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

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.

💡 Research Summary

The paper investigates how Lorentz invariance violation (LIV) described within the Standard‑Model Extension (SME) manifests when neutrinos propagate through the Earth’s matter potential. After a concise review of vacuum‑based LIV phenomenology, the authors introduce the full Hamiltonian H = H₀ + V + H_SME, where H₀ is the standard mass‑driven term, V is the usual MSW‑type matter potential proportional to the electron density, and H_SME encodes all possible LIV operators. They focus on the renormalizable CPT‑odd a^{(3)} and CPT‑even c^{(4)} coefficients, which act as 3 × 3 flavor‑space matrices and can be expanded in spherical harmonics to capture anisotropic (dipole, quadrupole, etc.) background fields.

A key observation is that the energy scaling of these operators differs dramatically from the standard 1/E dependence: a^{(3)} contributes a constant term (E⁰) while c^{(4)} grows linearly with energy (E¹). Consequently, at TeV–PeV energies the LIV Hamiltonian can dominate over the mass term, even for coefficients well below existing bounds. The authors illustrate this with example values for (a^{(3)}y)^{33} and (c^{(4)}{ty})^{33}, showing how the muon‑neutrino survival probability transitions from the familiar oscillatory pattern to a flat, LIV‑driven behavior above ∼100 GeV.

Because the SME background is fixed in a Sun‑centered inertial frame, its projection onto a detector’s local coordinates depends on the Earth’s rotation. For detectors away from the rotation axis (e.g., KM3NeT/ARCA), the zenith angle associated with a given celestial direction varies over a sidereal day, leading to a time‑dependent baseline and matter profile. This induces “sidereal modulations” not only in standard oscillations (through the L/E term) but also in LIV‑induced oscillations, which acquire an additional direction‑dependent modulation reflecting the anisotropic field strength. The paper visualizes these effects, showing null‑oscillation bands where the LIV field is orthogonal to the neutrino momentum and enhanced oscillations when the propagation direction aligns with the field.

The central novelty lies in the regime where the LIV Hamiltonian magnitude becomes comparable to the matter potential V. In this crossover region, the two terms interfere coherently, producing direction‑dependent resonant enhancements of oscillation probabilities. These resonances appear as “oscillation islands” on the right‑ascension/declination sky map, separated by null directions that persist regardless of baseline length. Such features are absent in pure vacuum or pure matter scenarios.

Furthermore, because CPT‑even c^{(4)} operators affect neutrinos and antineutrinos identically, while the matter potential flips sign between them, their combination breaks the ν–\barν symmetry on macroscopic scales. This leads to a measurable asymmetry in the antineutrino flux at high energies, offering a distinct experimental signature.

A third phenomenological consequence concerns the τ‑regeneration cycle. High‑energy ν_τ traversing the Earth produce τ leptons that quickly decay back into ν_τ. LIV‑induced flavor conversion from ν_μ or ν_e into ν_τ enhances the source term for this cycle, potentially increasing the ν_τ flux at the detector by an order of magnitude or more. This effect is especially relevant for IceCube and KM3NeT, which are sensitive to ν_τ‑induced cascade events.

To quantify these effects, the authors implement the full anisotropic LIV Hamiltonian in the nuSQuIDS neutrino‑propagation library and release a Python wrapper called DEIMOS on GitHub. Using the PREM Earth model, they compute realistic density profiles for a wide range of trajectories and demonstrate the impact on observable quantities such as flavor ratios, zenith‑angle distributions, and sidereal time spectra.

In conclusion, the study shows that neglecting the interplay between LIV and matter can miss sizable, direction‑dependent signatures. The three identified phenomena—direction‑dependent resonant enhancement, macroscopic ν–\barν asymmetry for CPT‑even operators, and a boosted ν_τ flux via τ regeneration—provide complementary avenues for future LIV searches with next‑generation neutrino telescopes (IceCube‑Gen2, KM3NeT‑ARCA). The publicly available code enables the community to explore higher‑dimensional operators, different flavor structures, and astrophysical neutrino sources, paving the way for comprehensive tests of Lorentz symmetry in the neutrino sector.