aainstitutetext: Department of Physics, California Institute of Technology,
1200 E California Blvd, Pasadena, CA 91125, U.S.A.bbinstitutetext: Max Planck
Institute for Gravitational Physics (Albert Einstein Institute),
Am Mühlenberg 1, D-14476 Potsdam, Germanyccinstitutetext: Department of
Physics and Astronomy, Seoul National University,
1 Gwanak-ro, Gwanak-gu, Seoul 08826, Koreaddinstitutetext: Center for
Theoretical Physics, Seoul National University,
1 Gwanak-ro, Gwanak-gu, Seoul 08826, Koreaeeinstitutetext: College of Liberal
Studies, Seoul National University,
1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea

# Massive twistor worldline in electromagnetic fields

Joon-Hwi Kim b Jung-Wook Kim c,d,e Sangmin Lee<EMAIL_ADDRESS>jung-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the (ambi-)twistor model for spinning particles interacting via
electromagnetic field, as a toy model for studying classical dynamics of
gravitating bodies including effects of both spins to all orders. We compute
the momentum kick and spin kick up to one-loop order and show precisely how
they are encoded in the classical eikonal. The all-orders-in-spin effects are
encoded as a dynamical implementation of the Newman-Janis shift, and we find
that the expansion in both spins can be resummed to simple expressions in
special kinematic configurations, at least up to one-loop order. We confirm
that the classical eikonal can be understood as the generator of canonical
transformations that map the in-states of a scattering process to the out-
states. We also show that cut contributions for converting worldline
propagators from time-symmetric to retarded amount to the iterated action of
the leading eikonal at one-loop order.

##  1 Introduction

Chandrasekhar has remarked that “(t)he black holes of nature are the most
perfect macroscopic objects there are in the universe: […] they are the
simplest objects as well.” Chandrasekhar:1985kt  Can we idealise these
simplest objects of the universe and make them even simpler? Since
electromagnetic interactions are simpler than gravitational interactions, let
us phrase this question more concretely in the context of electromagnetism.
What would be the description of the simplest charged, massive, spinning
(macroscopic) objects moving on a background electromagnetic field?

One class of charged spinning (macroscopic) objects that can be called
“simplest” is known in the literature as root-Kerr particles Arkani-
Hamed:2019ymq , which possess spin-induced multipole moments of Kerr-Newman
black holes Chung:2019yfs ; Scheopner:2023rzp . They can be called simplest in
the sense that they correspond to the classical spin limit of “minimal
coupling” defined by the high-energy limit Arkani-Hamed:2017jhn , and that all
multipole moments are generated by the Newman-Janis shift Newman:1965tw ,
where the position of the particle sourcing the gravitational/electromagnetic
field is complexified and shifted in the imaginary spin direction. In its
original formulation, the Newman-Janis shift only applies to stationary
solutions of the Einstein(-Maxwell) equations, therefore the answer to the
question posed in the previous paragraph would only be complete when the
Newman-Janis shift is generalised to dynamical worldlines of spinning bodies.

In this work, improving upon the ideas of ref. Guevara:2020xjx , we argue that
the twistorial description of relativistic spherical tops Kim:2021rda
qualifies as a complete answer. The authors showed in ref. Kim:2021rda  that
the spherical top model Hanson:1974qy  for a relativistic spinning particle is
equivalent to a massive twistor model (similar but not identical to ref.
Fedoruk:2014vqa ) in the absence of interactions. The attempt to couple the
twistor model to a background field was initiated in ref. Kim:2023aff . A
complete description of the twistor model minimally coupled to electromagnetic
field is given here.

We use the model to compute scattering observables, such as the momentum kick
and the spin kick, at low orders in perturbation theory while maintaining
exact spin-dependence. Following the nomenclature of ref. Bern:2023ccb , we
call the perturbation theory “post-Lorentzian” (PL) expansion, where $n$-PL
order terms are suppressed by $(q_{1}q_{2})^{n}$ where $q_{1}$, $q_{2}$ are
the electric charges of two interacting particles. When organised
diagrammatically through Feynman-like diagrams, $n$-PL order terms involve
$(n-1)$-loop momentum integrals, although the diagrams themselves have no
loops. This is a toy model for studying the gravitational case, where the
dynamics is organised in the post-Minkowskian (PM) expansion while keeping the
exact spin-dependence.

We stress that exact spin-dependence is not only of theoretical interest, but
is also of phenomenological interest. The previous sentence may sound odd to a
person familiar with post-Newtonian (PN) calculations: In the PN expansion
spin effects are formally counted as 1PN, since the corrections take the form
of $a/r$, and for compact objects such as black holes the spin length scales
as the horizon scale $a\sim Gm$. For computing the gravitational
waveforms—which are the quantities directly relevant for observations—the
4.5PN corrections seem to be good enough, at least for non-spinning quasi-
circular equal mass binaries; the 4.5PN contributions add less-than-a-radian
correction to the $\sim 10^{3-5}$ cumulative gravitational wave cycles in the
detector frequency bands Blanchet:2023bwj . The spin effect corrections to the
conservative dynamics has already been computed to 5PN order Kim:2021rfj ;
Kim:2022pou ; Kim:2022bwv ; Levi:2022dqm ; Levi:2022rrq ; Mandal:2022nty ;
Mandal:2022ufb . Why would we need all-orders-in-spin effects if they are
going to be smaller than what is already known, which already seems to be
sufficient for observations?111Caution: The PN expansion is known to converge
best for equal-mass quasi-circular orbits, and convergence of the best-case
scenario does not guarantee convergence in other regions of the parameter
space. For example, even 5PN may not be enough to reduce systematic errors
below the level of statistical errors Owen:2023mid .

One reason all-orders-in-spin results can be of interest is because in
practical applications the perturbative results need to be resummed for a
better accuracy Buonanno:1998gg ; Antonelli:2019ytb ; Khalil:2022ylj ;
Damour:2022ybd ; Rettegno:2023ghr ; Buonanno:2024vkx .222One may also recall
that the revival of interest in the PM expansion was partly kindled by the
search for alternative resummation schemes of the gravitational two-body
dynamics Damour:2016gwp . The resummations reorganise the perturbative
expansion by leveraging the knowledge of singularity structures that the non-
perturbative answer is expected to possess. Therefore, all-orders-in-spin
calculations may reveal singularity structures we can take advantage of in the
resummations, which were not visible at the lowest spin orders. This can be
used, for example, in improving effective-one-body based waveform models which
are known to perform worse for extremal black hole spins Ramos-Buades:2023ehm
; Pompili:2023tna , where spin effects are resummed as geodesic motion on a
deformed Kerr geometry Khalil:2023kep . Understanding the mechanism behind the
effectiveness of the resummation will be useful in motivating alternative
resummation schemes for spin effects, which may yield better accuracy.

From the viewpoint of resummations, an all-orders-in-spin result that is as
rigid as possible and as simple as possible while keeping essential features
of the dynamics will be the most useful, since we are interested in the
singularity structures of all-orders-in-spin dynamics; any additional
structures or free parameters may obscure the singularity structures that we
wish to dig up from all-orders-in-spin results. This motivation brings us back
to the question raised in the beginning of this manuscript; what is the
simplest spinning object that interacts with the background Maxwell field? The
motivation also limits the inputs of the theory to all-orders-in-spin
multipole moment information encoded by the Newman-Janis shift. The
expectation is that while the dynamics of the spinning particle may deviate
from that of physical black holes from $\mathcal{O}(q^{2})$, the singularity
structures of the all-orders-in-spin dynamics are still captured by the
twistor worldline model.

The twistor worldline model predicts surprisingly simple singularity
structures in special kinematic configurations, which we expect to be shared
by scattering dynamics of physical black holes. For example, the 2PL _aligned-
spin eikonal_ (260) resums to the simple expression

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

where the impact parameter $b^{\mu}\sim(x_{1}^{\mu}-x_{2}^{\mu})_{\perp}$ is
defined by the covariant spin supplementary conditions (SSC),
$a^{\mu}=a_{1}^{\mu}+a_{2}^{\mu}$ is the sum of the spin-length vectors, and
$\zeta$ is the ratio parameter defined by $a_{1}^{\mu}=\zeta a^{\mu}$. To the
best of authors’ knowledge, this is the first observation of spin effect
resummation in binary dynamics at the next-to-leading order (NLO) in the
coupling constant expansion, where the model is free of unphysical behaviour
and the spins of both constituents are included to all orders.333The known
spin-resummed NLO scattering angles reported in the literature Guevara:2018wpp
; Menezes:2022tcs  are based on the Compton amplitudes that develop unphysical
behaviour from cubic (electromagnetism) or quintic (gravity) order in spin.
The spin-resummed results reported by ref. Brandhuber:2024bnz  should be
considered as leading order effects in the $R^{3}$ coupling expansion.

In addition to the construction of the interacting twistor model, and
application of the model to compute observables to the 2PL order, another key
result of this work is the clarification of the _classical eikonal_ ’s role.
In the Hamiltonian formulation of binary dynamics, the classical eikonal is
defined as a suitable classical limit of the quantum eikonal phase that acts
as the generator of canonical transformations, mapping the incoming scattering
states to the outgoing states. The scattering states are defined in the phase
space of free particles, and the eikonal encodes the interactions such that it
produces all scattering observables through canonical transformations.

The manuscript is organised as follows. In section 2 we review the massive
twistor model and couple it to background electromagnetic fields. In section 3
we compute scattering observables using equations of motion. We set up WQFT
formulation of the model in section 4, and use it to compute Compton
amplitudes in section 5 and the classical eikonal in section 6. We conclude
our studies and propose future directions in section 7.

##### Note added

While this work was being completed, ref. Gonzo:2024zxo  appeared. Their
proposal for how to extract classical observables from the radial action
overlaps with our discussion on the classical eikonal as the scattering
generator in section 3.1.

##  2 Massive twistor in electromagnetic field

###  2.1 Free theory

In ref. Kim:2021rda , we proposed a massive twistor model and showed its
equivalence to the Hanson-Regge spherical top model at the free theory level.
A general discussion of how to couple the twistor model to background fields
was given in ref. Kim:2023aff . Here, we give a brief review of these two main
references and clarify some aspects of the twistor model before specialising
to the minimal coupling to a background electromagnetic field.

$\begin{array}[]{ccccc}\mbox{\bf Spherical Top}&&&&\mbox{\bf Massive
Twistor}\\\ \\\
\tilde{x}^{\mu},p^{\mu},\widetilde{S}^{\mu\nu},\widetilde{\Lambda}^{\mu}{}_{A=0,1,2,3}&&&&\lambda_{\alpha}{}^{I},\mu^{\dot{\alpha}I},\bar{\lambda}_{I\dot{\alpha}},\bar{\mu}_{I}{}^{\alpha}\\\
\\\ \scriptstyle{\mathbb{R}^{3}}\;\mbox{\small
redundancy}\;\;\downarrow&&&&\qquad\downarrow\;\;\scriptstyle{\mathrm{U(1)}}\;\mbox{\small
redundancy}\\\ \\\
p^{\mu},\Lambda^{\mu}{}_{a=1,2,3}&&\xrightarrow[]{\mathrm{\;unification\;}}&&\lambda_{\alpha}{}^{I}\bar{\lambda}_{J\dot{\alpha}}\\\
x^{\mu},S^{\mu\nu}&&\xrightarrow[]{\mathrm{\;unification\;}}&&\mu^{\dot{\alpha}I}(\lambda^{-1})_{I}{}^{\alpha}\\\
{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\scriptstyle{(p\cdot
S\cdot\Lambda)_{a}=0=(p\cdot\Lambda)_{a}}}&&&&\end{array}$
Figure 1: Equivalence between the spherical top model and the massive twistor
model.

A widely used model for describing relativistic spinning particles is the
Hanson-Regge spherical top Hanson:1974qy , which uses the variables
$(p_{\mu},\tilde{x}^{\mu},\tilde{\Lambda}^{\mu}{}_{A},\tilde{S}_{\mu\nu})$ to
describe a spinning body; the momentum $p_{\mu}$, the position
$\tilde{x}^{\mu}$, the body-fixed tetrad $\tilde{\Lambda}^{\mu}{}_{A}$
($A=0,1,2,3$) describing the orientation of the body, and the spin tensor
$\tilde{S}_{\mu\nu}$. The tilde notation emphasises the fact that under the
“spin-gauge” redundancy Steinhoff:2015ksa , the variables
$(\tilde{x},\tilde{\Lambda},\tilde{S})$ have some gauge dependency while $p$
is gauge-invariant. It was shown in ref. Kim:2021rda  that the gauge orbit is
$\mathbb{R}^{3}$ and the “origin” of the orbit corresponds to the covariant
gauge conditions:
$p^{\mu}S_{\mu\nu}\Lambda^{\nu}{}_{a}=0=p_{\mu}\Lambda^{\mu}{}_{a}$
($a=1,2,3$).

The twistor variables describe the same dynamics with less gauge redundancy.
Their global symmetry groups are the superconformal SU$(2,2)$ (to be broken by
the mass-shell condition) and the massive little-group SU$(2)$ . The gauge
redundancy on the twistor side is U(1): $(\lambda,\mu)\rightarrow
e^{i\theta}(\lambda,\mu)$, $(\bar{\lambda},\bar{\mu})\rightarrow
e^{-i\theta}(\bar{\lambda},\bar{\mu})$, $\theta\in\mathbb{R}$.

The equivalence of the two models is clearly articulated in terms of the
gauge-invariant coordinates on both sides. The twistor model unifies the
variables $(p_{\mu},\Lambda^{\mu}{}_{a})$ using the hermitian bi-linear
products of $(\lambda_{\alpha}{}^{I},\bar{\lambda}_{I\dot{\alpha}})$:

$\displaystyle
p^{\mu}=\frac{1}{2}\bar{\sigma}^{\mu\dot{\alpha}\alpha}\lambda_{\alpha}{}^{I}\bar{\lambda}_{I\dot{\alpha}}\,,\quad\Lambda_{(IJ)}^{\mu}=\frac{\bar{\sigma}^{\mu\dot{\alpha}\alpha}\lambda_{\alpha(I}\bar{\lambda}_{J)\dot{\alpha}}}{\sqrt{2}m}\,,$
(2)

in a similar vein to massive spinor-helicity variables Conde:2016izb ; Arkani-
Hamed:2017jhn . An advantage of the Hamiltonian formulation is that nothing
stops us from unifying momenta $p_{\mu}$ and “generalised positions”
$\Lambda^{\mu}{}_{a}$. The remaining variables $(x^{\mu},S_{\mu\nu})$ are
unified and mapped to a set of gauge-invariant complex variables
$\mu^{\dot{\alpha}I}(\lambda^{-1})_{I}{}^{\alpha}$ and their complex
conjugates. This is in contrast to the supersymmetric worldline description
Gibbons:1993ap ; Jakobsen:2021lvp ; Jakobsen:2021zvh  where the spin tensor
$S_{\mu\nu}$ is realised as a bi-linear in Grassmann variables
$(\Psi_{A},\bar{\Psi}_{A})$ and the tetrad $\Lambda^{\mu}{}_{a}$ is not
visible.

A prominent feature of the twistor model is that there is no spin-gauge
redundancy in the first place, so that no discussion of the spin supplementary
condition (SSC) is ever needed. The only constraint yet to be imposed is the
mass-shell condition $p^{2}+m^{2}=0$, which is common to the spherical top
model and the twistor model. Figure 1 summarises the mapping between the two
models before imposing the mass-shell constraint.

####  2.1.1 Constraints revisited

Our conventions for spinors and twistors are slightly different from those of
ref. Kim:2021rda ; see appendix A for details. Let us mention a few key
relations. The fundamental Poisson brackets of the spherical top model include

$\displaystyle\\{x^{\mu},p_{\nu}\\}=\delta^{\mu}_{\nu}\;\;\Rightarrow\;\;\\{x^{\dot{\alpha}\alpha},p_{\beta\dot{\beta}}\\}=-2\delta^{\dot{\alpha}}_{\dot{\beta}}\delta^{\alpha}_{\beta}\,.$
(3)

The fundamental Poisson brackets of the twistor model are

$\displaystyle\\{\bar{\mu}_{I}{}^{\alpha},\lambda_{\beta}{}^{J}\\}=\delta^{\alpha}_{\beta}\delta_{I}^{J}\,,\quad\\{\mu^{\dot{\alpha}I},\bar{\lambda}_{J\dot{\beta}}\\}=\delta_{\dot{\beta}}^{\dot{\alpha}}\delta^{I}_{J}\,.$
(4)

The incidence relations relating the two models read

$\displaystyle\mu^{\dot{\alpha}I}=\frac{1}{2}z^{\dot{\alpha}\beta}\lambda_{\beta}{}^{I}\,,\quad\bar{\mu}_{I}{}^{\alpha}=\frac{1}{2}\bar{\lambda}_{I\dot{\beta}}\bar{z}^{\dot{\beta}\alpha}\,,$
(5)

where the complexified position variable $z$ defined by the incidence relation
is mapped to the spherical top variables through

$\displaystyle z^{\mu}=x^{\mu}+iy^{\mu}\,,\quad
y^{\mu}=\frac{1}{m^{2}}\varepsilon^{\mu\nu\rho\sigma}p_{\nu}S_{\rho\sigma}\,.$
(6)

The imaginary part $y^{\mu}$ of the complexified position variable $z^{\mu}$
is related to the spin-length vector $a^{\mu}=s^{\mu}/m$ widely used in the
literature by $y^{\mu}=-a^{\mu}$. The extra sign is to respect the standard
relation of non-relativistic spins: $\\{s^{i},s^{j}\\}=\epsilon^{ijk}s^{k}$.

In our current conventions, the free action of ref. Kim:2021rda  reads

$\displaystyle\begin{split}&S_{\text{free}}=\int\left[\lambda_{\alpha}{}^{I}d\bar{\mu}_{I}{}^{\alpha}+\bar{\lambda}_{I\dot{\alpha}}d\mu^{\dot{\alpha}I}+\frac{1}{2}\left(\bar{\kappa}(\Delta-m)+\kappa(\bar{\Delta}-m)\right)d\sigma\right]\,,\\\
&\Delta={\rm
det}(\lambda)=-\frac{1}{2}\epsilon^{\alpha\beta}\epsilon_{IJ}\lambda_{\alpha}{}^{I}\lambda_{\beta}{}^{J}\,,\quad\bar{\Delta}={\rm
det}(\bar{\lambda})=\frac{1}{2}\epsilon^{IJ}\epsilon^{\dot{\alpha}\dot{\beta}}\bar{\lambda}_{I\dot{\alpha}}\bar{\lambda}_{J\dot{\beta}}\,.\end{split}$
(7)

The Lagrange multipliers $(\kappa,\bar{\kappa})$ enforce the conditions,

$\displaystyle\Delta=m=\bar{\Delta}\,,$ (8)

which in turn imply the mass-shell condition
$-p^{2}=\Delta\bar{\Delta}=m^{2}$. After a suitable gauge-fixing of
$(\kappa,\bar{\kappa})$, the “Hamiltonian”
$\mathrm{Re}(\kappa)(\Delta+\bar{\Delta})/2$ generates the worldline time-
evolution for all dynamical variables which matches the expectation from the
spherical top model.

Despite its success, it turns out that the free action in (7) is not suitable
for the transition to the interacting theory, and we propose an alternative
action:

$\displaystyle
S_{\mathrm{free}}=\int\left[\lambda_{\alpha}{}^{I}\dot{\bar{\mu}}_{I}{}^{\alpha}+\bar{\lambda}_{I\dot{\alpha}}\dot{\mu}^{\dot{\alpha}I}-\kappa^{0}\phi_{0}-\kappa^{1}\phi_{1}\right]d\sigma\,,$
(9)

where

$\displaystyle\phi_{0}=\frac{1}{2}(m^{2}-\Delta\bar{\Delta})=\frac{1}{2}(p^{2}+m^{2})\,,\quad\phi_{1}=\frac{1}{2i}(\bar{\lambda}_{I{\dot{\alpha}}}\mu^{{\dot{\alpha}}I}-\bar{\mu}_{I}{}^{\alpha}\lambda_{\alpha}{}^{I})=p\cdot
y\,.$ (10)

For later purposes, we also propose the gauge-fixing conditions,

$\displaystyle\chi^{0}=-\frac{1}{2}(\bar{\lambda}_{I{\dot{\alpha}}}\mu^{{\dot{\alpha}}I}+\bar{\mu}_{I}{}^{\alpha}\lambda_{\alpha}{}^{I})\,,\quad\chi^{1}=\frac{i}{2}\log(\Delta/\bar{\Delta})\,.$
(11)

We recognise $\phi_{0}$ as the mass-shell constraint in twistor variables, and
$\phi_{1}$ as the generator of the U(1) gauge orbit. Aside from how to
implement the mass-shell constraint, the main difference between the two
proposals is that (9) contains the U(1) gauge generator while (7) contains a
U(1) gauge-fixing condition in the form

$\displaystyle\frac{\mathrm{Im}(\kappa)}{2}(\Delta-\bar{\Delta})\,.$ (12)

To understand why the old proposal (7) is problematic and why it still yielded
the free equations of motion correctly, let us revisit the general theory of
constrained Hamiltonian dynamics, building upon appendix A of ref. Kim:2021rda
. The system consists of Hamiltonian $H$, symplectic form $\omega$, abelian
gauge generators $\phi_{A}$ and gauge fixing functions $\chi^{A}$. The minimal
requirements are

$\displaystyle\left\\{\phi_{A},H\right\\}=0,\quad\left\\{\phi_{A},\phi_{B}\right\\}=0\quad
C^{A}{}_{B}:=\left\\{\chi^{A},\phi_{B}\right\\},\quad\operatorname{det}(C)\neq
0\,.$ (13)

In general, the gauge-fixed action is written as

$\displaystyle S=\int
dt\left(p_{i}\dot{q}^{i}-H-\kappa^{A}\phi_{A}-\bar{\kappa}_{A}\chi^{A}\right)\,.$
(14)

The Lagrange multipliers $(\kappa^{A},\bar{\kappa}_{A})$ enforce the
constraints $\phi_{A}=0=\chi^{A}$. The variation of $S$ with respect to the
dynamical variables $(p_{i},q^{i})$ gives the equations of motion. The time
evolution of a generic function $f(p_{i},q^{i})$ is computed from the Poisson
bracket

$\displaystyle\frac{df}{dt}=\frac{\partial f}{\partial
q^{i}}\dot{q}^{i}+\frac{\partial f}{\partial
p_{i}}\dot{p}_{i}=\\{f,H+\kappa^{A}\phi_{A}+\bar{\kappa}_{A}\chi^{A}\\}\,.$
(15)

For consistency, the equations of motion should not induce change of
$\phi_{A}$ and $\chi^{A}$ in time:

$\displaystyle\frac{d\phi_{A}}{dt}=0=\frac{d\chi^{A}}{dt}\,.$ (16)

The vanishing of $d\phi_{A}/dt$ and the minimal requirements (13) imply
$\bar{\kappa}_{A}=0$. Hence, we do not see directly $\bar{\kappa}_{A}$ in the
final form of the equations of motion. The vanishing of $d\chi^{A}/dt$ implies
$\kappa^{A}=-(C^{-1})^{A}{}_{B}\\{\chi^{B},H\\}$. We can either choose some
$\chi^{A}$ to fix $\kappa^{A}$, or prescribe some $\kappa^{A}$ to fix
$\chi^{A}$ implicitly.

If some of the $\kappa^{A}$ multipliers can be set to zero without violating
the requirements (13), the corresponding $\phi_{A}$ will not directly
contribute to the equations of motion and it may look permissible to exchange
the roles of $\phi_{A}$ and $\chi^{A}$. That is precisely what happened to the
free twistor model. But, as soon as we add interaction terms in the action,
the distinction between the gauge generators $\phi_{A}$ and the gauge fixing
conditions $\chi^{A}$ becomes evident. By the very definition of gauge
redundancy, the interaction terms are required to Poisson-commute with all
$\phi_{A}$ in a sense to be specified below. On the contrary, there is no
reason for the interaction terms to commute with $\chi^{A}$.

To conclude, in view of the general theory of constrained Hamiltonian system
where the gauge generators and gauge-fixing conditions play different roles,
we need a new proposal for the free action (9) to incorporate interaction
terms.

####  2.1.2 Regge trajectory

Due to rotational kinetic energy, the mass of a spinning top is in general not
a constant, but rather a function of the spin-magnitude $W^{2}=(y\cdot
p)^{2}-y^{2}p^{2}$. The derivative $m^{\prime}=dm/d(W^{2})$ is colloquially
called the “Regge trajectory”. For a free spinning particle, the angular
velocity is $(2m^{\prime})$ times the spin, so $(2m^{\prime})^{-1}$ is the
relativistic rotational inertia Kim:2021rda .

In most applications, where we do not keep track of the angular velocity and
focus on the $(x,y,p)$ variables, the Regge trajectory does not affect the
dynamics. Specifically, the equations of motion for $(x,y,p)$ are independent
of $m^{\prime}$. In the rest of this section, we will see, from a few
different but related angles, how the Regge trajectory $m^{\prime}$ decouples
from the dynamics of $(x,y,p)$. From the next subsection on, we will set
$m^{\prime}=0$ to simplify computations.

####  2.1.3 Dirac bracket and effective phase space

To describe the physical phase space of the free twistor theory, it is
convenient to construct the Dirac bracket. The ingredients are the
unconstrained Poisson bracket (4), the gauge generators (10), and the gauge-
fixing functions (11).

Let us first consider the case of vanishing Regge trajectory ($m^{\prime}=0$).
Using

$\displaystyle
C^{A}{}_{B}=\\{\chi^{A},\phi_{B}\\}=\begin{pmatrix}\Delta\bar{\Delta}&0\\\
0&1\end{pmatrix}\,,$ (17)

we construct the Dirac bracket in the standard way,

$\displaystyle\\{f,g\\}_{*}=\\{f,g\\}-(C^{-1})^{A}{}_{B}\left(\\{f,\phi_{A}\\}\\{\chi^{B},g\\}-\\{g,\phi_{A}\\}\\{\chi^{B},f\\}\right)\,.$
(18)

The non-vanishing brackets among the twistor variables are

$\displaystyle\begin{split}\\{\bar{\mu}_{I}{}^{\alpha},\lambda_{\beta}{}^{J}\\}_{*}&=\delta^{\alpha}_{\beta}\delta_{I}^{J}-\frac{1}{2}\lambda_{\beta}{}^{J}(\lambda^{-1})_{I}{}^{\alpha}\,,\\\
\\{\mu^{\dot{\alpha}I},\bar{\lambda}_{J\dot{\beta}}\\}_{*}&=\delta_{\dot{\beta}}^{\dot{\alpha}}\delta^{I}_{J}-\frac{1}{2}\bar{\lambda}_{J{\dot{\beta}}}(\bar{\lambda}^{-1})^{{\dot{\alpha}}I}\,,\\\
\\{\bar{\mu}_{I}{}^{\alpha},\bar{\mu}_{J}{}^{\beta}\\}_{*}&=-\frac{1}{2}\left[\bar{\mu}_{I}{}^{\alpha}(\lambda^{-1})_{J}{}^{\beta}-\bar{\mu}_{J}{}^{\beta}(\lambda^{-1})_{I}{}^{\alpha}\right]\,,\\\
\\{\mu^{\dot{\alpha}I},\mu^{\dot{\beta}J}\\}_{*}&=-\frac{1}{2}\left[\mu^{\dot{\alpha}I}(\bar{\lambda}^{-1})^{\dot{\beta}J}-\mu^{\dot{\beta}J}(\bar{\lambda}^{-1})^{\dot{\alpha}I}\right]\,.\end{split}$
(19)

We can certainly use these to compute the Dirac brackets among the
U(1)-invariant composite variables $(x^{\mu},y^{\mu},p^{\mu})$. But, since we
are interested only in the dynamics of $(x^{\mu},y^{\mu},p^{\mu})$ variables
in the bulk of this paper, we can take a shortcut.

We can regard the 16-dimensional unconstrained phase space as a fibre bundle
with the base coordinates $(x,y,p)$ and the fibre SU(2)$\times$U(1). The
original Poisson bracket (4) projected onto the $(x,y,p)$ base space can be
written as

$\displaystyle\begin{split}\\{\;\;,\;\;\\}_{\circ}&=\eta^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\wedge\frac{\partial}{\partial
p^{\nu}}-\frac{y^{\mu}p^{\nu}+p^{\mu}y^{\nu}-\eta^{\mu\nu}(y\cdot
p)}{p^{2}}\frac{\partial}{\partial x^{\mu}}\wedge\frac{\partial}{\partial
y^{\nu}}\\\
&\quad-\frac{\epsilon^{\mu\nu}[y,p]}{2p^{2}}\left(\frac{\partial}{\partial
y^{\mu}}\wedge\frac{\partial}{\partial y^{\nu}}+\frac{\partial}{\partial
x^{\mu}}\wedge\frac{\partial}{\partial x^{\nu}}\right)\,.\end{split}$ (20)

The transition from the Poisson bracket to the Dirac bracket can be done
within this effective description. Since $(x,y,p)$ are all U(1) gauge
invariant, we can simply disregard the pair $(\phi_{1},\chi^{1})$ and take
into consideration

$\displaystyle\phi_{0}=\frac{1}{2}(p^{2}+m^{2})\,,\quad\chi^{0}=-x\cdot p\,.$
(21)

The resulting Dirac bracket on the $(x,y,p)$ base space is

$\displaystyle\begin{split}\\{\;\;,\;\;\\}_{*}&=\left(\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\frac{\partial}{\partial
x^{\mu}}\wedge\frac{\partial}{\partial
p^{\nu}}-\frac{y^{\mu}p^{\nu}-\eta^{\mu\nu}(y\cdot
p)}{p^{2}}\frac{\partial}{\partial x^{\mu}}\wedge\frac{\partial}{\partial
y^{\nu}}\\\
&\quad-\frac{\epsilon^{\mu\nu}[y,p]}{2p^{2}}\left(\frac{\partial}{\partial
y^{\mu}}\wedge\frac{\partial}{\partial y^{\nu}}+\frac{\partial}{\partial
x^{\mu}}\wedge\frac{\partial}{\partial
x^{\nu}}\right)-\frac{x^{[\mu}p^{\nu]}}{p^{2}}\frac{\partial}{\partial
x^{\mu}}\wedge\frac{\partial}{\partial x^{\nu}}\,.\end{split}$ (22)

It agrees perfectly with the full Dirac bracket (19) evaluated over $(x,y,p)$.

Let us summarise the content of the effective description. The phase space is
effectively nine dimensional: twelve coordinates $(x^{\mu},y^{\mu},p^{\mu})$
with three constraints,

$\displaystyle p^{2}+m^{2}=0\,,\quad p\cdot y=0\,,\quad p\cdot x=0\,.$ (23)

By construction, the Dirac brackets of the constraints automatically vanish:

$\displaystyle\\{f,p^{2}+m^{2}\\}_{*}=0\,,\quad\\{f,p\cdot
y\\}_{*}=0\,,\quad\\{f,p\cdot x\\}_{*}=0\,.$ (24)

Incidentally, the Dirac bracket is consistent with fixing the magnitude of
spin:

$\displaystyle\\{f,y^{2}\\}_{*}=0\,.$ (25)

So, the phase space is eight dimensional in a sense, although the conservation
of $y^{2}$ is of a dynamical origin and shouldn’t be put on an equal footing
as the true constraints.

In later sections, we will study scattering processes of the twistor
particles, where the interaction is (asymptotically) turned off at the
infinite past and future. It makes sense to use the (effective) free phase
space for each particle to describe the asymptotic scattering states.

For completeness, we also compute the Dirac bracket when the Regge trajectory
is non-trivial ($m^{\prime}\neq 0$). Remarkably, the $C$-matrix (17) is not
altered by $m^{\prime}$ at all. It remains to compute the
$(m^{\prime})$-dependent terms from $\\{f,\phi_{0}\\}\\{\chi^{0},g\\}$ in
(18).

In terms of the twistor variables, the function $\chi^{0}$ is given by

$\displaystyle\chi^{0}=-x\cdot
p=\frac{1}{2}\left(\mu^{{\dot{\alpha}}I}\bar{\lambda}_{I{\dot{\alpha}}}+\bar{\mu}_{I}{}^{\alpha}\lambda_{\alpha}{}^{I}\right)\,.$
(26)

It follows that

$\displaystyle\begin{aligned}
\\{\chi^{0},\lambda_{\alpha}{}^{I}\\}&=+\frac{1}{2}\lambda_{\alpha}{}^{I}\,,&\quad\\{\chi^{0},\mu^{{\dot{\alpha}}I}\\}&=-\frac{1}{2}\mu^{{\dot{\alpha}}I}\,,\\\
\\{\chi^{0},\bar{\lambda}_{I{\dot{\alpha}}}\\}&=+\frac{1}{2}\bar{\lambda}_{I{\dot{\alpha}}}&\quad\\{\chi^{0},\bar{\mu}_{I}{}^{\alpha}\\}&=-\frac{1}{2}\bar{\mu}_{I}{}^{\alpha}\,.\end{aligned}$
(27)

The $W$ tensor is defined as Kim:2021rda

$\displaystyle
W_{IJ}=-i\left(\mu^{\dot{\alpha}}{}_{(I}\bar{\lambda}_{J)\dot{\alpha}}-\lambda_{\alpha(I}\bar{\mu}_{J)}{}^{\alpha}\right)\,.$
(28)

It is normalized such that

$\displaystyle W^{2}=\frac{1}{2}W_{K}{}^{L}W_{L}{}^{K}=(y\cdot
p)^{2}-y^{2}p^{2}\,.$ (29)

The $(m^{\prime})$-dependent terms from $\\{f,\phi_{0}\\}$ come from

$\displaystyle\\{f,m(W^{2})^{2}/2\\}=mm^{\prime}\\{f,W^{2}\\}\,.$ (30)

It is useful to note that

$\displaystyle\begin{aligned}
\\{\lambda_{\alpha}{}^{I},W^{2}\\}&=-i\lambda_{\alpha}{}^{K}W_{K}{}^{I}\,,&\quad\\{\mu^{{\dot{\alpha}}I},W^{2}\\}&=-i\mu^{{\dot{\alpha}}K}W_{K}{}^{I}\,,\\\
\\{\bar{\lambda}_{I{\dot{\alpha}}},W^{2}\\}&=+iW_{I}{}^{K}\bar{\lambda}_{K{\dot{\alpha}}}\,,&\quad\\{\bar{\mu}_{I}{}^{\alpha},W^{2}\\}&=+iW_{I}{}^{K}\bar{\mu}_{K}{}^{\alpha}\,.\end{aligned}$
(31)

The full Dirac bracket consists of two parts,

$\displaystyle\\{f,g\\}_{*}=\\{f,g\\}^{0}_{*}+\\{f,g\\}^{\prime}_{*}\,,$ (32)

where $\\{X,Y\\}^{0}_{*}$ is the result for $m^{\prime}=0$ stated in (19) and
$\\{X,Y\\}^{\prime}_{*}$ denotes the $(m^{\prime})$ terms. Using (27) and
(31), it is straightforward to show that the $(m^{\prime})$ terms are

$\displaystyle\begin{aligned}
\\{\lambda_{\alpha}{}^{I},\lambda_{\beta}{}^{J}\\}^{\prime}_{*}&=\frac{im^{\prime}}{2m}\left[+(\lambda_{\alpha}{}^{K}W_{K}{}^{I})\lambda_{\beta}{}^{J}-\lambda_{\alpha}{}^{I}\left(\lambda_{\beta}{}^{K}W_{K}{}^{J}\right)\right]\,,\\\
\\{\lambda_{\alpha}{}^{I},\bar{\lambda}_{J\dot{\beta}}\\}^{\prime}_{*}&=\frac{im^{\prime}}{2m}\left[+(\lambda_{\alpha}{}^{K}W_{K}{}^{I})\bar{\lambda}_{J\dot{\beta}}+\lambda_{\alpha}{}^{I}(W_{J}{}^{K}\bar{\lambda}_{K\dot{\beta}})\right]\,,\\\
\\{\lambda_{\alpha}{}^{I},\mu^{\dot{\beta}J}\\}^{\prime}_{*}&=\frac{im^{\prime}}{2m}\left[-(\lambda_{\alpha}{}^{K}W_{K}{}^{I})\mu^{\dot{\beta}J}-\lambda_{\alpha}{}^{I}(\mu^{\dot{\beta}K}W_{K}{}^{J})\right]\,,\\\
\\{\lambda_{\alpha}{}^{I},\bar{\mu}_{J}{}^{\beta}\\}^{\prime}_{*}&=\frac{im^{\prime}}{2m}\left[-(\lambda_{\alpha}{}^{K}W_{K}{}^{I})\bar{\mu}_{J}{}^{\beta}+\lambda_{\alpha}{}^{I}(W_{J}{}^{K}\bar{\mu}_{K}{}^{\beta})\right]\,,\\\
\\{\mu^{\dot{\alpha}I},\mu^{\dot{\beta}J}\\}^{\prime}_{*}&=\frac{im^{\prime}}{2m}\left[-(\mu^{\dot{\alpha}K}W_{K}{}^{I})\mu^{\dot{\beta}J}+\mu^{\dot{\alpha}I}(\mu^{\dot{\beta}K}W_{K}{}^{I})\right]\,,\\\
\\{\mu^{\dot{\alpha}I},\bar{\mu}_{J}{}^{\beta}\\}^{\prime}_{*}&=\frac{im^{\prime}}{2m}\left[-(\mu^{\dot{\alpha}K}W_{K}{}^{I})\bar{\mu}_{J}{}^{\beta}-\mu^{\dot{\alpha}I}(W_{J}{}^{K}\bar{\mu}_{K}{}^{\beta})\right]\,,\end{aligned}$
(33)

and their complex conjugates. They agree perfectly with Dirac brackets
computed in ref. Kim:2021rda  despite the differences in the choice of
constraints.

Comparing (33) with (19), it may appear that the non-vanishing Regge
trajectory complicates the Dirac bracket significantly. Fortunately, when we
focus on the effective $(x,y,p)$ phase space, the complication disappears
completely and the Dirac bracket (22) remains valid. In essence, the reason is
that the $(m^{\prime})$-terms in (33) all involve infinitesimal shifts along
the SU(2) little-group, but the $(x,y,p)$ variables are little-group scalars.

###  2.2 Interacting theory

In this subsection, we explain how to couple the twistor particle to a
background electromagnetic field minimally using the Newman-Janis shift.

####  2.2.1 Symplectic perturbation theory

Given a Hamiltonian system defined by (13) and (14), we may consider two types
of deformations. A familiar way is to deform the Hamiltonian,

$\displaystyle H=H^{\circ}+q\,H^{\prime}\,,$ (34)

where $q$ is a continuous perturbation parameter. We demand that the gauge
generators are independent of the deformation. The requirement (13) implies
that

$\displaystyle\\{\phi_{A},H^{\circ}\\}=0=\\{\phi_{A},H^{\prime}\\}\,.$ (35)

Then all the statements around (16) remain valid for any value of $q$.

An alternative way to deform the theory, which we adopt for our twistor model,
is to keep the Hamiltonian fixed and deform the symplectic form Kim:2023aff :

$\displaystyle\omega=\omega^{\circ}+q\,\omega^{\prime}\,.$ (36)

This is equivalent to keeping the Hamiltonian fixed and deforming the Poisson
brackets used in equations of motion (EOM). For a non-spinning particle, a key
feature of the symplectic perturbation viewpoint is that
$\omega^{\prime}=F=dA$ is manifestly invariant under the gauge transformation
of the Maxwell field and that we do not need to distinguish the two notions of
momenta (canonical vs kinetic). This feature will generalise straightforwardly
to our twistor model.

Again, we demand that the gauge generators are independent of $q$. We should
make sure that the requirements $\\{\phi_{A},H\\}=0=\\{\phi_{A},\phi_{B}\\}$
hold with the deformed symplectic form. Perturbatively,

$\displaystyle\\{f,g\\}=\\{f,g\\}_{\circ}-q\\{f,\zeta^{m}\\}_{\circ}\omega^{\prime}_{mn}\\{\zeta^{n},g\\}_{\circ}+\mathcal{O}(q^{2})\,,$
(37)

where $\zeta^{m}$ is an arbitrary coordinate system on the phase space and
$\\{\bullet,\bullet\\}_{\circ}$ is the Poisson bracket defined by
$\omega^{\circ}$.

For a random choice of $\omega^{\prime}$, the perturbation term has no reason
to vanish:

$\displaystyle-q\\{\phi_{A},\zeta^{m}\\}_{\circ}\omega^{\prime}_{mn}\\{\zeta^{n},\phi_{B}\\}_{\circ}\neq
0\,.$ (38)

But, if we change coordinates (at least locally),
$\zeta^{m}\rightarrow(z^{i},w^{a})$ where $z^{i}$ are gauge-invariant while
$w^{a}$ are gauge-dependent, and take
$\omega^{\prime}=\frac{1}{2}\omega^{\prime}_{ij}dz^{i}\wedge dz^{j}$ with no
$dw^{a}$ components, it is immediately clear that
$\\{\phi_{A},H\\}=0=\\{\phi_{A},\phi_{B}\\}$ hold _exactly_.

Coming back to our twistor model, recall that the new free action (9) carries
two gauge generators $\phi_{0}$, $\phi_{1}$ in (10). Since the Poisson bracket
is antisymmetric, the only issue is whether $\\{\phi_{0},\phi_{1}\\}=0$
continues to hold in the interacting theory. But, as long as $\omega^{\prime}$
consists only of U$(1)$-gauge invariant coordinates such as $p_{\mu}$ and
$z^{\mu}=x^{\mu}+iy^{\mu}$, $\\{\phi_{0},\phi_{1}\\}=0$ would follow
trivially.

A common feature of a relativistic particle theory, shared by the spherical
top model, is that the mass-shell constraint $\phi_{0}=(p^{2}+m^{2})/2$ also
generates the worldline time-evolution. Ignoring other gauge generators
temporarily, the equation of motion of the free theory is

$\displaystyle\frac{df}{d\sigma}=\kappa^{0}\\{f,\phi_{0}\\}_{\circ}\,.$ (39)

There are two equivalent ways to express the equation of motion of the
interacting theory. One is based on the infinite series Kim:2023aff ,

$\displaystyle\begin{split}\omega^{-1}&=(\omega^{\circ})^{-1}-q(\omega^{\circ})^{-1}\omega^{\prime}(\omega^{\circ})^{-1}+q^{2}(\omega^{\circ})^{-1}\omega^{\prime}(\omega^{\circ})^{-1}\omega^{\prime}(\omega^{\circ})^{-1}+\cdots\\\
\Longrightarrow\quad\frac{1}{\kappa^{0}}\frac{df}{d\sigma}&=\\{f,\phi_{0}\\}_{\circ}-q\\{f,z^{i}\\}_{\circ}\omega^{\prime}_{ij}\\{z^{j},\phi_{0}\\}_{\circ}\\\
&\hskip
56.9055pt+q^{2}\\{f,z^{i}\\}_{\circ}\omega^{\prime}_{ij}\\{z^{j},z^{k}\\}_{\circ}\omega^{\prime}_{kl}\\{z^{l},\phi_{0}\\}_{\circ}+\cdots\,.\end{split}$
(40)

The other is more compact,

$\displaystyle\begin{split}\omega^{-1}&=(\omega^{\circ})^{-1}-q(\omega^{\circ})^{-1}\omega^{\prime}\omega^{-1}\\\
\Longrightarrow\quad\frac{1}{\kappa^{0}}\frac{df}{d\sigma}&=\\{f,\phi_{0}\\}_{\circ}-q\\{f,z^{i}\\}_{\circ}\omega^{\prime}_{ij}\\{z^{j},\phi_{0}\\}\,,\end{split}$
(41)

but less explicit in that the last term involves the Poisson bracket of the
interacting theory. Of course, (41) can be iterated to reproduce (40).

####  2.2.2 Minimal coupling via Newman-Janis shift

The unification of $(x^{\mu},S_{\mu\nu})$ into the complex coordinates
$(z,\bar{z})$ defined by (5) and (6) opens up an avenue for implementing the
Newman-Janis shift Newman:1965tw  into the dynamics. It led one of the authors
to the concept of “spinspacetime” Kim:2023vgb  which unifies spacetime and
spin at the fundamental level, which has refined Newman’s early idea
Newman:1974fr . Our twistor model is a new member of existing attempts to
incorporate the Newman-Janis shift into the dynamics Guevara:2020xjx ;
Alessio:2023kgf .

In the twistor model, the root-Kerr particle is defined by the minimal
extension via Newman-Janis (NJ) shift from Minkowski spacetime to
spinspacetime Kim:2023aff ; Kim:2023vgb . To introduce the minimal extension,
we begin with the usual coupling of a charged scalar particle to the Maxwell
field and split it into two pieces,

$\displaystyle S_{\text{int}}=q\int A_{\mu}(x)dx^{\mu}=q\int
A^{+}_{\mu}(x)dx^{\mu}+q\int A^{-}_{\mu}(x)dx^{\mu}\,,$ (42)

where $q$ is the charge and the convention fixed by $A_{\mu}=(-\phi,\vec{A})$.
The superscripts $(\pm)$ denote the self-dual and anti-self-dual parts in the
sense that, under the hodge dual $*$ on the field-strengths,

$\displaystyle F^{\pm}=dA^{\pm}\,,\quad*F^{\pm}=\pm iF^{\pm}\,.$ (43)

The minimal NJ shift correlates (anti-)self-duality and holomorphy of the
complexified spacetime as Guevara:2020xjx

$\displaystyle S_{\text{int}}=q\int A^{+}_{\mu}(z)dz^{\mu}+q\int
A^{-}_{\mu}(\bar{z})d\bar{z}^{\mu}=-\frac{q}{2}\int
A^{+}_{\alpha\dot{\alpha}}(z)dz^{\dot{\alpha}\alpha}-\frac{q}{2}\int
A^{-}_{\dot{\alpha}\alpha}(\bar{z})d\bar{z}^{\dot{\alpha}\alpha}\,.$ (44)

In the parlance of the symplectic perturbation theory,

$\displaystyle\begin{split}\omega^{\prime}&=(\omega^{\prime})^{+}+(\omega^{\prime})^{-}=\frac{q}{2}F^{+}_{\mu\nu}(z)dz^{\mu}\wedge
dz^{\nu}+\frac{q}{2}F^{-}_{\mu\nu}(\bar{z})d\bar{z}^{\mu}\wedge
d\bar{z}^{\nu}\,.\end{split}$ (45)

Compared to the non-spinning case where
$\omega^{\prime}=\textstyle{\frac{1}{2}}F_{\mu\nu}(x)dx^{\mu}\wedge dx^{\nu}$,
the Newman-Janis shift introduces a complicated non-linear deformation
depending on the $y^{\mu}$ coordinate. But, the invariance under the Maxwell
gauge transformation is still manifest.

As usual, the coupling (44) plays a dual role; it enters the equation of
motion of the charged particle, and it acts as a source term in the Maxwell’s
equations. We focus on the former aspect in this and the next section, leaving
the latter aspect to later sections.

####  2.2.3 Equations of motion

Adding up the free part (9) and the interaction part (44), the full action of
the root-Kerr particle becomes

$\displaystyle
S=\int\left[\lambda_{\alpha}{}^{I}\dot{\bar{\mu}}_{I}{}^{\alpha}+\bar{\lambda}_{I\dot{\alpha}}\dot{\mu}^{\dot{\alpha}I}-\frac{\kappa^{0}}{2}(m^{2}-\Delta\bar{\Delta})-\kappa^{1}W_{0}+qA^{+}_{\mu}(z)\dot{z}^{\mu}+qA^{-}_{\mu}(\bar{z})\dot{\bar{z}}^{\mu}\right]d\sigma\,.$
(46)

The variation of the action with the vanishing Regge trajectory
($m^{\prime}=0$) gives

$\displaystyle\begin{aligned} \delta
S&=\delta\lambda_{\alpha}{}^{I}\left[\dot{\bar{\mu}}_{I}{}^{\alpha}+\frac{\kappa^{0}\Delta\bar{\Delta}}{2}(\lambda^{-1})_{I}{}^{\alpha}+\frac{\kappa^{1}}{2i}\bar{\mu}_{I}{}^{\alpha}\right]\\\
&\phantom{=}+\delta\bar{\lambda}_{I\dot{\alpha}}\left[\dot{\mu}^{\dot{\alpha}I}+\frac{\kappa^{0}\Delta\bar{\Delta}}{2}(\bar{\lambda}^{-1})^{\dot{\alpha}I}-\frac{\kappa^{1}}{2i}{\mu}^{\dot{\alpha}I}\right]\\\
&\phantom{=}+\delta\mu^{\dot{\alpha}I}\left[-\dot{\bar{\lambda}}_{I\dot{\alpha}}-\frac{\kappa^{1}}{2i}{\bar{\lambda}}_{I\dot{\alpha}}\right]+\delta\bar{\mu}_{I}{}^{\alpha}\left[-\dot{\lambda}_{\alpha}{}^{I}+\frac{\kappa^{1}}{2i}{\lambda}_{\alpha}{}^{I}\right]\\\
&\phantom{=}+\delta
z^{\mu}\left[qF_{\mu\nu}^{+}(z)\dot{z}^{\nu}\right]+\delta\bar{z}^{\mu}\left[qF_{\mu\nu}^{-}(\bar{z})\dot{\bar{z}}^{\nu}\right]\\\
&=\delta\lambda_{\alpha}{}^{I}\left[\dot{\bar{\mu}}_{I}{}^{\alpha}+\frac{\kappa^{0}\Delta\bar{\Delta}}{2}(\lambda^{-1})_{I}{}^{\alpha}+\frac{\kappa^{1}}{2i}\bar{\mu}_{I}{}^{\alpha}+qF_{\mu\nu}^{+}(z)\dot{z}^{\nu}\sigma^{\mu}_{\beta\dot{\beta}}\mu^{\dot{\beta}J}(\lambda^{-1})_{J}{}^{\alpha}(\lambda^{-1})_{I}{}^{\beta}\right]\\\
&\phantom{=}+\delta\bar{\lambda}_{I\dot{\alpha}}\left[\dot{\mu}^{\dot{\alpha}I}+\frac{\kappa^{0}\Delta\bar{\Delta}}{2}(\bar{\lambda}^{-1})^{\dot{\alpha}I}-\frac{\kappa^{1}}{2i}{\mu}^{\dot{\alpha}I}+qF_{\mu\nu}^{-}(\bar{z})\dot{\bar{z}}^{\nu}\sigma^{\mu}_{\beta\dot{\beta}}\bar{\mu}_{J}{}^{\beta}(\bar{\lambda}^{-1})^{\dot{\beta}I}(\bar{\lambda}^{-1})^{\dot{\alpha}J}\right]\\\
&\phantom{=}-\delta\mu^{\dot{\alpha}I}\left[\dot{\bar{\lambda}}_{I\dot{\alpha}}+\frac{\kappa^{1}}{2i}{\bar{\lambda}}_{I\dot{\alpha}}+qF_{\mu\nu}^{+}(z)\dot{z}^{\nu}\sigma^{\mu}_{\alpha\dot{\alpha}}(\lambda^{-1})_{I}{}^{\alpha}\right]\\\
&\phantom{=}-\delta\bar{\mu}_{I}{}^{\alpha}\left[\dot{\lambda}_{\alpha}{}^{I}-\frac{\kappa^{1}}{2i}\lambda_{\alpha}{}^{I}+qF_{\mu\nu}^{-}(\bar{z})\dot{\bar{z}}^{\nu}\sigma^{\mu}_{\alpha\dot{\alpha}}(\bar{\lambda}^{-1})^{{\dot{\alpha}}I}\right]\,.\end{aligned}$
(47)

In the last step, we used the incidence relations to express
$\delta(z,\bar{z})$ in terms of $\delta(\mu,\lambda,\bar{\mu},\bar{\lambda})$.
To specify the equations of motion completely, we have to fix the Lagrange
multipliers $\kappa^{0,1}$. A convenient choice to be used throughout this
paper is

$\displaystyle\kappa^{0}=\frac{1}{m}\,,\quad\kappa^{1}=0\,.$ (48)

With this choice, the EOM for the twistor variables are

$\displaystyle\begin{aligned}
\dot{\bar{\mu}}_{I}{}^{\alpha}&=-\frac{m}{2}(\lambda^{-1})_{I}{}^{\alpha}-qF_{\mu\nu}^{+}(z)\dot{z}^{\nu}\sigma^{\mu}_{\beta\dot{\beta}}\mu^{\dot{\beta}J}(\lambda^{-1})_{J}{}^{\alpha}(\lambda^{-1})_{I}{}^{\beta}\,,\\\
\dot{\mu}^{\dot{\alpha}I}&=-\frac{m}{2}(\bar{\lambda}^{-1})^{\dot{\alpha}I}-qF_{\mu\nu}^{-}(\bar{z})\dot{\bar{z}}^{\nu}\sigma^{\mu}_{\beta\dot{\beta}}\bar{\mu}_{J}{}^{\beta}(\bar{\lambda}^{-1})^{\dot{\beta}I}(\bar{\lambda}^{-1})^{\dot{\alpha}J}\,,\\\
\dot{\bar{\lambda}}_{I\dot{\alpha}}&=-qF_{\mu\nu}^{+}(z)\dot{z}^{\nu}\sigma^{\mu}_{\alpha\dot{\alpha}}(\lambda^{-1})_{I}{}^{\alpha}\,,\\\
\dot{\lambda}_{\alpha}{}^{I}&=-qF_{\mu\nu}^{-}(\bar{z})\dot{\bar{z}}^{\nu}\sigma^{\mu}_{\alpha\dot{\alpha}}(\bar{\lambda}^{-1})^{{\dot{\alpha}}I}\,.\end{aligned}$
(49)

The equations for the U$(1)$ gauge invariant variables $(p,z,\bar{z})$ are, in
the vector notation,

$\displaystyle\begin{aligned}
\dot{p}_{\mu}&=qF^{+}_{\mu\nu}(z)\dot{z}^{\nu}+qF^{-}_{\mu\nu}(\bar{z})\dot{\bar{z}}^{\nu}\,,\\\
\dot{z}^{\mu}&=\frac{p^{\mu}}{m}+\frac{2iq}{m^{2}}\left[y^{\mu}p^{\nu}+p^{\mu}y^{\nu}+i\epsilon^{\mu\nu\rho\sigma}y_{\rho}p_{\sigma}\right]F^{-}_{\nu\lambda}(\bar{z})\dot{\bar{z}}^{\lambda}\,,\\\
\dot{\bar{z}}^{\mu}&=\frac{p^{\mu}}{m}-\frac{2iq}{m^{2}}\left[y^{\mu}p^{\nu}+p^{\mu}y^{\nu}-i\epsilon^{\mu\nu\rho\sigma}y_{\rho}p_{\sigma}\right]F^{+}_{\nu\lambda}({z})\dot{{z}}^{\lambda}\,,\end{aligned}$
(50)

where we applied the constraints $y\cdot p=0$, $\Delta\bar{\Delta}=m^{2}$
after deriving the equations. We may turn on the Regge trajectory
$(m^{\prime}\neq 0)$ and repeat deriving the equations. Not surprisingly, the
twistor equations receive new terms proportional to $(m^{\prime})$, but (50)
remains unchanged.

The appearance of time-derivatives on the RHS of (50) may look peculiar. But,
it is a generic feature of the symplectic perturbation theory explained
earlier. It is easy to see how (50) fits with the general structure of (41):

$\displaystyle\\{f,\Delta\bar{\Delta}\\}=\\{f,\Delta\bar{\Delta}\\}_{\circ}-q\\{f,z^{\nu}\\}_{\circ}F^{+}_{\nu\lambda}\\{z^{\lambda},\Delta\bar{\Delta}\\}-q\\{f,\bar{z}^{\nu}\\}_{\circ}F^{-}_{\nu\lambda}\\{\bar{z}^{\lambda},\Delta\bar{\Delta}\\}\,.$
(51)

Finally, turning on the Regge trajectory ($m^{\prime}\neq 0$) modifies (49)
slightly, but leaves (50) unchanged. In what follows, we take (50) as the
fundamental EOM for the twistor particle and use it to compute physical
observables.

Unless the background takes very special values, it would be impossible to
solve the EOM exactly. We approach the problem as a perturbative expansion in
$q$. The EOM truncated up to the 2PL order is given by

$\displaystyle\begin{split}\dot{x}^{\mu}&=\frac{p^{\mu}}{m}-i\frac{q}{m}(F^{+}-F^{-})^{\mu}{}_{\nu}y^{\nu}\\\
&\quad+\frac{4q^{2}}{m^{3}}p^{\mu}(yF^{+}F^{-}y)+\frac{2q^{2}}{m^{3}}y^{\mu}\left[(pF^{+}F^{-}y)+(pF^{-}F^{+}y)\right]+\mathcal{O}(q^{3})\,,\\\
\dot{y}^{\mu}&=\frac{q}{m}(F^{+}+F^{-})^{\mu}{}_{\nu}y^{\nu}-i\frac{4q^{2}}{m^{3}}\left[(pF^{-}y)(F^{+})^{\mu}{}_{\nu}-(pF^{+}y)(F^{-})^{\mu}{}_{\nu}\right]y^{\nu}+\mathcal{O}(q^{3})\,,\\\
\dot{p}^{\mu}&=\frac{q}{m}(F^{+}+F^{-})^{\mu}{}_{\nu}p^{\nu}+\mathcal{O}(q^{3})\,.\end{split}$
(52)

Here the scalar product of vectors and tensors are defined as, for example,

$\displaystyle(pF^{+}F^{-}y)=p_{\mu}(F^{+})^{\mu}{}_{\nu}(F^{-})^{\nu}{}_{\rho}y^{\rho}\,,\quad(pF^{-}y)=p_{\mu}(F^{-})^{\mu}{}_{\nu}y^{\nu}\,.$
(53)

##  3 Scattering observables

We compute the scattering observables (velocity kick and spin kick) of the
massive spinning bodies in the conservative sector up to the 2PL order; 2PL is
the lowest order where our model may disagree with other models implementing
the (dynamical) Newman-Janis shift. Before we compute the observables of our
twistor model, we present a general description of the scattering observables
that is valid in any (Hamiltonian) worldline model.

###  3.1 Classical eikonal as scattering generator

The eikonal phase has proved useful in quantum and classical scattering in
field theory, gravity and string theory; see ref. DiVecchia:2023frv  for a
comprehensive review. In the context of amplitudes-based methods for classical
gravitational scattering, several variants of the eikonal phase are available
in the literature such as the HEFT phase Brandhuber:2021eyq , the radial
action Bern:2021dqo , and the exponential representation Damgaard:2021ipf ;
Damgaard:2023ttc  just to mention a few.

Here, we introduce the notion of “classical eikonal” which is a particular
classical avatar of the eikonal phase. Simply put, the classical eikonal is
the generator of the canonical transformation between the initial states and
the final states of the scattering problem. It is motivated by the classical
limit of an $S$-matrix, but in can be defined purely within classical
mechanics. The definition is valid in any worldline model in Hamiltonian
formulation. The relevant phase space is that of free particles, just as the
scattering states of a quantum scattering process are defined on the free
Hilbert space.

##### Scattering generator

We first recall the KMOC Kosower:2018adc  method of extracting a classical
observable from a quantum theory:

$\displaystyle\Delta O=\lim_{\hbar\rightarrow
0}\left[\langle\psi|\hat{S}^{\dagger}\hat{O}\hat{S}|\psi\rangle-\langle\psi|\hat{O}|\psi\rangle\right]\,.$
(54)

We can trade the unitary operator $\hat{S}$ for a hermitian operator
$\hat{\chi}$ as Damgaard:2021ipf ; Damgaard:2023ttc

$\displaystyle\hat{S}=e^{i\hat{\chi}/\hbar}\,.$ (55)

Forgetting about the state $|\psi\rangle$ and working in the Heisenberg
picture, we have

$\displaystyle\begin{split}\hat{O}^{\prime}=\hat{S}^{\dagger}\hat{O}\hat{S}&=e^{-i\hat{\chi}/\hbar}\hat{O}e^{i\hat{\chi}/\hbar}\\\
&=\hat{O}+\frac{1}{i\hbar}[\hat{\chi},\hat{O}]+\frac{1}{2(i\hbar)^{2}}[\hat{\chi},[\hat{\chi},\hat{O}]]+\cdots\,.\end{split}$
(56)

Following Dirac’s correspondence,

$\displaystyle\lim_{\hbar\rightarrow
0}\frac{1}{i\hbar}[\hat{X},\hat{Y}]=\\{X,Y\\}\,,$ (57)

we deduce that the classical limit of (56) should give

$\displaystyle
O^{\prime}=O+\\{\chi,O\\}+\frac{1}{2}\\{\chi,\\{\chi,O\\}\\}+\cdots\,.$ (58)

The classical quantity $\chi$, which we call “scattering generator”, is to be
understood as a function on the phase space of a Hamiltonian system. The
scattering generator defines a canonical transformation that maps a free phase
space at past infinity to another free phase space at future infinity. For a
constrained Hamiltonian system with gauge generators and gauge-fixing
conditions, $\chi$ must be gauge-invariant. If we also demand that $O$ is
gauge-invariant, the difference between the Dirac bracket and the Poisson
bracket shown in (18) vanishes, so we may work with the Poisson bracket (20).

We used the classical limit of quantum mechanics to motivate the existence of
$\chi$. But, it is certainly possible to argue for its existence purely within
classical mechanics. The Hamiltonian time evolution generates infinitesimal
canonical transformation at each moment in time. Integrating the time
evolution from past infinity to future infinity would result in a finite
canonical transformation. Provided that the general relation between a Lie
group and its Lie algebra holds for the (infinite dimensional) group of all
canonical transformations on the free phase space, any finite canonical
transformation could be written in a “conjugation” form as in (58).

The master formula (58) can compute any scattering observable. Perturbatively,
with

$\displaystyle\chi=\chi_{(1)}+\chi_{(2)}+\chi_{(3)}+\cdots\,,$ (59)

it produces systematically all $n$-PL impulse formulas

$\displaystyle\begin{split}\Delta_{(1)}O&=\\{\chi_{(1)},O\\}\,,\\\
\Delta_{(2)}O&=\\{\chi_{(2)},O\\}+\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},O\\}\\}\,,\\\
\Delta_{(3)}O&=\\{\chi_{(3)},O\\}+\frac{1}{2}\\{\chi_{(2)},\\{\chi_{(1)},O\\}\\}+\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(2)},O\\}\\}\\\
&\hskip
85.35826pt+\frac{1}{6}\\{\chi_{(1)},\\{\chi_{(1)},\\{\chi_{(1)},O\\}\\}\\}\,.\end{split}$
(60)

An important feature of the master formula (58) is that, once all the
constraints of the system are taken into account by a suitable Dirac bracket,
the preservation of the constraints is guaranteed to all order in perturbation
theory. For the twistor model, recall from section 2.1.3 that the Dirac
bracket satisfies

$\displaystyle\\{f,p^{2}\\}_{*}=\\{f,y^{2}\\}_{*}=\\{f,y\cdot p\\}_{*}=0\,.$
(61)

When $f$ is gauge invariant, the Dirac bracket can be replaced by the Poisson
bracket. Setting $f=\chi=\chi_{(1)}+\chi_{(2)}+\cdots$, since each
$\chi_{(n)}$ scale differently with the coupling constant, we deduce that at
each $n$,

$\displaystyle p\cdot\\{\chi_{(n)},p\\}=0\,,\quad
y\cdot\\{\chi_{(n)},y\\}=0\,,\quad
y\cdot\\{\chi_{(n)},p\\}+p\cdot\\{\chi_{(n)},y\\}=0\,.$ (62)

From a perturbative point of view, the iteration terms in (60) are essential
for consistency. For example, at 2PL, the iteration term in (60) ensures the
mass-shell condition:

$\displaystyle\begin{split}2p\cdot\Delta_{(2)}p&=2p_{\mu}\\{\chi_{(2)},p^{\mu}\\}+p_{\mu}\\{\chi_{(1)},\\{\chi_{(1)},p^{\mu}\\}\\}\\\
&=\\{\chi_{(1)},p_{\mu}\\{\chi_{(1)},p^{\mu}\\}\\}-\\{\chi_{(1)},p_{\mu}\\}\\{\chi_{(1)},p^{\mu}\\}=-(\Delta_{(1)}p)^{2}\,,\end{split}$
(63)

where we used (62) and the Leibniz rule for the Poisson brackets. In view of
(62) and (63), when we compute the 2PL observables later in this section, we
will call $\\{\chi_{(2)},O\\}$ and
$\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},O\\}\\}$ “transverse” and
“longitudinal” (or “iteration”), respectively.

####  3.1.1 Non-spinning example

Arguing for the existence of $\chi$ is one thing, giving an algorithm to
compute $\chi$ is another. It seems plausible that the “WQFT eikonal”
$\chi_{\mathrm{WQFT}}$ Mogull:2020sak  coincides with our classical eikonal
$\chi$. We will verify this expectation up to 2PL by explicit computations.
Proving the equivalence to all orders in perturbation theory is an open
question, which requires incorporation of bremsstrahlung effects.

Before we compute the scattering generator and observables of the twistor
model, to illustrate the ideas in a simpler setting, we review the binary
dynamics of non-spinning particles in electromagnetism following ref.
Kosower:2018adc , which shows how to separate the transverse term
$\\{\chi_{(2)},p\\}$ from the longitudinal term
$\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},p\\}\\}$. The solution of the EOM
involves retarded Green functions on the worldline. When the solution is split
into the time-symmetric part and the time-anti-symmetric part, the former
gives the transverse term while the latter gives the iteration term.

The 1PL momentum kick is well known:

$\displaystyle\Delta_{(1)}p_{1}^{\mu}=-q_{1}q_{2}\gamma\int_{k}(ik^{\mu})\frac{e^{ik\cdot
b}}{k^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k)\,.$ (64)

where the velocity vectors are defined as $v_{a}^{\mu}:=p_{a}^{\mu}/m_{a}$ and
the relative boost is defined as $\gamma:=-(v_{1}\cdot v_{2})$. The impact
parameter $b^{\mu}$ is the projection of the relative position
$x_{12}^{\mu}=x_{1}^{\mu}-x_{2}^{\mu}$ onto the plane transverse to $v_{1}$
and $v_{2}$.

The 1PL momentum kick (64) satisfies (60) rather trivially as

$\displaystyle\Delta_{(1)}p_{1}^{\mu}=\eta^{\mu\nu}\frac{\partial\chi_{(1)}}{\partial
x_{1}^{\nu}}=\\{\chi_{(1)},p_{1}^{\mu}\\}\,,\quad\chi_{(1)}=-q_{1}q_{2}\gamma\int_{k_{\perp}}\frac{e^{ik\cdot
b}}{k^{2}}\,,$ (65)

where the integral over the transverse plane is defined as

$\displaystyle\int_{k_{\perp}}=\int
d\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{4}k\,\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k)\,.$ (66)

As we proceed to the 2PL, it is useful to recall that the computation can be
organized according to the mass ratio:

$\displaystyle\Delta_{(2)}p_{1}^{\mu}=\frac{1}{m_{1}}K^{\mu}_{1,1}+\frac{1}{m_{2}}K^{\mu}_{1,2}\,,\quad\Delta_{(2)}p_{2}^{\mu}=\frac{1}{m_{1}}K^{\mu}_{2,1}+\frac{1}{m_{2}}K^{\mu}_{2,2}\,.$
(67)

The vectors $K^{\mu}_{a,b}$ are integrals independent of the masses. The
conservation of $(p_{1}^{\mu}+p_{2}^{\mu})$ requires that

$\displaystyle K^{\mu}_{1,1}+K^{\mu}_{2,1}=0\,,\quad
K^{\mu}_{1,2}+K^{\mu}_{2,2}=0\,.$ (68)

The exchange symmetry between the two particles relate the integrals by

$\displaystyle K^{\mu}_{1,1}\leftrightarrow K^{\mu}_{2,2}\,,\quad
K^{\mu}_{1,2}\leftrightarrow K^{\mu}_{2,1}\,.$ (69)

So, it suffices to compute $K^{\mu}_{1,1}$. To do so, we may work in the probe
limit $(m_{2}\rightarrow\infty)$ Vines:2018gqi . In the rest of this section,
we will work in the probe limit until further notice.

The 2PL EOM of a spinless particle in the probe limit is given by

$\displaystyle\dot{v}_{(2)}^{\mu}$
$\displaystyle=\frac{q}{m}F^{\mu}{}_{\nu}\dot{x}_{(1)}^{\nu}+\frac{q}{m}x_{(1)}^{\lambda}\partial_{\lambda}F^{\mu}{}_{\nu}v_{(0)}^{\nu}\,,$
(70)

where $F$ means $F_{(0)}$. Using the Bianchi identity, we may rewrite it as

$\displaystyle\dot{v}_{(2)}^{\mu}$
$\displaystyle=\frac{q}{m}(\partial^{\mu}F_{\lambda\nu})x_{(1)}^{\lambda}v_{(0)}^{\nu}+\frac{d}{dt}\left(\frac{q}{m}F^{\mu}{}_{\nu}x_{(1)}^{\nu}\right)\,.$
(71)

The total derivative on the RHS does not contribute to the momentum kick, so
that

$\displaystyle\begin{split}\Delta_{(2)}p_{1}^{\mu}&=q\int
d\sigma(\partial^{\mu}F_{\lambda\nu})x_{(1)}^{\lambda}v_{(0)}^{\nu}\\\
&=\frac{q^{2}}{m}\int_{-\infty}^{\infty}d\sigma(\partial^{\mu}F_{\lambda\nu})v_{(0)}^{\nu}\left[\int_{-\infty}^{\sigma}d\sigma^{\prime}\int_{-\infty}^{\sigma^{\prime}}d\sigma^{\prime\prime}F^{\lambda}{}_{\rho}v_{(0)}^{\rho}\right]\,.\end{split}$
(72)

Performing Fourier transform of ($F_{\lambda\nu}$, $F^{\lambda}{}_{\rho}$) by
($e^{i\ell\cdot x}$, $e^{ik\cdot x}$) respectively, and switching to
$q=k+\ell$, we obtain

$\displaystyle\Delta_{(2)}p_{1}^{\mu}=\frac{(q_{1}q_{2})^{2}}{m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{\ell^{2}(q-\ell)^{2}}(i\ell^{\mu})\left[1+\gamma^{2}\frac{\ell\cdot(q-\ell)}{(v_{1}\cdot\ell+i0^{+})^{2}}\right]\,.$
(73)

The $i0^{+}$ prescription comes from the retarded Green function on the
worldline. To separate the time-symmetric parts from the time-anti-symmetric
parts, as explained in ref. Kosower:2018adc , we apply the exchange
$\ell\leftrightarrow q-\ell$ to get

$\displaystyle\Delta_{(2)}p_{1}^{\mu}=\frac{(q_{1}q_{2})^{2}}{m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{\ell^{2}(q-\ell)^{2}}(iq^{\mu}-i\ell^{\mu})\left[1+\gamma^{2}\frac{\ell\cdot(q-\ell)}{(v_{1}\cdot\ell-i0^{+})^{2}}\right]\,.$
(74)

The next step is to take the average of (73) and (74) and write the result as

$\displaystyle\Delta_{(2)}p_{1}^{\mu}=\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{tr}}+\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{iter}}\,.$
(75)

The time-symmetric part carries an integrand proportional to $q^{\mu}$ so that

$\displaystyle\begin{split}\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{tr}}&=\frac{(q_{1}q_{2})^{2}}{2m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{\ell^{2}(q-\ell)^{2}}(iq^{\mu})\left[1+\gamma^{2}\frac{\ell\cdot(q-\ell)}{(v_{1}\cdot\ell)^{2}}\right]\,.\end{split}$
(76)

It yields the 2PL eikonal via
$\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{tr}}=\\{\chi_{(2)},p_{1}^{\mu}\\}$.
Omitting the $i0^{+}$ prescription that is no longer needed, and restoring the
symmetry between the two particles, we obtain

$\displaystyle\begin{split}\chi_{(2)}&=\frac{(q_{1}q_{2})^{2}}{2m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{\ell^{2}(q-\ell)^{2}}\left[1+\gamma^{2}\frac{\ell\cdot(q-\ell)}{(v_{1}\cdot\ell)^{2}}\right]+(1\leftrightarrow
2)\,.\end{split}$ (77)

It agrees perfectly with the WQFT eikonal Wang:2022ntx . The time-anti-
symmetric part of (75) is

$\displaystyle\begin{split}\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{iter}}&=\frac{(q_{1}q_{2})^{2}\gamma^{2}}{2m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{k^{2}\ell^{2}}(k\cdot\ell)\left[\frac{i\ell^{\mu}}{(v_{1}\cdot\ell+i0^{+})^{2}}-\frac{i\ell^{\mu}}{(v_{1}\cdot\ell-i0^{+})^{2}}\right]\\\
&=-\frac{(q_{1}q_{2})^{2}\gamma^{2}}{2m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{k^{2}\ell^{2}}(k\cdot\ell)\ell^{\mu}\,\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(v_{1}\cdot\ell)\,,\end{split}$ (78)

where $k=q-\ell$ and we used the identity,

$\displaystyle\frac{i}{x+i0^{+}}-\frac{i}{x-i0^{+}}=\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(x)\quad\Longrightarrow\quad\frac{i}{(x+i0^{+})^{2}}-\frac{i}{(x-i0^{+})^{2}}=-\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(x)\,.$ (79)

We may compare it with the iteration term. From (65) and (64), we find

$\displaystyle\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},p_{1}^{\mu}\\}\\}=\frac{i}{2}(q_{1}q_{2})^{2}\left\\{\gamma\int_{k_{\perp}}\frac{e^{ik\cdot
b}}{k^{2}},\gamma\int_{\ell_{\perp}}\ell^{\mu}\frac{e^{i\ell\cdot
b}}{\ell^{2}}\right\\}\,,$ (80)

In computing the bracket in (80), we note that

$\displaystyle\\{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot
k)e^{ik\cdot(x_{1}-x_{2})},\gamma\\}=\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot
k)e^{ik\cdot(x_{1}-x_{2})}\frac{ik\cdot\left(-p_{2}+p_{1}\right)}{m_{1}m_{2}}=0\,.$
(81)

So, the only non-vanishing contributions come from

$\displaystyle\\{e^{ik\cdot(x_{1}-x_{2})},\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot\ell)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)\\}\quad\mbox{and}\quad\\{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k),e^{i\ell\cdot(x_{1}-x_{2})}\\}\,.$ (82)

Collecting the contributions, we obtain (in the probe limit)

$\displaystyle\begin{split}&\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},p_{1}^{\mu}\\}\\}=-\frac{1}{2}(q_{1}q_{2})^{2}\gamma^{2}\int_{k,\ell}e^{i(k+\ell)\cdot
b}\frac{k\cdot\ell}{k^{2}\ell^{2}}\mathcal{F}(k,\ell)\ell^{\mu}\,,\\\
&\mathcal{F}(k,\ell)=\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{1}\cdot k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot
k)\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(v_{1}\cdot\ell)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{m_{1}}-(k\leftrightarrow\ell)\,.\end{split}$ (83)

Using the identities,

$\displaystyle\delta(x+y)\delta(y)=\delta(x)\delta(y)\,,\quad\delta(x+y)\delta^{\prime}(y)=\delta(x)\delta^{\prime}(y)-\delta^{\prime}(x)\delta(y)\,,$
(84)

we can simplify (83) a bit further and obtain

$\displaystyle\begin{split}\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},p_{1\mu}\\}\\}=-\frac{(q_{1}q_{2})^{2}\gamma^{2}}{2m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{k\cdot\ell}{k^{2}\ell^{2}}\ell_{\mu}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(v_{1}\cdot\ell)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)\,,\end{split}$ (85)

in perfect agreement with (78).

In the paragraphs above, we manipulated the integrands of the Fourier
integrals to separate the iteration term. In the non-spinning case, it is easy
to perform the Fourier integral and compute the brackets in position space. We
write the impact parameter as

$\displaystyle b^{\mu}$ $\displaystyle=x_{12}^{\mu}+\frac{\gamma(x_{12}\cdot
v_{2})-(x_{12}\cdot v_{1})}{\gamma^{2}-1}v_{1}^{\mu}+\frac{\gamma(x_{12}\cdot
v_{1})-(x_{12}\cdot v_{2})}{\gamma^{2}-1}v_{2}^{\mu}\,,\,$ (86)

where $x_{12}^{\mu}=x_{1}^{\mu}-x_{2}^{\mu}$. The 1PL non-spinning eikonal is,
after the Fourier integral,

$\displaystyle\chi_{(1)}$
$\displaystyle=\frac{q_{1}q_{2}\gamma}{4\pi\sqrt{\gamma^{2}-1}}\left[\frac{1}{\epsilon}+\log\left(\frac{b^{2}}{b_{0}^{2}}\right)\right]\,,$
(87)

where $\epsilon$ is the dimensional regularisation parameter $D=4-2\epsilon$
and $b_{0}^{2}$ is a scheme-dependent constant. The precise values of
$\epsilon$ and $b_{0}$ are irrelevant. The relevant brackets are

$\displaystyle\\{b^{2},\gamma\\}=0\,,\quad\\{b^{2},p_{1}^{\mu}\\}\doteq
2b^{\mu}\,,\quad\\{b^{2},b^{\mu}\\}\doteq 2b^{2}\left(\frac{\gamma
v_{2}^{\mu}-v_{1}^{\mu}}{(\gamma^{2}-1)m_{1}}-\frac{\gamma
v_{1}^{\mu}-v_{2}^{\mu}}{(\gamma^{2}-1)m_{2}}\right)\,,$ (88)

where $\doteq$ denotes equivalence up to mass-shell constraint
$p_{i}^{2}+m_{i}^{2}=0$. Thus,

$\displaystyle\begin{split}\\{\chi_{(1)},\\{\chi_{(1)},p_{1}^{\mu}\\}\\}&=\left[\frac{q_{1}q_{2}\gamma}{4\pi\sqrt{\gamma^{2}-1}}\right]^{2}\\{\log
b^{2},\\{\log b^{2},p_{1}^{\mu}\\}\\}\\\
&=\left[\frac{q_{1}q_{2}\gamma}{4\pi\sqrt{\gamma^{2}-1}}\frac{1}{b^{2}}\right]^{2}\\{b^{2},\\{b^{2},p_{1}^{\mu}\\}\\}\doteq
2\left[\frac{q_{1}q_{2}\gamma}{4\pi\sqrt{\gamma^{2}-1}}\frac{1}{b^{2}}\right]^{2}\\{b^{2},b^{\mu}\\}\\\
&\doteq\left[\frac{q_{1}q_{2}\gamma}{2\pi\sqrt{\gamma^{2}-1}}\frac{1}{\sqrt{b^{2}}}\right]^{2}\times\left(\frac{\gamma
v_{2}^{\mu}-v_{1}^{\mu}}{(\gamma^{2}-1)m_{1}}-\frac{\gamma
v_{1}^{\mu}-v_{2}^{\mu}}{(\gamma^{2}-1)m_{2}}\right)\\\
&=\left|\\{\chi_{(1)},p_{1}^{\mu}\\}\right|^{2}\times\left(\frac{\gamma
v_{2}^{\mu}-v_{1}^{\mu}}{(\gamma^{2}-1)m_{1}}-\frac{\gamma
v_{1}^{\mu}-v_{2}^{\mu}}{(\gamma^{2}-1)m_{2}}\right)\,.\end{split}$ (89)

This term generates the longitudinal impulse needed to preserve the mass-shell
condition,

$\displaystyle p_{1\mu}\,\\{\chi_{(1)},\\{\chi_{(1)},p_{1}^{\mu}\\}\\}$
$\displaystyle=-\\{\chi_{(1)},p_{1}^{\mu}\\}^{2}\,.$ (90)

###  3.2 Momentum kick and spin kick from EOM

In the scattering problem of two spinning particles, the momentum kick and the
spin kick are the main scattering observables. In this subsection, we compute
them up to the 2PL order by solving the EOM perturbatively. The perturbation
computes the deviation from the free (straight line, constant spin) trajectory
for each particle,

$\displaystyle\begin{split}z^{\mu}(\sigma)&=(x_{(0)}^{\mu}+iy_{(0)}^{\mu}+v_{(0)}^{\mu}\sigma)+\delta
z^{\mu}(\sigma)\,,\\\ p^{\mu}(\sigma)&=m(v_{(0)}^{\mu}+\delta
v^{\mu}(\sigma))\,.\end{split}$ (91)

Without loss of generality, and taking constraints into consideration, we
demand that the constant parameters satisfy

$\displaystyle v_{(0)}\cdot v_{(0)}=-1\,,\quad(x_{(0)}+iy_{(0)})\cdot
v_{(0)}=0\,.$ (92)

The perturbative solution is arranged in the PL order as

$\displaystyle\begin{split}\delta
z^{\mu}&=z_{(1)}^{\mu}+z_{(2)}^{\mu}+\cdots\,,\\\ \delta
v^{\mu}&=v_{(1)}^{\mu}+v_{(2)}^{\mu}+\cdots\,,\end{split}$ (93)

where $X_{(n)}$ is proportional to $q^{n}$. To avoid clutter, we will often
omit the subscript $X_{(0)}$ from background values.

####  3.2.1 1PL

At 1PL, the equations of motion are reduced to

$\displaystyle\dot{v}_{(1)}^{\mu}$
$\displaystyle=\frac{q}{m}(F_{(0)}^{+}+F_{(0)}^{-})^{\mu}{}_{\nu}v_{(0)}^{\nu}\,,$
(94) $\displaystyle\dot{z}_{(1)}^{\mu}$
$\displaystyle=v_{(1)}^{\mu}+\frac{2iq}{m}(F_{(0)}^{-})^{\mu}{}_{\nu}y_{(0)}^{\nu}\,.$
(95)

It is understood that $F^{\pm}_{(0)}$ here are evaluated along the background
worldline (91).

In computing the impulse of particle 1 to the 1PL order, we may treat particle
2 as a stationary source. The field-strength measured at the position of
particle 1 is given by

$\displaystyle\begin{split}F^{+}_{\mu\nu}(z_{1}(\sigma_{1}))=i\frac{q_{2}}{2}\int_{k}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k)\frac{(k\wedge
v_{2})_{\mu\nu}-i\epsilon_{\mu\nu}[k,v_{2}]}{k^{2}}e^{ik\cdot(z_{1}(\sigma_{1})-\bar{z}_{2}(0))}\,,\\\
F^{-}_{\mu\nu}(\bar{z}_{1}(\sigma_{1}))=i\frac{q_{2}}{2}\int_{k}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot k)\frac{(k\wedge
v_{2})_{\mu\nu}+i\epsilon_{\mu\nu}[k,v_{2}]}{k^{2}}e^{ik\cdot(\bar{z}_{1}(\sigma_{1})-z_{2}(0))}\,,\end{split}$
(96)

where the worldline time-dependence from
$z_{2}^{\mu}(\sigma_{2})=(x_{2}^{\mu}+iy_{2}^{\mu}+v_{2}^{\mu}\sigma_{2})$ has
been integrated out to leave
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(v_{2}\cdot
k)$ behind. The wedge and the epsilon notations mean

$\displaystyle\begin{split}&(a\wedge
b)_{\mu\nu}=a_{\mu}b_{\nu}-b_{\mu}a_{\nu}\,,\\\
&\epsilon_{\mu\nu}[a,b]=\epsilon_{\mu\nu\rho\sigma}a^{\rho}b^{\sigma}\,,\quad\epsilon_{\mu}[a,b,c]=\epsilon_{\mu\nu\rho\sigma}a^{\nu}b^{\rho}c^{\sigma}\,,\quad\epsilon[a,b,c,d]=\epsilon_{\mu\nu\rho\sigma}a^{\mu}b^{\nu}c^{\rho}d^{\sigma}\,.\end{split}$
(97)

Contracting $F^{\pm}_{\mu\nu}$ with $v_{1}^{\nu}$ and integrating (94) over
$\sigma_{1}$, we obtain the velocity kick,

$\displaystyle\begin{aligned}
\Delta_{(1)}v_{1}^{\mu}&=-\frac{q_{1}q_{2}}{2m_{1}}\int_{k_{\perp}}\frac{i\gamma
k^{\mu}+\epsilon^{\mu}[k,v_{1},v_{2}]}{k^{2}}e^{ik\cdot(b+iy)}\\\
&\qquad-\frac{q_{1}q_{2}}{2m_{1}}\int_{k_{\perp}}\frac{i\gamma
k^{\mu}-\epsilon^{\mu}[k,v_{1},v_{2}]}{k^{2}}e^{ik\cdot(b-iy)}\\\
&=-\frac{q_{1}q_{2}}{m_{1}}\int_{k_{\perp}}\left[(ik^{\mu})\cosh(k\cdot
y)\gamma-\epsilon^{\mu}[k,v_{1},v_{2}]\sinh(k\cdot y)\right]\frac{e^{ik\cdot
b}}{k^{2}}\,.\end{aligned}$ (98)

The argument of the exponential is decomposed as

$\displaystyle
z_{1}^{\mu}-\bar{z}_{2}^{\mu}=(x_{1}^{\mu}-x_{2}^{\mu})+i(y_{1}^{\mu}+y_{2}^{\mu})=:x_{12}^{\mu}+iy^{\mu}\,,$
(99)

and $x_{12}^{\mu}$ is projected onto the impact parameter vector $b^{\mu}$ by
$\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(v_{1}\cdot
k)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(v_{2}\cdot
k)$. Note how the spin sum $(y_{1}^{\mu}+y_{2}^{\mu})$ arises from the
_difference_ between complex spinspacetime coordinates.

Similarly, we can compute the spin kick and find

$\displaystyle\begin{aligned}
\Delta_{(1)}y_{1}^{\mu}&=\frac{q_{1}q_{2}}{m_{1}}\int_{k_{\perp}}\left[i(k\wedge
v_{2})^{\mu}{}_{\nu}y_{1}^{\nu}\cosh(k\cdot
y)-\epsilon^{\mu}[k,v_{2},y_{1}]\sinh(k\cdot y)\right]\frac{e^{ik\cdot
b}}{k^{2}}\,.\end{aligned}$ (100)

It is easy to verify orthogonality and conservation of magnitudes of
velocity/spin at 1PL:

$\displaystyle\begin{aligned} \Delta(y_{1}\cdot v_{1})&=(\Delta y_{1}\cdot
v_{1})+(y_{1}\cdot\Delta v_{1})=0\,,\\\ \Delta(v_{1}^{2})&=2(\Delta v_{1}\cdot
v_{1})=0\,,\\\ \Delta(y_{1}^{2})&=2(\Delta y_{1}\cdot
y_{1})=0\,.\end{aligned}$ (101)

The 1PL observables can be compared with predictions of QED in the classical
limit. We use the results of ref. Bern:2023ity  as the reference. When
truncated to linear order in $y$, (98) and (100) are found to be consistent
with (4.45) and (4.46) of ref. Bern:2023ity , under the conditions $C_{i}=1$,
$D_{i}=0$, and the covariant spin supplementary condition. Note that the spin
tensor kick $\Delta S^{\mu\nu}$ reported by ref. Bern:2023ity  receives
contributions from both (98) and (100).

The 1PL observables can be reproduced by the classical eikonal as
Guevara:2019fsj

$\displaystyle\begin{split}\Delta_{(1)}p_{1}^{\mu}&=\\{\chi_{(1)},p_{1}^{\mu}\\}=\eta^{\mu\nu}\frac{\partial}{\partial
x_{1}^{\nu}}\chi_{(1)}\,,\\\
\Delta_{(1)}y_{1}^{\mu}&=\\{\chi_{(1)},y_{1}^{\mu}\\}=\frac{1}{m_{1}}\left[v_{1}^{\mu}y_{1}^{\nu}\frac{\partial}{\partial
x_{1}^{\nu}}+\epsilon^{\mu\nu}[v_{1},y_{1}]\frac{\partial}{\partial
y_{1}^{\nu}}\right]\chi_{(1)}\,.\end{split}$ (102)

An explicit form of the 1PL eikonal is

$\displaystyle\chi_{(1)}$
$\displaystyle=-q_{1}q_{2}\int_{k_{\perp}}\left[\cosh(k\cdot
y)\gamma-i\frac{\sinh(k\cdot y)}{k\cdot
y}\epsilon[k,v_{1},v_{2},y]\right]\frac{e^{ik\cdot b}}{k^{2}}\,.$ (103)

To reproduce (98) and (100) from (103) via (102), we need the identity,

$\displaystyle\frac{k^{\mu}\epsilon[k,v_{1},v_{2},y]}{k\cdot
y}\approx-\epsilon^{\mu}[k,v_{1},v_{2}]\,.$ (104)

The approximate equality ($\approx$) means that we may impose the
transversality condition $k\cdot v_{1}=0=k\cdot v_{2}$ and ignore ultra-local
($\propto k^{2}/k^{2}$) terms inside the integral. With this understanding,
the identity above can be derived from the 4d Schouten identity,

$\displaystyle\begin{split}&a^{\mu}\epsilon[b,c,d,e]+b^{\mu}\epsilon[c,d,e,a]+c^{\mu}\epsilon[d,e,a,b]\\\
&\qquad+d^{\mu}\epsilon[e,a,b,c]+e^{\mu}\epsilon[a,b,c,d]=0\,.\end{split}$
(105)

In what follows, we will not distinguish the approximate equality from the
strict equality. Inside the integrands of the Fourier integrals, we will take
the liberty to set

$\displaystyle k^{2}=0\,,\quad\ell^{2}=0\,,\quad k\cdot v_{2}=0=\ell\cdot
v_{2}\,,\quad(k+\ell)\cdot v_{1}=0\,.$ (106)

####  3.2.2 2PL

The 2PL equations directly relevant for the impulse computation are

$\displaystyle\begin{split}\dot{v}_{(2)}^{\mu}&=\frac{q}{m}(F^{+}_{(0)}+F^{-}_{(0)})^{\mu}{}_{\nu}v_{(1)}^{\nu}+\frac{q}{m}(F^{+}_{(1)}+F^{-}_{(1)})^{\mu}{}_{\nu}v_{(0)}^{\nu}\,,\\\
\dot{y}_{(2)}^{\mu}&=\frac{q}{m}(F^{+}_{(0)}+F^{-}_{(0)})^{\mu}{}_{\nu}y_{(1)}^{\nu}+\frac{q}{m}(F^{+}_{(1)}+F^{-}_{(1)})^{\mu}{}_{\nu}y_{(0)}^{\nu}\\\
&\quad-i\frac{4q^{2}}{m^{2}}\left[(vF^{-}y)(F^{+})^{\mu}{}_{\nu}-(vF^{+}y)(F^{-})^{\mu}{}_{\nu}\right]_{(0)}y_{(0)}^{\nu}\,.\end{split}$
(107)

##### 2PL velocity kick

As we saw in the non-spinning case, it is sufficient to work in the probe
limit $(m_{1}/m_{2}\rightarrow 0)$, and we can use the Bianchi identity and
discard a total derivative to get

$\displaystyle\Delta_{(2)}p^{\mu}$ $\displaystyle=q\int
d\sigma(\partial^{\mu}F^{+}_{\lambda\nu})z_{(1)}^{\lambda}v_{(0)}^{\nu}+q\int
d\sigma(\partial^{\mu}F^{-}_{\lambda\nu})\bar{z}_{(1)}^{\lambda}v_{(0)}^{\nu}\,.$
(108)

Next, using (95), we can turn $(z_{(1)},\bar{z}_{(1)})$ into integrals,

$\displaystyle\begin{split}\Delta_{(2)}p^{\mu}&=\Delta_{(2v)}p^{\mu}+\Delta_{(2y)}p^{\mu}\,,\\\
\Delta_{(2v)}p^{\mu}&=\frac{q^{2}}{m}\int_{-\infty}^{\infty}d\sigma\partial^{\mu}(F^{+}+F^{-})_{\lambda\nu}v_{(0)}^{\nu}\left[\int_{-\infty}^{\sigma}d\sigma^{\prime}\int_{-\infty}^{\sigma^{\prime}}d\sigma^{\prime\prime}(F^{+}+F^{-})^{\lambda}{}_{\rho}v_{(0)}^{\rho}\right]\,,\\\
\Delta_{(2y)}p^{\mu}&=\frac{2iq^{2}}{m}\int_{-\infty}^{\infty}d\sigma(\partial^{\mu}F^{+}_{\lambda\nu})v^{\nu}_{(0)}\int_{-\infty}^{\sigma}d\sigma^{\prime}(F^{-})^{\lambda}{}_{\rho}y_{(0)}^{\rho}\\\
&\quad-\frac{2iq^{2}}{m}\int_{-\infty}^{\infty}d\sigma(\partial^{\mu}F^{-}_{\lambda\nu})v^{\nu}_{(0)}\int_{-\infty}^{\sigma}d\sigma^{\prime}(F^{+})^{\lambda}{}_{\rho}y_{(0)}^{\rho}\,.\end{split}$
(109)

We divided the computation into two parts. The “y-part”
($\Delta_{(2y)}p^{\mu}$) is linear in $y_{1}$ aside from the $y$-dependence in
the exponential factors. The “v-part” ($\Delta_{(2v)}p^{\mu}$) is independent
of $y_{1}$ aside from the exponential factors. While we replace
$v^{\mu}_{(1)}$ by $\dot{z}_{(1)}$ or $\dot{\bar{z}}_{(1)}$, we also encounter
terms proportional to
$(F^{+})^{\mu}{}_{\nu}(F^{-})^{\nu}{}_{\lambda}-(F^{-})^{\mu}{}_{\nu}(F^{+})^{\nu}{}_{\lambda}$,
but they vanish identically. Let us analyse the two parts one by one.

For the y-part, after using the field-strengths (96) and integrating over the
worldline, we obtain

$\displaystyle\Delta_{(2y)}p_{1}^{\mu}$
$\displaystyle=\frac{(q_{1}q_{2})^{2}}{m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)(i\ell^{\mu})}{\ell^{2}k^{2}(ik\cdot
v_{1}+0^{+})}\left(\mathrm{ch}\\!\boxminus C_{y}+\mathrm{sh}\\!\boxminus
S_{y}\right)\,,$ (110)

where we set $q=k+\ell$ as before and introduce shorthand notations,

$\displaystyle\begin{split}\mathrm{ch}\boxminus=\cosh[(k-\ell)\cdot
y]\,,&\quad\mathrm{sh}\boxminus=\sinh[(k-\ell)\cdot y]\,,\\\
\mathrm{ch}\boxplus=\cosh[(k+\ell)\cdot
y]\,,&\quad\mathrm{sh}\boxplus=\sinh[(k+\ell)\cdot y]\,.\end{split}$ (111)

The functions $C_{y}$, $S_{y}$ are

$\displaystyle\begin{split}C_{y}&=-\epsilon[k,\ell,y_{1},v_{1}]+2(v_{2}\cdot
y_{1})\epsilon[k,\ell,v_{1},v_{2}]\,,\\\ S_{y}&=i(\ell\cdot
v_{1})[(k-\ell)\cdot y_{1}]+2i\gamma(v_{2}\cdot
y_{1})(k\cdot\ell)\,.\end{split}$ (112)

We divide the v-part further into the same helicity contribution
($\Delta_{(2vs)}p^{\mu}$) and the opposite helicity contribution
($\Delta_{(2vo)}p^{\mu}$). The opposite helicity part is

$\displaystyle\begin{split}\Delta_{(2vo)}p_{1}^{\mu}&=\frac{(q_{1}q_{2})^{2}}{m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)(i\ell^{\mu})}{\ell^{2}k^{2}(ik\cdot
v_{1}+0^{+})^{2}}\left(\mathrm{ch}\\!\boxminus\,C_{vo}+\mathrm{sh}\\!\boxminus\,S_{vo}\right)\,,\\\
&\quad C_{vo}=-(\gamma^{2}-\textstyle{\frac{1}{2}})(k\cdot\ell)+(k\cdot
v_{1})(\ell\cdot v_{1})\,,\quad
S_{vo}=i\gamma\,\epsilon[k,l,v_{1},v_{2}]\,.\end{split}$ (113)

The same helicity part is quite simple:

$\displaystyle\Delta_{(2vs)}p_{1}^{\mu}$
$\displaystyle=\frac{(q_{1}q_{2})^{2}}{m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)(i\ell^{\mu})}{\ell^{2}k^{2}(ik\cdot
v_{1}+0^{+})^{2}}(-\textstyle{\frac{1}{2}})(k\cdot\ell)\mathrm{ch}\\!\boxplus\,.$
(114)

In the non-spinning limit $(y\rightarrow 0)$, it cancels against the
$+\frac{1}{2}(k\cdot\ell)$ term of $C_{vo}$ in (113), in agreement with (73).

##### 2PL eikonal

As in the non-spinning example, we can extract the 2PL eikonal from the 2PL
momentum kick. The key idea Kosower:2018adc  is to apply the exchange
$\ell\leftrightarrow k=q-\ell$ to the integrand,

$\displaystyle\begin{split}\Delta_{(2)}p_{1}^{\mu}=\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}(i\ell^{\mu})\,\mathcal{J}(k,\ell)=\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}(iq^{\mu}-i\ell^{\mu})\,\mathcal{J}(\ell,k)\,.\end{split}$ (115)

Since $q\cdot v_{1}=k\cdot v_{1}+\ell\cdot v_{1}=0$, the exchange flips the
$i0^{+}$ prescription for the worldline Green function. Taking the average of
the two expressions and taking the term proportional to $q^{\mu}$ in the
integrand, we separate the transverse part of the momentum kick, from which we
read off the 2PL eikonal equipped with the time-symmetric $i0^{+}$,

$\displaystyle\begin{split}\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{tr}}&=\frac{1}{2}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}(iq^{\mu})\,\mathcal{J}(\ell,k)=\\{\chi_{(2)},p_{1}^{\mu}\\}=\eta^{\mu\nu}\frac{\partial}{\partial
x^{\nu}_{1}}\chi_{(2)}\\\
&\quad\Longrightarrow\quad\chi_{(2)}=\frac{1}{2}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\mathcal{J}(\ell,k)\,.\end{split}$ (116)

The remaining $\ell^{\mu}$ terms in the integrand should be matched against
the iteration term,

$\displaystyle\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{iter}}=\frac{1}{2}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}(i\ell^{\mu})\,\left[\mathcal{J}(k,\ell)-\mathcal{J}(\ell,k)\right]=\frac{1}{2}\\{\chi_{(1)},\\{\chi_{(1)},p_{1}^{\mu}\\}\\}\,.$
(117)

We have computed $\Delta_{(2)}p_{1}^{\mu}$ from the EOM. It is straightforward
to split it into $\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{tr}}$ and
$\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{iter}}$, and then read off
$\chi_{(2)}$ from $\left.\Delta_{(2)}p_{1}^{\mu}\right|_{\mathrm{tr}}$. The
final result for $\chi_{(2)}$ is

$\displaystyle\begin{split}\chi_{(2)}&=\chi_{(2)1}+\chi_{(2)2}\,,\\\
\chi_{(2)1}&=-\frac{(q_{1}q_{2})^{2}}{2m_{1}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{k^{2}\ell^{2}(\ell\cdot
v_{1})^{2}}\left[\mathcal{V}_{1}+\mathcal{Y}_{1}\right]\,,\\\
\mathcal{V}_{1}&=\left[-(\gamma^{2}-\textstyle{\frac{1}{2}})(k\cdot\ell)+(k\cdot
v_{1})(\ell\cdot v_{1})\right]\cosh[(k-\ell)\cdot y]\\\
&\quad-\textstyle{\frac{1}{2}}(k\cdot\ell)\cosh[(k+\ell)\cdot
y]+i\gamma\,\epsilon[k,\ell,v_{1},v_{2}]\sinh[(k-\ell)\cdot y]\,,\\\
\mathcal{Y}_{1}&=-i(\ell\cdot
v_{1})\left[-\epsilon[k,\ell,y_{1},v_{1}]+2(v_{2}\cdot
y_{1})\epsilon[k,\ell,v_{1},v_{2}]\right]\cosh[(k-\ell)\cdot y]\\\
&\quad+(\ell\cdot v_{1})\left[(\ell\cdot v_{1})[(k-\ell)\cdot
y_{1}]+2\gamma(v_{2}\cdot y_{1})(k\cdot\ell)\right]\sinh[(k-\ell)\cdot
y]\,.\end{split}$ (118)

The other half of the answer, $\chi_{(2)2}$, can be obtained from
$\chi_{(2)1}$ by the exchange of particle labels $(1\leftrightarrow 2)$.
Verifying the iteration relation (117) is also straightforward, but involves a
lengthy computation; see appendix C for details.

##### 2PL spin kick

To complete the 2PL story, let us compute the 2PL spin kick from the EOM and
confirm that it can be reproduced by the 2PL eikonal we obtained earlier. We
begin with a copy of the 2PL EOM for spin from (107),

$\displaystyle\begin{split}\dot{y}_{(2)}^{\mu}&=\frac{q}{m}(F^{+}_{(0)}+F^{-}_{(0)})^{\mu}{}_{\nu}y_{(1)}^{\nu}+\frac{q}{m}(F^{+}_{(1)}+F^{-}_{(1)})^{\mu}{}_{\nu}y_{(0)}^{\nu}\\\
&\quad-i\frac{4q^{2}}{m^{2}}\left[(vF^{-}y)(F^{+})^{\mu}{}_{\nu}-(vF^{+}y)(F^{-})^{\mu}{}_{\nu}\right]_{(0)}y_{(0)}^{\nu}\,.\end{split}$
(119)

Using the 1PL EOM, we turn the terms on the RHS to integrals,

$\displaystyle\begin{split}\Delta_{(2)}y^{\mu}&=\Delta_{(2a)}y^{\mu}+\Delta_{(2b)}y^{\mu}+\Delta_{(2c)}y^{\mu}+\Delta_{(2d)}y^{\mu}\,,\\\
\Delta_{(2a)}y^{\mu}&=\frac{q^{2}}{m^{2}}\int_{-\infty}^{\infty}d\sigma(F^{+}+F^{-})^{\mu}{}_{\nu}\int_{-\infty}^{\sigma}d\sigma^{\prime}(F^{+}+F^{-})^{\nu}{}_{\lambda}y_{(0)}^{\lambda}\,,\\\
\Delta_{(2b)}y^{\mu}&=\frac{q^{2}}{m^{2}}\int_{-\infty}^{\infty}d\sigma\partial_{\lambda}(F^{+}+F^{-})^{\mu}{}_{\nu}y_{(0)}^{\nu}\left[\int_{-\infty}^{\sigma}d\sigma^{\prime}\int_{-\infty}^{\sigma^{\prime}}d\sigma^{\prime\prime}(F^{+}+F^{-})^{\lambda}{}_{\rho}v_{(0)}^{\rho}\right]\,,\\\
\Delta_{(2c)}y^{\mu}&=\frac{2iq^{2}}{m^{2}}\int_{-\infty}^{\infty}d\sigma\partial_{\lambda}(F^{+})^{\mu}{}_{\nu}y_{(0)}^{\nu}\int_{-\infty}^{\sigma}d\sigma^{\prime}(F^{-})^{\lambda}{}_{\rho}y_{(0)}^{\rho}\\\
&\quad-\frac{2iq^{2}}{m^{2}}\int_{-\infty}^{\infty}d\sigma\partial_{\lambda}(F^{-})^{\mu}{}_{\nu}y_{(0)}^{\nu}\int_{-\infty}^{\sigma}d\sigma^{\prime}(F^{+})^{\lambda}{}_{\rho}y_{(0)}^{\rho}\,,\\\
\Delta_{(2d)}y^{\mu}&=-i\frac{4q^{2}}{m^{2}}\int_{-\infty}^{\infty}d\sigma\left[(vF^{-}y)(F^{+})^{\mu}{}_{\nu}-(vF^{+}y)(F^{-})^{\mu}{}_{\nu}\right]_{(0)}y_{(0)}^{\nu}\,.\end{split}$
(120)

Both (b) and (c) terms come from the second term on the RHS of (119).

Again, it suffices to work in the probe limit; we keep using the Fourier
integral (96) of the field-strength produced by a fixed source. After
worldline time integrals, we reach an expression of the form

$\displaystyle\Delta_{(2)}y_{1}^{\mu}=\frac{(q_{1}q_{2})^{2}}{m_{1}^{2}}\int_{q_{\perp}}e^{iq\cdot
b}\int_{\ell}\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(v_{2}\cdot\ell)}{k^{2}\ell^{2}(ik\cdot
v_{1}+0^{+})^{2}}\mathcal{N}^{\mu}\,.$ (121)

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.

$\displaystyle\begin{split}\mathcal{N}^{\mu}_{(2ao)}=\left(\mathrm{ch}\\!\boxminus
C_{ao}+\mathrm{sh}\\!\boxminus
S_{ao}\right)^{\mu}\,,&\quad\mathcal{N}^{\mu}_{(2as)}=\left(\mathrm{ch}\\!\boxplus
C_{as}+\mathrm{sh}\\!\boxplus S_{as}\right)^{\mu}\,,\\\
\mathcal{N}^{\mu}_{(2bo)}=\left(\mathrm{ch}\\!\boxminus
C_{bo}+\mathrm{sh}\\!\boxminus
S_{bo}\right)^{\mu}\,,&\quad\mathcal{N}^{\mu}_{(2bs)}=\left(\mathrm{ch}\\!\boxplus
C_{bs}+\mathrm{sh}\\!\boxplus S_{bs}\right)^{\mu}\,,\\\
\mathcal{N}^{\mu}_{(2c)}=\left(\mathrm{ch}\\!\boxminus
C_{c}+\mathrm{sh}\\!\boxminus
S_{c}\right)^{\mu}\,,&\quad\mathcal{N}^{\mu}_{(2d)}=\left(\mathrm{ch}\\!\boxminus
C_{d}+\mathrm{sh}\\!\boxminus S_{d}\right)^{\mu}\,,\end{split}$ (122)

It is straightforward to evaluate all $C$, $S$ functions in (122). The
intermediate steps involve many terms, but after some cancellations, the final
results are often quite simple. For instance, the same helicity sector gives

$\displaystyle\begin{split}C^{\mu}_{as}+C^{\mu}_{bs}&=-\frac{i}{2}(k\cdot\ell)\left[v^{\mu}_{1}(\ell\cdot
y_{1})-y^{\mu}_{1}(\ell\cdot v_{1})\right]\,,\\\
S^{\mu}_{as}+S^{\mu}_{bs}&=-\frac{1}{2}(k\cdot\ell)\epsilon^{\mu}[v_{1},y_{1},\ell]\,.\end{split}$
(123)

After collecting and simplifying the terms, using the symmetrisation of the
$i0^{+}$ prescription, we extract the transverse part of the spin kick and
check whether the result agrees with

$\displaystyle\begin{split}\\{\chi_{(n)},y_{1}^{\mu}\\}&=\frac{1}{m_{1}}\left[v_{1}^{\mu}y_{1}^{\nu}\frac{\partial}{\partial
x_{1}^{\nu}}+\epsilon^{\mu\nu}[v_{1},y_{1}]\frac{\partial}{\partial
y_{1}^{\nu}}\right]\chi_{(n)}\,.\end{split}$ (124)

Again, the procedure is straightforward, and we confirm the agreement, but the
calculations tend to be lengthy. We give some details in appendix C.2.

##  4 Twistor WQFT

Worldline Quantum Field Theory (WQFT) Mogull:2020sak  is a means of organising
classical equations of motion in a diagrammatic way that resembles Feynman
diagrams of particle physics. The simplest WQFT model action consists of two
parts; the bulk part which describes field degrees of freedom (DOFs) on the
background spacetime, and one-dimensional sigma model that takes the
background spacetime as the target space, where the latter is interpreted as
the worldline action of a point particle moving on the background spacetime.
The DOFs are decomposed into the background value (which satisfies the
classical equations of motion) and fluctuations from the background value, and
diagrammatic techniques developed for perturbative QFT calculations are
applied to the field fluctuations. The background value for the background
spacetime is usually taken to be the flat Minkowski spacetime, and the
background value for the worldline is usually taken to be the straight
trajectory of a free particle; $x^{\mu}(\sigma)=b^{\mu}+v^{\mu}\sigma+\delta
x^{\mu}(\sigma)$. The fluctuation DOFs are evaluated as a perturbative series
of the coupling constant, which in the gravitational case is taken to be the
Newton’s constant $G$.

###  4.1 Worldline Feynman rules

When we apply the WQFT method to our twistor model, one novelty is that the
twistor variables $(\lambda,\bar{\lambda},\mu,\bar{\mu})$ are the fundamental
variables, and the target space of the worldline is the twistor space
indirectly related to the background spacetime by the incidence relations. It
is natural to express the propagators and vertex factors in terms of the
twistor variables. But, it is often convenient to use the vector variables
$(z,\bar{z})$ in intermediate steps. We will use the incidence relations to
switch between the twistor variables (“twistor picture”) and the vector
variables (“spacetime picture”) whenever necessary.

##### Classical limit

When $\hbar$ is restored, the fundamental variables have the dimensions

$\displaystyle[\lambda_{\alpha}{}^{I}]=[\bar{\lambda}_{I\dot{\alpha}}]=[M^{1/2}]\,,\,[\mu^{\dot{\alpha}I}]=[\bar{\mu}_{I}{}^{\alpha}]=[M^{1/2}L]\,,\,[A_{\mu}]=[M^{1/2}L^{-1/2}]\,,$
(125)

and the coupling becomes dimensionful: $[q]=[M^{1/2}L^{1/2}]$. The action also
becomes $iS\to iS/\hbar$, therefore the interaction vertices are weighted by
$\hbar^{-1}$ while the propagators (both $\langle\lambda\bar{\mu}\rangle$ type
and $\langle\mu\mu\rangle$ type) are weighted by $\hbar$. We have no other
$\hbar$ scaling if we only use frequency $\omega$ and wavenumber four-vector
$k^{\mu}$ in momentum space, which is allowed because there is no “mass” in
any of the fluctuation fields. Therefore, at a given $q$ order (which
determines the number of vertices coupling to the photon field), the classical
contribution is determined from the diagrams with the least number of
propagators that makes the diagram connected, which is equivalent to the
statement that tree diagrams determine the classical physics. The tree
diagrams, however, generally have momentum integrals similar to loop integrals
of quantum field theory.

####  4.1.1 Background-fluctuation expansion

In the non-spinning WQFT, the expansion around a straight line trajectory is
done by

$\displaystyle x^{\mu}(\sigma)=b_{0}^{\mu}+v^{\mu}\sigma+\delta
x^{\mu}(\sigma)\,,\quad v^{2}=-1\,,\quad b_{0}\cdot v=0\,.$ (126)

After the NJ-shift, the expansion is generalised as

$\displaystyle z^{\mu}(\sigma)=b_{0}^{\mu}+iy_{0}^{\mu}+v^{\mu}\sigma+\delta
z^{\mu}(\sigma)\,,\quad y_{0}\cdot v=0\,.$ (127)

We should rephrase the background-fluctuation expansion in terms of twistor
variables. As we explained in section 2.1.2, we may assume a flat Regge
trajectory $m^{\prime}=0$. The resulting equations of motion for the free
action (9) are

$\displaystyle\begin{aligned} \frac{\delta
S_{\text{free}}}{\delta\bar{\mu}_{I}{}^{\alpha}}&=0=-\frac{d\lambda_{\alpha}{}^{I}}{d\sigma}+\frac{\kappa^{1}}{2i}{\lambda}_{\alpha}{}^{I}\\\
\frac{\delta
S_{\text{free}}}{\delta{\mu}^{\dot{\alpha}I}}&=0=-\frac{d\bar{\lambda}_{I\dot{\alpha}}}{d\sigma}-\frac{\kappa^{1}}{2i}{\bar{\lambda}}_{I\dot{\alpha}}\\\
\frac{\delta
S_{\text{free}}}{\delta{\lambda}_{\alpha}{}^{I}}&=0=\frac{d\bar{\mu}_{I}{}^{\alpha}}{d\sigma}+\frac{\kappa^{0}\Delta\bar{\Delta}}{2}(\lambda^{-1})_{I}{}^{\alpha}+\frac{\kappa^{1}}{2i}\bar{\mu}_{I}{}^{\alpha}\\\
\frac{\delta
S_{\text{free}}}{\delta\bar{\lambda}_{I\dot{\alpha}}}&=0=\frac{d{\mu}^{\dot{\alpha}I}}{d\sigma}+\frac{\kappa^{0}\Delta\bar{\Delta}}{2}(\bar{\lambda}^{-1})^{\dot{\alpha}I}-\frac{\kappa^{1}}{2i}{\mu}^{\dot{\alpha}I}\,.\end{aligned}$
(128)

Just like in the previous section, we fix the Lagrange multipliers as
$\kappa^{0}=1/m$ and $\kappa^{1}=0$. We introduce the background values
$\ell_{\alpha}{}^{I}$ and $\bar{\ell}_{I\dot{\alpha}}$ satisfying the
conditions

$\displaystyle{\rm det}(\ell)={\rm
det}(\bar{\ell})=m\,,\quad\ell_{\alpha}{}^{I}\bar{\ell}_{I\dot{\alpha}}=-mv_{\alpha\dot{\alpha}}\,,$
(129)

where $v^{\mu}$ is the normalised velocity vector introduced in (127). The
following relations satisfied by inverse matrices are useful in calculations.

$\displaystyle(\ell^{-1})_{I}{}^{\alpha}=-\frac{\epsilon_{IJ}\epsilon^{\alpha\beta}\ell_{\beta}{}^{J}}{{\rm
det}(\ell)}\,,\quad(\bar{\ell}^{-1})^{\dot{\alpha}I}=\frac{\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{IJ}\bar{\ell}_{J\dot{\beta}}}{{\rm
det}(\bar{\ell})}\,,\quad(\bar{\ell}^{-1})^{\dot{\alpha}I}(\ell^{-1})_{I}{}^{\alpha}=-\frac{v^{\dot{\alpha}\alpha}}{m}\,.$
(130)

The twistor variables are expanded as

$\displaystyle\begin{aligned}
\lambda_{\alpha}{}^{I}&\to\ell_{\alpha}{}^{I}+\lambda_{\alpha}{}^{I}(\sigma)\,,\\\
\bar{\lambda}_{I\dot{\alpha}}&\to\bar{\ell}_{I\dot{\alpha}}+\bar{\lambda}_{I\dot{\alpha}}(\sigma)\,,\\\
\mu^{\dot{\alpha}I}&\to
c^{\dot{\alpha}I}-\frac{m}{2}(\bar{\ell}^{-1})^{\dot{\alpha}I}\sigma+\mu^{\dot{\alpha}I}(\sigma)\,,\\\
\bar{\mu}_{I}{}^{\alpha}&\to\bar{c}_{I}{}^{\alpha}-\frac{m}{2}(\ell^{-1})_{I}{}^{\alpha}\sigma+\bar{\mu}_{I}{}^{\alpha}(\sigma)\,.\end{aligned}$
(131)

The background-fluctuation expansion for $z^{\mu}$ and $\bar{z}^{\mu}$ can be
determined from the incidence relations (274),

$\displaystyle\begin{aligned}
z^{\dot{\alpha}\alpha}&\to+2c^{\dot{\alpha}I}(\ell^{-1})_{I}{}^{\alpha}-m(\bar{\ell}^{-1})^{\dot{\alpha}I}(\ell^{-1})_{I}{}^{\alpha}\sigma+\delta
z^{\dot{\alpha}\alpha}(\sigma)=z_{0}^{\dot{\alpha}\alpha}+v^{\dot{\alpha}\alpha}\sigma+\delta
z^{\dot{\alpha}\alpha}(\sigma)\,,\\\
\bar{z}^{\dot{\alpha}\alpha}&\to+2(\bar{\ell}^{-1})^{\dot{\alpha}I}\bar{c}_{I}{}^{\alpha}-m(\bar{\ell}^{-1})^{\dot{\alpha}I}(\ell^{-1})_{I}{}^{\alpha}\sigma+\delta\bar{z}^{\dot{\alpha}\alpha}(\sigma)=\bar{z}_{0}^{\dot{\alpha}\alpha}+v^{\dot{\alpha}\alpha}\sigma+\delta\bar{z}^{\dot{\alpha}\alpha}(\sigma)\,.\end{aligned}$
(132)

The relation between the fluctuation fields are determined from the incidence
relations.

$\displaystyle\begin{aligned}
2\mu^{\dot{\alpha}I}(\sigma)&=\left(z_{0}^{\dot{\alpha}\alpha}+v^{\dot{\alpha}\alpha}\sigma\right)\lambda_{\alpha}{}^{I}(\sigma)+\delta
z^{\dot{\alpha}\alpha}(\sigma)\ell_{\alpha}{}^{I}+\delta
z^{\dot{\alpha}\alpha}(\sigma)\lambda_{\alpha}{}^{I}(\sigma)\,,\\\
2\bar{\mu}_{I}{}^{\alpha}(\sigma)&=\bar{\lambda}_{I\dot{\alpha}}(\sigma)\left(\bar{z}_{0}^{\dot{\alpha}\alpha}+v^{\dot{\alpha}\alpha}\sigma\right)+\bar{\ell}_{I\dot{\alpha}}\delta\bar{z}^{\dot{\alpha}\alpha}(\sigma)+\bar{\lambda}_{I\dot{\alpha}}(\sigma)\delta\bar{z}^{\dot{\alpha}\alpha}(\sigma)\,.\end{aligned}$
(133)

We use the positive frequency expansion,

$\displaystyle\phi(x)=\int_{k}\phi(k)e^{ikx}\,,\quad
f(\sigma)=\int_{\omega}f(\omega)e^{-i\omega\sigma}\,,$ (134)

which relates the frequency space coefficients to annihilation modes and
incoming momenta. The frequency space expression for the incidence relation
turns out to be more useful

$\displaystyle\begin{aligned} \delta
z^{\dot{\alpha}\alpha}(\omega)&=\left[2\mu^{\dot{\alpha}I}-z_{0}^{\dot{\alpha}\beta}\lambda_{\beta}{}^{I}+iv^{\dot{\alpha}\beta}\frac{\partial\lambda_{\beta}{}^{I}}{\partial\omega}-\int_{\omega^{\prime}}\delta
z^{\dot{\alpha}\beta}(\omega^{\prime})\lambda_{\beta}{}^{I}(\omega-\omega^{\prime})\right](\ell^{-1})_{I}{}^{\alpha}\,,\\\
\delta\bar{z}^{\dot{\alpha}\alpha}(\omega)&=(\bar{\ell}^{-1})^{\dot{\alpha}I}\left[2\bar{\mu}_{I}{}^{\alpha}-\bar{\lambda}_{I\dot{\beta}}\bar{z}_{0}^{\dot{\beta}\alpha}+i\frac{\partial\bar{\lambda}_{I\dot{\beta}}}{\partial\omega}v^{\dot{\beta}\alpha}-\int_{\omega^{\prime}}\bar{\lambda}_{I\dot{\beta}}(\omega-\omega^{\prime})\delta\bar{z}^{\dot{\beta}\alpha}(\omega^{\prime})\right]\,,\end{aligned}$
(135)

where we have suppressed the $\omega$ dependence whenever it is obvious.

The free action in terms of the fluctuation fields becomes

$\displaystyle\begin{aligned}
S_{\text{free}}&=\int\left[\lambda_{\alpha}{}^{I}d\bar{\mu}_{I}{}^{\alpha}+\bar{\lambda}_{I\dot{\alpha}}d\mu^{\dot{\alpha}I}+\frac{{\rm
det}(\lambda)+{\rm
det}(\bar{\lambda})+m(\ell^{-1})^{\alpha}{}_{I}\lambda_{\alpha}{}^{I}(\bar{\ell}^{-1})^{\dot{\alpha}J}\bar{\lambda}_{J\dot{\alpha}}}{2}d\sigma\right.\\\
&\phantom{=}\left.\phantom{\frac{asdf}{2}}+\frac{{\rm
det}(\lambda)(\bar{\ell}^{-1})^{\dot{\alpha}I}\bar{\lambda}_{I\dot{\alpha}}+(\ell^{-1})_{I}{}^{\alpha}\lambda_{\alpha}{}^{I}{\rm
det}(\bar{\lambda})}{2}d\sigma+\frac{{\rm det}(\lambda){\rm
det}(\bar{\lambda})}{2m}d\sigma+\cdots\right]\,.\end{aligned}$ (136)

where the ellipsis denotes constant and total derivative terms irrelevant for
Feynman rules. The first line determines the 2pt functions, while the second
line generates cubic and quartic vertices.

####  4.1.2 Twistor propagators

The quadratic part of the free action in frequency space can be written as

$\displaystyle\begin{aligned}
iS_{2}&=\frac{i}{2}\int_{\omega,\omega^{\prime}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\begin{pmatrix}\lambda_{\alpha}{}^{I}(\omega^{\prime})&\bar{\mu}_{I}{}^{\alpha}(\omega^{\prime})&\bar{\lambda}_{I\dot{\alpha}}(\omega^{\prime})&{\mu}^{\dot{\alpha}I}(\omega^{\prime})\end{pmatrix}\\\
&\phantom{=a}\times\begin{pmatrix}-\frac{1}{2}\epsilon^{\alpha\beta}\epsilon_{IJ}&-i\omega\delta_{I}^{J}\delta^{\alpha}_{\beta}&\frac{m}{2}(\ell^{-1})_{I}{}^{\alpha}(\bar{\ell}^{-1})^{\dot{\beta}J}&0\\\
+i\omega\delta^{I}_{J}\delta_{\alpha}^{\beta}&0&0&0\\\
\frac{m}{2}(\bar{\ell}^{-1})^{\dot{\alpha}I}(\ell^{-1})_{J}{}^{\beta}&0&+\frac{1}{2}\epsilon^{IJ}\epsilon^{\dot{\alpha}\dot{\beta}}&-i\omega\delta^{I}_{J}\delta^{\dot{\alpha}}_{\dot{\beta}}\\\
0&0&+i\omega\delta_{I}^{J}\delta_{\dot{\alpha}}^{\dot{\beta}}&0\end{pmatrix}\begin{pmatrix}\lambda_{\beta}{}^{J}(\omega)\\\
\bar{\mu}_{J}{}^{\beta}(\omega)\\\ \bar{\lambda}_{J\dot{\beta}}(\omega)\\\
{\mu}^{\dot{\beta}J}(\omega)\end{pmatrix}\,,\end{aligned}$ (137)

where we used the delta support to convert $\omega^{\prime}\to-\omega$.
Inverting the quadratic action leads to the twistor propagators in the
straight line background:

$\displaystyle\begin{aligned}
\langle\lambda_{\alpha}^{~{}I}(\omega^{\prime})\bar{\mu}_{J}^{~{}\beta}(\omega)\rangle&=-\frac{\delta_{\alpha}^{\beta}\delta_{J}^{I}}{\omega}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\quad\langle\bar{\mu}_{I}^{~{}\alpha}(\omega^{\prime})\bar{\mu}_{J}^{~{}\beta}(\omega)\rangle=+\frac{i}{2}\frac{\epsilon_{IJ}\epsilon^{\alpha\beta}}{\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle\bar{\lambda}_{I\dot{\alpha}}(\omega^{\prime}){\mu}^{\dot{\beta}J}(\omega)\rangle&=-\frac{\delta_{\dot{\alpha}}^{\dot{\beta}}\delta_{I}^{J}}{\omega}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\quad\langle{\mu}^{\dot{\alpha}I}(\omega^{\prime}){\mu}^{\dot{\beta}J}(\omega)\rangle=-\frac{i}{2}\frac{\epsilon^{IJ}\epsilon^{\dot{\alpha}\dot{\beta}}}{\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle{\mu}^{\dot{\alpha}I}(\omega^{\prime})\bar{\mu}_{J}^{~{}\beta}(\omega)\rangle&=-\frac{im}{2}\frac{(\bar{\ell}^{-1})^{\dot{\alpha}I}(\ell^{-1})^{\beta}{}_{J}}{\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,.\end{aligned}$ (138)

An $i0^{+}$ prescription is needed to determine the causality flow of the 2pt
functions; e.g. for $\omega\to\omega+i0^{+}$ causality flows from
$\omega^{\prime}$ to $\omega$. Otherwise stated, we use time-symmetric
$i0^{+}$ prescription in the calculations, which is the prescription relevant
for computing the eikonal Jakobsen:2021zvh . We also remark that all position
type 2pt functions (i.e. $\langle\mu\mu\rangle$,
$\langle\bar{\mu}\bar{\mu}\rangle$, and $\langle\mu\bar{\mu}\rangle$) should
be understood as

$\displaystyle\begin{aligned}
\langle\bar{\mu}_{I}^{~{}\alpha}(\omega^{\prime})\bar{\mu}_{J}^{~{}\beta}(\omega)\rangle&=+\frac{i\kappa^{0}{\rm
det}(\bar{\ell})}{2}\frac{\epsilon_{IJ}\epsilon^{\alpha\beta}}{\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle{\mu}^{\dot{\alpha}I}(\omega^{\prime}){\mu}^{\dot{\beta}J}(\omega)\rangle&=-\frac{i\kappa^{0}{\rm
det}(\ell)}{2}\frac{\epsilon^{IJ}\epsilon^{\dot{\alpha}\dot{\beta}}}{\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle{\mu}^{\dot{\alpha}I}(\omega^{\prime})\bar{\mu}_{J}^{~{}\beta}(\omega)\rangle&=-\frac{i\kappa^{0}{\rm
det}(\ell){\rm
det}(\bar{\ell})}{2}\frac{(\bar{\ell}^{-1})^{\dot{\alpha}I}(\ell^{-1})^{\beta}{}_{J}}{\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\end{aligned}$ (139)

when we remove the gauge-fixing condition $\kappa^{0}=\frac{1}{m}$ and
background value determinant conditions ${\rm det}(\ell)={\rm
det}(\bar{\ell})=m$. This will become relevant in discussions of causality
cuts applied to the twistor model Kim:causality-cut .

Let us introduce a graphical notation for the propagators (138). Without the
mass-shell constraint, the worldline propagator would be simply

$\fmffile{worldline-propagator-unconstrained}\parbox{80.0pt}{
\fmfgraph*(50,50) \fmfstraight\fmfbottom{i1,o1} \fmftop{i2,o2}
\fmfv{decor.shape=circle,decor.filled=empty,decor.size=10,label=$Z_{A}{}^{I}\;$,label.angle=180}{v2}
\fmfv{decor.shape=circle,decor.filled=shaded,decor.size=10,label=$\bar{Z}_{J}{}^{B}\;$,label.angle=180}{v1}
\fmf{phantom}{i1,v1,o1} \fmf{phantom}{i2,v2,o2} \fmf{plain,tension=0.4}{v1,v2}
}\hskip-42.67912pt=\;\;-\frac{\delta_{A}{}^{B}\delta_{J}{}^{I}}{\omega}\,,$
(140)

which includes $\langle\lambda\bar{\mu}\rangle$ and
$\langle\bar{\lambda}\mu\rangle$ propagators. The mass-shell constraint
introduces additional propagators. We denote the $\langle\mu\mu\rangle$,
$\langle\bar{\mu}\bar{\mu}\rangle$ propagators as

$\fmffile{worldline-propagator-constrained}\parbox{80.0pt}{ \fmfgraph*(50,50)
\fmfstraight\fmfbottom{i1,o1} \fmftop{i2,o2}
\fmfv{decor.shape=circle,decor.filled=empty,decor.size=10,label=$\mu^{{\dot{\alpha}}I}\;$,label.angle=180}{a2}
\fmfv{decor.shape=circle,decor.filled=empty,decor.size=10,label=$\mu^{{\dot{\beta}}J}\;$,label.angle=180}{a1}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=5}{t0}
\fmf{phantom}{i1,a1,o1} \fmf{phantom}{i2,a2,o2}
\fmf{plain,tension=0.4}{a1,t0,a2}
}\hskip-42.67912pt=\;\;-\frac{i}{2}\frac{\epsilon^{IJ}\epsilon^{{\dot{\alpha}}{\dot{\beta}}}}{\omega^{2}}\,,\hskip
28.45274pt\parbox{80.0pt}{ \fmfgraph*(50,50) \fmfstraight\fmfbottom{i1,o1}
\fmftop{i2,o2}
\fmfv{decor.shape=circle,decor.filled=shaded,decor.size=10,label=$\bar{\mu}_{I}{}^{\alpha}\;$,label.angle=180}{a2}
\fmfv{decor.shape=circle,decor.filled=shaded,decor.size=10,label=$\bar{\mu}_{J}{}^{\beta}\;$,label.angle=180}{a1}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=5}{t0}
\fmf{phantom}{i1,a1,o1} \fmf{phantom}{i2,a2,o2}
\fmf{plain,tension=0.4}{a1,t0,a2}
}\hskip-42.67912pt=\;\;+\frac{i}{2}\frac{\epsilon_{IJ}\epsilon^{\alpha\beta}}{\omega^{2}}\,.$
(141)

The black dots in the middle remind us of the fact that these propagators
originate from the $(\bar{\lambda}\bar{\lambda})$ and $(\lambda\lambda)$
vertices in the Lagrangian.444This is a valid interpretation of the 2pt
functions; we may only regard (140) as the fundamental 2pt functions and
understand the position type 2pt functions, (141) and (142), as insertions of
2pt vertices between products of fundamental 2pt functions. See appendix D for
regularisation of the divergences related to the symmetric $i0^{+}$
prescription of the 2pt functions. Finally, we denote the
$\langle\mu\bar{\mu}\rangle$ propagator by

$\fmffile{worldline-propagator-last}\parbox{80.0pt}{ \fmfgraph*(50,50)
\fmfstraight\fmfbottom{i1,o1} \fmftop{i2,o2}
\fmfv{decor.shape=circle,decor.filled=empty,decor.size=10,label=$\mu^{{\dot{\alpha}}I}\;$,label.angle=180}{a2}
\fmfv{decor.shape=circle,decor.filled=shaded,decor.size=10,label=$\bar{\mu}_{J}{}^{\beta}\;$,label.angle=180}{a1}
\fmfv{decor.shape=square,decor.filled=full,decor.size=5}{t0}
\fmf{phantom}{i1,a1,o1} \fmf{phantom}{i2,a2,o2}
\fmf{plain,tension=0.4}{a1,t0,a2}
}\hskip-42.67912pt=\;\;-\frac{im}{2}\frac{(\bar{\ell}^{-1})^{\dot{\alpha}I}(\ell^{-1})^{\beta}{}_{J}}{\omega^{2}}\,.$
(142)

The black square in the middle is to show that this propagator comes from the
$m(\ell^{-1})(\bar{\ell}^{-1})$ vertex in the Lagrangian.

####  4.1.3 Vector 2-point functions

Turning to the “spacetime picture” where we organize diagrams in terms of
${\delta}z$, ${\delta}\bar{z}$, the following 2pt functions will play a
crucial role.

$\displaystyle\begin{aligned} \langle\delta
z^{\dot{\alpha}\alpha}(\omega^{\prime})\delta
z^{\dot{\beta}\beta}(\omega)\rangle&=-\frac{2i\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{m\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega^{\prime})\delta\bar{z}^{\dot{\beta}\beta}(\omega)\rangle&=-\frac{2i\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{m\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega^{\prime})\delta
z^{\dot{\beta}\beta}(\omega)\rangle&=-\frac{2}{m}\left[\left(\frac{iv^{\dot{\alpha}\alpha}v^{\dot{\beta}\beta}}{\omega^{2}}+\frac{v^{\dot{\alpha}\beta}z_{0}^{\dot{\beta}\alpha}}{\omega^{\prime}}+\frac{v^{\dot{\alpha}\beta}\bar{z}_{0}^{\dot{\beta}\alpha}}{\omega}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\right.\\\
&\phantom{=}\left.\phantom{\frac{v^{\dot{\alpha}\beta}z_{0}^{\dot{\beta}\alpha}}{\omega^{\prime}}}-iv^{\dot{\alpha}\beta}v^{\dot{\beta}\alpha}\left(\frac{1}{\omega^{\prime}}+\frac{1}{\omega}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(\omega^{\prime}+\omega)\right]\,,\end{aligned}$ (143)

where $\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(x)=\frac{d}{dx}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(x)$ is the delta derivative. Note that terms proportional to
$\omega^{-2}$ in (143) are contributions from the 2pt functions given in
(139). We have neglected the loop contribution to the
$\langle\delta\bar{z}\delta z\rangle$ 2pt function,

$\displaystyle\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega^{\prime})\delta
z^{\dot{\beta}\beta}(\omega)\rangle\supset-\frac{4v^{\dot{\alpha}\alpha}v^{\dot{\beta}\beta}}{m^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\int_{\omega_{1}}\frac{1}{\omega_{1}(\omega-\omega_{1})}\to
0\,,$ (144)

based on two reasons. First, $\hbar$ counting from dimensional analysis
requires an extra $\hbar$ factor for this loop contribution compared to tree
contributions given in (143). Second, the loop integral evaluates to zero if
we assume invariance under shifts of the integration variable:555Although
widely used in dimensional regularisation, this is not a trivial assumption;
for example, ABJ anomalies evaluate to zero under this assumption for
(divergent) loop integrals.

$\displaystyle\int_{\omega^{\prime}}\frac{1}{\omega^{\prime}(\omega-\omega^{\prime})}=\frac{1}{\omega}\int_{\omega^{\prime}}\left[\frac{1}{\omega^{\prime}}-\frac{1}{\omega^{\prime}-\omega}\right]\,.$
(145)

The delta derivative contribution can be simplified by

$\displaystyle\begin{aligned}
\left(\frac{1}{\omega^{\prime}}+\frac{1}{\omega}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt^{\prime}(\omega^{\prime}+\omega)&=\frac{\partial}{\partial\omega}\left[\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)}{\omega^{\prime}}\right]+\frac{\partial}{\partial\omega^{\prime}}\left[\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)}{\omega}\right]\\\
&=\frac{\partial}{\partial\omega}\left[-\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)}{\omega}\right]+\frac{\partial}{\partial\omega^{\prime}}\left[\frac{\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)}{\omega}\right]\\\
&=\frac{1}{\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\end{aligned}$ (146)

leading to

$\displaystyle\begin{aligned}
\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega^{\prime})\delta
z^{\dot{\beta}\beta}(\omega)\rangle&=-\frac{2}{m}\left[\frac{i(v^{\dot{\alpha}\alpha}v^{\dot{\beta}\beta}-v^{\dot{\alpha}\beta}v^{\dot{\beta}\alpha})}{\omega^{2}}+\frac{v^{\dot{\alpha}\beta}(\bar{z}_{0}^{\dot{\beta}\alpha}-z_{0}^{\dot{\beta}\alpha})}{\omega}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\\\
&=-\frac{2}{m}\left[\frac{i\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{\omega^{2}}+\frac{v^{\dot{\alpha}\beta}(\bar{z}_{0}^{\dot{\beta}\alpha}-z_{0}^{\dot{\beta}\alpha})}{\omega}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\end{aligned}$ (147)

so that the delta derivative contribution vanishes. We have used $v^{2}=-1$ to
simplify the second line;
$v^{\dot{\alpha}\alpha}v^{\dot{\beta}\beta}-v^{\dot{\alpha}\beta}v^{\dot{\beta}\alpha}=-v^{2}\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}=\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}$.
Note that the term proportional to $\omega^{-1}$ implies propagation of spin
degrees of freedom $y^{\mu}\propto z^{\mu}-\bar{z}^{\mu}$.

In the vector notation, the $\delta z$ and $\delta\bar{z}$ 2pt functions take
the following form,

$\displaystyle\begin{aligned} \langle\delta z^{\mu}(\omega^{\prime})\delta
z^{\nu}(\omega)\rangle&=\frac{i\eta^{\mu\nu}}{m\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle\delta\bar{z}^{\mu}(\omega^{\prime})\delta\bar{z}^{\nu}(\omega)\rangle&=\frac{i\eta^{\mu\nu}}{m\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle\delta\bar{z}^{\mu}(\omega^{\prime})\delta
z^{\nu}(\omega)\rangle&=\frac{i}{m}\left[\frac{\eta^{\mu\nu}}{\omega^{2}}+\frac{2(v^{\mu}y_{0}^{\nu}+y_{0}^{\mu}v^{\nu}+i\varepsilon^{\mu\nu\lambda\sigma}v_{\lambda}y_{0\sigma})}{\omega}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,.\end{aligned}$ (148)

It is useful to present the 2-point functions pictorially. To distinguish them
from the twistor propagators (140)-(142), we denote $\delta z^{\mu}$,
$\delta\bar{z}^{\mu}$ by squares:

$\fmffile{zz-or-zbarzbar-2pt}\parbox{80.0pt}{ \fmfgraph*(50,50)
\fmfstraight\fmfbottom{i1,o1} \fmftop{i2,o2}
\fmfv{decor.shape=square,decor.filled=empty,decor.size=10, label=$\delta
z^{\mu}\;$,label.angle=180}{a2}
\fmfv{decor.shape=square,decor.filled=empty,decor.size=10, label=$\delta
z^{\nu}\;$,label.angle=180}{a1}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=0}{t0}
\fmf{phantom}{i1,a1,o1} \fmf{phantom}{i2,a2,o2}
\fmf{plain,tension=0.4}{a1,t0,a2} }\hskip-45.52458pt=\;\langle\delta
z^{\mu}\delta z^{\nu}\rangle\,,\hskip 28.45274pt\parbox{80.0pt}{
\fmfgraph*(50,50) \fmfstraight\fmfbottom{i1,o1} \fmftop{i2,o2}
\fmfv{decor.shape=square,decor.filled=shaded,decor.size=10,
label=$\delta\bar{z}^{\mu}\;$,label.angle=180}{a2}
\fmfv{decor.shape=square,decor.filled=shaded,decor.size=10,
label=$\delta\bar{z}^{\nu}\;$,label.angle=180}{a1}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=0}{t0}
\fmf{phantom}{i1,a1,o1} \fmf{phantom}{i2,a2,o2}
\fmf{plain,tension=0.4}{a1,t0,a2}
}\hskip-45.52458pt=\;\langle\delta\bar{z}^{\mu}\delta\bar{z}^{\nu}\rangle\,,\hskip
28.45274pt\parbox{80.0pt}{ \fmfgraph*(50,50) \fmfstraight\fmfbottom{i1,o1}
\fmftop{i2,o2} \fmfv{decor.shape=square,decor.filled=shaded,decor.size=10,
label=$\delta\bar{z}^{\mu}\;$,label.angle=180}{a2}
\fmfv{decor.shape=square,decor.filled=empty,decor.size=10,
label=$\delta\bar{z}^{\nu}\;$,label.angle=180}{a1}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=0}{t0}
\fmf{phantom}{i1,a1,o1} \fmf{phantom}{i2,a2,o2}
\fmf{plain,tension=0.4}{a1,t0,a2}
}\hskip-45.52458pt=\;\langle\delta\bar{z}^{\mu}\delta z^{\nu}\rangle\,.$ (149)

Finally, separating the position $x$ and the spin-length $y$, we get

$\displaystyle\begin{aligned} \langle\delta x^{\mu}(\omega^{\prime})\delta
x^{\nu}(\omega)\rangle&=\left[\frac{i\eta^{\mu\nu}}{m\omega^{2}}-\frac{1}{m\omega}\varepsilon^{\mu\nu\lambda\sigma}v_{\lambda}y_{0\sigma}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\ \langle\delta
y^{\mu}(\omega^{\prime})\delta
y^{\nu}(\omega)\rangle&=-\frac{1}{m\omega}\varepsilon^{\mu\nu\lambda\sigma}v_{\lambda}y_{0\sigma}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\ \langle\delta
x^{\mu}(\omega^{\prime})\delta
y^{\nu}(\omega)\rangle&=\frac{1}{m\omega}\left(v^{\mu}y_{0}^{\nu}+y_{0}^{\mu}v^{\nu}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)=-\langle\delta
y^{\mu}(\omega^{\prime})\delta x^{\nu}(\omega)\rangle\,.\end{aligned}$ (150)

####  4.1.4 Higher order correlators

For computations at 2PL or higher orders, we will need to evaluate the higher
order correlators. We write the 2pt correlators as

$\displaystyle\begin{aligned} \langle\delta
z^{\dot{\alpha}\alpha}(\omega^{\prime})\delta
z^{\dot{\beta}\beta}(\omega)\rangle&=-\frac{2i\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{m\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega^{\prime})\delta\bar{z}^{\dot{\beta}\beta}(\omega)\rangle&=-\frac{2i\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{m\omega^{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega^{\prime})\delta
z^{\dot{\beta}\beta}(\omega)\rangle&=-\frac{2i}{m}\left(\frac{\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{\omega^{2}}-\frac{2v^{\dot{\alpha}\beta}y_{0}^{\dot{\beta}\alpha}}{\omega}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,.\end{aligned}$ (151)

The connected part of the higher point correlators can be computed using the
recursive substitutions

$\displaystyle\begin{aligned} \delta
z^{\dot{\alpha}\alpha}(\omega)&\to\int_{\omega^{\prime}}\delta
z^{\dot{\alpha}\beta}(\omega^{\prime})\times\left[-\lambda_{\beta}{}^{I}(\omega-\omega^{\prime})(\ell^{-1})_{I}{}^{\alpha}\right]\,,\\\
\delta\bar{z}^{\dot{\alpha}\alpha}(\omega)&\to\int_{\omega^{\prime}}\left[-(\bar{\ell}^{-1})^{\dot{\alpha}I}\bar{\lambda}_{I\dot{\beta}}(\omega-\omega^{\prime})\right]\times\delta\bar{z}^{\dot{\beta}\alpha}(\omega^{\prime})\,,\end{aligned}$
(152)

the 2pt correlators

$\displaystyle\begin{aligned}
-(\ell^{-1})_{I}{}^{\alpha}\langle\lambda_{\beta}{}^{I}(\omega^{\prime})\delta\bar{z}^{\dot{\gamma}\gamma}(\omega)\rangle&=-\frac{2v^{\dot{\gamma}\alpha}\delta_{\beta}^{\gamma}}{m\omega}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\
-(\bar{\ell}^{-1})^{\dot{\alpha}I}\langle\bar{\lambda}_{I\dot{\beta}}(\omega^{\prime})\delta{z}^{\dot{\gamma}\gamma}(\omega)\rangle&=-\frac{2v^{\dot{\alpha}\gamma}\delta_{\dot{\beta}}^{\dot{\gamma}}}{m\omega}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\end{aligned}$ (153)

additional vertices from the free action

$\displaystyle\begin{aligned} iS_{\text{free},3}&=\frac{i}{2}\int
d\sigma\left[{\rm
det}(\lambda)(\bar{\ell}^{-1})^{\dot{\alpha}I}\bar{\lambda}_{I\dot{\alpha}}+(\ell^{-1})_{I}{}^{\alpha}\lambda_{\alpha}{}^{I}{\rm
det}(\bar{\lambda})\right]\\\
&=\frac{i}{2}\int_{\omega^{\prime},\omega}\left((\bar{\ell}^{-1})^{\dot{\alpha}I}\bar{\lambda}_{I\dot{\alpha}}(\omega^{\prime}){\rm
det}(\lambda)[\omega]+(\ell^{-1})_{I}{}^{\alpha}\lambda_{\alpha}{}^{I}(\omega^{\prime}){\rm
det}(\bar{\lambda})[\omega]\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\\\ iS_{\text{free},4}&=\frac{i}{2m}\int
d\sigma\left[{\rm det}(\lambda){\rm
det}(\bar{\lambda})\right]=\frac{i}{2m}\int_{\omega^{\prime},\omega}{\rm
det}(\lambda)[\omega^{\prime}]{\rm
det}(\bar{\lambda})[\omega]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega^{\prime}+\omega)\,,\end{aligned}$ (154)

where

$\displaystyle\begin{aligned} {\rm
det}(\lambda)[\omega]&=-\frac{\epsilon^{\alpha\beta}\epsilon_{IJ}}{2}\int_{\omega^{\prime}}\lambda_{\alpha}{}^{I}(\omega-\omega^{\prime})\lambda_{\beta}{}^{J}(\omega^{\prime})\,,\\\
{\rm
det}(\bar{\lambda})[\omega]&=\frac{\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{IJ}}{2}\int_{\omega^{\prime}}\bar{\lambda}_{I\dot{\alpha}}(\omega-\omega^{\prime})\bar{\lambda}_{J\dot{\beta}}(\omega^{\prime})\,,\end{aligned}$
(155)

and determinant insertions to the 2pt correlators

$\displaystyle\begin{aligned}
\langle\\!\langle\delta{z}^{\dot{\alpha}\alpha}(\omega_{0})\delta{z}^{\dot{\beta}\beta}(\omega_{1}){\rm
det}(\bar{\lambda})[\omega_{2}]\rangle\\!\rangle&=\frac{4\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{m\omega_{0}\omega_{1}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\,,\\\
\langle\\!\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega_{0})\delta\bar{z}^{\dot{\beta}\beta}(\omega_{1}){\rm
det}({\lambda})[\omega_{2}]\rangle\\!\rangle&=\frac{4\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{m\omega_{0}\omega_{1}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\,.\end{aligned}$ (156)

For example, the following 3pt correlator can be computed as

$\displaystyle\begin{aligned}
\langle\\!\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega_{0})\delta{z}^{\dot{\beta}\beta}(\omega_{1})\delta{z}^{\dot{\gamma}\gamma}(\omega_{2})\rangle\\!\rangle&=-{(\bar{\ell}^{-1})^{\dot{\alpha}I}}\int_{\omega^{\prime}}\langle\bar{\lambda}_{I\dot{\delta}}(\omega_{0}-\omega^{\prime})\delta{z}^{\dot{\beta}\beta}(\omega_{1})\rangle\langle\delta\bar{z}^{\dot{\delta}\alpha}(\omega^{\prime})\delta{z}^{\dot{\gamma}\gamma}(\omega_{2})\rangle\\\
&\phantom{=}-{(\ell^{-1})_{I}{}^{\beta}}\int_{\omega^{\prime}}\langle\lambda_{\delta}{}^{I}(\omega_{1}-\omega^{\prime})\delta\bar{z}^{\dot{\alpha}\alpha}(\omega_{0})\rangle\langle\delta{z}^{\dot{\beta}\delta}(\omega^{\prime})\delta{z}^{\dot{\gamma}\gamma}(\omega_{2})\rangle\\\
&\phantom{=asdf}+\left(\omega_{1}\leftrightarrow\omega_{2}\,,\,\beta\leftrightarrow\gamma\,,\,\dot{\beta}\leftrightarrow\dot{\gamma}\right)\\\
&\phantom{=}+\frac{i}{2}\int_{\omega^{\prime}}{(\ell^{-1})_{I}{}^{\delta}}\langle\lambda_{\delta}{}^{I}(-\omega^{\prime})\delta\bar{z}^{\dot{\alpha}\alpha}(\omega_{0})\rangle\\\
&\phantom{=asdfasdfasdf}\times\langle\\!\langle\delta{z}^{\dot{\beta}\beta}(\omega_{1})\delta{z}^{\dot{\gamma}\gamma}(\omega_{2}){\rm
det}(\bar{\lambda})[\omega^{\prime}]\rangle\\!\rangle\,,\end{aligned}$

where the first three lines come from the expansion (152) and the last line
comes from the insertion (156). The result partially simplifies to

$\displaystyle\begin{aligned}
\langle\\!\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega_{0})\delta{z}^{\dot{\beta}\beta}(\omega_{1})\delta{z}^{\dot{\gamma}\gamma}(\omega_{2})\rangle\\!\rangle&=-\frac{4iv^{\dot{\alpha}\beta}}{m^{2}}\left(\frac{\epsilon^{\dot{\beta}\dot{\gamma}}\epsilon^{\alpha\gamma}}{\omega_{0}\omega_{1}\omega_{2}}+\frac{2v^{\dot{\beta}\gamma}y_{0}^{\dot{\gamma}\alpha}}{\omega_{1}\omega_{2}}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\\\
&\phantom{=asdf}+\left(\omega_{1}\leftrightarrow\omega_{2}\,,\,\beta\leftrightarrow\gamma\,,\,\dot{\beta}\leftrightarrow\dot{\gamma}\right)\\\
&\phantom{=}+\frac{4iv^{\dot{\alpha}\alpha}\epsilon^{\dot{\beta}\dot{\gamma}}\epsilon^{\beta\gamma}}{m^{2}\omega_{0}\omega_{1}\omega_{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\,,\end{aligned}$ (157)

where the frequency exchange is only present for manifest symmetry. In
vectorial notation the last term coming from determinant insertion cancels and
simplifies to

$\displaystyle\begin{aligned}
\langle\\!\langle\delta\bar{z}^{\mu}(\omega_{0})\delta{z}^{\nu}(\omega_{1})\delta{z}^{\lambda}(\omega_{2})\rangle\\!\rangle&=-\frac{4i}{m^{2}\omega_{1}\omega_{2}}\left[y_{0}^{\mu}(\eta^{\nu\lambda}+2v^{\nu}v^{\lambda})+v^{\mu}(v^{\nu}y_{0}^{\lambda}+y_{0}^{\nu}v^{\lambda})\right.\\\
&\phantom{=}\left.\phantom{as}+i\epsilon^{\mu\nu}[v,y_{0}]v^{\lambda}+i\epsilon^{\mu\lambda}[v,y_{0}]v^{\nu}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\,.\end{aligned}$ (158)

A similar calculation for the conjugate 3pt correlator yields

$\displaystyle\begin{aligned}
\langle\\!\langle\delta\bar{z}^{\dot{\alpha}\alpha}(\omega_{0})\delta\bar{z}^{\dot{\beta}\beta}(\omega_{1})\delta{z}^{\dot{\gamma}\gamma}(\omega_{2})\rangle\\!\rangle&=\frac{4iv^{\dot{\alpha}\gamma}}{m^{2}}\left(\frac{\epsilon^{\dot{\beta}\dot{\gamma}}\epsilon^{\alpha\beta}}{\omega_{0}\omega_{1}\omega_{2}}+\frac{2v^{\dot{\beta}\alpha}y_{0}^{\dot{\gamma}\beta}}{\omega_{0}\omega_{1}}\right)\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\\\
&\phantom{=asdf}+\left(\omega_{0}\leftrightarrow\omega_{1}\,,\,\alpha\leftrightarrow\beta\,,\,\dot{\alpha}\leftrightarrow\dot{\beta}\right)\\\
&\phantom{=}+\frac{4iv^{\dot{\gamma}\gamma}\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}}{m^{2}\omega_{0}\omega_{1}\omega_{2}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\,,\end{aligned}$ (159)

which, in the vectorial notation, simplifies to

$\displaystyle\begin{aligned}
\langle\\!\langle\delta\bar{z}^{\mu}(\omega_{0})\delta\bar{z}^{\nu}(\omega_{1})\delta{z}^{\lambda}(\omega_{2})\rangle\\!\rangle&=\frac{4i}{m^{2}\omega_{0}\omega_{1}}\left[y_{0}^{\lambda}(\eta^{\mu\nu}+2v^{\mu}v^{\nu})+v^{\lambda}(v^{\mu}y_{0}^{\nu}+y_{0}^{\mu}v^{\nu})\right.\\\
&\phantom{=}\left.\phantom{as}-i\epsilon^{\lambda\mu}[v,y_{0}]v^{\nu}-i\epsilon^{\lambda\nu}[v,y_{0}]v^{\mu}\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\,.\end{aligned}$ (160)

The following correlator may be of interest,

$\displaystyle\langle\\!\langle\delta y^{\mu}(\omega_{0})\delta
y^{\nu}(\omega_{1})\delta y^{\lambda}(\omega_{2})\rangle\\!\rangle$
$\displaystyle=-\frac{1}{m^{2}}\left[\frac{y^{\mu}(\eta^{\nu\lambda}+v^{\nu}v^{\lambda})}{\omega_{1}\omega_{2}}+(\text{cyc.})\right]\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(\omega_{0}+\omega_{1}+\omega_{2})\,,$ (161)

where $(\text{cyc.})$ denotes cyclic permutation.

Note that $\delta z$ and $\delta\bar{z}$ variables do not obey Wick
factorisation, e.g. $\langle(\delta
z)(\delta\bar{z})(\delta\bar{z})\rangle\neq 0$. However, purely
holomorphic/anti-holomorphic correlators such as $\langle(\delta z)(\delta
z)\cdots(\delta z)\rangle$ do obey Wick factorisation, since $\delta
z$($\delta\bar{z}$) is at most linear in $\mu$($\bar{\mu}$) and the
correlators reduce to the correlators of the form
$\langle\mu\mu\cdots\mu\rangle$ or
$\langle\bar{\mu}\bar{\mu}\cdots\bar{\mu}\rangle$.

####  4.1.5 Photon coupling vertex rules

We begin with the spacetime picture where the vertex rules take a simple form
that are easy to compare with other worldline models. Inserting the mode
expansion of fluctuations into the interaction (44), we get666Empty sum is
zero and empty product is unity, i.e. $\sum_{i=1}^{0}\\#=0$ and
$\prod_{i=1}^{0}\\#=1$.

$\displaystyle q\int_{k,\\{\omega\\}}A_{\mu}^{+}(k)(v^{\mu})e^{ik\cdot
z_{0}}\sum_{n=0}^{\infty}\frac{1}{n!}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt\left((k\cdot
v)-\sum_{i=1}^{n}\omega_{i}\right)\prod_{i=1}^{n}(ik)\cdot\delta
z(\omega_{i})$ (162)
$\displaystyle+q\int_{k,\\{\omega\\}}A_{\mu}^{+}(k)(-i\omega_{0})\delta
z^{\mu}(\omega_{0})e^{ik\cdot
z_{0}}\sum_{n=0}^{\infty}\frac{1}{n!}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt\left((k\cdot
v)-\sum_{i=0}^{n}\omega_{i}\right)\prod_{i=1}^{n}(ik)\cdot\delta
z(\omega_{i})$ $\displaystyle\phantom{asdf}+\left(A_{\mu}^{+}(k)\to
A_{\mu}^{-}(k)\,,\,z_{0}^{\mu}\to\bar{z}_{0}^{\mu}\,,\,\delta
z(\omega_{i})\to\delta\bar{z}(\omega_{i})\right)\,.$

This expression is exact in $z_{0}$, $\bar{z}_{0}$. To obtain a result at a
fixed order in the background spin-length $y_{0}$, we may simply set
$z_{0}^{\mu}=b_{0}^{\mu}+iy_{0}^{\mu}$ and
$\bar{z}_{0}^{\mu}=b_{0}^{\mu}-iy_{0}^{\mu}$ and expand in $y_{0}$.

To compute Compton amplitudes or the 2PL eikonal, we only need terms up to
linear order in fluctuations,

$\displaystyle\begin{aligned}
S_{\text{int}}&=q\int_{k}A_{\mu}^{+}(k)v^{\mu}e^{ik\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt(k\cdot
v)\\\
&\phantom{=}+iq\int_{k,\omega}A_{\mu}^{+}(k)\left[v^{\mu}k_{\nu}-\omega\delta^{\mu}_{\nu}\right]\delta
z^{\nu}(\omega)e^{ik\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt[(k\cdot
v)-\omega]\\\ &\phantom{asdf}+\left[A_{\mu}^{+}(k)\to
A_{\mu}^{-}(k)\,,\,z_{0}^{\mu}\to\bar{z}_{0}^{\mu}\,,\,\delta
z(\omega_{i})\to\delta\bar{z}(\omega_{i})\right]\,.\end{aligned}$ (163)

Aside from the leading term proportional to $A^{\pm}\cdot v$, we can write the
Feynman rules in terms of gauge-invariant field-strengths $F^{\pm}$, since the
vertex rules with at least one $\delta z^{\mu}(\sigma)$ fluctuation field777We
argue using the time domain Feynman rules because the proof is simpler. can be
read out from the variational derivative

$\displaystyle\frac{\delta S_{\text{int}}}{\delta[\delta z^{\mu}(\sigma)]}$
$\displaystyle=qF_{\mu\nu}^{+}[z_{0}+v\sigma+\delta
z(\sigma)]\left(v^{\nu}+\frac{d(\delta z^{\nu}(\sigma))}{d\sigma}\right)\,,$
(164)

which only depends on the field-strength $F^{+}$. The same argument trivially
generalises to the anti-holomorphic sector. As a demonstration, we write the
interaction terms up to quadratic order in worldline perturbations as

$\displaystyle\begin{aligned} S_{\text{int}}&=q\int_{k}(A_{k}^{+}\cdot
v)e^{ik\cdot z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt(k\cdot v)+q\int_{k,\omega}(\delta z_{\omega}\cdot F_{k}^{+}\cdot
v)e^{ik\cdot z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern
1.99997pt[(k\cdot v)-\omega]\\\
&\phantom{=}+\frac{q}{4}\int_{k,\omega_{1},\omega_{2}}\hskip-15.0pt\left\\{\left[(\delta
z_{1}\cdot F^{+}_{k}\cdot v)(ik\cdot\delta z_{2})+(1\leftrightarrow
2)\right]+i(\omega_{1}-\omega_{2})(\delta z_{1}\cdot F^{+}_{k}\cdot\delta
z_{2})\right\\}\\\ &\phantom{=asdfasdfasdfasdf}\times e^{ik\cdot
z_{0}}\delta\kern-1.99997pt\makebox[0.0pt][l]{$\bar{}$}\kern 1.99997pt[(k\cdot
v)-\omega_{1}-\omega_{2}]\\\ &\phantom{asdf}+\left(A_{\mu}^{+}(k)\to
A_{\mu}^{-}(k)\,,\,F_{\mu\nu}^{+}(k)\to
F_{\mu\nu}^{-}(k)\,,\,z_{0}^{\mu}\to\bar{z}_{0}^{\mu}\,,\,\delta
z(\omega_{i})\to\delta\bar{z}(\omega_{i})\right)\,.\end{aligned}$ (165)

The expansion (165) is more useful than the expansion (162) since photon
propagators can be chosen to be free of Dirac string singularities. See
section 4.2 for more discussions on the photon propagator.

###  4.2 Photon propagator

A worldline model of a charged particle provides a localised source for the
electromagnetic field. Away from the sources, the photon propagates freely and
the photon propagator is independent of the worldline model. However, since
the NJ shift forces us to separate the self-dual and anti-self-dual parts of
the photon field, we find it useful to recall some facts regarding how to
split the propagator according to self-duality, which translates to the
helicity of the photon at the quantum level.

In our twistor model, the photon field couples to the particle worldline via
the NJ shift (44) which we copy here:

$\displaystyle S_{\text{int}}=q\int A^{+}_{\mu}(z)dz^{\mu}+q\int
A^{-}_{\mu}(\bar{z})d\bar{z}^{\mu}\,.$ (166)

This coupling may look unfamiliar to the readers. To gain some intuition, let
us expand it to the quadratic order in $y$. The zeroth order term reproduces
the standard minimal coupling for a non-spinning particle. The linear order
term is

$\displaystyle
S_{\mathrm{int}}^{(1)}=q\int\left[\tilde{A}_{\mu}\dot{y}^{\mu}+y^{\mu}(\partial_{\mu}\tilde{A}_{\nu})\dot{x}^{\nu}\right]d\sigma\,.$
(167)

The appearance of $\dot{y}$ is a notable feature of the root-Kerr coupling. To
linear order, we can remove it by integration by parts. Up to a total
derivative, we find

$\displaystyle
S_{\mathrm{int}}^{(1)}=q\int\left[\tilde{F}_{\mu\nu}y^{\mu}\dot{x}^{\nu}\right]d\sigma=\frac{q}{2}\int\left[\varepsilon_{\mu\nu\rho\sigma}y^{\mu}\dot{x}^{\nu}F^{\rho\sigma}\right]d\sigma\,.$
(168)

Starting from the quadratic order, it is impossible to remove all $\dot{y}$
factors. Up to a total derivative, we find

$\displaystyle
S_{\mathrm{int}}^{(2)}=\frac{q}{2}\int\left[\dot{y}^{\mu}y^{\nu}F_{\mu\nu}-(\partial_{\mu}F_{\nu\rho})y^{\mu}y^{\nu}\dot{x}^{\rho}\right]d\sigma\,.$
(169)

We can continue this expansion and express all $S^{(n)}_{\mathrm{int}}$
$(n\geq 1)$ as Lorentz invariant products of $y^{\mu}$, $\dot{y}^{\mu}$,
$\dot{x}^{\mu}$, $F_{\mu\nu}$ and $\varepsilon_{\mu\nu\rho\sigma}$, with no
reference to $A^{\pm}_{\mu}$ at all. With this form of the action, the usual
propagator for the photon field will suffice for all perturbative
computations.

The beauty of the NJ shift (166) is that we can perform computations exactly
in $y$ without ever expanding in powers of $y$. A small price to pay is that
we should use less familiar propagators written in terms of $A^{\pm}_{\mu}$.

Our discussion is inspired by Zwanziger’s (electromagnetic-duality covariant)
two-potential formalism Zwanziger:1970hk  (see also refs. Gubarev:1998ss ;
Shnir:2011zz ; Terning:2020dzg ; Moynihan:2020gxj ). But, we will not directly
follow Zwanziger’s formalism in that we never use two potentials or consider
sources with net magnetic charges. We are interested in the long-distance
interaction between two spatially localised sources. The interaction is
captured by the integral,

$\displaystyle I_{12}=\int J_{1}^{\mu}(x)\langle A_{\mu}(x)A_{\nu}(y)\rangle
J_{2}^{\nu}(y)\,.$ (170)

We are doing classical physics, but we can use the propagator (Green’s
function) in a QFT notation, where $\langle A_{\mu}A_{\nu}\rangle$ is the
2-point function, which we take to be time-ordered for concreteness.

Let us temporarily ignore the net (electric or magnetic) charges and focus on
the dipole or higher multipole moments. For a magnetic dipole, it is well
known that a long-distance observer cannot distinguish an Ampèrian dipole
(electric current loop) from a Gilbertian dipole (two opposing magnetic
monopole charges). A similar story holds for an electric dipole and all higher
electric/magnetic multipole moments. So, as far as the long-distance
interaction is concerned, we can describe the _same_ source using _either_ an