Neutrino masses and oscillations

Choose $|\nu_\alpha\rangle=\sum_i U_{\alpha i}^*|\nu_i\rangle$, where $U$ is the leptonic mixing matrix. For an ultrarelativistic beam, expanding the energy gives a relative propagation phase controlled by $m_i^2L/(2E)$ in natural units. A common overall phase cancels from probabilities. Production and detection must not resolve which mass component participated; otherwise the coherent flavor-state description needs modification. A wave-packet treatment makes that requirement more precise than drawing a point particle with changing color.

The detection amplitude is the sum of $U_{\beta i}U_{\alpha i}^*$ times each propagation phase. Square the complete sum, rather than independently squaring every term and adding, because the interference is the phenomenon. With a unitary mixing matrix and all accessible active flavors included, the vacuum probabilities sum to one. If all masses were equal, the common phase would remove any baseline-dependent flavor change even if a mixing matrix had been written in an arbitrary basis.

With two flavors and a real mixing rotation, the appearance probability separates into a mixing-depth factor and an oscillatory phase. The practical coefficient 1.267 converts a mass-squared difference in electronvolt squared, baseline in kilometers, and energy in gigaelectronvolts. It already incorporates the factors needed to convert the phase; inserting another factor of four would double-count the derivation. The sign of the mass-squared difference drops out of this two-flavor vacuum probability.

The first appearance maximum occurs when the sine-squared argument is $\pi/2$. Changing energy or baseline moves the beam through maxima and minima, but the detector observes a distribution of energies and baselines, not one perfect phase. Finite energy resolution and broad spectra average the oscillatory term. Long-baseline analyses must fold predicted probabilities with flux, interaction cross sections, acceptance, and the migration from true energy to reconstructed energy before comparing a binned spectrum with data.

Real active-neutrino phenomenology uses three flavors, two independent mass-squared splittings, three mixing angles, and a possible oscillation CP phase. Propagation through matter adds flavor-dependent forward-scattering potentials and can make neutrino and antineutrino probabilities differ even without intrinsic CP violation. Matter profiles and mass ordering therefore matter when interpreting experimental asymmetries. The two-flavor vacuum expression remains a useful limiting model, rather than a universal fit formula for every oscillation experiment.

Oscillations establish that at least some neutrino masses differ, but adding the same constant to all squared masses leaves the vacuum interference unchanged. Absolute-mass information instead comes from other observables, such as beta-decay endpoint kinematics, with their own assumptions. Neutrinoless double-beta searches probe lepton-number violation and require nuclear theory for interpretation. Loss of coherence or unresolved rapid oscillations also differs conceptually from a projective measurement at each point along the flight path.