Confining Dinstanton background in an external electric field
Abstract
Using holography, we discuss the effects of an external static electric field on the D3/Dinstanton theory at zerotemperature, which is a quasiconfining theory, with confined quarks and deconfined gluons. We introduce the quarks to the theory by embedding a probe D7brane in the gravity side, and turn on an appropriate U(1) gauge field on the flavor brane to describe the electric field. Studying the embedding of the D7brane for different values of the electric field, instanton density and quark masses, we thoroughly explore the possible phases of the system. We find two critical points in our considerations. We show that beside the usual critical electric field present in deconfined theories, there exists another critical field, with smaller value, below which no quark pairs even the ones with zero mass are produced and thus the electric current is zero in this (insulator) phase. At the same point, the chiral symmetry, spontaneously broken due to the gluon condensate, is restored which shows a first order phase transition. Finally, we obtain the full decay rate calculating the imaginary part of the DBI action of the probe brane and find that it becomes nonzero only when the critical value of the electric field is reached.
1 Introduction
The use of the AdS/CFT correspondence and in general gauge/gravity duality [1, 2, 3, 4, 5] have led to major advances in understanding different aspects of stronglycoupled gauge theories. An important example is how such systems respond to external electromagnetic fields. Among other things, this study can be useful to understand the confinement/deconfinement phase transition in the RHIC and LHC experiments, where strong electromagnetic fields are created due to heavy ion collisions [6].
It is known that an external electric field can make the vacuum unstable against the production of particleantiparticle pairs. This phenomenon known as the Schwinger effect is ubiquitous in any quantum field theory coupled with a U(1) gauge field and is also generalized to nonAbelian gauge theories such as the quantum chromodynamics (QCD). The Schwinger effect has been evaluated first under weakcoupling and weakfield conditions [7] and then generalized to the case of arbitrary coupling [8]. These calculations show that beyond a critical electric field \(E_c\) the vacuum decays catastrophically, i.e., the potential barrier for the pairs vanishes and they are produced without any obstruction. The value obtained for \(E_c\) is far beyond the weakfield approximation. However, the existence of such a critical electric field was verified later by studying the Schwinger effect beyond the weakfield approximation using the AdS/CFT correspondence [9], motivated by its connection to the string theory which predicts the existence of an upper critical electric field [10, 11]. Since then, there has been a growing interest in studying the holographic Schwinger effect in different situations, and in both deconfined and confined phases [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In these problems, they consider a probe D3brane in an intermediate position in the bulk to describe the quarks of finite mass.
The study of the Schwinger effect in confining backgrounds reveals that there exists another critical electric field, \(E_s\), below which no pairs even the massless ones can be produced. For deconfining backgrounds \(E_s=0\), i.e., the pair production occurs in the presence of any nonzero external electric field.
In [23] we employed a quasiconfining gauge theory to elaborate more on the response of the theories with confinement to external electric fields. We chose the D3/Dinstanton background first suggested in [24], where they consider the near horizon limit of the Dinstanton charge uniformly distributed on the D3brane at zero temperature. This geometry is dual to the \(\mathcal{N}=4\) super YangMills theory with constant gluon condensate. This theory is called “quasiconfining”, since it has been verified using AdS/CFT [24] that it is partially confining with confined quarks and deconfined gluons. The quark–antiquark potential in this theory is found to be linearly rising for large distances [25], showing the confinement of the quarks. At finite temperature, however, this theory becomes deconfined [25, 26], i.e., the transition to the deconfined phase occurs at \(T=0\), unlike the QCD. In [23] we investigated this theory at both zero and nonzero temperatures and found the critical electric fields \(E_c\) and \(E_s\). As expected, \(E_s\) is only nonzero for the zerotemperature case, i.e., the confined case.
The present paper is devoted to considering the effect of the electric field on this confining theory using a completely different method. Our approach here is to introduce the fundamental quarks to the theory by embedding a probe D7brane in the dual gravity background. To describe the external electric field, we turn on a nontrivial U(1) gauge field in an appropriate direction of the D7brane worldvolume and find the possible solutions of the DBI action. Using this setup, we are able to investigate the behavior of our confining system under external electric fields and explore its possible phases. Our calculations show that the quark condensate can be regarded as an order parameter of phase transitions due to the presence of the electric field. We moreover compare the results with those of [23]. We also evaluate the rate of the decay caused by the presence of the electric field on the theory at zero temperature. This can be done by calculating the imaginary part of the effective DBI action of the probe brane, which can be served as the effective Lagrangian of the theory, using the gauge/gravity duality (see for example [27, 28]). We should mention here that using a D7brane probing N D3branes , one can study the full decay rate due to turning on the electric field. In contrast, in [23] and the previous similar calculations where a D3brane is probing N D3branes, the studies are only in the Coulomb branch with the gauge group \(U(N+1)\) spontaneously broken to \(U(1)\times U(N)\) and contain just the leading exponent corresponding to the onshell action of the instanton.
In the next section we consider solutions of a D7 probe brane embedded in the D3/Dinstanton theory and briefly discuss about the chiral symmetry in this system. Then, we turn on an external electric field in Sect. 3 and consider in detail the response of the system to this field. Section 4 is devoted to the calculation of the imaginary part of the effective Lagrangian for the massless case. We finally summarize and draw some conclusions in Sect. 5.
2 Review on probe D7brane in Dinstanton background
In this section we present the setup of a D7 probe brane in the background geometry of our interest and quickly review the effect of the gluon condensate on the chiral symmetry of the system.
3 Response to an external electric field
3.1 Phase transitions under the effect of the electric field
Now, let us elaborate further on the behavior of the quark condensate of the system under the changes of the electric field. In the left graph of Fig. 4 we draw the quark condensate versus the electric field for a fixed q and massless quarks. We can observe that by increasing the value of the electric field, the condensate for massless quarks decreases and right when the electric field approaches the critical electric field it suddenly vanishes, that is the chiral symmetry is restored in the conducting phase, as stated before.

Insulator phase (\(E<E_s\)): As stated above, for \(E<E_s\) there is no singular region and therefore only one kind of embedding (usually called Minkowski embedding) is possible for all the quark masses. Hence, the electric current is zero although the electric field is present. The existence of such a critical electric field, below which no pairs even the ones of zero mass can be produced, is a common feature of confining theories.
In the left graph of Fig. 5 we depict the embeddings corresponding to quarks of different masses for given values of q and E (where \(E<E_s\)). The right graph shows the quark condensate as a function of the quark mass for the same values of q and E. As can be seen, even the embedding for the zeromass quark is nonflat and nontrivial, indicating a nonzero condensation, as is also obvious from the right graph. This shows the spontaneous breaking of the chiral symmetry in this phase. Note that the graph of the condensate versus the quark mass shows no jump or special behavior for \(E<E_s\). Thus, we find no phase transition in the insulator phase, in contrast to the result given in [29]. They keep the Chern–Simons term in the DBI action of the D7brane. This leads to two results different from ours. According to their calculations the flat embedding is preferred and chiral symmetry is not broken in this phase. This result is not expected, since the zero mode of the fermions in the Dinstanton background requests chiral symmetry breaking. They also find a phase transition at some value of the quark mass, for which there is no reasonable and convincing physical interpretation. As remarked before, the Chern–Simons term is shown [26] to become locally total derivative and consequently it cannot cancel the effect of the dilaton.

Conductor phase (\(E>E_s\)): For \(E>E_s\) there exists a vanishing radius in the bulk and hence two kinds of embeddings are possible, depending on the corresponding quark mass. The solutions for large enough quark masses do not cross the vanishing shell and the electric current is zero, that is although we are in the conductor phase, the repulsive force due to the electric field is not enough to overcome the color force between these heavy quarks. However, lighter quarks can be produced. Solutions for these quarks reach the vanishing shell and the electric current for them is not zero.^{2} A sample of the solutions for given values of q and E in the conductor phase is represented in Fig. 6. The left graph shows some of the solutions with nonzero (zero) electric currents, which reach (do not reach) the singular shell. In the right graph the quark condensate versus the quark mass is shown. As can be seen, near the transition between the two types of the solutions, a multivaluedness of the condensate is observed. This result implies that there happens a first order phase transition, which is the aforementioned transition occurred at \(E_c\) (see the right graph of Fig. 4).
4 Vacuum instability for the massless quarks
It is expected that an abrupt application of an electric field to a deconfined gauge theory causes its vacuum to become unstable. In confined QCDlike theories, however, one expects a threshold electric field for the quarks even the massless ones be liberated and the Schwinger effect starts. This is so, since a competing force is needed to overcome the confining force between quarks and antiquarks in a meson bound state. It seems interesting to check this problem for the confined theory studied here.
5 Summary and conclusion
The response of a confining field theory with nonzero gluon condensate to a background electric field has been considered using holography. The theory of our interest is a quasiconfining theory with confined quarks and deconfined gluons at zero temperature and becomes deconfined even for a small nonzero temperature. Adding a probe D7brane to the gravity side, we have introduced the quarks to the theory and switched on a nontrivial U(1) gauge field on the brane to describe the electric field in the dual field theory.
Changing the value of the electric field while keeping the other parameters fixed, we found two critical points in this theory. One of them occurs when the electric field equals the string tension, i.e., the electric repulsion force between the quarks overcomes the confining color force between them. We observe that below this value, called \(E_s\), no quark pairs of any masses are produced and therefore the electric current is zero. Above this value the production of the quarks becomes possible. Therefore, there is an insulator/conductor phase transition. Moreover, another transition occurs at exactly the same point, which is the chiral symmetry restoration. The theory at zero electric field has nonzero condensate even for zeromass quarks, since instantons tend to repel the D7brane from the origin. Increasing the electric field from zero, decreases the condensate, since the electric field acts exactly in the opposite way. In fact, the phase transition is the result of the interplay between these two effects. When the electric field reaches its critical value, the chiral condensate for zeromass quarks abruptly jumps to zero, showing a first order phase transition.
The other phase transition occurs at a higher value of the electric field \(E_c\). This one depends on the mass of the quarks as well as the instanton density of the system. The phase transition at this point is revealed by studying the condensate for each quark mass as the electric field changes. The behavior of this order parameter shows a first order phase transition at \(E_c\). Below this value for each quark mass, the production of these quarks is forbidden. However, lighter quarks can be produced, provided \(E>E_s\).
In [23], we found critical electric fields of the same system using a different approach based on the string worldsheet. Calculating the total potential of quark–antiquark pairs of a given mass, we obtained two critical electric fields with an exactly same description as the ones obtained here using the DBI action. We found the values of \(E_c\) obtained from two approaches, however, to be different, while \(E_s\)s are exactly the same. This result seems to be reasonable, since the string worldsheet and DBI action are different approximations of the string theory. Some particular \(\alpha '\) corrections are ignored in the DBI action and only a part of corrections are kept. However, worldsheet calculations could capture different parts of the \(\alpha '\) corrections. They only may coincide in specific BPS cases or when some hidden symmetry is present, which are not the case here. Therefore, the difference between the values of the critical electric field is indeed expected. Since \(E_s\) is determined by the string tension, it is found to be the same using two approaches. However, \(E_c\) as we found in our considerations, depends on the details of the theory and the quark mass, and thus it can be altered by the approximations that are used.
We have completed the study of the holographic Schwinger effect for our theory by calculating the full decay rate due to the onset of the instability caused by the presence of the external electric field. This has been done by calculating the imaginary part of the effective DBI action of the probe brane by taking the solution to the embedding equation obtained in the absence of the electric field, as the probe brane configuration. As expected from the other parts of our study, the decay rate grows from zero only after the critical value of the electric field, \(E_s\), is reached. Also, the decay rate is found to be independent of the gluon condensate as the electric field becomes large enough.
The study of quantum quenches in the context of quantum field theories is an important and challenging area of research. Since there is lack of other theoretical techniques to deal with this problem, holographic techniques have been vastly used to investigate the time evolution of systems driven out of equilibrium under quantum quenches. Recently there has been a growing interest in studying the quantum quenches in the confined phase (see e.g. [31, 32, 33]), where they have found some evidence that the equilibrium (formation of a black hole in the gravity side) may not be the final fate of a system driven far from equilibrium by applying a quench. In this regard, it would be interesting to study the evolution of a confining theory like the one in the present paper under electric field quenches, as an important class of quenches. We would follow this path using appropriate numerical techniques [18, 34, 35] in our future work.
Footnotes
 1.
In [25], they keep the Chern–Simons term in their calculations, which leads to a completely different result. According to their calculations, the Chern–Simons term cancels the effect of the dilaton, resulting in flat D7brane embedding and no chiral symmetry breaking.
 2.
Strictly speaking, there are two types of these solutions for the D7brane; those that satisfy \(w(\rho =0)\ne 0\), which have a conical singularity and those with \(w(\rho =0)=0\), which reach the origin without any singularity. These solutions are explained in many other cases, and first addressed in [30].
Notes
Acknowledgements
The authors would like to thank K. Hashimoto and R. Meyer for useful discussions. The work of Farid Charmchi has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM) under research project No. 1/602562.
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