# Violation of vacuum stability by inverse square electric fields

## Abstract

In the framework of QED with a strong background, we study particle creation (the Schwinger effect) by a time-dependent inverse square electric field. To this end corresponding exact in- and out-solutions of the Dirac and Klein–Gordon equations are found. We calculate the vacuum-to-vacuum probability and differential and total mean numbers of pairs created from the vacuum. For electric fields varying slowly in time, we present detailed calculations of the Schwinger effect and discuss possible asymptotic regimes. The obtained results are consistent with universal estimates of the particle creation effect by electric fields in the locally constant field approximation. Differential and total quantities corresponding to asymmetrical configurations are also discussed in detail. Finally, the inverse square electric field is used to imitate switching on and off processes. Then the case under consideration is compared with the one where an exponential electric field is used to imitate switching on and off processes.

## 1 Introduction

*T*-constant electric field [10, 24], in a periodic alternating in time electric field [25, 26], in an and exponentially growing and decaying electric fields [27, 28, 29] (see Ref. [30] for the review), and in several constant inhomogeneous electric fields of similar forms where the time

*t*is replaced by the spatial coordinate

*x*. An estimation of the role of switching on and off effects for the pair creation effect was done in Ref. [31].

In the present article we study the vacuum instability in an inverse square electric field (an electric field that is inversely proportional to time squared); see its exact definition in the next section. This behavior is characteristic for an effective mean electric field in graphene, which is a deformation of the initial constant electric field by backreaction due to the vacuum instability; see Ref. [32]. From the technical point of view, it should be noted that the problem of the vacuum instability caused by a constant electric field in the de Sitter space considered in Refs. [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43] shares some similarities to the above problem in the Minkowski space-time. In addition, an inverse square electric field is useful to study the one-loop Heisenberg–Euler effective action in the framework of a locally constant field approximation [44]. At last, results of our study allow one to better understand the role of switching on and off effects in the violation of the vacuum stability. In Sect. 2 we present, for the first time, exact solutions of the Dirac and Klein–Gordon equations with the inverse square electric field in the Minkowski space-time. With the help of these solutions, we study in detail the vacuum instability in such a background in the framework of QED with *t*-electric potential steps, using notation and some technical results of our review article [30]. In particular, differential and total mean numbers of particles created from the vacuum are calculated in Sect. 3 within the slowly varying approximation. The case of an asymmetric configuration of the inverse square electric field is discussed in Sect. 4. In Sect. 5, the inverse square electric fields is used to imitate switching on and off processes. The obtained results are compared with the case when the form of switching on and off is exponential. Sect. 6 contains some concluding remarks.

## 2 Solutions of wave equations with the background under consideration

*D*spatial dimensions), that switches on at the infinitely remote past \( t=-\infty \), switches off at the infinitely remote future \(t=+\infty \) and it is inversely proportional to time squared. In what follows, we call such a field inverse square electric field. The field is homogeneously distributed over space, directed along the axis \(x^{1}=x,\) i.e., \({\mathbf {E}} =(E\left( t\right) ,0,\ldots ,0),\ E^{i}=0\), \(i=2,\ldots ,D\),

*t*-electric potential steps [30]. It is parameterized by two constants \(\tau _{1,2}\) which play the role of time scales for the pulse durations, respectively. The electric field (1) and its potential (2) are pictured on Fig. 1 for some values of \( \tau _{1,2}\).

^{1}

^{2}[45, 46, 49]

^{3}[49],

*D*-dimensional Euclidean space.

*via*linear transformations, for instance\(\ ^{\zeta }\psi _{n}\left( x\right) =\sum _{\zeta ^{\prime }}g\left( _{\zeta ^{\prime }}|^{\zeta }\right) \, _{\zeta ^{\prime }}\psi _{n}\left( x\right) \), where coefficients \(g\left( _{\zeta }|^{\zeta ^{\prime }}\right) \) are diagonal \( \left( \, _{\zeta }\psi _{n},\ ^{\zeta ^{\prime }}\psi _{n^{\prime }}\right) =g\left( _{\zeta }|^{\zeta ^{\prime }}\right) \delta _{nn^{\prime }}\) and obey the properties

^{4}

## 3 Quantities characterizing the vacuum instability

*g*’s coefficients allow us to find differential mean numbers \(N_{n}^{ \mathrm {cr}}\) of pairs created from the vacuum, the total number

*N*and the vacuum-to-vacuum transition probability \(P_{v}\):

### 3.1 Slowly varying field regime

#### 3.1.1 Differential mean numbers

*t*-electric steps. First, since the electric field is homogeneously directed along the

*x*-direction only, it creates pairs with a wider range of values of \(p_{x}\) instead \({\mathbf {p}}_{\perp }\), once they are accelerated along the direction of the field. Accordingly, one may consider a restricted range of values to \({\mathbf {p}}_{\perp }\), namely \(\sqrt{\lambda }<K_{\perp }\), in which \(K_{\perp }\) is any number within the interval \(\min \left( eE\tau _{1}^{2},eE\tau _{2}^{2}\right) \gg K_{\perp }^{2}\gg \max \left( 1,m^{2}/eE\right) \). As for the longitudinal momentum \(p_{x}\), we restrict subsequent considerations to \(p_{x}\) negative and generalize results for \( p_{x}\) positive using the properties discussed at the end of Sect. 2. Thus, as \(p_{x}\) admits values within the half-infinite interval \(-\infty <p_{x}\le 0\), the kinetic momentum \(\pi _{1}\) varies from large and positive to large and negative values \(eE\tau _{1}\ge \pi _{1}>-\infty \). However, differential mean numbers \(N_{n}^{\mathrm {cr}}\) are significant only in the range \(-\left| \pi _{\perp }\right| \beta _{1}\le \pi _{1}\le eE\tau _{1}\), whose main contributions lies in four specific subranges

^{5}This result coincides with differential number of created particles in a constant electric field [2, 3].

*E*equal to the critical Schwinger value \(E=E_{\mathrm {c}}=m^{2}/e\). In addition, we include the approximations given by Eq. (25) for the same values to the pulses durations \(\tau _{j}\) and amplitude

*E*. In these plots, we set \({\mathbf {p}}_{\perp }=0\) and select for convenience a system of units, in which \(\hslash =c=m=1\). In this system, the reduced Compton wavelength Open image in new window is one unit of length, the Compton time Open image in new window one unit of time and electron’s rest energy \(mc^{2}=1\) one unit of energy. In the plots below, the pulse durations \(\tau _{j}\) and the quantum numbers \(p_{x}\) are dimensionless quantities, relative to electron’s rest mass \(p_{x}/m\) and \(m\tau _{j}\).

According to the above results, the mean number of pairs created \(N_{n}^{ \mathrm {cr}}\) tend to the uniform distribution \(e^{-\pi \lambda }\) as the pulses duration \(\tau _{j}\) increases. This is consistent with the fact that the inverse square electric field (1) tends to a constant electric field (or a *T*-constant field with *T* sufficiently large) as the pulses duration \(\tau _{j}\) increases, whose mean numbers are uniform over a sufficiently wide range of values to the longitudinal momentum \(p_{x}\). Therefore, the exact distributions (16) are expected approach to the uniform distribution for sufficiently large values of the pulses duration \( \tau _{j}\). Moreover, it is seen that the exact distributions tends to the uniform distribution for sufficiently small values to the longitudinal momentum \(p_{x}\). This is also in agreement with the asymptotic estimate given by Eq. (22), obtained for \(p_{x}\) sufficiently small. Finally, comparing asymptotic approximations (dashed lines) with exact distributions (solid lines), we conclude that the accuracy of the approximations (25) increase as \(m\tau \) increases. This results from the fact that as \( m\tau \) increases, the parameter \(eE\tau ^{2}\) increases as well. Thus, larger values to \(m\tau \) present a better accuracy. For the values considered above, the lines (a), (b) and (c) correspond to \(eE\tau ^{2}=100\), 2500 and 10,000, respectively.

#### 3.1.2 Total numbers

*t*-electric potential steps, the total number of pairs created is proportional to the space time volume

^{6}

*t*-electric potential steps given by in Ref. [15]. Explicitly, one can use the universal form of the dominant density given by Eq. (3.6) in Ref. [15],

*T*-constant electric field [10, 24] and a peak electric field [28], whose dominant densities are proportional to the corresponding total increment of the longitudinal kinetic momentum in the slowly varying regime. Recalling the definitions of the

*T*-constant electric field and the peak electric field [10, 28, 30]

*E*and same longitudinal kinetic momentum increments \(\Delta U_{ \mathrm {p}}=\Delta U_{\mathrm {T}}\), we have shown in [15, 24, 30] that the peak electric field is equivalent to a

*T*-constant electric field in pair production, provided that it acts on the vacuum over an effective time duration

*T*-constant field. In other words, a

*T*-constant field acting over the time interval \(T=T_{\mathrm {eff}}\) is equivalent to the peak electric field in pair production. Extending these considerations to the case of the inverse square electric field (1), we obtain the following effective time duration

*T*-constant electric field acting on the vacuum over the same effective time duration \(T=T_{\mathrm {eff}}\) is equivalent to the inverse square electric field (1) in pair production.

*E*, namely

*d*. For strong amplitudes \(E\gg m^{2}/e\) though, one can restrict to the leading-order approximation of \(G\left( \alpha ,z\right) \) with small argument, \(G\left( \alpha ,z\right) \approx \alpha ^{-1}\), \(z\rightarrow 0\), to show that the latter relation does depend on the space-time dimensions \(\tau /2\approx \left( 1-d^{-1}\right) k^{-1}\). As a result, we see that \(\tau \approx k^{-1}\) for the lowest space-time dimension \(d=2\) and conclude that the relation between \(\tau \) and

*k*varies within the interval \(k^{-1}\le \tau \le 2k^{-1}\), for any amplitude

*E*or space-time dimensions

*d*, provided that both fields acts over the same effective time duration \(T_{\mathrm {eff}}\).

## 4 Asymmetric configuration

In the previous section, the inverse square electric field (1) was treated in a somewhat symmetrical manner, once the pulses duration \(\tau _{1} \) and \(\tau _{2}\) were considered large, approximately equal and with a fixed ratio \(\tau _{1}/\tau _{2}\). Here we supplement the above study with an essentially asymmetrical configuration for the electric field, characterized by a very sharp pulse duration in the first interval \(\mathrm {I }\) while remaining arbitrary in the second interval \({\mathrm {II}}\). In this way, the electric field is mainly defined on the positive half-interval. The present consideration provides insights on switching on or off effects by inverse square electric fields, as shall be discussed below.

*g*-coefficient \(g\left( _{-}|^{+}\right) \) given by Eq. (12) in the limit \(\sqrt{eE}\tau _{1}\rightarrow 0\) with the one computed directly for the inverse square decreasing electric field (41). To this end, one may repeat the same considerations as in Sect. 2 and take into account that the only essential difference between the fields (1) and (41) lies on the interval I, whose exact solutions of Eq. (4) are now plane waves,

*g*-coefficient \(g\left( _{-}|^{+}\right) \)

^{7}As a result, the influence from the first interval \({\mathrm {I}}\) appears only as next-to-leading order corrections, which means that we can study pair creation by the inverse square decreasing electric field (41) rather than by the inverse square field (1) with \(eE\tau _{1}^{2}\) obeying the conditions (40), in leading-order approximation. Therefore, without loss of generality, we shall study particle creation by the field (41). Note that from the property of the differential mean numbers \(N_{n}^{\mathrm {cr}}\) under the exchanges \(p_{x}\rightleftarrows -p_{x}\) and \(\tau _{1}\rightleftarrows \tau _{2}\), the present discussion can be easily generalized to a configuration in which the field is arbitrary during the first interval \({\mathrm {I}}\) but sharp during the second interval \( {\mathrm {II}}\).

*E*considered in Sect. 3.1. As before, we set \( {\mathbf {p}}_{\perp }=0\) and select the system in which \(\hslash =c=m=1\).

According to the graphs above, the mean number of pairs created \(N_{n}^{ \mathrm {cr}}\) tends to the uniform distribution \(e^{-\pi \lambda }\) as \(\tau _{2}\) increases. This is not unexpected since the inverse square decreasing electric field (41) tends to a constant field in the limit \(\tau _{2}\rightarrow \infty \); hence the exact mean numbers should approach to the uniform distribution as \(\tau _{2}\) increases. Moreover, for \(\tau _{2}\) fixed, the mean numbers approach to the uniform distribution as the amplitude *E* increases, as it can be seen comparing the results from Fig. 4 with those of Fig. 5. This is related with the extend of the dimensionless parameter \(eE\tau _{2}^{2}\) and its comparison to the threshold value \(\max \left( 1,m^{2}/eE\right) \): the greater the parameter \( eE\tau _{2}^{2}\) is in comparison to \(\max \left( 1,m^{2}/eE\right) \), the closer the mean numbers \(N_{n}^{\mathrm {cr}}\) approach to the uniform distribution \(e^{-\pi \lambda }\), which is characteristic to constant electric fields (or a *T*-constant electric field varying slowly in time).

For \(p_{x}\) sufficiently large, the exact results agree with the asymptotic approximation given by Eq. (47), as it can be observed comparing solid and dashed lines. This is a consequence of the fact that there are values of finite longitudinal kinetic momentum \(\pi _{2}\) (\(p_{x}\) finite, range \(\left( {\tilde{c}}\right) \)) in which the mean numbers tend to the asymptotic forms (47) in slowly varying regime. On the other hand, in the range of sufficiently small \(p_{x}\) (or sufficiently large \(\pi _{2}\)), there are deviations between the exact mean numbers and the asymptotic approximations. Such deviations are expected and usually occurs in the range of small \(p_{x}\), as in the case inverse square electric field (1), displayed in Figs. 2 and 3, or peak electric field [29], displayed in Fig. 4 of this reference. We conclude that the approximation of slowly varying regime does not apply uniformly throughout all values of \(p_{x}\) for values of \(eE\tau _{2}^{2}\) considered in the plots above. To be applicable uniformly, larger values of parameters are needed.

The most striking feature of the results displayed above is the presence of oscillations, an absent feature in the case of the inverse square electric field (1); compare Figs. 2, 3 with 4. These oscillations are consequences of an “abrupt” switching on process near \(t=0\) and frequently occurs in these cases, as reported recently by us in [31]. In this work, oscillations around the uniform distribution were found and discussed for the case of a *T*-constant electric field (that switches-on and off “abruptly” at definite time instants) and an electric field composed by independent intervals, one exponentially increasing, another constant over the duration *T* and a third one exponentially decreasing. This is an universal feature of “abrupt” switching on or off processes. Moreover, comparing the results displayed in Figs. 4 and 5 we conclude that the oscillations decrease in magnitude as the parameter \( eE\tau _{2}^{2}\) increases. As a result, the mean numbers are expected to become “rectangular” in the limit \(eE\tau _{2}^{2}\rightarrow \infty \).

## 5 Switching on and off by inverse square electric fields

As an application of the above results, we consider in this section an electric field of special configuration in which inverse square increasing and decreasing electric fields simulate switching on and off processes. This consideration allow us to compare effects with recent results [31], in which a composite electric field of similar form was regarded to study the influence of switching on and off processes in the vacuum.

^{8}

*g*-coefficients:

*T*-constant field in the slowly varying regime [10, 30], we see if the parameters satisfy

*g*-coefficients (57) according to the definition (16). Hence, in what follows we present mean numbers \(N_{n}^{\mathrm {cr}}\) of pairs created from the vacuum by the composite field (51) as a function of the longitudinal momentum \(p_{x}\) for some values of the parameters \(\sqrt{eE}\tau _{j}\) and \(\sqrt{eE}T\). Moreover, in order to compare switching on and off effects with an another composite electric field [31]

*T*-constant field [10, 30] (in which switching on and off processes are absent) we include, in each graph below, mean numbers of pairs created by the field (62) and the

*T*-constant field for some values of the parameters \(\sqrt{eE}k_{j}^{-1}\) and \(\sqrt{eE}T\). As in the previous sections, we set \({\mathbf {p}} _{\perp }=0\) and select the system in which \(\hslash =c=m=1\).

According to the graphs above, the differential mean numbers oscillate around the uniform distribution \(e^{-\pi \lambda }\), irrespective the electric field in consideration. This is consistent to asymptotic predictions for the *T*-constant field, in the sense that the differential mean numbers \(N_{n}^{\mathrm {cr}}\) stabilizes to the uniform distribution \( e^{-\pi \lambda }\) as soon as \(\sqrt{eE}T\) is sufficiently larger than the characteristic values \(\max \left( 1,m^{2}/eE\right) \). Thus, the larger the value of \(\sqrt{eE}T\), the smaller the magnitude of the oscillations. This explains why oscillations are larger in Fig. 6 in which \(\sqrt{eE} T=5\) in comparison to the ones in Fig. 9, in which \(\sqrt{eE}T=10 \sqrt{3}\). Moreover, one can see that the magnitude of oscillations decrease if a constant field is accompanied by switching on and off processes; compare solid and dashed lines. This decrease in the amplitude of the oscillations is a consequence of smoother switching on and off processes. In the case of the composite field (51), the mean numbers are approximated given by the first and third lines of Eq. (60) while the composite field (62), \(N_{n}^{\mathrm {cr}}\approx \exp \left( -2\pi \Xi _{1}^{-}\right) \) for \(p_{x}/\sqrt{eE}\le -\sqrt{eE}T/2\) and \(N_{n}^{\mathrm {cr}}\approx \exp \left( -2\pi \Xi _{2}^{+}\right) \) for \( p_{x}/\sqrt{eE}\ge \sqrt{eE}T/2\), in which \(\Xi _{j}^{\pm }=k_{j}^{-1}\left( \sqrt{{\tilde{\Pi }}_{j}^{2}+\pi _{\perp }^{2}}\pm \tilde{\Pi }_{j}\right) \), \({\tilde{\Pi }}_{j}=p_{x}-\left( -1\right) ^{j}eEk_{j}^{-1}\left( 1+k_{j}T/2\right) \). Accordingly, the exact mean numbers oscillate around these approximations, whose amplitudes decrease as \( eE\tau _{j}^{2}\), \(eEk_{j}^{-2}\) increases. At last, but not least, we see that the mean numbers of pairs created by the composite field (62) oscillate around the uniform distribution less than by the composite field (51), given the same longitudinal kinetic momentum increment of both switching on and off processes, for all values of the parameters under consideration. Based on the values chosen for the parameters, we conclude that the slowly varying regime provides a better approximation to the composite field (62) than for the field (51). However, assuming the same value for *E* for both composite fields, it is clear that for \(\tau \) sufficiently larger than \(k^{-1}\) (that is, longitudinal kinetic momentum increment of the inverse square fields is larger than one of exponential fields), the opposite situation occurs. The composite electric field (51) and its peculiarities supply our previous studies [31] on the role of switching on and off processes in the vacuum instability.

## 6 Some concluding remarks

In addition to few known exactly solvable cases in QED with external backgrounds, an inverse square electric field represents one more example where nonperturbative calculations of particle creation effect can be performed exactly. We have presented in detail consistent calculations of zero order quantum effects in the inverse square electric field as well as in a composite electric field of a special configuration, in which the inverse square electric field simulates switching on and off processes. In all these cases we find corresponding in and out exact solutions of the Dirac and Klein–Gordon equations. Using these solutions, we calculate differential mean numbers \(N_{n}^{\mathrm {cr}}\) of Fermions and Bosons created from the vacuum. Differential quantities are considered both exactly and approximately (within the slowly varying regime). In the second case, we studied these distributions as functions on the particle momenta, establishing ranges of dominant contributions and finding corresponding asymptotic representations. In order to be able to compare visually approximate results with exact ones, we compute and analyze plots of differential mean numbers \(N_{n}^{\mathrm {cr}}\) as functions of \(p_{x}\) for some values to the pulse durations \(\tau _{j}\) and for electric field magnitude *E* equal to the Schwinger’s critical value. The asymptotic representations agree substantially with exact results as the pulse durations increase. Using the asymptotic representations for differential quantities, we compute the total number \(N^{\mathrm {cr}}\) of created pairs the probability \(P_{v}\) for the vacuum remain the vacuum. The results are consistent with universal estimates in the locally constant field approximation. Moreover, comparing the results with dominant densities of pairs created by the *T*-constant and peak electric fields, we derive an effective time duration of the inverse square electric field and establish relations by which they are equivalent in pair production effect. Assuming that the peak and the inverse square electric fields act on the vacuum over the same effective time, we relate both fields and conclude that the relation between their pulses varies as \(k^{-1}\le \tau \le 2k^{-1}\), for any amplitude *E* or space-time dimensions *d*.

To complete the pictures, we consider in Sect. 4 the case of an asymmetrical configuration, in which the field presents a sharp pulse for \( t<0\). In the limit \(\tau _{1}\rightarrow 0\) the corresponding *g* -coefficients are consistent with *g*-coefficients calculated in the symmetric case. Analyzing plots of exact calculations, we see that the mean numbers oscillate around their asymptotic approximate values in contrast to the symmetric case were such oscillations are absent; compare Figs. 2, 3 with Fig. 4. These oscillations are attributed to the asymmetrical time dependence of the electric field or, in other words, to the existence of an “abrupt” switching on process near \(t=0\). Moreover, this feature does not depend on the form of external electric field, they can be observed in other cases, for instance in *T*-constant electric field (see Figs. 6, 7, 8, 9). Thus, we may conclude that the oscillations are universal features of “abrupt” switching on or off processes.

Considering an electric field composed by three parts, two of which are represented by inverse square fields, we calculate relevant *g*-coefficients for particle creation and discuss approximate expressions for differential quantities. To understand better switching on and off effects, we compare the above case with the case where switching on and off configurations have exponential behavior. Doing this we consider a configuration in which the duration *T* of the intermediate *T* -constant electric fields is greater than the duration of the characteristic pulses \(\tau _{j}\) and \(k_{j}^{-1}\). This configuration allows us to analyze how the differential distributions differ from their asymptotic form \(e^{-\pi \lambda }\). According to Figs. 6, 7, 8 and 9, we conclude that the way of switching on and off is essential for application of slowly varying regime approximation. For example, comparing results in the *T*-constant electric field (dashed lines) for Fermions with ones for composite fields (solid lines) in Fig. 8, we see they are close to results obtained in the slow variation approximation if parameters of composite fields satisfy the condition \(\sqrt{ eE}T\ge 5\sqrt{3}\), \(\sqrt{eE}\tau =\sqrt{eE}k^{-1}=1.\) At the same time, in the case of a *T*-constant field with \(\sqrt{eE}T=5\sqrt{3}\) it is not true and the corresponding mean numbers \(N_{n}^{\mathrm {cr}}\) deviate substantially from the uniform distribution \(e^{-\pi \lambda }\). For Bosons, one can see that composite fields with \(\sqrt{eE}T=5\sqrt{3}\), \(\sqrt{eE} \tau =\sqrt{eE}k^{-1}=1\) does not allow application of the slow variation approximation, whereas the condition \(\sqrt{eE}T\ge 5\sqrt{3}\) is close to the threshold condition for composite fields for Fermions. One can also see that differential quantities are quite sensitive to the form of switching on and off. For all configurations displayed in Figs. 6, 7, 8 and 9, we see that exponential switching on and off causes smaller oscillations around the uniform distribution in comparison to the inverse square switching on and off.

## Footnotes

- 1.
\(\psi (x)\) is a \(2^{[d/2]}\)-component spinor ([

*d*/ 2] stands for the integer part of*d*/ 2),*m*denotes the electron mass and \(\gamma ^{\mu }\) are Dirac matrices in*d*dimensions. We use the relativistic units \(\hslash =c=1\), in which the fine structure constant is \( \alpha =e^{2}/\hslash c=e^{2}\). - 2.
Hereafter, the index \(j=\left( 1,2\right) \) distinguish quantities associated with the first interval I \(\left( j=1\right) \) from the second interval II \(\left( j=2\right) \).

- 3.
Originally, Whittaker [45, 46] wrote this asymptotic form for a different domain in the

*z*-complex plane, namely \(\left| \arg z\right| \le \pi -0^{+}\), by expanding the binomial inside of his integral representation for \(W_{\kappa ,\mu }\left( z\right) \). However, as discussed in [48], the domain changes to \(\left| \arg \left( z\right) \right| \le 3\pi /2-0^{+}\) by rotating the path of integration over an angle near \(\pi /2\) in any direction. - 4.
We conveniently introduce an auxiliary constant \(\kappa \) to extend results to scalar QED, in which \(\kappa =-1\). It should not be confused with the parameters of the Whittaker functions \(\kappa _{j}\), defined by Eq. (7).

- 5.
Here and in what follows, we use the symbol “\(\approx \) ” to denote an asymptotic relation truncated in leading-order approximation, under the understanding that the condition (18) is satisfied.

- 6.
In Eq. (26), the sum over the quantum numbers \({\mathbf {p}}\) was transformed into an integral and the total number of spin polarizations \( J_{\left( d\right) }=2^{\left[ d/2\right] -1}\) factorizes out from the density, since \(N_{n}^{\mathrm {cr}}\) does not depend on spin variables.

- 7.
A similar demonstration can be carried out for the Klein–Gordon case.

- 8.

## Notes

### Acknowledgements

The authors acknowledge support from Tomsk State University Competitiveness Improvement Program and the partial support from the Russian Foundation for Basic Research (RFBR), under the project no. 18-02-00149. T.C.A. also thanks the Advanced Talents Development Program of the Hebei University, project no. 801260201271, for the partial support. D.M.G. is also supported by the Grant no. 2016/03319-6, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), and permanently by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.

## References

- 1.J. Schwinger, Phys. Rev.
**82**, 664 (1951)ADSMathSciNetCrossRefGoogle Scholar - 2.A.I. Nikishov, Zh. Eksp. Teor. Fiz.
**57**, 1210 (1969) [Transl. Sov. Phys. JETP**30**, 660 (1970)]Google Scholar - 3.A.I. Nikishov, Quantum electrodynamics of phenomena in intense fields, in
*Proceedings of P.N. Lebedev Physical Institute*, vol. 111 (Nauka, Moscow, 1979), p. 153Google Scholar - 4.D.M. Gitman, J. Phys. A
**10**, 2007 (1977)ADSCrossRefGoogle Scholar - 5.E.S. Fradkin, D.M. Gitman, Fortschr. Phys.
**29**, 381 (1981)CrossRefGoogle Scholar - 6.E.S. Fradkin, D.M. Gitman, S.M. Shvartsman,
*Quantum Electrodynamics with Unstable Vacuum*(Springer, Berlin, 1991)CrossRefGoogle Scholar - 7.W. Greiner, B. Müller, J. Rafelsky,
*Quantum Electrodynamics of Strong Fields*(Springer, Berlin, 1985)CrossRefGoogle Scholar - 8.N.D. Birrell, P.C.W. Davies,
*Quantum Fields in Curved Space*(Cambridge University Press, Cambridge, 1982)CrossRefGoogle Scholar - 9.A.A. Grib, S.G. Mamaev, V.M. Mostepanenko,
*Vacuum Quantum Effects in Strong Fields*(Friedmann Laboratory Publishing, St. Petersburg, 1994)Google Scholar - 10.S.P. Gavrilov, D.M. Gitman, Phys. Rev. D
**53**, 7162 (1996)ADSCrossRefGoogle Scholar - 11.S.P. Gavrilov, D.M. Gitman, J.L. Tomazelli, Nucl. Phys. B
**795**, 645 (2008)ADSCrossRefGoogle Scholar - 12.R. Ruffini, G. Vereshchagin, S. Xue, Phys. Rep.
**487**, 1 (2010)ADSCrossRefGoogle Scholar - 13.F. Gelis, N. Tanji, Prog. Part. Nucl. Phys.
**87**, 1 (2016)ADSCrossRefGoogle Scholar - 14.S.P. Gavrilov, D.M. Gitman, Phys. Rev. D
**93**, 045002 (2016)ADSMathSciNetCrossRefGoogle Scholar - 15.S.P. Gavrilov, D.M. Gitman, Phys. Rev. D
**95**, 076013 (2017)ADSCrossRefGoogle Scholar - 16.G.V. Dunne, Eur. Phys. J. D
**55**, 327 (2009)ADSCrossRefGoogle Scholar - 17.A. Di Piazza, C. Müller, K.Z. Hatsagortsyan, C.H. Keitel, Rev. Mod. Phys.
**84**, 1177 (2012)ADSCrossRefGoogle Scholar - 18.G. Mourou, T. Tajima, Eur. Phys. J. Spec. Top.
**223**, 979 (2014)CrossRefGoogle Scholar - 19.G.V. Dunne, Eur. Phys. J. Spec. Top.
**223**, 1055 (2014)CrossRefGoogle Scholar - 20.B.M. Hegelich, G. Mourou, J. Rafelski, Eur. Phys. J. Spec. Top.
**223**, 1093 (2014)CrossRefGoogle Scholar - 21.D. Das Sarma, S. Adam, E.H. Hwang, E. Rossi, Rev. Mod. Phys.
**83**, 407 (2011)ADSCrossRefGoogle Scholar - 22.O. Vafek, A. Vishwanath, Annu. Rev. Condens. Matter Phys.
**5**, 83 (2014)ADSCrossRefGoogle Scholar - 23.N.B. Narozhny, A.I. Nikishov, Yad. Fiz.
**11**, 1072 (1970) [Transl. Sov. J. Nucl. Phys. (USA)**11**, 596 (1970)]Google Scholar - 24.V.G. Bagrov, D.M. Gitman, S.M. Shvartsman, Zh. Eksp. Teor. Fiz.
**68**, 392 (1975) [Transl. Sov. Phys. JETP**41**, 191 (1975)]Google Scholar - 25.N.B. Narozhny, A.I. Nikishov, Sov. Phys. JETP
**38**, 427 (1974)ADSGoogle Scholar - 26.V.M. Mostepanenko, V.M. Frolov, Sov. J. Nucl. Phys. (USA)
**19**, 451 (1974)Google Scholar - 27.T.C. Adorno, S.P. Gavrilov, D.M. Gitman, Phys. Scr.
**90**, 074005 (2015)ADSCrossRefGoogle Scholar - 28.T.C. Adorno, S.P. Gavrilov, D.M. Gitman, Eur. Phys. J. C
**76**, 447 (2016)ADSCrossRefGoogle Scholar - 29.T.C. Adorno, R. Ferreira, S.P. Gavrilov, D.M. Gitman, Russ. Phys. J.
**60**, 417 (2017)CrossRefGoogle Scholar - 30.T.C. Adorno, S.P. Gavrilov, D.M. Gitman, Int. J. Mod. Phys. A
**32**, 1750105 (2017)ADSCrossRefGoogle Scholar - 31.T.C. Adorno, R. Ferreira, S.P. Gavrilov, D.M. Gitman, Int. J. Mod. Phys.
**33**, 1850060 (2018)ADSCrossRefGoogle Scholar - 32.S.P. Gavrilov, D.M. Gitman, N. Yokomizo, Phys. Rev. D
**86**, 125022 (2012)ADSCrossRefGoogle Scholar - 33.N.D. Birrell, J. Phys. A Math. Gen.
**12**, 337 (1979)ADSMathSciNetCrossRefGoogle Scholar - 34.J. Garriga, Phys. Rev. D
**49**, 6327 (1994)ADSMathSciNetCrossRefGoogle Scholar - 35.S. Haouat, R. Chekireb, Phys. Rev. D
**87**, 088501 (2013)ADSCrossRefGoogle Scholar - 36.R.-G. Cai, S.P. Kim, JHEP
**9**, 072 (2014)ADSCrossRefGoogle Scholar - 37.M.B. Fröb et al., JCAP
**04**, 009 (2014)ADSMathSciNetCrossRefGoogle Scholar - 38.T. Kobayashi, N. Ashfordi, JHEP
**10**, 166 (2014)ADSCrossRefGoogle Scholar - 39.C. Stahl, E. Strobel, S.-S. Xue, Phys. Rev. D
**93**, 025004 (2016)ADSMathSciNetCrossRefGoogle Scholar - 40.E. Bavarsad, C. Stahl, S.-S. Xue, Phys. Rev. D
**94**, 104011 (2016)ADSMathSciNetCrossRefGoogle Scholar - 41.E. Bavarsad, S.P. Kim, C. Stahl, S.-S. Xue, Phys. Rev. D
**97**, 025017 (2018)ADSCrossRefGoogle Scholar - 42.T. Hayashinaka, T. Fujita, J. Yokoyama, JCAP
**07**, 010 (2016)ADSCrossRefGoogle Scholar - 43.R. Sharma, S. Singh, Phys. Rev. D
**96**, 025012 (2017)ADSMathSciNetCrossRefGoogle Scholar - 44.F. Karbstein, Phys. Rev. D
**95**, 076015 (2017)ADSMathSciNetCrossRefGoogle Scholar - 45.E.T. Whittaker, Bull. Am. Math. Soc.
**10**, 125 (1903)CrossRefGoogle Scholar - 46.E.T. Whittaker, G.N. Watson,
*A Course of Modern Analysis*, 4th edn. (Cambridge University Press, Cambridge, 1950)zbMATHGoogle Scholar - 47.A. Erdelyi et al. (ed.),
*Higher Transcendental Functions (Bateman Manuscript Project)*, vols. 1 and 2 (McGraw-Hill, New York, 1953)Google Scholar - 48.H. Buchholz,
*The Confluent Hypergeometric Function with Special Emphasis on Its Applications*(Springer, New York, 1969)CrossRefGoogle Scholar - 49.NIST Digital Library of Mathematical Functions. Version 1.0.16. http://dlmf.nist.gov/,2015-08-07
- 50.G. Dunne, T. Hall, Phys. Rev. D
**58**, 105022 (1998)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}