# Rise and fall of laser-intensity effects in spectrally resolved Compton
process

U. Hernandez Acosta1,2,3, A. I. Titov4, B. Kämpfer1,2 1Helmholtz-Zentrum
Dresden-Rossendorf, 01314 Dresden, Germany  2Institut für Theoretische Physik,
TU Dresden, 01062 Dresden, Germany  3Center for Advanced Systems
Understanding, Untermarkt 20, 02826 Görlitz, Germany  4Bogoliubov Laboratory
of Theoretical Physics, JINR, Dubna 141980, Russia

(August 28, 2024)

###### Abstract

The spectrally resolved differential cross section of Compton scattering,
$d\sigma/d\omega^{\prime}|_{\omega^{\prime}=const}$, rises from small towards
larger laser intensity parameter $\xi$, reaches a maximum, and falls towards
the asymptotic strong-field region. Expressed by invariant quantities:
$d\sigma/du|_{u=const}$ rises from small towards larger values of $\xi$,
reaches a maximum at $\xi_{max}=\frac{4}{9}{\cal K}um^{2}/k\cdot p$, ${\cal
K}={\cal O}(1)$, and falls at $\xi>\xi_{max}$ like
$\propto\xi^{-3/2}\exp\left(-\frac{2um^{2}}{3\xi\,k\cdot p}\right)$ at $u\geq
1$. [The quantity $u$ is the Ritus variable related to the light-front
momentum-fraction $s=(1+u)/u=k\cdot k^{\prime}/k\cdot p$ of the emitted photon
(four-momentum $k^{\prime}$, frequency $\omega^{\prime}$), and $k\cdot
p/m^{2}$ quantifies the invariant energy in the entrance channel of electron
(four-momentum $p$, mass $m$) and laser (four-wave vector $k$).] Such a
behavior of a differential observable is to be contrasted with the laser
intensity dependence of the total probability, $\lim_{\chi=\xi k\cdot
p/m^{2},\xi\to\infty}\mathbb{P}\propto\alpha\chi^{2/3}m^{2}/k\cdot p$, which
is governed by the soft spectral part.

We combine the hard-photon yield from Compton with the seeded Breit-Wheeler
pair production in a folding model and obtain a rapidly increasing
$e^{+}e^{-}$ pair number at $\xi\lesssim 4$. Laser bandwidth effects are
quantified in the weak-field limit of the related trident pair production.

non-linear Compton scattering, nonlinear Breit-Wheeler pair production,
strong-field QED

###### pacs:

12.20.Ds, 13.40.-f, 23.20.Nx

##  I Introduction

Quantum Electro-Dynamics (QED) as pillar of the standard model (SM) of
particle physics possesses a positive $\beta$ function Peskin:1995ev  which
makes the running coupling strength $\alpha(\mathfrak{s})$ increasingly with
increasing energy/momentum scale $\mathfrak{s}$ Abbiendi:2005rx . In contrast,
Quantum Chromo-Dynamics (QCD) as another SM pillar possesses a negative
$\beta$ function due to the non-Abelian gauge group Peskin:1995ev , giving
rise to the asymptotic freedom,
$\lim_{\mathfrak{s}\to\infty}\alpha_{QCD}(\mathfrak{s})\to 0$, i.e. QCD has a
truly perturbative limit. In contrast, $\lim_{\mathfrak{s}\to
0}\alpha(\mathfrak{s})\to 1/137.0359895(61)$ is not such a strict limit,
nevertheless, QED predictions/calculations of some observables agree with
measurements within 13 digits, see Aoyama:2020ynm ; Arapoglou:2019taq ;
QED_prec1  for some examples. The situation in QED becomes special when
considering processes in external (or background) fields: One can resort to
the Furry (or bound-state) picture, where the (tree-level) interactions of an
elementary charge (e.g. an electron) with the background are accounted for in
all orders, and the interactions with the quantized photon field remains
perturbatively in powers of $\alpha$. However, the Ritus-Narozhny (RN)
conjecture RN1 ; RN2 ; RN3 ; RN4 ; RN5  argues that the effective coupling
becomes $\alpha\chi^{2/3}$, meaning that the Furry picture expansion beaks
down at $\alpha\chi^{2/3}>1$ Mironov:2020gbi ; Heinzl:2021mji ;
Edwards:2020npu ; Fedotov:2016afw  (for the definition of $\chi$ see below)
and one enters a genuinely non-perturbative regime. The latter requires
adequate calculation procedures, as the lattice regularized approaches, which
are standard since many years in QCD, e.g. in evaluations of observables in
the soft sector where $\alpha_{QCD}>1$, cf. Brambilla:2014jmp . (In QED
itself, an analog situation is meet in the Coulomb field of nuclear systems
with proton numbers $Z>Z_{crit}\approx 173$: if $\alpha Z_{crit}>1$ the QED
vacuum beak-down sets in; cf. Popov:2020xmd  for the actual status of that
field).

With respect to increasing laser intensities the quest for the possible break-
down of the Furry picture expansion in line with the RN conjecture becomes,
besides its principal challenge, also of “practical” interest, whether one can
explore experimentally this yet uncharted regime of QED. (For other
configurations, e.g. beam-beam interactions, cf. Yakimenko:2018kih ). A
prerequisite would be to find observables which display the typical dependence
$\propto\alpha\chi^{2/3}$, where we denote by $\alpha$ the above quoted fine-
structure constant at $\mathfrak{s}\to 0$. In doing so we resort here to the
lowest-order QED processes, that is nonlinear Compton and nonlinear Breit-
Wheeler. Both processes seem to be investigated theoretically in depth in the
past, however, enjoy currently repeated re-considerations w.r.t. refinements
DiPiazza:2020wxp ; Seipt:2020diz ; King:2020btz ; Ilderton:2020dhs ;
Titov:2018bgy ; Granz:2019sxb , establishing approximation schemes
Blackburn:2021rqm ; Heinzl:2020ynb ; DiPiazza:2018bfu ; Blackburn:2018sfn ;
Ilderton:2018nws  to be implemented in simulation codes Shvets:2018iar ;
Blackburn:2017dpn ; Gonoskov:2014mda  or to use them as building blocks in
complex processes Torgrimsson:2020gws , with the starting point at trident
Torgrimsson:2020wlz ; Dinu:2019wdw ; Dinu:2019pau .

Beginning with Compton scattering ($p$ and $p^{\prime}$ are the $in$\- and
$out$-electron four-momenta, $k$ the laser four-momentum, and $k^{\prime}$ the
$out$-photon’s four-momentum, respectively) the relevant (Lorentz and gauge
invariant) variables of the entrance channel are DiPiazza:2011tq
\- (i) the classical intensity parameter of the laser: $\xi=|e|{\cal
E}/(m\omega)$, here expressed by quantities in the lab.: ${\cal E}$ \-
electric laser field strength, $\omega$ \- the central laser frequency; $-|e|$
and $m$ stand for the electron charge and mass, respectively, and
$e^{2}/4\pi=\alpha$,111We employ natural units with $\hbar=c=1$.
\- (ii) the available energy squared: $k\cdot p/m^{2}=(\hat{s}/m^{2}-1)/2$
with $\hat{s}$ as Mandelstam variable,
\- (iii) the quantum nonlinearity parameter: $\chi=\xi k\cdot p/m^{2}$.
The latter quantity is often considered as the crucial parameter since in some
limits it determines solely the probability of certain processes. $\chi$ plays
also a prominent role in the above mentioned discussion of the RN conjecture,
where the Furry picture expansion of QED is argued to break down for
$\alpha\chi^{2/3}>1$. References Ilderton:2019kqp ; Podszus:2018hnz  point out
that the large-$\chi$ limits, facilitated by either large $\xi$ (the high-
intensity limit) or large $k\cdot p/m^{2}$ (the high-energy limit), are
distinctively different, with implications for approximation schemes in
simulation codes.

Figure 1 exhibits a few selected curves $\chi=const$ over the $\xi$ vs.
$k\cdot p/m^{2}$ landscape to illustrate the current situation w.r.t.
facilities where laser beams and electron beams are (or can be) combined. One
has to add the options at E-320 at FACET-II/SLAC E_320 ; Meuren:2020nbw  and
electron beams which are laser-accelerated to GeV scales, e.g. Poder:2018ifi ;
Cole:2017zca ; Keitel:2021gcq .

Usually, $\xi=1$ is said to mark the the onset of strong-field effects, and
the corresponding processes at $\xi>1$ are termed by the attribute
“nonlinear”. In this respect, the parameters provided by LUXE
Abramowicz:2019gvx ; Abramowicz:2021zja  and FACET-II E_320 ; Meuren:2020nbw
are interesting: $\chi={\cal O}(1)$ and above and $\xi>1$ as well.222 Despite
high intensities at XFELs, e.g. $I\to 10^{22}$ W/cm2, the intensity parameter
$\xi=\frac{7.5{\rm eV}}{\omega}\sqrt{\frac{I}{10^{20}{\rm W/cm}^{2}}}$
DiPiazza:2011tq  is small due to the high frequency, e.g. $\omega=$ 1 - 25
keV. In the following we consider the LUXE kinematics (see Fig. 1), $k\cdot
p/m^{2}\approx\omega E_{e}(1-\cos\Theta)=0.2078$ in head-on collisions
($\cos\Theta=-1$). Our aim is to quantify a simple observable as a function of
$\xi$. To be specific, we select the invariant differential cross section
$d\sigma/du$, where $u$ is the light-cone momentum-transfer of the
$in$-electron to the $out$-photon, related to light-front momentum-fraction of
the $out$-photon $s=\frac{u}{1+u}=\frac{k\cdot k^{\prime}}{k\cdot p}$. (The
mapping $u\mapsto\omega^{\prime}$ and $d\sigma/du\mapsto
d\sigma/d\omega^{\prime}$ is discussed in Acosta:2020agu ; Kampfer:2020cbx .)
To make the meaning of the Ritus variable $u$ more explicit let us mention the
relation
$u=\frac{e^{-\zeta}\nu^{\prime}(1-\cos\Theta^{\prime})}{1-e^{-\zeta}\nu^{\prime}(1-\cos\Theta^{\prime})}$,
where $\nu^{\prime}\equiv\omega^{\prime}/m$, and the electron energy $E_{e}$
in lab. determines the rapidity $\zeta$ via $E_{e}=m\cosh\zeta$ and
$\Theta^{\prime}$ denotes the polar lab. angle of the $out$-photon. We call
$d\sigma/du$ a spectrally resolved observable.

Figure 1: Curves of $\chi=const$ over the $\xi$ vs. $k\cdot p/m^{2}$ plane for
$\chi=0.001,$ 0.01, 0.1, 1 and 10. The double-line depicts the curve
$\alpha\chi^{2/3}=1$. Vertical thin lines are for maximum values of $k\cdot
p/m^{2}$ in reach at various electron accelerators (HZDR Jochmann:2013toa :
$E_{e}=33$ MeV, ELI Turcu:2016dxm : $E_{e}=600$ MeV, LUXE Abramowicz:2019gvx ;
Abramowicz:2021zja : $E_{e}=17.5$ GeV) in combination with a high-intensity
optical laser (we use $\omega=1.55$ eV as representative frequency). The
vertical dotted line indicates a possible combination of the European XFEL
($\omega=10$ keV) with a laser-accelerated electron beam ($E_{e}=10$ MeV)
available in the high-energy density cave of the HIBEF collaboration HIBEF .
The horizontal delineation line $\xi=1$ is thought to highlight the onset of
the strong-field region above.

Our note is organized as follows. In section II we briefly recall a few
approximations of the laser beam. Section III is devoted to an analysis of the
invariant differential cross section $d\sigma/du$ and its dependence on the
laser intensity parameter $\xi$ in nonlinear Compton scattering. That is, we
are going up and down on the vertical dashed line with label LUXE in Fig. 1
around the point $\chi=1$ or $\xi=1$. The discussion section IV (i) relates
the cross section to the probability and (ii) considers a folding model which
uses the hard-photon spectrum emerging from Compton scattering as seed for
subsequent Breit-Wheeler pair production; a brief discussion of (iii)
bandwidth effects relevant for sub-threshold trident pair production
complements this section. We conclude in section V. The appendix A recalls a
few basic elements of the one-photon Compton process.

##  II Laser models

In plane-wave approximation333Due to the high symmetry, the exact solutions of
the Dirac equation in a plane-wave background are in a comfortably compact
form enabling an easy processing in evaluations of matrix elements. Without
such a high symmetry, much more attempts are required DiPiazza:2021rum ;
DiPiazza:2016tdf ; Heinzl:2017zsr . For some useful parameterizations of laser
beams, see Karbstein:2020gzg ; Blinne:2018nbd ; Karbstein:2017jgh  and further
citations therein. The relevance of the Fourier-zero mode of (non-)unipolar
planar fields is mentioned in Dinu:2012tj . the laser (circular polarization)
can be described by the four-potential in axial gauge, $A=(0,\vec{A})$, with

$\vec{A}=f(\phi)\left(\vec{a}_{x}\cos\phi+\vec{a}_{y}\sin\phi\right)$ (1)

where $\vec{a}_{x}^{2}=\vec{a}_{y}^{2}=m^{2}\xi^{2}/e^{2}$; the polarization
vectors $\vec{a}_{x}$ and $\vec{a}_{y}$ are mutually orthogonal. We ignore a
possible non-zero value of the carrier envelope phase and focus on symmetric
envelope functions $f(\phi)$ w.r.t. the invariant phase $\phi=k\cdot x$. One
may classify the such a model class as follows.
\- 1) Laser pulses: $\lim_{\phi\to\pm\infty}f(\phi)=0$,
\- 2) Monochromatic beam: $f(\phi)=1$,
\- 3) Constant cross field (ccf): $\vec{A}=\phi\,\vec{a}_{x}$.
The probabilities for the constant cross field option 3) coincide with certain
limits of the plane-wave model 2) Ritus ; in Kampfer:2020cbx  they are related
to the large-$\xi$ limit. Item 1) could be divided into several further sub-
classes, such as 1.1): finite support region of the pulse, i.e.
$f(|\phi|>\phi_{pulse\,length})=0$, $\phi_{pulse\,length}<\infty$, and 1.2):
far-extended support region, i.e. $\lim_{\phi\to\pm\infty}f(\phi)\to 0$,
together with non-zero carrier envelope phase, asymmetric pulse shape,
frequency chirping, polarization gating etc. A specific class is 1.1) with
flat-top section, e.g. a box envelope $\sqcap$ (cf. King:2020hsk  for a recent
explication) belonging to $C^{0}$, or $\cos^{2}\otimes\,\sqcap$ belonging to
$C^{2}$ or the construction in Otto:2014ssa  belonging to $C^{\infty}$.
Examples for item 1.2) are Gauss, super-Gauss (employed in Otto:2018jbs , for
instance), symmetrized Fermi function Titov:2015pre ; Titov:2014usa ,
$1/\cosh$ etc. The monochromatic beam, item 2), corresponds formally to an
infinitely long flat-top “pulse”, abbreviated hereafter by IPA as acronym of
infinite pulse approximation. It may be considered as special case of 1.1)
with $\phi_{pulse\,length\to\infty}$. FPA stands henceforth for the finite
pulse-length plane-wave approximation.

To be specific, we employ here 1.2) with $f(\phi)=1/\cosh(\phi/\pi N)$, where
$N$ characterizes the number of oscillations in that pulse interval, where
$f(\phi)$ is significantly larger than zero (see Kampfer:2020cbx ;
Titov:2015pre ; Titov:2019kdk  for the formalism), and IPA from 2). The laser
model of class 1.1) is employed in subsection IV.3.

##  III Compton: differential invariant cross section

Let us consider the above pulse envelope function $f(\phi)=1/\cosh(\phi/(\pi
N))$ to elucidate the impact of a finite pulse duration and contrast it later
on with the monochromatic laser beam model and some approximations thereof.
Differential spectra $d\sigma/du$ as a function of $u$ are exhibited in Fig. 2
in the region $u\leq 3$ for several values of $\xi\leq 1$ for the FPA (dashed
curves) and IPA (solid curves) models recalled below. This complements figures
1 – 3 in Kampfer:2020cbx . One observes that the harmonic structures (which
would become more severe for linear polarization, see Kampfer:2020cbx ) fade
away at larger values of $u$ and $\xi$. Therefore, we are going to analyze
that region in parameter space. There, IPA results represent reasonably well
the trends of the more involved FPA calculations.

###  III.1 Numerical results

Figure 2: Invariant differential cross section $d\sigma(u,\xi,k\cdot
p/m^{2})/du$ as a function of $u$ for several values of $\xi$, $\xi=0.2$
(magenta), 0.4 (blue), 0.7 (green) and 1 (black). The dashed curves (label
FPA) are for pulses according to Eq. (1) with $f(\phi)=1/\cosh(\phi/(\pi N))$
for $N=10$. The IPA results for a monochromatic laser beam are depicted by
solid curves (with the limit $d\sigma_{IPA}/du|_{u\to 0}=2\pi\alpha^{2}/k\cdot
p$ independent of $\xi$). The pronounced harmonic structures around the Klein-
Nishina (KN) edge $u_{KN}\approx 0.416$ are irrelevant for the subsequent
discussion and, therefore, do not need a detailed representation. For
$\chi=\xi k\cdot p/m^{2}=0.2078\,\xi$.

In particular, we consider now $d\sigma(u,\xi,k\cdot p/m^{2})/du$ as a
function of $\xi$ for several constant values of $u$, $u=0.5,$ 1, 2, 4 and 8,
for $k\cdot p/m^{2}=0.2078$ – a value which is motivated by the LUXE
opportunities Abramowicz:2019gvx ; Abramowicz:2021zja . The solid, dotted and
dashed curves in Fig. 3 are based on easily accessible models Ritus :
$\bullet$ monochromatic model (IPA) (cf. equations (15, 16) in Kampfer:2020cbx
):

$\displaystyle\frac{d\sigma_{IPA}}{du}$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha^{2}}{k\cdot
p}\frac{1}{(1+u)^{2}}\sum_{n=1}^{\infty}\,\Theta(u_{n}-u)\,F_{n}(z_{n}),$ (2)
$\displaystyle F_{n}$ $\displaystyle=$
$\displaystyle-\frac{2}{\xi^{2}}J_{n}^{2}(z_{n})$ $\displaystyle+$
$\displaystyle
A\left(J_{n+1}^{2}(z_{n})+J_{n-1}^{2}(z_{n})-2J_{n}^{2}(z_{n})\right)$

with $A=1+\frac{u^{2}}{2(1+u)}$,
$z_{n}(u,u_{n})=\frac{2n\xi}{\sqrt{1+\xi^{2}}}\sqrt{\frac{u}{u_{n}}(1-\frac{u}{u_{n}})}$,
and $u_{n}=\frac{2nk\cdot p}{m^{2}(1+\xi^{2})}$ ($J_{n}$’s denote Bessel
functions of first kind);
$\bullet$ IPA–large-$\xi$ approximation (cf. equation (21) in Kampfer:2020cbx
):

$\displaystyle\frac{d\sigma_{large-\xi}}{du}$ $\displaystyle=$
$\displaystyle-\frac{4\pi\alpha^{2}}{m^{2}\xi}\frac{1}{\chi}\frac{1}{(1+u)^{2}}{\cal
F}_{C}$ (4) $\displaystyle{\cal F}_{C}$ $\displaystyle=$
$\displaystyle\int_{z}^{\infty}dy\,\Phi(y)+\frac{2}{z}\left[1+\frac{u^{2}}{2(1+u)}\right]\Phi^{\prime}(z)$
(5)

where $\Phi(z)$ and $\Phi(z)^{\prime}$ stand for the Airy function and its
derivative with arguments $z=(u/\chi)^{2/3}$ where $\chi=\xi k\cdot p/m^{2}$;
$\bullet$ large-$\xi$-large-$u$ approximation (cf. equation (22) in
Kampfer:2020cbx ):

$\frac{d\sigma_{large-\xi,\,large-u}}{du}=\frac{2\sqrt{\pi}\alpha^{2}}{m^{2}\xi}\chi^{-1/2}u^{-3/2}\exp\left(-\frac{2u}{3\chi}\right).$
(6)

Figure 3: Invariant differential cross section $d\sigma(u=const,\xi,k\cdot
p/m^{2})/du$ as a function of $\xi$ for several values of $u$, $u=0.5,$ 1, 2,
4 and 8; for $k\cdot p/m^{2}=0.2078$. The asterisks depict results of short
laser pulses with envelope function $f(\phi)=1/\cosh(\phi/(\pi N))$ for
$N=10$, $d\sigma_{FPA}/du$, Eq. (7). The solid curves are for the
monochromatic laser beam model $d\sigma_{IPA}/du$, Eq. (2), while dashed
(dotted) curves are based on $d\sigma_{large-\xi}/du$, Eq. (4)
($d\sigma_{large-\xi,\,large-u}/du$, Eq. (6)).

The asterisks in Fig. 3 depict results of the pulse model (FPA) described in
Kampfer:2020cbx ; Titov:2015pre  (cf. equations (5 - 14) in Kampfer:2020cbx ):

$\displaystyle\frac{d\sigma_{FPA}}{du}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}}{k\cdot
p}\frac{1}{(1+u)^{2}}\int\limits_{0}^{\infty}d\ell\,\Theta\left(\frac{um^{2}}{2k\cdot
p}-\ell\right)w(\ell),$ (7) $\displaystyle w(\ell,z)$ $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}d\phi_{e^{\prime}}\left[-\frac{2}{\xi^{2}}|\tilde{Y}_{\ell}(z_{\ell})|^{2}\right.$
(8)
$+A\left.\left(|Y_{\ell-1}(z_{\ell})|^{2}+|Y_{\ell+1}(z_{\ell})|^{2}-2\mbox{Re}\tilde{Y}_{\ell}(z_{\ell})X_{\ell}^{*}(z_{\ell}))\right)\right]$

with $A$ as above,
$z_{\ell}(u,\ell)={2\ell\xi}\sqrt{\frac{u}{u_{\ell}}(1-\frac{u}{u_{\ell}})}$
and $u_{\ell}=\frac{2\ell k\cdot p}{m^{2}}$. The basic functions
$Y_{\ell},\tilde{Y}_{\ell},X_{\ell}$ and their $\phi_{e^{\prime}}$ dependence
are spelled out in Kampfer:2020cbx ; Titov:2015pre ; they depend crucially on
the pulse envelope function $f(\phi)$. (Due to numerical accuracy reasons in
integrating highly oscillating functions, these evaluations are constrained
presently to not too large values of $\xi$.) The common basis of these models
is sketched in Appendix A.

We consider in Fig. 3 only $u>u_{KN}=2k\cdot p/m^{2}$, since at the Klein-
Nishina (KN) edge the harmonic structures become severe, as seen in Fig. 2.
The striking feature seen in Fig. 3 is the pronounced $\cap$ shape, which we
coin “rise and fall” of laser intensity effects. At small $\xi$, the realistic
FPA ($N=10$) results (asterisks) and the IPA model (solid curves) follow the
same trends, consistent with Fig. 2. The large-$\xi$ and large-$\xi$-large-$u$
approximations (dashed and dotted curves) are not supposed to apply in that
region. However, they become useful representatives at large $\xi$.

###  III.2 The rise

Some guidance of the rising parts of the FPA results in Fig. 3 can be gained
by the monochromatic model. Casting Eq. (III.1) in the form

$\displaystyle F_{n}$ $\displaystyle=$
$\displaystyle-\frac{2(1-\Xi^{2})}{\Xi^{2}}J_{n}^{2}(\Xi x_{n})$
$\displaystyle+$ $\displaystyle A\left(J_{n+1}^{2}(\Xi x_{n})+J_{n-1}^{2}(\Xi
x_{n})-2J_{n}^{2}(\Xi x_{n})\right)$

with $\Xi^{2}=\xi^{2}/(1+\xi^{2})$ and
$x_{n}(u)=2n\sqrt{\frac{u}{u_{n}}(1-\frac{u}{u_{n}})}$ and expanding in powers
of $\Xi$ yields for the first terms

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2A-x_{1}^{2})-\frac{x_{1}^{2}}{8}(8A-x_{1}^{2}-4)\Xi^{2}+\frac{x_{1}^{4}}{384}(90A-5x_{1}^{2}-48)\Xi^{4}+{\cal
O}(\Xi^{6}),$ (10) $\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle\hskip
79.6678pt\frac{x_{2}^{2}}{32}(8A-x_{2}^{2})\Xi^{2}-\quad\frac{x_{2}^{4}}{192}(18A-x_{2}^{2}-6)\Xi^{4}+{\cal
O}(\Xi^{6}),$ (11) $\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle\hskip
179.25244pt\frac{x_{3}^{4}}{1152}(18A-x_{3}^{2})\Xi^{4}+{\cal O}(\Xi^{6}).$
(12)

Due to the Heavyside $\Theta$ function in Eq. (2), the leading-order power in
$\Xi$ depends on the value of $u$, e.g. for $u<u_{1}$, the series starts with
${\cal O}(\Xi^{0})$ and the coefficients of higher orders accordingly sum up.
Higher values of $u$ facilitate higher orders of the leading terms, i.e. the
rise becomes steeper since the respective leading-order term is
$\propto\Xi^{2\lfloor u/u_{1}\rfloor}$. This statement is based on the
structures in Eqs. (10 \- 12), suggesting
$F_{n}=\sum_{i=n-1}F_{n}^{(2i)}\,\Xi^{2i}$ for the first terms $F_{n}^{(2i)}$,
and $\sum_{n=1}^{\infty}\Theta(u_{n}-u)\,F_{n}=\sum_{n_{min}}^{\infty}F_{n}$
with $n_{min}=1+\lfloor u/u_{1}\rfloor$. ($\lfloor\cdot\rfloor$ is the floor
operation.)

The series expansion of (III.2) in powers of $\Xi$ ignores the sub-leading
$\Xi$ dependence in $x_{n}$ via $u_{n}=2n\frac{k\cdot p}{m^{2}}(1-\Xi^{2})$
but is suitable for $k\cdot p=const$.

Analogously considerations apply to the pulse model, cf. section III.C in
Titov:2015pre . An essential role is played by the Fourier transform of the
pulse envelope in the limit $\xi\ll 1$. It bridges to the IPA for long pulses.

###  III.3 The fall

The maximum of the curves exhibited in Fig. 3 at $u={\cal O}(1)$ is attained
at $\xi={\cal O}(1)$, but moves towards larger values of $\xi$ with increasing
values of $u$. Remarkably, the often discredited large-$\xi$ and
large-$\xi$–large-$u$ approximations (dotted and dashed curves) deliver
results in fair agreement with the IPA results (solid curves) for $u>1$. That
is, when being interested in the high-energy photon tails, the simple
large-$\xi$–large-$u$ formula Eq. (6) Ritus  represents a fairly accurate
description supposed $\xi$ is sufficiently large. Obviously, at $\xi<1$ and
$u<2$, such an approximation fails quantitatively. (In particular, at $u<1$
the harmonic structures in IPA become severe.) Nevertheless, some estimate of
the maximum position is provided by $\xi_{max}\approx\frac{4}{9}um^{2}/k\cdot
p$ yielding
$d\sigma/du(\xi_{max})\approx\frac{2\sqrt{\pi}\alpha^{2}}{m^{2}}\frac{k\cdot
p}{m^{2}}\left(\frac{3}{2}\right)^{3}e^{-3/2}\,u^{-3}$. The asymptotic fall is
governed by $\frac{d\sigma}{du}\propto\frac{1}{m^{2}}\frac{m^{2}}{k\cdot
p}\xi^{-3/2}u^{-3/2}\exp\left(-\frac{2um^{2}}{3\xi k\cdot p}\right)$ from Eq.
(6).

We emphasize the same pattern of rise and fall of
$d\sigma/d\omega^{\prime}|_{\omega^{\prime}=const}$, see Fig. 4, again for
sufficiently large values of $\omega^{\prime}$. The spectrally resolved
differential cross section $d\sigma/d\omega^{\prime}|_{\omega^{\prime}=const}$
is directly accessible in experiments. In the strong-field asymptotic region
it displays a funneling behavior, i.e. the curves are squeezed into a narrow
corridor already in the non-asymptotic $\xi$ region, in contrast to
$d\sigma/du|_{u=const}$ in Fig. 3.

Figure 4: The same as in Fig. 3 but for $d\sigma/d\omega^{\prime}$ as a
function of laser intensity parameter $\xi$ for several constant values of
$\omega^{\prime}$.

##  IV Discussion

###  IV.1 Cross section vs. probability

The cross section $\sigma$ and Ritus probability rate $W$ are related as
$W=\frac{m^{4}}{4\pi\alpha}\frac{\xi\,\chi}{q_{0}}\sigma$ with $q_{0}$
denoting the energy component of the quasi-momentum of the $in$-electron. This
relation holds true for circular polarization and applies to respective
differential quantities too. The different normalization modifies in
particular the $\xi$ dependence: The above emphasized “rise and fall” of
$d\sigma/du$ corresponds to a monotonously rising probability $dW/du$ as a
function of $\xi$. Having in mind Ritus’ remark “the cross-sectional concept
becomes meaningless” since, at $\xi\to\infty$ we have $\sigma\to 0$ while $W$
remains finite Ritus , we turn in this sub-section to the probability. In
doing so we remind the reader of the subtle Ritus notation
$W(\chi)=\frac{1}{\pi}\int_{0}^{\pi}d\psi P(\chi\sin\psi)$ in distinguishing
the probabilities $W$ and $P$.

We also stress the varying behavior of total (cf. appendix A in Titov:2010ps )
and differential probabilities. Equation (6) emerges as a certain limit of the
constant cross field probability Ritus
$\frac{dP(\chi,u)}{du}=-V(1+u)^{-2}\,{\cal F}_{C}(z(\chi,u),u)$ where the
prefactor reads $V=\alpha m^{2}/\pi q_{0}$ and ${\cal F}_{C}$ is defined in
Eq. (5). With respect to the argument $z=(u/\chi)^{2/3}$ of the Airy functions
in Eq. (5), the $\chi$ \- $u$ plane can be divided into the regions I (where
$u>\chi$) and II (where $u<\chi$). Furthermore, the combination $1+u$ suggests
a splitting into $u<1$ and $u>1$ sub-domains. Picking up the leading-order
terms in the related series expansions one arrives at

$\displaystyle\frac{dP}{du}|_{u\gg\chi}$ $\displaystyle=$
$\displaystyle\frac{V}{2\sqrt{\pi}}\chi^{1/2}\exp\left(-\frac{2u}{3\chi}\right)$
(15) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{l}u^{-1/2}\quad\mbox{for}\quad u\ll
1,\quad(I^{<})\\\ u^{-3/2}\quad\mbox{for}\quad u\gg 1,\quad(I^{>})\\\
\end{array}\right.$ $\displaystyle\frac{dP}{du}|_{u\ll\chi}$ $\displaystyle=$
$\displaystyle-V\Phi^{\prime}(0)\chi^{2/3}$ (18) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{l}2u^{-2/3}\quad\mbox{for}\quad u\ll
1,\quad(II^{<})\\\ u^{-5/3}\,\,\,\quad\mbox{for}\quad u\gg 1.\quad(II^{>})\\\
\end{array}\right.$

It is the soft $u$-part $II^{<}$ of the differential probability in Eq. (18)
which essentially determines the total probability upon $u$-integration,444
The value $\chi=1$ is special since the small-$u$ region of II and the
large-$u$ region of I must be joint directly, $II^{<}\otimes I^{>}$, while for
$\chi<1$ the small-$u$ region II and the small-$u$ region of I must be joint
followed by the large-$u$ region of I, $II^{<}\otimes I^{<}\otimes I^{>}$. In
the opposite case of $\chi>1$, the small-$u$ region of II must be joint with
the large-$u$ region of II followed by the large-$u$ region of I,
$II^{<}\otimes II^{>}\otimes I^{>}$. This distinction becomes also evident
when inspecting the differently shaped sections of the continuous curves
$dP/du$ as a function of $u$ for several values of $\chi$.  i.e. the
celebrated result $P\propto\alpha\chi^{2/3}$ in Ritus notation Ritus  (stated
as $\lim_{\chi,\xi\to\infty}\mathbb{P}\propto\alpha\chi^{2/3}m^{2}/k\cdot p$
in Ilderton:2019kqp ), while the hard $u$-part $I^{>}$ in Eq. (15) is at the
origin of Eq. (6) when converting to cross section. In other words, one has to
distinguish the $\xi$ (or $\chi$) dependence either of integrated or
differential observables. In relation to the LUXE plans Abramowicz:2019gvx  we
mention the photon detector developments Fleck:2020opg  which should enable in
fact the access to the differential spectra.

###  IV.2 Secondary processes: Breit-Wheeler

Instead transferring these considerations to the Breit-Wheeler (BW) process
per se (cf. Kaminski:2006xlq ; Heinzl:2010vg ; Krajewska:2012eb ;
Krajewska:2014ssa ; Jansen:2016gvt ; Jansen:2016crq ; Golub:2020kkc ;
Titov:2020taw  and further citations below), we estimate now the BW pair
production seeded by the hard photons from the above Compton (C) process. The
following folding model C$\otimes$BW is a pure two-step ansatz on the
probabilistic level which mimics in a simple manner some part of the trident
process555For recent work on the formalism, see Torgrimsson:2020wlz ;
Dinu:2019wdw ; Dinu:2019pau ; King:2018ibi ; Mackenroth:2018smh ;
Acosta:2019bvh  which advance earlier investigations Baier:1972vc ;
Ritus:1972nf .
$e_{L}^{-}\to{e_{L}^{-}}^{\prime}+{e_{L}^{-}}^{\prime\prime}+e_{L}^{+}$ by
ignoring (i) the possible off-shell effects and non-transverse components of
the intermediate photon (that would be the one-step contribution) and (ii) the
anti-symmetrization of the two electrons in the final state (that would be the
exchange contribution). Such an ansatz is similar to the one in Hartin:2018sha
, where however bremsstrahlung$\otimes$BW has been analyzed. An analog
approach has been elaborated in Titov:2009cr  for di-muon production.

Specifically, we consider the two-step cascade process where a GeV-Compton
photon with energy $\omega^{\prime}$ is produced in the first step, and, in
the next step, that GeV-photon interacts with the same laser field producing a
BW-$e^{+}e^{-}$ pair. We estimate the number of pairs produced in one pulse by

$N^{e^{+}e^{-}}=\int\limits_{0}^{\infty}d\omega^{\prime}\,\int\limits_{0}^{T_{C}}dt\,\frac{d\Gamma_{C}(\omega^{\prime},t)}{d\omega^{\prime}}\int\limits_{t}^{T_{BW}}dt^{\prime}\,\Gamma_{BW}(\omega^{\prime},t^{\prime}),$
(19)

where $d\Gamma_{C}(\omega^{\prime},t)/d\omega^{\prime}$ is the rate of photons
per frequency interval $d\omega^{\prime}$ (which corresponds to $dP/du$ in
IV.1) emerging from Compton at time $t\in[0,T_{C}]$, and
$\Gamma_{BW}(\omega^{\prime},t^{\prime})$ is the rate of Breit-Wheeler pairs
generated by a probe photon of frequency $\omega^{\prime}$ at lab. frame time-
distance $t^{\prime}\in[t,T_{BW}]$. The underlying picture is that of an
electron traversing a laser pulse in head-on geometry near to light cone. The
passage time of an undisturbed electron would be $NT_{0}/2$,
$T_{0}=2\pi/\omega$. Neglecting spatio-temporal variations within the pulse,
the final formula becomes
$N^{e^{+}e^{-}}=F_{t}\int_{0}^{E_{e}}d\omega^{\prime}\,N^{e}_{0}\frac{d\Gamma_{C}(\omega^{\prime})}{d\omega^{\prime}}\,\Gamma_{BW}(\omega^{\prime})$
upon the restriction $\omega^{\prime}<E_{e}$ and
$F_{t}=T_{C}(T_{BW}-T_{C}/2)$. A crucial issue is the choice of the formation
time(s) Baier:1972vc ; Ritus:1972nf . When gluing C$\otimes$BW on the
amplitude level, such a time appears linearly in the pair rate Hu:2010ye 666
In laser pulses such an additional parameter is not needed Ilderton:2010wr .
or quadratically in the net probability via overlap light-cone times (cf.
King:2018ibi ; Mackenroth:2018smh  for instance). The time-ordered double
integral over the C and BW probabilities, yielding the cascade approximation
(cf. equation (43) in Mackenroth:2018smh ), is analog to our above formula by
folding two rates, which facilitates $F_{t}\propto T_{0}^{2}$, if
$T_{C,\,BW}\propto T_{0}$. We attach to the Compton rate the number
$N^{e}_{0}=6\times 10^{9}$ of primary electrons per bunch.

In the numerical evaluation we employ the following convenient approximations:

$\displaystyle\frac{d\Gamma_{C}}{d\omega^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{\alpha m^{2}}{\pi\,E_{e}^{2}}{\cal F}_{C}(z,u),$ (20)
$\displaystyle\Gamma_{BW}(\omega^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\alpha m^{2}}{\omega^{\prime}}{\cal F}_{BW},$ (21)
$\displaystyle{\cal F}_{BW}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3^{3}}{2^{9}}}\kappa\exp\left[-\frac{8}{3\kappa}\left(1-\frac{1}{15\xi^{2}}\right)\right],$
(22)

where $u=\kappa/(\chi-\kappa)$, $\chi=(k\cdot p)\xi/m^{2}$, $\kappa=(k\cdot
k^{\prime})\xi/m^{2}$, and ${\cal F}_{C}(z,u)$ with $z=(u/\chi)^{2/3}$ is
defined in Eq. (5). Note the rise of $\Gamma_{BW}$ and the fall of
$d\Gamma_{C}/d\omega^{\prime}$ as a function of $\omega^{\prime}$ at fixed
values of $\xi$ and $k\cdot p$ and laser frequency $\omega$, see Fig. 5.
Increasing values of $\xi$ lift $d\Gamma_{C}/d\omega^{\prime}$ somewhat and
make it flatter in the intermediate-$\omega^{\prime}$ region, thus sharpening
the drop at $\omega^{\prime}\to E_{e}$. Remarkable is the strong impact of
increasing $\xi$ on the BW rate.

Figure 5: The BW rate $\Gamma_{BW}$ from Eqs. (21, IV.2) (solid curves) and
the dimensionless differential C rate $d\Gamma_{C}/d\omega^{\prime}$ (dashed
curves) from Eqs. (5, 20) as a function of $\omega^{\prime}$ for $\xi=1$ (red)
and 3 (blue).

One may replace Eq. (5) in (20) by the sum over harmonics expressed by Bessel
functions Ritus  to get an improvement of accuracy in the
small-$\omega^{\prime}$ region:777Strictly speaking, imposing a finite
duration in the monochromatic laser beam model 2) turns it into a flat-top
laser pulse model of class 1.1), exemplified in King:2020hsk  and applied to
the nonlinear Compton process.

$\displaystyle\frac{d\Gamma_{C}^{IPA}}{d\omega^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{d\sigma_{C}^{IPA}}{d\omega^{\prime}}\frac{\xi^{2}m^{2}}{4\pi\alpha}\frac{k\cdot
p}{q_{0}},$ (23) $\displaystyle\frac{d\sigma_{C}^{IPA}}{d\omega^{\prime}}$
$\displaystyle=$ $\displaystyle\frac{\pi r_{e}^{2}m^{2}}{\xi^{2}k\cdot
p}\sum_{n=1}^{\infty}\frac{\Theta(u_{n}-u)}{P_{n}}\left[-4J_{n}^{2}\right.$
$\displaystyle+$
$\displaystyle\left.\xi^{2}(2+\frac{u^{2}}{1+u})\left(J_{n-1}^{2}+J_{n+1}^{2}-2J_{n}^{2}\right)\right]$

with $q_{0}=\sqrt{E_{e}^{2}-m^{2}}+\beta_{p}\omega$,
$\beta_{p}=\xi^{2}\frac{m^{2}}{2q\cdot k}$, $q\cdot k=k\cdot p$, and arguments
$z=\frac{2n\xi}{\sqrt{1+\xi^{2}}}\frac{1}{u_{n}}\sqrt{u(u_{n}-u)}$ of the
Bessel functions $J_{n}$ as well as $u_{n}=2n\frac{k\cdot
p}{m^{2}}\frac{1}{1+\xi^{2}}$,
$u=\frac{n\omega-\omega^{\prime}}{\kappa_{n}-n\omega+\omega^{\prime}}$,
$\kappa_{n}=n\omega-\frac{1}{2}me^{\zeta}+\frac{1}{2}m(1+\xi^{2})e^{-\zeta}$,
$P_{n}=m|n\frac{\omega}{m}-\sinh\zeta+\frac{\xi^{2}}{2}e^{-\zeta}|$. These
equations make the dependence $u(\omega^{\prime})$ explicit and relate again
the differential cross section $d\sigma/d\omega^{\prime}$ with the
differential rate $d\Gamma/d\omega^{\prime}$. Since the BW rate is exceedingly
small at small $\omega^{\prime}$ (see Fig. 5), improvements of the Compton
rate by catching the details of harmonic structures there are less severe for
the pair number Eq. (19).

Equation (22) of the BW rate, however, is inappropriate at smaller values of
$\xi$ Hartin:2018sha  and needs improvement. Instead of using the series
expansion in Bessel functions Ritus , a convenient formula is

$\displaystyle{\cal F}_{BW}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{n=n_{min}}^{\infty}\int_{1}^{u_{n}}\frac{du}{u^{3/2}\sqrt{1-u}}$
$\displaystyle\times$ $\displaystyle\exp\left\\{-2n(a-\tanh
a\right\\}\frac{1+2\xi^{2}(2u-1)\sinh^{2}a}{a\tanh a}$

with $n_{min}=2\xi(1+\xi^{2})/\chi$ and $u_{n}=n/n_{min}$. This representation
emerges from the large-$n$ approximation of Bessel functions,
$J_{n}(z)\approx\exp\left\\{-n(a-\tanh a)\right\\}/\sqrt{2n\pi\tanh a}$ and
$\tanh a=\sqrt{1-z^{2}/n^{2}}$. In the large-$\xi$ limit, one may replace the
summation over $n$ by an integration to arrive, via a double saddle point
approximation, at the famous Ritus expression (22), which in turn is a
complement of Eq. (6), see Ritus .

Figure 6: The yield of $e^{+}e^{-}$ pairs as a function of $\xi$ for electron
energies 45 GeV (magenta dash-dotted curve), 17.5 GeV (blue solid curve) and 8
GeV (red dashed curve) according to the probabilistic folding model
C$\otimes$BW Eq. (19). For $N_{0}^{e}=6\times 10^{9}$ electrons per bunch and
per laser shot of duration $NT_{0}$. The special normalization
$F_{t}=T_{0}^{2}/2$ is chosen, as realized by
$T_{BW}=(T_{0}^{2}+T_{C}^{2})/(2T_{C})$. The choice $T_{C}\approx T_{0}$
facilitates a Compton spectrum
$dN_{C}/d\omega^{\prime}=T_{C}\,d\Gamma_{C}/d\omega^{\prime}$ which agrees,
for $\xi={\cal O}(1)$ and in the region $\omega^{\prime}<10$ GeV, with a
bremsstrahlung spectrum generated by electrons of the same energy impinging on
a foil with $X/X_{0}=0.01$ Hartin:2018sha . It is the $\xi$ dependence of the
Compton spectrum (see Fig. 5) which makes the pair yield more rapidly rising
with $\xi$ than the pair yield of the bremsstrahlung$\otimes$BW model in
Hartin:2018sha .  Figure 7: The power $p$ as a function of $\xi$ for
$E_{e}=45$ GeV (magenta dash-dotted curve), 17.5 GeV (blue solid curve) and 8
GeV (red dashed curve). The results exhibited in Fig. 6 are described by
$N^{e^{+}e^{-}}(\xi,E_{e})=N^{e^{+}e^{-}}_{0}(E_{e})\,\xi^{p(\xi,E_{e})}$.

Numerical results are exhibited in Fig. 6 for $E_{e}=45$ GeV, 17.5 GeV and 8
GeV. One observes a stark rise of $N^{e^{+}e^{-}}$ up to $\xi\sim 4$, which
turns for larger values of $\xi$ into a modest rise. To quantify that rise one
can employ the ansatz
$N^{e^{+}e^{-}}(\xi,E_{e})=N^{e^{+}e^{-}}_{0}(E_{e})\,\xi^{p(\xi,E_{e})}$.
Note that, by such a quantification of the $\xi$ dependence, one gets rid of
the normalization $F_{t}$. For $E_{e}=17.5$ GeV we find $p(\xi\approx
1)\approx 20$ dropping to $p(\xi\approx 20)\approx 2$, see Fig. 7. Larger
values of $E_{e}$ reduce $p$, e.g. $p(\xi\approx
1)|_{E_{e}=45\,\rm{GeV}}\approx 10$ in agreement with Hu:2010ye , while
$p(\xi\approx 1)|_{E_{e}=8\,\rm{GeV}}>40$. At $\xi\to 20$, a universal value
of $p\approx 2$ seems to emerge. The extreme nonlinear sensitivity of the pair
number on the laser intensity parameter $\xi$ at $\xi<10$, and in particular
at $\xi\approx 1$, points to the request of a refined and adequately realistic
modeling beyond schematic approaches.

###  IV.3 Bandwidth effects in linear trident

The threshold for linear trident,
$e^{-}+\gamma(1.55\,{\mbox{e}V})\to{e^{-}}^{\prime}+{e^{-}}^{\prime\prime}+e^{+}$,
is at $E_{e}=337$ GeV, i.e. the LUXE kinematics is in the deep sub-threshold
regime, where severe multi-photon effects build up the nonlinearity. However,
also bandwidth effects can promote pair production in the sub-threshold region
Titov:2012rd ; Nousch:2012xe , even at $\xi\to 0$. The key is the cross
section of linear trident $\sigma_{ppT}(\sqrt{\hat{s}},\Delta\phi)$, which
depends on the invariant energy $\sqrt{\hat{s}}=m\sqrt{1+2k\cdot p/m^{2}}$ and
the pulse duration $\Delta\phi$ for a given laser pulse. The quantity
$\sigma_{ppT}(\sqrt{\hat{s}},\Delta\phi)$ is exhibited in Fig. 8 as a function
of $\sqrt{\hat{s}}$ for several values of $\Delta\phi$. For definiteness, we
employ the laser pulse model of class 1.1) with parameterization
$\vec{A}=f_{ppT}(\phi)\,\vec{a}_{x}\,\cos\phi$ and envelope function
$f_{ppT}=\cos^{2}\left(\frac{\pi\phi}{2\Delta\phi}\right)\sqcap(\phi,2\Delta\phi)$,
i.e. the number of laser-field oscillations within the pulse is
$N=\Delta\phi/\pi$. In contrast to the presentation above, we deploy results
in this sub-section for linear polarization and the $\cos^{2}$ envelope.

We employ the formalism in Acosta:2019bvh  and its numerical implementation,
that is “pulsed perturbative QED” in the spirit of Furry picture QED in a
series expansion in powers of $\xi$. Applied to trident, the pulsed
perturbative trident (ppT) arises from the diagrams $e^{+}$$e^{-}$ $-$
$e^{-}$$e^{+}$ (double lines: Volkov wave functions, vertical lines: photon
propagator) as leading-order term surviving $\xi\to 0$.

The scaled number of pairs is $N_{tot}/N_{0}^{e}=2\pi\Gamma_{tot}/\pi$, where
the probability rate is given by

$\Gamma_{tot}=\sigma_{ppT}\frac{\omega}{2}\frac{m^{2}\xi^{2}}{4\pi\alpha}\int_{-\infty}^{\infty}d\phi\,f^{2}_{ppT}(\phi).$
(26)

The chosen pulse implies
$\int_{-\infty}^{\infty}d\phi\,f^{2}_{ppT}(\phi)=\frac{3}{2}\Delta\phi$ and
$N_{tot}/N_{0}^{e}=\sigma_{ppT}\frac{3m^{2}\xi^{2}}{16\alpha}\Delta\phi$. Note
the $\xi^{2}$ dependence from the “target density” already entering in Eq.
(26) (cf. Seipt:2010ya ; Titov:2012rd  for analog relations). This is in
contrast to Fig. 6, where genuine nonlinear effects are at work and mix with a
stronger $\xi$ dependence for C$\otimes$BW. The $\xi^{2}$ dependence is
characteristic for pair production by probe photons provided by an “external
target”, such as in the bremsstrahlung-laser configuration of LUXE, cf.
Hartin:2018sha .

Figure 8: Pulsed perturbative trident cross section $\sigma_{ppT}$ as a
function of $\sqrt{\hat{s}}/m$ for several pulse lengths $\Delta\phi$. In the
IPA limit, i.e. a monochromatic laser beam or $\Delta\phi\to\infty$, the
threshold is at $\sqrt{\hat{s}}=3m$. Above the threshold, the dots depict a
few points (cf. table 1 in Haug:1975bk ) from perturbative trident without
bandwidth effects. Bandwidth effects enable the pair production in the sub-
threshold region $\sqrt{\hat{s}}<3m$.

For the long laser pulses used in E-144 Bamber:1999zt ; Burke:1997ew ;
Bula:1996st , such bandwidth effects are less severe.

##  V summary

In summary, inspired by the renewed interest in the Ritus-Narozhny conjecture
and the new perspectives offered by the experimental capabilities of LUXE and
E-320, we recollect a few features of elementary QED processes within the
essentially known formalism. In particular, we focus on the $\xi$ dependence.
For nonlinear Compton scattering, we point out that, in the non-asymptotic
region $\chi={\cal O}(1)$, $k\cdot p\lesssim m^{2}$, the spectrally resolved
cross section $d\sigma/du|_{u=const}$ as a function of the laser intensity
parameter $\xi$ displays a pronounced $\cap$ shape for $u>u_{KN}$ (the “rise
and fall”). This behavior is in stark contrast with the monotonously rising
integrated probability
$\lim_{\chi,\xi\to\infty}\mathbb{P}\propto\alpha\chi^{2/3}m^{2}/k\cdot p$.
That is, in different regions of the phase space, also different sensitivities
of cross sections/rates/probabilities on the laser intensity impact can be
observed. The soft (small-$u$) part, which determines the integrated cross
section/probability, may behave completely different than the hard (large-$u$)
contribution.888An analog situation is known in QCD Horowitz:2010yg : Tree-
level diagrams are calculated primarily with constant $\alpha_{QCD}$.
Renormalization improvement means then replacing $\alpha_{QCD}$ by
$\alpha_{QCD}(\mathfrak{s})$ which accounts at the same time for all vertices
in the considered diagram by one global scale $\mathfrak{s}$. Inserting
explicitly loop corrections leads finally to scale dependent couplings
$\alpha_{QCD}(\mathfrak{s}_{v})$ specific for each vertex $v$ separately.
Transferred to certain approximations used in simulation codes such a behavior
implies that one should test differentially where the conditions for
applicability are ensured.

The hard photons, once produced by Compton process in a laser pulse, act as
seeds for secondary processes, most notably the Breit-Wheeler process. A
folding model of type Compton$\otimes$Breit-Wheeler on the probabilistic level
points to a rapidly increasing rate of $e^{+}e^{-}$ production in the region
$\xi\lesssim 4$, when using parameters in reach of the planned LUXE set-up.
The actual plans (see figure 2.10 in LUXE CDR Abramowicz:2021zja ) uncover
$\xi=2$ (40 TW, 8 $\mu$m laser), 6 (40 TW, 3 $\mu$m) and 16 (300 TW, 3
$\mu$m), and E-320 envisages $\xi=10$. The folding model may be utilized as
reference to identify the occurrence of the wanted one-step trident process in
this energy-intensity regime. Furthermore, bandwidth effects in the trident
process are isolated by considering the weak-field regime $\xi\to 0$.

##  Appendix A Basics of nonlinear Compton process

Following SeiptDaniel:2012tua  we recall the basics of the underlying
formalism of the nonlinear Compton process. Within the Furry picture the
lowest-order, tree-level $S$ matrix element for the one-photon (four-momentum
$k^{\prime}$, four-polarization $\epsilon^{\prime}$) decay of a laser-dressed
electron $e_{L}$ in the background field (1) $e_{L}(p)\to
e_{L}(p^{\prime})+\gamma(k^{\prime},\epsilon^{\prime})$ reads with suitable
normalizations of the wave functions

$S_{fi}=-ie\int d^{4}x\,J\cdot{\epsilon^{*}}^{\prime}\,\exp\\{ik^{\prime}\cdot
x\\}$ (27)

where the current $J_{\mu}(x)=\bar{\Psi}_{p^{\prime}}\gamma_{\mu}\Psi_{p}$ is
built by the Volkov wave function $\Psi_{p}=E_{p}u_{p}\exp\\{-ip\cdot
x\\}\exp\\{-if_{p}\\}$ (spin indices are suppressed) and its adjoint
$\bar{\Psi}$ with Ritus matrix $E_{p}=1+\frac{e}{2k\cdot p}\not{k}\not{A}$ and
phase function $f_{p}(\phi)=\frac{e}{k\cdot
p}\int_{0}^{\phi}d\phi^{\prime}\left[p\cdot A-\frac{e}{2}A\cdot A\right]$. We
employ Feynman’s slash notation and denote scalar products by the dot between
four-vectors; $u_{p}$ is the free Dirac bi-spinor. Exploiting the symmetry of
the background field, $A(\phi=k\cdot x)$, Eq. (27) can be manipulated (cf.
SeiptDaniel:2012tua  for details) to arrive at

$S_{fi}=-ie(2\pi)^{3}\frac{2}{k_{-}}\,\delta^{(3)}(\underline{p}-\underline{p^{\prime}}-\underline{k^{\prime}})\,{\cal
M}(\ell),$ (28)

where $\ell\equiv(k^{\prime}_{-}+p^{\prime}_{-}-p_{-})/k_{-}=k^{\prime}\cdot
p/k\cdot p^{\prime}$ accomplishes the balance equation $p+\ell
k-p^{\prime}-k^{\prime}=0$. (See Titov:2015pre  for a formulation with
$S_{fi}=-ie(2\pi)^{4}\int\frac{d\ell}{2\pi}\,\delta^{(4)}(p+\ell
k-p^{\prime}-k^{\prime})\,{\cal M}(\ell)$.) Light-cone coordinates are useful
here, e.g. $k_{-}=k^{0}-k^{3}$, $k_{+}=k^{0}+k^{3}$,
$k_{\perp}=(k^{1},k^{2})$, and $\underline{k}=(k_{+},k_{\perp})$. Imposing
gauge invariance yields the matrix element

${\cal M}=\textstyle\sum_{i=1}^{3}J^{(i)}S^{(i)}$ (29)

with the pieces of the electron current

$\displaystyle\hat{J}^{(0)}$ $\displaystyle=$
$\displaystyle\bar{u}_{p^{\prime}}{\not{\epsilon}^{\prime}}^{*}u_{p},$ (30)
$\displaystyle\hat{J}^{(1,2)}$ $\displaystyle=$
$\displaystyle\bar{u}_{p^{\prime}}\left(d_{p^{\prime}}{\not{\epsilon}_{\pm}}\not{k}{\not{\epsilon}^{\prime}}^{*}+d_{p}{\not{\epsilon^{\prime}}}^{*}\not{k}{\not{\epsilon}_{\pm}}\right)u_{p},$
(31) $\displaystyle\hat{J}^{(3)}$ $\displaystyle=$ $\displaystyle
4k\cdot{\epsilon^{\prime}}^{*}d_{p^{\prime}}d_{p}\,\bar{u}_{p^{\prime}}\not{k}u_{p},$
(32)

which combine to
$J^{(1,2)}=\hat{J}^{(1,2)}+\frac{\alpha_{\pm}}{2\ell}\hat{J}^{(0)}$,
$J^{(3)}=\hat{J}^{(3)}+\frac{\beta}{\ell}\hat{J}^{(0)}$. The following
abbreviations are used: $d_{p}=\frac{m\xi}{4k\cdot p}$ and
$d_{p^{\prime}}=\frac{m\xi}{4k\cdot p^{\prime}}$,
$\alpha_{\pm}=m\xi\left(\frac{\epsilon_{\pm}\cdot p}{k\cdot
p}-\frac{\epsilon_{\pm}\cdot p^{\prime}}{k\cdot p^{\prime}}\right)$ with
$\vec{\epsilon}_{\pm}=(\vec{a}_{x}\pm i\vec{a}_{y})e^{2}/m^{2}\xi^{2}$ and
$\beta=\frac{m^{2}\xi^{2}}{4}\left(\frac{1}{k\cdot p}-\frac{1}{k\cdot
p^{\prime}}\right)$ as well.

The phase integrals $S^{(i)}$ are the remainders of the integration
$d^{4}x=d\phi\,dx_{-}\,d^{2}x_{\perp}/k_{-}$ in Eq. (27):

$\displaystyle S^{(1,2)}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\phi\,f(\phi)\,\exp\\{i(\ell\pm
1)\phi-i(f_{p}(\phi)-f_{p^{\prime}}(\phi))\\},$ $\displaystyle S^{(3)}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\phi\,f^{2}(\phi)\exp\\{i\ell\phi-i(f_{p}(\phi)-f_{p^{\prime}}(\phi))\\}$
$\displaystyle\times\left[1+\cos 2\phi\right].$

For a few special non-unipolar (plane-wave) fields and their envelopes
$f(\phi)$, the phase integrals can be processed exactly by analytic means
Seipt:2016rtk ; Dinu:2013hsd , but in general a numerical evaluation is
needed. The very special IPA case of $f(\phi)=1$ allows for a simple
representation of $f_{p}(\phi)$ with subsequent decomposition of $S^{(i)}$
into Bessel functions, yielding final expressions as in Eqs. (2, IV.2).

For identifying the weak-field limit, $\xi\to 0$, it is necessary to recognize
in Eq. (29) $J^{(0)}\propto\xi^{0}$, and
$\alpha_{\pm},\,J^{(1,2)}\propto\xi^{1}$, and $\beta,\,J^{(3)}\propto\xi^{2}$.
Thus, $\lim_{\xi\to 0}{\cal M}={\cal M}_{1}\xi+{\cal M}_{2}\xi^{2}+\cdots$. We
emphasize the Fourier transform of the pulse envelope, $\lim_{\xi\to
0}S^{(1,2)}=\int_{-\infty}^{\infty}d\phi\,f(\phi)\exp\\{i(\ell\pm 1)\phi\\}$,
entering ${\cal M}_{1}$, where only $S^{(1)}$ with $\ell\geq 0$ contributes to
the wanted one-photon emission:999 The same reasoning applies in IV.3 for one-
pair emission, i.e. the finite-width Fourier transform of $f(\phi)$ at
$\Delta\phi<\infty$ enables the sub-threshold pair production $e_{L}(p)\to
e_{L}(p^{\prime})+e_{L}(p^{\prime\prime})+\bar{e}_{L}(p^{\prime\prime\prime})$.
${\cal
M}_{1}(\ell)=J^{(1)}\int_{-\infty}^{\infty}d\phi\,f(\phi)\exp\\{i(\ell-1)\phi\\}$,
$J^{(1)}(\ell)=\frac{m}{2}\epsilon_{\mu}^{\prime}{}^{*}\epsilon_{+\nu}\bar{u}_{p^{\prime}}\left[\frac{\gamma^{\mu}\not{k}_{\ell}\gamma^{\nu}+2\gamma^{\mu}p^{\nu}}{2k_{\ell}\cdot
p}+\frac{\gamma^{\nu}\not{k}^{\prime}\gamma^{\mu}-2\gamma^{\nu}p^{\mu}}{2k^{\prime}\cdot
p}\right]u_{p}$ with $k_{\ell}\equiv\ell k$. For pulses with broad support,
$f(\phi)\to 1$, within the interval $\Delta\phi\gg 1$, one arrives at the
standard Compton (Klein-Nishina) expression in leading order by using $\xi\to
2e/m$. The appearance of $\delta(\ell-1)$ combined with the definition of
$\ell$ below Eq. (28) leads to the famous Compton formula via
$\delta\left(\omega^{\prime}-\frac{\omega
m}{m+\omega(1-\cos\Theta^{\prime})}\right)$ in the electron’s rest frame by
the subsequent phase space integration(s).

The discussion of the large-$\xi$ limit needs some care in general, cf.
Ilderton:2018nws ; DiPiazza:2018bfu . Useful limits are obtained for the IPA
case Ritus  under the side condition $\xi^{2}(1-z^{2}/n^{2})=const$ (cf. Eqs.
(23, IV.2)), e.g. $|{\cal M}|^{2}|_{u\ll 1}\propto(\xi/u)^{2/3}$ and $|{\cal
M}^{2}|_{u\gg 1}\propto
u^{-1}(\xi/u)^{1/2}\exp\\{-\frac{2u}{3\xi}\frac{m^{2}}{k\cdot
p}\\}\to\sqrt{\xi}$ (cf. Eqs. (18) and (15)), At $\xi\gg 1$, the $\xi$ and $u$
dependencies of resulting probabilities for circular polarization and constant
cross field backgrounds coincide.

These limits are at the heart of the “rise and fall”.

The differential emission probability per $in$-electron and per laser pulse
follows from (28) by partial integration over the $out$-phase space,

$\frac{d\mathbb{P}}{d\omega^{\prime}d\Omega^{\prime}}=\frac{e^{2}\omega^{\prime}}{64\pi^{3}\,k\cdot
p\,k\cdot p^{\prime}}|{\cal M}|^{2},$ (34)

and may be transformed to other coordinates, e.g. $u$ or $\ell$ etc. Having in
mind the IPA limit, one should turn to the dimensionless differential rate
Titov:2015pre

$\frac{d\Gamma_{C}}{d\omega^{\prime}d\Omega^{\prime}}=\frac{e^{2}\omega^{\prime}}{32\pi^{2}\,q_{0}\,k\cdot
p^{\prime}}|{\cal M}|^{2}.$ (35)

Spin averaging of the $in$-electron, and spin summation of the $out$-electron
and summation over the $out$-photon polarizations leads to $\overline{|{\cal
M}|^{2}}$, unless one is interested in polarization effects as in
Seipt:2020diz ; King:2020btz ; Ivanov:2004vh . The above expressions (29 \- A)
can be further processed for special field envelopes, or $|{\cal M}|^{2}$ is
numerically accessible, via Eq. (29), as mod-squared sum of complex number
products provided by Eqs. (30 \- A) which need afterwards explicit (numerical)
spin and polarization summation/averaging to arrive at $\overline{|{\cal
M}|^{2}}$.

The cross section is obtained by normalization on the integrated laser photon
flux: $d\sigma=d\Gamma_{C}\frac{q_{0}}{k\cdot p}\frac{\omega}{n_{L}}$ with
$n_{L}=\frac{m^{2}}{e^{2}}\xi^{2}\omega N_{L}$, where $N_{L}=1$ (IPA, quasi-
momentum $q_{0}$) or
$N_{L}=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\phi\,f(\phi)$ (FPA, $q_{0}\equiv
p_{0}$). The circularly polarized laser background (1) is supposed in these
relations.

###### Acknowledgements.

The authors gratefully acknowledge the collaboration with D. Seipt, T. Nousch,
T. Heinzl, and useful discussions with A. Ilderton, K. Krajewska, M. Marklund,
C. Müller, S. Rykovanov, and G. Torgrimsson. A. Ringwald and B. King are
thanked for explanations w.r.t. LUXE. The work is supported by R. Sauerbrey
and T. E. Cowan w.r.t. the study of fundamental QED processes for HIBEF.

## References

  * (1) M. E. Peskin and D. V. Schroeder, “An Introduction to Quantum Field Theory,” Addison-Wesley, Reading (1995).
  * (2) G. Abbiendi et al. [OPAL Collaboration], “Measurement of the running of the QED coupling in small-angle Bhabha scattering at LEP,” Eur. Phys. J. C 45, 1 (2006) [hep-ex/0505072].
  * (3) I. Arapoglou et al., “g Factor of Boronlike Argon Ar4013+,” Phys. Rev. Lett. 122, no. 25, 253001 (2019) [arXiv:1906.00881 [physics.atom-ph]].
  * (4) S. Alighanbari, G. S. Giri, F. L. Constantin, et al., “Precise test of quantum electrodynamics and determination of fundamental constants with HD+ ions,” Nature 581, 152–158 (2020).
  * (5) T. Aoyama et al., “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887, 1 (2020) [arXiv:2006.04822 [hep-ph]].
  * (6) V. Ritus, “Radiative Effects and Their Enhancement in an Intense Electromagnetic Field ,” Sov. Phys. JETP 30, 1181 (1970).
  * (7) D. Morozov and N. Narozhny, “Elastic scattering of photons in an intense field and the photoproduction of a pair and a photon,” Sov. Phys. JETP 45, 23 (1977).
  * (8) N. Narozhny, “Radiation corrections to quantum processes in an intense electromagnetic field,” Phys. Rev. D 20, 1313 (1979).
  * (9) N. Narozhny, “Expansion parameter of perturbation theory in intense-field quantum electrodynamics,” Phys. Rev. D 21, 1176 (1980).
  * (10) D. Morozov, N. Narozhny and V. I. Ritus, “Vertex function of an electron in a constant electromagnetic field,” Sov. Phys. JETP 53, 1103 (1981).
  * (11) A. A. Mironov, S. Meuren and A. M. Fedotov, “Resummation of QED radiative corrections in a strong constant crossed field,” Phys. Rev. D 102, no. 5, 053005 (2020) [arXiv:2003.06909 [hep-th]].
  * (12) T. Heinzl, A. Ilderton and B. King, “Classical resummation and breakdown of strong-field QED,” arXiv:2101.12111 [hep-ph].
  * (13) J. P. Edwards and A. Ilderton, “Resummation of background-collinear corrections in strong field QED,” Phys. Rev. D 103, no. 1, 016004 (2021) [arXiv:2010.02085 [hep-ph]].
  * (14) A. M. Fedotov, “Conjecture of perturbative QED breakdown at $\alpha\chi^{2/3}\gtrsim 1$,” J. Phys. Conf. Ser. 826, no. 1, 012027 (2017) [arXiv:1608.02261 [hep-ph]].
  * (15) N. Brambilla et al., “QCD and Strongly Coupled Gauge Theories: Challenges and Perspectives,” Eur. Phys. J. C 74, no. 10, 2981 (2014) [arXiv:1404.3723 [hep-ph]].
  * (16) R. V. Popov et al., “How to access QED at a supercritical Coulomb field,” Phys. Rev. D 102, no. 7, 076005 (2020) [arXiv:2008.05005 [hep-ph]].
  * (17) V. Yakimenko et al., “Prospect of Studying Nonperturbative QED with Beam-Beam Collisions,” Phys. Rev. Lett. 122, no. 19, 190404 (2019) [arXiv:1807.09271 [physics.plasm-ph]].
  * (18) A. Di Piazza, “Unveiling the transverse formation length of nonlinear Compton scattering,” Phys. Rev. A 103, no. 1, 012215 (2021) [arXiv:2009.00526 [hep-ph]].
  * (19) D. Seipt and B. King, “Spin- and polarization-dependent locally-constant-field-approximation rates for nonlinear Compton and Breit-Wheeler processes,” Phys. Rev. A 102, no. 5, 052805 (2020) [arXiv:2007.11837 [physics.plasm-ph]].
  * (20) B. King and S. Tang, “Nonlinear Compton scattering of polarized photons in plane-wave backgrounds,” Phys. Rev. A 102, no. 2, 022809 (2020) [arXiv:2003.01749 [hep-ph]].
  * (21) A. I. Titov, B. Kämpfer and H. Takabe, “Nonlinear Breit-Wheeler process in short laser double pulses,” Phys. Rev. D 98, no. 3, 036022 (2018) [arXiv:1807.04547 [physics.plasm-ph]].
  * (22) A. Ilderton, B. King and S. Tang, “Toward the observation of interference effects in nonlinear Compton scattering,” Phys. Lett. B 804, 135410 (2020) [arXiv:2002.04629 [physics.atom-ph]].
  * (23) L. F. Granz, O. Mathiak, S. Villalba-Chávez and C. Müller, “Electron-positron pair production in oscillating electric fields with double-pulse structure,” Phys. Lett. B 793, 85 (2019) [arXiv:1903.06000 [physics.plasm-ph]].
  * (24) T. G. Blackburn, A. J. MacLeod and B. King, “From local to nonlocal: higher fidelity simulations of photon emission in intense laser pulses,” arXiv:2103.06673 [hep-ph].
  * (25) T. Heinzl, B. King and A. J. Macleod, “The locally monochromatic approximation to QED in intense laser fields,” Phys. Rev. A 102, 063110 (2020) [arXiv:2004.13035 [hep-ph]].
  * (26) A. Ilderton, B. King and D. Seipt, “Extended locally constant field approximation for nonlinear Compton scattering,” Phys. Rev. A 99, no. 4, 042121 (2019) [arXiv:1808.10339 [hep-ph]].
  * (27) T. G. Blackburn, D. Seipt, S. S. Bulanov and M. Marklund, “Benchmarking semiclassical approaches to strong-field QED: nonlinear Compton scattering in intense laser pulses,” Phys. Plasmas 25, no. 8, 083108 (2018) [arXiv:1804.11085 [physics.plasm-ph]].
  * (28) A. Di Piazza, M. Tamburini, S. Meuren and C. H. Keitel, “Improved local-constant-field approximation for strong-field QED codes,” Phys. Rev. A 99, no. 2, 022125 (2019) [arXiv:1811.05834 [hep-ph]].
  * (29) V. N. Khudik et al., “Towards realistic simulations of QED cascades: Non-ideal laser and electron seeding effects,” Phys. Plasmas 25, no. 8, 083104 (2018) [arXiv:1807.09747 [physics.plasm-ph]].
  * (30) T. G. Blackburn, A. Ilderton, C. D. Murphy and M. Marklund, “Scaling laws for positron production in laser–electron-beam collisions,” Phys. Rev. A 96, no. 2, 022128 (2017) [arXiv:1708.00298 [physics.plasm-ph]].
  * (31) A. Gonoskov et al., “Extended particle-in-cell schemes for physics in ultrastrong laser fields: Review and developments,” Phys. Rev. E 92, no. 2, 023305 (2015) [arXiv:1412.6426 [physics.plasm-ph]].
  * (32) G. Torgrimsson, “Loops and polarization in strong-field QED,” arXiv:2012.12701 [hep-ph].
  * (33) G. Torgrimsson, “Nonlinear trident in the high-energy limit: Nonlocality, Coulomb field and resummations,” Phys. Rev. D 102, no. 9, 096008 (2020) [arXiv:2007.08492 [hep-ph]].
  * (34) V. Dinu and G. Torgrimsson, “Trident process in laser pulses,” Phys. Rev. D 101, no. 5, 056017 (2020) [arXiv:1912.11017 [hep-ph]].
  * (35) V. Dinu and G. Torgrimsson, “Approximating higher-order nonlinear QED processes with first-order building blocks,” Phys. Rev. D 102, no. 1, 016018 (2020) [arXiv:1912.11015 [hep-ph]].
  * (36) A. Di Piazza, C. Müller, K. Z. Hatsagortsyan and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177 (2012) [arXiv:1111.3886 [hep-ph]].
  * (37) A. Ilderton, “Note on the conjectured breakdown of QED perturbation theory in strong fields,” Phys. Rev. D 99, no. 8, 085002 (2019) [arXiv:1901.00317 [hep-ph]].
  * (38) T. Podszus and A. Di Piazza, “High-energy behavior of strong-field QED in an intense plane wave,” Phys. Rev. D 99, no. 7, 076004 (2019) [arXiv:1812.08673 [hep-ph]].
  * (39) S. Meuren, “Probing Strong-field QED at FACET-II (SLAC E-320) (2019)”,

        https://conf.slac.stanford.edu/facet-2-2019/sites/
    facet-2-2019.conf.slac.stanford.edu/files/
    basic-page-docs/sfqed_2019.pdf.

  * (40) S. Meuren et al., “On Seminal HEDP Research Opportunities Enabled by Colocating Multi-Petawatt Laser with High-Density Electron Beams,” arXiv:2002.10051 [physics.plasm-ph].
  * (41) K. Poder et al., “Experimental Signatures of the Quantum Nature of Radiation Reaction in the Field of an Ultraintense Laser,” Phys. Rev. X 8, no. 3, 031004 (2018) [arXiv:1709.01861 [physics.plasm-ph]].
  * (42) J. M. Cole et al., “Experimental evidence of radiation reaction in the collision of a high-intensity laser pulse with a laser-wakefield accelerated electron beam,” Phys. Rev. X 8, no. 1, 011020 (2018) [arXiv:1707.06821 [physics.plasm-ph]].
  * (43) C. Keitel et al., “Photo-induced pair production and strong field QED on Gemini,” arXiv:2103.06059 [physics.plasm-ph].
  * (44) H. Abramowicz et al., “Letter of Intent for the LUXE Experiment,” arXiv:1909.00860 [physics.ins-det].
  * (45) H. Abramowicz et al., “Conceptual Design Report for the LUXE Experiment,” arXiv:2102.02032 [hep-ex].
  * (46) U. Hernandez Acosta, A. Otto, B. Kämpfer and A. I. Titov, “Non-perturbative signatures of non-linear Compton scattering,” Phys. Rev. D 102, 024914 (2020) [arXiv:2001.03986 [hep-ph]].
  * (47) B. Kämpfer and A. I. Titov, “Impact of laser polarization on q-exponential photon tails in non-linear Compton scattering,” Phys. Rev. A 103, 033101 (2021) [arXiv:2012.07699 [hep-ph]].
  * (48) A. Jochmann et al., “High Resolution Energy-Angle Correlation Measurement of Hard X Rays from Laser-Thomson Backscattering,” Phys. Rev. Lett. 111, no. 11, 114803 (2013).
  * (49) I. C. E. Turcu et al., “High field physics and QED experiments at ELI-NP,” Rom. Rep. Phys. 68, no. Supplement, S145 (2016).
  * (50) HIBEF: Helmholtz International Beam Line for Extreme Fields, cf.

        https://www.hzdr.de/db/Cms?pOid=50566&pNid=694

  * (51) A. Di Piazza, “WKB electron wave functions in a tightly focused laser beam,” Phys. Rev. D 103, no. 7, 076011 (2021) [arXiv:2102.06692 [hep-ph]].
  * (52) A. Di Piazza, “First-order strong-field QED processes in a tightly focused laser beam,” Phys. Rev. A 95, no. 3, 032121 (2017) [arXiv:1612.04132 [hep-ph]].
  * (53) T. Heinzl and A. Ilderton, “Exact classical and quantum dynamics in background electromagnetic fields,” Phys. Rev. Lett. 118, no. 11, 113202 (2017) [arXiv:1701.09166 [hep-ph]].
  * (54) F. Karbstein and E. A. Mosman, “Enhancing quantum vacuum signatures with tailored laser beams,” Phys. Rev. D 101, no. 11, 113002 (2020) [arXiv:2004.04268 [quant-ph]].
  * (55) A. Blinne, H. Gies, F. Karbstein, C. Kohlfürst and M. Zepf, “All-optical signatures of quantum vacuum nonlinearities in generic laser fields,” Phys. Rev. D 99, no. 1, 016006 (2019) [arXiv:1811.08895 [physics.optics]].
  * (56) F. Karbstein and E. A. Mosman, “Photon polarization tensor in pulsed Hermite- and Laguerre-Gaussian beams,” Phys. Rev. D 96, no. 11, 116004 (2017) [arXiv:1711.06151 [hep-ph]].
  * (57) V. Dinu, T. Heinzl and A. Ilderton, “Infra-Red Divergences in Plane Wave Backgrounds,” Phys. Rev. D 86, 085037 (2012) [arXiv:1206.3957 [hep-ph]].
  * (58) V. I. Ritus, “Quantum effects of the interaction of elementary particles with an intense electromagnetic field,” J. Sov. Laser Res. (United States) 6, 497 (1985).
  * (59) B. King, “Pulse envelope effect on nonlinear Compton scattering in electron-laser collisions,” Phys. Rev. D 103, no. 3, 036018 (2021) [arXiv:2012.05920 [hep-ph]].
  * (60) A. Otto, D. Seipt, D. Blaschke, B. Kämpfer and S. A. Smolyansky, “Lifting shell structures in the dynamically assisted Schwinger effect in periodic fields,” Phys. Lett. B 740, 335 (2015) [arXiv:1412.0890 [hep-ph]].
  * (61) A. Otto, H. Oppitz and B. Kämpfer, “Assisted Vacuum Decay by Time Dependent Electric Fields,” Eur. Phys. J. A 54, no. 2, 23 (2018) [arXiv:1801.09943 [hep-ph]].
  * (62) A. I. Titov, B. Kämpfer, A. Hosaka and H. Takabe, “Quantum processes in short and intensive electromagnetic fields,” Phys. Part. Nucl. 47, no. 3, 456 (2016) [arXiv:1512.07987 [hep-ph]].
  * (63) A. I. Titov, B. Kämpfer, T. Shibata, A. Hosaka and H. Takabe, “Laser pulse-shape dependence of Compton scattering,” Eur. Phys. J. D 68, no. 10, 299 (2014) [arXiv:1408.1040 [hep-ph]].
  * (64) A. I. Titov, A. Otto and B. Kämpfer, “Multi-photon regime of non-linear Breit-Wheeler and Compton processes in short linearly and circularly polarized laser pulses,” Eur. Phys. J. D 74, no. 2, 39 (2020) [arXiv:1907.00643 [physics.plasm-ph]].
  * (65) A. I. Titov, B. Kämpfer, H. Takabe and A. Hosaka, “Neutrino pair emission off electrons in a strong electromagnetic wave field,” Phys. Rev. D 83, 053008 (2011) [arXiv:1011.4860 [hep-ph]].
  * (66) K. Fleck, N. Cavanagh and G. Sarri, “Conceptual Design of a High-flux Multi-GeV Gamma-ray Spectrometer,” Sci. Rep. 10, no. 1, 9894 (2020).
  * (67) J. Z. Kamiński, K. Krajewska and F. Ehlotzky, “Monte Carlo analysis of electron-positron pair creation by powerful laser-ion impact,” Phys. Rev. A 74, no. 3, 033402 (2006).
  * (68) T. Heinzl, A. Ilderton and M. Marklund, “Finite size effects in stimulated laser pair production,” Phys. Lett. B 692, 250 (2010) [arXiv:1002.4018 [hep-ph]].
  * (69) K. Krajewska and J. Z. Kaminski, “Breit-Wheeler Process in Intense Short Laser Pulses,” Phys. Rev. A 86, 052104 (2012) [arXiv:1209.2394 [hep-ph]].
  * (70) K. Krajewska and J. Z. Kamiński, “Coherent combs of antimatter from nonlinear electron-positron-pair creation,” Phys. Rev. A 90, no. 5, 052108 (2014) [arXiv:1407.4101 [hep-ph]].
  * (71) M. J. A. Jansen, J. Z. Kamiński, K. Krajewska and C. Müller, “Strong-field Breit-Wheeler pair production in short laser pulses: Relevance of spin effects,” Phys. Rev. D 94, 013010 (2016) [arXiv:1605.03476 [hep-ph]].
  * (72) M. J. A. Jansen and C. Müller, “Strong-Field Breit-Wheeler Pair Production in Two Consecutive Laser Pulses with Variable Time Delay,” Phys. Lett. B 766, 71 (2017) [arXiv:1612.07137 [quant-ph]].
  * (73) A. I. Titov and B. Kämpfer, “Non-linear Breit–Wheeler process with linearly polarized beams,” Eur. Phys. J. D 74, no. 11, 218 (2020) [arXiv:2006.04496 [hep-ph]].
  * (74) A. Golub, S. Villalba-Chávez, H. Ruhl and C. Müller, “Linear Breit-Wheeler pair production by high-energy bremsstrahlung photons colliding with an intense X-ray laser pulse,” Phys. Rev. D 103, no. 1, 016009 (2021) [arXiv:2010.03871 [hep-ph]].
  * (75) A. Hartin, A. Ringwald and N. Tapia, “Measuring the Boiling Point of the Vacuum of Quantum Electrodynamics,” Phys. Rev. D 99, no. 3, 036008 (2019) [arXiv:1807.10670 [hep-ph]].
  * (76) A. I. Titov, B. Kämpfer and H. Takabe, “Dimuon production by laser-wakefield accelerated electrons,” Phys. Rev. ST Accel. Beams 12, 111301 (2009) [arXiv:0907.3038 [physics.acc-ph]].
  * (77) B. King and A. M. Fedotov, “Effect of interference on the trident process in a constant crossed field,” Phys. Rev. D 98, no. 1, 016005 (2018) [arXiv:1801.07300 [hep-ph]].
  * (78) F. Mackenroth and A. Di Piazza, “Nonlinear trident pair production in an arbitrary plane wave: a focus on the properties of the transition amplitude,” Phys. Rev. D 98, no. 11, 116002 (2018) [arXiv:1805.01731 [hep-ph]].
  * (79) U. Hernandez Acosta and B. Kämpfer, “Laser pulse-length effects in trident pair production,” Plasma Phys. Control. Fusion 61, no. 8, 084011 (2019) [arXiv:1901.08860 [hep-ph]].
  * (80) V. N. Baier, V. M. Katkov and V. M. Strakhovenko, “Higher-order effects in external field - pair production by a particle,” Yad. Fiz. 14, 1020 (1971).
  * (81) V. I. Ritus, “Vacuum polarization correction to elastic electron and muon scattering in an intense field and pair electro- and muoproduction,” Nucl. Phys. B 44, 236 (1972).
  * (82) H. Hu, C. Muller and C. H. Keitel, “Complete QED theory of multiphoton trident pair production in strong laser fields,” Phys. Rev. Lett. 105, 080401 (2010) [arXiv:1002.2596 [physics.atom-ph]].
  * (83) A. Ilderton, “Trident pair production in strong laser pulses,” Phys. Rev. Lett. 106, 020404 (2011) [arXiv:1011.4072 [hep-ph]].
  * (84) A. I. Titov, H. Takabe, B. Kämpfer and A. Hosaka, “Enhanced subthreshold electron-positron production in short laser pulses,” Phys. Rev. Lett. 108, 240406 (2012) [arXiv:1205.3880 [hep-ph]].
  * (85) T. Nousch, D. Seipt, B. Kämpfer and A. I. Titov, “Pair production in short laser pulses near threshold,” Phys. Lett. B 715, 246 (2012).
  * (86) E. Haug, “Bremsstrahlung and Pair Production in the Field of Free Electrons,” Z. Naturforsch. A 30, 1099 (1975).
  * (87) D. Seipt and B. Kämpfer, “Non-Linear Compton Scattering of Ultrashort and Ultraintense Laser Pulses,” Phys. Rev. A 83, 022101 (2011) [arXiv:1010.3301 [hep-ph]].
  * (88) C. Bamber et al., “Studies of nonlinear QED in collisions of 46.6-GeV electrons with intense laser pulses,” Phys. Rev. D 60, 092004 (1999).
  * (89) D. L. Burke et al., “Positron production in multi - photon light by light scattering,” Phys. Rev. Lett. 79, 1626 (1997).
  * (90) C. Bula et al. [E144 Collaboration], “Observation of nonlinear effects in Compton scattering,” Phys. Rev. Lett. 76, 3116 (1996).
  * (91) W. A. Horowitz and Y. V. Kovchegov, “Running Coupling Corrections to High Energy Inclusive Gluon Production,” Nucl. Phys. A 849, 72 (2011) [arXiv:1009.0545 [hep-ph]].
  * (92) D. Seipt, “Strong-Field QED Processes in Short Laser Pulses,” PhD thesis, TU Dresden (2012).
  * (93) D. Seipt, V. Kharin, S. Rykovanov, A. Surzhykov and S. Fritzsche, “Analytical results for nonlinear Compton scattering in short intense laser pulses,” J. Plasma Phys. 82, no. 2, 655820203 (2016) [arXiv:1601.00442 [hep-ph]].
  * (94) V. Dinu, “Exact final state integrals for strong field QED,” Phys. Rev. A 87, no. 5, 052101 (2013) [arXiv:1302.1513 [hep-ph]].
  * (95) D. Y. Ivanov, G. L. Kotkin and V. G. Serbo, “Complete description of polarization effects in e+e- pair production by a photon in the field of a strong laser wave,” Eur. Phys. J. C 40, 27 (2005) [hep-ph/0412032].