Potential divergence in tracing μ and τ 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.