# High-Energy Neutrino Flavor State Transition Probabilities
## 1 Introduction  
As neutrinos oscillate along their path, their quantum mechanical wave packets develop phase shifts that change how they combine to produce a superposition of the three flavors \(v_{e},v_{\mu}\) and \(v_{\tau}\). Thus, what starts as an electron neutrino may be detected in a detector as a muon or tau neutrino. On Earth, the electron neutrino, \(v_{e}\), is the most often detected. This makes sense as this state is what we would generally expect from solar neutrinos [1]. Let us first recognize that the flavor states can accommodate an ultra-relativistic left-handed neutrino. We then allow this neutrino to have a flavor \(\alpha\) (\(\alpha=e,\mu,\tau\)) and assume the neutrino to have a 4-momentum \(P^{\mu}\) with spatial component \(\vec{p}\). Following [2], we write flavor states of the newly created neutrino  
\[\ket{v_{\alpha}}=\Sigma_{k}U_{ak}^{*}\ket{v_{k}}\,, \tag{1}\]  
where \(U_{ak}\) is a unitary mixing matrix. According to [PERSON] and [PERSON] [2], observations show that our acquired state, \(\ket{v_{k}}\), is a massive neutrino state, with its momentum \(\vec{p}\) being an eigenstate of the vacuum Hamiltonian, \(\mathcal{H}_{0}\), which can be written as  
\[\mathcal{H}_{0}\ket{v_{k}}=E_{k}\ket{v_{k}} \tag{2}\]  
where \(E_{k}=\sqrt{\vec{p}^{2}+m_{k}^{2}}\)  
The total Hamiltonian in matter can then be written as  
\[\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{1} \tag{3}\]  
where \(\mathcal{H}_{1}\ket{v_{\alpha}}=V_{\alpha}\ket{v_{\alpha}}\)[2]. This tells us that our effective potential from our initial neutrino, \(v_{\alpha}\), is then described by the component \(V_{\alpha}\)[2]. We have seen that propagationof neutrinos is of quantum mechanical interest due to the phenomenon of state mixing; the production of neutrinos, on the other hand, garners appeal due to its implications for astrophysical environments. The interior of a celestial body produces its own magnetic field through currents created by its rotation, which significantly influences the behavior of charged particles on the body's outer surface. When examining a star's collapse, it is essential to consider the acceleration of protons within the core of the collapsing star. This acceleration primarily results from the increasing pressure during the infall phases of the collapse. Additionally, cosmic rays, which are a characteristic of a planet's magnetosphere, may also contribute to the acceleration of particles from the core [3]. As part of this study, we investigate the potential generation of neutrinos from these cosmic rays. Several elements contribute to understanding the acceleration of particles within a magnetosphere. This study focuses on a few of these elements, specifically the observation and analysis of distant cosmic rays, as well as first-order and second-order Fermi acceleration. Fermi acceleration describes the process by which a charged particle acquires energy due to shock speed or the movement of magnetic mirrors--with the order indicating how the energy gained relates to velocity. In the first-order process, magnetic fluctuations within shockwaves facilitate the acceleration. An example of first-order Fermi acceleration can be seen during a star's collapse, particularly during the bounce phase. Just before the material rebounds after collapsing inward, a shockwave is generated, aiding in the acceleration of charged particles within the magnetosphere itself [4]. Supernova remnant shocks serve as a significant astrophysical environment for first-order Fermi acceleration (diffusive shock acceleration) and are examined here as a potential site for neutrino production [3]. During second-order Fermi acceleration, stochastic acceleration by magnetic mirrors will have a role in answering each of our questions as well as playing a major part in the solution with regard to the probability of a complete flavor transformation. Magnetic mirrors assume the role of bouncing a particle back and forth, accelerating it outward. The magnetic mirrors are in the form of a magnetic cloud pushing the charged particles along [4].