Past exam of the particle physics course of the University of Oxford / 2021 / 3 / a

A mass eigenstate $|{\nu }_i⟩$ has a definite neutrino mass $m_i$ and acquires a definite propagation phase at fixed momentum. An interaction eigenstate is the neutrino flavor selected by a weak charged current: ${\nu }_e$ is produced or detected together with an electron, and ${\nu }_{\mu }$ together with a muon. The two bases are related by a unitary matrix, here the real rotation matrix $U$.

Inverting the supplied transformation gives $|{\nu }_e⟩=c|{\nu }_1⟩+s|{\nu }_2⟩$ and $|{\nu }_{\mu }⟩=-s|{\nu }_1⟩+c|{\nu }_2⟩$, where $c=\cos \theta$, $s=\sin \theta$. If $\theta =0$, these bases coincide: each produced neutrino flavor propagates as one mass eigenstate, and there is no neutrino oscillation, even if the two masses differ. Conversely, equal masses also prevent vacuum neutrino oscillation, even when a nonzero basis-rotation angle is written down.