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### Solar neutrinos, astronaut ice cream, and flavor physics

I’ve been thinking of a good way to introduce flavor physics—a subject which can be surprisingly subtle—to a general audience. Here’s my best shot at it.

An invitation: the solar neutrino problem.

By the 1960s physicists thought they had a pretty good understanding of the nuclear reactions that caused the sun to shine. One of the many predictions of their model is the number of neutrinos emitted by the sun.

A scientific model is only as good as its the experimental verification of its predictions, so the next step was to actually count these solar neutrinos. Of course, our readers already know that neutrinos are very weakly interacting and this makes them very hard to detect.

Well, when a couple of enterprising astrophysicists set up such an experiment at the Homestake Mine in the 60s (which is still a template for modern neutrino detectors), they were shocked to find that they only counted only a third of the expected neutrino flux.

At this point a good scientist will go back and check their experiment, look for systematic errors, and then go back to check the assumptions of the underlying model. Let’s gloss over these rather important steps and just state that this discrepancy could not be explained by any known effect and would be referred to as the solar neutrino problem.

What gives??

From solar neutrinos to astronaut ice cream

Before answering this puzzle, let’s take a detour and fast forward many decades to my first visit to the National Air and Space Museum when I was about 10. I remember thinking that the big airplane displays were pretty cool… but nothing compared to a discovery I made in the gift shop: astronaut ice cream. (Sometimes I look back and wonder how I ever became a scientist.)

For those who aren’t familiar, astronaut ice cream is just freeze-dried ice cream that has a uniquely chalky texture. My favorite variety was Neapolitan, which was a combination of strawberry, chocolate, and vanilla. The bars looked something like this:

The Neapolitan astronaut ice cream bar will be a very useful analogy in what follows, so bear with me. Ordinarily one would expect the bars to come in a single flavor: strawberry, chocolate, or vanilla. Instead, a Neapolitan bar is a mixture of all three.

In fact, to properly set up the analogy, we should imagine that there are three types of Neapolitan bars so that if we took one of each bar, we would have the same amount of each flavor as we would if we had one of each single-flavor bar. Thus the three Neapolitan bars are just a mixture of the three single-flavor bars.

Now here’s the crux of the matter: even though the Neapolitan bar is packaged as a mix of three flavors, when you bite into it you only get to taste one flavor at a time.

Okay, maybe you can mix two flavors if you take a bite along the seam—but let’s forget about those cases because they break the careful analogy I’m trying to put together. 🙂

What this all has to do with neutrinos

Now let’s connect this to the solar neutrino problem. The incorrect assumption associated with the solar neutrino problem turned out to be that neutrinos are more like Neapolitan bars rather than single-flavor bars. The “flavor” in question is the identity of the neutrino as either electron-like, muon-like, or tau-like.

In other words, the “pre-packaged” neutrinos that propagate between the sun and Earth are a mixture of electron/muon/tau-like neutrinos. What we mean by this is that they are a quantum superposition of these three different flavors, in precisely the same way that Schrodinger’s cat is a superposition of different corporeal states.

Now here’s the neat part: even though the neutrinos propagate as Neapolitan bars, they only interact as definite flavors (electron, muon, or tau). In other words, when the neutrinos are produced in the sun, they are produced with a definite flavor. They are also detected on Earth with a definite flavor. But everywhere in between when they’re propagating on their own, they are a mixture of all three flavors.

Physicists will say that there is an “interaction basis” (electron, muon, tau neutrinos) and a “mass basis” (propagating superpositions).

We can now work out the resolution of the solar neutrino problem. The nuclear reactions in the sun involve electrons (not muons or taus) and so produce electron-neutrinos. Similarly, the detectors on Earth only detect electron-neutrinos since are composed molecules made up of electrons. In between, however, the neutrinos travel a long enough distance that they get all mixed up into Neapolitan admixtures of all three flavors. Thus when the solar neutrinos reach the detectors, only one third of them are detectable, explaining the deficit of neutrino counts!

Actually, this explanation for the factor of 1/3 is a big fat lie… it’s just a cute numerical coincidence. The point is that mixing causes one to only observe a fraction of the total neutrinos, but the specific fraction depends on many things. We’ll discuss this below.

Neapolitan Neutrinos and their relation to mass

Of course, this resolution came from decades of progress in theory and experiment, including many red-herring directions which we won’t discuss (but is a key part of doing real science!). One important a fact that from our understanding of quantum field theory is particularly important:

Particles which propagate through any appreciable distance are states of definite mass.

For more advanced readers, the reason for this is that the mass term is part of the quadratic part of the action which can be expilcitly solved and about which we perform perturbation theory.

The reason why neutrinos propagate as Neapolitan mixtures is that those are the mixtures that have definite mass. A purely electron-flavored neutrino turns out not to have a definite mass, but rather a ‘quantum superposition’ of masses. Conservation of energy requires that only a single mass state should be allowed to travel over long (i.e. non-quantum) distances.

Thus the discovery of neutrino mixing (and hence the resolution of the solar neutrino problem) only came hand-in-hand with the discovery that neutrinos have tiny but non-zero masses in 1998. This discovery, at the joint US/Japan Super-Kamiokande detector in Japan, is a great science story for another day.

Update (3 Aug 2010): as a commenter pointed out, the definitive solution to the solar neutrino problem actually only came with data from the joint US/UK/Canada Solar Neutrino Observatory (SNO) in Ontario. In 2001, SNO detected a 1/3 of the expected solar neutrinos while Super-K detected 1/2. The difference between the two experiments is that SNO is sensitive only to electron-neutrinos, while Super-K also has some sensitivity to muon- and tau-neutrinos. By combining the information from the two experiments, SNO researchers were able to extrapolate the total number of neutrinos (of all flavors) and found that this number matched the total neutrino flux expected from the sun. These solar neutrinos were all produced as electron-neutrinos, but “oscillated” into other flavors while propagating as mass-states. For a more detailed but accessible account of this story written by one of its heroes, see John Bachall’s contribution to the Nobel eMuseum.

Revised Feynman Rules

Recall that the W boson mediates flavor-changing effects. In that previous post, readers mori and Stephen correctly point out that I was being a little misleading about the W interactions. This was a deliberate choice to avoid this “flavor vs. mass” state issue. Now that we’re familiar with the difference between neutrino flavor states (electron, muon, tau) versus neutrino mass states (Neapolitan mixtures which we’ll just call 1, 2, and 3), however, we can revise our W boson Feynman rules to be more accurate.

Let’s start in the flavor basis. For clarity I will associate electron-neutrinos with strawberry ice cream. These single-flavor states are the actual states that interact with other particles. In particular, electrons will only interact with electron-neutrinos. In terms of these interacting-states, the Feynman rules are simple:

We’re only drawing the electron interactions. There are also interactions with muons which only interact with muon-neutrinos (chocolate flavored), and similarly for taus (vanilla). However, although the Feynman rules are simple, the flavor basis isn’t so useful since these states only exist at the instant of interaction. The moment the neutrino flies off, it settles into one of three mass states, which we will call neutrino-1, neutrino-2, and neutrino-3. We’ll represent these as Neapolitan ice cream bars.

Let us draw the Feynman rules in terms of these mass states. In other words, we’re drawing the Feynman rules with the assumption that the particles are given a chance to travel some distance. Now an electron can interact with any of the three mass states:

The reason for this is that the electron only interacts with electron-neutrinos, i.e. strawberry flavor; but each of the three mass states (ν1, ν2, ν3) contain some electron (strawberry). This is where flavor mixing really shows up in the W interactions: the e doesn’t only interact with ν1, but all of the mass eigenstate neutrinos.

How much mixing?

There’s no reason to believe that the mass-state neutrinos all have an equal amount of each flavor. In fact, the particular mixtures look something more like this:

These ratios are set by the particular values of the neutrino masses.

• ν1 is about 2/3 electron-neutrino and 1/6 each of muon/tau-neutrino
• ν2 is about and equal mixture of all three
• ν3 is mostly an even split between muon and tau neutrinos

Note that this may lead you to wonder why it was that the original Homestake experiment detected 1/3 of the expected neutrinos, since this is the value we would expect if each mass state had an equal fraction of each flavor. The answer: this is a coincidence!

The particular fraction of the total number of detected neutrinos depends on a lot of factors in a rather involved equation. These factors include:

• The differences between the neutrino masses
• The distance between the Earth and the sun
• The energy (or rather the energy spectrum) of neutrinos emitted by the sun
• How the neutrinos interact within the sun
• The range of energies to which our neutrino detectors are sensitive

Different solar neutrino detection experiments have found a range of different values for the number of detected neutrinos, but once these effects are taken into account, they are all consistent and shed light on the fundamental parameters that govern the neutrino sector.

Analogy to quark mixing

I haven’t yet properly introduced the Feynman rules quarks, but it turns out that you can obtain the interactions of the quarks with the photon, W, and Z by simply taking our lepton Feynman rules and replacing charged leptons with up-type quarks and neutrinos with down-type quarks.

In particular, there are three up/down-type flavor pairs:

• up quark and down quark
• charm quark and strange quark
• top quark and bottom quark

The W boson again causes mixing between these families, while all other interactions only stay within an up/down pair. It turns out that the mixing between quarks is not as dramatic as that between leptons, but because of hadronic effects (i.e. the strong force) measurements of quark flavor can be notoriously difficult. (For experts: See this post at Resonaances for an update on a recent interesting quark flavor storyline at the D0 detector in Fermilab and this post from ICHEP by the same author for a broader status report.)

Closing Remarks

• This pattern of neutrino mixing has a fancy name, tri-bimaximal mixing, and one interesting line of research is to understand where this structure comes from. (It seems to be related to the symmetries of the tetrahedron.)
• Because the amount of detected mixing depends on so many experimental parameters, there are many different neutrino experiments that differ by baseline (distance between source and observer). Since we can’t change the distance between the sun and the Earth, a good alternative is to detect neutrinos coming from nuclear reactors by setting up detectors at fixed distances.
• Yet another source of neutrinos come from the atmosphere, when cosmic rays interact with molecules in the upper atmosphere (some at LHC energies!) and send a shower of particles down to Earth.
• Here’s a really, really good question that may even stump a few physicists: why is it that the neutrinos mix while the charged leptons don’t? (Alternately, why do down quarks mix but not up quarks?) Shouldn’t they somehow behave similarly? The answer turns out to be somewhat technical, but the punchline is that they do, but the time scales involved make the effect irrelevant. I refer those with a technical background to arXiv:0706.1216.

That’s all for now!
Flip, US/LHC

• Anonymous

If I may please correct a rather egregious error in the historical facts of your otherwise fine post … Super-Kamiokande did NOT solve the solar neutrino problem. Super-K did demonstrate that atmospheric neutrinos oscillate, but this in no way meant that oscillation had to be the correct explanation of the solar neutrino problem. Rather, the excellent work of the Sudbury Neutrino Observatory solved the solar neutrino problem in 2001/2002 by directly detecting non-electron type neutrinos coming from the Sun. Credit where credit is due!

• Hi Anon! Absolutely correct, my apologies to the SNO team (as well as the many other pioneers in this field whom I swept under the rug due to either ignorance or brevity).

I’ve updated the post accordingly.

Thanks for pointing this out!

• Lisa

Great article! Fantastic description. Thanks for taking the time to explain such a complex topic in such an easy way. (Viva Neapolitan ice cream!) 🙂

• Martin Pavlicek

Maybe I was not reading carefully enough, but why is the detected neutrino type depending on travelled distance? My understanding from the article is that the mass eigenstate is a constant mix of neutrino flavours (i.e. the size of the flavour bars does not change over time). That would mean that the detected neutrino type does not depend on distance from the source. The distance dependency would be there only if the flavour mixture would change over time inside one mass eigenstate. I am sorry if I misunderstood something.

• Hi Martin! Good question. This is an interference effect that I swept under the rug because I didn’t want to get into the details. [In what follows I’m assuming you have some background in quantum mechanics because of the way you asked your question—please comment again if you want a less technical (or more technical) answer.]

The shady statement that I made was to say that the neutrinos propagate as mass eigenstates. This is correct, but one also has to account for the fact that when they are produced they are produced as a flavor eigenstate. Thus the correct thing to do is to decompose the flavor eigenstate (which was observed in principle) into a sum of mass eigenstates and let each mass eigenstate propagate and interfere with one another. Because they have different masses these mass-states have different time-evolution.

Thus the distance-traveled (or time-elapsed) dependence of the neutrino oscillations is really a statement about the interference of the three mass eigenstates which each have different flavor compositions.

I think Wikipedia turns out to have a rather nice discussion if you want to see a few equations sketched out:

http://en.wikipedia.org/wiki/Neutrino_oscillation#Propagation_and_interference

Cheers!
Flip

• Martin Pavlicek

Thank you, Flip! This is what I was missing.

• alex

What was it about the third proton-proton reaction- the high energy one- that was so important when it came to solving the problem?