Superluminal constraints from ultra-high-energy neutrino events
The $220^{+570}_{-100}\,$PeV neutrino detected by KM3NeT marks the beginning of ultra-high-energy neutrino astronomy and provides a powerful probe of Lorentz Invariance Violation (LIV). In superluminal scenarios, neutrinos can decay through vacuum $e^-e^+$ pair emission or neutrino splitting. Previous analyses of the KM3-230213A event relied on simplified survival-probability estimates and, in some cases, used inaccurate decay-width expressions or neglected redshift and threshold effects. In this work we present a unified and self-consistent framework that corrects these issues and applies to both the energy-independent ($n=0$) and quadratic ($n=2$) superluminal cases. We collect and recast the decay-width and threshold expressions, clarify their flavor dependence, and include a consistent treatment of cosmological propagation. We also assess the impact of cascade regeneration and show that cascade effects are negligible for the purpose of setting LIV bounds. The survival-probability approximation adopted in previous works is therefore justified, while our framework provides a coherent basis for future analyses of superluminal neutrino constraints, which should consistently include possible time-delay signatures.