# Quantum corrections to frame-dragging in scattering amplitudes

Jung-Wook Kim (김정욱<EMAIL_ADDRESS>Centre for Theoretical Physics,
Department of Physics and Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, United States

###### Abstract

Frame-dragging effect manifests itself as polarization direction rotation when
linearly polarized electromagnetic/gravitational wave scatters from a spinning
point source through gravitational interactions. Treating general relativity
as an effective field theory, the polarization rotation angle and its quantum
corrections are computed using scattering amplitudes. While the classical
angle is universal as expected from the equivalence principle, the quantum
corrections are found to be different between electromagnetic and
gravitational waves, supporting earlier studies that some formulations of the
equivalence principle seem to be violated in the quantum regime.

††preprint: QMUL-PH-22-20

##  I Introduction

Quantum gravity is incomplete in the sense that no known theory remains valid
at all energy scales. However, it is known that any theory of gravitons
(defined as self-interacting massless spin-2 particles) obeying certain
physical principles such as unitarity, Lorentz invariance, and locality must
reduce to general relativity (GR) at sufficiently low energy or curvature
scales [1, 2, 3]. Moreover, GR as a quantum theory is a perfectly valid
effective theory which gives consistent predictions below the cut-off scale,
where the cut-off is conjectured to be of the order of the Planck mass [4, 5,
6]. Therefore, it is still possible to make definite predictions in quantum
gravity which do not depend on the ultraviolet completion of the theory. The
aim of this study is probing quantum corrections to frame-dragging, also known
as the Lense-Thirring effect.

Frame-dragging is a feature of GR which its Newtonian counterpart does not
exist. Thus, the study of quantum corrections to frame-dragging is a study of
quantum GR, which is qualitatively different from studies of quantum effects
in Newtonian gravity; the former cannot be reduced to the latter. This makes
quantum frame-dragging an interesting subject of study as usual studies of
quantum effects focus on phenomena having Newtonian analogues [4, 5, 7, 8, 9,
10, 11].

Frame-dragging is sourced by angular momentum, and localized distribution of
angular momentum can be approximated as a point source with spin. Spin effects
in quantum GR have already been studied in the literature; as corrections to
the metric [12, 13, 14, 15], or as corrections to the potential [16, 17].
However, both the metric and the potential are gauge-dependent quantities in
GR, which renders them incomplete as observable effects. Moreover, the spins
of the point sources considered in the works were quantum spins having values
of $s=1/2$ or $s=1$. There is a conceptual gap between quantum spin (having
values $s=1/2,1,3/2,\cdots$) and classical spin, and closure of the gap seems
necessary to make connections with potentially realizable experiments; the
frame-dragging effect sourced by quantum spins would be too small to have any
observable consequences. Fortunately, methods to circumvent both limitations
have already been studied in the literature.

The limitation of gauge-dependent quantities can be overcome by studying
scattering observables measured at asymptotic infinity, which are free of
gauge ambiguities. The approach has already been used to argue breakdown of
the equivalence principle in the quantum realm [10, 11, 18, 19]. A scattering
observable that is a direct consequence of frame-dragging effect is the
gravitational Faraday rotation (GFR), where polarization direction of linearly
polarized light (or gravitational wave) is rotated due to the ambient
gravitomagnetic field sourced by angular momentum [20, 21, 22, 23, 24, 25, 26,
27, 28]. The rotation angle can also be computed using scattering amplitudes
of quantum field theory [29], which makes it an ideal observable for studying
quantum frame-dragging.

The conceptual gap between quantum and classical spin has been studied in the
context of scattering amplitude approaches to the classical gravitational two-
body problem [5, 8, 9, 30, 31, 32, 33, 34]. While it is known that certain
classical spin effects can be captured by quantum spins [16, 35, 36, 37, 38],
the description fails to capture terms that are affected by identities
dependent on spin representations of the spinning particle, such as trace
identities. Therefore, higher-spin ($s>2$) particle descriptions are required
to capture the full dynamics of classical spinning particles [39, 40, 41, 42,
43], which has met with great success in determining the classical two-body
dynamics in GR [39, 40, 44, 45, 43, 46, 47, 48, 49, 50, 51]. In this work the
semiclassical Compton amplitude [48], which is defined as the Compton
amplitude in the classical spin limit ($s\to\infty$ with $S=s\hbar$ fixed), is
used. Note that amplitude computations at tree level [39, 40, 44, 41, 45, 52]
and one-loop level [53, 49, 51, 54] with spins in the classical spin limit has
been checked against corresponding classical computations, and found to
reproduce classical spin dependence to all orders in spin, which can be viewed
as an evidence for the validity of the approach.

The GFR angle $\alpha$ and its leading quantum corrections at linear order in
spin are computed to be

$\displaystyle\alpha_{X}(b)$ $\displaystyle=\frac{5\pi
G^{2}m^{2}(\hat{k}\cdot\vec{a})}{4b^{3}}+c_{X}\frac{G^{2}m\hbar(\hat{k}\cdot\vec{a})}{\pi
b^{4}}\,,$ (1a) where $m$ is the mass of the gravitating point source,
$\vec{a}=\vec{S}/m$ is its spin-length vector, $b$ is the impact parameter,
$\hat{k}=\vec{k}/\omega$ is the propagation direction of the
electromagnetic/gravitational wave, and $X=\gamma,h$ denotes particle species.
$\displaystyle\begin{aligned}
c_{\gamma}&=-\frac{994}{15}+32\log\frac{b}{b_{0}}&&&&\text{photon}\\\
c_{h}&=-60+24\log\frac{b}{b_{0}}&&&&\text{graviton}\end{aligned}$ (1b)

The leading classical term of eq.(1a) agrees with known geometric optics limit
results [20, 28, 29]. Similar to the deflection angle [10, 11, 18, 19], the
quantum corrections to the GFR angle differ for different particle species.

In terms of the deflection angle
$\theta=\frac{4Gm}{b}(1+{\mathcal{O}}(\frac{Gm}{b}))$, the GFR angle can be
written as

$\displaystyle\begin{aligned}
\alpha_{X}(\theta)&=\frac{5\pi}{256}(\hat{k}\cdot\vec{\chi})\,\theta^{3}+\tilde{c}_{X}\frac{m_{\text{Pl}}^{2}}{m^{2}}(\hat{k}\cdot\vec{\chi})\,\theta^{4}\,,\\\
\tilde{c}_{X}&=\left\\{\begin{aligned}
\tilde{c}_{\gamma}&=-\frac{497}{240}-\log\frac{\theta}{\theta_{0}}&&&&\text{photon}\\\
\tilde{c}_{h}&=-\frac{15}{8}-\frac{3}{4}\log\frac{\theta}{\theta_{0}}&&&&\text{graviton}\end{aligned}\right.\end{aligned}$
(2)

where $\vec{\chi}=\vec{S}/Gm^{2}$ is the dimensionless spin parameter and
$m_{\text{Pl}}=\sqrt{\hbar/8\pi G}$ is the reduced Planck mass. This
expression has the advantage that all quantities on both sides can (in
principle) be directly measured at asymptotic infinity, and that ill-
definedness of $b$ at the wavepacket scale can be avoided 111A particle’s
position is ill-defined at the wavepacket scale. Another approach would be to
reverse-engineer $b$ from the deflection angle, which is the approach of
eq.(2).. It would be interesting to study whether measurements at asymptotic
infinity open up loopholes that can be used to bypass the known no-go results
for experimental observation of quantum gravity effects [56, 57].

##  II Computation set-up

Consider a massive-massless $2\to 2$ scattering $p_{1}k_{2}\to
p_{1}^{\prime}k_{2}^{\prime}$, where $p$ denotes massive momenta with mass $m$
and $k$ denotes massless momenta, while
$q=k_{2}^{\prime}-k_{2}=p_{1}-p_{1}^{\prime}$ is the transfer momentum. The
massive particle has spin $S^{\mu}$, causing GFR on massless particle’s
polarization through frame-dragging.

The GFR angle $\alpha$ is obtained from the difference of phase gain between
two helicity states of the massless particle. The difference in phase gain is
identified with the difference in eikonal phases $\delta$ of helicity-
preserving scattering amplitudes [29],

$\displaystyle\alpha:=-\frac{\delta_{+}-\delta_{-}}{2|h|}\,,$ (3)

where the eikonal phase $\delta_{\pm}$ is the logarithm of the scattering
amplitude in impact parameter space (IPS) [58, 59, 60, 61, 62, 63, 64, 65].

$\displaystyle\begin{aligned}
e^{i\delta_{\pm}}&:=\int\frac{d^{D}q}{(2\pi)^{D}}P(q)\langle
p_{1}^{\prime},k_{2}^{\prime}(\pm)|S|p_{1},k_{2}(\pm)\rangle\,,\\\
P(q)&=e^{-ib\cdot
q}\hat{\delta}[q\cdot(p_{1}+p_{1}^{\prime})]\hat{\delta}[q\cdot(k_{2}+k_{2}^{\prime})]\,.\end{aligned}$
(4)

Here $\hat{\delta}(x)=2\pi\delta(x)$, and $\pm$ is the massless particle’s
helicity. The classical terms of the eikonal phase satisfy exponentiation
under the adopted projection condition $P(q)$ [66, 67]. Only the non-analytic
terms of the scattering amplitude contribute to finite $b$ eikonal phase,
which are constructible from unitarity-based methods [68, 69, 70].

$M_{L}$$M_{R}$$p_{1}$$p_{4}$$k_{2}$$k_{3}$$\ell_{1}$$\ell$ Figure 1: The two-
graviton cut for the one-loop amplitude
$M_{4}(p_{1},k_{2}^{+h},k_{3}^{-h},p_{4})$. Solid lines denote massive semi-
classical spinning particle, dashed lines denote massless helicity $h$
particle, and wavy lines denote internal exchanged gravitons. $M_{L}$ is the
semi-classical gravitational spinning Compton amplitude eq.(6), and $M_{R}$ is
the two graviton-two helicity $h$ amplitude eq.(7). Figure drawn with package
TikZ-Feynman [71].

The relevant $S$-matrix elements $iM$ can be obtained from the $4\to 0$
amplitude $M_{4}(p_{1},k_{2}^{+h},k_{3}^{-h},p_{4})$ using crossing symmetry.
The relevant tree-level amplitudes were found to not contribute to GFR [29],

$\displaystyle\begin{aligned}
M_{4}^{(0)}&=M_{4,s=0}^{(0)}\times\exp\left(\frac{i[{2}|A|{3}\rangle}{[{2}|p_{1}|{3}\rangle}\right)\,,\\\
\frac{M_{4,s=0}^{(0)}}{(\kappa/2)^{2}}&=\left\\{\begin{aligned}
&\phantom{a}-\frac{[{2}|p_{1}|{3}\rangle^{2}}{t}&&h=1\\\
&\frac{[{2}|p_{1}|{3}\rangle^{4}}{(s-m^{2})(u-m^{2})t}&&h=2\end{aligned}\right.\end{aligned}$
(5)

where $s=(p_{1}+k_{2})^{2}$, $t=(k_{2}+k_{3})^{2}$, $u=(p_{1}+k_{3})^{2}$, and
$\kappa=\sqrt{32\pi G}$. The case $h=2$ corresponds to the semi-classical
spinning Compton amplitude for Kerr black holes valid to quartic order in spin
[39, 44, 72, 42, 48], which also enters the two-graviton cut fig.1 as $M_{L}$.
The vector $A^{\mu}=(k_{2}+k_{3})_{\nu}S^{\mu\nu}(p_{1})$ was introduced to
simplify the notations 222The vector $A^{\mu}$ is manifestly invariant under
the shift $S^{\mu}\to S^{\mu}+\xi q^{\mu}/q^{2}$ of ref.[50] where
$q^{\mu}=-k_{2}^{\mu}-k_{3}^{\mu}$. The invariance of the amplitude under the
shift is violated because the terms dependent on $p_{1}\cdot S=0$ are dropped
when products of Levi-Civita tensors are reduced, while $p_{1}\cdot q\neq 0$.
The exact shift invariance would be restored if all $p_{1}\cdot S$ were kept.
Further discussions can be found in updated versions of ref.[29].. The spin
tensor $S^{\mu\nu}(p_{1})$ is the Lorentz generator $J^{\mu\nu}$ restricted to
the little group space of the incoming particle of momentum $p_{1}$,
satisfying the covariant spin supplementary condition $p_{1\mu}S^{\mu\nu}=0$
and related to the spin vector by
$S^{\mu\nu}=-\frac{1}{m}\epsilon^{\mu\nu\alpha\beta}p_{1\alpha}S_{\beta}$. The
spin vector is the mass rescaled Pauli-Lubanski pseudovector
$S^{\mu}=\frac{-1}{2m}\epsilon^{\mu\alpha\beta\gamma}p_{1\alpha}J_{\beta\gamma}$
[43, 46, 48]. Only the terms up to linear order in spin are considered, but
the extension to higher orders would be straightforward.

The relevant terms of the one-loop amplitude $M_{4}^{(1)}$ can be constructed
from the two-graviton cut fig.1. The cut has non-vanishing contributions only
when internal graviton states have opposite helicities [11], e.g. $\ell^{+}$
and $\ell_{1}^{+}$. For this helicity configuration the subamplitudes are
given as 333The spinor notations follow that of ref.[40] where incoming
positive helicity states are associated with $|{p}]$ spinors. Normalization
for the Levi-Civita tensor is $\epsilon^{0123}=+1$.

$\displaystyle M_{L}^{\ell^{+},-\ell_{1}^{-}}$
$\displaystyle=\frac{\left(-{\kappa}/{2}\right)^{2}[{\ell}|p_{1}|{\ell_{1}}\rangle^{4}}{-4t(k_{2}\cdot\ell)(k_{2}\cdot\ell_{1})}\exp\left(\frac{i[{\ell}|A|{\ell_{1}}\rangle}{[{\ell}|p_{1}|{\ell_{1}}\rangle}\right)\,,$
(6) $\displaystyle M_{R}^{-\ell^{-},\ell_{1}^{+}}$
$\displaystyle=\frac{\left(-{\kappa}/{2}\right)^{2}[{\ell_{1}}|2|{\ell}\rangle^{4}}{-4t(k_{2}\cdot\ell)(k_{2}\cdot\ell_{1})}\left(\frac{\langle
3\ell\rangle}{\langle 2\ell\rangle}\right)^{2h}\,,$ (7)

where $M_{L}=M_{\text{Comp}}(p_{1},\ell^{+2},-\ell_{1}^{-2},p_{4})$ is the
semi-classical spinning Compton amplitude eq.(5), and
$M_{R}=M_{XhhX}(k_{2}^{+h},\ell_{1}^{+2},-\ell^{-2},k_{3}^{-h})$ is the two
graviton-two helicity $h$ amplitude valid for $h\leq 2$. The cases $h=1,2$ are
considered, which correspond to the Einstein-Maxwell theory ($h=1$) and GR
($h=2$).

Using unitarity-based methods [68, 69, 70], the one-loop integrand for
$M_{4}(p_{1},k_{2}^{+h},k_{3}^{-h},p_{4})$ is constructed from the two-
graviton cut as 444The use of four dimensional helicity scheme [90, 91] in
dimensional regularisation of the loop integral is implicit, since four
dimensional spinor variables were used to construct the integrand.

$\displaystyle iM_{4}^{(1)}$
$\displaystyle=\int\hskip-4.0pt\frac{d^{D}\ell}{(2\pi)^{D}}\frac{\left[M_{L}^{\ell^{+},-\ell_{1}^{-}}\hskip-2.0ptM_{R}^{-\ell^{-},\ell_{1}^{+}}\right]+(\ell\leftrightarrow-\ell_{1})}{2\ell^{2}\ell_{1}^{2}}\,,$
(8)

where the “numerator” $M_{L}\times M_{R}$ captures all momentum dependence of
the full one-loop integrand subject to the cut conditions $\ell^{2}=0$ and
$\ell_{1}^{2}=(\ell-k_{2}-k_{3})^{2}=0$, which determines the non-analytic
$t$-dependence of the full amplitude relevant to long-distance physics [9].
The second term $(\ell\leftrightarrow-\ell_{1})$ in the numerator of eq.(8)
denotes opposite helicity configuration for the gravitons. Explicit evaluation
of the first term
$M_{L}^{\ell^{+},-\ell_{1}^{-}}M_{R}^{-\ell^{-},\ell_{1}^{+}}$ for $n$-th
order in spin yields

$\displaystyle\begin{aligned}
\left.\frac{M_{L}M_{R}}{(\kappa/4)^{4}}\right|_{{\mathcal{O}}(S^{n})}\hskip-9.0pt&=\frac{i^{n}}{n!}\frac{[\text{tr}_{-}(\not{k_{2}}\not{\ell_{1}}\not{p_{1}}\not{\ell})]^{4-n}[\text{tr}_{-}(\not{k_{2}}\not{\ell_{1}}\not{A}\not{\ell})]^{n}}{t^{2}(p_{1}\cdot\ell)(p_{1}\cdot\ell_{1})(k_{2}\cdot\ell)(k_{2}\cdot\ell_{1})}\\\
&\phantom{=a}\times\frac{1}{[{3}|p_{1}|{2}\rangle^{2h}}\left(\frac{\text{tr}_{-}(\not{k_{2}}\not{\ell}\not{k_{3}}\not{p_{1}})}{2(k_{2}\cdot\ell)}\right)^{2h}\,,\end{aligned}$
(9)

where
$\text{tr}_{-}(\not{a}\not{b}\cdots)=\text{tr}(\frac{1-\gamma_{5}}{2}\not{a}\not{b}\cdots)$.
The vector $A^{\mu}$ is expanded on the basis formed from external momenta

$\displaystyle
A^{\mu}=a_{1}p_{1}^{\mu}+a_{2}k_{2}^{\mu}+a_{3}k_{3}^{\mu}+a_{4}\epsilon^{\alpha\beta\gamma\mu}p_{1\alpha}k_{2\beta}k_{3\gamma}\,,$
(10)

where the coefficients $a_{i}$ are rational functions of Mandelstam invariants
and spin vector Lorentz invariants

$\displaystyle\\{\,(k_{2}\cdot S)\,,\,(k_{3}\cdot
S)\,,\,\epsilon_{\alpha\beta\gamma\mu}p_{1}^{\alpha}k_{2}^{\beta}k_{3}^{\gamma}S^{\mu}\,\\}\,.$
(11)

All loop momentum dependent Levi-Civita contractions take the form
$\epsilon_{\alpha\beta\gamma\mu}p_{1}^{\alpha}k_{2}^{\beta}k_{3}^{\gamma}\ell^{\mu}$
under the expansion eq.(10). Any expression linear in this Levi-Civita
contraction vanishes after loop momentum integration [19], so they can be
discarded before performing reduction to master integrals.

Reduction to master one-loop integrals was performed using the package LiteRed
v1.83 [76, 77], where the one-loop amplitude is expanded on the basis of
scalar bubble ($I_{2}(t)$), massive ($I_{3}(t,m)$) and massless ($I_{3}(t,0)$)
triangle, and two box ($I_{4}(s,t)$ and $I_{4}(u,t)$) integrals. The results
for ${\mathcal{M}}_{4}(h,n)$ defined by the relations

$\displaystyle\frac{\kappa^{4}{\mathcal{M}}_{4}(h,n)}{[{3}|p_{1}|{2}\rangle^{2h}}=\left.iM_{4}^{(1)}(p_{1},k_{2}^{+h},k_{3}^{-h},p_{4})\right|_{{\mathcal{O}}(S^{n})}\,,$
(12)

expanded in terms of master integrals

$\displaystyle\begin{aligned}
{\mathcal{M}}_{4}(h,n)&=b_{1}I_{4}(s,t)+b_{2}I_{4}(u,t)+t_{1}I_{3}(t,m)\\\
&\phantom{=asdf}+t_{2}I_{3}(t,0)+bI_{2}(t)\,,\end{aligned}$ (13)

are presented in the ancillary file integralcoeffs.m for $h=0,1,2$ and
$n=0,1$, where the coefficients ($b_{1}$, $b_{2}$, $t_{1}$, $t_{2}$, and $b$)
are rational functions of Mandelstam invariants and polynomials of spin
Lorentz invariants eq.(11). The integral coefficients were checked against
overlapping known results in the literature [11, 19, 29] and found to agree.

The eikonal phase eq.(4) is evaluated using the center of momentum (COM) frame
kinematics parametrised as

$\displaystyle\begin{gathered}p_{1}^{\mu}=(\sqrt{m^{2}+\omega^{2}},-\vec{k}+\vec{q}/2)\,,\,k_{2}^{\mu}=(\omega,\vec{k}-\vec{q}/2)\,,\\\
p_{1}^{\prime\mu}=(\sqrt{m^{2}+\omega^{2}},-\vec{k}-\vec{q}/2)\,,\,k_{2}^{\prime\mu}=(\omega,\vec{k}+\vec{q}/2)\,,\end{gathered}$
(16)

where $k:=\frac{k_{2}+k_{2}^{\prime}}{2}$ and $q:=k_{2}^{\prime}-k_{2}$. On-
shell conditions require $\vec{k}\cdot\vec{q}=0$ and
$\omega^{2}=\vec{k}^{2}+\vec{q}^{2}/4$. The $\hbar$ factors are restored
through the rules [78, 36, 79, 29]

$\displaystyle
G\to\frac{G}{\hbar}\,,\,S^{\mu}\to\frac{S^{\mu}}{\hbar}\,,\,\omega\to\hbar\omega\,,\,\vec{k}\to\hbar\vec{k}\,,\,\vec{q}\to\hbar\vec{q}\,,$
(17)

and the classical expansion parameters are chosen as

$\displaystyle\left\\{Gm|\vec{q}|\sim\frac{Gm}{b}\,,\,\frac{|\vec{q}||\vec{S}|}{m}\sim\frac{S}{mb}\,,\,\frac{|\vec{q}|}{\omega}\sim\frac{1}{\omega
b}=\frac{\lambda}{b}\right\\}\,,$ (18)

where $b=|\vec{b}|$ is the impact parameter and $\lambda:=\omega^{-1}$ is the
wavelength of the massless particle. Only the leading terms in $(\omega
b)^{-1}$ are considered, limiting the analyses to the geometric optics regime
[79, 29]. $\hbar=1$ is used unless explicit $\hbar$ counting is necessary. In
the COM frame, the components of the covariant spin vector are

$\displaystyle
S^{\mu}=\left(\frac{\vec{p}_{1}\cdot\vec{S}}{m},\vec{S}+\frac{\vec{p}_{1}(\vec{p}_{1}\cdot\vec{S})}{m(m+E_{1})}\right)\,,$
(19)

where $E_{1}=\sqrt{m^{2}+\omega^{2}}$ and $\vec{p}_{1}=-\vec{k}+\vec{q}/2$.
The following basis of spin Lorentz invariants are used in the eikonal phase
where
$n_{\mu}=\epsilon_{\alpha\beta\gamma\mu}p_{1}^{\alpha}k^{\beta}q^{\gamma}$.

$\displaystyle\begin{aligned} (k\cdot
S)&=-\left\\{\frac{m+\omega}{m}(\vec{k}\cdot\vec{S})-\frac{\omega(\vec{q}\cdot\vec{S})}{2m}\right\\}[1+{\mathcal{O}}(\hbar^{2})]\,,\\\
(q\cdot S)&=-(\vec{q}\cdot\vec{S})[1+{\mathcal{O}}(\hbar^{2})]\,,\\\ (n\cdot
S)&=(m+\omega)\left\\{\vec{q}\cdot(\vec{k}\times\vec{S})\right\\}[1+{\mathcal{O}}(\hbar^{2})]\,.\end{aligned}$

The expressions were expanded to first subleading order in $\hbar$. The little
group factor $[{2^{\prime}}|p_{1}|{2}\rangle$ is

$\displaystyle\begin{aligned}
[{2^{\prime}}|p_{1}|{2}\rangle&=[{2}|p_{1}|{2^{\prime}}\rangle=\sqrt{\text{tr}_{-}(\not{k_{2}}\not{p_{1}}\not{k_{2}^{\prime}}\not{p_{1}})}\\\
&=2(m+\omega)\,\omega\left[1+{\mathcal{O}}((q/\omega)^{2},\hbar^{2})\right]\,.\end{aligned}$
(20)

The $\epsilon^{-1}$ pole subtracted scalar integrals ($D=4-2\epsilon$) are
used when converting to IPS, where $t=-\vec{q}^{2}$ and relevant terms for
leading quantum corrections were kept. The integrals were taken from ref.[80].

$\displaystyle\begin{gathered}I_{2}(t)=\frac{-i}{16\pi^{2}}\log(\vec{q}^{2})\,,\,I_{3}(t,0)=\frac{-i}{32\pi^{2}}\frac{\log^{2}(\vec{q}^{2})}{\vec{q}^{2}}\,,\\\
I_{3}(t,m)=\frac{-i}{32}\left(\frac{1}{m\sqrt{\vec{q}^{2}}}+\frac{\log(\vec{q}^{2})}{\pi^{2}m^{2}}\right)\,,\\\
I_{4}(s,t)+I_{4}(u,t)=\frac{-(m-\omega)}{16\pi
m^{2}\omega}\frac{\log(\vec{q}^{2})}{\vec{q}^{2}}\,,\\\
I_{4}(s,t)-I_{4}(u,t)=\frac{-1}{16\pi
m\omega}\frac{\log(\vec{q}^{2})}{\vec{q}^{2}}\,.\end{gathered}$ (25)

The table for IPS Fourier transform is

$\displaystyle\begin{aligned}
\int\hskip-4.0pt\frac{d^{2}q}{(2\pi)^{2}}e^{i\vec{q}\cdot\vec{b}}\,\frac{1}{\vec{q}^{2}}&=-\frac{\log(b)}{2\pi}\,,\\\
\int\hskip-4.0pt\frac{d^{2}q}{(2\pi)^{2}}e^{i\vec{q}\cdot\vec{b}}\,\frac{1}{\sqrt{\vec{q}^{2}}}&=\frac{1}{2\pi
b}\,,\\\
\int\hskip-4.0pt\frac{d^{2}q}{(2\pi)^{2}}e^{i\vec{q}\cdot\vec{b}}\,\log(\vec{q}^{2})&=-\frac{1}{\pi
b^{2}}\,,\\\
\int\hskip-4.0pt\frac{d^{2}q}{(2\pi)^{2}}e^{i\vec{q}\cdot\vec{b}}\,\frac{\log(\vec{q}^{2})}{\vec{q}^{2}}&=\frac{\log^{2}(b)}{2\pi}\,,\\\
\int\hskip-4.0pt\frac{d^{2}q}{(2\pi)^{2}}e^{i\vec{q}\cdot\vec{b}}\,\frac{\log^{2}(\vec{q}^{2})}{\vec{q}^{2}}&=-\frac{2\log^{3}(b)}{3\pi}\,,\end{aligned}$
(26)

where regulator-dependent terms (e.g. $\log(4\pi)$, $\gamma_{E}$, $\zeta_{n}$,
etc.) were dropped from the ${\mathcal{O}}(\epsilon^{0})$ terms computed from
the master integral eq.(3.14) of ref.[81] ($D-2=2-2\epsilon$).

##  III Results

The eikonal phase eq.(4) is decomposed into the average part $\bar{\delta}$
and the GFR angle $\alpha$ eq.(3) as

$\displaystyle\delta_{\pm\to\pm}=\bar{\delta}\mp|h|\alpha\,,$ (27)

which is further decomposed into five pieces,

$\displaystyle\delta=\delta^{\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}}+\delta^{\Box-\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}^{2}}+\delta^{\bigtriangleup}+\delta^{\bigtriangledown}+\delta^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}\,,$ (28)

where $\delta$ stands for $\bar{\delta}$, $\alpha$, or $\delta_{\pm}$. The
decomposition is based on the contributing integral;
$\delta^{\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}}$ is the
tree, $\delta^{\bigtriangleup}$($\delta^{\bigtriangledown}$) is the
massive(massless) triangle, $\delta^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$ is the massless
bubble, and $\delta^{\Box-\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}^{2}}$ is the
remainder that captures failure of exponentiation by the box contributions,
e.g. 555Since helicity non-preserving amplitudes are present at tree level for
graviton scattering, the eikonal should rather be treated as a matrix with
$\delta_{\pm}$ as diagonal entries and remainder terms should be calculated
using matrix algebra [92, 89]. Separate treatment of $\delta_{+}$ and
$\delta_{-}$ is justified as helicity non-preserving off-diagonal terms are
suppressed at the considered order in $(\omega b)^{-1}$ [29].

$\displaystyle i\delta_{+}^{\Box-\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}^{2}}=\left[\int\hskip-4.0pt\frac{d^{2}q}{(2\pi)^{2}}\frac{\kappa^{4}[b_{1}I_{4}(s,t)+b_{2}I_{4}(u,t)]}{[{2}|p_{1}|{2^{\prime}}\rangle^{2h}}\right]-\frac{[\delta_{+}^{\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}}]^{2}}{2!}\,.$

The remainder term is also known as the quantum remainder [65, 83, 84], but is
known to contribute classically at first subleading order in $(\omega b)^{-1}$
[29]. When writing the eikonal phase, it is convenient to define the vectors

$\displaystyle\hat{k}:=\frac{\vec{k}}{\omega}\,,\,\vec{a}:=\frac{\vec{S}}{m}\,,\,\vec{d}:=\frac{\vec{k}\times\vec{S}}{m\omega}=\hat{k}\times\vec{a}\,.$
(29)

The vector $\hat{k}$ has unit length to the considered approximation order, is
directed along the massless particle’s propagation direction, and is
orthogonal to the IPS.

The tree eikonal phase from eq.(5) is independent of helicity,

$\displaystyle\bar{\delta}^{\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}}=\delta_{\pm}^{\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}}=-4G(m+\omega)\,\omega\left[\log(b)+\frac{\vec{d}\cdot\vec{b}}{b^{2}}\right]\,,$
(30)

and has quantum terms 666As a curiosity, it is remarked that the quantum terms
can be absorbed by redefining $m\to m^{\prime}=m+\omega$, cancelling the
quantum remainder terms of the spinless sector at the same time. This may have
connections to the difference between the on-shell mass and the classical mass
[66]. However, the shift fails to cancel the quantum remainder terms of spin-
linear sector. which did not appear in the previous study, traced back to
different IPS projection condition $P(q)$ and different definition of $\omega$
[29].

The remainder and massive triangle contributions are found to be universal;
they are the same for $h=0,1,2$.

$\displaystyle i\bar{\delta}^{\Box-\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}^{2}}$
$\displaystyle=\frac{2G^{2}m^{2}}{b^{2}}\left[1-\frac{6(\vec{d}\cdot\vec{b})}{b^{2}}+\frac{2\omega\hbar}{m}\left[1-\frac{4(\vec{d}\cdot\vec{b})}{b^{2}}\right]\right]\,,$
$\displaystyle i\alpha^{\Box-\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}^{2}}$
$\displaystyle=\frac{8G^{2}m^{2}(\hat{k}\cdot\vec{a})}{\omega
b^{4}}\left(1+\frac{\omega\hbar}{m}\right)\,,$
$\displaystyle\bar{\delta}^{\bigtriangleup}$ $\displaystyle=\frac{15\pi
G^{2}m^{2}\omega}{4b}\left(1+\frac{\omega\hbar}{m}\right)\left[1-\frac{4}{3}\frac{(\vec{d}\cdot\vec{b})}{b^{2}}\right]$
$\displaystyle\phantom{=a}-\frac{15}{2\pi}\left(\frac{Gm}{b}\right)^{2}\left(\frac{\omega\hbar}{m}\right)\left[1-\frac{8}{3}\frac{(\vec{d}\cdot\vec{b})}{b^{2}}\right]\,,$
(31) $\displaystyle\alpha^{\bigtriangleup}$ $\displaystyle=\frac{5\pi
G^{2}m^{2}(\hat{k}\cdot\vec{a})}{4b^{3}}\left[1-\frac{8}{\pi^{2}}\frac{\hbar}{mb}\right]\,.$
(32)

The remainder terms are purely imaginary, and the classical remainder terms
are $(\omega b)^{-1}$ subleading compared to the classical triangle terms
[29]. Since it is unclear how to interpret the imaginary terms, the remainder
terms are neglected in the final analysis.

The remaining contributions differ for external scalar($\varphi$),
photon($\gamma$), and graviton($h$). The massless triangle contributions have
a universal scaling

$\displaystyle\bar{\delta}_{X}^{\bigtriangledown}$
$\displaystyle=\frac{8G^{2}m\omega\hbar}{\pi
b^{2}}\left[\frac{(\vec{d}\cdot\vec{b})}{b^{2}}+\log(b)\left[1-\frac{2(\vec{d}\cdot\vec{b})}{b^{2}}\right]\right]c_{1,X}^{\bigtriangledown}\,,$
(33) $\displaystyle\alpha_{X}^{\bigtriangledown}$
$\displaystyle=\frac{8G^{2}m\hbar(\hat{k}\cdot\vec{a})}{\pi
b^{4}}\left[-1+\log(b)\right]c_{2,X}^{\bigtriangledown}\,,$ (34)

where

$\displaystyle c_{1,\varphi}^{\bigtriangledown}$ $\displaystyle=3\,,\,$
$\displaystyle c_{1,\gamma}^{\bigtriangledown}$ $\displaystyle=3\,,$
$\displaystyle c_{1,h}^{\bigtriangledown}$ $\displaystyle=4\,,$ (35)
$\displaystyle c_{2,\gamma}^{\bigtriangledown}$ $\displaystyle=4\,,\,$
$\displaystyle c_{2,h}^{\bigtriangledown}$ $\displaystyle=3\,,$ (36)

and $X=\varphi,\gamma,h$ denotes particle species. The bubble contributions
are not as orderly;

$\displaystyle\bar{\delta}_{X}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=\frac{G^{2}m\omega\hbar}{\pi
b^{2}}c_{3,X}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}+\frac{G^{2}m\omega\hbar(\vec{d}\cdot\vec{b})}{\pi
b^{4}}c_{4,X}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}\,,$ (37)
$\displaystyle\alpha_{X}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=\frac{G^{2}m\hbar(\hat{k}\cdot\vec{a})}{\pi
b^{4}}c_{5,X}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}\,.$ (38)

The coefficients are found to be;

$\displaystyle c_{3,\varphi}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=\frac{3}{10}\,,\,$ $\displaystyle
c_{3,\gamma}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=-\frac{161}{30}\,,\,$ $\displaystyle
c_{3,h}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=-\frac{29}{2}\,,$ (39) $\displaystyle
c_{4,\varphi}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=\frac{52}{5}\,,\,$ $\displaystyle
c_{4,\gamma}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=\frac{326}{15}\,,\,$ $\displaystyle
c_{4,h}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=40\,,$ (40) $\displaystyle
c_{5,\gamma}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=-\frac{364}{15}\,,\,$ $\displaystyle
c_{5,h}^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$
$\displaystyle=-26\,.$ (41)

Collecting the results ($\alpha^{\bigtriangleup}$,
$\alpha^{\bigtriangledown}$, and $\alpha^{\mathrel{\mathchoice{\hbox{\hbox
to1.0pt{$\displaystyle($\hss}$\displaystyle)$}}{\hbox{\hbox
to1.0pt{$\textstyle($\hss}$\textstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptstyle($\hss}$\scriptstyle)$}}{\hbox{\hbox
to1.0pt{$\scriptscriptstyle($\hss}$\scriptscriptstyle)$}}}}$) yields the GFR
angle eq.(1), or eq.(2) in dimensionless variables, after restoring reference
scales ($b_{0}$ or $\theta_{0}$) in the logarithms. The difference of the GFR
angle between photons and gravitons is

$\displaystyle\begin{aligned}
\alpha_{\gamma}-\alpha_{h}&=-\left(\frac{94}{15}-8\log\frac{b}{b_{0}}\right)\frac{G^{2}m\hbar(\hat{k}\cdot\vec{a})}{\pi
b^{4}}\\\
&=-\left(\frac{47}{240}+\frac{1}{4}\log\frac{\theta}{\theta_{0}}\right)\frac{m_{\text{Pl}}^{2}}{m^{2}}(\hat{k}\cdot\vec{\chi})\,\theta^{4}\,.\end{aligned}$
(42)

While the remainder term $\alpha^{\Box-\mathrel{\mathchoice{\hbox{\hbox
to0.0pt{$\displaystyle\bot$\hss}$\displaystyle\top$}}{\hbox{\hbox
to0.0pt{$\textstyle\bot$\hss}$\textstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptstyle\bot$\hss}$\scriptstyle\top$}}{\hbox{\hbox
to0.0pt{$\scriptscriptstyle\bot$\hss}$\scriptscriptstyle\top$}}}^{2}}$ is
nonzero, the difference eq.(42) is independent of it because its contribution
is universal.

##  IV Discussions

Measuring classical frame-dragging effect is already a technological challenge
[86], and measuring its quantum corrections does not seem to be possible in
the near future if not impossible (for reference,
$(m_{\text{Pl}}/M_{\odot})^{2}\sim 10^{-78}$ for solar mass $M_{\odot}$).
Nevertheless, the quantum corrections to the GFR angle are interesting from a
theoretical perspective, as the result provides a window into how the
equivalence principle—the founding principle of GR which can be bootstrapped
from physical requirements [87]—should be modified when quantum loops are
present.

In the classical geometric optics approximation the GFR angle is computed by
integrating the Levi-Civita connection along the ray’s trajectory [20], which
is independent of particle’s species and can be understood as a manifestation
of the equivalence principle [28, 29]. But in the quantum regime different
particle species experience different polarization rotation, which can be
attributed to; the difference in the trajectories [10], the difference in the
frame-dragging rate, or both. Any of the suggested options are in tension with
the usual formulation of the equivalence principle, which demands that
massless particles travel on a universal trajectory and experience a universal
frame-dragging rate.

One interpretation is that the difference eq.(42) is a tidal effect induced by
the finite size of the wavepacket, for which the equivalence principle need
not hold. For this interpretation to be correct, the difference should depend
on the size of the wavepacket, which in turn is determined by the massless
particle’s wavelength $\lambda=\omega^{-1}$. However, the quantum suppression
factor $\frac{\hbar}{mb}$ does not depend on $\lambda$, unless the wavelength
dependence is hidden in the reference scale $b_{0}$. This possibility seems
unlikely as the difference is due to massless triangle and bubble
coefficients, and the logarithmic dependence is due to master integrals which
only depend on $t=-\vec{q}^{2}$. It would be desirable to understand the
criteria for determining the reference scale ($b_{0}$ or $\theta_{0}$) of the
logarithms.

As a final remark, the difference of the GFR angle eq.(42) is robust against
redefinitions; it is invariant under the mass redefinition $m\to
m\,(1+\\#\frac{\omega\hbar}{m})$ or the impact parameter redefinition $b\to
b\,(1+\\#\frac{Gm}{b}+\\#\frac{\hbar}{mb})$, where $\\#$ are
${\mathcal{O}}(1)$ $c$-numbers. The former is relevant since on-shell mass and
classical mass can be different [66]. The latter is relevant since the
definition of $b$ may change due to geometry of the kinematics [88], and since
it is ill-defined at the wavepacket scale ($\frac{\hbar}{m}$ is the Compton
wavelength). Whether the conclusions of this work, which are based on
calculations of a single quantum of electromagnetic/gravitational wave
scattering from a point source, still apply for classical waves described by
coherent states [67] would be a subject of future studies.

###### Acknowledgements.

##  V Acknowledgements

The author would like to thank Manuel Accettulli Huber, Andreas Brandhuber,
Gang Chen, Wei-Ming Chen, Stefano De Angelis, Yu-Tin Huang, and Gabriele
Travaglini for discussions and valuable feedback, and especially Manuel
Accettulli Huber for sharing calculations of ref.[89] as an example of using
the package LiteRed. The author would also like to thank Roman N. Lee for help
on using the package LiteRed. The author was supported by the Science and
Technology Facilities Council (STFC) Consolidated Grant ST/T000686/1
“Amplitudes, Strings and Duality”. The author thanks KITP for their
hospitality during the stay at the programme “High-Precision Gravitational
Waves”, where this work was initiated. This research was supported in part by
the National Science Foundation under Grant No. NSF PHY-1748958.

## References

  * Weinberg [1965] S. Weinberg, Phys. Rev. 138, B988 (1965).
  * Feynman [1996] R. P. Feynman,  _Feynman lectures on gravitation_ , edited by F. B. Morinigo, W. G. Wagner, and B. Hatfield (1996).
  * Benincasa and Cachazo [2007] P. Benincasa and F. Cachazo, (2007), arXiv:0705.4305 [hep-th] .
  * Donoghue [1994a] J. F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994a), arXiv:gr-qc/9310024 .
  * Donoghue [1994b] J. F. Donoghue, Phys. Rev. D 50, 3874 (1994b), arXiv:gr-qc/9405057 .
  * Burgess [2004] C. P. Burgess, Living Rev. Rel. 7, 5 (2004), arXiv:gr-qc/0311082 .
  * Khriplovich and Kirilin [2002] I. B. Khriplovich and G. G. Kirilin, J. Exp. Theor. Phys. 95, 981 (2002), arXiv:gr-qc/0207118 .
  * Bjerrum-Bohr  _et al._ [2003a] N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Phys. Rev. D 67, 084033 (2003a), [Erratum: Phys.Rev.D 71, 069903 (2005)], arXiv:hep-th/0211072 .
  * Bjerrum-Bohr  _et al._ [2014] N. E. J. Bjerrum-Bohr, J. F. Donoghue, and P. Vanhove, JHEP 02, 111, arXiv:1309.0804 [hep-th] .
  * Bjerrum-Bohr  _et al._ [2015] N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Planté, and P. Vanhove, Phys. Rev. Lett. 114, 061301 (2015), arXiv:1410.7590 [hep-th] .
  * Bjerrum-Bohr  _et al._ [2016] N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Plante, and P. Vanhove, JHEP 11, 117, arXiv:1609.07477 [hep-th] .
  * Donoghue  _et al._ [2002] J. F. Donoghue, B. R. Holstein, B. Garbrecht, and T. Konstandin, Phys. Lett. B 529, 132 (2002), [Erratum: Phys.Lett.B 612, 311–312 (2005)], arXiv:hep-th/0112237 .
  * Bjerrum-Bohr  _et al._ [2003b] N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Phys. Rev. D 68, 084005 (2003b), [Erratum: Phys.Rev.D 71, 069904 (2005)], arXiv:hep-th/0211071 .
  * Holstein [2006a] B. R. Holstein, Phys. Rev. D 74, 084030 (2006a), arXiv:gr-qc/0607051 .
  * Holstein [2006b] B. R. Holstein, (2006b), arXiv:gr-qc/0607054 .
  * Holstein and Ross [2008] B. R. Holstein and A. Ross, (2008), arXiv:0802.0716 [hep-ph] .
  * Fröb [2016] M. B. Fröb, JHEP 10, 051, [Erratum: JHEP 11, 176 (2016)], arXiv:1607.03129 [hep-th] .
  * Bai and Huang [2017] D. Bai and Y. Huang, Phys. Rev. D 95, 064045 (2017), arXiv:1612.07629 [hep-th] .
  * Chi [2019] H.-H. Chi, Phys. Rev. D 99, 126008 (2019), arXiv:1903.07944 [hep-th] .
  * Ishihara  _et al._ [1988] H. Ishihara, M. Takahashi, and A. Tomimatsu, Phys. Rev. D 38, 472 (1988).
  * Nouri-Zonoz [1999] M. Nouri-Zonoz, Phys. Rev. D 60, 024013 (1999), arXiv:gr-qc/9901011 .
  * Sereno [2004] M. Sereno, Phys. Rev. D 69, 087501 (2004), arXiv:astro-ph/0401295 .
  * Brodutch and Terno [2011] A. Brodutch and D. R. Terno, Phys. Rev. D 84, 121501 (2011), arXiv:1107.1274 [gr-qc] .
  * Farooqui  _et al._ [2014] A. Farooqui, N. Kamran, and P. Panangaden, Adv. Theor. Math. Phys. 18, 659 (2014), arXiv:1306.6292 [math-ph] .
  * Shoom [2021] A. A. Shoom, Phys. Rev. D 104, 084007 (2021), arXiv:2006.10077 [gr-qc] .
  * Deriglazov [2021] A. A. Deriglazov, Phys. Rev. D 104, 025006 (2021), arXiv:2103.07794 [gr-qc] .
  * Chakraborty [2022] C. Chakraborty, Phys. Rev. D 105, 064072 (2022), arXiv:2106.03520 [gr-qc] .
  * Li  _et al._ [2022] Z. Li, W. Zhao, and X. Er, (2022), arXiv:2204.10512 [gr-qc] .
  * Chen  _et al._ [2022] W.-M. Chen, M.-Z. Chung, Y.-t. Huang, and J.-W. Kim, (2022), arXiv:2205.07305 [hep-th] .
  * Neill and Rothstein [2013] D. Neill and I. Z. Rothstein, Nucl. Phys. B 877, 177 (2013), arXiv:1304.7263 [hep-th] .
  * Cheung  _et al._ [2018] C. Cheung, I. Z. Rothstein, and M. P. Solon, Phys. Rev. Lett. 121, 251101 (2018), arXiv:1808.02489 [hep-th] .
  * Bern  _et al._ [2019a] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019a), arXiv:1901.04424 [hep-th] .
  * Bern  _et al._ [2019b] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, JHEP 10, 206, arXiv:1908.01493 [hep-th] .
  * Bern  _et al._ [2021a] Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 126, 171601 (2021a), arXiv:2101.07254 [hep-th] .
  * Vaidya [2015] V. Vaidya, Phys. Rev. D 91, 024017 (2015), arXiv:1410.5348 [hep-th] .
  * Maybee  _et al._ [2019] B. Maybee, D. O’Connell, and J. Vines, JHEP 12, 156, arXiv:1906.09260 [hep-th] .
  * Chiodaroli  _et al._ [2022] M. Chiodaroli, H. Johansson, and P. Pichini, JHEP 02, 156, arXiv:2107.14779 [hep-th] .
  * Febres Cordero  _et al._ [2022] F. Febres Cordero, M. Kraus, G. Lin, M. S. Ruf, and M. Zeng, (2022), arXiv:2205.07357 [hep-th] .
  * Guevara  _et al._ [2019a] A. Guevara, A. Ochirov, and J. Vines, JHEP 09, 056, arXiv:1812.06895 [hep-th] .
  * Chung  _et al._ [2019] M.-Z. Chung, Y.-T. Huang, J.-W. Kim, and S. Lee, JHEP 04, 156, arXiv:1812.08752 [hep-th] .
  * Arkani-Hamed  _et al._ [2020] N. Arkani-Hamed, Y.-t. Huang, and D. O’Connell, JHEP 01, 046, arXiv:1906.10100 [hep-th] .
  * Aoude  _et al._ [2020] R. Aoude, K. Haddad, and A. Helset, JHEP 05, 051, arXiv:2001.09164 [hep-th] .
  * Bern  _et al._ [2021b] Z. Bern, A. Luna, R. Roiban, C.-H. Shen, and M. Zeng, Phys. Rev. D 104, 065014 (2021b), arXiv:2005.03071 [hep-th] .
  * Guevara  _et al._ [2019b] A. Guevara, A. Ochirov, and J. Vines, Phys. Rev. D 100, 104024 (2019b), arXiv:1906.10071 [hep-th] .
  * Chung  _et al._ [2020a] M.-Z. Chung, Y.-T. Huang, and J.-W. Kim, JHEP 09, 074, arXiv:1908.08463 [hep-th] .
  * Kosmopoulos and Luna [2021] D. Kosmopoulos and A. Luna, JHEP 07, 037, arXiv:2102.10137 [hep-th] .
  * Aoude and Ochirov [2021] R. Aoude and A. Ochirov, JHEP 10, 008, arXiv:2108.01649 [hep-th] .
  * Chen  _et al._ [2021] W.-M. Chen, M.-Z. Chung, Y.-t. Huang, and J.-W. Kim, (2021), arXiv:2111.13639 [hep-th] .
  * Aoude  _et al._ [2022a] R. Aoude, K. Haddad, and A. Helset, (2022a), arXiv:2203.06197 [hep-th] .
  * Bern  _et al._ [2022] Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, and F. Teng, (2022), arXiv:2203.06202 [hep-th] .
  * Aoude  _et al._ [2022b] R. Aoude, K. Haddad, and A. Helset, (2022b), arXiv:2205.02809 [hep-th] .
  * Cangemi and Pichini [2022] L. Cangemi and P. Pichini, (2022), arXiv:2207.03947 [hep-th] .
  * Chung  _et al._ [2020b] M.-Z. Chung, Y.-T. Huang, and J.-W. Kim, JHEP 12, 103, arXiv:1911.12775 [hep-th] .
  * Menezes and Sergola [2022] G. Menezes and M. Sergola, (2022), arXiv:2205.11701 [hep-th] .
  * Note [1] A particle’s position is ill-defined at the wavepacket scale. Another approach would be to reverse-engineer $b$ from the deflection angle, which is the approach of eq.(2).
  * Rothman and Boughn [2006] T. Rothman and S. Boughn, Found. Phys. 36, 1801 (2006), arXiv:gr-qc/0601043 .
  * Dyson [2013] F. Dyson, Int. J. Mod. Phys. A 28, 1330041 (2013).
  * Cheng and Wu [1969] H. Cheng and T. T. Wu, Phys. Rev. Lett. 22, 666 (1969).
  * Abarbanel and Itzykson [1969] H. D. I. Abarbanel and C. Itzykson, Phys. Rev. Lett. 23, 53 (1969).
  * Levy and Sucher [1969] M. Levy and J. Sucher, Phys. Rev. 186, 1656 (1969).
  * Amati  _et al._ [1987] D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 197, 81 (1987).
  * Amati  _et al._ [1988] D. Amati, M. Ciafaloni, and G. Veneziano, Int. J. Mod. Phys. A 3, 1615 (1988).
  * Kabat and Ortiz [1992] D. N. Kabat and M. Ortiz, Nucl. Phys. B 388, 570 (1992), arXiv:hep-th/9203082 .
  * Akhoury  _et al._ [2021] R. Akhoury, R. Saotome, and G. Sterman, Phys. Rev. D 103, 064036 (2021), arXiv:1308.5204 [hep-th] .
  * Koemans Collado  _et al._ [2019] A. Koemans Collado, P. Di Vecchia, and R. Russo, Phys. Rev. D 100, 066028 (2019), arXiv:1904.02667 [hep-th] .
  * Brandhuber  _et al._ [2021] A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, JHEP 10, 118, arXiv:2108.04216 [hep-th] .
  * Cristofoli  _et al._ [2021a] A. Cristofoli, R. Gonzo, N. Moynihan, D. O’Connell, A. Ross, M. Sergola, and C. D. White, (2021a), arXiv:2112.07556 [hep-th] .
  * Bern  _et al._ [1994] Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys. B 425, 217 (1994), arXiv:hep-ph/9403226 .
  * Bern  _et al._ [1995] Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys. B 435, 59 (1995), arXiv:hep-ph/9409265 .
  * Bern  _et al._ [1998] Z. Bern, L. J. Dixon, and D. A. Kosower, Nucl. Phys. B 513, 3 (1998), arXiv:hep-ph/9708239 .
  * Ellis [2017] J. Ellis, Comput. Phys. Commun. 210, 103 (2017), arXiv:1601.05437 [hep-ph] .
  * Bautista and Guevara [2019] Y. F. Bautista and A. Guevara, (2019), arXiv:1903.12419 [hep-th] .
  * Note [2] The vector $A^{\mu}$ is manifestly invariant under the shift $S^{\mu}\to S^{\mu}+\xi q^{\mu}/q^{2}$ of ref.[50] where $q^{\mu}=-k_{2}^{\mu}-k_{3}^{\mu}$. The invariance of the amplitude under the shift is violated because the terms dependent on $p_{1}\cdot S=0$ are dropped when products of Levi-Civita tensors are reduced, while $p_{1}\cdot q\neq 0$. The exact shift invariance would be restored if all $p_{1}\cdot S$ were kept. Further discussions can be found in updated versions of ref.[29].
  * Note [3] The spinor notations follow that of ref.[40] where incoming positive helicity states are associated with $|{p}]$ spinors. Normalization for the Levi-Civita tensor is $\epsilon^{0123}=+1$.
  * Note [4] The use of four dimensional helicity scheme [90, 91] in dimensional regularisation of the loop integral is implicit, since four dimensional spinor variables were used to construct the integrand.
  * Lee [2012] R. N. Lee, (2012), arXiv:1212.2685 [hep-ph] .
  * Lee [2014] R. N. Lee, J. Phys. Conf. Ser. 523, 012059 (2014), arXiv:1310.1145 [hep-ph] .
  * Kosower  _et al._ [2019] D. A. Kosower, B. Maybee, and D. O’Connell, JHEP 02, 137, arXiv:1811.10950 [hep-th] .
  * Cristofoli  _et al._ [2021b] A. Cristofoli, R. Gonzo, D. A. Kosower, and D. O’Connell, (2021b), arXiv:2107.10193 [hep-th] .
  * Ellis and Zanderighi [2008] R. K. Ellis and G. Zanderighi, JHEP 02, 002, arXiv:0712.1851 [hep-ph] .
  * Kim and Shim [2021] J.-W. Kim and M. Shim, JHEP 02, 217, arXiv:2010.14347 [hep-th] .
  * Note [5] Since helicity non-preserving amplitudes are present at tree level for graviton scattering, the eikonal should rather be treated as a matrix with $\delta_{\pm}$ as diagonal entries and remainder terms should be calculated using matrix algebra [92, 89]. Separate treatment of $\delta_{+}$ and $\delta_{-}$ is justified as helicity non-preserving off-diagonal terms are suppressed at the considered order in $(\omega b)^{-1}$ [29].
  * Di Vecchia  _et al._ [2021] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, JHEP 07, 169, arXiv:2104.03256 [hep-th] .
  * Haddad [2022] K. Haddad, Phys. Rev. D 105, 026004 (2022), arXiv:2109.04427 [hep-th] .
  * Note [6] As a curiosity, it is remarked that the quantum terms can be absorbed by redefining $m\to m^{\prime}=m+\omega$, cancelling the quantum remainder terms of the spinless sector at the same time. This may have connections to the difference between the on-shell mass and the classical mass [66]. However, the shift fails to cancel the quantum remainder terms of spin-linear sector.
  * Everitt  _et al._ [2011] C. W. F. Everitt _et al._ , Phys. Rev. Lett. 106, 221101 (2011), arXiv:1105.3456 [gr-qc] .
  * Weinberg [1964] S. Weinberg, Phys. Rev. 135, B1049 (1964).
  * Bern  _et al._ [2020] Z. Bern, H. Ita, J. Parra-Martinez, and M. S. Ruf, Phys. Rev. Lett. 125, 031601 (2020), arXiv:2002.02459 [hep-th] .
  * Accettulli Huber  _et al._ [2020] M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, Phys. Rev. D 102, 046014 (2020), arXiv:2006.02375 [hep-th] .
  * Bern and Kosower [1992] Z. Bern and D. A. Kosower, Nucl. Phys. B 379, 451 (1992).
  * Bern  _et al._ [2002] Z. Bern, A. De Freitas, L. J. Dixon, and H. L. Wong, Phys. Rev. D 66, 085002 (2002), arXiv:hep-ph/0202271 .
  * Camanho  _et al._ [2016] X. O. Camanho, J. D. Edelstein, J. Maldacena, and A. Zhiboedov, JHEP 02, 020, arXiv:1407.5597 [hep-th] .