Three Flavours and the PMNS Matrix

Nature uses three neutrino flavours, so the two-flavour rotation becomes a rotation in three dimensions. The object that performs it is the Pontecorvo–Maki– Nakagawa–Sakata (PMNS) matrix — the leptonic counterpart of the CKM matrix that mixes quarks.

The matrix

Each flavour state is a superposition of the three mass states, with the coefficients collected in the unitary PMNS matrix $U$:

$$\begin{pmatrix} \nu_e \\ \nu_\mu \\ \nu_\tau \end{pmatrix} = \begin{pmatrix} U_{e1} & U_{e2} & U_{e3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{pmatrix} \begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{pmatrix}.$$

A general 3×3 unitary mixing matrix of this kind is parameterised by three mixing angles and one CP-violating phase:

• $\theta_{12}$ — the “solar” angle, dominant in solar-neutrino oscillation;
• $\theta_{23}$ — the “atmospheric” angle, dominant for atmospheric and accelerator muon neutrinos;
• $\theta_{13}$ — the smallest angle, the last to be measured, which links the other two sectors and gates access to the phase;
• $\delta_{\mathrm{CP}}$ — the CP-violating phase, which can make neutrinos and antineutrinos oscillate differently.

(If neutrinos are their own antiparticles, two further “Majorana phases” exist, but they do not affect oscillation.)

Two mass splittings

Three masses define two independent mass-squared differences:

• $\Delta m^2_{21} \approx 7.5\times 10^{-5}\,\mathrm{eV}^2$ — the small “solar” splitting;
• $\Delta m^2_{31} \approx 2.5\times 10^{-3}\,\mathrm{eV}^2$ — the large “atmospheric” splitting, about thirty times larger.

The hierarchy of scales is what allows the two-flavour approximation to work so well in practice: most experiments are sensitive to one splitting at a time. The measured angles are large compared with their quark counterparts — leptonic mixing is “big,” a fact still not fully understood.

Why θ₁₃ mattered so much

For years $\theta_{13}$ was consistent with zero, and its smallness was a genuine obstacle. It matters because the CP-violating phase $\delta_{\mathrm{CP}}$ only shows up in oscillation multiplied by $\theta_{13}$: if $\theta_{13}$ were zero, leptonic CP violation would be inaccessible through oscillation. The 2012 discovery that $\theta_{13}$ is comfortably non-zero (Part 5) therefore did more than complete the table of angles — it opened the door to searching for the matter–antimatter asymmetry in neutrinos.

CP violation in one line

CP violation means a difference between $P(\nu_\mu \rightarrow \nu_e)$ and $P(\bar\nu_\mu \rightarrow \bar\nu_e)$. In the PMNS framework this difference is controlled by $\delta_{\mathrm{CP}}$ and is only non-zero if all three angles are non-zero. It is of deep interest because a matter–antimatter asymmetry in the lepton sector could, through the mechanism of leptogenesis, help explain why the universe is made of matter at all.

References

1. 1.S. Navas et al. (Particle Data Group), Review of Particle Physics — Neutrino Masses, Mixing, and Oscillations, Phys. Rev. D (2024). DOI: 10.1103/PhysRevD.110.030001.
2. 2.Z. Maki, M. Nakagawa, S. Sakata, Remarks on the Unified Model of Elementary Particles, Prog. Theor. Phys. (1962). DOI: 10.1143/PTP.28.870.