A comment on generalized Schwinger effect

Karthik Rajeev1∗, Sumanta Chakraborty2†and T. Padmanabhan1‡

1IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411007, India2Department of Theoretical Physics, Indian Association for the Cultivation of Science, Kolkata-700032, India

December 20, 2017

Abstract

A spatially homogeneous, time-dependent, electric field can produce charged particle pairs from thevacuum. When the electric field is constant, the mean number of pairs which are produced depends onthe electric field and the coupling constant in a non-analytic manner, showing that this result cannotbe obtained from the standard perturbation theory of quantum electrodynamics. When the electricfield varies with time and vanishes asymptotically, the result may depend on the coupling constanteither analytically or non-analytically. We investigate the nature of this dependence in detail. Weshow that the dependence of particle production on coupling constant is non-analytic for a class oftime-dependent electric fields which vanish asymptotically when a specific condition is satisfied. Wealso demonstrate that for another class of electric fields, which vary rapidly, the dependence of particleproduction on coupling constant is analytic.

1 Introduction

Production of particles by a non-trivial classical background source is a ubiquitous effect in quantumfield theory (QFT). Hawking radiation from a black-hole, the production of particles in an expandinguniverse and the Schwinger effect are some of the most studied examples of this phenomenon [1–15]. Eventhough the physical mechanism of particle creation depends on the specific system we are considering,there are certain common features. For instance, when the classical background is spatially homogeneous,the analysis of particle production in many interesting scenarios reduces to the study of a time-dependentharmonic oscillator. Because of this similarity in the mathematical description, from the study of a specificQFT system, one can gain important insights about several other systems (for example, see [16]). In thispaper, we will focus on the mechanism by which charged particle pairs can be produced from vacuumin a spatially homogeneous, time-dependent electric field [17–26], which could be called the generalizedSchwinger effect.

Besides its mathematical simplicity, there are several features of the Schwinger effect that makes ita promising model for understanding various aspects of QFT. An important such feature is its non-perturbative nature. The rate of particle production in a homogeneous time-independent electric fieldbackground depends on the field strength and coupling constant in a non-analytic manner (see [15] for

∗karthik@iucaa.in†sumantac.physics@gmail.com‡paddy@iucaa.in

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a recent review). This shows that one cannot obtain this result from the standard perturbation theoryof quantum electrodynamics (QED). While the predictions of perturbative approaches in QFT has beenverified to very high precisions [27–30], the verification of non-perturbative results remains as a majorchallenge in experimental physics. Schwinger effect provides us such an opportunity to understand thisrelatively unexplored regime of QFT in a better (and possibly deeper) way. Therefore, recently there havebeen several studies regarding the experimental verification of Schwinger effect [31–34]. Even though thestrength of the electric field that is required to test this phenomenon is beyond the current state-of-the-art,it will be accessible to some of the upcoming laser facilities like Extreme Light Infrastructure (ELI) [35]and European X-ray Free-Electron Laser (European XFEL) [36].

While the Schwinger mechanism is an exactly solvable problem, it is practically impossible to realize aspatially homogeneous, constant electric field in an experimental setup. In a laboratory setting, one canonly produce electric fields that vanish asymptotically in time though it could be spatially homogeneousto a necessary level of approximation. A well studied configuration of electric field with this property inwhich the pair production can be analytically computed is the so-called Sauter type electric field given byE = (E0 sech 2(αt), 0, 0), where α is a dimension-full constant [37–44]. The study of pair production in theSauter type field reveals an interesting feature of the generalized Schwinger effect. It turns out that themean number of particles nk, produced with a given momentum k in the presence of a Sauter type field candepend on the coupling constant q and field strength E0 either analytically or non-analytically dependingon whether the field varies rapidly or adiabatically in time, respectively. In particular, the expressionfor nk when the field is adiabatically varying can be shown to approach the corresponding expression forthe Schwinger case. Therefore, one expects that the mean number of particles produced for a genericelectric field configuration E(t) may depend on the coupling constant and field strength either analyticallyor non-analytically depending on certain conditions of the relevant parameters involved. Motivated bythis, we seek for two general classes of electric field configurations such that the mean number of particlesproduced exhibit, either non-analytic dependence on the coupling constant (and field strength) when aspecific condition is satisfied or, exhibit analytic dependence on the coupling constant (and field strength).

The paper is organized as follows: In Section 2, we review the instability of the vacuum of QED inthe presence of a Sauter type electric field. It turns out that the nature of dependence of the particlecontent on the coupling constant under Sauter type field depends on a parameter γ ≡ mα/(|qE0|) wherem and q are the mass and the charge of the field respectively. This parameter measures the degree ofadiabaticity of the electric field in its time variation. We demonstrate that the mean number of producedparticles nk as well as the probability of pair creation P depend analytically on |qE0|, when γ � 1 andare non-analytic functions of |qE0|, when γ � 1. In Section 3, we focus on a class of electric fields of theform E = (E0f(αt), 0, 0), where f(s) vanishes in the limit |s| → ∞. We examine the conditions for thevalidity of perturbative analysis for this system. Further, we show explicitly that, when the aforementionedconditions are met, the mean number of particles produced is an analytic function of |qE0|. For the electricfield of the form E = (E0f(αt), 0, 0), in Section 3.2, we consider the scenario in which perturbative analysisfails. We show explicitly that, in this non-perturbative regime, the mean number of particles producedhas a factor which is non-analytic in |qE0|, when the integral of f(s) has a certain asymptotic behaviour.Throughout the paper, we use a system of units in which c = ~ = 1. We work with (+,−,−,−) signaturefor the metric tensor.

2

2 Warm-up: Vacuum instability in Sauter type potential

Let us rapidly review the Schwinger effect in the Sauter type electric field to identify the two regimes inwhich the effect is perturbative or non-perturbative. Consider a spatially homogeneous electric field alongthe x direction, with the time dependence:

E(t) = E0 sech 2αt . (1)

The associated vector potential could be taken as Aµ = (0, A(t), 0, 0) with

A(t) = −∫E(t)dt = −E0

αtanhαt . (2)

For α→∞ the electric field would oscillate rapidly. In this case, the particle production rate is known tobe analytic in |qE0|. While, for α→ 0, we have γ � 1 and hence we can expand the vector potential A(t)in a Taylor series with the following leading order behaviour: A(t) ≈ −(E0/α)αt = −E0t, which mimicsthe standard Schwinger effect. Thus, in this limit, mean number of particles produced is expected to benon-analytic in |qE0|.

In order to study the vacuum instability in the presence of this field, one may compute the vacuumpersistence probability P. It can be shown that, to the leading order, this probability is given by P =exp(−SE) where SE is the Euclidean action evaluated for an instanton solution of the equation of motiondp/dt = qE, where p is the momentum [41, 42, 45–47]. This procedure works whenever, the solutionexhibits periodicity in the Euclidean time, which happens for the Sauter potential. Let us now apply thisprocedure to obtain the vacuum persistence probability.

We first need to determine the classical action for the charged particle. The equation of motion for acharged particle with charge q and mass m corresponds to dp/dt = qE, which can be integrated for theabove electric field yielding,

px(t) =qE0

αtanhαt = −qAx , (3)

while all the other components, e.g., py and pz are taken to be vanishing. The energy associated with theabove trajectory is:

E =√p2x + p2y + p2z +m2 =

m

γ

√γ2 + tanh2 αt; γ =

mα

|qE0|. (4)

(The parameter γ is always positive, depending on only |qE0|. But, throughout our discussion we willassume that qE0 is positive, so that the modulus will not be explicitly displayed hereafter.) To get thevacuum persistence amplitude (VPA) we need to evaluate the trajectory of the particle. Using the factthat E = m(dt/dτ) we obtain,

τ = γ

∫coshαt dt√

sinh2 αt+ γ2 cosh2 αt. (5)

The above integral can be performed in a straight forward manner by substituting z = sinhαt, whichultimately results in,

sinhαt =γ√

1 + γ2sinh

(√1 + γ2

γατ

). (6)

3

Further using the fact that px = m(dx/dτ) and the above result connecting t and τ we obtain,

dx

dτ=

1

γtanhαt =

1

γ

γ sinh

(√1+γ2

γ ατ

)√

1 + γ2 cosh2

(√1+γ2

γ ατ

) , (7)

which can be integrated by the substitution z = cosh(√

1 + γ2ατ/γ) resulting into,

sinh(√

1 + γ2 αx)

= γ cosh

(√1 + γ2

γατ

). (8)

To obtain the VPA, we need to analytically continue the trajectory into Euclidean time. The correspondingtrajectory in the Euclidean plane can be obtained by the transformation: t→ itE and τ → iτE, giving,

sinαtE =γ√

1 + γ2sin

(√1 + γ2

γατE

), (9)

sinh(√

1 + γ2 αx)

= γ cos

(√1 + γ2

γατE

). (10)

Therefore in the Euclidean time τE we have a periodicity with the period τpE:

τpE =2πγ

α√

1 + γ2≡ 2π

α. (11)

The action for the particle in the Lorentzian sector is given by

S = −m∫dτ + q

∫A(t)dx ,

= −m∫dτ −m

∫dτ

(dx

dτ

)2

,

= −m∫dτ −m

∫sinh2 ατ dτ

1 + γ2 cosh2 ατ. (12)

Transforming to the Euclidean sector and integrating over a complete period in the imaginary time, weobtain,

SE ≡ iS = m

∫dτE

cos2(ατE)

1− γ2

1+γ2 sin2(ατE). (13)

Integration yields,

SE =πm

α

2γ

1 +√

1 + γ2. (14)

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We can compute the probability for vacuum persistence, P ≈ exp(−SE) in the two limits: (a) γ � 1 and(b) γ � 1. As mentioned earlier, we expect P to be non-analytic in the field strength in the case γ � 1,(which includes the case of a constant electric field), while it could be analytic for γ � 1. For γ � 1 case,from Eq. (14) it follows that the probability for pair production becomes,

P = e−2πmα

(1 +

2πm

αγ+O(γ−2)

)≈ e− 2πm

α

(1 +

2πqE0

α2

). (15)

Thus P is independent of the field strength to leading order in 1/γ. Hence in this limit the pair productionprobability is analytic in the field strength. On the other hand, when γ � 1, the pair production probabilitytakes the following form

P = exp

{−πm

2

qE0

(1− γ2

4

)+O

(γ3)}≈ exp

{−πm

2

qE0

(1− m2α2

4(qE0)2

)}. (16)

Thus it is clear that for small γ, the pair production probability is non-analytic, as anticipated. This ismainly due to the fact that γ � 1 corresponds to α → 0 limit and hence is similar to a constant electricfield.

2.1 Mean number of particles produced

It is useful to rework the above result more explicitly in terms of mode functions and Bogoliubov coefficientswhich will allow us to determine the mean number of particles produced in either limit. To do this wewill start with the explicit solution for complex scalar field φ in a Sauter type potential. Given the spatialhomogeneity, it is convenient to work in the Fourier space. The Fourier transform φk of a complex scalarfield φ of charge q and mass m in the Sauter type electric field introduced in Section 2 satisfies the followingdifferential equation

φk +

[m2 + |k⊥|2 +

(kx −

qE0

αtanhαt

)2]φk = 0 . (17)

The solution to this equation which corresponds to the positive frequency modes at asymptotic past canbe written in terms of hypergeometric functions as follows

ξ∗in(t) = C1e−iω−t(1 + e2αt)

i2α (ω−−ω+)F(a, b, c; y) , (18)

where,

y =1

2(1 + tanh αt); ω± =

√m2 + |k⊥|2 +

(kx ∓

qE0

α

)2

, (19)

a =

(i

2α

)(2qE0

α+ ω+ − ω−

); b = 1 +

(i

2α

)(−2qE0

α+ ω+ − ω−

); c = 1− iω−

α, (20)

and C1 is an arbitrary normalization constant. Using the asymptotic behaviour of hypergeometric functionswe can determine its form as t→∞ to be:

ξ∗in(t) ≈ A∗e−iω+t + Beiω+t , (21)

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where, the mean number of particles produced at future infinity is given by [43,44]:

nk = |B|2 =

cosh2

[π

√(qE0

α2

)2− 1

4

]+ sinh2

[π2α (ω+ − ω−)

]sinh

(πω−α

)sinh

(πω+

α

) . (22)

The two limits discussed in the previous section corresponds to the following two limits in this particularcase: (a) α�

√qE0, which corresponds to γ � 1 in the previous scenario, and (b) α� (qE0/

√m2 + k2),

equivalent to γ � 1. Here k2 = k2x + |k⊥|2. When α�√qE0, it immediately follows that

nk = exp

(−π(m2 + |k⊥|2)

qE0

), (23)

which coincides with the pair production probability derived in the previous section associated with γ � 1.As evident and anticipated the mean number of particles produced is non-analytic in the field strength. Theprobability P(k) that a particle with momentum k and an anti-particle with momentum −k is producedis given by P(k) = nk/(1 + nk). When the electric field is weak (i.e., qE0/m � 1), so that nk � 1, theprobability of pair production becomes P(k) ≈ nk. Therefore, for kz = ky = 0 and the case of weak E,which was the case considered in Section 2, the probability of pair production to leading order in qE0

becomes P(k) = exp(−πm2/qE0), which is in agreement with Eq. (16).The opposite limit, corresponding to α� (qE0/

√m2 + k2), can also be easily obtained from Eq. (22),

leading to,

nk ≈

(πqE0kxα2ε

)2sinh2

(πα

√k2 +m2

) , (24)

where, ε2 = k2 +m2. Clearly, nk is analytic in qE0 in conformity with the result in the previous section.Again, we see that the particle production is non-analytic in the coupling constant for small α, while isanalytic for large α. In the first case, the mean number of particles produced, to the leading order coincideswith that of the standard Schwinger effect.

3 Analytic and non-analytic dependences in a general context

We will now generalise the previous results — which were obtained for a specific form of the electric field,viz. the Sauter field — to a more general configuration. For this analysis we will consider an electric fieldalong the x-axis of the form [19]

E(t) = E0f(αt) , (25)

where, E0 is a constant. The vector potential can then be chosen to be (0, A, 0, 0), where

A(t) = −E0

αF (αt) , (26)

where F is defined through dF (s)/ds = f(s). Since a physically realizable electric field vanishes as t→∞,in Eq. (25), we impose the condition that f(αt) vanishes as we approach the asymptotic past and future

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times. Therefore, the function F satisfies

lim|s|→∞

dF

ds= 0 . (27)

In this external background electric field, the Fourier transform φk of a complex scalar field of mass mand charge q satisfies the following time dependent harmonic oscillator equation:

φk + ω2kφk = 0 , (28)

where the time dependent frequency ωk is given by

ω2k(t) = ε2 +

2kxmF (αt)

γ+m2F (αt)2

γ2. (29)

and ε2 = k2 +m2. We will now study this equation and its solutions in the two appropriate limits.

3.1 Perturbative limit

In this subsection we will assume that the function F is bounded as follows:

Condition 1 : Fmin < F < Fmax . (30)

Moreover, from Eq. (29), we see that the field strength dependent part of ω2k can be treated as a pertur-

bation whenever,

F− < F (t) < F+ , (31)

where, F± = γ(−kx ±√k2x + ε2) are the roots of the equation 2kxmF/γ + m2F 2/γ2 = ε2. Therefore,

the field strength dependent term in Eq. (28), viz. (2kxmF/γ + m2F 2/γ2)φk can be always treated as aperturbation if the following conditions are satisfied,

Condition 2: Fmax � F+ and |Fmin| � |F−| . (32)

When this condition is satisfied, we can solve Eq. (28) perturbatively in powers of 1/γ. Therefore, if anelectric field of the form given by Eq. (25) satisfies Condition 1 and 2, we can take the solution to Eq. (28)to be of the following form

φk = φk(0) +1

γφk(1) +

1

γ2φk(2) + · · · . (33)

The O(1/γ) solution which approaches a positive frequency mode at t→ −∞ can be easily found to be

φk = eiεt − 2mkxγε

∫ t

−∞dt′ sin[ε(t− t′)]F (αt′)eiεt

′+O(γ−2) . (34)

Due to the appearance of sin[ε(t − t′)], we see that a positive frequency mode in the asymptotic pastevolves into a linear combination of both positive and negative frequency modes. That is

φk ≈

{eiεt, t→ −∞Aeiεt + B∗e−iεt, t→∞

(35)

7

where, A = 1 +O(1/γ) and

B∗ =

(πkxqE0

iαε2

)f

(2ε

α

)+O(γ−2) , (36)

where, f(s) is the Fourier transform of f(τ). Therefore, the mean number of particles produced is givenby

nk(∞) =|πkx(qE0)f

(2εα

)|2

α2ε4+O(γ−4) . (37)

Note that this expression is a Taylor series in qE0 and hence analytic in the same. So we obtain a simple,closed form expression for the number of particles produced in this case.

We will now provide an alternate derivation of Eq. (37). To do this, it is convenient to consider thefollowing quantity.

z =B∗(t)A∗(t)

e2iρ , (38)

where, ρ = ωk, along with A and B being the time dependent Bogoliubov coefficients. It can be shownthat z satisfies the following differential equation

z + 2iωkz +ωk

2ωk

(z2 − 1

)= 0 . (39)

The time dependent particle number nk in terms of z is given by

nk =|z|2

1− |z|2. (40)

The variable z was introduced in [48,49] for the analysis of particle production in an external background.It is worth mentioning that z is also related to the classicality of the φk and used extensively in thecontext of cosmology to understand the transition to classicality [50–53]. For γ � 1, we have |z| � 1 andωk ≈ ε + mkxF/(εγ), so that the differential equation for z presented in Eq. (39) reduces, in the leadingorder to,

d(ze2iεt)

dt=kx(qE0)

2εe2iεtf(αt) . (41)

This implies that |z(t→∞)| is given by

|z(∞)| =∣∣∣∣πkx(qE0)

αε2f

(2ε

α

)∣∣∣∣ . (42)

We can easily see from this that the asymptotic value of mean number of particles produced is given byEq. (37). Therefore, we have shown in this section that: if the electric field of the form given by Eq. (25)satisfies Conditions 1 and Condition 2, the asymptotic value of the mean number of particles produced nkis an analytic function of |qE0|.

As a consistency check, let us apply this to the case of Sauter type electric field that we discussed inSection 3. In this case the relevant f(τ) is given by sech 2(ατ), whose Fourier transform evaluated at

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2ε/α turns out to be α−1ε csch(πα/ε). Therefore, from Eq. (37), the mean number of particles producedto leading order in γ is given by

nk =

(πqE0kxα2ε

)2sinh2

(πα

√k2 +m2

) +O(γ−4) , (43)

which is in complete agreement with Eq. (24).

3.2 Non-Perturbative Limit

In this subsection we will consider electric field configurations of the form Eq. (25), which violates Condition1 and 2, presented in Eq. (30) and Eq. (32) respectively. This corresponds to the case in which the fielddependent part of ω2

k is much larger than ε2 in some range of time. We focus on the situation whichsatisfies the following conditions:

Condition 3: |F (t)| � F+, |F−| when |t| � tc , (44)

where tc is some critical time. This implies that the perturbation theory cannot be used to find a solutionto Eq. (28) valid for all times t. However, since we have assumed the field to vanish asymptotically, fromEq. (27) we see that the WKB approximation becomes valid at late and early times. The simplest exampleof this case is given by a constant field. To set the stage, we will briefly review this special case beforedoing a general analysis.

The Fourier modes φk of a complex scalar field in a constant electric field is given by E = (E0, 0, 0)satisfies the following harmonic oscillator equation, such that,

φk + ω2k(t)φk = 0 (45)

where, for the choice of gauge Aµ = (0,−E0t, 0, 0),

ω2k(t) = m2 + |k⊥|2 + (qE0t+ kx)2 . (46)

The exact solutions of Eq. (45) are known in terms of parabolic cylinder functions. But, the exact numberof particles produced at t = +∞ can be found using the WKB solutions of Eq. (45) in the limit |t| → ∞.Let us denote by ξk(in), the ‘in-modes’, which are solutions to Eq. (45) that behave as positive frequencymodes at t→ −∞. Similarity, we define the ‘out-modes’ ξk(out), which are the positive frequency solutionsof Eq. (45) at t → ∞. From Eq. (46), we see that (|ωk|/ω2

k) � 1 as |t| → ∞. Hence, in the asymptoticpast and future the WKB approximation is valid. In these regions, we have the following approximationfor ξk(in) and ξk(out)

As t→ −∞: ξk(in) ≈1√|ωk|

exp

(i

∫|ωk|dt

)≈ (−qE0t)

−iλ/2−1/2 exp

(− iqE0t

2

2

), (47)

As t→∞: ξk(out) ≈1√|ωk|

exp

(i

∫|ωk|dt

)≈ (qE0t)

iλ/2−1/2 exp

(iqE0t

2

2

), (48)

where, λ = (|k⊥|2 + m2)/(qE0). Let us consider the evolution of ξ∗k(in) from t → −∞ to t → ∞. Since

{ξk(out), ξ∗k(out)} is a complete set of solutions of Eq. (45), we can write

ξ∗k(in) = Aξ∗k(out) + B∗ξk(out) (49)

9

where, A and B are the standard Bogoliubov coefficients. The mean number of particles produced nk isthen given by |B|2. One can use the asymptotic expansions of the parabolic cylinder functions to computeB. But there is simpler and more elegant procedure (originally due to Landau), which can be used toobtain B and hence, nk. To follow this procedure, we start with the WKB approximation for ξ∗k(in) givenby

ξ∗k(in) =

(−qE0t)iλ/2−1/2 exp

(iqE0t

2

2

), t→ −∞

A(qE0t)−iλ/2−1/2 exp

(− iqE0t

2

2

)+ B∗(qE0t)

iλ/2−1/2 exp(iqE0t

2

2

), t→∞

(50)

Now, in the asymptotic expression for ξ∗k(in) at t → −∞, if we treat t as a complex variable and rotate

t in the complex plane from arg[t] = 0 to arg[t] = π, the solution nicely gets mapped to the asymptoticexpression for ξk(out) at t→∞. Therefore the Bogoliubov coefficient B is just:

B = e−iπ/2e−πλ/2 . (51)

Hence, the mean number of particles produced is given by nk = e−πλ, which is the standard result.To generalise this result we note that the time dependent frequency ωk has the following series expansion

near |t| → ∞ :

ω(t) ≈ κ|t|(

1 +ω20

2κ2|t|2− ω4

0

8κ4|t|4+ ...

)(52)

=

1∑n=−∞

Cn(κ)|t|2n−1 , (53)

where, κ = (qE0) and ω20 = (m2 + |k⊥|2). Note that there are only odd powers of |t| in the expansion of

ω(t) near |t| =∞. We observe that the mean number of particles produced, found from rotating t in thecomplex plane is related to C0 as

n = e−2πC0 , (54)

and the non analyticity comes from the fact that C0 ∝ 1/(qE0). We will now show that these featurescontinue to hold even in a more general context.

Motivated by the above observation, let us consider a class of electric fields satisfying Condition 3 as inEq. (44), for which the mean number of particles produced is non-analytic in the coupling constant. Thisclass is characterized by the following properties:

• For, |τ | → ∞, f(τ) is symmetric under τ → −τ .

• F (τ) diverges as a power series, as |τ | → ∞.

The first condition implies that F (|τ |) has the following expansion near infinity

F (|τ |) ≈∞∑

n=−∞Cn|τ |2n−1 . (55)

10

For later use, let us also note the asymptotic series for 1/F (|τ |),

1

F (|τ |)=

∞∑n=−∞

Cn|τ |2n−1 , (56)

where, the coefficients Cn and Cn satisfies,

∞∑n=−∞

CnCl−n =

{1, l = 1

0, ∀l ∈ Z− {1}(57)

The Fourier modes in this electric field satisfies the equation of motion of a harmonic oscillator of unitmass and frequency is given by Eq. (29). For |t| → ∞, the frequency ωk is given by

ωk(t) ≈ mF (αt)

γ+ kx +

γ(|k⊥|2 +m2)

2mF (αt)+O

(F (αt)−2

). (58)

Proceeding exactly as in our analysis of the standard Schwinger effect, the WKB approximation for the‘in’ and ‘out’ modes can be found to be

As t→ −∞: ξk(in) ≈ (|qE0F (αt)|)−1/2 exp

(iqE0

2α

∫dtF (αt) + ikxt+

γ(k2⊥ +m2)

2m

∫dt

F (αt)

)(59)

As t→∞: ξk(out) ≈ (|qE0F (αt)|)−1/2 exp

(−i qE0

2α

∫dtF (αt)− ikxt−

γ(k2⊥ +m2)

2m

∫dt

F (αt)

).

(60)

Again, since ξ∗k(in) can be written as a linear combination of ξk(out) and ξ∗k(out) we have,

ξ∗k(in) =

{(|qE0F (αt)|)−1/2 exp

(i qE0

2α

∫dtF (αt) + ikxt+ γ(|k⊥|2+m2)

2m

∫dt

F (αt)

), t→ −∞

Aξ∗k(out) + B∗ξk(out), t→∞(61)

where, we have to use Eq. (60) for the WKB approximation of ξk(out) near t→∞. Let us now treat t as acomplex variable and rotate t in the complex plane from arg[t] = 0 to arg[t] = π. As in the previous case,the WKB solution for ξ∗k(in) at t → −∞ maps to the asymptotic expression for ξk(out) at t → ∞. TheBogoliubov coefficient B to leading order in qE0 is then given by

B = e−iπ/2e−πλC0/2 , (62)

where, λ = (|k⊥|2 + m2)/(qE0). Therefore, we see that the mean number of particles produced has thefollowing non perturbative part

nk(non-pert) = e−π(m2+|k⊥|

2)C0qE . (63)

Therefore, we have shown that: For the class of electric field with the asymptotic properties mentionedabove, the asymptotic value of the number of particle produced, nk, has a non-perturbative factor given bynk(non-pert) = exp{−π(m2 + k2⊥)C0/qE0}.

11

As an example to this result and as a consistency check, consider the asymptotic expansion of thevector potential for the Sauter type potential in the limit α� 1/

√qE0. This is given by:

ASauter(t) ≈ E0t+O[(αt)2] . (64)

Therefore the corresponding expansion of 1/F for this case reduces to

1

F (|τ |)≈ 1

|τ |, (65)

implying that, C0 = 1. So the particle number is given by

nk(non-pert) ≈ e−π(m2+|k⊥|

2)

qE0 , (66)

which reproduces the result for mean number of particles produced of the standard Schwinger effect case,as expected.

4 Conclusion

The particle production in an external electric field serves as a toy model to understand various featuresof a generic QFT in an external classical background. An important feature of the particle production inthe case of a constant electric field (Schwinger effect) is that it is a non-perturbative phenomenon. This isreflected in the fact that, the mean number of particles produced with a particular momentum is a non-analytic function of the coupling constant and the field strength. However, when the external electric fieldis time-dependent, the mean number of particles produced could be either analytic or non-analytic in thecoupling constant. A simple example which covers both the limits is provided by the Sauter type electricfield in which case this behaviour is governed by a specific parameter in the potential. Motivated by this,we have investigated a more general class of field configurations to contrast analytic versus non-analyticdependence in the coupling constant. We have considered two fairly general classes of electric fields, suchthat, for one class the particle production is analytic while for the other it is non-analytic in qE0. Wehave identified the conditions for this behaviour in a general setting. The results and the techniques haveimplications in other contexts, like for example, particle production in an expanding universe, which wehope to investigate in a future publication.

Acknowledgements

Research of S.C. is supported by the SERB-NPDF grant (PDF/2016/001589) from SERB, Government ofIndia. T.P’s research is partially supported by the J.C.Bose Research Grant of DST, India.

A Perturbative Analysis of Schwinger effect

We have seen that one cannot apply perturbation theory to calculate the mean number of particles pro-duced, if Condition 1 and 2 are not satisfied by the external electric field. The vector potential Aµ

corresponding to a constant electric field along x-axis may be taken as Aµ = (0,−E0t, 0, 0). One canidentify that, in this case, F (αt) = αt and hence, Conditions 1 and 2 are clearly violated. Therefore, the

12

mean number of particles for the constant electric field case cannot be obtained from the perturbationanalysis given in Section 3.1. However, just out of curiosity, one can ask: what does a naive perturbativeanalysis give for particle production in a constant electric field? To answer this, we start by noticing fromEq. (29) that the time dependent frequency ωk in for the Schwinger case has the following form

ω2k(t) = κ2t2 + ω2

0 , (67)

where, κ = qE0 and ω20 = m2 + k2y + k2z and we have taken kx = 0. Let us expand the Bogoliubov

coefficients in powers of ε ≡ κ2/ω40 , as,

Figure 1: n(t) for ω0 = 1 and κ = 0.01. Brown graph is for n(t) calculated to leading order in ε and Bluegraph is for the next to leading order calculation.

A = 1 +A1 +A2 + ... (68)

B = B1 + B2 + ... (69)

where, An and Bn are proportional to εn, for n = 1, 2, · · · . The following expressions, in this context, arealso useful,

ωk

2ωke2iρ = C1 + C2 + ... (70)

ωk

2ωke−2iρ = D1 +D2 + ... (71)

where,

C1 =κ2te2itω0

2ω20

; C2 =iκ4t3e2itω0(tω0 + 3i)

6ω40

; · · · (72)

D1 =κ2te−2itω0

2ω20

; D2 = − iκ4t3e−2itω0(tω0 − 3i)

6ω40

; · · · (73)

The differential equation satisfied by the Bogoliubov co-efficients to O(ε2) can be written as,

A1 = 0; A2 = C1B1 (74)

B1 = 0; B2 = D2 +D1A1 (75)

13

where the initial conditions are Ai(0) = Bi(0) = 0. Note that this is different from the initial conditionstaken in the main body of this work. That is, the initial condition Ai(0) = Bi(0) = 0 implies that thequantum state at t = 0 is the instantaneous vacuum of the Hamiltonian of the system. On the other hand,we assumed the quantum state to be the vacuum at t = −∞ in the main body. We have chosen the initialconditions Eq. (74) and Eq. (75) just for convenience. The aim of this section is to demonstrate that naiveperturbation theory fails when applied to the study particle production in constant electric field. Eitherof the initial conditions can be used to demonstrate the same hence, we have chosen the more convenientone here. The solution to second order is found to be

A1 = 0; A2(t) =κ4(8it3ω3

0 + 6t2ω20 + 6itω0e

2itω0 − 3e2itω0 + 3)

192ω80

(76)

B1(t) = −κ2e−2itω0

(−2itω0 + e2itω0 − 1

)8ω4

0

; (77)

B2(t) = −κ4e−2itω0

(−4t4ω4

0 + 20it3ω30 + 30t2ω2

0 − 30itω0 + 15e2itω0 − 15)

48ω80

(78)

Therefore, the mean number of particles n(t) is given by

n(t) =κ4

1152ω160

(κ4(8t8ω8

0 + 80t6ω60 − 90t4ω4

0 + 225)− 36κ2ω4

0

(6t4ω4

0 − 5t2ω20 − 5

))+

κ4

1152ω160

[− 3

(5κ2 + 2ω4

0

){2tω0

(5κ2

(3− 2t2ω2

0

)+ 6ω4

0

)sin(2tω0)

+(κ2(4t4ω4

0 − 30t2ω20 + 15

)+ 6ω4

0

}cos(2tω0)

) ]+

κ4

1152ω160

(36(2t2ω10

0 + ω80

))(79)

The mean number of particles produced clearly diverges, as can also be seen from Fig. 1. The divergingbehaviour of the mean number of particles is due to the fact that the perturbation theory fails for thiscase. To be more specific, Condition 1 and 2 are not satisfied. Therefore, one has to employ other methodsto compute the particle production on the constant electric field case. One such method is illustrated onSection 3.2.

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