The PMNS Mixing Matrix: Structure of Neutrino Mixing

Once it was established that neutrinos oscillate, the field needed a language for the mismatch between the states that are produced by the weak interaction and the states that propagate freely through space. That language is the PMNS matrix: a 3×3 unitary matrix that rotates flavor eigenstates into mass eigenstates. Its entries are the mixing amplitudes. Its parameters are the mixing angles and the CP-violating phase. It is the central object of modern neutrino phenomenology, and its measurement has been the organizing goal of more than a dozen experiments over the past two decades.

What the matrix does

A neutrino produced by a charged-current weak interaction emerges in a definite flavor eigenstate — ${\nu }_e$, ${\nu }_{\mu }$, or ${\nu }_{\tau }$ — set by the charged lepton it was produced alongside. A neutrino travelling freely through space, however, must be in an eigenstate of the Hamiltonian, which for a relativistic particle means an eigenstate of mass.

The two bases are connected by a linear transformation: $|{\nu }_{\alpha }⟩={\sum }_i{U}_{\alpha i}^{\ast }|{\nu }_i⟩$ where $\alpha =e,\mu ,\tau$ labels flavor and $i=1,2,3$ labels mass. The matrix $U$ with entries $U_{\alpha i}$ is the PMNS matrix. Unitarity — $U^{†}U=1$ — is the statement that the two bases are both complete and orthonormal. Probability is conserved: the nine $|U_{\alpha i}{|}^2$ entries in each row or column sum to one.

The standard parametrization

Any 3×3 unitary matrix can be written in terms of three rotation angles and phases. The standard PDG parametrization of the PMNS matrix uses three mixing angles ${\theta }_{12}$, ${\theta }_{13}$, ${\theta }_{23}$ and a single Dirac CP-violating phase ${\delta }_{CP}$:

$U=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_{23}& s_{23}\\ 0& -s_{23}& c_{23}\end{array}\right)\left(\begin{array}{ccc}{c}_{13}& 0& s_{13}{e}^{-i\delta }\\ 0& 1& 0\\ -s_{13}{e}^{i\delta }& 0& c_{13}\end{array}\right)\left(\begin{array}{ccc}{c}_{12}& s_{12}& 0\\ -s_{12}& c_{12}& 0\\ 0& 0& 1\end{array}\right)$

where $c_{ij}=\cos {\theta }_{ij}$ and $s_{ij}=\sin {\theta }_{ij}$. The three matrices correspond, from left to right, to the atmospheric, reactor, and solar mixing sub-rotations — and each is anchored experimentally in a different set of measurements.

If neutrinos are Majorana particles, two further phases ${\alpha }_1,{\alpha }_2$ appear in a diagonal matrix applied on the right. These Majorana phases do not affect oscillation probabilities and can only be probed by lepton-number-violating processes, most notably neutrinoless double-beta decay.

Measured values

Three decades of oscillation experiments have produced precise measurements of the three angles and the two mass-squared differences:

Solar sector. ${\theta }_{12}$ is the solar mixing angle, measured by SNO’s neutral-current/charged-current comparison and confirmed by KamLAND’s reactor-antineutrino measurement. Current world-average value: ${\sin }^2{\theta }_{12}\approx 0.307$, giving ${\theta }_{12}\approx 33.5°$.

Atmospheric sector. ${\theta }_{23}$ is the atmospheric mixing angle, measured by Super-Kamiokande from the zenith-angle dependence of atmospheric ${\nu }_{\mu }$ disappearance, and now refined by accelerator experiments T2K and NOvA and by atmospheric-neutrino analyses at IceCube-DeepCore. Current value: ${\sin }^2{\theta }_{23}\approx 0.55$, giving ${\theta }_{23}\approx 48°$ — close to maximal, with the “octant” (whether it is above or below 45°) not yet definitively resolved.

Reactor sector. ${\theta }_{13}$ is the reactor mixing angle, measured in 2012 by Daya Bay, RENO, and Double Chooz from short-baseline ${\bar{\nu }}_e$ disappearance at reactor sites. Current value: ${\sin }^2{\theta }_{13}\approx 0.0221$, giving ${\theta }_{13}\approx 8.6°$. It is the smallest of the three but is non-zero at better than 10 standard deviations — a finding that opened the door to CP-violation measurements.

CP phase. ${\delta }_{CP}$ is measured by comparing ${\nu }_{\mu }\to {\nu }_e$ and ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_e$ appearance rates in accelerator experiments. T2K and NOvA have published preliminary fits preferring ${\delta }_{CP}$ near $-\pi /2$, but the combined global significance is still below five sigma. JUNO, DUNE, and Hyper-Kamiokande will settle it in the next decade.

Mass-squared differences. Oscillation measures two independent $\Delta m_{ij}^2=m_i^2-m_j^2$: $\Delta m_{21}^2\approx 7.4\times 10^{-5}{\text{eV}}^2$ $|\Delta m_{32}^2|\approx 2.5\times 10^{-3}{\text{eV}}^2$ The sign of $\Delta m_{32}^2$ — the mass ordering — is the one remaining unknown. “Normal ordering” has $m_3>m_2>m_1$; “inverted ordering” has $m_2>m_1>m_3$. JUNO’s 2025 reactor-spectrum measurement is on track to settle this in the near term.

Why the structure matters

Three features of the PMNS matrix have no counterpart in the quark-mixing CKM matrix, and each tells us something about the deeper structure of the lepton sector.

Two angles are large. In the quark sector the analogous angles are all small — the largest, the Cabibbo angle, is about 13°. In the neutrino sector ${\theta }_{12}\approx 34°$ and ${\theta }_{23}\approx 48°$ are both very large, and ${\theta }_{13}$ is the only small one. No compelling flavor-model explanation of this striking contrast has yet emerged.

CP violation is potentially large. The Jarlskog invariant, the basis-independent measure of CP violation, is $J_{CP}=\sin {\theta }_{12}\cos {\theta }_{12}\sin {\theta }_{23}\cos {\theta }_{23}{\sin }^2{\theta }_{13}\sin {\delta }_{CP}$ With the measured angles, $J_{CP}$ can be as large as $\sim 0.034$ — more than thirty times the quark-sector value of $\sim 3\times 10^{-5}$. If ${\delta }_{CP}$ turns out to be near $-\pi /2$, the leptonic CP violation is close to maximal — a hint that the matter-antimatter asymmetry of the universe may trace to lepton-sector physics via leptogenesis.

Unitarity is a testable assumption. The PMNS matrix is unitary in the Standard Model because the three active neutrinos exhaust the light-neutrino spectrum. A fourth, sterile, species would make the 3×3 sub-block non-unitary. Unitarity triangles in the lepton sector — analogous to the quark-sector triangles used to test the CKM matrix — are the subject of active experimental programs at short-baseline reactor and accelerator experiments.

Measuring CP violation

The observable for leptonic CP violation in long-baseline experiments is the difference between neutrino and antineutrino appearance probabilities: $\Delta P=P({\nu }_{\mu }\to {\nu }_e)-P({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)$ This asymmetry depends on ${\delta }_{CP}$ through $\sin {\delta }_{CP}$, with matter effects contributing an additional asymmetry of the same sign as $\Delta m_{31}^2$ — which means the CP and mass-ordering measurements are entangled. DUNE, with its 1300 km baseline and broad energy beam, will have enough matter sensitivity to break the degeneracy. Hyper-Kamiokande, with a shorter baseline, will complement the measurement with cleaner CP sensitivity and less matter-ordering ambiguity.

The target is a five-sigma determination of the sign of $\sin {\delta }_{CP}$ — the first observation of CP violation in the lepton sector.

Open questions

Even after the PMNS framework has been measured with sub-percent precision on most of its parameters, three open questions define the next decade:

Mass ordering. JUNO and DUNE each have independent pathways to the sign of $\Delta m_{32}^2$. A determination is expected within a few years.

Octant of ${\theta }_{23}$. Whether ${\theta }_{23}$ is above or below 45° discriminates between flavor-model classes. NOvA and T2K have mild tensions that future long-baseline data will resolve.

CP phase. The value of ${\delta }_{CP}$ — and its implications for whether leptonic CP violation could account for the matter-antimatter asymmetry — is the central measurement goal of DUNE and Hyper-Kamiokande.

The PMNS matrix was written down in 1962 as a formal possibility, dismissed for decades as phenomenologically irrelevant, and then — between 1998 and 2012 — measured element by element. The next decade will complete it.