# Master Formulae for $N$-photon tree level amplitudes in plane wave
backgrounds

Patrick Copinger<EMAIL_ADDRESS>Centre for Mathematical
Sciences, University of Plymouth, Plymouth, PL4 8AA, UK  James P. Edwards
<EMAIL_ADDRESS>Centre for Mathematical Sciences, University of
Plymouth, Plymouth, PL4 8AA, UK  Anton Ilderton<EMAIL_ADDRESS>Higgs
Centre, School of Physics and Astronomy, University of Edinburgh, EH9 3FD, UK
Karthik Rajeev<EMAIL_ADDRESS>Higgs Centre, School of Physics and
Astronomy, University of Edinburgh, EH9 3FD, UK

###### Abstract

The presence of strong electromagnetic fields adds huge complexity to QED
Feynman diagrams, such that new methods are required to calculate higher-loop
and higher-multiplicity scattering amplitudes. Here we use the worldline
formalism to present ‘Master Formulae’ for all tree level amplitudes of two
massive particles and an arbitrary number of photons, in a plane wave
background, in both scalar and spinor QED. The plane wave is treated without
approximation throughout, meaning in particular that our formulae are valid in
the strong-field regime of current theoretical and experimental interest. We
check our results against literature expressions obtainable at low
multiplicity via direct Feynman diagram calculations.

##  I Introduction

Strong fields can generate nonlinear and non-perturbative effects in particle
interactions. Strong electromagnetic fields may be generated terrestrially by
several means, including by ultra-intense lasers Strickland and Mourou (1985);
Nob . QED processes in the presence of these fields acquire an intensity-
dependence characterised by a coupling which typically exceeds unity, and
which must therefore be treated without recourse to perturbation theory.
Several upcoming experiments aim to observe nonlinear effects in the
scattering of electrons ELI ; Abramowicz  _et al._ (2021); Clarke  _et al._
(2022) and photons Schlenvoigt  _et al._ (2016); Karbstein (2022) on intense
lasers.

The standard theory approach to ‘strong field QED’ is based on the Furry
expansion, or background field perturbation theory. The strong (e.g. laser)
field is described as a fixed background, the coupling of which to matter is
treated exactly. Interactions between particles scattering on this background
are then treated in perturbation theory as usual, see Fedotov  _et al._ (2023)
for a recent review. There are however several topics in strong field QED
which require the development of new theoretical methods.

First, the majority of progress to date has been made for the special, highly
symmetric laser model of a plane wave background, for which the Furry
expansion can be practically realised. It is a long-standing challenge to
account analytically for realistic pulse geometry, and the new phenomenology
this brings Fedotov  _et al._ (2023). Second, while plane wave results can be
extended to realistic fields via local approximations (e.g. Di Piazza  _et
al._ (2019); Ilderton  _et al._ (2019); Heinzl  _et al._ (2020)), and so
implemented in numerical codes, those codes must still be benchmarked against
theory. This has been performed for first-order (i.e. low multiplicity)
processes, but benchmarking higher-order processes is made challenging by, in
part, a lack of analytic results; the state-of-the-art in the plane wave model
is, at tree level, only _four_ -point scattering. Third, if we consider higher
loop corrections, it has been conjectured Ritus (1970); Narozhnyi (1980);
Fedotov (2017) that at very high background field strengths the loop expansion
must be resummed in order to provide reliable physical predictions (at least
in the low frequency, ‘constant crossed field’ limit). Doing so is a
formidable challenge Heinzl  _et al._ (2021); Mironov and Fedotov (2022);
Torgrimsson (2021).

To attack these problems one can use approximations that do not rely on weak
coupling Di Piazza (2016), develop exactly solvable models which capture some
physics of interest Heinzl and Ilderton (2017), or use alternative methods to
simplify Furry-picture quantities. One potential method is the worldline
formalism, which casts QFT in terms of path integrals over relativistic point
particle trajectories. Its roots can be traced back to Feynman Feynman (1950,
1951), though its use as a serious alternative to the standard QFT formalism
was first advocated by Strassler Strassler (1992), following Bern and Kosower
(1991, 1992). One of the main advantages of the worldline approach is that it
automatically sums over all Feynman diagrams which contribute at fixed
multiplicity and loop order, thus greatly simplifying the combinatorics which
comes with higher numbers of scatterers and/or loops.

The worldline formalism was initially developed for one-loop (and then higher
loop) processes in vacuum and in background fields, and a common output of the
approach is ‘Master Formulae’; these are _all_ -multiplicity formulae for
correlation functions of a chosen set of fields, at fixed loop order. Such
Master Formulae, which would be extremely challenging to reproduce using
Feynman diagrams, have been obtained for processes in vacuum Strassler (1992);
Schmidt and Schubert (1993); Corradini  _et al._ (2015), in constant
electromagnetic backgrounds Shaisultanov (1996); Adler and Schubert (1996);
Reuter  _et al._ (1997); Dittrich and Shaisultanov (2000); Schubert (2000);
McKeon and Sherry (1994) and in plane wave backgrounds Edwards and Schubert
(2021, 2022); Schubert and Shaisultanov (2023). The worldline approach has
also been applied to the calculation of effective actions in background fields
via numerical implementations Gies and Langfeld (2002), the Casimir effect
Gies  _et al._ (2003), vacuum birefringence Ilderton and Torgrimsson (2016),
tadpole corrections Ahmadiniaz  _et al._ (2019, 2017); Edwards and Schubert
(2017), and nonlinear Breit-Wheeler pair production Degli Esposti and
Torgrimsson (2022). A long-standing focus of the approach has been the
investigation of non-perturbative effects via worldline instantons Affleck
_et al._ (1982); Srinivasan and Padmanabhan (1999); Kim and Page (2002); Dunne
and Schubert (2005); Dunne  _et al._ (2006); Dumlu and Dunne (2011); Ilderton
_et al._ (2015). For reviews see Schubert (2001); Edwards and Schubert (2019).

Only recently has much attention been paid to worldline Master Formulae for
processes with external matter lines, or processes at tree level Ahmadiniaz
_et al._ (2020, 2022); Bhattacharya (2017); Ahmad  _et al._ (2017); Ahmadiniaz
_et al._ (2016); Corradini and Esposti (2021). Furthermore, while external
photon lines typically appear in the worldline formalism already LSZ-
amputated, matter lines do not, and it has not yet been fully established how
one should perform the required LSZ amputation which turns correlation
functions into amplitudes.

We fill in some missing pieces of this puzzle in this paper, which is
organised as follows. In Sect. II we construct worldline Master Formulae for
all tree level $N+2$-point correlation functions describing the emission of
$N$ photons from a massive particle in a background plane wave, in both scalar
and spinor QED. In Sect. III we turn to the LSZ amputation of the master
formula, converting it into an all-multiplicity formula for the corresponding
$N$-photon emission/absorption amplitudes from a massive particle in a plane
wave background. Example calculations in which we compare with known
literature results at low multiplicity are presented in Sect. IV. We conclude
in Sect. V. The appendices contain additional checks on our results.

#### Conventions:

We set $\hbar=c=1$. We work throughout in Minkowski space with lightfront
coordiantes, so that
${\mathrm{d}}s^{2}={\mathrm{d}}x^{\scriptscriptstyle+}{\mathrm{d}}x^{\scriptscriptstyle-}-{\mathrm{d}}x^{\scriptscriptstyle\perp}{\mathrm{d}}x^{\scriptscriptstyle\perp}$
where $x^{\scriptscriptstyle\perp}=(x^{1},x^{2})$ are the ‘transverse’
directions. We introduce a null vector $n_{\mu}$ which projects onto the
‘lightfront time’ direction, that is $n\cdot x=x^{\scriptscriptstyle+}$. The
covariant derivative is $D_{\mu}=\partial_{\mu}+ieA_{\mu}$.

##  II Master Formulae for $2+N$–point tree level correlators in plane wave
backgrounds

Figure 1:  We consider tree level scattering amplitudes of 2 massive charges
and $N$ photons, as illustrated on the _right_ (for scalar QED). The double
line represents the presence of a plane wave background, the coupling to which
is treated exactly. Amplitudes are obtained by LSZ reduction of the
corresponding correlation functions. In the worldline approach, a natural
starting pointing is the partially amputated correlator, or ‘dressed
propagator’, in which the photons are already reduced out, but the matter
fields are not. This is illustrated on the _left_. Thus LSZ reduction is still
required for the external matter lines.

The goal of this section is to write down and evaluate the worldline path
integral Master Formulae for tree level _correlation functions_ of $N$ photons
and two charged particles in the presence of a plane wave background, valid
for arbitrary $N$. We will do this in both scalar and spinor QED.

Our plane wave background may be described by the potential
$eA_{\mu}(x)=a_{\mu}(x^{\scriptscriptstyle+})=\delta_{\mu}^{\scriptscriptstyle\perp}a_{\scriptscriptstyle\perp}(x^{\scriptscriptstyle+})$,
a transverse function of lightfront time $x^{{\scriptscriptstyle+}}$. We may
always choose $a_{\scriptscriptstyle\perp}(-\infty)=0$, but then
$a_{\scriptscriptstyle\perp}(\infty)\eqqcolon
a_{\scriptscriptstyle\perp}^{\infty}$ is in general non-zero (and carries an
electromagnetic memory effect Dinu  _et al._ (2012); Bieri and Garfinkle
(2013); Cristofoli  _et al._ (2023).). The corresponding field strength is
$f_{\mu\nu}(x^{\scriptscriptstyle+})=n_{\mu}a^{\prime}_{\nu}(x^{\scriptscriptstyle+})-n_{\nu}a^{\prime}_{\mu}(x^{\scriptscriptstyle+})$,
where a prime denotes an $x^{\scriptscriptstyle+}$-derivative.

###  II.1 Scalar QED

In the Master Formulae we derive in this section, the $N$ external photons
will be LSZ-amputated, but the matter lines not, and thus our correlation
functions carry spacetime indices $x$ and $x^{\prime}$, as well as a
dependence on the $N$ photon momenta $\\{k_{i}\\}$ and polarisations
$\\{\varepsilon_{i}\\}$. We hide the latter dependencies, denoting the
partially reduced correlators, or ‘dressed propagators’ as they are called in
the worldline literature, by $\mathcal{D}_{N}^{x^{\prime}x}$; see Fig. 1. We
take all photons to be _outgoing_ ; other configurations are trivially
obtained by sending $k\to-k$.

The worldline representation of such correlation functions is given in terms
of a path integral over relativistic point particle trajectories, denoted
$x^{\mu}(\tau)$ with $\tau$ the proper time of the trajectory. The
trajectories obey Dirichlet boundary conditions $x^{\mu}(T)=x^{\prime\mu}$,
$x^{\mu}(0)=x^{\mu}$, corresponding to the spacetime dependence of the dressed
propagator. The trajectories have length $T$, which is ultimately also
integrated out, respecting reparameterisation invariance of the path integral
Polyakov (1987); Mansfield (1990). To write down this path integral, we start
from the worldline action that minimally couples a relativistic point particle
to an arbitrary gauge field $A_{\mu}$, namely

$S_{\mathrm{WL}}[x(\tau),A]=-\int_{0}^{T}\\!{\mathrm{d}}\tau\,\Big{[}\frac{\dot{x}^{2}}{4}+e{A(x(\tau))\cdot\dot{x}(\tau)}\Big{]}\;,$
(1)

where over-dots denote proper time derivatives, and where the unusual
normalisation of the kinetic term has become standard in the worldline
literature, so we preserve it here. $S_{\mathrm{WL}}$ enters the path integral
for the scalar field propagator, call it $\mathcal{D}^{x^{\prime}x}$, via

$\mathcal{D}^{x^{\prime}x}=\int_{0}^{\infty}{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T}\,\int^{x(T)=x^{\prime}}_{x(0)=x}\mathcal{D}x(\tau)\,\mathrm{e}^{iS_{\mathrm{WL}}[x(\tau),A]}\;.$
(2)

Note that $A_{\mu}$ is not integrated over, rather it appears as a given field
– it is well known (see, for example Itzykson and Zuber (1980)) that
correlation functions with $N$ external photons _in vacuum_ can be extracted
from (2) by fixing $A_{\mu}$ to be a sum over asymptotic photon wavefunctions
with momenta $k_{i}$ and polarisations $\varepsilon_{i}$:

$A_{\mu}(x)\to
A^{\gamma}_{\mu}(x)=\sum_{i=1}^{N}\varepsilon_{\mu\,i}\mathrm{e}^{ik_{i}\cdot
x}\,,$ (3)

and then expanding the dressed propagator (2) to multi-linear order in the
polarisation vectors. The additional complication here is the presence of the
background gauge potential in (6). This is however easily included; we simply
split the gauge field into a semi-classical part representing the plane wave
background and a ‘quantised’ part representing scattering photons:

$eA_{\mu}(x)\to a_{\mu}(x)+eA^{\gamma}_{\mu}(x)\,.$ (4)

Inserting this into (2) and expanding to multi-linear order, the path integral
to be performed is

$\mathcal{D}^{x^{\prime}x}_{N}=(-ie)^{N}\int_{0}^{\infty}{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T}\,\int^{x(T)=x^{\prime}}_{x(0)=x}\mathcal{D}x(\tau)\,\mathrm{e}^{iS_{\mathrm{B}}[x(\tau),a]}\prod_{i=1}^{N}V^{x^{\prime}x}[\varepsilon_{i},k_{i}]\;,$
(5)

in which the weight is now given by the reduced action

$S_{\mathrm{B}}[x(\tau),a]=-\int_{0}^{T}\\!{\mathrm{d}}\tau\,\Big{[}\frac{\dot{x}^{2}}{4}+{a(x(\tau))\cdot\dot{x}(\tau)}\Big{]}\;,$
(6)

while the $N$ external photons appear (following the expansion to multi-linear
order) through the vertex functions

$V^{x^{\prime}x}[\varepsilon,k]\coloneqq\int_{0}^{T}d\tau\,\varepsilon\cdot\dot{x}(\tau)\,\mathrm{e}^{ik\cdot
x(\tau)}\,.$ (7)

(We leave implicit a causal and IR convergence factor $\exp(-\epsilon T)$
under the ${\mathrm{d}}T$ integral in (5).)

Our task is to evaluate the integrals in (5). Let us first consider the
$x^{\mu}$ integrals, and in particular the Dirichlet BCs. To deal with these
we follow the standard procedure used for the evaluation of such integrals in
vacuum, and expand $x^{\mu}(\tau)$ into a straight line trajectory and a
fluctuation $q(\tau)$ according to

$x^{\mu}(\tau)=x^{\mu}+z^{\mu}\frac{\tau}{T}+q^{\mu}(\tau)\,,\qquad
z^{\mu}:=x^{\prime\mu}-x^{\mu}\;.$ (8)

The fluctuation must satisfy the homogeneous Dirichlet BCs $q(0)=q(T)=0$ (with
measure $\mathcal{D}x(\tau)\rightarrow\mathcal{D}q(\tau)$). For the analogue
problem in vacuum ($a(x^{{\scriptscriptstyle+}})\rightarrow 0$) the path
integral is Gaussian in $q_{\mu}$ and can thus be computed analytically111This
is also the case for a constant background in Fock-Schwinger gauge Ahmad  _et
al._ (2017).. Here, however, the fluctuation appears _inside_ the background
field
$a(x^{\scriptscriptstyle+}(\tau))=a(x^{\scriptscriptstyle+}+z^{\scriptscriptstyle+}\tau/T+q^{\scriptscriptstyle+})$,
and this has an arbitrary functional form. At first glance this seems to
destroy the Gaussianity of the path integral, and prohibit its evaluation.
However, it has been shown for one-loop photon-scattering processes (meaning
no external matter lines, and a path integral with periodic rather than
Dirichlet BCs) that the properties of the plane wave background mean the
integral is still effectively Gaussian Ilderton and Torgrimsson (2016);
Edwards and Schubert (2021); *Edwards:2021uif. It is thus crucial to
demonstrate that the hidden Gaussianity of the path integral is also present
here.

To do so we follow the approach of Schubert and Shaisultanov (2023),
introducing a Lagrange multiplier $\chi(\tau)$ and auxiliary field $\xi(\tau)$
into the path integral through the equality

$\mathrm{e}^{-i\int{\mathrm{d}}\tau\,a(x^{\scriptscriptstyle+}(\tau))\cdot\dot{q}}=\mathrm{e}^{-i\int{\mathrm{d}}\tau\,a(x^{\scriptscriptstyle+}+z^{\scriptscriptstyle+}\frac{\tau}{T}+q^{\scriptscriptstyle+})\cdot\dot{q}}=\int\\!\mathcal{D}\xi\mathcal{D}\chi\,\mathrm{e}^{i\int{\mathrm{d}}\tau\,\big{[}\chi(\xi-q^{\scriptscriptstyle+})-a(x^{\scriptscriptstyle+}+z^{\scriptscriptstyle+}\frac{\tau}{T}+\xi)\cdot\dot{q}\big{]}}\,.$
(9)

These auxiliary integrals render that over $q(\tau)$ to be Gaussian. The
crucial point, as we show below, is that after evaluating the $q$-integral,
the remaining integrals over $\xi$ and $\chi$ can still be evaluated, for a
plane wave background.

We now compute the fluctuation integral. As is usual in this ‘string-inspired’
approach, it is convenient to manipulate the vertex operators as follows. We
exponentiate the polarisation-dependent factor, so that it appears linearly in
an exponent in the operator, with the understanding that the result should
later be expanded to linear order in (each of) the $\varepsilon_{i}$, so we
write

$V^{x^{\prime}x}[\varepsilon,k]\to\int_{0}^{T}\\!{\mathrm{d}}\tau\,\mathrm{e}^{ik\cdot
x+\varepsilon\cdot\dot{x}}\Big{|}_{\mathrm{lin.}\,\varepsilon}\;.$ (10)

The result of this is that all dependence on the particle trajectory
$x(\tau)$, or rather the fluctuation $q(\tau)$ to be integrated out, now
appears _linearly_ under the path integral. The integrals to be evaluated are
now

$\mathcal{D}_{N}^{x^{\prime}x}=(-ie)^{N}\\!{\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T-i\frac{z^{2}}{4T}}}\prod_{i=1}^{N}\int_{0}^{T}\\!{\mathrm{d}}\tau_{i}\,\mathrm{e}^{\,\sum\limits_{j=1}^{N}ik_{j}\cdot(x+z\frac{\tau_{j}}{T})+\varepsilon_{j}\cdot\frac{z}{T}}\int\\!\mathcal{D}\xi\,\mathcal{D}\chi\,\int_{q(0)=0}^{q(T)=0}\\!\mathcal{D}q(\tau)\,\mathrm{e}^{i\int{\mathrm{d}}\tau[-\frac{\dot{q}^{2}}{4}-\mathcal{J}\cdot
q]}\Big{|}_{\mathrm{lin.}\;\varepsilon_{1}\ldots\varepsilon_{N}}\,,$

in which $\mathcal{J}^{\mu}(\tau)$ is an effective (operator valued) source

$\mathcal{J}^{\mu}(\tau):=a^{\mu}(x^{\scriptscriptstyle+}+z^{\scriptscriptstyle+}\tau/T+\xi)\frac{{\mathrm{d}}}{{\mathrm{d}}\tau}+\chi(\tau)n^{\mu}+i\sum_{i=1}^{N}\big{(}ik_{i}^{\mu}-\varepsilon_{i}^{\mu}\frac{{\mathrm{d}}}{{\mathrm{d}}\tau}\big{)}\delta(\tau-\tau_{i})\;,$
(11)

Since the fluctuation integral is now Gaussian, it is easily computed in terms
of the worldline Green’s function $\Delta(\tau_{i},\tau_{j})$, that is the
inverse of $2{\mathrm{d}}^{2}/{\mathrm{d}}\tau^{2}$ with Dirichlet BCs, which
is found to be

$\Delta_{ij}\coloneqq\Delta(\tau_{i},\tau_{j})=\frac{1}{2}|\tau_{i}-\tau_{j}|-\frac{1}{2}(\tau_{i}+\tau_{j})+\frac{\tau_{i}\tau_{j}}{T}\,.$
(12)

It is easily checked that Dirichlet BCs hold:
$\Delta(0,\tau_{i})=\Delta(T,\tau_{i})=\Delta(\tau_{j},0)=\Delta(\tau_{j},T)=0$.
With this, the fluctuation integral becomes

$\int_{q(0)=0}^{q(T)=0}\mathcal{D}q(\tau)\,\mathrm{e}^{i\int_{0}^{T}{\mathrm{d}}\tau[-\frac{\dot{q}^{2}}{4}-\mathcal{J}\cdot
q]}={-i(4\pi
T)^{-2}}\,\exp\bigg{[}-i\int_{0}^{T}{\mathrm{d}}\tau_{i}{\mathrm{d}}\tau_{j}\,\mathcal{J}_{\mu}(\tau_{i})\Delta_{ij}\mathcal{J}^{\mu}(\tau_{j})\bigg{]}\;.$
(13)

This defines the fundamental contraction for the fluctuation variable,

$\langle
q^{\mu}(\tau)q^{\nu}(\tau^{\prime})\rangle=2i{\eta^{\mu\nu}}\Delta(\tau,\tau^{\prime})\,,$
(14)

and the free path integral normalisation is recovered by setting
$\mathcal{J}=0$. To proceed, we wish to write out the exponent in (13)
explicitly. Note, though, that $\Delta_{ij}$ is not proper time-translation
invariant due to the boundary conditions Ahmadiniaz  _et al._ (2020), hence
left and right proper time-derivatives must be distinguished. We denote these
as follows:

${}^{\bullet}\\!\Delta_{ij}\coloneqq\frac{{\mathrm{d}}}{{\mathrm{d}}\tau_{i}}\Delta_{ij}\;,\qquad\Delta^{\hskip-1.42262pt\bullet}_{ij}\coloneqq\frac{{\mathrm{d}}}{{\mathrm{d}}\tau_{j}}\Delta_{ij}\;,\qquad{}^{\bullet\bullet}\\!\Delta_{ij}\coloneqq\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}\tau_{i}^{2}}\Delta_{ij}\;,\qquad\mathrm{etc.}$
(15)

With this, we write out the exponent of (13), using that the background is
transverse and on-shell ($n\cdot a=0$ and $n^{2}=0$) to simplify. We find,
writing $a_{i}\equiv
a(x^{\scriptscriptstyle+}+z^{\scriptscriptstyle+}\tau_{i}/\tau+\xi(\tau_{i}))$,

$\displaystyle\int\mathcal{J}\cdot\Delta\cdot\mathcal{J}$
$\displaystyle=\int\\!{\mathrm{d}}\tau_{i}{\mathrm{d}}\tau_{j}\,a_{i}\cdot
a_{j}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}_{ij}+2i\sum_{j=1}^{N}\int\\!{\mathrm{d}}\tau_{i}\,({}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}_{ij}\,a_{i}\cdot\varepsilon_{j}+i{}^{\bullet}\\!\Delta_{ij}a_{i}\cdot
k_{j})$
$\displaystyle+2i\sum_{j=1}^{N}\int\\!{\mathrm{d}}\tau_{i}\,\,\chi_{i}[\Delta^{\hskip-1.42262pt\bullet}_{ij}\varepsilon_{j}^{{\scriptscriptstyle+}}+i\Delta_{ij}k_{j}^{{\scriptscriptstyle+}}]-\sum_{i,j=1}^{N}[{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}+2i{}^{\bullet}\\!\Delta_{ij}\varepsilon_{i}\cdot
k_{j}-\Delta_{ij}k_{i}\cdot k_{j}]\,.$ (16)

The trivial dependence on $\chi$ means that this field can now be integrated
out, yielding a $\delta$-functional:

$\int\mathcal{D}\xi\mathcal{D}\chi\,\mathrm{e}^{i\int
d\tau\,\chi\big{[}\xi-2i\sum_{j=1}^{N}(\Delta^{\hskip-1.42262pt\bullet}_{\tau\tau_{j}}\varepsilon_{j}^{{\scriptscriptstyle+}}+i\Delta_{\tau\tau_{j}}k_{j}^{{\scriptscriptstyle+}})\big{]}}=\int\mathcal{D}\xi\,\delta\big{[}\xi(\tau)-2\sum_{j=1}^{N}(i\Delta^{\hskip-1.42262pt\bullet}_{\tau\tau_{j}}\varepsilon_{j}^{{\scriptscriptstyle+}}-\Delta_{\tau\tau_{j}}k_{j}^{{\scriptscriptstyle+}})\big{]}\,.$
(17)

This $\delta$-functional has the effect of shifting the argument of the
background field, such that from here on we have

$a^{\mu}_{i}\equiv a^{\mu}(\tau_{i})\equiv
a^{\mu}\Bigl{(}x^{{\scriptscriptstyle+}}+z^{{\scriptscriptstyle+}}\frac{\tau_{i}}{T}+2\sum_{j=1}^{N}\big{[}-\Delta_{ij}k_{j}^{{\scriptscriptstyle+}}+i\Delta^{\hskip-1.42262pt\bullet}_{ij}\varepsilon_{j}^{{\scriptscriptstyle+}}\big{]}\Bigr{)}\,.$
(18)

The dynamical fluctuation is thus replaced by a coupling of the plane wave to
the $N$ scattering photons Ilderton and Torgrimsson (2016); Edwards and
Schubert (2021); *Edwards:2021uif. This is particular to plane wave
backgrounds because (a) for $n^{2}\neq 0$ equation (16) picks up a
contribution quadratic in $\chi$, whilst (b) for $n\cdot a\neq 0$ there is an
additional term linear in $\chi$ that depends on the background; instead of
(18) one would have obtained via (17) only an implicit equation for $a^{\mu}$.

All remaining background-dependent terms in (17) may be expressed in terms of
just two worldline structures, namely the worldline average and the periodic
integral

$\langle\\!\langle f\rangle\\!\rangle\coloneqq
T^{-1}\int_{0}^{T}\\!{\mathrm{d}}\tau\,f(\tau)\;,\qquad
I_{\mu}(\tau)\coloneqq\int_{0}^{\tau}\\!{\mathrm{d}}\tau^{\prime}\big{[}a_{\mu}(\tau^{\prime})-\langle\\!\langle
a_{\mu}\rangle\\!\rangle\big{]}\;,$ (19)

respectively. These would have to be computed for a given background once the
functional form of $a_{\mu}$ has been fixed. At this stage the path integral
has (at least formally) been computed. Gathering everything together we obtain
our Master Formulae for the $N$-photon dressed propagator

$\begin{split}\mathcal{D}_{N}^{x^{\prime}x}={i}(-e)^{N}\int_{0}^{\infty}\\!{\mathrm{d}}T\,&(4i\pi
T)^{-2}\mathrm{e}^{-i\frac{z^{2}}{4T}}{\prod_{i=1}^{N}\int_{0}^{T}\\!{\mathrm{d}}\tau_{i}}\\\
&\mathrm{e}^{-iM^{2}(a)T}\bar{\mathfrak{P}}^{x^{\prime}x}(\varepsilon_{1},\ldots\,\varepsilon_{N})\,\mathrm{e}^{-iz\cdot\langle\\!\langle
a\rangle\\!\rangle+i\sum_{j=1}^{N}(x+\frac{z}{T}\tau_{j}-2I(\tau_{j}))\cdot
k_{j}-i\sum_{i,j=1}^{N}\Delta_{ij}k_{i}\cdot
k_{j}}{\Big{|}_{\mathrm{lin.}\;\varepsilon_{1}\ldots\varepsilon_{N}}}\;,\end{split}$
(20)

in which $M^{2}(a)\coloneqq m^{2}-\langle\\!\langle
a^{2}\rangle\\!\rangle+\langle\\!\langle a\rangle\\!\rangle^{2}$ is analogous
to the Kibble ‘mass’ Kibble  _et al._ (1975) which typically appears in pulsed
plane waves Harvey  _et al._ (2012), while $\bar{\mathfrak{P}}^{x^{\prime}x}$
is defined by

$\bar{\mathfrak{P}}^{x^{\prime}x}(\varepsilon_{1},\ldots\,\varepsilon_{N})\coloneqq
i^{N}e^{\sum_{i=1}^{N}\varepsilon_{i}\cdot\frac{z}{T}+2\sum_{i=1}^{N}(\langle\\!\langle
a\rangle\\!\rangle-
a_{i})\cdot\varepsilon_{i}+i\sum_{i,j=1}^{N}[2i{}^{\bullet}\\!\Delta_{ij}\varepsilon_{i}\cdot
k_{j}+\varepsilon_{i}\cdot\varepsilon_{k}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}_{ij}]}\,.$
(21)

We emphasise that this Master Formula holds for any multiplicity $N\geq 0$; it
would be extremely challenging to obtain this starting from the Feynman rules.
Evaluating in specific cases we can check against the literature; for $N=0$,
for example, we recover a one-parameter (proper-time) representation of the
scalar Volkov propagator:

${\mathcal{D}_{0}^{x^{\prime}x}}={i}\mathrm{e}^{-iz\cdot\langle\\!\langle
a\rangle\\!\rangle}\int_{0}^{\infty}\\!{\mathrm{d}}T\,(4i\pi
T)^{-2}\mathrm{e}^{-iM^{2}(a)T}e^{-i\frac{z^{2}}{4T}}\,.$ (22)

Observe that in this case $a_{\mu}(\tau)\equiv
a_{\mu}\big{(}x^{+}+z^{+}\frac{\tau}{T}\big{)}$ so that, changing variables to
$u=\frac{\tau}{T}$, the worldline average becomes $T$–independent and can be
taken outside the $T$-integral. It may be written as a _spacetime_ average
(see Ilderton and Torgrimsson (2016)),

$\langle\\!\langle
a_{\mu}\rangle\\!\rangle=\int_{0}^{1}\\!{\mathrm{d}}u\,a_{\mu}\big{(}x^{{\scriptscriptstyle+}}+z^{{\scriptscriptstyle+}}u\big{)}=\frac{1}{x^{\prime+}-x^{{\scriptscriptstyle+}}}\int_{x^{{\scriptscriptstyle+}}}^{x^{\prime+}}\\!{\mathrm{d}}y\,a_{\mu}(y)\equiv\langle
a_{\mu}\rangle\,,$ (23)

and as such $M^{2}(a)=m^{2}-\langle a^{2}\rangle+\langle a\rangle^{2}$ now
corresponds exactly to the Kibble mass.

Equation (22) is equivalent to the standard momentum-integral representation
of the Volkov propagator, and offers a concise version of the position space
propagator in Schwinger (1951); Borghardt and Karpenko (1998). For $N=1$ we
recover the (2-scalar 1-photon) three point function, and so on. Since the
correlators themselves are not of immediate interest, we will present these
checks later, implicitly, as part of our checks on the corresponding formula
for _scattering amplitudes_.

The actual computation of the dressed propagator (and, later, the amplitudes)
is greatly simplified by observing that we can choose the gauge
$n\cdot\varepsilon=\varepsilon^{\scriptscriptstyle+}=0$. This removes the
polarisation vectors from the argument of $a_{\mu}$, and thus extraction of
the multi-linear piece of (24) reduces to the expansion of
$\bar{\mathfrak{P}}(\varepsilon_{1},\ldots\,\varepsilon_{N})$ alone. We adopt
this gauge from here on in order to present the simplest possible expressions
and also match to the strong field QED literature, where this gauge is common.
Doing so, then, we can write the Master Formula in this gauge as

$\mathcal{D}_{N}^{x^{\prime}x}={i}{(-e)^{N}}\int_{0}^{\infty}\\!{\mathrm{d}}T\,(4i\pi
T)^{-2}\mathrm{e}^{-i\frac{z^{2}}{4T}}{\prod_{i=1}^{N}\int_{0}^{T}{\mathrm{d}}\tau_{i}}\,\mathrm{e}^{-iM^{2}(a)T}\bar{\mathfrak{P}}_{N}^{x^{\prime}x}\mathrm{e}^{-iz\cdot\langle\\!\langle
a\rangle\\!\rangle+i\sum_{i=1}^{N}(x+\frac{z}{T}\tau_{i}-2I(\tau_{i}))\cdot
k_{i}-i\sum_{i,j=1}^{N}\Delta_{ij}k_{i}\cdot k_{j}}\;,$ (24)

where the polynomial $\bar{\mathfrak{P}}^{x^{\prime}x}_{N}$ is defined by the
expansion of the polarisation-dependent terms to multi-linear order:

${\bar{\mathfrak{P}}_{N}^{x^{\prime}x}}\coloneqq
i^{N}\mathrm{e}^{\sum_{i=1}^{N}\varepsilon_{i}\cdot\frac{z}{T}+2\sum_{i=1}^{N}(\langle\\!\langle
a\rangle\\!\rangle-
a_{i})\cdot\varepsilon_{i}+i\sum_{i,j=1}^{N}[2i{}^{\bullet}\\!\Delta_{ij}\varepsilon_{i}\cdot
k_{j}+{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}_{ij}\varepsilon_{i}\cdot\varepsilon_{j}]}{\Big{|}_{\mathrm{lin.}\;\varepsilon_{1}\ldots\varepsilon_{N}}\,.}$
(25)

These polynomials generalise those defined for closed worldlines in vacuum
($P_{N}$) in Schubert (2001), for open lines in vacuum ($\bar{P}_{N}$) in
Schubert (2000), and for the closed loop in a background field
($\mathfrak{P}_{N}$) in Edwards and Schubert (2022) (in position space for the
time being). For convenience let us write out the first few terms:

$\displaystyle\bar{\mathfrak{P}}_{0}^{x^{\prime}x}$ $\displaystyle=1\,,$ (26)
$\displaystyle\bar{\mathfrak{P}}_{1}^{x^{\prime}x}$
$\displaystyle=i\Bigl{[}\frac{z}{T}+2(\langle\\!\langle a\rangle\\!\rangle-
a_{1})-2{}^{\bullet}\\!\Delta_{11}k_{1}\Bigr{]}\cdot\varepsilon_{1}\,,$ (27)
$\displaystyle\bar{\mathfrak{P}}_{2}^{x^{\prime}x}$
$\displaystyle=-\Bigl{[}\frac{z}{T}+2(\langle\\!\langle a\rangle\\!\rangle-
a_{1})-2{}^{\bullet}\\!\Delta_{11}k_{1}-2{}^{\bullet}\\!\Delta_{12}k_{2}\Bigr{]}\cdot\varepsilon_{1}$
$\displaystyle\hphantom{=}\,\times\Bigl{[}\frac{z}{T}+2(\langle\\!\langle
a\rangle\\!\rangle-
a_{2})-2{}^{\bullet}\\!\Delta_{21}k_{1}-2{}^{\bullet}\\!\Delta_{22}k_{2}\Bigr{]}\cdot\varepsilon_{2}-2i{}^{\bullet}\\!\Delta_{12}\\!{}^{\bullet}\varepsilon_{1}\cdot\varepsilon_{2}\,.$
(28)

###  II.2 Spinor QED

We now turn to the computation of the analogous $N$-photon dressed propagators
in spinor QED, denoting these by $\mathcal{S}^{x^{\prime}x}_{N}$. Due to the
spin degrees of freedom this is a Dirac matrix-valued function, but we
suppress the corresponding indices for brevity. Referring the reader to
Fradkin and Gitman (1991); Ahmadiniaz  _et al._ (2020) for details, we begin
by writing down the analogue of the ‘propagator’ (2) in an arbitrary
background, but now accounting for the spin of the fermion:

$\displaystyle\mathcal{S}^{x^{\prime}x}$
$\displaystyle=\big{(}-i\not{D}_{x^{\prime}}-m\big{)}\mathcal{K}^{x^{\prime}x}(a)$
(29) $\displaystyle\mathcal{K}^{x^{\prime}x}(a)$
$\displaystyle=\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T}\\!\int_{x(0)=x}^{x(T)=x^{\prime}}\mathcal{D}x(\tau)\,\mathrm{e}^{iS_{\mathrm{WL}}[x(\tau),A]}\,{2^{-\frac{D}{2}}}\mathrm{symb}^{-1}\oint_{\textrm{A/P}}\mathcal{D}\psi(\tau)\,\mathrm{e}^{i\widetilde{S}_{\mathrm{WL}}[\psi(\tau),x(\tau),A]}$
(30) $\displaystyle\widetilde{S}_{\mathrm{WL}}[\psi(\tau),x(\tau),A]$
$\displaystyle=\int_{0}^{T}{\mathrm{d}}\tau\Big{[}\frac{i}{2}\psi\cdot\dot{\psi}+ie\big{(}\psi(\tau)+\eta\big{)}\cdot
F(x(\tau))\cdot\big{(}\psi(\tau)+\eta\big{)}\Big{]}\,.$ (31)

The kernel $\mathcal{K}^{x^{\prime}x}$ contains an integral over relativistic
particle trajectories, as for the scalar case, and also a path integral over
Grassmann–valued fields $\psi(\tau)$, obeying anti-periodic (A/P) BCs
$\psi(0)=-\psi(T)$. These represent the spin degrees of freedom of the fermion
and are minimally coupled to $A$ through its field strength $F(x(\tau))$
appearing in the action ${\tilde{S}}_{\text{WL}}$. An additional Grassmann
variable $\eta$ also appears; the Dirac matrix structure of the propagator is
produced by acting on this variable by the (inverse of) the _symbolic map_ ,
defined by

${\mathrm{symb}\Bigl{\\{}\gamma^{[\mu_{1}}\gamma^{\mu_{2}}...\gamma^{\mu_{n}]}\Bigr{\\}}=(-i\sqrt{2})^{n}\eta^{\mu_{1}}\eta^{\mu_{2}}\ldots\eta^{\mu_{n}}\,.}$
(32)

This map converts between antisymmetric combinations of Dirac matrices (a
combinatorial factor of $1/n!$ factor is assumed) and products of Grassmann
variables $\eta$. Use of the symbol map avoids lengthy Dirac-matrix algebra as
it automatically produces the kernel in the (even sub-algebra of the) Clifford
basis of the Dirac algebra. Note that all $\eta$-dependence in (34) or any of
our expressions vanishes after evaluation of the inverse map; it is therefore
pragmatic to state once and for all the results relevant to us in
(3+1)-dimensions as

$\displaystyle\mathrm{symb}^{-1}\bigl{\\{}1\bigr{\\}}=\mathbb{I}_{4}\,\qquad\mathrm{symb}^{-1}\bigl{\\{}\eta^{\mu}\eta^{\nu}\bigr{\\}}=-\frac{1}{2}\gamma^{[\mu}\gamma^{\nu]}=-\frac{1}{4}[\gamma^{\mu},\gamma^{\nu}]$
$\displaystyle\mathrm{symb}^{-1}\bigl{\\{}\eta^{\mu}\eta^{\nu}\eta^{\alpha}\eta^{\beta}\bigr{\\}}=\frac{1}{4!}\Bigl{[}\\{\gamma^{[\mu}\gamma^{\nu]},\gamma^{[\alpha}\gamma^{\beta]}\\}-\\{\gamma^{[\mu}\gamma^{\alpha]},\gamma^{[\nu}\gamma^{\beta]}\\}+\\{\gamma^{[\mu}\gamma^{\beta]},\gamma^{[\nu}\gamma^{\alpha]}\\}\Bigr{]}=i\gamma_{5}\epsilon^{\mu\nu\alpha\beta}\,.$
(33)

Now, taking $A$ as in (3) to introduce both our background plane wave and the
$N$ external photons, we expand (29) to multi-linear order in the photon
polarisations to obtain the $N$-photon dressed propagator

$\displaystyle\mathcal{S}^{x^{\prime}x}_{N}$
$\displaystyle=(-i\not{\partial}_{x^{\prime}}+\not{a}(x^{\prime{\scriptscriptstyle+}})-m){\mathcal{K}}_{N}^{x^{\prime}x}(a)+e\not{A}^{\gamma}(x^{\prime}){\mathcal{K}}_{N-1}^{x^{\prime}x}(a)\,,$
(34) $\displaystyle{\mathcal{K}}_{N}^{x^{\prime}x}(a)$
$\displaystyle=(-ie)^{N}\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T}\\!\int_{x(0)=x}^{x(T)=x^{\prime}}\mathcal{D}x(\tau)\,\mathrm{e}^{iS_{\mathrm{B}}[x(\tau),a]}\,2^{-\frac{D}{2}}\mathrm{symb}^{-1}\oint_{\textrm{A/P}}\mathcal{D}\psi(\tau)\,\mathrm{e}^{i\widetilde{S}_{\mathrm{B}}[\psi(\tau),x(\tau),a]}\prod_{i=1}^{N}V_{\eta}^{x^{\prime}x}[\varepsilon_{i},k_{i}]\,,$

where $\widetilde{S}_{\mathrm{B}}[\psi(\tau),x(\tau),a]$ is given by replacing
$eF(x(\tau))$ in $\widetilde{S}_{\mathrm{WL}}[\psi(\tau),x(\tau),A]$ with
$f(x(\tau))$. In the ‘$N$-photon kernel’ $\mathcal{K}_{N}^{x^{\prime}x}(a)$,
the proper time and bosonic integrals are the same as in the scalar case –
these represent the orbital degrees of freedom which remain unchanged. In the
so-called subleading term involving ${\mathcal{K}}_{N-1}^{x^{\prime}x}$, for
each term in the sum in $\not{A}^{\gamma}(x^{\prime})$ we remove the
corresponding photon from the kernel to maintain the projection onto the
multi-linear sector. Finally, writing
$\tilde{f}_{i\mu\nu}=k_{i\mu}\varepsilon_{i\nu}-k_{i\nu}\varepsilon_{i\mu}$
for the linearised field strength associated with the $i^{\text{th}}$ photon,
the vertex operator is now given by

$V_{\eta}^{x^{\prime}x}[\varepsilon_{i},k_{i}]\coloneqq\int_{0}^{T}{\mathrm{d}}\tau\,\Big{[}\varepsilon_{i}\cdot\dot{x}(\tau_{i})+\big{(}\psi(\tau_{i})+\eta\big{)}\cdot\tilde{f}_{i}\cdot\big{(}\psi(\tau_{i})+\eta\big{)}\Big{]}\mathrm{e}^{ik_{i}\cdot
x(\tau_{i})}\,,$ (35)

in which the second term represents the spin coupling of the external photons
to the particle trajectories.

Despite the obvious added complexity from the spin coupling to the photon
fields, we stress that the same hidden Gaussianity is present here as in the
scalar case. Consider again the path integral over $x^{\mu}$; we treat it as
we did above, introducing auxiliary fields to yield a Gaussian path integral
in the fluctuation $q^{\mu}$. While there is now an additional dependence on
the background $f_{\mu\nu}$ introduced by the spin factor, this behaves in the
same way as above when integrating out the auxiliary fields, i.e. $f$ in the
spin factor is ultimately evaluated at a shifted argument,

$f^{\mu\nu}_{i}\equiv f^{\mu\nu}(\tau_{i})\equiv
f^{\mu\nu}\Bigl{(}x^{{\scriptscriptstyle+}}+z^{{\scriptscriptstyle+}}\frac{\tau_{i}}{T}-2\sum_{j=1}^{N}\Delta_{ij}k_{j}^{{\scriptscriptstyle+}}\Bigr{)}\,,$
(36)

just for $a_{\mu}$ earlier (recall we have gauged
$\varepsilon^{\scriptscriptstyle+}_{i}=0$ for conveniences). In short, and as
is natural, the only real difference compared to the scalar case lies in the
evaluation of the Grassmann path integral, which is the focus of the remainder
of this section.

Observe that the vertex operators (35) introduce factors of
$\psi_{\eta}(\tau)\equiv(\psi(\tau)+\eta)$ under the Grassmann integral. This
motivates us to introduce the following functions,

$\displaystyle\mathfrak{W}_{\eta}(\tilde{f}_{i_{1}};$
$\displaystyle\ldots;\tilde{f}_{i_{S}})\coloneqq\Big{\langle}\psi_{\eta}(\tau_{i_{1}})\cdot\tilde{f}_{i_{1}}\cdot\psi_{\eta}(\tau_{i_{1}})\ldots\psi_{\eta}(\tau_{i_{S}})\cdot\tilde{f}_{i_{S}}\cdot\psi_{\eta}(\tau_{i_{S}})\Big{\rangle}$
(37)
$\displaystyle=2^{-\frac{D}{2}}\oint_{\mathrm{A/P}}\mathcal{D}\psi(\tau)\,\psi_{\eta}(\tau_{i_{1}})\cdot\tilde{f}_{i_{1}}\cdot\psi_{\eta}(\tau_{i_{1}})\ldots\psi_{\eta}(\tau_{i_{S}})\cdot\tilde{f}_{i_{S}}\cdot\psi_{\eta}(\tau_{i_{S}})\,\mathrm{e}^{i\int_{0}^{T}{\mathrm{d}}\tau\,\big{[}\frac{i}{2}\psi\cdot\dot{\psi}+i\psi_{\eta}(\tau)\cdot
f(\tau)\cdot\psi_{\eta}(\tau)\big{]}}\,,$ (38)

which generalise the expectation values of the spin-part of the vertex
operator introduced in vacuum ($W(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})$
on the loop in Schubert (2001) and
$W_{\eta}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})$ for open lines in
Ahmadiniaz  _et al._ (2020)) and for one-loop amplitudes in the plane wave
background ($\mathfrak{W}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})$ in
Edwards and Schubert (2021)). We generate the insertions under the path
integral by derivatives w.r.t. a fictitious Grassmann source $\theta$ (anti-
commuting with $\psi$ and $\eta$), from which follows

$\mathfrak{W}_{\eta}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})=\frac{\delta}{\delta\theta_{i_{1}}}\cdot\tilde{f}_{i_{1}}\cdot\frac{\delta}{\delta\theta_{i_{1}}}\cdots\frac{\delta}{\delta\theta_{i_{S}}}\cdot\tilde{f}_{i_{S}}\cdot\frac{\delta}{\delta\theta_{i_{S}}}\,2^{-\frac{D}{2}}\oint_{\mathrm{A/P}}\\!\mathcal{D}\psi(\tau)\,\mathrm{e}^{i\int_{0}^{T}{\mathrm{d}}\tau\,[\frac{i}{2}\psi\cdot\dot{\psi}+i\psi_{\eta}\cdot
f\cdot\psi_{\eta}+i\theta\cdot\psi_{\eta}]}\bigg{|}_{\theta=0}\,,$ (39)

and the corresponding spin factor is produced through

$\mathrm{Spin}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})\,{\coloneqq}\,\,\mathrm{symb}^{-1}\,\mathfrak{W}_{\eta}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})\,.$
(40)

To compute the integral in (39) we require the (spinor) worldline propagator
in the field, $\mathfrak{G}^{\mu\nu}{(\tau,\tau^{\prime})}$. This will define
the fundamental contraction between the Grassmann fields,

$\big{\langle}\psi^{\mu}(\tau)\psi^{\nu}(\tau^{\prime})\big{\rangle}=\frac{1}{2}\mathfrak{G}^{\mu\nu}(\tau,\tau^{\prime})\,.$
(41)

From the quadratic part of the operator appearing in the path integral action,
$\mathfrak{G}$ must obey

$\Big{(}\frac{1}{2}\eta_{\mu\sigma}\frac{{\mathrm{d}}}{{\mathrm{d}}\tau}+f_{\mu\sigma}{(\tau)}\Big{)}\mathfrak{G}^{\sigma\nu}{(\tau,\tau^{\prime})}=\eta_{\mu}{}^{\nu}\,\delta(\tau-\tau^{\prime})\;,$
(42)

as well as anti-periodic boundary conditions
$\mathfrak{G}(0,\tau^{\prime})=-\mathfrak{G}(T,\tau^{\prime})$ and
$\mathfrak{G}(\tau,0)=-\mathfrak{G}(\tau,T)$. Observe that $\mathfrak{G}$ has
the anti-symmetric property
$\mathfrak{G}^{\mu\nu}(\tau,\tau^{\prime})=-\mathfrak{G}^{\nu\mu}(\tau^{\prime},\tau)$.
The general homogeneous solution of (42) for _arbitrary_ $f(\tau)$ is written
conveniently in terms of an auxiliary function
$\mathcal{O}(\tau,\tau^{\prime})$, which takes care of the ordering of $\tau$
and $\tau^{\prime}$, defined by

$\mathcal{O}(\tau,\tau^{\prime})=\mathcal{P}^{\star}\,e^{-2\int_{\tau^{\prime}}^{\tau}d\sigma
f(\sigma)}\,,$ (43)

where $\Theta$ is the Heaviside step function,
$\mathcal{P}^{\star}\equiv\mathcal{P}^{\star}(\tau,\tau^{\prime})=\Theta(\tau-\tau^{\prime})\mathcal{P}+\Theta(\tau^{\prime}-\tau)\bar{\mathcal{P}}$
with $\mathcal{P}$ ($\bar{\mathcal{P}}$) denoting (anti-)path ordering in
proper time and we have made use of a matrix form for the Lorentz indices
(with respect to which $\mathcal{O}$ is orthogonal). With the homogeneous
solution, we can then find the general solution to (42) with appropriate anti-
periodic boundary conditions as

$\mathfrak{G}(\tau,\tau^{\prime})=\mathrm{sgn}(\tau-\tau^{\prime})\mathcal{O}(\tau,\tau^{\prime})+\mathcal{O}(\tau,0){\frac{1-\mathcal{O}(T,0)}{1+\mathcal{O}(T,0)}}\mathcal{O}(0,\tau^{\prime})\,,$
(44)

However, there are notable simplifications in our particular case that $f$ is
a plane wave because, as is well known, the field strength is then nilpotent
of order three. Further, $f$ evaluated at different $\tau$ commute. The Green
function thus reduces to222This is an alternative way of writing the Green
function given in equation (45) of Edwards and Schubert (2021), with the
advantage of being manifestly gauge invariant. There $\mathfrak{G}^{\mu\nu}$
was written in terms of periodic integrals of the derivative of $a(\tau)$
which made its anti-periodicity easier to see.

$\displaystyle\mathfrak{G}(\tau,\tau^{\prime})$
$\displaystyle=\mathrm{e}^{-2\int_{\tau^{\prime}}^{\tau}d\sigma\,f(\sigma)}\Big{[}\mathrm{sgn}(\tau-\tau^{\prime})+\tanh\Big{(}\int_{0}^{T}d\sigma\,f(\sigma)\Big{)}\Big{]}$
(45)
$\displaystyle=\mathrm{sgn}(\tau-\tau^{\prime})\Bigl{[}1-2\int_{\tau^{\prime}}^{\tau}\\!{\mathrm{d}}\sigma\,f(\sigma)+2\Big{(}\int_{\tau^{\prime}}^{\tau}\\!{\mathrm{d}}\sigma\,f(\sigma)\Big{)}^{2}\Bigr{]}+T\langle\\!\langle
f\rangle\\!\rangle\Bigl{[}1-2\int_{\tau^{\prime}}^{\tau}\\!{\mathrm{d}}\sigma\,f(\sigma)\Bigr{]}\,.$
(46)

Equipped with the Green function, we compute the integral in (39) by
completing the square, using the shift
$\tilde{\psi}(\tau)=\psi(\tau)+\int\\!{\mathrm{d}}\tau^{\prime}\mathfrak{G}(\tau,\tau^{\prime})\cdot(f(\tau^{\prime})\cdot\eta+\frac{1}{2}{\theta}(\tau^{\prime}))$
The integral over $\tilde{\psi}$ then generates the determinant
$\mathrm{Det}(\frac{1}{2}\frac{{\mathrm{d}}}{{\mathrm{d}}\tau}+f)$ (for anti-
periodic boundary conditions) which because of the nilpotency of $f$ simply
gives a factor of $2^{\frac{D}{2}}$, being the number of degrees of freedom of
the fermion in $D$ (even) space-time dimensions (this should be contrasted
with the constant field case, where the normalisation picks up a non-trivial
field dependence Reuter  _et al._ (1997); Shaisultanov (1996)).

Gathering all of the above together, the Grassmann integral as defined in (39)
becomes

$\mathfrak{W}_{\eta}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})=\frac{\delta}{\delta\theta_{i_{1}}}\cdot\tilde{f}_{i_{1}}\cdot\frac{\delta}{\delta\theta_{i_{1}}}\cdots\frac{\delta}{\delta\theta_{i_{S}}}\cdot\tilde{f}_{i_{S}}\cdot\frac{\delta}{\delta\theta_{i_{S}}}\mathrm{e}^{-\int_{0}^{T}\\!{\mathrm{d}}\tau\,[\eta\cdot
f(\tau)\cdot\eta+{\theta(\tau)}\cdot\eta]-\int_{0}^{T}\\!{\mathrm{d}}\tau{\mathrm{d}}\tau^{\prime}[\eta\cdot
f(\tau)\cdot\mathfrak{G}(\tau,\tau^{\prime})\cdot\theta(\tau^{\prime})+\frac{1}{4}\theta(\tau)\cdot\mathfrak{G}(\tau,\tau^{\prime})\cdot\theta(\tau^{\prime})]}\Big{|}_{\theta=0}\,.$
(47)

The Grassmann path integral is therefore formally computed. In particular,

$\displaystyle\mathfrak{W}_{\eta}(\emptyset)$
$\displaystyle=e^{-\int_{0}^{T}d\tau\,\eta\cdot f(\tau)\cdot\eta}\,,$ (48)
$\displaystyle\mathfrak{W}_{\eta}(\tilde{f}_{i_{1}})$
$\displaystyle=\Bigl{\\{}-\tfrac{1}{2}\mathrm{tr}[\tilde{f}(\tau_{i_{1}})\cdot\mathfrak{G}(\tau_{i_{1}},\tau_{i_{1}})]+\eta\cdot\mathfrak{G}^{\mathbb{T}}(\tau_{i_{1}})\cdot\tilde{f}(\tau_{i_{1}})\cdot\mathfrak{G}(\tau_{i_{1}})\cdot\eta\Bigr{\\}}e^{-\int_{0}^{T}d\tau\,\eta\cdot
f(\tau)\cdot\eta}\,,$ (49)
$\displaystyle\mathfrak{W}_{\eta}(\tilde{f}_{i_{1}};\tilde{f}_{i_{2}})$
$\displaystyle=\Biggl{\\{}\bigl{[}-\tfrac{1}{2}\mathrm{tr}[\tilde{f}(\tau_{i_{2}})\cdot\mathfrak{G}(\tau_{i_{2}},\tau_{i_{2}})]+\eta\cdot\mathfrak{G}^{\mathbb{T}}(\tau_{i_{2}})\cdot\tilde{f}(\tau_{i_{2}})\cdot\mathfrak{G}(\tau_{i_{2}})\cdot\eta\bigr{]}\times\bigl{[}\tau_{i_{2}}\to\tau_{i_{1}}\bigr{]}$
$\displaystyle\hskip
25.00003pt-\tfrac{1}{2}\mathrm{tr}[\tilde{f}(\tau_{i_{1}})\cdot\mathfrak{G}(\tau_{i_{1}},\tau_{i_{2}})\cdot\tilde{f}(\tau_{i_{2}})\cdot\mathfrak{G}(\tau_{i_{2}},\tau_{i_{1}})]$
$\displaystyle\hskip
25.00003pt+2\eta\cdot\mathfrak{G}^{\mathbb{T}}(\tau_{i_{2}})\cdot\tilde{f}(\tau_{i_{2}})\cdot\mathfrak{G}(\tau_{i_{2}},\tau_{i_{1}})\cdot\tilde{f}(\tau_{i_{1}})\cdot\mathfrak{G}(\tau_{i_{1}})\cdot\eta\Biggr{\\}}e^{-\int_{0}^{T}d\tau\,\eta\cdot
f(\tau)\cdot\eta}\,,$ (50)

where
$\mathfrak{G}_{\mu\nu}(\tau_{i})\coloneqq\eta_{\mu\nu}-\int_{0}^{T}d\tau[\mathfrak{G}(\tau_{i},\tau)\cdot
f(\tau)]_{\mu\nu}$ and $\mathbb{T}$ denotes the transpose in Lorentz indices –
in particular we have
$\mathfrak{G}^{\mathbb{T}}{}_{\mu\nu}(\tau_{i})=\eta_{\mu\nu}-\int_{0}^{T}d\tau[f(\tau)\cdot\mathfrak{G}(\tau,\tau_{i})]_{\mu\nu}$.

Putting all of this together, the $N$-photon dressed propagator can be written
in a ‘spin-orbit decomposition’ by summing over assignation of the $N$
external photons to either the spin or bosonic part of the vertex Edwards and
Schubert (2021), as follows:

$\displaystyle\mathcal{S}_{N}^{x^{\prime}x}$
$\displaystyle=(-i\not{\partial}_{x^{\prime}}+\not{a}(x^{\prime{\scriptscriptstyle+}})-m){\mathcal{K}}_{N}^{x^{\prime}x}(a)+e\not{A}^{\gamma}(x^{\prime}){\mathcal{K}}_{N-1}^{x^{\prime}x}(a)\,,$
(51) $\displaystyle\mathcal{K}_{N}^{x^{\prime}x}(a)$
$\displaystyle=\sum_{S=0}^{N}\sum_{\\{i_{1}:i_{S}\\}}\mathcal{K}_{NS}^{\\{i_{1}:i_{S}\\}x^{\prime}x}(a)\,,$
(52) $\displaystyle\mathcal{K}_{NS}^{\\{i_{1}:i_{S}\\}x^{\prime}x}(a)$
$\displaystyle={i}(-e)^{N}\int_{0}^{\infty}\\!{\mathrm{d}}T\,(4\pi
iT)^{-2}\,\mathrm{e}^{-iM^{2}(a)T-i\frac{z^{2}}{4T}-iz\cdot\langle\\!\langle
a\rangle\\!\rangle}$
$\displaystyle\times\prod_{i=1}^{N}\int_{0}^{T}\\!{\mathrm{d}}\tau_{i}\,\mathrm{Spin}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})\,\bar{\mathfrak{P}}_{NS}^{\\{i_{1}:i_{S}\\}x^{\prime}x}\,\,\mathrm{e}^{i\sum_{i=1}^{N}\big{[}x+\frac{z}{T}\tau_{i}-2I(\tau_{i})\big{]}\cdot
k_{i}-i\sum_{i,j=1}^{N}\Delta_{ij}k_{i}\cdot k_{j}}\,.$ (53)

The sum on the second line runs over the allocation of $S$, out of the $N$,
photons to the spin part of the vertex operator, $V_{\eta}^{x^{\prime}x}$,
which subsequently appear in
$\mathrm{Spin}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})$. Then the
remaining $N-S$ photons appear in the polynomial
$\bar{\mathfrak{P}}_{NS}^{\\{i_{1}:i_{S}\\}x^{\prime}x}$, defined by

$\bar{\mathfrak{P}}_{NS}^{\\{i_{1}:i_{S}\\}x^{\prime}x}\coloneqq
i^{{N-S}}\mathrm{e}^{\sum_{i=1}^{N}\varepsilon_{i}\cdot\frac{z}{T}+2\sum_{i=1}^{N}\big{[}(\langle\\!\langle
a\rangle\\!\rangle-
a_{i})\cdot\varepsilon_{i}\big{]}+i\sum_{i,j=1}^{N}\big{[}\varepsilon_{i}\cdot\varepsilon_{k}{}^{\bullet}\\!\Delta^{\hskip-1.42262pt\bullet}_{ij}+2i{}^{\bullet}\\!\Delta_{ij}\varepsilon_{i}\cdot
k_{j}\big{]}}\Big{|}_{\varepsilon_{i_{S+1}}...\varepsilon_{i_{N}}}^{\varepsilon_{i_{1}}...\varepsilon_{i_{S}}=0}\,.$
(54)

where the notation on the far right means that the polarisation vectors
$\varepsilon_{i_{1}}$ to $\varepsilon_{i_{S}}$ should be put to zero before
the remaining expression is expanded to multi-linear order in the
$\varepsilon_{i_{S+1}}$ to $\varepsilon_{i_{N}}$. These polynomials generalise
those introduced in vacuum ($\bar{P}_{NS}^{\\{i_{1};i_{S}\\}}$) in Ahmadiniaz
_et al._ (2020) and satisfy

$\displaystyle\bar{\mathfrak{P}}_{N0}^{\\{\\}x^{\prime}x}=\bar{\mathfrak{P}}_{N}^{x^{\prime}x}\,,\qquad\bar{\mathfrak{P}}_{NN}^{\\{1:N\\}x^{\prime}x}=1\,.$
(55)

Again, these are position space expressions, but below we shall transform to
momentum space for the purpose of evaluating scattering amplitudes. Although
this Master Formula appears lengthy, it is important to emphasise that it
represents a formal evaluation of the path integral for an arbitrary number of
photons inserted along the background-dressed propagator, conveniently split
into contributions from the vertex function representing orbital interactions
(in $\bar{\mathfrak{P}}_{NS}^{\\{i_{1}:i_{S}\\}x^{\prime}x}$) and spin
interactions (in $\mathrm{Spin}(\tilde{f}_{i_{1}};\ldots;\tilde{f}_{i_{S}})$).
All of these insertions are integrated along the particle trajectories, so
that the Master Formula represents a sum over all Feynman diagrams
contributing to the dressed propagator that differ by permutation of the
external photons. Obtaining such a formula from the standard formalism (Furry
picture, say) of strong field QED would be a significantly more complicated
task.

For completeness, we note that the $N=0$ case provides a worldline
representation of the well-known Volkov propagator as a one-parameter integral

$\mathcal{S}^{x^{\prime}x}={i}\big{(}-i(\not{\partial}_{x^{\prime}}+i\not{a}(x^{\prime+}))-m\big{)}\mathrm{e}^{-iz\cdot\langle
a\rangle}\int_{0}^{\infty}\\!{\mathrm{d}}T\,(4\pi
iT)^{-2}\mathrm{e}^{-iM^{2}(a)T-i\frac{z^{2}}{4T}+{\frac{T}{z^{+}}}\big{[}\not{n}\not{a}(x^{\prime+})+\not{a}(x^{+})\not{n}\big{]}}\,,$
(56)

where we used $\mathrm{Spin}(\not{0})=1+{\frac{T}{2}}\gamma\cdot\langle
f\rangle\cdot\gamma={1+{T}\not{n}\langle\not{a}^{\prime}\rangle}$, computed
the integral in the average explicitly and re-exponentiated using
$\not{n}^{2}=0$. This is again equivalent to other representations of the
Volkov propagator Schwinger (1951); Borghardt and Karpenko (1998); Fedotov
_et al._ (2023).

##  III LSZ for scattering amplitudes

The objective of this section is to take the Master Formulae for the dressed
propagators $D^{x^{\prime}x}_{N}$ and $S_{N}^{x^{\prime}x}$ above and produce
from them equivalent Master Formulae for (2-scalar) $N$-photon scattering
amplitudes (for $N\geqslant 1$). To do so we must perform LSZ reduction on the
two massive, external legs of the dressed propagators.

In previous worldline literature, amputation was often done ‘by-hand’, by
obtaining the $N$-point correlation functions in momentum space and then –
once the proper-time integral had been computed – removing external legs with
the appropriate inverse matter propagators Ahmadiniaz  _et al._ (2020, 2022).
Only then could the external particles be taken on-shell – the proper-time
integral produces the pole structure of the correlation functions with respect
to external matter legs and so is divergent in the on-shell limit. This is a
notable example where the Feynman diagram prescription to omit external
propagators had appeared less trivial from a worldline perspective. Recently,
however, Bonocore (2021); Mogull  _et al._ (2021) showed how amputation can be
achieved under the proper-time integral for scalar matter legs, with the
inverse propagators simply modifying the bounds on the proper-time and
parameter integrals. This exposes the on-shell residue of the correlation
functions without the need to carry out amputation by hand. We will here
generalise this approach to spinor theories, and also show it is unspoiled by
the plane-wave background.

To perform LSZ we draw the external legs out to asymptotic times and Fourier
transform. Alternatively, we can Fourier transform to momentum space and find
the residues of the dressed propagator as the momenta are taken onto the mass-
shell. Starting with scalar QED, the amplitude takes the form

$\displaystyle\mathcal{A}_{N}^{p^{\prime}p}$ $\displaystyle={-\lim_{p^{\prime
2},p^{2}\rightarrow
m^{2}}}\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\mathrm{e}^{i(p^{\prime}+a^{\infty})\cdot
x^{\prime}-ip\cdot
x}\,{\left[(\partial_{x^{\prime}}+ia^{\infty})^{2}+m^{2}\right]}{\big{[}\partial^{2}_{x}+m^{2}\big{]}}\,\mathcal{D}_{N}^{x^{\prime}x}\,,$
(57) $\displaystyle=\lim_{p^{\prime 2},p^{2}\rightarrow m^{2}}-(p^{\prime
2}-m^{2})(p^{2}-m^{2})\mathcal{D}_{N}^{\tilde{p}^{\prime}p}\,,$ (58)

where in the second line we defined $\tilde{p}^{\prime}=p+a^{\infty}$ and
introduced the momentum-space propagator $\mathcal{D}_{N}^{p^{\prime}p}$,
defined by

$\displaystyle\mathcal{D}_{N}^{p^{\prime}p}$
$\displaystyle:=\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\mathrm{e}^{ip^{\prime}\cdot
x^{\prime}-ip\cdot x}\,\mathcal{D}^{x^{\prime}x}_{N}\;.$ (59)

The expression (57) is (almost) textbook-standard LSZ in position space but to
compensate for the fact that our potential becomes pure gauge in the far
future, the on-shell, outgoing momentum $p^{\prime}$ in the Fourier kernel is
shifted to ${\tilde{p}}^{\prime}=p^{\prime}+a^{\infty}$ Kibble  _et al._
(1975); Dinu  _et al._ (2012). The expression (58) makes it clear that the
amplitude $\mathcal{A}_{N}^{p^{\prime}p}$ is the residue of
$\mathcal{D}^{\tilde{p}^{\prime}p}_{N}$ at on-shell momenta. In our
conventions $\mathcal{A}_{N}^{p^{\prime}p}$ describes $N$-photon _emission_
from a particle traversing the plane wave. Absorption and pair-
production/annihilation amplitudes are of course obtained by crossing.

Similarly for the spinor case, starting from the master formula for the
dressed propagator (51), we can extract the spin-polarised amplitude
$\mathcal{M}_{Ns^{\prime}s}^{p^{\prime}p}$ as

$\mathcal{M}_{Ns^{\prime}s}^{p^{\prime}p}=i{\lim_{p^{\prime
2},p^{2}\rightarrow
m^{2}}}\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\mathrm{e}^{i\tilde{p}^{\prime}\cdot
x^{\prime}-ip\cdot
x}\bar{u}_{s^{\prime}}(p^{\prime})(i\not{\partial}_{x^{\prime}}-\not{a}^{\infty}-m)\mathcal{S}_{N}^{x^{\prime}x}(-i\overleftarrow{\not{\partial}}_{x}-m)u_{s}(p)\,,$
(60)

in which $\bar{u}_{s^{\prime}}(p^{\prime})$ and $u_{s}(p)$ are free Dirac
spinors. We now proceed to perform the LSZ reduction explicitly, starting with
scalar QED.

###  III.1 Scalar QED

We begin by evaluating the momentum-space propagator via direct Fourier
transform of the Master Formula (24):

$\mathcal{D}_{N}^{\tilde{p}^{\prime}p}=\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\mathrm{e}^{i{\tilde{p}}^{\prime}\cdot
x^{\prime}-ip\cdot x}\,\mathcal{D}^{x^{\prime}x}_{N}\;.$ (61)

The integrals over $x^{\prime{\scriptscriptstyle\perp},{\scriptscriptstyle-}}$
and $x^{{\scriptscriptstyle\perp},{\scriptscriptstyle-}}$ generate333To
evaluate similar integrals in the existing literature it was found to be
convenient to change variables to endpoint centre of mass and relative
separation ($z$). However, for our later LSZ amputation of the external legs
it is more useful to integrate separately with respect to the endpoint
coordinates., as in the vacuum case, four $\delta$-functions, explicitly
${\delta^{3}_{\perp,{\scriptscriptstyle-}}\big{(}\tilde{p}^{\prime}+K-p\big{)}}$$\delta(x^{+}-x^{\prime+}+2g^{+}+2p^{\prime+}T)$,
where we write $K=\sum_{i=1}^{N}k_{i}$ to compactify notation. The first three
$\delta$-functions describe the (expected) conservation of lightfront three-
momentum in the plane wave background. The final $\delta$-function allows us
to trivially perform e.g. the $x^{\prime{\scriptscriptstyle+}}$ integral, so
that we can replace $x^{\prime+}\rightarrow x^{+}+2g^{+}+2p^{\prime+}T$ in
what remains; in particular, the classical trajectory on which the gauge field
depends throughout $\mathcal{D}_{N}^{x^{\prime}x}$, as in (18), is modified
to, where $g\equiv
g(\\{\tau_{i}\\}):=\sum_{i=1}^{N}(k_{i}\tau_{i}-i\epsilon_{i})$,

$x^{\scriptscriptstyle+}_{cl}(\tau)=x^{\scriptscriptstyle+}+g^{{\scriptscriptstyle+}}+(p^{\prime}+p)^{\scriptscriptstyle+}\tau-\sum_{i=1}^{N}k_{i}^{{\scriptscriptstyle+}}|\tau-\tau_{i}|\,.$
(62)

Thus we can do all but one of the Fourier integrals, which eventually yield

$\displaystyle\mathcal{D}_{N}^{\tilde{p}^{\prime}p}$
$\displaystyle=(-ie)^{N}(2\pi)^{3}{\delta^{3}_{\perp,{\scriptscriptstyle-}}\big{(}\tilde{p}^{\prime}+K-p\big{)}}\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{i(p^{\prime
2}-m^{2}+i0^{{\scriptscriptstyle+}})T}\int_{-\infty}^{\infty}{\mathrm{d}}x^{\scriptscriptstyle+}\,\mathrm{e}^{i(p^{\prime}_{{\scriptscriptstyle+}}+K_{{\scriptscriptstyle+}}-p_{{\scriptscriptstyle+}})x^{{\scriptscriptstyle+}}}$
(63) $\displaystyle\times\prod_{i=1}^{N}\int_{0}^{T}{\mathrm{d}}\tau_{i}\,\
\mathrm{e}^{-2ig\cdot\langle\\!\langle
a\rangle\\!\rangle-2iTp^{\prime}\cdot\langle\\!\langle\delta
a\rangle\\!\rangle+iT\langle\\!\langle\delta
a^{2}\rangle\\!\rangle-2i\sum_{i=1}^{N}\left[k_{i}\cdot
I(\tau_{i})-i\varepsilon_{i}\cdot I^{\prime}(\tau_{i})\right]}$
$\displaystyle\hskip
50.00008pt\times\mathrm{e}^{ig\cdot(2\tilde{p}^{\prime}+K)-i\sum_{i,j=1}^{N}\left(\frac{|\tau_{i}-\tau_{j}|}{2}k_{i}\cdot
k_{j}-i\,\textrm{sign}(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot
k_{j}+\delta(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot\varepsilon_{j}\right)}\Big{|}_{\textrm{lin.
}\varepsilon_{1}\ldots\varepsilon_{N}}\,,$

in which we have defined $\delta a(x^{+}):=a(x^{+})-a^{\infty}$ and
$a(\tau)\equiv a(x^{{\scriptscriptstyle+}}_{\rm cl}(\tau))$. Note that in the
vacuum limit $a_{\mu}\rightarrow 0$ we can carry out the $\hat{x}^{+}$
integral to complete the conservation of $4$-momentum and so recover one
version of the Master Formula given in Ahmadiniaz  _et al._ (2020) and
Shaisultanov (1996).

To convert (63) into a master formula for the amplitudes, we have to perform
LSZ on each massive scalar leg (these are produced by the parameter and proper
time integrals). To do so we observe that (58) has, using (63), the following
form, writing down only the relevant structures:

$-i({p^{\prime}}^{2}-m^{2}+i0^{\scriptscriptstyle+})\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{i({p^{\prime}}^{2}-m^{2}+i0^{\scriptscriptstyle+})T}F(T)\;.$
(64)

The on-shell limit ${p^{2}\to m^{2}-i0^{+}}$ therefore returns the residue of
the mass-shell pole of the function defined by the integral. To isolate this
pole we proceed as in Laenen  _et al._ (2009); Bonocore (2021); Mogull  _et
al._ (2021) where LSZ was considered for, e.g., the $N$-graviton-dressed
propagator in vacuum444We note in passing that the same ‘trick’ is useful in
exposing the connection between gauge invariance and infra-red behaviour of
amplitudes in background plane waves Ilderton and MacLeod (2020). We integrate
by parts (off-shell) in order to expose the residue, as so:

$\begin{split}-i({p^{\prime}}^{2}-m^{2}+i0^{\scriptscriptstyle+})\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{i({p^{\prime}}^{2}-m^{2}+i0^{\scriptscriptstyle+})T}F(T)=F(0)+\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{i({p^{\prime}}^{2}-m^{2}+i0^{\scriptscriptstyle+})T}\frac{{\mathrm{d}}}{{\mathrm{d}}T}F(T)\;.\end{split}$
(65)

We can now take ${p^{\prime}}^{2}\to m^{2}$ and $0^{\scriptscriptstyle+}\to 0$
(in either order), upon which the integral becomes exact, and we have

$\lim_{{p^{\prime}}^{2}\to
m^{2}}-i({p^{\prime}}^{2}-m^{2}+i0^{\scriptscriptstyle+})\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{i({p^{\prime}}^{2}-m^{2}+i0^{\scriptscriptstyle+})T}F(T)=F(\infty)\;.$
(66)

Ultimately, then, performing the first amputation on (63) is equivalent to
dropping the integral over proper time $T$ and its accompanying mass-shell
exponent, and taking the limit $T\rightarrow\infty$ of what remains (this is
the same argument as in vacuum, which we comment on further after performing
the second amputation, below). We thus find

$\displaystyle\lim_{p^{\prime 2}\rightarrow m^{2}}-i(p^{\prime
2}-m^{2}+i0^{\scriptscriptstyle+}){\mathcal{D}^{p^{\prime}p}_{N}}=(-ie)^{N}(2\pi)^{3}\delta^{3}_{{\scriptscriptstyle\perp},{\scriptscriptstyle-}}(\tilde{p}^{\prime}+K-p)\int_{-\infty}^{\infty}\\!{\mathrm{d}}x^{{\scriptscriptstyle+}}\mathrm{e}^{i\left(p^{\prime}_{\scriptscriptstyle+}+K_{\scriptscriptstyle+}-p_{\scriptscriptstyle+}\right)x^{\scriptscriptstyle+}}\prod_{i=1}^{N}\int_{0}^{\infty}\\!{\mathrm{d}}\tau_{i}\,$
(67) $\displaystyle\mathrm{e}^{-i\int_{0}^{\infty}[2p^{\prime}\cdot\delta
a(\tau)-\delta
a^{2}(\tau)]{\mathrm{d}}\tau-2i\sum\limits_{i=1}^{N}\left[\int_{0}^{\tau_{i}}k_{i}\cdot
a(\tau){\mathrm{d}}\tau-i\varepsilon_{i}\cdot
a(\tau_{i})\right]+ig\cdot(2\tilde{p}^{\prime}+K)-i\sum\limits_{i,j=1}^{N}(\frac{|\tau_{i}-\tau_{j}|}{2}k_{i}\cdot
k_{j}-i\,\mathrm{sgn}(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot
k_{j}+\delta(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot\varepsilon_{j})}\bigg{|}_{\textrm{lin.
}\varepsilon_{i}\ldots\varepsilon_{N}}\,.$

We note that all terms with worldline averages have ultimately been replaced
with (convergent) integrals over $\mathbb{R}^{+}$. This was the advantage of
having computed the Fourier integrals with respect to the individual endpoints
as discussed above. Equation (67) is the one-side amputated propagator.

Turning to the amputation with respect to $p$, at this stage it is
advantageous to introduce the mean and deviation proper time variables as
follows:

$\tau_{0}\coloneqq\frac{1}{N}\sum_{i=1}^{N}\tau_{i}\;,\qquad\bar{\tau}_{i}\coloneqq\tau_{i}-\tau_{0}\;.$
(68)

The reason for this change of variable is that it allows us to re-express (67)
in a form which renders the _second_ LSZ amputation immediate. To achieve
this, we first rewrite the proper time integrals appearing in (67) in terms of
the new variables as (note the factor of $\frac{1}{N}$ in the
$\delta$-function is missing in (3.18) of Mogull  _et al._ (2021))

$\prod_{i=1}^{N}\int_{0}^{\infty}\\!{\mathrm{d}}\tau_{i}=\int_{-\infty}^{\infty}\\!{\mathrm{d}}\tau_{0}\prod_{i=1}^{N}\int_{-\infty}^{\infty}\\!{\mathrm{d}}\bar{\tau}_{i}\,\delta\bigg{(}\sum_{j=1}^{N}\frac{\bar{\tau}_{j}}{N}\bigg{)}\;,$
(69)

We also make a change of variable for the
$x^{\scriptscriptstyle+}$-integration,
$\bar{x}^{\scriptscriptstyle+}:={x^{\scriptscriptstyle+}+(p^{\prime}+p+K)^{\scriptscriptstyle+}\tau_{0}+g^{\scriptscriptstyle+}(\\{\bar{\tau}_{i}\\})}$,
and it is convenient to change variables in all ${\mathrm{d}}\tau$ integrals
from $\tau$ to $\bar{\tau}\coloneqq\tau-\tau_{0}$, such that the background
gauge field now appears as

$\displaystyle a(\bar{\tau})\equiv
a\big{(}\bar{x}^{{\scriptscriptstyle+}}+(p^{\prime}+p)^{\scriptscriptstyle+}\,{\bar{\tau}}-\sum_{i=1}^{N}k_{i}^{{\scriptscriptstyle+}}|{\bar{\tau}}-\bar{\tau}_{i}|\big{)}\;.$
(70)

In terms of the shifted variables
$\\{\bar{x}^{{\scriptscriptstyle+}},\tau_{0},\bar{\tau}_{i}\\}$, the once-
amputated propagator (67) takes the form

$(-ie)^{N}(2\pi)^{3}\delta_{{\scriptscriptstyle\perp},{\scriptscriptstyle-}}(\tilde{p}^{\prime}+K-p)\\!\int_{-\infty}^{\infty}\\!{\mathrm{d}}\bar{x}^{\scriptscriptstyle+}\,\mathrm{e}^{i(K+p^{\prime}-p)_{\scriptscriptstyle+}\bar{x}^{{\scriptscriptstyle+}}}\int_{0}^{\infty}\\!{\mathrm{d}}\tau_{0}\,\mathrm{e}^{i{(p^{2}-m^{2})}\tau_{0}}\int_{-\infty}^{\infty}\prod_{i=1}^{N}{\mathrm{d}}\bar{\tau}_{i}\,\delta\bigg{(}\sum_{i=1}^{N}\frac{\bar{\tau}_{i}}{N}\bigg{)}G(\tau_{0})\;,$
(71)

in which the function appearing in the factor is

$\displaystyle G(\tau_{0})$
$\displaystyle=\mathrm{e}^{-i(2p^{\prime}+a^{\infty})\cdot
a^{\infty}\tau_{0}-i\int_{-\tau_{0}}^{\infty}\\!{\mathrm{d}}\bar{\tau}\,[2p^{\prime}\cdot\delta
a(\bar{\tau})-\delta
a^{2}(\bar{\tau})]{-2i\sum_{i=1}^{N}\left[\int_{-\tau_{0}}^{\bar{\tau}_{i}}\\!{\mathrm{d}}\bar{\tau}\,k_{i}\cdot
a(\bar{\tau})-i\varepsilon_{i}\cdot a(\bar{\tau}_{i})\right]}}$ (72)
$\displaystyle\times\mathrm{e}^{i(\tilde{p}^{\prime}+p)\cdot
g-i\sum_{i,j=1}^{N}\left(\frac{|\bar{\tau}_{i}-\bar{\tau}_{j}|}{2}k_{i}\cdot
k_{j}-i\,\mathrm{sgn}(\bar{\tau}_{i}-\bar{\tau}_{j})\varepsilon_{i}\cdot
k_{j}+\delta(\bar{\tau}_{i}-\bar{\tau}_{j})\varepsilon_{i}\cdot\varepsilon_{j}\right)}\;.\bigg{|}_{\textrm{lin.
}\varepsilon_{1},\ldots\varepsilon_{N}}$

Note that the factor $-i(2p^{\prime}+a^{\infty})\cdot a^{\infty}\tau_{0}$ in
the exponential diverges in the $\tau_{0}\rightarrow\infty$ limit, but can be
absorbed into the Volkov-like term, also divergent in the same limit, to yield
the convergent factor $-i\int_{-\tau_{0}}^{0}[2\tilde{p}^{\prime}\cdot
a(\bar{\tau})-a^{2}(\bar{\tau})]{\mathrm{d}}\bar{\tau}-i\int_{0}^{\infty}[2p^{\prime}\cdot\delta
a(\bar{\tau})-\delta a^{2}(\bar{\tau})]{\mathrm{d}}\bar{\tau}$. After this
rearrangement, one finds that the dependence on $\\{p^{2}-m^{2},\tau_{0}\\}$
in (71)–(72) exactly mirrors the dependence on $\\{p^{\prime 2}-m^{2},T\\}$ in
the original expression, before the first amputation. Thus we can simply
repeat the previous LSZ argument but applied to $\\{p^{2}-m^{2},\tau_{0}\\}$
in order to extract the pole at the _incoming_ mass-shell; effectively this
removes the integral over $\tau_{0}$ and takes $\tau_{0}\rightarrow\infty$ in
the remainder, yielding our final master formula for the 2-scalar $N$-photon
scattering amplitudes:

$\displaystyle\mathcal{A}^{p^{\prime}p}_{N}$
$\displaystyle=(-ie)^{N}(2\pi)^{3}\delta_{\perp,-}({\tilde{p}}^{\prime}+K-p)\int_{-\infty}^{\infty}\\!{\mathrm{d}}x^{{\scriptscriptstyle+}}e^{i(K+p^{\prime}-p)_{\scriptscriptstyle+}x^{{\scriptscriptstyle+}}}\int_{-\infty}^{\infty}\prod_{i=1}^{N}{\mathrm{d}}\tau_{i}\,\delta\bigg{(}\sum_{j=1}^{N}\frac{\tau_{j}}{N}\bigg{)}$
$\displaystyle\times\mathrm{e}^{-i\int_{-\infty}^{0}[2\tilde{p}^{\prime}\cdot
a(\tau)-a^{2}(\tau)]{\mathrm{d}}\tau-i\int_{0}^{\infty}\\![2p^{\prime}\cdot\delta
a(\tau)-\delta
a^{2}(\tau)]{\mathrm{d}}\tau-2i\sum_{i=1}^{N}[\int_{-\infty}^{\tau_{i}}k_{i}\cdot
a(\tau){\mathrm{d}}\tau-i\varepsilon_{i}\cdot a(\tau_{i})]}$
$\displaystyle\times e^{i({\tilde{p}}^{\prime}+p)\cdot
g-i\sum_{i,j=1}^{N}\bigl{(}\frac{|\tau_{i}-\tau_{j}|}{2}k_{i}\cdot
k_{j}-i\,\mathrm{sgn}(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot
k_{j}+\delta(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot\varepsilon_{j}\bigr{)}}\bigg{|}_{\textrm{lin.
}\varepsilon}\;,$ (73)

where $a(\tau)$ is as in (70), and we have simply relabelled
${\bar{x}}^{\scriptscriptstyle+}\to x^{\scriptscriptstyle+}$, and
$\bar{\tau},\bar{\tau}_{i}\to\tau,\tau_{i}$.

There are several features of this all-orders formula worth discussing. First,
as a consistency check, it is straightforward to check that in the vacuum
limit ($a\to 0$) the $x^{\scriptscriptstyle+}$ integral can again be performed
and one recovers the known results in Daikouji  _et al._ (1996); Ahmad  _et
al._ (2017); Mogull  _et al._ (2021). Second, similarly to Mogull  _et al._
(2021), a short set of rules summarises the LSZ reduction. The first three are
shared with the vacuum case Mogull  _et al._ (2021): (i) drop the $T$
integral, (ii) insert $\delta(\sum_{j=1}^{N}\tau_{j}/N)$ and (iii) take the
${\mathrm{d}}\tau_{i}$ and ${\mathrm{d}}\tau$ integrals over $\mathbb{R}$.
Here, beyond the vacuum case, there are additional rules: iv) drop all
worldline averages and (v) ‘introduce’ the divergent factor
$\int_{-\infty}^{0}-2i\tilde{p}^{\prime}\cdot a^{\infty}d\tau$ into the
exponential, which ensures that the proper-time integral is convergent in the
asymptotic past – we stress that this ‘by hand’ addition only occurs at the
level of these rules, it emerges naturally as part of LSZ reduction, as
described above.

Third, the change in integration range for the
${\mathrm{d}}\tau_{i}$-integrals can be understood as manifesting the fact
that $\mathcal{A}^{p^{\prime}p}_{N}$ is an asymptotic quantity, while the
purpose of $\delta(\sum_{j=1}^{N}\tau_{j}/N)$ is to ‘gauge’ the proper-time
translational symmetry of the system. Clearly neither of these features should
be particular to any choice of background that tends to at most a constant
asymptotically, and indeed they are the same in our plane wave background as
in vacuum.

Finally, we observe that $x_{cl}^{{\scriptscriptstyle+}}(\tau)$ in (70) solves
the classical worldline equation of motion with the boundary conditions
$\tfrac{1}{4}\dot{x}^{{\scriptscriptstyle+}}(-\infty)=p_{{\scriptscriptstyle-}}$
and
$\tfrac{1}{4}\dot{x}^{{\scriptscriptstyle+}}(\infty)=p^{\prime}_{{\scriptscriptstyle-}}$.
It is natural for this solution to appear in the amplitudes because, although
it may not be obvious, the stated boundary conditions are (particular
components of) those in play for the momentum space propagator, from which the
amplitude is constructed. We will show this in the following subsection, in
which we briefly digress from the master formula in order to investigate how
the Volkov wavefunctions arise from worldline path integrals.

###  III.2 Mixed boundary conditions and the Volkov wavefunction

Before moving on to the spinor case, we remark that one can, in fact, compute
the momentum-space propagator without going _explicitly_ via the position-
space representation. Returning to the original expression (5) for
$\mathcal{D}_{N}^{x^{\prime}x}$, we immediately perform the Fourier transform
(59). Now, the exponent $p^{\prime}\cdot x^{\prime}-p\cdot x$ in the Fourier
kernel is, under the path integral, the same as $p^{\prime}\cdot x(T)-p\cdot
x(0)$, and the spacetime integrals
${\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x$ can be interpreted as
${\mathrm{d}}^{4}x(T){\mathrm{d}}^{4}x(0)$. Hence, taking the Fourier
transform of (5) is equivalent to performing a path integral with a free
boundary, i.e. no _apparent_ restriction on the endpoints of the worldline.
There is though an alternative, but equivalent, perspective; consider the
change of the total action, $\delta S$, under the variations of the endpoints
of the worldline, $x(0)\rightarrow x(0)+\delta x_{0}$ and $x(T)\rightarrow
x(T)+\delta x_{T}$:

$\displaystyle\delta S$ $\displaystyle\equiv\delta
S_{B}+\delta\big{(}p^{\prime}.x(T)-p.x(0)\big{)}$
$\displaystyle=\left[\frac{1}{2}\dot{x}(0)+a(x(0))-p\right]\cdot\delta
x_{0}-\left[\frac{1}{2}\dot{x}(T)+a(x(T))-p^{\prime}\right]\cdot\delta x_{T}$
(74)

Integrating over $\delta x_{T}$ and $\delta x_{0}$ therefore returns delta
functions which impose the vanishing of the terms in square brackets of (74);
these are _Robin_ boundary conditions which relate the worldline end-point
momenta $\dot{x}$ to the end-point positions $x$ and the external asymptotic
momenta. It follows that the momentum-space propagator can be computed
alternatively from the path integral expression

$\displaystyle\mathcal{D}^{p^{\prime}p}_{N}=(-ie)^{N}\\!\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T}\,\int^{\dot{x}(T)2+a(x(T))=2p^{\prime}}_{\dot{x}(0)+2a(x(0))=2p}\mathcal{D}x(\tau)\,\mathrm{e}^{iS_{\mathrm{B}}[x(\tau)]}\prod_{i=1}^{N}V[\varepsilon_{i}\,k_{i}]\;.$
(75)

In the previous section we carried out the Fourier transform of
$\mathcal{D}^{x^{\prime}x}_{N}$ literally, to obtain
$\mathcal{D}^{p^{\prime}p}_{N}$. Expression (75) shows a more ‘direct’
approach to deriving the master formula in (63), through a modification of the
boundary conditions on the path integral. This fits in more naturally with the
‘worldline philosophy’ of incorporating all information into the worldline
path integral. Note that evaluation of (74) requires a worldline propagator
with different boundary conditions. Indeed, this helps explain a puzzle
arising in Corradini  _et al._ (2015) (Section 3, footnote 3), where a version
of the momentum space Master Formula was given that involves a Green function
with mixed boundary conditions: by expanding about a suitable reference
trajectory, (75) can be cast into a path integral for the fluctuation variable
that must satisfy the mixed boundary conditions $\dot{q}(0)=0=q(T)$.

This discussion prompts us to study the propagator $\mathcal{D}_{N}^{xp}$ with
mixed boundary conditions which, examining (75), is given by the integral

$\displaystyle\mathcal{D}^{xp}_{N}=(-ie)^{N}\\!\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T}\,\int^{x(T)=x}_{\dot{x}(0)+2a(x(0))=2p}\mathcal{D}x(\tau)\,\mathrm{e}^{iS_{\mathrm{B}}[x(\tau)]}\prod_{i=1}^{N}V[\varepsilon_{i}\,k_{i}]\;.$
(76)

To see the significance of the mixed propagator, consider the case $N=0$, that
is the tree level two-point function for the scalar field, with mixed boundary
conditions. In Feynman diagram language, this is just an external leg, Fourier
transformed at one end. Taking the momentum at this end onto the mass-shell,
i.e. performing LSZ reduction, we must recover the scalar Volkov
wavefunctions. These are solutions of the Klein-Gordon equation in a plane
wave background which reduce to $e^{\pm ip.x}$ in the asymptotic past/future
and thus represent incoming and outgoing particles in scattering amplitudes.

To confirm this, we first compute the path integral in (76) for $N=0$ (we drop
the product of vertex operators). We do not dwell on this step; the entire
integral turns out, unsuprisingly given the nature of the Volkov solutions and
hidden Gaussianity of the worldline path integral, to be equal to its semi-
classical value $\exp[iS_{cl}(T)]$, i.e. the exponential of the classical
action evaluated on the classical path obeying the mixed boundary conditions,
which is

$\displaystyle S_{cl}(T)=(p^{2}-m^{2}+i0^{{\scriptscriptstyle+}})T-p\cdot
x-\int_{x^{{\scriptscriptstyle+}}-4p_{-}T}^{x^{{\scriptscriptstyle+}}}\\!{\mathrm{d}}s\,\frac{2p\cdot
a(s)-a^{2}(s)}{4p_{\scriptscriptstyle-}}\;.$ (77)

The final step is to take $p^{2}\to m^{2}$ and identify the on-shell residue
via

$\lim_{p^{2}\to
m^{2}}-i(p^{2}-m^{2}+i0^{\scriptscriptstyle+})\int_{0}^{\infty}\\!{\mathrm{d}}T\,\mathrm{e}^{-im^{2}T}\,\mathrm{e}^{iS_{cl}(T)}\;.$
(78)

Of course it is clear from the preceding calculations how to proceed; we
perform the same manipulations as for the master formula, in particular taking
the $T\to\infty$ limit, immediately finding

$\displaystyle\lim_{p^{2}\rightarrow
m^{2}}-i(p^{2}-m^{2})\mathcal{D}^{xp}=\exp\bigg{[}-ip\cdot
x-i\int_{-\infty}^{x^{\scriptscriptstyle+}}\\!{\mathrm{d}}s\,\frac{2p\cdot
a(s)-a(s)^{2}}{4p_{{\scriptscriptstyle-}}}\bigg{]}{\equiv\varphi^{\text{\rm
in}}_{p}(x)}.$ (79)

The right-hand side is precisely the incoming scalar Volkov wavefunction
$\varphi^{\text{\rm in}}_{p}(x)$ which reduces to $e^{-ip\cdot x}$ in the
asymptotic past. A similar amputation of the propagator $\mathcal{D}^{px}_{0}$
(where the boundary conditions are swapped) yields the outgoing Volkov
wavefunctions, i.e. those which reduce to $e^{+i{\tilde{p}}^{\prime}\cdot x}$
in the asymptotic future. Of course the same procedure can be applied to the
spinor propagator, wherein the path integral with mixed boundary conditions
produces the spinor Volkov wavefunctions. Worldline path integrals analogous
to (76), with mixed boundary conditions, have also been used before, in a
similar context, to recover the exact solutions of the Klein-Gordon equation
in a constant external electromagnetic field Rajeev (2021). For numerical
studies of open line instantons see Degli Esposti and Torgrimsson (2022).

###  III.3 Spinor QED

Turning to LSZ reduction in spinor QED, we proceed from (60), writing
$\mathcal{S}_{N}^{x^{\prime}x}$ in terms of the kernels appearing in (51) and
evaluating the $\not{\partial}_{x^{\prime}}$, $\not{\partial}_{x}$ derivatives
(using integration by parts) in (60) to find

$\displaystyle\mathcal{M}_{Ns^{\prime}s}^{p^{\prime}p}$
$\displaystyle=i\lim_{p^{\prime 2},p^{2}\rightarrow
m^{2}}\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\mathrm{e}^{i\tilde{p}^{\prime}\cdot
x^{\prime}-ip\cdot
x}\bar{u}_{s^{\prime}}(p^{\prime})(\not{p}^{\prime}-m)\Bigl{\\{}(-\not{p}^{\prime}{+\delta\not{a}(x^{\prime{\scriptscriptstyle+}})}-m)\,\mathcal{K}_{N}^{x^{\prime}x}+e\sum_{i=1}^{N}\not{\varepsilon}_{i}e^{ik_{i}\cdot
x^{\prime}}\,\mathcal{K}_{N-1}^{x^{\prime}x}\Bigr{\\}}(\not{p}-m)u_{s}(p)\,.$
(80)

Next, following Ahmadiniaz  _et al._ (2022) we use the _on-shell_ relation
$\bar{u}_{s^{\prime}}(p^{\prime})(\not{p}^{\prime}+m)^{-1}=\bar{u}_{s^{\prime}}(p^{\prime})(2m)^{-1}$,
(which is allowed since it does not remove the associated pole, or affect the
final expression), and likewise for $(\not{p}+m)^{-1}u_{s}(p)$ to find

$\displaystyle\mathcal{M}_{Ns^{\prime}s}^{p^{\prime}p}=i\lim_{p^{\prime
2},p^{2}\rightarrow m^{2}}\frac{1}{2m}\int
d^{4}x^{\prime}d^{4}x\,e^{i\tilde{p}^{\prime}\cdot x^{\prime}-ip\cdot
x}\,\bar{u}_{s^{\prime}}(p^{\prime})$ $\displaystyle(p^{\prime
2}-m^{2})\Bigl{\\{}\Bigl{[}-1{+\frac{1}{2m}\delta\not{a}(x^{\prime+}))}\Bigr{]}\,\mathcal{K}_{N}^{x^{\prime}x}$
$\displaystyle\hskip
40.00006pt+\frac{e}{2m}\sum_{i=1}^{N}\not{\varepsilon}_{i}e^{ik_{i}\cdot
x^{\prime}}\,\mathcal{K}_{N-1}^{x^{\prime}x}\Bigr{\\}}(p^{2}-m^{2})u_{s}(p)\,.$
(81)

Due to the worldline approach being based on the second order formalism of
QED, the exponent under the proper time integral of the spinor amplitude
contains the same terms as for the scalar amplitude – in particular the
parameter and proper time integrals produce (free) scalar propagators. Hence
it suffices to revise the scalar case for this argument. The difference lies
in the spin factor of the kernel, the subleading contibutions (those
proportional to $\mathcal{K}_{N-1}$), and the $\delta
a(x^{{\scriptscriptstyle+}\prime})$ factor from the covariant derivative.
However the differences do not impede processing the $T$, and later
$\tau_{0}$, proper time integrals as for scalars. The result is that the LSZ
amputation is realised in precisely the same way, by taking
$T,\tau_{0}\rightarrow\infty$ as in equations (64)–(69). Moreover, after
taking the Fourier transform, the conservation of momenta enforced by
$\delta(x^{+}-x^{\prime+}+2g^{+}+2p^{\prime+}T)$ sends

$a(x^{\prime+})\to
a\bigl{(}2Tp^{\prime+}+x^{{\scriptscriptstyle+}}+2g^{\scriptscriptstyle+}\bigr{)}\,.$
(82)

The LSZ truncation projects onto asymptotic late time, taking
$a(x^{\prime+})\to a^{\infty}$ when $T\rightarrow\infty$, cancelling the field
dependent term in square brackets of (III.3). One may then express (III.3) in
terms of the momentum space kernel

$\mathcal{M}_{Ns^{\prime}s}^{p^{\prime}p}=i\lim_{p^{\prime 2},p^{2}\rightarrow
m^{2}}\frac{1}{2m}\bar{u}_{s^{\prime}}(p^{\prime})(p^{\prime
2}-m^{2})\Bigl{\\{}-\mathcal{K}_{N}^{{\tilde{p}}^{\prime}p}+\frac{e}{2m}\sum_{i=1}^{N}\not{\varepsilon}_{i}\,\mathcal{K}_{N-1}^{({\tilde{p}}^{\prime}+k_{i})p}\Bigr{\\}}(p^{2}-m^{2})u_{s}(p)\,.$
(83)

Now we address the subleading terms. These are seen to have poles not in the
required mass-shell $p^{\prime 2}-m^{2}$, but rather in
$((p^{\prime}+k_{i})^{2}-m^{2})$. Contributions involving these shifted poles
hence vanish after taking the on-shell limit of $(p^{\prime
2}-m^{2})/((p^{\prime}+k_{i})^{2}-m^{2})$. This is a remarkable generalisation
of the vacuum case Ahmadiniaz  _et al._ (2022). We can be more precise with
how this cancellation comes about. In the kernel of the subleading terms,
$\mathcal{K}_{N-1}^{(\tilde{p}^{\prime}+k_{i})p}$, one must first remove an
$\varepsilon_{i}$ and $k_{i}$, and then replace $a^{\infty}$ with
$a^{\infty}+k_{i}$ in (III.1). This operation leaves $\tilde{p}^{\prime}+K$
invariant, but it does affect the term
$\int^{\infty}_{0}d\tau\,p^{\prime}\cdot\delta a(\tau)$, which was convergent
as $\tau\rightarrow\infty$, but now produces a rapidly oscillating phase;
noting that the proper time integral calculates the Laplace transform of the
function $F(T)$ in (64), the Abelian final value theorem can be invoked to
confirm that the subleading contributions must vanish.

Since the manipulations are similar to the scalar case, let us simply record
the spinor amplitude in its final form as

$\displaystyle\mathcal{M}_{Ns^{\prime}s}^{p^{\prime}p}$
$\displaystyle={\sum_{S=1}^{N}\sum_{\\{i_{1}:i_{S}\\}}\mathcal{M}_{NSs^{\prime}s}^{\\{i_{1}:i_{S}\\}p^{\prime}p}}$
(84)
$\displaystyle{\mathcal{M}_{NSs^{\prime}s}^{\\{i_{1}:i_{S}\\}p^{\prime}p}}$
$\displaystyle=(-ie)^{N}(2\pi)^{3}\delta_{\perp,-}({\tilde{p}}^{\prime}+K-p)\int_{-\infty}^{\infty}\\!{\mathrm{d}}x^{{\scriptscriptstyle+}}e^{i(K+p^{\prime}-p)_{\scriptscriptstyle+}x^{{\scriptscriptstyle+}}}\int_{-\infty}^{\infty}\prod_{i=1}^{N}{\mathrm{d}}\tau_{i}\,\delta\bigg{(}\sum_{j=1}^{N}\frac{\tau_{j}}{N}\bigg{)}$
(85)
$\displaystyle\times\mathrm{e}^{-i\int_{-\infty}^{0}\bigl{[}2\tilde{p}^{\prime}\cdot
a(\tau)-a^{2}(\tau)\bigr{]}d\tau-i\int_{0}^{\infty}\bigl{[}2p^{\prime}\cdot\delta
a(\tau)-\delta
a^{2}(\tau)\bigr{]}{\mathrm{d}}\tau-2i\sum_{i=1}^{N}\bigl{[}\int_{-\infty}^{\tau_{i}}k_{i}\cdot
a(\tau){\mathrm{d}}\tau-i\varepsilon_{i}\cdot a(\tau_{i})\bigr{]}}$
$\displaystyle\times e^{i({\tilde{p}}^{\prime}+p)\cdot
g-i\sum_{i,j=1}^{N}\bigl{(}\frac{|\tau_{i}-\tau_{j}|}{2}k_{i}\cdot
k_{j}-i\,\mathrm{sgn}(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot
k_{j}+\delta(\tau_{i}-\tau_{j})\varepsilon_{i}\cdot\varepsilon_{j}\bigr{)}}\bigg{|}_{\varepsilon_{i_{S+1}}...\varepsilon_{i_{N}}}^{\varepsilon_{i_{1}}...\varepsilon_{i_{S}}=0}$
$\displaystyle\times\frac{1}{2m}\bar{u}_{s^{\prime}}(p^{\prime})\mathrm{Spin}(\tilde{f}_{i_{1}:i_{S}})u_{s}(p)\,.$

After LSZ reduction, the argument of the exponential in the spin factor, (47),
takes the following form

$-\int_{-\infty}^{\infty}\\!{\mathrm{d}}\tau\,[\eta\cdot
f\cdot\eta+{\theta}\cdot\eta]-\int_{-\infty}^{\infty}\\!{\mathrm{d}}\tau\int_{-\infty}^{\infty}{\mathrm{d}}\tau^{\prime}\bigl{[}\eta\cdot
f(\tau)\cdot\mathfrak{G}(\tau,\tau^{\prime})\cdot\theta(\tau^{\prime})+\tfrac{1}{4}\theta(\tau)\cdot\mathfrak{G}(\tau,\tau^{\prime})\cdot\theta(\tau^{\prime})\bigr{]}\,;$
(86)

the worldline average in the fermion Green function is also now understood to
be $T\langle\\!\langle f\rangle\\!\rangle=\int^{\infty}_{-\infty}d\tau
f(\tau)$. Also, the background gauge potential, $a$, and field strength, $f$,
are understood to be functions of the classical solution
$x_{cl}^{{\scriptscriptstyle+}}(\tau)$ as shown in (70). Finally, the sums in
the first line of (84) are – as usual – over the assignation of $S$ photons
out of $N$ to the spin part of the vertex operator.

##  IV Examples

In this section we provide checks on our amplitude Master Formulae (III.1) and
(84), showing by comparison with the existing literature that they are
consistent with results expected from Furry picture perturbation theory.

###  IV.1 $N=1$, nonlinear Compton scattering in scalar QED

The case $N=1$ describes single photon emission from a (scalar) electron in a
plane wave background, which is the well-studied process of ‘nonlinear Compton
scattering’. In this case, several parts of the Master Formulae (III.1)
simplify immediately. First, the delta function fixes $\tau_{1}=0$. Next, the
gauge field is evaluated as

$\displaystyle
a(\tau)=\begin{cases}a(x^{{\scriptscriptstyle+}}+2p^{{\scriptscriptstyle+}}\tau)\;,&\tau<0\\\
a(x^{{\scriptscriptstyle+}}+2p^{\prime{\scriptscriptstyle+}}\tau)\;,&\tau>0\;.\end{cases}$
(87)

This form facilitates an easy conversion of integrals over proper time $\tau$
to integrals over lightfront time $x^{{\scriptscriptstyle+}}$, which are
expected in the standard formalism (see also Ilderton and Torgrimsson (2016)).
Specifically, we can conveniently treat the positive and negative $\tau$
regions separately. The field-dependent terms in the exponent of the master
formula then reduce to

$\displaystyle-i\int_{-\infty}^{0}\\!{\mathrm{d}}\tau[2\tilde{p}^{\prime}\cdot
a(\tau)-a^{2}(\tau)]-i\int_{0}^{\infty}\\!{\mathrm{d}}\tau[2p^{\prime}\cdot\delta
a(\tau)-\delta a^{2}(\tau)]-2i\int_{-\infty}^{0}\\!\
{\mathrm{d}}\tau\,k_{1}\cdot a(\tau)$ (88)
$\displaystyle=-i\int_{-\infty}^{x^{{\scriptscriptstyle+}}}\\!{\mathrm{d}}s^{\scriptscriptstyle+}\,\frac{2p\cdot
a(s^{{\scriptscriptstyle+}})-a^{2}(s^{{\scriptscriptstyle+}})}{2p^{{\scriptscriptstyle+}}}-i\int_{x^{{\scriptscriptstyle+}}}^{\infty}\\!{\mathrm{d}}s^{{\scriptscriptstyle+}}\,\frac{2p^{\prime}\cdot\delta
a(s^{{\scriptscriptstyle+}})-\delta
a^{2}(s^{{\scriptscriptstyle+}})}{2p^{\prime{\scriptscriptstyle+}}}\;,$ (89)

in which we simply inserted (87) and used momentum conservation in the
transverse directions to eliminate $k_{1}$ in favour of $p^{\prime}$ and $p$.
With this, expanding (III.1) for $N=1$ to linear order in $\varepsilon_{1}$,
and using the Fourier representation of the momentum conserving
$\delta$-functions shows that the amplitude is equivalent to

$\displaystyle\mathcal{A}^{p^{\prime}p}_{1}=-ie\int{\mathrm{d}}^{4}x\,\left\\{\tilde{p}^{\prime}_{\mu}+p_{\mu}-2a_{\mu}(x^{{\scriptscriptstyle+}})\right\\}\varepsilon^{\mu}_{1}e^{ik_{1}.x}\varphi_{p^{\prime}}^{\text{out}}(x)\varphi^{\text{\rm
in}}_{p}(x)\,,$ (90)

where $\varphi^{\text{\rm in}}_{p}$ is the incoming scalar Volkov wavefuncion
of (79) while $\varphi^{\text{\rm out}}_{p^{\prime}}$ is the outgoing
wavefunction,

$\displaystyle\varphi^{\text{out}}_{p^{\prime}}(x)={\mathrm{e}^{i\tilde{p}^{\prime}\cdot
x}}\exp\left[-i\int_{x^{{\scriptscriptstyle+}}}^{\infty}\\!{\mathrm{d}}s^{{\scriptscriptstyle+}}\,\frac{2p^{\prime}\cdot\delta
a(s^{{\scriptscriptstyle+}})-\delta
a^{2}(s^{{\scriptscriptstyle+}})}{2p^{\prime{\scriptscriptstyle+}}}\right].$
(91)

Expression (90) is precisely the expected result for nonlinear Compton
scattering in scalar QED, providing a positive check on our master formula.

We stress that the method we employed above to process the worldline integrals
was meant only to allow direct comparison with existing results. It is _not_
the approach we wish to take in future work; instead, we will use the
worldline representation to deal _directly_ with the $\tau$-integrals. Since
the major advantages of the worldline approach include that (a) one does not
have to split amplitudes into sectors according to permutations of external
legs, and (b) internal momentum integrals are recast in terms of the proper
time integral, we expect this to provide some advantage over the standard
formalism, at least in various physical limits of interest. This will be
discussed elsewhere.

###  IV.2 $N=1$, nonlinear Compton scattering in spinor QED

Let us now confirm the $N=1$ case for spinor QED, which requires expanding the
Master Formula (84) to linear order in $\varepsilon_{1}$. Since the field
dependence of the exponent in for spinor QED contains that of scalar QED one
may write the resulting amplitude using the scalar Volkov wavefunctions, (91),
as

$\mathcal{M}_{1s^{\prime}s}^{p^{\prime}p}=-ie\frac{1}{2m}\int
d^{4}x\,e^{ik_{1}\cdot
x}\varphi_{p^{\prime}}^{\text{out}}(x)\varphi_{p}^{\text{\rm
in}}(x)\bar{u}_{s^{\prime}}(p^{\prime})\Bigl{[}(\tilde{p}^{\prime}+p-2a(x^{{\scriptscriptstyle+}}))\cdot\varepsilon_{1}\mathrm{Spin}(\emptyset)+\mathrm{Spin}(\tilde{f_{1}})\Bigr{]}u_{s}(p)\,,$
(92)

requiring only the evaluation of the spin factor (we have again used the
Fourier representation of the $\delta$-functions). Before embarking upon the
comparison to the standard formalism, we should emphasise that the approach
outlined here, namely writing in terms of spacetime averages with steps to
follow, is necessary to make the connection to the perturbative Furry picture
with Volkov wavefunctions. However, this would be inefficient for practical
worldline calculations.

The spin factors are determined using (48) and (49) under the LSZ reduction
(86) and the inverse symbol map, (II.2). Because of the nilpotency of $f$ one
has, _under the inverse symbol map_ ,
$\exp(-\int^{\infty}_{-\infty}\\!{\mathrm{d}}\tau\,\eta\cdot
f\cdot\eta)=1-\int^{\infty}_{-\infty}\\!{\mathrm{d}}\tau\eta\cdot f\cdot\eta$,
and therefore the factor without photon insertion is readily determined to be

$\mathrm{Spin}(\emptyset)=\Bigl{[}1-\frac{1}{2p^{\prime+}}\not{n}\delta\not{a}(x^{{\scriptscriptstyle+}})\Bigr{]}\Bigl{[}1+\frac{1}{2p^{+}}\not{n}\not{a}(x^{+})\Bigr{]}\,,$
(93)

where we have already transformed the parameter integral to a spacetime
average and computed its value. This is simply the Dirac matrix structure
necessary to construct the spinor Volkov wavefunctions.

Let us next treat the single photon spin factor,
$\mathrm{Spin}(\tilde{f}_{1})$. Beginning with the Grassmann integral with one
photon insertion, provided in (49) we apply the inverse symbolic map in (II.2)
and realise the LSZ reduction according to (86). The various worldline
averages are then transformed into their corresponding spacetime averages as
was done in the $N=1$ scalar case, to find

$\displaystyle\mathrm{Spin}(\tilde{f}_{1})$
$\displaystyle=-\frac{1}{2}[\not{k}_{1},\not{\varepsilon}_{1}]+k_{1}^{+}\varepsilon_{1}\cdot\Bigl{(}-\frac{\delta
a(x^{+})}{2p^{\prime+}}+\frac{a(x^{+})}{2p^{+}}\Bigr{)}+\varepsilon_{1}\cdot\Bigl{(}\frac{\delta
a(x^{+})}{2p^{\prime+}}+\frac{a(x^{+})}{2p^{+}}\Bigr{)}\frac{1}{2}[\not{k}_{1},\not{n}]$
$\displaystyle+k_{1}^{+}\frac{1}{2}\Bigl{[}\not{\varepsilon}_{1},\frac{\delta\not{a}(x^{+})}{2p^{\prime+}}+\frac{\not{a}(x^{+})}{2p^{+}}\Bigr{]}+\Bigl{[}k_{1}\cdot\Bigl{(}\frac{\delta
a(x^{+})}{2p^{\prime+}}+\frac{a(x^{+})}{2p^{+}}\Bigr{)}+2k_{1}^{+}\Bigl{(}\frac{\delta
a(x^{+})}{2p^{\prime+}}\cdot\frac{a(x^{+})}{2p^{+}}\Bigr{)}\Bigr{]}\not{n}\not{\varepsilon}_{1}$
$\displaystyle+\frac{2k_{1}^{+}}{2p^{\prime+}2p^{+}}\varepsilon_{1}\cdot\big{[}a(x^{+})\delta\not{a}(x^{+})+\delta
a(x^{+})\not{a}(x^{+})\big{]}\not{n}+(k_{1}+a^{\infty})_{\mu}\varepsilon_{1\nu}n_{\alpha}\Bigl{(}\frac{\delta
a(x^{+})}{2p^{\prime+}}-\frac{a(x^{+})}{2p^{+}}\Bigr{)}_{\beta}i\gamma_{5}\epsilon^{\mu\nu\alpha\beta}\,.$
(94)

Next, we express the photon momentum, $k_{1}$, in terms of the electron
momenta and asymptotic value of the background field. For the ${+,\perp}$
components we can use momentum conservation,
$k_{1}^{+,\perp}=(p-\tilde{p}^{\prime})^{+,\perp}$. The $k_{1}^{-}$ component
requires us to carry out an an integration by parts with respect to $x^{+}$.
We illustrate this step, to be applied to the various $\not{k}_{1}$ terms in
(IV.2), with the following manipulation:

$\int\\!{\mathrm{d}}^{4}x\,\mathrm{e}^{ik_{1}\cdot
x}{k_{1}^{\mu}}\varphi_{p^{\prime}}^{\text{out}}(x)\varphi_{p}^{\text{\rm
in}}(x)=\int\\!{\mathrm{d}}^{4}x\,\mathrm{e}^{ik_{1}\cdot
x}\Bigl{[}\Bigl{(}\frac{2p\cdot
a(x^{+})-a(x^{+})^{2}}{2p^{+}}-\frac{2p^{\prime}\cdot\delta a(x^{+})-\delta
a(x^{+})^{2}}{2p^{\prime+}}\Bigr{)}{n^{\mu}}+{p^{\mu}}-{\tilde{p}^{\prime\mu}}\Bigr{]}\varphi_{p^{\prime}}^{\text{out}}(x)\varphi_{p}^{\text{\rm
in}}(x)\,,$ (95)

In fact, if additional factors of $a(x^{+})$ appear under the above integral,
in turns out that the additional derivatives produced by integrating by parts
always contract away. Therefore (95) can be used throughout (IV.2). Moreover,
applying the above procedure to $k_{1}$ in the $\gamma_{5}$ term of (IV.2),
one can see that in effect $k_{1}^{\mu}\to p^{\mu}-\tilde{p}^{\prime\mu}$,
since the two $n^{\mu}$ contract to zero against the Levi-Civita tensor. In
fact the only term in which the $n^{\mu}$ part of (95) survives after these
replacements is the first term on the RHS of (IV.2).

Last, since we are taking the on-shell limit we may use the Dirac equation for
the sandwiching spinors so as to send their corresponding $\not{p}$ and
$\not{p}^{\prime}$ to $m$, anti-commutating where necessary. Again,
illustrating this step with the $\gamma_{5}$ term in (IV.2) we rewrite
$\gamma_{5}$ in terms of products of four matrices using (II.2). After acting
on the spinor solutions at most three matrices will remain. After this
process, the $\gamma_{5}$ term, as it appears in the amplitude (92), becomes

$\displaystyle(k_{1}+a^{\infty})_{\mu}\varepsilon_{1\nu}n_{\alpha}\Bigl{(}\frac{\delta
a(x^{+})}{2p^{\prime+}}-$
$\displaystyle\frac{a(x^{+})}{2p^{+}}\Bigr{)}_{\beta}i\gamma_{5}\epsilon^{\mu\nu\alpha\beta}=(p^{+}+p^{\prime+})\frac{1}{2}\Bigl{[}\frac{\delta\not{a}(x^{+})}{2p^{\prime+}}-\frac{\not{a}(x^{+})}{2p^{+}},\not{\varepsilon}_{1}\Bigr{]}+(p+p^{\prime})\cdot\varepsilon_{1}\not{n}\Bigl{(}\frac{\delta\not{a}(x^{+})}{2p^{\prime+}}-\frac{\not{a}(x^{+})}{2p^{+}}\Bigr{)}$
$\displaystyle+(p+p^{\prime})\cdot\Bigl{(}\frac{\delta
a(x^{+})}{2p^{\prime+}}-\frac{a(x^{+})}{2p^{+}}\Bigr{)}\not{\varepsilon}_{1}\not{n}-m\Big{\\{}\not{\varepsilon}_{1},\not{n}\Bigl{(}\frac{\delta\not{a}(x^{+})}{2p^{\prime+}}-\frac{\not{a}(x^{+})}{2p^{+}}\Bigr{)}\Big{\\}}\,.$
(96)

Using the above steps to replace $k_{1}^{\mu}$ in the remaining terms of
(IV.2), after some algebra one may gather terms to find that

$\bar{u}_{s^{\prime}}(p^{\prime})\Bigl{\\{}(\tilde{p}^{\prime}+p-2a(x^{+}))\cdot\varepsilon_{1}\mathrm{Spin}(\emptyset)+\mathrm{Spin}(\tilde{f_{1}})\Bigr{\\}}u_{s}(p)=2m\bar{u}_{s^{\prime}}(p^{\prime})\Bigl{\\{}\not{\varepsilon}_{1}-\frac{1}{2p^{\prime+}}\not{n}\delta\not{a}(x^{+})\not{\varepsilon}_{1}+\frac{1}{2p^{+}}\not{\varepsilon}_{1}\not{n}\not{a}(x^{+})\Bigr{\\}}u_{s}(p)\,,$
(97)

and hence

${\mathcal{M}^{p^{\prime}p}_{1s^{\prime}s}}=-ie\int\\!{\mathrm{d}}^{4}x\,\mathrm{e}^{ik\cdot
x}\Psi_{p^{\prime},s^{\prime}}^{\text{out}}(x)\not{\varepsilon}_{1}\Psi_{p,s}^{\text{\rm
in}}(x)\,,$ (98)

where we have used the spinor Volkov wavefunctions, which read

$\displaystyle\Psi_{p,s}^{\text{in}}(x)$
$\displaystyle=\Bigl{[}1+\frac{1}{2p^{+}}\not{n}\not{a}(x^{+})\Bigr{]}u_{s}(p)\varphi_{p}^{\text{\rm
in}}(x)\,,$ (99) $\displaystyle\Psi_{p^{\prime},s^{\prime}}^{\text{out}}(x)$
$\displaystyle=\bar{u}_{s^{\prime}}(p^{\prime})\Bigl{[}1-\frac{1}{2p^{\prime+}}\not{n}\delta\not{a}(x^{+})\Bigr{]}\varphi_{p^{\prime}}^{\text{out}}(x)\,.$
(100)

This successfully verifies that the worldline approach reproduces the known
amplitude for the $N=1$ process.

###  IV.3 $N=2$, double nonlinear Compton scattering in scalar QED

To complete our discussion of the relevant structures in scalar QED we must
also consider the case $N=2$, where the so-called seagull vertex (the four-
point scalar-photon-photon-scalar vertex) first appears. We will describe the
way this works briefly here, as the calculations proceed largely as for $N=1$,
leaving the details for the Appendix. Expanding (III.1), there are now two
$\tau$-integrals, with one, say $\tau_{2}$, fixed by the worldline delta
function in (III.1), and the other, say $\tau_{1}$, remaining. The mapping
onto Feynman diagrams is most natural: the contributions from $\tau_{1}>0$ and
$\tau_{1}<0$ recover one each of the expected contributions from the two
diagrams with two 3-point vertices, with $\tau_{1}$ being mapped to the
lightfront time of one vertex. The seagull contribution is picked up from the
term in (III.1) which goes like $\varepsilon_{1}\cdot\varepsilon_{2}$; this
comes with a delta function with support at exactly $\tau_{1}=0$, hence
leaving only a single unevaluated integral, as expected. The full calculation
is presented in Appendix A.

##  V Conclusions

We have presented worldline Master Formulae for all-multiplicity tree level
scattering amplitudes of 2 massive charged particles and $N$ photons, in a
plane wave background, in both scalar and spinor QED. The background field may
have arbitrary strength and functional profile, and is treated without
approximation throughout. This is particularly relevant as the target
application of our results is to laser-matter interactions in the _high
intensity_ regime where the field is characterised by a dimensionless strength
(the coupling to matter) larger than unity, and hence must be treated without
recourse to perturbation theory.

Our Master Formulae have been derived using the worldline approach to quantum
field theory. While several previous publications have derived wordline Master
Formulae for various correlation functions in vacuum, or even at higher loop
level in background fields, our focus here has been on scattering amplitudes
involving external matter. As such it was necessary to identify the _worldline
description_ of LSZ reduction in a plane wave background. We found this to be
a fairly direct generalisation of the known worldline prescription for LSZ
amplitudes in vacuum Bonocore (2021); Mogull  _et al._ (2021). A second
notable generalisation from known results in vacuum holds for the spinor case:
namely that in the second order formalism, which implies a split into
‘leading’ and ‘subleading’ terms, only the former survives the on-shell limit
once the LSZ prescription is imposed. Furthermore, the background field
dependent part of this leading term also drops out in the asymptotic limit.
This allows for a large number of terms to be discarded (and in the vacuum
case allowed for the gauge invariance of the amplitudes to be manifest).

We have checked our results against the existing literature, which contains
only _low_ -multiplicity amplitudes derived using Feynman rules. Explicitly,
these are the cases $N=1$ and $N=2$, or single and double nonlinear Compton
scattering. Moving beyond scattering amplitudes, we have also seen how to
recover _off-shell_ quantities, in particular the scalar and spinor
correlation functions dressed by the background and the Volkov wavefunctions,
from worldline path integrals. The latter is a particularly interesting case
as it exposes the relevance of mixed boundary conditions; the relevant path
integrals carry Dirichlet conditions at one limit, representing the local
spacetime argument of the wavefunction, and Robin boundary conditions at the
other limit, encoding the asymptotic momentum characterising the Volkov
solution.

It is fair to say that the Master Formulae for amplitudes we have derived here
still require, for a chosen number of photons $N$, some processing in order to
extract all their physical content. In future work we will pursue methods of
evaluating the remaining proper time integrals in an efficient manner, or in
an approximate manner relevant to interesting physical regimes. Here, benefit
should be gained by not breaking the parameter integrals into ordered sectors
corresponding to photon permutations, which will maximally exploit the
calculational efficiency. Constructing observables from our amplitudes at
$N>2$ (which are lacking in the literature) will help to benchmark numerical
codes which approximate multi-photon processes using sequential single-photon
emissions. It would be revealing to compare our expressions with the compact
all-multiplicity results of Adamo  _et al._ (2020, 2023). We also plan to
generalise our results to higher loop orders, in order to pursue the Ritus-
Narozhny conjecture on the behaviour of loop corrections at very high
intensity, see Fedotov (2017); Fedotov  _et al._ (2023) for reviews.

###### Acknowledgements.

The authors are supported by the EPSRC Standard Grants EP/X02413X/1 (PC, JPE)
and EP/X024199/1 (AI, KR), and the STFC consolidator grant ST/X000494/1 (AI).

##  Appendix A Master formula check for $N=2$

In this appendix we confirm that the master formula (III.1) correctly
reproduces, at $N=2$, the amplitude for ‘double nonlinear Compton scattering’
Seipt and Kampfer (2012); Mackenroth and Di Piazza (2013) in scalar QED, that
is the emission of two photons from a particle in a plane wave background. (By
crossing symmetry this is directly related to the amplitude for the Compton
effect in the background.) Recall that in scalar QED, the standard approach
would require evaluation of three separate Feynman diagrams – conveniently
combined into one calculation on the worldline – one of which contains the
four-point seagull vertex.

Starting from (III.1) with $N=2$, the LSZ factor
$\delta(\tau_{1}/2+\tau_{2}/2)$ means that we have only one non-trivial
proper-time integral, over, say, $\tau_{1}$. It is convenient to split this
integral into three pieces and analyse each separately; we split the
integration range into $-\infty<\tau_{1}<0^{\scriptscriptstyle-}$,
$0^{\scriptscriptstyle-}<\tau_{1}<{0^{\scriptscriptstyle+}}$ and
$0^{\scriptscriptstyle+}<\tau_{1}<\infty$, and refer henceforth to the
corresponding contribution to the amplitudes as
$\mathcal{A}^{p^{\prime}p}_{2-}$, $\mathcal{A}^{p^{\prime}p}_{2\delta}$ and
$\mathcal{A}^{p^{\prime}p}_{2+}$, respectively.

###  A.1 $\tau_{1}\in(0,\infty)$

When $\tau_{1}>0$, the field-independent terms in the exponential of (III.1)
reduce to

$\displaystyle
i(\tilde{p}^{\prime}+p)\cdot(k_{1}-k_{2})\tau_{1}+\varepsilon_{1}\cdot(\tilde{p}^{\prime}+p-k_{2})+\varepsilon_{2}\cdot(\tilde{p}^{\prime}+p+k_{1})-2i\tau_{1}k_{1}\cdot
k_{2}+i\left(K_{{\scriptscriptstyle+}}+p^{\prime}_{{\scriptscriptstyle+}}-p_{{\scriptscriptstyle+}}\right)x^{{\scriptscriptstyle+}}\,.$
(101)

The gauge field at the interaction points $\pm\tau_{1}$ (indicating the
insertion point of photon with momentum $k_{1}$) takes the values

$\displaystyle a(\tau_{1})$
$\displaystyle=a(x^{{\scriptscriptstyle+}}+\tau_{1}(2p^{\prime{\scriptscriptstyle+}}+k_{1}^{\scriptscriptstyle+}-k_{2}^{\scriptscriptstyle+})\,,$
(102) $\displaystyle{a(-\tau_{1})}$
$\displaystyle=a(x^{{\scriptscriptstyle+}}-\tau_{1}(2p^{\scriptscriptstyle+}+k_{1}^{\scriptscriptstyle+}-k_{2}^{\scriptscriptstyle+})\,.$
(103)

This motivates us to make the change of variable $x^{{\scriptscriptstyle+}}\to
x^{{\scriptscriptstyle+}}-\tau_{1}(2p^{\scriptscriptstyle+}+k_{1}^{\scriptscriptstyle+}-k_{2}^{\scriptscriptstyle+})$,
such that the field-independent terms (101) transform to

$\displaystyle\mathcal{T}_{0}$ $\displaystyle\equiv
i\left(4(p_{+}+k_{1+})q^{{\scriptscriptstyle+}}-2q^{2}_{\perp}-2m^{2}+i0^{{\scriptscriptstyle+}}\right)\tau_{1}+\varepsilon_{1}\cdot(2\tilde{p}^{\prime}+k_{1})+\varepsilon_{2}\cdot(\tilde{p}^{\prime}+p+k_{1})+i\left(K_{{\scriptscriptstyle+}}+p^{\prime}_{{\scriptscriptstyle+}}-p_{{\scriptscriptstyle+}}\right)x^{{\scriptscriptstyle+}}$
(104) $\displaystyle-i(2p^{\prime}+a^{\infty})a^{\infty}\tau_{1}\,,$

where we have defined $q=p-k_{2}$ and used the fact the momenta are on-shell
to simplify. We shall shortly need the last term
$-i(2p^{\prime}+a^{\infty})a^{\infty}\tau_{1}$ to simplify some of the field
dependent terms. Before going into that, we return to the exponent of (III.1)
and note that the following field-dependent term is already sufficiently
simplified.

$\displaystyle\mathcal{T}_{1}$
$\displaystyle\equiv-2\sum_{i=1}^{N}\varepsilon_{i}\cdot
a(\tau_{i})\rightarrow-2\varepsilon_{1}\cdot
a(x^{{\scriptscriptstyle+}})-2\varepsilon_{2}\cdot
a(x^{{\scriptscriptstyle+}}+4q^{{\scriptscriptstyle+}}\tau_{1})\,.$ (105)

The rest of the field-dependent terms combine with
$-i(2p^{\prime}+a^{\infty})a^{\infty}\tau_{1}$ from (104) to yield

$\displaystyle\mathcal{T}_{2}-$ $\displaystyle
i(2p^{\prime}+a^{\infty})a^{\infty}\tau_{1}\equiv$ (106)
$\displaystyle-2i\sum_{i=1}^{N}\int_{-\infty}^{\tau_{i}}\\!{\mathrm{d}}\tau
k_{i}\cdot
a(\tau)-i\int_{-\infty}^{0}\\!{\mathrm{d}}\tau[2\tilde{p}^{\prime}\cdot
a(\tau)-a^{2}(\tau)]-i\int_{0}^{\infty}\\!{\mathrm{d}}\tau[2p^{\prime}\cdot\delta
a(\tau)-\delta a^{2}(\tau)]-i(2p^{\prime}+a^{\infty})\cdot a^{\infty}\tau_{1}$
$\displaystyle=-2i\sum_{i=1}^{N}\int_{-\infty}^{\tau_{i}}\\!{\mathrm{d}}\tau\,k_{i}\cdot
a(\tau)-i\int_{-\infty}^{\tau_{1}}\\!{\mathrm{d}}\tau\,[2\tilde{p}^{\prime}\cdot
a(\tau)-a^{2}(\tau)]-i\int_{\tau_{1}}^{\infty}\\!{\mathrm{d}}\tau[2p^{\prime}\cdot\delta
a(\tau)-\delta a^{2}(\tau)]\;.$

We now use the dependence of $a_{\mu}(x_{\rm{cl}}(\tau)$ on the classical
solution to transform the proper-time integrals into space-time integrals and
simplify the above terms as

$\displaystyle-2i\sum_{i=1}^{N}\int_{-\infty}^{\tau_{i}}\\!{\mathrm{d}}\tau\,k_{i}\cdot
a(\tau)-i\int_{-\infty}^{\tau_{1}}\\!{\mathrm{d}}\tau\,[2\tilde{p}^{\prime}\cdot
a(\tau)-a^{2}(\tau)]-i\int_{\tau_{1}}^{\infty}\\!{\mathrm{d}}\tau\,[2p^{\prime}\cdot\delta
a(\tau)-\delta a^{2}(\tau)]$ (107)
$\displaystyle=-i\int_{-\infty}^{x^{{\scriptscriptstyle+}}}\frac{2p.a(s)-a^{2}(s)}{2p^{\scriptscriptstyle+}}ds-i\int_{x^{{\scriptscriptstyle+}}}^{x^{{\scriptscriptstyle+}}+4q^{{\scriptscriptstyle+}}\tau_{1}}\\!{\mathrm{d}}s\,\frac{2q\cdot
a(s)-a^{2}(s)}{2q^{\scriptscriptstyle+}}-i\int_{x^{{\scriptscriptstyle+}}+4q^{{\scriptscriptstyle+}}\tau_{1}}^{\infty}\\!{\mathrm{d}}s\,\frac{2p^{\prime}\cdot\delta
a(s)-\delta a^{2}(s)}{2p^{\prime{\scriptscriptstyle+}}}\,,$ (108)

where we have used momentum conservation to replace
$\tilde{p}_{\perp}+K_{\perp}$ with $p_{\perp}$, and
$\tilde{p}_{\perp}+k_{1\perp}$ with $q_{\perp}$. The contribution
$\mathcal{A}^{p^{\prime}p}_{2+}$ to the amplitude from $\tau_{1}>0$ can then
be written as

$\displaystyle\mathcal{A}^{p^{\prime}p}_{2+}=2(-ie)^{2}(2\pi)^{3}\delta_{\perp,-}\left(\tilde{p}^{\prime}+K-p\right)\int_{-\infty}^{\infty}\\!{\mathrm{d}}x^{{\scriptscriptstyle+}}\int_{0}^{\infty}{\mathrm{d}}\tau_{1}\,\left.\mathrm{e}^{\mathcal{T}_{0}+\mathcal{T}_{1}+\mathcal{T}_{2}}\right\rvert_{\textrm{lin.
}\varepsilon}$ (109)

We are now going to show that the right hand side of the above expression is
equivalent to one of the three Feynman diagram contributions to double
nonlinear Compton, namely that containing two 3-point vertices in which photon
$k_{1}$ is emitted on the outgoing leg. The Feynman rules give this
contribution as

$\displaystyle(-ie)^{2}\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\mathrm{e}^{ik_{1}\cdot
x^{\prime}}\Big{[}\varphi^{\rm
out}_{p^{\prime}}(x^{\prime})\,(\varepsilon_{1}\cdot\stackrel{{\scriptstyle\leftrightarrow}}{{D_{x^{\prime}}}})\,G(x^{\prime},x)(\varepsilon_{2}\cdot\stackrel{{\scriptstyle\leftrightarrow}}{{D_{x}}})\,\varphi^{\rm
in}_{p}(x)\Big{]}e^{ik_{2}\cdot x}\,,$ (110)

where $D$ denotes the background-covariant derivative and
$G(x^{\prime},x)=\mathcal{D}^{x^{\prime}x}_{0}$ is the scalar particle
propagator in the plane wave background (the double arrow indicates the
right$-$left alternating derivative). We then observe that this is equivalent
to

$\displaystyle\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\varphi^{\rm
out}_{p^{\prime}}(x^{\prime}-i\varepsilon_{1})\,e^{ik_{1}\cdot
x^{\prime}-2\varepsilon_{1}\cdot
a(x^{\prime})}\,G(x^{\prime}+i\varepsilon_{1},x-i\varepsilon_{2})\,e^{ik_{2}\cdot
x-2\varepsilon_{2}\cdot a(x)}\,\varphi^{\rm
in}_{p}(x+i\varepsilon_{2})\bigg{|}_{\textrm{lin.
}\varepsilon_{1}\ldots\varepsilon_{N}}\;.$ (111)

Taking this expression, we start by using the Fourier representation of
$G(x^{\prime},x)$ to rewrite it as

$\displaystyle\int\\!{\mathrm{d}}^{4}x^{\prime}\,{\mathrm{d}}^{4}x\,\varphi^{\rm
out}_{p^{\prime}}(x^{\prime}-i\varepsilon_{1})\,\mathrm{e}^{ik_{1}\cdot
x^{\prime}-2\varepsilon_{1}\cdot
a(x^{\prime})}\,G(x^{\prime}+i\varepsilon_{1},x-i\varepsilon_{2})\,\mathrm{e}^{ik_{2}\cdot
x-2\varepsilon_{2}\cdot a(x)}\,\varphi^{\rm in}_{p}(x+i\varepsilon_{2})$ (112)
$\displaystyle=$
$\displaystyle\int\\!\frac{{\mathrm{d}}^{4}r}{(2\pi)^{4}}{\mathrm{d}}^{4}x^{\prime}\,{\mathrm{d}}^{4}x\,\varphi^{\text{out}}_{p^{\prime}}(x^{\prime}-i\varepsilon_{1})\,e^{ik_{1}\cdot
x^{\prime}-2\varepsilon_{1}\cdot
a(x^{\prime})}\frac{i\mathrm{e}^{-ir\cdot(x^{\prime}-x+i\varepsilon_{1}+i\varepsilon_{2})-i\int_{x^{{\scriptscriptstyle+}}}^{x^{\prime{\scriptscriptstyle+}}}\frac{2r\cdot
a(s)-a^{2}(s)}{4r_{-}}{\mathrm{d}}s}}{r^{2}-m^{2}+i0^{{\scriptscriptstyle+}}}\,e^{ik_{2}\cdot
x-2\varepsilon_{2}\cdot a(x)}\,\varphi^{\text{\rm
in}}_{p}(x+i\varepsilon_{2})$

We can easily evaluate the
$x^{\prime{\scriptscriptstyle-},{\scriptscriptstyle\perp}},x^{{\scriptscriptstyle-},{\scriptscriptstyle\perp}}$
and $r^{{\scriptscriptstyle-},{\scriptscriptstyle\perp}}$ integrals and
rewrite the propagator denominator using a standard Schwinger proper-time
integral to obtain

$\displaystyle(2\pi)^{3}\delta_{\perp,-}(\tilde{p}^{\prime}+K-p)\mathrm{e}^{p\cdot\epsilon_{1}+q.\varepsilon_{2}}\int_{-\infty}^{\infty}\\!{\mathrm{d}}x^{\prime{\scriptscriptstyle+}}\,\,\mathrm{e}^{i(p_{+}+k_{1+}-r_{+})x^{\prime{\scriptscriptstyle+}}-2\varepsilon_{1}\cdot
a(x^{\prime{\scriptscriptstyle+}})}\mathrm{e}^{-i\int_{x^{\prime{\scriptscriptstyle+}}}^{\infty}\frac{2p^{\prime}\cdot\delta
a(s)-\delta a^{2}(s)}{2p^{{\scriptscriptstyle+}}}ds}$ (113)
$\displaystyle\times
2\int_{-\infty}^{\infty}\\!{\mathrm{d}}x^{{\scriptscriptstyle+}}\mathrm{e}^{-2\varepsilon_{2}\cdot
a(x^{{\scriptscriptstyle+}})}e^{-ix^{{\scriptscriptstyle+}}q_{{\scriptscriptstyle+}}}\int_{0}^{\infty}d\tau_{1}\int\frac{{\mathrm{d}}r_{\scriptscriptstyle+}}{2\pi}\,\mathrm{e}^{ir_{+}(x^{{\scriptscriptstyle+}}-x^{\prime{\scriptscriptstyle+}}+4q^{{\scriptscriptstyle+}}\tau_{1})}e^{-2i\tau_{1}\left[q_{\perp}^{2}+m^{2}-i0^{{\scriptscriptstyle+}}\right]}$
$\displaystyle\times\mathrm{e}^{-i\int_{x^{{\scriptscriptstyle+}}}^{x^{\prime{\scriptscriptstyle+}}}\\!{\mathrm{d}}s\,\frac{2q\cdot
a(s)-a^{2}(s)}{2q^{{\scriptscriptstyle+}}}-i\int_{-\infty}^{x^{{\scriptscriptstyle+}}}\\!{\mathrm{d}}s\,\frac{2p\cdot
a(s)-a^{2}(s)}{2p^{{\scriptscriptstyle+}}}}\;.$

The $r_{\scriptscriptstyle+}$ integral can now be evaluated to give
$2\pi\delta(x^{{\scriptscriptstyle+}}-x^{\prime{\scriptscriptstyle+}}+8q_{{\scriptscriptstyle-}}\tau_{1})$.
The remaining $x^{\prime{\scriptscriptstyle+}}$ integral is therefore
trivialised and effects the replacement
$x^{\prime{\scriptscriptstyle+}}\rightarrow
x^{{\scriptscriptstyle+}}+8q_{{\scriptscriptstyle-}}\tau_{1}$. Taking the
multi-linear limit, one recovers precisely the right hand side of (109) as
promised.

###  A.2 $\tau_{1}\in(-\infty,0^{-})$

For $\tau_{1}<0$, one recovers the Feynman diagram contribution in which
photon $k_{2}$ is emitted from the outgoing leg. The proof of this follows
exactly the same steps as for $\mathcal{A}^{p^{\prime}p}_{2+}$ above. Hence we
simply state that

$\displaystyle\mathcal{A}^{p^{\prime}p}_{2-}=(-ie)^{2}\int\\!{\mathrm{d}}^{4}x^{\prime}{\mathrm{d}}^{4}x\,\mathrm{e}^{ik_{2}\cdot
x}\Big{[}\varphi^{\rm
out}_{p^{\prime}}(x^{\prime})\,(\varepsilon_{2}\cdot\stackrel{{\scriptstyle\leftrightarrow}}{{D_{x^{\prime}}}})\,G(x^{\prime},x)(\varepsilon_{1}\cdot\stackrel{{\scriptstyle\leftrightarrow}}{{D_{x}}})\,\varphi^{\rm
in}_{p}(x)\Big{]}\mathrm{e}^{ik_{1}\cdot x}\;.$ (114)

###  A.3 $\tau_{1}\in(0^{-},0^{{\scriptscriptstyle+}})$

In this range, the field-independent term in the exponent of (III.1) going
like $\delta(\tau_{1})\epsilon_{1}\cdot\epsilon_{2}$ cannot be neglected.
Noting that this term is already linear in both $\epsilon_{1}$ and
$\epsilon_{2}$, the corresponding contribution to the amplitude is immediately
seen to be proportional to the $\tau_{1}\rightarrow 0$ and
$\epsilon_{1,2}\rightarrow 0$ limit of the integrand of the proper-time
integral:

$\displaystyle\mathcal{A}^{p^{\prime}p}_{2\delta}$
$\displaystyle=-2(-ie)^{2}(2\pi)^{3}\delta_{{\scriptscriptstyle\perp},{\scriptscriptstyle-}}\left(\tilde{p}^{\prime}+K-p\right)$
(115)
$\displaystyle\times\int_{-\infty}^{\infty}dx^{{\scriptscriptstyle+}}(i\varepsilon_{1}\cdot\varepsilon_{2})e^{+i\left(K+p^{\prime}-p\right)_{{\scriptscriptstyle+}}x^{{\scriptscriptstyle+}}-i\int_{-\infty}^{0}[2\tilde{p}^{\prime}\cdot
a(\tau)-a^{2}(\tau)]{\mathrm{d}}\tau-i\int_{0}^{\infty}[2p^{\prime}\cdot\delta
a(\tau)-\delta a^{2}(\tau)]{\mathrm{d}}\tau-2i\int_{-\infty}^{0}K\cdot
a(\tau){\mathrm{d}}\tau}$

By inspection, this is equivalent to

$\displaystyle\mathcal{A}^{p^{\prime}p}_{2\delta}=-2i(-ie)^{2}\varepsilon_{1}\cdot\varepsilon_{2}\int\\!{\mathrm{d}}^{4}x\,\mathrm{e}^{i(k_{1}+k_{2})\cdot
x}\varphi^{\rm out}_{p^{\prime}}(x)\varphi^{\rm in}_{p}(x)\;,$ (116)

which is indeed the seagull vertex contribution to double nonlinear Compton
scattering. Summing (109), (114) and (116) recovers the full amplitude.

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