---
title: Neutrino Mass and the PMNS Matrix
module: Neutrino Physics
moduleNumber: 10
lessonNumber: 2
order: 1002
summary: >
  Three-flavour mixing promotes the single oscillation angle to the unitary
  Pontecorvo–Maki–Nakagawa–Sakata matrix, parametrised by three angles and a Dirac CP
  phase. This lesson decomposes the PMNS matrix into three rotations, records the
  measured angles and mass-squared splittings, lays out the normal and inverted mass
  orderings, contrasts the large leptonic mixing with the near-diagonal CKM matrix, and
  collects the absolute-mass bounds from beta decay and cosmology.
topics: [Neutrino Physics]
draft: false
sources:
  - book: Thomson
    ref: "Ch. 13 — Neutrinos and neutrino oscillations, §13.5–13.8"
  - book: Griffiths
    ref: "Ch. 11 — Gauge Theories, §11.7 (lepton mixing)"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §7.1.4–7.1.5 (the PMNS matrix, CP violation in the lepton sector)"
  - book: PDG
    ref: "Neutrino mixing angles, mass-squared differences, and ordering — pdg.lbl.gov"
---

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
$3\times3$ 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,

$$
\begin{pmatrix}\nu_e\\ \nu_\mu\\ \nu_\tau\end{pmatrix}
=
\begin{pmatrix}
U_{e1} & U_{e2} & U_{e3}\\
U_{\mu1} & U_{\mu2} & U_{\mu3}\\
U_{\tau1} & U_{\tau2} & U_{\tau3}
\end{pmatrix}
\begin{pmatrix}\nu_1\\ \nu_2\\ \nu_3\end{pmatrix}.
$$

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.[^tong-pmns]

The measured magnitudes are strikingly non-hierarchical,

$$
|U_{\rm PMNS}| \approx
\begin{pmatrix}
0.82 & 0.55 & 0.15\\
0.30 & 0.60 & 0.74\\
0.44 & 0.58 & 0.65
\end{pmatrix}.
$$

Every element is order unity except the small $U_{e3}$. Compared to the CKM matrix,
whose off-diagonal elements fall away as powers of $\lambda\approx0.22$, the contrast
is total: leptonic mixing is nearly as large as unitarity permits, and no ordering
principle behind it is known.[^pdg-pmns]

$$
% caption: The PMNS and CKM magnitudes as fill-intensity grids. The CKM matrix is
% nearly diagonal, its off-diagonal elements fading to near white. The PMNS matrix is
% close to uniform, every entry order unity except the small top-right element. Fill
% tracks magnitude; numerical values are in the text.
\begin{tikzpicture}[font=\footnotesize, scale=0.92]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- PMNS grid on the left ---
  \begin{scope}[shift={(0,0)}]
    \fill[acc!34] (0,2) rectangle (1,3);
    \fill[acc!20] (1,2) rectangle (2,3);
    \fill[acc!6]  (2,2) rectangle (3,3);
    \fill[acc!10] (0,1) rectangle (1,2);
    \fill[acc!24] (1,1) rectangle (2,2);
    \fill[acc!30] (2,1) rectangle (3,2);
    \fill[acc!16] (0,0) rectangle (1,1);
    \fill[acc!22] (1,0) rectangle (2,1);
    \fill[acc!26] (2,0) rectangle (3,1);
    \draw[black] (0,0) grid (3,3);
    \node[black, font=\scriptsize, below] at (1.5,-0.35) {PMNS (leptons)};
  \end{scope}
  % --- CKM grid on the right ---
  \begin{scope}[shift={(4.4,0)}]
    \fill[acc!34] (0,2) rectangle (1,3);
    \fill[acc!16] (1,2) rectangle (2,3);
    \fill[acc!4]  (2,2) rectangle (3,3);
    \fill[acc!16] (0,1) rectangle (1,2);
    \fill[acc!34] (1,1) rectangle (2,2);
    \fill[acc!8]  (2,1) rectangle (3,2);
    \fill[acc!4]  (0,0) rectangle (1,1);
    \fill[acc!8]  (1,0) rectangle (2,1);
    \fill[acc!34] (2,0) rectangle (3,1);
    \draw[black] (0,0) grid (3,3);
    \node[black, font=\scriptsize, below] at (1.5,-0.35) {CKM (quarks)};
  \end{scope}
\end{tikzpicture}
$$

## Three rotations and the Dirac phase

A general $3\times3$ 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:

$$
U =
\begin{pmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{pmatrix}
\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta}\\0&1&0\\-s_{13}e^{i\delta}&0&c_{13}\end{pmatrix}
\begin{pmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{pmatrix},
$$

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 $\sim100\,\mathrm{km}$.
- **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 $\sim1\,\mathrm{km}$ and the last of the three to be
  determined.

$$
% caption: The PMNS matrix as an ordered product of three rotations. The two-three
% rotation carries the atmospheric angle, the one-three rotation the small reactor
% angle and the Dirac phase, and the one-two rotation the solar angle. Each block acts
% in a different plane of the three mass states.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % three boxes with an equals-to-product layout
  \node[draw=black, thick, minimum width=1.9cm, minimum height=1.4cm] (a) at (0,0) {};
  \node[black, font=\scriptsize] at (0,0.28) {atmospheric};
  \node[black, font=\scriptsize] at (0,-0.28) {plane 2-3};
  \node[black, font=\large] at (1.35,0) {$\times$};
  \node[draw=acc, very thick, fill=acc!8, minimum width=1.9cm, minimum height=1.4cm] (b) at (2.7,0) {};
  \node[acc, font=\scriptsize] at (2.7,0.28) {reactor};
  \node[black, font=\scriptsize] at (2.7,-0.28) {plane 1-3};
  \node[black, font=\large] at (4.05,0) {$\times$};
  \node[draw=black, thick, minimum width=1.9cm, minimum height=1.4cm] (c) at (5.4,0) {};
  \node[black, font=\scriptsize] at (5.4,0.28) {solar};
  \node[black, font=\scriptsize] at (5.4,-0.28) {plane 1-2};
  \node[black, font=\scriptsize, below] at (2.7,-1.05) {phase sits on the reactor block};
\end{tikzpicture}
$$

## 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

$$
\sin^2\theta_{12}\approx0.31,\qquad
\sin^2\theta_{23}\approx0.55,\qquad
\sin^2\theta_{13}\approx0.022,
$$

corresponding to $\theta_{12}\approx34^\circ$, $\theta_{23}\approx48^\circ$, and
$\theta_{13}\approx8.5^\circ$. 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,

$$
\Delta m^2_{21} = m_2^2 - m_1^2 \approx 7.4\times10^{-5}\,\mathrm{eV}^2,
\qquad
|\Delta m^2_{32}| = |m_3^2 - m_2^2| \approx 2.5\times10^{-3}\,\mathrm{eV}^2.
$$

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.[^pdg-params]

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 $\Delta
m^2_{31}$, 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.[^tong-order]

$$
% caption: The two allowed mass orderings. Bars mark the three mass-squared levels;
% shading within each bar indicates flavour content, darkest for the largest electron
% component. In the normal ordering the closely spaced solar pair sits at the bottom
% and the electron-poor third state at the top; in the inverted ordering the pair sits
% at the top and the third state below.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- normal ordering (left) ---
  \begin{scope}[shift={(0,0)}]
    \fill[acc!30, draw=black, very thick] (0,0.0) rectangle (2.2,0.28);
    \node[black, font=\scriptsize, right] at (2.3,0.14) {mass 1};
    \fill[acc!22, draw=black, very thick] (0,0.55) rectangle (2.2,0.83);
    \node[black, font=\scriptsize, right] at (2.3,0.69) {mass 2};
    \fill[acc!8, draw=black, very thick] (0,2.7) rectangle (2.2,2.98);
    \node[black, font=\scriptsize, right] at (2.3,2.84) {mass 3};
    \node[black, font=\scriptsize, below] at (1.1,-0.35) {normal};
  \end{scope}
  % --- inverted ordering (right) ---
  \begin{scope}[shift={(5.6,0)}]
    \fill[acc!8, draw=black, very thick] (0,0.0) rectangle (2.2,0.28);
    \node[black, font=\scriptsize, right] at (2.3,0.14) {mass 3};
    \fill[acc!30, draw=black, very thick] (0,2.42) rectangle (2.2,2.70);
    \node[black, font=\scriptsize, right] at (2.3,2.56) {mass 1};
    \fill[acc!22, draw=black, very thick] (0,2.97) rectangle (2.2,3.25);
    \node[black, font=\scriptsize, right] at (2.3,3.11) {mass 2};
    \node[black, font=\scriptsize, below] at (1.1,-0.35) {inverted};
  \end{scope}
\end{tikzpicture}
$$

## 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
  $m \gtrsim \sqrt{|\Delta m^2_{32}|}\approx0.05\,\mathrm{eV}$, 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.[^tong-abs]

$$
% caption: Constraints on the neutrino mass scale on a logarithmic ruler. The
% oscillation floor sets a lower edge near a twentieth of an eV; the beta-decay
% endpoint and the cosmological sum bound set upper edges. The allowed window is the
% shaded band between the floor and the tighter upper bound.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, font=\scriptsize] {mass (log scale)};
  % tick labels
  \node[black, font=\scriptsize, below] at (1.2,-0.05) {0.01 eV};
  \node[black, font=\scriptsize, below] at (3.6,-0.05) {0.1 eV};
  \node[black, font=\scriptsize, below] at (6.0,-0.05) {1 eV};
  % allowed band between floor (x=2.6) and cosmology bound (x=3.9)
  \fill[acc!14] (2.6,0.3) rectangle (3.9,1.7);
  \draw[acc, very thick] (2.6,0.3) rectangle (3.9,1.7);
  \node[acc, font=\scriptsize] at (3.25,2.0) {allowed};
  % floor marker
  \draw[black, very thick, densely dashed] (2.6,0.1) -- (2.6,2.3);
  \node[black, font=\scriptsize, above] at (2.6,2.3) {minimum};
  % cosmology bound
  \draw[black, very thick, densely dashed] (3.9,0.1) -- (3.9,2.3);
  \node[black, font=\scriptsize, above] at (3.9,2.3) {cosmology};
  % beta-decay bound (weaker, further right)
  \draw[black, very thick, densely dotted] (5.3,0.1) -- (5.3,1.9);
  \node[black, font=\scriptsize, above] at (5.3,1.9) {beta decay};
\end{tikzpicture}
$$

## 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.[^tong-cp]

## 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, $\theta_{23}\approx48^\circ$,
$\theta_{12}\approx34^\circ$, $\theta_{13}\approx8.5^\circ$, unlike the small CKM
angles, and the two splittings $\Delta m^2_{21}\approx7.4\times10^{-5}\,\mathrm{eV}^2$
and $|\Delta m^2_{32}|\approx2.5\times10^{-3}\,\mathrm{eV}^2$ 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.

[^tong-pmns]: Tong, _The Standard Model_ (Cambridge Part III), §7.1.4 — the PMNS matrix as the leptonic mismatch between mass and flavour bases, the charged-current form, and the index-order difference from the CKM matrix. [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html)
[^pdg-pmns]: PMNS magnitudes and the contrast with the near-diagonal CKM matrix: Tong, §7.1.4, quotes $|U|\approx\big(\begin{smallmatrix}0.8&0.5&0.1\\0.3&0.5&0.7\\0.4&0.6&0.6\end{smallmatrix}\big)$; current global-fit values from the Particle Data Group neutrino review, [pdg.lbl.gov](https://pdg.lbl.gov).
[^pdg-params]: Mixing angles $\sin^2\theta_{ij}$ and mass-squared differences from the Particle Data Group global fit, [pdg.lbl.gov](https://pdg.lbl.gov); Tong, §7.1, gives $\Delta m^2_{21}\approx7.4\times10^{-5}\,\mathrm{eV}^2$ and $|\Delta m^2_{32}|\approx2.5\times10^{-3}\,\mathrm{eV}^2$; Thomson, §13.6.
[^tong-order]: Tong, _The Standard Model_ (Cambridge Part III), §7.2.3 (Neutrino Mass Differences) — the normal versus inverted ordering, the sign ambiguity in the large splitting, and the cosmological preference for the normal ordering; Thomson, §13.7.
[^tong-abs]: Absolute-mass constraints: Tong, §7.1, gives the oscillation floor $m_\nu\gtrsim0.05\,\mathrm{eV}$ and the cosmological bound $\sum m_\nu\lesssim0.25\,\mathrm{eV}$; the tritium beta-decay endpoint bound $m_\beta\lesssim0.8\,\mathrm{eV}$ from the Particle Data Group neutrino-mass review, [pdg.lbl.gov](https://pdg.lbl.gov).
[^tong-cp]: Tong, _The Standard Model_ (Cambridge Part III), §7.1.5 (CP Violation in the Lepton Sector) — the Dirac phase, the two extra Majorana phases and the parameter counting, and the possibility of larger leptonic CP violation because the mixing angles are not small; Thomson, §13.8.