Lower energy consequences of an anomalous high-energy neutrino cross section

A new strong interaction has been postulated for neutrinos above $\sim {10}^{19}\mathrm{eV}$ to explain the production of highest-energy cosmic-ray events. We derive a dispersion relation relating the hypothesized high-energy cross section to the lower-energy neutrino-nucleon elastic amplitude. Remarkably, we find that the real forward amplitude becomes anomalous seven orders of magnitude lower in energy than does the total cross section. We discuss possible measurable consequences of this early onset of new neutrino physics and conclude that a significantly enhanced elastic $\nu N$ scattering rate may occur for the neutrino beams available at Fermilab and CERN.