Neutrino flux ratios at neutrino telescopes: The role of uncertainties of neutrino mixing parameters and applications to neutrino decay

In this paper, we derive simple and general perturbative formulas for the flavor flux ratios $R_{\alpha \beta }={\varphi }_{{\nu }_{\alpha }}/{\varphi }_{{\nu }_{\beta }}$ that could be measured at neutrino telescopes. We discuss in detail the role of the uncertainties of the neutrino mixing parameters showing that they have to be seriously taken into account in any realistic discussion about flavor measurements at neutrino telescopes. In addition, we analyze the impact of such uncertainties in telling the standard neutrino oscillation framework from scenarios involving, e.g., neutrino decay and we find that the ratio $R_{e\mu }$ is the most sensitive one to “new physics” effects beyond the standard model. We also compute the more realistic muon-to-shower ratio for a particular configuration of the IceCube experiment, observing that using this experimental quantity a clear separation between standard and nonstandard neutrino physics cannot be obtained.

Figure 1

Comparison between the approximate formulas $R_{e\mu }^{\mathrm{first}}$ and $R_{e\mu }^{\mathrm{second}}$ as well as the exact formula for $R_{e\mu }$. The solid curve corresponds to the reference value 1, whereas the dotted and dashed curves correspond to the first- and second-order approximations, respectively.

Figure 2

The spread of the three flux ratios considered in this work due to the present uncertainties of the neutrino mixing parameters [see Eq. (20)], around the best-fit values ${\overline{\theta }}_{12}=33°$ and ${\overline{\theta }}_{23}=45°$, as a function of the mixing angle ${\theta }_{13}$. From above to below: $R_{e\tau }$, $R_{\mu \tau }$, and $R_{e\mu }$.

Figure 3

The parameter space giving a value $R_{e\mu }=1\pm 5\%$. The dotted, solid, and dashed curves correspond to ${\theta }_{23}=43°$, 45°, 47°, respectively. The mixing angle ${\theta }_{12}$ is fixed to 33°.

Figure 4

The two different neutrino decay scenarios. Upper-left picture: Neutrino decay I with normal hierarchy (${\alpha }_{11}=0\Rightarrow d_1=1$). Upper-right picture: Neutrino decay I with inverted hierarchy (${\alpha }_{33}=0\Rightarrow d_3=1$). Lower-left picture: Neutrino decay II with normal hierarchy (${\alpha }_{11}={\alpha }_{22}=0\Rightarrow d_1=d_2=1$). Lower-right picture: Neutrino decay II with inverted hierarchy (${\alpha }_{11}={\alpha }_{33}=0\Rightarrow d_1=d_3=1$).

Figure 5

The flux ratio $R_{e\mu }$ including neutrino decay as a function of the mixing angle ${\theta }_{13}$. The middle area corresponds to the standard neutrino oscillation framework, whereas the upper and lower areas correspond to neutrino decay with the lightest neutrino mass eigenstate being stable only for NH and IH, respectively. The case of the two lightest neutrino mass eigenstates being stable is visualized by the solid (NH) and dotted (IH) curves, respectively. The uncertainties of the neutrino mixing angles are those given in Eq. (20), around the best-fit values ${\overline{\theta }}_{12}=33°$ and ${\overline{\theta }}_{23}=45°$.

Figure 6

The flux ratio $R_{\mu \tau }$ including neutrino decay as a function of the mixing angle ${\theta }_{13}$. The innermost area corresponds to the standard neutrino oscillation framework, whereas the outermost and middle areas correspond to neutrino decay with the lightest neutrino mass eigenstate being stable only for NH and IH, respectively. The case of the two lightest neutrino mass eigenstates being stable is visualized by the solid (NH) and dotted (IH) curves, respectively. The uncertainties of the neutrino mixing angles are those given in Eq. (20), around the best-fit values ${\overline{\theta }}_{12}=33°$ and ${\overline{\theta }}_{23}=45°$.

Figure 7

The muon-to-shower ratio $\mathcal{R}$ as a function of the mixing angle ${\theta }_{13}$, evaluated for three of the scenarios discussed in the main text: standard neutrino framework (boxes), “neutrino decay I,” NH (diamonds), and “neutrino decay I,” IH (circles). Left plot: Results obtained considering 1 yr of data taking at IceCube. Right plot: Results obtained considering ten years of data taking at IceCube. In both plots, the uncertainties of the neutrino mixing angles are those given in Eq. (20).