Neutrino Oscillations

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 $\theta$,

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.¹

The two-flavour oscillation probability

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

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

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 $\Delta E=E_2-E_1$:

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

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

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 $L/E$:

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

The oscillation length — the distance over which the phase advances by $2\pi$ — is $L_{\mathrm{osc}}=4\pi E/\Delta m^2$. For $\Delta m^2\sim 10^{-3}{\mathrm{eV}}^2$ and $E\sim 1\mathrm{GeV}$ this is a few hundred kilometres, a length scale that a terrestrial baseline can span.²

The solar neutrino problem

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

The dominant chain produces neutrinos with energy below about $0.4\mathrm{MeV}$; rarer branches involving beryllium and boron reach up to $\sim 10\mathrm{MeV}$. 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:

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 $\sim 0.2\mathrm{MeV}$ 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 $\sim 8\mathrm{MeV}$ 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.³

The SNO resolution

The Sudbury Neutrino Observatory settled the question with a target of heavy water, ${\mathrm{D}}_2\mathrm{O}$, 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,

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

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.⁴

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,

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 $\sim 15\mathrm{km}$ thickness of the atmosphere, against those coming up through the Earth, which have travelled up to $\sim 1.3\times 10^4\mathrm{km}$.

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.⁵

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,

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

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

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, $V\approx 10^{-12}\mathrm{eV}$, is minuscule beside a neutrino mass, but multiplied by an MeV energy it becomes comparable to $\Delta m^2$. When

the effective mixing angle reaches ${\theta }_m=\pi /4$ — maximal mixing — regardless of how small the vacuum angle is. This is the resonance condition.⁶

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.