Revealing the Production Mechanism of High-energy Neutrinos from NGC 1068

The effective optical depth to the Bethe–Heitler pair production process in the disk photon field is estimated to be (e.g., Murase et al. 2020)

$$\begin{eqnarray}&&\begin{array}{l}{f}_{\mathrm{BH}}\approx {\tilde{n}}_{\mathrm{disk}}{\hat{\sigma }}_{\mathrm{BH}}R(c/V)\\ \,\simeq 7.0{\tilde{L}}_{\mathrm{disk},\ 44.7}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)}^{-1}\\ \,\times {\left(\displaystyle \frac{{\varepsilon }_{\mathrm{disk}}}{31.5\,\mathrm{eV}}\right)}^{-1}{\left(\displaystyle \frac{V}{0.1c}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 5 }$$

where ${\hat{\sigma }}_{\mathrm{BH}}\sim 0.8\times {10}^{-30}\,{\mathrm{cm}}^{2}$ and the typical proton energy causing pair production is ${\tilde{\varepsilon }}_{p}^{\mathrm{BH}-\mathrm{disk}}\approx 0.5{\bar{\varepsilon }}_{\mathrm{BH}}{m}_{p}{c}^{2}$/${\varepsilon }_{\mathrm{disk}}\simeq 1.5\times {10}^{5}\,\mathrm{GeV}\,{({\varepsilon }_{\mathrm{disk}}/31.5\,\mathrm{eV})}^{-1}$ and ${\bar{\varepsilon }}_{\mathrm{BH}}\approx 10\,\mathrm{MeV}$. In the photohadronic scenario, TeV neutrinos mostly come from interactions with X-ray photons, and the effective optical depth to the photomeson production process is estimated to be

$$\begin{eqnarray}&&\begin{array}{c}{f}_{p\gamma }\approx {\eta }_{p\gamma }{\hat{\sigma }}_{p\gamma }R(c/V){\tilde{n}}_{X}{\left(\frac{{\varepsilon }_{p}}{{\varepsilon }_{p}^{p\gamma -X}}\right)}^{{{\rm{\Gamma }}}_{\mathrm{cor}}-1}\\ \,\simeq 0.39\,{\eta }_{p\gamma }{\tilde{L}}_{\mathrm{cor},43.3}{\left(\frac{R}{10{R}_{S}}\right)}^{-1}{\left(\frac{{M}_{\mathrm{BH}}}{{10}^{7}{M}_{\odot }}\right)}^{-1}\\ \,\times {\left(\frac{V}{0.1c}\right)}^{-1}{\left(\frac{{\varepsilon }_{p}}{{\varepsilon }_{p}^{p\gamma -X}}\right)}^{{{\rm{\Gamma }}}_{\mathrm{cor}}-1},\end{array}\end{eqnarray} \tag{ 6 }$$

where η_{p γ} = 2/(1 + Γ_cor), ${\hat{\sigma }}_{p\gamma }\sim 0.7\times {10}^{-28}\,{\mathrm{cm}}^{2}$, and ${\varepsilon }_{p}^{p\gamma -{\rm{X}}}\simeq 7.9\times {10}^{4}\,{\rm{GeV}}\,{\left({\varepsilon }_{{\rm{X}}}/2{\rm{keV}}\right)}^{-1}$. We can see that, at ∼10 TeV, proton energy losses via the Bethe–Heitler pair production process are more significant than the photomeson production (as shown in Murase 2022), which is consistent with Figure 1.

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The optical depths to electron–positron and muon–antimuon pair production from the two-photon annihilation process are shown in Figure 1. For the electron–positron pair production, the threshold gamma-ray energy is ${\tilde{\varepsilon }}_{\gamma }^{{e}^{+}{e}^{-}-{\rm{X}}}\approx {m}_{e}^{2}{c}^{4}/{\varepsilon }_{{\rm{X}}}\simeq 0.13\,{\rm{GeV}}\,{\left({\varepsilon }_{{\rm{X}}}/2{\rm{keV}}\right)}^{-1}$, and the two-photon annihilation optical depth to electron–positron pair production in the GeV gamma-ray range is (Murase 2022)

$$\begin{eqnarray}&&\begin{array}{c}{\tau }_{\gamma \gamma \to {e}^{+}{e}^{-}}\approx {\eta }_{\gamma \gamma }{\sigma }_{{\rm{T}}}R{\tilde{n}}_{{\rm{X}}}{\left(\displaystyle \frac{{\varepsilon }_{\gamma }}{{\tilde{\varepsilon }}_{\gamma }^{{e}^{+}{e}^{-}-{\rm{X}}}}\right)}^{{{\rm{\Gamma }}}_{{\rm{cor}}}-1}\\ \,\simeq 44\,{\tilde{L}}_{{\rm{cor}},43.3}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{{\rm{BH}}}}{{10}^{7}{M}_{\odot }}\right)}^{-1}\\ \,\times {\left(\displaystyle \frac{{\varepsilon }_{\gamma }}{{\tilde{\varepsilon }}_{\gamma }^{{e}^{+}{e}^{-}-{\rm{X}}}}\right)}^{{{\rm{\Gamma }}}_{{\rm{cor}}}-1},\end{array}\end{eqnarray} \tag{ 7 }$$

where η_{γ γ} ≈ 0.12 is the coefficient (Svensson 1987) and σ_T ≈ 6.65 × 10⁻²⁵ cm² is the Thomson cross section. The two-photon annihilation optical depth to muon–antimuon pair production is

$$\begin{eqnarray}&&\begin{array}{l}{\tau }_{\gamma \gamma \to {\mu }^{+}{\mu }^{-}}\approx {\eta }_{\gamma \gamma }{\left({m}_{e}/{m}_{\mu }\right)}^{2}{\sigma }_{{\rm{T}}}R{\tilde{n}}_{{\rm{X}}}{\left(\displaystyle \frac{{\varepsilon }_{\gamma }}{{\tilde{\varepsilon }}_{\gamma }^{{\mu }^{+}{\mu }^{-}-X}}\right)}^{{{\rm{\Gamma }}}_{\mathrm{cor}}-1}\\ \quad \simeq 1.0\times {10}^{-3}\,{\tilde{L}}_{\mathrm{cor},\ 43.3}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)}^{-1}\\ \quad \times {\left(\displaystyle \frac{{\varepsilon }_{\gamma }}{{\tilde{\varepsilon }}_{\gamma }^{{\mu }^{+}{\mu }^{-}-X}}\right)}^{{{\rm{\Gamma }}}_{\mathrm{cor}}-1},\end{array}\end{eqnarray} \tag{ 8 }$$

where ${\tilde{\varepsilon }}_{\gamma }^{{\mu }^{+}{\mu }^{-}-{\rm{X}}}\approx {m}_{\mu }^{2}{c}^{4}/{\varepsilon }_{{\rm{X}}}\simeq 5.6\times {10}^{3}\,{\rm{GeV}}\,{\left({\varepsilon }_{{\rm{X}}}/2{\rm{keV}}\right)}^{-1}$. The above expressions for ${\tau }_{\gamma \gamma \to {e}^{+}{e}^{-}}$ and ${\tau }_{\gamma \gamma \to {\mu }^{+}{\mu }^{-}}$ are consistent with the numerical results shown in Figure 1. We can also see that the optical depth to the electron–positron pair production at ∼10 TeV is ∼10⁶ larger than that to the muon–antimuon pair production.

We show cascaded photon and all-flavor neutrino spectra in Figure 2 for different emission radii with R = 0.5R_S, 3R_S, 10R_S, and 30R_S, respectively. The neutrino spectra are shown with dashed lines, while the cascade photon spectra are shown with solid lines. We also depict the intrinsic flux of X-rays with dotted lines. Note that the cascade flux in the X-ray range is lower than the intrinsic X-ray flux from coronae, as shown in previous works (Murase et al. 2020; Murase 2022), so that constraints from X-ray observations are much weaker than those from gamma-ray observations. The AGN corona model itself predicts that the neutrino luminosity is approximately proportional to the X-ray luminosity (Murase & Waxman 2016; Murase et al. 2020), and the cosmic-ray luminosity is expected to be lower than the X-ray luminosity (Murase et al. 2016).

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The cascaded photon spectra vary with emission radii. For larger emission radii, e.g., R = 30R_S, the cascade photon spectra overshoot the Fermi-LAT data obtained from sub-GeV to GeV energies (Ajello et al. 2023), especially for ξ_B = 0.1 and ξ_B = 0.01. While our numerical results confirm those of Murase (2022), the gamma-ray data adopted in this work (Ajello et al. 2023) extend to lower and higher energies, which give tighter constraints on R. The synchrotron cascade with ξ_B ≳ 0.1 allows more parameter space for R, because more gamma rays appear at sub-GeV or lower energies.

Then, we obtain gamma-ray constraints on emission radii by requiring the cascade photon spectra from our simulations not to overshoot the measured gamma-ray data, especially the lowest-energy Fermi-LAT upper limit at the 95% confidence level. Our results are represented by the red and orange curves in Figure 3, which are calculated for ξ_B = 1.0 and ξ_B = 0.01, respectively. We find that the size of the emission region is constrained to R ≲ 15R_S for ξ_B = 1.0 and to R ≲ 3R_S for ξ_B = 0.01. The constraints are tighter for softer spectral indices of the injected proton spectrum. This is because a softer cosmic-ray spectrum results in more Bethe–Heitler pair production processes that give rise to larger cascaded photon fluxes.

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Our constraints on R for ξ_B ≲ 0.01 imply that the neutrino emission region is unlikely to be high-β plasma if NGC 1068 is a Schwarzschild black hole. This is because the upper limits on R are smaller than the innermost stable circular orbit (ISCO) for the nonrotating case. While it does not exclude ISCOs for a Kerr black hole with prograde orbits, given that NGC 1068 does not have a strong jet, an extreme Kerr black hole might also be disfavored, although details depend on the magnetic flux configuration.

In Figure 3, we show the required cosmic-ray proton luminosity as a function of R and s_CR. We find that the typical range of the injected cosmic-ray luminosity for our chosen parameter space is L_CR ∼ (3 − 30) × 10⁴³ erg s⁻¹, which is less than the bolometric and Eddington luminosities of NGC 1068. From Equation (6), we see that the efficiency for neutrino production is proportional to R⁻¹, which means that smaller radii are preferred for efficient neutrino production. Correspondingly, the required minimum cosmic-ray luminosity may decrease with emission radii.

On the other hand, harder spectral indices could alleviate the required cosmic-ray power. While the IceCube data provide the best-fit spectrum with a spectral index of 3.2 (Abbasi et al. 2022), hard spectra with a cutoff may explain the data (Murase 2022). For s_CR ≲ 2, protons with energy around ${\varepsilon }_{p}^{\max }$ are dominant. Therefore, the L_CR requirement changes drastically upon going from s_CR = 1 to s_CR = 2, as it mostly increases the density of lower-energy protons that do not contribute to neutrino production. For s_CR ≲ 2, protons with energy around ${\varepsilon }_{p}^{\min }$ are dominant. Therefore, further increasing the softness does not affect the required L_CR much.

The minimum cosmic-ray proton power is estimated to be

$$\begin{eqnarray}&&{L}_{\mathrm{CR}}\gtrsim (3-30)\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 11 }$$

which can be below the bolometric and Eddington luminosities, so we conclude that the photohadronic scenario is viable as a mechanism for neutrino emission from NGC 1068.

In Figure 4, we show cascaded photon and all-flavor neutrino spectra in the leptonic scenario (solid and dashed lines, respectively). The injected gamma-ray spectra are shown using dotted lines. The injected gamma rays interact with the disk and coronal photons, and some of them produce muon–antimuon pairs. Most of the gamma rays are cascaded down to MeV or lower energies and overwhelm the intrinsic coronal spectrum.

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We also examine cascade constraints as in the photohadronc scenario. In the leptonic scenario, we find that the cascaded photon spectra violate gamma-ray observations including the Fermi-LAT data (Ajello et al. 2023) for almost the entire region of parameter space that we test. In addition, the cascaded X-ray flux is much larger than the intrinsic X-ray flux inferred by NuSTAR observations (represented by the dotted–dashed line). We do not find a viable parameter space for the leptonic scenario to explain the observed IceCube data (Abbasi et al. 2022) without violating the intrinsic X-ray or gamma-ray data.

This situation can be understood as follows. At ε_γ ≳ a few GeV, the two-photon annihilation optical depth for electron–positron pair production is expected to be dominated by the interaction with disk photons or other optical and ultraviolet photons if they exist. The optical depth at ≳10−100 GeV is naively estimated by the following expression:

$$\begin{eqnarray}&&\begin{array}{l}{\tau }_{\gamma \gamma \to {e}^{+}{e}^{-}}\simeq 1.1\times {10}^{6}\,{\left(\displaystyle \frac{{\varepsilon }_{\gamma }^{{e}^{+}{e}^{-}-\mathrm{dm}}}{{\tilde{\varepsilon }}_{\gamma }^{{e}^{+}{e}^{-}-\mathrm{disk}}}\right)}^{-1/3}{\left(\displaystyle \frac{{\varepsilon }_{\gamma }}{{\tilde{\varepsilon }}_{\gamma }^{{e}^{+}{e}^{-}-\mathrm{dm}}}\right)}^{-1}\\ \quad \times {\tilde{L}}_{\mathrm{disk},\ 44.7}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)}^{-1}{{\rm{\Lambda }}}_{1},\end{array}\end{eqnarray} \tag{ 14 }$$

where ${\tilde{\varepsilon }}_{\gamma }^{{e}^{+}{e}^{-}-\mathrm{disk}}\approx {m}_{e}^{2}{c}^{4}/{\varepsilon }_{\mathrm{disk}}\simeq 8.3\,\mathrm{GeV}\,{({\varepsilon }_{\mathrm{disk}}/31.5\,\mathrm{eV})}^{-1}$, ${\tilde{\varepsilon }}_{\gamma }^{{e}^{+}{e}^{-}-\mathrm{dm}}\approx {m}_{e}^{2}{c}^{4}/{\varepsilon }_{\mathrm{dm}}\simeq 56\,\mathrm{GeV}\,{({\varepsilon }_{\mathrm{dm}}/4.7\,\mathrm{eV})}^{-1}$, and Λ is the logarithmic factor. In the leptonic scenario, neutrinos in the TeV range are mostly produced via interactions with coronal photons or other high-energy photons originating from electromagnetic cascades.

For given ε_X and ε_disk, with Γ_cor = 2, the ratio of the above two optical depths at sufficiently high energies is estimated by

$$\begin{eqnarray}&&\displaystyle \frac{{\tau }_{\gamma \gamma \to {e}^{+}{e}^{-}}}{{\tau }_{\gamma \gamma \to {\mu }^{+}{\mu }^{-}}}\sim 2\times {10}^{6}\,\left(\displaystyle \frac{{\tilde{L}}_{\mathrm{disk},\ 44.7}}{{\tilde{L}}_{\mathrm{cor},\ 43.3}}\right){{\rm{\Lambda }}}_{1},\end{eqnarray} \tag{ 15 }$$

which roughly agrees with the numerical result shown in Figure 1. This ratio is also consistent with that of the gamma-ray flux to the neutrino flux shown in Figure 4. For comparison, Hooper & Plant (2023) argued that the ratio is ∼0.1−0.01 for the optical depth at around 10 TeV assuming a blackbody spectrum with a temperature of 1 keV. For a realistic X-ray spectrum that is described by a power law, as shown in this work, the ratio is much larger, which is the reason why the contribution of neutrinos from muon–antimuon production was not usually shown in the previous calculations (Murase 2022).

As seen in Figure 5, we calculate the required power of injected gamma rays, which is L_γ ∼ (1−4) × 10⁴⁷ erg s⁻¹, to explain the observed neutrino flux in the leptonic scenario. These values are about two orders of magnitude greater than the bolometric luminosity. Our results on the required power of injected gamma rays are also about four orders larger than the required electron luminosity derived in Hooper & Plant (2023). The difference is that we adopt a realistic coronal photon field with a differential energy density of ${\tilde{U}}_{\mathrm{cor}}\simeq 5.9\times {10}^{4}\,\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\,{(R/10{R}_{{\rm{S}}})}^{-2}{\tilde{L}}_{\mathrm{cor},43.3}$, which is many orders smaller than the energy density of the blackbody photon field with a temperature of k_B T = 1 keV.

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Compared to the required cosmic-ray power in the photohadronic scenario, the required power of injected gamma rays in the leptonic scenario is insensitive to the size of the emission region, as shown in Figure 3. The reason is that, for environments that are optically thick to electron–positron pair production, the neutrino flux is determined by the ratio between muon–antimuon and electron–positron pair production. Analytically, considering ${\tau }_{\gamma \gamma \to {e}^{+}{e}^{-}}\gg 1$, the all-flavor neutrino flux is estimated by

$$\begin{eqnarray}&&{\varepsilon }_{\nu }\displaystyle \frac{{{dL}}_{{\varepsilon }_{\nu }}}{d{\varepsilon }_{\nu }}\approx \displaystyle \frac{2}{3}\displaystyle \frac{{\tau }_{\gamma \gamma \to {\mu }^{+}{\mu }^{-}}}{{\tau }_{\gamma \gamma \to {e}^{+}{e}^{-}}}{\varepsilon }_{\gamma }\displaystyle \frac{{{dL}}_{{\varepsilon }_{\gamma }}}{d{\varepsilon }_{\gamma }}.\end{eqnarray} \tag{ 16 }$$

Given that the ratio is ∼10⁵−10⁶, the minimum gamma-ray power is

$$\begin{eqnarray}&&{L}_{\gamma }\sim ({10}^{5}-{10}^{6}){L}_{\nu }\sim ({10}^{47}-{10}^{48})\,\mathrm{erg}\,{{\rm{s}}}^{-1}\gg {L}_{\mathrm{bol}}.\end{eqnarray} \tag{ 17 }$$

The required minimum gamma-ray luminosity is much larger than the Eddington luminosity, so that we conclude that the muon–antimuon pair production process is excluded as a mechanism for high-energy neutrinos from NGC 1068.

In Figure 6, we show cascaded photon and all-flavor neutrino spectra in the beta-decay scenario (solid and dashed lines, respectively). The injected cosmic-ray spectra are shown with dotted lines. The blue lines consider the photodisintegration process, the photomeson production process, the Bethe–Heitler pair production process, and neutron decay. For comparison purposes, we also show the case without considering the photomeson production process with red lines.

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The effective photodisintegration cross section for A ≤ 4 (including inelasticity with 1/A) is given by ${\hat{\sigma }}_{\mathrm{dis}}\approx 4.3\times {10}^{-28}\,{\mathrm{cm}}^{2}\,{(A/4)}^{-2.433}$ (Karakula et al. 1994; Murase & Beacom 2010a). Then, the effective optical depth to photodisintegration for helium nuclei is

$$\begin{eqnarray}&&\begin{array}{l}{f}_{\mathrm{dis}}\approx {\tilde{n}}_{\mathrm{disk}}{\hat{\sigma }}_{\mathrm{dis}}R(c/V)\\ \quad \simeq 3.8\times {10}^{-4}\,{\tilde{L}}_{\mathrm{disk},\ 44.7}{\left(\displaystyle \frac{R}{{10}^{8}{R}_{{\rm{S}}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)}^{-1}\\ \quad \times {\left(\displaystyle \frac{{\varepsilon }_{\mathrm{disk}}}{31.5\,\mathrm{eV}}\right)}^{-1}{\left(\displaystyle \frac{V}{0.1c}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 20 }$$

which can be almost unity at sufficiently small emission radii. Note that a fraction of energy carried by nucleons is only 1/A per collision, which should be considered for the calculations of neutrinos and gamma rays (e.g., Murase & Beacom 2010b; Zhang & Murase 2023). In addition, the average energy of neutrinos from neutron decay is 0.48 MeV in the neutron rest frame (e.g., Murase & Beacom 2010b), so the energy fraction carried by neutrinos is 0.48 MeV/(m_n c²) ∼ 1/(2000). The average kinetic energy of beta-decay electrons is 0.3 MeV (Zhang & Murase 2023).

One important point is that the Bethe–Heitler pair production process is unavoidable. This is because the threshold for electron–positron production, 2m_e c², is lower than that for nuclear disintegration. A similar efficiency issue exists for de-excitation gamma rays, although the energy range of de-excitation gamma rays is different from that of cascade gamma rays from the Bethe–Heitler pairs and beta-decay electrons (Aharonian & Taylor 2010; Murase & Beacom 2010a; Zhang & Murase 2023). When disk photons are main targets, the ratio of the Bethe–Heitler contribution to the beta-decay contribution is given by

$$\begin{eqnarray}&&\displaystyle \frac{6000{f}_{\mathrm{BH}}({Z}^{2}/A)}{{f}_{\mathrm{dis}}}\sim 10,\end{eqnarray} \tag{ 21 }$$

where Z²/A = 1 for helium and f_BH in Equation (5) is defined for protons. While we find that the cascaded photon spectrum only from beta-decay electrons overshoots the gamma-ray data including the Fermi-LAT data and MAGIC upper limits, this conclusion is robust in the presence of electron–positron pairs from the Bethe–Heitler process.

Note that the electromagnetic cascade is likely to be dominated by inverse Compton emission. The dust field is dominant over CMB and EBL for sufficiently small emission radii that assure efficient photodisintegration, at which the magnetic field is weaker than the equipartition magnetic field,

$$\begin{eqnarray}&&\begin{array}{l}{B}_{\mathrm{eq}}=\sqrt{\displaystyle \frac{8\pi {L}_{\mathrm{DT}}}{4\pi {R}^{2}c}}\\ \quad \simeq 0.24\,{\rm{G}}\,{(R/{R}_{\mathrm{DT}})}^{-1}(0.1\,\mathrm{pc}/{R}_{\mathrm{DT}}){L}_{\mathrm{DT},43.9}^{1/2}.\end{array}\end{eqnarray} \tag{ 22 }$$

We have also checked the results with a higher value of the magnetic field strength with B = 300 μG, and our conclusions remain unchanged. Even though the results remain similar, the cascade photon flux may also be enhanced because of the contribution from synchrotron emission.

Furthermore, the primary gamma-ray component from π⁰ decay must exist once the photomeson production is consistently included. The threshold energy for the pion production is higher, but due to the higher efficiency (see also Murase & Beacom 2010b, 2010a), we find that the contribution from the photomeson production is not negligible and violates the MAGIC upper limits independent of magnetic fields.

The minimum cosmic-ray luminosity required from the IceCube neutrino data is shown in Figure 7. We find that the cosmic-ray helium luminosity is L_CR ≳ 3 × 10⁴⁶ erg s⁻¹ for the whole parameter region. The required cosmic-ray luminosity almost linearly increases with the emission radius, while being less sensitive to the spectral index.

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Analytically, the differential muon neutrino luminosity after neutrino mixing is written as

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{\nu }\displaystyle \frac{{{dL}}_{{\varepsilon }_{{\nu }_{\mu }}}}{d{\varepsilon }_{\nu }}\approx \displaystyle \frac{2}{9}\displaystyle \frac{1}{2000}{\varepsilon }_{n}\displaystyle \frac{{{dL}}_{{\varepsilon }_{n}}}{d{\varepsilon }_{n}}\\ \quad \approx \displaystyle \frac{2}{9}\displaystyle \frac{1}{2000}\displaystyle \frac{1}{2}\min [1,{f}_{\mathrm{dis}}]{\varepsilon }_{A}\displaystyle \frac{{{dL}}_{{\varepsilon }_{A}}}{d{\varepsilon }_{A}},\end{array}\end{eqnarray} \tag{ 23 }$$

where ${\varepsilon }_{n}({{dL}}_{{\varepsilon }_{n}}/d{\varepsilon }_{n})$ is the differential neutron luminosity. Here we have considered that for the tribimaximal mixing of neutrinos from beta decay the flavor ratio on Earth becomes ν_e :ν_μ :ν_τ ≈ 5:2:2. This implies that, no matter what the effective optical depth is, the cosmic-ray power is larger than the muon neutrino luminosity by a factor of ∼10⁴ and we have

$$\begin{eqnarray}&&{L}_{\mathrm{CR}}\gtrsim {10}^{4}{L}_{\nu }\sim {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\gg {L}_{\mathrm{bol}},\end{eqnarray} \tag{ 24 }$$

which agrees with our numerical simulations shown in Figure 7. We also note that this conclusion about energetics constraints is insensitive to magnetic fields.

We have found that larger values of ξ_B more easily satisfy the Fermi-LAT gamma-ray constraints because of synchrotron cascades (Murase 2022). If the neutrino emission region coincides with the X-ray emission region, i.e., the hot corona, the ion plasma β is

$$\begin{eqnarray}&&\beta =\frac{8\pi {n}_{p}{k}_{{\rm{B}}}{T}_{p}}{{B}^{2}}\approx \frac{{\tau }_{T}{{GM}}_{\mathrm{BH}}{m}_{p}}{\sqrt{3}{\zeta }_{e}{\sigma }_{{\rm{T}}}{R}^{2}{U}_{\gamma }}{\xi }_{B}^{-1}\approx \left(\frac{{\tau }_{T}}{\sqrt{3}{\zeta }_{e}{\lambda }_{\mathrm{Edd}}}\right){\xi }_{B}^{-1}\end{eqnarray} \tag{ 25 }$$

where T_p ≈ GM_BH m_p /(3k_B R) is the virial temperature, n_p ≈ τ_T /(ζ_e σ_T H) is the nucleon density of the coronal plasma, H is the coronal scale height, ζ_e is the possible pair multiplication factor, and

$$\begin{eqnarray}&&\displaystyle \frac{{\tau }_{T}{{GM}}_{\mathrm{BH}}{m}_{p}}{\sqrt{3}{\zeta }_{e}{\sigma }_{{\rm{T}}}{R}^{2}{U}_{\gamma }}\approx \displaystyle \frac{4\pi {\tau }_{T}{{GM}}_{\mathrm{BH}}{m}_{p}c}{\sqrt{3}{\zeta }_{e}{\sigma }_{{\rm{T}}}{L}_{\mathrm{bol}}}=\displaystyle \frac{{\tau }_{T}}{\sqrt{3}{\zeta }_{e}{\lambda }_{\mathrm{Edd}}},\end{eqnarray} \tag{ 26 }$$

which is ∼0.1−1 for ion–electron plasma because of τ_T ∼ 0.1−1 for X-ray coronae, and $\sqrt{3}{\lambda }_{\mathrm{Edd}}\sim 1$ for NGC 1068. Thus, high-β plasma with β ≳ 10 means ξ_B ≲ 0.01–0.1, in which inverse Compton cascades are expected, in which R ≲ 3R_S should be satisfied. The synchrotron dominance for electromagnetic cascades occurs for ξ_B ≳ 0.1 (Murase et al. 2020), implying that β ≲ 1−10 is necessary to allow R ∼ 10R_S.

This favors the presence of low-β plasma if hot coronae are responsible for high-energy neutrino emission. ⁵ Together with our gamma-ray constraints on the emission radius, the most promising particle acceleration mechanisms would be magnetic dissipation, including magnetic reconnections (Kheirandish et al. 2021; Mbarek et al. 2024; Fiorillo et al. 2024) and associated stochastic acceleration in turbulence (Murase et al. 2020; Kimura et al. 2021; Eichmann et al. 2022) although some shock acceleration could be associated with magnetic reconnections (Murase 2022).

If the corona is not magnetically powered with high β, the emission radius has to be even more compact. Although we do not exclude the entire parameter space, at least we exclude shock models with R ∼ 10R_S−30R_S and weak magnetic fields, which are motivated by Atacama Large Millimeter/submillimeter Array (ALMA) data at the submillimeter band (Inoue et al. 2020). Note that our constraints are even more severe if the contribution of primary nonthermal electrons is included. On the other hand, we also note that this conclusion does not apply if the neutrino emission region is different from the hot corona responsible for X-rays. For example, failed outflows (Alvarez-Muniz & Mészáros 2004; Inoue et al. 2023) and the base of weak jets (Pe'er et al. 2009; Murase 2022) may be viable.

NGC 1068 is known to have a low-power jet with a luminosity of L_j ∼ 10⁴³ erg s⁻¹, as indicated by radio observations of e-Merlin and the Very Large Array (Mutie et al. 2024), as well as ALMA (Michiyama et al. 2022), through the empirical relationship (Cavagnolo et al. 2010). Knots within these jets may be responsible for gamma-ray emission in the range of ∼0.1−1 GeV (Salvatore et al. 2024). The jet efficiency of jet-quiet AGNs is small, which is L_j /L_bol ∼ 0.01−0.1 ≪ λ_Edd. It is believed that the efficiency of the jet launched via the Blandford–Znajek (BZ) mechanism (Blandford & Znajek 1977) for magnetically arrested disks is high. Based on this, Inoue et al. (2024) speculated that coronae of Seyferts such as NGC 1068 have β ≳ 10−100. However, such high-β coronae are strongly constrained by multimessenger observations, as shown in this work. Moreover, observationally, a relatively low jet efficiency of ∼0.01−0.1 has been inferred even for objects with powerful jets that are presumably launched by the BZ mechanism. The examples include Fanaroff–Riley II jets (Fan & Wu 2019), Fanaroff–Riley 0 jets (Khatiya et al. 2023), and gamma-ray burst jets (e.g., Suwa & Ioka 2011), which could be caused by the time dependence of the magnetic flux and configurations. Low-β coronae without powerful jets have also been inferred from numerical simulations (e.g., Jiang et al. 2019), and the accretion disk may also be strongly magnetized (Hopkins et al. 2024a, 2024b). Recent particle-in-cell (e.g., Grošelj et al. 2024) and magnetohydrodynamic simulations (e.g., Bambic et al. 2023) have also supported that magnetic dissipation in magnetically powered coronae is the most promising mechanism for electron heating.

Our results support low-β plasma in the coronal region or at the base of possible jets as an acceleration site and disfavor accretion shocks at least for R ≳ 10R_S. Among various possible acceleration processes, magnetic reconnections (e.g., Hoshino 2013, 2015; Guo et al. 2016; Werner et al. 2017; Ball et al. 2018) and stochastic acceleration (e.g., Lynn et al. 2014; Kimura et al. 2016, 2019; Comisso & Sironi 2018; Zhdankin et al. 2019; Lemoine & Malkov 2020; Wong et al. 2020; Lemoine et al. 2024) are promising.

The coronal plasma can be collisionless for ions, although it may not be for electrons, in which stochastic acceleration may operate in the presence of magnetic turbulence. Whether plasma is low β or high β, the timescale is given by

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{acc}}\approx {\eta }_{\mathrm{tur}}{\left(\displaystyle \frac{c}{{V}_{{\rm{A}}}}\right)}^{2}\displaystyle \frac{H}{c}{\left(\displaystyle \frac{{\varepsilon }_{p}}{e{\rm{B}}{\rm{H}}}\right)}^{2-q}\\ \quad \simeq 1.8\times {10}^{4}\,{\rm{s}}\,{\eta }_{\mathrm{tur},1}{\beta }_{-1}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{2}\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)\\ \quad \times {\left(\displaystyle \frac{{\varepsilon }_{p}}{e{\rm{B}}{\rm{H}}}\right)}^{2-q},\end{array}\end{eqnarray} \tag{ 27 }$$

where ${\eta }_{\mathrm{tur}}^{}$ is a parameter representing the efficiency of stochastic acceleration, q ∼ 1.5−2 describes the energy dependence of the diffusion coefficient, and ${V}_{{\rm{A}}}=B/\sqrt{4\pi {m}_{p}{n}_{p}}$ is the Alfvén velocity.

For cosmic-ray protons with energy around 100−1000 TeV, the dominant cooling timescale is the Bethe–Heitler pair production cooling, which is

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{BH}}\approx \displaystyle \frac{1}{{\tilde{n}}_{\mathrm{disk}}{\hat{\sigma }}_{\mathrm{BH}}c}\\ \quad \simeq 1.4\times {10}^{3}\,\,{\rm{s}}\,{\tilde{L}}_{\mathrm{disk},\ 44.7}^{-1}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{2}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)}^{2}\\ \quad \times \left(\displaystyle \frac{{\varepsilon }_{\mathrm{disk}}}{31.5\,\mathrm{eV}}\right),\end{array}\end{eqnarray} \tag{ 28 }$$

and the balance between acceleration and cooling timescales, t_acc ∼ t_BH, may give ${\varepsilon }_{p}^{\max }\sim { \mathcal O }(100)$ TeV.

Particles may also be accelerated by magnetic reconnections. The magnetization parameter is

$$\begin{eqnarray}&&\begin{array}{l}\sigma =\displaystyle \frac{{B}^{2}}{4\pi {n}_{p}{m}_{p}{c}^{2}}={\beta }^{-1}\left(\displaystyle \frac{2{k}_{{\rm{B}}}{T}_{p}}{{m}_{p}{c}^{2}}\right)={\beta }^{-1}\left(\displaystyle \frac{{R}_{{\rm{S}}}}{3R}\right)\\ \quad =\displaystyle \frac{2{\xi }_{B}{\zeta }_{e}{L}_{\mathrm{bol}}{\sigma }_{{\rm{T}}}}{\sqrt{3}{\tau }_{T}{m}_{p}{c}^{2}R}\\ \quad \simeq 4.7\times {10}^{-2}\displaystyle \frac{{\xi }_{B}{\zeta }_{e}}{{\tau }_{T}}{L}_{\mathrm{bol},\ 45}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{-1}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 29 }$$

Particle acceleration via magnetic reconnections is efficient and leads to hard spectra for σ > 1. For R ≲ 30R_S, this requires β ≲ 0.01−1, so very low β plasma environments are necessary, and extreme magnetization parameters such as σ ≳ 10⁴ have been considered (Mbarek et al. 2024; Fiorillo et al. 2024). The acceleration timescale is

$$\begin{eqnarray}&&\begin{array}{l}{t}_{\mathrm{acc}}\approx \eta \displaystyle \frac{{\varepsilon }_{p}}{{eBc}}\\ \quad \simeq (0.013\,{\rm{s}}){\eta }_{1}\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right){\xi }_{B}^{-1/2}{L}_{\mathrm{bol},\ 45}^{-1/2}\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)\\ \quad \times \left(\displaystyle \frac{{\varepsilon }_{p}}{100\,\mathrm{TeV}}\right),\end{array}\end{eqnarray} \tag{ 30 }$$

where η is a coefficient describing the efficiency of acceleration via magnetic reconnections. At higher energies, the dominant cooling process becomes the photomeson production, whose timescale is given by

$$\begin{eqnarray}&&\begin{array}{l}{t}_{p\gamma }\approx \displaystyle \frac{1}{{\tilde{n}}_{\mathrm{disk}}{\hat{\sigma }}_{p\gamma }c}\\ \quad \simeq 16\,\,{\rm{s}}\,{\tilde{L}}_{\mathrm{disk},\ 44.7}^{-1}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{2}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{7}\,{M}_{\odot }}\right)}^{2}\\ \quad \times \left(\displaystyle \frac{{\varepsilon }_{\mathrm{disk}}}{31.5\,\mathrm{eV}}\right),\end{array}\end{eqnarray} \tag{ 31 }$$

for cosmic-ray protons at sufficiently high energies. The condition t_acc ∼ t_{p γ} gives a maximum energy of ${\varepsilon }_{p}^{\max }\sim { \mathcal O }(10)$ PeV.

In the leptonic scenario, the acceleration timescale of electrons is

$$\begin{eqnarray}&&{t}_{\mathrm{acc}}\approx \eta \displaystyle \frac{{\varepsilon }_{e}}{{eBc}},\end{eqnarray} \tag{ 32 }$$

and the balance between the synchrotron and inverse Compton cooling gives the electron maximum energy,

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{e}^{\max }\approx {\left(\displaystyle \frac{6\pi e}{{\sigma }_{{\rm{T}}}B\eta (1+Y)}\right)}^{1/2}{m}_{e}{c}^{2}\\ \quad \simeq 0.64\,\,\mathrm{TeV}\,{[\eta (1+Y)]}^{-1/2}{\left(\displaystyle \frac{R}{10{R}_{{\rm{S}}}}\right)}^{1/2}{\xi }_{B}^{-1/4}\\ {L}_{\mathrm{bol},\ 45}^{-1/4},\end{array}\end{eqnarray} \tag{ 33 }$$

where Y denotes the inverse Compton Y parameter. The maximum energy required for the leptonic scenario is realized only if the magnetic field is sufficiently weak.

In the strongly magnetized plasma, the inverse Compton emission is highly suppressed. Note that the maximum synchrotron energy is limited as

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{syn}\gamma }^{\max }\approx \displaystyle \frac{3{heB}}{4\pi {m}_{e}c}{\left(\displaystyle \frac{{\varepsilon }_{e}^{\max }}{{m}_{e}{c}^{2}}\right)}^{2}\simeq 240\,\mathrm{MeV}{[\eta (1+{\rm{Y}})]}^{-1},\end{eqnarray} \tag{ 34 }$$

which is the so-called synchrotron burnoff limit (Guilbert et al. 1983; Aharonian et al. 2002; Zhang 2018), independent of the magnetic field strength. This energy is much below 10 TeV, being insufficient for the leptonic scenario.

In this work, we adopt V = 0.1c as the default velocity of the system. We note that the velocity can be smaller, which affects effective optical depths through t_dyn. For example, the infall velocity is estimated to be

$$\begin{eqnarray}&&{V}_{\mathrm{fall}}=\alpha {V}_{{\rm{K}}}\sim 0.01c{\alpha }_{-1}{\left(R/{R}_{{\rm{S}}}\right)}^{-1/2},\end{eqnarray} \tag{ 35 }$$

where α is the viscous parameter in the accreting flow and ${V}_{{\rm{K}}}=\sqrt{{{GM}}_{\mathrm{BH}}/R}=c/\sqrt{2(R/{R}_{{\rm{S}}})}$ is the Keplerian velocity. The effective optical depth to photomeson production is larger with smaller velocities, in which the cosmic-ray luminosity requirement can be relaxed, especially for large values of R. For the photohadronic scenario, in the limit that the system is calorimetric, the cosmic-ray luminosity is required to be ${L}_{\mathrm{CR}}\gtrsim {10}^{43}\,{f}_{\sup }^{-1}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, which is consistent with the conclusion by Murase (2022), although our constraint is tighter, due to the suppression factor by the Bethe–Heitler process. Similarly, for the beta-decay scenario, the efficiency is higher, so that the power requirement is relaxed for large radii, but it cannot be below ∼10⁴⁶ erg s⁻¹. On the other hand, the gamma-ray power requirement in the leptonic scenario is not affected because ${\tau }_{\gamma \gamma \to {e}^{+}{e}^{-}}\gg 1$ is satisfied.

Although we primarily focus on p γ scenarios, one may consider pp scenarios, where neutrinos are mostly produced via inelastic collisions between cosmic rays and plasma gas (Inoue et al. 2020; Murase et al. 2020; Eichmann et al. 2022). The effective optical depth for inelastic pp cooling rate is estimated to be

$$\begin{eqnarray}&&\begin{array}{l}{f}_{{pp}}\approx {n}_{p}{\hat{\sigma }}_{{pp}}R(c/V)\\ \quad \simeq 0.26\,({\tau }_{T}/0.5){\left(\displaystyle \frac{V}{0.1c}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 36 }$$

where ${\hat{\sigma }}_{{pp}}\sim 2\times {10}^{-26}\,{\mathrm{cm}}^{2}$ is the effective pp cross section including the inelasticity, n_p ≈ τ_T /(ζ_e σ_T H) is the nucleon density in the corona region, and τ_T ∼ 0.1−1. Based on the comparison with Equation (6), we see that the pp process can be more important than the p γ process at large emission radii. However, our conclusions for ξ_B ≲ 0.1 would remain the same because cascades from photohadronic processes are more relevant at small emission radii, while R = 30R_S is allowed for ξ_B ∼ 1 (Murase 2022; Ajello et al. 2023).

Finally, we comment on the flavor ratio. In the standard hadronic scenarios, the approximate flavor ratio observed on Earth is expected to be ν_e :ν_μ :ν_τ ≈ 1:1:1, which is assumed in the IceCube data shown in Figure 2. However, the flavor ratio varies among different mechanisms, and we have

$$\begin{eqnarray}&&\begin{array}{l}{\nu }_{e}:{\nu }_{\mu }:{\nu }_{\tau }\approx 1:1:1\,\,\,(\mathrm{hadronic})\\ {\nu }_{e}:{\nu }_{\mu }:{\nu }_{\tau }\approx 14:11:11\,\,\,(\mathrm{leptonic})\\ {\nu }_{e}:{\nu }_{\mu }:{\nu }_{\tau }\approx 5:2:2\,\,\,(\mathrm{beta}\,\mathrm{decay}).\end{array}\end{eqnarray} \tag{ 37 }$$

Correspondingly, in Figures 4 and 6, different flavor ratios are assumed when the IceCube data are depicted. The determination of neutrino flavors with future observations will also be relevant as an independent method to discriminate among different neutrino production mechanisms (e.g., Beacom et al. 2003; Shoemaker & Murase 2016; Bustamante & Ahlers 2019). The IceCube Collaboration has analyzed the flavor composition using all-sky neutrino data (Aartsen et al. 2015a, 2015b; Lad & Cowen 2023). It has been proposed that AGNs like NGC 1068 can account for the all-sky neutrino flux in the 10−100 TeV range (Murase et al. 2020), and the all-sky neutrino data can also be used to discriminate among different production mechanisms. In particular, the beta-decay scenario has already been ruled out as the dominant neutrino production mechanism (Bustamante & Ahlers 2019).