Neutrinos — just like the charged leptons (electron, muon, tau), the up-type quarks (up, charm, top), and the down-type quarks (down, strange, bottom) — come in three types. But there’s more than one way to divide them up. And thanks to the quantum nature of our world, you can only use one way at a time. In this article I’ll explain this remark, and how a very interesting and scientifically crucial phenomenon — neutrino oscillations — arises from this fact.
You probably think of particles as having a definite mass — all electrons have a mass-energy (E=m c2) of 0.000511 GeV — and from one point of view, the three types of neutrinos are no exception. We can classify the three neutrinos by their masses (still largely unknown, see below), and call them, from lightest to heaviest, neutrino-1, neutrino-2 and neutrino-3. We’ll call this the mass-classification, and these types of neutrinos as mass-type.
Fig. 1: At left, the mass-type neutrinos (neutrino-1, -2 and -3) have definite masses (still unknown, though some differences of the squares of their masses are known throguh the measurements described below.) At right: the weak-type neutrinos (the electron-, muon- and tau-neutrino) are named for the charged lepton that they accompany when they interact with a positively charged W particle, a carrier of the weak nuclear force. An electron neutrino is a mixture of the three mass-type neutrinos, while neutrino-3 is a mixture of the weak-type neutrinos.
But another way to classify the neutrinos is by how they are connected with the charged leptons (the electron, the muon and the tau.) [This shows up in my article on what the known particles would look like if the Higgs field were zero.] The best way to understand this is to focus on how the neutrinos are affected by the weak nuclear force, which is reflected in their interactions with the W particle. A W particle is very heavy, and if you make one, it can sometimes decay (see Figure 1) to one of the three charged anti-leptons and one of the three neutrinos. If the W decays to an anti-tau, the neutrino produced in association with it is a tau-neutrino. Similarly, if the W decays to an anti-muon, a muon-neutrino is emitted. (As is crucial in making a neutrino beam, a pion decays through the weak interactions, and positively-charged pions make an anti-muon and a muon-neutrino.) And if the W decays to a positron, out comes an electron-neutrino. We’ll call this the weak-classification, and these neutrinos weak-type, since it is the weak nuclear force that determines their details.
What’s the big deal? We use multiple classifications of people all the time. We talk about people being young, middle-aged and old; or we talk about them being tall, average-height and short. But with people, we can always separate them further, if we choose, into nine categories: young and tall, young and average-height, middle-aged and short, old and short, etc. Yet quantum mechanics prohibits us from doing the same for these neutrino classifications. There is no such thing as a neutrino that is both a muon-neutrino and a neutrino-1; there is no such thing as a tau-neutrino-3. If I tell you a neutrino’s mass (and therefore that it is neutrino-1, -2 or -3) I simply cannot tell you if it is an electron-, muon- or tau-neutrino. In fact, a neutrino of definite mass-type is a mixture, or “superposition”, of the three weak-type neutrinos. Each mass-type neutrino — neutrino-1, neutrino-2 and neutrino-3 — is a precise but different mixture of electron-, muon- and tau-neutrino.
The reverse is also true. If I see a pion decay to an anti-muon and a neutrino, I know immediately that the neutrino emitted was a muon-neutrino — but I can’t know its mass, because it is a mixture of neutrino-1, neutrino-2 and neutrino-3. The electron-neutrino and tau-neutrino are also precise but different mixtures of the three neutrinos of definite mass.
The relation between these mass-types and weak-types is more like (but not exactly like) the relation between classifying highways as “north/south” and “east/west”, (as the US government does, assigning odd numbers to N/S highways and even to E/W roads) versus describing them as “north-east/south-west” or “south-east/north-west”. There might be good reasons to use either one: the N/S-E/W classification is good if you want to focus on latitude and longitude, while NE/SW-SE/NW might be more convenient in the vicinity of a coastline that itself runs from southwest to northeast. But either way, you cannot use both classifications at the same time. A northeastward road is both partly a north road and partly an east road; you cannot say it is one or the other. And a northward road is a mixture of a northeast road and a northwest road. So it is with neutrinos: a mass-type neutrino is a mixture of weak-type neutrinos, and a weak-type neutrino is similarly a mixture of mass-type neutrinos. (Where this breaks down as an analogy is that you could, if you wanted, use a more refined four-category N/S-NE/SW-E/W-SE/NW classification for roads; the analogous option is not available for neutrinos.)
This inability to classify a neutrino as being of definite mass-type and of definite weak-type simultaneously is an example of the “Uncertainty Principle”, similar to the weirdness that makes it impossible to know the position of a particle exactly and simultaneously know the particle’s velocity exactly. If you know one of these quantities perfectly, you have no clear knowledge of the other. Or you can learn something, but not everything, about both. Quantum mechanics tells you exactly how well you can balance your knowledge against your ignorance. By the way, these issues aren’t special to neutrinos. They occur with some other particles too. But they are especially important for the behavior of neutrinos.
A few decades ago, times were simpler. Then it was thought neutrinos might be massless, in which case the weak-type neutrino classification would suffice. And if you look at old papers or at old books for lay-people, you’ll see only the names electron-neutrino, muon-neutrino and tau-neutrino. However, after the discoveries of the 1990s, this no longer suffices.
Now here’s where it starts to get interesting. Suppose you start with a high-energy neutrino of electron-type, so that it is a particular mixture of neutrino-one, neutrino-two and neutrino-three. The neutrino propagates through space, but within it, its three different mass-types travel at very, very slightly different velocities, all of them very close to the speed of light. Why is that? Because the speed of an object depends both on its energy and on its mass, and the three mass-types have different masses. [Click here for the relevant (simple) formulas.] The velocity difference is super tiny for any neutrinos we have a hope of measuring — it’s never been observed — but it has a surprising and big effect anyway!
And here’s where it goes from just interesting to very weird.
This very tiny difference in velocities causes the precise mixture of neutrino-1, neutrino-2 and neutrino-3 that makes an electron-neutrino to change gradually as the neutrino moves through space. And that means that the electron-neutrino that we start with is, after a while, no longer an electron-neutrino, which corresponded to one very particular mixture of the neutrino-1, -2, and -3. The different masses of the three mass-type neutrinos cause the original electron-neutrino, as it travels, to become a mixture of a electron-neutrino with a muon-neutrino and a tau-neutrino. The amount of mixing depends on the differences in velocities, and hence on the energy of the original neutrino and on the difference in the masses (actually the difference between the squares of the masses) of the neutrinos.
Fig. 2: An electron neutrino is a mixture of the three mass-type neutrinos, which, due to their differing masses, will travel at slightly different velocities. This means that the electron-neutrino will evolve into a mixture of the three weak-type neutrinos, and back into an electron neutrino. (Actually this is oversimplified, as the oscillation of three neutrino types typically has two frequencies, not one as shown here.) This effect, sensitive to differences in the squares of neutrino masses and to the mixing between mass-type and weak-type neutrinos, can be measured and serves as a powerful probe of neutrino properties.
Initially the effect increases as the neutrino travels. But curiously, as sketched in Figure 2, this effect doesn’t just grow and grow. It grows, and then it shrinks again, and then grows again, and shrinks, over and over, as the neutrino moves along. This is called neutrino oscillations. How exactly the oscillations happen depends on what the masses of the neutrinos are and how the mass-type neutrinos and the weak-type neutrinos are mixtures of one another.
The effect of the oscillation can be measured because an electron-neutrino, when it collides with a nucleus (which is how neutrinos can be detected) can turn into an electron, but not a muon or tau, while a muon-neutrino can turn into a muon but not an electron or tau. Thus if one starts with a muon-neutrino beam, and some of the neutrinos in the beam, after traveling some distance, strike a nucleus and are converted into an electron, then it means that oscillation is taking place and muon-neutrinos are turning into electron-neutrinos.
There’s one more really important effect that complicates and enriches the story. Because ordinary matter is made from electrons but not from muons and taus, electron-neutrinos have different interactions with ordinary matter than muon- and tau-neutrinos do. These interactions, which occur through the weak nuclear force, are really, really tiny. But if a neutrino passes through a great deal of matter (such as a substantial fraction of the earth or sun) these small effects can add up and have a big impact on the oscillations. Fortunately, we know enough about the weak nuclear force to predict this effect in detail, and to work backwards from what we measure in an experiment to figure out what the neutrinos’ properties must be.
All if this involves quantum mechanics. If it doesn’t seem intuitive to you, relax; it isn’t intuitive to me either. Any intuition I have I learned from the equations.
It turns out that careful measurements of neutrino oscillations are in fact the fastest way to learn about the properties of neutrinos! Nobel prizes have already been won for this work. Indeed, the whole story I’ve been telling you arose out of a classic interplay between experiment and theory that has stretched from the 1960s through to the present. Let me mention a number of important measurements that have been crucial.
For one thing, we can study electron-neutrinos produced in the center of the sun, in its well-understood nuclear furnace. These neutrinos travel outward through the sun and through empty space to earth. It has been found that by the time the neutrinos arrive at the earth, they are just about as likely to be of muon- or tau-type as of electron-type. That by itself is evidence for neutrino oscillation, and the detailed pattern gives us some precise information about the neutrinos.
We also have muon-neutrinos produced in the decay of pions that are in turn produced in cosmic rays. (High energy particles from space hit atomic nuclei high up in the atmosphere; the showers of debris include many pions, many of which decay to muon-neutrinos and anti-muons, or to muon-anti-neutrinos and muons.) Some of these neutrinos (and anti-neutrinos) are detected in our neutrino detectors, and we can see what fraction of them are electron-neutrinos (and anti-neutrinos) as a function of how much earth they traveled through to get to the detectors. Again, this gives us significant insights into how neutrinos behave.
These “solar” and “atmospheric” neutrinos have taught us much about neutrinos in the last twenty years (and the first hint of something interesting goes back nearly 50 years.) And these natural neutrino sources have been supplemented by numerous studies done with neutrino beams, such as the one used in the OPERA experiment, and with neutrinos from ordinary nuclear reactors. Each of these measurements has largely agreed with the standard interpretation of the solar and atmospheric neutrinos, and has allowed more precise measurements of the mixtures of the mass-type and weak-type neutrinos and the differences of the squares of the mass-type neutrinos’ masses.
As you would expect with any group of experiments, there are some small discrepancies from theoretical expectations, but none of them have been confirmed, and most if not all are probably just statistical accidents or experimental problems. So far there is nothing that has been confirmed by multiple experiments that contradicts this understanding of neutrinos and their behavior. But conversely, this picture is new enough and poorly-enough tested that it is possible, though perhaps unlikely, that there are completely different interpretations. Indeed some serious alternatives have been suggested. So clarifying the details of neutrino properties remains an area of active research, one in which a consensus has been emerging but in which some substantial questions remain open — including determining the neutrino masses once and for all.
Hi, great article. One thing I didn’t understand: the three mass components of an electron-neutrino travel at slightly different speeds, and you say this causes the electron neutrino to oscillate into a mixture of all three weak types. But over time, would not the three components separate out in space(due to their different speeds), so that after travelling a large enough distance, one would find three mass-neutrinos at different points in space?
A final question, is would an initial mass-type neutrino oscillate into a superposition of other stuff?
Eventually, yes, they would separate… but this is a subtle subject involving the notion of quantum decoherence.. I’d have to review some fairly detailed literature to make sure I made correct statements about this. For all current experiments this doesn’t play a role, but certainly my presentation relies on some approximations whose validity at some point will break down.
An initial mass-type neutrino would not oscillate in empty space, but could still oscillate in matter, since its electron-type part would behave differently in matter than its muon- and tau-type parts, meaning that it would become a mixture of different mass-types. But this doesn’t ever arise, because the only way to make neutrinos in practice is through the weak interactions, so we always end up starting with a weak-type neutrino.
“these issues aren’t special to neutrinos. They occur with some other particles too. But they are especially important for the behavior of neutrinos.”
By “these issues”, do you just mean that the Uncertainty Principle (discussed in the preceding paragraph) applies to other particles too, or are you saying that other types of particles can also (in principle) oscillate between types in this way (for example, that electrons could oscillate into muons or taus)?
If it’s only neutrinos that have these two conflicting classifications (mass vs. weak) and the resulting oscillation between types, is anything known/speculated about why they’re unique in this way?
Other types of particles do oscillate in this way, yes. In fact it is very common among composite particles made from quarks and anti-quarks. But this is a long story.
Similarly, conflicting classifications are very common too (but it’s a bit complicated.) For example, there is one conflict among the quarks. If we assert that we don’t want to have these conflicting classifications for the up-type quarks, then the down-type quarks have them; if we try to remove the conflict for the down-type quarks, then the up-type quarks have them. [In fact, we could remove the conflict from the neutrinos and give it to the charged leptons, if we wanted.] And EVERYTHING is known about why this happens; it’s a simple argument using linear algebra and counting. The effect is strange, but completely understood.
Conventionally, the conflict of mass-type versus weak-type is assigned to the neutrinos and the down-type quarks — but that’s an arbitrary choice.
The conflict of mass-type and weak-type in the quarks is essential in nature. This is why for instance, top quarks can decay to strange quarks by emitting a W particle even though at first glance you might think a W particle would only interact with a top quark and a bottom quark. The subtlety is that the top quark emits a W particle to become a weak-type bottom quark, but a weak-type bottom quark is a mixture of a mass-type bottom quark, a mass-type strange quark and a mass-type down quark. And so a mass-type top quark = weak-type top quark (remember we removed the conflict for the up, charm and top quarks) can decay to a mass-type strange quark. Similarly bottom quarks can decay to charm quarks, which would be impossible if all weak-type quarks and mass-type quarks were the same — so without this conflict there would be more stable particles than we observe.
Hi, very nice article. There is one point that I don’t understand. I thought that in relativity the mass squared commute with the hamiltonian. How come that it does not commute with the interaction terms giving all these weird oscillations?
thank you
Technical answer: Often one can write the Hamiltonian as a free part H0 and an interacting part H’. [This is not always true, but if there are nice particle states this is almost always the case.] Then H0 commutes with the masses (which means that particles that are not interacting with other particles will propagate with a definite mass) but H’ and H0 do not in general commute. So I think you were thinking of the free Hamiltonian. (Particle states are NOT exact eigenstates of the full Hamiltonian, a fact which has all sorts of important implications in quantum field theory. In fact, if they were eigenstates, then how could they ever decay?!)
As always, you are truly a good teacher. You made such a complicate issue an easy read. At least, it becomes easier for me to realize some questions which might never come to my mind before. Again, the issue is the nitty-gritty of semantic meanings of some terms used in physics, such as, oscillation, entity, classification, etc.. Let me use an analogy to ask my questions.
Case 1 — Mr. A (one person) has multiple personalities.
A1 — a nice guy.
A2 — a murder.
A3 — a cry baby.
Case 2 — there are “three” gentlemen.
Mr. B — a nice guy.
Mr. C — a murder.
Mr. D — a cry baby.
In case 1, Mr. A can oscillate among A1, A2 and A3. We can call this oscillation type I or OC (I).
In case 2, there is also an oscillation mechanism among Mr. B, Mr. C and Mr. D. We can call this oscillation type II or OC (II).
Question 1 — Is there any foundational difference between the case 1 and the case 2, if those gentlemen are neutrinos? If the answer of this question is a “definitely No”, then, no further question is needed. Is this what you mean “the conflict classification”? Note: my personal opinion, the conflict classification as you described will not make case 1 and case 2 becoming indistinguishable.
Question 2 — Can OC (I) be identical to OC (II), (in their internal mechanism)?
Case 3 — there exist one “uncertainty principle” which makes the OC (I) and OC (II) indistinguishable although the case 1 and the case 2 are different in essence. We can call this type oscillation the type III or OC (III).
Case 4 — the oscillation was observed as book-keeping data, without concerning its internal mechanism. We can call this type of oscillation the type (book-keeping) or OC (bk).
Question 3 — can the term “oscillation” be used for all these four types of oscillation without any confusion?
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A question regarding neutrino oscillations: Do the oscillations occur such that the heavier neutrino component becomes lighter and the lighter component becomes heavier so as to keep the neutrino from “separating” or “falling apart”, if you will allow me to use that phraseology? And if a beam of one type of neutrino were traveling unimpeded through space, do the oscillations occur in a coherent or in-phase manner, or is it rather random?
Neutrino oscillations are a coherent phenomenon. And if the distance traveled is long enough, the different components of the neutrino will separate in space because the heavier and lighter neutrinos have slightly different velocities. But this is actually quite a subtle point, still being argued in the literature, which doesn’t arise in real experiments due to the small neutrino masses. See for example http://arxiv.org/abs/arXiv:1006.2372
This goes back to that Mass and Energy article. The three neutrinos 123 have the same energy, but different masses. So they have different momentums. Particles are also waves, so they have different wavelengths (which is related to their momentums). Their crests would thus not coincide. Since the amplitude of the wave is the probabiltiy you find it, the probability of getting any one combonation of neutrinos 123 would change the further you move from the source, and its type would change. Is this correct?
Essentially, yes.
“The reverse is also true. If I see a pion decay to an anti-muon and a neutrino, I know immediately that the neutrino emitted was a muon-neutrino — but I can’t know its mass, because it is a mixture of neutrino-1, neutrino-2 and neutrino-3. The electron-neutrino and tau-neutrino are also precise but different mixtures of the three neutrinos of definite mass.”
If each of the weak-type neutrinos is a precise mixture of the three mass-type neutrinos, and the mass-type neutrinos each has a definite mass, doesn’t that mean that each weak-type neutrino has a definite mass? So, if we knew precisely the masses of each of the mass-type neutrinos we’d know the masses of the weak-type neutrinos?
No. This is quantum mechanics. The electron neutrino is a mixture of the three masstype states, which means it has all three masses with different probabilities. (Even that statement is not quite correct.) It does not have a definite mass: it doesn’t get an average of the three masses . If it did, there would be no oscillations.
Matt thank you for taking the time to answer our questions. We must seem like ignorant undergraduate students. I’m sure you are very busy since the holidays are over, but it’s very cool what you are doing! I’m sorry if I’m just restating when you are writing. Putting things in my own words is one way I learn.
I started reading about mixed states. In the definition of a mixed state, we don’t know which state the particle is in. It’s not even a super-position of states, because even that’s a definite state. (Is this correct? I’m not sure.) We can calculate the probability it’s in each state, but not know for certain. So for an electron neutrino, we wouldn’t know which mass it had BECAUSE it’s a mixture of those states. Likewise if we determined its mass, we wouldn’t know which type of neutrino it was: electron-type, muon-type, or tau-type. Like you said in the article, it’s like the uncertainty principle but for mass/flavor instead of momentum/position.
I’ve confused you, with wording: a “mixed state” is not what’s involved here, just a “super-position”.
Technical jargon: An electron-neutrino is a superposition of the three mass-type neutrinos; similarly, any mass-type neutrino is a superposition of the three weak-type neutrinos. The mass of the electron-neutrino is not well-defined, not because it is in a mixed state, but because it is a superposition of states that have well-defined but distinct masses.
Now, why did this confusion arise?
Part of my technical problem in running this website is that I have different audiences with different levels of experience and expectations. The term “super-position” doesn’t mean anything to a non-expert, but “mixture” does, as a word in English. So I chose (with some trepidation) to use the word “mixture” to replace the technical term “super-position”, to convey the intuitive idea of a blending together of two or more states. Of course the only way to fully understand what I mean by this is to learn the math, but still, the term at least conveys a certain idea.
However, in the term “mixed state” — which most non-experts never come across — the word “mixed” is used in its Physics-ese meaning. Much more technical and precise, and different, as you say, from superposition.
I apologize that this choice of mine causes some confusion here, but I still don’t see a good alternative for my pedagogical choice at the moment.
A superposition of states is still an ordinary quantum state, where the state is as precisely specified as is possible in quantum mechanics. A system is in a mixed state if quantum information is missing.
Thanks for clearing that up!
Insignificant nit-pick: In the US, North-South highways are given odd numbers, and East-West highways are given even numbers, at least for Interstate and US highways. You have it backwards in the analogy given in the article. Other than that, though, great analogy.
thanks for catching that!