Hadronic neutrinos require targets, thresholds and decay

For a proton striking a stationary proton, the lowest neutral-pion channel is pp→ppπ⁰. Its projectile kinetic threshold is about 280 MeV. Charged-pion neutrino channels have different final-state masses and slightly different thresholds; a neutral-pion threshold by itself is not a neutrino threshold. In photon-target production, evaluate the proton and photon energies in the same frame. A head-on ultrarelativistic proton colliding with a photon reaches a specified nucleon–pion final state when their energy product supplies its invariant mass.

The often-used delta-resonance approximation describes where a photohadronic cross section is enhanced, rather than the first allowed pion threshold. Broad photon spectra and collision angles require an interaction-rate integral; a monochromatic head-on estimate cannot replace it. Throughout, primes indicate a local plasma frame. Observed neutrino energies acquire the factor δ/(1+z) from that frame. A target photon energy measured by the observer must first be transformed before combining it with a comoving proton energy.

At high proton energy, an illustrative gas-target energy-loss time is the reciprocal of density times inelastic cross section times speed times inelasticity. With constant coefficients over a residence interval, the fractional proton energy lost is one minus an exponential. This energy-loss fraction is not the interaction probability, because inelasticity is smaller than one. Only a channel-dependent fraction of that lost energy enters charged pions. Their neutrino yield also depends on whether pions and muons decay before synchrotron, adiabatic or other losses modify them.

A rough characteristic neutrino energy of five percent of the parent proton energy assumes a representative pion energy partition and an unsuppressed decay chain. It is broad, energy dependent and inappropriate as an exact threshold relation. Neutral pion photons can pair-produce and cascade, while neutrinos usually escape the source with much less interaction. Consequently a gamma-ray upper limit is not automatically the same upper limit on emitted neutrino power. Conversely a leptonic gamma-ray source need not produce a corresponding hadronic neutrino flux.

The conventional charged-pion decay chain gives approximate source flavor numbers one electron to two muon neutrinos when particles and antiparticles are combined and muons decay without major cooling. Over astrophysical baselines, averaged vacuum mixing gives a composition near one to one to one, not an exact universal ratio. Muon damping, neutron decay or other source conditions change it. Neutrinos travel nearly straight, but detection probability depends strongly on energy and direction, and Earth absorption becomes relevant at sufficiently high energy. A measured track energy is not a unique incoming-neutrino energy.

For a point-source number fluence per energy, the expected selected event count is its energy integral weighted by the direction-dependent effective area, summed over flavors. That response includes interaction, propagation through Earth and the chosen event cuts; it is not merely geometric detector area. The Markov lab illustrates only repeated removal: Reset and set both controls p=0.1,q=0; Step twice for two transitions, predicting surviving state-zero probability 0.81. Reset and set p=0.2,q=0; Step twice, predicting 0.64. There is no energy-dependent hadronic cascade, flavor oscillation or neutrino detector in this two-state analogy.