# High-Energy Neutrino Flavor State Transition Probabilities
## 2 Neutrino Cross-Sections  
Since the first direct neutrino detection by [PERSON] and [PERSON] in 1956 [5], neutrino cross-sections have been notoriously small-concordant, with the weakly interacting nature of neutrino interactions with matter. When considering the cross-section as a product of energy gain, \(\langle\frac{AE}{E}\rangle=\frac{8}{3}(\frac{v}{c})^{2}\), we find that the individual interaction cross-sections are estimated to range from \(\sigma\equiv\sigma_{v_{h}-e-\mu-v_{e}}{\sim}1.33\times 10^{-38}\) cm\({}^{2}\) to \({\sim}2.67\times 10^{-38}\) cm\({}^{2}\), where \(v\) is the speed of the magnetic cloud and \(c\) describes the speed of the particle [3].  
In a lecture given at Fermilab in 2019 [6], it was shown that the simplest method for calculating cross-sections relative to the event rate is by finding that  
\[\frac{dN_{r}}{dt}=\sigma_{r}\cdot\frac{dN_{f}}{dt}\cdot n_{b}\cdot d \tag{4}\]  
where \(N_{r}\) is the number of reactions and \(n_{b}\cdot d\) is the number of targets per surface unit, with \(d\) being the individual target width. Here, \(N_{f}\) describes the number of particles within a beam [6; 7]. After integrating over \(t\), we find  
\[\sigma_{r}=\frac{N_{r}}{N_{f}}\frac{1}{n_{b}\cdot d} \tag{5}\]  
The cross-section for the interaction illustrated in Figure 1, in which a muon neutrino interacts with an electron to produce an electron neutrino and a muon, can be succinctly expressed as a function of neutrino energy by  
\[\sigma\approx\frac{2m_{e}G_{F}^{2}E_{v}}{\pi}=\frac{G_{F}^{2}s}{\pi} \tag{6}\]However, to compute cross-sections at high or ultra-high energies such as those of cosmic rays, we have that  
\[\begin{split}\frac{d\sigma(v_{e})}{dy}&=\frac{m_{e}G_{F }^{2}E_{\
u}}{2\pi}((g_{V}-g_{A})^{2}+\\ (g_{V}-g_{A}^{2})(1-y)^{2}-(g_{V}^{2}-g_{A}^{2})(\frac{m_{e}y}{E_ {\
u}}))\end{split} \tag{7}\]  
where \(G_{F}=\frac{g^{2}}{4\sqrt{2}M_{W}^{2}}\approx 1.166\times 10^{-5}\,\text{GeV}^{-2}\) and \(y=2m_{e}E_{\
u}\), which is the outgoing leptonic energy, bounded by \(0\leq y\leq y_{max}=1-\frac{m_{e}^{2}}{2m_{e}E_{\
u}+m_{e}^{2}}\). The coupling constants are given as \(g_{L}=+1/2,g_{R}=0,g_{V}=+1/2\) and \(g_{A}=+1/2\), in which case \(L,R,V\) and \(A\) are the left, right, vector and axial components, respectively [8].  
We can then determine approximate cross-sections for high- and ultra-high-energy cosmic rays. First, by substituting in our components \(m_{e}=9.11\times 10^{-31}\,\text{kg}\), with an energy of 300 GeV, we obtain a value for \(y\) of \(\sim\)5.466 \(\times\)\(10^{-28}\), which gives an approximate cross-section of 5.072 \(\times\)\(10^{-34}\) cm\({}^{2}\). Next, we can compare that with an energy level of 500 GeV or 0.5 TeV. This gives us a \(y\) value of \(\sim\)9.11 \(\times\)\(10^{-28}\) in which we deduce a cross-section of approximately 8.453 \(\times\)\(10^{-34}\) cm\({}^{2}\). Lastly, we will use an energy of \(E_{\
u}=1000\,\text{GeV}\) or 1 TeV, which is the energy level used in determining our cosmic ray probability calculations. The component \(y\) we find to be on the order of \(\sim\)1.822 \(\times\)\(10^{-27}\). This yields an approximate neutrino cross-section for cosmic rays as approximately 1.691 \(\times\)\(10^{-33}\) cm\({}^{2}\).