The Cosmic Horizon of Neutrinos

Abstract

The persistent discrepancy between the experimental measurement and the Standard Model (SM) prediction of the muon's anomalous magnetic moment $(g-2)_μ$ remains one of the most intriguing hints of physics beyond the SM. A well-motivated explanation involves a light $Z'$ gauge boson associated with a broken $U(1)_{L_μ- L_τ}$ symmetry. Such a boson not only resolves the $(g-2)_μ$ anomaly, but also induces resonant interactions between high-energy cosmic neutrinos and the cosmic neutrino background (C$ν$B), potentially shaping the observable neutrino flux at Earth. In this work, we explore the implications of such interactions for the cosmic propagation of high-energy neutrinos. We compute the optical depth for neutrino attenuation via $Z'$-mediated scattering, accounting for neutrino masses, hierarchies, and thermal distributions. We delineate the regions in $(m_{Z'}, m_ν)$ space where the optical depth exceeds unity, defining a ``neutrino cosmic horizon'' beyond which high-energy neutrinos are significantly attenuated. We confront these results with the parameter space required to simultaneously explain the muon $g-2$ anomaly and ease the Hubble tension via an additional contribution to the effective number of relativistic degrees of freedom, $ΔN_{\mathrm{eff}} \simeq 0.2-0.5$. Our analysis reveals a consistent region in parameter space where all three phenomena-$(g-2)_μ$, $N_{\mathrm{eff}}$, and high-energy neutrino attenuation-can be explained by the same light mediator. These findings motivate future searches for spectral features in IceCube and its next-generation successors as indirect probes of new physics in the neutrino sector.

Figures (4)

• Figure 1: Optical depth $\log_{10}(\tau_\nu)$ as a function of $m_{Z'}$ and $m_\nu$ for the normal (left) and inverted (right) neutrino mass hierarchies. we computed the optical depth for PeV-scale neutrinos ($E_\nu \sim 10^{15}$ eV), which corresponds to the energy range of the highest-energy neutrinos observed by IceCube. Yellow bands correspond to regions where $\tau_\nu \gtrsim 1$. The regions above the white dashed lines are ruled out by cosmology Planck:2018vyg.
• Figure 2: As in fig. \ref{['fig:Z-prime vs neutrino']}, but for an impinging neutrino energy of 0.1 PeV (left) and 10 PeV (right), for the normal hierarchy.
• Figure 3: Cutouts of the optical depth $\log_{10}(\tau_\nu)$ as a function of $\log_{10}(m_{Z'}\,[\mathrm{GeV}])$ for fixed values of the lightest neutrino mass. The left panel shows the normal hierarchy and the right panel the inverted hierarchy. Colored curves correspond to $\log_{10}(m_{\nu,\min}[\mathrm{GeV}])=-10.5,-11,-12,$ and $-13$, as indicated in the legend. The horizontal dashed line denotes $\tau_\nu = 1$. Sharp peaks arise when the resonance condition $m_{Z'}^2 \simeq 2 E p$ is satisfied within the thermally populated region of the cosmic neutrino background. For sufficiently small $m_{Z'}$, the resonance momentum lies below the thermal scale and the optical depth becomes large, whereas for larger $m_{Z'}$ the required resonance momentum exceeds the characteristic thermal momentum, leading to exponential suppression from the tail of the Fermi--Dirac distribution. The position and width of the peaks therefore reflect the interplay between the mediator mass, the neutrino energy, and the C$\nu$B thermal scale rather than a simple dependence on the neutrino rest mass alone.
• Figure 4: Parameter space in the $(\log_{10} m_{Z'}, \log_{10} g_{Z'})$ plane showing the $(g-2)_\mu$$2\sigma$ preferred region (red) and the region where $0.2 < \Delta N_{\rm eff} < 0.5$ (light blue). Superimposed vertical bands indicate the ranges of mediator mass $m_{Z'}$ for which the optical depth satisfies $\tau_\nu > 1$ for selected values of the lightest neutrino mass and representative neutrino energies. The band widths reflect the resonance condition and the full thermal averaging over the cosmic neutrino background. The overlap (or lack thereof) between these bands and the $(g-2)_\mu$ region illustrates the degree to which efficient high-energy neutrino attenuation can occur within phenomenologically viable couplings.