# Gravitational Waveform: A Tale of Two Formalisms

Donato Bini  Istituto per le Applicazioni del Calcolo M. Picone, CNR, I-00185
Rome, Italy  INFN, Sezione di Roma Tre, I-00146 Rome, Italy  Thibault Damour
Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France
Stefano De Angelis  Institut de Physique Théorique, CEA, CNRS, Université
Paris-Saclay, F–91191 Gif-sur-Yvette cedex, France  Andrea Geralico  Istituto
per le Applicazioni del Calcolo M. Picone, CNR, I-00185 Rome, Italy  Aidan
Herderschee  Institute for Advanced Study, Princeton, NJ 08540, USA  Radu
Roiban  Institute for Gravitation and the Cosmos, Pennsylvania State
University, University Park, PA 16802, USA  Fei Teng  Institute for
Gravitation and the Cosmos, Pennsylvania State University, University Park, PA
16802, USA

###### Abstract

We revisit the quantum-amplitude-based derivation of the gravitational
waveform emitted by the scattering of two spinless massive bodies at the third
order in Newton’s constant, $h\sim G+G^{2}+G^{3}$ (one-loop level), and
correspondingly update its comparison with its classically-derived multipolar-
post-Minkowskian counterpart. A spurious-pole-free reorganization of the one-
loop five-point amplitude substantially simplifies the post-Newtonian
expansion. We find complete agreement between the two results up to the fifth
order in the small velocity expansion after taking into account three subtle
aspects of the amplitude derivation: (1) in agreement with [arXiv:2312.07452
[hep-th]], the term quadratic in the amplitude in the observable-based
formalism [JHEP 02, 137 (2019)] generates a frame rotation by half the
classical scattering angle; (2) the dimensional regularization of the infrared
divergences of the amplitude introduces an additional $(d-4)/(d-4)$ finite
term; and (3) zero-frequency gravitons are found to contribute additional
terms both at order $h\sim G^{1}$ and at order $h\sim G^{3}$ when including
disconnected diagrams in the observable-based formalism.

##  I Introduction

The remarkable detection of gravitational waves by the LIGO and Virgo
collaborations [1, 2] sparked the development of novel quantum field theory
(QFT)-based approaches to the general relativistic two-body problem which
build on enormous advances in scattering amplitudes and effective field
theory. The scattering regime at large minimum separation is a theoretical
environment free of many subtleties related to the definition of initial and
final states, making it the ideal ground for thorough comparison and
cooperation with the well-tested traditional analytical approaches to this
problem. In this paper we revisit such a comparison between the amplitudes-
based prediction in the observable-based (KMOC) formalism [3, 4] for the
classical gravitational wave signal from the scattering of two non-spinning
masses and the corresponding waveform derived within the Multipolar-post-
Minkowskian (MPM) formalism [5, 6, 7].

A natural framework for discussing classical scattering from the QFT
perspective consists in taking the classical limit ($\hbar\to 0$) of the
perturbative QFT expansion of scattering amplitudes, which is an expansion in
powers of $\frac{Gm_{1}m_{2}}{Jc}$, where $J$ is the orbital angular momentum.
By collecting the appropriate terms contributing to classical observables, one
should obtain the classical post-Minkowskian (PM) expansion (in powers of
Newton’s constant $G$) of the considered classical observable (e.g. an
impulse, $\Delta p^{\mu}$, or the waveform $h_{\mu\nu}\equiv
g_{\mu\nu}-\eta_{\mu\nu}$). For pioneering computations of the classical PM
expansion of scattering observables, see [8, 9] (waveform) and [10, 11]
(impulse). Classical bodies are treated as point particles with additional
properties encoded in higher-dimension operators extending their minimal
interactions with gravity. The classical limit is identified in the spirit of
the correspondence principle, and Lorentz invariance of standard QFT
perturbation theory facilitates keeping the complete dependence on the
velocity of particles at every order in the $G$ expansion.

Extensive effort, utilizing advances in scattering amplitude methods such as
of generalized unitarity [12, 13, 14, 15, 16], the double-copy relations [17,
18, 19] and powerful integration methods [20, 21, 22, 23, 24, 25, 26, 27] and
complemented by worldline approaches [28, 29], explored and evaluted inclusive
scattering observables for interactions of spinning and spinless bodies, such
as scattering angles, impulse, etc., to remarkably high perturbative orders
[30, 31, 32, 33, 34]. Similarly, position-dependent (point-like) observables
such as the gravitational waveform emitted during a weak-field scattering
process have been discussed from different perspectives and approaches [4, 35,
36, 37]. Recently, aspects of radiation and reaction were discussed in KMOC
formalism at the third order in Newton’s constant in [38] and the scattering
waveform, $h_{\mu\nu}\sim G^{1}+G^{2}+G^{3}+\dots$, was constructed to this
order for spinless particles [39, 40, 41] (see also [42]), for spinning
particles at $O(G^{2})$ [43, 44, 45] and at $O(G^{3})$ [46] and various orders
in spin.

Gravitational wave emission from gravitating binaries has been studied with
traditional analytic methods over many years, with the purpose of constructing
analytical-based waveforms to be used in the detection and data-analysis of
gravitational wave signals. One of the most successful analytical methods for
computing the GW emission from generic sources has been the (post-Newtonian-
matched) Multipolar-Post-Minkowskian (MPM) formalism [5, 6, 7]. The MPM
formalism has been developed over the years to a high perturbative accuracy,
recently reaching the fourth post-Newtonian (4PN) accuracy [47, 48], i.e. the
N4LO level in a PN expansion (in powers of
$\frac{v^{2}}{c^{2}}+\frac{Gm}{rc^{2}}$) beyond the leading-order quadrupole
formula pioneered by Einstein [49]. This PN accuracy includes nonlinear terms
of order $O(G^{5})$ in $h_{\mu\nu}$.

A comparison of the MPM and amplitudes-based results [50] revealed substantial
differences. Ref. [50] also pointed out a remarkable structure: if the
amplitudes-based results are interpreted as being in a frame rotated by half
the classical scattering angle, then the differences are dramatically reduced.
The origin of this frame rotation was recently understood by Georgoudis,
Heissenberg and Russo [51] in terms of a certain cut contribution pointed out
by in Ref. [42], and demonstrated explicitly in a small-frequency
expansion.111In the version 2 update of [51], this relation between the cut
contribution and frame rotation was also tested to NNLO (1PN) in the PN
expansion.

In this paper we revisit several aspects of the amplitude- and observable-
based (KMOC) [4] derivation of the classical gravitational wave signal emitted
during the scattering of two non-spinning masses at the NNLO order,222In the
following we will mainly refer to this as “the EFT waveform” following the
nomenclature used in the original comparison of [50], but at times we will
also use “the amplitudes-based waveform” and the “KMOC waveform”.
$h_{\mu\nu}=G^{1}+G^{2}+G^{3}$, and of the comparison of the frequency-domain
result with the corresponding waveform derived within the Multipolar-post-
Minkowskian formalism. We find complete agreement to the available post-
Newtonian accuracy (see Secs. VIII, X and XI).

The main difficulty with carrying out the PN expansion of the amplitude-based
scattering waveform stems from the severe spurious singularities exhibited by
the available results [39, 41, 40]. We reorganize both the amplitude and the
cut expressions so that they are written in terms of functions free of such
spurious poles on the physical sheet (see App. C). This facilitates the PN
expansion, which we carry out here to 3PN accuracy333Ref. [50] had reached the
2.5PN accuracy for the $\phi$-even part of the waveform, without incorporating
the 2PN-level terms., in the equatorial ($\theta=\pi/2$) plane and for the
complete dependence on the azimuthal angle $\phi$. We demonstrate that the
frame rotation of [50] is equivalent to the connected part of the cut
contribution of the KMOC formalism through this order (see Sec. IX).

Gravitons of vanishing frequency and their contributions to observables are a
notoriously thorny subject, see e.g. [52]. Ignoring them in the KMOC
calculation leads to differences with the MPM results already at leading order
in Newton’s constant (in the time-independent Coulombic field
$h_{\mu\nu}=O(G)$ generated by the incoming particles). We will show that
including them resolves this difference (see Sec. V.1). They also have
nontrivial contributions at higher orders in Newton’s constant, where, through
the disconnected cut contributions, they yield terms that can be interpreted
as a Bondi-Metzner-Sachs (BMS) [53, 54] supertranslation of the result with no
contributions from such states (see Secs. XI and V.3). This BMS
supertranslation is the same one that clarifies [52] the existence of an
angular momentum loss at $O(G^{2})$ [55]; its importance in the amplitudes-MPM
comparison was initially observed in a low-frequency expansion at the first
non-universal order [51]. A last ingredient in our comparison is a subtle
$O(\epsilon/\epsilon)$ contribution which originates from the graviton being
described by a symmetric, traceless $(d-2)\times(d-2)$ matrix in $d$
dimensions (see Secs. V.2 and X) in our (retarded-time renormalization)
scheme.

We begin with a brief description of the MPM and amplitudes-based methods used
to evaluate the waveforms, emphasizing the new contributions in the latter,
and then proceed to detaling our comparison. We collect the list of notations
used in this paper in App. A.

#### Note added:

While this paper was being written up, we became aware of concurrent work by
Georgoudis, Heissenberg and Russo, which partly overlaps with aspects of our
analysis. We thank the authors for communication and coordination on the
submission.

##  II Introducing the Multipolar Post-Minkowskian (MPM) formalism

The two transverse-traceless components

$f_{+}=\lim_{r\to\infty}(r\,h_{+})\,,\qquad
f_{\times}=\lim_{r\to\infty}(r\,h_{\times})$ (1)

of the classical, time-domain asymptotic waveform are encoded in the complex
quantity

$\displaystyle W(T_{r},\theta,\phi)$
$\displaystyle=\frac{c^{4}}{4G}(f_{+}-if_{\times})$
$\displaystyle=\frac{c^{4}}{4G}\lim_{r\to\infty}\bar{m}^{\mu}\bar{m}^{\nu}r\,h_{\mu\nu}\,,$
(2)

where the normalization prefactor $\frac{c^{4}}{4G}$ is used to simplify the
PN expansion of $W$.444We indicate powers of $c$, or $\eta\equiv\frac{1}{c}$,
when they help to understand the PN order of various quantities. Otherwise, we
often set $c$ to 1 without warning. Here $T_{r}\simeq
t-\frac{r}{c}-2\frac{G\mathsf{M}}{c^{3}}\log\frac{r}{cb_{0}}$ denotes a
retarded time which contains a logarithmic shift, involving an arbitrary time
scale $b_{0}$, proportional to the total center-of-mass (c.m.) mass-energy of
the system, $\mathsf{M}\equiv\frac{E}{c^{2}}$, while $\bar{m}^{\mu}$ (defined
below) is a complex, null polarization vector tangent to the sphere at
infinity. The MPM formalism is usually set up in the (incoming) c.m. of the
system, and computes the components of the radiative multipole moments of the
binary system with respect to a frame ${\bar{e}}_{0}$, $e_{x}$, $e_{y}$,
$e_{z}$ anchored on the classical averaged momenta

$\bar{p}_{a}=\frac{1}{2}(p_{a}+p^{\prime}_{a})\,,\quad(a=1,2)\,,$ (3)

rather than the incoming momenta $p_{a}$. Here, $p^{\prime}_{a}$ are the
classical outgoing momenta, differing from the incoming ones $p_{a}$ by the
full classical scattering angle $\chi=O(G)$. The time axis ${\bar{e}}_{0}$ of
the c.m. frame is

${\bar{e}}_{0}^{\mu}=\frac{{\bar{p}}_{1}^{\mu}+{\bar{p}}_{2}^{\mu}}{|{\bar{p}}_{1}^{\mu}+{\bar{p}}_{2}^{\mu}|}\,.$
(4)

The vector $e_{y}$ lies in the spatial direction of ${\bar{p}}_{1}$ (i.e. the
bisector between the incoming and the outgoing spatial momentum of the first
particle in the c.m. frame),

${\bar{p}}_{1}={E}_{1}{\bar{e}}_{0}+{\bar{P}}_{\rm
c.m.}e_{y}\,,\qquad{\bar{p}}_{2}={E}_{2}{\bar{e}}_{0}-{\bar{P}}_{\rm
c.m.}e_{y}\,.$ (5)

Here, $E_{a}=\sqrt{m_{a}^{2}+P_{\text{c.m.}}^{2}}$, and $P_{\text{c.m.}}$ is
the magnitude of the spatial part of the incoming momenta. In principle,
${\bar{P}}_{\text{c.m.}}$ differs from $P_{\text{c.m.}}$ by ${\bar{P}}_{\rm
c.m.}={P}_{\rm c.m.}\cos\frac{1}{2}\chi$. However, the difference is only at
the order $\chi^{2}=O(G^{2})$ and will be negligible in our work. The axis
vector $e_{x}$ is in the plane of motion and orthogonal to $e_{y}$ (and
oriented from particle 2 towards particle 1). It is also the direction of the
relative impact parameter,

$b_{12}=b\,e_{x}\,.$ (6)

Here $b$ is an eikonal-type impact parameter, linked to the incoming-momenta
impact parameter $b_{\text{in}}$ by $b_{\text{in}}=b\,\cos\frac{1}{2}\chi$, in
which the difference $O(\chi^{2})=O(G^{2})$ is again negligible. The last
spatial axis vector $e_{z}$ is orthogonal to the plane of motion (and such
that $e_{x}$, $e_{y}$, $e_{z}$ is positively oriented). All vectors and
tensors are decomposed in the frame ${\bar{e}}_{0}$, $e_{x}$, $e_{y}$, $e_{z}$
and the angles $\theta,\phi$ are accordingly defined, so that

$\displaystyle{\bf n}(\theta,\phi)$
$\displaystyle=\sin\theta\cos\phi\,e_{x}+\sin\theta\sin\phi\,e_{y}+\cos\theta\,e_{z}\,,$
$\displaystyle k$ $\displaystyle={\omega}\left({\bar{e}}_{0}+{\bf
n}(\theta,\phi)\right),$ $\displaystyle\bar{m}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\partial_{\theta}{\bf
n}(\theta,\phi)-\frac{i}{\sin\theta}\partial_{\phi}{\bf
n}(\theta,\phi)\right].$ (7)

Here ${\bf n}$ is the spatial unit vector that characterizes the direction of
the gravitational wave propagation.

The Multipolar post-Minkowskian (MPM) formalism [5, 6, 7] computes the time-
domain waveform $W^{\rm MPM}(T_{r},\theta,\phi)$ as a sum over irreducible
multipolar contributions, keyed by their multipole order $\ell$ and their
spatial parity. Even-parity (radiative) multipoles are denoted $U_{\ell}$,
$\ell=2,3,4,\ldots$, while odd-parity ones are denoted $V_{\ell}$,
$\ell=2,3,4,\ldots$. It is convenient to keep track of the
$\eta\equiv\frac{1}{c}$ prefactors entering the multipole expansion of the
waveform emitted by a slow-motion source by writing $W$ as

$\displaystyle W^{\rm MPM}(T_{r},\theta,\phi)$ $\displaystyle=$ $\displaystyle
U_{2}+\eta(V_{2}+U_{3})+\eta^{2}(V_{3}+U_{4})$ (8) $\displaystyle+$
$\displaystyle\eta^{3}(V_{4}+U_{5})+\eta^{4}(V_{5}+U_{6})$ $\displaystyle+$
$\displaystyle\eta^{5}(V_{6}+U_{7})+\cdots\,.$

With this normalization, the PN expansion of each radiative multipole
$U_{\ell}$ or $V_{\ell}$ starts at the Newtonian order (in the sense that the
leading order (LO) value of each multipole is expressed in terms of Newtonian-
level quantities). In Eq. (8), each $U_{\ell}$ or $V_{\ell}$ is expressed in
terms of corresponding symmetric-trace-free (STF) Cartesian tensors of order
$\ell$, according to

$\displaystyle U_{\ell}(T_{r},\theta,\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{\ell!}\bar{m}^{i}\bar{m}^{j}n^{i_{1}}n^{i_{2}}\cdots
n^{i_{\ell-2}}U_{iji_{1}i_{2}\cdots i_{\ell-2}}(T_{r})\,,$ $\displaystyle
V_{\ell}(T_{r},\theta,\phi)$ $\displaystyle=$
$\displaystyle-\frac{1}{\ell!}\frac{2\ell}{\ell+1}\bar{m}^{i}\bar{m}^{j}n^{c}n^{i_{1}}n^{i_{2}}\cdots
n^{i_{\ell-2}}$ (9) $\displaystyle\times\;\epsilon_{cdi}V_{jdi_{1}i_{2}\cdots
i_{\ell-2}}(T_{r})\,.$

The two infinite sequences of radiative multipole moments $U_{i_{1}i_{2}\cdots
i_{\ell}}(t)$, $V_{i_{1}i_{2}\cdots i_{\ell}}(t)$ (where $t$ henceforth
denotes the retarded time variable $T_{r}$) fully, and uniquely, parametrize
the time, and angular, dependences of any gravitational waveform. They are
formally observable if we imagine surrounding the emitting system by
infinitely many GW detectors located on the sphere at infinity (in the c.m.
frame). Actually, only the time-dependent part of each multipole, e.g.,
${U}_{i_{1}i_{2}\cdots i_{\ell}}(t)-{U}_{i_{1}i_{2}\cdots
i_{\ell}}(t=-\infty)$, is directly observable when considering the LO
$O(\frac{1}{r})$ asymptotic gravitational field at future null infinity.
Perturbative computations have, however, shown that the presence of non-zero
initial values ${U}_{i_{1}i_{2}\cdots i_{\ell}}(t=-\infty)\neq 0$ of the
radiative multipoles have physical effects at future null infinity at higher
orders in $O(\frac{1}{r})$, via the back-scattering of gravitational waves on
spacetime curvature (“tails”) [56, 57]. We will come back to this issue below.

The (post-Newtonian(PN)-matched) MPM formalism computes each (time-domain)
radiative multipole moment $U_{i_{1}i_{2}\cdots i_{\ell}}(t)$,
$V_{i_{1}i_{2}\cdots i_{\ell}}(t)$ in terms of the stress-energy tensor of the
material source, $T^{\mu\nu}(x^{\lambda})$, in several steps. At the end of
the day, each radiative multipole at retarded time $t$ is given by a sum of
contributions involving both source variables at time $t$ and hereditary
integrals over the past behavior of the source, i.e. over times $-\infty\leq
t^{\prime}\leq t$. All those hereditary integrals are expressed in terms of
some source multipole moments, of mass-type, $I_{i_{1}i_{2}\cdots
i_{\ell}}(t)$, and spin-type, $J_{i_{1}i_{2}\cdots i_{\ell}}(t)$. Some of the
hereditary integrals are linear in the source moments (“linear tails”) while
others are nonlinear because they arise from nonlinear couplings between the
multipole signals in the zone exterior to the source. Finally, the source
moments are computed in terms of the dynamical variables of the source by
using a PN-expanded approach to the gravitational field in the near zone.

To give an idea of the structure of the expressions coming out of the MPM
formalism, let us display a few of the contributions to the radiative
quadrupole555In the formulas below the angular brackets $\langle\dots\rangle$
denote a STF projection over the enclosed indices. $U_{ij}(t)$ at the
fractional 2.5PN accuracy ($\eta^{5}$ beyond the classic Einstein quadrupole
formula):

$\displaystyle U_{ij}(t)$ $\displaystyle=$
$\displaystyle\frac{d^{2}I_{ij}(t)}{dt^{2}}$ $\displaystyle+$
$\displaystyle\frac{2G\mathsf{M}}{c^{3}}\int_{0}^{\infty}d\tau
I_{ij}^{(4)}(t-\tau)\left(\log\left(\frac{\tau}{2b_{0}}\right)+\frac{11}{12}\right)$
$\displaystyle+$ $\displaystyle\frac{G}{c^{5}}\left(\frac{1}{7}I_{a\langle
i}^{(5)}I_{j\rangle a}^{\vphantom{(5)}}-\frac{5}{7}I_{a\langle
i}^{(4)}I_{j\rangle a}^{(1)}-\frac{2}{7}I_{a\langle i}^{(3)}I_{j\rangle
a}^{(2)}\right)+\cdots$

in which $\mathsf{M}\equiv\frac{E}{c^{2}}$ is the total mass-energy of the
system, and one must insert the 2.5PN-accurate value of the source quadrupole
$I_{ij}(t)$.

In the MPM formalism, the successive contributions indicated in Eq. (II) have
different origins: (i) the source quadrupole moment $I_{ij}(t)$ computes (in a
PN-expanded approximation) the quadrupole moment of the effective stress-
energy tensor of the system, i.e. the sum of the $T^{\mu\nu}_{\rm pp}$ of the
point particles and of the effective $T^{\mu\nu}_{\rm g}$ of the near-zone
gravitational field (potential gravitons, plus some local retarded-field
effects); (ii) the second (hereditary) term is the linear tail contribution
generated by the back-scattering of quadrupolar waves on the external-zone
Coulomb field $\sim\frac{G\mathsf{M}}{r}$ generated by the total mass-energy
of the system; and (iii) is an example of the several (instantaneous)
nonlinear terms arising from cubic couplings between multipole moments (here
quadrupole $\times$ quadrupole $\times$ quadrupole coupling).

In the c.m. frame the 2.5PN-accurate source quadrupole $I_{ij}$ has the
following structure

$\displaystyle I_{ij}^{\leq 2.5PN}(t)$ $\displaystyle=$ $\displaystyle\nu
M\left(1+\frac{a_{2}}{c^{2}}+\frac{a_{4}}{c^{4}}-\frac{24}{7}\frac{\nu}{c^{5}}\frac{G^{2}M^{2}}{r^{2}}\dot{r}\right)x^{\langle
i}x^{j\rangle}$ (11) $\displaystyle+$ $\displaystyle{\rm
terms\;in}\;x^{\langle i}v^{j\rangle}\;{\rm and}\;v^{\langle
i}v^{j\rangle}\,.$

Here, $x^{i}(t)=x_{1}^{i}(t)-x_{2}^{i}(t)$ denotes the relative motion (in the
c.m. frame), $v^{i}(t)=\frac{dx^{i}}{dt}$ is the relative velocity,
$M=m_{1}+m_{2}$ is the total mass and $\nu$ denotes the symmetric mass ratio,
$\nu=m_{1}m_{2}/(m_{1}+m_{2})^{2}$. In addition, in order to get explicit
results for the radiative moments as a function of time, one must solve the
correspondingly PN-accurate equations of motion of the binary system. The
latter task is aided by the existence of a rather simple quasi-Keplerian
representation of the hyperbolic-like solution of the 2PN-level conservative
equations of motion (to which the effect of radiation reaction, which starts
at the $G^{2}/c^{5}$ level, must be added) [58, 59, 60].

As the asymptotic behavior of $x^{i}(t)$ in the incoming state is
$x^{i}(t)\approx v^{i}_{0}\,t+O(\log t)$, the source quadrupole moment is
found to behave as $I_{ij}(t)\approx\nu
M\text{STF}[v^{i}_{0}v^{j}_{0}]\left(1+O(\frac{v_{0}^{2}}{c^{2}})\right)t^{2}$.
One then checks that all the hereditary integrals defining $U_{ij}(t)$ are
convergent, and that $U_{ij}(t)$ has a nonzero value when $t\to-\infty$ of the
form

$U_{ij}(t=-\infty)=2\nu
M\,\text{STF}[v^{i}_{0}v^{j}_{0}]\left(1+O(\frac{v_{0}^{2}}{c^{2}})\right)\,.$
(12)

Note that the nonzero value of the radiative quadrupole moment at the
Newtonian approximation is a simple consequence of the classic Einstein
quadrupole formula saying that
$U_{ij}=\frac{d^{2}}{dt^{2}}(\sum_{a}\text{STF}(m_{a}x_{a}^{i}x_{a}^{j}))+O(\eta^{2})$.

Similar results hold for all the multipole moments. Indeed, the origin of all
these nonzero initial radiative multipole moments is the fact that the
waveform in the infinite past is simply given by the Coulombic field of the
incoming worldlines, namely

$f_{ij}(t=-\infty,\mathbf{n})=\sum_{a}4G\left[\frac{\left(p_{a}^{i}p_{a}^{j}\right)^{\rm
TT}}{E_{a}-\mathbf{n}\cdot\mathbf{p}_{a}}\right],$ (13)

leading to (with $n\equiv k/{\omega}$)

$W(t=-\infty)=\sum_{a}\frac{(\bar{m}\cdot
p_{a})^{2}}{E_{a}-\mathbf{n}\cdot\mathbf{p}_{a}}=\sum_{a}\frac{(\bar{m}\cdot
p_{a})^{2}}{-(n\cdot p_{a})}\,.$ (14)

In other words, the MPM waveform necessarily incorporates the nonzero initial
value, Eqs. (13), (14), given by the Coulombic field of the incoming
worldlines.

Finally, after having obtained some representation of the various time-domain
multipole moments, one must compute their Fourier transforms (over the
retarded time variable), namely

$\displaystyle U_{i_{1}i_{2}\cdots
i_{\ell}}({\omega})=\int_{-\infty}^{+\infty}dte^{i{\omega}t}U_{i_{1}i_{2}\cdots
i_{\ell}}(t)\,,$ $\displaystyle V_{i_{1}i_{2}\cdots
i_{\ell}}({\omega})=\int_{-\infty}^{+\infty}dte^{i{\omega}t}V_{i_{1}i_{2}\cdots
i_{\ell}}(t)\,.$ (15)

This leads to a Fourier-domain MPM waveform of the form

$\displaystyle W^{\rm MPM}({\omega},\theta,\phi)$ $\displaystyle\equiv$
$\displaystyle U_{2}({\omega},\theta,\phi)$ $\displaystyle+$
$\displaystyle\eta(V_{2}({\omega},\theta,\phi)+U_{3}({\omega},\theta,\phi))$
$\displaystyle+$
$\displaystyle\eta^{2}(V_{3}({\omega},\theta,\phi)+U_{4}({\omega},\theta,\phi))$
$\displaystyle+$
$\displaystyle\eta^{3}(V_{4}({\omega},\theta,\phi)+U_{5}({\omega},\theta,\phi))$
$\displaystyle+$
$\displaystyle\eta^{4}(V_{5}({\omega},\theta,\phi)+U_{6}({\omega},\theta,\phi))$
$\displaystyle+$
$\displaystyle\eta^{5}(V_{6}({\omega},\theta,\phi)+U_{7}({\omega},\theta,\phi))+\cdots$

Here, the explicit powers of $\eta=\frac{1}{c}$ indicate those associated with
the LO, Newtonian value of each radiative multipole. They serve as a reminder
that the knowledge of the full waveform $W^{\rm MPM}({\omega},\theta,\phi)$
at, say, the 2.5PN accuracy requires one to know: (i) $U_{2}$ at the
fractional 2.5PN accuracy ($\eta^{5}$); (ii) $V_{2}$ and $U_{3}$ at the
fractional 2PN accuracy ($\eta^{4}$); (iii) $V_{3}$ and $U_{4}$ at the
fractional 1.5PN accuracy ($\eta^{3}$); (iv) $V_{4}$ and $U_{5}$ at the
fractional 1PN accuracy ($\eta^{2}$); (v) $V_{5}$ and $U_{6}$ at the Newtonian
accuracy (because their first PN correction starts at 1PN) and, finally, (vi)
$V_{6}$ and $U_{7}$ at the Newtonian accuracy.

In addition, as we are comparing the MPM waveform to the one-loop-accuracy
amplitudes-based waveform, we only need to determine the $O(G)$ and $O(G^{2})$
parts of the $G-$expansion of the waveform (in its $W$ guise, ie. after
dividing $\lim_{r\to\infty}\bar{m}^{\mu}\bar{m}^{\nu}r\,h_{\mu\nu}$ by $4G$).
Finally, an MPM/EFT waveform comparison at the 2.5PN accuracy requires, on the
MPM side, the knowledge of:

$\displaystyle U_{2}({\omega})$ $\displaystyle\sim$
$\displaystyle(G+G^{2})(\eta^{0}+\eta^{2}+\eta^{3}+\eta^{4}+\eta^{5})$
$\displaystyle V_{2}({\omega})\;{\rm and}\;U_{3}({\omega})$
$\displaystyle\sim$
$\displaystyle(G+G^{2})(\eta^{0}+\eta^{2}+\eta^{3}+\eta^{4})$ $\displaystyle
V_{3}({\omega})\;{\rm and}\;U_{4}({\omega})$ $\displaystyle\sim$
$\displaystyle(G+G^{2})(\eta^{0}+\eta^{2}+\eta^{3})$ $\displaystyle
V_{4}({\omega})\;{\rm and}\;U_{5}({\omega})$ $\displaystyle\sim$
$\displaystyle(G+G^{2})(\eta^{0}+\eta^{2})$ $\displaystyle
V_{5}({\omega})\;{\rm and}\;U_{6}({\omega})$ $\displaystyle\sim$
$\displaystyle(G+G^{2})(\eta^{0})$ $\displaystyle V_{6}({\omega})\;{\rm
and}\;U_{7}({\omega})$ $\displaystyle\sim$
$\displaystyle(G+G^{2})(\eta^{0})\,.$ (17)

The time-dependent part (${\omega}\neq 0$) of each multipole starts at order
$G^{1}$. Only their static (initial) values contribute at order $G^{0}$, as
exhibited in Eq. (14).

In these expansions, the terms of order $\eta^{3}$ only arise at order $G^{2}$
and have two different origins. One origin is the linear tails which are all
of the form

$\displaystyle U_{\ell}^{G^{2}\rm tail}(t)$
$\displaystyle=\frac{2G\mathsf{M}}{c^{3}}\int_{0}^{\infty}d\tau\frac{d^{2}}{dt^{2}}U_{\ell}^{G}(t-\tau)$
$\displaystyle\hskip
56.9055pt\times\left(\log\left(\frac{\tau}{2b_{0}}\right)+\kappa_{\ell}\right)$
$\displaystyle V_{\ell}^{G^{2}\rm tail}(t)$
$\displaystyle=\frac{2G\mathsf{M}}{c^{3}}\int_{0}^{\infty}d\tau\frac{d^{2}}{dt^{2}}V_{\ell}^{G}(t-\tau)$
$\displaystyle\hskip
56.9055pt\times\left(\log\left(\frac{\tau}{2b_{0}}\right)+\sigma_{\ell}\right),$
(18)

with

$\displaystyle\kappa_{\ell}$ $\displaystyle=$
$\displaystyle\frac{2\ell^{2}+5\ell+4}{\ell(\ell+1)(\ell+2)}+\sum_{k=1}^{k=\ell-2}\frac{1}{k},$
$\displaystyle\sigma_{\ell}$ $\displaystyle=$
$\displaystyle\frac{\ell-1}{\ell(\ell+1)}+\sum_{k=1}^{k=\ell-1}\frac{1}{k},$
(19)

in the time domain, and

$\displaystyle U_{\ell}^{G^{2}\rm tail}({\omega})$ $\displaystyle=$
$\displaystyle\frac{2G\mathsf{M}{\omega}}{c^{3}}\left(\frac{\pi}{2}+i\left[\log(2{\omega}b_{0}e^{\gamma_{E}})-\kappa_{\ell}\right]\right)U_{\ell}^{G}({\omega}),$
$\displaystyle V_{\ell}^{G^{2}\rm tail}({\omega})$ $\displaystyle=$
$\displaystyle\frac{2G\mathsf{M}{\omega}}{c^{3}}\left(\frac{\pi}{2}+i\left[\log(2{\omega}b_{0}e^{\gamma_{E}})-\sigma_{\ell}\right]\right)V_{\ell}^{G}({\omega}),$

in the frequency domain.

Another origin of the $G^{2}\eta^{3}$ terms exists for $V_{3}$ and $U_{4}$.
They come from quadrupole$\times$quadrupole couplings. E.g., in $U_{4}$ there
are terms of the form

$\displaystyle U_{ijkl}^{\rm II}(t)$ $\displaystyle=$
$\displaystyle\frac{G}{c^{3}}\left(-\frac{21}{5}I^{\vphantom{(5)}}_{\langle
ij}I_{kl\rangle}^{(5)}-\frac{63}{5}I_{\langle
ij}^{(1)}I_{kl\rangle}^{(4)}\right.$ (21) $\displaystyle-$
$\displaystyle\left.\frac{102}{5}I_{\langle
ij}^{(2)}I_{kl\rangle}^{(3)}\right)\,.$

These quadrupole$\times$quadrupole couplings generate terms of order
$G^{2}/c^{5}$ in $U_{2}$ (see last line in Eq. (II)).

On the other hand, the terms of even order in $\eta$ come from PN corrections
to the source multipole moments, as exemplified by the terms
$\frac{a_{2}}{c^{2}}+\frac{a_{4}}{c^{4}}$ in Eq. (11).

##  III Structure and computation of the MPM waveform

Ref. [50] computed the multipoles needed to reach the absolute
$(G^{1}+G^{2})\eta^{5}$ accuracy in the even-in-$\phi$ projection of the
waveform. Because of the planar nature (in the $xy$ plane) of the binary
motion generating the waveform, it is easily seen that a parity-even multipole
$U_{l}$ contains (when decomposed in $e^{im\phi}$) $m$ values equal to
$m=l,l-2,l-4$ etc. By contrast, a parity-odd multipole $V_{l}$ contains
$e^{im\phi}$ contributions for $m=l-1,l-3,l-5$ etc. Therefore, the even-
in-$\phi$ projection of $W^{\rm MPM}$ is given, at accuracy $\eta^{5}$, by
$U_{2}$, $V_{3}$, $U_{4}$, $V_{5}$, and $U_{6}$. The latter multipoles were
computed in Ref. [50], and their values in the equatorial plane
($\theta=\frac{\pi}{2}$) were displayed there.

In the present work, we complete the MPM/EFT comparison by also computing some
of the multipoles contributing to the odd-in-$\phi$ part of the waveform. For
simplicity, we focus on only two terms in the PN expansion of the odd-
in-$\phi$ piece, namely, the terms of order
$(G^{1}+G^{2})(\eta^{1}+\eta^{4})$. These terms come from the Newtonian-level
terms in $V_{2}$ and $U_{3}$, together with the corresponding tail terms
(which are $G\eta^{3}$ smaller than the LO contribution). We present the full
angular dependence of the multipoles by giving the coefficients of the
decomposition of $W^{\rm MPM}({\omega})$ in spin-weighted (with spin-weight
$s=-2$) spherical harmonics (SWSH) ${}_{s}Y_{lm}(\theta,\phi)$, defined with
the convention spelled out in an ancillary file MPMmultipoles.nb. For
instance, denoting for clarity $s=-2$ as $\bar{2}$, and $m=-2$ as $\bar{2}$)

$\displaystyle{}_{{\bar{2}}}Y_{22}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{5}{4\pi}}e^{2i\phi}\cos^{4}\left(\frac{\theta}{2}\right),$
$\displaystyle{}_{{\bar{2}}}Y_{2{\bar{2}}}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{5}{4\pi}}e^{-2i\phi}\cos^{4}\left(\frac{\theta}{2}\right)\,.$
(22)

The $(l,m)$ SWSH coefficients of $W^{\rm MPM}$ are the sum of the SWSH
coefficients of $U_{l}$ and $V_{l}$ (respectively multiplied by $\eta^{l-2}$
and $\eta^{l-1}$),

$W^{\rm MPM}_{lm}=\eta^{l-2}U_{lm}+\eta^{l-1}V_{lm}\,,$ (23)

where the coefficients $U_{lm},V_{lm}$ are defined by

$\displaystyle U_{l}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\sum_{m=-l}^{l}U_{lm}\,{}_{\bar{2}}Y_{lm}(\theta,\phi),$
$\displaystyle V_{l}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\sum_{m=-l}^{l}V_{lm}\,{}_{\bar{2}}Y_{lm}(\theta,\phi).$ (24)

The coefficients $U_{lm},V_{lm}$ can be obtained by using the orthonormality
of the ${}_{s}Y_{lm}(\theta,\phi)$’s, e.g.

$U_{lm}=\int\sin\theta d\theta
d\phi\,U_{l}(\theta,\phi)\;{}_{\bar{2}}Y_{2,m}^{*}(\theta,\phi)\,.$ (25)

In the time-domain, the $U_{lm}$, and $V_{lm}$ are functions of the retarded
time $t$. When going to the frequency domain, they are functions of
${\omega}$, with, e.g.,

$U_{lm}({\omega})=\int_{-\infty}^{+\infty}dte^{i{\omega}t}\,U_{lm}(t)\,.$ (26)

In practice, the Fourier transforms of the radiative multipoles are more
conveniently performed on the Cartesian representation of $U_{i_{1}i_{2}\cdots
i_{l}}(t)$ and $V_{i_{1}i_{2}\cdots i_{l}}(t)$. As explained in Ref. [50], it
is convenient to Fourier transform the first time-derivatives of each
$U_{i_{1}i_{2}\cdots i_{l}}(t)$ and $V_{i_{1}i_{2}\cdots i_{l}}(t)$ (which
vanish like $O(1/t^{2})$ at $t\to-\infty$) and to divide the result by
$-i{\omega}$. We only consider strictly positive frequencies ${\omega}>0$
because the ${\omega}<0$ part is determined by the reality of
$U_{i_{1}i_{2}\cdots i_{l}}(t)$ and $V_{i_{1}i_{2}\cdots i_{l}}(t)$, while the
${\omega}=0$ part is fully described by the ingoing, asymptotic waveform
$W(t=-\infty)$. $W(t=-\infty)$ is exactly given by the $G^{0}$, linearized-
gravity result Eq. (14).

Let us illustrate the structure (and physical content) of $U_{lm}({\omega})$,
and $V_{lm}({\omega})$, by focussing on the quadrupolar contribution
$U_{2}=\frac{1}{2}\bar{m}^{i}\bar{m}^{j}U_{ij}$ which is the only one which
starts at the Newtonian order $\eta^{0}$.

We have sketched in Eq. (II) the various contributions to $U_{ij}(t)$ which
contribute at the fractional $\eta^{5}$ (2.5PN) level. The first term in Eq.
(II),

$U_{ij}^{I}(t)=\frac{d^{2}I_{ij}(t)}{dt^{2}}\,,$ (27)

has $\eta^{5}$ contributions coming both from an explicit 2.5PN contribution
in the source quadrupole moment [61],

$I_{ij}(t)\sim\text{STF}\Big{[}\sum_{a}m_{a}x_{a}^{i}x_{a}^{j}+\eta^{2}+\eta^{4}+\eta^{5}+\ldots\Big{]}\,,$
(28)

from the need to include $O(G^{2}/c^{5})$ radiation-reaction effects [62] in
the second time-derivative of the positions $x_{a}^{i}(t)$, and from the need
to compute the Fourier transform along the radiation-reacted hyperbolic
motions666 However, as explained in Ref. [50], this can be avoided by working
with $\frac{d}{dt}U_{ij}^{I}(t)$..

The second term in Eq. (II) is the (linear) tail contribution which starts at
order $G^{2}\eta^{3}$ and reads, in the frequency domain,

$\displaystyle U_{ij}^{G^{2}\rm tail}({\omega})$ $\displaystyle=\frac{2G{\sf
M}{\omega}}{c^{3}}$ (29)
$\displaystyle\quad\times\left[\frac{\pi}{2}+i\left(\log(2{\omega}b_{0}e^{\gamma_{E}})-\frac{11}{12}\right)\right]U_{ij}^{G}({\omega})\,,$

where we recall that ${\sf M}\equiv\frac{E}{c^{2}}$. Here $b_{0}$ is the time
scale used in the MPM formalism to adimensionalize the (Coulomb-like)
logarithmic divergence contained in the tail.

The last term in Eq. (II) is indicative of a sum of several nonlinear
contributions (generated by various couplings between the multipolar waves in
the zone exterior to the system). These terms are expressed as products of
derivatives of source (and gauge) multipole moments, all taken at the same
retarded time $t$ as $U_{ij}(t)$ itself. In $U_{ij}$ at order $G^{2}/c^{5}$,
there are three such nonlinear contributions: (i) one bilinear in $I_{ij}$
($U_{ij}^{\rm II}$, denoted $U_{ij}^{\rm QQ}$ in [50]); (ii) one bilinear in
$I_{ij}$ and the angular momentum $L_{k}$ ($U_{ij}^{\rm LI}\equiv U_{ij}^{\rm
LQ}$), and (iii) one bilinear in $I_{ij}$ and the $\ell=0$ $W_{i_{1}\ldots
i_{\ell}}$ gauge moment ($U_{ij}^{\rm WI}\equiv U_{ij}^{\rm WQ}$):

$\displaystyle U_{ij}^{\rm II+LI+WI}(t)$ $\displaystyle=$
$\displaystyle\frac{G}{c^{5}}\left(\frac{1}{7}I_{a\langle
i}^{(5)}I^{\vphantom{(5)}}_{j\rangle a}-\frac{5}{7}I_{a\langle
i}^{(4)}I_{j\rangle a}^{(1)}\right.$ (30)
$\displaystyle\qquad\left.-\;\frac{2}{7}I_{a\langle i}^{(3)}I_{j\rangle
a}^{(2)}+\ldots\right)\,.$

Though only one power of $G$ enters the prefactor of Eq. (30), $U_{ij}^{\rm
II+LI+WI}(t)$ actually starts at the level $G^{2}$ because of the high-order
($n\geq 3$) time-derivatives of the source quadrupole moment $I_{ij}(t)$ it
contains. [As said in Ref. [50] there is a final hereditary term in
$U_{ij}(t)$, the nonlinear memory contribution which, however, starts at order
$G^{3}$, i.e. at the two-loop order.]

When considering the frequency-domain version of $U_{2}$, we have (at one-loop
accuracy, i.e. modulo $O(G^{3})$ in $W$)

$U_{2}({\omega})=U_{2}^{G}({\omega})+U_{2}^{G^{2}}({\omega})\,,$ (31)

where both $U_{2}^{G}({\omega})$ and $U_{2}^{G^{2}}({\omega})$ start at
Newtonian order. The PN expansions of $U_{2}^{G}({\omega})$ and
$U_{2}^{G^{2}}({\omega})$ have the structure

$\displaystyle U_{2}^{G}$ $\displaystyle\sim$
$\displaystyle\frac{GM^{2}\nu}{p_{\infty}}\left(1+\eta^{2}p_{\infty}^{2}+\eta^{4}p_{\infty}^{4}+\eta^{6}p_{\infty}^{6}+\ldots\right)\,,$
$\displaystyle U_{2}^{G^{2}}$ $\displaystyle\sim$
$\displaystyle\left(\frac{GM}{bp_{\infty}^{2}}\right)\frac{GM^{2}\nu}{p_{\infty}}(1+\eta^{2}p_{\infty}^{2}+\eta^{3}p_{\infty}^{3}+\eta^{4}p_{\infty}^{4}$
(32)
$\displaystyle+\;\eta^{5}p_{\infty}^{5}+\eta^{6}p_{\infty}^{6}+\ldots)\,.$

Note that $\left(\frac{GM}{bp_{\infty}^{2}}\right)$ is dimensionless (when
considering that $p_{\infty}$ has the dimension of a velocity) and of the
order of the Newtonian scattering angle. When considering that $p_{\infty}$ is
a velocity, the PN expansion is encoded in powers of
$\eta\,p_{\infty}=\frac{p_{\infty}}{c}$. The tree-level waveform $W^{G}$
(being time-symmetric) contains only even powers of $\eta\,p_{\infty}$. By
contrast, the one-loop waveform $W^{G^{2}}$ starts having odd powers of
$\eta\,p_{\infty}$ in its fractional PN expansion at order
$\eta^{3}p_{\infty}^{3}$, which is the order where hereditary tail effects
arise (see Eq. (29)). The next odd power of $\eta\,p_{\infty}$ is
$\eta^{5}p_{\infty}^{5}$ which contains several types of contributions coming
from: (i) $\eta^{5}$ and radiation-reaction contributions to Eq. (28); (ii)
fractional 1PN corrections contained in the linear tail term Eq. (29); and
(iii) the nonlinear contributions to $U_{ij}$, Eq. (30).

Let us finally illustrate the explicit structure of the frequency-domain
waveform by displaying a few important contributions at the $O(G)$ and
$O(G^{2})$ levels. The Fourier transform is done at the level of the once-
time-differentiated, PN-expanded, Cartesian-component radiative multipoles
$\frac{d}{dt}U_{i_{1}i_{2}\cdots i_{l}}(t)$ and
$\frac{d}{dt}V_{i_{1}i_{2}\cdots i_{l}}(t)$. The time-dependence of these
quantities is conveniently obtained by using the explicit quasi-Keplerian
representation of the relative two-body motion (in the c.m. frame). The latter
representation is naturally expressed in the $e_{x},e_{y}$ vectorial frame
(which is anchored on the classical ${\bar{p}_{a}}$ momenta, rather than the
incoming momenta $p_{a}$). The polar coordinates $r,\varphi$ of the relative
motion,

$\displaystyle{\bf x}(t)$ $\displaystyle=$ $\displaystyle x(t)e_{x}+y(t)e_{y}$
(33) $\displaystyle=$ $\displaystyle
r(t)\left[\cos\varphi(t)e_{x}+\sin\varphi(t)e_{y}\right]\,,$

are given as explicit functions of an (hyperbolictype) “eccentric anomaly”
variable $v$ by quasi-Keplerian expressions ons of the form

$\displaystyle{\bar{n}}\,t$ $\displaystyle=$ $\displaystyle
e_{t}\sinh(v)-v+O(\eta^{4})\,,$ $\displaystyle r$ $\displaystyle=$
$\displaystyle{\bar{a}}_{r}\left(e_{r}\cosh(v)-1)\right)+O(\eta^{4})\,,$ (34)
$\displaystyle\varphi$ $\displaystyle=$ $\displaystyle
2K\arctan\left(\sqrt{\frac{e_{\varphi}+1}{e_{\varphi}-1}}\tanh\left(\frac{v}{2}\right)\right)+O(\eta^{4})\,.$

Here, the quasi-Keplerian quantities ${\bar{n}},e_{t},e_{r},e_{\varphi},K$ are
(PN-expanded) functions of the c.m. energy and angular momentum of the binary
system (see Refs. [58, 59, 60]).

The quasi-Keplerian representation Eq. (III) incorporates (in the conservative
case) a time symmetry around $t=0$, corresponding to the closest approach
between the two bodies. The asymptotic logarithmic drift of the two worldlines
is embodied in the $v$ parametrization involving hyperbolic functions.

Inserting the (PN-expanded) quasi-Keplerian representation Eq. (III) in, say,

$-i{\omega}U_{i_{1}i_{2}\cdots
i_{l}}({\omega})=\int_{-\infty}^{+\infty}dte^{i{\omega}t}\frac{d}{dt}U_{i_{1}i_{2}\cdots
i_{l}}(t)\,,$ (35)

leads to integrals which can all be obtained (at the $G+G^{2}$ accuracy) from
the master integrals (for $u>0$)

$\displaystyle\int_{-\infty}^{+\infty}dve^{i\left(u\sinh(v)-\mu\,v\right)}$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{dT}{\sqrt{1+T^{2}}}e^{iuT-i\mu\,{\rm
arcsinh}T}$ $\displaystyle=2e^{\mu\frac{\pi}{2}}K_{i\mu}(u)\,,$ (36)
$\displaystyle\int_{-\infty}^{+\infty}\frac{dv}{\cosh v}e^{iu\sinh(v)}$
$\displaystyle=\int_{-\infty}^{+\infty}\frac{dT}{1+T^{2}}e^{iuT}$
$\displaystyle=\pi e^{-u}\,.$ (37)

Actually, one only needs the large-eccentricity expansion (corresponding to a
PM expansion in powers of $G$, and to an expansion in powers of $\mu$) of the
master integral Eq. (III).

Using these integrals, all the SWSH components of the waveform at order
$G+G^{2}$ are expressed as linear combinations of modified Bessel functions of
order $0$ and $1$, $K_{0}(u)$ and $K_{1}(u)$, and (starting at the 1PN
fractional order) of the exponential $e^{-u}/u$, with argument given by the
following dimensionless version of the frequency:

$u\equiv\frac{{\omega}b}{p_{\infty}}\,.$ (38)

The coefficients of $K_{0}(u)$, $K_{1}(u)$ and $e^{-u}/u$ are
($\nu$-dependent) polynomials in $u$ whose degrees increase with the PN order.

As already said, when $l=2$ the $m$ values are only $m=+2,m=0$, and $m=-2$.

Here is a sample of the SWSH components of the quadrupolar piece of the
waveform. At order $G^{1}\eta^{0}$ we find

$\displaystyle U_{22}^{G^{1}\eta^{0}}$ $\displaystyle=$
$\displaystyle-\frac{GM^{2}\nu}{p_{\infty}}\frac{2\sqrt{5\pi}}{5}\Big{[}(2u+1)K_{0}(u)+2(u+1)K_{1}(u)\Big{]},$
$\displaystyle U_{20}^{G^{1}\eta^{0}}$ $\displaystyle=$
$\displaystyle-\frac{GM^{2}\nu}{p_{\infty}}\frac{\sqrt{30\pi}}{15}K_{0}(u),$
$\displaystyle U_{2\bar{2}}^{G^{1}\eta^{0}}$ $\displaystyle=$
$\displaystyle\frac{GM^{2}\nu}{p_{\infty}}\frac{2\sqrt{5\pi}}{5}\Big{[}(2u-1)K_{0}(u)-2(u-1)K_{1}(u)\Big{]}.$
(39)

At order $G^{1}\eta^{2}$ we have

$\displaystyle U_{22}^{G^{1}\eta^{2}}$ $\displaystyle=$
$\displaystyle\frac{\nu
GM^{2}p_{\infty}\sqrt{5\pi}}{105}\Big{\\{}[(-72\nu+38)u^{2}+(-36\nu-2)u-24\nu+78]K_{0}(u)$
$\displaystyle+\;[(-72\nu+38)u^{2}+(-72\nu+17)u-48\nu-138]K_{1}(u)\Big{\\}},$
$\displaystyle U_{20}^{G^{1}\eta^{2}}$ $\displaystyle=$ $\displaystyle\nu
GM^{2}p_{\infty}\frac{\sqrt{30\pi}}{105}\Big{[}(-8\nu+26)K_{0}(u)+u(-16\nu+59)K_{1}(u)\Big{]},$
$\displaystyle U_{2\bar{2}}^{G^{1}\eta^{2}}$ $\displaystyle=$
$\displaystyle\frac{\nu
GM^{2}p_{\infty}\sqrt{5\pi}}{105}\Big{\\{}[(-72\nu+38)u^{2}+(36\nu+2)u-24\nu+78]K_{0}(u)$
(40)
$\displaystyle+\;[(72\nu-38)u^{2}+(-72\nu+17)u+48\nu+138]K_{1}(u)\Big{\\}}.$

The $O(G^{2}\eta^{0})$ multipoles have a simple relation to the
$O(G^{1}\eta^{0})$ ones:

$U_{2m}^{G^{2}\eta^{0}}=\frac{\pi}{2}\frac{GM}{bp_{\infty}^{2}}\,u\,U_{2m}^{G^{1}\eta^{0}}\,.$
(41)

Proceeding to $O(G^{2}\eta^{2})$, we have instead

$\displaystyle U_{22}^{G^{2}\eta^{2}}$ $\displaystyle=$
$\displaystyle-\frac{G^{2}M^{3}\nu\pi^{3/2}\sqrt{5}}{210bp_{\infty}}\Big{\\{}u[u^{2}(72\nu-38)+u(78\nu-124)+45\nu-141]K_{0}(u)$
$\displaystyle+\;u[u^{2}(72\nu-38)+u(114\nu-143)+90\nu+12]K_{1}(u)+\frac{252(2u^{2}+2u+1)}{u}e^{-u}\Big{\\}}\,,$
$\displaystyle U_{20}^{G^{2}\eta^{2}}$ $\displaystyle=$
$\displaystyle-\frac{G^{2}M^{3}\nu\pi^{3/2}\sqrt{30}}{210bp_{\infty}}\Big{[}u(-47+15\nu)K_{0}(u)+u^{2}(-59+16\nu)K_{1}(u)\Big{]}\,,$
$\displaystyle U_{2\bar{2}}^{G^{2}\eta^{2}}$ $\displaystyle=$
$\displaystyle\frac{G^{2}M^{3}\nu\pi^{3/2}\sqrt{5}}{210bp_{\infty}}\Big{\\{}-u[u^{2}(72\nu-38)+u(-78\nu+124)+45\nu-141]K_{0}(u)$
(42)
$\displaystyle+\;u[u^{2}(72\nu-38)+u(-114\nu+143)+90\nu+12]K_{1}(u)+\frac{252}{u}e^{-u}\Big{\\}}\,.$

At 2PN the structure of $U_{2}^{G^{2}\eta^{4}}$ is similar to that of
$U_{2}^{G^{2}\eta^{2}}$ apart from a quadratic-in-$\nu$ dependence of the
coefficients and an overall multiplication by $p_{\infty}^{2}$.

Let us finally focus on the $G^{2}\eta^{5}$ contribution to $U_{2}$. Following
the decomposition presented above, each $lm$ component of $U_{2}$ is the sum
of two terms: (i) an “instantaneous" contribution, $U_{2m}^{\rm inst}$,
defined as

$U_{2m}^{\rm inst}=U_{2m}^{I}+U_{2m}^{\rm II}+U_{2m}^{\rm LI}+U_{2m}^{\rm
WI}\,,$ (43)

i.e. the sum of the source-moment-related term, $U_{2m}^{I}$, Eq. (28), and of
the nonlinear terms indicated in Eq. (30); and (ii) a (nonlocal in time) tail
term $U_{2m}^{\rm tail}$, Eq. (29) (here projected on its $G^{2}\eta^{5}$
contribution). They are respectively given by:

$\displaystyle U_{22}^{{\rm inst}\,G^{2}\eta^{5}}$ $\displaystyle=$
$\displaystyle\frac{2G^{2}M^{3}\nu^{2}\sqrt{5\pi}p_{\infty}^{2}}{525b}iu\Big{[}2(245u^{2}-57u-15)K_{0}(u)+(490u^{2}+131u-72)K_{1}(u)\Big{]}\,,$
$\displaystyle U_{20}^{{\rm inst}\,G^{2}\eta^{5}}$ $\displaystyle=$
$\displaystyle\frac{2G^{2}M^{3}\nu^{2}\sqrt{30\pi}p_{\infty}^{2}}{105b}iu\Big{[}-2K_{0}(u)+29uK_{1}(u)\Big{]}\,,$
$\displaystyle U_{2\bar{2}}^{{\rm inst}\,G^{2}\eta^{5}}$ $\displaystyle=$
$\displaystyle\frac{2G^{2}M^{3}\nu^{2}\sqrt{5\pi}p_{\infty}^{2}}{525b}iu\Big{[}2(245u^{2}+57u-15)K_{0}(u)-(490u^{2}-131u-72)K_{1}(u)\Big{]}\,,$
(44)

and

$\displaystyle U_{22}^{{\rm tail}\,G^{2}\eta^{5}}$ $\displaystyle=$
$\displaystyle-\frac{2}{105}\frac{G^{2}M^{3}\nu\sqrt{5\pi}p_{\infty}^{2}}{b}\left(i\log(2b_{0}\omega
e^{\gamma_{E}})-i\frac{11}{12}+\frac{\pi}{2}\right)u\Big{\\{}[(72\nu-38)u^{2}+(78\nu+2)u+45\nu-78]K_{0}(u)$
$\displaystyle+\;[(72\nu-38)u^{2}+(114\nu-17)u+90\nu+138]K_{1}(u)\Big{\\}}\,,$
$\displaystyle U_{20}^{{\rm tail}\,G^{2}\eta^{5}}$ $\displaystyle=$
$\displaystyle-\frac{2}{105}\frac{G^{2}M^{3}\nu\sqrt{30\pi}p_{\infty}^{2}}{b}\left(i\log(2b_{0}\omega
e^{\gamma_{E}})-i\frac{11}{12}+\frac{\pi}{2}\right)u\Big{[}(15\nu-26)K_{0}(u)+u(16\nu-59)K_{1}(u)\Big{]}\,,$
$\displaystyle U_{2\bar{2}}^{{\rm tail}\,G^{2}\eta^{5}}$ $\displaystyle=$
$\displaystyle-\frac{2}{105}\frac{G^{2}M^{3}\nu\sqrt{5\pi}p_{\infty}^{2}}{b}\left(i\log(2b_{0}\omega
e^{\gamma_{E}})-i\frac{11}{12}+\frac{\pi}{2}\right)u\Big{\\{}[(72\nu-38)u^{2}+(-78\nu-2)u+45\nu-78]K_{0}(u)$
(45)
$\displaystyle+\;[(-72\nu+38)u^{2}+(114\nu-17)u-90\nu-138]K_{1}(u)\Big{\\}}\,.$

Let us finally give an example of the SWSH components of the $\phi$-odd (and
$(m_{1}-m_{2})$-odd) piece of the waveform

$\displaystyle V_{21}^{G^{1}\eta^{0}}$ $\displaystyle=$
$\displaystyle-4i\frac{\sqrt{5\pi}}{15}\nu
GM^{2}\frac{m_{1}-m_{2}}{M}u[K_{0}(u)+K_{1}(u)]\,,$ $\displaystyle
V_{2\bar{1}}^{G^{1}\eta^{0}}$ $\displaystyle=$
$\displaystyle-4i\frac{\sqrt{5\pi}}{15}\nu
GM^{2}\frac{m_{1}-m_{2}}{M}u[K_{0}(u)-K_{1}(u)]\,,$ (46)

with

$\displaystyle V_{21}^{G^{2}\eta^{0}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}\left(\frac{GM}{bp_{\infty}^{2}}\right)u\,V_{21}^{G^{1}\eta^{0}}\,,$
$\displaystyle V_{2\bar{1}}^{G^{2}\eta^{0}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}\left(\frac{GM}{bp_{\infty}^{2}}\right)u\,V_{2\bar{1}}^{G^{1}\eta^{0}}\,,$
(47)

as before. The corresponding tail contributions are given by

$\displaystyle V_{21}^{G^{2}\eta^{3}}$ $\displaystyle=$
$\displaystyle-\frac{4\sqrt{5\pi}}{15b}\frac{m_{1}-m_{2}}{M}\left(\frac{7}{3}+\pi
i-2\log(2b_{0}e^{\gamma_{E}}\omega)\right)p_{\infty}G^{2}M^{3}\nu
u^{2}[K_{0}(u)+K_{1}(u)]$ $\displaystyle=$
$\displaystyle-i\frac{GM}{b}p_{\infty}u\left(\frac{7}{3}i+i\pi-2i\log(2b_{0}e^{\gamma_{E}}\omega)\right)V_{21}^{G^{1}\eta^{0}},$
$\displaystyle V_{2\bar{1}}^{G^{2}\eta^{3}}$ $\displaystyle=$
$\displaystyle-\frac{4\sqrt{5\pi}}{15b}\frac{m_{1}-m_{2}}{M}\left(\frac{7}{3}+\pi
i-2\log(2b_{0}e^{\gamma_{E}}\omega)\right)p_{\infty}G^{2}M^{3}\nu
u^{2}[K_{0}(u)-K_{1}(u)]$ (48) $\displaystyle=$
$\displaystyle-i\frac{GM}{b}p_{\infty}u\left(\frac{7}{3}+\pi
i-2i\log(2b_{0}e^{\gamma_{E}}\omega)\right)V_{2\bar{1}}^{G^{1}\eta^{0}}\,.$

All the SWSH components of the waveform $W^{\rm MPM}$ that we computed are
available in the ancillary file MPMmultipoles.nb attached to this paper. They
reach (in $W$) $(G^{1}+G^{2})(1+\eta^{2}+\eta^{3}+\eta^{4}+\eta^{5})$ accuracy
for the even $m$ terms, and the $(G^{1}+G^{2})(\eta+\eta^{4})$ accuracy for
the odd $m$ terms.

##  IV The Observable-based formalism

Scattering waveform can also be computed using field-theory based methods. In
particular, the observable-based formalism [3] relates directly final state
observables with $S$-matrix elements in the classical limit. Before we start
to introduce the formalism, we note that in field theory, it is customary to
define metric perturbation as
$g_{\mu\nu}=\eta_{\mu\nu}+\kappa\mathsf{h}_{\mu\nu}$, where $\kappa^{2}=32\pi
G$. It differs from $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ used in the MPM
calculation by a factor of $\kappa$. We will stick to the field-theory
convention when computing $S$-matrix elements. We will compensate the factor
of $\kappa$ when performing the Fourier transform to impact parameter space,
see Eq. 54 below. We will also set $c=1$ in all the field-theory related
calculations.

###  IV.1 General setup

The observable-based formalism [3] constructs quantum scattering observables
by comparing the expectation value of the corresponding hermitian operator
$\mathbb{O}$ between the initial and final state of the scattering process

$\langle\mathbb{O}\rangle=\langle\psi_{\textrm{out}}|\mathbb{O}|\psi_{\textrm{out}}\rangle-\langle\psi_{\textrm{in}}|\mathbb{O}|\psi_{\textrm{in}}\rangle\
,$ (49)

where $\mathbb{O}$ could be, e.g., the momentum transferred during the
scattering process or the curvature at large distances. Observables are
assumed to be measured at infinity, where the gravitational field is
effectively free. The evaluation of the expectation value in Eq. (49) proceeds
by constructing the final state $|\psi_{\text{out}}\rangle$ as the time-
evolution of the initial state $|\psi_{\text{in}}\rangle$,
$|\psi_{\text{out}}\rangle=\mathbb{S}|\psi_{\text{in}}\rangle$, and then
inserting a complete set of final states next to ${\mathbb{O}}$. The result is
an expression of $\langle{\mathbb{O}}\rangle$ in terms products of $S$-matrix
elements suitably integrated over the final-state phase space.

For the scattering problem, we construct the initial two-particle states as
the tensor product of two localized single-particle states,

$\displaystyle|\psi_{\text{in}}\rangle$
$\displaystyle=\int\text{d}\Phi_{p_{1}}\text{d}\Phi_{p_{2}}\varphi(p_{1})\varphi(p_{2})e^{-i(p_{1}\cdot
b_{1}+p_{2}\cdot b_{2})}|p_{1}p_{2}\rangle\,,$ $\displaystyle\text{d}\Phi_{p}$
$\displaystyle=\frac{\text{d}^{4}p}{(2\pi)^{4}}\Theta(p^{0})(2\pi)\delta(p^{2}+m^{2})\,,$
(50)

where the measure $\text{d}\Phi_{p}$ imposes the on-shell condition, and the
wavepacket is normalized as $\int\text{d}\Phi_{p}|\varphi(p)|^{2}=1$.

To compute the scattering waveform we may choose the operator ${\mathbb{O}}$
to either be the linearized Riemann tensor777The linearization is a
consequence of the operator being inserted in the final state, i.e. at
infinity. The LSZ reduction then projects out all Green’s functions with more
than a single graviton. or to be the transverse-traceless components of the
asymptotic metric fluctuations about Minkowski space at infinity. In the
former approach, the metric is extracted from the expectation value of the
asymptotic Newman-Penrose scalar
$\Psi_{4}\propto{\ddot{f}}_{+}-i{\ddot{f}}_{\times}$, by integrating twice
over time and determining the two integration constants from physical
considerations. One integration constant is fixed by the boundary condition
$\lim_{t\rightarrow\infty}\dot{f}(t)\sim t^{-2}$. The other corresponds to the
EFT analog of the Coulombic field in Eq. (13).

The latter choice yields directly the waveform. It also gives a matrix-element
interpretation of the integration constants needed to reconstruct the
asymptotic metric from the Newman-Penrose scalar. In particular, the
integration constant that is time-independent is determined by the parts of
Eq. (49) that have support only on vanishing outgoing-graviton frequency
$\omega$. Such $\delta(\omega)$-supported matrix elements yield nontrivial
time-dependent contributions to the waveform at higher orders in $G$ through
iteration, as we will see in Sec. V.3.

The observable-based formalism yields final-state observables in the complete
quantum theory. The building block of the momentum-space expectation value of
the classical asymptotic metric is

$\displaystyle\mathscr{M}$
$\displaystyle(\varepsilon,k,p_{1},p_{2},q_{1},q_{2})$ $\displaystyle\equiv
i\langle p_{3}p_{4}|\hat{a}(k)\mathbb{T}|p_{1}p_{2}\rangle+\langle
p_{3}p_{4}|\mathbb{T}^{\dagger}\hat{a}(k)\mathbb{T}|p_{1}p_{2}\rangle$
$\displaystyle=i\langle p_{3}p_{4}k|\mathbb{T}|p_{1}p_{2}\rangle+\langle
p_{3}p_{4}|\mathbb{T}^{\dagger}\hat{a}(k)\mathbb{T}|p_{1}p_{2}\rangle\,,$ (51)

where $\hat{a}$ creates a physical graviton with polarization vector
$\varepsilon$ when acting to the left. The momentum lost by each matter
particle is

$\displaystyle q_{1}=p_{1}-p_{4}\,,\qquad q_{2}=p_{2}-p_{3}\,,$ (52)

and momentum conservation requires that $q_{1}+q_{2}$ be the outgoing graviton
momentum $k$. The first term in $\mathscr{M}$ is the five-point amplitude with
one graviton in the final state; the second term is a unitarity cut that gives
a particular discontinuity of the quantum amplitude. We shall refer to this
bilinear-in-$T$ term as the “unitarity cut term”, or simply, the “cut term”.
The matrix element $\mathscr{M}$ is a Lorentz invariant function of the
various scalar products of $(\varepsilon,k,p_{1},p_{2},q_{1}-q_{2})$. For
notational convenience, in the following we will (partially) suppress the
arguments of $\mathscr{M}$.

The absence of local diffeomorphism-invariant observables in general
relativity requires that one be careful with the effect of large gauge
transformations on asymptotic observables. The asymptotic Newman-Penrose
scalar and the transverse-traceless components of the asymptotic metric are
invariant under decaying-at-infinity coordinate transformations, $\delta
h_{\mu\nu}=\partial_{(\mu}\xi_{\nu)}$, but are not invariant under BMS
transformations. However, as we will see in Section V.3, both the KMOC and MPM
formalisms (implicitly) use the same BMS vacuum, referred to as “intrinsic
gauge” in Ref. [52].888 Perhaps the term “intrinsic frame” might be more
appropriate as it should be possible to discern its properties through
gravitational wave measurements.

###  IV.2 Classical limit

As we reviewed above, the observable-based formalism establishes the relation
between observables with complete velocity dependence and the matrix element
$\mathscr{M}$ in the quantum theory. The correspondence principle, stating
that the classical regime emerges in the limit of large (macroscopic)
conserved charges, provides a means to extract the classical observables from
their quantum mechanical counterpart. In our case these charges are the
particles’ masses, momenta and orbital angular momenta $J$, i.e. $J\gg\hbar=1$
and $m\gg M_{\rm Pl}$. It then follows [63, 3, 64] that the impact parameter
$b$ is much larger than the Compton wave length $\lambda_{c}=p^{-1}$ or,
equivalently, that the momentum transfer $q$ is much smaller than the typical
external momenta $p$ 999 In later sections we will compare MPM and amplitudes-
based predictions in the PN expansion. To be in the regime of validity of both
this and of the PM expansions originally used in the amplitudes computation,
we must have the stronger condition on the impact parameter, $b\gg
v^{-2}R_{s}$, corresponding to a small scattering angle $\chi\sim
GM/(bv^{2})$. and that observables are series in the effective coupling
$Gm_{1}m_{2}/J$.

In this limit, certain parts of loop-level expressions contribute to classical
observables. The precise dependence on Newton’s constant $G$ and momentum
transfer follows from the observation that, like Newton’s potential itself,
powers of Newton’s potential are also classical; in impact parameter space, a
classical $L$-loop contribution is proportional $(G/b)^{L+1}$. The inclusion
of outgoing (gravitational) radiation with energy $\omega$ commensurate with
the momentum transfer, $\omega\sim q$, does not alter this dependence because
there is no suppression for the emission of classical (gravitational)
radiation. Quantum scattering amplitudes are however more singular in the
large impact parameter (small transferred momentum) expansion than their
classical counterparts, behaving $G^{L+1}/b^{l<L+1}$, with additional mass-
dependent factors. This can be understood by recalling the structure of
potential scattering in non-relativistic quantum mechanics, which at $N$-th
order in perturbation theory contains multiple insertions of the lower-order
potential. The cancellation of these classically singular (or superclassical)
terms in the matrix element $\mathscr{M}$ is a test of the calculation.

It is extremely convenient to take the classical limit as early as possible
[63, 65, 64], before the Feynman loop integrals are evaluated, as this
substantially reduces the complexity of calculations. Each of the internal
graviton exchanged between matter particles should mediate a long-range
interaction; thus, in this regime all loop momenta $\ell$ should be of the
same order as the total exchanged momentum $q$. The method of regions [66] in
dimensional regularization is a systematic method to select the classical
contributions to $\mathscr{M}$. The integration domain is split into two
regions

$\begin{split}\textrm{Hard Region}:\quad(q_{i},k,\ell)&\rightarrow(\lambda
q_{i},\lambda k,\lambda^{0}\ell)\ ,\\\ \textrm{Soft
Region}:\quad(q_{i},k,\ell)&\rightarrow(\lambda q_{i},\lambda k,\lambda\ell)\
,\end{split}$ (53)

and the latter generates all classical terms. To evaluate an integral in only
one of the regions one simply expands the integrand according to the indicated
scaling in $\lambda$ and integrates over the entire integration domain. The
full integral is given by the sum of the integrals in the two regions [66];
there is no overcounting because the integrand in one region yields only
scaleless integrals when further expanded in the other region, and such
integrals vanish in dimensional regularization.

The momentum space KMOC matrix element $\mathscr{M}$ is related to the
frequency-domain EFT waveform $\mathcal{W}$ (here normalized as the MPM
waveform $W^{\rm MPM}$) through a Fourier transform to impact-parameter space,

$\displaystyle\mathcal{W}(\varepsilon,k,p_{1},p_{2},b_{1},b_{2})$ (54)
$\displaystyle=\frac{2i}{\kappa}\int\mu(k,q_{1},q_{2})e^{-i(q_{1}\cdot
b_{1}+q_{2}\cdot b_{2})}\mathscr{M}(\varepsilon,k,p_{1},p_{2},q_{1},q_{2})\,,$

and we choose the (incoming) impact parameter $b_{12}=b_{1}-b_{2}$ to be
orthogonal to both $p_{1}$ and $p_{2}$,

$p_{1}\cdot b_{12}=p_{2}\cdot b_{12}=0\,.$ (55)

The integration measure enforces momentum conservation and on-shell conditions
for the matter particles. Following Sec. IV.1, we can write the measure as

$\displaystyle\int\text{d}\Phi_{p_{1}}\text{d}\Phi_{p_{2}}\text{d}\Phi_{p_{3}}\text{d}\Phi_{p_{4}}\varphi^{*}(p_{3})\varphi^{*}(p_{4})\varphi(p_{1})\varphi(p_{2})$
$\displaystyle\hskip
95.3169pt\times\hat{\delta}^{4}(p_{1}+p_{2}-p_{3}-p_{4}-k)$
$\displaystyle=\int\mu(k,q_{1},q_{2})+(\text{quantum})\,,$ (56)

and the classical contribution is

$\displaystyle\mu(k,q_{1},q_{2})=\frac{\text{d}^{4}q_{1}}{(2\pi)^{4}}\frac{\text{d}^{4}q_{2}}{(2\pi)^{4}}$
$\displaystyle\hat{\delta}(2p_{1}\cdot q_{1})\hat{\delta}(2p_{2}\cdot q_{2})$
(57) $\displaystyle\times\hat{\delta}^{4}(q_{1}+q_{2}-k)\,,$

where $\hat{\delta}(x)=2\pi\delta(x)$ and
$\hat{\delta}^{d}(x)=(2\pi)^{d}\delta^{d}(x)$. We now sketch the derivation of
Eq. 57. In the classical limit, the Compton wavelength $\ell_{\text{c}}$ is
much smaller than the spread of the wavepacket $\ell_{\phi}$ in the position
space, which is in turn much smaller than the impact parameter
$|b_{1}-b_{2}|$, namely, $\ell_{\text{c}}\ll\ell_{\phi}\ll|b_{1}-b_{2}|$.
Consequently, in a classical scattering process, the wavepackets
$\varphi(p_{1})$ and $\varphi^{*}(p_{4})$ exactly overlap up to quantum
corrections, which allows us to integrate out the wavepackets,
$\int\text{d}\Phi_{p_{1}}\varphi(p_{1})\varphi^{*}(p_{4})\simeq 1$ [same for
$\varphi(p_{2})$ and $\varphi^{*}(p_{3})$]. Then in $\text{d}\Phi_{p_{3}}$ and
$\text{d}\Phi_{p_{4}}$ we change the variables to $q_{1}$ and $q_{2}$ while
keeping only the leading term in the on-shell condition,

$\displaystyle\delta(p_{4}^{2}+m_{1}^{2})$
$\displaystyle\simeq\delta(2p_{1}\cdot q_{1})\,,$
$\displaystyle\delta(p_{3}^{2}+m_{2}^{2})$
$\displaystyle\simeq\delta(2p_{2}\cdot q_{2})\,.$ (58)

Note that we can drop the positive energy constraint in the measure because it
is already implicit in the above delta function constraints.

###  IV.3 IR divergence and time shift

The matrix element $\mathscr{M}$ is infrared-divergent [39, 38, 41, 40, 42],
with contributions from both terms in Eq. (IV.1), which we regularize in
dimensional regularization with $d=4-2\epsilon$. Even at the one-loop order we
are focusing on, the finite waveform integrand depends on whether internal
states are four- or $d$-dimensional. This is in contrast with scattering angle
calculations, where this subtlety appears only at $O(G^{4})$ [31], and is due
here to the presence of divergences which are effectively (additively)
renormalized. To avoid possible issues probably due to unconventional forms of
dimensional regularization (see e.g. [67, 68]), we use conventional
dimensional regularization (CDR) [69], where all states and momenta are
uniformly continued to $d=4-2\epsilon$ dimensions. In the final expressions we
restrict the external momenta to be four dimensional and use special four-
dimensional relations such as vanishing Gram determinants for further
simplifications. This regularization scheme was important in the comparison of
the ${O}(G^{4})$ scattering angle [31] and the corresponding post-Newtonian
calculations of Ref. [70]; this will be the case here as well.

The IR divergences in $\mathscr{M}$ are due to virtual soft gravitons and are
expected to exponentiate [71],

$\displaystyle\mathscr{M}=\exp\left[-i\omega\frac{GE}{\epsilon}(1-\Gamma/2)\right]\mathscr{M}^{\text{fin}}\,,$
(59)

where $\mathscr{M}^{\text{fin}}$ is IR-finite but dependent of $\epsilon$. The
first term in the exponent originates from the Weinberg soft factor from the
five-point amplitude [71], whose proof can be generalized to all orders in the
dimensional regulator. The second term comes from the cut term [42] and
$\Gamma$ is defined as

$\displaystyle\Gamma=\frac{3y-2y^{3}}{(y^{2}-1)^{3/2}}\ .$ (60)

The factor $\omega E$ in the exponent of Eq. 59 can be written covariantly as

$\displaystyle m_{1}w_{1}+m_{2}w_{2}=\omega E\,,$ (61)

where $w_{1}$ and $w_{2}$ are

$\displaystyle w_{a}=-u_{a}\cdot k=-\frac{p_{a}\cdot k}{m_{a}}\,.$ (62)

To understand the fate of the IR divergence, it is useful to consider the
time-domain waveform, which is related to the frequency-domain waveform (54)
by the Fourier-transform

$\displaystyle\mathcal{W}(t)=\int_{-\infty}^{+\infty}\frac{\text{d}\omega}{2\pi}\,\mathcal{W}(\omega)\,e^{-i\omega
t}\,.$ (63)

The IR divergent phase can now be absorbed in the (re)definition of the
retarded time,

$\displaystyle t\rightarrow t+\delta t_{\text{IR}}\,,\quad\delta
t_{\text{IR}}=\frac{GE}{\epsilon}(1-\Gamma/2)\,.$ (64)

Equivalently, we can choose to use the IR finite matrix element in Eq. 54 when
computing the waveform,

$\displaystyle\mathscr{M}\rightarrow\mathscr{M}^{\text{fin}}=\exp\left[i\omega\frac{GE}{\epsilon}(1-\Gamma/2)\right]\mathscr{M}\,.$
(65)

The result for this IR-finite matrix element at $L$ loops depends on the
${O}(\epsilon^{1\leq n\leq L})$ part of $\mathscr{M}$, which in turn depends
on the dimensionality of the internal states at lower-loop orders. In
particular, the tree-level term in $\mathscr{M}$ contains such ${O}(\epsilon)$
contributions which enforce that the regularized theory contains only
$d$-dimensional graviton states, which we will refer to as
$\epsilon\mathcal{M}^{(0)}_{\text{extra}}$. While they are unimportant for the
$O(G)$ waveform, they interfere with the $O(G/\epsilon)$ argument of the phase
and yield finite contributions at $O(G^{2})$ proportional to
$\mathcal{M}^{(0)}_{\text{extra}}$. At one-loop order inclusion of these terms
amounts to a choice of scheme, and the one used here is different from that
used in [39, 38, 41, 40, 42]. Their inclusion however is natural in light of
the fact that Weinberg’s exponentiation of IR divergences holds to all orders
in dimensional regulator in CDR suggesting that they are required for the
exponentiation at two and higher loops. Furthermore, this scheme led to a
successful comparison of amplitudes and PN predictions for the ${O}(G^{4})$
conservative scattering angle, where similar $\epsilon/\epsilon$ effects
played an important role.

##  V Waveform from observable-based formalism

Based on their origin in the $T$-matrix element, we can divide momentum-space
matrix element $\mathscr{M}$ into three categories,

$\displaystyle\mathscr{M}(k,q_{1},q_{2})$
$\displaystyle=\mathscr{M}_{\text{const}}(k,q_{1},q_{2})+\mathscr{M}_{\text{conn}}(k,q_{1},q_{2})$
$\displaystyle\quad+\mathscr{M}_{\text{disc}}(k,q_{1},q_{2})\,.$ (66)

The first term $\mathscr{M}_{\text{const}}$ is contributed by matrix elements
that have the external graviton attached to an on-shell three-point vertex.
Due to momentum conservation, the graviton must have zero energy. Thus
$\mathscr{M}_{\text{const}}$ corresponds to a constant gravitational
background. The second term $\mathscr{M}_{\text{conn}}$ contains the
amplitudes and unitarity cuts contributed from connected $T$-matrix elements
with generic momentum, while $\mathscr{M}_{\text{disc}}$ contains the
unitarity cuts containing disconnected $T$-matrix elements.

In this section, the matter momentum $p_{a}$ and four-velocity $u_{a}$ are to
be understood as the “barred variables” in amplitude literature (see e.g. [72,
73]). They differ from the true momentum $\mathsf{p}_{a}$ (used e.g. in App.
D) by half of the momentum transfer $q_{a}$, i.e.
$p_{a}=\mathsf{p}_{a}+q_{a}/2$. In the classical limit, the difference between
$p_{a}$ and $\mathsf{p}_{a}$ is quantum so they are indistinguishable. In
contrary, the $\bar{p}_{a}$ used in the MPM calculation is the “(classical)
averaged momentum”, defined as $\bar{p}_{a}=p_{a}+\Delta p_{a}/2$, where
$\Delta p_{a}$ is the classical impulse.

###  V.1 Constant background

Exactly-zero energy excitations are an interesting feature of quantum field
theories with massless particles. On the one hand, the inclusion of an
arbitrary number of such excitations does not change the energy of any state,
such as the Higgs field in spontaneously-broken gauge theories [74, 75]. On
the other, they, and their off-shell extensions, may condense into nontrivial
time-independent backgrounds, such as the Coulomb field of a point charge or
the Schwarzschild metric [76, 77]. From the perspective of on-shell scattering
amplitudes around flat space, only the asymptotic zero-energy states appear to
be relevant. It has been suggested in Ref. [78] that their effects may be
incorporated in a vacuum with no such excitations through a coherent state.

It was emphasized in Ref. [52] that the constant part of the waveform is
dependent on the choice of BMS supertranslation frame. The so-called
“canonical frame” is defined to have a vanishing time-independent component
and the “intrinsic frame” having the non-vanishing value given in Eq. (13). A
BMS supertranslation connects the waveforms in the two frames; in light of
Ref. [78] we may expect that this supertranslation is captured by coherent
state of low/vanishing-energy gravitons. Instead of exploring the all-order
nature of this expectation, we will study the effect of including zero-energy
gravitons in the KMOC computation.

The leading order contribution to $\mathscr{M}_{\text{const}}$ comes from the
diagram shown in Fig. 1a: the momentum of one matter particle is conserved
while the other emits a graviton. The relevant amplitudes are

$\displaystyle\mathcal{M}_{3}(k,p_{1},p_{4})$
$\displaystyle=-i\kappa(\varepsilon\cdot p_{1})^{2}\,,$
$\displaystyle\mathcal{M}_{3}(k,p_{2},p_{3})$
$\displaystyle=-i\kappa(\varepsilon\cdot p_{2})^{2}\,.$ (67)

For on-shell and real momenta, this process is supported at the origin of
phase space, at which the emitted graviton has exactly zero energy.

Since one matter particle does not participate in the scattering, we need to
use the three-particle phase space to perform the Fourier transform,

$\displaystyle\mathcal{W}_{\text{const}}(k)\Big{|}_{m_{1}}$
$\displaystyle=\frac{2i}{\kappa}\int\text{d}\Phi_{p_{4}}\hat{\delta}^{4}(p_{1}-p_{4}-k)e^{-i(p_{1}-p_{4})\cdot
b_{1}}$ $\displaystyle\hskip 56.9055pt\times\mathcal{M}_{3}(k,p_{1},p_{4})$
$\displaystyle=m_{1}(\varepsilon\cdot u_{1})^{2}\hat{\delta}(u_{1}\cdot k)\,.$
(68)

It is easy to see by going to the rest frame of the particle of mass $m_{1}$
that the delta function has a unique solution for which the graviton has
exactly zero energy.

Thus, if exacty zero energy gravitons are excluded from the spectrum of
states, $\mathcal{W}_{\text{const}}(k)=0$ and we recover the expected result
in the BMS canonical frame. Consistently excluding such gravitons from higher-
order calculations amounts to excluding all kinematic configurations which
constrain at least one graviton to have exactly zero energy.

Conversely, allowing such contributions reproduces the MPM Coulombic
background given in Eqs. 13 and 14.101010We note that the “constant
background” here is different from $\mathcal{M}^{\text{lin}}(k)$ given in Eq.
(5.2) of [50]. Here the background is chosen to be the pure time-independent
piece of the waveform that is present at $t=-\infty$, while
$\mathcal{M}^{\text{lin}}(k)$ there was defined as the average of the time-
independent contributions at $t=\pm\infty$. In other words,
$\mathcal{M}^{\text{lin}}(k)$ included some time-dependent evolutions that
contribute to the gravitational-wave memory.111111The observation that zero-
energy gravitons reproduce the Coulombic background was also made recently in
a related context in [79]. At higher orders, it requires us to consistently
include special kinematic configurations localizing at least one graviton at
the origin of phase space. Furthermore, we may expect that these terms may be
obtained from those supported on finite-energy configurations through the same
supertranslation as the one connecting the intrinsic and canonical frames.

As we have discussed in Sec. IV.1, the Coulombic background serves to fix the
initial value of the waveform. Since the three-point amplitudes are sufficient
to match the initial condition with the MPM calculation, other diagrams
involving a zero energy graviton are not expected to contribute to the
constant background. We will show in App. D that it is indeed the case.

$p_{1}$$p_{2}$$p_{4}$$p_{3}$$k$ (a)

$p_{1}$$p_{2}$$p_{4}$$p_{3}$$k$ (b)

Figure 1: The contribution to the KMOC matrix element from the disconnected
components of the $T$-matrix.  Figure 2: The connected contribution to the
KMOC matrix element. Each blob here is a connected component of the
$T$-matrix. The first two graphs are respectively the tree and one-loop
amplitude, and the last graph is a unitarity cut. The external particles are
labeled in the same way as Fig. 1.

###  V.2 Amplitudes and connected cuts

The contribution of all the connected (generic momentum) $T$-matrix elements
to $\mathscr{M}$ have been evaluated in [39, 38, 41, 40] to $O(\kappa^{5})$.
We can write it as

$\displaystyle\mathscr{M}_{\text{conn}}(k,q_{1},q_{2})$
$\displaystyle=\mathcal{M}^{\text{tree}}+\mathcal{M}^{\text{1
loop}}+\mathcal{S}^{\text{1 loop}}+O(\kappa^{7})\,,$ (69)

where $\mathcal{M}^{\text{tree}}$ and $\mathcal{M}^{\text{1 loop}}$ are
respectively the tree-level and one-loop amplitude contributed from the first
term (linear in $T$) of Sec. IV.1, while the cut contribution
$\mathcal{S}^{\text{1 loop}}$ comes from the second term of Sec. IV.1.
Diagrammatically they are represented by Fig. 2.

For later convenience, we separate out the $\kappa$ and mass dependence from
$\mathcal{M}^{\text{tree}}$,

$\displaystyle\mathcal{M}^{\text{tree}}=\kappa^{3}m_{1}^{2}m_{2}^{2}\mathcal{M}^{(0)}_{d}\,.$
(70)

The reduced tree-level amplitude is given by

$\displaystyle\mathcal{M}^{(0)}_{d}$ $\displaystyle=$ $\displaystyle
i\left[-F_{2}^{2}\frac{y^{2}}{q_{1}^{2}q_{2}^{2}w_{2}^{2}}\right.$
$\displaystyle+$ $\displaystyle
F_{3}^{2}\left(-\frac{1}{q_{1}^{2}q_{2}^{2}}+\frac{y}{q_{2}^{2}w_{1}w_{2}}+\frac{y^{2}}{4w_{1}^{2}w_{2}^{2}}-\frac{q_{1}^{2}y^{2}}{4q_{2}^{2}w_{1}^{2}w_{2}^{2}}\right)$
$\displaystyle+$
$\displaystyle\left.F_{2}F_{3}\left(-\frac{2y}{q_{1}^{2}q_{2}^{2}w_{2}}+\frac{y^{2}}{q_{2}^{2}w_{1}w_{2}^{2}}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{i}{d-2}\left[\frac{F_{3}^{2}(q_{1}^{2}-q_{2}^{2})}{4q_{2}^{2}w_{1}^{2}w_{2}^{2}}+\frac{F_{2}^{2}}{q_{1}^{2}q_{2}^{2}w_{2}^{2}}-\frac{F_{2}F_{3}}{q_{2}^{2}w_{1}w_{2}^{2}}\right].$

where the graviton polarizations are encoded in the gauge invariant variables

$\displaystyle F_{1}=F_{\mu\nu}q_{1}^{\mu}u_{1}^{\nu}\,,\quad
F_{2}=F_{\mu\nu}q_{2}^{\mu}u_{2}^{\nu}\,,\quad
F_{3}=F_{\mu\nu}u_{1}^{\mu}u_{2}^{\nu}\,,$ (72)

and $F_{\mu\nu}=k_{\mu}\varepsilon_{\nu}-k_{\nu}\varepsilon_{\mu}$ is the
linearized field strength. For $d=4-2\epsilon$, we have,

$\displaystyle\mathcal{M}_{d=4-2\epsilon}^{(0)}=\mathcal{M}^{(0)}_{d=4}+\epsilon\mathcal{M}^{(0)}_{\text{extra}}+O(\epsilon^{2})\,,$
(73)

where the extra dimensional contribution comes from the $\frac{1}{d-2}$ factor
in Eq. 73,

$\displaystyle{\mathcal{M}}^{(0)}_{\rm
extra}=\frac{i[4F_{2}^{2}w_{1}^{2}-4F_{2}F_{3}w_{1}q_{1}^{2}+F_{3}^{2}q_{1}^{2}(q_{1}^{2}-q_{2}^{2})]}{8(q_{1}^{2}q_{2}^{2}w_{2}^{2}w_{1}^{2})}\,.$
(74)

Although the external states are kept in exactly four dimensions, the
$\epsilon$ dependence appears here to ensure that only graviton states
propagate in $d=4-2\epsilon$ dimensions.

This is a consequence of the dimensional regularization scheme we choose. Of
course, we can set $d=4$ when computing the tree-level waveform, and
$\mathcal{M}^{(0)}_{\text{extra}}$ does not contribute.

Similarly, we can write the one-loop amplitude and cut contribution as

$\displaystyle\mathcal{M}^{\text{one loop}}$
$\displaystyle=\kappa^{5}m_{1}^{3}m_{2}^{2}\mathcal{M}_{1}^{(1)}+\kappa^{5}m_{1}^{2}m_{2}^{3}\mathcal{M}_{2}^{(1)}\,,$
$\displaystyle\mathcal{S}^{\text{one loop}}$
$\displaystyle=\kappa^{5}m_{1}^{3}m_{2}^{2}\mathcal{S}_{1}^{(1)}+\kappa^{5}m_{1}^{2}m_{2}^{3}\mathcal{S}_{2}^{(1)}\,.$
(75)

The reduced one-loop amplitude $\mathcal{M}_{1}^{(1)}$ is given by

$\displaystyle\mathcal{M}_{1}^{(1)}=-\frac{iw_{1}}{32\pi\epsilon}\mathcal{M}^{(0)}_{d=4}+\widetilde{\mathcal{M}}_{1}^{(1)}\,,$
(76)

where the IR finite part $\widetilde{\mathcal{M}}_{1}^{(1)}$ is the same as
Eq. (5.6) of [50]. The expression of $\widetilde{\mathcal{M}}_{1}^{(1)}$ can
be found in App. B. We note that $\mathcal{M}_{2}^{(1)}$ in Sec. V.2 can be
obtained from $\mathcal{M}_{1}^{(1)}$ by exchanging the particle labels $1$
and $2$. On the other hand, the cut contribution can be written as

$\displaystyle\mathcal{S}^{(1)}_{1}$
$\displaystyle=\frac{iw_{1}\Gamma}{64\pi\epsilon}\mathcal{M}^{(0)}_{d=4}+\widetilde{\mathcal{S}}_{1}^{(1)}\,,$
(77)

where the finite part $\widetilde{\mathcal{S}}_{1}^{(1)}$ is given in App. B,
and $\mathcal{S}_{2}^{(1)}$ is given by the same particle relabeling. Note
that we have removed a local UV divergence from $\mathcal{S}_{1}^{(1)}$. Such
terms do not affect the long-range interactions that we are interested in, and
they are projected out by the Fourier transform to impact parameter space.

The full IR divergence of the one-loop waveform is

$\displaystyle(\mathcal{M}^{\text{1 loop}}+\mathcal{S}^{\text{1
loop}})\Big{|}_{\text{IR}}=-\frac{i\omega\kappa^{2}E(2-\Gamma)}{64\pi\epsilon}\mathcal{M}^{\text{tree}}_{d=4}\,,$
(78)

Following Eq. 65 and the discussion below, we can absorb the IR divergence
into an infinite shift in the retarded time $\delta
t_{\text{IR}}=\frac{GE}{\epsilon}(1-\Gamma/2)$ to obtain the IR finite
momentum-space waveform,

$\displaystyle\mathscr{M}^{\text{fin}}$
$\displaystyle=\exp\left[i\omega\frac{GE}{\epsilon}(1-\Gamma/2)\right]\mathscr{M}$
$\displaystyle=\mathcal{M}^{\text{tree}}+\mathcal{M}^{\text{1
loop}}+\mathcal{S}^{\text{1 loop}}$
$\displaystyle\quad+\frac{i\omega\kappa^{2}E(2-\Gamma)}{64\pi\epsilon}\mathcal{M}^{\text{tree}}+O(\kappa^{7})$
(79) $\displaystyle=\mathcal{M}^{\text{tree}}+\mathcal{M}^{\text{1 loop
fin}}+\mathcal{S}^{\text{1 loop fin}}+O(\kappa^{7})\,.$

Crucially, the time shift introduces a term proportional to the
$d$-dimensional tree amplitude $\mathcal{M}^{\text{tree}}$. While this term
cancels the IR divergence (78), it brings an additional finite contribution
proportional to the extra dimensional component (74). Organizing the finite
one-loop amplitude and cut by the mass dependence,

$\displaystyle\mathcal{M}^{\text{1 loop fin}}$
$\displaystyle=\kappa^{3}m_{1}^{3}m_{2}^{2}\mathcal{M}_{1}^{(1)\text{
fin}}+\kappa^{3}m_{1}^{2}m_{2}^{3}\mathcal{M}_{2}^{(1)\text{ fin}}\,,$
$\displaystyle\mathcal{S}^{\text{1 loop fin}}$
$\displaystyle=\kappa^{3}m_{1}^{3}m_{2}^{2}\mathcal{S}_{1}^{(1)\text{
fin}}+\kappa^{3}m_{1}^{2}m_{2}^{3}\mathcal{S}_{2}^{(1)\text{ fin}}\,,$ (80)

we get

$\displaystyle\mathcal{M}_{1}^{(1)\text{ fin}}$
$\displaystyle=\widetilde{\mathcal{M}}_{1}^{(1)}+\frac{iw_{1}}{32\pi}\mathcal{M}^{(0)}_{\text{extra}}\,,$
$\displaystyle\mathcal{S}_{1}^{(1)\text{ fin}}$
$\displaystyle=\widetilde{\mathcal{S}}_{1}^{(1)}-\frac{iw_{1}\Gamma}{64\pi}\mathcal{M}^{(0)}_{\text{extra}}\,.$
(81)

Comparing with Eqs. 76 and 77 we see that, while the divergence is removed by
absorbing the $1/\epsilon$ into a time shift, this shift introduces a new
contribution proportional to the extra dimensional component of the five-point
tree amplitude defined in Eq. 73. Thus the additional terms proportional to
$\mathcal{M}^{(0)}_{\text{extra}}$ come from the $\epsilon/\epsilon$
cancellation, and they are the direct consequence of the regularization scheme
chosen in Sec. IV.3. While at this loop order the inclusion of this term may
be regarded as a scheme dependence (and was not included in previous amplitude
calculations or comparisons), its presence here is justified by the expected
higher-loop structure of IR divergences as well as by its importance in
related contexts [31]. As we will see in Sec. VIII, its presence substantially
improves the comparison between MPM and amplitudes-based results.

###  V.3 Cuts with disconnected matrix elements and BMS transformation

If zero-energy gravitons are excluded, then all cut contributions containing a
three-point S-matrix element or an $n+1$-point matrix element in the
$1\rightarrow n$ configuration vanish identically. All disconnected
contributions to the cut term in Eq. (IV.1) are of this type. If zero-energy
gravitons are kept in the spectrum, the (albeit degenerate) three-point
amplitude generates, at higher-order in Newton’s constant, time-dependent
terms through the cut term in this equation. The relevant contribution at one-
loop order is shown in Fig. 1b; the discussion above implies that, at $L$
loops, the six-point tree amplitude factor is replaced by the corresponding
$(L-1)$-loop six-point amplitude with no changes to the disconnected
factor.121212In the full quantum theory the three-point Green’s function is UV
divergent; this divergence arises from loop momenta outside the soft region,
and do not contribute to the classical observables we are discussing here.

The evaluation of this cut makes use of the fact that the cut massless
momentum $\ell$ must have vanishing frequency, so only the leading soft limit
of the six-point tree amplitude (the left blob of Fig. 1b) with two external
gravitons of momenta $k$ and $\ell$ is relevant,

$\displaystyle{\cal M}_{6}(p_{1\ldots 4},k,\ell)$
$\displaystyle\simeq-\kappa\,\mathcal{M}_{5}(p_{1\ldots 4},k)$ (82)
$\displaystyle\quad\times\left(\sum_{i=a}^{4}\frac{\eta_{a}\varepsilon(\ell)_{\mu\nu}p_{a}^{\mu}p_{a}^{\nu}}{2\ell\cdot
p_{a}-i\epsilon\eta_{a}}+\frac{\varepsilon(\ell)_{\mu\nu}k^{\mu}k^{\nu}}{2\ell\cdot
k-i\epsilon}\right),$

where $\eta_{a}=\pm 1$ for outgoing and incoming particles, respectively, and
$\mathcal{M}_{5}(p_{1\ldots 4},k)$ is the five-point amplitude with one
external graviton of momentum $k$.

Sewing Eq. 82 onto the disconnected five-point amplitude, we find that the
four summed terms in Eq. 82 that involve massive matter momenta will lead to
integrals of the form

$\displaystyle\int\text{d}^{d}\ell\,\hat{\delta}(u_{3}\cdot\ell)\hat{\delta}(\ell^{2})\Theta(\ell^{0})\,,$
$\displaystyle\int\text{d}^{d}\ell\,\hat{\delta}(u_{3}\cdot\ell)\hat{\delta}(\ell^{2})\hat{\delta}(u_{4}\cdot\ell)\Theta(\ell^{0})\,,$
(83)

both of which are integrated to zero in $d=4-2\epsilon$. Therefore, a nonzero
contribution can only come from the last term in this equation, which
corresponds to the soft graviton being attached to the outgoing graviton with
momentum $k$. We can replace $\mathcal{M}_{5}(p_{1...4},k)$ by its classical
limit given in Eq. 70. It then yields the following contribution to
$\mathscr{M}_{\text{disc}}$ at ${O}(\kappa^{5})$ (which makes use of the on-
shell condition for the outgoing graviton):

$\displaystyle\mathscr{M}_{\text{disc}}$
$\displaystyle=-i\kappa^{2}\mathcal{M}^{\text{tree}}\left[m_{2}^{2}w_{2}^{2}I(u_{3},k)+m_{1}^{2}w_{1}^{2}I(u_{4},k)\right]\,,$
(84)

where we used that in the classical limit $-p_{3,4}\cdot k=-p_{2,1}\cdot
k+{O}(q^{2})\simeq m_{2,1}w_{2,1}$ and $I(u_{3},k)$ is defined as

$\displaystyle I(u_{3},k)$
$\displaystyle=\frac{1}{m_{2}}\int\frac{\text{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2u_{3}\cdot\ell)\hat{\delta}(\ell^{2})\Theta(\ell^{0})}{2\ell\cdot
k}$
$\displaystyle=\frac{1}{m_{2}\omega}\int\frac{\text{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2u_{3}\cdot\ell)\hat{\delta}(\ell^{2})\Theta(\ell^{0})}{2\ell\cdot{\hat{k}}}\
,$ (85)

where ${\hat{k}}=k/\omega$, and $I(u_{4},k)$ is obtained simply by relabeling.
We did not include $\Theta(p_{3}^{0}+\ell^{0})$ for the matter cut line
because $|\ell^{0}|\ll p_{3}^{0}$ so $\Theta(p_{3}^{0}+\ell^{0})=1$. We have
also suppressed the $i\epsilon$ in the propagator since it does not affect the
integral. However, to properly include the zero energy graviton contribution
across the cut, we need to introduce additional regularizations.

Since the zero energy graviton is supported only at the origin of phase space,
we must regularize it so that this point, $|\ell|=0$, remains in the
integration domain.131313This is analogous with the determination of the snail
diagrams in quantum scattering amplitudes, see e.g. [80]. We may do this by
relaxing the on-shell condition $u_{3}^{2}=-1$ by introducing
$u_{3}\rightarrow\tilde{u}_{3}=u_{3}-\beta{\hat{k}}$ with arbitrary $\beta$.
In the presence of $\hat{\delta}(\ell^{2})$, this modifies the cut matter
propagator to be

$\displaystyle 2u_{3}\cdot\ell=(u_{3}+\ell)^{2}+1$
$\displaystyle\rightarrow(\tilde{u}_{3}+\ell)^{2}+1$ $\displaystyle\simeq
2u_{3}\cdot\ell-2\beta u_{3}\cdot\hat{k}\,,$ (86)

where we dropped the term proportional to ${\hat{k}}\cdot\ell$ as it leads to
integrals proportional the first of Eq. (V.3) and thus vanishes. We therefore
consider instead the integral

$\displaystyle\tilde{I}(u_{3},k)$
$\displaystyle=\frac{e^{\gamma_{\text{E}}\epsilon}}{m_{2}\omega}\int\frac{\text{d}^{d}\ell}{(2\pi)^{d}}\frac{\hat{\delta}(2u_{3}\cdot\ell-2\beta
u_{3}\cdot\hat{k})\hat{\delta}(\ell^{2})\Theta(\ell^{0})}{2\ell\cdot\hat{k}}$
$\displaystyle=\frac{\Theta(\beta)}{16\pi
m_{2}w_{2}}\left[\frac{1}{\epsilon}-\log\frac{w_{2}^{2}}{\omega^{2}}-\log\frac{\beta^{2}}{\pi}\right],$
(87)

and we keep only the $O(\epsilon^{0})$ terms. The original integral
$I(u_{3},k)$ is recovered at $\beta=0$. Using $\Theta(0)=\frac{1}{2}$, we
get,141414Alternatively, we can obtain $I(u_{3},k)$ directly as a
discontinuity of an off-shell triangle integral, which was computed in Eq.
(C.4) of Ref. [81] (see Ref. [82] for comments on similar integrals), with the
same off-shell continuation as discussed here.

$\displaystyle I(u_{3},k)=\frac{1}{32\pi
m_{2}w_{2}}\left[\frac{1}{\epsilon}-\log\frac{w_{2}^{2}}{\omega^{2}}-\log\frac{\beta^{2}}{\pi}\right]\
,$ (88)

where the divergent constants $\log\beta$ and $1/\epsilon$ will be absorbed by
a time shift. We note that both integrals in Sec. V.3 remain zero even if we
used the above off-shell regularization.

Putting together Eqs. 84 and 88, we find that the cut in Fig. 1b gives a
finite contribution $\mathcal{S}^{\text{1 loop disc}}$,

$\displaystyle\mathscr{M}_{\text{disc}}=\mathcal{S}^{\text{1 loop
disc}}-i\omega
GE\left[\frac{1}{\epsilon}-\log\frac{\beta^{2}}{\pi}\right]\mathcal{M}^{\text{tree}}\,,$
$\displaystyle\mathcal{S}^{\text{1 loop
disc}}=iG\left[m_{1}w_{1}\log\frac{w_{1}^{2}}{\omega^{2}}+m_{2}w_{2}\log\frac{w_{2}^{2}}{\omega^{2}}\right]\mathcal{M}^{\text{tree}}\,.$
(89)

The second term of $\mathscr{M}_{\text{disc}}$ can be removed by a time shift
that supplements the one in Eq. 64. The first term, $\mathcal{S}^{\text{1 loop
disc}}$, which may be interpreted as a direction-dependent time shift, is in
fact the BMS supertranslation identified in [52] as relating the “canonical”
and “intrinsic” BMS frames, thus explaining the angular momentum loss at
${O}(G^{2})$ [55], as well as the transformation needed to relate the first
nonuniversal term in the soft expansion of the amplitude-based [51, 83] and
MPM waveforms [50].

We therefore see that zero-energy gravitons generate on the one hand a time-
independent background metric at ${O}(G^{0})$ (see Sec. V.1) and, on the other
hand, a time-dependent waveform at higher orders in $G$ and moreover that both
of them can be removed by the same BMS supertranslation that connects the
intrinsic and canonical frames. This reinforces the close connection between
zero-energy gravitons and the choice of BMS frame, which echoes features in
the angular momentum loss calculations [52, 84, 85], and suggests that one may
consistently ignore the disconnected cut contributions and restore them
through the BMS transformation of Ref. [52]. It would be interesting to
attempt to formulate the observable-based formalism in the “intrinsic” BMS
frame in terms of a vacuum dressed with a coherent state of zero-energy
gravitons in the spirit of Ref. [78]. Such a formulation, in which the
contribution of zero-energy particles (and the associated subtleties) will
emerge because of this dressing, may give a means to argue for an all-order
connection between the disconnected cut contributions and the choice of BMS
frame.

##  VI PN and soft expansions of KMOC waveform

The one-loop amplitude part of the $\mathscr{M}$ matrix element computed in
[39, 40, 41] exhibits spurious poles (SP). Some of these poles cancel
manifestly when the amplitude is expanded in special kinematic limits such as
at low velocity, in the soft limit, etc. However, manifesting such
cancellations for general kinematic configurations is, at first sight, highly
non-trivial and appears to require the use of Gram determinant and Bianchi
identities among a huge number of terms. Algorithms for addressing such
problems have been developed in Ref. [86] (see also Ref. [87]) and applied to
the current problem in Ref. [46].

In App. C we take a different route and reorganize the one-loop five-point
scattering amplitude (76) and the cut term to make manifest the cancellation
of the spurious poles on the physical sheet. By direct inspection of Feynman
integrals involving non-trivial numerators, we find combinations of the square
roots and logarithms appearing in (76) such that they have high-order zeros in
correspondence with the singularities of the rational functions multiplying
them. It turns out that the same functions are sufficient to render both the
amplitude and the cut contributions to $\mathscr{M}$ free of spurious poles.
Their expressions after this reorganization are

$\displaystyle\mathcal{M}_{1}^{(1)\text{ fin}}$
$\displaystyle=\left[\frac{iw_{1}}{32\pi}\log\frac{w_{1}^{2}}{\mu_{\text{IR}}^{2}}+\frac{w_{1}}{32}+\frac{\Gamma
w_{1}}{64}\right]\mathcal{M}^{(0)}_{d=4}$ (90)
$\displaystyle\quad+\frac{iw_{1}}{32\pi}\mathcal{M}^{(0)}_{\text{extra}}+c_{1}I_{1}+c_{2}I_{2}+l^{\prime}_{q}L_{q}+l^{\prime}_{w}L_{w}$
$\displaystyle\quad+R+l_{q}\log\frac{q_{1}^{2}}{q_{2}^{2}}+l_{w}\log\frac{w_{2}^{2}}{w_{1}^{2}}+l_{y}\frac{\operatorname{arccosh}y}{\sqrt{y^{2}-1}}\
,$ $\displaystyle\mathcal{S}_{1}^{(1)\text{ fin}}$
$\displaystyle=\frac{iw_{1}\Gamma}{64\pi}\mathcal{M}^{(0)}_{d=4}\log\frac{w_{1}w_{2}\mu_{\text{IR}}^{2}}{q_{1}^{2}q_{2}^{2}(y^{2}-1)}$
(91)
$\displaystyle\quad-\frac{iw_{1}\Gamma}{64\pi}\mathcal{M}^{(0)}_{\text{extra}}+c_{\text{rat}}+c_{\text{ratio}}\log\frac{w_{1}^{2}w_{2}^{2}}{q_{1}^{2}q_{2}^{2}}$
$\displaystyle\quad+c_{y}\operatorname{arccosh}y+c_{q}L_{q}+\mathcal{S}_{1}^{(1)\text{
local}}\,,$

where $\\{I_{1},I_{2},L_{q},L_{w}\\}$ are SP-free combinations. Their explicit
forms and construction are given in App. C. Here $\mathcal{S}_{1}^{(1)\text{
local}}$ includes all terms which have no singularities in $q_{1,2}^{2}$
(including $c_{w}L_{w}$) and therefore integrate to delta-function-supported
terms in impact parameter space. Thus $\mathcal{S}_{1}^{(1)\text{ local}}$ can
be ignored. The explicit expressions of the $c$ and $l$ coefficients in Eqs.
(90) and (91) are included in the ancillary file WaveformReorganisation.nb.

###  VI.1 Impact-parameter-space waveform

With the reorganized parts of the momentum space $\mathscr{M}$ matrix element,
we now discuss the Fourier transform (54) to obtain the impact-parameter-space
(and frequency-space) waveform $\mathcal{W}(k,b_{1},b_{2})$. Here we will
focus on the $\omega\neq 0$ (time-dependent) contribution up to $O(G^{2})$. As
$\mathscr{M}(k,q_{1},q_{2})$ is a function of the kinematic data of the
incoming state, the waveform $\mathcal{W}(k,b_{1},b_{2})$ is naturally given
in the incoming c.m. frame $(e_{0},e_{1},e_{2},e_{3})$ anchored on the
incoming momenta $p_{1}$ and $p_{2}$. The unit time vector $e_{0}$ can be
chosen as

$\displaystyle
e_{0}^{\mu}=\frac{p_{1}^{\mu}+p_{2}^{\mu}}{|p_{1}^{\mu}+p_{2}^{\mu}|}\,,$ (92)

while the incoming momenta are along the $e_{2}$ direction,

$\displaystyle p_{1}=E_{1}\,e_{0}+P_{\text{c.m.}}\,e_{2}\,,\quad
p_{2}=E_{2}\,e_{0}-P_{\text{c.m.}}\,e_{2}\,,$ (93)

where $E_{a}=\sqrt{P_{\text{c.m.}}^{2}+m_{a}^{2}}$. Note that we can express
$E_{a}$ and $P_{\text{c.m.}}$ in terms of $y$ as

$\displaystyle P_{\text{c.m.}}$
$\displaystyle=\frac{m_{1}m_{2}\sqrt{y^{2}-1}}{E}\,,$ $\displaystyle E$
$\displaystyle=\sqrt{m_{1}^{2}+2ym_{1}m_{2}+m_{2}^{2}}\,.$ (94)

We then choose the impact parameter $b_{1}$ and $b_{2}$ to satisfy
$\frac{E_{1}}{E}b_{1}^{\mu}+\frac{E_{2}}{E}b_{2}^{\mu}=0$ and be orthogonal to
$e_{0}$. We define the relative transferred momentum $q^{\mu}$ by

$\displaystyle
q^{\mu}=q_{1}^{\mu}-\frac{E_{1}}{E}k^{\mu}=-q_{2}^{\mu}+\frac{E_{2}}{E}k^{\mu}\,,$
(95)

such that the exponential factor in Eq. 54 becomes

$\displaystyle e^{-i(q_{1}\cdot b_{1}+q_{2}\cdot
b_{2})}=e^{-iq\cdot(b_{1}-b_{2})}=e^{-iq\cdot b_{12}}\,,$ (96)

and thus $\mathcal{W}$ depends only on $b_{12}$. The impact parameter $b_{12}$
is orthogonal to both of the incoming momenta $p_{1}$ and $p_{2}$. Thus we can
choose it to be along the $e_{1}$ direction,151515We note that
$b_{\text{in}}=b+O(G^{2})$. See the discussion below Eq. 6.

$\displaystyle b_{12}=b_{\text{in}}\,e_{1}\,.$ (97)

The momentum $k$ and polarization vector $\varepsilon$ for the emitted
graviton are given by

$\displaystyle\mathbf{\tilde{n}}(\theta,\phi)$
$\displaystyle=\sin\theta\cos\phi\,e_{1}+\sin\theta\sin\phi\,e_{2}+\cos\theta\,e_{3}\,,$
$\displaystyle k$
$\displaystyle={\omega}\left(e_{0}+{\bf\tilde{n}}(\theta,\phi)\right),$
$\displaystyle\varepsilon$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\partial_{\theta}{\bf\tilde{n}}(\theta,\phi)-\frac{i}{\sin\theta}\partial_{\phi}{\bf\tilde{n}}(\theta,\phi)\right].$
(98)

They are formally the same as Eq. 7, but the polar angles here $(\theta,\phi)$
are defined instead with respect to the spatial frame $(e_{1},e_{2},e_{3})$.
Therefore, the Fourier transform of $\mathscr{M}(k,q_{1},q_{2})$ in Eq. 54 now
becomes

$\displaystyle\text{FT}_{b}[f]\equiv\int\mu\Big{(}k,q+\frac{E_{1}}{E}k,-q+\frac{E_{2}}{E}k\Big{)}e^{-iq\cdot
b_{12}}f(q)\,,$
$\displaystyle\mathcal{W}(\omega,\theta,\phi)=\frac{2i}{\kappa}\,\text{FT}_{b}\left[\mathscr{M}\\!\left(k,q+\frac{E_{1}}{E}k,-q+\frac{E_{2}}{E}k\right)\right].$
(99)

The resulting impact parameter space waveform $\mathcal{W}$ is in principle a
Lorentz invariant function consisting of scalar products of
$(\varepsilon,k,p_{1},p_{2},b_{12})$. However, performing this Fourier
transform in complete generality is challenging. In the following, we will
proceed in the incoming c.m. frame, such that the frequency-domain waveform is
given as a function $\mathcal{W}(\omega,\theta,\phi)$, and consider its PN
expansion.

###  VI.2 PN expansion

For the PN expansion, we will stick to the conventions of reference [50], also
already used here in sections II and III. It is an expansion in powers of the
velocity parameter

$p_{\infty}\equiv\sqrt{y^{2}-1}\ .$ (100)

Here $p_{\infty}$ is taken as dimensionless (i.e. we can replace
$\eta=\frac{1}{c}$, used in Secs. II and III by $1$).

Following Ref. [50], one rewrites the kinematic variables in terms of the
expansion parameter $p_{\infty}$, and expands the integrand before computing
the Fourier transform to impact-parameter space. This gives rise to a rather
simple integrand whose integral can be expressed solely in terms of Bessel K
functions and exponentials.

One starts by decomposing $q^{\mu}$ in terms of the external vectors (a
covariant form of this decomposition, using $u_{1},u_{2},b$ and a fourth
vector orthogonal on them, was introduced in Ref. [4])

$q^{\mu}=q^{0}e_{0}^{\mu}+\Omega\,Q_{x}e_{1}^{\mu}+q_{y}e_{2}^{\mu}+\Omega\,Q_{z}e_{3}^{\mu}\,,$
(101)

where $e_{1,2,3}^{\mu}$ denote the direction of impact parameter, of the c.m.
momentum and of the orthogonal direction to the scattering plane, respectively
– as defined in Section II. Moreover, $\Omega$ is the frequency of the emitted
radiation rescaled by the expansion parameter:

$\omega=\Omega\,p_{\infty}\,.$ (102)

The integration over $q_{0,y}$ gives the following Jacobian:

$\int\\!\mu\left(k,q_{1},q_{2}\right)e^{-iq\cdot
b_{12}}=\frac{\Omega^{2}}{4m_{1}m_{2}p_{\infty}}\int\frac{\text{d}Q_{x}\text{d}Q_{z}}{(2\pi)^{2}}e^{-iuQ_{x}}\,,$
(103)

where $u$ is the dimensionless frequency variable

$u\equiv\Omega\,b\,\equiv\frac{\omega\,b}{p_{\infty}},$ (104)

and the delta functions fix the components $q^{0},q_{y}$:

$\begin{split}q^{0}&=-\frac{\Omega\,p_{\infty}}{E^{2}}\left(m_{1}^{2}-m_{2}^{2}-m_{1}m_{2}p_{\infty}\sin\theta\sin\phi\right)\,,\\\
q_{y}&=\frac{\Omega}{E^{2}}\left(m_{1}\sqrt{p_{\infty}^{2}+1}+m_{2}\right)\left(m_{2}\sqrt{p_{\infty}^{2}+1}+m_{1}\right)\,.\end{split}$
(105)

These relations correspond to Eqs. (5.9) in Ref. [50], except that the latter
equations were expressed in the the barred com frame
$(\bar{e}_{0},e_{x},e_{y},e_{z})$.

For completeness, let us give also the expressions of $w_{1,2}$ in the c.m.
frame:

$\begin{split}w_{1}&=\frac{\Omega\,p_{\infty}}{E}\left[m_{1}+m_{2}\left(\sqrt{p_{\infty}^{2}+1}-p_{\infty}\sin\theta\sin\phi\right)\right]\,,\\\
w_{2}&=\frac{\Omega\,p_{\infty}}{E}\left[m_{2}+m_{1}\left(\sqrt{p_{\infty}^{2}+1}+p_{\infty}\sin\theta\sin\phi\right)\right]\,.\\\
\end{split}$ (106)

The original form of the amplitude [39, 40, 41] made it challenging to compute
its PN expansion because of the presence of spurious poles, though Ref. [50]
succeeded to carry out the expansion up to the 2.5PN approximation, and to
evaluate the Fourier transform, thereby obtaining the EFT waveform as a
function of ${\omega}$, $\theta$ and $\phi$. The SP-free version of the
amplitude presented above significantly simplifies the task of PN-expanding
the amplitude. The leading orders of the various terms in Eq. (90) in a
small-$p_{\infty}$ expansion are

$\displaystyle c_{1}I_{1}\sim p_{\infty}^{0}$ $\displaystyle
l^{\prime}_{q}L_{q}\sim p_{\infty}^{2}$ $\displaystyle R\sim p_{\infty}^{-5}$
$\displaystyle c_{2}I_{2}\sim p_{\infty}^{-5}$ $\displaystyle
l^{\prime}_{w}L_{w}\sim p_{\infty}^{2}$ (107) $\displaystyle
l_{q}\log\frac{q_{1}^{2}}{q_{2}^{2}}\sim p_{\infty}^{0}\;$ $\displaystyle
l_{w}\log\frac{w_{2}^{2}}{w_{1}^{2}}\sim p_{\infty}^{2}\;$ $\displaystyle
l_{y}\frac{\operatorname{arccosh}y}{\sqrt{y^{2}-1}}\sim p_{\infty}^{-1}\,.$

Thus, in this organization three of the eight terms start contributing only at
2PN order. Moreover, the cancellation of terms $p_{\infty}^{n\leq-2}$ is quite
straightforward to observe. This allowed us to reach the 3PN accuracy. Even
though we have so far only computed the 3PN-accurate EFT waveform in the
equatorial plane, i.e. when taking $\theta=\frac{\pi}{2}$, there is no
technical obstacle to derive it for arbitrary values of $\theta$. Although
most integrals in $Q_{z}$ and $Q_{x}$ are absolutely convergent, some of the
contributions from the cut term must be interpreted as Fourier transforms of
tempered distributions.

As explained in [50] the PN expansion of the amplitude yields an integrand
containing only integer or half-integer powers of

$D_{0}=1+Q_{x}^{2}+Q_{z}^{2}\,,$ (108)

with numerators polynomial in $Q_{z}$ and $Q_{x}$. The Fourier transforms of
all these terms are computed from the general formula

$\int\frac{\text{d}Q_{x}\text{d}Q_{z}}{(2\pi)^{2}}e^{-iuQ_{x}}\frac{Q_{x}^{n}}{D_{0}^{m}}=\frac{i^{n}2^{-m}}{\pi\,\Gamma[m]}\,\partial_{u}^{n}\\!\left[u^{m-1}K_{m-1}(u)\right]\,,$
(109)

However, the cut terms (new with this work) involve the first power of $\log
D_{0}$ in the numerator. Such terms can be integrated by using the formula

$\displaystyle\int\frac{\text{d}Q_{x}\text{d}Q_{z}}{(2\pi)^{2}}e^{-iuQ_{x}}\frac{Q_{x}^{n}\,\log\\!D_{0}}{D_{0}^{m}}=$
(110)
$\displaystyle=-\partial_{\rho}\left\\{\frac{i^{n}2^{-m-\rho}}{\pi\,\Gamma[m+\rho]}\,\partial_{u}^{n}\\!\left[u^{m+\rho-1}K_{m+\rho-1}(u)\right]\right\\}\Bigg{|}_{\rho=0}\,.$

involving a derivative with respect to the order of the Bessel $K_{\nu}(u)$
(around an integer value of $\nu$).

The final result can be further reduced using relations among modified Bessel
functions of the second kind to linear combinations of $K_{0}(u)$, $K_{1}(u)$
and $e^{-u}$ with coefficients that are rational functions of $u$. The results
of this expansion up to 3PN order are presented in the ancillary file
PNexpandedEFT.nb.161616For the analysis of the PN expansion of the various
terms contributing to the amplitudes, we refer to Section V of Ref. [50].

###  VI.3 Soft-graviton expansion

We also computed the soft-graviton expansion of the waveform in impact
parameter space:171717This PM calculation was carried out independently in
Ref. [51] and both results were presented for the first time at [83, 88]. The
calculation was done to 2.5 PN order in Ref. [50].

$\displaystyle\mathcal{W}\simeq\frac{\mathcal{A}}{\omega}+\mathcal{B}\log\omega+\mathcal{C}\,\omega(\log\omega)^{2}+\mathcal{D}\,\omega\log\omega\
.$ (111)

Care must be take when carrying out the soft-graviton expansion at the level
of the waveform integrand. As one quickly discovers, a naive expansion leads
to uncontrolled infrared divergences at sufficiently high orders in the
subsequent Fourier transform in momentum transfer at sufficiently high orders.

Thus, the soft-graviton expansion is intertwined with the Fourier transform to
impact parameter space. The presence of the small parameter $\omega$ indicates
that the appropriate framework for the evaluation of the latter is the method
of regions. 181818Note that, while analogous in spirit, is a distinct from the
method of regions used to isolate the classical contribution of loop
integrals, which have already been been evaluated here. . There are two
distinct contributing regions, $q_{1}\rightarrow\omega^{0}q_{1}$ and
$q_{1}\rightarrow\omega q_{1}$, and each of them has a different
small-$\omega$ expansion which must be added to obtain the correct result. In
both regions the resulting integrals can be evaluated in closed form, as done
in Ref. [4].

The soft-graviton expansion corresponds to the large time expansion of the
time-domain waveform. The behavior of the scattering waveform in this regime
exhibits universality as emphasized in Refs. [89, 90]. In particular, when
expressed in terms of the incoming and outgoing momentum, which differ by the
impulse $\Delta p$, the functions $\mathcal{A}$, $\mathcal{B}$ and
$\mathcal{C}$ are independent of the specifics of the hard scattering process.
We extracted these coefficients from the EFT waveform and reproduced the
predictions of Ref. [89].

We have also computed the non-universal $\mathcal{D}$ coefficient and found
exact agreement with Eq. (9.11) of [50]. This is the first order in the soft-
graviton expansion where the second region mentioned above, in which the
Fourier integration variable scales as $q_{1}\rightarrow\omega q_{1}$, first
contributes. It is also the first order at which the disconnected cut (89)
contributes crucially to the match with [50]. The necessity of this term for a
successful comparison with the MPM waveform was first suggested in [51] as
part of a BMS transformation. In Secs. V.1 and V.3 we understood its
connection to the disconnected cuts of the observable-based formalism if zero-
energy gravitons are part of the spectrum. On the other hand, the
$\epsilon/\epsilon$ term in Sec. V.2 does not contribute to any of the
coefficients in Eq. 111 up to a time shift.

##  VII Transforming the KMOC waveform to the barred momentum frame

The MPM waveform $W_{\rm MPM}({\omega},\theta,\phi)$ is computed in a spatial
frame anchored on the classical barred momenta (3), while the EFT waveform
$\mathcal{W}({\omega},\theta,\phi)$ is originally obtained in a spatial frame
anchored on the incoming momenta. Both spatial frames are of the c.m. type,
with a negligible $O(G^{3})$ difference (due to the radiated momentum) between
their time-axes, namely

$\displaystyle\bar{e}_{0}^{\mu}=e_{0}^{\mu}+O(G^{3})\,.$ (112)

At order $G^{2}$ the only significant difference between the barred c.m. frame
spanned by $(\bar{e}_{0},e_{x},e_{y},e_{z})$ (defined in Sec. II) and the
frame $({e}_{0},e_{1},e_{2},e_{3})$ is a spatial rotation of $\chi/2$ mapping
the unit vectors $e_{1},e_{2}$ into $e_{x},e_{y}$. [Note that in both frames
we use a polarization vector respectively anchored on $e_{1},e_{2}$ or
$e_{x},e_{y}$, as per Eqs. (7) and (98)].

Consequently, the function of $\theta$ an $\phi$ describing the EFT waveform
referred to the barred momenta frame, say
$W_{\text{EFT}}(\omega,\theta,\phi)$, is related to the function of $\theta$
an $\phi$ describing the (original) EFT waveform
$\mathcal{W}(\omega,\theta,\phi)$ referred to the incoming momenta frame by

$\displaystyle W_{\text{EFT}}(\omega,\theta,\phi)$
$\displaystyle=\mathcal{W}(\omega,\theta,\phi)$ (113)
$\displaystyle\quad+\frac{\chi_{\text{1PM}}}{2}\frac{\partial}{\partial\phi}\mathcal{W}^{G^{1}}(\omega,\theta,\phi)+O(G^{3})\,.$

Here, as indicated, it is sufficient to insert the 1PM [$O(G^{1})$]
approximation to the scattering angle, namely

$\displaystyle\chi_{\text{1PM}}=\frac{2GE}{b}\frac{2y^{2}-1}{y^{2}-1}\,.$
(114)

Thus, using Eq. (113), the comparison between the MPM and EFT waveform is
reduced to comparing the function $W_{\rm MPM}({\omega},\theta,\phi)$ to the
function $W_{\text{EFT}}(\omega,\theta,\phi)$.

##  VIII Comparison between MPM and KMOC waveforms

When working in the barred momenta frame, the comparison between the MPM
waveform and the EFT one amounts to comparing the following two functions of
${\omega},\theta,\phi$ (and of the impact parameter $b$): $W^{\rm
MPM}({\omega},\theta,\phi)$ on the one hand, and
$W_{\text{EFT}}({\omega},\theta,\phi)$ on the other hand. A complete physical
agreement between these two waveforms would mean that these two functions
differ at most by the effect of an angular-independent time shift $\delta t$.
As discussed above, when adding to the EFT waveform the (linearized gravity)
contribution coming from the disconnected, zero-frequency diagrams displayed
on the left of Fig. 1, the $O(G^{0})$ static background (Coulomb-like)
contributions in the two waveforms agree with each other [compare Eq. 14 with
Sec. V.1]. In the following, we therefore focus on the frequency-dependent
parts (${\omega}\neq 0$) of the two waveforms, which both start at order
$G^{1}$.

Summarizing the results so far, the function
$W_{\text{EFT}}({\omega},\theta,\phi)$ can be written (at the one loop
accuracy) as the sum of six contributions, namely

$\displaystyle W_{\text{EFT}}(\omega,\theta,\phi)$
$\displaystyle=\mathcal{W}^{G^{1}}(\omega,\theta,\phi)+\mathcal{W}^{\text{1
loop (0)}}(\omega,\theta,\phi)$ $\displaystyle+\mathcal{W}^{\text{1 loop
}\,\epsilon/\epsilon}(\omega,\theta,\phi)+\mathcal{W}^{\text{1 loop
cut}}(\omega,\theta,\phi)$ $\displaystyle+\mathcal{W}^{\text{1 loop
disc}}(\omega,\theta,\phi)+\delta^{\text{rot}}\mathcal{W}(\omega,\theta,\phi)$
$\displaystyle+O(G^{3})\,.$ (115)

Here, each term is defined as follows.

The $O(G)$ waveform $\mathcal{W}^{G^{1}}$ comes from the tree-level matrix
element $\mathcal{M}^{\text{tree}}$ in Eq. 70,

$\displaystyle\mathcal{W}^{G^{1}}(\omega,\theta,\phi)=\frac{2i}{\kappa}\text{FT}_{b}\Big{[}\mathcal{M}^{\text{tree}}\Big{]}.$
(116)

The $O(G^{2})$ (one-loop) contributions to the EFT waveform indicated in Eq.
(VIII) denote, respectively:

(i) the contribution from the (linear in $T$) 5-point amplitude obtained by
dropping the IR $\frac{1}{\epsilon}$ divergent terms in the results of Refs.
[41, 39, 40], as displayed in Eq. (5.6) of [50], namely, in the notation
introduced in Eq. (VIII) above:

$\mathcal{W}^{\text{1
loop}(0)}=\frac{2i}{\kappa}\,\text{FT}_{b}\Big{[}\kappa^{5}m_{1}^{3}m_{2}^{2}\widetilde{\mathcal{M}}_{1}^{(1)}+(1\leftrightarrow
2)\Big{]}\,;$ (117)

(ii) the extra, $\frac{\epsilon}{\epsilon}$ contribution from the (linear in
$T$) 5-point amplitude generated after absorbing the IR $\frac{1}{\epsilon}$
divergence into a time shift as discussed in Secs. IV.3 and V.2, namely, in
the notation introduced in Eq. (VIII) above:

$\displaystyle\mathcal{W}^{\text{1 loop }\,\epsilon/\epsilon}$
$\displaystyle=$
$\displaystyle\frac{2i}{\kappa}\,\text{FT}_{b}\Big{[}\kappa^{5}m_{1}^{3}m_{2}^{2}\frac{iw_{1}}{32\pi}\mathcal{M}^{(0)}_{\text{extra}}+(1\leftrightarrow
2)\Big{]}$ (118) $\displaystyle=$
$\displaystyle\frac{-2\kappa^{4}}{32\pi}\,m_{1}^{2}m_{2}^{2}E{\omega}\text{FT}_{b}\Big{[}\mathcal{M}^{(0)}_{\text{extra}}\Big{]}\,;$

(iii) the oneloop cut contribution from connected $T$ elements discussed in
Sec. V.2, namely, in the notation introduced in Eq. (VIII) above:

$\mathcal{W}^{\text{1 loop
cut}}=\frac{2i}{\kappa}\,\text{FT}_{b}\Big{[}\mathcal{S}^{\text{1 loop
fin}}\Big{]}\,;$ (119)

(iv) the oneloop cut contribution involving disconnected $T$ elements
discussed in Sec. V.3, namely, in the notation introduced in Eq. (VIII) above:

$\mathcal{W}^{\text{1 loop
disc}}=\frac{2i}{\kappa}\,\text{FT}_{b}\Big{[}\mathcal{S}^{\text{1 loop
disc}}\Big{]}\,;$ (120)

and finally, (v) the contribution from the spatial rotation between the
barred-$p$ frame and the incoming $p$ one discussed in Sec. VII, namely, in
the notation introduced in Eq. (VIII) above:

$\delta^{\text{rot}}\mathcal{W}=+\frac{\chi_{\text{1PM}}}{2}\frac{\partial}{\partial\phi}\mathcal{W}^{G^{1}}(\omega,\theta,\phi)\,.$
(121)

Ref. [50] computed, at the $G^{2}\eta^{5}$ accuracy, the (even in $\phi$
sector of the) difference

$\delta^{\rm old}W(\omega,\theta,\phi)\equiv W^{\rm
EFT^{\prime}}(\omega,\theta,\phi)-W^{\rm MPM}(\omega,\theta,\phi),$ (122)

between the EFT waveform, as it was understood in the first versions of Refs.
[41, 39, 40], and interpreted in the barred $p$ frame, and the MPM one. In the
notation of the present work, the quantity $W^{\rm
EFT^{\prime}}(\omega,\theta,\phi)$ considered in [50] is equal to

$W^{\rm
EFT^{\prime}}(\omega,\theta,\phi)=\mathcal{W}^{G^{1}}(\omega,\theta,\phi)+\mathcal{W}^{\text{1
loop}(0)}(\omega,\theta,\phi)\,.$ (123)

The result of [50] for the difference $\delta^{\rm old}W(\omega,\theta,\phi)$,
Eq. (122), was that it started at the $G^{2}/c^{5}$ (2.5PN) level, and
contained (at this order) both terms linear in the symmetric mass ratio,
$\nu$, and terms quadratic in $\nu$. The explicit expression for the
difference

$\delta^{\rm
old}W(\omega,\theta,\phi)=\delta^{\nu}W(\omega,\theta,\phi)+\delta^{\nu^{2}}W(\omega,\theta,\phi)$
(124)

was provided in [50] in tensorial form, and in the time domain. See Eqs.
(8.10)-(8.13) there. The equivalent result in SWSH form, and in the frequency
domain, contains only $l=2^{+}$ contributions at order $\nu^{1}$, and both
$l=2^{+}$, $l=3^{-}$ and $l=4^{+}$ contributions at order $\nu^{2}$. Namely,

$\displaystyle\delta^{\nu}W(\omega,\theta,\phi)$ $\displaystyle=$
$\displaystyle\sum_{m=2,0,-2}\delta^{\nu}U_{2m}\;{}_{\bar{2}}Y_{lm}(\theta,\phi)\,,$
(125) $\displaystyle\delta^{\nu^{2}}W(\omega,\theta,\phi)$ $\displaystyle=$
$\displaystyle\sum_{m=2,0,-2}\delta^{\nu^{2}}U_{2m}\;{}_{\bar{2}}Y_{2m}(\theta,\phi)$
$\displaystyle+$
$\displaystyle\sum_{m=1,-1}\delta^{\nu^{2}}V_{3m}\;{}_{\bar{2}}Y_{3m}(\theta,\phi)$
$\displaystyle+$
$\displaystyle\sum_{m=4,2,0,-2,-4}\delta^{\nu^{2}}U_{4m}\;{}_{\bar{2}}Y_{4m}(\theta,\phi)\,,$

with

$\displaystyle\delta^{\nu}U_{22}$
$\displaystyle=-\frac{4i\sqrt{5\pi}}{5}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu
u[K_{0}(u)-2K_{1}(u)]\,,$ $\displaystyle\delta^{\nu}U_{20}$
$\displaystyle=-\frac{4i\sqrt{30\pi}}{15}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu
u[K_{0}(u)+2uK_{1}(u)]\,,$ $\displaystyle\delta^{\nu}U_{2\bar{2}}$
$\displaystyle=-\frac{4i\sqrt{5\pi}}{5}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu
u[K_{0}(u)+2K_{1}(u)]\,,$ (127) $\displaystyle\delta^{\nu^{2}}U_{22}$
$\displaystyle=\frac{8i\sqrt{5\pi}}{105}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[(13u+6)K_{0}(u)$
$\displaystyle\quad+13(u+1)K_{1}(u)]\,,$ $\displaystyle\delta^{\nu^{2}}U_{20}$
$\displaystyle=\frac{8i\sqrt{30\pi}}{105}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[2K_{0}(u)-uK_{1}(u)]\,,$
$\displaystyle\delta^{\nu^{2}}U_{2\bar{2}}$
$\displaystyle=\frac{8i\sqrt{5\pi}}{105}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[(-13u+6)K_{0}(u)$
$\displaystyle\quad+13(u-1)K_{1}(u)]\,,$ (128)
$\displaystyle\delta^{\nu^{2}}V_{32}$
$\displaystyle=-\frac{2i\sqrt{7\pi}}{21}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[uK_{0}(u)$
$\displaystyle\quad+(1+u)K_{1}(u)]\,,$ $\displaystyle\delta^{\nu^{2}}V_{30}$
$\displaystyle=-\frac{2i\sqrt{210\pi}}{105}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[uK_{0}(u)+K_{1}(u)]\,,$
$\displaystyle\delta^{\nu^{2}}V_{3\bar{2}}$
$\displaystyle=-\frac{2i\sqrt{7\pi}}{21}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[uK_{0}(u)$
$\displaystyle\quad+(1-u)K_{1}(u)]\,,$ (129)
$\displaystyle\delta^{\nu^{2}}U_{44}$
$\displaystyle=-\frac{i\sqrt{7\pi}}{21}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[(1+2u)K_{0}(u)$
$\displaystyle\quad+2(u+1)K_{1}(u)],$ $\displaystyle\delta^{\nu^{2}}U_{42}$
$\displaystyle=-\frac{2i\sqrt{\pi}}{21}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u(u+1)[K_{0}(u)+K_{1}(u)],$
$\displaystyle\delta^{\nu^{2}}U_{40}$
$\displaystyle=-\frac{i\sqrt{10\pi}}{105}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[3K_{0}(u)+2uK_{1}(u)],$
$\displaystyle\delta^{\nu^{2}}U_{4\bar{2}}$
$\displaystyle=-\frac{2i\sqrt{\pi}}{21}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u(1-u)[K_{0}(u)-K_{1}(u)],$
$\displaystyle\delta^{\nu^{2}}U_{4\bar{4}}$
$\displaystyle=-\frac{i\sqrt{7\pi}}{21}\frac{G^{2}M^{3}}{b}p_{\infty}^{2}\nu^{2}u[(1-2u)K_{0}(u)$
$\displaystyle\quad+2(u-1)K_{1}(u)]\,.$ (130)

Note that, $W^{\rm EFT^{\prime}}$ did not (on purpose191919Indeed, one of the
important findings of [50] was the need to rotate the EFT to the barred $p$
frame, otherwise the MPM/EFT mismatch would start at the Newtonian order,
instead of being relegated to the 2.5PN order.) contain the $\frac{\chi}{2}$
rotation contribution, $\delta^{\text{rot}}\mathcal{W}$, nor the three new
contributions discussed in the present work, namely $\mathcal{W}^{\text{1 loop
}\,\epsilon/\epsilon},\mathcal{W}^{\text{1 loop cut}}$ and
$\mathcal{W}^{\text{1 loop disc}}$. Therefore, updating the comparison done in
[50] leads to a new difference

$\delta^{\rm new}W(\omega,\theta,\phi)\equiv W^{\rm
EFT}(\omega,\theta,\phi)-W^{\rm MPM}(\omega,\theta,\phi),$ (131)

given by

$\displaystyle\delta^{\rm new}W$ $\displaystyle=$ $\displaystyle\delta^{\rm
old}W+\mathcal{W}^{\text{1 loop }\,\epsilon/\epsilon}+\mathcal{W}^{\text{1
loop cut}}$ (132) $\displaystyle+\;\mathcal{W}^{\text{1 loop
disc}}+\delta^{\text{rot}}\mathcal{W}\,.$

As we shall discuss next, the additional contributions (new with this work)
displayed in Eq. (132) are such that they precisely cancel the two types
($O(\nu^{1})$ and $O(\nu^{2})$) of discrepancies that were present in Eq.
(122), modulo an angular-independent time-shift between the two waveforms.

##  IX KMOC cut and frame rotation

It was first noticed in [50] by directly comparing the MPM waveform with the
amplitude contribution to the KMOC waveform that, up to 2PN order, there is a
remarkable agreement between the two when the latter is evaluated in the frame
rotated by half the scattering angle relative to the incoming frame (i.e. in
the barred frame). As mentioned earlier, this raised the possibility that the
cut contribution could be absorbed into this frame change, which is further
confirmed in the soft graviton expansion discussed in Sec. VI.3.

By Fourier-transforming the one-loop connected cut contribution to impact-
parameter space, Ref. [51] showed that $\mathcal{W}^{\text{1 loop cut}}$ can
be written as a certain differential operator acting on the tree-level
waveform up to a further finite redefinition of the the retarded time. In the
coordinates used here this differential operator acts as a rotation by
$\chi/2$. Using the explicit expression of the connected cut [39, 41, 91], we
explicitly verified this relation,

$\displaystyle\mathcal{W}^{\text{1 loop
cut}}=-\frac{\chi_{\text{1PM}}}{2}\frac{\partial}{\partial\phi}\mathcal{W}^{\text{tree}}+i\omega\delta
t^{\text{cut}}\mathcal{W}^{\text{tree}}\,,$ (133)

up to 3PN order. The $\epsilon/\epsilon$ term in the second Eq. (V.2) plays a
crucial role.202020As our results were written up, we became aware that this
point was also explicitly noted in the version 2 update of Ref. [51].
Resumming the 3PN-accurate expansion of the time shift (which starts at
Newtonian order) yields the following all-order-in-$p_{\infty}$ expression

$\displaystyle\delta t^{\text{cut}}=\frac{\kappa^{2}E}{32\pi}$
$\displaystyle\left[\Gamma\log\frac{\mu_{\text{IR}}be^{\gamma_{\text{E}}}}{2}-\frac{y}{2(y^{2}-1)^{3/2}}\right.$
$\displaystyle\;\left.+\;\frac{E_{1}E_{2}(2y^{2}-1)}{m_{1}m_{2}(y^{2}-1)^{3/2}}\right].$
(134)

To compare with Eq. (3.48) of the version 1 of Ref. [51], we include the IR-
divergent term in Eq. 77 to the left hand side (to obtain
$\text{FT}_{b}[\mathcal{S}^{\text{1 loop}}]$) and the time shift $\delta
t_{\text{IR}}^{\text{cut}}=-\frac{GE\Gamma}{2\epsilon}$ that cancels it to the
right hand side. This gives,

$\displaystyle\text{FT}_{b}\Big{[}\mathcal{S}^{\text{1 loop}}\Big{]}$
$\displaystyle=-\frac{\chi_{\text{1PM}}}{2}\frac{\partial}{\partial\phi}\,\text{FT}_{b}\Big{[}\mathcal{M}^{\text{tree}}\Big{]}$
$\displaystyle\quad+i\omega\delta
T^{\text{cut}}\,\text{FT}_{b}\Big{[}\mathcal{M}^{\text{tree}}\Big{]},$ (135)

where $\delta T^{\text{cut}}=\delta t^{\text{cut}}+\delta
t_{\text{IR}}^{\text{cut}}$. Through $O(\epsilon^{0})$ our time shift agrees
with the result of the version 1 of Ref. [51]

$\displaystyle\delta T^{\text{cut}}=\frac{\kappa^{2}E}{32\pi}$
$\displaystyle\left[\frac{y(y^{2}-\frac{3-4\epsilon}{2-2\epsilon})}{(y^{2}-1)^{3/2}}\frac{e^{\epsilon\gamma_{\text{E}}}\Gamma(-\epsilon)}{(\mu_{\text{IR}}^{2}b^{2}/4)^{-\epsilon}}\right.$
$\displaystyle\;\left.+\;\frac{E_{1}E_{2}(2y^{2}-1)}{m_{1}m_{2}(y^{2}-1)^{3/2}}\right]\,.$
(136)

Together with the observation that in our c.m. frame the operator
$\bar{\delta}$ in [51] becomes $-(\chi_{\text{1PM}}/2)\partial_{\phi}$, we
find exact agreement with the prediction of Ref. [51] (up to a redefintion of
the arbitrary scale $\mu_{\text{IR}}$), which we have thus verified through
direct evaluation up to ${O}(p_{\infty}^{3})$, or 3PN order.

In light of this agreement, let us comment on the technical differences
between the setup discussed here and that of Ref. [51]. Certain steps in our
evaluation and simplification of the cut term, such as removal of terms
proportional to the Gram determinant of five vectors, assumed that the
external kinematics, including the momentum transfer, were four-dimensional.
Consequently, all our Fourier transforms were strictly in four dimensions, as
enforced by the decomposition of the momentum transfer in Eq. (101). In
contrast, the setup in Ref. [51] assumes that the Fourier transform is
$d$-dimensional. Interestingly, this does not appear to have any consequences
in the comparison, such as a difference in the $\epsilon/\epsilon$
contributions that build on the IR-divergent terms.

The analysis outlined in this section indicates that when comparing the EFT
and MPM result in the barred c.m. frame, we can effectively ignore the
connected cut contribution $\mathcal{S}^{\text{1 loop}}$ altogether, as it is
cancelled by the frame rotation up to a time shift. In other words, we can
rewrite Eq. (132) in the simplified form

$\displaystyle\delta^{\rm new}W$ $\displaystyle=$ $\displaystyle\delta^{\rm
old}W+\mathcal{W}^{\text{1 loop }\,\epsilon/\epsilon}+$ (137) $\displaystyle+$
$\displaystyle\mathcal{W}^{\text{1 loop disc}}+{\rm time\,shift}\ .$

We now proceed to discuss the remaining two terms, $\mathcal{W}^{\text{1 loop
}\,\epsilon/\epsilon}$ and $\mathcal{W}^{\text{1 loop disc}}$.

##  X $\mathcal{W}^{\text{1 loop }\,\epsilon/\epsilon}$ and $\delta^{\nu}W$

The additional $\epsilon/\epsilon$ contribution is given by Eq. (118).
Remembering the presence of a factor $1/(m_{1}m_{2})$ in the Jacobian of
$\text{FT}_{b}$, $\mathcal{W}^{\text{1 loop }\,\epsilon/\epsilon}$ is seen to
be proportional to $m_{1}m_{2}$, i.e. to be proportional to $\nu^{1}$. In
addition, as it contains a $O(\eta^{3})$ factor $GE{\omega}=\frac{G{\sf
M}{\omega}}{c^{3}}$, and as $\mathcal{M}^{(0)}_{\text{extra}}$ is easily seen
to start to differ from its full-gravity tree-level counterpart
$\mathcal{M}^{(0)}_{d=4}$ only at the 1PN, $O(\eta^{2})$, level, we conclude
that $\mathcal{W}^{\text{1 loop }\,\epsilon/\epsilon}$ effectively (i.e.
modulo a time shift) starts contributing to $W$ at the $G^{2}\eta^{5}$ level.

An easy computation of the impact-parameter transform of the PN expansion of
$\mathcal{M}^{(0)}_{\text{extra}}$ actually shows that, modulo a time-shift
contribution, $i{\omega}\delta tW^{G^{1}}$, the even-in-$\phi$ projection of
$\mathcal{W}^{\text{1 loop }\,\epsilon/\epsilon}$ precisely cancels the
$O(\nu)$, $G^{2}\eta^{5}$-level, contribution
$\delta^{\nu}W(\omega,\theta,\phi)$ found (in [50]) to be present in $\delta
W(\omega,\theta,\phi)$, and displayed in Eq. (VIII). Namely,

$\delta^{\nu}W+\left[\mathcal{W}^{\text{1 loop
}\,\epsilon/\epsilon}+i\,{\omega}\,\delta
t^{\epsilon/\epsilon}\,W^{G^{1}}\right]^{\phi-{\rm
even}}=O\left(\frac{G^{2}}{c^{6}}\right)\,,$ (138)

with $\delta
t^{\epsilon/\epsilon}=\frac{G\,E}{c^{5}}=\frac{GM}{c^{3}}(1+\frac{\nu}{2}p_{\infty}^{2}+O(p_{\infty}^{4}))$.

##  XI  $\mathcal{W}^{\text{1 loop disc}}$ and $\delta^{\nu^{2}}W$

The final terms to discuss (at the highest PN accuracy explicitly studied in
[50]) are the (even-in-$\phi$ projection of the) disconnected cut contribution
to $\mathcal{W}$ and the $O(\nu^{2})$, $G^{2}\eta^{5}$-level, contribution
$\delta^{\nu^{2}}W(\omega,\theta,\phi)$ displayed in Eqs. (VIII), (VIII) and
(VIII).

It was suggested in Ref. [51] that the difference
$\delta^{\nu^{2}}W(\omega,\theta,\phi)$ is genuinely present in the EFT-MPM
difference and corresponds to the fact that the EFT waveform does not contain
some of the zero-frequency-graviton effects contained in the MPM waveform,
because it works in the canonical BMS frame of [52] (in which
$W(t=-\infty)=0$) rather than in the intrinsic BMS frame selected both by
standard classical PM theory and the MPM formalism. However, we found in Sec.
V that disconnected contributions to the cut term account for this. They lead
to the additional term $\mathcal{W}^{\text{1 loop disc}}$ in the EFT waveform
whose explicit form is

$\mathcal{W}^{\text{1 loop
disc}}=+iG\left[m_{1}w_{1}\log\frac{w_{1}^{2}}{\omega^{2}}+m_{2}w_{2}\log\frac{w_{2}^{2}}{\omega^{2}}\right]\mathcal{W}^{G^{1}}.$
(139)

This term is $O(\nu^{2})$, and its PN expansion starts at the $G^{2}/c^{5}$
level. An easy computation shows that its $\phi$-even projection exactly
cancels, at this order, all the $l=2^{+},l=3^{-}$ and $l=4^{+}$ pieces of
$\delta^{\nu^{2}}W(\omega,\theta,\phi)$ displayed in Eqs. (VIII), (VIII) and
(VIII):

$\delta^{\nu^{2}}W+\left[\mathcal{W}^{\text{1 loop disc}}\right]^{\phi-{\rm
even}}=O\left(\frac{G^{2}}{c^{6}}\right)\,,$ (140)

without the need of an additional time-shift.

In other words, after factoring out the effect of the $\chi/2$ rotation, the
total time-shift needed to align the MPM waveform (defined as in [50]) and the
EFT one (completed by the $\epsilon/\epsilon$ and the disconnected
contributions) is $\delta t^{\rm tot}=G\mathsf{M}/c^{3}$. This time-shift can
be absorbed in a modification of the link between the time scale $b_{0}$
entering the tail logarithms of the MPM formalism and the frequency scale
$\mu_{\text{IR}}$ of the amplitudes-based formalism. Namely, Eq. (8.8) of [50]
is modified into

$\log(2b_{0}\mu_{\text{IR}})+\gamma_{\text{E}}=0\,.$ (141)

Note the curious feature that the additional term, Eq. 139, which reads,

$\mathcal{W}^{\text{1 loop disc}}=i{\omega}\delta t^{\rm
VV}(\theta,\phi)W^{G^{1}},$ (142)

with the angle-dependent time shift

$\delta t^{\rm
VV}(\theta,\phi)=2G\left(m_{1}\frac{w_{1}}{{\omega}}\log\frac{w_{1}}{{\omega}}+m_{1}\frac{w_{2}}{{\omega}}\log\frac{w_{2}}{{\omega}}\right)\,,$
(143)

doubles, rather than cancels, the contribution to the EFT waveform coming from
the last two lines in Eq. (5.22) of Ref. [50]. See further discussion of this
term below.

The results (IX), (138) and (140) allow us to conclude that the inclusion of
the new contributions discussed in this work to the EFT waveform resolves all
the mismatches with the even-in-$\phi$ projection of the
$G^{2}\eta^{5}$-accurate MPM waveform computed in [50].

To complete the EFT-MPM comparison on the odd-in-$\phi$ sector, we have also
compared the odd-in-$\phi$ projection of $W^{\rm MPM}$, at the
$G^{2}(\eta^{1}+\eta^{4})$ accuracy (see Eqs. (III) and (III)) with its EFT
counterpart (given, in the equatorial plane, in the ancillary file
PNexpandedEFT.nb). We found perfect agreement between the two (with the same
time-shift needed for the $\phi$-even part, i.e. the same link Eq. (141)).
Though our checks in the odd-in-$\phi$ sector probe nonlinear contributions to
the waveform less deeply than our even-in-$\phi$ checks, they still include
the leading-order tail contributions, and are sensitive to the precise
relation Eq. (141). The explicit comparison between the MPM and amplitudes-
based waveforms is presented in the ancillary file MPMEFTcomparison.nb.

##  XII Conclusions

We revisited the amplitude- and observable-based (KMOC [3]) derivation of the
classical gravitational wave signal emitted during the scattering of two non-
spinning masses at the NNLO order $h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}\sim
G^{1}+G^{2}+G^{3}$, and compared our results for the frequency-domain,
$G$-rescaled waveform quantity
$W({\omega},\theta,\phi)\equiv\frac{c^{4}}{4G}\varepsilon^{\mu}\varepsilon^{\nu}h_{\mu\nu}\sim
G^{0}+G^{1}+G^{2}$, to the corresponding waveform derived within the (post-
Newtonian-matched) Multipolar-post-Minkowskian (MPM) formalism. The recent
computation of $W^{\rm MPM}({\omega},\theta,\phi)$ [50] had pointed out the
presence of large discrepancies (starting at the Newtonian level) between
$W^{\rm MPM}({\omega},\theta,\phi)$ and the (finite part of the) amplitude-
based waveform derived in the (first versions of) Refs. [41, 39, 40], say
$W^{\rm EFT^{\prime}}({\omega},\theta,\phi)$. Ref. [50] had further pointed
out that the EFT/MPM mismatch was drastically reduced from the Newtonian level
$G^{2}\eta^{0}$ (where $\eta\equiv\frac{1}{c}$) to the 2.5 PN level
$G^{2}\eta^{5}$ by rotating (by half the classical scattering angle $\chi/2$)
the EFT waveform from the incoming momenta frame to the (classical) averaged
momenta frame. Part of the puzzle posed by the EFT/MPM mismatch was recently
clarified by the eikonal-inspired approach of Georgoudis, Heissenberg and
Russo [51] who argued that the cut contribution needed [42] in the KMOC
framework (and missing in the first derivations of the EFT waveform) was
precisely implementing the needed rotation between the incoming-momenta frame
(naturally tuned to $W^{\rm EFT}({\omega},\theta,\phi)$) and the (classical)
averaged-momenta frame (conveniently used to compute $W^{\rm
MPM}({\omega},\theta,\phi)$) for all final-states that do not contain zero-
energy gravitons. Ref. [51] compared only the soft expansion of $W^{\rm
EFT}({\omega},\theta,\phi)$ to the soft expansion of $W^{\rm
MPM}({\omega},\theta,\phi)$ and could indeed verify that the $\chi/2$
rotational effect of the cut term reduced the EFT/MPM mismatch between these
expansions to the $G^{2}\eta^{5}$ level. However, the soft limit is a rather
blunt tool which is not sensitive to the details of the EFT/MPM mismatch (as
discussed, e.g., in Section IX of [50]).

Indeed, after factoring out the effect of the $\chi/2$ rotation, the EFT-MPM
mismatch found in [50] still contained significant terms of order
$\nu^{1}G^{2}\eta^{5}$ and of order $\nu^{2}G^{2}\eta^{5}$. The terms
$O(\nu^{1}G^{2}\eta^{5})$ effectively disappear in the soft limit (because
they are equivalent to an unobservable time shift). We showed in the present
work that a subtle additional $\epsilon/\epsilon$ contribution to the
amplitude result (coming, in our scheme, from the tracelessness of the
physical graviton in $d$ dimensions) precisely cancelled the
$O(\nu^{1}G^{2}\eta^{5})$ EFT-MPM mismatch (which could not be seen in the
soft limit). Concerning the last remaining $O(\nu^{2}G^{2}\eta^{5})$ EFT-MPM
mismatch, it was argued by Georgoudis et al. [51] that it remained and that,
in view of Eqs. (139), (143), it was to be physically interpreted as the
presence of the particular $O(G)$ supertranslation (i.e. an angle-dependent
time shift) $\delta t^{\rm VV}(\theta,\phi)$, Eq. (143), pointed out by
Veneziano and Vilkovisky [52], for its role in transforming the “intrinsic”
BMS frame (naturally used both in classical, worldline PM perturbation theory,
and in the MPM formalism) into the “canonical” BMS frame (defined so that the
initial value of the waveform vanishes in the infinite past). The
interpretation of the necessary presence of the supertranslation (143) was
that this supertranslation removes the effect of static modes (zero-frequency
gravitons) in the waveform, and that the amplitude approach naturally computes
a waveform without static modes. Our new results suggest, however, a different
picture. Static modes can be incorporated in the KMOC waveform if one includes
disconnected $T$-matrix elements, both at the linear-in-$T$ level and at the
bilinear-in-$T$ level (cut term). We have explicitly shown (see Eq. (V.1))
that the inclusion of disconnected diagrams (pictured in Fig. 1a) to the
linear-in-$T$ waveform was precisely generating the initial, zero-frequency
contribution which is naturally present in the MPM waveform (see Eq. (13)).
The situation concerning the effect of disconnected diagrams in the bilinear-
in-$T$ term is more subtle. In a natural regularization which takes the matter
velocity slightly off-shell, $u_{3}\rightarrow\tilde{u}_{3}$, we showed both
by direct evaluation and by interpreting it as the discontinuity of a triangle
integral that the cut pictured in Fig. 1b yields an additional contribution to
$W_{\text{EFT}}$ that precisely cancels the effect of the supertranslation
(143), thereby removing the last remaining $O(\nu^{2}G^{2}\eta^{5})$ EFT-MPM
mismatch. We have also shown in App. D that higher-loop contributions would
not modify our finding. It would clearly be interesting to confirm in a
different way our conclusion that the inclusion of disconnected diagrams in
the KMOC formalism does incorporate all the physical effects linked to the
static modes (which are included in the MPM formalism). It would also be
interesting to know whether a higher-order KMOC calculation, systematically