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

The spectrally resolved differential cross section of Compton scattering, $d \sigma / d \omega' \vert_{\omega' = const}$, rises from small towards larger laser intensity parameter $\xi$, reaches a maximum, and falls towards the asymptotic strong-field region. Expressed by invariant quantities: $d \sigma /du \vert_{u = const}$ rises from small towards larger values of $\xi$, reaches a maximum at $\xi_{max} = \frac49 {\cal K} u m^2 / k \cdot p$, ${\cal K} = {\cal O} (1)$, and falls at $\xi>\xi_{max}$ like $\propto \xi^{-3/2} \exp \left (- \frac{2 u m^2}{3 \xi \, k \cdot p} \right )$ at $u \ge 1$. [The quantity $u$ is the Ritus variable related to the light-front momentum-fraction $s = (1 + u)/u = k \cdot k' / k \cdot p$ of the emitted photon (four-momentum $k'$, frequency $\omega'$), and $k \cdot p/m^2$ quantifies the invariant energy in the entrance channel of electron (four-momentum $p$, mass $m$) and laser (four-wave vector $k$).] Such a behavior of a differential observable is to be contrasted with the laser intensity dependence of the total probability, $\lim_{\chi = \xi k \cdot p/m^2, \xi \to \infty} \mathbb{P} \propto \alpha \chi^{2/3} m^2 / k \cdot p$, which is governed by the soft spectral part. We combine the hard-photon yield from Compton with the seeded Breit-Wheeler pair production in a folding model and obtain a rapidly increasing $e^+ e^-$ pair number at $\xi \lesssim 4$. Laser bandwidth effects are quantified in the weak-field limit of the related trident pair production.


I. INTRODUCTION
Quantum Electro-Dynamics (QED) as pillar of the standard model (SM) of particle physics possesses a positive β function [1] which makes the running coupling strength α(s) increasingly with increasing energy/momentum scale s [2]. In contrast, Quantum Chromo-Dynamics (QCD) as another SM pillar possesses a negative β function due to the non-Abelian gauge group [1], giving rise to the asymptotic freedom, lim s→∞ α QCD (s) → 0, i.e. QCD has a truly perturbative limit. In contrast, lim s→0 α(s) → 1/137.0359895 (61) is not such a strict limit, nevertheless, QED predictions/calculations of some observables agree with measurements within 13 digits, see [3][4][5] for some examples. The situation in QED becomes special when considering processes in external (or background) fields: One can resort to the Furry (or bound-state) picture, where the (tree-level) interactions of an elementary charge (e.g. an electron) with the background are accounted for in all orders, and the interactions with the quantized photon field remains perturbatively in powers of α. However, the Ritus-Narozhny (RN) conjecture [6][7][8][9][10] argues that the effective coupling becomes αχ 2/3 , meaning that the Furry picture expansion beaks down at αχ 2/3 > 1 [11][12][13][14] (for the definition of χ see below) and one enters a genuinely non-perturbative regime. The latter requires adequate calculation procedures, as the lattice regularized approaches, which are standard since many years in QCD, e.g. in evaluations of observables in the soft sector where α QCD > 1, cf. [15]. (In QED itself, an analog situation is meet in the Coulomb field of nuclear systems with proton numbers Z > Z crit ≈ 173: if αZ crit > 1 the QED vacuum beak-down sets in; cf. [16] for the actual status of that field).
With respect to increasing laser intensities the quest for the possible break-down of the Furry picture expansion in line with the RN conjecture becomes, besides its principal challenge, also of "practical" interest, whether one can explore experimentally this yet uncharted regime of QED. (For other configurations, e.g. beam-beam interactions, cf. [17]). A prerequisite would be to find observables which display the typical dependence ∝ αχ 2/3 , where we denote by α the above quoted fine-structure constant at s → 0. In doing so we resort here to the lowest-order QED processes, that is nonlinear Compton and nonlinear Breit-Wheeler. Both processes seem to be investigated theoretically in depth in the past, however, enjoy currently repeated re-considerations w.r.t. refinements [18][19][20][21][22][23], establishing approximation schemes [24][25][26][27][28] to be implemented in simulation codes [29][30][31] or to use them as building blocks in complex processes [32], with the starting point at trident [33][34][35].
Beginning with Compton scattering (p and p are the in-and out-electron four-momenta, k the laser fourmomentum, and k the out-photon's four-momentum, respectively) the relevant (Lorentz and gauge invariant) variables of the entrance channel are [36] -(i) the classical intensity parameter of the laser: ξ = |e|E/(mω), here expressed by quantities in the lab.: E -electric laser field strength, ω -the central laser frequency; −|e| and m stand for the electron charge and mass, respectively, and e 2 /4π = α, 1 -(ii) the available energy squared: k · p/m 2 = (ŝ/m 2 − 1)/2 withŝ as Mandelstam variable, -(iii) the quantum nonlinearity parameter: χ = ξk·p/m 2 . The latter quantity is often considered as the crucial parameter since in some limits it determines solely the probability of certain processes. χ plays also a prominent role in the above mentioned discussion of the RN conjecture, where the Furry picture expansion of QED is argued to break down for αχ 2/3 > 1. References [37,38] point out that the large-χ limits, facilitated by either large ξ (the high-intensity limit) or large k · p/m 2 (the high-energy limit), are distinctively different, with implications for approximation schemes in simulation codes. Figure 1 exhibits a few selected curves χ = const over the ξ vs. k·p/m 2 landscape to illustrate the current situation w.r.t. facilities where laser beams and electron beams are (or can be) combined. One has to add the options at E-320 at FACET-II/SLAC [39,40] and electron beams which are laser-accelerated to GeV scales, e.g. [41][42][43].
Usually, ξ = 1 is said to mark the the onset of strongfield effects, and the corresponding processes at ξ > 1 are termed by the attribute "nonlinear". In this respect, the parameters provided by LUXE [44,45] and FACET-II [39,40] are interesting: χ = O(1) and above and ξ > 1 as well. 2 In the following we consider the LUXE kinematics (see Fig. 1), k · p/m 2 ≈ ωE e (1 − cos Θ) = 0.2078 in head-on collisions (cos Θ = −1). Our aim is to quantify a simple observable as a function of ξ. To be specific, we select the invariant differential cross section dσ/du, where u is the light-cone momentum-transfer of the in-electron to the out-photon, related to light-front momentum-fraction of the out-photon s = u 1+u = k·k k·p . (The mapping u → ω and dσ/du → dσ/dω is discussed in [46,47].) To make the meaning of the Ritus variable u more explicit let us mention the relation u = e −ζ ν (1−cos Θ ) 1−e −ζ ν (1−cos Θ ) , where ν ≡ ω /m, and the electron energy E e in lab. determines the rapidity ζ via E e = m cosh ζ and Θ denotes the polar lab. angle of the out-photon. We call dσ/du a spectrally resolved observable.
Our note is organized as follows. In section II we briefly recall a few approximations of the laser beam. Section III is devoted to an analysis of the invariant differential cross section dσ/du and its dependence on the laser intensity parameter ξ in nonlinear Compton scattering. That is, we are going up and down on the vertical dashed line with label LUXE in Fig. 1 around the point χ = 1 or ξ = 1. The discussion section IV (i) relates the cross section to the probability and (ii) considers a folding model which uses the hard-photon spectrum emerging from Compton 1 We employ natural units with = c = 1. 2 Despite high intensities at XFELs, e.g. I → 10 22 W/cm 2 , the intensity parameter ξ = 7.5eV ω I 10 20 W/cm 2 [36] is small due to the high frequency, e.g. ω = 1 -25 keV.  [44,45]: Ee = 17.5 GeV) in combination with a high-intensity optical laser (we use ω = 1.55 eV as representative frequency). The vertical dotted line indicates a possible combination of the European XFEL (ω = 10 keV) with a laser-accelerated electron beam (Ee = 10 MeV) available in the high-energy density cave of the HIBEF collaboration [50]. The horizontal delineation line ξ = 1 is thought to highlight the onset of the strong-field region above.
scattering as seed for subsequent Breit-Wheeler pair production; a brief discussion of (iii) bandwidth effects relevant for sub-threshold trident pair production complements this section. We conclude in section V. The appendix A recalls a few basic elements of the one-photon Compton process.

II. LASER MODELS
In plane-wave approximation 3 the laser (circular polarization) can be described by the four-potential in axial gauge, A = (0, A), with where a 2 x = a 2 y = m 2 ξ 2 /e 2 ; the polarization vectors a x and a y are mutually orthogonal. We ignore a possible non-zero value of the carrier envelope phase and focus on symmetric envelope functions f (φ) w.r.t. the invariant phase φ = k · x. One may classify the such a model class as follows. -1) Laser pulses: lim φ→±∞ f (φ) = 0, -2) Monochromatic beam: f (φ) = 1, -3) Constant cross field (ccf): A = φ a x . The probabilities for the constant cross field option 3) coincide with certain limits of the plane-wave model 2) [58]; in [47] they are related to the large-ξ limit. Item 1) could be divided into several further sub-classes, such as 1.1): finite support region of the pulse, i.e. f (|φ| > φ pulse length ) = 0, φ pulse length < ∞, and 1.2): far-extended support region, i.e. lim φ→±∞ f (φ) → 0, together with non-zero carrier envelope phase, asymmetric pulse shape, frequency chirping, polarization gating etc. A specific class is 1.1) with flat-top section, e.g. a box envelope (cf. [59] for a recent explication) belonging to C 0 , or cos 2 ⊗ belonging to C 2 or the construction in [60] belonging to C ∞ . Examples for item 1.2) are Gauss, super-Gauss (employed in [61], for instance), symmetrized Fermi function [62,63], 1/ cosh etc. The monochromatic beam, item 2), corresponds formally to an infinitely long flat-top "pulse", abbreviated hereafter by IPA as acronym of infinite pulse approximation. It may be considered as special case of 1.1) with φ pulse length→∞ . FPA stands henceforth for the finite pulse-length plane-wave approximation.
To be specific, we employ here 1.2) with f (φ) = 1/ cosh(φ/πN ), where N characterizes the number of oscillations in that pulse interval, where f (φ) is significantly larger than zero (see [47,62,64] for the formalism), and IPA from 2). The laser model of class 1.1) is employed in subsection IV C.

III. COMPTON: DIFFERENTIAL INVARIANT CROSS SECTION
Let us consider the above pulse envelope function f (φ) = 1/ cosh(φ/(πN )) to elucidate the impact of a finite pulse duration and contrast it later on with the monochromatic laser beam model and some approximations thereof. Differential spectra dσ/du as a function of u are exhibited in Fig. 2 in the region u ≤ 3 for several values of ξ ≤ 1 for the FPA (dashed curves) and IPA (solid curves) models recalled below. This complements figures 1 -3 in [47]. One observes that the harmonic structures (which would become more severe for linear polarization, see [47]) fade away at larger values of u and ξ. Therefore, we are going to analyze that region in parameter space. There, IPA results represent reasonably well the trends of the more involved FPA calculations. (1) with f (φ) = 1/ cosh(φ/(πN )) for N = 10. The IPA results for a monochromatic laser beam are depicted by solid curves (with the limit dσIP A/du|u→0 = 2πα 2 /k · p independent of ξ). The pronounced harmonic structures around the Klein-Nishina (KN) edge uKN ≈ 0.416 are irrelevant for the subsequent discussion and, therefore, do not need a detailed representation. For χ = ξk · p/m 2 = 0.2078 ξ.
We consider in Fig. 3 only u > u KN = 2k · p/m 2 , since at the Klein-Nishina (KN) edge the harmonic structures become severe, as seen in Fig. 2. The striking feature seen in Fig. 3 is the pronounced ∩ shape, which we coin "rise and fall" of laser intensity effects. At small ξ, the realistic FPA (N = 10) results (asterisks) and the IPA model (solid curves) follow the same trends, consistent with Fig. 2. The large-ξ and large-ξ-large-u approximations (dashed and dotted curves) are not supposed to apply in that region. However, they become useful representatives at large ξ.

B. The rise
Some guidance of the rising parts of the FPA results in Fig. 3 can be gained by the monochromatic model. Casting Eq. (3) in the form with Ξ 2 = ξ 2 /(1 + ξ 2 ) and x n (u) = 2n u un (1 − u un ) and expanding in powers of Ξ yields for the first terms Due to the Heavyside Θ function in Eq. (2), the leadingorder power in Ξ depends on the value of u, e.g. for u < u 1 , the series starts with O(Ξ 0 ) and the coefficients of higher orders accordingly sum up. Higher values of u facilitate higher orders of the leading terms, i.e. the rise becomes steeper since the respective leading-order term is ∝ Ξ 2 u/u1 . This statement is based on the structures in Eqs. (10 -12), The series expansion of (9) in powers of Ξ ignores the sub-leading Ξ dependence in x n via u n = 2n k·p m 2 (1 − Ξ 2 ) but is suitable for k · p = const.
Analogously considerations apply to the pulse model, cf. section III.C in [62]. An essential role is played by the Fourier transform of the pulse envelope in the limit ξ 1. It bridges to the IPA for long pulses.

C. The fall
The maximum of the curves exhibited in Fig. 3 at u = O(1) is attained at ξ = O(1), but moves towards larger values of ξ with increasing values of u. Remarkably, the often discredited large-ξ and large-ξ-large-u approximations (dotted and dashed curves) deliver results in fair agreement with the IPA results (solid curves) for u > 1. That is, when being interested in the high-energy photon tails, the simple large-ξ-large-u formula Eq. (6) [58] represents a fairly accurate description supposed ξ is sufficiently large. Obviously, at ξ < 1 and u < 2, such an approximation fails quantitatively. (In particular, at u < 1 the harmonic structures in IPA become severe.) Nevertheless, some estimate of the maximum position is provided by ξ max ≈ 4 The asymptotic fall is governed from Eq. (6). We emphasize the same pattern of rise and fall of dσ/dω | ω =const , see Fig. 4, again for sufficiently large values of ω . The spectrally resolved differential cross section dσ/dω | ω =const is directly accessible in experiments. In the strong-field asymptotic region it displays a funneling behavior, i.e. the curves are squeezed into a narrow corridor already in the non-asymptotic ξ region, in contrast to dσ/du| u=const in Fig. 3.

A. Cross section vs. probability
The cross section σ and Ritus probability rate W are related as W = m 4 4πα ξ χ q0 σ with q 0 denoting the energy component of the quasi-momentum of the in-electron. This relation holds true for circular polarization and applies to respective differential quantities too. The different normalization modifies in particular the ξ dependence: The above emphasized "rise and fall" of dσ/du corresponds to a monotonously rising probability dW/du as a function of ξ. Having in mind Ritus' remark "the cross-sectional concept becomes meaningless" since, at ξ → ∞ we have σ → 0 while W remains finite [58], we turn in this sub-section to the probability. In doing so we remind the reader of the subtle Ritus notation W (χ) = 1 π π 0 dψP (χ sin ψ) in distinguishing the probabilities W and P .
We also stress the varying behavior of total (cf. appendix A in [65]) and differential probabilities. Equation (6) emerges as a certain limit of the constant cross field probability [58] where the prefactor reads V = αm 2 /πq 0 and F C is defined in Eq. (5). With respect to the argument z = (u/χ) 2/3 of the Airy functions in Eq. (5), the χ -u plane can be divided into the regions I (where u > χ) and II (where u < χ). Furthermore, the combination 1 + u suggests a splitting into u < 1 and u > 1 sub-domains. Picking up the leading-order terms in the related series expansions one arrives at It is the soft u-part II < of the differential probability in Eq. (14) which essentially determines the total probability upon u-integration, 4 i.e. the celebrated result P ∝ αχ 2/3 in Ritus notation [58] (stated as lim χ,ξ→∞ P ∝ αχ 2/3 m 2 /k · p in [37]), while the hard u-part I > in Eq. (13) is at the origin of Eq. (6) when converting to cross section. In other words, one has to distinguish the ξ (or χ) dependence either of integrated or differential observables. In relation to the LUXE plans [44] we mention the photon detector developments [66] which should enable in fact the access to the differential spectra.

B. Secondary processes: Breit-Wheeler
Instead transferring these considerations to the Breit-Wheeler (BW) process per se (cf. [67][68][69][70][71][72][73][74] and further citations below), we estimate now the BW pair production seeded by the hard photons from the above Compton (C) process. The following folding model C⊗BW is a pure two-step ansatz on the probabilistic level which mimics in a simple manner some part of the trident process 5 e − L → e − L + e − L + e + L by ignoring (i) the possible off-shell effects and non-transverse components of the intermediate photon (that would be the one-step contribution) and (ii) the anti-symmetrization of the two electrons in the final state (that would be the exchange contribution). Such an ansatz is similar to the one in [75], where however bremsstrahlung⊗BW has been analyzed. An analog approach has been elaborated in [76] for di-muon production.
Specifically, we consider the two-step cascade process where a GeV-Compton photon with energy ω is produced in the first step, and, in the next step, that GeVphoton interacts with the same laser field producing a BW-e + e − pair. We estimate the number of pairs produced in one pulse by where dΓ C (ω , t)/dω is the rate of photons per frequency interval dω (which corresponds to dP/du in IV A) emerging from Compton at time t ∈ [0, T C ], and Γ BW (ω , t ) is the rate of Breit-Wheeler pairs generated by a probe photon of frequency ω at lab. frame time-distance t ∈ 4 The value χ = 1 is special since the small-u region of II and the large-u region of I must be joint directly, II < ⊗ I > , while for χ < 1 the small-u region II and the small-u region of I must be joint followed by the large-u region of I, II < ⊗ I < ⊗ I > . In the opposite case of χ > 1, the small-u region of II must be joint with the large-u region of II followed by the large-u region of I, II < ⊗ II > ⊗ I > . This distinction becomes also evident when inspecting the differently shaped sections of the continuous curves dP/du as a function of u for several values of χ. 5 For recent work on the formalism, see [33][34][35][77][78][79] which advance earlier investigations [80,81].
[t, T BW ]. The underlying picture is that of an electron traversing a laser pulse in head-on geometry near to light cone. The passage time of an undisturbed electron would be N T 0 /2, T 0 = 2π/ω. Neglecting spatiotemporal variations within the pulse, the final formula becomes N e + e − = F t Ee 0 dω N e 0 dΓ C (ω ) dω Γ BW (ω ) upon the restriction ω < E e and F t = T C (T BW − T C /2). A crucial issue is the choice of the formation time(s) [80,81]. When gluing C⊗BW on the amplitude level, such a time appears linearly in the pair rate [82] 6 or quadratically in the net probability via overlap light-cone times (cf. [77,78] for instance). The time-ordered double integral over the C and BW probabilities, yielding the cascade approximation (cf. equation (43) in [78]), is analog to our above formula by folding two rates, which facilitates F t ∝ T 2 0 , if T C, BW ∝ T 0 . We attach to the Compton rate the number N e 0 = 6 × 10 9 of primary electrons per bunch.
In the numerical evaluation we employ the following convenient approximations: Note the rise of Γ BW and the fall of dΓ C /dω as a function of ω at fixed values of ξ and k · p and laser frequency ω, see Fig. 5. Increasing values of ξ lift dΓ C /dω somewhat and make it flatter in the intermediate-ω region, thus sharpening the drop at ω → E e . Remarkable is the strong impact of increasing ξ on the BW rate. One may replace Eq. (5) in (16) by the sum over harmonics expressed by Bessel functions [58] to get an improvement of accuracy in the small-ω region: 7 , q · k = k · p, and arguments z = 2nξ √ 1+ξ 2 1 un u(u n − u) of the Bessel func- 6 In laser pulses such an additional parameter is not needed [83]. 7 Strictly speaking, imposing a finite duration in the monochromatic laser beam model 2) turns it into a flat-top laser pulse model of class 1.1), exemplified in [59] and applied to the nonlinear Compton process. tions J n as well as u n = 2n k·p m 2 1 1+ξ 2 , u = nω−ω κn−nω+ω , κ n = nω− 1 2 me ζ + 1 2 m(1+ξ 2 )e −ζ , P n = m|n ω m −sinh ζ + ξ 2 2 e −ζ |. These equations make the dependence u(ω ) explicit and relate again the differential cross section dσ/dω with the differential rate dΓ/dω . Since the BW rate is exceedingly small at small ω (see Fig. 5), improvements of the Compton rate by catching the details of harmonic structures there are less severe for the pair number Eq. (15).
Equation (18) of the BW rate, however, is inappropriate at smaller values of ξ [75] and needs improvement. Instead of using the series expansion in Bessel functions [58], a convenient formula is × exp {−2n(a − tanh a} 1 + 2ξ 2 (2u − 1) sinh 2 a a tanh a with n min = 2ξ(1 + ξ 2 )/χ and u n = n/n min . This representation emerges from the largen approximation of Bessel functions, J n (z) ≈ exp {−n(a − tanh a)} / √ 2nπ tanh a and tanh a = 1 − z 2 /n 2 . In the large-ξ limit, one may replace the summation over n by an integration to arrive, via a double saddle point approximation, at the famous Ritus expression (18), which in turn is a complement of Eq. (6), see [58].
Numerical results are exhibited in Fig. 6 for E e = 45 GeV, 17.5 GeV and 8 GeV. One observes a stark rise of N e + e − up to ξ ∼ 4, which turns for larger values of ξ into a modest rise. To quantify that rise one can employ the ansatz N e + e − (ξ, E e ) = N e + e − 0 (E e ) ξ p(ξ,Ee) . Note that, by such a quantification of the ξ dependence, one gets rid of the normalization F t . For E e = 17.5 GeV we find p(ξ ≈ 1) ≈ 20 dropping to p(ξ ≈ 20) ≈ 2, see Fig. 7.  (15). For N e 0 = 6 × 10 9 electrons per bunch and per laser shot of duration N T0. The special normalization Ft = T 2 0 /2 is chosen, as realized by TBW = (T 2 0 + T 2 C )/(2TC ). The choice TC ≈ T0 facilitates a Compton spectrum dNC /dω = TC dΓC /dω which agrees, for ξ = O(1) and in the region ω < 10 GeV, with a bremsstrahlung spectrum generated by electrons of the same energy impinging on a foil with X/X0 = 0.01 [75]. It is the ξ dependence of the Compton spectrum (see Fig. 5) which makes the pair yield more rapidly rising with ξ than the pair yield of the bremsstrahlung⊗BW model in [75]. Larger values of E e reduce p, e.g. p(ξ ≈ 1)| Ee=45 GeV ≈ 10 in agreement with [82], while p(ξ ≈ 1)| Ee=8 GeV > 40. At ξ → 20, a universal value of p ≈ 2 seems to emerge. The extreme nonlinear sensitivity of the pair number on the laser intensity parameter ξ at ξ < 10, and in particular at ξ ≈ 1, points to the request of a refined and adequately realistic modeling beyond schematic approaches.

C. Bandwidth effects in linear trident
The threshold for linear trident, e − + γ(1.55 eV ) → e − + e − + e + , is at E e = 337 GeV, i.e. the LUXE kinematics is in the deep sub-threshold regime, where severe multi-photon effects build up the nonlinearity. However, also bandwidth effects can promote pair production in the sub-threshold region [84,85], even at ξ → 0. The key is the cross section of linear trident σ ppT ( √ŝ , ∆φ), which depends on the invariant energy √ŝ = m 1 + 2k · p/m 2 and the pulse duration ∆φ for a given laser pulse. The quantity σ ppT ( √ŝ , ∆φ) is exhibited in Fig. 8 as a function of √ŝ for several values of ∆φ. For definiteness, we employ the laser pulse model of class 1.1) with parameterization A = f ppT (φ) a x cos φ and envelope function f ppT = cos 2 πφ 2∆φ (φ, 2∆φ), i.e. the number of laserfield oscillations within the pulse is N = ∆φ/π. In contrast to the presentation above, we deploy results in this sub-section for linear polarization and the cos 2 envelope.
We employ the formalism in [79] and its numerical implementation, that is "pulsed perturbative QED" in the spirit of Furry picture QED in a series expansion in powers of ξ. Applied to trident, the pulsed perturbative trident (ppT) arises from the diagrams r r (double lines: Volkov wave functions, vertical lines: photon propagator) as leading-order term surviving ξ → 0. The scaled number of pairs is N tot /N e 0 = 2πΓ tot /π, where the probability rate is given by The chosen pulse implies ∞ −∞ dφ f 2 ppT (φ) = 3 2 ∆φ and N tot /N e 0 = σ ppT 3m 2 ξ 2 16α ∆φ. Note the ξ 2 dependence from the "target density" already entering in Eq. (22) (cf. [84,87] for analog relations). This is in contrast to Fig. 6, where genuine nonlinear effects are at work and mix with a stronger ξ dependence for C⊗BW. The ξ 2 dependence is characteristic for pair production by probe photons provided by an "external target", such as in the bremsstrahlung-laser configuration of LUXE, cf. [75].
For the long laser pulses used in E-144 [88][89][90], such bandwidth effects are less severe. /m for several pulse lengths ∆φ. In the IPA limit, i.e. a monochromatic laser beam or ∆φ → ∞, the threshold is at √ŝ = 3m. Above the threshold, the dots depict a few points (cf. table 1 in [86]) from perturbative trident without bandwidth effects. Bandwidth effects enable the pair production in the sub-threshold region √ŝ < 3m.
cesses within the essentially known formalism. In particular, we focus on the ξ dependence. For nonlinear Compton scattering, we point out that, in the non-asymptotic region χ = O(1), k · p m 2 , the spectrally resolved cross section dσ/du| u=const as a function of the laser intensity parameter ξ displays a pronounced ∩ shape for u > u KN (the "rise and fall"). This behavior is in stark contrast with the monotonously rising integrated probability lim χ,ξ→∞ P ∝ αχ 2/3 m 2 /k · p. That is, in different regions of the phase space, also different sensitivities of cross sections/rates/probabilities on the laser intensity impact can be observed. The soft (small-u) part, which determines the integrated cross section/probability, may behave completely different than the hard (large-u) contribution. 8 Transferred to certain approximations used in simulation codes such a behavior implies that one should test differentially where the conditions for applicability are ensured.
The hard photons, once produced by Compton process in a laser pulse, act as seeds for secondary processes, most notably the Breit-Wheeler process. A folding model of type Compton⊗Breit-Wheeler on the probabilistic level points to a rapidly increasing rate of e + e − production in the region ξ 4, when using parameters in reach of the planned LUXE set-up. The actual plans (see figure  2.10 in LUXE CDR [45]) uncover ξ = 2 (40 TW, 8 µm laser), 6 (40 TW, 3 µm) and 16 (300 TW, 3 µm), and E-320 envisages ξ = 10. The folding model may be utilized as reference to identify the occurrence of the wanted one-step trident process in this energy-intensity regime. Furthermore, bandwidth effects in the trident process are isolated by considering the weak-field regime ξ → 0.
Appendix A: Basics of nonlinear Compton process Following [92] we recall the basics of the underlying formalism of the nonlinear Compton process. Within the Furry picture the lowest-order, tree-level S matrix element for the one-photon (four-momentum k , fourpolarization ) decay of a laser-dressed electron e L in the background field (1) e L (p) → e L (p ) + γ(k , ) reads with suitable normalizations of the wave functions where the current J µ (x) =Ψ p γ µ Ψ p is built by the Volkov wave function Ψ p = E p u p exp{−ip · x} exp{−if p } (spin indices are suppressed) and its adjointΨ with Ritus We employ Feynman's slash notation and denote scalar products by the dot between four-vectors; u p is the free Dirac bi-spinor. Exploiting the symmetry of the background field, A(φ = k · x), Eq. (A1) can be manipulated (cf. [92] for details) to arrive at where ≡ (k − + p − − p − )/k − = k · p/k · p accomplishes the balance equation p + k − p − k = 0. (See [62] for a formulation with S f i = −ie(2π) 4 d 2π δ (4) (p + k − p − k ) M( ).) Light-cone coordinates are useful here, e.g. k − = k 0 − k 3 , k + = k 0 + k 3 , k ⊥ = (k 1 , k 2 ), and k = (k + , k ⊥ ). Imposing gauge invariance yields the matrix element with the pieces of the electron current The phase integrals S (i) are the remainders of the integration d 4 x = dφ dx − d 2 x ⊥ /k − in Eq. (A1): For a few special non-unipolar (plane-wave) fields and their envelopes f (φ), the phase integrals can be processed exactly by analytic means [93,94], but in general a numerical evaluation is needed. The very special IPA case of f (φ) = 1 allows for a simple representation of f p (φ) with subsequent decomposition of S (i) into Bessel functions, yielding final expressions as in Eqs. (2,20).
These limits are at the heart of the "rise and fall". The differential emission probability per in-electron and per laser pulse follows from (A2) by partial integration over the out-phase space, dP dω dΩ = e 2 ω 64π 3 k · p k · p |M| 2 , and may be transformed to other coordinates, e.g. u or etc. Having in mind the IPA limit, one should turn to the dimensionless differential rate [62] dΓ C dω dΩ = e 2 ω 32π 2 q 0 k · p |M| 2 . (A9) Spin averaging of the in-electron, and spin summation of the out-electron and summation over the out-photon polarizations leads to |M| 2 , unless one is interested in polarization effects as in [19,20,95]. The above expressions (A3 -A7) can be further processed for special field envelopes, or |M| 2 is numerically accessible, via Eq. (A3), as mod-squared sum of complex number products provided by Eqs. (A4 -A7) which need afterwards explicit (numerical) spin and polarization summation/averaging to arrive at |M| 2 . The cross section is obtained by normalization on the integrated laser photon flux: dσ = dΓ C q0 k·p ω n L with n L = m 2 e 2 ξ 2 ωN L , where N L = 1 (IPA, quasi-momentum q 0 ) or N L = 1 2π ∞ −∞ dφ f (φ) (FPA, q 0 ≡ p 0 ). The circularly polarized laser background (1) is supposed in these relations.