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Hindawi Advances in High Energy Physics Volume 2023, Article ID 8685867, 15 pages https://doi.org/10.1155/2023/8685867 Research Article 1,2 Udit Narayan Chowdhury Saha Institute of Nuclear Physics, Block-AF, Sector-1, Salt Lake, Kolkata 700064, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India Correspondence should be addressed to Udit Narayan Chowdhury; chowdhury.udit@gmail.com Received 26 October 2022; Revised 25 February 2023; Accepted 4 April 2023; Published 29 April 2023 Academic Editor: Theocharis Kosmas Copyright © 2023 Udit Narayan Chowdhury. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP . We study pair production of particles in the presence of an external electric field in a large N non-supersymmetric Yang-Mills theory using the holographic duality. The dual geometry we consider is asymptotically AdS and is effectively parametrized by pffiffiffiffiffiffiffi two parameters, u and − 5/2 < δ ≤ 0, both of which can be related to the effective mass of quark/antiquark for non- supersymmetric theories. We numerically calculate the interquark potential profile and the effective potential to study pair production and analytically find out the threshold electric field beyond which one gets catastrophic pair creation by studying rectangular Wilson loops using the holographic method. We also find out the critical electric field from DBI analysis of a probe brane. Our initial investigations reveal that the critical electric field necessary for spontaneous pair production increases or decreases w.r.t. its non-supersymmetric value depending on the parameter δ. Ultimately, we find out the pair production rate of particles in the presence of an external electric field by evaluating circular Wilson loops using perturbative methods. From the later investigation, we note the resemblance with our earlier prediction. However, we also see that for and below another certain value of the parameter δ, the pair production rate of particle/antiparticle pairs blows up as the external electric field is taken to zero. We thus infer that the vacuum of the non-SUSY gauge theory is unstable for a range of non- supersymmetric parameter δ and that the geometry/non-SUSY field theory under consideration has quite different characteristics than earlier reported. 1. Introduction D3 branes has two parameters δ and u , (i.e.., the field the- ory dual is a two-parameter deformation over usual N =4 For the last few decades, the AdS/CFT correspondence [1–4] super Yang-Mills) and has certain features which makes it (relating N =4 superconformal Yang-Mills in 4 space-time an attractive dual for large N QCD studies via holography dimensions to quantum gravity in asymptotic AdS ⊗ S [9]. To the best of our knowledge, an explicit interpretation spaces), and some of its modifications are one of the exem- of these two parameters in terms of dual field theory mea- plary ideas in theoretical physics. AdS/CFT chiefly comes surables lacks till date. It had been previously reported that from black hole thermodynamics [5], and type IIB string [7–9] those include running coupling+confinement in the theory [6] is thus inherently supersymmetric in nature. This infrared and absence of both SUSY and conformal symmetry is a strong-weak duality meaning; strong coupling in the (thus, the two-parameter deformation commented above field theory side corresponds to weak coupling in the quan- violates both SUSY and conformal symmetry). Part of what tum gravity side and vice versa. However even after so many makes AdS/CFT alluring is that when the field theory cou- years, no trace of supersymmetry has been found by experi- pling is high, the corresponding coupling in the quantum ments, and again, conformal symmetry is not found quite gravity side is low, and thus, we are left with classical gravity much in nature. Thus, it is necessary to formulate a modifi- which is easily computable. cation of AdS/CFT without supersymmetry and conformal The coupling constant in field theory is usually used as a symmetry yet respecting its string theory/supergravity ori- perturbative parameter, and observables are expressed in a gins. Such a solution is obtained in [7, 8]. This solution for series w.r.t. this parameter; this is called perturbative field 2 Advances in High Energy Physics theory. However, there are quite some effects in quantum tion does not happen at all. In this work, we want to study field theory which cannot be explained as such, i.e., nonper- the Schwinger effect for non-supersymmetric gauge theories turbative effects. Amongst them, the Schwinger effect stands via holographic methods using both of this methods, our its ground. The vacuum of QED or any gauge theory inter- chief interest being twofold. On the one hand, we like to acting with charged matter is full of virtual particles and see the theoretical effect absence of supersymmetry yields antiparticles (henceforth, qq). In the presence of an external on the value of critical electric field at least for large N electric field/gauge field, these particles get the required Yang-Mills theories (and if such a relation can be reframed energy and become real particles. There is no magic involved to be an indirect experimental evidence towards the presence in this. In realistic situations, the energy of the real qq pairs or absence of supersymmetry in real-world nature). We also is obtained from the electric field. Schwinger calculated [10] like to demonstrate the effect of confinement (as reported the pair production rate for this process in U(1) gauge the- earlier for large N non-supersymmetric YM theories via ory and obtained holography) towards holographic Schwinger particle decay and look into exotic results if any. For our purpose, the vir- tual qq pairs are imagined to be endpoint of a string in the eE ðÞ 2 −πm /eE Γ = e : ð1Þ boundary. We calculate the rectangular Wilson loop in 2π ðÞ space-time direction to find out the interquark potential. To account for an external electric field, we add an extra The exponential suppression hints that pair productions term. We analytically find out the critical electric field from can be modeled as a tunneling process. Assuming that the the same. We also plot figures to illustrate the tunneling phe- virtual qq pair has a separation x, the potential on a virtual nomenon. Next, we move on to finding out the critical elec- quark in the presence on an external electric field is given as tric field from analysis of the DBI action using the fact that the action should be real valued. Then, we move on to find- V = − − eEx +2m: ð2Þ ing out the circular Wilson loop. It is impossible to do so eff without any simplification. We thus expand the expressions to first order of the non-SUSY deformation parameter ðu Þ . Imagine this to be the potential barrier though which the Doing so, we explicitly find out the profile of circular Wilson qq pairs tunnel out in the opposite direction and become Loop up to the first order of ðu Þ from which we find out real. For E < m /eα, there exists two zero points of the poten- 0 the pair production rate. tial, and V is positive for intermediate values of x. That eff This paper is organized as follows: in Section 2, we recap means there is a potential barrier, and quarks have to tunnel non-SUSY D3 branes and their decoupling limit from super- out through them justifying the exponential factor stated 2 gravity. We also show that the non-SUSY solution goes over above. However for E > m /eα, the potential becomes nega- to usual AdS when appropriate limits are taken. In Section 3, tive all along and stops putting up a potential barrier, indi- we show the derivation of pair production in theory with cating a catastrophic instability of vacuum where the qq U(1) gauge field coupled to charged matter. Relevant expres- are produced spontaneously. The value of electric field for sion for large N gauge theory is also given. In Section 4, we which the potential stops putting up a tunneling barrier is carry on potential analysis of virtual qq pairs from which the called “critical/threshold electric field” E . critical electric field is derived both by analytical and numer- The Schwinger effect in holographic setting was first cal- ical means. In Section 5, we use the DBI action and find out culated in [11] (see [12] for an even earlier work) wherein the critical electric field using the fact that the action should the pair modified pair production rate was found to be be real valued. In Section 6, we use perturbative analysis to 2 3 sffiffiffiffiffi ! find out the profile for circular Wilson loop when the string pffiffiffi rffiffiffiffiffi 2 λ E E 2πm ends at a finite position (u ). Using this, we find the critical 4 5 b Γ ~ exp − − , E = pffiffiffi : ð3Þ 2 E E electric field and pair production rate and make some com- ments about the later. We close this paper with conclusions in Section 7. This formula matches with the one above for low electric field (much lower than E ). For field much higher than E , c c we do not see an exponential suppression anymore hinting 2. Non-SUSY Dp Branes and Their at catastrophic decay. The chief idea of this work was to Decoupling Limit place the probe brane at a finite position unlike what is done usually (placing the probe brane at the conformal boundary In this section, we will take a brief recap of non- of AdS) and then to calculate the circular Wilson loop. supersymmetric Dp brane solutions [16] and show how to Another approach was pioneered in [13] which calculated recover the BPS Dp brane solutions from them. Then, we the rectangular Wilson loop for virtual qq pair and relate it will state the decoupling limit of non-SUSY D3 branes by to interquark potential and then find the critical electric field analogy with the BPS case and make sure that the decou- from the same. Holographic Schwinger effect for confining pling goes over to the BPS brane decoupling limit when gauge theories has also been studied in literature [14, 15], SUSY is restored [7]. In addition, we also show by taking and the confinement manifests itself in the presence of suitable coordinate transformation that the decoupled throat another “threshold” electric field, below which pair produc- geometry is actually identical with two-parameter solution Advances in High Energy Physics 3 obtained previously by Constable and Myres in which super- Q has dimensions of four volume, and others are dimension- symmetry and conformal symmetry are both broken [8]. We less. One should further note from (9) that the solution start with the action for ten-dimensional type II supergravity given above has a naked singularity at ρ =0 and the physical which in addition to the string frame metric g has a dila- region is given by ρ >0. In the context of string theory, one μν hopes that quantum fluctuations modify the behavior of the tion ϕ field and a ð8 − pÞ RR from gauge field F . ½8−p 2ϕ solution near the singularity point. As e is the effective qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi string coupling, for the supergravity description to remain 1 1 1 10 μ S = d x −det g R − ∂ ϕ∂ ϕ − F : μν μ ½ 8−p valid, one needs the parameter δ to be less or equal to zero 16πG 2 28ðÞ − p ! so as to make the string coupling small. The parameters of ð4Þ the solutions are not all independent but satisfy some consis- tency relations like We will be looking for solutions of the above using the ansatz. α = β, 2 2ArðÞ 2 2 2 Q=2αρ sinh 2θ, ds = e −dt + dx +⋯+dx str 1 p ð11Þ ð5Þ 2BrðÞ 2 2 2 2 2 + e dr + r dΩ , α + δ = : 8−p ÀÁ F = QVol Ω : ð6Þ 8−p 8−p In arbitrary dimensions, the solutions and the con- ½ straints are a bit complicated and are given in [18]. Just like In the above, the metric has an ISOðp, 1Þ ×SOð9 − pÞ the BPS D3 brane solution, the non-SUSY solution too is isometry and represents a magnetically charged p brane in asymptotically flat. One can recover the BPS solution from 10 dimensions with magnetic charge Q. It can be shown that the non-SUSY solution given above by considering the limits the above solution conserves supersymmetry, i.e., saturates 2 2 ρ ⟶ 0 and θ⟶ ∞ keeping α/2ρ ðcosh θ + sinh θÞ⟶ 0 0 the BPS bound if [17] R = fixed. Under this scaling, one has GðρÞ⟶ 1 and Fðρ 4 4 4 Þ⟶ 1+ ðR /ρ Þ and Q⟶ 4R under which the standard p +1 Br +7 − p Ar =0: ð7Þ ðÞ ðÞ ðÞ ðÞ BPS solution is regained. The decoupling limit is a low energy limit in which inter- Solution of equations of (4) compatible with (5)–(7) actions between the bulk theory and theory living on the leads to usual BPS p branes. We will be interested in super- brane vanish. To work out the decoupling limit and hence- gravity solutions which defy the condition (7) and hence forth the throat geometry, one needs to make a change of donot saturate the BPS bound , thus breaks spacetime super- variables in analogy with the BPS D3 brane. symmetries. In the rest of the paper, we will be concerned with non-supersymmetric D3 brane solution and thus will consider the case where p =3. The non-supersymmetric D3 ′ ρ = α u, brane solution is given as ρ = α u , 0 0 Âà 2 −1/2 δ/4 2 2 2 2 ð12Þ ds = F ρ G ρ −dt + dx + dx + dx λ ðÞ ðÞ 1 2 3 2 α cosh θ = , dρ ′ α u 1/2 1+δ /4 ðÞ 2 2 + FðÞ ρ GðÞ ρ + ρ dΩ , GðÞ ρ α ⟶ 0: ð8Þ 2ϕ 2 δ e = g GðÞ ρ , In the above, u and u have the dimensions of energy 1 0 F = pffiffiffi 1+⋆ Q Vol Ω : ðÞ ðÞ and are kept fixed. From (11) and (12), it can be shown that ½5 5 Q/α ≫ 1 implying that the curvature of space-time in string units must be very small for the supergravity descrip- In the above the functions, FðρÞ and GðρÞ are given as tion to be valid. A justification of the above decoupling limit is given explicitly in [7, 18]. Under the above said limit, α/2 2 −β/2 2 ð9Þ F ρ = G ρ cosh θ − G ρ sinh θ, ðÞ ðÞ ðÞ 4 u ρ 0 GðÞ ρ ⟶Gu ðÞ =1+ = fixed, G ρ =1+ : ð10Þ ðÞ 4 4 u ð13Þ ~ ~ FðÞ ρ ⟶ Fu ðÞ = Fu ðÞ: It can be shown that the non-SUSY solution (8) violates condition (7) and thus breaks space-time supersymmetries. 2ϕ In the above, e is the effective string coupling constant α/2 −α/2 and the solution is characterized by six parameters, i.e., α, In the above, FðuÞ =1/αu ðGðuÞ − GðuÞ Þ, and the β, δ, θ, ρ , and Q, of which ρ has the dimensions of length, non-SUSY D3 brane throat geometry in the decoupling limit 0 0 4 Advances in High Energy Physics mentioned above becomes tions involved), and it is very difficult to invert u in terms of m. Thus, we express our results in this work with pffiffiffi formula for mass (m )of N =4 SYM. −1/2 δ/4 2 μ ν 0 ds = α λ Fu ðÞ Gu ðÞ η dx dx μν pffiffiffi ð14Þ du 1/2 ðÞ 1+δ /4 2 2 + Fu Gu + u dΩ , m = u : ð19Þ ðÞ ðÞ 5 0 b Gu 2π ðÞ 2ϕ 2 δ ð15Þ e = g Gu : ðÞ 3. Pair Production in Presence of In the above, the space-time coordinates have been External Fields pffiffiffi i i rescaled as ðt, x Þ⟶ λðt, x Þ, where λ is the ‘t Hooft cou- In this section, we will revisit the concept of pair produc- pling. In the limit u ⟶ 0, one has GðuÞ⟶ 1 and FðuÞ tion in the presence of external electric fields. i.e., the 4 4 4 8 8 4 =1/αu ½ðαu /u Þ + Oðu /u Þ ≈ u . In this limit, the non- 0 0 0 “Schwinger effect.” We will demonstrate the effect using SUSY throat geometry (14) goes over to the known AdS Euclidean version of the electromagnetic action [20] and ×S , and the effective string coupling becomes constant. generalize to large N gauge theories. The Euclidean ver- To check the relation of solution (14) with that of the previ- sion of U(1) gauge theory coupled to a massive complex ously known one by Constable and Myres [8] which was scalar field is given by conjectured to be dual to some non-supersymmetric field theory, one has to rewrite the solution in the Einstein frame. 4 2 ex 2 2 S = d x F + ∂ + ieA + iea ϕ + m jj ϕ : ð20Þ μν μ μ μ pffiffiffi 4 2 −1/2 α/4 μ ν ds = α λ Hu Gu η dx dx ðÞ ðÞ E μν ð16Þ du 1/2 ðÞ 1−α /4 2 2 In the above, A refers to the dynamical U(1) gauge +Hu ðÞ Gu ðÞ + u dΩ , ex Gu ðÞ field and a refers to the external value of (constant) elec- tromagnetic field. The pair production rate, Γ, can be writ- 2ϕ 2 δ ð17Þ e = g Gu : ðÞ ten as [21] ð ð In the above, the function HðuÞ is defined by HðuÞ = G −S −S α/2 α eff VΓ = −2 Imln DADϕ e = −2 Imln DAe , ð21Þ ðuÞ FðuÞ = GðuÞ − 1. Now, one has to make a coordinate −1/4 4 4 4 4 transformation like u = rð1+ ðω /r ÞÞ , where ω = u /4. 4 4 Under this transformation, GðuÞ⟶ ð1+2ðω /r ÞÞ and H 2α 4 2 ex 2 4 4 where S = 1/4 d xF + trln½−ð∂ + ieA + iea Þ + m . ðuÞ⟶ ð1+2ðω /r ÞÞ − 1. From these relations and (16), eff μν μ μ μ one can exactly produce the two-parameter family of solu- For leading order calculations, one can ignore the coupling tions as found in [8] in which both supersymmetry and of the dynamical gauge field with the scalar field. Thus, conformal symmetry are broken. The solution in [8] also the expression above reduces to exhibits QCD-like behavior like running gauge coupling and confinement in the infrared. The geometry (16) exhibits a naked singularity at u =0 and thus should be ex 2 VΓ = −2 Im trln − ∂ + iea + m : ð22Þ corrected by stringy corrections which should become dominant at low length scales. Moreover, the proper dis- tance (spatial) from the exterior (say u = u ) to the interior is finite (which says that stringy corrections are a must). −AT Using the relation, trlnðAÞ = − ðdT/TÞtre and In holography, the proper distance is identified with mass evaluating the trace in position basis, one can rewrite the of the string hanging from the boundary to the interior above expression to [19]. To find the same, we have to choose a gauge of the form: x = t, u = s, and all others = constant. With this ð ð gauge, the mass is given by dT 2 −m T/2 4 VΓ =Im e d x sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð23Þ pffiffiffi /4 ðÞ 2δ−3 u 4 2 λ u ex Á x exp −T − ∂ + iea x : m = du 1+ μ ð18Þ 2π u = finite and positive for all allowed values of δ: Note that the integrand under d x is synonymous to the path integral of a nonrelativistic particle under the The integral can indeed be done in closed form. How- ex ever, the result is very complicated (hypergeometric func- influence of the Hamiltonian H = 1/2½P + ea . Using μ Advances in High Energy Physics 5 quantum mechanical path integral representation [22], one 23], the calculation of which is indeed complicated. The 2 3 can write modified prefactor is given by ðeEÞ /ð2πÞ . Thus, we see that the pair production rate goes to zero if the external ð ð dT 2 electric field is switched off. In arbitrary coupling, one −m T/2 VΓ =Im e Dx exp T can no longer neglect the effect of the dynamical fields 0 x 0 =xT ðÞ ðÞ ð þ and one has to include contribution from the Wilson 2 ex Á − dτx_ + ie a dx loops. μ μ ð24Þ ð ð ð ð ð ∞ ∞ 1 dT dT 2 1 −m T/2 2 =Im Dx exp VΓ = −2Im e Dx exp − dτx_ T 2T 0 xðÞ 0 =xðÞ 1 0 0 ð29Þ þ þ ð þ 1 2 1 m T ex 2 ex + ie a dx exp ie A dx : Á − dτx_ − + ie a dx , μ μ μ μ μ μ 2T 2 The pair production rate gets modified to [15, 20] where in the last line, we have rescaled τ⟶ 1/Tτ.We 2 2 assume m dτx_ ≫ 1 (a condition signifying heavy mass) 0 2 ∞ n+1 2 2 eE −1 πm e ðÞ ðÞ and note that the integration over T has the form of a Γ = 〠 exp −n − : ð30Þ 3 2 n eE 4 ðÞ 2π modified Bessel function K ðxÞ = ðdt/tÞ exp ð−t − ðx /4t n=1 pffiffiffiffiffiffiffiffiffi −x ÞÞ with the asymptotic behavior, K ðxÞ ≃ π/2xe , for From the above, one can work out that the pair pro- large x. Thus, the above integral becomes duction rate is not exponentially suppressed once the value sffiffiffiffiffiffi ð of electric field exceeds the so-called critical value E =4π Âà 1 2π 2 3 m /e , beyond which the vacuum becomes unstable. VΓ =Im Dx exp −S : ð25Þ m T 0 To implement this argument for AdS/CFT like theo- ries, one faces a number of problems. Firstly, the field the- qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi Ð Ð ory in those circumstances is a conformal one, and one 2 2 In the above, T =1/m dτx_ and S = m dτx_ − 0 p cannot get a mass term a priori. Moreover in the dual ex ex ie a dx and a = −iEx (signifying constant electric μ μ 1 0 gauge theory, matter fields exist in the adjoint representa- field of value E in x direction, iota comes in due to tion of SU(N) gauge group. To evade these issues, one Euclidean signature). We like to evaluate the above inte- uses the Higgs mechanism to break the symmetry group gral by the method of steepest descent. The argument from SUðN+1Þ⟶ SUðNÞ ⊗ Uð1Þ. Because of this split- within the exponential is the action for a relativistic parti- ting, one has 5 massive W bosons transforming in funda- ex cle executing a periodic motion under influence of a . mental representation of SU(N) and interacting with the background Yang-Mills theory. Now, the pair production The equation of motion for it is given by rate in the presence on an external electric field is given 1 by [15, 24] ex qffiffiffiffiffiffiffiffi m€x = eF x_ : ð26Þ μ ν μν x_ ð ð ð 1 1 pffiffiffiffiffi 2 ðÞ E Γ ~ −5N Dx exp −m dτ x_ + i dτa x_ Wx , hi ½ μ μ 0 0 Keeping in mind the periodic boundary conditions x ex ð31Þ ð0Þ = x ð1Þ and F = E, one has the following classical μ 01 solution: where W½x is the SU(N) Wilson loop and can be calcu- cl x = RðÞ 0, 0, cos 2πτ, sin 2πτ , lated by holographic means. R= , ð27Þ 4. Pair Production in Non-Supersymmetric eE Theories via Holography πm cl S = : eE The ideal way to argue the Schwinger effect [11, 25] is to cal- pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi culate the expectation value of circular Wilson loops and Using the above values, one has 1/m 2π/T = eE/m. 0 relate it to the decay rate. However, one can alternatively Thus, decay rate can be approximated as view the vacuum to be made of virtual qq pairs in the pres- ence of an attractive potential and study the influence of an pffiffiffiffiffi eE 2 external electric field [13]. This basically amounts to calcu- −πm /eE VΓ ≈ e ð28Þ lating the interquark potential which one does by consider- ing the rectangular Wilson loop. In doing so, one has to Ideally one should go around calculating the one loop make some additional approximations. One considers that prefactor and complete the steepest descent process [20, the time scale associated with the Wilson loop is much lesser 6 Advances in High Energy Physics than the length scale. Intuitively, one thinks that the quark pffiffiffi du ðÞ in ðÞ 2δ−3 /4 −det G = α λ Gus antiquark pairs are separated in the far past and unite in ðÞ ðÞ ab ds the far future. In holography, the Wilson loop is given by fol- ð35Þ !# lowing formula [26, 27]: 3/4 −1 +GusðÞ ðÞ FuðÞ ðÞs : −SX½ ,h hi W½ C = DXDh e : ð32Þ ab Vol ∂X=C It is not possible to carry on analysis without some sim- plification. We therefore assume that ðu /uÞ ≪ 1, and with S½X, h is the Wick rotated action of the fundamental the mentioned, simplify the area, i.e., on-shell Nambu- string [6] with endpoints ending at contour C situated on Goto action, to the probe brane. In the classical limit (α ⟶ 0), the extre- ð ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mal value of the string action dominates, and thus, the Wil- T /2 L in ðÞ son loop is the extremal area of string world-sheet ending on S = dt ds −det G ng ab 2πα −T /2 −L the contour. To study the rectangular Wilson loop, we take sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi L 2 4 4 the quark antiquark dipole to be aligned in the x direction. λ du u u 0 0 = T ds 1+ A + u 1+ B , The string action whose on-shell value we are interested with 4 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2π ds u u −L ðinÞ is the Nambu-Goto action S = 1/2πα dtds det G NG ab ð36Þ ðinÞ μ a ν b with G ≡ G ð∂x /∂s Þð∂x /∂s Þ which has two diffeo- ab μν morphism symmetries. We exploit those to choose the fol- wherein lowing gauge: δ +1 B = , x ðÞ s, t = t, 2δ − 3 ð37Þ x ðÞ s, t = s, A = , ð33Þ usðÞ , t =us ðÞ, A + = B: 1,2 x =0, ΘðÞ s, t = constant: Crudely speaking, this can be seen as treating the non- SUSY theory as perturbation over the N =4 supersymmetric Yang-Mills. Since the expression (36) does not explicitly For present purposes, x ≡ s is assumed to range between depend on the parameter s, the corresponding “Hamilto- ½−L, L and temporal direction x ≡ t is ranged between ½− nian,” Q, is conserved. T , T with the assumption that T ≫ L. 2L indicates the interquark separation on the probe brane with the boundary dL condition uð±LÞ = u , where u indicates the position of the du ng b b Q = − + L ng probe brane along the holographic direction (see Figure 1). ds ddðÞ u/ds Finally another word about the configuration, it is possible ð38Þ 4 4 u + Bu to consider the qq pairs at a velocity in the x direction. = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ÀÁÀÁ ÀÁÀÁ 2 4 4 4 4 4 However in the present case where the virtual particles in ðÞ du/ds 1+Au /u + u 1+Bu /u 0 0 vacuum are modeled as qq dipoles, such a configuration seems hardly sensible. The induced metric as per the above A is indicated in [28]; the fundamental string is assumed gauge choice reads (14) and (33). to carry charges at two of its endpoints and is otherwise symmetric about its origin. From the above expression, we in ðÞ a b −1/2 δ/4 2 see that du/ds has both positive and negative signs. Appeal- pffiffiffi G ds ds = −FuðÞ ðÞs GusðÞ ðÞ dt ab ′ ing to its symmetric nature, there exists a point, namely, α λ turning point (with string parameter s ), such that 2 −1/2 δ/4 1/2 + ds Fu s Gus + Fu s ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ du u =0: ð39Þ ðÞ 2 t ds 1 du 1+δ /4 ðÞ Á GusðÞ ðÞ : GusðÞ ðÞ ds Using the above expression in (38), the value of the con- ð34Þ served Hamiltonian is found in terms of the turning point qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi From the above, we have the determinant of the induced 4 4 Q = u + Bu : ð40Þ t 0 metric to be Advances in High Energy Physics 7 Boundary Probe (u ) ‘‘adiabatic switching of ’’ q q x q – ‘‘adiabatic switching on’’ u u ∞ (u ) 0 B (a) (b) Figure 1: This figure illustrates the setup used. The probe brane is placed at a finite position (u ) on the holographic direction as in (b). On the probe brane, the placement of the Wilson loop is shown in (a); the arrows indicate the contour of the loop (not the propagation of the string). For adiabatic interactions, one can neglect the effects of the dotted lines and the string profile becomes static. Putting the above value in (38), we get 0.0 2 4 6 8 10 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁÀÁ 4 4 L 4 4 du u − u u + Bu 2 t 0 = u ÀÁÀÁ : ð41Þ 4 4 4 ds u + Bu u + Au t 0 0 The length of the (virtual) dipole can be calculated to be (see Figure 1) –0.5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L/2 u b 4 u + Au 4 4 0 L = dx = u + Bu du qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 3 t 0 ÀÁÀÁ 2 4 4 −L/2 u 4 4 t u u − u u + Bu ð42Þ –1.0 From (41) and (36), we can find the on-shell value of interquark potential. pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S ÀÁÀÁ λ 1 ng 4 4 4 4 U = = du u + Au u + Bu pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : PE+SE 0 0 2 4 T 2π u u u − u t t –1.5 ð43Þ Notice from (42) that when u ⟶ u , the value of the t b interquark separation becomes small. But as said earlier, we are in an approximation where ðu /uÞ ≪ 1. Thus, the calculations in this section are trustable for large interquark separation. Now, the expression in (43) (see Figure 2 for the –2.0 plot) does not take the presence of an external electric field Figure 2: This is the graph of U vs. L. The rest mass has been into account. Thus, we define an effective potential as PE duly subtracted. Note that for small values of L, the graph is approximately linear and for large L coulombic behavior is V = U − E:L=1 − r E :L +Gu L : ð44Þ ðÞ ðÞ ðÞ eff PE+SE c t mimicked. Deviation from usual coulombic behavior is evident. The values used are δ = −0:75, u /u =0:01, and λ =4π . 0 b In the above, we have assumed the presence on a critical electric field E , above which the effective interquark force becomes repulsive for all values of the interquark separation. PE 8 Advances in High Energy Physics The quantity Gðu Þ is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 4 u + Au GuðÞ = U − E L = du pffiffiffiffiffiffiffiffiffiffiffiffiffiffi t PE+SE c u u u − u t t "# pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð45Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 λ u + Bu t 0 Á u + Bu − E pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 0 c 2π 4 u + Bu The parameter r is the ratio of applied electric field to its critical value. The slope of the effective potential is given as dV du dGðÞ u eff t t 0.0 0.2 0.4 0.6 0.8 1.0 =1ðÞ − r E + : ð46Þ dL dL du u t We now proceed to find the value of the critical electric field. Note that at u = u , the interquark separation (42) and Figure 3: This is the graph of L vs. u . Note that the function is an t t b isomorphism. The values used are δ = −0:75 and u /u =0:01. the interquark potential (43) vanish (see Figure 3). At the 0 b st critical value of the electric field r =1,the 1 term of (46) ceases to contribute, and the behavior of the interquark force will be completely governed by the second term of (46). Crit- 1.5 icality demands that the potential ceases to put up a tunnel- ing barrier for all values of interquark separation (see red line in Figure 4). Given that Gðu ðLÞÞ vanishes at L =0,we need to show that Gðu ðLÞÞ is a monotonically decreasing 1.0 function with respect to L whose slope vanishes at L =0 (this is because critical electric field is the least one for which pair production happens spontaneously). From (42), we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 0.5 dL u + Au t 0 = − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ du 4 2 4 4 u u + ϵ − u t t t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð47Þ ÀÁÀÁ 4 4 ð 4 4 u 3 u + Au u + Bu 0 0 0.0 +2 du qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 4 6 8 10 2 ÀÁ ÀÁ u 3 4 4 4 u +ϵ 4 t u − u u + Bu L t t 0 Similarly, we have from (45) –0.5 "# pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 dG u + Au λ t 0 4 4 u = − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u + Bu − E ðÞ t 0 c du 2π 2 4 4 u ðÞ u + ϵ − u t t t –1.0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁÀÁ ð 4 4 4 4 u + Au u + Bu 0 0 +2u du pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u +ϵ 2 4 u u − u –1.5 "# pffiffiffi λ E Á − Figure 4: The plot indicates the effective potential (in the presence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 2π u + Bu of external electric field) vs. the interquark separation. Imagine this 0 "# pffiffiffi to be then potential through which qq tunnels out. The green line qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ dL 4 4 indicates r =0:25 and blue line for r =0:75. The parameter r is = u + Bu − E : 0 c 2π du the ratio of the applied field to its threshold value. The red line which exhibits the threshold behavior, i.e., no potential barrier, ð48Þ stands for r =1:0 and cyan for r =1:75 shows catastrophic decay of vacuum. Note that at the threshold/critical value, the slope of Thus, we get the potential vanishes at L =0 and is negative for nonzero value of L which is precisely the conditions we have used to analytically "# pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi find out the value of E . Also note that none of the above plots c dV λ eff 4 4 =1ðÞ − r E + u + Bu − E : ð49Þ exhibit confining behavior as earlier reported in literature. The c t 0 c dL 2π values used are δ = −0:75, u /u =0:01, and λ =4π . 0 b L Advances in High Energy Physics 9 At threshold condition, the slope of the potential should for small values of interquark separation? The answer to this be zero at when interquark separation vanishes, i.e., u = u . question will be found in the next section. t b Implementing the same in (49), we get It so happens that analytical solutions to (42), (43), and (44) cannot be found out in a closed from via Mathematica. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 2 4 Thus, we resort to numerical methods. Some plots to illus- λ δ +1 u 2π λ u 2 0 2 0 E = u 1+ = pffiffiffi m 1+ ðÞ δ +1 : trate the situation are given. c b 0 4 4 2π 2 u 32π m ð50Þ 5. DBI Analysis of Critical Electric Field In this section, we look to find out the critical electric field We thus have from analysis of the DBI action of the probe brane in the "# pffiffiffi pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi presence of an external electric field. In due course, we will dV λ λ eff 4 4 4 4 =1ðÞ − r E + u + Bu − u + Bu : also answer the question raised in Section 4. c t 0 0 dL 2π 2π As earlier, we imagine the probe brane situated at u = u (see Figure 1) in the holographic dual with an electric field ð51Þ switched on at the brane position. The DBI action is given as From Figure 3, we see that L increases as u decreases rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi using which we can say from (51) that dV /dL is a mono- eff S = d x −det Pg½ + B +2πα F : tonically decreasing function of L at r =1. It can be easily DBI μν μν μν ðÞ 2π g α u=u s b understood that from r >1, the effective potential is totally ð52Þ repulsive. Thus, we establish the existence of a critical elec- tric field with value given by (50). We see that as δ switches over −1, the critical electric field increases and decreases, In the above, P½g is the pullback of the curved metric μν respectively, compared to the supersymmetric value. Not on the probe brane, B is the NS 2-form which is zero in the μν even that, just at δ = −1, the critical field has the same value present case, and F is the Faraday tensor which we set to as that of the supersymmetric theory. Will this kind of μν behavior remain when one considers higher orders? How the value F = E, to indicate the presence of an external much of the calculation in this section should be trusted electric field. Evaluating the above from (14), we have 0 pffiffiffi 1 −1/2 δ/4 −α′ λFu Gu 00 2πα′E ðÞ ðÞ b b B C pffiffiffi B C −1/2 δ/4 B 0 α λFuðÞ GuðÞ 00 C b b B C Pg½ +2πα F = : μν pffiffiffi μν B C −1/2 δ/4 B ′ C 00 α λFuðÞ GuðÞ 0 b b @ A pffiffiffi −1/2 δ/4 ′ ′ −2πα E 00 α λFu Gu ðÞ ðÞ b b ð53Þ Thus, the DBI action becomes The functions FðuÞ and GðuÞ have been defined before in (9). One can check that up to Oðu /u Þ , (55) reduces to (50). 0 b However in finding (55), we have refrained from using pertur- α λ 4 −1/2 δ/4 S = d xF u Gu ðÞ ðÞ bations of any sort, and thus, (55) is the exact value. Let us DBI b b 2π g ðÞ u=u b check the behavior of it with respect to the parameter δ. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð54Þ We see from Figure 5 that the critical electric field is the 2πE 1 /4 − δ Á FuðÞ GuðÞ − pffiffiffi : same as that of its supersymmetric cousin somewhere b b around δ = −0:85, which matches more or less with our per- turbative analysis in the last section. For values of δ >0:85, the critical electric field is greater than the supersymmetric Thus, we see that (54) is not real for all values of the counterpart, and for δ <0:85, the critical field is lesser. Thus, external electric field and there is an upper limit of the same. the question raised in the last section is answered in the affir- This limiting value is nothing but the critical electric field is mative. The calculation in Section 4 will not be affected dras- pffiffiffi tically for small values of interquark separation. (This is −1/2 δ/4 because it is the small separation behavior that decides the E = FuðÞ GuðÞ : ð55Þ c b b critical value.) 2π 10 Advances in High Energy Physics x = r σ cos τ, ðÞ 0.40 x = r σ sin τ, ð56Þ ðÞ 0.35 u = u σ : ðÞ 0.30 In the above, all other coordinates have been put to be constants as circular symmetry would imply. The parameter τ ranges from ð0, 2πÞ while the parameter σ is still arbitrary. –1.5 –1.0 -0.5 0.0 There exists a diffeomorphism invariance of the Nambu- Goto action with which we can set u = uðσÞ to a function 0.20 of our choosing. Putting the ansatz (56) in (14), we have the induced metric to be Figure 5: This is the plot of critical electric field E vs. the non- pffiffiffi supersymmetric parameter δ. In this figure, we have set λ/2π = " pffiffiffi 1 and u /u = 1/0:5. According to this values, the supersymmetric dr 2 −1/2 δ/4 0 b ds = α λ Fu ðÞ Gu ðÞ critical field would have been 0.25. dσ du 1/2 ðÞ δ−3 /4 2 + Fu ðÞ Gu ðÞ dσ ð57Þ dσ 6. Holographic Pair Production Rate for Non- Supersymmetric Theories 2 −1/2 δ/4 2 + r Fu Gu dτ : ðÞ ðÞ In this section, we calculate the pair production rate by using the method of circular Wilson loops. As indicated earlier in From the above, one can get the Nambu-Goto action to (31) and (32), to find the pair production rate, we need to find the on-shell value of the Nambu-Goto action with string be of the form endpoints ending on a circular contour at the probe brane pffiffiffi ð ð (u = u ). For pure AdS, the calculation of the same has been α λ 2π σ presented in [11, 15, 29]. However, it is not possible to find S = dτ dσ ng 2πα 0 0 exact solutions to the relevant equation of motions for the ð58Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 present case (14). Thus, we will resort to perturbative treat- /2 1 3/4 δ − − ′ ′ Á r Gu ðÞ Fu ðÞ r +Gu ðÞ u : ments like that of [30] to calculate the circular Wilson loop, and hence, the decay rate to first order of the non-SUSY deformation parameter (u ). Since the metric (14) enjoys For purposes of calculation, we expand the function Fð circular symmetry, we start by making an ansatz. uÞ and GðuÞ in their leading order to the non- supersymmetric deformation parameter, and we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 4 4 4 8 pffiffiffi b 2 2 δ u 1 u 3 u u 0 0 0 o 2 4 ′ ′ S = λ dσ r 1+ r u 1+ + u 1 − + O ng 4 4 4 8 2 u 2 u 4 u u ð59Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 4 4 pffiffiffi b 2 2 u u 0 0 2 4 ′ ′ ≈ λ dσ r r u 1+ A + u 1+ B , 4 4 u u wherein bation over the regular N =4 SYM. In this paper, we limit ourselves to the first-order perturbations. Recall that we still had one diffeomorphism invariance left as mentioned δ +1 B = , before, with the help of which we set duðσÞ/dσ =1. Thus, (59) is simplified to 2δ − 3 ð60Þ A = , sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 pffiffiffi dr ÀÁ rÀÁ 4 4 5 2 4 S = λ du r u + Au + u + Bu : ð61Þ ng 0 0 A + = B: du u 4 t The above binomial expansion and all the others that We would like to find out the function r = rðuÞ which follow are simply treating the non-SUSY theory as a pertur- extremizes (61). Extremizing the same, one has to encounter c Advances in High Energy Physics 11 the equation Thus, the full solution is 2 4 ÀÁ dρ ÀÁ dρ 1 A − B u 4 4 4 4 4 7 2 4 0 u u + Au 2 u + Bu − u r ðÞ u = ρðÞ u = u K − + 0 0 0 2 6 6 du du u 5u u 1 A − B u ÀÁ ÀÁ dρ 2 0 3 8 4 4 8 4 4 ð65Þ ≡ K − + − 4ρ u ABu +3Bu u +2u − u + Bu 2 6 0 0 0 u 5 u du 1 1 u ÀÁÀÁ d ρ = K − − , 4 4 4 4 4 2 6 − 2u u + Au u + Bu ρ =0, u 4 u 0 0 du ð62Þ where a redefinition of constant has been made. Now, it is 2 time to relate the constant K to physical parameters. At u where ρ = r . The above equation is very hard to solve in = u , the value of r is the radius of the Wilson loop R. Thus, closed form. Thus, we adopt perturbative techniques like we have that of [24]. To do so, we decompose the solution to (62) 4 2 as ρ = ρ + u ρ in which ρ = −1/u , and ρ indicates the 0 0 1 0 1 1 A − B u 2 0 perturbation. From (62), the equation for ρ to the leading K = R + − : ð66Þ 2 6 4 5 u u order of u is b b From the above, we can also find the value of the turning dρ d ρ 2 7 1 8 1 point u , since at the turning point, the radius rðu Þ =0. t t 2u 6 B − A +2u + u =0: ð63Þ ðÞ du du Thus, the equation which determines the turning point is 1 A − B u One can check that the above is solved by K = 1 − : ð67Þ 2 4 u u t t A − B Now, we proceed to calculate the on-shell value of the ρ u = + K : ð64Þ ðÞ 5u Nambu-Goto action (61) on the solution (65). We have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 4 pffiffiffi b ÀÁ ÀÁ 1 drðÞ u 4 4 2 0 S = λ du u + Au + r 1+ B ng 0 4 du u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 4 4 pffiffiffi ÀÁ 1 4 3 u u 1 A − B u 0 4 0 0 4 ð68Þ = λ du · 1 − A − B u + Au +1+ B K − + ðÞ 6 4 4 2 6 4 u 5 u u u 5 u rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 pffiffiffi b ÀÁ 0 8 = λ du K + KB + O u : We neglect the Oðu Þ term in the above. The integral of Happily, there is a way out of this mess. Recall that our theme has been to work in the leading order of u and the the remaining part cannot be done in closed form by using Mathematica. So we resort to perturbative methods again last two terms of (69) come with a u of their own. Thus and write to leading order, we may substitute the usual AdS relations (relating K to u and u ) in the last term of (69), but use t b u 4 pffiffiffi b pffiffiffiffi ÀÁ B u the non-SUSY relations (66) and (67) in the first term of 0 8 S = λ du K 1+ + O u ng 0 the same. Doing so, we have 2 u 0 1 "# pffiffiffiffi ð69Þ hi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffi b 2B Ku 0 4 @ A pffiffiffi ÀÁ ≈ λ Ku − : A − B u 2 2 u u S = λ R u +1 − t ng b 4 u 5 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "# pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 2 So far so good, however, the reader may agree that work- A − B u 2B 2B R u +1 0 4 b − 1 − + u − ing with (69) is still daunting given that we now have to sub- 4 4 4 5 u u u t t b stitute the highly nonlinear relations (66) and (67) into it. 12 Advances in High Energy Physics sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi u 2ðÞ A +19B A − B ÀÁ A − B u 0 2 2 1 − 2E + + − 2B =0: ð73Þ ≈ λ R u +1 − c 4 5 10 5 u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ Thus, A − B u 0 2 − 1 − R u +1 u pffiffiffi λ 1 u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 E = u 1+ 23 + 24δ u ÀÁ ðÞ c b 0 2 2 4 2 2 2π 8 + 2BR u +1 − 2B R u +1 u b b b u "# ð74Þ 2 4 2π λ u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 0 ÀÁ ÀÁ u A +19B 2 = pffiffiffi m 1+ ðÞ 23 + 24δ : 0 2 2 2 2 4 4 ≈ λ R u +1 − 1+ R u +1 128π m b b 4 λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A − B We see that like (50) and (55), the value of the critical 2 2 − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2B R u +1 : electric field is greater than the supersymmetric value for 10 R u +1 the value δ = −23/24 and less than the supersymmetric ð70Þ cousin otherwise. Our perturbative analysis has even shown that the value of parameter δ for which this phase transition In the second line of the above, we have used the usual occurs is slightly bigger than -1 as can be seen from the non- AdS relations for the term 1/u to leading order as it is perturbative DBI analysis. Now to find the expression of the accompanied by a u . Again in the third line, we have used pair production rate, we need to solve (72) for y. As can be a binomial expansion and retained terms of leading order seen, that is not analytically possible. We thus resort to per- 4 4 in u /u . Now, in the presence of an electric field, the effec- turbative treatments again and write 0 b tive action of the string has an extra piece, S = T dσdτ B 0 μ ν B ∂ x ∂ x . Specializing to constant electric field, the con- u μν σ τ 0 y = y + y : ð75Þ 0 4 1 tribution of S is a pure boundary term with on-shell value πR E, where R is the radius of the Wilson loop, E = B , and all other components of the B are set to zero. The y is the usual AdS solution, i.e., u =0 in (72). The value μν 0 0 of y is 1/2E. We put the above relation in the equation in effective action is given by 4 4 (72) to get up to leading order in u /u . 0 b pffiffiffi pffiffiffi u A +19B 0 2 1 A +19B 1 2 A − B ðÞ S = S + S = λ x − 1+ x 2 ef f ng B 4 y = + E − 2B : ð76Þ 10 1 u 3 b 2E 20 5 ð71Þ A − B pffiffiffi − pffiffiffi − 2B x − Ex + E : Now, we put (75) and (76) in (71), i.e., find out the on- 10 x 4 4 shell action. Retaining terms in leading order of u /u leads 0 b us to pffiffiffi 2 2 In the above, x = R u +1 and E = ð λu /πÞE. Thus, the pffiffiffi b b λ 1 u radius RðxÞ is a free parameter in expression (71). Following on‐shell 0 S = − 2+2E + eff [11, 25], the radius should be set to an extremum of (71). 2 2E u Instead of extremizing w.r.t. R, we extremize the action A +19B 1 B − A 2B pffiffiffi Á + E − (71) w.r.t. the parameter y = x. Doing so, we find 80 10 E pffiffiffi ð77Þ 4 λ 1 u pffiffiffi dS u 0 eff 0 = − 2+2E + 0= = λ 1 − 2Ey + 4 4 2 2E u dy u b ð72Þ 40δ +39 1 1 δ +1 2 A +19B A − B ðÞ Á + E − : Á y + − 2B : 4 320 40 E 2 E 5 10y The pair production rate of quark antiquark pairs per The radius R should be set to be the solution of (72); pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi unit volume per unit time is given by the formula Γ ~ 2 2 recall y = R u +1. Thus, the value of y in the above equa- b on‐shell −S eff e . Note that we are using reduced parameter E = ðπ/ tion is constrained and should always be greater than 1. This pffiffiffi λu ÞE, in terms of which the pure AdS pair production is because the radius of the Wilson loop should be a real number. A subtle point is that the range of parameter y rate (per unit volume per unit time) is given by [11, 25, 29] should be restricted to half of the real line, because the radius "# ! pffiffiffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi R is nonnegative. The critical electric field E is the one for λ 1/2 E Γ ~ exp − − : ð78Þ which the radius R =0, i.e., y =1. Setting so in the above, SUSY 2 E 1/2 we see Advances in High Energy Physics 13 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 Figure 6: The figure on the left illustrates relation between the pair production rate and applied electric field for pure N =4 SYM (78). In our units, the critical electric field is at E = 1/2. The right figure is for non-supersymmetric case (77) for the value δ = −0:75. Almost no change in the profile is found. 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 pffiffiffi Figure 7: We plot the pair production rate for non-supersymmetric Yang-Mills for the value δ = − 2. On the right, we have ðu /u Þ = 0 b 4 4 4 ð1/5Þ and on the left ðu /u Þ = ð1/3:75Þ . 0 b us recall that the prefactor of the pair production rate is For the pure AdS/Supersymmetric scenario, the critical pffiffiffi 2 3 given by field theoretic calculations to be ðeEÞ /ð2πÞ electric field is E = 1/2, i.e., E = λu /2π. Now unlike c c the supersymmetric case, the pair production rate cannot (see (30)). Although the holographic calculation of the be brought in closed form. We will have to resort to fluctuation prefactor is currently a mystery, it should def- numerical calculations. We present the plots of pair pro- initely match with field theoretic calculation for low elec- duction rate. Computation of the fluctuation prefactor tric field. For small applied electric fields, the production 2 3 rate shoots up signaling in nonperturbative instability of (i.e., the ðeEÞ /ð2πÞ term in (30)) is somewhat an open question in holography, which is the reason we have plot- the vacuum. We say “nonperturbative” because the −S on‐shell Schwinger effect is by itself a nonperturbative phenome- ted e instead of Γ. non. We see that the limit δ = −0:975 approx. is for more Physical interpretation of the plots: as commented in interesting than earlier imagined. the caption, no stark contrast is found between the SUSY Let us end by writing down the pair production rate per and non-SUSY case in Figure 6. This is the case when the unit spatial volume per unit time for non-SUSY Yang-Mills. parameter is greater than (somewhere around) -0.975. The reason for this can be seen from (77). For low electric 4 4 field, the pair production rate is dominated by the ðu /u 0 b pffiffiffi pffiffiffi Þðð40δ +39Þ/320Þð1/E Þ term. Above δ = −0:975, the effec- λ 2πm 1 λ Γ ≈ exp − − 2+ E pffiffiffi non−SUSY tive correction at the low electric field limit is positive. At 2 2 E 2πm the limit of high electric field limit, the correction of pair 2 4 4 8 production rate due to the non-SUSY deformation param- λ u 4π mðÞ 40δ +39 1 0 0 ð79Þ eter is always positive. This is reason that at the high elec- 4 2 4 16π m E 5λ tric field limit, the behavior of non-SUSY pair production )!# pffiffiffi rate is same as its supersymmetric cousin for all values of λ 4πm δ +1 1 ðÞ + E − pffiffiffi : parameters. Startling effects happen when the parameter 160πm E δ < −0:975, for which the plot is shown in Figure 7. Let exp (–S ) susy exp (–S ) non–susy exp (–S ) non–susy exp (–S ) non–susy 14 Advances in High Energy Physics 7. Conclusion References In this paper, we have studied pair production (Schwinger [1] J. 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Advances in High Energy Physics – Hindawi Publishing Corporation
Published: Apr 29, 2023
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