Plane-wave model of neutrino oscillations revisited

The phenomenology of massive neutrinos -- flavour mixing in the lepton sector causing oscillations between different neutrino-types along their propagation over macroscopic distances in vacuum -- aims at relating observable quantities (oscillation frequency or, equivalently, oscillation length) to the neutrino properties: mixing angles $\theta_{ij}$ and mass-squared differences $\Delta m_{ij}^2$. Calculation of the probabilities for a given neutrino-type either to survive or to mutate into another type, as functions of momentum $p$ and travelling distance $L$, are properly based on wave-packet models of varying complexity. Approximations neglecting subtle effects like decoherence result in the standard oscillation formulae with terms proportional to $\sin^2(\Delta m_{ij}^2 L / 4 p)$. The same result may also be derived by a simple plane-wave model as shown in most textbooks. However, those approaches rely on unphysical a-priory assumptions: either "equal energy" or "equal velocity" or "equal momentum" in the phases of different mass eigenstates -- which are refuted elsewhere. In addition, some assume tacitly that interference occurs at time $t = L$. This study re-examines the plane-wave model. No unphysical assumption is necessary for deriving the standard formulae: a heuristic approach relies only on carefully defining interference at time $t = L / \beta$, and is justified by coherence arguments based in a qualitative way on wave-packets.