Neutrino Mass and the PMNS Matrix

Two flavours capture the mechanism of oscillation but not the real neutrino sector, which has three. With three generations the mixing between flavour and mass bases is a $3\times 3$ unitary matrix, the leptonic analogue of the CKM matrix. Its structure came as a surprise: where quark mixing is nearly diagonal, leptonic mixing is close to maximal. This lesson builds the Pontecorvo–Maki–Nakagawa–Sakata matrix from three rotations, records the parameters oscillation experiments have measured, and sets out what is known and unknown about the absolute mass scale and its ordering.

The PMNS matrix

The charged-current interaction couples each charged lepton to a neutrino. Written in the flavour basis, each of ${\nu }_e,{\nu }_{\mu },{\nu }_{\tau }$ pairs with its charged lepton $e,\mu ,\tau$. In the mass basis ${\nu }_1,{\nu }_2,{\nu }_3$ the coupling carries a unitary mixing matrix $U$, and the flavour states are its combinations,

This $U$ is the PMNS matrix. It arises exactly as the CKM matrix does: the charged leptons and the neutrinos have their mass matrices diagonalised by separate unitary rotations, and $U$ is the leftover mismatch that survives in the charged current. As with quarks, its entries cannot be predicted from within the Standard Model; they are measured.¹

The measured magnitudes are strikingly non-hierarchical,

Every element is order unity except the small $U_{e3}$. Compared to the CKM matrix, whose off-diagonal elements fall away as powers of $\lambda \approx 0.22$, the contrast is total: leptonic mixing is nearly as large as unitarity permits, and no ordering principle behind it is known.²

Three rotations and the Dirac phase

A general $3\times 3$ unitary matrix has nine parameters. Rephasing the lepton fields removes five, leaving three real mixing angles and one physical complex phase — the same counting as the CKM matrix, provided the neutrinos carry Dirac masses. The standard parametrisation factors $U$ into three rotations, in the planes $(2,3)$, $(1,3)$, and $(1,2)$, with the phase $\delta$ riding on the $(1,3)$ rotation:

with $c_{ij}=\cos {\theta }_{ij}$ and $s_{ij}=\sin {\theta }_{ij}$. The three angles are measured in complementary channels, so the decomposition is not merely formal: each rotation is fixed by a different class of experiment.

• The $(1,2)$ rotation, ${\theta }_{12}$ — the solar angle, measured by the energy-dependent suppression of solar ${\nu }_e$ and confirmed by reactor ${\bar{\nu }}_e$ disappearance over $\sim 100\mathrm{km}$.
• The $(2,3)$ rotation, ${\theta }_{23}$ — the atmospheric angle, measured by the disappearance of atmospheric and accelerator ${\nu }_{\mu }$.
• The $(1,3)$ rotation, ${\theta }_{13}$ — the small angle, measured by reactor ${\bar{\nu }}_e$ disappearance over $\sim 1\mathrm{km}$ and the last of the three to be determined.

Measured angles and splittings

Global fits to all oscillation data give the three angles and the two independent mass-squared differences. In terms of ${\sin }^2{\theta }_{ij}$ the angles are

corresponding to ${\theta }_{12}\approx 34^{\circ }$, ${\theta }_{23}\approx 48^{\circ }$, and ${\theta }_{13}\approx 8.5^{\circ }$. The atmospheric angle is close to maximal ($45^{\circ }$), the solar angle is large but not maximal, and only the reactor angle is small. The two mass-squared differences differ by a factor of about thirty,

The hierarchy of splittings, small solar versus large atmospheric, is what allows a two-flavour approximation to work in each experiment: over a solar baseline the large splitting has averaged out, and over an atmospheric baseline the small one has not yet turned on.³

The magnitudes translate into a plain statement about how much of each flavour each mass state carries. The lightest-mixing pattern is roughly

• ${\nu }_1$ — about two-thirds electron neutrino, the rest muon and tau.
• ${\nu }_2$ — a nearly even third of each flavour.
• ${\nu }_3$ — close to an even split of muon and tau, with only about ten percent electron neutrino, the small ${\sin }^2{\theta }_{13}$.

Mass ordering

Oscillations measure only differences of squared masses, and only the sign of the small solar splitting is fixed, by the matter effect inside the Sun, which requires $m_2>m_1$. The sign of the large splitting is unknown, leaving two possible orderings:

• Normal ordering — $m_1<m_2<m_3$, with the closely spaced solar pair at the bottom and ${\nu }_3$, poor in electron flavour, at the top.
• Inverted ordering — $m_3<m_1<m_2$, with the solar pair at the top and the electron-poor ${\nu }_3$ at the bottom.

Resolving the ordering requires an experiment sensitive to the sign of $\Delta m_{31}^2$, through matter effects in long-baseline beams or in atmospheric neutrinos. Cosmological bounds on the mass sum, together with the latest global fits, currently favour the normal ordering.⁴

The absolute mass scale

Oscillations fix the splittings but not the overall scale: adding a common mass to all three states leaves every $\Delta m^2$ unchanged. Three independent handles bound the absolute masses.

• Beta-decay endpoint. The shape of the electron spectrum near the endpoint of tritium beta decay is distorted by a nonzero neutrino mass. The observable is the effective electron-neutrino mass $m_{\beta }=\sqrt{{\sum }_i|U_{ei}{|}^2{m}_i^2}$. Direct searches set $m_{\beta }\lesssim 0.8\mathrm{eV}$, a purely kinematic bound independent of the Dirac-or-Majorana question.
• Cosmology. Neutrinos in the early universe suppress the growth of structure on small scales and leave an imprint on the microwave background. This bounds the sum of the masses, ${\sum }_i{m}_i\lesssim 0.12\text{–}0.25\mathrm{eV}$, the tightest constraint but a model-dependent one.
• Oscillation floor. The splittings alone force at least one neutrino to satisfy $m\gtrsim \sqrt{|\Delta m_{32}^2|}\approx 0.05\mathrm{eV}$, so the masses are not all zero.

The three constraints frame a narrow window: the heaviest neutrino lies between about $0.05$ and a few tenths of an $\mathrm{eV}$, six orders of magnitude below the electron. Explaining that gap is the subject of the next lesson.⁵

CP violation in the lepton sector

The Dirac phase $\delta$ plays the role the CKM phase plays for quarks: it makes the matrix complex and allows CP violation, here an asymmetry between $P({\nu }_{\alpha }\to {\nu }_{\beta })$ and $P({\bar{\nu }}_{\alpha }\to {\bar{\nu }}_{\beta })$. Two features distinguish the leptonic case. First, because none of the PMNS angles is small — in particular ${\theta }_{13}$, while the smallest, is far larger than the corresponding CKM element — leptonic CP violation could be substantially larger than in the quark sector. Second, if the neutrinos are their own antiparticles the counting changes: a Majorana mass adds two further phases, ${\alpha }_1$ and ${\alpha }_2$, that do not affect oscillations but do affect lepton-number-violating processes. The value of $\delta$ is being measured by comparing neutrino and antineutrino oscillations in long-baseline beams; the Majorana phases, if they exist, are inaccessible to oscillation and require the experiments of the next lesson.⁶

Summary

Three-flavour mixing is encoded in the unitary PMNS matrix, the leptonic partner of the CKM matrix. It decomposes into three rotations — the atmospheric ${\theta }_{23}$, the reactor ${\theta }_{13}$, and the solar ${\theta }_{12}$ — with a Dirac phase $\delta$ on the $(1,3)$ block. The measured angles are large, ${\theta }_{23}\approx 48^{\circ }$, ${\theta }_{12}\approx 34^{\circ }$, ${\theta }_{13}\approx 8.5^{\circ }$, unlike the small CKM angles, and the two splittings $\Delta m_{21}^2\approx 7.4\times 10^{-5}{\mathrm{eV}}^2$ and $|\Delta m_{32}^2|\approx 2.5\times 10^{-3}{\mathrm{eV}}^2$ differ by a factor of thirty. The sign of the large splitting — normal versus inverted ordering — is not yet fixed. Oscillations leave the absolute scale free; beta-decay endpoints, cosmology, and the oscillation floor bracket the heaviest neutrino between $0.05\mathrm{eV}$ and a few tenths of an eV. Whether the neutrino is a Dirac or a Majorana particle sets how many CP phases live in the matrix, and that question drives the experiments taken up next.