Potential divergence in tracing $\mu$ and $\tau$ flavors of astrophysical neutrinos

We derive general formulas for three flavor fractions $({\eta }_e^{},{\eta }_{\mu }^{},{\eta }_{\tau }^{})$ of the high-energy neutrinos originating from a remote astrophysical source by using their flavor ratios $(f_e^{},f_{\mu }^{},f_{\tau }^{})$ observed at a neutrino telescope, and diagnose a potential divergence associated with ${\eta }_{\mu }^{}$ and ${\eta }_{\tau }^{}$ as an unavoidable consequence of the $\mu$-$\tau$ interchange symmetry exhibiting in the $3\times 3$ lepton flavor mixing matrix $U$. We present a complete set of analytical expressions for $({\eta }_e^{},{\eta }_{\mu }^{},{\eta }_{\tau }^{})$ as functions of two typical $\mu$-$\tau$ symmetry breaking parameters in the standard parametrization of $U$, and apply it to the recent IceCube all-sky neutrino flux data ranging from 5 TeV to 10 PeV in the assumption that the relevant sources have a common flavor composition. We also explain why only ${\eta }_e^{}$ and ${\eta }_{\mu }^{}+{\eta }_{\tau }^{}$ can be extracted from a precision measurement of $f_e^{}$ and $f_{\mu }^{}=f_{\tau }^{}$ in the exact $\mu$-$\tau$ flavor symmetry limit.