  In this work we analyze the scenario where a MeV scale $L_\mu - L_\tau$ gauge
boson can explain the deficit in the diffuse ultra high energy (UHE)
astrophysical neutrino spectrum observed in IceCube, as well as the discrepancy
between experimental and $e^+e^-$ dispersion data driven SM calculations of the
muon anomalous magnetic moment. We map the parameter space of the model where
the elastic resonant s-channel scattering of UHE neutrinos with the cosmic
neutrino background, mediated by the new Z', can improve the description of the
observed cascade and track spectra over the no-scattering hypothesis. Comparing
to recent NA64$\mu$ results, we find that some part of the parameter space
remains unexplored, but at a data volume of $10^{11}$ muons on target NA64$\mu$
will completely probe this region.
