Potential divergence in tracing μ and τ flavors of astrophysical neutrinos

We derive general formulas for three flavor fractions ( η e , η μ , η τ ) of the high-energy neutrinos originating from a remote astrophysical source by using their flavor ratios ( f e , f μ , f τ ) observed at a neutrino telescope, and diagnose a potential divergence associated with η μ and η τ as an unavoidable consequence of the μ − τ interchange symmetry exhibiting in the 3 × 3 lepton flavor mixing matrix U . We present a complete set of analytical expressions for ( η e , η μ , η τ ) as functions of two typical μ − τ 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 η e and η μ + η τ can be extracted from a precision measurement of f e and f μ = f τ in the exact μ − τ flavor symmetry limit.