# High-Energy Neutrino Flavor State Transition Probabilities
## 6 Results  
We are now in a position to use our expression for neutrino transitional probabilities to estimate the neutrino transition fractions (among three species) at various distances. First, we turn to our most important neutrino source, the Sun. The transitional probabilities from \(\
u_{e}\rightarrow\
u_{\mu}\) and \(\
u_{\mu}\rightarrow\
u_{\tau}\) are shown in Table 1 for solar neutrinos. For Table 2, we maintain the parameters used above and apply them to neutrinos detected from supernova events. We assume a median supernova neutrino energy level of \(\sim\)10 MeV. \begin{table}
\begin{tabular}{c c c c} \hline
**Distance \(L[m]\)** & \(\mathbf{P}(\
u_{e}\rightarrow\
u_{\mu})\) & \(\mathbf{P}(\
u_{\mu}\rightarrow\
u_{\tau})\) & \(\mathbf{P}(\
u_{\tau}\rightarrow\
u_{\lambda})\) \\ \hline \(10^{5}\) & 0.358 & 0.602 & 0.140 \\ \(10^{10}\) & 0.548 & 0.919 & 0.399 \\ \(10^{15}\) & 0.205 & 0.268 & 0.041 \\ \(10^{20}\) & 0.220 & 0.692 & 0.792 \\ \(10^{25}\) & 0.868 & 0.117 & 0.355 \\ \(10^{27}\) & 0.595 & 0.117 & 0.822 \\ \hline \end{tabular}
\end{table} 
Table 2: Oscillation probabilities from supernova neutrinos at various distances from the source of neutrino production. Finally, in Table 3, we consider a significant increase in energy. This is partly due to the first- and second-order Fermi processes within the magnetized clouds accelerating the cosmic rays. With this in mind, and again utilizing our base parameters, we now have an increased cosmic ray energy of \(\sim\)\(1\times 10^{6}\) MeV. From the above tables, we have shown the distance needed to oscillate from the first flavor to the second flavor, with a consistently high probability of transition. Once we move into the second, third and \(n\)-th flavors, we become less certain about our flavor transition probabilities. From the above two tables, we can infer why this is the case. We can clearly see that our probabilities for the higher flavors (\(\mu\to\tau,\tau\to\lambda\)) are the lowest when the probability for the transition from the first to the second flavor (\(e\to\mu\)) is at its highest. For example, for a cosmic ray neutrino, using our transition from \(v_{\tau}\to v_{\lambda}\) at distances \(10^{5}\) m and \(10^{10}\) m, we obtained probabilities of \(\sim\)0.00% and \(\sim\)12.6%, respectively. When comparing this with our probability from \(v_{e}\to v_{\mu}\) at the same distances, we find a probability of 99.9% but a significant decrease to \(\sim\)35.8% when \(v_{\mu}\to v_{\tau}\) is \(\sim\)60.2%. However, in Tables 1 and 3 there are distances from which our first flavor transition is approximately equal to our third flavor transition. In the case of solar neutrinos, at \(L[m]\)\(\sim\)\(10^{20}\) m the probabilities for first and third transitions are \(\sim\)44.9% and \(\sim\)45.7%, respectively. For the case with cosmic rays, we have that at \(L[m]\)\(\sim\)\(10^{27}\) m our first transition is \(\sim\)60.5% while our third transition is \(\sim\)66.7%. If we consider our distance of travel to be on the order of 1 AU Unit (AU), we find that our probability would be \(\sim\)0.261 for flavor oscillation into the second state and \(\sim\)0.111 for higher-order flavor transitions. When we compare this to known detection rates, we find approximately \(\frac{2}{3}\) of detected neutrinos from the Sun to be in either the \(v_{e}\) or \(v_{\mu}\) flavor state, which is consistent with our findings in the range of \(\sim\)26.1% found in the \(v_{e}\) flavor state [13]. A similar assumption can be made for atmospheric neutrinos. As cosmic rays bombard the atmosphere of Earth, protons strike the nuclei of atoms within our atmosphere, forming pions. These short-lived particles decay into muons which then leave behind an electron, anti-electron and a neutrino, the \(v_{\mu}\). This flavor, according to Fermilab, makes up approximately \(\frac{2}{3}\) of the detected neutrinos on Earth, leaving behind \(\frac{1}{3}\) to be detected as \(v_{e}\)[13].