Neutrinos are constantly being emitted by the sun in prodigious numbers. About 10^14 go through your body each second (maybe more if you supersize yourself like my old buddy Mario did). More technically, anti-neutrinos are created when neutrons decay into a proton + electron + anti-neutrino, and neutrinos are created when protons absorb electrons in the reverse process: proton+electron -> neutron + neutrino. These sorts of things happen to various atoms of the sun, depending on what sort of decay they are subject to, (beta decay or reverse beta decay, respectively), as well as proton-proton collisions that produce a positron a deuteron (PN), and a neutrino. However, not enough neutrinos seem to arrive on earth. This was known quite some time ago, see The Solar Neutrino Problem for the technical details as of 1998.
The explanation for the deficit in neutrinos is now called “neutrino oscillation.” In the theory of neutrino oscillation, these solar neutrinos are “electron neutrinos”, and in their passage to the earth, they change form and become “muon neutrinos” or “tau neutrinos.” For those students who take their physics on faith, this is not a problem. But every now and then someone learns their theory just a little too well and begins to have doubts about how it can be that a particle can transform itself into another particle in vacuum. Yes, neutrino oscillation is a little strange, but it can be explained much more clearly, and kept in context with the rest of particle physics, by analyzing the problem as neutrino interference. This way neutrino oscillation can be described in a way that doesn’t confuse students. And such is the topic of this post.
First, let’s write down the interaction that is at the heart of the detection of solar neutrinos. The solar neutrino deficit was first discovered by the chlorine detector at the Homestake gold mine. This detector consisted of as much chlorine as one could reasonably pack into a small volume, and placed as deep underground (and protected from cosmic rays and other radiation)as possible. The expectation was that a neutrino would arrive and convert a Cl-37 atom to a Ar-37 atom, plus an electron. That is, a neutron in the nucleus of the chlorine atom is converted into a proton, plus an electron. In short, this is the interaction . The atomic mass is unchanged. The neutrino was created in the sun by a reverse sort of interaction, an electron capture: . There is other stuff going on, but this is enough to establish the problem and its solution.
So what happens is this: A proton and an electron in the sun are combined into a neutron. On the earth, a neutron is split into a proton and an electron. And a neutrino is exchanged. This is illustrated in the figure above. At a deeper level of detail, what is happening is that an up quark in the sun’s proton is being converted into a down quark, and on the earth, a down quark in the neutron is being converted into an up quark. This further complication doesn’t change the picture much, we now have:
And modern theory no longer treats the above interaction as two 4-body interactions; it will be better if we follow convention and split the neutrino emission and absorption up by putting a W boson between the quarks and leptons. And I’ll drop the cute earth and sun pictures:
Now the problem with all this was that the number of neutrinos being detected on the right did not seem to equal the number of neutrinos that were pretty much known to be emitted on the left. Something was happening to them.
The CKM Matrix
A clue was given by the up quark and down quark interactions on the far left and far right of the above drawing. On the left, an up quark turns into a W+ boson and a down quark. On the right, a down quark absorbs a W+ and turns into an up quark. It turns out that these interactions are not quite so simple. While there is only one W+ boson, the quarks come in three generations. The three generations have different masses but otherwise have the same quantum numbers (depending on what you call a quantum number).
So when an up quark emits a W+ boson it can turn into other quarks than just the down quark. Electric charge has to be conserved, so the other possible quarks, besides the down, with charge -1/3, are the strange and bottom. Now over the long term, since the strange and bottom quarks weigh more than the up quark, the interaction will end up with just the down quark (the strange and bottom quarks will convert to a down quark, or be reabsorbed, or do something cool).
But on a short enough time scale, any of the down quarks (d,s,b) can end up created along with that W+. Like anything in quantum mechanics, there are different probabilities of these things happening. And, like anything in quantum mechanics, there are different phases, which can cause interference. Our sum total knowledge of what happens in this interaction is held in the Cabibo-Kobayashi-Masakawa (or CKM) matrix. In order to provide phase information, the matrix entries are amplitudes. The entries of the matrix specify an up type quark (u,c,t) and a down type quark (d,s,b). It has 3×3 = 9 entries, running from “ud” to “tb”; each entry “xy” defines the amplitude that when a the W+ emitted by an “x” quark with charge +2/3, the resulting quark with charge -1/3 will be the “y” quark.
According to the latest Wikipedia data, the entry for the CKM matrix is 0.97383. It’s an amplitude, so to get a probability we have to square it; the resulting probability for the down quark is about 94.83%. So this isn’t much of a cause for a neutrino deficit; and as I mentioned above, this doesn’t matter anyway, a higher generation quark would just decay down to the d anyway.
So the CKM matrix can’t explain the loss of those neutrinos. You might say that the neutrinos were there but we weren’t detecting them, but the detection rates were found by comparing the number of detections of neutrinos from nearby nuclear reactors; physicists had a good idea how many should be detected.
The MNS Matrix
But there is a way to arrange for fewer than expected interactions, if the neutrinos have mass, and if they have a generation structure like the quarks (and the charged leptons, that is, the electron and its cohorts, the muon and tau). This is quite a stack of ifs, but making this assumption makes the neutrinos look more like the other leptons and quarks. We will call the three neutrinos as is done in the literature. At this time, it is not known which of these neutrinos is the lightest.
If there are three neutrinos, then there is a mixing matrix for how the W+ interacts with them. The matrix is called the Maki-Nakagawa-Sakata matrix or MNS matrix. (Why doesn’t the MNS matrix get its own wikipedia entry like the CKM matrix does? Non-neutrality?) Like the CKM matrix, there are 3×3 = 9 entries in the matrix. The first entry is and designates the charged lepton that absorbs the W+. The second entry, 1,2,3, designates the neutrino.
In quantum mechanics, one computes probabilities by taking the squared magnitudes of amplitudes. The amplitudes are obtained by adding up all the possible amplitudes that can contribute to the problem. Since we now have three neutrinos, there are now three amplitudes contributing the probability that we detect that “neutrino.” The total amplitude A is given by the sum of the amplitudes for the three neutrinos:
Now when we do these sorts of calculations, we have to assume an energy for the neutrino. If we’re not sure of the energy, then we do the calculation for each energy that is possible, and then compute an average. But for a given energy, in quantum mechanics the de Broglie frequency and wavelength of a particle depends on the mass. Thus the three neutrinos will pick up different relative phases, and the three amplitudes, will interfere with each other; that is, the sum will change in magnitude as the individual contributions go into and out of phase with each other.
So when the neutrinos are first created a W+ and an electron, they are just perfect for converting back into an electron and a W+. As you get farther and farther away from the source, the three types of neutrinos begin interfering with each other more and more. Eventually their phases become random. The result is a decreased probability of detection. The solar neutrinos are there, but we don’t detect all of them because their relative phases are inconvenient for our detectors.
In the historical developement of particle physics, at first it was assumed that neutrinos could not interfere. So they called the neutrinos by names defined by what they did. In the literature still today it is common to call the three neutrinos . But these “flavor eigenstate” neutrinos require three regular Feynman diagrams to describe. They are not mass eigenstates and they are not solutions of a Dirac equation (which requires a mass like any other fermion).
Every now and then you will see people talking about the mass of the “electron neutrino”. In this case, they are talking about an average mass, after translating the electron neutrino into the true neutrinos, the . And this is another source of confusion for the student. It would be much easier if they used the same conventions with the neutrinos as are used for most other elementary particle and talked about the mass eigentates as the particles.
What amazes me about neutrino interference is how quantum mechanics works even though the intermediate (virtual) states travel the distance from the sun to the earth. The trip takes several minutes, and yet, the three neutrinos manage to interfere with each other when they arrive. I think that’s awesome. This universe is a very strange place.
So I’d like this to be a resource for students wishing to understand neutrino oscillation. If you have questions, add them to the end and I’ll try to answer them. And corrections, of course, I’ve typed this up in just two hours and it is undoubtedly filled with errors. And surely I’ve left something off here.
Research Going On Around Here
Of course I am responsible for extending Koide’s mass formula for the charged leptons to the neutrinos. The oldest paper is The Lepton Masses (2006). My formula for the neutrino masses was mentioned by Alexei Smirnov at a plenary lecture at the Neutrinos-2008 meeting as a non perturbative method of looking at the generation structure of the neutrinos. So if I’m off the deep end, at least I’m not completely alone in the water. Most of my efforts right now are on extending the Koide formulas to the quarks, which I’m writing up here. Kea is busily working on the mixing matrices. She’s starting with the MNS (lepton) matrix which is quite regular and fits in with stuff I’m doing. I’m hoping that she will tackle the CKM matrix and help explain the quarks.