Hidden Cores of Active Galactic Nuclei as the Origin of Medium-Energy Neutrinos:Critical Tests with the MeV Gamma-Ray Connection

Hidden Cores of Active Galactic Nuclei as the Origin of Medium-Energy Neutrinos:
Critical Tests with the MeV Gamma-Ray Connection

Kohta Murase Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Center for Particle and Gravitational Astrophysics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Yukawa Institute for Theoretical Physics, Kyoto, Kyoto 606-8502 Japan    Shigeo S. Kimura Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Center for Particle and Gravitational Astrophysics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA    Peter Mészáros Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Center for Particle and Gravitational Astrophysics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

The cores of active galactic nuclei (AGNs) are among the candidate sources of the IceCube neutrinos, but the underlying cosmic-ray acceleration processes are unclear. Based on the standard disk-corona picture of AGNs, we present a phenomenological model, in which protons are stochastically accelerated by turbulence from the magnetorotational instability. We show that this model can explain a large diffuse flux of about 30 TeV neutrinos if the cosmic rays carry a few percent of the coronal thermal energy. We find that the Bethe-Heitler process plays a crucial role in connecting these neutrinos and cascaded MeV gamma rays, and point out that the gamma-ray flux can be even enhanced by reacceleration of secondary pairs. Critical tests of the model are given by its prediction that a significant fraction of the MeV gamma-ray background correlates with the 10 TeV neutrino background, and nearby Seyfert galaxies should be seen by future MeV gamma-ray telescopes.


I Introduction

The origin of cosmic neutrinos observed in IceCube is a major enigma Aartsen et al. (2013a, b), which has been deepened by the latest results on high- and medium-energy starting events and shower events Aartsen et al. (2015a, b, 2017). The atmospheric background of high-energy electron neutrinos is much lower than that of muon neutrinos, allowing us to investigate the data below 100 TeV Beacom and Candia (2004); Laha et al. (2013). The comparison with the extragalactic gamma-ray background (EGB) measured by Fermi indicates that the extragalactic neutrino background (ENB) at  TeV energies originates from hidden sources preventing the escape of GeV-TeV gamma rays Murase et al. (2016).

Active galactic nuclei (AGNs) are major contributors to the energetics of high-energy cosmic radiations Murase and Fukugita (2019); radio quiet (RQ) AGNs are dominant in the extragalactic x-ray sky Fabian and Barcons (1992); Ueda et al. (2003); Hasinger et al. (2005); Ajello et al. (2008); Ueda et al. (2014), and radio loud (RL) AGNs including blazars give dominant contributions to the EGB Costamante (2013); Inoue (2014); Fornasa and Sánchez-Conde (2015). AGNs may also explain the MeV gamma-ray background whose origin has been under debate (e.g., Inoue et al. (2008); Ajello et al. (2009); Lien and Fields (2012)).

High-energy neutrino production in the vicinity of supermassive black holes (SMBHs) were discussed early on Eichler (1979); Berezinskii and Ginzburg (1981); Begelman et al. (1990); Stecker et al. (1991), in particular to explain x-ray emission by cosmic-ray (CR) induced cascades assuming the existence of high Mach number accretion shocks at the inner edge of the disk Kazanas and Ellison (1986); Zdziarski (1986); Sikora et al. (1987); Stecker et al. (1991). However, cutoff features evident in the x-ray spectra of Seyfert galaxies and the absence of electron-positron annihilation lines ruled out the simple cascade scenario for the x-ray origin (e.g., Di Matteo (1999); Ricci et al. (2018)). In the standard scenario, the observed x rays are attributed to thermal Comptonization of disk photons Shapiro et al. (1976); Sunyaev and Titarchuk (1980); Zdziarski et al. (1996); Poutanen and Svensson (1996); Haardt et al. (1997), and electrons are presumably heated in the coronal region Liang and Price (1977); Galeev et al. (1979). There has been significant progress in our understanding of accretion disks with the identification of the magnetorotational instability (MRI) Balbus and Hawley (1991, 1998), which can result in the formation of a corona above the disk as a direct consequence of the accretion dynamics and magnetic dissipation (e.g., Miller and Stone (2000); Merloni and Fabian (2001); Liu et al. (2002); Blackman and Pessah (2009); Io and Suzuki (2014); Suzuki and Inutsuka (2014); Jiang et al. (2014)).

Turbulence is also important for particle acceleration Lazarian et al. (2012). The roles of nonthermal particles have been studied in the context of radiatively inefficient accretion flows (RIAFs; Narayan and Yi (1994); Yuan and Narayan (2014)), in which the plasma is often collisionless because Coulomb collisions are negligible for protons (e.g., Takahara and Kusunose (1985); Mahadevan et al. (1997); Mahadevan and Quataert (1997); Kimura et al. (2014); Lynn et al. (2014); Ball et al. (2018)). Recent studies based on numerical simulations of the MRI Kimura et al. (2016, 2019) support the idea that high-energy ions might be stochastically accelerated by the ensuing magnetohydrodynamic (MHD) turbulence.

The vicinity of SMBHs is often optically thick to GeV-TeV gamma rays, where CR acceleration cannot be directly probed by these photons, but high-energy neutrinos can be used as a unique probe of the physics of AGN cores. In this work, we present a concrete model for their high-energy emissions (see Fig. 1), in which spectral energy distributions (SEDs) are constructed from the data and from empirical relations. We compute neutrino and gamma-ray spectra, by solving both CR transport equations with the relevant energy losses and the resulting electromagnetic cascades of secondaries. We demonstrate the importance of future MeV gamma-ray observations for revealing the origin of IceCube neutrinos.

Figure 1: Schematic picture of the AGN disk-corona scenario. Protons are accelerated by turbulence generated by the MRI in coronae, and produce high-energy neutrinos and cascaded gamma rays via interactions with matter and radiation.

Ii Phenomenological Prescription of AGN Disk Coronae

We construct a phenomenological disk-corona model based on the existing data. SEDs of Seyfert galaxies have been extensively studied, which consist of several components; radio emission (see Ref. Panessa et al. (2019)), infrared emission from a dust torus Netzer (2015), optical and ultraviolet components from an accretion disk Koratkar and Blaes (1999), and x rays from a corona Sunyaev and Titarchuk (1980).

The averaged SEDs are provided in Ref. Ho (2008) as a function of the Eddington ratio, , where and are bolometric and Eddington luminosities, respectively. The well-known “blue” bump is attributed to multicolor blackbody emission from the geometrically thin, optically thick disk Shakura and Sunyaev (1973). The spectrum is expected to have an exponential cutoff at , where is the maximum effective temperature of the disk (e.g., Pringle (1981)). Here, is the SMBH mass, is the mass accretion rate, and is the Schwarzschild radius. Assuming a standard disk, we use with a radiative efficiency of . Although the spectra calculated by Ref. Ho (2008) extend to low energies, we only consider photons with eV because infrared photons would come from a dust torus.

X rays are produced via Compton upscattering by thermal electrons with  K. The spectrum can be modeled by a power law with an exponential cutoff. The photon index, , is correlated with the Eddington ratio as  Trakhtenbrot et al. (2017). The cutoff energy is also given by  keV Ricci et al. (2018). The electron temperature is written as for an optically thin corona. Then, assuming a slab geometry, the Thomson optical depth is given by  Ricci et al. (2018). The x-ray luminosity is converted into following Ref. Hopkins et al. (2007), and the SMBH mass can be estimated by  Mayers et al. (2018). The thus constructed SEDs are shown in Fig. 2.

We expect the disk coronae to be characterized by two temperatures, i.e.,  Di Matteo et al. (1997); Cao (2009) (see Appendix). We assume that the thermal protons are at the virial temperature, , where is the coronal size and is the normalized radius. The normalized proton temperature is . With the sound speed and Keplerian velocity , the scale height is written as , leading to a nucleon target density, . The magnetic field is estimated by , where is the plasma beta.

We summarize our model parameters in Table 1. Note that most of the physical quantities can be estimated from the observational correlations. Thus, for a given , and are the only remaining parameters. They are also constrained in a certain range by observations Jin et al. (2012); Morgan et al. (2012) and numerical simulations Io and Suzuki (2014); Jiang et al. (2014). For example, recent MHD simulations show that in the coronae can be as low as (e.g., Miller and Stone (2000); Suzuki and Inutsuka (2014)). We assume , and adopt throughout this work.

Figure 2: Disk-corona SEDs and CR proton differential luminosities for , , , , (from bottom to top).
42.0 43.0 6.51 1.72 0.27 0.59 13.5 10.73 0.27
43.0 44.2 7.25 1.80 0.23 0.52 14.2 9.93 0.54
44.0 45.4 8.00 1.88 0.20 0.46 15.0 9.13 0.94
45.0 46.6 8.75 1.96 0.16 0.41 15.7 8.33 1.54
46.0 47.9 9.49 2.06 0.12 0.36 16.4 7.53 2.34
Table 1: Parameters used in this work. Units are [erg ] for and , [] for , [cm] for , [] for , and [%] for the ratio of the CR pressure to the thermal pressure.

Iii Stochastic Acceleration and Secondary Production in Coronae

For the disk coronae considered here, the infall and dissipation time scales are estimated to be and , where is the viscosity parameter Shakura and Sunyaev (1973). The electron relaxation time via Coulomb collisions, , is always shorter than . The proton relaxation time is much longer, which can ensure two temperature coronae (see Appendix). These collisionallity arguments imply that turbulent acceleration is promising for not electrons but protons (although fast acceleration by small-scale reconnections might occur Hoshino (2015); Li et al. (2015)). The situation is somewhat analogous to that in RIAFs, for which nonthermal signatures have been studied (e.g., Ozel et al. (2000); Kimura et al. (2015); Ball et al. (2016)).

We expect that protons are accelerated in the MHD turbulence. We compute steady state CR spectra by solving the following Fokker-Planck equation (e.g., Becker et al. (2006); Stawarz and Petrosian (2008); Chang and Cooper (1970); Park and Petrosian (1996)),


where is the CR distribution function, is the diffusion coefficient in energy space, is the total cooling rate, is the escape rate, and is the injection function (see Appendix 111We consider meson production processes by () and () interactions, as well as the Bethe-Heitler pair production (), proton synchrotron radiation (), diffusive escape (), and infall losses ().). The stochastic acceleration time is given by , where is the Alfvén velocity and is the inverse of the turbulence strength Dermer et al. (1996, 2014). We adopt , which is consistent with the recent MHD simulations Kimura et al. (2019), together with . Because the dissipation rate in the coronae is expected to be proportional to , we assume that the injection function linearly scales as . To explain the ENB, the CR pressure required for turns out to be % of the thermal pressure, which is reasonable. We plot in Fig. 2, where is the volume.

While the CRs are accelerated, they interact with matter and radiation modeled in the previous section, and produce secondary particles. Following Ref. Murase (2018); Murase et al. (2019), we solve the kinetic equations taking into account electromagnetic cascades. In this work, secondary injections by the Bethe-Heitler and processes are approximately treated as , , and . The resulting cascade spectra are broad, being determined by synchrotron and inverse Compton emission.

In general, stochastic acceleration models naturally predict reacceleration of secondary pairs populated by cascades Murase et al. (2012). The critical energy of the pairs, , is consistently determined by the balance between the acceleration time and the electron cooling time . We find that whether the secondary reacceleration occurs or not is rather sensitive to and . For example, with and , the reaccelerated pairs can upscatter x-ray photons up to , which may form a gamma-ray tail. However, if  MeV (for and ), reacceleration is negligible, and small-scale turbulence is more likely to be dissipated at high  Howes et al. (2011).

Iv Neutrino background and MeV gamma-ray connection

Figure 3: EGB and ENB spectra in our RQ AGN core model. The data are taken from Swift-BAT Ajello et al. (2014) (green), Nagoya balloon Fukada et al. (1975) (blue), SMM Watanabe et al. (1997) (purple), COMPTEL Weidenspointner et al. (2000) (gray), Fermi-LAT Ackermann et al. (2015) (orange), and IceCube Aartsen et al. (2017) for shower (black) and upgoing muon track (blue shaded) events. A possible contribution of reaccelerated pairs is indicated (thin solid).

We calculate neutrino and gamma-ray spectra for different source luminosities, and obtain the EGB and ENB through Eq. (31) of Ref. Murase et al. (2014). We use the x-ray luminosity function , given by Ref. Ueda et al. (2014), taking into account a factor of 2 enhancement by Compton thick AGNs. Results are shown in Fig. 3. Our RQ AGN core model can explain the ENB at  TeV energies if the CR pressure is % of the thermal pressure.

In the vicinity of SMBHs, high-energy neutrinos are produced by both and interactions. The disk-corona model indicates (see Table 1), which leads to the effective optical depth . Note that is a function of (and ). X-ray photons from coronae provide target photons for the photomeson production, whose effective optical depth Murase et al. (2008, 2016) is , where , is the attenuation cross section,  GeV, , and is used. The total meson production optical depth is given by , which always exceeds unity in our model.

Importantly,  TeV neutrinos originate from CRs with  PeV. Different from previous studies explaining the IceCube data Stecker (2013); Kalashev et al. (2015), disk photons are irrelevant for the photomeson production because its threshold energy is . However, CRs in the 0.1-1 PeV range should efficiently interact with disk photons via the Bethe-Heitler process because the characteristic energy is , where  MeV (Chodorowski et al., 1992; Stepney and Guilbert, 1983). Approximating the number of disk photons by , the Bethe-Heitler effective optical depth Murase and Beacom (2010) is estimated to be , where . The dominance of the Bethe-Heitler process is a direct consequence of the observed disk-corona SEDs, implying that the medium-energy neutrino flux is suppressed by .

The ENB flux is analytically estimated to be


where and for and interactions, respectively, and represents the redshift evolution of RQ AGNs. Eq. (2) is consistent with the numerical results presented in Fig. 3. Here is the conversion factor from bolometric to differential luminosities, is the CR loading parameter defined against the x-ray luminosity, and corresponds to in our model. We find that the ENB and EGB are dominated by AGNs with , at which the local number density is  Murase and Waxman (2016).

The , and Bethe-Heitler processes all initiate electromagnetic cascades, whose emission appears in the MeV range. Thanks to the dominance of the Bethe-Heitler process, RQ AGNs responsible for the medium-energy ENB should contribute to % of the MeV EGB. Possible reacceleration can enhance the MeV gamma-ray flux, and the MeV EGB could be explained if % of the pairs is injected into the reacceleration process. For comparison, models for RL AGNs (Inoue (2011); Ajello et al. (2015) for the EGB and Fang and Murase (2018) for the ENB) are also shown in Fig. 3. This demonstrates that in principle the dominant portions of the EGB and ENB from MeV to PeV energies can be explained by the combination of RQ AGNs and RL AGNs. However, we also caution that other possibilities such as starburst galaxies are still viable Murase et al. (2013).

V Multimessenger Tests

Figure 4: Point source fluxes of all flavor neutrinos and gamma rays from a nearby RQ AGN. A possible effect of secondary reacceleration is indicated (thin solid). For eASTROGAM De Angelis et al. (2017) and AMEGO Moiseev and Team (2018) sensitivities, the observation time of  s is assumed. The IceCube eight-year sensitivity Aartsen et al. (2019) and the 5 times better case Aartsen et al. (2014) are shown.

Detecting MeV signals from individual Seyferts would be crucial for testing the model, which is challenging for existing gamma-ray telescopes. However, this would be feasible with future telescopes like eASTROGAM De Angelis et al. (2017), GRAMS Aramaki et al. (2019), and AMEGO Moiseev and Team (2018) (see Fig. 4).

For luminous Seyfert galaxies, the fact that x rays come from thermal Comptonization suggests that the photon energy density is larger than the magnetic field energy density. In the scenario to explain 10-100 TeV neutrinos, secondary pairs are injected in the 100-300 GeV range and form a fast cooling spectrum down to MeV energies in the steady state. Thus, in the simple inverse Compton cascade scenario, the cascade spectrum is extended up to the break energy due to . In reality, both synchrotron and inverse Compton processes can be important. The characteristic frequency of synchrotron emission by Bethe-Heitler pairs is given by  Murase et al. (2019). Because disk photons lie in the  eV range, the Klein-Nishina effect is moderately important at the injection energies. The synchrotron cascade is dominant if the photon energy density is smaller than , i.e., . In either synchrotron or inverse Compton cascades, MeV gamma rays are expected.

The ENB and EGB are dominated by AGNs with . AMEGO’s differential sensitivity suggests that point sources with  Mpc are detectable, and the number of the sources within this horizon is . Detections or nondetections of the MeV gamma-ray counterparts will support or falsify the AGN core model as the origin of  TeV neutrinos. Note that the predicted neutrino flux shown in Fig. 4 is below the current IceCube sensitivity. Nearby Seyferts may be seen as point sources with IceCube-Gen2, but stacking analyses are more promising.

Vi Summary and discussion

We presented the results of a concrete model for RQ AGNs which can explain the medium-energy neutrino data. The disk-corona SEDs have been well studied, and known empirical relations enabled us to estimate model parameters, with which we solved the relevant transport equations and computed subsequent cascades consistently. The model is not only motivated from both observations and theories but it also provides clear predictions. In particular, the dominance of the Bethe-Heitler process is a direct consequence of the observed SEDs, leading to a robust MeV gamma-ray connection. Nearby Seyferts will be promising targets for future MeV gamma-ray telescopes such as eASTROGAM and AMEGO. A good fraction of the MeV EGB may come from RQ AGNs especially in the presence of secondary reacceleration, in which gamma-ray anisotropy searches should be powerful tools Inoue et al. (2013). Neutrino multiplet and stacking searches with IceCube-Gen2 are also promising Murase and Waxman (2016).

The suggested tests are crucial for unveiling nonthermal phenomena in the vicinity of SMBHs. For low-luminosity AGNs, where the plasma density is low, direct acceleration may occur Levinson (2000) and TeV gamma rays can escape Aleksic et al. (2014). However, in Seyferts, the plasma density is so high that a gap is not expected, and GeV-TeV gamma rays are blocked. Only MeV gamma rays can escape from the core region, and neutrinos serve as a smoking gun.

Our results strengthen the importance of further theoretical studies of disk-corona systems. Simulations on turbulent acceleration in coronae and particle-in-cell computations of acceleration via magnetic reconnections are encouraged in order to understand the CR acceleration in the disk-corona system. Global MHD simulations will also be relevant to examine other postulates such as accretion shocks Kazanas and Ellison (1986); Stecker et al. (1991); Szabo and Protheroe (1994); Stecker and Salamon (1996) or colliding blobs Alvarez-Muniz and Mészáros (2004) and to reveal the origin of low-frequency emission that could come from the outer region of coronae Inoue and Doi (2014, 2018).

This work is supported by Alfred P. Sloan Foundation and NSF Grant No. PHY-1620777 (K.M.), JSPS Oversea Research Fellowship and the IGC post-doctoral fellowship program (S.S.K.), and NASA NNX13AH50G as well as the Eberly Foundation (P.M.). While we were finalizing this project, we became aware of a related work by Inoue et al. (arXiv:1904.00554). We thank for Yoshiyuki Inoue for discussions. Both works are independent and complementary, and there are notable differences. We consider stochastic acceleration by turbulence based on the disk-corona model rather than by accretion shocks. Also, we focus on the origin of 10-100 TeV neutrinos, for which the Bethe-Heitler suppression is critical. Third, we calculate CR-induced electromagnetic cascades, which is critical for testing the scenario for IceCube neutrinos. K.M. also thanks for the invitation to the AMEGO Splinter meeting held in January 2019, in which preliminary results of cascade emission were presented.


.1 Time Scales for Thermal Particles

To accelerate particles through stochastic acceleration in turbulence, the Coulomb relaxation time scales of particles at their injection energy should be longer than the dissipation time scale. Also, in order to form the two-temperature corona that is often discussed in the literature (e.g., Di Matteo et al. (1997); Cao (2009)), the dissipation time scale should be shorter than the proton-electron relaxation time. We discuss these plasma time scales in this subsection. The infall time scale is expected to be similar to that of the advection dominated accretion flow Narayan and Yi (1994); Yuan and Narayan (2014):


where is the viscous parameter in the accretion flow Shakura and Sunyaev (1973). Assuming that the dissipation in the corona is related to some magnetic process like reconnections, the dissipation time scale can be expressed as


The relaxation times for electrons and protons are estimated to be Takahara and Kusunose (1985); Kimura et al. (2014)


where ( or ), is the Coulomb logarithm, and we may consider a proton-electron plasma Ricci et al. (2018). We plot these time scales as a function of in Fig. S1 for and . We see that among the five time scales and are the shortest and longest, respectively. This means that electrons are easily thermalized while nonthermal protons are naturally expected and could be accelerated through stochastic acceleration. Also, because is satisfied for the range of our interest, one may expect the two-temperature corona to be formed.

.2 Time Scales for High-Energy Protons

Nonthermal proton spectra are determined by the balance among the acceleration, cooling, and escape processes. We consider stochastic acceleration by turbulence, and take account of infall and diffusion as escape processes. We also treat inelastic collisions, photomeson production, Bethe-Heitler pair production, and synchrotron radiation as cooling processes.

The stochastic acceleration is modeled as a diffusion phenomenon in momentum or energy space (e.g., Lynn et al. (2014); Kimura et al. (2016, 2019). Assuming gyro-resonant scattering through the turbulence with a power spectrum of , the acceleration time is written as Dermer et al. (1996); Stawarz and Petrosian (2008); Murase et al. (2012); Dermer et al. (2014); Kimura et al. (2015)


where is the turbulence strength parameter and is the Alfvén velocity. The infall time is given by Eq. (S1). Using the same scattering process for the stochastic acceleration, the diffusive escape time is estimated to be Dermer et al. (1996); Stawarz and Petrosian (2008); Kimura et al. (2015)


The cooling rate by inelastic collisions is estimated to be


where and are the cross section and inelasticity for interactions, as implemented in Refs. Murase (2018); Murase et al. (2019). The photomeson production energy loss rate is calculated by


where is the proton Lorentz factor,  MeV is the threshold energy for the photomeson production, is the photon energy in the proton rest frame, and are the cross section and inelasticity, respectively, and the normalization is given by . We utilize the fitting formula based on GEANT4 for and , which are used in Ref. Murase et al. (2008). The Bethe-Heitler cooling rate is written in the same form of Eq. (S9) by replacing the cross section and inelasticity with and , respectively, where we use the fitting formula given in Refs. Chodorowski et al. (1992) and Stepney and Guilbert (1983). Finally, the synchrotron time scale for protons is given by


We plot the times scales in Fig. S2 with a parameter set of , , , , and . We can see that particle acceleration is limited by interactions with photons except for . For the lowest-luminosity case, the photomeson production, the Bethe-Heitler process, the reaction, and the diffusive escape rates are comparable to the acceleration rate around  GeV, while the Bethe-Heitler process hinders the acceleration for the other cases at  GeV due to a softer spectrum for a higher-luminosity Seyfert galaxy that has a lower maximum energy due to its larger photon number density.

Figure S1: The infall time and Coulomb relaxation time scales as a function of .
Figure S2: Comparison of time scales for high-energy protons in the cases with (top), (middle), and (bottom).

.3 Spectra of Nonthermal Protons

To obtain the spectrum of nonthermal protons, we solve the Fokker-Planck equation given in Eq. (1) using the Chang-Cooper method Chang and Cooper (1970); Park and Petrosian (1996). The resulting spectra are shown in Fig. 2. We tabulate the critical energy for protons, , at which the acceleration balances with loss processes in Table S1. For a lower value of , the critical energy is higher owing to their lower loss rates. We consider two cases, which give similar proton spectra.

The dissipation rate in the coronae is expected to be proportional to , so we may write the injection function as,


where is the injection energy and is the injection fraction. The values of and do not affect the resulting spectral shape as long as . For example, if we use , the resulting spectrum is shown in Fig. 2 corresponds to . Note that is larger as is higher. For , the stochastic acceleration mechanism predicts a very hard spectrum, .

For , owing to the inefficient escape processes, the accelerated particles pile up around , which creates a hardening feature around and a strong cutoff above . Note that in our model the energy density of these nonthermal protons are much lower than that of the thermal protons: is of the order of for all the cases, where and . Hence, these nonthermal protons do not affect the dynamical structure Kimura et al. (2014).

w.o. secondary reacceleration [] [s] [TeV] [MeV] [kG] 42.0 4.87 200 3.4 43.0 5.61 150 1.3 44.0 6.36 120 0.53 45.0 7.11 84 0.21 46.0 7.85 60 0.085 w. secondary reacceleration [] [s] [TeV] [MeV] [kG] 42.0 4.87 370 22 1.9 43.0 5.61 260 21 0.77 44.0 6.36 200 16 0.31 45.0 7.11 140 11 0.12 46.0 7.85 100 6 0.049

Table S1: Physical quantities related to nonthermal particles for a given .
Figure S3: Comparison of time scales for high-energy electrons and positrons in the cases with (left), (middle), and (right).

.4 Time Scales for High-Energy Pairs

Even if primary electrons are not accelerated through the turbulence due to their efficient Coulomb losses, electron-positron pairs injected via hadronic processes and populated via electromagnetic cascades processes can be accelerated to higher energies without suffering from Coulomb losses Murase et al. (2012). Such high-energy pairs may rapidly cool down by synchrotron and inverse Compton emissions. In Seyferts, the energy of the dominant target photons for inverse Compton scatterings is around  eV, implying that the Klein-Nishina effect is unimportant at  GeV. If the secondary pairs can be accelerated by the turbulence, the acceleration and cooling is expected to balance at the critical energy, , and this effect is relevant if is higher than the energy of the thermal protons. The time scale of the reacceleration by the turbulence is given by Eq. (S6) above the thermal energy of protons, MeV. Below this energy, the turbulent power spectrum should become steeper due to kinetic effects Howes et al. (2011), and the reacceleration time is considerably longer than that by Eq. (S6).

The synchrotron cooling time scale is


where is the electron Lorentz factor. Then, the inverse Compton cooling time in the Thomson limit is estimated to be


where is the target photon energy density. Note that the Klein-Nishina effect is taken into account in the calculations of photon spectra presented in the main text.

We plot the time scales for high-energy pairs in Fig. S3, where thin and thick lines are for the cases with and without reacceleration, respectively. For the case without reacceleration,  MeV. When this energy is lower than the thermal energy of protons, the waves are expected to be dissipated, and the stochastic acceleration of electrons is unlikely in all the range of . For the case with reacceleration, the electrons can be maintained with energies between MeV, depending on (see Table S1). For , the synchrotron cooling is more likely to be important, while for , the inverse Compton cooling is dominant. The critical electron energy, , at which the cooling and reacceleration balance with each other is lower for a higher value of due to the efficient inverse Compton cooling. When the critical energy is higher than the thermal proton temperature, the reacceleration can occur.

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