PMNS mixing and vacuum oscillations

Use $|\nu_\alpha\rangle=\sum_iU_{\alpha i}^*|\nu_i\rangle$, with alpha labeling electron, muon, or tau flavor and i the mass eigenstate. For ultrarelativistic neutrinos of common characteristic energy E, remove an irrelevant common phase and evolve the mass components with $e^{-im_i^2L/(2E)}$ in natural units. Projecting on flavor beta gives an amplitude with $U_{\beta i}U_{\alpha i}^*$, not two unconjugated entries. Unitarity makes the zero-baseline amplitude the appropriate Kronecker delta and conserves total probability across the three active flavors.

Choose $U=R_{23}U_{13}(\delta)R_{12}P_M$, with real rotations having positive sine above and negative sine below the relevant diagonal. The 13 complex entry is $s_{13}e^{-i\delta}$, and $P_M=\operatorname{diag}(1,e^{i\alpha_{21}/2},e^{i\alpha_{31}/2})$ contains possible Majorana phases. Each column phase cancels between the two U factors in an ordinary oscillation amplitude. For antineutrinos in vacuum replace U by its complex conjugate. A three-flavor CP asymmetry can depend on delta, whereas these Majorana phases cannot be extracted from ordinary flavor oscillation probabilities.

In a real two-flavor reduction, define the signed splitting $\Delta m^2=m_2^2-m_1^2$ and mixing angle theta through the same real rotation convention. Subtracting the two propagation amplitudes yields a sine of half their relative phase. The transition probability therefore contains $\sin^2(\Delta m^2L/(4E))$, not the sine squared of the full relative phase. Squaring makes this two-flavor vacuum probability insensitive to the sign of the splitting. It cannot by itself establish a mass ordering or exhibit intrinsic CP violation.

For practical units, the sine argument is 1.267 times the splitting in electronvolts squared times baseline in kilometres divided by energy in GeV. Using metres and MeV gives the same numerical coefficient because their ratios are equal. An extra conversion factor of a thousand would be an error. Oscillation wavelengths grow with energy and shrink with splitting magnitude. The first transition maximum sets the phase to pi over two, while the full probability period corresponds to a phase increase of pi. Always identify which of these distances a quoted oscillation length denotes.

A detector generally records a distribution of production points, energies, and reconstructed resolutions. Averaging a rapidly varying sine squared gives one half when the phase is well sampled, leaving a transition probability of one half times the mixing prefactor. This need not be one half for nonmaximal mixing. Such averaging is different from a proof that each neutrino lost microscopic coherence. A sufficiently resolved spectrum can retain oscillatory information even when a coarse total count does not. The wavepacket lesson separates these physical and statistical mechanisms.

For three-flavor naming choose $\Delta m_{21}^2>0$: normal ordering means $m_1<m_2<m_3$, and inverted ordering means $m_3<m_1<m_2$. This labeling does not determine the lightest mass. Several splittings, appearance channels, and neutrino-antineutrino comparisons constrain the angles and CP phase jointly. Matter effects must also enter long-baseline analyses. The T2K and NOvA joint analysis released in October 2025 reported no statistically significant preference between mass orderings; its strongest CP evidence was conditional on inverted ordering. This dated result illustrates why a preferred fit value is not a settled ordering or an unconditional CP discovery. The elementary two-state formulas here explain phase scales, not that full experimental inference.