Pair production: the view from the lightfront
PPair production: the view from the lightfront
Florian Hebenstreit,
1, 2, ∗ Anton Ilderton, † and Mattias Marklund ‡ Department of Physics, Ume˚a University, SE-901 87 Ume˚a, Sweden Institut f¨ur Physik, Karl-Franzens Universit¨at Graz, A-8010 Graz, Austria
We give an exact, analytic, and manifestly gauge invariant account of pair production in combinedlongitudinal and transverse electromagnetic fields, both depending arbitrarily on lightfront time.The instantaneous, nonperturbative probability of pair creation is given explicitly along with thespectra of the final particle yield. Our results are relevant to high-intensity QED experiments nowbeing planned for future optical and x-ray free electron lasers.
I. INTRODUCTION
The use of laser light sources in examining the highintensity regime of QED continues to draw attention,prompted by the advent of a new generation of both x-rayand optical laser facilities such as the European XFELand the Extreme Light Infrastructure project [1, 2].Dominating the theoretical activity in this area is thepursuit of (Schwinger) pair production in strong back-ground fields [3].In this paper we apply the Dirac-Heisenberg-Wigner(DHW) formalism [4–6] to the problem of pair produc-tion in background fields. Within this approach, whichwas developed in the early 1990’s [7, 8] and has gainedincreased attention in recent years [9–12], one studiesphase space distribution functions instead of the usualS-matrix elements. Since the DHW formalism deals es-sentially with quasi-probabilities, interpretation can bechallenging. Nevertheless, quite some progress can bemade by considering simple, but nontrivial, cases.An advantage of this approach is that the key object,called the DHW function, which is essentially the fielddensity in an appropriate state, can in principle be ob-tained by solving a single partial differential equation.This is true at least when the gauge field is external:going beyond this is notoriously hard within the DHWapproach [13], while perturbation theory is obviouslystraightforward in field theory. In order to ensure thatthe obtained solution is physical, one must use sensibleinitial data which corresponds to calculating the DHWfunction in a physical state. Alternatively, one could tryto construct the DHW function directly. This is chal-lenging since it requires finding the exact solutions of theDirac equation in the chosen background field and thenquantising the theory.In this paper we continue the investigation of the DHWfunction started in [10] using lightfront methods (see [14–16] for applications of related methods to QED in a va-riety of strong external fields). Our previous results fo-cussed on plane wave backgrounds, i.e. transverse, or-thogonal electric and magnetic fields of equal magnitude, ∗ fl[email protected] † [email protected] ‡ [email protected] depending on lightfront time x + . While we made progressin understanding the effective mass of a particle in anarbitrary pulse, we were of course unable to study pairproduction since single plane waves cannot produce pairs.In this paper we retain the plane wave fields but add alongitudinal electric field. Both of our fields will dependarbitrarily on lightfront time, allowing us to model mod-ern short-duration laser pulses [14, 17–19].To obtain the DHW function, we follow the secondapproach described above: we will therefore presentnew solutions of the Dirac equation in a combinationof longitudinal and transverse fields, quantise the the-ory and calculate the lightfront DHW function directly.This function, as we will show, can be interpreted as a(quasi)probabilistic measure of electron/positron occu-pation numbers. This will give us a clear signal of paircreation as we will be able to see, in a gauge invariantmanner, the filling of states as particles are produced.We also confirm the results given in [20–23], namely thatfrom the infinitely boosted lightfront frame one sees onlythe created positrons (modulo the choice of field) sincethe electrons decouple from the theory after creation.We begin in Sect. II by reviewing the DHW approachand presenting some basic results. In Sect. III we givethe required solution of the Dirac equation in our chosenbackground. The quantisation of the theory and con-struction of the DHW function is not too hard but theexpressions involved can become quite lengthy, and aretherefore relegated to the appendices. We give the exactDHW function in Sect. IV and analyse pair creation onthe lightfront, using explicit examples of both short andlong pulses. We also reconstruct the final particle spec-trum in the lab frame. Conclusions are given in Sect. V.Our lightfront conventions are quite standard but we en-courage the reader not familiar with lightfront methodsto consult the appendices for details. II. THE LIGHTFRONT DHW FUNCTION
The DHW function is essentially given by the Fouriertransform of the fermion field density in a chosen state,usually the vacuum. Our background fields will dependon lightfront time x + ≡ ( x + x ) / √ ψ split into dynamical fields ψ + , and a r X i v : . [ h e p - ph ] S e p constrained fields ψ − (this is reviewed below, details arenot needed here). It is therefore convenient, and simpler,to study the DHW function defined by the density of thedynamical fields rather than the full Dirac spinor.The equal lightfront time DHW function begins withthe dynamical fermion density U in, say, the vacuum | (cid:105) .Noting that we use a sans-serif font to denote the spatiallightfront variables and momenta, i.e. x ≡ { x − , x ⊥ } and p ≡ { p − , p ⊥ } , this density is U αβ ≡ (cid:104) | (cid:2) ψ + α ( x + , x ) , ψ † + β ( x + , x ) (cid:3) | (cid:105) , (1)where α and β are spin indices. Setting x ≡ x + y / x ≡ x − y /
2, the DHW function is defined by Fouriertransforming with respect to the relative co-ordinate y : W + αβ ( x + ; x , p ) = √ (cid:90) d y e i p . y + ie (cid:82) d z.A ( z ) U αβ , (2)where gauge invariance of W + is ensured by the Wil-son line in the exponent. The line integral is taken overthe straight path from x to x which corresponds tominimally coupling the free DHW function by replac-ing ∂ → D [24]. The factor of √ A. Free theory
All our DHW functions will be proportional to thelightfront projector Λ + ≡ γ − γ + , so we write W + αβ ≡ Λ + αβ W . (3)The DHW function (2) for free fermions is easily found bywriting down the mode expansion, calculating the expec-tation value and performing the Wigner transformation(without Wilson line). One finds W ( x + ; x , p ) = Sign( p − ) , (4)which is spatially homogeneous and displays only a sim-ple dependence on the lightfront momentum p − . Thisbehaviour is due to the existence of both positrons andelectrons. To see why, replace | (cid:105) in (1) with | full (cid:105) , inwhich every positron and electron state is occupied. TheDHW function becomes W ( x + ; x , p ) = Sign( − p − ) , filled vacuum . (5)Similarly, filling all the electron or positron states, oneobtains instead W ( x + ; x , p ) = − , electrons filled , = +1 , positrons filled . (6)These results are shown in Fig. 1. The region p − > p − < FIG. 1. The free W as a function of p − , calculated in the ordi-nary, empty vacuum (black), in the state filled with positrons(red, dashed), the state filled with electrons (red, dotted) andthe completely filled state (blue). The region p − < p − > | full (cid:105) . is a matter of convention (it does not refer to negativeenergy) which follows from the choice of exponent in thetransform (2), since the mode expansion for ψ looks like ψ ∼ ∞ (cid:90) d k − k − (cid:90) d k ⊥ ( e − ik.x u sk b sk + e ik.x v sk d s † k ) . (7)Following this, the DHW variable p may be associatedwith the momentum of an electron, or minus the mo-mentum of a positron. Consider also a mixed state, | mixed (cid:105) = √ − P | (cid:105) + √ P | full (cid:105) , (8)which has probability P of being full and 1 − P of beingempty. The DHW function is easily found to be W = Sign( p − ) + 2 P Sign( − p − ) , (9)so that, for example, the p − < P relative to that in thevacuum, see Fig. 2. The DHW function therefore gives usa (quasi)probabilistic measure of the occupation numbersof electrons and positrons. The above points are worthkeeping in mind for later, as they will aid our interpreta-tion of the DHW function in the interacting theory. B. Plane wave backgrounds
The DHW function for both scalars and spinors in anarbitrary plane wave background was calculated in [10].That paper also considered the covariant DHW func-tion in which the lightfront times are also separated,i.e. one works with the density (cid:2) ψ + ( x ) , ψ † + ( x ) (cid:3) with x µ , = x µ ± y µ /
2. The DHW function is then definedby an integral over d y , and was found to beSign( p − ) (cid:90) d y + exp (cid:20) iy + (cid:18) p + − p ⊥ + M ( x + , y + )2 p − (cid:19)(cid:21) , (10)where M is clearly an effective mass (see [10, 25] for de-tails) which extends the intensity-dependent mass shiftfrom purely periodic plane waves [26–28], to arbitraryplane wave backgrounds. Integrating out p + , one recov-ers, by definition, our W : this is precisely the same asin the free theory (4). The natural interpretation of thisresult is the well known statement that plane waves donot create pairs.In the following sections we will examine what happensto the DHW function when a pair-creating longitudinalelectric field is added to the plane waves. We will con-tinue to focus on the simpler W rather than the covariantDHW function as this is the more common approach inthe literature, and because, following the above, any de-viation in W from the free vacuum result (4) must bedue (at least in part) to the longitudinal field. III. LONGITUDINAL AND TRANSVERSEFIELDS
We consider a laser pulse moving up the x -axis. Thisdefines our ‘longitudinal’ direction, while x , x are thetransverse directions. A variety of models for the laserfield can be found in the literature: the case of a con-stant, longitudinal electric field is of course covered bySchwinger’s classic result [29]. The models for whichmost analytic progress can be made (in terms of calculat-ing scattering amplitudes) are plane waves depending on x + [26–28]. The combination of a constant longitudinalelectric field and periodic plane waves was described in[28]. The case of a purely longitudinal electric field E ( x + )depending arbitrarily on x + was covered by [20, 21] (andincludes Schwinger’s result as a particular case). Thelightfront methods used in those papers can be extendedto cover the case of longitudinal E ( x + ) with a parallelB-field also depending on x + [30]. For fields dependingon both x + and x − see [31, 32]. Purely longitudinal elec-tric fields E ( x ) depending arbitrarily on (instant) time x are widely used in the literature. Such fields modelthe focus of counter propagating laser pulses in which themagnetic field components cancel. They have been usedto investigate pair production for oscillating fields [33–35], pulsed fields [36, 37] and pulsed fields with sub-cyclestructure [38–40]. Here we further extend the above results, covering thecase of longitudinal electric and transverse electromag-netic plane wave fields, both depending arbitrarily on x + .We work in ‘anti-lightcone’ gauge A − ≡ A + = 0 (theusual lightcone gauge is A − = 0). The remaining com-ponents of the potential are given by, A − = − x + (cid:90) d y E (cid:107) ( y ) , A ⊥ = √ x + (cid:90) d y E ⊥ ( y ) , (11)where we assume for simplicity that the fields turn on at x + = 0. This can, and will, be relaxed below. A. Solutions of the Dirac equation
Defining the projectors Λ ± ≡ γ ∓ γ ± the fermion fielddecomposes into ψ ≡ ψ + + ψ − with ψ ± ≡ Λ ± ψ . TheDirac equation then separates into i∂ + ψ + = ( iγ ⊥ D ⊥ + m ) γ − ψ − , (12) iD − ψ − = ( iγ ⊥ D ⊥ + m ) γ + ψ + . (13)We immediately take the Fourier transform of the trans-verse coordinates, i∂ ⊥ → k ⊥ , which replaces iD ⊥ → k ⊥ − eA ⊥ ( x + ) ≡ π ⊥ ( x + ) . (14)One solves the Dirac equation by first observing that ψ − is a constrained field, since it can be expressed in termsof ψ + using (12): ψ − ≡ γ ⊥ π ⊥ + mω iγ + ∂ + ψ + , (15)where the mode frequency is defined by ω ( x ⊥ ) ≡ π ⊥ ( x + ) + m = k ⊥ + m + e A ⊥ ( x + ) − ek ⊥ A ⊥ ( x + ) , (16)in which we recognise the Volkov exponent [41]. Substi-tuting (15) into (13), and noting that D − and ∂ + do notcommute, one obtains a simple equation for ψ + : D − ∂ + ψ + = − ω ψ + . (17)If we try to Fourier transform i∂ − → k − , we see thatsolving (17) requires inverting k − − eA − ( x + ) , (18)which can clearly vanish, possibly multiple times, for agiven k − . This is the zero-mode problem of lightfrontfield theory [42], but made time-dependent by the exter-nal field. The physics of the zero-mode in the currentcontext is as follows. An electron with momentum k − attime x + = 0 acquires (as follows from solving the Lorentzequation) a momentum k (cid:48) − = k − − eA − ( x + ) at later times. FIG. 3. The domain of our solution (19), x + > x − > − L . The finite-duration background fields depend on x + ∈ . . . x + f , as is also illustrated, along with the behaviour ofelectrons and positrons created within the field. We have e <
0, so if we imagine that E (cid:107) is positive, then k (cid:48) − will vanish at some later lightfront time, which meansthe electron moves parallel to the x − axis (reaches thespeed of light) at this instant: it therefore vanishes fromthe theory since it cannot be seen at any subsequent light-front time [22, 23]. Note that a positron’s momentum,on the other hand, only increases in the above circum-stances. Hence the positrons remain in the theory. Thiswill be useful for later.We are now ready to give the solution of the Diracequation. We follow the method of [20, 21]. The ideais to turn the fields on at x + = 0 (for convenience, thiscan be relaxed, see below), and solve the Dirac equationin the semi-infinite region x + > x − > − L for somepositive L , in terms of initial data. This gives, as wewill see, a prescription for handling the singularity at k − − eA − = 0. In the end the limit L → ∞ is taken.The domain of our solution, together with an illustrationof our fields and the motion of particles within them, isshown in Fig. 3.It may be checked directly that the solution to (12)-(13) in x + > x − > − L is ψ + ( x + , x − ) = ∞ (cid:90) − L d y − ψ + (0 , y − ) ¯ D − ( y ) G (cid:0) x + , x − , y − (cid:1) − x + (cid:90) d y + ∂ψ + ∂y + ( y + , − L ) G (cid:0) x + , y + ; x − , − L (cid:1) , (19)with ψ − given by (15). We consider the various terms.First, ψ + ’s dependence on the boundary data is explicit:the solution depends on ψ + on the characteristic x + = 0and ∂ + ψ + ∼ ψ − on the characteristic x − = − L , since i∂ + ψ + = 12 ( /π + m ) γ − ψ − , (20) from (15). The function G is G (cid:0) x + , y + ; x − , y − (cid:1) = − i (cid:90) d k − π e i ( y − − x − )( k − + i/L ) k − − eA − ( y + ) + i/L E k − ( y + , x + ) , (21)and E is defined by E k − ( x + , y + ) = exp (cid:20) − i x + (cid:90) y + d s ω ( s ) k − − eA − ( s ) + i/L (cid:21) . (22)It is worth considering this function in a little detail, asit exhibits the essential difference between the transverseand longitudinal fields. The transverse plane wave fieldsenter just as in the Volkov solution, in the numerator ofthe exponent [41]. These terms may therefore be recov-ered by resumming all orders of perturbation theory inthe plane wave coupling eA ⊥ . The longitudinal field, onthe other hand, appears in the denominator and exhibitsa singularity on the real line, regulated by the factors of i/L : when L → ∞ this leads to an essential singularityin the coupling, as in Schwinger’s result.An advantage of the approach we adopt is that thedifferences between the types of field, and the importantstructures, are laid bare. Nothing is hidden inside thebehaviour of special functions, as is frequently the casein the instant-form approach: the equal x (instant time)DHW function is expressed in terms of parabolic cylin-der functions for E ( x ) = E , constant, and in terms ofhypergeometric functions for E ( x ) = E sech ( ωx ) [9].However, a disadvantage of our approach is that expres-sions quickly become lengthy. For this reason, the quan-tisation of (19) and the calculation of the DHW functionare left to the appendix. Related calculations are explic-itly performed in [21], which the reader may consult forfurther examples. The final result for the DHW functionin the limit L → ∞ is, however, extremely compact, andwe turn to it now. IV. PAIR PRODUCTION
If one considers only a longitudinal electric field de-pending on x + , one finds that not only the vacuum per-sistence amplitude but also the pair production rate maybe calculated instantaneously as a function of x + . Thederivation of this latter result requires a careful interpre-tation of the Heisenberg operators in order to identifythe pair production probability [20, 21]. We can providea (positive) check of that interpretation using the DHWfunction. Moreover, our approach is manifestly gaugeinvariant.It was found in [28] that the addition of a plane waveto a constant electric field (Schwinger’s case) does notchange the vacuum persistence amplitude. We will seefor our fields that this remains true: the plane wave hasno impact on the creation of particles, nor the propertiesthey are created with. Rather neatly, though, our resultsmake clear the post-creation effect of the plane waves onthe particles. A. The DHW function
Our solution of the Dirac equation (and its quantisa-tion, as described in the appendix) is valid for arbitrarilylongitudinal and plane wave fields. We present here re-sults for the case in which E (cid:107) is assumed to be positive,so that eA − is positive and increasing, as this is when thesingularities in (22) have the simplest structure in phasespace. Our longitudinal fields therefore model subcyclepulses, which are of considerable current interest [43].The plane wave fields remain arbitrary.We can now give the DHW function W . At x + = 0, W is that of the free theory, see (4). Once the longitudi-nal fields turn off at x + = x + f , the theory becomes stableagainst pair production and we find that the DHW func-tion again becomes constant in lightfront time, matchingits final value in the pulse. For the duration of the pulse,i.e. 0 < x + < x + f , the DHW function is W ( x + ; x , p ) = Sign( p − ) + 2 P θ ( − p − ) θ (cid:0) eA − ( x + ) + p − (cid:1) . (23)The first term is the DHW function of the empty vacuum.The second term contains the effects of the backgroundfields, and has a form similar to that in (9), though it isrestricted to p − <
0. The description and investigationof this term, in particular P , will occupy the remainderof the paper.It is important before embarking on this to give theinterpretation of the DHW variable p . In the free the-ory, see Sect. II, p ( − p ) is the kinetic momentum of anelectron (positron). This also holds once the backgroundfields turn off and the theory again becomes free (the fi-nal particle spectrum is of course what we would be inter-ested in experimentally). The DHW function smoothlyconnects the initial and final distributions in a gauge in-variant manner. Furthermore, (2) shows that (canoni-cal!) momentum dependence on, say, k in the densitywill be set equal to p + e A in the Wigner function, sothat p is naturally interpreted as a kinematic momen-tum. From here on we therefore associate p − < π − ≡ − p − >
0, as in the free theory:from (23), this is clearly the region of interest.
B. From dynamics to probabilities
In order to give the most compact and intutive expres-sion for P it is useful to recall some results on the motionof particles in our background fields. The Lorentz equa-tion for a positron with kinematic momentum π µ andcharge − e > π µ = − eF µν d x ν . Suppose then, thata positron is created with momentum π − = 0 at some initial time x + i . From the Lorentz equation, it will at alater time x + have momentum π − = eA − ( x + ) − eA − ( x + i ) , (24)using (11). Since eA − is positive and increasing for theduration of the pulse, it has a unique inverse X p suchthat eA − ( X p ) = p and X eA − ( x ) = x . It follows that, onobservation of a positron with momentum π − at time x + ,the ‘initial time’ could be reconstructed from (24): eA − ( x + i ) = eA − ( x + ) − π − = ⇒ x + i ( π − ) ≡ X eA − ( x + ) − π − . (25)(We suppress the dependence of x + i on x + for compact-ness.) If the positron also has zero transverse momentumat the initial time, its later transverse momentum is π ⊥ ( π − ) ≡ eA ⊥ ( x + ) − eA ⊥ ( x + i ( π − )) . (26)(Again suppressing dependence of π ⊥ on x + .) It isclear from the integral expressions (11) that the results(24)–(26) simply describe the energy transferred to thepositron from the background fields over the elapsedtime. This is illustrated in Fig. 4.With these definitions we can give a very simple ex-pression for P : P = exp (cid:20) − πm + π (cid:2) p ⊥ + π ⊥ ( − p − ) (cid:3) | e | E (cid:107) (cid:0) x + i ( − p − ) (cid:1) (cid:21) . (27)where x + i and π ⊥ are defined in (25) and (26). We recog-nise a similar structure as found in Schwinger’s results,but for more general fields, and also depending instan-taneously on lightfront time. Comparing (23) and (9), P is naturally interpreted as the probability that thepositron states with momentum π − = − p − have beenfilled by time x + , since particles are being created by thebackground fields. Using the dynamics discussed above, x + i x + { π ⊥ , π − } { , } FIG. 4. A particle is created at time x + i , with probabilitydetermined by the electric field strength at that time. Theparticle has zero longitudinal momentum, and transverse mo-mentum normally distributed around zero. It is observed ata later time x + , after which it has acquired longitudinal andtransverse momenta π − and π ⊥ . a more precise statement is the following: P gives theprobability of observing positrons with momenta π − and π ⊥ at time x + , such that these positrons were createdat x + i with π − = 0 and transverse momentum normallydistributed about π ⊥ = 0. Note that because the argu-ment of the pair creating field in P is not x + but x + i , theprobability of observing a positron is dependent on theelectric field strength at the moment of creation x + i , andnot on the ‘observation’ time x + , see also Fig. 4. This isa neat and physically sensible result.We now explain this interpretation in more detail andreinforce it with a series of examples. In order to keep thepresentation as clear as possible, we begin by droppingdown to 1+1 dimensions, turning off all transverse depen-dence. This will be reinstated below. The first obviousquestion, and obvious difference between (23) and (9), iswhy do we not see electron states being filled? Why isthere no change to the DHW function for p − > p − = 0 as time evolves. This is also where the singu-larities in the Dirac equation live in the L → ∞ limit,and we have already seen that P describes particles withzero initial longitudinal momentum. Put together, thismeans that pairs are created travelling at the speed oflight. The distinction is that the electrons, being acceler-ated down the x -axis by the positive field, travel parallelto the x − direction and so, from the perspective of theinfinitely boosted lightfront frame, immediately vanish.The positrons, on the other hand, are accelerated up the x -axis and therefore acquire positive π − , see also (24),and remain visible in the lightfront frame. We thereforeconfirm the result of [20, 21]: from the lightfront perspec-tive, one only sees the positrons.The final piece of (23) to consider is the second stepfunction. This states simply that the argument of E (cid:107) ,that is X eA ( x + ) − π − , must be positive; in other words,sufficient time must have elapsed for a particle with zerolongitudinal momentum to have acquired π − by time x + .Together, the two theta functions therefore imply, using(24)-(26), that 0 < x + i < x + , (28)which is a simple statement of causality: observed pairsmust have been created at earlier times, but after thelongitudinal field turns on. We conclude that the DHWfunction shows us pair production from the vacuum, inreal lightfront time, with P the probability of pair cre-ation. The positrons appearing at time x + with a fi-nite range of momenta π − are subject to the (natural)constraint that sufficient time must have elapsed for thepositron to have absorbed this momenta from the fields,starting from π − = 0. We now move on to explicit ex-amples, staying in 1 + 1 dimensions for the moment. (cid:45) (cid:45) p (cid:96) (cid:45) (cid:45) W Π Π Π Π x (cid:96) (cid:43) FIG. 5. The DHW function in the subcycle pulse (29), plot-ted as a function of p − , for zero transverse momentum. Themaximum allowed momentum, see (31) is also shown in the(ˆ p − , ˆ x + ) plane. C. Example: finite pulse duration
We begin with the electric field E (cid:107) ( x + ) = E sin( ωx + ) (29)for 0 ≤ ωx + ≤ π and zero otherwise, modelling a half-cycle of the laser. The DHW function W is plotted inFig. 5. When plotting, we use rescaled variables (cid:15) ≡ | e | E m , ˆ x + ≡ ωx + , ˆ p − = p − ω | e | E , (30)which measures the electric field strength in units of theSchwinger field and p − in units of (as we are about tosee) half its maximum value. We have chosen (cid:15) = π for our plot, which means our electric field strength isroughly three times higher than the Schwinger limit: thiscompensates fully for the damping factor in the exponentof (27) and allows us to clearly see the behaviour of theDHW function. We consider other field strengths below.In Fig. 5, we see that W matches the free theory result(4) at x + = 0. As time evolves, W becomes both p − and x + dependent, with the deviation from vacuum spreadingout from p − = 0 at x + = 0. At time x + the function isexplicitly limited in extent by the theta-functions in (23),which give eA − ( x + ) + p − > ⇐⇒ x + > X − p − , (31)The DHW function eventually stabilises as the fieldswitches off at x + = x + f , upon which it is straightforwardto extract properties of the final positron distribution.The final range of possible positron momenta is dictatedby, following the above, − p − > − p − < eA − ( x + f ). (cid:45) (cid:45) p (cid:96) (cid:45) (cid:45) (cid:45) (cid:45) W (cid:45) x (cid:96) (cid:43) FIG. 6. The DHW function in the sech pulse (37) withpeak amplitude (cid:15) = 1, see (30), plotted as a function of p − ,zero transverse momentum. As before, the range of allowedmomentum, see (31) is shown in the (ˆ p − , ˆ x + ) plane: there isa smoother falloff than in the previous example. In our current example, ωx + f ≡ π , so the final range ofpositron momenta is0 < π − < | e | E ω ≡ ma , (32)where, in the final equality, we have introduced the peakfield intensity a [45], a = | e | E max ωm . (33)The momentum distribution is peaked around that valueof π − such that the electric field is maximal at the instantof creation . Let x + be the time at which E (cid:107) ( x + ) is max-imal, then the most probable kinetic momentum (cid:104) π − (cid:105) isthe solution of the equation x + = X −(cid:104) π − (cid:105) + eA − ( x + f ) . (34)For our current example ωx + = π/ (cid:104) π − (cid:105) = | e | E ω ≡ ma , (35)which is also clearly seen from Fig. 5. Since the momentaare on-shell in the free theory, we have (cid:104) π + (cid:105) = m / (cid:104) π − (cid:105) and so we can easily convert these expressions back tocartesian co-ordinates to find the likely energy and z -component of the momentum, which are √ (cid:104) π (cid:105) = m a + ma , √ (cid:104) π (cid:105) = m a − ma . (36)(Note that the probability P for producing very high en-ergy particles π − (cid:39) D. Example: adiabatic switching
In the light of recent literature results, which we dis-cuss below, it is worthwhile pointing out that thereis nothing to stop us turning our fields on arbitrarilysmoothly starting from arbitrary initial times, withoutaffecting the essential properties of our solutions (19) orour DHW function (23): our choice of switching the fieldson at x + = 0 was for convenience. We therefore considera field which, while qualitatively similar to our previousexample, falls off quickly but smoothly at ±∞ , namely E (cid:107) ( x + ) = E sech ( ωx + ) , (37)where E gives the peak intensity and 1 /ω the effectiveduration of the pulse. The corresponding gauge potentialis given by the integral from x + = −∞ of this function,and is therefore eA − ( x + ) = | e | E ω (cid:0) ωx + ) (cid:1) , (38)where the constant term follows from the definition (11),with the initial time translated to x + = −∞ . The result-ing DHW function is plotted in Fig. 6 for a peak fieldstrength equal to the Schwinger field. The DHW func-tion is shown over the temporal range − / < ωx + < < π − < ma .The probability P is peaked, in the limit x + → ∞ , around π − = ma . Note that even at the Schwinger limit, theprobability of pair production remains small. E. Example: ever-increasing field
We note that the factor of 2 in (23) can be understoodfrom a second perspective. Suppose we consider an elec-tric field which increases without bound, for example E (cid:107) ( x + ) = E ωx + . (39)As time increases, the electric field becomes overcriticaland we expect the probability of creating particles withany given momentum (in the allowed range, which alsoexpands in time) to approach unity. This means that thestate should become filled with positrons, and we expectto recover, from (6), W = +1 in the region p − <
0. Thisis precisely what the factor of 2 ensures: if P →
1, theDHW function approaches W = 1 for p − <
0. This isshown explicitly in Fig. 7: the DHW function transfersfrom − p − range as timeevolves. (cid:45) (cid:45) p (cid:96) (cid:45) (cid:45) W x (cid:96) (cid:43) FIG. 7. DHW function in the field (39), with (cid:15) = 0 .
5, whichincreases without bound. As time evolves, all positron statesare filled with unit probability.
F. Comparison with the instant-form approach
Our results share some similarities with investigationsof particle creation in x –dependent electric fields, withinthe usual (instant-form) DHW formalism. In that ap-proach, a certain combination of instant-form DHWspinor components can be interpreted as a particle num-ber density [44], and its behaviour is as follows. As theelectric field grows with time x , energy is transferred tothe Dirac field such that a peak around p = 0 develops.At intermediate (non-asymptotic) times this peak is in-terpreted as being composed of virtual electron-positronpairs. As time evolves, only a part of these virtual par-ticles become real particles which are then acceleratedby the electric field and spread out from p = 0. Atasymptotic times x → ∞ , the particle number densityof real particles stabilises whereas the virtual electron-positron peak around p = 0 disappears again. An ex-ample is shown in Fig. 8 for the asymptotically switchedfield E ( x ) = E sech ( ωx ). For more details see [44].One difference between these results and our own isthat the lightfront DHW function does not exhibit theintermediate virtual particle peak or oscillatory struc-ture seen in the instant form. The reason for this seemsto be that pair production on the lightfront is an in-stantaneous event, occurring at the instant when a givenfermion mode can produce a particle of zero longitudinalmomentum, see (18). This is confirmed by our expres-sion for the pair creation probability P : it is expressedentirely in terms of classical particle trajectories.This is quite intriguing, as it may be related to the‘triviality’ of the lightfront vacuum. The instant–formvacuum is filled with virtual pairs which can be pulledonto the mass–shell by the external field. Recall thatthe Schwinger field strength can be obtained by equatingthe electron rest mass with the work done by the elec-
20 15 10 5 0 p ! ! " x ! FIG. 8. Particle number density N for the x -dependentsech pulse within the instant–form DHW formalism, plottedas a function of ˆ p and zero transverse momentum. Peakamplitude (cid:15) = 1. tric field over the lifetime of a virtual pair: in this sense,there is a time scale involved in Schwinger pair produc-tion. The lightfront vacuum, on the other hand, is oftenreferred to as ‘trivial’, which is the statement that it iscompletely empty of particles, both real and virtual [46].Moreover, it is stable. In this picture, then, pairs are cre-ated from the energy pumped into the system, not fromvirtual particles being pulled on-shell, and the Schwinger‘time-scale’ is absent. This is an investigation for anothertime, though. We now return to properties of the DHWfunction. G. Nonperturbative dependence
Recent results on pair production in x -dependent elec-tric fields (in the usual instant time DHW formalism),find a purely perturbative dependence on a particularelectric field which switches on adiabatically in the infi-nite past [47]. Moreover, it is stated that the essentialsingularity of Schwinger’s results must therefore be dueto the unphysical nature of a constant, ever-present elec-tric field.Let us reconsider our results in this light. We have seenexplicitly that for both sharply and smoothly switchedfields, the pair production probability is basically de-scribed by a factor exp( − mπ/ | e | E (cid:107) ), which retains theessential singularity in the coupling from Schwinger’s re-sult. It is therefore clear that the nonperturbative natureof pair creation is not due to some unphysical assump-tion about when, or how smoothly, the fields turn on oroff. The dependence of our results on e is of course alittle more complex than that, since E (cid:107) is evaluated at x + i ( − p − ), see (27).It is useful to examine the form of the (final) probabil-ity P when the fields turn off. We work with the physicalmomentum π − here, and turn off the plane wave fieldsfor simplicity, setting also π ⊥ = 0. We begin with thefield (29), for which the final probability is − log P (cid:12)(cid:12) ωx + = π = πm /ω (cid:112) π − (2 ma − π − ) . (40)Note that the denominator is positive because of the fi-nite allowed π − range. We can examine this probabilityfor, for example, small momenta (which corresponds toextremely energetic particles in the lab frame) by expand-ing in π − : πm (cid:112) | e | E ωπ − (cid:18) ωπ − | e | E + . . . (cid:19) (41)We clearly see the 1 / | e | E dependence. What happensfor the adiabatically switched sech pulse? For this field,the final probability islim x + →∞ − log P = π | e | E m | e | ωE π − − ω π − . (42)Again, the denominator is positive, and we can make thesame small π − expansion as above, findinglim x + →∞ − log P (cid:39) πm ωπ − (cid:18) ωπ − | e | E + . . . (cid:19) . (43)This displays a different dependence on the various pa-rameters. In particular, the dominant term is indepen-dent of the field strength E . Does this correspond to aperturbative dependence on the field strength? The an-swer is no: not only does the second term in the expan-sion contain explicitly nonperturbative (Schwinger-like)terms, but the leading term of (43) actually contains ahidden nonperturbative dependence. To see this, notethat the leading term, despite not being explicitly de-pendent on E , does not survive the limit E → π − is finite, being limited by eA − , so that tak-ing E → π − →
0, and this kills P as thefields turn off.What this result really shows is only that, and as isnot surprising, the distribution of the produced parti-cles depends on the geometry of the field, for examplewhether the field turns off sharply or smoothly. We havenot been able to identify a regime where the results maybe expressed as a perturbation in E . We stress that thisholds at least on the lightfront, for fields depending on x + : there are differences between this and the instant-form approach, see Sect. IV F, above.It is, though, entirely possible for the effective action toexhibit both perturbative and Schwinger-like nonpertur-bative behaviour when the electric field depends on x (and is even adiabatically switched), depending on therelative sizes of the parameters involved. This is shownin [48], which discusses many deep connections between perturbative and nonperturbative physics. One is leadto conclude that the perturbative dependence found in[47], while very interesting, is not inconsistent with theexistence of nonperturbative behaviour. Combining thiswith our own results, we do not believe that any doubtis cast on the validity of Schwinger’s result. H. Transverse dependence
Finally, it is time to return to 3 + 1 dimensions proper,and allow for plane wave fields. Consider the full ex-pression (27) for the probability. The plane wave contri-butions do not appear in the step functions, thus theydo not affect the constraints dictating the momentumranges. Nor do the plane waves enter into the argumentof the pair creating field E (cid:107) .It is clear that without the plane waves, the probabilityfor production of pairs with nonzero transverse momentais normally distributed around π ⊥ = 0. Turning on theplane waves, it may seem strange at first glance that thepeak of this distribution is shifted to nonzero values: isthe plane wave affecting the probability of pair produc-tion? The answer is no: recalling (24)-(26), the distribu-tion in (27) is obtained for positrons created with trans-verse momentum normally distributed around π ⊥ = 0,and which at the subsequent time x + must have acquiredtransverse momentum x + (cid:90) x i ( π − ) d s √ eE ⊥ ( s ) ≡ π ⊥ ( π − ) , (44)from the plane wave fields, using the Lorentz equation.The DHW function therefore takes into account bothwhat happens at the instant of creation, but also whatwould subsequently be observed.To summarise, the plane waves do of course influencethe particles after they are created, and so it is no sur-prise that they appear in the DHW function and thefinal particle distribution. The plane waves do have noinfluence, though, on whether particles are created ornot. This is reaffirmed by integrating over momenta toobtain, for example, the total probability of pair produc-tion or the vacuum persistence amplitude: one finds thatall dependence on the plane wave fields vanishes because π ⊥ ( − p − ) can be absorbed into p ⊥ by a change of vari-able. Thus the plane wave fields, and in particular anyeffective mass they may generate, do not influence theprobability of vacuum decay [20, 21, 28]. V. CONCLUSIONS
We have investigated the phenomenon of non-perturbative pair creation in background electromagneticfields, within the lightfront DHW formalism. We calcu-lated the DHW function by solving the Dirac equation0in a combination of longitudinal electric and transverseplane wave fields which both depend arbitrarily on light-front time. This extends the work of [20, 21, 26] to aneven wider class of fields.As shown in [10], the DHW function W is not alteredby a single plane wave field, since a plane wave can notproduce pairs. Switching on an additional pair-creatingelectric field, however, one observes a deviation from thevacuum result which signals pair creation. The pair cre-ation probability itself is exponentially suppressed by afactor exp( − mπ/ | e | E (cid:107) ) which retains the essential sin-gularity in the electromagnetic coupling e in Schwinger’sresult, but is valid for much more general fields, andin particular is independent of how the electric field isswitched on and off. This may be contrasted with recentresults in the instant–form approach [47].Notably, we have seen that the value of the DHW func-tion W is altered only in its positron sector, whereas theelectron content remains unchanged. This confirms pre-vious results that, from the lightfront perspective, onlyone of the particle species remains in the theory followingcreation [20, 21].All in all, the DHW function can be a powerfultool in analysing quantum physics in background fields(particularly pair production), and in a language whichis essentially classical. We have given an elegant andphysical interpretation in terms of the pair creationprobability and the subsequent dynamics of the parti-cles. This makes it clear that we observe real particleproduction in the lightfront formalism. ACKNOWLEDGMENTS
A. I. gratefully thanks Richard Woodard for correspon-dence, and Tom Heinzl for discussions. Fig. 1 to Fig. 4created using JaxoDraw [49, 50].F. H. is supported by the Baltic Foundations. A. I. andM. M. are supported by the European Research Coun-cil Contract number 204059-QPQV. This work was per-formed under the
Light in Science and Technology
StrongResearch Environment, Ume˚a University.
Appendix A: Notation.
Our lightcone directions are x ± = ( x + x ) / √
2. Weprefer momenta to carry covariant indices and so p − isa spatial momentum conjugate to x − , while p + is thelightfront energy, conjugate to lightfront time x + . Themetric has determinant − v ∓ = v ± forarbitrary vectors v . We use a sans-serif font to denotethe ‘spatial’ variables and momenta, i.e. x ≡ { x − , x ⊥ } and p ≡ { p − , p ⊥ } . Our Fourier conventions are f ( p ) = (cid:90) d x e ipx f ( x ) , f ( x ) = (cid:90) d p π e − ipx f ( p ) , (A1)and we Fourier transform before taking conjugates so f † ( p ) = (cid:90) d x e − ipx f † ( x ) , f † ( x ) = (cid:90) d p π e ipx f † ( p ) . (A2) Appendix B: The quantum theory
We now wish to quantise our solution (19). Quantisation is performed by imposing canonical commutation relationson the initial data. As in [20, 21] one finds that the quantised spinor fields obey the desired commutation relations,returning briefly to full co-ordinate space, (cid:8) ψ + ( x ) , ψ † + ( y ) (cid:9)(cid:12)(cid:12) x + = y + = 1 √ + δ ( x ⊥ − y ⊥ ) δ ( x − − y − ) , (B1)provided that the initial data obeys (cid:8) ψ + ( x ) , ψ † + ( y ) (cid:9)(cid:12)(cid:12) x + = y + =0 = 1 √ + δ ( x ⊥ − y ⊥ ) δ ( x − − y − ) , (cid:8) ψ − ( x ) , ψ † − ( y ) (cid:9)(cid:12)(cid:12) x − = y − = − L = 1 √ − δ ( x ⊥ − y ⊥ ) δ ( x + − y + ) , (B2)What we are really doing here is solving a Cauchy problem with initial data on two lightlike characteristics: this is notquite what one usually does on the lightfront (where the operators on the x − characteristic are not needed explicitly)but is necessitated by the time-dependence introduced into the zero-mode problem. Note that without the operatorson x − = − L one does not recover known results such as Schwinger’s vacuum persistence amplitude in the L → ∞ limit. Moreover, the approach used here has been verified by alternative methods [51]. (The question of whetherthere is a method to recover the results of this approach in ordinary lightfront quantisation without the operators at x − = − L has not, to our knowledge, been addressed.)From here one can construct the Hamiltonian and states of the theory as normal. Computationally, it is useful towork in what becomes Fourier space in the L → ∞ limit: considering the k − -integrations inside the functions G , (21),1we define the ‘almost’ Fourier transform ˜ ψ + by ψ + ( x + , x − ) ≡ (cid:90) d k − π e − ix − ( k − + i/L ) ˜ ψ + ( x + , k − ) . (B3)Dependence on k ⊥ will not be written unless it is needed. Explicitly, our new field is˜ ψ + ( x + , k − ) = ∞ (cid:90) − L d y − e iy − ( k − + i/L ) E k − (0 , x + ) ψ + (0 , y − ) + x + (cid:90) d y + e − iL ( k − + i/L ) k − − eA − ( y + ) + i/L E k − ( y + , x + ) i∂ + ψ + ( y + , − L ) . (B4)(The operators ˜ ψ + become canonically normalised creation (and annihilation) operators as L → ∞ , in the interpreta-tion of [20, 21], up to a factor of 2 / : hence the factor of √ free vacuum, since what we wish to do is begin in this state at x + = 0, apply our externalfields and see what happens as time evolves. Since we are working in the Heisenberg picture, in which the states aretime-independent, the state we need is precisely the free vacuum state. This means that the density we need, (1), iscalculated by first expressing ψ + in terms of the free initial data using (B4), and then evaluating the density of thesefree fields in the free vacuum state. This calculation is performed using the ordinary free-field mode expansion. Forexample, a free-field calculation easily gives us (cid:104) | ψ + ( x ) ψ † + ( y ) | (cid:105) (cid:12)(cid:12) x + = y + =0 = 1 √ + ∞ (cid:90) d k − π e − ik − ( x − x ) − δ ( x ⊥ − y ⊥ ) , (B5)which we will use below. Appendix C: Calculating the DHW function
We now wish to calculate (1) and (2). To do so we make an assumption on the longitudinal electric field, namelythat it is positive, e.g. modelling a subcycle pulse. As a result, eA − is a positive function increasing from 0 at x + = 0. This approximation can be relaxed, but doing so means that the analytic structure in the functions E becomes significantly more complex, since the kinematic momentum π − ( x + ) = k − − eA − ( x + ) may have multiple zeros.The ‘subcycle assumption’ means that this function has at most one zero for a given k − . With this assumption, wecan define the inverse function X ( k − ) ≡ X k − by k − − eA − ( X k − ) ≡ . (C1)This function exists for the duration of the longitudinal field E (cid:107) . What happens after the fields turn off is explainedbelow. To proceed, we express the density U in terms of the modes ˜ ψ + ( x + , k − ). The first term of the commutator is (cid:104) | ψ + ( x + , x − ) ψ † + ( x + , x − ) | (cid:105) = (cid:90) d l − π f ( l − ) e − i ( l − + iL ) x − (cid:90) d q − π f ∗ ( q − ) e i ( q − − iL ) x − (cid:104) | ˜ ψ + ( x + , l − ) ˜ ψ † + ( x + , q − ) | (cid:105) , (C2)where we have included test functions under the momentum integrals in order to take the L → ∞ limit morerigourously. Since ψ + itself contains two terms, the overlap in (C2) contains four. Only one of these, the product ofthe first term in (B4) with its conjugate, ultimately contributes, so we present only this calculation. Using (B4) towrite down this term, and the free-field expression (B5), one finds that the overlap in (C2) is1 √ + ∞ (cid:90) d k − π ∞ (cid:90) − L d v e i ( l − + i/L ) v ∞ (cid:90) − L d z e − i ( q − − i/L ) v E l − (0 , x + ) E ∗ q − (0 , x + ) e − ik − ( v − z ) = 1 √ + ∞ (cid:90) d k − π e − iL ( l − + i/L ) e iL ( q − − i/L ) E l − (0 , x + ) l − − k − + i/L E ∗ q − (0 , x + ) q − − k − − i/L , (C3)carrying out the v and z integrations to arrive at the second line. We now bring in the momentum integrals andchange variables l − → a = L ( l − − k − ) and q − → b = L ( q − − k − ):(C2) = 1 √ + ∞ (cid:90) d k − π e − ip ( x − − x − ) (cid:20) (cid:90) d a π f ( aL + k − ) E a/L + k − (0 , x + ) a + i e − i ( a + i )(1+ x − /L ) (cid:21)(cid:20) (cid:90) d b π f ∗ ( bL + k − ) E ∗ b/L + k − (0 , x + ) b − i e i ( b − i )(1+ x − /L ) (cid:21) . (C4)2Our task now is to take the L → ∞ limit. The behaviour of the E functions in this limit depends crucially on therelative values of x + and k − , as these determine whether or not we hit the singularity. Explicitly, we have E a/L + k − (0 , x + ) = exp (cid:20) − i x + (cid:90) d s ω ( s ) k − − eA − ( x + ) + a + iL (cid:21) . (C5)We have k − >
0. It is therefore clear that if k − − eA − ( x + ) > L → ∞ limit.Although we cannot carry out the s -integral exactly we know that E becomes a phase, and this will cancel betweenthe two large bracketed terms in (C4). We can therefore write part of the solution immediately: (cid:104) | ψ + ( x + , x − ) ψ † + ( x + , x − ) | (cid:105) → √ + αβ ∞ (cid:90) eA − ( x + ) d k − π | f ( k − ) | e − ik − ( x − − x − ) (cid:90) d a π e − i ( a + i ) a + i (cid:90) d b π e i ( b − i ) b − i + . . . = 1 √ + αβ ∞ (cid:90) eA − ( x + ) d k − π | f ( k − ) | e − ik − ( x − − x − ) + . . . (C6)The integral over the range k − ∈ . . . eA − ( x + ) requires more care. In this case there is always a value of s such thatthe integral in (C5) acquires an imaginary part which does not drop out of (C4). To identify the imaginary part wechange variables eA − ( s ) ≡ t , i.e. s = X t and then expand the resulting t dependence in the numerator: E a/L + k − (0 , x + ) = exp (cid:20) − i eA − ( x + ) (cid:90) d t ω ( X t ) X (cid:48) t k − − t + a + iL (cid:21) = exp (cid:20) − iλ ( k − ) eA − ( x + ) (cid:90) d tk − − t + a + iL + . . . (cid:21) (C7)where we have shown only the first term in the series and defined the function λ ( p ) = ω ( X p )2 | e | E (cid:107) ( X p ) . (C8)Carrying out the integrals and then taking the limit L → ∞ one finds E a/L + k − (0 , x + ) → exp (cid:20) − iλ ( k − ) (cid:0) − log[ | − eA − ( x + ) /k − | ] − iπ + real (cid:1)(cid:21) , (C9)where the real logarithm and iπ come from the first term of the Taylor expansion, and all other terms give real, a -independent contributions. (This behaviour is different from that in the second commutator term calculated in [21],where the singularity always lies precisely at one of the integral’s limits, rather than between them, and the a and b integrals are more complex.) Thus, each E function contributes the exponential of − πλ and the rather simple, finalexpression for the first term of the density (C2) is (cid:104) | ψ + ( x + , x − ) ψ † + ( x + , x − ) | (cid:105) = 1 √ + ∞ (cid:90) eA − ( x + ) d k − π | f ( k − ) | e − ik − ( x − − x − ) + eA − ( x + ) (cid:90) d k − π | f ( k − ) | e − ik − ( x − − x − ) e − πλ ( k − ) . (C10)Before moving on to the DHW function itself, it is worth checking this result. The second commutator term has beencalculated independently in [21], see equations (4.2) and (4.9) therein. That calculation is in two dimensions, so if weturn off our transverse dependences completely, one finds, adding their result to our own, (C10), then we correctlyobtain the (1+1) field anticommutator (B1). Given this positive result, we return to four dimensions, and calculatethe second term of the density U from (B1). The final expression for the density is, removing the test functions, U = e − ik ⊥ y ⊥ √ + (cid:20) ∞ (cid:90) −∞ d k − π Sign (cid:0) k − − eA − ( x + ) (cid:1) e − ik − y − + 2 eA − ( x + ) (cid:90) d k − π e − ik − y − e − πλ (cid:21) , (C11)where we have written x ≡ x + y / , x ≡ x − y / as in Sect. II, and λ in this expression is λ = m + [ k ⊥ − eA ⊥ ( X k − )] | e | E (cid:107) ( X k − ) . (C12)3We now perform the Fourier transform in (2). Including the Wilson line, the transformation sets p ⊥ + eA ⊥ ( x + ) = k ⊥ , and p − + eA − ( x + ) = k − . 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