The Mixing Angle
Field note: the angles that bind flavor to mass.
What a mixing angle is
The PMNS matrix has three mixing angles: theta_12, theta_23, and theta_13. They are not angles in any geometric sense. They parameterize the unitary matrix that connects the flavor eigenstates of the neutrino to its mass eigenstates. The flavor states are what you produce and detect. The mass states are what propagate. The mixing angles tell you how much of each flavor is composed of each mass state.
They are the most fundamental numbers in neutrino physics. Everything — oscillation probabilities, interaction rates, the oscillation pattern — derives from them.
The measured values
Theta_12 is the largest, at approximately 33 degrees. It dominates solar neutrino oscillation. It is measured by SNO, Super-Kamiokande, Borexino, and confirmed by KamLAND.
Theta_23 is close to 45 degrees — maximal mixing. It is between 39 and 49 degrees depending on mass ordering. It governs atmospheric neutrino oscillation, the oscillation that first proved neutrinos have mass.
Theta_13 is the smallest, approximately 8.5 degrees. It was the hardest to measure. Daya Bay, RENO, and Double Chooz used reactor antineutrinos at baselines of one to two kilometers. Their 2012 measurements showed theta_13 was non-zero — a relief and a disappointment. A zero angle would have simplified the analysis. A non-zero angle means there is a CP-violating phase to measure.
What the angles mean together
The three angles combine in the oscillation probability, which depends on the mass-squared differences, the baseline, the energy, and the CP phase. The angles set the amplitudes. The mass differences set the frequencies. The CP phase, if non-zero, sets the asymmetry between neutrinos and antineutrinos.
The hierarchy is unusual. Quark mixing angles are small — the Cabibbo angle is about 13 degrees. Lepton mixing angles are large. Two of three are big. This is one of the most striking and least understood features of the Standard Model's flavor structure.
The future
JUNO, DUNE, and Hyper-Kamiokande are refining these measurements. Each tightening of the angle constraints tightens the constraints on the theory. The angles are measured to a few percent at best. The CP phase is unknown. The hierarchy is unknown. The angles are the most solid numbers we have, measured for something that weighs less than an electronvolt.
These angles bind flavor to mass. They are the connection. They are the reason the neutrino changes as it travels.