Theory Deep Dives — Part 3: The PMNS matrix and three-flavor mixing

This is the third part of the Theory Deep Dives series. We turn to the mathematical framework that lets us calculate oscillation probabilities and the empirical question of what its parameters are: the PMNS matrix.

Flavor vs mass

In the Standard Model, neutrinos are produced and detected in flavor eigenstates: $\nu_e$, $\nu_\mu$, $\nu_\tau$. Each flavor is defined by the charged lepton it couples to in weak interactions. An electron-neutrino is the one produced together with a positron in beta-plus decay, or the one that produces an electron when it scatters off a nucleus.

But neutrinos propagate as mass eigenstates: $\nu_1$, $\nu_2$, $\nu_3$. These are the states with definite masses $m_1, m_2, m_3$, which evolve in time according to definite phase factors $e^{-i E_i t}$, where $E_i = \sqrt{m_i^2 c^4 + p^2 c^2}$.

A neutrino produced in a flavor eigenstate is a linear superposition of mass eigenstates. The PMNS matrix tells us the coefficients: $$|\nu_\alpha\rangle = \sum_i U^*_{\alpha i} |\nu_i\rangle$$

where $\alpha = e, \mu, \tau$ labels flavor and $i = 1, 2, 3$ labels mass eigenstate.

The matrix structure

The 3×3 PMNS matrix can be parametrized as a product of three rotations:

$$U_{\text{PMNS}} = R_{23}(\theta_{23}) , R_{13}(\theta_{13}, \delta_{CP}) , R_{12}(\theta_{12}) , \text{diag}(1, e^{i\alpha_1/2}, e^{i\alpha_2/2})$$

The three angles parametrize the mixing magnitudes. The single Dirac CP phase $\delta_{CP}$ multiplies the smallest mixing angle’s rotation. The two Majorana phases $\alpha_1, \alpha_2$ appear in a diagonal matrix and affect only Majorana-relevant processes.

The history of the name:

• Pontecorvo in 1957 proposed two-flavor oscillation.
• Maki, Nakagawa, Sakata in 1962 extended to three flavors.

Hence the name PMNS — Pontecorvo-Maki-Nakagawa-Sakata.

What we know about each parameter

$\theta_{12} \approx 33.4°$ (solar mixing angle). Determined primarily by solar neutrino data (SNO, Super-K, Borexino) and KamLAND reactor antineutrinos. Known to about ±1°. This is the “large mixing angle” associated with the solar splitting.

$\theta_{13} \approx 8.6°$ (the smallest angle). Long believed to be zero or very small. Measured at high significance in 2012 by Daya Bay (China) and RENO (Korea), with subsequent refinement by Double Chooz. Known to about ±0.2°.

$\theta_{23} \approx 49°$ (atmospheric mixing angle). Determined by atmospheric neutrino data (Super-K, IceCube) and long-baseline beam experiments (T2K, NOvA, MINOS). Close to maximal mixing (45°). Current uncertainty about ±3°, with a slight preference for the upper octant ($\theta_{23} > 45°$).

$\delta_{CP}$ (the CP-violating phase). Best fits near $3\pi/2$ (close to maximal violation), but the constraints are weak — uncertainty about ±50°. The first 3σ CP-violation measurement is expected by the late 2020s from T2K + NOvA combined; definitive 5σ measurement from DUNE + Hyper-K in the 2030s.

Majorana phases $\alpha_1, \alpha_2$. Completely unconstrained. Will probably remain unknown unless 0νββ is observed multiple times with sufficient precision to constrain them.

The probability of oscillation

Starting from the PMNS matrix, you can compute the oscillation probability $P(\nu_\alpha \to \nu_\beta)$ for a neutrino of energy $E$ traveling a distance $L$:

$$P(\nu_\alpha \to \nu_\beta) = \delta_{\alpha\beta} - 4 \sum_{i > j} \text{Re}(U^{\alpha i} U{\beta i} U_{\alpha j} U^{\beta j}) \sin^2\left(\frac{\Delta m^2{ij} L}{4E}\right)$$ $$\qquad + 2 \sum_{i > j} \text{Im}(U^{\alpha i} U{\beta i} U_{\alpha j} U^{\beta j}) \sin\left(\frac{\Delta m^2{ij} L}{2E}\right)$$

The expression depends on:

• The PMNS matrix elements (mixing angles and phases).
• The mass-squared differences $\Delta m^2_{ij}$.
• The L/E ratio.

Different L/E ranges probe different splittings:

• $L/E \sim 10$ km/GeV → solar oscillation through $\Delta m^2_{21}$.
• $L/E \sim 500$ km/GeV → atmospheric oscillation through $\Delta m^2_{32}$.

The CP-violating term

The imaginary part in the oscillation probability — the term proportional to $\sin(\Delta m^2 L / 2E)$ — encodes CP violation. It changes sign between neutrinos and antineutrinos. The asymmetry $P(\nu_\mu \to \nu_e) - P(\bar\nu_\mu \to \bar\nu_e)$ is sensitive to $\delta_{CP}$.

Specifically: $$P(\nu_\mu \to \nu_e) - P(\bar\nu_\mu \to \bar\nu_e) \propto \sin \delta_{CP} \cdot \sin 2\theta_{12} \cdot \sin 2\theta_{13} \cdot \sin 2\theta_{23} \cdot \sin(\Delta m^2_{21} L / 4E) \cdot \sin(\Delta m^2_{31} L / 4E) \cdot \sin(\Delta m^2_{32} L / 4E)$$

The asymmetry is maximal when all three mass-squared splittings produce maximal oscillation simultaneously — typically at very long baselines and specific energies.

The CKM analogue

The PMNS matrix is the leptonic analogue of the CKM matrix in the quark sector. The CKM matrix has the same structural form but with very different parameter values: small mixing angles and a small CP-violating phase. The PMNS matrix has large mixing angles — two of them near maximal, the third still substantial — and a large CP-violating phase if current best fits hold up.

This contrast between large lepton-sector mixing and small quark-sector mixing is one of the persistent puzzles of flavor physics. It might be a clue to the structure of the underlying flavor theory at high scales.

Why this framework is so powerful

The PMNS matrix lets you make quantitative predictions for any oscillation experiment:

• Disappearance: $P(\nu_\alpha \to \nu_\alpha)$ — fraction of neutrinos staying the same flavor.
• Appearance: $P(\nu_\alpha \to \nu_\beta)$ for $\alpha \neq \beta$ — fraction converting.

Given the matrix elements and the mass splittings, all probabilities are calculable. Conversely, experiments constrain the matrix elements and splittings by fitting to data.

The framework has been spectacularly successful. From the 1998 atmospheric oscillation discovery to the 2012 θ_13 measurement, every new experiment has been consistent with the standard three-flavor PMNS picture. No statistically significant deviation has yet been established. This is the simplest possible three-neutrino theory that’s compatible with all experimental data.

The next part of this series turns to a critical subtlety — what happens to the oscillation pattern when neutrinos travel through dense matter: the MSW effect.

Neutrino Times Editorial Team. (2026, April 5). Theory Deep Dives — Part 3: The PMNS matrix and three-flavor mixing. Neutrino Times. https://neutrino-times.com/articles/theory-deep-dives-part-3-pmns-matrix/

Neutrino Times Editorial Team. "Theory Deep Dives — Part 3: The PMNS matrix and three-flavor mixing." Neutrino Times, April 5, 2026. https://neutrino-times.com/articles/theory-deep-dives-part-3-pmns-matrix/.

Neutrino Times Editorial Team. "Theory Deep Dives — Part 3: The PMNS matrix and three-flavor mixing." Neutrino Times, 5 Apr. 2026, https://neutrino-times.com/articles/theory-deep-dives-part-3-pmns-matrix/.

@misc{neutrino-times-theory-deep-dives-part-3-pmns-matrix,
  author       = {Neutrino Times Editorial Team},
  title        = {Theory Deep Dives — Part 3: The PMNS matrix and three-flavor mixing},
  howpublished = {Neutrino Times},
  year         = {2026},
  month        = {apr},
  url          = {https://neutrino-times.com/articles/theory-deep-dives-part-3-pmns-matrix/},
  note         = {Accessed: 2026-04-05}
}

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