Past exam of the particle physics course of the University of Oxford / 2022 / 6 / a

The lepton mixing matrix is unitary, so each row has unit norm. Explicitly its electron row obeyswhere $c_{ij}=\cos {\theta }_{ij}$ and $s_{ij}=\sin {\theta }_{ij}$. This is also the normalization needed for unit electron neutrino survival at $L=0$.

Use $\mathrm{Re}(1-e^{ix})=1-\cos x=2{\sin }^2(x/2)$ and define ${\Delta }_{ij}=({\varphi }_i-{\varphi }_j)/2=\Delta m_{ij}^{2}L/(4E)$ in natural units. The exact vacuum neutrino oscillation result becomesThe given mass-squared differences satisfy $|\Delta m_{12}^2/\Delta m_{23}^2|\simeq 0.027\ll 1$, so the two fast oscillation frequencies are close: ${\Delta }_{13}={\Delta }_{23}+{\Delta }_{12}$. Approximating their phases as equal combines their coefficients intowhile $4c_{12}^2{s}_{12}^2{c}_{13}^4=c_{13}^4{\sin }^{2}2{\theta }_{12}$. HenceThe complex CP phase cancels from every $|U_{ei}{|}^2$. The phase-replacement approximation is uniformly reliable for $|{\Delta }_{12}|\ll 1$, especially at short reactor baselines; closeness of the mass-squared differences alone does not guarantee close sine values at arbitrarily large $L/E$. At medium baselines the distinct fast frequencies can beat, so the exact three-frequency expression is preferable if their detailed pattern matters.