# High-Energy Neutrino Flavor State Transition Probabilities
## 8 Discussion  
In this paper, we have shown that we can determine the probabilities of neutrino flavor states by first calculating the product of flavor eigenstate and then mass eigenstate probabilities. We found that the greater the distance from the source, the higher the probability of finding the neutrino in a different state than that in which it was produced. We found that for solar neutrinos, \(10^{5}\) m from the source, the neutrino has a 99.9% chance of being detected in the state \(\
u_{\mu}\) and a 46.4% chance of oscillating into a fourth flavor or simply reverting backwards. We find that at around \(10^{27}\) m this process starts over again. For supernova neutrinos at the same distances, we expect to find \(\
u_{\tau}\) with a 60.2% chance and \(\
u_{e}\) with a 35.8% chance. This time at around \(10^{25}\) m, we find the process starting again. For the higher-energy cosmic rays, we find a similarity between their probabilities and those of the solar neutrinos. When comparing at the same distances, we have a 99.9% chance of detection in the state \(\
u_{e}\) and a 0.00% chance of higher flavor transitions. Again, back to our distance of \(10^{27}\) m, we find the process starts again. Figure 8: The above image shows the fraction of predicted cosmic ray neutrino flavor detection given an oscillation distance of \(\sim\)10\({}^{11}\) m.  
Figure 7: The above image shows the fraction of predicted supernova neutrino flavor detection given an oscillation distance of \(1.6\times 10^{21}\) m.  
In repeating the same process, but for a distance of one AU unit, or \(1.5\times 10^{11}\) m, we found that our neutrino flux should be comprised of approximately 50% electron neutrinos, 26.7% muon neutrinos and 23% tau neutrinos. When we extend this distance a bit further for supernova, we found approximately 48%, 32% and 20% electron, muon and tau neutrinos, respectively. When considering higher-energy cosmic rays at a distance of approximately \(10^{11}\) m, we found a surprising split between 36%, 33% and 31% for the electron, muon and tau neutrinos. We have shown with minimal assumptions on a well-established neutrino transition probability function that neutrino population statistics differ significantly among astrophysical sources. Both solar and supernova neutrino populations detected on Earth are relatively \(v_{e}\)-dominant, whereas higher-energy cosmic ray neutrinos that are candidates for Ice Cube, Super Kamiokande and future planned detectors have a relatively uniform flavor distribution. So far, only three flavor states have been detected; however, we have shown that the likelihood of oscillating into a fourth flavor is equal to the oscillation back through the previous flavors, \(\
u_{\tau}\to\
u_{\mu}\to\
u_{e}\). This raises an important question: since we know the flavors are linear combinations of the mass eigenstates, is it possible for a fourth flavor transition or would the neutrino's transformation cease at \(v_{\tau}\) and simply revert back to the first flavor state? To put it more concisely, measuring in the flavor state prevents us from identifying the mass state, except as a linear combination of all possible masses. In future research, we may explore whether the assumption of non-locality suggests that mass eigenstates can also be represented as flavor eigenstates. We determined that when examining the probabilities of detected states, we can make predictions based on the distance traveled since their creation. We discovered that a neutrino has a nearly certain probability not only of oscillating further but also of oscillating back into earlier states. This opens up opportunities for further research to focus on mass eigenstates and investigate if combinations of masses can replicate other flavors, and vice versa. Conceptualization, J.H. ; data curation, J.H. ; formal analysis, J.H. ; funding acquisition, R.A.; investigation, J.H. ; methodology, J.H. ; resources, J.H. and R.A.; software, J.H.