Neutrino oscillation

Neutrino oscillation is coherent conversion between neutrino flavors: different mass eigenstates acquire different propagation phases. It measures squared-mass differences and mixing, rather than an absolute neutrino mass. The Super-Kamiokande atmospheric-neutrino experiment tests path-dependent disappearance; the Sudbury Neutrino Observatory compares reactions with complementary neutrino flavor sensitivity. Solar neutrino propagation also requires matter effects; the two-flavour vacuum survival model is only an illustration.

A long-baseline tau-appearance experiment with the CERN-to-Gran-Sasso muon neutrino beam. Lead targets and nuclear emulsion resolve a short tau lepton decay after a tau neutrino charged current interaction.

A long-baseline neutrino oscillation experiment using a Fermilab beam, near and far detectors, and magnetized iron/scintillator sampling calorimeters. A charged current muon neutrino event gives a negative muon and a hadronic shower; their reconstructed energies estimate the incident neutrino energy.

Coherent forward weak scattering in ordinary matter changes neutrino propagation phases. The charged-current electron contribution gives the displayed potential difference, with the opposite sign for antineutrinos. A common neutral-current term can be removed from active-flavour oscillations. A sufficiently large effect requires propagation with the matter Hamiltonian rather than the vacuum formula.

Matter changes the effective mixing angle and level splitting of propagating neutrinos. The electron-flavour charged current potential is $\sqrt{2}{G}_F{N}_e$ relative to the other active flavours. In the two-flavour Hamiltonian, equality of its two diagonal entries gives the displayed resonance condition. Adiabatic evolution through a varying density can therefore cause a large flavour change even when the local vacuum mixing angle is smaller. The equality alone does not guarantee adiabatic propagation.

Flavor conversion in a slowly varying medium when an occupied instantaneous propagation eigenstate follows its eigenvalue branch without transitions to the other branch. For positive electron-neutrino matter potential and normal two-state ordering, a high-density electron neutrino follows the upper branch into the vacuum mass eigenstate ${\nu }_2$. Detection in the electron flavor then has probability ${\sin }^2\theta$, rather than the phase-averaged vacuum value $1-\frac{1}{2}{\sin }^{2}2\theta$. High production density alone does not guarantee adiabatic evolution.

A neutrino state selected by the charged lepton accompanying a weak charged current production or detection process. It is generally a coherent quantum superposition of mass eigenstates, rather than a state of definite propagating mass.

For coherent relativistic vacuum propagation and a unitary lepton mixing matrix, square ${\sum }_i|U_{ei}{|}^2\exp [-i{m}_i^{2}L/(2E)]$ to obtain this probability. Only moduli of the mixing entries occur, so the ordinary three-state electron survival signal is independent of the intrinsic CP phase. Replacing two close fast phases by one is useful at short baselines but may fail over many periods.

The difference between the squared masses of two propagation states. In vacuum its contribution to a neutrino oscillation probability has phase $\Delta m_{ij}^{2}L/(4E)$ in natural units. The exact identity $\Delta m_{13}^2=\Delta m_{12}^2+\Delta m_{23}^2$ constrains any proposed three-state numerical inputs.

The normal ordering has the third light neutrino above the solar pair in mass; the inverted ordering has it below that pair. Oscillation experiments determine differences of squared masses, not the common absolute mass scale. A pure vacuum electron-survival expression uses squared sines and does not by itself fix the signs of individual phase differences.

The matrix relating weak-interaction neutrino flavour fields to mass eigenfields, ${\nu }_{\alpha L}={\sum }_i{U}_{\alpha i}{\nu }_{iL}$. In the standard three-active-neutrino description it is unitary. Oscillation probabilities involve products of entries and mass-squared differences; lepton-number-violating amplitudes can depend on additional Majorana phases through products such as $U_{ei}^2{m}_i$.