Cascading Constraints from Neutrino-emitting Blazars: The Case of TXS 0506+056

The neutrino flux from the 2014/15 neutrino flare of TXS 0506+056 was best described by a spectrum between 32 TeV and 3.6 PeV of the form Φ_ν(E_ν) = Φ₀ ${E}_{14}^{-2.1}$ with Φ₀ = 2.2 × 10⁻¹⁵ cm⁻² s⁻¹ TeV⁻¹ and ${E}_{14}\equiv {E}_{\nu }/(100\,\mathrm{TeV})$ (Aartsen et al. 2018b). For a redshift of TXS 0506+056 as measured by Ajello et al. (2014) and Paiano et al. (2018), z = 0.3365, which corresponds to a luminosity distance of d_L ≈ 1.8 Gpc ≈ 5.5 × 10²⁷ cm, the IceCube neutrino flux corresponds to an isotropic-equivalent luminosity in muon neutrinos of L_ν,iso ≈ 5.8 × 10⁴⁶ erg s⁻¹.

We assume that the neutrinos are produced by relativistic hadrons in the high-energy emission region (sometimes referred to as the "blob") of (jet-frame) size R', moving relativistically along the jet with constant Lorentz factor Γ ≡ 10 Γ₁₀ and viewing angle θ_obs, resulting in relativistic Doppler boosting by the Doppler factor $D={\left({\rm{\Gamma }}\left[1-{\beta }_{{\rm{\Gamma }}}\cos {\theta }_{\mathrm{obs}}\right]\right)}^{-1}\equiv 10\,{D}_{1}$ with β_Γc the bulk velocity. In that case, the total neutrino luminosity (comprising all neutrino flavors and assuming complete flavor mixing) produced in the comoving frame of the emission region is reduced to ${L}_{\nu }^{{\prime} }\approx 1.7\times {10}^{43}\,{D}_{1}^{-4}$ erg s⁻¹.

To study the electromagnetic behavior of TXS 0506+056 during the neutrino flare we collect data in the optical, X-ray, and γ-ray bands. Optical V-band and g-band data are obtained from the All-Sky Automated Survey for Supernovae (ASAS-SN; Shappee et al. 2014; Kochanek et al. 2017). Data are publicly available from the ASAS-SN website⁷ and available between 2014 February 2 and 2018 September 25. At X-ray energies, we use data from the Burst Alert Telescope (BAT), on board the Neil Gehrels Swift Observatory, to derive a constraint at keV energies for a time interval simultaneous with the neutrino flare. Fermi-LAT has been monitoring the whole sky for almost 10 years, including the relevant period of the neutrino flare as described in Section 2.2.2.

2.2.1. Swift-BAT Data Analysis

2.2.2. Fermi-LAT Data

In this work we consider 10 years of Fermi-LAT data, collected between 2008 August 4 and 2018 September 29 (MJD 54682–58390). Events are selected in a 10° × 10° square centered on the position of TXS 0506+056, from 100 MeV up to 300 GeV. To minimize the contamination from γ-rays produced in the Earth's atmosphere, we apply a zenith angle cut of θ < 90°. Time periods during which the LAT detected bright solar flares and γ-ray bursts were excluded. We perform a binned likelihood analysis with the package fermipy⁹ (v0.17.3), based on the standard LAT ScienceTools¹⁰ (v11-07-00) and the P8R2_SOURCE_V6 instrument response functions. The model of the region accounts for all sources included in the Fermi-LAT preliminary eight year source list,¹¹ and contained in a 15° region from the position of TXS 0506+056, as well as the isotropic and Galactic diffuse γ-ray emission models (iso_P8R2_SOURCE V6 v06.txt and gll_iem_v06.fits). The spectral parameters of the bright γ-ray object located 12 from TXS 0506+056 and associated with the blazar PKS 0502+049, are left free to vary in the fit. As in Garrappa et al. (2018), the residual map of the region does not show evidence for significant structures, indicating that our best-fit model adequately describes the region.

To investigate the γ-ray data of TXS 0506+056 during the neutrino flare, we adopt as definition for the time interval the "box window" identified by Aartsen et al. (2018b), i.e., 158 days (MJD 56937 to MJD 57096, that is 2014 October 7–2015 March 15). For this period, the γ-ray SED of TXS 0506+056 is well modeled with a power-law spectral shape. A maximum likelihood fit yields the best-fit (>100 MeV) flux value of (3.3 ± 0.9) × 10⁻⁸ cm⁻² s⁻¹, with a photon spectral index of 1.9 ± 0.1, and the source is detected at a significance of more than 12σ. This gives an isotropic equivalent γ-ray luminosity in the LAT-energy range during the neutrino flare of 1.9 × 10⁴⁵ erg s⁻¹, or $1.9\,\times {10}^{41}{D}_{1}^{-4}$ erg s⁻¹ in the comoving frame of the jet.

Downturns or upturns in the MeV–GeV spectrum of TXS 0506+056 could be interesting to confirm/rule out features generated by the cascade scenarios. A previous work reported a hint for a hardening of the TXS 0506+056 spectrum in the >2 GeV LAT band (Padovani et al. 2018) during the 2014/15 "neutrino flare" interval. Issues arising from source confusion at the lower LAT energies prevented the same authors from conducting a full investigation of the blazar spectrum in the >100 MeV range. In our analysis we are able to overcome these issues by accounting for the nearby known bright γ-ray blazar PKS 0502+049 in our model for the region of interest, and allowing its spectral parameters to vary in the fit. During the neutrino flare, we find that the source flux is consistent with a quiescent state and the spectral shape with the average one in the >100 MeV range (see also Garrappa et al. 2018). Including this full broadband information provided by the LAT allows us to rule out a large variety of cascade scenarios, as explained in more detail in Section 3.2.

Figure 1 shows the γ-ray and optical light curves. The Fermi-LAT light curve is computed with an eight week binning, integrating energies above 300 MeV. The time bin edges are chosen to conveniently overlap with the neutrino flare, highlighted by the green shaded area in 1. Notably, the detection of the high-energy neutrino IC170922A (red line) coincides with the major γ-ray outburst experienced by TXS 0506+056 in the 10 years of γ-ray monitoring, reaching a peak flux of (1.20 ± 0.07) × 10⁻⁷ cm⁻² s⁻¹. The more recent 2018 data highlight the fading trend of the flare, while the flux remains still at higher-than-average values. The behavior in the optical band overall matches quite well the major γ-ray variations, displaying a lower-than-average optical flux during times coincident with the neutrino flare. The brightest prolonged variations are coincident with the major γ-ray flare and IC170922A.

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Neutrinos are produced through hadronic interactions of relativistic protons of (jet-frame) energy ${E}_{{\rm{p}}}^{{\prime} }={\gamma }_{{\rm{p}}}^{{\prime} }\,{m}_{{\rm{p}}}{c}^{2}$ with target photons of (jet-frame) energy ' when the center-of-momentum frame energy $\sqrt{s}$ of the interaction is above threshold, i.e., $\sqrt{s}\gt \sqrt{{s}_{\mathrm{thr}}}=({m}_{{\rm{p}}}+{m}_{\pi }){c}^{2}\simeq 1.08\,\mathrm{GeV}$. The nucleon energy required to produce neutrinos at hundreds of TeV energies, E_ν ≡ 100 E₁₄ TeV is ${E}_{{\rm{p}}}^{{\prime} }$ ≃ 200E₁₄/(D₁ξ_0.05) TeV (i.e., ${\gamma }_{{\rm{p}}}^{{\prime} }$ = ${E}_{{\rm{p}}}^{{\prime} }$/m_pc² ≃ 2 × 10⁵E₁₄/(D₁ξ_0.05)) where ξ ≡ 0.05 ξ_0.05 is the average neutrino energy per injected nucleon energy in photohadronic interactions (Mücke et al. 2000a). Protons of this energy are very inefficiently emitting synchrotron radiation in the magnetic fields typically assumed in hadronic blazar models (B ∼ 10–100 G) with observed synchrotron cooling timescales of ${t}_{\mathrm{psy}}^{\mathrm{obs}}\,\sim 25\,{D}_{1}^{-1}\,{B}_{1}^{-2}\,{(\gamma /[6\times {10}^{6}])}^{-1}\,\mathrm{yr}$, where B₁ ≡ B/(10 G). The Larmor radius of protons with such energy is ${r}_{L}\sim 2\,\times {10}^{12}\,(\gamma /[6\times {10}^{6}]){B}_{1}^{-1}$ cm, indicating that they are expected to be well confined within the emission region and can plausibly be accelerated by standard mechanisms, such as diffusive shock acceleration.

With the threshold condition $\epsilon ^{\prime} {E}_{{\rm{p}}}^{{\prime} }$ ≥ 0.14 GeV², one finds the minimum energy of the target photons required to produce neutrinos at tens of TeV energies, $\epsilon ^{\prime} \geqslant 0.7{D}_{1}{\xi }_{0.05}/{E}_{14}$ keV. For pion (and neutrino) production at the Δ⁺ resonance, where $s={D}_{{{\rm{\Delta }}}^{+}}^{2}={(1232\mathrm{MeV})}^{2}$, target photons of ' ≥ 1.6D₁ξ_0.05/E₁₄ keV are required. In order to produce neutrinos in a broad spectral range we consider the injection of relativistic protons with a power-law distribution ${{dN}}_{{\rm{p}}}^{{\prime} }/{{dE}}_{{\rm{p}}}^{{\prime} }\propto {E}_{{\rm{p}}}^{{\prime} -{\alpha }_{{\rm{p}}}}$ in the range ${E}_{{\rm{p}}}^{{\prime} }$ = [m_pc², ${E}_{{\rm{p}},\max }^{{\prime} }$] in a target photon field of photon energy density ${\text{}}{u}_{t}^{{\prime} }$, and use the SOPHIA Monte Carlo code (Mücke et al. 2000b) to calculate all resulting yields. We parameterize the target field using a simple power law for its differential photon spectrum ${n}_{t}^{{\prime} }\propto \epsilon {{\prime} }^{-{\alpha }_{t}}$ in the energy range $\epsilon ^{\prime} =[{\epsilon }_{\min }^{{\prime} },{\epsilon }_{\max }^{{\prime} }]$ and re-construct (using the SOPHIA code) the minimal target photon field, irrespective of its origin, that is required to explain the observed neutrino spectrum upon injection of a power-law proton distribution.¹²

Figure 3 shows the resulting muon neutrino (dotted and dashed-triple-dotted black lines) and electron neutrino (dashed and dashed-dotted black lines) spectra at production for a "blob" that is moving with D = 10. The spectra are produced assuming that protons with a power-law spectrum with index α_p = 2 and energies up to ${E}_{{\rm{p}},\max }^{{\prime} }=30$ PeV interact with a distribution of target photons with spectral index α_t = 1, between energies ${\epsilon }_{\min }^{{\prime} }=10\,\mathrm{keV}$ and ${\epsilon }_{\max }^{{\prime} }=60\,\mathrm{keV}$. Neutron-decay neutrinos would appear at ∼10⁻³ smaller energies, and are not considered here. We adopted somewhat higher target photon energies than naively estimated so that the simulated neutrino spectrum matches the best-fit spectrum for the 2014/15 neutrino flare reported by the IceCube collaboration. Looking at the SED of TXS 0506+056, the required energy of the target photon field coincides with the rising part of the second, high-energy hump, where the electromagnetic spectrum follows roughly dN/dE ∝ E⁻¹. This motivates the choice of α_t = 1. The observational uncertainties of the neutrino spectrum are well captured by varying the proton spectral index, in the range ${\alpha }_{{\rm{p}}}=2.0{\pm }_{0.2}^{0.1}$ along with a corresponding proton spectrum normalization variation by ±10% (see Figure 4). The calculation of the total neutrino spectrum (black solid lines in Figures 3 and 4) takes into account neutrino oscillations, and is compared to the IceCube observations (violet bow tie) of the neutrino flare. Once the target photon spectral shape and Doppler factor are fixed, the neutrino flux normalization depends further on the injected proton energy density ${u}_{{\rm{p}}}^{{\prime} }$ (mildly coupled to α_p) and, through the photomeson production opacity τ_pγ, on the target photon energy density ${u}_{t}^{{\prime} }$ and the size R' of the emission region (see Figure 2). The combination of these parameters is chosen to fit the observed neutrino flux (and hence is determined by observables), while each parameter individually is not required to be fixed. In the case of target photon fields external to the jet, particle–photon collisions may become anisotropic in the comoving jet frame, which in turn affects the collision rates. The observed neutrino flux level would then be matched by adjusting the injected proton energy density correspondingly.

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Secondary pairs are produced along with the neutrinos, and are depicted as the blue dashed-line bump at high energies (∼0.1 TeV–10 PeV) in Figure 3, while the produced γ-ray spectrum from meson decay is shown by the solid blue line. The ratio of the secondary pair and photon number density to that of the neutrinos remains constant, even in the case of scatterings involving target photon populations that are anisotropic in the jet frame. Electromagnetic proton–photon interactions (Bethe–Heitler pair production) are guaranteed at sites where photohadronic interactions occur, and are therefore taken into account here (see Figure 2).

We simulate Bethe–Heitler pair production in the same target photon field using the Monte Carlo code of Protheroe & Johnson (1996), and using the same proton injection distribution and flux normalization. The resulting pair yields upon production are shown as the blue dashed line lower-energy (10 MeV–30 GeV) hump in Figure 3 for a Doppler factor D = 10 as an example, and its spread due to the observational uncertainties of the neutrino spectrum is depicted in Figure 4. Note that the ratio of the pair-to-pion production rates in the same target photon field (see Figure 5) is fixed by the nature of the interaction.

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Figure 5 depicts interaction rates for various processes. It illustrates the dominance of losses due to Bethe–Heitler pair production for low proton energies while, above the threshold for meson production, photohadronic interactions dominate. While the minimal energy range of the target photon field is fixed to some extent by the energy range of the observed neutrino spectrum (modulo the Doppler factor in jetted neutrino sources) its shape is less constrained. Indeed, by choosing softer (e.g., α_t = 2–3) or harder (e.g., α_t = 0–0.5) power-law target photon fields we were able to find equally good representations of the observed neutrino spectrum, with only very small adjustments of the remaining parameters (e.g., α_p, etc.). We attribute this behavior to the narrowness of the observed neutrino spectrum. Changing the target photon spectral index in the considered narrow energy range is expected to cause only minor changes in the associated pair cascade spectrum (see Section 3.2), well within the uncertainties implied by the observed neutrino spectrum. The situation changes somewhat when extending sufficiently peaked target photon spectra to a broader energy range. For example, by extending the α_t = 1 target photon spectrum down to ${\epsilon }_{\min }^{{\prime} }=1\,\mathrm{eV}$, we were also able to find a representation of the observed neutrino spectrum for α_p = 2.2 and ${E}_{{\rm{p}},\max }^{{\prime} }\,=20$ PeV. While the corresponding impact on the synchrotron-supported cascades turns out minor (except for a broadening of the absorption troughs), the Compton-supported cascades will become broader (see Section 3.2.1).

Finally, varying the Doppler factor shifts the energy range of the minimal target photon field and injected proton spectrum correspondingly, while the required flux normalization to reach the observed neutrino flux level can be adjusted by changing the injected proton energy density accordingly. We scanned from D = 1 (suitable for misaligned jetted AGNs) up to D = 50 (suggested by minute-timescale γ-ray flux variations found in some blazars; e.g., Begelman et al. 2008) to find satisfactory representations of the observed neutrino spectrum using ${\epsilon }_{\min }^{{\prime} }=1\ldots 170\,\mathrm{keV}$, ${\epsilon }_{\max }^{{\prime} }=4\ldots 1000\,\mathrm{keV}$, and ${E}_{{\rm{p}},\max }^{{\prime} }\,=3\ldots 300$ PeV.

To summarize, Figure 3 shows the distribution of all secondary pairs and γ-rays that are inevitably produced along with the (observed) neutrinos. By varying the proton spectral index ${\alpha }_{{\rm{p}}}=2.0{\pm }_{0.2}^{0.1}$ we propagate the observed uncertainties of the neutrino spectrum to the corresponding pair and γ-ray distributions. These pairs and γ-rays typically initiate electromagnetic cascades (see Figure 2) which re-distribute their power to energies where the produced photons can eventually escape the source. The γγ-pair production optical depth and its energy dependence determine the spectral escape probability of the photons, and is therefore central for calculating the emerging photon spectrum associated with the neutrino spectrum.

The high-energy pairs and γ-rays that were produced along with the neutrino flux in the minimal target radiation field are injected into a pair-cascading code. In a radiative and magnetized environment the electromagnetic cascade develops rapidly in the target photon field through photon–photon pair production, and is either driven by mainly inverse Compton scattering in the same radiation field ("Compton-supported cascades") if ${u}_{t}^{{\prime} }\gg {u}_{B}^{{\prime} }$ (with ${u}_{B}^{{\prime} }$ the magnetic field energy density), by synchrotron radiation ("synchrotron-supported cascades") if ${u}_{t}^{{\prime} }\ll {u}_{B}^{{\prime} }$, or by both ("synchrotron–Compton cascades") in cases where the magnetic field energy density turns out to be comparable to the target photon energy density (see Figure 2). Once the cascade photons reach down to energies too low for pair production, the cascade development ceases, and the remaining pairs lose their energy via inverse Compton and/or synchrotron emission. The emerging photon spectrum is governed by both the energy-dependent photon–photon pair production optical depth and the dominating radiative dissipation process.

The pair-cascading code employs the matrix multiplication method described by Protheroe & Stanev (1993) and Protheroe & Johnson (1996). Monte Carlo programs for photon–photon pair production and inverse Compton scattering are used to calculate the mean interaction rates (see Figure 5) and the secondary particle and photon yields due to the interaction with the target radiation field. Note that inverse Compton scattering during the cascading proceeds in the Klein–Nishina regime, and transitions to the Thomson regime only below the threshold for pair production. The calculation of the synchrotron yields follows Protheroe (1990; see also Pacholczyk 1970). The yields are then used to build up transfer matrices which describe the change in the electron and photon spectra after propagating a given time step δt, which we chose as the dynamical timescale of the problem. The escape probability for photons is calculated using the formula given in Osterbrock & Ferland (2006) for a spherical region, while charged particles are assumed not to escape. For steady-state spectra we continue the transfer process until convergence is reached. In each time step energy conservation is verified.

In the following, all the photon models are generated with a normalization that yields the observed neutrino flux.

3.2.1. Compton-supported Cascades

If inverse Compton scattering supports the cascade (${u}_{t}^{{\prime} }\gg {u}_{B}^{{\prime} }$), the emerging photon spectra are fully determined once the opacity due to photon–photon pair production is fixed, because τ_γγ ∝ τ_IC (with τ_IC the inverse Compton optical depth). This energy-dependent opacity in turn is related to the optical depth of photon–proton interactions, which itself is linked to the neutrino spectral flux (see Section 3.1). Compton-supported cascading on the minimal target photon field (as determined in Section 3.1)¹³ therefore provides the corresponding minimal cascade flux for ${u}_{t}^{{\prime} }$ ≫ ${u}_{B}^{{\prime} }$. In the following we choose to vary the maximum of the energy-dependent photon–photon opacity, τ_γγ,max (which here serves as a proxy for (R' · ${u}_{t}^{{\prime} }$)) to calculate the corresponding cascade spectra. Figures 6–9 show our results for Doppler factors D = 1, 10, and 50 and $\mathrm{log}({\tau }_{\gamma \gamma ,\max })=-6,-5,\,\ldots ,\,0,1,2$, covering the optically thin and thick cases, and compares them to the MWL SED of TXS 0506+056 during the period of the neutrino flare. The shaded areas represent the spread of the cascade spectra (shown for selected τ_γγ,max = 10⁻⁵, 1, 100 as examples) due to the uncertainties in the observed neutrino spectrum.

The cascade spectra at source (thin lines) are then corrected for absorption in the extragalactic background light (EBL) using the model of Franceschini & Rodighiero (2017). EBL-corrected cascade spectra are shown as thick lines. For τ_γγ,max ≪ 1 (e.g., Figure 6), internal absorption can be neglected, and the emerging inverse Compton flux increases with increasing opacity since τ_γγ ∝ τ_IC. None of these cascade spectra reaches the flux level of the MWL (in particular, Fermi-LAT) data. Therefore, in these cases, most of the photon flux has to be produced by emission processes occurring in the source that were not initiated by proton–photon interactions. When increasing the opacity further, τ_γγ ≫ 1 (e.g., Figure 7), deep absorption troughs in the GeV range appear. Any photons in the GeV range produced co-spatially to the neutrino flux will inevitably suffer from internal absorption and will subsequently be cascaded to lower energies, independent of their production process. In this case of an efficient neutrino producer, the observed GeV flux has to be produced either in a region within the jet of TXS 0506+056 that has no dense radiation fields at keV energies, or in a different source altogether. As a consequence, there is no causal connection between the observed neutrino flux and observed GeV flux. These findings are similar for all values of the Doppler factor inspected (see Figures 8 and 9).

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As pointed out in Section 3.1 broader target radiation fields may also lead to photohadronically produced neutrino spectra that fit the observations (e.g., extending the target field ${n}_{t}^{{\prime} }\propto {\epsilon }^{{\prime} -1}$ to lower energies, e.g., ${\epsilon }_{\min }^{{\prime} }=1$ eV). Inspecting the associated Compton-supported cascade spectra shows broader SEDs than for the case of narrow target photon fields (see Figure 10). This is because the photon energy associated with the lowest-energy radiating electron is lower for ${\epsilon }_{\min }^{{\prime} }=1\,\mathrm{eV}$ (broad target field) than for ${\epsilon }_{\min }^{{\prime} }=10\,\mathrm{keV}$ (narrow target field). In addition, the overall cascade flux increases notably below the GeV regime when increasing the energy range of the target photon field. We note that in such cases X-ray data may be able to place stringent constraints on models.

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The Compton optical depth ${\tau }_{\mathrm{IC}}$ may be altered when scatterings sample photon fields that are anisotropic in the comoving frame. Since τ_IC is directly linked to τ_γγ (see Figure 5) and the emerging cascade spectra presented here are proxied by τ_γγ, no significant changes of the Compton cascade spectra for a given τ_γγ are expected in such cases.

3.2.2. Synchrotron-supported Cascades

If the magnetic energy density ${u}_{B}^{{\prime} }$ is much larger than the target photon field energy density ${u}_{t}^{{\prime} }$ the electromagnetic cascades are supported by synchrotron radiation. In blazar jet environments the magnetic field strength B (assumed constant here) is typically $\ll {B}_{\mathrm{cr}}=2\pi {m}_{e}^{2}{c}^{3}/({eh})=4.4\times {10}^{13}$ G, the critical magnetic field strength for radiating pairs, which is therefore also our assumption here. As an example we show in Figures 11–13 the case where ${u}_{B}^{{\prime} }=10\,{u}_{t}^{{\prime} }$ with the above specified target photon spectrum, again corrected for absorption in the EBL. Synchrotron-self absorption is not taken into account. The uncertainties in the IceCube flux are propagated again to the corresponding cascade spectra, and depicted in Figures 11–13 as shaded areas for the cases τ_γγ,max = 10⁻⁶, 1, 100. In the optically thin case the standard synchrotron cooled spectra for a two-component (meson decay and the lower energy Bethe–Heitler pairs) pair population is recovered, with increasing cooling for increasing field strength (see Figure 14). Note that for B ≪ B_cr the energy loss rate is $\propto {\gamma }_{e}^{2}$ for all energies whereas for Compton-loss-dominated pair cascades the interaction probability changes from ∝γ_e in the Klein–Nishina regime to $\propto {\gamma }_{e}^{2}$ in the Thomson regime. This is reflected in the resulting shape of the cascade spectra. Because the secondary pair distribution resulting from proton–photon interactions is fixed by the observed neutrino spectrum the corresponding synchrotron-supported cascade emission in the optically thin case is expected to be unaffected by a possible anisotropy of the collisions. In the optically thick case, τ_γγ ≫ 1, the number of pairs produced from synchrotron photons increases dramatically in the cascade. The amount of reprocessing is given by τ_γγ, and the resulting cascade flux for a given τ_γγ is therefore independent of whether it is produced by isotropic or anisotropic scatterings, as long as the contribution to the scattering rate per unit scattering angle is energy-independent. Also, the produced pairs are more energetic in environments with high magnetic fields than in sites of low magnetization (see Figure 15). As for the optically thick Compton-supported cascades, we note deep absorption troughs at GeV energies leading to the same implications for the association between the neutrino and GeV source, i.e., the lack of a causal connection between the observed neutrino and GeV flux. Note that the hump-like feature at high energies (∼1 GeV–1 TeV) in the optically thick cases disappears by sufficiently broadening the target photon spectrum to lower energies. As compared to the Compton-supported case, the synchrotron-supported cascades extend down to much lower energies: whereas the inverse Compton photon spectrum extends down to $\propto {\gamma }_{\min }^{2}{\epsilon }_{t,\min }^{{\prime} }$, for the synchrotron spectrum the lowest energy reached is $\propto {\gamma }_{\min }^{2}(B/{B}_{\mathrm{cr}}){m}_{e}{c}^{2}$ with typically $(B/{B}_{\mathrm{cr}}){m}_{e}{c}^{2}\ll {\epsilon }_{t,\min }^{{\prime} }$ for an observed neutrino spectrum extending down to tens of TeV.

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When compared to the quasi-simultaneous observed SED we note that deep optical-to-X-ray observations are able to provide sensitive constraints, in particular on the efficiency of neutrino production. For the present case we find that the BAT upper limit constrains the neutrino-producing environment to τ_γγ,max < 100 for D ≤ 10, whereas the Fermi-LAT spectrum excludes τ_γγ,max < 10.

3.2.3. Synchrotron–Compton Cascades

Generally in environments where ${u}_{B}^{{\prime} }\sim {u}_{t}^{{\prime} }$, both synchrotron and Compton losses have to be considered when following electromagnetic cascades. In the high-energy regime, down to the threshold for pair production, the synchrotron–Compton-supported (see Figures 16–18) and synchrotron-supported cascade SEDs are very similar. This is because Compton-scattering proceeds in the Klein–Nishina regime here at a rate much smaller than the synchrotron loss rate, and the resulting spectra are determined by synchrotron radiation and pair production. At lower energies, photon production takes place through synchrotron and inverse Compton scattering on an equal footing, which is reflected in the resulting cascade spectra. With increasing opacity the number of (cascade) pairs rises. At MeV energies the resulting SED is dominated by inverse Compton photons from the primary Bethe–Heitler pairs and cascade pairs, and synchrotron photons from the π^±-decay pairs, while the SED at lower energies is due to synchrotron radiation from the Bethe–Heitler and cascade pairs.

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By comparing also here with the quasi-simultaneous observed SED we find again the BAT upper limit to place constraints on the neutrino production efficiency τ_pγ: Only τ_γγ,max ∼ a few tens for small Doppler factors D < 10, τ_γγ,max ∼ 10 if D = 10, and τ_γγ,max ∼ a few hundreds for large D ≫ 10 do not violate the minimal cascade fluxes computed, similar to the synchrotron-supported cascades.

Since the cross section maximum for photon–photon pair production (near its threshold) compares to the maximum cross section value for photomeson production (at the Δ₁₂₃₂-resonance near threshold) as σ_γγ,max ≈ 300 σ_pγ,max, we expect in typical AGN environments the maximum optical depth of the respective interactions in the same target photon field to approximately scale correspondingly. In particular, for the case of an efficient neutrino producer, i.e., τ_pγ > 1, one yields τ_γγ ≫ 1, and deep absorption troughs appear in the SED at energies ${E}_{\gamma }^{{\prime} }$ where γγ-pair production sets in (see also discussions in, e.g., Begelman et al. 1990; Dermer et al. 2007; Murase et al. 2016):

$$\begin{eqnarray}&&{E}_{\gamma ,\mathrm{thr}}^{{\prime} }=\displaystyle \frac{{s}_{\gamma \gamma ,\mathrm{thr}}}{2\epsilon ^{\prime} (1-\cos \theta )}=\displaystyle \frac{4{m}_{e}^{2}{c}^{4}}{2\epsilon ^{\prime} (1-\cos \theta )}\end{eqnarray} \tag{ 1 }$$

where $\sqrt{{s}_{\gamma \gamma ,\mathrm{thr}}}$ is the center-of-momentum frame threshold energy of the interaction, ' the target photon energy and θ the interaction angle. Considering that neutrinos are produced in the same target photon field via photomeson production dominantly in the Δ-resonance region, the energy of the target photons is related to the observed neutrino energy E_ν,obs through

$$\begin{eqnarray}&&\epsilon ^{\prime} \gtrsim \displaystyle \frac{{s}_{{\rm{\Delta }}}-{m}_{{\rm{p}}}^{2}{c}^{4}}{2{E}_{{\rm{p}}}^{{\prime} }(1-\cos \theta )}=\displaystyle \frac{({s}_{{\rm{\Delta }}}-{m}_{{\rm{p}}}^{2}{c}^{4})\xi D}{2{E}_{\nu ,\mathrm{obs}}(1-\cos \theta )}\end{eqnarray} \tag{ 2 }$$

with s_Δ ≃ (1.232 GeV)² and assuming β_p = 1. Note that the interaction angle for proton–photon and photon–photon interactions is in this case expected to be similar for highly relativistic protons. By combining Equations (1) and (2) one can then relate the (observer frame) photon energy E_γ,obs to the detected neutrino energy through

$$\begin{eqnarray}&&{E}_{\gamma ,\mathrm{obs}}\lesssim \displaystyle \frac{4{m}_{e}^{2}{c}^{4}{E}_{\nu ,\mathrm{obs}}}{({s}_{{\rm{\Delta }}}-{m}_{{\rm{p}}}^{2}{c}^{4})\xi }\approx 3.3\times {10}^{-5}{E}_{\nu ,\mathrm{obs}}{\xi }_{0.05}^{-1}.\end{eqnarray} \tag{ 3 }$$

If neutrinos are very efficiently produced photohadronically, the associated cascade photons therefore can only escape the source at energies ≲E_γ,obs. For a neutrino flare spectrum in the 30 TeV–3 PeV range, the corresponding cascade spectrum is hence expected at ≪1 GeV. This applies also to >GeV photons from alternative radiation processes. As a consequence, one does not expect a causal connection between the GeV flux detected by the LAT from TXS 0506+056 and the observed "flare" IceCube neutrinos for an efficient neutrino producer, irrespective of the dominant dissipation process.

As in Section 3 we consider a proton spectrum of the form ${N}_{{\rm{p}}}({\gamma }_{{\rm{p}}})={N}_{0}{\gamma }_{{\rm{p}}}^{-{\alpha }_{{\rm{p}}}}$ with α_p = 2. The comoving neutrino luminosity (see Section 2.1) can be related to the proton energy loss rate due to photopion production, given by Kelner & Aharonian (2008):

$$\begin{eqnarray}&&{\dot{\gamma }}_{{\rm{p}},{\rm{p}}\gamma }\approx -c\,\langle {\sigma }_{{\rm{p}}\gamma }f\rangle \,{n}_{\mathrm{ph}}^{{\prime} }({\epsilon }_{t}^{{\prime} })\,{\epsilon }_{t}^{{\prime} }\,{\gamma }_{{\rm{p}}}\end{eqnarray} \tag{ 4 }$$

where ${\epsilon }_{t}^{{\prime} }$ = '/(m_ec²) and $\langle {\sigma }_{{\rm{p}}\gamma }f\rangle \approx {10}^{-28}$ cm² is the inelasticity-weighted pγ interaction cross section. The factor ${n}_{\mathrm{ph}}^{{\prime} }({\epsilon }_{t}^{{\prime} })\,{\epsilon }_{t}^{{\prime} }$ provides a proxy for the comoving energy density of the target photon field, ${u}_{t}^{{\prime} }\approx {m}_{e}{c}^{2}\,{n}_{\mathrm{ph}}^{{\prime} }({\epsilon }_{t}^{{\prime} })\,{\epsilon }_{t}^{{\prime} }$. Considering that the energy lost by protons in pγ interactions is shared approximately equally between photons and neutrinos, the 30 TeV–3 PeV neutrino luminosity is given by

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\nu }^{{\prime} } & \approx & \displaystyle \frac{1}{2}{N}_{0}\,{m}_{{\rm{p}}}{c}^{2}\,{\displaystyle \int }_{{\gamma }_{1}}^{{\gamma }_{2}}{\gamma }_{{\rm{p}}}^{-{\alpha }_{{\rm{p}}}}\left|{\dot{\gamma }}_{{\rm{p}},{\rm{p}}\gamma }\right|\,d{\gamma }_{{\rm{p}}}\\ & \approx & \ 1.3\times {10}^{-14}\,{N}_{0}\,{u}_{t}^{{\prime} }\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}\end{array}\end{eqnarray} \tag{ 5 }$$

for α_p = 2. The limits in the integral in Equation (5) are given by γ₁ = 6.4 × 10⁴(D₁ξ_0.05)⁻¹ and γ₂ = 6.4 × 10⁶(D₁ξ_0.05)⁻¹, and α_p = 2. Setting the neutrino luminosity from Equation (5) equal to the comoving neutrino luminosity inferred from IceCube observations, yields

$$\begin{eqnarray}&&{N}_{0}\,{u}_{t}^{{\prime} }\approx 1.3\times {10}^{57}\,{D}_{1}^{-4}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 6 }$$

We now have two independent constraints on the target photon field. First is the direct observational constraint that the observed photon flux corresponding to the target photon field cannot exceed the observed flux in the relevant energy range. This provides a direct upper limit on ${u}_{t}^{{\prime} }$, and will be used in Sections 4.2 and 4.3.

Second is the limit on the γγ opacity provided by the target photon field. In Section 3.2, we had found that, in order not to violate constraints from the observed optical, X-ray, and/or Fermi-LAT flux, the system needs to be either in a regime where cascades are Compton dominated (as the minimal level of the Compton cascade emission turned out to lie always below the observed SED, irrespective of the value of τ_γγ,max), or synchrotron (or synchrotron–Compton) dominated for τ_γγ,max ∼ a few tens (hundreds) for small D ≤ 10 (large D ≫ 10) Doppler factor. Compton-supported cascades will occur if

$$\begin{eqnarray}&&{{u}_{t}^{{\prime} }}^{\mathrm{Comp}}\gg {u}_{B}^{{\prime} }\approx 4\,{B}_{1}^{2}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 7 }$$

in the comoving frame with B = B₁ 10 G the magnetic field strength.

Using a simple δ-function approximation for the γγ pair production cross section, we can estimate ${\tau }_{\gamma \gamma ,\max }$ ∼ ${\sigma }_{{\rm{T}}}\,{u}_{t}^{{\prime} }\,R^{\prime} /(3\,{m}_{e}{c}^{2})$ ∼ $2.7\times {10}^{-3}{u}_{0}\,{R}_{16}$, where u₀ = ${u}_{t}^{{\prime} }$/(erg cm⁻³) parameterizes the target photon energy density, and R₁₆ = R'/(10¹⁶ cm) is the size of the emission region. Thus, the condition τ_γγ,max ∼ a few tens (hundreds) for D ≤ 10 (D ≫ 10) translates into a limit of

$$\begin{eqnarray}\begin{array}{rcl}{{u}_{t}^{{\prime} }}^{\mathrm{Syn}} & \lesssim & {10}^{4}\,{R}_{16}^{-1}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\ {\rm{for}}\ D\leqslant 10,\\ {{u}_{t}^{{\prime} }}^{\mathrm{Syn}} & \lesssim & {10}^{5}\,{R}_{16}^{-1}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\ {\rm{for}}\ D\gg 10.\end{array}\end{eqnarray} \tag{ 8 }$$

Combining with ${u}_{t}^{{\prime} }$ ≤ ${u}_{B}^{{\prime} }$ ≈ 4 ${B}_{1}^{2}$ erg cm⁻³ for synchrotron (synchrotron–Compton) cascades one gets

$$\begin{eqnarray}&&{R}_{16}{B}_{1}^{2}\geqslant 2.5\times {10}^{3}(2.5\times {10}^{4})\ {\rm{for}}\ D\leqslant 10(D\gg 10).\end{eqnarray} \tag{ 9 }$$

This indicates the need for large field strengths and/or emission regions in the case of synchrotron (synchrotron–Compton) cascades operating in the source.

A limit on the proton kinetic power can then be derived by combining Equation (6) with (8) in the case where synchrotron (synchrotron–Compton) cascades determine the dissipation process. The comoving relativistic proton energy density, assuming that the proton spectrum extends as an unbroken power law with index α_p = 2 from γ_min = 1 to γ₂ ≫ γ_min, is calculated by

$$\begin{eqnarray}\begin{array}{rcl}{u}_{{\rm{p}}}^{{\prime} } & = & {m}_{{\rm{p}}}{c}^{2}{V}^{{\prime} -1}{N}_{0}{\displaystyle \int }_{1}^{{\gamma }_{2}}d{\gamma }_{{\rm{p}}}{\gamma }_{{\rm{p}}}^{1-{\alpha }_{{\rm{p}}}}\\ & \approx & \ 3.6\times {10}^{-52}\,{N}_{0}\,{R}_{16}^{-3}\,(15.7-\eta )\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\end{array}\end{eqnarray} \tag{ 10 }$$

with η = ln(D₁ξ_0.05), and V' the comoving volume of the emission region. Since typically η ≈ 0 with only small deviations expected, we will neglect the η-term in the following. The relativistic proton kinetic power L_p = 2πR^'2cΓ²${u}_{{\rm{p}}}^{{\prime} }$ carried by the jet in highly magnetized environments where pair synchrotron losses cannot be neglected can then be evaluated as

$$\begin{eqnarray}&&\begin{array}{l}{L}_{{\rm{p}}}\gg 1.4(14)\times {10}^{47}{\left({{\rm{\Gamma }}}_{1}/{D}_{1}^{2}\right)}^{2}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\\ \quad \ {\rm{for}}\ D\gg 10(D\leqslant 10),\end{array}\end{eqnarray} \tag{ 11 }$$

with Γ the bulk Lorentz factor Γ ≡ 10 Γ₁, and a magnetic field power (using Equation (9)) of

$$\begin{eqnarray}&&\begin{array}{l}{L}_{B}\gg 0.2(2)\times {10}^{50}{{\rm{\Gamma }}}_{1}^{2}{R}_{16}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\\ \quad \ {\rm{for}}\ D\leqslant 10(D\gg 10).\end{array}\end{eqnarray} \tag{ 12 }$$

Hence, the total jet power turns out to be in this case at least of order ∼10^49...50 erg s⁻¹ (or higher), independent of the origin of the target photon field.

We now use the observed SED in the energy range of the target photons (here ${\epsilon }_{\min }^{{\prime} }$ = 1...170 keV, ${\epsilon }_{\max }^{{\prime} }=4\ldots 1000\,\mathrm{keV}$ for D = 1 ...50) to derive a direct limit on ${u}_{t}^{{\prime} }$ which will then be combined with the previous constraints.

If the target photon field is comoving with the emission region, the required target photon energy corresponds to hard X-rays at an observed energy of ${\epsilon }_{\mathrm{obs}}\geqslant 16{D}_{1}^{2}{\xi }_{0.05}/{E}_{14}$ keV. The observed X-ray flux from TXS 0506+056 is of the order of ${F}_{X}^{\mathrm{obs}}$ ∼ 10⁻¹² erg cm⁻² s⁻¹, implied from archival data. The flux received directly from the target photon field, present in an assumed spherical emission region of size ${R}_{t}^{{\prime} }\equiv {10}^{16}\,{R}_{t,16}$ cm, is ${F}_{a}^{\mathrm{obs}}=\tfrac{{R}_{t}^{{\prime} 2}}{{d}_{L}^{2}}\,c\,{u}_{t}^{{\prime} }\,{D}^{4}$ and must be equal to or less than the observed value. This constrains the comoving target photon energy density to be

$$\begin{eqnarray}&&{u}_{t}^{{\prime} }\lesssim 9\times {10}^{-4}\,{R}_{t,16}^{-2}\,{D}_{1}^{-4}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\end{eqnarray} \tag{ 13 }$$

and in turn the proton-spectrum normalization factor in Equation (6):

$$\begin{eqnarray}&&{N}_{0}\gtrsim 1.5\times {10}^{60}\,{R}_{t,16}^{2}.\end{eqnarray} \tag{ 14 }$$

The corresponding total kinetic power carried by the jet can then be evaluated as

$$\begin{eqnarray}&&{L}_{{\rm{p}}}\gtrsim 1.55\times {10}^{55}\,{{\rm{\Gamma }}}_{1}^{2}\,{R}_{t,16}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 15 }$$

which appears unreasonably high to be powered by an AGN accretion flow, as it exceeds the Eddington luminosity of even the most massive known supermassive black holes (M_bh ≲ 10¹⁰ M_⊙) by several orders of magnitude.

The photon energy density limit from Equation (13) can also be used to calculate the proton cooling timescale due to photopion production:

$$\begin{eqnarray}\begin{array}{rcl}{t}_{{\rm{p}}\gamma }^{\mathrm{obs}} & = & \displaystyle \frac{{m}_{e}{c}^{2}}{D\,c\,\langle {\sigma }_{{\rm{p}}\gamma }f\rangle \,{u}_{t}^{{\prime} }}\gtrsim 3\times {10}^{13}\,{R}_{t,16}^{2}\,{D}_{1}^{3}\,{\rm{s}}\\ & \sim & \ 9.5\times {10}^{5}\,{R}_{t,16}^{2}{D}_{1}^{3}\,\mathrm{yr}\end{array}\end{eqnarray} \tag{ 16 }$$

to compare with proton synchrotron cooling. The importance of proton synchrotron radiation in the context of hadronic AGN models was first noticed by Mücke & Protheroe (2000, 2001) and Aharonian (2000). Equation (16) indicates that photopion production is several orders of magnitude less efficient than proton synchrotron radiation with a cooling timescale of

$$\begin{eqnarray}&&{t}_{{\rm{p}}{\rm{s}}{\rm{y}}{\rm{n}}}^{{\rm{o}}{\rm{b}}{\rm{s}}}\simeq 9\times {10}^{8}\,{B}_{1}^{-2}\,{D}_{1}^{-1}\left(\displaystyle \frac{{\gamma }_{{\rm{p}}}}{6\times {10}^{6}}\right)\,{\rm{s}},\end{eqnarray} \tag{ 17 }$$

even for protons of moderate Lorentz factors of γ_p ∼ 6 × 10⁶. In fact, this allows us to calculate the relative radiative power output in proton synchrotron- versus photopion-induced emissions:

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{{\rm{p}}{\rm{s}}{\rm{y}}}}{{L}_{{\rm{p}}\gamma }}\sim \displaystyle \frac{{t}_{{\rm{p}}\gamma }^{{\rm{{\prime} }}}}{{t}_{{\rm{p}}{\rm{s}}{\rm{y}}}^{{\rm{{\prime} }}}}\sim 4\times {10}^{4}\,{R}_{t,16}^{2}\,{D}_{1}^{4}\,{B}_{1}^{2}\left(\displaystyle \frac{{\gamma }_{{\rm{p}}}}{6\times {10}^{6}}\right)\end{eqnarray} \tag{ 18 }$$

with proton-synchrotron emission peaking at

$$\begin{eqnarray}&&{\nu }_{{\rm{p}}{\rm{s}}{\rm{y}}}^{{\rm{o}}{\rm{b}}{\rm{s}}}\sim 9\times {10}^{16}\,{B}_{1}{\left(\displaystyle \frac{{\gamma }_{{\rm{p}}}}{6\times {10}^{6}}\right)}^{2}\,{D}_{1}\,{\rm{H}}{\rm{z}}\end{eqnarray} \tag{ 19 }$$

i.e., in the extreme UV/soft X-ray band. This implies that, in this scenario, the protons producing the 0.03–3 PeV neutrino flux are expected to produce proton synchrotron emission in the X-ray regime at a νF_ν flux value larger than the corresponding νF_ν neutrino flux value by a factor given by Equation (18). Reducing this to the limit based on the observed X-ray flux (L_psy/L_pγ ≲ 0.1) would require an unusually low magnetic field value (B ≲ 1 G) and/or a very low Doppler factor D ∼ 1. This is demonstrated in Figures 6 and 13 where we have added the proton synchrotron radiation on top of the minimal cascade component for various model parameters.

Based on these results, we may confidently rule out a scenario in which the IceCube neutrino flare flux from TXS 0506+056 is photohadronically produced in a target photon field that is comoving with the emission (and neutrino production) region.

In the case of a target photon field that is stationary in the AGN rest frame, the target photon energy (in the AGN frame) should be _obs ≥ 0.16D₁ξ_0.05/E₁₄ keV, and the AGN-frame radiation energy density is boosted into the comoving frame as ${u}_{t}^{{\prime} }$ ≈ Γ² u_t. In the external photon field case, the spatial extent of the photon field may be scaled in units of a typical broad-line region size, ${R}_{t}\equiv {10}^{17}{R}_{t,17}$ cm (AGN frame), and the corresponding flux received is ${F}_{\mathrm{UV}}^{\mathrm{obs}}=\tfrac{{R}_{t}^{2}}{{d}_{L}^{2}}\,c\,{u}_{t}^{{\prime} }\,{{\rm{\Gamma }}}^{-2}$.

For the Compton-dominated cascade case we now insert the energy density from Equation (7) to derive

$$\begin{eqnarray}&&{F}_{\mathrm{UV}}^{\mathrm{obs}}\gg 4.3\times {10}^{-13}{\left({B}_{1}{R}_{t,17}/{{\rm{\Gamma }}}_{1}\right)}^{2}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 20 }$$

which is below the observed SED from archival data for not too large field strengths. Inspecting now the case for a highly magnetized environment where synchrotron losses become non-negligible we derive a UV–soft X-ray flux of

$$\begin{eqnarray}&&{F}_{\mathrm{UV}}^{\mathrm{obs}}\gg ({10}^{-9}\ldots {10}^{-8}){R}_{16}^{-1}{\left({R}_{t,17}/{{\rm{\Gamma }}}_{1}\right)}^{2}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 21 }$$

for D ≤ 10 (D ≫ 10) by making use of Equation (8). This flux level seems several orders of magnitude higher than the observed SED from archival data implies, and hence gives strong arguments to rule out an environment of the neutrino emission region where synchrotron (synchrotron–Compton) cascades operate.

From the observed SED a conservative upper limit on the νF_ν flux at UV–soft X-rays may be placed at νF_ν ≲ 10⁻¹¹ erg cm⁻² s⁻¹. The comoving target photon energy density can then be constrained as

$$\begin{eqnarray}&&{u}_{t}^{{\prime} }\lesssim 94\,{{\rm{\Gamma }}}_{1}^{2}\,{R}_{t,17}^{-2}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}\end{eqnarray} \tag{ 22 }$$

giving a corresponding pair production optical depth of

$$\begin{eqnarray}&&{\tau }_{\gamma \gamma ,\max }\lesssim 0.25\,{R}_{16}\,{{\rm{\Gamma }}}_{1}^{2}\,{R}_{t,17}^{-2}\end{eqnarray} \tag{ 23 }$$

and a photohadronic optical depth of

$$\begin{eqnarray}&&{\tau }_{{\rm{p}}\gamma ,\max }\lesssim 8\times {10}^{-4}\,{R}_{16}\,{{\rm{\Gamma }}}_{1}^{2}\,{R}_{t,17}^{-2}\end{eqnarray} \tag{ 24 }$$

above the thresholds of the interactions. For moderate magnetic fields, B ≲ 10 G, this may lead to Compton-supported cascades, thus not violating limits on the multiwavelength cascade flux. Equation (6) then constrains the proton-spectrum normalization to

$$\begin{eqnarray}&&{N}_{0}\gtrsim 1.4\times {10}^{55}\,{D}_{1}^{-4}\,{{\rm{\Gamma }}}_{1}^{-2}\,{R}_{t,17}^{2}\end{eqnarray} \tag{ 25 }$$

yielding a required proton kinetic jet power of

$$\begin{eqnarray}&&{L}_{{\rm{p}}}\gtrsim 1.5\times {10}^{50}\,{D}_{1}^{-4}\,{R}_{t,17}^{2}\,{R}_{16}^{-1}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 26 }$$

Due to the strong inverse dependence on the Doppler factor, for D significantly exceeding 10 (and/or a larger target photon field radius or smaller emission region), this may be within the range plausibly powered by accretion onto a supermassive black hole of mass ∼3 × 10⁸ M_⊙ (Padovani et al. 2019). The relativistic proton power from Equation (26) may be compared to the power carried along the jet in magnetic fields,

$$\begin{eqnarray}&&{L}_{B}=3.8\times {10}^{45}\,{R}_{16}^{2}\,{{\rm{\Gamma }}}_{1}^{2}\,{B}_{1}^{2}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 27 }$$

Thus, for B ≲ 10 G, the jet would have to be strongly dominated by the kinetic power of the relativistic protons.

For the target photon field of Equation (22), the proton photopion cooling timescale is, analogous to Equation (16),

$$\begin{eqnarray}&&{t}_{{\rm{p}}\gamma }^{\mathrm{obs}}\gtrsim 2.9\times {10}^{9}\,{{\rm{\Gamma }}}_{1}^{-2}\,{R}_{17}^{2}\,{\rm{s}}\sim 92\,{{\rm{\Gamma }}}_{1}^{-2}\,{R}_{17}^{2}\,\mathrm{yr}.\end{eqnarray} \tag{ 28 }$$

Also in this case, proton–synchrotron radiation, at X-ray frequencies according to Equation (19), cannot be neglected, and the ratio of proton–synchrotron- to photopion-induced radiative (and neutrino) output will be

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{{\rm{p}}{\rm{s}}{\rm{y}}}}{{L}_{{\rm{p}}\gamma }}\gtrsim 0.4\,{{\rm{\Gamma }}}_{1}^{-2}\,{D}_{1}\,{R}_{t,17}^{2}\,{B}_{1}^{2}\left(\displaystyle \frac{{\gamma }_{{\rm{p}}}}{6\times {10}^{6}}\right)\end{eqnarray} \tag{ 29 }$$

thus predicting a X-ray proton–synchrotron emission component with a flux of $\nu {F}_{\nu }(X)\gtrsim {10}^{-10}\,{{\rm{\Gamma }}}_{1}^{-2}\,{D}_{1}\,{R}_{t,17}^{2}\,{B}_{1}^{2}\,({\gamma }_{{\rm{p}}}/[6\times {10}^{6}])$ erg cm⁻² s⁻¹. For a magnetic field of B ∼ a few G, this does not seem to violate observational limits (see also Figure 6).

We thus conclude that a scenario appears feasible in which an external soft X-ray photon field provides the targets for photopion reactions producing the IceCube neutrinos of TXS 0506+056 in an environment where dominantly Compton scattering supports pair cascading. Equations (24) and (23) imply in this case still very inefficient neutrino production, and a corresponding pair cascade flux that lies significantly below the γ-ray flux level observed by the LAT during the neutrino flare (see Figures 6–10). Further high-energy radiation processes are therefore needed to explain the observed γ-ray SED, which is unlikely proton-initiated, during the neutrino flare.