Superluminal constraints from ultra-high-energy neutrino events

TL;DR

The paper develops a unified, self-consistent framework to constrain Lorentz Invariance Violation using ultra-high-energy neutrinos, valid for both energy-independent ($n=0$) and quadratic ($n=2$) dispersion relations. It refines decay-width calculations by incorporating VPE and NSpl with full flavor dependence, proper thresholds, and cosmological propagation, and it clarifies when cascade regeneration can be neglected. Applying the method to the KM3-230213A event, the authors obtain robust bounds: for $n=0$, $ obreak\delta$ is constrained more stringently at cosmological distances (e.g., $z\sim1-3$ yields $\delta\lesssim 10^{-22}-10^{-23}$), while for $n=2$, the LIV scale satisfies $\Lambda/ E_{Pl} \gtrsim (3\times10^{-2} - 5\times10^{1})$ depending on $z$. They also relate these stability bounds to potential time-of-flight signatures, finding that current timing effects are sub-second and typically well below detection thresholds, thereby providing a coherent framework for interpreting future UHE neutrino observations and their LIV implications.

Abstract

The $\sim 100\,$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.

Figures (4)

• Figure 1: Constraints on the LIV parameter $\delta=\Delta_0$ as a function of the source redshift $z$ (bottom axis) and the corresponding source distance $L$ in meters (top axis). The left vertical axis shows the upper bound on $\delta$ obtained from Eqs. \ref{['eq:boundaboveth']} and \ref{['eq:boundbelowth']} for different detected energies, each represented by a different color as indicated in the legend. The right vertical axis displays the corresponding VPE threshold energy $E^\text{th}_{0}$ associated with the values of $\delta$ shown on the left axis.
• Figure 2: Heat–map representation of the constraints in the $(E_d,\delta,z)$ space for the $n=0$ case. The bottom axis shows the LIV parameter, while the top axis gives the corresponding VPE threshold energy, which is also shown as a dashed line in the plot. The left vertical axis corresponds to the detected energy $E_d$. Colors encode the constraint derived from the survival--probability condition in the range from $L=10^{17}\,\text{m}$ up to $z=10$. A detected event corresponds to a horizontal line; its intersection with the colored boundary yields the upper bound on $\delta$ at each $z$.
• Figure 3: Constraints on the LIV parameter $\Lambda$ in units of the Planck energy $E_\text{Pl}$ as a function of the source redshift $z$ (bottom axis) and the corresponding source distance $L$ (top axis), for fixed detected energy $E_d$. The left vertical axis shows the bound on $\Lambda/E_\text{Pl}$ obtained from the survival--probability analysis, while the right vertical axis indicates the associated VPE threshold energy. As in the $n=0$ case, the inclusion of cosmic expansion modifies the slope of the curves at large $z$, and the bound on $\Lambda$ rapidly deteriorates.
• Figure 4: Heat–map representation of the constraints in the $(E_d,\Lambda/E_\text{Pl},z)$ space for the $n=2$ case. The bottom axis shows the LIV parameter, while the top axis gives the corresponding VPE threshold energy, which is also shown as a dashed line in the plot. The left vertical axis corresponds to the detected energy $E_d$. Colors encode the constraint derived from the survival--probability condition in the range from $L=10^{17}\,\text{m}$ up to $z=10$. A detected event corresponds to a horizontal line; its intersection with the colored boundary yields the lower bound on $\Lambda$ at each $z$.