High-energy cosmic neutrinos as a probe of the vector mediator scenario
  in light of the muon $g-2$ anomaly and Hubble tension
  In light of the recent Muon $g-2$ experiment data from Fermilab, we
investigate the implications of a gauged $L_{\mu} - L_{\tau}$ model for high
energy neutrino telescopes. It has been suggested that a new gauge boson at the
MeV scale can both account for the Muon $g-2$ data and alleviate the tension in
the Hubble parameter measurements. It also strikes signals at IceCube from the
predicted resonance scattering between high-energy neutrinos and the cosmic
neutrino background. We revisit this model based on the latest IceCube shower
data, and perform a four-parameter fit to find a preferred region. While the
data are consistent with the absence of resonant signatures from secret
interactions, we find the preferred region consistent with the muon $g-2$
anomaly and Hubble tension. We demonstrate that future neutrino telescopes such
as IceCube-Gen2 can probe this unique parameter space, and point out that
successful measurements would infer the neutrino mass with $0.05~{\rm
eV}\lesssim \Sigma m_\nu\lesssim 0.3~{\rm eV}$.
