Aspects of the flavor triangle for cosmic neutrino propagation

Abstract

Over cosmic distances, astrophysical neutrino oscillations average out to a classical flavor propagation matrix $\mathcal{P}$. Thus, flavor ratios injected at the cosmic source ${W}_{e},{W}_{\ensuremath{\mu}},{W}_{\ensuremath{\tau}}$ evolve to flavor ratios at Earthly detectors ${w}_{e},{w}_{\ensuremath{\mu}},{w}_{\ensuremath{\tau}}$ according to $\stackrel{\ensuremath{\rightarrow}}{w}=\mathcal{P}\stackrel{\ensuremath{\rightarrow}}{W}$. The unitary constraint reduces the Euclidean octant to a ``flavor triangle.'' We prove a theorem that the area of the Earthly flavor triangle is proportional to $\text{Det}(\mathcal{P})$. One more constraint would further reduce the dimensionality of the flavor triangle at Earth (two) to a line (one). We discuss four such constraints. The first is the possibility of a vanishing determinant for $\mathcal{P}$. We give a formula for a unique $\ensuremath{\delta}({\ensuremath{\theta}}_{ij})$ that yields the vanishing determinant. Next, we consider the thinness of the Earthly flavor triangle. We relate this thinness to the small deviations of the two angles ${\ensuremath{\theta}}_{32}$ and ${\ensuremath{\theta}}_{13}$ from maximal mixing and zero, respectively. Then we consider the confusion resulting from the tau-neutrino decay topologies, which are showers at low energy, ``double-bang'' showers in the PeV range, and a mixture of showers and tracks at even higher energies. We examine the simple low-energy regime, where there are just two topologies: ${w}_{\text{shower}}={w}_{e}+{w}_{\ensuremath{\tau}}$ and ${w}_{\text{track}}={w}_{\ensuremath{\mu}}$. We apply the statistical uncertainty to be expected from IceCube to this model. Finally, we consider ramifications of the expected lack of ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ injection at cosmic sources. In particular, this constraint reduces the Earthly triangle to a boundary line of the triangle. Some tests of this ``no ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ injection'' hypothesis are given.