Neutrino Oscillation ParametersEdit

Neutrino oscillation parameters encode how neutrinos of different flavors transform into one another as they propagate. In the widely used three-neutrino framework, flavor states are superpositions of mass states, connected by the Pontecorvo–Maki–Nakagawa–Sakata matrix Pontecorvo–Maki–Nakagawa–Sakata matrix. The pattern of oscillations is governed by three mixing angles (θ12, θ23, θ13), a CP-violating phase (δ_CP), and two independent mass-squared differences (Δm^2_21 and Δm^2_31, or equivalently Δm^2_21 and Δm^2_32). The study of these parameters rests on observations from a broad range of experiments, including solar neutrinos solar neutrino, atmospheric neutrinos atmospheric neutrino, reactor neutrinos reactor neutrino, and accelerator neutrinos accelerator neutrino; many effects also involve matter-enhanced oscillations described by the Mikheyev–Smirnov–Wolfenstein (MSW) mechanism MSW effect.

The extraction of oscillation parameters has transformed our understanding of particle physics. The existence of nonzero neutrino masses, inferred from oscillations, requires physics beyond the original Standard Model. The parameters themselves are not just numbers; they encode the structure of lepton mixing and place important constraints on theories of flavor, grand unification, and possible new physics at high scales. The current picture is robust in many respects, yet it remains incomplete in several respects, including the ordering of the neutrino mass states, the size and origin of CP violation in the lepton sector, and whether additional (sterile) neutrino states exist beyond the three active flavors.

The parametric framework

Mixing and the PMNS matrix

In the three-neutrino description, flavor eigenstates (νe, νμ, ν_τ) are linear combinations of mass eigenstates (ν_1, ν_2, ν_3) connected by the PMNS matrix. The parameters θ12, θ23, and θ13 set the mixing angles; δ_CP governs CP-violating differences in oscillation probabilities; and the arrangement of mass eigenvalues is described by the mass-squared differences Δm^2_21 and Δm^2_31 (or Δm^2_32). See Pontecorvo–Maki–Nakagawa–Sakata matrix and neutrino oscillation for the formalism.

Mixing angles

θ12 is largely determined by solar and long-baseline reactor experiments and controls the dominant oscillation channel in solar neutrinos.
θ23 is mainly constrained by atmospheric and long-baseline accelerator data and is associated with νμ ↔ ντ mixing; its precise value has a degeneracy known as the octant problem (whether θ23 is less than or greater than 45°).
θ13 is the smallest of the three mixing angles, but its nonzero value is essential for observing CP violation in neutrinos; it was first measured with reactor experiments and long-baseline data.

CP-violating phase

δ_CP is the phase that would produce differences between neutrino and antineutrino oscillations if CP symmetry is violated in the lepton sector. Current global analyses show a preference for sizable CP violation, but the exact value of δ_CP remains imprecise and is correlated with the mass-ordering choice and θ23 octant.

Mass-squared differences and mass ordering

Δm^2_21 (the solar mass-squared difference) is well measured from solar and reactor data and sets the scale for solar-driven oscillations.
|Δm^2_31| (or |Δm^2_32|) sets the atmospheric-driven oscillations; the sign of Δm^2_31 distinguishes the normal ordering (NO: m1 < m2 < m3) from the inverted ordering (IO: m3 < m1 < m2). As of the latest global analyses, NO is favored but IO remains a live question, with uncertainties that keep both possibilities in play.
The absolute mass scale of the neutrinos is not set by oscillation experiments alone; complementary information from beta decay, neutrinoless double-beta decay searches, and cosmology is essential neutrino mass.

Experimental landscape

Solar and reactor data

Solar neutrino experiments and long-baseline reactor experiments (such as KamLAND) provide the cleanest measurements of θ12 and Δm^2_21, and they constrain the MSW matter effect experienced by neutrinos traversing dense solar material KamLAND.

Atmospheric and accelerator data

Atmospheric neutrino observations (notably from Super-Kamiokande) and accelerator experiments (such as T2K and NOvA) constrain θ23 and Δm^2_31 and are central to probing δ_CP and the mass ordering. These data sets probe different energies and baselines, helping to break parameter degeneracies.

Global fits and current values

Global analyses synthesize information from solar, atmospheric, reactor, and accelerator data to provide the most precise estimates of the oscillation parameters. While exact numbers evolve with new data, the community generally reports:
- θ12 is large but not maximal, reflecting substantial mixing between ν_e and the other flavors.
- θ13 is nonzero and about 8–9 degrees, enabling CP-violation tests in accelerator experiments.
- θ23 is near maximal mixing, with ongoing efforts to determine whether it lies below or above 45° (the octant).
- Δm^2_21 is well known and positive in the standard convention; |Δm^2_31| is known with growing precision, while the sign (normal vs inverted ordering) is still under active investigation.
- δ_CP is the least well-determined of the leading parameters, with hints for CP violation but significant uncertainty remaining.
The MSW effect remains a key aspect of interpreting solar and long-baseline data, as matter can modify oscillation probabilities in a way that depends on the sign of Δm^2_31 and on the energy and density profile encountered by the neutrinos.

Controversies and open questions

Mass ordering

The question of whether the neutrino mass spectrum is normally ordered or inverted ordered has persisted for years. While recent data show a preference for normal ordering in some global fits and analyses, the result is not yet universally decisive, and future experiments (including long-baseline projects and atmospheric neutrino studies) aim to settle the ordering with high significance.

CP violation and δ_CP

A primary scientific objective is to measure δ_CP with precision sufficient to establish leptonic CP violation and, if possible, determine its exact value. Current measurements favor substantial CP violation, but large uncertainties remain, and degeneracies with the mass ordering and θ23 complicate the extraction.

Sterile neutrinos and anomalies

Several short-baseline anomalies (historically the LSND and more recently MiniBooNE results) have prompted discussions of additional sterile neutrino states. The interpretation of these signals is contested, with some experiments finding no confirmation, while others report persistent anomalies. A coherent, globally consistent picture incorporating sterile neutrinos has not yet emerged, and dedicated experiments continue to test these possibilities.

Absolute mass scale and cosmology

Oscillation data do not fix the absolute mass of the neutrinos; cosmological observations and laboratory experiments seeking the absolute mass scale (via beta decay kinematics and neutrinoless double-beta decay) provide complementary constraints. The interplay between particle physics and cosmology remains a dynamic area of study.