A Primer on Particle Physics, Part 7: Leptons and Neutrino Oscillations

Well, it’s been a month since part 6 of this series, and I can only apologize for the delay.  But at last it’s back, and today we’re going to talk about my favorite part of the Standard Model (if only because it’s the part I study): the neutrinos.  I could go on about neutrinos at much greater length than the present article, and I’ve had to restrain myself from going into detail about how neutrinos are detected, showing diagrams for all their interactions, and talking about some of the other areas of physics neutrinos overlap with.  Maybe someday I’ll write another article specifically on neutrinos (and of course I’d be happy to talk more about this stuff in the comments).

But let’s back up first.  I told you that fermions (the particles that make up matter) come in two familes, the quarks and the leptons. So far we’ve talked a lot about quarks, but the only lepton we’ve met is the electron. So today we’ll introduce the rest of the leptons, and in particular we’ll talk about the most mysterious particles in the Standard Model, the neutrinos.

To put a big picture overview first, we’ll find that as with the quarks, the leptons come in three generations, each containing two particles. Whereas with the quarks, each generation has a positively charged (+2/3) and a negatively charged (-1/3) member, with the leptons each generation contains a charged lepton (with a charge of -1) and a neutrino, with a charge of zero.

The Charged Leptons

We’ve already talked about the good old electron, familiar to us through being, you know, the foundation of all chemistry and, as if that wasn’t enough, also electricity. The electron has a mass of about 511 keV, which makes it about a quarter as massive as the lightest of the quarks. And it has a negative electric charge, which in particle physics we conventionally define as -1 (in other words, we measure all electric charges in units of the electron’s charge).

As we talked about a few weeks ago, all you really “need” to make up most of the universe as we experience it are the electron, the lightest two quarks, the gluon, and the photon. From the quarks and gluons, you can build protons and neutrons and pions, and from these you can build atomic nuclei. With the photon, you can bind electrons to those nuclei and make atoms. So by the 1930s, physicists more or less thought they knew all the subatomic particles – they had no knowledge of quarks, yet, but they knew about protons and neutrons and electrons and photons, and they had an inkling of the pion, and that seemed to be all that was needed. But in 1936, in an experiment that was studying cosmic radiation (particles originating outside the solar system that hit the Earth), they noticed something strange. Like most experiments of the day, and many now, this one used a magnetic field to measure the properties of the particles it was detecting. Charged particles curve into spiral trajectories in a magnetic field, with positively charged particles curving in one direction and negatively charged particles in the other. How sharply the particle curves depends on the magnitude of its charge and on the particle’s mass. In this experiment, Carl Anderson and Seth Neddermeyer noticed that, along with all the massive protons that spiralled very slowly in one direction, and the light electrons that spiralled much more sharply in the other direction, they would occasionally see a third pattern – a particle that curved in the same direction as the electron, but with a curvature in between that of the electron and the proton. They concluded, correctly, that they were seeing a previously unknown particle, one with the same charge as the electron, but much heavier. This was the first hint of the existence of particles beyond the ones needed to make up atoms, and this discovery is the one that prompted Nobel laureate I.I. Rabi’s reaction, “Who ordered that?”

The particle they had discovered is the one now known as the muon, or sometimes the “mu lepton”1. The muon has exactly the same charge as the electron (at least to the level of precision we’ve been able to measure), but it is over two hundred times heavier, with a mass of 105.7 MeV. In the 1970s, the third generation of leptons was discovered with the tau lepton2. Again, the tau has the same charge as the electron, but its mass is about 1777 MeV. The muon and the tau can both more or less be thought of as heavy electrons. Like the electron, they do not couple to the strong interaction that plays so large a role in the lives of the quarks, but they do couple to the electromagnetic and weak forces.

Both the muon and the tau are unstable (as one would expect of heavy particles). The muon’s lifetime is about 2.2×10^-6 seconds, while the tau’s is about 3×10^-13 seconds. Those both might sound like really short lifetimes, but the muon actually sticks around long enough to do some interesting things. For one thing, muons are among the particles generated when cosmic rays hit the atmosphere (this is how they were first observed, after all), and because they are generated with such large energies, these muons are travelling at a significant fraction of the speed of light. If you’ve ever learned anything about Special Relativity, you’ll know that this means they experience time dilation, so that 2.2 microsecond lifetime gets stretched out, and this is what allows those muons to survive long enough to reach the Earth’s surface. A muon can penetrate many meters of solid material before its random collisions with atoms in that material slow it down and it decays. This is one of the main reasons that a lot of high-sensitivity physics experiments are built deep underground; you need a lot of rock between the Earth’s atmosphere and your experiment, or else you’ll just be swamped with these cosmic muons.

The fact that muons can penetrate so deeply into matter makes them useful for tomography.  We’re all familiar with x-ray tomography, which uses the scattering of high energy photons to create images of the interior of solid matter.  But muons are even more deeply penetrating than x-rays, so they can be used in a similar fashion for much larger objects.  For instance, muon imaging has been used on a number of Egyptian pyramids and has even revealed a previously unknown chamber in the Great Pyramid of Khafre.

The Neutrinos

In the 1920s, physicists were grappling with a thorny problem. By that time, many experiments had demonstrated the phenomenon called “beta decay”, a radioactive process wherein a neutron in an atomic nucleus decays into a proton (changing the atomic number of the nucleus) and emits an electron.

The puzzling thing about it was this. If the neutron is decaying into just two particles – a relatively heavy proton and a relatively light electron – then for momentum to be conserved, the proton that is created will have to be nearly stationary, while almost all of the kinetic energy is carried off by the electron. And if energy is conserved, then (since the masses of the neutron, proton, and electron are fixed constants), there should always be exactly the same amount of kinetic energy, which is just the leftover energy when we subtract the masses of the proton and electron from that of the neutron. In other words, if we measure the energy of the electrons produced by beta decays, we should always get the same answer – with just a tiny bit of variation due to the electron potentially scattering off of other atoms before it reaches our detector.

In other words, what physicists expected to see was something like this:

discrete spectrum

But instead, these beta decay experiments found that the emitted electron could have a wide variety of energies. Instead of a single spike, the observed energies formed a continuous spectrum:

continuous spectrum

This was quite bewildering. The idea that energy could just disappear – not being converted into some other form, but just vanishing from the universe – was deeply counterintuitive, but the problem was greater than that. Conservation of energy and momentum was a core concept in every theory of physics, from Newtonian mechanics to Electrodynamics to Quantum physics. If experiments demonstrated that energy was not conserved, it would mean the rejection of a lot of otherwise very successful physical theories, and it was not at all clear how Quantum Mechanics could be revised to accommodate this. Nonetheless, the idea that energy conservation might have to be abandoned was seriously considered, as clearly something that wasn’t understood was going on with beta decay. No less a personage than Niels Bohr proposed that perhaps the conservation of energy was not a strict law of physics, but only a “statistical” one; energy might be created or destroyed on a small scale, but on average, those gains and losses would balance out.

Wolfgang Pauli proposed an alternative, which he called a “desperate remedy”, to save energy conservation. What if the beta decay produced not just a proton and an electron, but also a third particle? Then, instead of the kinetic energy being almost entirely carried off by the electron, it could be split between the electron and this new particle – and, importantly, it could be split between them in any way; sometimes, the electron might get almost all the excess energy, sometimes the other particle, sometimes it would be split more evenly between them. This new particle would have to be electrically neutral, in order for charge conservation to still work, so it was soon named the “neutrino”3.

But in order for this to make any sense, Pauli had to postulate that the neutrino was, essentially, undetectable. When a beta decay happens, we can see the electron; we can detect it, measure its energy, etc. But in no beta decay experiment was an additional particle ever seen. Pauli was, then, proposing a particle that he believed could not be detected; he was putting forward a hypothesis that appeared to have no empirical consequences and not to be subject to experimental testing. This, understandably, deeply troubled him for philosophical reasons. Nonetheless, existence of this new particle would very neatly and very simply explain the beta decay observations4, and it was a more attractive proposal than abandoning energy conservation.

Fortunately, Pauli’s hypothesized particle turned out not to be undetectable, but only very difficult to detect. As the theory describing the weak interaction took shape, it became clear that the same coupling between the neutrino and the weak bosons that allowed neutrinos and antineutrinos to be produced in beta decays and other nuclear processes should also allow them to interact with particles like protons, neutrons, and electrons.

Why, then, is the neutrino created in a beta decay not detected? It’s because those weak interactions are the only ones available to the neutrino, and the weak interaction is weak. I mentioned that muons are deeply penetrating particles compared to electrons or protons, but they’ve got nothing on neutrinos. A muon can, on average, penetrate many meters of rock before its electromagnetic-mediated collisions with atoms in that rock slow it down and stop it. On the other hand, a neutrino could, on average, pass through a light year’s thickness of lead before its weak-mediated collisions with atoms in the lead slowed it down and stopped it. The chances that any particular neutrino will interact within a detector are essentially nil. That means that in an experiment where we’re looking at processes that produce neutrinos, those neutrinos are always invisible.

There are lots of neutrinos in the universe, though. Hold out your hand for three seconds. About one trillion neutrinos just went through it. We’re able to build successful neutrino detectors simply because there are so many of them; even if only a tiny fraction of them happen to interact with atoms within our detector, a tiny fraction of a trillion can still be a lot. Neutrinos were finally detected in 1956, putting to bed the discomfort with the “undetectable” particle.

Neutrinos come from many sources, as they are produced in all sorts of nuclear reactions. The fusion processes that power the sun generate huge numbers of them, which form the bulk of the neutrinos that pass through the Earth. Nuclear reactors also produce neutrinos in their fission reactions. A background of neutrinos created immediately after the Big Bang suffuses the universe. And at particle accelerators, physicists can produce high-intensity neutrino beams.

As with the charged leptons, the neutrinos come in three flavors: the electron neutrino, the muon neutrino, and the tau neutrino. The weak force couples each neutrino with its corresponding charged lepton (much as it couples the positive and negative quark in each generation).  That means that a vertex like this is allowed in a Feynman diagram, with an electron neutrino either emitting or absorbing a W boson and converting into an electron:

Neutrino electron W vertex



However, this vertex would not be allowed if the neutrino in question were a <i>muon neutrino</i> instead of an electron neutrino.  Thus a scattering process like this one is allowed:

neutrino-electron charged currentWhile the corresponding diagram with a muon neutrino scattering off of an electron would not work.  Of course, the process would be allowed if it were a muon neutrino scattering off of a muon.  That’s what we mean by neutrino flavor, and it’s useful to remember this definition later when we talk about neutrino oscillations; the three different flavors of neutrino are defined by which charged lepton they can form weak interaction vertices with.

All experiments aimed at directly measuring the mass of these neutrinos have turned up results of zero to within the precision of the measurement. For a long time, it was thought that the neutrinos’ masses were exactly zero, like the photon or the gluon. In the late ’90s, though, indirect evidence showed that neutrinos do in fact have mass – albeit very small, less than about 0.1 eV. Remember, the mass of the electron (the lightest of the elementary particles we’ve run into so far) is 511 keV – that is, the electron is at least five million times heavier than the neutrinos are.

So how do we know that the neutrino’s mass is non-zero? Well, to answer that, we’ll have to talk about neutrino oscillation.

Neutrino Mixing and Oscillation

In the 1960s, Ray Davis and John Bahcall designed the “Homestake experiment”, intended to study solar neutrinos – the neutrinos produced by the sun’s fusion reactions, which, just like the light that the sun produces, stream outward in all directions into the solar system. Bahcall had contributed to the development of the “standard solar model”, a mathematical model for the structure of, and nuclear reactions in, the sun. Using this model, he calculated the number of neutrinos that the sun’s thermonuclear engine should produce, from which he was able to predict the number of neutrinos that should be detected in the Homestake experiment.

The fusion reactions in the sun all generate electron neutrinos. When an electron neutrino interacts with an atomic nucleus, one process that can occur is that the neutrino is converted into an electron, while one of the neutrons in the nucleus is converted into a proton. In the Homestake experiment, chlorine was used as the target material; when one of the neutrons in a chlorine atom is converted into a proton, the chlorine nucleus turns into a radioactive isotope of argon. These argon nuclei can then be collected and counted to determine the number of neutrino interactions that have occurred.

Much to everyone’s surprise, the Homestake experiment ended up observing only about one third as many neutrinos as predicted. It was, naturally, thought that there must be a mistake either in Bahcall’s calculation or in Davis’s experimental methodology; either the model predicting the number of neutrinos must be wrong, or the experiment must be flawed. This was called the “solar neutrino problem”. In subsequent years, several other experiments also observed just a third of the predicted solar neutrinos – and yet, no one was really sure how solar model could be adjusted to account for these observations.

But there was a third possibility: that there was nothing wrong either with the experiments or with the standard solar model, but that something was happening to the neutrinos between when they were produced in the sun and when they passed through the detector on Earth. It wasn’t until the late ’90s that this “something” was discovered, when the Super-Kamiokande experiment, looking at neutrinos produced by cosmic rays in the atmosphere, saw evidence that neutrinos could change flavor. What was happening with the solar neutrinos, then, was that while all of them were produced as electron neutrinos, about two thirds of them had turned into muon or tau neutrinos by the time they reached the Earth. Since Homestake and similar experiments could only detect electron neutrinos, they saw a deficit compared to what was expected. In 2001, this explanation was confirmed, when the SNO experiment, which unlike its predecessors was able to detect all flavors of neutrino, found that when you count all the neutrinos, the results agree well with the standard solar model.

The mechanism for this flavor-changing behavior is something we’ve already encountered. We saw earlier that quark mixing allows quarks of one generation to decay into quarks of another generation. What allows this is that the flavor states (the states that the weak force “sees”) of the quarks are not quite identical to their mass states. Thus, say, a quark that is a bottom quark in terms of its mass state has a small probability of acting like a strange quark or a down quark when it undergoes a weak interaction.

Something very similar is going on with the leptons, but there are some important differences as well. When we talk about electron neutrinos, muon neutrinos, and tau neutrinos, we are talking about the neutrinos’ flavor states; an electron neutrino, for instance, means a neutrino that couples to the electron through the weak force. As with the quarks, it turns out that these flavor states are not the same as the neutrinos’ mass states. But whereas with the quarks, we saw that the mass states are nearly the same as the flavor states, with the neutrinos, the two sets of states are completely different. With the quarks, for instance, 95% of the down flavor state comes from one mass state, with just small contributions from the other two mass states. With the neutrinos, the flavor states consist of much more even mixes of the three mass states.

Here’s what I mean. Below are visualizations of the “mixing matrices” for the quarks and the neutrinos, showing how much each of the flavor states (the rows) comes from each of the different mass states (the columns). For the quarks, each flavor state is made mostly of a single mass state, but for the neutrinos they’re all mixed up.

Mixing matrices

Another obvious difference is that while quarks either decay or form composite particles immediately after being produced, neutrinos are stable – meaning they don’t decay – and they very rarely interact. So when a neutrino is produced, it will generally zip off in some direction and just keep going and going.

Now here’s where I’m going to have to get a little hand-wavey unless we want to go deep into the math, but hopefully this will still make sense. We talked way back about the fact that an elementary particle is actually an excitation of a quantum field, and that its motion through space is that of a wave moving through the field. We’re somewhat used to thinking of photons in this way – we talk about the wavelength and frequency of light or radio waves, for instance. But of course a neutrino, too, is really a wave moving through the underlying neutrino field that fills the universe. As a wave, it also has a frequency. And in Quantum Field Theory, it turns out that the frequency of this wave depends on the mass of the particle. The upshot of all this is that neutrinos with different masses will move through space as waves with different frequencies.

So what happens when, say, an electron neutrino is produced by a fusion reaction in the sun? The electron neutrino state is a particular combination of the three different neutrino mass states. And each of those mass states will propagate with a slightly different frequency. This means that the different mass states will gradually become “out of phase” with each other.

Out of phase

The sum of those mass states will no longer add up to exactly the combination that makes up an electron neutrino. If we were to write the neutrino’s state out in terms of flavor states now, it would also have contributions from the muon neutrino and tau neutrino flavors. And that means that if the neutrino undergoes a weak interaction – say, colliding with an atom in a detector on Earth – there is a chance it will interact as one of those other flavors. This, then, is what happened in the “solar neutrino problem”. Each neutrino coming from the sun started as an electron neutrino – a particular combination of the three different mass states. But as it travelled to the Earth, those three different components shifted into a different combination, and by the time it reached the Homestake detector, the particular combination they were in had only about a one in three chance of interacting as an electron neutrino.

The chance that a neutrino that began life as one flavor will interact as a different flavor (the “oscillation probability”) depends on a few things. First of all, it depends on what particular combination of mass states each flavor state is made of – in other words, what the values are in that matrix we saw above. As with quark mixing, we can write these matrix elements in terms of a set of three mixing angles and something called a CP-violating phase. The oscillation probability also depends on the mass differences between the three mass states, since these determine how quickly the three waves will move out of phase. And finally, it depends on the distance that the neutrino has travelled since being created5, since that determines how far out of phase (or, eventually, back into phase) the different mass states will have gotten.

Note that in order for any of this to happen, the neutrinos have to have three different mass states. This is why the observation of neutrino oscillations implies that neutrinos have mass – if the masses were all zero, there’d be no mixing, because there wouldn’t be different mass states to mix, and there wouldn’t be different frequencies to get out of phase with each other. (Technically, all that we know is that the three masses are different – it’s still possible that one of them could be zero, but that would be pretty strange).

But the thing we can measure when we study neutrino oscillations is just the number of neutrinos of one flavor that have turned into another flavor, and as I said, that only depends on the differences between the masses6. And that means that we can’t actually measure the absolute masses of the neutrinos through oscillation experiments, only the differences between them. Moreover, we can’t tell which direction those differences go in – we can’t tell which mass state is heavier and which is lighter. Experiments have found that two of the mass states are only about 0.000075 eV apart, and the other mass state is about 0.0024 eV away from these two, but we don’t know whether it’s heavier or lighter7:

Mass hierarchies

In any event, as mentioned earlier, experiments that have tried to directly determine the masses of the neutrinos have so far only found that they must be less than about 0.1 eV8. The fact that the neutrinos’ masses are so tiny, and yet not zero, suggests that perhaps there is something different about those masses come about compared to the other elementary particles. There is a big philosophical question surrounding whether we can reasonably be surprised at the value of any given fundamental constant – at some level, one could argue, the masses of the particles just are what they are and there’s no “why” about it; that’s just the way the universe is. Nonetheless, it does seem, in some sense, unlikely that the neutrino masses would just happen to be so small, when there’s no reason they couldn’t have had masses on the same scale as the other fermions. There’s a sort of intuition one has that there ought to be something deeper that explains why the neutrino masses are so small. This is what physicists call a “fine-tuning problem”.

Neutrinos Beyond the Standard Model

A lot of ideas relating to neutrino mass have been put forward, and we’re in an exciting era where we’re just starting to be able to test some of them. To talk in any depth about these ideas would require its own article, but there are two main ideas I want to mention here. The first is that the neutrinos might have what is called a “Majorana mass”, as opposed to the other fermions which all have only a “Dirac mass”. I hope to get into this difference a little more in the next article, but one of the upshots of this would be that neutrinos are their own antiparticles – a neutrino and an antineutrino would be interchangeable, in which case they would be the only known fermions for which this is true.

The other idea, which turns out only to be possible if the neutrino ends up being a Majorana fermion, goes by the amusing name of the “Seesaw Mechanism” and is a possible explanation for the neutrinos’ masses being so small. This proposal posits that in addition to the three neutrinos we know of, there are one or more very heavy “sterile neutrinos”. These would be neutrinos that do not interact even via the weak force, and are thus invisible to our detectors, but that could still participate in neutrino oscillations. The masses of these heavy neutrinos would be inversely related to the masses of the light neutrinos – the heavier the sterile neutrinos are, the lighter our familiar light neutrinos would be, hence “seesaw”. Thus, the fact that the light neutrinos have masses that are so much smaller than the other elementary particles would be explained by the large masses of the sterile neutrinos.

When we talked about quark mixing, we saw that one of the physical constants that governs the mixing is a “CP-violating phase”. This essentially dictates to what degree the mixing works differently for quarks than it does for antiquarks. As I mentioned then, the CP-violating phase for the quarks has been measured to be non-zero (meaning there is some asymmetry between quarks and antiquarks), but small. Too small, in fact, to explain the asymmetry we see in the universe between matter and antimatter. All the stuff of the universe, after all, from stars to planets to humans, is made of matter; but without CP-violation, equal amounts of matter and antimatter would have been created in the Big Bang, and would have annihilated with each other in the early universe. The very fact that we exist, then, is a bit of a puzzle, and as we saw last time, the degree of CP-violation in the quarks is too small to account for the asymmetry.

Now that we know about neutrino mixing, though, a possible solution presents itself, since the neutrinos have their own CP-violating phase. If this turns out to be large enough, then it may explain why more matter than antimatter was produced in the early universe. To be sure, generating the observed asymmetry in the leptons and quarks requires a bit more than just a large CP-violating phase for the neutrinos; there would also have to be mechanisms that convert that asymmetry into an asymmetry in the other particles. As a matter of fact, the Seesaw Mechanism, which was proposed to explain the neutrinos’ tiny masses, might do the trick and explain the preponderance of matter through asymmetries in the decays of the heavy sterile neutrinos into other particles.

Granted, for this explanation to work, a lot of things are needed: neutrinos have to be Majorana particles, there have to be heavy sterile neutrinos, and the neutrinos’ CP-violating phase has to be large. All of these questions are still up in the air at the moment, but experiments that are running now or will be running in the near future will start to answer them. Neutrino oscillation experiments have already found evidence that the CP-violating phase is not zero, and as they continue to collect data, more precise measurements of the phase will be made. And a number of experiments are searching for a tell-tale process called “neutrinoless double beta decay” that, if observed, would mean that neutrinos have a Majorana mass.  So it’s very possible that in the coming years, neutrino experiments will be the first to point the way toward a new understanding of particle physics that goes beyond the Standard Model.

I plan for there to be two more articles in this series, and hopefully I can get them done in a more timely fashion.  The next one will talk about the one big remaining piece of the Standard Model that we haven’t talked about yet: the Higgs mechanism and the unification of electromagnetism and the weak force into the “electroweak” force.

If you’ve made it this far, thanks for reading (and good job)!  And as always, I’d be happy to answer any questions or talk further about this stuff in the comments.