Neutrino Telescopes as a Probe of Broken $\mu$-$\tau$ Symmetry

It is known that neutrino oscillations may map $\phi_e : \phi_\mu : \phi_\tau = 1 : 2 : 0$, the initial flavor ratios of ultrahigh-energy neutrino fluxes produced from a distant astrophysical source, into $\phi^D_e : \phi^D_\mu : \phi^D_\tau = 1 : 1 : 1$ at the detector of a neutrino telescope. We remark that this naive expectation is only valid in the $\mu$-$\tau$ symmetry limit, in which two neutrino mixing angles satisfy $\theta_{13} = 0$ and $\theta_{23} = \pi/4$. Allowing for the slight breaking of $\mu$-$\tau$ symmetry, we find $\phi^D_e : \phi^D_\mu : \phi^D_\tau = (1 -2 \Delta) : (1 +\Delta) : (1 +\Delta)$ with $\Delta$ characterizing the combined effect of $\theta_{13} \neq 0$ and $\theta_{23} \neq \pi/4$. Current neutrino oscillation data indicate $-0.1 \lesssim \Delta \lesssim +0.1$. We also look at the possibility to probe $\Delta$ by detecting the $\bar{\nu}_e$ flux of $E_{\bar{\nu}_e} \approx 6.3 {\rm PeV}$ via the Glashow resonance channel $\bar{\nu}_e e \to W^- \to anything$.