Interplay of Lorentz Invariance Violation and Earth's Matter Potential in High-Energy Neutrinos

Abstract: Searches for Lorentz invariance violation (LIV) in the neutrino sector have traditionally focused on non-standard neutrino oscillations induced by LIV in vacuum. In this work, however, we study anisotropic LIV in matter. First, we review vacuum LIV phenomenology, explaining the energy and direction dependence of sidereal modulations for anisotropic coefficients in the Standard Model Extension. We then demonstrate that for high-energy neutrinos, the interplay between anisotropic LIV operators and the Earth's matter potential produces, distinct, observable signatures absent in the vacuum case. We identify a crossover regime where the energy-dependent LIV Hamiltonian becomes comparable to the matter potential, leading to strong interference effects. By analyzing the propagation of neutrinos through a realistic Earth model, we establish three key phenomenological consequences: (1) direction-dependent resonant enhancements of oscillation probabilities, (2) a macroscopic breakdown of neutrino-antineutrino symmetry for CPT-even operators, and (3) a significant increase of the $ν_τ$ flux due to LIV-driven injection of high-energy neutrinos into the $τ$ regeneration cycle. These results highlight that accounting for the interplay between LIV and matter is essential for future LIV searches at large-scale neutrino telescopes.