The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix relates the neutrino flavor eigenstates (nu_e, nu_mu, nu_tau) to the mass eigenstates (nu_1, nu_2, nu_3). It is parameterized by three mixing angles (theta_12, theta_13, theta_23), one Dirac CP phase (delta_CP), and potentially two additional Majorana phases. Unlike the CKM matrix, which is nearly diagonal, the PMNS matrix has two large mixing angles -- a qualitative difference that may hint at different underlying physics governing quark and lepton masses.
The PMNS matrix is the leptonic analog of the CKM matrix, describing how the three neutrino flavor eigenstates relate to the three mass eigenstates. It is conventionally parameterized as a product of three rotation matrices (by angles theta_12, theta_13, theta_23) with one Dirac CP phase delta_CP, multiplied by a diagonal matrix containing two Majorana phases. The standard parameterization is U = R_23(theta_23) * diag(1,1,e^{-i*delta}) * R_13(theta_13) * diag(1,e^{i*delta},1) * R_12(theta_12) * diag(1, e^{i*alpha_1/2}, e^{i*alpha_2/2}).
The three mixing angles have been measured by different types of experiments. theta_12 (~34 degrees) was determined from solar neutrino oscillations (SNO, Super-K) and confirmed by the KamLAND reactor experiment. theta_23 (~49 degrees) was measured from atmospheric neutrino oscillations (Super-K) and confirmed by long-baseline accelerator experiments (K2K, MINOS, T2K, NOvA). theta_13 (~8.5 degrees) was measured by reactor experiments (Daya Bay, RENO, Double Chooz) in 2012, a breakthrough that opened the door to measuring the CP phase delta_CP, since CP violation in oscillations requires all three angles to be nonzero.
The CP phase delta_CP is the most important unmeasured parameter in the PMNS matrix. Current hints from T2K and NOvA suggest delta_CP may be near -pi/2 (maximal CP violation), but the statistical significance is insufficient for a definitive claim. DUNE (using a 1300 km baseline from Fermilab to South Dakota) and Hyper-Kamiokande (using a 295 km baseline from J-PARC to Kamioka) are designed to measure delta_CP with sufficient precision to establish CP violation at 5 sigma significance for a large fraction of possible values. Leptonic CP violation is of profound interest because it could be connected to the matter-antimatter asymmetry of the universe through leptogenesis.
The overall structure of the PMNS matrix -- two large angles and one small angle -- is strikingly different from the CKM matrix and suggests different organizing principles for the quark and lepton sectors. Numerous models based on discrete flavor symmetries (A_4, S_4, etc.) have been proposed to explain the pattern. The tribimaximal mixing pattern (sin^2(theta_12) = 1/3, sin^2(theta_23) = 1/2, theta_13 = 0) was a leading Ansatz until the discovery of nonzero theta_13. Modified patterns that accommodate theta_13 while preserving the approximate structure remain active areas of theoretical research.