Neutrinos and what oscillations measure
Weak interactions produce and detect electron, muon and tau neutrino flavours. Propagation is described by mass eigenstates ν₁, ν₂ and ν₃. The PMNS matrix connects these bases. In the convention used here,
The star means complex conjugation. Different mass components accumulate different phases, so the probability of detecting a flavour can change with travel distance and energy. Oscillations measure mass-squared differences and mixing, rather than the absolute mass of each neutrino directly.
For appearance of a different flavour (α ≠ β), in an effective two-flavour, coherent, ultra-relativistic vacuum approximation,
L is in km, E is in GeV and Δm² is in eV². The numerical factor converts these units; θ is a mixing angle. The argument of the sine is dimensionless. The three-flavour theory includes additional angles, phases and two independent splittings. Matter effects can change propagation; this vacuum teaching formula is not a complete fit model for every experiment.
At least two neutrinos must have nonzero mass to produce the two observed independent mass-squared splittings. Whether neutrinos are Dirac or Majorana particles remains open. Tritium beta-decay measurements constrain an effective electron-neutrino mass; cosmological limits depend on the cosmological model. These are complementary measurements, not interchangeable labels.
Try it: at fixed Δm², doubling both L and E leaves this vacuum probability unchanged. Oscillations alone do not establish an absolute mass scale.