11institutetext: Max Planck Institute for Gravitational Physics (Albert
Einstein Institute), Am Mühlenberg 1, Potsdam, 14476, Germany22institutetext:
Department of Physics, University of Maryland, College Park, MD 20742,
USA33institutetext: Higgs Centre for Theoretical Physics, University of
Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh,
EH9 3FD, UK44institutetext: Department of Physics, Pennsylvania State
University, University Park, PA 16802, USA55institutetext: Mani L. Bhaumik
Institute for Theoretical Physics, University of California at Los Angeles,
Los Angeles, CA 90095, USA

# Snowmass White Paper: Gravitational Waves and Scattering Amplitudes

Alessandra Buonanno 1,2 Mohammed Khalil 3 Donal O’Connell 4 Radu Roiban 5
Mikhail P. Solon 3 Mao Zeng

###### Abstract

We review recent progress and future prospects for harnessing powerful tools
from theoretical high-energy physics, such as scattering amplitudes and
effective field theory, to develop a precise and systematically improvable
framework for calculating gravitational-wave signals from binary systems
composed of black holes and/or neutron stars. This effort aims to provide
state-of-the-art predictions that will enable high-precision measurements at
future gravitational-wave detectors. In turn, applying the tools of quantum
field theory in this new arena will uncover theoretical structures that can
transform our understanding of basic phenomena and lead to new tools that will
further the cycle of innovation. While still in a nascent stage, this research
direction has already derived new analytic results in general relativity, and
promises to advance the development of highly accurate waveform models for
ever more sensitive detectors.

##  1 Executive Summary

The ambitious future of gravitational-wave (GW) science calls for invigorating
the theoretical framework for precision calculations of GW signals, thus
galvanizing a new approach that harnesses the cutting-edge tools of
theoretical high-energy physics such as on-shell methods, double copy,
advanced multiloop integration, and effective field theory (EFT). These are
the engines that drive modern calculations of scattering amplitudes in
particle theory, and integrating them together for application to GWs has
recently led to new results in the perturbative, analytic solution of the two-
body problem in general relativity. Theorists developing waveform models for
the LIGO-Virgo-KAGRA (LVK) collaboration LIGOScientific:2014pky ;
VIRGO:2014yos ; KAGRA:2020agh  have performed initial studies
Antonelli:2019ytb ; Khalil:2022  of these early results, see Figure 1, and
have strongly encouraged further developments. Indeed, if these calculations
are pushed to higher orders, and are extended to include all physical effects
(i.e., spins and tides), they can be used, in combination with other analytic
methods Regge:1957td ; Zerilli:1970se ; Vishveshwara:1970cc ; Teukolsky:1973ha
; Buonanno:1998gg ; Buonanno:2000ef 111See also Blanchet:2013haa ;
Porto:2016pyg ; Schafer:2018kuf ; Barack:2018yvs  for recent reviews. and with
numerical-relativity (NR) simulations Pretorius:2005gq ; Campanelli:2005dd ;
Baker:2005vv ; Foucart:2022iwu , to provide highly accurate waveform models of
binary systems composed of black holes and/or neutron stars. Another aspect of
this program that has drawn significant interest from both the high-energy
physics and general relativity communities is the exploration of theoretical
structures that emerge in the classical limit of scattering amplitudes.

This new research direction in theoretical high-energy physics is an
opportunity to deploy the classic and modern tools of quantum field theory
(QFT) in a new arena, thereby impacting an important experimental frontier and
uncovering rich theoretical structure that can lead to new tools. The program
is in a nascent stage, and significant progress will come in the next several
years, building towards the vision that QFT tools will advance the computation
of gravitational waveforms. In particular, they will address the need for high
precision in upcoming LVK runs, in space-based detectors such as LISA
LISA:2017pwj , and in future ground-based detectors such as LIGO-India
Saleem:2021iwi , Cosmic Explorer Reitze:2019iox  and Einstein Telescope
Punturo:2010zz . High-precision waveform models will be crucial for maximizing
the discovery potential and extracting the best science with GW observations
of ever more sensitive future detectors LISA:2017pwj ; Sathyaprakash:2019yqt ;
Maggiore:2019uih ; Kalogera:2021bya ; Berti:2022wzk .

Figure 1:  (left) The binding energy (in units of the reduced mass) versus the
orbital frequency of an equal-mass non-spinning binary black hole following an
adiabatic quasi-circular orbit towards merger. The horizontal axis is also
given as the number of orbits before merger. (right) The scattering angle
versus impact parameter of two equal-mass non-spinning black holes following
hyperbolic trajectories with initial relative velocity $v=0.4$. These plots
are adapted from Khalil:2022 , where the authors compared predictions for
post-Minkowskian (PM) conservative dynamics obtained using QFT tools (solid
lines of increasing accuracy in Newton’s constant $G$, i.e., in the PM
approximation) with NR Damour:2014afa ; Ossokine:2017dge  and effective-one-
body (EOB) results, here at third post-Newtonian (PN) order, which are the
benchmarks for building waveform models in the LVK collaboration.

##  2 Introduction

The emergence of GW science Abbott:2016blz ; TheLIGOScientific:2017qsa ;
LIGOScientific:2021qlt  has already transformed multiple domains of astronomy,
cosmology, and particle physics, yet this represents only a small fraction of
its future potential LISA:2017pwj ; Kalogera:2021bya . Space- and ground-based
observatories of the coming decade will map out and characterize millions of
merger events per year with sensitivity well beyond that of current LIGO/Virgo
facilities (see, e.g., Favata:2013rwa ; Samajdar:2018dcx ; Purrer:2019jcp ;
Huang:2020pba ; Gamba:2020wgg ). One of the key challenges will be advancing
the theoretical modeling of compact binary coalescences to produce accurate
gravitational waveforms.

Theoretical modeling of GW sources is challenging due to multiple physical
scales that are nonlinearly coupled through general relativity. Notably, these
complications — multiple scales and nonlinearity — are exactly what drove
breakthroughs in QFT in the last few decades, leading to the modern scattering
amplitudes program. Scattering amplitudes have revealed mathematical
structures in gauge theory and gravity, leading to new physical insights,
efficient methods for computation, and the seeds for even bolder ideas.

The Parke-Taylor formula Parke:1986gb  reduced pages of Feynman calculus to a
half-line expression describing gluon scattering, famously heralding the
enormous value of understanding theoretical structures lying at the heart of
scattering amplitudes. In recent decades, major advances have been driven by
two parallel developments. First are new methods that formulate QFT without
explicit quantum fields, thus focusing on physical quantities. These “on-shell
methods”, reinvigorated by twistor string ideas Witten:2003nn ; Roiban:2004yf
; Gukov:2004ei ; Cachazo:2004kj , have become efficient mainstream tools for
tree-level Britto:2005fq  and loop-level Bern:1994zx ; Fusing ;
TripleCuteeJets ; BCFUnitarity  calculations in gauge and gravity theories;
see BDKUniarityReview ; ElvangHuangReview ; Elvang:2015rqa ; JJHenrikReview ;
BernHuangReview  for reviews. Second, is a radically new perspective on
gravity: gravitational scattering amplitudes ${\cal M}_{\rm gravity}$ can be
realized as a “double copy” of gauge theory amplitudes ${\cal M}_{\rm gauge}$
BCJ ; BCJLoop ,

${\cal M}_{\rm gauge}\times{\cal M}_{\rm gauge}\sim{\cal M}_{\rm gravity}\,.$
(1)

This structure builds on the relation between tree-level open and closed
string scattering KLT , its field theory limit Bern:1993wt , and structures
gleaned in explicit higher-loop calculations CompactThree ; Bern:2009kd . It
extends vanilla examples of related gauge and gravity theories to a veritable
web of theories that share common building blocks, and has been applied to the
exploration of a number of new directions, such as the ultraviolet properties
of supergravity theories up to five loops GravityThree ; Bern:2012cd ;
N4GravFourLoop ; N5GravFourLoop ; UVFiveLoops , and the nonperturbative
structure of classical solutions of Einstein’s equations with sources such as
Schwarzschild Monteiro2014cda ; see BCCJRReview  for a recent review and also
the dedicated Snowmass White Paper doublecopyWhitePaper2022 .

In the past few years there has been a flurry of activity in applying both on-
shell methods and the double copy, in combination with advanced multiloop
integration techniques and EFT, to develop new tools for state-of-the-art
predictions of GW signals. This development was encouraged by the general
relativity community Damour:2017zjx , and has led to a number of new results
in general relativity; for example see Figures 1 and 2.

The new approach based on tools from theoretical high-energy physics aims to
complement, and has greatly benefited from, decades of successful work using
traditional approaches to solve the relativistic two-body problem, including
the post-Newtonian (PN) approximation Einstein:1938yz ; Einstein:1940mt ;
Ohta:1973je ; Jaranowski:1997ky ; Damour:1999cr ; Blanchet:2000nv ;
Damour:2001bu ; Damour:2014jta ; Jaranowski:2015lha , the gravitational self-
force formalism Mino:1996nk ; Quinn:1996am , the effective–one-body (EOB)
formalism Buonanno:1998gg ; Buonanno:2000ef , the nonrelativistic general-
relativity (NRGR) formalism Goldberger:2004jt , the post-Minkowskian (PM)
approximation Bertotti:1956pxu ; Kerr:1959zlt ; Bertotti:1960wuq ;
Portilla:1979xx ; Westpfahl:1979gu ; Portilla:1980uz ; Bel:1981be ;
Westpfahl:1985tsl ; Damour:2016gwp ; Damour:2017zjx , and NR Pretorius:2005gq
; Campanelli:2005dd ; Baker:2005vv  (see, e.g., Blanchet:2013haa ;
Porto:2016pyg ; Schafer:2018kuf ; Barack:2018yvs ; Levi:2018nxp  for recent
reviews). In particular, the seminal work Goldberger:2004jt  introduced
nonrelativistic EFT ideas from particle physics to the worldline approach to
binary dynamics, and has led to a number of landmark results; for recent
developments see the dedicated Snowmass White Paper NRGRWhitePaper2022  on
this subject.

Figure 2: Map of perturbative corrections to Newton’s potential, where $G$ is
Newton’s constant and $v$ is the relative velocity of the binary constituents.
New results through $O(G^{4})$ were recently obtained using QFT tools (red
box). They are valid to all orders in velocity, and overlap with the state-of-
the-art from the PN expansion (dark triangle) and the contributions required
by future detectors (light triangle) (see, e.g., Favata:2013rwa ;
Samajdar:2018dcx ; Purrer:2019jcp ; Huang:2020pba ; Gamba:2020wgg ).

The tools of theoretical particle physics have been honed in a wide variety of
intricate quantum calculations such as the Higgs gluon-fusion cross section at
N3LO Anastasiou:2015vya , NNLO corrections to $e^{+}e^{-}$ event shapes
Gehrmann-DeRidder:2007vsv , cusp anomalous dimension at four loops
Henn:2019swt , electron $g-2$ at four loops Laporta:2017okg , ${\cal N}=8$
supergravity four-point amplitude at five loops Bern:2018jmv , and ${\cal
N}=4$ super-Yang-Mills four-point amplitude at six loops Carrasco:2021otn .
Leveraging these tools for GW physics yields several advantages. First, the
structure of perturbation theory is vastly simplified by special relativity,
on-shell methods, and the double copy, leading to compact expressions that
make theoretical structures manifest. Second, the technology and deep
knowledge base for integration of loops in QFT, which have seen decades of
development for collider physics applications, are directly transferred,
including integration-by-parts systems Chetyrkin:1981qh ; Laporta:2000dsw ;
Smirnov:2008iw  and differential equations Kotikov:1990kg ; Bern:1993kr ;
Remiddi:1997ny ; Gehrmann:1999as ; Henn:2013pwa ; Henn:2013nsa ; Parra-
Martinez:2020dzs . Finally, EFT efficiently and systematically targets
contributions to various processes in the classical limit. Figure 3
illustrates these tools in action.

Figure 3: (top) Classical binary dynamics is encoded in scattering amplitudes
of massive particles (thick lines) interacting through gravitons (wavy lines):
(A) four-point scattering encodes higher-order corrections to conservative
binary dynamics, (B) five-point scattering encodes radiative effects due to
graviton emission, and (C) higher-dimension operators (solid circle) encode
tidal deformation of neutron stars. Spinning black holes can be described by
higher-spin representations in QFT. (bottom) A sample calculational pipeline
using the tools of theoretical high-energy physics: Starting from tree-level
gauge theory amplitudes (a), corresponding gravitational amplitudes (b) are
obtained using the double-copy. These are then fused into loop amplitudes (c)
using generalized unitarity. The integrated amplitude (d) is obtained using
advanced multiloop integration methods developed in high-energy physics, in
combination with EFT. The amplitude can then be mapped, using a variety of
methods, to the EOB Hamiltonian used for producing waveforms (e). The
classical limit is applied at every stage, leading to vast simplifications.

Aside from providing state-of-the-art predictions, the new approach using
particle physics tools aims to explore theoretical structures that emerge in
the classical limit of scattering amplitudes. These structures may be familiar
from particle physics, but are not manifest in traditional approaches to the
two-body problem in general relativity, or they may be rooted in the classical
regime and are yet to be explored through the lens of QFT. Examples include
universality in the high-energy limit, the interplay between conservative and
dissipative effects, nonperturbative connections to classical solutions, and
perturbation theory in curved backgrounds.

The upshot is that scattering amplitudes are highly effective tools for
understanding and precise modeling of GW sources, in ways similar to their
application for interactions of fundamental particles. The recent progress and
future goals of this program revolve around three objectives, broadly defined:

  * •

Provide state-of-the-art predictions for the dynamics and gravitational
radiation of compact binaries composed of black holes and neutron stars, which
can be used, in combination with other analytic methods and NR simulations, to
produce highly accurate waveforms. This includes modeling spin, tidal, and
radiative effects.

  * •

Develop the mathematical and physical tools of theoretical high-energy physics
for application to GWs. This involves tailoring existing tools to improve
scalability, and formulating new tools and approaches to the binary problem.

  * •

Explore the theoretical landscape of scattering amplitudes in the classical
regime. Examples include the universality of high-energy scattering,
nonperturbative relations between scattering amplitudes and classical
dynamics, and connections to exact spacetime geometries.

In the rest of this white paper, we will discuss the classical limit, and then
describe each of the three topics above, summarizing recent progress. Then we
will describe future directions.

##  3 Classical Limit

The correspondence principle states that classical physics emerges from the
quantum theory in the limit of macroscopic conserved charges such as masses,
electric charges, spins, orbital angular momenta, etc. Perturbations around
such a configuration, which are subleading in the large charges, can be
systematically included and have the natural interpretation of quantum
corrections.222This perspective was used to great effect in string theory
tests of the asymptotic Bethe ansatz for the anomalous dimensions of single-
trace operators in ${\cal N}=4$ super-Yang-Mills theory, see e.g.
Gubser:2002tv ; Kruczenski:2004kw ; Giombi:2009gd  and Beisert:2010jr  for a
review.

For the application to GWs, the classical limit of scattering amplitudes is
defined by two properties that distinguish compact binaries from their quantum
counterparts:

  * •

Bound compact objects have large angular momentum $J\gg\hbar$, as opposed to
$J\sim\hbar\equiv 1$ for quantum bound states.

  * •

Compact objects, such as black holes or neutron stars, have large
gravitational charges $M_{\odot}/M_{\rm Planck}\sim 10^{38}$, as opposed to
$e/Q_{\rm Planck}\sim 10^{-1}$ for electric charges of elementary particles.
Here $M_{\odot}$ and $e$ denote the solar mass and electron charge, while
$M_{\rm Planck}$ and $Q_{\rm Planck}$ are the Planck mass and charge,
respectively.

In other words, for a system of two gravitationally interacting spinless
bodies with masses $m_{1}$ and $m_{2}$, the classical regime emerges in the
limit $J\gg\hbar$ and $m_{1},m_{2}\gg M_{\rm Planck}$. This limit also
coincides with the limit in which the de Broglie wave length $\lambda$ of the
particles is much smaller than the particles’ separation $|\bm{b}|$, which is
conjugate to the momentum transfer $\bm{q}$. Thus, the kinematic regime of
classical physics is when the momentum transfer is much smaller than the
incoming momenta, $|\bm{p}|\gg|\bm{q}|$, similar to the Regge limit. Other
charges that may characterize classical particles also have a similar scaling
in the classical limit; for example, the spin $S$ and finite size $R$ scale as
$|\bm{q}|R\sim{\cal O}(1)$, $|\bm{q}||\bm{S}|\sim m_{1,2}$. These scalings are
consistent with the classical nature of Newton’s potential, $|\bm{b}|\gg Gm$,
and fixes the general form of four-point amplitudes as well as of generating
functions of observables.

Another perspective on the classical limit was taken in Kosower:2018adc ,
which together with the associated framework is usually referred to as the
KMOC formalism. Quantum mechanical computation of observables yields the
classical result in the correspondence limit which is arranged in two steps:
(1) a suitable restoration of Planck’s constant $\hbar$, which effectively
plays a role similar to the momentum transfer $\bm{q}$, and expansion at small
$\hbar$, and (2) incorporation from the outset that from a quantum-mechanical
perspective, particles are described by wavepackets having finite width in
both position- and momentum-space rather than plane waves. This width must be
negligible in the classical limit, leading to inequalities which constrain the
parameters of the scattering, analogous to those coming from the large-$J$
limit.

One implication of the classical limit is that loop amplitudes involving
massive particles may contain classical contributions. The classical limit can
be taken at the earliest stages of calculations, yielding vast simplifications
prior to integration. This enables the calculation of loop amplitudes that
would otherwise be beyond the reach of current technology if evaluated
including quantum effects. Properties of amplitudes in an expansion around the
classical limit were previously unexplored systematically. The application to
GW physics provides the motivation to fill this gap, and have led to
interesting results Cristofoli:2021jas ; Britto:2021pud .

##  4 State-of-the-Art Predictions

Waveform models for the inspiral stage of a binary system are built from the
conservative and dissipative two-body dynamics. Therefore a direct path for
amplitudes methods to have an impact is to compute state-of-the-art
conservative two-body potentials, including the effects of spin, and tidal
deformation for modeling neutron stars Blanchet:2013haa ; Punturo:2010zz ;
LISA:2017pwj ; Reitze:2019iox . For instance, the general relativity community
had urged amplitudes experts to compute the ${\cal O}(G^{3})$ contributions to
conservative binary dynamics for spinless compact objects Damour:2017zjx .

Potentials are naturally encoded in four-point scattering amplitudes, and can
be extracted from the latter via a variety of methods such as an amplitude
matching calculation CRS , direct mapping to an EOB parameterization
Damour:2019lcq ; Damgaard:2021rnk , using the eikonal phase as a generating
functional Amati:1987wq ; tHooft:1987vrq ; Muzinich:1987in ; Amati:1987uf ,
and analytic continuation Cho:2018upo ; Kalin:2019rwq ; Kalin:2019inp ;
Cho:2021arx . However, at ${\cal O}(G^{4})$, the conservative two-body
Hamiltonian is no longer universal to both bound and unbound trajectories. The
effect of back-scattered radiation on the conservative dynamics — in
particular the so-called tail effects Thorne:1980ru ; Blanchet:1987wq ;
Blanchet:1993ec  — leads to nonlocal-in-time contributions to the Hamiltonian
that depend on the specific trajectory. Understanding the mapping between
bound and unbound orbits in the presence of radiation is an important open
problem that needs to be solved to maximally leverage scattering amplitudes
for deriving classical bound state dynamics.

Higher-Order Corrections to Conservative Binary Dynamics. Conservative binary
dynamics at ${\cal O}(G)$ and ${\cal O}(G^{2})$ and to all orders in
velocity333The ${\cal O}(G^{n})$ result to all orders in velocity is also
referred to as the $n$-th order post-Minkowskian result or $n$PM result. In
this White Paper we will often simply refer to this as the ${\cal O}(G^{n})$
result, following the conventional nomenclature of perturbative orders in
particle physics. were derived from scattering amplitudes in Ref. CRS . While
these results were known in general relativity using classical methods, the
method employed in Ref. CRS  laid a basic path for extending to higher orders
using scattering amplitudes.

Figure 4: The binding energy (in units of the reduced mass) versus the orbital
frequency of an equal-mass non-spinning binary black hole following an
adiabatic quasi-circular orbit towards merger. The horizontal axis is also
given as the number of orbits before merger. The plot is adapted from
Khalil:2022 , where the authors compared predictions for post-Minkowskian (PM)
conservative dynamics obtained using QFT tools (solid lines of increasing
accuracy in Newton’s constant $G$, i.e., in the PM approximation) to NR
results Ossokine:2017dge . In contrast to Figure 1, the predictions shown here
are obtained by incorporating the QFT results into the EOB formalism,
resulting in better agreement with NR towards merger.

Conservative binary dynamics at ${\cal O}(G^{3})$ was computed in Refs. 3PM ;
3PMLong  using amplitudes techniques, and has since been verified using a
variety of methods (see, e.g., Blumlein:2020znm ; Cheung:2020gyp ;
Kalin:2020mvi ). The result of Refs. 3PM ; 3PMLong  for conservative binary
dynamics has been extended to include dissipative effects in a number of
studies DiVecchia:2020ymx ; Damour:2020tta ; DiVecchia:2021bdo ; Bjerrum-
Bohr:2021din ; Damgaard:2021ipf ; Brandhuber:2021eyq ; Herrmann:2021lqe ;
Herrmann:2021tct ; DiVecchia:2021ndb ; Heissenberg:2021tzo .

Conservative binary dynamics at ${\cal O}(G^{4})$ was computed in Refs.
Bern:2021dqo ; Bern:2021yeh  using amplitudes techniques. Radiative
contributions to the conservative dynamics first appear at ${\cal O}(G^{4})$,
and complicates the analysis due to difficulties in precisely defining
conservative processes in the presence of radiation. This represents one of
the main obstacles for advancing the theoretical modeling of GW sources to
higher orders. Interestingly, some of the issues are sidestepped in the QFT-
based approach where everything follows from well-established properties of
scattering amplitudes. In particular, four-point scattering amplitudes should
fully capture two-body conservative dynamics for hyperbolic orbits and is
equivalent to using time-symmetric graviton propagators Wheeler:1949hn ;
Damour:1995kt ; Damour:2016gwp .

The result of Refs. Bern:2021dqo ; Bern:2021yeh  reproduces the sixth-order PN
result in Ref. Bini:2021gat . Moreover, due to Lorentz invariance and
dimensional analysis, the result exhibits a simple mass dependence, confirming
arguments made in Damour:2019lcq . This mass dependence puts strong
constraints on radiative contributions to conservative binary dynamics
Bini:2021gat . For instance, the result of Refs. Bern:2021dqo ; Bern:2021yeh
also agrees with the fifth-order PN result of Ref. Blumlein:2021txe  up to a
single term that does not have the expected mass dependence. The origin of
such contributions requires further study. The result of Refs. Bern:2021dqo ;
Bern:2021yeh  to all orders in velocity has also been partially verified
Dlapa:2021npj ; Dlapa:2021vgp . Initial studies of these results were
performed by theorists developing waveform models for the LVK collaboration
with encouraging conclusions; see Figure 1. Quite interestingly, the results
derived from amplitudes and EFT can be included in the EOB framework in a way
that improves the comparison with NR towards merger; see Figure 4. While these
results at 4PM order are in an early stage with assumptions 444The results at
4PM order, i.e. at ${\cal O}(G^{4})$, have been obtained from the Hamiltonian
computed for scattering orbits in Bern:2021dqo ; Bern:2021yeh ; Dlapa:2021npj
; Dlapa:2021vgp . that require further investigation, the key message is that
results from scattering amplitudes are promising and further developments are
strongly encouraged.

Spin Effects. From a theoretical perspective, inclusion of the spin of the
binary components in amplitudes methods faces the difficulty that fields of
arbitrary spins are needed while no-go theorems Shamaly:1972zu ;
Hortacsu:1974bm ; Deser:2000dz ; Porrati:2008gv ; Camanho:2014apa ; Afkhami-
Jeddi:2018apj  show that under certain assumptions such field theories have
unphysical features. Five proposals Arkani-Hamed:2017jhn ; Bern:2020buy ;
Chiodaroli:2021eug ; Aoude:2020onz ; Jakobsen:2021zvh  have been put forth to
construct classical spin-dependent two-body Hamiltonians. Their results are
consistent with each other, and also with results obtained through standard
general-relativity methods, when the latter are available.

The stress tensor of a Kerr black hole Vines:2017hyw  was obtained from
higher-spin interactions in Guevara:2019fsj  and Bern:2020buy , the latter
also obtaining the stress tensor of more general spinning compact objects
Levi:2015msa . The all-orders-in-spin ${\cal O}(G)$ and quartic-in-spin ${\cal
O}(G^{2})$ scattering angle for an aligned-spin configuration (i.e., spins
that are aligned or anti-aligned with the orbital angular momentum) was found
in Guevara:2018wpp . The ${\cal O}(G^{2})$ Hamiltonians expected to describe
Kerr black hole binaries for general spin configurations through fourth power
of the spin were derived in Bern:2020buy ; Kosmopoulos:2021zoq ; Chung:2019duq
; Chung:2020rrz ; Chen:2021qkk , and a general direct relation between
amplitudes and observables was conjectured in Bern:2020buy . The change in
momentum and in the spin at ${\cal O}(G^{3})$ to quadratic order in spin have
been computed in Jakobsen:2022fcj , together with the radiation-reaction
effects due to the radiated momentum at 2PM order Jakobsen:2021smu .

Proceeding past fourth order in spin at ${\cal O}(G^{2})$ had been an
important open problem for standard general-relativity methods. The neutron
star Hamiltonian quintic in spin was predicted in spin5  for arbitrary spin
orientation, and the Kerr scattering amplitude to eighth order in spin in
spin5competition . Refs. spin5 ; spin5competition  proposed that the Kerr
black hole corresponds to a particular subclass of spin structures that can
appear in a general scattering amplitude, which was explained in spin5
through a shift symmetry of the spin vector reminiscent of reparametrization
invariance in heavy-particle EFT Luke:1992cs ; Heinonen:2012km .

Relatedly, the dynamics of spin is technically very similar to the dynamics of
color in classical (infrared-free) Yang-Mills theories; using similar
techniques, the 2PM Hamiltonian for color-charged matter was found in
delaCruz:2020bbn ; delaCruz:2021gjp .

Tidal Effects. Recent detections of GWs from the merger of neutron stars
already constrain the equation of state of matter at nuclear densities
TheLIGOScientific:2017qsa ; LIGOScientific:2020aai , and more accurate
measurements at future detectors strongly motivate calculations at higher
precision Favata:2013rwa ; Samajdar:2018dcx ; Purrer:2019jcp ; Huang:2020pba ;
Gamba:2020wgg . Similar to traditional approaches using classical methods
Goldberger:2004jt ; Bini:2020flp ; Kalin:2020mvi , the approach based on
scattering amplitudes uses higher-dimension operators to model the rigidity of
the body and the susceptibility of its shape to change in response to a tidal
potential Cheung:2020sdj . The ${\cal O}(G^{3})$ contributions from the
leading tidal operators were computed using scattering amplitudes in
Cheung:2020sdj , and was verified using a worldline approach in Kalin:2020lmz
. Other amplitudes-based calculations of tidal effects have been pursued in
Refs. Haddad:2020que ; Aoude:2020ygw ; Bern:2020uwk ; Cheung:2020gbf ,
including results for infinite classes of tidal operators as well as all
orders in $G$ results in the probe limit.

Aside from calculating predictions for tidal effects for neutron stars, tidal
effects for black holes have also been the subject of many recent studies
within the particle physics community Chia:2020yla ; Hui:2020xxx ;
Charalambous:2021mea . Black holes have vanishing static, conservative tidal
responses, and a symmetry explanation for this has recently been put forth
Charalambous:2021kcz ; Hui:2021vcv .

Beyond General Relativity. Two-body Hamiltonians capturing specific models of
physics beyond general relativity have also been computed using scattering
amplitudes methods Emond:2019crr ; Cristofoli:2019ewu ;
AccettulliHuber:2019jqo ; AccettulliHuber:2020dal ; Bern:2020uwk ;
Cheung:2020sdj ; AccettulliHuber:2020oou ; Carrillo-Gonzalez:2021mqj .

Radiative and Absorptive Effects. As mentioned above, waveform models rely on
both the conservative and dissipative dynamics. Calculations of radiative
effects have so far focused on hyperbolic scattering trajectories, in
particular the scattering angle DiVecchia:2020ymx ; DiVecchia:2021bdo ;
Bjerrum-Bohr:2021din ; Damgaard:2021ipf ; Brandhuber:2021eyq ;
Herrmann:2021lqe ; Herrmann:2021tct ; DiVecchia:2021ndb ; Heissenberg:2021tzo
, as well as the loss of linear Herrmann:2021tct ; Herrmann:2021lqe ;
Riva:2021vnj  and angular momentum Damour:2020tta ; Bini:2021gat ;
Gralla:2021qaf ; Jakobsen:2021smu ; Jakobsen:2021lvp ; Mougiakakos:2021ckm ;
Manohar:2022dea , and the energy flux from higher-order tail effects
Bini:2021qvf ; Edison:2022cdu . A proposal for deriving radiation-reaction
forces from scattering amplitudes was recently given in Manohar:2022dea .

So far, absorptive effects have not been incorporated in a QFT framework, see
Goldberger:2019sya  for a worldline EFT approach.

##  5 Theoretical Structures

In this section we give an overview of mathematical and physical structures
that are relevant in the classical regime of scattering amplitudes. Some of
these structures, such as nonperturbative properties of amplitudes and their
direct connections to classical solutions, were exposed through explicit
calculations of different observables at higher orders. Others, such as the
eikonal phase and universality of high-energy scattering, have been studied in
the context of theoretical high energy physics but only recently applied for
GWs. We highlight examples where theoretical structures bring insight to the
phenomenology of GWs, and can be leveraged to develop calculational tools.

High Energy Limit. Even for classical scattering, the high energy (ultra-
relativistic) limit exposes interesting structures such as the interplay
between exclusive and inclusive observables Bloch:1937pw ; Kinoshita:1962ur ;
Lee:1964is , connections to soft graviton theorems DiVecchia:2021ndb , and
universality among gravitational theories with and without supersymmetry
tHooft:1987vrq ; DiVecchia:2020ymx ; Parra-Martinez:2020dzs .

For example, the four-point classical potential scattering amplitude at ${\cal
O}(G^{3})$ develops a singularity in the high-energy limit, $s\to\infty$:

$\displaystyle{\cal M}\to-8\pi G^{3}s^{2}\log(-t)\log\left(m_{1}m_{2}\over
s\right)\,,$ (2)

where $s$ and $t$ are Mandelstam variables, and $m_{i}$ are the masses of the
scattering black holes. On general grounds, such singularities cannot exist in
complete amplitudes Akhoury:2011kq . The finite inclusive observable,
including contributions from radiation modes, can be derived using soft
graviton theorems to describe graviton emission DiVecchia:2021ndb  or by a
linear response analysis Damour:2020tta . Moreover, the double logarithmic
structure can be understood from the Regge limit, and is given by the cusp
anomalous dimension Henn:2012qz , or by the heavy-heavy current anomalous
dimension in Heavy Quark Effective Theory Manohar:2000dt .

These connections illustrate that amplitudes in simplifying limits, even in
supersymmetric versions of gravity, can carry useful information about
classical binary dynamics. More generally, there is a wealth of knowledge on
four- and five-point amplitudes in theoretical high energy physics, and many
of the tools and physical insights can be transplanted to classical binary
dynamics. For example, quantum electrodynamics (QED) as well as supersymmetric
cousins of gravity, such as ${\cal N}=8$ supergravity, offer controlled
laboratory settings for sharpening tools and dissecting basic phenomena.

Eikonal Phase. It has long been known that in the classical limit the eikonal
phase provides a good description of elastic four-point amplitudes in gauge
and gravity theories, which effectively exponentiate as a consequence of
unitarity Levy:1969cr ; Abarbanel:1969ek ; Cheng:1969eh ; Amati:1990xe ;
Kabat:1992tb ; Saotome:2012vy ; Akhoury:2013yua

$i{\cal M}=e^{i\delta}-1\ .$ (3)

It is moreover expected that the dominant contributions at high energies to
the eikonal $\delta$ come from the exchange of the highest-spin state in the
theory tHooft:1987vrq ; Amati:1990xe ; Muzinich:1987in , and therefore that it
is universal in gravitational theories Bellini:1992eb .

Modern evidence for these properties was obtained in recent work on massive
theories Parra-Martinez:2020dzs ; DiVecchia:2020ymx ; Bjerrum-Bohr:2021din ,
as well as massless theories with various amounts of supersymmetry, including
general relativity Bern:2020gjj . This led to the idea that inclusion of all
contributions from exchanged gravitons with momenta of the order of the
transferred momentum is a necessary (and possibly sufficient) condition for
universality of the eikonal in the ultrarelativistic limit and for a smooth
interpolation to the nonrelativistic limit DiVecchia:2020ymx . A self-
contained treatment of the real and imaginary parts of the eikonal in the
entire soft region at ${\cal O}(G^{3})$ in massive ${\cal N}=8$ and in general
relativity was presented in DiVecchia:2021bdo  and confirmed in Bjerrum-
Bohr:2021vuf , giving another demonstration of the universality properties.

The eikonal form of the S matrix is quite similar to that of the amplitude-
radial action relation (4), but they differ in the definition of the iteration
terms Bern:2021dqo . Similar to the radial action, the eikonal is a generating
function of scattering observables Amati:2007ak , such as the scattering angle
or the time delay. See Sec. 6 for further discussion.

Eikonal with Spin. The unitarity argument for the exponentiation of elastic
amplitudes holds also for spinning external states. This was demonstrated
explicitly for spin-1/2 particles in Haddad:2021znf  and for arbitrary spin in
the classical limit from the perspective of minimization of the spin variance
in Cristofoli:2021jas .

The eikonal continues to be a generating function for observables when
scattering spinning particles. This was verified explicitly through ${\cal
O}(S_{1}S_{2})$ in Bern:2020buy  where the change in momentum (also referred
to as the “impulse”) and the change in spin (also referred to as the “spin
kick”) obtained from the eikonal were compared with the results of Hamilton’s
equations. A conjecture was also put forth Bern:2020buy , connecting eikonal
and scattering observables to all orders in spin. This relation was
successfully tested through fourth order in spin Kosmopoulos:2021zoq ;
Chen:2021qkk  at ${\cal O}(G^{2})$. It is quite interesting and surprising
that a single function can capture the complicated three-dimensional dynamics
of scattering of spinning particles.

Nonperturbative Structures. EFT methods can be used to connect scattering
amplitudes and classical binary dynamics through a matching calculation
between gravity and a low energy EFT that describes particles interacting
through an effective potential. An outcome of this matching calculation is a
beautiful relation between the classical limit of the scattering amplitude
${\cal M}$ in impact parameter space and the radial action $I_{r}$ of
Hamilton-Jacobi theory:

$\displaystyle i{\cal M}=e^{iI_{r}}-1.$ (4)

This means that the scattering amplitude directly determines the radial action
$I_{r}$, which in turn determines orbital trajectories. This equation was
first derived in Bern:2021dqo , and is dubbed the “amplitude-action relation”.
The exponential form of the amplitude-action relation is reminiscent of the
eikonal phase (3), and has an important practical implication for
calculations: a large class of integrals come from exponentiation of lower-
order contributions and can be systematically dropped prior to integration.
The amplitude-action relation was studied to all orders in perturbation theory
for a probe in the Schwarzschild or pure-NUT gravitational backgrounds as well
as for a probe interacting with point-charges and monopoles in Kol:2021jjc .

The amplitude-action relation in Eq. (4) is an example of remarkable
nonperturbative structures that appear in the classical limit. Another example
is the following relation between the classical scattering amplitude in
position space ${\cal M}(r)$ and the local center-of-mass momentum in a
hyperbolic orbit,

$\displaystyle{\cal M}(r)={p(r)^{2}-p(\infty)^{2}\over 2E}\,.$ (5)

Here $p(r)^{2}$ is the squared center-of-mass momentum at position $r$, and
$E$ is the total center-of-mass energy. This was first noticed and used in 3PM
; 3PMLong , and further developed and formalized in Kalin:2019rwq  (where it
was dubbed “impetus formula”) and also Bjerrum-Bohr:2019kec . This relation
can be used to derive amplitudes to all orders in $G$. For instance, the
diagrams in Figure 5 describe a probe particle in a Schwarzschild background,
and can be resummed by determining $p(r)$ from geodesic motion. One can also
derive nonperturbative amplitudes involving higher-dimensional operators that
describe tidal effects Cheung:2020gbf .

Another nonperturbative relation builds on the Newman-Janis Newman:1965tw
relation between the Schwarzschild and Kerr solutions of Einstein’s equations.
This relation, employing a certain complex shift, has been used in Arkani-
Hamed:2019ymq  to obtain the impulse for spinning particles from that of
spinless particles at leading post-Minkowskian order. It remains an intriguing
open question whether the relation holds at higher orders and how to exploit
it Guevara:2020xjx .

Figure 5: The resummation of these diagrams to all orders in $G$ yields the
amplitude for a probe particle in a Schwarzschild background.

Gravitational Self-Force. Extreme-mass-ratio inspirals are binary systems
consisting of compact bodies with masses $m_{1}$ and $m_{2}$ with $m_{2}\gg
m_{1}$. The limiting case is described by a probe particle orbiting in a
background spacetime such as Schwarzschild or Kerr. Beyond this, the particle
interacts with its own gravitational field, giving rise to an effective “self-
force”, which is computed as an expansion in $m_{1}/m_{2}$ but to all orders
in $G$. The gravitational self-force for generic bound geodesics in
Schwarzschild and Kerr spacetimes were found in Barack:2010tm  and
vandeMeent:2017bcc , respectively. The precision of LISA will require the
second-order self force in the conservative and dissipative dynamics, i.e.,
corrections of ${\cal O}(m_{1}^{2}/m_{2}^{2})$ and all orders in $G$, which is
not completely solved. However, see Ref. Pound:2019lzj  for significant recent
progress.

Interestingly, scattering amplitudes have revealed a connection between
perturbative corrections to binary dynamics and self-force corrections. The
mass dependence of the $n$-loop scattering amplitude in the classical limit
follows simply from dimensional analysis, and implies that it can probe the
${\cal O}(G^{n+1})$ contribution to the $\lfloor\frac{n}{2}\rfloor$-th order
self-force correction. In other words, the $n$-loop scattering amplitude
contains contributions of the form $\sim
G^{n+1}(m_{1}/m_{2})^{\lfloor\frac{n}{2}\rfloor}$. This structure is leveraged
in a powerful new method, dubbed “Tutti-Frutti”, for extracting perturbative
corrections to binary dynamics from self-force calculations BDG1 ; BDG2 ; BDG3
; Damour:2019lcq ; Antonelli:2020aeb ; Antonelli:2020ybz ; Khalil:2021fpm .
This mass dependence, which ultimately traces back to Lorentz invariance, also
has strong implications for the radiative contribution to conservative
dynamics Damour:2019lcq . In particular, it implies nontrivial cancellations
among contributions from source multipole moments and imposes strong
consistency checks on perturbative calculations. Notably, there are now
numerical and theoretical efforts to extend gravitational self-force
calculations to hyperbolic trajectories in order to make contact with results
from scattering amplitudes; see, e.g., Gralla:2021qaf .

Coherent States and Waveforms. The gravitational waveform during a scattering
event can itself be directly computed from amplitudes Cristofoli:2021vyo . At
lowest perturbative order it is given by an integral of a five-point tree
scattering amplitude. The relation to the intuitive picture, in which a GW
consists of a large number of gravitons, is interesting: GWs are described by
coherent states (with very large occupation numbers) of the gravitational
field. Gravitons building up this state are emitted independently
Cristofoli:2021jas ; Britto:2021pud , and each emission is described by the
same five-point amplitude.

Coherent states also play a role in understanding the emergence of classical
spin and color from QFT Bern:2020buy ; delaCruz:2020bbn ; Aoude:2021oqj , and
in understanding the emergence of classical physics from quantum mechanics
quite generally Yaffe:1981vf .

Eikonal methods have been generalized to include such coherent outgoing
radiation Cristofoli:2021jas , building on earlier work of Ciafaloni:2018uwe
and Cristofoli:2021vyo . In particular, Ref. Cristofoli:2021jas  argued that
the eikonal is effectively extended by a coherent state operator, which
creates arbitrarily many outgoing massless particles and that the
nonperturbative radiation field is described by the waveshape parameter
defining the coherent state, which in turn is determined by the five-point
amplitude.

Soft Gravitons. Recently it has become clear that there is a beautiful
relation between the soft limit in quantum theories and memory effects in
classical dynamics Strominger:2014pwa . This can be understood in the KMOC
formalism (see Sec. 6) by studying the radiated momentum. In the long-
wavelength limit, the scattering amplitude involved in the radiation
simplifies as a soft factor times a lower point amplitude, recovering the
impulse Bautista:2019tdr . Classically, this impulse is the “memory” of the
step-change in the field described by its very low frequency Fourier
components. More generally, the KMOC formalism reveals a rich interplay
between classical physics, soft or low-frequency radiation, and scattering
amplitudes Laddha:2018rle ; Laddha:2018myi ; Sahoo:2018lxl ; Manu:2020zxl ;
Bautista:2021llr .

Analytic Continuation and Time Nonlocality. The integrability of the two-body
equations of motion with Newton’s potential guarantees that their solutions
are uniquely specified by integrals of motion – the total energy and the
orbital angular momentum. Hyperbolic and elliptic motions are mapped into each
other by analytic continuation of the boundary conditions. While integrability
is not known to exist for two-body conservative Hamiltonians even at 2PM order
Caron-Huot:2018ape , Refs. Cho:2018upo ; Kalin:2019rwq ; Kalin:2019inp ;
Cho:2021arx  argued that bound state observables can be obtained through a
suitable analytic continuation in energy and in angular momentum from
scattering observables. This procedure was dubbed “Boundary-to-Bound” or “B2B”
and formalizes the fact that such Hamiltonians can be constructed by matching,
e.g. the scattering angle, and then subsequently used for bound motion by
changing the boundary conditions. Physically, this can be understood as a
consequence of the time locality of the potential generated by potential-
region gravitons.

However, this approach fails for conservative radiation-reaction effects,
beginning at ${\cal O}(G^{4})$ with the tail effects Cho:2021arx . Given a
scattering angle, it is always possible to construct a local Hamiltonian that
reproduces it. Fundamentally however, because it captures effects of radiation
modes propagating over long periods before being reabsorbed by the binary, the
“off-shell” Hamiltonian has both an instantaneous component and a non-local in
time one Damour:2014jta . While the analytic continuation of the local and
universal (logarithmic) part of the nonlocal Hamiltonian is straightforward,
obtaining the non-universal part of the bound Hamiltonian from the unbound one
is an important open problem.

Exploring Structure in Simpler Theories. Even in the classical limit
gravitational interactions are complicated. Before proceeding to full-fledged
gravitational calculations, it is useful to test ideas and tools and search
for theoretical structures in much simpler settings such as gauge theory.
Since color in the classical limit becomes essentially abelian, it suffices to
study QED. Calculations designed to explore some of the subtleties that appear
at ${\cal O}(G^{3})$ in gravitational calculations were carried out in gauge
theories, both using scattering amplitudes delaCruz:2020bbn ; Bern:2021xze ,
and classical gravity methods Saketh:2021sri ; Buonanno:2000qq .

Gravitational theories with additional symmetries, like supersymmetry are
another possibility. As discussed above, results obtained in supergravity
theories confirmed universality properties of the eikonal function and led to
an understanding of the role of radiative corrections in classical scattering.

Relations Between Amplitude Fragments. The essential difference between
classical and quantum measurements is the variance: it is nonzero for quantum
measurements and must become negligible in the classical limit. Defining
measurements in terms of expectation values of operators, Ref.
Cristofoli:2021jas  showed that the condition of zero variance (effectively
requiring factorization of expectation values of products of operators)
implies an infinite set of relations between different amplitude “fragments”
(that is different orders in an amplitude’s expansion in momentum transfer)
with different numbers of loops and legs. For four-point amplitudes they are
the relations required for eikonal exponentiation. At five points, higher-loop
fragments are related to lower-loop five-point and four-point amplitude
fragments. Among the implications of the zero-variance condition are detailed
predictions for the momentum transfer dependence of amplitudes and a novel
understanding of the eikonal in the presence of outgoing radiation mentioned
above.

##  6 Developing Tools

In this section we describe the tools that enable state-of-the-art
calculations, such as those summarized in Sec. 4, as well as, recently
developed tools that were inspired by new theoretical structures or by the
challenges of pushing the cutting edge. These tools are forged by streamlining
existing calculations, testing on toy theories such as QED or supersymmetric
cousins of gravity, and direct forays into new calculations in Einstein’s
gravity. We highlight synergies between scattering amplitudes and other
particle physics tools such as EFT and advanced multiloop integration
techniques, as well as, with methods in general relativity such as
gravitational self-force and the worldline formalism.

Effective Field Theory. Compact binaries share many of the essential
characteristics of bound states of elementary particles such as positronium,
hydrogen, or quarkonia. Therefore, the nonrelativistic EFT techniques
developed for QED and QCD Caswell:1985ui ; Isgur:1989vq ; Bodwin:1994jh ;
Beneke:1997zp ; Manohar:2000dt ; Manohar:2000hj ; Manohar:2006nz  are well-
suited for application to compact binaries. This is underscored by the many
landmark calculations that were enabled by the NRGR framework
Goldberger:2004jt , see the dedicated Snowmass White Paper NRGRWhitePaper2022
for recent developments on this subject. These physical systems are
characterized by scales that define the following modes or regions:

(10)

Here $(E,p)$ denotes the scalings of the energy and three-momentum of the
modes, $m$ is the mass of the binary constituents, $v$ is the relative
velocity, and $J$ is the angular momentum. As discussed in Sec. 3, the
classical limit corresponds to the limit of large angular momentum.

The new approach based on scattering amplitudes adapts these EFT tools, and
closely related methods such as the method of regions Beneke:1997zp , to
efficiently extract classical contributions from the various modes described
above. In particular, power-counting and factorization allows contributions
from potential and radiation modes to be systematically and separately
considered in scattering amplitudes, and then resummed to all orders in the
velocity $v$ using differential equations.

Similar to applications in QED and QCD Manohar:2000hj , another basic role of
EFT is to connect scattering amplitudes and classical binary dynamics through
a matching calculation between gravity and a low energy EFT that describes
particles interacting through an effective potential Neill:2013wsa ;
Vaidya:2014kza ; CRS . For the case of conservative dynamics, the basic
framework has now been applied and extended for application at higher orders
CRS ; 3PM ; 3PMLong ; Bern:2021dqo ; Bern:2021yeh , for including spin and
tidal effects Cheung:2020sdj ; Bern:2020buy ; Kosmopoulos:2021zoq ;
Haddad:2020que ; Cheung:2020gbf ; Bern:2020uwk ; Aoude:2020ygw , and for
gravitational theories with supersymmetry and in arbitrary dimensions Caron-
Huot:2018ape ; Parra-Martinez:2020dzs ; Cristofoli:2020uzm . In Ref.
Bern:2021dqo , this EFT matching procedure was used to derive the amplitude-
action relation (4).

Advanced Multiloop Integration. The quest for high-precision calculations in
particle physics, e.g. for colliders and for theoretical explorations of
various field theories, has led to the development of advanced technology for
integration of loop contributions. This can be transplanted for application to
GWs, where the evaluation of integrals is a challenge common to all field-
theory–based approaches, including traditional approaches in general
relativity.

Integration by parts reduction as implemented in automated programs, such as
AIR Anastasiou:2004vj , FIRE Smirnov:2008iw ; Smirnov:2013dia , Kira
Maierhoefer:2017hyi , Reduze Studerus:2009ye , and LiteRED Lee:2012cn ;
Lee:2013mka , can be used to obtain a set of master integrals, which are then
evaluated using the method of differential equations Kotikov:1990kg ;
Bern:1993kr ; Remiddi:1997ny ; Gehrmann:1999as , perhaps in canonical form
Henn:2013pwa ; Henn:2013nsa . The results are given in terms of multiple
polylogarithms, their elliptic generalizations and perhaps more complicated
functions. A judicious choice of variables is crucial Landshoff:1969yyn ;
Parra-Martinez:2020dzs . Integrals that appear in general-relativity–based
approaches to PN calculations are also amenable to these methods, as was
demonstrated in Bini:2020uiq ; Bini:2020rzn . These methods mesh well with the
expansion around the classical limit, and extraction of potential and
radiation contributions as implemented through the method of regions
Beneke:1997zp .

Generating Functions. The radial action $I_{r}$, defined as the integral of
the radial momentum along the trajectory, contains the entire classical
central-field dynamics. Observables such as the scattering angle, redshift,
and time delay, can be derived from the radial action via thermodynamic-type
relations, e.g. $dI_{r}=\frac{\theta}{2\pi}dL+\tau dE+\sum_{a}\langle
z_{a}\rangle dm_{a}$, where $\theta$ is the scattering angle, $L$ is the
orbital angular momentum, $\tau$ is the time delay, and $\langle z\rangle$ is
the redshift. Ref. Bern:2021dqo  demonstrated a direct relation (4) between
elastic four-point scattering amplitudes and the radial action for the
scattering trajectory of the two particles. The radial action can thus be
directly obtained from particular contributions to the four-point scattering
amplitude Bern:2021dqo ; Bjerrum-Bohr:2021wwt . Ref. Damgaard:2021ipf  argued
that the radial action constructed this way is related to the WKB
approximation to the scattering process, which establishes a connection with
the eikonal exponentiation of amplitudes (3). The amplitude-action relation
has also been used to explore non-perturbative results in the probe limit
Kol:2021jjc .

From a field theory point of view, the eikonal function provides another
generating function for observables. Indeed, the kinematics that selects the
classical part of a four-point amplitude, $s\gg t=-q^{2}$, is the same as the
Regge kinematics, so properties of amplitudes in this regime, such as its
exponentiation in terms of the eikonal function, can be used to gain insight
into classical physics. The original framework has been extended to capture
the exponentiation of amplitudes of spinning particles and of five-point
amplitudes, under suitable assumptions regarding the coherence of the emitted
gravitons Cristofoli:2021jas . While structurally similar, the radial action
and the eikonal functions differ in the details of the definition of the
exponential or, equivalently, in the definition of the iteration terms.

KMOC – A Formalism for Observables from Amplitudes. Another approach to
extracting classical physics from amplitudes focuses on determining physical
observables which are well defined in both the classical and quantum theories.
This observables-based approach was first discussed in Kosower:2018adc , and
is sometimes referred to as the KMOC formalism. It is a quantum-first
treatment of observables, which incorporates key aspects of classical physics
from the beginning. By arranging a system to be under the purview of the
correspondence principle, then quantum effects are negligible and it must be
the case that the quantum mechanical computation of the observables will yield
the classical result.

In the KMOC formalism observables are evaluated as expectation values of
operators in the quantum field theory, $\langle\psi|{\cal O}|\psi\rangle$.
Scattering amplitudes enter through time evolution which, given an initial
state $\ket{\psi}$ in the far past, produces a final state $S\ket{\psi}$ in
the far future, and scattering amplitudes are the matrix elements of $S$. This
strategy is somewhat reminiscent to QCD event shapes Basham:1978bw ;
Basham:1978zq ; Richards:1983sr ; Korchemsky:1999kt ; Dokshitzer:1999sh ;
Lee:2006fn ; Bauer:2008dt , such as the energy or charge flow Belitsky:2001ij
and energy-energy correlators Basham:1978bw ; Basham:1978zq .

A whole range of observables can be computed using the KMOC formalism. A
particularly simple example is the impulse. The impulse is the change in the
expectation value of the momentum operator $\mathbbm{P}^{\mu}$ of the quantum
field describing the massive particle and is closely related to the classical
potential. There is a direct connection between the impulse and the scattering
angle: once the final momentum is known, it is straightforward to determine
the direction of motion relative to the initial momentum. On the other hand if
the scattering angle is known the final momentum can be reconstructed,
assuming no energy is radiated away.

The amount of momentum radiated is itself another simple observable, arising
as the expectation value of the momentum operator $\mathbb{K}^{\mu}$ of
whatever field is transporting the momentum. For example, in gravitational
scattering, it is the momentum operator of the gravitational field. Because
conservation of momentum is built into the underlying quantum field theory,
the KMOC formalism naturally incorporates the effects of radiation reaction.
It also has natural generalizations that include the spin of particles
Maybee:2019jus ; Guevara:2019fsj ; Arkani-Hamed:2019ymq  and capture the
dynamics of color in the context of classical (infrared-free) Yang-Mills
theories delaCruz:2020bbn ; delaCruz:2021gjp .

The gravitational waveform emitted during a scattering event follows directly
from amplitudes Cristofoli:2021vyo ; the observable of interest is the Riemann
curvature operator $\mathbbm{R}_{\mu\nu\rho\sigma}(x)$ in linearized gravity.
The linearized approximation is appropriate as the GWs travel over extremely
large spatial distances and retains only the leading term in inverse distance.
The waveform itself is an integral of an appropriate component of the
expectation
$\bra{\psi}S^{\dagger}\mathbbm{R}_{\mu\nu\rho\sigma}(x)S\ket{\psi}$. At lowest
perturbative order, it is an integral of a five-point tree scattering
amplitude. The relation with the intuitive picture in which a wave is composed
of a large number of gravitons is established by describing the wave as a
coherent state of the gravitational field. Interestingly, vanishing variance
in the classical limit Cristofoli:2021jas  implies that correlators of
products of operators, ${}_{\text{out}}\langle\psi|{\cal O}_{1}{\cal
O}_{2}|\psi\rangle_{\text{out}}$ factorize as
${}_{\text{out}}\langle\psi|{\cal
O}_{1}|\psi\rangle_{\text{out}}{}_{\text{out}}\langle\psi|{\cal
O}_{2}|\psi\rangle_{\text{out}}$; they nevertheless contain new classical
information as demonstrated in Manohar:2022dea .

Amplitude Building-Blocks in the Classical Limit. Through generalized
unitarity, tree-level amplitudes are the building blocks for all higher-order
calculations. The application of scattering amplitudes to GWs has motivated
reorienting the structure of these building blocks to take advantage of
simplifications in the classical limit. A number of approaches are based on
heavy-particle effective theory Damgaard:2019lfh , and the double-copy in such
theories Brandhuber:2021kpo ; Brandhuber:2021eyq ; Brandhuber:2021bsf , which
build on ongoing efforts to understand the kinematic algebra behind
color/kinematics duality, see e.g. Chen:2019ywi ; Chen:2021chy . Other
approaches focus on reorganizing the soft expansion to reduce the number of
master integrals by combining terms related by permutations of graviton legs
Bjerrum-Bohr:2021vuf .

Another approach uses generalized unitarity to first construct gauge-theory
amplitudes and then applies the double-copy to obtain the corresponding
gravitational amplitudes Carrasco:2020ywq . Various methods are available to
project out the dilaton and axion Johansson:2014zca ; Luna:2017dtq  that
naturally appear in this approach. It will be interesting to explore how to
take the classical limit before the double-copy, and use gauge theory tree-
level amplitudes adapted for this purpose.

Many developments focused on amplitudes for higher-spin particles. The
amplitudes obtained in the classical limit avoid the strong constraints
imposed by no-go theorems Shamaly:1972zu ; Hortacsu:1974bm ; Deser:2000dz ;
Porrati:2008gv ; Camanho:2014apa ; Afkhami-Jeddi:2018apj . Ref. Arkani-
Hamed:2017jhn  constructed three-point amplitudes for arbitrary-spin massive
particles through standard amplitudes methods and, by demanding good high-
energy properties, identified the so-called “minimal amplitudes” which
correspond to the linear response of Kerr black holes to an external
gravitational field Vines:2017hyw . Using Lagrangian methods, Ref.
Bern:2020buy ; spin5  constructed an EFT that described the classical
interactions of higher-spin fields at large impact parameter,
$|\bm{b}|\gg|\bm{p}|^{-1}$, which includes arbitrary spin-induced multipoles
and whose three-point amplitudes exhibit a double-copy relation to higher-spin
fields coupled with gluons. The heavy-particle approach in Ref.
Damgaard:2019lfh  was extended for higher-spin fields with double-copy
properties in Aoude:2020onz ; Haddad:2020tvs ; spin5competition . For suitable
values of the parameters, the three-point amplitudes following from these
Lagrangians reproduce the Kerr stress tensor Vines:2017hyw . A gravitational
Compton amplitude for spin-$5/2$ massive particles that is free of spurious
poles was found in Chiodaroli:2021eug , demonstrating that the methods
employed can produce consistent higher-point amplitudes for higher-spin
particles. Another construction Bjerrum-Bohr:2020syg ; Bjerrum-Bohr:2019nws
makes use of the scattering equation formalism to provide diagrammatic and
recursive tools for finding covariant expressions for $D$-dimensional tree-
level $n$-point amplitudes with pairs of spinning massive particles.

Synergy with Traditional Approaches. The new approach based on the tools of
theoretical high-energy physics has benefited immensely from traditional
methods, in terms of both conceptual guidance from existing frameworks and
practical guidance from explicit results. In turn, the emergence of new
results derived from scattering amplitudes has spurred new developments within
traditional methods, as well as, hybrid approaches that meld traditional
methods and particle-physics tools. Such an extensive exchange between various
approaches in general relativity and particle physics has led to the rapid
advance of recent years, and will continue to drive breakthroughs in the
future.

For instance, one of the main tools for describing binary dynamics within the
PN approach to general relativity is the worldline approach Goldberger:2004jt
; Foffa:2019yfl ; Foffa:2019rdf ; Blumlein:2019zku ; Foffa:2019hrb ;
Foffa:2011ub ; Gilmore:2008gq ; Levi:2018nxp ; Kol:2007bc . Recently, inspired
by amplitudes methods, this has been extended to a PM framework Kalin:2020mvi
to derive results to all orders in velocity. Calculations in this approach
share many similarities with amplitudes-based methods, such as the use of
integration technology imported from particle physics. Other worldline-based
approaches having aspects of scattering amplitudes have also been recently
developed for calculations of tail effects in a nonrelativistic EFT for a
binary Edison:2022cdu  and of PM observables Jakobsen:2022fcj ;
Jakobsen:2021smu ; Mogull:2020sak , including spin Jakobsen:2022fcj ;
Jakobsen:2021zvh ; Cho:2021mqw  and radiative Jakobsen:2022fcj ;
Jakobsen:2021lvp  effects.

As discussed in Sec. 5, new results from scattering amplitudes have exposed a
number of theoretical structures. One feature that was hidden in traditional
non-relativistic calculations was the simple mass dependence of scattering
amplitudes, or equivalently the radial action and scattering angle. Indeed, it
was noticed Vines:2018gqi , and then rigorously proved Damour:2019lcq , that
the scattering angle exhibits a particular dependence in the symmetric mass
ratio $m_{1}m_{2}/(m_{1}+m_{2})^{2}$, leading to the so-called “good mass
polynomiality” rule. The latter is the basis of the “Tutti-Frutti” method BDG1
; BDG2 ; BDG3 ; Damour:2019lcq , which establishes the overlap between the
self-force expansion and the PM expansion. This powerful method has derived a
number of new results, which have in turn provided important guidance for new
amplitudes-based calculations. The “Tutti-Frutti” method has also been
extended for systems with spins Antonelli:2020aeb ; Antonelli:2020ybz ;
Khalil:2021fpm .

The B2B map is another instance in which patterns revealed from explicit
amplitudes calculations 3PM ; 3PMLong  were combined with the application of
analytic continuation Cho:2018upo  to develop efficient new tools
Kalin:2019rwq ; Kalin:2019inp ; Cho:2021arx .

##  7 Future directions and challenges

In this section we summarize several broad directions that are as essential
for long-term progress and fruitful synergy between particle and GW physics.

New Perspectives on Observables. Scattering amplitudes are naturally defined
by data at asymptotic infinity and can offer insights on this class of
observables in general relativity, which can be subtle due to general
coordinate invariance. Of particular significance are questions regarding
radiative contributions, which are not only of theoretical interest, but also
of practical importance for obtaining high-precision waveforms.

One open question is: how are observables in bound and unbound orbits
generally related? In the absence of radiative effects, both types of dynamics
can be straightforwardly derived from the same Hamiltonian and are related by
analytic continuation. However, this is no longer true in the presence of
radiative effects, which introduce correlations over arbitrary time intervals,
e.g., due to tail effects Thorne:1980ru ; Blanchet:1987wq ; Blanchet:1993ec .
This leads to a nonlocal-in-time Hamiltonian that depends on the particular
trajectory Damour:2014jta  (see also Foffa:2011np  for a more recent
perspective). This non-universality prevents a simple analytic continuation
Cho:2021arx .

It is imperative to address this question in order to fully leverage
scattering amplitudes, which are naturally associated with unbound motion, for
describing the dynamics of bound compact objects. A complete framework for
connecting bound and unbound trajectories will not only apply to conservative
dynamics but also to the radiated energy, momentum, and angular momentum, and
perhaps to the waveforms themselves. Interestingly, there is evidence for
double copy in GW emission in both bound and unbound systems
Goldberger:2016iau ; Goldberger:2017vcg ; Shen:2018ebu .

Relatedly, it would be useful to carefully formulate important concepts such
as conservative vs. dissipative dynamics, as well as inclusive vs. exclusive
or IR safe vs. unsafe observables. For instance, recent differences in the
conservative dynamics at ${\cal O}(G^{4})$ are due to difficulties in
precisely defining conservative dynamics in the presence of radiative effects.
To this end, it may be fruitful to compute new observables, beyond those
traditionally considered in binary dynamics, that can serve as toy models for
understanding these issues. Such new observables may also be of
phenomenological interest. For example, it would be interesting to detect
waveforms from hyperbolic encounters. Although such events are expected to be
rare and have a short-duration signal, their theoretical predictions are well-
understood and can be obtained directly with scattering amplitude methods
without need of analytic continuation.

Observables and unexpected structures may be further revealed by formulations
of gravitational interactions without direct reference to geometry, in analogy
with structure exposed by formulations of quantum field theory without quantum
fields. The classical double copy Monteiro2014cda  provides an example, as do
recent attempts to define the Kerr black hole from a QFT perspective spin5 ;
spin5competition . While currently perturbative, such theoretical tools may
help fully exploit GW observations in the quest to understand gravitating
compact objects and black-hole horizons.

Still-Higher Perturbative Orders. The precision demands of future experiments
KAGRA:2013rdx ; Saleem:2021iwi ; Reitze:2019iox ; Punturo:2010zz ;
LISA:2017pwj  are quite challenging since, depending on the source, GWs from
binary systems will be observed with a signal-to-noise ratio that is one or
two orders of magnitude higher than with current detectors
LIGOScientific:2014pky ; VIRGO:2014yos ; KAGRA:2020agh . To keep the modeling
systematics below the expected statistical errors Favata:2013rwa ;
Samajdar:2018dcx ; Purrer:2019jcp ; Huang:2020pba ; Gamba:2020wgg , the
accuracy of current state-of-the-art waveform models Hinderer:2016eia ;
Dietrich:2017aum ; Nagar:2018plt ; Nagar:2018zoe ; Cotesta:2018fcv ;
Varma:2018mmi ; Dietrich:2019kaq ; Varma:2019csw ; Ossokine:2020kjp ;
Pratten:2020ceb ; Thompson:2020nei ; Matas:2020wab ; Estelles:2021gvs ;
Gamba:2021ydi , which are built from combining analytic and numerical
relativity, needs to be improved by, say, two orders of magnitude. This would
require up to ${\cal O}(G^{7})$ and/or ${\cal O}(v^{12})$, as shown in Figure
2 for the potential. Moreover, higher-order calculations are needed for the
dissipative sector, the multipolar waveforms, and other physical effects such
as spins, tides, and eccentricity.

While amplitude-based methods are powerful, achieving this will still require
a significant build-up of technology over several years of cycling between
performing state-of-the-art calculations and developing new tools to solve
bottlenecks. For example, current pipelines can be significantly improved by
streamlining the construction of classical scattering amplitudes and
efficiently removing iteration terms that are already determined by lower-
order calculations. In the same spirit, it will be important to exploit the
nontrivial interplay between different graph topologies that lead to vast
simplifications in the integrand, as demonstrated in Akhoury:2013yua  and
further explored in the context of GW physics in Brandhuber:2021eyq ;
Brandhuber:2021kpo . Similarly, it will be interesting to understand and
utilize the consequences of the infinite number of relationships between
multi-point, multi-loop amplitudes in the classical limit Cristofoli:2021jas ,
which suggest that various terms in higher-point and/or higher loop amplitudes
can be predicted from lower loop, lower point amplitudes. It also would be
interesting to examine whether this phenomenon could be useful in contexts
outside of GW physics, for example in collider experiments.

Apart from applications to precision GW physics, higher-order calculations
also bring new perspectives on known phenomena. For example, the calculation
of conservative binary dynamics at ${\cal O}(G^{5})$ including radiative
effects can give a better understanding of the role of GW memory in binary
dynamics.

Higher-order calculations contain a wealth of information about the analytic
structure of the two-body Hamiltonian and of observables. Up to rational
functions, they are given by the nontrivial loop Feynman integrals that can
appear at a given order in perturbation theory. A classification of the
functionally independent integrals can directly constrain observables from a
general knowledge of their analytic properties or their behavior in various
limits. Moreover, these functions can be matched to NR simulations, which, for
the time being, provide the most accurate description of the final moments of
a binary inspiral. This fitting procedure may offer further analytic insight
into this period of the binary’s evolution and, conversely, on analytic
approaches to nonperturbative properties of gravitational interactions.

Integration Challenges. Evaluation of higher-loop Feynman-type integrals is a
challenge shared by QFT applications to both particle physics and to
gravitational and GW physics.

The integration-by-parts (IBP) reduction Chetyrkin:1981qh ; Laporta:2000dsw ;
Laporta:1996mq ; Smirnov:2008iw ; Maierhoefer:2017hyi ; Studerus:2009ye ;
Lee:2012cn ; Lee:2013mka  used to reduce the expanded Feynman integrals to
basis of integrals, referred to as master integrals, becomes computationally
demanding as the number of loops increase. Cutting edge developments that
increase the efficiency of IBP reduction Gluza:2010ws ; Schabinger:2011dz ;
Ita:2015tya ; Georgoudis:2016wff ; Ita:2016oar ; Zhang:2016kfo ; Bern:2017gdk
; Mastrolia:2018uzb ; Frellesvig:2019uqt , perhaps with further improvements
related to the specific structure of the integrals that appear in the
classical limit, will be vital for field theory-based approaches to reach high
perturbative orders in the post-Minkowskian and the post-Newtonian expansion
demanded by future observations. Further improvements may come from the
development of an algorithmic choice of master integral basis tailored for the
two-body general-relativity problem.

The master integrals relevant for the two-body problem depend only on the
relative velocity of the two bodies and can be evaluated through a system of
linear first-order ODEs Kotikov:1990kg ; Bern:1993kr ; Remiddi:1997ny ;
Gehrmann:1999as  in this velocity. Expansion in the static limit,
$v\rightarrow 0$, yields the integrals that appear in the post-Newtonian
expansion and the leading terms serve as boundary conditions for the ODE
system. The differential equations themselves simplify when a “canonical
basis” of master integrals is chosen Henn:2013pwa ; Henn:2013nsa . New
algorithms for finding such canonical bases, improving upon those in
Lee:2014ioa ; Argeri:2014qva ; Henn:2014qga ; Gituliar:2016vfa ;
Gituliar:2017vzm ; Prausa:2017ltv ; Chicherin:2018old ; Dlapa:2020cwj ;
Chen:2020uyk ; Dlapa:2021qsl , and the study of the special functions that
solve the resulting differential equations Brown:2009ta ; Brown:2009rc ;
BrownArxiv ; Panzer:2014caa ; Broedel:2017kkb ; Bourjaily:2018yfy  will be
important for further progress, see also the dedicated Snowmass White Paper
integrationWhitePaper2022 . For specific practical applications it will also
be profitable to explore other methods for evaluating master integrals, such
as Mellin-Barnes representations Smirnov:2004ym , direct Feynman parameter
integration Panzer:2014caa ; Bourjaily:2018aeq , and difference equations from
dimensional recurrence Tarasov:1996br ; Lee:2009dh ; Lee:2012te . These
methods apply equally well to conservative calculations and to the calculation
of observables involving outgoing gravitational radiation, such as the total
energy radiated during a scattering event. The collider method of reverse
unitarity Anastasiou:2002yz ; Anastasiou:2002qz ; Anastasiou:2003yy ;
Anastasiou:2015yha  allows systematic evaluation of the phase space integrals
appearing in the KMOC formalism with techniques similar to those used for loop
integrals, as demonstrated in Herrmann:2021tct ; Herrmann:2021lqe .

Higher-Order Structure, Resummation, and Beyond Perturbation Theory. Exact
results in interacting theories, even in the classical regime, are rare.
Patterns and structure exposed by explicit higher perturbative orders may hold
the key to understanding the structure of perturbation theory and eventually
lead to its resummation, as illustrated e.g. by the BDS/ABDK conjectures
Anastasiou:2003kj ; Bern:2005iz  for the resummation of the planar MHV
amplitudes in ${\cal N}=4$ super-Yang-Mills theory. This is an important
theoretical justification, which complements the practical one emphasized
earlier, for further in-depth higher-order exploration. The complete velocity
dependence gives us access to the analytic structure of the Hamiltonian – and
of the observables derived from it – and thus can inform on the structure at
even higher orders, possibly providing sufficient information to resum the
entire perturbation theory.

A proof of principle is the conjectured expression relating scattering
observables of spinning bodies to the eikonal of the corresponding scattering
amplitude Bern:2020buy . While expected from the motion of spinless bodies,
the existence a single function – the eikonal, or the radial action – that
potentially captures all conservative classical observables of such processes
is surprising. Tests of such conjectures and the identification of novel,
bolder ones may exploit special configurations of particles, new perturbative
expansions which access different regions of parameter space, etc.

When present, symmetries are powerful means to extrapolate fixed-order
properties of observables. The double copy hints at a relationship between the
asymptotic symmetry group of Yang-Mills theory and of general relativity.
Since the expectation value of the field itself can be computed (e.g., using
scattering amplitudes and the KMOC formalism), it may be possible to test this
idea directly. Quantum mechanical deformations of the symmetry group, relating
for example to infrared divergences, should be accessible by studying
appropriate observables and may reveal all-order structures.

Another path to exact results proceeds through enhancements of perturbative
calculations. For instance, the EOB approach Buonanno:1998gg ; Buonanno:2000ef
provides a framework that can incorporate not only perturbative results from
PN, PM and gravitational self-force but also exact results available in the
probe limit. In general, it is critical that information from distinct regions
of parameter space and from different physical phenomena be included in the
same formalism. The amount of physics contained in all-order results is
remarkable and includes, among others, possible analytic access to horizon
formation in the binary coalescence.

Yet another systematic strategy is the gravitational self-force approach
Mino:1996nk ; Quinn:1996am ; Barack:2018yvs , in which one expands in the mass
ratio of the binary. It makes contact with QFT in curved space, which we will
return to shortly. It is a challenge for the future to develop the tools
necessary to carry out such calculations with QFT methods.

Apart from having been instrumental for driving our understanding of gravity,
iconic classical solutions of general relativity such as the Schwarzschild and
Kerr solutions also are at the foundation of the EOB theory and of the self-
force approach. Scattering amplitudes and the KMOC formalism applied to three-
point amplitudes Monteiro:2020plf  can access these via analytic continuation
either to complex momenta or to spacetimes with an exotic signature. This led,
for example, to a new understanding of the Newman-Janis shift Arkani-
Hamed:2019ymq  which relates the Kerr solution in a very simple way to the
Schwarzschild solution. Extending this observation beyond leading order should
establish a connection between scattering amplitudes and higher-order terms in
the self-force expansion to all orders in Newton’s constant, beyond the “good
mass polynomiality” already manifest in perturbative scattering amplitudes and
associated observables. It could also be interesting to examine this
connection in higher dimensions where the space of known classical solutions
if far richer than in four dimensions.

Many-Body Dynamics from Amplitudes. While there is a large effort directed
towards understanding compact binaries, systems of three or more massive
bodies also occur in our Universe and may source GWs detectable in the future.
These systems may probe scenarios of sequential and hierarchical mergers
Naoz:2012bx ; Lim:2020cvm ; Martinez:2020lzt ; Fragione:2020gly , and have
qualitatively different dynamics Boekholt:2021ifv , including chaotic dynamics
with interesting effects at relatively high velocities Zwart:2021qxe . The
current state-of-the-art Hamiltonian is the 2PN order SCHAFER1987336 ;
Chu:2008xm ; Will:2013cza  obtained through general-relativity methods. A
formal 2PM expression was obtained in Loebbert:2020aos  using the worldline
formalism and can be used to generate higher-order PN terms of the form
$G^{2}v^{2n}$.

Scattering amplitude methods used for two-body dynamics can also be applied
for $N$-body dynamics, and exploring this may uncover novel features of
gravitational interactions. This will require extending tools for constructing
$2N$-point matter amplitudes, identifying the classical limit, evaluation of
integrals, etc. Further assumptions about the hierarchy of scales in $N$-body
dynamics establish connections with multi-Regge kinematics in particle
physics.

Ringdown from Amplitudes. The focus of the recent effort to understand and
explore binary coalescence with particle physics methods has been the inspiral
phase of the process, which can be treated perturbatively around flat space.
The GW signal contains information about the masses and spins of the
constituents, the shape of the orbit (eccentricity and mean anomaly), and in
late stages also about their tidal deformability. For binary black holes, the
final phase of a merger, known as the ringdown, appears as a superposition of
quasi-normal modes of the merged remnant Vishveshwara:1970cc ; Press:1971wr .
The frequency and decay time of each mode are determined in general relativity
by the remnant’s mass and spin, while their relative amplitude contains
information about the properties of the binary’s components and geometry,
including the spin orientations (see, e.g., Kamaretsos:2011um ; Hughes:2019zmt
; Li:2021wgz ).

This poses an interesting challenge and an equally interesting opportunity: to
develop field theory methods to describe the emission of gravitons from the
remnant of a binary merger. A naive expectation is that the relevant framework
is QFT in a time-dependent curved space which is perhaps a classical double
copy and that the relevant observables are final-state correlation functions
akin to cosmological correlators. While still in its infancy, see e.g.
Adamo:2017sze ; Adamo:2018mpq ; Adamo:2020qru ; Adamo:2020yzi ; Adamo:2021rfq
; Diwakar:2021juk ; Cheung:2022pdk ; Herderschee:2022ntr ; Arkani-
Hamed:2018kmz ; Baumann:2020dch ; Benincasa:2018ssx  and the Snowmass White
Paper Baumann:2022jpr , the generalization of the amplitudes program to curved
space together with the gauge theory realization of classical solutions of
general relativity Monteiro2014cda ; Luna2015paa ; Luna2016due ; Luna2016hge ;
CarrilloGonzalez:2019gof ; CarrilloGonzalez:2018ejf ; Carrillo-
Gonzalez:2017iyj  may offer a new perspective on the ringdown phase, as well
as, new means to explore analytically the correlation between initial
parameters and intrinsic properties of the remnant, including its tidal
deformability, etc. Clearly, a systematic development of these methods will
have wider applications, establishing connections to other areas of physics,
such as gauge/string duality. Curved space methods will also make contact with
the self-force approach to the two-body problem Mino:1996nk ; Quinn:1996am ;
Barack:2018yvs , which can be interpreted as the back-reacted propagation of a
light massive particle in the curved space generated by a heavy massive
particle, thus providing another path to analytic resummation of the two-body
Hamiltonian or two-body observables. See Adamo:2022rmp  for a related
application of the KMOC formalism.

Probes of General Relativity, Quantum Gravity and New Physics. Quantum gravity
remains one of the biggest mysteries of fundamental physics of this century.
While indirect evidence for it abound, its precise formulation is elusive, and
there is currently no observed deviations from predictions of classical
general relativity in GW observations LIGOScientific:2021sio . Future more
precise observations with also a much larger number of events will provide a
means for more efficient probes of physics beyond general relativity
Barack:2018yly , or even of quantum gravity and horizon-scale physics (e.g.,
Mathur:2005zp ; Mathur:2008nj ; Almheiri:2012rt ; Almheiri:2013hfa ); see also
the Snowmass White Paper Berti:2022wzk  on fundamental physics and beyond the
standard model. More accurate waveforms, covering long-time evolution, as well
as various possibilities for new physics, is a step towards this goal, as are
the identification of new observables tailored for such physics. A
complementary approach is the development of model-independent studies,
covering both classical extensions of general relativity and its quantization.
From an EFT perspective there is of course a certain overlap between them, so
identifying methods to lift this degeneracy will be important.

Further developments in amplitude methods, e.g. extending the work in
Carrasco:2021ptp ; Bonnefoy:2021qgu ; Chi:2021mio ; Endlich:2017tqa  to
systematically include and classify all possible higher-dimension/higher-
derivative operators that can appear as counterterms in gravitational
theories, including those discussed using standard GR methods in e.g.
deRham:2020ejn ; Cano:2020cao ; Cano:2021myl , will aid both model-specific
and model-independent studies of GW signals, both with regard to the inspiral
and the ringdown phase, see also Snowmass White Paper deRham:2022hpx  on UV
constraints and IR physics.

##  8 Coda

In its short history, the program to apply particle physics methods to
gravitational physics in general and gravitational-wave physics in particular
has achieved remarkable success by uncovering rich theoretical structures,
developing powerful new tools, and producing predictions for future precision
gravitational-wave detectors. Close synergy with the EFT and gravitational-
wave communities has been essential, and provided crucial guidance regarding
the theoretical needs for waveform modeling and the necessary tools to achieve
them.

This white paper provides a snapshot of the current status of this vibrant
field and of its progress to date. There are many open questions and avenues
to explore, ranging from the very formal to the very practical. Some can be
addressed with existing methods whose full potential is not yet realized,
while others require novel ideas that may even open up completely unexpected
directions that will further our knowledge of gravitational interactions,
binary dynamics, black-hole physics and perhaps even quantum field theory.

## Acknowledgements

D.O.C. is supported by the U.K. Science and Technology Facilities Council
(STFC) grant ST/P000630/1. R.R. is supported by the US Department of Energy
under Grant No. DE-SC00019066. M.P.S. is grateful to the Mani L. Bhaumik
Institute for Theoretical Physics for support. M.Z. is supported by the U.K.
Royal Society through Grant URF\R1\20109.

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