aainstitutetext: The Institute of Mathematical Sciences
IV Cross Road, C.I.T. Campus, Taramani, Chennai 600 113, Indiabbinstitutetext:
Center for High Energy Physics, Indian Institute of Science,
C.V. Raman Avenue, Bangalore 560012, Indiaccinstitutetext: Homi Bhabha
National Institute
Training School Complex, Anushakti Nagar, Mumbai 400 094, India

# Classical Observables using Exponentiated Spin factors: Electromagnetic
Scattering

Samim Akhtar b Arkajyoti Manna a,c and Akavoor Manu<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In Arkani-Hamed:2019ymq , the authors argued that the Newman-Janis symmetry on
the space of classical solutions in general relativity and electromagnetism
could be used in the space of scattering amplitudes to map an amplitude with
external scalar states to an amplitude associated to the scattering of
“infinite spin particles”. The minimal coupling of these particles to the
gravitational or Maxwell field is equivalent to the classical coupling of the
Kerr blackhole with linearized gravity or the so-called $\sqrt{\text{Kerr}}$
charged state with the electromagnetic field. The action of the Newman-Janis
symmetry on scattering amplitudes was then used to compute the linear impulse
at first post-Minkowskian (1PM) order, via the Kosower, Maybee, O’Connell
(KMOC) formalism. In this paper, we continue with the idea of using the
Newman-Janis symmetry on the space of scalar QED amplitudes to compute
classical observables such as the radiative gauge field and the angular
impulse. We show that for tree-level amplitudes, the Newman-Janis symmetry
action can be reinterpreted as a dressing of the photon propagator. This turns
out to be an efficient way to compute these classical observables. Along the
way, we highlight a subtlety that arises in proving the conservation of
angular momentum for scalar $-\sqrt{\text{Kerr}}$ scattering.

##  1 Introduction

The array of tools available at our disposal in computing observables in
gravitational and electromagnetic scattering has seen a remarkable increase in
the last few years. A central premise in this resurgence is the realization
that on-shell amplitudes in gauge theories and gravity can be deployed to
compute classical observables ranging from scattering angle to radiative flux
in Post Minkowskian scattering Bern:2019nnu ; Bern:2019crd ; Bern:2020gjj ;
Bern:2020buy ; Bern:2020uwk ; Bern:2021dqo ; Bern:2021yeh ; Bern:2021xze ;
Bern:2022kto ; Bern:2022jvn ; Barack:2023oqp ; Bern:2023ccb ; Manohar:2022dea
; KoemansCollado:2019ggb ; Cristofoli:2020uzm ; DiVecchia:2021ndb ;
DiVecchia:2021bdo ; Alessio:2022kwv ; DiVecchia:2022piu ; DiVecchia:2022owy ;
Bjerrum-Bohr:2018xdl ; Cristofoli:2019neg ; Bjerrum-Bohr:2019kec ; Bjerrum-
Bohr:2021vuf ; Bjerrum-Bohr:2021din ; Bjerrum-Bohr:2021wwt ; Menezes:2022tcs ;
Kalin:2020mvi ; Liu:2021zxr ; Kalin:2022hph . One of the most potent
formalisms which uses the quantum S-matrix to generate classical observables
was proposed in a seminal paper by Kosower, Maybee, and O’Connell (KMOC)
Kosower:2018adc ; Cristofoli:2021vyo ; Maybee:2019jus ; Aoude:2021oqj ;
Adamo:2022rmp . The KMOC formalism uses scattering amplitudes to compute a set
of “in-in” asymptotic quantities, whose classical limits are the observables
of interest.

In addition to a large class of state of the art techniques which have been
developed in the last several decades to analyze perturbative amplitudes,
there are a few non-perturbative results in the study of amplitudes (such as
soft factorization theorems Sen:2017nim ; Laddha:2017ygw ; Chakrabarti:2017ltl
; Laddha:2018rle ; Laddha:2018myi ; Sahoo:2018lxl ; Laddha:2019yaj ;
Saha:2019tub ; Ghosh:2021bam ; Sahoo:2020ryf ) which can be used to
significantly simplify the computation of the same.

Among the latter category of tools, a technique was proposed in Arkani-
Hamed:2019ymq  which “spins” the external states in scalar QED or scalar-GR
amplitudes from scalars to massive infinite spin particles that are coupled
minimally to photons or gravitons. These states are known as
$\sqrt{\textrm{Kerr}}$ or Kerr particles respectively. The “spinning”
technique discovered in Arkani-Hamed:2019ymq  was inspired by the well-known
Newman-Janis (NJ) algorithm in classical general relativity and
electromagnetism. The NJ algorithm implements a remarkable symmetry on the
space of solutions of general relativity/electromagnetism. As Newman and Janis
showed, a complex coordinate transformation can be used to derive the metric
of a Kerr black hole from the metric of the Schwarzschild solution nj .
Remarkably, one would get the Kerr solution by simply deforming the radial
coordinate $r$ in the Schwarzschild solution with an imaginary factor
involving the ring radius of the Kerr black hole. Thereafter, various kinds of
Newman-Janis shifts have been explored at great length Erbin:2016lzq \-
providing specific ways to implement such complex deformation to derive
classical solutions for different spinning objects. In their original work,
Newman and Janis also noted that the same shift on the Coulombic field
produces, what is now known, as the $\sqrt{\text{Kerr}}$ field. The features
of this solution were further expanded upon in Lynden-Bell:2002dvr .

Motivated by the ideas in Alessio:2023kgf , we propose a “spin dressed” photon
propagator and use this to compute the following observables in a scattering
process involving a scalar and a $\sqrt{\text{Kerr}}$ particle. All our
computations are at leading order in the coupling.111This is the analog of
leading order (LO) post-Minkowskian expansion in gravity.

  1. 1.

The electromagnetic radiation emitted by the scalar particle. This computation
shows the power of the NJ algorithm even in the non-conservative sector. We
show that just as for the linear impulse, the radiation emitted by scalar
particle can also be obtained via complexification of the impact parameter
$\vec{b}\,\rightarrow\,\vec{b}+i\vec{a}$.

  2. 2.

We use the NJ shift to compute the angular impulse for the scalar and
$\sqrt{\text{Kerr}}$ particles. We also highlight an important subtlety that
underlies the computation of angular impulse for the spinning particle. This
subtlety is crucially tied to the use of spin tensor ($S^{\mu\nu}$) as opposed
to the spin pseudovector ($a^{\mu}$) as the fundamental variable. These are
related via the following duality relation

$\displaystyle
a^{\mu}=\frac{1}{2m^{2}}\epsilon^{\mu\nu\rho\sigma}p_{\nu}S_{\rho\sigma}\,.$
(1)

  3. 3.

We show that to linear order in the initial spin parameter, the total angular
impulse (of the scalar-$\sqrt{\text{Kerr}}$ system) is consistent with
classical results Bern:2023ity  so long as the initial coherent state of the
spinning particle is parametrized in terms of $S_{\mu\nu}$ and $p^{\mu}$.

The plan of the paper is as follows. In section 2, we review the KMOC
formalism used to compute classical observables from scattering amplitudes. In
section 3, we give a brief review of the Newman-Janis algorithm used as a
classical solution-generating technique and its manifestation at the level of
three-point amplitudes. In section 4, we reformulate the NJ algorithm for the
case of scattering amplitudes via a specific deformation of the photon
polarisation data. We then use this version of the NJ algorithm to compute the
radiation emitted by the scalar particle in section 5. In section 6 we compute
the orbital angular impulse suffered by the scalar particle, the spin angular
momentum change for the $\sqrt{\text{Kerr}}$ particle and highlight an
important subtlety involved in computing the orbital angular momentum of the
$\sqrt{\text{Kerr}}$ particle. We conclude by discussing some open questions
in section 7. In the appendices we state our conventions, provide the
classical calculations for the observables and evaluate some integrals that
are used in the main text.

##  2 KMOC Formalism in a nut-shell

The KMOC formalism Kosower:2018adc ; Cristofoli:2021vyo  is a framework that
is used to compute classical observables from on-shell scattering amplitudes
for large impact parameter scattering222The scattering setups in which the
particles don’t deviate from the initial trajectories very much are known as
large impact parameter scattering. Naturally, the characteristic length scale
is set by the impact parameter. Recently, in Cristofoli:2021vyo , the
formalism has been extended beyond this regime for incoming waves scattering
off massive particles.. The procedure to compute classical observables is as
follows: we start with an initial coherent state, compute the change in the
expectation value of a self-adjoint quantum mechanical operator, and then take
an appropriate classical limit to obtain the classical result. The main
feature of the formalism is that the classical limit is taken before
evaluating the full amplitude because of which the computation becomes
significantly simpler. Additionally, radiation reaction effects are naturally
inbuilt within the framework. For a short sample of the results obtained with
the formalism, we refer the reader to Bautista:2021llr ; Elkhidir:2023dco ;
Bern:2021xze . In this section, we shall highlight some of the features of the
formalism relevant to us.

We start by describing the initial state,

$\ket{\Psi}=\int\prod_{i=1}^{2}d\Phi(p_{i})e^{ip_{2}\cdot
b/\hbar}\phi_{i}(p_{i})\zeta_{i}\ket{\vec{p}_{1};\vec{p}_{2}},$ (2)

where

$d\Phi(p)=\frac{d^{4}p}{(2\pi)^{4}}\hat{\delta}(p^{2}-m^{2})\Theta(p^{0}),\
\int d\Phi(p)\ |\phi(p)|^{2}=1,$ (3)

$\phi_{i}(p_{i})$s are the minimum uncertainty wave packets (in momentum
space) and $\zeta_{i}$ are the coherent spin state wave function for the
particles. The wavepacket of the second particle is translated, with respect
to the first particle’s wavepacket, by a distance of $b$ \- the impact
parameter. Since the initial particles are described by coherent states we
have

$\braket{\mathbb{P}^{\mu}_{i}}=m_{i}u^{\mu}_{i}+\mathcal{O}(\hbar),\qquad\frac{\sigma_{i}^{2}}{m_{i}^{2}}=\frac{(\braket{\mathbb{P}^{2}_{i}}-\braket{\mathbb{P}_{i}}^{2})}{m_{i}^{2}}\xrightarrow{\hbar\rightarrow
0}0,$ (4)

where $\sigma_{i}^{2}$ is the variance and $m_{i}$s are the masses of the
particles. Here the expectation value of the momentum operator is with respect
to the initial state in eq.(2).

The spin of a particle in quantum field theory is given by the expectation
value of the Pauli-Lubanski vector Maybee:2019jus ,

$\displaystyle\mathbb{W}^{\mu}=\frac{1}{m}\epsilon^{\mu\nu\rho\sigma}\mathbb{P}_{\nu}\mathbb{S}_{\rho\sigma}\,.$
(5)

Hence, it is the expectation value of the above operator which gives the
classical spin pseudovector,

$\displaystyle\braket{\mathbb{W}^{\mu}_{i}}=s_{i}^{\mu}\,+\mathcal{O}(\hbar).$
(6)

The variance for the spin is also small, as for the momentum of the particle.
For a more detailed construction of these wavefunctions, we refer the reader
to Aoude:2021oqj .
We now move on to describe the construction of the classical observables. The
basic idea is to compute the change in the expectation value of a quantum
mechanical operator as this is what is relevant from a classical perspective.
So we write

$\displaystyle\braket{\Delta\mathbb{O}^{A}}=\bra{\Psi}S^{{\dagger}}\mathbb{O}^{A}S\ket{\Psi}-\bra{\Psi}\mathbb{O}^{A}\ket{\Psi}.$
(7)

where $S=I+iT$ is the S-matrix. For the linear impulse,
$\mathbb{O}^{A}=\mathbb{P}^{\mu}$, the momentum operator,
$\mathbb{O}^{A}=\mathbb{J}^{\mu\nu}$ for angular impulse,
$\mathbb{O}^{A}=\mathbb{W}^{\mu}/m$ for the spin kick and
$\mathbb{O}^{A}=\mathbb{A}^{\mu}(x)$, the gauge field operator, from which we
read off the radiation kernel. The expression in eq.(7) can be simplified
using the on-shell completeness relation and unitarity of the S-matrix
Kosower:2018adc . For the linear impulse, we get

(8)

and for the angular impulse, we have

$\begin{split}\langle\Delta J^{\mu\nu}\rangle=&\hbar\int\prod_{i=1}^{2}\
\hat{d}^{4}q_{i}\ \hat{\delta}(2p_{i}\cdot q_{i}+q_{i}^{2})\ e^{-iq_{2}\cdot
b/\hbar}\\\
&\scalebox{0.95}{$\sum_{j=1}^{2}\bigg{(}\bigg{(}p_{j}\wedge\frac{\partial}{\partial
p_{j}}\bigg{)}^{\mu\nu}+\bigg{(}(p_{j}+q_{j})\wedge\frac{\partial}{\partial(p_{j}+q_{j})}\bigg{)}^{\mu\nu}\bigg{)}\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+q_{1},p_{2}+q_{2})\ \,+\mathcal{O}(T^{{\dagger}}T)$},\end{split}$ (9)

We do not display the higher order contributions as in this work we will only
be interested in calculating the observables to leading order in coupling.
Similarly, for the gauge field, we can read off the radiation kernel
Cristofoli:2021vyo

$\displaystyle\langle\mathcal{R}^{\mu}(k)\rangle$ $\displaystyle=\
\hbar^{3/2}\int\prod_{i=1}^{2}\hat{d}^{4}q_{i}\ \hat{\delta}(2p_{i}\cdot
q_{i}+q_{i}^{2})\ e^{-iq_{2}\cdot b/\hbar}$ (10)
$\displaystyle\qquad\qquad\hat{\delta}^{(4)}(q_{1}+q_{2}-k)\
\mathcal{A}_{5}(p_{1}+q_{1},p_{2}+q_{2}\rightarrow p_{1},p_{2,},k)\ +\
\mathcal{O}(T^{{\dagger}}T)\,.$ (11)

The spin kick, to leading order in the coupling, is given by Maybee:2019jus

$\displaystyle\langle\Delta a^{\mu}\rangle$ $\displaystyle=\int\hat{d}^{4}q\
\hat{\delta}(2p_{1}\cdot q+q^{2})\hat{\delta}(2p_{2}\cdot q-q^{2})\ e^{iq\cdot
b/\hbar}$ (12)
$\displaystyle\qquad\qquad\qquad\Big{(}a^{\mu}(p+q)\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+q,p_{2}-q)-\mathcal{A}_{4}(p_{1},p_{2}\rightarrow p_{1}+q,p_{2}-q)\
a^{\mu}(p)\Big{)}\,.$ (13)

The classical limit is taken at the level of integrand, by expressing massless
momenta in terms of their wave numbers (e.g. $q_{i}=\bar{q}_{i}\hbar$),
appropriately rescaling the dimensionful couplings and keeping leading order
terms in $\hbar$. In QED, the dimensionful coupling is obtained by333The
dimensionless coupling is the fine structure constant
$\alpha=\frac{e^{2}}{\hbar}$. $e\rightarrow e/\sqrt{\hbar}$.

For spinning external states, in the classical limit the final state spin
pseudovector can be written as

$\displaystyle a_{i}^{\mu}(p_{i}+\hbar\bar{q}_{i})$
$\displaystyle=a_{i}^{\mu}(p_{i})+\Delta a_{i}^{\mu}(p_{i})\,,$ (15)
$\displaystyle\Delta a_{i}^{\mu}(p_{i})$ $\displaystyle=\omega^{\mu}_{\
\nu}(p_{i};\bar{q}_{i})a^{\nu}(p)\,,$ (16) $\displaystyle\omega^{\mu}_{\
\nu}(p_{i};\bar{q}_{i})$
$\displaystyle=-\frac{\hbar}{m^{2}}(p_{i}\wedge\bar{q})^{\mu}_{\ \nu}\,,$ (17)

where $\omega^{\mu\nu}(p;\bar{q})$ is the infinitesimal boost parameter. With
these expressions in hand, we can write down the classical limit of the linear
impulse, the radiation kernel, and the spin kick, at leading order in the
coupling. The expression for the former is

$\displaystyle\Delta p^{\mu}=\Big{\llangle}\int\hat{d}^{4}\bar{q}\
\hat{\delta}(2p_{1}\cdot\bar{q})\hat{\delta}(2p_{2}\cdot\bar{q})\
e^{i\bar{q}\cdot b}\ \bar{q}^{\mu}\
(\hbar^{2}\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q},p_{2}-\hbar\bar{q}))\Big{\rrangle}\,.$ (18)

Here $\Big{\llangle}f(p_{1},p_{2},q\ldots)\Big{\rrangle}$ denotes the
integration over the minimum uncertainty wave packets which localizes the
momenta and spin onto their classical values. Similarly, for the radiation
kernel, we get

$\displaystyle\mathcal{R}^{\mu}(\bar{k})=\Big{\llangle}\int\prod_{i=1}^{2}\hat{d}^{4}\bar{q}_{i}\
\hat{\delta}(2p_{i}\cdot\bar{q}_{i})$ $\displaystyle\ e^{-i\bar{q}_{2}\cdot
b}\ \hat{\delta}^{(4)}(\bar{q}_{1}+\bar{q}_{2}-\bar{k})$ (19)
$\displaystyle\hskip
28.45274pt\big{(}\hbar^{2}\mathcal{A}_{5}(p_{1}+\hbar\bar{q}_{1},p_{2}+\hbar\bar{q}_{2}\rightarrow
p_{1},p_{2,},\hbar\bar{k})\big{)}\Big{\rrangle}\,,$ (20)

and for the spin kick, we get

$\displaystyle\Delta a^{\mu}$
$\displaystyle=\Big{\llangle}i\hbar^{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(2p_{1}\cdot\bar{q})\hat{\delta}(2p_{2}\cdot\bar{q})e^{i\bar{q}\cdot
b}\Big{\\{}\left[a^{\mu}(p),\mathcal{A}_{4}\right]+\frac{\hbar}{m}(a\cdot\bar{q})u^{\mu}\mathcal{A}_{4}\Big{\\}}\Big{\rrangle}\,.$
(21)

In all of the above expressions, we have taken out the $\hbar-$ scaling of the
coupling constant. For the orbital angular impulse, we shall derive the
corresponding expression in Section 6 as it is slightly more detailed.
The KMOC formalism has been generalized to describe different types of
scattering. For instance, in Cristofoli:2021vyo  the formalism has been
extended to include incoming waves in the initial state. It has also been
extended to include additional internal degrees of freedom like color charges
in delaCruz:2020bbn . Finally, it has been generalized to describe scattering
in curved backgrounds Adamo:2021rfq ; Adamo:2022rmp .

##  3 The Newman-Janis algorithm for three-point amplitudes

The Newman-Janis (NJ) algorithm has been known for a long time as a classical
solution-generating technique, primarily used in the context of General
relativity. In their original work nj , Newman and Janis showed that one can
“derive” the Kerr metric from the Schwarzschild solution (when written in the
so-called Kerr-Schild coordinates) by doing a complex transformation of the
radial coordinate, with the parameter by which it transforms interpreted as
the spin of the Kerr black hole solution. Interestingly, they observed that
there exists a similar mapping between solutions of the free Maxwell’s
equations as well. In electrodynamics, the NJ algorithm generates the so-
called $\sqrt{\text{Kerr}}$ field from the Coulombic field of charged point
particle sitting at the origin Lynden-Bell:2002dvr .
For static electromagnetic fields in vacuum, one can define the magnetostatic
potential exactly as done for electrostatic solutions, since
$\vec{\nabla}\times\vec{B}=0\ \Rightarrow B_{i}=\partial_{i}\chi$. For a
static point charge at the origin, then we have

$\Phi(\vec{x})=\phi+i\chi=\frac{Q}{r},$ (22)

where $\phi$ and $\chi$ are the electrostatic and magnetostatic potential,
respectively. Now, just as was done for the Kerr solution in GR, we do a
complex transformation on the radial coordinate. We get

$\Phi(\vec{x})=\frac{Q}{r}\rightarrow\frac{Q}{\sqrt{(\vec{x}-i\vec{a})^{2}}}=\phi+i\chi.$
(23)

Here $\vec{a}$ is to be interpreted as the ring radius of the field, the
radius at which there is a ring singularity. From the above expression, we can
compute the $\sqrt{\text{Kerr}}$ electromagnetic field Lynden-Bell:2002dvr ,

$\vec{F}=-\vec{\nabla}\Phi=\vec{E}+i\vec{B}$ (24)

In the recent past, there have been investigations in understanding the
Newman-Janis algorithm in effective field theory (EFT). As shown in
Guevara:2020xjx , the $\sqrt{\text{Kerr}}$ field in EFT can be thought of as
being generated by a conserved current. The conserved current then defines a
classical $\sqrt{\text{Kerr}}$ point particle with an infinite number of
multipole moments described solely in terms of its mass ($m$), charge ($q$)
and spin ($a$) Chung:2019yfs . From the conserved current, we can compute the
gauge field created by this configuration,

$A^{\mu}(x)=\int d^{4}x^{\prime}\ G_{r}(x,x^{\prime})\
J^{\mu}(x^{\prime})=\sum_{n\geq 0}D^{\mu}_{n}(m,q,a)\frac{1}{r^{n}}$ (25)

where $D_{n}^{\mu}(m,q,a)$ are the multipole moments. This is the
electromagnetic analog of the Kerr black hole, studied in Vines:2017hyw .
Remarkably, this recent understanding of $\sqrt{\text{Kerr}}$ field as a
particle can also be understood, within the EFT framework Guevara:2020xjx , as
the classical limit of the three-point amplitude of a massive spin - $S$
particle interacting with a photon. We shall review this now.

Consider the three-point amplitude for a generic massive spin - $S$ particle
of mass $m_{2}$ “minimally” 444Here the term “minimally” means that the three-
point amplitude is well behaved in the high energy limitArkani-Hamed:2017jhn .
coupled to a photon. In the massive spinor helicity formalism Arkani-
Hamed:2017jhn , the amplitude is given by

$\displaystyle\mathcal{A}_{3}[\textbf{2}^{S},\textbf{2}^{\prime S},q^{+}]$
$\displaystyle=\ i\sqrt{2}Q_{2}\
x\frac{\braket{\mathbf{2}\mathbf{2^{\prime}}}^{2S}}{m_{2}^{2S-1}}\,,$ (26)
$\displaystyle\mathcal{A}_{3}[\textbf{2}^{S},\textbf{2}^{\prime S},q^{-}]$
$\displaystyle=i\sqrt{2}Q_{2}x^{-1}\frac{[\mathbf{2}\mathbf{2^{\prime}}]^{2S}}{m_{2}^{2S-1}}\,.$
(27)

Here the massive spinor helicity variables $|\textbf{2}\rangle$ and
$|\textbf{2}^{\prime}\rangle$ are defined w.r.t incoming momentum $p_{2}$ and
outgoing momentum $p^{\prime}_{2}$, respectively. The $x-$factor, which is a
hallmark of a minimally coupled amplitude is defined via the photon
polarization: $x=\frac{1}{m_{2}}(\varepsilon^{+}(q)\cdot p_{2})$. We now take
the classical limit as described in the previous section. For the above
amplitude, we replace $q=\hbar\bar{q}$ and $Q_{2}\rightarrow
Q_{2}/\sqrt{\hbar}$. From the three particle kinematics,
$2p_{2}\cdot\bar{q}=-\hbar\bar{q}^{2}$, keeping the leading term in $\hbar$,
we obtain

$\displaystyle\frac{1}{m_{2}}\langle\mathbf{2}\mathbf{2}^{\prime}\rangle=\mathbb{I}+\frac{1}{2Sm_{2}}\bar{q}\cdot
s_{2}+\mathcal{O}(\hbar)\,,$ (28)

where we have suppressed all the SU(2) indices. Here $s_{2}^{\mu}$ is the
Pauli-Lubanski pseudovector associated with the spin-$S$ particle

$s_{2}^{\mu}=\frac{S\hbar}{m_{2}}\langle\mathbf{2}|\sigma^{\mu}|\mathbf{2}]\,.$
(29)

It was shown in Arkani-Hamed:2019ymq  that if we take
$S\rightarrow\infty,\hbar\rightarrow 0$ such that $S\hbar=\text{constant}$,
then the above three-point amplitude exponentiates

$\displaystyle\mathcal{A}_{3,\sqrt{\text{Kerr}}}^{\pm}$
$\displaystyle=iQ_{2}\sqrt{2}m_{2}\ x^{\pm 1}e^{\pm\bar{q}\cdot a_{2}}\,,$
(30) $\displaystyle=\mathcal{A}^{\pm}_{3,\text{scalar}}\ e^{\pm\bar{q}\cdot
a_{2}},$ (31)

where $a_{2}^{\mu}=\frac{s_{2}^{\mu}}{m_{2}}$ is the rescaled spin of the
$\sqrt{\text{Kerr}}$ particle. We note that the classical limit of the massive
spin - $S$ particle has thus “spun” the three-point amplitude of a minimally
coupled scalar (in the classical limit). This exponentiation is the
realization of the Newman - Janis algorithm for three-point amplitudes, for
scalars minimally coupled to the photon.

##  4 Spin dressing of the photon propagator

In this section, we interpret the exponentiation of the three-point amplitude,
in eq.(30), as a “spin dressing” of the photon propagator. This is motivated
by the simple observation that the three-point amplitude in eq.(30) can be
written as

$\begin{split}\mathcal{A}_{3,\sqrt{\text{Kerr}}}^{\pm}&=iQ_{2}m_{2}\
(\varepsilon^{\pm}(\bar{q})\cdot p_{2})\ e^{\pm\bar{q}\cdot a_{2}},\\\
&=iQ_{2}m_{2}(\varepsilon^{\prime\pm}(\bar{q},a_{2})\cdot p_{2})\end{split}$
(32)

where
$\varepsilon^{\prime\mu\pm}(\bar{q},a_{2})=\varepsilon^{\mu\pm}(\bar{q})e^{\pm
a_{2}\cdot\bar{q}}$. This was first observed in Alessio:2023kgf . Building on
this observation, we move on to the construction of the four-point scattering
amplitude involving a scalar particle of charge $Q_{1}$ and mass $m_{1}$ and a
$\sqrt{\text{Kerr}}$ particle, mediated by photon. The incoming momenta for
the particles are $(p_{1},p_{2})$ and the outgoing momenta are
$(p_{1}+q,p_{2}-q)$. Diagramatically, the four-point amplitude is represented
in Figure 1.

$q$$\mathbf{2}^{a_{2}}$$\mathbf{2}^{\prime
a_{2}}$$\mathbf{1}^{0}$$\mathbf{1^{\prime}}^{0}$ Figure 1: The four-point
scalar-$\sqrt{\text{Kerr}}$ amplitude with photon exchange. Here ‘$a_{2}$’
denotes the rescaled spin of the $\sqrt{\text{Kerr}}$ particle.

The amplitude is then

$\displaystyle\mathcal{A}_{4}\left[p_{1},p_{2}\rightarrow
p^{\prime}_{1},p^{\prime}_{2}\right]$
$\displaystyle=\mathcal{A}_{3,\sqrt{\text{Kerr}}}^{\mu}\left[p^{\prime}_{2},p_{2},\hbar\bar{q}\right]\frac{\mathcal{P}_{\mu\nu}}{\hbar^{2}\bar{q}^{2}}\mathcal{A}_{3,\text{scalar}}^{\nu}\left[p^{\prime}_{1},p_{1},-\hbar\bar{q}\right]\,,$
(33)

where $p^{\prime}_{1}=p_{1}+\hbar\bar{q}$, $p^{\prime}_{2}=p_{2}-\hbar\bar{q}$
and
$\mathcal{P}^{\mu\nu}\mathrel{\mathop{\ordinarycolon}}=\sum_{h=\pm}\varepsilon_{h}^{\mu}\
\varepsilon^{\nu}_{-h}$. Using the three-point amplitudes in eq.(30), we note
that eq.(33) can be written as

$\displaystyle\mathcal{A}_{4}\left[p_{1},p_{2}\rightarrow
p^{\prime}_{1},p^{\prime}_{2}\right]$
$\displaystyle=\mathcal{A}_{3,\text{scalar}}^{\mu}\left[p^{\prime}_{2},p_{2},\hbar\bar{q}\right]\frac{\widetilde{\mathcal{P}}_{\mu\nu}(\bar{q})}{\hbar^{2}\bar{q}^{2}}\mathcal{A}_{3,\text{scalar}}^{\nu}\left[p^{\prime}_{1},p_{1},\hbar\bar{q}\right]\,,$
(34)

where $\mathcal{A}_{3,\text{scalar}}^{\mu}$ is the three-point minially
coupled scalar amplitude. We now deform the internal photon projector
$\mathcal{P}^{\mu\nu}$ as follows

$\displaystyle\mathcal{P}^{\mu\nu}\rightarrow\widetilde{\mathcal{P}}^{\mu\nu}(\bar{q})$
$\displaystyle\mathrel{\mathop{\ordinarycolon}}=e^{\bar{q}\cdot
a_{2}}\varepsilon_{+}^{\mu}(\bar{q})\varepsilon_{-}^{\nu}(\bar{q})+e^{-\bar{q}\cdot
a_{2}}\varepsilon_{+}^{\nu}(\bar{q})\varepsilon_{-}^{\mu}(\bar{q})$ (35)
$\displaystyle=\cosh(a_{2}\cdot\bar{q})\eta^{\mu\nu}+\sinh(a_{2}\cdot\bar{q})\Pi^{\mu\nu}(\bar{q})\,.$
(36)

Here we have used $\varepsilon^{(\mu}_{+}\varepsilon_{-}^{\nu)}=\eta^{\mu\nu}$
and define the anti-symmetric part of the projector as
$\Pi^{\mu\nu}(\bar{q})\mathrel{\mathop{\ordinarycolon}}=\varepsilon^{[\mu}_{+}\varepsilon_{-}^{\nu]}(\bar{q})$.
Since the anti-symmetric part555Usually the anti-symmetric part does not
appear in the projector. In this case, we get it due to the helicity
dependence of the exponentiation of the massive spin - $S$ amplitude in
eq.(30). of the projector is ambiguous up to a residual gauge, we shall choose
an expression for $\Pi^{\mu\nu}(\bar{q})$ that can be used in the computation
of all the physical observables. We choose

$\displaystyle\Pi^{\mu\nu}(\bar{q})=\frac{i}{(a_{2}\cdot\bar{q})}\epsilon^{\mu\nu\rho\sigma}a_{2\rho}\bar{q}_{\sigma}\,,$
(37)

and substitute in equation eq.(34) to obtain the amplitude 666 A similar
construction of spin dressed photon propagator was obtained in Alessio:2023kgf
.

$\displaystyle\mathcal{A}_{4,\text{scalar}-\sqrt{\text{Kerr}}}$
$\displaystyle=\frac{4Q_{1}Q_{2}}{\hbar^{2}\bar{q}^{2}}\left[(p_{1}\cdot
p_{2})\cosh(\bar{q}\cdot a_{2})+i\frac{\sinh(\bar{q}\cdot
a_{2})}{(\bar{q}\cdot a_{2})}\epsilon(p_{1},p_{2},a_{2},\bar{q})\right]\,,$
(38)

with
$\epsilon(p_{1},p_{2},a_{2},\bar{q})\mathrel{\mathop{\ordinarycolon}}=\epsilon_{\mu\nu\rho\sigma}p_{1}^{\mu}p_{2}^{\nu}a_{2}^{\rho}\bar{q}^{\sigma}$.
The amplitude depends on the external momenta of the scattering particles as
well as the (classical) spin vector $a^{\mu}$. It is related to the spin
tensor $S^{\mu\nu}$ via the dual relation, eq.(1). It is rather natural to
interpret $S^{\mu\nu}$ as the independent spin tensor which can be thought of
as an “intrinsic” spin angular momentum of a classical particle. In this case

$\displaystyle a_{2}^{\mu}\,=\,a_{2}^{\mu}(S_{2},p_{2})\,.$ (39)

We will denote the projection of $S_{2}^{\mu\nu}$ orthogonal to the time-like
vector $p_{2}^{\mu}$ as $S_{2}^{\perp\mu\nu}$. Thus we will interpret the spin
pseudovector as a function of $S_{2}^{\perp\mu\nu}$ and $p_{2}^{\mu}$. We will
not explicitly indicate the dependence of $a_{2}^{\mu}$ on $S_{2}^{\mu\nu}$
except in section 6, when we derive the angular impulse.777A moment of
reflection reveals that for orbital angular impulse of $\sqrt{\text{Kerr}}$
particle, the choice of $a_{2}^{\mu}$ versus $S_{2}^{\mu\nu}$ as independent
variable in ${\cal A}_{4}$ will produce inequivalent results as
$L_{2}^{\mu\nu}=(p_{2}\,\wedge\ \partial_{p_{2}})^{\mu\nu}$.

Using the above amplitude, the linear impulse for the scalar particle is

$\begin{split}\Delta p_{1}^{\mu}&=iQ_{1}Q_{2}\int\hat{d}^{4}\bar{q}\
\hat{\delta}(\bar{q}\cdot p_{1})\ \hat{\delta}(\bar{q}\cdot
p_{2})\frac{e^{i\bar{q}\cdot b}}{\bar{q}^{2}}\ \bar{q}^{\mu}\\\ &\hskip
85.35826pt\Big{[}\cosh(a_{2}\cdot\bar{q})\ (p_{1}\cdot
p_{2})+i\frac{\sinh(a_{2}\cdot\bar{q})}{(a_{2}\cdot\bar{q})}\epsilon(p_{1},p_{2},a_{2},\bar{q})\Big{]}\,\end{split}$
(40)

Using the identities

$\displaystyle\bar{q}_{\mu}\sinh{w}=i\epsilon_{\mu\nu\rho\sigma}u_{1}^{\nu}u_{2}^{\rho}\bar{q}^{\sigma}\,,\
\ \sinh w=\sqrt{\gamma^{2}-1}\,,\quad\gamma=(u_{1}\cdot u_{2})\,,$ (41)

and rewriting the $\cosh(a_{2}\cdot\bar{q})$ and $\sinh(a_{2}\cdot\bar{q})$
terms as exponential functions, we obtain

$\Delta p_{1}^{\mu}=\frac{Q_{1}Q_{2}}{2\pi\gamma\beta}\leavevmode\nobreak\
\text{Re}\leavevmode\nobreak\ \Big{[}\frac{\gamma(b+i\Pi
a_{2})^{\mu}-i\epsilon^{\mu}(b+i\Pi a_{2},u_{1},u_{2})}{(b+i\Pi
a_{2})^{2}}\Big{]}.$ (42)

This is the expression obtained in Arkani-Hamed:2019ymq . Hence, we see that
the NJ algorithm, within the EFT for a $\sqrt{\text{Kerr}}$ particle, can also
be interpreted as a deformation on the photon data rather than on the impact
parameter as shown in Arkani-Hamed:2019ymq .

##  5 Radiation Kernel to all order in spin

In this section, we use the spin-dressed photon propagator (36) to compute the
leading order radiative gauge field emitted by a scalar particle as it
scatters in the background of a $\sqrt{\text{Kerr}}$ particle. The basic
ingredient for computing the radiative field via KMOC formalism is the
inelastic five-point amplitude.

$\mathbf{2}^{a_{2}}$$\mathbf{\bar{2}}^{a_{2}}$$\mathbf{\bar{1}}^{0}$$\mathbf{1}^{0}$$k$
Figure 2: The five-point amplitude appearing in the radiation kernel at
leading order in coupling.

We will only compute the radiation emitted from the scalar particle using the
NJ algorithm. It is straightforward to compute the five-point amplitude when
the photon is emitted from the scalar particle since, in this case, the
complexity due to the spin is completely contained within the three-point
amplitude involving the $\sqrt{\text{Kerr}}$ particle. The other sub-amplitude
needed to obtain the full amplitude is then the ordinary scalar-Compton
amplitude as indicated in Figure 3. Following the construction of the four-
point massive amplitude in section 4, we use the deformed internal photon
projector (36)

$\displaystyle\widetilde{\mathcal{P}}^{\mu\nu}(\bar{q}_{2})=\cosh(a_{2}\cdot\bar{q}_{2})\eta_{\mu\nu}+i\frac{\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon^{\mu\nu\rho\sigma}a_{2\rho}\bar{q}_{2\sigma}\,.$
(43)

to obtain the five-point amplitude as follows

$\displaystyle\mathcal{A}^{\delta}_{5}$
$\displaystyle=\frac{1}{q_{2}^{2}}\mathcal{A}^{\mu}_{3,\sqrt{\text{Kerr}}}\mathcal{P}_{\mu\nu}\mathcal{A}_{4,\text{Scalar-
Compton}}^{\nu\delta}$ (44)
$\displaystyle=\frac{1}{q_{2}^{2}}\mathcal{A}^{\mu}_{3,\text{Scalar}}\widetilde{\mathcal{P}}_{\mu\nu}\mathcal{A}_{4,\text{Scalar-
Compton}}^{\nu\delta}\,.$ (45)

Here $\mathcal{A}^{\mu}_{3,\text{Scalar}}$ and $\mathcal{A}_{4,\text{Scalar-
Compton}}^{\nu\delta}$ are the three-point massive scalar-photon and the
scalar-Compton amplitude in scalar QED. To derive the amplitude, we’ll be
using the following momentum convention

$\displaystyle(\bar{p}_{1},\bar{p}_{2})\rightarrow(p_{1},p_{2},k)\,,\quad\text{with}\quad\bar{p}_{1}=p_{1}+q_{1}\,,\quad\bar{p}_{2}=p_{2}+q_{2}\,,\quad
k=(q_{1}+q_{2})\,,$ (46)

as indicated in Figure 3. There are three scattering diagrams. We start with
diagram I in Figure 3. It is given by

$\displaystyle\mathcal{A}^{\delta}_{5,I}$
$\displaystyle=4Q_{1}^{2}Q_{2}\frac{(2p_{2}+q_{2})^{\mu}p_{1}^{\nu}(p_{1}-q_{2})^{\delta}}{q_{2}^{2}[(p_{1}-q_{2})^{2}-m_{1}^{2}]}\left[\cosh(a_{2}\cdot\bar{q}_{2})\eta_{\mu\nu}+\frac{i\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon^{\mu\nu\rho\sigma}a_{2\rho}\bar{q}_{2\sigma}\right]$
(47)
$\displaystyle=\frac{4Q_{1}^{2}Q_{2}(p_{1}-q_{2})^{\delta}}{q_{2}^{2}[(p_{1}-q_{2})^{2}-m_{1}^{2}]}\left[\cosh(a_{2}\cdot\bar{q}_{2})(2p_{1}\cdot
p_{2}+k\cdot
p_{1})+2i\frac{\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon(p_{2},p_{1},a_{2},\bar{q}_{2})\right]\,.$
(48)

Similarly, from diagram II, we obtain

$\displaystyle\mathcal{A}^{\delta}_{5,II}=\frac{4Q_{1}^{2}Q_{2}}{q_{2}^{2}(k\cdot
p_{1})}p_{1}^{\delta}\Big{[}\cosh(a_{2}\cdot\bar{q}_{2})$
$\displaystyle\left(p_{1}\cdot p_{2}+k\cdot p_{2}+{\frac{k\cdot
p_{1}}{2}+\frac{k\cdot q_{2}}{2}}\right)$ (50)
$\displaystyle+\frac{i\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon(p_{2},p_{1}+k,a_{2},\bar{q}_{2})\Big{]}\,.$
(51)

We use the usual Feynman rule for the scalar-photon four-point vertex in
scalar QED theory to obtain the contribution from diagram III in Figure 3

$\displaystyle\mathcal{A}_{5,III}^{\delta}$
$\displaystyle=-2Q_{1}^{2}Q_{2}\eta^{\rho\delta}\frac{1}{q_{2}^{2}}(2p_{2}+q_{2})^{\nu}\left[\cosh(a_{2}\cdot\bar{q}_{2})\eta_{\rho\nu}+\frac{i\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon_{\nu\rho\sigma\alpha}a^{\sigma}_{2}\bar{q}^{\alpha}_{2}\right]\,,$
(52)
$\displaystyle=-4Q_{1}^{2}Q_{2}\frac{1}{q_{2}^{2}}\left[\cosh(a_{2}\cdot\bar{q}_{2})\left(p_{2}+\frac{q_{2}}{2}\right)^{\delta}-\frac{i\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon^{\delta}(p_{2},a_{2},\bar{q}_{2})\right]\,.$
(53)

$q_{2}$$\mathbf{2}$$\mathbf{\bar{2}}$$\mathbf{1}$$\mathbf{\bar{1}}$$k$I

$q_{2}$$\mathbf{2}^{a_{2}}$$\mathbf{\bar{2}}^{a_{2}}$$\mathbf{1}^{0}$$\mathbf{\bar{1}}^{0}$$k$II

$q_{2}$$\mathbf{2}^{a_{2}}$$\mathbf{\bar{2}}^{a_{2}}$$\mathbf{\bar{1}}^{0}$$\mathbf{\bar{1}}^{0}$$k$III

Figure 3: Diagrams contributing to the tree level five-particle amplitude with
a photon emitted from the scalar particle.

Next, we scale the massless momentum $q_{2}^{\mu}$ as $\hbar\bar{q}_{2}^{\mu}$
and collect the terms of $\mathcal{O}(\hbar^{-2})$ needed to compute the
leading order radiation kernel. But unlike the four-point case, individual
diagrams in this tree-level amplitude contain superclassical terms. As
expected, they cancel after summing up all the diagrams. To see this first of
all we rewrite the massive propagator in diagram I as

$\displaystyle[(p_{1}-q_{2})^{2}-m_{1}^{2}]^{-1}=-\frac{1}{2\hbar\bar{k}\cdot
p_{1}}\left(1-\hbar\frac{\bar{k}\cdot\bar{q}_{2}}{\bar{k}\cdot
p_{1}}\right)^{-1}\,,$ (54)

where we set $k^{2}=0$ and $p_{2}\cdot q_{2}=-\frac{q_{2}^{2}}{2}$. The second
condition is due to one of the two on-shell delta functions present in the
radiation kernel. Expanding the contribution from diagram I upto
$\mathcal{O}(\hbar^{-2})$, we find

$\displaystyle\mathcal{A}^{\delta}_{5,I,\mathcal{O}(\hbar^{-2})}=\frac{4Q_{1}^{2}Q_{2}}{\hbar^{2}\bar{q}_{2}^{2}(\bar{k}\cdot
p_{1})}\left[\cosh(a_{2}\cdot\bar{q}_{2})(p_{1}\cdot
p_{2})+i\frac{\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon(p_{2},p_{1},a_{2},\bar{q}_{2})\right]$
(55)
$\displaystyle\times\left(\bar{q}_{2}^{\delta}-\frac{p_{1}^{\delta}(\bar{k}\cdot\bar{q}_{2})}{\bar{k}\cdot
p_{1}}\right)-\frac{2Q_{1}^{2}Q_{2}}{\hbar^{2}\bar{q}_{2}^{2}}\cosh(a_{2}\cdot\bar{q}_{2})p_{1}^{\delta}\,,$
(56)

and

$\displaystyle\mathcal{A}^{\delta}_{5,I,\mathcal{O}(\hbar^{-3})}=\frac{4Q_{1}^{2}Q_{2}p_{1}^{\delta}}{\hbar^{3}\bar{q}_{2}^{2}(\bar{k}\cdot
p_{1})}\left[(-p_{1}\cdot
p_{2})\cosh(a_{2}\cdot\bar{q}_{2})-i\frac{\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon(p_{2},p_{1},a_{2},\bar{q}_{2})\right]$
(57)

Similarly, from diagram II, we get both $\mathcal{O}(\hbar^{-2})$ and
$\mathcal{O}(\hbar^{-3})$ terms but the latter cancels with the contribution
from I.

$\displaystyle\mathcal{A}^{\delta}_{5,II,\mathcal{O}(\hbar^{-2})}$
$\displaystyle=\frac{4Q_{1}^{2}Q_{2}p_{1}^{\delta}}{\hbar^{2}\bar{q}_{2}^{2}(\bar{k}\cdot
p_{1})}\left[\cosh(a_{2}\cdot\bar{q}_{2})\left(\bar{k}\cdot
p_{2}+\frac{\bar{k}\cdot
p_{1}}{2}\right)+i\frac{\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon(p_{2},\bar{k},a_{2},\bar{q}_{2})\right]\,,$
(58) $\displaystyle\mathcal{A}^{\delta}_{5,II,\mathcal{O}(\hbar^{-3})}$
$\displaystyle=\frac{4Q_{1}^{2}Q_{2}p_{1}^{\delta}}{\hbar^{3}\bar{q}_{2}^{2}(\bar{k}\cdot
p_{1})}\left[(p_{1}\cdot
p_{2})\cosh(a_{2}\cdot\bar{q}_{2})+i\frac{\sinh(a_{2}\cdot\bar{q}_{2})}{a_{2}\cdot\bar{q}_{2}}\epsilon(p_{2},p_{1},a_{2},\bar{q}_{2})\right]$
(59)

The $\mathcal{O}(\hbar^{-2})$ terms from diagram III can be found trivially.
We collect all the terms of $\mathcal{O}(\hbar^{-2})$ below

$\displaystyle\mathcal{A}^{\delta}_{5,\mathcal{O}(\hbar^{-2})}=\frac{4Q_{1}^{2}Q_{2}}{\hbar^{2}\bar{q}_{2}^{2}}\frac{m_{1}m_{2}}{\bar{k}\cdot
p_{1}}\Big{[}\cosh(a_{2}\cdot\bar{q}_{2})\big{\\{}\gamma\bar{q}_{2}^{\delta}-u_{2}^{\delta}(\bar{k}\cdot
u_{1})-\frac{p_{1}^{\delta}}{\bar{k}\cdot
p_{1}}\left(\gamma(\bar{k}\cdot\bar{q}_{2})-(\bar{k}\cdot u_{2})(\bar{k}\cdot
u_{1})\right)\big{\\}}$ (60)
$\displaystyle+i\sinh(a_{2}\cdot\bar{q}_{2})\big{\\{}\epsilon^{\delta}(u_{2},u_{1},\bar{q}_{2})-\frac{p_{1}^{\delta}}{\bar{k}\cdot
p_{1}}\epsilon(\bar{k},u_{2},u_{1},\bar{q}_{2})\big{\\}}\Big{]}\,.$ (61)

Using the formula of eq. (19), we obtain the radiation kernel as

$\displaystyle\mathcal{R}^{\mu}_{1}(\bar{k},a_{2})$
$\displaystyle=Q_{1}^{2}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}[u_{1}\cdot(\bar{q}-\bar{k})]\hat{\delta}(u_{2}\cdot\bar{q})\frac{e^{-i\bar{q}\cdot
b}}{\bar{q}^{2}}\frac{1}{\bar{k}\cdot p_{1}}$ (62)
$\displaystyle\times\Big{[}\cosh(a_{2}\cdot\bar{q})\\{\gamma\bar{q}^{\mu}-u_{2}^{\mu}(u_{1}\cdot\bar{k})\\}+i\sinh(a_{2}\cdot\bar{q})\epsilon^{\mu}(u_{2},u_{1},\bar{q})$
(63) $\displaystyle-\frac{p_{1}^{\mu}}{\bar{k}\cdot
p_{1}}\big{\\{}\cosh(a_{2}\cdot\bar{q})\left(\gamma(\bar{k}\cdot\bar{q})-(\bar{k}\cdot
u_{2})(u_{1}\cdot\bar{k})\right)+i\sinh(a_{2}\cdot\bar{q})\epsilon(\bar{k},u_{2},u_{1},\bar{q})\big{\\}}\Big{]}\,.$
(64)

This expression agrees with the result in (LABEL:radker_scalar_exacta), which
is obtained using classical equations of motion.
The radiation kernel for the scalar can also be implemented as a complex shift
in the impact parameter space, just as it was done for linear impulse in
Arkani-Hamed:2019ymq . Consider the radiation kernel for scalar-scalar
scattering in electrodynamics

$\displaystyle\mathcal{R}^{\mu}_{1}(\bar{k})=Q_{1}^{2}Q_{2}\int\hat{d}^{4}\bar{q}$
$\displaystyle\hat{\delta}(u_{2}\cdot\bar{q})\hat{\delta}[u_{1}\cdot(\bar{q}-\bar{k})]\frac{e^{-i\bar{q}\cdot
b}}{\bar{q}^{2}}\frac{1}{\bar{k}\cdot u_{1}}$ (66)
$\displaystyle\times\left[\bar{k}_{\alpha}(u_{1}\wedge\partial_{u_{1}})^{\mu\alpha}(u_{1}\cdot
u_{2})+(u_{1}\cdot
u_{2})\left\\{\bar{q}^{\mu}-\frac{(\bar{k}\cdot\bar{q})u_{1}^{\mu}}{\bar{k}\cdot
u_{1}}\right\\}\right]\,.$ (67)

Inside this expression for the radiation kernel, we complexify
$\mathfrak{b}=b+ia_{2}$ and re-write the expression as follows.

$\displaystyle\mathcal{R}^{\mu}_{1}(\bar{k},a_{2})$
$\displaystyle=Q_{1}^{2}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(u_{2}\cdot\bar{q})\hat{\delta}[u_{1}\cdot(\bar{q}-\bar{k})]\frac{1}{\bar{q}^{2}(\bar{k}\cdot
u_{1})}$ (68)
$\displaystyle\times\Big{[}k_{\alpha}(u_{1}\wedge\partial_{u_{1}})^{\mu\alpha}(e^{-i\bar{q}\cdot\mathfrak{b}^{\star}}e^{w}+e^{-i\bar{q}\cdot\mathfrak{b}}e^{-w})+(e^{-i\bar{q}\cdot\mathfrak{b}^{\star}}e^{w}+e^{-i\bar{q}\cdot\mathfrak{b}}e^{-w})\left\\{\bar{q}^{\mu}-\frac{(\bar{k}\cdot\bar{q})u_{1}^{\mu}}{\bar{k}\cdot
u_{1}}\right\\}\Big{]}\,,$ (69)

with $u_{1}\cdot u_{2}=\cosh w=\gamma$. We can now factor out the overall
$e^{-i\bar{q}\cdot b}$ in the second line

$\displaystyle(e^{-i\bar{q}\cdot\mathfrak{b}^{\star}}e^{w}+e^{-i\bar{q}\cdot\mathfrak{b}}e^{-w})=e^{-i\bar{q}\cdot
b}\left[\gamma\cosh(\bar{q}\cdot a_{2})-\sqrt{\gamma^{2}-1}\sinh(\bar{q}\cdot
a_{2})\right]\,,$ (71)

where we use $\gamma=\cosh w$ and $\sqrt{\gamma^{2}-1}=\sinh w$. We can now
evaluate the derivatives with respect to initial velocity $u_{1}$,

$\displaystyle(u_{1}\wedge\partial_{u_{1}})^{\mu\nu}\left[\gamma\cosh(\bar{q}\cdot
a_{2})-\sqrt{\gamma^{2}-1}\sinh(\bar{q}\cdot a_{2})\right]$ (72)
$\displaystyle=(u_{1}\wedge
u_{2})^{\mu\nu}\left[\cosh(a_{2}\cdot\bar{q})-\frac{1}{\beta}\sinh(a_{2}\cdot\bar{q})\right]\,,$
(73)

with $\sinh w=\beta\gamma$. Using the following identity, consistent with the
two on-shell delta function constraints: $u_{2}\cdot\bar{q}=0$ and
$u_{1}\cdot\bar{q}=\bar{k}\cdot u_{1}$, we find that

$\displaystyle\bar{q}^{\mu}\sinh w=\frac{\bar{k}\cdot u_{1}}{\sinh w}(\gamma
u_{2}^{\mu}-u_{1}^{\mu})+i\epsilon^{\mu}(\bar{q},u_{1},u_{2})\,,$ (74)

and we recover the radiation kernel in scalar-$\sqrt{\text{Kerr}}$ scattering
when the photon is emitted from the scalar particle in (62).

$q_{1}$$\mathbf{2}^{a_{2}}$$\mathbf{\bar{2}}^{a_{2}}$$\mathbf{1}^{0}$$\mathbf{\bar{1}}^{0}$$k$I

$q_{1}$$\mathbf{2}^{a_{2}}$$\mathbf{\bar{2}}^{a_{2}}$$\mathbf{1}^{0}$$\mathbf{\bar{1}}^{0}$$k$II

$q_{1}$$\mathbf{2}^{a_{2}}$$\mathbf{\bar{2}}^{a_{2}}$$\mathbf{\bar{1}}^{0}$$\mathbf{\bar{1}}^{0}$$k$III

Figure 4: Diagrams contributing to the tree level five-point amplitude with a
photon emitted from the $\sqrt{\text{Kerr}}$ particle.

We note that to compute the total radiative flux (which includes the radiation
emitted by the $\sqrt{\text{Kerr}}$ particle), we require the expression for
the five-point amplitude where the incoming $\sqrt{\text{Kerr}}$ and scalar
states are scattered into $\sqrt{\text{Kerr}}$, scalar and a photon. A
diagrammatic representation of this amplitude is in Fig. 4.

One avenue to compute the inelastic amplitude ${\cal
A}_{5}(\sqrt{\text{Kerr}}+\textrm{scalar}\,\rightarrow\,\sqrt{\text{Kerr}}+\textrm{scalar}+\gamma)$
is via an EFT computation of the Compton sub-amplitude which has been pursued
extensively in the literature recently Chung:2018kqs ; Guevara:2018wpp ;
Guevara:2017csg ; Alessio:2023kgf ; Aoude:2020onz ; Haddad:2020tvs ;
Chiodaroli:2021eug ; Aoude:2022trd ; Bern:2022kto ; Aoude:2022thd ;
Chen:2022clh ; Cangemi:2022abk ; Saketh:2022wap ; Bjerrum-Bohr:2023jau ;
Bjerrum-Bohr:2023iey ; Haddad:2023ylx ; Brandhuber:2023hhy ; Aoude:2023vdk ;
Bern:2023ity ; Brandhuber:2023hhl ; Scheopner:2023rzp . Hence a possible
strategy to compute the leading order radiation kernel in the present case
would be to use the Compton sub-amplitude to evaluate all the diagrams in Fig.
4.

However, a more natural route is to start with a gauge invariant bare
Lagrangian of QED with $\sqrt{\text{Kerr}}$ charged matter and compute the
five-point amplitude directly using the resulting Feynman rules. Given that
the three-point coupling of $\sqrt{\text{Kerr}}$ with a photon is known, one
can in principle use gauge invariance to fix all the higher point couplings.
In Alessio:2023kgf , it was shown that just as in the case of scalars and
fermions, a gauge-invariant Lagrangian which describes the minimal coupling of
a $\sqrt{\text{Kerr}}$ particle with the electromagnetic field is simply the
scalar QED lagrangian in which the gauge covariant derivative is replaced by a
twisted covariant derivative,
$D_{\mu}^{(a)}=\partial_{\mu}-ie\exp{\\{\epsilon^{\nu}{}_{\mu}(a,\partial)\\}}A_{\nu}$.
It would be rather natural to simply use this Lagrangian and compute the
radiative field emitted during scalar$-\sqrt{\text{Kerr}}$ scattering. However
as the author emphasizes in Alessio:2023kgf , such a Lagrangian is not
consistent with all the Compton amplitudes, and as a result, it is unclear how
it would lead to the correct answer for classical radiation. We stress that
the computation of the complete radiation kernel emitted during
scalar$-\sqrt{\text{Kerr}}$ scattering using the KMOC formalism has the
potential to unravel the full power of the NJ algorithm. This will be pursued
elsewhere Akhtar:2024rzp .

##  6 Leading order angular impulse

In this section, we use the NJ algorithm to compute the leading order angular
impulse $\Delta J^{\mu\nu}$ for the scalar and $\sqrt{\text{Kerr}}$ particles.
Angular impulse is an intriguing observable in $D=4$ dimensions. Already for
the scattering of scalar particles, it was shown in Gralla:2021eoi  that the
net angular impulse of the particles (in a $2\rightarrow\,2$ scattering) does
not add up to zero even at leading order in the coupling. The missing
contribution is due to angular momentum stored in the late-time Coloumbic
modes. In Gralla:2021eoi  this contribution was called electromagnetic scoot.
We will denote the scoot as $\delta^{\mu\nu}_{\textrm{scalar-scoot}}$, where
the subscript indicates that the scoot has been computed for the case of
scalar-scalar scattering.

The NJ algorithm offers a powerful tool to compute angular impulse for the
scalar-$\sqrt{\text{Kerr}}$ system. We will denote the angular impulse for the
scalar particle as $\Delta L^{\mu\nu}_{1}$. It can be computed to all orders
in spin using the NJ algorithm and the final result is given in eq.(96). The
computation of total angular impulse (i.e. change in orbital angular momentum,
$\Delta L_{2}^{\mu\nu}$ plus the change in spin angular momentum, $\Delta
S_{2}^{\perp\mu\nu}$) for the $\sqrt{\text{Kerr}}$ particle can also be done
using the NJ algorithm. We will denote it as $\Delta J_{2}^{\mu\nu}$. A result
(in the integral form) for $\Delta L_{2}^{\mu\nu}$ and $\Delta
S_{2}^{\perp\mu\nu}$ appears in eq.(109) and (133), respectively. In
principle, this completes the computation of angular impulse for the
scalar-$\sqrt{\text{Kerr}}$ system to leading order in coupling.

To test our results, we compute the net angular impulse of the scattering
particles and subject it to the conservation law.

Based on Gralla:2021eoi  we deduce that to leading order in the coupling,

$\displaystyle\Delta L^{\mu\nu}_{1}+\Delta
J^{\mu\nu}_{2}+\delta^{\mu\nu}_{\textrm{scoot}}\,=\,0$ (75)

On general grounds, we expect that the entire contribution to the
electromagnetic scoot is independent of the spin of the particles as it simply
arises due to the late-time Coloumbic effects which do not depend on the spin,

$\displaystyle\delta^{\mu\nu}_{\textrm{scoot}}\,=\,\delta^{\mu\nu}_{\textrm{scalar-
scoot}}\,.$ (76)

We thus expect that

$\displaystyle\Delta L^{\mu\nu}_{1}+\Delta
J^{\mu\nu}_{2}+\delta^{\mu\nu}_{\textrm{scalar-scoot}}\,=\,0\,.$ (77)

We verify the conservation to next to leading (i.e. linear) order in
$S_{2}^{\mu\nu}$ in a perturbative expansion which is valid when
$|a_{2}|\ll|b|$. As we will argue, verifying conservation for finite spin
$|a_{2}|\sim\,|b|$ is rather subtle and will be pursued elsewhere
Akhtar:2024rzp .
We start with the computation of the orbital angular impulse for the scalar -
$\sqrt{\text{Kerr}}$ scattering, to leading order in coupling.

###  6.1 Orbital Angular Impulse

The leading order orbital angular impulse in the KMOC formalism is given by

$\displaystyle\Delta
L_{i}^{\mu\nu}=\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}_{1}\hat{d}^{4}\bar{q}_{2}\hat{\delta}(p_{1}\cdot\bar{q}_{1})\hat{\delta}(p_{2}\cdot\bar{q}_{2})e^{-i(b\cdot\bar{q}_{2})}\bigg{[}\bigg{(}\tilde{p}_{i}\wedge\frac{\partial}{\partial\tilde{p}_{i}}\bigg{)}^{\mu\nu}+\bigg{(}p_{i}\wedge\frac{\partial}{\partial
p_{i}}\bigg{)}^{\mu\nu}\bigg{]}$ (78)
$\displaystyle\hat{\delta}^{(4)}(\bar{q}_{1}+\bar{q}_{2})\
\mathcal{A}_{4}(p_{1},p_{2}\rightarrow\tilde{p}_{1},\tilde{p}_{2})\,,$ (79)

where $p_{i}$’s are initial momenta and we denote the final momenta as
$\tilde{p}_{i}=p_{i}+\hbar\bar{q}_{i}$. Here
$\mathcal{A}_{4}(p_{1},p_{2}\rightarrow\tilde{p}_{1},\tilde{p}_{2})$ is the
four-point scalar$-\sqrt{\text{Kerr}}$ scattering amplitude given in (38).
Since we express the amplitude as a function of $(p_{i},\bar{q}_{i})$, we
shall treat them as independent variables and consider the transformation
$(p_{i},\tilde{p}_{i})\rightarrow(p^{\prime}_{i},q^{\prime}_{i})$ and then set
$p^{\prime}_{i}=p_{i}$ to obtain the correct differential operator for the
angular impulse. With

$\displaystyle p^{\prime}_{i}=p_{i}\,,\qquad
q^{\prime}_{i}=\tilde{p}_{i}-p_{i}\,,$ (80)

we obtain the differential operators in new variables by treating
$p_{i}=p_{i}(p^{\prime}_{i},q^{\prime}_{i})$ and
$\tilde{p}_{i}=\tilde{p}_{i}(p^{\prime}_{i},q^{\prime}_{i})$. We find

$\displaystyle\partial_{p_{i}}^{\mu}\mathrel{\mathop{\ordinarycolon}}=\frac{\partial}{\partial
p_{i\mu}}=\partial_{p^{\prime}_{i}}^{\mu}-\partial_{q^{\prime}_{i}}^{\mu}\,,\qquad\partial_{\tilde{p}_{i}}^{\mu}\mathrel{\mathop{\ordinarycolon}}=\frac{\partial}{\partial\tilde{p}_{i\mu}}=\partial_{q^{\prime}_{i}}^{\mu}\,.$
(81)

Using these transformations, we write the orbital angular impulse where we
treat $p^{\prime}_{i}=p_{i}$ and $q^{\prime}_{i}$ as independent
variables888We have for convenience replaced $p^{\prime}_{i}=p_{i}$ and
$q^{\prime}_{i}=q_{i}$ from now on.

$\displaystyle\Delta
L_{i}^{\mu\nu}=\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}_{1}\hat{d}^{4}\bar{q}_{2}\hat{\delta}(p_{1}\cdot\bar{q}_{1})\hat{\delta}(p_{2}\cdot\bar{q}_{2})e^{-ib\cdot\bar{q}_{2}}\left[(p_{i}\wedge\partial_{p_{i}})^{\mu\nu}+(\bar{q}_{i}\wedge\partial_{\bar{q}_{i}})^{\mu\nu}\right]$
(82)
$\displaystyle\left\\{\hat{\delta}^{(4)}(\bar{q}_{1}+\bar{q}_{2})\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q}_{1},p_{2}+\hbar\bar{q}_{2})\right\\}\,.$ (83)

####  6.1.1 Orbital angular impulse of the scalar particle

For the scalar particle, we do integration by parts on the second term in the
first line in eq.(82) and integrate over $\bar{q}_{2}$ to obtain

$\displaystyle\Delta L_{1}^{\mu\nu}=\Delta L_{1,I}^{\mu\nu}+\Delta
L_{1,II}^{\mu\nu}\,,$ (84)

with

$\displaystyle\Delta L_{1,I}^{\mu\nu}$
$\displaystyle=\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot
b}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\left(p_{1}\wedge\frac{\partial}{\partial
p_{1}}\right)^{\mu\nu}\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q},p_{2}-\hbar\bar{q})$ (85) $\displaystyle\Delta
L_{1,II}^{\mu\nu}$
$\displaystyle=-\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot b}\
\hat{\delta}^{{}^{\prime}}(p_{1}\cdot\bar{q})\
\hat{\delta}(p_{2}\cdot\bar{q})(\bar{q}\wedge p_{1})^{\mu\nu}\
\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q},p_{2}-\hbar\bar{q})\,.$ (86)

where ${\cal A}_{4}$ is given in eq. (38). As

$\displaystyle\frac{\partial}{\partial
p_{j}^{\mu}}\,p_{i}^{\alpha}\,=\,\delta_{i}^{j}\,\delta_{\mu}^{\alpha},\qquad\partial_{p_{1}^{\mu}}\,a_{2}^{\alpha}=0\,,$

we will suppress the explicit dependence of $a_{2}^{\mu}$ on $p_{2}^{\mu}$ as
in the radiation kernel derivation.

It is straightforward to evaluate the expression in $\Delta L_{1,I}^{\mu\nu}$.
Since $p_{1}$ and $\bar{q}$ are independent, we get

$\displaystyle\scalebox{0.97}{$\Delta
L_{1,I}^{\mu\nu}=Q_{1}Q_{2}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot
b}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{1}{\bar{q}^{2}}\left[(p_{1}\wedge
p_{2})^{\mu\nu}\cosh(a_{2}\cdot\bar{q})+i\frac{\sinh(a_{2}\cdot\bar{q})}{(a_{2}\cdot\bar{q})}p_{1}^{[\mu}\epsilon^{\nu]}(p_{2},a_{2},\bar{q})\right]$}\,.$
(87)

Evaluation of $\Delta L_{1,II}^{\mu\nu}$ on the other hand needs a more
careful analysis since it involves derivative of the on-shell delta function.
In order to simplify this, we shall decompose the momentum $\bar{q}^{\mu}$
along $p_{1,2}$ and in the transverse direction

$\displaystyle\bar{q}^{\mu}=\alpha_{1}p_{1}^{\mu}+\alpha_{2}p_{2}^{\mu}+\bar{q}_{\perp}^{\mu}\,,\qquad
p_{i}\cdot\bar{q}_{\perp}=0\,,$ (88)

where the coefficients are given by

$\displaystyle\alpha_{1}=\frac{1}{\mathcal{D}}[(p_{1}\cdot
p_{2})x_{2}-m_{2}^{2}x_{1}]\,,\quad\alpha_{2}=\frac{1}{\mathcal{D}}[(p_{1}\cdot
p_{2})x_{1}-m_{1}^{2}x_{2}]\,,$ (89)

with $x_{1,2}\mathrel{\mathop{\ordinarycolon}}=(p_{1,2}\cdot\bar{q})$. Due to
this change of variables, the measure transforms as follows

$\displaystyle\hat{d}^{4}\bar{q}=\frac{1}{\sqrt{\mathcal{D}}}\hat{d}^{2}\bar{q}_{\perp}dx_{1}dx_{2}\,,\qquad\mathcal{D}=(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}\,.$ (90)

In terms of $x_{1,2}$ and $\bar{q}_{\perp}$ variables, we rewrite

$\displaystyle\Delta
L_{1,II}^{\mu\nu}=-\frac{Q_{1}Q_{2}}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}\bar{q}_{\perp}dx_{1}\hat{\delta}^{\prime}(x_{1})\frac{e^{i\bar{q}_{\perp}\cdot
b}}{\bar{q}^{2}}(\bar{q}\wedge p_{1})^{\mu\nu}\Big{\\{}(p_{1}\cdot
p_{2})\cosh(a_{2}\cdot\bar{q})$ (91)
$\displaystyle+i\frac{\sinh(a_{2}\cdot\bar{q})}{(a_{2}\cdot\bar{q})}\epsilon(p_{1},p_{2},a_{2},\bar{q})\Big{\\}}\,,$
(92)

where we have done the $x_{2}$ integral and used $b\cdot p_{1,2}=0$. Next, we
perform an integration by parts in $x_{1}$. At this stage, we note that from
eq. (88) we can write

$\displaystyle
q^{2}=\alpha_{1}^{2}p_{1}^{2}+\alpha_{2}^{2}p_{2}^{2}+q_{\perp}^{2}+2\alpha_{1}\alpha_{2}(p_{1}\cdot
p_{2})$ (93)

Therefore any first order derivative of $\frac{1}{\bar{q}^{2}}$ w.r.t $x_{1}$
or $x_{2}$ vanishes due to the on-shell delta function constraints:
$x_{1}=x_{2}=0$. Using
$\partial_{x_{1}}(a_{2}\cdot\bar{q})=-\frac{m_{2}^{2}}{\mathcal{D}}(a_{2}\cdot
p_{1})$, we get

(94)
$\displaystyle\qquad\scalebox{0.97}{$+\frac{m_{2}^{2}}{\mathcal{D}}(a_{2}\cdot
p_{1})(p_{1}\wedge\bar{q}_{\perp})^{\mu\nu}\left\\{(p_{1}\cdot
p_{2})\sinh(a_{2}\cdot\bar{q}_{\perp})+i\mathcal{Y}\epsilon(p_{1},p_{2},a_{2},\bar{q}_{\perp})\right\\}\Big{]}\,$},$
(95)

where
$\mathcal{Y}=\left[\frac{\cosh{(a_{2}\cdot\bar{q}_{\perp})}}{(a_{2}\cdot\bar{q}_{\perp})}-\frac{\sinh{(a_{2}\cdot\bar{q}_{\perp})}}{(a_{2}\cdot\bar{q}_{\perp})^{2}}\right]$.
Summing the two expressions $\Delta L_{1,I}^{\mu\nu}$ and $\Delta
L_{1,II}^{\mu\nu}$, we obtain the orbital angular impulse of the scalar
particle

$\displaystyle\Delta L_{1}^{\mu\nu}=Q_{1}Q_{2}$
$\displaystyle\int\hat{d}^{4}\bar{q}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{[}(p_{1}\wedge
p_{2})^{\mu\nu}\cosh(a_{2}\cdot\bar{q})\left(1-\frac{(p_{1}\cdot
p_{2})^{2}}{\mathcal{D}}\right)$ (96)
$\displaystyle+i\frac{\sinh(a_{2}\cdot\bar{q})}{(a_{2}\cdot\bar{q})}\left(p_{1}^{[\mu}\epsilon^{\nu]}(p_{2},a_{2},\bar{q})+\frac{p_{1}\cdot
p_{2}}{\mathcal{D}}\epsilon(p_{1},p_{2},a_{2},\bar{q})(p_{2}\wedge
p_{1})^{\mu\nu}\right)$ (97)
$\displaystyle+\frac{m_{2}^{2}}{\mathcal{D}}(a_{2}\cdot
p_{1})(p_{1}\wedge\bar{q})^{\mu\nu}\left\\{(p_{1}\cdot
p_{2})\sinh(a_{2}\cdot\bar{q})+i\mathcal{Y}\epsilon(p_{1},p_{2},a_{2},\bar{q})\right\\}\Big{]}\,.$
(98)

In Appendix B.2 we have verified the result in classical theory.

We use the results of the integrals from appendix C and obtain the orbital
angular impulse as follows

$\displaystyle\Delta L_{1}^{\mu\nu}$
$\displaystyle=\frac{Q_{1}Q_{2}}{2\pi\sqrt{\mathcal{D}}}\leavevmode\nobreak\
\text{Re}\leavevmode\nobreak\
\Bigg{[}\frac{1}{\gamma^{2}\beta^{2}}(p_{2}\wedge
p_{1})^{\mu\nu}\Big{\\{}\Big{(}1+\sum_{n=1}\frac{(-a_{2}\cdot
i\partial_{b})^{2n}}{(2n)!}\Big{)}\log|\mu_{1}b|\Big{\\}}$ (99)
$\displaystyle+\Big{(}p_{1}^{[\mu}\epsilon^{\nu]\sigma\alpha\beta}p_{2\alpha}a_{2\beta}+\frac{p_{1}\cdot
p_{2}}{\mathcal{D}}\epsilon^{\rho\sigma\alpha\beta}p_{1\rho}p_{2\alpha}a_{2\beta}(p_{2}\wedge
p_{1})^{\mu\nu}\Big{)}\mathcal{X}_{\sigma}$ (100)
$\displaystyle+\frac{m_{2}^{2}}{\mathcal{D}}(a_{2}\cdot
p_{1})\Big{\\{}i(p_{1}\wedge(b+i\Pi a_{2}))^{\mu\nu}\frac{(p_{1}\cdot
p_{2})}{(b+i\Pi
a_{2})^{2}}+\epsilon^{\rho\sigma\alpha\beta}p_{1\rho}p_{2\alpha}a_{2\beta}\Big{(}p_{1}\wedge\frac{\partial}{\partial
a_{2}}\Big{)}^{\mu\nu}\mathcal{X}_{\sigma}\Big{\\}}\Bigg{]}\,.$ (101)

Here we have defined

$\displaystyle\mathcal{X}_{\sigma}\mathrel{\mathop{\ordinarycolon}}=\frac{b_{\sigma}}{b^{2}+i(b\cdot
a_{2})}+i\frac{(\Pi a_{2})_{\sigma}}{(\Pi a_{2})^{2}}\Big{[}\frac{\Pi
a_{2}}{\Pi a_{2}-ib}+\log\Big{|}\frac{b}{b+i\Pi a_{2}}\Big{|}\Big{]}\,,$ (103)

where $\Pi^{\nu}{}_{\rho}$ is the projector into the plane orthogonal to both
$u_{1}$ and $u_{2}$ Maybee:2019jus ,

$\displaystyle\Pi^{\nu}{}_{\rho}=\delta^{\nu}{}_{\rho}+\frac{1}{\gamma^{2}\beta^{2}}[u_{1}^{\nu}(u_{1\rho}-\gamma
u_{2\rho})+u_{2}^{\nu}(u_{2\rho}-\gamma u_{1\rho})]\,,$ (104)

with $\Pi a_{2}=\sqrt{\Pi a_{2}\cdot\Pi a_{2}}$ and $b=\sqrt{-b^{2}}$. In
eq.(99), $\mu_{1}$ is the infrared (IR) cut-off. Note that the spin dependent
terms in first line are not IR divergent as it involves derivative over $b$
and the scalar term contributes to the electromagnetic scoot.

The angular impulse for the scalar particle to linear order in spin written in
terms of $S_{2}^{\perp\mu\nu}$ is

$\displaystyle\Delta
L_{1}^{\mu\nu}=\frac{Q_{1}Q_{2}}{2\pi\sqrt{\mathcal{D}}}\Big{[}\frac{1}{\beta^{2}\gamma^{2}}(p_{2}\wedge
p_{1})^{\mu\nu}\log|\mu_{1}b|+\frac{1}{b^{2}}\Big{(}p_{1}^{[\mu}S_{2}^{\perp\nu]\rho}b_{\rho}+(p_{2}\wedge
p_{1})^{\mu\nu}\frac{(p_{1}\cdot
p_{2})}{\mathcal{D}}S_{2}^{\perp\rho\sigma}p_{1\rho}b_{\sigma}\Big{)}\Big{]}$
(105) $\displaystyle+\mathcal{O}(S_{2}^{\perp 2})\,.$ (106)

####  6.1.2 Orbital angular impulse of the $\sqrt{\text{Kerr}}$ particle

The integral expression for the leading order orbital angular impulse of
$\sqrt{\text{Kerr}}$ particle can be written as,

(107)
$\displaystyle\scalebox{0.97}{$-\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot
b}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}^{{}^{\prime}}(p_{2}\cdot\bar{q})\
(\bar{q}\wedge p_{2})^{\mu\nu}\ \mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q},p_{2}-\hbar\bar{q})$}.$ (108)

We use the formula for linear impulse to rewrite the above integral as follows

$\displaystyle\Delta L_{2}^{\mu\nu}=-(b\wedge\Delta p_{2})^{\mu\nu}+\Delta
L_{2,I}^{\mu\nu}+\Delta L_{2,II}^{\mu\nu}\,,$ (109)

where

$\displaystyle\Delta L_{2,I}^{\mu\nu}$
$\displaystyle=\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot
b}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\left(p_{2}\wedge\frac{\partial}{\partial
p_{2}}\right)^{\mu\nu}\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q},p_{2}-\hbar\bar{q})\,,$ (110) $\displaystyle\Delta
L_{2,II}^{\mu\nu}$
$\displaystyle=-\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot b}\
\hat{\delta}(p_{1}\cdot\bar{q})\
\hat{\delta}^{{}^{\prime}}(p_{2}\cdot\bar{q})\ (\bar{q}\wedge p_{2})^{\mu\nu}\
\mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q},p_{2}-\hbar\bar{q})\,.$ (111) $\displaystyle\Delta
p^{\mu}_{2}$
$\displaystyle=\frac{\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\
e^{i\bar{q}\cdot b}\ (-i\bar{q}^{\mu})\ \mathcal{A}_{4}(p_{1},p_{2}\rightarrow
p_{1}+\hbar\bar{q},p_{2}-\hbar\bar{q})$ (112)

The evaluation of $\Delta L_{2,I}^{\mu\nu}$ is rather subtle as for any
function $f(a_{2},\bar{q})$ we obtain terms involving
$\frac{\partial}{\partial p_{2}^{\mu}}\,f|_{a_{2}(S_{2}^{\perp},\,p_{2})}$.

$\displaystyle\frac{\partial}{\partial
p_{2}^{\mu}}\,f(a_{2},\bar{q})\,=\,-\frac{1}{2m_{2}^{2}}\frac{\partial
f(a_{2},\bar{q})}{\partial
a_{2}^{\alpha}}\epsilon_{\mu}{}^{\alpha\rho\sigma}S_{2\rho\sigma}^{\perp}\,,$
(113)

where we have used

$\displaystyle\frac{\partial a_{2}^{\alpha}}{\partial
p_{2}^{\mu}}=\frac{1}{2m_{2}^{2}}\epsilon^{\alpha\beta\rho\sigma}S_{2\rho\sigma}^{\perp}\delta_{\mu\beta}\,,$

using the dual relation (1). The derivation of $\Delta L_{2,I}^{\mu\nu}$ (and
hence $\Delta L_{2}^{\mu\nu}$) (for $|a_{2}|\,\sim\,|b|$) will be pursued
elsewhere Akhtar:2024rzp . In this paper, we simply evaluate the orbital
angular impulse to linear order in $S_{2}^{\perp\mu\nu}$.

Using the dual relation (1), we obtain

$\displaystyle\Delta L_{2}^{\mu\nu}$
$\displaystyle=Q_{1}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
p_{1})\hat{\delta}(\bar{q}\cdot p_{2})e^{i\bar{q}\cdot
b}\frac{1}{\bar{q}^{2}}\Big{[}\frac{1}{\beta^{2}\gamma^{2}}(p_{1}\wedge
p_{2})^{\mu\nu}-(b\wedge\bar{q})^{\mu\nu}\
(S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\sigma})$ (114) $\displaystyle\hskip
170.71652pt+i(p_{1}\wedge p_{2})^{\mu\nu}\frac{(p_{1}\cdot
p_{2})}{\mathcal{D}}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\sigma}\Big{]}$
(115)
$\displaystyle=\frac{Q_{1}Q_{2}}{2\pi\sqrt{\mathcal{D}}}\Big{[}\frac{1}{\beta^{2}\gamma^{2}}(p_{1}\wedge
p_{2})^{\mu\nu}\log|\mu_{2}b|-\frac{1}{b^{2}}\Big{(}b^{[\mu}S_{2}^{\perp\nu]\rho}p_{1\rho}-(p_{1}\wedge
p_{2})^{\mu\nu}\frac{(p_{1}\cdot
p_{2})}{\mathcal{D}}S_{2}^{\perp\rho\sigma}p_{1\rho}b_{\sigma}\Big{)}\Big{]}\,,$
(116)

where $\mu_{2}$ is the IR cutoff. This matches with the result obtained in
classical theory given in Appendix B.3.

###  6.2 Spin Angular Impulse

The computation of the spin angular impulse $\Delta S_{2}^{\perp\mu\nu}$ is
rather straightforward via the NJ algorithm. Using the inverse of the dual
relation in eq.(1),

$S^{\perp\mu\nu}=\epsilon^{\mu\nu\rho\sigma}p_{\rho}a_{\sigma}$ (117)

we obtain Luna:2023uwd

$\displaystyle\Delta S_{2}^{\perp\mu\nu}=\epsilon^{\mu\nu\rho\sigma}\Delta
p_{2\rho}a_{2\sigma}+\epsilon^{\mu\nu\rho\sigma}p_{2\rho}\Delta
a_{2\sigma}\,,$ (118)

where $\Delta a_{2}^{\mu}$ is known as the spin kick. We note that although
$S_{2}^{\perp\mu\nu}$ is the fundamental spin degree of freedom, the NJ
algorithm lets us directly compute the spin kick which can then be used to
deduce $\Delta S_{2}^{\perp\mu\nu}$. Since the linear impulse doesn’t receive
any radiative contribution at leading order in the coupling, the expression
for the linear impulse $\Delta p_{2}^{\mu}$ is exactly opposite to $\Delta
p_{1}^{\mu}$, derived in section 4 and it is given by

$\displaystyle\Delta
p_{2}^{\mu}=-iQ_{1}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{[}\gamma\cosh{(a_{2}\cdot\bar{q})}\bar{q}^{\mu}+i\sinh(a_{2}\cdot\bar{q})\epsilon^{\mu}(\bar{q},u_{1},u_{2})\Big{]}\,,$
(119)

where we rewrite the $\sinh(a_{2}\cdot\bar{q})$ term using the identity

$\displaystyle(a_{2}\cdot\bar{q})\epsilon_{\nu}(u_{1},u_{2},\bar{q})=\bar{q}_{\nu}\epsilon(u_{1},u_{2},a_{2},\bar{q})\,.$
(120)

We study the leading order spin kick using the following formula
Maybee:2019jus

$\displaystyle\Delta a_{2}^{\mu}$
$\displaystyle=\Big{\llangle}\frac{i\hbar^{2}}{4}\int\hat{d}^{4}\bar{q}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})e^{i\bar{q}\cdot
b}\Big{\\{}\left[a_{2}^{\mu}(p_{2}),\mathcal{A}_{4}\right]+\frac{\hbar}{m_{2}}(a_{2}\cdot\bar{q})u_{2}^{\mu}\mathcal{A}_{4}\Big{\\}}\Big{\rrangle}\,.$
(121)

The commutator in the first term is defined in the SU(2) little group space.
The SU(2) indices are left implicit under the double angle bracket notation,
explained in section 2. Note that, the formula (121) appears to be non-uniform
in the order of $\hbar$, however, the commutator term also includes an
additional factor of $\hbar$ and it is given by

$\displaystyle[a_{2,IK}^{\mu},a_{2,KJ}^{\nu}]=\frac{i\hbar}{m_{2}}\epsilon^{\mu\nu\rho\sigma}u_{2\rho}a_{2\sigma,IJ}\,,$
(122)

where we display the SU(2) indices $(I,J,K)$. From hereafter, we shall drop
the SU(2) indices and the double angle bracket notation altogether. Using
(122), we get

$\displaystyle[a_{2}^{\mu},\cosh(a_{2}\cdot\bar{q})]=\frac{i\hbar}{m_{2}}\sinh{(a_{2}\cdot\bar{q})}\epsilon^{\mu}(\bar{q},u_{2},a_{2})\,.$
(123)

Next we consider the following commutator

$\displaystyle\Big{[}a_{2}^{\mu},\frac{\sinh(a_{2}\cdot\bar{q})}{(a_{2}\cdot\bar{q})}\epsilon(u_{1},u_{2},a_{2},\bar{q})\Big{]}$
$\displaystyle=\frac{i\hbar}{m_{2}}\mathcal{Y}\epsilon^{\mu}(\bar{q},u_{2},a_{2})\epsilon(u_{1},u_{2},a_{2},\bar{q})$
(124)
$\displaystyle+\frac{i\hbar}{m_{2}}\frac{\sinh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})}\epsilon^{\mu\nu}(u_{2},a_{2})\epsilon_{\nu}(u_{1},u_{2},\bar{q})\,.$
(125)

where we defined
$\mathcal{Y}\mathrel{\mathop{\ordinarycolon}}=\Big{(}\frac{\cosh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})}-\frac{\sinh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})^{2}}\Big{)}$.
Using eq. 150 it can be shown that,

$\displaystyle(a_{2}\cdot\bar{q})\epsilon_{\nu}(u_{1},u_{2},\bar{q})=\bar{q}_{\nu}\epsilon(u_{1},u_{2},a_{2},\bar{q})\,,$
(126)

to get

$\displaystyle\Big{[}a_{2}^{\mu},\frac{\sinh(a_{2}\cdot\bar{q})}{(a_{2}\cdot\bar{q})}\epsilon(u_{1},u_{2},a_{2},\bar{q})\Big{]}$
$\displaystyle=\frac{i\hbar}{m_{2}}\frac{\cosh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})}\epsilon^{\mu}(\bar{q},u_{2},a_{2})\epsilon(u_{1},u_{2},a_{2},\bar{q})\,.$
(127)

Next, we use the identity

$\displaystyle\epsilon^{\mu}(\bar{q},u_{2},a_{2})\epsilon(u_{1},u_{2},a_{2},\bar{q})=(a_{2}\cdot\bar{q})\left[(a_{2}\cdot\bar{q})(u_{1}^{\mu}-\gamma
u_{2}^{\mu})-\bar{q}^{\mu}(a_{2}\cdot u_{1})\right]\,,$ (128)

where we have set $(a_{2}\cdot u_{2})=0=(u_{1,2}\cdot\bar{q})$ and
$(u_{1}\cdot u_{2})=\gamma$, to finally get

$\displaystyle\Big{[}a_{2}^{\mu},\frac{\sinh(a_{2}\cdot\bar{q})}{(a_{2}\cdot\bar{q})}\epsilon(u_{1},u_{2},a_{2},\bar{q})\Big{]}=\frac{i\hbar}{m_{2}}\cosh{(a_{2}\cdot\bar{q})}[-\gamma
u_{2}^{\mu}(a_{2}\cdot\bar{q})+u_{1}^{\mu}(a_{2}\cdot\bar{q})-\bar{q}^{\mu}(a_{2}\cdot
u_{1})]\,.$ (129)

Substituting various commutator expressions, we obtain the leading order spin
kick as

$\displaystyle\Delta
a_{2}^{\mu}=\frac{iQ_{1}Q_{2}}{m_{2}}\int\hat{d}^{4}\bar{q}\frac{1}{\bar{q}^{2}}\hat{\delta}(u_{1}\cdot\bar{q})\hat{\delta}(u_{2}\cdot\bar{q})e^{i\bar{q}\cdot
b}\Big{\\{}\cosh{(a_{2}\cdot\bar{q})}[u_{1}^{\mu}(a_{2}\cdot\bar{q})-\bar{q}^{\mu}(a_{2}\cdot
u_{1})]$ (130)
$\displaystyle-i\sinh{(a_{2}\cdot\bar{q})}\epsilon^{\mu}(u_{1},a_{2},\bar{q})\Big{\\}}\,,$
(131)

where we make use of the identity

$\displaystyle
u_{2}^{\mu}\epsilon(u_{1},u_{2},a_{2},\bar{q})=\gamma\epsilon^{\mu}(u_{2},a_{2},\bar{q})-\epsilon^{\mu}(u_{1},a_{2},\bar{q})\,.$
(132)

This matches with the spin kick obtained in Guevara:2020xjx  using classical
equations of motion. Plugging these expressions into (118), we obtain the
following expression for spin angular impulse

$\displaystyle\Delta S_{2}^{\perp\mu\nu}$
$\displaystyle=-Q_{1}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{[}\cosh{(a_{2}\cdot\bar{q})}\Big{\\{}iu_{1}^{[\mu}\epsilon^{\nu]}(\bar{q},u_{2},a_{2})$
(133)
$\displaystyle-i\bar{q}^{[\mu}\epsilon^{\nu]}(u_{1},u_{2},a_{2})\Big{\\}}+\frac{1}{m_{2}}\sinh(a_{2}\cdot\bar{q})\Big{\\{}u_{2}^{[\mu}u_{1}^{\nu]}(a_{2}\cdot\bar{q})-u_{2}^{[\mu}\bar{q}^{\nu]}(a_{2}\cdot
u_{1})+\gamma a_{2}^{[\mu}\bar{q}^{\nu]}\Big{\\}}\Big{]}\,.$ (134)

Using the integral results, we get

$\displaystyle\Delta S^{\perp\mu\nu}_{2}$
$\displaystyle=\frac{Q_{1}Q_{2}}{2\pi\gamma\beta}\text{Re}\leavevmode\nobreak\
\Big{[}\frac{1}{(b+i\Pi
a_{2})^{2}}\Big{\\{}b^{[\mu}\epsilon^{\nu]}(u_{1},u_{2},a_{2})-u_{1}^{[\mu}\epsilon^{\nu]}(u_{2},a_{2},b)+u_{2}^{[\mu}u_{1}^{\nu]}a_{2}^{2}$
(136)
$\displaystyle+\frac{1}{\gamma^{2}\beta^{2}}u_{2}^{[\mu}a_{2}^{\nu]}(a_{2}\cdot
u_{1})+\frac{(a_{2}\cdot
u_{1})}{\gamma\beta^{2}}a_{2}^{[\mu}u_{1}^{\nu]}+i\Big{(}a_{2}^{[\mu}\epsilon^{\nu]}(u_{1},u_{2},a_{2})-\frac{(a_{2}\cdot
u_{1})}{\gamma\beta^{2}}u_{2}^{[\mu}\epsilon^{\nu]}(u_{1},u_{2},a_{2})$ (137)
$\displaystyle-u_{2}^{[\mu}u_{1}^{\nu]}(a_{2}\cdot
b)+u_{2}^{[\mu}b^{\nu]}(a_{2}\cdot u_{1})+\gamma
a_{2}^{[\mu}b^{\nu]}\Big{)}\Big{\\}}\Big{]}\,.$ (138)

It can be verified that the RHS of eqn.(136) is $\Delta S_{2}^{\perp\mu\nu}$
as it satisfies the SSC constraint to leading order in the coupling.

$\displaystyle\Delta S_{2}^{\perp\mu\nu}p_{2\nu}+S_{2}^{\perp\mu\nu}\Delta
p_{2\nu}=0\,.$ (139)

At linear order in $S_{2}^{\perp\mu\nu}$, the spin angular impulse can be
evaluated.

$\displaystyle\Delta S_{2}^{\perp\mu\nu}$
$\displaystyle=-iQ_{1}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})e^{i\bar{q}\cdot
b}\frac{1}{\bar{q}^{2}}\Big{[}p_{1}^{[\mu}S_{2}^{\perp\nu]\sigma}\bar{q}_{\sigma}-\bar{q}^{[\mu}S_{2}^{\perp\nu]\sigma}p_{1\sigma}\Big{]}+\mathcal{O}(S_{2}^{\perp
2})$ (140) $\Delta
S_{2}^{\perp\mu\nu}=\frac{Q_{1}Q_{2}}{2\pi\sqrt{\mathcal{D}}\
b^{2}}\big{(}b^{[\mu}S^{\perp\nu]\alpha}p_{1\alpha}-p_{1}^{[\mu}S^{\perp\nu]\alpha}b_{\alpha}\big{)}.$
(141)

This expression is in agreement with the result of Bern:2023ity , which was
derived using the classical equations of motion.

We now have all the expressions to compute the total angular impulse, $\Delta
J^{\mu\nu}$ in eq.(75) for the the scalar-$\sqrt{\text{Kerr}}$ scattering to
linear order in spin. This is given by the sum of eqs.(105),(114) and (140).
We obtain the following result

$\displaystyle\Delta J^{\mu\nu}$
$\displaystyle=\frac{Q_{1}Q_{2}}{2\pi\sqrt{\mathcal{D}}}\frac{1}{\beta^{2}\gamma^{2}}(p_{1}\wedge
p_{2})^{\mu\nu}\log\Big{|}\frac{\mu_{2}}{\mu_{1}}\Big{|}+\delta^{\mu\nu}_{\textrm{scalar-
scoot}}=0\,,$ (142)

where

$\displaystyle\delta^{\mu\nu}_{\textrm{scalar-
scoot}}=-\frac{Q_{1}Q_{2}}{2\pi\sqrt{\mathcal{D}}}\frac{1}{\beta^{2}\gamma^{2}}(p_{1}\wedge
p_{2})^{\mu\nu}\log\Big{|}\frac{\tau_{1}}{\tau_{2}}\Big{|}\,.$ (143)

The IR cutoffs are related to the proper times of the two particles via
$\frac{\mu_{2}}{\mu_{1}}=\frac{\tau_{1}}{\tau_{2}}$ DiVecchia:2022piu ;
Bhardwaj:2022hip . Hence, the total angular momentum for scalar -
$\sqrt{\text{Kerr}}$ scattering is conserved, to linear order in spin. The
study of conservation of angular momentum to all orders in spin is under
investigation Akhtar:2024rzp .

##  7 Discussion

Building upon the synthesis of the Newman-Janis (NJ) algorithm with the KMOC
formalism in Arkani-Hamed:2019ymq , in this work, we have used the NJ
algorithm to compute classical observables beyond the linear impulse for
electromagnetic scattering involving $\sqrt{\text{Kerr}}$ particles. As is
well known, the real power of the NJ algorithm lies in all orders in spin
results for classical observables. We hope that by combining the on-shell
methods along with the NJ algorithm one can build “loop integrands” associated
with the scattering of $\sqrt{\text{Kerr}}$ particles such that our analysis
can be extended beyond leading order in the coupling. For some early attempts
in this direction, see Menezes:2022tcs .

Our main idea is to simply re-interpret the three-point coupling involving
$\sqrt{\text{Kerr}}$ as a “spin dressing” of the photon polarisation data
while computing higher point amplitudes. Our work shows that the resulting
“spin-dressed” photon propagator is particularly useful for constructing the
five-point amplitude where the photon is emitted from the external scalar
state. We used this five-point amplitude to derive the radiation emitted by
the scalar to all orders in the spin and found perfect agreement with its
calculation using equations of motion.

We then used the four-point amplitude computed using the spin-dressed photon
propagator to compute the angular impulse of the scalar as well as
$\sqrt{\text{Kerr}}$ particles. Here we encountered a subtlety. We used the
spin tensor $S_{2}^{\mu\nu}$ instead of the spin-vector $a_{2}^{\mu}$ as the
fundamental spin degree of freedom. With this choice, we found that the result
for the angular impulse of the $\sqrt{\text{Kerr}}$ particle is consistent
with the angular momentum conservation. As explained in Section 6 the reason
for this choice becomes evident if one recalls that the calculation of the
angular impulse via the KMOC formalism involves the expectation value of
differential operators acting on amplitudes. Choosing the spin to be
parametrized by either $a_{2}$ or $S_{2}$ leads to different results as a
consequence. To leading order in $S_{2}$ (valid as long as $|a_{2}|\ll|b|$) we
have checked that our results are consistent with Bern:2023ity .

Using the NJ algorithm, we also calculated the leading-order orbital angular
impulse of a scalar particle to all order in the spin of the
$\sqrt{\text{Kerr}}$ particle. In addition we gave a closed form expression
for the total angular impulse of the $\sqrt{\text{Kerr}}$ particle to leading
order in spin. An all-order-in spin evaluation of the total angular momentum
of the $\sqrt{\text{Kerr}}$ particle is beyond the scope of this paper and
will be pursued in Akhtar:2024rzp .
Our broader goal is to compute classical gravitational observables for the
Kerr black hole. In this context, the double copy will be an important tool
Bern:2019prr . For conservative observables (at leading order in coupling),
the double copy of the three-point amplitudes is the only ingredient that is
needed. It has been observed in Arkani-Hamed:2017jhn  that the three-point
amplitude for a massive spin - $S$ minimally coupled to gravity can be
obtained by simply squaring the ‘$x$’ factors in eq.(26). Using this in
Arkani-Hamed:2019ymq , the 1PM linear impulse for the scalar-Kerr system was
obtained. It will be interesting to use this double copy to study the angular
impulse for the same system and check the conservation of angular momentum.
However, to study radiation from gravitational scattering involving Kerr black
holes, we need the double copy of the non-abelian counterpart of the
$\sqrt{\text{Kerr}}$ solution. Since we do not have a consistent bare
Lagrangian for the latter we leave the question of computing gravitational
radiation for Kerr black holes from amplitudes, to future work. In the
meantime, it might be instructive to study gravitational radiation from Kerr
black holes directly using the equations of motion derived in
Scheopner:2023rzp .

###### Acknowledgements.

We are grateful to Alok Laddha for suggesting the problem and numerous
insightful discussions. We thank him for carefully going through the draft and
providing valuable suggestions and comments on it. We thank Sujay Ashok for
constant encouragement and valuable comments on the draft. We thank all the
participants of ‘Amplitudes @ Chennai’ workshop at the Chennai Mathematical
Institute for interesting discussions. SA would like to thank the organizers
of the ‘Indian Strings Meeting 2023’ at the Indian Institute of Technology
Bombay for an opportunity to present a part of this work as a poster, and the
participants for their feedback. AM thanks Aninda Sinha for support and
acknowledges financial support from DST through the SERB core grant
CRG/2021/000873.

##  Appendix A Conventions

Throughout the paper, we will use the metric signature as $(+,-,-,-)$, unless
otherwise stated. So, the on-shell condition is $p^{2}=m^{2}$. Since the
impact parameter is spacelike we have $-b^{2}>0$. We will use
$\epsilon_{0123}=+1$. The rescaled delta functions appearing in section 2 are
defined as

$\hat{\delta}(p\cdot q)\mathrel{\mathop{\ordinarycolon}}=2\pi\delta(p\cdot
q),\ \ \ \ \
\hat{\delta}^{(4)}(p+q)\mathrel{\mathop{\ordinarycolon}}=(2\pi)^{4}\delta^{(4)}(p+q).$
(144)

where $p^{\mu}$ and $q^{\mu}$ are generic four vectors. We also absorb the
$2\pi$ factor in the measure $d^{4}q$ and define the rescaled the rescaled
measure as

$\hat{d}^{4}q\mathrel{\mathop{\ordinarycolon}}=\frac{d^{4}q}{(2\pi)^{4}}.$
(145)

The anti-symmetric bracket in all the expressions are defined as

$\displaystyle(A\wedge
B)^{\mu\nu}=A^{[\mu}B^{\nu]}=A^{\mu}B^{\nu}-B^{\mu}A^{\nu}\,.$ (146)

We use the following compact notations in the main text for convenience

$\displaystyle\epsilon^{\mu\nu}(A,B)$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}A_{\alpha}B_{\beta}$ (147)
$\displaystyle\epsilon^{\mu}(A,B,C)$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}A_{\nu}B_{\alpha}C_{\beta}$ (148)
$\displaystyle\epsilon(A,B,C,D)$
$\displaystyle=\epsilon^{\mu\nu\alpha\beta}A_{\mu}B_{\nu}C_{\alpha}D_{\beta}\,,$
(149)

where $(A^{\mu},B^{\mu},C^{\mu},D^{\mu})$ are generic 4-vectors. The following
identity is used throughout this paper.

$\displaystyle A^{[\mu}\epsilon^{\nu]}(B,C,D)$ $\displaystyle=-(A\cdot
B)\epsilon^{\mu\nu}(C,D)-(A\cdot
C)\epsilon^{\mu\nu}(D,B)-(A.D)\epsilon^{\mu\nu}(B,C)\,.$ (150)

##  Appendix B Classical calculations

In this appendix, we present the classical calculations of angular impulse and
the radiation kernel for scalar in scalar-$\sqrt{\text{Kerr}}$ scattering to
leading order in coupling.

###  B.1 Equations of motion

We present all the equations of motion that will be used to derive the
physical observables discussed in the main text. The equation of motion for
the scalar particle in scalar-$\sqrt{\text{Kerr}}$ scattering Guevara:2020xjx
is

$\displaystyle\frac{dp_{1}^{\mu}}{d\tau}=Q_{1}\text{Re}F_{2,+}^{\mu\nu}(x+ia_{2})u_{1\nu}\,,$
(151)

where $F_{+}^{\mu\nu}(x+ia_{2})$ is the self-dual part of the electromagnetic
field strength of the $\sqrt{\text{Kerr}}$ particle. The self dual and anti-
self dual field strengths are defined w.r.t to Minkowski metric as follows

$\displaystyle
F_{\pm}^{\mu\nu}(x+ia_{2})=F^{\mu\nu}\pm\frac{i}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}\,.$
(152)

Note that, the self dual and anti-self dual fields are related to each other
via complex conjugation 999We assume that (for all $c_{n}$ be real)
$\displaystyle F_{\mu\nu}(x\pm ia)=\sum_{n}c_{n}(x\pm
ia)^{n}\Rightarrow[F_{\mu\nu}(x\pm ia)]^{\dagger}=F_{\mu\nu}(x\mp ia)\,.$
(153)

$\displaystyle
F^{-}_{\mu\nu}(x-ia)=\left[F^{+}_{\mu\nu}(x+ia)\right]^{\dagger}\,.$ (154)

Therefore we can rewrite the real part of the self dual field strength as

$\displaystyle
2\text{Re}F_{2,+}^{\mu\nu}(x_{1}+ia_{2})=F_{+}^{\mu\nu}(x_{1}+ia_{2})+F_{-}^{\mu\nu}(x_{1}-ia_{2})$
(155)

We then use the definition (152) to express the equation of motion as follows

$\displaystyle\frac{dp_{1}^{\mu}}{d\tau}=Q_{1}\left[\cos(a_{2}\cdot\partial)F^{\mu\nu}(x)-\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\sin(a_{2}\cdot\partial)F_{\rho\sigma}(x)\right]u_{1\nu}\,.$
(156)

Note that, the field strength appearing here is due to a charged scalar
particle! We now use the following expression for the field strength in
momentum space

$\displaystyle F_{2}^{\mu\nu}(\bar{q})$
$\displaystyle=(iQ_{2})e^{-i\bar{q}\cdot
b}\hat{\delta}(u_{2}\cdot\bar{q})\frac{1}{\bar{q}^{2}}(\bar{q}^{\mu}u_{2}^{\nu}-\bar{q}^{\nu}u_{2}^{\mu})\,,$
(157)

to substitute in (156) to obtain an expression for $\dot{p}_{1}^{\mu}$ to all
order in spin $a_{2}^{\mu}$

$\displaystyle\dot{p}_{1}^{\mu}(\tau)=iQ_{1}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(u_{2}\cdot\bar{q})e^{-i\bar{q}\cdot
x(\tau)}\frac{e^{-i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{[}\cosh(a_{2}\cdot\bar{q})\\{\gamma\bar{q}^{\mu}-u_{2}^{\mu}(u_{1}\cdot\bar{q})\\}$
(158)
$\displaystyle+i\sinh(a_{2}\cdot\bar{q})\epsilon^{\mu\nu\rho\sigma}u_{1\nu}\bar{q}_{\rho}u_{2\sigma}\Big{]}\,.$
(159)

###  B.2 Orbital angular impulse for the scalar particle

Classically, the orbital angular impulse is defined as

$\displaystyle\frac{dL^{\mu\nu}}{d\tau}=(x\wedge\dot{p})^{\mu\nu}\,,$ (160)

where $p^{\mu}=m\dot{x}^{\mu}$. Now using the parametrization of the classical
trajectory of particle 1 $x_{1}^{\mu}(\tau)=u^{\mu}_{1}\tau\,,$ the orbital
angular momentum impulse to leading order in coupling is

$\displaystyle\frac{dL_{1}^{\mu\nu}}{d\tau}=\tau(u_{1}\wedge\dot{p}_{1})^{\mu\nu}\,.$
(161)

We use the following identity from appendix 150:

$\displaystyle(a_{2}\cdot\bar{q})\epsilon^{\mu}(p_{1},p_{2},\bar{q})=\bar{q}^{\mu}\epsilon(p_{1},p_{2},a_{2},\bar{q})-(p_{1}\cdot\bar{q})\epsilon^{\mu}(a_{2},\bar{q},p_{2})\,,$
(162)

to rewrite (159) as follows

$\displaystyle\dot{p}_{1}^{\mu}(\tau)=\frac{iQ_{1}Q_{2}}{m_{1}}\int\hat{d}^{4}\bar{q}\hat{\delta}(p_{2}\cdot\bar{q})$
$\displaystyle\frac{e^{+i\bar{q}\cdot b}}{\bar{q}^{2}}e^{-i(\bar{q}\cdot
p_{1})\frac{\tau}{m_{1}}}\Big{[}\cosh(a_{2}\cdot\bar{q})\\{(p_{1}\cdot
p_{2})\bar{q}^{\mu}-(p_{1}\cdot\bar{q})p_{2}^{\mu}\\}$ (163)
$\displaystyle+i\frac{\sinh(a_{2}\cdot\bar{q})}{a_{2}\cdot\bar{q}}\\{(\bar{q}\cdot
p_{1})\epsilon^{\mu}(a_{2},p_{2},\bar{q})+\bar{q}^{\mu}\epsilon(a_{2},\bar{q},p_{1},p_{2})\\}\Big{]}\,,$
(164)

to derive the LO expression for the orbital angular impulse for the scalar
particle

$\displaystyle\Delta L_{1}^{\mu\nu}$
$\displaystyle=\frac{iQ_{1}Q_{2}}{m_{1}}\int
d\tau\hat{d}^{4}\bar{q}\hat{\delta}(p_{2}\cdot\bar{q})\frac{e^{+i\bar{q}\cdot
b}}{\bar{q}^{2}}\left(i\partial_{p_{1}\cdot\bar{q}}e^{-i(\bar{q}\cdot
p_{1})\frac{\tau}{m_{1}}}\right)\Big{[}\cosh(a_{2}\cdot\bar{q})\\{(p_{1}\cdot
p_{2})(p_{1}\wedge\bar{q})^{\mu\nu}$ (165)
$\displaystyle-(p_{1}\cdot\bar{q})(p_{1}\wedge
p_{2})^{\mu\nu}\\}+i\frac{\sinh(a_{2}\cdot\bar{q})}{a_{2}\cdot\bar{q}}\\{(\bar{q}\cdot
p_{1})p_{1}^{[\mu}\epsilon^{\nu]}(a_{2},p_{2},\bar{q})+(p_{1}\wedge\bar{q})^{\mu\nu}\epsilon(a_{2},\bar{q},p_{1},p_{2})\\}\Big{]}\,.$
(166)

Here we need the derivative w.r.t $u_{1}\cdot\bar{q}$ due to the factor of
$\tau$ in (161). Now in the classical computation, we shall replace

$\displaystyle\bar{q}^{\mu}=\alpha_{1}p_{1}^{\mu}+\alpha_{2}p_{2}^{\mu}+\bar{q}_{\perp}^{\mu}\,,$
(167)

where the coefficients are given in equation (88). Again, we can do the
$x_{2}=p_{2}\cdot\bar{q}$ integral in the above integral and write

$\displaystyle\Delta L_{1}^{\mu\nu}$
$\displaystyle=-\frac{Q_{1}Q_{2}}{m_{1}\sqrt{\mathcal{D}}}\int d\tau
d^{2}\bar{q}_{\perp}dx_{1}\frac{e^{+i\bar{q}_{\perp}\cdot
b}}{\bar{q}^{2}}\left(\partial_{x_{1}}e^{-i\frac{x_{1}\tau}{m_{1}}}\right)\Big{[}\cosh(a_{2}\cdot\bar{q})\\{(p_{1}\cdot
p_{2})(p_{1}\wedge\bar{q})^{\mu\nu}$ (168)
$\displaystyle-(p_{1}\cdot\bar{q})(p_{1}\wedge
p_{2})^{\mu\nu}\\}+i\frac{\sinh(a_{2}\cdot\bar{q})}{a_{2}\cdot\bar{q}}\\{x_{1}p_{1}^{[\mu}\epsilon^{\nu]}(a_{2},p_{2},\bar{q})+(p_{1}\wedge\bar{q})^{\mu\nu}\epsilon(a_{2},\bar{q},p_{1},p_{2})\\}\Big{]}\,,$
(169)

where $x_{1}=p_{1}\cdot\bar{q}$. Integrating by parts in $x_{1}$ variable and
then completing the $\tau$ integral, we obtain

$\displaystyle\Delta L_{1}^{\mu\nu}=\frac{Q_{1}Q_{2}}{\sqrt{\mathcal{D}}}$
$\displaystyle\int
d^{2}\bar{q}_{\perp}dx_{1}\hat{\delta}(x_{1})\frac{e^{i\bar{q}_{\perp}\cdot
b}}{\bar{q}_{\perp}^{2}}\Big{[}\frac{1}{\beta^{2}\gamma^{2}}\cosh(a_{2}\cdot\bar{q})(p_{1}\wedge
p_{2})^{\mu\nu}$ (171)
$\displaystyle-i\frac{\sinh(a_{2}\cdot\bar{q})}{a_{2}\cdot\bar{q}}\left\\{p_{1}^{[\mu}S_{2}^{\nu]\sigma}\bar{q}_{\sigma}+\frac{p_{1}\cdot
p_{2}}{\mathcal{D}}(p_{2}\wedge
p_{1})^{\mu\nu}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\perp\sigma}\right\\}$
(172) $\displaystyle-\frac{m_{2}^{2}(a_{2}\cdot
p_{1})}{\mathcal{D}}\left[\sinh(a_{2}\cdot\bar{q})(p_{1}\cdot
p_{2})+i\mathcal{Y}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\perp\sigma}\right](p_{1}\wedge\bar{q}_{\perp})^{\mu\nu}\Big{]}$
(173)

To linear order in spin, we find

$\displaystyle\Delta
L_{1,\mathcal{O}(S_{2})}^{\mu\nu}=-\frac{iQ_{1}Q_{2}}{\sqrt{\mathcal{D}}}\int
d^{2}\bar{q}_{\perp}\frac{e^{+i\bar{q}_{\perp}\cdot
b}}{\bar{q}_{\perp}^{2}}\left\\{p_{1}^{[\mu}S_{2}^{\nu]\sigma}\bar{q}_{\sigma}+\frac{p_{1}\cdot
p_{2}}{\mathcal{D}}(p_{2}\wedge
p_{1})^{\mu\nu}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\perp\sigma}\right\\}$
(174)

###  B.3 Angular impulse for $\sqrt{\text{Kerr}}$ particle

The orbital angular impulse for the $\sqrt{\text{Kerr}}$ particle is given by

$\displaystyle\Delta L_{2}^{\mu\nu}=(b\wedge\Delta p_{2})^{\mu\nu}+\int
d\tau\tau(u_{2}\wedge\dot{p}_{2})^{\mu\nu}=(b\wedge\Delta
p_{2})^{\mu\nu}+I^{\mu\nu}\,.$ (175)

Using the equation of motion for the $\sqrt{\text{Kerr}}$ particle to linear
order in spin

$\displaystyle\dot{p}_{2}^{\rho}$
$\displaystyle=Q_{1}F_{1}^{\rho\sigma}p_{2\sigma}-\frac{Q_{1}}{2}S_{2}^{\perp\mu
c}\partial^{\rho}F_{1,\mu c}\,,$ (176)

we obtain the following integral expression for $\dot{p}_{2\rho}$

$\displaystyle\dot{p}_{2\rho}=-\frac{iQ_{1}Q_{2}}{m_{2}}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot
b}e^{i(p_{2}\cdot\bar{q})\frac{\tau}{m_{2}}}\frac{\hat{\delta}(p_{1}\cdot\bar{q})}{\bar{q}^{2}}\Big{[}(p_{1}\cdot
p_{2})\bar{q}_{\rho}-(p_{2}\cdot\bar{q})p_{1\rho}+i\bar{q}_{\rho}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\sigma}\Big{]}\,.$
(177)

We use this expression in (175) to obtain

$\displaystyle I^{\mu\nu}=-\frac{Q_{1}Q_{2}}{m_{2}}\int\hat{d}^{4}\bar{q}d\tau
e^{i\bar{q}\cdot
b}\frac{\partial}{\partial(p_{2}\cdot\bar{q})}\Big{(}e^{i(p_{2}\cdot\bar{q})\frac{\tau}{m_{2}}}\Big{)}\frac{\hat{\delta}(p_{1}\cdot\bar{q})}{\bar{q}^{2}}\Big{[}(p_{1}\cdot
p_{2})(p_{2}\wedge\bar{q})^{\mu\nu}-(p_{2}\cdot\bar{q})(p_{2}\wedge
p_{1})^{\mu\nu}$ (178)
$\displaystyle+i(p_{2}\wedge\bar{q})^{\mu\nu}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\sigma}\Big{]}\,.$
(179)

Again, in order to evaluate this integral we use the decomposition in (167)
and following similar steps as we did in evaluating $\Delta L_{1}^{\mu\nu}$.
Finally, we expand $\Delta p_{2}^{\mu}$ to $\mathcal{O}(S_{2})$ and get the
orbital angular impulse as

$\displaystyle\Delta
L_{2}^{\mu\nu}=\frac{Q_{1}Q_{2}}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}\bar{q}_{\perp}e^{i\bar{q}_{\perp}\cdot
b}\frac{1}{\bar{q}_{\perp}^{2}}\Big{[}\frac{1}{\beta^{2}\gamma^{2}}+i\frac{(p_{1}\cdot
p_{2})}{\mathcal{D}}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\perp\sigma}\Big{]}(p_{2}\wedge
p_{1})^{\mu\nu}$ (180)
$\displaystyle-i\frac{Q_{1}Q_{2}}{\sqrt{\mathcal{D}}}\int\hat{d}^{2}\bar{q}_{\perp}e^{i\bar{q}_{\perp}\cdot
b}\frac{1}{\bar{q}_{\perp}^{2}}(b\wedge\bar{q}_{\perp})^{\mu\nu}\Big{[}(p_{1}\cdot
p_{2})+iS_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\perp\sigma}\Big{]}\,.$ (181)

###  B.4 Radiation from the scalar particle

The classical current from particle 1 in momentum space is given by

$\displaystyle J_{1}^{\mu}(x)=Q_{1}\int d\tau_{1}e^{i\bar{k}\cdot
x_{1}(\tau_{1})}\frac{i}{\bar{k}\cdot
p_{1}}\left[\dot{p}_{1}^{\mu}-\frac{\bar{k}\cdot\dot{p}_{1}}{\bar{k}\cdot
p_{1}}p_{1}^{\mu}\right]\,.$ (182)

Using the expression in (159), we obtain the current to all order in spin

$\displaystyle J_{1}^{\mu}(\bar{k},a_{2})$
$\displaystyle=Q_{1}^{2}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(u_{1}\cdot\bar{q}-u_{1}\cdot\bar{k})\hat{\delta}(u_{2}\cdot\bar{q})\frac{e^{-i\bar{q}\cdot
b}}{\bar{q}^{2}}\frac{1}{\bar{k}\cdot p_{1}}$ (183)
$\displaystyle\times\Big{[}\cosh(a_{2}\cdot\bar{q})\\{\gamma\bar{q}^{\mu}-u_{2}^{\mu}(u_{1}\cdot\bar{q})\\}+i\sinh(a_{2}\cdot\bar{q})\epsilon^{\mu\nu\rho\sigma}u_{1\nu}\bar{q}_{\rho}u_{2\sigma}$
(184) $\displaystyle-\frac{p_{1}^{\mu}}{\bar{k}\cdot
p_{1}}\big{\\{}\cosh(a_{2}\cdot\bar{q})\left(\gamma(\bar{k}\cdot\bar{q})-(\bar{k}\cdot
u_{2})(u_{1}\cdot\bar{q})\right)+i\sinh(a_{2}\cdot\bar{q})\epsilon(\bar{k},u_{1},\bar{q},u_{2})\big{\\}}\Big{]}\,.$
(185)

We now take a soft limit of the current and show that, on comparing with the
classical sub-leading soft factor, it reproduces the angular impulse for the
scalar particle. We use the following identity from Appendix 150:

$\displaystyle(a_{2}\cdot\bar{q})\epsilon^{\mu}(u_{1},u_{2},\bar{q})=\bar{q}^{\mu}\epsilon(u_{1},u_{2},a_{2},\bar{q})-(u_{1}\cdot\bar{k})\epsilon^{\mu}(a_{2},\bar{q},u_{2})\,.$
(187)

The classical current is then rewritten as

$\displaystyle J^{\mu}_{1}(\bar{k},a_{2})$
$\displaystyle=Q_{1}^{2}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}[u_{1}\cdot(\bar{q}-\bar{k})]\hat{\delta}(u_{2}\cdot\bar{q})\frac{e^{-i\bar{q}\cdot
b}}{\bar{q}^{2}}\frac{1}{\bar{k}\cdot p_{1}}$ (188)
$\displaystyle\times\Big{[}\cosh(a_{2}\cdot\bar{q})\Big{\\{}\gamma\bar{q}^{\mu}-u_{2}^{\mu}(u_{1}\cdot\bar{k})-\frac{p_{1}^{\mu}}{p_{1}\cdot\bar{k}}\Big{(}\gamma(\bar{k}\cdot\bar{q})-(\bar{k}\cdot
u_{2})(\bar{k}\cdot u_{1})\Big{)}\Big{\\}}$ (189)
$\displaystyle-i\frac{\sinh(a_{2}\cdot\bar{q})}{a_{2}\cdot\bar{q}}\Big{\\{}\bar{q}^{\mu}\epsilon(u_{1},u_{2},a_{2},\bar{q})-(u_{1}\cdot\bar{k})\epsilon^{\mu}(a_{2},\bar{q},u_{2})$
(190) $\displaystyle-\frac{p_{1}^{\mu}}{\bar{k}\cdot
p_{1}}\Big{(}(\bar{q}\cdot\bar{k})\epsilon(u_{1},u_{2},a_{2},\bar{q})-(u_{1}\cdot\bar{k})\epsilon(\bar{k},a_{2},\bar{q},u_{2})\Big{)}\Big{\\}}\Big{]}\,.$
(191)

Next, we do a soft expansion in $\bar{k}$ where $k^{\mu}=\omega(1,\hat{n})$.
We expand the delta function as follows

$\displaystyle\hat{\delta}(p_{1}\cdot\bar{q}-p_{1}\cdot\bar{k})=\hat{\delta}(p_{1}\cdot\bar{q})-(p_{1}\cdot\bar{k})\hat{\delta}^{\prime}(p_{1}\cdot\bar{q})+\mathcal{O}(\bar{k}^{2})\,,$
(193)

and write the sub-leading soft terms of the classical current as follows

$\displaystyle J^{\mu}_{1}(\bar{k},a_{2})|_{\mathcal{O}(\omega^{0})}$
$\displaystyle=Q_{1}^{2}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{e^{-i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{[}\cosh(a_{2}\cdot\bar{q})\Big{\\{}-p_{2}^{\mu}+p_{1}^{\mu}\frac{(\bar{k}\cdot
p_{2})}{(\bar{k}\cdot p_{1})}\Big{\\}}$ (194)
$\displaystyle\qquad\qquad-i\frac{\sinh(a_{2}\cdot\bar{q})}{a_{2}\cdot
q}\Big{\\{}-\epsilon^{\mu}(a_{2},\bar{q},p_{2})+\frac{p_{1}^{\mu}}{(\bar{k}\cdot
p_{1})}\epsilon(\bar{k},a_{2},\bar{q},p_{2})\Big{\\}}\Big{]}$ (195)
$\displaystyle-
Q_{1}^{2}Q_{2}\int\hat{d}^{4}\bar{q}\hat{\delta}^{\prime}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{e^{-i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{[}\cosh(a_{2}\cdot\bar{q})(p_{1}\cdot
p_{2})\Big{\\{}\bar{q}^{\mu}-\frac{p_{1}^{\mu}}{p_{1}\cdot\bar{k}}(\bar{k}\cdot\bar{q})\Big{\\}}$
(196)
$\displaystyle\qquad\qquad-i\frac{\sinh(a_{2}\cdot\bar{q})}{a_{2}\cdot\bar{q}}\Big{\\{}\bar{q}^{\mu}\epsilon(p_{1},p_{2},a_{2},\bar{q})-\frac{p_{1}^{\mu}}{\bar{k}\cdot
p_{1}}(\bar{q}\cdot\bar{k})\epsilon(p_{1},p_{2},a_{2},\bar{q})\Big{\\}}\Big{]}\,.$
(197)

Comparing with the classical sub-leading soft factor Manu:2020zxl ,

$\displaystyle S_{1}^{(1)\mu}=Q_{1}\Big{[}\frac{\Delta
J_{1}^{\mu\nu}\bar{k}_{\nu}}{(p_{1}\cdot\bar{k})}-\frac{(\Delta
p_{1}\cdot\bar{k})}{(p_{1}\cdot\bar{k})^{2}}J_{-1}^{\mu\nu}\bar{k}_{\nu}\Big{]}\,,$
(199)

where $\Delta p_{1}^{\mu}$ is the LO linear impulse of the scalar particle,
$\Delta J_{1}^{\mu\nu}$ is the LO total angular impulse of the scalar particle
and $J_{-1}^{\mu\nu}$ is the initial angular momentum tensor, we obtain

$\displaystyle\Delta J_{1}^{\mu\nu}$
$\displaystyle=Q_{1}Q_{2}\int\hat{d}^{4}\bar{q}e^{-i\bar{q}\cdot
b}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{1}{\bar{q}^{2}}\Big{[}(p_{1}\wedge
p_{2})^{\mu\nu}\cosh(a_{2}\cdot\bar{q})-i\frac{\sinh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})}p_{1}^{[\mu}S_{2}^{\perp\nu]\sigma}\bar{q}_{\sigma}\Big{]}$
(200) $\displaystyle-Q_{1}Q_{2}\int\hat{d}^{4}\bar{q}e^{-i\bar{q}\cdot
b}\hat{\delta}^{\prime}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{1}{\bar{q}^{2}}(\bar{q}\wedge
p_{1})^{\mu\nu}\Big{[}\cosh{(a_{2}\cdot\bar{q})}(p_{1}\cdot p_{2})$ (201)
$\displaystyle\qquad\qquad\qquad-i\frac{\sinh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\sigma}\Big{]}\,.$
(202)

Replacing $\bar{q}\rightarrow-\bar{q}$, we get

$\displaystyle\Delta J_{1}^{\mu\nu}$
$\displaystyle=Q_{1}Q_{2}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot
b}\hat{\delta}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{1}{\bar{q}^{2}}\Big{[}(p_{1}\wedge
p_{2})^{\mu\nu}\cosh(a_{2}\cdot\bar{q})+i\frac{\sinh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})}p_{1}^{[\mu}S_{2}^{\perp\nu]\sigma}\bar{q}_{\sigma}\Big{]}$
(203) $\displaystyle-Q_{1}Q_{2}\int\hat{d}^{4}\bar{q}e^{i\bar{q}\cdot
b}\hat{\delta}^{\prime}(p_{1}\cdot\bar{q})\hat{\delta}(p_{2}\cdot\bar{q})\frac{1}{\bar{q}^{2}}(\bar{q}\wedge
p_{1})^{\mu\nu}\Big{[}\cosh{(a_{2}\cdot\bar{q})}(p_{1}\cdot p_{2})$ (204)
$\displaystyle\qquad\qquad\qquad+i\frac{\sinh{(a_{2}\cdot\bar{q})}}{(a_{2}\cdot\bar{q})}S_{2}^{\perp\rho\sigma}p_{1\rho}\bar{q}_{\sigma}\Big{]}\,,$
(205)

which matches with the computation of angular impulse of the scalar particle
of eq. (85). Here $J_{-1}^{\mu\nu}=(b_{1}\wedge p_{1})^{\mu\nu}=0$ in our
setup as $b_{1}=0,b_{2}=b$.

##  Appendix C Evaluation of Integrals

In this appendix, we perform the integrals required to calculate the angular
impulse for the scalar$-\sqrt{\text{Kerr}}$ scattering. To obtain an
expression for $\Delta L_{1}^{\mu\nu}$ from eq.(96), we need to evaluate a
series of integrals which are discussed below.

  * •

We start with the following integral

$\displaystyle I_{1}$
$\displaystyle=\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\cosh(a_{2}\cdot q)$ (206)
$\displaystyle=\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{(}1+\sum_{n=1}\frac{(a_{2}\cdot q)^{2n}}{(2n)!}\Big{)}$
(207)
$\displaystyle=\frac{1}{2\pi\gamma\beta}\Big{[}1+\sum_{n=1}\frac{(-a_{2}\cdot
i\partial_{b})^{2n}}{(2n)!}\Big{]}\log|\mu b|\,,$ (208)

where $\mu$ is the infrared (IR) cutoff. Note that the spin dependent terms
are not IR divergent as it involves derivative over $b$.

  * •

Next, we consider

$\displaystyle I_{2,\sigma}$
$\displaystyle=i\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\frac{\sinh{(a_{2}\cdot q)}}{(a_{2}\cdot q)}q_{\sigma}$ (209)
$\displaystyle=\frac{i}{2}\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\int_{0}^{1}d\lambda\Big{(}e^{\lambda(a_{2}\cdot
q)}+e^{-\lambda(a_{2}\cdot q)}\Big{)}q_{\sigma}$ (210)
$\displaystyle=\leavevmode\nobreak\ \text{Re}\leavevmode\nobreak\
\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\int_{0}^{1}d\lambda
e^{i\bar{q}\cdot(b+i\lambda a_{2})}\frac{i}{\bar{q}^{2}}q_{\sigma}$ (211)
$\displaystyle=\frac{1}{2\pi\gamma\beta}\leavevmode\nobreak\
\text{Re}\int_{0}^{1}d\lambda\frac{(b+i\lambda\Pi
a_{2})_{\sigma}}{b^{2}+2i\lambda(b\cdot\Pi a_{2})-\lambda^{2}(\Pi
a_{2})^{2}}\,.$ (212)

Now we do the $\lambda$ integral as follows

$\displaystyle\int_{0}^{1}d\lambda\frac{(b+i\lambda\Pi
a_{2})_{\sigma}}{b^{2}+2i\lambda(b\cdot\Pi a_{2})-\lambda^{2}(\Pi a_{2})^{2}}$
(213)
$\displaystyle=b_{\sigma}\int_{0}^{1}\frac{d\lambda}{b^{2}+2i\lambda(b\cdot\Pi
a_{2})-\lambda^{2}(\Pi a_{2})^{2}}+i(\Pi
a_{2})_{\sigma}\int_{0}^{1}\frac{\lambda d\lambda}{b^{2}+2i\lambda(b\cdot\Pi
a_{2})-\lambda^{2}(\Pi a_{2})^{2}}$ (214)
$\displaystyle=\frac{b_{\sigma}}{(b^{2}+i(b\cdot\Pi a_{2}))}+i\frac{(\Pi
a_{2})_{\sigma}}{(\Pi a_{2})^{2}}\Big{(}\frac{\Pi a_{2}}{\Pi
a_{2}-ib}+\log\Big{|}\frac{b}{b+i\Pi a_{2}}\Big{|}\Big{)}\,.$ (215)

Therefore, we get

$\displaystyle I_{2,\sigma}$
$\displaystyle=\frac{1}{2\pi\gamma\beta}\leavevmode\nobreak\
\text{Re}\Big{[}\frac{b_{\sigma}}{(b^{2}+i(b\cdot\Pi a_{2}))}+i\frac{(\Pi
a_{2})_{\sigma}}{(\Pi a_{2})^{2}}\Big{(}\frac{\Pi a_{2}}{\Pi
a_{2}-ib}+\log\Big{|}\frac{b}{b+i\Pi a_{2}}\Big{|}\Big{)}\Big{]}\,.$ (216)

  * •

We consider the integral

$\displaystyle I_{3}^{\alpha}$ $\displaystyle=\int\hat{d}^{4}qe^{iq\cdot
b}\hat{\delta}(u_{1}\cdot q)\hat{\delta}(u_{2}\cdot
q)\frac{1}{q^{2}}\sinh(a_{2}\cdot q)q^{\alpha}$ (217)
$\displaystyle=\frac{1}{2}\int\hat{d}^{4}qe^{iq\cdot b}\hat{\delta}(u_{1}\cdot
q)\hat{\delta}(u_{2}\cdot q)\frac{1}{q^{2}}q^{\alpha}\Big{(}e^{(a_{2}\cdot
q)}-e^{-(a_{2}\cdot q)}\Big{)}$ (218)
$\displaystyle=\frac{1}{2\pi\gamma\beta}\leavevmode\nobreak\
\text{Re}\leavevmode\nobreak\ \Big{[}\frac{i(b+i\Pi a_{2})^{\alpha}}{(b+i\Pi
a_{2})^{2}}\Big{]}\,.$ (219)

  * •

Lastly, we evaluate the integral

$\displaystyle I_{4}^{\alpha\beta}$ $\displaystyle=i\int\hat{d}^{4}qe^{iq\cdot
b}\hat{\delta}(u_{1}\cdot q)\hat{\delta}(u_{2}\cdot
q)\frac{1}{q^{2}}\mathcal{Y}q^{\alpha}q^{\beta}$ (220)
$\displaystyle=i\int\hat{d}^{4}qe^{iq\cdot b}\hat{\delta}(u_{1}\cdot
q)\hat{\delta}(u_{2}\cdot q)\frac{1}{q^{2}}\Big{(}\frac{\cosh{(a_{2}\cdot
q)}}{(a_{2}\cdot q)}-\frac{\sinh{(a_{2}\cdot q)}}{(a_{2}\cdot
q)^{2}}\Big{)}q^{\alpha}q^{\beta}$ (221)
$\displaystyle=i\frac{\partial}{\partial
a_{2\alpha}}\int\hat{d}^{4}qe^{iq\cdot b}\hat{\delta}(u_{1}\cdot
q)\hat{\delta}(u_{2}\cdot q)\frac{1}{q^{2}}\frac{\sinh{(a_{2}\cdot
q)}}{(a_{2}\cdot q)}q^{\beta}\,.$ (222)

Using the result for $I_{2,\sigma}$ in (216), we find

$\displaystyle I_{4}^{\alpha\beta}$
$\displaystyle=\frac{1}{2\pi\gamma\beta}\,\frac{\partial}{\partial
a_{2\alpha}}\text{Re}\Big{[}\frac{b^{\beta}}{(b^{2}+i(b\cdot\Pi
a_{2}))}+i\frac{(\Pi a_{2})^{\beta}}{(\Pi a_{2})^{2}}\Big{(}\frac{\Pi
a_{2}}{\Pi a_{2}-ib}+\log\Big{|}\frac{b}{b+i\Pi
a_{2}}\Big{|}\Big{)}\Big{]}\,.$ (223)

We use the results of these integrals to derive the expression for $\Delta
L_{1}^{\mu\nu}$ in (99). For the spin angular impulse presented in eq.(133),
we need the integral in (217) and evaluate the following integral

$\displaystyle I_{5}^{\mu\nu}$
$\displaystyle=i\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot u_{2})\frac{e^{i\bar{q}\cdot
b}}{\bar{q}^{2}}\Big{[}q^{[\mu}S_{2}^{\nu]\rho}u_{1\rho}-u_{1}^{[\mu}S_{2}^{\nu]\sigma}q_{\sigma}\Big{]}\cosh(a_{2}\cdot
q)$ (224) $\displaystyle=\leavevmode\nobreak\ \text{Re}\leavevmode\nobreak\
\int\hat{d}^{4}\bar{q}\hat{\delta}(\bar{q}\cdot
u_{1})\hat{\delta}(\bar{q}\cdot
u_{2})e^{i\bar{q}\cdot(b+ia_{2})}\frac{i}{\bar{q}^{2}}\Big{[}q^{[\mu}S_{2}^{\nu]\rho}u_{1\rho}-u_{1}^{[\mu}S_{2}^{\nu]\sigma}q_{\sigma}\Big{]}$
(225) $\displaystyle=\frac{1}{2\pi\gamma\beta}\leavevmode\nobreak\
\text{Re}\leavevmode\nobreak\ \Big{[}\frac{(b+i\Pi
a_{2})^{[\mu}S_{2}^{\nu]\rho}u_{1\rho}-u_{1}^{[\mu}S_{2}^{\nu]\sigma}(b+i\Pi
a_{2})_{\sigma}}{(b+i\Pi a_{2})^{2}}\Big{]}\,.$ (226)

We use this and (217) to obtain the expression for $\Delta S_{2}^{\mu\nu}$ in
(136).

Note that in all of the above integrals, $\Pi^{\nu}{}_{\rho}$ is the projector
into the plane orthogonal to both $u_{1}$ and $u_{2}$,

$\displaystyle\Pi^{\nu}{}_{\rho}=\delta^{\nu}{}_{\rho}+\frac{1}{\gamma^{2}\beta^{2}}[u_{1}^{\nu}(u_{1\rho}-\gamma
u_{2\rho})+u_{2}^{\nu}(u_{2\rho}-\gamma u_{1\rho})]\,,$ (227)

with $\Pi a_{2}=\sqrt{\Pi a_{2}\cdot\Pi a_{2}}$ and $b=\sqrt{-b^{2}}$.

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