# Nonlinear Compton scattering and nonlinear Breit-Wheeler pair production
including the damping of particle states

T. Podszus<EMAIL_ADDRESS>Max Planck Institute for Nuclear Physics,
Saupfercheckweg 1, D-69117 Heidelberg, Germany  V. Dinu<EMAIL_ADDRESS>Department of Physics, University of Bucharest, P.O. Box MG-11, Măgurele
077125, Romania  A. Di Piazza<EMAIL_ADDRESS>Max Planck Institute for
Nuclear Physics, Saupfercheckweg 1, D-69117 Heidelberg, Germany

###### Abstract

In the presence of an electromagnetic background plane-wave field, electron,
positron, and photon states are not stable, because electrons and positrons
emit photons and photons decay into electron-positron pairs. This decay of the
particle states leads to an exponential damping term in the probabilities of
single nonlinear Compton scattering and nonlinear Breit-Wheeler pair
production. In this paper we investigate analytically and numerically the
probabilities of nonlinear Compton scattering and nonlinear Breit-Wheeler pair
production including the particle states’ decay. For this we first compute
spin- and polarization-resolved expressions of the probabilities, provide some
of their asymptotic behaviors and show that the results of the total
probabilities are independent of the spin and polarization bases. Then, we
present several plots of the total and differential probabilities for
different pulse lengths and for different spin and polarization quantum
numbers. We observe that it is crucial to take into account the damping of the
states in order for the probabilities to stay always below unity and we show
that the damping factors also scale with the intensity and pulse duration of
the background field. In the case of nonlinear Compton scattering we show
numerically that the total probability behaves like a Poissonian distribution
in the regime where the photon recoil is negligible. In all considered cases,
the kinematic conditions are such that the final particles momenta transverse
to the propagation direction of the plane wave are always much smaller than
the particles longitudinal momenta and the main spread of the momentum
distribution on the transverse plane is along the direction of the plane-wave
electric field.

##  I Introduction

With the ongoing progress in laser technology toward higher intensities,
lasers represent a promising tool to test QED in a regime where quantum
effects induced by the laser field play an important role. In QED the vacuum
is _pictorially_ described as being populated by fluctuations of virtual
electron-positron pairs which can be polarized by a sufficiently large
electromagnetic field. The so called “critical” field of QED
$F_{\text{cr}}=m^{2}/|e|=1.3\times 10^{16}\ \text{V}/\text{cm}=4.4\times
10^{13}\ \text{G}$ (here $m$ and $e<0$ are the electron mass and charge,
respectively, and we use units where $\epsilon_{0}=\hbar=c=1$) [1, 2, 3]
determines the typical field scale of QED. In the presence of an electric
field of the order of $F_{\text{cr}}$ the vacuum becomes unstable under
electron-positron pair production and the interaction energy of a Bohr
magneton with a magnetic field of strength $F_{\text{cr}}$ is of the order of
$m$. The field strength $F_{\text{cr}}$ corresponds to a critical laser
intensity $I_{\text{cr}}\sim 10^{29}\ \text{W}/\text{cm}^{2}$, which is far
from being reached by available lasers. Indeed, today’s record for the laser
peak intensity $I_{0}$ is about $1.1\times 10^{23}\ \text{W}/\text{cm}^{2}$
[4], and even upcoming laser facilities are aiming for intensities of the
order of $I_{0}\sim 10^{23}-10^{24}\ \text{W}/\text{cm}^{2}$ [5, 6, 7, 8, 9].
However, due to the Lorentz-invariance of QED, physical observables like
transition probabilities depend on the electromagnetic field only via Lorentz-
and gauge-invariant parameters, such that the interesting regime where field-
induced quantum effects dominate the dynamics can be efficiently entered
experimentally already with today’s technology. For an electron (photon) of
four-momentum $p^{\mu}=(\epsilon,\bm{p})$ ($q^{\mu}=(\omega,\bm{q})$) moving
in a background field, represented by the field tensor
$F_{0}^{\mu\nu}=(\bm{E}_{0},\bm{B}_{0})$ in the laboratory frame, the
probability of a physical process depends on the so-called quantum
nonlinearity parameter
$\chi_{0}=\sqrt{\left|(F_{0}^{\mu\nu}p_{\nu})^{2}\right|}/mF_{\text{cr}}$
($\kappa_{0}=\sqrt{\left|(F_{0}^{\mu\nu}q_{\nu})^{2}\right|}/mF_{\text{cr}}$),
with the metric tensor $\eta^{\mu\nu}=\text{diag}(+1,-1,-1,-1)$ [10, 11, 12,
13, 14, 15, 16, 17]. In case of an electron or positron this parameter
corresponds to the field strength that the electron or positron experiences in
its rest frame in units of the critical field.

First experiments probing laser-electron interactions in the regime
$\chi_{0}\lesssim 1$ were performed in the late 90s at SLAC [18, 19, 20] and
recently two experiments have been carried out close to the $\chi_{0}\sim 1$
regime by using an all-optical setup where the electron beam was generated via
laser wake-field acceleration [21, 22]. Further experiments for testing the
strong-field regime of QED with intense lasers are planned at DESY [23] and at
SLAC [24].

Already at intensities lower than $I_{\text{cr}}$ classical nonlinear effects
due to the interaction with the background field become significant and
complicate the theoretical description of the electron and positron dynamics.
These nonlinear effects are controlled by the so-called classical nonlinearity
parameter $\xi_{0}=|e|E_{0}/m\omega_{0}$, where $E_{0}$ is the electric field
amplitude and $\omega_{0}$ is the central angular frequency of the laser
pulse. For $\xi_{0}\gtrsim 1$ the energy that an electron or a positron gains
by the acceleration in the background field in one laser wavelength is
comparable to its rest energy and the interaction of the charge with the
background field cannot be treated perturbatively anymore [10, 11, 12, 13, 14,
15, 16, 17]. Indeed, for optical lasers the parameter $\xi_{0}$ exceeds unity
already at intensities of the order of $10^{18}\ \text{W}/\text{cm}^{2}$ and
the interaction with the background field has to be treated exactly in the
calculations in this case. This problem is commonly solved by working in the
so-called Furry picture [25]. Here the background field is taken into account
in the quantization procedure of the fermion field [2, 1], such that the
corresponding Dirac equation includes the interaction with the background
field. An analytical solution of the Dirac equation can be found in the case
of a plane-wave background field [26] (see also Ref. [1]) and the
corresponding states are known as Volkov states.

Two elementary processes which have been thoroughly investigated by employing
Volkov states are the emission of a photon by an electron or positron
(nonlinear Compton scattering) and the decay of a photon into an electron-
positron pair (nonlinear Breit-Wheeler pair production), which, among others,
were studied in Refs. [27, 28, 11, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] (nonlinear Compton
scattering) and Refs. [54, 28, 55, 56, 13, 57, 58, 59, 60, 61, 62, 63, 64, 52,
65] (nonlinear Breit-Wheeler pair production).

Considering an electron (photon) moving in a plane wave laser pulse, it turns
out that the leading-order in the fine-structure constant
$\alpha=e^{2}/4\pi\approx 1/137$ total probability of nonlinear Compton
scattering (nonlinear Breit-Wheeler pair production) exceeds unity for a
sufficiently long total phase duration $\Phi_{L}$ of the laser pulse (or for a
sufficiently large laser intensity). Probabilities larger than unity are of
course unphysical and in contradiction with the unitarity of the $S$-matrix.
Instead, for nonlinear Compton scattering it can be interpreted as the average
number of photons emitted by the electron in the classical limit rather than
as a probability [66]. In general, from the QED perspective the reason behind
this apparent contradiction is that higher-order loop corrections and
processes become significant. This is intuitively clear for nonlinear Compton
scattering since the emission of several photons by an electron due to
nonlinear multiple Compton scattering processes becomes sizable with an
increasing pulse length [67, 68, 69, 70, 71, 72]. A first investigation to
taking into account these effects was carried out in Ref. [73] for
$\chi_{0}\lesssim 1$. The probability for an arbitrary number of consecutive
incoherent photon emissions by an electron was calculated, taking into account
the recoil at each photon emission. Each probability was then “renormalized”
by imposing that the total probability of emitting either no photons or an
arbitrary number of photons was unity. The renormalization ensures that the
final probability of nonlinear (multiphoton) Compton scattering stays below
unity even for large phase lengths of the laser pulse. These results where
confirmed in Refs. [74, 75] by means of a kinetic approach, which also
included the effects of nonlinear Breit-Wheeler pair production [75]. In Refs.
[74, 75] also inclusive quantities like average momenta are computed and an
approach to obtain the momentum expectation values of an electron including
multiple photon emissions and loops has been put forward in Ref. [76].

In Ref. [77] the probability of an electron emitting an arbitrary number of
photons was derived via the following considerations. The probability of an
electron emitting $N$ photons was calculated by first combining the
probability of the electron to emit $N-1$ photons until a certain time $t$,
with the probabilities to emit one photon between $t$ and $t+dt$, with $dt$
being an infinitesimal positive time interval, and with the probability of
emitting no photons from time $t+dt$ on, and finally by integrating the result
over all possible times $t$ (a similar method was used in Ref. [78] to compute
the pair-production yield as an observable to diagnose the intensity of the
laser beam producing the pairs). Additionally, the photon recoil was taken
into account. The obtained recursive equation for nonlinear multiphoton
Compton scattering contains an exponential damping term describing the “decay”
of the electron state by emitting a photon. This damping term depends on time
and on the energy of the electron, and ensures that the total probability of
emitting either no or an arbitrary number of photons is unity. The results in
the regime $\chi_{0}\lesssim 1$ were in agreement with those in Ref. [73].

Figure 1: The amplitude of nonlinear Compton scattering computed with the
exact electron and photon states (thick straight and wiggly lines,
respectively) is equal to the resummation of all one-particle reducible
diagrams with corrections to the one-loop mass operator (M) on the Volkov-
electron states (thin double lines) and to the one-loop polarization operator
(P) on the photon states (thin wiggly lines) (see Ref. [79]).

In Ref. [79] the probability of nonlinear Compton scattering and of nonlinear
Breit-Wheeler pair production was computed from first principles. The derived
expressions are equivalent to the resummation of all one-particle reducible
diagrams containing an arbitrary number of corrections to the electron and
photon states by the one-loop mass and polarization operators, respectively
(see Fig. 1 for the case of nonlinear Compton scattering). This was achieved
by calculating the $S$-matrix with the “exact” electron and photon states,
which are the solutions of the Schwinger-Dyson equation. This work and Refs.
[73, 74, 75, 76, 77] were framed within the so-called locally-constant-field
approximation (LCFA). In the LCFA probabilities of QED processes reduce to the
corresponding probabilities in a constant crossed field averaged over the
phase-dependent plane-wave profile. This is reasonable in the limit of low-
frequency plane waves with fixed electric-field amplitude, since here the
formation length of QED processes is much smaller than the typical wavelength
of the plane wave [11]. The assumption is valid if $\xi_{0}\gg 1$ at
$\chi_{0},\kappa_{0}\sim 1$, which was assumed throughout the derivation (we
did not consider the problems which occur for the LCFA at low photon energies
in the case of nonlinear Compton scattering [50, 53]). It can be shown that
the solution of the Schwinger-Dyson equation intrinsically includes the
resummation of all corrections by the one-loop mass and polarization operator
for the exact electron or positron and photon state, respectively. The final
probabilities for nonlinear Compton scattering and nonlinear Breit-Wheeler
pair production comprise an exponential damping term, describing the “decay”
of the electron states by emitting a photon and the decay of the photon state
into an electron-positron pair. This decay of the particle states turned out
to become significant if $\alpha\xi_{0}\Phi_{L}\gtrsim 1$. It is interesting
to notice that the result for the probability of nonlinear Compton scattering
is structurally similar to the single photon emission probability derived by
the probabilistic approach in Ref. [77]. However, the probability in Ref. [79]
additionally includes spin and polarization effects as well as the decay of
the photon state into an electron-positron pair.

In Ref. [72] an approach has been developed to investigate higher-order QED
processes also beyond the LCFA but for sufficiently long plane-wave pulses
that the dynamics is dominated by the so-called cascade channel. In the
cascade channel the higher-order process occurs as a sequence of the
elementary building blocks represented by nonlinear Compton scattering and
nonlinear Breit-Wheeler pair production.

The aim of the present work is to present new analytical insights and
numerical examples on the probabilities of nonlinear Compton scattering and
nonlinear Breit-Wheeler pair production including the particle states decay
derived in Ref. [79]. The paper is organized as follows. First, we give a
detailed analytical evaluation of the differential and the total
probabilities, together with the asymptotic behavior for one final particle
getting all or none of the initial light-cone energy. Further, we prove that
the results of the total probabilities are independent of the spin and
polarization basis that we use in the calculations. Then, we pass to the
numerical evaluation of the results. We present plots of the differential and
the total probabilities at two different pulse lengths for both nonlinear
Compton scattering and nonlinear Breit-Wheeler pair production. In the case of
nonlinear Compton scattering we compare the results with a Poissonian
distribution. For the differential probability we show different plots
corresponding to different combinations of spin and polarization of the
initial and final particles. Finally, the main conclusions of the paper are
presented. An appendix contains the explicit computation of the traces of
Dirac matrices, which are not essential for the understanding of the main
results of the paper.

##  II Analytical calculations

###  II.1 Notation

As indicated in the introduction, we investigate the probability of nonlinear
Compton scattering and nonlinear Breit-Wheeler pair production including the
particle states decay, which was presented in Ref. [79] and for better
comparison we employ the same notation here.

We consider a plane-wave background field with central photon four-momentum
$k_{0}^{\mu}=\omega_{0}n^{\mu}$, where we introduced the quantity
$n^{\mu}=(1,\bm{n})$, with the three-dimensional unit vector $\bm{n}$ pointing
along the direction of propagation of the background field. Additionally, we
introduce the quantity $\tilde{n}^{\mu}=(1,-\bm{n})/2$ such that
$(n\tilde{n})=1$. Since the background field is a plane wave, its four-
potential $A^{\mu}(\phi)=(A^{0}(\phi),\bm{A}(\phi))$ only depends on the
light-cone time $\phi=(nx)=t-\bm{n}\cdot\bm{x}$. Further, it is a solution of
the free wave equation $\partial_{\mu}\partial^{\mu}A^{\nu}=0$, as we assume
that it fulfills the Lorentz gauge condition $\partial_{\mu}A^{\mu}(\phi)=0$.
By additionally fixing the gauge such that $A^{0}(\phi)=0$, the vector
potential is perpendicular to the direction of propagation of the plane wave,
i.e., $\bm{n}\cdot\bm{A}(\phi)=0$, if $\bm{A}(\phi)\to 0$ at
$\phi\to\pm\infty$, which we also assume in the following.

By introducing the two four-vectors $a_{j}^{\mu}=(0,\bm{a}_{j})$ with $j=1,2$,
which obey the relations
$(na_{j})=-2(\tilde{n}a_{j})=-\bm{n}\cdot\bm{a}_{j}=0$ and
$(a_{j}a_{j^{\prime}})=-\bm{a}_{j}\cdot\bm{a}_{j^{\prime}}=-\delta_{jj^{\prime}}$,
with $j,j^{\prime}=1,2$, the vector potential of the plane wave can be
expressed as $\bm{A}(\phi)=\psi_{1}(\phi)\bm{a}_{1}+\psi_{2}(\phi)\bm{a}_{2}$.
Here, $\psi_{j}(\phi)$ denotes the $j$th pulse shape function and it vanishes
for $\phi\to\pm\infty$. In the following we will only consider the case of a
linearly polarized plane wave, such that we choose without loss of generality
$\psi_{2}(\phi)=0$ and $\psi_{1}(\phi)=A_{0}\psi(\phi)$, with $A_{0}<0$ being
related to the amplitude of the electric field of the plane wave. The
functions $\psi(\phi)$ and $\psi^{\prime}(\phi)$ are assumed to be such that
$|\psi(\phi)|,|\psi^{\prime}(\phi)|\lesssim 1$. Here and in the following a
prime at a function denotes the derivative of the function with respect to its
argument. The electromagnetic field tensor is then given by
$F^{\mu\nu}(\phi)=n^{\mu}A^{\prime\nu}(\phi)-n^{\nu}A^{\prime\mu}(\phi)=A_{0}^{\mu\nu}\psi^{\prime}(\phi)$,
where we have introduced the notation
$A_{0}^{\mu\nu}=A_{0}(n^{\mu}a_{1}^{\nu}-n^{\nu}a_{1}^{\mu})$. The dual of the
field tensor is
$\tilde{F}^{\mu\nu}(\phi)=\tilde{A}_{0}^{\mu\nu}\psi^{\prime}(\phi)$, where
$\tilde{A}_{0}^{\mu\nu}=(1/2)\varepsilon^{\mu\nu\lambda\rho}A_{0,\lambda\rho}$
and we define the four-dimensional Levi-Civita tensor as
$\varepsilon^{0123}=+1$ (note that in the chosen reference frame it is
$\tilde{A}_{0}^{\mu\nu}=A_{0}(n^{\mu}a_{2}^{\nu}-n^{\nu}a_{2}^{\mu})$). Since
the four-potential and the field tensor always occur multiplied by the
electron charge, we introduce the notation
$\mathcal{A}^{\mu}(\phi)=eA^{\mu}(\phi)$, $\mathcal{A}_{0}=eA_{0}$, and
$\mathcal{F}^{\mu\nu}(\phi)=eF^{\mu\nu}(\phi)$.

The quantities $n^{\mu}$, $\tilde{n}^{\mu}$, and $a_{j}^{\mu}$ fulfill the
relation
$\eta^{\mu\nu}=n^{\mu}\tilde{n}^{\nu}+\tilde{n}^{\mu}n^{\nu}-a_{1}^{\mu}a_{1}^{\nu}-a_{2}^{\mu}a_{2}^{\nu}$.
It is useful to employ light-cone coordinates and for an arbitrary four-vector
$v^{\mu}=(v_{0},\bm{v})$ the components in light-cone coordinates are
$v_{-}=(nv)=v_{0}-\bm{n}\cdot\bm{v}$,
$v_{+}=(\tilde{n}v)=(v_{0}+\bm{n}\cdot\bm{v})/2$, and
$\bm{v}_{\perp}=(v_{\perp,1},v_{\perp,2})=-((va_{1}),(va_{2}))=(\bm{v}\cdot\bm{a}_{1},\bm{v}\cdot\bm{a}_{2})$.
Further we introduce the notation $\hat{v}=\gamma^{\mu}v_{\mu}$, where
$\gamma^{\mu}$ are the Dirac-matrices, and we define
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$.

###  II.2 Nonlinear Compton Scattering

In the case of nonlinear Compton scattering, we assume the incoming (outgoing)
electron to have four-momentum $p^{\mu}=(\varepsilon,\bm{p})$
($p^{\prime\mu}=(\varepsilon^{\prime},\bm{p}^{\prime})$), with energy
$\varepsilon=\sqrt{m^{2}+\bm{p}^{2}}$
($\varepsilon^{\prime}=\sqrt{m^{2}+\bm{p}^{\prime 2}}$), and an asymptotic
spin quantum number $s=\pm 1$ ($s^{\prime}=\pm 1$). The spin quantization axis
of the incoming (outgoing) electron is chosen along the four-vector
$\zeta^{\mu}=-\tilde{A}_{0}^{\mu\nu}p_{\nu}/(p_{-}A_{0})$
($\zeta^{\prime\,\mu}=-\tilde{A}_{0}^{\mu\nu}p^{\prime}_{\nu}/(p^{\prime}_{-}A_{0})$),
which corresponds to the three-dimensional spin vector $\bm{\zeta}$
($\bm{\zeta}^{\prime}$) pointing in the same direction of the magnetic field
in the case of a constant crossed field and in the rest frame of the incoming
(outgoing) electron. For the outgoing photon the four-momentum is
$q^{\mu}=(\omega,\bm{q})$, with energy $\omega=|\bm{q}|$ and we define its two
transverse polarization states, identified by the index $j=1,2$, along the
four-vector $\Lambda_{1}^{\mu}(q)=A_{0}^{\mu\nu}q_{\nu}/(q_{-}A_{0})$ and the
pseudo-four-vector
$\Lambda_{2}^{\mu}(q)=\tilde{A}_{0}^{\mu\nu}q_{\nu}/(q_{-}A_{0})$, which
fulfill the relation
$(\Lambda_{j}(q)\Lambda_{j^{\prime}}(q))=-\delta_{jj^{\prime}}$ with
$j,j^{\prime}=1,2$.

With these definitions the probability of nonlinear Compton scattering
including the damping of particle states within the LCFA, which was derived in
Ref. [79], is given by the expression

$\begin{split}P^{(e^{-}\to
e^{-}\gamma)}_{j,s,s^{\prime}}=&\int\frac{d^{3}q}{16\pi^{2}}\frac{\alpha}{p_{-}p^{\prime}_{-}\omega}\int
d\phi_{+}e^{2\text{Im}\left\\{\frac{m}{p_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
M_{s}(p,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi\left[\frac{m}{p^{\prime}_{-}}M_{s^{\prime}}(p^{\prime},\varphi)+\frac{m}{q_{-}}P_{j}(q,\varphi)\right]\right\\}}\\\
&\times\int
d\phi_{-}e^{i\frac{m^{2}}{2p_{-}}\frac{q_{-}}{p^{\prime}_{-}}\left\\{[1+\bm{\pi}_{\perp,e}^{2}(\phi_{+})]\phi_{-}+\frac{\bm{\mathcal{E}}^{2}(\phi_{+})}{m^{2}}\frac{\phi_{-}^{3}}{12}\right\\}}T_{j,s,s^{\prime}},\end{split}$
(1)

where we introduced the trace

$\begin{split}T_{j,s,s^{\prime}}=\frac{1}{4}\text{tr}&\left\\{\left[1-\frac{\hat{n}[\hat{\mathcal{A}}(\phi_{+})+\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p^{\prime}_{-}}\right]\hat{\Lambda}_{j}(q)\left[1+\frac{\hat{n}[\hat{\mathcal{\mathcal{A}}}(\phi_{+})+\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p_{-}}\right]\right.\\\
&\left.\times(\hat{p}+m)(1+s\gamma^{5}\hat{\zeta})\left[1-\frac{\hat{n}[\hat{\mathcal{A}}(\phi_{+})-\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p_{-}}\right]\hat{\Lambda}_{j}(q)\right.\\\
&\left.\times\left[1+\frac{\hat{n}[\hat{\mathcal{A}}(\phi_{+})-\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p^{\prime}_{-}}\right](\hat{p}^{\prime}+m)(1+s^{\prime}\gamma^{5}\hat{\zeta}^{\prime})\right\\},\end{split}$
(2)

the transverse momentum

$\bm{\pi}_{\perp,e}(\phi)=\frac{\bm{p}_{\perp}}{m}-\frac{p_{-}}{q_{-}}\frac{\bm{q}_{\perp}}{m}-\frac{\bm{\mathcal{A}}_{\perp}(\phi)}{m},$
(3)

and the plane-wave electric field (times the electron charge)
$\bm{\mathcal{E}}(\phi)=-\bm{\mathcal{A}}^{\prime}(\phi)$. Due to energy-
momentum conservation the minus component and the perpendicular component of
the outgoing electron momentum are fixed to $p^{\prime}_{-}=p_{-}-q_{-}$ and
$\bm{p}^{\prime}_{\perp}=\bm{p}_{\perp}-\bm{q}_{\perp}$, respectively. As we
explained in the introduction, the probability comprises a damping term due to
the particle states decay, which is the first exponential function in Eq. (1).
The exponent contains the imaginary parts of the mass and polarization
operator, more precisely, the one-loop mass operator within the LCFA, given by
the expression [80, 81, 82]

$M_{s}(p,\phi)=\frac{\alpha
m}{2\pi}\int_{0}^{\infty}du\int_{0}^{\infty}\frac{dv}{(1+v)^{3}}e^{-iu\left[1+\frac{1}{3}\frac{\chi^{2}_{p}(\phi)}{v^{2}}u^{2}\right]}\left[\frac{5+7v+5v^{2}}{3}\frac{\chi^{2}_{p}(\phi)}{v^{2}}u-is\chi_{p}(\phi)\right],$
(4)

for $s=\pm 1$, with the $\phi$-dependent quantum nonlinearity parameter
defined as $\chi_{p}(\phi)=-(p_{-}/m)A_{0}\psi^{\prime}(\phi)/F_{\text{cr}}$,
and the transverse part of the one-loop polarization operator within the LCFA,
given by [83, 84, 85]

$P_{j}(q,\phi)=\frac{\alpha}{48\pi}mj\kappa_{q}^{2}(\phi)\int_{0}^{\infty}du\,u\int_{0}^{1}dve^{-iu\left[1+\frac{(1-v^{2})^{2}}{48}\kappa_{q}^{2}(\phi)u^{2}\right]}(1-v^{2})[3-(-1)^{j}v^{2}],$
(5)

for $j=1,2$, with
$\kappa_{q}(\phi)=-(q_{-}/m)A_{0}\psi^{\prime}(\phi)/F_{\text{cr}}$.

According to the optical theorem, the quantity
$-(2m/p_{-})\text{Im}[M_{s}(p,\phi)]$ is equal to the total probability per
unit $\phi$ that an electron of four-momentum $p$ and spin quantum number $s$
emits a photon [80] and the quantity $-(2m/q_{-})\text{Im}[P_{j}(q,\phi)]$ is
equal to the total probability per unit $\phi$ that a photon of four-momentum
$q$ and polarization quantum number $j$ decays into an electron-positron pair
[84]. Hence, this exponential damping can be understood as the electron and
photon states not being stable in the background field but decaying, where the
electron “decays” by emitting a photon and the photon decays into an electron-
positron pair.

Further, we notice that the damping exponential depends on the spin of the
incoming and outgoing electrons and on the polarization of the outgoing
photon. This prevents one from employing the commonly used spin and
polarization sum rules when solving the trace in Eq. (2), such that the spin-
and polarization-resolved traces have to be calculated. A complete analytical
derivation can be found in the appendix. Here, we only present the main steps
of the calculations. With our choice of the spin and polarization basis, the
trace for the two polarization states $j=1,2$ reduces in a linearly-polarized
field to

$\begin{split}T_{1,s,s^{\prime}}&=(1+ss^{\prime})\bigg{[}(pp^{\prime})-m^{2}-\frac{1}{2}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\left(p_{1}-\frac{p_{-}}{q_{-}}q_{1}\right)^{2}\\\
&-\left(2+\frac{1}{2}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\right)\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})\frac{\phi_{-}^{2}}{4}\\\
&+\left(2+\frac{1}{2}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\right)\left(p_{1}-\frac{p_{-}}{q_{-}}q_{1}+\mathcal{A}_{0}\psi(\phi_{+})\right)^{2}\bigg{]}\\\
&-i(s+s^{\prime})\frac{m}{2}\mathcal{A}_{0}\psi^{\prime}(\phi_{+})\phi_{-}\frac{q_{-}}{p_{-}}\left(2+\frac{q_{-}}{p_{-}-q_{-}}\right)-ss^{\prime}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}\end{split}$
(6)

and

$\begin{split}T_{2,s,s^{\prime}}&=(1-ss^{\prime})\bigg{[}(pp^{\prime})-m^{2}-\frac{1}{2}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\left(p_{1}-\frac{p_{-}}{q_{-}}q_{1}\right)^{2}\\\
&-\frac{1}{2}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})\frac{\phi_{-}^{2}}{4}\\\
&+\frac{1}{2}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\left(p_{1}-\frac{p_{-}}{q_{-}}q_{1}+\mathcal{A}_{0}\psi(\phi_{+})\right)^{2}\bigg{]}\\\
&+(1+ss^{\prime})2\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}+ss^{\prime}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}\\\
&+i(s-s^{\prime})\frac{m}{2}\mathcal{A}_{0}\psi^{\prime}(\phi_{+})\phi_{-}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}.\end{split}$
(7)

We observe that the traces depend on the pulse shape function
$\psi(\phi_{+})$. However, it should be possible to express the dependence on
$\psi(\phi_{+})$ in a manifestly gauge-invariant way. In order to achieve
this, we consider the transverse momentum $\bm{\pi}_{\perp,e}(\phi_{+})$
defined in Eq. (3). After some simplifications all the dependence on
$\psi(\phi_{+})$ and $q_{1}$ turns out to be in the electron quasi-momentum
$\bm{\pi}_{\perp,e}(\phi_{+})$, which can be removed by performing the
integral in $\phi_{-}$ and using the properties of the Airy functions. Indeed,
terms proportional to the derivative in $\phi_{-}$ of the second exponential
function in Eq. (1), i.e., terms proportional to
$\left[1+\bm{\pi}^{2}_{\perp,e}(\phi_{+})+(\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})/m^{2})(\phi_{-}^{2}/4)\right]$, vanish when performing the
integral over $\phi_{-}$. By performing the appropriate substitutions, the
traces ultimately depend only on the derivative $\psi^{\prime}(\phi_{+})$ of
the pulse shape function and the probability is therefore manifestly gauge-
invariant. Ignoring the corresponding vanishing terms, the traces can be
equivalently written as

$\begin{split}T_{1,s,s^{\prime}}&=-2(1+ss^{\prime})m^{2}-(1+ss^{\prime})\left(4+\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\right)\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})\frac{\phi_{-}^{2}}{4}\\\
&-i(s+s^{\prime})\frac{m}{2}\mathcal{A}_{0}\psi^{\prime}(\phi_{+})\phi_{-}\frac{q_{-}}{p_{-}}\left(2+\frac{q_{-}}{p_{-}-q_{-}}\right)\\\
&-\left(2+2ss^{\prime}+ss^{\prime}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\right)\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}\end{split}$
(8)

and

$\begin{split}T_{2,s,s^{\prime}}&=-(1-ss^{\prime})\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})\frac{\phi_{-}^{2}}{4}\\\
&+i(s-s^{\prime})\frac{m}{2}\mathcal{A}_{0}\psi^{\prime}(\phi_{+})\phi_{-}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\\\
&+\left(2+2ss^{\prime}+ss^{\prime}\frac{q_{-}}{p_{-}}\frac{q_{-}}{p_{-}-q_{-}}\right)\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}.\end{split}$
(9)

We transform the integral in the photon momentum into light-cone coordinates
using the relation $d^{3}q=(\omega/q_{-})dq_{-}d^{2}q_{\perp}$ and introduce
the notation

$\tilde{T}_{j,s,s^{\prime}}=-\frac{1}{4\pi^{2}m^{2}}\frac{p_{-}}{q_{-}p^{\prime}_{-}}\int
d\phi_{-}\int
d^{2}q_{\perp}e^{i\frac{m^{2}q_{-}}{2p_{-}p^{\prime}_{-}}\left\\{[1+\bm{\pi}_{\perp,e}^{2}(\phi_{+})]\phi_{-}+\frac{\bm{\mathcal{E}}^{2}(\phi_{+})}{m^{2}}\frac{\phi_{-}^{3}}{12}\right\\}}T_{j,s,s^{\prime}}.$
(10)

Now, the integral in the perpendicular photon momentum $\bm{q}_{\perp}$ can be
computed analytically by using the two basic integrals [86]

$\displaystyle\int
d^{2}q_{\perp}e^{i\frac{m^{2}q_{-}}{2p_{-}p^{\prime}_{-}}\bm{\pi}^{2}_{\perp,e}(\phi_{+})\phi_{-}}=2\pi
i\frac{q_{-}p^{\prime}_{-}}{p_{-}(\phi_{-}+i0)},$ (11) $\displaystyle\int
d^{2}q_{\perp}\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}e^{i\frac{m^{2}q_{-}}{2p_{-}p^{\prime}_{-}}\bm{\pi}^{2}_{\perp,e}(\phi_{+})\phi_{-}}=-2\pi\frac{p^{\prime
2}_{-}}{(\phi_{-}+i0)^{2}}.$ (12)

For the integral in $\phi_{-}$ we use the integral representation
$\text{Ai}(z)=\int_{-\infty}^{\infty}(d\tilde{\phi}/2\pi)\exp(iz\tilde{\phi}+i\tilde{\phi}^{3}/3)$
of the Airy function [86]. With the substitutions
$\tilde{\phi}=\left[q_{-}\bm{\mathcal{E}}^{2}(\phi_{+})/(8p_{-}p^{\prime}_{-})\right]^{1/3}\phi_{-}$
and $z=\left[q_{-}/(p^{\prime}_{-}\chi_{p}(\phi_{+}))\right]^{2/3}$, the
integral in $\phi_{-}$ can be taken. Hence, the probability of nonlinear
Compton scattering including the damping of particle states is finally given
by

$\begin{split}P^{(e^{-}\to e^{-}\gamma)}_{j,s,s^{\prime}}=&-\frac{\alpha
m^{2}}{4p_{-}^{2}}\int_{0}^{p_{-}}dq_{-}\int d\phi_{+}\
\tilde{T}_{j,s,s^{\prime}}\\\ &\times
e^{2\text{Im}\left\\{\frac{m}{p_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
M_{s}(p,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi\left[\frac{m}{p^{\prime}_{-}}M_{s^{\prime}}(p^{\prime},\varphi)+\frac{m}{q_{-}}P_{j}(q,\varphi)\right]\right\\}},\end{split}$
(13)

where

$\begin{split}\tilde{T}_{1,s,s^{\prime}}=&\left[1+ss^{\prime}\left(1-\frac{q_{-}^{2}}{2p_{-}(p_{-}-q_{-})}\right)\right]\text{Ai}_{1}(z)\\\
&+\left[3+\frac{q_{-}^{2}}{p_{-}(p_{-}-q_{-})}+ss^{\prime}\left(3+\frac{q_{-}^{2}}{2p_{-}(p_{-}-q_{-})}\right)\right]\frac{\text{Ai}^{\prime}(z)}{z}\\\
&+(s+s^{\prime})\left(2\frac{q_{-}}{p_{-}}+\frac{q_{-}^{2}}{p_{-}(p_{-}-q_{-})}\right)\frac{\text{Ai}(z)}{\sqrt{z}}\text{sgn}(\psi^{\prime}(\phi_{+}))\end{split}$
(14)

and

$\begin{split}\tilde{T}_{2,s,s^{\prime}}=&\left[1+ss^{\prime}\left(1+\frac{q_{-}^{2}}{2p_{-}(p_{-}-q_{-})}\right)\right]\text{Ai}_{1}(z)\\\
&+\left[1+\frac{q_{-}^{2}}{p_{-}(p_{-}-q_{-})}+ss^{\prime}\left(1-\frac{q_{-}^{2}}{2p_{-}(p_{-}-q_{-})}\right)\right]\frac{\text{Ai}^{\prime}(z)}{z}\\\
&+(s^{\prime}-s)\frac{q_{-}^{2}}{p_{-}(p_{-}-q_{-})}\frac{\text{Ai}(z)}{\sqrt{z}}\text{sgn}(\psi^{\prime}(\phi_{+})),\end{split}$
(15)

with $\text{Ai}_{1}(z)=\int_{z}^{\infty}dx\text{Ai}(x)$ and with
$\text{sgn}(\psi^{\prime}(\phi_{+}))$ denoting the sign of
$\psi^{\prime}(\phi_{+})$. Note that without the exponential damping term the
results reduce to the expressions of the spin- and polarization-resolved
probabilities of nonlinear Compton scattering, which can be found in Ref.
[87]. This observation can be used to prove analytically that the probability
$P^{(e^{-}\to e^{-}\gamma)}_{s}=\sum_{j,s^{\prime}}P^{(e^{-}\to
e^{-}\gamma)}_{j,s,s^{\prime}}$ is always smaller than unity. In fact, since
the damping exponentials are smaller or equal to unity, it is

$\begin{split}P^{(e^{-}\to e^{-}\gamma)}_{s}&<-\frac{\alpha
m^{2}}{4p_{-}^{2}}\sum_{j,s^{\prime}}\int_{0}^{p_{-}}dq_{-}\int d\phi_{+}\
\tilde{T}_{j,s,s^{\prime}}e^{2\text{Im}\frac{m}{p_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
M_{s}(p,\varphi)}\\\ &=\int d\phi_{+}\frac{\partial
P^{\text{NC}}_{s,p}}{\partial\phi_{+}}e^{-\int_{-\infty}^{\phi_{+}}d\varphi\frac{\partial
P^{\text{NC}}_{s,p}}{\partial\varphi}}=-\int
d\phi_{+}\frac{\partial}{\partial\phi_{+}}e^{-\int_{-\infty}^{\phi_{+}}d\varphi\frac{\partial
P^{\text{NC}}_{s,p}}{\partial\varphi}}\\\
&=1-e^{-\int_{-\infty}^{\infty}d\varphi\frac{\partial
P^{\text{NC}}_{s,p}}{\partial\varphi}}<1,\end{split}$ (16)

where $\partial P^{\text{NC}}_{s,p}/\partial\phi$ indicated the probability
without damping of nonlinear Compton scattering per unit of light-cone time
$\phi$.

At this point we also want to investigate the asymptotic behavior of the
differential probability for the two cases $q_{-}\ll p_{-}$ and
$p_{-}-q_{-}\ll p_{-}$. In the first case, the photon recoil is negligible,
whereas in the second case, almost all light-cone energy of the incoming
electron goes into the photon. The differential probability is obtained from
Eq. (13) and it is given by

$\begin{split}\frac{\partial P^{(e^{-}\to
e^{-}\gamma)}_{j,s,s^{\prime}}}{\partial q_{-}}=&-\frac{\alpha
m^{2}}{4p_{-}^{2}}\int
d\phi_{+}e^{D^{\text{NC}}_{j,s,s^{\prime}}}\tilde{T}_{j,s,s^{\prime}},\end{split}$
(17)

where we have renamed the exponent of the exponential damping function as

$\begin{split}D^{\text{NC}}_{j,s,s^{\prime}}=2\text{Im}\left\\{\frac{m}{p_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
M_{s}(p,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi\left[\frac{m}{p^{\prime}_{-}}M_{s^{\prime}}(p^{\prime},\varphi)+\frac{m}{q_{-}}P_{j}(q,\varphi)\right]\right\\}.\end{split}$
(18)

As already mentioned, according to the optical theorem, the probabilities of
nonlinear Compton scattering and nonlinear Breit-Wheeler pair production are
related to the imaginary part of the mass and polarization operator,
respectively. Hence, the exponent of the damping term,
$D^{\text{NC}}_{j,s,s^{\prime}}$, is equal to minus the sum of the probability
of nonlinear Compton scattering between $-\infty$ and $\phi_{+}$ for the
incoming electron with light-cone energy $p_{-}$ and spin quantum number $s$,
and of the probabilities of nonlinear Compton scattering of the outgoing
electron with light-cone energy $p^{\prime}_{-}$ and spin quantum number
$s^{\prime}$ and of nonlinear Breit-Wheeler pair production of the outgoing
photon with light-cone energy $q_{-}$ and polarization quantum number $j$,
both between $\phi_{+}$ and $+\infty$ [80, 84], i.e.,

$\begin{split}D^{\text{NC}}_{j,s,s^{\prime}}=-\int_{-\infty}^{\phi_{+}}d\varphi\frac{\partial
P^{\text{NC}}_{s,p}}{\partial\varphi}-\int_{\phi_{+}}^{\infty}d\varphi\left(\frac{\partial
P^{\text{NC}}_{s^{\prime},p^{\prime}}}{\partial\varphi}+\frac{\partial
P^{\text{NBW}}_{j,q}}{\partial\varphi}\right).\end{split}$ (19)

####  II.2.1 Asymptotic expression for $q_{-}\ll p_{-}$

We first analyze the asymptotic expression of the differential probability in
the asymptotic region $q_{-}\ll p_{-}$. Also, we assume that the quantum
nonlinearity parameter $\chi_{p}(\varphi)$ of the electron is fixed, such that
the absolute value of the quantum nonlinearity parameter
$\kappa_{q}(\varphi)=(q_{-}/p_{-})\chi_{p}(\varphi)$ of the photon is much
smaller than unity (if $|\chi_{p}(\varphi)|$ is larger than unity, the ratio
$q_{-}/p_{-}$ is assumed to be sufficiently small that
$|\kappa_{q}(\varphi)|\ll 1$). Thus, we can use in the damping term the
corresponding asymptotic expression for the probability of nonlinear Breit-
Wheeler pair production: [11]

$\frac{\partial
P^{\text{NBW}}_{j,q}}{\partial\varphi}\overset{\kappa_{q}(\varphi)\ll
1}{\approx}\sqrt{\frac{3}{2}}\frac{\alpha
m^{2}|\kappa_{q}(\varphi)|j}{8q_{-}}e^{-\frac{8}{3|\kappa_{q}(\varphi)|}}.$
(20)

Since it is exponentially suppressed in the limit $\kappa_{q}(\varphi)\to 0$,
we neglect it below. Furthermore, due to the conservation of the minus
component of the four-momentum, if $q_{-}\ll p_{-}$ then
$p^{\prime}_{-}\approx p_{-}$, and the damping function reduces to

$\begin{split}D^{\text{NC}}_{j,s,s^{\prime}}\overset{q_{-}\ll
p_{-}}{\approx}\frac{2m}{p_{-}}\text{Im}\left[\int_{-\infty}^{\phi_{+}}d\varphi
M_{s}(p,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi
M_{s^{\prime}}(p,\varphi)\right].\end{split}$ (21)

Now, we will shortly see that for $q_{-}\ll p_{-}$ the spin-dependent terms of
the probability of nonlinear Compton scattering can be neglected [29], such
that the damping exponent effectively reduces to a constant, which is equal to
minus the probability of an electron of momentum $p_{-}$ emitting a photon
between phase $-\infty$ and $+\infty$ averaged over the electron spin.

In the functions $\tilde{T}_{j,s,s^{\prime}}$ we can expand the Airy functions
for
$z=\left(\frac{q_{-}}{p^{\prime}_{-}\chi_{p}(\phi_{+})}\right)^{2/3}\approx\left(\frac{q_{-}}{p_{-}\chi_{p}(\phi_{+})}\right)^{2/3}\ll
1$ (we assume that $\chi_{p}(\phi_{+})$ is fixed and that the ratio
$q_{-}/p_{-}$ is much smaller than $1/|\chi_{p}(\varphi)|$ if
$|\chi_{p}(\varphi)|<1$) and we obtain

$\displaystyle\tilde{T}_{1,s,s}$ $\displaystyle\overset{z\ll
1}{\approx}-\frac{2\times 3^{2/3}}{\Gamma\left(\frac{1}{3}\right)z},$
$\displaystyle\tilde{T}_{1,s,-s}\overset{z\ll
1}{\approx}-\frac{q_{-}^{2}}{2p^{2}_{-}}\frac{1}{3^{1/3}\Gamma\left(\frac{1}{3}\right)z},$
(22) $\displaystyle\tilde{T}_{2,s,s}$ $\displaystyle\overset{z\ll
1}{\approx}-\frac{2}{3^{1/3}\Gamma\left(\frac{1}{3}\right)z},$
$\displaystyle\tilde{T}_{2,s,-s}\overset{z\ll
1}{\approx}-\frac{q_{-}^{2}}{2p^{2}_{-}}\frac{3^{2/3}}{\Gamma\left(\frac{1}{3}\right)z}.$
(23)

As expected in the classical limit where the photon recoil is small, we see
that the probability of spin flip is substantially suppressed as compared to
the case $s=s^{\prime}$. Further, the damping function in Eq. (21) in the
dominant case $s=s^{\prime}$ is independent of the integration variables and
corresponds in the classical limit to the mean number of photons emitted by an
electron. Hence, within this limit the total probability of emitting a single
photon is in agreement with that obtained from the Poissonian distribution.

####  II.2.2 Asymptotic expression for $p_{-}-q_{-}\ll p_{-}$

Now, we evaluate the asymptotic expression of the differential probability in
Eqs. (17) and (18) in the region $p^{\prime}_{-}=p_{-}-q_{-}\ll p_{-}$. In
this case, since we again assume $\chi_{p}(\varphi)$ to be fixed, the absolute
value of the quantum nonlinearity parameter
$\chi_{p^{\prime}}(\varphi)=(p^{\prime}_{-}/p_{-})\chi_{p}(\varphi)$ of the
outgoing electron is assumed to be smaller than unity. We use the
corresponding asymptotic expression for the probability of nonlinear Compton
scattering in Eq. (19), which is independent of $p^{\prime}$ and given by [11]

$\frac{\partial
P^{\text{NC}}_{s^{\prime},p^{\prime}}}{\partial\varphi}\overset{\chi_{p^{\prime}}(\varphi)\ll
1}{\approx}\frac{5}{2\sqrt{3}}\frac{\alpha m^{2}|\chi_{p}(\varphi)|}{p_{-}}.$
(24)

The damping function is then

$\begin{split}D^{\text{NC}}_{j,s,s^{\prime}}\overset{p^{\prime}_{-}\ll
p_{-}}{\approx}2\,\text{Im}\left[\frac{m}{p_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
M_{s}(p,\varphi)+\frac{m}{p_{-}}\int_{\phi_{+}}^{\infty}d\varphi
P_{j}(p,\varphi)\right]-\int_{\phi_{+}}^{\infty}d\varphi\frac{5}{\sqrt{3}}\frac{\alpha
m^{2}|\chi_{p}(\varphi)|}{p_{-}}.\end{split}$ (25)

In the functions $\tilde{T}_{j,s,s^{\prime}}$ we assume that the ratio
$p^{\prime}_{-}/p_{-}$ is sufficiently small that
$z=\left(\frac{q_{-}}{p^{\prime}_{-}\chi_{p}(\phi_{+})}\right)^{2/3}\approx\left(\frac{p_{-}}{p^{\prime}_{-}\chi_{p}(\phi_{+})}\right)^{2/3}\gg
1$. We obtain for photon polarization $j=1$ and identical spin quantum numbers
$(s=s^{\prime})$

$\begin{split}\tilde{T}_{1,s,s}\overset{z\gg
1}{\approx}&-\frac{1}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\bigg{[}\frac{p_{-}}{p^{\prime}_{-}}\left(1-s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right)+2\frac{p^{\prime}_{-}}{p_{-}}s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\bigg{]}\\\
&-\frac{1}{96\sqrt{\pi}}z^{-9/4}e^{-\frac{2}{3}z^{3/2}}\left(124+20s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right)\\\
&-\frac{1}{9216\sqrt{\pi}}z^{-15/4}e^{-\frac{2}{3}z^{3/2}}\frac{p_{-}}{p^{\prime}_{-}}\left(3938-770s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right).\end{split}$ (26)

Note that here in the special case of $s=\text{sgn}(\psi^{\prime}(\phi_{+}))$
compensations occur and this is why higher-order terms have been reported in
the expression above. However, by keeping in mind that the functions
$\tilde{T}_{j,s,s^{\prime}}$ are ultimately integrated over the light-cone
time $\phi_{+}$ to compute the emission probability, the function
$\text{sgn}(\psi^{\prime}(\phi_{+}))$ takes both values $+1$ and $-1$ (we
implicitly assume here that the plane wave describes an oscillating laser
wave). Therefore, the scaling of the probability will be determined by the
term in $\tilde{T}_{1,s,s}$ scaling as $z^{-3/4}/p^{\prime}_{-}$ and we can
approximate

$\begin{split}\tilde{T}_{1,s,s}\overset{z\gg
1}{\approx}&-\frac{2}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\frac{p_{-}}{p^{\prime}_{-}},\quad\text{for
$s=-\text{sgn}(\psi^{\prime}(\phi_{+}))$}.\end{split}$ (27)

For opposite spin quantum numbers $(s=-s^{\prime})$ the asymptotic expression
is

$\begin{split}\tilde{T}_{1,s,-s}\overset{z\gg
1}{\approx}-\frac{1}{4\sqrt{\pi}}z^{-9/4}e^{-\frac{2}{3}z^{3/2}}\frac{p_{-}}{p^{\prime}_{-}}.\end{split}$
(28)

With photon polarization $j=2$ we have for identical spin quantum numbers
$(s=s^{\prime})$

$\begin{split}\tilde{T}_{2,s,s}\overset{z\gg
1}{\approx}-\frac{1}{4\sqrt{\pi}}z^{-9/4}e^{-\frac{2}{3}z^{3/2}}\frac{p_{-}}{p^{\prime}_{-}}\end{split}$
(29)

and for opposite spin quantum numbers $(s=-s^{\prime})$

$\begin{split}\tilde{T}_{2,s,-s}\overset{z\gg
1}{\approx}&-\frac{1}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\frac{p_{-}}{p^{\prime}_{-}}\left[1+s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right]\\\
&-\frac{1}{9216\sqrt{\pi}}z^{-15/4}e^{-\frac{2}{3}z^{3/2}}\frac{p_{-}}{p^{\prime}_{-}}\left[3938+770s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right].\end{split}$ (30)

Analogously as above, the scaling of the emission probability will be
determined by the approximated expression

$\tilde{T}_{2,s,-s}\overset{z\gg
1}{\approx}-\frac{2}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\frac{p_{-}}{p^{\prime}_{-}},\quad\text{for
{\hbox{s=\text{sgn}(\psi'(\phi_+))}}}.$ (31)

###  II.3 Nonlinear Breit-Wheeler pair production

Now, we pass to the probability of nonlinear Breit-Wheeler pair production
including the particle states decay.

Here, the incoming photon has four-momentum $q^{\mu}=(\omega,\bm{q})$, with
polarization quantum number $j=1,2$ and the outgoing positron (electron) has
four-momentum $p^{\mu}=(\varepsilon,\bm{p})$
($p^{\prime\mu}=(\varepsilon^{\prime},\bm{p}^{\prime})$) and spin quantum
number $s=\pm 1$ ($s^{\prime}=\pm 1$). The symbols of quantum numbers of the
particles have been chosen in order to exploit the crossing symmetry between
the amplitudes of nonlinear Compton scattering and of nonlinear Breit-Wheeler
pair production. The expression of the probability was computed in Ref. [79]
and it is given by

$\begin{split}P^{(\gamma\to
e^{-}e^{+})}_{j,s,s^{\prime}}=&\int\frac{d^{3}p}{16\pi^{2}}\frac{\alpha}{q_{-}p^{\prime}_{-}\varepsilon}\int
d\phi_{+}e^{2\text{Im}\left\\{\frac{m}{q_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
P_{j}(q,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi\left[\frac{m}{p^{\prime}_{-}}M_{s^{\prime}}(p^{\prime},\varphi)+\frac{m}{p_{-}}M_{s}(-p,\varphi)\right]\right\\}}\\\
&\times\int
d\phi_{-}e^{i\frac{m^{2}}{2p_{-}}\frac{q_{-}}{p^{\prime}_{-}}\left\\{[1+\bm{\pi}_{\perp,p}^{2}(\phi_{+})]\phi_{-}+\frac{\bm{\mathcal{E}}^{2}(\phi_{+})}{m^{2}}\frac{\phi_{-}^{3}}{12}\right\\}}G_{j,s,s^{\prime}},\end{split}$
(32)

with the trace

$\begin{split}G_{j,s,s^{\prime}}=\frac{1}{4}\text{tr}&\left\\{\left[1-\frac{\hat{n}[\hat{\mathcal{A}}(\phi_{+})+\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p^{\prime}_{-}}\right]\hat{\Lambda}_{j}(q)\left[1-\frac{\hat{n}[\hat{\mathcal{\mathcal{A}}}(\phi_{+})+\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p_{-}}\right]\right.\\\
&\left.\times(\hat{p}-m)(1+s\gamma^{5}\hat{\zeta})\left[1+\frac{\hat{n}[\hat{\mathcal{A}}(\phi_{+})-\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p_{-}}\right]\hat{\Lambda}_{j}(q)\right.\\\
&\left.\times\left[1+\frac{\hat{n}[\hat{\mathcal{A}}(\phi_{+})-\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p^{\prime}_{-}}\right](\hat{p}^{\prime}+m)(1+s^{\prime}\gamma^{5}\hat{\zeta}^{\prime})\right\\}\end{split}$
(33)

and with

$\bm{\pi}_{\perp,p}(\phi)=\frac{\bm{p}_{\perp}}{m}-\frac{p_{-}}{q_{-}}\frac{\bm{q}_{\perp}}{m}+\frac{\bm{\mathcal{A}}_{\perp}(\phi)}{m}.$
(34)

The trace can be simplified and the integrals in the perpendicular positron
momentum and $\phi_{-}$ can be taken analogously as in the case of nonlinear
Compton scattering. An important difference is however that the relations for
momentum conservation are instead
$\bm{p}^{\prime}_{\perp}=\bm{q}_{\perp}-\bm{p}_{\perp}$ and
$p^{\prime}_{-}=q_{-}-p_{-}$.

By performing the trace over the matrix contractions (see the appendix for the
derivation), the expressions for the two polarization quantum numbers $j=1$
and $j=2$ become

$\begin{split}G_{1,s,s^{\prime}}&=(1+ss^{\prime})\bigg{[}\frac{1}{2}\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\left(m^{2}+m^{2}\bm{\pi}_{\perp,p}^{2}(\phi_{+})-\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})\frac{\phi_{-}^{2}}{4}\right)\\\
&\qquad\qquad\quad-2\left(m^{2}\bm{\pi}_{\perp,p}^{2}(\phi_{+})-\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})\frac{\phi_{-}^{2}}{4}\right)\bigg{]}\\\
&+i(s+s^{\prime})\frac{m}{2}\mathcal{A}_{0}\psi^{\prime}(\phi_{+})\phi_{-}\frac{q_{-}}{p_{-}}\left(2-\frac{q_{-}}{p^{\prime}_{-}}\right)\\\
&+\left(2+2ss^{\prime}-ss^{\prime}\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\right)\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}\end{split}$
(35)

and

$\begin{split}G_{2,s,s^{\prime}}&=(1-ss^{\prime})\frac{1}{2}\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\left(m^{2}+m^{2}\bm{\pi}_{\perp,p}^{2}(\phi_{+})-\mathcal{A}_{0}^{2}\psi^{\prime
2}(\phi_{+})\frac{\phi_{-}^{2}}{4}\right)\\\
&+i(s-s^{\prime})\frac{m}{2}\mathcal{A}_{0}\psi^{\prime}(\phi_{+})\phi_{-}\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\\\
&-\left(2+2ss^{\prime}-ss^{\prime}\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\right)\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}.\end{split}$
(36)

Here we have already expressed the pulse shape function in terms of
$\bm{\pi}_{\perp,p}(\phi_{+})$. The dependence on
$\bm{\pi}_{\perp,p}(\phi_{+})$ can in turn be removed by using the fact that
the integral over terms proportional to
$\left[1+\bm{\pi}_{\perp,p}^{2}(\phi_{+})+({\mathcal{A}^{2}_{0}\psi^{\prime
2}(\phi_{+})}/{m^{2}})({\phi_{-}^{2}}/{4})\right]$ vanish when performing the
integral in $\phi_{-}$, due to the properties of the Airy functions. In this
way, by adding and subtracting suitable terms, the traces can be written in a
manifestly gauge-invariant way. Furthermore, we transform the integral in the
positron momentum into light-cone coordinates by using that
$d^{3}p=(\varepsilon/p_{-})dp_{-}d^{2}p_{\perp}$ and we employ the notation

$\tilde{G}_{j,s,s^{\prime}}=-\frac{1}{4\pi^{2}m^{2}}\frac{q_{-}}{p_{-}p^{\prime}_{-}}\int
d\phi_{-}\int
d^{2}p_{\perp}e^{i\frac{m^{2}q_{-}}{2p_{-}p^{\prime}_{-}}\left\\{[1+\bm{\pi}_{\perp,p}^{2}(\phi_{+})]\phi_{-}+\frac{\bm{\mathcal{E}}^{2}(\phi_{+})}{m^{2}}\frac{\phi_{-}^{3}}{12}\right\\}}G_{j,s,s^{\prime}}.$
(37)

The integral over the transverse positron momentum $\bm{p}_{\perp}$ is taken
by employing the two Gaussian integrals [86]

$\displaystyle\int
d^{2}p_{\perp}e^{i\frac{m^{2}q_{-}}{2p_{-}p^{\prime}_{-}}\bm{\pi}^{2}_{\perp,p}(\phi_{+})\phi_{-}}=2\pi
i\frac{p_{-}p^{\prime}_{-}}{q_{-}(\phi_{-}+i0)},$ (38) $\displaystyle\int
d^{2}p_{\perp}\left(p_{2}-\frac{p_{-}}{q_{-}}q_{2}\right)^{2}e^{i\frac{m^{2}q_{-}}{2p_{-}p^{\prime}_{-}}\bm{\pi}^{2}_{\perp,p}(\phi_{+})\phi_{-}}=-2\pi\left[\frac{p_{-}p^{\prime}_{-}}{q_{-}(\phi_{-}+i0)}\right]^{2},$
(39)

and the integral in $\phi_{-}$ results again in Airy functions. In this way,
the probability of nonlinear Breit-Wheeler pair production including the
damping of particle states finally reads

$\begin{split}P^{(\gamma\to e^{-}e^{+})}_{j,s,s^{\prime}}=&-\frac{\alpha
m^{2}}{4q_{-}^{2}}\int_{0}^{q_{-}}dp_{-}\int
d\phi_{+}\tilde{G}_{j,s,s^{\prime}}\\\ &\times
e^{2\text{Im}\left\\{\frac{m}{q_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
P_{j}(q,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi\left[\frac{m}{p^{\prime}_{-}}M_{s^{\prime}}(p^{\prime},\varphi)+\frac{m}{p_{-}}M_{s}(-p,\varphi)\right]\right\\}},\end{split}$
(40)

with

$\begin{split}\tilde{G}_{1,s,s^{\prime}}=&\left[-(1+ss^{\prime})-ss^{\prime}\frac{q_{-}^{2}}{2p_{-}p^{\prime}_{-}}\right]\text{Ai}_{1}(z)\\\
&+\left[-3(1+ss^{\prime})+\left(1+\frac{ss^{\prime}}{2}\right)\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\right]\frac{\text{Ai}^{\prime}(z)}{z}\\\
&-(s+s^{\prime})\left(\frac{q_{-}}{p_{-}}-\frac{q_{-}}{p^{\prime}_{-}}\right)\frac{\text{Ai}(z)}{\sqrt{z}}\text{sgn}(\psi^{\prime}(\phi_{+}))\end{split}$
(41)

and

$\begin{split}\tilde{G}_{2,s,s^{\prime}}=&\left[-(1+ss^{\prime})+ss^{\prime}\frac{q_{-}^{2}}{2p_{-}p^{\prime}_{-}}\right]\text{Ai}_{1}(z)\\\
&+\left[-(1+ss^{\prime})+\left(1-\frac{ss^{\prime}}{2}\right)\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\right]\frac{\text{Ai}^{\prime}(z)}{z}\\\
&+(s^{\prime}-s)\frac{q_{-}^{2}}{p_{-}p^{\prime}_{-}}\frac{\text{Ai}(z)}{\sqrt{z}}\text{sgn}(\psi^{\prime}(\phi_{+})).\end{split}$
(42)

Without the exponential damping term the above probability reduces to the
result of the spin- and polarization-resolved probability of nonlinear Breit-
Wheeler pair production calculated in Ref. [87]. Analogously as in the case of
nonlinear Compton scattering, it can be proved analytically that the
probability in Eq. (40) is always smaller than unity [see the discussion below
Eq. (15)].

Now, we investigate the asymptotic behavior in the two regions $q_{-}-p_{-}\ll
q_{-}$ and $p_{-}\ll q_{-}$ and for the differential probability of nonlinear
Breit-Wheeler pair production

$\begin{split}\frac{\partial P^{(\gamma\to
e^{-}e^{+})}_{j,s,s^{\prime}}}{\partial p_{-}}=&-\frac{\alpha
m^{2}}{4q_{-}^{2}}\int
d\phi_{+}e^{D^{\text{NBW}}_{j,s,s^{\prime}}}\tilde{G}_{j,s,s^{\prime}},\end{split}$
(43)

with

$\begin{split}D^{\text{NBW}}_{j,s,s^{\prime}}=2\text{Im}\left\\{\frac{m}{q_{-}}\int_{-\infty}^{\phi_{+}}d\varphi
P_{j}(q,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi\left[\frac{m}{p^{\prime}_{-}}M_{s^{\prime}}(p^{\prime},\varphi)+\frac{m}{p_{-}}M_{s}(-p,\varphi)\right]\right\\},\end{split}$
(44)

which we obtained from Eq. (40). According to the optical theorem, the
exponent of the damping term is here equal to minus the sum of the total
probability of nonlinear Breit-Wheeler pair production between $-\infty$ and
$\phi_{+}$ for the incoming photon with light-cone energy $q_{-}$ and
polarization quantum number $j$ and the total probabilities of nonlinear
Compton scattering between $\phi_{+}$ and $+\infty$ for the outgoing electron
with light-cone energy $p^{\prime}_{-}$ and spin quantum number $s^{\prime}$
and for the outgoing positron with light-cone energy $p_{-}$ and spin quantum
number $s$ [80, 84], i.e.

$\begin{split}D^{\text{NBW}}_{j,s,s^{\prime}}=-\int_{-\infty}^{\phi_{+}}d\varphi\frac{\partial
P^{\text{NBW}}_{j,q}}{\partial\varphi}-\int_{\phi_{+}}^{\infty}d\varphi\left(\frac{\partial
P^{\text{NC}}_{s^{\prime},p^{\prime}}}{\partial\varphi}+\frac{\partial
P^{\text{NC}}_{s,p}}{\partial\varphi}\right).\end{split}$ (45)

####  II.3.1 Asymptotic expression for $q_{-}-p_{-}\ll q_{-}$

First, we consider the asymptotic region $p^{\prime}_{-}=q_{-}-p_{-}\ll
q_{-}$. Thus, by assuming that the quantum nonlinearity parameter
$\kappa_{q}(\varphi)$ of the photon is fixed, the absolute value of the
quantum nonlinearity parameter of the electron
$\chi_{p^{\prime}}(\varphi)=(p^{\prime}_{-}/q_{-})\kappa_{q}(\varphi)$ is much
smaller than unity (if $\kappa_{q}(\varphi)$ is larger than unity, the ratio
$p^{\prime}_{-}/q_{-}$ is assumed to be sufficiently small that
$|\chi_{p^{\prime}}(\varphi)|\ll 1$) and we use for the damping term in Eq.
(45) the corresponding asymptotic expression for the probability of nonlinear
Compton scattering, which is independent of $p^{\prime}$ and given in Eq.
(24). The exponent of the damping term becomes (we use the fact that
$\chi_{p}(\varphi)=(p_{-}/q_{-})\kappa_{q}(\varphi)$)

$\begin{split}D^{\text{NBW}}_{j,s,s^{\prime}}\overset{p^{\prime}_{-}\ll
q_{-}}{\approx}\frac{2m}{q_{-}}\text{Im}\left[\int_{-\infty}^{\phi_{+}}d\varphi
P_{j}(q,\varphi)+\int_{\phi_{+}}^{\infty}d\varphi
M_{s}(-q,\varphi)\right]-\int_{\phi_{+}}^{\infty}d\varphi\frac{5}{\sqrt{3}}\frac{\alpha
m^{2}|\kappa_{q}(\varphi)|}{q_{-}}.\end{split}$ (46)

Concerning the function $\tilde{G}_{j,s,s^{\prime}}$, we can expand the Airy
functions for
$z=\left(\frac{q_{-}}{p^{\prime}_{-}\chi_{p}(\phi_{+})}\right)^{2/3}\approx\left(\frac{q_{-}}{p^{\prime}_{-}\kappa_{q}(\phi_{+})}\right)^{2/3}\gg
1$. In the case of photon polarization $j=1$ and identical spin quantum
numbers $(s=s^{\prime})$, we obtain the expression

$\begin{split}\tilde{G}_{1,s,s}\overset{z\gg
1}{\approx}&-\frac{1}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\left[\frac{q_{-}}{p^{\prime}_{-}}\left(1-s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right)+2\frac{p^{\prime}_{-}}{q_{-}}s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right]\\\
&+\frac{1}{96\sqrt{\pi}}z^{-9/4}e^{-\frac{2}{3}z^{3/2}}\left[124+20s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right]\\\
&-\frac{1}{9216\sqrt{\pi}}z^{-15/4}e^{-\frac{2}{3}z^{3/2}}\frac{q_{-}}{p^{\prime}_{-}}\left[3938-770s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right].\end{split}$ (47)

As in the case of nonlinear Compton scattering, the scaling of the probability
is determined by the case $s=-\text{sgn}(\psi^{\prime}(\phi_{+}))$, where

$\begin{split}\tilde{G}_{1,s,s}\overset{z\gg
1}{\approx}&-\frac{2}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\frac{q_{-}}{p^{\prime}_{-}}.\end{split}$
(48)

For opposite spin quantum numbers $(s=-s^{\prime})$ we have

$\begin{split}\tilde{G}_{1,s,-s}\overset{z\gg
1}{\approx}-\frac{1}{4\sqrt{\pi}}z^{-9/4}e^{-\frac{2}{3}z^{3/2}}\frac{q_{-}}{p^{\prime}_{-}}.\end{split}$
(49)

With photon polarization $j=2$ the asymptotic expansion is for identical spin
quantum numbers $(s=s^{\prime})$

$\begin{split}\tilde{G}_{2,s,s}\overset{z\gg
1}{\approx}-\frac{1}{4\sqrt{\pi}}z^{-9/4}e^{-\frac{2}{3}z^{3/2}}\frac{q_{-}}{p^{\prime}_{-}}\end{split}$
(50)

and for opposite spin quantum numbers $(s=-s^{\prime})$

$\begin{split}\tilde{G}_{2,s,-s}\overset{z\gg
1}{\approx}&-\frac{1}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\frac{q_{-}}{p^{\prime}_{-}}\left[1+s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right]\\\
&-\frac{1}{9216\sqrt{\pi}}z^{-15/4}e^{-\frac{2}{3}z^{3/2}}\frac{q_{-}}{p^{\prime}_{-}}\left[3938+770s\
\text{sgn}(\psi^{\prime}(\phi_{+}))\right].\end{split}$ (51)

The scaling of the probability of nonlinear Breit-Wheeler pair production is
determined by the case $s=\text{sgn}(\psi^{\prime}(\phi_{+}))$, i.e.,

$\begin{split}\tilde{G}_{2,s,-s}\overset{z\gg
1}{\approx}&-\frac{2}{\sqrt{\pi}}z^{-3/4}e^{-\frac{2}{3}z^{3/2}}\frac{q_{-}}{p^{\prime}_{-}}.\end{split}$
(52)

####  II.3.2 Asymptotic expression for $p_{-}\ll q_{-}$

Due to the symmetry of the probability of pair production under the exchanges
$p_{-}\leftrightarrow p^{\prime}_{-}$, $s\leftrightarrow s^{\prime}$, and
$\psi^{\prime}(\phi_{+})\leftrightarrow-\psi^{\prime}(\phi_{+})$ [see Eqs.
(40)-(42)], the asymptotic expressions of the differential probability of
nonlinear Breit-Wheeler pair production in the region $p_{-}\ll q_{-}$ can be
easily obtained from those derived in the region $p^{\prime}_{-}\ll q_{-}$ via
the corresponding substitution rules.

###  II.4 Arbitrary spin and polarization basis

So far, both here and in Ref. [79], the calculations were performed by
employing a special direction for the electron or positron spin and for the
photon polarization. This choice of the spin and polarization four-vectors has
the advantage that the mass and polarization operator become diagonal.
Consequently, the equations including the damping of the states can be solved
in a relatively straightforward way. However, if the final probabilities are
summed over a spin and/or a polarization quantum number, the results should be
in the end independent of the choice of the corresponding spin and
polarization basis, which we would like to prove explicitly below.

In the case of the spin state of an electron the two free, positive-energy
spinors $u_{1}(p)$ and $u_{-1}(p)$ form a spin basis [1]. They are normalized
as $u^{{\dagger}}_{s}(p)u_{s^{\prime}}(p)=2\varepsilon\delta_{ss^{\prime}}$,
with $s,s^{\prime}=\pm 1$ and fulfill the relation
$\gamma^{5}\hat{\zeta}u_{s}(p)=su_{s}(p)$ (for the positron the spin basis is
formed by the two free negative-energy spinors $v_{1}(p)$ and $v_{-1}(p)$,
normalized as
$v^{{\dagger}}_{s}(p)v_{s^{\prime}}(p)=2\varepsilon\delta_{ss^{\prime}}$) [1].
A spinor corresponding to an arbitrary spin direction, indicated here as
$u_{+}(p)$, can be expressed by a linear combination of the two basis spinors
$u_{1}(p)$ and $u_{-1}(p)$ as

$u_{+}(p)=\beta_{1}u_{1}(p)+\beta_{-1}u_{-1}(p),$ (53)

where $\beta_{1},\beta_{-1}$ are two complex numbers such that
$|\beta_{1}|^{2}+|\beta_{-1}|^{2}=1$. The two coefficients $\beta_{1}$ and
$\beta_{-1}$ are related to the polar angle $\theta$ and the azimuthal angle
$\varphi$ between the spin vector $\bm{\zeta}$ and the new spin axis [1]. The
spinor $u_{+}(p)$ in Eq. (53) forms a basis together with the spinor

$u_{-}(p)=\beta^{*}_{-1}u_{1}(p)-\beta^{*}_{1}u_{-1}(p),$ (54)

which is perpendicular to $u_{+}(p)$. Now, in Ref. [79] the probabilities were
calculated by using the exact electron, positron and photon states, which were
obtained by solving the corresponding Schwinger-Dyson equations. For the
electron out-state $\Psi_{e}^{(\text{out})}(x)$ the Schwinger-Dyson equation
is
$\\{\gamma^{\mu}[i\partial_{\mu}-\mathcal{A}_{\mu}(\phi)]-m\\}\Psi_{e}^{(\text{out})}(x)=\int
d^{4}y\,\bar{M}_{L}(y,x)\Psi_{e}^{(\text{out})}(y)$, with
$\bar{M}_{L}(y,x)=\gamma_{0}M_{L}^{\dagger}(y,x)\gamma_{0}$ and $M_{L}(y,x)$
being the mass operator in a plane wave. This equation is linear in the spin
basis $u_{1}(p)$ and $u_{-1}(p)$, such that we can decompose an arbitrary
electron state in terms of the two states $\Psi_{e,1}^{(\text{out})}(x)$ and
$\Psi_{e,-1}^{(\text{out})}(x)$, which are solutions of the Schwinger-Dyson
equation constructed via the spinors $u_{1}(p)$ and $u_{-1}(p)$, respectively.
The electron out-state $\Psi_{e,b}^{(\text{out})}(x)$ of an arbitrary spin
direction $b=\\{+,-\\}$, given by the solution of the Schwinger-Dyson equation
for the spinor $u_{b}(p)$, can be expressed then by the linear combination

$\begin{split}\Psi^{(\text{out})}_{e,+}(x)=&\beta_{1}\Psi^{(\text{out})}_{e,1}(x)+\beta_{-1}\Psi^{(\text{out})}_{e,-1}(x),\\\
\Psi^{(\text{out})}_{e,-}(x)=&\beta^{*}_{-1}\Psi^{(\text{out})}_{e,1}(x)-\beta^{*}_{1}\Psi^{(\text{out})}_{e,-1}(x).\end{split}$
(55)

The physical meaning of the states with the above choice of $\beta_{1}$ and
$\beta_{-1}$ is that at $x^{0}\to\infty$ in the rest frame of the electron the
spin axis points along the chosen axis and fulfill the above-mentioned
relations. Considering now the process of nonlinear Compton scattering, the
$S$-matrix element for an incoming electron of spin quantum number $s$, an
outgoing photon of polarization $j$, and an outgoing electron with spin
quantum number $b$ is proportional to

$S^{(e^{-}\to e^{-}\gamma)}_{s,j,+}=\beta^{*}_{1}S^{(e^{-}\to
e^{-}\gamma)}_{s,j,1}+\beta^{*}_{-1}S^{(e^{-}\to e^{-}\gamma)}_{s,j,-1}$ (56)

and

$S^{(e^{-}\to e^{-}\gamma)}_{s,j,-}=\beta_{-1}S^{(e^{-}\to
e^{-}\gamma)}_{s,j,1}-\beta_{1}S^{(e^{-}\to e^{-}\gamma)}_{s,j,-1}.$ (57)

By using these $S$-matrix elements, we then easily obtain the probability of
nonlinear Compton scattering for the two spin quantum numbers $+$ and $-$ of
the final electron as

$\begin{split}P^{(e^{-}\to
e^{-}\gamma)}_{s,j,+}&=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,+}|^{2}\\\
&=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\Big{[}|\beta_{1}|^{2}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,1}|^{2}+|\beta_{-1}|^{2}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,-1}|^{2}\\\ &\qquad\qquad\ \ \
+\beta_{1}\beta^{*}_{-1}S^{(e^{-}\to e^{-}\gamma)\,*}_{s,j,1}S^{(e^{-}\to
e^{-}\gamma)}_{s,j,-1}+\beta^{*}_{1}\beta_{-1}S^{(e^{-}\to
e^{-}\gamma)}_{s,j,1}S^{(e^{-}\to e^{-}\gamma)\,*}_{s,j,-1}\Big{]}\end{split}$
(58)

and

$\begin{split}P^{(e^{-}\to
e^{-}\gamma)}_{s,j,-}&=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,-}|^{2}\\\
&=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\Big{[}|\beta_{-1}|^{2}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,1}|^{2}+|\beta_{1}|^{2}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,-1}|^{2}\\\ &\qquad\qquad\ \ \
-\beta_{1}\beta^{*}_{-1}S^{(e^{-}\to e^{-}\gamma)\,*}_{s,j,1}S^{(e^{-}\to
e^{-}\gamma)}_{s,j,-1}-\beta^{*}_{1}\beta_{-1}S^{(e^{-}\to
e^{-}\gamma)}_{s,j,1}S^{(e^{-}\to
e^{-}\gamma)\,*}_{s,j,-1}\Big{]}.\end{split}$ (59)

Taking now the sum of the probabilities for both spin states $b=\\{+,-\\}$,
i.e., taking the sum of Eqs. (58) and (59), we obtain

$\begin{split}\sum_{b=\\{+,-\\}}P^{(e^{-}\to
e^{-}\gamma)}_{s,j,b}&=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\Big{[}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,+}|^{2}+|S^{(e^{-}\to e^{-}\gamma)}_{s,j,-}|^{2}\Big{]}\\\
&=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{d^{3}p^{\prime}}{(2\pi)^{3}}\Big{[}|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,1}|^{2}+|S^{(e^{-}\to
e^{-}\gamma)}_{s,j,-1}|^{2}\Big{]}=\sum_{s^{\prime}=\\{1,-1\\}}P^{(e^{-}\to
e^{-}\gamma)}_{s,j,s^{\prime}},\end{split}$ (60)

which is identical to the analogous result obtained for the original spin
axis. Hence, the probability summed over a spin quantum number is independent
of the used quantization axis. It can easily be shown that the same holds for
electron in-states and for the positron in- and out-states.

In order to describe the photon polarization states, we have chosen the two
four-vectors $\Lambda^{\mu}_{1}(q)$ and $\Lambda^{\mu}_{2}(q)$. An arbitrary
polarization basis is given by the two four-vectors

$\begin{split}\Lambda^{\mu}_{+}(q)=b_{1}\Lambda^{\mu}_{1}(q)+b_{2}\Lambda^{\mu}_{2}(q)\\\
\Lambda^{\mu}_{-}(q)=b_{2}\Lambda^{\mu}_{1}(q)-b_{1}\Lambda^{\mu}_{2}(q)\end{split}$
(61)

with $b_{1},b_{2}$ being two real numbers such that $b_{1}^{2}+b_{2}^{2}=1$.
Analogously to the case of the spin, one can show that also in this case the
probability summed over the polarization indexes does not depend on the
polarization basis.

##  III Numerical calculations

###  III.1 Methods

The results in the following section have been produced using two types of
integration: 1) a purely numerical one in order to obtain probability
densities, fully differential with respect to an outgoing particle’s momentum
and 2) a quadrature which takes advantage of the fact that in lightfront
coordinates the transverse momenta yield analytical Gaussian integrals, thus
leaving just one longitudinal momentum integral to be done numerically in
order to derive the total probability of the process.

We successfully compared the two methods against each other, by integrating
the results of method 1 with respect to the transverse momenta and leaving
aside the longitudinal integral in method 2. In fact, since the longitudinal
momentum is practically proportional to the energy for the ultrarelativistic
particles we consider, the latter is a good method of obtaining the energy
spectrum. In both methods, before proceeding to perform the $\phi_{+}$
integral in Eqs. (1) and (32) we expressed the integrals over $\phi_{-}$
through Airy functions and stored them in an interpolation table in
logarithmic form. Then, we also produced interpolation tables that store, for
different spin and polarization numbers and longitudinal momentum values, the
mass and polarization integrals in Eqs. (4) and (5), respectively. These were
then used in computing differential probabilities and total ones. All the
numerical quadratures were performed using adaptive Gauss-Konrod rules.

An important step for the production of fully differential spectra was to
divide the $\phi_{+}$ integration domain into some well-chosen subintervals so
that the adaptive integrator did not miss any relevant region. For large
values of $\xi_{0}$ the integrand has some very narrow peaks apart from which
it is almost vanishing and, unless the integrator is led to those peaks, it
would stop short of finding them, yielding a negligible result for the whole
integral. To find the peaks we first had to identify and store the intervals
of monotonicity of the vector potential and then identify the possible peaks
in each such interval using a nonlinear equation solver. As the positions of
these sharp peaks change with the transverse momenta, a prior analytical
(sharp) Gaussian integration with respect to the latter yielded a $\phi_{+}$
integral that is easy to handle without any subdivision of intervals. For the
Compton case the longitudinal momentum adaptive quadrature has been made
faster through a deformation of its variable, to reduce the number of
subdivisions needed to get a good precision at very small $q_{-}$, while
keeping in mind, though, that the behavior of the LCFA result is different
from the exact one in this limit.

###  III.2 Results

In this section we present plots based on numerical implementations of the
analytical results obtained above. We use the linearly-polarized (along the
$x$ direction) plane-wave laser pulse with Gaussian envelope described by the
vector potential

$\begin{split}{\bm{A}}(\phi)=A_{0}e^{-(\phi/\tau)^{2}}\sin(\omega_{0}\phi){\bm{a}}_{1}.\end{split}$
(62)

All plots have been made for a carrier angular frequency $\omega_{0}$
corresponding to $1.55\ \text{eV}$ in our units, which also corresponds to a
wavelength of $0.8\ \mu\text{m}$. The parameter $\tau$ describes the length of
the pulse and we have chosen the two values $\tau=5$ fs and $\tau=20$ fs.
Within the full width at half maximum (FWHM) of the intensity the pulse
contains about 2.2 cycles for $\tau=5$ fs and 8.8 cycles for $\tau=20$ fs.
Since our analytical results are valid within the LCFA, we consider below
amplitudes of the vector potential corresponding to $\xi_{0}=5$ and we also
restrict the parameters $\eta_{0}=\chi_{0}/\xi_{0}=(k_{0}p)/m^{2}$ for
nonlinear Compton scattering and $\rho_{0}=\kappa_{0}/\xi_{0}=(k_{0}q)/m^{2}$
for nonlinear Breit-Wheeler pair production, with
$k^{\mu}_{0}=\omega_{0}n^{\mu}$ to values not exceeding (approximately) unity,
by setting an upper bound of $100\ \text{GeV}$ for the incoming particle’s
energy. For larger values of $\xi_{0}$ one can further relax the condition on
$\eta_{0}$ and $\rho_{0}$, allowing them to take even larger values.

We present both probability densities, differential with respect to the
momentum of one of the outgoing particles (which determines the momentum of
the other outgoing particle through the conservation laws) and total
probabilities. Concerning the probabilities integrated over the unconstrained
momentum of one of the final particles, we observe the following. The meaning
of the spin and polarization quantum numbers implicitly depends on the momenta
of the particles, because the spin and polarization four-vectors depend on the
particles’ momenta. Thus, in general, once one integrates over the
unconstrained momentum of one of the final particles, the physical meaning of
the discrete quantum numbers of the final particles is unclear. However, we
will always consider head-on collisions with the incoming particle having an
energy much larger than $m\xi_{0}$, such that the angular spread of the
produced particles is small. In this case, one can then conclude that the spin
and polarization linearly-independent directions approximately correspond to
the directions of the electric and magnetic field of the (linearly-polarized)
background plane wave in the laboratory frame. Within this approximated
framework one can then investigate, for instance, the occurrence of electron
spin flip for nonlinear Compton scattering and the distinction between same-
spin and different-spin states of the pair produced in nonlinear Breit-Wheeler
process. It is interesting to discriminate between such cases, for their
probabilities can be very different.

It is convenient to introduce the notation $P^{(e^{-}\to
e^{-}\gamma)}_{s}=\sum_{j,s^{\prime}}P^{(e^{-}\to
e^{-}\gamma)}_{j,s,s^{\prime}}$ and $P^{(\gamma\to
e^{-}e^{+})}_{j}=\sum_{s,s^{\prime}}P^{(\gamma\to
e^{-}e^{+})}_{j,s,s^{\prime}}$ for nonlinear Compton scattering and nonlinear
Breit-Wheeler pair production, respectively.

In Figs. 2 and 3 we show the total probability $P^{(e^{-}\to
e^{-}\gamma)}_{s}$ of nonlinear Compton scattering (top panels) and the one-
event value of the Poisson distribution corresponding to the average
$P^{\text{NC}}_{s,p}$ given by the first-order probability of nonlinear
Compton scattering, computed by ignoring the decay of particles states, i.e.
[see also the discussion below Eq. (15)], $P^{\text{NC}}_{s,p}=-({\alpha
m^{2}})/({4p_{-}^{2}})\sum_{j,s^{\prime}}\int_{0}^{p_{-}}dq_{-}\int d\phi_{+}\
\tilde{T}_{j,s,s^{\prime}}$ (bottom panels). For sufficiently small values of
$\eta_{0}$, such that $\chi_{0}\ll 1$, the photon recoil is negligible and we
expect that the emissions are independent of each other. In Ref. [66] it was
shown that in this classical limit the probability of emitting an arbitrary
number of photons follows a Poissonian distribution. Indeed, we observe that
at such low values of $\chi_{0}$ the Poissonian distribution well approximates
the full QED results [see also the discussion below Eq. (21)]. As $\chi_{0}$
increases, important differences start to be seen.

Figure 2: Nonlinear Compton scattering total probability $P^{(e^{-}\to
e^{-}\gamma)}_{s}$ including the decay of the wave functions (top panel), as
compared to the result obtained from a Poissonian distribution whose average
photon number is the total “undamped emission probability”
$P^{\text{NC}}_{s,p}$ (bottom panel). The pulse length corresponds to
$\tau=5\;\text{fs}$ and the initial electron spin corresponds to $s=1$.

Figure 3: Same as in Fig. 2, but for $\tau=20\;\text{fs}$.

Two pulse length parameters $\tau$ have been considered: $\tau=5$ fs (Fig. 2)
and $\tau=20$ fs (Fig. 3). In the first case the probability increases as
$\xi_{0}$ increases, reaches a maximum at $\xi_{0}$ smaller than about 10, and
then decreases. In the classical regime of low values of $\chi_{0}$, the
maximum value is $e^{-1}$, as predicted by the Poissonian
$P^{\text{NC}}_{s,p}e^{-P^{\text{NC}}_{s,p}}$. For the longer pulse used in
Fig. 3, the whole plot is in the region where the probability decays with
$\xi_{0}$. Indeed, the decay of the states is now stronger and essentially any
increase of $\xi_{0}$ reduces the single-photon emission probability, as
multiple photon emissions become favored over the single-photon emission. As
mentioned in the introduction, the decay of the particle states becomes
significant if the quantity $\alpha\xi_{0}\Phi_{L}$ is of the order of unity
or larger. This corresponds to values of $\xi_{0}\gtrsim 9.9$ and
$\xi_{0}\gtrsim 2.5$ for a pulse with $\tau=5\ \text{fs}$ (Fig. 2) and
$\tau=20\ \text{fs}$ (Fig. 3), respectively. Here, we estimated the total
phase duration by the FWHM of the intensity, i.e., we set
$\Phi_{L}=\sqrt{2\ln{2}}\ \omega_{0}\tau$. We see that for larger values of
$\tau$ the damping effect becomes significant already at smaller values of
$\xi_{0}$.

For the probability of nonlinear Breit-Wheeler pair production we show plots
similar to the aforementioned ones, in Figs. 4 (for a pulse duration
corresponding to $\tau=5$ fs) and 5 (for a pulse duration corresponding to
$\tau=20$ fs). A logarithmic scale is used for the energy of the incoming
particle in both cases, but in the present case we started from higher values
than for nonlinear Compton scattering, due to the exponential suppression of
the process that occurs at low $\kappa_{0}=\rho_{0}\xi_{0}$. In both cases the
probability shows a maximum in $\xi_{0}$ (see also Ref. [88]). This feature is
observable in the plots only at high incoming photon energies for the shorter
pulse, but also at lower incoming photon energies for the longer pulse,
indicating that the size of the effect also depends on the damping of the
states.

Figure 4: The probability $P^{(\gamma\to e^{-}e^{+})}_{j}$ of nonlinear Breit-
Wheeler pair production in a short, $\tau=5\;\text{fs}$ pulse, by a photon
with polarization quantum number $j=1$ (left panel) and $j=2$ (right panel).
Figure 5: Same as in Fig. 4, but for $\tau=20\;\text{fs}$.

It can be seen from Fig. 6 that, as $\rho_{0}$ increases toward unity, the
probability for $(j=1,s=s^{\prime})$ reduces to around $80\%$ of the total
nonlinear Breit-Wheeler pair production probability (implying that for
$(j=1,s=-s^{\prime})$ it grows to around $20\%$), whereas the probability for
$(j=2,s=s^{\prime})$ stays smaller than $3\permil$. Analogous behaviors can be
observed in nonlinear Compton scattering, according to the crossing symmetry
existing between the two processes.

Figure 6: Fraction of the total probability of nonlinear Breit-Wheeler pair
production corresponding to same-spin pairs, $(\sum_{s}P^{(\gamma\to
e^{-}e^{+})}_{j,s,s})/P^{(\gamma\to e^{-}e^{+})}_{j}$ for $j=1$ (left panel)
and for $j=2$ (right panel). In both cases, the pulse duration corresponds to
$\tau=20\;\text{fs}$.

Now, we present sections of the fully differential distribution of the photon
momentum for the nonlinear Compton scattering effect (Figs. 7 and 8) and of
the momentum of the positron produced in the nonlinear Breit-Wheeler pair
production process (Figs. 9 \- 12). In all cases we fix the energy of the
incoming particle to $10\ \text{GeV}$ (recall that the incoming particle is
counterpropagating with respect to the plane wave) and we consider two laser
intensities corresponding to $\xi_{0}=10$ and $50$, which are equivalent to
$2.1\times 10^{20}\ \text{W}/\text{cm}^{2}$ and $5.4\times 10^{21}\
\text{W}/\text{cm}^{2}$, respectively. The sections are plotted as functions
of $q_{x}=-q_{1}$ and $q_{y}=-q_{2}$ for nonlinear Compton scattering and of
$p_{x}=-p_{1}$ and $p_{y}=-p_{2}$ for nonlinear Breit-Wheeler pair production,
by keeping the light-cone energy of the emitted photon and of the produced
positron fixed (which also means, with very good approximation, to keep fixed
$q_{3}$ or $q_{-}$ and $p_{3}$ or $p_{-}$, respectively). Indeed, the plots
show that the outgoing particles are most probably produced with transverse
momenta much smaller than $|q_{3}|$ and $|p_{3}|$, respectively (see that the
scale of the sections is in MeV units, instead of the GeV units used for the
longitudinal momentum). As expected, the sections are asymmetric, spreading
more along the direction of the electric field of the plane wave, i.e., the
$x$ direction.

Figure 7: Sections through the nonlinear Compton scattering probability
distribution in GeV-3, for $\tau=20\;\text{fs}$, $\varepsilon=10\;\text{GeV}$,
$\xi_{0}=10$, and incoming electron spin quantum number $s=1$. The color
levels indicate percentages of the maximum reached by the distribution in that
section. The approximate value of that maximum is shown at the upper right
corner of each subplot. Figure 8: Sections through the nonlinear Compton
scattering probability distribution in GeV-3, for $\tau=20\;\text{fs}$,
$\varepsilon=10\;\text{GeV}$, $\xi_{0}=50$, and incoming electron spin quantum
number $s=1$. The color levels indicate percentages of the maximum reached by
the distribution in that section. The approximate value of that maximum is
shown at the upper right corner of each subplot.

Both in the case of nonlinear Compton scattering and nonlinear Breit-Wheeler
pair production, for some quantum number combinations the spectra vanish if
the component of the momentum of the particle along the magnetic field of the
wave, i.e., along the $y$ direction, vanishes. This can be explained
analytically looking at Eqs. (8) and (9) for nonlinear Compton scattering and
at Eqs. (35) and (36) for nonlinear Breit-Wheeler pair production. These
equations show that for both processes the probabilities vanish in the cases
$(j=1,s=-s^{\prime})$ and $(j=2,s=s^{\prime})$ for $q_{2}=0$ and $p_{2}=0$,
respectively. Now, since the incoming particle is assumed to counterpropagate
with respect to the plane wave for both processes, we have that $p_{2}=0$ in
nonlinear Compton scattering and $q_{2}=0$ in nonlinear Breit-Wheeler pair
production. Thus, if in addition $q_{2}=0$ in nonlinear Compton scattering and
$p_{2}=0$ in nonlinear Breit-Wheeler pair production, the corresponding
probability vanishes for the mentioned spin and polarization combinations.
Furthermore, we see in Figs. 7 \- 12 that the maximum of each section is found
at the origin for the complementary combinations $(j=1,s=s^{\prime})$ and
$(j=2,s=-s^{\prime})$. For the examples of nonlinear Breit-Wheeler pair
production processes of Figs. 9 \- 12 the aforementioned cases
$(j=1,s=s^{\prime})$ and $(j=2,s=-s^{\prime})$ also have an altogether larger
probability than the complementary ones $(j=1,s=-s^{\prime})$ and
$(j=2,s=s^{\prime})$, as it can be recognized also noticing the different
scales of the panels.

For nonlinear Compton scattering we only show results for the initial spin
state defined by $s=1$ (see Figs. 7 and 8) as the $s=-1$ case gives very
similar results, provided one changes the sign of $s^{\prime}$, too. The
different scales in the transverse photon momenta for each individual plot
show a general tendency of broadening of the probability distribution on the
transverse plane for increasing energies of the emitted photon.

Finally, for nonlinear Breit-Wheeler pair production the probability density
always vanishes for the energy of any final particle approaching the energy of
the incoming photon. In addition, for ($j=2$, $s=s^{\prime}$) it also vanishes
when the final particles both have half of the total available energy (see
Figs. 10 and 12).

Figure 9: Sections through the nonlinear Breit-Wheeler pair production
probability distribution in GeV-3, for $\tau=20\;\text{fs}$,
$\omega=10\;\text{GeV}$, $\xi_{0}=10$, and incoming photon polarization state
$j=1$. The horizontal axes correspond to $p_{x}$ [MeV] and the vertical ones
correspond to $p_{y}$ [MeV]. Figure 10: Same as in Fig. 9, but for incoming
photon polarization state $j=2$. Figure 11: Sections through the nonlinear
Breit-Wheeler pair production probability distribution in GeV-3, for
$\tau=20\;\text{fs}$, $\omega=10\;\text{GeV}$, $\xi_{0}=50$, and incoming
photon polarization state $j=1$. The horizontal axes correspond to $p_{x}$
[MeV] and the vertical ones correspond to $p_{y}$ [MeV]. Figure 12: Same as in
Fig. 11, but for incoming photon polarization state $j=2$.

##  IV Conclusions

In conclusion, we presented analytical expressions and numerical evaluations
of the probabilities for nonlinear Compton scattering and nonlinear Breit-
Wheeler pair production including the particle states decay. The probabilities
take into account that in a plane-wave background field the electron and
photon states are not stable because electrons and positrons emit photons and
because photons themselves decay into electron-positron pairs. Within our
approach based on the locally-constant field approximation, the decay of the
states leads to an exponential damping term in the expressions of the
probabilities, which depends on the plane-wave light-cone time as well as on
the light-cone energies and discrete quantum numbers of the participating
particles.

In the analytical part, we first calculated the spin- and polarization-
resolved traces and took the integrals over the transverse momenta and phase
differences. The final probabilities depend on the spin and polarization
quantum numbers, on the incoming particle four-momentum, as well as on the
quantum nonlinearity parameter. Furthermore, we computed the asymptotic
expressions for the probabilities differential in the light-cone momentum of
one final particle for arbitrary spin and polarization quantum numbers in the
limit of one of the outgoing particles gaining all or none of the light-cone
energy of the incoming particle. All calculations were carried out using
particular spin and polarization four-vectors and we proved that the results
for the total probabilities are independent of the chosen spin and
polarization bases.

In the numerical part we presented plots of the total and differential
probabilities for nonlinear Compton scattering and nonlinear Breit-Wheeler
pair production by considering two different pulse lengths. Due to the
particle states decay, the total probabilities stay below unity in all cases.
The damping becomes important for $\alpha\xi_{0}\Phi_{L}\gtrsim 1$ at values
of the quantum nonlinearity parameter of the order of unity, such that it
increases more rapidly with $\xi_{0}$ for larger values of the pulse phase
length $\Phi_{L}$. For nonlinear Compton scattering we saw that the
probability behaves like a Poissonian distribution for low values of
$\eta_{0}$ such that photon recoil is negligible, whereas important
differences arise when $\eta_{0}$ increases. For the differential probability
we found it has its maximum at vanishing perpendicular momentum in the case of
polarization $j=1$ and same spins $s=s^{\prime}$, and in the case of
polarization $j=2$ and opposite spins $s=-s^{\prime}$, otherwise it vanishes
at $q_{y}=0$. The same behavior is observed for nonlinear Breit-Wheeler pair
production for the same spin and polarization combination and for $p_{y}=0$,
i.e., for the component of the positron momentum along the magnetic field of
the wave. For nonlinear Breit-Wheeler pair production we have also seen that
the differential probability vanishes for all spin and polarization
combinations if the light-cone energy of the incoming photon all goes to
either the electron or the positron. Also, for ($j=2$, $s=s^{\prime}$) the
differential probability vanishes if the electron and the positron have the
same light-cone energy, i.e., half the light-cone energy of the incoming
photon. Finally, in all the cases we investigated numerically, the kinematic
conditions were such that 1) the transverse momenta of produced particles are
much smaller than the corresponding longitudinal momentum and 2) the main
spread of the transverse momenta is along the direction of the plane-wave
electric field.

###### Acknowledgements.

This publication was also supported and T. P. is funded by the Collaborative
Research Centre 1225 funded by Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - Project-ID 273811115 - SFB 1225 and for this work V. D.
was supported by a grant from the Romanian National Authority for Scientific
Research, CNCS-UEFISCDI, the Project no. PN-III-P4-ID-PCCF-2016-0164.
Furthermore, this article comprises parts of the Ph.D. thesis work of T. P.,
submitted to Heidelberg University, Germany.

##  Appendix A Computation of the trace

We present here the computation of the traces for nonlinear Compton scattering
and nonlinear Breit-Wheeler pair production given in Eq. (2) and Eq. (33),
respectively. For this we first simplify the expressions by introducing the
quantity

$\begin{split}Q_{p,s}(\phi_{+},\phi_{-})=&\left[1+\frac{\hat{n}[\hat{\mathcal{\mathcal{A}}}(\phi_{+})+\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p_{-}}\right](\hat{p}+m)(1+s\gamma^{5}\hat{\zeta})\\\
&\times\left[1-\frac{\hat{n}[\hat{\mathcal{\mathcal{A}}}(\phi_{+})-\hat{\mathcal{A}}^{\prime}(\phi_{+})\phi_{-}/2]}{2p_{-}}\right].\end{split}$
(63)

With this notation the trace for nonlinear Compton scattering is given by

$T_{j,s,s^{\prime}}=\frac{1}{4}\text{tr}\left\\{\hat{\Lambda}_{j}(q)Q_{p,s}(\phi_{+},\phi_{-})\hat{\Lambda}_{j}(q)Q_{p^{\prime},s^{\prime}}(\phi_{+},-\phi_{-})\right\\}$
(64)

and for nonlinear Breit-Wheeler pair production the trace reads

$G_{j,s,s^{\prime}}=-\frac{1}{4}\text{tr}\left\\{\hat{\Lambda}_{j}(q)Q_{-p,s}(\phi_{+},\phi_{-})\hat{\Lambda}_{j}(q)Q_{p^{\prime},s^{\prime}}(\phi_{+},-\phi_{-})\right\\}.$
(65)

Now the function $Q_{p,s}(\phi_{+},\phi_{-})$ can be decomposed into a linear
combination of the matrices $1_{4\times 4}$, $\gamma^{5}$, $\gamma^{\mu}$,
$i\gamma^{\mu}\gamma^{5}$,
$\sigma^{\mu\nu}=(i/2)(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$
such that

$Q_{p,s}(\phi_{+},\phi_{-})=c_{1}1_{4\times
4}+c_{5}\gamma^{5}+c_{\mu}\gamma^{\mu}+c_{5\mu}i\gamma^{\mu}\gamma^{5}+c_{\mu\nu}\sigma^{\mu\nu},$
(66)

where the coefficients are obtained by solving the following traces

$\displaystyle c_{1}=$ $\displaystyle\frac{1}{4}\text{Tr}\left[1_{4\times
4}Q_{p,s}(\phi_{+},\phi_{-})\right],$ (67) $\displaystyle c_{5}=$
$\displaystyle\frac{1}{4}\text{Tr}\left[\gamma^{5}Q_{p,s}(\phi_{+},\phi_{-})\right],$
(68) $\displaystyle c_{\mu}=$
$\displaystyle\frac{1}{4}\text{Tr}\left[\gamma_{\mu}Q_{p,s}(\phi_{+},\phi_{-})\right],$
(69) $\displaystyle c_{5\mu}=$
$\displaystyle\frac{1}{4}\text{Tr}\left[i\gamma_{\mu}\gamma^{5}Q_{p,s}(\phi_{+},\phi_{-})\right],$
(70) $\displaystyle c_{\mu\nu}=$
$\displaystyle\frac{1}{8}\text{Tr}\left[\sigma_{\mu\nu}Q_{p,s}(\phi_{+},\phi_{-})\right].$
(71)

These traces can be calculated and we present them here for a linear polarized
plane wave background field

$c_{1}=m-i\frac{s}{4p_{-}}\epsilon^{\alpha\beta\gamma\delta}\zeta_{\alpha}\mathcal{F}_{\beta\gamma}\psi^{\prime}(\phi_{+})\phi_{-}p_{\delta},$
(72) $c_{5}=0,$ (73)
$\begin{split}c_{\mu}=&p_{\mu}-\mathcal{A}_{\mu}(\phi_{+})-\frac{1}{p_{-}}n_{\mu}\bm{p}_{\perp}\cdot\bm{\mathcal{A}}_{\perp}(\phi_{+})\\\
&+\frac{1}{2p_{-}}n_{\mu}\left[\bm{\mathcal{A}}_{\perp}^{2}(\phi_{+})-\bm{\mathcal{A}}^{\prime
2}_{\perp}(\phi_{+})\frac{\phi_{-}^{2}}{4}\right]\\\
&-i\frac{ms}{4p_{-}}\eta_{\mu\nu}\epsilon^{\alpha\beta\delta\nu}\zeta_{\alpha}\mathcal{F}_{\beta\delta}\psi^{\prime}(\phi_{+})\phi_{-},\end{split}$
(74)
$\begin{split}c_{5\mu}=&ims\zeta_{\mu}+\frac{1}{4p_{-}}\eta_{\mu\nu}\epsilon^{\alpha\beta\delta\nu}p_{\alpha}\mathcal{F}_{\beta\delta}\psi^{\prime}(\phi_{+})\phi_{-},\end{split}$
(75)

and

$\begin{split}c_{\mu\nu}=&\frac{1}{2}\bigg{\\{}-i\frac{m}{2p_{-}}\mathcal{F}_{\mu\nu}\psi^{\prime}(\phi_{+})\phi_{-}-s\epsilon_{\mu\nu\rho\sigma}p^{\rho}\zeta^{\sigma}\\\
&+\frac{s}{2p_{-}}(\eta_{\nu\rho}\epsilon_{\mu\sigma\tau\delta}-\eta_{\mu\rho}\epsilon_{\nu\sigma\tau\delta})p^{\rho}\zeta^{\sigma}\mathcal{F}^{\tau\delta}\psi(\phi_{+})\\\
&+\frac{s}{2p_{-}}\epsilon_{\mu\nu\tau\rho}n^{\tau}\zeta^{\rho}\left[\mathcal{A}_{\delta}(\phi_{+})\mathcal{A}^{\delta}(\phi_{+})-\mathcal{A}^{\prime}_{\delta}(\phi_{+})\mathcal{A}^{\prime\delta}(\phi_{+})\frac{\phi_{-}^{2}}{4}\right]\bigg{\\}}.\end{split}$
(76)

The total trace of nonlinear Compton scattering, given in Eq. (64), contains
also the function $Q_{p^{\prime},s^{\prime}}(\phi_{+},-\phi_{-})$ for which we
introduce the notation

$Q_{p^{\prime},s^{\prime}}(\phi_{+},-\phi_{-})=c_{1}^{\prime}1_{4\times
4}+c_{5}^{\prime}\gamma^{5}+c_{\tau}^{\prime}\gamma^{\tau}+c_{5\tau}^{\prime}i\gamma^{\tau}\gamma^{5}+c_{\tau\lambda}^{\prime}\sigma^{\tau\lambda},$
(77)

where the primed coefficients are obtained analogously to Eqs. (67)-(71). With
this the trace for nonlinear Compton scattering can be reduced to the form

$\begin{split}T_{j,s,s^{\prime}}=&-c_{1}c_{1}^{\prime}+c_{5}c_{5}^{\prime}+(c_{\mu}c_{\tau}^{\prime}-c_{5\mu}c_{5\tau}^{\prime})(2\Lambda_{j}^{\mu}(q)\Lambda_{j}^{\tau}(q)+\eta^{\mu\tau})\\\
&-2c_{\mu\nu}c_{\tau\lambda}^{\prime}\eta^{\nu\lambda}(4\Lambda_{j}^{\mu}(q)\Lambda_{j}^{\tau}(q)+\eta^{\mu\tau}).\end{split}$
(78)

The trace turns out to only depend on contractions of the primed and the
corresponding not primed coefficients. These contractions can be calculated
and they are given by

$\begin{split}c_{1}c^{\prime}_{1}=&m^{2}-i\frac{m}{4}\epsilon^{\alpha\beta\gamma\delta}\mathcal{F}_{\beta\gamma}\psi^{\prime}(\phi_{+})\phi_{-}\left[\frac{s}{p_{-}}\zeta_{\alpha}p_{\delta}-\frac{s^{\prime}}{p^{\prime}_{-}}\zeta^{\prime}_{\alpha}p^{\prime}_{\delta}\right]\\\
&+\frac{1}{16}\frac{ss^{\prime}}{p_{-}p^{\prime}_{-}}\epsilon^{\alpha\beta\gamma\delta}\mathcal{F}_{\beta\gamma}\zeta_{\alpha}p_{\delta}\epsilon^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}\mathcal{F}_{\beta^{\prime}\gamma^{\prime}}\zeta^{\prime}_{\alpha^{\prime}}p^{\prime}_{\delta^{\prime}}\psi^{\prime
2}(\phi_{+})\phi_{-}^{2},\end{split}$ (79) $c_{5}c^{\prime}_{5}=0,$ (80)
$\begin{split}c_{\mu}c^{\prime}_{\tau}&\left(2\Lambda_{j}^{\mu}(q)\Lambda_{j}^{\tau}(q)+\eta^{\mu\tau}\right)=(pp^{\prime})+(\bm{p}_{\perp}+\bm{p}^{\prime}_{\perp})\cdot\bm{\mathcal{A}}_{\perp}(\phi_{+})\\\
&-\left[\frac{p^{\prime}_{-}}{p_{-}}\bm{p}_{\perp}+\frac{p_{-}}{p^{\prime}_{-}}\bm{p}^{\prime}_{\perp}\right]\cdot\bm{\mathcal{A}}_{\perp}(\phi_{+})+\frac{1}{2}\left(\frac{p^{\prime}_{-}}{p_{-}}+\frac{p_{-}}{p^{\prime}_{-}}\right)\left[\bm{\mathcal{A}}_{\perp}^{2}(\phi_{+})-\bm{\mathcal{A}}^{\prime
2}_{\perp}(\phi_{+})\frac{\phi_{-}^{2}}{4}\right]\\\
&+\mathcal{A}_{\mu}(\phi_{+})\mathcal{A}^{\mu}(\phi_{+})-i\frac{m}{4}\epsilon^{\alpha\beta\gamma\delta}\mathcal{F}_{\beta\gamma}\psi^{\prime}(\phi_{+})\phi_{-}\left[\frac{s}{p_{-}}\zeta_{\alpha}p^{\prime}_{\delta}-\frac{s^{\prime}}{p^{\prime}_{-}}\zeta^{\prime}_{\alpha}p_{\delta}\right]\\\
&+2\left(p\Lambda_{j}(q)\right)\left(p^{\prime}\Lambda_{j}(q)\right)-2[p_{\mu}+p^{\prime}_{\mu}]\Lambda_{j}^{\mu}(q)\left(\Lambda_{j}(q)\mathcal{A}(\phi_{+})\right)\\\
&+2\left(\Lambda_{j}(q)\mathcal{A}(\phi_{+})\right)^{2},\end{split}$ (81)

$\begin{split}c_{5\mu}c^{\prime}_{5\tau}&\left(2\Lambda_{j}^{\mu}(q)\Lambda_{j}^{\tau}(q)+\eta^{\mu\tau}\right)=-m^{2}ss^{\prime}\left[2\left(\zeta\Lambda_{j}(q)\right)\left(\zeta^{\prime}\Lambda_{j}(q)\right)+(\zeta\zeta^{\prime})\right]\\\
&+i\frac{m}{4}\epsilon^{\alpha\beta\gamma\delta}\mathcal{F}_{\beta\gamma}\psi^{\prime}(\phi_{+})\phi_{-}\left[\frac{s^{\prime}}{p_{-}}p_{\alpha}\zeta^{\prime}_{\delta}-\frac{s}{p^{\prime}_{-}}p^{\prime}_{\alpha}\zeta_{\delta}\right]\\\
&+i\frac{m}{2}\epsilon^{\alpha\beta\gamma\delta}\mathcal{F}_{\beta\gamma}\psi^{\prime}(\phi_{+})\phi_{-}\left[\frac{s^{\prime}}{p_{-}}p_{\alpha}\left(\zeta^{\prime}\Lambda_{j}(q)\right)-\frac{s}{p^{\prime}_{-}}p^{\prime}_{\alpha}\left(\zeta\Lambda_{j}(q)\right)\right]\Lambda_{j,\delta}(q)\\\
&-\frac{1}{16p_{-}p^{\prime}_{-}}\epsilon^{\alpha\beta\gamma\delta}p_{\alpha}\mathcal{F}_{\beta\gamma}\epsilon^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}p^{\prime}_{\alpha^{\prime}}\mathcal{F}_{\beta^{\prime}\gamma^{\prime}}\left[\eta_{\delta\delta^{\prime}}+2\Lambda_{j,\delta}(q)\Lambda_{j,\delta^{\prime}}(q)\right]\psi^{\prime
2}(\phi_{+})\phi_{-}^{2},\end{split}$ (82)

and

$\begin{split}2c_{\mu\nu}&c^{\prime}_{\tau\lambda}\eta^{\nu\lambda}\left(4\Lambda_{j}^{\mu}(q)\Lambda_{j}^{\tau}(q)+\eta^{\mu\tau}\right)=-ss^{\prime}\left[(p\zeta^{\prime})(p^{\prime}\zeta)-(pp^{\prime})(\zeta\zeta^{\prime})\right]\\\
&-2ss^{\prime}\Big{[}(p\zeta^{\prime})(p^{\prime}\Lambda_{j}(q))(\zeta\Lambda_{j}(q))+(\zeta
p^{\prime})(\zeta^{\prime}\Lambda_{j}(q))(p\Lambda_{j}(q))\\\
&\qquad\qquad-(pp^{\prime})(\zeta\Lambda_{j}(q))(\zeta^{\prime}\Lambda_{j}(q))-(\zeta\zeta^{\prime})(p\Lambda_{j}(q))(p^{\prime}\Lambda_{j}(q))\Big{]}\\\
&+i\frac{m}{4}\psi^{\prime}(\phi_{+})\phi_{-}\epsilon^{\mu\nu\rho\sigma}\left[\mathcal{F}_{\mu\nu}+4\Lambda_{j,\mu}(q)\left(\Lambda_{j}^{\tau}(q)\mathcal{F}_{\tau\nu}\right)\right]\left[\frac{s^{\prime}}{p_{-}}p^{\prime}_{\rho}\zeta^{\prime}_{\sigma}-\frac{s}{p^{\prime}_{-}}p_{\rho}\zeta_{\sigma}\right]\\\
&-\frac{ss^{\prime}}{2}\left[\mathcal{A}_{\delta}(\phi_{+})\mathcal{A}^{\delta}(\phi_{+})-\mathcal{A}^{\prime}_{\delta}(\phi_{+})\mathcal{A}^{\prime\delta}(\phi_{+})\frac{\phi_{-}^{2}}{4}\right]\\\
&\qquad\times\left[(\zeta\zeta^{\prime})+2(\zeta\Lambda_{j}(q))(\zeta^{\prime}\Lambda_{j}(q))\right]\left(\frac{p_{-}}{p^{\prime}_{-}}+\frac{p^{\prime}_{-}}{p_{-}}\right)\\\
&+ss^{\prime}\psi(\phi_{+})(\zeta\zeta^{\prime})\left[\frac{p_{-}-p^{\prime}_{-}}{p_{-}p^{\prime}_{-}}(p^{\prime}\mathcal{F}p)-2\frac{(p^{\prime}\Lambda_{j}(q))}{p^{\prime}_{-}}(p\mathcal{F}\Lambda_{j}(q))-2\frac{(p\Lambda_{j}(q))}{p_{-}}(p^{\prime}\mathcal{F}\Lambda_{j}(q))\right]\\\
&-\frac{ss^{\prime}}{4p_{-}p^{\prime}_{-}}\psi^{2}(\phi_{+})(\epsilon^{\rho^{\prime}\sigma\gamma\delta}p^{\prime}_{\rho^{\prime}}\zeta_{\sigma}\mathcal{F}_{\gamma\delta})(\epsilon^{\rho\sigma^{\prime}\gamma^{\prime}\delta^{\prime}}p_{\rho}\zeta^{\prime}_{\sigma^{\prime}}\mathcal{F}_{\gamma^{\prime}\delta^{\prime}}).\end{split}$
(83)

Inserting Eqs. (79)-(83) into Eq. (78) and simplifying the expression for the
two polarization states $j=1$ and $j=2$ leads finally to Eqs. (6) and (7),
respectively.

The trace for nonlinear Breit-Wheeler pair production given in Eq. (65) can be
derived directly from the result of nonlinear Compton scattering in Eqs.
(78)-(83) by multiplying Eq. (78) by an overall minus sign and by changing the
sign of the four-momentum $p^{\mu}$ in the Eqs. (79)-(83). With that one
finally obtains Eqs. (35) and (36) for the two polarization states $j=1$ and
$j=2$, respectively.

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