# Dynamically assisted Schwinger effect at strong coupling with its holographic extension

## Abstract

At strong-coupling and weak-field limit, the scalar Schwinger effect is studied by the field-theoretical method of worldline instantons for dynamic fields of single-pulse and sinusoidal types. By examining the Wilson loop along the closed instanton path, corrections to the results obtained from weak-coupling approximations are discovered. They show that this part of contribution for production rate becomes dominant as Keldysh parameter increases, it makes the consideration at strong coupling turn out to be indispensable for dynamic fields. Moreover a breaking of weak-field condition similar to constant field also happens around the critical field, defined as a point of vacuum cascade. In order to make certain whether the vacuum cascade occurs beyond the weak-field condition, following Semenoff and Zarembo’s proposal, the Schwinger effects of dynamic fields are studied with an \({\mathcal {N}}=4\) supersymmetric Yang–Mills theory in the Coulomb phase. With the help of the gauge/gravity duality, the vacuum decay rate is evaluated by the string action with instanton worldline as boundary, which is located on a probe D3–brane. The corresponding classical worldsheets are estimated by perturbing the integrable case of a constant field.

## 1 Introduction

The extreme light infrastructure (ELI) is designed to produce the highest power and intense laser worldwide [1, 2]. It has a potential of reaching ultra-relativistic intensities, challenging the Schwinger limit \(E_\text {s} :=m^2/e \approx 1.32\times 10^{18} \,\mathrm { V\, m^{-1}}\). As laser field approaches this value, vacuum becomes unstable, and a large amount of charged particles produces in pairs, so that laser loses the energy, and its intensity stays within the upper limit. However, Schwinger effect is not only a phenomenon in electromagnetism, but a universal aspect of quantum vacuum in the presence of a \({\mathsf {U}}(1)\) gauge field with a classical background, see e.g. [3, 4, 5, 6, 7].

The pair-production rate in the constant electric field had been pioneered by Sauter, Heisenberg and Euler [8, 9], and the corresponding *Effect* was named after Schwinger [10], who did the calculation based on field-theoretical approaches, see e.g. [11, 12] for a current review. A semiclassical approach called worldline instantons has been introduced more recently to the study of constant and inhomogeneous fields in the small-coupling and weak-field approximations [13, 14], where the production rate, in Wick-rotated Euclidean space, is represented by a worldline path integral. The so-called *worldline instantons* are the periodic saddle points relevant for a calculation of integral by the steepest descent method. The extension of inhomogeneous fields originates from more practical purpose. As analysed in [14], 1-D dynamic electric fields reduce the critical value of Schwinger effect, such that pair production from vacuum is more close to experimentally observable conditions.

The Schwinger effect at arbitrary coupling in constant field has also been studied in [13] at the weak-field condition, which is considered originally in order to overcome some obstacles from direct application of Schwinger’s approach. However the later observation (e.g. [15]) found that the weak-field condition is broken around the critical field, defined as a point of vacuum cascade, such that the mechanism in weak-field condition loses its prediction for this vacuum phenomenon. Inspired by the similar existence of critical value of electric field in string theory, it is of a possibility to clarify the vacuum cascade in the Coulomb phase beyond the weak-field condition with the help of gauge/gravity duality [7, 16, 17]. This is also known as the Semenoff–Zarembo construction, where the production rate has been obtained by calculating the classical action of a bosonic string, which is attached to a probe D3–brane and coupled to a Kalb–Ramond field.

Since the instanton action in the production rate is equivalent to the string action, which is proportional to the area, calculation of the rate is related to integrating the classical equations of motion for bosonic string in the given external field. In other words, duality converts the problem to evaluating the area of a *minimal surface* [18, 19] in Euclidean \(\mathrm {AdS}_3\), the boundary of which is assumed to be the trajectory of the worldline on the probe D3–brane. In mathematics, such a Dirichlet problem is known as the *Plateau’s problem* [20].

In this work, we consider the scalar pair production in a dynamic external field of single-pulse and sinusoidal types at strong coupling and weak-field limit based on the method of worldline instantons, which is explained in Sects. 2 and 3. We first show by a field-theoretical approach, that besides the enhancement due to the dynamics of electric fields, a further contribution to the production rate arises from the Wilson loop, and it becomes dominant in production rate as Keldysh parameter increases. However, such a correction seems to diverge as Keldysh adiabaticity parameter \(\gamma \rightarrow \infty \) based on our estimated formula; and it leads to a contradiction to the weak-field condition, so that near the critical field, the method itself breaks down, and the prediction of a vacuum cascade becomes unclear. To overcome the problem of breaking weak-field condition, we then follow Semenoff and Zarembo’s proposal in Sect. 4, applying the gauge/gravity duality to the Schwinger effect in the Coulomb phase of an \({\mathcal {N}}=4\) supersymmetric Yang–Mills theory. The classical solution of the corresponding string worldsheets are estimated by perturbing the solvable case of a constant external electric field.

## 2 Worldline instantons at strong coupling

*weak-field condition*

*W*

*C*is along the 2D trajectory of worldline instanton. In the Feynman gauge of the \({\mathsf {U}}(1)\) field, the Wilson loop becomes [24]

## 3 Wilson loops along worldline instanton paths

*x*approaches

*y*. A regulator \(\varepsilon \) has been introduced in [25], such that Eq. (9) becomes

*s*and

*t*can be understood as angular coordinates for the instanton.

*a*is defined in the same way as in [14]. Hence up to second order of

*t*, one has

*subtraction term*

*perimeter law*, depicting the behaviour of the Wilson loop in Euclidean space [24, ch. 82]. Since \({x'}\mathopen {}\left( s\right) \mathclose {}^2 =a^2\) is independent of integration variable, the subtraction term can be worked out as

In our practice with the dynamic fields, \({\mathcal {A}}_\varepsilon \) has yet to be worked out in a closed form, and its estimation is to be discussed in Sect. 3.1.

### 3.1 Estimation of \({\mathcal {A}}_\varepsilon \) and \({\mathcal {A}}_\text {phy}\)

*t*-

*expansion*up to second order of

*t*, which follows straightforwardly from the separation of divergent term. This method belongs to rational approximation. Furthermore in our application in dynamic fields, this expansion can be worked out in a closed form easily. Also note the \(-4\) term in Eq. (22), which will be mentioned again later with example. The validity of

*t*-expansion is closely related to the uniform convergence of integrand, and demonstrated in Appendix A. If one uses diagonal Padé approximant for integrand rather than

*t*-expansion, the convergence is obvious [26, 27, 28].

Yet another way of estimating \({\mathcal {A}}_\text {phy}\) is numerical integration, in which the regulator \(\varepsilon \) is still needed. There are polynomial contributions of \(\varepsilon \) in the bare term \({\mathcal {A}}_\varepsilon \), and one might think taking a small \(\varepsilon \) would give a good result. However, \({\mathcal {A}}_\varepsilon \) and the counter term \(\updelta \!{\mathcal {A}}\) both diverges like \(\varepsilon ^{-1}\) as \(\varepsilon \rightarrow 0\). A small \(\varepsilon \) leads to a numerically dissatisfying operation, in which two big numbers cancels, yielding a small result and a great loss of significance. This problem becomes catastrophic for the dynamic fields when \(\gamma \rightarrow 0^+\). In order to overcome the potentially catastrophic cancellation, we use linear extrapolation near \(\varepsilon = 0\), in which for each \(\gamma \), \({\mathcal {A}}_\text {reg}\) is numerically calculated for several different values of \(\varepsilon \). The limit of \(\varepsilon \rightarrow 0\) is then obtained by linearly extrapolate the series of results with respect to \(\varepsilon \). In this approach, the error due to extrapolation can also be obtained by estimation of the parameters in linear regression. Furthermore, in our application the numerator and denominator in Eq. (13) scales as \(\gamma ^{-2}\) when \(\gamma \rightarrow +\infty \), so that for a fixed \(\varepsilon \), at large \(\gamma \) the regularised integrand is dominated by the regulator on the denominator. This is overcome by scaling \(\varepsilon \) accordingly, such that the subtraction term in Eq. (13) remains constant with respect to \(\gamma \).

### 3.2 Constant electric field

### 3.3 Single-pulse field \(E(t)=E {{\,\mathrm{sech}\,}}^2(\omega t)\)

*enhancement factor*with respect to the case of a constant external field in Eq. (30). This name of \(\lambda (\gamma )\) is seen to be appropriate from the fact that \(\lambda (\gamma )\) is a monotonically decreasing function and tends to unit at adiabatic limit \(\gamma \rightarrow 0\).

*t*-expansion becomes

*t*-expansion, which is caused by accuracy of estimation method, thus it approaches to zero as the approximation order increases, see Fig. 1. In other words, the emergence of finite terms are expected for each orders, and the higher-order contribution should cancel \(2/\uppi ^2\) from the 2nd-order. Based on this consideration we remove \(2/\uppi ^2\) derivation directly in the final results, which should not be confused with the counter term Eq. (19).

*t*-expansion up to \({O}\mathopen {}\left( t^2\right) \mathclose {}\), Eq. (13) reads

*t*-expansion. If the critical field is defined as saddle point, at which the exponential suppression is precisely zero, then one could have

*E*its contribution for production rate becomes dominant as \(\gamma \) increases. In addition, both estimation Eq. (41) (or Eq. (39)) and numerical result seem to be divergent as \(\gamma \) approaches \(+\infty \), even if the pre-exponential factor of Feynman integral were taken into account [30]. It implies that Wilson loop ought to be of a pole at \(\gamma \rightarrow +\infty \), and loses its meaning at this point, where the instanton trajectory collapses to a singular point.

### 3.4 Sinusoidal field \(E(t)=E \cos (\omega t)\)

*t*-expansion reads

From above three examples, one may note that, first, the weak-field condition in the non-perturbative ranges \(\gamma <1\) is inevitably broken at strong coupling, which makes the vacuum cascade around the critical field ambiguous [15]; and second, the correction due to the Wilson loop in dynamic fields is a monotonically increasing function with respect to \(\gamma \) and diverges as \(\gamma \rightarrow \infty \).

## 4 Holographic Schwinger effect with dynamic field

In order to answer the question, if the vacuum cascade for strong coupling happens as the strength of time-dependent field goes close to the critical limit [15], we consider a similar effect in the context of gauge/gravity duality, where the gauge field theory refers to an \({\mathcal {N}}=4\) \({{\mathsf {S}}}{{\mathsf {U}}}(N+1)\) supersymmetric Yang–Mills theory on the 4D boundary of an \(\mathrm {AdS}_5\times S^5\) space, and the quantum gravity is a type IIB superstring theory in the bulk of the \(\mathrm {AdS}_5\times S^5\). The same as the case with constant field, we expect that the string theory could shed some light on the catastrophic vacuum cascade through the duality principle.

In the Semenoff–Zarembo construction, the worldsheet ends on the probe D3–brane with a boundary, taking the same shape as the worldline instanton. Hence the essential problem is converted to compute the on-shell action of string in Euclidean \(\mathrm {AdS}_5\) with the given boundary. Note that the Nambu–Gotō action is proportional to the worldsheet area, and extremising the area leads to a minimal surface. In other words, calculation of the exponential factor now corresponds to a Plateau’s problem in the framework of gauge/gravity duality.

### 4.1 Constant electric field

*J*is the Jacobian. Therefore, the orientation of the Kalb–Ramond coupling is also reversed. The Nambu–Gotō action in our parameterisation becomes

*R*can be fixed by extremising the total action, yielding

### 4.2 Estimation of single-pulse field \(E(t)=E {{\,\mathrm{sech}\,}}^2(\omega t)\)

*u*, i.e.

*r*. The simplicity of Semenoff–Zarembo construction for constant field led us to speculate that the similar production rates could have been obtained by repeating above procedure. However, it is not anything like worldline instanton, the string worldsheets are not exactly integrable for dynamic fields in our cases. Thus to make an effect estimation, we expand the instanton at adiabatic limit, i.e. \(\gamma \rightarrow 0\),

*r*component can be estimated by noting that

### 4.3 Estimation of sinusoidal field \(E(t)=E \cos (\omega t)\)

## 5 Conclusion and discussion

In this paper, the scalar Schwinger effect for dynamic fields at strong coupling and weak-field limit has been studied, by first using the field-theoretical method of worldline instantons. A non-trivial contribution to the production rate is discovered by evaluating the Wilson loop along the instanton path, which depends on the Keldysh adiabaticity parameter \(\gamma \). Thus one may expect that such correction may save the weak-field condition in strong coupling. However after computations, we find that the introduction of the correction term also leads to a contradiction to the weak-field condition near the critical field strength.

We note also that the correction from Wilson loop is a monotonically increasing function with respect to \(\gamma \), which makes the contribution for production rate from Wilson loop become dominant as \(\gamma \) increases. Moreover both *t*-expansion and numerical calculation suggest a divergent value as \(\gamma \) approaches infinity, even if the pre-exponential factor of Feynman integral is considered. One possible explanation is that the Wilson loop loses its meaning at \(\gamma \rightarrow \infty \), because the instanton trajectory collapses to a singular point.

In order to clarify the vacuum cascade beyond the weak-field condition, in the context of an \({\mathcal {N}}=4\) supersymmetric Yang–Mills theory, the production rate is calculated by the gauge/gravity duality, according to which the instanton action has a string counterpart of the classical string action in Euclidean \(\mathrm {AdS}_3\), where the boundary on the probe D3–brane is given by the instanton path. Thus the problem is converted to solving the classical motion of string with Dirichlet boundary conditions. However the string worldsheets for dynamic fields are not integrable as in the worldline instantons. To provide an explicit estimation, we treat the specific worldsheets as perturbations around the one with circle boundary, which had been solved exactly. Such an expansion is an adiabatic approximation, it is practical and realistic, because only low-frequency laser (comparing with electron mass) is currently operational. The obtained decay rates in the two examples with dynamic fields are similar concave functions as in the case with constant field, but the critical fields increase considerably. In other words, up to \(O(\gamma ^2)\) the correction due to the dynamics of electric field suppresses the pair production, which is opposite to cases of worldline instantons.

## Notes

### Acknowledgements

Y.-F.W. is grateful to Claus Kiefer and Nick Kwidzinski (Cologne), Chao Li (Princeton) and Ziping Rao (Vienna). H.G. is supported by ELI–ALPS project, co-financed by the European Union and the European Regional Development Fund No. GINOP-2.3.6-15-2015-00001. Y.-F.W. is supported by the Bonn-Cologne Graduate School for Physics and Astronomy (BCGS).

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