The PMNS Matrix: Three Angles That Describe How Neutrinos Mix
When a neutrino is created it has a definite flavour: electron, muon or tau. When it travels it does so as a combination of three mass states. Those are two different descriptions of the same particle, and the matrix that translates between them is named after Bruno Pontecorvo, who proposed neutrino mixing, and Ziro Maki, Masami Nakagawa and Shoichi Sakata, who formalised it. Everything experiments have learned about neutrino oscillation over fifty years is stored in four numbers inside it - and one of those numbers is still unmeasured.
Two Descriptions of the Same Particle
A neutrino produced in beta decay is an electron neutrino by definition: it is the partner of the electron in that interaction. A neutrino travelling through space propagates as a mass state, because mass is what determines how quantum phase accumulates. These two ways of describing the particle do not line up.
The PMNS matrix is the translation. Each flavour state is a specific superposition of the three mass states, with coefficients given by the matrix entries. Because those coefficients are not zero or one, a particle created as one flavour arrives as a mixture, and that is oscillation.
A three-by-three unitary matrix of this kind can be written using three rotation angles and one complex phase. The angles are conventionally labelled by which pair of mass states they connect, and each is measured by a different kind of experiment.
The reason this matters beyond bookkeeping is that the values are not predicted by any theory. They are inputs, measured one at a time, and their pattern is one of the few empirical clues available about physics beyond the Standard Model.
How Each Parameter Was Pinned Down
The solar angle, around 33 degrees, was established by solar neutrino experiments culminating in the Sudbury Neutrino Observatory, and then confirmed with reactor antineutrinos over a long baseline. It governs the oscillation that resolved the solar neutrino problem.
The atmospheric angle, close to 49 degrees and therefore near maximal, came from Super-Kamiokande in 1998 and has since been refined by accelerator experiments. Near-maximal mixing means an almost complete conversion at the right distance and energy, which is why the atmospheric deficit was so pronounced.
The third angle, about 8.6 degrees, was long consistent with zero and was the last to be measured. Reactor experiments settled it in 2012 by comparing antineutrino rates at different distances from nuclear plants. Had it been zero, leptonic CP violation would be unobservable in oscillation, and the entire long-baseline programme would have had no target.
The phase remains the open parameter. It is what DUNE and the next generation of experiments are built to measure, and its value determines whether neutrinos and antineutrinos oscillate differently.
Why It Looks Nothing Like the Quark Version
Quarks mix too, described by an equivalent matrix. But there the mixing angles are small - a few degrees at most - so the matrix is nearly diagonal and each quark is overwhelmingly its own mass state.
The neutrino matrix is the opposite. Two of its angles are large, one near maximal, and the mixing is thorough. A muon neutrino is not mostly one mass state with small corrections; it is a genuine blend.
This contrast is one of the sharpest unexplained patterns in particle physics. Both matrices describe the same kind of relationship for the two families of fundamental fermions, and they look completely different. Any theory that explains the origin of fermion masses has to account for both, and none convincingly does.
Various patterns have been proposed and then ruled out by better data - several elegant schemes required the third angle to be exactly zero, which the 2012 measurements killed. That history is a useful reminder that appealing symmetry arguments are not evidence.
The Phases Nobody Can See
If the neutrino is a Majorana particle, the matrix carries two additional phases beyond the one that affects oscillation. These are physically real but they cancel out of every oscillation probability, so no oscillation experiment can ever measure them.
They do affect neutrinoless double beta decay, which is one of several reasons the interpretation of that experiment is complicated. The Majorana phases can suppress the rate through cancellation even when the masses would otherwise permit a detectable signal.
This is a genuine limit rather than a temporary one. Some parameters of the lepton sector are inaccessible to the most powerful technique available for studying it, and reaching them requires a completely different kind of measurement.
For the same reason, statements about how well the neutrino sector is understood should be read carefully. Three angles are known to a few percent; one phase is being measured; two more may exist and would be invisible to the whole oscillation programme.
Frequently asked questions
What does the PMNS matrix actually describe?
The relationship between the three neutrino flavours and the three neutrino mass states. Each flavour is a specific superposition of mass states, and the matrix entries are the coefficients.
How many parameters does it contain?
Three mixing angles and one CP-violating phase for a Dirac neutrino. If the neutrino is Majorana, two further phases exist which oscillation experiments cannot measure.
Why is the comparison with quarks interesting?
Because the quark mixing matrix is nearly diagonal with small angles, while the neutrino one has large, thorough mixing. Both describe the same kind of relationship, and no theory explains why they look so different.
Why did the smallest angle matter so much?
Because if it had been exactly zero, leptonic CP violation would be invisible in oscillation experiments. Its measurement in 2012 is what made the current long-baseline programme worth building.
Are all the parameters known?
The three angles are measured to a few percent. The CP phase is not yet determined and is the target of DUNE and the next generation. The two possible Majorana phases are inaccessible to oscillation entirely.
