aainstitutetext:  Mani L. Bhaumik Institute for Theoretical Physics,
Department of Physics and Astronomy, University of California Los Angeles, Los
Angeles, CA 90095, USA

# Absorptive Effects and Classical Black Hole Scattering

Callum R. T. Jones<EMAIL_ADDRESS>a Michael S. Ruf
<EMAIL_ADDRESS>
###### Abstract

We describe an approach to incorporating the physical effects of the
absorption of energy by the event horizon of black holes in the scattering
amplitudes based post-Minkowskian, point-particle effective description.
Absorptive dynamics are incorporated in a model-independent way by coupling
the usual point-particle description to an invisible sector of gapless
internal degrees-of-freedom. The leading order dynamics of this sector are
encoded in the low-energy expansion of a spectral density function obtained by
matching an absorption cross section in the ultraviolet description. This
information is then recycled using the scattering amplitudes based Kosower-
Maybee-O’Connell in-in formalism to calculate the leading absorptive
contribution to the impulse and change in rest mass of a Schwarzschild black
hole scattering with a second compact body sourcing a massless scalar,
electromagnetic or gravitational field. The results obtained are in complete
agreement with previous worldline Schwinger-Keldysh calculations and provide
an alternative on-shell scattering amplitudes approach to incorporating
horizon absorption effects in the gravitational two-body problem.

##  1 Introduction

The detection of gravitational waves by the LIGO and Virgo collaborations has
inaugurated a new era of gravitational wave astrophysics
LIGOScientific:2016aoc ; LIGOScientific:2017vwq . Beyond the celebrated
initial discovery, the next generation of ground- and space-based
gravitational wave detectors Punturo:2010zz ; 2017arXiv170200786A ;
Reitze:2019iox  are expected to increase sensitivity by two orders-of-
magnitude, beginning an era of precision gravitational wave science. The
success of this experimental program will require commensurate advances in
theoretical waveform predictions. A powerful approach to generating these
predictions is given by the effective one-body (EOB) resummation
Buonanno:1998gg ; Buonanno:2000ef , taking as input information from a variety
of different physical regimes including non-perturbative numerical relativity
simulation Pretorius:2005gq ; baumgarte_shapiro_2010 ; Damour:2014afa ;
Hopper:2022rwo , self-force expansion Mino:1996nk ; Quinn:1996am ;
Poisson:2011nh ; Barack:2018yvs  and perturbative weak-field calculations
based on the post-Newtonian (PN) Einstein:1938yz ; Ohta:1973je  and post-
Minkowskian (PM) expansions Bertotti:1956pxu ; Kerr:1959zlt ; Bertotti:1960wuq
; Westpfahl:1979gu ; Portilla:1980uz ; Bel:1981be .

Recent years have seen tremendous advances in these weak-field perturbative
approaches. In particular, new and powerful calculational methods based on
relativistic scattering amplitudes Cheung:2018wkq ; Bern:2019nnu ;
Bern:2019crd ; DiVecchia:2020ymx ; DiVecchia:2021bdo ; Bjerrum-Bohr:2021din ;
Brandhuber:2021eyq ; Herrmann:2021lqe ; DiVecchia:2021ndb ;
Heissenberg:2021tzo ; Bern:2021dqo ; Bern:2021yeh ; Jones:2022aji  and
worldline effective field theories (EFT) Kalin:2020mvi ; Kalin:2020fhe ;
Dlapa:2021npj ; Mogull:2020sak ; Jakobsen:2021smu ; Jakobsen:2022psy ;
Loebbert:2020aos , have allowed for rapid progress in understanding the PM
scattering regime of the gravitational two-body problem. These approaches
share the common strategy of initially framing the problem in quantum
mechanical language, and then using the eventual classical or
$\hbar\rightarrow 0$ limit as an additional expansion leading to vast
simplifications in the required calculations. This quantum-first perspective
leads to a natural synthesis with techniques developed in modern high-energy
physics including the systematic use of EFTs Goldberger:2004jt ; Neill:2013wsa
; Cheung:2018wkq ; Damgaard:2019lfh ; Goldberger:2022rqf , unitarity-based
methods Bern:1994zx ; Bern:1994cg ; Bern:1995db ; Bern:1997sc ; Britto:2004nc
, double-copy Kawai:1985xq ; Bern:2008qj ; Bern:2010ue ; Bern:2019prr  and
methods for advanced multi-loop integration Chetyrkin:1981qh ; Laporta:2000dsw
; Kotikov:1990kg ; Bern:1993kr ; Remiddi:1997ny ; Gehrmann:1999as . Further
details can be found in the review Buonanno:2022pgc .

Physically, the PM scattering regime is an expansion based on a separation of
scales between the intrinsic size of the compact bodies $R$
($=2G_{\mathrm{N}}M$ for a Schwarzschild black hole) and the impact parameter
$b$. In the limit $R\ll b$ the problem admits an effective description as the
scattering of point-particles interacting by exchanging gravitons (and
possibly other massless force-mediating particles). At low orders in this
long-distance expansion it is sufficient to calculate scattering observables
from a model in which the compact bodies are mathematically represented as
structureless elementary particles minimally coupled to the mediator fields.
At higher-orders, finite size effects will begin to contribute; some of these
effects can be captured by modifying the effective description with non-
minimal couplings (permanent multipole moments or tidal Love numbers)
Cheung:2020sdj ; Bern:2020uwk , but some cannot. As emphasized long-ago
Goldberger:2005cd , even in the long-distance regime, classical macroscopic
bodies, including black holes and neutron stars, differ qualitatively from
elementary particles due to the existence of gapless internal degrees-of-
freedom PhysRev.83.34 ; Goldberger:2022rqf .

As a motivating illustration, consider a lump of some material modelled as an
Einstein solid, where the internal degrees-of-freedom are approximated as a
large number of independent quantum harmonic oscillators
https://doi.org/10.1002/andp.19063270110 . The quantum mechanical spectrum of
this model is discrete with level spacing of
$\mathcal{O}\mathopen{}\mathclose{{}\left(\hbar}\right)$. In the classical
limit $\hbar\rightarrow 0$, this model has an effectively continuous spectrum
of excited states extending all the way to the ground state threshold without
a gap. Even though a precise microscopic description of a Schwarzschild black
hole is not known, we expect that it shares these qualitative features; from
explicit black hole perturbation theory calculations it is known that a
Schwarzschild black hole can absorb radiation of arbitrarily low frequency
Starobinskil:1974nkd ; Page:1976df , necessitating the existence of
(classically) gapless excited states.

In a two-body scattering event, no matter how small the momentum transfer
between the bodies, there will necessarily exist near-threshold excited states
that can go on-shell and therefore cannot be integrated out of the point-
particle effective theory. Since we may have neither direct experimental
access to these states, nor a precise microscopic theoretical description, our
approach to incorporating their physical effects is to regard them as
constituting an invisible sector into which energy and other quantum numbers
may be absorbed. As our choice of language suggests, this problem shares many
formal similarities with model-independent approaches to describing
interactions between the Standard Model and a hidden sector Contino:2020tix ;
Arina:2021nqi ; Klose:2022vro . From this perspective, the problem of
scattering black holes and other compact macroscopic bodies is to be treated
as the evolution of an open quantum system.

In this paper we develop a scattering amplitudes based formalism for
incorporating the physical effects of these near-threshold excited states,
closely modelled on the worldline formalism described in Goldberger:2005cd ;
Porto:2007qi ; Goldberger:2019sya ; Goldberger:2020wbx ; Goldberger:2020fot .
At leading PM order the effects of absorption are parametrized by the low-
energy expansion of a spectral density function obtained by explicitly
matching with an absorption cross section calculated in black hole
perturbation theory Starobinskil:1974nkd ; Page:1976df . We then, in an EFT
sense, recycle this information to calculate distinct low-energy observables.
In particular, in this paper we calculate the leading PM absorptive
contributions to the impulse of a Schwarzschild black hole scattering with a
second compact body sourcing a massless scalar (65), electromagnetic (67) or
gravitational field (69). The contribution to the impulse from graviton
absorption agrees perfectly with previous worldline calculations
Goldberger:2020wbx ; as far as we are aware the expressions for the impulse
due to scalar and photon absorption are new.

Previous theoretical studies of horizon absorption effects include
Poisson:1994yf ; Alvi:2001mx ; Poisson:2004cw ; Hughes:2001jr ; Nagar:2011aa ;
Bernuzzi:2012ku ; Taracchini:2013wfa ; Goldberger:2005cd ; Porto:2007qi ;
Endlich:2016jgc ; Goldberger:2019sya ; Goldberger:2020wbx ; Goldberger:2020fot
; Saketh:2022xjb ; Aoude:2023fdm ; Tagoshi:1997jy ; Chatziioannou:2016kem ;
Chia:2020yla ; Saketh:2023bul . The problem of calculating the absorptive
contribution to the impulse in two-body scattering was approached in
Goldberger:2020wbx  using off-shell worldline based methods, where in-in
observables are naturally calculated using the Schwinger-Keldysh or closed-
time-path formalism Schwinger:1960qe ; Keldysh:1964ud ; Galley:2009px . In
this paper we approach this problem from an on-shell scattering amplitudes
perspective, where the natural in-in formalism is given by Kosower, Maybee and
O’Connell (KMOC) Kosower:2018adc . As emphasized recently Damgaard:2023vnx
these formalisms are closely related, but organize the calculation in very
different ways. Even though we are calculating an in-in observable, KMOC takes
ordinary in-out scattering amplitudes as building blocks, allowing us to
retain some of the power and simplicity of modern amplitudes methods and has a
natural synthesis with the previously mentioned scattering amplitudes based
approaches to the gravitational two-body problem.

This paper is organized as follows. In Section 2 we introduce a general
framework for incorporating horizon absorption by coupling the point-particle
effective description to an invisible sector. In Section 2.2 we demonstrate
that the leading-order low-energy dynamics of this sector are encoded in the
expansion of a spectral density function and obtain this information by
explicitly matching with an absorption cross section. The absorptive
contribution to the impulse of a black hole during a two-body scattering event
is then obtained in Section 3. We explain how absorption of energy by the
invisible sector degrees-of-freedom can be naturally incorporated in the KMOC
formalism in Section 3.1. We then describe in Sections 3.2 and 3.3, how the
“heavy” and “light” invisible sector degrees of freedom manifest as Love
number type contact contributions and absorptive effects respectively. In 3.4
the necessary box diagrams are constructed using unitarity-based methods and
the resulting impulses and mass-shifts calculated for the absorption of
scalars, photons and gravitons. Finally, in Section 3.5 we generalize the
discussion to generic compact bodies including neutron stars, parametrizing
the leading-order impulse and mass-shifts in terms of dissipation numbers.

##  2 Classical Black Holes as Open Quantum Systems

###  2.1 Visible and Invisible Degrees-of-Freedom

We will consider the scattering of a generic compact body $\phi_{1}$ with a
Schwarzschild black hole $\phi_{2}$ with masses $m_{1}$ and $m_{2}$
respectively. In addition to the gravitational field, body-1 may source an
electromagnetic field $A_{\mu}$ with electric charge $Q_{e}$ and a massless
scalar field $\psi$ with scalar charge $Q_{s}$. These degrees-of-freedom
constitute what we will call the visible sector, and are described by an
action

$\displaystyle S_{\text{vis}}$
$\displaystyle=\int\text{d}^{4}x\sqrt{-g}\mathopen{}\mathclose{{}\left[-\frac{2}{\kappa^{2}}R-\frac{1}{4}F_{\mu\nu}^{2}+\frac{1}{2}\mathopen{}\mathclose{{}\left(\nabla_{\mu}\psi}\right)^{2}+|D_{\mu}\phi_{1}|^{2}-m_{1}^{2}|\phi_{1}|^{2}}\right.$
$\displaystyle\hskip
76.82243pt\mathopen{}\mathclose{{}\left.+\frac{1}{2}\mathopen{}\mathclose{{}\left(\nabla_{\mu}\phi_{2}}\right)^{2}-\frac{1}{2}m_{2}^{2}\phi_{2}^{2}+Q_{s}\psi|\phi_{1}|^{2}}\right]+S_{\text{HD}}+S_{\text{GF}},$
(1)

where $\kappa\coloneqq\sqrt{32\pi G_{\mathrm{N}}}$,
$D_{\mu}\coloneqq\nabla_{\mu}+\mathrm{i}Q_{e}A_{\mu}$ and
$F_{\mu\nu}\coloneqq\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$; $S_{\text{GF}}$
are gauge fixing terms and $S_{\text{HD}}$ denotes non-minimal (tidal Love
number) interactions Cheung:2020sdj ; Bern:2020uwk . As discussed in Section
1, at low orders in the PM expansion this action captures the relevant physics
of scattering compact bodies at long distances. At some order in
$G_{\mathrm{N}}$ (to be determined), the physical effects of horizon
absorption become important; physical transitions between the black hole
ground state and an excited state become possible and these cannot be
described by an action of the form (2.1).

The Hilbert space in this problem takes a factored form
$\mathcal{H}_{\text{vis}}\otimes\mathcal{H}_{\text{inv}}$, corresponding to
visible states (ground state black holes and radiation) and invisible states
(excited state black holes). There are many different approaches to describing
the dynamics of this system, we could explicitly trace over
$\mathcal{H}_{\text{inv}}$ and describe the scattering problem as the non-
unitary time evolution of a reduced density matrix in
$\mathcal{H}_{\text{vis}}$. To retain some of the simplicity of unitary time
evolution we instead choose to work on the full Hilbert space, calculating in-
in observables where the initial state is in the visible sector. The invisible
states, that we will collectively denote as $X$, appear only as internal
states to be summed over.

Our formalism for incorporating these invisible sector states is modelled on
the worldline formalism described in Goldberger:2005cd ; Porto:2007qi ;
Goldberger:2019sya ; Goldberger:2020wbx ; Goldberger:2020fot . The complete
action for the system consists of the the visible sector (2.1), the unknown
(possibly strongly coupled) self-interactions of the invisible $X$-states
denoted $S_{\text{inv}}$ and portal couplings between the visible and
invisible sectors. For the latter, the $X$-states are encoded in the form of
abstract, composite, local operators $\mathcal{O}_{i}(x)$; for which we assume
the usual properties of Poincaré invariance, gauge/diffeomorphism invariance
and locality. Without loss of generality we exclude quadratic portal couplings
e.g. $\phi_{2}\mathcal{O}$; such interactions can always be removed by
redefining the visible sector field basis. To leading order in
$G_{\mathrm{N}}$ the portal couplings are given by

$\displaystyle S_{\text{portal}}$
$\displaystyle=\int\text{d}^{4}x\sqrt{-g}\mathopen{}\mathclose{{}\left[\kappa\phi_{2}\psi\mathcal{O}_{0}+\kappa\phi_{2}F_{\mu\nu}\mathcal{O}_{1}^{\mu\nu}+\phi_{2}C_{\mu\nu\rho\sigma}\mathcal{O}_{2}^{\mu\nu\rho\sigma}+\cdots}\right],$
(2)

where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor. The idea is then to calculate
observables by evaluating the path integral for the visible sector, treating
the portal couplings perturbatively in $G_{\mathrm{N}}$ and leaving the
invisible sector path integral unevaluated. As an illustrative example,
consider the calculation of the off-shell Green’s function

$\langle\phi_{2}(x_{1})\psi(x_{2})\phi_{2}(x_{1}^{\prime})\psi(x_{2}^{\prime})\rangle=\int[\mathcal{D}\phi_{2}][\mathcal{D}\psi][\mathcal{D}X]\phi_{2}(x_{1})\psi(x_{2})\phi_{2}(x_{1}^{\prime})\psi(x_{2}^{\prime})e^{\mathrm{i}S_{\text{vis}}+\mathrm{i}S_{\text{inv}}+\mathrm{i}S_{\text{portal}}}.$
(3)

Diagrammatically the leading connected contribution from the $X$-states takes
the form:

$x_{2}$$x_{1}$$x_{2}^{\prime}$$x_{1}^{\prime}$$+$$\mathopen{}\mathclose{{}\left(x_{2}\;\;\leftrightarrow\;\;x_{2}^{\prime}}\right),$

where the internal doubled line corresponds to the (non-perturbative)
invisible sector 2-point function

$\langle\mathcal{O}_{0}(x)\mathcal{O}_{0}(0)\rangle_{X}\coloneqq\int[\mathcal{D}X]\mathcal{O}_{0}(x)\mathcal{O}_{0}(0)e^{\mathrm{i}S_{\text{inv}}}.$
(4)

It is natural to rewrite this in Källén-Lehmann (KL) form

$\langle\mathcal{O}_{0}(x)\mathcal{O}_{0}(0)\rangle_{X}=\int\hat{\text{d}}^{D}k\;e^{\mathrm{i}k\cdot
x}\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\frac{\mathrm{i}\rho_{0}(\mu^{2})}{k^{2}-\mu^{2}+\mathrm{i}0},$
(5)

where the unknown invisible sector physics is contained in the spectral
density $\rho_{0}(\mu^{2})$. In this form, momentum space calculations
involving internal $X$-states can be performed using a small modification of
standard Feynman rules, summarized in Appendix D, incorporating weighted
integration over the spectral parameter $\mu^{2}$.

###  2.2 UV Matching: Absorption Cross Section

To determine the spectral density we match an absorption cross section
calculated in both the UV (full General Relativity) and the IR (the point-
particle effective description). To leading order in $\omega\rightarrow 0$,
where $\omega$ is the energy of the absorbed radiation, the necessary
Schwarzschild black hole absorption cross sections were calculated long ago
Starobinskil:1974nkd ; Page:1976df

$\sigma_{\text{abs}}\mathopen{}\mathclose{{}\left(\omega}\right)\sim\begin{cases}16\pi
G_{\mathrm{N}}^{2}m_{2}^{2}&\text{for scalars,}\\\
\frac{64\pi}{3}G_{\mathrm{N}}^{4}m_{2}^{4}\omega^{2}&\text{for photons,}\\\
\frac{256\pi}{45}G_{\mathrm{N}}^{6}m_{2}^{6}\omega^{4}&\text{for
gravitons.}\end{cases}$ (6)

For the photon and graviton cases the above cross sections correspond to an
incoming polarized state with helicity $\pm 1$ and $\pm 2$ respectively, the
explicit expression is independent of the polarization state. In the point-
particle effective description, the corresponding absorption cross sections
are calculated perturbatively; the general procedure is illustrated first for
the absorption of a massless scalar, and then the particular complications of
photon and graviton absorption described separately.

#### Scalar Absorption Cross Section

For scalar absorption we first calculate the Compton amplitude

$\mathcal{M}\mathopen{}\mathclose{{}\left(\phi_{2}\mathopen{}\mathclose{{}\left(-p_{2}}\right)\psi\mathopen{}\mathclose{{}\left(-k_{1}}\right)\rightarrow\phi_{2}\mathopen{}\mathclose{{}\left(p_{3}}\right)\psi\mathopen{}\mathclose{{}\left(k_{4}}\right)}\right),$
(7)

where the external momenta are lablelled in the all-outgoing convention.
Similar to the off-shell Green’s function (3) the contribution of the visible
sector states is calculated perturbatively, while the invisible $X$-states
contribute through the KL “propagator” (5). In the forward limit kinematics

$-{p}_{2}^{\mu}=p_{3}^{\mu}=\mathopen{}\mathclose{{}\left(m_{2},0,0,0}\right),\hskip
28.45274pt-{k}_{1}^{\mu}=k_{4}^{\mu}=\mathopen{}\mathclose{{}\left(\omega,0,0,\omega}\right),$
(8)

the optical theorem relates the imaginary part of the Compton amplitude to the
total cross section

$\sigma_{\text{total}}\mathopen{}\mathclose{{}\left(\omega}\right)=\frac{\text{Im}\mathcal{M}^{\text{forward}}\mathopen{}\mathclose{{}\left(\omega}\right)}{2m_{2}\omega}.$
(9)

At leading order the absorptive part of the cross section can be disentangled
from the elastic part by considering the contribution of the diagram

$k_{1}$$p_{2}$$p_{4}$$k_{3}$,

in the forward kinematics (8) this corresponds to

$\mathcal{M}^{(\psi)}\biggr{|}_{\text{forward}}=-\kappa^{2}\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\frac{\rho_{0}(\mu^{2})}{m_{2}^{2}+2m_{2}\omega-\mu^{2}+\mathrm{i}0}.$
(10)

Using the distributional identity

$\text{Im}\mathopen{}\mathclose{{}\left(\frac{1}{x+\mathrm{i}0}}\right)=-\pi\delta(x),$
(11)

we obtain the imaginary part

$\text{Im}\mathcal{M}^{(\psi)}\biggr{|}_{\text{forward}}=\pi\kappa^{2}\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\rho_{0}(\mu^{2})\delta\mathopen{}\mathclose{{}\left(m_{2}^{2}+2m_{2}\omega-\mu^{2}}\right)=\pi\kappa^{2}\rho_{0}\mathopen{}\mathclose{{}\left(m_{2}^{2}+2m_{2}\omega}\right),$
(12)

and therefore the absorption cross section

$\sigma_{\text{abs}}^{(\psi)}\mathopen{}\mathclose{{}\left(\omega}\right)=\frac{\pi\kappa^{2}\rho_{0}\mathopen{}\mathclose{{}\left(m_{2}^{2}+2m_{2}\omega}\right)}{2m_{2}\omega}.$
(13)

Matching this with the UV cross section (6) gives the near-threshold
asymptotic expansion of the spectral density

$\rho_{0}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\sim\frac{G_{\mathrm{N}}m_{2}^{2}}{2\pi}\mathopen{}\mathclose{{}\left(\mu^{2}-m_{2}^{2}}\right),\hskip
14.22636pt\text{as}\hskip 14.22636pt\mu^{2}\rightarrow m_{2}^{2}.$ (14)

#### Photon Absorption Cross Section

For photon absorption, the composite operator $\mathcal{O}_{1}^{\mu\nu}$ that
appears in (2) transforms non-trivially under the Lorentz group and therefore
has a more complicated KL form

$\langle\mathcal{O}_{1}^{\mu\nu}(x)\mathcal{O}_{1}^{\rho\sigma}(0)\rangle_{X}=\int\hat{\text{d}}^{D}k\;e^{\mathrm{i}k\cdot
x}\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\frac{\mathrm{i}\Pi_{1}^{\mu\nu\rho\sigma}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right)}{k^{2}-\mu^{2}+\mathrm{i}0}.$
(15)

In general the projector $\Pi_{1}$ is constructed from the available Lorentz
tensors $k^{\mu}$, $\eta^{\mu\nu}$ and $\epsilon^{\mu\nu\rho\sigma}$ and
constrained by the symmetry of the operators. Without loss of generality we
can assume that $\mathcal{O}_{1}^{\mu\nu}=-\mathcal{O}_{1}^{\nu\mu}$ and we
will impose that the 2-point function is parity invariant. Together with the
exchange symmetry

$\Pi_{1}^{\mu\nu\rho\sigma}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right)=\Pi_{1}^{\rho\sigma\mu\nu}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right),$
(16)

we find there are two tensor structures compatible with these assumptions

$\displaystyle\Pi_{1}^{\mu\nu\rho\sigma}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right)$
$\displaystyle=\rho_{1}^{(1)}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\mathopen{}\mathclose{{}\left[\eta^{\mu\rho}\eta^{\nu\sigma}-\eta^{\mu\sigma}\eta^{\nu\rho}}\right]$
$\displaystyle\hskip
14.22636pt+\rho_{1}^{(2)}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\mathopen{}\mathclose{{}\left[\frac{k^{\nu}k^{\rho}\eta^{\mu\sigma}}{k^{2}}+\frac{k^{\mu}k^{\sigma}\eta^{\nu\rho}}{k^{2}}-\frac{k^{\mu}k^{\rho}\eta^{\nu\sigma}}{k^{2}}-\frac{k^{\nu}k^{\sigma}\eta^{\mu\rho}}{k^{2}}}\right].$
(17)

As discussed further in Section 3.5 for a generic macroscopic body capable of
absorbing electromagnetic radiation (e.g. a neutron star or a dielectric
sphere with complex permittivity) there are two independent spectral functions
that must be matched to the UV description. As described in Endlich:2016jgc ;
Aoude:2023fdm  the independent contributions can be obtained by separately
matching contributions to the absorption cross section from incoming spinning
partial waves. For Schwarzschild black holes in $d=4$ this is actually
unnecessary due to the presence of a hidden symmetry, the electromagnetic
self-duality of the Einstein-Maxwell equations Deser:1976iy . Since this is a
symmetry of the UV model we impose that it is also a symmetry of the effective
point-particle description. Imposing duality invariance of the 2-point
function

$\Pi_{1}^{\mu\nu\rho\sigma}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right)=\frac{1}{4}{\epsilon^{\mu\nu}}_{\alpha\beta}{\epsilon^{\rho\sigma}}_{\gamma\delta}\Pi_{1}^{\alpha\beta\gamma\delta}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right),$
(18)

gives a relation between the spectral functions $\rho_{1}^{(1)}$ and
$\rho_{1}^{(2)}$. The unique duality invariant tensor structure is found to be

$\Pi_{1}^{\mu\nu\rho\sigma}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right)=\rho_{1}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\;\hat{\Pi}_{1}^{\mu\nu\rho\sigma}\mathopen{}\mathclose{{}\left(k}\right),$
(19)

where

$\hat{\Pi}_{1}^{\mu\nu\rho\sigma}\mathopen{}\mathclose{{}\left(k}\right)\coloneqq\frac{1}{2}\mathopen{}\mathclose{{}\left(\eta^{\mu\rho}\eta^{\nu\sigma}-\eta^{\mu\sigma}\eta^{\nu\rho}}\right)+\frac{1}{k^{2}}\mathopen{}\mathclose{{}\left(k^{\nu}k^{\rho}\eta^{\mu\sigma}+k^{\mu}k^{\sigma}\eta^{\nu\rho}-k^{\mu}k^{\rho}\eta^{\nu\sigma}-k^{\nu}k^{\sigma}\eta^{\mu\rho}}\right).$
(20)

In the context of modern unitarity-based methods Bern:1994zx ; Bern:1994cg ;
Bern:1995db ; Bern:1997sc ; Britto:2004nc , the natural object to consider is
not the off-shell 2-point function, but rather the on-shell Compton amplitude

$k_{1}$$p_{2}$$k_{4}$$p_{3}$$\pm 1$$\pm 1$
$\displaystyle\mathcal{M}^{(\gamma)}_{+-}$
$\displaystyle=-\frac{4\kappa^{2}}{m_{2}^{2}}[1|p_{2}|4\rangle^{2}\int_{m_{2}^{2}}^{\infty}\mathrm{d}\mu^{2}\frac{\rho_{1}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)}{(k_{1}+p_{2})^{2}-\mu^{2}+\mathrm{i}0}\;\;\;+\;\;\;\mathopen{}\mathclose{{}\left(2\leftrightarrow
3}\right),$ $\displaystyle\mathcal{M}^{(\gamma)}_{++}$
$\displaystyle=\mathcal{M}^{(\gamma)}_{--}=0.$ (21)

From this on-shell perspective, we see that the somewhat abstruse
electromagnetic self-duality constraint (18) manifests as the vanishing of
helicity violating Compton amplitudes Rosly:2002jt ; Agullo:2016lkj ;
Novotny:2018iph ; Agullo:2018nfv ; Agullo:2018iya . We then proceed as in the
scalar case above, calculating the imaginary part of the forward Compton
amplitude leading to the absorption cross section

$\sigma^{(\gamma)}_{\text{abs}}\mathopen{}\mathclose{{}\left(\omega}\right)=\frac{2\pi\kappa^{2}\omega}{m_{2}}\rho_{1}\mathopen{}\mathclose{{}\left(m_{2}^{2}+2m_{2}\omega}\right).$
(22)

Matching this with the UV cross section (6) gives the near-threshold spectral
density

$\rho_{1}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\sim\frac{G_{\mathrm{N}}^{3}m_{2}^{4}}{6\pi}\mathopen{}\mathclose{{}\left(\mu^{2}-m_{2}^{2}}\right),\hskip
14.22636pt\text{as}\hskip 14.22636pt\mu^{2}\rightarrow m_{2}^{2}.$ (23)

#### Graviton Absorption Cross Section

For graviton absorption, the analysis of the relevant 2-point function

$\langle\mathcal{O}_{2}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(x)\mathcal{O}_{2}^{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}(0)\rangle_{X}=\int\hat{\text{d}}^{D}k\;e^{\mathrm{i}k\cdot
x}\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\frac{\mathrm{i}\Pi_{2}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right)}{k^{2}-\mu^{2}+\mathrm{i}0}.$
(24)

is somewhat more involved due to the proliferation of Lorentz indices. Without
loss of generality we can assume that this operator has the same symmetry and
tracelessness properties as the Weyl tensor

$\mathcal{O}_{2}^{\mu\nu\rho\sigma}=-\mathcal{O}_{2}^{\nu\mu\rho\sigma},\hskip
14.22636pt\mathcal{O}_{2}^{\mu\nu\rho\sigma}=\mathcal{O}_{2}^{\rho\sigma\mu\nu},\hskip
14.22636pt\mathcal{O}_{2}^{\mu\nu\rho\sigma}+\mathcal{O}_{2}^{\mu\rho\sigma\nu}+\mathcal{O}_{2}^{\mu\sigma\nu\rho}=0,\hskip
14.22636pt\eta_{\mu\rho}\mathcal{O}_{2}^{\mu\nu\rho\sigma}=0.$ (25)

In addition, as for photon absorption, the 2-point function is further
constrained by a hidden gravitational self-duality symmetry. The symmetry in
this case is the duality invariance of the linearized Einstein equations
Henneaux:2004jw  on a Petrov type-D background Porto:2007qi . The implications
of this symmetry for black hole perturbation theory, in particular the quasi-
normal mode isospectrality of axial and polar perturbations of Schwarzschild
black holes was first pointed out by Chandrasekhar Chandrasekhar:1975nkd .
Following Goldberger:2005cd  we impose self-duality as a constraint on the
2-point function in the form

$\Pi_{2}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right)=\frac{1}{4}{\epsilon^{\mu_{1}\mu_{2}}}_{\alpha_{1}\alpha_{2}}{\epsilon^{\nu_{1}\nu_{2}}}_{\beta_{1}\beta_{2}}\Pi_{2}^{\alpha_{1}\alpha_{2}\mu_{3}\mu_{4}\beta_{1}\beta_{2}\nu_{3}\nu_{4}}\mathopen{}\mathclose{{}\left(\mu^{2},k}\right).$
(26)

We find that there is a unique solution to the combined constraints (25) and
(26), the somewhat complicated expression is given in (D) and (D). As for the
case of photon absorption above, the natural on-shell object to consider is
the graviton Compton amplitude

$k_{1}$$p_{2}$$k_{4}$$p_{3}$$\pm 2$$\pm 2$
$\displaystyle\mathcal{M}^{(h)}_{+-}$
$\displaystyle=-\frac{64\kappa^{2}}{m_{2}^{4}}[1|p_{2}|4\rangle^{4}\int_{m_{2}^{2}}^{\infty}\mathrm{d}\mu^{2}\frac{\rho_{2}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)}{(k_{1}+p_{2})^{2}-\mu^{2}+\mathrm{i}0}\;\;\;+\;\;\;\mathopen{}\mathclose{{}\left(2\leftrightarrow
3}\right),$ $\displaystyle\mathcal{M}^{(h)}_{++}$
$\displaystyle=\mathcal{M}^{(h)}_{--}=0,$ (27)

where the vanishing of the helicity violating amplitudes is again the on-shell
manifestation of self-duality. From this expression we calculate the
absorption cross section

$\sigma^{(h)}_{\text{abs}}\mathopen{}\mathclose{{}\left(\omega}\right)=32\pi\kappa^{2}m_{2}^{2}\omega^{3}\rho_{2}\mathopen{}\mathclose{{}\left(m_{2}^{2}+2m_{2}\omega}\right),$
(28)

matching this with the UV cross section (6) gives the near-threshold spectral
density

$\rho_{2}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\sim\frac{G_{\mathrm{N}}^{5}m_{2}^{6}}{360\pi}\mathopen{}\mathclose{{}\left(\mu^{2}-m_{2}^{2}}\right),\hskip
14.22636pt\text{as}\hskip 14.22636pt\mu^{2}\rightarrow m_{2}^{2}.$ (29)

##  3 Absorptive Impulse

###  3.1 In-In Scattering Observables and KMOC

Our goal is now, in the spirit of EFT, to recycle the information contained in
the near-threshold expanded spectral functions (14), (23), (29) to calculate
distinct low-energy observables. Specifically, we calculate the leading
contribution to the impulse on a Schwarzschild black hole scattering with a
second compact body sourcing a scalar, electromagnetic or gravitational field.
In this paper we approach this problem from an on-shell scattering amplitudes
perspective, where the natural in-in formalism was formulated by KMOC
Kosower:2018adc .

We begin with an executive summary of the KMOC formalism, see Kosower:2018adc
; Herrmann:2021tct  for a more comprehensive description. In a general quantum
mechanical system admitting an S-matrix describing time evolution from
$t=-\infty$ to $t=+\infty$, the asymptotic change in the expectation value of
an observable $\mathbb{O}$, initially in some state $|\text{in}\rangle$, is
given by the formal expression

$\Delta
O=\langle\text{in}|S^{\dagger}\mathbb{O}S|\text{in}\rangle-\langle\text{in}|\mathbb{O}|\text{in}\rangle.$
(30)

In the present context, $S$ corresponds to the S-matrix on the complete
Hilbert space $\mathcal{H}_{\text{vis}}\otimes\mathcal{H}_{\text{inv}}$ and is
therefore unitary: $S^{\dagger}S=1$. By making the standard definition
$S\coloneqq 1+\mathrm{i}T$ and using the unitarity relation

$T^{\dagger}=T-\mathrm{i}T^{\dagger}T,$ (31)

the above can be rewritten as

$\Delta
O=\mathrm{i}\langle\text{in}|[\mathbb{O},T]|\text{in}\rangle+\langle\text{in}|T^{\dagger}[\mathbb{O},T]|\text{in}\rangle,$
(32)

where the two terms on the right-hand-side are sometimes referred to as the
virtual and real contributions respectively. The different representations of
the KMOC formula have complementary virtues: (30) manifests the fact that if
$\mathbb{O}$ is Hermitian then the change in the expectation value is real-
valued, while (32) manifests the fact that if $\mathbb{O}$ is a generator of a
symmetry and commutes with $T$ then the expectation value is time independent.

We will now apply this formalism to calculate the asymptotic impulse $\Delta
p_{2}^{\mu}$ of a black hole undergoing 2-to-2 scattering with a second
compact body. The in-state is chosen to be

$|\text{in}\rangle\coloneqq\int\hat{\text{d}}^{D}p_{1}\hat{\text{d}}^{D}p_{2}\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(p_{1}^{2}-m_{1}^{2}}\right)\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(p_{2}^{2}-m_{2}^{2}}\right)\psi_{1}(p_{1})\psi_{2}(p_{2})e^{-\mathrm{i}p_{1}\cdot
b_{1}}e^{-\mathrm{i}p_{2}\cdot b_{2}}|p_{1},p_{2}\rangle,$ (33)

where the wave-packets $\psi_{i}$, defined explicitly in Kosower:2018adc , are
chosen to localize particle-$i$ around the classical free particle trajectory
with 4-velocity $u_{i}^{\mu}$ and impact parameter $b_{i}^{\mu}$. In the
$\hbar\rightarrow 0$ limit, the quantum mechanical uncertainty in position and
momentum vanishes and the corresponding asymptotic change in the expectation
value of the operator $\mathbb{P}_{2}^{\mu}$ reduces to the corresponding
classical impulse. To extract the classical contributions efficiently, the
resulting expressions are expanded before loop integration using the method of
regions Beneke:1997zp . In particular we expand to leading non-trivial order
in the soft region Herrmann:2021tct  defined by the scaling

$q^{\mu}\sim\ell^{\mu}\ll u_{i}^{\mu}\sim m_{i}.$ (34)

After simplifying Kosower:2018adc , the KMOC formula for the classical impulse
takes the form

$\Delta
p_{2}^{\mu}=\frac{1}{4m_{1}m_{2}}\int\hat{\text{d}}^{D}q\;\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot
q}\right)\hat{\delta}\mathopen{}\mathclose{{}\left(u_{2}\cdot
q}\right)e^{-\mathrm{i}q\cdot
b}\mathopen{}\mathclose{{}\left[\mathcal{I}_{v}^{\mu}+\mathcal{I}_{r}^{\mu}}\right],$
(35)

where $b^{\mu}\coloneqq b_{1}^{\mu}-b_{2}^{\mu}$ is the relative impact
parameter, without loss of generality defined to satisfy $u_{i}\cdot b=0$. The
functions $\mathcal{I}^{\mu}_{v,r}$ are referred to as the virtual/real KMOC
kernels, corresponding to the respective terms in (32). The virtual kernel is
straightforwardly related to an elastic scattering amplitude

$\mathcal{I}^{\mu}_{v}=\mathrm{i}q^{\mu}\mathcal{M}\mathopen{}\mathclose{{}\left(\phi_{1}(m_{1}u_{1})\phi_{2}(m_{2}u_{2})\rightarrow\phi_{1}(m_{1}u_{1}+q)\phi_{2}(m_{2}u_{2}-q)}\right).$
(36)

The real kernel is given by inserting a complete set of states in the second
term of (32). Restricting to the black hole ground state

$\mathds{1}\supset\int\hat{\mathrm{d}}^{D}r_{1}\hat{\mathrm{d}}^{D}r_{2}\;\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(r_{1}^{2}-m_{1}^{2}}\right)\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(r_{2}^{2}-m_{1}^{2}}\right)|\phi_{1}(r_{1})\phi_{2}(r_{2})\rangle\langle\phi_{1}(r_{1})\phi_{2}(r_{2})|,$
(37)

gives the conservative contribution. In this paper we are interested in the
leading absorptive contribution, this corresponds to an insertion of the
$X$-states

$\mathds{1}\supset\sum_{i}\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\rho_{i}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\int\hat{\mathrm{d}}^{D}r_{1}\hat{\mathrm{d}}^{D}r_{2}\;\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(r_{1}^{2}-m_{1}^{2}}\right)\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(r_{2}^{2}-\mu^{2}}\right)|\phi_{1}(r_{1})X(r_{2})\rangle\langle\phi_{1}(r_{1})X(r_{2})|,$
(38)

where the sum over $i$ includes all internal quantum numbers of the excited
states including spin. The resulting contribution to the real kernel is then
given by

$\displaystyle\mathcal{I}^{\mu}_{r}$
$\displaystyle=\sum_{i}\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\rho_{i}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\int\hat{\text{d}}^{D}\ell\;\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left((m_{1}u_{1}-\ell)^{2}-m_{1}^{2}}\right)\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left((m_{2}u_{2}+\ell)^{2}-\mu^{2}}\right)$
(39) $\displaystyle\hskip
119.50157pt\times\ell^{\mu}\times\mathcal{M}\mathopen{}\mathclose{{}\left(\phi_{1}(m_{1}u_{1})\phi_{2}(m_{2}u_{2})\rightarrow\phi_{1}(m_{1}u_{1}-\ell)X(m_{2}u_{2}+\ell)}\right)$
$\displaystyle\hskip
142.26378pt\times\mathcal{M}^{*}\mathopen{}\mathclose{{}\left(\phi_{1}(m_{1}u_{1}+q)\phi_{2}(m_{2}u_{2}-q)\rightarrow\phi_{1}(m_{1}u_{1}-\ell)X(m_{2}u_{2}+\ell)}\right).$

Since we are interested in the leading-order PM contribution to the absorptive
impulse we can make use of the unitarity relation (31) to replace the
conjugated amplitude $\mathcal{M}^{*}$ with the unconjugated time-reversed
process. This expression can be rewritten in the compact diagrammatic weighted
cut notation defined in Appendix C:

$\mathcal{I}^{\mu}_{r}$$=$$m_{2}u_{2}$$m_{1}u_{1}$$m_{2}u_{2}-q$$\ell$$-\ell^{\mu}$.
(40)

Finally, at leading order there is no mixing between radiation and horizon
absorption; the leading absorptive impulse on the small body can then be
obtained trivially using conservation of momentum

$\mathopen{}\mathclose{{}\left(\Delta
p_{1}^{\mu}}\right)_{\text{abs}}=-\mathopen{}\mathclose{{}\left(\Delta
p_{2}^{\mu}}\right)_{\text{abs}}.$ (41)

###  3.2 Heavy Modes and Love Numbers

By assumption the state created by $\phi_{2}$ is a stable ground state,
meaning $\rho\mathopen{}\mathclose{{}\left(\mu^{2}<m_{2}^{2}}\right)=0$.
Before proceeding it is therefore useful to define a shifted spectral
parameter and spectral density

$\rho\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\coloneqq\tilde{\rho}(s),\hskip
28.45274pts\coloneqq\frac{\mu^{2}-m_{2}^{2}}{2m_{2}},$ (42)

where $\tilde{\rho}(s<0)=0$. In this notation, both the virtual (36) and real
(40) absorptive contributions to the KMOC formula (35) require evaluating a
spectral integral from $0<s<\infty$. Naively this is problematic as it would
require detailed knowledge of the spectral function
$\tilde{\rho}\mathopen{}\mathclose{{}\left(s}\right)$ at all energy scales.
However, there is clearly an important distinction between those $X$-states
that can go on-shell during the scattering event and those that cannot. To
illustrate the distinction in a physically transparent manner, we introduce a
cutoff $\Lambda$, splitting the spectral integral into two contributions:

$\displaystyle\textit{Light modes:}\hskip 28.45274pt0<s<\Lambda\hskip
14.22636pt\Leftrightarrow\hskip 14.22636pts\sim q$ $\displaystyle\textit{Heavy
modes:}\hskip 22.76219pt\Lambda<s<\infty\hskip 9.95845pt\Leftrightarrow\hskip
14.22636pts\gg q,$ (43)

where $q$ is the momentum transfer, assumed to be small in the soft expansion
(34). In the calculation of the virtual kernel (36), the contribution to the
$X$-state propagator from the heavy modes can never be on-shell; expanding to
leading order in the soft region

$\int_{\Lambda}^{\infty}\text{d}s\;\frac{\tilde{\rho}\mathopen{}\mathclose{{}\left(s}\right)}{2m_{2}(u_{2}\cdot\ell)+\ell^{2}-2m_{2}s+\mathrm{i}0}\approx-\frac{1}{2m_{2}}\int_{\Lambda}^{\infty}\text{d}s\;\frac{\tilde{\rho}\mathopen{}\mathclose{{}\left(s}\right)}{s}.$
(44)

That is, we find that for the heavy modes the $X$-state propagator pinches,
generating an effective contact diagram shown in figure 1.

Figure 1: Effective contact vertex obtained from the heavy modes in the KL
representation.

The contact term can be interpreted as a new effective operator contribution
to the visible sector action (2.1). For example, for massless scalar
absorption the leading-order operator takes the form

$S_{\text{vis}}\supset\int\text{d}^{4}x\sqrt{-g}\mathopen{}\mathclose{{}\left[c\psi^{2}\phi_{2}^{2}+...}\right],\hskip
14.22636pt\text{where}\hskip
14.22636ptc\sim\int_{\Lambda}^{\infty}\text{d}s\;\frac{\tilde{\rho}\mathopen{}\mathclose{{}\left(s}\right)}{s}.$
(45)

The Wilson coefficients of such higher-dimension operators are, in this
context, usually referred to as tidal Love numbers Binnington:2009bb ;
Cheung:2020sdj ; Bern:2020uwk . Clearly, we should never have included the
heavy modes to begin with, they are always off-shell and so can be
consistently integrated out of the point-particle effective description. We
will therefore assume that this has been done and the associated Love number
contributions to observables calculated as part of the conservative
contribution Cheung:2020sdj ; Bern:2020uwk .

By contrast, the light modes (gapless modes in the language of Section 1)
cannot be integrated out since they can go on-shell during the scattering
event. To calculate their contribution to the impulse we only need the
spectral function in a small neighbourhood of the threshold value
$s\rightarrow 0$, and as demonstrated in Section 2.2, this information can
obtained by matching the absorption cross section expanded around
$\omega\rightarrow 0$.

In practice the crude (though physically transparent) cutoff $\Lambda$ can be
replaced by an analytic regularization of the spectral integral for the light
modes

$\int_{0}^{\Lambda}\text{d}s\;\tilde{\rho}(s)\hskip
14.22636pt\Rightarrow\hskip
14.22636pt\int_{0}^{\infty}\text{d}s\;s^{\alpha}\tilde{\rho}(s).$ (46)

In this perspective the spectral integral should be taken together with the
dimensionally regularized loop integral as the object to be expanded using the
method of regions. The soft region is then defined by

$s\sim q^{\mu}\sim\ell^{\mu}\ll u_{i}^{\mu}\sim m_{i},$ (47)

and therefore only the leading-order term in the Taylor expansion of
$\tilde{\rho}(s)$ around $s=0$ contributes to leading-order in the soft
expansion. By analogy with dimensional regularization, after expansion in the
soft region the analytically regularized $s$-integration domain is extended to
cover the entire range $0<s<\infty$, evaluated for convergent values of
$\alpha$ and then analytically continued to $\alpha=0$. This form of
regularization has many familiar advantages, scaleless $s$-integrals evaluate
to zero, power-law UV divergences are absent and logarithmic UV divergences
show up as finite order poles at $\alpha=0$.

###  3.3 Light Modes and Absorption

To simplify the calculation of the absorptive impulse, it is useful to first
decompose the kernel into scalar contributions

$\mathcal{I}^{\mu}=q^{\mu}\mathcal{I}_{q}+\check{u}_{1}^{\mu}\mathcal{I}_{\check{u}_{1}}+\check{u}_{2}^{\mu}\mathcal{I}_{\check{u}_{2}},$
(48)

where our kinematic conventions and notation are defined in Appendix A. To
calculate these contributions to the real kernel we decompose the loop
momentum insertion in (40) as111The component perpendicular to the scattering
plane integrates to zero.

$\ell^{\mu}\rightarrow\mathopen{}\mathclose{{}\left(\frac{\ell\cdot
q}{q^{2}}}\right)q^{\mu}+\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)\check{u}_{1}^{\mu}+\mathopen{}\mathclose{{}\left(u_{2}\cdot\ell}\right)\check{u}_{2}^{\mu}.$
(49)

Because of the cut condition for particle-1, $u_{1}\cdot\ell=0$, we trivially
find that

$\mathcal{I}_{r,\check{u}_{1}}=0.$ (50)

The remaining non-vanishing contributions $\mathcal{I}_{r,q}$ and
$\mathcal{I}_{r,\check{u}_{2}}$ will be denoted transverse and longitudinal
respectively. The transverse contribution can be simplified using

$\ell\cdot
q=\frac{1}{2}\mathopen{}\mathclose{{}\left(\ell+q}\right)^{2}-\frac{1}{2}\ell^{2}-\frac{1}{2}q^{2},$
(51)

the first two terms pinch mediator propagators and therefore produce vanishing
scaleless integrals. For the longitudinal contribution we use the cut
condition for particle-2, $u_{2}\cdot\ell=s$, where we use the shifted
spectral parameter $s$ defined in (42). All together we find the effective
decomposition

$\ell^{\mu}\rightarrow\frac{1}{2}q^{\mu}+s\check{u}_{2}^{\mu},$ (52)

or diagrammatically:

$\mathcal{I}_{r,q}$$=$$m_{2}u_{2}$$m_{1}u_{1}$$m_{2}u_{2}-q$$\ell$$-\frac{1}{2}$,
(53)
$\mathcal{I}_{r,\check{u}_{2}}$$=$$m_{2}u_{2}$$m_{1}u_{1}$$m_{2}u_{2}-q$$\ell$$-s$.
(54)

We can re-express these in a compact form by defining a partial amplitude
$\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(s,q^{2}}\right)$ corresponding
to the contributions of $X$-states with fixed invariant mass to the elastic
amplitude $\mathcal{M}\mathopen{}\mathclose{{}\left(q^{2}}\right)$; by
definition

$\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(q^{2}}\right)=\int_{m_{2}^{2}}^{\infty}\text{d}\mu^{2}\;\rho\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(\mu^{2},q^{2}}\right)=2m_{2}\int_{0}^{\infty}\text{d}s\;\tilde{\rho}(s)\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(s,q^{2}}\right).$
(55)

The transverse and longitudinal contributions to the real kernel then take the
form

$\displaystyle\mathcal{I}_{r,q}=\text{Im}\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(q^{2}}\right),\hskip
28.45274pt\mathcal{I}_{r,\check{u}_{2}}=4m_{2}\int_{0}^{\infty}\text{d}s\;s\;\tilde{\rho}(s)\;\text{Im}\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(s,q^{2}}\right).$
(56)

The amplitudes appearing in these expressions are assumed to be expanded to
leading order in the generalized soft limit (47). In the corresponding one-
loop contribution to the conservative impulse, the leading soft contribution
is super-classical and cancels when the real and virtual kernels are summed
Kosower:2018adc . In the absorptive case however, the leading soft
contribution is the classical contribution due to an $\hbar/\hbar$
cancellation; the super-classical scaling of the box diagram is compensated by
the “quantum” scaling of the $X$-state effective propagator.

Next we consider the virtual kernel (36), which has only a transverse
contribution. Decomposing this into real and imaginary parts

$\mathcal{I}_{v,q}=-\text{Im}\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(q^{2}}\right)+\mathrm{i}\,\text{Re}\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(q^{2}}\right),$
(57)

we find that the transverse part of the real kernel is exactly cancelled by a
corresponding contribution from the virtual kernel222Each of these pieces
individually produce imaginary contributions to the impulse after Fourier
transforming; this cancellation is therefore a consequence of the non-manifest
reality of the observable..

Combining (56) and (57) we find that the transverse impulse (proportional to
$b^{\mu}$) receives contributions only from the real part of the elastic
amplitude. In general this takes the form

$\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(q^{2}}\right)=\int\hat{\text{d}}^{D}\ell\;\mathcal{N}\mathopen{}\mathclose{{}\left[(u_{2}\cdot\ell),q^{2},y}\right]\frac{\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)}{\ell^{2}(\ell+q)^{2}}\int_{0}^{\infty}\text{d}s\;\frac{s^{1+\alpha}}{u_{2}\cdot\ell-s+\mathrm{i}0},$
(58)

where $\mathcal{N}$ is a model dependent polynomial and as above we have
dropped scaleless contributions that pinch mediator propagators. The cut
propagator for particle-1 arises from interference between the box and
crossed-box diagrams in the soft limit, which are shown in Figure 2.

$m_{2}u_{2}$$m_{1}u_{1}$$m_{2}u_{2}-q$$l$$m_{2}u_{2}$$m_{1}u_{1}$$m_{2}u_{2}-q$$l$
Figure 2: The box and the crossed-box diagram are the two Feynman diagrams
nescessary for the computation of the leading order mass-shift. Wiggly lines
represent massless force carriers: scalars, photons or gravitons.

using the distributional identity

$\frac{1}{u_{1}\cdot\ell-\mathrm{i}0}-\frac{1}{u_{1}\cdot\ell+\mathrm{i}0}=\mathrm{i}\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right).$
(59)

Importantly, since we have expanded to leading-order in the soft limit (47)
the $\mathcal{N}$ polynomial is homogeneous

$\mathcal{N}\mathopen{}\mathclose{{}\left[\lambda(u_{2}\cdot\ell),\lambda^{2}q^{2},y}\right]=\lambda^{k}\mathcal{N}\mathopen{}\mathclose{{}\left[(u_{2}\cdot\ell),q^{2},y}\right],$
(60)

where $k=0$ for scalars, $k=2$ for photons and $k=4$ for gravitons. Since $k$
is even in each case, $\mathcal{N}$ must only contain even powers of
$(u_{2}\cdot\ell)$ and is therefore symmetric under $\ell\rightarrow-\ell-q$.
This becomes significant once we restrict to the real part of the spectral
integral given by the Cauchy principal value (PV); in the analytic
regularization described in Section 3.2 we calculate333We can alternatively
calculate this integral using the cutoff regularization described in Section
3.2. This leads to an additional power law divergent contribution that is even
under $l\rightarrow-\ell-q$ and therefore non-vanishing after loop
integration. Physically, we interpret this as an ambiguity in the definition
of the scale $\Lambda$ separating the light and heavy modes in the spectral
integral. Importantly, since the logarithmically divergent contributions
vanish after loop integration, there is no associated classical RG running of
the leading (static) Love numbers and therefore no tension with their observed
vanishing in $d=4$ and recently discovered associated symmetries
Charalambous:2022rre ; Ivanov:2022qqt ; Charalambous:2023jgq ; Saketh:2023bul
; Hui:2020xxx ; Hui:2021vcv ; Hui:2022vbh .

$\text{PV}\int_{0}^{\infty}\text{d}s\;\frac{s^{1+\alpha}}{u_{2}\cdot\ell-s}=\frac{u_{2}\cdot\ell}{\alpha}+(u_{2}\cdot\ell)\log\mathopen{}\mathclose{{}\left|u_{2}\cdot\ell}\right|+\mathcal{O}\mathopen{}\mathclose{{}\left(\alpha}\right).$
(61)

We therefore find, for the contribution of the light modes, the real part of
the integrand (58) is odd under the change of variables
$\ell\rightarrow-\ell-q$ and integrates to zero. Note that, as discussed in
Section 3.2, the contributions of the heavy modes pinch the propagator (44)
giving an expression that is even under this change of variables and therefore
a non-vanishing real part.

All together we conclude that, to leading non-trivial PM order, the absorptive
contribution to the impulse is purely longitudinal, while the conservative
contribution (including Love number operators) is purely transverse. The
leading absorptive impulse is therefore completely determined by the mass-
shift, the change in the black hole rest mass during scattering, explicitly

$\mathopen{}\mathclose{{}\left(\Delta
p_{2}^{\mu}}\right)_{\text{abs}}=\mathopen{}\mathclose{{}\left(\Delta
m_{2}}\right)\check{u}_{2}^{\mu},$ (62)

where

$\Delta
m_{2}=\frac{1}{m_{1}}\int\hat{\text{d}}^{D}q\;\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot
q}\right)\hat{\delta}\mathopen{}\mathclose{{}\left(u_{2}\cdot
q}\right)e^{-\mathrm{i}q\cdot
b}\int_{0}^{\infty}\text{d}s\;s\;\tilde{\rho}(s)\;\text{Im}\mathcal{M}_{4}\mathopen{}\mathclose{{}\left(s,q^{2}}\right).$
(63)

In Goldberger:2020wbx  it was noted that, since for Schwarzchild black holes
the area of the event horizon is $A=16\pi G_{\mathrm{N}}^{2}m_{2}^{2}$, a
decrease in mass during scattering would imply a violation of the Hawking area
theorem Hawking:1971vc . An appealing feature of the above simple formula is
that it manifests the positivity of the mass-shift. Unitarity relates the
imaginary part of the partial amplitude to the strictly positive absorptive
cross section where the final $X$-state has fixed invariant mass. Together
with the fact that the spectral density is non-negative and, crucially,
vanishes for $\mu^{2}<m_{2}^{2}$, positivity of the mass-shift is a trivial
corollary.

###  3.4 Calculation of the Mass-Shift

In the explicit cases of scalar, photon and graviton absorption, the mass-
shift can now be straightforwardly calculated from the imaginary parts of the
relevant box diagrams depicted in Figure 2. These can be obtained without
difficulty either using the Feynman rules in Appendix D or, as discussed in
more detail in Section 3.5, by directly sewing together the triangle cut using
the Compton amplitudes calculated in Section 2.2. There are then three
remaining integrals to calculate ($\int\text{d}s$, $\int\text{d}^{D}\ell$ and
$\int\text{d}^{D}q$), the necessary master integrals are collected in Appendix
E. The final results for the mass-shifts in each case are summarized in the
following paragraphs.

#### Scalar Absorption

The imaginary part of the box diagram for scalar absorption

$\text{Im}\mathcal{M}_{4}^{(\psi)}\mathopen{}\mathclose{{}\left(s,q^{2}}\right)=\frac{4\pi
G_{\mathrm{N}}Q_{s}^{2}}{m_{1}m_{2}}\int\hat{\text{d}}^{D}\ell\frac{\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)\hat{\delta}\mathopen{}\mathclose{{}\left(u_{2}\cdot\ell-s}\right)}{\ell^{2}(\ell+q)^{2}},$
(64)

together with the near-threshold spectral density (14) gives the corresponding
mass-shift

$\boxed{\mathopen{}\mathclose{{}\left(\Delta
m_{2}}\right)^{(\psi)}=\frac{G_{\mathrm{N}}^{2}Q_{s}^{2}m_{2}^{2}}{32m_{1}^{2}|b|^{3}}\sqrt{y^{2}-1}.}$
(65)

#### Photon Absorption

The imaginary part of the box diagram for photon absorption

$\text{Im}\mathcal{M}_{4}^{(\gamma)}\mathopen{}\mathclose{{}\left(s,q^{2}}\right)=-\frac{16\pi
G_{\mathrm{N}}Q_{e}^{2}m_{1}}{m_{2}}\int\hat{\text{d}}^{D}\ell\mathopen{}\mathclose{{}\left[4s^{2}+\mathopen{}\mathclose{{}\left(2y^{2}-1}\right)|q|^{2}}\right]\frac{\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)\hat{\delta}\mathopen{}\mathclose{{}\left(u_{2}\cdot\ell-s}\right)}{\ell^{2}(\ell+q)^{2}},$
(66)

together with the near-threshold spectral density (23) gives the corresponding
mass-shift

$\boxed{\mathopen{}\mathclose{{}\left(\Delta
m_{2}}\right)^{(\gamma)}=\frac{3G_{\mathrm{N}}^{4}Q_{e}^{2}m_{2}^{4}}{32|b|^{5}}\mathopen{}\mathclose{{}\left(5y^{2}-1}\right)\sqrt{y^{2}-1}.}$
(67)

#### Graviton Absorption

The imaginary part of the box diagram for graviton absorption

$\displaystyle\text{Im}\mathcal{M}_{4}^{(h)}\mathopen{}\mathclose{{}\left(s,q^{2}}\right)$
$\displaystyle=\frac{256\pi^{2}G_{\mathrm{N}}^{2}m_{1}^{3}}{m_{2}}\int\hat{\text{d}}^{D}\ell\mathopen{}\mathclose{{}\left[16s^{4}+8\mathopen{}\mathclose{{}\left(4y^{2}-1}\right)s^{2}|q|^{2}+\mathopen{}\mathclose{{}\left(8y^{4}-8y^{2}+1}\right)|q|^{4}}\right]$
$\displaystyle\hskip
113.81102pt\times\frac{\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)\hat{\delta}\mathopen{}\mathclose{{}\left(u_{2}\cdot\ell-s}\right)}{\ell^{2}(\ell+q)^{2}},$
(68)

together with the near-threshold spectral density (29) gives the corresponding
mass-shift

$\boxed{\mathopen{}\mathclose{{}\left(\Delta m_{2}}\right)^{(h)}=\frac{5\pi
G_{\mathrm{N}}^{7}m_{1}^{2}m_{2}^{6}}{16|b|^{7}}\mathopen{}\mathclose{{}\left(21y^{4}-14y^{2}+1}\right)\sqrt{y^{2}-1}.}$
(69)

###  3.5 Neutron Stars and Dissipation Numbers

The results of the previous section for Schwarzschild black hole absorption
were obtained after explicitly matching with a UV cross section and exploiting
the additional simplification of duality invariance as explained in Section
2.2. For generic compact bodies, such as neutron stars, this UV information
may not be available. In such a case this formalism is still predictive, since
the leading-order absorption depends only on the near-threshold Taylor series
coefficients of the spectral density, we can proceed in the spirit of bottom
up EFT and use these unknown coefficients as a parametrization of our
ignorance. As usual, at higher-orders finding a non-redundant parametrization
is highly non-trivial; on-shell amplitudes based approaches are very well-
suited to this problem, directly providing gauge invariant and non-redundant
expressions by construction (for a relevant example in the context of Standard
Model EFT see e.g. Li:2020gnx ).

$\mathcal{M}_{c}$$m_{2}u_{2}$$m_{1}u_{1}$$m_{2}u_{2}-q$$k_{1}$$k_{4}$ Figure
3: The Compton cut is the only unitary cut needed in order to compute leading-
order classical observables. In the explicit Compton amplitudes given below
the mediator momenta are labelled as shown, after sewing we identify
$k_{1}=-l$, $k_{4}=l+q$.

As we have seen explicitly, the input required to compute the mass-shift is
the absorptive contribution to the 2-to-2 elastic amplitude. Since we are only
interested in long-distance effects, this quantity can be determined using
standard on-shell unitarity-based methods, by computing the triangle cut of
the amplitude depicted in Figure 3. Absorptive effects are parametrized by the
most general allowed form of the Compton amplitude. This task is somewhat
complicated by the fact that, due to the unfamiliar $s$-integrals, the
absorptive amplitudes are non-analytic functions of the external kinematics.

Based on the general discussion in Section 2.2, it is convenient to define a
Compton kernel $\mathcal{M}_{c}(s)$ as the integrand of the (low-energy)
spectral integral, where this quantity is expanded in the soft limit (47). Let
us first consider the case of the scalar-force model. The Compton kernel for
scattering a scalar off a black hole takes the general form

$\displaystyle\mathcal{M}^{(\psi)}(s)$ $\displaystyle=-\frac{1}{u_{2}\cdot
k_{1}-s+\mathrm{i}0}\mathopen{}\mathclose{{}\left[\kappa^{2}s\,c^{(\psi)}+\kappa^{4}s^{2}c_{2}^{(\psi)}+\kappa^{4}(k_{1}\cdot
k_{4})c_{3}^{(\psi)}+\dots}\right]+\mathopen{}\mathclose{{}\left(1\leftrightarrow
4}\right)\,,$ (70)

where the ellipsis denotes terms with higher orders in $G_{\mathrm{N}}$.
Notice that terms including $u_{2}\cdot k_{1}$ and $u_{2}\cdot k_{4}$ are
absent, since $u_{2}\cdot(k_{1}+k_{4})=-\frac{k_{1}\cdot k_{4}}{m_{2}}$ and
pinch terms correspond to vanishing scaleless $s$-integrals. The coefficients
$c_{i}^{(\psi)}$ provide an on-shell, gauge invariant, definition of so-called
dissipation numbers Saketh:2023bul . In the following we work to leading order
and keep only the term proportional to $c^{(\psi)}$, higher-order terms may be
important for sub-leading absorptive effects.

For gravity and electrodynamics there are two independent Compton amplitudes
corresponding to helicity conserving and helicity violating interactions;
accordingly we have to parametrize two kernels $\mathcal{M}_{+\mp}(s)$
independently. For electrodynamics the most general form is

$\displaystyle\mathcal{M}_{+-}^{(\gamma)}(s)$
$\displaystyle=-\frac{[1|u_{2}|4\rangle^{2}}{u_{2}\cdot
k_{1}-s+\mathrm{i}0}\mathopen{}\mathclose{{}\left[\kappa^{2}s\,c_{+-}^{(\gamma)}+\dots}\right]+\mathopen{}\mathclose{{}\left(1\leftrightarrow
4}\right)\,,$ (71) $\displaystyle\mathcal{M}_{++}^{(\gamma)}(s)$
$\displaystyle=-\frac{[14]^{2}}{u_{2}\cdot
k_{1}-s+\mathrm{i}0}\mathopen{}\mathclose{{}\left[\kappa^{2}s\,c_{++}^{(\gamma)}+\dots}\right]+\mathopen{}\mathclose{{}\left(1\leftrightarrow
4}\right)\,.$ (72)

The spinor prefactors are fixed by the little-group covariance of the
amplitudes. Again, the ellipsis denotes higher orders in $G_{\mathrm{N}}$ and
we will focus on the leading-order term. The analysis for the gravitational
case is similar

$\displaystyle\mathcal{M}_{+-}^{(h)}(s)$
$\displaystyle=-\frac{[1|u_{2}|4\rangle^{4}}{u_{2}\cdot
k_{1}-s+\mathrm{i}0}\mathopen{}\mathclose{{}\left[\kappa^{2}s\,c_{+-}^{(h)}+\dots}\right]+\mathopen{}\mathclose{{}\left(1\leftrightarrow
4}\right)\,,$ (73) $\displaystyle\mathcal{M}_{++}^{(h)}(s)$
$\displaystyle=-\frac{[14]^{4}}{u_{2}\cdot
k_{1}-s+\mathrm{i}0}\mathopen{}\mathclose{{}\left[\kappa^{2}s\,c_{++}^{(h)}+\dots}\right]+\mathopen{}\mathclose{{}\left(1\leftrightarrow
4}\right)\,.$ (74)

We see the clear simplification compared with the off-shell the construction
of $\mathcal{O}$ operators. For completeness, these on-shell expression can be
mapped onto a specific set of operators that are explicitly given in Appendix
B.

The helicity violating Compton amplitudes $\mathcal{M}_{++}$ vanish in the
forward limit and therefore matching against a cross section fixes the
coefficient $c_{+-}$ uniquely, while leaving $c_{++}$ undetermined. In order
to determine all parameters another observable is required. In
Bautista:2021wfy ; Bautista:2022wjf  the Compton amplitudes themselves were
directly matched with black hole perturbation theory, while in Aoude:2023fdm
the absorption cross sections for individual spinning partial waves were used.
It might also be possible distinguish the contributions by matching
forward/backward asymmetry of the Compton differential cross section
Holstein:2013kia . Explicitly performing this matching for generic compact
bodies is an interesting open problem; for the relevant case of a neutron star
this information is ultimately a prediction of nuclear theory.

Starting from the Compton amplitudes, we construct the 2-to-2 elastic
amplitude by sewing generalized unitarity cuts. In our conventions the
necessary 3-particle amplitudes are: for the scalar

$\mathcal{M}_{3}\mathopen{}\mathclose{{}\left(1_{\psi},2_{\phi_{1}},3_{\overline{\phi}_{1}}}\right)=Q_{s}\,,$
(75)

for electrodynamics

$\mathcal{M}_{3}\mathopen{}\mathclose{{}\left(1_{\gamma}^{+1},2_{\phi_{1}},3_{\overline{\phi}_{1}}}\right)=-\sqrt{2}Q_{e}\frac{[1|p_{2}|\xi\rangle}{\langle
1\xi\rangle},\hskip
14.22636pt\mathcal{M}_{3}\mathopen{}\mathclose{{}\left(1_{\gamma}^{-1},2_{\phi_{1}},3_{\overline{\phi}_{1}}}\right)=\sqrt{2}Q_{e}\frac{\langle
1|p_{2}|\xi]}{[1\xi]},$ (76)

and for gravity

$\mathcal{M}_{3}\mathopen{}\mathclose{{}\left(1_{h}^{+2},2_{\phi_{1}},3_{\overline{\phi}_{1}}}\right)=-\frac{\kappa}{2}\frac{[1|p_{2}|\xi\rangle^{2}}{\langle
1\xi\rangle^{2}},\hskip
14.22636pt\mathcal{M}_{3}\mathopen{}\mathclose{{}\left(1_{h}^{-2},2_{\phi_{1}},3_{\overline{\phi}_{1}}}\right)=-\frac{\kappa}{2}\frac{\langle
1|p_{2}|\xi]^{2}}{[1\xi]^{2}},$ (77)

where $|\xi\rangle$ and $|\xi]$ are arbitrary auxiliary spinors. The case of
the scalar is trivial and the resulting numerator, defined in (58), resulting
from the sewing procedure is

$\mathcal{N}_{4}^{(\psi)}(s)=\frac{\kappa^{2}Q_{s}^{2}c^{(\psi)}}{2m_{1}}\,.$
(78)

For photons and gravitons, there are two different types of cuts contributing:

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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{60.92932pt}{-22.06961pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{104.23166pt}{-22.41684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{53.01662pt}{-36.64322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$+$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{112.14438pt}{-36.29599pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$-$}}
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\;\;\;,$ (79)

the two remaining cuts $\mathcal{C}_{--}$ and $\mathcal{C}_{+-}$ are related
by conjugation. Using the Compton amplitudes in (71) and (72) as well as the
three-point amplitudes in (76) and making use of spinor identities listed in
Appendix A, for electrodynamics we calculate

$\displaystyle\mathcal{C}_{++}^{(\gamma)}(s)={}$
$\displaystyle-\frac{2\kappa^{2}Q_{e}^{2}c_{++}^{(\gamma)}m_{1}^{2}s|q|^{2}}{u_{2}\cdot\ell-s+\mathrm{i}0}+(u_{2}\to-
u_{2})\,,$ (80) $\displaystyle\mathcal{C}_{-+}^{(\gamma)}(s)={}$
$\displaystyle-\frac{2\kappa^{2}Q_{e}^{2}c_{-+}^{(\gamma)}m_{1}^{2}s}{u_{2}\cdot\ell-s+\mathrm{i}0}\frac{\mathopen{}\mathclose{{}\left(y|q|^{2}+2\mathrm{i}\varepsilon(\ell,q,u_{1},u_{2})}\right)^{2}}{|q|^{2}}+(u_{2}\to-
u_{2})\,,$ (81)

with
$\varepsilon(\ell,q,u_{1},u_{2})\coloneqq\varepsilon_{\mu\nu\rho\sigma}\ell^{\mu}q^{\nu}u_{1}^{\rho}u_{2}^{\sigma}$.
For gravity we find a simple double-copy-like form

$\displaystyle\mathcal{C}_{++}^{(h)}(s)={}$
$\displaystyle-\frac{1}{4}\frac{\kappa^{4}c_{++}^{(h)}m_{1}^{4}s|q|^{4}}{u_{2}\cdot\ell-s+\mathrm{i}0}+(u_{2}\to-
u_{2})\,,$ (82) $\displaystyle\mathcal{C}_{-+}^{(h)}(s)={}$
$\displaystyle\frac{\kappa^{4}c_{-+}^{(h)}m_{1}^{4}s}{u_{2}\cdot\ell-s+\mathrm{i}0}\frac{\mathopen{}\mathclose{{}\left(y|q|^{2}+2\mathrm{i}\varepsilon(\ell,q,u_{1},u_{2})}\right)^{4}}{|q|^{4}}+(u_{2}\to-
u_{2})\,.$ (83)

Summing over the four different cuts, yields the following numerators, up to
pinch terms

$\displaystyle\mathcal{N}^{(\gamma)}_{4}(s)={}$ $\displaystyle
2\kappa^{2}Q_{e}^{2}m_{1}\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left[4s^{2}+(2y^{2}-1)|q|^{2}}\right]c_{+-}^{(\gamma)}-|q|^{2}c_{++}^{(\gamma)}}\right\\}\,,$
(84) $\displaystyle\mathcal{N}^{(h)}_{4}(s)={}$
$\displaystyle-\frac{\kappa^{4}m_{1}^{3}}{4}\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left[16s^{4}{+}8(4y^{2}-1)s^{2}|q|^{2}{+}(8y^{4}{-}8y^{2}{+}1)|q|^{4}}\right]c_{+-}^{(h)}+|q|^{4}c_{++}^{(h)}}\right\\}\,.$
(85)

Starting from these expressions and using the mass-shift formula (63) and the
integrals (125)–(125), it is straightforward to derive the leading-order mass-
shifts for generic bodies

$\displaystyle\mathopen{}\mathclose{{}\left(\Delta m_{2}}\right)^{(\psi)}$
$\displaystyle=\frac{\pi
G_{\mathrm{N}}Q_{s}^{2}c^{(\psi)}}{32m_{1}^{2}m_{2}|b|^{3}}\sqrt{y^{2}-1}\,$
(86) $\displaystyle\mathopen{}\mathclose{{}\left(\Delta
m_{2}}\right)^{(\gamma)}$ $\displaystyle=\frac{9\pi
G_{\mathrm{N}}Q_{e}^{2}}{32m_{2}|b|^{5}}\mathopen{}\mathclose{{}\left[c_{+-}^{(\gamma)}\mathopen{}\mathclose{{}\left(5y^{2}-1}\right)-4c_{++}^{(\gamma)}}\right]\sqrt{y^{2}-1}\,,$
(87) $\displaystyle\mathopen{}\mathclose{{}\left(\Delta m_{2}}\right)^{(h)}$
$\displaystyle=\frac{225\pi^{2}G_{\mathrm{N}}^{2}m_{1}^{2}}{16m_{2}|b|^{7}}\mathopen{}\mathclose{{}\left[c_{+-}^{(h)}\mathopen{}\mathclose{{}\left(21y^{4}-14y^{2}+1}\right)+8c_{++}^{(h)}}\right]\sqrt{y^{2}-1}\,.$
(88)

These expressions reduce to the Schwarzschild mass-shifts (65), (67) and (69)
for the choice of dissipation numbers

$c^{(\psi)}=\frac{G_{\mathrm{N}}m_{2}^{3}}{\pi},\;\;\;\;c_{+-}^{(\gamma)}=\frac{G_{\mathrm{N}}^{3}m_{2}^{5}}{3\pi},\;\;\;\;c_{++}^{(\gamma)}=0,\;\;\;\;c_{+-}^{(h)}=\frac{G_{\mathrm{N}}^{5}m_{2}^{7}}{45\pi},\;\;\;\;c_{++}^{(h)}=0,$
(89)

where as discussed in Section 2.2, the vanishing of the helicity violating
amplitudes is a manifestation of (linearized) self-duality.

The general result (86) agrees with the contribution of the electric and
magnetic components that can be read of from Eq. (3.20) in Goldberger:2020wbx
. Let us point out the trivial fact that given that we have computed the
absorption for models with scalar, vector and tensor exchange, we can also
make predictions for more general theories including supergravity.444The
contributions from fermionic exchanges are a bit more complicated and
necessitate a process where the light particle transitions from a spin 0 to a
higher-spin state. In this context it is interesting to note that no matter
which mediation fields are present in the theory, the helicity-conserving
contribution of the graviton will always be dominant in the high-energy limit.
This implies a universal high-energy behaviour for the mass-shift, which
closely mirrors similar findings for the scattering angle discussed
Bern:2020gjj ; Parra-Martinez:2020dzs .

##  4 Discussion

The main results of this paper are the expressions (65), (67) and (69) for the
change of rest mass of a Schwarzschild black hole scattering with a second
compact body sourcing a massless scalar, electromagnetic or gravitational
field. From these expressions we trivially obtain the 4-momentum impulse on
either body using (62) and (41). The gravitational mass-shift (69) was
previously calculated in Goldberger:2020wbx  using a Schwinger-Keldysh in-in
path integral based on the worldline formalism introduced in Goldberger:2004jt
; Goldberger:2005cd ; our result obtained using the scattering amplitudes
based KMOC formalism Kosower:2018adc  is in complete agreement. As far as we
are aware, the expressions for the impulse due to scalar (65) and photon (67)
absorption are new.

The case of massless scalar absorption is of some importance for recent
comparisons between PM methods and (a simplified scalar toy-model of) the
self-force expansion Barack:2022pde ; Barack:2023oqp . In the model considered
in Barack:2022pde ; Barack:2023oqp , beginning at
$\mathcal{O}\mathopen{}\mathclose{{}\left(G_{\mathrm{N}}^{2}Q_{s}^{2}}\right)$,
there is energy loss in the form of both radiation and horizon absorption.
Combining the previous PM calculation of the two-loop radiative energy loss
Barack:2023oqp  with the scalar absorption result of this paper (65), we find
excellent numerical agreement with the predicted energy loss from scalar self-
force calculations at this order. Details of this comparison will be presented
in a forthcoming paper.

Beyond these specific results, in this paper we have developed an approach to
incorporating horizon absorption based on familiar in-out scattering
amplitudes. As explained in Section 3.4, prior to integration, the non-trivial
part of the calculation reduces to an application of standard unitarity
methods for constructing integrands Bern:1994zx ; Bern:1994cg ; Bern:1995db ;
Bern:1997sc ; Britto:2004nc . Furthermore, as emphasized in Section 3.2, the
near-threshold expansion of the spectral integrals is most naturally treated
as part of the expansion of the loop integrand in the soft region (34). Taken
together, the approach described in this paper can be incorporated quite
naturally into the existing workflow for scattering amplitudes based methods
applied to the PM gravitational two-body problem.

An important application of these methods will be the incorporation of spin
degrees-of-freedom and the calculation of absorptive effects in the scattering
of Kerr black holes Porto:2007qi ; Goldberger:2022rqf . As observed in
Chung:2018kqs ; Guevara:2018wpp , the static Kerr solution corresponds to a
3-particle on-shell amplitude previously identified as possessing uniquely
soft UV behavior Arkani-Hamed:2017jhn . There has been considerable effort in
attempting to generalize this correspondence to 4-particle Compton amplitudes
Chung:2018kqs ; Guevara:2018wpp ; Bautista:2019tdr ; Aoude:2020onz ;
Saketh:2022wap ; Chiodaroli:2021eug ; Bern:2022kto ; Cangemi:2022bew ;
Aoude:2022trd ; Aoude:2022thd ; Aoude:2023vdk ; Bjerrum-Bohr:2023iey . As
emphasized recently Bautista:2021wfy ; Bautista:2022wjf , direct extraction of
the Kerr Compton amplitude from the Teukolsky equation is complicated at
higher-orders in spin by non-trivial absorption effects. We may hope that an
intrinsically amplitudes based approach to horizon absorption will be useful
in clarifying the situation.

While the discussion in this paper is presented having in mind the case of
compact astronomical objects such as black holes and neutron stars, we note
that the discussion should equally apply to elementary processes, in
particular the low energy scattering of photons off heavy ions and pions.
Compton scattering serves as an important probe in low energy QCD and the
polarizabilities, e.g. of pions constitute key quantities to be computed from
nuclear models (see e.g. Schumacher:2005an ; Dattoli:1977wg ; Hecking:1981fsu
; Weiner:1985ih ) and are measured e.g. at the COMPASS experiment at CERN
COMPASS:2007rjf .

Finally, in this paper we have described the calculation of the leading
absorptive effects in two-body scattering. A significant simplification in
this case was the fact that only 2-point functions of the $\mathcal{O}$
operators contributed, and therefore, as discussed in Section 3.5, our
ignorance of the invisible sector could be parametrized in terms of a finite
set of dissipation numbers arising from the near-threshold expansion of the
spectral density. At higher-orders we would naively expect higher-point
functions to contribute, and it is less clear how we can treat these
quantities in a model-independent way. In Goldberger:2019sya  it was
conjectured that the hidden sector dynamics is approximately Gaussian and that
higher-point correlators factor into products of 2-point functions. To test
this conjecture it may be useful to perform the EFT matching in the context of
a toy model where the quantum mechanics of the black hole horizon is
explicitly calculable Mertens:2022irh . We leave this and other important open
questions to future work.

Acknowledgements

We would like to thank Leor Barack, Zvi Bern, Enrico Herrmann, Dimitrios
Kosmopoulos, Oliver Long and Chia-Hsien Shen for useful discussions and
comments on the draft. CRTJ and MR are supported by the Department of Energy
under Award Number DE-SC0009937, and gratefully acknowledge the continued
support of the Mani L. Bhaumik Institute for Theoretical Physics. We would
also like to acknowledge the hospitality of Nordita during the program
Amplifying Gravity at All Scales where part of this work was completed.

##  Appendix A Conventions

We will use the mostly-minus metric convention, in $D$ spacetime dimensions

$\eta_{\mu\nu}=\text{diag}(+1,\underbrace{-1,...,-1}_{D-1}).$ (90)

Loop and Fourier integrals are evaluated using dimensional regularization with
$D=4-2\epsilon$.

Our kinematic conventions for elastic scattering amplitudes will be to denote
the incoming (outgoing) momenta $p_{i}$ ($p_{i}^{\prime}$) using the
4-velocities $u_{i}$ and momentum transfer $q$ as

$p_{1}^{\mu}=m_{1}u_{1}^{\mu},\hskip
14.22636ptp_{2}^{\mu}=m_{2}u_{2}^{\mu},\hskip
14.22636pt{p_{1}^{\prime}}^{\mu}=m_{1}u_{1}^{\mu}+q^{\mu},\hskip
14.22636pt{p_{2}^{\prime}}^{\mu}=m_{2}u_{2}^{\mu}-q^{\mu}.$ (91)

It is also convenient to introduce the dual vectors

$\check{u}_{1}^{\mu}\coloneqq\frac{u_{1}^{\mu}-yu_{2}^{\mu}}{1-y^{2}},\hskip
28.45274pt\check{u}_{2}^{\mu}\coloneqq\frac{u_{2}^{\mu}-yu_{1}^{\mu}}{1-y^{2}},\hskip
28.45274pty\coloneqq u_{1}\cdot u_{2},$ (92)

defined to satisfy $u_{i}\cdot\check{u}_{j}=\delta_{ij}$.

Following Kosower:2018adc  we absorb many factors of $2\pi$ by defining

$\hat{\text{d}}x\coloneqq\frac{\text{d}x}{2\pi},\hskip
28.45274pt\hat{\delta}\mathopen{}\mathclose{{}\left(x}\right)\coloneqq
2\pi\delta\mathopen{}\mathclose{{}\left(x}\right).$ (93)

Our scattering conventions mostly follow the conventions of Peskin-Schroeder
Peskin:1995ev . Scattering amplitudes are related to the formal S-matrix
operator by

$\langle\text{out}|T|\text{in}\rangle\coloneqq\hat{\delta}^{(D)}\mathopen{}\mathclose{{}\left(p_{\text{out}}-p_{\text{in}}}\right)\mathcal{M}\mathopen{}\mathclose{{}\left(\text{in}\rightarrow\text{out}}\right),$
(94)

where $S\coloneqq 1+\mathrm{i}T$. All of the amplitudes needed in the paper
can be calculated using the Feynman rules enumerated in Appendix D. We use the
standard phase convention, these rules directly applied calculate
$\mathrm{i}\mathcal{M}$.

Our spinor-helicity conventions mostly follow Elvang-Huang Elvang:2013cua ,
modified to align with the choice of metric signature. All Pauli matrix
identities and definitions are chosen to align with Dreiner:2008tw . We choose
$|p]_{\alpha}$ and $\langle p|_{\dot{\alpha}}$ to carry (outgoing) helicity
weights $+1/2$ and $-1/2$ respectively and for real momenta
$\mathopen{}\mathclose{{}\left(|p]_{\alpha}}\right)^{*}=\langle
p|_{\dot{\alpha}}$. We use the convention in which the helicity spinors and
bispinor momenta are related as

$p_{\mu}\sigma^{\mu}_{\alpha\dot{\alpha}}\coloneqq
p_{\alpha\dot{\alpha}}\coloneqq-|p]_{\alpha}\langle p|_{\dot{\alpha}}.$ (95)

A consistent choice of (outgoing) spin-1 polarization vectors is given by

$\varepsilon_{+}^{*\mu}\mathopen{}\mathclose{{}\left(p}\right)\coloneqq\frac{[p|\sigma^{\mu}|\xi\rangle}{\sqrt{2}\langle
p\xi\rangle},\hskip
28.45274pt\varepsilon_{-}^{*\mu}\mathopen{}\mathclose{{}\left(p}\right)\coloneqq-\frac{\langle
p|\overline{\sigma}^{\mu}|\xi]}{\sqrt{2}[p\xi]},$ (96)

where $|\xi\rangle$ and $|\xi]$ are arbitrary auxiliary vectors; this choice
is consistent with the property
$\mathopen{}\mathclose{{}\left(\varepsilon_{+}^{\mu}}\right)^{*}=\varepsilon_{-}^{\mu}$
as well as the normalization and completeness conditions

$\varepsilon_{\lambda}^{*}\mathopen{}\mathclose{{}\left(p}\right)\cdot\varepsilon_{\lambda^{\prime}}\mathopen{}\mathclose{{}\left(p}\right)=-\delta_{\lambda\lambda^{\prime}},\,\quad\hskip
28.45274pt\sum_{\lambda=\pm}(\varepsilon_{\lambda}^{\mu}\mathopen{}\mathclose{{}\left(p}\right))^{*}\varepsilon_{\lambda}^{\nu}\mathopen{}\mathclose{{}\left(p}\right)=-\eta^{\mu\nu}+\frac{p^{\mu}\xi^{\nu}+p^{\nu}\xi^{\mu}}{p\cdot\xi}.$
(97)

##  Appendix B Duality-Organized Operator Basis

By adopting an on-shell approach as we have done in Section 3.5, we never have
to introduce an off-shell effective action. However for the convenience of the
reader and in order to facilitate comparison with the literature, we
explicitly give the operators that when added to the minimal action give the
corresponding Compton amplitudes. The photon Compton amplitudes in (71) and
(72) can be obtained from

$\displaystyle S^{(\gamma)}_{\text{portal}}={}$ $\displaystyle
2\kappa\int\mathrm{d}^{4}x\,\mathopen{}\mathclose{{}\left[\frac{\partial_{\mu}\phi_{2}}{m_{2}}F_{\mathrm{SD}}^{\mu\nu}\mathcal{O}_{\nu}^{\mathrm{SD}}+\frac{\partial_{\mu}\phi_{2}}{m_{2}}F_{\mathrm{ASD}}^{\mu\nu}\mathcal{O}_{\nu}^{\mathrm{ASD}}}\right]\,,$
(98)

where the self-dual and anti-self-dual field strengths are given by

$F_{\mathrm{SD}}^{\mu\nu}=\frac{1}{2}\mathopen{}\mathclose{{}\left(F^{\mu\nu}+\frac{\mathrm{i}}{2}\epsilon^{\mu\nu}_{\phantom{\mu\nu}\alpha\beta}F^{\alpha\beta}}\right)\,,\quad
F_{\mathrm{ASD}}^{\mu\nu}=\frac{1}{2}\mathopen{}\mathclose{{}\left(F^{\mu\nu}-\frac{\mathrm{i}}{2}\epsilon^{\mu\nu}_{\phantom{\mu\nu}\alpha\beta}F^{\alpha\beta}}\right)\,.$
(99)

The fact that this basis is particularly amplitude friendly is well-known and
frequently used in the context of EFT (see e.g. Dixon:2004za ). The
correlation functions are given by

$\displaystyle\langle\mathcal{O}_{\mathrm{SD}}^{\mu}(x)\mathcal{O}_{\mathrm{ASD}}^{\nu}(0)\rangle={}$
$\displaystyle-i\int\mathrm{d}^{D}p\,e^{\mathrm{i}p\cdot
x}\int_{m_{2}^{2}}^{\infty}\mathrm{d}\mu^{2}\frac{c_{+-}^{(\gamma)}(\mu^{2}-m_{2}^{2})}{p^{2}-\mu^{2}+\mathrm{i}0}\Pi_{1}^{\mu\nu}(p)+\dots$
(100)
$\displaystyle\langle\mathcal{O}_{\mathrm{SD}}^{\mu}(x)\mathcal{O}_{\mathrm{SD}}^{\nu}(0)\rangle={}$
$\displaystyle-i\int\mathrm{d}^{D}p\,e^{\mathrm{i}p\cdot
x}\int_{m_{2}^{2}}^{\infty}\mathrm{d}\mu^{2}\frac{c_{++}^{(\gamma)}(\mu^{2}-m_{2}^{2})}{p^{2}-\mu^{2}+\mathrm{i}0}\Pi_{1}^{\mu\nu}(p)+\dots$
(101)

The mixed other correlators are related by parity. In this we have introduced
the projector

$\Pi_{1}^{\mu\nu}(p)=\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}\,.$ (102)

Where as before the ellipsis denote higher-order-in-$G_{\mathrm{N}}$ and
quantum terms. The operators for the gravitational case look very similar and
are explicitly

$\displaystyle S_{\text{portal}}^{(h)}={}$
$\displaystyle\int\text{d}^{4}x\mathopen{}\mathclose{{}\left[\frac{\partial^{\alpha}\partial^{\beta}\phi_{2}}{m_{2}^{2}}C_{\mu\alpha\nu\beta}^{\mathrm{SD}}\mathcal{O}_{\mathrm{SD}}^{\mu\nu}+\frac{\partial^{\alpha}\partial^{\beta}\phi_{2}}{m_{2}^{2}}C_{\mu\alpha\nu\beta}^{\mathrm{ASD}}\mathcal{O}_{\mathrm{ASD}}^{\mu\nu}}\right]\,.$
(103)

The self-and anti-self-dual projections of the Weyl tensor are

$C_{\mu\nu\rho\sigma}^{\mathrm{SD}}=\frac{1}{2}\mathopen{}\mathclose{{}\left(C_{\mu\nu\rho\sigma}+\frac{\mathrm{i}}{2}\epsilon_{\mu\nu\alpha\beta}C^{\alpha\beta}_{\phantom{\mu\nu}\rho\sigma}}\right)\,,\quad
C_{\mu\nu\rho\sigma}^{\mathrm{ASD}}=\frac{1}{2}\mathopen{}\mathclose{{}\left(C_{\mu\nu\rho\sigma}-\frac{\mathrm{i}}{2}\epsilon_{\mu\nu\alpha\beta}C^{\alpha\beta}_{\phantom{\mu\nu}\rho\sigma}}\right)\,.$
(104)

The correlators are

$\displaystyle\langle\mathcal{O}_{\mathrm{SD}}^{\mu\nu}(x)\mathcal{O}_{\mathrm{ASD}}^{\alpha\beta}(0)\rangle={}$
$\displaystyle-\mathrm{i}\int\mathrm{d}^{D}p\,e^{\mathrm{i}p\cdot
x}\int_{m_{2}^{2}}^{\infty}\mathrm{d}\mu^{2}\frac{c_{+-}^{(h)}(\mu^{2}-m_{2}^{2})}{p^{2}-\mu^{2}+\mathrm{i}0}\Pi_{2}^{\mu\nu\alpha\beta}(p)+\dots\,,$
(105)
$\displaystyle\langle\mathcal{O}_{\mathrm{SD}}^{\mu\nu}(x)\mathcal{O}_{\mathrm{SD}}^{\alpha\beta}(0)\rangle={}$
$\displaystyle-\mathrm{i}\int\mathrm{d}^{D}p\,e^{\mathrm{i}p\cdot
x}\int_{m_{2}^{2}}^{\infty}\mathrm{d}\mu^{2}\frac{c_{++}^{(h)}(\mu^{2}-m_{2}^{2})}{p^{2}-\mu^{2}+\mathrm{i}0}\Pi_{2}^{\mu\nu\alpha\beta}(p)+\dots\,,$
(106)

once again the remaining correlators are related by parity; the projector is
given by555 In practice the correlator will always contract into an conserved
current, so we can replace $\Pi_{1}^{\mu\nu}\to\eta^{\mu\nu}$ and
$\Pi^{\mu\nu\alpha\beta}_{2}\to\eta^{\mu\alpha}\eta^{\nu\beta}$.

$\Pi^{\mu\nu\alpha\beta}_{2}(p)=\frac{1}{2}\Pi_{1}^{\mu\alpha}(p)\Pi_{1}^{\nu\beta}(p)+\frac{1}{2}\Pi_{1}^{\mu\beta}(p)\Pi_{1}^{\nu\alpha}(p)-\frac{1}{D-1}\Pi_{1}^{\mu\nu}(p)\Pi_{1}^{\alpha\beta}(p)\,.$
(107)

##  Appendix C Weighted Cuts

It is convenient to define a diagrammatic weighted cut as a generalization of
the standard Cutkosky cutting rules Cutkosky :

$\vdots$$\vdots$$l_{1}$$l_{2}$$F(l_{i})$$\coloneqq$
$\mathrm{i}\mathcal{M}\biggr{|}_{\frac{1}{[d_{1}+\mathrm{i}0][d_{2}+\mathrm{i}0]}\rightarrow(-\mathrm{i})^{2}F(l_{i})\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(d_{1}}\right)\hat{\delta}^{(+)}\mathopen{}\mathclose{{}\left(d_{2}}\right)},$
(108)

where $d_{i}\coloneqq l_{i}^{2}-m_{i}^{2}$ and $F(l_{i})$ is an arbitrary
function we will refer to as the weight of the cut. Choosing $F(l_{i})=1$
gives an ordinary Cutkosky cut that calculates the imaginary part of the
amplitude Cutkosky :

$\vdots$$\vdots$$l_{1}$$l_{2}$$1$ $=\hskip
25.60747pt\text{Disc}\mathopen{}\mathclose{{}\left(\mathrm{i}\mathcal{M}}\right)\hskip
25.60747pt=\hskip 25.60747pt-2\;\text{Im}\mathcal{M}.$ (109)

##  Appendix D Feynman Rules

### Scalar Model

$k$ $=\hskip 34.1433pt\frac{\mathrm{i}}{k^{2}-m_{1}^{2}+\mathrm{i}0}.$

$k$ $=\hskip 34.1433pt\frac{\mathrm{i}}{k^{2}+\mathrm{i}0}.$

$=\hskip 34.1433pt\mathrm{i}Q_{s}.$

$k$ $=\hskip
34.1433pt\int_{0}^{\infty}\text{d}\mu^{2}\frac{\mathrm{i}\rho_{0}(\mu^{2})}{k^{2}-\mu^{2}+\mathrm{i}0}.$

$=\hskip 34.1433pt\mathrm{i}\kappa.$

### Scalar QED

$\mu$$\nu$$k$ $=\hskip
34.1433pt\frac{-\mathrm{i}\eta^{\mu\nu}}{k^{2}+\mathrm{i}0}.$

$p$$p^{\prime}$$\mu$ $=\hskip
34.1433pt-\mathrm{i}Q_{e}\mathopen{}\mathclose{{}\left(p^{\mu}+p^{\prime\mu}}\right).$

$k$$\mu\nu$$\rho\sigma$ $=\hskip
34.1433pt\int_{0}^{\infty}\text{d}\mu^{2}\rho_{1}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\frac{\mathrm{i}\hat{\Pi}_{1}^{\mu\nu\rho\sigma}(k)}{k^{2}-\mu^{2}+\mathrm{i}0},$

where $\hat{\Pi}_{1}$ is given in (20).

$\mu$$\nu\rho$$p$ $=\hskip
34.1433pt-\kappa\mathopen{}\mathclose{{}\left(\eta^{\mu\nu}p^{\rho}-\eta^{\mu\rho}p^{\nu}}\right).$

### General Relativity

$\mu\nu$$\rho\sigma$$k$ $=\hskip
34.1433pt\frac{\frac{\mathrm{i}}{2}\mathopen{}\mathclose{{}\left(\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho}-\eta^{\mu\nu}\eta^{\rho\sigma}}\right)}{k^{2}+\mathrm{i}0}.$

$p$$p^{\prime}$$\mu\nu$ $=\hskip
34.1433pt\frac{\mathrm{i}\kappa}{2}\mathopen{}\mathclose{{}\left(\eta^{\mu\nu}((p\cdot
p^{\prime})-m_{1}^{2})-p^{\mu}{p^{\prime}}^{\nu}-p^{\nu}{p^{\prime}}^{\mu}}\right).$

$k$$\mu_{1}\mu_{2}\mu_{3}\mu_{4}$$\nu_{1}\nu_{2}\nu_{3}\nu_{4}$ $=\hskip
34.1433pt\int_{0}^{\infty}\text{d}\mu^{2}\rho_{2}\mathopen{}\mathclose{{}\left(\mu^{2}}\right)\frac{\mathrm{i}\hat{\Pi}_{2}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\nu_{1}\nu_{2}\nu_{3}\nu_{4}}(k)}{k^{2}-\mu^{2}+\mathrm{i}0},$

where $\hat{\Pi}_{2}\coloneqq\mathcal{P}\cdot\Sigma(k)$, with

$\displaystyle\mathcal{P}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\nu_{1}\nu_{2}\nu_{3}\nu_{4}}_{\rho_{1}\rho_{2}\rho_{3}\rho_{4}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}$
$\displaystyle\coloneqq\frac{1}{8}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\delta^{[\mu_{1}}_{\rho_{1}}\delta^{\mu_{2}]}_{\rho_{2}}\delta^{[\mu_{3}}_{\rho_{3}}\delta^{\mu_{4}]}_{\rho_{4}}+\delta^{[\mu_{3}}_{\rho_{1}}\delta^{\mu_{4}]}_{\rho_{2}}\delta^{[\mu_{1}}_{\rho_{3}}\delta^{\mu_{2}]}_{\rho_{4}}}\right)\mathopen{}\mathclose{{}\left(\delta^{[\nu_{1}}_{\sigma_{1}}\delta^{\nu_{2}]}_{\sigma_{2}}\delta^{[\nu_{3}}_{\sigma_{3}}\delta^{\nu_{4}]}_{\sigma_{4}}+\delta^{[\nu_{3}}_{\sigma_{1}}\delta^{\nu_{4}]}_{\sigma_{2}}\delta^{[\nu_{1}}_{\sigma_{3}}\delta^{\nu_{2}]}_{\sigma_{4}}}\right)}\right.$
$\displaystyle\hskip
42.67912pt+\mathopen{}\mathclose{{}\left.\mathopen{}\mathclose{{}\left(\delta^{[\nu_{1}}_{\rho_{1}}\delta^{\nu_{2}]}_{\rho_{2}}\delta^{[\nu_{3}}_{\rho_{3}}\delta^{\nu_{4}]}_{\rho_{4}}+\delta^{[\nu_{3}}_{\rho_{1}}\delta^{\nu_{4}]}_{\rho_{2}}\delta^{[\nu_{1}}_{\rho_{3}}\delta^{\nu_{2}]}_{\rho_{4}}}\right)\mathopen{}\mathclose{{}\left(\delta^{[\mu_{1}}_{\sigma_{1}}\delta^{\mu_{2}]}_{\sigma_{2}}\delta^{[\mu_{3}}_{\sigma_{3}}\delta^{\mu_{4}]}_{\sigma_{4}}+\delta^{[\mu_{3}}_{\sigma_{1}}\delta^{\mu_{4}]}_{\sigma_{2}}\delta^{[\mu_{1}}_{\sigma_{3}}\delta^{\mu_{2}]}_{\sigma_{4}}}\right)}\right],$
(110)

and

$\displaystyle\Sigma^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\nu_{1}\nu_{2}\nu_{3}\nu_{4}}(k)$
$\displaystyle=\frac{8}{3}\mathopen{}\mathclose{{}\left(2\eta^{\mu_{1}\nu_{4}}\eta^{\mu_{2}\nu_{3}}\eta^{\mu_{3}\nu_{2}}\eta^{\mu_{4}\nu_{1}}+2\eta^{\mu_{1}\nu_{4}}\eta^{\mu_{2}\nu_{2}}\eta^{\mu_{3}\nu_{3}}\eta^{\mu_{4}\nu_{1}}-\eta^{\mu_{1}\mu_{4}}\eta^{\mu_{2}\mu_{3}}\eta^{\nu_{1}\nu_{4}}\eta^{\nu_{2}\nu_{3}}}\right)$
$\displaystyle\hskip
14.22636pt-\frac{32}{k^{2}}\mathopen{}\mathclose{{}\left(2k^{\mu_{1}}k^{\mu_{3}}\eta^{\mu_{2}\nu_{4}}\eta^{\mu_{4}\nu_{2}}\eta^{\nu_{1}\nu_{3}}+2k^{\mu_{1}}k^{\nu_{1}}\eta^{\mu_{2}\nu_{4}}\eta^{\mu_{3}\nu_{3}}\eta^{\mu_{4}\nu_{2}}-2k^{\mu_{3}}k^{\nu_{1}}\eta^{\mu_{1}\nu_{4}}\eta^{\mu_{2}\nu_{3}}\eta^{\mu_{4}\nu_{2}}}\right.$
$\displaystyle\hskip
56.9055pt\mathopen{}\mathclose{{}\left.+k^{\mu_{1}}k^{\mu_{3}}\eta^{\mu_{2}\mu_{4}}\eta^{\nu_{1}\nu_{4}}\eta^{\nu_{2}\nu_{3}}+2k^{\mu_{1}}k^{\nu_{1}}\eta^{\mu_{2}\mu_{4}}\eta^{\mu_{3}\nu_{4}}\eta^{\nu_{2}\nu_{3}}}\right)$
$\displaystyle\hskip
14.22636pt+\frac{128k^{\mu_{1}}k^{\mu_{3}}k^{\nu_{1}}k^{\nu_{3}}}{(k^{2})^{2}}\mathopen{}\mathclose{{}\left(2\eta^{\mu_{2}\nu_{4}}\eta^{\mu_{4}\nu_{2}}-\eta^{\mu_{2}\mu_{4}}\eta^{\nu_{2}\nu_{4}}}\right).$
(111) $\alpha\beta$$\mu\nu\rho\sigma$$p$ $\displaystyle=\hskip 22.76219pt$
$\displaystyle-\frac{\mathrm{i}\kappa}{4}\mathopen{}\mathclose{{}\left[\eta_{\mu\alpha}\eta_{\sigma\beta}p_{\nu}p_{\rho}-\eta_{\nu\alpha}\eta_{\sigma\beta}p_{\mu}p_{\rho}+\eta_{\nu\alpha}\eta_{\rho\beta}p_{\mu}p_{\sigma}}\right.$
$\displaystyle\hskip
28.45274pt\mathopen{}\mathclose{{}\left.-\eta_{\mu\alpha}\eta_{\rho\beta}p_{\nu}p_{\sigma}+\eta_{\mu\beta}\eta_{\sigma\alpha}p_{\nu}p_{\rho}-\eta_{\nu\beta}\eta_{\sigma\alpha}p_{\mu}p_{\rho}}\right.$
$\displaystyle\hskip
28.45274pt\mathopen{}\mathclose{{}\left.+\eta_{\nu\beta}\eta_{\rho\alpha}p_{\mu}p_{\sigma}-\eta_{\mu\beta}\eta_{\rho\alpha}p_{\nu}p_{\sigma}}\right].$

##  Appendix E Master Integrals

To calculate the mass-shift (63) there are three integrals to evaluate, which
can in principle be performed in any order. The integrals in question, in
general take the form

$I_{k,n}=\int_{0}^{\infty}\mathrm{d}s\,s^{k}\int\hat{\mathrm{d}}^{4}q\,\hat{\delta}(u_{2}\cdot
q)\hat{\delta}(u_{1}\cdot q)e^{\mathrm{i}q\cdot
b}|q|^{2n}\int\hat{\mathrm{d}}^{D}\ell\frac{\hat{\delta}(u_{2}\cdot\ell-s)\hat{\delta}(u_{1}\cdot\ell)}{\ell^{2}(\ell+q)^{2}}\,,$
(112)

where we set $D=4$ since all integrals that are required in the main text are
finite. Furthermore we will only need the case where $k=2r$,
$r\in\mathds{Z}_{\geq 0}$. The approach we will take is to first evaluate the
spectral integral using

$\int_{0}^{\infty}\text{d}s\;s^{2r}\hat{\delta}\mathopen{}\mathclose{{}\left(u_{2}\cdot\ell-s}\right)=2\pi(u_{2}\cdot\ell)^{2r}\theta\mathopen{}\mathclose{{}\left(u_{2}\cdot\ell}\right)\,.$
(113)

By making a change of variables $\ell\rightarrow-\ell-q$ and using the trivial
identity $\theta(x)+\theta(-x)=1$, the resulting loop integrals can then be
simplified using

$\int\hat{\text{d}}^{4}\ell\mathopen{}\mathclose{{}\left[(u_{2}\cdot\ell)^{2r}\theta\mathopen{}\mathclose{{}\left(u_{2}\cdot
l}\right)}\right]\frac{\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)}{\ell^{2}(\ell+q)^{2}}=\frac{1}{2}\int\hat{\text{d}}^{D}\ell\frac{(u_{2}\cdot\ell)^{2r}\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)}{\ell^{2}(\ell+q)^{2}}.$
(114)

Next we need scalar loop integrals of the form

$\int\hat{\text{d}}^{4}\ell\frac{(u_{2}\cdot\ell)^{2r}\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)}{\ell^{2}(\ell+q)^{2}}\;\;=\;\;\frac{(-1)^{r}(2r)!}{2^{4r+3}(r!)^{2}}\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{r}\mathopen{}\mathclose{{}\left(-q^{2}}\right)^{\frac{2r-1}{2}},\hskip
14.22636ptr\in\mathds{Z}_{\geq 0}.$ (115)

This formula is derived by specializing to the rest frame of particle-1

$u_{1}^{\mu}=(1,0,\boldsymbol{0}),\hskip
28.45274ptu_{2}^{\mu}=(y,\sqrt{y^{2}-1},\boldsymbol{0}),\hskip
28.45274ptq^{\mu}=(0,0,\boldsymbol{q}),$ (116)

where

$y=u_{1}\cdot u_{2},\hskip
28.45274pt|\boldsymbol{q}|=\mathopen{}\mathclose{{}\left(-q^{2}}\right)^{1/2},$
(117)

giving

$\displaystyle\int\hat{\text{d}}^{4}\ell\;\frac{\mathopen{}\mathclose{{}\left(u_{2}\cdot\ell}\right)^{2r}\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot\ell}\right)}{\ell^{2}(\ell+q)^{2}}$
$\displaystyle=\int\hat{\text{d}}\ell_{0}\hat{\text{d}}\ell_{1}\hat{\text{d}}^{2}\boldsymbol{\ell}\;\frac{\mathopen{}\mathclose{{}\left(y\ell_{0}-\sqrt{y^{2}-1}\ell_{1}}\right)^{2r}\hat{\delta}\mathopen{}\mathclose{{}\left(\ell_{0}}\right)}{[\ell_{0}^{2}-\ell_{1}^{2}-\boldsymbol{\ell}^{2}][\ell_{0}^{2}-\ell_{1}^{2}-(\boldsymbol{\ell}+\boldsymbol{q})^{2}]}$
$\displaystyle=\frac{\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{r}}{\pi}\int_{0}^{\infty}\text{d}\ell_{1}\;\ell_{1}^{2r}\int\hat{\text{d}}^{2}\boldsymbol{\ell}\;\frac{1}{[\boldsymbol{\ell}^{2}+\ell_{1}^{2}][(\boldsymbol{\ell}+\boldsymbol{q})^{2}+\ell_{1}^{2}]}$
$\displaystyle=\frac{\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{r}}{\pi}\int_{0}^{\infty}\text{d}\ell_{1}\;\ell_{1}^{2r}\mathopen{}\mathclose{{}\left[\frac{\text{arcsinh}\mathopen{}\mathclose{{}\left(\frac{|\boldsymbol{q}|}{2\ell_{1}}}\right)}{\pi|\boldsymbol{q}|\sqrt{\boldsymbol{q}^{2}+4\ell_{1}^{2}}}}\right]$
$\displaystyle=\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{r}\mathopen{}\mathclose{{}\left(-q^{2}}\right)^{\frac{2r-1}{2}}\times\frac{1}{\pi^{2}}\int_{0}^{\infty}\text{d}x\;x^{2r}\mathopen{}\mathclose{{}\left[\frac{\text{arcsinh}\mathopen{}\mathclose{{}\left(\frac{1}{2x}}\right)}{\sqrt{1+4x^{2}}}}\right].$
(118)

The remaining integral is just a number, but diverges for
$r\in\mathds{Z}_{>0}$; we evaluate it for $r\in\mathds{R}$ in the range
$-1/2<r<1/2$ and analytically continue the result. Finally we take the Fourier
transform to impact parameter space. The derivation of the general result is
well-known, for example Appendix A of Herrmann:2021tct , and is repeated here
for convenience

$\displaystyle\int\hat{\text{d}}^{4}q\;\hat{\delta}\mathopen{}\mathclose{{}\left(u_{1}\cdot
q}\right)\hat{\delta}\mathopen{}\mathclose{{}\left(u_{2}\cdot
q}\right)e^{-\mathrm{i}q\cdot
b}\mathopen{}\mathclose{{}\left(-q^{2}}\right)^{-\alpha}=\frac{1}{\sqrt{y^{2}-1}}\frac{\Gamma\mathopen{}\mathclose{{}\left(1-\alpha}\right)}{2^{2\alpha}\pi\Gamma(\alpha)}|b|^{2\alpha-2},$
(119)

where by assumption $u_{1}\cdot b=u_{2}\cdot b=0$. For completeness we cases
needed in the main text

$\displaystyle I_{2,0}={}$
$\displaystyle\frac{1}{128}\frac{\sqrt{y^{2}-1}}{|b|^{3}}\,,$ (120)
$\displaystyle I_{4,0}={}$
$\displaystyle\frac{27}{2048}\frac{\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{3/2}}{|b|^{5}}\,,$
(121) $\displaystyle I_{2,1}={}$
$\displaystyle-\frac{9}{128}\frac{\sqrt{y^{2}-1}}{|b|^{5}}\,,$ (122)
$\displaystyle I_{6,0}={}$
$\displaystyle\frac{1125}{16384}\frac{\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{5/2}}{|b|^{7}}\,,$
(123) $\displaystyle I_{4,1}={}$
$\displaystyle-\frac{675}{2048}\frac{\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{3/2}}{|b|^{7}}\,,$
(124) $\displaystyle I_{2,2}={}$
$\displaystyle\frac{225}{128}\frac{\sqrt{y^{2}-1}}{|b|^{7}}\,.$ (125)

The integral $I_{2,0}$ is needed in the scalar computation, $I_{4,0},I_{2,1}$
are needed for the photon and $I_{6,0},I_{4,1},I_{2,2}$ are needed for the
graviton.

Let us also give an alternative derivation. First note that we can set $n=0$,
all other integrals are obtained through the usual trick of taking derivatives
with respect of $b$. Specifying to the rest frame of particle-1 in (116) and
performing the $\ell_{0}$ and $\ell_{1}$ integrations which sets $\ell_{0}=0$
and $\ell_{1}=-\frac{s}{\sqrt{y^{2}-1}}$, we find

$\displaystyle
I_{k,0}=\frac{1}{(y^{2}-1)}\int_{0}^{\infty}\mathrm{d}s\,s^{k}{}$
$\displaystyle\int\hat{\mathrm{d}}^{2}\boldsymbol{q}\,e^{\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{b}}\int\frac{\hat{\mathrm{d}}^{2}\boldsymbol{\ell}}{[\boldsymbol{\ell}^{2}+s^{2}/(y^{2}-1)][(\boldsymbol{\ell}+\boldsymbol{q})^{2}+s^{2}/(y^{2}-1)]}.$
(126)

We notice that this the Fourier transform of a convolution, such that

$\displaystyle I_{k,0}={}$
$\displaystyle\frac{1}{(y^{2}-1)}\int_{0}^{\infty}\mathrm{d}s\,s^{k}{}\mathopen{}\mathclose{{}\left[\int\frac{\hat{\mathrm{d}}^{2}\boldsymbol{q}\,e^{\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{b}}}{[\boldsymbol{q}^{2}+s^{2}/(y^{2}-1)]}}\right]^{2}\,.$
(127)

The Fourier transform is well-known,

$\int\frac{\hat{\mathrm{d}}^{2}\boldsymbol{q}\,e^{\mathrm{i}\boldsymbol{q}\cdot\boldsymbol{b}}}{\boldsymbol{q}^{2}+a^{2}}=\frac{1}{2\pi}K_{0}(a|b|)\,,$
(128)

where $K_{0}$ is a modified Bessel function of the second kind. It is perhaps
interesting to note that this implies that the Fourier transform of
$\operatorname{Im}\mathcal{M}(s,q^{2})$ decays exponentially as $e^{-2s|b|}$,
thus rendering the integrals finite assuming $\tilde{\rho}(s)$ grows at most
polynomial which can be arranged using proper subtraction terms in the KL
representation. Performing the Fourier integral, we find

$\displaystyle I_{k,0}={}$
$\displaystyle\frac{1}{4\pi^{2}(y^{2}-1)}\int_{0}^{\infty}\mathrm{d}s\,s^{k}K_{0}\mathopen{}\mathclose{{}\left[\frac{sb}{\sqrt{y^{2}-1}}}\right]^{2}=\frac{\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{\frac{k-1}{2}}\Gamma\mathopen{}\mathclose{{}\left(\frac{k+1}{2}}\right)^{3}}{16\pi^{3/2}\Gamma\mathopen{}\mathclose{{}\left(\frac{k}{2}+1}\right)}\frac{1}{|b|^{k+1}}\,.$
(129)

Taking repeated derivatives yields the desired result

$I_{k,n}=\frac{(-1)^{n}4^{n-2}\Gamma\mathopen{}\mathclose{{}\left(\frac{k+1}{2}}\right)^{3}\mathopen{}\mathclose{{}\left(\frac{k+1}{2}}\right)_{n}^{2}}{\pi^{3/2}\Gamma\mathopen{}\mathclose{{}\left(\frac{k}{2}+1}\right)}\frac{\mathopen{}\mathclose{{}\left(y^{2}-1}\right)^{\frac{k-1}{2}}}{|b|^{2n+1+k}},$
(130)

where $(x)_{n}$ is the rising factorial.

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