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Cosmic Ray Neutrino Annihilation on Relic Neutrinos Revisited: A Mechanism for Generating Air Showers above the Greisen-Zatsepin-Kuzmin Cut-off

If neutrinos are a significant contributor to the matter density of the universe, then they should have $\sim$ eV mass and cluster in galactic (super) cluster halos, and possibly in galactic halos as well. It was noted in the early 1980's that cosmic ray neutrinos with energy within $δE/E_R=Γ_Z/M_Z \sim 3%$ of the peak energy $E_R=4 (eV/m_ν)\times 10^{21}$ eV will annihilate on the nonrelativistic relic antineutrinos (and vice versa) to produce the Z-boson with an enhanced, resonant cross section of ${\cal O}(G_F)\sim 10^{-32}{cm}^2$. The result of the resonant neutrino annihilation is a hadronic Z-burst 70% of the time, which contains, on average, thirty photons and 2.7 nucleons with energies near or above the GZK cutoff energy of $5\times 10^{19}$ eV. These photons and nucleons produced within our Supergalactic halo may easily propagate to earth and initiate super-GZK air showers. Here we show that the probability for each neutrino flavor at its resonant energy to annihilate within the halo of our Supergalactic cluster is likely within an order of magnitude of 1%, with the exact value depending on unknown aspects of neutrino mixing and relic neutrino clustering. The absolute lower bound in a hot Big Bang universe for the probability to annihilate within a 50 Mpc radius (roughly a nucleon propagation distance) of earth is 0.036%. From fragmentation data for Z-decay, we estimate that the nucleons are more energetic than the photons by a factor $\sim 10$. Several tests of the hypothesis are indicated.