Holographic Schwinger Eﬀect in Anisotropic Media

Using the guage/gravity correspondence, we study the holographic Schwinger eﬀect in anisotropic background. First, the separate length of the particle-antiparticle pairs is computed in anisotropic background and the anisotropy of the background is parameterized by dynamical exponent ν . It is found that the maximum separate length x decreases with the increase of dynamical exponent ν which indicates the virtual particles become real ones more easily. Subsequently, we ﬁnd that the potential barrier is reduced by dynamical exponent ν , warp factor coeﬃcient c and chemical potential µ at small separate distance. Moreover, we also ﬁnd the critical electric ﬁeld is reduced by the chemical potential and dynamical exponent, but enhanced by the warp factor coeﬃcient.


I. INTRODUCTION
It is known that the pair production of electron and positron under a strong external electric field is named as Schwinger effect [1].This phenomenon shows a general feature of vacuum instability in the presence of the external field.A qualitative understanding of this phenomenon can be obtained by looking at the potential energy of the pair in the presence of an electric field E [2] V where α s ≃ 1/137 is the fine-structure constant and virtual pairs are separated by a distance x.The pair production is described as a tunneling process which creates a particleantiparticle pair.From above formula, one can find that the potential barrier decreases with the increase of electric field, and vanishes at a certain critical electric field E c .It is instructive to mention the formulas calculated by Schwinger [1] and Affleck-Alvarez-Manton(AAM) [3], for arbitrary coupling and weak field.
Here, e is charge of the particle pairs and m is the mass of particle pairs.Comparing with Schwinger case, one can find that there exists a critical field eE c = (4π/e 2 ) m 2 in the AAM case.Obviously, the critical value does not satisfy the weak-field condition eE ≪ m 2 .
AdS/CFT(Anti-de Sitter/Conformal Field Theory) correspondence [4][5][6][7]provides a way to study the Schwinger effect at the strong coupling and there is no constraint for the values of external fields [8,9].The holographic Schwinger effect was formally proposed in Ref. [9] and they studied the particles produced in the N =4 super-Yang Mills theory which is dual to the N D3-brane with a probe D3-brane placed at finite radial position in the bulk [10].
In the usual studies, the test particles are assumed to be heavy quark limit.To avoid pair creation suppressed by the divergent mass, the location of the probe D3-brane is at finite radial position rather than at the AdS boundary [9,11].Thus, the production rate of particles with mass m can be estimated as and the critical electric-field can be written as which is consistent with the Dirac-Born-Infeld(DBI) result.Following this idea, lots of works have been carried out to study the holographic Schwinger effect [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].In Ref. [11,22], they discussed the holographic Schwinger effect in the extreme condition created in the high energy physics experiment.In addition to high temperature, large chemical potential and strong magnetic field, partonic system generated in an ultra-relativistic heavy-ion collisions cannot be homogeneous and isotropic at the very early time of collision [27].Asymptotic weak-coupling enhances the longitudinal expansion substantially than the radial expansion so the system becomes colder in the longitudinal direction than in the transverse direction [28].Besides, some results of lattice QCD in anisotropic medium has been put forward in Refs.[29,30].
Some interesting results of anisotropy through holographic method have been carried out in recent years.For example, the themodynamics and instabilities of anisotropic plasma is discussed in Ref. [31].In Refs.[32,33], they study the jet quenching and drag force in anisotropic plasma.Thermal photon production in the anisotropic plasma was also researched in Ref. [34].The quarkonium dissociation was discussed in Ref. [35].In particular, the anisotropic background discloses a more abundant structure than that in the isotropic case with the small/large black holes phase transition [36].Other related works can be found in Refs.[37][38][39][40][41][42][43][44].
In this work, we mainly focus on the holographic Schwinger effect in the anisotropic 5-dimensional Einstein-dilaton-two-Maxwell system [45] and the anisotropic background is parameterized by dynamical exponent ν.The remainder of this paper is organized as follows: In Sec.II, we introduce the 5-dimensional Einstein-dilaton-two-Maxwell system.In Sec.III, we mainly focus on the potential analysis in anisotropic background.In Sec.IV, we study the potential analysis in finite chemical potential and different warp factor coefficient.The conclusion can be found in Sec.V.

II. BACKGROUND GEOMETRY
The 5-dimensional Einstein-dilaton-two-Maxwell system was introduced in Ref. [45], which describes a anisotropic background parameterized by the dynamical exponent ν.This back-ground can give the total multiplicity dependence on energy, which is agree with the experimental data [45].And the action in the Einstein frame is given by where F 1 is Maxwell field with field strength tensor is F (1) and F 2 is the other Maxwell field with field strength tensor is F (2) µν = qdy 1 ∧ dy 2 .f 1 (φ), f 2 (φ) are the gauge functions which correspond to the two Maxwell fields.V (φ) is the scalar potential.
The metric ansatz of the black brane solution in the anisotropic background is where b(z) = e cz 2 /2 is the warp factor, and c represents the deviation from conformality.
g(z) is the blackening function.For convenience, we set the AdS radius L to be one.And all the quantities are dimensionless units.Through solving the equation of motion obtained from above action, the function g(z) can be calculated as and Then the temperature can be given as

III. POTENTIAL ANALYSIS IN ANISOTROPIC BACKGROUND
The coordinates of the particle pairs can be written as Here we only choose the y 1 direction.Then the Nambu-Goto action reads where T F is the string tension and g αβ is the induced metric.The Lagrangian density can be written as L does not rely on σ, so it must satisfy the follow equation Moreover, the boundary condition gives here one should note that the probe D3 brane locates at z = z 0 .So, the conserved quantity can be evaluated as Combining Eq. 14 and Eq. 16, one finds then the separating length of the test particle pairs can be obtained by integrating Eq. 17, The Coulomb potential and static energy are In order to obtain the critical electric field, one should compute the the DBI action of the probe D3 brane, namely If the electric field is along x direction, then one will find So, one can rewrite Eq.21 at z = z 0 as If the equation has a physical meaning, then we require By simple calculating, one can find that the critical electric field is If we define a dimensionless parameter β ≡ E Ec , then the total potential of the particleantiparticle pair will be In this part, we mainly focus on the Schwinger effect in the anisotropic background.First of all, we calculate the separating length x which is given by Eq. 18.The dependence of separating length x on z c is shown in Fig. 1.Then one can find that the U-shape string is unstable at small z c , but stable existence at large z c for different values of dynamical exponent ν.From the picture, we can find that the maximum value of x decreases with the increase of dynamical exponent ν.Then it may indicate that the Schwinger effect will occur easily at larger dynamical exponent ν.Using the Eq. 20, we show the dependence of total potential of particle-antiparticle pair on the separating length x in the Fig. 2. Then one can find that the potential barrier decreases with the increasing of external electric-field.Moreover, we find that the potential barrier are reduced by dynamical exponent ν at small x, which is consistent with Schwinger effect.In addition, the dynamical exponent ν enlarges the width of the potential barrier and weakens the Schwinger effect in large distance x.In fact, the potential barrier exists when β < 1, and the particle production is understood as the tunneling process.When β ≥ 1, the particles are easier to create because of the increasing external field.We also plot E c versus ν in Fig. 3. Then one can find that E c decreases with the increasing ν.

IV. POTENTIAL ANALYSIS WITH CHEMICAL POTENTIAL AND WARP FACTOR COEFFICIENT
The effect of the chemical potential on the total potential in different external electric fields is studied in Fig. 4. One can find that the chemical potential weakens the total potential in small distance x with β = 0.8.But the effect of the chemical potential on the width of the potential barrier is obvious in the large distance x.It is found that the chemical potential enlarges the width of the potential barrier and weakens the Schwinger effect in large distance x.Moreover, we can also find that the external critical field is a deceasing function of the chemical potential in Fig. 5 The effect of the warp factor coefficient on the total potential in different external electric-field is plotted in Fig. 6.Then one can find that the warp factor coefficient reduces the height of the total potential in small distance x with β = 0.8.In fact, it also implies that the warp factor coefficient can enlarge the width of the potential barrier and weaken the Schwinger effect in large distance.We also plot E c versus c in Fig. 7.
Surprisingly, we find the critical field increases with the increasing of warp factor coefficient, which is different from the previous two cases.

V. SUMMARY AND CONCLUSIONS
In this paper, we study the Schwinger effect in Einstein-dilaton-two-Maxwell-scalar system in a anisotropic background.The isotropic models can reappear the main properties of QCD.Then it is natural to study how these properties are changed in the anisotropic one.
The separate length of the particle-antiparticle pair in the anisotropic background is computed.We find that the separate length decreases with the increasing ν.In this case, the U-shape string is unstable at small z c , but stable at large z c .We obtain the critical electric field E c via the DBI action, and calculate the total potential.Then one can find the warp factor coefficient, chemical potential and dynamical exponent ν reduce the potential barrier.
This means that they can increase the production rate of the real particle-antiparticle pairs.
We also find the critical electric field is reduced by the chemical potential and dynamical exponent, but enhanced by the warp factor coefficient.In particular, dynamical exponent, the warp factor coefficient and chemical potential also can enlarge the width of the potential barrier, and weaken the Schwinger effect at large distance of x.
Since the Schwinger effect is an important mechanism to create a plasma of gluons and quarks from initial color-electric flux tubes [46], we hope that the Schwinger effect in the anisotropic background could provide some new insights on the understanding of the QGP.
Moreover, the potential analysis in the holographic shock wave model may be worth discussing [47].We leave it in the future work.

FIG. 1 .
FIG. 1.The separating length x as a function of z c .The chemical potential µ = 3, the temperature T = 0.58 and the warp factor coefficient c = −0.3.The blue line is ν = 4.3, black line is ν = 4.4 and red line is ν = 4.5, respectively.

FIG. 2 .
FIG. 2. (a)The total potential V tot as a function of separating length x for different dynamical exponent ν.The chemical potential µ = 3, the temperature T = 0.58 and the warp factor coefficient c = −0.3.The red line is ν = 4.5, black line is ν = 4.4 and blue line is ν = 4.3, respectively.(b) The total potential V tot against separating length x at different β.The red line is β = 0.4, black line is β = 0.6 and blue line is β = 1.1.

FIG. 4 .FIG. 5 .
FIG. 4. The total potential V tot as a function of separating length x at difference chemical potential.The temperature T = 0.58, β = 0.8 and the warp factor c = −0.3.The red line is µ = 3, black line is µ = 2.98 and blue line is µ = 2.96, respectively.

FIG. 6 .FIG. 7 .
FIG. 6.The total potential V tot versus separate length x for different warp factor coefficient.The temperature T = 0.58, β = 0.8 and chemical potential µ = 3.The red line is c = −0.3,black line is c = −0.2 and blue line is c = −0.1,respectively.