEFT}\,,\end{split}$ (8.14)

where

$\displaystyle\delta_{\rm
HEFT}\coloneqq\int\\!d\mu^{(D)}e^{iq_{\cdot}b}\,\big{(}-i\mathcal{M}_{4}^{\rm
HEFT}\big{)}(q)\,,$ (8.15)

and $S_{\rm W}^{\rm HEFT}$ is given in (8.6), and we also recall that
$k{=}\omega\hat{k}$. Note that $\delta_{\rm HEFT}$ is real because of the
property (8.9).

In the spinless case, one can further simplify this result by noticing that

$\displaystyle\frac{\partial}{\partial
b_{\mu}}{=}-P\hat{b}^{\mu}\frac{\partial}{\partial J}\,,$ (8.16)

where $J=P\sqrt{-b^{2}}$, and the relation between the scattering angle and
the real part of the HEFT phase

$\displaystyle-\frac{\partial}{\partial J}\,{\rm Re}\,\delta_{\rm
HEFT}=\chi,\,$ (8.17)

which itself is already a real quantity at tree level. Using these one finds

$\displaystyle S_{\rm W}^{\rm HEFT}\Big{(}\hat{k},-i\frac{\partial}{\partial
b_{\mu}};h\Big{)}\,\delta_{\rm HEFT}=-iPS_{\rm W}^{\rm
HEFT}(\hat{k},\hat{b}^{\mu};h)\chi\,,$ (8.18)

leading to the compact relation, valid in the spinless case,

$\displaystyle\begin{split}\Delta(h^{\infty}_{+}\pm
ih^{\infty}_{\times})&=-\frac{\kappa^{2}}{2}P\varepsilon_{\rho\lambda}^{\pm\pm}s^{\rho\lambda}(\hat{k},\hat{b})\,\chi\,,\end{split}$
(8.19)

where we have set

$\displaystyle\begin{split}S_{\rm W}^{\rm
HEFT}&\coloneqq\frac{\kappa}{2}\varepsilon_{\rho\lambda}^{\pm\pm}s^{\rho\lambda}(\hat{k},\hat{b})\\\
s^{\mu\nu}(\hat{k},\hat{b})&=-\Big{[}\frac{p_{1}^{\mu}\hat{b}^{\nu}+p_{1}^{\nu}\hat{b}^{\mu}}{p_{1}{\cdot}\hat{k}}-p_{1}^{\mu}p_{1}^{\nu}\frac{\hat{b}{\cdot}\hat{k}}{(p_{1}{\cdot}\hat{k})^{2}}\,-\,1\leftrightarrow
2\Big{]}\,,\end{split}$ (8.20)

and we recall that $\hat{k}=(1,\mathbf{\hat{x}})$ and
$\hat{b}=b/\sqrt{-b^{2}}$. In the spinning case we do not have a simple
relation such as (8.17) and we will instead make use of (8.14). To compute the
gravitational memory in the spinning case we will then use (8.14) and (8.15).

We now move on to compute the tree-level-four-point amplitude that features in
(8.15).

###  8.2 Four-point two-to-two spinning amplitude

In this section we derive the tree-level amplitude for the two-to-two
scattering of two heavy objects with spin vectors $a_{1}$ and $a_{2}$ to all
orders in the spin. We will then compute its Fourier transform to impact
parameter space needed in (8.15).

We can derive the four-point amplitude using the HEFT BCFW method first
described in [62], to which we refer the reader for further details. There is
a single diagram in the $q^{2}$-channel for which we glue two of the three
point amplitudes (3.2) with the BCFW-shifted momenta described in [62]. We
find that the four-point tree-level amplitude $\mathcal{M}_{4}$ is then

$\displaystyle\begin{split}\mathcal{M}_{4}&=-{i\frac{\kappa^{2}}{q^{2}}}m_{1}^{2}m_{2}^{2}\Big{[}\Big{(}y^{2}-\frac{1}{2}\Big{)}\cosh\left(a{\cdot}q\right)\\\
&+\,i\,y\Big{(}\sinh\left(a_{2}{\cdot}q\right)\cosh\left(a_{1}{\cdot}q\right)\frac{\epsilon\left(a_{2}qv_{1}v_{2}\right)}{a_{2}{\cdot}q}+\sinh\left(a_{1}{\cdot}q\right)\cosh\left(a_{2}{\cdot}q\right)\frac{\epsilon\left(a_{1}qv_{1}v_{2}\right)}{a_{1}{\cdot}q}\Big{)}\Big{]}\\\
&+\mathcal{M}_{4,c}\,,\end{split}$ (8.21)

where $a{\coloneqq}a_{1}+a_{2}$, and the contact term $\mathcal{M}_{4,c}$ is

$\mathcal{M}_{4,c}\coloneqq-i\kappa^{2}m_{1}^{2}m_{2}^{2}\Big{[}y\,a_{1}{\cdot}v_{2}\,a_{2}{\cdot}v_{1}-a_{1}{\cdot}a_{2}\Big{(}y^{2}-{1\over
2}\Big{)}\Big{]}{\frac{\sinh(a_{1}{\cdot}q)}{a_{1}{\cdot}q}}{\frac{\sinh(a_{2}{\cdot}q)}{a_{2}{\cdot}q}}\,.$
(8.22)

We note however that contact terms play no role for the computation of the
memory, since they only contribute delta-function supported terms after
Fourier transforming to impact parameter space. We will then drop them from
now on (denoting the contact terms as $\mathcal{O}(1)$).

We now simplify the expression (8.21) for the four-point amplitude making use
of the new spin vectors [43, 123]

$\displaystyle\mathfrak{a}^{\mu}_{i}\coloneqq\frac{\epsilon(a_{i}\mu
v_{1}v_{2})}{\sqrt{y^{2}-1}}\,,\quad i=1,2,$ (8.23)

which are orthogonal to both $v_{1}$ and $v_{2}$. These quantities also
satisfy the following Gram determinant relations

$\displaystyle(a_{i}{\cdot}q)^{2}=({i\mathfrak{a}_{i}{\cdot}q})^{2}+\mathcal{O}(q^{2})\,,$
(8.24)

which are proven in Appendix A, and their “square rooted” form

$\displaystyle a_{i}{\cdot}q=\pm{i\mathfrak{a}_{i}{\cdot}q}\,,$ (8.25)

valid up to terms of $\mathcal{O}(q^{2})$, that is $q$ on-shell and so
necessarily complex. Furthermore, as both $\cosh(a_{i}{\cdot}q)$ and
${\sinh(a_{i}{\cdot}q)\over a_{i}{\cdot}q}$ are parity-even functions of
$a_{i}{\cdot}q$, the sign ambiguity drops out and the amplitude can be
simplified to

$\displaystyle\begin{split}\mathcal{M}_{4}&=-{i\kappa^{2}\over
q^{2}}{m_{1}^{2}m_{2}^{2}}\Big{\\{}\Big{(}y^{2}-\frac{1}{2}\Big{)}\cosh({i\mathfrak{a}{\cdot}q})+y{\sqrt{y^{2}-1}}\sinh({i\mathfrak{a}{\cdot}q})\Big{\\}}+\mathcal{O}(1)\,\\\
&=-{i\kappa^{2}\over q^{2}}{m_{1}^{2}m_{2}^{2}\over
2}\Big{\\{}\Big{(}y^{2}-\frac{1}{2}+y{\sqrt{y^{2}-1}}\Big{)}e^{{i\mathfrak{a}{\cdot}q}}+\Big{(}y^{2}-\frac{1}{2}-y{\sqrt{y^{2}-1}}\Big{)}e^{-{i\mathfrak{a}{\cdot}q}}\Big{\\}}\\\
&+\mathcal{O}(1)\,,\end{split}$ (8.26)

where

$\displaystyle\mathfrak{a}{\coloneqq}\mathfrak{a}_{1}{+}\mathfrak{a}_{2}\ .$
(8.27)

Note the nontrivial fact that at tree level the pole part of the amplitude
that we have considered so far depends only on the sum $\mathfrak{a}$ of the
spins of the two heavy objects. We also remark that the contact term (8.22)
does not have this property.

###  8.3 Fourier transform to impact parameter space

Having cast the amplitude (up to contact terms) in the form (8.26), we can
perform the Fourier transform to impact parameter space to all orders in the
spin, which will trivially shift $b^{\mu}\to b^{\mu}\pm\mathfrak{a}$, as was
seen in [123]. We have

$\displaystyle\widetilde{\mathcal{M}}_{4}=\int\\!\frac{d^{D}q}{(2\pi)^{D-2}}\delta(2p_{1}{\cdot}q)\delta(2p_{2}{\cdot}q)\,e^{iq{\cdot}b}\,\mathcal{M}_{4}=\frac{1}{4m_{1}m_{2}\sqrt{y^{2}-1}}\int\\!\frac{d^{D-2}q_{\perp}}{(2\pi)^{D-2}}e^{-i\vec{q}_{\perp}{\cdot}\vec{b}}\,{\mathcal{M}}_{4}\,,$
(8.28)

where $q_{\perp}{\cdot}p_{1,2}{=}0$ and

$\displaystyle\mathcal{M}_{4}=f_{+}(y)\frac{e^{i\mathfrak{a}{\cdot}q}}{q^{2}}+f_{-}(y)\frac{e^{-i\mathfrak{a}{\cdot}q}}{q^{2}}\,,$
(8.29)

with

$\displaystyle f_{\pm}(y)\coloneqq\
-i\,\frac{\kappa^{2}m_{1}^{2}m_{2}^{2}}{2}\left(y^{2}-\frac{1}{2}\pm
y\sqrt{y^{2}-1}\right)\,.$ (8.30)

Thus, we have to compute the Fourier transform

$\displaystyle\widetilde{\mathcal{M}}_{4}=\int\\!\frac{d^{D}q}{(2\pi)^{D-2}}\delta(2p_{1}{\cdot}q)\delta(2p_{2}{\cdot}q)\,\left[e^{iq{\cdot}(b+\mathfrak{a})}f_{+}(y)+e^{iq{\cdot}(b-\mathfrak{a})}f_{-}(y)\right]\,.$
(8.31)

We use

$\displaystyle\int\\!\frac{d^{d}q}{(2\pi)^{d}}e^{-i\vec{q}{\cdot}\vec{b}}|\vec{q}\,|^{p}=\frac{2^{p}\pi^{-\frac{d}{2}}\Gamma\left(\frac{d+p}{2}\right)}{\Gamma\left(-\frac{p}{2}\right)}\frac{1}{|\vec{b}\,|^{d+p}}\,,$
(8.32)

which in our case gives the result, as $D\to 4$,

$\displaystyle\int\\!\frac{d^{D-2}q}{(2\pi)^{D-2}}\frac{e^{-i\vec{q}{\cdot}\vec{b}}}{\vec{q}^{\,2}}=\frac{\Gamma\left(\frac{D-4}{2}\right)}{4\pi^{\frac{D-2}{2}}|\vec{b}\,|^{D-4}}\xrightarrow[D\to
4]{}-\frac{1}{2\pi}\log(|\vec{b}\,|)+\cdots\,,$ (8.33)

where the dots stand for $b$-independent terms. This leads to

$\displaystyle\widetilde{\mathcal{M}}_{4}=\frac{1}{8\pi
m_{1}m_{2}\sqrt{y^{2}-1}}\Big{[}f_{+}(y)\log(|b+\mathfrak{a}|)+f_{-}(y)\log(|b-\mathfrak{a}|)\Big{]}+\cdots\,,$
(8.34)

where we observe that the vector
$\mathfrak{a}{=}\mathfrak{a}_{1}+\mathfrak{a}_{2}$ lives in the same two-
dimensional subspace orthogonal to $p_{1}$ and $p_{2}$ as $b$. (8.34) agrees
with (51) of [123].

###  8.4 Result for the gravitational memory

Finally, to compute the gravitational memory we use (8.14) and (8.15).
Introducing the two vectors

$\displaystyle b_{\pm}\coloneqq
b\pm\mathfrak{a}\,,\qquad\text{with}\qquad\hat{b}_{\pm}\coloneqq\frac{b_{\pm}}{|b_{\pm}|}\,,$
(8.35)

we have at once

$\displaystyle\begin{split}\Delta(h^{\infty}_{+}\pm
ih^{\infty}_{\times})&=-\frac{i\kappa}{8\pi
m_{1}m_{2}\sqrt{y^{2}-1}}\Big{[}S_{\rm W}^{\rm
HEFT}(\hat{k},\hat{b}_{+};\pm)\frac{f_{+}(y)}{|b_{+}|}+S_{\rm W}^{\rm
HEFT}(\hat{k},\hat{b}_{-};\pm)\frac{f_{-}(y)}{|b_{-}|}\Big{]}\end{split},$
(8.36)

which is the final result for the memory, with $S_{\rm W}^{\rm HEFT}$ defined
in (8.20) and $f_{\pm}$ in (8.30). This result is exact to all orders in the
spin vector $a$. One can expand it to various order in $a$, and doing so one
finds perfect agreement with the result of [151] for the memory in the aligned
spin case up to $\mathcal{O}(a^{2})$.

We also note that in the spinless case, the previous formula becomes

$\displaystyle\begin{split}\left.\Delta(h^{\infty}_{+}\pm
ih^{\infty}_{\times})\right|_{a=0}&=-\frac{i\kappa}{8\pi
m_{1}m_{2}\sqrt{y^{2}-1}}(-i\kappa^{2}m_{1}^{2}m_{2}^{2})\frac{1}{|b|}\Big{(}y^{2}-\frac{1}{2}\Big{)}S_{\rm
W}^{\rm HEFT}(\hat{k},\hat{b};\pm)\\\
&=-\frac{\kappa^{3}}{8\pi}\frac{m_{1}m_{2}}{\sqrt{y^{2}-1}}\frac{1}{|b|}\Big{(}y^{2}-\frac{1}{2}\Big{)}S_{\rm
W}^{\rm HEFT}(\hat{k},\hat{b};\pm)\,,\end{split}$ (8.37)

in agreement with known results (see e.g. [151]).

## Acknowledgements

We would like to thank Massimo Bianchi, N. Emil J. Bjerrum-Bohr, Stefano De
Angelis, Thibault Damour, Claudio Gambino, Fabio Riccioni and Marcos Skowronek
for several interesting conversations. GT thanks the Physics Department at the
University of Rome “Tor Vergata” for their warm hospitality and support. This
work was supported by the Science and Technology Facilities Council (STFC)
Consolidated Grants ST/P000754/1 “String theory, gauge theory & duality” and
ST/T000686/1 “Amplitudes, strings & duality”. The work of GRB and JG is
supported by an STFC quota studentship. 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”. No new data were
generated or analysed during this study.

##  Appendix A Simplifying the four-point amplitude

In the main text we have defined a new spin vector (8.23), and
$\mathfrak{a}\coloneqq\mathfrak{a}_{1}+\mathfrak{a}_{2}$ which are all
orthogonal to both $v_{1}$ and $v_{2}$. Now from the square of the Levi-Civita
tensor we obtain a Gram determinant, for $i=1,2$,

$\displaystyle\big{[}\epsilon(a_{i}qv_{1}v_{2})\big{]}^{2}={(a_{i}{\cdot}q)^{2}\over
1-y^{2}}+q^{2}\Big{[}(a_{i}{\cdot}v_{1})^{2}+(a_{i}{\cdot}v_{2})^{2}-2y\,a_{i}{\cdot}v_{1}a_{i}{\cdot}v_{2}+a_{i}^{2}(y^{2}-1)\Big{]}\,,$
(A.1)

valid with the HEFT constraints $v_{1,2}{\cdot}q{=}0$. Using that
$v_{i}{\cdot}a_{i}{=}0$ we then find

$\displaystyle\begin{split}\big{[}\epsilon\left(a_{1}qv_{1}v_{2}\right)\big{]}^{2}&=-{(a_{1}{\cdot}q)^{2}\over
y^{2}-1}+q^{2}\Big{(}(a_{1}{\cdot}v_{2})^{2}+a_{1}^{2}(y^{2}-1)\Big{)}\,,\\\
\big{[}\epsilon\left(a_{2}qv_{1}v_{2}\right)\big{]}^{2}&=-{(a_{2}{\cdot}q)^{2}\over
y^{2}-1}+q^{2}\Big{(}(a_{2}{\cdot}v_{1})^{2}+a_{2}^{2}(y^{2}-1)\Big{)}\,.\end{split}$
(A.2)

In the calculation in impact parameter space, $\mathcal{O}(q^{2})$ terms do
not contribute, giving (8.24) in the main text.

To simplify the four-point amplitude we actually used the square root of the
above relations, that is

$\displaystyle\frac{\epsilon\left(a_{2}qv_{1}v_{2}\right)}{a_{2}{\cdot}q}=\frac{\epsilon\left(a_{1}qv_{1}v_{2}\right)}{a_{1}{\cdot}q}=\mp
i\sqrt{y^{2}-1}+\mathcal{O}(q^{2})\,,$ (A.3)

which are again valid up to terms order $\mathcal{O}(q^{2})$. Since the
amplitude is parity even, the sign ambiguity in these relations drops out.

One might ask what determines the sign on the right-hand side of (A.3). A
simple way to answer this question is to go to the rest frame of particle one,
and show that the $\mp$ sign in (A.3) follows the particular choice of the on-
shell momentum $q$. We can set

$\displaystyle v_{1}=(1,0,0,0)\,,\qquad v_{2}=(y,0,0,\sqrt{y^{2}-1})\,.$ (A.4)

Now, $q$ is on-shell, $q^{2}{=}0$, and satisfies the usual constraints
$q{\cdot}v_{1}{=}q{\cdot}v_{2}=0$, hence it must have the form

$\displaystyle q=(0,r,\pm ir,0)\,.$ (A.5)

Finally, using $a_{1}{\cdot}v_{1}=0$ and $a_{2}{\cdot}v_{2}=0$, the spin
vectors are of the form

$\displaystyle\begin{split}a_{1}&=(0,a_{1x},a_{1y},a_{1z})\,,\\\
a_{2}&=\Big{(}{s\over
y},a_{2x},a_{2y},{s\over\sqrt{y^{2}-1}}\Big{)}\,,\end{split}$ (A.6)

and hence we can evaluate, for $i=1,2$,

$\displaystyle\begin{split}\epsilon(v_{1}v_{2}a_{i}q)&=\epsilon(\vec{v}_{2}\vec{a}_{i}\vec{q})=\pm\sqrt{y^{2}-1}\,r\,(ia_{ix}\mp
a_{iy})\,,\\\ a_{i}{\cdot}q&=ir(ia_{ix}\mp a_{iy})\,.\end{split}$ (A.7)

In conclusion

$\displaystyle\frac{\epsilon(v_{1}v_{2}qa_{i})}{a_{2}{\cdot}q}=\mp
i\sqrt{y^{2}-1}\,,$ (A.8)

with the same plus or minus sign appearing for $a_{1}$ or $a_{2}$ and
following from the solution (A.5) chosen for the on-shell momentum $q$.

##  Appendix B More on the integrand

The coefficients in the $q^{2}_{1}$-channel are listed below:

$\displaystyle\begin{split}&c_{1}\to-\frac{1}{2}\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2},\
c_{2}\to\frac{1-2y^{2}}{4w_{1}^{2}},\ c_{3}\to\frac{iy}{2w_{1}^{2}},\
c_{4}\to\frac{iyv_{1}{\cdot}F_{k}{\cdot}v_{2}}{2w_{1}^{2}},\\\
&c_{5}\to-\frac{iyv_{1}{\cdot}F_{k}{\cdot}v_{2}}{2w_{1}^{2}},\
c_{6}\to\frac{yv_{1}{\cdot}F_{k}{\cdot}v_{2}}{w_{1}},\
c_{7}\to-\frac{iv_{1}{\cdot}F_{k}{\cdot}v_{2}}{2w_{1}},\
c_{8}\to-\frac{i\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}}{2w_{1}},\\\
&c_{9}\to-\frac{\left(2y^{2}-1\right)\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}}{4w_{1}^{2}w_{2}^{2}},\
c_{10}\to-\frac{\left(2y^{2}-1\right)v_{1}{\cdot}F_{k}{\cdot}v_{2}}{2w_{1}^{2}w_{2}},\
c_{11}\to\frac{iy\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}}{2w_{1}^{2}w_{2}^{2}},\\\
&c_{12}\to\frac{iyv_{1}{\cdot}F_{k}{\cdot}v_{2}}{w_{1}^{2}w_{2}},\
c_{13}\to\frac{iy\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}}{2w_{1}^{2}w_{2}},\
c_{14}\to\frac{y\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}}{w_{1}w_{2}},\\\
&c_{15}\to-\frac{i\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}}{2w_{1}w_{2}},\
c_{16}\to-\frac{iw_{2}y}{2w_{1}^{2}},\
c_{17}\to\frac{iw_{2}v_{1}{\cdot}F_{k}{\cdot}v_{2}}{2w_{1}}\,.\end{split}$
(B.1)

The coefficients in the $q^{2}_{2}$-channel are listed below:

$\displaystyle c_{2}$
$\displaystyle\to-\frac{1}{2}a_{1}{\cdot}a_{1}\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)^{2}$
$\displaystyle c_{4}$
$\displaystyle\to\left(a_{1}{\cdot}v_{2}\right){}^{2}\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}-\frac{1}{4}a_{1}{\cdot}a_{1}a_{1}{\cdot}F_{k}{\cdot}v_{1}\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}-2a_{1}{\cdot}v_{2}v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)$
$\displaystyle c_{5}$
$\displaystyle\to\frac{1}{16}(-3)ia_{1}{\cdot}k\left(a_{1}{\cdot}a_{1}-2\left(a_{1}{\cdot}v_{2}\right){}^{2}\right)\text{tr}\left(F_{k}{\cdot}S_{1}\right),$
$\displaystyle c_{6}$
$\displaystyle\to\frac{3}{16}i\left(a_{1}{\cdot}a_{1}-2\left(a_{1}{\cdot}v_{2}\right){}^{2}\right)\text{tr}\left(F_{k}{\cdot}S_{1}\right)a_{1}{\cdot}F_{k}{\cdot}v_{1},$
$\displaystyle c_{7}$
$\displaystyle\to\frac{y\left(a_{1}{\cdot}v_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}+a_{1}{\cdot}a_{1}v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)}{2w_{1}^{3}},$
$\displaystyle c_{10}$
$\displaystyle\to\frac{a_{1}{\cdot}F_{k}{\cdot}v_{1}\left(2ya_{1}{\cdot}ka_{1}{\cdot}v_{2}+w_{1}\left(\left(a_{1}{\cdot}v_{2}\right){}^{2}+\left(y^{2}-1\right)a_{1}{\cdot}a_{1}\right)\right)}{4w_{1}^{3}},$
$\displaystyle c_{11}$
$\displaystyle\to-\frac{\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}+2a_{1}{\cdot}v_{2}v_{1}{\cdot}F_{k}{\cdot}v_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}+2a_{1}{\cdot}a_{1}\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\right){}^{2}}{4w_{1}^{2}},$
$\displaystyle c_{12}$
$\displaystyle\to\frac{i\left(2y^{2}-1\right)\text{tr}\left(F_{k}{\cdot}S_{1}\right)}{8w_{1}^{2}},~{}~{}~{}~{}c_{13}\to\frac{\left(1-2y^{2}\right)\text{tr}\left(F_{k}{\cdot}S_{1}\right)a_{1}{\cdot}F_{k}{\cdot}v_{1}}{16w_{1}^{2}},$
$\displaystyle c_{14}$
$\displaystyle\to-\frac{w_{1}a_{1}{\cdot}k\left(\left(a_{1}{\cdot}v_{2}\right){}^{2}+\left(y^{2}-1\right)a_{1}{\cdot}a_{1}\right)+y\left(a_{1}{\cdot}k\right){}^{2}a_{1}{\cdot}v_{2}+w_{1}^{2}ya_{1}{\cdot}a_{1}a_{1}{\cdot}v_{2}}{4w_{1}^{3}},$
$\displaystyle c_{15}$
$\displaystyle\to\frac{a_{1}{\cdot}F_{k}{\cdot}v_{1}\left(ya_{1}{\cdot}k+w_{1}a_{1}{\cdot}v_{2}\right)}{4w_{1}^{2}},$
$\displaystyle c_{16}$
$\displaystyle\to\frac{ya_{1}{\cdot}F_{k}{\cdot}v_{1}\left(2a_{1}{\cdot}v_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}+a_{1}{\cdot}a_{1}v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)}{2w_{1}},$
$\displaystyle c_{19}$
$\displaystyle\to-\frac{i\left(w_{1}a_{1}{\cdot}v_{2}v_{1}{\cdot}F_{k}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)+a_{1}{\cdot}F_{k}{\cdot}v_{1}k{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}\right)}{4w_{1}},$
$\displaystyle c_{20}$
$\displaystyle\to\frac{\text{tr}\left(F_{k}{\cdot}S_{1}\right)\left(\left(2y^{2}-1\right)a_{1}{\cdot}k+2w_{1}ya_{1}{\cdot}v_{2}\right)}{16w_{1}^{2}},$
$\displaystyle c_{21}$
$\displaystyle\to\frac{\text{tr}\left(F_{k}{\cdot}S_{1}\right)\left(2ya_{1}{\cdot}ka_{1}{\cdot}v_{2}-w_{1}\left(a_{1}{\cdot}a_{1}-2\left(a_{1}{\cdot}v_{2}\right){}^{2}\right)\right)}{16w_{1}},$
$\displaystyle c_{22}$
$\displaystyle\to-\frac{ya_{1}{\cdot}v_{2}\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}}{4w_{1}^{3}},c_{25}\to\frac{iya_{1}{\cdot}F_{k}{\cdot}v_{1}}{2w_{1}^{2}},c_{26}\to-\frac{y\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\right)^{2}}{4w_{1}^{2}},$
$\displaystyle c_{29}$
$\displaystyle\to-\frac{iyv_{1}{\cdot}F_{k}{\cdot}S_{1}{\cdot}v_{2}}{2w_{1}^{2}},c_{32}\to\frac{a_{1}{\cdot}F_{k}{\cdot}v_{1}}{8w_{1}},c_{33}\to-\frac{ia_{1}{\cdot}F_{k}{\cdot}v_{1}}{4w_{1}},c_{34}\to-\frac{a_{1}{\cdot}ka_{1}{\cdot}F_{k}{\cdot}v_{1}}{8w_{1}},$
$\displaystyle c_{35}$
$\displaystyle\to-\frac{\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}}{8w_{1}},c_{40}\to\frac{v_{1}{\cdot}F_{k}{\cdot}v_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}}{4w_{1}},c_{41}\to-\frac{iv_{1}{\cdot}F_{k}{\cdot}v_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}}{2w_{1}},$
$\displaystyle c_{43}$
$\displaystyle\to\frac{iv_{1}{\cdot}F_{k}{\cdot}v_{2}v_{1}{\cdot}F_{k}{\cdot}S_{1}{\cdot}v_{2}}{2w_{1}},c_{44}\to\frac{3ia_{1}{\cdot}ka_{1}{\cdot}F_{k}{\cdot}v_{1}}{8w_{1}},c_{45}\to\frac{3ia_{1}{\cdot}ka_{1}{\cdot}v_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}}{4w_{1}},$
$\displaystyle c_{46}$
$\displaystyle\to\frac{3i\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\right){}^{2}}{8w_{1}},c_{47}\to-\frac{3ia_{1}{\cdot}v_{2}\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\right)^{2}}{4w_{1}},c_{48}\to\frac{iya_{1}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)}{4w_{1}},$
$\displaystyle c_{49}$
$\displaystyle\to-\frac{ya_{1}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)a_{1}{\cdot}F_{k}{\cdot}v_{1}}{8w_{1}},c_{50}\to\frac{yv_{1}{\cdot}F_{k}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)}{8w_{1}},c_{51}\to-\frac{iyv_{1}{\cdot}F_{k}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)}{4w_{1}},$
$\displaystyle c_{52}$
$\displaystyle\to\frac{3iya_{1}{\cdot}ka_{1}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)}{8w_{1}},c_{53}\to-\frac{3iya_{1}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)a_{1}{\cdot}F_{k}{\cdot}v_{1}}{8w_{1}},$
$\displaystyle c_{57}$
$\displaystyle\to\frac{1}{4}a_{1}{\cdot}F_{k}{\cdot}v_{1}\left(a_{1}{\cdot}a_{1}\left(a_{1}{\cdot}k+2w_{2}a_{1}{\cdot}v_{2}\right)-4a_{1}{\cdot}k\left(a_{1}{\cdot}v_{2}\right){}^{2}\right),$
$\displaystyle c_{58}$
$\displaystyle\to-\left(a_{1}{\cdot}v_{2}\right)a_{1}{\cdot}F_{k}{\cdot}v_{1}\left(w_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}+a_{1}{\cdot}kv_{1}{\cdot}F_{k}{\cdot}v_{2}\right),$
$\displaystyle c_{59}$
$\displaystyle\to-\frac{1}{4}a_{1}{\cdot}a_{1}v_{1}{\cdot}F_{k}{\cdot}v_{2}\left(w_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}+a_{1}{\cdot}kv_{1}{\cdot}F_{k}{\cdot}v_{2}\right),$
$\displaystyle c_{60}$
$\displaystyle\to\frac{1}{16}\Big{(}a_{1}{\cdot}k\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}\left(\text{tr}\left(F_{k}{\cdot}S_{1}\right)-\frac{2k{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}}{w_{1}}\right)-2a_{1}{\cdot}v_{2}v_{1}{\cdot}F_{k}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)\right)$
$\displaystyle-2w_{2}a_{1}{\cdot}v_{2}\text{tr}\left(F_{k}{\cdot}S_{1}\right)a_{1}{\cdot}F_{k}{\cdot}v_{1}\Big{)},$
$\displaystyle c_{61}$
$\displaystyle\to\frac{3i\left(\text{tr}\left(F_{k}{\cdot}S_{1}\right)\left(2w_{2}a_{1}{\cdot}v_{2}a_{1}{\cdot}F_{k}{\cdot}v_{1}-a_{1}{\cdot}k\left(a_{1}{\cdot}F_{k}{\cdot}v_{1}-2a_{1}{\cdot}v_{2}v_{1}{\cdot}F_{k}{\cdot}v_{2}\right)\right)\right)}{16}$
$\displaystyle+\frac{3i\text{tr}\left(F_{k}{\cdot}S_{1}\right)\left(2a_{1}{\cdot}ka_{1}{\cdot}F_{k}{\cdot}v_{1}k{\cdot}S_{1}{\cdot}F_{k}{\cdot}v_{1}\right)}{16w_{1}},$
$\displaystyle c_{62}$
$\displaystyle\to-\frac{y\left(a_{1}{\cdot}ka_{1}{\cdot}v_{2}+a_{1}{\cdot}a_{1}\left(w_{1}y-w_{2}\right)\right)}{2w_{1}^{3}},c_{64}\to\frac{\left(w_{1}-2w_{2}y\right)a_{1}{\cdot}F_{k}{\cdot}v_{1}-2ya_{1}{\cdot}kv_{1}{\cdot}F_{k}{\cdot}v_{2}}{4w_{1}^{3}},$
$\displaystyle c_{63}$
$\displaystyle\to\frac{a_{1}{\cdot}F_{k}{\cdot}v_{1}\left(v_{1}{\cdot}F_{k}{\cdot}v_{2}\left(ya_{1}{\cdot}k+w_{1}a_{1}{\cdot}v_{2}\right)+w_{2}ya_{1}{\cdot}F_{k}{\cdot}v_{1}\right)}{4w_{1}^{3}},$