The locally monochromatic approximation to QED in intense laser fields
TThe locally monochromatic approximation to QED in intense laser fields
T. Heinzl, ∗ B. King, † and A. J. MacLeod ‡ Centre for Mathematical Sciences, University of Plymouth, Plymouth, PL4 8AA, United Kingdom (Dated: April 29, 2020)We derive an approximation to QED effects in strong background fields which can be employedto improve numerical simulations of laser-particle collisions. Treating the laser as a plane waveof arbitrary intensity, we split the wave into fast (carrier) and slow (envelope) modes. We solvethe interaction dynamics exactly for the former while performing a local expansion in the latter.This yields a ‘locally monochromatic’ approximation (LMA), which we compare with exact spectrafor nonlinear Compton scattering and nonlinear Breit-Wheeler pair production, for realistic laserpulse durations. We show that, as an improvement over the commonly used ‘locally-constant field’approximation, the LMA exactly reproduces the correct low energy behaviour of particle spectra,as well as the position and amplitude of harmonic features.
I. INTRODUCTION
There is a growing interest in experimentally verify-ing the predictions of quantum electrodynamics (QED)in the strong field, high-intensity, regime. To accessthis regime in experiment, two requirements must bemet: (i) an electromagnetic field is present which is suf-ficiently intense so that many field quanta participatein a given process; (ii) the momentum transfer (recoil)in scattering is large enough that the quantum natureof processes is manifest. Upcoming laser facilities suchas ELI-Beamlines [1], ELI-NP [2], and SEL (see [3] foran overview) will reach field strengths to fulfil require-ment (i). One way to fulfil (ii) is to use laser wakefieldaccelerated particles, recent successes of which includethe generation of positron beams in the lab [4] and mea-surement of quantum signals of radiation reaction [5, 6].Background electromagnetic field strength can bequantified using an intensity parameter, ξ , equivalent tothe work done by the background over a Compton wave-length, in units of the background photon energy. When ξ ∼ O (1), the standard approach of treating the back-ground in perturbation theory fails, because this assumesthat processes are more probable when fewer backgroundphotons are involved. When ξ (cid:29)
1, an alternative ap-proximation is often employed, in which the instanta-neous rate for processes in a constant (‘crossed’) planewave background (treated without recourse to perturba-tion theory) is integrated over the classical trajectories ofthe scattered particles. This “locally-constant field ap-proximation” (LCFA) [7–10] has the particular advantagethat it can be applied to arbitrary external fields. There-fore, when used in conjunction with a classical Maxwellfield equation solver, it can be employed in situations forinhomogeneous backgrounds. The locally-constant fieldapproximation is almost exclusively the method by whichQED processes in intense fields are added to laser-plasma ∗ [email protected] † [email protected] ‡ [email protected] simulation codes [11–22]. It has recently been extendedin several respects, by including higher derivative cor-rections [23–25], analysing simple, non-constant, fields inSchwinger pair production [26] and extending it to pre-viously neglected processes [27, 28].An alternative approach to probe the strong-fieldregime of QED is to use a conventional particle accel-erator to fulfil the energy condition (ii), and a less in-tense laser to fulfil the field condition (i). This wasdemonstrated by the landmark E144 experiment [29]which investigated photon emission [30] and pair pro-duction [31, 32] in the weakly nonlinear regime. Usingmodern high-intensity laser systems, this form of exper-iment will be performed at E320 at FACET-II and atLUXE [33] at DESY, to measure QED in the highly non-linear, non-perturbative regime, which was out of reachfor E144. These experiments will access the intermedi-ate intensity regime ξ ∼ O (1), where the locally-constantfield approximation breaks down and fails to capture ex-perimental observables such as the harmonic structure inspectra [34–36].To address this problem we derive here, from QED, the“locally monochromatic approximation” (LMA). Thisis particularly well suited to the analysis of experi-ments using high-energy, conventionally-accelerated par-ticle beams and moderately intense, hence nearly plane-wave, lasers, in which both focussing effects and backre-action [37, 38] can be neglected to a first approximation.The LMA treats the fast dynamics related to the carrierfrequency of the plane wave exactly, but uses a local ex-pansion to describe the slow dynamics associated withthe pulse envelope. This combines the slowly-varying en-velope approximation [39–43] with the locally-constantfield approximation, improving upon both. It capturesfeatures to which the locally-constant field approxima-tion is blind, yet because it is still an explicitly local ap-proximation, it can be added to single-particle simulationcodes. Furthermore, by benchmarking the LMA againstexact calculations in pulses, an additional feature in themid-IR region of nonlinear Compton scattering will be-come apparent, which may provide an additional signalto be searched for in experiment. a r X i v : . [ h e p - ph ] A p r The paper is organised as follows. In Sec. II we outlinethe key steps in deriving the LMA for a general first-orderstrong field QED process. In Sec. III we give an out-line of the numerical methods that form the basis of ourbenchmarking against finite-pulse results. The LMA fornonlinear Compton scattering is then compared to QEDin circularly and linearly polarised pulse backgrounds inSec. IV. We demonstrate the validity of the LMA for non-linear Breit-Wheeler pair production in Sec. V. We con-clude in Sec. VI. In Appendix A, a detailed derivation ofthe LMA for nonlinear Compton scattering in a circularlypolarised background is presented and in Appendix B weinclude an alternative derivation of the infra-red limit ofnonlinear Compton scattering, demonstrating also thatthe correct limit is trivially reproduced from the LMA.Finally, in Appendix C, we show that the locally-constantfield approximation can be recovered as a high-intensitylimit of the LMA. II. OUTLINE OF THE LOCALLYMONOCHROMATIC APPROXIMATION
Let the gauge potential of the background, a µ ( ϕ ), de-pend only on the phase ϕ = k · x , with k being thewave four-vector. We will work in lightfront coordinates x = ( x + , x − , x ⊥ ) where x ± = x ± x and x ⊥ = ( x , x ).Here x + is lightfront time while x − and x ⊥ are calledthe longitudinal and perpendicular directions, respec-tively [44]. With this notation, the wave vector of thebackground k µ = δ + µ k + , and ϕ = k + x + . The scatter-ing amplitude, S fi , for an incoming electron with on-shell momentum p , p = m , is then calculated using theVolkov wavefunction [45],Ψ p ( x ) = (cid:18) /k/a ( ϕ )2 k · p (cid:19) u p e − iS p ( x ) . (1)In the exponent, S p ( x ) is the classical action for an elec-tron in a plane wave background, S p ( x ) = p · x + (cid:90) ϕ −∞ p · a ( t ) − a ( t )2 k · p dt. (2)The scattering amplitude S fi in a plane wave back-ground can then be written as S fi =(2 π ) δ − , ⊥ ( p in − p out ) M , (3)with an invariant amplitude M . Due to the non-trivialstructure of the background, overall momentum conser-vation (encoded in the delta functions) only holds in threedirections, {− , ⊥} . Here and throughout, we use ‘infra-red’ to denote low lightfront energy n · P , for n the laser propagation direction and P anygiven particle momentum. This is a natural variable in planewave calculations. A closed form solution for phase integrals such as (2)is only known for some special cases of the backgroundfield, for example infinite “monochromatic” plane waves(see e.g. [8] for extensive applications). Beyond thesesolutions, one can turn to a numerical approach or em-ploy an approximation. The slowly varying envelope ap-proximation is known to simplify the classical action (2)occurring in the exponent and hence make the phase in-tegrations tractable [39–43]. It is applied as follows. Letthe pulse a µ ( ϕ ) have the form a µ ( ϕ ) = m ξ f (cid:16) ϕ Φ (cid:17)(cid:0) ε µ cos δ cos ϕ + ¯ ε µ sin δ sin ϕ (cid:1) , (4)where ξ is the dimensionless Lorentz and gauge invari-ant measure of the field intensity [46], f ( ϕ/ Φ) is thepulse envelope with phase duration Φ and ε µ , and ¯ ε µ are polarisation directions satisfying ε = ¯ ε = − ε · ¯ ε = k · ε = k · ¯ ε = 0. The parameter δ ∈ (0 , π/ δ = 0 for linearpolarisation along ε , δ = π/ ε and δ = π/ . We consider thepulse envelope to be asymptotically switched on and off,lim φ →±∞ f ( φ ) = 0.The slowly varying envelope approximation assumesthat the pulse duration Φ is sufficiently long that termsof order O (Φ − ) can be neglected. (Higher orders canin principle be included in the approximation but theywill lead to a more complicated result that takes longerto numerically evaluate and, as we shall see, the leadingorder terms will already be sufficient to reproduce themain features of spectra.) As a result, derivatives of theenvelope with respect to the phase can be neglected, be-cause they are of the form df ( ϕ/ Φ) /dϕ ∼ Φ − f (cid:48) ( ϕ/ Φ).In other words, the envelope varies slowly compared tothe fast dynamics of the carrier frequency. The practicalbenefit of this is that we can simplify the classical action(2). More explicitly, the classical action will have termsboth linear and quadratic in the field envelope. In allterms involving both fast and slow oscillations, we inte-grate by parts, picking up terms of order O (Φ − ) whichwe neglect, and so remove the integrals from (2). Thisgives us, for the possible linear terms arising, (cid:90) ϕ −∞ d ψ f (cid:16) ψ Φ (cid:17)(cid:8) cos ψ, sin ψ (cid:9) (cid:39) f (cid:16) ϕ Φ (cid:17)(cid:8) sin ϕ, − cos ϕ (cid:9) , (5)and for the possible quadratic terms (cid:90) ϕ −∞ d ψ f (cid:16) ψ Φ (cid:17)(cid:8) cos ψ, sin ψ (cid:9) (cid:39) f (cid:16) ϕ Φ (cid:17)(cid:110)(cid:0) ϕ + sin ϕ cos ϕ (cid:1) , (cid:0) ϕ − sin ϕ cos ϕ (cid:1)(cid:111) . (6) We make implicit a normalisation factor in the gauge potential(4) such that Max[ a µ ( ϕ ) / ( mξ )] = 1. For the particular case of a circularly polarised back-ground, there arises a term containing only slow oscilla-tions (the integral of f without trigonometric functions),which must be approximated by different means (seebelow). With these approximations, the background-dependent parts of the classical action can always be putin the form S p ( x ) (cid:39) G (cid:16) ϕ, ϕ Φ (cid:17) + 12 α (cid:16) ϕ Φ (cid:17)(cid:2) u ( ϕ ) − u − ( ϕ ) (cid:3) + 12 β (cid:16) ϕ Φ (cid:17)(cid:2) v ( ϕ ) − v − ( ϕ ) (cid:3) . (7)The functions α and β are purely slowly-varying func-tions of the phase ϕ . The functions u ( ϕ ) and v ( ϕ ) areof the form exp( icϕ ), for c ∈ { , } . Note the similarityof the form of the exponent with the generating functionfor the Bessel function of the first kind,exp (cid:110) z (cid:16) ϕ Φ (cid:17) (cid:2) u ( ϕ ) − u − ( ϕ ) (cid:3)(cid:111) = (cid:88) n ∈ Z u n ( ϕ ) J n (cid:104) z (cid:16) ϕ Φ (cid:17)(cid:105) . (8)This was recognised and exploited in [39] and essen-tially gives a generalisation of the infinite monochromaticfield results [8, 47] to the case where the argument ofthe Bessel function now depends slowly on the phase.There will also appear rapidly oscillating terms in thepre-exponent, but these can be incorporated by differen-tiating (8) with respect to z and combining terms. Thescattering amplitude will thus be defined in terms of har-monics, represented by the sum over integers n in (8).So far everything has been typical for the application ofthe slowly-varying envelope approximation in the strong-field QED literature [39–43]. It is at this point that wetake the further step of performing a local expansion inthe phase variables to arrive at a local “rate” which canbe implemented in one-particle numerical simulations.To define the local expansion, we will concentrate onsingle (dressed) vertex “one-to-two” processes: nonlin-ear Compton scattering and nonlinear Breit-Wheeler pairproduction. The amount of literature on these processeshas become too large to be cited here in full; regard-ing nonlinear Compton scattering see [7, 48, 49] for theoriginal papers, [8, 50] for reviews and [41, 51–54] for aselection of more recent results. Nonlinear Breit-Wheelerpair creation was first discussed in [7, 55, 56], while thestudy of finite size effects was initiated in [57]. Both pro-cesses were observed (at mildly nonlinear intensities) bythe SLAC E144 experiment [29, 30, 32]. For the two ex-amples to be considered, the reduced amplitude M in(3) will have one phase integral, and after applying theslowly-varying envelope approximation, will be defined interms of an infinite sum over the harmonic order n , i.e. M = ∞ (cid:88) n = −∞ (cid:90) d ϕ M n ( ϕ ) . (9)Squaring the amplitude for the probability, we will have something of the form, P ∼ ∞ (cid:88) n,n (cid:48) = −∞ (cid:90) dΩ LIPS (cid:90) d ϕ d ϕ (cid:48) M † n ( ϕ ) M n (cid:48) ( ϕ (cid:48) ) , (10)i.e., a double infinite sum over harmonic orders, twophase integrals, and an integration over the Lorentz in-variant phase space of the process, dΩ LIPS .Now we perform a local expansion of the probability, inanalogy to the locally-constant field approximation (seee.g. [7–10]). We make a change of variables to the sumand difference of phases, φ = 12 (cid:0) ϕ + ϕ (cid:48) (cid:1) , θ = ϕ − ϕ (cid:48) . (11)Terms in the probability are then expanded in a Taylorseries in θ (cid:28)
1, and the slowly-varying envelope approx-imation is then applied to all derivatives of the pulseenvelope, giving f (cid:16) ϕ Φ (cid:17) ≈ f (cid:16) ϕ (cid:48) Φ (cid:17) ≈ f (cid:16) φ Φ (cid:17) . (12)This allows the d θ integrals to be performed, and theprobability takes the form P = (cid:90) d φ R ( φ ) , (13)where R ( φ ) is interpreted as a local “rate” . For theprocesses of nonlinear Compton scattering and nonlinearBreit-Wheeler pair production we can write: P LMA ≈ (cid:90) d φ R mono [ ξf ( φ/ Φ)] , (14)where R mono is the probability per unit phase of the pro-cess in a monochromatic (infinitely long) plane wave. Fora circularly polarised background the LMA is exactlyequal to the integral on the right-hand side of (14). Fora linearly polarised background, it is not so straightfor-ward, as interference between different harmonic orders isincluded, but we will find that, to a good approximation,both sides of (14) are equal.To conclude this outline of the LMA, we reiterate thatthe LMA is simply the application of two well-knownapproximations in the strong-field QED literature, theslowly varying envelope approximation and the “local”expansion in the relative phase variable, θ , carried out In general R ( φ ) will contain infinite sums over harmonic orders,and a number of final state momentum integrals. The aim isto do as many of these final state momentum integrals as possi-ble. Despite the added complexity which arises from retaining aslowly varying dependence on the phase variable φ , the numberof final state integrals that can be performed is the same in theLMA as for a first order process in an infinite monochromaticplane wave [8, 47] (see appendix A). at the level of the probability. Although the approxima-tion has been used before for a circularly polarised back-ground [58], as far as we are aware, this is the first explicitderivation and benchmarking with the direct calculationfrom QED for a plane-wave pulse. The monochromaticresult is obtained from the LMA by taking the infinitepulse limit Φ → ∞ , i.e. f → III. DIRECT CALCULATION FROM QED FORA PULSED BACKGROUND
We wish to benchmark the LMA against the numericalevaluation of exact expressions from high-intensity QED.We provide here the details of the integration schemeused. For both nonlinear Compton scattering and nonlin-ear Breit-Wheeler pair production in a plane-wave pulse,one can write the total probability in the form P = α I /η ,where α denotes the fine structure constant, η = k · P/m is the energy parameter of the incoming particle (where k is the light-like wave vector of the plane wave back-ground, P is the four-momentum of the incoming par-ticle) and I is a triple integral. I involves two phaseintegrals, φ , θ , and an integral over s , the fraction of theincoming particle’s light-front momentum, P − , carriedaway by the emitted particle. For nonlinear Comptonscattering, this is of the form: I = (cid:90) ∞−∞ d φ (cid:90) d s (cid:110) − π (cid:90) ∞ d θθ [1 + h ( a, s )] sin [ g ( s ) θµ ( φ, θ )] (cid:27) . (15)For the numerical calculation of the exact QED result, weare using the “ i(cid:15) ” regularisation at the level of the prob-ability (see e.g. [59]), as evidenced by the π/ a de-fined in (4) resides in both h ( a, s ) and in the Kibble mass[60, 61] normalised by the electron mass: µ ( φ, θ ) = 1 − θ (cid:90) φ + θ/ φ − θ/ a m + (cid:18) θ (cid:90) φ + θ/ φ − θ/ am (cid:19) . (16)In what follows we will outline some manipulations al-lowing for a straightforward numerical integration of I .The phase integration plane ( φ, θ ) can be split natu-rally into subregions where the integrand in (15) takes aspecific form according to the following two observations:First, the field-dependent function h ( a, s ) only has sup-port for a (cid:54) = 0. Second, the Kibble mass becomes phase-independent when φ ± θ/ f ( ϕ/ Φ), whichis symmetric about the origin with support | ϕ | < L/ f = cos , where the phase duration is L = π Φ and thepulse length parameter, Φ, can be related to the number ϕ / πΦ θ / π Φ FIG. 1. Overview of the regions integrated over in the φ - θ plane. The non-striped regions are inside of the pulse: | ϕ (cid:48) | < π Φ. The dark subregion in the area covered by I signifies | ϕ | < π Φ. of cycles, N , via Φ = 2 N . Using the symmetry of theintegrand, we only have to consider the first quadrant inthe ( φ, θ )-plane, which splits into the sub-regions shownin Fig. 1 such that I = (cid:82) ds (cid:80) k =1 I k .To deal with the infinite numerical integation of a non-linearly oscillating pure phase term, we first rewrite theregularisation factor as π (cid:90) ∞ d θθ sin Kθ, which is independent of the choice of the constant factor K . In order to make for a simpler numerical evaluation,we choose K = g ( s ), allowing us to combine it with theother infinite phase term in (15). (Other choices are use-ful in other circumstances, see for example [62] and (B1)in the appendix.) Using this trick, we find that the firstintegral vanishes, I = (cid:90) ∞ π Φ d σ (cid:90) φ − π Φ)0 dθθ {− sin [ g ( s ) θ ]+ sin [ g ( s ) θµ ( φ, θ )] } = 0 . This can be shown by noting that lim a → µ ( φ, θ ) = 1,and in this phase region the pulse has no support. Thisis because terms depending on the potential, a ( ϕ ), a ( ϕ (cid:48) )are zero unless: | ϕ | = | φ + θ/ | < π Φ or | ϕ (cid:48) | = | φ − θ/ | < π Φ . In contrast, the integral I over the region where thepulse is yet to pass through, is non-zero: I = (cid:90) ∞ d φ (cid:90) ∞ π Φ+ φ ) d θθ {− sin [ g ( s ) θ ]+ sin [ g ( s ) θµ ( φ, θ )] } (cid:54) = 0 . Nevertheless, it may be calculated analytically by not-ing that the combination θµ ( φ, θ ) accumulates a constanttotal phase, θµ → θ + θ ∞ , when the probe particle tra-verses the pulse and continues to propagate in vacuum.Explicitly, one finds for both nonlinear Compton andBreit-Wheeler processes that θ ∞ = 3 πc ε ξ Φ /
2, where c ε = 1 ( c ε = 1 /
2) for a circularly (linearly) polarisedbackground. This finally leads to I = 4 π Φ sin X [cos X Ci Y − sin X Si Y + π X − Y sin ( X + Y ) (cid:21) , (17)where X = θ ∞ g ( s ) / Y = 4 π Φ g ( s ). This is relatedto recent studies of interference effects in a double-pulsebackground [63, 64].The remaining integral, I , collects the contributionswhere the average phase φ is outside the pulse, while thephase difference θ is large enough that φ − θ/ I = (cid:90) ∞ π Φ d φ (cid:90) φ +2 π Φ)2( φ − π Φ) d θθ {− sin [ g ( s ) θ ]+ sin [ g ( s ) µ ( φ, θ )] } , The integrand oscillates with a slowly decaying ampli-tude for φ > π Φ outside the pulse. As the oscillationsare regular, they can be handled by using many datapoints. We also expect (and will show later) that contri-butions from outside the pulse are important mainly inthe infra-red region of the spectrum, where we have ananalytical expression for the limit.Finally, I is just the evaluation of the full integral inEq. (15), for φ ∈ [0 , π Φ], θ ∈ [0 , π Φ+ φ )], i.e. “on topof” the pulse. As this is a well-defined, finite integrationrange, convergence can be assured by simply increasingthe sampling resolution of the integrand. ξ = η = - - ℐ / ds a ) I I I ℐ - ℐ / ds b ) FIG. 2. A demonstrative plot showing how different parts ofthe integration region contribute to the spectrum (here, for alinearly polarised pulse) using a) a log scale and b) a linearscale.
The contribution of each part of the phase integrationplane ( φ, θ ) to the spectrum is shown, for example param-eters, in Fig. 2. This demonstrates that in the infra-redlimit, s →
0, the integral I from (15) is dominated by by the sub-integral I , i.e. by contributions from phaseregions located outside the pulse. On the one hand, thisagrees with intuition based on the uncertainty principle—the lowest photon energies require the longest interactionof the electron with the background as has already beenpointed out in the literature for nonlinear Compton scat-tering [9]. On the other hand, when studying the infra-red, one should take into account soft contributions fromhigher-order processes [59]. IV. NONLINEAR COMPTON SCATTERINGA. Circularly polarised plane wave ξ = η = - cycle4 - cycle8 - cycle16 - cycle32 - cycle0.05 0.10 0.15 0.20 0.25 0.30 0.35 s d ℐ / ds a ) s s d ℐ / ds b ) s d ℐ / ds c ) s d ℐ / ds d ) FIG. 3. The photon spectrum from nonlinear Comptonscattering in a circularly polarised background, in the high-energy, weakly nonlinear regime, normalised by N/ N . The locally-constantfield approximation (dotted red line) poorly approximates thespectrum, whereas the LMA (dashed black line) captures theharmonic structure and becomes more accurate as the lengthof the pulse increases. Plotted left-to-right is: a) the yieldspectrum; b) the energy spectrum; c) the IR part of the spec-trum (log-linear); d) the UV part of the spectrum (log). Thevertical solid lines here and in the following figures correspondto the positions of the harmonic edges calculated for an infi-nite monochromatic plane wave. Having evaluated the full QED integrals for a pulse,we can now compare with the LMA. The latter is nu-merically more efficient, but also implies enhanced ana-lytical control as it typically results in well-known specialfunctions. Beyond these immediate advantages, our mo- ξ = η = - cycle4 - cycle8 - cycle16 - cycle32 - cycle0.02 0.04 0.06 0.08 0.10 s d ℐ / ds a ) s d ℐ / ds b ) - s d ℐ / ds c ) s d ℐ / ds d ) FIG. 4. The photon spectrum from nonlinear Compton scat-tering in a circularly -polarised background, in the high-energynonlinear regime, normalised by half the number of laser cy-cles. The locally-constant field approximation (dotted redline) approximates the spectrum well for values of s corre-sponding to higher harmonics. The LMA (dashed black line)captures both the harmonic structure and the large- s be-haviour and becomes more accurate as the length of the pulseincreases. Plotted left-to-right is: a) the yield spectrum; b)the energy spectrum; c) the IR part of the spectrum (log-linear); d) the UV part of the spectrum (log). tivation to improve standard literature approximationsis three-fold: (i) to have a locally defined rate whichcould in principle be implemented in numerical simula-tion codes; (ii) to be able to resolve the harmonic struc-tures present in the exact QED probabilities with this ap-proximation; and (iii) to be able to work in the moderateintensity regime, ξ ∼
1, relevant for current state-of-the-art laser facilities. By construction, item (i) is readilyprovided by the LMA. To test the LMA for the othertwo goals, we will benchmark it against numerically inte-grated exact QED probabilities, beginning with the pro-cess of nonlinear Compton scattering.Consider the interaction of an electron, initial invariantenergy parameter η e = k · p/m , with the plane wave a µ ( φ ) = mξ cos (cid:16) φ Φ (cid:17)(cid:0) ε µ cos φ + ¯ ε µ sin φ (cid:1) , (18)which has circular polarisation and envelope f ∼ cos .The LMA to the nonlinear Compton spectrum in thissetup is given in (A21). In Fig. 3 we compare the pho-ton spectrum predicted by the LMA with the exact QEDresult, for the parameters ξ = 0 . η e = 0 .
1, and vari-ous pulse lengths Φ. This is the low intensity, high-energy regime which will be probed at, for example, LUXE [33].In this regime, the locally-constant field approximation,valid for ξ /η e (cid:29) N and the pulse duration Φ are related by Φ = 2 N .)The numerically integrated exact QED spectra have beennormalised by N/ s →
1, regime. This is char-acteristic of the locally-constant field approximation for ξ < ξ = 2 .
5. We are now ina regime where the locally-constant field approximationis able to more accurately capture at least the s → < s <
1, defined in relation to the position ofthe first harmonic/Compton edge, which for a monochro-matic plane wave is located at s = 2 η e / (1 + ξ + 2 η e ).There is the far infra-red sector where 0 < s (cid:28) s , theharmonic range where s > s which includes all of theharmonic structure of the spectrum, and the intermediateregime where s (cid:46) s . In both the far infra-red and theharmonic range the LMA gives a very good agreementwith the numerically integrated exact QED spectrum,outperforming the locally-constant field approximationin both cases. One of the most striking improvements inthis regard is the agreement between the LMA and theexact QED spectrum in the far infra-red, s → s → s (cid:46) s . It turns out that this sector ofthe spectrum contains features which, to the best of ourknowledge, have not been extensively commented on inthe literature. Most numerical investigations of the exactQED spectrum/probability are compared to the locally-constant field approximation, which is well known (i) tonot capture harmonic structure and (ii) to diverge to-wards the infra-red. The LMA, however, yields the cor-rect infra-red limit, s →
0, and very good agreement inthe harmonic range, but does not capture the full struc-ture of the spectrum in the intermediate range. In eachof the spectra coming from the numerically integrated ex-act QED results there is a clear “bump” in the range justbefore the first harmonic. This same feature can be seenin various other works in the literature, see for example[23, 24, 67].A qualitative explanation for these additional peaks isthat a pulse profile introduces additional frequency scalesin the dynamics, analogous to the usual harmonics foundat locations determined by the carrier frequency scaleof the background, see e.g. [41, 52, 53]. For the cur-rent choice of a cos pulse envelope, we found that theapproximate position of these peaks can be determinedas follows. One first introduces a rescaled frequency,˜ k = k / I , where k is the carrier wave frequency and I is the integral of the pulse profile, f . One then cal-culates the position of the first harmonic/Compton edge, s , using the rescaled energy parameter η e → η e / I . Aspulse duration increases, the additional broad peaks getpushed further back into the infra-red and are smoothedout, eventually disappearing in the infinite plane wave(monochromatic) limit. The amplitude of these peaksalso decreases significantly as ξ falls below unity.Fig. 4 also shows that in the UV range, s →
1, there isgood agreement between the LMA and the locally con-stant field approximation for ξ = 2 .
5. However, this isno longer true when ξ = 0 . − s ) − . Following this route, though, isbeyond the scope of our present discusion.The case of a circularly polarised plane wave pulsegives the simplest form of the LMA due to the additionalsymmetries of the choice of background. The approachcan, however, still be used for the case of linear polarisa-tion, to which we now turn. For circular polarisation, i.e. the choice (18), one finds I = π Φ / k = k / √ π Φ. ξ = η = - cycle4 - cycle8 - cycle16 - cycle32 - cycle0.05 0.10 0.15 0.20 0.25 0.30 s d ℐ / ds a ) s d ℐ / ds b ) - s d ℐ / ds c ) s × - d ℐ / ds d ) FIG. 5. The photon spectrum from nonlinear Compton scat-tering in a linearly polarised background, in the high-energy,weakly nonlinear regime, normalised by half the number oflaser cycles. The agreement of the LMA (dashed black line)and disagreement of the locally-constant field approximation(dotted red line) with the numerically exact results is simi-lar to the circularly polarised case. The dashed gray line isthe spectrum acquired by taking the LMA for a circularly polarised background and rescaling the intensity parameter ξ → ξ/ √ B. Linearly polarised plane wave
As above, we compare the LMA for a linearly po-larised background field with the numerically integratedexact result for a fixed electron energy η e = 0 . ξ = 0 . ξ = 2 . ξ → ξf , as could be done inthe circularly polarised case. In principle, the additionalstructure of a double-harmonic sum allows for the possi-bility of interference effects between the harmonics. How-ever, in the intermediate intensity, high-energy regime,we did not find any appreciable contribution from thisinterference.For weak fields, ξ <
1, the low-energy part of the spec-trum, i.e. the region s (cid:46) s below the first harmonic, s , is well approximated by the perturbative contribu-tion from the squared potential, a . In this case, the lin-early polarised LMA turns out to be well-approximatedby taking the circularly polarised LMA and making thereplacement ξ → ξ/ √
2, as is demonstrated in (6). Be-cause of this, rescaling the circularly polarised result isa method which has been used to implement rates forlinear polarisation in numerical codes.However, this method fails for ξ >
1. In this regime,higher harmonics, proportional to a n for the n th har-monic, contribute to the spectrum and can no longer beobtained through a simple modification of the circularly ξ = η = - cycle4 - cycle8 - cycle16 - cycle32 - cycle0.00 0.02 0.04 0.06 0.08 0.10 s d ℐ / ds a ) s d ℐ / ds b ) - s d ℐ / ds c ) s d ℐ / ds d ) FIG. 6. The photon spectrum from nonlinear Compton scat-tering in a linearly polarised background, in the high-energy,nonlinear regime, normalised by half the number of laser cy-cles. The agreement of the LMA (dashed black line) andthe locally-constant field approximation (dotted red line) withthe exact pulsed results is similar to the circularly polarisedcase. The dashed gray line is the spectrum acquired by takingthe LMA for a circularly polarised background and replacingthe intensity parameter ξ → ξ/ √
2. Unlike in the weak-fieldregime, the linearly polarised LMA is not well approximatedby rescaling the intensity parameter in the circularly polarisedLMA. polarised LMA. This impact of the background polari-sation at higher values of the field strength is demon-strated in Fig. 6. Although the position of the harmon-ics is still correctly predicted by the rescaled circularlypolarised LMA, their amplitude is not, nor is the overallshape of the spectrum correctly captured: the rescaledcircularly polarised result gives an underestimate for thesmallest values of s , but an overestimate for larger values.Hence, the linearly polarised LMA proper, rather thanthe rescaled circularly polarised LMA, must be used inthe intensity regime of upcoming experiments [33].From both the circular and linear polarisation casesjust discussed one notes that the higher the field strength ξ , the better the agreement between LMA and locally-constant field approximation in the ultra-violet (large- s )regime. In appendix C we show explicitly that this isnot just some numerical accident. Indeed, we will derivethe locally-constant field approximation as the high-fieldlimit of the LMA. V. NONLINEAR BREIT-WHEELER
So far our focus has been on implementing andanalysing the LMA for nonlinear Compton scattering.In principle, however, the LMA can be applied to anyQED scattering process in a plane wave background.As another example, consider nonlinear Breit-Wheelerpair production, where an initial photon decays intoan electron-positron pair. The derivation of the LMAfor this process follows the same route as for nonlinearCompton scattering (see appendix A), and we again findthat in the case of a circularly polarised plane wave thefinal differential probability is simply the textbook re-sult in a monochromatic plane wave [47] with a locali-sation of the field strength, ξ → ξf , see (A36) in theappendix. A well known feature of the nonlinear Breit-Wheeler process is the strict lower bound, n (cid:63) , on theharmonic number contributing for a given field strengthand initial photon energy. This is because the outgoingparticle states are massive, so that their production canonly proceed above an energy threshold.For a monochromatic plane wave, the lower bound isgiven by n mono (cid:63) = 2(1 + ξ ) /η γ , where η γ = k · (cid:96)/m isthe energy parameter for the incident photon with four-momentum (cid:96) . Comparing this to (A37), we can see thatfor a pulse there are points along the phase for which theminimum harmonic n (cid:63) < n mono (cid:63) for the same ξ and η γ .At first glance, this would appear to mean that at thewings of the pulse, as f →
0, the minimum harmonic con-tributing would decrease, and since Bessel harmonics oflower order are typically greater in magnitude, that theprocess would be more probable at lower field strengths.One has to keep in mind, though, that the argument z ( φ )of the Bessel function depends on the pulse profile f andvanishes in the limit f →
0. The only Bessel functionsurviving this limit is J . However, since the harmonicsum in (A36) is over strictly positive n > (cid:1) - (cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:6) - (cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:7) - (cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:8)(cid:9) - (cid:2)(cid:3)(cid:2)(cid:4)(cid:5)(cid:10)(cid:1) - (cid:2)(cid:3)(cid:2)(cid:4)(cid:5) d ℐ / ds FIG. 7. The spectrum of electrons produced in nonlinearBreit-Wheeler pair production for ξ = 1, η γ = 3. A compar-ison of the locally-constant field approximation (dotted redline) and the LMA (dashed black line) with the exact numer-ical result in pulses of varying duration (coloured solid lines). excludes J , there is no contribution to the probabilityfor f →
0. Hence, in comparison to nonlinear Comp-ton scattering, the nonlinear Breit-Wheeler process willstill require either very high field strengths, for which thelocally-constant field approximation should be a good ap-proximation, or very high initial photon energies.For both the Compton and Breit-Wheeler processes,the momentum taken from the field increases with fieldstrength, and the harmonic structure becomes less welldefined. In order to demonstrate the LMA for the Breit-Wheeler process, the centre-of-mass energy should beclose to the pair rest-energy in order that only veryfew laser photons are required for pair production totake place. In Fig. 7, we demonstrate such a situation,where we present the spectrum of electrons produced bya head-on collision of a 250 GeV photon ( η γ = 3) witha laser pulse of intensity ξ = 1. We note that the har-monic structure of the spectrum for long pulses is well-approximated by the LMA, whereas the locally-constantfield approximation both misses the harmonics in thespectrum and under-predicts the yield. VI. SUMMARY
Motivated by the need to improve the theoretical toolsrequired for supporting state-of-the-art laser experimentsprobing the high-intensity regime of QED, we have in-troduced here the locally monochromatic approximation(LMA).This technique treats the quickly- and slowly-oscillating components of laser field profiles differently, inorder to improve on the accuracy of the existing locally-constant field approximation, which essentially treats allfield components as slowly varying. Oscillations due tothe carrier frequency of the laser field are treated exactly, while the slowly-varying field envelope degrees of freedomare treated in a local expansion.We have shown that this approach offers several qual-itative and quantitative advantages over the locally-constant field approximation. Namely, the LMA canresolve harmonic structure in particle spectra (whichare experimental observables [35, 36]), works for mod-erate values of the laser intensity, and also captures thefar infra-red limit of nonlinear Compton scattering: thelocally-constant field approximation is well-known to failin each of these regards.As a caveat we mention that the LMA in its presentformulation, despite being local in the phase variable,cannot be employed straightforwardly in a PIC simula-tion involving a plasma environment. This is becausethe LMA relies on the presence of structures particularto laser fields, essentially a central frequency and an en-velope, which normally are absent in a plasma. Never-theless, we have demonstrated that the LMA is a pow-erful tool in the study of the interaction between lasersand particle beams, especially in the regime of moder-ate laser intensity and high particle energy, which is thetarget parameter regime of upcoming experiments [33].In this paper we have considered the first-order pro-cesses of nonlinear Compton scattering and nonlinearBreit-Wheeler pair production, but the LMA could alsobe extended to higher-order processes such as trident pairproduction (see e.g. [70–75]) and double nonlinear Comp-ton scattering (see e.g. [76–81]) to name just a few.
ACKNOWLEDGMENTS
The authors thank Anton Ilderton for many usefuldiscussions and a careful reading of the manuscript.B.K. and A.J.M. are supported by the EPSRC grantEP/S010319/1.
Appendix A: Detailed derivation of the LMA
In Sec. II we presented only the key steps involved incalculating the LMA for a high-intensity QED process.To be more explicit, we turn to an example derivation forthe process of nonlinear Compton scattering in a planewave pulse. We will give a thorough account of the calcu-lation for a circularly polarised pulse, and provide detailsabout the technical differences in the linearly polarisedcase. We note that the calculation of the LMA for non-linear Breit-Wheeler pair production discussed in Sec. Vfollows a completely analogous procedure. Hence, it willbe sufficient to simply quote the final result below. Oncethe slowly-varying envelope approximation has been ap-plied to the exponent of the scattering amplitude, theremaining analysis amounts to a generalisation of thecalculation in a purely monochromatic plane wave (seefor example [47] for the circularly polarised case).0Nonlinear Compton scattering is the process by whichan electron of 4-momentum p scatters off a backgroundplane wave pulse to emit a photon of momentum (cid:96) (cid:48) andpolarisation (cid:15) ∗ (cid:96) (cid:48) , e − ( p ) → e − ( p (cid:48) ) + γ ( (cid:96) (cid:48) ). The amplitudeis given by the standard S -matrix element S NLC = − ie (cid:90) d x ¯Ψ p (cid:48) ( x ) /(cid:15) ∗ (cid:96) (cid:48) Ψ p ( x ) e i(cid:96) (cid:48) · x . (A1)The explicit representation of the Volkov wavefunc-tions (1) and some trivial integrations lead to the rep-resentation (3) with reduced amplitude M NLC = − ie (cid:90) d ϕ S ( ϕ ) exp (cid:18) i (cid:90) ϕ −∞ (cid:96) (cid:48) · π p k · ( p − (cid:96) (cid:48) ) (cid:19) . (A2)The integrand involves a spin structure S ( ϕ ) = ¯ u p (cid:48) (cid:18) /a ( ϕ ) /k k · p (cid:48) (cid:19) /(cid:15) ∗ (cid:96) (cid:48) (cid:18) /k/a ( ϕ )2 k · p (cid:19) u p . (A3)and an exponential given in terms of the kinetic momen-tum of a classical electron in a plane wave background, π µp ( ϕ ) = p µ − a µ ( ϕ ) + 2 p · a ( ϕ ) − a ( ϕ )2 k · p k µ . (A4)We proceed now to the particular case of a circularlypolarised pulse.
1. Circularly polarised plane wave pulse
The circularly polarised plane wave pulse is given by(4) with δ = π/ √ max ( | a µ /mξ | ) = 1, a µ ( ϕ ) = mξ f (cid:16) ϕ Φ (cid:17)(cid:0) ε µ cos ϕ + ¯ ε µ sin ϕ (cid:1) . (A5)The term quadratic in the gauge potential in (A4) onlycontains the slow timescale in ϕ : a ( ϕ ) = − m ξ f (cid:16) ϕ Φ (cid:17) , (A6)and so the slowly-varying envelope approximation (5)need only be applied to the other terms linear in a µ .This results in (cid:90) ϕ −∞ (cid:96) (cid:48) · π p k · ( p − (cid:96) (cid:48) ) (cid:39) G ( ϕ ) − α c (cid:16) ϕ Φ (cid:17) sin ϕ + α s (cid:16) ϕ Φ (cid:17) cos ϕ. (A7) The function G ( ϕ ) is slowly varying with ϕ , G ( ϕ ) = s η e (1 − s ) (cid:20)(cid:18) | (cid:96) (cid:48) ⊥ − s p ⊥ | s m (cid:19) ϕ + (cid:90) ϕ −∞ d ψ ξ f (cid:16) ψ Φ (cid:17)(cid:21) , (A8)and depends on η e = k · p/m , the normalised measure ofthe electron’s light-front momentum, and s = k · (cid:96) (cid:48) /k · p ,the light-front momentum fraction of the outgoing pho-ton. The rapidly oscillating terms { cos ϕ, sin ϕ } eachhave a slowly-varying pre-factor, α c (cid:16) ϕ Φ (cid:17) = ξf ( ϕ Φ ) η e m (1 − s ) ( (cid:96) (cid:48) − sp ) · ε ,α s (cid:16) ϕ Φ (cid:17) = ξf ( ϕ Φ ) η e m (1 − s ) ( (cid:96) (cid:48) − sp ) · ¯ ε , (A9)respectively, and depend on a 4-vector L = (cid:96) (cid:48) − sp , pro-jected onto the polarisation directions, ε and ¯ ε , of thebackground. Defining an angle ϑ via the ratio α s α c = tan − ϑ , (A10)allows us to write α c (cid:16) ϕ Φ (cid:17) = z ( ϕ ) cos ϑ , α s (cid:16) ϕ Φ (cid:17) = z ( ϕ ) sin ϑ , (A11)such that z ( ϕ ) = (cid:112) α c + α s = (cid:115) ξ f ( φ Φ ) η e m (1 − s ) | (cid:96) (cid:48) − s p | . (A12)This drastically simplifies the exponent (A7), and thereduced amplitude (A2), which becomes M NLC = − ie (cid:90) d ϕ S ( ϕ ) e i { G ( ϕ ) − z ( ϕ ) sin( ϕ − ϑ ) } . (A13)The probability is now calculated in the usual wayby averaging/summing over incoming/outgoing spins andpolarisations and integrating over the outgoing particlephase space with the result1 P (circ)NLC = α π ( k · p ) (cid:90) d ϕ (cid:90) d ϕ (cid:48) (cid:90) d ss (1 − s ) (cid:90) d | L ⊥ | (cid:90) d ϑ T NLC ( ϕ, ϕ (cid:48) ) × exp (cid:2) iG ( ϕ ) − iG ( ϕ (cid:48) ) − iz ( ϕ ) sin (cid:0) ϕ − ϑ ) + iz ( ϕ (cid:48) ) sin (cid:0) ϕ (cid:48) − ϑ ) (cid:3) . (A14)Here, we have introduced the fine structure constant α and the auxiliary quantity T NLC ( ϕ, ϕ (cid:48) ) = − m + (cid:18) s − s ) (cid:19)(cid:0) a ( ϕ ) − a ( ϕ (cid:48) ) (cid:1) , (A15)(up to a factor) representing the trace, tr ¯ SS , from thespin sum/average. For the perpendicular photon momen-tum integrals one has d (cid:96) (cid:48)⊥ = d L ⊥ or, in polar coordi-nates, (cid:90) d (cid:96) (cid:48)⊥ = 12 (cid:90) d | L ⊥ | (cid:90) d ϑ . (A16) The trace term (A15) also depends on the gauge po-tential. However, the rapidly oscillating parts of both theexponential and the pre-exponential can always be com-bined into the Bessel generating function (8). Doing so,we expand each term in the probability into sums overBessel harmonics, writing, in this expression, z ≡ z ( ϕ )and z (cid:48) ≡ z ( ϕ (cid:48) ), T NLC ( ϕ, ϕ (cid:48) ) e − iz sin( ϕ − ϑ )+ iz (cid:48) sin( ϕ (cid:48) − ϑ ) = ∞ (cid:88) n,n (cid:48) = −∞ e − inϕ + in (cid:48) ϕ (cid:48) + i ( n − n (cid:48) ) ϑ (cid:26) − m J n ( z ) J n (cid:48) ( z (cid:48) ) (A17) − m ξ (cid:18) s − s ) (cid:19)(cid:20)(cid:18) f (cid:16) ϕ Φ (cid:17) + f (cid:16) ϕ (cid:48) Φ (cid:17)(cid:19) J n ( z ) J n (cid:48) ( z (cid:48) ) − f (cid:16) ϕ Φ (cid:17) f (cid:16) ϕ (cid:48) Φ (cid:17)(cid:104) J n +1 ( z ) J n (cid:48) +1 ( z (cid:48) ) + J n − (cid:0) z ) J n (cid:48) − ( z (cid:48) ) (cid:105)(cid:21)(cid:27) . Observe that the only dependence on the angle ϑ isthrough the term exp[ i ( n − n (cid:48) ) ϑ ] (recall G ( ϕ ) is also in-dependent of ϑ , c.f. (A8)). The integral over this anglecan then be performed, giving a δ -function which meansthe probability only has support on n = n (cid:48) , reducing thecomplexity from a doubly infinite sum to a single one.(It is interesting to note that in the calculation for amonochromatic plane wave [47] this factor setting n = n (cid:48) comes instead from a phase integral.)The probability (A14) still has a complicated form,with two phase integrals and an integral over the trans-verse momentum variable | L ⊥ | which resides in the ar-gument z ( ϕ ) of the Bessel functions. As the integralscannot be done analytically, the route forward now is tointroduce a local expansion. We switch to the sum anddifference variables (11) and expand in θ = ϕ − ϕ (cid:48) (cid:28) f ( ϕ/ Φ). Then we have f ( ϕ/ Φ) ≈ f ( ϕ (cid:48) / Φ) ≈ f ( φ/ Φ),and consequently z ( ϕ ) ≈ z ( ϕ (cid:48) ) ≈ z ( φ ) . (A18)Finally, after setting n = n (cid:48) as discussed above, the re-maining terms in the exponential are given by G ( ϕ ) − G ( ϕ (cid:48) ) − nϕ + n (cid:48) ϕ (cid:48) = (cid:20) s η e (1 − s ) (cid:18) | L ⊥ | s m + ξ f (cid:16) φ Φ (cid:17)(cid:19) − n (cid:21) θ . (A19)The only dependence on the phase variable θ now comesfrom (A19), which appears in the exponent of the inte-grand. The integral over θ yields another δ -function, so P (circ)NLC = − αη e (cid:90) d φ (cid:90) d s (cid:90) d | L ⊥ | ∞ (cid:88) n = −∞ δ (cid:18) | L ⊥ | − m (cid:104) η e (1 − s ) sn − s (cid:16) ξ f (cid:16) φ Φ (cid:17)(cid:17)(cid:105)(cid:19) × (cid:26) J n (cid:0) z ( φ ) (cid:1) + 12 ξ f (cid:16) φ Φ (cid:17)(cid:18) s − s ) (cid:19)(cid:104) J n (cid:0) z ( φ ) (cid:1) − J n +1 (cid:0) z ( φ ) (cid:1) − J n − (cid:0) z ( φ ) (cid:1)(cid:105)(cid:27) . (A20)2The remaining momentum integral is now trivial, giving the final result ofd P (circ)NLC d s (cid:39) − αη e ∞ (cid:88) n =1 (cid:90) d φ (cid:26) J n [ z ( φ )] + 12 ξ f (cid:16) φ Φ (cid:17)(cid:18) s (1 − s ) (cid:19)(cid:20) J n [ z ( φ )] − J n +1 [ z ( φ )] − J n − [ z ( φ )] (cid:21)(cid:27) , (A21)in which z ( φ ) = 2 nξ (cid:12)(cid:12) f (cid:0) φ Φ (cid:1)(cid:12)(cid:12)(cid:113) ξ f (cid:0) φ Φ (cid:1) (cid:20) ss n (1 − s ) (cid:18) − ss n (1 − s ) (cid:19)(cid:21) / (A22)and s n = 2 nη e ξ f (cid:0) φ Φ (cid:1) , η e = k · pm , s = k · (cid:96) (cid:48) k · p . (A23)(We have suppressed the argument of s n = s n ( φ/ Φ) forbrevity.) Momentum conservation leads to the condition n ≥ s -integrationof s n / (1 + s n ). So for a circularly polarised plane wavefield, (A5), the LMA gives a direct generalisation of theinfinite monochromatic plane wave result (see e.g. [47]),where the field strength has been localised, i.e. turnedinto a function of phase, φ , by replacing ξ → ξf ( φ/ Φ).As mentioned in the main text, this ad-hoc replacementhas been used in the literature [58], but to the best ofour knowledge, the necessary approximations required,and in fact the validity of the approach, has not beenstudied. In [58] this trick of localising the field strengthin the monochromatic result is also used for the case ofa linearly polarised plane wave pulse. However, we willsee below that the validity of this replacement may notbe applicable in all cases.
2. Linearly polarised plane wave pulse
The derivation of the LMA for a linearly polarisedplane wave pulse mostly follows the same path as forcircular polarisation, and so we just point out the keydifferences, most importantly the reasons why it is notpossible to simply take the standard results for the caseof a linearly polarised monochromatic plane wave (seee.g. [8]) and “localise” the field strength ξ .We will assume the pulse to be linearly polarised inthe ε direction by adopting (4) with δ = 0, hence a µ ( ϕ ) = mξf (cid:16) ϕ Φ (cid:17) ε µ cos ϕ . (A24)Choosing a linearly polarised background leads to addi-tional structure in the final expression for the probability.The source of this is quite simple: in a circularly polarisedpulse the term (A6) quadratic in the gauge potential is slowly-varying with ϕ , but in the linear case containsrapidly oscillating parts, a ( ϕ ) = − m ξ f (cid:16) ϕ Φ (cid:17) cos ϕ . (A25)The appearance of the cos ϕ term means that we mustimplement both slowly-varying envelope approximations,(5) and (6), for the linear terms. The exponent in (A2)is then (cid:90) ϕ −∞ (cid:96) (cid:48) · π p k · ( p − (cid:96) (cid:48) ) (cid:39) G ( ϕ ) − α ( ϕ ) sin ϕ + β ( ϕ ) sin 2 ϕ , (A26)where we have introduced the function G ( ϕ ) = s η e (1 − s ) (cid:20)(cid:18) | (cid:96) (cid:48) ⊥ − s p ⊥ | s m (cid:19) + ξ f (cid:16) ϕ Φ (cid:17)(cid:21) ϕ , (A27)and the abbreviations α ( ϕ ) = ξf ( ϕ Φ ) η e m (1 − s ) (( (cid:96) (cid:48) − sp ) · ε ) ,β ( ϕ ) = s η e (1 − s ) ξ f (cid:16) φΦ (cid:17) . (A28)Notice that α ( · ) ≡ α c ( · ) (see (A9)) and thus dependson the projection of the photon momentum along ε ,but that β ( ϕ ) is independent of the perpendicular direc-tions. These simple observations have far reaching con-sequences. Most notably, the two trigonometric termsin (A26) cannot be combined as was done in (A13).Therefore, each of the oscillating terms in the exponen-tial will have to be expanded individually, once they havebeen put into the form of the Bessel generating function(8). Furthermore, after implementing the expansion intoBessel harmonics, the probability will depend on termslike J n [ α ( ϕ )], the argument of which depends on boththe magnitude | L ⊥ | and the angle ϑ (using the notationof the circularly polarised case). As such, only one ofthe two integrals coming from the perpendicular compo-nents of the outgoing photon momentum can be done, In the monochromatic limit, f →
1, the use of linear polarisationis also well known to add some additional complexity to theprobability as the quadratic term in a circularly polarised pulseis constant , while it varies with the phase in the linear case.Compare e.g. the results of [8] (linear) with [47] (circular) fornonlinear Compton scattering. | L ⊥ | ). Rememberthat for circular polarisation the simple dependence onthe angle ϑ in (A17) meant that the integral over ϑ couldbe performed, and the probability then only had supporton n = n (cid:48) . This is not the case for linear polarisation,and one finds that the number of harmonic sums can- not be reduced to the same amount as for the case of aninfinite monochromatic plane wave.With all this in mind we jump ahead to the probability,expand in the phase difference variable, θ = ϕ − ϕ (cid:48) . andperform all the remaining integrals which can be doneanalytically. Defining the combinations of Bessel func-tionsΓ ,n ( φ ) ≡ ∞ (cid:88) r = −∞ J n +2 r [ α ( φ )] J r [ β ( φ )] , (A29)Γ ,n ( φ ) ≡ ∞ (cid:88) r = −∞ { J n +2 r +1 [ α ( φ )] + J n +2 r − [ α ( φ )] } J r [ β ( φ )] , (A30)Γ ,n ( φ ) ≡ ∞ (cid:88) r = −∞ { J n +2 r +2 [ α ( φ )] + J n +2 r − [ α ( φ )] + 2 J n +2 r [ α ( φ )] } J r [ β ( φ )] , (A31)with arguments α ( φ ) = − ( n + n (cid:48) ) ξ (cid:12)(cid:12) f (cid:0) φ Φ (cid:1)(cid:12)(cid:12) cos ϑ (cid:113) ξ f (cid:0) φ Φ (cid:1) (cid:114) w n + n (cid:48) (cid:16) − w n + n (cid:48) (cid:17) , β ( φ ) = ξ f (cid:0) φ Φ (cid:1) η e s − s (A32)and abbreviations w n + n (cid:48) = ss n + n (cid:48) (1 − s ) , s n + n (cid:48) = ( n + n (cid:48) ) η e ξ f (cid:0) φ Φ (cid:1) , (A33)the probability for linearly polarised nonlinear Compton scattering finally becomes P ( lin ) NLC (cid:39) α πη e ∞ (cid:88) n =1 ∞ (cid:88) n (cid:48) =1 (cid:90) s n + n (cid:48) d s (cid:90) π d ϑ (cid:90) d φ exp (cid:0) − i (cid:0) n − n (cid:48) (cid:1) φ (cid:1) × (cid:26) − Γ ,n (cid:0) φ (cid:1) Γ ,n (cid:48) (cid:0) φ (cid:1) − ξ f (cid:16) φ Φ (cid:17)(cid:18) s − s ) (cid:19)(cid:20) Γ ,n (cid:0) φ (cid:1) Γ ,n (cid:48) (cid:0) φ (cid:1) + Γ ,n (cid:0) φ (cid:1) Γ ,n (cid:48) (cid:0) φ (cid:1) − ,n (cid:0) φ (cid:1) Γ ,n (cid:48) (cid:0) φ (cid:1)(cid:21)(cid:27) . (A34)Due to the additional structure and the infinite sum-mations it is not possible to simply take the standardmonochromatic plane wave result and “localise” the fieldstrength, as could be done in the circularly polarised case.In principle, the appearance of this extra structure opensup the possibility of interference between different har-monics. However, in the parameter regime investigatedin the main text we did not find a case where this inter-ference was significant.
3. Nonlinear Breit-Wheeler Pair Production
Nonlinear Breit-Wheeler pair production [31, 32, 55] isthe decay of a photon, of momentum (cid:96) and polarisation ε , into an electron-positron pair, with momenta p (cid:48) and q (cid:48) respectively: γ ( (cid:96) ) → e − ( p (cid:48) ) + e + ( q (cid:48) ). In vacuum, thisprocess is forbidden as it violates energy-momentum con-servation, but here is made possible through the interac-tion with a background electromagnetic field. Again, theamplitude is given in terms of the Volkov wave functions(1), namely S BW = − ie (cid:90) d x ¯Ψ p (cid:48) ( x ) /(cid:15) (cid:96) Ψ − q (cid:48) ( x ) e − i(cid:96) · x . (A35)The derivation of the LMA is exactly the same as fornonlinear Compton scattering, and so we do not give anydetails here. Instead, we simply state the final resultfor the case of the circularly polarised plane wave (A5),namely4d P ( circ ) BW d r (cid:39) αη γ ∞ (cid:88) n>n (cid:63) (cid:90) d φ (cid:26) J n (cid:0) z ( φ ) (cid:1) − ξ f (cid:16) φ Φ (cid:17)(cid:18) r (1 − r ) − (cid:19)(cid:20) J n (cid:0) z ( φ ) (cid:1) − J n +1 (cid:0) z ( φ ) (cid:1) − J n − (cid:0) z ( φ ) (cid:1)(cid:21)(cid:27) (A36)where we have employed the abbreviations z ( φ ) = 2 nξ (cid:12)(cid:12) f (cid:0) φ Φ (cid:1)(cid:12)(cid:12)(cid:113) ξ f (cid:0) φ Φ (cid:1) (cid:20) r n (1 − r ) r (cid:18) − r n (1 − r ) r (cid:19)(cid:21) / , r n = 2 nη γ ξ f , η γ = k · (cid:96)m , n (cid:63) = 2(1 + ξ f ) η γ . (A37) Appendix B: Infra-red limit ( s → ) of nonlinearCompton scattering A well known discrepancy between the locally-constantfield approximation and exact QED results is the fail-ure of the former in the “infra-red”, s →
0, limit of theemitted photon spectrum. This is a consequence of per-forming a local expansion in θ = ϕ − ϕ (cid:48) (cid:28) s spectrum is dominatedby contributions from large θ [9]. Here we present a newderivation of this limit from the full QED probability inan arbitrary plane wave pulse, and show that the samelimit can be obtained trivially in the LMA.The probability of nonlinear Compton scattering canbe expressed in differential form as (see e.g. [23, 62]), d P ds = − απη e (cid:90) ∞−∞ d φ (cid:90) ∞ d θ sin (cid:18) θµ η e s − s (cid:19) (cid:26) µ ∂µ∂θ + g ( s ) θ (cid:2) a ( φ + θ/ − a ( φ − θ/ (cid:3) (cid:27) . (B1)Here we employ the Kibble effective mass µ introducedin (16) for the gauge potential a µ = (0 , a ), whence µ ( φ, θ ) = 1 + 1 θ (cid:90) φ + θ/ φ − θ/ a − (cid:18) θ (cid:90) φ + θ/ φ − θ/ a (cid:19) . (B2)Rescaling the phase difference variable, θ = ts , (B3)the Kibble effective mass (B2) becomes, to leading order in small s , lim s → µ ∼ st (cid:90) ∞−∞ a , (B4)which is independent of the phase variable φ . The deriva-tive of the Kibble mass appearing in (B1) is then triviallyof order s . Using also that, as s → g ( s ) → /
2, andreplacing 1 / (1 − s ) → d P ds ∼ απη e (cid:90) ∞−∞ d φ (cid:90) ∞ d tt sin (cid:18) t η e (cid:19) (cid:2) a ( φ + t/ s ) − a ( φ + t/ s ) · a ( φ − t/ s ) (cid:3) , (B5)in which we have also shifted integration variables tocompactify the expression. To proceed, we Fourier trans-form the gauge potentials, make use of (cid:90) d φ e iω ( φ + t/ s ) e iν ( φ ± t/ s ) = 2 πδ ( ω + ν ) e it ( ω ∓ ν ) / s , (B6) to get rid of the integral over φ , and put the differentialprobability in the formd P d s ∼ απη e (cid:90) d ω a ( ω ) · a (cid:63) ( ω ) × (cid:90) ∞ d tt sin (cid:18) t η e (cid:19) (cid:20) − cos (cid:18) tωs (cid:19)(cid:21) . (B7)5The remaining integral over t can now be performedexactly, (cid:90) ∞ d tt sin (cid:18) t η e (cid:19) (cid:20) − cos (cid:18) tωs (cid:19)(cid:21) = − π (cid:20) − (cid:18) − | ω | s (cid:19) + sign (cid:18) | ω | s (cid:19)(cid:21) . (B8)In the limit s → s → (cid:90) ∞ d tt sin (cid:18) t η e (cid:19) (cid:20) − cos (cid:18) tωs (cid:19)(cid:21) = π , (B9)so that the infra-red limit finally becomeslim s → d P d s = α η e (cid:90) d ω a ( ω ) · a (cid:63) ( ω ) . (B10)This agrees with the result found in [10].Now, the locally-constant field approximation is wellknown to fail in predicting the correct value for the s → z ( φ ), defined in (A22),has leading order behaviour z ( φ ) → s → z → z → J m ( z ) = (cid:26) m = 00 for m (cid:54) = 0 . (B11)So the only term that remains non-zero in the s → ∝ J n − ( z ) in (A21), with n = 1. Then, afterFourier transforming the remaining terms and calculat-ing the resulting trivial integrals, one recovers precisely(B10). The same argument carries through for linear po-larisation as well. Appendix C: High-field limit ( ξ (cid:29) ) of the LMA We noted in the main text that the locally-constantfield approximation can be derived as the high-field limitof the LMA; we show this explicitly here. We will focuson the simplest case, nonlinear Compton scattering in acircularly polarised background field, c.f. (A21).We begin by considering the behaviour of the argumentof the Bessel function (A22) for ξ (cid:29) s , there is a minimum value of theharmonic number given by n min = ¯ ξ χ e s − s (cid:20) ξ (cid:21) , (C1) where we use the shorthand ¯ ξ = ξf and defined χ e = ¯ ξη e .We note that as ¯ ξ → ∞ , n ∼ ¯ ξ /χ e , and hence thecorresponding harmonic order at a fixed value of s be-comes very large. In this limit, the behaviour of theBessel function terms may be determined as follows. Let v = s/s n, ∗ where s n, ∗ = s n / (1 + s n ) is the edge of the n th harmonic. This removes any dependency on n fromthe the s integration range: (cid:90) s n, ∗ d s → s n, ∗ (cid:90) d v. Then the argument of the Bessel functions becomes: z = 2 n ¯ ξ (cid:112) ξ (cid:20) s n, ∗ s n v − vs n, ∗ (cid:18) − s n, ∗ s n v − vs n, ∗ (cid:19)(cid:21) / . (C2)Recalling that s n = 2 nη e / (1 + ¯ ξ ), we see that, in thelimit of ¯ ξ → ∞ , keeping all other variables fixed, z → nζ ( v ), where ζ is independent of n . In the high-fieldlimit, ¯ ξ (cid:29)
1, the function ζ ( v ) tends tolim ¯ ξ (cid:29) ζ ( v ) (cid:39) v (1 − v )] / . (C3)In this limit, the main contribution to the probabilitycomes from the vicinity of ζ ∼ v ∼ /
2. Inother words, the main contribution comes from the re-gion where z ∼ n . Using the high field limit of n , theargument of the Bessel functions, z , can be shown toapproach the finite value z → ¯ ξ χ e s − s . (C4)To proceed we follow the approach of Ritus [8] andintroduce a new parameter, τ = ¯ ξ (cid:34)(cid:18) zχ e ¯ ξ (1 − s ) s (cid:19) − (cid:35) , (C5)which characterises the difference between z and its lim-iting high-field value at the points of maximum contri-bution to the probability (with a normalisation set forlater convenience). Then, using the relationship between z and the harmonic number n , we can express n in termsof τ , n = ¯ ξ χ e s − s (cid:18) τ ¯ ξ (cid:19) + n min . (C6)In the probability one now exchanges the order of sum-mation (over n ) and integration (over s ). In the high-fieldlimit, one can replace the summation by an integrationover τ such that,6 P NLC (cid:39) − αη e (cid:90) d φ (cid:90) ∞ d s (cid:90) ∞− ¯ ξ/ d τ (cid:18) ¯ ξ χ e s − s (cid:19) (cid:26) J n ( nζ ) − ¯ ξ (cid:18) s (1 − s ) (cid:19)(cid:20) − ζ ζ J n ( nζ ) + J (cid:48) n ( nζ ) (cid:21)(cid:27) . (C7)As noted previously, in the high field limit, the minimumvalue n min for the harmonic number becomes large, andthe main contribution to the probability comes from theregions where ζ ∼
1. These two conditions allow us to use the Watson representation [83] of the Bessel functions, J n ( nζ ) (cid:39) (cid:16) n (cid:17) / Ai( y ) , y = (cid:16) n (cid:17) / (cid:16) − ζ (cid:17) , (C8)where Ai( y ) is an Airy function. Implementing the Wat-son representation, expanding around ¯ ξ (cid:29) u = s/ (1 − s ) we can approximate y (cid:39) (cid:16) u χ e (cid:17) / (1 + τ ) , (C9)such that the probability turns into P NLC (cid:39) − αη e (cid:90) d φ (cid:90) ∞ d u (1 + u ) (cid:90) ∞ d τ (cid:16) uχ e (cid:17) / (cid:26) Ai ( y ) − (cid:16) χ e u (cid:17) / (cid:18) u (1 + u ) (cid:19)(cid:104) y Ai ( y ) + Ai (cid:48) ( y ) (cid:105)(cid:27) . (C10)In (C10) we made use of the fact that the only depen-dence of the probability on τ is through (C9), and so theintegration in τ is symmetric in the ¯ ξ (cid:29) T = ( u/ χ e ) / τ , use the two Airy function identities [8, 83] y Ai ( y ) + Ai (cid:48) ( y ) = 12 d d y Ai ( y ) , (C11) (cid:90) ∞ d T √ T Ai ( A + T ) = 12 (cid:90) ∞ / A d x Ai( x ) , (C12)and define¯ z = (cid:16) uχ e (cid:17) / Ai (¯ z ) = (cid:90) ∞ ¯ z d x Ai( x ) , (C13)to cast the probability into the form P NLC (cid:39) − αη e (cid:90) d φ (cid:90) ∞ d u (1 + u ) (cid:26) Ai (¯ z ) + 2¯ z (cid:18) u (1 + u ) (cid:19) Ai (cid:48) (¯ z ) (cid:27) (C14)Finally, changing variables back to s = u/ (1 + u ) the probability can be put in the form P NLC (cid:39) − αη e (cid:90) d φ (cid:90) d s (cid:26) Ai (¯ z ) + (cid:18) z + sχ √ ¯ z (cid:19) Ai (cid:48) (¯ z ) (cid:27) (C15)This is exactly the locally-constant field approximation, cf. (A.14) of [23], where χ γ ≡ sχ . 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