The PMNS Matrix

The Pontecorvo–Maki–Nakagawa–Sakata matrix, usually denoted $U_{\text{PMNS}}$ or simply $U$, is the $3\times 3$ unitary matrix that relates the three neutrino flavor eigenstates — ${\nu }_e$, ${\nu }_{\mu }$, ${\nu }_{\tau }$ — to the three mass eigenstates ${\nu }_1$, ${\nu }_2$, ${\nu }_3$. It is the single mathematical object that encodes the entire phenomenon of neutrino oscillation and every measurement of neutrino mixing that has been made.

In terms of left-handed flavor fields: $\left(\begin{array}{c}{\nu }_e\\ {\nu }_{\mu }\\ {\nu }_{\tau }\end{array}\right)=U\left(\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\end{array}\right)$ or component-wise, ${\nu }_{\alpha }={\sum }_i{U}_{\alpha i}{\nu }_i$ for flavor index $\alpha \in \{e,\mu ,\tau \}$ and mass index $i\in \{1,2,3\}$.

The matrix has nine complex entries, but unitarity ($U^{†}U=1$) and a rephasing freedom reduce the number of physical parameters to four: three mixing angles and one CP-violating phase for Dirac neutrinos, with two additional phases if neutrinos are Majorana particles. Over the past twenty-five years, nearly every entry of this matrix has been measured directly by oscillation experiments. The PMNS matrix is now one of the most quantitatively characterised objects in particle physics — yet it still hides the deepest open questions in the Standard Model.

Historical origin

The concept emerged in three stages. In 1957, Bruno Pontecorvo — a physicist trained in Fermi’s Rome group, then working at Dubna — published a one-page note in JETP suggesting that if the neutrino had mass, it could oscillate between particle and antiparticle states, analogous to the recently discovered $K^0\leftrightarrow {\bar{K}}^0$ oscillations. This proposal was motivated by a pure formal analogy, with no experimental evidence of neutrino mass available at the time.

Pontecorvo’s 1957 picture was “two-component” oscillation between $\nu$ and $\bar{\nu }$, not between flavors. The modern picture — mixing between flavor and mass eigenstates — was formulated in 1962 by Ziro Maki, Masami Nakagawa, and Shoichi Sakata at Nagoya, in response to a different puzzle: the discovery of the muon neutrino as distinct from the electron neutrino. The MNS paper introduced the explicit $2\times 2$ mixing matrix between ${\nu }_e$ and ${\nu }_{\mu }$ and wrote down an oscillation probability formula that is, modulo an additional flavor, the formula still used today.

The full three-flavor framework with the phase structure used in modern fits was assembled through the 1980s, notably in work by Kobayashi and Maskawa (who had already developed the analogous quark mixing machinery for CP violation) and by several groups unifying the phenomenological conventions. The experimental confirmation of neutrino oscillation at Super-Kamiokande (1998) and SNO (2001) — rather than the theoretical framework — was what converted the PMNS matrix from a formal construction to an empirical object.

The standard parameterisation

Every $3\times 3$ unitary matrix can be written as a product of three rotation matrices and a phase. The Particle Data Group convention factors the PMNS matrix as $U=R_{23}({\theta }_{23})\cdot R_{13}({\theta }_{13},{\delta }_{CP})\cdot R_{12}({\theta }_{12})$ where the rotations are $R_{23}=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_{23}& s_{23}\\ 0& -s_{23}& c_{23}\end{array}\right),R_{13}=\left(\begin{array}{ccc}{c}_{13}& 0& s_{13}{e}^{-i\delta }\\ 0& 1& 0\\ -s_{13}{e}^{i\delta }& 0& c_{13}\end{array}\right),R_{12}=\left(\begin{array}{ccc}{c}_{12}& s_{12}& 0\\ -s_{12}& c_{12}& 0\\ 0& 0& 1\end{array}\right)$ with $c_{ij}\equiv \cos {\theta }_{ij}$ and $s_{ij}\equiv \sin {\theta }_{ij}$. The three mixing angles — ${\theta }_{12}$ (the “solar”), ${\theta }_{13}$ (the “reactor”), ${\theta }_{23}$ (the “atmospheric”) — plus the Dirac CP-violating phase ${\delta }_{CP}$ are the four physical parameters for Dirac neutrinos.

The naming of the angles reflects the experimental regimes in which each is dominant:

• ${\theta }_{12}$: Solar-neutrino oscillations and long-baseline reactor experiments ($L\sim 10000$ km/MeV) are the primary probes. It was determined by SNO’s neutral-current measurement and confirmed by KamLAND.
• ${\theta }_{13}$: Short-baseline reactor experiments at $L/E\sim 500$ km/MeV provide the cleanest measurement. It was established at 5σ by Daya Bay, RENO, and Double Chooz in 2012 — the last of the three angles to be measured.
• ${\theta }_{23}$: Atmospheric neutrinos and accelerator long-baseline ($L/E\sim 500$ km/GeV) are the probes. It was the first non-zero mixing angle established, via Super-Kamiokande’s 1998 atmospheric ${\nu }_{\mu }$ disappearance result.

If neutrinos are Majorana particles — that is, identical to their own antiparticles — two additional phases enter the matrix, conventionally written as a diagonal factor $P=\text{diag}(1,e^{i{\alpha }_{21}/2},e^{i{\alpha }_{31}/2})$. These Majorana phases do not affect oscillation probabilities (they cancel in the rephasing involved in defining oscillation amplitudes) but they do contribute to neutrinoless double beta decay amplitudes and are therefore experimentally accessible through $0\nu \beta \beta$ searches.

Measured values

Global fits to the worldwide oscillation data yield these central values and 1σ uncertainties, as of 2024 compilations:

Translated into angles, ${\theta }_{12}\approx 33.4°$, ${\theta }_{13}\approx 8.6°$, ${\theta }_{23}\approx 42°$ (or $49°$ in the upper octant). The Dirac CP phase prefers a value in the third quadrant near $-\pi /2$, though with large uncertainty and compatible with CP conservation at the 2σ level.

The squared-mass differences tell us that two of the three mass eigenstates are very close together (splitting $\sim 7\times 10^{-5}$ eV²) and one is further away (splitting $\sim 2.5\times 10^{-3}$ eV²). The sign of $\Delta m_{31}^2$ — the mass ordering — is not yet known.

Oscillation probability

In vacuum, the probability for a neutrino produced as flavor $\alpha$ to be detected as flavor $\beta$ after traversing distance $L$ at energy $E$ is $P({\nu }_{\alpha }\to {\nu }_{\beta })={\delta }_{\alpha \beta }-4{\sum }_{i>j}\text{Re}(U_{\alpha i}^{\ast }{U}_{\beta i}{U}_{\alpha j}{U}_{\beta j}^{\ast }){\sin }^2\left(\frac{\Delta m_{ij}^{2}L}{4E}\right)+2{\sum }_{i>j}\text{Im}(U_{\alpha i}^{\ast }{U}_{\beta i}{U}_{\alpha j}{U}_{\beta j}^{\ast })\sin \left(\frac{\Delta m_{ij}^{2}L}{2E}\right)$ The first correction term is CP-even (identical for neutrinos and antineutrinos). The second is CP-odd — the imaginary part carries the sign flip between $\nu$ and $\bar{\nu }$ and is the observable signature of leptonic CP violation.

Because the three squared-mass differences differ by nearly two orders of magnitude, a given experiment typically operates in a regime where one $\Delta m^2$ dominates:

• Solar neutrinos ($L\sim 10^8$ km, $E\sim$ MeV) probe $\Delta m_{21}^2$
• Long-baseline reactor (KamLAND: $L\sim 180$ km, $E\sim$ MeV) also probes $\Delta m_{21}^2$
• Short-baseline reactor (Daya Bay: $L\sim 2$ km, $E\sim$ MeV) probes $|\Delta m_{31}^2|$ through ${\theta }_{13}$
• Atmospheric ($L\sim 10^4$ km, $E\sim$ GeV) probes $|\Delta m_{31}^2|$
• Accelerator long-baseline (T2K: $L\sim 300$ km, DUNE: $L\sim 1300$ km, $E\sim$ GeV) probes $|\Delta m_{31}^2|$ and accesses ${\delta }_{CP}$ through appearance channels

This hierarchy is why neutrino mixing is not measured by a single “grand experiment” but by a network of facilities each tuned to a specific corner of parameter space.

Matter effects and the MSW mechanism

In vacuum, the above formula is complete. In matter — the Sun, the Earth, a supernova — the ${\nu }_e$ component acquires an extra coherent-forward-scattering potential from elastic scattering off electrons, via the charged current. The effective Hamiltonian changes, and the effective mixing angles and masses differ from their vacuum values.

At the densities of the solar core, this Mikheyev–Smirnov–Wolfenstein (MSW) effect is so large that ${\nu }_e$ is almost an instantaneous eigenstate of the effective Hamiltonian. As the neutrino propagates out through the Sun’s density gradient, the Hamiltonian evolves adiabatically, and the neutrino emerges predominantly as ${\nu }_2$. This is the resolution of the solar neutrino problem: high-energy solar ${\nu }_e$ arrive at Earth as a coherent mixture of ${\nu }_2$ components across the PMNS matrix, with the ${\nu }_e$ survival probability reduced to roughly 30%.

The MSW effect is also responsible for the matter-induced asymmetry in long-baseline accelerator appearance, where neutrinos and antineutrinos pass through Earth’s crust. This asymmetry, which grows with baseline, is entangled with the intrinsic CP-violating asymmetry from ${\delta }_{CP}$. Disentangling them is the core experimental challenge for DUNE and Hyper-Kamiokande.

CP violation in the lepton sector

CP violation in the lepton sector requires four conditions simultaneously:

1. All three mixing angles to be non-zero (satisfied: ${\theta }_{13}$ was measured non-zero in 2012)
2. All three squared-mass differences to be non-zero (satisfied: measured)
3. The Dirac phase ${\delta }_{CP}$ to differ from $0$ and $\pi$ (measurements favour $\sim -\pi /2$ but not yet at 5σ)
4. Neutrinos to be distinguishable from antineutrinos at the level of oscillation (automatic if the PMNS is Dirac; for Majorana, still present in standard oscillation modes)

A basis-invariant measure of CP violation is the Jarlskog invariant: $J_{CP}=\text{Im}(U_{e2}{U}_{\mu 2}^{\ast }{U}_{e3}^{\ast }{U}_{\mu 3})=\frac{1}{8}\cos {\theta }_{13}\sin 2{\theta }_{12}\sin 2{\theta }_{13}\sin 2{\theta }_{23}\sin {\delta }_{CP}$ At current central values, $J_{CP}\approx 0.033\sin {\delta }_{CP}$. If ${\delta }_{CP}=-\pi /2$, the leptonic $J_{CP}$ is about 30 times larger than the quark-sector Jarlskog invariant $J_{CKM}\approx 3\times 10^{-5}$.

Observing CP violation experimentally requires comparing $P({\nu }_{\mu }\to {\nu }_e)$ with $P({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)$ in a long-baseline beam experiment. T2K and NOvA together have accumulated a few hundred ${\nu }_e$ and ${\bar{\nu }}_e$ appearance events and find a combined preference for ${\delta }_{CP}$ in the lower half-plane, with T2K alone excluding CP conservation at approximately $95\%$ confidence but still shy of discovery significance.

DUNE (1300 km baseline, broadband beam) and Hyper-Kamiokande (295 km baseline, narrowband beam from J-PARC) are designed to reach 5σ sensitivity on the sign of $\sin {\delta }_{CP}$ within 5–7 years of first data. If the observed tendency toward ${\delta }_{CP}\sim -\pi /2$ holds, definitive discovery of leptonic CP violation could come by the early 2030s.

The mass ordering

The sign of $\Delta m_{31}^2$ determines whether ${\nu }_3$ is the heaviest mass eigenstate (normal ordering, NO: $m_1<m_2<m_3$) or the lightest (inverted ordering, IO: $m_3<m_1<m_2$). The implications extend well beyond neutrino physics:

• Neutrinoless double beta decay: the expected effective Majorana mass $⟨m_{\beta \beta }⟩$ reaches a cosmological floor ($\sim 1$ meV) in the normal ordering but is bounded below by approximately 15 meV in the inverted ordering. A null result from ton-scale $0\nu \beta \beta$ experiments at the meV level essentially eliminates IO if neutrinos are Majorana.
• Cosmology: the sum $\sum m_{\nu }$ has a minimum of 0.06 eV in NO and 0.10 eV in IO. Planck + BOSS already constrain $\sum m_{\nu }<0.12$ eV; with DESI and the Simons Observatory this pushes into sensitivity for IO exclusion.
• Supernova neutrinos: the ordering shapes the flavor transformations in the dense supernova interior, leaving distinct signatures in the spectra of ${\nu }_e$, ${\bar{\nu }}_e$, and the non-electron flavors that a nearby galactic supernova would resolve.

Three measurement strategies are in progress:

• JUNO (Jiangmen Underground Neutrino Observatory): 20-kt liquid scintillator at 53 km from multiple reactor cores, using the interference pattern between the $\Delta m_{21}^2$ and $\Delta m_{31}^2$ oscillation frequencies in the detected antineutrino spectrum. Expected 3σ determination within 6 years of first data (2023); potentially 4σ with atmospheric+reactor combination.
• DUNE: atmospheric neutrinos through its 40-kt liquid-argon TPC, resolving the matter-asymmetry enhancement of NO or IO.
• KM3NeT/ORCA: Mediterranean water Cherenkov, sensitive to atmospheric-neutrino oscillations in Earth’s mantle.

Combined global fits currently prefer NO at approximately 2σ. A multi-experiment, multi-method 5σ resolution is expected by the late 2020s.

Why the mixing pattern matters

Quark mixing (CKM) is nearly diagonal: $|V_{us}|\sim 0.22$, $|V_{cb}|\sim 0.04$, $|V_{ub}|\sim 0.004$. Lepton mixing is strikingly different: $\sin {\theta }_{12}\approx 0.55$ and $\sin {\theta }_{23}\approx 0.7$ are of order unity, while $\sin {\theta }_{13}\approx 0.15$ is the “small” angle — but still eight times larger than $|V_{cb}|$.

The contrast is not a minor curiosity. In the Standard Model and minimal extensions, quark and lepton masses come from the same type of Yukawa coupling, so one might naively expect similar mixing patterns. The difference has motivated two decades of flavor-symmetry model-building:

• Discrete non-Abelian flavor symmetries ($A_4$, $S_4$, ${\Delta }_{27}$, $T^{\prime }$): groups that naturally produce specific mixing patterns at leading order. The “tri-bimaximal” pattern (${\sin }^2{\theta }_{12}=1/3$, ${\sin }^2{\theta }_{23}=1/2$, ${\sin }^2{\theta }_{13}=0$) was the historical benchmark; the measurement of non-zero ${\theta }_{13}$ in 2012 required modifications to the original models.
• Grand unified theories (GUTs): embed quark and lepton mixing in a single framework. Typical GUT predictions tie ${\theta }_{13}$ to the Cabibbo angle, with residual freedom from the heavy-neutrino sector.
• Anarchy: the hypothesis that PMNS entries are drawn from a random distribution with no underlying symmetry. Remarkably, anarchic predictions for the distribution of mixing angles fit the observed pattern surprisingly well.

No consensus has emerged. The four measured PMNS parameters are not obviously “special” values in any predictive sense — unlike, for example, ${\sin }^2{\theta }_W$, which is fixed by the electroweak gauge structure. Mixing patterns remain an open invitation for beyond-Standard-Model physics.

Unitarity tests

Like the CKM matrix, the PMNS matrix is required by Standard Model internal consistency to be unitary. Any evidence of non-unitarity — deviations from ${\sum }_i|U_{\alpha i}{|}^2=1$ — would indicate mixing with additional neutrino states (for example, sterile neutrinos) or new physics at high scales.

Current tests of PMNS unitarity come from combining oscillation measurements with independent determinations of individual matrix elements. The most stringent bounds are on first-row unitarity: $|U_{e1}{|}^2+|U_{e2}{|}^2+|U_{e3}{|}^2=1\pm 0.02$ at 90% CL. Row and column unitarity at the few-percent level are satisfied. Improving this requires cross-correlating oscillation measurements with very-short-baseline searches for sterile neutrinos, which is an active programme at reactors (PROSPECT, STEREO, DANSS) and stopped-muon sources.

Open questions and the next decade

The PMNS matrix is the most thoroughly measured mixing matrix in the Standard Model. What remains to be determined:

• Sign of $\Delta m_{31}^2$ (mass ordering) — JUNO, DUNE, Hyper-K, ORCA by late 2020s
• Octant of ${\theta }_{23}$ (above or below $\pi /4$?) — DUNE, Hyper-K
• Precise ${\delta }_{CP}$ — DUNE, Hyper-K at 5σ
• Majorana phases — only accessible through $0\nu \beta \beta$ (LEGEND-1000, nEXO)
• Unitarity — short-baseline sterile searches + improved long-baseline precision
• Absolute mass scale — KATRIN and Project 8 for the kinematic bound, cosmology for the sum, $0\nu \beta \beta$ for the Majorana effective mass

When all these are nailed down, the PMNS matrix will be specified to the same precision as the CKM matrix is today. That will not explain why the matrix looks the way it does — but it will give theorists the complete empirical target to match.

References

1. Pontecorvo, B. (1957). “Mesonium and anti-mesonium.” Soviet Physics JETP 6, 429. Soviet Physics archive link
2. Maki, Z., Nakagawa, M., Sakata, S. (1962). “Remarks on the unified model of elementary particles.” Prog. Theor. Phys. 28, 870. doi:10.1143/PTP.28.870
3. Fukuda, Y. et al. (Super-Kamiokande Collaboration, 1998). “Evidence for oscillation of atmospheric neutrinos.” Phys. Rev. Lett. 81, 1562. arXiv:hep-ex/9807003
4. Ahmad, Q. R. et al. (SNO Collaboration, 2002). “Direct evidence for neutrino flavor transformation from neutral-current interactions in the Sudbury Neutrino Observatory.” Phys. Rev. Lett. 89, 011301. arXiv:nucl-ex/0204008
5. An, F. P. et al. (Daya Bay Collaboration, 2012). “Observation of electron-antineutrino disappearance at Daya Bay.” Phys. Rev. Lett. 108, 171803. arXiv:1203.1669
6. Esteban, I., González-García, M. C., Maltoni, M., et al. (2020). “The fate of hints: updated global analysis of three-flavor neutrino oscillations.” JHEP 09, 178. arXiv:2007.14792
7. Abi, B. et al. (DUNE Collaboration, 2020). “Long-baseline neutrino oscillation physics potential of the DUNE experiment.” Eur. Phys. J. C 80, 978. arXiv:2006.16043
8. Abe, K. et al. (T2K Collaboration, 2020). “Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations.” Nature 580, 339. arXiv:1910.03887

Current global-fit parameters

The following table summarises the present best-fit values for the three mixing angles, the Dirac CP phase, and the two squared-mass differences. Values are drawn from the NuFIT 5.2 global analysis and the Daya Bay 2022 final θ₁₃ result.

Interactive calculator

Explore oscillation probabilities by adjusting mixing angles, squared-mass differences, and the L/E range. Presets set the scale to the regimes of specific experiments.

Frequently asked

Why is it called the PMNS matrix?

After Bruno Pontecorvo, who first proposed neutrino oscillations in 1957 by analogy with kaon mixing, and Ziro Maki, Masami Nakagawa, and Shoichi Sakata, who wrote down the explicit flavor-mass mixing formalism in 1962. The acronym honors both the physical idea (Pontecorvo) and the algebraic framework (MNS).

How is the PMNS matrix different from the CKM matrix of the quark sector?

Both are 3×3 unitary mixing matrices with three angles and one CP phase. Numerically they are strikingly different: quark mixing is near-diagonal with the largest off-diagonal element (the Cabibbo angle) at about 13°. Lepton mixing has two angles near 35° and 45° and one small angle at 8.6°. The 'anarchic' lepton pattern has stimulated two decades of flavor-symmetry model building.

Has CP violation in the lepton sector been observed?

Not yet at 5σ significance. The T2K and NOvA long-baseline experiments both see hints favoring maximal CP violation with δ_CP near -π/2, but their combined significance remains at the 2–3σ level. DUNE and Hyper-Kamiokande are designed to reach 5σ within their first decade of data.

What is the mass ordering, and has it been resolved?

The mass ordering is the question of whether m₃ is the heaviest mass eigenstate ('normal ordering') or the lightest ('inverted ordering'). Oscillation measurements determine only squared-mass differences, not signs. Current data slightly prefers normal ordering at ~2σ; definitive resolution is expected from JUNO (by 2028), DUNE, and atmospheric-neutrino measurements at Hyper-Kamiokande and KM3NeT/ORCA.