On high-energy particles in accretion disk coronae of supermassive black holes: implications for MeV gamma rays and high-energy neutrinos from AGN cores

TL;DR

This paper explores how particle acceleration in AGN coronae can produce observable MeV gamma rays and neutrinos, linking recent observations to cosmic background radiation and high-energy neutrino flux.

Contribution

It proposes a model where diffusive shock acceleration in AGN coronae explains non-thermal emissions and neutrino fluxes, supported by observational parameters.

Findings

Non-thermal electrons can generate MeV gamma rays via inverse Compton scattering.

The model reproduces the cosmic X-ray and gamma-ray background up to 10 MeV.

Protons accelerated in coronae can account for diffuse neutrino flux below 300 TeV.

Abstract

Recent observations with ALMA have revealed evidence for non-thermal synchrotron emission from the core regions of two nearby Seyfert galaxies. This suggests that the coronae of accretion disks in active galactic nuclei (AGNs) can be conducive to the acceleration of non-thermal electrons, in addition to the hot, thermal electrons responsible for their X-ray emission through thermal Comptonization. Here we investigate the mechanism of such particle acceleration, based on observationally inferred parameters for AGN disk coronae. One possibility to account for the observed non-thermal electrons is diffusive shock acceleration, as long as the gyrofactor $\eta_g$ does not exceed $\sim10^6$. These non-thermal electrons can generate gamma rays via inverse Compton scattering of disk photons, which can appear in the MeV band, while those with energies above $\sim100$ MeV would be attenuated via internal $\gamma\gamma$ pair production. The integrated emission from all AGNs with thermal and non-thermal Comptonization can reproduce the observed cosmic background radiation in X-rays as well as gamma-rays up to $\sim 10$ MeV. Furthermore, if protons are accelerated in the same conditions as electrons and $\eta_g\sim30$, our observationally motivated model is also able to account for the diffuse neutrino flux at energies below 100-300 TeV. The next generation of MeV gamma-ray and neutrino facilities can test these expectations by searching for signals from bright, nearby Seyfert galaxies such as NGC 4151 and IC 4329A.

Equations

$$n_e$$

$$t_{\mathrm{fall}}=R_c/v_{\text{mbox}ff}\simeq 2.5\times 10^5{\left(\frac{r_c}{40}\right)}^{1/2}\left(\frac{M_{\text{mbox}BH}}{10^8{M}_{\odot }}\right)[\mathrm{s}].$$

$$t_{\mathrm{syn},e}({\gamma }_e)$$

$$t{}_{\text{mbox}IC}({\gamma }_e)=\frac{3m_{e}c}{4{\sigma }_{\text{mbox}T}}{\left[\underset{0}{\overset{\infty }{\int }}dϵf{}_{\text{mbox}KN}(\tilde{b})\frac{U_{\mathrm{ph}}(ϵ)}{ϵ}\right]}^{-1}{\gamma }_e^{-1},$$

$$U_{\mathrm{ph},}{}_{\mathrm{tot}}=\int dϵU_{\mathrm{ph}}(ϵ)$$

$$\sim 5\times 10^3\frac{L_{\mathrm{ph},\mathrm{bol}}}{2\times 10^{45}\mathrm{erg}{\mathrm{s}}^{-1}}{\left(\frac{r_c}{40}\right)}^{-2}{\left(\frac{M_{\text{mbox}BH}}{10^8{M}_{\odot }}\right)}^{-2}[\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-3}].$$

$$t{}_{\text{mbox}IC}{}_{/syn,p}=\frac{3}{4}{\left(\frac{m_p}{m_e}\right)}^3\frac{m_e{c}^2}{c{\sigma }_{\text{mbox}T}{U}_{ph/B}}{\gamma }_p^{-1},$$

$$t_{\mathrm{syn},p}\simeq 4.8\times 10^{14}{\left(\frac{B}{10\mathrm{G}}\right)}^{-2}{\left(\frac{{\gamma }_p}{100}\right)}^{-1}[\mathrm{s}].$$

$$t_{pp}$$

$${\sigma }_{pp}$$

$$t_{p\gamma }^{-1}=\frac{c}{2{\gamma }_p^2}\underset{{\bar{\epsilon }}_{\mathrm{thr}}}{\overset{\infty }{\int }}d\bar{\epsilon }{\sigma }_{p\gamma }(\bar{\epsilon })K_{p\gamma }(\bar{\epsilon })\bar{\epsilon }\underset{\bar{\epsilon }/(2{\gamma }_p)}{\overset{\infty }{\int }}dϵ\frac{U_{\mathrm{ph}}(ϵ)}{{ϵ}^4},$$

$$t_{\mathrm{BH}}^{-1}$$

$$t{}_{\text{mbox}DSA}\simeq \frac{{\eta }_{\mathrm{acc}}D(E{}_{\text{mbox}CR})}{{v_{\text{mbox}sh}}^2},$$

$$D(E{}_{\text{mbox}CR})\simeq \frac{{\eta }_{g}cE{}_{\text{mbox}CR}}{3eB},$$

$$t{}_{\text{mbox}DSA}\simeq \frac{10}{3}\frac{{\eta }_{g}c{R}_g}{{v_{\text{mbox}sh}}^2},$$

$$\simeq 7.6\times 10^{-3}\left(\frac{{\eta }_g}{100}\right)\left(\frac{m_{p/e}}{m_e}\right)\left(\frac{r_c}{40}\right){\left(\frac{B}{10\mathrm{G}}\right)}^{-1}\left(\frac{{\gamma }_{p/e}}{100}\right)[\mathrm{s}].$$

$$\log L_{2\mathrm{keV}}=0.760\log L_{2500\mathring{\mathrm{A}}}+3.508$$

$$\Gamma =\sqrt{\frac{9}{4}+\frac{1}{{\theta }_e[{\tau }_{\text{mbox}T}({\tau }_{\text{mbox}T}+1/3)]}}-\frac{1}{2},$$

$$B\propto M_{\text{mbox}BH}^{-1/2},$$

$$M_{\text{mbox}BH}=2\times 10^7{M}_{\odot }{\left[\frac{L_{2-10\mathrm{keV}}}{1.155\times 10^{43}\mathrm{erg}{\mathrm{s}}^{-1}}\right]}^{0.746}.$$

$${ϵ}_{\mathrm{peak}}\simeq \frac{3.5m_e^2{c}^4}{E_{\gamma }}\simeq 1\left(\frac{1\mathrm{TeV}}{E_{\gamma }}\right)\mathrm{eV}.$$

$${\sigma }_{\gamma \gamma }(E_{\gamma },ϵ,\theta )=\frac{3{\sigma }_{\text{mbox}T}}{16}(1-{\beta }^2)$$

$$\times \left[2\beta ({\beta }^2-2)+(3-{\beta }^4)\ln \left(\frac{1+\beta }{1-\beta }\right)\right],$$

$$\beta \equiv \sqrt{1-\frac{2m_e^2{c}^4}{ϵE_{\gamma }(1-\mu )}};\mu \equiv \cos \theta .$$

$${\tau }_{\gamma \gamma }(E_{\gamma })=\underset{-1}{\overset{1}{\int }}d\mu \underset{{ϵ}_{\mathrm{th}}}{\overset{\infty }{\int }}dϵ\frac{1-\mu }{2}\frac{U_{\mathrm{ph}}(ϵ)}{{ϵ}^2}{\sigma }_{\gamma \gamma }(E_{\gamma },ϵ,\theta )R_c$$

$${ϵ}_{\mathrm{th}}=\frac{2m_e^2{c}^4}{E_{\gamma }(1-\mu )}.$$

$${\sigma }_{\gamma \gamma }$$

$$\frac{\partial }{\partial \gamma }\left({\dot{\gamma }}_{\mathrm{cool}}n\right)+\frac{n}{t_{\mathrm{fall}}}=Q(\gamma ),$$

$$n=\frac{1}{{\dot{\gamma }}_{\mathrm{cool}}}\underset{\gamma }{\overset{\infty }{\int }}Q({\gamma }^{\prime })e^{-T(\gamma ,{\gamma }^{\prime })}d{\gamma }^{\prime },$$

$$T({\gamma }_1,{\gamma }_2)=\frac{1}{t_{\mathrm{fall}}}\underset{{\gamma }_1}{\overset{{\gamma }_2}{\int }}\frac{d\gamma }{{\dot{\gamma }}_{\mathrm{cool}}}$$

$$P_{\text{mbox}sh}$$

$$f_{pp}=\frac{t_{\mathrm{fall}}}{t_{pp}}\simeq 0.16\left(\frac{{\tau }_{\text{mbox}T}}{1.1}\right){\left(\frac{r_c}{40}\right)}^{-1/2}.$$

Full text

accretion, accretion disks — black hole physics — galaxies: active — (galaxies:) quasars: supermassive black holes — acceleration of particles — neutrinos

††journal: ApJ

\DeclareAcronym

agn short = AGN, long = active galactic nucleus, short-plural = s, long-plural-form = active galactic nuclei, class = astro, first-style = default

\DeclareAcronymebl short = EBL , long = extragalactic background light , class = astro, first-style = default

\DeclareAcronymsmbh short = SMBH , long = supermassive black hole , class = astro , first-style = default

\DeclareAcronymdsa short = DSA , long = diffusive shock acceleration , class = astro , first-style = default

\DeclareAcronymsto short = StA , long = stochastic acceleration , class = astro , first-style = default

\DeclareAcronymic short = IC , long = inverse Compton , class = astro , first-style = default

\DeclareAcronymkn short = KN , long = Klein–Nishina , class = astro , first-style = default

\DeclareAcronymec short = EC , long = elastic Coulomb , class = astro , first-style = default

\DeclareAcronympp short = $pp$ , long = hadronuclear , class = astro , first-style = default

\DeclareAcronympg short = $p\gamma$ , long = photomeson , class = astro , first-style = default

\DeclareAcronymsed short = SED , long = spectral energy distribution , class = astro , first-style = default

\DeclareAcronymalma short = ALMA , long = Atacama Large Millimeter/submillimeter Array , class = instruments , first-style = default

1 Introduction

\Acp

agn are powered by mass accretion onto \acpsmbh. They emit intense electromagnetic radiation in broad range of frequencies. Measurements of X-ray spectra of \acpagn allow us to study various aspect of \acpsmbh such as black hole spins (e.g., Reynolds, 2014), geometrical structures (e.g., Ramos Almeida & Ricci, 2017), and cosmological evolution (e.g., Ueda et al., 2014).

A key for understanding these phenomena is primary X-ray radiation of the accretion disk which arises from Comptonization of disk photons in moderately thick thermal plasma, namely coronae, above an accretion disk (see, e.g., Katz, 1976; Bisnovatyi-Kogan & Blinnikov, 1977; Pozdniakov et al., 1977; Galeev et al., 1979; Takahara, 1979; Sunyaev & Titarchuk, 1980). X-ray observations have indicated the coronal temperature of $\sim 10^{9}$ K and the Thomson scattering opacity of $\gtrsim 1$ (e.g. Zdziarski et al., 1994; Fabian et al., 2015). However, the nature of \acagn coronae is still veiled in mystery.

Very recently, Inoue & Doi (2018) has reported the detection of coronal radio synchrotron emission from two nearby Seyferts (e.g., Di Matteo et al., 1997; Inoue & Doi, 2014; Raginski & Laor, 2016) utilizing the \acalma. The inferred coronal magnetic field strength was $\sim 10$ G with a size of $40R_{s}$, where $R_{s}$ is the Schwartzschild radius, for both active \acpsmbh with a mass of $\sim 10^{8}M_{\odot}$. It is also found that coronae of Seyferts contain both thermal and non-thermal electrons. This implies that acceleration of high energy particles happens in AGN coronae.

High energy particles in the nuclei of Seyferts have been discussed for a long time111High energy particles in the coronae of X-ray binaries have been also discussed in literature (e.g., Bhattacharyya et al., 2003, 2006).. In the past, it was argued that primary X-ray emission comes from pair cascades induced by high energy particles accelerated in and/or around accretion flows (e.g., Zdziarski, 1986; Kazanas & Ellison, 1986; Ghisellini et al., 2004). In the pair cascade model, particles are accelerated by shock dissipation in accretion flows (e.g., Cowsik & Lee, 1982; Protheroe & Kazanas, 1983; Zdziarski, 1986; Kazanas & Ellison, 1986; Sikora et al., 1987; Begelman et al., 1990). However, the detection of the \acagn spectral cutoffs (e.g., Madejski et al., 1995; Zdziarski et al., 2000) and non-detection of Seyfert \acpagn in the gamma-ray band (e.g., Lin et al., 1993) ruled out the pair cascade scenario as a dominant source for the primary X-ray emission222TeV gamma rays are measured from the Galactic center (HESS Collaboration et al., 2016). This detection indicated possible particle acceleration in accretion flow, even though accretion rate in the Galactic center is several orders of magnitude lower than that in standard disks..

In this paper, we investigate the production mechanism of the observed high energy particles in \acagn coronae. As an example, we consider those high energy particles are supplied by \acdsa processes (e.g., Drury, 1983; Blandford & Eichler, 1987) in the coronae. Contrary to the previously discussed AGN accretion shock models, the required shock power is much lower in order to explain the observed non-thermal species and to be in concordance with the current picture of coronal X-ray emission. Moreover, previous studies of high energy particles in \acagn accretion disks have treated as free parameters corona size and magnetic field, which are important parameters for the understandings of particle acceleration. The \acalma observations allowed us to determine both of them (Inoue & Doi, 2018). Most critically, the observationally determined strength of the magnetic field appeared to be significantly smaller than the one previously considered in the literature. We take into account these newly determined coronal parameters.

Thermal coronal emission from Seyferts is known to explain the entire cosmic X-ray background radiation (e.g., Ueda et al., 2014). In contrast, the origin of the cosmic MeV background radiation from 0.1 MeV to several tens MeV is still unknown (see e.g., Inoue, 2014). Here, the non-thermal electrons in coronae seen by ALMA will invoke power-law MeV gamma-ray emission via Comptonization of disk photons. Such non-thermal emission is suggested as a possible explanation for the cosmic MeV gamma-ray background radiation (Inoue et al., 2008). However, non-thermal electron species in the previous work were included in an ad hoc way. In this work, we revisit the contribution of Seyferts to the MeV gamma-ray background radiation by considering the particle acceleration of non-thermal populations in coronae together with the latest X-ray luminosity function of Seyferts (Ueda et al., 2014).

High energy particles around accretion disks of \acpagn also generate intense neutrino emission through \acpp and \acpg interaction processes by interacting accreting gas and photon fields (e.g., Eichler, 1979; Begelman et al., 1990; Stecker et al., 1992; Alvarez-Muñiz & Mészáros, 2004). Although these originally predicted fluxes have been significantly constrained by high energy neutrino observations (The IceCube Collaboration, 2005), recent studies have revisited the estimated fluxes and found that \acagn core models are still viable (Stecker, 2005, 2013; Kalashev et al., 2015). However, normalization of neutrino fluxes from \acpagn and acceleration properties of high energy particles in those models are assumed to match with the observation. In this work, we also discuss the possible contribution from \acagn cores given our \acalma observations and investigate the required parameter spaces for the explanation of the IceCube diffuse neutrino fluxes.

We describe general particle acceleration processes in \acagn coronae in § 2. The broadband emission spectrum of the central region of \acpagn and physical properties of \acagn coronae are presented in § 3. Relevant timescales and steady-state particle spectra are discussed in § 4 and § 5, respectively. § 6 and § 7 present the results of the expected gamma-ray and neutrino fluxes from individual AGN cores and the cosmic gamma-ray and neutrino background fluxes from \acagn cores, respectively. Discussion including other possible particle acceleration mechanism is given in § 8, and conclusions are in § 9. Throughout this paper, we adopt the standard cosmological parameters of $(h,\Omega_{M},\Omega_{\Lambda})=(0.7,0.3,0.7)$.

2 Particle Acceleration in Nuclei of Seyferts

As non-thermal coronal synchrotron emission is seen in nearby Seyferts (Inoue & Doi, 2018), particle acceleration should occur in AGN coronae, even though thermal populations are energetically dominant. Particle acceleration mechanism in the coronae is highly uncertain. Various acceleration mechanisms can take place in the coronae such as \acdsa mechanism (e.g., Drury, 1983; Blandford & Eichler, 1987), turbulent acceleration (e.g., Zhdankin et al., 2018), magnetosphere acceleration (e.g., Beskin et al., 1992; Levinson, 2000), and magnetic reconnection (e.g., Hoshino & Lyubarsky, 2012). In this work, for simplicity, we consider the \acdsa as the fiducial particle acceleration process. We discuss the other possible acceleration processes in § 8.3.

In order to investigate particle acceleration mechanism of the observed non-thermal electrons, we consider the interaction of locally injected relativistic particles with the matter, photons, and magnetic field in the infalling coronae. Although the location of shock sites is uncertain, for simplicity, we assume that shocks occur inside of the coronae. The shock accelerates a part of inflow plasma to high energies. As the energy loss timescale of high energy protons is in general longer than the free-fall timescale, a sufficiently high energy density of relativistic particles is maintained to provide pressure to support a standing shock around a \acsmbh (Protheroe & Kazanas, 1983).

Coronae are assumed to be spherical with a radius of $R_{c}\equiv r_{c}R_{s}$. $r_{c}$ is the dimensionless parameter of the corona size and $R_{s}=2GM_{\mbox{\tiny{BH}}}/c^{2}$, where $G$ is the gravitational constant, $M_{\mbox{\tiny{BH}}}$ is the mass of the central \acsmbh, $c$ is the speed of light. Coronae are also set to be in a steady state. We also do not consider positrons in coronae. Thus, the proton number density $n_{p}$ is equal to the electron density $n_{e}$ in this work, which gives the maximum number of protons in coronae. $n_{e}$ is defined through the Thomson scattering opacity in coronae, $\tau_{\mbox{\tiny{T}}}$ as

[TABLE]

where $\sigma_{\mbox{\tiny{T}}}$ is the Thomson scattering cross section.

2.1 Dynamical Timescale

The gas is assumed to be spherically accreted on to the \acsmbh with free-fall velocity ${v_{\mbox{\tiny{ff}}}}=\sqrt{2GM_{\mbox{\tiny{BH}}}/R_{c}}$. The free-fall timescale from the coronal region is estimated to be

[TABLE]

2.2 Radiative Cooling

High energy particles loose their energies through radiative cooling processes. In \acagn coronae, high-energy electrons mainly lose their energies via synchrotron and \acic radiation. The synchrotron cooling rate for an electron with a Lorentz factor of $\gamma_{e}$ is

[TABLE]

where $m_{e}$ is the electron rest mass and $U_{\rm B}=B^{2}/8\pi$ is the magnetic field energy density of magnetic field strength $B$.

The inverse Compton cooling rate including the \ackn cross section (Jones, 1968; Moderski et al., 2005; Khangulyan et al., 2014) is

[TABLE]

where $\tilde{b}\equiv 4\gamma_{e}\epsilon/m_{e}c^{2}$ and $f{}_{\mbox{\tiny{KN}}}\simeq 1/(1.0+\tilde{b})$ (Moderski et al., 2005). $\epsilon$ is the target photon energy and $U_{\rm ph}$ is the photon energy density given as $U_{\rm ph}(\epsilon)=L_{\rm ph}(\epsilon)/4\pi R_{c}^{2}c$. The total \acagn disk luminosity, $L_{\rm ph}$, which includes contribution from the accretion disk and corona, is defined in § 3.1. For simplicity, we consider a uniform photon density in the coronae. If the coronae has spatially homogeneous emissivity rather uniform emission, the mean photon density inside the source is enhanced by a factor of $\sim 2.24$ on average (Atoyan & Aharonian, 1996).

For the typical characteristics of the coronae, the energy density of the photon field is

[TABLE]

For the magnetic field strength inferred with \acalma, $B\simeq 10\rm\,G$ for $M_{\mbox{\tiny{BH}}}=10^{8}M_{\odot}$ SMBHs, the energy density of the photon field exceeds the magnetic field energy density if $L_{\rm ph,bol}\geq 2\times 10^{42}\rm\,erg\,s^{-1}$. We note that the dominance of photon fields over magnetic field does not necessary prevents particle acceleration as such conditions are met in some efficient non-thermal sources, e.g., in gamma-ray binary systems (Aharonian et al., 2006; Khangulyan et al., 2008). Moreover, high density of target photons can enable the converter acceleration mechanism if a relativistic velocity jump present in the system (Derishev et al., 2003).

Relativistic protons are predominately cooled though inelastic \acpp interactions, \acpg reactions, and proton \acic/synchrotron channels. Since only the Thomson regime might be relevant for the proton \acic cooling, the proton synchrotron and \acic cooling time-scales are

[TABLE]

where $m_{p}$ is the proton rest mass and $\gamma_{p}$ is the proton Lorentz factor. In the case of the synchrotron losses, this yields

[TABLE]

Given the higher energy density of the photon field, the \acic cooling time can be up to $\sim 10^{4}$ times faster. These electrodynamic cooling channels are inefficient as compared to the hardronic mechanisms below. Hereinafter, we do not consider proton \acic/synchrotron coolings.

The \acpp cooling time can be expressed as

[TABLE]

where $\kappa_{pp}\sim 0.5$ is the proton inelasticity of the process and we adopt $\sigma_{pp}=3\times 10^{-26}\ {\rm cm}^{2}$. Below we adopt the formalism developed by Kelner et al. (2006). The total cross section of the inelastic \acpp process $\sigma_{pp}$ is represented as a function of the proton energy $E_{p}=\gamma_{p}m_{p}c^{2}$,

[TABLE]

for $E_{p}\geq E_{pp,\rm thr}$, where $1\ {\rm mb}=10^{-27}\ {\rm cm}^{2}$, $L=\log(E_{p}/1\,\rm TeV)$, and $E_{pp,\rm thr}=1.22$ GeV (Kelner et al., 2006).

The \acpg cooling time via photomeson interactions is

[TABLE]

where $\bar{\varepsilon}$ and $\epsilon$ are the photon energy in the proton rest frame and the black hole frame, respectively, $U_{\rm ph}$ is the energy density of the photon target, and $\bar{\varepsilon}_{\rm thr}=145$ MeV. For numerical calculation we follow the formalism suggested by Kelner & Aharonian (2008).

The \acpg interaction also generates pairs, so-called the Bethe-Heitler pair production process and its cooling timescale is approximated as (Gao et al., 2012)

[TABLE]

where $\alpha_{f}$ is the fine-structure constant.

2.3 Acceleration

In the frame work of \acdsa (e.g., Drury, 1983; Blandford & Eichler, 1987), the acceleration time scale can be approximated as

[TABLE]

where $D$ is the diffusion coefficient, $E{}_{\mbox{\tiny{CR}}}$ is the particle energy, and ${v_{\mbox{\tiny{sh}}}}$ is the shock speed. $\eta_{\rm acc}$ is a numerical factor that depends on the shock compression ratio and the spatial dependence of $D$ (Drury, 1983). We set $\eta_{\rm acc}=10$. Assuming a Bohm-like diffusion,

[TABLE]

where $e$ is the electric charge and $\eta_{g}$ is the gyrofactor which is the mean free path of a particle in units of the gyroradius. $\eta_{g}$ characterizes the efficiency of the acceleration. $\eta_{g}=1$ corresponds to the Bohm limit case. The \acdsa time can be written as

[TABLE]

where $R_{g}$ is the gyro radius and ${v_{\mbox{\tiny{sh}}}}$ is set as ${v_{\mbox{\tiny{ff}}}}(R_{c})$. $\eta_{g}$ varies in different astrophysical environments. $\eta_{g}\sim 1$ is possibly seen in a Galactic supernova remnant (Uchiyama et al., 2007), while $\eta_{g}\sim 10^{4}$ is seen in the case of blazars in the framework of one-zone leptonic models (e.g., Inoue & Takahara, 1996; Finke et al., 2008; Inoue & Tanaka, 2016).

3 Properties of Active Supermassive Black Holes

In this section, we summarize the general observational properties of the central region of \acpagn related to high-energy particles in coronae.

3.1 Broadband Emission from the Core Region

Emission from the \acagn core region mainly arises from two components (Elvis et al., 1994). First is the geometrically thin and optically thick standard accretion disks (Shakura & Sunyaev, 1973). This standard accretion disk generates a big blue bump from optical to UV attributed by multi-color blackbody radiation. Second is the Comptonized accretion disk photons from the coronal regions above the accretion disk (Katz, 1976; Bisnovatyi-Kogan & Blinnikov, 1977; Pozdniakov et al., 1977; Sunyaev & Titarchuk, 1980). This Comptonized emission appears in the X-ray band together with emission reprocessed by the surrounding cold materials, a so-called Compton reflection component (e.g., Lightman & White, 1988; Magdziarz & Zdziarski, 1995; Ricci et al., 2011).

In this work, for the primary X-ray emission from coronae, we assume a cut-off power-law model in the form of $E^{-\Gamma}\exp(E/E_{c})$, where we set $\Gamma=1.9$ and $E_{c}=300$ keV (Ueda et al., 2003, 2014). For the Compton reflection component, we use the pexrav model Magdziarz & Zdziarski (1995) assuming a solid angle of $2\pi$, an inclination angle of $\cos i=0.5$, and the solar abundance for all elements. Since we consider the photons only around the core regions, we ignore the absorption by torus.

The optical-UV accretion-disk \acpsed are taken from Elvis et al. (1994). Here, the primary 2 keV X-ray disk luminosity is connected to the accretion-disk luminosity at 2500 Å as

[TABLE]

based on the study of 545 X-ray selected type 1 \acpagn from the XMM-COSMOS survey (Lusso et al., 2010). Between UV and X-ray, following Lusso et al. (2010), we linearly connect the UV luminosity at 500 Å  to the luminosity at 1 keV. Figure 1 shows the broadband \acagn SED arising from the core region for various X-ray luminosities. \acagn core \acpsed typically have a spectral peak at $\sim 30$ eV corresponding to $\sim 10^{5}$ K (Fig. 1), which corresponds to the emission radius at around $\sim 10R_{s}$.

3.2 Physical Properties of Coronae

X-ray spectral studies allow us to determine some of the coronal parameters such as the coronal electron temperature $kT_{e}$ and the Thomson scattering optical depth $\tau_{\mbox{\tiny{T}}}$ (e.g., Brenneman et al., 2014). $k$ is the Boltzmann constant and $T_{e}$ is the electron temperature in Kelvin. The spectral cutoff at $\sim 300$ keV of \acagn core spectra corresponds to the electron temperature of $kT_{e}\sim 100$ keV. The process of Comptonization by thermal plasma is described by the Kompaneets equation (Kompaneets, 1957). Here, the photon index of the primary emission is assumed to be 1.9 in this work. This corresponds to $\tau_{\mbox{\tiny{T}}}\sim 1.1$ based on the solution to the Kompaneets equation (Zdziarski et al., 1996) as

[TABLE]

where the dimensionless electron temperature $\theta_{e}\equiv kT_{e}/m_{e}c^{2}$. Therefore, in this work, we adopt $kT_{e}=100$ keV and $\tau_{\mbox{\tiny{T}}}=1.1$. These values are consistent with the results from detailed X-ray spectral analysis (e.g., Fabian et al., 2015).

Recently, utilizing X-ray and radio data, Inoue & Doi (2018) found that the coronal magnetic field strength $B$ is approximately $10$ Gauss on scales of $\sim 40R_{s}$ from the \acpsmbh for two nearby Seyferts whose BH masses are $\sim 10^{8}~{}M_{\odot}$333Contrary to this observational result, recent numerical simulations of the hot accretion flows (e.g., Kimura et al., 2019) shows the magnetic field enhanced more by the magnetorotational instability (MRI; Balbus & Hawley, 1991, 1998).. This coronal size is consistent with optical–X-ray spectral fitting studies (Jin et al., 2012) and micorolensing observation (Morgan et al., 2012). Thus, in this paper, we set the coronal size as $40R_{s}$ for all \acpsmbh and $B=10$ G for $10^{8}~{}M_{\odot}$ \acpsmbh.

Inoue & Doi (2018) also suggested that the coronae are likely to be advection heated hot accretion flows (Kato et al., 2008; Yuan & Narayan, 2014) rather than magnetically heated corona (Haardt & Maraschi, 1991; Liu et al., 2002) because the measured magnetic field strength is too weak to keep the coronae hot and is rather consistent with the value based on the self-similar solutions of hot accretion flows (Kato et al., 2008; Yuan & Narayan, 2014). Thus, we assume that coronal magnetic field strength scales as

[TABLE]

following the self-similar solution for the hot accretion flow (Yuan & Narayan, 2014) where we ignore dependence on accretion rate and other parameters for simplicity.

Mayers et al. (2018) have recently investigated a relation between the intrinsic 2–10 keV X-ray luminosity and the mass of central \acpsmbh using \acpagn from the XMM-Newton Cluster Survey. The empirical relation found in Mayers et al. (2018) is given as

[TABLE]

Using this relation, we can convert X-ray luminosities to masses of central \acpsmbh.

3.3 Internal Gamma-ray Attenuation in Coronae

Accelerated electrons and protons in coronae would emit gamma rays (see §3.1). However, high energy gamma-ray photons are attenuated by photon-photon pair production interactions ($\gamma\gamma\rightarrow e^{+}e^{-}$) with low-energy photons. For isotropic target photons the pair production cross section achieves its maximum of $\approx 0.2\sigma_{\mbox{\tiny{T}}}$ when a gamma-rays of energy $E_{\gamma}$ interacts with a low-energy photon with energy (see, e.g., Aharonian, 2004)

[TABLE]

In terms of wavelength, $\lambda_{\rm peak}\simeq 1.4(E_{\gamma}[{\rm TeV}])\ \mu{\rm m}$.

Abundant photons are emitted from the \acagn core region (Fig. 1). From the SED of \acagn core regions as given in § 3.1, we can compute the optical depth for high-energy gamma rays to $\gamma\gamma$ pair production interactions. The cross section for this process is (Breit & Wheeler, 1934; Heitler, 1954)

[TABLE]

where $\beta$ is

[TABLE]

where $\theta$ is the angle between the colliding photons’ momenta.

For a photon with an energy of $E_{\gamma}$, the $\gamma\gamma$ optical depth is

[TABLE]

where $\epsilon_{\rm th}$ is the pair production threshold energy,

[TABLE]

Integration over the interaction angle in Eq. (22) can be performed analytically resulting in the angle averaged $\gamma\gamma$ cross section (Aharonian, 2004):

[TABLE]

where $s=E_{\gamma}\epsilon/m_{e}^{2}c^{4}$.

Figure 2 shows the internal gamma-ray optical depth in the core region for various X-ray luminosities. The core region is expected to be optically thick against gamma-ray photons above 10–100 MeV depending on disk luminosities. Such high optical thicknesses against pair production in \acagn coronae are well known (e.g., Bonometto & Rees, 1971; Done & Fabian, 1989; Fabian et al., 2015) based on the compactness parameter argument (Guilbert et al., 1983).

4 Timescales

Given the observed properties of AGN core regions, we can estimate the various timescales of high energy particles in the coronae. Figure 3 shows the cooling rates of electrons in the coronae for different energy-loss processes, together with the acceleration rate and the free-fall timescale following § 2 and parameters presented in § 3. We set $\eta_{g}=30$ in the figure, which reproduces the IceCube neutrino background fluxes as discussed later in § 7. Each panel corresponds to 2-10 keV X-ray luminosity of $10^{42}$, $10^{44}$, $10^{46}~{}{\rm erg\ s^{-1}}$.

Due to the intense broadband radiation field, the cooling is dominated by the Compton cooling. However, at higher energy regions, the main cooling channel is replaced by synchrotron cooling because of the \ackn effect. The more luminous AGNs tend to have more efficient \acic cooling effect, as the target photon density increases. When we assume $\eta_{g}=30$, electron acceleration up to $\gamma_{e}\sim 10^{5}$ ($\sim 50$ GeV) is feasible in \acagn coronae at various luminosities. Therefore, synchrotron radiation through coronal magnetic fields and gamma-ray emission by Comptonization of disk photons are naturally expected in AGN coronae.

\ac

alma spectra of two nearby Seyferts, whose X-ray luminosities are about $10^{44}\ {\rm erg\ s^{-1}},$ extends their radio synchrotron power-law spectra at least up to 230 GHz, which corresponds to $\gamma_{e}\sim 80$ given the magnetic field strength of $10$ G (Inoue & Doi, 2018)444This frequency limit is due to the instrumental coverage of the \acalma band-6 receiver. Therefore, the emission itself is likely to extend to higher frequencies, even though those emission signals would be buried in thermal dust emission.. As shown in the top right panel (the case of $\log L_{X}=44$) in Fig. 3, relativistic electrons with $\gamma_{e}\sim 80$ seen by \acalma can be easily accelerated in AGN coronae. Notably, such electrons can be accelerated even by a low efficiency acceleration process, e.g., with $\eta_{g}\sim 10^{6}$. For this energy, Compton cooling is the dominant energy loss process. As the cooling timescale for $\gamma_{e}\sim 80$ is about 100 s, flux variability in the radio synchrotron emission is expected, some Seyferts are already known to show a flux variation at least in day scales (Baldi et al., 2015). Further dense light curve observations may see shorter timescale variabilities.

Similar to Fig. 3 for electrons, Fig. 4 shows the timescales for high energy for various luminosities. As in Fig. 3, we set $\eta_{g}=30$. Since synchrotron and Compton cooling are not effective for protons in our case, we do not show these timescales in the figure.

It is evident that protons can be accelerated up to $\gamma_{p}\sim 10^{6}$ ($\sim 1$ PeV) in \acagn coronae for various luminosities. Maximum attainable energy is controlled by different processes for different luminosity AGNs due to SED and size dependence. For low-luminosity Seyferts ($L_{X}<10^{44}\ {\rm erg\ s^{-1}}$), acceleration is limited by the dynamical timescale rather than radiative cooling, while it becomes limited by the Bethe-Heitler cooling for higher luminosity objects. As the luminosity increases, \acpg and Bethe-Heitler cooling effects become more prominent. At higher luminosities, the Bethe-Heitler processes dominate the energy loss process of high energy particles. Therefore, in cases of high luminosity objects, resulting hadronic gamma-ray and neutrino spectra in the TeV band will show spectral suppression due to the Bethe-Heitler processes (see e.g., Murase, 2008, for the cases of gamma-ray burst).

5 Particle Spectrum

The steady state particle distributions $n=dN/d\gamma$ can be derived from the solution of the transport equation (Ginzburg & Syrovatskii, 1964)

[TABLE]

where $\dot{\gamma}_{\rm cool}$ is the total cooling rate, $Q(\gamma)$ is the injection function, which describes phenomenologically some acceleration process, e.g., \acdsa. The injection function for non-thermal protons and electrons is set as $Q(\gamma)=Q_{0}\gamma^{-p_{\rm inj}}\exp(-\gamma/\gamma_{\rm max})$. Here, $\gamma_{\rm max}$ is the maximum Lorentz factor determined by balancing the acceleration and cooling time scales (Figures. 3 and 4). The corresponding solution is

[TABLE]

where

[TABLE]

By solving Equation. 26, we obtain a steady-state spectrum of the non-thermal particles.

Fig. 5 shows the steady-state non-thermal electron spectrum obtained for the injection spectral index of $p_{\rm inj}=2.0$ together with the observationally determined electron spectral distribution for IC 4329A (Inoue & Doi, 2018). \acalma observed non-thermal synchrotron radiation between 90.5 GHz and 231 GHz which corresponds to the electron Lorentz factors between 50 and 80, respectively. The corresponding region is shown as the shaded region in the Fig. 5.

For the calculation of the steady-state spectrum, we set $M_{\mbox{\tiny{BH}}}=10^{8}M_{\odot}$, $r_{c}=40$, $B=10$ G, $kT_{e}=100$ keV, $\tau_{\mbox{\tiny{T}}}=1.1$, and $\eta_{g}=30$. The synthetic electron distribution obtained for $p_{\rm inj}=2.0$ nicely reproduces the observationally determined electron spectrum in the energy range constrained by the observations. This injection index is naturally expected in a simple \acdsa scenario for a strong shock.

The resulting particle spectrum at $\gamma_{e}>10^{4}$ becomes softer than observationally determined index at $50\lesssim\gamma_{e}\lesssim 80$. This is because of the influence of the cutoff imposed by the particle cooling. Therefore, if we consider the high energy synchrotron or \acic spectral shapes, the cooling effects should be taken into account accurately. Even though the electron spectrum extends down to lower energies, it is hard to see the corresponding synchrotron emission due to synchrotron self-absorption effect (Inoue & Doi, 2014).

The calculated electron spectrum is renormalized to agree with the observationally determined spectrum, which is achieved if the non-thermal electrons contains $f_{\rm nth}=0.03$ of the energy in thermal leptons. We note that, in order to define the energy content in the non-thermal particles, we formally integrate above $\gamma_{e}=1$ in this study. We keep this fraction for non-thermal electron energy fixed in calculations below for all Seyferts.

The energy fraction of non-thermal electrons was fixed to $\xi_{\rm nth}=0.04$ in Inoue & Doi (2018). $\xi_{\rm nth}$ is defined beyond the break electron Lorentz factor, while $f_{\rm nth}$ is above $\gamma_{e}=1$. That amount of non-thermal electrons overproduces the MeV background flux given the measured electron spectral index (see §. 7). To be consistent with the observed cosmic MeV gamma-ray background flux, we set $\xi_{\rm nth}=0.015$ in this work, which corresponds to $f_{\rm nth}=0.03$. The obtained best fit parameters with this fraction for the radio spectrum of IC 4329A is $p=2.9\pm 0.9$, $B=11.4\pm 5.6$ G, and $r_{c}=42.7\pm 7.8$, which are very similar to those obtained for the case of $\xi_{\rm nth}=0.04$. We adopt these parameters for the observationally determined electron distribution in the Fig. 5. Fitting results for the other parameters were also the same as those with $\xi_{\rm nth}=0.04$.

Here, the total shock power $P_{\mbox{\tiny{sh}}}$ can be estimated as

[TABLE]

For objects with $L_{\rm X}=10^{44}~{}{\rm erg\ s^{-1}}$, $f_{\rm nth}=0.03$ corresponds to $\sim 5$% of the shock power is injected into acceleration of electrons. This high value implies that if \acdsa is responsible for particle acceleration in \acagn coronae then processes regulating injection of electrons into \acdsa are very efficient. For example in the case of \acdsa in supernovae remnants non-thermal electrons obtain only $\sim 1$% of energy transferred to non-thermal protons (Ackermann et al., 2013). Detailed consideration of the reasons of this unusually high efficiency of electron acceleration is beyond the scope of this paper, however we note that a significant presence of positrons may affect the ratio (see, e.g., Park et al., 2015). Given these uncertainties, for protons we set that the same energy injection rate is achieved as for electrons. This power appears to be sufficient to explain the observed IceCube neutrino fluxes.

For the other object, NGC 985, the observed electron spectral index is $2.11\pm 0.28$ (Inoue & Doi, 2018), which is hard considering the radiative cooling effect. Cascade components would have such a hard spectrum below the threshold energy (see, e.g., Aharonian & Plyasheshnikov, 2003). In addition, due to the quality of data at low frequencies, we could not precisely determine the other components such as free-free emission and synchrotron emission from star formation activity, and synchrotron emission from the jet. Those uncertainties may resulted in a less reliable measurement of the corona emission spectrum slope. Further observations are required to determine the radio spectral properties in NGC 985 precisely.

6 Gamma Rays and Neutrinos from AGN Coronae

Accelerated electrons and protons in AGN coronae generate gamma-ray and neutrino emission through \acic scattering, $pp$ interaction, and $p\gamma$ interaction. Adopting a steady-state particle spectrum, we calculate the resulting gamma-ray and neutrino spectra from AGN coronae. We follow Blumenthal & Gould (1970) for the gamma-ray emission due to the \acic scattering by non-thermal electrons. We calculate the gamma-ray and neutrino emission induced by hadronic interactions following Kelner et al. (2006) for \acpp interactions and Kelner & Aharonian (2008) for \acpg interactions. For simplicity, we do not take into account \acic scattered emission by secondary electrons and positrons. For the thermal Comptonization spectra, we adopt the AGN SED shown in Fig. 1 which takes into account reflection components but does not account for attenuation by torus. The torus attenuation is mainly relevant for $\lesssim 30$ keV, which is below the range of our interest.

Figure 6 shows the resulting gamma-ray and neutrino spectra for two cases. The neutrino flux is shown in the form of per flavour. The left panel of the figure shows the case with a 2-10 keV luminosity of $10^{43}~{}{\rm erg\ s^{-1}}$ at a distance of 14 Mpc, while the right panel shows the case with a luminosity of $10^{44}~{}{\rm erg\ s^{-1}}$ at a distance of 69 Mpc. The former and the latter roughly corresponds to NGC 4151 and IC 4329A, respectively. NGC 4151 is the brightest Seyfert in the X-ray sky (Oh et al., 2018). For the comparison, the overall fluxes of both panels are renormalized to match with the Swift/BAT flux of NGC 4151 and IC 4329A, respectively, at 14-195 keV (Oh et al., 2018). We note that we do not calculate the detailed X-ray spectra of each objects, which is beyond the scope of this paper.

We set the injection spectral index of $p_{\rm inj}=2.0$ and the gyrofactor of $\eta_{g}=30$ for both electrons and protons (See § 5). We also set the same injection power into protons and electrons as described in §. 5. The target photon density for \acic scatterings and \acpg interactions is defined as $U_{\rm ph}(\epsilon)$ (See § 3.1). Since we assume a uniform spherical source, gamma-ray photons are attenuated by internal photon field by a factor of $3u_{\rm int}(\tau_{\rm int})/\tau_{\rm int}$, where $u_{\rm int}(\tau)=1/2+\exp(-\tau)/\tau-[1-\exp(-\tau)]/\tau^{2}$ (See Sec. 7.8 in Dermer & Menon, 2009), where $\tau_{\rm int}$ is the internal gamma-ray optical depth (See §. 3.3). Gamma rays are also attenuated by the \acebl during the propagation in the intergalactic space. We adopt Inoue et al. (2013a) for the \acebl attenuation.

For the comparison, we also show the expected sensitivity curve of planned MeV missions: COSI-X (300 days)555COSI collaboration website (The Compton Spectrometer and Imager http://cosi.ssl.berkeley.edu/, e-ASTROGM (3 yrs, De Angelis et al., 2017)666e-ASTROGAM collaboration website (enhanced ASTROGAM http://eastrogam.iaps.inaf.it/, GRAMS (35 days, Aramaki et al., 2019), and GRAMS (3 yrs, Aramaki et al., 2019). 10-yr sensitivity of Fermi/LAT777Fermi/LAT collaboration website (The Large Area Telescope http://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm is also shown. We also plot the sensitivity of neutrino detectors: IceCube888IceCube collaboration website (https://icecube.wisc.edu/ and IceCube-Gen2 (van Santen & IceCube-Gen2 Collaboration, 2017). For the left panel, we assume the declination $\delta$ of $30^{\circ}$, while $-30^{\circ}$ for the right panel.

Since the spectral index of electrons is $\sim 3$ after radiative cooling, the resulting non-thermal gamma-ray spectrum is flat in $\nu F_{\nu}$ in the MeV band which appears after the thermal cutoff. Given the cooling limited maximum energy $\gamma_{e}\sim 10^{5}$, the intrinsic \acic spectrum can extend up to $\sim 100$ GeV. However, due to the strong internal gamma-ray attenuation effect, the spectra will have a cutoff around 100 MeV in both cases. In the sub-MeV band, the spectrs show super-thermal tails due to the combination of thermal and non-thermal components and a spectral hardening at $\sim 1$ MeV. These superthermal and flat spectral tails should be tested by future MeV gamma-ray missions. Ballon flights with such as GRAMS (Aramaki et al., 2019) and SMILE (Takada et al., 2011; Komura et al., 2017)999SMILE collaboration website (The Sub-MeV gamma-ray Imaging Loaded-on-balloon Experiment http://www-cr.scphys.kyoto-u.ac.jp/research/MeV-gamma/wiki/wiki.cgi?page=Top_en may be able to catch this superthermal tail. And, satellite-class MeV missions such as e-ASTROGAM (De Angelis et al., 2017), AMEGO101010AMEGO collaboration website (The All-sky Medium Energy Gamma-ray Observatory https://asd.gsfc.nasa.gov/amego/, and GRAMS (Aramaki et al., 2019) will be able to see also the non-thermal power-law tail. For the case of NGC 4151, Fermi/LAT may be able to see the signature with its 10 yrs survey. However, the expected flux is almost at the sensitivity limit. Thus, it may need further exposures for Fermi/LAT to see the coronal emission.

The \acpp and \acpg production efficiency is given by the ratio between the dynamical timescale (Eq. 2) and the interaction timescales (Eqs. 2.2 and 10). The \acpp production efficiency is analytically given as

[TABLE]

Gamma rays and neutrinos induced by hadronic interactions carry $1/3$ and $1/6$ of those interacted hadron powers. Therefore, hadronic gamma-ray and neutrino luminosity is expected to be $\sim 5$% and $\sim 3$% of the intrinsic proton luminosity, respectively. Since we assume the same energy injection to electrons and protons and the coronal Thomson scattering optical depth is 1.1, before the attenuation, we have hadronic gamma-ray and neutrino fluxes are $\sim 5$% and $\sim 3$% of the \acic gamma-ray fluxes.

The \acpp and \acpg induced gamma rays are also mostly attenuated by the internal photon fields. Thus, we do not expect any $\gtrsim$GeV gamma-ray emission from Seyferts. Moreover, the intrinsic gamma-ray energy fluxes due to hadronic interactions is about a factor of 10 less than that by primary electrons because of radiative efficiency differences between protons and electrons. This implies that gamma rays produced by secondary pairs should not significantly alter the resulting spectra. Therefore, we can safely ignore the cascade contribution.

On the contrary to gamma rays, neutrinos induced by hadronic interactions can escape from the system without any attenuation. Since we adopt the same $p_{\rm inj}=2$ for protons as for electrons, we expect a flat $\nu F_{\nu}$ spectrum for neutrinos, to which \acpp makes dominant contribution. At higher energies, especially in the case of IC 4329A, \acpp and \acpg spectra are suppressed due to the Bethe-Heitler cooling process. The exact position of the cutoff energy depends on the assumed $\eta_{g}$. Here, as described later, we set $\eta_{g}=30$ in order to be consistent with the IceCube background flux measurements. This gyrofactor results in a neutrino spectral cutoff around 100 TeV. Although it is difficult to see neutrino signals from individual Seyferts with the current generation of IceCube, it would be possible to see bright Seyferts in the northern hemisphere in the era of IceCube-Gen2 (see also Murase & Waxman, 2016, for more general arguments). Therefore, even though Seyferts are faint in the GeV gamma-ray band, future MeV gamma-ray and TeV neutrino observations can test our scenario.

7 Cosmic Gamma-ray and Neutrino Background Fluxes From High Energy Particles in AGN Coronae

In this section, we calculate the cosmic gamma-ray and neutrino background spectra from AGN coronae. For the cosmological evolution of \acpagn, we follow Ueda et al. (2014) in which the evolutionary functions are defined at 2–10 keV intrinsic X-ray luminosity. We briefly review their formalism here.

Based on the luminosity-dependent density evolution model, the \acagn X-ray luminosity function at a given luminosity $L_{X}$ and a given redshift $z$ is defined as

[TABLE]

where ${d\Phi_{\rm X}(L_{\rm X},0)}/{d{\rm log}L_{\rm X}}$ is the luminosity function in the local universe defined as

[TABLE]

where $A$ is the normalization and $L_{*}$ is the break luminosity. $e(z,L_{\rm X})$ is the evolution factor represented as

[TABLE]

Here the luminosity dependence for the $p1$ parameter is considered as

[TABLE]

where we set ${\rm log}L_{\rm p}=44$. Both cutoff redshifts are given by power law functions of $L_{\rm X}$ as

[TABLE]

and

[TABLE]

The parameters are summarized in Table. 4 in Ueda et al. (2014). There is also a substantial fraction of Compton-thick AGNs in the universe (e.g., Ueda et al., 2003; Ricci et al., 2015). In order to take into account this population, we multiply the normalization factor by a factor of 1.5 (see Ueda et al., 2014, for details).

The cosmic gamma-ray background fluxes are calculated as

[TABLE]

where $E^{\prime}=(1+z)E$ and $L_{\gamma}(E,L_{\rm X})$ is the gamma-ray luminosity at energy $E$ for a given X-ray luminosity of $L_{\rm X}$. The redshift and luminosity ranges are selected to be the same as in Ueda et al. (2014). $\tau_{\rm int}$ and $\tau{}_{\mbox{\tiny{EBL}}}$ is the gamma-ray optical depth due to the internal photon field and the \acebl. We do not consider the cascade gamma-ray photons (e.g., Inoue & Ioka, 2012) because the gamma-ray energy fluxes due to hadronic interactions is already subdominant compairing to that by primary electrons.

The neutrino background fluxes can be also calculated in the same manner ignoring the gamma-ray attenuation terms and replacing $L_{\gamma}(E,L_{\rm X})$ with $L_{\nu}(E,L_{\rm X})$. $L_{\nu}(E,L_{\rm X})$ is the neutrino intensity at an energy of $E$ for a given X-ray luminosity of $L_{\rm X}$.

Figure 7 shows the cosmic X-ray/gamma-ray and neutrino background spectra from \acagn coronae assuming the case of $p_{\rm inj}=2.0$ and $\eta_{g}=30$. We also plot the observed background spectrum data by HEAO-1 A2 (Gruber et al., 1999), INTEGRAL (Churazov et al., 2007), HEAO-1 A4 (Kinzer et al., 1997), Swift-BAT (Ajello et al., 2008), SMM (Watanabe et al., 1997), Nagoya–Ballon (Fukada et al., 1975), COMPTEL (Weidenspointner et al., 2000), Fermi-LAT (Ackermann et al., 2015), and IceCube (Aartsen et al., 2015).

Figure 8 shows the cosmic MeV gamma-ray background spectrum only from Figure 7. By setting $f_{\rm nth}=0.03$, the gamma-ray fluxes from \acpagn coronae due to \acic scattering by thermal and non-thermal electrons can nicely explain the observed cosmic MeV gamma-ray background radiation in an extension from the cosmic X-ray background radiation, which is known to be explained by Seyferts (Ueda et al., 2014). Since the spectral index of non-thermal electrons in the coronae is $\sim 3$, the resulting MeV gamma-ray background spectrum becomes flat in $E^{2}dN/dE$ (See Fig. 8). Here, the cosmic X-ray background spectrum by Seyferts has a spectral cutoff above $\sim 300$ keV because of temperature of thermal electrons $\sim 100$ keV (Ueda et al., 2014). By summing up these two thermal and non-thermal components, superthermal tail appears in the sub-MeV band as observed by Fukada et al. (e.g., 1975); Kinzer et al. (e.g., 1997); Watanabe et al. (e.g., 1997). Since the dominant \acic contributors switches from thermal electrons to non-thermal electrons at around $1$ MeV, the MeV background spectrum may have spectral hardening feature at $\sim 1$ MeV. In the figure, we set $\eta_{g}=30$. The result does not significantly change as far as $\eta_{g}<1000$. If $\eta_{g}>1000$, we may require lower $f_{\rm nth}$.

Due to the internal gamma-ray attenuation effect, these non-thermal gamma rays can not contribute to the emission above GeV. Because of the same reason, most of hadronic gamma-ray photons are attenuated by internal photon fields, resulting in generation of multiple secondary particles. Since calculation of those populations are beyond the scope of this paper, we ignore those populations in our estimate. Moreover, as we describe above, the intrinsic hadronic fluxes are already an order of magnitude below the leptonic fluxes. Thus, pairs induced by hadronic cascades will not significantly change our results.

Here, \acic emission due to non-thermal electrons also contribute in the X-ray band. Their contribution is about $\sim 5$% at 30 keV of the observed cosmic X-ray background flux, which may reduce the required number of the Compton-thick population of AGNs.

The model curve at $\sim 10$ keV slightly overproduces the measured background spectrum. This is because we do not take into account X-ray attenuation by torus. However, the treatment of those soft X-ray photons does not affect our results at all.

For neutrinos, the combination of \acpp and \acpg interactions can nicely reproduce the IceCube fluxes below 100–300 TeV by assuming $\eta_{g}=30$ and about 5% of the shock power into proton acceleration, same as electrons. \acpp interactions dominate the flux at $\lesssim 10$ TeV, while \acpg interactions prevail above this energy. Because of the target photon field SED, \acpg is subdominant in the GeV-TeV band. If we inject more powers into protons, it inevitably overproduces the IceCube background fluxes. As $\gtrsim$ GeV gamma rays are internally attenuated, AGN coronae emission will not be seen in GeV gamma-rays, even though they can make the IceCube neutrino fluxes. Such hidden cosmic-ray accelerators are suggested as a possible origin of the IceCube neutrinos (see Murase et al., 2016, for a general argument).

Figure 9 shows the cosmic neutrino background spectra from \acagn cores with various gyro factors ranging from 1 (Bohm limit) to $10^{3}$. It is clear that if $\eta_{g}\ll 30$, the resulting neutrino fluxes overproduce the measured fluxes. On the contrary, if $\eta_{g}\gg 30$, AGN coronae can not significantly contribute to the observed neutrino background fluxes. Thus, in order to explain the IceCube neutrino background fluxes by AGN cores, $\eta_{g}\sim 30$ is required. However, we note that these estimates are based on the assumed energy injection fraction to protons. Recent particle-in-cell simulations of proton-electron plasma considering radiatively inefficient accretion flows (RIAFs) showed that protons will carry have several factors more energies than electrons (Zhdankin et al., 2018). If this is the case, larger $\eta_{g}$ is favored.

8 Discussion

8.1 Comparison with Previous works on High Energy Neutrinos

In literature, it has been argued that high energy particles in the core of \acpagn generate intense neutrino emission (e.g., Eichler, 1979; Begelman et al., 1990; Stecker et al., 1992; Alvarez-Muñiz & Mészáros, 2004). These originally predicted fluxes have been ruled out by high energy neutrino observations (The IceCube Collaboration, 2005). However, recent studies have revisited the estimated fluxes and found that \acagn core models can account for the whole measured fluxes (Stecker, 2013; Kalashev et al., 2015). In this section, we would like to compare our results with those recent studies (Stecker, 2013; Kalashev et al., 2015).

The model suggest by Stecker (2013) is very similar to the originally proposed one (Stecker et al., 1992), but the background flux is assumed to be lower by a factor of 20. The original model is motivated by the models explaining \acagn X-ray spectra by the electromagnetic cascade emission of secondary particles (Zdziarski, 1986; Kazanas & Ellison, 1986), which is not the case based on current X-ray and gamma-ray observational results. The shock radius and the magnetic field strength was assumed to be $10R_{s}$ and $10^{3}$ G in the model by Stecker et al. (1992).

The model in Kalashev et al. (2015) is an extension of Stecker et al. (1992) taking into radial emission profile in the standard accretion disk for the consideration of the \acpg cooling processes. In our modeling, we do not take into account such anisotropic radiation field. However, given the observationally determined corona size, the dominant photon targets are likely to be generated in the inner region of the coronae. The particle spectra in Kalashev et al. (2015) are fixed to match with the IceCube data.

Neutrino fluxes or cosmic-ray spectra are fixed to match with the latest IceCube data in Stecker (2013); Kalashev et al. (2015). In this work, we take more physical approach. Corona plasma density, corona size, and magnetic field strength are determined from observations (Inoue & Doi, 2018) in our work. For example, we set $R_{c}=40R_{s}$ and $B=10$ G based on ALMA observations (Inoue & Doi, 2018). With those parameters, we can follow the acceleration processes in coronae in the framework of \acdsa. We found the AGN coronae can explain the IceCube neutrino background in the TeV band, if the gyrofactor is $\eta_{g}=30$ and about 5% of the shock energy goes into proton acceleration. We also predict that next generation MeV gamma-ray and neutrino experiments can test our model by observing nearby bright Seyferts such as NGC 4151 and IC 4329A.

8.2 Plasma Condition in Coronae

Considering the plasma density in the accreting coronae, high energy particles may have sufficient time to redistribute their kinetic energy through thermalization by elastic Coulomb (EC) collisions before the gas reaches the event horizon (Takahara & Kusunose, 1985; Mahadevan & Quataert, 1997). In this section, we discuss thermalization timescales of electrons and protons in the AGN coronae.

First, the electron thermalization timescale in the non-relativistic regime is estimated to be (Spitzer, 1962; Stepney, 1983)

[TABLE]

where $\ln\Lambda\approx 20$ is the Coulomb logarithm. For relativistic electrons with Lorentz factors $\gamma_{e}\gg 1+\theta_{e}$ the thermalization timescale due to interactions with the background plasma becomes (Dermer & Liang, 1989)

[TABLE]

where $K_{n}$ is the modified Bessel function of order $n$, and parameter $u_{\rm ee}=(\gamma_{e}/\gamma^{\prime}_{e}+\gamma^{\prime}_{e}/\gamma_{e})/2\theta_{e}$. This equation can be approximated as

[TABLE]

This is a good analytic approximation at $\theta_{e}\gtrsim 0.3$ and $\gamma_{e}\gtrsim 2$ (Dermer & Liang, 1989).

Second, the proton-proton relaxation timescale in the non-relaticistic regime is estimated to be (Spitzer, 1962; Stepney, 1983)

[TABLE]

where $\theta_{p}\equiv kT_{p}/m_{p}c^{2}$ is the dimensionless proton temperature. At high kinetic energies, nuclear interaction becomes important (see Gould, 1982, for details). In the mildly relativistic case, the elastic proton-proton relaxation timescale approximately becomes (Gould, 1982)

[TABLE]

where $\sigma_{h}\sim 2.3\times 10^{-26}\ {\rm cm}^{2}$. This approximation is valid at $70\ {\rm MeV}\lesssim(\gamma-1)m_{p}c^{2}\lesssim 500$ MeV. Above 500 MeV, inelastic processes start to dominate.

Lastly, the proton-electron thermalization timescale due to \acec collisions in the non-relativistic regime is estimated to be (Spitzer, 1962; Stepney, 1983)

[TABLE]

where we assume $\theta_{p}=\theta_{e}$. The temperature of a hot accretion can roughly reach to virial temperature $T_{p}\simeq GM_{\mbox{\tiny{BH}}}m_{p}/3kR\sim 3\times 10^{12}(R/R_{s})^{-1}$ K. At such higher temperature, $t_{\rm\mbox{\tiny{EC}},ep}$ becomes longer. In the case of relativistic protons, the energy loss timescale through \acec interactions is given as (Mannheim & Schlickeiser, 1994; Dermer et al., 1996)

[TABLE]

where $\beta_{p}=\sqrt{1-1/\gamma_{p}^{2}}$. At $\gamma_{p}\gg 1$ and $\theta_{e}\ll 1$, the relativistic \acec scattering relaxation time can be approximated as

[TABLE]

Fig. 10 shows EC thermalization timescales for electrons and protons for the luminosity of $L_{X}=10^{44}~{}{\rm erg\ s^{-1}}$. Since EC thermalization is effective at low energy particles, the horizontal axis is shown in $\gamma\beta$.

Around $\gamma_{e}\beta_{e}\sim 2$, $t_{\mbox{\tiny{EC}},ee}$ shows a sharp feature, which is related to the temperature of the background plasma, $kT_{e}=100$ keV. At this temperature, the electron distribution has a peak around $\sim 3kT_{e}$ corresponding to $\gamma_{e}\beta_{e}\sim 1.2$. Thus, around this energy, mean energy transfer is small. We note that below this energy, electrons gain energies from the background plasma through elastic $ee$ scatterings rather than loosing their energies (Dermer & Liang, 1989), however, this energy gain process is not considered in our work, since it is not relevant for our energy range of interest. As seen in the Fig. 10, the energy loss process of electrons is dominated by the Compton cooling at $\gamma_{e}\beta_{e}\gtrsim 1$.

Following Gould (1982), we calculate the elastic $pp$ timescale in the mildly relativistic regime. Since it assumes an incident proton has much higher kinetic energy than background plasma, we combine the non-relativistic $t_{\mbox{\tiny{EC}},pp}$ (Equation. 44) and that from Gould (1982). As discussed above, inelastic processes start to dominate at the kinetic energies of $\gtrsim 500$ MeV ($\gamma_{p}\beta_{p}\gtrsim 1.2$). For the comparison, we also show inelastic $pp$ interaction timescale $t_{pp}$.

As the proton-electron Coulomb timescale ($t_{\mbox{\tiny{EC}},pe}$) is longer than $t_{\rm fall}$, protons and electrons may not be in the thermal equilibrium in AGN coronae. The proton temperature of a hot accretion can roughly reach to virial temperature $T_{p}\simeq GM_{\mbox{\tiny{BH}}}m_{p}/3kR\sim 3\times 10^{12}(R/R_{s})^{-1}$ K, which is $\gg T_{e}$. And, the existence of pairs in coronae can reduce $n_{p}$. Moreover, the shock heated proton temperature becomes $kT_{p}\sim 3m_{p}v_{\rm sh}^{2}\sim 4(r_{c}/40)^{-1}~{}{\rm MeV}$. Those shock heated protons and electrons also gain and loose their energies through the processes and would contribute as a thermal population in the coronae. These electrons are heated and cooled through EC proton-electron thermalization and Comptonization, respectively (e.g., Katz et al., 2011; Murase et al., 2011). The heating rate can be written as

[TABLE]

assuming $\theta_{e}\gg\theta_{p}$. The cooling rate through Comptonization is

[TABLE]

By equating these two heating and cooling rates of thermal electrons, the shock heating electron temperature is estimated to be

[TABLE]

where we assume $L_{\rm ph,bol}\propto M_{\rm BH}$. This temperature is close to the measured coronal temperature. Therefore, such shock heating mechanism may be able to explain the current observed coronal temperature. For the understanding the detailed nature of thermal coronae, further studies including thermodynamical processes are required.

8.3 Other Particle Acceleration Mechanisms

In this paper, we consider the \acdsa as fiducial acceleration mechanism. However, other acceleration mechanisms such as turbulent acceleration, magnetosphere acceleration, and magnetic reconnection can also operate in AGN coronae. We briefly discuss these processes here.

First, turbulent acceleration is considered for low-accretion rate objects such as low-luminosity \acpagn (e.g., Kimura et al., 2015; Zhdankin et al., 2017, 2018; Wong et al., 2019). In this scenario, particles are accelerated stocastically by turbulence and magnetic reconnection in accretion disk or coronae. Recently, Zhdankin et al. (2018) investigated electron-ion plasma energization via turbulent dissipation in RIAFs using particle-in-cell simulations for the ion temperature $T_{i}$ in the range of $m_{e}c^{2}\lesssim k_{B}T_{i}\lesssim m_{p}c^{2}$. Turbulent electron-ion plasma driven by MRIs generate power-law spectra for both species and the indices depends on the initial ion temperature. The fraction of the kinetic energy in the non-thermal ions and electrons are $\sim 60$% and $6$% for ions and electrons at $k_{B}T_{i}\sim m_{e}c^{2}$, respectively. The fraction in non-thermal electrons is close to the required value for the MeV background (See § 7).

We briefly follow the stochastic acceleration in the AGN coronae case. According to the quasi-linear theory, the diffusion coefficient in the momentum space is (e.g., Dermer et al., 1996)

[TABLE]

where $k_{\rm min}\sim R_{c}^{-1}$ is the minimum wave number of turbulence spectrum (corresponding to the size of the corona), ${v_{\mbox{\tiny{A}}}}=B/\sqrt{4\pi m_{p}n_{p}}$ is the Alfvén speed, $r_{\mbox{\tiny{L}}}=m_{p}c^{2}/eB$ is the Larmor radius, and $\zeta=\delta B^{2}/B^{2}$ is the ratio of strength of turbulence fields against the background. Then, the acceleration timescale is estimated to be

[TABLE]

Assuming the Kolomogorov spectrum for the turbulent ($q=5/3$) and $\zeta=1$, the timescale becomes

[TABLE]

Thus, stochastic acceleration appears to be inefficient as compared to the typical cooling rates. This is caused by the measured weak magnetic fields, which results in small Alfvén speed. If the magnetic fields are amplified by MRIs, more efficient acceleration can be realized (e.g., Zhdankin et al., 2018)111111After we submitted our paper to the journal and arXiv, similar study on AGN coronae by Murase et al. (2019) appeared on arXiv. Both studies are independent and the most different point is the assumed particle acceleration processes. In our paper, we consider \acdsa, while Murase et al. (2019) consider stochastic acceleration motivated by recent numerical simulations (Kimura et al., 2019). However, as we discussed in this section, stochastic acceleration may not work given the ALMA results of weak coronal magnetic field..

Second, magnetosphere acceleration can also accelerate particles in the vicinity of \acpsmbh (e.g., Beskin et al., 1992; Levinson, 2000; Neronov & Aharonian, 2007; Levinson & Rieger, 2011; Rieger, 2011). At low accretion rates, the injection of charges into the BH magnetosphere is not sufficient for a full screening of the electric field induced by the rotation of the compact object. The regions with unscreened electric field, so-called gaps, are able to accelerate charged particles effectively.

In order to have gaps, the maximum allowed accretion rate is (Levinson & Rieger, 2011; Aleksić et al., 2014; Aharonian et al., 2017)

[TABLE]

where $\dot{m}$ is the accretion rate in the Eddington units. Since we are considering the standard accretion disk regime $\dot{m}\gtrsim 0.01$, particle acceleration by gaps will not be operated in our case.

Lastly, magnetic reconnection would accelerate particles (see e.g., Hoshino & Lyubarsky, 2012, for reviews). Reconnection would naturally happens in coronae as they are magnetized and radiative magnetic reconnection is suggested as an origin of the X-ray emission seen in accreting black hole systems (Beloborodov, 2017). However, even in the case of solar flares, particle acceleration mechanisms in magnetic reconnection is still uncertain (e.g., Liu et al., 2008; Nishizuka & Shibata, 2013). Although quantitative discussion is not easy here, the available energy injection power can estimated as

[TABLE]

This power is not sufficient for providing the non-thermal particle energies. For detailed estimation, we may need to consider spatial distribution fo magnetic field. However, such information is not currently available.

8.4 Cosmic MeV Gamma-ray Background Radiation

It is known that Seyferts generate the cosmic X-ray background radiation (Ueda et al., 2014). The cosmic gamma-ray background at 0.1–820 GeV is believed to be explained by three components: blazars (e.g., Inoue & Totani, 2009; Ajello et al., 2015), radio galaxies (Inoue, 2011), and star-forming galaxies (Ackermann et al., 2012a), even though the contributions of radio galaxies and star-forming galaxies are still uncertain due to a small number of gamma-ray detected samples. On the contrary to the cosmic X-ray and GeV background radiation, the origin of the cosmic MeV gamma-ray background radiation is still veiled in mystery.

As a possible scenario, non-thermal \acic emission from coronae in Seyferts has been suggested (Inoue et al., 2008). The MeV tail extended from the X-ray background spectrum is generated by non-thermal electrons with very soft spectral index (Inoue et al., 2008). However, non-thermal electrons are included in an ad hoc way. In our work, we consider the particle acceleration and cooling processes given the latest observations. The tail is due to the superposition of thermal Comptonization cut-off spectrum and $\gamma\gamma$ attenuated flat non-thermal \acic component. We can distinguish these two scenarios by observing individual objects in radio and X-ray bands.

Not only Seyferts, but also blazars are considered as a candidate as the origin of the MeV background (Ajello et al., 2009). In order to distinguish Seyferts and blazars, we need to resolve the MeV sky. However, it is not easy even with future MeV instruments (Inoue et al., 2015). Here, it is suggested that anisotropy measurements may distinguish these two scenarios (Inoue et al., 2013b) because blazar background should feature stronger Poisson fluctuations. Future MeV gamma-ray anisotropy observations will be important to understand the particle acceleration in coronae and the origin of the MeV gamma-ray background radiation.

8.5 Gamma-ray Observations toward Seyferts

Gamma rays from Seyfert galaxies are not robustly detected yet (Lin et al., 1993; Teng et al., 2011; Ackermann et al., 2012b). Possible signature of gamma-ray emission above 0.1 GeV have been reported for ESO 323-G077 and NGC 6814 (Ackermann et al., 2012b), whose X-ray luminosities are about $10^{43}\ {\rm erg\ s^{-1}}$. The required luminosity ratio between X-ray and gamma-ray $L_{\rm 0.1-10~{}GeV}/L_{\rm 14-195~{}keV}$ for these sources is about 0.1 (Ackermann et al., 2012b). Our model estimates this ratio as $\sim 0.01$. Therefore, coronal gamma-ray emission is most-like not able to account for the observed gamma-ray fluxes from those Seyfert galaxies.

Although gamma rays from other Seyferts have not been detected yet, Fermi/LAT has set upper limits on their gamma-ray fluxes (Teng et al., 2011; Ackermann et al., 2012b). Based on the analysis of the first 2-3 years data, $L_{\rm 0.1-10~{}GeV}/L_{\rm 14-195~{}keV}<0.1$ in the 95% confidence level is obtained in most cases, which is consistent with our model estimate. The most stringent observational constraint is derived for NGC 4151, in which $L_{\rm 0.1-10~{}GeV}/L_{\rm 14-195~{}keV}<0.0025$, even though the limit can vary with an assumed spectral shape. Following our models, the current 10 yrs survey data of Fermi/LAT may be able to see NGC 4151 (Figure. 1), even though the expected flux is almost at the sensitivity limit.

8.6 Fraction of Non-thermal Electrons

We set the energy fraction of non-thermal electrons in AGN coronae as $f_{\rm nth}=0.03$ because it nicely reproduces the observed MeV gamma-ray background radiation. As discussed in Inoue & Doi (2018), $f_{\rm nth}$, $B$, and $R_{c}$ are closely tied, current radio and X-ray data do not allow us to solve these three parameters simultaneously without decoupling thermal and non-thermal components.

Observationally, $f_{\rm nth}$ is constrained as $<0.3$ in order not to violate X-ray data based on NuSTAR observations (Fabian et al., 2017). If $f_{\rm nth}$ is significantly lower, it becomes difficult for Seyfert to explain the MeV gamma-ray background radiation. However, too much lower $f_{\rm nth}$ contradicts with other observations since it requires a bigger $R_{c}$ based on the radio spectral fitting. If we set $f_{\rm nth}=10^{-3}$ and $10^{-4}$, $R_{c}$ becomes $\sim 70R_{s}$ and $\sim 100R_{s}$, respectively. The size of coronae is also constrained as an order of $\sim 10R_{s}$ by optical–X-ray spectral fitting studies (Jin et al., 2012) and micorolensing observation (Morgan et al., 2012). Therefore, $f_{\rm nth}$ can not become much smaller than the adopted value.

8.7 Nuclear Spallation in AGNs

Given the \acalma results, particle accelerations occurs in AGN coronae. As we demonstrated, high energy protons are easily accelerated in coronae. These high energy protons can be also traced by future high-resolution calorimeter spectroscopy in the X-ray band such as XRISM (Tashiro et al., 2018) and Athena (Nandra et al., 2013)121212The Athena X-ray observatory website (https://www.the-athena-x-ray-observatory.eu/. As narrow line features are seen in AGN X-ray disk spectra, there are abundant metal elements in AGN cores. Accelerated protons also interact with those nuclei and induce nuclear spallation. The nuclear spallation in AGN disks will result in enhancement of emission lines from Mn, Cr, V, and Ti (Gallo et al., 2019). Those signatures will be another clue for the test of our model.

9 Conclusion

Recently, Inoue & Doi (2018) has reported the coronae of Seyferts are composed of both thermal and non-thermal electrons based on \acalma observations, which implies that particle acceleration occurs in AGN coronae. In order to investigate the production mechanism of those high energy particles, we study the particle acceleration process in AGN coronae. We consider particle acceleration by the \acdsa process in the coronae as an example. By taking into account the observationally determined coronal properties, such as temperature, density, size, and magnetic field strength, we found that standard \acdsa processes can easily reproduce the observed non-thermal electron in the coronae with an injection electron spectral index of $p_{\rm inj}=2$. Even in low acceleration efficiency cases ($\eta_{g}\sim 10^{6}$), such populations can be realized in coronae. Given the observed magnetic field strength of 10 G and accretion rates, we also found that other possible acceleration mechanisms such as turbulent acceleration, magnetosphere acceleration, and magnetic reconnection confront difficulty in reproducing the observed non-thermal electrons.

The accelerated non-thermal electron populations will generate a MeV gamma-ray power-law spectrum in the AGN SEDs up to $\sim 0.1$ GeV, which is limited by internal gamma-ray attenuation. In the sub-MeV band, the spectrum shows a superthermal tail due to the combination of thermal and non-thermal components and spectral flattening occurs at $\sim 1$ MeV. These superthermal and flat spectral tails should be tested by future MeV gamma-ray missions.

We also study the contribution of \acagn coronae to the cosmic gamma-ray background radiation. By setting the energy fraction of non-thermal electrons $f_{\rm nth}\sim 3$%, corresponding to $\sim 5$% of the shock energy in electron acceleration, \acagn coronae can explain the MeV background in an extension of the X-ray background contribution of Seyferts. Due to a strong internal gamma-ray attenuation effect, the contribution of \acagn coronae to the GeV background is negligible.

Accelerated particles would also result in neutrino production through hadronic processes. Intense neutrino emission has been expected to be produced in \acagn coronae once hadrons are accelerated together (e.g., Begelman et al., 1990; Stecker et al., 1992; Alvarez-Muñiz & Mészáros, 2004). Recent studies have proposed that these \acagn core models could reproduce the high energy neutrino fluxed measured by IceCube (Stecker, 2005, 2013; Kalashev et al., 2015). However, normalization of neutrino fluxes from \acpagn and acceleration properties of high energy particles in those models are assumed to match with the observation.

We found that \acagn coronae can explain the diffuse neutrino fluxes below 100–300 TeV under specific parameters of energy injection rates in protons and gyro factors. The allowed parameter regions are quite narrow. Protons and electrons should have the same energy injection rate and the gyro factor $\eta_{g}$ should be $\sim 30$. IceCube Gen-2 will be able to test this scenario by searching the neutrino signal from nearby Seyfert galaxies such as NGC 4151 and IC 4329A.

In summary, Seyfert coronae are feasible sites for particle acceleration. If the energy injection rate is 5% for both protons and electrons and the gyro factor is $\eta_{g}=30$, they may be able to simultaneously explain the cosmic X-ray, MeV gamma-ray, and TeV neutrino background radiation. Future MeV gamma-ray and TeV neutrino observations will be able to test this scenario by observations of nearby bright Seyferts.

We thank the anonymous referee for his/her helpful comments which improved the manuscript. We also would like to thank Tsuguo Aramaki, Mitch Begelman, Norita Kawanaka, Shigeo Kimura, Ari Laor, Kohta Murase, Satomi Nakahara, and Marek Sikora for useful discussions and comments. YI is supported by JSPS KAKENHI Grant Number JP16K13813, JP19K14772, program of Leading Initiative for Excellent Young Researchers, MEXT, Japan, and RIKEN iTHEMS Program. DK is supported by JSPS KAKENHI Grant Numbers JP18H03722, JP24105007, and JP16H02170.