11institutetext: Centre for Research in String Theory, School of Physics and
Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS, United
Kingdom22institutetext: Department of Physics and Astronomy, Seoul National
University, Seoul 08826, Korea33institutetext: Department of Physics and
Research Institute of Basic Science, Kyung Hee University,
Seoul 02447, Korea

# Gravitational Dyonic Amplitude at One-Loop and its Inconsistency with the
Classical Impulse

Jung-Wook Kim 3 Myungbo Shim<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The recent proposal Huang:2019cja ; Emond:2020lwi  of implementing electric-
magnetic duality rotation at the level of perturbative scattering amplitudes
and its generalisation to gravitational context where usual gravitational mass
is rotated to the NUT parameter of the Taub-NUT spacetime opens up an
interesting avenue for studying NUT-charged objects as dynamical entities, in
contrast to the usual approach where NUT-charged objects are considered as a
static background. We extend the tree-order analysis to one-loop order, and
find a disagreement between geodesic motion on Taub-NUT background and impulse
computation of scattering amplitudes. As a by-product of our analysis, we find
a relation between tidal response parameters and resonance excitation
parameters in the language of quantum field theory scattering amplitudes.

QMUL-PH-20-30

##  1 Introduction

During the expedition for a solution of Einstein’s equation having angular
momentum, Taub-NUT spacetime had been discovered independently by Taub and
Newman, Unti, and Tamburino Taub:1950ez ; Newman:1963yy . The solution has an
additional parameter mistaken for angular momentum at the time of discovery,
the so-called NUT charge, and the spacetime can be considered as a simple
generalisation of the Schwarzschild solution Misner:1963fr . Interestingly,
Taub-NUT solutions cannot be extrapolated or constructed from summing over
higher order corrections of post-Newtonian expansion or modified gravitational
theories Ramaswamy81 ; Ramaswamy:1986kf ; Kol:2020zth , but it is one of the
simplest, Hamilton-Jacobi soluble Carter:1968ks , and exact solutions of the
vacuum Einstein equations. Although the spacetime with angular momentum has
later been identified as the Kerr solution, the physical meaning and effects
of NUT charges has still been studied, and the NUT parameter has later been
identified as a source of locally rotating frame or gravitomagnetic force
which becomes the Coriolis force in (post)-Newtonian regime bonnor69 ;
Dowker74 , being the charge of dual supertranslation symmetry Huang:2019cja ;
Alawadhi:2019urr ; Kol:2019nkc . It was also found that angular momentum of
Kerr spacetime and NUT parameter of Taub-NUT spacetime can coexist, leading to
what is known as the Kerr-Taub-NUT spacetime osti_4444917 .

Studies on physical implications of NUT charges can be found in the
literature, especially in the context of astrophysical observations. The
investigations based on geodesic analysis are: Lensing and atomic spectra
LyndenBell:1996xj , gravitational lensing NouriZonoz:1998va ; Wei:2011nj ,
microlensing Rahvar:2002es ; Rahvar:2003fh , particle acceleration Shen:2003pi
; Liu:2010ja , circular geodesics Pradhan:2014zia , and electromagnetic
radiation from near horizon region in Kerr-Taub-NUT spacetime Long:2018tij . A
complete analysis of geodesics, singularities, and causal structures of Taub-
NUT spacetime has also been done recently Kagramanova:2010bk . Perfect fluids
that can source NUT charge have been studied in ref.GarciaReyes:2004qn . For a
possibility of actual astrophysical objects having NUT charges,
ref.Chakraborty:2017nfu  suggested the stellar-mass black hole-candidate X-ray
binary, GRO J1655–40, as a possible NUT-charged object. Since horizons exist
in Taub-NUT spacetime, black hole thermodynamics can be applied to it as well.
This has been studied in recently papers such as refs. Bordo:2019rhu ;
Durka:2019ajz ; Kubiznak:2019yiu ; Bordo:2020kxm ; Awad:2020dhy , although the
thermodynamics of their Euclidean cousins have been studied much earlier
Gibbons:1979xm .

In the aforementioned studies the NUT parameter was predominantly assigned to
the static background, and to best of authors’ knowledge any detailed study on
consequences of dynamical objects (in the sense that they can be accelerated
or decelerated by external influences) having NUT charges has not yet
appeared. This is where the recent proposals of understanding the NUT charges
as "electric-magnetic duality transformations" of graviton coupling in the
context of scattering amplitudes Huang:2019cja ; Emond:2020lwi  become
interesting, as the proposal can be used to study dynamical properties of NUT-
charged particles.

The works, refs.Huang:2019cja ; Emond:2020lwi ; Moynihan:2020gxj , have
appeared as attempts to reconcile electric-magnetic duality of gauge theory
Montonen:1977sn ; Seiberg:1994rs ; Seiberg:1994aj ; Olive:1995sw ;
Harvey:1996ur ; Olive:1997fg  with double copy relations of gauge theory and
gravity Kawai:1985xq ; Bern:2008qj ; Bern:2010ue ; Bern:2010yg ;
Monteiro:2014cda ; this is the same motivation behind related works,
refs.Luna:2015paa ; Alawadhi:2019urr ; Banerjee:2019saj ; Bahjat-Abbas:2020cyb
, but the latter is restricted to the solutions of classical field
equations111Consult refs. Ridgway:2015fdl ; Carrillo-Gonzalez:2017iyj ;
Goldberger:2017vcg ; Bahjat-Abbas:2017htu ; Lee:2018gxc ; Kim:2019jwm ;
delaCruz:2020bbn  and references therein for details on classical double
copy.. The works were based on the observation that a phase rotation to the
kinematic factor $x$ appearing in on-shell three-point kinematics of equal
mass-massless($m_{1}=m_{2}=m,m_{3}=0$) scattering, defined as the
proportionality constant Arkani-Hamed:2017jhn ,

$\displaystyle mx\lambda^{\alpha}$
$\displaystyle=\bar{\lambda}_{\dot{\alpha}}p_{1}^{\dot{\alpha}\alpha}\,,$ (1)

where $\lambda^{\alpha}$ and $\bar{\lambda}_{\dot{\alpha}}$ are spinor-
helicity variables of the massless leg, generates electric-magnetic duality
transformations on Maxwell’s theory.

From the perspectives of an on-shell photon, electric and magnetic charges are
distinguished by how they couple to distinct helicity modes of the photon;
electric charges couple with the same sign, while magnetic charges couple with
the opposite sign Weinberg:1965rz . Reconciling the sign choices with CPT
symmetry forces magnetic couplings to have imaginary coupling constants
Weinberg:1965rz , and the phase rotation to the kinematic factor $x$
introduced in ref.Huang:2019cja  can be understood as a phase rotation to the
coupling constant that generates magnetic couplings from electric couplings.
Pushing the analogy to gravitons, an ordinary mass would couple to both
helicity modes of the gravitons with the same sign while a "magnetic mass"
would couple with opposite signs. However, no known current could be found in
Weinberg’s analysis that would "magnetically" couple with the graviton and be
consistent with CPT symmetry Weinberg:1965rz , and for this reason the
possibility had been ruled out. Nevertheless, if such a coupling exists, it
would correspond to the NUT parameter of (Kerr-)Taub-NUT spacetime
Huang:2019cja ; Emond:2020lwi , at least to leading Newton’s constant order
($G^{1}$) in perturbation theory. The goal of present work is to examine this
relation at next-to-leading order ($G^{2}$) in perturbation theory.

The manuscript is structured as follows. The motion of a test particle on
Taub-NUT spacetime determined by general relativity will be reviewed in
section 2 to the second order in Newton’s constant, referred to as the second
post-Minkowskian (2PM) order in the literature. The _deflection angle_ 222The
terminology _scattering angle_ is also used in the literature, but the name
will be reserved for a related observable. This observable is free of gauge
ambiguities (general covariance) of general relativity. will be the main
observable that will be compared with quantum field theory computations. The
scattering amplitude computation of the equivalent perturbation order will be
performed in section 3. In section 3.4 we find that the widely-used
approximation tool, the _eikonal approximation_ , is not valid at one-loop
order when electric-magnetic duality transformation has been performed.
Nonetheless, the imaginary part of the impulse computations cancel and a
sensible deflection angle can be obtained. Based on a naïve expectation for
the Compton amplitude, we obtain the deflection angle in section 3.5 and find
a disagreement with the predictions of general relativity. Possible
resolutions for closing the gap are considered in sections 3.5.2 and 3.5.3,
which turn out to be insufficient for restoring coherence. As a by-product of
our analysis, we obtain a relation between tidal response parameters and
resonance excitation parameters, the equations (109) and (110). We conclude
our study in section 4.

##  2 The scattering angle from GR geodesic motion

The Taub-NUT metric is given as

$\displaystyle ds^{2}=-$ $\displaystyle
f(r;M,N)(dt+2N\cos{\theta}d\varphi)^{2}+\frac{dr^{2}}{f(r;M,N)}+(r^{2}+N^{2})d\Omega_{S^{2}}^{2},$
(2) $\displaystyle f(r;M,N)\equiv\frac{r^{2}-2Mr-N^{2}}{r^{2}+N^{2}},\qquad
r_{h}=M+\sqrt{M^{2}+N^{2}},$ (3)

where $M$, $N$, and $r_{h}$ are ADM mass, NUT parameter(charge), and the
location of the outer horizon, respectively. The Cauchy horizon lies outside
the domain $r\in[0,\infty)$ and therefore can be ignored for our purposes.
Regularity of spacetime requires periodicity of $2\pi/N$ for the time-like $t$
coordinates, but the condition can be loosened when dual supertranslation
symmetry is promoted to a gauge symmetry Kol:2020ucd . The Euclidean version
of the solutions known as _Euclidean Taub-NUT_ and _Taub-Bolt_ solutions,
which appear in the context of gravitational instantons Gibbons:1979xm , are
obtained from the above metric by a Wick rotation, $t\to-i\tau$ and $N\to-iN$,
and their differences arise from allowed range for the radial coordinate $r$.

###  2.1 Killing fields, integrals of motion, and geodesics

Killing vector fields are the generators of the Taub-NUT spacetime isometry
group, which in the coordinate patch of (2) take the following form
Visinescu:1999zs .

$\displaystyle\begin{split}K^{(0)}=\partial_{t},\\\
K^{(3)}=\partial_{\varphi},\end{split}$
$\displaystyle\begin{split}\left(\begin{array}[]{c}K^{(1)}\\\
K^{(2)}\end{array}\right)=\left(\begin{array}[]{c}\cos{\varphi}\\\
\sin{\varphi}\end{array}\right)\left[2N\frac{1}{\sin{\theta}}\partial_{t}-\cot{\theta}\partial_{\varphi}\right]+\partial_{\varphi}\left(\begin{array}[]{c}\cos{\varphi}\\\
\sin{\varphi}\end{array}\right)\partial_{\theta}\,.\end{split}$ (4)

Taking the inner product with the tangent vector $\partial_{\lambda}$ gives
the integrals of motion for the geodesics.

$\displaystyle{\mathcal{E}}$
$\displaystyle=-g(\partial_{\lambda},K^{(0)})=f\frac{dt}{d\lambda}+2Nf\cos{\theta}\frac{d\varphi}{d\lambda},$
$\displaystyle J_{\varphi}$
$\displaystyle=g(\partial_{\lambda},K^{(3)})=-2N{\mathcal{E}}\cos{\theta}+(r^{2}+N^{2})\sin^{2}{\theta}\dot{\varphi},$
(5) $\displaystyle C$
$\displaystyle=\left(\begin{array}[]{c}g(\partial_{\lambda},K^{(1)})\\\
g(\partial_{\lambda},K^{(2)})\end{array}\right)=\left[-\frac{2N{\mathcal{E}}+J_{\varphi}\cos{\theta}}{\sin{\theta}}+(r^{2}+N^{2})\frac{d\theta}{d\lambda}\partial_{\varphi}\right]\left(\begin{array}[]{c}\cos{\varphi}\\\
\sin{\varphi}\end{array}\right).$ (10)

The relations can be inverted to find the tangent vector,

$\displaystyle\dot{\varphi}=\frac{2N{\mathcal{E}}\cos{\theta}+J_{\varphi}}{(r^{2}+N^{2})\sin^{2}{\theta}},$
$\displaystyle\dot{t}=\frac{{\mathcal{E}}}{f}-2N\cos{\theta}\dot{\varphi},$
$\displaystyle\dot{\theta}^{2}=\frac{{\mathcal{C}}^{2}\sin^{2}{\theta}-(2N{\mathcal{E}}+J_{\varphi}\cos{\theta})^{2}}{(r^{2}+N^{2})^{2}\sin^{2}{\theta}},$
(11)

where ${\mathcal{C}}^{2}=C^{2}_{1}+C^{2}_{2}$. The geodesic equation for a
particle of unit mass then reduces to

$\displaystyle{\mathcal{E}}\frac{dt}{d\lambda}-J_{\varphi}\frac{d\varphi}{d\lambda}-(r^{2}+N^{2})(\frac{d\theta}{d\lambda})^{2}-\frac{1}{f}(\frac{dr}{d\lambda})^{2}=1,$
(12)

which can be further reduced to the one-dimensional radial equation,

$\displaystyle\left(\frac{dr}{d\lambda}\right)^{2}={\mathcal{E}}^{2}-V_{\textrm{eff}},$
$\displaystyle
V_{\textrm{eff}}=\frac{r^{2}-2Mr-N^{2}}{r^{2}+N^{2}}\left(\frac{{\mathcal{C}}^{2}+J_{\varphi}^{2}-4N^{2}{\mathcal{E}}^{2}}{(r^{2}+N^{2})}+1\right).$
(13)

The parameters can be chosen without loss of generality such that the geodesic
lies on an _orbital cone_ 333The conditions ${\mathcal{C}}>0$ and
${\mathcal{C}}=0$ correspond to linear and spiral scattering, respectively
Kagramanova:2010bk . of $\theta=\theta_{0}$ determined as

$\displaystyle\cos{\theta_{0}}=-\frac{2N{\mathcal{E}}}{J_{\varphi}}.$ (14)

Unfortunately this coordinate system is inconvenient for the Hamilton-Jacobi
analysis as the method of separation of variables cannot be applied. Doing a
gauge transformation allows application of the method for geodesics lying on
the orbital cone $\theta=\theta_{0}$.

###  2.2 Gauge transformation and action-angle variables

The Hamilton-Jacobi analysis can be made more tractable by the gauge
transformation,

$\displaystyle t\rightarrow t_{{\rm oc}}=t+2N\cos{\theta_{0}}\varphi_{{\rm
oc}},$ $\displaystyle\varphi\rightarrow\varphi_{{\rm oc}}=\varphi.$ (15)

The metric (2) is then transformed to

$\displaystyle ds^{2}=-f(r;M,N)(dt_{{\rm
oc}}+2N(\cos{\theta}-\cos{\theta_{0}})d\varphi)^{2}+\frac{dr^{2}}{f(r;M,N)}+(r^{2}+N^{2})d\Omega_{S^{2}}^{2}\,,$
(16)

while the Killing fields (4) become

$\displaystyle\begin{split}K^{(0)}_{{\rm oc}}&=\partial_{t_{{\rm oc}}},\\\
K^{(3)}_{{\rm oc}}&=\partial_{\varphi_{{\rm oc}}},\end{split}$
$\displaystyle\begin{split}K_{{\rm
oc}}^{(1)}=2N\cos{\varphi}\frac{(1-\cos{\theta}\cos{\theta_{0}})}{\sin{\theta}}\partial_{t}-\cos{\varphi}\cot{\theta}\partial_{\varphi}-\sin{\varphi}\partial_{\theta},\\\
K_{{\rm
oc}}^{(2)}=2N\sin{\varphi}\frac{(1-\cos{\theta}\cos{\theta_{0}})}{\sin{\theta}}\partial_{t}-\sin{\varphi}\cot{\theta}\partial_{\varphi}+\cos{\varphi}\partial_{\theta}\,.\end{split}$
(17)

The old Killing fields (4) and the new Killing fields (17) are related by the
relations,

$\displaystyle\begin{split}K^{(0)}&=K_{{\rm oc}}^{(0)},\\\
K^{(3)}&=K^{(3)}_{{\rm oc}}+2N\cos{\theta_{0}}K^{(0)}_{{\rm oc}},\end{split}$
$\displaystyle\begin{split}K^{(1)}&=K_{{\rm oc}}^{(1)},\\\ K^{(2)}&=K_{{\rm
oc}}^{(2)}.\end{split}$ (18)

Integrals of motion from $K_{{\rm oc}}$ are given as

$\displaystyle\begin{split}{\mathcal{E}}&=f\dot{t}_{{\rm
oc}}+2Nf(\cos{\theta}-\cos{\theta_{0}})\dot{\varphi},\\\
J&=-2N(\cos{\theta}-\cos{\theta_{0}}){\mathcal{E}}+(r^{2}+N^{2})\sin^{2}{\theta}\dot{\varphi},\\\
C&=\left[-\frac{2NE(1-\cos{\theta}\cos{\theta_{0}})+J_{{\rm
oc}}\cos{\theta}}{\sin{\theta}}+(r^{2}+N^{2})\frac{d\theta}{d\lambda}\partial_{\varphi}\right]\left(\begin{array}[]{c}\cos{\varphi}\\\
\sin{\varphi}\end{array}\right).\end{split}$ (19)

The tangent vector components444The condition ${\mathcal{C}}=0$ needs to be
imposed for non-straight-line scattering motion. transform as

$\displaystyle\begin{split}\dot{\varphi}&=\frac{2N{\mathcal{E}}(\cos{\theta}-\cos{\theta_{0}})+J}{(r^{2}+N^{2})\sin^{2}{\theta}},\\\
\dot{t}_{{\rm
oc}}&=\frac{{\mathcal{E}}}{f}-2Nf(\cos{\theta}-\cos{\theta_{0}})\dot{\varphi},\end{split}$
$\displaystyle\dot{\theta}^{2}=-\left(\frac{2N{\mathcal{E}}(1-\cos{\theta}\cos{\theta_{0}})+J\cos{\theta}}{(r^{2}+N^{2})\sin{\theta}}\right)^{2},$
(20)

and the geodesic equation reduces to

$\displaystyle
f^{-1}{\mathcal{E}}^{2}-\frac{J^{2}}{(r^{2}+N^{2})\sin^{2}{\theta_{0}}}-f^{-1}\dot{r}^{2}=1.$
(21)

We remind the reader that the geodesic lies on the orbital cone,
$\theta=\theta_{0}$, and the metric can be effectively considered as three-
dimensional;

$\displaystyle ds^{2}$ $\displaystyle=-fdt_{{\rm
oc}}^{2}+\frac{dr^{2}}{f}+(r^{2}+N^{2})\sin^{2}{\theta_{0}}d\varphi^{2}.$ (22)

The radial equation (13) becomes

$\displaystyle\dot{r}^{2}={\mathcal{E}}^{2}-V_{{\rm eff}},$ $\displaystyle
V_{{\rm
eff}}=f\left(\frac{J^{2}}{(r^{2}+N^{2})\sin^{2}{\theta_{0}}}+1\right).$ (23)

The opening angle of the orbital cone in terms of the parameters of the
geodesic, (14), needs to be expressed using new integrals of motion (19). We
solve the pair of equations,

$\displaystyle\cos{\theta_{0}}=-\frac{2N{\mathcal{E}}}{J_{\varphi}},$
$\displaystyle J_{\varphi}=J-2N{\mathcal{E}}\cos{\theta_{0}},$ (24)

for $J_{\varphi}$ and obtain

$\displaystyle J_{\varphi}=\frac{J+\sqrt{J+16N^{2}{\mathcal{E}}^{2}}}{2}\,,$
(25)

with the sign choices $J_{\varphi}>0$, $J>0$, $N>0$, and $\cos{\theta_{0}}<0$.
The opening angle $\theta_{0}$ in terms of ${\mathcal{E}}$, $J$ and $N$ is
then given as

$\displaystyle\cos{\theta_{0}}=-\frac{4N{\mathcal{E}}}{J+\sqrt{J^{2}+16N^{2}{\mathcal{E}}^{2}}},$
$\displaystyle\sin^{2}{\theta_{0}}=\frac{2J}{\sqrt{J^{2}+16N^{2}{\mathcal{E}}^{2}}+J}.$
(26)

###  2.3 Hamilton-Jacobi analysis on the orbital cone

The action functional in terms of action-angle variables on the orbital cone
can be written using the effective three-dimensional metric.

$\displaystyle{\mathcal{S}}=-{\mathcal{E}}t_{{\rm oc}}+J\varphi+\int
P(r^{\prime})dr^{\prime}=-\mu\int d\lambda,$ (27)

where $\mu$ is the mass of the probe particle moving along the geodesic. The
relativistic Hamilton-Jacobi equation is

$\displaystyle
g^{\mu\nu}\partial_{\mu}\mathcal{S}\partial_{\nu}{\mathcal{S}}=-\mu^{2},$ (28)

which coincides with the geodesic equation on the orbital cone.

$\displaystyle-f^{-1}{\mathcal{E}}^{2}+\frac{J^{2}}{(r^{2}+N^{2})\sin^{2}{\theta_{0}}}+fP(r)^{2}=-\mu^{2}.$
(29)

Solving for the radial momentum $P(r)$ gives

$\displaystyle
P(r)=\pm\left[f(r;M,N)^{-2}{\mathcal{E}}^{2}-\frac{\mu^{2}}{f(r;M,N)}-\frac{J^{2}}{f(r;M,N)(r^{2}+N^{2})\sin^{2}{\theta_{0}}}\right]^{\frac{1}{2}}.$
(30)

Note that $f^{2}P^{2}={\mathcal{E}}^{2}-V_{{\rm eff}}$.

###  2.4 Scattering at second post-Minkowskian order

Since we are interested in effects up to second post-Minkowskian order (2PM)
or to second order in Newton’s constant $G$, the expansion parameter $G$ will
be restored to $GM$ and $GN$ in this section. Standard Hamilton-Jacobi
analysis can be applied to compute the angle variables. The change in the
azimuthal angle during scattering, $\Delta\varphi$, is

$\displaystyle\Delta\varphi=-2\int_{r_{\textrm{min}}}^{\infty}\frac{\partial
P(r)}{\partial J}dr=-2\int_{u_{\textrm{min}}}^{\infty}\frac{\partial
P(r(u))}{\partial J}du=2(\varphi(\infty)-\varphi(b)),$ (31)

where the $r_{\min}$ is the largest positive root of the equation
${\mathcal{E}}^{2}-V_{{\rm eff}}=0$. Expanding $r_{\min}$ to $G^{2}$ order,

$\displaystyle
r_{\min}=\frac{J}{\sqrt{\mathcal{E}^{2}-\mu^{2}}}+\frac{GM\mathcal{E}^{2}}{\mu^{2}-\mathcal{E}^{2}}+\frac{G^{2}\left(M^{2}\left(4\mu^{2}\mathcal{E}^{2}-3\mathcal{E}^{4}\right)+N^{2}\left(\mathcal{E}^{4}-\mu^{4}\right)\right)}{2J\left(\mathcal{E}^{2}-\mu^{2}\right)^{3/2}}+{\mathcal{O}}(G^{3}).$
(32)

When evaluating the integral (31) to $G^{2}$ order, simply expanding the
integrand and lower/upper limits to $G^{2}$ order and effecting the integral
mechanically will not give correct results. This is because the perturbations
of integral limits can ruin the structure of the integral. Reparametrising the
integration variable so that the integral limits are not perturbed will make
the perturbation integral well-behaved.

The change in azimuthal angle during scattering is then given as

$\displaystyle\Delta\varphi$
$\displaystyle=\pi+\frac{2M\left(2{\mathcal{E}}^{2}-\mu^{2}\right)}{\sqrt{{\mathcal{E}}^{2}-\mu^{2}}}\left(\frac{G}{J}\right)$
$\displaystyle\qquad-\frac{\pi\left(3\mu^{2}\left(M^{2}+N^{2}\right)+{\mathcal{E}}^{2}\left(N^{2}-15M^{2}\right)\right)}{4}\left(\frac{G}{J}\right)^{2}+O\left(G^{3}\right).$
(33)

Due to the presence of NUT parameter $N$, the geodesic motion lies on the
orbital cone of $\theta=\theta_{0}$ and $\Delta\varphi$ should not be
considered as the scattering angle. Instead, the _scattering angle_ $\Phi$ and
the _deflection angle_ $\vartheta$ are given as

$\displaystyle\cos{\Phi}=\cos^{2}{\theta_{0}}+\sin^{2}{\theta_{0}}\cos{\Delta\varphi},$
$\displaystyle\vartheta=\Phi-\pi.$ (34)

Solving the above equation to $G^{2}$ order yields the 2PM scattering angle

$\displaystyle\Phi$
$\displaystyle=\pi+\vartheta=\pi+2\sqrt{4{\mathcal{M}}_{{\rm
TN}}^{2}{\mathcal{E}}^{2}+\frac{M^{2}\mu^{4}}{{\mathcal{E}}^{2}-\mu^{2}}}\left(\frac{G}{J}\right)$
$\displaystyle\qquad+\pi\frac{M\left(2{\mathcal{E}}^{2}-\mu^{2}\right)\left(16M^{2}{\mathcal{E}}^{2}-{\mathcal{M}}_{{\rm
TN}}^{2}\left(3\mu^{2}+{\mathcal{E}}^{2}\right)\right)}{4\sqrt{\mu^{4}M^{2}+4{\mathcal{M}}_{{\rm
TN}}^{2}{\mathcal{E}}^{2}({\mathcal{E}}-\mu)({\mathcal{E}}+\mu)}}\left(\frac{G}{J}\right)^{2}+{\mathcal{O}}(G^{3}).$
(35)

where ${\mathcal{M}}_{{\rm TN}}^{2}=M^{2}+N^{2}$. In the Schwarzschild limit,
$N\to 0$, the orbital cone opening angle $\theta_{0}$ becomes $\pi/2$ and
geodesic motion becomes equatorial. In this limit $\Delta\varphi$ coincides
with $\Phi$.

$\displaystyle\Delta\varphi_{{\rm Sch}}$ $\displaystyle=\Phi_{{\rm
Sch}}=\pi+\frac{2M\left(2{\mathcal{E}}^{2}-\mu^{2}\right)}{\sqrt{{\mathcal{E}}^{2}-\mu^{2}}}\left(\frac{G}{J}\right)+\frac{3\pi
M^{2}\left(5{\mathcal{E}}^{2}-\mu^{2}\right)}{4}\left(\frac{G}{J}\right)^{2}+{\mathcal{O}}\left(G^{3}\right).$
(36)

The deflection angle $\vartheta$ is vectorial in this set-up, therefore we
define the _deflection angle vector_ $\vec{\vartheta}$ as $\hat{n}_{\rm
out}-\hat{n}_{\rm in}$. This is the main observable that will be compared with
scattering amplitude source-probe limit computations.

$\displaystyle\vec{\vartheta}=\left(\begin{array}[]{c}-\cos^{2}{\Delta\theta_{0}}\left(\cos{\vartheta_{\varphi}}-1\right)-2\sin^{2}{\Delta\theta_{0}}\\\
-\cos{\Delta\theta_{0}}\sin{\vartheta_{\varphi}}\\\
-2\sin{\Delta\theta_{0}}\cos{\Delta\theta_{0}}(1+\sin^{2}{\vartheta_{\varphi}})\end{array}\right),$
(40)

where $\Delta\theta_{0}=\theta_{0}-\pi/2$,
$\vartheta_{\varphi}=\Delta\varphi-\pi$. The deflection angle $\vartheta$ is
given as the size of the deflection angle vector
$\vartheta={\left|\vec{\vartheta}\right|}$. In amplitude computations, the
incoming test particle will be considered as directed along the $x$-axis,
which is _not_ the incoming direction of the geodesic analysis. Therefore we
rotate the reference frames to match the normalisations of amplitude
computations.

$\displaystyle\begin{split}\vec{\vartheta}&=\hat{n}^{\rm amp}_{\rm
out}-\hat{n}^{\rm amp}_{\rm
in}={\mathcal{R}}_{\Delta\theta_{0}}\vec{\vartheta}_{\rm GR},\\\
\vec{\vartheta}_{\rm GR}&=\hat{n}^{\rm GR}_{\rm out}-\hat{n}^{\rm GR}_{\rm
in},\end{split}$ $\displaystyle\begin{split}\hat{n}^{\rm amp}_{\rm
in}&={\mathcal{R}}_{\Delta\theta_{0}}\hat{n}^{\rm GR}_{\rm in}=-\hat{x},\\\
\hat{n}^{\rm amp}_{\rm out}&={\mathcal{R}}_{\Delta\theta_{0}}\hat{n}^{\rm
GR}_{\rm out}.\end{split}$ (41)

The vectors and matrices can be explicitly evaluated.

$\displaystyle{\mathcal{R}}_{\Delta\theta_{0}}=\left(\begin{array}[]{ccc}\cos{\Delta\theta_{0}}&0&\sin{\Delta\theta_{0}}\\\
0&0&0\\\ -\sin{\Delta\theta_{0}}&0&\cos{\Delta\theta_{0}}\end{array}\right),$
$\displaystyle\hat{n}^{\rm GR}_{\rm
in}=\left(\begin{array}[]{c}-\cos{\Delta\theta_{0}}\\\ 0\\\
\phantom{-}\sin{\Delta\theta_{0}}\end{array}\right),$
$\displaystyle\hat{n}^{\rm GR}_{\rm
out}=\left(\begin{array}[]{c}-\cos{\Delta\theta_{0}}\cos{\vartheta_{\varphi}}\\\
-\cos{\Delta\theta_{0}}\sin{\vartheta_{\varphi}}\\\
-\sin{\Delta\theta_{0}}\end{array}\right).$ (51)

The 2PM deflection angle vector in the transformed frame is then given as

$\displaystyle\vec{\vartheta}=\left(\begin{array}[]{c}0\\\
\frac{2M\left(\mu^{2}-2\mathcal{E}^{2}\right)}{\sqrt{\mathcal{E}^{2}-\mu^{2}}}\\\
-4N\mathcal{E}\end{array}\right)\left(\frac{G}{J}\right)+\left(\begin{array}[]{c}-4\left(4{\mathcal{M}}_{{\rm
TN}}^{2}\mathcal{E}^{2}-8M^{2}\mathcal{E}^{2}-\frac{M^{2}\mu^{4}}{\mathcal{E}^{2}-\mu^{2}}\right)\\\
\pi\frac{{\mathcal{M}}_{{\rm
TN}}^{2}\left(\mathcal{E}^{2}+3\mu^{2}\right)-16M^{2}\mathcal{E}^{2}}{4}\\\
0\end{array}\right)\left(\frac{G}{J}\right)^{2}+{\mathcal{O}}\left(G^{3}\right)\,.$
(58)

The Schwarzschild limit can be recovered by the limit $N\to 0$. The first
component of the vector is directed along the direction of the incoming test
particle, and is irrelevant for the perturbation order we are interested in.
Note that while electric ($M$) and magnetic ($N$) masses deflect the particle
in orthogonal directions at 1PM order ($G^{1}$), the deflections are aligned
at 2PM order($G^{2}$).

##  3 The classical deflection angle from QFT amplitudes

###  3.1 Momentum parametrisation and the holomorphic classical limit

A generic one-loop amplitude can be expanded on a basis of integrals known as
_scalar integrals_. Unitarity methods Bern:1994zx ; Bern:1994cg  aim to
compute the coefficients of the basis scalar integrals by examining analytic
structures of the loop integral, often building up the integrand from lower
perturbation order on-shell amplitudes. We resort to unitarity methods for the
analysis of one-loop amplitude, as the analysis of refs.Huang:2019cja ;
Emond:2020lwi  was based on on-shell tree amplitudes whose continuation to
off-shell kinematics555while retaining Lorentz invariance and locality is
unclear; the textbook method of constructing the loop integrand from Feynman
rules requires off-shell data.

Classical physics is dictated by long-distance physics, or physics at large
impact parameters. Since impact parameter $b^{\mu}$ is the Fourier dual of
transferred momentum $q^{\mu}$ on the impact parameter space, the expansion in
large impact parameter can be traded for an expansion in small transferred
momentum. Dimensional analysis in Planck’s constant $\hbar$ shows that only
the scalar box integrals and the scalar triangle integrals have the relevant
dimensions to contribute to classical physics Holstein:2004dn ; Neill:2013wsa
; Guevara:2017csg ; Bern:2019crd .

A computationally efficient means of probing the leading contributions is the
_holomorphic classical limit_ (HCL) Guevara:2017csg , where transfer momentum
$q^{\mu}$ is complexified to a non-zero complex null vector. The subdominant
$q^{2}$ contributions drop out in the computations as $q^{\mu}$ is a null
vector, but as $q^{\mu}$ is non-zero certain information such as spin
dependence can be reconstructed in this limit Chung:2018kqs ; Chung:2019duq ;
Chung:2019yfs ; Chung:2020rrz ; Guevara:2018wpp ; Guevara:2019fsj ;
Moynihan:2019bor . Cut momenta parametrisation will be given in this section
for later reference; consult ref.Guevara:2017csg  for derivation of the cut
momenta parametrisations.

The parametrisation for external momenta is given as

$\displaystyle\begin{aligned}
p_{1}&=(E_{a},\vec{p}+\vec{q}/2)=|{\hat{\eta}}]\langle{\hat{\lambda}}|+|{\hat{\lambda}}]\langle{\hat{\eta}}|\,,\\\
p_{1}^{\prime}&=(E_{a},\vec{p}-\vec{q}/2)=\beta^{\prime}|{\hat{\eta}}]\langle{\hat{\lambda}}|+\frac{1}{\beta^{\prime}}|{\hat{\lambda}}]\langle{\hat{\eta}}|+|{\hat{\lambda}}]\langle{\hat{\lambda}}|\,,\\\
p_{2}&=(E_{b},-\vec{p}-\vec{q}/2)=|{\eta}]\langle{\lambda}|+|{\lambda}]\langle{\eta}|\,,\\\
p_{2}^{\prime}&=(E_{b},-\vec{p}+\vec{q}/2)=\beta|{\eta}]\langle{\lambda}|+\frac{1}{\beta}|{\lambda}]\langle{\eta}|+|{\lambda}]\langle{\lambda}|\,,\\\
q&=p_{1}-p_{1}^{\prime}=(0,\vec{q})\,,\end{aligned}$ (59)

where spinor brackets are normalised by the conditions666$m_{a}$ and $m_{b}$
are inertial masses. The notation is summarised in Table 1.,
$\langle\hat{\lambda}\hat{\eta}\rangle=[\hat{\lambda}\hat{\eta}]=m_{a}$ and
$\langle\lambda\eta\rangle=[\lambda\eta]=m_{b}$. The limit $\beta\to 1$
corresponds to the HCL, and following definitions for the variables will be
used.

Inertial Mass | ${\mathcal{M}}^{2}=M^{2}+N^{2}$ | $m_{a}\to{\mathcal{M}}_{a}$ | $m_{b}\to{\mathcal{M}}_{b}$
---|---|---|---
Electric Mass | $M$ | $M_{a}$ | $M_{b}$
Magnetic Mass | $N$ | $N_{a}$ | $N_{b}$
Table 1: Gravitational charges

$\displaystyle\begin{aligned}
u=[{\lambda}|p_{1}|{\eta}\rangle\,,\,v=[{\eta}|p_{1}|{\lambda}\rangle\,,\,\sigma=\frac{p_{1}\cdot
p_{2}}{m_{a}m_{b}}\,.\end{aligned}$ (60)

The momentum $p_{1}$ can be expanded in terms of unhatted spinors using above
variables777The expansion can be derived using eq.(6.10) of ref.Chung:2018kqs
..

$\displaystyle\begin{aligned}
p_{1}&=|{\eta}]\frac{u}{m_{b}^{2}}\langle{\lambda}|+|{\lambda}]\frac{v}{m_{b}^{2}}\langle{\eta}|+\frac{|{\lambda}]\beta(uv-
m_{a}^{2}m_{b}^{2})\langle{\lambda}|}{A{m_{b}^{2}}}+\frac{|{\eta}]A\langle{\eta}|}{\beta{m_{b}^{2}}}\,,\\\
A&=(\beta-1)(\beta v-u+(\beta-1)m_{b}^{2})\,.\end{aligned}$ (61)

Figure 1: The $b$-topology triangle cut with cut conditions
$k_{3}^{2}=k_{4}^{2}=L^{2}-m_{b}^{2}=0$. The $a$-topology cut is defined as
the mirror topology obtained by exchanging $m_{a}\leftrightarrow m_{b}$.

For the $b$-topology triangle cut, figure 1, the cut momenta are given as

$\displaystyle\begin{aligned}
|{k_{3}}\rangle=\frac{1}{\beta+1}\left((\beta^{2}-1)|{\eta}\rangle-\frac{1+\beta
y}{y}|{\lambda}\rangle\right)&\,,\,|{k_{3}}]=\frac{1}{\beta+1}\left((\beta^{2}-1)y|{\eta}]+(1+\beta
y)|{\lambda}]\right)\,,\\\
|{k_{4}}\rangle=\frac{1}{\beta+1}\left(\frac{\beta^{2}-1}{\beta}|{\eta}\rangle+\frac{1-y}{y}|{\lambda}\rangle\right)&\,,\,|{k_{4}}]=\frac{1}{\beta+1}\left(-\beta(\beta^{2}-1)y|{\eta}]+(1-\beta^{2}y)|{\lambda}]\right)\,.\end{aligned}$
(62)

The $b$-topology contribution to the amplitude can be computed as

$\displaystyle\begin{aligned}
i{\mathcal{A}}^{\bigtriangleup^{(b)}}&=-I_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}\left[\sum_{h_{3},h_{4}}{\mathcal{A}}_{4}\times{\mathcal{A}}_{3}\times{\mathcal{A}}_{3}\right](y)\,,\\\
I_{\bigtriangleup^{(b)}}&=\frac{-i}{32m_{b}\sqrt{-q^{2}}}+\cdots\,,\end{aligned}$
(63)

where integration contour $\Gamma$ for $y$ encircles $y=\infty$, which can be
traded for reading off $y^{0}$ coefficient of the expression’s Laurent
expansion at $y=\infty$ Forde:2007mi . This scalar triangle integral
contribution captures the ${\left|\vec{q}\right|}^{-1}\sim r^{-2}$ dependence
of the classical Hamiltonian in position space and
${\left|\vec{q}\right|}^{-1}\sim b^{-1}$ dependence of the eikonal phase in
impact parameter space.

###  3.2 The impulse formula

Define _pre-eikonal phase_ as

$\displaystyle\begin{aligned}
\chi[{\mathcal{A}}](b,p_{1},p_{2})&:=\int\hat{d}^{4}q\hat{\delta}(2p_{1}\cdot
q+q^{2})\hat{\delta}(2p_{2}\cdot
q-q^{2})\Theta(p_{1}^{0}+q^{0})\Theta(p_{2}^{0}-q^{0})\\\
&\phantom{asdfasdfasdf}\times e^{-ib\cdot q}{\mathcal{A}}(p_{1},p_{2}\to
p_{1}+q,p_{2}-q)\,,\end{aligned}$ (64)

where

$\displaystyle\hat{d}^{n}p:=\frac{d^{n}p}{(2\pi)^{n}}\,,\,\hat{\delta}(x):=2\pi\delta(x)\,,$
(65)

are the normalised measure and delta constraint in momentums space. While the
above integral kernel is widely used to compute eikonal amplitudes, the
terminology pre-eikonal phase has been adopted because diagrams that do not
contribute to the eikonal phase will also be effected by this kernel. In the
COM frame with small momentum mismatch $q^{\mu}\sim w_{1}^{\mu}\sim 0$, the
delta constraints can be simplified as

$\displaystyle\chi[{\mathcal{A}}](b,p_{1},p_{2})$
$\displaystyle\stackrel{{\scriptstyle q^{2}\to
0}}{{\simeq}}\frac{1}{4{\mathcal{E}}_{real}{\left|\vec{p}_{{\rm
com}}\right|}}\int\hat{d}^{2}q\,e^{i\vec{b}\cdot\vec{q}}{\mathcal{A}}(p_{1},p_{2}\to
p_{1}+q,p_{2}-q)\,,$ (66)

where ${\mathcal{E}}_{real}=\sqrt{s}$ is the COM frame total energy and
$\vec{p}_{{\rm com}}$ is the spatial momentum in the COM frame. The above
integral is performed in the Euclidean 2-plane dual to the impact parameter
space $\vec{b}$. Note that

$\displaystyle{\left|\vec{p}_{{\rm com}}\right|}$
$\displaystyle=\frac{m_{a}m_{b}\sqrt{\sigma^{2}-1}}{\sqrt{s}}\,,$ (67)

therefore the normalisation factor for the deflection angle is given as
Guevara:2018wpp

$\displaystyle\frac{1}{4{\mathcal{E}}_{real}{\left|\vec{p}_{{\rm
com}}\right|}^{2}}$
$\displaystyle=\frac{{\mathcal{E}}_{real}}{(2m_{a}m_{b}\sqrt{\sigma^{2}-1})^{2}}\,.$
(68)

The impulse formula eq.(4.31) and eq.(4.32) of ref. Kosower:2018adc  can be
approximated to

$\displaystyle\Delta p_{1}^{\mu}$
$\displaystyle\simeq-\frac{\partial\chi[{\mathcal{A}}](b,p1,p2)}{\partial
b_{\mu}}+i\chi[{\mathcal{A}}^{\ast}](-b,p1,p2)\frac{\partial\chi[{\mathcal{A}}](b,p1,p2)}{\partial
b_{\mu}}\,,$ (69)

when production of extra particles is neglected, i.e. $X=\varnothing$, and the
full expression is assumed to be dominated by small momentum mismatch
$q^{\mu}\sim w_{1}^{\mu}\sim 0$. This approximation is valid up to $G^{2}$
order in the classical limit, and the impulse up to this order can be written
as;

$\displaystyle\Delta p_{1}^{\mu}$
$\displaystyle=-\frac{\partial\chi[{\mathcal{A}}^{(0)+(1)}](b)}{\partial
b_{\mu}}+i\chi[({\mathcal{A}}^{(0)})^{\ast}](-b)\frac{\partial\chi[{\mathcal{A}}^{(0)}](b)}{\partial
b_{\mu}}+{\mathcal{O}}(G^{3})\,,$ (70)

where momenta of hard particles have been suppressed and the superscript on
${\mathcal{A}}$ denotes loop-order of the amplitude; e.g.
${\mathcal{A}}^{(n)}$ is the $n$-loop amplitude. Scaling the impulse by
${\left|\vec{p}_{\text{com}}\right|}$ gives the deflection angle, which is
reliable up to $G^{2}$ order888The reliability relies on two facts; (1) extra
particle production can only contribute at $G^{3}$ or higher orders, and (2)
the small angle approximation $2\sin(\theta/2)\simeq\theta$ is exact up to
$\theta^{2}$ order, where leading scaling is $\theta\propto G$..

$\displaystyle\begin{aligned} \vec{\vartheta}&=\frac{\Delta
p_{1}^{\mu}}{{\left|\vec{p}_{\text{com}}\right|}}+{\mathcal{O}}(G^{3})\\\
&=\frac{{\mathcal{E}}_{real}}{m_{a}m_{b}\sqrt{\sigma^{2}-1}}\left[\frac{\partial\chi[{\mathcal{A}}^{(0)+(1)}](b)}{\partial\vec{b}}-i\chi[({\mathcal{A}}^{(0)})^{\ast}](-b)\frac{\partial\chi[{\mathcal{A}}^{(0)}](b)}{\partial\vec{b}}\right]+{\mathcal{O}}(G^{3})\,.\end{aligned}$
(71)

The master integral for (66) is

$\displaystyle\begin{aligned}
I_{n,m}(\vec{b}):=\int\hat{\delta}^{n}q{\left|q\right|}^{m}e^{i\vec{b}\cdot\vec{q}}&=\frac{2^{2-n}}{\pi^{\frac{n-3}{2}}\Gamma[\frac{n-1}{2}]{\left|b\right|}^{n+m}}\int_{0}^{\infty}\frac{dx}{2\pi}x^{n+m-1}J_{0}(x)\\\
&=\frac{2^{m}\Gamma[\frac{m+n}{2}]}{\pi^{\frac{n-1}{2}}\Gamma[\frac{2-m-n}{2}]\Gamma[\frac{n-1}{2}]{\left|b\right|}^{m+n}}\,,\end{aligned}$
(72)

where $n=2-2\epsilon$ is the dimensional regulator for UV/IR divergences. The
integral is convergent for $0<m+n<3/2$. Differentiating by $m$,

$\displaystyle\begin{aligned} \frac{\partial I_{n,m}(\vec{b})}{\partial
m}&=\int\hat{\delta}^{n}q{\left|q\right|}^{m}\log{\left|q\right|}e^{i\vec{b}\cdot\vec{q}}\\\
&=\frac{I_{n,m}(\vec{b})}{2}\left[-2\log{\left|b\right|}+\psi^{(0)}\left(\frac{2-m-n}{2}\right)+\psi^{(0)}\left(\frac{m+n}{2}\right)+\log(4)\right]\,.\end{aligned}$
(73)

Following results will be used in the computations.

$\displaystyle\left(\frac{4\mu^{2}}{\pi
e^{\gamma}}\right)^{\epsilon}I_{2-2\epsilon,-2}$
$\displaystyle=-\frac{1}{4\pi\epsilon}-\frac{\log{\left|b\mu\right|}}{2\pi}+O\left(\epsilon^{1}\right),$
(74) $\displaystyle I_{2,-1}$
$\displaystyle=\frac{1}{2\pi{\left|b\right|}}\,,$ (75)
$\displaystyle\left(\frac{4\mu^{2}}{\pi
e^{\gamma}}\right)^{\epsilon}\left.\frac{\partial I_{2-2\epsilon,m}}{\partial
m}\right|_{m=-2}$
$\displaystyle=-\frac{1}{8\pi\epsilon^{2}}+\frac{\gamma-\log(2\mu)}{4\pi\epsilon}$
$\displaystyle\phantom{=}+\frac{16(\gamma-\log(2))\log{\left|b\mu\right|}+8\log^{2}{\left|b\right|}-8\log^{2}(\mu)+\pi^{2}}{32\pi}+O\left(\epsilon^{1}\right).$
(76)

###  3.3 The impulse at tree order

This analysis has already been presented in ref.Huang:2019cja , which has been
included for completeness. The on-shell three-point amplitudes are given as
Arkani-Hamed:2017jhn

$\displaystyle{\mathcal{A}}_{\phi\bar{\phi}h}^{+}=\frac{\kappa
m^{2}(e^{i\varphi}x)^{2}}{2}\,,\,{\mathcal{A}}_{\phi\bar{\phi}h}^{-}=\frac{\kappa
m^{2}}{2(e^{i\varphi}x)^{2}}\,,$ (77)

where $\kappa=\sqrt{32\pi G}$, ${\mathcal{A}}^{+}$(${\mathcal{A}}^{-}$) is the
coupling to positive(negative) helicity graviton, and $x$ is the kinematic
factor carrying little group weights of the graviton.

$\displaystyle x$ $\displaystyle=\frac{[{3}|p_{1}|{\xi}\rangle}{m\langle
3\xi\rangle}=\sqrt{2}p_{1}\cdot\varepsilon^{(+)}(p_{3})\,,$ (78)

where $\varepsilon^{(+)}_{\mu}(p_{3})$ is the positive helicity polarisation
vector for a massless spin-1 particle with momentum $p_{3}$. The phase
rotation $\varphi$ generates magnetic coupling Huang:2019cja . The
$m_{a}m_{b}\to m_{a}m_{b}$ tree amplitude can be obtained by gluing the on-
shell three-point amplitudes, and when particle $a$ is subject to phase
rotation $\varphi$ the amplitude is given as;

$\displaystyle\begin{aligned} {\mathcal{A}}^{(0)}&\stackrel{{\scriptstyle
q^{2}\to 0}}{{\simeq}}\frac{8\pi
Gm_{a}^{2}m_{b}^{2}}{q^{2}}\left[\left(\frac{x_{1}}{x_{2}}\right)^{2}e^{2i\varphi}+\left(\frac{x_{2}}{x_{1}}\right)^{2}e^{-2i\varphi}\right]\\\
&=\frac{16\pi
Gm_{a}^{2}m_{b}^{2}}{q^{2}}\left[(2\sigma^{2}-1)\cos(2\varphi)+2i\sigma\sqrt{\sigma^{2}-1}\sin(2\varphi)\right]\,,\\\
\chi[{\mathcal{A}}^{(0)}]&\simeq\frac{2Gm_{a}m_{b}}{\sqrt{\sigma^{2}-1}}\left[(2\sigma^{2}-1)\cos(2\varphi)+2i\sigma\sqrt{\sigma^{2}-1}\sin(2\varphi)\right]\log{\left|b\right|}\,.\end{aligned}$
(79)

The sign difference comes from
$q^{2}=-\vec{q}^{2}\leftrightarrow\frac{\log{\left|b\mu\right|}}{2\pi}$. The
deflection angle from (pre-)eikonal phase $\chi$ is given as;

$\displaystyle\begin{aligned}
\vec{\vartheta}_{1}&=\frac{{\mathcal{E}}_{real}}{m_{a}m_{b}\sqrt{\sigma^{2}-1}}\frac{\partial\chi[{\mathcal{A}}^{(0)}](b)}{\partial\vec{b}}\\\
&=\frac{{\mathcal{E}}_{real}}{(2\sqrt{\sigma^{2}-1})^{2}}\frac{8G}{{\left|b\right|}}\left[-(2\sigma^{2}-1)\cos(2\varphi)\hat{y}+2\sigma\sqrt{\sigma^{2}-1}\sin(2\varphi)\hat{z}\right]\,,\end{aligned}$
(80)

where imaginary part of the impulse has been re-interpreted as
$i\sqrt{\sigma^{2}-1}q^{\mu}=\pm\epsilon^{\mu}(u_{1},u_{3},q)$, with sign
choice fixed by matching to the Newtonian limit Huang:2019cja . This result
can be matched to general relativity predictions (58) using the parameter
mappings of table 2, yielding

$\displaystyle\vec{\vartheta}^{\rm Amp}_{1}$
$\displaystyle=-\left[\frac{2M(2{\mathcal{E}}^{2}-\mu^{2})}{(\sqrt{{\mathcal{E}}^{2}-\mu^{2}})}\hat{y}+4{\mathcal{E}}N\hat{z}\right]\left(\frac{G}{J}\right)\,,$
(81)

consistent with (58) to this order.

Amplitude variables | ${\mathcal{E}}_{real}$ | $\sigma$ | $b\sqrt{\sigma^{2}-1}$ | $\cos{(2\varphi)}$ | $\sin{(2\varphi)}$
---|---|---|---|---|---
Geodesic variables | ${\mathcal{M}}_{{\rm TN}}$ | ${\mathcal{E}}/\mu$ | $J/\mu$ | $M/{\mathcal{M}}_{{\rm TN}}$ | $N/{\mathcal{M}}_{{\rm TN}}$
Table 2: Parameter mappings between amplitude variables and geodesic
variables. The mapping is based on the relations
${\mathcal{E}}_{real}={\mathcal{M}}_{{\rm
TN}}+{\mathcal{E}}\simeq{\mathcal{M}}_{{\rm TN}}$, $p_{1}\cdot
p_{2}\simeq{\mathcal{M}}_{{\rm TN}}{\mathcal{E}}$, and
$J=b{\left|\vec{p}_{{\rm com}}\right|}$.

###  3.4 The one-loop amplitude : Boxes and eikonal (non-)exponentiation

The box topology contribution to the one-loop amplitude in the HCL is;

$\displaystyle i{\mathcal{A}}^{\square}$ $\displaystyle\stackrel{{\scriptstyle
q^{2}\to
0}}{{\simeq}}64\pi^{2}G^{2}m_{a}^{4}m_{b}^{4}(16\sigma^{4}-16\sigma^{2}+2+e^{4i\varphi}+e^{-4i\varphi})I_{\square}\,,$
(82) $\displaystyle i{\mathcal{A}}^{\triangleright\hskip-1.0pt\triangleleft}$
$\displaystyle\stackrel{{\scriptstyle q^{2}\to
0}}{{\simeq}}64\pi^{2}G^{2}m_{a}^{4}m_{b}^{4}(16\sigma^{4}-16\sigma^{2}+2+e^{4i\varphi}+e^{-4i\varphi})I_{\triangleright\hskip-1.0pt\triangleleft}\,,$
(83) $\displaystyle I_{\square}$
$\displaystyle=\frac{-i}{8\pi^{2}}\frac{\log(-q^{2})}{q^{2}}\frac{1}{2m_{a}m_{b}\sqrt{\sigma^{2}-1}}\left[\log{\left|\frac{\sigma-1-\sqrt{\sigma^{2}-1}}{\sigma-1+\sqrt{\sigma^{2}-1}}\right|}+i\pi\right]\,,$
(84) $\displaystyle I_{\triangleright\hskip-1.0pt\triangleleft}$
$\displaystyle\stackrel{{\scriptstyle q^{2}\to
0}}{{\simeq}}\frac{-i}{8\pi^{2}}\frac{\log(-q^{2})}{q^{2}}\frac{1}{2m_{a}m_{b}\sqrt{\sigma^{2}-1}}\left[\log{\left|\frac{\sigma+1+\sqrt{\sigma^{2}-1}}{\sigma+1-\sqrt{\sigma^{2}-1}}\right|}\right]\,,$
(85)

where $\sigma=\frac{p_{1}\cdot p_{2}}{m_{a}m_{b}}>1$ has been applied to the
imaginary part of the scalar integrals. The scalar integral coefficients can
be obtained by imposing the fourth cut condition on the triple-cut solutions
(62). Adding up, the logarithmic terms in square brackets cancel and only the
imaginary part of the scalar integrals survive.

$\displaystyle\begin{aligned}
i{\mathcal{A}}^{\square+\triangleright\hskip-1.0pt\triangleleft}&\stackrel{{\scriptstyle
q^{2}\to 0}}{{\simeq}}\frac{4\pi
G^{2}m_{a}^{3}m_{b}^{3}(16\sigma^{4}-16\sigma^{2}+2+e^{4i\varphi}+e^{-4i\varphi})}{\sqrt{\sigma^{2}-1}}\frac{\log(-q^{2})}{q^{2}}\,.\end{aligned}$
(86)

The pre-eikonal phase is computed as;

$\displaystyle\chi[{\mathcal{A}}^{\square+\triangleright\hskip-1.0pt\triangleleft}]$
$\displaystyle\stackrel{{\scriptstyle q^{2}\to
0}}{{\simeq}}i\frac{2G^{2}m_{a}^{2}m_{b}^{2}(16\sigma^{4}-16\sigma^{2}+2+e^{4i\varphi}+e^{-4i\varphi})}{4(\sigma^{2}-1)}\log^{2}{\left|b\right|}+\cdots\,,$
(87)

where only the leading $b$ dependence has been kept. Setting the phase
rotation zero $\varphi=0$,

$\displaystyle\chi[{\mathcal{A}}^{\square+\triangleright\hskip-1.0pt\triangleleft}]$
$\displaystyle\stackrel{{\scriptstyle q^{2}\to
0}}{{\simeq}}\frac{i}{2}\left[\frac{2Gm_{a}m_{b}(2\sigma^{2}-1)}{\sqrt{\sigma^{2}-1}}\log{\left|b\right|}\right]^{2}+\cdots\,,$
(88) $\displaystyle\chi[{\mathcal{A}}^{(0)}]$
$\displaystyle\simeq\frac{2Gm_{a}m_{b}(2\sigma^{2}-1)}{\sqrt{\sigma^{2}-1}}\log{\left|b\right|}\,,$
(89)

which is consistent with exponentiation of the eikonal phase;
$\chi[{\mathcal{A}}^{\square+\triangleright\hskip-1.0pt\triangleleft}]=\frac{i}{2}\chi[{\mathcal{A}}^{(0)}]^{2}$.
This relation obviously breaks down when non-zero phase rotation $\varphi\neq
0$ is introduced, rendering eikonal approximation for dyonic amplitudes
questionable. However, _their impulse contributions cancel each other_ ,

$\displaystyle\frac{\partial\chi[{\mathcal{A}}^{\square+\triangleright\hskip-1.0pt\triangleleft}](b)}{\partial\vec{b}}-i\chi[({\mathcal{A}}^{(0)})^{\ast}](-b)\frac{\partial\chi[{\mathcal{A}}^{(0)}](b)}{\partial\vec{b}}$
$\displaystyle=0\,,$ (90)

since

$\displaystyle\begin{aligned}
(16\sigma^{4}-16\sigma^{2}+2+e^{4i\varphi}+e^{-4i\varphi})-4{\left|(2\sigma^{2}-1)\cos(2\varphi)+2i\sigma\sqrt{\sigma^{2}-1}\sin(2\varphi)\right|}^{2}&=0\,.\end{aligned}$

Therefore,

$\displaystyle\begin{aligned}
\vec{\vartheta}_{2}&=\frac{{\mathcal{E}}_{real}}{m_{a}m_{b}\sqrt{\sigma^{2}-1}}\left[\frac{\partial\chi[{\mathcal{A}}^{\bigtriangleup+\bigtriangledown+\square+\triangleright\hskip-1.0pt\triangleleft}](b)}{\partial\vec{b}}-i\chi[({\mathcal{A}}^{(0)})^{\ast}](-b)\frac{\partial\chi[{\mathcal{A}}^{(0)}](b)}{\partial\vec{b}}\right]\\\
&=\frac{{\mathcal{E}}_{real}}{m_{a}m_{b}\sqrt{\sigma^{2}-1}}\left[\frac{\partial\chi[{\mathcal{A}}^{\bigtriangleup+\bigtriangledown}](b)}{\partial\vec{b}}\right]\,.\end{aligned}$
(91)

Thus, only the triangle contributions contribute to the classical deflection
angle.

###  3.5 The one-loop amplitude : Triangles

####  3.5.1 Analysis from the naïve Compton amplitude

The gravitational Compton amplitude is needed when evaluating the triangle
contribution for the loop amplitude, as can be seen on the left side of figure
1. The gravitational Compton amplitudes are given as Arkani-Hamed:2017jhn ;
Caron-Huot:2018ape ; Chung:2018kqs ; Johansson:2019dnu ; Chung:2019duq

$\displaystyle\begin{aligned} {\mathcal{A}}^{++}_{\phi
hh\bar{\phi}}&=\frac{(\kappa/2)^{2}m^{4}[23]^{4}}{(s-m^{2})t(u-m^{2})}\,,\\\
{\mathcal{A}}^{+-}_{\phi
hh\bar{\phi}}&=\frac{(\kappa/2)^{2}[2|p_{1}|3\rangle^{4}}{(s-m^{2})t(u-m^{2})}\,,\end{aligned}$
(92)

where Mandelstam invariants are defined as999All-outgoing convention is used.
$s=(p_{1}+p_{2})^{2}$, $t=(p_{1}+p_{4})^{2}$, and $u=(p_{1}+p_{3})^{2}$. As in
(77), NUT charges can be generated by phase rotation
$e^{2i\varphi}$($e^{-2i\varphi}$) for each positive(negative) helicity
graviton Huang:2019cja , which results in the expressions,

$\displaystyle\begin{aligned} {\mathcal{A}}^{++}&\to
e^{4i\varphi}{\mathcal{A}}^{++}=e^{4i\varphi}\frac{(\kappa/2)^{2}m^{4}[23]^{4}}{(s-m^{2})t(u-m^{2})}\,,\\\
{\mathcal{A}}^{+-}&\to{\mathcal{A}}^{+-}=\frac{(\kappa/2)^{2}[2|p_{1}|3\rangle^{4}}{(s-m^{2})t(u-m^{2})}\,.\end{aligned}$
(93)

As noted in ref.Caron-Huot:2018ape , these two amplitudes do _not_ satisfy the
$t$-channel factorisation conditions101010The $t$-channel pole does not exist
for the electromagnetic Compton amplitude, therefore the analogous rotation
generates a consistent dyonic amplitude for electromagnetism.. This is the
reason they are referred to as _naïve Compton amplitudes_. Nevertheless, the
impulse and the deflection angle will be analysed using the naïve amplitudes
(93) and the corrections due to the difference (97) will be analysed in the
following section.

Combining (62), (77), and (93) into (63), it can be shown that

$\displaystyle\begin{aligned}
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{++}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}&=0\,,\\\
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{+-}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{+}&=\frac{3G^{2}\pi^{2}m_{a}^{2}m_{b}^{3}(5\sigma^{2}-1)}{\sqrt{-t}}\,,\end{aligned}$
(94)

to leading order in HCL. Summing over all possible intermediate graviton
states is effectively equivalent to summing over these two terms and doubling
the result. Summing over $a$-topology by exchanging $m_{a}\leftrightarrow
m_{b}$, the $G^{2}$-order deflection angle is then given as

$\displaystyle\begin{aligned}
\vec{\vartheta}_{2}&=\frac{{\mathcal{E}}_{real}}{m_{a}m_{b}\sqrt{\sigma^{2}-1}}\left[\frac{\partial\chi[{\mathcal{A}}^{\bigtriangleup+\bigtriangledown}](b)}{\partial\vec{b}}\right]\\\
&=\frac{{\mathcal{E}}_{real}}{(2\sqrt{\sigma^{2}-1})^{2}}\frac{3\pi
G^{2}(m_{a}+m_{b})}{{\left|b\right|}^{2}}\left[-(5\sigma^{2}-1)\hat{y}\right]\,,\end{aligned}$
(95)

which, using the mapping of table 2 and making the approximation
$m_{a}+m_{b}\simeq m_{a}={\mathcal{M}}_{{\rm TN}}$ yields a result
inconsistent with geodesic computations (58) for non-zero $\varphi$.

$\displaystyle\vec{\vartheta}^{\rm Amp}_{2}$
$\displaystyle=-\frac{3\pi{\mathcal{M}}^{2}_{{\rm
TN}}}{4}(5{\mathcal{E}}^{2}-\mu^{2})\left(\frac{G}{J}\right)^{2}\hat{y}\,.$
(96)

There are a few notable points to these results.

  1. 1.

The $(-t)^{-1/2}$ dependence solely comes from the scalar triangle integral
$I_{\bigtriangleup^{(b)}}$. This means the residue integral
$\oint\frac{dy}{y}{\mathcal{A}}_{4}{\mathcal{A}}_{3}{\mathcal{A}}_{3}$ must
scale as $(\beta-1)^{0}$ to be consistent with the geodesic computations. The
three-point amplitudes ${\mathcal{A}}_{3}$ can be shown to scale as
$(\beta-1)^{0}$ through direct computations, therefore the condition can be
refined to $(\beta-1)^{0}$ scaling on the Compton amplitude
${\mathcal{A}}_{4}$.

  2. 2.

When the deflection angle (95) is evaluated using the naïve amplitude (93),
only the helicity configurations invariant under the phase rotation
contributes to the impulse. In other words, _one-loop contributions to the
impulse are indifferent to electric-magnetic rotations_. This is in stark
contrast to the 2PM result of geodesic motion on a Taub-NUT background (58),
where $G^{2}$ order deflection angle had a dependence in the rotation angle
$\varphi$ through the combination $M={\mathcal{M}}_{\rm TN}\cos(2\varphi)$.

  3. 3.

The $b$-topology contribution (94) scales as $\propto m_{a}^{2}m_{b}^{3}$. Put
differently, _incorrectness of the Compton amplitude ( 93) for NUT-charged
particles is inconsequential for background-probe analysis_ where NUT-charged
particle acts as the background, because the contribution from the triangle
topology where massive propagator on the background side has been cut
dominates over the other triangle topology111111If the Compton amplitude
${\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}$ scales as $\propto
m_{a}^{3}/m_{b}$, it would provide a loophole to this argument. Whether
manifestation of Lorentz covariance breaking (the Dirac-Misner strings) can
generate such a scaling will be left for future work..

For completeness, possible resolutions for closing the gap between geodesic
motion computations and amplitude results will be surveyed, to no avail.

####  3.5.2 Compton hot-fix and impulse non-fix

The naïve amplitude (93) is obviously wrong. As argued in the previous
section, closing this loophole is unlikely to close the gap between geodesic
computations and amplitude results. Nontheless, let us analyse the effects of
naïve amplitude corrections to the impulse.

Although the naïve amplitudes (93) do not satisfy the $t$-channel
factorisation conditions, they do satisfy the factorisation conditions on the
$s$\- and $u$-channels. This implies that the difference between the correct
amplitude and the naïve amplitude can only have poles on the $t$-channel,

$\displaystyle{\mathcal{A}}_{\text{correct}}-{\mathcal{A}}_{\text{na\"{i}ve}}$
$\displaystyle=\frac{f}{t}+\text{(polynomials)}\,,$ (97)

where the residue on the $t$-channel is proportional to the product of on-
shell three-point amplitudes,

$\displaystyle f\stackrel{{\scriptstyle t\to
0}}{{\propto}}{\mathcal{A}}_{\phi\bar{\phi}h}{\mathcal{A}}_{hhh}\,,$ (98)

and the sum over exchanged $t$-channel graviton states has been suppressed.
Although the exact expression for the difference (97) is not available, the
exactness of the expression is unimportant as leading HCL contributions
dominate classical physics; when studying classical contributions from the
$b$-topology cut of figure 1, the leading behaviour of the expression (97) is
all that matters121212If triangle coefficient computation and HCL commutes;
look at section 3.5.3 for a counterexample. However, all computations seem to
imply that noncommutativity arises at subleading $(\beta-1)$ orders..

The Mandelstam invariant $t$ for the Compton amplitude inside the $b$-topology
cut is

$\displaystyle(k_{3}+k_{4})^{2}=q^{2}=-\frac{(\beta-1)^{2}m_{b}^{2}}{\beta}\,,$
(99)

so the pole contributions dominate in the HCL $\beta\to 1$. Therefore, the
difference of the real amplitude and the naïve amplitude can be approximated
as products of on-shell three-point amplitudes sitting on the $t$-channel
pole.

$\displaystyle\left.({\mathcal{A}}_{\text{correct}}-{\mathcal{A}}_{\text{na\"{i}ve}})\right|_{\text{HCL}}\simeq\sum_{\eta=\pm}c_{\eta}\frac{\left.{\mathcal{A}}_{\phi\bar{\phi}h}^{+\eta}\right|_{\text{HCL}}\left.{\mathcal{A}}_{hhh}^{-\eta}\right|_{\text{HCL}}}{t}\,,$
(100)

where the superscript $\eta$ denotes the helicity of the exchanged graviton
and $c_{\eta}$ is the proportionality constant for that exchange
contribution131313This approximation relies on the observation that _any_
polynomial Lorentz invariants constructed from the momenta and spinors of
external kinematics satisfying the little group weight constraints of
gravitons vanish in the HCL. An obvious loophole is when spinors parametrising
Dirac-Misner string degrees of freedom can be used to form Lorentz invariants;
e.g. the invariant $[k_{3}\xi]^{4}[k_{4}\xi]^{4}$ for ${\mathcal{A}}^{++}$
will not vanish in the HCL when the spinor parametrising the Dirac-Misner
string $|{\xi}]$ is independent of HCL, unless $|{\xi}]\propto|{\lambda}]$..
The on-shell three-point amplitudes can be approximated as

$\displaystyle\begin{aligned}
\left.{\mathcal{A}}_{\phi\bar{\phi}h}^{\pm}\right|_{\text{HCL}}&=\left.{\mathcal{A}}_{\phi\bar{\phi}h}^{\pm}(-p_{1},p_{1}^{\prime},-(k_{3}+k_{4}))\right|_{\text{HCL}}\end{aligned}$
(101)

for the on-shell three-point amplitude of NUT-charged particle coupling to a
graviton, and

$\displaystyle\begin{aligned}
\left.{\mathcal{A}}_{hhh}^{\mp}\right|_{\text{HCL}}&=\lim_{\beta\to
1}\left[{\mathcal{A}}_{hhh}^{\mp}(\left.(k_{3}+k_{4})\right|_{\text{HCL}},-k_{3},-k_{4})\right]\end{aligned}$
(102)

for the on-shell three-graviton amplitude. While the amplitude (101) can be
evaluated at the strict HCL, evaluating the amplitude (102) requires some
care; the amplitude is first evaluated at momentum conservation violating
kinematics where only $k_{3}+k_{4}$ is taken to the HCL, and then the
resulting expression is taken to the HCL. The reason for taking such a
convoluted sequence of limits is because the strict HCL is the collinear limit
$k_{3}\propto k_{4}$; ${\mathcal{A}}_{hhh}=0$ at HCL.

Direct computation of the HCL approximation (100) to the difference (97)
yields

$\displaystyle\begin{aligned}
\left.({\mathcal{A}}_{\text{correct}}^{++}-{\mathcal{A}}_{\text{na\"{i}ve}}^{++})\right|_{\text{HCL}}&\simeq
c_{+}^{++}\frac{\left.{\mathcal{A}}_{\phi\bar{\phi}h}^{+}\right|_{\text{HCL}}\left.{\mathcal{A}}_{hhh}^{-}\right|_{\text{HCL}}}{t}=-c_{+}^{++}\frac{u^{2}y^{2}}{m_{b}^{2}}\,,\\\
\left.({\mathcal{A}}_{\text{correct}}^{+-}-{\mathcal{A}}_{\text{na\"{i}ve}}^{+-})\right|_{\text{HCL}}&\simeq\sum_{\eta=\pm}c_{\eta}^{+-}\frac{\left.{\mathcal{A}}_{\phi\bar{\phi}h}^{+\eta}\right|_{\text{HCL}}\left.{\mathcal{A}}_{hhh}^{-\eta}\right|_{\text{HCL}}}{t}=-\frac{c_{+}^{+-}u^{2}y^{2}+c_{-}^{+-}v^{2}y^{2}}{m_{b}^{2}}\,.\end{aligned}$
(103)

Depending on helicity, the three-point amplitudes of the $b$-topology cut
scales as $y^{\pm 2}$. This means augmentation to the Compton amplitude must
scale as $y^{0}$ or $y^{\pm 4}$ to contribute to the triangle coefficient
(94), but (103) do not have the required scaling; fixing the naïve Compton
amplitude will not fix the mismatch of scattering angle computations.

####  3.5.3 Exciting internal structures and NUT-exciting results

As argued in appendix A, there is a strong tension in treating electrically
charged and magnetically charged particles simultaneously as particles without
internal structures, at least for abelian gauge theory. Although the same
argument does not apply to gravity, the argument alludes to the possibility
that NUT-charged particles cannot be structureless. This motivates the
following question; will the disagreement between geodesic computations and
amplitude computations be resolved when effects of NUT-charged particle’s
internal structures are taken into account?

One way to model internal structure excitations of the particle is to consider
transmutation of quantum numbers such as mass or spin. This process can be
captured by on-shell three-point amplitudes.

  1. 1.

Equal masses; $\phi\to\phi^{\prime}h$ where $m_{\phi}=m_{\phi^{\prime}}$ and
spin of $\phi^{\prime}$ can be $s^{\prime}=0,1,2$.

  2. 2.

Unequal masses; $\phi\to\phi^{\prime}h$ where $m_{\phi}\neq m_{\phi^{\prime}}$
and spin of $\phi^{\prime}$ is $s^{\prime}=2$.

The on-shell three-point amplitudes for the possibilities enumerated above can
be kinematically determined as demonstrated in appendix B. Introduction of
these processes affect the $b$-topology cut, figure 1, through alterations in
the Compton amplitude or in the pair of three-point amplitudes. The equal mass
case $m_{\phi}=m_{\phi^{\prime}}=m$ will be studied first, where cut momentum
parametrisation (62) can still be used.

The modifications to the Compton amplitude from transmutation processes will
always accompany $s$\- or $u$-channel poles, therefore their primary effect
appears on the four-particle cut; their effect on the three-particle cut is
subleading in $(\beta-1)$ compared to the naïve answer (94). As an explicit
demonstration, when approximations to the modifications of the Compton
amplitude similar to the previous section are adopted, computation of
$b$-topology contributions to ${\mathcal{A}}^{\bigtriangleup^{(b)}}$ from
intermediate spin-1 (spin-2) particle contains additional powers of
$(\beta-1)$ and yields the scaling $\propto q^{0}$($\propto q^{2}$), which are
contact interactions not affecting the long-distance physics141414These
results must be taken with a grain of salt; the scaling implies scalar
integral coefficients are _not_ rational functions of external kinematics. For
example, a proper computation using spin-2 exchange modification (106) with
$m_{\phi}=m_{\phi^{\prime}}$ results in vanishing integral coefficients as can
be seen in (111).. For this reason, only the modifications to the pair of
three-point amplitudes appearing on the $b$-topology cut, figure 1, will be
studied in detail for the equal mass case.

For the equal helicity exchange channel, the sum over intermediate states for
the pair of three-point amplitude yields

$\displaystyle\sum_{\phi^{\prime}}{\mathcal{A}}_{\phi\phi^{\prime}h}^{-}(k_{3}){\mathcal{A}}_{\phi^{\prime}\phi
h}^{-}(k_{4})$ $\displaystyle=\left\\{\begin{aligned}
&\left(\frac{\gamma_{0}\kappa
m^{2}}{2}\right)^{2}\frac{1}{y^{4}\beta^{2}}&&s^{\prime}=0\\\
&\left(\frac{\gamma_{1}\kappa}{2}\right)^{2}\frac{m^{4}(\beta-1)^{2}}{y^{4}\beta^{3}}&&s^{\prime}=1\\\
&\left(\frac{\gamma_{2}\kappa}{2m^{2}}\right)^{2}\frac{m^{8}(\beta-1)^{4}}{y^{4}\beta^{4}}&&s^{\prime}=2\end{aligned}\right.$
(104)

which implies
$\oint\frac{dy}{y}{\mathcal{A}}^{++}{\mathcal{A}}^{-}{\mathcal{A}}^{-}=0$; the
equal helicity exchange channel still does not contribute to the $b$-topology
contribution ${\mathcal{A}}^{\bigtriangleup^{(b)}}$. On the other hand, for
the mixed helicity exchange channel the same computations lead to

$\displaystyle\sum_{\phi^{\prime}}{\mathcal{A}}_{\phi\phi^{\prime}h}^{-}(k_{3}){\mathcal{A}}_{\phi^{\prime}\phi
h}^{+}(k_{4})$ $\displaystyle=\left\\{\begin{aligned}
&\left(\frac{\gamma_{0}\kappa m^{2}}{2}\right)^{2}\beta^{2}&&s^{\prime}=0\\\
&\left(\frac{\gamma_{1}\kappa}{2}\right)^{2}m^{4}(\beta-1)^{2}\beta&&s^{\prime}=1\\\
&\left(\frac{\gamma_{2}\kappa}{2m^{2}}\right)^{2}m^{8}(\beta-1)^{4}&&s^{\prime}=2\end{aligned}\right.$
(105)

which implies that
$\gamma^{2}(\beta-1)^{2s^{\prime}}\simeq\gamma^{2}\left(\frac{-t}{m_{b}^{2}}\right)^{s^{\prime}}$
factor is attached to the original $b$-topology contribution
${\mathcal{A}}^{\bigtriangleup^{(b)}}$. An interesting observation point is
that for $s^{\prime}=2$ transmutations, the modification to the $b$-topology
contribution $\Delta{\mathcal{A}}^{\bigtriangleup^{(b)}}$ scales as
$\propto{\left|q\right|}^{3}\sim r^{-6}$, which is known to be the $1/r$
dependence for tidal effect corrections Cheung:2020sdj . While the resulting
$\sigma$-dependence for $m_{\phi}=m_{\phi^{\prime}}$ case is inconsistent with
the results of ref.Cheung:2020sdj , the conclusions are different for the case
of $m_{\phi}\neq m_{\phi^{\prime}}$.

For unequal mass case $m_{\phi^{\prime}}\neq m_{\phi}$, the Compton amplitudes
are modified by inclusion of $\phi^{\prime}$ exchange on the $s$\- and
$u$-channels as

$\displaystyle\begin{aligned} \Delta{\mathcal{A}}_{\phi
hh\bar{\phi}}^{++}&=-\frac{\gamma^{\prime
2}(\kappa/2)^{2}m_{\phi^{\prime}}^{4}[23]^{4}}{m_{\phi}^{4}}\left(\frac{1}{s-m_{\phi^{\prime}}^{2}}+\frac{1}{u-m_{\phi^{\prime}}^{2}}\right)\,,\\\
\Delta{\mathcal{A}}_{\phi hh\bar{\phi}}^{+-}&=-\frac{\gamma^{\prime
2}(\kappa/2)^{2}[{2}|p_{1}|{3}\rangle^{4}}{m_{\phi}^{4}}\left(\frac{1}{s-m_{\phi^{\prime}}^{2}}+\frac{1}{u-m_{\phi^{\prime}}^{2}}\right)\,,\end{aligned}$
(106)

where the three-point amplitudes (117) have been glued on the channels as in
ref.Arkani-Hamed:2017jhn  to arrive at the results. The absolute square of the
coupling ${\left|\gamma^{\prime}\right|}^{2}$ is related to the characteristic
decay rate of resonance excitations and
$m_{\phi^{\prime}}^{2}-m_{\phi}^{2}\simeq 2m_{\phi}\Delta m$ is related to the
characteristic frequency of the resonance excitations $\hbar\omega=\Delta m$.

Interestingly, the order of taking the HCL and effecting the
$\oint\frac{dy}{y}$ integral matters. When constituent tree amplitudes are
taken to the HCL before $\oint\frac{dy}{y}$ integration, the modifications to
$b$-topology contribution are computed as

$\displaystyle\begin{aligned}
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}\left[\Delta{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{++}\right]{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}&=G^{2}\pi^{2}m_{b}^{3}\left[\frac{4\gamma_{a}^{\prime
2}(m_{a^{\prime}}/m_{a})^{4}}{m_{a^{\prime}}^{2}-m_{a}^{2}}\right]\sqrt{-t}^{3}\,,\\\
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}\left[\Delta{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{+-}\right]{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{+}&=G^{2}\pi^{2}m_{b}^{3}\left[\frac{\gamma_{a}^{\prime
2}(3-30\sigma^{2}+35\sigma^{4})}{2(m_{a^{\prime}}^{2}-m_{a}^{2})}\right]\sqrt{-t}^{3}\,,\end{aligned}$
(107)

which scales as $\propto{\left|q\right|}^{3}\sim r^{-6}$ and can be recast as

$\displaystyle\begin{aligned}
\Delta{\mathcal{A}}^{\bigtriangleup^{(b)}}&=G^{2}\pi^{2}m_{b}^{3}\sqrt{-t}^{3}\left[\gamma_{a}^{\prime
2}\frac{8(m_{a^{\prime}}/m_{a})^{4}+(3-30\sigma^{2}+35\sigma^{4})}{m_{a^{\prime}}^{2}-m_{a}^{2}}\right]\\\
&\simeq G^{2}\pi^{2}m_{b}^{3}\sqrt{-t}^{3}\left[\gamma_{a}^{\prime
2}\frac{(11-30\sigma^{2}+35\sigma^{4})+32(\Delta m_{a}/m_{a})}{2m_{a}\Delta
m_{a}}\right]\,,\end{aligned}$ (108)

where lowest order expansion in $\Delta m_{a}=m_{a^{\prime}}-m_{a}$ has been
performed on the last line. This has an interpretation as tidal effects;
comparing with eq.(4) of ref.Cheung:2020sdj , the tidal response coefficients
$\lambda$ and $\eta$ can be related to $\gamma^{\prime}$ and $\Delta m$
by151515If there are more than one resonance modes, the relation becomes a sum
over different resonance modes. A quantum mechanical version of the relation
has been considered in ref.Brustein:2020tpg .

$\displaystyle\lambda=\frac{4\gamma^{\prime
2}}{m^{2}}\,,\,\eta=\frac{16\gamma^{\prime 2}}{m\Delta m}\,,$ (109)

which in turn relates them to the worldline tidal operator coefficients
through eq.(12) of the same reference.

$\displaystyle\sigma^{(2)}=-\frac{12\gamma^{\prime
2}}{m^{3}}\,,\,\mu^{(2)}=\frac{16\gamma^{\prime 2}}{m^{2}\Delta
m}+\frac{96\gamma^{\prime 2}}{3m^{3}}\,.$ (110)

The above equation relates the parameters of dissipation through resonance
excitations to that of tidal responses, which is reminiscent of Kramers-Kronig
relations.

On the other hand, performing the $\oint\frac{dy}{y}$ integration and then
taking the HCL results in

$\displaystyle\begin{aligned}
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}\left[\Delta{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{++}\right]{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}&=0\,,\\\
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}\left[\Delta{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{+-}\right]{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\bar{\phi}_{b}h}^{+}&=-\frac{4\gamma_{a}^{\prime
2}G^{2}\pi^{2}m_{a}^{2}m_{b}^{3}}{\sqrt{-t}}\left[\frac{m_{a^{\prime}}^{2}-m_{a}^{2}}{m_{a}^{2}}\right]^{3}\,,\end{aligned}$
(111)

which has the wanted scaling $\propto{\left|q\right|}^{-1}\sim r^{-2}$.
However (111) cannot resolve the discrepancy between geodesic computations and
amplitude computations as the mass scaling is $\propto m_{a}^{2}m_{b}^{3}$,
similar to naïve Compton amplitude impulse computation (94); if the particle
$a$ having internal structures is considered as the background, then this
contribution is suppressed by the mass ratio $m_{b}/m_{a}$ compared to the
$a$-topology contribution.

When the massive propagator of the $\phi^{\prime}$ particle is cut as in
figure 1 with the condition $m_{\phi^{\prime}}\neq m_{\phi}$, the leading
small $q^{2}$ contributions to $\Delta{\mathcal{A}}^{\bigtriangleup^{(b)}}$
are computed as161616The cut momentum parametrisations (62) are no longer
valid, and the new cut conditions
$k_{3}^{2}=k_{4}^{2}=L^{2}-m_{b^{\prime}}^{2}=0$ were solved to arrive at the
result (112).

$\displaystyle\begin{aligned}
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{++}{\mathcal{A}}_{\phi_{b}\phi^{\prime}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\phi^{\prime}_{b}h}^{-}&=0\,,\\\
iI_{\bigtriangleup^{(b)}}\oint_{\Gamma}\frac{dy}{2\pi
iy}{\mathcal{A}}_{\phi_{a}hh\bar{\phi}_{a}}^{+-}{\mathcal{A}}_{\phi_{b}\phi^{\prime}_{b}h}^{-}{\mathcal{A}}_{\phi_{b}\phi^{\prime}_{b}h}^{+}&=\frac{\gamma_{b}^{\prime
2}G^{2}\pi^{2}m_{a}^{2}m_{b}^{5}(3\sigma^{2}-1)}{\sqrt{-t}^{3}}\left[\frac{m_{b}}{m_{b^{\prime}}}\right]\left[\frac{m_{b^{\prime}}^{2}-m_{b}^{2}}{m_{b}^{2}}\right]^{6}\,.\end{aligned}$
(112)

The interpretation of this result is unclear: Although the overall coefficient
$(\Delta m_{b}/m_{b})^{6}$ highly suppresses the result, the above scales as
${\left|q\right|}^{-3}\sim\log r$ which actually _grows_ at long distances. If
the limit $m_{b^{\prime}}\to m_{b}$ is taken before the limit $q^{2}\to 0$
then the resulting cut computation reduces to (105) instead, therefore it is
unlikely that computations leading to the above result are wrong. How to
correctly interpret this result will be left for a future work.

In conclusion, exciting the NUT charges, modelled as transmutations to
distinct particle species, cannot close the gap between geodesic computations
of general relativity and amplitude computations at one-loop. While the
results are not exciting from the viewpoint of original motivations, the
relation (109) between internal excitation parameters and tidal deformation
parameters obtained from (108) is an interesting by-product of the analysis
that calls for future studies.

##  4 Conclusion

In this study the classical impulse at the second post-Minkowskian order
($G^{2}$) was analysed from Hamilton-Jacobi analysis of general relativity on
Taub-NUT spacetime and quantum field theory one-loop amplitude calculations.
Although the computations agreed at 1PM, the computations were shown to
disagree at 2PM. Possible modifications to one-loop amplitude computations
that could resolve the disagreement had been analysed, but all considered
scenarios had failed to close the gap.

There are a few loopholes to the amplitude analysis, one being possible
modifications to the amplitudes due to presence of Dirac-Misner strings. As
mentioned in footnote 11, the argument that only the $b$-topology cut of
figure 1 with $m_{b}\gg m_{a}$ is relevant for 2PM scattering angle analysis
in the background-probe limit is based on the expectation that the Compton
amplitude ${\mathcal{A}}_{\phi_{b}hh\bar{\phi}_{b}}$, which appears in the
$a$-topology cut, knows nothing about the mass of particle $a$. If the degrees
of freedom parametrising the Dirac-Misner string attached to particle $b$
somehow knows about $m_{a}$ then the argument based on (94) that $a$-topology
contribution is negligible would break down. The approximations to the
difference of correct and naïve Compton amplitude (100) also contains a
similar loophole as explained in footnote 13.

It is also possible that application of unitarity methods found in the
literature need generalisations when magnetic charges are involved. For
example, it is unclear how pairwise little group introduced in
ref.Csaki:2020inw , which demands introduction of a new quantum number
(pairwise helicity) for each dyonic pair of particles, can be applied to
unitarity methods. When unitarity methods are applied to loop effects for
electric charge-monopole scattering, the cuts can be performed in a way that
all resulting tree diagrams only involve one type of charges. This results in
an awkward situation where none of the constituent subdiagrams require
pairwise helicity but the diagram as a whole does.

It would be an interesting exercise to ponder the ramifications of the
relation (109) and its generalisations. Black holes of general relativity in
four space-time dimensions are known to be inert to tidal forces, expressed in
the form of vanishing Love numbers Porto:2016zng . The relation implies that
vanishing of tidal Love numbers poses a constraint on quantum mechanical
excitations of the black hole Kim:2020dif . We also remark that the relation
is similar in spirit to the Fermi four-fermion interactions being completed to
weak interactions of the standard model.

###### Acknowledgements.

We would like to thank to Yu-Tin Huang for technical discussions, Seok Kim for
helpful suggestions, and our Ph.D. advisors, Sangmin Lee and Nakwoo Kim, for
feedbacks. JK and MS also thank for the hospitality of Kyung Hee University
and Seoul National University during their respective visits. JK was supported
in part by the Science and Technology Facilities Council (STFC) Consolidated
Grant ST/T000686/1 _“Amplitudes, Strings and Duality”_ , and in part by
National Research Foundation (NRF) of Korea grant NRF-2019R1A2C2084608. MS was
supported in a part by National Research Foundation (NRF) of Korea grant
NRF-2019R1A2C2004880 and NRF-2018R1A2A3074631, and a scholarship from Hyundai
Motor Chung Mong-Koo Foundation.

##  Appendix A The elementarity tension between electric charges and monopoles

It is well known that Lagrangian field theory description of a system
including electrically charged particles and magnetically charged particles as
elementary degrees of freedom in its spectrum simultaneously can only be
attained at the price of (1) manifest Lorentz invariance or (2) manifest local
description Hagen:1965zz ; Zwanziger:1970hk ; Schwarz:1993vs ; Dirac:1948um ;
Schwinger:1966nj . Nevertheless, the following question may be asked; _is it
eligible to describe electrically charged particles and magnetically charged
particles as "elementary particles" in an effective theory whose cut-off is
much higher than the particles’ masses?_ Here, "elementary particle" refers to
particles whose internal structures can be ignored and regarded as point
particles. For example, in sufficiently low energy experiments protons and
pions can be regarded as "elementary particles".

A simple argument shows this is unlikely. The _classical radius_ $r_{c}$ is
the scale where point particle description of the body breaks down in the
classical theory, and the _Compton wavelength_ $\lambda_{C}$ is the scale
where classical description of the body breaks down and quantum theory must be
introduced. If $r_{c}>\lambda_{C}$, there is an intermediate length scale $L$,
$r_{c}>L>\lambda_{C}$, such that its non-point-particle-like structure must
become manifest because it cannot be clouded by quantum effects. Therefore,
_for a particle to be described as an "elementary particle" its Compton
wavelength must be greater than its classical radius; $\lambda_{C}>r_{c}$_. A
slightly stronger condition $\lambda_{C}\gg r_{c}$ cannot be imposed on
electric and magnetic charges simultaneously due to Dirac-Schwinger-Zwanziger
quantisation conditions.

Ignoring $O(1)$ numerical coefficients, the classical radius and the Compton
wavelength of an electrically charged particle of mass $m$ and unit charge $e$
is given as;

$\displaystyle
r_{c}\simeq\frac{e^{2}}{mc^{2}}\,,\,\lambda_{C}\simeq\frac{\hbar}{mc}\;\Rightarrow\;\frac{r_{c}}{\lambda_{C}}\simeq\frac{e^{2}}{\hbar
c}=\alpha_{e}\,.$ (113)

Likewise, the ratio for a magnetically charged particle of unit charge is
given as $r_{c}/\lambda_{C}\simeq g^{2}/\hbar c=\alpha_{g}$. The least
restrictive Dirac-Schwinger-Zwanziger quantisation condition demands $eg\simeq
O(1)$, which translates to $\alpha_{e}\alpha_{g}\simeq O(1)$; the "elementary
particle" conditions $\alpha_{e}\ll 1$ and $\alpha_{g}\ll 1$ are not
compatible with the quantisation conditions, unless the effective theory cut-
off scale $\Lambda_{\text{EFT}}$ screens the classical radius and allows
"elementary particle" descriptions. In the latter case, the condition
$\Lambda_{\text{EFT}}<m$ must be satisfied by the heavier of the
charges171717Essentially the same argument appears in magnetic version of the
weak gravity conjecture ArkaniHamed:2006dz ..

Note that the "classical radius" for gravitational interactions is also a
scale where Newtonian gravity becomes inadequate; the radius can be defined as
the scale where Newtonian gravitational self-energy becomes comparable to its
rest mass. Up to $O(1)$ numerical coefficients,

$\displaystyle
mc^{2}\simeq\frac{Gm^{2}}{r_{c}}\;\Rightarrow\;r_{c}\simeq\frac{Gm}{c^{2}}\,,$
(114)

which is the scale of the body’s Schwarzschild radius; general relativity must
be introduced at this scale.

##  Appendix B All on-shell three-point amplitudes corresponding to
transmutation

All kinematicly allowed gauge-invariant on-shell three-point amplitudes of the
type ${\mathcal{A}}_{\phi\phi^{\prime}h}$ are enumerated in this appendix,
which are special cases of the classification initiated in ref.Arkani-
Hamed:2017jhn . The massive spinor-helicity variables of ref.Arkani-
Hamed:2017jhn  is used for massive spinning particles and conventions for the
variables follow that of ref.Chung:2018kqs .

For sake of simplicity, only the coupling to positive helicity graviton will
be considered. The coupling to negative helicity can be achieved by
exchanging181818Couplings are assumed to be real. square and angle spinor
brackets $\langle pq\rangle\leftrightarrow[pq]$. The little group constraint
on ${\mathcal{A}}_{\phi\phi^{\prime}h}^{+}$ for the graviton leg is

$\displaystyle|{3}]^{4-n}|{3}\rangle^{-n}\subset{\mathcal{A}}_{\phi\phi^{\prime}h}^{+}\,,$
(115)

where $|{3}]$ and $|{3}\rangle$ are spinors for the graviton leg and the
exponent shows how many times each spinor appears on the numerator/denominator
of the amplitude.

###  B.1 Equal mass

When $m_{\phi}=m_{\phi^{\prime}}=m$, the kinematic factor $x$ of (78)
introduced in ref.Arkani-Hamed:2017jhn  can be used to provide additional
little group weight for the graviton leg. Three possibilities can be
considered.

$\displaystyle{\mathcal{A}}_{\phi\phi^{\prime}h}^{+}=\left\\{\begin{aligned}
&\gamma_{0}\frac{\kappa m^{2}}{2}x^{2}&&s^{\prime}=0\\\
&\gamma_{1}\frac{\kappa}{2}x[3\mathbf{2}]^{2}&&s^{\prime}=1\\\
&\gamma_{2}\frac{\kappa}{2m^{2}}[3\mathbf{2}]^{4}&&s^{\prime}=2\end{aligned}\right.$
(116)

In the above expression, the transmuted particle $\phi^{\prime}$ has been
given the leg number 2 and the couplings $\gamma_{i}$’s have been scaled to be
dimensionless using $\kappa$ and $m$.

###  B.2 Unequal mass

The kinematic factor $x$ of (78) is gauge-invariant only if
$m_{\phi}=m_{\phi^{\prime}}$, where gauge-invariance denotes invariance of the
expression under the gauge transform
$\varepsilon^{\mu}(p_{3})\sim\varepsilon^{\mu}(p_{3})+cp_{3}^{\mu}$. Gauge
invariance enforces the condition $p_{1}\cdot p_{3}=0$, which in turn enforces
$p_{1}^{2}=p_{2}^{2}$; the existence of $x$-factor demands191919$m>0$ is
implicitly assumed. $m_{\phi}=m_{\phi^{\prime}}$. This means four more spinors
must be introduced to form a Lorentz invariant from four
$|{3}]^{\dot{\alpha}}$ spinors of the graviton leg. The only possibility is to
introduce at least four spinors of the $\phi^{\prime}$ leg, meaning the
transmuted particle $\phi^{\prime}$ must have spin $s^{\prime}\geq 2$.
Saturating the bound $s^{\prime}=2$ yields the amplitude

$\displaystyle{\mathcal{A}}_{\phi\phi^{\prime}h}^{+}=\gamma^{\prime}\frac{\kappa}{2m_{\phi}^{2}}[3\mathbf{2}]^{4}\,.$
(117)

This result is consistent with angular momentum conservation and can be
understood as a gravitational analogue of the selection rule $\Delta L=\pm 1$
for dipole transitions of the hydrogen atom; when $m_{\phi}\neq
m_{\phi}^{\prime}$, a real process of heavier of the two decaying into the
lighter of the two by emitting a graviton is allowed.

Assuming $m_{\phi}>m_{\phi}^{\prime}$, the decay process
$\phi\to\phi^{\prime}h$ can be analysed at the rest frame of $\phi$. In this
frame the total angular momentum of the initial state $\phi$ is zero, which
must be conserved after the decay. Since orbital angular momentum of the final
state $\phi^{\prime}h$ cannot contribute to the total angular momentum
directed along the direction of the emitted graviton, the spin component along
this direction for the decay products must sum up to zero. The graviton has
spin 2 along this direction, so the decay product $\phi^{\prime}$ also must
have spin 2 along this direction with opposite signs. Extension of the
analysis for the case $m_{\phi}<m_{\phi}^{\prime}$ is straightforward.

##  Appendix C Dyonic generalisation of effective one-body formalism

In Newtonian mechanics, the two-body problem can be reduced to a one-body
problem using reduced mass. The _effective one-body_ (EOB) formalism is an
attempt to generalise the reduction to relativistic systems, where the two-
body motion is mapped to an effective one-body geodesic motion on a background
metric Buonanno:1998gg . The formalism often reduces to finding a map between
the real energy ${\mathcal{E}}_{\rm real}$ of the two-body system and the
effective energy ${\mathcal{E}}_{{\rm eff}}$ of the one-body geodesic motion,
which smoothly interpolates between background-probe limit and comparable mass
case. In the extreme mass-ratio limit the effective metric often reduces to
known black hole solutions. To find the energy map, one first sets the action-
angle variables of the real two-body system and the effective one-body system
to be the same, and then inverts the relation between energy and action-angle
variables to obtain a relation between the energy of the real system and the
energy of the effective system.

We will attempt a generalisation of effective one-body formalism for
scattering states Damour:2016gwp  to cases involving "magnetic mass", using
amplitude computations as the input for real dynamics. The dynamics obtained
from amplitudes defies an interpretation as the motion on a Taub-NUT
background with backreaction of the probe particle taken into account, as the
extreme mass-ratio limit is not the geodesic motion on a Taub-NUT background.
Therefore, an interesting problem would be to find the effective background
metric the dynamics corresponds to. We adopt the usual _total mass_
${\mathcal{M}}_{total}$ and _effective mass_ $\mu$ relations for inertial
masses.

$\displaystyle{\mathcal{M}}^{2}\equiv M^{2}+N^{2},$
$\displaystyle{\mathcal{M}}_{total}={\mathcal{M}}_{a}+{\mathcal{M}}_{b},,$
$\displaystyle\mu=\frac{{\mathcal{M}}_{a}{\mathcal{M}}_{b}}{{\mathcal{M}}_{total}}\,.$
(118)

The electric and magnetic masses are given as

$\displaystyle\cos(2\varphi)=\frac{M}{\mathcal{M}_{{\rm
TN}}}\leftrightarrow\frac{M_{{\rm eff}}}{\mathcal{M}_{total}},$
$\displaystyle\sin(2\varphi)=\frac{N}{\mathcal{M}_{{\rm
TN}}}\leftrightarrow\frac{N_{{\rm eff}}}{\mathcal{M}_{total}},$ (119)
$\displaystyle b{\left|\vec{p}_{{\rm com}}\right|}$
$\displaystyle=\frac{m_{a}m_{b}b\sqrt{\sigma^{2}-1}}{\sqrt{s}}=J\Leftrightarrow
b\sqrt{\sigma^{2}-1}=\frac{J{\mathcal{E}}_{real}}{m_{a}m_{b}}\,.$ (120)

If both particles undergo dyonic rotation, only the relative rotation angle
$\varphi=\varphi_{a}-\varphi_{b}$ affects the dynamics. The angles can be
written using the masses as given below.

$\displaystyle\cos{2\varphi_{a}}=\frac{M_{a}}{{\mathcal{M}}_{a}},$
$\displaystyle\cos{2\varphi_{b}}=\frac{M_{b}}{{\mathcal{M}}_{b}},$
$\displaystyle\cos{2\varphi}=\frac{M_{{\rm
eff}}}{{\mathcal{M}}_{total}}=\frac{M_{a}M_{b}+N_{a}N_{b}}{{\mathcal{M}}_{a}{\mathcal{M}}_{b}},$
(121) $\displaystyle\sin{2\varphi_{a}}=\frac{N_{a}}{{\mathcal{M}}_{a}},$
$\displaystyle\sin{2\varphi_{b}}=\frac{N_{b}}{{\mathcal{M}}_{b}},$
$\displaystyle\sin{2\varphi}=\frac{N_{{\rm
eff}}}{{\mathcal{M}}_{total}}=\frac{M_{a}N_{b}-M_{b}N_{a}}{{\mathcal{M}}_{a}{\mathcal{M}}_{b}}\,.$
(122)

The effective dynamics will depend on the effective electric(magnetic) mass
$M_{{\rm eff}}$($N_{{\rm eff}}$), which can be written using the reduced mass
$\mu$.

$\displaystyle M_{{\rm eff}}=\frac{M_{a}M_{b}+N_{a}N_{b}}{\mu},$
$\displaystyle N_{{\rm eff}}=\frac{M_{a}N_{b}-M_{b}N_{a}}{\mu}\,.$ (123)

For the rest of this section, we drop the subscripts on $M_{{\rm eff}}$ and
$N_{{\rm eff}}$ for simplicity.

###  C.1 The deflection angle of effective dynamics

Let us consider an effective metric202020$\beta$ denotes deformation
parameters. which generalises the deformed Schwarzschild metric in
ref.Damour:2016gwp  up to second Post-Minkowskian order.

$\displaystyle ds^{2}_{{\rm eff}}$
$\displaystyle=-f_{1}dt^{2}+f_{2}dr^{2}+(r^{2}+G^{2}N^{2})d\Omega_{2},$ (124)

where

$\displaystyle f_{1}=1-\frac{2GM}{r}-\frac{2G^{2}N^{2}}{r^{2}},$
$\displaystyle
f_{2}(\beta)=1+\beta_{1}\frac{2GM}{r}+\beta_{2}\frac{(2GM)^{2}}{r^{2}}+\beta_{3}\frac{2(GN)^{2}}{r^{2}}.$
(125)

The corresponding effective Hamilton-Jacobi equation on the cone is

$\displaystyle-f_{1}^{-1}{\mathcal{E}}_{{\rm
eff}}^{2}+\frac{J^{2}_{\varphi}}{(r^{2}+N^{2})}+f_{2}^{-1}P(r)^{2}=-\mu^{2},$
(126)

or, equivalently,

$\displaystyle P_{\pm}(r)=\pm\left[\frac{f_{2}}{f_{1}}{\mathcal{E}}_{{\rm
eff}}^{2}-f_{2}\mu^{2}-f_{2}\frac{J^{2}_{\varphi}}{(r^{2}+N^{2})}\right]^{\frac{1}{2}}.$
(127)

The effective deflection angle can be obtained as

$\displaystyle\vartheta_{{\rm eff}}$
$\displaystyle=2\sqrt{\frac{M^{2}\left(\left(\beta_{1}+1\right){\mathcal{E}}_{{\rm
eff}}^{2}-\beta_{1}\mu^{2}\right)^{2}}{{\mathcal{E}}_{{\rm
eff}}^{2}-\mu^{2}}+4N^{2}{\mathcal{E}}_{{\rm
eff}}^{2}}\left(\frac{G}{J}\right)$
$\displaystyle\qquad+\pi\frac{M\left(\left(\beta_{1}+1\right){\mathcal{E}}_{{\rm
eff}}^{2}-\beta_{1}\mu^{2}\right)}{4\sqrt{M^{2}\left(\left(\beta_{1}+1\right){\mathcal{E}}_{{\rm
eff}}^{2}-\beta_{1}\mu^{2}\right){}^{2}+4N^{2}{\mathcal{E}}_{{\rm
eff}}^{2}\left({\mathcal{E}}_{{\rm eff}}^{2}-\mu^{2}\right)}}$
$\displaystyle\qquad\qquad\qquad\times\Bigl{[}M^{2}\left(\left(\beta_{1}^{2}-4\beta_{1}-4\beta_{2}-8\right){\mathcal{E}}_{{\rm
eff}}^{2}-\left(\beta_{1}^{2}-4\beta_{2}\right)\mu^{2}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+N^{2}\left(\left(2\beta_{3}+1\right)\mu^{2}+\left(3-2\beta_{3}\right){\mathcal{E}}_{{\rm
eff}}^{2}\right)\Bigr{]}\left(\frac{G}{J}\right)^{2}.$ (128)

###  C.2 The 1PM mapping

Since the action-angle variables are matched in the effective one-body
formalism, we set the deflection angles equal; $\vartheta_{{\rm
eff}}=\vartheta_{\rm real}$. This results in

$\displaystyle\sqrt{\frac{1}{(\mu^{2}\sigma^{2}-\mu^{2})}(2\mu^{2}\sigma^{2}-\mu^{2})^{2}M^{2}+4\mu^{2}\sigma^{2}N^{2}}=\sqrt{\frac{M^{2}\left(\left(\beta_{1}+1\right)\mathcal{E}_{{\rm
eff}}^{2}-\beta_{1}\mu^{2}\right){}^{2}}{\mathcal{E}_{{\rm
eff}}^{2}-\mu^{2}}+4N^{2}\mathcal{E}_{{\rm eff}}^{2}}.$ (129)

Adopting new dimensionless variables, $u_{r}^{2}=(\sigma^{2}-1)$ and
$u_{e}^{2}=\left({\mathcal{E}}^{2}/\mu^{2}-1\right)$, the relation is recast
into

$\displaystyle
g_{1,r}(u_{r})M^{2}+g_{2,r}(u_{r})4N^{2}=g_{1,e}(u_{e})M^{2}+g_{2,e}(u_{e})4N^{2},$
(130)

where

$\displaystyle\begin{split}g_{1,r}(u_{r})&=(2u_{r}+u_{r}^{-1})^{2},\\\
g_{2,r}(u_{r})&=(u_{r}^{2}+1),\end{split}$
$\displaystyle\begin{split}g_{1,e}(u_{e})&=(\left(\beta_{1}+1\right)u_{e}+u_{e}^{-1})^{2},\\\
g_{2,e}(u_{e})&=(u_{e}^{2}+1).\end{split}$ (131)

The parameter $\beta_{1}$ can be determined as in ref.Damour:2016gwp , where
the criteria for fixing $\beta_{1}$ is given as

$\displaystyle\begin{split}g_{1,r}(u_{r})\sim g_{1,e}(u_{e})&\Rightarrow
u^{\min(g_{2,r})}_{r}=\frac{1}{\sqrt{2}},u^{\min(g_{2,e})}_{e}=\frac{1}{\sqrt{\beta_{1}+1}},\\\
g_{2,r}(u_{r})\sim g_{2,e}(u_{e})&\Rightarrow
u^{\min(g_{2,r})}_{r}=0,\phantom{aa}u^{\min(g_{2,e})}_{e}=0.\end{split}$ (132)

Solving for the relations yields

$\displaystyle\beta_{1}=1,$ $\displaystyle\sigma^{2}=\frac{{\mathcal{E}}_{{\rm
eff}}^{2}}{\mu^{2}}=\frac{{\mathcal{E}}_{\rm
real}^{2}-{\mathcal{M}}_{a}^{2}-{\mathcal{M}}_{b}^{2}}{2{\mathcal{M}}_{a}{\mathcal{M}}_{b}}.$
(133)

Combining everything, we arrive at the effective one-body energy mapping at
1PM order.

$\displaystyle{\mathcal{E}}_{{\rm eff}}=\frac{{\mathcal{E}}_{\rm
real}^{2}-{\mathcal{M}}_{a}^{2}-{\mathcal{M}}_{b}^{2}}{2{\mathcal{M}}_{total}}.$
(134)

Taking the limit $N_{i}\to 0$ reduces the above relation to the known answers
of ref.Damour:2016gwp .

###  C.3 The 2PM mapping

Using $\beta_{1}=1$ from first order matching, the second order deflection
angle with undetermined parameters $\\{\beta\\}$ of the effective metric is
given as

$\displaystyle\left|\vec{\vartheta}^{{\rm eff}}_{G^{2}}\right|$
$\displaystyle=-\pi\frac{M\mu^{2}\left(2\frac{{\mathcal{E}}_{{\rm
eff}}^{2}}{\mu^{2}}-1\right)}{4\sqrt{M^{2}+4{\mathcal{M}}_{{\rm
TN}}^{2}\frac{{\mathcal{E}}_{{\rm
eff}}^{2}}{\mu^{2}}(\frac{{\mathcal{E}}_{{\rm eff}}^{2}}{\mu^{2}}-1)}}$
$\displaystyle\qquad\times\Bigl{[}M^{2}\left(\left(14+4\beta_{2}-2\beta_{3}\right)\frac{{\mathcal{E}}_{{\rm
eff}}^{2}}{\mu^{2}}+\left(2-4\beta_{2}+2\beta_{3}\right)\right)$
$\displaystyle\qquad\qquad-{\mathcal{M}}_{{\rm
TN}}^{2}\left(\left(3-2\beta_{3}\right)\frac{{\mathcal{E}}_{{\rm
eff}}^{2}}{\mu^{2}}+\left(2\beta_{3}+1\right)\right)\Bigr{]}\left(\frac{G}{J}\right)^{2}\,.$
(135)

In the Schwarzschild limit, the effective deflection angle reduces to

$\displaystyle{\left|\vartheta_{Sch}\right|}^{{\rm eff}}_{G^{2}}$
$\displaystyle=-\frac{\pi{\mathcal{M}}_{total}^{2}\mu^{2}}{4}\Bigl{[}\left(\left(4\beta_{2}+11\right)\frac{{\mathcal{E}}_{{\rm
eff}}^{2}}{\mu^{2}}+\left(1-4\beta_{2}\right)\right)\Bigr{]}\left(\frac{G}{J}\right)^{2}.$
(136)

This is in contrast to the second order result from amplitude computations,
which does not depend on the phase $\varphi$ but only on the inertial masses.

$\displaystyle{\left|\vartheta\right|}^{\rm Amp}_{2}$
$\displaystyle=-\frac{\pi{\mathcal{M}}_{total}^{2}\mu^{2}}{4}(15\sigma^{2}-3)\left(\frac{G}{J}\right)^{2}.$
(137)

Apparently (137) cannot be reproduced from (135) by adjusting the $\beta_{2}$
and $\beta_{3}$, but can be matched to (136) under absence of NUT charges. It
seems our expectation of finding a consistent effective metric of EOB
formalism that matches amplitude computations cannot be met by the
generalisation considered.

Using the same dimensionless variables, $u$, we can directly compare the both
and determine $\beta_{2}$ for the $N\to 0$ limit.

$\displaystyle 15u_{r}^{2}+12=(4\beta_{2}+11)u^{2}_{e}+12$
$\displaystyle\Leftrightarrow$ $\displaystyle
g_{3,r}(u_{r})=g_{3,e}(u_{e};\beta_{2}).$ (138)

It is suffice to solve for the zeros of the functions to be the same,
$u_{r,0}=u_{e,0}$.

$\displaystyle
u_{r,0}^{2}=-\frac{12}{15}=u_{e,0}^{2}=-\frac{12}{4\beta_{2}+11}\Rightarrow\beta_{2}=1.$
(139)

The two solutions, $\beta_{1}=1$ and $\beta_{2}=1$, correspond to the second
order post-Minkowskian expansion of the Schwarzschild solution. This gives us
the same energy map (134) obtained from first order analysis.

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