Past exam of the particle physics course of the University of Oxford / 2017 / 3 / b / Solution

Use $\mathrm{Re}(e^{ia}-1)=\cos a-1=-2{\sin }^2(a/2)$. With $k=1/4$, the source's ${\Delta }_{ij}=2k({\varphi }_i-{\varphi }_j)$ is precisely $({\varphi }_i-{\varphi }_j)/2=(m_i^2-m_j^2)L/(4E)$. The electron-row probabilities are $c_{12}^2{c}_{13}^2$, $s_{12}^2{c}_{13}^2$, and $s_{13}^2$. ThereforeNeither ${\theta }_{23}$ nor the Dirac phase appears in vacuum electron-flavour survival. The same vacuum survival expression applies to reactor electron antineutrinos. A sufficiently large matter effect would require a different propagation calculation, but is negligible at the short reactor baselines used for measuring ${\theta }_{13}$.