Neutrinos, Cosmic Rays, and Black-Hole Jets

A neutrino arriving from a jet direction is evidence for hadronic interactions only after direction, time, energy, background, and trials are combined. Even then, it does not by itself identify the emitting zone, prove that the accompanying photons are hadronic, or show that charged cosmic rays escape. A persuasive case must join event association to an energy-and-opacity calculation and then survive source-class stacking. Instrument-specific gravitational-wave likelihood methods are outside this page’s scope.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon and spin scales; Relativistic Matter and Compact Objects supplies relativistic particles and radiation; Black-Hole Astrophysics supplies jet launching and spectral components; Gravitational-Wave Astronomy supplies multimessenger association logic; Relativistic Cosmology supplies redshift and luminosity-distance conversions.

From accelerated protons to observable neutrinos

Electrons radiate efficiently but do not make high-energy neutrinos. A jet must accelerate protons or nuclei and expose them to gas ($pp$) or photons ($p\gamma$). Primes denote quantities in the emitting plasma’s rest frame. For an ultrarelativistic proton and a photon meeting at angle $\psi'$, the squared center-of-mass energy is

$$s\simeq m_p^2+2E'_p\epsilon'_\gamma(1-\cos\psi').$$

The $\Delta$ resonance occurs near $s=m_\Delta^2$. With $m_pc^2=0.938$ GeV and $m_\Delta c^2=1.232$ GeV, a head-on collision therefore has $E'_p\epsilon'_\gamma\simeq0.16\ {\rm GeV}^2$. Replacing $1-\cos\psi'$ by a representative value of one gives the often-used estimate $0.3\ {\rm GeV}^2$. These are angle-dependent resonance estimates, not the lower kinematic threshold for producing a pion. A useful estimate for the neutrino energy is

$$E_{\nu,\rm obs}\simeq \frac{0.05\,\delta}{1+z}E'_p ,$$

where $\delta$ is the Doppler factor. Thus an observed neutrino energy selects a combination of target-photon energy, collision angle, and beaming; it does not locate the target field uniquely. For confinement in a source of radius $R'$ and magnetic field $B'$, an ultrarelativistic nucleus of charge $Ze$ needs $r'_L=E'/(ZeB')\lesssim R'$ in Gaussian electromagnetic units. If its acceleration time is $t'_{\rm acc}\simeq r'_L/(\beta_{\rm acc}c)$ and the available time is $R'/c$, this gives

$$E'_{\max}\lesssim ZeB'R'\beta_{\rm acc}, \qquad 0\lt\beta_{\rm acc}\leq1.$$

Here $\beta_{\rm acc}$ describes acceleration efficiency, not the bulk jet speed. An escaping particle’s observed energy also requires the boost associated with its escape direction and a factor $(1+z)^{-1}$. This Hillas-type bound is necessary, but synchrotron, adiabatic, photohadronic, and escape losses can impose tighter limits. The relevant acceleration and pion-production regimes are reviewed in Blandford, Meier, and Readhead 2019, §§2.5, 4.1, and 4.2.2.5–4.2.2.6.

Let $f_{p\gamma}\leq1$ be the fraction of proton energy lost to pion production and $f_{\rm ch}$ the fraction of pion energy carried by charged pions. If pions and muons decay before appreciable cooling, about three quarters of the charged-pion energy becomes neutrinos. An order-of-magnitude all-flavor relation is then

$$E_\nu L_{E_\nu}\simeq \frac{3}{4}f_{\rm ch}f_{p\gamma}\,E_pL_{E_p}, \qquad E_\nu\simeq0.05E_p.$$

Both energy bins and luminosities must be compared in the same frame; $E L_E$ is power per logarithmic energy interval. The familiar coefficient $3/8$ assumes $f_{\rm ch}\simeq1/2$; it is not universal, and a pure $\Delta$-resonance branching approximation gives a different charged-pion fraction. Infer the required cosmic-ray power with the appropriate yield and require it to remain below the beaming-corrected jet power. Neutral pions and secondary pairs carry comparable electromagnetic energy. If $\gamma$ rays escape, their cascade may overproduce the measured X-ray-to-GeV spectrum; if $\tau_{\gamma\gamma}\gg1$, the energy must reappear at lower frequency or heat the flow. This yields two physically distinct regimes: a transparent one-zone emitter with a visible cascade, and a hidden hadronic zone plus a separate radiating zone.

Event association is a four-dimensional likelihood

For event $i$, a standard unbinned mixture has the form

$$\mathcal L(n_s,\gamma)= \prod_i\left[ \frac{n_s}{N}S_i(\Omega_i,E_i,t_i;\gamma) +\left(1-\frac{n_s}{N}\right)B_i(\delta_i,E_i,t_i) \right].$$

The signal density includes the directional point-spread function, detector acceptance, an energy spectrum, and a predeclared time model. The background density is measured from data as a function of declination, energy proxy, and time. The test statistic compares the best signal-plus-background likelihood with $n_s=0$. Every scanned source, time window, spectrum, and flare definition contributes to the trial correction.

IceCube-170922A provides the canonical example. The through-going track arrived on 2017-09-22 from a direction consistent with gamma-ray blazar TXS 0506+056 during enhanced emission. Its quoted $\sim290$ TeV is a most-probable neutrino energy conditional on the assumed spectrum, not a calorimetric measurement; the detector sees a secondary muon. The calibration and conditional energy distribution are documented in IceCube et al. 2018, Supplementary Materials, pp. S17–S20, Figures S1–S2. The chance-correlation statistic explicitly combines 2,257 Fermi-LAT sources, the point-spread function, detector acceptance, and alternative gamma–neutrino weightings in IceCube et al. 2018, Supplementary Materials, pp. S36–S41, Equations S1–S3.

An independent, prompted search of 9.5 years found a 2014–2015 excess from the same direction: $13\pm5$ events in the best 158-day box window, with 3.5-sigma post-trial significance. IceCube Collaboration 2018, pp. 147–151, Table 1 and Figures 1–4 separates the time-dependent from time-integrated searches; its Supplementary Materials, Equations S1–S9 and Figures S2–S3, show the likelihood and look-elsewhere correction. This strengthens the source-direction case, but the absence of an analogous gamma-ray flare in that interval argues against a universal one-zone gamma–neutrino relation.

Data release, calibration, and reproduction

Use the IceCube 2008–2017 TXS 0506+056 data release, DOI 10.21234/B4QG92, including its 2019-10-22 addition of tabulated Figures S4–S5. It supplies six sample intervals, effective areas, and events within $3^\circ$ of the source: MJD, J2000 coordinates, angular uncertainty, and reconstructed muon-energy proxy. That proxy differs across samples and is only approximately in GeV; it is not an event-by-event neutrino-energy estimate.

A reproduction must preserve the sample-specific effective area and selection, detector livetime, angular calibration and ice-systematic point-spread function, time standard, energy-proxy definition, and alert revisions. Recompute the time-integrated and time-dependent likelihoods, scramble right ascension or time as specified for the background, and repeat every scanned window and spectrum. Report local and post-trial probabilities separately. Then close the source model with electromagnetic light curves, upper limits, jet power, $p\gamma$ or $pp$ efficiency, $\gamma\gamma$ opacity, and cascade transfer.

Charged ultra-high-energy cosmic rays require a different association strategy. Galactic and intergalactic magnetic deflection scales roughly as $Z/E$ and induces path-dependent time delays, so a narrow prompt window is generally inappropriate. Composition, anisotropy, source density, and energy-loss horizon become population observables. A neutrino coincidence can establish hadronic activity without showing that the same particles escape as the observed cosmic rays.

Results and limitations through 2026-08-18

Results after 2026-08-18 are outside this assessment. Significances and population fractions remain conditional on the stated source list, spectrum, weighting, and trials.

Evidence or result	Hadronic interpretation	Scope and limitations	Counterexample or independent check
IceCube-170922A plus 2017 TXS flare	A high-energy neutrino is directionally and temporally compatible with an active blazar	One alert; energy is conditional; association weighting and trials are model-dependent	Independent 2014–2015 search and electromagnetic cascade closure
2014–2015 TXS directional excess	$13\pm5$ fitted events in 158 days give 3.5-sigma post-trial evidence from the source direction	No matching bright gamma flare; emission site and spectrum remain uncertain	Reproduce from B4QG92 with both box and Gaussian windows
NGC 1068 ten-year point-source analysis	An obscured AGN can be neutrino bright without being gamma-ray bright	This Seyfert result does not establish a relativistic jet origin	Compare coronal and jet populations under the same likelihood
2026 public-data AGN population analysis	Two-sigma upper limits are 14.5% for all gamma-ray-bright blazars and 15.6% for the non-variable subset in the least restrictive tested weighting/spectrum cases	These limits include the analysis’s completeness correction; they do not constrain every possible faint or hidden jet population	Repeat with alternative weights and held-out catalogs

The last row refers to the Fermi 4LAC-DR3 source lists and the tested gamma-ray-flux and geometric weights in Jain, Hooper, and Halzen 2026, § 4.2, Tables 4–5, and Appendix B. The event data, reconstructed directions, detector selections, and released effective areas are durable. The target zone, baryon loading, dominant interaction channel, relation to gamma flares, fraction of the diffuse neutrino sky supplied by jets, and connection to escaping ultra-high-energy cosmic rays remain unresolved.

Competing positions and parameter regimes

A hadronic jet can produce neutrinos and a constrained cascade. A purely leptonic flare can explain the electromagnetic brightening while the neutrino is unrelated. A hidden compact zone can produce neutrinos while another, more transparent zone makes the observed gamma rays. Finally, the directional and temporal match can be a background fluctuation. The parameter regime axes are $pp$ versus $p\gamma$, transparent versus gamma-obscured, steady versus flaring, one-zone versus multi-zone, and individual-object versus population inference. No result may be moved along those axes without recalculating efficiency, opacity, selection, and trials.

Limitations and open questions

The current evidence is sparse at the individual-source level, angular errors enclose multiple possible emitters, and source-class limits depend on unobserved faint populations. Jet power and cascade constraints further inherit Doppler factor, geometry, target-field, and magnetic-field uncertainty.

What could change the conclusion. The interpretation could change when a high-energy alert has dense contemporaneous electromagnetic coverage, an event reconstruction or release changes, a source-class stacking analysis materially shifts the blazar fraction, cosmic-ray anisotropy constrains the same population, or a transfer calculation closes both neutrino and cascade energy budgets.

Formulation-choice diagnostic

A one-zone fit requires $L_\nu=3\times10^{46}\ {\rm erg\,s^{-1}}$, has $f_{p\gamma}=0.05$, and permits at most $L_{\rm jet}=5\times10^{47}\ {\rm erg\,s^{-1}}$. Take $f_{\rm ch}=1/2$, negligible pion and muon cooling, and luminosities integrated over corresponding particle-energy ranges in the same frame. Ignoring beaming changes, can the zone pass the proton-power test, and what additional observation is decisive?

Solution

Using $L_\nu\simeq(3/8)f_{p\gamma}L_p$ gives

$$L_p\simeq\frac{8L_\nu}{3f_{p\gamma}} =1.6\times10^{48}\ {\rm erg\,s^{-1}},$$

more than three times the permitted jet power. The one-zone formulation fails before detailed spectral fitting. Increasing $f_{p\gamma}$ lowers the proton demand, but also increases accompanying pion-cascade power and often requires a denser target field with larger $\gamma\gamma$ opacity. The decisive check is therefore simultaneous X-ray-to-gamma coverage and upper limits passed through a cascade calculation. A hidden second zone might evade the visible gamma constraint, but that is a new multi-zone formulation with reprocessing and total-power requirements, not a rescue of the original fit.

References

• Blandford, R., Meier, D., and Readhead, A. (2019), “Relativistic Jets from Active Galactic Nuclei,” Annual Review of Astronomy and Astrophysics 57, 467–509, doi:10.1146/annurev-astro-081817-051948.
• IceCube Collaboration (2018), “Neutrino Emission from the Direction of the Blazar TXS 0506+056 Prior to the IceCube-170922A Alert,” Science 361, 147–151, doi:10.1126/science.aat2890; associated data release, doi:10.21234/B4QG92.
• IceCube Collaboration (2022), “Evidence for Neutrino Emission from the Nearby Active Galaxy NGC 1068,” Science 378, 538–543, doi:10.1126/science.abg3395.
• IceCube Collaboration et al. (2018), “Multimessenger Observations of a Flaring Blazar Coincident with High-Energy Neutrino IceCube-170922A,” Science 361, eaat1378, doi:10.1126/science.aat1378.
• Jain, S., Hooper, D., and Halzen, F. (2026), “Evaluating the Contribution of Active Galactic Nuclei to the Diffuse High-Energy Neutrino Flux,” Journal of Cosmology and Astroparticle Physics 2026(05), 045, doi:10.1088/1475-7516/2026/05/045.