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 $\theta_{23} = \pi/4$\mathrm{.}Allowingfortheslightbreakingof$\mu$-$\tau$symmetry,wefind$\phi^D_e : \phi^D_\mu : \phi^D_\tau = (1 -2 \Delta) : (1 +\Delta) : (1 +\Delta)$with$\Delta$characterizingthecombinedeffectof$\theta_{13} \neq 0$and$\theta_{23} \neq \pi/4$\mathrm{.}Currentneutrinooscillationdataindicate$-0.1 \lesssim \Delta \lesssim +0.1$\mathrm{.}Wealsolookatthepossibilitytoprobe$\Delta$bydetectingthe$\bar{\nu}e$fluxof$E{\bar{\nu}_e} \approx 6.3 {\rm PeV}$viatheGlashowresonancechannel$\bar{\nu}_e e \to W^- \to anything$.