# Neutrino Oscillations

by Miriam Diamond

A few years ago, I gave a PSIminar entitled “Neutrinos, and Why I’m Obsessed with Them”. I have been obsessed with neutrinos for nearly a decade, and am still obsessed with them to this day. It began with my work for the Sudbury Neutrino Observatory (SNO), a collaboration of neutrino-obsessives that shared the 2016 Breakthrough Prize in Fundamental Physics and whose Director is among 2015's Nobel laureates. And I doubt it will ever end, as neutrinos keep presenting us with mystery after mystery to unravel.

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Left: Outside of view of Sudbury Neutrino Observatory (SNO) photomultiplier tube support structure. Right: Arthur McDonald who shared the 2015 Physics Nobel Prize with Takaaki Kaijta for leading the experiments that lead to the discovery of neutrino oscillations.

What is it about these shy evasive particles that drove me all the way down a 6800-foot-deep mineshaft? Well, they’re endearingly mischievous little fellows. They first dropped a hint of their existence by running off with some of the beta-decay energy that we thought should rightfully belong to electrons. They enjoy a long-standing career of stealing transverse energy from collision events in particle accelerators. And then, in the spirit of all tricky identity-changing bandits, there’s their ability to oscillate.

Neutrinos come in three flavour eigenstates and three mass eigenstates. But the flavour and mass eigenstates are mismatched, related to one another by a 3x3 matrix. Known as the PMNS matrix, this is the lepton sector equivalent to the quark sector CKM matrix. Quantum field theory tells us that a neutrino produced in a given flavour eigenstate has a certain probability of spontaneously oscillating into a different flavour eigenstate, where we can calculate this probability from the neutrino energy, mass eigenstate splitting, distance travelled in vacuum, and PMNS matrix elements. (When travelling through matter, the oscillation probabilities are modified by the MSW Effect, which stems from coherent forward scattering of neutrinos off of particles in the medium.) It turns out that the off-diagonal terms in PMNS are much larger than in CKM, meaning there is an oddly high amount of mixing between neutrino flavours. As I said, mischievous little fellows.

Neutrino oscillations were the source of the so-called “solar neutrino problem”, which SNO was built to solve. Back in the 1960s, the first solar neutrino detector, the “Homestake Experiment” operated by Ray Davis and John Bahcall, detected only one-third to one-half the number of neutrinos predicted by the standard solar model. There were three possible explanations: results from the fairly primitive detector (basically a giant container of chlorine) were wrong, the standard solar model (dealing with the nuclear fusion processes in the core of the Sun) was wrong, or the Standard Model of particle physics (which at the time assumed massless neutrinos) was wrong. The solar model predictions were based on detailed calculations of the proton-proton chain reaction that produces alpha particles, positrons, gamma rays, and neutrinos – specifically, electron neutrinos. Since helioseismology observations were sparse at the time, it was conceivable that the temperature and pressure values used as input to these calculations were incorrect. But in 1968, nuclear physicist Bruno Pontecorvo pointed out that if neutrinos were, in fact, massive, and the mass eigenstates were mismatched with the flavour eigenstates, electron neutrinos from the Sun could oscillate into other flavours on their way to Earth. And the Homestake Experiment was sensitive only to electron neutrinos.

More advanced experiments over the course of the next few decades confirmed the solar electron neutrino deficit, while helioseismology advances backed up the standard solar model. It was not until 1998, however, that the first experimental evidence of neutrino oscillations emerged, courtesy of the Super-Kamiokande (SK) collaboration in Japan. The SK results showed a deficit in muon neutrinos produced in cosmic-ray showers in Earth’s atmosphere, which could be explained by oscillations into tau neutrinos that the detector was not sensitive to. SNO was the first detector with sensitivity to all three neutrino flavours, and it was able to distinguish between electron neutrinos and the other two flavours. The 2001 SNO results demonstrated that the electron flavour made up only about 35% of the solar neutrinos, with the total number of neutrinos across all flavours agreeing well with the standard solar model predictions. The successful resolution of the solar neutrino problem, and ensuing modification of the Standard Model to include neutrino masses and oscillations, is what this year’s Nobel fuss was all about.

In current neutrino literature, PMNS is conveniently parameterized in terms of three mixing angles (θ12, θ13, θ23) and a complex phase (δ).  θ12 is often called the “solar” mixing angle and θ23 the “atmospheric” angle, based on the source of the neutrinos best-suited to providing constraints on each of them. While we have made remarkable progress in pinning down these angles over the past few decades (as evidenced by the shrinking uncertainties listed for them in the Particle Data Book), the uncertainty on θ13 is still fairly large, and we are utterly at a loss about δ.  And δ is perhaps the most fascinating of all the PMNS parameters, as it controls the amount of CP violation in the lepton sector. I remind you, once again, that neutrinos are mischievous little fellows.

Amongst the newest types of neutrino detector to join our experimental arsenal, complementary to the SNO and SK-style atmospheric and solar “telescopes”, are the so-called “long baseline experiments”. These rely on man-made neutrino sources, such as particle accelerators and nuclear reactors, and measure the mix of neutrino flavours at near-detectors (close to the sources) compared to far-detectors (located underground several kilometres away). Such experiments are valuable for measuring θ13 and δ, as they allow us to control the neutrino fluxes and flavours produced, and to optimize the detector distance to target an oscillation maximum.  The NOvA experiment, which came online last year, is expected to be the first to provide constraints on δ.

While I have focused on neutrino oscillation parameters here, there are yet more unsolved mysteries posed by my favourite subatomic bandits. For example, while we have fairly precise measurements of the absolute values of the neutrino mass splittings, we still do not know the sign of ∆m23. In other words, we do not know whether the mass eigenstate with the most tau flavour component is the heaviest (“normal hierarchy”) or the lightest (“inverted hierarchy”). And we don’t know whether neutrinos are Dirac or Majorana, a conundrum that several experiments hope to resolve by observing (or ruling out) the process of neutrinoless double beta decay. And we don’t know whether a long-hypothesized fourth flavour of neutrino, a right-handed or “sterile” neutrino that would be much more massive than the active flavours, actually exists. If we have learned anything from our past discoveries about neutrinos, it’s that they love to surprise us.

Has some of my neutrino obsession perhaps rubbed off on you? There are plenty of opportunities to get in on the action in this exciting field. Stay tuned for more neutrino Nobel Prizes in the future!

It was not until 1998, however, that the first experimental evidence of neutrino oscillations emerged, courtesy of the Super-Kamiokande (SK) collaboration in Japan. The SK results showed a deficit in muon neutrinos produced in cosmic-ray showers in Earth’s atmosphere, which could be explained by oscillations into tau neutrinos that the detector was not sensitive to. SNO was the first detector with sensitivity to all three neutrino flavours, and it was able to distinguish between electron neutrinos and the other two flavours.

Miriam Diamond is a former PSI student, a neutrino enthusiast and a Ph.D. student at the University of Toronto.