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 $W_e,W_{\mu },W_{\tau }$ evolve to flavor ratios at Earthly detectors $w_e,w_{\mu },w_{\tau }$ according to $\vec{w}=\mathcal{P}\vec{W}$. 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 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 motivated such constraints. The first is the possibility of a vanishing determinant for $\mathcal{P}$. We give a formula for a unique $\delta ({\theta }_{ij}$'s) 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 ${\theta }_{32}$ and ${\theta }_{13}$ 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, $w_{\mathrm{shower}}=w_e+w_{\tau }$ and $w_{\mathrm{track}}=w_{\mu }$. We apply the statistical uncertainty to be expected from IceCube to this model. Finally, we consider ramifications of the expected lack of ${\nu }_{\tau }$ injection at cosmic sources. In particular, this constraint reduces the Earthly triangle to a boundary line of the triangle. Some tests of this "no ${\nu }_{\tau }$ injection" hypothesis are given.

arXiv comment22 pages, 5 figures. Accepted by Phys. Rev. D

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