aainstitutetext: Perimeter Institute for Theoretical Physics, Waterloo, ON N2L
2Y5, Canadabbinstitutetext: Department of Physics and Astronomy, York
University, Toronto, Ontario, M3J 1P3, Canadaccinstitutetext: Society of
Fellows, Harvard University, Cambridge, MA 02138, USAddinstitutetext: Max
Planck Institute for Gravitational Physics (Albert Einstein Institute), Am
Mühlenberg 1, Potsdam 14476, Germany

# From Scattering in Black Hole Backgrounds to Higher-Spin Amplitudes: Part I

Yilber Fabian Bautista<EMAIL_ADDRESS>a,c Alfredo Guevara
<EMAIL_ADDRESS>d Chris Kavanagh<EMAIL_ADDRESS>d and Justin Vines<EMAIL_ADDRESS>
###### Abstract

The scattering of massless waves of helicity $|h|=0,\frac{1}{2},1$ in
Schwarzschild and Kerr backgrounds is revisited in the long-wavelenght regime.
Using a novel description of such backgrounds in terms of gravitating massive
particles, we compute classical wave scattering in terms of $2\to 2$ QFT
amplitudes in flat space, to all orders in spin. The results are Newman-
Penrose amplitudes which are in direct correspondence with solutions of the
Regge-Wheeler/Teukolsky equation. By introducing a precise prescription for
the point-particle limit, in Part I of this work we show how both agree for
$h=0$ at finite values of the scattering angle and arbitrary spin orientation.

Associated classical observables such as the scattering cross sections, wave
polarizations and time delay are studied at all orders in spin. The effect of
the black hole spin on the polarization and helicity of the waves is found in
agreement with previous analysis at linear order in spin. In the particular
limit of small scattering angle, we argue that wave scattering admits a
universal, point-particle description determined by the eikonal approximation.
We show how our results recover the scattering eikonal phase with spin up to
second post-Minkowskian order, and match it to the effective action of null
geodesics in a Kerr background. Using this correspondence we derive classical
observables such as polar and equatorial scattering angles.

This study serves as a preceding analysis to Part II, where the Gravitational
Wave ($h=2$) case will be studied in detail.

##  1 Introduction

Since the early days of General Relativity, the stability of spacetimes as
dictated by Einstein’s field equations has been an important open problem with
far-reaching consequences from mathematics to phenomenology. In their seminal
work Regge:1957td  (see also Chandrasekhar:579245 ) Regge and Wheeler (RW)
proved that the Schwarzschild black hole was stable under small perturbations
by introducing the differential equation

$\frac{d^{2}R_{\ell}(r)}{dr_{*}^{2}}+(E^{2}-V_{\ell}(r))R_{\ell}(r)=0\,,\quad
V_{\ell}(r)=\left(1-\frac{2GM}{r}\right)\left[\frac{\ell(\ell+1)}{r^{2}}+\frac{2GM(1-h^{2})}{r^{3}}\right]\,,$
(1)

governing the radial dependence of a perturbation with spin $h$ and orbital
angular momentum $\ell\geq|h|$. Here $r_{*}$ is the usual tortoise radial
coordinate. The Regge-Wheeler equation sets on the same footing the cases of a
scalar ($h=0$), neutrino ($|h|=1/2$), electromagnetic ($|h|=1$) and
gravitational ($|h|=2$) waves. In a more modern language, the full
perturbation is easily encoded in a Newman-Penrose scalar for spin $h$111In
the more standard notation of e.g. Newman:1961qr , this corresponds to the top
scalar $\boldsymbol{\Psi}_{(-2h)}$ for a helicity-$h$ field.

$\boldsymbol{\Psi}_{h}(t,r,\theta,\phi)\propto\frac{1}{r}e^{-iEt}\sum_{l=0}^{\infty}D_{r}^{h}R_{\ell}(r)\,\,{}_{h}Y_{\ell
0}(\theta,\phi)\,,$ (2)

where ${}_{h}Y_{\ell m}$ are spherical harmonics of spin-weight/helicity $h$
Goldberg:1966uu  and $D_{r}^{h}$ is a differential operator trivial for $h=0$.
Although closed-form solutions to (1) are rare, numerical and analytical
methods can be implemented to gain insight into rich black hole physics, see
e.g. Berti:2004md ; Fiziev:2005ki . In particular, for scattering situations
one is interested in the radiative content of $\boldsymbol{\Psi}_{h}$, which
is encoded in the asymptotic behaviour of $R_{\ell}(r)$ as $r\to\infty$
Newman:1968uj ; futterman88 . In fact, one can gain insight by thinking of
$\boldsymbol{\Psi}_{h}$ as a certain effective wavefunction and identifying
(1) as its associated time-independent Schrödinger equation. This hints that
Quantum Mechanics (QM) can be used effectively to describe the perturbations
$\boldsymbol{\Psi}_{h}$ provided a certain classical prescription is
implemented.

On the other hand, it is well known that scattering solutions in Quantum
Mechanics can be easily recovered by a more modern, second-quantized,
scattering amplitude in Quantum Field Theory (QFT). This suggests that
scattering governed by (1) can be described by a QFT amplitude featuring
massless states and no particle production. In PhysRevD.16.237  this intuition
was confirmed, and it was shown that such S-matrix is reproduced in the Born
approximation by the scattering of scalars ($h=0$), photons ($h=\pm 1$) and
gravitons ($h=\pm 2$) off a background current associated to the Schwarzschild
metric.

What is the difference between the amplitude computed from the Schrodinger-
like equation (1) and from the QFT picture? For a given order in $GM/r$,
corrections to (1) can only appear through $\frac{E}{M}\sim\frac{1}{rM}$,
where $r$ is conjugate to the momenta. Thus, we can disregard these
corrections in the QFT amplitude by implementing the ‘classical limit’
$E\to\hbar\omega$, $r\to r/\hbar$ and taking $\hbar\to 0$. This is equivalent
to a multiple soft limit where the momenta for both real and virtual massless
particles scales as $k_{i}=\hbar\hat{k}_{i}\to 0$. A multiple soft limit
approach has been used in diverse contexts to obtain classical observables,
see e.g. Strominger:2017zoo ; Guevara:2018wpp ; Guadagnini:2008ha ;
Laddha:2018rle , where the classical radiative field usually emerges as a
coherent state made of on-shell massless particles.

A purely second-quantized approach would also demand the source in (1), i.e.
the Schwarzschild black hole, to be represented by an interacting particle in
a scattering amplitude. This resonates with a remarkable fact, already
suggested in the seminal analysis of Duff:1973zz , that classical dynamics of
black holes can be obtained by representing them as fundamental massive
particles. Such correspondence has recently (re)emerged to gain deep insight
into the binary black hole problem Damour_2016 ; Cheung:2018wkq ;
Cheung:2020gyp ; Bern:2019nnu ; Bern:2019crd ; Bern:2021dqo ; Bjerrum-
Bohr:2018xdl ; Cristofoli:2019neg ; Bjerrum-Bohr:2019kec ; Bjerrum-
Bohr:2021din ; Bjerrum-Bohr:2021vuf ; DiVecchia:2020ymx ; DiVecchia:2021ndb ;
DiVecchia:2021bdo ; Kosower:2018adc ; Mogull:2020sak ; Kalin:2019rwq ;
Kalin:2019inp ; Levi:2020kvb ; Levi:2020uwu ; Kalin:2020mvi ; Kalin:2020fhe ;
Damour:2020tta ; Damour:2019lcq ; Maybee:2019jus ; Arkani-Hamed:2019ymq ;
Bern:2020buy ; Chung:2019duq ; Chung:2020rrz ; Cachazo:2017jef ;
Guevara:2017csg ; Guevara:2018wpp ; Guevara:2019fsj ; Aoude:2020onz ;
Goldberger:2020fot ; Goldberger:2017vcg ; Goldberger:2017ogt ;
Bautista:2019tdr ; Bautista:2019evw ; Herrmann:2021lqe ; Herrmann:2021tct ;
Jakobsen:2021smu ; Bern:2020uwk ; Jakobsen:2021lvp ; Mougiakakos:2021ckm ;
Cristofoli:2020uzm ; Vines:2018gqi . In particular, it has been shown that the
perturbative Schwarzschild metric can be directly mapped to a three-point
amplitude $A_{3}$ for a massive particle emitting a graviton Donoghue:1994dn ;
Bjerrum-Bohr:2018xdl ; Jakobsen:2020ksu ; PhysRevD.68.084005 ;
Mougiakakos:2020laz , where the latter encodes the gauge invariant on-shell
modes of the metric. The situation is even more interesting in the case of
spinning black holes, where it has been found that the linearized Kerr metric
can be mapped to $A_{3}$ for a massive particle of infinite quantum spin
Guevara:2018wpp ; Chung:2018kqs ; Arkani-Hamed:2019ymq ; Chung:2019yfs . The
fact that the higher-spin particle effectively models the Kerr black hole in a
classical regime can be phrased as the property of being minimally coupled to
gravity, in the sense that it has a smooth high-energy limit $M\to 0$ Arkani-
Hamed:2017jhn .

Higher multiplicity amplitudes, associated to a massive particle emitting
$n>1$ gravitons or massless particles, have also recently gathered attention
in connection to higher perturbative orders of the two-black hole problem.
More precisely, four-point amplitudes emerge naturally in unitarity cuts when
evaluating both conservative Guevara:2018wpp ; Guevara:2019fsj ; Bern:2019crd
; Bern:2019nnu ; Bern:2021dqo ; Bern:2020buy ; Bjerrum-Bohr:2013bxa ;
Cachazo:2017jef  and radiation Bautista:2019tdr ; Herrmann:2021lqe ;
Herrmann:2021tct  two-body effects. In that context, a concern was raised in
Arkani-Hamed:2017jhn ; Guevara:2018wpp ; Chung:2018kqs ; Chung:2019duq ;
Aoude:2020onz  regarding the appropriate higher-spin amplitude $A_{4}$ that
could model the interaction of spinning black holes. In fact, at four points
the aforementioned notion of minimal coupling breaks down, as any description
of higher-spin particles contains $1/M$ singularities.222In other words such
amplitudes hold only at low energies energies $E\ll M$. This has been observed
to be intimately tied to causality violation in classical regimes
PhysRevD.46.3529 ; Deser:2001dt ; Deser:2000dz ; Cucchieri:1994tx . Because of
this, no obvious candidate arises to encode the Kerr black hole in a four-
point amplitude.

In this two-part work we shed light on the the four-point amplitudes
associated to Schwarzschild and Kerr black holes. For this, we first define a
suitable classical limit of the four-point QFT amplitudes suitable for the
wave scattering process. We then establish a map from such classical amplitude
to the radiative NP scalar appearing in (1). Furthermore, to tackle the case
of spinning black holes/particles we appeal to the generalization of the
Regge-Wheeler equation for the Kerr black hole, introduced long ago by
Teukolsky 1973ApJ…185..635T ; articleteu2 ; 1974ApJ…193..443T . This equation
expresses a remarkable simplification in the treatment of Kerr perturbations
in contrast with generic backgrounds, and stands as a cornerstone of the
modern black hole perturbation theory (BHPT) framework (see e.g. Pound:2021qin
for a recent review). In this paper, Part I, we will consider the scattering
of waves of helicity $h=0,1/2,1$ and extend previous BHPT results
PhysRevD.16.237 ; Westervelt:1971pm ; Doran:2001ag ; Dolan:2007ut ;
Matzner1968 ; Chrzanowski:1976jb ; PhysRevD.13.775  to all orders in spin, by
introducing novel Feynman diagrammatic rules for our QFT higher-spin
amplitudes. The matching sheds new light on different observable effects that
are well-known for scattering in Kerr: For instance, we derive how the spin of
the black hole induces polarization on the waves and we provide a new argument
for classical conservation of their helicity. Furthermore, unlike the $h=2$
case which will be studied in Part II, these amplitudes do not contain $1/M$
singularities and are in a sense directly connected to the minimal-coupling
three-point vertices, thereby naturally extending such a notion to four-
points.

To match the NP scalars to amplitudes at weak coupling we will consider the
long-wavelength regime $GM\omega=\pi r_{S}/\lambda\ll 1$ in which the black
hole behaves as a point-like object. In the spinning case, we find that a
natural prescription in such limit is to consider the leading order in
$GM/a\to 0$ of solutions to the Teukolsky equation, which introduces an
analytic continuation of the physical region $GM/a\geq 1$ . The result then
yields a series of multipole corrections in powers of $a\omega=2\pi
a/\lambda$, reflecting that the scattered wave can indeed probe the internal
structure of the spinning body.

As an important consistency check we will consider the geometrical optics,
i.e. eikonal, approximation tHooft:1987vrq ; Amati:1987wq ; Amati:1987uf ;
Amati:1990xe ; Kabat:1992tb ; Verlinde:1991iu ; Camanho:2014apa ;
Akhoury:2013yua ; Melville:2013qca ; DiVecchia:2019kta ; brittin1959lectures ,
arising at small scattering angle and large orbital angular momentum in our
classical amplitudes. It can already be argued from the potential in (1) that
the $\ell\to\infty$ regime of the wave scattering process is universal
(independent of $h$) and is controlled by the effective potential of a null
geodesic. Indeed, we will show that by exponentiating the QFT amplitudes for
$h\leq 2$ in the eikonal limit we can recover the scattering observables of a
null geodesic, such as Shapiro time delay, establishing a wave-particle
duality on the classical side. This provides a three-way crosscheck: On the
one hand we match the results from the Teukolsky equation to the eikonal phase
from spinning QFT amplitudes, up to 1-loop. On the other hand, from the
eikonal phase we derive two different scattering angles, namely polar and
equatorial, that then match those computed from null geodesics of a Kerr
background.

This paper is organized as follows. In section 2 we consider the scattering of
waves of $h=0,1/2,1,2$ in the Schwarzschild background. We define the
classical limit of 4-point amplitudes and establish the link with the Newman-
Penrose scalar. In section 3 we then extend the construction to the Kerr
background/higher-spin amplitudes. We also revisit observable classical
effects such as spin-induced polarization of the wave and helicity
violation/conservation. In section 4 we introduce the BHPT framework and
present the results for the infinite series of partial waves in the
aforementioned point-like limit. After projecting the previous NP scalars into
such partial waves we obtain a perfect match. Finally, section 5 introduces
the particle-wave duality in the eikonal regime and presents the matching to
null geodesics. In Appendix A we revisit the solutions of the Teukolsky
equation using BHPT, whereas Appendix B provides a derivation of eikonal
universality from 4-pt amplitudes.

##  2 Scattering of Waves in Schwarzschild Backgrounds

From the perspective of scattering amplitudes in QFT, the Schwarzschild Black
Hole (SBH) of $r_{S}=2GM$ can be modeled by a scalar particle of mass $M$
which is minimally coupled to gravitons Duff:1973zz . More generally, it has
been observed the scalar, neutrino, electromagnetic and gravitational wave
perturbations in such background can be modeled by the classical limit of
massless particles of helicity $h\leq 2$ Guadagnini:2008ha ; PhysRevD.16.237 .

In order to define the classical piece of the QFT amplitude and link it to
wave scattering we proceed as follows. The null momenta of the massless
particles, $k_{i}$, is to be identified with the classical wavenumbers,
$\hat{k}_{i}$, corresponding to the direction of wave propagation. Thus, this
leads us to write

$k_{i}=\hbar\hat{k}_{i}\,,[\hat{k_{i}}]=[1/L]\,,$ (3)

as $\hbar\to 0$. This scaling will be sufficient for the QFT scattering
amplitudes we are interested in, involving a single matter line (see also
Kosower:2018adc ; Bautista:2019evw ; Bautista:2019tdr ). For such case this
also implies that internal massless momentum $q=\sum_{i}k_{i}$ has the same
behaviour, $q=\hbar\hat{q}$. This is precisely the $\hbar$ deformation
explained in the introduction: The scaling $E\to\hbar\omega$ corresponds to
the frequency of external momentum whereas $r\to\hbar^{-1}r$ corresponds to
the scaling (massless) momentum transfer $q\sim 1/r$.

Consider now the object $A_{4}^{h}$ representing the $D=4$ four-point
scattering amplitude333The scattering amplitude is defined in such a way that
the momentum conservation delta function is striped away, and overall factors
of $i$ are removed. We work in the mostly minus signature for the metric,
$\eta=\rm{diag}(1,-1,-1,-1)$. of two massive scalar legs of momenta $p_{1}$
and $p_{4}$ and two massless legs of momenta $k_{2}$ and $k_{3}$ (with certain
helicity $\pm h$). In the classical interpretation, the massive momenta will
be associated to initial and final states of the black hole, whereas the
massless momenta represent the incident and scattered wave. The classical
limit is the leading order term in the $\hbar\rightarrow 0$ expansion of the
amplitude,

$\langle A_{4}\rangle:=\lim_{\hbar\rightarrow 0}A_{4}\,.$ (4)

To make contact with BHPT we choose to evaluate it in the reference frame for
which the black hole (BH) is initially at rest and the scattering process is
restricted to the $x-z$ plane. By adopting the scaling given in (3) and the
rest frame for $p_{1}$, the momenta of the particles read

$\begin{split}p_{1}^{\mu}&=(M,0,0,0),\\\ k_{2}^{\mu}&=\hbar\omega(1,0,0,1),\\\
k_{3}^{\mu}&=\frac{\hbar\omega(1,\sin\theta,0,\cos\theta)}{1+2\frac{\hbar\omega}{M}\sin^{2}(\theta/2)},\\\
p_{4}^{\mu}&=p_{1}^{\mu}+k_{2}^{\mu}-k_{3}^{\mu},\end{split}$ (5)

where the form of the energy of the outgoing wave $k_{3}$ is fixed by the on-
shell condition for the outgoing massive particle. The independent kinematic
invariants are

$\begin{split}s&=(p_{1}+k_{2})^{2}=M^{2}\left(1+2\frac{\hbar\omega}{M}\right)\,,\\\
t&=(k_{3}-k_{2})^{2}=-\frac{4\hbar^{2}\omega^{2}\sin^{2}\left(\theta/2\right)}{1+2\frac{\hbar\omega}{M}\sin^{2}(\theta/2)}\,.\end{split}$
(6)

It will also be convenient to introduce the optical parameter:

$\xi^{-1}:=-\frac{M^{2}t}{(s-M^{2})(u-M^{2})}=\sin^{2}(\theta/2)\,.$ (7)

We can now distinguish the following expansions arising in the scattering of
waves off a spinless massive object:

  * •

The classical expansion in the dimensionless combination $\epsilon=2GM\omega$.
This is in correspondence to the perturbative (loop) expansion in the QFT
amplitude and classically corresponds to a weak-field expansion. We will work
at leading order in $GM\omega$, which corresponds to tree-level amplitudes. As
anticipated, the frequency $\omega$ is also the energy of the momentum
transfer $k_{3}-k_{2}\sim 1/r$: This means that the classical combination is
$GM\omega\sim GM/r$ and hence BH-horizon effects should not be relevant in the
weak-coupling limit.

  * •

Quantum corrections suppressed by $\frac{\hbar\omega}{M}$. These corrections
appear in $A_{4}^{h}$ as powers of $\frac{(s-M^{2})}{M^{2}}$ or
$\frac{t}{M^{2}}$ and will be dropped for our matching. We see that the
classical deformation $\frac{\hbar\omega}{M}\to 0$ can be understood as a
limit in which massless particles become soft or equivalently the massive
particle becomes very heavy. Note that the opposite (high-energy/massless)
limit $\frac{\hbar\omega}{M}\to\infty$ for which our effective description can
break down as explained in the introduction, simply lies outside the classical
regime.

  * •

The eikonal (geometrical optics) expansion as $\xi^{-1}\sim\theta\to 0$, which
will be studied in detail in Section 5. The eikonal has been recently
exploited to extract classical observables of the two-body problem via a WKB-
type analysis Bjerrum-Bohr:2016hpa ; AccettulliHuber:2020oou ;
Cristofoli:2020hnk ; Guevara:2018wpp  and we will obtain related results that
link wave scattering to unbounded null geodesics. In this regime $t\ll
s-M^{2}\ll M^{2}$ and hence only the t-channel factorization
$A^{h}_{4}\sim\frac{1}{t}A_{3}A^{\prime h}_{3}$ contributes to the amplitude.
Nevertheless, note that for generic wave scattering we are able to extract the
classical piece at all orders in $\xi^{-1}$ and not only the eikonal, which
demands to include all channels and contact terms in the evaluation of (4).
This non-linear dependence in $\theta$ is crucial to match the partial
amplitudes $A_{\ell m}Y_{\ell m}(\theta,\phi)$ resulting from the BHPT
formalism.

Further considerations are required for the spin expansion, see Section 3. In
a nutshell, we shall adopt the classical limit simply by setting $\hbar=1$ and
then taking the "multisoft" limit $\omega\to 0$ at a fixed power of $G$. From
a classical perspective, this corresponds to the long-wavelength regime
$2GM\omega=\pi r_{S}/\lambda\ll 1\,$ 1964PhDT……..64H ; Matzner1968 ;
Mashhoon1973zz ; PhysRevD.10.1059 , for which the internal structure of the
classical object (including e.g. dispersive effects/quasinormal modes) cannot
be probed at leading order. Such information would in principle require
corrections in $GM\omega$ corresponding to loop diagrams. Nevertheless, for
the case of spinning bodies we will see that there are indeed tree-level
corrections $(a\omega)\sim 1/\lambda$ that effectively probe the internal
structure.

The classical amplitude as defined in (4) transforms covariantly under a
$U(1)$ phase, corresponding to the little group of the massless $|h|\leq 2$
states. We will take care of the phase ambiguity when performing the matching
to classical scattering. As suggested by related results in quantum and
classical scattering Bjerrum-Bohr:2016hpa ; 1964PhDT……..64H ; Matzner1968 ;
Mashhoon1973zz ; PhysRevD.10.1059  the matching may also be performed
unambiguously in terms of the unpolarized classical differential cross
section. For that we use the well known formula for the differential cross
section in QFT, and then proceed to take its classical limit according to our
prescription. Let us assume that the incoming massless particles have fixed
helicity $h>0$, whereas the outgoing massless particle can in general have a
different helicity $h^{\prime}=\pm h$. Then, the unpolarized differential
cross section will be given by

$d\sigma=\sum_{h^{\prime}}\frac{|A_{4}(h\rightarrow h^{\prime})|^{2}d{\rm
LIPS}_{2}}{2E_{1}2E_{2}|\vec{v}_{1}-\vec{v}_{2}|}\,,$ (8)

where the sum runs over all the polarization states for the outgoing massless
particle, and the two particle Lorentz invariant phase space has the simple
form

$d{\rm LIPS}_{2}=\frac{s-M^{2}}{32\pi^{2}s}d\Omega\,.$ (9)

Noting that in the classical limit
$k_{3}^{\mu}\rightarrow\omega(1,\sin\theta,0,\cos\theta)$, the differential
cross section simply becomes

$\frac{d\langle\sigma\rangle}{d\Omega}=\sum_{h^{\prime}}\frac{|\langle
A_{4}(h\rightarrow h^{\prime})\rangle|^{2}}{64\pi^{2}M^{2}}\,.$ (10)

The impinging wave can also be unpolarized. In such case, the helicity states
for both in and out waves allow us to define the elements of the scattering
matrix as follows

$A_{4}^{h}=\left[\begin{array}[]{cc}A_{++}^{h}&A_{+-}^{h}\\\
A_{-+}^{h}&A_{--}^{h}\end{array}\right],$ (11)

where the sub-indices denote the polarization of the incoming and outgoing
wave respectively, and $h$ denotes the nature of the wave444 It is understood
that for $h=0$, the scattering matrix is one dimensional.. We associate $+h$
($-h$) states with circular left (right) wave polarizations. Motivated by the
discussion of wave polarization in the next sections, we will refer to the
diagonal elements $A^{h}_{++},A^{h}_{--}$ as helicity preserving amplitudes,
and to the off diagonals $A^{h}_{+-},A^{h}_{-+}$ as helicity reversing. An
important caveat here is that the helicity of particle $k_{3}$ appears flipped
with respect to somewhat standard conventions: As $k_{3}$ is outgoing with
helicity $h^{\prime}$ it is equivalent to an incoming particle with helicity
$-h^{\prime}$. Writing the scattering matrix in this fashion will become
useful when we discuss the effect of the BH on the polarization of the
impinging wave, as we will do in the next sections.

###  2.1 Scalar Waves

To illustrate how the computation of the amplitude works, let us proceed to
reproduce the result for the classical scattering of a massless scalar wave
off the SBH. To compute the scattering amplitudes throughout this section we
will use the Feynman rules derived in the harmonic gauge, see e.g.
Holstein:2008sx . The metric perturbation is defined as

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}\,,$ (12)

with $\kappa^{2}=32\pi G$. The gauge condition is

$\partial^{\nu}h_{\mu\nu}-\frac{1}{2}\partial_{\mu}h^{\nu}_{\,\,\nu}=0\,.$
(13)

To leading order in $G$ there is only one Feynman diagram contributing to the
amplitude for the scattering the wave off the SBH. This diagram corresponds to
the one graviton exchange between one massive and one massless scalar. To
compute this amplitude we use the minimal coupling 3-vertex for two scalars
legs and one graviton

$V_{\phi\phi
h}^{\mu\nu}=-i\kappa\left[\mathcal{P}_{\mu\nu\alpha\beta}p_{1}^{\alpha}p_{4}^{\beta}+\frac{M}{2}\eta^{\mu\nu}\right],$
(14)

where $\mathcal{P}_{\mu\nu\alpha\beta}$ is the trace-reverser operator555It
satisfies
$\mathcal{P}_{\mu\nu\alpha\beta}\mathcal{P}^{\alpha\beta\rho\sigma}=\delta_{\mu}^{(\rho}\delta_{\nu}^{\sigma)}$.

$\mathcal{P}_{\mu\nu\alpha\beta}=\frac{1}{2}[\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-\eta_{\mu\nu}\eta_{\alpha\beta}]\,.$
(15)

The vertex (14) can be also used for the massless scalar by setting $M=0$. We
will also need the graviton propagator,

$D_{\mu\nu\alpha\beta}=\frac{i\mathcal{P}_{\mu\nu\alpha\beta}}{q^{2}+i\epsilon}\,.$
(16)

When working with tree-level amplitudes we can ignore the contour $i\epsilon$
prescription. The computation of the tree-level scattering amplitude is
straightforward and leads to

$A_{4}^{h=0}=\frac{i\kappa^{2}p_{1}{\cdot}k_{2}}{2q^{2}}\left[2p_{1}{\cdot}k_{2}+q^{2}\right]\,,$
(17)

where $q=k_{3}-k_{2}$ is the momentum transfer. Using the kinematics
$(\ref{eq:kinematics})$, the classical amplitude $(\ref{eq:classical
amplitude})$ takes the simple form

$\langle
A_{4}^{h=0}\rangle=\frac{i\kappa^{2}(p_{1}{\cdot}k_{2})^{2}}{q^{2}}=\frac{\kappa^{2}M^{2}}{4\sin^{2}\left(\theta/2\right)},$
(18)

In the next section we will show that the Newman-Penrose amplitude, obtained
from the scalar (2) as $r\to\infty$, indeed takes the form

$\boldsymbol{\Psi}_{0}\sim\frac{1}{r}\langle A_{4}^{h=0}\rangle
e^{-i\omega(t-r)}\,.$ (19)

This definition then will naturally extend to waves carrying helicity.

By virtue of equation $(\ref{eq:classical cross section})$ we can directly
obtain the classical differential cross section for the scattering of a scalar
wave off the SBH:

$\frac{d\langle\sigma\rangle}{d\Omega}=\frac{G^{2}M^{2}}{\sin^{4}\left(\theta/2\right)}\,.$
(20)

This is nothing but the gravitational version of the standard Rutherford
cross-section. It agrees with the seminal derivation given in Matzner1968 ;
Chrzanowski:1976jb ; PhysRevD.13.775 , see also section 4.1. The divergence
$\approx\frac{16G^{2}M^{2}}{\theta^{4}}$ for forward scattering $\theta\to 0$
is a universal feature in all the examples we will study. It follows from the
long-range nature of gravity. Indeed, it is well known that Newtonian
interaction yields the classical scattering angle $\theta=4GM/b$ as a function
of the impact parameter $b$. This means that the scattering cross-section
should behave as

$\displaystyle d\langle\sigma\rangle$ $\displaystyle=bdbd\phi\,,$
$\displaystyle\approx\frac{16G^{2}M^{2}}{\theta^{3}}d\theta d\phi\,,$
$\displaystyle\approx\frac{16G^{2}M^{2}}{\theta^{4}}d\Omega\,.$ (21)

To close this section, we note that the classical amplitude (18) for the
scattering of scalar waves can be reproduced solely by its the residue as
$t\to 0$, corresponding to a three-point factorization. This in general does
not hold true for waves of higher helicity $|h|>0$. Roughly, this is because
in such case the factor $(s-m^{2})^{2}$ can appear in the residue as $t\to 0$,
for which the combination $(s-m^{2})^{2}/t$ has the same order in $\hbar$ as a
contact term $\mathcal{O}(t^{0})$. This leads to the important conclusion that
contact terms in the momentum transfer $q$, which in position space are
localized along the BH singularity at $r=0$, can indeed contribute to wave
scattering.666Dropping contact terms in $t$ raised some doubts about the gauge
invariance of the result of PhysRevD.16.237  when considering the scattering
of gravitational waves off the SBH. It also led to discrepancies in the
literature results for the induced polarization for the scattering of a
gravitational wave (GW) off the Kerr black hole to linear order in the spin
Guadagnini:2008ha ; Dolan:2008kf . We will comment on this issue in section
3.3 and in BCGV .

###  2.2 Relation to Scattering in Schwarzschild

Before we study the case of waves with helicity it is instructive to
demonstrate how the above computation relates to a scalar wave scattering off
the Schwarzschild black hole.

For a scalar wave the RW equation (1) given in the introduction is nothing but
the Klein-Gordon equation defined on a Schwarzschild background. The Born
approximation of the RW equation ($GM\omega\to 0$), which should connect to
tree-level amplitudes, can therefore be obtained from the Klein-Gordon
equation using the metric of the form (12) (which is linearized in $\kappa$).
Using the harmonic gauge (13) the scalar wave equation becomes

$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi=-\kappa
h^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi+\mathcal{O}(\kappa^{3})\,,$ (22)

In contrast with the previous approach here $h_{\mu\nu}$ is not dynamical, but
instead the fixed background whose explicit form will not be needed. Regarding
the RHS as an interaction potential, scattering can be studied to leading
order in $\kappa$ by implementing the corresponding Lippmann-Schwinger
equation. That is, we expand the field into a combination of an incoming plane
wave and a scattered wave as follows. Setting $k^{2}=0,k^{0}=\omega$, we write

$\psi(x)=e^{ik\cdot x}\phi^{\textrm{PW}}+e^{-i\omega
t}\phi^{\textrm{S}}(x_{i})\,,$ (23)

so that we obtain the three-dimensional equation

$(\omega^{2}+\vec{\nabla}^{2})\phi^{\textrm{S}}(\vec{x})=\kappa
k^{\mu}k^{\nu}h_{\mu\nu}(\vec{x})e^{i\vec{k}{\cdot}\vec{x}}\phi^{\textrm{PW}}+\mathcal{O}(\kappa^{3})\,.$
(24)

This is the leading Born approximation. Setting $\phi^{\textrm{PW}}=1$ for
simplicity, we define the classical current as given by the Fourier transform
of the source in the RHS

$\displaystyle j(\vec{k}^{\prime})=$
$\displaystyle\int\frac{d^{3}\vec{y}}{(2\pi)^{3}}e^{-i\vec{k}^{\prime}{\cdot}\vec{y}}\times\left(\kappa
k^{\mu}k^{\nu}h_{\mu\nu}(\vec{y})e^{i\vec{k}{\cdot}\vec{y}}\right)\,,$
$\displaystyle=$ $\displaystyle\kappa
k^{\mu}k^{\nu}\bar{h}_{\mu\nu}(\vec{q})\,,$ (25)

where $\vec{q}=\vec{k}^{\prime}-\vec{k}$ is the 3-momentum transfer and
$\bar{h}_{\mu\nu}(\vec{q})$ is the three-dimensional Fourier transform of the
linearized Schwarzschild metric. It can be read off from equations (67) -(69)
below with $u=(1,0,0,0)$ and $q^{0}=0$:

$\bar{h}_{\mu\nu}(q)=\frac{\kappa
M^{2}}{q^{2}}\mathcal{P}_{\mu\nu\rho\sigma}u^{\rho}u^{\sigma}\,.$ (26)

It then becomes clear that this can be interpreted as an effective 3-point
vertex including the graviton propagator, and that (25) is nothing but the
t-channel diagram leading to the classical amplitude (18)

$j(k^{\prime})=\kappa k^{\mu}k^{\nu}\bar{h}_{\mu\nu}(q)=\langle
A_{4}^{h=0}\rangle\,,$ (27)

with $p_{1}=Mu$. The condition $q^{0}=0$ is just the on-shell condition
$M^{2}=(p_{1}+q)^{2}\Rightarrow p_{1}\cdot q=0$ in the classical limit. It
implies that (26) is indeed in the harmonic gauge
$q^{\mu}\bar{h}_{\mu\nu}=\frac{1}{2}q_{\nu}\bar{h}$. Because
$\bar{h}_{\mu\nu}$ is off-shell, we note that (26) has an ambiguity given by
contact terms which are regular in $q^{2}$, as anticipated. More precisely,
such terms Fourier transform to $\delta^{3}(\vec{x})$ terms in position space
and therefore are invisible in the Schwarzschild metric. These terms are
important and can indeed contribute to the 4-point amplitude, as we will
discuss, but for now we assume they are absent.

Having elucidated the relation between the classical current and the
amplitude, we proceed to compute the Newman-Penrose field. We define it as the
scattered part of the scalar:

$\boldsymbol{\Psi}_{0}:=e^{-i\omega t}\phi^{S}(\vec{x})\,,$ (28)

where $\phi^{S}(x_{i})$ solves the wave equation (24). Using the source (25),
the solution is given by

$\boldsymbol{\Psi}_{0}(\vec{x},t)=e^{-i\omega
t}\phi^{S}(x_{i})=\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}e^{i(\vec{k}{\cdot}\vec{x}-\omega
t)}\frac{j(k)}{\omega^{2}-\vec{k}^{2}}\,.$ (29)

As expected, this is just the standard relation between the current and the
sourced field (we can assume $j(k)$ to be a function of the 4-vector
$(\omega,\vec{k})$). Note that integration in $\omega$ is not performed since
we are studying scattering in frequency domain, not in time domain. The
radiative data encoded in $\boldsymbol{\Psi}_{0}(\vec{x},t)$ can be extracted
by evaluating it at null infinity via $r=|\vec{x}|\to\infty$. In such limit
the integral is customary and easily evaluated using one of its saddle-points
(see e.g. Exercise 4 in Strominger:2017zoo ) we obtain

$\displaystyle\boldsymbol{\Psi}_{0}^{\textrm{rad}}(\vec{x},t)\sim\frac{e^{i\omega(r-t)}}{r}\times
j(k)\,.$ (30)

where $\vec{k}=\omega\vec{x}/r$ is fixed as pointing to a direction in the
celestial sphere. This is the fundamental relation between the NP components
and the amplitude: It links the radiative components of the field, given by
the NP scalar, to the phase space of massless particles given by the classical
amplitude (see Frolov:1977bp ; Ashtekar:1987tt ; Strominger:2013lka  for
appropriate characterizations of the phase space). Here we have identified
$j(k)$ as the classical limit of a quantum current.777Indeed, this is closely
related to the construction of Strominger:2013lka  and references therein. The
identification of quantum and classical currents there is obtained by
considering coherent states, see also CGKO . In the next sections we will deal
with its natural extension to spinning waves.

Some comments are in order. On the one hand, the argument can be easily seen
to make no assumption about the specific form of the metric, which in
particular could have been that of a (linearized) spinning body. Thus, our
argument relating the scalar amplitude to the NP scalar indeed holds for a
Kerr background as well. On the other hand, the Born approximation used here
does not take into account several effects that are indeed present in the RW
equation: First, it ignores the long-distance drag of the gravitational force,
signaled by the presence of $r^{*}=r+\mathcal{O}(G)$ in (1). Indeed, as
explained in eq. 90 the fall-off of the scattered wave should be $\phi^{S}\sim
e^{i\omega r*}/r$ instead of (30). This however does not spoil the relation of
the scattering amplitude to the classical current, which is independent of
$r$. Second, the Born approximation ignores an overall IR phase known as the
Newtonian (or Coulombian) phase. This phase does not contribute to the cross-
section, but can be easily reintroduced through the eikonal framework as done
in section 5. In the end, our aim is to show that even though these effects
are built-in in the RW or Teukolsky equations, we can still match their
solutions to our QFT amplitudes if we define a suitable ‘Born limit’
$GM\omega\to 0$ for the former. This will be particularly powerful for waves
with spin (such as GW) as the corresponding non-linear wave equations turn out
to be considerably more complicated than the Klein-Gordon equation, whereas
the RW/Teukolsky system has essentially the same complexity for all
helicities.

###  2.3 Spinning Waves of $h\leq 1$

Our next aim is to relate the scattering of spinning waves off the
Schwarzschild black hole to the classical limit of the corresponding QFT
amplitudes. We will use the electromagnetic case $h=1$ as our guiding example.
Since the observable amplitude carries helicity dependence, it is convenient
to project it into helicity components. These are precisely given by the NP
scalars. We will also incorporate the limiting procedure of the previous
section to present the correspondence between quantum and classical helicity
amplitudes.888A related comprehensive treatment of the classical limit in the
context of the Kosower-Maybee-O’Connell formalism is to be given in CGKO .

To begin with, we note that we can decompose the full metric (12) into the
null tetrad

$g_{\mu\nu}=-2m_{(\mu}\bar{m}_{\nu)}+2l_{(\mu}n_{\nu)}\,,$ (31)

where the only non-vanishing inner products are $l\cdot n=-m\cdot\bar{m}=1$.
In the linearized regime $G\to 0$, such that $g_{\mu\nu}\to\eta_{\mu\nu}$, the
null tetrad becomes

$\begin{split}l^{\mu}&\to(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\,,\\\
n^{\mu}&\to{\color[rgb]{0,0,0}-}\frac{1}{2}(-1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\,,\\\
m^{\mu}&\to\frac{1}{\sqrt{2}}(0,\cos\theta\cos\phi-i\sin\phi,\cos\theta\sin\phi+i\cos\phi,-\sin\theta)\,,\\\
\bar{m}^{\mu}&\to\frac{1}{\sqrt{2}}(0,\cos\theta\cos\phi+i\sin\phi,\cos\theta\sin\phi-i\cos\phi,-\sin\theta)\,.\end{split}$
(32)

Note that there is an isometry subgroup $U(1)\subset SO(3,1)$ in
$\eqref{eq:gnullex}$ which rotates

$m^{\mu}\to e^{i\alpha}m^{\mu},\bar{m}^{\mu}\to e^{-i\alpha}\bar{m}^{\mu}\,,$
(33)

This transformation is the starting point of the formalism of Newman and
Penrose Newman_1962 , and characterizes the helicity of the states defined at
null infinity. Furthermore, if we identify the flat limit of $l^{\mu}$ with a
normalized massless momentum, $\hat{k}^{\mu}$, the conjugate pair
$m^{\mu},\bar{m}^{\mu}$ may be interpreted as the corresponding polarization
vectors $\epsilon^{+},{\color[rgb]{0,0,0}-}\epsilon^{-}$ of helicity $\pm 1$
respectively, satisfying

$\hat{k}\cdot\epsilon^{+}=\hat{k}\cdot\epsilon^{-}=0,\,\,\,\,\epsilon^{+}\cdot\epsilon^{-}=1\,.$
(34)

Following the argument of the previous section, we now consider the asymptotic
EM field for a mode with $k=\omega\hat{k}$999Here we can ignore the Coulomb
dragging which appears in the presence of long-range interactions. We will
come back to this point in section 4.1 and section 5.1 .

$A^{\textrm{rad}}_{\mu}(x)\propto j_{\mu}(k)e^{ik\cdot x}\,,$ (35)

where as explained only the on-shell modes, i.e. those with $k^{2}=0$, will
yield radiation flux and hence can be associated with the phase space of
massless particles. Again, as we approach future null infinity $t+r\to\infty$
only the modes with 3-momentum $\vec{k}$ aligned with $\vec{x}$ will
contribute to such flux. That is to say we will take

$\frac{\vec{k}}{\omega}=\frac{\vec{x}}{|\vec{x}|}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\Longrightarrow
k^{\mu}\to\omega l^{\mu}\,,$ (36)

in the $G\to 0$ limit, as anticipated. The classical scattering amplitude is
described by NP scalar in (2), which in this case is defined as futterman88

$\displaystyle\boldsymbol{\Psi}_{-1}^{\textrm{rad}}$ $\displaystyle:=$
$\displaystyle-\bar{m}^{\mu}n^{\nu}(\partial_{\mu}A_{\nu}^{\textrm{rad}}-\partial_{\nu}A_{\mu}^{\textrm{rad}})\,,$
(37)

carrying weight $h=-1$ under the helicity transformation (33), with the
conjugate case defined similarly. For gravitational scattering we know that
$A_{\mu}^{\textrm{rad}}=\mathcal{O}(G)$, hence hereafter we can consider the
null tetrad at the strict flat limit (32). Further using (35-36) we have

$\displaystyle\boldsymbol{\Psi}_{-1}^{\textrm{rad}}$ $\displaystyle\propto$
$\displaystyle-\epsilon_{-}^{\mu}n^{\nu}(k_{\mu}j_{\nu}(k)-k_{\nu}j_{\mu}(k))e^{\omega
l\cdot x}=\omega\,\epsilon_{-}^{\mu}j_{\mu}(k)e^{i\omega(t-r)}\,.$ (38)

The component $\epsilon_{-}^{\mu}j_{\mu}(k)$ is nothing but the EM current
projected onto a helicity state. It is related to the quantum current,
$J^{\mu}$ by our classical limit prescription. Following LSZ reduction we can
identify the latter with the scattering amplitude for the process under
consideration,

$\epsilon_{-}^{\mu}j_{\mu}(k)=\epsilon_{-}^{\mu}\langle
J_{\mu}(k)\rangle=\langle A_{4}^{h=-1}\rangle\,.$ (39)

In order to match (38) with the result of the Teukoslky equation we need to
perform an expansion into spin-weighted spherical harmonics carrying helicity
$h=-1$ under the transformation (33). This will be relegated to Part II, where
we will provide a simple representation of such harmonics together with a
projection method.

We are now in good position to evaluate the classical amplitude (39). First,
by choosing $\phi=0$ in our null tetrad (32) we note that the vector $l$ for
the scattered wave is precisely aligned with $k_{3}$ in (5). Thus we identify
$\theta$ as the scattering angle. The corresponding polarization directions
are read off from (32):

$\begin{split}\epsilon_{3}^{+}=&m=\frac{1}{\sqrt{2}}(0,\cos\theta,i,-\sin\theta)\,,\\\
\epsilon_{3}^{-}=&-\bar{m}=-\frac{1}{\sqrt{2}}(0,\cos\theta,-i,-\sin\theta)\,.\end{split}$
(40)

An analogous construction holds for the incoming wave. We set the incoming
momentum $k_{2}$ in (5) as corresponding to the same configuration with
$\theta=0$, namely

$\begin{split}k_{2}=&(1,0,0,1)\,,\\\
\epsilon_{2}^{+}=&\frac{1}{\sqrt{2}}(0,1,i,0)\,,\\\
\epsilon_{2}^{-}=&-\frac{1}{\sqrt{2}}(0,1,-i,0)\,.\end{split}$ (41)

To evaluate the graviton exchange diagram we will need the three-vertex of two
photons legs and one graviton. In the harmonic gauge it reads Holstein:2008sx
:

$\begin{split}V_{\gamma\gamma
h}^{\alpha,\beta,\mu\nu}&=-\frac{i\kappa}{2}\left[\eta^{\mu\nu}\left(k_{2}{\cdot}k_{3}\eta^{\alpha\beta}-k_{2}^{\beta}k_{3}^{\alpha}\right)-2I_{\,\,\,\,\kappa\lambda}^{\mu\nu}\left(k_{2}{\cdot}k_{3}I^{\alpha\beta,\kappa\lambda}+k_{2}^{(\kappa}k_{3}^{\lambda)}\right)\eta^{\alpha\beta}\right.\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.-\left(k_{2}^{\kappa}k_{3}^{\alpha}\eta^{\lambda\beta}+k_{3}^{\kappa}k_{2}^{\beta}\eta^{\lambda\alpha}\right)\right],\,\quad
I^{\alpha\beta,\gamma\delta}:=\frac{1}{2}\left(\eta^{\alpha\gamma}\eta^{\beta\delta}+\eta^{\alpha\delta}\eta^{\beta\gamma
s}\right).\end{split}$ (42)

The tree-level scattering amplitude is then obtained by gluing this vertex to
(14). It becomes

$\begin{split}A_{4}^{h=1}&=-\frac{\kappa^{2}}{4q^{2}}\left[p_{1}{\cdot}k_{2}\left(4p_{1}{\cdot}k_{2}+2q^{2}\right)\epsilon_{2}{\cdot}\epsilon_{3}^{*}+\left(2k_{2}{\cdot}\epsilon_{3}^{*}q{\cdot}\epsilon_{2}{+}q^{2}\epsilon_{2}{\cdot}\epsilon_{3}^{*}\right)M^{2}\right.\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.-2q^{2}p_{1}{\cdot}\epsilon_{2}\left(p_{1}{+}k_{2}\right){\cdot}\epsilon_{3}^{*}-4p_{1}{\cdot}k_{2}\left(p_{1}{\cdot}\epsilon_{2}k_{2}{\cdot}\epsilon_{3}^{*}{+}p_{1}{\cdot}\epsilon_{3}^{*}q{\cdot}\epsilon_{2}\right)\right],\end{split}$
(43)

where $2p_{1}\cdot k_{2}=(s-M^{2}),q^{2}=t$. As explained, in the classical
limit we need to keep all the terms that scale as $\omega^{0}$. This means
that the $q^{2}$ terms in the numerator, i.e. contact terms, will indeed
contribute at the same footing as the t-channel residue. The full classical
result is then

$\begin{split}\langle
A_{4}^{h=1}\rangle&=-\frac{\kappa^{2}}{4q^{2}}\left[\left(2p_{1}{\cdot}k_{2}\right)^{2}\epsilon_{2}{\cdot}\epsilon_{3}^{*}+\left(2k_{2}{\cdot}\epsilon_{3}^{*}q{\cdot}\epsilon_{2}{+}q^{2}\epsilon_{2}{\cdot}\epsilon_{3}^{*}\right)M^{2}\right.\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left.-2q^{2}p_{1}{\cdot}\epsilon_{2}p_{1}{\cdot}\epsilon_{3}^{*}-4p_{1}{\cdot}k_{2}\left(p_{1}{\cdot}\epsilon_{2}k_{2}{\cdot}\epsilon_{3}^{*}{+}p_{1}{\cdot}\epsilon_{3}^{*}q{\cdot}\epsilon_{2}\right)\right].\end{split}$
(44)

We now consider different helicity combinations, thereby evaluating the
elements of the scattering matrix (11). For the helicity preserving
combination,

$\langle A_{4,++}^{h=1}\rangle=\langle
A_{4,--}^{h=1}\rangle=\frac{\kappa^{2}M^{2}}{4}\frac{\cos^{2}\left(\frac{\theta}{2}\right)}{\sin^{2}\left(\frac{\theta}{2}\right)}\,.$
(45)

On the other hand, we find that in the classical limit the helicity reversing
amplitudes, $\langle A_{4,+-}^{h=1}\rangle,\langle A_{4,-+}^{h=1}\rangle$,
vanish. This is consistent with the intuition that in the soft limit, photons
$k_{2},k_{3}$ behave as a matter line of definite chirality, which sources a
classical stress-energy tensor (here represented by the exchanged soft
graviton Weinberg:1965nx ). At subleading order $\mathcal{O}(\hbar\omega)$
such soft behaviour is modified and the helicity of the photon can flip. This
picture is true as long as $h<2$, as for $h=2$ there emerge different Feynman
diagrams for graviton interactions leading to a helicity flip even at the
classical level $(\hbar\omega)^{0}$, as we will see in section 2.4.

To confirm the above picture about conservation of helicity, we briefly
discuss here the scattering of neutrino ($\pm h=\pm 1/2$) waves futterman88 .
For this we translate (32) into spinor-helicity variables, see Elvang:2013cua
for conventions. Using
$v_{\alpha\dot{\alpha}}=\sigma^{\mu}_{\alpha\dot{\alpha}}v_{\mu}$ to convert
Lorentz vectors $v^{\mu}$ into spinors, we obtain

$\displaystyle k_{\alpha\dot{\alpha}}=\omega
l_{\alpha\dot{\alpha}}=|\lambda\rangle_{\alpha}[\tilde{\lambda}|_{\dot{\alpha}}\,$
, $\displaystyle
n_{\alpha\dot{\alpha}}=|\mu\rangle_{\alpha}[\tilde{\mu}|_{\dot{\alpha}}\,,$
$\displaystyle
m_{\alpha\dot{\alpha}}=\epsilon^{+}_{\alpha\dot{\alpha}}=\sqrt{2}\frac{|\mu\rangle_{\alpha}[\tilde{\lambda}|_{\dot{\alpha}}}{\langle\lambda\mu\rangle}\,$
,
$\displaystyle\bar{m}_{\alpha\dot{\alpha}}=-\epsilon^{-}_{\alpha\dot{\alpha}}=\sqrt{2}\frac{|\lambda\rangle_{\alpha}[\tilde{\mu}|_{\dot{\alpha}}}{[\tilde{\mu}\tilde{\lambda}]}\,.$
(46)

The relation $l\cdot n=-m\cdot\bar{m}=1$ now becomes
$|\langle\lambda\mu\rangle|=|[\tilde{\lambda}\tilde{\mu}]|=\sqrt{2\omega}$.
The Weyl spinors $|\lambda\rangle$, $|\lambda]$ carry helicity weight
$-1/2,+1/2$ under (33), respectively.101010The spinorial description of
spacetime (31) has been developed in Penrose:1987uia , and this
parametrization will play a crucial role for GW scattering in Part II of this
work. Spinors have been recently used in Chacon:2021wbr  to expand the
classical double copy construction Monteiro:2014cda ; Monteiro:2015bna .
Fixing the little group we can write them explicitly as

$\begin{split}|\mu\rangle&=\left(e^{-i\phi/2}\cos\theta/2,e^{i\phi/2}\sin\theta/2\right),\,\,\,|\lambda\rangle=\sqrt{2\omega}\left(-e^{-i\phi/2}\sin\theta/2,e^{i\phi/2}\cos\theta/2\right),\\\
[\tilde{\mu}|&=\left(e^{i\phi/2}\cos\theta/2,e^{-i\phi/2}\sin\theta/2\right),\,\,\,[\tilde{\lambda}|=\sqrt{2\omega}\left(-e^{i\phi/2}\sin\theta/2,e^{-i\phi/2}\cos\theta/2\right).\end{split}$
(47)

Note that, nicely, these spinors can be used to construct both solutions of
the massless Dirac equation for momentum $k=\omega l$:

$u^{-}(k)=\left(|\lambda\rangle_{\alpha},0,0\right)^{T}\,,\,\,u^{+}(k)=\left(0,0,|\tilde{\lambda}]^{\dot{\alpha}}\right)^{T}\,.$
(48)

The states associated to $k_{3},k_{2}$ are again recovered by setting $\phi=0$
(and $\theta=0$ for the latter) in (47). Plugging these into the corresponding
four-point graviton exchange amplitude, which is given by Holstein:2008sx

$A_{4}^{h=\frac{1}{2}}=\frac{\kappa^{2}}{8q^{2}}\left[4p_{1}{\cdot}k_{2}+q^{2}\right]\bar{u}_{3}\not{p}_{1}u_{2}\,,$
(49)

and taking the classical limit, we obtain

$\langle A_{4,++}^{h=\frac{1}{2}}\rangle=\langle
A_{4,--}^{h=\frac{1}{2}}\rangle=-\frac{\kappa^{2}M^{2}}{4}\frac{\cos\left(\frac{\theta}{2}\right)}{\sin^{2}\left(\frac{\theta}{2}\right)}\,,$
(50)

whereas the helicity-reversing elements vanish as expected.

Finally, collecting the results (20),(45),(50), the classical cross-sections
(10) can be written in the compact formula

$\boxed{\frac{d\langle\sigma^{h}\rangle}{d\Omega}=G^{2}M^{2}\frac{\cos^{4h}\left(\theta/2\right)}{\sin^{4}\left(\theta/2\right)}}\,,$
(51)

for $h=0,1/2,1$, in perfect agreement with the results in PhysRevD.16.237 ;
Westervelt:1971pm ; Doran:2001ag ; Dolan:2007ut . Of course, this result is
physical and independent of our choice of little-group fixing. An interesting
‘double copy’ structure is manifest, with the numerator for $h=1$ being the
square of the $h=1/2$ case. This can be easily seen at the level of the
amplitudes by plugging spinor-helicity variables (2.3) into the photon result
(44). We now show how the pattern in (51) breaks down for $h=2$ due to the
fact that the helicity-violating amplitudes do not vanish.

###  2.4 Gravitational Waves

Even though the case of GW will be extensively covered in Part II for the Kerr
background, let us here present the classical prescription leading to GW
scattering in Schwarzschild. We will follow an analogous route and relate the
graviton Compton amplitude $A_{4}^{h=2}$ to the corresponding NP scalar. In
this case the Weyl scalar reads Newman_1962

$\boldsymbol{\Psi}^{\textrm{rad}}_{-2}=-\bar{m}^{\mu}n^{\nu}\bar{m}^{\rho}n^{\sigma}C^{\textrm{rad}}_{\mu\nu\rho\sigma}\,.$
(52)

Here the radiative piece of the Weyl tensor is obtained from a linearized
metric perturbation sourced by the on-shell modes of the stress-energy tensor

$h_{\mu\nu}^{\textrm{rad}}\propto T_{\mu\nu}(k)e^{ik\cdot x}\,,$ (53)

with $k^{2}=0$. The conservation condition $k^{\mu}T_{\mu\nu}=0$ is
equivalent, through the gauge fixing (13), to the fact that the metric
perturbation is transverse-traceless (TT):

$\partial^{\mu}h_{\mu\nu}^{\textrm{rad}}=\eta^{\mu\nu}h_{\mu\nu}^{\textrm{rad}}=0\,.$
(54)

We use this to conclude that all contractions of the linearized Riemann tensor

$R^{\textrm{rad}}_{\alpha\beta\mu\nu}=-\frac{1}{2}(\partial_{\alpha\mu}h^{\textrm{rad}}_{\beta\nu}+\partial_{\beta\nu}h^{\textrm{rad}}_{\alpha\mu}-\partial_{\alpha\nu}h^{\textrm{rad}}_{\beta\mu}-\partial_{\beta\mu}h^{\textrm{rad}}_{\alpha\nu})\,,$
(55)

vanish. This means that
$C^{\textrm{rad}}_{\alpha\beta\mu\nu}=R^{\textrm{rad}}_{\alpha\beta\mu\nu}$
for the radiative modes. The Weyl NP scalar (52) then becomes

$\boldsymbol{\Psi}^{\textrm{rad}}_{-2}\propto\frac{1}{2}\bar{m}^{\alpha}n^{\beta}\bar{m}^{\mu}n^{\nu}\left(k_{\alpha}k_{\mu}T_{\beta\nu}+k_{\beta}k_{\nu}T_{\alpha\mu}-2k_{\alpha}k_{\nu}T_{\beta\mu}\right)e^{ik\cdot
x}\,,$ (56)

and using the saddle-point approximation (36) we obtain

$\boldsymbol{\Psi}^{\textrm{rad}}_{-2}\propto\frac{\omega^{2}}{2}\bar{m}^{\mu}\bar{m}^{\nu}T_{\mu\nu}(k)e^{i\omega(t-r)}\,.$
(57)

Identifying the graviton polarizations

$\epsilon^{\mu\nu}_{+}=\epsilon_{+}^{\mu}\epsilon_{+}^{\nu}=m^{\mu}m^{\nu}\,,\,\,\epsilon^{\mu\nu}_{-}=\epsilon_{-}^{\mu}\epsilon_{-}^{\nu}=\bar{m}^{\mu}\bar{m}^{\nu}\,,$
(58)

we can finally relate the NP scalar to the classical limit of the scattering
amplitude

$\bar{m}^{\mu}\bar{m}^{\nu}T_{\mu\nu}(k)=\langle A_{4}^{h=2}\rangle\,.$ (59)

In order to evaluate this let us note a crucial difference with previous
cases. The classical amplitude $\langle A_{4}^{h=2}\rangle$ is controlled not
only by the single graviton exchange diagram, but actually all three channels
contribute. Indeed, using the factorized states (58), the amplitude can be
written as the double copy of the scalar-QED Compton amplitude Bjerrum-
Bohr:2013bxa

$A_{4}^{h=2}=-\kappa^{2}\frac{\left(2p_{1}{\cdot}F_{2}{\cdot}F_{3}{\cdot}p_{1}\right)^{2}}{\left(s-M^{2}\right)\left(u-M^{2}\right)t}\,,$
(60)

where $F_{i}^{\mu\nu}=2k_{i}^{[\mu}\epsilon_{i}^{\nu]}$.

The gravitational amplitude contains poles corresponding to $s,u$ and
$t$-channels. The residues of these get mixed under gauge transformations of
the external gravitons. Moreover, under the classical limit (6) and finite
scattering angle $\xi$, we learn that the $s$ and $u$ channels form a double
pole in $s-M^{2}\sim M^{2}-u$, with the same classical scaling as
$t\sim(\hbar\omega)^{2}$. This means that the contributions from all diagrams
are entangled in the classical limit, in contrast with previous gauge-
dependent approaches Guadagnini:2008ha ; Westervelt:1971pm .111111Indeed, it
was argued in Guadagnini:2008ha ; Westervelt:1971pm  that the classical
amplitude for the scattering of gravitational waves off classical matter has
only the contribution of the graviton exchange diagram. However, we have
checked that this fact turns out to be true only in the spatial gauge
$\epsilon^{0\mu}_{i}=0$ of the gravitons. This trick also breaks down for the
rotating case, so we choose instead to simply work in a manifestly Lorentz
covariant gauge.

Using the kinematics $(\ref{eq:kinematics})$, and the basis for polarization
vectors $(\ref{eq:photon polarization1}),(\ref{eq:photon polarization2})$ it
is now easy to show that the diagonal elements of the classical scattering
matrix (11) are given by

$\langle A_{4,++}^{h=2}\rangle=\langle
A_{4,--}^{h=2}\rangle=\frac{\kappa^{2}}{4}\frac{\cos^{4}\left(\frac{\theta}{2}\right)}{\sin^{2}\left(\frac{\theta}{2}\right)}\,,$
(61)

whereas for the off-diagonal elements we have

$\langle A_{4,+-}^{h=2}\rangle=\langle
A_{4,-+}^{h=2}\rangle=\frac{\kappa^{2}}{4}\frac{\sin^{4}\left(\frac{\theta}{2}\right)}{\sin^{2}\left(\frac{\theta}{2}\right)}\,.$
(62)

We therefore conclude that helicity is not preserved for GW scattering. For
the case of an incoming gravitational wave with fixed positive helicity, the
unpolarized differential cross section (10) becomes 121212Since (60) is a
double copy amplitude, it is easy to check that for single copy case, the
classical differential cross section is given by
$\frac{d\langle\sigma^{\rm{QED}}\rangle}{d\Omega}=\left(\frac{e^{2}}{4\pi
M}\right)^{2}\left[\cos^{4}\left(\frac{\theta}{2}\right)+\sin^{4}\left(\frac{\theta}{2}\right)\right]$,
which recovers the unpolarized differential cross section for the Thompson
scattering Jackson:100964 .

$\frac{d\langle\sigma^{h=2}\rangle}{d\Omega}=\frac{G^{2}M^{2}}{\sin^{4}\left(\frac{\theta}{2}\right)}\left[\cos^{8}\left(\frac{\theta}{2}\right)+\sin{}^{8}\left(\frac{\theta}{2}\right)\right]$
(63)

in agreement with PhysRevD.16.237 ; Westervelt:1971pm ; Doran:2001ag ;
Dolan:2007ut . Then as announced above, the GW scattering cross section
differs from the lower-helicity result (51). There is an extra sine term in
the unpolarized differential cross section for $h=2$ arising from the opposite
helicities configuration of the Gravitational Waves (62). In the black hole
perturbation theory framework Dolan:2008kf , this was explained from the
parity dependence of the GW-BH interaction as we will outline and expand in
Part II.

This new term does not alter the universal $\theta\to 0$ Newtonian behaviour
(2.1). Moreover, we will confirm that GW scattering leads to the same eikonal
behaviour as the lower-helicity waves in section 5.

To close this section, let us discuss the effect of the BH on the polarization
of the scattered wave. We note that the cross-section (63), as well as the
previous examples, does not depend on the helicity $\pm h$ of the incoming
wave. This means that for a given scattering angle $\theta$ in a detector,
left and right polarized beams will scatter with the same probability and the
average polarization is preserved. This does not hold if the black hole is
spinning as one helicity will be favoured, as we show in the next section.

##  3 Massless wave scattering off a Kerr background

In this section we will extend the previous framework to study the scattering
of massless plane waves off the Kerr BH. In the 1970’s, motivated by the
stability problem of Kerr, preliminary analyses found that even for slowly
spinning bodies there are interesting observable effects. For instance, the
spin of the black hole can induce a partial polarization of electromagnetic or
gravitational waves after the scattering process PhysRevD.16.237 ;
Dolan:2008kf . We will review and extend these results.

Our tool will be the relation between scattering of spinning black holes and
higher-spin amplitudes. A connection between classical rotating bodies and
massive particles with spin was first pointed out in Vaidya:2014kza  and
extended in Guevara:2017csg . It was suggested that QFT amplitudes of massive
particle of spin $S$ emitting gravitons encode the gravitational coupling to
$2S$ classical multipoles, namely the powers $a^{j}$ for $j\leq 2S$. The main
evidence for this comes from the multipoles of a single Kerr black hole
metric, which are encoded in the on-shell modes of its effective stress-energy
tensor Vines:2017hyw , and can be mapped to three-point amplitudes by
introducing the QFT notion of minimal-coupling. This correspondence was tested
by studying the scattering of two black holes to leading order in $G$
Cachazo:2017jef ; Guevara:2017csg ; Guevara:2018wpp ; Guevara:2019fsj ;
Maybee:2019jus ; Chung:2018kqs ; Chung:2019duq ; Chung:2020rrz

In the previous section, wave scattering in the Schwarzschild background was
modeled using the off-shell three-point vertex (14) for a quantum massive
scalar coupled to gravity, uniquely defined assuming minimal coupling and the
harmonic gauge. For spinning particles these considerations are not enough: On
the one hand, there is no clear notion of minimal coupling at four-points, so
even though three-point amplitudes for Kerr can indeed be fixed by minimal
coupling as in Guevara:2017csg , this prescription breaks down for four-point
amplitudes Ferrara:1992yc .131313At the quantum level, massive higher-spin
particles can exhibit diverse inconsistencies related to its bad UV behaviour
Ferrara:1992yc ; Cucchieri:1994tx ; Arkani-Hamed:2017jhn . Even though these
are tamed at three-points by introducing certain criteria for minimal coupling
Deser:2000dz ; Deser:2001dt ; Arkani-Hamed:2017jhn , reflecting the fact that
their massless limit is still consistent at three-points Benincasa:2007xk ,
they do persist at four-points leading to acausality/superluminality
Deser:2001dt ; Afkhami-Jeddi:2018apj ; Camanho:2014apa ; Bautista:2019evw . On
the other hand, given that four-point amplitudes can be constructed from
Feynman diagrams we may be tempted to extend the aforementioned three-point
amplitude to an (off-shell) vertex for a higher-spin quantum particle. Such an
off-shell extension is however not unique even assuming minimal coupling and
carries the ambiguities of the multipole decomposition of higher-spin objects,
see e.g Lorce:2009br  and subsequent work.

In this section we extend the correspondence to the four point amplitudes
representing the scattering of a massless particle/wave off the the Kerr black
hole, as an infinite-spin massive particle. In order to sort out the above
complications at four-points we will instead take a more indirect approach. We
assume the existence of a QFT three-point vertex, containing all multipoles
$\sim a^{j}$, such that in the classical limit it reduces to the off-shell
Kerr metric. More precisely, assuming the harmonic gauge, the classical limit
of the vertex should be equivalent to the energy-momentum tensor presented in
a compact form in Vines:2017hyw . This extends to all spin-multipoles the
seminal approach of PhysRevD.16.237  to wave scattering, which used a
background gravitational field containing a spin monopole/dipole as source.
Also, by extending the classical rules given in the previous section, we will
check that our infiite-spin vertex indeed has classical scaling and can be
used for our amplitudes.

Let us remark that in this part of the work we will use the described Feynman
rules to compute the amplitudes for $h<2$. These amplitudes inherit the
exponential structure of the three-point amplitude and hence are free of high-
energy divergences $M\to 0$. In Part II, however, we will see that the GW case
($h=2$) indeed contain such inconsistencies and can only be constructed via an
on-shell ansatz since there are no natural choice of Feynman rules for
couplings $R^{2}\sim G^{2}$.

###  3.1 Classical Expansion

To begin establishing the correspondence we first define the classical
variables into play. It was explained in section 2 that the classical limit is
equivalent to the scaling $r\to\hbar^{-1}r$, for a scattering radius (i.e.
impact parameter) $r$. This is equivalent to an expansion in large orbital
angular momentum, $L\sim r\times p$,

$L=\hbar^{-1}\hat{L}\,.$ (64)

This perspective has been detailed in e.g. Damour:2017zjx ; Cheung:2018wkq .
We can now extend the scaling rule to the intrinsic part of the angular
momentum. Let $a$ be the norm of the Pauli-Lubanski vector of a massive
particle. Then, the intrinsic angular momentum is given by $S=Ma$ and the rule
(64) is extended by considering Maybee:2019jus

$a=\hbar^{-1}\hat{a}\,.$ (65)

The classical variable $\hat{a}$ can be identified with the radius of the
ring-singularity in Kerr Guevara:2018wpp . As a consequence of this scaling,
the expansions considered in Section 2 can be extended by an expansion in the
following dimensionless quantities:

${\color[rgb]{0,0,0}a^{\star}}=\frac{a}{GM}\,,\quad a\omega\,,$ (66)

where we recall that $\omega=\hbar\hat{\omega}$ is the energy of the massless
momenta. Both dimensionless combinations scale as $\hbar^{0}$ and hence are
classical (we know that e.g. $a\omega=\hat{a}\hat{\omega}$, thus the hat
notation can be omitted hereafter). As their ratio is given by
$a\omega/{\color[rgb]{0,0,0}a^{\star}}=GM\omega=\epsilon/2$, corresponding to
our classical weak-field parameter, only one new expansion is needed to
incorporate the full spin dependence. As we are working with tree-level
amplitudes at leading order in $G$, it is more natural to perform an expansion
in $a\omega<1$. It is easy to see that this corresponds to a soft expansion in
the external massless momenta, as advocated in Guevara:2018wpp ;
Bautista:2019evw . The expansion involves corrections in $a\omega\sim
1/\lambda$ that can probe the internal structure of the spinning body,
although as explained the orders $(a\omega)^{0}$ and $(a\omega)^{1}$
correspond to spin monopole and dipole respectively, and are universal. More
generally, the $2S+1$ classical spin multipoles of the scattering amplitude
$\langle A^{S,h}_{4}\rangle$, for a spin-S particle, are in one-to-one
correspondence with the soft expansion up to order $(a\omega)^{2S}$.

###  3.2 Scattering of waves of helicity $h<2$

As anticipated, for lower helicity waves we will not model the Kerr BH by a
three point vertex between a massive quantum particle and a graviton, which
was our approach in the previous section. Instead, we will directly consider
the off-shell classical source given in Vines:2017hyw . Motivated by the
approach in PhysRevD.16.237  we will use this source as a background field to
compute the scattering amplitude with a wave of helicity $h$, denoted by
$\langle A_{4}^{S\to\infty,h}\rangle$. It is given by a single graviton
exchange diagram with a massless particle of helicity $h$, where the internal
graviton is required to be off-shell in order to capture the contact terms at
finite scattering angle $\theta$. Here $S\to\infty$ means that we are
considering the full infinite series in $(a\omega)$, as opposed to the linear
in spin results of PhysRevD.16.237 .

To begin with, consider the linearized form of the Kerr metric as in
Vines:2017hyw ; Harte:2016vwo , satisfying the harmonic gauge condition
(13)141414This form of the vertex differs by ultralocal pieces, e.g.
$\delta(r)$, from the compact expression (31) given in Vines:2017hyw . This
pieces will lead to $\mathcal{O}(q^{0})$ corrections in eq. (69), which were
irrelevant in Vines:2017hyw  but do contribute to our amplitude at finite
scattering angle. We consider the form in (67) to be accurate at leading order
in $G$ as it follows essentially from the Newman-Janis shift Harte:2016vwo .

$h_{\mu\nu}(x)=\mathcal{P}_{\mu\nu\alpha\beta}\left(u^{\alpha}u^{\beta}\cos{a{\cdot}\partial}{+}\frac{\sin{a{\cdot}\partial}}{a{\cdot}\partial}u^{(\alpha}\epsilon_{\,\,\rho\sigma\delta}^{\beta)}u^{\rho}a^{\sigma}\partial^{\delta}\right)\frac{4GM}{r},$
(67)

where $u$ describes the four-velocity of the black-hole, in the rest frame
$u=(1,0,0,0)$. In general we will identify $u=p_{1}/M$. As we further
elaborate below, its spin is encoded in the constant pseudovector $a^{\mu}$
satisfying $a\cdot u=0$. Noting that $\partial\sim 1/r\sim\omega$ we note that
this form corresponds to an infinite expansion in $a\omega$, while only
considering the leading order in $GM/r\sim GM\omega$ as desired (recall that
both combinations are classical). To construct the scattering amplitude we map
this form to momentum space writing the retarded green function as
Vines:2017hyw

$\frac{1}{4\pi r}:=\int d\tau G_{ret}(x,u\tau)=\int
d\tau\int\frac{d^{4}q}{(2\pi)^{4}q^{2}}e^{iq\cdot(x-u\tau)}=\int\frac{d^{4}q}{(2\pi)^{4}q^{2}}(4\pi)\delta(2q\cdot
u)e^{iq\cdot x}\,.$ (68)

where we have ignored the contour ($i\epsilon$) prescription. Plugging this
into (67) we can read off, up to a normalization, the three-point off-shell
vertex corresponding to the Kerr background in momentum space:

$\tilde{h}_{\mu\nu}(q)=\frac{\kappa}{q^{2}}\delta(2q\cdot
p_{1})\mathcal{P}_{\mu\nu\alpha\beta}\left(p_{1}^{\alpha}p_{1}^{\beta}\cosh{a{\cdot}q}{+}i\frac{\sinh{a{\cdot}q}}{a{\cdot}q}p_{1}^{(\alpha}\epsilon_{\,\,\rho\sigma\delta}^{\beta)}p_{1}^{\rho}a^{\sigma}q^{\delta}\right),$
(69)

where $p_{1}^{2}=M^{2}$, and we note that $\delta(2q\cdot p_{1})$ is nothing
but the classical limit of the delta function $\delta(p_{4}^{2}-m^{2}=2q\cdot
p_{1}+q^{2})$,which will be omitted in our scattering amplitude. Note also
that the graviton momentum $q^{\mu}$ is still off-shell.

Regarded as a Feynman vertex, the form (69) extends to all orders in spin the
scalar vertex (14) including graviton propagator (16) (note however that the
classical limit is not required as this is a background field and not a
quantum particle). The scalar piece, i.e. ‘spin monopole’, corresponds to the
first term $p_{1}^{\alpha}p_{1}^{\beta}$, whereas the universal spin dipole is
given by the coupling
$p_{1}^{(\alpha}\epsilon_{\,\,\rho\sigma\delta}^{\beta)}p_{1}^{\rho}a^{\sigma}q^{\delta}$.
The remaining multipoles $\sim(a\cdot q)^{n}$ are fixed completely by the
exponential structures.

In order to compute the scattering amplitudes we will follow the same logic as
in the previous section. Indeed, the argument leading to e.g. (37)-(38),
relating the radiative part of the Weyl scalar $\boldsymbol{\Psi}^{h}$ to the
classical amplitude $\langle A^{h}_{4}\rangle$ holds irrespective of the
scattering source and spin effects.

Using the variables given in (5), let us further choose a generic orientation
for the spin of the black hole,

$a^{\mu}=(0,a_{x},a_{y},a_{z}).$ (70)

The computation of $\langle A_{4}^{h}\rangle$ for waves of helicity $h<2$
follows from a single graviton-exchange diagram as in the previous section. We
use the vertex $(\ref{eq:justin vertex classical})$ for the massive line and
the Feynman rules from Holstein:2008sx  for the massless particles.

Scalar scattering. For the scattering of scalar waves, the classical amplitude
reads

$\langle
A_{4}^{S\to\infty,h=0}\rangle=\frac{\kappa^{2}p_{1}{\cdot}k_{2}}{q^{2}}\left[p_{1}{\cdot}k_{2}\cosh{a{\cdot}q}+iM\vec{a}{\cdot}\vec{q}{\times}\vec{k}_{2}\frac{\sinh{a{\cdot}q}}{a{\cdot}q}\right].$
(71)

After using the kinematics $(\ref{eq:kinematics})$ and $(\ref{eq:polar spin})$
this amplitude takes the form

$\langle
A_{4}^{S\to\infty,h=0}\rangle=\frac{M^{2}\kappa^{2}}{4\sin^{2}\left(\frac{\theta}{2}\right)}\left[\cosh{a{\cdot}q}-i\omega
a_{y}\sin\theta\frac{\sinh{a{\cdot}q}}{a{\cdot}q}\right],$ (72)

where $q{\cdot}a=-\omega(a_{x}\sin(\theta)-2a_{z}\sin^{2}(\theta/2))$.

Neutrino scattering. For the scattering of neutrino waves it follows that

$\langle
A_{4}^{S\to\infty,h=1/2}\rangle{=}\frac{\kappa^{2}}{2q^{2}}\bar{u}_{3}\left[p_{1}{\cdot}k_{2}\not{p}_{1}\cosh{a{\cdot}q}{-}iM\left(p_{1}{\cdot}k_{2}\vec{a}{\cdot}\vec{\gamma}{\times}\vec{q}{-}\vec{a}{\cdot}\vec{q}{\times}\vec{k}_{2}\not{p}_{1}\right)\frac{\sinh{a{\cdot}q}}{2a{\cdot}q}\right]u_{2},$
(73)

and in terms of the kinematics $(\ref{eq:kinematics})$, the basis of spinors
(48), and the spin vector $(\ref{eq:polar spin})$, the diagonal elements of
the scattering matrix (11) read

$\begin{split}\langle
A_{4,++}^{S\to\infty,h=1/2}\rangle&=\frac{M^{2}\kappa^{2}\cos\left(\frac{\theta}{2}\right)}{4\sin^{2}\left(\frac{\theta}{2}\right)}\biggl{[}\cosh{a{\cdot}q}{-}\omega\tan\left(\theta/2\right)\times\\\
&\hskip
48.36958pt\left(2a_{x}\sin^{2}\left(\theta/2\right){+}2ia_{y}\left(1{+}\cos^{2}\left(\theta/2\right)\right){+}a_{z}\sin\theta\right)\frac{\sinh{a{\cdot}q}}{2a{\cdot}q}\bigg{]}\\\
\langle A_{4,--}^{S\to\infty,h=1/2}\rangle&=\left[\langle
A_{4,++}^{\rm{K},h=1/2}\rangle^{*}\right]_{\omega\rightarrow-\omega},\end{split}$
(74)

whereas as for the spin-less case, the off-diagonal elements vanish.

Photon scattering. Finally, for the scattering of electromagnetic waves, the
classical amplitude is given by

$\langle A_{4}^{S\to\infty,h=1}\rangle{=}\langle
A_{4}^{h=1}\rangle\cosh{a{\cdot}q}{+}\frac{iM\kappa^{2}}{2q^{2}}\left[2\epsilon_{2}{\cdot}\epsilon_{3}^{*}\vec{a}{\cdot}\vec{q}{\times}\vec{k}_{2}{-}\epsilon_{3}^{*}{\cdot}k_{2}\vec{a}{\cdot}\vec{q}{\times}\vec{\epsilon}_{2}{-}\epsilon_{2}{\cdot}k_{2}\vec{a}{\cdot}\vec{q}{\times}\vec{\epsilon}_{3}^{*}\right]\frac{\sinh{a{\cdot}q}}{a{\cdot}q},$
(75)

where $\langle A_{4}^{h=1}\rangle$ is the classical amplitude for the
scattering of electromagnetic waves off the SBH, given in (44). For this case,
the elements of (11) are

$\begin{split}\langle
A_{4,++}^{S\to\infty,h=1}\rangle&=\frac{M^{2}\kappa^{2}\cos^{2}\left(\frac{\theta}{2}\right)}{4\sin^{2}\left(\frac{\theta}{2}\right)}\biggl{[}\cosh{a{\cdot}q}{-}\omega\tan\left(\theta/2\right)\left(2a_{x}\sin^{2}\left(\theta/2\right){+}i2a_{y}{+}a_{z}\sin\theta\right)\frac{\sinh{a{\cdot}q}}{a{\cdot}q}\bigg{]},\\\
\langle A_{4,--}^{S\to\infty,h=1}\rangle&=\left[\langle
A_{4,++}^{S\to\infty,h=1}\rangle^{*}\right]_{\omega\rightarrow-\omega},\end{split}$
(76)

and similar to the previous case, the off-diagonal elements vanish.

We have checked that to linear order in spin, amplitudes (71), (73) and (74)
recover the result for the scattering of scalar, neutrino and electromagnetic
waves off classical rotating matter respectively, computed in Barbieri:2005kp
. However, as we mentioned in previous section the scalar and spin dipole
terms are universal and follow from the soft behaviour of gravitational
interactions. Therefore the wave cannot probe the nature of the compact object
to this order in spin, which explains why the same result is obtained for a
rotating star PhysRevD.16.237 . On the other hand, for higher multipoles our
results show the genuine signature of the Kerr black hole in the
exponentiation of $a\cdot q$, a remnant of the Newman-Janis shift Arkani-
Hamed:2019ymq . In section 4.3 we explicitly show the matching of the scalar
amplitude (72), to the result from the BHPT computation, up to order
$(a\omega)^{5}$.

We can easily obtain the corresponding cross sections (10) as before. For
simplicity let us focus on the polar scattering, where the momentum of the
incoming wave is aligned with the spin direction. 151515For general spin
orientation, the cross section can be easily computed using the amplitudes
(72), (74) and (76), for the different helicity scenarios. , where the spin
3-vector is aligned with the impinging wave, e.g. $a_{x}=a_{y}=0$. In this
scenario, the scattering enjoys cylindrical symmetry and the cross-section
simplifies drastically at finite scattering angle. We obtain

$\frac{d\sigma}{d\Omega}=G^{2}M^{2}\frac{\cos^{4h}(\theta/2)}{\sin^{4}(\theta/2)}\left[\cosh{\left(2a_{z}\omega\sin^{2}(\theta/2)\right)}-h\sinh{\left(2a_{z}\omega\sin^{2}(\theta/2)\right)}\right]^{2}\,,h\leq
1\,.$ (77)

Notice that for $a_{z}=0$, we recover the amplitudes for the scattering of
waves off the SBH (51).

Let us now study the behaviour of our classical amplitudes in the small
scattering angle, e.g. $\theta\to 0$ regime, which we refer to as eikonal
limit. It can easily be shown that in this limit, for any fixed incoming
helicity $h$ the scattering amplitude behave as

$\boxed{\langle
A_{4,++}^{S\to\infty,h}\rangle=\frac{M^{2}\kappa^{2}}{\theta^{2}}\left[\cosh(a_{x}\omega\theta)-\frac{ia_{y}}{a_{x}}\sinh(a_{x}\omega\theta)+\ldots\right]}$
(78)

where $\ldots$ stands for terms of the form $\omega^{n}\theta^{m}$ with $m>n$,
which are non-universal and generically depend on $h$. Moreover, as in all
cases we found $\langle A_{4,--}\rangle=\left[\langle
A_{4,++}\rangle^{*}\right]_{\omega\rightarrow-\omega}$, we conclude that the
eikonal limit is universal. Thus, this extends the Newtonian universality
found in (2.1) to all orders in the spin for Kerr background. We will show
that this behaviour is indeed controlled by null geodesics in the Kerr
background in section 5.

As in the previous section, the soft theorem of gravitons also guarantees that
the off-diagonal elements of the scattering matrix (11) vanish. This means
that helicity is conserved in the scattering process. However, the fact that
the diagonal elements $\langle A_{++}\rangle$,$\langle A_{--}\rangle$ are not
equal induces a ‘shear’ in the $2\times 2$ scattering matrix. We now
characterize this phenomena which emerges due to spin: The spinning motion of
the Kerr black hole induces a partial polarization of the waves.

###  3.3 Spin-induced Polarization

As anticipated, the induced polarization after the scattering will be
reflected in the difference between elements of the scattering matrix (11),
which in turn shows how waves of opposite circular polarization, scatter
different from the Kerr BH. To see this explicitly we compare the scattering
cross-sections for a left ($+$) and right ($-$) circularly polarized incoming
wave:

$\begin{split}64\pi^{2}M^{2}\frac{d\langle\sigma_{+}\rangle}{d\Omega}&=\langle
A_{++}\rangle\langle A_{++}\rangle^{*}+\langle A_{+-}\rangle\langle
A_{+-}\rangle^{*}\\\
64\pi^{2}M^{2}\frac{d\langle\sigma_{-}\rangle}{d\Omega}&=\langle
A_{--}\rangle\langle A_{--}\rangle^{*}+\langle A_{-+}\rangle\langle
A_{-+}\rangle^{*}\end{split}$ (79)

We have found in the previous examples that opposite helicity amplitudes are
related via $\langle A\rangle\to\langle A\rangle^{*}$, accompanied by the
expected time reversal $\omega\to-\omega$, map. This is more transparent in
the spinor-helicity formalism, and can be seen as a consequence of
CPT/crossing symmetry: Opposite helicities are related by chiral (i.e.
complex) conjugation in the amplitude BCGV . This induces a parity
transformation which flips the sign of $a^{\mu}$, which corresponds to a
pseudovector as it describes the orientation of the rotating black hole. Due
to the fact that spin only enters through the combination $a\omega$ the map
$a^{\mu}\to-a^{\mu}$ is of course equivalent to $\omega\to-\omega$.

From the above discussion, using (79), we easily conclude that

$\frac{d\langle\sigma_{+}\rangle}{d\Omega}=\left[\frac{d\langle\sigma_{-}\rangle}{d\Omega}\right]_{(\omega\to-\omega)}$
(80)

Following PhysRevD.16.237 ; Dolan:2008kf ; Barbieri:2005kp  we also introduce
the polarization measurement

$\mathcal{P}=\frac{\frac{d\langle\sigma_{+}\rangle}{d\Omega}-\frac{d\langle\sigma_{-}\rangle}{d\Omega}}{\frac{d\langle\sigma_{+}\rangle}{d\Omega}+\frac{d\langle\sigma_{-}\rangle}{d\Omega}}\,.$
(81)

According to (80) the numerator of the polarization depends only on odd powers
of $a\omega$ in the cross-section, and in particular vanishes for the
Schwarzschild case. For the polar scattering case considered in (77) we find

$\mathcal{P}^{(h)}=-h\sinh{\left(4a_{z}|\omega|\sin^{2}(\theta/2)\right)}\left[\cosh^{2}{\left(2a_{z}|\omega|\sin^{2}(\theta/2)\right)}+h^{2}\sinh^{2}{\left(2a_{z}|\omega|\sin^{2}(\theta/2)\right)}\right]^{-1},$
(82)

which for $\theta\rightarrow 0$ becomes
$\mathcal{P}^{(h)}=-ha_{z}|\omega|\theta^{2}$ and thus recovers the results in
PhysRevD.16.237 ; Barbieri:2005kp . Naively we might expect that equation (82)
also holds true for $h=2$, as was miss-concluded in Guadagnini:2008ha ;
Barbieri:2005kp  to linear order in spin. This would be the case if to compute
the scattering amplitude, we consider only the graviton exchange diagram
between the Kerr background and the gravitational wave. In such a case
however, the resulting amplitude fails to be gauge invariant, and therefore
the additional diagrams need to be consider. We expand on this issue in Part
II.

A non-trivial polarization implies that different helicity waves scatter by
different angles. For instance, linearly polarized incoming waves become
elliptically polarized after the scattering process PhysRevD.16.237 . We can
obtain the scattering angles by integrating the cross section as in (2.1):
Writing $d\Omega=\sin\theta d\theta d\phi$ we have

$\sin(\theta)\frac{d\langle\sigma_{\pm}\rangle}{d\Omega}=b_{\pm}(\theta,\phi)\frac{\partial
b_{\pm}}{\partial\theta}(\theta,\phi)\Longrightarrow
b^{2}_{\pm}(\theta,\phi)=2\int^{\pi}_{\theta}\frac{d\langle\sigma_{\pm}\rangle}{d\Omega}\sin(\theta^{\prime})d\theta^{\prime}\,.$
(83)

The angular splitting $\delta$ between both helicities is then computed by
solving the condition

$b^{2}_{+}(\theta,\phi)=b^{2}_{-}(\theta+\delta,\phi)=b^{2},$ (84)

for $\delta\ll\theta\approx\frac{4GM}{b}\ll 1$ (we also recall that our
coordinate system is chosen so that $\phi=0$). To linear order in the spin of
the black hole we find

$\delta^{(h)}=2\left(\frac{4GM}{b}\right)^{3}\left[a_{z}\omega
h\log\left(\frac{b}{2GM}\right)+\frac{\omega}{2}\left(\frac{\pi}{4}a_{x}-ha_{z}\right)\left(\gamma_{E}+\psi^{(0)}(1+2h)\right)\right],\,\,\,h\leq
1\,.$ (85)

where $\psi^{(0)}(z)=\frac{d}{dz}\log(\Gamma(z))$ is the polygamma function.
The result recovers that of PhysRevD.16.237  for $h=1$ and extends it to all
helicities. In section 5 we will study the classical scattering angle $\theta$
in the eikonal regime, as controlled by geodesics of point-particles in Kerr.
It will then be evident that the splitting (85) is a correction of order
$\epsilon=GM\omega$ to the subleading eikonal angle $\sim G^{2}M^{2}a/b^{3}$.
This reflects that the angular splitting does not admit a point-particle
interpretation, indeed the corrections are non-universal and depend on the
helicity of the wave. If one insists in such a particle-like description one
should include the so-called Papatetrou-type forces 10.2307/98893 .

##  4 Classical plane wave scattering from BHPT

By implementing our classical limit, we have written down the generic form of
the 4-point scattering amplitudes to all orders in spin and helicities
$h=0,1/2,1$. Our aim in this section is to match the associated Newman-Penrose
scalars to the classical solutions of the wave equation in BH backgrounds. We
will focus on the $h=0$ case to show how the QFT amplitudes can be matched
perfectly. In Part II of this work we will obtain similar results for the case
$h=2$.

We use black hole perturbation theory (BHPT) to compute the classical
scattering amplitudes and the differential cross section for a plane wave
scattering off both the Schwarzschild and the Kerr black hole, considering the
weak-coupling regime $GM\omega\ll 1$ (but not necessarily only the leading
order). As mentioned, in this limit the black hole size is neglectable with
respect to the scattering wavelength and we expect the singularity to behave
as point-like compact object. Further considerations required to treat the
spinning singularity are given in the next sections.

###  4.1 Scalar Waves on the Schwarzschild Black hole

We begin by reviewing the case of scattering of scalar waves in a
Schwarzschild spacetime. This will illustrate the partial wave approach
without the encumbering technical details of Kerr spacetime. For more details
on the Schwarzschild problem see Glampedakis:2001cx .

The dynamics of a scalar field $\psi$ is governed by the scalar wave equation:

$\displaystyle\nabla^{\mu}\nabla_{\mu}\psi=0.$ (86)

In Schwarzschild spacetime this can be reduced to a radial ODE with the
separability ansatz

$\displaystyle\psi(t,r,\vartheta,\varphi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}e^{-i\omega
t}Y_{lm}(\vartheta,\varphi)R_{lm}(r)$ (87)

where $Y_{lm}(\vartheta,\varphi)$ are the spherical harmonics, and the radial
mode functions $R_{lm}(r)$ are decoupled and satisfy the $h=0$, Regge-Wheeler
equation (1)

$\displaystyle\frac{d^{2}R_{lm}}{dr_{*}^{2}}+(\omega^{2}-V_{l}(r))R_{lm}=0.$
(88)

Here $r_{*}=r+2GM\log(\frac{r}{2GM}-1)$ is the tortoise coordinate and the
effective potential is
$V_{l}(r)=\frac{r-2GM}{r}\left(\frac{l(l+1)}{r^{2}}+\frac{2GM}{r^{3}}\right)$.
We seek a vacuum solution $\psi$ to the scalar wave equation consisting of an
incoming plane wave $\psi^{\rm PW}$ and an outgoing scattered wave $\psi^{\rm
S}$: $\psi=\psi^{\rm PW}+\psi^{\rm S}$. We choose the plane wave to move along
the $z$-axis of the spacetime so that asymptotically it takes the form

$\displaystyle\psi^{\rm PW}\sim e^{-i\omega(t-z)}.$ (89)

Crucially, in this expression $z=r_{*}\cos\vartheta$, which corresponds to the
expression for a plane wave in flat spacetime with the replacement
$r\rightarrow r_{*}$, accounting for the long range gravitational potential
from the black hole Matzner1968 . The scattered wave will asymptotically be a
purely outgoing wave, which written on a basis of spherical harmonics takes
the form

$\displaystyle\psi^{\rm S}\sim r^{-1}\sum_{lm}A_{lm}^{\rm
S}Y_{lm}(\vartheta,\varphi)e^{-i\omega(t-r_{*})}\,.$ (90)

The key observable from the scattering processes is the differential cross
section, which measures the angular profile of the flux from the scattered
wave. For scalar waves it is given by futterman88

$\displaystyle\frac{d\sigma}{d\Omega}$
$\displaystyle=\lim_{r\rightarrow\infty}r^{2}|\psi^{S}|^{2}$ (91)
$\displaystyle=|\sum_{lm}A_{lm}^{\rm
S}Y_{lm}(\vartheta,\varphi)|^{2}\equiv|f(\vartheta,\varphi)|^{2}$ (92)

Determining the amplitude function $f(\vartheta,\varphi)$ and the differential
cross section requires determining $A^{\rm S}_{lm}$, the asymptotic amplitude
modes of the scattered wave.
By projecting onto spherical harmonics, the plane wave (89) can be written as

$\displaystyle\psi^{\rm PW}\sim r^{-1}\sum_{l=0}^{\infty}e^{-i\omega
t}Y_{l0}(\vartheta,\varphi)\left(A^{-}_{l}e^{-i\omega
r_{*}}+A^{+}_{l}e^{i\omega r_{*}}\right),$ (93)

where $A^{-}_{l}=2\pi i\omega^{-1}(-1)^{l}Y_{l0}(0,0)$ and $A^{+}_{l}=-2\pi
i\omega^{-1}Y_{l0}(0,0)$. Since the plane wave only has modes with $m=0$ the
scattered wave will also only be nonzero for $m=0$, i.e. $A^{\rm S}_{l,m\neq
0}=0$ and we will write $A^{\rm S}_{l}\equiv A^{\rm S}_{l0}$. The total
solution can then be written

$\displaystyle\psi$ $\displaystyle=\psi^{\rm PW}+\psi^{\rm S}$ (94)
$\displaystyle\sim r^{-1}\sum_{l=0}^{\infty}e^{-i\omega
t}Y_{l0}(\vartheta,\varphi)\left(A^{-}_{l}e^{-i\omega r_{*}}+(A^{+}_{l}+A^{\rm
S}_{l})e^{i\omega r_{*}}\right)$ (95)
$\displaystyle=r^{-1}\sum_{l=0}^{\infty}e^{-i\omega
t}Y_{l0}(\vartheta,\varphi)\left(A^{-}_{l}e^{-i\omega
r_{*}}+A^{+}_{l}e^{2i\delta_{l}}e^{i\omega r_{*}}\right).$ (96)

In the last line we have introduced the phase shift function $\delta_{l}$,
which is nothing but the particular form of the S-matrix in this basis

$e^{2i\delta_{l}}=1+A^{\rm S}_{l}/A^{+}_{l}$ (97)

Note in particular that the imaginary part of $\delta_{l}$, for which the
scattered wave will have dissipation in (96), is related to absorption into
the black hole Sanchez:1976xm , and that if $\delta_{l}$ were purely real,
there would be no absorption and the scattering would be purely elastic.

The total solution must be a homogeneous solution which has the physical
boundary condition of being a purely ‘ingoing’ at the horizon. This condition
fixes the total solution up to a normalization constant (see Appendix A), and
in particular at radial infinity it can be written as

$\displaystyle\psi\sim r^{-1}\sum_{l=0}^{\infty}Z_{l}e^{-i\omega
t}Y_{l0}(\vartheta,\varphi)\left(B^{\rm inc}_{l0}e^{-i\omega r_{*}}+B^{\rm
ref}_{l0}e^{i\omega r_{*}}\right)$ (98)

where $B^{\rm inc/ref}_{lm}$ are the incidence and reflection coefficients
which can be calculated with standard techniques, see (189)-(190). Here
$Z_{l}$ is the normalization which we will determine presently. Equating (96)
and (98) gives

$\displaystyle Z_{l}$ $\displaystyle=\frac{A^{-}_{l}}{B^{\rm inc}_{l0}},$ (99)
$\displaystyle A^{\rm S}_{l}$ $\displaystyle=\frac{A^{-}_{l}B^{\rm
ref}_{l0}}{B^{\rm inc}_{l0}}-A^{+}_{l}$ (100)
$\displaystyle=\frac{2\pi}{i\omega}Y_{l0}(0,0)\left(e^{2i\delta_{l}}-1\right),$
(101)

where $e^{2i\delta_{l}}=(-1)^{l+1}\frac{B^{\rm ref}_{l0}}{B^{\rm inc}_{l0}}$,
and the amplitude function is

$f(\vartheta,\varphi)=\frac{2\pi}{i\omega}\sum_{l=0}^{\infty}Y_{l0}(0,0)Y_{l0}(\vartheta,\varphi)\left(e^{2i\delta_{l}}-1\right).$
(102)

This will be the fundamental equation to match in the case of the
Schwarzschild black hole. Next we will show how to compute this infinite sum
in the low energy limit.

####  4.1.1 Long wavelength scattering

We now impose that the wavelength of the scattered wave is large relative to
the natural length scales of the problem, namely $\lambda\gg GM$, or in terms
of the frequency $\omega\ll\frac{1}{GM}$. In this limit standard techniques
(see e.g. Sasaki:2003xr ) lead to series expansions for the phase function
$e^{2i\delta_{lm}}$ in the small parameter $\epsilon=2GM\omega$. As will be
appreciated in the following, this is not the Born (tree-level) approximation
in QFT but also encompasses certain loop effects.

As $\epsilon\to 0$, the phase function can be written in a well known form,
see for instance Andersson:2000tf ,

$\displaystyle e^{2i\delta_{lm}}\sim
e^{i\Phi}\frac{\Gamma(l+1-i\epsilon)}{\Gamma(l+1+i\epsilon)}+\mathcal{O}(\epsilon^{2})$
(103)

where $\Phi=2\epsilon\log(2\epsilon)-\epsilon\to 0$ is an overall irrelevant
phase (independent of $l$). The contribution is the ‘Newtonian’ piece in the
sense that it dominates in the limit $\theta\to 0$, containing the expected
divergence as we now show.

We need to compute the spherically symmetric sum

$\displaystyle
f^{N}(\vartheta)=\frac{2\pi}{i\omega}\sum_{l=0}^{\infty}\frac{2l+1}{4\pi}\frac{\Gamma(l+1-i\epsilon)}{\Gamma(l+1+i\epsilon)}P_{l}(\cos\vartheta),$
(104)

where the $m=0$ spherical harmonic reduced to the Legendre function
$P_{l}(x)$. Note that only the phase term contributes in (102). This sum has
closed form

$\displaystyle
f^{N}(\vartheta)=GM\frac{\Gamma(1-i\epsilon)}{\Gamma(1+i\epsilon)}\sin\left(\frac{\vartheta}{2}\right)^{-2+i2\epsilon}.$
(105)

As anticipated, this agrees with the Born approximation (18), up to a phase
depending on $\epsilon$ (which expectedly vanishes as $\epsilon\to 0$). The
factor

$\frac{\Gamma(1-i\epsilon)}{\Gamma(1+i\epsilon)}\,,$ (106)

is termed ‘Newtonian phase’ in analogy with its electromagnetic counterpart.
Crucially, it is not a phase for $\omega\in\mathbb{C}$. Its poles located at
$\epsilon:=2GM\omega=in\,,n\in\mathbb{N}$ are nothing but the spectrum of
bounded states of the Newtonian problem. We will see how this phase can be
recovered from tree-level amplitudes in section 5.1.

###  4.2 Scalar Waves on the Kerr Black Hole

We are now ready to tackle the problem of scalar waves in Kerr via partial
waves methods. Here the fundamental fact is the separability of the scalar
wave equation into the so-called Teukolsky equations and its solution based on
BHPT as we review in Appendix A.

In order to obtain analytic solutions we will need the following
considerations. In the spinning case there are two dimensionless quantities
which appear in our analysis: $GM\omega$ and $a/GM$. The expansion we are
going to adopt should correspond to a point-particle setup in which only the
intrinsic structure of the BH, i.e. its spin multipoles, is preserved. For
this we consider the black hole characteristic size to be small as compared to
both the wavelength of the waves $1/\omega$ and the ‘spin’ radius $a$. That is
to say we take small $GM\omega$ and large $a/GM$. Of course, the latter of
these requirements naturally reveals the ring-singularity as its radius $a$
grows bigger than the horizon itself, leading to a superextremal Kerr metric.
But we expect these effects to be unobservable from large distances as the
singularity becomes essentially point-like, so we will further consider
$\omega\sim 1/r\ll 1/a$. Thus, in synthesis, we assume

$\omega\ll\frac{1}{a}\ll\frac{1}{GM},\qquad\lambda\gg a\gg GM.$ (107)

In practice we will determine the differential cross section as an expansion
in small frequency at fixed $a/GM<1$, where BHPT is applicable. We will then
analytically continue all functions to $a/GM>1$.

Throughout this section we will use $(t,r,\vartheta,\varphi)$ to denote
standard Boyer-Lindquist coordinates. The main alteration to the previous
discussion is that the separability ansatz now relies on the spheroidal
harmonics

$\displaystyle\psi(t,r,\vartheta,\varphi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}e^{-i\omega
t}S_{lm}(\vartheta,\varphi;a\omega)R_{lm}(r)$ (108)

and the radial functions satisfy the $s=0$ radial Teukolsky equation (185).
The spheroidal harmonics admit a low frequency expansion whose leading order
is simply the spherical harmonics

$\displaystyle S_{lm}(\vartheta,\varphi;a\omega)\sim
Y_{lm}(\vartheta,\varphi)+\mathcal{O}(a^{2}\omega^{2}).$ (109)

but whose subleading orders contain, as expected, corrections related to the
internal structure of the spinning body.

Let us first comment on the polar scattering  scenario. In this case the plane
wave moves up the BH axis of rotation, which is aligned with z-axis of the
coordinate system. All other details carry through as in Schwarzschild
scattering and the amplitude function is now

$\displaystyle
f(\vartheta,\varphi)=\frac{2\pi}{i\omega}\sum_{l=0}^{\infty}S_{l0}(0,0;a\omega)S_{l0}(\vartheta,\varphi;a\omega)\left(e^{2i\delta_{l}}-1\right),$
(110)

with $e^{2i\delta_{l}}=(-1)^{l+1}\frac{B^{\rm ref}_{l0}}{B^{\rm inc}_{l0}}$,
where all dependence on the BH spin is now included in the coefficients
$B^{\rm ref/inc}_{lm}$.

On the other hand, when the plane wave is incident along a vector in the
$x$-$z$ plane at an angle $\gamma$ from the $z$-axis, the calculation follows
the same procedure, with some more technical complications (see Appendix A of
Glampedakis:2001cx ). The end result is that the amplitude function is

$\displaystyle
f(\vartheta,\varphi)=\frac{2\pi}{i\omega}\sum_{l=0}^{\infty}\sum_{m=-\infty}^{\infty}S_{lm}(\gamma,0;a\omega)S_{lm}(\vartheta,\varphi;a\omega)\left(e^{2i\delta_{lm}}-1\right),$
(111)

with $e^{2i\delta_{lm}}=(-1)^{l+1}\frac{B^{\rm ref}_{lm}}{B^{\rm inc}_{lm}}$.
Thus the phases acquire and additional dependence on the azimuthal quantum
number $m$ conjugate to the angle $\phi$. This will play an important role in
the eikonal description of scattering, where it will be identified with the
angular momentum component $L_{z}$.

####  4.2.1 Higher order scattering

Writing the phase function in the factorised form

$\displaystyle
e^{2i\delta_{lm}}=e^{i\Phi}\frac{\Gamma(l+1-i\epsilon)}{\Gamma(l+1+i\epsilon)}\beta_{lm},$
(112)

higher order corrections in the spin of the BH are captured within the
function $\beta_{lm}$. These take the form

$\displaystyle\beta_{lm}=1+\beta_{lm}^{(2)}\epsilon^{2}+\beta_{lm}^{(3)}\epsilon^{3}+\beta_{lm}^{(4)}\epsilon^{4}+\ldots$
(113)

For generic modes the $\beta_{lm}^{(i)}$ are polynomials of $a^{\star}=a/GM$.
As mentioned above eq. (107), we will only be concerned with the $a^{\star}\gg
1$ expansion of these functions, and so we give here only the leading order
terms in the large-$a^{\star}$ expansion. For generic values of $l$ (see
below), they are given by

$\displaystyle\beta_{lm}^{(2)}$
$\displaystyle\sim-\frac{im}{l(l+1)}a^{\star}+\mathcal{O}(1),$ (114)
$\displaystyle\beta_{lm}^{(3)}$
$\displaystyle\sim-\frac{i\left(l(l+1)-3m^{2}\right)}{2l(l+1)(2l-1)(2l+3)}a^{\star
2}+\mathcal{O}(a^{\star}),$ (115) $\displaystyle\beta_{lm}^{(4)}$
$\displaystyle\sim-\frac{im\left(3l^{2}+3l-1-5m^{2}\right)}{6(l-1)l(l+1)(l+2)(2l-1)(2l+3)}a^{\star
3}+\mathcal{O}(a^{\star 2}),$ (116) $\displaystyle\beta_{lm}^{(5)}$
$\displaystyle\sim\frac{i}{8l(l+1)(2l-3)(2l-1)^{3}(2l+3)^{3}(2l+5)}\bigg{(}l(l+1)(12l^{4}+24l^{3}-l^{2}-13l+15$
$\displaystyle-2\left(100l^{4}+200l^{3}+13l^{2}-87l+45\right)m^{2}+9\left(28l^{2}+28l-5\right)m^{4})\bigg{)}a^{\star
4}+\mathcal{O}(a^{\star 3}),$ (117) $\displaystyle\beta_{lm}^{(6)}$
$\displaystyle\sim-\frac{im}{40(l-2)(l-1)l(l+1)(l+2)(l+3)(2l-3)(2l-1)^{3}(2l+3)^{3}(2l+5)}$
$\displaystyle\bigg{(}140l^{8}+560l^{7}+35l^{6}-1855l^{5}-958l^{4}+1829l^{3}+364l^{2}-953l+138$
$\displaystyle-10\left(84l^{6}+252l^{5}-135l^{4}-690l^{3}-28l^{2}+359l-192\right)m^{2}$
$\displaystyle+\left(828l^{4}+1656l^{3}-2009l^{2}-2837l-438\right)m^{4}\bigg{)}a^{\star
5}+\mathcal{O}(a^{\star 4}),$ (118)

Since the $\beta_{lm}^{(i)}$ are simply polynomials in $a^{\star}$, these
expressions are identical to the highest power of $a^{\star}$ in the full
expressions. We highlight here the important feature that in this limit each
$\beta_{lm}^{(i)}$ is purely imaginary, which results in the phase function
$\delta_{lm}$ being purely real. As mentioned, this corresponds to the absence
of horizon absorption Sanchez:1976xm ; Dolan:2008kf . While we do not have a
definition of horizon absorption in the limit $a^{\star}\gg 1$, we assume the
analytic continuation from $a^{\star}<1$ preserves these features.

These results are valid for generic values of $l$, meaning that they are
indeed incorrect for some $l<l_{\textrm{min}}$. In fact it can be seen that
they contain poles at such low values of $l$, reminiscent of anomalous
contributions in Regge theory Gribov:2003nw . As we will see in section 5, and
as will be preponderant in Part II, generic values of $l$ are associated to
factorization diagrams, whereas certain small values of $l$ (or anomalous)
shall be associated to contact terms in the QFT amplitude. We now discuss the
anomalous contributions.

#### Anomalous Terms and Absorption

The generic-$l$ expressions need to be supplemented by explicit computation of
low-$l$ values which contain anomalous terms. In particular (114)-(118) are
valid for all $l\geq 3$. For $l=2$: $\beta_{lm}^{(i\geq 5)}$ are incorrect,
for $l=1$: $\beta_{lm}^{(i\geq 3)}$ are incorrect and for $l=0$ none are
correct. While the full expressions for the anomalous values will be provided
in the supplemental files

we will examine closely particular terms to illustrate the complications which
arise. For example, consider the first two terms for $l=m=0$:

$\displaystyle\beta_{00}^{(2)}$
$\displaystyle=-(1+\hat{\kappa}-\frac{11}{6}i\pi),$ (119)
$\displaystyle\beta_{00}^{(3)}$
$\displaystyle=\frac{i}{18}(3{a^{\star}}^{2}+(-33+18i\pi)\hat{\kappa}-36\log(2\epsilon\hat{\kappa})+11\pi^{2}+18i\pi-36\gamma_{\mathrm{E}}+51),$
(120)

where $\hat{\kappa}=\sqrt{1-a^{\star 2}}$ and $\gamma_{\mathrm{E}}$ is Euler’s
constant. We see immediately the dependence on $a^{\star}$ is no longer purely
polynomial due to the appearance of $\hat{\kappa}$. For $\beta_{00}^{(2)}$ the
leading order contribution in the limit $a^{\star}\gg 1$ comes from the
$\hat{\kappa}$ term, and notably is real when $a^{\star}<1$. Thus we interpret
it as being an absorption effect. Anticipating the matching to the quantum
calculation which manifestly has a conservation of incoming and outgoing
radiation (the tree-level amplitudes are real away from singular kinematics)
we will remove the absorption.

Our procedure to remove absorption is simply to take the real part of
$\delta_{lm}$161616In practice we will ignore the factor of
$\frac{\Gamma(l+1-i\epsilon)}{\Gamma(l+1+i\epsilon)}$ in (112) in this
procedure, since it does not affect the leading-order-in-G.. This is expected
to be equivalent to setting reflective boundary conditions on the horizon but
we have not explicitly checked this. Once absorption is removed in this
manner, we can analytically continue the result to the $a^{\star}\gg 1$ limit.
Defining the new ‘conservative’ scattering by $\tilde{\beta}_{lm}$, our
procedure has the desired effect on the examples given above, removing the
dependence on $\hat{\kappa}$, and moreover having the leading order term in
the $a^{\star}\gg 1$ limit come purely from polynomial dependence in
$a^{\star}$:

$\displaystyle\tilde{\beta}_{00}^{(2)}$ $\displaystyle=\frac{11}{6}i\pi,$
(121) $\displaystyle\tilde{\beta}_{00}^{(3)}$
$\displaystyle=\frac{1}{18}i\left(3a^{\star
2}-33\hat{\kappa}-36\log(2\epsilon\hat{\kappa})+11\pi^{2}-36\gamma_{\mathrm{E}}+51\right).$
(122)

When $m\neq 0$, more complicated functions arise changing this simple
behaviour. For example,

$\displaystyle\beta_{11}^{(4)}$
$\displaystyle=-\frac{1}{360}i\left(2{a^{\star}}^{3}-65i{a^{\star}}^{2}+160{a^{\star}}-10i\right)+\frac{12870ia^{\star
2}+65100{a^{\star}}+78037i}{189000}\pi+\frac{i{a^{\star}}}{9}\gamma_{\mathrm{E}}$
$\displaystyle+\frac{1}{18}i{a^{\star}}\log(2\epsilon\hat{\kappa})-\left(\frac{361}{1800}+\frac{7i{a^{\star}}}{90}\right)\pi^{2}+\frac{1}{36}\hat{\kappa}+\frac{1}{18}ia^{\star}\psi^{(0)}\left(i{a^{\star}}/\hat{\kappa}\right).$
(123)

where $\psi^{(0)}(z)$ is the digamma function. We have not found a simple
prescription to take the large $a^{\star}$ limit of the polygamma function:
Since $\psi^{(n)}(z)$ diverges at $z=-1$ but is finite at $z=1$, the polygamma
function can contribute or not to the limit depending on the branch that we
choose for $\hat{\kappa}$, as the argument becomes
$i{a^{\star}}/\hat{\kappa}\to\pm 1$. Explicitly, the two branches give

$\displaystyle\frac{1}{36}i{a^{\star}}\psi^{(0)}\left(i{a^{\star}}/\hat{\kappa}_{+}\right)\sim-\frac{1}{36}i\gamma
a+\mathcal{O}({a^{\star}}^{-1}),$ (124)
$\displaystyle\frac{1}{36}i{a^{\star}}\psi^{(0)}\left(i{a^{\star}}/\hat{\kappa}_{-}\right)\sim\frac{i{a^{\star}}^{3}}{18}+\mathcal{O}({a^{\star}}),$
(125)

Notably, giving a real cotribution to the phase, the second branch can
explicitly contribute to the ${a^{\star}}^{3}$ order term in the scattering
amplitude. To account for this ambiguity let us write171717Of course,
according to our prescription we should take the real part
$\psi^{(0)}\left(i{a^{\star}}/\hat{\kappa}\right)+\psi^{(0)}\left(-i{a^{\star}}/\hat{\kappa}\right)$
before the large ${a^{\star}}$ limit, which can certainly account for the
contribution of both branches. However, even this combination does not
contribute to the limit $|a^{\star}|\to\infty$ in a generic (complex)
direction, as opposed to the polynomial expansions we have obtained
previously, so it still appears natural to remove it.

$\displaystyle\beta_{11}^{(4)}$
$\displaystyle=\beta_{11,\textrm{reg}}^{(4)}+\frac{1}{18}i{a^{\star}}\psi^{(0)}\left(i{a^{\star}}/\hat{\kappa}\right).$
(126)

We find similar behaviour with the polygamma functions for $m=-1$, as well as
at higher powers of $\epsilon$ for different $l$-values, where we define the
regularized values $\beta_{lm,\textrm{reg}}$ by simply excluding the
contributions from the polygamma functions $\psi^{(n)}$. As it turns out, we
show in the next section how the contribution from the regularized values
$\beta_{lm,\textrm{reg}}$, or more precisely their real parts
$\tilde{\beta}_{lm,\textrm{reg}}$, provides a perfect match to the Born
amplitudes obtained in section 3. On the other hand, the contributions from
$\psi^{(n)}$ lead to additional contact terms, hence can also be added to the
Born amplitudes without spoiling their factorization properties.

###  4.3 Matching procedure: BHPT and QFT amplitudes

In order to match our QFT amplitudes with the results from BHPT it is
convenient to now project the previous amplitude function (111) onto spherical
harmonics

$f(\vartheta,\varphi)=\sum_{lm}Y_{lm}(\vartheta,\varphi)f_{lm}(\gamma),$ (127)

where $f_{lm}=f^{N}+f^{(1)}_{lm}z+f^{(2)}_{lm}z^{2}+\ldots$ and
$z=a^{\star}\epsilon={{\color[rgb]{0,0,0}2}}a\omega$. While there are some
subtleties in the projection as discussed in Sec. 4.3.1. of Dolan:2008kf , we
will omit such details here.

An important feature emerges for polar scattering, as described around (110),
which is obtained by setting $\gamma=0$. One finds that only the modes
$f^{(i)}_{l0}$ are non-trivial and

$f^{(i)}_{l0}(0)=0\,,\,\,\textrm{ for }\,\,i=2k+1\,\,\textrm{ or for }i\leq
l.$ (128)

This means that, other than the Newtonian term, for polar scattering the
infinite sum over the spherical harmonics reduces to a finite sum for each
power of $z$. This has been noted in Dolan:2008kf  for the case of GW
scattering in Kerr and will be proved in Part II of this work starting from
the QFT framework.

In the off-axis case where $\gamma\neq 0$, no such simplification occurs, and
we find it convenient to compare with the 4-pt amplitudes in section 3, more
precisely the scalar amplitude (71), by working mode-by-mode. To do this we
first need to align our coordinates by a rotation.

Let us see how this works. The amplitude function (127) is written as a sum
over spherical harmonics $Y_{lm}(\vartheta,\varphi)$, where
$(\vartheta,\varphi)$ is the direction of the outgoing wave in a coordinate
system where the spin direction is $\vartheta=0$, i.e. the $+Z$ direction, and
the incoming wave is in the direction $(\vartheta,\varphi)=(\gamma,0)$ (in the
$X$-$Z$ plane), so that $\gamma$ is the angle between the spin and the
incoming wave.

Now let us rotate our $(\vartheta,\varphi)$–$(X,Y,Z)$ coordinate system about
the $Y$ axis (the same as the new $Y^{\prime}$-axis) by an angle $\gamma$, to
bring the incoming wave direction to the new $+Z^{\prime}$ direction, and call
the new coordinates
$(\theta^{\prime},\phi^{\prime})$–$(X^{\prime},Y^{\prime},Z^{\prime})$. This
is still not the same as the $(\theta,\phi)$–$(x,y,z)$ coordinate system used
in section 2, but now the $z$-axes are the same. The rotation of spherical
harmonics is known to be accomplished by

$Y_{lm}(\theta^{\prime},\phi^{\prime})=\sum_{m^{\prime}}D^{l*}_{mm^{\prime}}(\gamma)\;Y_{lm^{\prime}}(\vartheta,\varphi)\,,$
(129)

where $D^{l*}_{mm^{\prime}}$ is the (complex conjugate) Wigner $D$-matrix with
Euler angles $(0,\gamma,0)$,

$D^{l*}_{mm^{\prime}}(\gamma)=D^{l*}_{mm^{\prime}}(0,\gamma,0)=(-1)^{m^{\prime}}\sqrt{\frac{4\pi}{2l+1}}{}_{-m^{\prime}}Y_{lm}(\gamma,0).$
(130)

Now the amplitude (127) takes the form

$f=\sum_{lm}Y_{lm}(\theta^{\prime},\phi^{\prime})f^{\prime}_{lm}(\gamma),$
(131)

with

$f^{\prime}_{lm}(\gamma)=\sum_{m^{\prime}}D^{l*}_{m^{\prime}m}(\gamma)\;f_{lm^{\prime}}(\gamma),$
(132)

where we relabeled $m\leftrightarrow m^{\prime}$ after substituting. Now, in
this new $(\theta^{\prime},\phi^{\prime})$ coordinate system, with
corresponding $(X^{\prime},Y^{\prime},Z^{\prime})$, the spin vector is
$\vec{a}=a(-\sin\gamma,0,\cos\gamma)$, the incoming wave vector is
$\vec{k}_{2}=(0,0,\omega)$, and the outgoing wave vector is
$\vec{k}_{3}=\omega(\sin\theta^{\prime}\cos\phi^{\prime},\sin\theta^{\prime}\sin\phi^{\prime},\cos\theta^{\prime})$.

Finally, we have the third $(\theta,\phi)$–$(x,y,z)$ coordinate system used in
section 2, where $\vec{k}_{2}$ is in the $z$ direction (same as $Z^{\prime}$
direction) and $\vec{k}_{3}$ is in the $x$-$z$ plane at an angle $\theta$ from
$\vec{k}_{2}$, and this is the same $\theta=\theta^{\prime}$ from the second
coordinates. To translate the result (72) into the
$(\theta^{\prime},\phi^{\prime})$ coordinates, we use $\theta=\theta^{\prime}$
and

$a_{z}=a\cos\gamma,\quad a_{x}=-a\sin\gamma\cos\phi^{\prime},\quad
a_{y}=-a\sin\gamma\sin\phi^{\prime},$ (133)

as can be confirmed by comparing the values of $\vec{a}\cdot\vec{k}_{2}$,
$\vec{a}\cdot\vec{k}_{3}$ and $\vec{k}_{2}\cdot\vec{k}_{3}$.

It is most convenient to compare our amplitudes results from section 3 using
$(\theta,\phi)$ to our BHPT results from section 4.2.1 using
$(\vartheta,\varphi)$ by comparing the amplitudes of their spherical harmonic
modes in the intermediate $(\theta^{\prime},\phi^{\prime})$ coordinates as in
(131).

We recall that the amplitude function $f$ at the leading order in $\epsilon$
(at fixed $a\omega=a^{\star}\epsilon/2$), coming from the tree-level
scattering amplitude (72), is

$f=\frac{1}{2\sin^{2}\left(\frac{\theta}{2}\right)}\left[\cosh{a{\cdot}q}-i\omega
a_{y}\sin\theta\frac{\sinh{a{\cdot}q}}{a{\cdot}q}\right],$ (134)

with $q{\cdot}a=-\omega(a_{x}\sin(\theta)-2a_{z}\sin^{2}(\theta/2))$, which is
expressed in terms of $(\theta^{\prime},\phi^{\prime})$ by using
$\theta=\theta^{\prime}$ and (133). Its mode amplitudes $f^{\prime}_{lm}$ from
(131) are given by integrals over the 2-sphere,

$f^{\prime}_{lm}(\gamma)=\int
d\Omega^{\prime}\,Y^{*}_{lm}(\theta^{\prime},\phi^{\prime})f(\gamma,\theta^{\prime},\phi^{\prime}),$
(135)

and depend only on the angle $\gamma$ between the incoming momentum and the
spin, and on the parameters $\epsilon$ and $a^{\star}$. In general, let us
write these as an expansion in $\epsilon$, focusing on the leading order in
the large $a^{\star}$ expansion at each order in $\epsilon$,

$f^{\prime}_{lm}=\sum_{n=0}^{\infty}\epsilon^{n}\Big{[}f^{\prime}_{lm,n}a^{\star}{}^{n}+\mathcal{O}(a^{\star}{}^{(n-1)})\Big{]}.$
(136)

We find that this pattern also holds for the analytically continued BHPT
amplitudes in the large $a^{\star}$ expansion. At linear order in spin, from
(134), we find

$\displaystyle\begin{aligned} f^{\prime}_{00,1}&=0,\\\
\\{f^{\prime}_{1m,1}\\}&=\frac{1}{2}\sqrt{\frac{3\pi}{2}}\sin\gamma\\{1,0,1\\},\\\
\\{f^{\prime}_{2m,1}\\}&=\frac{1}{2}\sqrt{\frac{5\pi}{6}}\sin\gamma\\{0,1,0,1,0\\},\\\
\\{f^{\prime}_{3m,1}\\}&=\frac{1}{4}\sqrt{\frac{7\pi}{3}}\sin\gamma\\{0,0,1,0,1,0,0\\},\end{aligned}$
(137)

and so on, with $m=\\{-l,\ldots,l\\}$. From (134) at quadratic order in spin,
we find

$\displaystyle\begin{aligned} f^{\prime}_{00,2}&=\frac{\sqrt{\pi}}{16}(3+\cos
2\gamma),\\\ \\{f^{\prime}_{1m,2}\\}&=\\{\frac{1}{4}\sqrt{\frac{\pi}{6}}\sin
2\gamma,-\frac{1}{16}\sqrt{\frac{\pi}{3}}(1+3\cos
2\gamma),-\frac{1}{4}\sqrt{\frac{\pi}{6}}\sin 2\gamma\\},\\\
\\{f^{\prime}_{2m,1}\\}&=\frac{1}{16}\sqrt{\frac{5\pi}{6}}\sin^{2}\gamma\\{1,0,0,0,1\\},\\\
\\{f^{\prime}_{3m,1}\\}&=\frac{1}{16}\sqrt{\frac{7\pi}{30}}\sin^{2}\gamma\\{0,1,0,0,0,1,0\\},\\\
\\{f^{\prime}_{4m,1}\\}&=\frac{1}{16}\sqrt{\frac{\pi}{10}}\sin^{2}\gamma\\{0,0,1,0,0,0,1,0,0\\},\end{aligned}$
(138)

and so on. Remarkably, also continuing to large values of $l$, we find that
all of these $(\epsilon a^{\star})^{1}$ and $(\epsilon a^{\star})^{2}$ terms
in the mode amplitudes from the (minimal) tree-level scattering amplitude
(134) precisely match those computed from the analytically continued (and
absorption-removed) BHPT theory amplitudes as described in section 4.2.1.

At the $(\epsilon a^{\star})^{3}$ and $l\leq 2$ the BHPT computation leads to
anomalous terms as explained around eq. (126). Taking
$\beta_{lm}\to\beta_{lm,\textrm{reg}}$, that is, discarding the extra terms
arising from polygamma functions we obtain a perfect match to the Born
amplitudes.

Since the Born amplitudes can be obtained directly by solving the classical
Klein-Gordon equation in the limit $GM\omega\to 0$, it makes sense to regard
our prescription as a recipe for obtaining the point-particle limit. However,
it is also of interest to understand the contributions from the polygamma
functions more in detail. As it turns out, the fact that the corrections
appear only for $l\leq 2$ enables us to write down a deformation of the Born
amplitude (134) that resums these terms

$f_{\textrm{anomalous}}=\frac{1}{2\sin^{2}\left(\frac{\theta}{2}\right)}\left[\cosh{a{\cdot}q}-i\omega
a_{y}\sin\theta\left(\frac{\sinh{a{\cdot}q}}{a{\cdot}q}+\Delta
f\right)+\mathcal{O}(a^{7})\right].$ (139)

where the anomalous contributions can be found to be

$\Delta
f=\frac{q^{2}a^{2}}{3}\left(1{-}\cot^{2}(\theta/2)\frac{q^{2}a^{2}}{15}{+}\frac{(a{\cdot}q)^{2}}{10}\right)\,\,\,,\qquad
q^{2}=-4\omega^{2}\sin^{2}(\theta/2)\,.$ (140)

As anticipated, the correction corresponds to a contact term, with no
factorization residues in $q^{2}$. As we will see when discussing the eikonal
approximation, this just reflects the fact that the anomalous contribution is
not present for large $l$. On the other hand, contact terms can be added to
the four-point amplitude without spoiling the three-point on-shell structures
that have been previously matched to the Kerr metric. In fact, as anticipated
below (26), the first term $\sim a^{3}q^{3}$ can be easily incorporated in the
effective vertex as a $\mathcal{O}(q^{2})$ correction.

##  5 A classical Wave-Particle duality and the eikonal in Kerr

Unlike the Schwarzschild case, the results (71)-(76) for the scattering of
waves of diverse helicities in the Kerr background do not share a universal
pattern. Moreover, for scattering of gravitational waves, it will become clear
in Part II that diverse effective operators emerge and lead to an even more
exotic result for $h=2$.

However, we have found in equation (78) that a universality class arises for
the Kerr black hole in the forward limit $\theta\to 0$, which we refer to as
the high-energy, eikonal or geometrical optics limit.181818The eikonal limit
has been studied with renewed interested for both massive and massless
particles since it allows for a clean passage to obtain classical observables
AccettulliHuber:2020oou ; DiVecchia:2021bdo ; DiVecchia:2020ymx ;
DiVecchia:2019myk ; Chen:2021huj ; Saotome:2012vy ; DiVecchia:2019kta ;
Bern:2020gjj . It is interesting however that in the case of wave scattering
we are able to define a classical limit for finite values of $\theta$. A
complementary picture in this context of wave scattering is to be provided in
CGKO . This holds for any helicity configuration, including gravitational
waves as we will show. The reason is that, as we will see, such regime
corresponds to a particle-like behaviour of the scattered waves and the phases
computed in the previous section encode classical trajectories in the sense of
the WKB/optical approximation. Therefore, the universality reflects that,
classically, massless particles follow null geodesics irrespective of their
internal structure. From the QFT perspective, the eikonal is controlled by
factorization channels where gravitons couple to the massless particle in a
universal way, reflecting the equivalence principle Guevara:2018wpp .

In this section we aim to make the previous discussion precise. We first show
how the partial wave series reduces to the eikonal amplitude. The eikonal
phase is obtained from the partial waves in the large $l$ limit, or
equivalently ‘high-energies’ $\omega b\to\infty$ for an impact parameter $b$.
At leading order in $\epsilon=GM\omega\to 0$ the eikonal is linked directly to
Born amplitude computed previously, but further provides information to all
orders in $G$ such as the Newtonian phase. Moreover, the partial waves
computed in the previous section also incorporate corrections in $\epsilon$:
We match such corrections to the subleading eikonal phases computed recently
in Guevara:2017csg ; Liu:2021zxr ; Kosmopoulos:2021zoq ; Hinderer:2008dm  from
loop amplitudes and higher orders in spin. This provides an important cross-
check of different results.

As will be argued below, the eikonal phase, even at subleading (loop) orders
in $\epsilon$, is determined solely from the radial action of null geodesics
in Kerr in the spirit of the WKB approximation. Because of this, using
Hamilton-Jacobi theory, it can be used to compute two important classical
observables: the scattering angle and the Shapiro time delay, which reflects
the underlying unitarity of the effective theory. We incorporate spin
corrections to these observable in two regimes: Polar and equatorial
orientations of the incoming wave.

###  5.1 From Partial Waves to Eikonal

The eikonal amplitude can be easily obtained from the $\theta\to 0$ limit of
partial waves, and is dominated by the contributions at large angular orbital
momentum $l$. Let us illustrate this with the spinless clase $a=0$ for
simplicity.191919This closely follows more detailed discussions that can be
found in e.g. FORD1959259 ; brittin1959lectures ; Gribov:2003nw ; BETHE1951 ;
Glampedakis:2001cx ; Andersson:2000tf ; Jauch:1976ava , where the argument
holds at least to subleading order in $\theta\to 0$. Consider the partial wave
amplitude (102),

$f(\theta)=\frac{1}{2i\omega}\sum_{l=0}^{\infty}(2l+1)(e^{2i\delta_{l}}-1)P_{l}(\cos(\theta))\,,$
(141)

In the $\theta\to 0$ limit the scattered momenta is close to the axis of the
incoming wave. Spherical waves are then approximated by cylindrical waves via

$P_{l}(\cos(\theta))=J_{0}\left((2l+1)\sin(\theta/2)\right)+\mathcal{O}(\sin^{2}(\theta/2))$
(142)

Such relation is most accurate at large $l$ which will provide the dominant
contribution to (141). Using the integral representation of the Bessel
function we obtain

$\displaystyle P_{l}(\cos(\theta))\to$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}d\phi
e^{i(2l+1)\sin(\theta/2)\cos\phi}$ (143) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}d\phi e^{i\vec{q}\cdot\vec{b}}$
(144)

where we have introduced the 2-vectors $\vec{q},\vec{b}$ defined by

$|\vec{q}|=2\omega\sin(\theta/2)\,,\,\,\,b=|\vec{b}|=\frac{1}{\omega}(l+1/2)$
(145)

with $\phi$ as a relative angle. Thus we will identify $\vec{q}$ as the
momentum transfer vector and $l+1/2=\omega b$ as the classical angular
momentum in the orbital plane Andersson:2000tf . So far no assumption has been
made regarding $\omega$. In order to approximate the infinite sum over
$l=1,2,\ldots$ by integration over $b$ we will take $\Delta
b/b_{c}=\frac{1}{b_{c}\omega}\to 0$ for some characteristic impact parameter
for which we expect the integral to localize. This high-energy regime is
characteristic of the eikonal: as the wavelength is neglectable with respect
to the classical value of the impact parameter it leads to a point-particle
behaviour. Note that this still overlaps with the Born approximation as long
as $GM\omega\to 0$.

Putting everything together, (141) can be written as

$if(\vec{q})=\frac{\omega}{2\pi}\int
bdbd\phi(e^{i\chi(b)}-1)e^{i\vec{q}\cdot\vec{b}}\,,$ (146)

where the eikonal phase is given by

$\chi(b):=2\delta_{l}=2\delta_{\omega b-1/2}$ (147)

Equivalently, we can write the inverse relation as

$\displaystyle(e^{i\chi(b)}-1)=$
$\displaystyle\frac{i}{\omega}\int\frac{d^{2}\vec{q}}{(2\pi)}f(\vec{q})e^{-i\vec{q}\cdot\vec{b}}\,$
(148) $\displaystyle=$ $\displaystyle i8\pi
M\int\frac{d^{4}q}{(2\pi)^{2}}e^{iq\cdot b}\delta(2p_{1}\cdot
q)\delta(2k_{2}\cdot q)f(q)$ (149)

where we have covariantized the first line by making use of the kinematics
given in section 2. We can now implement the relation between QFT amplitudes
and the classical NP scalars, $8\pi Mf(q)\to\langle A_{4}(q)\rangle$. Taking
the leading Born order $GM\omega\to 0$ on both sides gives:

$\chi_{\textrm{LO}}(b)=\int\frac{d^{4}q}{(2\pi)^{2}}e^{iq\cdot
b}\delta(2p_{1}\cdot q)\delta(2k_{2}\cdot q)\langle A_{4}(q)\rangle,$ (150)

Notice that we have written the amplitude in the classical limit, i.e.
$(s-M^{2})^{2}\sim tM^{2}\ll M^{4}$. However, as contact terms in $t=q^{2}$
will lead to ultralocal functions of the impact parameter $b$, it becomes
clear that we can drop powers of
${\color[rgb]{0,0,0}M}^{2}t/(s-M^{2})^{2}=\sin^{2}\theta/2$. This regime, i.e.

${\color[rgb]{0,0,0}M}^{2}t\ll(s-M^{2})^{2}\ll M^{4},$ (151)

is precisely what we assumed to derive equation (146). The regime has been
recently considered in the eikonal approximation for for massless particles
scattering off heavy ones Bjerrum-Bohr:2016hpa ; AccettulliHuber:2020oou .
Remarkably, it is contained as a limiting case of the classical prescription
of Section 2, i.e. $\theta\to 0$.

In order to illustrate the comparison to partial waves let us evaluate the
eikonal amplitude (146) using the leading approximation $\chi_{\textrm{LO}}$.
This is computed from the tree-level amplitude (150). As we review below in
equation (172), in the Schwarzschild case ($a=0$) for any helicity of the
wave, it is given by

$\chi_{\rm{LO}}(b)=-4GM\omega\log\omega b+\Phi_{\textrm{IR}},$ (152)

where $\Phi_{\textrm{IR}}$ contains a $b$-independent infrared divergent
phase. In our computation using partial waves, eq. (105), such a divergence is
regulated by the form of asymptotic wavefunctions (90), so we will ignore it
here. Plugging this into (146), and assuming $b\neq 0$, the integral evaluates
to

$if(q)=iGM\frac{\Gamma[1-i\epsilon]}{\Gamma[1+i\epsilon]}\left(\sin(\theta/2)\right)^{-2+2i\epsilon}$
(153)

where we have used $q^{2}=-4\sin^{2}(\theta/2)\omega^{2}$. This result is
precisely the Newtonian approximation obtained in (105), which dominates in
the forward limit $\theta\to 0$ Andersson:2000tf . The fact that the leading
eikonal can be used to derive the Newtonian phase was noted in Kabat:1992tb ,
where it was used to further study the spectrum of bound states associated to
the Newtonian potential. Bound states are characterized by the singularities
of (153) which appear at imaginary frequencies $\epsilon=i,2i,3i,\ldots$ as
expected.

###  5.2 Relation to Null Geodesics

In order to derive observables from the eikonal approximation it is important
to understand its direct relation to null geodesics in Kerr. This does not
only reflect the universality of the optical limit of wave scattering, but
also shows explicitly that it admits a point-particle description. Here we
present an argument for the leading eikonal (tree-level), whereas in section
5.4 we will check it holds to subleading (loop) orders.

In the harmonic gauge (13), we consider the momentum covector
$k_{\mu}=g_{\mu\nu}k^{\nu}$, where $k^{\mu}$ is a null four-velocity. Nicely,
in terms of the covector the (linearized) geodesic equation for an affine
parameter $\lambda$ takes a very simple form:

$\frac{dk_{\mu}}{d\lambda}=\frac{\kappa}{2}\partial_{\mu}h_{\alpha\beta}(x)k^{\alpha}k^{\beta}\,.$
(154)

Interpreting $k_{\mu}$ as canonical momenta, we can already see from this form
that the contraction $h_{\alpha\beta}(x)k^{\alpha}k^{\beta}$ corresponds to an
effective potential, and in fact is nothing but the four-point amplitude. Let
us be more explicit. Integrating this equation along the trajectory
$x^{\mu}(\lambda)=k_{2}^{\mu}\lambda+b^{\mu}+\mathcal{O}(G)$ (at the
linearized level we ignore the Coulomb drag) we obtain

$k_{2}^{\mu}(+\infty)-k_{2}^{\mu}(-\infty)=\Delta k^{\mu}=\frac{\kappa}{2}\int
d\lambda\partial^{\mu}h_{\alpha\beta}(k_{2}\lambda+b)k^{\alpha}_{2}k^{\beta}_{2}\,.$
(155)

Note that at $\lambda\to\pm\infty$ we have raised the indices of $k_{\mu}$
using the flat metric $\eta^{\mu\nu}$. This is because geodesics reach null
infinity where the metric is (asymptotically) flat, hence we can identify
$k_{\mu}$ with the momentum vector in the scattering amplitude.

We now plug in the Fourier representation of the linearized gravitational
field generated by a source with momentum $p_{1}$ (cf. equations (67)-(69))

$h_{\mu\nu}(x)=\int\frac{d^{4}q}{(2\pi)^{3}}\frac{\delta(2q\cdot
p_{1})}{q^{2}+i\epsilon}e^{iq\cdot x}\hat{h}_{\mu\nu}(q)\,.$ (156)

Recall that the field component $\hat{h}_{\mu\nu}$ (which can be read off from
(69) in the case of the Kerr metric) has classical scaling and can be
interpreted as an off-shell three-point vertex including the harmonic
projector. From (155) we obtain the momentum deflection

$\displaystyle\Delta k^{\mu}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}\frac{\kappa}{2}}\int\frac{d^{4}q}{(2\pi)^{3}}\frac{\delta(2q\cdot
p_{1})}{q^{2}+i\epsilon}q^{\mu}\hat{h}_{\alpha\beta}(q)k^{\alpha}_{2}k^{\beta}_{2}\int
d\lambda e^{iq\cdot(k_{2}\lambda+b)}$ (157) $\displaystyle=$
$\displaystyle\kappa\int\frac{d^{4}q}{(2\pi)^{2}}e^{iq\cdot b}\delta(2q\cdot
p_{1})\delta(2q\cdot
k_{2})q^{\mu}\frac{\hat{h}_{\alpha\beta}(q)k^{\alpha}_{2}k^{\beta}_{2}}{q^{2}+i\epsilon}\,.$

As we explained in section section 2.2, the tensor
$k_{2}^{\alpha}k_{2}^{\beta}$ corresponds to the massless three-point vertex
for a scalar emitting a graviton. Moreover, it turns out that in the on-shell
limit $q^{2}\to 0$ this is not only the scalar-graviton amplitude but also the
3-pt amplitude for any massless particle emitting a graviton, up to a phase.
More precisely, note that from gauge invariance the amplitude for a massless
scalar emitting a graviton is simply

$\langle A_{3}^{0,\mu\nu}\rangle\propto k_{2}^{\mu}k_{2}^{\nu}\,.$ (158)

Furthermore, as in general the three-point amplitude is completely fixed by
helicity weights, it is easy to see that the case $h\neq 0$ simply yields an
helicity factor

$\langle A_{3}^{h,-h,\mu\nu}\rangle\propto
k_{2}^{\mu}k_{2}^{\nu}(-i\mathcal{N}^{X})^{2h}\,.$ (159)

where $(-i\mathcal{N}^{X})^{2h}$ turns into a phase when evaluated in the real
momenta of the 4-pt amplitude. It can be absorbed by a little-group
transformation of the massless states, setting $-i\mathcal{N}^{X}=1$. We show
this explicitly in Appendix appendix B, with three-point amplitudes given in
(196) in terms of spinor-helicity variables (parametrized such that
$-i\mathcal{N}^{X}=1$). Then, as $q^{2}\to 0$ we can replace the 4-pt
amplitude for any helicity scattering by its 3-pt factorization:

$\langle
A_{4}^{h}(q)\rangle\to\kappa\frac{\hat{h}_{\alpha\beta}(q)k^{\alpha}_{2}k^{\beta}_{2}}{q^{2}+i\epsilon}$
(160)

where the fact that the RHS is independent of the helicity confirms that the
$q^{2}\to 0$ limit of $\langle A_{4}^{h}(q)\rangle\,$ is indeed universal, see
also appendix B.

Moreover, using the fact that we only require the leading term in $q^{2}$ in
the integrand of (157), we use the replacement (160) and obtain

$\displaystyle\Delta k^{\mu}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{2}}e^{iq\cdot b}\delta(2q\cdot
p_{1})\delta(2q\cdot k_{2})q^{\mu}\langle A_{4}^{h}(q)\rangle\,.$ (161)

which is nothing but the massless analog of the KMO formula derived from QFT
in Kosower:2018adc . Here we have derived it from the geodesic equation,
showing that the eikonal limit of wave scattering is directly related to
particle motion. More precisely, the latter the equation can be written as

$\Delta k^{\mu}=\frac{\partial}{\partial b_{\mu}}\chi(b),$ (162)

where we have now used the eikonal phase (150). This means that the eikonal
phase can be used to derive classical observables associated to a null
geodesic. In particular, as the momentum deflects in the direction of the
impact parameter (at leading order) the previous formula is equivalent to

$-\theta=\frac{\hat{b}^{\mu}\Delta
k_{\mu}}{|\vec{k}|}=\frac{1}{\mathcal{\omega}}\frac{\partial}{\partial
b}\chi=\frac{\partial}{\partial J}\chi,$ (163)

where we used $\hat{b}^{\mu}=\frac{\partial b^{\mu}}{\partial b}$. Here
$\theta$ is the classical (eikonal) scattering angle characterizing the
scattering of the particle off the Kerr background. In the last equation,
$J=\omega b$ is the orbital angular momentum of the particle.

Equation (162) reveals that $\chi(b)$ is naturally interpreted as the radial
action part of the action of a null geodesic. It corresponds to the Hamilton-
Jacobi equation for endpoint observables (or in QFT terms to classical the
saddle point of (141)). As such, it holds up to subleading eikonal orders as
we will check explicitly by comparing to both geodesics in Kerr and QFT
results. This contrasts with the massive eikonal case as studied recently
Kalin:2020fhe  where the radial action is not a geodesic action but instead
describes the effective relative motion of the two bodies.202020In the
massless limit of the background, i.e. boosted Schwarzschild or Kerr metric,
the correspondence to null geodesics is well known at least at leading eikonal
order, see e.g. tHooft:1987vrq ; Amati:1987uf ; Camanho:2014apa ;
Cristofoli:2020hnk .

Following Hamilton-Jacobi theory, the conjugate equation to (163) gives the
Shapiro time delay associated to the null trajectory:

$\displaystyle t_{S}$
$\displaystyle=\frac{\partial}{\partial\mathcal{\omega}}\chi.$ (164)

Next we carry out explicitly the leading eikonal approximation from our
previous QFT amplitudes with spin, and obtain the corresponding observables in
the polar and equatorial configurations.

###  5.3 Leading Eikonal in Kerr

We will illustrate the computation with a scalar wave. That this leads to the
same result independently of helicities is concluded in appendix B. Consider
then the scalar amplitude evaluated in the kinematics (5)

$\langle
A_{4}^{S\to\infty,h=0}\rangle=\frac{\kappa^{2}M^{2}\omega}{q^{2}}\left[\omega\cosh{a{\cdot}q}+i\vec{a}{\cdot}\vec{q}{\times}\vec{k}_{2}\frac{\sinh{a{\cdot}q}}{a{\cdot}q}\right],$
(165)

in the $t\to 0$ regime. A way to treat the above limit is precisely to take
$\theta\to 0$ in the parametrization (5)-(70). This leads to

$\vec{a}{\cdot}\vec{q}{\times}\vec{k}_{2}\to-\omega\theta a_{y}\,,\quad a\cdot
q\to\omega\theta a_{x}\,.$ (166)

and thus matches the universal form (78) as promised. However, in order to
evaluate the eikonal integral (150) we should first massage the expression
using the support of $t\to 0$. To do this, note that the four-point amplitude
behaves as $1/t$ times the residue, which we can evaluate at the strict $t=0$
limit. We thus need to pick a branch since $t=\braket{23}[32]$. Each choice of
vanishing brackets represents the exchange of a graviton of opposite helicity
and as such are related by parity conjugation. Moreover, on the support of the
$t-$channel residues, the argument of the trigonometric functions in (165)
becomes

$\displaystyle q{\cdot}a\big{|}_{[23]=0}$
$\displaystyle=i\frac{\epsilon(q,a,p_{1},k_{2})}{p_{1}\cdot
k_{2}}=i\vec{q}{\cdot}\hat{k}_{2}\times\vec{a}$ (167) $\displaystyle
q{\cdot}a\big{|}_{\braket{23}=0}$
$\displaystyle=-i\frac{\epsilon(q,a,p_{1},k_{2})}{p_{1}\cdot
k_{2}}=-i\vec{q}{\cdot}\hat{k}_{2}\times\vec{a},$ (168)

After inserting this into (165) we observe that both residues give the same
contribution, and thus we can choose either branch. With this in mind, and
restoring the overall factor of $i$ in the amplitude we can write

$\langle
A_{4}^{K,h=0}\rangle=\frac{i\kappa^{2}\left(M\mathcal{\omega}\right)^{2}}{\vec{q}^{2}}e^{i\vec{q}{\cdot}\hat{k}_{2}\times\vec{a}}+\mathcal{O}(t^{0})\,.$
(169)

As expected, we note that the $h=0$ amplitude is invariant under the helicity
flip (i.e. time reversal) operation introduced in section 3.3, namely by
taking $\langle A_{4}^{K,h=0}\rangle\to\langle A_{4}^{K,h=0}\rangle^{*}$ and
$a\to-a$. However, such invariance is not preserved once we compute the
eikonal phase which selects a time direction, i.e. we define our incoming
momenta.

The integral $(\ref{eq:eikonal phase})$ has been done explicitly in
Guevara:2019fsj , and we recap its derivation here. First we split the
4-dimensional momentum transfer into the longitudinal and transverse
directions as

$q=\alpha_{1}\,p_{1}+\alpha_{2}\,k_{2}+q_{\perp},$ (170)

Next we can use the delta functions to do the integrals in the longitudinal
directions, which translate into taking $\alpha_{1}=\alpha_{2}=0$. In
addition, using $p_{1}{\cdot}k_{2}=M\omega$, and replacing (169) into (150),
we arrive at the integral

$\chi(b)=8\pi
GM\omega\int\frac{d^{2}q_{\perp}}{\left(2\pi\right)^{2}}\frac{e^{i\vec{q}_{\perp}{\cdot}(\vec{b}+\hat{k}_{2}\times\vec{a})}}{\vec{q}_{\perp}^{2}},$
(171)

which can be evaluated to get

$\chi(b)=-4GM\omega\log|\omega\vec{b}+\vec{k}_{2}\times\vec{a}|+\Phi_{\textrm{IR}},$
(172)

where we have performed the 2d integration noting that $\vec{b}$ and
$\vec{k}_{2}\times\vec{a}$ are both orthogonal to $p_{1}$ and $k_{2}$. The IR
piece is independent of both the impact parameter and the spin, and can be
given as

$\Phi_{\textrm{IR}}=4GM\omega\log\omega/\mu=-4GM\omega\left[\frac{1}{2\varepsilon}+\frac{\gamma_{E}}{2}-\log\omega\right]$
(173)

where in the second form we used dimensional regularization with the dim-reg
parameter $\varepsilon$. This was expected since the massless amplitude is
known to contain an IR divergent phase.212121Indeed, eikonal methods have
proven powerful in evaluating the IR-divergent phase for massless amplitudes,
see e.g. Naculich:2011ry .

We can make contact with the results of the previous section, based on BHPT,
as follows. Using the coordinate system described around eq. (127) we can
translate the eikonal phase into the form

$\chi(b,m)-\Phi_{\textrm{IR}}=-2GM\omega\log\left[b^{2}\omega^{2}+a^{2}\omega^{2}\sin^{2}\gamma+2a\omega
m\right]=2\,\delta_{lm}^{\textrm{eik}}(\gamma)$ (174)

where we recall the orbital angular momentum is
$(b\omega)^{2}=l(l+1)\approx(l+1/2)^{2}$ and the azimuthal component (in the
direction of $\vec{a}$) is given by $m$. In deriving this result we have used