PMNS

The PMNS matrix, named for Pontecorvo, Maki, Nakagawa, and Sakata, is the lepton flavor mixing matrix in the three-neutrino framework. It relates the neutrino flavor eigenstates (electron, muon, and tau neutrinos) to the neutrino mass eigenstates (ν1, ν2, ν3) through a unitary transformation. In the standard convention, |να> = Σi Uαi |νi>, where α = e, μ, τ and i = 1, 2, 3. This mixing allows neutrinos produced with a given flavor to be detected later as a superposition of flavors, a phenomenon known as neutrino oscillation.

The PMNS matrix is typically parameterized by three mixing angles (θ12, θ23, θ13) and a CP-violating phase δ. In addition, if neutrinos are Majorana particles, two extra phases can be included, but these Majorana phases do not affect oscillations. The commonly used form expresses U in terms of sines and cosines of the angles and the phase δ, with small corrections to accommodate the exact ordering of rows and columns.

Experimentally, the three mixing angles and two independent mass-squared differences Δm21^2 and Δm31^2 (or Δm32^2) have been measured through solar, atmospheric, reactor, and accelerator neutrino experiments. Current values place θ12 around 33°, θ23 around 42°–49° (the octant is still under study), and θ13 about 8.5°. The mass-squared differences are Δm21^2 ≈ 7.4×10^-5 eV^2 and |Δm31^2| ≈ 2.5×10^-3 eV^2, with the ordering of the mass eigenstates (normal or inverted) remaining an open question.

The PMNS matrix underpins observable neutrino oscillations and CP-violating effects in the lepton sector. It remains a central object in debates about the absolute neutrino mass scale, the mass ordering, and the potential Majorana nature of neutrinos.