Measurements of natCd(γ, x) reaction cross sections and isomer ratio of 115m,gCd with the bremsstrahlung end-point energies of 50 and 60 MeV

Muhammad Nadeem; Md. Shakilur Rahman; Muhammad Shahid; Guinyun Kim; Haladhara Naik; Nguyen Thi Hien

doi:10.1088/1674-1137/ac256b

Chinese Physics C> 2021, Vol. 45> Issue(12) : 124002 DOI: 10.1088/1674-1137/ac256b

Measurements of ^natCd(γ, x) reaction cross sections and isomer ratio of ^115m,gCd with the bremsstrahlung end-point energies of 50 and 60 MeV

1.
Department of Physics, Kyungpook National University, Daegu 41566, Korea
2.
Institute of Nuclear Science and Technology, Bangladesh Atomic Energy Commission, Savar, Dhaka, Bangladesh
3.
Radiochemistry Division, Bhabha Atomic Research Centre, Mumbai 400085, India

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Abstract：
The flux-weighted average cross sections of ^natCd(γ, xn)^{115g,m,111m,109,107,105,104}Cd and ^natCd(γ, x)^{113g,112,111g,110m}Ag reactions were measured at the bremsstrahlung end-point energies of 50 and 60 MeV. The activation and off-line γ-ray spectrometric technique was carried out using the 100 MeV electron linear accelerator at the Pohang Accelerator Laboratory, Korea. The ^natCd(γ, xn) reaction cross sections as a function of photon energy were theoretically calculated using the TALYS-1.95 and the EMPIRE-3.2 Malta codes. Then, the flux-weighted average cross sections were obtained from the theoretical values of mono-energetic photons. These values were compared with the flux-weighted values from the present study and were found to be in general agreement. The measured experimental reaction cross-sections and integral yields were described for cadmium and silver isotopes in the ^natCd(γ, xn)^{115g,m,111m,109,107,105,104}Cd and ^natCd(γ, x)^{113g,112,111g,110m}Ag reactions. The isomeric yield ratio (IR) of ^115g,mCd in the ^natCd(γ, xn) reaction was determined for the two bremsstrahlung end-point energies. The measured isomeric yield ratios of ^115g,mCd in the ^natCd(γ, xn) reaction were also compared with the theoretical values of the nuclear model codes and previously published literature data of the ¹¹⁶Cd(γ, n) and ¹¹⁶Cd(n, 2n) reactions. It was found that the IR value increases with increasing projectile energy, which demonstrates the characteristic of excitation energy. However, the higher IR value of ^115g,mCd in the ¹¹⁶Cd(n, 2n) reaction compared to that in the ¹¹⁶Cd(γ, n) reaction indicates the role of compound nuclear spin alongside excitation energy.
- ^natCd(γ, xn) reaction cross sections ,
- isomer yield ratio of ^115g,mCd ,
- off-line γ-ray spectrometric technique ,
- TALYS-1.95 ,
- TENDL-2019 ,
- EMPIRE-3.2 Malta

I. INTRODUCTION

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

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

However, previous studies mainly focused on analyzing spontaneous pair production in the near horizon region of black holes in an asymptotically flat spacetime. A charged black hole in AdS spacetime has an additional AdS symmetry at the asymptotical boundary. From the holographic point of view, the CFT description of pair production has been revealed only in the near horizon region in terms of AdS₂/CFT₁ (or warped AdS₃/CFT₂). Although particle pairs produced in the near horizon region of black holes indeed provide important contributions to those in full spacetime, an understanding of the whole picture is still lacking. In the present paper, we extend the study of pair production to a full near extremal RN-AdS₅ black hole background, which possesses an AdS₅ geometry at the asymptotic spatial boundary as well as an AdS₂ structure in the near horizon region. It is shown that the radial equation of the charged scalar field propagating in this spacetime can be transformed into a Heun-like differential equation and thus be solved by matching its solutions in the near and far spacetime regions, using the low temperature limit. Consequently, analytical forms of the full solutions for the pair production rate, the absorption cross section ratio, and the retarded Green's functions are obtained, and they are shown to have concise relations with their counterparts calculated in the near horizon region. Based on these concise relations, numerical analysis can easily be performed, and the pair production rate in full spacetime is shown to be smaller than that in the near horizon region, which is consistent with the assumption that pair production mainly comes from the black hole near horizon region.

A near extremal RN-AdS₅ black hole is also a very useful background for studying holographic dualities. As the near horizon AdS₂ (or warped AdS₃) spacetime is dual to a 1D CFT (or chiral CFT₂), while asymptotical 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 renormalization group (RG) flow along the radial direction [25-27]. For example, it has been shown that a near extremal RN-AdS black hole acts as a holographic model in describing typical properties of a (non)Fermi liquid at the quantum critical point [28-31]. It is thus natural and interesting to find holographic descriptions of pair production in an RN-AdS₅ black hole both in the IR CFT₁ in the near horizon region and the UV CFT₄ at the asymptotical AdS₅ boundary. We show that the picture in the IR CFT₁ is very similar to those in the near extremal RN and KN black holes, and that the pair production rate and the absorption cross section ratio calculated from the AdS₂ spacetime can be matched with those from the dual IR CFT. Regarding the UV 4D CFT, a direct comparison of calculations between the bulk and the boundary in terms of the AdS₅/CFT₄ is not made due to a lack of information on the dual finite temperature CFT₄ side. However, from the bulk gravity side, the condition for pair production in the full near extremal RN-AdS₅ spacetime is the violation of the Breitenlohner-Freedman (BF) bound [32, 33] in AdS₅ spacetime. This, on the dual 4D CFT side, corresponds to a complex conformal weight for the scalar operator dual to the bulk charged scalar field, which indeed indicates instabilities for the scalar operator on the boundary and is consistent with the situation in the IR CFT. Furthermore, we determined an interesting relation between the full pair production rate and the absorption cross section ratio via changing the roles of sources and operators simultaneously both in the IR and the UV CFTs.

The rest of the paper is organized as follows. In Sec. II, we provide a brief review of the bulk theory and consider the near horizon geometry of an RN- ${\rm{AdS}}_{d+1}$ black hole and the EoMs of the probe charged scalar field. In Sec. III, spontaneous pair production in the near horizon region of near extremal RN- ${\rm{AdS}}_{d+1}$ black holes is discussed, and the 2-point functions of the charged scalar field (such as the retarded Green's function), pair production rate, and absorption cross section ratio are calculated. In Sec. IV, the full analytical solution for the radial equation of the charged scalar field in RN-AdS₅ black holes is obtained by applying the matching technique. Consequently, the full analytical forms of the pair production rate, absorption cross section ratio, and retarded Green's function are found, and the connections with their counterparts in the near horizon region of the black hole are discussed. Then, in Sec. V, the dual CFTs descriptions of spontaneous pair production are both analyzed in terms of the AdS₂/CFT₁ correspondence in the IR region and the AdS₅/CFT₄ correspondence in the UV region, and their connections are also revealed. Finally, the conclusion and physical implications are provided in Sec. VI.

II. BULK THEORY

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

A RN- ${\rm{AdS}}_{d+1}$ black hole

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

The $d+1$ dimensional Einstein-Maxwell theory has an action (in units of $c = \hbar = 1$ ) as

$I = \int {\rm d}^{d+1}x \sqrt{-g} \left[ \frac{1}{16 \pi G_{d+1}} \left( R + \frac{d(d-1)}{L^2} \right) - \frac1{g_{\mathrm s}^2} F_{\mu\nu} F^{\mu\nu} \right],$

(1)

where L is the curvature radius of the asymptotical ${\rm{AdS}}_{d+1}$ spacetime, and $g_{\mathrm s}$ is the dimensionless coupling constant of the $U(1)$ gauge field. The dynamical equations

$\begin{aligned}[b] R_{\mu\nu} \!-\! \frac12 g_{\mu\nu} R \!-\! \frac{d(d-1)}{2L^2} g_{\mu\nu} & \!=\! \frac{8 \pi G_{d+1}}{g_{\mathrm s}^2} \left( 4 F_{\mu\lambda} F_{\nu}{}^{\lambda} \!-\! g_{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right), \\ \partial_\mu \left( \sqrt{-g} F^{\mu\nu} \right) & = 0, \end{aligned}$

(2)

admit the Reissner-Nordström-Anti de Sitter (RN- ${\rm{AdS}}_{d+1}$ ) black brane (or the planar black hole) solution [34]

$\begin{aligned}[b] {\rm d}s^2 & = \frac{L^2}{r^2 f(r)} {\rm d}r^2 + \frac{r^2}{L^2} \left( -f(r) {\rm d}t^2 + {\rm d}x_i^2 \right), \\ A & = \mu \left( 1 - \dfrac{r_{\mathrm o}^{d-2}}{r^{d-2}} \right) {\rm d}t, \end{aligned}$

(3)

with

$\begin{aligned}[b] f(r) =& 1 - \frac{G_{d+1} L^2 M}{r^d} + \frac{G_{d+1} L^2 Q^2}{r^{2d-2}}, \qquad {} \\ \mu =& \sqrt{\frac{d-1}{2(d-2)}} \frac{g_{\mathrm s} Q}{r_{\mathrm o}^{d-2}},\end{aligned}$

(4)

where $r_{\mathrm o}$ is the radius of the outer horizon ( $f(r_{\mathrm o}) = 0$ ), $\mu$ is the chemical potential with dimension $[\mu] = \mathrm{length}^{-(d-1)/2}$ , M is the mass, and Q is the charge of the black brane. We may find an explicit expression of $r_{\mathrm o}$ for $d = 4$ from a solution of the cubic equation, which is complicated, but $r_\mathrm{o}$ has a general expression in the extremal case, i.e., $r_*$ in IIB. The condition $f(r_{\mathrm o}) = 0$ gives $M = \dfrac{r_{\mathrm o}^d}{G_{d+1} L^2} + \dfrac{Q^2}{r_{\mathrm o}^{d-2}}$ (which is the Smarr-like relation related to the first law of thermodynamics of the black brane); temperature T and “surface” entropy density s of the black brane are, respectively,

$\begin{aligned}[b] T =& \frac{r_{\mathrm o} \, d}{4\pi L^2} \left( 1 - \frac{d-2}{d} \frac{G_{d+1} L^2 Q^2}{r_{\mathrm o}^{2d-2}} \right), \qquad {\rm{}} \\ s = &\frac{1}{4G_{d+1}} \left( \frac{r_{\mathrm o}}{L} \right)^{d-1}. \end{aligned}$

(5)

Moreover, the first law of thermodynamics of the dual boundary d-dimensional quantum field is

$\delta \epsilon = T \delta s + \mu \delta\rho_{\mathrm c},$

(6)

where the “surface” energy and charge densities are, respectively,

$\begin{aligned}[b]& \epsilon = \frac{d - 1}{16 \pi L^{d-1}} M, \qquad {\rm{}} \qquad \\&\rho_{\mathrm c} = \frac{\sqrt{2(d-1)(d-2)}}{8 \pi g_{\mathrm s} L^{d-1}} Q. \end{aligned}$

(7)

Then, it is straightforward to check the Euler relation

$\left( \frac{d}{d-1} \right) \epsilon = \epsilon + p = T s + \mu \rho_{\mathrm c},$

(8)

where the pressure is $p = \dfrac{\epsilon}{d-1}$ , which shows that the dual d-dimensional quantum field theory on the asymptotic boundary is conformal, as expected.

B Near-horizon near-extremal geometry

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

To make the following analysis convenient, let us introduce the length scale $r_*^{2d-2} \equiv \dfrac{d-2}{d} G_{d+1} L^2 Q^2$ ; then, the temperature can be rewritten as

$T = \frac{r_{\mathrm o} d}{4\pi L^2} \left( 1 - \frac{r_*^{2d-2}}{r_{\mathrm o}^{2d-2}} \right).$

(9)

Note that $r_*$ may be treated as the “effective” radius of the inner black hole horizon though $f(r_*) \neq 0$ in general and $r_* < r_{\mathrm o}$ . The extremal condition for a degenerate horizon at $r_\mathrm{o} = r_*$ is $M = M_0 \equiv \dfrac{2 (d-1)}{d-2} \dfrac{r_*^d}{G_{d+1} L^2}$ . The near extremal limit of the near horizon is obtained by taking the limit $\varepsilon \to 0$ of the transformations

$\begin{aligned}[b]& M - M_0 = \frac{d (d-1) r_*^{d-2}}{G_{d+1} L^2} \varepsilon^2 \rho_\mathrm{o}^2, \\& r_{\mathrm o} - r_* = \varepsilon \rho_{\mathrm o}, \quad r - r_{\mathrm o} = \varepsilon (\rho - \rho_0), \quad t = \frac{\tau}{\varepsilon}, \end{aligned}$

(10)

where in general $\rho_{\mathrm o}$ is finite and $\rho \in [\rho_0, \infty)$ .

Expanding $f(r)$ around $r = r_{\mathrm o}$ , we have

$f(r) \simeq \frac{{d(d - 1)}}{{r_{\rm{o}}^2}}({\rho ^2} - \rho _{\rm{o}}^2){\varepsilon ^2} + O\left( {{\varepsilon ^3}} \right),$

(11)

the near horizon geometry is given by

$\begin{aligned}[b] {\rm d}s^2 & = - \frac{\rho^2 - \rho_{\mathrm o}^2}{\ell^2} {\rm d}\tau^2 + \frac{\ell^2 {\rm d}\rho^2}{\rho^2 - \rho_{\mathrm o}^2} + \frac{r_{\mathrm o}^2}{L^2} {\rm d}x_i^2, \\ A & = \frac{(d-2) \mu}{r_{\mathrm o}} (\rho - \rho_{\mathrm o}) {\rm d}\tau, \end{aligned}$

(12)

where $\ell^2 \equiv \dfrac{L^2}{d (d - 1)}$ is defined as the square of the curvature radius of the effective AdS₂ geometry. The limit $\rho_{\mathrm o} \to 0$ yields the extremal limit.

The solution in Eq. (12) can also be written in the Poincaré coordinates in terms of $\xi = \ell^2/\rho$ , ( $|\xi| \leqslant \xi_{\mathrm o} =$ $\ell^2/\rho_\mathrm{o}$ ),

$\begin{aligned}[b] {\rm d}s^2 & = \frac{\ell^2}{\xi^2} \left( - \left( 1 - \frac{\xi^2}{\xi_{\mathrm o}^2} \right) {\rm d}\tau^2 + \frac{{\rm d}\xi^2}{1-\dfrac{\xi^2}{\xi_{\mathrm o}^2}} \right) + \frac{r_{\mathrm o}^2}{L^2} {\rm d}x_i^2, \\ A & = \frac{(d-2) \mu \ell^2}{r_{\mathrm o}} \left( \frac{1}{\xi} - \frac{1}{\xi_{\mathrm o}} \right) {\rm d}\tau. \end{aligned}$

(13)

The above geometry is a black brane with both local and asymptotical topology ${\rm{AdS}}_2 \times {{R}}^{d-1}$ (AdS₂ has the $SL(2,R)_R$ symmetry). The horizons of the new black brane are located at $\xi = \pm \xi_{\mathrm o}$ , and its temperature is $T_{ {{n}}} = \dfrac{1}{2 \pi \xi_{\mathrm o}}$ . Note that if we adopt the new coordinates $z \equiv \xi/\xi_{\mathrm o}$ with $|z| \leqslant 1$ and $\eta = \tau/\xi_{\mathrm o}$ , the metric becomes

${\rm d}s^2 = \frac{\ell^2}{z^2} \left( - (1 - z^2) {\rm d}\eta^2 + \frac{{\rm d}z^2}{1-z^2} \right) + \frac{r_{\mathrm o}^2}{L^2} {\rm d}x_i^2,$

(14)

and the temperature associated with the inverse period of $\eta$ is normalized to $\tilde{T}_{ n} = \dfrac{1}{2\pi}$ .

C Charged scalar field probe

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

The action of a bulk probe charged scalar field $\Phi$ with mass m and charge q is

$S = \int {{{\rm d}^{d + 1}}} x\sqrt { - g} \left( { - \frac{1}{2}{D_\alpha^* }{\Phi ^*}{D^\alpha }\Phi - \frac{1}{2}{m^2}{\Phi ^*}\Phi } \right),$

(15)

where $D_{\alpha} \equiv \nabla_{\alpha} - {\rm i} q A_{\alpha}$ with $\nabla_\alpha$ being the covariant derivative in curved spacetime. The corresponding Klein-Gordon (KG) equation is

$(\nabla_\alpha - {\rm i} q A_\alpha) (\nabla^\alpha - {\rm i} q A^\alpha) \Phi = m^2 \Phi.$

(16)

Moreover, the radial flux of the probe field is

${\cal F} = {\rm i}\sqrt { - g} {g^{rr}}(\Phi D_r^*{\Phi ^*} - {\Phi ^*}{D_r}\Phi ).$

(17)

In the RN- ${\rm{AdS}}_{d+1}$ background (3), assuming $\Phi(t, \vec{x}, r) =$ $\phi(r) \mathrm{e}^{-{\rm i} \omega t + {\rm i} \vec{k} \cdot \vec{x}}$ , the KG Eq. (16) has the radial form

$\begin{aligned}[b]& \left(\frac{L}{r}\right)^{d-1} \partial_r \left( \frac{r^{d+1}}{L^{d+1}} f(r) \partial_r \right) \phi(r) \\& + \left(\frac{L^2 (\omega + q A_t)^2}{r^2 f(r)} - m^2 - \frac{L^2}{r^2} \vec{k}^2 \right) \phi(r) = 0. \end{aligned}$

(18)

The solutions to Eq. (18) cannot be directly found in terms of special functions in the full spacetime region. In what follows, we solve it in different regions and match these solutions to obtain the full solution.

III. PAIR PRODUCTION IN THE INNER ADS₂

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

A Near-horizon solutions

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

Firstly, we analyze the near horizon, near extreme region (13) and solve the KG Eq. (16) by expanding the scalar field as

$\Phi(\tau, \vec{x}, \xi) = \phi(\xi) \mathrm{e}^{-{\rm i} w \tau + {\rm i} \vec{k} \cdot \vec{x}}.$

(19)

Then, the KG equation reduces to ^①

$\begin{aligned}[b] \xi^2 \left( 1 - \frac{\xi^2}{\xi_{\mathrm o}^2} \right) \phi''(\xi) - \frac{2\xi^3}{\xi_{\mathrm o}^2} \phi'(\xi) + \xi^2 \frac{(w + q A_\tau)^2}{1 - \dfrac{\xi^2}{\xi_{\mathrm o}^2}} \phi(\xi) = m_\mathrm{eff}^2 \ell^2 \phi(\xi), \end{aligned}$

(20)

where the effective mass square is defined as $m_\mathrm{eff}^2 =$ $m^2 + \dfrac{L^2 \vec{k}^2}{r_{\mathrm o}^2}$ , or the KG equation can be expressed in the z coordinate as

$\begin{aligned}[b] z^2 (1 - z^2) \phi''(z) - 2 z^3 \phi'(z) &+ \frac{z^2}{1 - z^2} \left[ \left( w \xi_{\mathrm o} + q_\mathrm{eff} \ell \frac{1 - z}{z} \right)^2\right. \\&\left.- m_\mathrm{eff}^2 \ell^2 \frac{1 - z^2}{z^2} \right] \phi(z) = 0, \end{aligned}$

(21)

where the effective charge of the probe field is $q_\mathrm{eff} \equiv$ $(d-2) \dfrac{\mu \ell}{r_{\mathrm o}} q$ . The singularities of Eq. (21) are located at $z = 0, z = \pm 1$ and $z = \infty$ .

To find the solutions, we determine the indices at each singular point. For $z \to 0$ , setting $\phi(z) \sim z^{\bar{\alpha}}$ , the leading terms in Eq. (21) are

$z^2 \phi''(z) + (q_\mathrm{eff}^2 - m_\mathrm{eff}^2) \ell^2 \phi(z) = 0,$

(22)

which gives

$\begin{aligned}[b] \bar{\alpha} = &\frac12 \pm \frac12 \sqrt{1 + 4 ( m_\mathrm{eff}^2 - q_\mathrm{eff}^2 ) \ell^2} \\\equiv &\frac12 \pm \frac12 \sqrt{1 + 4 \tilde{m}_\mathrm{eff}^2 \ell^2} \equiv \frac12 \pm \nu. \end{aligned}$

(23)

For $z \to -1$ , setting $\phi(z) \sim (1 + z)^{\bar{\beta}}$ , Eq. (21) reduces to

$2 (1 + z) \phi''(z) + 2 \phi'(z) + \frac{(w \xi_{\mathrm o} - 2 q_\mathrm{eff} \ell)^2}{2 (1 + z)} \phi(z) = 0,$

(24)

and the index is

$\bar{\beta} = \pm {\rm i} \left( \frac{w \xi_{\mathrm o}}{2} - q_\mathrm{eff} \ell \right) = \pm {\rm i} \left( \frac{w}{4 \pi T_{\mathrm n}} - q_\mathrm{eff} \ell \right).$

(25)

Finally, for $z \to 1$ , setting $\phi(z) \sim (1 - z)^{\bar{\gamma}}$ , Eq. (21) reduces to

$2 (1 - z) \phi''(z) - 2 \phi'(z) + \frac{(w \xi_{\mathrm o})^2}{2 (1 - z)} \phi(z) = 0,$

(26)

from which

$\bar{\gamma} = \pm {\rm i} \frac{w \xi_{\mathrm o}}{2} = \pm {\rm i} \frac{w}{4 \pi T_{\mathrm n}} = \pm {\rm i}\frac{\omega/\varepsilon}{4\pi /(2\pi \xi_{\mathrm o})} = \pm {\rm i}\frac{\omega}{2\varepsilon\rho_{\mathrm o}/\ell^2} = \pm {\rm i}\frac{\omega}{4\pi T}$

(27)

is obtained. Further, imposing the ingoing boundary condition at the black brane horizon $z = 1$ requires $\bar{\gamma} =$ $-{\rm i} \dfrac{w \xi_{\mathrm o}}{2} = -{\rm i} \dfrac{w}{4\pi T_{\mathrm n}}$ .

Also, note that Eq. (21) can be rewritten in a more explicit form as

$\begin{aligned}[b] \phi''(z) + \left( \frac{1}{z+1} + \frac{1}{z-1} \right) \phi'(z) + \left( \frac{\tilde{m}_\mathrm{eff}^2 \ell^2}{z} + \frac{\dfrac{1}{2} (w \xi_{\mathrm o} - 2 q_\mathrm{eff} \ell)^2}{z+1} + \frac{\dfrac12 w^2 \xi_{\mathrm o}^2}{z-1} \right) \frac{\phi(z)}{z (z+1) (z-1)} = 0, \end{aligned}$

(28)

which becomes the Fuchs equation with three canonical singularities $a_1$ , $a_2$ and $a_3$ , as follows:

$\begin{aligned}[b] \phi''(z) &+ \left( \frac{1-\bar{\alpha}_1-\bar{\alpha}_2}{z-a_1} + \frac{1-\bar{\beta}_1-\bar{\beta}_2}{z-a_2} + \frac{1-\bar{\gamma}_1-\bar{\gamma}_2}{z-a_3} \right) \phi'(z) + \left( \frac{\bar{\alpha}_1 \bar{\alpha}_2 (a_1-a_2) (a_1-a_3)}{z-a_1} + \frac{\bar{\beta}_1 \bar{\beta}_2 (a_2-a_3) (a_2-a_1)}{z-a_2}\right. \\&\left.+ \frac{\bar{\gamma}_1 \bar{\gamma}_2 (a_3-a_1) (a_3-a_2)}{z-a_3} \right) \frac{\phi(z)}{(z-a_1)(z-a_2)(z-a_3)} = 0, \end{aligned}$

(29)

where $a_1 = 0$ , $a_2 = -1$ , and $a_3 = 1$ and

$\begin{aligned}[b]& \bar{\alpha}_1 = \frac12 \pm \nu, \quad \bar{\alpha}_2 = \frac12 \mp \nu, \quad \bar{\beta}_1 = - \bar{\beta}_2 = \pm {\rm i} \frac{w \xi_{\mathrm o} - 2 q_\mathrm{eff} \ell}2, \\& \bar{\gamma}_1 = - \bar{\gamma}_2 = \pm {\rm i} \frac{w \xi_{\mathrm o}}2, \end{aligned}$

(30)

and $\bar{\alpha}_1 + \bar{\alpha}_2 + \bar{\beta}_1 + \bar{\beta}_2 + \bar{\gamma}_1 + \bar{\gamma}_2 = 1$ is satisfied. The Fuchs Eq. (29) can be transformed into the standard hypergeometric function

$\zeta (1 - \zeta) \psi''(\zeta) + \left[ \tilde{\gamma} - (1 + \tilde{\alpha} + \tilde{\beta}) \zeta \right] \psi'(\zeta) - \tilde{\alpha} \tilde{\beta} \psi(\zeta) = 0,$

(31)

via the conformal coordinate transformation

$\zeta = \frac{(a_2-a_3)(z-a_1)}{(a_2-a_1)(z-a_3)}, \quad {\rm{}} \;\; \phi(z) = \left(\frac{z-a_1}{z-a_3}\right)^{\bar{\alpha}_1} \left(\frac{z-a_2}{z-a_3}\right)^{\bar{\beta}_1} \psi(\zeta),$

(32)

where $\tilde{\alpha} = \bar{\alpha}_1 + \bar{\beta}_1 + \bar{\gamma}_1, \tilde{\beta} = \bar{\alpha}_1 + \bar{\beta}_1 + \bar{\gamma}_2$ and $\tilde{\gamma} = 1 + \bar{\alpha}_1 - \bar{\alpha}_2$ . (Note that one can freely choose the indices $i = 1,\; 2$ for $\bar{\alpha}_i$ , $\bar{\beta}_i$ and $\bar{\gamma}_i$ .)

For Eq. (28), we have $\zeta = 2z/(z-1)$ , $\tilde{\alpha} = \dfrac12 \pm \nu +$ ${\rm i} w \xi_{\mathrm o} - {\rm i} q_\mathrm{eff} \ell, \;\;\;\;\tilde{\beta} = \dfrac12 \pm \nu - {\rm i} q_\mathrm{\rm eff} \ell, \;\;\;\; \tilde{\gamma} = 1 \pm 2\nu$ . Therefore, the explicit solutions in the near horizon near extreme region are

$\begin{aligned}[b] \phi(z) =& c_1 \left(\frac{z}{z-1}\right)^{\frac12 + \nu} \left(\frac{z+1}{z-1}\right)^{{\rm i} \frac{w \xi_{\mathrm o}}{2} - {\rm i} q_\mathrm{eff} \ell} {_2F_1}\left(\frac12 + \nu + {\rm i} w \xi_{\mathrm o} - {\rm i} q_\mathrm{eff} \ell, \frac12 + \nu - {\rm i} q_\mathrm{eff} \ell; 1 + 2 \nu; \frac{2z}{z-1} \right) \\ & + c_2 \left(\frac{z}{z-1}\right)^{\frac12 - \nu} \left(\frac{z+1}{z-1}\right)^{{\rm i} \frac{w \xi_{\mathrm o}}{2} - {\rm i} q_\mathrm{eff} \ell} {_2F_1}\left(\frac12 - \nu + {\rm i} w \xi_{\mathrm o} - {\rm i} q_\mathrm{eff} \ell, \frac12 - \nu - {\rm i} q_\mathrm{eff} \ell; 1 - 2 \nu; \frac{2z}{z-1} \right).\\ \end{aligned}$

(33)

B 2-point correlators from AdS₂

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

At the horizon of the AdS₂ black brane, $z = 1$ , Eq. (33) is expanded as follows:

$\phi(z) = c_H^{(\mathrm {in})}(1-z)^{-{\rm i} \frac{w}{4\pi T_{ n}}} + c_H^{(\mathrm {out})} (1-z)^{{\rm i} \frac{w}{4\pi T_{n}}},$

(34)

where

$\begin{aligned}[b] c_H^{(\mathrm {in})} = c_1 (-)^{-\frac12 - \nu - {\rm i} \frac{w}{2 \pi T_{ n}} + {\rm i} q_\mathrm{eff} \ell} \, 2^{-\frac12 - \nu + {\rm i} \frac{w}{4\pi T_{ n}}} \frac{\Gamma\left(1 + 2\nu\right) \Gamma\left({\rm i} \dfrac{w}{2\pi T_{ n}}\right)}{\Gamma\left(\dfrac12 + \nu + {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 + \nu + {\rm i} \dfrac{w}{2\pi T_{ n}} - {\rm i} q_\mathrm{eff} \ell\right)}\end{aligned}$

$\begin{aligned}[b]+ c_2 (-)^{-\frac12 + \nu - {\rm i} \frac{w}{2 \pi T_{ n}} + {\rm i} q_\mathrm{eff} \ell} \, 2^{-\frac12 + \nu + {\rm i} \frac{w}{4\pi T_{ n}}} \frac{\Gamma\left(1 - 2\nu\right) \Gamma\left({\rm i} \dfrac{w}{2\pi T_{ n}}\right)}{\Gamma\left(\dfrac12 - \nu + {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 - \nu + {\rm i} \dfrac{w}{2\pi T_{ n}} - {\rm i} q_\mathrm{eff} \ell\right)}, \end{aligned}$

(35)

and

$\begin{aligned}[b] c_H^{(\mathrm {out})} =& c_1 (-)^{-\frac12 - \nu - {\rm i} \frac{w}{2 \pi T_{ n}} + {\rm i} q_\mathrm{eff} \ell} \, 2^{-\frac12 - \nu - {\rm i} \frac{w}{4 \pi T_{ n}}} \frac{\Gamma\left(1 + 2\nu\right) \Gamma\left(-{\rm i} \dfrac{w}{2\pi T_{ n}}\right)}{\Gamma\left(\dfrac12 + \nu - {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 + \nu - {\rm i} \dfrac{w}{2\pi T_{n}} + {\rm i} q_\mathrm{eff} \ell\right)} \\ &+ c_2 (-)^{-\frac12 + \nu - {\rm i} \frac{w}{2 \pi T_{ n}} + {\rm i} q_\mathrm{eff} \ell} \, 2^{-\frac12 + \nu - {\rm i} \frac{w}{4\pi T_{ n}}} \frac{\Gamma\left(1 - 2\nu\right) \Gamma\left(-{\rm i} \dfrac{w}{2\pi T_{ n}}\right)}{\Gamma\left(\dfrac12 - \nu - {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 - \nu - {\rm i} \dfrac{w}{2\pi T_{ n}} + {\rm i} q_\mathrm{eff} \ell\right)}. \end{aligned}$

(36)

In contrast, at the AdS₂ boundary, $z \rightarrow 0$ , the asymptotic expansion of Eq. (33) is

$\begin{aligned}[b] \phi(z) =& c_2 (-)^{\frac12 - \nu + {\rm i} \frac{w}{4\pi T_{ n}} - {\rm i} q_\mathrm{eff} \ell} z^{\frac{1}{2}- \nu} + c_1 (-)^{\frac12 + \nu + i \frac{w}{4\pi T_{\mathrm n}} - {\rm i} q_\mathrm{eff} \ell} z^{\frac{1}{2}+ \nu} \\=& {\cal A}(w, \vec{k})z^{\frac{1}{2}- \nu}+{\cal B}(w, \vec{k})z^{\frac{1}{2}+ \nu},\end{aligned}$

(37)

where ${\cal A}$ is the source of the charged scalar field in the bulk AdS₂, while ${\cal B}$ is the response or the operator ${\cal \hat{O}}(w,\vec{k})$ (in the momentum space) of the boundary CFT₁ (i.e., the IR CFT) dual to the charged scalar field in the bulk AdS₂ background. Note that in order to obtain the propagating modes, $\nu$ should be purely imaginary, which can be set as $\nu \equiv {\rm i} |\nu|$ , i.e., $\phi(z) = c_B^{(\mathrm {out})} z^{\frac{1}{2} - {\rm i}|\nu|} + c_B^{(\mathrm {in})} z^{\frac{1}{2} + {\rm i}|\nu|}$ . It was shown in [3] that the condition of an imaginary $\nu$ is equivalent to the violation of the BF bound in AdS₂ spacetime, namely

$\begin{aligned} \tilde{m}_\mathrm{eff}^2 < -\frac{1}{4\ell^2}, \end{aligned}$

(38)

which corresponds to a complex conformal weight of the scalar operator in the dual IR CFT.

1 Pair production rate and absorption cross section ratio

The Schwinger pair production rate $|\mathfrak{b}|^2$ and the absorption cross section ratio $\sigma_{\mathrm{abs}}$ can be calculated from the radial flux by imposing different boundary conditions

$\begin{align} {\cal F} = {\rm i} \left(\frac{r_{\mathrm o}}{L}\right)^{d-1} (1-z^2) (\Phi \partial_z \Phi^* - \Phi^* \partial_z \Phi), \end{align}$

(39)

which gives

$\begin{aligned}[b] {\cal F}_B^{(\mathrm {in})} =& 2 |\nu| \left(\frac{r_{\mathrm o}}{L}\right)^{d-1} |c_B^{(\mathrm {in})}|^2,\\ {\cal F}_B^{(\mathrm {out})} =& -2 |\nu| \left(\frac{r_{\mathrm o}}{L}\right)^{d-1} |c_B^{(\mathrm {out})}|^2, \\ {\cal F}_H^{(\mathrm {in})} =& \frac{w}{2\pi T_{\mathrm n}} \left(\frac{r_{\mathrm o}}{L}\right)^{d-1} |c_H^{(\mathrm {in})}|^2, \\ {\cal F}_H^{(\mathrm {out})} =& -\frac{w}{2\pi T_{\mathrm n}} \left(\frac{r_{\mathrm o}}{L}\right)^{d-1} |c_H^{(\mathrm {out})}|^2, \end{aligned}$

(40)

where ${\cal F}_B^{(\mathrm {in})}$ and ${\cal F}_B^{(\mathrm {out})}$ are the ingoing and outgoing fluxes at the AdS₂ boundary, while ${\cal F}_H^{(\mathrm {in})}$ and ${\cal F}_H^{(\mathrm {out})}$ are the ingoing and outgoing fluxes at the AdS₂ black brane horizon, respectively.

The Schwinger pair production rate ${{{\left| {{\mathfrak{b}^{{\text{Ad}}{{\text{S}}_{\text{2}}}}}} \right|}^{\text{2}}}}$ can be computed either by choosing the inner boundary condition or the outer boundary condition, which gives the same result [3], e.g., by adopting the outer boundary condition, i.e., ${\cal F}_B^{(\mathrm {in})} = 0$ , ( $c_B^{(\mathrm {in})} = 0 \Rightarrow c_1 = 0$ ),

$\begin{aligned}[b] {{{\left| {{\mathfrak{b}^{{\text{Ad}}{{\text{S}}_{\text{2}}}}}} \right|}^{\text{2}}}}& = \frac{{\cal F}_B^{(\mathrm {out})}}{{\cal F}_H^{(\mathrm {in})}} = \frac{4 \pi T_{ n} |\nu|}{w} \left|\frac{c_B^{(\mathrm {out})}}{c_H^{(\mathrm {in})}}\right|^2 = \frac{ 8 \pi T_{ n} |\nu|}{w} \left| \frac{\Gamma\left(\dfrac12 - {\rm i}|\nu| + {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 - {\rm i}|\nu| + {\rm i} \dfrac{w}{2\pi T_{ n}} -{\rm i} q_\mathrm{eff} \ell\right)}{\Gamma\left(1 - 2{\rm i}|\nu|\right) \Gamma\left({\rm i} \dfrac{w}{2\pi T_{ n}}\right)}\right|^2\\ & = \frac{2\sinh\left(2\pi|\nu| \right)\sinh\left(\dfrac{w}{2T_{ n}}\right)}{\cosh\pi\left(|\nu|-q_\mathrm{eff} \ell\right)\cosh\pi\left(|\nu|-\dfrac{w}{2\pi T_{ n}}+q_\mathrm{eff} \ell\right)}. \end{aligned}$

(41)

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

$\begin{aligned}[b] \sigma _{{\text{abs}}}^{{\text{Ad}}{{\text{S}}_{\text{2}}}}& = \frac{{\cal F}_B^{(\mathrm {out})}}{{\cal F}_H^{(\mathrm {out})}} = \frac{4 \pi T_{ n} |\nu|}{w} \left|\frac{c_B^{(\mathrm {out})}}{c_H^{(\mathrm {out})}}\right|^2 = \frac{ 8 \pi T_{ n} |\nu|}{w} \left| \frac{\Gamma\left(\dfrac12 - {\rm i}|\nu| - {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 - {\rm i}|\nu| - {\rm i} \dfrac{w}{2\pi T_{ n}} +{\rm i} q_\mathrm{eff} \ell\right)}{\Gamma\left(1 - 2{\rm i}|\nu|\right) \Gamma\left(-{\rm i} \dfrac{w}{2\pi T_{ n}}\right)}\right|^2\\ & = \frac{2\sinh\left(2\pi|\nu| \right)\sinh\left(\dfrac{w}{2T_{ n}}\right)}{\cosh\pi\left(|\nu|+q_\mathrm{eff} \ell\right)\cosh\pi\left(|\nu|+\dfrac{w}{2\pi T_{ n}}-q_\mathrm{eff} \ell\right)}. \end{aligned}$

(42)

The pair production rate and the absorption cross section ratio are connected by the simple relation

${{{\left| {{\mathfrak{b}^{{\text{Ad}}{{\text{S}}_{\text{2}}}}}} \right|}^{\text{2}}}} =-\sigma_{\mathrm{abs}}(|\nu|\rightarrow -|\nu|).$

(43)

It was shown that the abovementioned relation also holds for a charged scalar field [11] and for a charged spinor field [4], both in a four-dimensional near extremal RN black hole.

2 Retarded Green's function

The two-point retarded Green's function of the boundary operator dual to the bulk charged scalar field is computed through

$\begin{aligned}[b] G_R^{\mathrm {AdS_2}}(w, \vec{k}) \equiv & \langle {\cal \hat{O}} {\cal \hat{O}} \rangle_R\\ =& -2 {\cal F}|_{z\rightarrow 0} \sim \frac{{\cal B}(w, \vec{k})}{{\cal A}(w, \vec{k})} \\&+ \rm{contact\; terms} \end{aligned}$

(44)

by taking the inner boundary condition, i.e., ${\cal F}_H^{(\mathrm {out})} = 0$ , which gives

$\frac{c_2}{c_1} = (-)^{1 - 2\nu} \, 2^{-2\nu} \frac{\Gamma\left(1 + 2\nu\right) \Gamma\left(\dfrac12 - \nu - {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 - \nu - {\rm i} \dfrac{w}{2\pi T_{\mathrm n}} + {\rm i} q_\mathrm{eff} \ell\right)}{\Gamma\left(1 - 2\nu\right) \Gamma\left(\dfrac12 + \nu - {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 + \nu - {\rm i} \dfrac{w}{2\pi T_{\mathrm n}} + {\rm i} q_\mathrm{eff} \ell\right)}.$

(45)

Thus, the two-point retarded Green's function is

$G_R^{\mathrm {AdS_2}}(w, \vec{k}) \sim \frac{{\cal B}(\omega, \vec{k})}{{\cal A}(\omega, \vec{k})} = (-)^{2\nu} \frac{c_1}{c_2} = (-)^{4\nu-1} \, 2^{2\nu} \frac{\Gamma\left(1 - 2\nu\right) \Gamma\left(\dfrac12 + \nu - {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 + \nu - {\rm i} \dfrac{w}{2\pi T_{\mathrm n}} + {\rm i} q_\mathrm{eff} \ell\right)}{\Gamma\left(1 + 2\nu\right) \Gamma\left(\dfrac12 - \nu - {\rm i} q_\mathrm{eff} \ell\right) \Gamma\left(\dfrac12 - \nu - {\rm i} \dfrac{w}{2\pi T_{\mathrm n}} + {\rm i} q_\mathrm{eff} \ell\right)}.$

(46)

In addition, the corresponding boundary condition ( ${\cal F}_B^{(\mathrm {in})} = 0$ and ${\cal F}_H^{(\mathrm {out})} = 0$ ) is used to obtain the quasinormal modes of the charged scalar field in AdS₂ spacetime, which correspond to the poles of the retarded Green's function of dual operators (with complex conformal weight $h_R = \dfrac12+\nu$ ) in the IR CFT, namely

$\begin{aligned}[b]& \frac12 +\nu - {\rm i} \frac{w}{2\pi T_{ n}} + {\rm i} q_\mathrm{eff} \ell = -N \Rightarrow w\\ =& 2\pi T_{ n}\left(q_\mathrm{eff} \ell-{\rm i}N-{\rm i}h_R\right),\quad N = 0,1,\cdots. \end{aligned}$

(47)

Eq. (47) gives the quasinormal modes of the charged scalar field perturbation.

IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

In this section, we describe our study of the pair production for the whole spacetime of RN-AdS₅. Like before, we need to solve the corresponding radial Klein equation for the scalar field.

To find the solution in the full region, we focus on $d = 4$ and the near extremal cases. By introducing the coordinate transformation $\varrho = \dfrac{{{r^2}}}{{M'}}$ (and denoting $M'{\text{ = }}$ ${{\text{G}}_{d + 1}}{L^2}M$ , $\varrho_{\mathrm o} = \dfrac{r_{\mathrm o}^2}{M'}$ and $\varrho _* = \dfrac{r_*^2}{M'}$ ), the radial Eq. (18) can be expressed as

$\phi ''(\varrho ) + \left( {\frac{1}{{\varrho - {\varrho _1}}} + \frac{1}{{\varrho - {\varrho _2}}} + \frac{1}{{\varrho - {\varrho_{\mathrm o} }}}} \right)\phi '\left( \varrho \right) + \left( {\frac{{\varrho {{\left( {\tilde \omega \varrho - \tilde q\mu {\varrho_{\mathrm o} }} \right)}^2}}}{{{{\left( {\varrho - {\varrho _1}} \right)}^2}{{\left( {\varrho - {\varrho _2}} \right)}^2}{{\left( {\varrho - {\varrho_{\mathrm o} }} \right)}^2}}} - \frac{{{{\tilde m}^{\rm{2}}}\varrho + {{\tilde k}^2}}}{{\left( {\varrho - {\varrho _1}} \right)\left( {\varrho - {\varrho _2}} \right)\left( {\varrho - {\varrho_{\mathrm o} }} \right)}}} \right)\phi \left( \varrho \right) = 0,$

(48)

where the parameters are $\tilde \omega = \dfrac{{{L^2}(\omega + q\mu )}}{{2\sqrt {M'} }}$ , $\tilde q = \dfrac{{{L^2}q}}{{2\sqrt {M'} }}$ , $\tilde m = \dfrac{Lm}{2}$ , $\tilde k = \dfrac{{{L^2}\left| {\vec k} \right|}}{{2\sqrt {M'} }}$ and

$\begin{aligned}[b] \varrho _1 & = - \frac{1}{2}{\varrho_{\mathrm o} } - \frac{1}{2}\sqrt {{\varrho_{\mathrm o} }^{\text{2}} + 8\frac{{{\varrho _*}^{\text{3}}}}{{{\varrho_{\mathrm o} }}}}, \\ \varrho _2 & = - \frac{1}{2}{\varrho_{\mathrm o} }{\text{ + }}\frac{1}{2}\sqrt {{\varrho_{\mathrm o} }^{\text{2}} + 8\frac{{{\varrho _*}^{\text{3}}}}{{{\varrho_{\mathrm o} }}}}. \end{aligned}$

(49)

Further, defining another coordinate

$y\equiv \frac{{\varrho - {\varrho_{\mathrm o} }}}{{{\varrho_{\mathrm o} }}},\quad a\equiv \frac{\varrho _2-\varrho _{\mathrm{o}}}{\varrho _{\mathrm{o}}},\quad {} b\equiv \frac{\varrho _1-\varrho _{\mathrm{ o }}}{\varrho _{\mathrm{ o }}},$

(50)

the metric of the RN-AdS₅ black hole becomes

$\begin{aligned}[b] {\rm d}s^2& = \frac{L^2 {\rm d}y^2}{4(1+y)^2 f(y)}+\frac{r_{\mathrm{o}}^2}{L^2}(1+y)\left(-f(y){\rm d}t^2+{\rm d}x_i^2 \right),\\ A& = \frac{\mu y}{1+y}{\rm d}t, \end{aligned}$

(51)

where

$\begin{aligned}[b] f(y) =& 1-\frac{M'}{r_{\mathrm{o}}^4}(1+y)^{-2}+\frac{Q'^2}{r_{\mathrm{o}}^6}(1+y)^{-3}, \\ {{Q'}^2} =& {{\text{G}}_{d + 1}}{L^2}{Q^2}. \end{aligned}$

(52)

Moreover, Eq. (48) transforms into

$\begin{aligned}[b] \phi ''(y) &+ \left( {\frac{1}{y} + \frac{1}{{y - a}} + \frac{1}{{y - b}}} \right)\phi '\left( y \right) \\&+ \left( \frac{{{{\left( {\tilde \omega (y + 1) - \tilde q\mu } \right)}^2}(y + 1)}}{{{y^2}{{\left( {y - a} \right)}^2}{{\left( {y - b} \right)}^2}}}\right.\\&\left. - \frac{{{{\tilde m}^2}(y + 1){\varrho_{\mathrm o} } + {{\tilde k}^2}}}{{y\left( {y - a} \right)\left( {y - b} \right)}} \right)\frac{{\phi \left( y \right)}}{{{\varrho_{\mathrm o} }}} = 0.\end{aligned}$

(53)

To solve $\phi(y)$ , first, we determine its exponents at the corresponding singularities $0$ , a, b, and $\infty$ , which are $\alpha _{1,2}$ , $\beta _{1,2}$ , $\gamma _{1,2}$ , and $\delta _{1,2}$ , respectively,

$\begin{aligned}[b] &\alpha _{1,2} = \pm {\rm i}\frac{(\tilde \omega - \tilde q\mu) }{ab\sqrt {\varrho_{\mathrm o}}} = \pm\frac{{\rm i}\omega}{4\pi T},\\ &\beta _{1,2} = \pm {\rm i}\frac{{(\tilde \omega (1 + b) - \tilde q\mu )\sqrt {1 + b} }}{{(a - b)b\sqrt {{\varrho_{\mathrm o} }} }},\\& \gamma _{1,2} = \pm {\rm i}\frac{{(\tilde \omega (1 + a) - \tilde q\mu )\sqrt {1 + a} }}{{(b - a)a\sqrt {{\varrho_{\mathrm o} }} }}, \end{aligned}$

(54)

where the index “1” corresponds to the “ $+$ ” sign, and the index “2” corresponds to the “ $-$ ” sign. Then, decomposing $\phi \left( y \right)$ as

$\phi \left( y \right) = {\left( {\frac{y}{{y - b}}} \right)^{{\alpha _1}}}{\left( {\frac{{y - a}}{{y - b}}} \right)^{{\gamma _1}}}R(y),$

(55)

we obtain

$\begin{aligned}[b] R''(y) &+ \left( {\frac{1}{y} + \frac{1}{{y - a}} + \frac{1}{{y - b}} - \frac{{2b{\alpha _1}}}{{y(y - b)}} + \frac{{2(a - b){\gamma _1}}}{{(y - a)(y - b)}}} \right)R'\left( y \right) \\&+ {V_2}R(y) = 0, \end{aligned}$

(56)

where

${V_2} \equiv - \frac{{(2 + 3a - {a^2}y){{\tilde \omega }^{\rm{2}}} - {\rm{4}}\left( {a + 1} \right)\tilde \omega \tilde q\mu {\rm{ + }}\left( {a + {\rm{2}}} \right){{\tilde q}^{\rm{2}}}{\mu ^{\rm{2}}}}}{{{\varrho _{\rm{o}}}{a^2}y\left( {y - a} \right){{\left( {y - b} \right)}^2}}} - {M_1},$

(57)

and

$\begin{aligned}[b] M_1 =& \frac{{\left( {b - a} \right){\gamma _1}}}{{y\left( {y - a} \right)(y - b)}} + \frac{{2b\left( {a - b} \right){\alpha _1}{\gamma _1}}}{{y\left( {y - a} \right){{(y - b)}^2}}} + \frac{{{{\tilde m}^2}(y + 1){\varrho_{\mathrm o} } + {{\tilde k}^2}}}{{{\varrho_{\mathrm o} }y\left( {y - a} \right)\left( {y - b} \right)}}\\& + \frac{{b{\alpha _1}}}{{y\left( {y - a} \right)(y - b)}}. \end{aligned}$

(58)

A Solutions in the near and far regions

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

We divide the regions into a near region

$y = \frac{{\varrho - {\varrho_{\mathrm o} }}}{{{\varrho_{\mathrm o} }}} \ll 1,$

(59)

and a far region

$y = \frac{{\varrho - {\varrho_{\mathrm o} }}}{{{\varrho_{\mathrm o} }}}\gg -a,$

(60)

and an overlapping region, in which

$-a \ll 1.$

(61)

The physical reasoning of $-a \ll 1$ relies on the observation that the temperature of a black hole is

$T = \frac{{{r_{\mathrm o} }}}{{\pi {L^2}}}\left( {1 - \frac{{{\varrho _*}^3}}{{{\varrho_{\mathrm o} }^3}}} \right),$

(62)

which gives $-a \to 0$ for $T \to 0$ , as

$- a = \frac{3}{2} - \frac{1}{2}\sqrt {9 - \frac{8\pi L^2 T}{r_{\mathrm o} }}.$

(63)

We want to point out that the matching condition in Eq. (61) indicates that the near extremal condition is essential for matching the solutions in the near and far regions. It is not necessary for the frequency to be infinitely small; however, the frequency should definitely not be very large compared with the temperature T; otherwise, the backreaction to the background geometry cannot be ignored.

Now we find the approximate solutions in different regions. First, by using the near region condition ( $y \ll 1$ ), Eq. (53) reduces to

$\begin{aligned}[b] \phi ''(y) &+ \left( {\frac{1}{y} + \frac{1}{{y - a}}} \right)\phi '\left( y \right) \\&+ \left( { - \frac{{{{(a{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell y)}^2}}}{{{y^{\rm{2}}}{{\left( {y - a} \right)}^{\rm{2}}}}} + \frac{{{{\tilde m}^2}{\varrho _{\rm{o}}} + {{\tilde k}^2}}}{{{\varrho _{\rm{o}}}by(y - a)}}} \right)\phi (y) = 0. \end{aligned}$

(64)

Obviously Eq. (64) can be solved by the hypergeometric function as

$\begin{aligned}[b] \phi \left( y \right) =& {\left( {y - a} \right)^{{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell}}\bigg( {{c_3}{y^{{\alpha _1}}}_2{F_1}\left( {\alpha ,\beta ;\gamma ;\frac{y}{a}} \right)} \\&+ {c_4}{y^ - }^{{\alpha _1}}{}_2{F_1}\left( {1 - \gamma + \alpha ,1 - \gamma + \beta ;2 - \gamma ;\frac{y}{a}} \right) \bigg), \end{aligned}$

(65)

where $\alpha = \dfrac{1}{2} + \nu + 2{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell$ , $\beta = \dfrac{1}{2} - \nu + 2{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell$ , and $\gamma = 1 + 2{\alpha _1}$ , and $\ell = L/\sqrt{12}$ is the radius of the effective AdS₂ geometry in the near horizon region of the RN-AdS₅ black hole. Second, in the far region, by using the condition ( $y \gg -a$ ), Eq. (56) can turn into

$R''(y) + \left( {\frac{2}{y} + \frac{1}{{y - b}} - \frac{{4b{\alpha _1}}}{{y(y - b)}}{\rm{ + }}\frac{{2{\rm i}b{q_{{\rm{eff}}}}\ell }}{{y(y - b)}}} \right)R'\left( y \right) + {V_3}R(y) = 0,$

(66)

where

$\begin{aligned}[b] {V_3} \equiv & \frac{{{{\tilde \omega }^2}}}{{{\varrho _{\rm{o}}}y{{(y - b)}^2}}} + \frac{{4{b^2}{\alpha _1}\left( {{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell } \right)}}{{{y^2}{{(y - b)}^2}}} \\&- \frac{{b\left( {2{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell } \right)}}{{{y^2}(y - b)}} - \frac{{{{\tilde m}^2}(y + 1){\varrho _{\rm{o}}} + {{\tilde k}^2}}}{{{\varrho _{\rm{o}}}{y^2}(y - b)}},\end{aligned}$

(67)

(where the relation ${\gamma _1} \approx {\alpha _1} + \dfrac{{{\rm i}\tilde q\mu }}{{b\sqrt {{\varrho _{\rm{o}}}} }} = {\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell$ when $\left| a \right| \ll 1$ is used). Similarly, Eq. (66) has a solution in terms of the hypergeometric function as

$\begin{aligned}[b] \phi \left( y \right) =& {\left( {\frac{y}{b} - 1} \right)^\lambda}\bigg( {{c_5}{y^{\nu - \frac{1}{2}}}_2{F_1}\left( {\alpha ',\beta ';\gamma ';\frac{y}{b}} \right)} \\&+ {c_6}{y^{ - \nu - \frac{1}{2}}}_2{F_1}\left( {1 - \gamma ' + \alpha ',1 - \gamma ' + \beta ';2 - \gamma ';\frac{y}{b}} \right) \bigg) \end{aligned}$

(68)

in which $\alpha ' = \dfrac{1}{2} + \nu + \Delta + \lambda$ , $\beta ' = \dfrac{1}{2} + \nu - \Delta + \lambda$ , $\gamma ' = 1 + 2\nu$ , $\Delta = \sqrt {1 + {{\tilde m}^2}}$ , and $\lambda = \sqrt {{{\left( {{\rm i}{q_{{\rm{eff}}}}\ell } \right)}^2} - \dfrac{{{{\tilde \omega }^2}}}{{{\varrho _{\rm{o}}}b}}}$ .

B Near-far matching

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

In the overlapping region, one has the inequalities

$- a \ll y \ll 1 < -b$

(69)

( $1 < - b$ since $- b = \dfrac{3}{2}{\text{ + }}\dfrac{1}{2}\sqrt {9 - \dfrac{{8\pi {L^2}T}}{{{r_{\mathrm o} }}}} \to {\text{3}}$ , as $T \to {\text{0}}$ ), which means $\left| {\dfrac{a}{y}} \right| \to 0$ and $\left| {\dfrac{y}{b}} \right| \to 0$ , which transforms Eqs. (65) and (68) into the following forms: the near regionsolution

$\begin{aligned}[b] \phi (y) =& \left( {{\left( { - 1} \right)}^{ - \alpha }}{c_3}\frac{{\Gamma \left( \gamma \right)\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( \beta \right)\Gamma \left( {\gamma - \alpha } \right)}}\right.\\&\left. + {{\left( { - 1} \right)}^{ - 1 + \gamma - \alpha }}{c_4}\frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( {1 - \gamma + \beta } \right)\Gamma \left( {1 - \alpha } \right)}} \right){y^{ - \frac{1}{2} - \nu }} \\ &+ \left( {{\left( { - 1} \right)}^{ - \beta }}{c_3}\frac{{\Gamma \left( \gamma \right)\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( \alpha \right)\Gamma \left( {\gamma - \beta } \right)}} \right.\\&+\left. {{\left( { - 1} \right)}^{ - 1 + \gamma - \beta }}{c_4}\frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( {1 - \gamma + \alpha } \right)\Gamma \left( {1 - \beta } \right)}} \right){y^{ - \frac{1}{2} + \nu }} \end{aligned}$

(70)

and the far region solution

$\phi (y) \to {\left( { - 1} \right)^\lambda}{c_5}{y^{ - \frac{1}{2} + \nu }} + {\left( { - 1} \right)^\lambda}{c_6}{y^{ - \frac{1}{2} - \nu }}.$

(71)

Comparing these two identities, one finds the connection relations

$\begin{aligned}[b] {c_5} =& {( - 1)^{ - \lambda}}\left( {{\left( { - 1} \right)}^{ - \beta }}{c_3}\frac{{\Gamma \left( \gamma \right)\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( \alpha \right)\Gamma \left( {\gamma - \beta } \right)}}\right.\\&\left. + {{\left( { - 1} \right)}^{ - 1 + \gamma - \beta }}{c_4}\frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( {1 - \gamma + \alpha } \right)\Gamma \left( {1 - \beta } \right)}} \right), \end{aligned}$

(72)

$\begin{aligned}[b]{c_6} =& {( - 1)^{ - \lambda}}\left( {{\left( { - 1} \right)}^{ - \alpha }}{c_3}\frac{{\Gamma \left( \gamma \right)\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( \beta \right)\Gamma \left( {\gamma - \alpha } \right)}}\right.\\&\left. + {{\left( { - 1} \right)}^{ - 1 + \gamma - \alpha }}{c_4}\frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( {1 - \gamma + \beta } \right)\Gamma \left( {1 - \alpha } \right)}} \right).\end{aligned}$

(73)

B 2-point correlators from AdS₅

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

1 Pair production and absorption cross section

Now we denote the radial flux of the charged scalar field in metric (51) as ${\cal D}$ :

${\cal D} = \frac{2{\rm i}r_{\mathrm{o}}^4(1+y)^3 f(y)}{L^5}\bigg(\phi(y)\partial_y \phi^*(y)- \phi^*(y)\partial_y \phi(y) \bigg).$

(74)

In the near horizon limit, i.e., $y \to 0$ , Eq. (65) reduces to

$\phi (y) = {c_3}{y^{{\alpha _1}}}{\left( {y - a} \right)^{{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell }} + {c_4}{y^{ - {\alpha _1}}}{\left( {y - a} \right)^{{\alpha _1} - {\rm i}{q_{{\rm{eff}}}}\ell }},$

(75)

where the first part is the outgoing mode, and the second part is the ingoing mode. Further, the asymptotic form of $\phi(y)$ at the boundary ( $y\to \infty$ ) of the AdS₅ spacetime results in the form

$\phi(y) = A(\tilde{\omega}, \tilde{k})y^{- 1 + \Delta}+B(\tilde{\omega}, \tilde{k})y^{- 1 - \Delta},$

(76)

where $A(\tilde{\omega}, \tilde{k})$ is the source of the charged scalar field in the bulk RN-AdS₅ black hole, while $B(\tilde{\omega}, \tilde{k})$ is the response (the operator) of the boundary CFT₄ (i.e., the UV CFT) dual to the charged scalar field in the bulk. As in the case of the AdS₂ spacetime, the condition for the propagating modes requires an imaginary $\Delta$ , i.e., $\Delta = {\rm i}\left| \Delta \right|$ , which means

$m^2\leqslant -\frac{4}{L^2},$

(77)

namely, the violation of the BF bound in AdS₅ spacetime.

Therefore, the corresponding outgoing and ingoing fluxes at the horizon and the boundary of the near extremal RN-AdS₅ black brane are

$\begin{aligned}[b] {\cal D}_H^{(\mathrm{out})} & = \frac{{4\pi \omega T{r_{\mathrm{o}}}^2}}{{abL}}|{c_3}{|^2} = \frac{2r_{\mathrm{o}}^3\omega}{L^3}|c_3|^2 ,\\{\kern 1pt} {\cal D}_H^{(\mathrm{in})} &= - \frac{{4\pi \omega T{r_{\rm{o}}}^2}}{{abL}}|{c_4}{|^2} = -\frac{2r_{\mathrm{o}}^3\omega}{L^3}|c_4|^2, \\ {\cal D}_B^{({\mathrm{out}})} & = \frac{{4|\Delta |{r_{\rm{o}}}^4}}{{{L^5}}}\left| A( {\tilde \omega ,\tilde k})\right|^2,\\ {\cal D}_B^{({\mathrm{in}})}& = - \frac{{4|\Delta |{r_{\rm{o}}}^4}}{{{L^5}}}\left|B( {\tilde \omega ,\tilde k})\right|^2. \end{aligned}$

(78)

The absorption cross section ratio $\sigma _{\mathrm{abs}}^{\mathrm{AdS_5}}$ and the Schwinger pair production rate $\left|\mathfrak{b}^{\mathrm{AdS_5}}\right|^2$ can be calculated by choosing the inner boundary condition ${\cal D}_H^{(\mathrm{out})} = 0$ and ( $c_3 = 0$ ) and are given by

$\sigma _{{\rm{abs}}}^{{\rm{Ad}}{{\rm{S}}_{\rm{5}}}} = \left| {\frac{{{\cal D}_H^{({\rm{in}})}}}{{{\cal D}_B^{({\rm{in}})}}}} \right| = \frac{{{\rm{2}}T{L^{\rm{2}}}\left| \nu \right|\sinh \left( {2\pi \left| \Delta \right|} \right)}}{{{r_{\rm{o}}}{{\left| {H\left( {\nu ;\Delta ;\lambda } \right){{\left( {G_R^{{\rm{Ad}}{{\rm{S}}_{\rm{2}}}}} \right)}^{ - 1}} + H\left( { - \nu ;\Delta ;\lambda } \right)} \right|}^2}}}{\sigma _{{\rm{abs}}}},$

(79)

and

${\left| {{\mathfrak{b}^{{\text{Ad}}{{\text{S}}_{\text{5}}}}}} \right|^2} \!\!=\!\! \left| {\frac{{{\cal D}_H^{({\text{in}})}}}{{{\cal D}_B^{({\text{out}})}}}} \right| \!\!=\!\! \frac{{{\text{2}}T{L^{\text{2}}}\left| \nu \right|\sinh \left( {2\pi \left| \Delta \right|} \right)}}{{{r_{\rm{o}}}{{\left| {H\left( {\! -\! \nu ; \!-\! \Delta ;\lambda } \right)G_R^{{\text{Ad}}{{\text{S}}_{\text{2}}}} \!+\! H\left( {\nu ;\! -\! \Delta ;\lambda } \right)} \right|}^2}}}{\left| {{\mathfrak{b}^{{\text{Ad}}{{\text{S}}_{\text{2}}}}}} \right|^2},$

(80)

where $H\left( {x;y;z} \right)$ denotes a function

$H\left( {x;y;z} \right) \equiv {\left( { - 1} \right)^{2x}}{2^x}\frac{{\Gamma \left( {1 + {\rm{2}}x} \right)}}{{\Gamma \left( {\dfrac{1}{2} + x - y + z} \right)\Gamma \left( {\dfrac{1}{2} + x - y - z} \right)}},$

(81)

and

$G_R^{\rm{AdS_2}} = {\left( { - 1} \right)^{4\nu - 1}}{2^{2\nu }}\frac{{\Gamma \left( {1 - 2\nu } \right)}}{{\Gamma \left( {1 + 2\nu } \right)}}\frac{{\Gamma \left( {\dfrac{1}{2} + \nu - {\rm i}{q_{{\rm{eff}}}}\ell } \right)}}{{\Gamma \left( {\dfrac{1}{2} - \nu - {\rm i}{q_{{\rm{eff}}}}\ell } \right)}}\frac{{\Gamma \left( {\dfrac{1}{2} + \nu - {\rm i}\dfrac{\omega }{{2\pi T}} + {\rm i}{q_{{\rm{eff}}}}\ell } \right)}}{{\Gamma \left( {\dfrac{1}{2} - \nu - {\rm i}\dfrac{\omega }{{2\pi T}}{\rm{ + }}{\rm i}{q_{{\rm{eff}}}}\ell } \right)}},$

(82)

which is exactly the retarded Green's function in Eq. (46) of the IR CFT in the near horizon, near extremal region. Furthermore, ${\sigma _{\mathrm{abs}}}$ and ${\left| {{\mathfrak{b}^{{\text{Ad}}{{\text{S}}_{\text{2}}}}}} \right|^2}$ are exactly the absorption cross section ratio and the mean number of produced pairs of the corresponding IR CFT obtained from Eqs. (42) and (41). We find a relationship

$\left|\mathfrak{b}^{\mathrm{AdS_5}}\right|^2 = -\sigma _{\mathrm{abs}}^{\mathrm{AdS_5}}\left(\left| \nu \right| \to - \left| \nu \right|, \left| \Delta \right| \to - \left| \Delta \right|\right),$

(83)

which is similar to Eq. (43) except for a combined change in signs in both $|\nu|$ and $|\Delta|$ .

With Eq. (80) at hand we can easily investigate the relationship between the pair production rate in the near horizon and that for the whole spacetime of RN-AdS₅. As shown in Fig. 1, we can see that the mean number of produced pairs for the whole spacetime is less than that from near horizon region. Moreover, with increasing charge of the scalar field, the corresponding ratio becomes smaller, which is consistent with previous assumptions stating that the Schwinger effect mainly occurs in the near horizon region.

Figure 1

Figure 1. (color online) Ratio of mean number of produced pairs for the whole spacetime to that in the near horizon region as a function of

$\omega {L^2}/{r_{\rm{o}}}$ for different values of

${q_{{\rm{eff}}}}\ell$ with

$T{L^{\text{2}}}/{{r_{\text{o}}}} = 0.001$ ,

$\nu = 0.1{\rm i}$ , and

$\Delta = 0.1{\rm i}$ (left);

$T{L^{\text{2}}}/{{r_{\text{o}}}} = 0.001$ ,

$\nu = 0.01{\rm i}$ , and

$\Delta = 0.01{\rm i}$ (middle);

$T{L^{\text{2}}}/{{r_{\text{o}}}} = 0.01$ ,

$\nu = 0.1{\rm i}$ , and

$\Delta = 0.1i$ (right).

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2 Retarded Green's function

To calculate the retarded Green's function, an ingoing boundary condition is required, namely $c_3 = 0$ . Then, from Eqs. (72) and (73), the connection relations are

${c_5} = {\left( { - 1} \right)^{ - 1 + \gamma - \beta - \lambda}}{c_4}\frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( {1 - \gamma + \alpha } \right)\Gamma \left( {1 - \beta } \right)}},$

(84)

${c_6} = {\left( { - 1} \right)^{ - 1 + \gamma - \alpha - \lambda}}{c_4}\frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( {1 - \gamma + \beta } \right)\Gamma \left( {1 - \alpha } \right)}}.$

(85)

Substituting Eqs. (84) and (85) into Eq. (68) and taking $y \to \infty$ , namely the boundary of the AdS₅ spacetime, one obtains

$\begin{aligned}[b] A(\tilde \omega ,\tilde k) = & \left( - 1 \right)^{{\rm i} q_{\rm{eff}}\ell - 2\lambda - 1 + \Delta }c_4\bigg( \frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( {1 - \gamma + \alpha } \right)\Gamma \left( {1 - \beta } \right)}}\frac{{\Gamma \left( {\gamma '} \right)\Gamma \left( {\alpha ' - \beta '} \right)}}{{\Gamma \left( {\alpha '} \right)\Gamma \left( {\gamma ' - \beta '} \right)}}+ \frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( {1 - \gamma + \beta } \right)\Gamma \left( {1 - \alpha } \right)}}\frac{{\Gamma \left( {2 - \gamma '} \right)\Gamma \left( {\alpha ' - \beta '} \right)}}{{\Gamma \left( {1 - \gamma ' + \alpha '} \right)\Gamma \left( {1 - \beta '} \right)}} \bigg),\\ \\ B(\tilde{\omega}, \tilde{k}) = & \left( - 1 \right)^{{\rm i} q_{\rm{eff}}\ell - 2\lambda - 1 - \Delta }c_4\bigg( \frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( {1 - \gamma + \alpha } \right)\Gamma \left( {1 - \beta } \right)}}\frac{{\Gamma \left( {\gamma '} \right)\Gamma \left( {\beta ' - \alpha '} \right)}}{{\Gamma \left( {\beta '} \right)\Gamma \left( {\gamma ' - \alpha '} \right)}}+ \frac{{\Gamma \left( {2 - \gamma } \right)\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( {1 - \gamma + \beta } \right)\Gamma \left( {1 - \alpha } \right)}}\frac{{\Gamma \left( {2 - \gamma '} \right)\Gamma \left( {\beta ' - \alpha '} \right)}}{{\Gamma \left( {1 - \gamma ' + \beta '} \right)\Gamma \left( {1 - \alpha '} \right)}} \bigg). \end{aligned}$

(86)

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

$G_R^{\mathrm{AdS_5}}\sim\frac{B(\tilde{\omega}, \tilde{k})}{A(\tilde{\omega}, \tilde{k})} = {\left( { - 1} \right)^{ - 2\Delta }}\frac{{\dfrac{{\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( {1 - \gamma + \alpha } \right)\Gamma \left( {1 - \beta } \right)}}\dfrac{{\Gamma \left( {\gamma '} \right)\Gamma \left( {\beta ' - \alpha '} \right)}}{{\Gamma \left( {\beta '} \right)\Gamma \left( {\gamma ' - \alpha '} \right)}} + \dfrac{{\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( {1 - \gamma + \beta } \right)\Gamma \left( {1 - \alpha } \right)}}\dfrac{{\Gamma \left( {2 - \gamma '} \right)\Gamma \left( {\beta ' - \alpha '} \right)}}{{\Gamma \left( {1 - \gamma ' + \beta '} \right)\Gamma \left( {1 - \alpha '} \right)}}}}{{\dfrac{{\Gamma \left( {\alpha - \beta } \right)}}{{\Gamma \left( {1 - \gamma + \alpha } \right)\Gamma \left( {1 - \beta } \right)}}\dfrac{{\Gamma \left( {\gamma '} \right)\Gamma \left( {\alpha ' - \beta '} \right)}}{{\Gamma \left( {\alpha '} \right)\Gamma \left( {\gamma ' - \beta '} \right)}} + \dfrac{{\Gamma \left( {\beta - \alpha } \right)}}{{\Gamma \left( {1 - \gamma + \beta } \right)\Gamma \left( {1 - \alpha } \right)}}\dfrac{{\Gamma \left( {2 - \gamma '} \right)\Gamma \left( {\alpha ' - \beta '} \right)}}{{\Gamma \left( {1 - \gamma ' + \alpha '} \right)\Gamma \left( {1 - \beta '} \right)}}}},$

(87)

which is further simplified into

$G_R^{\rm{AdS_5}} \sim {\left( { - 1} \right)^{ - 2\Delta }}\frac{{\Gamma \left( { - 2\Delta } \right)}}{{\Gamma \left( {2\Delta } \right)}}\frac{{H\left( {\nu ;\Delta ;\lambda} \right) + H\left( { - \nu ;\Delta ;\lambda} \right)G_R^{\rm{AdS_2}}}}{{H\left( {\nu ; - \Delta ;\lambda} \right) + H\left( { - \nu ; - \Delta ;\lambda} \right)G_R^{\rm{AdS_2}}}}.$

(88)

V. CFTS DESCRIPTIONS IN IR AND UV REGIONS

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

From the AdS/CFT correspondence, the IR CFT₁ in the near horizon, near extremal limit and the UV CFT₄ at the asymptotic boundary of the RN-AdS₅ black hole can be connected by the holographic RG flow [26, 27]. The CFT description of the Schwinger pair production in the IR region of charged black holes has been systematically studied in a series of previous works [3, 4, 6, 7]. Herein, we address the dual CFTs descriptions in the UV region and compare them with those in the IR region.

The IR CFT₁ of the RN-AdS black hole is very similar to that of the RN black hole in an asymptotically flat spacetime, as CFT₁ can be viewed as a chiral part of CFT₂, which has the universal structures in its correlation functions. For instance, the absorption cross section of a scalar operator ${\cal O}$ in 2D CFT has the universal form

$\begin{aligned}[b] \sigma \sim & \frac{(2 \pi T_{\rm L})^{2h_{\rm L}-1}}{\Gamma(2 h_{\rm L})} \frac{(2 \pi T_{\rm R})^{2 h_{\rm R}-1}}{\Gamma(2 h_{\rm R})} \sinh\left( \frac{\omega_{\rm L} - q_{\rm L} \Omega_{\rm L}}{2 T_{\rm L}} + \frac{\omega_{\rm R} - q_{\rm R} \Omega_{\rm R}}{2 T_{\rm R}} \right) \\ & \times \left| \Gamma\left( h_{\rm L} + {\rm i} \frac{\omega_{\rm L} - q_{\rm L} \Omega_{\rm L}}{2 \pi T_{\rm L}} \right) \right|^2 \left| \Gamma\left( h_{\rm R} + {\rm i} \frac{\omega_{\rm R} - q_{\rm R} \Omega_{\rm R}}{2 \pi T_{\rm R}} \right) \right|^2, \end{aligned}$

(89)

where $(h_{\rm L}, h_{\rm R})$ , $(\omega_{\rm L}, \omega_{\rm R})$ , and $(q_{\rm L}, q_{\rm R})$ are the left- and right-hand conformal weights, excited energies, charges associated with operator ${\cal O}$ , respectively, while $(T_{\rm L}, T_{\rm R})$ and $(\Omega_{\rm L}, \Omega_{\rm R})$ are the temperatures and chemical potentials of the corresponding left- and right-hand sectors of the 2D CFT. Further identifying the variations in the black hole area entropy with those of the CFT microscopic entropy, namely $\delta S_{\rm BH} = \delta S_{\rm CFT}$ , one derives

$\frac{\delta M}{T_H} - \frac{\Omega_H \delta Q}{T_H} = \frac{\omega_{\rm L} - q_{\rm L} \Omega_{\rm L}}{T_{\rm L}} + \frac{\omega_{\rm R} - q_{\rm R} \Omega_{\rm R}}{T_{\rm R}},$

(90)

where the left hand side of Eq. (90) is calculated with coordinate (14), for which $\delta M = \xi_{\mathrm o}w$ , $\delta Q = q$ , $T_H = \tilde{T}_{ n}$ , and $\Omega_H = 2\mu\ell^2/r_{\mathrm o}$ , and thus, it is equal to $w/T_{ n}-2\pi q_{\mathrm{eff}}\ell$ . Moreover, the violation of the BF bound in AdS₂ makes the conformal weights of the scalar operator ${\cal O}$ dual to $\phi$ a complex, which can be chosen as $h_{\rm L} = h_{\rm R} = \dfrac 1 2+{\rm i}|\nu|$ , even without further knowledge about the central charge and $(T_{\rm L}, T_{\rm R})$ of the IR CFT dual to the near extremal RN-AdS₅ black hole. One can also see that the absorption cross section ratio (42) in the AdS₂ spacetime has the form of Eq. (89) up to some coefficients, depending on the mass and charge of the scalar field.

In contrast, the absorption cross section and retarded Green's functions in a general 4D finite temperature CFT cannot be as easily calculated in momentum space as in the 2D CFT. Thus, it is not straightforward to compare the calculations between the bulk gravity and the boundary CFT sides. Nevertheless, from Eqs. (79) and (80), both the absorption cross section ratio $\sigma _{\mathrm{abs}}^{\mathrm{AdS_5}}$ and the Schwinger mean number of produced pairs $\left|\mathfrak{b}^{\mathrm{AdS_5}}\right|^2$ calculated from the bulk near extremal RN-AdS₅ black hole have a simple proportional relation with their counterparts in the near horizon region. Moreover, the violation of the BF bound (77) in AdS₅ spacetime indicates the complex conformal weights $\bar{\Delta} = 2+2{\rm i}|\Delta|$ of the scalar operator $\bar{{\cal O}}$ in the UV 4D CFT at the asymptotic spatial boundary of the RN-AdS₅ black hole, which also indicates that, to have pair production in the full bulk spacetime, the corresponding operators in the UV CFT should be unstable. Interestingly, Eq. (83) shows that under the interchange between the roles of source and operator both in the IR and UV CFTs at the same time, namely $h_{\rm L,R} = \dfrac12+$ ${\rm i}|\nu| \to \dfrac12-{\rm i}|\nu|$ and $\bar{\Delta} = 2+2{\rm i}|\Delta|\to 2-2{\rm i}|\Delta|$ , the full absorption cross section ratio $\sigma _{\mathrm{abs}}^{\mathrm{AdS_5}}$ and the Schwinger pair production rate $\left|\mathfrak{b}^{\mathrm{AdS_5}}\right|^2$ are interchanged with each other only up to a minus sign. Note that both the charge and the mass of the scalar particle contribute to the conformal weights $h_{\rm L,R}$ of the scalar operator in the dual IR CFT; however, only the mass contributes to the conformal weight $\bar{\Delta}$ of the scalar operator in the dual UV CFT. Actually, it can be seen from the expressions of the conformal weights that the non-zero charge and mass for the scalar field are crucial for the violation of the BF bound in the corresponding AdS spacetimes and hence guarantee the existence of the Schwinger pair production. However, when the charge of the particle is zero, there will be no Schwinger effect, except for an exponentially suppressed Hawking radiation in near extremal black holes.

VI. SUMMARY AND DISCUSSION

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

In this paper, we describe our study of the spontaneous scalar pair production in a near extremal RN-AdS₅ black hole that possesses an AdS₂ structure in the IR region and an AdS₅ geometry in the UV region.

We firstly calculated the mean number of produced pairs (see Eq. (41)) in the near horizon region, which has an AdS₂ structure. The retarded Green's function (see Eq. (46)) has also been obtained for this region. Then, we solved the equation for the whole spacetime of the near extremal RN-AdS₅ black hole by using the matching technique. The matching condition we chose is the low temperature limit, i.e., the near extremal limit of the black hole. Therefore, the greybody factor in Eq. (79) and the mean number of produced pairs in Eq. (80) for the whole spacetime are not merely valid for the low frequency limit, and one can easily apply our calculation to the RN-dS black hole, which was recently described in the low frequency limit [35]. Moreover, the retarded Green's function for an RN-AdS black hole has been calculated (see Eq. (88)), which again is valid at finite frequency, and its corresponding value has only been investigated in the low frequency limit [31] before. Interestingly, we found that there exists a very explicit relationship between the mean number of produced pairs (see also Eq. (80)) for the whole spacetime and that in the near horizon region, which enables us to easily compare the pair production rates of these two regions. We showed that, for an near-extremal RN-AdS₅ black hole, the dominant contribution to the pair production rate mainly comes from the near horizon region, as expected.

Moreover, the CFT descriptions of the pair production are investigated both from the AdS₂/CFT₁ correspondence in the IR and the AdS₅/CFT₄ duality in the UV regions, and consistent results and new connections between the pair production rate and the absorption cross section ratio are found, although the related information computed from the finite temperature 4D CFT is incomplete. This work has successfully generalized the study of pair production in charged black holes to the full spacetime and provided new insights for a complete understanding of the pair production process in curved spacetime.

ACKNOWLEDGEMENT

I. INTRODUCTION
II. BULK THEORY
     A RN- ${\rm{AdS}}_{d+1}$ black hole
     B Near-horizon near-extremal geometry
     C Charged scalar field probe
III. PAIR PRODUCTION IN THE INNER ADS2
     A Near-horizon solutions
     B 2-point correlators from AdS2
IV. PAIR PRODUCTION IN THE ASYMPTOTICAL ${\rm{AdS}}_{5}$
     A Solutions in the near and far regions
     B Near-far matching
     B 2-point correlators from AdS5
V. CFTS DESCRIPTIONS IN IR AND UV REGIONS
VI. SUMMARY AND DISCUSSION
ACKNOWLEDGEMENT

We would like to thank Shu Lin, Rong-Xin Miao, and Yuan Sun for useful discussions.

References

[1]	M. Berglund and M. E. Wieser, Pure Appl. Chem. 83, 397 (2011) doi: 10.1351/PAC-REP-10-06-02
[2]	F. Tárkányi, B. Király, F. Ditrói et al., Nucl. Instrum. Methods B 245, 379 (2006) doi: 10.1016/j.nimb.2005.12.004
[3]	D. Filossofov, A. Novgorodov, N. Korolev et al., J. Appl. Radiat. Isot. 57, 437 (2002) doi: 10.1016/S0969-8043(02)00122-7
[4]	I. Hossain, N. Sharip, and K. Viswanathan, Sci. Res. Essays 7, 86 (2012)
[5]	J. Luo, F. Tuo, X. Kong et al., J. Appl. Radiat. Isot. 66, 1920 (2008) doi: 10.1016/j.apradiso.2008.04.017
[6]	A. Hermanne, S. Takács, F. Tárkányi et al., Nucl. Instrum. Methods B 217, 193 (2004) doi: 10.1016/j.nimb.2003.09.038
[7]	S. M. Qaim, F. T. Tárkányi, P. Obložinský et al., J. Nucl. Sci. Tech. 39, 1282 (2002) doi: 10.1080/00223131.2002.10875338
[8]	V. Zheltonozhskij, V. Mazur, D. Symochko et al., Yad. Fyizik. Energ. 44, 140 (2012)
[9]	V. Mazur, Z. Byigan, and D. Simochko, Ukrayins' kij Fyizichnij Zhurnal (Kyiv) 52, 746 (2007)
[10]	D. Kolev, E. Dobreva, N. Nenov et al., Nucl. Instrum. Methods A 356, 390 (1995) doi: 10.1016/0168-9002(94)01319-5
[11]	N. Demekhina, A. Danagulyan, and G. Karapetyan, Phys. At. Nucl. 65, 365 (2002) doi: 10.1134/1.1451955
[12]	A. Belov, Y. P. Gangrsky, A. Tonchev et al., Phys. At. Nucl. 59, 553 (1996)
[13]	T. D. Thiep, T. T. An, N. T. Khai et al., Phys. Partic. Nucl. Lett. 6, 126 (2009) doi: 10.1134/S1547477109020058
[14]	M. Davydov, V. Magera, A. Trukhov et al., At. Energ. 58, 47 (1985)
[15]	M. S. Rahman, M. Lee, K.-S. Kim et al., Nucl. Instrum. Methods B 276, 44 (2012) doi: 10.1016/j.nimb.2012.01.037
[16]	R. Prestwood and B. Bayhurst, Phys. Rev. 121, 1438 (1961) doi: 10.1103/PhysRev.121.1438
[17]	A. Koning, S. Hilaire, and S. Goriely, TALYS-1.95. A Nuclear Reaction Program. A New Edition, December 2019
[18]	M. Herman, R. Capote, M. Sin et al., EMPIRE-3.2 Malta Modular system for nuclear reaction calculations. 2013, Tech. Rep. INDC (NDS)-0642, BNL-101378-2013
[19]	M. S. Rahman, K. Kim, G. Kim et al., Eur. Phys. J. A 52, 194 (2016) doi: 10.1140/epja/i2016-16194-x
[20]	V. D. Nguyen, D. K. Pham, T. T. Kim et al., J. Korean Phys. Soc. 50, 417 (2007) doi: 10.3938/jkps.50.417
[21]	R. B. Firestone and L. P. Ekstrom, Table of radioactive isotopes, version 2.1. 2004, Lawrence Berkeley National Laboratory. available at http://ie.lbl.gov/toi/index.asp
[22]	NuDat 2.6, National Nuclear Data Center, Brookhaven National Laboratory, updated 2011; available at http://www.nndc.bnl.gov/
[23]	S. Agostinelli, J. Allison, K. Amako et al., Nucl. Instrum. Methods A 506, 250 (2003) doi: 10.1016/S0168-9002(03)01368-8
[24]	H. Naik, G. Kim, K. Kim et al., Nucl. Phys. A 948, 28 (2016) doi: 10.1016/j.nuclphysa.2016.01.015
[25]	H. Naik, G. Kim, R. Schwengner et al., Eur. Phys. J. A 50, 83 (2014) doi: 10.1140/epja/i2014-14083-0
[26]	J. C. Sublet, A. Koning, D. Rochman et al., TENDL-2019: Delivering Both Completeness and Robustness, https://tendl.web.psi.ch/tendl_2019/tendl2019.html
[27]	H. Thierens, D. De Frenne, E. Jacobs et al., Phys. Rev. C 14, 1058 (1976) doi: 10.1103/PhysRevC.14.1058
[28]	M. Nadeem, G. N. Kim, K. Kim et al., J. Radioanal. Nucl. Chem. 313, 1 (2017) doi: 10.1007/s10967-017-5290-2
[29]	J. K. Tuli, Nuclear Wallet Cards, BNL, Upton, New York 11973-5000, USA.January 2000
[30]	M. Tatari, G. N. Kim, H. Naik et al., J. Radioanal. Nucl. Chem. 300, 269 (2014) doi: 10.1007/s10967-014-3008-2
[31]	J. P. François, R. Gijbels, and J. Hoste, J. Radioanal. Nucl. Chem. 5, 251 (1970) doi: 10.1007/BF02513842

References

[1]	M. Berglund and M. E. Wieser, Pure Appl. Chem. 83, 397 (2011) doi: 10.1351/PAC-REP-10-06-02
[2]	F. Tárkányi, B. Király, F. Ditrói et al., Nucl. Instrum. Methods B 245, 379 (2006) doi: 10.1016/j.nimb.2005.12.004
[3]	D. Filossofov, A. Novgorodov, N. Korolev et al., J. Appl. Radiat. Isot. 57, 437 (2002) doi: 10.1016/S0969-8043(02)00122-7
[4]	I. Hossain, N. Sharip, and K. Viswanathan, Sci. Res. Essays 7, 86 (2012)
[5]	J. Luo, F. Tuo, X. Kong et al., J. Appl. Radiat. Isot. 66, 1920 (2008) doi: 10.1016/j.apradiso.2008.04.017
[6]	A. Hermanne, S. Takács, F. Tárkányi et al., Nucl. Instrum. Methods B 217, 193 (2004) doi: 10.1016/j.nimb.2003.09.038
[7]	S. M. Qaim, F. T. Tárkányi, P. Obložinský et al., J. Nucl. Sci. Tech. 39, 1282 (2002) doi: 10.1080/00223131.2002.10875338
[8]	V. Zheltonozhskij, V. Mazur, D. Symochko et al., Yad. Fyizik. Energ. 44, 140 (2012)
[9]	V. Mazur, Z. Byigan, and D. Simochko, Ukrayins' kij Fyizichnij Zhurnal (Kyiv) 52, 746 (2007)
[10]	D. Kolev, E. Dobreva, N. Nenov et al., Nucl. Instrum. Methods A 356, 390 (1995) doi: 10.1016/0168-9002(94)01319-5
[11]	N. Demekhina, A. Danagulyan, and G. Karapetyan, Phys. At. Nucl. 65, 365 (2002) doi: 10.1134/1.1451955
[12]	A. Belov, Y. P. Gangrsky, A. Tonchev et al., Phys. At. Nucl. 59, 553 (1996)
[13]	T. D. Thiep, T. T. An, N. T. Khai et al., Phys. Partic. Nucl. Lett. 6, 126 (2009) doi: 10.1134/S1547477109020058
[14]	M. Davydov, V. Magera, A. Trukhov et al., At. Energ. 58, 47 (1985)
[15]	M. S. Rahman, M. Lee, K.-S. Kim et al., Nucl. Instrum. Methods B 276, 44 (2012) doi: 10.1016/j.nimb.2012.01.037
[16]	R. Prestwood and B. Bayhurst, Phys. Rev. 121, 1438 (1961) doi: 10.1103/PhysRev.121.1438
[17]	A. Koning, S. Hilaire, and S. Goriely, TALYS-1.95. A Nuclear Reaction Program. A New Edition, December 2019
[18]	M. Herman, R. Capote, M. Sin et al., EMPIRE-3.2 Malta Modular system for nuclear reaction calculations. 2013, Tech. Rep. INDC (NDS)-0642, BNL-101378-2013
[19]	M. S. Rahman, K. Kim, G. Kim et al., Eur. Phys. J. A 52, 194 (2016) doi: 10.1140/epja/i2016-16194-x
[20]	V. D. Nguyen, D. K. Pham, T. T. Kim et al., J. Korean Phys. Soc. 50, 417 (2007) doi: 10.3938/jkps.50.417
[21]	R. B. Firestone and L. P. Ekstrom, Table of radioactive isotopes, version 2.1. 2004, Lawrence Berkeley National Laboratory. available at http://ie.lbl.gov/toi/index.asp
[22]	NuDat 2.6, National Nuclear Data Center, Brookhaven National Laboratory, updated 2011; available at http://www.nndc.bnl.gov/
[23]	S. Agostinelli, J. Allison, K. Amako et al., Nucl. Instrum. Methods A 506, 250 (2003) doi: 10.1016/S0168-9002(03)01368-8
[24]	H. Naik, G. Kim, K. Kim et al., Nucl. Phys. A 948, 28 (2016) doi: 10.1016/j.nuclphysa.2016.01.015
[25]	H. Naik, G. Kim, R. Schwengner et al., Eur. Phys. J. A 50, 83 (2014) doi: 10.1140/epja/i2014-14083-0
[26]	J. C. Sublet, A. Koning, D. Rochman et al., TENDL-2019: Delivering Both Completeness and Robustness, https://tendl.web.psi.ch/tendl_2019/tendl2019.html
[27]	H. Thierens, D. De Frenne, E. Jacobs et al., Phys. Rev. C 14, 1058 (1976) doi: 10.1103/PhysRevC.14.1058
[28]	M. Nadeem, G. N. Kim, K. Kim et al., J. Radioanal. Nucl. Chem. 313, 1 (2017) doi: 10.1007/s10967-017-5290-2
[29]	J. K. Tuli, Nuclear Wallet Cards, BNL, Upton, New York 11973-5000, USA.January 2000
[30]	M. Tatari, G. N. Kim, H. Naik et al., J. Radioanal. Nucl. Chem. 300, 269 (2014) doi: 10.1007/s10967-014-3008-2
[31]	J. P. François, R. Gijbels, and J. Hoste, J. Radioanal. Nucl. Chem. 5, 251 (1970) doi: 10.1007/BF02513842

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Muhammad Nadeem, Md. Shakilur Rahman, Muhammad Shahid, Guinyun Kim, Haladhara Naik and Nguyen Thi Hien. Measurements of ^natCd(γ, x) reaction cross-sections and isomer ratio of ^115m,gCd with the bremsstrahlung end-point energies of 50- and 60-MeV[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac256b

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通讯作者: 陈斌, bchen63@163.com

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沈阳化工大学材料科学与工程学院沈阳 110142

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I. INTRODUCTION

Measurements of photon induced reaction cross-sections of natural cadmium are connected with different fields of science, such as the production of medically useful radioisotopes and yield measurements of long-lived radioactive products for radioactive waste handling and dose estimations. Cadmium isotopes are also used in nuclear technology as an important material to make bearings and alloys as well as for electroplating. Therefore, its activation can be used to estimate the radiation dose deposited inside materials for industrial or even medical purposes. In ^natCd, there are eight different isotopes [1]: ¹¹⁶Cd (7.49%), ¹¹⁴Cd (28.73%), ¹¹³Cd (12.22%), ¹¹²Cd (24.13%), ¹¹¹Cd (12.8%), ¹¹⁰Cd (12.49%), ¹⁰⁸Cd (0.89%), and ¹⁰⁶Cd (1.25%). The proton induced reactions of ^natCd are good sources for the production of medically important indium radioisotopes [2]. The radioisotopes ^111mCd and ¹¹¹In play a crucial role in time dependent perturbed angular correlation (TDPAC) studies for the investigation of material properties [3]. The radioisotope ¹⁰⁹Cd is frequently used for detector calibration due to its long half-life and high γ-ray abundance [4]. Cadmium isotopes are also used to enhance the coherence length and output power of HeCd metallic lasers [5]. Important radioactive isotopes of Cd can be produced from the photon-induced reactions of ^natCd. Similarly, the photon-induced reactions of ^natCd produce different radioisotopes of Ag. Among them, ^110mAg is frequently used as a γ-ray reference source. The radioisotopes ^111gAg and ^110mAg are also medically important radioisotopes for radiotherapy and imaging purposes [2, 6].

To maintain an optimal radioisotope production database, the addition of new, reliable experimental data is always important [7]. Previous studies were conducted in the giant dipole resonance (GDR) energy region for the production of ^115mCd [8] and ^111mCd [8, 9] using a γ-ray spectrometry technique; however, the [8, 9] results were higher than the estimated values. No further literature data have been found for any of the nuclides of interest, even in the GDR region. In a nuclear reaction, daughter products may have ground and meta-stable states with different nuclear spins. The ratio between the reaction cross sections of isomers with high spin ${(\sigma }_{\rm H}$ ) and low spin $\left({\sigma }_{\rm L}\right)$ is known as the isomeric yield ratio $\left({{\sigma }_{\rm H}}/{{\sigma }_{\rm L}}\right)$ [10]. There are few earlier experimental studies on the IR of ^115m,gCd produced from the ¹¹⁶Cd(γ, n)^115m,gCd reaction in the giant dipole resonance region [11-14], based on mono-energetic photon beams. However, the measured IR of ^g,m115Cd in the ¹¹⁶Cd(γ, n) reaction using bremsstrahlung endpoint energies of 50, 60, and 70 MeV is available in literature [15], and the IR at high neutron energies for the ¹¹⁶Cd(n, 2n)^115m,gCd reactions [16] is also cited.

Based on these data, in this study, well-established radiation activation and off-line γ-ray spectrometrywere employed to determine the nuclear reaction cross-sections and integral yields of the ^{115g,m,111m,109,107,105,104}Cd and ^{113g,112,111g,110m}Ag radionuclides produced from ^natCd with the bremsstrahlung end-point energies of 50 and 60 MeV. The IR of ^115m,gCd produced in the ^natCd(γ, n) reactions was also measured using the activation technique with these bremsstrahlung end-point energies. For comparison, the ^natCd(γ, xn)^{115g,m,111m,109,107,105,104}Cd and ^natCd(γ, pxn)^{113g,112,111g,110m}Ag reaction cross sections were also theoretically calculated by employing the computer codes TALYS-1.95 [17] and EMPIRE-3.2 Malta [18]. The IR values from the present study with the bremsstrahlung end-point energies of 50 and 60 MeV are compared with the calculated values [17, 18] as well as with the available literature data [11-15].

V. CONCLUSION

The flux-weighted average photon induced nuclear reaction cross-sections for the ^natCd(γ, xn; x=1-6)^{115g,m,111m,109,107,105,104}Cd and ^natCd(γ, xnyp; y=1 x=1-5)^{113g,112,111g,110m}Ag reactions as well as the isomeric yield ratios of ^115g,mCd in the ¹¹⁶Cd(γ, n) reaction were measured with the bremsstrahlung end-point energies of 50- and 60- MeV. Integral yield was also measured to assess the activities produced in the nuclear reactions. The photon induced formation reaction cross sections of ^natCd and the IR value of ¹¹⁵Cd were calculated using the TALYS-1.95 and EMPIRE-3.2 codes and the evaluated data taken from the TENDL-2019 library. In most of the cases, calculated values from the TALYS and EMPIRE codes matched each other and the experimental value. It was also found that the flux-weighted average cross sections increased sharply in the GDR region due to photo absorption and then decreased slightly in QDR region due to the opening of particle emission reaction channels. The isomeric yield ratio of ^115g,mCd in the ¹¹⁶Cd(γ, n) reaction from this study and previously published data was compared with literature data for the ¹¹⁶Cd(n, 2n) reaction. It was found that the experimental IR values of ^115g,mCd in the ^natCd(γ, xn) reaction agree with the calculations but were well below the IR values due to the ¹¹⁶Cd(n, 2n) reaction. The isomeric yield ratio previously measured in the ¹¹⁶Cd(n, 2n) reaction was lower than the calculated values for the same reaction; this is most likely due to the use of default parameters in the theoretical calculations. It was also observed that the theoretical and experimental values increased with excitation energy. However, at the same excitation energy, the IR values in the ¹¹⁶Cd(n, 2n) reaction were significantly higher than those in the ¹¹⁶Cd(γ, n) reaction due to the higher spin of the compound nucleus in the former. This indicates the role of compound nucleus spin alongside the effect of excitation energy.

ACKNOWLEDGEMENT

The authors express their sincere thanks to the staff of the Pohang Accelerator Laboratory (PAL), Pohang, Korea, for the excellent operation and their support during the experiment.

Reference (31)

Nuclides (spin & parity)	Half-life	Decay mode (%)	g-ray energy /keV	γ-ray abundance (%)	Reactions	Q-value /MeV	Threshold /MeV
^115gCd (1/2)⁺	53.46 h	β^- : 100.00	336.24	45.9	¹¹⁶Cd(γ, n)	−8.699	8.700
			336.24	45.9	Decay of ^115mAg (T_1/2 = 18 s) by β^-(79%)
			527.90	27.45	Decay of ^115gAg (T_1/2 = 20 m) by β^-(100%)
^115mCd (11/2)^-	44.56 d	β^-: 100.00	158.03	0.02	¹¹⁶Cd(γ, n)	−8.699	8.700
			484.47	0.29	Decay of ¹¹⁵Ag (T_1/2 = 20 m) by β^-(100%)
			933.80	2	Decay of ¹¹⁵Ag (T_1/2 = 20 m) by β^-(100%)
^111mCd (11/2)^-	48.5 m	IT : 100	150.82	29.1	¹¹¹Cd(γ, g’)	0.0	0.0
					¹¹²Cd(γ, n)	−9.394	9.394
					¹¹³Cd(γ, 2n)	−15.934	15.935
			245.39	94	¹¹⁴Cd(γ, 3n)	−24.977	24.980
			245.39	94	¹¹⁶Cd(γ, 5n)	−39.817	39.824
¹⁰⁹Cd (5/2)⁺	461.9 d	EC: 100	88.03	3.64	¹¹⁰Cd(γ, n)	−9.915	9.915
					¹¹¹Cd(γ, 2n)	−16.891	16.892
					¹¹²Cd(γ, 3n)	−26.285	26.288
					¹¹³Cd(γ, 4n)	−32.824	32.829
					¹¹⁴Cd(γ, 5n)	−41.867	41.875
					¹¹⁶Cd(γ,7n)	−56.707	56.722
¹⁰⁷Cd (5/2)⁺	6.5 h	EC :100	93.12	4.7	¹⁰⁸Cd(γ, n)	−10.334	10.334
					¹¹⁰Cd(γ, 3n)	−27.572	27.575
					¹¹¹Cd(γ, 4n)	−34.547	34.553
					¹¹²Cd(γ, 5n)	−43.941	43.950
			828.93	0.163	¹¹³Cd(γ, 6n)	−50.481	50.493
					¹¹⁴Cd(γ, 7n)	−59.524	59.541
					¹¹⁶Cd(γ, 9n)	−74.364	74.390
¹⁰⁵Cd (5/2)⁺	55.5 m	EC :100	346.87	4.2	¹⁰⁶Cd(γ, n)	−10.870	10.870
			346.87	4.2	¹⁰⁸Cd(γ, 3n)	−29.133	29.137
			433.24	2.81	¹¹⁰Cd(γ, 5n)	−46.371	46.381
			433.24	2.81	¹¹¹Cd(γ, 6n)	−53.346	53.360
			961.84	4.7	¹¹²Cd(γ, 7n)	−62.740	62.759
			961.84	4.7	¹¹³Cd(γ, 8n)	−69.280	69.303
¹⁰⁴Cd (0)⁺	57.7 m	EC :100			¹⁰⁶Cd(γ, 2n)	−19.306	19.308
			66.6	2.4	¹⁰⁸Cd(γ, 4n)	−37.569	37.576
			83.5	47	¹¹⁰Cd(γ, 6n)	−54.808	54.822
			709.3	19.5	¹¹¹Cd(γ, 7n)	−61.783	61.802
					¹¹²Cd(γ, 8n)	−71.177	71.201
^113gAg (1/2)^-	5.37 h	β^- : 100.00	258.72	1.64	¹¹⁴Cd(γ, p)	−10.277	10.278
			258.72	1.64	¹¹⁶Cd(γ, p2n)	−25.117	25.120
			298.6	10	Decay of ^113mAg
			298.6	10	(T_1/2 = 62 s, spin=7/2⁺) by IT(64%)
Continued on next page

Table 1-continued from previous
Nuclides (spin & parity)	Half-life	Decay mode (%)	g-ray energy /keV	γ-ray abundance (%)	Reactions	Q-value /MeV	Threshold /MeV
¹¹²Ag (2)^-	3.13 h	β^- : 100.00	606.82	3.1	¹¹³Cd(γ, p)	−9.749	9.749
			617.52	43	¹¹⁴Cd(γ, pn)	−18.792	18.793
			694.87	2.9	¹¹⁶Cd(γ, p3n)	−33.632	33.637
^111gAg (1/2)^-	7.45 d	β^- : 100.00	245.40	1.24	¹¹²Cd(γ, p)	−9.648	9.649
					¹¹³Cd(γ, pn)	−16.188	16.189
					¹¹⁴Cd(γ, p2n)	−25.231	25.234
			342.13	7	¹¹⁶Cd(γ, p4n)	−40.071	40.079
			342.13	7	Decay of ^111mAg (T_1/2 = 1.08 m, spin=7/2+) by IT (99.3%)
^110mAg (6)⁺	249.83 d	IT : 1.33	657.76	95.61	¹¹¹Cd(γ, p)	−9.084	9.084
					¹¹²Cd(γ, pn)	−18.478	18.479
					¹¹³Cd(γ, p2n)	−25.018	25.021
		β- : 98.67	884.67	75	¹¹⁴Cd(γ, p3n)	−34.061	34.066
		β- : 98.67	884.67	75	¹¹⁶Cd(γ, p5n)	−48.901	48.912
^196gAu (2)^-	6.18 d	EC : 92.8 β- : 7.2	355.68	87.0	¹⁹⁷Au(γ, n)	−8.072	8.073

Produced Nuclei	Reaction	E_th/MeV	E_e =50 MeV		E_e =60 MeV
Produced Nuclei	Reaction	E_th/MeV	${Y_{i,j}}\left( {{E_{\rm e}}} \right)$	${F_{i,j}}\left( {{E_{\rm e}}} \right)$	${Y_{i,j}}\left( {{E_{\rm e}}} \right)$	${F_{i,j}}\left( {{E_{\rm e}}} \right)$
^115gCd	¹¹⁶Cd(γ, n)	8.70	Y_116,115= 100	F_116,115= 0.934	Y_116,115 = 100	F_116,115= 0.940
^115mCd	¹¹⁶Cd(γ, n)	8.70	Y_116,115= 100	F_116,115= 0.934	Y_116,115 = 100	F_116,115= 0.940
^111mCd	¹¹¹Cd(γ, g’)	0.0	Y_111,111= 1.70	F_111,111= 2.910	Y_111,111 =1.40	F_111,111= 2.710
	¹¹²Cd(γ, n)	9.39	Y_112,111= 64.9	F_112,111= 0.890	Y_112,111=62.6	F_112,111= 0.901
	¹¹³Cd(γ, 2n)	15.94	Y_113,111= 18.5	F_113,111= 0.554	Y_113,111= 19.0	F_113,111= 0.592
	¹¹⁴Cd(γ, 3n)	24.96	Y_114,111= 14.5	F_114,111= 0.306	Y_114,111= 16.0	F_114,111= 0.359
	¹¹⁶Cd(γ, 5n)	39.82	Y_116,111= 0.40	F_116,111= 0.080	Y_116,111= 1.00	F_116,111= 0.148
¹⁰⁹Cd	¹¹⁰Cd(γ, n)	9.92	Y_110,109= 75.7	F_110,109= 0.848	Y_110,109= 74.1	F_110,109= 0.863
	¹¹¹Cd(γ, 2n)	16.89	Y_111,109= 18.7	F_111,109= 0.518	Y_111,109= 18.7	F_111,109= 0.560
	¹¹²Cd(γ, 3n)	26.29	Y_112,109= 4.60	F_112,109= 0.278	Y_112,109= 5.00	F_112,109= 0.333
	¹¹³Cd(γ, 4n)	32.83	Y_113,109= 0.80	F_113,109= 0.168	Y_113,109= 1.10	F_113,109= 0.231
	¹¹⁴Cd(γ, 5n)	41.86	Y_114,109= 0.20	F_114,109= 0.059	Y_114,109= 1.10	F_114,109= 0.127
¹⁰⁷Cd	¹⁰⁸Cd(γ, n)	10.33	Y_108,107= 67.5	F_108,107= 0.823	Y_108,107= 59.2	F_108,107= 0.840
	¹¹⁰Cd(γ, 3n)	27.58	Y_110,107= 24.7	F_110,107= 0.255	Y_110,107= 24.5	F_110,107= 0.316
	¹¹¹Cd(γ, 4n)	34.55	Y_111,107= 7.30	F_111,107= 0.145	Y_111,107= 10.1	F_111,107= 0.210
	¹¹²Cd(γ, 5n)	43.95	Y_112,107= 0.50	F_112,107= 0.039	Y_112,107= 5.80	F_112,107= 0.108
	¹¹³Cd(γ, 6n)	50.49	Y_113,107= 0.00	F_113,107= 0.00	Y_113,107= 0.40	F_113,107= 0.052
¹⁰⁵Cd	¹⁰⁶Cd(γ, n)	10.87	Y_106,105= 98.6	F_106,105= 0.786	Y_106,105= 96.4	F_106,105= 0.806
	¹⁰⁸Cd(γ, 3n)	29.14	Y_108,105= 1.40	F_108,105= 0.227	Y_108,105= 1.60	F_108,105= 0.286
	¹¹⁰Cd(γ, 5n)	46.38	Y_110,105= 0.00	F_110,105= 0.020	Y_110,105= 1.90	F_110,105= 0.086
	¹¹¹Cd(γ, 6n)	53.36	Y_111,105= 0.00	F_111,105= 0.000	Y_110,105= 0.10	F_111,105= 0.033
¹⁰⁴Cd	¹⁰⁶Cd(γ, 2n)	19.31	Y_106,104= 98.2	F_106,104= 0.443	Y_106,104= 96.2	F_106,104= 0.488
	¹⁰⁸Cd(γ, 4n)	37.58	Y_108,104= 1.80	F_108,104= 0.107	Y_108,104= 3.60	F_108,104= 0.174
	¹¹⁰Cd(γ, 6n)	54.82	Y_110,104= 0.00	F_110,104= 0.000	Y_110,104= 0.20	F_110,104= 0.016
¹¹³Ag	¹¹⁴Cd(γ, p)	10.28	Y_114,113= 88.1	F_114,113= 0.823	Y_114,113= 92.9	F_114,113= 0.840
¹¹³Ag	¹¹⁶Cd(γ, p2n)	25.12	Y_116,113= 11.9	F_116,113= 0.302	Y_116,113= 7.10	F_116,113= 0.356
¹¹²Ag	¹¹³Cd(γ, p)	9.75	Y_113,112= 27.0	F_113,112= 0.862	Y_113,112= 21.9	F_113,112= 0.875
	¹¹⁴Cd(γ, pn)	18.79	Y_114,112= 71.8	F_114,112= 0.460	Y_114,112= 74.3	F_114,112= 0.505
	¹¹⁶Cd(γ, p3n)	33.64	Y_116,112= 1.20	F_116,112= 0.156	Y_116,112= 3.80	F_116,112= 0.234
¹¹¹Ag	¹¹²Cd(γ, p)	9.65	Y_112,111= 56.9	F_112,111= 0.862	Y_112,111= 49.9	F_112,111= 0.875
	¹¹³Cd(γ, pn)	16.19	Y_113,111= 23.1	F_113,111= 0.546	Y_113,111= 23.0	F_113,111= 0.585
	¹¹⁴Cd(γ, p2n)	25.23	Y_114,111= 19.9	F_114,111= 0.297	Y_114,111= 26.5	F_114,111= 0.352
	¹¹⁶Cd(γ, p4n)	40.08	Y_116,111= 0.10	F_116,111= 0.078	Y_116,111= 0.60	F_116,111= 0.146
^110mAg	¹¹¹Cd(γ, p)	9.08	Y_111,110= 17.5	F_111,110= 0.905	Y_111,110= 10.7	F_111,110= 0.914
	¹¹²Cd(γ, pn)	18.48	Y_112,110= 58.7	F_112,110= 0.466	Y_112,110= 49.2	F_112,110= 0.511
	¹¹³Cd(γ, p2n)	25.02	Y_113,110= 16.4	F_113,110= 0.301	Y_113,110= 18.3	F_113,110= 0.356
	¹¹⁴Cd(γ, p3n)	34.06	Y_114,110= 7.40	F_114,110= 0.151	Y_114,110= 21.6	F_114,110= 0.215
	¹¹⁶Cd(γ, p5n)	48.91	Y_116,110= 0.00	F_116,110= 0.004	Y_116,110= 0.20	F_116,110= 0.065

Nuclear reactions	Total correction factors ( $C_j^T\left( { {E_{\rm e}} } \right)$ )
	Bremsstrahlung end-point energy, E_e/MeV
	50	60
^natCd(γ, n)^115gCd	0.934	0.941
^natCd(γ, n)^115mCd	0.934	0.941
^natCd(γ, xn)^111mCd	0.774	0.774
^natCd(γ, xn)¹⁰⁹Cd	0.753	0.765
^natCd(γ, xn)¹⁰⁷Cd	0.629	0.602
^natCd(γ, xn)¹⁰⁵Cd	0.778	0.783
^natCd(γ, xn)¹⁰⁴Cd	0.437	0.476
^natCd(γ, pxn)^113gAg	0.761	0.806
^natCd(γ, pxn)¹¹²Ag	0.565	0.576
^natCd(γ, pxn)^111gAg	0.676	0.665
^natCd(γ, pxn)^110mAg	0.492	0.416

Reaction	Bremsstrahlung end-point energy/MeV	Flux-weighted average cross-section $\left\langle {{\sigma _i}} \right\rangle$ /mb
		Present work	Theoretical calculations
		Present work	TALYS 1.95 [17]	Empire 3.2 Malta [18]
^natCd(γ, xn)^115gCd	50	2.432 ± 0.345	2.521	3.631
^natCd(γ, xn)^115gCd	60	2.123 ± 0.261	2.330	3.544
^natCd(γ, xn)^115mCd	50	0.5 ± 0.071	0.534	0.778
^natCd(γ, xn)^115mCd	60	0.446 ± 0.059	0.493	0.760
^natCd(γ, xn)^111mCd	50	1.461 ± 0.219	0.458	0.811
^natCd(γ, xn)^111mCd	60	1.413 ± 0.212	0.449	0.785
^natCd(γ, xn)¹⁰⁹Cd	50	12.346 ± 1.786	9.082	8.381
^natCd(γ, xn)¹⁰⁹Cd	60	10.210 ± 1.501	8.466	7.724
^natCd(γ, xn)¹⁰⁷Cd	50	0.733 ± 0.101	0.788	0.656
^natCd(γ, xn)¹⁰⁷Cd	60	0.681 ± 0.094	0.834	0.711
^natCd(γ, xn)¹⁰⁵Cd	50	0.43 ± 0.065	0.669	0.702
^natCd(γ, xn)¹⁰⁵Cd	60	0.385 ± 0.059	0.632	0.651
^natCd(γ, xn)¹⁰⁴Cd	50	0.105 ± 0.015	0.121	0.112
^natCd(γ, xn)¹⁰⁴Cd	60	0.087 ± 0.011	0.112	0.086
^natCd(γ, xn)^113g+mAg	50	0.534 ± 0.075	0.057	0.108
^natCd(γ, xn)^113g+mAg	60	0.501 ± 0.061	0.057	0.110
^natCd(γ, xn)¹¹²Ag	50	0.318 ± 0.043	0.068	0.159
^natCd(γ, xn)¹¹²Ag	60	0.367 ± 0.046	0.081	0.172
^natCd(γ, xn)^111g+mAg	50	0.126 ± 0.019	0.0425	0.121
^natCd(γ, xn)^111g+mAg	60	0.123 ± 0.018	0.045	0.119
^natCd(γ, xn)^110mAg	50	0.026 ± 0.005	0.017	0.032
^natCd(γ, xn)^110mAg	60	0.027 ± 0.004	0.026	0.035

Measurements of ^natCd(γ, x) reaction cross sections and isomer ratio of ^115m,gCd with the bremsstrahlung end-point energies of 50 and 60 MeV

Abstract：

1 Pair production rate and absorption cross section ratio

2 Retarded Green's function

1 Pair production and absorption cross section

Figure 1

2 Retarded Green's function

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