One-photon pair-annihilation in pulsed plane-wave backgrounds
OOne-photon pair-annihilation in pulsed plane-wave backgrounds
S. Tang, ∗ A. Ilderton, and B. King
Centre for Mathematical Sciences, University of Plymouth, Plymouth, PL4 8AA, United Kingdom
We study the 2 → → I. INTRODUCTION
At the single-vertex level, the stimulated QED pro-cesses that can occur in a background electromagneticfield may be divided into the 1 → → → ∼ × cm − ) is about 10 times denser thansolid density ( ∼ cm − ). However, this should beverified by e.g. calculation and simulation, and further-more there are several situations in laser-plasma physicswhere high particle densities can occur. For example,at the boundary of an irradiated solid target [26–30], anextremely dense electron foil is compressed by ultrarel-ativistic lasers. Also, at very high values of the laserfield intensity parameter ξ = eE/mω l ≈ O (10 ) (where e and m denote the positron charge and mass and E and ω l are the laser electric field amplitude and frequencyrespectively), QED cascades comprising chains of NLCand NBW processes are predicted to occur [31–37]. Insuch cascades, electron-positron plasmas are produced ∗ [email protected] pq pq ll l FIG. 1. Feynman diagram for pair annihilation. Left: Two-photon process in vacuum. right: One-photon process in afield background. Double lines denote dressed Volkov states. and could be compressed to densities much higher thanthe plasma relativistic critical density.In this paper we derive a numerical implementation ofone-photon pair annihilation and investigate its relevanceto the above situations. The paper is organised as follows.In Sec. II, we calculate the probability of annihilation toone photon, derive its locally constant field approxima-tion (LCFA) [23] and benchmark against exact resultsfor a circularly-polarised monochromatic field. We thenpresent a study of the dependency of parameters for theprobability and a comparison with the background two-photon process. Approximate scaling arguments are alsoobtained. In Sec. III we demonstrate numerical imple-mentations of our results and investigate the relevance ofone-photon pair annihilation to laser-plasma and cascadescenarios. We conclude in Sec. IV.
II. ONE-PHOTON PAIR-ANNIHILATION
Electron-positron annihilation in vacuum yields atleast two photons [38, 39]. However, in the presence ofa background field, annihilation to one photon becomeskinematically accessible; see Fig. 1. At low backgroundintensities, the leading-order process is again two-photonemission, but with one of the photons emitted into thebackground. Therefore, when we consider one-photonpair-annihilation, we are also summing over processesthat are degenerate with it, such as the (unobservable)emission back to the field [40].Here we briefly outline the derivation of one-photonpair-annihilation in pulsed plane wave backgrounds,modelling intense laser pulses. We highlight only thoseparts of the calculation that differ relative to the morestandard 1 → a r X i v : . [ phy s i c s . p l a s m - ph ] S e p detailed example of our derivation can be found inRef. [23]). We use natural units (cid:126) = c =1 throughout andthe QED coupling constant is α = e ≈ / p µ ( q µ )annihilates to a photon with momentum l µ in a back-ground laser field as shown in (a). However, an exper-imental scenario will likely involve high particle densityof one species as a target, which we choose to be theelectrons, and consider the positrons to be in a beamas shown in (b). The laser field is modelled by the po-tential eA µ ( φ ) = a µ ( φ ) = (0 , a ( φ ) , a ( φ ) , φ = k · x , with k = ω l (1 , , ,
1) being the laser wavevec-tor. The electron is described as a standard Volkov wave-function [41], Ψ e − ( p ) = (cid:114) m V p (cid:18) /k/a k · p (cid:19) u p,σ e − i (cid:104) p · x + (cid:82) φ dφ (cid:48) (cid:16) p · ak · p − a k · p (cid:17)(cid:105) , (1)where the spinor u p,σ satisfies the relation: (cid:80) σ u pσ u pσ =( /p + m ) / (2 m ). The positron is also described by aVolkov wavefunction, but we include a momentum-spacewavepacket ρ ( q ) to represent a beam of positrons. Writ-ing Ψ e + ( q ) for the positron Volkov wavefunction, thepositron is described byΦ e + = (cid:90) d q (2 π ) mq ρ ( q ) Ψ e + ( q ) , (2)where ρ obeys the normalisation condition (cid:90) d q (2 π ) mq | ρ ( q ) | = 1 . (3)The S -matrix element for annihilation is S fi = ie (cid:114) πl V (cid:90) d x Φ e + /(cid:15) e il · x Ψ e − . (4)where (cid:15) µ is the polarisation of the produced photon,obeying (cid:15) · (cid:15) = − l · (cid:15) = 0. The probability P for annihilation is then P = (cid:90) V d l (2 π ) (cid:88) pol , spin | S fi | , (5)where we sum over the polarisation of the outgoing pho-ton and average over the spins of the incoming pair. Fordetails of the calculation see e.g. [40]. We find: P = αλ c π V mp (cid:90) d q (2 π ) mq | ρ ( q ) | η q η l (cid:90) dφ dφ (cid:40) a ( φ ) − a ( φ )] m η p + η q η q η p (cid:41) e − i ηl (cid:82) φ φ dφ (cid:48) π lm , (6)where η p = k · p/m , η q = k · q/m , η l = η p + η q , and λ c = 2 π/m is the electron Compton wavelength. We 𝑒 + 𝑒 − 𝑒 + 𝜃 𝑝 𝜃 𝑞 𝑘 𝑥 𝑧 𝑦 𝐴 𝜇 𝑧 𝑦 𝑥 𝑒 − 𝜃 𝑞 𝐴 𝜇 𝑘 FIG. 2. Scheme of one-photon pair annihilation. (a) An elec-tron ( e − ) beam and a positron ( e + ) beam collide with a laserpulse ( A µ ). (b) A positron beam impinges a dense electrontarget. define the shorthand π l ( φ ) = π p ( φ ) + π q ( φ ), and π µp = p µ + a µ − (2 p · a + a ) k µ / (2 k · p ) ( π µq = q µ − a µ + (2 q · a − a ) k µ / (2 k · q )) is the instantaneous four-momentumof the electron (positron) in a plane-wave. Note that theprobability contains the leading density factor λ c /V . A. LCFA
To derive the LCFA, we follow the usual procedure ofrewriting the external-field phases in terms of an averagephase ψ = ( φ + φ ) / ϑ = φ − φ [23], expanding the exponent to order ϑ : (cid:90) φ φ dφ (cid:48) π l → ϑπ l ( ψ ) + ϑ
24 [ π l ( ψ )] (cid:48)(cid:48) , (7)and the pre-exponent up to order ϑ through the replace-ment [ a ( φ ) − a ( φ )] → − m ϑ ξ ( ψ ), where we definethe normalised electric field ξ through a (cid:48) /m = (0 , ξ ).This allows us to integrate Eq. (6) over ϑ . The probabil-ity becomes P = αλ c πV (cid:90) d q m (2 π ) q | ρ ( q ) | g ( p, q, ξ ) , (8)which has the form of an incoherent average over thepositron wavepacket | ρ ( q ) | and the probability for one-photon annihilation of a pair with definite momenta p and q , which is encoded in g . The dependence on theparticle momenta and the field ξ is described by g ( p, q, ξ ) = mp η q (cid:90) dψ f ( p, q, ξ ) , (9)in which f ( p, q, ξ ) = (cid:32) χ / q χ / p χ / l + χ p + χ q χ l z (cid:33) Ai( z ) , (10)where all χ variables depend on the average phase ψ via χ p = η p | ξ ( ψ ) | , χ q = η q | ξ ( ψ ) | , χ l = χ p + χ q and Ai( z ) isthe Airy function with argument z = ( π p + π q ) m χ l (cid:18) χ q χ p χ l (cid:19) . (11)Observe that the LCFA result depends not only on thequantum nonlinearity parameter χ p,q , but also on thelocal momenta of the two particles, π p,q . B. LCFA Benchmarking
To benchmark the LCFA result, we consider one-photon pair-annihilation in a circularly polarisedmonochromatic field: ξ ( ψ ) = ξ [cos ψ, − sin ψ, → → | ρ ( q ) | = (2 π ) q − m ν ( q − ) 4 ln(2) π ∆ m e − m | q ⊥ − q ⊥ i | , (12)which is Gaussian distributed in transverse momentumwith full width at half maximum ∆ m , while the longitu-dinal wavepacket ν ( q − ) satisfies the normalisation condi-tion: (cid:82) ∞ dq − ν ( q − ) = 1. This ansatz for the wavepacketfacilitates the intended comparison by matching well withthe symmetries of the plane wave background. The ex-plicit form of ν ( q − ) will not be required, as we will focuson transverse momentum dependence.Inserting the above wavepacket into Eq. (8), we canobtain the probability: P lcfa ν = 32 π ln(2) δ (0) αV k p ∆ (cid:90) dv ν ( q − )(1 − v ) h l ( v ) , (13)where δ (0) = (cid:82) dψ/ (2 π ), v = η q /η l and h l ( v ) = 12 π (cid:90) d r (cid:90) π dψ e − u | r − r i | × v (cid:20) u ( vχ p ) / + (1 + u ) z (cid:21) Ai( z ) , (14)with the definitions: r = q ⊥ m √ u − p ⊥ m (cid:114) u , r i = q ⊥ i m √ u − p ⊥ m (cid:114) u , where u = η p /η q .We find the annihilation probability in a circularly po-larised monochromatic wave to be: P mono ν = 32 π ln(2) δ (0) αV k p ∆ (cid:90) dv ν ( p − )(1 − v ) h a ( v ) , (15) in which h a contains a sum over harmonics h a ( v ) = ∞ (cid:88) n ≥ n v T n (16)with the lower harmonic bound n v = (1 + ξ ) / (2 η p v ) andT n ( v ) = (cid:90) π − π dϕ e − u ( r n − r i ) H n (17)where ϕ is the angle between r n and r i , cos( ϕ ) = r n · r i / ( r n r i ), r i := | r i | is a function of only initial vari-ables and r n = | r n | = 2 nη p v − v ∗ v (1 − v ) , v ∗ = 1 + ξ nη p , and n is the harmonic numberH n = ξ (cid:20) n − s n s n J n ( s n ) + J (cid:48) n ( s n ) (cid:21) − v uv u + J n ( s n )2 , where J n ( s n ) is the Bessel function of the first kind, withargument s n = ξ (cid:112) nη p ( v − v ∗ ) χ p v We compare in Fig. 3 the LCFA result Eq. (14) withthe exact monochromatic result Eq. (16), for various pa-rameters and wavepackets of different widths. In the fig-ure, the LCFA is represented by dashed lines and themonochromatic result by solid lines.In Fig. 3 (a), the LCFA result for a flat wavepacketmatches well with the exact calculation when the fieldintensity is relatively strong; notice that the LCFA can-not reproduce the low- n harmonic structure visible ate.g. ξ = 1 (red lines), as expected from other inves-tigations of the LCFA [42], see also [14, 23, 43]. Forpair-annihilation, we find a similar dependency on theminimum harmonic as for the time-reversed process ofNBW pair-creation [1]. There is a lower bound n v onthe harmonic number which increases with intensity; asthe LCFA does better at reproducing results where largenumbers of higher-harmonics contribute [14, 23, 42, 43],the quality of the LCFA improves quickly here, being ex-tremely accurate already for ξ = 4 (magenta lines). Con-sequently, a similar effect results from decreasing η p : thisalso raises the harmonic lower bound, leading to weakerharmonic structure, meaning that the LCFA gives a bet-ter approximation of the monochromatic result even atlow laser intensities. This is confirmed in Fig. 3 (b) and(c).We also highlight the behaviour at v = 1 in Fig. 3(a) for the flat wavepacket, where h l , h a → ∞ . Thisdivergence comes from the superposition of an infinitenumber of states with the same longitudinal momentum.However, this behaviour is different when the includedwavepacket has a finite momentum bandwidth. As wecompare, in order, Figs. 3 (a)-(d), the wavepacket be-comes narrower, and the exact result oscillates rapidlyas v →
1, with the oscillating structure spreading tolower v as the wavepacket narrows. These rapid oscilla-tions result from an interplay between the harmonics andthe Gaussian wavepacket: the contribution of each har-monic is effectively localised by the narrow wavepacket.The dominant contribution from each harmonic origi-nates from the condition ( r i − r n ) = 0, as can be seenfrom the exponent in Eq. (17). This can be solved for v ,showing that the n th harmonic H n will be restricted tocontribute around v (cid:39) v n where v n := 1 + 2 ξ η p n . (18)We have used here that p ⊥ = − q ⊥ i and | p ⊥ | = ξ as inFig. 3. To see these effects explicitly we zoom in to thepeak structure for ξ = 1 (red solid line) in Fig. 3 (d)and highlight, in Fig. 4, the contribution from differentharmonics. As predicted, the n th harmonic is highly lo-calised around the point v = v n . Also, the harmonicsthat significantly contribute are substantially above thelower bound n v . It is clear from Fig. 4 that the separationof the harmonic contributions, due to the wavepacket, isresponsible for the appearance of the oscillatory struc-ture as the wavepacket width decreases. Furthermore,the LCFA result fails to manifest this peak structure atall. As v decreases, the harmonic peaks overlap and canbe matched better by the LCFA, but this agreement isagain lost as the wavepacket continues to narrow and theharmonic peaks become much sharper, as in Fig. 3 (c)and (d). (We also find the oscillation frequency increaseswith the increase of the laser intensity, Fig. 3 (d).) Weconclude that the LCFA is unable to reproduce the 2 → C. Phenomenology
In this section, we study the dependence of one-photon pair-annihilation on the incident particle pa-rameters, assuming plane-wave initial states. Theparticle momenta are expressed in spherical po-lar co-ordinates as depicted in Fig. 2 (a): p = − ( E p − m ) / [sin θ p cos ϕ p , sin θ p sin ϕ p , cos θ p ], and q = − ( E q − m ) / [sin θ q cos ϕ q , sin θ q sin ϕ q , cos θ q ] where E p , θ p , ϕ p ( E q , θ q , ϕ q ) are the incident energy, polar andazimuthal angle of electron (positron). We analyse onecycle of a monochromatic field with i) linear polarisation: a µ ( ψ ) = mξ [0 , cos ψ, , a µ ( ψ ) = mξ [0 , sin ψ, cos ψ, ϕ p = ϕ q = 0).It is helpful for what follows to understand where thedominant contributions to f ( p, q, ξ ) in Eq. (10) comefrom, in terms of phase ψ and as a function of the par-ticle momenta. The pair should have similar energy (a) (b) (c) (d) FIG. 3. LCFA vs exact result. (a) ∆ = ∞ , η p = 2. (b)∆ = 20, η p = 1. (c) ∆ = 2, η p = 0 .
5. (d) ∆ = 0 . η p = 0 .
2. Dashed (Solid)lines are for LCFA (exact) result.Red lines: ξ = 1; Green lines: ξ = 2, Magenta lines: ξ = 4,and Blue lines: ξ = 8. In (b), (c), (d), q i,x = − p x = ξ and q i,y = p y = 0. FIG. 4. Harmonic peak structure. We zoom in the peakstructure of the red line for ξ = 1 in Fig. 3 (d), and sepa-rate the contribution from different order of harmonics (cyclesand diamonds). The black dash-dotted lines denote v n , as inEq. (18), corresponding to each harmonic. E p ≈ E q , and we find that f ( p, q, ξ ) exhibits one (two)sharp peaks per laser cycle for circular (linear) polari-sation. These peaks appear at the points where πππ p ( ψ )is parallel to πππ q ( ψ ). For linear polarisation: if θ p = θ q ,the peaks appear at the points where a ( ψ ) = 0, and if θ p = − θ q = θ , the peaks appear at mξ cos ψ = E p sin θ .For circular polarisation: a peak appears at a ( ψ ) = E p [sin θ cos ψ, sin θ sin ψ, p ⊥ + q ⊥ = 0. Becausethe EM field rotates in a circularly-polarised background,these acceptance peaks are much narrower than for alinearly-polarised background. Thus one-photon pair-annihilation is much more effective in a linearly polarisedlaser. FIG. 5. Parametric dependency of g ( p, q, ξ ) on the laseramplitude ξ and the particles’ incident angle θ p , θ q . (a) θ p = − θ q =: θ . (b) θ p = θ q =: θ . The red dashed line in (a) is ξ = | p | sin( θ ) /m , and the green dashed line in (b) correspondsto η p ξ = 4 /
3. The particle energy is fixed E p = E q = 2000 m ,and a laser is linearly polarised, with frequency ω l = 1 .
24 eV.
Fig. 5 shows the dependency of the integrated expres-sion g ( p, q, ξ ) from Eq. (9), on the laser amplitude ξ andthe particles’ incident angle θ . In Fig. 5 (a), we cansee that, in order that the process is not strongly sup-pressed, the laser intensity must be increased for largervalues of incident collision angle θ . The reason for this,is that the laser field must be strong enough to make thelocal momenta of the pair particles, πππ p and πππ q , parallelto one another. If the pair particles propagate parallelto one another and collide head-on with the laser pulse,there is an optimal value of ξ , above which the proba-bility then decreases. This is because, even though thestrength of the interaction is increased, there is a sup-pression at high intensities due to a narrowing of theeffective phase width in the integrand f ( p, q, ξ ). Fora high enough intensity, the most probable set-up forone-photon pair-annihilation is actually when the col-lision is not directly head-on, as shown in Fig. 5 (b).This is because one-photon pair-annihilation achieves thelargest probability if the quantum nonlinearity parameter χ = 4 / ∼ [1 + cos( θ )] ξ (this will be further commentedon in the approximations section and Eq. (22)).In Fig. 6, we show the dependency of g ( p, q, ξ ) on theincident parameters ( E q , θ q ) of the positron for the givenlaser amplitude ξ = 100 and electron incident parameters[ E p = 1360 m , θ p = 0 in (a) and θ p = π/ θ p ≈ θ q , E p ≈ E q ). With a relative larger incident angle in Fig. 6 (b),the process would be less effective because χ p ∼ θ )is smaller.In Fig. 7, we consider the dependency of g ( p, q, ξ ) onthe particle energy and laser amplitude with head-on col-lisions θ p = θ q = 0. As shown in Fig. 7 (a), a strongerlaser field could induce larger annihilation probabilityand also decrease the requirement for the particle energy(see the red dotted line). This is because the probabil-ity has a maximum at χ p = 2 / FIG. 6. Parametric dependency of g ( p, q, ξ ) on the incidentparameters of the positron. (a) θ p = 0, E p = 1360 m . (b) θ p = π/ E p = 1360 m . The laser amplitude is ξ = 100, andthe other parameters are same in Fig. 5.FIG. 7. (a) Parametric dependency of g ( p, q, ξ ) on the particleenergy E p = E q and laser amplitude ξ . The red dotted linedenotes the particle energy giving the largest g ( p, q, ξ ). (b)Largest g ( p, q, ξ ) for a given laser intensity ξ . The red linecorresponds to the red dashed line in (a) and the blue linescomes from the approximation Eq. (23). Head on collision( θ p = θ q = 0) is applied, and the other parameters are samein Fig. 5. ability for a given laser field ξ increases with a strongerlaser field. D. Approximations
To understand the dependency of one-photon pair-annihilation on experimental parameters, it is useful toapproximate the phase integral in Eq. (9).To simplify the calculation, in this section we con-sider two cases, corresponding to Fig. 5 (a) and (b)respectively. First of all we assume a head-on colli-sion θ p = θ q = 0 (Fig. 5 (a)) with a linearly-polarisedmonochromatic laser field ξ ( ψ ) = ξ [sin( ψ ) , , z , of the Airy function, becomes: z = z m [1 + ξ cos ( ψ )] sin − / ( ψ ) , (19)where z m = ( η p + η q ) / / ( η q η p ξ ) / . The leading con-tribution comes when a ( ψ ) ≈ z is at a minimum(and the Airy functions are at their maximum). If weTaylor-expand z in ψ to order ψ around the correspond-ing points at ψ = π/ , π/
2, we can integrate over ψ inEq. (9) and arrive at: g ( p, q, ξ ) ≈ mp η q ( η p + η p ) − ( η p η q ) ξ − π √ z m (cid:20)(cid:18) − ξ (cid:19) Ai (cid:16) − z m (cid:17) + 1 + 2Γ + 6 ξ Γ3 ξ z m Ai (cid:48) (cid:16) − z m (cid:17)(cid:21) , (20)where Γ = Γ( p, q ) = ( η p + η q ) / (2 η q η p ). If 2 − z m (cid:29) g ( p, q, ξ ) ≈ mp η q η p η q ( η p + η q ) ξ exp (cid:18) − z m (cid:19) , (21)and if p = q , g ( p, q, ξ ) can be further simplified: g ( p, q, ξ ) ≈ m p χ m exp (cid:18) − χ m (cid:19) , (22)where χ m = η p ξ . (This is reminiscent of the famousexp( − / χ l ) scaling of the time-reversed process of NBWpair-creation in a constant crossed field in the asymptoticlimit χ l (cid:28) θ p = θ q = θ (Fig. 5 (b)). Using the same approximation as inEq. (22), we see that, for a given particle energy E p = E q , g ( p, q, ξ ) has a maximum if χ m = 4 / ξ , g ( p, q, ξ ) has a maximum if χ m = 2 /
3, (seethe blue line in Fig. 7 (b)). g ( p, q, ξ ) then takes the value: g m = 278 ω l ξm e − , (23)(where we have made use of the relation k · p ≈ ω l p if p (cid:29) a ( ψ i ) = 0: g ( p, q, ξ ) ≈ m p η p ( η p + η q ) (cid:88) i | ξ ( ψ i ) | e − z / m ( ψ i ) . (24)To demonstrate the validity of the approximation, weshow in Fig. 8 the comparison between the numericalcalculation of Eq. (9) and the approximation Eq. (21).As we can see, the approximation works well in a broadparameter region, with the discrepancy growing in theextremely high field and high energy region as z m (cid:38) E. Comparison with zero-field two-photonpair-annihilation
Based on Eq. (8) and the definition of the cross section, σ = (1 / | v rel | n e − ) d P /dt [44], where t is time and | v rel | =
500 1000 150000.20.40.60.81 10 -4 (a)
500 1000 150000.511.5 10 -4 (b)
500 1000 150000.511.5 10 -3 (c)
500 1000 150000.511.52 10 -3 (d) FIG. 8. Comparison between the numerical calculationof Eq. (9) and the approximation Eq. (21) for a head-oncollision. In (a) and (b), the results for one cycle of amonochromatic laser pulse ξ ( ψ ) = ξ [sin( ψ ) , ,
0] are dis-played. In (c) and (d) the cases of a long laser pulse ξ ( ψ ) = ξ [sin( ψ ) , ,
0] sech [ ψ/ ( ω l T )], ω l = 1 .
24 eV, T = 5 T l , T l = 2 π/ω l are presented. In (a) and (c), E p = E q = 2000 m ;in (b) and (d), E p = 2000 m , E q = 1000 m . ( p q ) − (cid:112) ( p · q ) − m is the relative velocity betweenthe pair particles, we can easily calculate the cross sectionfor one-photon pair-annihilation: σ = 2 αλ c √ κ − κ πN l (cid:90) dψ f ( p, q, ξ ) , (25)where κ = ( p + q ) /m is the scaled Mandelstam invari-ant, N l is the number of laser cycles and we replace thevolume factor 1 /V in Eq. (8) with the electron density n e − . (Here, an “evening-out” of the instantaneous crosssection is performed by averaging over the phase of the in-cident laser pulse.) The cross section for the two-photonannihilation process in vacuum is calculated in Ref. [39]: σ = α λ c π κ − (cid:34) − κ + 4 κ (cid:114) κ − κ +ln (cid:32) κ −
22 + (cid:114) κ − κ (cid:33) κ + 4 κ − κ (cid:35) . (26)In Fig. 9, we compare the ratio σ /σ of the cross sec-tions for the two processes. As we can see, with smallincident angle θ (cid:28)
1, the laser assisted one-photon pair-annihilation can be more probable than the two-photonannihilation in vacuum, especially when we have head-oncollision θ = 0 with the laser pulse. To measure this inexperiment, we see that one would have to resolve theangular spectra of annihilation photons, where, in thesmall-angle region, one-photon annihilation from withinthe pulse can exceed two-photon zero-field annihilation. FIG. 9. Ratio between σ and σ . (a) θ p = θ q = 0, E p = E q .(b) θ p = − θ q = θ , E p = E q = 10 m , ω l = 4 .
65 eV. Thegreen dotted lines denote σ = σ . N l = 1, and the otherparameters are same in Fig. 5. III. NUMERICAL IMPLEMENTATION
In this section, we combine our analytical calculationswith numerical implementation. To consider the numberof one-photon pair-annihilation events in realistic situa-tions, we specify the positron momentum distribution tobe | ρ ( q ) | = (2 π ) ( q /m ) δ (3) ( q − q i ) which clearly fulfillsthe normalisation condition Eq. (3). The number N a ofpositron annihilation events in the interaction of N e + in-cident positrons with a dense electron target and a laserpulse is then: N a = N e + n e − λ c α π mp η q (cid:90) dψ f ( p, q, ξ ) . (27)This number of events is suppressed by the electron den-sity factor, which is small unless there is, on average, oneelectron per Compton wavelength cubed. (This wouldcorrespond to a density of ∼ × cm − , more than10 times higher than solid density ∼ cm − .) Inthe following, we consider two example applications ofone-photon pair-annihilation. A. QED Cascade and Laser Plasma Interaction
We first consider one-photon pair-annihilation in QEDcascades. In Ref. [31] the typical particle density in aQED cascade was given as approximately equal to therelativistic critical density n e + = n e − ≈ ξn c , in which n c = ω l m/ π is the plasma critical density. The typicalparticle energy in the cascade is around E p ≈ E q ≈ mξ .Given these parameters, we show in Fig. 10 (a) the num-ber of pair annihilations in the volume of one laser wave-length cubed. In the calculation, the number of positronsis N e + ≈ ξn c λ l . As we can see, the number of annihi-lations is at best six orders of magnitude smaller thanthe initial positron number. We thus conclude that one-photon annihilation will have a negligible effect on QEDcascades.Another scenario in which a high electron density canarise is the irradiation of a solid plasma with an intense
200 400 600 800 1000050100150 (a)
200 400 600 800 10000246810 (b)
FIG. 10. Number of annihilation events in (a) QED cas-cades, (b) a laser-plasma interaction. A linearly polarisedlaser pulse: ξ ( ψ ) = ξ sech [ ψ/ ( ω l T )] [sin( ψ ) , ,
0] is employedwhere T = 5 T l , ω l = 1 .
55 eV, T l = 2 π/ω l . laser pulse [27, 30]. At the plasma surface, an extremedensity electron foil, with the typical density n e − ∼ ξ n c and energy E p ≈ ξm , can be compressed. We considerthe number of annihilations when a beam of N e + ≈ positrons with E q = 2000 m is fired at the electron foil.Fig. 10 (b) shows calculation results for the number ofannihilation events during this laser-plasma interaction.Again, this number is many orders of magnitude lowerthan the initial number of positrons, and as for cascadeswe conclude that one-photon pair-annihilation is negligi-ble.We note that our calculations neglect the influenceof the particle direction and assume all the particlesmove head-on with the laser pulse, in order to con-sider the most optimistic situation for one-photon pair-annihilation. When more experimentally-realisable pa-rameters are considered, the number of one-photon pair-annihilation events could be much smaller than the esti-mated numbers. B. Incorporation in PIC
The 1 → → d P dt = αλ c n e − m π p π q f ( p, q, ξ ) , (28)which can be implemented in the standard PIC-algorithmas it depends only on local parameters. In each time step∆ t , the probability for one positron annihilated in the j thpseudo-positron is P j = w j ∆ t (cid:88) i w i ∆ V αλ c m π p i π q j f ( p i , q j , ξ ) = (cid:88) i P i,j , (29) FIG. 11. Numerical simulations. Black line: theoretical num-ber of annihilations. Red line with error bars: simulation re-sults for the number of annihilations, averaged over 40 runs,with the error bar denoting the standard deviation. The laserpulse is same as in Fig. 10 except ω l = 4 .
65 eV. where we sum over all the electrons i in the samegrid cell as the j th pseudo-positron, ∆ V is the vol-ume of the cell, and w i,j are the particle weights. AMonte Carlo method is applied to describe the one-photon pair-annihilation process semi-classically. Tworandom numbers r and r in [0 ,
1] are generated to de-termine whether an annihilation occurs and to choosethe momentum of the photon. For each pseudo-positron, j , an annihilation event is accepted if r < P j , andthen the momentum of the produced photon is calcu-lated using the momentum of the k th pseudo-electron,if (cid:80) i 88 at ξ = 100 in the black line in Fig. 11.This method can also be simply extended to realis-tic situations with specific momentum distributions be-cause of the way the momentum part of the wavepacketfactorises into the total expression see Eq. (8). If weconsider a simulation with the positron momentum dis-tribution | ρ ( q ) | , we can, in principle, split it into aset of simulations with different positron momenta q andnumber N e + | ρ ( q ) | , and then sum the results N a in eachcase. Based on this point, even though rapid oscilla-tions appear in the exact result in Fig. 3, it is reasonableto implement the LCFA result in a standard PIC-code,because the LCFA effectively averages across these oscil-lations when implemented in this way. IV. CONCLUSION We have analysed one-photon electron-positron pairannihilation in a plane wave background. We derivedthe locally constant field approximation (LCFA) for thisprocess and benchmarked it against the exact result for acircularly polarised monochromatic background. As onemay expect on the basis of LCFA results for NLC, theLCFA was found to be incapable of reproducing harmonicstructure. However, a new shortcoming of the LCFA wasidentified: the LCFA result cannot reproduce the physicsof narrow wavepackets, which here manifested as a highlyoscillatory structure in the high-energy region.We obtained simple scaling relations for annihilationin various setups, and compared the one-photon annihi-lation cross section in a plane wave with the cross sec-tion of two-photon pair-annihilation in vacuum. The one-photon process can be dominant for small-angle scatter-ing in the head-on configuration.Using numerical simulations based on the LCFA wewere able to confirm that one-photon pair-annihilationwill have a negligible effect on QED cascades and cer-tain laser-plasma interactions at realisable particle den-sities. We also showed that annihilation can be includedin large-scale numerical simulation frameworks, bench-marking our results against a Particle-In-Cell (PIC) sim-ulation. V. ACKNOWLEDGMENTS We thank A. J. 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