Neutrino

A neutrino is an electrically neutral spin-one-half lepton with very weak ordinary interactions. The three active flavors are associated with the electron, muon and tau in weak charged current production and detection. A flavor state is generally a superposition of neutrino mass eigenstates, permitting neutrino oscillations during propagation.

Detection infers a neutrino from its interaction products rather than observing an ionizing track from a free neutrino. A charged-current event can produce the corresponding charged lepton, while a neutral-current event leaves a neutrino in the final state and deposits energy in the target. Small interaction probabilities motivate large target masses and careful control of backgrounds. Neutrality does not imply zero mass; oscillations constrain differences of squared masses, rather than fixing the absolute mass scale by themselves.

A neutrino is a spin-one-half fermion. Its spin projection along its momentum defines neutrino helicity.

A neutrino with energy much larger than its mass. At fixed momentum, expanding the relativistic energy-momentum relation gives $E_i\simeq p+m_i^2/(2p)$, so differences of propagation phases are governed by $\Delta m^2/(2E)$.

A neutrino with distinct particle and antiparticle states and a Dirac mass term connecting left- and right-chiral fields. If conventional lepton number is conserved, such a mass does not cause neutrinoless double beta decay. Oscillation evidence for nonzero neutrino masses does not alone distinguish the Dirac and Majorana possibilities.

A neutrino participating in the weak interactions of the Standard Model, in contrast to a sterile state. Its thermal cosmological abundance and momentum distribution depend on weak decoupling and subsequent cosmic history.

For a massless incident neutrino and a stationary target electron, the invariant squared center-of-mass energy is $s=m_e^2+2m_e{E}_{\nu }$ in natural units. Production of a charged lepton of mass $m_{\ell }$ and a massless neutrino requires $s\ge m_{\ell }^2$, giving the displayed threshold. For the electron flavour the threshold is zero; for heavier charged leptons it is enhanced by the small stationary-target mass.

A neutral fermion without the ordinary active-neutrino weak gauge couplings. Mixing can still let it interact or decay. Its possible dark-matter role depends on lifetime, production and free streaming, rather than on Majorana character alone.

A proposed sterile neutrino component of dark matter must survive for cosmological times. If its decay rate is $\Gamma$, its lifetime $\tau =1/\Gamma$ must greatly exceed the time over which the required abundance is to persist. Otherwise the assumed present-day dark-matter density would be depleted and the decay products could alter cosmological observations.

Mixing with active neutrinos can permit decays even when the sterile state has no direct ordinary weak gauge coupling. The same mixing can therefore affect both its production and its survival. A lifetime check must use the allowed channels for the stated mass and model, rather than inferring stability from electrical neutrality or Majorana fermion character. Cosmological abundance and constraints on observable decay products impose additional requirements beyond longevity alone.

A right-handed neutrino can transform as $(1,1,0)$ under colour, weak isospin and hypercharge. It has no direct Standard Model gauge coupling, but may mix with active neutrinos. Sterility under the low-energy group does not imply sterility under an enlarged unbroken gauge group.

A Majorana neutrino is its own antiparticle. A corresponding mass term can violate conventional lepton number by two. Observing a wrong-helicity charged-current process is nevertheless suppressed by the very small neutrino mass relative to its energy, rather than having the ordinary charge-conjugate branching fraction.

An atmospheric neutrino is produced in a cosmic-ray air shower, principally through decays of secondary mesons and muons. For example, a charged pion can produce a muon and a muon-flavor neutrino; the later muon decay produces additional electron- and muon-flavor neutrinos or antineutrinos.

The direction determines a broad range of propagation distances: downward events originate relatively nearby, whereas upward events can arrive through Earth from the opposite atmosphere. Comparing event distributions by direction and energy therefore tests flavor conversion over different baselines. The 1998 Super-Kamiokande analysis found a direction-dependent muon-neutrino deficit consistent with oscillations, using a large water detector to observe charged interaction products.^[ref] The incident spectrum and interaction probabilities remain part of the inference.

The electron neutrino is the neutrino flavour associated with the electron in the weak charged current.

The neutrino flavor associated with the tau lepton in the weak charged current. Its direct detection through interactions producing a tau lepton was announced by the DONUT experiment in 2000.^[ref]

For a nearly massless incident tau neutrino and stationary target of mass $M_T$, the initial Mandelstam variables give $s=M_T^2+2M_T{E}_{\nu }$. At threshold the final tau lepton and the lightest allowed recoil system have no relative momentum, so $s=(m_{\tau }+M_X{)}^2$. This proves the formula. A free neutron recoil channel gives approximately $3.45\mathrm{GeV}$; an electron target has a different recoil and a threshold near $3.09\mathrm{TeV}$.

The antiparticle associated with the tau-flavour neutrino. A tau lepton of positive charge emits it in an ordinary charged current decay; a negative tau lepton instead emits a tau neutrino.

A Fermilab experiment that directly observed tau neutrino charged current interactions using nuclear emulsion and a prompt neutrino beam from an $800\mathrm{GeV}$ proton dump. Ds meson decays and the subsequent tau lepton decays supply the main tau-flavour component.

The mass of a neutrino mass eigenstate. A neutrino flavor need not have one definite mass; neutrino oscillation probes differences of squared masses.

A neutrino label defined by the associated charged lepton in weak charged current production or detection. A produced flavour state can evolve through neutrino oscillation.

If all neutrinos are exactly massless, arbitrary unitary rotations among them preserve both their kinetic terms and their zero mass matrix. After diagonalizing the charged-lepton Yukawa matrix, choose the neutrino basis to coincide with that rotation. The resulting weak charged current has no physical lepton mixing matrix. With nondegenerate neutrino masses this extra basis freedom is lost, permitting observable neutrino oscillations and lepton mixing.

The muon neutrino is the neutral lepton coupled to a muon by the weak charged current. It carries lepton number $+1$; its antineutrino has the opposite lepton number.

An antineutrino is the antiparticle of a neutrino and carries the opposite additive lepton number.

The antineutrino associated with the muon flavour of the weak charged current has zero electric charge and lepton number $-1$ in the usual production-and-detection description. A negative muon decays to an electron, an electron antineutrino and a muon neutrino; the charge-conjugate process instead produces the muon antineutrino.

The antineutrino associated with the electron flavour of the weak charged current. It has zero electric charge and lepton number $-1$ in the usual production-and-detection description.

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 $U_{ei}^2{m}_i$.