Total yield of electron-positron pairs produced from vacuum in strong electromagneticfields: validity of the locally constant field approximation

D. G. Sevostyanov,1 I. A. Aleksandrov,1, 2, ∗ G. Plunien,3 and V. M. Shabaev1

1Department of Physics, St. Petersburg State University,7/9 Universitetskaya Naberezhnaya, Saint Petersburg 199034, Russia

2Ioffe Institute, Politekhnicheskaya 26, Saint Petersburg 194021, Russia3Institut fur Theoretische Physik, Technische Universitat Dresden,

Mommsenstrasse 13, Dresden D-01062, Germany

The widely-used locally constant field approximation (LCFA) can be utilized in order to derivea simple closed-form expression for the total number of particles produced in the presence of astrong electromagnetic field of a general spatio-temporal configuration. A usual justification forthis approximate approach is the requirement that the external field vary slowly in space and time.In this investigation, we examine the validity of the LCFA by comparing its predictions to theresults obtained by means of exact nonperturbative numerical techniques. To benchmark the LCFAin the regime of small field amplitudes and low frequencies, we employ a semiclassical approach.As a reference, we consider a standing electromagnetic wave oscillating both in time and spaceas well as two spatially uniform field configurations: Sauter pulse and oscillating electric field.Performing a thorough numerical analysis, we identify the domain of the field parameters where theapproximation is well justified. In particular, it is demonstrated that the Keldysh parameter is nota relevant quantity governing the accuracy of the LCFA.

I. INTRODUCTION

One of the most remarkable phenomena predictedby quantum electrodynamics (QED) is the process ofelectron-positron pair production in strong externalfields [1–4]. Even in the absence of real particles, vac-uum fluctuations of the quantum electron-positron fieldmay be galvanized due to the interaction with a strongexternal background, so the vacuum state decays pro-ducing e+e− pairs. In a constant uniform electric fieldE0, the probability of this striking phenomenon is pro-portional to exp(−πEc/E0), where Ec = m2c3/(|e|~) ≈1.3 × 1016 V/cm is a critical (Schwinger) value of theelectric field strength. This fundamental process can-not be described by means of perturbation theory. Asthe amplitudes of the most intense laser pulses that canbe generated nowadays are far below Ec, one may ex-pect that the Schwinger pair production will not be ex-perimentally observed within the near future. On theother hand, in nonstationary backgrounds the temporaloscillations of the field can effectively enhance the prob-ability of the process, even if its nonperturbative natureis still preserved. Furthermore, taking into account thepre-exponential factor could also boost the particle yieldwithin the theoretical estimations. To obtain reliablequantitative predictions, one basically strives to considerrealistic field configurations. However, due to the lim-ited computational resources, it is unfeasible to performnonperturbative calculations for general inhomogeneousbackgrounds in (3+1) dimensions. To reduce the compu-tational time when studying complex scenarios, one can

∗ i.aleksandrov@spbu.ru

employ approximate methods instead of carrying out rig-orous calculations.

A widely-used approach is based on the idea that thespatio-temporal scale of quantum processes is commonlymuch smaller than the characteristic parameters of theexternal background. It allows one to treat the field aslocally constant and utilize simple results derived in thecase of space-time-independent backgrounds. The ac-tual implementation of this general approach should bediscussed once the physical process is specified. For in-stance, in the context of nonlinear Compton scattering(NCS) [5–7], i.e., emission of a photon by an electronor positron in a strong field, and the so-called nonlin-ear Breit-Wheeler (NBW) mechanism [7–9], i.e., decayof a photon into an electron-positron pair, the externalfield can be considered locally constant if the “forma-tion length” of the quantum processes is much smallerthan the length scale of the electromagnetic background,e.g., laser wavelength. Moreover, one can employ theclosed-form expressions for the corresponding rates de-rived in the case of constant crossed fields [5, 10]. Thisapproach provides a powerful tool for studying strong-field QED phenomena in arbitrary space-time-dependentbackgrounds. Note that the applicability of the locallyconstant field approximation (LCFA) depends not onlyon the external field parameters but also on the initialstate of the electron (for NCS) or photon (for NBW). Thevalidity of the LCFA in the context of NCS and NBWis discussed in recent papers [11–15] (see also referencestherein).

In the present study, we will examine the LCFA beingapplied to the process of vacuum pair production. If oneaims to approximately calculate the momentum distri-butions of particles created, a consistent formulation ofthe LCFA exists in the case of spatially uniform external

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fields [16], whereas for taking into account both temporaland spatial inhomogeneities, there is no rigorous LCFAprocedures, so one can only propose ad hoc approxi-mate methods such as, e.g., the local dipole approxima-tion [16, 17] treating locally only the spatial dependenceof the field. In contrast, the total number of pairs canbe evaluated within the LCFA by using the exact expres-sion for the pair-creation rate in a constant field [18, 19],where space and time are considered in the same way.Since this expression is valid for constant electric andmagnetic fields of arbitrary direction and magnitude, onecan simply integrate it over space and time taking intoaccount the inhomogeneities of the external backgroundunder consideration (see, e.g., Refs. [20–24]). As a con-stant electromagnetic field produces pairs in the tunnel-ing regime, one can expect the LCFA to be well justifiedonly in the domain of slowly varying fields with smallKeldysh parameters γ = (Ec/E0)[~ω/(mc2)], where ω isthe frequency of the external field. The condition γ � 1is equivalent to the requirement that the formation length2mc2/(|e|E0) be much smaller than the laser wavelengthλ = 2πc/ω. The main goal of the present study is toinvestigate the accuracy of the LCFA once γ 6= 0.

To this end, one has to compare the LCFA predic-tions with the exact results obtained by means of rigor-ous QED techniques. Note that we are interested in theprocess of vacuum pair production in a classical back-ground treated nonperturbatively within the zeroth orderin the fine-structure constant, i.e., we neglect the radia-tive corrections, so “exact methods” are those dealingwith the spatio-temporal inhomogeneities of the exter-nal field without any additional approximations. First,we will employ the so-called quantum kinetic equations(QKE) [25–27] to analyze spatially uniform scenarios. Totake into account the coordinate dependence in the caseof a standing electromagnetic wave, we will use our nu-merical technique [17, 28, 29] based on the Furry-picturequantization of the electron-positron field [30]. Integrat-ing the momentum distributions of electrons created, wecompute the total number of pairs in order to bench-mark the LCFA and deduce general patterns concern-ing its justification. Unfortunately, numerical methodscan be effectively applied only within the domain of suf-ficiently large field amplitudes (E0 & 0.1Ec) and highfrequencies (~ω & 0.1mc2). To assess the accuracy ofthe LCFA also in the case of smaller E0 and ω, we willperform a semiclassical analysis based on the imaginary-time method [31–35]. This allows us to gain a completepicture of where the LCFA is valid.

The paper is organized as follows. In Sec. II we discussthe implementation of the LCFA for computing the totalnumber of pairs and provide the explicit formulas for twospecific scenarios: oscillating electric field and standingelectromagnetic wave. In Sec. III we outline the exacttechniques utilized in the study. Section IV contains themain findings in the case of spatially homogeneous fieldconfigurations. In Sec. V we perform a semiclassical anal-ysis of an oscillating electric field. In Sec. VI we examine

a standing electromagnetic wave varying both in timeand space. Finally, we conclude in Sec. VII.

Throughout the article, we assume ~ = c = 1 and theelectron charge e < 0.

II. LOCALLY CONSTANT FIELDAPPROXIMATION

The idea of the LCFA is to employ the exact closed-form expression for the total number of pairs created ina constant uniform electromagnetic field. Let us firstrecall the main results concerning pair production in thissimple external background.

Due to the interaction with the external classical elec-tromagnetic field, the initial vacuum state of the quan-tized electron-positron field may not remain a vacuumstate after the interaction if the amplitude of the vacuum-vacuum transition has an absolute value less than unity.Let us denote this amplitude by eiW . If the effective ac-tion W has a nonzero imaginary part, then the vacuumdecay probability is also nonzero and reads

Pdecay = 1− |eiW |2 = 1− e−2ImW . (1)

In the case of a constant external field, the effective ac-tion is a product of the spatio-temporal volume V T andthe effective Lagrangian L. The latter is the one-loop ef-fective Lagrangian L(1) [2–4] since we disregard the quan-tized part of the electromagnetic field. The correspond-ing imaginary part [4] is given by

2 ImL(1)[E ,H] =e2EH4π2

∞∑

n=1

1

ncoth

nπHE exp

(− nπm2

|e|E

),

(2)where E and H are defined via

E =

√√F2 + G2 + F , (3)

H =

√√F2 + G2 −F . (4)

Here F = (E2−H2)/2, G = E·H, and the vectorsE andH are the electric and magnetic field in a given (labora-tory) frame. The invariant quantities E and H coincidewith the magnitudes of the fields E and H, respectively,if these vectors are parallel to each other. According toEq. (2), the imaginary part is negligible unless the electricfield strength E is close to the critical value Ec = m2/|e|.

The vacuum instability appears due to the process ofelectron-positron pair production. In the present study,we are interested in calculating the total number of pairsN , which in the case of a constant background can befound exactly [18] (see also Ref. [19]). If the externalfield does not vary in space and time, it is the number ofpairs per unit volume and time which yields a finite value,N/(V T ). One can also view it as a differential quantitydN/(dtdx), and its explicit form reads [18, 19]

dN

dtdx[E ,H] =

e2EH4π2

cothπHE exp

(− πm2

|e|E

). (5)

3

Note that this expression is the first term of the series (2)and its physical meaning differs from that of 2 ImL (see,e.g., Refs. [30, 36, 37]). Nevertheless, if the external fieldproduces a small number of pairs according to Eq. (5),then Eqs. (1), (2), and (5) yield indistinguishable values.As we are interested in computing the total number ofpairs, in what follows we will employ Eq. (5).

If the external field slowly varies in space and time,one can approximately treat it as locally constant, i.e., as-sume that the particle number arising from a small space-time region of size ∆V∆T around the point x = (t,x)can be evaluated by means of Eq. (5) where E and H arereplaced with E(x) and H(x), respectively. Multiplyingthe density by ∆V∆T and summing such individual con-tributions over space and time, one immediately receivesa Riemann sum which for ∆V , ∆T → 0 yields

N (LCFA) =

∫d4x

dN

dtdx[E(x),H(x)]. (6)

The LCFA treats time and the spatial coordinates in auniform manner. If the external field exists in a finitespace-time domain, the integral in Eq. (6) provides a fi-nite dimensionless value, which is a total number of pairsproduced.

Our aim is to clarify what the expression “slowly vary-ing field” actually means, i.e., to find out when the LCFAformula (6) is justified. In what follows, we will discusshow this approximate method is used in the case of twospecific field configurations.

A. Standing electromagnetic wave

First, we will discuss how the LCFA approach can beimplemented in the case of a standing wave. The externalfield is described in the gauge A0 = 0 by the followingvector potential:

A(t, z) =E0

ωF (t) sinωt cos kzz ex, (7)

where kz = ω, ex is a unit vector along the x direction,and F (t) is a smooth envelope function vanishing for t→±∞. The electric and magnetic fields then read

E(t, z) = −E0

[F (t) cosωt+

F ′(t)ω

sinωt

]cos kzz ex, (8)

H(t, z) = −E0F (t) sinωt sin kzz ey. (9)

Plugging them into the definition of F and G, we obtainG = 0 and

F(t, z) =E2

0

2F 2(t) cos(ωt+ kzz) cos(ωt− kzz) +

E20

2

F ′(t)ω

[F (t) sin 2ωt+

F ′(t)ω

sin2 ωt

]cos2 kzz. (10)

We choose the envelope in the following form:

F (t) =

sin2 πNc − |ωt|2

if π(Nc − 1) 6 |ωt| < πNc,

1 if |ωt| < π(Nc − 1),

0 otherwise,

(11)so it contains a flat plateau of Nc − 1 carrier cycles andtwo switching on/off parts of half a cycle each. In whatfollows, we will use Nc = 5.

Let us now present the explicit form of Eq. (6) in thecase of the standing wave (7)–(9). First, since G = 0, oneof the two invariants E and H is always zero:

E =√

2F , H = 0 for F > 0, (12)

E = 0, H =√−2F for F < 0. (13)

The space-time regions where F is negative do not con-tribute to N (LCFA). For F > 0, we take the limit H → 0in Eq. (5). Second, integration over the xy plane leads toa factor S representing the cross section of the system,which is assumed to tend to infinity. Third, as the ex-ternal field is also infinite in the z direction and periodicwith a period of 2π/kz = 2π/ω, integration over z shouldbe performed according to the prescription

∫dz =

Lz/2∫

−Lz/2

dz =

2π/ω∫

0

dzωLz2π

, (14)

where Lz is the size of the system in the z direction, soV = SLz. From this, it follows that the total number ofpairs can be evaluated as

N (LCFA) =ωV

8π4

∫dt

2π/ω∫

0

dz θ(E2 −H2) e2(E2 −H2) exp

(− πm2

|e|√E2 −H2

), (15)

where θ(x) is the Heaviside step function and the fields E and H are given by Eqs. (8) and (9), respectively. In the

4

final estimations, the full volume of the system V can bereplaced with the interaction volume depending on howthe external laser pulses are focused. In order to evaluatethe integrals in Eq. (15), it is convenient to use the sub-stitutions φ = ωt and ψ = ωz. It immediately becomesclear that the product ωN (LCFA) is completely indepen-dent of ω, provided the envelope function depends onlyon ωt (e.g., the pulse duration is governed by the numberof cycles which is independent of ω). Thus we will factorout the ω dependence as well as the volume V presentingthe results in terms of the following quantity:

ν(LCFA) =ω

VN (LCFA). (16)

To benchmark the LCFA results, we will also performexact calculations of the analogous function ν = ωN/Vdepending on ω beyond the LCFA.

B. Spatially homogeneous oscillating electric field

We will also consider a purely time-dependent field ofthe following form:

A(t) =E0

ωF (t) sinωt ex. (17)

The field invariant F can be obtained by setting z = 0in Eq. (10) and it is given by

F(t) =E2

0

2F 2(t) cos2 ωt

+E2

0

2

F ′(t)ω

[F (t) sin 2ωt+

F ′(t)ω

sin2 ωt

].(18)

In Eq. (6) we replace E with E(t) = −A′x(t) and againtake the limit H → 0 in Eq. (5). The number of pairsper unit volume is then determined via

N (LCFA)

V=

e2

4π3

∫dtE2(t) exp

(− πm2

|eE(t)|

). (19)

This expression being multiplied by ω is also independentof ω, so we present the results in terms of the functionν(LCFA) defined by Eq. (16).

III. EXACT APPROACHES

To explore the validity of the LCFA, we benchmarkthis approximation against the exact methods. In thissection, we briefly describe two different approaches. Thefirst one is based on the so-called kinetic description ofvacuum pair production in the case of spatially homo-geneous fields. The second method rests on the Furry-picture quantization of the electron-positron field in thepresence of a space-time-dependent external background.If the field depends solely on time, the two methods yieldidentical results, whereas the latter approach allows usto analyze the pair-creation process in a standing elec-tromagnetic wave taking into account its spatial inhomo-geneity.

A. Quantum kinetic equations

In the case of a spatially homogeneous external field,a nonperturbative kinetic approach was developed inRefs. [25–27] and subsequently generalized in Refs. [38,39] allowing one to consider uniform fields of arbitrarypolarization (see also recent review [40] and referencestherein). Here we will recap the basics of the approachin the case of a linearly polarized electric field.

We assume that the external field is directed along thex axis:

A(t) = A(t)ex, E(t) = E(t)ex. (20)

The field vanishes outside the interval t ∈ [tin, tout]. Forinstance, Eq. (11) implies tout = −tin = πNc/ω.

The process of pair production is described in terms ofthe electron number density in momentum space,

f(p) =(2π)3

V

dNp,s

dp. (21)

It does not carry the spin quantum number s because inthe linearly-polarized field the particle distribution doesnot depend on s. To obtain the total number of pairs,one should integrate f(p) over p and multiply the resultby 2 due to the spin degeneracy. Within the quantumkinetic approach, one introduces a time-dependent dis-tribution function f(p, t) which obeys f(p, tout) = f(p).At the initial time instant t = tin, we employ the vacuumcondition f(p, tin) = 0. The evolution of the distributionfunction is governed by the following integro-differentialequation:

f(p, t) = λ(p, t)

t∫

tin

dt′ λ(p, t′)

[1

2− f(p, t′)

]cos 2θ(t, t′),

(22)where

λ(p, t) =eE(t)µ

ω2(p, t), (23)

µ =√m2 + p2⊥, (24)

ω(p, t) =√µ2 + [p‖ − eA(t)]2, (25)

θ(t, t′) =

t∫

t′

ω(p, t′′) dt′′. (26)

Due to the azimuthal symmetry, the distribution functiondepends only on the longitudinal momentum projection

p‖ = px and the transverse one p⊥ =√p2y + p2z. Alterna-

tively, one can recast Eq. (22) into the following Cauchyproblem:

f(p, t) =1

2λu(p, t), (27)

u(p, t) =[1− 2f(p, t)

]− 2ωv(p, t), (28)

v(p, t) = 2ωu(p, t). (29)

5

Here λ and ω depend on p and t, and we assumef(p, tin) = u(p, tin) = v(p, tin) = 0 for all p. The sys-tem (27)–(29) is usually referred to as the quantum ki-netic equations (QKE).

The QKE can be solved numerically for given p, i.e.,p‖ and p⊥. Varying p, one obtains the momentum spec-tra of the electrons created. In this study, we integratethe distribution function over p (dp → 2πp⊥dp⊥dp‖) tocompute the total number of pairs.

B. Furry picture in momentum space

To incorporate both the temporal and spatial inhomo-geneities of the external field, one should employ moregeneral methods than that discussed in the previous sub-section. For instance, the so-called Dirac-Heisenberg-Wigner formalism is a kinetic approach allowing one, inprinciple, to investigate arbitrary backgrounds in (3+1)dimensions [41, 42]. However, we will use a general tech-nique based on the Furry-picture quantization with theaid of in/out solutions [30] and discuss its implementa-tion within momentum space [28].

Let us consider the Dirac equation including the inter-action with a classical background Aµ = (A0,A):

(γµ[i∂µ − eAµ(t,x)

]−m

)ψ(t,x) = 0. (30)

We define two sets of solutions ζψn(t,x) and ζψn(t,x)which satisfy the following conditions:

ζψn(tin,x) = ζψ(0)n (x), ζψn(tout,x) = ζψ(0)

n (x). (31)

Here ζψ(0)n (x) and ζψ

(0)n (x) form two complete sets of

the Hamiltonian eigenfunctions at t = tin and t = tout,respectively, with the sign of the energy denoted by ζ =±. The functions ζψn(t,x) and ζψn(t,x) are called in-and out-solutions, respectively. The number density ofelectrons created with quantum numbers l then reads [30]

n−l =∑

n

G(+|−)lnG(−|+)nl, (32)

where the G matrices are defined as the following innerproducts:

G(ζ |κ)ln = (ζψl, κψn), (33)

G(ζ |κ)ln = (ζψl,κψn). (34)

Note that they do not depend on time because the DiracHamiltonian is Hermitian. According to Eq. (32), the ex-ternal field creates pairs only if it permits transitions be-tween the positive-energy and negative-energy continua.

The asymptotic behavior of the in- and out-solutionsis given by the simple plane-wave solutions of the Diracequation for a free particle. These states are determinedby momentum p and spin quantum number s = ±1, i.e.,l = {p, s}. Instead of evaluating the inner products (33)and (34) in coordinate space, one can evolve the Fourier

components of the electron wavefunction which directlyyield the elements of the corresponding G matrices. Thisapproach is described in detail in Ref. [28].

The method is particularly efficient in the case of spa-tially periodic backgrounds [17, 28, 29]. If the exter-nal field is described by the vector potential A(t,x) =Ax(t, z) ex, and Ax(t, z) is a periodic function of z withperiod 2π/ω, then the out-solution can be represented inthe following form:

+ψp,s(t,x) =eipx

(2π)3/2χp,s(t, z), (35)

where

χp,s(t, z) =

+∞∑

j=−∞eiωjz wjp,s(t). (36)

In terms of the four-component time-dependent functionswjp,s(t), the Dirac equation reads

iwjp,s = (α · p+ αzωj + βm)wjp,s − eαx∞∑

l=−∞al−j(t)w

lp,s,

(37)where al are the Fourier components of the external field,

al(t) =ω

2π

∫ 2π/ω

0

Ax(t, z)e−iωlzdz. (38)

Taking into account the “initial” conditions wjp,s(tout) =up,sδj0, where up,s is a constant bispinor correspond-ing to the positive-energy states, we evolve the functionswjp,s(t) backwards in time and obtain the density (21)via

f(p) =

∞∑

j=−∞

∑

s′=±1

∣∣v†px,py,pz+ωj, s′wjp,s(tin)

∣∣2. (39)

Here we have used Eqs. (32)–(34); vp,s is a constantbispinor regarding the negative-energy wavefunction.Equation (39) yields the same result no matter whatvalue of s is chosen. A detailed comparison betweenthis numerical approach and that based on the Dirac-Heisenberg-Wigner formalism can be found in Ref. [17].

IV. RESULTS: SPATIALLY HOMOGENEOUSEXTERNAL FIELDS

In this section, we will benchmark the LCFA againstthe exact results obtained with the aid of the QKEand Furry-picture formalism for two different scenariosinvolving spatially homogeneous backgrounds: Sauterpulse and oscillating electric field.

6

A. Sauter pulse

The external field is purely electric and reads

Ex(t) =E0

cosh2(t/τ), Ey = Ez = 0. (40)

A crucial feature of this field configuration is that thenumber density of electrons produced can be derived an-alytically [43]:

(2π)3

V

dNp,s

dp=

sinh[12πτ(2eE0τ + ω− − ω+)

]sinh

[12πτ(2eE0τ + ω+ − ω−)

]

sinh(πω+τ) sinh(πω−τ), (41)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

mτ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

E0/E

c

1.0

4.0

8.0

mτ(E0/Ec) = η

mτ(E0/Ec)3/2 = η

0%

20%

40%

60%

80%

100%

Figure 1. Ratio [N (LCFA)/N ] × 100% as a function of the amplitude E0 of a Sauter pulse (40) and its duration τ . The threecolor-coded pairs of curves correspond to η = 1.0, 4.0, and 8.0, where η is defined in the legend separately for solid and dashedlines.

where ω± =√µ2 + (P‖ ∓ eE0τ)2, µ2 = m2 + p2⊥, and

P‖ = p‖−eE0τ . Numerically integrating this expression,we evaluated the total particle yield and tested our nu-merical procedure developed for treating arbitrary time-dependent external fields.

The LCFA is accurate only if the Sauter pulse has asufficiently large duration τ and amplitude E0. For in-stance, the requirement that the Keldysh parameter bemuch smaller than unity, γ � 1, is equivalent to thefollowing condition:

|eE0|τ � m. (42)

We assume that the characteristic frequency of the Sauterpulse is ω = 1/τ . Nevertheless, as was demonstrated inRef. [16], the relevant condition of the LCFA applicabilityhas a different form:

|eE0|3/2τ � m2. (43)

This result was obtained with the assumption mτ � 1,which is completely realistic since in relativistic units

m−1 = 1.3 × 10−21 s. To demonstrate which of theinequalities (42) and (43) is relevant to the LCFA jus-tification, we directly compare the approximate resultsfor the number of pairs with the exact values.

As the LCFA incorporates only the tunneling mecha-nism completely neglecting multiphoton processes, it al-ways underestimates the particle yield, i.e., N (LCFA) <N , so it is convenient to present the results in terms of thequantity [N (LCFA)/N ]×100% (see Fig. 1). We also depictthe three pairs of curves determined by η = mτ(E0/Ec)(dashed lines) and η = mτ(E0/Ec)

3/2 (solid lines), whereη = 1.0, 4.0, and 8.0 for the black, green, and gray curves,respectively. The heatmap confirms that the relevantcondition is given by Eq. (43) as, unlike the dashed line,the solid line “1.0” is very close to a contour line of thegraph (as was indicated above, we are not interested insmall values of mτ). The naive condition (42) is weakerthan (43), so the LCFA may not be valid even if theKeldysh parameter is small.

7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

E0/Ec

10−18

10−14

10−10

10−6

10−2

ν/m

4 LCFA

bosons (×2)

ω = 1.0m

ω = 0.6m

ω = 0.2m

Figure 2. Parameter ν = ωN/V as a function of the amplitudeof the oscillating background (17) calculated within the LCFA(solid light-blue line) and computed exactly for various valuesof the field frequency ω (dashed lines). The dash-dotted lineshows the exact results obtained in the case of scalar particles(the data was multiplied by 2 in order to balance the Fermispin factor).

B. Oscillating electric field

Here we will discuss the LCFA in the case of a spa-tially homogeneous oscillating electric field (17). As waspointed out above, the results will be presented in termsof the function ν = ωN/V generally depending on E0

and ω. The LCFA yields, however, ω-independent re-sults. We compare them with the exact predictions inFig. 2. The LCFA accurately reproduces the results ofcomplete quantum simulations if the external field is suf-ficiently strong and its frequency is low. Evaluating theKeldysh parameter, we observe that the green dashedline (ω = 0.2m) is far from the LCFA solid line atE0 = 0.2Ec, i.e., γ = 1.0, whereas the results almostcoincide at E0 = 0.4Ec, i.e., γ = 0.5. Moreover, othervalues of ω lead to different threshold values of γ, so thecondition γ � 1 seems again irrelevant to the problem ofthe LCFA justification.

As was mentioned above, the LCFA is likely to un-derestimate the particle yield. Nevertheless, the greenline in Fig. 2 (ω = 0.2m) intersects the LCFA curve. Tomake it evident, we display the same graph with a lin-ear scale (see Fig. 3). We observe that the LCFA resultsbecome greater than the exact values for E0 & 0.85Ec.The physical explanation for this behavior lies in the factthat the LCFA does not take into account the Pauli exclu-sion principle at all, while the distribution function f(p)computed exactly cannot exceed unity. For sufficientlylarge field amplitudes, many of the positive-energy elec-tron states become occupied, which inhibits the pair-production process. To confirm this statement, we per-formed the analogous calculations in the case of bosons,i.e., within scalar QED (see the gray dash-dotted line inFigs. 2 and 3). First, note that the LCFA in the caseof bosons leads to precisely the same expressions (5) and(6), provided the boson yield is multiplied by the Fermi

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

E0/Ec

0.00

0.25

0.50

0.75

1.00

1.25

ν/m

4[ 1

0−

2]

LCFA

bosons (×2)

ω = 1.0m

ω = 0.6m

ω = 0.2m

Figure 3. Parameter ν = ωN/V versus the amplitude of theoscillating background (17) evaluated within the LCFA (solidlight-blue line) and computed exactly for various ω (dashedlines). The dash-dotted line shows the exact results obtainedin the case of scalar particles.

spin factor 2 (see, e.g., Ref. [30]), although the imaginarypart of the effective Lagrangian is different [3, 4, 30]. Sec-ond, the QKE method should be modified according tothe following prescription [25, 27]: one has to change thesign of f(p, t′) in the square brackets in Eq. (22) and de-fine λ = eE(t)[p‖−eA(t)]/ω2. According to our data, theLCFA results never exceed the total number of bosons asshown in Figs. 2 and 3.

Finally, we underline that the exact values of the totalnumber of pairs decreases much more slowly with de-creasing E0 compared to the LCFA predictions. More-over, one should also keep in mind that the interactionvolume in the real experimental setup may yield a hugeadditional factor since the Compton volume m−3 cor-responds to 5.76 × 10−38 m3. For instance, one cubicmicrometer, which seems a realistic interaction volume,is 19 orders of magnitude larger. This suggests that theactual threshold of pair production can be considerablylower than Ec (see, e.g., Refs. [21, 22]).

In the regime of weaker fields and lower frequencies,where E0 � Ec and ω � m, we are not able to carry outthe exact computations due to the technical limitations.On the other hand, one can invoke semiclassical methodsto benchmark the LCFA, which will be discussed in thenext section.

V. SEMICLASSICAL ANALYSIS

Semiclassical methods are expected to be accurate onceE0 � Ec and ω � m. In the case of a spatially homo-geneous oscillating background E(t) = E0 sinωt ex, theproblem was analyzed, e.g., in Refs. [32, 35] by means ofthe so-called imaginary time method [31, 33, 34]. In whatfollows, we will first consider the limit ω → 0 for givenE0 and compare the LCFA results with the semiclassicalpredictions. Second, we will turn to the case of nonzeroω to determine to which extent one can rely on the LCFA

8

if pair production is no longer a pure tunneling process.

A. Zero-frequency limit

Here in Sec. V, we do not introduce the envelope func-tion as it is inessential for the present analysis. In thiscase, evaluating the LCFA expression (19), we obtain thefollowing number of pairs per unit spatio-temporal vol-ume:

N(LCFA)∞V T

=m4

2π4

(E0

Ec

)2π/2∫

0

dφ cos2 φ exp

(− πEc

E0 cosφ

),

(44)where the subscript “∞” indicates that the external pulseis infinite in time, i.e., there is no envelope. Note thatthe expression (44) does not depend on ω at all, so it canbe considered as the limit of the exact result as ω → 0for given E0 as long as the Pauli principle is insignificant,i.e., the field strength E0 is not too large.

It is convenient to use the Keldysh parameter,

γ =mω

|eE0|=Ec

E0

ω

m. (45)

The zero-γ limit of the exact results is given by Eq. (44).Let us compare it with the semiclassical estimates. Inthe limit γ → 0, the imaginary-time method (ITM)yields [32, 35] (see also Ref. [44])

N(SC)∞V T

=m4

23/2π4

(E0

Ec

)5/2

exp

(−πEc

E0

). (46)

Since we are interested in the total number of pairs, wecan exploit the ITM expression, although it does not takeinto account the interference effects that are pronouncedin the momentum spectra of particles produced in laserpulses with a subcycle structure [45, 46]. The result (46)should coincide with Eq. (44) once E0 � Ec. To explic-itly verify this, we first express the integral in Eq. (44)as J (πEc/E0), where

J (z) =z

2

∞∫

1

(1 +

1

x2

)K0(zx)dx. (47)

This representation is derived in the Appendix. Assum-ing then z � 1, i.e., E0 � Ec, one can deduce the asymp-totic expansion of the LCFA result. It is convenient touse the dimensionless function µ∞(E0) defined by

N∞V T

= m4

(E0

Ec

)2

exp

(−πEc

E0

)µ∞(E0). (48)

Then we receive

µ(SC)∞ (E0) =

1

23/2π4

√E0

Ec, (49)

µ(LCFA)∞ (E0) =

1

2π4exp

(πEc

E0

)J(πEc

E0

). (50)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

E0/Ec

0.0

0.1

0.2

0.3

0.4

µ∞

(E0)[ 1

0−

2]

LCFA

SC

Figure 4. Function µ∞(E0) defined in Eq. (48) and computedwithin the LCFA [Eq. (50), solid line] and by means of thesemiclassical expression (49) (dashed line).

The asymptotic behavior of the LCFA expression (50) forE0 � Ec reads (see the Appendix)

µ(LCFA)∞ (E0)

µ(SC)∞ (E0)

= 1− 13

8π

E0

Ec+

657

128π2

E20

E2c

+O(E3

0

E3c

), (51)

so for E0 � Ec we recover the semiclassical prediction.The constant term in Eq. (51) can also be identifiedby expanding 1/ cosφ ≈ 1 + φ2/2 in the exponentialin Eq. (44) and integrating then the Gaussian functionover φ ∈ (0, ∞). Note that the derivation presentedin the seminal paper of Brezin and Itzykson [47] yieldsµ∞ = 1/(8π2), so it does not properly capture the pre-exponential factor.

To find out how strict the condition E0 � Ec is, weplot Eqs. (49) and (50) versus E0 (see Fig. 4). Notethat the total particle yield is proportional directly toµ∞(E0) according to Eq. (48), so these two quantitieshave the same relative uncertainties. In Fig. 4 we observethat the semiclassical approach is quite accurate even ifE0 is greater than 0.1Ec. If E0 = 0.1Ec, the relativeerror of the semiclassical result amounts to 4.9%. Notethat taking into account the terms up to (E0/Ec)

2 inEq. (51), one obtains the LCFA result for E0 = 0.1Ec

with a relative uncertainty of 0.06%.Having examined the validity of the semiclassical ap-

proach in the limit γ → 0, i.e. ω → 0, we now turn tothe analysis of the LCFA justification for nonzero γ.

B. Nonzero field frequency

The semiclassical approach also allows one to obtainthe total number of pairs produced for nonzero valuesof the Keldysh parameter γ. For instance, if γ � 1,instead of Schwinger’s exponential exp (−πEc/E0), onereceives a power-law multiphoton behavior of the parti-cle yield, which can be deduced by means of perturba-tion theory. Since the LCFA takes into account only the

9

tunneling mechanism of pair production, one can expectthis approximation to be accurate only within a certainvicinity of γ = 0 [recall that the LCFA prediction (44) isγ-independent for given E0]. In order to determine thesize of this vicinity, we expand the general semiclassicalexpressions given in Ref. [35] in terms of γ. Instead ofEq. (49), we obtain

µ(SC)∞ (E0, γ) = µ(SC)

∞ (E0)[1 +O(γ2)

]

× exp

[πEc

E0

γ2

8

{1 +O(γ2)

}]. (52)

In the limiting case, µ(SC)∞ (E0, 0) = µ

(SC)∞ (E0). We as-

sume that E0 < 0.1Ec, so µ(LCFA)∞ (E0) ≈ µ(SC)

∞ (E0) witha relative uncertainty of less than 5%. To make sure that

the field frequency is sufficiently low, so µ(SC)∞ (E0, γ) ≈

µ(SC)∞ (E0) and the LCFA is applicable, we should require

Ec

E0γ2 � 1, (53)

which is equivalent to

|eE0|3/2ω

� m2. (54)

This condition is exactly the same as Eq. (43) for τ =1/ω. Note that the inequality (54) is, in fact, quite weakdue to extracting the square root of Eq. (53) as well as thefactor π/8 in the exponential in Eq. (52). For instance, if

we require µ(SC)∞ (E0, γ) < 1.1µ

(SC)∞ (E0), i.e., the relative

uncertainty does not exceed 10%, then the field ampli-tude and its frequency should obey |eE0|3/2 > 2.0ωm2.

To summarize, if E0 < 0.1Ec, then the LCFA is valid,provided Eq. (54) is satisfied. This condition is consid-erably stronger than the naive requirement γ � 1 onceE0 . 0.1Ec. If E0 > 0.1Ec, the LCFA predictions canbe compared with the exact numerical results as was dis-cussed in the previous section.

VI. STANDING ELECTROMAGNETIC WAVE

Here as in Sec. IV B we will compare the approximateω-independent results with those obtained by means ofthe approach described in Sec. III B in the case of astanding-wave background (7). This specific scenarioproperly takes into account the magnetic component ofthe external field according to Maxwell’s equations, soit represents more realistic setup which mimics a com-bination of two counterpropagating laser pulses. Be-sides, as the external background is a periodic functionof z, the computations can be carried out very efficiently,cf. Ref. [48].

The results in terms of the function ν = ωN/V are dis-played in Fig. 5. The overall behavior of the curves is very

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

E0/Ec

10−19

10−17

10−13

10−9

10−5

10−2

ν/m

4

LCFA

ω = 1.0m

ω = 0.6m

ω = 0.2m

Figure 5. Parameter ν = ωN/V as a function of the amplitudeof the standing wave (7) evaluated within the LCFA (solidlight-blue line) and computed exactly for various values ofthe field frequency ω (dashed lines).

similar to that shown in Fig. 2. First, the LCFA is ac-curate for sufficiently low field frequencies and large fieldamplitudes, while at small E0 the discrepancy is tremen-dous. Second, although it is not clear from the graph,the LCFA begins to overestimate the particle yield forsufficiently large values of the field amplitude. The onsetof this overestimation is shifted compared to the case ofan oscillating electric field since the spatial cosine-profileof a standing wave makes the field effectively weaker, sothe Pauli principle plays a significant role for larger val-ues of the parameter E0. In other words, in the case ofa homogeneous background, the root mean square valueof the field strength coincides with E0, while in the caseof a standing wave, it is

√2 times smaller than E0. We

find that for these two scenarios, the dashed curve forω = 0.2m intersects the LCFA one at different values ofE0, whose ratio approximately matches

√2. Third, one

can directly verify that according to our findings, thevalidity of the LCFA is not governed by the conditionγ � 1.

The similarity between Figs. 2 and 5 indicates that theapplicability of the LCFA is not sensitive to the specificform of the field profile, which is quite natural as weonly count the number of particles without following theirdynamics in the external field (the latter is extremelyimportant for predicting the momentum spectra [16, 17]).

Let us finally provide a simple example of how onecan assess the applicability of the LCFA. To this end,we consider a field configuration involving multiple laserpulses [23]. The characteristic field amplitude chosen inRef. [23] is E0 ' (0.1–0.2)Ec, while the field frequency isvery small, ω ∼ 10−6m. The direct comparison betweenthe LCFA and exact calculations displayed in Figs. 2 and5 clearly indicates that for such low frequencies the LCFAshould be justified. On the other hand, at E0 ∼ 0.1Ec,one can also employ the requirement (54), which in ourcase demands 104 � 1. Since this condition is also satis-fied, the LCFA analysis performed in Ref. [23] is evidentlyreliable.

10

VII. CONCLUSION

In the present investigation, we scrutinized the validityof the locally constant field approximation by comparingits predictions with the exact values of the total num-ber of e+e− pairs. In the region of large field amplitudes(E0 & 0.1Ec) and high frequencies (ω & 0.1m), we per-formed numerical computations by means of two nonper-turbative techniques. It was shown that the LCFA mayindeed be very accurate even if the Keldysh parameter isclose to γ = 1/2, i.e., the condition γ � 1 is not satisfied.On the other hand, for E0 . 0.1Ec and ω . 0.1m, eventhe requirement γ � 1 does not necessarily justify theLCFA. In fact, one has to take care that the parameter|eE0|3/2/(ωm2) = (E0/Ec)

3/2(m/ω) is sufficiently large.This condition was deduced in the case of a Sauter pulseand that of an oscillating electric field, which suggeststhat the accuracy of the LCFA is governed by one uni-versal parameter which differs from γ.

Besides, we carried out exact calculations in the case ofa standing electromagnetic wave depending both on timeand one of the spatial coordinates. It was demonstratedthat the main patterns revealed for an oscillating electricbackground hold also for a standing wave. This also cor-roborates that the LCFA justification is not that sensitiveto the details of the spatio-temporal shape of the externalfield, although the particle dynamics could change drasti-cally. In this study, we considered a standing wave whose“temporal” frequency ω matches the frequency regard-ing the coordinate dependence kz according to Maxwell’sequations. One may expect that in a hypothetical sce-nario involving a standing wave with kz 6= ω, one shouldsimply choose the higher frequency in order to find outwhether the LCFA is applicable. This issue is, however,beyond the scope of the present study.

Our investigation provides a theoretical basis whichallows one to employ the LCFA and perform very efficientcomputations as soon as the parameters of the externalfield configuration are chosen within the proper domainidentified in this paper.

ACKNOWLEDGMENTS

This work was supported by Russian Foundationfor Basic Research (RFBR) (Grant No. 20-52-12017)and by Deutsche Forschungsgemeinschaft (DFG) (GrantNo. PL 254/10-1). D.G.S. acknowledges also the supportfrom the St. Petersburg University Alumni Association.

Appendix: LCFA expression in the case of anoscillating electric field

Here we will represent the integral in the LCFA expres-sion (44) in the form (47) and derive the asymptotic ex-pansion (51). First, we use the substitution 1/ cosφ = y

to obtain the following formula:

N(LCFA)∞V T

=m4

2π4

(E0

Ec

)2

J(πEc

E0

), (A.1)

where

J (z) =

∞∫

1

e−zy

y3√y2 − 1

dy. (A.2)

This function can be represented as

J (z) =

∞∫

1

e−zy

y√y2 − 1

dy −∞∫

1

√y2 − 1

y3e−zy dy. (A.3)

Using the identity

e−zy

y=

∞∫

z

e−yx dx, (A.4)

we write the first term in Eq. (A.3) in the form

I1(z) =

∞∫

z

K0(x)dx, (A.5)

where Kν(x) are the modified Bessel functions of the sec-ond kind. The second integral in Eq. (A.3) can be rewrit-ten via integration by parts:

I2(z) = −z2

∞∫

1

√y2 − 1

y2e−zy dy +

1

2I1(z). (A.6)

Integrating by parts one more time and using Eq. (A.4),we arrive at

I2(z) =z2

2

∞∫

z

K1(x)

xdx− z

2K0(z) +

1

2I1(z). (A.7)

Now using the identity K1(x) = −K ′0(x), we obtain

I2(z) = −z2

2

∞∫

z

K0(x)

x2dx+

1

2I1(z). (A.8)

Combining Eqs. (A.5) and (A.8), we present J (z) =I1(z)− I2(z) in the form

J (z) =1

2

∞∫

z

(1 +

z2

x2

)K0(x)dx (A.9)

or, equivalently,

J (z) =z

2

∞∫

1

(1 +

1

x2

)K0(zx)dx, (A.10)

11

which coincides with Eq. (47). In order to identify theasymptotic behavior (51), we assume that z � 1 accord-ing to Eq. (A.1) and employ the asymptotic expansion ofthe modified Bessel function K0,

K0(x) =

√π

2xe−x[1− 1

8x+

9

128x2+O

(1

x3

)]. (A.11)

Performing then integration in Eq. (A.10), one receivesan expression involving incomplete Gamma functionsΓ(a, z). By means of the recurrence relation

Γ(a, z) = e−zza−1 + (a− 1) Γ(a− 1, z) (A.12)

or using directly the formula

Γ(a, z) = za−1 e−z[1− 1− a

z+

(2− a)(1− a)

z2+O

(1

z3

)],

(A.13)we collect the terms and obtain

J (z) =

√π

2ze−z[1− 13

8z+

657

128z2+O

(1

z3

)]. (A.14)

Taking into account Eqs. (A.1), (A.14), (48), and (49),one immediately arrives at Eq. (51).

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