# Pontecorvo–Maki–Nakagawa–Sakata matrix
The **Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix** is the [unitary matrix](https://www.edgepedia.org/unitary-matrix) that connects the three neutrino flavor states, produced together with the charged leptons e, μ, and τ in weak interactions, with the three neutrino mass eigenstates that actually propagate. A neutrino born in flavor state \( \nu_\alpha \) is the superposition \( |\nu_\alpha\rangle = \sum_{i=1}^{3} U^{*}_{\alpha i}\,|\nu_i\rangle \), and the mismatch between flavor and mass is what makes neutrino oscillation possible[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf). The matrix plays the same role in the lepton sector that the Cabibbo–Kobayashi–Maskawa (CKM) matrix plays for quarks[2](https://indico.cern.ch/event/1010096/contributions/4238833/attachments/2192170/3822293/NeutrinoOscilations.pdf).
| Key fact | Detail |
|---|---|
| Physical parameters | Three mixing angles \( \theta_{12}, \theta_{13}, \theta_{23} \), one Dirac CP phase \( \delta_{CP} \), and, if neutrinos are Majorana particles, two additional Majorana phases[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf) |
| Origin | Two-neutrino mixing proposed by Ziro Maki, Masami Nakagawa, and Shoichi Sakata, Progress of Theoretical Physics 28, 870 (1962); flavor-oscillation idea due to Bruno Pontecorvo (1957, 1967)[3](https://www.minicod.com/papers/remarks-on-the-unified-model-of-elementary-particles-caa98c08) • [4](https://www.neutrino-physics.com/research/papers/pontecorvo-1957/) |
| Measured angles | Lepton mixing has two angles near 35° and 45° and one small angle at 8.6°, in sharp contrast to the near-diagonal CKM matrix[5](https://www.neutrino-physics.com/oscillations/pmns-matrix/) |
| Mass splittings | \( \Delta m^{2}_{21} = 7.42 \times 10^{-5}\ \mathrm{eV^2} \) and \( |\Delta m^{2}_{31}| = 2.515 \times 10^{-3}\ \mathrm{eV^2} \) (NuFIT 5.2 global fit)[5](https://www.neutrino-physics.com/oscillations/pmns-matrix/) |
| CP phase status | \( \delta_{CP} \) is still unknown; for normal mass ordering the global fit is consistent with CP conservation within 1σ[6](https://link.springer.com/article/10.1007/JHEP12%282024%29216) |
| Majorana phases | They cancel out of the oscillation probability; the most sensitive probe is neutrinoless double beta decay[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf) |
| Unitarity | For three light neutrinos U is unitary for all practical purposes; seesaw-induced violation is of order \( O(M_D/M_N) \) and severely constrained experimentally[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf) |
## Role in neutrino oscillation
Neutrino flavor and mass states do not coincide, so a neutrino produced in flavor state \( \nu_\alpha \) may be detected as a different flavor \( \nu_\beta \) after propagating a distance L. In vacuum the oscillation probability follows from the evolution operator \( S = e^{-iH_0 L} \), with the vacuum Hamiltonian
\[ H_0 = \frac{1}{2E_\nu}\, U \begin{pmatrix} 0 & 0 & 0 \\ 0 & \Delta m^{2}_{21} & 0 \\ 0 & 0 & \Delta m^{2}_{31} \end{pmatrix} U^\dagger \]
so the probabilities depend on the PMNS elements and on the two independent mass-squared differences, but not on the absolute masses[7](https://cds.cern.ch/record/2764480/files/2104.06852.pdf). The oscillation phase is \( X_{ij} = 1.267\,\Delta m^{2}_{ij}(\mathrm{eV^2})\, L/E\,(\mathrm{m/MeV}) \), and the CP-odd term of the probability changes sign between neutrinos and antineutrinos, \( U \to U^{*} \)[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf).
The matrix became an empirical object through experiment rather than theory: the confirmation of oscillations at [Super-Kamiokande](https://www.edgepedia.org/super-kamiokande) in 1998 and SNO in 2001 established its elements as measured quantities[5](https://www.neutrino-physics.com/oscillations/pmns-matrix/). In the solar case, high-energy \( \nu_e \) arrive at Earth as a coherent mixture of mass components, with the \( \nu_e \) survival probability reduced to roughly 30%[5](https://www.neutrino-physics.com/oscillations/pmns-matrix/).
## Parametrization
The community has agreed on a standard parametrization as a product of three rotation matrices and a phase matrix:
\[ U = \begin{pmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{pmatrix} \begin{pmatrix} c_{13} & 0 & s_{13}e^{-i\delta_{CP}} \\ 0 & 1 & 0 \\ -s_{13}e^{i\delta_{CP}} & 0 & c_{13} \end{pmatrix} \begin{pmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} e^{i\eta_1} & 0 & 0 \\ 0 & e^{i\eta_2} & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
where \( c_{ij} \equiv \cos\theta_{ij} \) and \( s_{ij} \equiv \sin\theta_{ij} \)[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf). The angles lie in the first quadrant, \( \theta_{ij} \in [0, \pi/2] \), and \( \delta_{CP} \in [0, 2\pi] \); values of \( \delta_{CP} \) different from 0 and \( \pi \) imply [CP violation](https://www.edgepedia.org/cp-violation) in oscillations in vacuum[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf). For three Dirac neutrinos the two [Majorana phases](https://www.edgepedia.org/majorana-phases) \( \eta_1, \eta_2 \) can be absorbed into the neutrino states, leaving one physical phase, as in the CKM case; for Majorana neutrinos they cannot be removed, giving six independent parameters in total[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf).
**Majorana phases and oscillations.** The Majorana phases cancel out of the oscillation probability, which is expected because flavor oscillation conserves total lepton number[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf). They are also hard to measure for a second reason: any process involving them is suppressed by a power of \( m_\nu/E \), and the most sensitive experimental probe is the rate of neutrinoless double beta decay[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf).
## History
The possibility of mixing between neutrino and antineutrino states was proposed by [Bruno Pontecorvo](https://www.edgepedia.org/bruno-pontecorvo) in 1957, and he considered flavor oscillations later, in 1967[4](https://www.neutrino-physics.com/research/papers/pontecorvo-1957/) • [8](https://www.jstage.jst.go.jp/article/pjab/96/6/96_PJA9606B-02/_html/-char/ja). Soon after the discovery of the muon neutrino, Ziro Maki, Masami Nakagawa, and [Shoichi Sakata](https://www.edgepedia.org/shoichi-sakata) proposed a mixing scheme between \( \nu_e \) and \( \nu_\mu \) with two massive "true" neutrinos \( \nu_1 \) and \( \nu_2 \), within the Nagoya Model, in their 1962 paper in Progress of Theoretical Physics[3](https://www.minicod.com/papers/remarks-on-the-unified-model-of-elementary-particles-caa98c08) • [9](https://www.sciencedirect.com/science/article/pii/S055032131600105X). With the later discovery of the tau neutrino at Fermilab by the DONUT experiment, the framework extended to three flavors, and the mixing matrix acquired its present four-person name[9](https://www.sciencedirect.com/science/article/pii/S055032131600105X).
## Measured values and current unknowns
Global fits determine the matrix elements from solar, atmospheric, reactor, and long-baseline accelerator data. The NuFIT 5.2 global analysis gives \( \sin^2\theta_{12} = 0.303 \), \( \sin^2\theta_{13} = 0.02225 \), \( \sin^2\theta_{23} = 0.451 \) or 0.570 (a two-fold octant ambiguity), \( \delta_{CP} \approx 1.2\pi \), \( \Delta m^{2}_{21} = 7.42 \times 10^{-5}\ \mathrm{eV^2} \), and \( |\Delta m^{2}_{31}| = 2.515 \times 10^{-3}\ \mathrm{eV^2} \)[5](https://www.neutrino-physics.com/oscillations/pmns-matrix/). The three mixing angles have all been measured, and current efforts focus on precision; \( \delta_{CP} \) remains unknown[7](https://cds.cern.ch/record/2764480/files/2104.06852.pdf).
The NuFIT 6.0 analysis of data through September 2024 reports relative 3σ precisions of about 13% for \( \theta_{12} \), 8% for \( \theta_{13} \), 15% for \( \Delta m^{2}_{21} \), and 6% for \( |\Delta m^{2}_{3\ell}| \). The \( \theta_{23} \) octant ambiguity persists, and the CP-phase determination depends on the mass ordering: for inverted ordering, CP-violating values of \( \delta_{CP} \) around 270° are favored against CP conservation at more than 3.6σ, while for normal ordering the fit is consistent with CP conservation within 1σ. Adding Super-Kamiokande atmospheric data tips the global fit to normal ordering with \( \Delta\chi^2 = 6.1 \)[6](https://link.springer.com/article/10.1007/JHEP12%282024%29216).
**JUNO precision results.** The JUNO reactor experiment, using the first 59.1 days of data after detector completion in August 2025, made the first simultaneous high-precision determination of the solar parameters, \( \sin^2\theta_{12} = 0.3092 \pm 0.0087 \) and \( \Delta m^{2}_{21} = (7.50 \pm 0.12) \times 10^{-5}\ \mathrm{eV^2} \), improving the precision by a factor of 1.6 over the combination of all previous measurements[10](https://doi.org/10.1038/s41586-026-10538-z). A subsequent JUNO analysis reported \( \Delta m^{2}_{31} = +2.509^{+0.027}_{-0.025} \times 10^{-3}\ \mathrm{eV^2} \) for normal ordering and \( \sin^2\theta_{12} = 0.3036 \pm 0.0064 \), a relative precision of about 2.1%[11](https://arxiv.org/abs/2609.37895).
## Unitarity and physics beyond three flavors
Unitarity of the 3×3 PMNS matrix, \( U^\dagger U = 1 \), is what guarantees the cleanness of three-flavor oscillation analyses, but it is an assumption rather than a directly proven fact[12](https://www.neutrino-physics.com/blog/pmns-unitarity-tests-non-unitarity/). If additional heavy neutral leptons mix with the three active ones, as the seesaw mechanism requires, the matrix measured at low energies is only the upper-left 3×3 corner of a larger unitary matrix, and the same happens with kinematically accessible sterile neutrinos[12](https://www.neutrino-physics.com/blog/pmns-unitarity-tests-non-unitarity/) • [1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf).
In the seesaw case the unitarity violation is of order \( O(M_D/M_N) \), where \( M_N \) is the heavy-neutrino mass scale, and it is severely constrained experimentally, so for all practical purposes the 3ν mixing matrix is treated as unitary whether neutrinos are Dirac or Majorana particles[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf). Deviations from the unitarity condition \( \sum_i |U_{\alpha i}|^2 = 1 \) would indicate mixing with additional neutrino states or new physics at high scales, and several processes probe different combinations of the deviation matrix entries; the invisible Z width measured at LEP, which fixes the effective number of light neutrino species, would be reduced from three to \( 3 - \sum_\alpha \eta_{\alpha\alpha} \) in a non-unitary scenario[5](https://www.neutrino-physics.com/oscillations/pmns-matrix/) • [12](https://www.neutrino-physics.com/blog/pmns-unitarity-tests-non-unitarity/). The measured value \( N_\nu = 2.984 \pm 0.008 \) already implies that any extension of the [Standard Model](https://www.edgepedia.org/standard-model) contains three, and only three, light active neutrinos[1](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf).
## References
1. [M.C. Gonzalez-Garcia and M. Yokoyama, "Neutrino Masses, Mixing, and Oscillations," Review of Particle Physics (Particle Data Group), Phys. Rev. D 110, 030001 (2024)](https://pdg.lbl.gov/2024/reviews/rpp2024-rev-neutrino-mixing.pdf)
2. [Neutrino Oscillations lecture slides, CERN Indico](https://indico.cern.ch/event/1010096/contributions/4238833/attachments/2192170/3822293/NeutrinoOscilations.pdf)
3. [Z. Maki, M. Nakagawa, S. Sakata, "Remarks on the Unified Model of Elementary Particles," Progress of Theoretical Physics 28, 870 (1962)](https://www.minicod.com/papers/remarks-on-the-unified-model-of-elementary-particles-caa98c08)
4. [Pontecorvo, "Mesonium and Anti-Mesonium" (1957), neutrino-physics.com](https://www.neutrino-physics.com/research/papers/pontecorvo-1957/)
5. [The PMNS Matrix, neutrino-physics.com](https://www.neutrino-physics.com/oscillations/pmns-matrix/)
6. [I. Esteban et al., "NuFit-6.0: updated global analysis of three-flavor neutrino oscillations," JHEP 2024, 216 (2024)](https://link.springer.com/article/10.1007/JHEP12%282024%29216)
7. [The PMNS matrix and neutrino oscillation parameters, CERN Document Server](https://cds.cern.ch/record/2764480/files/2104.06852.pdf)
8. [The Sun, neutrinos and Super-Kamiokande, Proceedings of the Japan Academy](https://www.jstage.jst.go.jp/article/pjab/96/6/96_PJA9606B-02/_html/-char/ja)
9. [The discovery of the appearance of νμ − ντ oscillations, Nuclear Physics B](https://www.sciencedirect.com/science/article/pii/S055032131600105X)
10. [Measurement of reactor neutrino oscillation with the first JUNO data, Nature](https://doi.org/10.1038/s41586-026-10538-z)
11. [Measurement of Neutrino Oscillation Parameters at JUNO and Indication of the Neutrino Mass Ordering, arXiv](https://arxiv.org/abs/2609.37895)
12. [Is the PMNS Matrix Really Unitary? Tests of the Three-Flavor Closure, neutrino-physics.com](https://www.neutrino-physics.com/blog/pmns-unitarity-tests-non-unitarity/)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Particles and nuclei › Particle physics › Neutrino physics › Neutrino mixing parameters and measurements*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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