Neutrinos, PMNS Matrix, CKM Matrix, Gravitation, Electromagnetic Force, and Weak Force

The PMNS Matrix: Flavor Mixing in the Leptonic Sector -

A first look

by Stefan Geier et al.

Abstract

The Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix is a fundamental component of the Standard Model of particle physics, governing the mixing of neutrino flavor states into mass eigenstates. This paper reviews the mathematical formulation of the PMNS matrix, evaluates its critical parameters against its quark-sector counterpart—the Cabibbo–Kobayashi–Maskawa (CKM) matrix—and discusses its role in explaining the phenomenon of neutrino oscillation and CP violation.

1. Introduction and Historical Context

a) Geier Stefan: https://advancedphysicsinthe21stcenturysg.blogspot.com/2026/03/hypothesis-quarks-follow-geiers.html .
b) The concept of particle mixing was pioneered by Nicola Cabibbo in 1963 and expanded to three generations by Makoto Kobayashi and Toshihide Maskawa in 1973, establishing the CKM matrix. Concurrently, the leptonic analogue evolved over decades. Bruno Pontecorvo hypothesized neutrino-antineutrino oscillations in 1957, which was later adapted to flavor mixing by Ziro Maki, Masami Nakagawa, and Shoichi Sakata in 1962. Following the experimental discovery of neutrino mass and oscillation at the turn of the 21st century, the PMNS matrix took its place as the definer of leptonic flavor dynamics.
c) Geier Stefan et al.: Fibonacci, Lucas and Φ structures in neutrino physics - Structural connections and testable predictions within the Geier programme: An analytical review with reproducible mathematical and numerical results. October 2026 DOI: 0.13140/RG.2.2.31802.96962 .
d) Wikipedia: https://en.wikipedia.org/wiki/Electroweak_interaction .

2. Mathematical Formalism

In the Standard Model, neutrinos are produced via weak interactions in specific flavor eigenstates (\(\vert{}\nu_e\rangle, \vert{}\nu_\mu\rangle, \vert{}\nu_\tau\rangle\)). However, they propagate through space as mass eigenstates (\(\vert{}\nu_1\rangle, \vert{}\nu_2\rangle, \vert{}\nu_3\rangle\)). The PMNS matrix, denoted as \(U\), acts as the unitary transformation relating these two bases:

\(\left(\begin{matrix}|{}\nu _{e}\rangle \\ |{}\nu _{\mu }\rangle \\ |{}\nu _{\tau }\rangle \end{matrix}\right)=\left(\begin{matrix}U_{e1}&U_{e2}&U_{e3}\\ U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\ U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{matrix}\right)\left(\begin{matrix}|{}\nu _{1}\rangle \\ |{}\nu _{2}\rangle \\ |{}\nu _{3}\rangle \end{matrix}\right)\)

By standard convention, \(U\) is parameterized by three mixing angles (\(\theta_{12}, \theta_{23}, \theta_{13}\)) and one Dirac CP-violating phase (\(\delta _{CP}\)):

\(U=\left(\begin{matrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{CP}}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{CP}}&s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{CP}}&c_{23}c_{13}\end{matrix}\right)\)

(where \(c_{ij} = \cos\theta_{ij}\) and \(s_{ij} = \sin\theta_{ij}\))

If neutrinos are Majorana particles (meaning they are their own antiparticles), an additional diagonal matrix containing two Majorana phases (\(\alpha_{1}, \alpha_{2}\)) is appended to the right side of \(U\):

\(P=\text{diag}(1,e^{i\alpha _{1}/2},e^{i\alpha _{2}/2})\)

These Majorana phases do not affect neutrino oscillations but are critical for processes like neutrinoless double beta decay (\(0\nu\beta\beta\)).

3. Comparative Analysis: CKM vs. PMNS

The primary divergence between quarks and leptons lies in the structural topology of their respective matrices. The CKM matrix is nearly diagonal, exhibiting a hierarchical architecture with very small off-diagonal elements (e.g., \(\theta_{12} \approx 13^\circ, \theta_{23} \approx 2.4^\circ\)). Quarks rarely cross generational boundaries during weak decay.

Conversely, the PMNS matrix is anarchic. The mixing angles are large, reflecting heavy generation blending.

• \(\theta _{12}\) (The Solar Angle): Measures at \(\approx 33.4^{\circ }\), driving solar neutrino oscillations.
• \(\theta _{23}\) (The Atmospheric Angle): Measures at \(\approx 42^\circ - 49^\circ\), demonstrating near-maximal or maximal mixing between the second and third generations.
• \(\theta _{13}\) (The Reactor Angle): Long assumed to be zero, it was measured in 2012 to be \(\approx 8.6^{\circ }\). While small, it is notably larger than the largest mixing angle found in quarks.

This stark structural contrast points to fundamentally different mass-generation mechanisms, such as the See-saw Mechanism for neutrinos versus the traditional Higgs Mechanism for quarks.

4. Current Horizons and CP Violation

The nonzero value of \(\theta _{13}\) opened the door to measuring the leptonic CP-violating phase (\(\delta _{CP}\)). Current global fits point to a significantly large CP phase (\(\delta_{CP} \sim 200^\circ\)). This could yield CP asymmetries in the lepton sector several orders of magnitude larger than those observed in quarks. Discovering substantial leptonic CP violation is a prerequisite for leptogenesis, a leading theoretical framework that explains why the universe is dominated by matter rather than antimatter.

5. Conclusion

The PMNS matrix is the definitive tool for studying leptonic flavor dynamics. Though mathematically identical in construction to the CKM matrix, its wide, non-hierarchical mixing angles suggest that the lepton sector operates under distinct physical laws or symmetries yet to be fully uncovered. Ongoing long-baseline neutrino experiments, such as DUNE and Hyper-Kamiokande, will continue to refine these parameters, probing the boundaries of the Standard Model. In addition, the 3x2 structure of the PMNS matrix provides evidence for our "GEIER programme" with or 3x2 Rule derived by GEIER's equations. In additon, the weak force can be seen in association with the elctromagnetic force (i.e. electroweak force).

Critique welcome!
Stefan Geier, Haidholzen