Aspects of the flavor triangle for cosmic neutrino propagation

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 $\overrightarrow{w}=\mathcal{P}\overrightarrow{W}$. 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 $\delta ({\theta }_{ij})$ 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 ${\theta }_{32}$ 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: $w_{\text{shower}}=w_e+w_{\tau }$ and $w_{\text{track}}=w_{\mu }$. 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.

Figure 1

The analog of the Earthly triangle is shown (red interior triangle) for the mixing angles that relate quark flavors and masses. Also shown is the centroid point, labeled “C.”

Figure 2

The Earthly triangles for the best values (from left to right) of the normal hierarchy with ${\theta }_{32}$ in the first octant, the normal hierarchy with ${\theta }_{32}$ in the second octet, and the inverted hierarchy.

Figure 3

The left panel shows the flavor triangle at Earth, for the best-fit values of the normal hierarchy with ${\theta }_{32}$ in the first octant (also shown in Fig. 2), and the right panel shows the straight line that results from the assumption that ${\nu }_{\tau }$ production at the source is negligible. Note that the “no ${\nu }_{\tau }$ injected” line is a boundary of the full triangle, and therefore does not contain the centroid (shown) at the point ($\frac{1}{3},\frac{1}{3},\frac{1}{3}$). The small red line is the $2\sigma$ statistical error at the centroid values, assuming 140 total events. We note that the statistical error bar is comparable to the width of the Earthly triangle, shown in the left panel.

Figure 4

The $2\sigma$ ($m=2$, 95% C.L.) standard deviation from Eq. (20) for $N_{\text{Total}}=140$ events, versus the measured $w^0$ (the same for $w_{\text{track}}$ and $w_{\text{shower}}$). The dots denote the special values of $w^0=1/2$ where the standard deviation has a maximum, and $w^0=1/3$, the centroid value expected for neutrino flavors from the pion decay chain.

Figure 5

Source triangle partitioned into six subtriangles, each characterized by a distinct ordering of the injection ratios $\{W_e,W_{\mu },W_{\tau }\}$ as well as the Earthly ratios $\{w_e,w_{\mu },w_{\tau }\}$.