# High-Energy Neutrino Flavor State Transition Probabilities
## 5 Methods  
We have previously shown the high probability of detecting our neutrino to be in states \(\
u_{e}\) or \(\
u_{\mu}\). To consider a higher flavor transition, we must first consider the transition from \(\
u_{\mu}\) to \(\
u_{\tau}\), whose probability can be shown as [2]  
\[P(\
u_{\mu}\rightarrow\
u_{\tau})=\sin^{2}(2\theta_{32})\sin^{2}\Biggl{(}\frac {1.27\Delta m_{32}^{2}[eV]L[m]}{E[MeV]}\Biggr{)} \tag{19}\]  
We now need to expand the PMNS matrix [10] to include a fourth flavor by considering that  
\[ \begin{split}|m_{ee}|_{4v}&=|\Sigma|U_{ej}|^{2}e^ {i\phi_{j}}m_{j}|=\\ |c_{13}^{2}c_{12}^{2}c_{14}^{2}m_{1}+c_{13}^{2}c_{12}^{2}c_{14} ^{2}e^{ika}m_{2}\\ &+s_{13}^{2}c_{14}^{2}e^{i\beta}m_{3}+s_{14}^{2}e^{i\gamma}m_{4} \end{split}  \tag{20}\]  
\[m_{ee}=|\Sigma|U_{ej}|^{2}e^{i\phi_{j}}m_{j}|=\]  
\[ \begin{split}|c_{13}^{2}c_{12}^{2}c_{14}^{2}m_{1}+c_{13}^{2}c_{1 2}^{2}c_{14}^{2}e^{ika}\sqrt{m_{1}^{2}+\Delta_{21}}\\ +s_{13}^{2}c_{14}^{2}e^{i\beta}\sqrt{m_{1}^{2}+\Delta_{31}}+s_{14} ^{2}e^{i\gamma}\sqrt{m_{1}^{2}+\Delta_{41}}\\ \end{split}  \tag{21}\]  
\[ \begin{split}\begin{pmatrix}\
u_{e}\\ \
u_{\mu}\\ \
u_{\tau}\\ \
u_{\lambda}\end{pmatrix}&=\begin{pmatrix}U_{e1}&U_{e2}&U_{e3}&U_{e4}\\ U_{\mu 1}&U_{\mu 2}&U_{\mu 3}&U_{\mu 4}\\ U_{\tau 1}&U_{\tau 2}&U_{\tau 3}&U_{\tau 4}\\ U_{\lambda 1}&U_{\lambda 2}&U_{\lambda 3}&U_{\lambda 4}\\ \end{pmatrix}\end{pmatrix}\end{split}  \tag{22}\]  
where \(U=R_{34}R_{24}R_{14}R_{23}R_{13}R_{12}P\) so that [10]  
\[ \begin{split} R_{14}=\left(\begin{array}{cccc}c_{14}&0&0&-s _{14}e^{i\phi_{14}}\\ 0&1&0&0\\ 0&0&1&0\\ -s_{14}e^{i\phi_{14}}&0&0&c_{14}\\ \end{array}\right)\text{and}\\ R_{34}=\left(\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&c_{34}&s_{34}\\ 0&0&-s_{34}&c_{34}\\ \end{array}\right)\end{split}  \tag{23}\]  
We can now define the probability for a higher-order transition as  
\[P(\
u_{\tau}\rightarrow\
u_{\lambda})=\sin^{2}(2\theta_{34})\sin^{2}\Biggl{(} \frac{1.27\Delta m_{43}^{2}[eV]L[m]}{E[MeV]}\Biggr{)} \tag{24}\]where \(L[m]\) is the distance from the source, \(E[MeV]\) is the energy and \(\Delta m\) is the mass difference between the flavors during the oscillation. This indicates that the probability of finding a neutrino in the fourth flavor state should be equal to the probability of finding it in the third flavor state. Taking what we know from our probability functions, Equations (12) and (19), we can now use this to determine the probability for an \(n\)-th flavor. If we assume an oscillation into a fourth flavor, the sterile neutrino, is probable, then we can assume an equal probability of continuous oscillation into \(n\) flavors. We can now determine the theoretical travel distance for oscillation from \(\
u_{e}\rightarrow\
u_{\mu}\) by assuming our mixing angle \(\theta\) to be \(\sim\)0.846 and our \(|\Delta m_{21}|^{2}\) to be approximately \(7.4\times 10^{-5}\,\left(\text{eV}/\text{c}^{2}\right)^{2}\), and we can assume an approximate energy \(E\) of \(10^{8}\,\text{eV}\), assuming a supernova neutrino creation. We can show a travel distance of approximately \(10^{21}\,\text{m}\) or around \(105,700\) light years, with a probability of \(\sim\)1 or \(\sim\)100% of the transition from electron neutrino, \(\
u_{e}\), to muon neutrino, \(\
u_{\mu}\). Given this amount of travel distance, the neutrino would have definitively transitioned from its electron state to its muon state. To ascertain the pattern of transitional probabilities, a Python 3.9.6 program was developed (see Appendix A) to compare transitions between three widely detected neutrino types: solar, supernova and cosmic rays. For our first simulation, the following parameters were used for solar neutrinos:  
For the first flavor change we used for our mixing angle \(\sin^{2}\theta\approx 0.846\) and a mass difference of \(\Delta m^{2}_{21}\approx 7.40\times 10^{-5}\,\text{eV}^{2}\). For our second flavor change, \(\sin^{2}\theta\approx 0.92\) and \(\Delta m^{2}_{32}\approx 2.40\times 10^{-3}\,\text{eV}^{2}\) were used.