# High-Energy Neutrino Flavor State Transition Probabilities
## 4 Sterile Neutrinos  
As we consider the neutrino's ability to oscillate among \(v_{e},v_{\mu},v_{\tau}\), it is theoretically possible to oscillate into a heavier fourth flavor, without undergoing further oscillations, provided that this flavor interacts solely through the gravitational force. Due to this property, this fourth flavor has received the name sterile neutrino. It is essential to recognize that the sterile neutrino functions independently from the three active flavors. When a neutrino is generated, it can be described as a superposition of the three active flavors, each associated with their respective masses, while the sterile neutrino remains excluded from this interaction. Like other neutrinos, the sterile neutrino is electrically neutral and possesses a half-integer spin. As noted, this type of neutrino is anticipated to be heavier than the three known flavors, with theoretical mass estimates ranging from 10 GeV to 1 eV [12].  
When we consider the existence of a sterile neutrino, we must also consider the probability of it being categorized as a Majorana particle. A fermion is considered Majorana when it is found to be its own antiparticle. Typical fermions are categorized as Dirac fermions at low energies, with the exception of the neutrino. Dirac fermions do not possess their own antiparticles. At this point, which category the sterile neutrino fits is unknown. If sterile neutrinos are Majorana, when two sterile neutrinos collide, they would annihilate each other with an enormous amount of energy.  
We classify these particles as fermions, i.e., with a half-integer spin. In the context of neutrinos, this means their spin may either be \(1/2\) or \(-1/2\) depending on the spin quantum number \(s\) quantifying the direction of intrinsic rotation. Neutrinos are typically considered to be \"left-handed\" fermions, i.e., with Dirac spinor representations that can be expressed as projections by the left-handed projection operator \(P_{L}=\frac{1-\gamma^{5}}{2}\). Moreover, the \(W^{\pm}\) bosons mediating weak interactions involving neutrinos only interact with left-handed particles. When considering sterile neutrinos, they are assumed to be \"right-handed\" fermions, thus confirming the existence of a positive-spin neutrino.  
If we take the mass of one \"hand\" multiplied by the other \"hand\" we would deduce a constant mass.  
So, we can show that  
\[(RightHandMass)\times(LeftHandMass)=Constant \tag{15}\]  
Therefore, we can now assume that if we were to increase the mass of one hand, that is, left or right, the other hand must become smaller. This results in a process referred to as the \"Seesaw Mechanism\" [12]  
\[M_{(v)}=\begin{pmatrix}v_{L}f&vq\\ vy&v_{R}f\end{pmatrix} \tag{16}\]  
where \(f\) directly corresponds to neutrinos of heavier mass; however, a mixture of \(f\) results in neutrinos of lighter mass [12]. This yields the mass eigenstate of  
\[M_{(v)}=\begin{pmatrix}0&M_{D}\\ M_{D}&M_{NHL}\end{pmatrix} \tag{17}\]in which the mass hierarchy can be shown as \(M_{v}>>M_{D}>>M_{NHL}\), where \(M_{v}\) is the neutrino mass, \(M_{D}\) is the mass of the corresponding deuteron and \(M_{NHL}\) represents the neutral-heavy lepton [12]. By the Seesaw Mechanism, this yields an approximate neutrino mass of  
\[M_{v}=\frac{M_{D}^{2}}{M_{NHL}} \tag{18}\]  
Once again, the Seesaw Mechanism shows that if these right-handed sterile neutrinos exist, they could in fact be categorized as Majorana, which could help explain the difficulty in detecting them. This mechanism gives us a matrix theorizing six neutrino fields, three of which would have masses \(<1\) eV, while the other three would have masses \(>1\) eV. Unlike other fermions, these Majorana particles do not exhibit intrinsic electric or magnetic moments. Instead, they would possess toroidal moments in which the field of the solenoid is bent into a torus.