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Trident pairproduction instrong laserpulses Anton Ilderton1,∗ 1Department of Physics, Umea˚ University, SE-901 87 Umea˚, Sweden Wecalculatethetridentpairproductionamplitudeinastronglaserbackground. Weallowforfinitepulsedura- tion,whilestilltreatingthelaserfieldsnonperturbativelyinstrong-fieldQED.Ourapproachrevealsexplicitly theindividualcontributionsoftheone-stepandtwo-stepprocesses. Wealsoexposetherolegaugeinvariance playsintheamplitudesanddiscusstherelationbetweenourresultsandtheopticaltheorem. lasEerl,ecistrocnu-rpreonstiltyroanptoapiricproofdcuocntisoind,earatbthleeifnotceursesotfdauneintotenthsee e−(p2) e−(p3) e+(p4) development of extreme light sources such as ELI [1]. As γ(k′) −(p2↔p3) is typical for particle scattering experiments, many different processesmaycontributeto the finalyield ofpairs. Itis im- e−(p1) portant to be able to judge the relative importance of these 1 processes,anddistinguishtheircontributionsfromeachother. FIG.1:Thetridentprocess.Doublelinesrepresentthefermionprop- 1 In the SLAC E144 experiment [2], pairs were produced by agator in the background field. The second term accounts for the 0 collidinga(lowintensity)laserwiththeSLACelectronbeam; exchangeofindistinguishablefermions. 2 highenergyphotonsradiatedbytheelectronsthencombined n withphotonsinthelasertoproducepairs. Inthatexperiment, a aprocesscalled‘trident’mayalsohaveproducedpairs.How- J ofsomek·xrange. ever, its contributioncouldonly be evaluatedapproximately, 4 sincenoexactexpressionforthetridentamplitudewasavail- Fig. 1 actually describes two processes. The first is ‘one 1 able,norhasonebeengiventodate[3]. Accountingtheoret- step’, in which the intermediatephotonis virtual: this is the ] ically for all relevant effects in modern laser experiments is process traditionally referred to as trident. The second pro- h a formidable challenge. As well as contributions from pro- cessis‘twostep’,inwhicharealphotonisscatteredfromthe p cesses suchastrident(alongwith, forexample,vacuumpair incomingelectron(allowedinabackgroundfieldvianonlin- - p production [4] and cascades [5]), one would like to include ear Compton scattering [6]), and this real photon then goes e effects due to the properties of the laser itself, such as their on to create a pair via stimulated pair production [7]. This h ultra-highintensity(currently1022 W/cm2),ultra-shortdura- was the process of interest in the SLAC experiment. Since [ tion(measuredinfemtosecondsorevenattoseconds)andtight theFeynmandiagraminFig.1makesnodistinctionbetween 2 focus(focaldiameteroftheorderof10microns). theseprocesses,wewillrefertothefulldiagramas‘trident’, v andusethe notionsofone-andtwo-stepprocessesto distin- In this Letter, we present the first complete calculation of 2 guishthe two contributions. The tridentS-Matrixelementis 7 thetridentprocess. Weincludehigh-intensityeffectsbytreat- 0 ingthe laser field nonperturbativelyand exactly. We include easily written down, using strong-fieldQED, in termsof the 4 finite size effects by allowing finite pulse duration. We be- photonpropagatorGmn andVolkovsolutionsy j carryingthe . asymptoticmomenta p inFig.1[8]: 1 ginbydescribingtheessentialfeaturesofthetridentprocess, j 1 modellingthelaserasanullfieldandusingVolkovsolutions 0 todescribethescatteredparticles.Keyaspectsofexistingcal- :1 culationsarediscussed,andshowntobeunphysicalduetovi- Sfi=e2 d4x d4yy¯2(x)g m y 1(x)Gmn (x−y)y¯3(y)g n y 4+(y) v olationsoftheWardidentitiesofQED.Wethencalculatethe Z Z Xi scatteringamplitude,emphasisingitsphysicalcontent,andre- −(p2↔p3). (1) late our results to the opticaltheorem. We use gaugeinvari- r a ancetosimplifythefinalexpressionfortheemissionrate,and TheVolkovsolutionsalsocarryadependenceonk·x,andk·y, discussapproximationsusefultothehighintensityregime. which describes the “dressing” of the particles by the back- We consideran electron, incidentupona laser field, emit- ground. One trades this dependence, via Fourier transform, ting a photon. This photon then combines with photons in for new variables r and s: in the periodic plane wave case, the laser to produce an electron-positron pair. The relevant the dressingis responsiblefor the shiftedelectron mass, and Feynman diagram is shown in Fig. 1. Double lines repre- the Fouriertransformreducesthe S-Matrixelementto a dis- sentthefermionpropagatorinthebackgroundlaserfield. We cretesumoverprocessesinvolvingdifferentnumbersof‘laser take this backgroundto be a plane wave, with field strength photons’[9,10]. Ourcontinuousrandsaretheanaloguesof Fmn ≡ Fmn (k·x), where the momentum km is lightlike, so photon number for pulsed plane waves, see [7]. After this, k2 =0. Thelaserfrequencyisw =|k|. Ourtreatmentholds the x and y integrals trivially yield two delta functions con- forarbitraryk·xdependence:inparticular,wehavefinitepulse servingfour-momentum. Oneofthesefixestheintermediate durationwhenFmn vanishes, orgoesrapidlyto zero, outside photonmomentumk′,andwehave,writingd p≡p1−p2and 2 p ≡p +p +p fromhereon, The resolution of the above problem is simple: in a com- out 2 3 4 pletetreatment,thereisnodivergence. Itappearsonlywhen S = (2p )2e2 dr dsM(r,s)d 4(p −p −rk−sk) oneneglectsthepoleprescriptionalreadycontainedinFeyn- fi out 1 Z Z manpropagators.Thephotonpropagatorisreally1/(k′2+ie ) × 1 −(2↔3). (2) andonemustremembertotaketheinfinitesimale →0+. To k′2+ie (cid:12)k′=d p+rk showthereisnodivergence,wereturnto(2),wheretheprop- (cid:12) agatorisevaluatedatk′=d p+rk. Thiscancertainlygoon- (cid:12) The expression for M will(cid:12)be given soon: it is lengthy and shell,sok′2=0,forcertainr. Thisbehaviourmaybeexposed we do not need it yet. The Fourier transformed (Feynman byrecallingthedistributionalresult: gauge) photon propagator is gmn /(k′2+ie ). The remaining deltafunction,above,seestheasymptoticmomentaandhence 1 e→=0+−ipd (k′2)+P 1 , (5) the free electron mass, not the shifted mass of the infinite k′2+ie k′2 planewavecalculations;thisisaconsequenceofallowingfi- whichclearlyseparatescontributionsfrom,respectively,real nite pulse duration, just as is observed in [7] for stimulated (on-shell) and virtual (off-shell) photons. We will reaffirm pairproductionandin[11]fornonlinearComptonscattering this statement below. Here, we perform the essential step, in pulsed fields. From the delta function in (2) it becomes inserting (5) into (2). In the first term, the delta-function is clear that r and s parameterise the energy-momentumtaken d (k′2)=d (2rk·d p+d p2)andisthereforeeliminatedbyper- fromthelaser. formingtherintegral.Thisfixes In the current literature, a divergence has been identified in (1), arising from the pole in the photon propagatorin the d p2 two-step process, i.e. when enoughenergyis taken from the r=− ≡r′, (6) backgroundtoputthephotonon-shellandk′2=0.Thisdiver- 2k·d p gence was attributedto the infinite temporo-spatialextentof so the propagator’s pole fixes, quite naturally, the energy the (periodic plane wave) backgroundconsidered. Based on transfer from the laser such that the photon goes on-shell, this, itwassuggestedthatthedivergencecouldbedealtwith i.e. such that the two-step process occurs. There is no di- bymodifyingthephotonpropagatoraccordingto[12] vergence here, nor any need to deform the theory along the linesof(3). (Note,thedenominatorofr′ isnonzerobecause 1 1 →! , (3) k·d p=k·(p +p )>0, using the momentum conservation k′2 k′2+2i|k′|/T 3 4 0 law in (2).) To proceed, one eliminates the s integral using whichdampsthepropagatoroutsideofachosen,fixed,tempo- one component of the delta function in (2). This is easily ralrangeT,inordertomodelfinitepulseduration. However, done in lightfrontco-ordinatesx± :=x0±x3, x⊥ :={x1,x2}, itiswellknownthatthestructureofpropagatorsandvertices andq± =(q0±q3)/2for momenta. Since the laser momen- inQED isseverelyrestrictedbygaugeinvariance,violations tumkm islightlike,wemaychoosek+ tobeitsonlynonzero ofwhicharemeasuredbytheWard identities[13]. Thepre- component.Hence,sappearsonlyintheq+componentofthe scription(3)manifestlyviolatestheWardidentity[14] deltafunction,whicheliminatesthesintegralandsets k′m hAm (k′)An (q)i= kn′ d 4(k′+q), (4) s= p2out−m2 −r≡sr. (7) k′2 2k·p1 which is sufficient to reject (3). We briefly state some addi- Three components of the delta function remain, conserving tionalphysicalreasons.First,theoriginofthedivergencecan- momentuminthe−and⊥directions,transversetothelaser. notbetheinfiniteextentofthebackground.Anybackground Writingd lf(q)≡d 2(q )d (q )/k ,theS-Matrixelementis ⊥ − + whichpermitsthetwo-stepprocess,producinganon-shellin- termediatephoton,willadmitthek′2=0divergence,indepen- Sfi=2e2p 2d lf(pout−p1)× (8) dentofthebackground’sspacetimesupport(evenifthetheory −ip 1 containsanaturalanalogueofthecutoffscaleT,forexample). 2k·d pM(r′,sr′)+ drM(r,sr)P(d p+rk)2−(2↔3) . Secondly, there is also no reason why any chosen T should (cid:20) Z (cid:21) be preferred by the theory over any other. The prescription ThiscompletesthecalculationofS . Itshouldbeclearfrom fi (3) also breaksmanifestLorentz invariance. While this may (5) that the off/on-shell parts of (8) correspond precisely to notbeconsideredtooserious,giventhatthebackgroundfield theone/two-stepprocesses. Nevertheless,letusconfirmthis, alreadyintroducespreferreddirectionsinspacetime,thesedi- which will also serve as a check on S beforemovingon to fi rections are lightlike, not timelike as (3) implies. Using (3) the cross section. Consider first the one-step process, with also gives a T dependent correction to the divergence-free overall momentum conservation given by the explicit delta- one-step process, and leaves unphysical factors of T in the functionin(2). Squaringthe argumentofthe delta function, weakfieldlimitwhereallbackgroundeffectsshouldbecome one finds that r+s>4m2/(k·p ), which states that the to- 1 negligible. Itisthereforedifficulttoascribeaphysicalmean- talincomingenergy(thatoftheinitialelectronandthattaken ingtoresultsfollowingfrom(3). from the laser), must be sufficient to producethree particles 3 of rest mass m. This is the only constraintin S if the pho- orderedSbySˆ): fi ton is off-shell, k′2 6=0, and we pick up the principal value G m (k·x):=u¯ Sˆ(p ,k·x)g m S(p ,k·x)u term. This is the ‘trident’ contribution in the old nomencla- p2 2 1 p1 ture.(Again,thereisnodependenceontheshiftedmassfound ×exp[iJ(p ,k·x,¥ )+iJ(p ,−¥ ,k·x)], 2 1 (12) inaperiodicplanewave.)Considernowthetwo-stepprocess, D m (k·y):=u¯ Sˆ(p ,k·y)g m S(−p ,k·y)v whichhastwomomentumconservationrelations p3 3 4 p4 ×exp[iJ(p ,k·y,¥ )+iJ(−p ,¥ ,k·y)]. 3 4 p +rk=p +k′ nonlinearComptonscattering, 1 2 (9) The limits in J are prescribedby the LSZ reductionformula (k′+sk=p3+p4 stimulatedpairproduction. for the trident amplitude. The product of the Fourier trans- formsofthesepartsgivesustheamplitudeM(r,s)fromabove, Squaring,onefindsthat,fortheindividualprocessestooccur, 2m2 M(r,s) = df G m (f )eirf dj D m (j )eisj −(2↔3) r>0 and s> . (10) k·k′ ≡ ZG˜m (r)D˜m (s)−(2Z↔3). (13) Usingtheoveralldeltafunctionin(2),itisstraightforwardto showthatthefixedparametersr′ ands whichappearinour To expose the further role of gauge invariance in our result, ok·nd-sph=ellkp·kar′teovbaleuyarte′d>o0nashnedllr,s′w>er2emco2rv/′ekr·d(1p0.)R. Oecuarllsionlguttihoant wwheigcihveontheemeaxyplwicriittefodromwonfG˜thmebpyaicrrporsosidnugctsiyomnmpaerttryD˜)m: (from thuscontainsthecorrectkinematicsofboththeone-andtwo- a2m2km stepprocesses,whicharedescribedpreciselybytheoff-and D˜m (s)=u¯ g m B (s)− /kB (s) p3 0 2k·p k·p 3 on-shell parts of the decomposition (5). Thus, our solution (cid:20) 3 4 (14) allowsthecontributionsoftheseprocessestobeindividually (cid:229) 2 e a/j/kg m g m /ka/j + − B (s) v . calculatedandcompared. 2 k·p k·p j p4 j=1 (cid:18) 3 4 (cid:19) (cid:21) The identity (5) also corresponds to a split into real and imaginaryparts. Fromtheopticaltheoremforscatteringam- Alldependenceontheparameters(a‘photonnumber’)iscon- plitudes (that is, unitarity of the S-Matrix), one expects the tainedinfourfunctionsB0...B3. For j=1,2,3,andwriting appearance of imaginary, or absorptive, parts to correspond f3≡ f12+f22,thesearedefinedby(sumovern=1,2,3) totheexcitationofrealratherthanvirtualintermediatestates ¥ [16, 17]. Indeed, we have seen that the imaginary part of B (s):= dz f (z)exp isz+ia dw f (w) , (15) (5) corresponds to the intermediate photon becoming real. j j n n The existence of this imaginary part is entirely due to the Z (cid:20) Zz (cid:21) dressing of the fermions by the background: in cutting lan- wherethecoefficientsa maybereadofffrom(11)–(13): n guage,thedressingallowsonetoperformacutthroughFig.1 p p which yields physical, non-zeroscattering amplitudes; those a =ean· 4 − 3 n=1,2, n fornonlinearComptonscatteringandstimulatedpairproduc- (cid:18)k·p4 k·p3(cid:19) (16) tion. (Thetridentamplitudeitselfmaybeobtainedbycutting a2m2 1 1 a =− + . the two-loop fermion propagator[18].) Our result therefore 3 2 k·p k·p appearstocontainarathernovelexampleoftheopticaltheo- (cid:18) 4 3(cid:19) remattreelevel,madepossiblebythebackground. Wehope The functions (15) are finite, provided the fj vanish asymp- toinvestigatethisfurtherinthefuture. totically, asis thecase for pulsedfields. We assume thisbe- In the remainder of this Letter we give the complete cal- haviourfromhereon. (Theperiodicplanewavecasemaybe culation of the emission rate. We will also show how gauge recoveredinasuitablelimit,see[15].)ThefourthfunctionB0 invariancesimplifiesthefinalresults. isdefinedasin(15)butwithoutanydampingfactorof f un- Our background field has potential Am (k·x)= fj(k·x)aj, dertheintegral. Consequently,B0 istheFouriertransformof where j is summed over the transverse directions. The po- a pure phase, and some prescriptionis required for calculat- larisationvectorsobeyai·k=0,ai·aj =−m2a2/e2d ij which ingit,atleastnumerically.Regularisationsofthisintegral(or definesaninvariant,dimensionlessamplitudea.Fromthecor- itsequivalentinthenonlinearComptonpartoftheamplitude) respondingVolkovsolutions,wedefine were suggested in [19, 20], but there is a more fundamental methodofdefiningB :oneappealstogaugeinvariance.Mak- 0 c J(p,b,c)=− 1 df 2eA(f )·p−e2A2(f ), inga(quantum)gaugetransformationin(2),onefindsthatSfi isgaugeinvariantprovidedthat(sumovern=1,2,3,) 2k·p Z (11) b e sB (s)=a B (s), (17) S(p,k·x)= + A/(k·x)/k. 0 n n 4 1 2k·p andsimilarlyforanalogousfunctionsinG m . Hence,bydefin- Thesearecombinedinto‘nonlinearCompton’,G m , and‘pair ingB in termsof thewell-behavedB , gaugeinvariancere- 0 j production’, D m , parts as follows (we denote the reverse- ducesthenumberoffunctionsinplayfromfourtothree.This 4 result should be comparedwith equations(A3) in [9], (6) in Inconclusion,wehavegiventhefirstfullcalculationoftri- [10],and(23)in[11]: thoseexpressionswereobtainedeither dentpair productionin a laser field, using strong-fieldQED. asidentitiesofspecialfunctionsorbyregularisation,butthey Wehaveincludedfinitesizeeffectsduetotheultra-shortdu- areunifiedandexplainedphysicallybygaugeinvariance. rationofmodernpulses. Boththeone-stepandtwo-steppro- Onenowhasallthenecessaryingredientstocalculate(8). cessesinvolvedhavebeenexplicitlyidentified,andourresults Toobtainthefullrate,onetakes|S |2anddividesoutthevol- are in agreementwith the opticaltheorem. We havealso re- fi ume transverse to the pulse, as usual. The remaining steps vealedtherolegaugeinvarianceplays,notonlyinthetrident aretosum/averageoverspinsandperformthefinalstateinte- process,butalsoinnonlinearComptonscatteringandstimu- grations, where againit is naturalto use lightfrontvariables. lated pair production, through the relation (17). We remark There are nine integrals, corresponding to three momentum that our approach is equally valid for laser assisted Møller components for three outgoing particles. Three of these are scattering,asthisisjustthecrossedprocessoftridentpairpro- eliminatedbytheremainingd lf. WritingK forthebracketed duction: the appropriate S-Matrix element is obtained from termin(8),oneobtains,witha ≡e2/(4p ), (8) by taking the outgoing positron to be an incoming electron. ¥ R= a 2 (cid:213) d2p⊥ dp− q (p2−) (cid:229) |K|2 . Contraryto recentclaimsin the literature,we haveshown 4w 2p3,p4Z (2p )3Z0 2p− p2− spins (cid:12)(cid:12)shell that there is no divergencein the trident amplitude. Despite   (cid:12) (18) this,thenumericalmethodspreviouslyemployedtocalculate (cid:12) The instruction‘shell’indicatesthat each p is evaluatedon the amplitude, from which a great deal of information was + shell,i.e. p =(p2+m2)/(4p ),and p ,whichiseliminated obtained,areequallyappropriatehere. Inparticular,onemay + ⊥ − 2 by momentum conservation, obeys p = (p −p −p ) noweasilycomparethecontributionsoftheone-stepandtwo- 2⊥ 1 3 4 ⊥ and p =(p −p −p ) . Theremainingintegralsareover stepprocesses,aswellastesttheoldWeizsa¨cker-Williamsap- 2− 1 3 4 − themomentaoftheproducedelectron-positronpair. Thefull proximationfortheformer[3].Itwillbeextremelyinteresting emission probability is a function of the laser amplitude, a, toseewhattheseinvestigationsreveal. thepulsegeometry,andtheincomingmomentawhichappear A. I. thanks Thomas Heinzl, Martin Lavelle and Mattias throughk·p . Thiscompletesthecalculation. 1 Marklundforusefuldiscussions.A.I.issupportedbytheEu- Despitethelengthoftheexpressionsinvolved,ourfinalre- ropeanResearchCouncilunderContractNo. 204059-QPQV. sult for the rate is not more complicatedthan that of the periodic plane wave case studied in [12]. Let us compare the expressionsthereinwithourownresults. Therearethreereal differences. The first is that the discrete sums over photon number are replaced by Fourier integrals over r and s. The ∗ [email protected] secondisthattheshiftedmassplays,ingeneral,noroleinour [1] http://www.extreme-light-infrastructure.eu/ expressions.Finally,ourS-Matrixcontainstwodistinctterms [2] C.Bamberetal.,Phys.Rev.D60(1999)092004. correspondingtotheone-andtwo-stepprocesses. [3] C.BulaandK.T.McDonald,arXiv:hep-ph/0004117. Animportantstepinthecalculationoftherateistheevalua- [4] G.V.Dunne,Eur.Phys.J.D55(2009)327. [5] N.V.Elkinaetal.,arXiv:1010.4528[hep-ph]. tionofthefunctions(15).Ingeneral,thesemustbecalculated [6] C.Harvey, T.HeinzlandA.Ilderton, Phys.Rev.A79(2009) numerically. Invariouslimits,though,analyticalapproxima- 063407. tionsexist. Theweak field limitis anobviousexample: one [7] T. Heinzl, A. Ilderton and M. Marklund, Phys. Lett. B 692 simplyexpandseverythinginpowersofthefieldamplitudea. (2010)250. Whatis moreinterestingformodernexperimentsisthe high [8] D.Volkov,Z.Phys.94,250(1935). intensity limit, which amounts to a≫1. This is studied in [9] A.I.NikishovandV.I.Ritus,Sov.Phys.JETP19(1964)529. [11]inthecontextofnonlinearComptonscattering,usingan [10] A.I.NikishovandV.I.Ritus,Sov.Phys.JETP19(1964)1191. [11] F.MackenrothandA.DiPiazza,arXiv:1010.6251[hep-ph]. asymptotic expansion of the (equivalents of) the B . Let us j [12] H.Hu,C.Mu¨llerandC.H.Keitel,Phys.Rev.Lett.105(2010) adaptthistoourtridentprocess: theideaistolookforpoints 080401. ofstationaryphasein(15). However,onecanshowfromthe [13] L.F.Abbott,ActaPhys.Polon.B13(1982)33. kinematics that s−a jfj(k·x)6=0∀k·x and hence no points [14] Thegaugefixingparameterisx =1inFeynmangauge. ofstationaryphaseexist. Thus,onecantrytoinsteaddeform [15] T.W.B.Kibble,Phys.Rev.138(1965)B740. the contours into the complex plane and use the method of [16] R.J.Eden,P.V.Landshoff,D.I.Olive,J.C.Polkinghorne,The steepestdescent.Thisisawellunderstoodtechniqueandmay analyticS-Matrix,CambridgeUniversityPress(1966). [17] The contribution of the imaginary part also implies that the beappliedimmediatelytothetridentamplitude. Itwouldbe propagator’s pole becomes a branch cut in the scattering am- interesting to try and understand the physics behind this ap- plitude,seee.g.Chapter7.3ofPeskinandSchroeder,AnIntro- proximationinmoredetail: asafirststep,onecouldreturnto ductionToQuantumFieldTheory. theinfiniteplanewavecaseandattempttoestablishaconnec- [18] V.I.Ritus,Nucl.Phys.B44(1972)236. tion between the saddle pointsin the complex plane and the [19] M.BocaandV.Florescu,Phys.Rev.A80(2009)053403. photonnumber,ortheeffectiveelectronmass. [20] D.SeiptandB.Kampfer,arXiv:1010.3301[hep-ph].