Aspects of the Flavor Triangle for Cosmic Neutrino Propagation

TL;DR

This paper investigates the geometric properties of the flavor triangle for cosmic neutrino propagation, establishing a link between the flavor triangle's area and the propagation matrix determinant, and explores constraints affecting flavor ratios at Earth.

Contribution

It proves a theorem relating the flavor triangle area to the determinant of the propagation matrix and discusses multiple physical constraints that reduce the flavor ratio space at Earth.

Findings

The flavor triangle area is proportional to Det(𝒫).

Vanishing determinant occurs at specific mixing angles.

Constraints like no tau injection reduce the flavor space.

Abstract

Over cosmic distances, astrophysical neutrino oscillations average out to a classical flavor propagation matrix $\mathscr{P}$. 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 $\vec{w}=\mathscr{P} \vec{W}$. 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$(\mathscr{P})$. 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 $\mathscr{P}$. 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, $w_{\rm shower}=w_e+w_\tau$ and $w_{\rm track}=w_\mu$. 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.