Sauter-Schwinger Pair Creation Dynamically Assisted by a Plane Wave

PHYSICAL REVIEW D 97, 096004 (2018)

Sauter-Schwinger pair creation dynamically assisted by a plane wave

Greger Torgrimsson,1,2,3 Christian Schneider,1 and Ralf Schützhold1 1Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, 47057 Duisburg, Germany 2Theoretisch-Physikalisches Institut, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany 3Helmholtz Institute Jena, Fröbelstieg 3, 07743 Jena, Germany

(Received 11 January 2018; published 7 May 2018)

We study electron-positron pair creation by a strong and constant electric field superimposed with a weaker transversal plane wave which is incident perpendicularly (or under some angle). Comparing the fully nonperturbative approach based on the world-line instanton method with a perturbative expansion into powers of the strength of the weaker plane wave, we find good agreement—provided that the latter is carried out to sufficiently high orders. As usual for the dynamically assisted Sauter-Schwinger effect, the additional plane wave induces an exponential enhancement of the pair-creation probability if the combined Keldysh parameter exceeds a certain threshold.

DOI: 10.1103/PhysRevD.97.096004

I. INTRODUCTION is strongly enhanced by adding a weaker time-dependent field to the strong field E. So far, most studies of this As one of the most striking consequences of quantum enhancement mechanism have been devoted to purely field theory, extreme external conditions can tear apart time-dependent fields [15]. quantum vacuum fluctuations and thereby create real As a step towards a more realistic field configuration, we particles. Already in 1939, Schrödinger predicted that consider a propagating plane wave superimposed to the the rapid expansion of the Universe could induce such a constant field E in the following. Plane waves propagating process [1]. As another example, the strong gravitational parallel to the constant electric field were already considered field around a black hole can tear apart quantum vacuum in [17–20], for example. It was found that such transversal fluctuations leading to Hawking radiation, i.e., black hole planewaves do not enhance the pair creation probability [18]. evaporation [2,3]. In analogy to the gravitational force, a Further, for longitudinal parallel waves, E ðt þ zÞ, the pair strong electric field can have a similar effect and create z creation probability is given by the locally constant field electron-positron pairs out of the quantum vacuum—the approximation [21–23], which implies that the enhancement Sauter-Schwinger effect [4–7]. For a constant electric field is comparably small. Both results can be understood by E, the pair creation probability (per unit time and volume) considering a Lorentz boost along the direction of the strong scales as (ℏ ¼ c ¼ 1) field which leaves the strong field invariant but reduces the m2 Peþe− ∼ exp −π ; ð1Þ frequency of the plane wave, see also [24]. In the transversal qE case, the field strength of the plane wave is reduced as well where q and m are the elementary charge and the mass of while the longitudinal wave retains its field strength. the positron/electron, respectively. In contrast to the parallel scenarios discussed in [18–23], Unfortunately, this fundamental prediction of quantum we consider the case of a transversal plane wave propa- field theory has not been directly verified experimentally yet gating perpendicular to the strong field E because the required field strength is very large. This E e ε Ω − e motivates the quest for ways to enhance the pair-creation ðt; xÞ¼E z þ E cosð ½t xÞ z; ð2Þ probability or, equivalently, to lower the required field corresponding to the vector potential (in temporal strength. One option is the dynamically assisted Sauter- gauge) A ðt; xÞ¼Et þ εE sinðΩ½t − xÞ=Ω. Schwinger effect [8–14], where the pair-creation probability z This scenario has several advantages: since such a transversal wave cannot create electron-positron pairs on its own (due to a similar Lorentz boost argument as above), Published by the American Physical Society under the terms of pair creation can only occur in cooperation with the strong the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to field E, which retains the nonperturbative character of this the author(s) and the published article’s title, journal citation, effect. Furthermore, the above profile (2) represents a and DOI. Funded by SCOAP3. vacuum solution to the Maxwell equations and could be

2470-0010=2018=97(9)=096004(9) 096004-1 Published by the American Physical Society TORGRIMSSON, SCHNEIDER, and SCHÜTZHOLD PHYS. REV. D 97, 096004 (2018) a reasonable approximation for an experimental setup the exponentials in PN grow according to Eq. (4).Asa where E represents the focus of an optical laser while the result, there could be a dominant order N which yields the plane wave is generated by an x-ray free-electron laser maximum contribution to the sum (3). In order to study this (XFEL). Finally, we found that this scenario (2) yields the question, we approximately treat N as a continuous variable N maximum enhancement of the pair creation probability. and apply the saddle point method to the term ε PN, i.e., Other profiles, polarizations and propagation directions will 2 2 pffiffiffiffiffiffiffiffiffiffiffiffi be discussed below in Sec. III and Appendix A. Note that this d m⊥ 2 −Njlnεj− ðarccosΣ − Σ 1 − Σ Þ ¼ 0: ð6Þ profile (2) was already considered in [25] using first-order dN qE perturbation theory in ε, whereas we are going to consider higher orders as well as a fully nonperturbative approach. This yields the dominant order N as solution of the transcendental equation II. PERTURBATIVE APPROACH Ω 4 γ 1 − N m qE ε crit 4 arctan Ω ¼ Ω jln j¼ γ ; ð7Þ At first, we employ a perturbative expansion of the total m N m pair creation probability Peþe− in powers of the relative strength ε of the plane wave, which is supposed to be small where we have introduced the (combined) Keldysh param- γ Ω γ ε ε ≪ 1 eter ¼ m =ðqEÞ and its threshold crit ¼jln j. We only X∞ obtain real solutions N if the right-hand side is less than N γ γ Peþe− ¼ ε PN; ð3Þ unity, i.e., if exceeds the threshold crit. At the threshold, 0 γ γ 0 N¼ ¼ crit, we find N ¼ which implies the original Schwinger result (1).Forγ>γ and Ω ≪ m, however, the where the contributions PN can be derived via the world-line crit formalism, for an introduction see [26,27] and references dominant order N can be quite large (which justifies γ 3γ therein. The zeroth order N ¼ 0 reproduces the original the continuum approximation). For example, for ¼ crit, ∼ 2 ε Sauter-Schwinger effect in Eq. (1), and odd orders vanish in the dominant order N scales as N m =ðqEj ln jÞ which this situation (but not always [28]). can be a large number for electric fields E well below the 2 The lowest-order term N ¼ 2 corresponding to the one- Schwinger limit ES ¼ m =q. γ γ ≫ 1 photon contribution has already been calculated in [25]. In the limit = crit , we may approximate the solution Deriving the exponential dependence for the higher-order of the transcendental equation (7) via terms, it turns out that the exponent for two photons rffiffiffiffiffiffiffiffiffiffi 4m γ (N ¼ 4) with frequency Ω is the same as that for a single γ ≫ γ ≈ Nð critÞ Ω 3γ ; ð8Þ photon (N ¼ 2) with twice the frequency 2Ω, and so on for crit more photons (see Appendix A). which will also be a large number unless the frequency Ω Consequently, we find far exceeds the electron mass m. Inserting this approximate 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi m⊥ 2 N P ∼ exp − ðarccos Σ − Σ 1 − Σ Þ ; ð4Þ solution for the dominant order back into the exponent N qE (4), we find Σ rffiffiffiffiffiffiffi where the function of in the exponentp isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi already known 2 γ m crit 2 2 − ∼ −8 from [14]. The effective mass m⊥ ¼ m þðNΩ=4Þ Peþe exp : ð9Þ qE 3γ reflects momentum conservation in x-direction, where the momentum NΩ=2 of the N=2 photons has to be In contrast to the dynamically assisted Sauter-Schwinger transferred to the electron-positron pair. As a result, the effect with a purely time-dependent field, we see that the effective mass m⊥ is higher than the original mass m, and exponent still crucially depends on the strong field E, hence the pair creation probability is lower than in the case which demonstrates the nonperturbative character of Σ of a purely time-dependent field. Finally, describes the this effect even for γ=γ ≫ 1. As mentioned in the 2 crit relative contribution of the energy of the N= photons in Introduction, this is a consequence of the fact that a plane 2 comparison to the effective mass gap m⊥ wave alone cannot create electron-positron pairs out of the Ω 2 Ω vacuum. Σ N = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN ¼ ¼ 2 2 : ð5Þ 2m⊥ 4 m þðNΩ=4Þ B. Improved approximation Ω↓0 Σ↓0 ↓ In the limit of , i.e., , where m⊥ m, we recover In the following, we try to improve the accuracy of the Eq. (1), as expected. approximation outlined in the previous section. The above estimate of the leading order N was based on the A. Dominant order competition between the factor εN and the exponent (4). Inspecting the terms in the sum (3) we find that the However, the prefactor in front of this exponent will also prefactors εN decrease as N increases (due to ε ≪ 1) while depend on N and thereby slightly modify the scaling

096004-2 SAUTER-SCHWINGER PAIR CREATION DYNAMICALLY … PHYS. REV. D 97, 096004 (2018) with N. Thus, in order to improve our approximation, we f() make an educated guess for the scaling of that prefactor with N. Obviously, each additional power of ε must be accom- panied by a factor of qE since this governs the coupling to the fermionic field. Recalling the structure of the QED μ interaction (vertex) term ψ¯ qAμγ ψ, it seems quite reasonable to suppose a scaling with εqE=Ω since Aμ ∝ E=Ω. Finally, in view of dimensionality arguments, we arrive at the following rough estimate for the scaling of the prefactor: qE N εNP ∼ ε × const N mΩ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2m⊥ × exp − ðarccos Σ − Σ 1 − Σ2Þ ; ð10Þ qE

f()

3.0

2.5

2.0

1.5

e⊥ FIG. 2. Same as Fig. 1, but for perpendicular polarization P as 20 40 60 80 100 discussed in Sec. III. Again, the constant factor in Eq. (11) is f() obtained by fitting and gives 1.9 for ε ¼ 10−2 (top) and 2 for ε ¼ 10−3 (bottom). We observe that perpendicular polarization 3.0 yields a lower pair-creation probability Peþe−.

where const is a (so far undetermined) constant—or, more 2.5 precisely, a factor which should only weakly depend on the involved parameters. 2.0 The saddle point of the above expression gives a slightly shifted dominant order N as a solution of the transcendental equation with a modified right-hand side 1.5 Ω 4 1 − N m qE ε qE 20 40 60 80 100 4 arctan Ω ¼ Ω ln Ω × const : ð11Þ m N m m

FIG. 1. Plot of the exponent of the pair-creation probability Peþe− as a function of γ for perpendicular incidence and parallel polari- Comparison with fully nonperturbative results obtained zation (2) with ε ¼ 10−2 (top) and ε ¼ 10−3 (bottom). The exponent with the world-line instanton method as described in has been multiplied by qE=m2, i.e., the plot shows fðγÞ such that Sec. IV shows that this modification is indeed an improve- 2 Peþe− ∼ expf−fðγÞm =½qEg. The circles represent the numerical ment of our approximation and leads to good agreement, world-line instanton results from Sec. IV,andthedashedcurve see Figs. 1 and 2. corresponds to the large-γ approximation in (9). The solid curve shows the result of our improved analytical approximation obtained III. OTHER DIRECTIONS by inserting the dominant order N from Eq. (11) into Eq. (4).The constant factor in Eq. (11) has been chosen in order to match the So far, we have considered the case of perpendicular world-line instanton results, which gives a factor of 8 for ε ¼ 10−2 incidence and a plane wave with the electric field compo- (top) and a factor of9.5 forε ¼ 10−3 (bottom). With thesevalues,we nent parallel to the strong field (2), which yields the observe good agreement between our improved analytical approxi- maximum enhancement. Now let us briefly discuss more mation and the numerical world-line instanton results. general angles

096004-3 TORGRIMSSON, SCHNEIDER, and SCHÜTZHOLD PHYS. REV. D 97, 096004 (2018)

2 2 Eðt; rÞ¼Eez þ εE cosðΩt − K · rÞeP; ð12Þ mẍμðτÞ¼aqiFμνðxðτÞÞx_νðτÞ;a¼ x_ ¼ const; ð16Þ where the corresponding vector potential reads A0 ¼ 0 and and evaluate their action which gives the exponential A r e ε e Ω − K r Ω ðt; Þ¼Et z þ E P sinð t · Þ= . Without loss of dependence of the pair production rate. The subleading K e e ≥ 0 generality, we may set ¼ Kk z þ K⊥ x where K⊥ . prefactor is given by quadratic fluctuations around such e K e 0 The polarization vector P obeys · P ¼ , for example solutions. For simple fields, iFμν is real (the Euclidean e⊥ e P ¼ y. potential is purely imaginary) and (16) can often be restricted Inserting this more general field profile (12), we obtain to a 2D-plane, sometimes even solved analytically [30,31]. formally the same results as in the previous section with Ω In slightly more complicated fields, solutions can be being replaced by K⊥. Since the enhancement of the pair- found using a shooting method, numerically integrating (16) creation probability is monotonic in Ω (i.e., now K⊥), we using initial conditions that are varied until the periodicity 0 Ω see that the perpendicular case with Kk ¼ and K⊥ ¼ is condition is met [32]. This is not feasible here, as the indeed optimal, see also [25]. instantons are genuinely three-dimensional in the parallel Note that, while the exponents do not depend on the polarization case and four-dimensional for other polariza- polarization vector eP, the prefactors do depend on eP, tions. Furthermore, they are not even purely real: we choose cf. [25], which can also generate a slight polarization the Euclidean four potential (for parallel polarization) dependence of the dominant order N via the constant εE factor in (11). This is consistent with the results of the next iA4 ¼ Ex3;iA3 ¼ i sin ðΩðx1 − ix4ÞÞ: ð17Þ section, which show that the world-line instantons, and Ω e their actions do also depend on the polarization vector P. Without the x1-dependence, this would give real instanton equations (as considered in [12,33]), but in this case real and IV. WORLD-LINE INSTANTONS imaginary parts mix, see Fig. 3. To robustly find instantons and evaluate both the We can express the probability for pair creation using the exponent and the prefactor in such fields, we employ a vacuum persistence amplitude 0 0 and thus the h outj ini method that will be discussed in detail elsewhere [34] and Γ 0 0 iΓ effective action with h outj ini¼e only provide the basic ideas here. Instead of a numerical 2 −2ℑΓ integration of (16) (arising in a saddle point approximation − 1 − 0 0 1 − ≈ 2ℑΓ Peþe ¼ jh outj inij ¼ e : ð13Þ The world-line instanton method is a semiclassical evaluation of the world-line path integral for the Euclidean effective action [29–31] (Euclidean because we replace 1.0 time by imaginary time x4 ¼ it, it is related to the Minkowskian quantity by Γ ¼ iΓE [31]), Z Z ∞ 2 dT −m ΓE ¼ e 2 T DxðτÞΦ½x 0.5 0 T Z T x_2 − τ E _ × exp d 2 þ iqA · x ; ð14Þ 0 x4 0.0 where the paths xðτÞ are (in general four-dimensional) closed trajectories parametrized by the proper time τ with a E period T, Aμ ðxðτÞÞ is the Euclidean four-potential evaluated –0.5 1.0 on the trajectory and 0.5 R 1 1 T E 0.0 dτσμνiqFμνðxðτÞÞ x Φ P 4 0 –1.0 3 ½x¼2 trΓ e ð15Þ –0.5 –0.5 σ 1 γ γ 0.0 is the spin factor with μν ¼ 2 ½ μ; ν,trΓ denoting the trace –1.0 over spinor indices and Pe… the path ordered exponential. Im(x1) 0.5 In this section we will only work with Euclidean quantities, E FIG. 3. Worldline instantons (in units of m=qE) for the case of so we omit the superscript for brevity. parallel polarization with ε ¼ 10−2 and γ ¼ mΩ=qE ranging A saddle point evaluation for both the T- and the path from 0 to 15. The purple circle in the x1 ¼ 0-plane is the circular integral consists of finding periodic solutions that satisfy the instanton in the limit γ → 0. For increasing values of γ the Euler-Lagrange equations corresponding to the exponent in instantons shrink (predominantly in the x3-direction) and rotate in (14), that is, the ℑðx1Þ − x4-plane.

096004-4 SAUTER-SCHWINGER PAIR CREATION DYNAMICALLY … PHYS. REV. D 97, 096004 (2018) of the continuous path integral), we discretize the paths in Im( ) (14) into N points from the beginning and perform the saddle point method on the resulting N × d-dimensional integral, where d is the number of space-time dimensions. 10–5 The equations of motion (16) are then replaced by a system of N × d nonlinear equations in N × d unknowns, which –10 can be solved efficiently using a Newton-Raphson scheme, 10 provided we choose a sufficiently close initial guess. In our γ → 0 Parallel case, the limit corresponds to a static, homogeneous 10–15 field so we can start with the known circular instanton, solve Perp. Spinor γ for the instanton at a small, finite and use that as initial guess Perp. Scalar for the next value. This process is called natural parameter continuation [35] and is essentially what was used in [36].We 10 20 30 40 can improve on this using a more sophisticated predictor- corrector algorithm, also detailed in [34]. FIG. 4. Pair production probabilities for different values of the γ 30 Note that the question of whether instanton solutions combined Keldysh parameter , strong field strength E ¼ ES= ≈ 5 1026 2 exist for a given (and possibly complicated) field configu- (corresponding to a laser intensity of I × W=cm ) and weak field ε ¼ 10−2. We have assumed a spacetime volume of ration can be nontrivial. Here, we address this issue by 1 μ 4 starting with a known instanton solution and then changing m . As shown before, the case of parallel polarization yields stronger enhancement. Further, the spin factor does not contribute the parameters gradually. Then, via analyticity arguments, in the parallel case, while it enhances pair production in the one would expect continuously modified instanton solu- Ω 500 Ω 1 — perpendicular case. The values ¼ keV or ¼ MeV tions to exist unless one hits a critical point in parameter considered in [25], for example, would correspond to γ ¼ 30 or space where the instanton develops a singularity. (For γ ¼ 60, respectively, and hence result in a drastic enhancement. example, if the electrostatic potential difference drops Unfortunately, however, these values are probably outside the below the mass gap, the instanton grows infinitely large range of near future XFELs. Lower values such as 25 keV [39] γ 3 2 30 and pair creation stops.) In the present case, one does not correspond to ¼ = (when E ¼ ES= ) and are thus not encounter such difficulties, and we have found instanton sufficient for an exponential enhancement. On the other hand, for 100 solutions for all considered parameters. lower field strengths such as E ¼ ES= , the same frequency of Having found instantons for different values of the 25 keV would correspond to γ ¼ 5 where we start to see Keldysh parameter γ we can evaluate the instanton action exponential enhancement. However, the total probabilities for 100 to obtain the leading exponential contribution to the effective E ¼ ES= are much lower and thus very hard to detect. action. We can also evaluate the prefactor, which is just given by the inverse square root of the Hessian matrix H.Wedo and without including the spin factor. While there is no need to regularize zero modes arising in the integral, due to spin dependence in the parallel case, the spin factor reparametrization and translational invariance. We deal with further enhances pair production for perpendicular polari- them using the Faddeev-Popov method [37,38], exponenti- zation. The nonmonotonic behavior for small γ in the ating the Dirac delta function, which modifies the Hessian to perpendicular case is probably a sign that the saddle point remove zero eigenvalues (details, again, in [34]). The final approximation breaks down, as the instanton is not con- semiclassical result is then fined strongly enough in the x4-direction. This has already

rffiffiffiffiffiffi E been verified in the purely time dependent case in [33], N0 Nd cl − SA½xcl Γ E 2 2π N 2 Φ½x e E ≈ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð18Þ where a comparison with the (numerical) solution of the N0 cl cl cl VN0 m ES a a det H½x Riccati equation could be made. For the volume factor we have assumed that the strong field ranges over a four- where N0 is the number of invariant directions, V the 4 N0 volume of 1 μm , which is completely covered by the plane cl corresponding volume factor, x the discrete instanton wave. It does not matter how much further the plane wave cl (the collection of points) and a its velocity. The trajectory actually extends, as it cannot produce any pairs without the and a are made dimensionless by rescaling with m=qE. strong field. We thus hold e.g. x1ð0Þ, x2ð0Þ and x3ð0Þ or The expression (18) has the advantage that it is appli- their center of mass fixed, giving the three-volume V3 in cable for every electromagnetic background field, yielding (18) and sum over the instantons located at each maximum the correct prefactor including spin effects without having Ω 2π of the wave giving a factor of Ninst ¼ T = . to compare limiting cases to determine normalization factors. Also the instantons are independent of the field V. CONCLUSIONS strength, so as soon as they are computed, we can evaluate (18) for many different values of E=ES. As an example for the dynamically assisted Sauter- Figure 4 shows evaluations of (18) for both parallel and Schwinger effect, we studied electron-positron pair creation perpendicular polarization, in the perpendicular case with due to a strong and constant electric field E superimposed

096004-5 TORGRIMSSON, SCHNEIDER, and SCHÜTZHOLD PHYS. REV. D 97, 096004 (2018) Z I Z ∞ Dψ by a weaker transversal plane wave with a frequency Ω and dT − T Γ ¼ 2 e i2 Dx field strength εE. In analogy to other examples, we found 0 T 4 Z an exponential enhancement of the pair-creation probability 1 x_2 i i γ Ω − τ _ − ψψ_ − ψ ψ if the combined Keldysh parameter ¼ m =ðqEÞ exceeds × exp i d 2 þ Ax 2 2 TF ; ðA1Þ γ γ ∼ 0 T a threshold value crit which scales in the same way crit ε j ln j as for a purely time-dependent sinusoidal field. ∂ − ∂ However, the exponential enhancement above the threshold where Fμν ¼ μAν νAμ. In this appendix we consider γ γ the superposition of a strong, constant field, E, and a plane > crit is reduced in comparison to a purely time- wave with an arbitrary field shape, given by the potential dependent sinusoidal field due to the effective mass m⊥ ≥m A0 ¼ 0 and Aðt; rÞ¼Etez þ εEePηðnxÞ, where the wave stemming from momentum conservation. 2 In order to treat this genuinely space-time dependent and polarization vectors satisfy nμ ¼ð1; nÞ, n ¼ 1, e2 1 n e 0 n r field, we employed an analytical approach based on a P ¼ , · P ¼ , nx ¼ t þ · , and where fðnxÞ¼ perturbative expansion (3) into powers εN of the weaker η0ðnxÞ is, at this point, an arbitrary function; ηðnxÞ¼ plane wave (while taking into account the strong field E sinðΩnxÞ=Ω gives the field consideredP in the main text. ε ≪ 1 Γ ∞ εNΓ nonperturbatively). The exponential dependence (4) of the With , we expand ¼ N 0 N and express the ¼ ˜ Nth order allows us to infer via (7) a dominant order N, weak field in terms of its Fourier transform fðωÞ. The which yields the strongest contribution and can be quite center of mass part of the worldline path integral gives an ≫ 1 γ large N , see also Appendix B. For large , we obtain “energy conserving” delta function the asymptotic expressions (8) and (9). Finally, we compare Z these analytical results to a fully nonperturbative numerical XN XN 4 − 2πδ ω method based on the world-line instanton technique and d xcm exp i kixcm ¼ V3 i ; ðA2Þ 1 1 find good agreement, e.g., regarding the dominant order N i¼ i¼ (see Appendix B) and the pair-creation exponent in Figs. 1 ω ω and 2, especially after inserting an improved approximation where ki;μ ¼ inμ and i are the Fourier frequencies (10) based on an ansatz for the subleading scaling of the corresponding to the N factors of f˜. The rest of the path prefactor (including one fitting parameter, fixed by the integral is Gaussian and can be performed as in [26,27,40] numerical results). for N-photon amplitudes in constant fields. This involves A Note that the world-line instanton action inst, which the worldline Green’s functions GB and GF for the x and ψ yields the exponent in the pair-creation probability path integrals, respectively. For the exponential part of the − ∼ −A Peþe expf instg, depends on the polarization and probability we only need GB, which is given by propagation direction. We find that perpendicular incidence 1 with parallel polarization (2) yields maximum enhance- B i 2 ⊥ GμνðτÞ¼− s 2½jτj− τ − gμν ment, see Fig. 4 and [25]. Furthermore, the prefactor in 2E 3 front of the exponential contains the volume scaling. Since i cos½sð1 − 2jτjÞ 1 k the plane wave is supposed to be filling the whole volume, − − gμν 4 2E sins s this prefactor scales with L instead of L as for a single photon, cf. [25]. ϵðτÞ sin½sð1 − 2jτjÞ ˆ þ − ð1 − 2jτjÞ Fμν; ðA3Þ 2E sins ACKNOWLEDGMENTS where s ¼ iET=2 and where the vector structure is deter- G. Torgrimsson is supported by the Alexander von k mined by the direction of the strong field, i.e. gμν ¼ Humboldt foundation. 0 0 3 3 ⊥ 1 1 2 2 ˆ 0 3 3 0 δμδν − δμδν, gμν ¼ −δμδν − δμδν, and Fμν ¼ δμδν − δμδν. This Green’s function is the Minkowski version of the ’ APPENDIX A: WORLD-LINE FORMALISM corresponding Euclidean Green s function, which can be found in [26,40]. In terms of this Green’s function, we find Here in the Appendix, we use for convenience units with that the dominant contribution to the Nth order is given by m ¼ 1 and absorb the charge into the definition of the field → Z Z Z strength, i.e. qE E. YN XN ∞ 1 YN N ˜ Our starting point is again the worldline representation of ε PN ¼ Im dωifðωiÞδ ωi ds dτi 0 0 the effective action. The spin factor can either be expressed i¼1 i¼1 i¼1 in terms of a path-ordered exponential as in Sec. IV or as a s i XN path integral over an anticommuting Grassmann variable, … exp − − k ½G ðτ − τ Þ − G ð0Þk ; E 2 i B i j B j ψ μðτÞ, with antisymmetric boundary conditions, ψð1Þ¼ i;j¼1 −ψð0Þ, ðA4Þ

096004-6 SAUTER-SCHWINGER PAIR CREATION DYNAMICALLY … PHYS. REV. D 97, 096004 (2018) where the ellipses stand for subleading prefactor terms. The APPENDIX B: DOMINANT ORDER last term in (A3) does not contribute, because all ki are ˆ In this appendix we will present two methods for obtaining parallel and kiFkj ¼ 0. The two terms proportional estimates of the dominant order, N, in the instanton ⊥ k 2 ω ω to g and g both lead to terms proportional to n⊥ i j ¼ formalism. These methods allow us to confirm the dominant ⊥ −kig kj, so, n⊥ωi gives an effective Fourier frequency. order found using the approach in Appendix A. Since jn⊥ωij ≤ jωij, the exponential is therefore maxi- Our starting point is the worldline representation of the mized by plane waves traveling perpendicular to the strong effective action (A1). In the first method we focus on the field, i.e. for nz ¼ 0. It follows from the delta function (A2) scalar part of (A1), i.e. the part without Grassmann that we necessarily have both positive and negative variables or Dirac matrices; we will show below using frequencies. We label the frequencies such that ωi > 0 the second method that the spin factor does not signifi- 1 … ω 0 1 … ∝ ε for i ¼ ; ;J and i < for i ¼ J þ ; ;N. Consider cantly affect N. Let aμ be the weak field. For the fields each term in the sum in the exponent of (A4) separately. we focus on in this paper, aμ is a plane wave, but the ω ω The term proportional to i j is maximized by methods we present here work also for more general field jτi − τjj¼0, 1 for ωiωj > 0 and by jτi − τjj¼1=2 for shapes. We expand the exponent in the weak field, ωiωj < 0. Similar to the saddle point method, we obtain the dominant exponential contribution by substituting these Z X∞ Z “ ” τ 1 1 1 N maximizing values of i into (A4). This gives the exp −i ax_ ¼ −i ax_ ; ðB1Þ 0 N! 0 following exponential for the s-integral: N¼0 2 2m⊥ s s exp − − Σ2 tan ; ðA5Þ E 2 2 where we have defined

XJ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi p⊥ 1 2 Σ ≔ p⊥ ≔ ki⊥ m⊥ ≔ 1 þ p⊥: ðA6Þ m⊥ 2 i¼1

With the saddle point given by s ¼ 2 arccos Σ, we find the general result Z YN XN N ˜ ε PN ∼ dωifðωiÞδ ωi 1 1 i¼ i¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2m⊥ × exp − ðarccos Σ − Σ 1 − Σ2Þ E Z YN XN ˜ ¼ dωifðωiÞδ ωi i¼1 i¼1 2 2 1 × exp − −p⊥ þ m⊥ arctan : ðA7Þ E p⊥

With only one photon, N ¼ 2,wehavep⊥ ¼ k⊥=2, and the exponential in (A7) reduces to that in Eq. (5) in [25],as expected. Note that the exponential in (A7) has the same functional dependence of Σ as for the longitudinal, purely time-dependent fields we considered in [14], see Eq. (3.4) in [14]. Comparing Σ in (A6) with the corresponding quantity in Eq. (3.3) in [14], we see that the main difference FIG. 5. These plots show our three estimates of the dominant order for perpendicular polarization. The dashed lines are in going from the purely time-dependent fields in [14] to obtained from Eq. (11), the solid lines show (B2), and the circles the plane waves considered here is the appearance of a show (B4), in which Peþe− is the total probability for spinor QED, heavy effective mass, i.e. m → m⊥ >m(c.f. [41]), due to including the prefactor. The third approach gives negative N the spatial components of the wave vector. This means that below the threshold, but that is just another sign (c.f. Fig. 4) that plane waves will in general lead to less exponential the instanton prediction of the prefactor breaks down in that enhancement than a purely time-dependent weak field. regime [42].

096004-7 TORGRIMSSON, SCHNEIDER, and SCHÜTZHOLD PHYS. REV. D 97, 096004 (2018) and obtain an estimate of the dominant order from the A more direct way of estimating the dominant order is to “saddle point” of the sum over N. Assuming that the calculate the logarithmic derivative of the probability with dominant order is “large”, we use Stirling’s approximation respect to ε, i.e., ln N! ≈ Nðln N − 1Þ and find Z − 1 d log Peþe − _ N ¼ ε : ðB4Þ N ¼ i ax: ðB2Þ d log 0 At this order we recover the original exponent, i.e. This expression is motivated by the fact that in a regime − ∼ εN0 Z Z where Peþe , (B4) gives N ¼ N0. In Fig. 5 we 1 1 N 1 − _ ≈ − _ evaluate (B4) within the instanton formalism and show that ! i ax exp i ax : ðB3Þ N 0 0 (B4) agrees quite well with the simpler estimate of (B2). One advantage of (B4) is that it is general and does not The instantons and the resulting exponential part of the − depend on how we calculate Peþe . For example, for purely probability will therefore be exactly the same as before, i.e. time dependent fields, like the ones we studied in [14], one as without the additional steps (B1) to (B3). The point is can obtain the exact probability by solving the Riccati that substituting the instantons into (B2) gives us a simple equation numerically, and then (B4) gives an exact estimate of the dominant order in the instanton formalism. description of how the probability depends on ε. Note that, while we assume that aμ is weak, the integral of All the plots in Fig. 5 show qualitatively the same γ ≈ 0 aμ in (B2) gives N, which is supposed to be large. So, in behavior as a function of : below the threshold N (for (B1) we expand the exponent in a parameter which is ε sufficiently small compared to E) [42], where the actually large. That is of course not a problem as it only probability is given by Schwinger’s constant field result. means that we have to sum up many terms (the Taylor series After the threshold, N quickly reaches a maximum and for the exponential has an infinite radius of convergence). then slowly decreases as γ → ∞. So, pair creation actually In fact, we want this expansion parameter to be large becomes less “multiphoton” as γ increases beyond the because in regimes where it is small, the dominant con- maximum. As Fig. 5 shows, the maximum dominant order tribution comes from N ¼ 0 and then there is no significant can be quite large. However, it is not large for all relevant enhancement of the probability. As shown in Fig. 5, the parameters. For e.g. E ¼ 1=30 and ε ¼ 1=100, the maxi- ∼ 4 instanton estimate (B2) agrees with the previous estimate mum dominant order is only N , which suggests that it based on (11). So, (B2) seems to give a good estimate of the might be feasible in this case to actually calculate also the dominant order in dynamical assistance. preexponential contributions to all important orders.

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