Holographic Schwinger Effect in Anisotropic Media
aa r X i v : . [ h e p - t h ] J a n Holographic Schwinger Effect in Anisotropic Media
Jing Zhou ∗ and Jialun Ping † Department of Physics, Nanjing Normal University, Nanjing, Jiangsu 210097, China
Abstract
Using the guage/gravity correspondence, we discuss the holographic Schwinger effect inanisotropic backgrond. First of all, we compute the separating length of the particle-antiparticlepairs at different anisotropic background which is specified by dynamical exponent ν with theisotropic case is ν = 1. Then it is found that the maximum separating length x decreases with theincreasing of dynamical exponent ν . This can be regarded as the virtual particles become real onesmore easily. Subsequently, we find that the potential barrier is reduced by dynamical exponent ν ,warp factor coefficient c and chemical potential µ at small distance. Moreover, we also find thecritical electric field is reduced by the chemical potential and dynamical exponent, but enhancedby the warp factor coefficient. PACS numbers: 11.25.Tq, 25.75.Nq ∗ † [email protected]; corresponding author . INTRODUCTION The pair production of electron and positron under a strong external electric field isknown as Schwinger effect [1]. However, this phenomenon is not restricted to QED, but hasa general properties of vacuum in the quantum field theory, which contains virtual particle.In the Schwinger effect, the production rate of particle-antiparticle pair was obtained bysolving the Euler-Heisenberg Lagrangian and was extended to the case of weak-field laterby Affleck-Alvarez-Manton [2] Γ ∼ exp (cid:18) − πm eE + e (cid:19) , (1)where m is the mass of particle, e is the charge, and E represents the the external electric-field. Then one can find that there exist a critical field at eE c = (4 π/e ) m . In fact, thepotential energy of virtual particles can be the real particles is, V ( x ) = 2 m − Ex − α s x , (2)where the first term is particle and anti-particle mass, the second term is the energy of theexternal electric field and the third one is the Coulomb potential of the particles. So, one canfind that the the potential barrier decreases with the increasing electric field, and vanishesat critical electric field E c , where the system is unstable.However, this is not the whole story. One can also apply the Ads/CFT [3–6] to study theSchwinger effect and the critical electric field, known as holographic Schwinger effect[7, 8].This method was formally proposed in Ref.[8], and they study the particles produced inthe N = 4 super-Yang Mills theory, which is dual to the D3 brane with a probe D3-braneplaced at finite radial position in the bulk AdS space. So, one can find that the mass of theparticles is finite, which also can avoid particles production suppressed in this Schwingereffect. Then at large N and strong t’Hooft coupling limit, the production rate of particleswith mass m can be estimate asΓ ∼ exp − √ λ r E c E − r EE c ! , (3)and the critical electric-field can be written as E c = 2 πm √ λ , (4)2hich is consistent with the Dirac-Born-Infeld(DBI) result.Here, we focus on the Schwinger effect in a holographic setup with potential analysis,the particle-antiparticle pairs creation describing as tunneling process. In order to be realparticle, a pair of virtual particle-antiparticle need to obtain energy from the strong externalelectric field. The potential analysis provide new insight to study the Schwinger effect withevaluating the vacuum expectation value((VEV) of the rectangular Wilson loop instead ofcircular Wilson loop. Later, some related work have been carried out by this holographicmethod. For example, the Schwinger effect in the magnetized background was studied inRef.[9]. The potential analysis with gluon condense effect was analysed in Ref.[10]. Theholographic Schwinger effect in the WSS model with D0-D4 background is discussed inRef.[11]. The potential analysis with a moving D3-brane is also investigated in Ref.[12].Further studies can be found in Refs.[13–22].It is believed that the so called quark-gluon plasma has been created in the RHIC and LHC[23–26]. Moreover, it may indicate that the QGP is in the anisotropic state which is about0.04 fm/c after the initial collision[27]. Some results of lattice QCD with anisotropic has beenput forword in Refs.[28, 29]. However, here, we mainly focus on the anisotropic QGP withholographic setup which is different from the lattice results. The static holographic modelshave reproduced some main properties of the real QGP. Then it is natural to ask how theseproperties are performed in anisotropic backgrounds, which may give different views on thereal QGP. This can also be considered as a promotion of the original D3 brane backgroud.Actually, some interesting results have been carried out. For example, the themodynamicsand instabelities of anisotropic plasma is discussed in Ref.[30]. In Refs.[31, 32], they studythe jet quenching and drag force in anisotropic plasma. Thermal photon production in theanisotropic plasma is also researched in Ref.[33]. The quarkonium dissociation is discussedin Ref.[34]. Other important work, see [35–42].In this work, we consider the holographic Schwinger effect in the anisotropic 5- dimen-sional Einstein-dilaton-two-Maxwell system [43]. And the anisotropic background is specifiedby dynamical exponent ν with the isotropic case is ν = 1. In particular, the anisotropicbackground discloses a more abundant structure than that in the isotropic case with thesmall/large black holes phase transition [44]. The paper is organized as follows. In Sec. II ,we introduce the 5- dimensional Einstein-dilaton-two-Maxwell system. In Sec. III, we mainlyfocus on the potential analysis in anisotropic background. In Sec. IV, we study the potential3nalysis in finite chemical potential and different warp factor coefficient. Further discussionand conclusion can be found in Sec. V. II. BACKGROUND GEOMETRY
The 5-dimensional Einstein-dilaton-two-Maxwell system was introduced in Ref.[43], andthe action in the Einstein frame is given by S = Z d x πG q − det ( g µν ) (cid:20) R − f ( φ )4 F − f ( φ )4 F − ∂ µ φ∂ µ φ − V ( φ ) (cid:21) , (5)where F is Maxwell field with field strength tensor is F (1) µν = ∂ µ A ν − ∂ ν A µ , and F is theother Maxwell field with field strength tensor is F (2) µν = qdy ∧ dy . f ( φ ), f ( φ ) are thegauge 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 ds = L b ( z ) z (cid:20) − g ( z ) dt + dx + z − ν (cid:0) dy + dy (cid:1) + dz g ( z ) (cid:21) (6) φ = φ ( z ) , A (1) µ = A t ( z ) δ µ (7) F (2) µν = qdy ∧ dy , (8)where b ( z ) = e cz / 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. Andall the quantities are dimensionless units. The function g ( z ) is given as g ( z ) = 1 − z ν z ν h G (cid:0) cz (cid:1) G (cid:0) cz h (cid:1) − µ cz ν e cz h (cid:18) − e cz h (cid:19) G (cid:0) cz (cid:1) + µ cz ν e cz h (cid:18) − e cz h (cid:19) G (cid:0) cz (cid:1) G (cid:0) cz h (cid:1) G (cid:0) cz h (cid:1) , (9)and G ( x ) = ∞ X n =0 ( − n x n n ! (cid:0) n + ν (cid:1) . (10)Then the temperature can be T ( z h , µ, c, ν ) = g ′ ( z h )4 π = e − cz h πz h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G (cid:0) cz h (cid:1) + µ cz ν h e czh (cid:18) − e cz h (cid:19) − e cz h G ( cz h ) G (cid:0) cz h (cid:1) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (11)4 II. POTENTIAL ANALYSIS IN ANISOTROPIC BACKGROUND
The coordinates of the particle pairs can be written as t = τ, y = σ, x = y = 0 , z = z ( σ ) , (12)then the Nambu-Goto action reads S = T F Z dσdτ L = T F Z dσdτ p det g αβ , (13)here T F is the string tension and g αβ is the induced metric. In fact, the lagrangian densitycan be written as L = p det g αβ = b ( z ) z q g ( z ) z − ν + ˙ z . (14) L does not rely on σ , so it must satisfy the follow equation L − ∂ L ∂ ˙ z ˙ z = C. (15)Moreover, the boundary condition gives dzdσ = 0 , z = z c ( z h < z c < z ) , (16)here one should note that the probe D3 brane locates at z = z . So, the conserved quantitycan be evaluated as C = b ( z c ) z c q g ( z c ) z − ν c . (17)Combining Eq. 15 and Eq. 17, one finds˙ z = dzdσ = vuuut g ( z ) z − ν ( b ( z ) g ( z ) z − ν z b ( z c ) g ( z c ) z − νc z c − , (18)then the separating length of the test particle pairs can be obtained by integrating Eq. 18, x = 2 Z z z c dz vuut g ( z ) z − ν ( b ( z )2 g ( z ) z − νz b ( zc )2 g ( zc ) z − νcz c − . (19)The Coulomb potential and static energy are V ( CP + SE ) = 2 T F Z x dσ L (20)= 2 T F Z z z c dz q b ( z ) g ( z ) z − ν z b ( z ) z r b ( z ) g ( z ) z − ν z − b ( z c ) g ( z c ) z − νc z c . (21)5n order to obtain the critical electric field, one should compute the the DBI action of theprobe D3 brane, namely S DBI = − T D Z d x q − det ( G µν + F µν ) . (22)If the electric field is along x direction, then one will find G µv + F µv = − g ( z ) b ( z ) z πα ′ E − πα ′ E b ( z ) z b ( z ) z z z − ν
00 0 0 b ( z ) z z − ν . (23)So, one can rewrite Eq. 22 at z = z as S DBI = − T D Z d x q b ( z ) z − − ν q − b ( z ) g ( z ) + (2 πα ′ z ) E . (24)If the equation has a physical meaning, then we will require − b ( z ) g ( z ) + (cid:0) πα ′ z (cid:1) E ≥ . (25)By simple calculating, one can find that the critical electric field is E c = T F s b ( z ) g ( z ) z . (26)If we define a dimensionless parameter β ≡ EE c , then the total potential of the particle-antiparticle pair will be V tot = V ( CP + SE ) − Ex (27)= 2 T F Z z z c dz q b ( z ) g ( z ) z − ν z b ( z ) z r b ( z ) g ( z ) z − ν z − b ( z c ) g ( z c ) z − νc z c − β Z z z c dz q − b ( z ) g ( z ) z vuut g ( z ) z − ν ( b ( z )2 g ( z ) z − νz b ( zc )2 g ( zc ) z − νcz c − . (28)In this part, we mainly focus on the Schwinger effect in the anisotropic background. Firstof all, we calculate the separating length x which is given by Eq. 19.6 .4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.5 z c x 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 = − .
3. The blue line is ν = 4 .
3, black line is ν = 4 . ν = 4 .
5, respectively.
The dependence of separating length x on z c is shown in Fig. 1. Then one can findthat the U-shape string is unstable at small z c , but stable existence at large z c for differentdynamical exponent ν . From the picture, we can find that the maximum value of x decreaseswith the increasing of dynamical exponent ν . Then it may indicate that the Schwinger effectwill occur easily at larger dynamical exponent ν . - - - - x V t o t ( a ) - - x V t o t ( b ) FIG. 2. (a)The total potential V tot as a function of separating length x for different dynamicalexponent ν . The chemical potential µ = 3, the temperature T = 0 .
58 and the warp factor coefficient c = − .
3. The red line is ν = 4 .
5, black line is ν = 4 . ν = 4 .
3, respectively. (b)The total potential V tot against separating length x at different β . The red line is β = 0 .
4, blackline is β = 0 . β = 1 . Using the Eq. 21, we show the dependence of total potential of particle-antiparticle pair7n the separating length x in the Fig. 2. Then one can find that the potential barrierdecreases with the increasing of external electric-field. Moreover, we find that the potentialbarrier are reduced by dynamical exponent ν at small x , which is consistent with Schwingereffect. In addition, the dynamical exponent ν enlarges the width of the potential barrier andweakens 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, theparticles 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 . ν E c FIG. 3. E c versus dynamical exponent ν . The temperature T = 0 .
58, warp factor coefficient c = 0 . µ = 3. V. POTENTIAL ANALYSIS WITH CHEMICAL POTENTIAL AND WARPFACTOR COEFFICIENT - - - - x V t o t FIG. 4. The total potential V tot as a function of separating length x at difference chemical potential.The temperature T = 0 . β = 0 . c = − .
3. The red line is µ = 3, blackline is µ = 2 .
98 and blue line is µ = 2 .
96, respectively. μ E c FIG. 5. E c versus chemical potential µ when the temperature T = 0 .
58, warp factor coefficient c = 0 . ν = 4 . The effect of the chemical potential on the total potential in different external electric-field is studied in Fig. 4. One can find that the chemical potential weakens the total potentialin small distance x with β = 0 .
8. But the effect of the chemical potential on the width of thepotential barrier is obvious in the large distance x . It is found that the chemical potentialenlarges the width of the potential barrier and weakens the Schwinger effect in large distance9 . Moreover, we can also find that the external critical field is a deceasing function of thechemical potential in Fig. 5 - - - - - x V t o t FIG. 6. The total potential V tot versus separate length x for different warp factor coefficient. Thetemperature T = 0 . β = 0 . µ = 3. The red line is c = − .
3, black lineis c = − . c = − .
1, respectively.
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 heightof the total potential in small distance x with β = 0 .
8. In fact, it also implies that the warpfactor coefficient can enlarge the width of the potential barrier and weaken the Schwingereffect in large distance. We also plot E c versus c in Fig. 7. Surprisingly, we find the criticalfield increases with the increasing of warp factor coefficient, which is different from theprevious two cases. c E c FIG. 7. E c versus warp factor coefficient c . The chemical potential µ = 3, the temperature T = 0 . ν = 4 . . SUMMARY AND CONCLUSIONS In this paper, we study the Schwinger effect in Einstein-dilaton-two-Maxwell-scalar sys-tem with a anisotropic background. The isotropic models reappear the main properties ofthe potential analysis. Then it is natural to study how these properties are changed in theanisotropic one.The separating length of the particle-antiparticle pair in the anisotropic background iscomputed. We find that the separating length decreases with the increasing ν . In this case,the U-shape string is unstable at small z c , but stable existence at large z c . We obtain thecritical electric field E c via the DBI action, and calculate the total potential. Then one canfind the warp factor coefficient, chemical potential and dynamical exponent ν reduce thepotential 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 potentialand dynamical exponent, but enhanced by the warp factor coefficient. In particular, thethe warp factor coefficient, chemical potential also can enlarge the width of the potentialbarrier, and weaken the Schwinger effect at large distance of x .Since the Schwinger effect is an important mechanism to create a plasma of gluons andquarks from initial color-electric flux tubes [45], we hope that the anisotropic backgroundon the Schwinger effect could provide some new insights on the understanding of the QGP. ACKNOWLEDGMENTS
This work is partly supported by the National Natural Science Foundation of China underContract Nos. 11775118, 11535005.
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