Gamma-ray and Neutrino Signals from Accretion Disk Coronae of Active Galactic Nuclei

To explain X-ray spectra of active galactic nuclei (AGN), non-thermal activity in AGN coronae such as pair cascade models has been extensively discussed in the past literature. Although X-ray and gamma-ray observations in the 1990s disfavored such pair cascade models, recent millimeter-wave observations of nearby Seyferts establish the existence of weak non-thermal coronal activity. Besides, the IceCube collaboration reported NGC 1068, a nearby Seyfert, as the hottest spot in their 10-yr survey. These pieces of evidence are enough to investigate the non-thermal perspective of AGN coronae in depth again. This article summarizes our current observational understandings of AGN coronae and describes how AGN coronae generate high-energy particles. We also provide ways to test the AGN corona model with radio, X-ray, MeV gamma-ray, and high-energy neutrino observations.


Introduction
Back in 2013, the IceCube collaboration reported the evidence for astrophysical neutrinos with energies in the 30 TeV-1 PeV range [1]. The detected neutrinos show isotropic distribution with the flux level of ≈ 10 −8 GeV cm −2 s −1 sr −1 [2]. This discovery indicates that TeV-PeV neutrinos travel to the Earth from the distant Universe. Although various theoretical models have attempted to explain the cosmic TeV-PeV neutrino fluxes even before the operation of IceCube [see, e.g., 3,4, for reviews], the lack of clear source identification hampers our understandings of the origin of the TeV-PeV neutrinos.
In 2018, the IceCube collaboration, together with the other electromagnetic wave observatories, reported a possible spatial and temporal coincidence of a neutrino event with a blazar flare TXS 0506+056 [5,6]. This observation provides the first evidence of a blazar as a cosmic neutrino factory. However, stacking analysis of blazars show that they can afford only up to ∼ 30% of the measured cosmic TeV-PeV neutrino background flux [7]. Therefore, even after the first possible source identification, the origin of the cosmic neutrino background flux was still veiled in mystery.
Very recently, the accumulation of 10-yr IceCube survey data revealed the existence of a neutrino hot spot toward the direction of NGC 1068 with a 2.9σ confidence level [8].  [38], Hickox and Alexander [39].
lowering the expected neutrino fluxes from Seyferts significantly. Therefore, high energy signals from AGN coronae have not been extensively investigated in the community in the late 1990s and 2000s.

Properties of AGN Coronae Revealed by X-ray Observations
Today, it is widely believed that the AGN X-ray emission is primarily from the Comptonized accretion disk photons from moderately thick thermal plasma, namely coronae, above an accretion disk [45][46][47][48][49][50]. This Comptonized emission appears together with emission reprocessed by the surrounding cold materials, a so-called Compton reflection component [e.g., [51][52][53]. Figure 1 shows a current schematic representation of the AGN spectral energy distribution (SED) without obscuration [see 38,39, for details]. We note that the origin of soft X-ray excess is still under debate [54,55].
X-ray spectral studies allow us to determine some of the coronal parameters such as the coronal electron temperature, T e , and the Thomson scattering optical depth τ T n e σ T R c [e.g., 58]. Here n e is the electron number density, σ T is the Thomson scattering cross section, and R c is the coronal size. Dadina [56] reported that local bright Seyferts typically have the spectral cutoff at E c ≈ 300 keV (Fig. 2). This cutoff corresponds to the electron temperature of kT e ≈ 100 keV (here k is the Boltzmann constant). The process of Comptonization by thermal plasma is described by the Kompaneets equation [59]. The photon index of the primary X-ray emission of Seyferts is typically Γ ≈ 1.9 [57] (Fig. 3). This corresponds to τ T ≈ 1.1 based on the solution to the Kompaneets equation [60] as Γ = (9/4 + 1/[θ e τ T (1 + τ T /3)]) 1/2 − 1/2, where the dimensionless electron temperature θ e ≡ kT e /m e c 2 . It should be also mentioned that simultaneous optical-X-ray spectral fitting studies [61] and microlensing observation [62] suggested the corona size R c ∼ 10R s , where R s is the Schwarzschild radius.
Hard X-ray observations of Seyferts allow us to investigate the AGN coronae further. It has constrained the amount of non-thermal particles in the AGN coronae from the nondetection of the non-thermal features. In order not to violate the NuSTAR observations, the latest constraint on the non-thermal electron energy fraction is obtained as f nth < 0.3 [63].    Together with the success of the Comptonized corona model, population studies of Seyferts revealed that Seyferts dominate the cosmic X-ray background radiation up to ∼ 300 keV [e.g., 65,66]. Here, as shown in Fig. 4, the cosmic MeV gamma-ray background radiation spectrum smoothly extends from the cosmic X-ray background spectrum [67][68][69]. Because of a spectral cutoff in individual Seyferts, integrated Seyfert X-ray photons would not explain the cosmic MeV gamma-ray background radiation. Several theoretical models proposed that a small fraction of non-thermal electrons in AGN coronae may be enough to explain the MeV background radiation as well [70,71]. The required fraction was f nth ∼ 0.03. This amount indicated the possible existence of weak non-thermal activity in AGN coronae. However, it was an ad hoc way to have a simultaneous explanation on the X-ray and MeV backgrounds. 1

Properties of AGN Coronae Revealed by Millimeter Observations
Theoretically, it is natural to expect that AGN coronae are magnetized [74,75]. Then, coronal non-thermal electrons, as expected for the MeV background radiation, inevitably generate coronal synchrotron emission [76][77][78], which should appear as an excess in the millimeter (mm) band. Inconclusive signs of such a new component in the radio spectra of several Seyfert galaxies have been reported in literature [79][80][81][82]. However, a paucity of multi-band data and the contamination of extended dust emission hamper the investigation.
Recently, Inoue and Doi [37] reported the detection of power-law coronal radio synchrotron emission from two nearby Seyferts, IC 4329A and NGC 985, utilizing ALMA (See Figs. 5), which enabled multi-band observations with high enough angular resolution to exclude the galactic contamination. These observations provided the first determination of the fundamental physical parameters of the AGN coronae: magnetic field strength and its size. The inferred coronal magnetic field strength B c was ∼ 10 G with a size R c of 40R s for both Seyferts with a central black hole mass of ∼ 10 8 M .
Coronal synchrotron emission is also reported for NGC 1068 in Inoue et al. [17]. Fig.  6 shows the cm-mm spectrum of NGC 1068, which shows a mm-excess similar to IC 4329A and NGC 985. The coronal synchrotron emission can reproduce the mm excess of NGC 1068 with parameters of B c = 100 G, R c = 10 R s , and the spectral index of non-thermal electrons p = 2.7.
Power-law coronal synchrotron emission also suggests that AGN coronae contain both thermal and non-thermal electrons, i.e., hybrid coronae. Therefore, acceleration of high-energy particles should happen in AGN coronae. However, the exact f nth can not be determined by the current measurements because of degeneracy among R c , B c , and f nth . Here, significantly low f nth 10 −3 requires a relatively large R c based on the radio spectral fitting, which contradicts with optical-X-ray spectral fitting studies [61] and microlensing observations [62]. Thus, we may have weak but non-negligible non-thermal coronal activity at the level of 10 −3 < f nth < 0.3.   Figure 6. The cm-mm spectrum of NGC 1068. The data points from VLBA [83] and ALMA [84][85][86] are shown in green and orange, respectively. The open points represent the newly analyzed ALMA data. The size of circles corresponds to the beam sizes as indicated in the figure. We also show the archival mm-cm data having large beam sizes as upper limits in purple [12,87,88]. The error bars correspond to 1-σ uncertainties. The blue-dashed and green-dotted lines show the coronal synchrotron and pc-scale free-free component, respectively. The solid black line shows the sum of these two components. Taken from Inoue et al. [17].
Inoue and Doi [37] also suggested that the coronae are likely to be advection heated hot accretion flows [89,90] rather than magnetically heated coronae [74,75] because the measured magnetic field strength (i) is too weak to keep the coronae hot and (ii) is consistent with the value based on the self-similar solutions of hot accretion flows. Thus, we may assume that coronal magnetic field strength scales as B ∝ M −1/2 BH [90], where we ignore its dependence on the accretion rate and other parameters for simplicity.

Generation of High Energy Particles in AGN Coronae
Previous non-thermal coronal models have overestimated the non-thermal activity strength since various coronal parameters are not well determined. As described above, current X-ray and radio observations now allow us to determine the corona size R c , the electron density n e , the magnetic field B c , non-thermal electrons' spectral index p, and even the non-thermal electron energy fraction f nth . With that knowledge, we can investigate the high-energy particle production processes in AGN coronae again. This section overviews the energy loss and acceleration processes of high-energy particles in the AGN coronae.  The geometry of black hole coronae is still under debate. A recent polarization study revealed the extended coronal structure in a nearby black hole binary [91]. Such extended coronal geometry is also naturally expected in AGNs. The detailed structure of extended coronae can be a spherical, slab, or patchy. For simplicity, in this Review, coronae are assumed to be spherical with a radius of R c ≡ r c R s , where r c is the dimensionless corona size.
Coronae are also set to be in a steady state. The proton number density n p is set to be equal to n e , which gives the maximum number of protons in coronae. n e is defined through τ T as For simplicity, the gas is assumed to be accreted on to the SMBH with free-fall velocity v ff = √ 2GM BH /R c . The free-fall timescale from the coronal region is estimated to be

Energy Loss Processes
High energy particles loose their energies through radiative cooling processes. In AGN coronae, high-energy electrons mainly lose their energies via synchrotron and inverse Compton (IC) radiation. The synchrotron cooling rate for an electron with a Lorentz factor of γ e is t syn,e (γ e ) = 3 4 where m e is the electron rest mass and U B = B 2 c /8π is the magnetic field energy density of magnetic field strength B. The inverse Compton cooling rate including the Klein-Nishina cross section [92][93][94] is whereb ≡ 4γ e /m e c 2 and f KN 1/(1.0 +b) [93]. The photon energy density, U ph , is given as U ph ( ) = L ph ( )/4πR 2 c c, where L ph is the AGN core (disk + corona) luminosity and is the photon energy.
Relativistic protons are predominately cooled through inelastic pp interactions and pγ reactions. Proton synchrotron and IC cooling channels are inefficient as compared to the hardronic mechanisms. Hereinafter, we do not consider proton IC/synchrotron coolings. The pp cooling time can be expressed as where κ pp ∼ 0.5 is the proton inelasticity of the process and we adopt σ pp = 3 × 10 −26 cm 2 and n p ≈ n e . Below we follow the formalism developed by Kelner et al. [95]. The pγ cooling time via photomeson interactions is where γ p is the proton Lorentz factor, andε and are the photon energy in the proton rest frame and the black hole frame, respectively, U ph is the energy density of the photon target, andε thr = 145 MeV [96]. The pγ interaction also generates secondary leptons and enable pair production via the so-called Bethe-Heitler process. Cooling timescale for Bethe-Heitler process is approximated as [97] where m p is the proton rest mass and α f is the fine-structure constant.

Diffusive Shock Acceleration
In the frame work of DSA [e.g., 98,99], the acceleration time scale can be approximated as where D is the diffusion coefficient, E CR is the particle energy, and v sh is the shock speed. η acc is a numerical factor that depends on the shock compression ratio and the spatial dependence of D [98]. We set η acc = 10. Assuming a Bohm-like diffusion, where e is the electric charge and η g is the gyrofactor which is the mean free path of a particle in units of the gyroradius. η g characterizes the efficiency of the acceleration. η g = 1 corresponds to the Bohm limit case. The DSA time can be written as t DSA 10 3 where R g = E CR /eB c is the gyro radius. v sh is set as v ff . Thus, r c appears in Eq. 10. η g varies in different astrophysical environments. η g ∼ 1 is possibly seen in a Galactic supernova remnant [104] and a microquasar jet [105], while η g ∼ 10 4 is seen in the case of blazars in the framework of one-zone leptonic models [e.g., 106-108].

Stochastic Acceleration
Stochastic (turbulent) acceleration has been considered for low-accretion rate objects such as low-luminosity AGN [e.g., 100,[109][110][111]. In this scenario, particles are accelerated stochastically by turbulence and magnetic reconnection in accretion disk or coronae. 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., 112] where k min ∼ R −1 c is the minimum wave number of turbulence spectrum (corresponding to the size of the corona), v A = B c / 4πm p n p is the Alfvén speed, r g = m p c 2 /eB c is the gyro radius, ζ = δB 2 c /B 2 c is the ratio of strength of turbulence fields against the background, and q describes the spectrum of the turbulence. Then, the acceleration timescale is estimated to be Assuming the Kolomogorov spectrum for the turbulence (q = 5/3) and ζ = 1, the timescale becomes Thus, stochastic acceleration appears to be inefficient compared to the typical cooling rates because of the measured weak magnetic fields, which results in slow Alfvén speed.

Magnetosphere Acceleration
Magnetosphere acceleration can also accelerate particles in the vicinity of SMBH [e.g., 101,102,[113][114][115]. 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 the unscreened electric field, so-called gaps, can accelerate charged particles effectively.
In order to have gaps, the maximum allowed accretion rate is [114,116,117] whereṁ is the accretion rate in the Eddington units. Since we are considering the standard accretion disk regimeṁ 0.01, particle acceleration by gaps does not operate in our case.

Reconnection Acceleration
Magnetic reconnection would accelerate particles [see e.g., 103, for reviews]. Reconnection would naturally happen in magnetized coronae, and radiative magnetic reconnection is also suggested as a possible origin of the X-ray emission seen in accreting black hole systems [118]. However, even in the case of solar flares, particle acceleration mechanisms in magnetic reconnection are still uncertain [e.g., 119,120]. Therefore, a quantitative discussion is not straightforward.
Here, given the magnetic field measurements, we can estimate the available energy injection power by global magnetic activity as This power seems not sufficient for providing the non-thermal particle energies under onezone estimates. However, highly non-homogeneous configurations of magnetic field, such as by a local magneto-rotational instability [121], may provide enough energy to produce non-thermal particles.

Comparison of Timescales
Given the observed properties of AGN core regions, one can estimate the radiative cooling and acceleration timescales of high-energy particles in the coronae. Here, we take the DSA as a fiducial acceleration process.
Left panel of Fig. 7 shows the timescales of high energy electrons for log L X = 43 [see 34, for details] With η g = 30, electron acceleration up to γ e ∼ 10 5 (∼ 50 GeV) is feasible in AGN coronae. Because of the intense radiation field, Compton cooling dominates the cooling. However, at higher energy regions, the main cooling channel is replaced by synchrotron cooling due to the Klein-Nishina effect. We note that the dominance of photon fields over the magnetic field does not necessarily prevent particle acceleration as such conditions are met in some efficient non-thermal sources, e.g., in gamma-ray binary systems [122,123]. Moreover, the high density of target photons can enable the converter acceleration mechanism if a relativistic velocity jump present in the system [124].
Right panel of Fig. 7 shows the timescales for protons [see 34, for details]. Protons can be accelerated up to γ p ∼ 10 6 (∼ 1 PeV) in AGN coronae. Maximum attainable energy is controlled by different processes for different luminosity AGNs. For low-luminosity Seyferts (L X < 10 44 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.

Particle Spectrum
The steady state particle distributions n = dN/dγ can be derived from the solution of the transport equation [125] ∂ ∂γ (γ cool n)   whereγ cool is the total cooling rate, Q(γ) is the injection function, which describes phenomenologically some acceleration process. Q(γ) is set as Q 0 γ −p inj exp(−γ/γ max ). Here, γ max is the maximum Lorentz factor determined by balancing the acceleration and cooling time scales. The corresponding solution is where By solving Equation. 17, we can obtain a steady-state spectrum of the non-thermal particles. Fig. 8 shows the steady-state non-thermal electron spectrum obtained for the injection spectral index of p inj = 2.0 together with the observationally determined electron spectral distribution for IC 4329A [37]. The synthetic electron distribution obtained for p 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 DSA scenario for a strong shock.

Energy Injection
The total shock power P sh can be estimated as f nth = 0.03 corresponds to ∼ 5% of the shock power is injected into the acceleration of electrons. Moreover, to explain the observed IceCube neutrino fluxes, the same energy injection rate is achieved for protons [34]. This high value implies that if DSA is responsible for particle acceleration in AGN coronae, then processes regulating the injection of electrons into DSA are very efficient. For example, in the case of DSA in supernovae remnants, nonthermal electrons obtain only ∼ 1% of energy transferred to non-thermal protons [126]. Detailed consideration of the reasons for this unusually high efficiency of electron acceleration is needed to be studied in the future. However, recent particle-in-cell simulations of protonelectron plasma considering radiatively inefficient accretion flows showed that the energy ratio depends on the proton temperature. Higher proton temperature will result in higher electron energy fraction [100]. A further detailed investigation in the corona cases is required.

Internal Gamma-ray Attenuation in Coronae
Accelerated electrons and protons in AGN coronae generate gamma-ray and neutrino emission through IC scattering, pp interaction, and pγ interaction. Those high energy gammaray photons are attenuated by photon-photon pair production interactions (γγ → e + e − ) with low-energy photons [127][128][129]. We can compute the optical depth for high-energy gamma rays to γγ pair production interactions from the SED of AGN core regions. Figure 9 shows the internal gamma-ray optical depth (τ γγ ) in the core region. 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 AGN coronae are well known [e.g., 36,130,131] based on the compactness parameter argument [132]. For the gamma-ray attenuation in AGNs, we can consider two cases. One is the "uniform" emissivity case, while the other is the "screened" case. In the uniform emissivity case, gammarays and target photons are uniformly distributed. Gamma rays are attenuated by a factor of In the screened case, gamma-rays are assumed to be generated in the inner part of the corona, and the dominant attenuating photon field surrounds it. Since the disk and corona temperature depends on the disk radius [134], such configuration can be realized. Then, gamma-rays are attenuated by a factor of exp(−τ γγ ). Figure 10 shows the resulting steady-state gamma-ray and neutrino spectra from AGN coronae for the case assuming IC 4329A (L X = 10 44 erg s −1 at a distance of 69 Mpc). The neutrino flux is shown in the form of per flavor. In this figure, the injection spectral index is p inj = 2.0, and the gyrofactor is η g = 30 for both electrons and protons. We also set the same power injected in protons and electrons. For the gamma-ray attenuation, the uniform emissivity model is adopted.

General SED Pictures
Since the spectral index of electrons becomes ∼ 3 after radiative cooling, the resulting non-thermal gamma-ray spectrum is flat in νF ν in the MeV band after the thermal cutoff at 1 MeV. Due to the internal gamma-ray attenuation effect, the spectra will have a cutoff of around 100 MeV.
Gamma rays and neutrinos induced by hadronic interactions carry 1/3 and 1/6 of those interacted hadron energies. Also, the pp and pγ production efficiency are given by the ratio between the dynamical timescale and the interaction timescales. The pp production efficiency is analytically given as f pp = t fall /t pp 0.16(τ T /1.1)(r c /40) −0.5 . Therefore, hadronic gammaray and neutrino luminosity is expected to be ∼ 5% and ∼ 3% of the intrinsic proton luminosity. Since we assume the same energy injection to electrons and protons, hadronic gamma-ray and neutrino fluxes are ∼ 5% and ∼ 3% of the IC gamma-ray flux.
Contrary to gamma rays, neutrinos induced by hadronic interactions can escape from the system without attenuation. Since we adopt the same p inj = 2 for protons as for electrons, we expect a flat spectrum for neutrinos, to which pp makes the dominant contribution. The exact position of the cutoff energy depends on the assumed η g . Here, as described later, we set η g = 30 in order to be consistent with the IceCube background flux measurements. This gyrofactor results in a neutrino spectral cutoff around 100 TeV.

Application to NGC 1068
A nearby Seyfert galaxy NGC 1068 was reported as the hottest neutrino spot with a 2.9-σ confidence level in 10-yr all-sky survey observations of IceCube [8]. As we describe above, the hybrid corona model is one possible solution. Thus, understanding the required physical parameters to explain the neutrino in NGC 1068 is crucial for the coronal model test. Fig. 11 shows the expected gamma-ray and neutrino signals from NGC 1068 together with the observed gamma-ray data [16,135,136] and the IceCube data [8]. We follow the assumptions on the coronal parameters as described above except for the gyrofactor and parameters determined by the coronal synchrotron model explaining the mm-excess (See §. 2 and Fig. 6). Considering the neutrino measurement uncertainty, in the figure, we plot the model curve region in the range of 30 ≤ η g ≤ 3 × 10 4 for each curve. The darker region corresponds to lower η g , in which models extend to higher energies. Further detailed neutrino spectrum will narrow down the range of allowed η g . Regarding the gamma-ray measurements, in the screened gamma-ray attenuation case, the model can explain the preliminary neutrino signals above several TeV without violating the gamma-ray data. On the other hand, the uniform emissivity model violates the low-energy gamma-ray data. This implies a further detailed study of coronal geometry is necessary. In either case, it is not easy for the corona model to explain the entire observed gamma-ray flux data up to 20 GeV, requiring another mechanism to explain gamma-rays above 100 MeV such as star formation activity [15], jet [137], or disk wind [138]. Therefore, the coronal model can explain the IceCube neutrino signal without violating the gamma-ray data.  Figure 11. The gamma-ray and neutrino spectrum of NGC 1068. The circle, square, and triangle data points are from The Fermi-LAT collaboration [135], Ajello et al. [136], and Acciari et al. [16], respectively. The green shaded regions represent the 1, 2, and 3σ regions on the spectrum measured by IceCube [8].
The expected gamma-ray and neutrino spectrum from the corona are shown for 30 ≤ η g ≤ 3 × 10 4 . The darker region corresponds to lower η g . The blue region shows the expected neutrino spectrum. The orange and magenta shaded region shows the gamma-ray spectrum for the uniform case and the screened case, respectively. Taken from Inoue et al. [17].
An important question is what differentiates NGC 1068 from other nearby Seyfert galaxies. NGC 1068 is not the brightest X-ray Seyfert [139]. Its observed hard X-ray flux is a factor of ∼ 16 fainter than the one of the brightest Seyfert, NGC 4151. NGC 1068 is a type-2 Seyfert galaxy, and obscured by the materials up to the neutral hydrogen column density of N H ∼ 10 25 cm −2 [10,11]. If we correct this attenuation effect to understand the intrinsic X-ray radiation power, NGC 1068 appears to be the intrinsically brightest Seyfert. For example, intrinsically, it would be by a factor of ∼ 3.6 brighter than NGC 4151 in X-ray. As the dusty torus does not obscure coronal neutrino emission, which can scale with accretion power, NGC 1068 might be the brightest source in the neutrino sky. This could be why NGC 1068 appears as the hottest spot in the IceCube map rather than other Seyfert galaxies.
In NGC 1068, the jets are prominent and extend for several kpcs in both directions. In the central ∼ 14 − 70 pc region, the downstream jet emission dominates in the centimeter regime [83,87]. These jets can also be the production site of the reported neutrinos. However, gamma-ray attenuation is not significant in these far side regions from the nucleus. Therefore, these jets may not be the dominant neutrino production sites.
By setting f nth = 0.03, AGN coronae can nicely explain the cosmic MeV gamma-ray background in an extension from the cosmic X-ray background radiation. Since the spectral index of non-thermal electrons in the coronae is ∼ 3, the resulting MeV gamma-ray background spectrum becomes flat in E 2 dN/dE. Since the dominant IC contributors switch from thermal electrons to non-thermal electrons at around 1 MeV, the MeV background spectrum may have a spectral hardening feature at ∼ 1 MeV. In Fig. 12, we set η g = 30. The result for the MeV background does not significantly change as far as η g < 1000. If η g > 1000, we may require lower f nth . Here, IC emission due to non-thermal electrons also contributes to the X-ray band. Their contribution is about ∼ 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. Due to the internal gamma-ray attenuation effect, these non-thermal gamma rays can not contribute to the emission above GeV, where blazars, radio galaxies, and star-forming galaxies dominate [see, e.g., 145].
For neutrinos, the combination of pp and pγ interactions can nicely reproduce the IceCube fluxes below 100-300 TeV. pp interactions dominate the flux at 10 TeV, while pγ interactions prevail above this energy. Because of the target photon field SED, pγ is subdominant in the GeV-TeV band. If we inject more powers into protons, it inevitably overproduces the IceCube background fluxes. Figure 13 shows the cosmic neutrino background spectra from AGN cores with various gyro factors ranging from 1 (Bohm limit) to 10 3 . It is clear that if η g 30, the resulting neutrino fluxes overproduce the measured fluxes. On the contrary, if η g 30, AGN coronae can not significantly contribute to the observed neutrino background fluxes. Thus, to explain the IceCube neutrino background fluxes by AGN cores, η g ∼ 30 is required.  IceCube Figure 13. The cosmic neutrino background spectrum per flavour from AGN coronae. The dashed, dotted, solid, dot-dashed, and double-dot-dashed curve shows the contribution with η g =1 (Bohm limit), 10, 30, 10 2 , and 10 3 , respectively. The square data points correspond to the cosmic neutrino background spectrum by the IceCube [144]. Taken from Inoue et al. [34]. a factor of 10 less than that by primary electrons (see Fig. 10) because of radiative efficiency differences between protons and electrons. Therefore, hadronically induced secondary pairs should not be energetically important. Pairs generated through γγ annihilation of primary leptonic fluxes may also contribute such as to the MeV gamma-ray spectrum. However, considering the intrinsic photon index Γ ph = 2 and acceleration limit, the resulting pair contribution will not be higher than primaries. Thus, the secondary particle component would not significantly alter the resulting spectra.
Here, hadronic power can be much higher than we assume. Then, secondary leptons can energetically dominate the primary leptons. However, as seen in Fig. 12, such models can not explain the cosmic MeV gamma-ray and TeV-PeV neutrino background fluxes.

Comparison of current available models
In literature, it has been argued that high energy particles in the AGN coronae generate intense neutrino emission [e.g., 26,35,147]. These originally predicted fluxes have been ruled out by the IceCube observations [148]. However, recent studies have revisited the estimated fluxes [18,33,34,146]. In this section, we clarify the differences in recent AGN corona models for high-energy neutrinos. Figure 14 shows the resulting cosmic neutrino background spectrum from those papers. These corona models predict different neutrino spectra, although they consider the same neutrino production region. This section briefly describes the differences among those models.
Stecker [146] considered a similar model of the originally proposed one [35], but the background flux is assumed to be lower by a factor of 20 to match with the IceCube flux. The shock radius and the magnetic field strength were assumed to be 10R s and 10 3 G in the model by Stecker et al. [35]. The particle spectral index was also assumed to be 2 in the framework of the DSA.
Kalashev et al. [33] followed the treatment in Stecker et al. [35], Stecker [146], but accounting for the radial emission profile in the standard accretion disk in their consideration of the pγ cooling processes. The particle spectra in Kalashev et al. [33] are normalized to match with the IceCube data. Given the observationally determined corona size R c ∼ 40R s , the dominant photon target is likely to be generated in the inner region of the coronae.   Figure 14. The cosmic neutrino background spectrum per flavour from AGN coronae, showing the currently available models [18,33,34,146]. The original prediction by Stecker et al. [27,35] is also shown.
Inoue et al. [34] took into account both X-ray and radio measurements, which allowed us to derive R c = 40R s and B c = 10 G [37]. And, to explain the cosmic MeV background by Seyferts, f nth is set as 0.03. The proton spectral index is assumed to be the same as that of electrons, determined through the radio measurement. However, the non-thermal electron/proton energy ratio is set to be K ep = 1, and η g = 30 are assumed to reproduce the cosmic neutrino background fluxes. We note that it can be 30 ≤ η g ≤ 3 × 10 4 for the explanation of NGC 1068. R c , B c , p, and f nth are, for the first time, observationally determined, while K ep and η g are still assumed to reproduce the IceCube measurements. Besides, the exact acceleration process is not well determined, as they follow the radio observation results.
Murase et al. [18] consider stochastic acceleration motivated by recent numerical simulations for low-luminosity AGNs [149]. The required cosmic ray pressure to explain the IceCube data is about 1-10% of thermal pressure, which is similar to Inoue et al. [34]. Their model assumed R c = 30R s and B c 10 3 G. This high magnetic field contradicts the radio measurements. As discussed above, the measured magnetic field may not be high enough to accelerate particles in the stochastic acceleration scheme efficiently.
Very recently, Gutiérrez et al. [19] also model high energy signals from AGN coronae taking into account X-ray and radio observations. They basically follow Inoue et al. [34]. However, they consider lower K ep than in Inoue et al. [34]. This assumption is more consistent with K ep seen in nearby supernova remnants. Because of this low K ep , secondary leptons make the dominant contribution in the MeV band. Although they have not estimated the integrated MeV gamma-ray and TeV-PeV neutrino background fluxes in their paper yet, such a study based on their model will be helpful for future comparison.

Future test of models
Although non-thermal AGN coronal models have failed to explain the X-ray data in the 1990s, millimeter radio observations found weak coronal activity in nearby Seyferts in 2018, and then the possible detection of NGC 1068 in 2020 shed light again on the AGN corona model. However, as described above, the models still have profound ambiguity. We require multi-messenger tests on that. Below we list possible ways to test the AGN corona scenario.

Radio Synchrotron Emission
As non-thermal particles are accelerated in the magnetized coronae, we should expect coronal synchrotron emission. ALMA observations have already grasped the evidence of the coronal synchrotron emission. Therefore, we need to consider the expected radio synchrotron emission for each model and compare it with the ALMA measurements. As the ALMA data are already available for nearby Seyferts, this test can be easily performed.

Nuclear Spallation Effect Appearing in X-ray
High energy protons can be traced by future high-resolution calorimeter spectroscopy in the X-ray band such as XRISM [150] and Athena [151]. As narrow line features are seen in AGN X-ray disk spectra, there are abundant metal elements in AGN cores. Accelerated protons interact with those nuclei and induce nuclear spallation. The nuclear spallation in AGN disks will enhance emission lines from Mn, Cr, V, and Ti [152]. Those signatures will be another clue for the test of the corona model.

MeV power-law tail
In the hybrid AGN corona model, the non-thermal gamma-ray should appear after the thermal cutoff (∼ 300 keV). The expected MeV gamma-ray flux is about 5% of the intrinsic Xray flux. Due to the strong internal gamma-ray attenuation effect, the spectra will have a cutoff around 10-100 MeV depending on the photon distribution. These MeV gamma-ray photons should be seen by future MeV gamma-ray observatories. If not, this means that the amount of non-thermal population is much less than required for the MeV gamma-ray background radiation, which means the corona model can not explain the neutrino background.

Further Neutrino Observations
The neutrino measurement can provide a critical test on AGN corona scenarios and robustly constrain acceleration of protons. Currently, only NGC 1068 is reported as the possible neutrino production site. Further neutrino observations should see more nearby Seyferts following the AGN corona model [19,34,153]. Additionally, neutrino observations may pin down the acceleration mechanism in the AGN corona by taking the neutrino spectrum from individual objects.

Summary
In this Review, we overview our current understanding of the AGN coronae from X-ray and mm radio observations. We show that these measurements contain critical information for constraining particle acceleration in the AGN coronae. AGN coronae are feasible sites for particle acceleration. If the energy injection rate is adequate, AGN coronae can explain the neutrino signals from NGC 1068 and a part of the diffuse neutrino fluxes. Future radio, X-ray, MeV gamma-ray, and TeV neutrino observations will be able to test this scenario by observations of nearby bright Seyferts.