# High-Energy Neutrino Flavor State Transition Probabilities
## 3 Oscillation and the Mean Free Path  
Using the conversion probability function, we can ascertain the travel distance for oscillation and anticipate the state of our neutrino at a specific position and time. To simplify this process, it helps to conceptualize oscillating neutrinos as wave packets existing in some arbitrary state, \(|\psi\rangle\). This arbitrary state is our initial condition, in which we have \(|\psi_{a}\rangle\), where \(\alpha\) is the flavor eigenstate pre-measurement (or detection), \(|v_{e},v_{\mu},v_{\tau}\rangle\). Similarly, \(|\psi_{a}^{\prime}\rangle\) is our final state after detection. We must also consider the probability of measurement of the mass eigenstate. For the sake of simplification, we use the definition for neutrino flavor states as explained by [PERSON] et al. (2022) [9], utilizing the \(3\times 3\) Pontecorvo-Maki-Nakagawa-Sakata [PMNS] unitary mixing matrix [10], \(\mathbf{U}(\bar{\theta},\delta)\) as follows:  
\[|\psi^{(f)}\rangle=\mathbf{U}(\bar{\theta},\delta)\;|\psi^{(m)}\rangle \tag{8}\]  
Figure 1: Feynman diagram depicting the scattering of an incoming muon neutrino and an at-rest electron. Figure used with permission of author [8]. It is imperative to note that, in the above equation, we must explicitly show that there are two possible eigenstates: \(\ket{\underline{\
u}^{(f)}}\) is our flavor eigenstate representing the transpose of \((\ket{v_{e}},\ket{v_{\mu}},\ket{v_{\tau}})\). The, we have that \(\ket{\underline{\
u}^{(m)}}\) is our mass eigenstate, which is represented by the transpose of \((\ket{v_{1}},\ket{v_{2}},\ket{v_{3}})\). The probability for the flavor transition \(v_{\alpha}\to v_{\beta}\)[2] can be defined as  
\[P_{v_{\alpha}\to v_{\beta}}(t)=|\bra{v_{\beta}}|\ket{v_{\alpha}(t)}|^{2}=| \tilde{\mathbf{U}}_{\alpha\beta}(t)|^{2} \tag{9}\]  
This indicates the flavor transition at time \(t=0\), with the contribution of [9]. Here, we can assume from earlier that our variables, \(\alpha\) and \(\beta\), can represent any of the flavors \((e,\mu,\tau)\), and we have that \(\tilde{\mathbf{U}}(t=0)\) is normalized [9]. We know from above that our neutrino flavor mixing matrix can be written as  
\[ \begin{pmatrix}v_{e}\\ v_{\mu}\end{pmatrix} = \begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}v_{1}\\ v_{2}\end{pmatrix}  \tag{10}\]  
We can now show this as a unitary mixing matrix  
\[\mathbf{U}(\theta)= \begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}  \tag{11}\]  
Assuming that neutrinos have the ability to oscillate into three, four or even an infinite number of flavors, we can illustrate the mixing phenomenon by examining a state involving two flavors. It is important to note that in the previous discussion, neutrinos were regarded as flavor states rather than as a combination of mass eigenstates, \(v_{1},v_{2},v_{3}\). With this in mind, we can consider two weak eigenstates, \(v_{e}\) and \(v_{\mu}\). Their plane wave forms [2] can be expressed as  
\[\ket{v_{1}(t)}=\ket{v_{1}}e^{i(\mathbf{p}_{1}:x-E_{1}t)}=\ket{v_{1}}e^{-ip_{1} \cdot x} \tag{12}\]  
and also  
\[\ket{v_{2}(t)}=\ket{v_{2}}e^{i(\mathbf{p}_{2}:x-E_{2}t)}=\ket{v_{2}}e^{-ip_{2} \cdot x} \tag{13}\]  
If we let \(A\) be the flavor eigenstate consisting of \(e,\mu,\tau\) and let \(B\) be the neutrino mass eigenstate consisting of \(m_{1},m_{2},m_{3}\), then we must consider that \(A\times B\
eq B\times A\). Assuming this constraint, we can deduce that \(A\times B=\{(e,m_{1}),(e,m_{2}),(e,m_{3}),(\mu,m_{1}),(\mu,m_{2}),(\mu,m_{3}),\\ (\tau,m_{1}),(\tau,m_{2}),(\tau,m_{3})\}\). Similarly, \(B\times A=\{(m_{1},e),(m_{2},e),(m_{3},e),(m_{1},\mu),(m_{2},\mu),(m_{3},\mu),\\ (m_{1},\tau),(m_{2},\tau),(m_{3},\tau)\}\). Again, we must stress the importance that for this assumption, the probability of \(A\) does not depend on the probability of \(B\) and the probability of \(B\) certainly does not depend on the probability of \(A\).