electric current _or_ a magnetic current.

To switch between the two pictures, we recall that Maxwell’s equations with
both electric and magnetic sources read

$\displaystyle d^{\dagger}F=J\,,\quad
d^{\dagger}({\smash{{*}^{-1}F}\vphantom{F}})={\smash{J^{\star}}\vphantom{J}}\,,\quad
d^{\dagger}F:=(\partial^{\nu}F_{\mu\nu})\hskip 1.00006ptdx^{\mu}\,.$ (171)

Electric-magnetic duality states that this set of equations is invariant under

$\displaystyle\text{EMD}\quad:\quad\,F\,\,\mapsto\,\,{\smash{{*}^{-1}F}\vphantom{F}}\,,\quad(J,{\smash{J^{\star}}\vphantom{J}})\,\,\mapsto\,\,({\smash{J^{\star}}\vphantom{J}},-J)\,.$
(172)

It is natural to use the complex combinations of $F$ and
${\smash{{*}^{-1}F}\vphantom{F}}$ that are eigenstates of $*$,

$\displaystyle F^{\pm}:={\textstyle\frac{1}{2}}\Big{(}{F\pm i\hskip
1.00006pt{\smash{{*}^{-1}F}\vphantom{F}}}\Big{)}\,,\quad
J^{\pm}:={\textstyle\frac{1}{2}}\Big{(}{J\pm
i{\smash{J^{\star}}\vphantom{J}}}\Big{)}\quad\Longrightarrow\quad
d^{\dagger}F^{\pm}=J^{\pm}\,.$ (173)

For a given multipole, in the electric picture, we solve

$\displaystyle d^{\dagger}F=J\,,\quad
d^{\dagger}({\smash{{*}^{-1}F}\vphantom{F}})=0\,,$ (174)

while in the magnetic picture, we solve

$\displaystyle d^{\dagger}F=0\,,\quad
d^{\dagger}({\smash{{*}^{-1}F}\vphantom{F}})={\smash{J^{\star}}\vphantom{J}}\,.$
(175)

The two pictures are related such that $F$ away from the source is exactly the
same. In other words, for a “point-like” source, the difference between the
two pictures is ultra-local (delta function supported).

Depending on which picture we choose for each of the two sources, the integral
(170) can take different forms,

$\displaystyle\begin{split}I_{12}&=\int J_{1}(x)\langle A(x)A(y)\rangle
J_{2}(y)=\int{\smash{J^{\star}}\vphantom{J}}_{1}(x)\langle
A^{\star}(x)A^{\star}(y)\rangle{\smash{J^{\star}}\vphantom{J}}_{2}(y)\\\
&=\int J_{1}(x)\langle
A(x)A^{\star}(y)\rangle{\smash{J^{\star}}\vphantom{J}}_{2}(y)=\int{\smash{J^{\star}}\vphantom{J}}_{1}(x)\langle
A^{\star}(x)A(y)\rangle J_{2}(y)\,,\end{split}$ (176)

where we suppressed the vector indices to avoid clutter. If we call $\langle
AA\rangle$ “electric-electric” propagator, we may call $\langle
A^{\star}A^{\star}\rangle$ “magnetic-magnetic”, $\langle AA^{\star}\rangle$
“electric-magnetic”, etc. In the QFT approach to the propagators, which we
will review shortly, we split the mode expansion according to the photon’s
helicity such that888The notation of this section is related to those in
appendix A as $\tilde{A}=-A^{\star}$. The minus sign originates from
${{*}^{-1}}F=-*F$.

$\displaystyle A=A^{+}+A^{-}\,,\quad A^{\star}=-i(A^{+}-A^{-})\,.$ (177)

Since $\langle A^{+}A^{+}\rangle=0=\langle A^{-}A^{-}\rangle$, it follows that

$\displaystyle\begin{split}&\langle AA\rangle=\langle
A^{+}A^{-}\rangle+\langle A^{-}A^{+}\rangle=\langle
A^{\star}A^{\star}\rangle\,,\\\ &\langle AA^{\star}\rangle=+i\left[\langle
A^{+}A^{-}\rangle-\langle A^{-}A^{+}\rangle\right]=-\langle
A^{\star}A\rangle\,.\end{split}$ (178)

So far, our discussion has been general. Now let us focus on the multipole
moments of a root-Kerr particle. As we saw in (168) and (169), we find
electric moments at $\mathcal{O}(y^{2k})$ and magnetic moments at
$\mathcal{O}(y^{2k+1})$. This splitting is expected to be a generic feature of
any parity-preserving spinning charged particle. The NJ shift (166) suggests a
hybrid approach which uses the electric picture for the electric multipoles
and the magnetic picture of the magnetic multipoles. We denote the currents by
$J_{e}$ and ${\smash{J^{\star}}\vphantom{J}}_{o}$ where $e$ and $o$ stand for
even and odd, respectively.

Contributions from different multipole moments simply add up to give

$\displaystyle\begin{split}I_{12}&=\int J_{1,e}(x)\langle A(x)A(y)\rangle
J_{2,e}(y)+\int{\smash{J^{\star}}\vphantom{J}}_{1,o}(x)\langle
A^{\star}(x)A^{\star}(y)\rangle{\smash{J^{\star}}\vphantom{J}}_{2,o}(y)\\\
&\quad+\int J_{1,e}(x)\langle
A(x)A^{\star}(y)\rangle{\smash{J^{\star}}\vphantom{J}}_{2,o}(y)+\int{\smash{J^{\star}}\vphantom{J}}_{1,o}(x)\langle
A^{\star}(x)A(y)\rangle J_{2,e}(y)\,.\end{split}$ (179)

Rewriting it in terms of $\langle A^{\pm}A^{\mp}\rangle$, we find

$\displaystyle I_{12}=\int J_{1,+}(x)\langle A^{+}(x)A^{-}(y)\rangle
J_{2,-}(y)+\int J_{1,-}(x)\langle A^{-}(x)A^{+}(y)\rangle J_{2,+}\,,$ (180)

where, for each source,

$\displaystyle J_{\pm}=J_{e}\mp i{\smash{J^{\star}}\vphantom{J}}_{o}\,.$ (181)

Comparing this with (173), we note a slightly non-trivial “metric” in the
complex basis,

$\displaystyle J_{\pm}=2J^{\mp}\,.$ (182)

Applying it to the root-Kerr particle, we have

$\displaystyle\begin{split}\frac{1}{2}(J_{-})^{\mu}(x)=(J^{+})^{\mu}(x)&=\frac{1}{2}q\int\dot{\bar{z}}^{\mu}(\sigma)\delta^{4}(x-\bar{z}(\sigma))d\sigma\,,\\\
\frac{1}{2}(J_{+})^{\mu}(x)=(J^{-})^{\mu}(x)&=\frac{1}{2}q\int\dot{z}^{\mu}(\sigma)\delta^{4}(x-z(\sigma))d\sigma\,.\end{split}$
(183)

To summarize, we took a long route to explain how the NJ shift (166) can be
understood in conventional descriptions of multipole moments, only to motivate
a less familiar method; the most efficient way to compute the interaction
between two root-Kerr particles is to use the “helicity propagators” $\langle
A^{\pm}A^{\mp}\rangle$.

####  4.2.1 Helicity propagator

Let us present the result first and review the derivation. In terms of self-
dual and anti-self-dual fields, the propagators are Weinberg:1965rz

$\displaystyle\begin{split}\Delta^{+-}_{\mu\nu}(k)&:=\langle
A_{\mu}^{+}(k)A_{\nu}^{-}(-k)\rangle=\frac{i}{k^{2}-i0^{+}}\left[\frac{2k_{(\mu}n_{\nu)}-(k\cdot
n)\eta_{\mu\nu}+i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}n^{\beta}}{2(k\cdot
n)}\right]\,,\\\
\Delta^{-+}_{\mu\nu}(k)&=\Delta^{+-}_{\nu\mu}(k)=[\Delta^{+-}_{\mu\nu}(k)]^{*}\,,\quad\Delta^{++}_{\mu\nu}(k)=\Delta^{--}_{\mu\nu}(k)=0\,,\end{split}$
(184)

where $n_{\mu}$ is an auxiliary reference vector, which we call a “Dirac
string”. The spinor notation offers a more compact expression. Pictorially, we
denote the propagator as

$\fmffile{photon-propagator}\parbox{80.0pt}{ \fmfgraph*(50,30)
\fmfstraight\fmfleft{i1,o1} \fmfright{i2,o2} \fmf{phantom}{i1,v1,o1}
\fmf{phantom}{i2,v2,o2} \fmffreeze\fmf{wiggly,label=$k$,tension=0}{v1,v2}
\fmfv{decor.shape=square,decor.filled=empty,decor.size=10,
label=$A^{+}_{{\smash{\alpha{\dot{\alpha}}}\vphantom{\beta}}}\;$,label.angle=180}{v1}
\fmfv{decor.shape=square,decor.filled=shaded,decor.size=10,
label=$\;A^{-}_{\smash{\beta{\dot{\beta}}}\vphantom{\beta}}$,label.angle=0}{v2}
}\;=\;\Delta^{+-}_{{\smash{\alpha{\dot{\alpha}}\beta{\dot{\beta}}}\vphantom{\beta}}}(k)\;=\;\frac{n_{{\smash{\alpha{\dot{\beta}}}\vphantom{\beta}}}\hskip
0.24994ptk_{{\smash{\beta{\dot{\alpha}}}\vphantom{\beta}}}}{n{\hskip
1.00006pt\cdot\hskip 1.00006pt}k}\frac{i}{k^{2}-i0^{+}}\,.$ (185)

For later purposes, we also note that

$\displaystyle\begin{split}\Delta^{+-}_{\mu\nu}(k)+\Delta^{-+}_{\mu\nu}(k)&=\frac{-i}{k^{2}-i0^{+}}\left(\eta_{\mu\nu}-\frac{2k_{(\mu}n_{\nu)}}{k\cdot
n}\right)\,,\\\
i\Delta^{+-}_{\mu\nu}(k)-i\Delta^{-+}_{\mu\nu}(k)&=\frac{-i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}n^{\beta}}{(k^{2}-i0^{+})(k\cdot
n)}\,.\end{split}$ (186)

The chiral photon field $A_{\mu}^{\pm}$ frequently appears in the form of the
field strength tensor
$F_{\mu\nu}^{\pm}=\partial_{\mu}A_{\nu}^{\pm}-\partial_{\nu}A_{\mu}^{\pm}$. We
consider the combinations

$\displaystyle\langle F_{\mu\nu}^{\pm}(k)A_{\lambda}^{\mp}(-k)\rangle
v^{\lambda}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)\,,\quad\langle
F_{\mu\nu}^{\pm}(k)F_{\lambda\sigma}^{\mp}(-k)\rangle\,.$

These 2pt functions can be expressed without the auxiliary reference vector
$n_{\mu}$, since they can be constructed from the non-chiral photon propagator
(186) using the (anti-)self-dual projectors

$\displaystyle(P^{\pm})_{\mu\nu}^{\alpha\beta}:=\frac{\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}-\delta_{\mu}^{\beta}\delta_{\nu}^{\alpha}\mp
i\epsilon_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}}{4}\,,\quad
F_{\mu\nu}^{\pm}=(P^{\pm})_{\mu\nu}^{\alpha\beta}F_{\alpha\beta}=:(P^{\pm}\cdot
F)_{\mu\nu}\,.$ (187)

resulting in

$\displaystyle\begin{aligned} \langle
F^{\pm}_{\mu\nu}(k)A^{\mp}_{\alpha}(-k)\rangle
v^{\alpha}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)&=(P^{\pm})_{\mu\nu}^{\lambda\sigma}\langle
F_{\lambda\sigma}(k)A_{\alpha}(-k)\rangle\rangle
v^{\alpha}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)\\\ &=\frac{k_{\mu}v_{\nu}-v_{\mu}k_{\nu}\mp
i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}v^{\beta}}{2(k^{2}-i0^{+})}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)\,,\end{aligned}$ (188)

and

$\displaystyle\begin{aligned} \langle
F^{+}_{\mu\nu}(k)F^{-}_{\alpha\beta}(-k)\rangle&=(P^{+})_{\mu\nu}^{\lambda\sigma}(P^{-})_{\alpha\beta}^{\gamma\delta}\langle
F_{\lambda\sigma}(k)F_{\gamma\delta}(-k)\rangle\\\
&=\frac{-i[\eta_{\mu\alpha}k_{\nu}k_{\beta}-\eta_{\nu\alpha}k_{\mu}k_{\beta}-\eta_{\mu\beta}k_{\nu}k_{\alpha}+\eta_{\nu\beta}k_{\mu}k_{\alpha}]}{2(k^{2}-i0^{+})}\\\
&\phantom{=as}+\frac{-\left(k_{\mu}\epsilon_{\nu\alpha\beta\lambda}-k_{\nu}\epsilon_{\mu\alpha\beta\lambda}\right)k^{\lambda}+\left(k_{\alpha}\epsilon_{\beta\mu\nu\lambda}-k_{\beta}\epsilon_{\alpha\mu\nu\lambda}\right)k^{\lambda}}{4(k^{2}-i0^{+})}\,,\end{aligned}$
(189)

where we have dropped the ultra-local (non-pole-possessing) terms. These 2pt
functions can also be derived from mode expansions of chiral photon fields.

We remark that all scattering observables and the classical eikonal (except
for the 1PL eikonal) can be computed from the 2pt functions (188) and (189),
therefore the dependence on the “Dirac string” of (184) is only superficial.

####  4.2.2 Mode expansion for the propagator

We can obtain the helicity propagator (184) through an off-shell extension of
the on-shell mode expansion in QFT. We define the polarisation vectors as
($k^{0}>0,n^{0}>0$)

$\displaystyle\begin{aligned}
\varepsilon_{\mu}^{\pm}(+k^{0},\vec{k})=\varepsilon_{\mu}^{\pm}(\vec{k})\,,&\quad\varepsilon_{\mu}^{\pm}(-k^{0},\vec{k})=-\varepsilon_{\mu}^{\pm}(-\vec{k})\,,\\\
\varepsilon_{\mu}^{+}(k)=\frac{[k|\sigma_{\mu}|n\rangle}{\sqrt{2}\langle
kn\rangle}\,,&\quad\varepsilon_{\mu}^{-}(k)=\frac{\langle
k|\sigma_{\mu}|n]}{\sqrt{2}[kn]}\,,\end{aligned}$ (190)

such that $[\varepsilon_{\mu}^{\pm}(k)]^{\ast}=\varepsilon_{\mu}^{\mp}(-k)$.
The mode expansion of the chiral photon fields $A_{\mu}^{\pm}(x)$ are

$\displaystyle\begin{aligned}
A_{\mu}^{\pm}(x)&=\int_{\vec{k}}\frac{1}{2k^{0}}\left[\varepsilon_{\mu}^{\pm}(\vec{k})a_{\vec{k},\pm}e^{-ik^{0}t+i\vec{k}\cdot\vec{x}}+\left[\varepsilon_{\mu}^{\mp}(\vec{k})\right]^{\ast}(a_{\vec{k},\mp})^{\dagger}e^{+ik^{0}t-i\vec{k}\cdot\vec{x}}\right]\\\
&=\int_{\vec{k}}\frac{\varepsilon_{\mu}^{\pm}(k)}{2k^{0}}\left[a_{\vec{k},\pm}e^{-ik^{0}t+i\vec{k}\cdot\vec{x}}-(a_{\vec{k},\mp})^{\dagger}e^{+ik^{0}t-i\vec{k}\cdot\vec{x}}\right]\,,\end{aligned}$
(191)

where $k^{0}=|\vec{k}|$ and $[A_{\mu}^{\pm}(x)]^{\dagger}=A_{\mu}^{\mp}(x)$.
The creation-annihilation operators satisfy

$\displaystyle\left[a_{\vec{k},h},(a_{\vec{k}^{\prime},h^{\prime}})^{\dagger}\right]=2k^{0}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{(3)}(\vec{k}-\vec{k}^{\prime})\delta_{h,h^{\prime}}\,,\quad
h,h^{\prime}=\pm\,.$ (192)

The usual photon field and the dual photon field are given as (see appendix A)

$\displaystyle\begin{aligned}
A_{\mu}(x)&=A_{\mu}^{+}(x)+A_{\mu}^{-}(x)\,,\,\\\
\tilde{A}_{\mu}(x)&=iA_{\mu}^{+}(x)-iA_{\mu}^{-}(x)\,.\end{aligned}$ (193)

The time-ordered 2pt functions are

$\displaystyle\langle A_{\mu}^{\pm}(x)A_{\nu}^{\pm}(0)\rangle$
$\displaystyle=0\,,$ (194) $\displaystyle\langle
A_{\mu}^{+}(x)A_{\nu}^{-}(0)\rangle$
$\displaystyle=\int_{\vec{k}}\frac{\varepsilon_{\mu}^{+}(\vec{k})[\varepsilon_{\nu}^{+}(\vec{k})]^{\ast}\Theta(t)e^{-i|\vec{k}|t+i\vec{k}\cdot\vec{x}}+[\varepsilon_{\mu}^{-}(\vec{k})]^{\ast}\varepsilon_{\nu}^{-}(\vec{k})\Theta(-t)e^{+i|\vec{k}|t-i\vec{k}\cdot\vec{x}}}{2|\vec{k}|}$
$\displaystyle=\int_{\vec{k}}\frac{\Theta(t)e^{-i|\vec{k}|t+i\vec{k}\cdot\vec{x}}+\Theta(-t)e^{+i|\vec{k}|t-i\vec{k}\cdot\vec{x}}}{2|\vec{k}|}\times\left[-\varepsilon_{\mu}^{+}(k)\varepsilon_{\nu}^{-}(k)\right]$
$\displaystyle=\int_{k^{\mu}}\frac{i[\varepsilon_{\mu}^{+}(k)\varepsilon_{\nu}^{-}(k)]}{k^{2}-i0^{+}}e^{ik\cdot
x}$
$\displaystyle=\int_{k^{\mu}}\frac{i}{k^{2}-i0^{+}}\left[\frac{2k_{(\mu}n_{\nu)}-(k\cdot
n)\eta_{\mu\nu}+i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}n^{\beta}}{2(n\cdot
k)}\right]e^{ik\cdot x}\,,$ (195)

where we used the identity

$\displaystyle\begin{aligned} \int\frac{d\omega}{2\pi}\frac{e^{-i\omega
t}}{-\omega^{2}+\vec{k}^{2}-i0^{+}}&=\int\frac{d\omega}{2\pi}\frac{e^{-i\omega
t}}{(|\vec{k}|-i0^{+}-\omega)(|\vec{k}|-i0^{+}+\omega)}\\\
&=\frac{+i}{2|\vec{k}|}\left[\Theta(t)e^{-i|\vec{k}|t}+\Theta(-t)e^{+i|\vec{k}|t}\right]\,.\end{aligned}$
(196)

The computation can be repeated for 2pt functions of field strength tensors,
which can be used to justify the propagators (188) and (189). For example,
(188) can be computed from the substitution

$\displaystyle\varepsilon_{\mu}^{+}(k)\varepsilon_{\nu}^{-}(k)\to
2i[k_{[\mu}\varepsilon_{\nu]}^{+}(k)\varepsilon_{\lambda}^{-}(k)]v^{\lambda}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot
v)=\frac{ik_{[\mu}[k|\sigma_{\nu]}|n\rangle[n|v|k\rangle}{\langle
kn\rangle[nk]}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)\,,$

in the second line of (195) before using the identity (196), where
$k^{\mu}=(|\vec{k}|,\vec{k})$ satisfies the on-shell condition $k^{2}=0$.
Using the delta constraint, we can recast the numerator as

$\displaystyle\frac{ik_{[\mu}[k|\sigma_{\nu]}|n\rangle[n|v|k\rangle}{\langle
kn\rangle[nk]}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)$
$\displaystyle=-\frac{i}{4}[k|\sigma_{\mu}\sigma_{\nu}v|k\rangle\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)$
$\displaystyle=-\frac{i}{2}\left(k_{\mu}v_{\nu}-k_{\nu}v_{\mu}-i\epsilon_{\mu\nu\alpha\beta}k^{\alpha}v^{\beta}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)\,,$

which leads to the 2pt function (188) after off-shell continuation $k^{2}\neq
0$ using (196).

With the modified definitions for the mode operators ($k^{0}>0$)

$\displaystyle
a_{k^{\mu},\pm}=a_{\vec{k},\pm}\,,\,a_{-k^{\mu},\pm}=-a_{-\vec{k},\mp}^{\dagger}\,,\,k^{\mu}=(k^{0},\vec{k})\,,$
(197)

we can rewrite the mode expansion (191) as

$\displaystyle A_{\mu}^{\pm}(x)$
$\displaystyle=\int_{k}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k^{2})\,\varepsilon_{\mu}^{\pm}(k)\,a_{k,\pm}\,e^{ik\cdot
x}\quad\stackrel{{\scriptstyle\text{off-
shell}}}{{\longrightarrow}}\quad\int_{k}A_{\mu}^{\pm}(k)\,e^{ik\cdot x}\,,$
(198)

which is the off-shell continued form used to obtain Feynman rules.

##  5 Compton amplitude

In this section, we compute the classical Compton amplitudes for a root-Kerr
particle, and compare them with similar results in the literature. We find
perfect agreement to the linear order in spin (at $g=2$), whereas we find
model-dependent discrepancy starting from the quadratic order in spin.

##### 3-point coupling

The shift (166) induces the 3-point coupling of an incoming positive helicity
photon,

$\displaystyle iA_{3}\sim iq(\varepsilon^{+}\cdot v)e^{-k\cdot y}e^{ik\cdot
b}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v)\,,$ (199)

where $k^{\mu}$ is photon momentum. This agrees with the black hole 3-point
coupling in the literature. For example, appendix B. of Chen:2021kxt  gives
the minimal coupling as

$\displaystyle M_{3}^{\eta,s}$
$\displaystyle=M_{3}^{\eta,s=0}\times\exp[-\eta\kappa_{0}(k\cdot a)]\,,$ (200)

where $\eta=\pm 1$ is the helicity sign of the incoming massless quanta of
momentum $k^{\mu}$, $a^{\mu}$ is the spin-length vector $(y^{\mu}=-a^{\mu})$,
and $\kappa_{0}=\eta_{00}$ is the metric convention parameter.

###  5.1 Computation

The Compton amplitudes will first be computed using the Feynman rules derived
from the interaction term expansion (162) to parallel the WQFT computations in
the literature Wang:2022ntx ; Kim:2023drc , which will be reorganised into a
form that connects more naturally to the Feynman rules of the alternative
expansion (165).

compton-a (40,70) i1,a1,b1,o1 i2,a2,b2,o2
decor.shape=square,decor.filled=empty,decor.size=10a1
decor.shape=square,decor.filled=empty,decor.size=10b1
decor.shape=square,decor.filled=shaded,decor.size=10a2
decor.shape=square,decor.filled=shaded,decor.size=10b2 dotsi1,a1 dotsb1,o1
plain,width=1a1,b1 phantomi2,a2,b2,o2 wiggly,tension=0a1,a2
wiggly,tension=0b1,b2

(a) same helicity

compton-c (40,70) i1,a1,b1,o1 i2,a2,b2,o2
decor.shape=square,decor.filled=empty,decor.size=10a1
decor.shape=square,decor.filled=shaded,decor.size=10b1
decor.shape=square,decor.filled=shaded,decor.size=10a2
decor.shape=square,decor.filled=empty,decor.size=10b2 dotsi1,a1 dotsb1,o1
phantomi2,a2,b2,o2 plain,width=1a1,b1 wiggly,tension=0a1,a2
wiggly,tension=0b1,b2

(b) opposite helicity

Figure 2: Diagrams for Compton amplitudes.

####  5.1.1 Same helicity

The diagram in Figure 2(a) gives the same helicity amplitude. We need to
compute the following 2-point function of linear $A^{+}$-fluctuation coupling
terms (163), where we substitute the $A^{+}_{\mu}$ fields by the polarisation
vectors of the external photons.

$\displaystyle\begin{split}iA_{4}^{++}&=\left\langle
V^{+}(\varepsilon_{3},k_{3})\times
V^{+}(\varepsilon_{4},k_{4})\right\rangle\,,\\\
V^{+}(\varepsilon,k)&=-q\int_{k,\omega}\varepsilon_{\mu}^{+}(k)\left[\kappa
v^{\mu}k_{\nu}-\omega\delta^{\mu}_{\nu}\right]\delta z^{\nu}(\omega)e^{ik\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[\kappa(k\cdot v)-\omega]\,.\end{split}$ (201)

Using the $\langle\delta z\delta z\rangle$ 2-point function (148) and
contracting the tensor indices, we get

$\displaystyle iA_{4}^{++}$
$\displaystyle=\frac{iq^{2}}{m}e^{i(k_{3}+k_{4})\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot v]$ (202)
$\displaystyle\phantom{=}\times\left\\{\frac{(k_{3}\cdot
k_{4})(\varepsilon_{3}^{+}\cdot v)(\varepsilon_{4}^{+}\cdot v)}{(k_{4}\cdot
v)^{2}}+\frac{(\varepsilon_{3}^{+}\cdot k_{4})(\varepsilon_{4}^{+}\cdot
v)-(\varepsilon_{3}^{+}\cdot v)(\varepsilon_{4}^{+}\cdot k_{3})}{(k_{4}\cdot
v)}-(\varepsilon_{3}^{+}\cdot\varepsilon_{4}^{+})\right\\}\,.$

This is a simple shift of the non-spinning sector results Kim:2023drc  by an
exponential spin factor. The sign difference compared to (3.26) of the same
reference comes from metric conventions. We may also write the amplitude in a
gauge-invariant form as

$\displaystyle iA_{4}^{++}$ $\displaystyle=\frac{iq^{2}}{m}\frac{(v\cdot
F_{3}^{+}\cdot F_{4}^{+}\cdot v)}{(k_{4}\cdot v)^{2}}e^{i(k_{3}+k_{4})\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot v]\,,$ (203)

where
$F_{\mu\nu}^{\pm}=i(k_{\mu}\varepsilon_{\nu}^{\pm}-k_{\nu}\varepsilon_{\mu}^{\pm})$
is the on-shell field strengths of the external photons and $(v\cdot
F_{1}\cdot
F_{2}\cdots)=v_{\mu_{1}}F_{1~{}~{}\mu_{2}}^{~{}\mu_{1}}F_{2~{}~{}\mu_{3}}^{~{}\mu_{2}}\cdots$
is a shorthand notation for a concatenation of tensor contractions. The
expression (203) can be obtained directly from the Feynman rules corresponding
to the alternative interaction term expansion (165).

####  5.1.2 Opposite helicity

The opposite helicity Compton amplitude comes from the diagram in Figure 2(b).
We may simplify the relevant expression in (148) as

$\displaystyle\langle\delta\bar{z}^{\mu}(\omega^{\prime})\delta
z^{\nu}(\omega)\rangle$
$\displaystyle\approx\frac{2i}{m}\left[\frac{\eta^{\mu\nu}}{2\omega^{2}}+\frac{i\epsilon^{\mu\nu\lambda\sigma}v_{\lambda}y_{0\sigma}}{\omega}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,.$ (204)

We discarded terms proportional to $v^{\mu}$ using the fact that the vertex
rules at linear order in perturbations satisfy the “Ward identity” and vanish
under the substitution $\delta z^{\mu}\to v^{\mu}$.

$\displaystyle\begin{aligned} &iq\int_{k,\omega}A_{\mu}^{+}(k)\left[\kappa
v^{\mu}k_{\nu}-\omega\delta^{\mu}_{\nu}\right]\delta z^{\nu}(\omega)e^{ik\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[\kappa(k\cdot v)-\omega]\\\ &=-i\kappa
q\int_{k,\omega}A_{\mu}^{+}(k)(k\cdot
v)\left[\delta^{\mu}_{\nu}-\frac{v^{\mu}k_{\nu}}{k\cdot v}\right]\delta
z^{\nu}(\omega)e^{ik\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[\kappa(k\cdot v)-\omega]\,.\end{aligned}$ (205)

Introducing the notation
$a^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}v_{\alpha}y_{0\beta}$, the resulting
Compton amplitude becomes

$\displaystyle\begin{split}iA_{4}^{+-}&=\frac{iq^{2}}{m}e^{i(k_{3}\cdot
z_{0}+k_{4}\cdot\bar{z}_{0})}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot v]\\\ &\phantom{=}\times\left\\{\frac{(k_{3}\cdot
k_{4})(\varepsilon_{3}^{+}\cdot v)(\varepsilon_{4}^{-}\cdot v)}{(k_{4}\cdot
v)^{2}}+\frac{(\varepsilon_{3}^{+}\cdot k_{4})(\varepsilon_{4}^{-}\cdot
v)-(\varepsilon_{3}^{+}\cdot v)(\varepsilon_{4}^{-}\cdot k_{3})}{(k_{4}\cdot
v)}\right.\\\
&\phantom{=}\left.\phantom{asdf}-(\varepsilon_{3}^{+}\cdot\varepsilon_{4}^{-})+\frac{2i(k_{3}\cdot
a\cdot k_{4})(\varepsilon_{3}^{+}\cdot v)(\varepsilon_{4}^{-}\cdot
v)}{(k_{4}\cdot v)}-2i(\varepsilon_{3}^{+}\cdot
a\cdot\varepsilon_{4}^{-})(k_{4}\cdot v)\right.\\\
&\phantom{=}\left.\phantom{\frac{(k_{3}\cdot a\cdot k_{4})}{(k_{4}\cdot
v)}}+2i(\varepsilon_{3}^{+}\cdot a\cdot k_{4})(\varepsilon_{4}^{-}\cdot
v)-2i(k_{3}\cdot a\cdot\varepsilon_{4}^{-})(\varepsilon_{3}^{+}\cdot
v)\right\\}\,.\end{split}$ (206)

Similar to (203), the amplitude can be written in a gauge-invariant form as

$\displaystyle iA_{4}^{+-}$
$\displaystyle=\frac{iq^{2}}{m}\left\\{\frac{(v\cdot F_{3}^{+}\cdot
F_{4}^{-}\cdot v)}{(k_{4}\cdot v)^{2}}+2i\frac{(v\cdot F_{3}^{+}\cdot a\cdot
F_{4}^{-}\cdot v)}{(k_{4}\cdot v)}\right\\}e^{i(k_{3}\cdot
z_{0}+k_{4}\cdot\bar{z}_{0})}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot v]\,,$ (207)

which is more natural when the alternative interaction term expansion (165) is
used for the Feynman rules.

###  5.2 Comparison

We compare our classical Compton amplitudes with existing results in the
literature, setting $g=2$. The linear-in-spin amplitude should agree, since it
is the universal part captured by the Thomas-Bargmann-Michel-Telegdi (TBMT)
equation. We may find model-dependent discrepancies starting from the
quadratic order.

The deviation at quadratic order in spin is an analogue of possible $R^{2}$
type couplings at $\mathcal{O}(S^{4})$ in the gravitational case Bern:2022kto
. Such curvature-squared type couplings ($R^{2}$ or $F^{2}$) have an
interpretation as contributions from induced multipole moments, requiring
dimensionful coefficients for their correct normalisation; $[ML^{4}]$ for
gravity and $[M^{-1}L^{2}]$ for electromagnetism at the leading order. We can
introduce such operators without introducing any additional length scale in
the case of spinning objects, since the spin-length vector $a^{\mu}$ provides
the necessary length scale, the spin order being $\mathcal{O}(S^{4})$ for
gravity and $\mathcal{O}(S^{2})$ for electromagnetism for the leading order
curvature-squared operators.

####  5.2.1 Comparison with SUSY WQFT calculations

To compare the two results (202) and (206) with those of ref. Kim:2023drc , we
use the explicit polarisation vectors

$\displaystyle\varepsilon_{3\mu}^{+}=\frac{[3|\bar{\sigma}_{\mu}|4\rangle}{\sqrt{2}\langle
34\rangle}\,,\quad\varepsilon_{4\mu}^{-}=\frac{[3|\bar{\sigma}_{\mu}|4\rangle}{\sqrt{2}[43]}\,,\quad\varepsilon_{4\mu}^{+}=\frac{[4|\bar{\sigma}_{\mu}|3\rangle}{\sqrt{2}\langle
43\rangle}\,,$ (208)

and the complex conjugation conditions

$\displaystyle\left(\lambda_{\alpha}\right)^{\ast}=\text{sgn}(p^{0})\bar{\lambda}_{\dot{\alpha}}\,.$
(209)

We use $\text{sgn}(k_{3}^{0})\text{sgn}(k_{4}^{0})=-1$ because one of the
massless photons has to be ingoing and the other has to be outgoing. In the
rest frame of $v^{\mu}=(1,\vec{0})$ where $\omega=k_{3}^{0}$ is the energy of
the photon, $k^{\mu}=k_{3}^{\mu}+k_{4}^{\mu}$ is the transfer momentum, and

$\displaystyle\begin{gathered}(k_{4}\cdot v)=-(k_{3}\cdot
v)=\omega\,,\,|\langle 34\rangle|^{2}=\langle
34\rangle[34]={k^{2}}=4\omega^{2}\sin^{2}(\theta/2)=|[34]|^{2}\,,\,\\\
|[3|v|4\rangle|^{2}={-[3|v|4|v|3\rangle}={4(v\cdot
k_{4})^{2}-k^{2}}=4\omega^{2}(1-\sin^{2}(\theta/2))=|[4|v|3\rangle|^{2}\,,\end{gathered}$
(212)

where $\theta$ is the scattering angle and we localised onto
$(k_{3}+k_{4})\cdot v=0$ for the second expression.

The same helicity amplitude becomes

$\displaystyle iA_{4}^{++}$ $\displaystyle=\frac{iq^{2}}{m}e^{-k\cdot
y_{0}}\sin^{2}(\theta/2)$ (213)

which is simple to evaluate because the $\langle 34\rangle^{-2}$ factors out
in the calculations, which we substitute by $|\langle 34\rangle|^{-2}$. For
the opposite helicity amplitude, we get

$\displaystyle\begin{aligned}
iA_{4}^{+-}&=\frac{iq^{2}}{m}e^{(k_{4}-k_{3})\cdot
y_{0}}\frac{1}{-2k^{2}}\left\\{\frac{k^{2}(n\cdot
v)^{2}}{2\omega^{2}}+\frac{2i\epsilon[k_{3},k_{4},v,y_{0}](n\cdot
v)^{2}}{\omega}-2i(k\cdot a\cdot n)(n\cdot v)\right\\}\,,\end{aligned}$ (214)

where $n^{\mu}=[3|\bar{\sigma}^{\mu}|4\rangle$ vector carries the helicity
weights. Note that this form is _manifestly shift-symmetric_ ; the expression
is invariant under the shift of the spin vector by

$\displaystyle S^{\mu}\to S^{\mu}+\xi k^{\mu}/k^{2}\,,$

where $\xi$ is an arbitrary parameter. The shift symmetry is one of the
conjectures for tensor structures of spinning black holes Aoude:2022trd ;
Bern:2022kto .

Now we multiply the factor $\frac{[4|v|3\rangle^{2}}{|[4|v|3\rangle|^{2}}$ to
compensate the helicity weights, and use the identity for
$f^{\mu}=-a^{\mu\nu}k_{\nu}$,

$\displaystyle\begin{aligned} \right)\\\ &=-4\omega(k_{3}\cdot a\cdot
k_{4})+2i\epsilon[f,k_{3},k_{4},v]\\\
&=-4\omega\epsilon[k_{3},k_{4},v,y_{0}]-ik^{2}(k_{3}-k_{4})\cdot
y_{0}\end{aligned}$

to obtain

$\displaystyle\begin{aligned}
iA_{4}^{+-}&=\frac{iq^{2}}{m}e^{(k_{4}-k_{3})\cdot
y_{0}}\left\\{-\cos^{2}(\theta/2)-(k_{3}-k_{4})\cdot
y_{0}+i\frac{\epsilon[k_{3},k_{4},v,y_{0}]}{\omega}\right\\}\\\
&=\frac{iq^{2}}{m}\left\\{-\cos^{2}(\theta/2)+(\cos^{2}(\theta/2)-1)(k_{3}-k_{4})\cdot
y_{0}+i\frac{\epsilon[k_{3},k_{4},v,y_{0}]}{\omega}+\mathcal{O}(y_{0}^{2})\right\\}\,.\end{aligned}$
(215)

Setting $g=2$, the results in (3.27) of ref. Kim:2023drc  for $h=1$ become999A
factor of 2 has been removed from the results of ref. Kim:2023drc ; the factor
is due to overcounting $s$-channel and $u$-channel diagrams.

$\displaystyle iA_{4}^{+-}$
$\displaystyle=\frac{iq^{2}}{m}\left[-\cos^{2}(\theta/2)+\sin^{2}(\theta/2)\kappa_{0}[(k_{1}-k_{4})\cdot
a]-\frac{i}{\omega}(-\kappa_{1})\epsilon[k_{1},k_{4},a,v]\right]\,,\,$ (216a)
$\displaystyle iA_{4}^{++}$
$\displaystyle=\frac{iq^{2}}{m}\sin^{2}(\theta/2)\left[1-\kappa_{0}[(k_{1}+k_{4})\cdot
a]\right]\,,$ (216b)

where $a^{\mu}=S^{\mu}/m$, and we have restored the convention parameters
$\kappa_{0}$ and $\kappa_{1}$; ref. Kim:2023drc  uses
$\kappa_{0}=\eta_{00}=+1$ and $\kappa_{1}=\epsilon_{0123}=-1$. The expressions
match perfectly when we set $k_{1}^{\mu}=k_{3}^{\mu}$ and
$y_{0}^{\mu}=-a^{\mu}$.

####  5.2.2 Comparison with amplitude calculations

For the spin squared coupling, we compare our result with the minimal coupling
amplitude constructed from BCFW recursion Chen:2021kxt

$\displaystyle\begin{aligned}
A_{4}^{+-}&\propto\frac{[3|p_{1}|4\rangle^{2}}{(s-m^{2})(u-m^{2})}\exp\left[-i\frac{k^{\mu}[3|\sigma^{\nu}|4\rangle
S_{\mu\nu}}{\kappa_{0}[3|p_{1}|4\rangle}\right]\,,\end{aligned}$ (217)

where we have restored the metric convention parameter $\kappa_{0}$. Matching
the overall normalisation101010As remarked in ref. Kim:2023drc , there is a
mass factor difference between QFT amplitude results and WQFT amplitude
results, which can be interpreted as the ratio of
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot v)$
to $\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(2k\cdot
p)$. we write the amplitude as

$\displaystyle iA_{4}^{+-}$ $\displaystyle=-\frac{iq^{2}}{m}\frac{(n\cdot
v)^{2}}{4(k_{4}\cdot
v)^{2}}\exp\left[i\frac{\epsilon[k_{3}+k_{4},n,v,a]}{(n\cdot v)}\right]\,,$
(218)

where we set $p_{1}^{\mu}=mv^{\mu}$ and take the classical limit for the
Mandelstam invariants as

$\displaystyle s-m^{2}=-(u-m^{2})+\mathcal{O}(\hbar^{2})=2m(k_{4}\cdot
v)+\mathcal{O}(\hbar^{2})\,.$ (219)

The exponent can be simplified to

$\displaystyle\frac{i\epsilon[k_{3}+k_{4},n,v,a]}{(n\cdot
v)}=(k_{3}-k_{4})\cdot a-\frac{[(k_{3}-k_{4})\cdot v](n\cdot a)}{(n\cdot v)}$
(220)

using the identity

$\displaystyle
i\varepsilon^{\mu\nu\lambda\sigma}=\frac{(\sigma^{\mu}\bar{\sigma}^{\nu}\sigma^{\lambda}\bar{\sigma}^{\sigma})_{\alpha}{}^{\alpha}-(\sigma^{\nu}\bar{\sigma}^{\lambda}\sigma^{\sigma}\bar{\sigma}^{\mu})_{\alpha}{}^{\alpha}}{-4}$
(221)

and Schouten identities. It leads to the form of the amplitude used in
HPET/HEFT Aoude:2022trd ; Aoude:2022thd ; Aoude:2023vdk  and BHPT
Bautista:2021wfy ; Bautista:2022wjf  approaches,

$\displaystyle i(A_{4}^{+-})_{\mathrm{HEFT}}$
$\displaystyle=-\frac{iq^{2}}{m}\frac{(n\cdot v)^{2}}{4(k_{4}\cdot
v)^{2}}e^{(k_{3}-k_{4})\cdot a}\exp\left[\frac{2(k_{4}\cdot v)(n\cdot
a)}{(n\cdot v)}\right]$ (222)

after localising onto $\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot v]$, where the amplitude is regular up to order
$\mathcal{O}(a^{2})$. We compare it to the classical Compton amplitude of our
twistor model,

$\displaystyle\begin{aligned}
iA_{4}^{+-}&=\frac{iq^{2}}{m}e^{(k_{3}-k_{4})\cdot a}\left\\{-\frac{(n\cdot
v)^{2}}{4(k_{4}\cdot v)^{2}}+\frac{i\epsilon[k_{3},k_{4},v,a](n\cdot
v)^{2}}{2(k_{3}\cdot k_{4})(k_{4}\cdot
v)}-\frac{i\epsilon[k_{3}+k_{4},n,v,a](n\cdot v)}{2(k_{3}\cdot
k_{4})}\right\\}\\\ &=-\frac{iq^{2}}{m}\frac{(n\cdot v)^{2}}{4(k_{4}\cdot
v)^{2}}e^{(k_{3}-k_{4})\cdot a}\left\\{1-\frac{2(k_{4}\cdot
v)^{2}}{(k_{3}\cdot k_{4})}\left[\frac{i\epsilon[k_{3},k_{4},v,a]}{(k_{4}\cdot
v)}-\frac{i\epsilon[k_{3}+k_{4},n,v,a]}{(n\cdot
v)}\right]\right\\}\end{aligned}$

where we used $y_{0}^{\mu}=-a^{\mu}$. We can use the identities
($k=k_{3}+k_{4}$)

$\displaystyle\begin{aligned} (v\cdot n)\epsilon[k,k_{4},v,a]&=(v\cdot
k)\epsilon[n,k_{4},v,a]+(v\cdot k_{4})\epsilon[k,n,v,a]\\\
&\phantom{=asdf}+(v\cdot v)\epsilon[k,k_{4},n,a]+(v\cdot
a)\epsilon[k,k_{4},v,n]\\\ &=(k_{4}\cdot
v)\epsilon[k,n,v,a]-\epsilon[k_{3},k_{4},n,a]\,,\end{aligned}$ (223)

and

$\displaystyle i\epsilon[k_{3},k_{4},n,a]$ $\displaystyle=(k_{3}\cdot
k_{4})(n\cdot a)\,,$ (224)

to write the twistor Compton amplitude as

$\displaystyle i(A_{4}^{+-})_{\mathrm{twistor}}$
$\displaystyle=-\frac{iq^{2}}{m}\frac{(n\cdot v)^{2}}{4(k_{4}\cdot
v)^{2}}e^{(k_{3}-k_{4})\cdot a}\left\\{1+\frac{2(k_{4}\cdot v)(n\cdot
a)}{(n\cdot v)}\right\\}\,.$ (225)

The difference between (222) and (225) is a non-pole-possessing term,

$\displaystyle i\Delta A_{4}^{+-}$
$\displaystyle=-\frac{iq^{2}}{m}e^{(k_{3}-k_{4})\cdot a}(n\cdot
a)^{2}+\mathcal{O}(a^{3})\,.$ (226)

This term can be reverse-engineered to find a $\mathcal{O}(a^{2})$ worldline
contact term that generates this contribution. From the definition of the
polarisation vector (208) we relate the $n^{\mu}$ vectors to the polarisation
vectors as

$\displaystyle n^{\mu}n^{\nu}=-4(k_{3}\cdot
k_{4})\varepsilon_{3}^{+\mu}\varepsilon_{4}^{-\nu}\,.$ (227)

To ensure the Ward identity, we use the substitution rule

$\displaystyle\begin{aligned} n^{\mu}n^{\nu}&\to-4\left[(k_{3}\cdot
k_{4})\varepsilon_{3}^{+\mu}\varepsilon_{4}^{-\nu}-(\varepsilon_{3}^{+}\cdot
k_{4})k_{3}^{\mu}\varepsilon_{4}^{-\nu}-(k_{3}\cdot\varepsilon_{4}^{-})\varepsilon_{3}^{+\mu}k_{4}^{\nu}+(\varepsilon_{3}^{+}\cdot\varepsilon_{4}^{-})k_{3}^{\mu}k_{4}^{\nu}\right]\end{aligned}$
(228)

with symmetrisation if necessary. We may also write this substitution as

$\displaystyle n_{\mu}n_{\nu}$ $\displaystyle\to
4\eta^{\alpha\beta}F_{\alpha\mu}^{+}(k_{3})F_{\beta\nu}^{-}(k_{4})\,,\,F_{\mu\nu}^{\pm}(k):=ik_{\mu}\varepsilon^{\pm}_{\nu}(k)-ik_{\nu}\varepsilon^{\pm}_{\mu}(k)\,,$
(229)

where $F^{\pm}_{\mu\nu}(k)$ is the momentum space mode coefficient of the
field strength 2-form $F^{\pm}=dA^{\pm}$. We can now attribute the difference
(226) to the worldline contact term

$\displaystyle iS_{\text{cont}}$
$\displaystyle=i\int\frac{4q^{2}}{m}\left[y\cdot F^{+}(z)\cdot
F^{-}(\bar{z})\cdot y\right]d\sigma+\mathcal{O}(y^{3})\,.$ (230)

As we observed in section 2.2.2, the combination $(y\cdot F^{+}\cdot
F^{-}\cdot y)$ is an inevitable consequence of the zig-zag symplectic
perturbation theory. Our twistor model differs from other models which do not
carry $(y\cdot F^{+}\cdot F^{-}\cdot y)$ terms.

The Compton amplitude (225) can also be compared to predictions of higher-spin
gauge symmetry Cangemi:2023ysz . In the notations of ref. Cangemi:2023ysz ,
the non-scalar part of (225) can be written as $e^{x}(1-w)$, which differs
from the result (6.61) of ref. Cangemi:2023ysz  reproduced below

$\displaystyle e^{x}\cosh z-we^{x}\frac{\sinh
z}{z}+\frac{w^{2}-z^{2}}{2}E(x,y,z)\,,$ (231)

where

$\displaystyle\begin{gathered}x=-(k_{4}-k_{3})\cdot
a\,,\,y=-(k_{3}+k_{4})\cdot a\,,\,z=-|a|v_{1}\cdot(k_{4}-k_{3})\,,\\\
w=-\frac{(n\cdot a)[v\cdot(k_{4}-k_{3})]}{(n\cdot v)}\,,\end{gathered}$

and

$\displaystyle E(x,y,z)$ $\displaystyle=\frac{e^{y}-e^{x}\cosh
z+(x-y)e^{x}\frac{\sinh z}{z}}{(x-y)^{2}-z^{2}}+(y\to-y)\,.$

While this amplitude is quite different from the twistor worldline prediction
(225), the amplitude shares the same $e^{x}$ factor conjectured to be
responsible for the singularity structure of the aligned-spin one-loop eikonal
(260).

##  6 Conservative dynamics from WQFT

In this section, we revisit the scattering observables of the binary system
from the WQFT perspective. We computed the 1PL and 2PL observables in section
3 by solving the equations of motion and extracted the classical eikonal along
the way. One advantage of the WQFT approach is that it allows us to compute
the eikonal before computing observables. Using two approaches to compute the
same eikonal serves as a consistency check. Besides, we specialise to the
aligned spin configurations and evaluate the Fourier integrals explicitly to
obtain the position space expressions for the classical eikonal up to the 2PL
order.

###  6.1 1PL observables and eikonal

Since the complex coordinate $z^{\mu}=x^{\mu}+iy^{\mu}$ includes the position
and the spin (recall the sign $a^{\mu}=-y^{\mu}$), we can compute the velocity
kick $\Delta v^{\mu}$ and the spin kick $\Delta y^{\mu}$ both from the
expectation values $\langle\\!\langle\delta
z_{1}^{\mu}(\omega)\rangle\\!\rangle$ and
$\langle\\!\langle\delta\bar{z}_{1}^{\mu}(\omega)\rangle\\!\rangle$. For
particle 1, the expectation value $\langle\\!\langle\delta
z_{1}^{\mu}(\omega)\rangle\\!\rangle$ is given by

$\displaystyle\begin{aligned} \langle\\!\langle\delta
z_{1}^{\mu}(\omega)\rangle\\!\rangle&=-q_{1}q_{2}\int_{k,k^{\prime},\omega^{\prime}}\left[\langle\delta
z_{1}^{\mu}(\omega)\delta z_{1}^{\nu}(\omega^{\prime})\rangle\langle
F_{\nu\lambda}^{+}(k)A_{\alpha}^{-}(k^{\prime})\rangle e^{ik\cdot
z_{1}+ik^{\prime}\cdot\bar{z}_{2}}\right.\\\
&\phantom{=asdfasdfasdfasdf}\left.{}+\langle\delta
z_{1}^{\mu}(\omega)\delta\bar{z}_{1}^{\nu}(\omega^{\prime})\rangle\langle
F_{\nu\lambda}^{-}(k)A_{\alpha}^{+}(k^{\prime})\rangle
e^{ik\cdot\bar{z}_{1}+ik^{\prime}\cdot{z}_{2}}\right]\\\
&\phantom{=asdfasdfasdfasdfasdfasdf}\times
v_{1}^{\lambda}v_{2}^{\alpha}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k\cdot
v_{1})-\omega^{\prime}]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k^{\prime}\cdot v_{2})\,,\end{aligned}$ (232)

where $z_{1,2}$, $\bar{z}_{1,2}$ in the exponents are understood as background
values. This expression is free of Dirac string singularity when we use the
propagator (188).

The velocity kick and the spin kick can be computed from the expectation
values as

$\displaystyle\Delta_{(1)}v_{1}^{\mu}$
$\displaystyle={\operatorname{Re}}\lim_{\tau\to\infty}\frac{d\delta
z_{1}^{\mu}(\tau)}{d\tau}={\operatorname{Re}}\int_{\tau}\frac{d^{2}\delta
z_{1}^{\mu}(\tau)}{d\tau^{2}}={\operatorname{Re}}\int_{\tau,\omega}(-\omega^{2})\delta
z_{1}^{\mu}(\omega)e^{-i\omega\tau}$
$\displaystyle={\operatorname{Re}}\lim_{\omega\to 0}(-\omega^{2})\delta
z_{1}^{\mu}(\omega)\,,$ (233) $\displaystyle\Delta_{(1)}y_{1}^{\mu}$
$\displaystyle={\operatorname{Im}}\lim_{\tau\to\infty}\delta
z_{1}^{\mu}(\tau)={\operatorname{Im}}\int_{\tau}\frac{d\delta
z_{1}^{\mu}(\tau)}{d\tau}={\operatorname{Im}}\int_{\tau,\omega}(-i\omega)\delta
z_{1}^{\mu}(\omega)e^{-i\omega\tau}$
$\displaystyle={\operatorname{Im}}\lim_{\omega\to 0}(-i\omega)\delta
z_{1}^{\mu}(\omega)\,.$ (234)

This may be viewed as the worldline version of the LSZ reduction formula
Lehmann:1954rq ; we can expect that the equivalent of the $S$-matrix
equivalence theorem Chisholm:1961tha ; Kamefuchi:1961sb  will also hold for
worldline observables in WQFT. After some algebra, we can write the velocity
kick as

$\displaystyle\begin{aligned}
\Delta_{(1)}v_{1}^{\mu}&=-\frac{q_{1}q_{2}}{2m_{1}}\int_{k}\frac{i\gamma
k^{\mu}-\epsilon^{\mu}[k,v_{1},v_{2}]}{k^{2}}e^{k\cdot y}e^{ik\cdot
b}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v_{2})\\\ &\phantom{=}-\frac{q_{1}q_{2}}{2m_{1}}\int_{k}\frac{i\gamma
k^{\mu}+\epsilon^{\mu}[k,v_{1},v_{2}]}{k^{2}}e^{-k\cdot y}e^{ik\cdot
b}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v_{2})\,.\end{aligned}$ (235)

Similarly, we can write the spin kick as

$\displaystyle\begin{aligned}
\Delta_{(1)}y_{1}^{\mu}&=+\frac{q_{1}q_{2}}{2m_{1}}\int_{k}\frac{i(v_{2}\cdot
y_{1})k^{\mu}-i(k\cdot
y_{1})v_{2}^{\mu}-v_{1}^{\mu}\epsilon[k,v_{1},v_{2},y_{1}]-\gamma\epsilon^{\mu}[k,v_{1},y_{1}]}{k^{2}}\\\
&\hskip 56.9055pt\times
e^{ik\cdot(b-iy)}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v_{2})\\\
&\phantom{=}+\frac{q_{1}q_{2}}{2m_{1}}\int_{k}\frac{i(v_{2}\cdot
y_{1})k^{\mu}-i(k\cdot
y_{1})v_{2}^{\mu}+v_{1}^{\mu}\epsilon[k,v_{1},v_{2},y_{1}]+\gamma\epsilon^{\mu}[k,v_{1},y_{1}]}{k^{2}}\\\
&\hskip 56.9055pt\times
e^{ik\cdot(b+iy)}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v_{2})\,.\end{aligned}$ (236)

Both $\Delta_{(1)}v_{1}^{\mu}$ and $\Delta_{(1)}y_{1}^{\mu}$ agree with the
results of section 3 as expected.

####  6.1.1 1PL eikonal

1PM-a (40,60) i1,a1,o1 i2,a2,o2
decor.shape=square,decor.filled=empty,decor.size=10a1
decor.shape=square,decor.filled=shaded,decor.size=10a2 dotsi1,a1,o1
dotsi2,a2,o2 wiggly,tension=0a1,a2 1PM-b (40,60) i1,a1,o1 i2,a2,o2
decor.shape=square,decor.filled=shaded,decor.size=10a1
decor.shape=square,decor.filled=empty,decor.size=10a2 dotsi1,a1,o1
dotsi2,a2,o2 wiggly,tension=0a1,a2

Figure 3: Diagrams contributing to the 1PL eikonal.

The eikonal is evaluated as the sum over the diagrams in Figure 3:

$\displaystyle\begin{aligned}
i\chi_{(1)}&=-q_{1}q_{2}\int_{k_{\perp}}v_{1}^{\mu}v_{2}^{\nu}\left[e^{ik\cdot(z_{1}-\bar{z}_{2})}\Delta^{+-}_{\mu\nu}(k)+e^{ik\cdot(\bar{z}_{1}-{z}_{2})}\Delta^{-+}_{\mu\nu}(k)\right]\,,\end{aligned}$
(237)

where $\Delta^{+-}_{\mu\nu}(k)$ and $\Delta^{-+}_{\mu\nu}(k)$ are the helicity
propagators (184). We write the integrand as

$\displaystyle\begin{aligned} &v_{1}^{\mu}v_{2}^{\nu}\left[e^{-k\cdot
y}\Delta^{+-}_{\mu\nu}(k)+e^{+k\cdot y}\Delta^{-+}_{\mu\nu}(k)\right]\\\
&\qquad=\cosh(k\cdot
y)v_{1}^{\mu}v_{2}^{\nu}\left[\Delta^{+-}_{\mu\nu}(k)+\Delta^{-+}_{\mu\nu}(k)\right]\\\
&\qquad\quad\phantom{=}+\frac{\sinh(k\cdot y)}{(k\cdot
y)}y^{\lambda}v_{1}^{\mu}v_{2}^{\nu}(ik_{\lambda})\left[i\Delta^{+-}_{\mu\nu}(k)-i\Delta^{-+}_{\mu\nu}(k)\right]\,.\end{aligned}$
(238)

Using (186), we can simplify the $(\cosh)$ term slightly and write

$\displaystyle\cosh(k\cdot
y)v_{1}^{\mu}v_{2}^{\nu}\left[\Delta^{+-}_{\mu\nu}(k)+\Delta^{-+}_{\mu\nu}(k)\right]=\cosh(k\cdot
y)(v_{1}\cdot v_{2})\frac{-i}{k^{2}}\,.$ (239)

We remind the readers that we are treating terms proportional to $(v_{1}\cdot
k)$, $(v_{2}\cdot k)$ or $k^{2}$ as zero. The $(\sinh)$ term is more
interesting.

$\displaystyle\begin{aligned}
&y^{\lambda}v_{1}^{\mu}v_{2}^{\nu}(ik_{\lambda})\left[i\Delta^{+-}_{\mu\nu}(k)-i\Delta^{-+}_{\mu\nu}(k)\right]\\\
&=y^{\lambda}v_{1}^{\mu}v_{2}^{\nu}(ik_{\lambda})\left[i\Delta^{+-}_{\mu\nu}(k)-i\Delta^{-+}_{\mu\nu}(k)\right]-y^{\lambda}v_{1}^{\mu}v_{2}^{\nu}(ik_{\mu})\left[i\Delta^{+-}_{\lambda\nu}(k)-i\Delta^{-+}_{\lambda\nu}(k)\right]\\\
&=y^{\lambda}v_{1}^{\mu}v_{2}^{\nu}\epsilon_{\lambda\mu}{}^{\alpha\beta}(ik_{\alpha})\left[\Delta^{+-}_{\beta\nu}(k)+\Delta^{-+}_{\beta\nu}(k)\right]=i\epsilon[k,v_{1},v_{2},y]\frac{-i}{k^{2}}\,.\end{aligned}$
(240)

The term added to the second line vanishes due to
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(v_{1}\cdot
k)$; its purpose is to anti-symmetrise in $\lambda$, $\mu$ indices. The
equality between the second line and the third line follows from (186) and the
4d Schouten identity (105). In the end, we obtain

$\displaystyle\chi_{(1)}$
$\displaystyle=-q_{1}q_{2}\int_{k_{\perp}}\left[\cosh(k\cdot
y)\gamma-i\frac{\sinh(k\cdot y)}{k\cdot
y}\epsilon[k,v_{1},v_{2},y]\right]\frac{e^{ik\cdot b}}{k^{2}}\,.$ (241)

in agreement with (103). It is free of the Dirac string ambiguity as expected.

Next, we perform the Fourier integral explicitly and obtain

$\displaystyle\begin{split}\chi_{(1)}&=\frac{q_{1}q_{2}\gamma}{4\pi\sqrt{\gamma^{2}-1}}\left[\frac{1}{\epsilon}+{\operatorname{Re}}\left(\log\frac{(b^{\mu}+iy_{\perp}^{\mu})^{2}}{b_{0}^{2}}\right)\phantom{+\left(\frac{y_{\perp}^{2}+\sqrt{y_{\perp}^{2}-(b\cdot
y_{\perp})^{2}}}{y_{\perp}^{2}-\sqrt{y_{\perp}^{2}-(b\cdot
y_{\perp})^{2}}}\right)}\right.\\\
&\phantom{=asdf}\left.\phantom{asdf}-\frac{\epsilon[b,v_{1},v_{2},y_{\perp}]}{2\gamma\sqrt{b^{2}y_{\perp}^{2}-(b\cdot
y_{\perp})^{2}}}\log\left(\frac{b^{2}+y_{\perp}^{2}+2\sqrt{b^{2}y_{\perp}^{2}-(b\cdot
y_{\perp})^{2}}}{b^{2}+y_{\perp}^{2}-2\sqrt{b^{2}y_{\perp}^{2}-(b\cdot
y_{\perp})^{2}}}\right)\right]\,,\end{split}$

where $D=4-2\epsilon$, $b_{0}^{2}$ is the dimensional regularisation parameter
absorbing all regularisation artefacts (factors of $\pi$ and $\gamma_{E}$,
etc.), and

$\displaystyle y_{\perp}^{\mu}=y^{\mu}+\left(\frac{\gamma(y\cdot
v_{2})-(y\cdot
v_{1})}{\gamma^{2}-1}\right)v_{1}^{\mu}+\left(\frac{\gamma(y\cdot
v_{1})-(y\cdot v_{2})}{\gamma^{2}-1}\right)v_{2}^{\mu}$ (242)

is the projection of $y^{\mu}$ onto the impact parameter space defined by
$b\cdot v_{1}=b\cdot v_{2}=0$. In the aligned spin configuration,
$y_{\perp}^{\mu}=y^{\mu}$ and $(b\cdot y)=0$, the eikonal simplifies even
further,

$\displaystyle\chi_{(1,\text{aligned})}$
$\displaystyle=\frac{q_{1}q_{2}\gamma}{4\pi\sqrt{\gamma^{2}-1}}\left[\frac{1}{\epsilon}+\log\frac{b^{2}-y^{2}}{b_{0}^{2}}-\frac{\epsilon[b,v_{1},v_{2},y]}{\gamma|b||y|}\log\left(\frac{|b|+|y|}{|b|-|y|}\right)\right]\,,$
(243)

where $|b|=\sqrt{b^{2}}$ and $|y|=\sqrt{y^{2}}$. Note that the aligned spin
eikonal develops a logarithmic singularity at $b^{2}=y^{2}$; the eikonal
“knows” that classical spin is a finite-size effect and the point particle
approximation breaks down when the two bodies are too close to each other.

###  6.2 2PL eikonal

The diagrams relevant for the 2PL eikonal are shown in Figure 4.

2PM-a (40,70) i1,a1,b1,o1 i2,a2,b2,o2
decor.shape=square,decor.filled=empty,decor.size=10a1
decor.shape=square,decor.filled=shaded,decor.size=10b1
decor.shape=square,decor.filled=shaded,decor.size=10a2
decor.shape=square,decor.filled=empty,decor.size=10b2 dotsi1,a1 dotsb1,o1
dotsi2,a2,b2,o2 plain,width=1a1,b1 wiggly,tension=0a1,a2 wiggly,tension=0b1,b2
2PM-b (40,70) i1,a1,b1,o1 i2,a2,b2,o2
decor.shape=square,decor.filled=empty,decor.size=10a1
decor.shape=square,decor.filled=empty,decor.size=10b1
decor.shape=square,decor.filled=shaded,decor.size=10a2
decor.shape=square,decor.filled=shaded,decor.size=10b2 dotsi1,a1 dotsb1,o1
plain,width=1a1,b1 dotsi2,a2,b2,o2 wiggly,tension=0a1,a2 wiggly,tension=0b1,b2

Figure 4: Diagrams contributing to the 2PL eikonal, up to the exchange of the
two particles and the overall flip of holomorphy/helicity.

##### Building up the eikonal integrand

The integrand can be constructed by replacing the field strength tensors
$F^{\pm}_{i\mu\nu}$ of the Compton amplitudes (203) and (207) by the
linearised source contribution from the other particle using the propagator
(188),

$\displaystyle\begin{aligned} F_{i\mu\nu}^{+}&\to\frac{(k\wedge
v_{2})_{\mu\nu}-i\epsilon_{\mu\nu}[k_{i},v_{2}]}{2k_{i}^{2}}\times
iq_{2}e^{-ik_{i}\cdot\bar{z}_{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k_{i}\cdot v_{2})\,,\\\ F_{i\mu\nu}^{-}&\to\frac{(k\wedge
v_{2})_{\mu\nu}+i\epsilon_{\mu\nu}[k_{i},v_{2}]}{2k_{i}^{2}}\times
iq_{2}e^{-ik_{i}\cdot{z}_{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k_{i}\cdot v_{2})\,,\end{aligned}$ (244)

attaching symmetry factors, integrating over photon momenta
$\int_{k_{3},k_{4}}$, summing over helicity configurations, and summing over
worldline permutation $1\leftrightarrow 2$.

$\displaystyle i\chi_{(2)}$
$\displaystyle=i\int_{k_{3},k_{4}}\left[I^{++}+2I^{+-}+I^{--}\right]+(1\leftrightarrow
2)\,,$ (245)

where we used the fact that $I^{+-}=I^{-+}$.

##### Same helicity integrand

The integrand turns out to be quite simple

$\displaystyle
I^{++}=-\frac{(q_{1}q_{2})^{2}}{8m_{1}}e^{i(k_{3}+k_{4})\cdot(z_{1}-\bar{z}_{2})}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k_{3}+v_{1}\cdot
k_{4})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot
k_{3})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k_{4})}{(v_{1}\cdot k_{3})(v_{1}\cdot
k_{4})}\frac{k_{3}\cdot k_{4}}{k_{3}^{2}k_{4}^{2}}\,.$ (246)

At the level of the integrand, it is clear that $I^{++}+I^{--}$ agrees with
the $\cosh[(k+\ell)\cdot y]$ term in (118) upon the identification
$(k_{3},k_{4})_{\mathrm{here}}\leftrightarrow(k,\ell)_{\mathrm{there}}$.

We can use Passarino-Veltman reduction to rewrite the integrand as

$\displaystyle\begin{aligned} \frac{k_{3}\cdot k_{4}}{(v_{1}\cdot
k_{3})(v_{1}\cdot
k_{4})k_{3}^{2}k_{4}^{2}}&=\frac{1}{2}\left[\frac{(k_{3}+k_{4})^{2}}{(v_{1}\cdot
k_{3})(v_{1}\cdot
k_{4})k_{3}^{2}k_{4}^{2}}-\frac{k_{3}^{2}+k_{4}^{2}}{(v_{1}\cdot
k_{3})(v_{1}\cdot k_{4})k_{3}^{2}k_{4}^{2}}\right]\\\
&=\frac{-1}{2}\left[\frac{k^{2}}{(v_{1}\cdot
k_{3})^{2}k_{3}^{2}(k-k_{3})^{2}}-\frac{2}{(v_{1}\cdot
k_{3})^{2}k_{3}^{2}}\right]\end{aligned}$ (247)

where we used the symmetry between $k_{3}$ and $k_{4}$, and then used the
condition $\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot v_{1}]$. We also set
$k^{\mu}=k_{3}^{\mu}+k_{4}^{\mu}$. We are left with evaluation of the integral
(in $D=4-2\epsilon$ dimensions for regularisation)

$\displaystyle\begin{aligned}
\mathcal{I}^{++}(k,v)&:=\int_{k_{3},k_{4}}\frac{k_{3}\cdot
k_{4}\,e^{i(k_{3}+k_{4})\cdot z}}{(v_{1}\cdot k_{3})(v_{1}\cdot
k_{4})k_{3}^{2}k_{4}^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot
v_{1}]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k_{3}+k_{4})\cdot
v_{2}]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k_{3}\cdot v_{2})\\\ &=-\frac{1}{2}\int_{k_{\perp}}\,e^{ik\cdot
z}\int_{k_{3}}\left[\frac{k^{2}}{(v_{1}\cdot
k_{3})^{2}k_{3}^{2}(k-k_{3})^{2}}-\frac{2}{(v_{1}\cdot
k_{3})^{2}k_{3}^{2}}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k_{3}\cdot v_{2})\,.\end{aligned}$ (248)

Then we separate the integral argument into $k_{3\parallel}$ and $k_{3\perp}$,
such that $k_{3\perp}\cdot v_{2}=0$. The $dk_{3\parallel}$ integral is trivial
due to $\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k_{3}\cdot v_{2})$, and we get

$\displaystyle\int d^{D-1}k_{3\perp}\left[\frac{k^{2}}{(v_{1\perp}\cdot
k_{3\perp})^{2}k_{3\perp}^{2}(k-k_{3\perp})^{2}}-\frac{2}{(v_{1\perp}\cdot
k_{3\perp})^{2}k_{3\perp}^{2}}\right]\,,$ (249)

where $v_{1\perp}^{\mu}=v_{1}^{\mu}+(v_{1}\cdot
v_{2})v_{2}^{\mu}=v_{1}^{\mu}-\gamma v_{2}^{\mu}$ is the projection of
$v_{1}^{\mu}$ onto the orthogonal space such that $v_{1\perp}\cdot v_{2}=0$.
The projection for $k^{\mu}$ is not needed due to
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v_{2})$ constraint. The remaining integrals evaluate to zero when using the
master integral (286), which is consistent with vanishing same helicity sector
contributions for the triangle coefficient in amplitude calculations
Guevara:2017csg ; Guevara:2018wpp ; Chung:2018kqs .

##### Opposite helicity integrand

Evaluating the relevant diagrams, and factoring out the common denominator as

$\displaystyle\begin{aligned}
I^{+-}&=-\frac{(q_{1}q_{2})^{2}}{8m_{1}}e^{i(k_{3}+k_{4})\cdot
b}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k_{3}+v_{1}\cdot
k_{4})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot
k_{3})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k_{4})}{k_{3}^{2}k_{4}^{2}(v_{1}\cdot k_{3})(v_{1}\cdot
k_{4})}\\\ &\phantom{=asdfasdf}\times
e^{(k_{4}-k_{3})\cdot(y_{1}+y_{2})}N[k_{3},k_{4},v_{1},v_{2},y_{1}]\,,\end{aligned}$
(250)

we find that the numerator, organized in powers of $\gamma$, is given as

$\displaystyle N$ $\displaystyle=2\gamma^{2}\left(i(v_{1}\cdot
k_{4})\epsilon[k_{3},k_{4},v_{1},y_{1}]+(k_{3}\cdot k_{4})\right)$ (251)
$\displaystyle\phantom{=}+2i\gamma\left(\epsilon[k_{3},k_{4},v_{1},v_{2}]-2i(k_{3}\cdot
k_{4})(v_{1}\cdot k_{4})(v_{2}\cdot y_{1})-(v_{1}\cdot
k_{4})^{2}\epsilon[(k_{3}+k_{4}),v_{1},v_{2},y_{1}]\right)$
$\displaystyle\phantom{=}-(k_{3}\cdot k_{4})+2(v_{1}\cdot
k_{4})^{2}\left[1+(y_{1}\cdot k_{3})-(y_{1}\cdot k_{4})\right]+2i(v_{1}\cdot
k_{4})(v_{2}\cdot y_{1})\epsilon[k_{3},k_{4},v_{1},v_{2}]\,.$

It is straightforward to show that it agrees with the opposite helicity
integrand in (118).

To evaluate the Fourier integrals, it is convenient to reorganise the
numerator as

$\displaystyle\begin{aligned}
N&=\gamma^{2}\left[k^{2}-k_{3}^{2}-k_{4}^{2}\right]+2i\gamma^{2}(v_{1}\cdot
k_{4})\epsilon[k_{3},k,v_{1},y_{1}]\\\
&\phantom{=}+2i\gamma\epsilon[k_{3},k,v_{1},v_{2}]+2\gamma(v_{2}\cdot
y_{1})(v_{1}\cdot k_{4})\left[k^{2}-k_{3}^{2}-k_{4}^{2}\right]\\\
&\phantom{=}+2i\gamma(v_{1}\cdot k_{3})(v_{1}\cdot
k_{4})\epsilon[k,v_{1},v_{2},y_{1}]\\\
&\phantom{=}-\frac{1}{2}\left[k^{2}-k_{3}^{2}-k_{4}^{2}\right]-2(v_{1}\cdot
k_{3})(v_{1}\cdot k_{4})\left[1+(y_{1}\cdot k_{3})-(y_{1}\cdot
k_{4})\right]\\\ &\phantom{=}+2i(v_{1}\cdot k_{4})(v_{2}\cdot
y_{1})\epsilon[k_{3},k,v_{1},v_{2}]\,.\end{aligned}$ (252)

Inspecting the master integral (286) we find that $k_{3}^{2}$ and $k_{4}^{2}$
of the numerator will evaluate to zero ($\lambda_{1}=0$ or $\lambda_{2}=0$
condition) and can be thrown away. We organise the integrand as

$\displaystyle\begin{aligned}
I^{+-}&=\frac{(q_{1}q_{2})^{2}}{8m_{1}}e^{ik\cdot(b-iy)}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k)\frac{e^{-2k_{3}\cdot
y}N[k_{3},k-k_{3},v_{1},v_{2},y_{1}]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k_{3})}{k_{3}^{2}(k-k_{3})^{2}(v_{1}\cdot
k_{3})^{2}}\,,\\\ y^{\mu}&=y_{1}^{\mu}+y_{2}^{\mu}\,,\end{aligned}$ (253)

where we use $k_{3}$ as the loop momentum. Performing the
$dk_{3\parallel}=dk_{3}^{0}$ integral we get

$\displaystyle\int\frac{d^{D}k_{3}}{(2\pi)^{D}}\frac{e^{-2k_{3}\cdot
y}N\,\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k_{3})}{k_{3}^{2}(k-k_{3})^{2}(v_{1}\cdot k_{3})^{2}}$
$\displaystyle=\int\frac{d^{D-1}k_{3\perp}}{(2\pi)^{D-1}}\frac{e^{-2k_{3}\cdot
y_{\perp}}N_{\perp}}{k_{3}^{2}(k-k_{3})^{2}(v_{1\perp}\cdot k_{3})^{2}}$ (254)

where $y_{\perp}^{\mu}=y_{1}^{\mu}+y_{2}^{\mu}+v_{2}^{\mu}(y_{1}\cdot v_{2})$
and the effective numerator is

$\displaystyle N_{\perp}$
$\displaystyle=\left[\gamma^{2}-1/2\right]k^{2}-2\gamma(v_{2}\cdot
y_{1})k^{2}(v_{1\perp}\cdot k_{3})$
$\displaystyle\phantom{=}+2\left[\frac{i\gamma\epsilon[k,v_{1},v_{2},y_{1}]}{\gamma^{2}-1}+1\right](v_{1\perp}\cdot
k_{3})^{2}$ $\displaystyle\phantom{=}+4(v_{1\perp}\cdot
k_{3})^{2}\left(\left[y_{1}-\frac{k(k\cdot y_{1})}{k^{2}}+v_{2}(y_{1}\cdot
v_{2})\right]\cdot k_{3}\right)$
$\displaystyle\phantom{=}-2i\left\\{\gamma^{2}\left(\left[\epsilon^{\mu}[k,v_{1},y_{1}]+\frac{v_{1\perp}^{\mu}\gamma\epsilon[k,v_{1},v_{2},y_{1}]}{\gamma^{2}-1}\right]+v_{2\mu}\epsilon[v_{2},k,v_{1},y_{1}]\right)\right.$
$\displaystyle\phantom{=asdfasdf}\left.\phantom{\left(\left[\frac{v_{1\perp}^{\mu}}{\gamma^{2}}\right]\right)}+(v_{2}\cdot
y_{1})\epsilon_{\mu}[k,v_{1},v_{2}]\right\\}(v_{1\perp}\cdot
k_{3})k_{3}^{\mu}$
$\displaystyle\phantom{=}+2i\gamma\epsilon_{\mu}[k,v_{1},v_{2}]k_{3}^{\mu}\,.$
(255)

The remaining integral can be evaluated using the list of integrals in
appendix B.1. We present the results in the ancillary file loopdata.dat.m.
Including the overall $e^{+k\cdot y}$ factor from (253), the integral is
consistent with the QED amplitude coefficients provided by ref. Bern:2023ity
to bilinear order in spins, under the conditions $C_{i}=1$, $D_{i}=0$, and
covariant SSC. We also present the full eikonal as a formal power series in
the ancillary file eikonaldata.dat.m. While the expressions by themselves do
not provide any insight, they greatly simplify in the aligned spin
configuration, which we present next.

####  6.2.1 Aligned spin

Let us simplify the expression by going to the aligned spin configuration. For
aligned spin we have the conditions $y_{1}^{\mu}\propto y_{2}^{\mu}$ and
$y\cdot v_{1,2}=0$. We introduce the ratio parameter $\zeta$ defined by
$y_{1}^{\mu}=\zeta y^{\mu}$; it follows that $y_{2}^{\mu}=(1-\zeta)y^{\mu}$
and $y_{\perp}^{\mu}=y^{\mu}$. This reduces the expression to a single
infinite sum and we get

$\displaystyle\begin{aligned}
&\int\frac{d^{4}k_{3}}{(2\pi)^{4}}\frac{e^{-2k_{3}\cdot
y}N\,\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k_{3})}{k_{3}^{2}(k-k_{3})^{2}(v_{1}\cdot
k_{3})^{2}}=\int\frac{d^{3}k_{3\perp}}{(2\pi)^{3}}\frac{e^{-2k_{3}\cdot
y_{\perp}}N_{\perp}}{k_{3}^{2}(k-k_{3})^{2}(v_{1\perp}\cdot k_{3})^{2}}\\\
&=\frac{e^{-k\cdot
y}}{4(k^{2})^{\frac{1}{2}}}\sum_{m=0}^{\infty}\frac{\left(-\frac{k^{2}y^{2}}{2}\right)^{m}}{m!}\left[\left(\frac{2\gamma^{2}-1}{\gamma^{2}-1}-\zeta\right)\frac{I_{m+1}(k\cdot
y)}{(k\cdot y)^{m-1}}\right.\\\
&\phantom{=}\left.\phantom{\left(\frac{2\gamma^{2}}{\gamma^{2}}\right)}+\left(\frac{\gamma^{2}(4m+1)-2m-1}{\gamma^{2}-1}-2m\zeta+\frac{i\gamma(\zeta-2)\epsilon[k,v_{1},v_{2},y]}{(\gamma^{2}-1)}\right)\frac{I_{m}(k\cdot
y)}{(k\cdot y)^{m}}\right]\,,\end{aligned}$ (256)

where $I_{n}(x)$ is the modified Bessel function of the first kind. The impact
parameter space integral can be organised as

$\displaystyle\chi^{+-}$
$\displaystyle=\int_{k_{3},k_{4}}I^{+-}=\frac{(q_{1}q_{2})^{2}}{32m_{1}}\int\frac{d^{4}k}{(2\pi)^{4}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k)\frac{\mathcal{I}^{+-}e^{ik\cdot b}}{\sqrt{k^{2}}}$
$\displaystyle=\frac{(q_{1}q_{2})^{2}}{32m_{1}\sqrt{\gamma^{2}-1}}\int\frac{d^{2}k_{E}}{(2\pi)^{2}}\frac{\mathcal{I}^{+-}e^{i(k_{E}\cdot
b)}}{\sqrt{k_{E}^{2}}}\,,$ (257) $\displaystyle\mathcal{I}^{+-}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{\left(-\frac{k^{2}y^{2}}{2}\right)^{m}}{m!}\left[\left(\frac{2\gamma^{2}-1}{\gamma^{2}-1}-\zeta\right)\frac{I_{m+1}(k\cdot
y)}{(k\cdot y)^{m-1}}\right.$
$\displaystyle\phantom{=}\left.\phantom{as}+\left(\frac{\gamma^{2}(4m+1)-2m-1}{\gamma^{2}-1}-2m\zeta+\frac{i\gamma(\zeta-2)\epsilon[k,v_{1},v_{2},y]}{(\gamma^{2}-1)}\right)\frac{I_{m}(k\cdot
y)}{(k\cdot y)^{m}}\right]\,,$ (258)

where the factor $i\epsilon[k,v_{1},v_{2},y]$ can be traded for the derivative
operator $\epsilon^{\mu}[v_{1},v_{2},y]\frac{\partial}{\partial b^{\mu}}$.
Imposing the additional constraint111111This additional constraint conforms to
the usage of “aligned spin” in the literature, where the orbital angular
momentum is also aligned with the spin direction. $y\cdot b=0$ simplifies the
expression further and yields,

$\displaystyle\chi^{+-}$
$\displaystyle=\frac{(q_{1}q_{2})^{2}}{64m_{1}\sqrt{\gamma^{2}-1}}\left(\frac{1}{\pi(b^{2}-y^{2})^{3/2}}\left[b^{2}+\frac{\gamma^{2}(1-\zeta)+\zeta}{\gamma^{2}-1}\,y^{2}\right]\right.$
$\displaystyle\left.\phantom{asdf\frac{1-(1+2\zeta)(\frac{y^{2}}{b^{2}})}{\pi\sqrt{b^{2}}}}+\frac{(\zeta-2)\gamma}{\pi(\gamma^{2}-1)}\epsilon^{\mu}[v_{1},v_{2},y]\frac{\partial}{\partial
b^{\mu}}\frac{1}{\sqrt{b^{2}-y^{2}}}\right)\,,$ (259)

which has a singularity structure
$(b^{2}-y^{2})^{-3/2}=(b^{2}-(a_{1}+a_{2})^{2})^{-3/2}$ that was not visible
in the original perturbative spin expansion. The full aligned-spin 2PL eikonal
is

$\displaystyle\chi_{(2,\text{aligned})}$
$\displaystyle=\frac{(q_{1}q_{2})^{2}\left(b^{2}-\frac{(\zeta-2)\gamma}{(\gamma^{2}-1)}\,\epsilon[b,v_{1},v_{2},y]+\frac{\gamma^{2}(1-\zeta)+\zeta}{\gamma^{2}-1}\,y^{2}\right)}{32\pi
m_{1}\sqrt{\gamma^{2}-1}\,(b^{2}-y^{2})^{3/2}}+\left(1\leftrightarrow
2\right)\,,$ (260)

where symmetrisation is implemented by $\\{m_{1}\to
m_{2}\,,\,v_{1}^{\mu}\leftrightarrow
v_{2}^{\mu}\,,\,b^{\mu}\to-b^{\mu}\,,\,\zeta\to 1-\zeta\\}$.

We remark that the singularity is still present in the spinless probe limit
$\zeta\to 0$, and since spin-dependence of the eikonal integrand (253) enters
only through the exponential factors $e^{+k\cdot y}e^{-2k_{3}\cdot y}$ in this
limit, the singularity structure $(b^{2}-y^{2})^{-3/2}$ seems to be a
consequence of the “Newman-Janis shift” of the integrand, which shifts the
displacement between the two worldlines by an imaginary spin sum vector $\pm
iy^{\mu}=\mp i(a_{1}^{\mu}+a_{2}^{\mu})$. Note that a similar singularity
structure of the form $a_{p}^{k}(b^{2}-a_{b}^{2})^{-3/2-k}$ was reported for
the 2PM gravitational aligned-spin spinning probe scattering where $a_{b}$ is
the spin parameter of the Kerr background and $a_{p}$ is the probe spin
($k\leq 2$) Damgaard:2022jem ,121212Ref. Damgaard:2022jem  reports the
scattering angle which scales as
$a_{p}^{k}(b^{2}-a_{b}^{2})^{-5/2-k}\sim\partial_{b}[a_{p}^{k}(b^{2}-a_{b}^{2})^{-3/2-k}]$.
which can be viewed as an artifact of expanding the singularity
$(b^{2}-(a_{b}+a_{p})^{2})^{-3/2}$. If the “Newman-Janis shift” of the
integrand persists at higher loop orders, we can conjecture that the
singularity structure of the spinless probe scattering
$(b^{2}-a_{b}^{2})^{-3n/2}$ from $n$-loop contributions reported by ref.
Damgaard:2022jem  generalises to the singularity structure
$(b^{2}-(a_{b}+a_{p})^{2})^{-3n/2}$. As remarked when comparing twistor
worldline Compton amplitudes with that of higher spin gauge symmetry
Cangemi:2023ysz , it would also be interesting to check whether same
singularity structures appear in the eikonal when the Compton amplitude has
exponential dependence on spin, a feature that is also shared by the Compton
amplitude construction in ref. Bjerrum-Bohr:2023iey .

We also consider axial scattering $y^{\mu}\propto b^{\mu}$, which is
independent of the sign of $y\cdot b$ because the Fourier integrand (258)
contains only even powers of $k\cdot y$. The $i\epsilon[k,v_{1},v_{2},y]$
contribution drops out due to the condition $y^{\mu}\propto b^{\mu}$, and the
Fourier transform (257) evaluates to

$\displaystyle\chi^{+-}$
$\displaystyle=\frac{(q_{1}q_{2})^{2}\,\sqrt{b^{2}}}{32\pi^{2}m_{1}(\gamma^{2}-1)^{3/2}}\left[\frac{\gamma^{2}(\zeta-1)-\zeta}{b^{2}}K\left(-\frac{y^{2}}{b^{2}}\right)-\frac{\gamma^{2}(\zeta-2)-(\zeta-1)}{b^{2}+y^{2}}E\left(-\frac{y^{2}}{b^{2}}\right)\right]\,,$
(261)

where $K(x)$ and $E(x)$ are the complete elliptic integrals of the first and
second kind. The full result is

$\displaystyle\chi_{(2,\text{axial})}$
$\displaystyle=\frac{(q_{1}q_{2})^{2}\,\sqrt{b^{2}}}{16\pi^{2}m_{1}(\gamma^{2}-1)^{3/2}}\left[\frac{\gamma^{2}(\zeta-1)-\zeta}{b^{2}}K\left(-\frac{y^{2}}{b^{2}}\right)-\frac{\gamma^{2}(\zeta-2)-(\zeta-1)}{b^{2}+y^{2}}E\left(-\frac{y^{2}}{b^{2}}\right)\right]$
$\displaystyle\phantom{=asdf}+(1\leftrightarrow 2)\,,$ (262)

which, unlike the aligned-spin case (260), develops a singularity at the
unphysical impact parameter $b^{2}=-y^{2}$. The results (260) and (262) can be
reproduced from the full eikonal given in the ancillary file eikonaldata.dat.m
by taking the corresponding configurations and resumming the series expansion
in $y^{2}/b^{2}$.

Before ending this section, we remark that the LSZ-like formulae (233) and
(234) can also be applied to 2PL scattering observables, where retarded
worldline propagators are used instead Jakobsen:2022psy . The 2PL observables
can be separated into the _eikonal part_ (the same diagrams with symmetric
worldline $i0^{+}$ prescription) and the _causality cut part_ (the
contributions from changing the worldline $i0^{+}$ prescription), where the
eikonal part computes $\\{\chi_{(2)},O\\}$ and the causality cut part computes
$\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},O\\}\\}$. This computation serves as a
consistency check of the calculations in section 3.2.2. The separation of the
observables into the eikonal part and the causality cut part can be shown to
be a more general phenomenon that holds in Hamiltonian worldline models
Kim:causality-cut .

##  7 Discussion

The (ambi-)twistor model for electromagnetically interacting spinning
particles was studied in this manuscript, which has the advantage that it is
one of the simplest descriptions of charged spinning particles where spin
effects can be tracked to arbitrarily high orders. Using (dynamical) Newman-
Janis shift as the only input for generating all-orders-in-spin interactions,
it was found that the spin effects can be resummed to simple expressions in
special kinematic configurations; in the aligned-spin case (260) and in the
axial scattering case (262). Also, the model was used to confirm the
interpretation, up to the 2PL order, of the classical eikonal as the generator
of canonical transformations that map the incoming scattering states to
outgoing scattering states.

Despite the disparities between electromagnetic and gravitational
interactions, the similarities between the singularity structures of the spin-
resummed electromagnetic eikonal (260),
$\chi_{(2)}\propto(b^{2}-(a_{1}+a_{2})^{2})^{-3/2}$, and the probe limit Kerr
scattering reported by ref. Damgaard:2022jem ,
$\theta_{(2)}\sim\partial_{b}\chi_{(2)}\propto\partial_{b}[(b^{2}-a_{b}^{2})^{-3/2}]$,
provides further evidence that using the total spin length vector
$a_{+}^{\mu}=a_{1}^{\mu}+a_{2}^{\mu}$ as the spin parameter of the effective
Kerr metric—an ansatz motivated by leading order PN Hamiltonian results
Vines:2016qwa —in the effective-one-body approach Khalil:2023kep  is the
preferable choice for resumming spin effects.131313This is not the unique
choice considered in the literature. A comparison of different choices for the
spin parameter of the effective Kerr metric can be found in ref.
Khalil:2020mmr . On the other hand, one-loop results only correspond to
leading order effects in the mass-ratio expansion Vines:2018gqi , therefore
the singularity structures resembling that of the background-probe calculation
Damgaard:2022jem  could be a coincidence of the leading order mass-ratio
expansion. Whether novel singularity structures arise at NLO in mass-ratio
expansion will only be answered by pushing the computations to two-loops and
higher orders, and may point us to new directions in resumming spin effects.
Of course, studying the gravitationally interacting case is also necessary to
confirm that such singularity structures are also present in gravitating
binary black holes.

When viewing the classical eikonal as the generator of canonical
transformations, it would be interesting to understand what it means to
analytically continue the scattering generator to bound dynamics. The
boundary-to-bound map for the radial action Kalin:2019rwq ; Kalin:2019inp
suggests that the continuation is a finite time-evolution generator that
advances the system by one radial period, e.g. the periastron passing is sent
to the next periastron passing. If this interpretation is correct, then we may
argue that separability of the Hamilton-Jacobi equations is not necessary for
the existence of the bound orbit counterpart of the classical eikonal,
although its determination by analytic methods may only be possible when
Hamilton-Jacobi equations are separable Gonzo:2023goe ; Gonzo:2024zxo .

Apart from the obvious future direction—massive twistor worldline in
gravitational fields—there are several other directions that would be
interesting to expand upon. One future research direction would be to explore
whether recent attempts to resum analytic results for gravitational scattering
of spinning black holes Rettegno:2023ghr ; Buonanno:2024vkx  can be improved
using the singularity structures of (260) and their conjectured generalisation
to higher loops $\chi_{(n)}\propto(b^{2}-(a_{1}+a_{2})^{2})^{-3n/2}$.

Another direction would be making the (WQFT approach to the) model live up to
its name; _quantisation_. Since the twistor model has a simple set of
constraints, the standard BRST-BFV methods should be applicable. For small
values of quantised spin, say $1/2$ or $1$, we expect the results to agree
with the standard QFT of massive spinning fields. The attempt to quantise the
model for higher spin may shed new light on the complication with massive
higher spin fields. Comparison of the approach with chiral models for massive
higher spin fields Ochirov:2022nqz  would also be an interesting study.

While the fundamental variables of our model are twistors, the physical
observables (and the classical eikonal) were given entirely in terms of the
gauge invariant $(x,y,p)$ variables. Some intermediate steps of the
computations, such as the ones in appendix C, tend to be quite lengthy and not
particularly illuminating. The computations may become vastly simplified when
full advantage of the twistor variables is taken. To do so, it would be
crucial to use massless twistor variables for the photon fields as well.
Bailey’s twistor propagator Bailey:1985a , and Guevara’s holomorphic classical
limit Guevara:2017csg  and twistor reconstruction Guevara:2021yud  could
provide clues for further progress.

We remark that iterated action of the classical eikonal can be understood as
causality cuts, which computes contributions associated to changing the
$i0^{+}$ prescription of the worldline propagators from time-symmetric to
retarded; in the WQFT formalism scattering observables are computed using
retarded propagators Jakobsen:2022psy , and changing the $i0^{+}$ prescription
of the worldline propagators from retarded to symmetric generates (nested)
Poisson brackets which reorganises the scattering observable $\Delta O$ as the
action of the scattering generator $e^{\\{\chi,\bullet\\}}O$ Kim:causality-cut
. A direct consequence is that the longitudinal impulse at 2PL order is
related to the $i0^{+}$ prescription of the worldline propagators, which could
be an interesting observation for understanding the $i0^{+}$ prescription
affecting the definition of the impact parameter used in one-loop waveform
results Brandhuber:2023hhy ; Herderschee:2023fxh ; Elkhidir:2023dco ;
Georgoudis:2023lgf ; Caron-Huot:2023vxl ; Bini:2023fiz ; Georgoudis:2023eke .

Finally, it would be interesting to generalise the concept of the classical
eikonal to massless fields. Such an extended eikonal would place massive
particles and massless fields on an equal footing, and may help us clarify to
what extent we can identify the eikonal as the classical shadow of the quantum
$S$-matrix.

###### Acknowledgements.

JWK would like to thank Alessandra Buonanno, Gustav Jakobsen, Gustav Mogull,
Raj Patil, Harald Pfeiffer, Lorenzo Pompili, and Jan Steinhoff for valuable
discussions. SL is grateful to Hojin Lee, Kanghoon Lee, Sungjay Lee, Chia-
Hsien Shen, Tianheng Wang and Piljin Yi for discussions. SL would also like to
thank Korea Institute for Advanced Study for hospitality where a large part of
this work was done. The work of SL is supported by the National Research
Foundation of Korea grant NRF-2019R1A2C2084608.

##  Appendix A Conventions

### Vector

Flat metric and Levi-Civita tensor,

$\displaystyle\eta_{\mu\nu}=\mathrm{diag}(-,+,+,+)\,,\quad\varepsilon_{0123}=+1\,.$
(263)

Electromagnetism without spin,

$\displaystyle\partial^{\mu}F_{\mu\nu}=-J_{\nu}\,,\quad
m\frac{du^{\mu}}{d\tau}=qF^{\mu\nu}u_{\nu}\,.$ (264)

Hodge star acting on a two-form,

$\displaystyle(*F)_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}\,.$
(265)

Self-dual and anti-self-dual parts of a two-form,

$\displaystyle F^{\pm}=\frac{1}{2}(F\mp
i\\!*\\!F)\quad\Longrightarrow\quad*(F^{\pm})=\pm i(F^{\pm})\,.$ (266)

We may also use $\tilde{F}_{\mu\nu}=(*F)_{\mu\nu}$. If we define $A^{\pm}$ and
$\tilde{A}$ by $F^{\pm}=dA^{\pm}$ and $\tilde{F}=d\tilde{A}$,

$\displaystyle
A_{\mu}=A^{+}_{\mu}+A^{-}_{\mu}\,,\quad\tilde{A}_{\mu}=i(A^{+}_{\mu}-A^{-}_{\mu})\,.$
(267)

### Spinor

We follow the conventions of ref. Kim:2021rda  to a large extent, where
$|\lambda\rangle$ spinors are associated to _incoming_ negative helicity
states. An important difference is that we define

$\displaystyle
v_{\alpha\dot{\alpha}}:=v_{\mu}\sigma^{\mu}_{\alpha\dot{\alpha}}\,,\quad
v^{\mu}=-\frac{1}{2}(\bar{\sigma}^{\mu})^{{\dot{\alpha}}\alpha}v_{\alpha{\dot{\alpha}}}$
(268)

for _all_ vectorial quantities, nullifying the exception for $x^{\mu}$ made in
ref. Kim:2021rda . To compare with references where the metric $\eta^{\mu\nu}$
and/or the Levi-Civita tensor $\varepsilon_{\mu\nu\rho\sigma}$ carry the
opposite sign (e.g. ref. Chung:2018kqs ), an invariant way to express
conversion between spinor and Lorentz indices is to introduce the parameters
$\kappa_{0}=\eta_{00}$ and $\kappa_{1}=\varepsilon_{0123}$:

$\displaystyle p_{\alpha\dot{\alpha}}$
$\displaystyle=p_{\mu}\sigma^{\mu}_{\alpha\dot{\alpha}}=\left\\{\begin{aligned}
&\kappa_{0}|p\rangle_{\alpha}[p|_{\dot{\alpha}}&&p^{2}=0\\\
&\kappa_{0}|p^{I}\rangle_{\alpha}[p_{I}|_{\dot{\alpha}}&&p^{2}=\kappa_{0}m^{2}\end{aligned}\right.$
(269) $\displaystyle v_{\alpha\dot{\alpha}}w^{\dot{\alpha}\alpha}$
$\displaystyle=2\kappa_{0}(v\cdot w)$ (270)
$\displaystyle(\sigma^{\mu}\bar{\sigma}^{\nu}\sigma^{\lambda}\bar{\sigma}^{\sigma})_{\alpha}{}^{\alpha}$
$\displaystyle=2(\eta^{\mu\nu}\eta^{\lambda\sigma}-\eta^{\mu\lambda}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\lambda}-i\kappa_{1}\varepsilon^{\mu\nu\lambda\sigma})$
(271)

The invariant tensor satisfies the complex conjugation relation

$\displaystyle\epsilon^{\dot{\alpha}\dot{\gamma}}\epsilon^{\beta\delta}\left[\sigma^{\mu}_{\gamma\dot{\delta}}\right]^{\ast}=\bar{\sigma}^{\mu\dot{\alpha}\beta}$
(272)

which is useful for evaluating complex conjugation of vectors.

### Twistor

The two major differences from ref. Kim:2021rda  are

$\displaystyle(x^{{\dot{\alpha}}\alpha})_{\mathrm{here}}=(-2)(x^{{\dot{\alpha}}\alpha})_{\mathrm{there}}\,,\quad(\mu^{\dot{\alpha}I},\bar{\mu}_{I}{}^{\alpha})_{\mathrm{here}}=-(\mu^{\dot{\alpha}I},\bar{\mu}_{I}{}^{\alpha})_{\mathrm{there}}\,.$
(273)

These changes propagate to all other equations. For example, the incidence
relations read

$\displaystyle\mu^{\dot{\alpha}I}=\frac{1}{2}z^{\dot{\alpha}\beta}\lambda_{\beta}{}^{I}\,,\quad\bar{\mu}_{I}{}^{\alpha}=\frac{1}{2}\bar{\lambda}_{I\dot{\beta}}\bar{z}^{\dot{\beta}\alpha}\,,$
(274)

where we define the complex conjugate relations as

$\displaystyle\bar{z}^{\mu}=\left[z^{\mu}\right]^{\ast}\Rightarrow\bar{z}^{\dot{\alpha}\beta}=\left[z^{\dot{\beta}\alpha}\right]^{\ast}\,,\quad\bar{\mu}_{I}{}^{\alpha}=\left[\mu^{\dot{\alpha}I}\right]^{\ast}\,,\,\bar{\lambda}_{I\dot{\alpha}}=\left[\lambda_{\alpha}{}^{I}\right]^{\ast}\,.$
(275)

The defining Poisson brackets are

$\displaystyle\begin{gathered}\\{x^{\mu},p_{\nu}\\}=\delta^{\mu}_{\nu}\;\;\Rightarrow\;\;\\{x^{\dot{\alpha}\alpha},p_{\beta\dot{\beta}}\\}=-2\delta^{\dot{\alpha}}_{\dot{\beta}}\delta^{\alpha}_{\beta}\,,\\\
\\{\bar{\mu}_{I}{}^{\alpha},\lambda_{\beta}{}^{J}\\}=\delta^{\alpha}_{\beta}\delta_{I}^{J}\,,\quad\\{\mu^{\dot{\alpha}I},\bar{\lambda}_{J\dot{\beta}}\\}=\delta_{\dot{\beta}}^{\dot{\alpha}}\delta^{I}_{J}\,.\end{gathered}$
(278)

The consistency of the defining brackets can be confirmed from the relations

$\displaystyle
x^{\dot{\alpha}\alpha}=\frac{z^{\dot{\alpha}\alpha}+\bar{z}^{\dot{\alpha}\alpha}}{2}\,,\quad
p_{\alpha\dot{\alpha}}=-\lambda_{\alpha}{}^{I}\bar{\lambda}_{I\dot{\alpha}}\,.$
(279)

To determine the relation between $y^{\mu}$ and the spin-length vector
$a^{\mu}=S^{\mu}/m$, we note the Poisson brackets of the rotation generators

$\displaystyle\\{J^{23},J^{31}\\}=J^{12}\;\;\Leftrightarrow\;\;\\{J^{1},J^{2}\\}=J^{3}$
(280)

and leverage the calculation to demand that

$\displaystyle\\{S^{\mu},S^{\nu}\\}_{\ast}=(-\kappa_{1})\varepsilon^{\alpha\mu\nu\lambda}\frac{(-p_{\alpha})}{m}S_{\lambda}\;\;\Leftrightarrow\;\;\\{a^{\mu},a^{\nu}\\}_{\ast}=\frac{\kappa_{1}}{m^{2}}\varepsilon^{\alpha\mu\nu\lambda}p_{\alpha}a_{\lambda}$
(281)

where $\\{\bullet,\bullet\\}_{\ast}$ is the Dirac bracket and
$\kappa_{1}=\varepsilon_{0123}$. The end result is the standard convention for
the orientation of $a^{\mu}$:

$\displaystyle\\{a^{1},a^{2}\\}_{*}=+\frac{a^{3}}{m}\,.$ (282)

On the twistor side, from the Poisson brackets we find

$\displaystyle\\{z^{\mu},\bar{z}^{\nu}\\}$
$\displaystyle=\frac{-2i}{m}\left[y^{\mu}v^{\nu}-\eta^{\mu\nu}(y\cdot
v)+v^{\mu}y^{\mu}+i\kappa_{1}\varepsilon^{\mu\nu\alpha\beta}y_{\alpha}v_{\beta}\right]$
(283)

which implies

$\displaystyle\\{y^{\mu},y^{\nu}\\}=\frac{\kappa_{1}}{m}\varepsilon^{\mu\nu\alpha\beta}y_{\alpha}v_{\beta}\;\;\Rightarrow\;\;\\{y^{1},y^{2}\\}=-\kappa_{1}\varepsilon^{1230}\frac{y^{3}}{m}=-\frac{y^{3}}{m}\,.$
(284)

Thus we have to set $y^{\mu}=-a^{\mu}$.

##  Appendix B List of integrals

###  B.1 Master one-loop integral

We compute the Euclidean loop integral $(k\cdot v=0)$

$\displaystyle\int\frac{d^{D}\ell_{E}\,\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})^{\lambda_{1}}[(k-\ell_{E})^{2}]^{\lambda_{2}}(2v\cdot\ell_{E}-i0^{+})^{\lambda_{3}}}\,,$

using the identities

$\displaystyle\begin{aligned}
\frac{1}{\alpha^{\lambda}}&=\frac{1}{\Gamma(\lambda)}\int_{0}^{\infty}dt\,t^{\lambda-1}e^{-\alpha
t}\,,\\\
\frac{1}{(\alpha-i0^{+})^{\lambda}}&=\frac{i^{\lambda}}{\Gamma(\lambda)}\int_{0}^{\infty}dt\,t^{\lambda-1}e^{-i(\alpha-i0^{+})t}\,.\end{aligned}$
(285)

After substitution, we have the integral

$\displaystyle\frac{i^{\lambda_{3}}}{\Gamma(\lambda_{1})\Gamma(\lambda_{2})\Gamma(\lambda_{3})}\int
dt_{1}dt_{2}dt_{3}\,t_{1}^{\lambda_{1}-1}t_{2}^{\lambda_{2}-1}t_{3}^{\lambda_{3}-1}\exp\left[-\frac{t_{1}t_{2}}{t_{1}+t_{2}}k^{2}-\frac{t_{3}^{2}}{t_{1}+t_{2}}v^{2}\right]$
$\displaystyle\phantom{asdfas}\times\exp\left[\frac{2t_{2}}{t_{1}+t_{2}}(k\cdot
a)+\frac{-2it_{3}}{t_{1}+t_{2}}(v\cdot
a)+\frac{a^{2}}{t_{1}+t_{2}}\right]\int_{\ell_{E}^{\prime}}e^{-(t_{1}+t_{2})(\ell_{E}^{\prime})^{2}}$

where $\ell_{E}^{\prime}$ is the shifted loop integration variable. We perform
the Gaussian integral and expand the exponential of the second line.
Evaluating the gamma function and beta function integrals, we get

$\displaystyle\int\frac{\mu^{2\epsilon}d^{D}\ell_{E}\,\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})^{\lambda_{1}}[(k-\ell_{E})^{2}]^{\lambda_{2}}(2v\cdot\ell_{E}-i0^{+})^{\lambda_{3}}}$
$\displaystyle=\frac{i^{\lambda_{3}}\pi^{D/2}\mu^{2\epsilon}}{2\Gamma(\lambda_{1})\Gamma(\lambda_{2})\Gamma(\lambda_{3})(k^{2})^{\lambda_{1}+\lambda_{2}+\frac{\lambda_{3}}{2}-\frac{D}{2}}(v^{2})^{\frac{\lambda_{3}}{2}}}$
$\displaystyle\phantom{=}\times\sum_{l,m,n=0}^{\infty}\frac{(2k\cdot
a)^{l}\left(-2i(v\cdot
a)\sqrt{\frac{k^{2}}{v^{2}}}\right)^{m}(k^{2}a^{2})^{n}}{l!m!n!}\frac{\Gamma(\lambda_{1}+\lambda_{2}+\frac{\lambda_{3}}{2}-\frac{m}{2}-n-\frac{D}{2})\Gamma(\frac{\lambda_{3}+m}{2})}{\Gamma(D-\lambda_{1}-\lambda_{2}-\lambda_{3}+l+m+2n)}$
$\displaystyle\phantom{=asdfasdf}\times\Gamma(\frac{D}{2}-\lambda_{2}-\frac{\lambda_{3}}{2}+\frac{m}{2}+n)\,\Gamma(\frac{D}{2}-\lambda_{1}-\frac{\lambda_{3}}{2}+l+\frac{m}{2}+n)\,,$
(286)

where $\mu^{2\epsilon}$ is the mass scale required for dimensional
regularisation $D=3-2\epsilon$. It is easy to verify that for $\lambda_{1}=0$
or $\lambda_{2}=0$ the integral vanishes, and for $\lambda_{3}=0$ that the
integral localises onto $m=0$.141414The value for $\lambda_{3}=0$ should be
understood as a limiting value $\lambda_{3}\to 0$, where
$\frac{\Gamma(\lambda_{3}/2)}{\Gamma(\lambda_{3})}\to 2$. Setting $a=0$ we
only keep $l=m=n=0$ of the sum, and recover the Euclidean version of (10.25)
of ref. Smirnov:2012gma . The divergence of the integral for non-positive
integral values of
$(\lambda_{1}+\lambda_{2}+\frac{\lambda_{3}}{2}-\frac{m}{2}-n-\frac{D}{2})\in\mathbb{Z}^{\leq
0}$ is harmless since the result formally becomes non-negative integral powers
of $k^{2}$, which vanishes under the impact parameter space integral
$\int_{k}e^{ik\cdot b}$ for $b^{\mu}\neq 0$. The master integral (286) can be
viewed as a _tensor integral generating function_ , e.g. the vector integral
can be evaluated as

$\displaystyle\int\frac{d^{D}\ell_{E}\,\,\ell_{E}^{\mu}\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})^{\lambda_{1}}[(k-\ell_{E})^{2}]^{\lambda_{2}}(2v\cdot\ell_{E}-i0^{+})^{\lambda_{3}}}=\frac{1}{2}\frac{\partial}{\partial
a_{\mu}}\int\frac{d^{D}\ell_{E}\,\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})^{\lambda_{1}}[(k-\ell_{E})^{2}]^{\lambda_{2}}(2v\cdot\ell_{E}-i0^{+})^{\lambda_{3}}}\,.$

This can be used to check consistency of (286), e.g.
$v^{\mu}\frac{\partial}{\partial
a^{\mu}}\Leftrightarrow(\lambda_{3}\to\lambda_{3}-1)$. Such consistency
relations could be used to bootstrap tensor integral generating functions
Feng:2022hyg .

In all the integrals listed in this appendix, we only keep the non-analytic
terms in $k^{2}$ and drop dimensional regularisation artefacts
($\mathcal{O}(\epsilon^{-1})$ and $\mathcal{O}(\epsilon)$). All
$\log(k^{2})$-dependent terms of the integrals vanish for time-symmetric
$i0^{+}$ prescription, which is equivalent to taking the real part of the
integrals.

##### Special cases: Scalar integrals

$\lambda_{1}=\lambda_{2}=1$, $\lambda_{3}=2$, and $D=3-2\epsilon$ with extra
$k^{2}$.

$\displaystyle
k^{2}\int\frac{d^{D}\ell_{E}\,\,\mu^{2\epsilon}e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})(k-\ell_{E})^{2}(2v\cdot\ell_{E}-i0^{+})^{2}}$
$\displaystyle=\frac{\pi^{3}}{2(k^{2})^{1/2}v^{2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n-\frac{1}{2})(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+\frac{1}{2})\Gamma(n+1)\Gamma(l+2m+2n-1)}$
$\displaystyle\phantom{=}+\frac{i\pi^{2}(v\cdot
a)\log(k^{2})}{2(v^{2})^{3/2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n)(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+1)\Gamma(n+1)\Gamma(l+2m+2n)}$
$\displaystyle=\frac{\pi^{5/2}}{2(k^{2})^{1/2}v^{2}}\sum_{l,m=0}^{\infty}\frac{\Gamma(l+m-\frac{1}{2})(2k\cdot
a)^{l}(-k^{2}a^{2})^{m}\,{}_{2}F_{1}(1,-m;\frac{1}{2};\frac{(v\cdot
a)^{2}}{v^{2}a^{2}})}{\Gamma(l+1)\Gamma(m+1)\Gamma(l+2m-1)}$
$\displaystyle\phantom{=}+\frac{i\pi^{2}(v\cdot
a)\log(k^{2})}{2(v^{2})^{3/2}}\sum_{l,n=0}^{\infty}\frac{\Gamma(l+n)(2k\cdot
a)^{l}(-k^{2}a^{2})^{n}\left(1-\frac{(v\cdot
a)^{2}}{a^{2}v^{2}}\right)^{n}}{\Gamma(l+1)\Gamma(n+1)\Gamma(l+2n)}\,.$ (287)

$\lambda_{1}=\lambda_{2}=\lambda_{3}=1$ and $D=3-2\epsilon$ with extra
$k^{2}$.

$\displaystyle
k^{2}\int\frac{d^{D}\ell_{E}\,\,\mu^{2\epsilon}e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})(k-\ell_{E})^{2}(2v\cdot\ell_{E}-i0^{+})}$
$\displaystyle=\frac{\pi^{3}(v\cdot
a)(k^{2})^{1/2}}{2v^{2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n+\frac{1}{2})(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+\frac{3}{2})\Gamma(n+1)\Gamma(l+2m+2n+1)}$
$\displaystyle\phantom{=}+\frac{i\pi^{2}\log(k^{2})}{2(v^{2})^{1/2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n)(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+1)\Gamma(n+1)\Gamma(l+2m+2n)}$
$\displaystyle=\frac{\pi^{5/2}(v\cdot
a)(k^{2})^{1/2}}{v^{2}}\sum_{l,m=0}^{\infty}\frac{\Gamma(l+m+\frac{1}{2})(2k\cdot
a)^{l}(-k^{2}a^{2})^{m}\,{}_{2}F_{1}(1,-m;\frac{3}{2};\frac{(v\cdot
a)^{2}}{v^{2}a^{2}})}{\Gamma(l+1)\Gamma(m+1)\Gamma(l+2m+1)}$
$\displaystyle\phantom{=}+\frac{i\pi^{2}\log(k^{2})}{2(v^{2})^{1/2}}\sum_{l,n=0}^{\infty}\frac{\Gamma(l+n)(2k\cdot
a)^{l}(-k^{2}a^{2})^{n}\left(1-\frac{(v\cdot
a)^{2}}{a^{2}v^{2}}\right)^{n}}{\Gamma(l+1)\Gamma(n+1)\Gamma(l+2n)}\,.$ (288)

$\lambda_{1}=\lambda_{2}=1$, $\lambda_{3}=0$, and $D=3$.

$\displaystyle\int\frac{d^{D}\ell_{E}\,\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})(k-\ell_{E})^{2}}$
$\displaystyle=\frac{\pi^{5/2}}{(k^{2})^{1/2}}\sum_{l,n=0}^{\infty}\frac{\Gamma(l+n+\frac{1}{2})(2k\cdot
a)^{l}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(n+1)\Gamma(l+2n+1)}\,.$ (289)

$\lambda_{1}=\lambda_{2}=1$, $\lambda_{3}=-1$, and $D=3$.

$\displaystyle\int\frac{d^{D}\ell_{E}\,\,(2v\cdot\ell_{E})\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})(k-\ell_{E})^{2}}$ $\displaystyle=-2\pi^{5/2}(v\cdot
a)(k^{2})^{1/2}\sum_{l,m=0}^{\infty}\frac{\Gamma(l+m+\frac{3}{2})(2k\cdot
a)^{l}(-k^{2}a^{2})^{m}}{\Gamma(l+1)\Gamma(m+1)\Gamma(l+2m+3)}\,.$ (290)

##### Special cases: Vector integrals

Assume $f\cdot k=f\cdot v=0$. $\lambda_{1}=\lambda_{2}=\lambda_{3}=1$ and
$D=3-2\epsilon$.

$\displaystyle\int\frac{d^{D}\ell_{E}\,\,\mu^{2\epsilon}(f\cdot\ell_{E})\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})(k-\ell_{E})^{2}(2v\cdot\ell_{E}-i0^{+})}$
$\displaystyle=-\frac{\pi^{3}(v\cdot a)(f\cdot
a)(k^{2})^{1/2}}{2v^{2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n+\frac{3}{2})(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+\frac{3}{2})\Gamma(n+1)\Gamma(l+2m+2n+3)}$
$\displaystyle\phantom{=}-\frac{i\pi^{2}(f\cdot
a)\log(k^{2})}{2(v^{2})^{1/2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n+1)(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+1)\Gamma(n+1)\Gamma(l+2m+2n+2)}$
$\displaystyle=-\frac{\pi^{5/2}(v\cdot a)(f\cdot
a)(k^{2})^{1/2}}{v^{2}}\sum_{l,m=0}^{\infty}\frac{\Gamma(l+m+\frac{3}{2})(2k\cdot
a)^{l}(-k^{2}a^{2})^{m}\,{}_{2}F_{1}(1,-m;\frac{3}{2};\frac{(v\cdot
a)^{2}}{v^{2}a^{2}})}{\Gamma(l+1)\Gamma(m+1)\Gamma(l+2m+3)}$
$\displaystyle\phantom{=}-\frac{i\pi^{2}(f\cdot
a)\log(k^{2})}{2(v^{2})^{1/2}}\sum_{l,n=0}^{\infty}\frac{\Gamma(l+n+1)(2k\cdot
a)^{l}(-k^{2}a^{2})^{n}\left(1-\frac{(v\cdot
a)^{2}}{a^{2}v^{2}}\right)^{n}}{\Gamma(l+1)\Gamma(n+1)\Gamma(l+2n+2)}\,.$
(291)

$\lambda_{1}=\lambda_{2}=1$, $\lambda_{3}=2$, and $D=3-2\epsilon$.

$\displaystyle\int\frac{d^{D}\ell_{E}\,\,\mu^{2\epsilon}(f\cdot\ell_{E})\,e^{2\ell_{E}\cdot
a}}{(\ell_{E}^{2})(k-\ell_{E})^{2}(2v\cdot\ell_{E}-i0^{+})^{2}}$
$\displaystyle=-\frac{\pi^{3}(f\cdot
a)}{2(k^{2})^{1/2}v^{2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n+\frac{3}{2})(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+\frac{1}{2})\Gamma(n+1)\Gamma(l+2m+2n+3)}$
$\displaystyle\phantom{=}-\frac{i\pi^{2}(f\cdot a)(v\cdot
a)\log(k^{2})}{2(v^{2})^{3/2}}\sum_{l,m,n=0}^{\infty}\frac{\Gamma(l+m+n+1)(2k\cdot
a)^{l}\left(k^{2}\frac{(v\cdot
a)^{2}}{v^{2}}\right)^{m}(-k^{2}a^{2})^{n}}{\Gamma(l+1)\Gamma(m+1)\Gamma(n+1)\Gamma(l+2m+2n+2)}$
$\displaystyle=-\frac{\pi^{5/2}(f\cdot
a)}{2(k^{2})^{1/2}v^{2}}\sum_{l,m=0}^{\infty}\frac{\Gamma(l+m+\frac{1}{2})(2k\cdot
a)^{l}(-k^{2}a^{2})^{m}\,{}_{2}F_{1}(1,-m;\frac{1}{2};\frac{(v\cdot
a)^{2}}{v^{2}a^{2}})}{\Gamma(l+1)\Gamma(m+1)\Gamma(l+2m+1)}$
$\displaystyle\phantom{=}-\frac{i\pi^{2}(f\cdot a)(v\cdot
a)\log(k^{2})}{2(v^{2})^{3/2}}\sum_{l,n=0}^{\infty}\frac{\Gamma(l+n+1)(2k\cdot
a)^{l}(-k^{2}a^{2})^{n}\left(1-\frac{(v\cdot
a)^{2}}{a^{2}v^{2}}\right)^{n}}{\Gamma(l+1)\Gamma(n+1)\Gamma(l+2n+2)}\,.$
(292)

###  B.2 Fourier transform integral

The Fourier transform to impact parameter space is given as

$\displaystyle\int\frac{d^{D}k_{E}}{(2\pi)^{D}}\frac{e^{i(k_{E}\cdot
b)}}{[k_{E}^{2}\mp i0^{+}]^{\lambda}}$
$\displaystyle=\frac{\Gamma(\frac{D}{2}-\lambda)}{2^{2\lambda}\pi^{\frac{D}{2}}\Gamma(\lambda)}\frac{1}{\left(b^{2}\pm
i0^{+}\right)^{\frac{D}{2}-\lambda}}\,,$ (293)

where we assumed $b^{2}\in\mathbb{R}$ and included $i0^{+}$ prescription for
convergence. The $i0^{+}$ prescription can be dropped since there is no branch
cut ambiguity for Euclidean signature.

Fourier transforms with numerators can be evaluated using differentiation;
$k_{E}^{\mu}\leftrightarrow-i\frac{\partial}{\partial b_{\mu}}$. Repeated
numerator factors can be computed as directional derivatives, i.e.

$\displaystyle\begin{aligned}
\int\frac{d^{D}k_{E}}{(2\pi)^{D}}\frac{(k_{E}\cdot a)^{l}e^{i(k_{E}\cdot
b)}}{[k_{E}^{2}]^{\lambda}}&=l!\left.\int\frac{d^{D}k_{E}}{(2\pi)^{D}}\frac{e^{i(k_{E}\cdot[b-ia])}}{[k_{E}^{2}]^{\lambda}}\right|_{\mathcal{O}(a^{l})}\\\
&=\left.\frac{l!\Gamma(\frac{D}{2}-\lambda)}{2^{2\lambda}\pi^{\frac{D}{2}}\Gamma(\lambda)}\frac{1}{\left[(b-ia)^{2}\right]^{\frac{D}{2}-\lambda}}\right|_{\mathcal{O}(a^{l})}\\\
&=\frac{l!\Gamma(\frac{D}{2}-\lambda)}{2^{2\lambda}\pi^{\frac{D}{2}}\Gamma(\lambda)}\frac{i^{l}C_{l}^{(\frac{D}{2}-\lambda)}(\frac{(a\cdot
b)}{(a^{2}b^{2})^{1/2}})(a^{2})^{l/2}}{\left(b^{2}\right)^{\frac{D}{2}-\lambda+\frac{l}{2}}}\,,\end{aligned}$
(294)

where $C_{n}^{(\lambda)}(x)$ is the ultraspherical/Gegenbauer polynomial.
Differentiation in $\lambda$ can be used to compute Fourier transform for
logarithms,

$\displaystyle\begin{aligned}
&\int\frac{d^{D}k_{E}}{(2\pi)^{D}}[k_{E}^{2}]^{\lambda}(k_{E}\cdot
a)^{l}\log(k_{E}^{2})e^{i(k_{E}\cdot
b)}=l!\left.\frac{\partial}{\partial\lambda}\int\frac{d^{D}k_{E}}{(2\pi)^{D}}[k_{E}^{2}]^{\lambda}e^{i(k_{E}\cdot[b-ia])}\right|_{\mathcal{O}(a^{l})}\\\
&\phantom{asdfasdf}=-\frac{(-4)^{\lambda}\lambda!\Gamma(\frac{D}{2}+\lambda)\Gamma(l+1)}{\pi^{\frac{D}{2}}}\frac{i^{l}C_{l}^{(\frac{D}{2}+\lambda)}(\frac{(a\cdot
b)}{(a^{2}b^{2})^{1/2}})(a^{2})^{l/2}}{\left(b^{2}\right)^{\frac{D}{2}+\lambda+\frac{l}{2}}}\,,\end{aligned}$
(295)

where $\lambda\in\mathbb{Z}^{\geq 0}$ is assumed.

##### Special cases: aligned spin

Only even powers of $l$ are relevant for the aligned spin configuration, where
$a\cdot b=0$. The master integral (294) reduces to

$\displaystyle\begin{aligned}
\int\frac{d^{D}k_{E}}{(2\pi)^{D}}\frac{(k_{E}\cdot a)^{2l}e^{i(k_{E}\cdot
b)}}{[k_{E}^{2}]^{\lambda}}&=\frac{\Gamma(\frac{D}{2}-\lambda)}{2^{2\lambda}\pi^{\frac{D}{2}}\Gamma(\lambda)}\frac{(\frac{D}{2}-\lambda)_{l}}{\left(b^{2}\right)^{\frac{D}{2}-\lambda}}\frac{(2l)!}{l!}\left(\frac{a^{2}}{b^{2}}\right)^{l}\,,\end{aligned}$
(296)

where $(a)_{n}$ is the Pochhammer symbol. For axial scattering we set $a\cdot
b=\pm\sqrt{a^{2}b^{2}}$, which leads to

$\displaystyle\begin{aligned}
\int\frac{d^{D}k_{E}}{(2\pi)^{D}}\frac{(k_{E}\cdot a)^{2l}e^{i(k_{E}\cdot
b)}}{[k_{E}^{2}]^{\lambda}}&=\frac{\Gamma(D+2l-2\lambda)}{2^{D-1}\pi^{\frac{D-1}{2}}\Gamma(\lambda)\Gamma(\frac{D+1}{2}-\lambda)}\frac{1}{\left(b^{2}\right)^{\frac{D}{2}-\lambda}}\left(\frac{-a^{2}}{b^{2}}\right)^{l}\,.\end{aligned}$
(297)

##  Appendix C 2PL computations

###  C.1 Longitudinal part of the momentum kick

Our goal is to show that the longitudinal part of $\Delta_{(2)}p_{1}^{\mu}$
agrees with

$\displaystyle\Delta_{(2)}p_{1}^{\mu}|_{\mathrm{iter}}=\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},p_{1}^{\mu}\\}\\}=\frac{1}{2}\\{\chi_{(1)},\Delta_{(1)}p_{1}^{\mu}\\}\,.$
(298)

##### Part 1

In the spin-less case, all iteration terms contain
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt^{\prime}$.
That is no longer true when spin is turned on. Let us first focus on the new
terms not containing $\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}$.

Using the following results as building blocks,

$\displaystyle\begin{split}\\{k\cdot b,\ell\cdot b\\}=\\{k\cdot y,\ell\cdot
y\\}&=\frac{\epsilon[k,\ell,y_{1},v_{1}]}{m_{1}}+\frac{\epsilon[k,\ell,y_{2},v_{2}]}{m_{2}}\,,\\\
\left\\{k\cdot
b,-\epsilon^{\mu}[\ell,v_{1},v_{2}]\right\\}&=\frac{\epsilon^{\mu}[k,\ell,v_{2}]}{m_{1}}+\frac{\epsilon^{\mu}[k,\ell,v_{1}]}{m_{2}}\,,\\\
\left\\{\epsilon[k,v_{1},v_{2},y],\ell\cdot
b\right\\}&=\frac{\epsilon[k,\ell,y,v_{2}]}{m_{1}}+\frac{\epsilon[k,\ell,y,v_{1}]}{m_{2}}\,,\\\
\left\\{\epsilon[k,v_{1},v_{2},y],\ell\cdot
y\right\\}&=(k\cdot\ell)\left(\frac{v_{2}\cdot y_{1}}{m_{1}}-\frac{v_{1}\cdot
y_{2}}{m_{2}}\right)\,,\end{split}$ (299)

we collect four contributions to (298):

$\displaystyle\begin{split}A_{1}^{\mu}&=(i\ell^{\mu})\gamma^{2}\left\\{\cosh(k\cdot
y)e^{ik\cdot b},\cosh(\ell\cdot y)e^{i\ell\cdot b}\right\\}\\\
&=(i\ell^{\mu})\gamma^{2}\\{k\cdot y,\ell\cdot y\\}\sinh(k\cdot
y)\sinh(\ell\cdot y)e^{iq\cdot b}\\\ &\quad-(i\ell^{\mu})\gamma^{2}\\{k\cdot
b,\ell\cdot b\\}\cosh(k\cdot y)\cosh(\ell\cdot y)e^{iq\cdot b}\,,\end{split}$
(300)
$\displaystyle\begin{split}A_{4}^{\mu}&=\left\\{-i\epsilon[k,v_{1},v_{2},y]\frac{\sinh(k\cdot
y)}{k\cdot y}e^{ik\cdot b},-\epsilon^{\mu}[\ell,v_{1},v_{2}]\sinh(\ell\cdot
y)e^{i\ell\cdot b}\right\\}\\\
&=i\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[\ell,v_{1},v_{2}]\\{k\cdot
y,\ell\cdot y\\}\left[\frac{\cosh(k\cdot y)}{k\cdot y}-\frac{\sinh(k\cdot
y)}{(k\cdot y)^{2}}\right]\cosh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad-i\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[\ell,v_{1},v_{2}]\\{k\cdot
b,\ell\cdot b\\}\frac{\sinh(k\cdot y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot
b}\\\
&\quad+\epsilon[k,v_{1},v_{2},y]\left(\frac{\epsilon^{\mu}[k,\ell,v_{2}]}{m_{1}}+\frac{\epsilon^{\mu}[k,\ell,v_{1}]}{m_{2}}\right)\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad-\left(\frac{\epsilon[k,\ell,y,v_{2}]}{m_{1}}+\frac{\epsilon[k,\ell,y,v_{1}]}{m_{2}}\right)\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+i(k\cdot\ell)\left(\frac{v_{2}\cdot y_{1}}{m_{1}}-\frac{v_{1}\cdot
y_{2}}{m_{2}}\right)\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\cosh(\ell\cdot y)e^{iq\cdot b}\,,\end{split}$ (301)
$\displaystyle\begin{split}A_{2}^{\mu}&=\gamma\left\\{\cosh(k\cdot
y)e^{ik\cdot b},-\epsilon^{\mu}[\ell,v_{1},v_{2}]\sinh(\ell\cdot
y)e^{i\ell\cdot b}\right\\}\\\
&=-\gamma\,\epsilon^{\mu}[\ell,v_{1},v_{2}]\\{k\cdot y,\ell\cdot
y\\}\sinh(k\cdot y)\cosh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+\gamma\,\epsilon^{\mu}[\ell,v_{1},v_{2}]\\{k\cdot b,\ell\cdot
b\\}\cosh(k\cdot y)\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+i\gamma\left(\frac{\epsilon^{\mu}[k,\ell,v_{2}]}{m_{1}}+\frac{\epsilon^{\mu}[k,\ell,v_{1}]}{m_{2}}\right)\cosh(k\cdot
y)\sinh(\ell\cdot y)e^{iq\cdot b}\,,\end{split}$ (302)
$\displaystyle\begin{split}A_{3}^{\mu}&=(i\ell^{\mu})\gamma\left\\{-i\epsilon[k,v_{1},v_{2},y]\frac{\sinh(k\cdot
y)}{k\cdot y}e^{ik\cdot b},\cosh(\ell\cdot y)e^{i\ell\cdot b}\right\\}\\\
&=\gamma\,\ell^{\mu}\epsilon[k,v_{1},v_{2},y]\\{k\cdot y,\ell\cdot
y\\}\left[\frac{\cosh(k\cdot y)}{k\cdot y}-\frac{\sinh(k\cdot y)}{(k\cdot
y)^{2}}\right]\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad-\gamma\,\ell^{\mu}\epsilon[k,v_{1},v_{2},y]\\{k\cdot b,\ell\cdot
b\\}\frac{\sinh(k\cdot y)}{k\cdot y}\cosh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+\gamma\,\ell^{\mu}(k\cdot\ell)\left(\frac{v_{2}\cdot
y_{1}}{m_{1}}-\frac{v_{1}\cdot y_{2}}{m_{2}}\right)\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+i\gamma\,\ell^{\mu}\left(\frac{\epsilon[k,\ell,y,v_{2}]}{m_{1}}+\frac{\epsilon[k,\ell,y,v_{1}]}{m_{2}}\right)\frac{\sinh(k\cdot
y)}{k\cdot y}\cosh(\ell\cdot y)e^{iq\cdot b}\,,\end{split}$ (303)

As usual, we may work in the probe limit $(m_{1}/m_{2}\rightarrow 0)$, where
we get

$\displaystyle\begin{split}m_{1}A_{1}^{\mu}&=-(i\ell^{\mu})\gamma^{2}\epsilon[k,\ell,y_{1},v_{1}]\cosh[(k-\ell)\cdot
y]e^{iq\cdot b}\,,\\\
m_{1}A_{4}^{\mu}&=-i\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[\ell,v_{1},v_{2}]\epsilon[k,\ell,y_{1},v_{1}]\frac{\sinh(k\cdot
y)}{(k\cdot y)^{2}}\cosh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+i\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[\ell,v_{1},v_{2}]\epsilon[k,\ell,y_{1},v_{1}]\frac{\cosh[(k-\ell)\cdot
y]}{k\cdot y}e^{iq\cdot b}\\\
&\quad+\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[k,\ell,v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad-\epsilon[k,\ell,y,v_{2}]\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+i(k\cdot\ell)(v_{2}\cdot
y_{1})\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot y)}{k\cdot
y}\cosh(\ell\cdot y)e^{iq\cdot b}\,,\end{split}$ (304)
$\displaystyle\begin{split}m_{1}A_{2}^{\mu}&=-\gamma\,\epsilon^{\mu}[\ell,v_{1},v_{2}]\epsilon[k,\ell,y_{1},v_{1}]\sinh[(k-\ell)\cdot
y]e^{iq\cdot b}\\\ &\quad+i\gamma\epsilon^{\mu}[k,\ell,v_{2}]\cosh(k\cdot
y)\sinh(\ell\cdot y)e^{iq\cdot b}\,,\\\
m_{1}A_{3}^{\mu}&=-\gamma\,\ell^{\mu}\epsilon[k,v_{1},v_{2},y]\epsilon[k,\ell,y_{1},v_{1}]\frac{\sinh(k\cdot
y)}{(k\cdot y)^{2}}\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad-\gamma\,\ell^{\mu}\epsilon[k,v_{1},v_{2},y]\epsilon[k,\ell,y_{1},v_{1}]\frac{\sinh[(k-\ell)\cdot
y]}{k\cdot y}e^{iq\cdot b}\\\ &\quad+\gamma\,\ell^{\mu}(k\cdot\ell)(v_{2}\cdot
y_{1})\frac{\sinh(k\cdot y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\\\
&\quad+i\gamma\,\ell^{\mu}\epsilon[k,\ell,y,v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\cosh(\ell\cdot y)e^{iq\cdot b}\,.\end{split}$ (305)

We can simplify $A_{2}^{\mu}$ a bit and write

$\displaystyle\begin{split}m_{1}A_{2}^{\mu}&=-\gamma\,\ell^{\mu}(k\cdot\ell)(v_{2}\cdot
y_{1})\sinh[(k-\ell)\cdot y]e^{iq\cdot b}\\\
&\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\;+i\gamma\epsilon^{\mu}[k,\ell,v_{2}]\cosh(k\cdot
y)\sinh(\ell\cdot y)e^{iq\cdot b}\;}\,,\end{split}$ (306)

Next, using (104), we try to remove $(k\cdot y)$ factors in the denominators
of $A_{3}^{\mu}$ and $A_{4}^{\mu}$:

$\displaystyle\begin{split}m_{1}A_{3}^{\mu}&=-\gamma\,\ell^{\mu}(k\cdot\ell)(v_{2}\cdot
y_{1})\sinh[(k-\ell)\cdot y]e^{iq\cdot b}\\\
&\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\;+i\gamma\,\ell^{\mu}\epsilon[k,\ell,y,v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\cosh(\ell\cdot y)e^{iq\cdot b}\;}\,,\\\
m_{1}A_{4}^{\mu}&=(i\ell^{\mu})(v_{2}\cdot
y_{1})\epsilon[k,\ell,v_{1},v_{2}]\cosh[(k-\ell)\cdot y]e^{iq\cdot b}\\\
&\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\;+\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[k,\ell,v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\;}\\\
&\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\;-\epsilon[k,\ell,y,v_{2}]\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)e^{iq\cdot b}\;}\,.\end{split}$ (307)

We have enumerated all terms _not_ containing
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt^{\prime}$.
Now we split them into two parts: the “y-part” and the “v-part”. The former is
linear in $y_{1}$ while the latter (marked red in the equations above) is
independent of $y_{1}$. The $k\cdot y$ factor in the denominator is to be
cancelled against an $\epsilon[\cdot,\cdot,\cdot,y]$ factor in the numerator.

The y-part gives fairly simple expressions:

$\displaystyle\begin{split}m_{1}A_{1^{+}4}^{\mu}|_{y}&=(i\ell^{\mu})(-\gamma^{2}\epsilon[k,\ell,y_{1},v_{1}]+(v_{2}\cdot
y)\epsilon[k,\ell,v_{1},v_{2}])\cosh[(k-\ell)\cdot y]e^{iq\cdot b}\,,\\\
m_{1}A_{2+3}^{\mu}|_{y}&=(i\ell^{\mu})(2i\gamma)(v_{2}\cdot
y_{1})(k\cdot\ell)\sinh[(k-\ell)\cdot y]e^{iq\cdot b}\,.\end{split}$ (308)

There agree perfectly with the longitudinal part of (110), (112) in the main
text.

##### Part 2

The v-part is more involved, as it gets combined with the
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt^{\prime}$
terms. In the probe limit, the
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt^{\prime}$
factor comes from

$\displaystyle\begin{split}\left\\{e^{ik\cdot
b}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v_{1}),e^{i\ell\cdot b}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\ell\cdot
v_{1})\right\\}\;\rightarrow\;\;&i(k\cdot\ell)\frac{e^{iq\cdot
b}}{m_{1}}\left[\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(\ell\cdot
v_{1})-\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(k\cdot
v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\ell\cdot v_{1})\right]\\\ &=i(k\cdot\ell)\frac{e^{iq\cdot
b}}{m_{1}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(q\cdot v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(\ell\cdot v_{1})\,.\end{split}$ (309)

This factor is to be multiplied by

$\displaystyle\begin{split}B^{\mu}&=\left[\gamma\cosh(k\cdot
y)-i\epsilon[k,v_{1},v_{2},y]\frac{\sinh(k\cdot y)}{k\cdot y}\right]\\\
&\quad\times\left[(i\ell^{\mu})\gamma\cosh(\ell\cdot
y)-\epsilon^{\mu}[\ell,v_{1},v_{2}]\sinh(\ell\cdot y)\right]\\\
&=B_{1}^{\mu}+B_{2}^{\mu}+B_{3}^{\mu}+B_{4}^{\mu}\,,\end{split}$ (310)

where

$\displaystyle\begin{split}B_{1}^{\mu}&=(i\ell^{\mu})\gamma^{2}\cosh(k\cdot
y)\cosh(\ell\cdot y)\,,\\\
B_{4}^{\mu}&=i\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)\,,\\\
B_{2}^{\mu}&=-\gamma\epsilon^{\mu}[\ell,v_{1},v_{2}]\cosh(k\cdot
y)\sinh(\ell\cdot y)\,,\\\
B_{3}^{\mu}&=\gamma\ell^{\mu}\epsilon[k,v_{1},v_{2},y]\frac{\sinh(k\cdot
y)}{k\cdot y}\cosh(\ell\cdot y)\,.\end{split}$ (311)

Multiplying them by $i(k\cdot\ell)$, and hiding $e^{iq\cdot
b}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(q\cdot
v_{1})\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(\ell\cdot v_{1})$ for now, we get

$\displaystyle\begin{split}D_{1}^{\mu}&=-\ell^{\mu}\gamma^{2}(k\cdot\ell)\cosh(k\cdot
y)\cosh(\ell\cdot y)\,,\\\
D_{4}^{\mu}&=-(k\cdot\ell)\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)\,,\\\
D_{2}^{\mu}&=-i\gamma(k\cdot\ell)\epsilon^{\mu}[\ell,v_{1},v_{2}]\cosh(k\cdot
y)\sinh(\ell\cdot y)\,,\\\
D_{3}^{\mu}&=i\gamma\ell^{\mu}(k\cdot\ell)\epsilon[k,v_{1},v_{2},y]\frac{\sinh(k\cdot
y)}{k\cdot y}\cosh(\ell\cdot y)\,.\end{split}$ (312)

Now, we bring the red colored terms from (306) and (307) and apply the
identity

$\displaystyle\delta(x)=-x\delta^{\prime}(x)\,.$ (313)

After pulling out some overall factor, we record the results as

$\displaystyle\begin{split}C_{1}^{\mu}&=0\,,\\\ C_{4}^{\mu}&=-(\ell\cdot
v_{1})\epsilon[k,v_{1},v_{2},y]\epsilon^{\mu}[k,\ell,v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)\\\ &\quad-(k\cdot
v_{1})\epsilon[k,\ell,y,v_{2}]\epsilon^{\mu}[\ell,v_{1},v_{2}]\frac{\sinh(k\cdot
y)}{k\cdot y}\sinh(\ell\cdot y)\,,\\\ C_{2}^{\mu}&=-i\gamma(\ell\cdot
v_{1})\epsilon^{\mu}[k,\ell,v_{2}]\cosh(k\cdot y)\sinh(\ell\cdot y)\,,\\\
C_{3}^{\mu}&=+i\gamma(k\cdot
v_{1})\ell^{\mu}\epsilon[k,\ell,y,v_{2}]\frac{\sinh(k\cdot y)}{k\cdot
y}\cosh(\ell\cdot y)\,.\end{split}$ (314)

For $C_{3}^{\mu}$ and the second line of $C_{4}^{\mu}$, we used
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v_{1}+\ell\cdot v_{1})$ to replace $(\ell\cdot v_{1})$ by $-(k\cdot v_{1})$.
Merging all the $C$-terms and the $D$-terms, we obtain the final result in
perfect agreement with the longitudinal part of $\Delta_{(2v)}p^{\mu}$ in
(113), (114) in the main text.

###  C.2 Transverse part of the spin kick

In section 3.2.2 of the main text, we showed how to compute the 2PL spin kick.
In this appendix, we give some details of the computation and confirm that the
transverse part of the spin kick agrees with the eikonal formula,

$\displaystyle\begin{split}\\{\chi_{(n)},y_{1}^{\mu}\\}&=\frac{1}{m_{1}}\left[v_{1}^{\mu}y_{1}^{\nu}\frac{\partial}{\partial
x_{1}^{\nu}}+\epsilon^{\mu\nu}[v_{1},y_{1}]\frac{\partial}{\partial
y_{1}^{\nu}}\right]\chi_{(n)}\,.\end{split}$ (315)

We begin with the overall structure of the 2PL spin kick (121):

$\displaystyle\Delta_{(2)}y_{1}^{\mu}=\frac{(q_{1}q_{2})^{2}}{m_{1}^{2}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{k^{2}\ell^{2}(ik\cdot
v_{1}+0^{+})^{2}}\mathcal{N}^{\mu}\,.$ (316)

The numerator $\mathcal{N}^{\mu}$ can be computed separately for each term in
(120). For (a) and (b) terms, we also distinguish the same/opposite helicity
contributions.

$\displaystyle\begin{split}\mathcal{N}^{\mu}_{(2ao)}=\left(\mathrm{ch}\\!\boxminus
C_{ao}+\mathrm{sh}\\!\boxminus
S_{ao}\right)^{\mu}\,,&\quad\mathcal{N}^{\mu}_{(2as)}=\left(\mathrm{ch}\\!\boxplus
C_{as}+\mathrm{sh}\\!\boxplus S_{as}\right)^{\mu}\,,\\\
\mathcal{N}^{\mu}_{(2bo)}=\left(\mathrm{ch}\\!\boxminus
C_{bo}+\mathrm{sh}\\!\boxminus
S_{bo}\right)^{\mu}\,,&\quad\mathcal{N}^{\mu}_{(2bs)}=\left(\mathrm{ch}\\!\boxplus
C_{bs}+\mathrm{sh}\\!\boxplus S_{bs}\right)^{\mu}\,,\\\
\mathcal{N}^{\mu}_{(2c)}=\left(\mathrm{ch}\\!\boxminus
C_{c}+\mathrm{sh}\\!\boxminus
S_{c}\right)^{\mu}\,,&\quad\mathcal{N}^{\mu}_{(2d)}=\left(\mathrm{ch}\\!\boxminus
C_{d}+\mathrm{sh}\\!\boxminus S_{d}\right)^{\mu}\,,\end{split}$ (317)

The same helicity terms are

$\displaystyle\begin{split}C^{\mu}_{as}&=\frac{1}{2}(ik\cdot
v_{1})\left[k^{\mu}(\ell\cdot y_{1})-\ell^{\mu}(k\cdot
y_{1})-y_{1}^{\mu}(k\cdot\ell)\right]\,,\quad S^{\mu}_{as}=\frac{i}{2}(ik\cdot
v_{1})\epsilon^{\mu}[k,\ell,y_{1}]\,,\\\
C^{\mu}_{bs}&=\frac{i}{2}\left[\ell^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right]\gamma(k\cdot\ell)+\frac{i}{2}\epsilon^{\mu}[\ell,v_{2},y_{1}]\epsilon[k,\ell,v_{1},v_{2}]\,,\\\
S^{\mu}_{bs}&=\frac{1}{2}\left[\ell^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right]\epsilon[k,\ell,v_{1},v_{2}]-\frac{1}{2}\epsilon^{\mu}[\ell,v_{2},y_{1}]\gamma(k\cdot\ell)\,.\end{split}$
(318)

Among the opposite helicity terms, (a) and (b) terms are linear in $y_{1}$:

$\displaystyle\begin{split}C^{\mu}_{ao}&=\frac{1}{2}(ik\cdot
v_{1})\left[-k^{\mu}(\ell\cdot y_{1})-\ell^{\mu}(k\cdot
y_{1})+y_{1}^{\mu}(k\cdot\ell)+2v_{2}^{\mu}(v_{2}\cdot
y_{1})(k\cdot\ell)\right]\,,\\\ S^{\mu}_{ao}&=\frac{i}{2}(ik\cdot
v_{1})\left[v_{2}^{\mu}\epsilon[k,\ell,y_{1},v_{2}]+\epsilon^{\mu}[k,\ell,v_{2}](v_{2}\cdot
y_{1})\right]\,,\\\ C^{\mu}_{bo}&=\frac{i}{2}\left[\ell^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right]\gamma(k\cdot\ell)-\frac{i}{2}\epsilon^{\mu}[\ell,v_{2},y_{1}]\epsilon[k,\ell,v_{1},v_{2}]\,,\\\
S^{\mu}_{bo}&=\frac{1}{2}\left[\ell^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right]\epsilon[k,\ell,v_{1},v_{2}]+\frac{1}{2}\epsilon^{\mu}[\ell,v_{2},y_{1}]\gamma(k\cdot\ell)\,,\end{split}$
(319)

whereas (c), (d) terms are quadratic in $y_{1}$:

$\displaystyle\begin{split}C^{\mu}_{c}&=(ik\cdot
v_{1})i\left[\left[\ell^{\mu}(v_{2}\cdot y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right]\epsilon[k,\ell,y_{1},v_{2}]-\epsilon^{\mu}[\ell,v_{2},y_{1}](k\cdot\ell)(v_{2}\cdot
y_{1})\right]\,,\\\ S^{\mu}_{c}&=(ik\cdot
v_{1})\left[\left[\ell^{\mu}(v_{2}\cdot y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right](k\cdot\ell)(v_{2}\cdot
y_{1})-\epsilon^{\mu}[\ell,v_{2},y_{1}]\epsilon[k,\ell,v_{2},y_{1}]\right]\,,\\\
C^{\mu}_{d}&=(ik\cdot v_{1})^{2}\left[(\ell\cdot v_{1})(v_{2}\cdot
y_{1})+\gamma(\ell\cdot y_{1})\right]\epsilon^{\mu}[k,v_{2},y_{1}]\\\
&\qquad+(ik\cdot
v_{1})^{2}\epsilon[\ell,v_{1},v_{2},y_{1}]\left[k^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(k\cdot y_{1})\right]\,,\\\ S^{\mu}_{d}&=(ik\cdot
v_{1})^{2}i\left[(\ell\cdot v_{1})(v_{2}\cdot y_{1})+\gamma(\ell\cdot
y_{1})\right]\left[k^{\mu}(v_{2}\cdot y_{1})-v_{2}^{\mu}(k\cdot
y_{1})\right]\\\ &\qquad-(ik\cdot
v_{1})^{2}i\epsilon[\ell,v_{1},v_{2},y_{1}]\epsilon^{\mu}[k,v_{2},y_{1}]\,.\end{split}$
(320)

##### Same helicity sector

The 2PL eikonal (118) contains a single term in the same helicity sector, so
its contribution to the spin kick should be also quite simple. Indeed, after a
lot of cancellations, we get

$\displaystyle\begin{split}C^{\mu}_{as}+C^{\mu}_{bs}&=-\frac{i}{2}(k\cdot\ell)\left[v^{\mu}_{1}(\ell\cdot
y_{1})-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}y^{\mu}_{1}(\ell\cdot
v_{1})}\right]\,,\\\
S^{\mu}_{as}+S^{\mu}_{bs}&=-\frac{1}{2}(k\cdot\ell)\epsilon^{\mu}[v_{1},y_{1},\ell]\,.\end{split}$
(321)

Not all terms contribute to the transverse part;
$\Delta_{(2)}y^{\mu}|_{\mathrm{tr}}$ should be orthogonal to $y_{1}^{\mu}$.
The non-orthogonal term, marked red in the equation above, is projected out
upon symmetrisation under the exchange operation $k\leftrightarrow\ell$,

$\displaystyle(\ell\cdot v_{1})(k\cdot\ell)\cosh\boxplus$
$\displaystyle\;\;\rightarrow\;\;(q\cdot v_{1})(k\cdot\ell)\cosh\boxplus=0\,.$
(322)

The equality hold in the $(q_{\perp})$ integral. What is left after the
symmetrisation is

$\displaystyle
C^{\mu}_{as+bs}|_{\mathrm{tr}}=-\frac{i}{2}(k\cdot\ell)v^{\mu}_{1}(\ell\cdot
y_{1})\,,\quad
S^{\mu}_{as+bs}|_{\mathrm{tr}}=-\frac{1}{2}(k\cdot\ell)\epsilon^{\mu}[v_{1},y_{1},\ell]\,.$
(323)

These match the expectation from the 2PL eikonal (315) perfectly.

##### Opposite helicity sector

We treat the (a)-(b) group and the (c)-(d) group separately. In the (a)-(b)
group, partial cancellations leave us with

$\displaystyle\begin{split}C^{\mu}_{ao}+C^{\mu}_{bo}&=i(k\cdot\ell)\gamma\left[\ell^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right]+\frac{i}{2}(k\cdot\ell)v_{1}^{\mu}(\ell\cdot y_{1})\\\
&\quad+i\ell^{\mu}(\ell\cdot v_{1})(k\cdot
y_{1})-\frac{i}{2}(k\cdot\ell)(\ell\cdot
v_{1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\left[2v_{2}^{\mu}(v_{2}\cdot
y_{1})+y_{1}^{\mu}\right]}\,,\\\
S^{\mu}_{ao}+S^{\mu}_{bo}&=\left[\ell^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(\ell\cdot
y_{1})\right]\epsilon[k,\ell,v_{1},v_{2}]+\frac{1}{2}(k\cdot\ell)\epsilon^{\mu}[\ell,v_{1},y_{1}]\\\
&\quad+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}v_{2}^{\mu}(\ell\cdot
v_{1})\epsilon[k,\ell,y_{1},v_{2}]}\,.\end{split}$ (324)

Again, the non-orthogonal terms, marked red in the equations above, are
projected out upon symmetrisation under the exchange $k\leftrightarrow\ell$.
We are left with

$\displaystyle\begin{split}C^{\mu}_{ao+bo}|_{\mathrm{tr}}&=i(k\cdot\ell)\gamma\left[q^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(q\cdot
y_{1})\right]+\frac{i}{2}(k\cdot\ell)v_{1}^{\mu}(q\cdot y_{1})\\\
&\quad+\frac{i}{2}[(k-\ell)\cdot v_{1}]\left[k^{\mu}(\ell\cdot
y_{1})-\ell^{\mu}(k\cdot y_{1})\right]\,,\\\
S^{\mu}_{ao+bo}|_{\mathrm{tr}}&=\left[q^{\mu}(v_{2}\cdot
y_{1})-v_{2}^{\mu}(q\cdot
y_{1})\right]\epsilon[k,\ell,v_{1},v_{2}]-\frac{1}{2}(k\cdot\ell)\epsilon^{\mu}[k-\ell,v_{1},y_{1}]\,.\end{split}$
(325)

On the other hand, the transverse spin kick derived from the eikonal by (315)
contains

$\displaystyle\begin{split}C^{\mu}_{\chi}|_{\mathrm{tr}}&=i(q\cdot
y_{1})v_{1}^{\mu}\left[-(\gamma^{2}-1/2)(k\cdot\ell)+(k\cdot v_{1})(\ell\cdot
v_{1})\right]\\\
&\quad+i\gamma\epsilon^{\mu}[k-\ell,v_{1},y_{1}]\epsilon[k,\ell,v_{1},v_{2}]\\\
&\quad-i[(k-\ell)\cdot
v_{1}]\epsilon^{\mu}[v_{1},v_{2},y_{1}]\epsilon[k,\ell,v_{1},v_{2}]\\\
&\quad-\frac{i}{2}[(k-\ell)\cdot
v_{1}]\epsilon^{\mu\nu}[v_{1},y_{1}]\epsilon_{\nu}[k,\ell,v_{1}]\,,\\\
S^{\mu}_{\chi}|_{\mathrm{tr}}&=-v_{1}^{\mu}(q\cdot
y_{1})\gamma\epsilon[k,\ell,v_{1},v_{2}]\\\
&\quad-(\gamma^{2}-1/2)(k\cdot\ell)\epsilon^{\mu}[k-\ell,v_{1},y_{1}]+\gamma[(k-\ell)\cdot
v_{1}](k\cdot\ell)\epsilon^{\mu}[v_{1},v_{2},y_{1}]\,.\end{split}$ (326)

Despite appearances, things do match as expected,

$\displaystyle
C^{\mu}_{ao+bo}|_{\mathrm{tr}}=C^{\mu}_{\chi}|_{\mathrm{tr}}\,.\quad
S^{\mu}_{ao+bo}|_{\mathrm{tr}}=S^{\mu}_{\chi}|_{\mathrm{tr}}\,.$ (327)

We can repeat the same exercise for the (c)-(d) group. The computations are
even lengthier and not particularly illuminating, so we omit the details here.

##  Appendix D Regularisation for the product of time-symmetric Green’s
functions

The usual time-symmetric $i0^{+}$ prescription for $\omega^{-2}$ propagators
is

$\displaystyle\frac{1}{2}\left(\frac{1}{(\omega+i0^{+})^{2}}+\frac{1}{(\omega-i0^{+})^{2}}\right)\quad\Leftrightarrow\quad-\frac{1}{2}|\sigma|\,,$
(328)

where we use the positive frequency expansion
$f(\sigma)=\int_{\omega}f(\omega)e^{-i\omega\sigma}$. The square of the time-
symmetric $\omega^{-1}$ propagator is given as

$\displaystyle\left[\frac{1}{2}\left(\frac{1}{\omega+i0^{+}}+\frac{1}{\omega-i0^{+}}\right)\right]^{2}\quad\Leftrightarrow\quad-\frac{1}{4}|\sigma|+\frac{e^{-0^{+}\times|\sigma|}}{40^{+}}=\frac{1}{40^{+}}-\frac{1}{2}|\sigma|+\mathcal{O}[(0^{+})^{1}]\,,$
(329)

where we expanded the expression as a Laurent series in the regulator $0^{+}$
and kept up to $\mathcal{O}[(0^{+})^{0}]$ terms. Employing the “minimal
subtraction” scheme for the regulator $0^{+}$, we throw out the divergent term
in $0^{+}$ and conclude that the propagators (328) and (329) are equivalent as
distributions.

The reason (329) has a divergent contribution compared to (328) is because it
should be understood as the convolution of the time-symmetric $\omega^{-1}$
propagator in the time domain,

$\displaystyle\frac{1}{2}\left(\frac{1}{\omega+i0^{+}}+\frac{1}{\omega-i0^{+}}\right)\quad\Leftrightarrow\quad-\frac{i}{2}\text{sgn}(\sigma)\,,$
(330)

where $\text{sgn}(\sigma)$ is the sign function. Unlike the retarded/advanced
Green’s functions given by the Heaviside step function, the convolution of
(330) with itself diverges due to the “infinite volume” of the real line,
which manifests itself as the $(0^{+})^{-1}$ divergence in (329).

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