PMNS Matrix Parameter Calculator
Compute the neutrino PMNS mixing matrix element probabilities from the three mixing angles and the CP-violating phase.
👻 What is the PMNS Matrix Parameter Calculator?
This PMNS matrix parameter calculator computes all 9 probability elements |U_αi|² of the neutrino mixing matrix from the three standard mixing angles (θ₁₂, θ₂₃, θ₁₃) and the CP-violating phase δ_CP, using the standard PDG parametrization.
The PMNS matrix is the mathematical bridge between how neutrinos are produced and detected (as flavor states ν_e, ν_μ, ν_τ) and how they propagate through space (as mass states ν₁, ν₂, ν₃). Since neutrinos are produced in one flavor but propagate as a superposition of mass states, this matrix is the foundation behind neutrino oscillation phenomena, including solar, atmospheric, reactor, and accelerator neutrino experiments.
A useful way to sanity-check any PMNS calculation is unitarity: since every flavor state must be built from a complete, normalized superposition of mass states (and vice versa), every row and every column of the |U_αi|² probability matrix must sum to exactly 100%. This calculator displays those sums directly so you can verify your inputs are being combined correctly.
This calculator is useful for particle and neutrino physics students exploring how the mixing angles shape the flavor-to-mass relationship, and for checking published best-fit global oscillation parameters against the resulting mixing matrix.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Current global best-fit values (normal ordering)
Example 2 - Maximal atmospheric mixing, no CP violation
Example 3 - Zero reactor angle (illustrative limit)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the PMNS matrix?
The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is the 3×3 unitary matrix that relates the three neutrino flavor eigenstates (ν_e, ν_μ, ν_τ) to the three neutrino mass eigenstates (ν₁, ν₂, ν₃). It is the neutrino-sector analog of the CKM matrix in the quark sector.
What are the three mixing angles in the PMNS matrix?
θ₁₂ (the 'solar' angle, ≈33.4°), θ₂₃ (the 'atmospheric' angle, close to 45°), and θ₁₃ (the 'reactor' angle, ≈8.6°), each measured independently through different neutrino oscillation experiments (solar, atmospheric, and reactor neutrinos respectively).
What is the formula for |U_e3|²?
|U_e3|² = sin²θ₁₃, the probability that a neutrino in mass state 3 would be measured as an electron neutrino. This is the smallest matrix element in the electron row, reflecting the small reactor mixing angle.
What is the CP-violating phase δ_CP?
δ_CP is a phase angle that, if nonzero (and not 180°), allows neutrino oscillations to behave differently from antineutrino oscillations, a form of CP violation. Its exact value is not yet precisely measured, with current global fits favoring values around 194°-232° depending on mass ordering.
Why do the rows and columns of the probability matrix sum to 100%?
The PMNS matrix is unitary, meaning each flavor state is a normalized (100% probability) superposition of the three mass states, and each mass state is a normalized superposition of the three flavor states. This calculator's row and column sums verify that property numerically.
Why does δ_CP not appear in the electron row?
|U_e1|², |U_e2|², and |U_e3|² depend only on θ₁₂ and θ₁₃, not on δ_CP or θ₂₃, in the standard PDG parametrization. The CP phase's effect only becomes visible in the muon and tau rows, which mix all three angles together.
What are today's best-fit values for the PMNS parameters?
Global neutrino oscillation fits (normal mass ordering) give approximately θ₁₂≈33.44°, θ₂₃≈49.2°, θ₁₃≈8.57°, and δ_CP≈194°, though these values are updated as new experimental data arrives and differ slightly for the inverted mass ordering.
What happens if δ_CP is set to 0° or 180°?
At δ_CP=0° or 180°, cos(δ_CP)=±1 and the matrix becomes real-valued (no complex phase), meaning there is no CP violation in neutrino oscillations at that specific value, this is the special case sometimes called 'CP conservation.'
Is this the same as the CKM matrix?
No, the CKM matrix describes quark flavor mixing (up-type to down-type quarks) and is nearly diagonal (small mixing angles), while the PMNS matrix describes neutrino flavor mixing and has much larger mixing angles, reflecting a qualitatively different mixing pattern between the two matrices.
Why is θ₂₃ called the 'atmospheric' angle?
θ₂₃ was first measured precisely using atmospheric neutrinos (produced by cosmic ray collisions in Earth's atmosphere) at experiments like Super-Kamiokande, which observed muon neutrino disappearance consistent with near-maximal ν_μ-ν_τ mixing.