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

U. Hernandez Acosta1,2,3, A. I. Titov4, B. Kampfer1,2

1Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany2Institut fur Theoretische Physik, TU Dresden, 01062 Dresden, Germany

3Center for Advanced Systems Understanding, Untermarkt 20, 02826 Gorlitz, Germany and4Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia

(Dated: May 26, 2021)

The spectrally resolved differential cross section of Compton scattering, dσ/dω′|ω′=const, risesfrom small towards larger laser intensity parameter ξ, reaches a maximum, and falls towards theasymptotic strong-field region. Expressed by invariant quantities: dσ/du|u=const rises from smalltowards larger values of ξ, reaches a maximum at ξmax = 4

9Kum2/k · p, K = O(1), and falls

at ξ > ξmax like ∝ ξ−3/2 exp(− 2um2

3ξ k·p

)at u ≥ 1. [The quantity u is the Ritus variable related

to the light-front momentum-fraction s = (1 + u)/u = k · k′/k · p of the emitted photon (four-momentum k′, frequency ω′), and k · p/m2 quantifies the invariant energy in the entrance channelof electron (four-momentum p, mass m) and laser (four-wave vector k).] Such a behavior of adifferential observable is to be contrasted with the laser intensity dependence of the total probability,limχ=ξk·p/m2,ξ→∞ P ∝ αχ2/3m2/k · p, which is governed by the soft spectral part.

We combine the hard-photon yield from Compton with the seeded Breit-Wheeler pair productionin a folding model and obtain a rapidly increasing e+e− pair number at ξ . 4. Laser bandwidtheffects are quantified in the weak-field limit of the related trident pair production.

PACS numbers: 12.20.Ds, 13.40.-f, 23.20.NxKeywords: non-linear Compton scattering, nonlinear Breit-Wheeler pair production, strong-field QED

I. INTRODUCTION

Quantum Electro-Dynamics (QED) as pillar of thestandard model (SM) of particle physics possesses apositive β function [1] which makes the running cou-pling strength α(s) increasingly with increasing en-ergy/momentum scale s [2]. In contrast, QuantumChromo-Dynamics (QCD) as another SM pillar pos-sesses a negative β function due to the non-Abeliangauge group [1], giving rise to the asymptotic freedom,lims→∞ αQCD(s) → 0, i.e. QCD has a truly perturba-tive limit. In contrast, lims→0 α(s)→ 1/137.0359895(61)is not such a strict limit, nevertheless, QED predic-tions/calculations of some observables agree with mea-surements within 13 digits, see [3–5] for some examples.The situation in QED becomes special when consider-ing processes in external (or background) fields: One canresort to the Furry (or bound-state) picture, where the(tree-level) interactions of an elementary charge (e.g. anelectron) with the background are accounted for in allorders, and the interactions with the quantized photonfield remains perturbatively in powers of α. However,the Ritus-Narozhny (RN) conjecture [6–10] argues thatthe effective coupling becomes αχ2/3, meaning that theFurry picture expansion beaks down at αχ2/3 > 1 [11–14] (for the definition of χ see below) and one enters agenuinely non-perturbative regime. The latter requiresadequate calculation procedures, as the lattice regular-ized approaches, which are standard since many years inQCD, e.g. in evaluations of observables in the soft sec-tor where αQCD > 1, cf. [15]. (In QED itself, an analogsituation is meet in the Coulomb field of nuclear systemswith proton numbers Z > Zcrit ≈ 173: if αZcrit > 1 the

QED vacuum beak-down sets in; cf. [16] for the actualstatus of that field).

With respect to increasing laser intensities the questfor the possible break-down of the Furry picture expan-sion in line with the RN conjecture becomes, besides itsprincipal challenge, also of “practical” interest, whetherone can explore experimentally this yet uncharted regimeof QED. (For other configurations, e.g. beam-beam in-teractions, cf. [17]). A prerequisite would be to find ob-servables which display the typical dependence ∝ αχ2/3,where we denote by α the above quoted fine-structureconstant at s → 0. In doing so we resort here to thelowest-order QED processes, that is nonlinear Comptonand nonlinear Breit-Wheeler. Both processes seem to beinvestigated theoretically in depth in the past, however,enjoy currently repeated re-considerations w.r.t. refine-ments [18–23], establishing approximation schemes [24–28] to be implemented in simulation codes [29–31] or touse them as building blocks in complex processes [32],with the starting point at trident [33–35].

Beginning with Compton scattering (p and p′ are thein- and out-electron four-momenta, k the laser four-momentum, and k′ the out-photon’s four-momentum, re-spectively) 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 fre-quency; −|e| and m stand for the electron charge and

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mass, respectively, and e2/4π = α,1

- (ii) the available energy squared: k · p/m2 = (s/m2 −1)/2 with s as Mandelstam variable,- (iii) the quantum nonlinearity parameter: χ = ξk·p/m2.The latter quantity is often considered as the crucial pa-rameter since in some limits it determines solely the prob-ability of certain processes. χ plays also a prominent rolein the above mentioned discussion of the RN conjecture,where the Furry picture expansion of QED is argued tobreak down for αχ2/3 > 1. References [37, 38] point outthat the large-χ limits, facilitated by either large ξ (thehigh-intensity limit) or large k · p/m2 (the high-energylimit), are distinctively different, with implications forapproximation schemes in simulation codes.

Figure 1 exhibits a few selected curves χ = const overthe ξ vs. k ·p/m2 landscape to illustrate the current situa-tion w.r.t. facilities where laser beams and electron beamsare (or can be) combined. One has to add the options atE-320 at FACET-II/SLAC [39, 40] and electron beamswhich are laser-accelerated to GeV scales, e.g. [41–43].

Usually, ξ = 1 is said to mark the the onset of strong-field effects, and the corresponding processes at ξ > 1 aretermed by the attribute “nonlinear”. In this respect, theparameters provided by LUXE [44, 45] and FACET-II[39, 40] are interesting: χ = O(1) and above and ξ > 1as well.2 In the following we consider the LUXE kinemat-ics (see Fig. 1), k · p/m2 ≈ ωEe(1 − cos Θ) = 0.2078 inhead-on collisions (cos Θ = −1). Our aim is to quantify asimple observable as a function of ξ. To be specific, we se-lect the invariant differential cross section dσ/du, where uis the light-cone momentum-transfer of the in-electron tothe out-photon, related to light-front momentum-fraction

of the out-photon s = u1+u = k·k′

k·p . (The mapping

u 7→ ω′ and dσ/du 7→ dσ/dω′ is discussed in [46, 47].)To make the meaning of the Ritus variable u more ex-

plicit let us mention the relation u = e−ζν′(1−cos Θ′)1−e−ζν′(1−cos Θ′)

,

where ν′ ≡ ω′/m, and the electron energy Ee in lab. de-termines the rapidity ζ via Ee = m cosh ζ and Θ′ denotesthe polar lab. angle of the out-photon. We call dσ/du aspectrally resolved observable.

Our note is organized as follows. In section II we brieflyrecall a few approximations of the laser beam. Section IIIis devoted to an analysis of the invariant differential crosssection dσ/du and its dependence on the laser intensityparameter ξ in nonlinear Compton scattering. That is,we are going up and down on the vertical dashed line withlabel LUXE in Fig. 1 around the point χ = 1 or ξ = 1.The discussion section IV (i) relates the cross section tothe probability and (ii) considers a folding model whichuses the hard-photon spectrum emerging from Compton

1 We employ natural units with ~ = c = 1.2 Despite high intensities at XFELs, e.g. I → 1022 W/cm2, the

intensity parameter ξ = 7.5eVω

√I

1020W/cm2 [36] is small due to

the high frequency, e.g. ω = 1 - 25 keV.

1 E - 4 0 . 0 0 1 0 . 0 1 0 . 1 1 1 0 1 0 00 . 0 1

0 . 1

1

1 0

1 0 0

ξ

p k / m 2

H Z D R E L I L U X E

χ = 1 010 . 1

0 . 0 10 . 0 0 1

α χ2/3 = 1

H I B E F

FIG. 1: Curves of χ = const over the ξ vs. k · p/m2 planefor χ = 0.001, 0.01, 0.1, 1 and 10. The double-line depictsthe curve αχ2/3 = 1. Vertical thin lines are for maximumvalues of k · p/m2 in reach at various electron accelerators(HZDR [48]: Ee = 33 MeV, ELI [49]: Ee = 600 MeV, LUXE[44, 45]: Ee = 17.5 GeV) in combination with a high-intensityoptical laser (we use ω = 1.55 eV as representative frequency).The vertical dotted line indicates a possible combination ofthe European XFEL (ω = 10 keV) with a laser-acceleratedelectron beam (Ee = 10 MeV) available in the high-energydensity cave of the HIBEF collaboration [50]. The horizontaldelineation line ξ = 1 is thought to highlight the onset of thestrong-field region above.

scattering as seed for subsequent Breit-Wheeler pair pro-duction; a brief discussion of (iii) bandwidth effects rel-evant for sub-threshold trident pair production comple-ments this section. We conclude in section V. The ap-pendix A recalls a few basic elements of the one-photonCompton process.

II. LASER MODELS

In plane-wave approximation3 the laser (circular polar-ization) can be described by the four-potential in axial

gauge, A = (0, ~A), with

~A = f(φ) (~ax cosφ+ ~ay sinφ) (1)

where ~a2x = ~a2

y = m2ξ2/e2; the polarization vectors ~axand ~ay are mutually orthogonal. We ignore a possible

3 Due to the high symmetry, the exact solutions of the Dirac equa-tion in a plane-wave background are in a comfortably compactform enabling an easy processing in evaluations of matrix ele-ments. Without such a high symmetry, much more attemptsare required [51–53]. For some useful parameterizations of laserbeams, see [54–56] and further citations therein. The relevanceof the Fourier-zero mode of (non-)unipolar planar fields is men-tioned in [57].

3

non-zero value of the carrier envelope phase and focus onsymmetric envelope functions f(φ) w.r.t. the invariantphase φ = k · x. One may classify the such a model classas follows.- 1) Laser pulses: limφ→±∞ f(φ) = 0,- 2) Monochromatic beam: f(φ) = 1,

- 3) Constant cross field (ccf): ~A = φ~ax.The probabilities for the constant cross field option 3)coincide with certain limits of the plane-wave model2) [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| > φpulse length) = 0, φpulse length < ∞, and 1.2):far-extended support region, i.e. limφ→±∞ f(φ)→ 0, to-gether with non-zero carrier envelope phase, asymmet-ric pulse shape, frequency chirping, polarization gatingetc. A specific class is 1.1) with flat-top section, e.g.a box envelope u (cf. [59] for a recent explication) be-longing to C0, or cos2⊗u belonging to C2 or the con-struction in [60] belonging to C∞. Examples for item1.2) are Gauss, super-Gauss (employed in [61], for in-stance), symmetrized Fermi function [62, 63], 1/ coshetc. The monochromatic beam, item 2), corresponds for-mally to an infinitely long flat-top “pulse”, abbreviatedhereafter by IPA as acronym of infinite pulse approx-imation. It may be considered as special case of 1.1)with φpulse length→∞. FPA stands henceforth for the fi-nite 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 os-cillations in that pulse interval, where f(φ) is significantlylarger than zero (see [47, 62, 64] for the formalism), andIPA from 2). The laser model of class 1.1) is employedin subsection IV C.

III. COMPTON: DIFFERENTIAL INVARIANTCROSS SECTION

Let us consider the above pulse envelope functionf(φ) = 1/ cosh(φ/(πN)) to elucidate the impact of afinite pulse duration and contrast it later on with themonochromatic laser beam model and some approxima-tions thereof. Differential spectra dσ/du as a function ofu are exhibited in Fig. 2 in the region u ≤ 3 for severalvalues of ξ ≤ 1 for the FPA (dashed curves) and IPA(solid curves) models recalled below. This complementsfigures 1 – 3 in [47]. One observes that the harmonicstructures (which would become more severe for linearpolarization, see [47]) fade away at larger values of u andξ. Therefore, we are going to analyze that region in pa-rameter space. There, IPA results represent reasonablywell the trends of the more involved FPA calculations.

FIG. 2: Invariant differential cross section dσ(u, ξ, k ·p/m2)/du as a function of u for several values of ξ, ξ = 0.2(magenta), 0.4 (blue), 0.7 (green) and 1 (black). The dashedcurves (label FPA) are for pulses according to Eq. (1) withf(φ) = 1/ cosh(φ/(πN)) for N = 10. The IPA results fora monochromatic laser beam are depicted by solid curves(with the limit dσIPA/du|u→0 = 2πα2/k · p independent ofξ). The pronounced harmonic structures around the Klein-Nishina (KN) edge uKN ≈ 0.416 are irrelevant for the subse-quent discussion and, therefore, do not need a detailed repre-sentation. For χ = ξk · p/m2 = 0.2078 ξ.

A. Numerical results

In particular, we consider now dσ(u, ξ, k · p/m2)/du asa function of ξ for several constant values of u, u = 0.5,1, 2, 4 and 8, for k · p/m2 = 0.2078 – a value whichis motivated by the LUXE opportunities [44, 45]. Thesolid, dotted and dashed curves in Fig. 3 are based oneasily accessible models [58]:• monochromatic model (IPA) (cf. equations (15, 16) in[47]):

dσIPAdu

=2πα2

k · p1

(1 + u)2

∞∑n=1

Θ(un − u)Fn(zn), (2)

Fn = − 2

ξ2J2n(zn) (3)

+ A(J2n+1(zn) + J2

n−1(zn)− 2J2n(zn)

)with A = 1 + u2

2(1+u) , zn(u, un) = 2nξ√1+ξ2

√uun

(1− uun

),

and un = 2nk·pm2(1+ξ2) (Jn’s denote Bessel functions of first

kind);• IPA–large-ξ approximation (cf. equation (21) in [47]):

dσlarge−ξdu

= −4πα2

m2ξ

1

χ

1

(1 + u)2FC (4)

FC =

∫ ∞z

dyΦ(y) +2

z

[1 +

u2

2(1 + u)

]Φ′(z)(5)

where Φ(z) and Φ(z)′ stand for the Airy function andits derivative with arguments z = (u/χ)2/3 where χ =

4

FIG. 3: Invariant differential cross section dσ(u = const, ξ, k ·p/m2)/du as a function of ξ for several values of u, u =0.5, 1, 2, 4 and 8; for k · p/m2 = 0.2078. The asterisksdepict results of short laser pulses with envelope functionf(φ) = 1/ cosh(φ/(πN)) for N = 10, dσFPA/du, Eq. (7).The solid curves are for the monochromatic laser beam modeldσIPA/du, Eq. (2), while dashed (dotted) curves are based ondσlarge−ξ/du, Eq. (4) (dσlarge−ξ, large−u/du, Eq. (6)).

ξk · p/m2;• large-ξ-large-u approximation (cf. equation (22) in[47]):

dσlarge−ξ, large−udu

=2√πα2

m2ξχ−1/2u−3/2 exp

(−2u

3χ

).

(6)The asterisks in Fig. 3 depict results of the pulse model

(FPA) described in [47, 62] (cf. equations (5 - 14) in [47]):

dσFPAdu

=α2

k · p1

(1 + u)2

∞∫0

d`Θ

(um2

2k · p − `)w(`),(7)

w(`, z) =

∫ 2π

0

dφe′

[− 2

ξ2|Y`(z`)|2 (8)

+A(|Y`−1(z`)|2 + |Y`+1(z`)|2 − 2ReY`(z`)X

∗` (z`))

)]with A as above, z`(u, `) = 2`ξ

√uu`

(1− uu`

) and u` =

2`k·pm2 . The basic functions Y`, Y`, X` and their φe′ depen-

dence are spelled out in [47, 62]; they depend cruciallyon the pulse envelope function f(φ). (Due to numericalaccuracy reasons in integrating highly oscillating func-tions, these evaluations are constrained presently to nottoo large values of ξ.) The common basis of these modelsis sketched in Appendix A.

We consider in Fig. 3 only u > uKN = 2k · p/m2, sinceat the Klein-Nishina (KN) edge the harmonic structuresbecome severe, as seen in Fig. 2. The striking featureseen in Fig. 3 is the pronounced ∩ shape, which wecoin “rise and fall” of laser intensity effects. At smallξ, the realistic FPA (N = 10) results (asterisks) andthe IPA model (solid curves) follow the same trends,consistent with Fig. 2. The large-ξ and large-ξ-large-uapproximations (dashed and dotted curves) are notsupposed to apply in that region. However, they becomeuseful representatives at large ξ.

B. The rise

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

Fn = −2(1− Ξ2)

Ξ2J2n(Ξxn) (9)

+ A(J2n+1(Ξxn) + J2

n−1(Ξxn)− 2J2n(Ξxn)

)with Ξ2 = ξ2/(1 + ξ2) and xn(u) = 2n

√uun

(1− uun

) and

expanding in powers of Ξ yields for the first terms

F1 =1

2(2A− x2

1)− x21

8(8A− x2

1 − 4)Ξ2 +x4

1

384(90A− 5x2

1 − 48)Ξ4 +O(Ξ6), (10)

F2 =x2

2

32(8A− x2

2)Ξ2 − x42

192(18A− x2

2 − 6)Ξ4 +O(Ξ6), (11)

F3 =x4

3

1152(18A− x2

3)Ξ4 +O(Ξ6). (12)

Due to the Heavyside Θ function in Eq. (2), the leading-order power in Ξ depends on the value of u, e.g. for u <u1, the series starts with O(Ξ0) and the coefficients ofhigher orders accordingly sum up. Higher values of u

facilitate higher orders of the leading terms, i.e. the risebecomes steeper since the respective leading-order termis ∝ Ξ2bu/u1c. This statement is based on the structuresin Eqs. (10 - 12), suggesting Fn =

∑i=n−1 F

(2i)n Ξ2i for

5

the first terms F(2i)n , and

∑∞n=1 Θ(un−u)Fn =

∑∞nmin

Fnwith nmin = 1 + bu/u1c. (b·c is the floor operation.)

The series expansion of (9) in powers of Ξ ignores the

sub-leading Ξ dependence in xn via un = 2nk·pm2 (1 − Ξ2)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 bythe Fourier transform of the pulse envelope in the limitξ 1. It bridges to the IPA for long pulses.

C. The fall

The maximum of the curves exhibited in Fig. 3 atu = O(1) is attained at ξ = O(1), but moves towardslarger values of ξ with increasing values of u. Remark-ably, the often discredited large-ξ and large-ξ–large-u ap-proximations (dotted and dashed curves) deliver resultsin fair agreement with the IPA results (solid curves) foru > 1. That is, when being interested in the high-energyphoton tails, the simple large-ξ–large-u formula Eq. (6)[58] represents a fairly accurate description supposed ξ issufficiently large. Obviously, at ξ < 1 and u < 2, suchan approximation fails quantitatively. (In particular, atu < 1 the harmonic structures in IPA become severe.)Nevertheless, some estimate of the maximum position isprovided by ξmax ≈ 4

9um2/k · p yielding dσ/du(ξmax) ≈

2√πα2

m2k·pm2

(32

)3e−3/2 u−3. The asymptotic fall is governed

by dσdu ∝ 1

m2m2

k·pξ−3/2u−3/2 exp

(− 2um2

3ξk·p

)from Eq. (6).

We emphasize the same pattern of rise and fall ofdσ/dω′|ω′=const, see Fig. 4, again for sufficiently largevalues of ω′. The spectrally resolved differential crosssection dσ/dω′|ω′=const is directly accessible in experi-ments. In the strong-field asymptotic region it displaysa funneling behavior, i.e. the curves are squeezed into anarrow corridor already in the non-asymptotic ξ region,in contrast to dσ/du|u=const in Fig. 3.

IV. DISCUSSION

A. Cross section vs. probability

The cross section σ and Ritus probability rate W are

related as W = m4

4παξ χq0σ with q0 denoting the energy

component of the quasi-momentum of the in-electron.This relation holds true for circular polarization and ap-plies to respective differential quantities too. The dif-ferent normalization modifies in particular the ξ depen-dence: The above emphasized “rise and fall” of dσ/ducorresponds to a monotonously rising probability dW/duas a function of ξ. Having in mind Ritus’ remark “thecross-sectional concept becomes meaningless” since, atξ → ∞ we have σ → 0 while W remains finite [58],we turn in this sub-section to the probability. In do-ing so we remind the reader of the subtle Ritus notation

W (χ) = 1π

∫ π0dψP (χ sinψ) in distinguishing the proba-

bilities W and P .

We also stress the varying behavior of total (cf. ap-pendix A in [65]) and differential probabilities. Equation(6) emerges as a certain limit of the constant cross field

probability [58] dP (χ,u)du = −V (1 + u)−2 FC(z(χ, u), u)

where the prefactor reads V = αm2/πq0 and FC is de-fined in Eq. (5). With respect to the argument z =(u/χ)2/3 of the Airy functions in Eq. (5), the χ - u planecan be divided into the regions I (where u > χ) andII (where u < χ). Furthermore, the combination 1 + usuggests a splitting into u < 1 and u > 1 sub-domains.Picking up the leading-order terms in the related seriesexpansions one arrives at

dP

du|uχ =

V

2√πχ1/2 exp

(−2u

3χ

)×u−1/2 for u 1, (I<)u−3/2 for u 1, (I>)

(13)

dP

du|uχ = −V Φ′(0)χ2/3

×

2u−2/3 for u 1, (II<)u−5/3 for u 1. (II>)

(14)

It is the soft u-part II< of the differential probabil-ity in Eq. (14) which essentially determines the total

FIG. 4: The same as in Fig. 3 but for dσ/dω′ as a function oflaser intensity parameter ξ for several constant values of ω′.

6

probability upon u-integration,4 i.e. the celebrated resultP ∝ αχ2/3 in Ritus notation [58] (stated as limχ,ξ→∞ P ∝αχ2/3m2/k · p in [37]), while the hard u-part I> inEq. (13) is at the origin of Eq. (6) when converting tocross section. In other words, one has to distinguish theξ (or χ) dependence either of integrated or differentialobservables. In relation to the LUXE plans [44] we men-tion the photon detector developments [66] which shouldenable 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. [67–74] and further ci-tations below), we estimate now the BW pair productionseeded by the hard photons from the above Compton (C)process. The following folding model C⊗BW is a puretwo-step ansatz on the probabilistic level which mimicsin a simple manner some part of the trident process5

e−L → e−L′+e−L

′′+e+

L by ignoring (i) the possible off-shelleffects and non-transverse components of the intermedi-ate photon (that would be the one-step contribution) and(ii) the anti-symmetrization of the two electrons in thefinal state (that would be the exchange contribution).Such an ansatz is similar to the one in [75], where how-ever bremsstrahlung⊗BW has been analyzed. An analogapproach has been elaborated in [76] for di-muon pro-duction.

Specifically, we consider the two-step cascade processwhere a GeV-Compton photon with energy ω′ is pro-duced in the first step, and, in the next step, that GeV-photon interacts with the same laser field producing aBW-e+e− pair. We estimate the number of pairs pro-duced in one pulse by

Ne+e− =

∞∫0

dω′TC∫0

dtdΓC(ω′, t)

dω′

TBW∫t

dt′ ΓBW (ω′, t′),

(15)where dΓC(ω′, t)/dω′ is the rate of photons per frequencyinterval dω′ (which corresponds to dP/du in IV A) emerg-ing from Compton at time t ∈ [0, TC ], and ΓBW (ω′, t′)is the rate of Breit-Wheeler pairs generated by a probephoton of frequency ω′ at lab. frame time-distance t′ ∈

4 The value χ = 1 is special since the small-u region of II and thelarge-u region of I must be joint directly, II< ⊗ I>, while forχ < 1 the small-u region II and the small-u region of I mustbe joint followed by the large-u region of I, II< ⊗ I< ⊗ I>. Inthe opposite case of χ > 1, the small-u region of II must bejoint with the large-u region of II followed by the large-u regionof I, II< ⊗ II> ⊗ I>. This distinction becomes also evidentwhen inspecting the differently shaped sections of the continuouscurves dP/du as a function of u for several values of χ.

5 For recent work on the formalism, see [33–35, 77–79] which ad-vance earlier investigations [80, 81].

[t, TBW ]. The underlying picture is that of an electrontraversing a laser pulse in head-on geometry near tolight cone. The passage time of an undisturbed elec-tron would be NT0/2, T0 = 2π/ω. Neglecting spatio-temporal variations within the pulse, the final formula

becomes Ne+e− = Ft∫ Ee

0dω′Ne

0dΓC(ω′)dω′ ΓBW (ω′) upon

the restriction ω′ < Ee and Ft = TC(TBW − TC/2). Acrucial issue is the choice of the formation time(s) [80, 81].When gluing C⊗BW on the amplitude level, such a timeappears linearly in the pair rate [82]6 or quadraticallyin the net probability via overlap light-cone times (cf.[77, 78] for instance). The time-ordered double integralover the C and BW probabilities, yielding the cascadeapproximation (cf. equation (43) in [78]), is analog toour above formula by folding two rates, which facilitatesFt ∝ T 2

0 , if TC,BW ∝ T0. We attach to the Comptonrate the number Ne

0 = 6 × 109 of primary electrons perbunch.

In the numerical evaluation we employ the followingconvenient approximations:

dΓCdω′

= −αm2

π E2e

FC(z, u), (16)

ΓBW (ω′) =αm2

ω′FBW , (17)

FBW =

√33

29κ exp

[− 8

3κ

(1− 1

15ξ2

)], (18)

where u = κ/(χ− κ), χ = (k · p)ξ/m2, κ = (k · k′)ξ/m2,and FC(z, u) with z = (u/χ)2/3 is defined in Eq. (5).Note the rise of ΓBW and the fall of dΓC/dω

′ as a functionof ω′ at fixed values of ξ and k · p and laser frequency ω,see Fig. 5. Increasing values of ξ lift dΓC/dω

′ somewhatand make it flatter in the intermediate-ω′ region, thussharpening the drop at ω′ → Ee. Remarkable is thestrong impact of increasing ξ on the BW rate.

One may replace Eq. (5) in (16) by the sum over har-monics expressed by Bessel functions [58] to get an im-provement of accuracy in the small-ω′ region:7

dΓIPAC

dω′=

dσIPAC

dω′ξ2m2

4πα

k · pq0

, (19)

dσIPAC

dω′=

πr2em

2

ξ2k · p∞∑n=1

Θ(un − u)

Pn

[−4J2

n (20)

+ ξ2(2 +u2

1 + u)(J2n−1 + J2

n+1 − 2J2n

)]with q0 =

√E2e −m2 +βpω, βp = ξ2 m2

2q·k , q ·k = k ·p, and

arguments z = 2nξ√1+ξ2

1un

√u(un − 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 monochro-

matic laser beam model 2) turns it into a flat-top laser pulsemodel of class 1.1), exemplified in [59] and applied to the non-linear Compton process.

7

FIG. 5: The BW rate ΓBW from Eqs. (17, 21) (solid curves)and the dimensionless differential C rate dΓC/dω

′ (dashedcurves) from Eqs. (5, 16) as a function of ω′ for ξ = 1 (red)and 3 (blue).

tions Jn as well as un = 2nk·pm21

1+ξ2 , u = nω−ω′κn−nω+ω′ , κn =

nω− 12me

ζ+ 12m(1+ξ2)e−ζ , Pn = m|n ωm−sinh ζ+ ξ2

2 e−ζ |.

These equations make the dependence u(ω′) explicit andrelate again the differential cross section dσ/dω′ with thedifferential rate dΓ/dω′. Since the BW rate is exceedinglysmall at small ω′ (see Fig. 5), improvements of the Comp-ton rate by catching the details of harmonic structuresthere are less severe for the pair number Eq. (15).

Equation (18) of the BW rate, however, is inappropri-ate at smaller values of ξ [75] and needs improvement.Instead of using the series expansion in Bessel functions[58], a convenient formula is

FBW =1

4π

∞∑n=nmin

∫ un

1

du

u3/2√

1− u (21)

× exp −2n(a− tanh a 1 + 2ξ2(2u− 1) sinh2 a

a tanh a

with nmin = 2ξ(1 + ξ2)/χ and un = n/nmin.This representation emerges from the large-n approximation of Bessel functions, Jn(z) ≈exp −n(a− tanh a) /

√2nπ tanh a and tanh a =√

1− z2/n2. In the large-ξ limit, one may replacethe summation over n by an integration to arrive, viaa double saddle point approximation, at the famousRitus expression (18), which in turn is a complement ofEq. (6), see [58].

Numerical results are exhibited in Fig. 6 for Ee =45 GeV, 17.5 GeV and 8 GeV. One observes a stark rise of

Ne+e− up to ξ ∼ 4, which turns for larger values of ξ intoa modest rise. To quantify that rise one can employ the

ansatz Ne+e−(ξ, Ee) = Ne+e−

0 (Ee) ξp(ξ,Ee). Note that,

by such a quantification of the ξ dependence, one getsrid of the normalization Ft. For Ee = 17.5 GeV we findp(ξ ≈ 1) ≈ 20 dropping to p(ξ ≈ 20) ≈ 2, see Fig. 7.

FIG. 6: The yield of e+e− pairs as a function of ξ for elec-tron energies 45 GeV (magenta dash-dotted curve), 17.5 GeV(blue solid curve) and 8 GeV (red dashed curve) accord-ing to the probabilistic folding model C⊗BW Eq. (15). ForNe

0 = 6× 109 electrons per bunch and per laser shot of dura-tion NT0. The special normalization Ft = T 2

0 /2 is chosen, asrealized by TBW = (T 2

0 +T 2C)/(2TC). The choice TC ≈ T0 fa-

cilitates a Compton spectrum dNC/dω′ = TC dΓC/dω

′ whichagrees, for ξ = O(1) and in the region ω′ < 10 GeV, with abremsstrahlung spectrum generated by electrons of the sameenergy impinging on a foil with X/X0 = 0.01 [75]. It is theξ dependence of the Compton spectrum (see Fig. 5) whichmakes the pair yield more rapidly rising with ξ than the pairyield of the bremsstrahlung⊗BW model in [75].

FIG. 7: The power p as a function of ξ for Ee = 45 GeV(magenta dash-dotted curve), 17.5 GeV (blue solid curve) and8 GeV (red dashed curve). The results exhibited in Fig. 6 are

described by Ne+e−(ξ, Ee) = Ne+e−0 (Ee) ξ

p(ξ,Ee).

Larger values of Ee reduce p, e.g. p(ξ ≈ 1)|Ee=45 GeV ≈ 10in agreement with [82], while p(ξ ≈ 1)|Ee=8 GeV > 40. Atξ → 20, a universal value of p ≈ 2 seems to emerge. Theextreme nonlinear sensitivity of the pair number on thelaser intensity parameter ξ at ξ < 10, and in particular at

8

ξ ≈ 1, points to the request of a refined and adequatelyrealistic 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 Ee = 337 GeV, i.e. the LUXE kine-

matics is in the deep sub-threshold regime, where severemulti-photon effects build up the nonlinearity. However,also bandwidth effects can promote pair production inthe sub-threshold region [84, 85], even at ξ → 0. The key

is the cross section of linear trident σppT (√s,∆φ), which

depends on the invariant energy√s = m

√1 + 2k · p/m2

and the pulse duration ∆φ for a given laser pulse. Thequantity σppT (

√s,∆φ) is exhibited in Fig. 8 as a func-

tion of√s for several values of ∆φ. For definiteness, we

employ the laser pulse model of class 1.1) with param-

eterization ~A = fppT (φ)~ax cosφ and envelope function

fppT = cos2(πφ

2∆φ

)u (φ, 2∆φ), i.e. the number of laser-

field oscillations within the pulse is N = ∆φ/π. In con-trast to the presentation above, we deploy results in thissub-section for linear polarization and the cos2 envelope.

We employ the formalism in [79] and its numerical im-plementation, that is “pulsed perturbative QED” in thespirit of Furry picture QED in a series expansion in pow-ers of ξ. Applied to trident, the pulsed perturbative tri-

dent (ppT) arises from the diagramsHHHH

e+e−rr − HHHH

e−e+rr

(double lines: Volkov wave functions, vertical lines: pho-ton propagator) as leading-order term surviving ξ → 0.

The scaled number of pairs is Ntot/Ne0 = 2πΓtot/π,

where the probability rate is given by

Γtot = σppTω

2

m2ξ2

4πα

∫ ∞−∞

dφ f2ppT (φ). (22)

The chosen pulse implies∫∞−∞ dφ f2

ppT (φ) = 32∆φ and

Ntot/Ne0 = σppT

3m2ξ2

16α ∆φ. Note the ξ2 dependence fromthe “target density” already entering in Eq. (22) (cf.[84, 87] for analog relations). This is in contrast toFig. 6, where genuine nonlinear effects are at work andmix with a stronger ξ dependence for C⊗BW. The ξ2 de-pendence is characteristic for pair production by probephotons provided by an “external target”, such as in thebremsstrahlung-laser configuration of LUXE, cf. [75].

For the long laser pulses used in E-144 [88–90], suchbandwidth effects are less severe.

V. SUMMARY

In summary, inspired by the renewed interest in theRitus-Narozhny conjecture and the new perspectives of-fered by the experimental capabilities of LUXE and E-320, we recollect a few features of elementary QED pro-

1.0 1.5 2.0 2.5 3.0 3.5 4.0√s/m

10−12

10−10

10−8

10−6

10−4

10−2

100

σ(√s)

[mb

]

∆φ

π

2π

3π

4π

25

50

250

500

∞

FIG. 8: Pulsed perturbative trident cross section σppT as a

function of√s/m for several pulse lengths ∆φ. In the IPA

limit, i.e. a monochromatic laser beam or ∆φ → ∞, thethreshold is at

√s = 3m. Above the threshold, the dots de-

pict a few points (cf. table 1 in [86]) from perturbative tridentwithout bandwidth effects. Bandwidth effects enable the pairproduction in the sub-threshold region

√s < 3m.

cesses within the essentially known formalism. In partic-ular, we focus on the ξ dependence. For nonlinear Comp-ton scattering, we point out that, in the non-asymptoticregion χ = O(1), k · p . m2, the spectrally resolvedcross section dσ/du|u=const as a function of the laser in-tensity parameter ξ displays a pronounced ∩ shape foru > uKN (the “rise and fall”). This behavior is in starkcontrast with the monotonously rising integrated proba-bility limχ,ξ→∞ P ∝ αχ2/3m2/k · p. That is, in differentregions of the phase space, also different sensitivities ofcross sections/rates/probabilities on the laser intensityimpact can be observed. The soft (small-u) part, whichdetermines the integrated cross section/probability, maybehave completely different than the hard (large-u) con-tribution.8 Transferred to certain approximations used insimulation codes such a behavior implies that one shouldtest differentially where the conditions for applicabilityare ensured.

The hard photons, once produced by Compton processin a laser pulse, act as seeds for secondary processes, mostnotably the Breit-Wheeler process. A folding model oftype Compton⊗Breit-Wheeler on the probabilistic levelpoints to a rapidly increasing rate of e+e− productionin the region ξ . 4, when using parameters in reach ofthe planned LUXE set-up. The actual plans (see figure2.10 in LUXE CDR [45]) uncover ξ = 2 (40 TW, 8 µm

8 An analog situation is known in QCD [91]: Tree-level diagramsare calculated primarily with constant αQCD. Renormalizationimprovement means then replacing αQCD by αQCD(s) which ac-counts at the same time for all vertices in the considered diagramby one global scale s. Inserting explicitly loop corrections leadsfinally to scale dependent couplings αQCD(sv) specific for eachvertex v separately.

9

laser), 6 (40 TW, 3 µm) and 16 (300 TW, 3 µm), andE-320 envisages ξ = 10. The folding model may be uti-lized as reference to identify the occurrence of the wantedone-step trident process in this energy-intensity regime.Furthermore, bandwidth effects in the trident process areisolated by considering the weak-field regime ξ → 0.

Appendix A: Basics of nonlinear Compton process

Following [92] we recall the basics of the underlyingformalism of the nonlinear Compton process. Withinthe Furry picture the lowest-order, tree-level S matrixelement for the one-photon (four-momentum k′, four-polarization ε′) decay of a laser-dressed electron eL inthe background field (1) eL(p)→ eL(p′) + γ(k′, ε′) readswith suitable normalizations of the wave functions

Sfi = −ie∫d4xJ · ε∗′ expik′ · x (A1)

where the current Jµ(x) = Ψp′γµΨp is built by the Volkovwave function Ψp = Epup exp−ip · x exp−ifp (spinindices are suppressed) and its adjoint Ψ with Ritusmatrix Ep = 1 + e

2k·p/k /A and phase function fp(φ) =ek·p∫ φ

0dφ′[p ·A− e

2A ·A]. We employ Feynman’s slash

notation and denote scalar products by the dot betweenfour-vectors; up is the free Dirac bi-spinor. Exploiting thesymmetry of the background field, A(φ = k ·x), Eq. (A1)can be manipulated (cf. [92] for details) to arrive at

Sfi = −ie(2π)3 2

k−δ(3)(p− p′ − k′)M(`), (A2)

where ` ≡ (k′−+ p′−− p−)/k− = k′ · p/k · p′ accomplishesthe balance equation p+ `k− p′ − k′ = 0. (See [62] for aformulation with Sfi = −ie(2π)4

∫d`2π δ

(4)(p + `k − p′ −k′)M(`).) Light-cone coordinates are useful here, e.g.k− = k0 − k3, k+ = k0 + k3, k⊥ = (k1, k2), and k =(k+, k⊥). Imposing gauge invariance yields the matrixelement

M =∑3i=1 J

(i)S(i) (A3)

with the pieces of the electron current

J (0) = up′/ε′∗up, (A4)

J (1,2) = up′(dp′/ε±/k/ε

′∗ + dp/ε′∗/k/ε±

)up, (A5)

J (3) = 4k · ε′∗dp′dp up′/kup, (A6)

which combine to J (1,2) = J (1,2) + α±2` J

(0), J (3) = J (3) +β` J

(0). The following abbreviations are used: dp = mξ4k·p

and dp′ = mξ4k·p′ , α± = mξ

(ε±·pk·p −

ε±·p′k·p′

)with ~ε± = (~ax±

i~ay)e2/m2ξ2 and β = m2ξ2

4

(1k·p − 1

k·p′)

as well.

The phase integrals S(i) are the remainders of the in-tegration d4x = dφ dx− d2x⊥/k− in Eq. (A1):

S(1,2) =

∫ ∞−∞

dφ f(φ) expi(`± 1)φ− i(fp(φ)− fp′(φ)),

S(3) =

∫ ∞−∞

dφ f2(φ) expi`φ− i(fp(φ)− fp′(φ)) (A7)

× [1 + cos 2φ] .

For a few special non-unipolar (plane-wave) fields andtheir envelopes f(φ), the phase integrals can be processedexactly by analytic means [93, 94], but in general a nu-merical evaluation is needed. The very special IPA caseof f(φ) = 1 allows for a simple representation of fp(φ)

with subsequent decomposition of S(i) into Bessel func-tions, yielding final expressions as in Eqs. (2, 20).

For identifying the weak-field limit, ξ → 0, itis necessary to recognize in Eq. (A3) J (0) ∝ ξ0,and α±, J (1,2) ∝ ξ1, and β, J (3) ∝ ξ2. Thus,limξ→0M = M1ξ + M2ξ

2 + · · · . We emphasize the

Fourier transform of the pulse envelope, limξ→0 S(1,2) =∫∞

−∞ dφ f(φ) expi(` ± 1)φ, entering M1, where only

S(1) with ` ≥ 0 contributes to the wanted one-photonemission:9 M1(`) = J (1)

∫∞−∞ dφ f(φ) expi(` − 1)φ,

J (1)(`) = m2 ε′µ∗ε+ν up′

[γµ/k`γ

ν+2γµpν

2k`·p + γν/k′γµ−2γνpµ

2k′·p

]up

with k` ≡ `k. For pulses with broad support, f(φ) → 1,within the interval ∆φ 1, one arrives at the standardCompton (Klein-Nishina) expression in leading order byusing ξ → 2e/m. The appearance of δ(` − 1) combinedwith the definition of ` below Eq. (A2) leads to the fa-

mous Compton formula via δ(ω′ − ωm

m+ω(1−cos Θ′)

)in the

electron’s rest frame by the subsequent phase space inte-gration(s).

The discussion of the large-ξ limit needs some care ingeneral, cf. [26, 28]. Useful limits are obtained for theIPA case [58] under the side condition ξ2(1 − z2/n2) =const (cf. Eqs. (19, 20)), e.g. |M|2|u1 ∝ (ξ/u)2/3

and |M2|u1 ∝ u−1(ξ/u)1/2 exp− 2u3ξ

m2

k·p →√ξ (cf.

Eqs. (14) and (13)), At ξ 1, the ξ and u dependen-cies of resulting probabilities for circular polarization andconstant cross field backgrounds coincide.

These limits are at the heart of the “rise and fall”.The differential emission probability per in-electron

and per laser pulse follows from (A2) by partial inte-gration over the out-phase space,

dPdω′dΩ′

=e2ω′

64π3 k · p k · p′ |M|2, (A8)

and may be transformed to other coordinates, e.g. u or` etc. Having in mind the IPA limit, one should turn to

9 The same reasoning applies in IV C for one-pair emission, i.e. thefinite-width Fourier transform of f(φ) at ∆φ < ∞ enables thesub-threshold pair production eL(p)→ eL(p′)+eL(p′′)+eL(p′′′).

10

the dimensionless differential rate [62]

dΓCdω′dΩ′

=e2ω′

32π2 q0 k · p′|M|2. (A9)

Spin averaging of the in-electron, and spin summationof the out-electron and summation over the out-photonpolarizations leads to |M|2, unless one is interested in po-larization effects as in [19, 20, 95]. The above expressions(A3 - A7) can be further processed for special field en-velopes, or |M|2 is numerically accessible, via Eq. (A3),as mod-squared sum of complex number products pro-vided 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 = dΓCq0k·p

ωnL

with

nL = m2

e2 ξ2ωNL, where NL = 1 (IPA, quasi-momentum

q0) or NL = 12π

∫∞−∞ dφ f(φ) (FPA, q0 ≡ p0). The circu-

larly polarized laser background (1) is supposed in theserelations.

Acknowledgments

The authors gratefully acknowledge the collaborationwith D. Seipt, T. Nousch, T. Heinzl, and useful dis-cussions with A. Ilderton, K. Krajewska, M. Marklund,C. Muller, S. Rykovanov, and G. Torgrimsson. A. Ring-wald and B. King are thanked for explanations w.r.t.LUXE. The work is supported by R. Sauerbrey andT. E. Cowan w.r.t. the study of fundamental QED pro-cesses for HIBEF.

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