Classical spinning soft factors from gauge theory amplitudes

Manu Aa,c, Debodirna Ghoshb,c

aInstitute of Physics, Sachivalaya Marg

Bhubaneswar, Odisha - 751005, India

bThe Institute of Mathematical Sciences

IV Cross Road, C.I.T. Campus

Taramani, Chennai - 600113, India

and

cHomi Bhabha National Institute

Training School Complex, Anushakti Nagar,

Mumbai - 400085, India

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

In this short note, we analyse low energy electromagnetic radiation for
spinning particles using the KMOC formalism[10] and the quantum soft theorems.
In particular, we study low energy electromagnetic radiation emitted by the
so-called $\sqrt{\text{Kerr}}$ object. The $\sqrt{\text{Kerr}}$ is a solution
of the free Maxwell’s equations with infinite multipole moments expressed
solely in terms of the charge(Q), mass(m) and spin(S) of the classical object
. We consider the scattering of two $\sqrt{\text{Kerr}}$ particles and using
the KMOC formalism generalised to spinning particles, we perturbatively prove
the classical subleading soft photon theorem to leading order in the low
deflection parameter and to $\mathcal{O}(S_{1}S_{2})$ in spin.

###### Contents

  1. 1 Introduction
  2. 2 Radiative gauge field from classical dynamics in the soft limit
    1. 2.1 Soft Radiative gauge field for $\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$
    2. 2.2 Classical sub leading soft factor
  3. 3 Review of KMOC formalism
  4. 4 Soft radiative gauge field for $\sqrt{\text{Kerr}_{1}}$-$\sqrt{\text{Kerr}_{2}}$ scattering
  5. 5 Conclusions
  6. 6 Acknowledgments
  7. A Matching of the $S^{(1)\mu}$ with the soft limit of the radiative gauge field
  8. B Notations and Feynman rules

##  1 Introduction

Classical soft theorems are exact statements about the long wavelength
radiation emitted in any classical scattering process. The theorems state that
in spacetimes greater than four ($D>4$) the radiative gauge / gravity field
measured at large distances is proportional to the “classical limit” of the
single soft factor with the momentum and angular momentum operators replaced
with their classical asymptotic values. They have been derived from the
quantum soft theorems using the saddle point approximation[1] as well as have
been given a classical derivation using classical equations of motion, which
holds for generic classical scattering processes[2]111The derivation from
quantum soft theorems is valid for certain classes of scattering processes
which can be characterised as large impact parameter scattering (low
deflection processes) and the probe-scatterer approximation (the ratio of the
probe to scatterer masses is small)..
In $D=4$ the situation is more complicated but more intriguing due to the
presence of infrared divergences in the $S$-matrix. The (quantum) soft
theorems receive non-analytic corrections starting from the subleading term,
$\ln\omega$[3, 4]. Classical low energy gravitational (and electromagnetic)
radiation have been analysed for generic classical scattering processes using
the classical equations of motion and consequently, classical soft theorems in
$D=4$ have been derived[5, 6]. They hold for spinning particles as well as
scalars. Unlike the situation in $D>4$, the proof of the classical soft
theorems in $D=4$ from their quantum counterparts is lacking. Given the
developments in the literature on computing classical observables from
scattering amplitudes, one can hope to prove the classical soft theorems in a
perturbative setting. For classical large impact parameter scattering, the
first steps towards this goal was undertaken in [7]. For such scattering
processes, using the KMOC formalism[8] and (quantum) subleading soft theorems
in $D=4$, the low energy gravitational and electromagnetic radiation was
analysed and subsequently, the classical soft theorems for scalar particles
were proved till Next-to-Leading order (NLO) in the coupling (low deflection
parameter)222Recently, the classical leading soft factor in electrodynamics
has been proved till NLO[9]..
The KMOC formalism is a framework which computes classical observables like
linear impulse, directly using on-shell scattering amplitudes. The basic idea
is to start with an initial coherent state of particles, with momenta peaked
around their classical values, in the far past, evolve it with the $S$-matrix
and then take the expectation value of a quantum observable. Classical limit
of the observable then gives the classical result. The small coupling
expansion of the scattering amplitude, in the classical limit, is then
identified as the large impact parameter expansion. The KMOC formalism has now
been generalised to include asymptotic states like spinning particles,
particles with colour as well as incoming radiation [10, 11, 12, 13]. The
formalism is one of the multiple methods used to compute classical observables
from scattering amplitudes, relevant for the binary black hole system. For a
selection of such results we refer the reader to[14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].
In this short note, we take the first steps to provide a perturbative proof of
the classical subleading soft theorems for spinning particles, from the
quantum subleading soft theorems. For this paper, we shall concentrate on
electromagnetic radiation and to Leading order (LO) in the low deflection
parameter. We will concentrate on the electromagnetic analog of the Kerr black
hole, the so-called $\sqrt{\text{Kerr}}$ particle333We shall provide the
relevant details of this classical object later in the paper.. Since, spinning
particles have internal angular momentum there is another length scale which
is called the spin expansion parameter, denoted by the spin of the
particle(S). Our main result is thus the computation of the radiative gauge
field for soft radiation to subleading order in the soft expansion for
$\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$ scattering upto
$\mathcal{O}(S_{1}S_{2})$ and leading order in the low deflection parameter.
This paper is organised as follows. In sect.2 we use the classical equations
of motion to derive the the soft radiative gauge field for
$\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$ scattering till leading order
in the coupling and $\mathcal{O}(S_{1}S_{2})$. In sect.3 we give a brief
summary of the KMOC and its generalisation to spinning particles and in sect.4
we compute the radiative gauge field created by two scattering
$\sqrt{\text{Kerr}}$s using the KMOC formalism to leading order int the
coupling and $\mathcal{O}(S_{1}S_{2})$, thus verifying the classical results
obtained. We end the paper with a few remarks and some open questions.

##  2 Radiative gauge field from classical dynamics in the soft limit

In this section we shall compute the radiative gauge field for low frequency
radiation in the scattering process of
$\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$. Before that we shall review
the necessary features of classical spinning particles relevant for us and
also give a brief summary of the $\sqrt{\text{Kerr}}$ solution.
Classical spinning particles in addition to mass also have internal spin
degrees of freedom which in the non - relativistic case can be described by
the internal coordinates fixed to the body, for eg. Euler angles. In the
covariant description the internal spin of a classical spinning particle can
either be described in terms of an anti-symmetric $2$-tensor, $S^{\mu\nu}$ or
a $4$-vector, $s^{\mu}$. Since soft theorems naturally involve the angular
momentum tensor, we shall be utilising the anti-symmetric $2$\- tensor,
$S_{\mu\nu}$. However, we note that $S^{\mu\nu}$ has 6 components whereas we
know that in the rest frame of the particle, we can describe the spin in terms
of the rotations about the three spatial axes. Hence, to obtain the correct
physical degrees of freedom, we impose the spin supplementary condition (SSC)

$S^{\mu\nu}(\tau)p_{1\nu}(\tau)=0.$ (2.1)

Here $\tau$ is the proper time along the path of the particle and the SSC has
to hold throughout the path of the particle. Also, the spin vector $s^{\mu}$
and $S^{\mu\nu}$ are related by

$S^{\mu\nu}=\frac{1}{m}\epsilon^{\mu\nu\rho\sigma}p_{\rho}s_{\sigma},\ \ \ \ \
\ \ \ \ s^{\mu}=\frac{1}{2m}\epsilon^{\mu\nu\rho\sigma}p_{\nu}S_{\rho\sigma}.$
(2.2)

We shall be analysing low frequency radiation for large impact parameter
scattering. To derive the equations of motion for the $\sqrt{\text{Kerr}}$
till leading order in the low deflection parameter, we need its conserved
current. This can be obtained by doing the single copy substitution on the
conserved stress tensor for the Kerr black hole444There is a hierarchy of
substitutions which can be used to go between classical solutions of Bi-
adjoint scalar theory, gauge theories and gravitational theories[33]. The
inverse map from gravitational solutions to gauge theory solutions is
generally known as single copy.. For the gravity case, the low deflection
parameter in $D=4$ is given by

$\frac{G_{N}E}{|\vec{b}|}\ll 1.$ (2.3)

From the Effective field theory perspective, the dynamical equations for
spinning particles interacting via gravity have been derived in [34, 35]. For
the special case of Kerr black holes, to leading order in the low deflection
parameter the equations can be obtained by using the conserved stress
tensor[36]

$T^{\mu\nu}(x)=\frac{1}{m}\int d\tau p^{(\mu}\ p^{\rho}(\tau)\
\exp{(a\star\partial)}^{\nu)}_{\ \rho}\ \hat{\delta}^{(4)}(x-r(\tau)).$ (2.4)

Here $a^{\mu}$ is the internal spin of the black hole and it is related to the
pseudo - spin vector as $a^{\mu}=s^{\mu}/m$. Also $(a\star\partial)^{\mu}_{\
\rho}=\epsilon^{\mu}_{\ \rho\alpha\beta}a^{\alpha}\frac{\partial}{\partial
x_{\beta}}$. From the above we can obtain the conserved current for the
$\sqrt{\text{Kerr}}$ by making the single copy substitution,
$p_{\mu}\rightarrow Q$. We get

$J^{\mu}(x)=\frac{Q}{m}\int d\tau\ p^{\rho}(\tau)\
\exp{(a\star\partial)}^{\mu}_{\ \rho}\ \hat{\delta}^{(4)}(x-r(\tau))$ (2.5)

This is the conserved current for the $\sqrt{\text{Kerr}}$ object to leading
order in the low deflection parameter and it is characterised by the charge
$Q$, mass $m$ and the spin, $a^{\mu}=s^{\mu}/m$. To cross check if this is the
correct conserved current for the $\sqrt{\text{Kerr}}$ particle, we will
evaluate the electrostatic and magnetostatic potential obtained from this
current to $\mathcal{O}(a^{2})$ and match it to the one obtained by doing the
complex deformation of the Coulomb solution[37]. The gauge field can be
calculated from the general solution of the Maxwell solution

$A^{\mu}(x)=\int d^{4}x^{\prime}\ G_{r}(x-x^{\prime})\ J^{\mu}(x^{\prime})$
(2.6)

where
$G_{r}(x-x^{\prime})=\delta\big{(}\frac{(x-x^{\prime})^{2}}{2}\big{)}\Theta(x-x^{\prime})$
is the standard retarted Green’s function. Plugging eq.(2.5) in the above
equation, expanding the exponential and doing by-parts we obtain the gauge
field,

$A^{\mu}(x)=\frac{Q}{m}\int d^{4}x^{\prime}\
\delta^{(4)}(x^{\prime}-r(\tau))\bigg{[}\delta^{\mu}_{\rho}+(a\star\partial)^{\mu}_{\
\rho}+\frac{1}{2!}(a\star\partial)^{\mu}_{\
\alpha}(a\star\partial)^{\alpha}_{\ \rho}\ +\ldots\bigg{]}\frac{p^{\rho}}{r}$
(2.7)

The above calculation is done along the same lines as the Kerr black hole in
[36]. We can now do the remaining integrals by going to the rest frame of the
particle, $p^{\mu}=(m,\vec{0})$ and putting $r^{\mu}(\tau)=\tau$. The SSC in
terms of the spin psuedo-vector is $a\cdot p=0$ which in the rest frame gives
$a^{0}=0$. The zeroth component of the gauge field is then

$A^{0}(x)=\phi(x)=Q\bigg{(}1-\frac{(\vec{a}\cdot\vec{\partial})^{2}}{2!}+\ldots\bigg{)}\frac{1}{r}$
(2.8)

and the vector potential is

$A^{i}(x)=Q\epsilon^{ijk}a^{j}\partial_{k}\bigg{(}1-\frac{(\vec{a}\cdot\vec{\partial})^{2}}{3!}+\ldots\bigg{)}\frac{1}{r}$
(2.9)

The derivatives can be evaluated straightforwardly and we get

$\phi(x)=\frac{Q}{r}\bigg{(}1-\frac{1}{2!}\bigg{(}-\frac{\vec{a}^{2}}{r^{2}}+3\frac{(\vec{a}\cdot\vec{x})^{2}}{r^{4}}\bigg{)}+\ldots\bigg{)},\
\ \ \ \ A^{i}(x)=Q\
\epsilon^{ijk}a^{j}\bigg{(}-\frac{x_{k}}{r^{3}}+\ldots\bigg{)}$ (2.10)

To obtain the magnetostatic potential555In regions where there are no sources,
the curl of the magnetic field is also zero which allows us to define a
magnetostatic potential, $B_{i}(x)=\partial_{i}\chi$. we calculate the
magnetic field $B_{i}=(\nabla\times\vec{A})_{i}$ and from it derive the
magnetostatic potential defined as $B_{i}=\partial_{i}\chi$. Doing this
computation, we get the magnetic field to be

$B_{i}(x)=-Q\frac{a_{i}}{r^{3}}+3Q\
x_{i}\frac{\vec{a}\cdot\vec{x}}{r^{5}}+\ldots=\partial_{i}\bigg{(}-Q\
\frac{\vec{a}\cdot\vec{x}}{r^{3}}+\ldots\bigg{)}$ (2.11)

Hence, from the conserved current we get the potentials to be

$\phi(x)=\frac{Q}{r}\bigg{(}1-\frac{1}{2!}\bigg{(}-\frac{\vec{a}^{2}}{r^{2}}+3\frac{(\vec{a}\cdot\vec{x})^{2}}{r^{4}}\bigg{)}+\ldots\bigg{)},\
\ \ \ \ \chi=-Q\ \frac{\vec{a}\cdot\vec{x}}{r^{3}}+\ldots$ (2.12)

Now as explained in [37] the potential created by the $\sqrt{\text{Kerr}}$
object can also be obtained by doing a complex deformation of the Coulomb
solution

$\Phi(x)=\frac{Q}{r}\rightarrow\frac{Q}{\sqrt{(\vec{x}-i\vec{a})^{2}}}=\phi+i\chi.$
(2.13)

Expanding the denominator in the $r=\sqrt{\vec{x}^{2}}\gg|\vec{a}|$ regime, we
get the above potential to be

$\Phi(x)=\frac{Q}{r}\bigg{(}1+\frac{1}{2}\frac{\vec{a}^{2}}{r^{2}}-i\frac{\vec{a}\cdot\vec{x}}{r^{2}}-\frac{3}{2}\frac{(\vec{a}\cdot\vec{x})^{2}}{r^{4}}+\ldots\bigg{)}$
(2.14)

From the above equation we can read off the electromagnetic and magnetostatic
potential and they are

$\phi(x)=\frac{Q}{r}+\ \frac{Q}{2}\frac{\vec{a}^{2}}{r^{3}}\
-\frac{3}{2}\frac{(\vec{a}\cdot\vec{x})^{2}}{r^{5}}+\ldots,\ \ \
\chi(x)=-Q\frac{\vec{a}\cdot\vec{x}}{r^{3}}+\ldots$ (2.15)

Hence, we see that there is an exact match with the same quantity calculated
using the conserved current in eq.(2.5). For more details on the
$\sqrt{\text{Kerr}}$ object we refer the reader to [37].
We will be interested in the spin expansion till $\mathcal{O}(S)$, so, we
expand the conserved current in eq.(2.5) till $\mathcal{O}(a)$ to get

$\begin{split}J^{\mu}(x)&=\ \frac{Q}{m}\int d\tau\ p^{\mu}(\tau)\
\hat{\delta}^{(4)}(x-r(\tau))\ +\frac{Q}{m}\int d\tau\
(a\star\partial)^{\mu}_{\ \rho}\ p^{\rho}\
\hat{\delta}^{(4)}(x-r(\tau))+\mathcal{O}(a^{2})\\\ &=Q\int d\tau\
v^{\mu}(\tau)\ \hat{\delta}^{(4)}(x-r(\tau))\ +\frac{Q}{m}\int d\tau\
S^{\mu\rho}\partial_{\rho}\
\hat{\delta}^{(4)}(x-r(\tau))+\mathcal{O}(S^{2})\end{split}$ (2.16)

In going from the first line to the second line, we have used the first
equation in eq.(2.2). So, this is the current of $\sqrt{\text{Kerr}}$ till
$\mathcal{O}(S)$. Due to the anti-symmetry of $S_{\mu\nu}$ the current is
conserved. We can also compute the gyromagnetic ratio, $g$, of the spinning
particle from the above current. We do this by taking the non-relativistic
limit of the interaction term which we obtain

$V=-\frac{Q}{m}\vec{S}\cdot\vec{B}$ (2.17)

From the above equation we see that the $\sqrt{\text{Kerr}}$ has $g=2$.
We can now compute the equations of motion by coupling the current to the
electromagnetic field. The equation for the linear momentum of the particle
comes out to be

$\frac{dp^{\mu}(\tau)}{d\tau}=QF^{\mu\nu}(x(\tau))v_{\nu}+\frac{Q}{2m}\
S^{\rho\sigma}\partial^{\mu}F_{\rho\sigma}(x(\tau))$ (2.18)

The equation for the spin can be derived by demanding that the SSC holds
throughout the path of the particle, taking the derivative of it

$\begin{split}&\dot{S}^{\mu\nu}p_{\nu}+S^{\mu\nu}\dot{p}_{\nu}=0,\\\
&\dot{S}^{\mu\nu}p_{\nu}+\frac{Q}{m}\ S^{\mu\nu}F_{\nu\alpha}p^{\alpha}=0,\\\
&\dot{S}^{\mu\nu}p_{\nu}+\frac{Q}{m}\ S^{\mu\alpha}F_{\alpha}^{\
\nu}p_{\nu}=0,\\\ &\bigg{(}\dot{S}^{\mu\nu}+\frac{2Q}{m}\
S^{[\mu\alpha}F_{\alpha}^{\ \nu]}\bigg{)}p^{\nu}=0\end{split}$ (2.19)

In the last step, we have anti-symmetrized in the $(\mu\nu)$ indices.
Therefore, the equation of motion for the $\sqrt{\text{Kerr}}$ till $O(S)$ and
LO is

$\displaystyle\frac{dp^{\mu}(\tau)}{d\tau}=QF^{\mu\nu}(x(\tau))v_{\nu}+\frac{Q}{2m}\
S^{\rho\sigma}\partial^{\mu}F_{\rho\sigma}(x(\tau))$ (2.20)
$\displaystyle\frac{dS^{\mu\nu}(\tau)}{d\tau}=\ -\frac{2Q}{m}\ F^{[\mu}_{\ \
\alpha}S^{\nu]\alpha}(\tau)$ (2.21)

Using these equations, the radiative gauge field can be computed using the
Fourier transform of the current,

$A_{\mu}(x)=\frac{Q}{2\pi r}\int d\omega\ e^{-i\omega t}\ \tilde{J}_{\mu}(k)$
(2.22)

where $k=\omega(1,\hat{n})=\omega(1,\frac{\vec{x}}{r})$. The Fourier transform
of the current is what we shall be computing using the equations of motion and
also comparing to the ones obtained from the KMOC formalism. Till
$\mathcal{O}(S)$ this is

$\tilde{J}^{\mu}(k)=iQ\ \int d\sigma\ e^{ik.x(\sigma)}\ \frac{1}{k\cdot
v(\sigma)}\bigg{[}a^{\mu}(\sigma)-\frac{k\cdot a(\sigma)}{k\cdot v(\sigma)}\
v^{\mu}(\sigma)\bigg{]}\\\ +\frac{Q}{m}\int d\sigma\ e^{ik\cdot
x(\sigma)}\frac{1}{k\cdot
v(\sigma)}\bigg{[}\dot{S}^{\mu\nu}(\sigma)-\frac{k\cdot a(\sigma)}{k\cdot
v(\sigma)}S^{\mu\nu}(\sigma)\bigg{]}k_{\nu}$ (2.23)

Here $a^{\mu}(\sigma)=\frac{d^{2}z^{\mu}(\sigma)}{d\sigma^{2}}$ is the
acceleration of the spinning particle, not be confused with the rescaled spin
vector.
We shall be studying $\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$
scattering in the low deflection regime (large impact parameter). This is the
same class of scattering processes studied in [38, 39]. The setup is that
initially in the far past the particles are assumed to be free, then they
undergo large impact scattering and in the process they emit soft radiation.
We take the path to be parametrised as

$x^{\mu}_{1}(\tau)=b_{1}^{\mu}+v_{1}\tau+z_{1}^{\mu}(\tau),\ \ \ \ \
x_{2}^{\mu}(\tau)=v_{2}^{\mu}\tau+z_{2}^{\mu}(\tau)$ (2.24)

Here $b_{1}^{\mu}$ is the impact parameter, $v_{i}$s are the initial
velocities and $z_{i}$s denote the deflection from the initial path. The spin
angular momentum will be parametrised as

$S^{\mu\nu}_{i}(\tau)=S^{\mu\nu}_{i}+\tilde{S}^{\mu\nu}_{i}(\tau)$ (2.25)

where $S_{i}^{\mu\nu}(\tau)$ is the initial spin of the two
$\sqrt{\text{Kerr}}$ particles and $\tilde{S}_{i}^{\mu\nu}$ denotes the change
in the initial spin angular momentum of the particles. Since the spinning
object has an intrinsic length scale, there will be two dimensionless
parameters. These are

$\frac{e^{2}}{Eb}\ll 1\ \ ;\ \ \frac{e^{2}a_{i}}{Eb^{2}}\ll 1.$ (2.26)

Here $b$ is the impact parameter of the scattering process, $E$ is the energy
of the massive spinning particle, $a_{i}$s are the characteristic length for
the $\sqrt{\text{Kerr}}$ objects and $e$ is the coupling constant. The first
inequality is obtained by requiring that the deflection should be much less
than the impact parameter and the second comes from the requirement that the
change in ’$a$’ be much smaller than the impact parameter. Additionally, it
can be checked that requiring low deflection ensures that the change in ’$a$’
is much smaller than ’$a$’.

###  2.1 Soft Radiative gauge field for
$\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$

In this subsection we compute the radiative gauge field in the soft limit for
$\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$ scattering process. We shall
restrict ourselves to $\mathcal{O}(S_{1}S_{2})$ in the spin expansion and to
leading order in the low deflection parameter.
Since the second particle is also a $\sqrt{\text{Kerr}}$, the EM field created
by it also has a contribution coming from its spin angular momentum. So, the
EM field

$F_{2}^{\mu\nu}(x)=iQ_{2}\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{2})\ e^{-il\cdot x}\ \frac{l^{\mu}}{l^{2}}\ \big{(}(l\wedge
v_{2})^{\mu\nu}-\frac{i}{m_{2}}(l\wedge
S_{2}^{-\alpha})^{\mu\nu}l_{\alpha}\big{)}$ (2.27)

Using this the acceleration of the first particle comes out to be

$\begin{split}a_{1}^{\mu}(\tau)&=\frac{iQ_{1}Q_{2}}{m_{1}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}(l\cdot v_{2})\ e^{-il\cdot b_{1}}\ e^{-i(l\cdot v_{1})\tau}\
\frac{1}{l^{2}}\\\ &\bigg{[}\ l^{\mu}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}-(l\cdot
v_{1})(v_{2}^{\mu}-\frac{i}{m_{2}}S_{2}^{\mu\alpha}l_{\alpha})\bigg{]}\end{split}$
(2.28)

Therefore the change in the linear impulse is given by

$\Delta p_{1}^{\mu}(\tau)=iQ_{1}Q_{2}\int d\mu(1,2,l)\ e^{-il\cdot b_{1}}\
\frac{l^{\mu}}{l^{2}}\bigg{(}v_{1}\cdot v_{2}-\frac{i}{m_{2}}(v_{1}\wedge
l)_{2}-\frac{i}{m_{1}}(l\wedge v_{2})_{1}+\frac{(l\cdot
S_{1}^{-\sigma})(l\cdot S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}$ (2.29)

Similarly the computation of the change in the spin angular momentum of the
first particle can also be computed and we obtain

$\begin{split}\dot{S}_{1}^{\mu\nu}(\tau)&=\frac{iQ_{1}Q_{2}}{m_{1}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}(l\cdot v_{2})\ e^{-il\cdot b_{1}}\ e^{-i(l\cdot v_{1})\tau}\
\frac{1}{l^{2}}\\\ &\bigg{[}(v_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}l_{\alpha}-(l\wedge
S_{1}^{-\alpha})^{\mu\nu}v_{2\alpha}-\frac{i}{m_{2}}(l\wedge
S_{1}^{-\alpha})^{\mu\nu}(l\cdot
S^{-}_{2\alpha})-\frac{i}{m_{2}}(S_{2}^{-\rho}\wedge
S_{1}^{-\alpha})^{\mu\nu}\ l_{\rho}l_{\alpha}\bigg{]}\end{split}$ (2.30)

The spin angular momentum impulse then becomes

$\begin{split}\Delta S_{1}^{\mu\nu}(\tau)&=\frac{iQ_{1}Q_{2}}{m_{1}}\int
d\mu(1,2,l)\ e^{-il\cdot b_{1}}\ \frac{1}{l^{2}}\\\ &\bigg{[}(v_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}l_{\alpha}-(l\wedge
S_{1}^{-\alpha})^{\mu\nu}v_{2\alpha}-\frac{i}{m_{2}}(l\wedge
S_{1}^{-\alpha})^{\mu\nu}(l\cdot
S^{-}_{2\alpha})-\frac{i}{m_{2}}(S_{2}^{-\rho}\wedge
S_{1}^{-\alpha})^{\mu\nu}\ l_{\rho}l_{\alpha}\bigg{]}\end{split}$ (2.31)

Finally using eq.(2.28) and eq.(2.42), we calculate the radiative gauge field
in the soft limit in the same manner as in the previous sections. We get the
radiative gauge field for the first $\sqrt{\text{Kerr}_{1}}$ for long
wavelength at the subleading order in soft frequency to be

$\begin{split}\tilde{J}_{1}^{\mu}(k)&=\frac{iQ_{1}^{2}Q_{2}}{m_{1}}\int
d\mu(1,2,l)\ e^{-il\cdot b_{1}}\ \frac{1}{l^{2}}\ \frac{b_{1}\cdot k}{k\cdot
v_{1}}\ \frac{k_{\nu}}{k\cdot v_{1}}(v_{1}\wedge l)^{\mu\nu}\\\ &\hskip
56.9055pt\bigg{(}v_{1}\cdot v_{2}-\frac{i}{m_{2}}(v_{1}\wedge
l)_{2}-\frac{i}{m_{1}}(l\wedge v_{2})_{1}+\frac{(l\cdot
S_{1}^{-\sigma})(l\cdot S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\\\
&-\frac{Q_{1}^{2}Q_{2}}{m_{1}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}^{\prime}(l\cdot v_{1})\hat{\delta}(l\cdot v_{2})\frac{1}{l^{2}}\
e^{-il\cdot b_{1}}\ \frac{k_{\nu}}{k\cdot v_{1}}\ (v_{1}\wedge l)^{\mu\nu}\
\bigg{(}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge v_{2})_{1}+\frac{(l\cdot
S_{1}^{-\sigma})(l\cdot S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\\\
&+\frac{Q_{1}^{2}Q_{2}}{m_{1}}\int d\mu(1,2,l)\ e^{-il\cdot b_{1}}\
\frac{1}{l^{2}}\ \bigg{[}\bigg{(}v_{2}^{\mu}-\frac{k\cdot v_{2}}{k\cdot
v_{1}}v_{1}^{\mu}\bigg{)}-\frac{i}{m_{2}}\bigg{(}S_{2}^{\mu\alpha}l_{\alpha}-\frac{(k\wedge
l)_{2}}{k\cdot v_{1}}v_{1}^{\mu}\bigg{)}\bigg{]}\\\
&+\frac{iQ_{1}^{2}Q_{2}}{m_{1}^{2}}\int d\mu(1,2,l)e^{-il\cdot b_{1}}\
\frac{1}{l^{2}}\\\ &\frac{k_{\nu}}{k\cdot v_{1}}\bigg{[}(v_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}l_{\alpha}-(l\wedge
S_{1}^{-\alpha})^{\mu\nu}v_{2\alpha}-\frac{i}{m_{2}}(l\wedge
S_{1}^{-\alpha})^{\mu\nu}(l\cdot
S^{-}_{2\alpha})-\frac{i}{m_{2}}(S_{2}^{-\rho}\wedge
S_{1}^{-\alpha})^{\mu\nu}\ l_{\rho}l_{\alpha}\\\ &\hskip
56.9055pt-\frac{k\cdot l}{k\cdot v_{1}}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}S_{1}^{\mu\nu}\bigg{]}\end{split}$
(2.32)

Using the change in linear impulse in eq.(2.29) and change in spin angular
momentum in eq.(2.31), the above expression can be written as

$\begin{split}\tilde{J}_{1}^{\mu}(k)&=\frac{Q_{1}}{m_{1}}\
\bigg{(}\frac{b_{1}\cdot k}{k\cdot v_{1}}\bigg{)}\ \frac{k_{\nu}}{k\cdot
v_{1}}(v_{1}\wedge\Delta
p_{1})^{\mu\nu}+\frac{Q_{1}}{m_{1}}\frac{k_{\nu}}{k\cdot v_{1}}\Delta
S_{1}^{\mu\nu}-\frac{Q_{1}}{m_{1}^{2}}\ \frac{k_{\nu}}{(k\cdot
v_{1})^{2}}(k\cdot\Delta p_{1})S_{1}^{\mu\nu}\\\
&-\frac{Q_{1}^{2}Q_{2}}{m_{1}}\frac{k_{\nu}}{k\cdot v_{1}}\
\bigg{(}v_{1}\wedge\frac{\partial}{\partial v_{1}}\bigg{)}^{\mu\nu}\ \int
d\mu(1,2,l)\ e^{-il\cdot b_{1}}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\\\ \end{split}$ (2.33)

We shall compute the radiative gauge field emitted by the second
$\sqrt{\text{Kerr}_{2}}$ particle. The EM field created by the first particle
is

$F_{1}^{\mu\nu}(x)=iQ_{1}\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{1})\ e^{-il\cdot x}\ e^{il\cdot b_{1}}\ \frac{l^{\mu}}{l^{2}}\
\big{(}(l\wedge v_{1})^{\mu\nu}-\frac{i}{m_{1}}(l\wedge
S_{1}^{-\alpha})^{\mu\nu}l_{\alpha}\big{)}$ (2.34)

Using this the acceleration of the second particle comes out to be

$\begin{split}a_{2}^{\mu}(\tau)&=\frac{iQ_{2}Q_{1}}{m_{2}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}(l\cdot v_{1})\ e^{il\cdot b_{1}}\ e^{-i(l\cdot v_{2})\tau}\
\frac{1}{l^{2}}\\\ &\bigg{[}\ l^{\mu}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{1}}(v_{2}\wedge l)_{1}-\frac{i}{m_{2}}(l\wedge
v_{1})_{2}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}-(l\cdot
v_{2})(v_{1}^{\mu}-\frac{i}{m_{1}}S_{1}^{\mu\alpha}l_{\alpha})\bigg{]}\end{split}$
(2.35)

Therefore the change in the linear impulse is given by

$\Delta p_{2}^{\mu}(\tau)=-iQ_{1}Q_{2}\int d\mu(1,2,l)\ e^{-il\cdot b_{1}}\
\frac{l^{\mu}}{l^{2}}\bigg{(}v_{1}\cdot v_{2}-\frac{i}{m_{2}}(v_{1}\wedge
l)_{2}-\frac{i}{m_{1}}(l\wedge v_{2})_{1}+\frac{(l\cdot
S_{1}^{-\sigma})(l\cdot S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}$ (2.36)

Similarly the computation of the change in the spin angular momentum of the
first particle can also be computed and we obtain

$\begin{split}\dot{S}_{2}^{\mu\nu}(\tau)&=-\frac{iQ_{1}Q_{2}}{m_{2}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}(l\cdot v_{1})\ e^{il\cdot b_{1}}\ e^{-i(l\cdot v_{2})\tau}\
\frac{1}{l^{2}}\\\ &\bigg{[}(l\wedge
S_{2}^{-\alpha})^{\mu\nu}v_{1\alpha}-(v_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}l_{\alpha}+\frac{i}{m_{2}}(l\wedge
S_{2}^{-\alpha})^{\mu\nu}(l\cdot
S^{-}_{1\alpha})+\frac{i}{m_{2}}(S_{1}^{-\rho}\wedge
S_{2}^{-\alpha})^{\mu\nu}\ l_{\rho}l_{\alpha}\bigg{]}\end{split}$ (2.37)

The spin angular momentum impulse then becomes

$\begin{split}\Delta S_{2}^{\mu\nu}&=\frac{iQ_{1}Q_{2}}{m_{2}}\int
d\mu(1,2,l)\ e^{-il\cdot b_{1}}\ \frac{1}{l^{2}}\\\ &\bigg{[}(l\wedge
S_{2}^{-\alpha})^{\mu\nu}v_{1\alpha}-(v_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}l_{\alpha}-\frac{i}{m_{2}}(l\wedge
S_{2}^{-\alpha})^{\mu\nu}(l\cdot
S^{-}_{1\alpha})-\frac{i}{m_{2}}(S_{1}^{-\rho}\wedge
S_{2}^{-\alpha})^{\mu\nu}\ l_{\rho}l_{\alpha}\bigg{]}\end{split}$ (2.38)

Using the eq.(2.35) and eq.(2.37), we compute the radiative gauge field for
the second $\sqrt{\text{Kerr}_{2}}$ in the soft limit at the subleading order.
We obtain

$\begin{split}\tilde{J}_{2}^{\mu}(k)&=-\frac{Q_{2}^{2}Q_{1}}{m_{2}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}(l\cdot v_{1})\hat{\delta}^{\prime}(l\cdot v_{2})e^{il\cdot
b_{1}}\frac{1}{l^{2}}\\\ &\hskip 28.45274pt\frac{k_{\nu}}{k\cdot
v_{2}}(v_{2}\wedge l)^{\mu\nu}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\\\ &\hskip
28.45274pt-\frac{Q_{2}^{2}Q_{1}}{m_{2}}\int d\mu(1,2,l)e^{il\cdot b_{1}}\
\frac{1}{l^{2}}\ \frac{k_{\nu}}{k\cdot v_{2}}\bigg{(}(v_{2}\wedge
v_{1})^{\mu\nu}-\frac{i}{m_{1}}(v_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}l_{\alpha}\bigg{)}\\\ &\hskip
85.35826pt+i\frac{Q_{2}^{2}Q_{1}}{m_{2}^{2}}\int d\mu(1,2,l)\ \frac{1}{l^{2}}\
e^{il\cdot b_{1}}\\\ &\frac{k_{\nu}}{k\cdot v_{2}}\bigg{[}(l\wedge
S_{2}^{-\alpha})^{\mu\nu}v_{1\alpha}-(v_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}l_{\alpha}+\frac{i}{m_{2}}(l\wedge
S_{2}^{-\alpha})^{\mu\nu}(l\cdot
S^{-}_{1\alpha})+\frac{i}{m_{2}}(S_{1}^{-\rho}\wedge
S_{2}^{-\alpha})^{\mu\nu}\ l_{\rho}l_{\alpha}\bigg{]}\\\
&-i\frac{Q_{2}^{2}Q_{1}}{m_{2}^{2}}\int d\mu(1,2,l)\ e^{il\cdot b_{1}}\
\frac{1}{l^{2}}\ \frac{k_{\nu}}{k\cdot v_{2}}\ \bigg{(}\frac{k\cdot l}{k\cdot
v_{2}}\bigg{)}\ S_{2}^{\mu\nu}\\\ &\hskip 56.9055pt\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{1}}(v_{2}\wedge l)_{1}-\frac{i}{m_{2}}(l\wedge
v_{1})_{2}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\end{split}$ (2.39)

Using eq.(2.36) and eq.(2.38), the above expression can be written in a
condensed manner as

$\begin{split}\tilde{J}_{2}^{\mu}(k)&=\
\frac{Q_{2}}{m_{2}}\frac{k_{\nu}}{k\cdot v_{2}}\Delta
S_{2}^{\mu\nu}-\frac{Q_{2}}{m_{2}^{2}}\ \frac{k_{\nu}}{(k\cdot
v_{2})^{2}}(k\cdot\Delta p_{2})S_{2}^{\mu\nu}\\\
&-\frac{Q_{2}^{2}Q_{1}}{m_{2}}\frac{k_{\nu}}{k\cdot
v_{2}}\bigg{(}v_{2}\wedge\frac{\partial}{\partial v_{2}}\bigg{)}^{\mu\nu}\int
d\mu(1,2,l)\ e^{-il\cdot b_{1}}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\end{split}$ (2.40)

The full radiative gauge field till $\mathcal{O}(S_{1}S_{2})$ and to leading
order in the coupling is

$\begin{split}\tilde{J}^{\mu}(k)&=\frac{Q_{1}}{m_{1}}\
\bigg{(}\frac{b_{1}\cdot k}{k\cdot v_{1}}\bigg{)}\ \frac{k_{\nu}}{k\cdot
v_{1}}(v_{1}\wedge\Delta
p_{1})^{\mu\nu}+\frac{Q_{1}}{m_{1}}\frac{k_{\nu}}{k\cdot v_{1}}\Delta
S_{1}^{\mu\nu}-\frac{Q_{1}}{m_{1}^{2}}\ \frac{k_{\nu}}{(k\cdot
v_{1})^{2}}(k\cdot\Delta p_{1})S_{1}^{\mu\nu}\\\
&-\frac{Q_{1}^{2}Q_{2}}{m_{1}}\frac{k_{\nu}}{k\cdot
v_{1}}\bigg{(}v_{1}\wedge\frac{\partial}{\partial v_{1}}\bigg{)}^{\mu\nu}\
\int d\mu(1,2,l)\ e^{-il\cdot b_{1}}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\\\
&+\frac{Q_{2}}{m_{2}}\frac{k_{\nu}}{k\cdot v_{2}}\Delta
S_{2}^{\mu\nu}-\frac{Q_{2}}{m_{2}^{2}}\ \frac{k_{\nu}}{(k\cdot
v_{2})^{2}}(k\cdot\Delta p_{2})S_{2}^{\mu\nu}\\\
&-\frac{Q_{2}^{2}Q_{1}}{m_{2}}\frac{k_{\nu}}{k\cdot
v_{2}}\bigg{(}v_{2}\wedge\frac{\partial}{\partial v_{2}}\bigg{)}^{\mu\nu}\int
d\mu(1,2,l)\ e^{-il\cdot b_{1}}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\end{split}$ (2.41)

Before we move to the quantum computation, we match this result with the
classical sub leading soft factor.

###  2.2 Classical sub leading soft factor

In this subsection we shall match the soft limit of the radiative kernel to
the classical subleading soft factor for spinning particles till leading order
in the low expansion parameter as done in [7]. For the first particle we get
the expression

$\begin{split}S^{(1)\mu}(\prod_{i=1}^{2}p_{i},S_{i},k)&=k_{\nu}\frac{J^{\mu\nu}_{1,+}}{p_{1,+}\cdot
k}-k_{\nu}\frac{J^{\mu\nu}_{1,-}}{p_{1,-}\cdot
k}=k_{\nu}\frac{(L^{\mu\nu}_{1,+}+S^{\mu\nu}_{1,+})}{p_{1,+}\cdot
k}-k_{\nu}\frac{(L^{\mu\nu}_{1,-}+S^{\mu\nu}_{1,-})}{p_{1,-}\cdot k}\\\
&=\frac{b_{1}\cdot k}{(p_{1}\cdot k)^{2}}(p_{1}\wedge\Delta
p_{1})^{\mu\nu}-k_{\nu}S_{1}^{\mu\nu}\ \frac{\Delta p_{1}\cdot k}{(p_{1}\cdot
k)^{2}}+\frac{k_{\nu}}{p_{1}\cdot k}\Delta
S_{1}^{\mu\nu}+\frac{k_{\nu}}{p_{1}\cdot k}(z_{1}(0)\wedge
p_{1})^{\mu\nu}\end{split}$ (2.42)

where the last term is obtained from the change in the orbital angular
momentum with the respect to the origin. In going from the first to the next
line we have used

$\begin{split}L_{1,+}^{\mu\nu}&=L_{1,-}^{\mu\nu}+(b_{1}\wedge\Delta
p_{1})^{\mu\nu}+(z_{1}(0)\wedge p_{1})^{\mu\nu}\\\ &=(b_{1}\wedge
p_{1})^{\mu\nu}+(b_{1}\wedge\Delta p_{1})^{\mu\nu}+(z_{1}(0)\wedge
p_{1})^{\mu\nu}\end{split}$ (2.43)

We can now compare the two expressions in eq.(2.42) and eq.(2.33). We see that
apart from the last term there is an exact match between the two expressions.
The last term is also consistent as we show in the appendixA. Hence, we can
write eq.(2.42) as

$\begin{split}&S^{(1)\mu}(\prod_{i=1}^{2}p_{i},S_{i},k)=\frac{b_{1}\cdot
k}{(p_{1}\cdot k)^{2}}(p_{1}\wedge\Delta
p_{1})^{\mu\nu}-k_{\nu}S_{1}^{\mu\nu}\ \frac{\Delta p_{1}\cdot k}{(p_{1}\cdot
k)^{2}}+\frac{k_{\nu}}{p_{1}\cdot k}\Delta S_{1}^{\mu\nu}\\\
&-\frac{Q_{1}^{2}Q_{2}}{m_{1}}\frac{k_{\nu}}{k\cdot
v_{1}}\bigg{(}v_{1}\wedge\frac{\partial}{\partial v_{1}}\bigg{)}^{\mu\nu}\
\int d\mu(1,2,l)\ e^{-il\cdot b_{1}}\bigg{(}v_{1}\cdot
v_{2}-\frac{i}{m_{2}}(v_{1}\wedge l)_{2}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\frac{(l\cdot S_{1}^{-\sigma})(l\cdot
S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\\\ \end{split}$ (2.44)

The matching of the subleading soft factor and the soft limit of the radiative
gauge field for the second particle proceeds in the same manner as we have
outlined for the first particle. Hence, we see that the two expressions match
for both the particles. We now turn to the quantum computation of the
radiative gauge field for the $\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$
scattering.

##  3 Review of KMOC formalism

In this section we shall first review the KMOC formalism for scalar particles
and then move to the spinning case. The essential idea is to start with
initial wave-packet with asymptotic data peaked around classical values,
evolve it with the S-matrix and then compute the expectation value of an
observable in this final state. The classical expression is obtained by taking
the classical limit. It is taken before any loop integrals are computed which
results in computing only a small number of Feynman diagrams, which is the
main advantage of the formalism.
The initial state is

$\ket{\Psi}=\int\prod_{i=1}^{2}\ d\Phi(p_{i})\ \phi(p_{i})\ e^{ib_{1}\cdot
p_{1}/\hbar}\ \ket{\vec{p}_{1},\vec{p}_{2}},\ \ \ \int d\Phi(p)\
|\phi(p)|^{2}=1$ (3.45)

The $\phi(p_{i})$ are gaussian wave functions which have peaks around their
classical values, $m_{i}u_{i}$ and the two particles are separated by a
distance of $|\vec{b}_{1}|$. The measure is the usual Lorentz invariant one,

$d\Phi(p_{i})=\frac{d^{4}p_{i}}{(2\pi)^{4}}\
\hat{\delta}^{(+)}(p_{i}^{2}-m_{i}^{2})=\frac{d^{4}p_{i}}{(2\pi)^{4}}\
\Theta(p_{i}^{0})\ \hat{\delta}(p_{i}^{2}-m_{i}^{2})$ (3.46)

The next step is construct a general observable by a suitable choice of an
operator,

$\hat{\delta}\hat{O}^{A}(p_{i},\ldots)=\hbar^{\alpha}\big{[}\bra{\Psi}\hat{S}\
\hat{O}^{A}\hat{S}\ket{\Psi}-\bra{\Psi}\hat{O}^{A}\ket{\Psi}\big{]}$ (3.47)

The above expression is the change in the expectation value of the operator,
$\hat{O}^{A}$, from the initial state. $\hat{S}$ is the S-matrix. The index
’$A$’ on $\hat{O}^{A}$ could be a colour index or spacetime indices. The dots
in the argument in the left hand side denote that the change can also depend
on other asymptotic data like colour or spin as we shall see. The
$\hbar^{\alpha}$ is a factor to account for the correct dimensions on both
sides. The above expression is then recast by using the unitarity of the
$S$-matrix, in terms of two terms, one which depend the $4$-pt amplitude and
another which has unitarity cuts stitching tree level amplitudes. The
classical limit of the observable is achieved by scaling the couplings,
$g\rightarrow g/\sqrt{\hbar}$ and massless exchanged momenta,
$q^{\mu}\rightarrow\bar{q}^{\mu}\hbar$. So, we shall be left with an integral
over the wavenumber of the field exchanged between the two incoming scattering
particles and this is the classical result.
The observable of interest to us is the Radiative kernel which in the
classical limit gives the radiative gauge / metric field. This observable can
be constructed by taking the expectation value of the gauge field operator,
$\hat{\mathbb{A}}^{\mu}(x)$[13]. Assuming no incoming radiation in the initial
state, the radiative kernel till leading order is

$\begin{split}\mathcal{R}^{\mu}(k)&=\braket{\braket{\ \hbar^{3/2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\mathcal{A}_{5}(p_{1},p_{2}\rightarrow\tilde{p}_{1},\tilde{p}_{2},k)\ }}\\\
&=\braket{\braket{\ \hbar^{3/2}\ \int d\hat{\mu}(1,2,q_{1},q_{2})\
e^{-iq_{1}\cdot b_{1}/\hbar}\ \hat{\delta}^{(4)}(q_{1}+q_{2}+k)\
A_{5}(p_{1},p_{2}\rightarrow\tilde{p}_{1},\tilde{p}_{2},k)\
}}+\mathcal{O}(T^{{\dagger}}T)\end{split}$ (3.48)

The $\braket{\braket{\ldots}}$ denotes the coherent state integration, the
result of which is plugging in the classical values. The
$\tilde{p}_{i}=p_{i}+q_{i}$ and $k$ is four momentum of the outgoing photon.
In the classical limit, $k^{\mu}=\hbar\bar{k}^{\mu}$ due to momentum
conservation. To leading order in the coupling, only the first term needs to
be kept. As a disclaimer, in practise we shall not use denote put a bar on the
wavenumber after taking the classical limit as this makes the expressions more
cluttered. However, we will state which steps we take the classical limit to
avoid any misunderstanding.
We review the spinning case now. The formalism was first extended to spinning
particles in [10]. The observable which gives the classical spin contribution
is the Pauli - Lubanski vector. This is motivated by the fact that the Pauli -
Lubanski vector is the generator of little group transformations and more
importantly, it isolates the spin contribution of any observable. Hence, we
have

$s^{\mu}_{ij}(p)=\frac{1}{m}\bra{\vec{p},i}\mathbb{W}^{\mu}\ket{\vec{p},j},\hskip
56.9055pts^{\mu}=\sum_{i,j}\int d\Phi(p)\ |\phi(p)|^{2}\ \zeta^{*}_{i}\
s^{\mu}_{ij}(p)\zeta_{j}$ (3.49)

Here $\zeta_{i}$ can be thought off as a schematic for the coherent state for
spin in the little group, $\mathbb{W}^{\mu}$ is the Pauli - Lubanski vector
and $\ket{\vec{p},i}$ is Fock space state with little group index ‘i’. In this
note we shall be studying only the lowest order in the classical spin
expansion, so we shall not discuss the construction of $\zeta_{i}$. This can
be found in [12]. In this note, we shall think of the classical spin vector as
the expectation of the Pauli - Lubanski vector in a coherent state and be
agnostic to the details of $\zeta_{i}$.
For a generic spinning particle, the $s^{\mu}_{ij}$ can be constructed using
the field expansions and the Lorentz representation generator. For $s=1/2$
particle, we get

$s^{\mu}_{ab}=\frac{\hbar}{4m}\bar{u}_{a}(p)\gamma^{5}\gamma^{\mu}u_{b}(p)$
(3.50)

where $a,b$ are the little group indices on the outgoing and incoming Dirac
fermion. The $u^{a}(p)$ is the Dirac spinor and $\gamma^{\mu}$ are the Dirac
matrices.
For a spinning particle, a general observable (like the linear impulse) will
depend on the final and initial $s^{\mu}_{ab}$. Since, the exchange momentum
is small the final $s^{\mu}_{ab}$ can be obtained from the initial one by an
infinitesimal boost,

$s^{\mu}_{ab}(p+\hbar\bar{q})=s^{\mu}_{1ab}(p)+\Delta s_{1ab}^{\mu}(p)$ (3.51)

Finally, the terms linear in $s^{\mu}_{ab}$ will give the contribution of the
classical pseudo-vector, $s^{\mu}$ after the use of eq.(3.49). In general for
spin-$s$ particle, the classical limit of an observable will contain a maximum
of $2s$-pt functions of the Pauli - Lubanski operators666From an EFT
perspective in the matching of the classical limit of the full amplitude, the
amplitude contains correlation functions of $\mathbb{W}^{\mu}$ which are
identified as contribution of spin operators in the low energy EFT[22, 40,
41].. Schematically then in taking the classical result, a general
contribution to an observable takes the form

$(A^{(0)}+A^{(1)}_{\mu_{1}}\braket{s^{\mu_{1}}}+A^{(2)}_{\mu_{1}\mu_{2}}\braket{s^{\mu_{1}}s^{\mu_{2}}}+\ldots+A^{(2s)}_{\mu_{1}\ldots\mu_{2s}}\braket{s^{\mu_{1}}\ldots
s^{\mu_{2s}}})$ (3.52)

The first contribution is the scalar term, the second is the linear in spin
vector and so on. The expectation value of the spin vectors is taken in the
spin coherent states and hence gives the classical result. In [10] the
classical observables like spin angular impulse and linear impulse for Kerr
black holes and the $\sqrt{\text{Kerr}}$ object were obtained for the first
few orders in the spin expansion using low finite spin particles. In the next
section we shall follow this same methodology to compute the soft radiative
gauge field to the first few orders of spin.

##  4 Soft radiative gauge field for
$\sqrt{\text{Kerr}_{1}}$-$\sqrt{\text{Kerr}_{2}}$ scattering

In this section we shall use the KMOC formalism generalised to include
spinning particles to compute classical soft electromagnetic radiation from
the scattering of two $\sqrt{\text{Kerr}}$ objects. From the works of [42,
43], classical objects like the $\sqrt{\text{Kerr}}$ can be thought of as the
classical limit of an infinite spin particle with
$S\rightarrow\infty,\hbar\rightarrow 0$ such that $S\hbar=const$, for tree
level scattering. This gives the classical result for a $\sqrt{\text{Kerr}}$
to all orders in spin. Since, we need to recover the radiative gauge field for
only the first few orders in the spin expansion, we shall work with low finite
spin particles. To ascertain the theory to be used, we note that the
gyromagnetic ratio for the $\sqrt{\text{Kerr}}$ is $g=2$. The Dirac fermion is
the lowest spinning particle with the same gyromagnetic ratio and so, we shall
work with it777The massive spin-1 particle is also another candidate, however,
preliminary investigations suggest there is an ambiguity with the classical
results obtained with the same.. It has also been shown that the linear
impulse till $\mathcal{O}(S)$ for the $\sqrt{\text{Kerr}}$ can be obtained
from the classical limit of the $4$-pt amplitude involving the Dirac
fermion[10]. For works at $1$-loop on the Kerr black hole and the
$\sqrt{\text{Kerr}}$, we refer the reader to [44, 45, 46, 47, 48].
We shall consider the scattering of two Dirac fermions. So we start with the
radiative kernel to leading order in the coupling, given in eq.(3.48)

$\begin{split}\mathcal{R}^{\mu}(k)&=\braket{\braket{\ \hbar^{3/2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\mathcal{A}_{5}^{\mu\
ab,cd}(p_{1},p_{2}\rightarrow\tilde{p}_{1},\tilde{p}_{2},k)\ }}\\\
&=\braket{\braket{\ \hbar^{3/2}\ \int d\hat{\mu}(1,2,q_{1},q_{2})\
e^{-iq_{1}\cdot b_{1}/\hbar}\ \hat{\delta}^{(4)}(q_{1}+q_{2}+k)\ A^{\mu\
ab,cd}_{5}(p_{1},p_{2}\rightarrow\tilde{p}_{1},\tilde{p}_{2},k)\
}}\end{split}$ (4.53)

The little group indices for the fermion of momentum
$\tilde{p}_{1}=p_{1}+q_{1}$ is $a$, for $p_{1}$ it is $b$, for
$\tilde{p}_{2}=p_{2}+q_{2}$ its $c$ and for $p_{2}$ it is $d$. We denote the
spin angular momentum of the first and second spin-1/2 particle as
$S_{1ab}^{\mu\nu}$ and $S_{2cd}^{\mu\nu}$, respectively.
The $5$-pt amplitude can be computed using the Feynman rules given in appendix
B and we get

$\begin{split}i&A^{\mu\
ab,cd}_{5}(p_{1},p_{2}\rightarrow\tilde{p}_{1},\tilde{p}_{2},k)\ =\\\
&-iQ^{2}_{1}Q_{2}\bigg{[}\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{\tilde{p}^{\mu}_{1}}{\tilde{p}_{1}\cdot k}\bigg{)}\
\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma^{\alpha}u^{b}(p_{1}))\
(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\ &\hskip
28.45274pt+i\ k_{\nu}\bigg{(}\
\frac{u^{a}(\tilde{p}_{1})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}u^{b}(p_{1})}{p_{1}\cdot
k}-\frac{u^{a}(\tilde{p}_{1})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}u^{b}(p_{1})}{\tilde{p}_{1}\cdot
k}\bigg{)}\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\bigg{]}\\\
&-iQ^{2}_{2}Q_{1}\bigg{[}\bigg{(}-\frac{p_{2}^{\mu}}{p_{2}\cdot
k}+\frac{\tilde{p}^{\mu}_{2}}{\tilde{p}_{2}\cdot k}\bigg{)}\
\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma^{\alpha}v^{d}(p_{2}))\
(\bar{u}^{a}(\tilde{p}_{1})\gamma_{\alpha}u^{b}(p_{1}))}{q_{1}^{2}}\\\ &\hskip
28.45274pt+i\ k_{\nu}\bigg{(}\
\frac{v^{a}(\tilde{p}_{2})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}v^{d}(p_{2})}{p_{2}\cdot
k}-\frac{v^{c}(\tilde{p}_{2})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}v^{d}(p_{2})}{\tilde{p}_{2}\cdot
k}\bigg{)}\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma_{\alpha}u^{b}(p_{1}))}{q_{1}^{2}}\bigg{]}\\\
\end{split}$ (4.54)

where $\hat{\Sigma}^{\mu\nu}$ is the Lorentz group generator in the $s=1/2$
representation. The first two lines are the contribution from the first
spin$-1/2$ and the rest is the contribution from the second particle. In
writing this we have used the on - shell conditions of the spinors and the
Clifford algebra. Plugging this into eq.(4.53), we get

$\begin{split}\mathcal{R}^{\mu}(k)&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\hat{\delta}^{(4)}(q_{1}+q_{2}+k)\\\
&\bigg{[}\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{\tilde{p}^{\mu}_{1}}{\tilde{p}_{1}\cdot k}\bigg{)}\
\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma^{\alpha}u^{b}(p_{1}))\
(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\ &\hskip
28.45274pt+i\ k_{\nu}\bigg{(}\
\frac{u^{a}(\tilde{p}_{1})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}u^{b}(p_{1})}{p_{1}\cdot
k}-\frac{u^{a}(\tilde{p}_{1})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}u^{b}(p_{1})}{\tilde{p}_{1}\cdot
k}\bigg{)}\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\bigg{]}\\\
&-\hbar^{3/2}\ Q_{2}^{2}Q_{1}\ \int d\hat{\mu}(1,2,q_{1},q_{2})\
e^{-iq_{1}\cdot b_{1}/\hbar}\ \hat{\delta}^{(4)}(q_{1}+q_{2}+k)\\\
&\bigg{[}\bigg{(}-\frac{p_{2}^{\mu}}{p_{2}\cdot
k}+\frac{\tilde{p}^{\mu}_{2}}{\tilde{p}_{2}\cdot k}\bigg{)}\
\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma^{\alpha}v^{d}(p_{2}))\
(\bar{u}^{a}(\tilde{p}_{1})\gamma_{\alpha}u^{b}(p_{1}))}{q_{1}^{2}}\\\ &\hskip
28.45274pt+i\ k_{\nu}\bigg{(}\
\frac{v^{a}(\tilde{p}_{2})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}v^{d}(p_{2})}{p_{2}\cdot
k}-\frac{v^{c}(\tilde{p}_{2})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}v^{d}(p_{2})}{\tilde{p}_{2}\cdot
k}\bigg{)}\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma_{\alpha}u^{b}(p_{1}))}{q_{1}^{2}}\bigg{]}\end{split}$
(4.55)

We have suppressed the expectation values in the coherent states here which is
$\braket{\braket{\ldots}}$. We shall compute the soft radiative gauge field
for the first particle. The contribution is

$\begin{split}\mathcal{R}_{1}^{\mu}(k)&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\hat{\delta}^{(4)}(q_{1}+q_{2}+k)\\\
&\bigg{[}\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{\tilde{p}^{\mu}_{1}}{\tilde{p}_{1}\cdot k}\bigg{)}\
\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma^{\alpha}u^{b}(p_{1}))\
(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\ &\hskip
28.45274pt+i\ k_{\nu}\bigg{(}\
\frac{u^{a}(\tilde{p}_{1})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}u^{b}(p_{1})}{p_{1}\cdot
k}-\frac{u^{a}(\tilde{p}_{1})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}u^{b}(p_{1})}{\tilde{p}_{1}\cdot
k}\bigg{)}\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\bigg{]}\\\
\end{split}$ (4.56)

We will split this into two types of terms,
$\mathcal{R}_{1}^{\mu}(k)=\mathcal{R}_{1a}^{\mu}(k)+\mathcal{R}_{1b}^{\mu}(k)$
where

$\begin{split}\mathcal{R}_{1a}^{\mu}(k)&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\hat{\delta}^{(4)}(q_{1}+q_{2}+k)\\\ &\hskip
56.9055pt\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{\tilde{p}^{\mu}_{1}}{\tilde{p}_{1}\cdot k}\bigg{)}\
\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma^{\alpha}u^{b}(p_{1}))\
(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\
\end{split}$ (4.57)

and

$\begin{split}\mathcal{R}_{1b}^{\mu}(k)&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\hat{\delta}^{(4)}(q_{1}+q_{2}+k)\\\ &\hskip 56.9055pti\ k_{\nu}\bigg{(}\
\frac{u^{a}(\tilde{p}_{1})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}u^{b}(p_{1})}{p_{1}\cdot
k}-\frac{u^{a}(\tilde{p}_{1})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}u^{b}(p_{1})}{\tilde{p}_{1}\cdot
k}\bigg{)}\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\
\end{split}$ (4.58)

We now compute the soft limit of the first term, eq.(4.57). We obtain

$\begin{split}\mathcal{R}_{1a}^{\mu}(k)&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\sum_{i=1}^{2}k\cdot\frac{\partial}{\partial
q_{i}}\hat{\delta}^{(4)}(q_{1}+q_{2})\\\ &\hskip
56.9055pt\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{\tilde{p}^{\mu}_{1}}{\tilde{p}_{1}\cdot k}\bigg{)}\
\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma^{\alpha}u^{b}(p_{1}))\
(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\
&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int d\hat{\mu}(1,2,q_{1},q_{2})\
e^{-iq_{1}\cdot b_{1}/\hbar}\
\big{[}\hat{\delta}^{(4)}(q_{1}+q_{2}+k)-\hat{\delta}^{(4)}(q_{1}+q_{2})\big{]}\\\
&\hskip 56.9055pt\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{\tilde{p}^{\mu}_{1}}{\tilde{p}_{1}\cdot k}\bigg{)}\
\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma^{\alpha}u^{b}(p_{1}))\
(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\
\end{split}$ (4.59)

We now integrate over with $q_{1}=-q-k$ and $q_{2}=q$ in both the integrals
and do a $q\rightarrow-q$ flip to obtain

$\begin{split}&\mathcal{R}_{1a}^{\mu}(k)=-Q_{1}^{2}Q_{2}\int\frac{d^{4}q}{(2\pi)^{4}}\
\hat{\delta}(2p_{2}\cdot q-q^{2})\ e^{-iq\cdot b/\hbar}\\\ &\bigg{\\{}\
\hat{\delta}(2p_{1}\cdot q+q^{2}-(2p_{1}\cdot k+2q\cdot k))\ e^{ik\cdot
b_{1}/\hbar}\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{(p_{1}+q)^{\mu}}{(p_{1}+q)\cdot
k}\bigg{)}\frac{\bar{u}^{a}(p_{1}+q-k)\gamma^{\alpha}u^{b}(p_{1})}{q^{2}}\\\
&\hskip 28.45274pt-\hat{\delta}(2p_{1}\cdot q+q^{2})\
\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{(p_{1}+q)^{\mu}}{(p_{1}+q)\cdot
k}\bigg{)}\frac{\bar{u}^{a}(p_{1}+q)\gamma^{\alpha}u^{b}(p_{1})}{q^{2}}\bigg{\\}}(v^{c}(p_{2}-q)\gamma_{\alpha}v^{d}(p_{2}))\end{split}$
(4.60)

So, keeping only the $\mathcal{O}(1)$ term in the soft limit we obtain

$\begin{split}&\mathcal{R}^{\mu}_{1a}(k)=-Q_{1}^{2}Q_{2}\int
d\hat{\mu}(1,2,q)\ e^{-iq\cdot b_{1}/\hbar}\\\ &\hskip
56.9055pti\bigg{(}\frac{b_{1}\cdot k}{\hbar}\bigg{)}\
\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{(p_{1}+q)^{\mu}}{(p_{1}+q)\cdot k}\bigg{)}\
\bar{u}^{a}(p_{1}+q)\gamma^{\alpha}u^{b}(p_{1})\
\frac{(v^{c}(p_{2}-q)\gamma_{\alpha}v^{d}(p_{2}))}{q^{2}}\\\ &+Q_{1}^{2}Q_{2}\
\int\frac{d^{4}q}{(2\pi)^{4}}\ \hat{\delta}^{\prime}(2p_{1}\cdot q+q^{2})\
\hat{\delta}(2p_{2}\cdot q-q^{2})\ e^{-iq\cdot b_{1}/\hbar}\\\ &\hskip
56.9055pt(p_{1}\cdot k+q\cdot k)\ \bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{(p_{1}+q)^{\mu}}{(p_{1}+q)\cdot k}\bigg{)}\
\bar{u}^{a}(p_{1}+q)\gamma^{\alpha}u^{b}(p_{1})\
\frac{(v^{c}(p_{2}-q)\gamma_{\alpha}v^{d}(p_{2}))}{q^{2}}\\\
&-Q_{1}^{2}Q_{2}\int d\hat{\mu}(1,2,q)\ e^{-iq\cdot b_{1}/\hbar}\\\ &\hskip
28.45274pt\bigg{(}-\frac{p_{1}^{\mu}}{p_{1}\cdot
k}+\frac{(p_{1}+q)^{\mu}}{(p_{1}+q)\cdot k}\bigg{)}\
\bigg{(}-k\cdot\frac{\partial}{\partial(p_{1}+q)}\bar{u}^{a}(p_{1}+q)\bigg{)}\gamma^{\alpha}u^{b}(p_{1})\
\frac{(v^{c}(p_{2}-q)\gamma_{\alpha}v^{d}(p_{2}))}{q^{2}}\end{split}$ (4.61)

The last term can be evaluated using the Laurent expansion of the spinor

$\bar{u}_{a}(p_{1}+q)=\bar{u}_{a}(p_{1})+\frac{p_{1}^{\rho}q_{1}^{\sigma}}{4m_{1}^{2}}\
\bar{u}_{a}(p_{1})[\gamma_{\rho},\gamma_{\sigma}]+\mathcal{O}(q^{2})$ (4.62)

and then evaluate the derivative as
$\frac{\partial}{\partial\tilde{p}_{1}^{\alpha}}q^{\mu}=\frac{\partial}{\partial\tilde{p}_{1}^{\alpha}}(\tilde{p}_{1}-p_{1})^{\mu}=\delta^{\mu}_{\alpha}$888Here
$\tilde{p}_{1}=p_{1}+q$.. Finally, we take the classical limit and we get

$\begin{split}&\mathcal{R}_{1a}^{\mu}(k)=-iQ_{1}\bigg{(}\frac{b_{1}\cdot
k}{p_{1}\cdot k}\bigg{)}\int d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\
\frac{k_{\nu}}{p_{1}\cdot k}\ (p_{1}\wedge q)^{\mu\nu}\
A_{4}^{\text{cl}}(p_{1},p_{2}\rightarrow p_{1}+q,p_{2}-q)\\\
&+Q_{1}\int\frac{d^{4}q}{(2\pi)^{4}}\ \hat{\delta}^{\prime}(2p_{1}\cdot
q)\hat{\delta}^{\prime}(2p_{2}\cdot q)e^{-iq\cdot b_{1}}\
\frac{k_{\nu}}{p_{1}\cdot k}\ (p_{1}\wedge q)^{\mu\nu}\
A_{4}^{\text{cl}}(p_{1},p_{2}\rightarrow p_{1}+q,p_{2}-q)\\\
&-Q_{1}^{2}Q_{2}\int d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\
\frac{k_{\nu}}{p_{1}\cdot k}(p_{1}\wedge q)^{\mu\nu}\frac{1}{q^{2}}\
\big{(}-4i(k\wedge p_{2})_{1}+\braket{(k\cdot S^{-\alpha}_{1})(q\cdot
S^{-}_{2\alpha})}\big{)}\end{split}$ (4.63)

Here the $\braket{S_{1}S_{2}}$ stand for the $2$-pt function of the two spin
tensors for the two particles. The $4$-pt amplitude for the scattering of two
Dirac fermions is

$iA_{4}(p_{1},p_{2}\rightarrow
p_{1}+q,p_{2}-q)=-iQ_{1}Q_{2}\frac{(\bar{u}^{a}(\tilde{p}_{1})\gamma^{\alpha}u^{b}(p_{1}))\
(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q^{2}}$ (4.64)

and the classical limit of which is

$A_{4}^{\text{cl}}(p_{1},p_{2}\rightarrow
p_{1}+q,p_{2}-q)=-Q_{1}Q_{2}\frac{4p_{1}\cdot p_{2}+4i(q\wedge
p_{1})_{2}-4i(q\wedge p_{2})_{1}+4\braket{(q\cdot S_{1}^{-\alpha})(q\cdot
S_{2\alpha}^{-})}}{q^{2}}$ (4.65)

We now evaluate the second term for the first Dirac fermion given in
eq.(4.58), which is

$\begin{split}\mathcal{R}_{1b}^{\mu}(k)&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\hat{\delta}^{(4)}(q_{1}+q_{2}+k)\\\ &\hskip 56.9055pti\ k_{\nu}\bigg{(}\
\frac{u^{a}(\tilde{p}_{1})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}u^{b}(p_{1})}{p_{1}\cdot
k}-\frac{u^{a}(\tilde{p}_{1})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}u^{b}(p_{1})}{\tilde{p}_{1}\cdot
k}\bigg{)}\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\
\end{split}$ (4.66)

The soft limit of the above expression is

$\begin{split}\mathcal{R}_{1b}^{\mu}(k)&=-\hbar^{3/2}\ Q_{1}^{2}Q_{2}\ \int
d\hat{\mu}(1,2,q_{1},q_{2})\ e^{-iq_{1}\cdot b_{1}/\hbar}\
\hat{\delta}^{(4)}(q_{1}+q_{2})\\\ &\hskip 56.9055pti\ k_{\nu}\bigg{(}\
\frac{u^{a}(\tilde{p}_{1})\gamma^{\alpha}\hat{\Sigma}^{\mu\nu}u^{b}(p_{1})}{p_{1}\cdot
k}-\frac{u^{a}(\tilde{p}_{1})\hat{\Sigma}^{\mu\nu}\gamma^{\alpha}u^{b}(p_{1})}{\tilde{p}_{1}\cdot
k}\bigg{)}\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\
&=-iQ_{1}^{2}Q_{2}\hbar^{3/2}\ \int d\hat{\mu}(1,2,q_{1},q_{2})\
e^{-iq_{1}\cdot b_{1}/\hbar}\ \hat{\delta}^{(4)}(q_{1}+q_{2})\\\ &\hskip
142.26378pt\frac{k_{\sigma}}{p_{1}\cdot k}\
\bar{u}^{a}(\tilde{p}_{1})[\gamma^{\alpha},\hat{\Sigma}^{\mu\sigma}]u^{b}(p_{1})\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\\\
&-iQ_{1}^{2}Q_{2}\hbar^{3/2}\ \int d\hat{\mu}(1,2,q_{1},q_{2})\
e^{-iq_{1}\cdot b_{1}/\hbar}\ \hat{\delta}^{(4)}(q_{1}+q_{2})\\\ &\hskip
142.26378pt\frac{k_{\sigma}}{(p_{1}\cdot k)^{2}}\
\bar{u}^{a}(\tilde{p}_{1})\hat{\Sigma}^{\mu\sigma}\gamma^{\alpha}u^{b}(p_{1})\
k_{\sigma}\
\frac{(\bar{v}^{c}(\tilde{p}_{2})\gamma_{\alpha}v^{d}(p_{2}))}{q_{2}^{2}}\end{split}$
(4.67)

These two terms can be straightforwardly evaluated using the following
identities

$\begin{split}&[\gamma^{\alpha},\hat{\Sigma}^{\mu\nu}]=i\
(\eta^{\alpha\mu}\gamma^{\nu}-\eta^{\alpha\nu}\gamma^{\mu})\\\
&[\gamma^{\mu},\gamma^{\nu}]\gamma^{\alpha}=2\eta^{\nu\alpha}\gamma^{\mu}-2\eta^{\mu\alpha}\gamma^{\nu}-2i\epsilon^{\mu\nu\alpha\rho}\gamma_{\rho}\gamma_{5}.\end{split}$
(4.68)

After taking the classical limit, we get

$\begin{split}\mathcal{R}_{1b}^{\mu}(k)&=4Q_{1}^{2}Q_{2}\ \int d\mu(1,2,q)\
e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\ \frac{k_{\nu}}{p_{1}\cdot k}\\\
&\big{[}(p_{2}\wedge p_{1})^{\mu\nu}+i(p_{2}\wedge
S_{1}^{-\sigma})^{\mu\nu}q_{\sigma}-i(S_{2}^{-\rho}\wedge
p_{1})^{\mu\nu}q_{\rho}+\braket{(S_{2}^{-\rho}\wedge
S_{1}^{-\sigma})^{\mu\nu}}q_{\rho}q_{\sigma}\big{]}\\\ &-iQ_{1}^{2}Q_{2}\
(4m_{1})\ \int d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\
\frac{k_{\nu}}{(p_{1}\cdot k)^{2}}(q\cdot k)\
\epsilon^{\mu\nu\alpha\rho}(s_{1\rho}\ p_{2\alpha}+i\
\braket{s_{1\rho}S_{2\rho\alpha}}q^{\rho})\end{split}$ (4.69)

We now use eq.(2.2) to express the spin vector in terms of the spin tensor. We
get

$\begin{split}\mathcal{R}_{1b}^{\mu}(k)&=4Q_{1}^{2}Q_{2}\ \int d\mu(1,2,q)\
e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\ \frac{k_{\nu}}{p_{1}\cdot k}\\\
&\big{[}-(p_{1}\wedge p_{2})^{\mu\nu}+i(p_{2}\wedge
S_{1}^{-\sigma})^{\mu\nu}q_{\sigma}+i(p_{1}\wedge
S_{2}^{-\rho})^{\mu\nu}q_{\rho}+\braket{(S_{2}^{-\rho}\wedge
S_{1}^{-\sigma})^{\mu\nu}}q_{\rho}q_{\sigma}\big{]}\\\ &-4iQ_{1}^{2}Q_{2}\int
d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\ \frac{k_{\nu}}{(p_{1}\cdot
k)^{2}}\ (q\cdot k)\\\ &\big{[}(p_{1}\wedge
S_{1}^{-\alpha})^{\mu\nu}p_{2\alpha}+(p_{1}\cdot p_{2})\ S_{1}^{\mu\nu}-i\
\braket{(p_{1}\wedge
S_{1}^{-\alpha})^{\mu\nu}S_{2\alpha\beta}}q^{\beta}-i\braket{(p_{1}\wedge
q)_{2}S_{1}^{\mu\nu}}\big{]}\end{split}$ (4.70)

Thus, by adding the eq.(4.63) and eq.(4.70) we get the soft radiative gauge
field for the first particle

$\begin{split}\mathcal{R}_{1}^{\mu}(k)&=-Q_{1}\bigg{(}\frac{b_{1}\cdot
k}{p_{1}\cdot k}\bigg{)}\ \frac{k_{\nu}}{p_{1}\cdot k}\ (p_{1}\wedge\Delta
p_{1})^{\mu\nu}\ +Q_{1}\frac{\Delta p_{1}\cdot k}{(p_{1}\cdot k)^{2}}\
k_{\nu}S_{1}^{\mu\nu}\\\ &+Q_{1}\frac{k_{\nu}}{p_{1}\cdot
k}\bigg{(}p_{1}\wedge\frac{\partial}{\partial p_{1}}\bigg{)}^{\mu\nu}\int\
d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\ A^{\text{cl}}_{4}(1,2,q)\\\
&\hskip 28.45274pt+iQ_{1}^{2}Q_{2}\int d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\
\frac{1}{q^{2}}\frac{k_{\nu}}{p_{1}\cdot k}\\\ &\big{[}\ (p_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}q_{\alpha}\ -(q\wedge
S_{1}^{-\alpha})^{\mu\nu}p_{2\alpha}-i(S_{2}^{-\alpha}\wedge
S_{1}^{-\beta})^{\mu\nu}q_{\alpha}q_{\beta}\ -i(q\wedge
S_{1}^{-\alpha})^{\mu\nu}(q\cdot S^{-}_{2\ \alpha})\big{]}\end{split}$ (4.71)

where the classical limit of the $4$-pt amplitude is given in eq.(4.65) and

$\Delta p_{1}^{\mu}=i\int d\mu(1,2,q)\ q^{\mu}\
A_{4}^{\text{cl}}(p_{1},p_{2}\rightarrow p_{1}+q,p_{2}-q)$ (4.72)

The soft radiative gauge field for the second particle can be obtained in the
exact manner as the first one. We get

$\begin{split}\mathcal{R}_{2}^{\mu}(k)&=Q_{2}\frac{k_{\nu}}{p_{2}\cdot
k}\bigg{(}p_{2}\wedge\frac{\partial}{\partial p_{2}}\bigg{)}^{\mu\nu}\int\
d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\
A^{\text{cl}}_{4}(p_{1},p_{2}\rightarrow p_{1}+q,p_{2}-q)\\\ &+Q_{2}\ k_{\nu}\
S_{2}^{\mu\nu}\ \frac{\Delta p_{2}\cdot k}{(p_{2}\cdot k)^{2}}\ +\
iQ_{2}^{2}Q_{1}\int d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\
\frac{1}{q^{2}}\frac{k_{\nu}}{p_{2}\cdot k}\\\ &\big{[}-(p_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}q_{\alpha}\ +(q\wedge
S_{2}^{-\alpha})^{\mu\nu}p_{1\alpha}-i(S_{1}^{-\alpha}\wedge
S_{2}^{-\beta})^{\mu\nu}q_{\alpha}q_{\beta}\ -i(q\wedge
S_{2}^{-\alpha})^{\mu\nu}(q\cdot S^{-}_{1\ \alpha})\big{]}\end{split}$ (4.73)

where $\Delta p^{\mu}_{2}=-\Delta p^{\mu}_{1}$. Adding the two radiative gauge
field in the soft limit we obtain the full radiative gauge field in the soft
limit from KMOC to be

$\begin{split}\mathcal{R}^{\mu}(k)&=-Q_{1}\bigg{(}\frac{b_{1}\cdot
k}{p_{1}\cdot k}\bigg{)}\ \frac{k_{\nu}}{p_{1}\cdot k}\ (p_{1}\wedge\Delta
p_{1})^{\mu\nu}\ +Q_{1}\frac{\Delta p_{1}\cdot k}{(p_{1}\cdot k)^{2}}\
k_{\nu}S_{1}^{\mu\nu}\\\ &+Q_{1}\frac{k_{\nu}}{p_{1}\cdot
k}\bigg{(}p_{1}\wedge\frac{\partial}{\partial p_{1}}\bigg{)}^{\mu\nu}\int\
d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\ A^{\text{cl}}_{4}(1,2,q)\\\
&\hskip 28.45274pt+iQ_{1}^{2}Q_{2}\int d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\
\frac{1}{q^{2}}\frac{k_{\nu}}{p_{1}\cdot k}\\\ &\big{[}\ (p_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}q_{\alpha}\ -(q\wedge
S_{1}^{-\alpha})^{\mu\nu}p_{2\alpha}-i(S_{2}^{-\alpha}\wedge
S_{1}^{-\beta})^{\mu\nu}q_{\alpha}q_{\beta}\ -i(q\wedge
S_{1}^{-\alpha})^{\mu\nu}(q\cdot S^{-}_{2\ \alpha})\big{]}\\\
&+Q_{2}\frac{k_{\nu}}{p_{2}\cdot k}\bigg{(}p_{2}\wedge\frac{\partial}{\partial
p_{2}}\bigg{)}^{\mu\nu}\int\ d\mu(1,2,q)\ e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\
A^{\text{cl}}_{4}(1,2,q)+Q_{2}k_{\nu}S_{2}^{\mu\nu}\ \frac{\Delta p_{2}\cdot
k}{(p_{2}\cdot k)^{2}}\\\ &\hskip 28.45274pt+iQ_{2}^{2}Q_{1}\int d\mu(1,2,q)\
e^{-iq\cdot b_{1}}\ \frac{1}{q^{2}}\frac{k_{\nu}}{p_{2}\cdot k}\\\
&\big{[}-(p_{1}\wedge S_{2}^{-\alpha})^{\mu\nu}q_{\alpha}\ +(q\wedge
S_{2}^{-\alpha})^{\mu\nu}p_{1\alpha}-i(S_{1}^{-\alpha}\wedge
S_{2}^{-\beta})^{\mu\nu}q_{\alpha}q_{\beta}\ -i(q\wedge
S_{2}^{-\alpha})^{\mu\nu}(q\cdot S^{-}_{1\ \alpha})\big{]}\end{split}$ (4.74)

Using eq.(4.72) for $\Delta p_{1}^{\mu}$ and $\Delta p_{2}^{\mu}$, we see that
there is an exact match with the classical result in eq.(2.41). Integrating
the final result using the integrals given in [8, 10] we get

$\begin{split}&\mathcal{R}^{\mu}(k)=-Q_{1}\bigg{(}\frac{b_{1}\cdot
k}{p_{1}\cdot k}\bigg{)}\ \frac{k_{\nu}}{p_{1}\cdot k}\ (p_{1}\wedge\Delta
p_{1})^{\mu\nu}\ +Q_{1}\frac{\Delta p_{1}\cdot k}{(p_{1}\cdot k)^{2}}\
k_{\nu}S_{1}^{\mu\nu}\\\ &-\ Q_{1}^{2}Q_{2}\ \frac{\ln(\omega
b_{1})}{2\pi}\frac{m_{1}^{2}m_{2}^{2}}{\\{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}\\}^{3/2}}\ \frac{k_{\nu}}{p_{1}\cdot
k}(p_{1}\wedge p_{2})^{\mu\nu}\\\ &+\frac{Q_{1}^{2}Q_{2}}{2\pi}\
\frac{1}{b_{1}^{2}}\ \frac{(p_{1}\cdot p_{2})}{\\{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}\\}^{3/2}}\ \frac{k_{\nu}}{p_{1}\cdot
k}(p_{1}\wedge p_{2})^{\mu\nu}\\\ &\hskip 28.45274pt\big{[}(b_{1}\wedge
p_{2})_{1}+(p_{1}\wedge b_{1})_{2}-(S_{1}\cdot S_{2})+\frac{2}{b_{1}^{2}}\
(b_{1}\cdot S_{1}^{-\alpha})(b_{1}\cdot S^{-}_{2\alpha})\big{]}\\\ &-\
\frac{Q_{1}^{2}Q_{2}}{2\pi}\frac{1}{b_{1}^{2}}\frac{1}{\sqrt{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\ \frac{k_{\nu}}{p_{1}\cdot k}\ (p_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}b_{1\alpha}\\\ &-\frac{Q_{1}^{2}Q_{2}}{2\pi}\
\frac{1}{b_{1}^{2}}\frac{1}{\sqrt{(p_{1}\cdot p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\
\frac{k_{\nu}}{p_{1}\cdot k}\ \big{[}\ (p_{2}\wedge
S_{1}^{\alpha})^{\mu\nu}b_{1\alpha}-(b_{1}\wedge
S_{1}^{-\alpha})^{\mu\nu}p_{2\alpha}\big{]}\\\ &+\frac{Q_{1}^{2}Q_{2}}{\pi}\
\frac{1}{b_{1}^{4}}\frac{1}{\sqrt{(p_{1}\cdot p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\
\frac{k_{\nu}}{p_{1}\cdot k}\ \big{[}(S_{2}^{-\alpha}\wedge
S_{1}^{-\beta})^{\mu\nu}b_{1\alpha}b_{1\beta}+(b_{1}\wedge
S_{1}^{-\alpha})^{\mu\nu}(b_{1}\cdot S^{-}_{2\alpha})\big{]}\\\
&-\frac{Q_{1}^{2}Q_{2}}{\pi}\ \frac{1}{b_{1}^{2}}\frac{1}{\sqrt{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\ \frac{k_{\nu}}{p_{1}\cdot k}\
(S_{2}^{-\alpha}\wedge S^{-}_{1\alpha})^{\mu\nu}\\\
&-\frac{Q_{1}^{2}Q_{2}}{2\pi}\ \frac{1}{b_{1}^{2}}\frac{1}{\\{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}\\}^{3/2}}\ \frac{k_{\nu}}{p_{1}\cdot k}\\\
&\big{[}-(S^{-\alpha}_{2}\wedge S^{-\beta}_{1})^{\mu\nu}\
p_{1\alpha}p_{2\beta}\ (p_{1}\cdot p_{2})-(p_{1}\cdot p_{2})\ (p_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}(p_{1}\cdot S^{-}_{2\alpha})+\ m_{2}^{2}(p_{1}\wedge
S_{1}^{-\alpha})^{\mu\nu}(p_{1}\cdot S^{-}_{2\alpha})\big{]}\\\
&+Q_{2}\frac{\Delta p_{2}\cdot k}{(p_{2}\cdot k)^{2}}\ k_{\nu}S_{2}^{\mu\nu}-\
Q_{2}^{2}Q_{1}\ \frac{\ln(\omega
b_{1})}{2\pi}\frac{m_{1}^{2}m_{2}^{2}}{\\{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}\\}^{3/2}}\ \frac{k_{\nu}}{p_{2}\cdot
k}(p_{2}\wedge p_{1})^{\mu\nu}\\\ &+\frac{Q_{2}^{2}Q_{1}}{2\pi}\
\frac{1}{b_{1}^{2}}\ \frac{(p_{1}\cdot p_{2})}{\\{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}\\}^{3/2}}\ \frac{k_{\nu}}{p_{2}\cdot
k}(p_{2}\wedge p_{1})^{\mu\nu}\\\ &\hskip 28.45274pt\big{[}(b_{1}\wedge
p_{2})_{1}+(p_{1}\wedge b_{1})_{2}-(S_{1}\cdot S_{2})+\frac{2}{b_{1}^{2}}\
(b_{1}\cdot S_{1}^{-\alpha})(b_{1}\cdot S^{-}_{2\alpha})\big{]}\\\ &+\
\frac{Q_{2}^{2}Q_{1}}{2\pi}\frac{1}{b_{1}^{2}}\frac{1}{\sqrt{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\ \frac{k_{\nu}}{p_{2}\cdot k}\ (p_{2}\wedge
S_{1}^{-\alpha})^{\mu\nu}b_{1\alpha}\\\ &-\frac{Q_{2}^{2}Q_{1}}{2\pi}\
\frac{1}{b_{1}^{2}}\frac{1}{\sqrt{(p_{1}\cdot p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\
\frac{k_{\nu}}{p_{2}\cdot k}\ \big{[}\ -(p_{1}\wedge
S_{2}^{\alpha})^{\mu\nu}b_{1\alpha}-(b_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}p_{1\alpha}\big{]}\\\ &+\frac{Q_{2}^{2}Q_{1}}{\pi}\
\frac{1}{b_{1}^{4}}\frac{1}{\sqrt{(p_{1}\cdot p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\
\frac{k_{\nu}}{p_{2}\cdot k}\ \big{[}(S_{1}^{-\alpha}\wedge
S_{2}^{-\beta})^{\mu\nu}b_{1\alpha}b_{1\beta}+(b_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}(b_{1}\cdot S^{-}_{1\alpha})\big{]}\\\
&-\frac{Q_{2}^{2}Q_{1}}{\pi}\ \frac{1}{b_{1}^{2}}\frac{1}{\sqrt{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}}}\ \frac{k_{\nu}}{p_{2}\cdot k}\
(S_{1}^{-\alpha}\wedge S^{-}_{2\alpha})^{\mu\nu}\\\
&-\frac{Q_{2}^{2}Q_{1}}{2\pi}\ \frac{1}{b_{1}^{2}}\frac{1}{\\{(p_{1}\cdot
p_{2})^{2}-m_{1}^{2}m_{2}^{2}\\}^{3/2}}\ \frac{k_{\nu}}{p_{2}\cdot k}\\\
&\big{[}-(S^{-\alpha}_{1}\wedge S^{-\beta}_{2})^{\mu\nu}\
p_{1\beta}p_{2\alpha}\ (p_{1}\cdot p_{2})-(p_{1}\cdot p_{2})\ (p_{1}\wedge
S_{2}^{-\alpha})^{\mu\nu}(p_{2}\cdot S^{-}_{1\alpha})+\ m_{1}^{2}(p_{2}\wedge
S_{2}^{-\alpha})^{\mu\nu}(p_{2}\cdot S^{-}_{1\alpha})\big{]}\end{split}$
(4.75)

This is the final result. We conclude this section with a few remarks.

  * •

We can identify the terms in the above terms with the various contributions in
the classical sub leading soft factor, in eq.(2.42). The two terms in the
first line are the contributions to the radiative gauge field coming from a
part of the change in the orbital angular momentum and the linear impulse. The
terms from the second line to the eight line arise from the $(z_{1}(0)\wedge
p_{1})^{\mu\nu}$ term in the orbital angular momentum. The terms in the ninth
and tenth line arise from the change in the spin tensor of the first particle.
Then the next line has a contribution from the linear impulse of the second
particle and till the second last line, they again arise from the
corresponding $(z_{2}(0)\wedge p_{2})^{\mu\nu}$ for the second particle. The
second last and last line come from the change in the spin angular momentum of
the second particle.

  * •

Just like scalar case[7], the above result is consistent with the classical
sub-leading soft factor in $D=4$[5] and we also see that the $\ln\omega$ term
is independent of the spin of the particle. As mentioned in [7], this is
because the $\ln\omega$ term comes from the integration region
$\omega\ll|\vec{l}|^{-1}\ll|\vec{a}|\ll|\vec{b}_{1}|$.

##  5 Conclusions

In this short note, we took the first steps to proving the classical
subleading soft photon theorem for spinning particles, for the
$\sqrt{\text{Kerr}}$ particle, from the quantum subleading soft photon
theorem. We did this by using the KMOC formalism generalised to describe
classical spinning particles. We took the soft limit of the radiative kernel
for $\sqrt{\text{Kerr}_{1}}-\sqrt{\text{Kerr}_{2}}$ scattering and showed that
in the classical limit it reproduces the classical subleading soft photon
theorem till $\mathcal{O}(S_{1}S_{2})$ and to LO in the coupling.
The obvious issue would be to extend this perturbative proof of the subleading
soft photon theorem at leading order in the coupling, to all orders in spin
and also study the analogous case of low energy gravitational radiation for
the Kerr black hole. It would also be interesting to extend this perturbative
proof to higher orders in the coupling. For spinning particles in gravity, the
leading spin dependent non-analytic term in $D=4$ has also been derived in
[6]. It would be interesting to relate this to the subsubleading soft factor
in gravity.

##  6 Acknowledgments

We would like to thank Alok Laddha for the many fruitful discussions during
the project and for his suggestions on the manuscript. We would also like to
thank Arkajyoti Manna and Samim Akhtar for discussions on some conceptual
points during the course of the project.

##  Appendix A Matching of the $S^{(1)\mu}$ with the soft limit of the
radiative gauge field

So, the starting expression is the acceleration of the first particle given in
eq.(2.28). We integrate twice and put $\tau=0$ to get

$\begin{split}&z_{1}^{\mu}(0)=-i\frac{Q_{1}Q_{2}}{m_{1}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}(l\cdot v_{2})\ e^{-il\cdot b_{1}}\
\frac{1}{l^{2}}\frac{1}{(l\cdot v_{1})^{2}_{+}}\\\ &\bigg{[}l^{\mu}(v_{1}\cdot
v_{2})-\ (l\cdot v_{1})v_{2}^{\mu}-\frac{i}{m_{2}}\ \\{l^{\mu}(v_{1}\wedge
l)_{2}-\ (l\cdot v_{1})\
S^{\mu\alpha}_{2}l_{\alpha}\\}+l^{\mu}\bigg{(}-\frac{i}{m_{1}}(l\wedge
v_{2})_{1}+\ \frac{(l\cdot S_{1}^{-\alpha})(l\cdot
S_{2\alpha}^{-})}{m_{1}m_{2}}\bigg{)}\bigg{]}\end{split}$ (A.76)

Here $(l\cdot v_{1})_{+}=l\cdot v_{1}+i\epsilon$. The $i\epsilon$ prescription
comes from requiring the initial condition that initially the particle is a
free particle that is, $z_{1}^{\mu}(\tau\rightarrow-\infty)\rightarrow 0$ as
discussed in [38]. So,

$\begin{split}&(z_{1}(0)\wedge
p_{1})^{\mu\nu}=iQ_{1}Q_{2}\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{2})\ e^{-il\cdot b_{1}}\ \frac{1}{l^{2}}\frac{1}{(l\cdot v_{1})^{2}_{+}}\\\
&\bigg{[}(v_{1}\wedge v_{2})^{\mu\nu}(l\cdot v_{1})-(v_{1}\wedge
l)^{\mu\nu}(v_{1}\cdot v_{2})+\frac{i}{m_{2}}\\{(v_{1}\wedge l)^{\mu\nu}\
(v_{1}\wedge l)_{2}-(l\cdot v_{1})\ (v_{1}\wedge
S^{-\alpha}_{2})^{\mu\nu}l_{\alpha}\\}\\\ &\hskip 184.9429pt-(v_{1}\wedge
l)^{\mu\nu}\bigg{(}-\frac{i}{m_{1}}(l\wedge v_{2})_{1}+\ \frac{(l\cdot
S_{1}^{-\alpha})(l\cdot S_{2\alpha}^{-})}{m_{1}m_{2}}\bigg{)}\bigg{]}\\\
&=iQ_{1}Q_{2}\bigg{(}v_{1}\wedge\frac{\partial}{\partial
v_{1}}\bigg{)}^{\mu\nu}\ \int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{2})\ e^{-il\cdot b_{1}}\ \frac{1}{l^{2}}\frac{1}{(l\cdot v_{1})_{+}}\\\
&\hskip 184.9429pt\bigg{(}v_{1}\cdot v_{2}-\frac{i}{m_{2}}(v_{1}\wedge
l)_{2}-\frac{i}{m_{1}}(l\wedge v_{2})_{1}+\frac{(l\cdot
S_{1}^{-\sigma})(l\cdot S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\end{split}$
(A.77)

We shall now do the partially do the integrals involved in the above
expression to show that this matches with the expression in eq.(2.44). There
are four types of integrals we need to study. The first of which is

$I_{1}=\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot v_{2})\ e^{-il\cdot
b_{1}}\ \frac{1}{(l\cdot v_{1})_{+}}\ \frac{1}{l^{2}}$ (A.78)

We choose the frame in which the first particle travels in the $z$\- direction
and the second particle is at rest,

$v_{2}^{\mu}=(1,\vec{0})\ \ \ \ \ \,v_{1}^{\mu}=\gamma(1,0,0,\beta).$ (A.79)

In this frame we can do the $l^{0}$ integral and we get

$I_{1}=\frac{1}{\gamma\beta}\int\frac{d^{2}l}{(2\pi)^{2}}\
e^{i\vec{l}_{\perp}\cdot\vec{b}_{1}}\int\frac{dl}{(2\pi)}\
\frac{1}{\vec{l}^{2}_{\perp}+\ l^{2}}\frac{1}{l-i\epsilon}$ (A.80)

where $l^{3}=l$. Since the impact parameter is perpendicular to both initial
velocities, it comes out of the integral. The above integral can be done using
a contour which closes in the lower half plane and we get

$I_{1}=-\frac{i}{\gamma\beta}\int\frac{d^{2}l_{\perp}}{(2\pi)^{2}}\
e^{i\vec{l}_{\perp}\cdot\vec{b}_{1}}\ \frac{1}{l_{\perp}^{2}}$ (A.81)

Now it can be shown quite easily by going to the frame in eq.(A.79) that

$I_{1}=i\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{1})\hat{\delta}(l\cdot v_{2})\ e^{-il\cdot b_{1}}\ \frac{1}{l^{2}}$ (A.82)

The second integral we have is

$I_{2}=-\frac{i}{m_{2}}\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{2})\ e^{-il\cdot b_{1}}\ \frac{1}{{l}^{2}}\frac{(v_{1}\wedge
l)_{2}}{(l\cdot v_{1})_{+}}$ (A.83)

In the frame we have chosen, we have $S_{2}^{\mu 0}=0$ coming from the SSC of
the second particle. Since, the first particle travels in the $z$-direction,
this means that $(v_{1}\wedge l)_{2}$ only depends on the $(x,y)$ coordinates
and thus by doing the $l^{0}$ and $l^{3}$ integrals as done in the previous
integral we get

$I_{2}=\frac{1}{m_{2}}\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{1})\hat{\delta}(l\cdot v_{2})\ e^{-il\cdot b_{1}}\ \frac{(v_{1}\wedge
l)_{2}}{l^{2}}$ (A.84)

Similarly the further two integrals also can be evaluated by arguing that only
the $l$-integrals which depend on $l_{perp}$ give non-zero answers and we get

$I_{3}=-\frac{i}{m_{1}}\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{2})\ e^{-il\cdot b_{1}}\ \frac{1}{{l}^{2}}\frac{(l\wedge
v_{2})_{1}}{(l\cdot v_{1})_{+}}=\frac{1}{m_{1}}\int\frac{d^{4}l}{(2\pi)^{4}}\
\hat{\delta}(l\cdot v_{1})\hat{\delta}(l\cdot v_{2})\ e^{-il\cdot b_{1}}\
\frac{(l\wedge v_{2})_{1}}{l^{2}}$ (A.85)

and finally

$I_{4}=\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot v_{2})\ e^{-il\cdot
b_{1}}\ \frac{1}{{l}^{2}}\frac{(l\cdot S_{1}^{-\alpha})(l\cdot
S_{2\alpha}^{-})}{m_{1}m_{2}}\frac{1}{(l\cdot v_{1})_{+}}\\\
=i\int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot v_{1})\hat{\delta}(l\cdot
v_{2})\ e^{-il\cdot b_{1}}\ \frac{1}{{l}^{2}}\frac{(l\cdot
S_{1}^{-\alpha})(l\cdot S_{2\alpha}^{-})}{m_{1}m_{2}}$ (A.86)

So, taking account all these integrals it can readily be checked that

$\begin{split}&(z_{1}(0)\wedge
p_{1})^{\mu\nu}=-Q_{1}Q_{2}\bigg{(}v_{1}\wedge\frac{\partial}{\partial
v_{1}}\bigg{)}^{\mu\nu}\ \int\frac{d^{4}l}{(2\pi)^{4}}\ \hat{\delta}(l\cdot
v_{1})\hat{\delta}(l\cdot v_{2})\ e^{-il\cdot b_{1}}\ \frac{1}{l^{2}}\\\
&\hskip 184.9429pt\bigg{(}v_{1}\cdot v_{2}-\frac{i}{m_{2}}(v_{1}\wedge
l)_{2}-\frac{i}{m_{1}}(l\wedge v_{2})_{1}+\frac{(l\cdot
S_{1}^{-\sigma})(l\cdot S^{-}_{{2}\sigma})}{m_{1}m_{2}}\bigg{)}\end{split}$
(A.87)

Plugging this expression into eq.(2.42) we get eq.(2.44).

##  Appendix B Notations and Feynman rules

We shall define some of the notations in this appendix that we use in the
paper. Most of the expressions will involve the spin tensor and to simplify
our lives we use the following notations:

$\begin{split}&(a\wedge b)^{\mu\nu}=a^{\mu}b^{\nu}-a^{\nu}b^{\mu},\\\
&(a\wedge b)_{1}=a_{\alpha}S^{\alpha\beta}b_{\beta},\\\ &a\cdot
S^{-\rho}=a_{\alpha}S^{\alpha\rho},\\\ &(a\wedge S^{-\rho})^{\mu\nu}(b\cdot
S^{-}_{\ \rho})=(a^{\mu}S^{\nu\rho}-a^{\nu}S^{\mu\rho})\
b_{\alpha}S^{\alpha}_{\ \rho}\end{split}$ (B.88)

The various measures are denoted by

$\begin{split}&d\mu(1,2,q_{1},q_{2})=\prod_{i=1}^{2}\ d^{4}q_{i}\
\hat{\delta}(2p_{i}\cdot q_{i})\\\ &d\mu(1,2,q)=d^{4}q\
\hat{\delta}(2p_{1}\cdot q)\ \hat{\delta}(2p_{2}\cdot q)\\\
&d\hat{\mu}(1,2,q_{1},q_{2})=\prod_{i=1}^{2}\ d^{4}q_{i}\
\hat{\delta}(2p_{i}\cdot q_{i}+q_{i}^{2})\end{split}$ (B.89)

The Feynman rules for the Dirac fermion are the following

$\displaystyle\text{interaction vertex}\ :\ -ie\gamma^{\mu}$ (B.90)
$\displaystyle\text{propagator}\ :\ i\frac{(\not{p}+m)}{p^{2}-m^{2}}$ (B.91)

and the Clifford algebra is

$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}\ \ ,\ \ \ \ \ \ \\\
\hat{\Sigma}^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}].$ (B.92)

The photon propagator is

$\text{photon propagator}\ :\ i\frac{\eta^{\mu\nu}}{q^{2}}$ (B.93)

## References

  * [1] A. Laddha and A. Sen, “Gravity Waves from Soft Theorem in General Dimensions,” JHEP 1809, 105 (2018) doi:10.1007/JHEP09(2018)105 [arXiv:1801.07719 [hep-th]].
  * [2] A. Laddha and A. Sen, “A Classical Proof of the Classical Soft Graviton Theorem in D $>$ 4,” 10.1103/PhysRevD.101.084011 Phys. Rev. D 101, no. 10, 084011 (2019)
  * [3] B. Sahoo and A. Sen, “Classical and Quantum Results on Logarithmic Terms in the Soft Theorem in four Dimensions,” JHEP 1902, 086 (2019) doi:10.1007/JHEP02(2019)086 [arXiv:1808.03288 [hep-th]].
  * [4] B. Sahoo, “Classical Sub-subleading Soft Photon and Soft Graviton Theorems in Four Spacetime Dimensions,” JHEP 12, 070 (2020) doi:10.1007/JHEP12(2020)070 [arXiv:2008.04376 [hep-th]].
  * [5] A. P. Saha, B. Sahoo and A. Sen, “Proof of the Classical Soft Graviton Theorem in D=4” [arXiv:1912.06413 [hep-th]]
  * [6] D. Ghosh and B. Sahoo, “Spin-dependent gravitational tail memory in $D=4$,” Phys. Rev. D 105, no.2, 025024 (2022) doi:10.1103/PhysRevD.105.025024 [arXiv:2106.10741 [hep-th]].
  * [7] A. Manu, D. Ghosh, A. Laddha and P. V. Athira, “Soft radiation from scattering amplitudes revisited,” JHEP 05, 056 (2021) doi:10.1007/JHEP05(2021)056 [arXiv:2007.02077 [hep-th]].
  * [8] D. A. Kosower, B. Maybee, D. O’Connell “Amplitudes, Observables, and Classical Scattering” doi:10.1007/JHEP02(2019)137
  * [9] Y. F. Bautista and A. Laddha, “Soft Constraints on KMOC Formalism,” [arXiv:2111.11642 [hep-th]].
  * [10] B. Maybee, D. O’Connell and J. Vines, “Observables and amplitudes for spinning particles and black holes,” JHEP 12, 156 (2019) doi:10.1007/JHEP12(2019)156 [arXiv:1906.09260 [hep-th]].
  * [11] L. de la Cruz, B. Maybee, D. O’Connell and A. Ross, “Classical Yang-Mills observables from amplitudes,” JHEP 12, 076 (2020) doi:10.1007/JHEP12(2020)076 [arXiv:2009.03842 [hep-th]].
  * [12] R. Aoude and A. Ochirov, “Classical observables from coherent-spin amplitudes,” JHEP 10, 008 (2021) doi:10.1007/JHEP10(2021)008 [arXiv:2108.01649 [hep-th]].
  * [13] A. Cristofoli, R. Gonzo, D. A. Kosower and D. O’Connell, “Waveforms from Amplitudes,” [arXiv:2107.10193 [hep-th]].
  * [14] G. Kalin, R. Porto, “From Boundary Data to Bound States”,
doi:10.1007/JHEP01(2020)072

  * [15] G. Kalin, R. Porto, “From Boundary data to Bound states II : Scattering Angles to Dynamical Invariants (with a twist)” doi:10.1007/JHEP02(2020)120
  * [16] G. Cho, G. Kälin and R. A. Porto, “From boundary data to bound states. Part III. Radiative effects,” JHEP 04, 154 (2022) [erratum: JHEP 07, 002 (2022)] doi:10.1007/JHEP04(2022)154 [arXiv:2112.03976 [hep-th]].
  * [17] N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Plante and P. Vanhove “Light-like Scattering in Quantum Gravity” doi:10.1007/JHEP11(2016)117
  * [18] N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Planté and P. Vanhove “General Relativity from Scattering Amplitudes” doi:10.1103/PhysRevLett.121.171601
  * [19] C. Cheung, I. Rothstein, M. Solon, “From Scattering Amplitudes to Classical Potentials in Post Minkowskian Expansion”, doi:10.1103/PhysRevLett.121.251101
  * [20] Ming-Zhi. Chung, Yu-tin. Huang, Jung-Wook. Kim and S. Lee “The simplest massive S-matrix: from minimal coupling to Black Holes” doi:10.1007/JHEP04(2019)156
  * [21] A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove “On Post-Minkowskian Hamiltonians in General Relativity” doi:10.1103/PhysRevD.100.084040
  * [22] A. Guevara, A. Ochirov and J. Vines, “Scattering of Spinning Black Holes from Exponentiated Soft Factors,” arXiv:1812.06895 [hep-th].
  * [23] Z. Bern, C. Cheung, R. Roiban, Chia-Hsien. Shen, M. P. Solon and M. Zeng “Black Hole Binary Dynamics from the Double Copy and Effective Theory” doi:10.1007/JHEP10(2019)206
  * [24] Ming-Zhi. Chung, Yu-tin. Huang, Jung-Wook. Kim “Classical potential for general spinning bodies” [arXiv:1908.08463 [hep-th, gr-qc]]
  * [25] Z. Bern, H. Ita, J. Parra-Martinez and M. S. Ruf “Universality in the classical limit of massless gravitational scattering” [arXiv :2002.02459 [hep-th]]
  * [26] C. Cheung, M. Solon, “Classical Gravitational Scattering at O($G^{3}$) from Feynman Diagrams” [arXiv:2003.08351 [hep-th]]
  * [27] M. Levi, A. Mcleod and M. V. Hippel, “NNNLO gravitational quadratic-in-spin interactions at the quartic order in G” [arXiv:2003.07890 [hep-th]]
  * [28] Ming-Zhi. Chung, Yu-tin. Huang, Jung-Wook. Kim and Sangmin. Lee “Complete Hamiltonian for spinning binary systems at first post-Minkowskian order” doi:10.1007/JHEP05(2020)105
  * [29] Z. Bern, J. Parra-Martinez, R. Roiban, E. Sawyer and C. H. Shen, “Leading Nonlinear Tidal Effects and Scattering Amplitudes,” JHEP 05, 188 (2021) doi:10.1007/JHEP05(2021)188 [arXiv:2010.08559 [hep-th]].
  * [30] Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C. H. Shen, M. P. Solon and M. Zeng, “Scattering Amplitudes and Conservative Binary Dynamics at ${\cal O}(G^{4})$,” Phys. Rev. Lett. 126, no.17, 171601 (2021) doi:10.1103/PhysRevLett.126.171601 [arXiv:2101.07254 [hep-th]].
  * [31] Z. Bern, J. P. Gatica, E. Herrmann, A. Luna and M. Zeng, “Scalar QED as a toy model for higher-order effects in classical gravitational scattering,” JHEP 08, 131 (2022) doi:10.1007/JHEP08(2022)131 [arXiv:2112.12243 [hep-th]].
  * [32] Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban and F. Teng, “Binary Dynamics Through the Fifth Power of Spin at $\mathcal{O}(G^{2})$,” [arXiv:2203.06202 [hep-th]].
  * [33] Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, “The Duality Between Color and Kinematics and its Applications,” [arXiv:1909.01358 [hep-th]].
  * [34] R. A. Porto, “Post-Newtonian corrections to the motion of spinning bodies in NRGR,” Phys. Rev. D 73, 104031 (2006) doi:10.1103/PhysRevD.73.104031 [arXiv:gr-qc/0511061 [gr-qc]].
  * [35] M. Levi and J. Steinhoff, “Spinning gravitating objects in the effective field theory in the post-Newtonian scheme,” JHEP 09, 219 (2015) doi:10.1007/JHEP09(2015)219 [arXiv:1501.04956 [gr-qc]].
  * [36] J. Vines, “Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings,” Class. Quant. Grav. 35, no.8, 084002 (2018) doi:10.1088/1361-6382/aaa3a8 [arXiv:1709.06016 [gr-qc]].
  * [37] D. Lynden-Bell, “A magic electromagnetic field,” [arXiv:astro-ph/0207064 [astro-ph]].
  * [38] W. D. Goldberger and A. K. Ridgway, “Radiation and the classical double copy for color charges,” Phys. Rev. D 95, no.12, 125010 (2017) doi:10.1103/PhysRevD.95.125010 [arXiv:1611.03493 [hep-th]].
  * [39] W. D. Goldberger, J. Li and S. G. Prabhu, “Spinning particles, axion radiation, and the classical double copy,” Phys. Rev. D 97, no.10, 105018 (2018) doi:10.1103/PhysRevD.97.105018 [arXiv:1712.09250 [hep-th]].
  * [40] Z. Bern, A. Luna, R. Roiban, C. H. Shen and M. Zeng, “Spinning black hole binary dynamics, scattering amplitudes, and effective field theory,” Phys. Rev. D 104, no.6, 065014 (2021) doi:10.1103/PhysRevD.104.065014 [arXiv:2005.03071 [hep-th]].
  * [41] C. Dlapa, G. Kälin, Z. Liu and R. A. Porto, “Dynamics of binary systems to fourth Post-Minkowskian order from the effective field theory approach,” Phys. Lett. B 831, 137203 (2022) doi:10.1016/j.physletb.2022.137203 [arXiv:2106.08276 [hep-th]].
  * [42] N. Arkani-Hamed, Y. t. Huang and D. O’Connell, “Kerr black holes as elementary particles,” JHEP 01, 046 (2020) doi:10.1007/JHEP01(2020)046 [arXiv:1906.10100 [hep-th]].
  * [43] N. Arkani-Hamed, T. C. Huang and Y. t. Huang, “Scattering amplitudes for all masses and spins,” JHEP 11, 070 (2021) doi:10.1007/JHEP11(2021)070 [arXiv:1709.04891 [hep-th]].
  * [44] A. Falkowski and C. S. Machado, “Soft Matters, or the Recursions with Massive Spinors,” JHEP 05, 238 (2021) doi:10.1007/JHEP05(2021)238 [arXiv:2005.08981 [hep-th]].
  * [45] M. Chiodaroli, H. Johansson and P. Pichini, “Compton black-hole scattering for s $\leq$ 5/2,” JHEP 02, 156 (2022) doi:10.1007/JHEP02(2022)156 [arXiv:2107.14779 [hep-th]].
  * [46] R. Aoude, K. Haddad and A. Helset, “Searching for Kerr in the 2PM amplitude,” JHEP 07, 072 (2022) doi:10.1007/JHEP07(2022)072 [arXiv:2203.06197 [hep-th]].
  * [47] R. Aoude, K. Haddad and A. Helset, “Classical gravitational spinning-spinless scattering at $\mathcal{O}(G^{2}S^{\infty})$,” [arXiv:2205.02809 [hep-th]].
  * [48] G. Menezes and M. Sergola, “NLO deflections for spinning particles and Kerr black holes,” [arXiv:2205.11701 [hep-th]].