# Electromagnetic instability and Schwinger effect in the Witten–Sakai–Sugimoto model with D0–D4 background

## Abstract

Using the Witten–Sakai–Sugimoto model in the D0–D4 background, we holographically compute the vacuum decay rate of the Schwinger effect in this model. Our calculation contains the influence of the D0-brane density which could be identified as the \(\theta \) angle or chiral potential in QCD. Under the strong electromagnetic fields, the instability appears due to the creation of quark–antiquark pairs and the associated decay rate can be obtained by evaluating the imaginary part of the effective Euler–Heisenberg action which is identified as the action of the probe brane with a constant electromagnetic field. In the bubble D0–D4 configuration, we find the decay rate decreases when the \(\theta \) angle increases since the vacuum becomes heavier in the present of the glue condensate in this system. And the decay rate matches to the result in the black D0–D4 configuration at zero temperature limit according to our calculations. In this sense, the Hawking–Page transition of this model could be consistently interpreted as the confined/deconfined phase transition. Additionally there is another instability from the D0-brane itself in this system and we suggest that this instability reflects to the vacuum decay triggered by the \(\theta \) angle as it is known in the \(\theta \)-dependent QCD.

## 1 Introduction

Recent years, there have been many advances in the researches on the strong electromagnetic field, especially in heavy-ion collision since it is expected that an extremely strong magnetic field is generated by the collision of the charged particles. In particular, the Schwinger effect should be one of the most interesting phenomena in the heavy-ion collision, because the pair creation of charged particles from the vacuum occurs under such an externally strong electromagnetic field. In the Schwinger effect, the creation rate of a pair of charged particles could be obtained by evaluating the imaginary part of Euler–Heisenberg Lagrangian [1, 2]. However, the result implies the Schwinger effect is a non-perturbative effect which shows up only under the strong electromagnetic field.

Although it is still challenging to evaluate the Schwinger effect under the electromagnetic field, the framework of gauge/gravity duality or AdS/CFT provides a powerful tool on studying the strongly coupled quantum field theories [3, 4, 5, 6]. It has been recognized that a (d+2)-dimensional classical gravity theory could correspond to a (d+1)-dimensional gauge theory as a weak/strong duality. So with this framework, various applications of studying the Schwinger effect holographically have been presented [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Particularly, in some top-down holographic approaches, e.g., the D3/D7 approach, since the dynamics of the flavors is described by the action of the probe flavor brane, the associated decay rate to the Schwinger effect could be evaluated by using the flavored action. Hence it implies this action could be identified as the holographic Euler–Heisenberg action and the creation rate of flavored quark–antiquark pairs (i.e., the vacuum decay rate) can be computed by the imaginary part of this action [14, 15, 16]. While this is a different method, it allows us to quantitatively explore the electromagnetic instability and Schwinger effect in holography.

*F*is the gauge field strength. While the experimental value of \(\theta \) is small, the Chern–Simons term leads to many observable phenomena such as chiral anomaly [19], chiral magnetic effect [20], deconfinement transition [21, 22] and effects of gluon condensate. Accordingly, to investigate the Schwinger effect with \(\theta \)-dependence in QCD would be significant since the creation rate of quark–antiquark pairs is affected by the \(\theta \) angle. So in this paper, we are motivated to study the electromagnetic instability and the Schwinger effect with such a topological term in holography.

^{1}However in the black D0–D4 configuration, the physical interpretation of D0-brane is less clear since D0-brane is not D-instanton. Nevertheless, we might identify the D0 charge as the chiral potential in the black D0–D4 configuration according to the phenomenal evidences presented in Refs. [38, 39, 40]. Besides, the flavors can be introduced by a stack of \(N_{f}\) D8 and anti-D8 branes (\(\mathrm {D}8/\overline{\mathrm {D}8}\)-branes) as probes into the D0–D4 background. Hence the chirally symmetric or broken phase of the dual field theory is represented by the various configurations of \(\mathrm {D}8/\overline{\mathrm {D}8}\)-branes in the bubble or black D0–D4 background.

In this paper, we will focus on the derivation of the effective Euler–Heisenberg Lagrangian first, then we could explore the electromagnetic instability and evaluate the creation rate of quark-antiquark pairs in the vacuum. The paper is organized as follows, in Sect. 2, we review the Witten–Sakai–Sugimoto model in the D0–D4 background with more details. In Sect. 3, we derive the the effective Euler–Heisenberg Lagrangian from the probe \(\mathrm {D}8/\overline{\mathrm {D}8}\)-branes action in the bubble and black D0–D4 background respectively. In Sect. 4, we evaluate the creation rate of quark–antiquark pairs in the vacuum. In the black D0–D4 background, we find the creation rate is finite at zero temperature limit which qualitatively coincides with the results from the bubble D0–D4 background. In this sense, we suggest that the Hawking–Page transition of this model is suitable to be identified as the confined/deconfined phase transition. And our numerical calculation shows the creation rate decreases when the D0 density (\(\theta \) angle) increases in the bubble D0–D4 case. This may be interpreted as that the vacuum becomes heavier due to the gluon condensate described by this model in terms of quantum field theory. The final section is the summary and discussion.

## 2 Review of the Witten–Sakai–Sugimoto model in the D0–D4 background

*R*of the bulk are given as follows,

*U*is the holographic radial direction, and \(U\rightarrow \infty \) corresponds to the boundary of the bulk. Therefore coordinate

*U*takes the values in the region \(U_{KK}\le U\le \infty \). In order to avoid a possible singularity at \(U=U_{KK}\), the coordinate

*U*satisfies the following periodic boundary condition [25],

^{2}

^{3}The various configurations of \(\mathrm {D}8/\overline{\mathrm {D}8}\)-branes are shown in Figs. 1 and 2.

## 3 Euler–Heisenberg Lagrangian of the D0–D4/D8 brane system

In this section, we are going to derive the Euler–Heisenberg Lagrangian in bubble and black D0–D4 brane background respectively. Then we can investigate the Schwinger effect or creation of quark–antiquark.

### 3.1 The bubble D0–D4 geometry

^{4}

^{5}For simplicity, we need to consider a single flavor \(N_{f}=1\) for the presence of an external electromagnetic field. We further require the electromagnetic field is non-dynamical (i.e., constant) and their components on the \(S^{4}\) are zero. Without losing generality, we could turn on the electric field on the \(X^{1}\) direction only and the magnetic fields could be introduced in \(X^{1},X^{2},X^{3}\) directions since the \(X^{i},\ i=1,2,3\) spacial space is rotationally symmetric. Inserting constant electromagnetic field with the induced metric (12) into the DBI action (14), the effective Lagrangian is obtained as,

^{6}

*d*and the current

*j*reads,

### 3.2 The black D0–D4 geometry

*U*, while \(X^{4}\) is a constant for parallel configuration. Then similar as done in the bubble case, we could obtain the following Euler–Heisenberg Lagrangian once the induced metic (21) and the constant electromagnetic fields are adopted,

*d*and current

*j*defined as,

## 4 Holographic pair creation of quark–antiquark

### 4.1 Imaginary part of the effective action at zero temperature

*U*: \(U_{0}\le U\le U_{*}\) , the Lagrangian is imaginary as shown in Fig. 3.

^{7}Moreover, we interestingly find that (39) may show an additionally possible instability because the second line of (39) could also be imaginary and it does not depend on the electromagnetic field. Since the bubble D0–D4/D8 system corresponds to a confining Yang–Mills theory with a topological Chern–Simons term, so the instability produced by \(\theta \) angle (i.e., D0 charge) could be holographically interpreted as the transition between the different \(\theta \) vacuum states, which has been very well-known in QCD. However in order to investigate the electromagnetic instability, we need to remove the \(\theta \)-instability from the vacuum. Hence we expand (39) by \(\zeta \) since \(\theta \) angle in QCD is very small. Therefore in the antipodal case (i.e., \(x_{4}^{\prime }=0\)) with small \(\zeta \) expansion, we obtain the following formula for the imaginary Lagrangian,

We furthermore look at the dependence on *E* with various \(\zeta \) in the case of a parallel/perpendicular magnetic field respectively. The numerical evaluation is summarized in Fig. 6. Accordingly, we could conclude that, in the bubble D0–D4 system, the electromagnetic instability is suppressed by the appearance of D0 charge (\(\theta \) angle in QCD), however with a fixed \(\zeta \), its behavior depends on the direction of the magnetic field relative to the electric field. Finally, we also plot the relation between the electric field with an arbitrary magnetic field to confirm our conclusion which is shown in Fig. 7.

### 4.2 Imaginary part of the action at finite temperature

*U*-integral is finite. Accordingly, in the presence of the electromagnetic field, the vacuum decay rate is finite at strong coupling in the zero temperature limit in our D0–D4/D8 system or the Witten–Sakai–Sugimoto model. Then let us compute it in details. Introducing the dimensionless variables as,

*Y*(or

*y*) in the neighborhood \(Y_{*}\simeq Y_{0}+\varepsilon \) where \(\varepsilon \ll Y_{0}\). Since in the connected configuration of the \(\mathrm {D}8/\overline{\mathrm {D}8}\)-branes, \(x_{4}\) is a function of

*y*, we obtain the following behavior of the imaginary Lagrangian in the small temperature limit \(\chi \rightarrow 0\),

## 5 Summary and discussion

In this paper, we have studied the electromagnetic instability by deriving the effective Euler–Heisenberg Lagrangian for the flavored quarks in the Witten–Sakai–Sugimoto model with the D0–D4 background. Since the dynamics of the flavored quarks is described by the DBI action of the probe \(\mathrm {D8/{\overline{D8}}}\)-branes, we identify its DBI action with the constant electromagnetic fields as the effective Euler–Heisenberg action. Then we explore the electromagnetic instability and evaluated the pair creation rate of quark–antiquark in the Schwinger effect. With the D0–D4/D8 model in string theory, our investigation contains the influence of the D0-brane density which could be interpreted as the \(\theta \) angle or chiral potential in QCD. In the bubble configuration, since the D4-branes with smeared D0-branes are wrapped on a cycle, it introduces a confining scale into this system. Therefore, we obtain a very different result from \({\mathcal {N}}=2\) supersymmetric QCD in the approach of D3/D7 [15, 16].

In order to investigate the electromagnetic instability, we assume the electromagnetic field is sufficient strong, then we find the \(\theta \)-dependent creation rate of flavored quark–antiquark obtained in the bubble D0–D4 background exactly coincides with [14] if setting \(\theta \) angle or \(\zeta =0\) (i.e. no D0-branes). Our numerical calculation also shows the creation rate decreases when \(\theta \) angle or \(\zeta \) increases and its behavior depends on the direction of the magnetic field relative to the electric field. To understand this, let us combine our results with [26, 28, 29, 30]. Since the critical electric field evaluated in (37) is in quantitative agreement with the mass spectrum in Refs. [26, 28, 29, 30] which describes the possible metastable states in the heavy-ion collision, our results imply that in the heavy-ion collision the metastable state could be created in the Schwinger effect then it soon decays to the true vacuum as discussed in Refs. [41, 42]. And because of the condensate of gluon, these metastable states become heavier, so the critical electric field increases while its associated decay rate decreases in the present of D0-branes, i.e., \(\theta \) angle.

Moreover, the creation rate in the black D0–D4 background has also been computed which remains to be finite while it is oppose to the D3/D7 approach [15, 16]. Nevertheless, our result would be significant. Since the Hawking-Page transition in the WSS model is usually interpreted as confined/deconfined phase transition in QCD [32, 36], it means the observables in the deconfined phase should return to confined case if the temperature goes to zero. So our results supports this statements qualitatively because it illustrates the creation rate obtained in the black D0–D4 background (at finite temperature) returns to the result from the bubble D0–D4 background (zero temperature).

In addition, if turning off the electromagnetic fields, the effective action (39) and (43) remains to include a vacuum instability in the present of D0-branes. We suggest that this vacuum instability might holographically describe the decay of the vacuum with various winding numbers triggered by the \(\theta \) angle or instantons in QCD since the D0-branes relates to the \(\theta \) angle thus could be identified as instantons.

However, there might be some issues concerning the instability of our holographic setup in this paper. For examples, first, our numerical calculations are all based on the small \(\theta \) angle or \(\zeta \) expansion, so it is natural to ask what if we keep all the orders of \(\theta \) angle or \(\zeta \)? How the electromagnetic and vacuum instability would be affected? Second, the electromagnetic field is non-dynamical in our calculations. So what about a dynamical case? Unfortunately, as opposed to the situation of the black D0–D4 background, it seems impossible to introduce an electric current directly in the bubble D0–D4 configuration since the flavor branes never end in the bulk. To solve this problem, one may need a baryon vertex. But it is less clear whether or not such a baryon vertex could be created. We leave these issues to a future study.

## Footnotes

- 1.
In [37], an alternative solution for the deconfinement phase has been proposed.

- 2.
In the black D0–D4 solution, we replace \(U_{KK}\) by \(U_{T}\) in the solution.

- 3.
- 4.
The low energy effective action of a D-brane consists of two parts: Dirac–Born–Infeld (DBI) action plus Chern–Simons (CS) term. In this paper, we do not need to consider the Chern–Simons term since it has nothing to do with the electromagnetic instability.

- 5.
For the antipodal case, we need to choose \(U_{0}=U_{KK}\).

- 6.
When both electric and magnetic fields are turned on, the Chern–Simons term in the action contributes to the equations of motion. It turns out that the equations of motion for \(A_{U}\) implies the chiral anomaly. For simplicity, we can ignore this anomaly effect by interpreting our outcome as the physical values measured at \(t=0\), at which \(A_{U}\) vanishes as an initial condition.

- 7.
In some limit, the contribution from \(x_{4}^{\prime }\) to the effective Lagrangian is not important. For example, if \(\varepsilon \rightarrow 0\) in (32), we obtain \(X^{4}\left( U_{0}+\varepsilon \right) \simeq X^{4}\left( U_{0}\right) \). Therefore in this limit, \(x_{4}^{\prime }\) does not contribute to the effective Lagrangian (32).

## Notes

### Acknowledgements

This work is inspired by our previous works [28, 29, 30, 31] in USTC, and [43] from our colleagues. We would like to thank Chao Wu and Shi Pu for valuable comments and discussions. Si-wen Li is supported by the research startup funds of Dalian Maritime University under Grant No. 02502608, the Fundamental Research Funds for the Central Universities under Grant No. 017192608 and partially by the National Natural Science Foundation of China under the Grant no. 11535012. Wenhe Cai is supported by the National Natural Science Foundation of China under the Grant no. 11805117.

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