Spin and polarisation dependent LCFA rates for nonlinear Compton and Breit-Wheeler processes
SSpin and polarisation dependent LCFA rates for nonlinear Compton andBreit-Wheeler processes
D. Seipt
1, 2, ∗ and B. King † Helmholtz Institut Jena, Fr¨obelstieg 3, 07743 Jena, Germany The G´erard Mourou Center for Ultrafast Optical Science,University of Michigan, Ann Arbor, Michigan 48109, USA Centre for Mathematical Sciences, University of Plymouth, Plymouth, PL4 8AA, United Kingdom
In this paper we derive and discuss the completely spin- and photon-polarisation dependentprobability rates for nonlinear Compton scattering and nonlinear Breit-Wheeler pair production.The local constant crossed field approximation, which is essential for applications in plasma–QEDsimulation codes, is rigorously derived from the strong-field QED matrix elements in the Furrypicture for a general plane-wave background field. We discuss important polarisation correlationeffects in the spectra of both processes. Asymptotic limits for both small and large values of χ arederived and their spin/polarisation dependence is discussed. I. INTRODUCTION
High-intensity laser experiments have now reached thepoint of being able to investigate the strong-field regime ofquantum electrodynamics (QED). In this novel regime, el-ementary particles, such as electrons and photons, interactnonperturbatively with extremely strong electromagneticfields. Recent measurements performed at the RAL-CLF’sGemini laser already hint at the relevance of quantumeffects in radiation reaction [1, 2]. The next generation ofmulti-PW high-power lasers [3–7] (for a review, see [8]),will allow a thorough exploration of this new regime.The two key strong-field QED processes to be investi-gated here are: the emission of a photon by an electron(or positron), known as nonlinear Compton scattering (NLC) [9–11] and the decay of a high-energy photon intoan electron-positron pair, known as the nonlinear Breit-Wheeler (NBW) process [11, 12]. For upcoming strong-field experiments it will be important not only to knowthe kinematic dependence and the particle spectra, butalso the spin and polarisation dependency of these pro-cesses. First, because polarised high-energy electrons andphotons find numerous applications such as nuclear spec-troscopy [13] and in being ideal probes for strong-field loopprocesses such as photon-photon scattering [14–17]. Sec-ond, to correctly model the coherent part of higher-ordereffects such as the trident process [18–27] (creation of anelectron-positron pair from a photon emitted by nonlin-ear Compton scattering) and double nonlinear Comptonscattering [28–32], the polarisation of the intermediateparticle must be taken into account. This naturally posesthe question of how important polarisation effects arein the modelling of strong-field electromagnetic cascades[33–41], which are, in general higher than second orderin multiplicity. Third, several processes in the exten-sion of strong-field QED to beyond-the-standard-modelphysics, are sensitive to particular polarisation channels, ∗ [email protected] † [email protected] such as axionic nonlinear Compton scattering, which, dueto the emission of a pseudoscalar, only proceeds witha spin-flip [42–44], and the decay of an axion into anelectron-positron pair, which has a preference for the spinof the particles produced [45].If the electromagnetic background is sufficiently weak,then spin and polarisation effects can be studied in per-turbative QED. For linear Compton scattering this hasbeen done in many works, with Klein and Nishina alreadystudying the effect of photon polarisation [46], and otherslooking at the role of spin and polarisation [47–50]. Simi-larly, the photon polarisation dependence of pair produc-tion in the collision of two photons was already consideredin the seminal work of Breit and Wheeler [51]. However,if the intensity of the background is strong enough that onaverage more than one photon interacts with an electron,one must consider nonlinear QED processes, typicallystudied in a plane-wave background .To investigate the relevance of the electron spin in theNLC and NBW processes several authors compared cal-culations for spin-1/2 Dirac particles with correspondingspin-0 Klein-Gordon particles [52–55]. The effect of theelectron being a spin-1/2 particle on the radiated lightspectrum was also studied experimentally for the caseof strong crystal fields (channeling radiation) [56]. Thedifference brought by the electron having a spin is thatthe spin—and its associated magnetic moment—can “flip”during the photon emission which is a quantum effect. Ina quantum treatment of radiation emission the photonspectrum has contribution from both the electric chargeand from the magnetic moment, and both of these con-tributions are present even if the incident particles areunpolarised and the final state polarisation remains un-observed. The contribution of spin-flips to NLC has beenstudied in a plane-wave pulse in comparison to classicalradiation calculations [57, 58]. Note the spin can also flipin a laser background due to a non-radiative process (e.g.involving the (dressed) mass operator), which has beenstudied in [59, 60]. The combination of radiative and non-radiative spin-flipping has been studied semiclassically ina constant magnetic field [61]. a r X i v : . [ phy s i c s . p l a s m - ph ] J u l In NLC, the emitted photon polarisation has beenstudied for an electron in a constant crossed field[18, 19, 31, 62, 63], in a monochromatic plane-wave back-ground [64] and recently in a plane-wave pulse [65, 66].The dependence of NLC on the incident electron spinhas been calculated in [67, 68] from the one-loop massoperator via the optical theorem (therefore yielding no in-formation about final state polarisation properties). Thespin-polarisation dependence of NLC in a monochromaticplane wave has been studied in [69] without consideringthe photon polarisation, in [70] including the photon po-larisation, and in [71] in a pulse. In [72] the electronspin-polarisation (averaged over photon polarisation) forNLC in a short pulse has been investigated using the den-sity matrix formalism where also the LCFA was calculated.For the case of a constant, homogeneous magnetic field,in which an electron produces (quantum) synchrotronradiation, there have also been several studies for thespin-polarised but photon-unpolarised case [73, 74], andall particles polarised [75, 76]. In some works specialemphasis was placed on radiation by the anomalous spinmagnetic moment [74, 77]. For a review on spin-polarisedparticle beams in synchrotrons see e.g. [78].For NBW pair production, the effect of photon polarisa-tion (but unobserved spin state of the pair) has been calcu-lated in a monochromatic plane wave [9, 18, 79, 80] and ina constant crossed field [63, 67, 80]. Similar calculationshave been performed also for constant magnetic fields [81]and arbitrary constant electromagnetic fields [82]. Thespin of electrons and positrons produced in NBW has alsobeen studied for a monochromatic background [64], andthe completely polarised NBW cross sections in a stronglinearly and circularly polarised monochromatic plane-wave have been calculated in [83]. Numerical results fora pulsed plane wave were obtained in [84]. Spin-resolvedpair production in a strong field has been calculated alsofor various different field configurations (and productionprocesses) [85–88].In the rest frame of an ultra-relativistic charge, an ar-bitrary strong electromagnetic field “looks” like a crossedfield (as shown by e.g. the Weizs¨acker-Williams approx-imation [89, 90]). If the field is sufficiently intense, thelength scale on which both NLC and NBW are “formed”,is much shorter than the length scale of the shortest in-homogeneity in the laser pulse, namely its wavelength.Hence, the probabilities can be calculated using a “locallyconstant” field approximation (LCFA). The significanceof strong-field quantum effects can be quantified usingthe quantum nonlinearity parameter , χ , which is definedfor electrons and photons as χ e,γ = | F µν p ν ( e,γ ) | / ( mE cr ),where F is the background field strength tensor, p ( e,γ ) is the probe particle momentum, and E cr = m / | e | isthe Sauter-Schwinger critical field of QED, with electronmass m and charge e <
0. By colliding a high-energyelectron beam with an intense laser pulse it is possible toreach the regime where χ e ∼ ξ (cid:29) ξ /χ (cid:29)
1, where ξ = ( m/κ )( E/E cr ), E is the field strength, and κ is the frequency of the background, which has the meaning ofa Keldysh-type parameter. For practical purposes, andwith χ ∼
1, the LCFA can be considered a reasonableapproximation for ξ (cid:38)
10 [72, 93], despite its known limi-tations [94, 95]. Monte-Carlo sampling of the LCFA rates[62, 95–98] is the central method by which strong-fieldQED effects are included in high-intensity laser-plasmasimulations [39, 99–101]. Some polarised LCFA rateshave been already implemented in (Monte Carlo) simu-lation codes to investigate the radiative self-polarisationof fermions in different field configurations [102–106] andto model photon polarisation effects [63, 107], as well aspolarised QED cascade formation [41].A reasonable amount of work has already been per-formed in investigating the role of polarisation and spinin different processes and different electromagnetic back-grounds. Yet, a systematic study of all spin and polar-isation effects of the NLC and NBW processes and aconsistent derivation of the LCFA is still lacking. Thisis achieved in this paper. The results for the completelypolarised LCFA rates presented in this paper are suitablefor a direct implementation in such numerical frameworks.In the current paper we present compact analytical ex-pressions for the fully polarised NLC and NBW processes.We calculate these processes in a plane-wave pulse, fromwhich the LCFA is derived. Asymptotic formulas for thefully polarised rates are given for small and large values ofthe seed particle’s quantum parameter, χ , and comparedquantitatively to the full LCFA. All polarisation channelsin each of the processes are visualised for various quantumparameter, and the relative ordering of each channel isexplained phenomenologically.The paper is organised as follows. In Sec. II we intro-duce the polarisation and spin bases and give an overviewof the kinematics, crossing symmetry and general struc-ture of the probabilities. Secs. III and IV present theresults for NLC and NBW respectively. Both sectionsinclude a presentation of the results for a plane waveand for the LCFA, for which the spin polarised asymp-totic scaling for large and small quantum nonlinearityparameter is given. Noteworthy aspects of the resultsare discussed at the end of each section. In Sec. III, wealso include an overview of the derivation. In Sec. Vthe paper is concluded. Appendices A and B containa detailed derivation of the results for NLC and NBWrespectively. Throughout the paper we employ naturalHeaviside-Lorentz units with (cid:126) = c = ε = 1. II. POLARISATION BASIS
We begin by introducing the polarisation states of vec-tor and spinor particles that will be appear throughoutthe calculation and in our final results for the fully po-larised nonlinear Compton (NLC) and nonlinear Breit-Wheeler (NBW) rates. We will concentrate on the caseof a linearly polarised plane-wave laser pulse of arbi-trary temporal shape. We introduce the laser polari-sation ε µ and four-wavevector κ µ , satisfying ε.ε = − κ.κ = 0 and ε.κ = 0. The normalised vector potentialof the background, a = eA with e <
0, depends onlyon the phase variable φ = κ.x , and can be given by a µ ( φ ) = mξε µ h ( φ ), with the classical nonlinearity param-eter ξ and an arbitrary shape function h ( φ ). In addition,it is useful to define the constant background field ten-sor f µν = κ µ ε ν − κ ν ε µ . Let us also define the magneticfield polarisation, β , satisfying β.β = − β.ε = β.κ = 0.The spatial components of the four-vectors ( ε, β, κ ) needto form a right-handed triad. For instance, in the labframe we can choose κ = ω (1 , , , ε = (0 , , , β = (0 , , , ω is the laser frequency. This en-sures that their spatial parts fulfill κ /ω = (cid:15) × β , i.e. κ agrees with the direction of the background field Poyntingvector. A. Photon polarisation Basis
With the triad of basis vectors ( ε, β, κ ), we can de-fine a (linear) polarisation basis for a photon with four-momentum k asΛ = ε − k.εk.κ κ ; Λ = β − k.βk.κ κ . (1)By construction the polarisation basis vectors fulfil k. Λ j =0 and Λ i . Λ j = − δ ij . An arbitrarily polarised photon (in apure state) with polarisation four-vector (cid:15) k can thereforebe written as the superposition (cid:15) k = c Λ + c Λ . (2)We will characterise the photon polarisation state usingthe Stokes parameter τ k = | c | − | c | , where τ k is, ingeneral, a real number and τ k ∈ [ − , τ k is chosen to be an integer and τ k ∈ {− , } , then thephoton is produced in an eigenstate of the polarisationoperator [108], and therefore the polarisation will notprecess as the photon propagates through the background(reviews of photon-photon scattering can be found in[109–111]). In this paper, we will consider the case that τ k ∈ {− , , } , where τ k = 0 indicates an unpolarisedphoton (in a mixed state), i.e. a polarisation average. (Forunobserved final state polarisation one has to multiplythe result by 2.) A photon in the Λ polarisation stateis polarised parallel to the laser polarisation direction ina frame in which k and κ are collinear. A photon in theΛ polarisation state is polarised perpendicular to thelaser. Hence, we may refer to these photons as (cid:107) - and ⊥ -polarised photons, respectively. B. Fermions
The spin basis can be chosen in a similar way to thephoton polarisation basis. It is useful to define a basisthat does not precess in the background field. Also, thebasis cannot depend on spacetime co-ordinates, otherwise we would be modifying the spacetime dependency of theVolkov solution, which would not fulfill the Dirac equationanymore. For linearly-polarised backgrounds in the ε direction our basis for the spin four-vector of an electronwith momentum p becomes: ζ p = β − p.βp.κ κ . (3)Then we see that ζ p .ζ p = − ζ p .p = 0, but also veryusefully: ζ p .κ = 0 and ζ p .ε = 0. The choice of this basisvector implies that we are looking specifically at light-fronttransverse polarisation, with the spin-vectors orientedalong the magnetic field in the rest frame of the particle.An important aspect of this choice of the spin-quantisationaxis is that then F.ζ p = 0, where F is the backgroundfield strength tensor. This fact immediately ensures thatthe spin-vector of the particles does not precess under theBargman-Michel-Telegdi (BMT) equation [112],d S µ d τ = eg e m F µν S ν − e ( g e − m u µ ( u.F.S ) , (4)where S µ is a general spin-polarisation vector, g e theelectron gyromagnetic ratio, and u µ its four-velocity. Al-though ζ p is defined using the asymptotic momentum, p , we see that we can replace, without loss of generality, p with the “instantaneous” classical kinetic momentum π p (= mu ) of the electron in a plane-wave background, π p ( φ ) = p − a + κ p.aκ.p − κ a.a κ.p , (5)and hence ζ π ≡ ζ p .The choice of the basis above S µ = ζ µ therefore ensuresthat d S µ / d τ = 0. Thus, the asymptotic polarisation stateof the particles agrees with the local values inside thestrong background field. This is a special choice of spinbasis. In general one could expand the spin vectors in adreibein: S µ = S ζ ζ µp + S η η µp + S κ κ µp , where η p and κ p aretwo additional space-like unit four-vectors perpendicularto p , and defined as η µp = ε µ − κ µ ( p.ε ) / ( p.κ ) and κ µp = mκ µ / ( κ.p ) − p µ /m , (noting F.η (cid:54) = 0 and F. κ (cid:54) = 0). Thus,the BMT equation would imply that a general spin vectorprecesses. It can be shown that the vectors ( ζ p , η p , κ p )are pointing in the direction of the background magneticfield, electric field, and wave-vector in the rest frame ofthe particle [72].The Dirac bi-spinors are defined using the spin basis ζ p , which is manifest in the density matrices [113]: u pσ p ¯ u pσ p = 12 ( /p + m )(1 + σ p γ /ζ p ) , (6) v pσ p ¯ v pσ p = 12 ( /p − m )(1 + σ p γ /ζ p ) , (7)where we explicitly introduce the spin index σ p = ± ↑ , σ p = +1) or anti-parallel (spin- ↓ , σ p = −
1) to ζ p . C. General considerations for thepolarisation-resolved probabilities p, σ p q, σ q k, j k, j q, σ q p, σ p FIG. 1. Feynman diagrams. Left: nonlinear Compton scat-tering (NLC). Right: nonlinear Breit-Wheeler (NBW) pairproduction.
Nonlinear Compton scattering (NLC) and nonlinearBreit-Wheeler (NBW) pair production are both 1 → p in , theoutgoing momentum of the particle under study as p out ,and the momentum of the outgoing particle we integrateover (i.e. its momentum is not observed, but its polarisa-tion state is) as the ancillary momentum q . We orient thecoordinate system in such a way that the laser propagatesalong the positive z -axis, i.e. κ + = 2 ω (where light-frontmomentum components are defined p ± = p ± p ) is theonly non-vanishing light-front component of κ µ . Then,for both processes, NLC and NBW, the light-front mo-mentum conservation can be expressed as p − in = p − out + q − , p ⊥ in = p ⊥ out + q ⊥ , (8)with p ⊥ = ( p , p ), and the exchange of “+” momen-tum between the particles and the background field doesnot yield a conservation law. In a plane-wave back-ground, one can write the S-matrix element using thefour-dimensional light-front delta function δ (4) ( P + (cid:96)κ ) =2 δ ( P + + (cid:96)κ + ) δ ( P − ) δ (2) ( P ⊥ ) = 2 δ ( P + ) δ l . f . ( P ) as: S = − ie (2 π ) (cid:90) d (cid:96) π δ (4) ( P + (cid:96)κ ) M , (9)where P = p in − p out − q , and the integral over (cid:96) takes intoaccount exactly the non-conservation of +-momentum.The amplitude M is specific to each process, and con-tains all the spin- and polarisation dependence. Thephase-space integrated probability for the process under consideration is then given by P = 12 p − in (cid:90) (cid:103) d q (cid:103) d p out | S | =: (cid:90) dΓ | S | (10)with the Lorentz-invariant on-shell phase space elementsunderstood in light-front coordinates, i.e. (cid:103) d q = d q − d q ⊥ (2 π ) q − .The conservation of three light-front momentum com-ponents in Eq. (8) allows one to completely integrateout the ancillary momentum q . The final particle phasespace of p out is conveniently parametrised by the nor-malised light-front momentum transfer s and transversemomentum r ⊥ : s := p − out p − in = κ.p out κ.p in , r ⊥ := p ⊥ out ms . (11)We thus can write the final particle phase space asdΓ = d q − d q ⊥ (cid:18) mp − (cid:19) s − s d s d r ⊥ π ) . (12)Moreover, for the squared S matrix we find | S | = (2 π ) e (cid:18) κ + (cid:19) δ l . f ( P ) | M | , (13)where we used the normalisation δ l . f . (0) = π ) . Inte-grating out the ancillary momentum q consumes the deltafunction and allows the total probability to be expressedas P = α π m b (cid:90) d s s − s (cid:90) d r ⊥ | M | , (14)with fine structure constant α = e / π , and quantumenergy parameter b = p in .κ/m . The squared amplitudeis given by a double integral over the laser phase, whichtakes the form | M | = (cid:90) d φ d φ (cid:48) e i Φ T j . (15)Here, the integrand is a product of a nonlinearly-oscillating factor, the trace of Dirac matrices T j =Λ µj T µν Λ νj containing the fermion spin structure in T µν ,and photon polarisation vectors Λ µj . The specific formof these expressions depends on the considered process.In the following sections, they are evaluated in a linearlypolarised plane wave laser background, first for nonlinearCompton scattering and then for nonlinear Breit-Wheelerpair production. From the general plane-wave results, wethen rigorously derive the locally constant field approxi-mation. III. NONLINEAR COMPTON SCATTERING
This section is devoted to the investigation of fullypolarised nonlinear Compton scattering, i.e. the emissionof a polarised photon by a spin-polarised electron, wherealso the spin-polarisation after the photon emission isobserved. We restrict the discussion to the case of allparticles being in polarisation eigenstates as discussedabove: Initial (final) electrons can be spin-polarised σ p = ± σ q = ±
1) along the axis ζ p ( ζ q ); photons are emittedin polarisation eigenstate Λ j , j = 1 , A. S-Matrix
We begin by recalling the basic properties of Volkovstates, which are solutions of the Dirac equation in aplane-wave background,( i /∂ − e /A − m )Ψ pσ p ( x ) = 0 , (16)and, with the normalised vector potential a = eA , givenbyΨ pσ p ( x ) = E p ( x ) u pσ p , (17) E p ( x ) = (cid:18) /κ/a p.κ (cid:19) exp (cid:26) − ip.x − (cid:90) d φ a.p − a.a κ.p (cid:27) , (18)where E p are the “Ritus matrices”, u pσ p are the Diracbi-spinors, and where σ p = ± ζ p . Because ζ p = ζ π they remainpolarised in that state during the interaction with thelaser prior to emitting a photon — and after.The normalised vector potential a of the backgrounddepends only on the phase variable φ = κ.x , and is rep-resented by a µ ( φ ) = mξε µ h ( φ ), where ξ is the classicalnonlinearity parameter [62], ε µ is the polarisation vectorobeying ε.ε = −
1, and h ( φ ) is an arbitrary shape func-tion. Examples of shape functions include h ( φ ) = cos φ ,for a linearly polarised infinite plane wave; and h ( φ ) = φ for a constant crossed field. We now write for the (nor-malised) field strength tensor F µν = mξf µν ˙ h ( φ ), where f µν = κ µ ε ν − κ ν ε µ is a constant tensor and ˙ h ( φ ) = d h/ d φ .The S-matrix element for this strong-field QED process,see Fig. 1 left, reads S NLC ( σ p , σ q , j ) = − ie (cid:90) d x ¯Ψ qσ q ( x ) / Λ j e ik.x Ψ pσ p ( x )= − ie (2 π ) (cid:90) d (cid:96) π δ (4) ( p + (cid:96)κ − q − k ) M NLC (19)with the amplitude M NLC ( σ p , σ q , j ) =Λ j,µ (cid:90) d φ e i (cid:82) dφ k.πp ( φ ) κ.q ¯ u qσ q J µ NLC ( φ ) u pσ p , (20)and the Dirac current which is independent of thepolarisation properties of all particles J µ NLC ( φ ) = γ µ + /a/κγ µ κ.q ) + γ µ /κ/a κ.p ) + /a/κγ µ /κ/a κ.p )( κ.q ) . (21) B. NLC Probability
With the results from Eq. (14) we can write the proba-bility as: P NLC ,j ( σ p , σ q ) = α π m b p (cid:90) d s s − s (cid:90) d r ⊥ | M NLC ( σ p , σ q , j ) | , (22)where the squared amplitude is given by a double phaseintegral over a dynamic phase factor, which is independentof the particle polarisation, multiplied by T j , which isthe Dirac trace T µν , contracted with the outgoing photonpolarisation vectors, T j = Λ µj T µν ( σ p , σ q )Λ νj . Explicitly, | M NLC ( σ p , σ q , j ) | = (cid:90) d θ d ϕ e iθ k. (cid:104) πp (cid:105) κ.q T j , (23)with θ = φ − φ (cid:48) , ϕ = ( φ + φ (cid:48) ) /
2, and the floating averagedefined by (cid:104) π p (cid:105) = (cid:104) π p (cid:105) ( ϕ, θ ) = 1 θ (cid:90) ϕ + θ/ ϕ − θ/ d φ (cid:48)(cid:48) π p ( φ (cid:48)(cid:48) ) . (24)The dynamic phase for Compton scattering is given by θ k. (cid:104) π p (cid:105) κ.q = sθ b p (1 − s ) [ µ + ( r ⊥ − (cid:104) π ⊥ p (cid:105) /m ) ] (25)with normalised Kibble’s mass µ = 1 + ξ (cid:104) h (cid:105) − ξ (cid:104) h (cid:105) , (26)energy parameter b p = κ.p/m , and s = κ.k/κ.p . Thespin trace T µν = 14 tr (cid:104) ( /q + m )(1 + σ q γ /ζ q ) J µ ( φ ) × ( /p + m )(1 + σ p γ /ζ p )¯ J ν ( φ (cid:48) ) (cid:105) . (27)can be decomposed into four parts: unpolarised ( UP ),initially polarised ( IP , depends only on the initial elec-tron polarisation), finally polarised ( FP , depends only onthe final electron polarisation), and polarisation correla-tion ( PC , depends on both the initial and final electronpolarisation). These terms are defined as follows: T µν ( σ p , σ q ) = UP µν + σ p IP µν + σ q FP µν + σ p σ q PC µν , (28)with the four contributions UP µν ≡
14 tr (cid:104) ( /q + m ) J µ ( φ ) ( /p + m ) ¯ J ν ( φ (cid:48) ) (cid:105) , (29) FP µν ≡
14 tr (cid:104) ( /q + m ) γ /ζ q J µ ( φ ) ( /p + m ) ¯ J ν ( φ (cid:48) ) (cid:105) , (30) IP µν ≡
14 tr (cid:104) ( /q + m ) J µ ( φ ) ( /p + m ) γ /ζ p ¯ J ν ( φ (cid:48) ) (cid:105) , (31) PC µν ≡
14 tr (cid:104) ( /q + m ) γ /ζ q J µ ( φ ) ( /p + m ) γ /ζ p ¯ J ν ( φ (cid:48) ) (cid:105) , (32)where the NLC current from Eq. (21) and its Dirac-adjoint¯ J = γ J † γ have to be inserted. ( FeynCalc [114, 115]was used to calculate the traces.) Then, the expressionfor the differential probability, can be written asd P NLC ,j d s ( σ p , σ q ) = α π m b p s − s (cid:90) d ϕ (cid:90) d θ (cid:90) d r ⊥ × e iθ k. (cid:104) πp (cid:105) κ.q [ UP j + σ p IP j + σ q FP j + σ p σ q PC j ] , (33)for photons emitted in a polarisation state j = 1 ,
2. In-troducing the Stokes parameter τ k of the emitted photon then yieldsd P d s ( σ p , σ q , τ k ) = 1 + τ k P d s + 1 − τ k P d s . (34)After evaluating the total of eight different traces, andanalytically performing the integration in r ⊥ , which isGaussian (the technical details of these steps are giving inAppendix A), and regularizing the resulting expressions(e.g. with an “ iε ” prescription [116]), one arrives at theexpression for the NLC spectrum in a plane-wave pulse :d P NLC d s = − α πb p (cid:90) d ϕ (cid:90) d θ − iθ e ix θµ I NLC , (35) I NLC = 1 + σ p σ q + (1 − g ) τ k σ p σ q − ξ ∆ h τ k (1 + gσ p σ q ) + ξ (cid:104) ˙ h (cid:105) θ g + σ p σ q ) − iθξ (cid:104) ˙ h (cid:105) (cid:20) sσ p + s − s σ q + τ k (cid:18) sσ q + s − s σ p (cid:19)(cid:21) , (36)where ∆ h = ( h ( φ ) −(cid:104) h (cid:105) )( h ( φ (cid:48) ) −(cid:104) h (cid:105) ), x = s/ [2 b p (1 − s )]and g = 1 + s / [2(1 − s )]. A numerical evaluation of thisexpression calls for an additional regularization of thatpart of I NLC not containing the laser pulse, i.e. being ∝ ξ . Several methods for this regularization have beendiscussed in the literature [23, 117, 118].The appearance of a pre-exponential term proportionalto 1 /θ , see e.g. Eqs. (A21) and (A30), is known frompolarised calculations in a plane wave [66]. In the expres-sion above it has already been treated using integration by parts, giving termsd( θµ )d θ = 1 + ξ ∆ h + θ ξ (cid:104) ˙ h (cid:105) . (37)To acquire the LCFA, and specifically a local rate , oneperforms an expansion of the exponent in Eq. (35) tocubic order in θ and each term in the pre-exponent toleading order θ . Then, the integrals over θ can be per-formed analytically. Let us define the probability rate R = d P / d ϕ as the probability for emission per unit laserphase. Combining (A52) and (A53), the differential NLCrate for all particles polarised is then given byd R NLC d s ( σ p , σ q , τ k ) = − α b p (cid:20) { σ p σ q + τ k σ p σ q (1 − g )) } Ai ( z )+ (cid:26) sσ p + s − s σ q + τ k (cid:18) s − s σ p + sσ q (cid:19)(cid:27) Ai ( z ) √ z sign ( ˙ h ( ϕ ))+ (cid:26) g + σ p σ q + τ k gσ p σ q (cid:27) (cid:48) ( z ) z (cid:21) . (38)The argument of the Airy function Ai ( · ), its deriva-tive Ai (cid:48) ( · ) and integral Ai ( z ) := (cid:82) ∞ z d x Ai ( x ) is z =( sχ e ( ϕ )(1 − s ) ) / and depends on the local value χ e ( ϕ ) = χ p | ˙ h ( ϕ ) | , where χ p = ξb p . The term sign ( ˙ h ( ϕ )) in thesecond line of (38) appears because of the oscillating na-ture of a plane wave pulse. It shows that this particular term switches its sign each half cycle of the wave togetherwith the direction of the magnetic field. Hence, in anoscillating field with many cycles one can expect that thisterm averages to zero when integrating the rate over thepulse if the field has a certain symmetry such that, inte-grated over a cycle (cid:82) d ϕ sign ( ˙ h ( ϕ ))Ai ( z ) / √ z ≈
0. Since z only depends on | ˙ h | this is the case if the field has some(generalised) parity property ˙ h ( φ ± φ ) ≈ − ˙ h ( φ ) for some φ . In order to efficiently radiatively polarise electronsthis symmetry needs to be broken, for instance using anultra-short sub-cycle pulse [72], or by a bi-chromatic (two-color) field [104]. By superimposing a 2nd harmonic withthe correct phase, e.g. ˙ h = cos φ + cos 2 φ , the (generalisedparity) symmetry is broken and it is impossible to find a φ such that − ˙ h ( φ ) ≈ ˙ h ( φ ± φ ). Similar arguments alsohold for NBW pair production [105].From this expression we can straightforwardly recoverliterature results for the partially polarised cases. Thecase for unobserved photon polarisation is acquired bysetting τ k = 0 and multiplying the result by 2 (for thesum over the final polarisation states)d R NLC d s ( σ p , σ q , τ k = 0) = − α b p (cid:20) (1 + σ p σ q )Ai ( z )+ (cid:18) sσ p + s − s σ q (cid:19) Ai ( z ) √ z sign ( ˙ h )+( g + σ p σ q ) 2Ai (cid:48) ( z ) z (cid:21) . (39)This result agrees with the diagonal elements of the spin-density matrix in Ref. [72].The rate for unpolarised final state particles, but po-larised initial electrons, had been calculated, e.g. by Ritusvia the imaginary part of the one-loop electron mass op-erator [67, 68]. We can obtain this from the generalexpression by setting σ q = τ k = 0 and multiplying by 4 totake into account the summation over final state particlesd R NLC d s ( σ p , σ q = 0 , τ k = 0) = − αb p (cid:20) Ai ( z ) + sσ p Ai ( z ) √ z sign ( ˙ h ) + g (cid:48) ( z ) z (cid:21) . (40)Finally, the case of unpolarised electrons, but polarisedphotons can be achieved by setting σ p = σ q = 0 andmultiplying by 2 (which is equivalent to performing anaverage over incoming spins and a sum over outgoingones) to achieve:d R NLC d s ( σ p = 0 , σ q = 0 , τ k ) = − α b p (cid:20) Ai ( z ) + (2 g + τ k ) Ai (cid:48) ( z ) z (cid:21) , (41)which agrees with literature results [63].Finally, the completely unpolarised nonlinear Comptonrate is obtained by setting σ p = σ q = τ k = 0 and multi-plying by 4 for the summation over the final electron spinand photon polarisation states, yielding [9, 10].d R NLC d s ( σ p = 0 , σ q = 0 , τ k = 0) = − αb p (cid:20) Ai ( z ) + 2 g Ai (cid:48) ( z ) z (cid:21) . (42) We can also make a connection to the expressions cal-culated by Sokolov and Ternov in a constant and homoge-neous magnetic field. Translating the Airy functions intomodified Bessel functions of the second kind and setting˙ h = 1 we get perfect agreement with the expressions fromthe literature [75]. C. Discussion of the Compton Rates
To discuss the relative and absolute weight of the eightdifferent polarisation channels, we plot the different NLCemission rates for a constant value of χ e = χ p in Fig. 2.We can make the following general remarks. For thetotal yield of photons due to each polarisation channel,shown in Fig. 2, we see the channels without a spin-flip are much larger than those with a spin-flip. Thedominant contribution is the non-flip transition whenthe polarisation of the emitted photon is in the (cid:107) state(which is approximately parallel to the background electricfield for a near head-on collision of electron and laserpulse). The non-flip channels with the photon emittedin the ⊥ polarisation state are next in the hierarchy ofrates. All spin-flip rates are much lower than the non-flip rates. Especially for χ p (cid:28) χ p (c.f. the discussion of theasymptotic behaviour below). The most probable spin-flip channel is the emission of a perpendicularly polarisedphoton during an ↑ to ↓ transition. − − χ p − − − − b p α χ p R σ p σ q τ k N L C ↑ ↑ k↑ ↑ ⊥ ↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥ ↓ ↑ k↓ ↑ ⊥ FIG. 2. Polarisation resolved total rates for nonlinear Comptonscattering as a function of the electron quantum parameter χ p . In order to visualise how the differential photon spec-trum comprises each polarisation channel, in Fig. 3 weselect four constant values of χ p at different orders of mag-nitude: χ p ∈ { . , , , } . In general, the hierarchy ofthe various polarisation channels can be different in thelow-energy infra-red part of the spectrum compared to thehigh-energy UV part of the spectrum. For small s → s is the fraction of photon light-front momentum) s − − b p α d R σ p σ q τ k N L C d s χ p = 0 . ↑ ↑ k↑ ↑ ⊥ ↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥ ↓ ↑ k↓ ↑ ⊥ s − − b p α d R σ p σ q τ k N L C d s χ p = 1 s − b p α d R σ p σ q τ k N L C d s χ p = 1010 − − − s − b p α d R σ p σ q τ k N L C d s χ p = 100 FIG. 3. Plots of the polarisation resolved differential Comp-ton spectra as functions of the normalised photon light-frontmomentum s = k − /p − for four different values of χ p . the rate of spin-flip channels go to zero, showing thatthe well known (integrable) infrared divergence of thepolarisation averaged LCFA rates originates solely in thenon-flip channels. For larger values of s the non-flip andspin-flip channels approach each other, and eventuallythe hierarchy even changes with certain spin-flip channelsbecoming larger than some non-flip channels. (In otherwords, it is not just the flip of the spin that determinesthe hierarchy of rates.) As χ p is increased, the part ofthe spectrum where the hierarchy between polarisation channels changes, moves to larger values of s . (Note thatby the conservation law Eq. (8) the final state electronnormalised light-front momentum is just 1 − s .)Also as χ p increases, a new spectral feature develops inthe high-energy part of the spectrum at s ≈
1. In Fig. 4we show the development of this “UV shoulder” in moredetail. Whilst the UV shoulder is known to exist and todevelop into a pronounced peak approximately locatedat s ∼ − / χ p for χ p (cid:29) (cid:107) -photon is emitted and the electron stays in a down state.For incident up electrons, a ⊥ -photon is emitted whilethe electron flips to a down state. Thus, by controllingthe incident electron polarisation one could control thepolarisation of the generated gamma rays in this highenergy feature of the spectrum. Because the photonshave very high energy, almost all of the incident electronenergy is transferred to the photon. The existence ofthe UV shoulder can be clearly seen in calculations oftwo-step part of second-order processes such as nonlineartrident (NLC followed by NBW) [22, 23, 25, 121], andits existence has been commented on as contributing tofree-particle “shower” type cascades [63].Although in this section we have thus far focussed on theNLC process for electrons , analogous arguments apply tothe NLC process for positrons . We note here the necessarychanges. First, in the classical kinetic momentum of theelectron in a plane wave π p ( φ ) from Eq. (5), a = eA ,where e < | e | >
0. Thus, the classical kinetic momentum of apositron differs from that of an electron. The correctexpression taking into account the change in the sign ofthe charge, is given by − π − p ( φ ).Moreover, the different sign of the charge for electronsand positrons implies that the vector of the magneticmoment and spin are parallel in one case and antiparallelin the other case. That means, the spin-field interactionhas the opposite sign for positrons. Thus, in order toemploy the electron NLC rates, Eq. (38), for positronsone also has to make the replacements: σ p → − σ p and σ q → − σ q . It is evident that this affects neither the termsin (38) containing the product σ p σ q nor does it affect thespin-averaged rates, Eqns. (41) and (42). D. Asymptotic Limits
For the discussion of the asymptotics of the rate forlarge and small values of χ p it is convenient to treat spinflip ( σ q = − σ p ) and non-flip ( σ q = σ p ) separately, aswe will find them to have different asymptotic behaviour.Here we choose the quantum parameters, χ e = χ p and χ γ = χ k , occurring in the LCFA, to take constant values,which is equivalent to considering the case of a constant . . . . . . s b p α d R σ p σ q τ k N L C d s χ p = 10 σ p σ q τ k ↑ ↑ k↑ ↑ ⊥↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥↓ ↑ k↓ ↑ ⊥ . . . . . . s b p α d R σ p σ q τ k N L C d s χ p = 100 σ p σ q τ k ↑ ↑ k↑ ↑ ⊥↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥↓ ↑ k↓ ↑ ⊥ FIG. 4. Differential Compton rate for χ p = 10, 100 on alinear scale, highlighting the formation of the UV shoulderat s (cid:39) χ p which has a very strong polarisationdependence. The electrons emerging from this interaction arestrongly down-polarised. There is also a strong correlationbetween the initial electron polarisation and the polarisationof the emitted photons in the shoulder. crossed field background. χ p (cid:28) For nonlinear Compton scattering, the χ k parameterof the emitted photon (which is bounded above by the χ p parameter of the initial electron) quantifies the recoil whenthe electron emits a photon. Furthermore, the incomingelectron parameter, χ p , is such that χ p , χ k ∝ (cid:126) . Therefore,the limit of χ p → R NLC for small χ p (cid:28) s to z (the argument of the Airy functions) and performing a systematic power series expansion in χ p , yielding R σ p ,σ p ,τ k NLC ∼ αχ p b p √ (cid:20)
52 + 32 τ k − (cid:18) σ p (1 + τ k ) + 4 + 3 τ k √ (cid:19) χ p + (cid:18) √ σ p (1 + τ k ) + 548 (75 + 62 τ k ) (cid:19) χ p (cid:21) , (43) R σ p , − σ p ,τ k NLC ∼ αχ p b p √ (cid:34) − τ k + √ σ p (1 − τ k ) (cid:35) , (44)as χ p →
0. In the non-flip rate, Eq. (43), the leading orderis O ( χ p ), and the leading order is independent of the spinstate of the incoming electron. A spin-splitting (differencebetween up and down incident electrons) only occurs inthe order O ( χ p ) and only for (cid:107) photon polarisation, τ k =+1 (there is no spin-splitting at all for the ⊥ polarisation).For the spin-flip rate, the leading term suppressed at O ( χ p ). Here, the leading term does show spin-splitting,but only for the ⊥ photon polarisation ( τ k = − χ p (cid:28) χ p (cid:29) The asymptotic expansion of the NLC rate for large χ p (cid:29) z . The resultingintegrals can be easily performed for the leading orderterms stemming from the Ai and Ai (cid:48) –terms, yielding R σ p ,σ p ,τ k NLC ∼ αχ / p b p Γ( )18 · / (cid:20) (cid:16) τ k (cid:17) − (3 χ p ) − / σ p (1 + τ k ) 72 Γ( )Γ( ) (cid:21) , (45) R σ p , − σ p ,τ k NLC ∼ αχ / p b p Γ( )18 · / (cid:20) − τ k χ p ) − / σ p (1 − τ k ) 52 Γ( )Γ( ) (cid:21) , (46)as χ p → ∞ . In this asymptotic limit, for both the spin-flip and non-flip rates the leading order term is O ( χ / p )and independent of the spin of the incident electron. Spindependence only occurs in the next to leading order, whichis O ( χ / p ). This term completely vanishes for unpolarisedelectrons, where the next non-vanishing term is O (1).To illustrate the asymptotic scaling of the relationsin Eqs. (43)–(46), and their accuracy, the χ p (cid:28) − − χ p − − − − − − b p α χ p R σ p σ q τ k N L C ↑ ↑ k↑ ↑ ⊥ ↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥ ↓ ↑ k↓ ↑ ⊥ FIG. 5. Comparison of the total nonlinear Compton rates(colored curves) with their asymptotic expansions (black dash-dotted curves) for χ p (cid:28) χ p (cid:29) χ p → χ p compared to the non-flipchannels. However, we also see that the value of χ p atwhich the asymptotic scaling reaches a prescribed level ofaccuracy, changes, depending on the order of the scaling.To make this manifest, in Fig. 5 (bottom) we plot therelative error of the asymptotic expression, as a functionof χ p . Generally speaking, to arrive at a given accuracy,the asymptotic relations for the spin-flip channels require χ p to be an order of magnitude more asymptotic, e.g.in the case χ p (cid:28)
1, an order of magnitude smaller thanfor the non-flip channels. For example, in the χ p → χ p ≈ .
1, whereas it requires χ p ≈ .
01 for thesame accuracy in the asymptotic relations of the spin-flipchannels. Likewise, it is remarkable that the asymptoticexpressions in the χ p → ∞ limit, see Fig. 6 (bottom), onlyreach an accuracy of 10% for χ p (cid:38) for the spin-flipchannels. For the spin-flip channel and emission of photoninto the ⊥ polarisation, this accuracy is reached at anorder of magnitude even larger than this. With present χ p − − b p α χ / p R σ p σ q τ k N L C ↑ ↑ k↑ ↑ ⊥ ↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥ ↓ ↑ k↓ ↑ ⊥ FIG. 6. Comparison of the (scaled) total nonlinear Comptonrates with their asymptotic expansions (black dash-dottedcurves) for χ p (cid:29) day laser and accelerator technology one can only reachvalues of χ p (cid:46)
10, and so large χ p asymptotic expressionscan only be used cautiously. IV. NONLINEAR BREIT-WHEELER PAIRPRODUCTIONA. S-matrix
To calculate the probability for nonlinear Breit-Wheelerpair production we need to utilise the Volkov state for an(outgoing) positron, which is given by [86, 122]Ψ ( − ) pσ p ( x ) = E − p ( x ) v pσ p , (47)with the Ritus matrices, Eq. (18), constant positron bi-spinors v pσ p , and where the superscipt “ − ” signifies thatthe positron Volkov state is a negative energy solution ofthe Dirac equation (16). Employing (47), the S-matrixelement of NBW, see Fig. 1 (right), can be expressed as1follows: S NBW ( kj → pσ p ; qσ q )= (cid:90) d x ¯Ψ qσ q ( x )[ − ie/(cid:15) j e − ik.x ]Ψ ( − ) pσ p ( x )= − ie (2 π ) (cid:90) d (cid:96) π δ (4) ( k + (cid:96)κ − q − p ) M NBW (48)We emphasise that p ( σ p ) is the four-momentum (spinindex) of the created positron and q ( σ q ) refers to theelectron. The nonlinear Breit-Wheeler amplitude M NBW ( σ p , σ q , j ) =Λ µ,j (cid:90) d φ e − i (cid:82) k.π − pκ.q d φ ¯ u qσ q J µ NBW ( φ ) v pσ p (49)can be expressed in terms of the current J µ NBW ( φ ) = γ µ + (cid:20) /a/κγ µ κ.q ) − γ µ /κ/a κ.p ) (cid:21) h ( φ ) − /a/κγ µ /κ/a κ.p )( κ.q ) h ( φ ) , (50)where the kinetic momentum of the positron is given by − π − p ( φ ), with π p ( φ ) given in Eq. (5). The Dirac trace for NBW is: T µν = 14 tr (cid:104) ( /q + m )(1 + σ q γ /ζ q ) J µ ( φ ) × ( /p − m )(1 + σ p γ /ζ p )¯ J ν ( φ (cid:48) ) (cid:105) (51)and using the current from (50), can be decomposedinto four parts: unpolarised ( UP ), electron polarised ( EP ),positron polarised ( PP ), and polarisation correlation ( PC ),which are defined as follows: T µν ( σ p , σ q ) = UP µν + σ q EP µν + σ p PP µν + σ p σ q PC µν , (52)with the four contributions UP µν ≡
14 tr (cid:104) ( /q + m ) J µ ( φ ) ( /p − m ) ¯ J ν ( φ (cid:48) ) (cid:105) , (53) EP µν ≡
14 tr (cid:104) ( /q + m ) γ /ζ q J µ ( φ ) ( /p − m ) ¯ J ν ( φ (cid:48) ) (cid:105) , (54) PP µν ≡
14 tr (cid:104) ( /q + m ) J µ ( φ ) ( /p − m ) γ /ζ p ¯ J ν ( φ (cid:48) ) (cid:105) , (55) PC µν ≡
14 tr (cid:104) ( /q + m ) γ /ζ q J µ ( φ ) ( /p − m ) γ /ζ p ¯ J ν ( φ (cid:48) ) (cid:105) . (56) B. Pair Production Probability
The evaluation of the traces for NBW pair production ispresented in detail in Appendix B. With those, and afterperforming the integration over the transverse momentumof the outgoing positron, we find the fully polarisationresolved NBW pair production probability in a linearlypolarised plane wave of arbitrary shape:d P NBW d s = α πb k (cid:90) d ϕ (cid:90) ∞−∞ d θ − iθ e iθµ ˜ x I NBW , (57) I NBW = 1 + σ p σ q + τ k σ p σ q (1 − ˜ g ) + ξ θ (cid:104) ˙ h (cid:105) g + σ p σ q ] − ξ ∆ h τ k [1 + ˜ gσ p σ q ] − iθξ (cid:104) ˙ h (cid:105) (cid:20) σ p s − σ q − s + τ k (cid:18) σ q s − σ p − s (cid:19)(cid:21) , (58)with the positron’s light-front momentum fraction s = p − /k − , ˜ g = 1 − s (1 − s ) , Kibble mass Eq. (26) and ˜ x defined in Eq. (B5). The initial photon is in a polarisationstate (cid:15) characterised by the Stokes parameter τ k = | c | −| c | , where (cid:15) = c Λ + c Λ . In addition, the definitionof ∆ h given below Eq. (35), as well as the statements about regularization apply here as well.Details of the derivation of the LCFA, including the inte-grals over the phase variable θ are collected in AppendixB. Here we present the final result for the completelypolarised NBW pair production rate within the LCFA2d R NBW d s ( σ p , σ q , τ k ) = α b k (cid:20) { σ p σ q + τ k σ p σ q (1 − ˜ g ) } Ai (˜ z )+ (cid:26) σ p s − σ q − s + τ k (cid:18) σ q s − σ p − s (cid:19)(cid:27) Ai (˜ z ) √ ˜ z sign ( ˙ h )+ (cid:26) (˜ g + σ p σ q ) + τ k gσ p σ q (cid:27) (cid:48) (˜ z )˜ z (cid:21) (59)where the argument of the Airy functions is given by˜ z = [ χ γ ( ϕ ) s (1 − s )] − / . The photon quantum parameter χ γ ( ϕ ) again refers to the local value in the field, givenby χ γ ( ϕ ) = χ k | ˙ h ( ϕ ) | , where χ k = ξb k . The quantumenergy parameter b k = k.κ/m is related to the center-of-mass energy of the incident photon colliding with theplane wave laser field. We emphasise again that s = p.κ/k.κ is the fractional light-front momentum of the positron in relation to the light-front momentum of theincident photon. Likewise, σ p refers to the spin state ofthe positron, and σ q to the spin-state of the electron.It is straightforward to recover expressions for totally orpartially unpolarised channels. For instance, for the decayof a polarised photon into an unpolarised pair we have tosum over all fermion polarisations, which is equivalent tosetting σ p = σ q = 0 and multiplying the result by 4:d R NBW d s ( σ p = 0 , σ q = 0 , τ k )= αb k (cid:20) Ai (˜ z ) + { g + τ k } Ai (cid:48) (˜ z )˜ z (cid:21) . (60)This agrees with expressions from the literature [63](Sometimes in the literature the Stokes parameter is ex-pressed as τ k = cos 2 ϑ , where ϑ is the angle of the photonpolarisation in relation to the laser polarisation, charac-terised by Λ ).To obtain the completely unpolarised NBW rate wehave to average Eq. (60) over the incoming photon polar-isation by setting τ k = 0:d R NBW d s ( σ p = 0 , σ q = 0 , τ k = 0)= αb k (cid:20) Ai (˜ z ) + 2˜ g Ai (cid:48) (˜ z )˜ z (cid:21) . (61)We can also find the result for the production of apolarised pair by unpolarised photons by just setting τ k = 0d R NBW d s ( σ p , σ q , τ k = 0) = α b k (cid:20) { σ p σ q } Ai (˜ z ) + { g + σ p σ q ) } Ai (cid:48) (˜ z )˜ z + (cid:26) σ p s − σ q − s (cid:27) Ai (˜ z ) √ ˜ z sign ( ˙ h ) (cid:21) . (62) C. Discussion of the Pair Production Rates − χ k − − − − − b k α χ k R σ p σ q τ k N B W ↑ ↑ k↑ ↑ ⊥ ↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥ ↓ ↑ k↓ ↑ ⊥ FIG. 7. Scaled total spin/polarisation resolved NBW pairproduction rates as a function of χ k . We illustrate the results for the various polarisationchannels of NBW pair production in a series of plots,starting with the total rates in Fig. 7. All of the eightchannels are strongly suppressed for small χ k . This is areflection of the fact that NBW pair production, unlikeNLC, is a pure quantum process that must vanish inthe classical limit as χ k →
0. (See also the detaileddiscussion of the asymptotic behaviour below.) Similarto NLC scattering, the plot of total NBW rates (see Fig.7) shows a certain hierarchy of the polarisation channelswhich does not change with χ k , apart from one particularchannel where a ⊥ -photon produces a pair with positronspin σ p = ↓ and electron spin σ q = ↑ . In this channel thepair is produced in its least favourable spin state since theelectron spin is aligned parallel to the magnetic field andthe positron is aligned anti-parallel to the magnetic field.This channel is one of the smallest contributions to theoverall rate for small χ k (cid:28)
1, but is one of the dominantones for χ k (cid:29)
1. In general, the most probable channelis the one in which a photon polarised in the ⊥ statedecays into a pair in which the spins are aligned such thattheir interaction energy with the background magneticfield is minimised, i.e. the electron (positron) is alignedantiparallel (parallel) with the field [76]. This can be seen3from the energy in the rest frame of the particle [123], U B = − µ · B , with µ = eg e s / m (and recalling e < s are the spatial components of thespin four-vector S = σ p ζ p in the electron rest frame. s . . . . b k α d R σ p σ q τ k N B W d s × − χ k = 0 . ↑ ↑ k↑ ↑ ⊥ ↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥ ↓ ↑ k↓ ↑ ⊥ s . . . . b k α d R σ p σ q τ k N B W d s χ k = 1 s b k α d R σ p σ q τ k N B W d s χ k = 100 . . . . . . s b k α d R σ p σ q τ k N B W d s χ k = 100 FIG. 8. Spin/polarisation resolved differential NBW pair pro-duction rates. Observe how photons of different polarisationproduce pairs with different symmetry properties. For thelargest contribution which comes from ⊥ -photons the spectraare symmetric around s = 1 /
2, and the pair has anti-parallelspins with both particles in the desired (lower energy) state.For (cid:107) -photons the pair has preferably parallel spins, withone high-energy particle and one low-energy particle beingproduced.
In Fig. 8 we plot the light-front momentum spectrum ofproduced positrons for a range of incoming photon quan-tum parameters, χ k ∈ { . , , , } . It is evident that,especially for smaller values of χ k , only a few channels aredominant. For increasing χ k we see new peak structuresappear close to s ∼ , s = 1 /
2, i.e. symmetric in the exchange of electron andpositron s ↔ − s . In Fig. 8 we clearly see that not allpolarisation resolved channels adhere to this symmetry.In particular the channels in which a (cid:107) -photon decays intoa pair with parallel spins (i.e. only one of the particles isin its preferred energy state), break the symmetry about s = 1 /
2, meaning that one of the particles is preferablycreated with a higher momentum than the other one.
D. Asymptotic Limits
Here we provide and discuss the asymptotic limits forsmall and large constant values of χ k for the spin andpolarisation dependent pair production rates. Here it isconvenient to distinguish the case of parallel spins σ q = σ p ,and anti-parallel spins σ q = − σ p . χ k (cid:28) For small χ k , the asymptotic scaling of the total NBWrate can be calculated by using the fact that for χ k (cid:28) z is always large. Per-forming an asymptotic expansion of the Airy functions forlarge ˜ z yields integrals with a factor e − z / / which canbe treated using Laplace’s method [124]. The exponentialterm turns into the e − / χ k suppression of the pair pro-duction rates which shows up in all combinations of spinand photon polarisation and reflects the fact that pairproduction behaves like a tunneling process in the semi-classical limit for small χ k (cid:28)
1. Distinguishing the caseof parallel spins and anti-parallel spins of the generatedpair we find R σ p ,σ p ,τ k NBW ∼ αb k χ k e − χk (cid:114) (cid:20) τ k + 13(1 + τ k )3 · χ k + 14677 + 11221 τ k · χ k (cid:21) , (63) R σ p , − σ p ,τ k NBW ∼ αb k χ k e − χk (cid:114) (cid:20) (1 + σ p )(1 − τ k )16+ 25 − τ k + 13 σ p (1 − τ k )3 · χ k −
707 + 5005 τ k − σ p (1 − τ k )3 · χ k (cid:21) , (64)as χ k → χ k (cid:28) explicitly depends on the incident photon polarisation. For all chan-nels there is an overall exponential suppression factor e − / χ k at small χ k (cid:28)
1, reflecting the tunnelling na-ture of the NBW pair production process for small χ k .It is quite interesting, however, that the exact leadingorder scaling is very much dependent on the specific chan-nel. For parallel spins, the leading order for (cid:107) -photons( τ k = +1) is ∝ χ k e − / χ k , and for ⊥ -photons it is muchsmaller ∝ χ k e − / χ k since the first two terms in (63) areproportional to 1+ τ k . Moreover, there is no spin-splitting,i.e. the cases ↑ ↑ and ↓ ↓ have the same rate. This is infact true not only for small χ k as can be seen for instancein Fig. 7.For anti-parallel spins the leading order of the rateis even more involved. For ⊥ -photons the leading or-der depends on the spin alignment of the positron. Forpositrons produced in the favourable ↑ state, the rateis large, ∝ χ k e − / χ k . However, for positrons producedin the (unfavourable) ↓ state, the leading order is muchsmaller at ∝ χ k e − / χ k . This asymptotic result recon-firms the dominance of the ↑ ↓ ⊥ channel in Fig. 8 for χ k = 0 .
01. For (cid:107) -photons the leading order for anti-parallel spins is ∝ χ k e − / χ k , independent of the spinalignment of the positron.It is know from the literature that in the limit χ k (cid:28) ⊥ -photons ( τ k = −
1) istwice as large as the rate of (cid:107) -photons [62]. Here wehave shown that the former case is dominated by thesingle spin-polarisation channel ↑ ↓ ⊥ . In contrast, for (cid:107) -photons two equally probable channels contribute. Itis also interesting to look at certain ratios of the pairproduction rates for specific incident photon polarisation.For instance, for (cid:107) -photons, τ k = +1, the probability togenerate the pair with anti-parallel spins is suppressedas R σ p , − σ p , +1NBW / R σ p ,σ p , +1NBW ∼ χ k / σ p . For ⊥ -photons, τ k = −
1, we have todistinguish two cases: R σ p ,σ p , − / R , − , − ∼ χ k /
512 and R σ p ,σ p , − / R − , , − ∼ / χ k (cid:28)
1, andin particular in the interesting range 0 . < χ k < χ k < . χ k (cid:29) The asymptotic expansion of NBW pair creation forlarge χ k is calculated in a similar manner as the corre- FIG. 9. Asymptotics for spin/polarisation resolved NBW pairproduction rates for small χ k (cid:28) sponding NLC expressions. The asymptotic expressionsbehave as R σ p ,σ p ,τ k NBW ∼ αχ / k b k · / · / Γ( )Γ( ) (cid:16) τ k (cid:17) , (65) R σ p , − σ p ,τ k NBW ∼ αχ / k b k / · / Γ( )Γ( ) (cid:20) (cid:16) − τ k (cid:17) + χ − / k σ p (1 − τ k ) 2 / · / Γ ( )Γ ( ) (cid:21) , (66)as χ k → ∞ .Here, the scalings with χ k are in principle the sameas for NLC, just the numerical factors are different. Themain difference is that there is no term at order χ / k forthe case of parallel spins. The asymptotic expressions forlarge χ k (cid:29)
1, Eqns. (65)–(66) are plotted in Fig. 10 (top)and the corresponding relative error (bottom).The asymptotic plots of the total yield in Fig. 9 and Fig.10 also display the behaviour of the “anomalous” channel, ↓ ↑ ⊥ . Firstly, it is the only polarisation channel to crossthe others, being the equal least probable channel in the χ k → χ k is increased until the5 χ k − − b k α χ / k R σ p σ q τ k N B W ↑ ↑ k↑ ↑ ⊥ ↓ ↓ k↓ ↓ ⊥ ↑ ↓ k↑ ↓ ⊥ ↓ ↑ k↓ ↑ ⊥ FIG. 10. Asymptotics for spin/polarisation resolved pair pro-duction rates (black dash-dotted curves) in comparison to thefull LCFA for large χ k (cid:29) χ k → ∞ limit where it is the equal most probable channel.It is remarkable that even by χ k as large as O (10 ), it hasnot yet reached its asymptotic value. This fact becomesparticularly clear by looking at the relative error of theasymptotic expansions in the bottom panels of Figs. 9and 10. We notice the same behaviour as in the NLC case,that the leading order asymptotic expressions are moreaccurate already at less extreme asymptotic parameter,whereas the less probable channels require much larger(smaller) values of χ k to reach a given accuracy in the χ k → ∞ ( χ k →
0) limits.
V. SUMMARY
In this paper we have given a comprehensive overviewof the rates of two of the most important strong-field QEDprocesses with the polarisation of all particles taken intoaccount. We introduced expressions for fully polarisednonlinear Compton scattering (NLC) and nonlinear Breit-Wheeler pair-creation (NBW) in a general plane-wavebackground and derived concise formulas for the fullypolarised locally constant field approximation (LCFA) of each process. The asymptotic scaling for each processand all of the eight polarisation channels has been derivedand presented in succinct expressions, and this scalinghas been benchmarked against the full LCFA result. Al-though some of these results exist in other works in theliterature, this is, to the best of our knowledge, the firstcomplete presentation and in-depth analysis of all polari-sation channels together. In doing so, we have been ableto resolve particle spectra by polarisation channel, andhave demonstrated that certain spectral features (such asthe appearance of a “UV shoulder/peak” at large quan-tum parameter), are particular to specific polarisationchannels. We have also identified ”anomalous“ channelsthat change in relative importance as the correspondingquantum parameter is increased.We note from our results that some polarisation chan-nels do not reach their large- χ asymptotic scaling until χ (cid:38) O (10 ). The Narozhny-Ritus conjecture predicts abreakdown of the QED perturbation expansion in dressedvertices when αχ / ∼ O (1) [125–128], i.e. χ ∼ O (10 ).Furthermore, polarised one-vertex tree-level processessuch as in NLC and NBW are necessary in order to cor-rectly factorise higher-order tree-level processes in thisperturbation expansion. Therefore it is likely that theresolution of the Narozhny-Ritus conjecture has implica-tions for the relative importance of polarisation channelsin NLC and NBW at large χ .All our results have been expressed in a polarisationbasis that respects the symmetry of the background field.However, depending on how polarisation is measured inexperiment, the polarisation of any “detector” must beborne in mind. For example, a measurement of high en-ergy photon polarisation has been suggested, which usesthe polarisation-dependent probabilities for Bethe-Heitlerpair-creation in a Coulomb field [16, 129]. Therefore itis the projection of our results onto the natural basis ofthe Bethe-Heitler polarimeter, which will play a role inany detection. The measurement of the spin-polarisationof high-energy electrons is often performed using Møllerpolarimeters [130, 131], which, however, are most sensi-tive to lontigudinal polarisation, or Compton polarime-ters [132, 133] which exploit angular asymmetries in thescattering spectra of linear Compton scattering. Someauthors also propose to use nonlinear QED processesthemselves for polarimetry applications [134, 135]. A re-view for existing and future electron beam polarimetrycan be found in Ref. [136].Even if the polarisation of the incoming or outgoingparticle is not measured, then the LCFA rates for theeight different polarisation channels we have presentedare still relevant for higher-order processes. The correctfactorisation of higher-order processes require a consis-tent polarisation of intermediate particles (propagators)between vertices. In this way, the polarised LCFA ratespresented here can be directly employed in numerical simu-lations of electromagnetic cascades in intense backgroundfields [41].6
Appendix A: Details of the Calculation of the LCFAfor Nonlinear Compton
We start by giving some important kinematic defini-tions: s ≡ κ.kκ.p , (A1) g = 1 + s − s ) (A2)With help of the auxiliary variable L , we can findsome useful kinematic relations for the incident electronmomentum p , outgoing electron momentum q and emittedphoton momentum k , p.q = m + Ls κ.p , (A3) q.k = L κ.p , (A4) p.k = L (1 − s ) κ.p , (A5)where L = s κ.p (1 − s ) (cid:34) m + X ε + X β s (cid:35) = x (cid:20) (cid:16) p ⊥ m − r ⊥ (cid:17) (cid:21) , (A6)and we introduced the normalised transverse momentumof the photon, r ⊥ = k ⊥ /ms , and the auxiliary variables X ε = k.ε − sp.ε and X β = k.β − sp.β . In addition, x = s b p (1 − s ) , (A7)with b p = κ.p/m .
1. NLC Traces
Here we list the expressions for all 8 Dirac traces fornonlinear Compton scattering, Eqs. (29)–(32). They areevaluated using
FeynCalc [114, 115]. Here we use theshort-hand notation h (cid:48) for h ( φ (cid:48) ) and h for h ( φ ), and alsowrite h − h (cid:48) = (cid:82) φφ (cid:48) ˙ h ( ϕ )d ϕ = θ (cid:104) ˙ h (cid:105) , with θ = φ − φ (cid:48) beingthe laser phase difference between the NLC amplitudeand its complex conjugate. UP = q.p − m − m ξ ( s − s − hh (cid:48) + mξ ( s − s − s X ε ( h + h (cid:48) ) + 2 X ε s , (A8) UP = − q.p − m + 2 k.qs + 2 (1 − s ) k.ps − m ξ s s − hh (cid:48) + mξs s − X ε ( h + h (cid:48) ) − s X ε , (A9) IP = iξm θ (cid:104) ˙ h (cid:105) s (2 − s )2(1 − s ) , (A10) IP = − iξm θ (cid:104) ˙ h (cid:105) s − s ) , (A11) FP = iξm θ (cid:104) ˙ h (cid:105) s (2 − s )2(1 − s ) , (A12) FP = iξm θ (cid:104) ˙ h (cid:105) s − s ) , (A13) PC = q.p − m − m ξ ( s − s − hh (cid:48) + mξ ( s − s − s X ε ( h + h (cid:48) ) + 2 s X ε + X β s − , (A14) PC = − q.p + m + 2 k.qs + 2(1 − s ) k.ps + m ξ s s − hh (cid:48) − mξs s − X ε ( h + h (cid:48) ) − s X ε − X β s − q.p − m = Lsκ.p .With these replacements it is straightforward to see thatall traces depend on the transverse photon momentumonly quadratically at most. Here we used that the light-front Levi-Cevita tensor (cid:15) + − xy = −
2, i.e. that Levi-Civitaterms occurring in traces with exactly one γ matrix canbe simplified as (cid:15) pβ(cid:15)κ = p.κ .
2. Gaussian Transverse Momentum Integrals
We find that the transverse momentum integrals over r ⊥ are all Gaussian for all 8 Compton traces. This facthas been customarily exploited in calculations of spin-averaged nonlinear Compton scattering, to analyticallyperform the transverse momentum integrals. Here, the7relevant integrals for polarised NLC read G = (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q = π − iθx e iθx µ , (A16) G ,ε = (cid:90) d r ⊥ X ε e iθ k. (cid:104) πp (cid:105) κ.q = msξ (cid:104) h (cid:105) G , (A17) G ,ε = (cid:90) d r ⊥ X ε e iθ k. (cid:104) πp (cid:105) κ.q = m s (cid:20) ξ (cid:104) h (cid:105) + 1 − iθx (cid:21) G , (A18) G ,β = (cid:90) d r ⊥ X β e iθ k. (cid:104) πp (cid:105) κ.q = 0 , (A19) G ,β = (cid:90) d r ⊥ X β e iθ k. (cid:104) πp (cid:105) κ.q = m s − iθx G , (A20)with x defined in Eq. (A7). With these, we find (we omitthe leading factor G here which has to be multiplied toall traces) UP → ( g − m + i gm θx + m ξ ( g + 1)( h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) , (A21) UP → ( g − m + i gm θx + m ξ ( g − h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) , (A22) IP → iξm θ (cid:104) ˙ h (cid:105) ( g − s ) , (A23) IP → − iξm θ (cid:104) ˙ h (cid:105) ( g − , (A24) FP → iξm θ (cid:104) ˙ h (cid:105) ( g − s ) , (A25) FP → iξm θ (cid:104) ˙ h (cid:105) ( g − , (A26) PC → ( g − m + i m θx + m ξ ( g + 1)( h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) , (A27) PC → − ( g − m + i m θx − m ξ ( g − h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) . (A28)
3. Short Coherence Interval Approximation and θ -integrals With the transverse momentum integrals done, the nextstep towards the LCFA is to expand the integrand of the θ -integral to lowest non-trivial order in the short coherenceinterval θ (cid:28)
1. This allows us to perform the θ -integralsanalytically. (Note that one can alternatively perform the θ -integral first, and not perform the r ⊥ integrals, whichleads to an angularly resolved LCFA, see for instanceRef. [137].) For the Kibble mass in the exponent thatmeans µ → µ = 1 + ξ ˙ h θ /
12 [94]. Furthermore, in thepre-exponential terms we use θ (cid:104) ˙ h (cid:105) → θ ˙ h = ξ ˙ h ( ϕ ) and( h (cid:48) − (cid:104) h (cid:105) )( h − (cid:104) h (cid:105) ) (cid:39) − θ h . (A29)Inserting the small- θ approximated prefactor G (cid:39) πb p − ss e iθx µ − iθ we obtain (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q UP (cid:39) πb p m − ss e iθx µ (cid:34) − gθ x + i ( g − θ − iθ ( g + 1) ˙ h ξ (cid:35) , (A30) (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q UP (cid:39) πb p m − ss e iθx µ (cid:34) − gθ x + i ( g − θ − iθ ( g −
1) ˙ h ξ (cid:35) , (A31) (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q IP (cid:39) − πm b p − ss e iθx µ ξ ˙ h ( g − s ) , (A32) (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q IP (cid:39) πm b p − ss e iθx µ ξ ˙ h ( g − , (A33) (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q FP (cid:39) − πm b p − ss e iθx µ ξ ˙ h ( g − s ) , (A34) (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q FP (cid:39) − πm b p − ss e iθx µ ξ ˙ h ( g − , (A35) (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q PC (cid:39) πb p m − ss e iθx µ (cid:34) − θ x + i ( g − θ − iθ ( g + 1) ˙ h ξ (cid:35) , (A36) (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q PC (cid:39) πb p m − ss e iθx µ (cid:34) − θ x − i ( g − θ + iθ ( g −
1) ˙ h ξ (cid:35) . (A37)8Next we perform the integrals over the phase variable θ yielding Airy functions (cid:90) d θ iθ e ix θ + i y θ = 2 π Ai (cid:48) ( z ) √ y , (A38) (cid:90) d θ e ix θ + i y θ = 2 π Ai ( z ) √ y , (A39) (cid:90) d θ − iθ e ix θ + i y θ = 2 π Ai ( z ) , (A40) (cid:90) d θ θ e ix θ + i y θ = 2 πx (cid:20) Ai ( z ) + Ai (cid:48) ( z ) z (cid:21) , (A41)where Ai ( z ) = (cid:82) ∞ z d x Ai ( x ) and Ai (cid:48) ( z ) = dAi ( z ) / d z .Here we have rewritten the exponential e iθx µ = e ix θ + i y θ with the definitions y = x ξ ˙ h , z = x √ y . (A42) In addition we use that √ y = √ zξ | ˙ h | / ξ ˙ h/ √ y = 2 ˙ h/ ( √ z | ˙ h | ) = 2 sign ( ˙ h ) / √ z .The first and second results follow by the integral defi-nition of the Airy function [138]. The third result can bederived in the following way: (cid:90) ∞−∞ d θθ e i ( rθ + c θ ) = lim ε → (cid:90) ∞−∞ d θθ + iε e i ( rθ + c θ ) = lim ε → − i (cid:90) ∞ d v (cid:90) ∞−∞ d θθ + iε e i (( r + v ) θ + c θ ) − ε vθ = − πi Ai (cid:20) r (3 c ) / (cid:21) , (A43)and the final result was derived in the appendix of Ref. [66].(It turns out the final result is equivalent to integrat-ing once by parts, ignoring the contribution from thepole in the evaluated term, and then using the standardSokhotsky-Weierstrass method to deal with the pole ofthe resulting 1 /θ integration.)Here is the collection of all 8 NLC traces after the θ -integrals have been performed: (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q UP (cid:39) − π m b p − ss (cid:20) Ai ( z ) + 2 g + 1 z Ai (cid:48) ( z ) (cid:21) , (A44) (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q UP (cid:39) − π m b p − ss (cid:20) Ai ( z ) + 2 g − z Ai (cid:48) ( z ) (cid:21) , (A45) (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q IP (cid:39) − π m b p − ss ( g − s ) 2Ai ( z ) √ z sign ( ˙ h ) , (A46) (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q IP (cid:39) − π m b p − ss (1 − g ) 2Ai ( z ) √ z sign ( ˙ h ) , (A47) (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q FP (cid:39) − π m b p − ss ( g − s ) 2Ai ( z ) √ z sign ( ˙ h ) , (A48) (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q FP (cid:39) − π m b p − ss ( g −
1) 2Ai ( z ) √ z sign ( ˙ h ) , (A49) (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q P C (cid:39) − π m b p − ss (cid:20) (2 − g ) Ai ( z ) + g + 2 z Ai (cid:48) ( z ) (cid:21) , (A50) (cid:90) d θ (cid:90) d r ⊥ e iθ k. (cid:104) πp (cid:105) κ.q PC (cid:39) − π m b p − ss (cid:20) g Ai ( z ) − g − z Ai (cid:48) ( z ) (cid:21) . (A51)By combining these results according to Eqn. (33), and by defining the differential probability rate per laser phase asd R / d s = d P / d s d ϕ we findd R d s ( σ p , σ q ) = − α b p (cid:20) (1 + σ p σ q (2 − g ))Ai ( z ) + 2( σ p + σ q )( g − s ) Ai ( z ) √ z sign ( ˙ h )+ (2 g + 1 + σ p σ q ( g + 2)) Ai (cid:48) ( z ) z (cid:21) , (A52)d R d s ( σ p , σ q ) = − α b p (cid:20) (1 + σ p σ q g )Ai ( z ) + 2( σ q − σ p )( g −
1) Ai ( z ) √ z sign ( ˙ h )+ (2 g − − σ p σ q ( g − (cid:48) ( z ) z (cid:21) . (A53)for a photon to be emitted in polarisation state Λ or Λ .9 Appendix B: Details of the Calculation of the LCFAfor Pair Production
For pair production, the incoming channel is charac-terised by the scalar product κ.k , with k the photonfour-momentum. The light-front momentum exchange isdefined here as s = p.κ/k.κ , where p refers to the positron momentum. Hence, for the electron momentum q we have q.κ = (1 − s ) q.κ . Moreover, we define ˜ g = 1 − s (1 − s ) .By introducing the auxiliary variable ˜ L , it is possibleto express p.q = ˜ Lk.κ − m , (B1) q.k = ˜ Lsk.κ , (B2) p.k = ˜ L (1 − s ) k.κ , (B3)where˜ L = ˜ x (cid:34) s (cid:18) r ⊥ − k ⊥ m (cid:19) (cid:35) = m + Y ε + Y β s (1 − s ) k.κ (B4)with ˜ x = 12 b k s (1 − s ) (B5)and Y ε = p.ε − sk.ε and Y β = p.β − sk.β , and the nor-malised transverse positron momentum r ⊥ = p ⊥ /ms . b k = k.κ/m is related to the squared centre-of-massenergy of the incident photons and can be related tothe kinematic pair production threshold of linear Breit-Wheeler via ˜ Lb k ≥
2, or ˜ L ≥ /b k .The dynamic phase of the pair production matrix ele-ment reads (without and with the floating average) − k.π − p κ.q = 12 k.κs (1 − s ) (cid:2) m + ( Y ε − mξh ) + Y β (cid:3) , (B6) − k. (cid:104) π − p (cid:105) κ.q = 12 k.κs (1 − s ) (cid:2) m µ + ( Y ε − mξ (cid:104) h (cid:105) ) + Y β (cid:3) = ˜ x (cid:2) µ + s ( r ⊥ + (cid:104) a ⊥ (cid:105) /s − u ⊥ ) (cid:3) , (B7)with the Kibble mass µ , Eq. (26).
1. NBW Traces
The NBW, Eqs. (53)–(56), traces are calculated in ananalogous way to the NLC traces, UP = q.p + m − m ξ (1 − s ) s − s hh (cid:48) + mξ (1 − s ) s − s ( h + h (cid:48) ) Y ε − Y ε , (B8) UP = − q.p + m + 2(1 − s ) k.p + 2 sk.q − m ξ s − s hh (cid:48) + mξ s − s ( h + h (cid:48) ) Y ε + 2 Y ε , (B9) PP = im ξ θ (cid:104) ˙ h (cid:105) s − s (1 − s ) , (B10) PP = − im ξ θ (cid:104) ˙ h (cid:105) s (1 − s ) , (B11) EP = im ξ θ (cid:104) ˙ h (cid:105) s − s (1 − s ) , (B12) EP = im ξ θ (cid:104) ˙ h (cid:105) s (1 − s ) , (B13) PC = q.p + m − m ξ (1 − s ) s − s hh (cid:48) + mξ (1 − s ) s − s ( h + h (cid:48) ) Y ε − Y ε + Y β ( s − s , (B14) PC = − q.p − m + 2(1 − s ) k.p + 2 sk.q + m ξ s − s hh (cid:48) − mξ s − s ( h + h (cid:48) ) Y ε + 2 Y ε − Y β ( s − s . (B15)By employing the kinematic relations from above it isstraightforward to see that all transverse momentum in-tegrals over the eight traces are Gaussian.
2. Gaussian Transverse Momentum Integrals forNBW
For NBW pair production, all transverse momentumintegrals over d r ⊥ are Gaussian as well. However, theexpression of the dynamic phase is slightly different, and0so are the results:˜ G = (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q = e iθ ˜ x µ π − iθ ˜ x s , (B16)˜ G ,ε = (cid:90) d r ⊥ Y ε e iθ − k. (cid:104) π − p (cid:105) κ.q = mξ (cid:104) h (cid:105) ˜ G , (B17)˜ G ,ε = (cid:90) d r ⊥ Y ε e iθ − k. (cid:104) π − p (cid:105) κ.q = (cid:20) m ξ (cid:104) h (cid:105) + m − i ˜ θx (cid:21) ˜ G , (B18)˜ G ,β = (cid:90) d r ⊥ Y β e iθ k. (cid:104) π (cid:105) κ.q = 0 , (B19)˜ G ,β = (cid:90) d r ⊥ Y β e iθ k. (cid:104) π (cid:105) κ.q = m − iθ ˜ x ˜ G , (B20)with ˜ x defined in Eq. (B5). Employing those Gaussianintegrals, the 8 NBW traces turn to the following expres- sions, omitting again the leading factor ˜ G : UP → (1 − ˜ g ) m − i ˜ gm θ ˜ x − (1 + ˜ g ) m ξ ( h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) , (B21) UP → (1 − ˜ g ) m − i ˜ gm θ ˜ x + (1 − ˜ g ) m ξ ( h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) , (B22) PP → − im ξ θ (cid:104) ˙ h (cid:105) (˜ g − s − ) , (B23) PP → im ξ θ (cid:104) ˙ h (cid:105) (˜ g − , (B24) EP → − im ξ θ (cid:104) ˙ h (cid:105) (˜ g − s − ) , (B25) EP → − im ξ θ (cid:104) ˙ h (cid:105) (˜ g − , (B26) PC → (1 − ˜ g ) m − im θ ˜ x − (1 + ˜ g ) m ξ ( h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) , (B27) PC → − (1 − ˜ g ) m − im θ ˜ x . − (1 − ˜ g ) m ξ ( h − (cid:104) h (cid:105) )( h (cid:48) − (cid:104) h (cid:105) ) (B28)
3. Short Coherence Interval Approximation and θ -Integrals The next step towards the LCFA for NBW is approxi-mating the integrand for short coherence interval θ (cid:28) θ ap-proximation of ˜ G (cid:39) e iθ ˜ x µ π − iθ ˜ x s = 2 πb k − ss e iθ ˜ x µ − iθ : (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q UP (cid:39) πb k m − ss e iθ ˜ x µ (cid:34) iθ ξ ˙ h g + 1) + i (1 − ˜ g ) θ + ˜ gθ ˜ x (cid:35) , (B29) (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q UP (cid:39) πb k m − ss e iθ ˜ x µ (cid:34) iθ ξ ˙ h g −
1) + i (1 − ˜ g ) θ + ˜ gθ ˜ x (cid:35) , (B30) (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PP (cid:39) πm b k − ss e iθ ˜ x µ ξ ˙ h (cid:18) ˜ g − s (cid:19) , (B31) (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PP (cid:39) πm b k − ss e iθ ˜ x µ ξ ˙ h (1 − ˜ g ) , (B32) (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q EP (cid:39) πm b k − ss e iθ ˜ x µ ξ ˙ h (cid:18) ˜ g − s (cid:19) , (B33) (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q EP (cid:39) πm b k − ss e iθ ˜ x µ ξ ˙ h (˜ g − , (B34) (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PC (cid:39) πb k m − ss e iθ ˜ x µ (cid:34) iθ ξ ˙ h g ) + i (1 − ˜ g ) θ + 1 θ ˜ x (cid:35) , (B35) (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PC (cid:39) πb k m − ss e iθ ˜ x µ (cid:34) iθ ξ ˙ h − ˜ g ) − i (1 − ˜ g ) θ + 1 θ ˜ x (cid:35) . (B36)Next we have to perform the integrals over θ which will yield the Airy functions.1The results of the θ -integration are formally the same as for Compton, Eqns. (A38)–(A41), but with the replacements x → ˜ x , y → ˜ y and z → ˜ z , where˜ y = ˜ x ξ ˙ h , ˜ z = ˜ x √ ˜ y = (cid:18) χ k | ˙ h | s (1 − s ) (cid:19) / . (B37)With these results we obtain for the 8 NBW pair production traces: (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q UP (cid:39) π m b k − ss (cid:20) Ai (˜ z ) + 2˜ g + 1˜ z Ai (cid:48) (˜ z ) (cid:21) , (B38) (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q UP (cid:39) π m b k − ss (cid:20) Ai (˜ z ) + 2˜ g − z Ai (cid:48) (˜ z ) (cid:21) , (B39) (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PP (cid:39) π m b k − ss Ai (˜ z ) √ ˜ z (cid:18) ˜ g − s (cid:19) sign ( ˙ h ) , (B40) (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PP (cid:39) π m b k − ss Ai (˜ z ) √ ˜ z − ˜ g ) sign ( ˙ h ) , (B41) (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q EP (cid:39) π m b k − ss Ai (˜ z ) √ ˜ z (cid:18) ˜ g − s (cid:19) sign ( ˙ h ) , (B42) (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q EP (cid:39) π m b k − ss Ai (˜ z ) √ ˜ z g −
1) sign ( ˙ h ) , (B43) (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PC (cid:39) π m b k − ss (cid:20) (2 − ˜ g ) Ai (˜ z ) + 2 + ˜ g ˜ z Ai (cid:48) (˜ z ) (cid:21) , (B44) (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q PC (cid:39) π m b k − ss (cid:20) ˜ g Ai (˜ z ) + 2 − ˜ g ˜ z Ai (cid:48) (˜ z ) (cid:21) . (B45)Combining these traces by plugging them intod R NBW ,j d s ( σ p , σ q ) = α π m b k s − s (cid:90) d θ (cid:90) d r ⊥ e iθ − k. (cid:104) π − p (cid:105) κ.q [ UP j + σ q EP j + σ p PP j + σ p σ q PC j ] , (B46)we get the LCFA expressions for the decay rate per unit laser phase of a polarised photon in a polarisation state Λ j , j = 1 ,
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