← Graph View All Domains

A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.

Physics › Particle Physics

Neutrino Mixing (PMNS Matrix)

Research Depth 187 in the knowledge graph ☐ I know this ☆ Set as goal
1topic build on this
1,268prerequisites beneath it
See this on the map →
Neutrino Masses and OscillationsCKM Matrix and Quark Mixing→→Lepton Flavor
pmns-matrix neutrino-mixing mixing-angles cp-phase-neutrino

Core Idea

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.

Explainer

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,ei*delta,1) * R_12(theta_12) * diag(1, ei*alpha_1/2, ei*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.

Practice Questions 3 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → One-to-One Correspondence → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Making 10 as an Addition Strategy → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Iterated Integrals and Fubini's Theorem → Double Integrals in Cartesian Coordinates → Double Integrals in Polar Coordinates → Double Integrals in Polar Coordinates → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Postulates of Special Relativity → Time Dilation → Length Contraction → Lorentz Transformation → Relativistic Velocity Addition → Relativistic Momentum and Energy → Mass-Energy Equivalence and E=mc² → Photons as Particles with Energy and Momentum → Planck-Einstein Relation: Energy and Frequency → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → The Measurement Problem → Interpretations of Quantum Mechanics → Postulates of Quantum Mechanics → Observables and Quantum Operators → Commutators and Commutation Relations → Quantum Angular Momentum → Quantum Mechanical Treatment of Hydrogen → Solving the Schrödinger Equation for Hydrogen Atom → Quantum Numbers → Electron Configuration → Periodic Trends → Covalent Bonding → Electronegativity and Bond Polarity → Ionic Bonding → Lewis Structures → VSEPR Theory and Molecular Geometry → Molecular Geometry and Electron Pair Geometry → Molecular Polarity and Dipole Moments → Intermolecular Forces → States of Matter and Phase Changes: Melting, Boiling, and Sublimation → Gas Laws and the Ideal Gas Equation → Gas Stoichiometry and Volume-Volume Calculations → Thermochemistry and Enthalpy → Heat Capacity and Calorimetry → Entropy and Molecular Disorder → Spontaneity and ΔG → Entropy and Gibbs Free Energy → Chemical Equilibrium → Statistical Mechanics: Ensembles and the Boltzmann Distribution → Partition Function: Definition and Properties → Helmholtz Free Energy → Gibbs Free Energy → Phase Transitions: First Order and Second Order → Critical Phenomena and Critical Exponents → Landau Theory of Phase Transitions → Symmetry Breaking and Phase Transitions → Goldstone's Theorem and Gapless Modes → Goldstone Theorem → Higgs Mechanism → Electroweak Unification → Standard Model Overview → Neutrino Masses and Oscillations → Neutrino Mixing (PMNS Matrix)

Longest path: 188 steps · 1268 total prerequisite topics

Prerequisites (2)

Neutrino Masses and Oscillationshard CKM Matrix and Quark Mixingsoft

Leads To (1)

Lepton Flavorhard