# High-Energy Neutrino Flavor State Transition Probabilities
## 7 Analysis of Results  
In Figures 2-4, we show a comparison between the probabilities of the flavor transitions \(\
u_{e}\to\
u_{\mu}\), \(\
u_{\mu}\to\
u_{\tau}\) and \(\
u_{\tau}\to\
u_{\lambda}\) for solar, supernova and cosmic ray neutrinos, where \(\lambda\) is any higher-order flavor transition from the 4-th...\(n\)-th flavor states. We can see from the probability distribution that the closer we are to the source, the lower the probability of a higher-order flavor transition, as expected. However, as the neutrino oscillates at greater distances, a pattern of fluctuating probabilities can be observed. We can infer from this that as the neutrino approaches \(\sim\)10\({}^{25}\) m from its source, the probabilities for all three flavors are all three approximations of each other, \(\pm\)0.10. With this understanding, we can then follow the flavor transition by reverting backward through each flavor culminating at an equivalent point for both \(\
u_{e}\) and \(\
u_{\mu}\). From there, and as the neutrino oscillates further, we can then assume a probability for the first flavor as it approaches the probability for the \(n\)-th flavor transition, in this case, the sterile neutrino. Figure 3: Scatter plot depicting the comparison of probabilities between supernova neutrino flavor transitions \(\
u_{e}\to\
u_{\mu}\), \(\
u_{\mu}\to\
u_{\tau}\) and \(\
u_{\tau}\to\
u_{\lambda}\). Where the blue, orange and green dots represent the predicted flavor for second, third and higher-order transitions, respectively. Figure 2: Scatter plot depicting the comparison of probabilities between solar neutrino flavor transitions \(\
u_{e}\to\
u_{\mu}\), \(\
u_{\mu}\to\
u_{\tau}\) and \(\
u_{\tau}\to\
u_{\lambda}\). Where the blue, orange and green dots represent the predicted flavor for second, third and higher-order transitions, respectively. From the data in the figures above, we can ascertain that at a certain point, the probability of a neutrino oscillating into a fourth flavor state after a certain distance approaches the probability that the neutrino is being found in its first flavor state. This further shows that each flavor can be thought of as a proper combination of all mass and flavor eigenstates oscillating back and forth between states. The probability of oscillating into a higher \(n\)-th flavor is approximately the same as that of the neutrino oscillating backward through the previous states and beginning the cycle. We can see in Figure 2, as the solar neutrino has traveled \(\sim\)10\({}^{27}\) m from its source, the probability of it being found in the electron state is approximately the probability of it being found in its lambda state. From Figure 3, we can see the same equivalency; however, with a supernova neutrino the distance is now \(\sim\)10\({}^{20}\) m. With higher-energy neutrinos, such as those in Figure 4, we can see a unique symmetry between all three flavors of neutrino states. Here, we can see where one flavor has a high probability, and the other two are at their lowest, and vice versa. When we consider the neutrino's mean free path of \(\sim\)105, 700 light years, or 10\({}^{21}\) m from its source, we can see that this fits perfectly with the neutrino energy stemming from supernova neutrinos. However, when we focus on the lower-energy solar neutrinos, we find that given a slightly further distance of travel (10\({}^{27}\) m), we arrive at a near 0.5 probability of detecting our neutrino in the first and fourth flavor states. An inference here tells us that the lower-energy neutrinos, given a longer distance of oscillation, have a higher probability of oscillating back through and beginning the cycle over again. Higher-energy levels tend to oscillate much faster, creating a uniquely consistent probability distribution for flavor transformations. From Figure 5, we can see the flavor fraction of detected neutrinos at the Reactor Experiment for Neutrino Oscillations (RENO) in South Korea [14].