Pair Production in Reissner-Nordström-Anti De Sitter Black Holes*

Chinese Physics C Vol. 45, No. 6 (2021) 065105

Pair production in Reissner-Nordström-Anti de Sitter black holes*

1† 1‡ 1§ 1♮ 2♯ Jun Zhang(张俊) Yi-Yu Lin(林乙裕) Hao-Chun Liang(梁灏淳) Ke-Jia Chi(匙可佳) Chiang-Mei Chen 3¶ 1♭ Sang Pyo Kim Jia-Rui Sun(孙佳睿) 1School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510275 2Department of Physics, and Center for High Energy and High Field Physics (CHiP), National Central University, Chungli 32001 3Department of Physics, Kunsan National University, Kunsan 54150

Abstract: We studied the pair production of charged scalar particles of a five-dimensional near extremal Reissner-

Nordström-Anti de Sitter (RN-AdS5) black hole. The pair production rate and the absorption cross section ratio in full spacetime are obtained and are shown to have a concise relation with their counterparts in the near horizon region. In addition, the holographic descriptions of the pair production, both in the IR CFT in the near horizon region

and the UV CFT at the asymptotic spatial boundary of the RN-AdS5 black hole, are analyzed in the AdS2/CFT1 and AdS5/CFT4 correspondences, respectively. This work gives a complete description of scalar pair production in a near extremal RN-AdS5 black hole. Keywords: black holes, pair production, AdS/CFT correspondence DOI: 10.1088/1674-1137/abf4f6

I. INTRODUCTION black hole [3-5] and the Kerr-Newman (KN) black hole [6, 7], in which the near horizon geometry is enhanced in- The Schwinger pair production of charged particles is to AdS2 or warped AdS3 in the near extremal limit. Ow- an important QED phenomenon that is related to the va- ing to the enhanced near horizon symmetry, the explicit cuum instability and persistence in the presence of strong forms of the pair production rate and other 2-point correl- external electromagnetic fields [1]. Another important ation functions have been obtained and their holographic spontaneous pair production phenomenon is the Hawk- descriptions have been found based on the RN/CFT [8- ing radiation from black holes, which can be viewed as a 13] and KN/CFTs dualities [14-16]. In addition to tunneling process through the black hole horizon [2]. A charged black hole backgrounds, pair production has also charged black hole thus provides a natural lab in which been investigated in pure AdS or dS spacetime, see, e.g., both the Schwinger pair production and the Hawking ra- [17-20], whereas in the absence of a gravitational field, diation can occur and mix with each other. Usually the the pure Schwinger effect has been efficiently analyzed equation of motions (EoMs) of quantum fields in a gener- by using the phase-integral method [21-24]. al black hole background is difficult to solve analytically However, previous studies mainly focused on analyz- in full spacetime. However, when the symmetry of the ing spontaneous pair production in the near horizon re- spacetime geometry is enhanced under some conditions, gion of black holes in an asymptotically flat spacetime. A the problem becomes manageable; for this reason, in a charged black hole in AdS spacetime has an additional series of recent studies, the spontaneous pair production AdS symmetry at the asymptotical boundary. From the of charged particles has been systematically studied in holographic point of view, the CFT description of pair near extremal charged black holes, including the RN production has been revealed only in the near horizon re-

Received 1 February 2021; Accepted 6 April 2021; Published online 11 May 2021 * Supported by the National Natural Science Foundation of China (11675272), the Ministry of Science and Technology of the R.O.C. under the grant MOST 108- 2112-M-008-007, National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1I1A3A01063183) † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] ♮ E-mail: [email protected] ♯ E-mail: [email protected] ¶ E-mail: [email protected] ♭ E-mail: [email protected] Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main- tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub- lishing Ltd

065105-1 Jun Zhang, Yi-Yu Lin, Hao-Chun Liang et al. Chin. Phys. C 45, 065105 (2021)

gion in terms of AdS2/CFT1 (or warped AdS3/CFT2). Al- boundary and is consistent with the situation in the IR though particle pairs produced in the near horizon region CFT. Furthermore, we determined an interesting relation of black holes indeed provide important contributions to between the full pair production rate and the absorption those in full spacetime, an understanding of the whole cross section ratio via changing the roles of sources and picture is still lacking. In the present paper, we extend the operators simultaneously both in the IR and the UV study of pair production to a full near extremal RN-AdS5 CFTs. black hole background, which possesses an AdS5 geo- The rest of the paper is organized as follows. In Sec. metry at the asymptotic spatial boundary as well as an II, we provide a brief review of the bulk theory and con- AdS2 structure in the near horizon region. It is shown that sider the near horizon geometry of an RN- AdSd+1 black the radial equation of the charged scalar field propagat- hole and the EoMs of the probe charged scalar field. In ing in this spacetime can be transformed into a Heun-like Sec. III, spontaneous pair production in the near horizon differential equation and thus be solved by matching its region of near extremal RN- AdSd+1 black holes is dis- solutions in the near and far spacetime regions, using the cussed, and the 2-point functions of the charged scalar low temperature limit. Consequently, analytical forms of field (such as the retarded Green's function), pair produc- the full solutions for the pair production rate, the absorption rate, and absorption cross section ratio are calculated. tion cross section ratio, and the retarded Green's func- In Sec. IV, the full analytical solution for the radial equa- tions are obtained, and they are shown to have concise re- tion of the charged scalar field in RN-AdS black holes is lations with their counterparts calculated in the near hori- 5 obtained by applying the matching technique. Con- zon region. Based on these concise relations, numerical sequently, the full analytical forms of the pair production analysis can easily be performed, and the pair production rate, absorption cross section ratio, and retarded Green's rate in full spacetime is shown to be smaller than that in function are found, and the connections with their coun- the near horizon region, which is consistent with the as- sumption that pair production mainly comes from the terparts in the near horizon region of the black hole are black hole near horizon region. discussed. Then, in Sec. V, the dual CFTs descriptions of spontaneous pair production are both analyzed in terms of A near extremal RN-AdS5 black hole is also a very useful background for studying holographic dualities. As the AdS2/CFT1 correspondence in the IR region and the AdS5/CFT4 correspondence in the UV region, and their the near horizon AdS2 (or warped AdS3) spacetime is connections are also revealed. Finally, the conclusion and dual to a 1D CFT (or chiral CFT2), while asymptotical

physical implications are provided in Sec. VI. AdS5 spacetime is dual to another 4D CFT, the former is called IR CFT, while the latter is called UV CFT, and they are connected with each other via the holographic II. BULK THEORY renormalization group (RG) flow along the radial direc- tion [25-27]. For example, it has been shown that a near A. RN- AdS + black hole extremal RN-AdS black hole acts as a holographic model d 1 in describing typical properties of a (non)Fermi liquid at The d + 1 dimensional Einstein-Maxwell theory has = = the quantum critical point [28-31]. It is thus natural and an action (in units of c h¯ 1) as interesting to find holographic descriptions of pair pro- ∫ [ ( ) ] duction in an RN-AdS black hole both in the IR CFT in √ − 5 1 d+1 1 d(d 1) 1 µν I = d x −g R + − FµνF , the near horizon region and the UV CFT4 at the asymp- π 2 2 16 Gd+1 L gs totical AdS5 boundary. We show that the picture in the IR (1) CFT1 is very similar to those in the near extremal RN and KN black holes, and that the pair production rate and the where L is the curvature radius of the asymptotical absorption cross section ratio calculated from the AdS2 AdSd+1 spacetime, and gs is the dimensionless coupling spacetime can be matched with those from the dual IR constant of the U(1) gauge field. The dynamical equa- CFT. Regarding the UV 4D CFT, a direct comparison of tions calculations between the bulk and the boundary in terms ( ) of the AdS5/CFT4 is not made due to a lack of informa- 1 d(d − 1) 8πGd+1 λ αβ Rµν− gµνR− gµν = 4FµλFν −gµνFαβF , tion on the dual finite temperature CFT4 side. However, 2 2L2 2 ( ) gs from the bulk gravity side, the condition for pair produc- √ µν ∂µ −gF = 0, tion in the full near extremal RN-AdS5 spacetime is the violation of the Breitenlohner-Freedman (BF) bound [32, (2) 33] in AdS5 spacetime. This, on the dual 4D CFT side, corresponds to a complex conformal weight for the scal- admit the Reissner-Nordström-Anti de Sitter (RN- ar operator dual to the bulk charged scalar field, which in- AdSd+1 ) black brane (or the planar black hole) solution deed indicates instabilities for the scalar operator on the [ 34]

065105-2 Pair production in Reissner-Nordström-Anti de Sitter black holes Chin. Phys. C 45, 065105 (2021)

L2 r2 ( ) B. Near-horizon near-extremal geometry ds2 = dr2 + − f (r)dt2 + dx2 , r2 f (r) L2 i ( ) To make the following analysis convenient, let us in- d−2 d − r 2d−2 ≡ 2 2 2 A = µ 1 − o dt, (3) troduce the length scale r∗ Gd+1L Q ; then, the rd−2 d temperature can be rewritten as ( ) 2d−2 with r d r∗ T = o 1 − . (9) π 2 2d−2 4 L ro G + L2 M G + L2Q2 f (r) =1 − d 1 + d 1 , rd r2d−2 √ Note that r∗ may be treated as the “effective” radius of d − 1 g Q the inner black hole horizon though f (r∗) , 0 in general µ = s , (4) − d−2 r∗ < r 2(d 2) ro and o . The extremal condition for a degenerate hori- d 2(d − 1) r∗ zon at ro = r∗ is M = M0 ≡ . The near ex- d − 2 + 2 where ro is the radius of the outer horizon ( f (ro) = 0), µ Gd 1L is the chemical potential with dimension tremal limit of the near horizon is obtained by taking the − − / ε → [µ] = length (d 1) 2 , M is the mass, and Q is the charge of limit 0 of the transformations the black brane. We may find an explicit expression of ro d−2 = d(d − 1)r∗ for d 4 from a solution of the cubic equation, which is M − M = ε2ρ2, 0 2 o complicated, but r has a general expression in the ex- Gd+1L o τ tremal case, i.e., r∗ in IIB. The condition f (ro) = 0 gives − = ερ , − = ε ρ − ρ , = , ro r∗ o r ro ( 0) t ε (10) rd Q2 M = o + 2 d−2 (which is the Smarr-like relation re- Gd+1L ro where in general ρo is finite and ρ ∈ [ρ0,∞). lated to the first law of thermodynamics of the black = Expanding f (r) around r r , we have brane); temperature T and “surface” entropy density s of o the black brane are, respectively, d(d − 1) ( ) f (r) ≃ (ρ2 − ρ2)ε2 + O ε3 , (11) ( ) r2 o r d d − 2 G + L2Q2 o T = o 1 − d 1 , π 2 2d−2 4 L d ro ( ) the near horizon geometry is given by 1 r d−1 s = o . (5) 4Gd+1 L ρ2 − ρ2 ℓ2dρ2 r2 ds2 = − o dτ2 + + o dx2, ℓ2 2 2 2 i ρ − ρo L Moreover, the first law of thermodynamics of the dual − µ = (d 2) ρ − ρ τ, boundary d-dimensional quantum field is A ( o)d (12) ro δϵ = Tδs + µδρ , (6) c L2 where ℓ2 ≡ is defined as the square of the d(d − 1) where the “surface” energy and charge densities are, re- curvature radius of the effective AdS2 geometry. The lim- spectively, it ρo → 0 yields the extremal limit. The solution in Eq. (12) can also be written in the d − 1 ξ = ℓ2/ρ |ξ| ⩽ ξ = ϵ = M, Poincaré coordinates in terms of , ( o πLd−1 ℓ2/ρ 16√ o ), 2(d − 1)(d − 2) ρ = Q. (7)   c 8πg Ld−1   s  ( )  ℓ2  ξ2 dξ2  r2 ds2 = − 1 − dτ2 +  + o dx2, ξ2  2 2  2 i Then, it is straightforward to check the Euler relation  ξ ξ  L  o −  1 2 ( ) ξo d ( ) ϵ = ϵ + p = T s + µρ , (8) (d − 2)µℓ2 1 1 d − 1 c A = − dτ. (13) ro ξ ξo ϵ where the pressure is p = , which shows that the d − 1 The above geometry is a black brane with both local × d−1 dual d-dimensional quantum field theory on the asymp- and asymptotical topology AdS2 R (AdS2 has the , totic boundary is conformal, as expected. SL(2 R)R symmetry). The horizons of the new black

065105-3 Jun Zhang, Yi-Yu Lin, Hao-Chun Liang et al. Chin. Phys. C 45, 065105 (2021)

−iwτ+i⃗k·⃗x brane are located at ξ = ξo , and its temperature is Φ τ,⃗x,ξ = ϕ ξ . 1 ( ) ( )e (19) T = . Note that if we adopt the new coordinates n 2πξ ≡ ξ/ξ o | | ⩽ η = τ/ξ 1)

z o with z 1 and o , the metric becomes Then, the KG equation reduces to ( ) ℓ2 2 2 ( ) 2 2 2 dz ro 2 ξ2 ξ3 + 2 ds = −(1 − z )dη + + dx , (14) 2 ′′ 2 ′ 2 (w qAτ) 2 2 2 − 2 2 i ξ 1 − ϕ (ξ) − ϕ (ξ) + ξ ϕ(ξ) = m ffℓ ϕ(ξ), z 1 z L ξ2 ξ2 ξ2 e o o − 1 2 ξo and the temperature associated with the inverse period of (20) 1 η is normalized to Tñ = . π 2 2 = where the effective mass square is defined as meff C. Charged scalar field probe L2⃗k2 m2 + 2 , or the KG equation can be expressed in the z The action of a bulk probe charged scalar field Φ with ro

coordinate as mass m and charge q is ( ) 2  − 2 ∫ ( ) 2 2 ′′ 3 ′ z  1 z √ z (1 − z )ϕ (z) − 2z ϕ (z) +  wξ + q ffℓ d+1 1 ∗ ∗ α 1 2 ∗ − 2 o e S = d x −g − DαΦ D Φ − m Φ Φ , (15) 1 z z 2 2 ] 1 − z2 −m2 ℓ2 ϕ(z) = 0, (21) eff z2 where Dα ≡ ∇α − iqAα with ∇α being the covariant deriv- ative in curved spacetime. The corresponding Klein-Gor- where the effective charge of the probe field is qeff ≡ don (KG) equation is µℓ (d − 2) q. The singularities of Eq. (21) are located at α α 2 ro (∇α − iqAα)(∇ − iqA )Φ = m Φ. (16) z = 0,z = 1 and z = ∞. To find the solutions, we determine the indices at α¯ Moreover, the radial flux of the probe field is each singular point. For z → 0, setting ϕ(z) ∼ z , the lead-

√ ing terms in Eq. (21) are F = − rr Φ ∗Φ∗ − Φ∗ Φ . i gg ( Dr Dr ) (17) 2ϕ′′ + 2 − 2 ℓ2ϕ = , z (z) (qeff meff) (z) 0 (22) In the RN-AdSd+1 background (3), assuming Φ(t,⃗x,r) = ϕ −iωt+i⃗k·⃗x (r)e , the KG Eq. (16) has the radial form which gives ( ) ( ) √ d−1 d+1 1 1 L r α = + 2 − 2 ℓ2 ∂ f (r)∂ ϕ(r) ¯ 1 4(m ff q ff) r r Ld+1 r 2 2 e e ( ) √ L2(ω + qA )2 L2 1 1 1 + t − m2 − ⃗k2 ϕ(r) = 0. (18) ≡ 1 + 4m ˜ 2 ℓ2 ≡ ν. (23) r2 f (r) r2 2 2 eff 2

→ − ϕ ∼ + β¯ The solutions to Eq. (18) cannot be directly found in For z 1, setting (z) (1 z) , Eq. (21) reduces to terms of special functions in the full spacetime region. In (wξ − 2q ﬀℓ)2 what follows, we solve it in different regions and match 2(1 + z)ϕ′′(z) + 2ϕ′(z) + o e ϕ(z) = 0, (24) 2(1 + z) these solutions to obtain the full solution.

and the index is III. PAIR PRODUCTION IN THE INNER ADS2 ( ) ( ξ ) β¯ = w o − ℓ = w − ℓ . A. Near-horizon solutions i qeﬀ i qeﬀ (25) 2 4πTn Firstly, we analyze the near horizon, near extreme region (13) and solve the KG Eq. (16) by expanding the Finally, for z → 1, setting ϕ(z) ∼ (1 − z)γ¯ , Eq. (21) reduces

scalar field as to

1) Note that from Eq. (10), one has ω = εw, which may indicate that if one considered the finite frequency probe field in the τ, ξ coordinates, the probe field in the original t, r coordinates is automatically driven into the low frequency region.

065105-4 Pair production in Reissner-Nordström-Anti de Sitter black holes Chin. Phys. C 45, 065105 (2021)

2 ′′ ′ (wξ ) is obtained. Further, imposing the ingoing boundary con- 2(1 − z)ϕ (z) − 2ϕ (z) + o ϕ(z) = 0, (26) 2(1 − z) dition at the black brane horizon z = 1 requires γ¯ = ξ − w o = − w from which i i . 2 4πTn wξ w ω/ε ω ω γ = o = = = = Also, note that Eq. (21) can be rewritten in a more ex- ¯ i i i i 2 i 2 4πTn 4π/(2πξo) 2ερo/ℓ 4πT (27) plicit form as

  ( )  1 2 1 2 2   2 ℓ2 (wξo − 2qeﬀℓ) w ξ  ′′ 1 1 ′ m˜ ﬀ o  ϕ(z) ϕ (z) + + ϕ (z) +  e + 2 + 2  = 0, (28) z + 1 z − 1  z z + 1 z − 1  z(z + 1)(z − 1) which becomes the Fuchs equation with three canonical singularities a , a and a , as follows: 1 2 3 ( ) ( 1 − α¯ − α¯ 1 − β¯ − β¯ 1 − γ¯ − γ¯ α¯ α¯ (a − a )(a − a ) β¯ β¯ (a − a )(a − a ) ϕ′′(z) + 1 2 + 1 2 + 1 2 ϕ′(z) + 1 2 1 2 1 3 + 1 2 2 3 2 1 z − a1 z − a2 z − a3 z − a1 z − a2 ) γ¯ γ¯ (a − a )(a − a ) ϕ(z) + 1 2 3 1 3 2 = 0, (29) z − a3 (z − a1)(z − a2)(z − a3)

= = − = via the conformal coordinate transformation where a1 0, a2 1, and a3 1 and

( )α ( )β¯ ξ − ℓ ¯ 1 1 1 1 w o 2qeff (a2 − a3)(z − a1) z − a1 z − a2 α¯ 1 = ν, α¯ 2 = ∓ ν, β¯1 = −β¯2 = i , ζ = , ϕ(z) = ψ(ζ), 2 2 2 (a2 − a1)(z − a3) z − a3 z − a3 wξo (32) γ¯ = −γ¯ = i , 1 2 2 (30) where α˜ = α¯ 1 + β¯1 + γ¯1,β˜ = α¯ 1 + β¯1 + γ¯2 and γ˜ = 1 + α¯ 1 − α¯ 2 . (Note that one can freely choose the indices i = 1, 2 for α + α + β¯ + β¯ + γ + γ = and ¯ 1 ¯ 2 1 2 ¯1 ¯2 1 is satisfied. The Fuchs α¯ i , β¯i and γ¯i .) Eq. (29) can be transformed into the standard hypergeo- 1 For Eq. (28), we have ζ = 2z/(z − 1),α˜ = ν+ metric function 2 1 iwξo − iqeffℓ, β˜ = ν − iqeffℓ, γ˜ = 1 2ν. Therefore, [ ] 2 ζ − ζ ψ′′ ζ + γ − + α + β˜ ζ ψ′ ζ − αβψ˜ ζ = , the explicit solutions in the near horizon near extreme re- (1 ) ( ) ˜ (1 ˜ ) ( ) ˜ ( ) 0 (31) gion are

( ) wξo ( ) ( ) 1 − ff ℓ +ν i 2 iqe z 2 z + 1 1 1 2z ϕ(z) =c F + ν + iwξ − iq ffℓ, + ν − iq ffℓ;1 + 2ν; 1 z − 1 z − 1 2 1 2 o e 2 e z − 1

( ) wξo ( ) ( ) 1 − ff ℓ −ν i 2 iqe z 2 z + 1 1 1 2z + c F − ν + iwξ − iq ffℓ, − ν − iq ffℓ;1 − 2ν; . 2 z − 1 z − 1 2 1 2 o e 2 e z − 1 (33)

(in) −i w (out) i w ϕ = − 4πTn + − 4πTn , B. 2-point correlators from AdS2 (z) cH (1 z) cH (1 z) (34) = At the horizon of the AdS2 black brane, z 1, Eq.

(33) is expanded as follows: where

( ) w Γ(1 + 2ν)Γ i 1 w 1 w 2πT (in) − −ν−i +iq ff ℓ − −ν+i n c = c (−) 2 2πTn e 2 2 4πTn ( ) ( ) H 1 1 1 w Γ + ν + iqeffℓ Γ + ν + i − iqeffℓ 2 2 2πTn

065105-5 Jun Zhang, Yi-Yu Lin, Hao-Chun Liang et al. Chin. Phys. C 45, 065105 (2021)

( ) w Γ(1 − 2ν)Γ i 1 w 1 w 2πT − +ν−i +iq ff ℓ − +ν+i n +c (−) 2 2πTn e 2 2 4πTn ( ) ( ), (35) 2 1 1 w Γ − ν + iqeffℓ Γ − ν + i − iqeffℓ 2 2 2πTn and ( ) w Γ(1 + 2ν)Γ −i 1 w 1 w 2πT (out) − −ν−i +iq ff ℓ − −ν−i n c =c (−) 2 2πTn e 2 2 4πTn ( ) ( ) H 1 1 1 w Γ + ν − iq ffℓ Γ + ν − i + iq ffℓ 2 e 2 2πT e ( n) w Γ(1 − 2ν)Γ −i 1 w 1 w 2πT − +ν−i +iq ff ℓ − +ν−i n + c (−) 2 2πTn e 2 2 4πTn ( ) ( ). (36) 2 1 1 w Γ − ν − iqeffℓ Γ − ν − i + iqeffℓ 2 2 2πTn

→ σ In contrast, at the AdS2 boundary, z 0, the asymp- sorption cross section ratio abs can be calculated from totic expansion of Eq. (33) is the radial flux by imposing different boundary conditions ( ) 1 w 1 1 w 1 d−1 −ν+ − ﬀ ℓ −ν +ν+ − ﬀ ℓ +ν r 2 i 4πT iqe 2 2 i 4πT iqe 2 o 2 ∗ ∗ ϕ(z) =c2(−) n z + c1(−) n z F = i (1 − z )(Φ∂ Φ − Φ ∂ Φ), (39) L z z ⃗ 1 −ν ⃗ 1 +ν =A(w,k)z 2 + B(w,k)z 2 , (37)

which gives where A is the source of the charged scalar field in the ( ) d−1 B Oˆ ,⃗ (in) ro (in) bulk AdS2, while is the response or the operator (w k) F =2|ν| |c |2, B L B (in the momentum space) of the boundary CFT1 (i.e., the ( ) d−1 ro IR CFT) dual to the charged scalar field in the bulk AdS2 F (out) = − |ν| | (out)|2, B 2 cB background. Note that in order to obtain the propagating ( L ) ν w r d−1 modes, should be purely imaginary, which can be set as F (in) = o |c(in)|2, (out) 1 −i|ν| (in) 1 +i|ν| H π H ν ≡ i|ν|, i.e., ϕ(z) = c z 2 + c z 2 . It was shown in 2 Tn L B B ( ) ν d−1 [3] that the condition of an imaginary is equivalent to F (out) = − w ro | (out)|2, H cH (40) the violation of the BF bound in AdS2 spacetime, namely 2πTn L

(in) (out) 2 1 where F and F are the ingoing and outgoing m˜ ﬀ < − , (38) B B e 4ℓ2 F (in) F (out) fluxes at the AdS2 boundary, while H and H are which corresponds to a complex conformal weight of the the ingoing and outgoing fluxes at the AdS2 black brane horizon, respectively. scalar operator in the dual IR CFT. 2 The Schwinger pair production rate bAdS2 can be 1. Pair production rate and absorption cross computed either by choosing the inner boundary condition or the outer boundary condition, which gives the section ratio same result [3], e.g., by adopting the outer boundary con- | |2 F (in) = (in) = ⇒ = The Schwinger pair production rate b and the ab- dition, i.e., B 0, (cB 0 c1 0),

( ) ( ) 2 1 1 w 2 Γ − |ν| + ℓ Γ − |ν| + − ℓ F (out) (out) i iqeff i i iqeff 2 4πTn|ν| c 8πTn|ν| 2 2 2πTn AdS2 = B = B = ( ) b F (in) w c(in) w w H H Γ(1 − 2i|ν|)Γ i 2πT ( ) n w 2sinh(2π|ν|)sinh 2T = ( n ). (41) w coshπ(|ν| − qeffℓ)coshπ |ν| − + qeffℓ 2πTn

065105-6 Pair production in Reissner-Nordström-Anti de Sitter black holes Chin. Phys. C 45, 065105 (2021)

Similarly, by adopting the outer boundary condition, the absorption cross section ratio is computed as

( ) ( ) 2 1 1 w 2 Γ − i|ν| − iq ffℓ Γ − i|ν| − i + iq ffℓ F (out) π |ν| (out) π |ν| e π e AdS B 4 Tn cB 8 Tn 2 2 2 Tn σ 2 = = = ( ) abs F (out) w c(out) w w H H Γ(1 − 2i|ν|)Γ −i 2πT ( ) n w 2sinh(2π|ν|)sinh 2T = ( n ). (42) w coshπ(|ν| + qeffℓ)coshπ |ν| + − qeffℓ 2πTn

The pair production rate and the absorption cross section The two-point retarded Green's function of the boundary operator dual to the bulk charged scalar field is com- ratio are connected by the simple relation

puted through

2 AdS2 b = −σabs(|ν| → −|ν|). (43) AdS2 ,⃗ ≡⟨OˆO⟩ˆ GR (w k) R B ,⃗ It was shown that the abovementioned relation also holds = − F | ∼ (w k) 2 z→0 ⃗ for a charged scalar field [11] and for a charged spinor A(w,k) field [4], both in a four-dimensional near extremal RN + contact terms (44)

black hole. F (out) = by taking the inner boundary condition, i.e., H 0, 2. Retarded Green's function which gives

( ) ( ) 1 1 w Γ(1 + 2ν)Γ − ν − iq ffℓ Γ − ν − i + iq ffℓ c 2 e 2 2πT e 2 = (−)1−2ν 2−2ν ( ) ( n ). (45) c1 1 1 w Γ(1 − 2ν)Γ + ν − iqeffℓ Γ + ν − i + iqeffℓ 2 2 2πTn Thus, the two-point retarded Green's function is

( ) ( ) Γ − ν Γ 1 + ν − ℓ Γ 1 + ν − w + ℓ ⃗ (1 2 ) iqeff i iqeff B(ω,k) ν c1 ν− ν 2 2 2πTn GAdS2 (w,⃗k) ∼ = (−)2 = (−)4 1 22 ( ) ( ). (46) R ⃗ A(ω,k) c2 1 1 w Γ(1 + 2ν)Γ − ν − iqeffℓ Γ − ν − i + iqeffℓ 2 2 2πTn

In addition, the corresponding boundary condition scalar field perturbation. F (in) = F (out) = ( B 0 and H 0) is used to obtain the quasinormal modes of the charged scalar field in AdS2 spacetime, IV. PAIR PRODUCTION IN THE which correspond to the poles of the retarded Green's ASYMPTOTICAL AdS function of dual operators (with complex conformal 5 1 In this section, we describe our study of the pair pro- weight hR = + ν) in the IR CFT, namely

2 duction for the whole spacetime of RN-AdS5. Like before, we need to solve the corresponding radial Klein 1 w + ν − i + iqeﬀℓ = −N ⇒ w equation for the scalar field. 2 2πT n To find the solution in the full region, we focus on =2πT (q ﬀℓ − iN − ih ), N = 0,1,··· . (47) n e R d = 4 and the near extremal cases. By introducing the co- r2 ordinate transformation ϱ = (and denoting M′ = Eq. (47) gives the quasinormal modes of the charged M′

065105-7 Jun Zhang, Yi-Yu Lin, Hao-Chun Liang et al. Chin. Phys. C 45, 065105 (2021)

2 2 ro r∗ G + L2 M, ϱ = and ϱ∗ = ), the radial Eq. (18) can be expressed as d 1 o M′ M′ ( ) ( ) 1 1 1 ϱ(ωϱ˜ − q˜µϱ )2 m˜ 2ϱ + k˜2 ϕ′′(ϱ) + + + ϕ′ (ϱ) + o − ϕ(ϱ) = 0, (48) ϱ − ϱ ϱ − ϱ ϱ − ϱ 2 2 2 ϱ − ϱ ϱ − ϱ ϱ − ϱ 1 2 o (ϱ − ϱ1) (ϱ − ϱ2) (ϱ − ϱo) ( 1)( 2)( o)

ω − µ ω L2(ω + qµ) L2q α = ( ˜ q˜ ) = i , ω = √ = √ 1,2 i √ where the parameters are ˜ , q˜ , ab ϱo 4πT 2 M′ 2 M′ L2 ⃗k √ Lm ω + − µ + m˜ = , k˜ = √ and ( ˜ (1 b) q˜ ) 1 b ′ β , = i √ , 2 1 2 2 M (a − b)b ϱo √ √ (ω ˜ (1 + a) − q˜µ) 1 + a 1 1 ϱ∗3 γ = √ , ϱ = − ϱ − ϱ 2 + 8 , 1,2 i (54) 1 o o (b − a)a ϱo 2 2 ϱo √ 1 1 ϱ∗3 where the index “1” corresponds to the “+” sign, and the ϱ = − ϱ ϱ 2 + . 2 o + o 8 (49) − 2 2 ϱo index “2” corresponds to the “ ” sign. Then, decompos- ϕ ing (y) as

( )α ( )γ Further, defining another coordinate y 1 y − a 1 ϕ(y) = R(y), (55) ϱ − ϱ ϱ − ϱ ϱ − ϱ y − b y − b y ≡ o , a ≡ 2 o , b ≡ 1 o , (50) ϱo ϱo ϱo

we obtain

the metric of the RN-AdS5 black hole becomes ( ) 1 1 1 2bα 2(a − b)γ R′′(y) + + + − 1 + 1 R′ (y) y y − a y − b y(y − b) (y − a)(y − b) L2dy2 r2 ( ) ds2 = + o (1 + y) − f (y)dt2 + dx2 , + = , 4(1 + y)2 f (y) L2 i V2R(y) 0 (56) µy A = dt, (51) 1 + y where

where (2 + 3a − a2y)ω ˜ 2 − 4(a + 1)ω˜ q˜µ+(a + 2)q˜2µ2 V2 ≡ − − M1, ϱ 2 − − 2 M′ Q′2 oa y(y a)(y b) f (y) =1 − (1 + y)−2 + (1 + y)−3, (57) 4 6 ro ro

′2 2 2 Q =Gd+1L Q . (52) and

(b − a)γ 2b(a − b)α γ m˜ 2(y + 1)ϱ + k˜2 Moreover, Eq. (48) transforms into 1 1 1 o M = + + 1 − − 2 ϱ − − ( ) y(y a)(y b) y(y − a)(y − b) oy(y a)(y b) ′′ 1 1 1 ′ bα ϕ (y) + + + ϕ (y) + 1 . y y − a y − b y(y − a)(y − b) ( (58) (ω˜ (y + 1) − q˜µ)2(y + 1) + y2(y − a)2(y − b)2 ) A. Solutions in the near and far regions 2 + ϱ + ˜2 ϕ −m˜ (y 1) o k (y) = . 0 (53) We divide the regions into a near region y(y − a)(y − b) ϱo ϱ − ϱ y = o ≪ 1, (59) To solve ϕ(y), first, we determine its exponents at the ϱo corresponding singularities 0, a, b, and ∞, which are α β γ δ 1,2 , 1,2 , 1,2 , and 1,2 , respectively, and a far region

065105-8 Pair production in Reissner-Nordström-Anti de Sitter black holes Chin. Phys. C 45, 065105 (2021)

ϱ − ϱ ( ) = o ≫ − , y a (60) ′′ 2 1 4bα1 2ibqeﬀℓ ′ ϱo R (y) + + − + R (y) + V R(y) = 0, y y − b y(y − b) y(y − b) 3 (66)

and an overlapping region, in which

where

−a ≪ 1. (61) 2 2 ω˜ 4b α1 (α1 − iqeffℓ) V3 ≡ + ϱ y(y − b)2 y2(y − b)2 − ≪ o The physical reasoning of a 1 relies on the observa- α − ℓ 2 + ϱ + ˜2 − b(2 1 iqeff ) − m˜ (y 1) o k , (67) tion that the temperature of a black hole is 2 2 y (y − b) ϱoy (y − b) ( ) r ϱ∗3 T = o − , (62) qµ 2 1 3 i˜ πL ϱo (where the relation γ1 ≈ α1 + √ = α1 − iqeffℓ when b ϱo | | ≪ − → → a 1 is used). Similarly, Eq. (66) has a solution in which gives a 0 for T 0, as

terms of the hypergeometric function as √ 2 3 1 8πL T ( )λ( ( ) −a = − 9 − . (63) y ν− 1 ′ ′ ′ y ϕ = − 2 α ,β γ 2 2 ro (y) 1 c5y 2F1 ; ; b ( b )) −ν− 1 ′ ′ ′ ′ ′ y + c y 2 F 1 − γ + α ,1 − γ + β ;2 − γ ; (68) We want to point out that the matching condition in 6 2 1 b Eq. (61) indicates that the near extremal condition is es- 1 1 sential for matching the solutions in the near and far re- in which α′ = + ν + ∆ + λ, β′ = + ν − ∆ + λ, γ′ = 1 + 2ν, gions. It is not necessary for the frequency to be infin- 2 √ 2 itely small; however, the frequency should definitely not √ ω˜ 2 ∆ = + 2 λ = ℓ 2 − be very large compared with the temperature T; other- 1 m˜ , and (iqeﬀ ) . ϱob wise, the backreaction to the background geometry cannot be ignored. B. Near-far matching

Now we find the approximate solutions in different In the overlapping region, one has the inequalities regions. First, by using the near region condition (y ≪ 1),

Eq. (53) reduces to −a ≪ y ≪ 1 < −b (69) ( ) √ ′′ 1 1 ′ 3 1 8πL2T ϕ (y) + + ϕ (y) (1 < −b since −b = + 9 − → 3, as T → 0), y y − a 2 2 r ( ) o 2 2 2 (aα − iq ﬀℓy) m˜ ϱ + k˜ a y + − 1 e + o ϕ y = . (64) → → 2 ( ) 0 which means 0 and 0, which transforms Eqs. y2(y − a) ϱoby(y − a) y b (65) and (68) into the following forms: the near region-

Obviously Eq. (64) can be solved by the hypergeometric solution function as ( Γ γ Γ β − α ϕ = − −α ( ) ( ) ( ( ) (y) ( 1) c3 Γ β Γ γ − α α − ℓ α y ( ) ( ) ϕ = − 1 iqeff 1 α,β γ (y) (y a) c3y 2F1 ; ; ) a Γ − γ Γ β − α ( )) −1+γ−α (2 ) ( ) − 1 −ν + − 2 −α1 y ( 1) c4 y + c4y 2F1 1 − γ + α,1 − γ + β;2 − γ; , (65) Γ(1 − γ + β)Γ(1 − α) a ( Γ γ Γ α − β + − −β ( ) ( ) 1 1 ( 1) c3 Γ α Γ γ − β where α = + ν + 2α − iq ffℓ, β = − ν + 2α − iq ffℓ, and ( ) ( ) 2 1 √ e 2 1 e ) γ = + α ℓ = / Γ − γ Γ α − β 1 2 1 , and L 12 is the radius of the effective −1+γ−β (2 ) ( ) − 1 +ν + − 2 ( 1) c4 Γ − γ + α Γ − β y (70) AdS2 geometry in the near horizon region of the RN- (1 ) (1 ) AdS5 black hole. Second, in the far region, by using the ≫ − condition (y a), Eq. (56) can turn into and the far region solution

065105-9 Jun Zhang, Yi-Yu Lin, Hao-Chun Liang et al. Chin. Phys. C 45, 065105 (2021)

λ − 1 +ν λ − 1 −ν ϕ(y) → (−1) c5y 2 + (−1) c6y 2 . (71) which means

4 Comparing these two identities, one finds the connec- m2 ⩽ − , (77) L2 tion relations

( namely, the violation of the BF bound in AdS spacetime. Γ(γ)Γ(α − β) 5 c =(−1)−λ (−1)−βc Therefore, the corresponding outgoing and ingoing 5 3 Γ(α)Γ(γ − β) ) fluxes at the horizon and the boundary of the near ex- − +γ−β Γ(2 − γ)Γ(α − β) +(−1) 1 c , (72) tremal RN-AdS5 black brane are 4 Γ(1 − γ + α)Γ(1 − β) πω 2 3ω 4 Tr 2r ( D(out) = o |c |2 = o |c |2, H 3 3 3 −λ −α Γ(γ)Γ(β − α) abL L c6 =(−1) (−1) c3 2 3 Γ(β)Γ(γ − α) 4πωTro 2r ω ) D(in) = − |c |2 = − o |c |2, Γ − γ Γ β − α H abL 4 L3 4 −1+γ−α (2 ) ( ) +(−1) c4 . (73) 4 Γ − γ + β Γ − α 4|∆|ro 2 (1 ) (1 ) D(out) = A(ω, ˜ k˜) , B L5 4 4|∆|r 2 D(in) = − o B(ω, ˜ k˜) . (78) B. 2-point correlators from AdS B L5 5

σAdS5 1. Pair production and absorption cross section The absorption cross section ratio abs and the 2 Schwinger pair production rate bAdS5 can be calculated Now we denote the radial flux of the charged scalar D by choosing the inner boundary condition D(out) = 0 and field in metric (51) as : H = ( c3 0) and are given by 4 + 3 ( ) 2iro(1 y) f (y) ∗ ∗ D = ϕ(y)∂ ϕ (y) − ϕ (y)∂ ϕ(y) . (74) (in) L5 y y D 2TL2 |ν|sinh(2π|∆|) σAdS5 = H = σ , abs ( ) abs D(in) −1 2 B ν ∆ λ AdS2 + −ν ∆ λ In the near horizon limit, i.e.,y → 0, Eq. (65) reduces ro H ( ; ; ) GR H ( ; ; ) to (79)

α α − ℓ −α α − ℓ ϕ = 1 − 1 iqeﬀ + 1 − 1 iqeﬀ , (y) c3y (y a) c4y (y a) (75) and where the first part is the outgoing mode, and the second D(in) 2 |ν| π|∆| 2 H 2TL sinh(2 ) 2 part is the ingoing mode. Further, the asymptotic form of bAdS5 = = bAdS2 , D(out) 2 ϕ(y) at the boundary (y → ∞) of the AdS spacetime res- −ν −∆ λ AdS2 + ν −∆ λ 5 B ro H ( ; ; )GR H ( ; ; ) ults in the form (80)

ϕ(y) = A(ω, ˜ k˜)y−1+∆ + B(ω, ˜ k˜)y−1−∆, (76) where H (x;y;z) denotes a function where A(ω, ˜ k˜) is the source of the charged scalar field in Γ(1 + 2x) the bulk RN-AdS black hole, while B(ω, ˜ k˜) is the re- H (x;y;z) ≡ (−1)2x2x ( ) ( ), (81) 5 1 1 sponse (the operator) of the boundary CFT4 (i.e., the UV Γ + x − y + z Γ + x − y − z CFT) dual to the charged scalar field in the bulk. As in 2 2 the case of the AdS2 spacetime, the condition for the propagating modes requires an imaginary ∆, i.e., ∆ = i|∆|, and

( ) ( ) 1 1 ω Γ + ν − iqeffℓ Γ + ν − i + iqeffℓ ν− ν Γ(1 − 2ν) 2 2 2πT GAdS2 = (−1)4 122 ( ) ( ) , (82) R Γ(1 + 2ν) 1 1 ω Γ − ν − iq ffℓ Γ − ν − i +iq ffℓ 2 e 2 2πT e

065105-10 Pair production in Reissner-Nordström-Anti de Sitter black holes Chin. Phys. C 45, 065105 (2021) which is exactly the retarded Green's function in Eq. (46) the Schwinger effect mainly occurs in the near horizon

of the IR CFT in the near horizon, near extremal region. region. 2 AdS2 Furthermore, σabs and b are exactly the absorption cross section ratio and the mean number of produced 2. Retarded Green's function pairs of the corresponding IR CFT obtained from Eqs. To calculate the retarded Green's function, an ingo- (42) and (41). We find a relationship ing boundary condition is required, namely c = 0. Then, 3 2 AdS bAdS5 = −σ 5 |ν| → −|ν|,|∆| → −|∆| , from Eqs. (72) and (73), the connection relations are abs ( ) (83)

Γ − γ Γ α − β which is similar to Eq. (43) except for a combined change −1+γ−β−λ (2 ) ( ) c5 = (−1) c4 , (84) in signs in both |ν| and |∆|. Γ(1 − γ + α)Γ(1 − β) With Eq. (80) at hand we can easily investigate the re-

lationship between the pair production rate in the near ho- Γ − γ Γ β − α rizon and that for the whole spacetime of RN-AdS . As = − −1+γ−α−λ (2 ) ( ) . 5 c6 ( 1) c4 Γ − γ + β Γ − α (85) shown in Fig. 1, we can see that the mean number of pro- (1 ) (1 ) duced pairs for the whole spacetime is less than that from near horizon region. Moreover, with increasing charge of Substituting Eqs. (84) and (85) into Eq. (68) and taking → ∞ the scalar field, the corresponding ratio becomes smaller, y , namely the boundary of the AdS5 spacetime, one which is consistent with previous assumptions stating that obtains

( Γ − γ Γ α − β Γ γ′ Γ α′ − β′ Γ − γ Γ β − α Γ − γ′ Γ α′ − β′ ) q ﬀ ℓ− λ− +∆ (2 ) ( ) ( ) ( ) (2 ) ( ) (2 ) ( ) A(ω, ˜ k˜) =(−1)i e 2 1 c + , 4 Γ(1 − γ + α)Γ(1 − β) Γ(α′)Γ(γ′ − β′) Γ(1 − γ + β)Γ(1 − α) Γ(1 − γ′ + α′)Γ(1 − β′)

( Γ − γ Γ α − β Γ γ′ Γ β′ − α′ Γ − γ Γ β − α Γ − γ′ Γ β′ − α′ ) q ﬀ ℓ− λ− −∆ (2 ) ( ) ( ) ( ) (2 ) ( ) (2 ) ( ) B(ω, ˜ k˜) =(−1)i e 2 1 c + . (86) 4 Γ(1 − γ + α)Γ(1 − β) Γ(β′)Γ(γ′ − α′) Γ(1 − γ + β)Γ(1 − α) Γ(1 − γ′ + β′)Γ(1 − α′)

Therefore, the retarded Green's function of the boundary CFT4 is given by

Γ α − β Γ γ′ Γ β′ − α′ Γ β − α Γ − γ′ Γ β′ − α′ ( ) ( ) ( ) + ( ) (2 ) ( ) ′ ′ ′ ′ ′ ′ B(ω, ˜ k˜) − ∆ Γ(1 − γ + α)Γ(1 − β) Γ(β )Γ(γ − α ) Γ(1 − γ + β)Γ(1 − α) Γ(1 − γ + β )Γ(1 − α ) GAdS5 ∼ = (−1) 2 , (87) R ω, ˜ Γ α − β Γ γ′ Γ α′ − β′ Γ β − α Γ − γ′ Γ α′ − β′ A( ˜ k) ( ) ( ) ( ) + ( ) (2 ) ( ) Γ(1 − γ + α)Γ(1 − β) Γ(α′)Γ(γ′ − β′) Γ(1 − γ + β)Γ(1 − α) Γ(1 − γ′ + α′)Γ(1 − β′) which is further simplified into

ν ∆ λ + −ν ∆ λ AdS2 − ∆ Γ(−2∆) H ( ; ; ) H ( ; ; )G AdS5 ∼ − 2 R . GR ( 1) (88) Γ(2∆) ν −∆ λ + −ν −∆ λ AdS2 H ( ; ; ) H ( ; ; )GR

Fig. 1. (color online) Ratio of mean number of produced pairs for the whole spacetime to that in the near horizon region as a function 2 2 2 of ωL /ro for different values of qeﬀ ℓ with TL /ro = 0.001, ν = 0.1i, and ∆ = 0.1i(left); TL /ro = 0.001, ν = 0.01i, and ∆ = 0.01i(middle); TL2/r = 0.01, ν = 0.1i, and ∆ = 0.1i(right). o

065105-11 Jun Zhang, Yi-Yu Lin, Hao-Chun Liang et al. Chin. Phys. C 45, 065105 (2021)

V. CFTS DESCRIPTIONS IN IR AND the calculations between the bulk gravity and the bound- UV REGIONS ary CFT sides. Nevertheless, from Eqs. (79) and (80), σAdS5 both the absorption cross section ratio abs and the From the AdS/CFT correspondence, the IR CFT1 in AdS 2 Schwinger mean number of produced pairs b 5 calcu- the near horizon, near extremal limit and the UV CFT4 at lated from the bulk near extremal RN-AdS5 black hole the asymptotic boundary of the RN-AdS5 black hole can be connected by the holographic RG flow [26, 27]. The have a simple proportional relation with their counter- CFT description of the Schwinger pair production in the parts in the near horizon region. Moreover, the violation of the BF bound (77) in AdS spacetime indicates the IR region of charged black holes has been systematically 5 complex conformal weights ∆¯ = 2 + 2i|∆| of the scalar op- studied in a series of previous works [3, 4, 6, 7]. Herein, O¯ we address the dual CFTs descriptions in the UV region erator in the UV 4D CFT at the asymptotic spatial boundary of the RN-AdS black hole, which also indic- and compare them with those in the IR region. 5 ates that, to have pair production in the full bulk space- The IR CFT of the RN-AdS black hole is very simil- 1 time, the corresponding operators in the UV CFT should ar to that of the RN black hole in an asymptotically flat be unstable. Interestingly, Eq. (83) shows that under the spacetime, as CFT can be viewed as a chiral part of 1 interchange between the roles of source and operator both CFT2, which has the universal structures in its correla- 1 tion functions. For instance, the absorption cross section in the IR and UV CFTs at the same time, namely hL,R = + O 2 of a scalar operator in 2D CFT has the universal form 1 i|ν| → − i|ν| and ∆¯ = 2 + 2i|∆| → 2 − 2i|∆|, the full absorp- 2 ( ) AdS5 − − tion cross section ratio σ and the Schwinger pair pro- 2hL 1 2hR 1 abs (2πTL) (2πTR) ωL − qLΩL ωR − qRΩR 2 σ ∼ sinh + duction rate bAdS5 are interchanged with each other only Γ(2hL) Γ(2hR) 2TL 2TR ( ) ( ) up to a minus sign. Note that both the charge and the ω − q Ω 2 ω − q Ω 2 × Γ + L L L Γ + R R R , mass of the scalar particle contribute to the conformal hL i π hR i π 2 TL 2 TR weights hL,R of the scalar operator in the dual IR CFT; (89) however, only the mass contributes to the conformal weight ∆¯ of the scalar operator in the dual UV CFT. Ac- where (hL,hR), (ωL,ωR), and (qL,qR) are the left- and tually, it can be seen from the expressions of the con- right-hand conformal weights, excited energies, charges formal weights that the non-zero charge and mass for the associated with operator O, respectively, while (TL,TR) scalar field are crucial for the violation of the BF bound and (ΩL,ΩR) are the temperatures and chemical poten- in the corresponding AdS spacetimes and hence guaran- tials of the corresponding left- and right-hand sectors of tee the existence of the Schwinger pair production. the 2D CFT. Further identifying the variations in the However, when the charge of the particle is zero, there black hole area entropy with those of the CFT microscop- will be no Schwinger effect, except for an exponentially δ = δ ic entropy, namely S BH S CFT , one derives suppressed Hawking radiation in near extremal black

holes. δM Ω δQ ω − q Ω ω − q Ω − H = L L L + R R R , (90) TH TH TL TR VI. SUMMARY AND DISCUSSION where the left hand side of Eq. (90) is calculated with co- In this paper, we describe our study of the spontaneous scalar pair production in a near extremal RN-AdS ordinate (14), for which δM = ξow, δQ = q, TH = T˜n , and 5 2 black hole that possesses an AdS structure in the IR re- ΩH = 2µℓ /ro , and thus, it is equal to w/Tn − 2πqeﬀℓ. 2 gion and an AdS5 geometry in the UV region. Moreover, the violation of the BF bound in AdS2 makes the conformal weights of the scalar operator O dual to ϕ a We firstly calculated the mean number of produced 1 pairs (see Eq. (41)) in the near horizon region, which has complex, which can be chosen as hL = hR = + i|ν|, even 2 an AdS2 structure. The retarded Green's function (see Eq. without further knowledge about the central charge and (46)) has also been obtained for this region. Then, we (TL,TR) of the IR CFT dual to the near extremal RN- solved the equation for the whole spacetime of the near AdS5 black hole. One can also see that the absorption extremal RN-AdS5 black hole by using the matching cross section ratio (42) in the AdS2 spacetime has the technique. The matching condition we chose is the low form of Eq. (89) up to some coefficients, depending on temperature limit, i.e., the near extremal limit of the black the mass and charge of the scalar field. hole. Therefore, the greybody factor in Eq. (79) and the In contrast, the absorption cross section and retarded mean number of produced pairs in Eq. (80) for the whole Green's functions in a general 4D finite temperature CFT spacetime are not merely valid for the low frequency lim- cannot be as easily calculated in momentum space as in it, and one can easily apply our calculation to the RN-dS the 2D CFT. Thus, it is not straightforward to compare black hole, which was recently described in the low fre-

065105-12 Pair production in Reissner-Nordström-Anti de Sitter black holes Chin. Phys. C 45, 065105 (2021)

quency limit [35]. Moreover, the retarded Green's func- pondence in the IR and the AdS5/CFT4 duality in the UV tion for an RN-AdS black hole has been calculated (see regions, and consistent results and new connections Eq. (88)), which again is valid at finite frequency, and its between the pair production rate and the absorption cross corresponding value has only been investigated in the low section ratio are found, although the related information frequency limit [31] before. Interestingly, we found that computed from the finite temperature 4D CFT is incom- there exists a very explicit relationship between the mean plete. This work has successfully generalized the study of number of produced pairs (see also Eq. (80)) for the pair production in charged black holes to the full space- whole spacetime and that in the near horizon region, time and provided new insights for a complete under- which enables us to easily compare the pair production standing of the pair production process in curved space-

rates of these two regions. We showed that, for an near- time. extremal RN-AdS5 black hole, the dominant contribution to the pair production rate mainly comes from the near ACKNOWLEDGEMENT horizon region, as expected. Moreover, the CFT descriptions of the pair produc- We would like to thank Shu Lin, Rong-Xin Miao, and tion are investigated both from the AdS2/CFT1 corres- Yuan Sun for useful discussions.

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