---
title: Neutrino Oscillations
module: Neutrino Physics
moduleNumber: 10
lessonNumber: 1
order: 1001
summary: >
  Neutrinos are produced and detected in flavour states, but they propagate as mass
  states, and the two bases are misaligned. A flavour therefore evolves coherently into
  a superposition of other flavours with a probability set by the mass-squared splitting
  and the ratio L/E. This lesson derives the two-flavour oscillation formula, applies it
  to the solar and atmospheric neutrino deficits, shows how the SNO neutral-current
  measurement resolved the solar problem, and works out the MSW resonance that amplifies
  mixing inside the Sun.
topics: [Neutrino Physics]
draft: false
sources:
  - book: Thomson
    ref: "Ch. 13 — Neutrinos and neutrino oscillations, §13.1–13.4"
  - book: Griffiths
    ref: "Ch. 11 — Gauge Theories, §11.7 (neutrino mass and oscillations)"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §7.2 (neutrino oscillations, matter effects, detection)"
  - book: PDG
    ref: "Neutrino masses, mixing, and oscillations review — pdg.lbl.gov"
---

A neutrino is produced in a definite flavour. Beta decay emits an electron neutrino;
a pion decays to a muon and a muon neutrino; the Sun's fusion chain releases electron
neutrinos. Each of these is defined by the charged lepton it partners at the $W$
vertex. Yet the states that propagate with definite energy and momentum are the mass
eigenstates, and these are not the flavour eigenstates. A neutrino born as one flavour
is a coherent superposition of mass states that dephase as they travel, so the
flavour content oscillates. The phenomenon requires only that at least one neutrino
carry mass and that the two bases be misaligned, and its discovery in solar and
atmospheric neutrinos was the first laboratory evidence for physics beyond the
original Standard Model.

## Flavour states and mass states

The neutrino that couples to the electron at a $W$ vertex is the **electron neutrino**
$\nu_e$; likewise $\nu_\mu$ and $\nu_\tau$ define the flavour basis. The states of
definite mass are labelled $\nu_1,\nu_2,\nu_3$. For two flavours the relation is a
single rotation through a mixing angle $\theta$,

$$
\begin{pmatrix}\nu_e\\ \nu_\mu\end{pmatrix}
=
\begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix}
\begin{pmatrix}\nu_1\\ \nu_2\end{pmatrix}.
$$

If the two mass states were degenerate the rotation would be unobservable — any
combination of degenerate states is again an eigenstate of the same mass. Oscillation
therefore demands both a nonzero mixing angle and a nonzero mass splitting. The
quark sector has the analogous misalignment in the CKM matrix, but there we track the
mass states (hadrons) and mixing appears as a decay; for neutrinos we hold a fixed
flavour and watch it evolve.[^tong-basis]

$$
% caption: The flavour states are a rotation of the mass states through the mixing
% angle theta. An electron neutrino is a coherent superposition of the two mass
% eigenstates; each carries a different phase as it propagates, so the superposition
% reassembles into a different flavour mixture downstream.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % mass axes
  \draw[->, black, thick] (0,0) -- (3.4,0) node[right, font=\scriptsize] {mass 1};
  \draw[->, black, thick] (0,0) -- (0,3.2) node[above, font=\scriptsize] {mass 2};
  % flavour axes rotated by theta
  \draw[->, acc, very thick] (0,0) -- (3.15,1.25) node[right, font=\scriptsize] {electron};
  \draw[->, acc, very thick] (0,0) -- (-1.25,3.05) node[above, font=\scriptsize] {muon};
  % angle arc
  \draw[acc, thick] (1.9,0) arc (0:21.6:1.9);
  \node[acc, font=\scriptsize] at (2.25,0.34) {angle};
\end{tikzpicture}
$$

## The two-flavour oscillation probability

Take an electron neutrino created at $t=0$ with energy $E$. In the mass basis,

$$
|\nu_e(0)\rangle = \cos\theta\,|\nu_1\rangle + \sin\theta\,|\nu_2\rangle .
$$

Each mass eigenstate is an energy eigenstate, so it advances in time by a phase
$e^{-iE_i t}$:

$$
|\nu_e(t)\rangle = e^{-iE_1 t}\cos\theta\,|\nu_1\rangle + e^{-iE_2 t}\sin\theta\,|\nu_2\rangle .
$$

Projecting back onto the flavour basis and computing the amplitude to find a muon
neutrino gives a probability that depends only on the phase difference $\Delta E =
E_2 - E_1$:

$$
P(\nu_e\to\nu_\mu) = \sin^2(2\theta)\,\sin^2\!\left(\frac{\Delta E\, t}{2}\right).
$$

The prefactor $\sin^2(2\theta)$ sets the amplitude of the oscillation — the maximum
transition probability — while the sine argument sets its wavelength. Both factors
must be present: $\theta=0$ or $\theta=\pi/2$ kills the amplitude, and $\Delta E = 0$
freezes the phase.

Neutrinos are ultrarelativistic, $|\vec p|\gg m$, so the dispersion relation expands
to

$$
E_i = \sqrt{\vec p_i^{\,2}+m_i^2}\approx |\vec p_i| + \frac{m_i^2}{2|\vec p_i|}.
$$

Taking the two mass states to share a common momentum and replacing $|\vec p|$ by the
energy $E$ in the denominator,

$$
\Delta E = \frac{m_2^2 - m_1^2}{2E} = \frac{\Delta m^2}{2E},
\qquad
\Delta m^2 \equiv m_2^2 - m_1^2 .
$$

The neutrinos travel at essentially the speed of light, so in natural units the elapsed
time equals the distance $L$. The oscillation probability becomes a function of the
experimentally controllable ratio $L/E$:

$$
P(\nu_e\to\nu_\mu) = \sin^2(2\theta)\,\sin^2\!\left(\frac{\Delta m^2\, L}{4E}\right).
$$

Restoring $\hbar$ and $c$ and inserting convenient units gives the working formula
used to design experiments,

$$
P(\nu_e\to\nu_\mu) = \sin^2(2\theta)\,
\sin^2\!\left(1.27\,\frac{\Delta m^2\,[\mathrm{eV}^2]\; L\,[\mathrm{km}]}{E\,[\mathrm{GeV}]}\right).
$$

The oscillation length — the distance over which the phase advances by $2\pi$ — is
$L_{\rm osc}=4\pi E/\Delta m^2$. For $\Delta m^2\sim10^{-3}\,\mathrm{eV}^2$ and
$E\sim1\,\mathrm{GeV}$ this is a few hundred kilometres, a length scale that a
terrestrial baseline can span.[^tong-osc]

> **Worked example.** For the atmospheric splitting $\Delta m^2 = 2.5\times10^{-3}\,\mathrm{eV}^2$
> and a $1\,\mathrm{GeV}$ muon neutrino, the first oscillation maximum
> ($1.27\,\Delta m^2 L/E = \pi/2$) sits at
>
> $$
> L = \frac{\pi/2}{1.27}\,\frac{E}{\Delta m^2}
> = \frac{1.24\times 1}{2.5\times10^{-3}}\;\mathrm{km}\approx 490\;\mathrm{km}.
> $$
>
> A neutrino coming straight down through $\sim15\,\mathrm{km}$ of atmosphere barely
> oscillates; one coming up through the $\sim1.3\times10^4\,\mathrm{km}$ diameter of the
> Earth has passed many oscillation lengths.

$$
% caption: The two-flavour survival and appearance probabilities against L over E.
% The appearance probability oscillates between zero and sin-squared of twice the
% mixing angle with a period fixed by the mass splitting; the survival probability is
% its complement. The first minimum of survival locates the oscillation maximum.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[right, font=\scriptsize] {L over E};
  \draw[->, black] (0,0) -- (0,3.0) node[above, font=\scriptsize] {probability};
  \draw[black, densely dotted] (0,2.4) -- (7.4,2.4);
  \node[black, left, font=\scriptsize] at (0,2.4) {peak};
  \node[black, left, font=\scriptsize] at (0,0) {0};
  % appearance: sin^2 curve, amplitude 2.4, period ~ 4.2 in x
  \draw[acc, very thick, domain=0:7.4, samples=160, smooth]
    plot (\x, {2.4*pow(sin(deg(\x*3.1416/4.2)),2)});
  % survival: complement
  \draw[black, very thick, dashed, domain=0:7.4, samples=160, smooth]
    plot (\x, {2.4*pow(cos(deg(\x*3.1416/4.2)),2)});
  \node[acc, font=\scriptsize] at (2.1,2.62) {appearance};
  \node[black, font=\scriptsize] at (4.9,2.62) {survival};
\end{tikzpicture}
$$

## The solar neutrino problem

The Sun generates energy by fusing hydrogen into helium, and the net reaction releases
electron neutrinos,

$$
4p \to {}^4\mathrm{He} + 2e^+ + 2\nu_e + 2\gamma .
$$

The dominant chain produces neutrinos with energy below about $0.4\,\mathrm{MeV}$;
rarer branches involving beryllium and boron reach up to $\sim10\,\mathrm{MeV}$. The
solar luminosity fixes the total neutrino flux precisely, so the count arriving at
Earth is a sharp prediction.

Starting in the late 1960s a series of experiments measured that flux through neutrino
capture, a charged-current process sensitive only to electron neutrinos:

$$
\nu_e + {}^{37}\mathrm{Cl}\to{}^{37}\mathrm{Ar}+e^-,
\qquad
\nu_e + {}^{71}\mathrm{Ga}\to{}^{71}\mathrm{Ge}+e^-.
$$

The chlorine experiment, with a threshold near $0.8\,\mathrm{MeV}$, saw about one third
of the predicted rate. The gallium experiments, reaching down to $\sim0.2\,\mathrm{MeV}$
and thus catching many more of the low-energy neutrinos, saw a smaller shortfall, near
$40\%$. Water-Cherenkov detectors reading the elastic scattering $\nu_x + e^-\to\nu_x+e^-$
above $\sim8\,\mathrm{MeV}$ found a deficit near $50\%$. The magnitude of the deficit
depended on energy, exactly as an oscillation would predict, but for two decades the
possibility that solar models were simply wrong could not be excluded.[^tong-solar]

$$
% caption: The solar-neutrino deficit and its resolution. Charged-current experiments,
% sensitive only to electron neutrinos, saw roughly a third to a half of the predicted
% flux. The SNO neutral-current channel counts all three flavours and recovers the
% full predicted total; the missing electron neutrinos have become muon and tau
% neutrinos.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % axis baseline
  \draw[black, ->] (0,0) -- (0,3.2) node[above, font=\scriptsize] {rate};
  \draw[black, densely dotted] (0,2.8) -- (6.6,2.8);
  \node[black, right, font=\scriptsize] at (6.6,2.8) {predicted total};
  % bar 1: charged current (electron only)
  \draw[very thick] (0.7,0) rectangle (1.9,1.0);
  \node[font=\scriptsize, below] at (1.3,-0.1) {charged};
  \node[font=\scriptsize, below] at (1.3,-0.55) {current};
  % bar 2: elastic scattering (electron plus a little)
  \draw[very thick] (2.7,0) rectangle (3.9,1.5);
  \node[font=\scriptsize, below] at (3.3,-0.1) {elastic};
  \node[font=\scriptsize, below] at (3.3,-0.55) {scatter};
  % bar 3: neutral current (all flavours) -> full height
  \fill[acc!14, draw=acc, very thick] (4.7,0) rectangle (5.9,2.8);
  \node[acc, font=\scriptsize, below] at (5.3,-0.1) {neutral};
  \node[acc, font=\scriptsize, below] at (5.3,-0.55) {current};
\end{tikzpicture}
$$

## The SNO resolution

The Sudbury Neutrino Observatory settled the question with a target of heavy water,
$\mathrm{D_2O}$, whose loosely bound deuteron can be broken by neutrinos in two ways.
The charged-current reaction produces an electron and so counts only electron
neutrinos,

$$
\nu_e + d\to p + p + e^-,
$$

while the neutral-current reaction proceeds through $Z$ exchange, which couples
identically to all three flavours,

$$
\nu_x + d\to p + n + \nu_x .
$$

The neutral-current rate measures the total neutrino flux of every flavour, independent
of any oscillation that has occurred. SNO found that the charged-current rate was again
low, but the neutral-current rate matched the solar-model prediction for the full flux.
The electron neutrinos missing from the charged-current channel had not disappeared;
they had transmuted into muon and tau neutrinos, which the neutral current still
registers. This was direct evidence that the deficit was flavour conversion, not a flaw
in the solar model.[^tong-sno]

## Atmospheric neutrinos

Cosmic-ray protons striking the upper atmosphere produce pions, which decay in a
chain that yields two muon-type neutrinos for every electron-type neutrino,

$$
\pi^+\to\mu^+ + \nu_\mu,\qquad \mu^+\to e^+ + \nu_e + \bar\nu_\mu ,
$$

with the charge-conjugate chain for $\pi^-$. These atmospheric neutrinos carry energies
around a GeV and arrive from all directions. A detector can compare neutrinos coming
straight down, which have travelled the $\sim15\,\mathrm{km}$ thickness of the
atmosphere, against those coming up through the Earth, which have travelled up to
$\sim1.3\times10^4\,\mathrm{km}$.

Super-Kamiokande found that the electron-neutrino flux matched expectation from every
direction, but the muon-neutrino flux showed a clear deficit for upward-going
trajectories. The downward flux was undiminished; the upward flux was suppressed. The
interpretation follows the oscillation formula directly: the long baseline gives the
muon neutrinos many oscillation lengths in which to convert, here predominantly into
tau neutrinos, while the short downward baseline gives almost none. The deficit's
dependence on zenith angle — equivalently on $L$ — is the signature that pins the
effect to oscillation rather than absorption, since neutrinos pass through the Earth
unimpeded.[^tong-atm]

$$
% caption: The atmospheric muon-neutrino flux against zenith angle. Downward-going
% neutrinos, at cosine near plus one, travel only the atmosphere thickness and arrive
% undiminished at the no-oscillation expectation. Upward-going neutrinos, at cosine
% near minus one, cross the Earth diameter and are suppressed as muon neutrinos
% oscillate into tau neutrinos.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.4,0) -- (3.6,0) node[right, font=\scriptsize] {zenith cosine};
  \draw[->, black] (-3.4,0) -- (-3.4,3.0) node[above, font=\scriptsize] {rate};
  \node[black, font=\scriptsize, below] at (-3.0,-0.05) {up};
  \node[black, font=\scriptsize, below] at (3.0,-0.05) {down};
  % no-oscillation expectation: flat
  \draw[black, very thick, dashed] (-3.4,2.5) -- (3.4,2.5);
  \node[black, font=\scriptsize] at (1.5,2.72) {no oscillation};
  % observed: suppressed on the left, recovering to the right
  \draw[acc, very thick, domain=-3.4:3.4, samples=120, smooth]
    plot (\x, {1.35 + 1.15/(1+exp(-2.2*(\x-0.2)))});
  \node[acc, font=\scriptsize] at (-1.7,1.15) {observed};
\end{tikzpicture}
$$

## Oscillations in matter and the MSW effect

Neutrinos crossing dense matter acquire an extra interaction that modifies the
oscillation. Every flavour scatters coherently and forward off electrons, protons, and
neutrons through $Z$ exchange, but only the electron neutrino has the additional
charged-current amplitude off electrons through $W$ exchange. The neutral-current piece
is common to all flavours and cannot affect a flavour difference, so only the
charged-current term matters. It adds a flavour-diagonal potential to the electron
neutrino,

$$
V = \sqrt{2}\,G_F\, n_e ,
$$

where $n_e$ is the electron number density and $G_F$ the Fermi constant. This term
enters the Hamiltonian like an energy shift for $\nu_e$ alone. In the flavour basis,
dropping an overall constant, the two-flavour Hamiltonian is

$$
H = \frac{\Delta m^2}{4E}
\begin{pmatrix}-\cos2\theta & \sin2\theta\\ \sin2\theta & \cos2\theta\end{pmatrix}
+
\begin{pmatrix}V & 0\\ 0 & 0\end{pmatrix}.
$$

A short diagonalisation gives the effective splitting and mixing angle in matter,

$$
(\Delta m^2_m)^2 = (\Delta m^2\cos2\theta - 2EV)^2 + (\Delta m^2\sin2\theta)^2,
$$

$$
\tan 2\theta_m = \frac{\sin2\theta}{\cos2\theta - 2EV/\Delta m^2}.
$$

The mixing in matter is not the vacuum mixing. What makes this dramatic is the
comparison it turns on: the relevant quantity is $m^2$ against $EV$, not $m$ against
$V$. The potential inside the Sun, $V\approx10^{-12}\,\mathrm{eV}$, is minuscule beside
a neutrino mass, but multiplied by an MeV energy it becomes comparable to $\Delta m^2$.
When

$$
\cos2\theta = \frac{2EV}{\Delta m^2},
$$

the effective mixing angle reaches $\theta_m=\pi/4$ — maximal mixing — regardless of
how small the vacuum angle is. This is the resonance condition.[^tong-msw]

$$
% caption: The effective in-matter mixing against the density-energy product. As the
% product two E V approaches Delta m squared times cosine of twice the vacuum angle,
% the effective mixing sine-squared of twice theta rises to one, a full resonance,
% even when the vacuum mixing is small. Above resonance the effective mixing falls
% again.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.2,0) node[right, font=\scriptsize] {density times energy};
  \draw[->, black] (0,0) -- (0,3.0) node[above, font=\scriptsize] {matter mixing};
  \draw[black, densely dotted] (0,2.5) -- (7.0,2.5);
  \node[black, left, font=\scriptsize] at (0,2.5) {full};
  % resonance peak (Lorentzian-like) centred at x=3.4
  \draw[acc, very thick, domain=0:7.0, samples=160, smooth]
    plot (\x, {2.5/(1+pow((\x-3.4)/1.2,2))});
  \draw[black, densely dashed] (3.4,0) -- (3.4,2.5);
  \node[black, font=\scriptsize, below] at (3.4,0) {resonance};
\end{tikzpicture}
$$

As a neutrino climbs out of the Sun the electron density falls smoothly from its
central value to zero, so $V$ sweeps through a range of values. A neutrino born deep in
the core will generically cross the point where the resonance condition holds. If the
density changes slowly on the scale of an oscillation length, the state follows the
instantaneous eigenstate adiabatically and emerges with a large conversion probability,
even for a small vacuum angle. This is the **Mikheyev–Smirnov–Wolfenstein (MSW)
effect**, and it governs the survival of the higher-energy solar neutrinos. For
antineutrinos the potential reverses sign, $V\to -V$, so the resonance shifts; matter
distinguishes neutrinos from antineutrinos, breaking CP and CPT locally because the
background is made of matter rather than antimatter.

## Summary

Neutrinos are created and detected in flavour states but propagate in mass states, and
the misalignment makes flavour oscillate. The two-flavour probability
$\sin^2(2\theta)\,\sin^2(\Delta m^2 L/4E)$ has an amplitude fixed by the mixing angle
and a phase fixed by the mass-squared splitting and the ratio $L/E$; oscillation
requires both a nonzero angle and a nonzero splitting. The solar-neutrino deficit,
seen in charged-current experiments sensitive only to $\nu_e$, was resolved by SNO's
neutral-current channel, which counts all flavours and recovers the full solar flux —
the missing electron neutrinos had become muon and tau neutrinos. The atmospheric
$\nu_\mu$ deficit for upward-going trajectories fixes the effect to the baseline $L$.
Matter adds a charged-current potential to $\nu_e$ that shifts the effective mixing and,
through the MSW resonance, can drive small vacuum mixing to maximal inside the Sun.
The next lesson promotes the single angle to the three-flavour PMNS matrix and its
measured parameters.

[^tong-basis]: Tong, _The Standard Model_ (Cambridge Part III), §7.1.4 and §7.2 — the flavour versus mass basis for neutrinos and the change of perspective relative to quark mixing. [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html)
[^tong-osc]: Tong, _The Standard Model_ (Cambridge Part III), §7.2.1 — the two-flavour derivation, the ultrarelativistic dispersion relation, and the $1.27\,\Delta m^2 L/E$ working formula; Thomson, §13.2, gives the same result. The $\sim1\,\mathrm{km}$ human scale of the oscillation length is emphasised there.
[^tong-solar]: Tong, _The Standard Model_ (Cambridge Part III), §7.2.3 (Solar Neutrinos) — the chlorine, gallium, and water-Cherenkov measurements and their energy-dependent deficits; Thomson, §13.3. Original solar-neutrino deficit: R. Davis Homestake result, reviewed by the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).
[^tong-sno]: Tong, _The Standard Model_ (Cambridge Part III), §7.2.3 — the SNO heavy-water charged- and neutral-current channels and the recovery of the full flux; Thomson, §13.3. SNO Collaboration flux measurement reviewed by the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).
[^tong-atm]: Tong, _The Standard Model_ (Cambridge Part III), §7.2.3 (Atmospheric Neutrinos) — the pion–muon decay chain, the two-to-one flavour ratio, and the upward $\nu_\mu$ deficit against zenith angle; Thomson, §13.4. Super-Kamiokande zenith-angle result reviewed by the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).
[^tong-msw]: Tong, _The Standard Model_ (Cambridge Part III), §7.2.2 (Oscillations in Matter) — the charged-current forward-scattering potential $V=\sqrt2\,G_F n_e$, the effective splitting and mixing angle, and the MSW resonance; Thomson, §13.5. Matter potential and resonance condition also in the Particle Data Group neutrino review, [pdg.lbl.gov](https://pdg.lbl.gov).