 OK, so let me pick up where we dropped off for the last time. So I was introducing to you the neutrinos that are produced in the sun. The features that you want to remember about this, which are most important, is that the energies are relatively low. So if you're familiar with high energy physics, this is in the very low energy end of high energy physics. So neutrinos produced in the sun have energies below about 10 MeV. And the other thing, which I didn't emphasize, but hopefully it's going to be clear in a little bit, is that as far as the production mechanisms are concerned, all of these neutrinos are produced via charge-current interactions. And because, and I'll try to make this a little bit more quantitative, at least in words, because the sun is relatively cold, as far as particle physics is concerned, charge-current interactions can only produce electron-type neutrinos and electrons. What this means is that there's not enough temperature in the sun to have a collision happen that produces like a muon. The sun is way, way, way, way too cold for that. So that's always something to keep in mind. So the things to keep in mind is the energies are relatively low. And as far as production is concerned, all of the neutrinos are electron neutrinos. And the third thing that I should emphasize as well is that these are all what we call neutrinos. There are no anti-neutrinos produced in the sun either. All of these statements are not exactly correct. If you go, for example, to very, very, very, very low energies, there are some very rare physics processes where you actually get to produce neutrinos via neutral-current processes. So you can get actually new mu's and new tau's as well. The fluxes are relatively small, and the energies are way smaller than the energies that I have here. So we've never seen those neutrinos, and we're not going to care about those neutrinos. OK? So based on that, I started telling you that the first measurement of neutrinos from the sun was done in this very large chlorine experiment. And the chlorine experiment is a charge-current reaction process for measuring neutrinos. The idea is very simple. You have an electron neutrino. It hits chlorine. It produces an electron plus argon. OK, so that's the reaction for that. And that's how neutrinos from the sun were measured for the very first time. There are two other kinds of experiments that are written up there. So these bars here, they indicate, first of all, what kind of neutrino energies these measurements are sensitive to. So what happens is this reaction here has a threshold. That means that for this reaction to happen, the neutrino energy has to be bigger than something. So these experiments here, or this experiment here, started off in the 1960s, and it ran for almost 40 years. So they took data for a ridiculously long amount of time, and they saw all kinds of interesting things that I'm not going to talk about in a lot of detail. But the key thing is, of course, they had a problem, which I will talk about in the next slide. But before I do that, I will tell you that this problem led people to think about other experiments on how to measure neutrinos from the sun. And the other kinds of experiments that came up, first of all, was this gallium experiment, and these are water Cherenkov experiments. So the gallium experiment is very, very similar in spirit to this one, because you're going to measure the neutrinos in a very similar way. You're going to have a big gallium detector. And it turns out that there's also a charge current reaction with that, where the gallium gets converted into germanium. These detectors were actually like a big solid detector, where you also have a very clever way of extracting the germanium out of the gallium. So this is how these kinds of experiments worked out. So in spirit, these two experiments are very similar. And I will mention now, and I'll say this again, the gallium experiments were invented because the threshold for that reaction is a lot lower. So that means that the chlorine experiments required neutrino energies above about an MEV or so, and the gallium experiments required neutrino energies above 0.2 MEV or so. And this turned out to be a big deal for these types of experiments, because they wanted to see the vast majority of the neutrinos produced in the sun. So just as a reminder, if you look at all these different lines here, they are associated to different reactions happening inside of the sun. And what I did say the last time is that the vast majority of the neutrinos are produced by this PP fusion reaction. So that's proton-proton going into deuteron plus positron plus an electron neutrino. And those energies are relatively low. So that's what these gallium experiments were for. The other class of experiments was this water Cherenkov experiments. And these were experiments that I'll talk about in a lot more detail in a minute as well. And what these experiments were measuring was a very, very different kind of physics. So again, it's kind of a fun fact if you've never thought about this. So you know what water is made of. It's hydrogen and oxygen. And you can ask yourself, what happens when a neutrino hits an oxygen atom? The answer is if the energy is low enough, nothing too exciting happens. There's no physics process you can write down other than some neutral current kind of process because it's very, very expensive energetically to convert oxygen into something else. So oxygen is one of these doubly magic nuclei if you've ever heard that expression before. If you've never heard about it before, don't worry about it. But it's a very, very tightly bound nucleus. So it's very hard to convert it into something else. And of course, the other thing that's in H2O is the H part. So there's also a bunch of protons. And then there's this other interesting fact, which is at very low energies, you cannot do a charged current reaction between an electron neutrino and a proton. You can think about it a little bit. I also have a homework problem that mentions this in passing. But you can try to write down the reaction electron neutrino proton going to something plus electron. And you can't think of what the reaction would be. So that means that electron neutrinos don't get to scatter off of protons either. So the question is, what's left if you have a big water tank and you're trying to measure neutrinos from the sun? The answer is the electrons. So in these very large water tanks, you can measure neutrino scattering on electrons. And that's what they measure. And I'll come back to this in just a second. So these were the kinds of reactions that people had in mind to measure neutrinos from the sun. And the thing that's very important is that all of these experiments have made measurements. I am making a time invariant statement. The first chlorine measurements are in the 60s. The first water measurements are in the late 80s. The gallium measurements are in the early 90s. Plus or minus three or four years, I'm kind of getting the dates a little bit wrong. But this is a summary of all the measurements. And if we ignore the last two bars, which have to do with what's called the snow experiment, this is a situation we were in by the end of the century. And this is a very busy plot that people don't show anymore. So I want to show it because it's kind of a cool plot. It actually has a combination of predictions and measurements in them. So that's what these bars indicate. So the blue bars are the different measurements of the flux of solar neutrinos from the sun. That was redundant. The flux of neutrinos from the sun. So that's what the blue bars indicate. This is measured in some weird units that we don't care about. And if you actually ask, these are defining what's called the solar neutrino units, which is totally useless except for solar neutrino experiments. So they were defined for that. And the idea is the chlorine experiment measured of flux, which is proportional to 2.5. These are the water experiments. They measure the flux, which is about 1.5. And these are the gallium experiments which measure the flux, which is about 70 in some random units. So the thing which, of course, we care most about is the other bar, which is the more colorful bar, which is not blue. And these are the expectations from theory. So we can calculate the flux of neutrinos coming from the sun. And the different colors indicate the different components that contribute to that particular measurement. The expectation, for example, for the chlorine experiment is 7.6. And most of these neutrinos come from what are called the boron-8 neutrinos. This is a beta plus decay of boron-8. And so this is this yellow fraction of the bar. Then there's some contribution from what's called the beryllium-7 neutrinos, and so on and so forth. And if you look at, say, the water-turenkov experiments, they only see these boron-8 neutrinos. And if you look at these agallium experiments, because the threshold is very low, they actually get to see different kinds of neutrinos, including these are very famous PP neutrinos. Now the message that we want to get out of this is that if you look at the predictions and if you look at the measurements, they all disagree by some healthy factor, like 2 or 3. So this is what was called the solar neutrino puzzle. And it was a puzzle that lasted for more than 40 years. And people stared at this. And the first time this showed up was in this measurement of neutrinos coming from the sun in these chlorine experiments, this Array Davis experiment, this big cleaning fluid tank in a mine in South Dakota in the United States. And when people looked at this, what you do is you ask yourself, what can be going wrong? That's what people did. And there are a couple of very obvious answers. One answer is, of course, that this is a really hard measurement. And the way in which this measurement is being done is super hard. You're trying to do something that's completely crazy. And if you're explaining what you're doing to your physics friends, there's a lot of chemistry involved. Physicists don't know any chemistry. So everybody thought that somehow this experiment was a screwing up a little bit somewhere. And nonetheless, it was pretty impressive that they got the right number of neutrinos from the sun to within a factor of three. So one possibility is that maybe the experiment is a little bit wrong. And by the way, this is what everybody would say. This is not just something that I'm making up. And this is a 40-year-old problem. So it existed for a very, very long time. The other possibility is to look at the prediction. And you can ask, how well can these theorists actually calculate this number? And again, at the time, everybody thought that this prediction was probably not completely trustworthy either. Calculating the flux of neutrinos from the sun is a very, very hard business because of lots of different reasons. One is, if you look at, for example, the composition of these neutrinos here, these are all of these reactions that are going on inside of the sun. And the boron-8 neutrinos are coming from this decay here. So you produce some boron-8 inside of the sun. And then the boron-8 will decay and will give you some spectrum. So of course, how do you make the prediction of how many neutrinos you get from that? First of all, you have to understand that decay, which is something you can try to do experimentally, and we don't want to go into the details of that. But of course, the other thing is you need to understand how much boron-8 there is inside of the sun. Now, boron-8 is not one of these things that you'll go out and find in the supermarket because it decays very quickly or relatively quickly. So it is a byproduct of the burning process in the sun. And it's not one of these main things that are there in the sun. If you look at all of these different steps, boron-8 is produced here. It's produced by Brillium-7 capturing a proton. And then you ask, OK, so where is this Brillium-7 coming from? And the Brillium-7 is coming from a reaction of a Helium-3 with a Helium-4 that gives you Brillium-7. And then you can say, OK, Helium-3 is also one of these weird things that don't show up everywhere. So where is that coming from? And it's coming from some other earlier process. So you notice that in order to get the boron-8 flux, you have to understand all of these intermediate steps very, very well. So this is a very non-trivial calculation that you have to do in order to calculate these fluxes very well. So the bottom line is, when people looked at this picture, at least most of the particle physics people for a very long time, they were not impressed by the fact that you were getting the answer wrong by about a factor of 3, because both the measurement was very difficult and the calculation was very difficult, OK? So that's what people did. But nonetheless, this inspired other experiments. And it is important to keep in mind that when I say inspired other experiments, it took more than 20 years for somebody else to do this experiment again. So it's not like we saw a problem, let's try to resolve it by doing another experiment and then you do another experiment the next year or the next five years. It took more than 20 years for that to happen. Now there are lots of reasons for that. These experiments are super hard. So you have to have different kinds of motivation for doing these types of experiments. OK, so the next kind of experiment are these water Cherenkov experiments. The water Cherenkov experiments are measuring neutrinos this way. And of course, they also see a suppression of the flux. The suppression is actually not as big. It's only 50% instead of a third. And that will be important for what I'll talk about in a minute. And finally, the last thing was this gallium experiments. And they also see a suppression by a little bit less than a half. And like I said, what people were very excited about with these gallium experiments was the fact that they could also measure these PP neutrinos. The PP neutrinos are very important because they come from this reaction here, which is the reaction that happens almost 100% of the time. It's the one thing that's, in quotes, easiest to calculate. And it also gives you the most number of neutrinos by far. And in particular, you can convince yourself, and that's almost true, that there's a relationship between the flux of neutrinos coming from the sun and the photons that are coming from the sun. And we have a good measurement of how many photons come from the sun. That's an easy measurement. We've done that for over 400 years or something like that. So that's a very robust number. This is a flux of PP neutrinos from the sun. And this, by the way, is included here. You notice there are some error bars here. We think we understand this flux to better than a percent. We don't claim that we understand this flux to better than 20%. And that's in the realm of the calculation. So if you don't believe the calculation itself, that's a different problem. But even if you believe the calculation, this flux is known super well. This is not known very well. So the Gallium experiments were interesting because the PP flux, which is the one that we claim we understand super well, saturates the measurement. So anything else is sort of a bonus. And that's why there's still a very robust problem that's going on here. So this is a solar neutrino problem. And again, like I said, there are two possibilities for what's going wrong. Either the calculation of the flux is incorrect or the measurements are bad. Notice that now we have the Gallium experiments are actually two different experiments. So we have three or four or five different experiments that are saying the same thing. So the likelihood that they're all wrong in the same way is less likely. The other possibility is that the calculation is incorrect. And it turns out that there is a third possibility, which is something that people started appreciating in the 80s, which is that the third possibility is that maybe there's something else among the hypotheses that I made that's incorrect. Or summarizing this in a different way is that maybe the neutrinos are wrong. It could be that the calculations are right, the measurements are right, but something goes wrong with the neutrinos. And one hypothesis which is very interesting is that I said that the sun can only produce electron neutrinos. Now let's imagine, just for the sake of argument, that even though the sun can only produce electron neutrinos, somehow that gets violated. And that some fraction of the neutrinos that are coming from the sun actually, when they get here, they behave like muon or tau neutrinos. Let's raise that hypothesis. So if you raise that hypothesis, you can actually explain a lot of these problems. For example, if you have a muon neutrino coming in and it hits a chlorine nucleus, it would have to produce a muon plus argon, but there's not enough energy for that. And the same is true for a tau neutrino. The same goes for the gallium experiment. If a muon neutrino comes in, it would want to produce a muon. And again, there's not enough energy for that. When I say there's not enough energy, remember, these neutrinos have energies below 10 MeV. And the muon mass is 105 MeV. So we have not enough energy to do any of that. The one thing which is cooler is that we can look at this electron scattering reaction. The electron scattering reaction is a little bit more interesting, because if a muon neutrino comes in, it actually knows how to hit the electron, which is kind of interesting. But there are lots of things you want to remind yourself. One is electron neutrino electron scattering has both a neutral current contribution and a charge current contribution. So you can think about the Feynman diagrams, and it's a useful exercise. There are two kinds of diagrams for that. For the muon neutrino scattering, there's only one diagram, which is a Z boson exchange. And it turns out, for these kinds of energies that we're talking about, this cross-section is about five or six times bigger than that cross-section. So that's just a fact, and this will be in one of the homework problems that I assign, which has to do with the snow experiment. But what it means is, if you're measuring neutrinos via neutrino electron scattering, if the neutrinos convert into some other flavor, you still get to see them, but you don't see them as efficiently. Instead of if you count the electron neutrino as one, as one unit of neutrino scattering, this guy is about 20% of an electron neutrino. So you would still see a suppression of the flux. And what's interesting is, if you make this hypothesis, you can explain all of these data on the left-hand side. So this is what the problem was about. So the interesting question you want to ask yourself is, so how do I test this hypothesis experimentally over what's called the neutrino flavor change? And the answer was, one of the answers was the snow experiment. And the snow experiment was a very, very clever experiment that used heavy water. Again, it's a classic experiment. It's a Nobel Prize-winning experiment. And what they did was very simple. So they wanted to say, let's figure out a way of measuring this flux of neutrinos from the sun that relies only on the neutral current. So that's what they wanted to do. And that's where the heavy water comes in. So it turns out that if you build a heavy water detector, instead of having protons, you have a deuteron, these proton-neutron bound states. And then you have a neutrino coming in. And the neutrino is going to hit this deuteron. And the one thing you want to remember about deuteron is that it's a very loosely bound nucleus. That means it's very easy to break it up. So if I have a neutrino coming in, I can break up the deuteron. And then I have another neutrino, which I can't see. It doesn't do anything. And then I have a proton. And then I have a neutron. And then if you're a very clever experimentalist, you figure out that measuring this proton here is absolutely hopeless. So it doesn't happen. But you can actually measure this neutron here. And that's the experiment that was designed. It's this heavy water experiment. It's called the SNO experiment for Sudbury neutrino observatory. This is a very deep mine in a Sudbury Canada, which is somewhere in Canada. Canada's very big. So this is that experiment. And actually, that experiment has a bonus because it also measures a couple of other reactions. So the nice thing about deuteron is that it also has neutrons in it. So you also get a reaction that goes like this. So this is another reaction that can happen. And this one here only happens for electron neutrinos. I forgot to say that this reaction here is a pure neutral current reaction. So it doesn't care whether you have electron muon or tau neutrinos. All of these reactions happen with the same rate. But they also got to measure this because they can see this electron here. I did this wrong, right, so it's not. Oh, no, this is right. OK, good. And finally, because this is a big heavy water tank, it also has a bunch of electrons in it. So you also get to measure neutrino electron scattering, which is the same thing that this other experiment here measured. So this is what the snow experiment was about. First results from the snow experiment came out around 2002 or so. And these are these bars that sit over here. So what this here represents is this measurement here of the flux of neutrinos from the sun. And again, you'll notice that the measurement is way, way less than the expectations. So again, it's about a third. But the last bar is the most exciting bar, which is what happens when you measure the flux of neutrinos coming from the sun using this reaction. And you notice that when you do that, or when they did that maybe about 10 years ago, the numbers changed a little bit, but it doesn't matter. You actually are exactly on top of what you expect. So this experiment here is considered to be unambiguous evidence that somehow there are muon and tau neutrinos coming from the sun, even though the sun only knows how to produce electron neutrinos. So this is a very, very non-trivial result. And here's a more sophisticated version of that picture. Because effectively what they can do is they can measure the flux of muon and tau neutrinos independently from the flux of electron neutrinos coming from the sun by combining these different reactions that I was telling you about. This is one of the homework problems that I ask you to do is to do this calculation here. But the conclusion is that they can assert that the flux of muon and tau neutrinos arriving from the sun for energies above something, above like 5 MeV or so, that that number is definitely not zero. It is a significantly far away from zero. This is 1, 2, and 3 sigma. So if you like sigma, this is like a 6 sigma evidence that there are muon and tau neutrinos coming from the sun. So that's the solar neutrino problem. And that's how the problem was resolved in terms of physics. The lesson is that somehow, even though the sun only knows how to produce electron neutrinos, by the time they get here, about 2 thirds of them choose to behave like muon or tau neutrinos. So that's what the sun is doing. Thought it was about maybe 7 sigma. Maybe the number is much bigger now. Did you remember a different number? It's not 100 sigma. And it's not 10 sigma either, but OK. And then the other thing which the calculators of the flux of neutrinos from the sun like to point out is this is what expectations were from these models. And you notice that this blue bar is right on top of it. And there's a long history associated with these calculations of neutrinos coming from the sun. And the people who were doing this calculation were very happy that they were vindicated in the end because they were sort of vilified for decades because people keep telling them, oh, you're doing your calculation wrong. You're doing your calculation wrong. But that turned out to be not the case. They were doing the calculation correctly. So let me talk about something completely different, which is the measurement of neutrinos produced in the atmosphere. So this is another thing that's very well known. The earth is bombarded by cosmic rays constantly. Most of the cosmic rays will hit the atmosphere. And they hit the atmosphere like a particle physics experiment. It's just like a fixed target experiment. You have a beam coming from somewhere. You hit a target. And what happens is you produce a lot of pions. So the pions are produced. They decay in the way that pions like to decay, which we already talked about. That's how the pion was discovered. And that's also how the muon was discovered. And the pion decays that way. And then depending on what the energetics of the problem are, sometimes the muon has time to decay as well. And if that happens, this is how the muon decays. So this is the experiment that you're doing. I'm not the experiment. This is happening for free. And then if you happen to have a very, very big detector that's deep underground, you are sensitive to these neutrinos. Now, this is a very complicated calculation to do. That is, it's very complicated to estimate how many neutrinos are actually produced. But there are a couple of things that are very, very robust about predictions that you can make. One thing that's super robust is if you look at the flavors of the neutrinos that are produced, and if you stare at the blackboard, and if I had done this correctly, I could do it this way, you will note that I have two muon neutrinos and one electron neutrino. So this means that I expect that for every electron neutrino that I see, I should see two muon neutrinos. Now, this is actually not completely correct. Because as we know, sometimes, especially as the energies get higher, the muons actually don't decay before they make it to the planet. So they actually get to the surface before they decay, in which case they will probably stop before they decay. So we're not interested in those decays. So what happens is, if the muons don't decay, you actually don't have this piece of the reaction here, which means you only get this muon neutrino from the pion decay. So a statement that I can make is that if I look at the ratio of muon neutrinos to electron neutrinos, I expect that ratio to be bigger than about two. So that was one very simple prediction that anybody can make without really understanding the details of how to calculate the flux of neutrinos produced in the atmosphere. There's another thing that you can say that's also very, very robust, which is, it's this picture here. So this is a picture of the Earth, and this is where the detector is. This is clearly not to scale for lots of reasons. One is that there's no way your detector is this big, and there's also no way that your detector is this deep. We don't know how to dig a hole that's this deep. That's way, way, way, way, way too deep. But anyway, this is the picture, and one question you can ask yourself is, if you look up and you count how many neutrinos are produced above you in some solid angle. Define some solid angle above you and count how many neutrinos are produced there. And then you look down and you ask how many neutrinos are produced that are coming to your detector on the other side of the planet. And so basically you can count how many neutrinos are produced here that will make it to your detector, and how many neutrinos are produced here that make it to your detector. If you assume that the flux of cosmic rays reaching the Earth is isotropic, the answer is that those two numbers have to be the same. If you never thought about this, I will leave it as an exercise, but you can quickly convince yourself that this statement is true as long as the Earth is a sphere and we live in three dimensions. I don't think the Earth even has to be a sphere. It's some variant of Gauss's law if you want to think about it that way. So that means you can predict that the number of neutrinos coming from above is the same as the number of neutrinos coming from below. That's a very, very, very robust prediction that only depends on the isotropy of the flux of cosmic rays hitting the Earth. Now you can ask yourself if that assumption is good, and the answer is it's kind of good, but it gets better with the energy, and you can ask why wouldn't it be a good assumption? And one thing you can ask is that maybe the universe has maybe the cosmic rays like to come from some direction, which is kind of true, but it's not true for the purposes of what we're doing. The other thing is of course the Earth has a magnetic field, and the magnetic field tends to guide the cosmic rays in a particular way when they come to hit the Earth, but if the energy is high enough, the magnetic fields are not particularly efficient. So if you take all of that into account, you can predict that this so-called up-down ratio is one. And I have to say these are super robust predictions. They don't rely on very much other than pi on decay, and the isotropy of cosmic rays hitting the atmosphere. If you do the measurement, and that's what people did, the most famous measure, the deciding measurement is from the super-camioconda experiment. They measure this up-down ratio to not be equal to one, but to be one-half with a very small error bar. And again, this is a huge effect, and it's virtually impossible to explain this effect with some vanilla physics, because again, the assumptions that we made were very, very simple. So something has to be breaking down at some very fundamental level. And again, if you had never heard about any of these things before, and you just woke up and you heard the story, you could yourself think about what could be going wrong, and of course, the one thing which is important is if you look at this up-down ratio, the only thing which we didn't worry about is the fact that for the neutrino to go from here to here, it has to propagate a much, much longer distance than the neutrino coming from there to here. And of course, this much longer distance is happening through the earth. So you can raise several hypotheses. One hypothesis is that maybe the neutrinos are being absorbed by the earth at some rate, which of course is a logical possibility, but we know enough about neutrinos to know that this would be totally crazy, because we understand neutrino interactions well enough to know that the mean free path of a one GV neutrino through the earth is much, much bigger than the earth. So there's no way that the neutrinos are being absorbed. And I'll talk about what the other problems are, but there are a couple of things I wanna mention. Technologically, the thing that made this possible were these very, very large water churranca of detectors. And again, this is just a very big pool of water that was sitting underground that was sensitive enough that they could see these are neutrinos coming from the atmosphere. Again, the same detector can also see the neutrinos from the sun. And the reason it's very easy to separate out those two components is the energy. The neutrinos produced in the atmosphere, they have energies above 100 MeV and much, much higher all the way to 100 GV and above that. And the neutrinos from the sun have energies below about 10 MeV. One thing that's important to appreciate historically is why were these detectors built to begin with? Why would anybody build a very, very large pool of water and have it deep underground? Do people know? Yeah. So the answer is in the name, if you know enough, which is, so this is called the super cameochanda experiment. The word super just means that there was another experiment before that was not super. So that was the cameochanda experiment. And cameochanda is actually not a name, but it's a pseudo acronym. So cameochas, the name of a small town in Japan where there's a mine, which is where they build the detector. And then the NDE is not just to make it sound nice, but it actually stands for something and it stands for a nucleon decay experiment. So cameochanda is the cameochanucleon decay experiment. And that experiment was designed to look for proton decay. Now why would anybody look for proton decay? The answer is because it's there. It's something you can look for. And of course the other answer is that it's driven by granjunified theories, which maybe you've all will talk about by Saturday. And so this very crazy theoretical idea of a granjunified theories allowed people to make a prediction for what the lifetime of the proton should be. And once you have a prediction and you're an experimentalist, your experiment gets to be much better. Because now you can say, hey, I wanna do this crazy thing which is look for proton decay. And by the way, my theory friend told me that the proton should decay in 10 to the 32 years. Which means that if you give me the money and I build a large enough tank, I should see proton decay. So that's why these experiments were built. And then you can ask why were they measuring the flux of neutrinos from the atmosphere? Again, one answer is because they're there and you wanna measure them. The real answer is that these atmospheric neutrinos are the background for proton decay. So because these detectors are so big, they actually have a significant rate of neutrinos from the atmosphere. And they have to understand that flux well enough because so that if a proton decay candidate shows up, they don't get confused. So that's what was going on here. Not clever enough. Oh, it is, okay. So you will also get to do a homework assignment on the detailed data, which is much, much better than the one I just showed before. They can actually measure the muon neutrinos and the electron neutrinos separately. And they don't just do an up-down ratio. They can actually measure the neutrino production as a function of the angle. So in these units here, cosine theta of one, these are neutrinos that are coming from above and cosine theta of minus one and neutrinos coming from below. And you notice that, and there are lots of different data sets. The data sets are characterized roughly by energy. So you have lower energy muon neutrinos, higher energy muon neutrinos, super high energy muon neutrinos, and the electron neutrinos. And you notice that the black, or the solid lines are the predictions and the points are the data points. And you notice that for many of these panels, especially almost all the panels that have to do with muon neutrinos, we get the wrong answer. And in particular, this factor of a half for the up-down asymmetry is mostly coming from this plot where you see that for neutrinos coming from above, the neutrinos are doing what we expect. And for neutrinos coming from all the way through the earth, the neutrinos are actually not behaving as expected. We're missing about half of them. There's a very cool statement you can make just by staring at this plot, which allows you to almost completely convince yourself that just by staring at these data, you have learned that the neutrinos have mass or the neutrinos have some weird interactions. Let's ignore the possibility of some weird interaction. What happens is you can tell just by staring at this plot that the neutrinos have non-zero masses. And the reason for that is very simple is that the only difference between these neutrinos here and those neutrinos there is that these neutrinos here traveled about, let's say 30 kilometers to get the earth detector and these neutrinos here traveled about 12,000 kilometers to get the earth detector. And somehow they're behaving differently just because they traveled a different distance. The conclusion you can reach from this is that somehow the neutrinos can tell time. They know that if they travel a little bit that's different from traveling a lot. So the other thing you wanna remember is massless particles can tell time. For them, there's no such thing as time. The life of a massless particle is very, very weird, okay? For them, in many quotes, everything happens at the same time. It's a very, very exciting life, okay? But if you can tell time, that means if you can tell whether you've traveled a little bit or a lot, you must have mass. Or Lorentz invariance has to be violated, which we don't believe in. So Lorentz invariance is good, okay? So this is the atmospheric neutrino puzzle. And again, flavor change can also explain that puzzle. And the idea is the following. Imagine that there's a non-zero probability that by the time these muon neutrinos get to the detector they actually are behaving like tau neutrinos. Or electron neutrinos. The reason we don't talk about electron neutrinos is because if we look over here the electron neutrinos are fine. They're not doing anything weird. We don't get more than we expected. We don't get less than we expected. We get about the right amount. The muon neutrinos are doing all the weirdness. So if you postulate that they have converted into tau neutrinos, that's enough to explain your data because all the measurements are being done by charge current reactions where you have a neutrino coming in and a muon being produced. So this neutrino flavor change hypothesis also gets to explain these data very, very well. So what I wanna tell you about now is why the postulation that neutrinos have mass allows you to explain all of these data via this neutrino flavor change. So again, the picture that I want you to have right now is that we have a solar neutrino data and atmospheric neutrino data and they can both be explained very well by postulating that if a neutrino that's born with a certain flavor behaves as a different flavor by the time it gets to the detector, that can explain all of the data. And the evidence is particularly overwhelming. So for the solar neutrino problem, you have the snow experiment that rules out all of the more boring explanations of what was going on. And for the atmospheric neutrinos, we don't have another data set, but this behavior here is super weird. I mean, there's no way for you to explain what's going on here by making a qualitatively different hypothesis. I will mention that there's actually another hypothesis you can make, which is kind of interesting, is again, the difference between these neutrinos and those neutrinos is that one travels a short distance, the other one travels a long distance. So again, if you had never heard about this story before, you would have said, hey, maybe by the time the neutrinos go from here to there, they decay. So imagine that that happened. That would explain why you're missing neutrinos. And you would actually get a curve that looks like this if you adjust the lifetime just right, which is kind of interesting. It does not violate the statement that I made before because if the neutrino can decay, it better have a mass because if you don't have mass, there's nothing for you to decay into. But it is another logical possibility that was actually ruled out by better data, by better data even from the same experiment. So now I wanna talk about neutrino masses and why neutrino masses can resolve all of these problems. And so let's start by raising the possibility that neutrinos have non-zero mass. Okay, so that's the only hypothesis I wanna make. And I'm gonna be talking about mixing, which is exactly the same kind of mixing that you just heard about in the last lecture, but I'm gonna be presenting it in a very different way. And the reason we do this is to confuse people. We wanna make sure that everything looks completely different. But it's exactly the same physics. And the reason I wanna present it in this way is because that's the language that everybody uses. And it turns out to be very, very useful because even though it's exactly the same physics, the values of the parameters are completely different. And that leads to interpreting the phenomenon slightly different ways becomes very, very useful as well. Okay, so let's do that. And I am then giving you the statement that neutrinos have non-zero masses. So of course, I have to tell you what the masses are. And we don't know what they are, so I will give them some names. So one mass is called M1, the other one is called M2, the other one is called M3. And if you have more neutrinos, you can keep going. And of course, the neutrino that has mass M1 is called Nu1, the neutrino that has mass M2 is called Nu2, and the neutrino that has mass M3 is called Nu3. And these are all of the official names of these neutrinos. You will notice these are super boring names. It's nothing like charm, strange, and top, and bottom, and truth, and beauty, and these things because I think we've run out of creativity. We don't know how to give particles names anymore. So that's the end of that story. So this is Nu1, Nu2, and Nu3, and they have well-defined masses. But of course, we also have a different way of labeling neutrinos that have to do with the interactions. We have what's called the electron neutrino, the muon neutrino, and the tau neutrino. So if I tell you that the neutrinos have mass, you're allowed to ask, so which one of these three is the electron neutrino and which one is the tau neutrino and which one is the muon neutrino? And the answer is yes. Okay? And what happens is, of course, this is quantum mechanics. So if I have an electron neutrino, the electron neutrino doesn't have to be any of these mass eigenstates. The electron neutrino is a linear superposition of the neutrinos with a well-defined mass, okay? That's the most general thing that I can say. And the same is true for the muon neutrino and the tau neutrino. Again, because the electron neutrino and the muon neutrino are different particles, those linear combinations of the mass eigenstates are gonna be orthogonal to one another. Because again, the probability that an electron neutrino is a muon neutrino is zero, okay? So if you're very, very familiar with quark physics and you wanna ask yourself, how do I explain this in quark language? Basically what we're saying is that the electron neutrino, the muon neutrino and the tau neutrino are kind of like the D prime, S prime, and B prime. Do you know what those are? Is every, have you ever heard about that? So if the answer is no, Yuval will tell you all about it, no. So the idea is that you can define these weak interaction eigenstates, which are not the strong, they're not the D, the B, and the D, the S, and the B, but there are these mixtures of the D, the S, and the B, and they're almost a one-to-one correspondent with one another because of the weirdness of the CKM matrix, but people then define these as prime states. I don't even know who came up with that, it might be from Cabebo or something like that. Yeah, there you go, yeah, so that's right. Yeah, the problem is that the mass is the one that has no index usually. So here it's the same story, but we don't say that, we just say the electron neutrino's a linear superposition of mass eigenstates. So what has this got to do with neutrino flavor change? And the idea is quantum mechanics. And what happens is when I have a neutrino production process invariably or for everything that we care about, the neutrino production process is gonna be a charge current weak process. That means that when a neutrino is produced, it's not gonna be produced as one of these mass eigenstates. It's gonna be produced as a linear combination of different mass eigenstates. So it's gonna be like an electron neutrino or a muon neutrino or a tau neutrino. Now, what you should be asking yourself is what does it mean to be a mass eigenstate? And again, you've also alluded to this. To be a mass eigenstate means that you are actually the real particle. You are an eigenstate of the free particle Hamiltonian, which is, again, there was a question that somebody asked. You're clearly not an eigenstate of the Hamiltonian. The Hamiltonian is super complicated and we don't even know what the eigenstates look like. But we can throw out all of the interactions and whatever is left has some eigenstates and the mass eigenstates are eigenstates of the free particle Hamiltonian. Now, the free particle Hamiltonian is the thing that governs particle propagation. That means that a mass eigenstate is also an eigenstate of the propagation Hamiltonian. It's the thing that tells you how you evolve in time when nobody's trying to hit you. Okay, so that's what you wanna think about. So this is what I wrote down over here. So if I produce a mass eigenstate, and let's say that I produce it at rest, and then I wait some time and then I ask what does my wave function look like? The wave function is the same as the one that I had before multiplied by a phase, which is what you learn in undergraduate quantum mechanics. Now, the mass eigenstate has an energy that is characteristic of whatever physics process is associated with the production of this mass eigenstate. But of course what's different from the different mass eigenstates is that their masses are different. And the way that this shows up is in this thing called the dispersion relation. It's something that tells how your energy and momentum are related to one another. And as far as we're concerned, everything that we care about is, if you produce a mass eigenstate and it evolves in space-time, it's going to acquire an overall global phase. If you produce a different mass eigenstate under the same circumstances, it's also gonna evolve in time and acquire a global phase. But that phase will be different than the phase that the other mass eigenstate got. Okay, and this is all we need to remember because now, if I produce an electron neutrino and I pretend that there are only two neutrino species, then that neutrino is a linear superposition of the mass eigenstates. And the coefficients here are conveniently called the cosine of theta and sine of theta because we know that this number squared plus that number squared is equal to one. And it's very convenient to write down numbers that add up to one as cosine theta and sine theta. The muon neutrino is also a linear superposition of mass eigenstates, which is different from that one. So it's orthogonal to that one, which is why you get a minus sign here and the sine and the cosine get flipped. So yes, so if you do the calculation in the right order, I don't think you have to worry about that. The mass is the, there's a physical mass, that's something that you measure and that's what defines your mass eigenstate. The mass is whatever the mass is. We're not talking about the mass parameter. We're talking about the quote unquote pole mass or the real mass of the thing that you can measure. If you could put it on a scale, okay, we don't put things, if you could hit it and measure the mass by kinematics, that's the mass. It's not the mass parameter. So the way you would think about this is you get to do all of your loop corrections before, you figure out where your masses are and then you go from there. So that's the way to think about that. Absolutely correct, yeah. So that will also tell you what the mixings are. So everything is set and then you'll go from there. That's one way of thinking about that. That's a good question, yeah. So again, so that's the picture that we have and by the way, these mixing parameters run, for example. So in all kinds of things you have to worry about. Again, this is the same that happens in the quark sector. Mixing parameters can run. They don't run very much, but they can and people have also played with that. But once you're doing a specific experiment where the kinematics are defined, nothing bad happens. It just means that what the running tells you is that if you do an experiment, I mean, I think you've all probably said this, the running just makes your life easier as far as comparing different experiments. And if you do one experiment at a certain kinematical regime, you measure all of your parameters and then you wanna repeat another experiment with a different kinematical regime and you'd like to use the result of the first experiment and apply it to the second one. The running allows you to do that very easily. So it basically says, one experiment at one energy measured all of the parameters. I wanna apply that to a different experiment at a different energy regime. And in order for me to make this comparison as painless as possible is I allow my parameters to run and then I make the measurement there. So that's another way of thinking about that, okay? All right, so the key thing for us is the fact if you stare at the electron neutrino, it's not an eigenstate of the Hamiltonian. And as you learned in undergraduate quantum mechanics, week one, is that if you're not an eigenstate of the Hamiltonian, upon time evolution, your state will change. You will no longer be the same state that you were before. That means you will not be the same state multiplied by a phase. You will get to be a different state. So that means that if you're born as an electron neutrino and this state here evolves, it will evolve into something that is not exactly an electron neutrino anymore. Now, because there are only two states, that means it will have a non-zero probability of being a muon neutrino at some instance in time. So that's all we need to know. So that's what neutrino oscillations are. The rest is details. We can do the calculation. They acquire phases. We're trying to do it in some relativistic way, and I'm not gonna go through the details because this is always very confusing, but the point is this is a calculation you have done before. In quantum mechanics, you do this in the context of a two-level system. So you must remember, if you don't remember anything from quantum mechanics, you have to remember the two-level system. Because that's the nicest system. It's very simple and it's not messy. So if you have a two-level system, so your Hamiltonian has two eigenstates or two eigenvalues, and your initial state is a linear superposition of those two eigenstates, and you calculate the time evolution of that, the probability that you are gonna be in the same state that you started in is gonna oscillate in time. So does everybody remember that? That's the only thing you have to remember from quantum mechanics. If you remember that, you know everything else. And on top of that, the frequency of the oscillation is proportional to the energy splitting. Okay, so you also, that's harder to remember, but that's proportional to the difference of the two energies that you have. That's exactly the problem that we're doing here. We're doing a relativistic version of that. And of course, one of the problems with the experiments that we do is that we don't know when the neutrino was produced. We know where the neutrino was produced. And we know where we detect the neutrino. We assume it's traveling at the speed of light. And of course, we know how to convert distance into time and vice versa. So at the end of the day, we can calculate the probability that a muon neutrino is measured as an electron neutrino or vice versa as a function of the distance. And the frequency of that, as I said, was the energy difference in the two level system. Here the frequency of that will be proportional to the difference of the neutrino mass is squared. So it's m one squared minus m two squared divided by the energy and also divided by a factor of four. This is a very simple calculation that I invite you to think about, but not too hard because people get very confused when they think about this too much. So don't think about it too much. And the key thing we wanna remember is that there's this conversion probability is proportional to these numbers. These are, they're called the mixing angles. And it's also proportional to the difference of the neutrino mass is squared multiplied by L divided by the neutrino energy. The L divided by the neutrino energy is something we could have guessed because in some sense, that's what's called a, that's proportional to the neutrino proper time. It's how much time the neutrino feels in its rest frame when you compare it with the distance in the lab frame. So the one over E is like a gamma factor, okay, from the Lorentz transformation. By the way, if we had assumed that the neutrinos decay and we asked that at what rate do we lose neutrinos as a function of distance, this would also scale like L over E. So it's a consequence of Lorentz invariance, okay? So we can make a plot of this. That means that the probability that you're born as an electron neutrino and measured as a muon neutrino will oscillate up and down like that. And it has some oscillation length, which is associated with the neutrino mass square difference and the neutrino energy. And there's some amplitude to the oscillation, which is proportional to these mixing parameters. So just to summarize, if you postulate that the neutrinos have mass, you expect that it is possible to be born as a certain flavor and to be detected as a different flavor. And what I try to say earlier was that that's what the data seemed to be telling us, is that it is possible that neutrinos produced in the sun are born as a certain flavor, but they're measured as a different flavor by the time they get here. And that the neutrinos produced in the atmosphere are born as muon neutrinos, but they behave as tau neutrinos by the time they get there. So the question is, does this hypothesis, which leads to a formula that's very simple and looks like that, can that explain the data? And I think I wanna skip this slide. No, let's talk about the slide. So for example, the survival probability. So again, the probability that you're born an electron neutrino and stay an electron neutrino, assuming that there are only two flavors, looks like this. And then you can ask yourself, when do I get to see this? What is required of my experiment for me to see this phenomenon? So for example, imagine that I do an experiment where I have a detector located a certain distance from a nuclear reactor, and then I'm counting electron anti-neutrinos, and then I ask if I get the same number that I expected. So somebody did this experiment about 20 years ago, and they got the right answer. They got as many as they expected with an error bar that said that this number is less than 5%. What does that mean in terms of understanding what's going on with the neutrino parameters? So if this number is very, very close to one, it means that this number here has to be very small. So how does that happen? I mean, this happens in lots of different ways. Maybe the sine squared of two theta is zero, so there's no mixing, so nothing happens. Or maybe this oscillation phase here is too small. So if the delta M squared L over four E is a very, very small number, you don't get to see this phenomenon either. This phenomenon requires that the neutrinos travel a certain distance. So there's a characteristic region of the parameter space where the delta M squared is big enough and the mixing angle is big enough where you expect to see something. And of course, if you had done the experiment, you would have said that these parameters are ruled out. And it's interesting to appreciate with these, so this is very, very common in neutrino experiments. People make plots that look like that a lot. And it's interesting to appreciate why this curve looks like that because it gives you some intuition about what the oscillation formula is doing as a function of different parameters. So of course your experiment has some characteristic L over E. So imagine that the delta M squared is super large compared with L over E in such a way that this number here is really, really big. The phase, the thing inside of the sine squared is a very big number. Now we have to appreciate the fact that the L over E is not a number, but it's a range of numbers. And we don't get to measure them perfectly, so we actually only see an average of values of L over E. So if I'm averaging over a very rapidly changing phase of the sine squared function, that will average out to about a half because that's what sine squared does. So that means that for delta M squared values that are very large, this PEE is proportional to one minus one half sine squared of two theta. So that means if sine squared of two theta times a half is bigger than something, your experiment should have seen an effect so you can rule that out. Let's go to the opposite part of the plot, which is when delta M squared is super small. When delta M squared is very small, this phase here is very small. So if it's too small, you don't see anything. Once it becomes small, but not super small, sine squared of something small is proportional to that something small, squared. So that means that in this region here, your parameter here looks like sine squared of two theta times delta M squared squared. So here's what a constant function of sine squared of two theta times delta M squared squared look like, it looks like this. So again, if you're everywhere above here, you're ruled out. So this is a very simple function over here. It's a very simple function over here. In the middle, something funny happens because you actually get to see oscillations and so on, and you're sensitive then to this region of the parameter space. So this is what the oscillation formula is doing. It's one way of thinking about that. I have a parentheses to make, which is since nobody asked. So this story that I told you here, that's a very, very fishy story. If you think about it just a little bit more, I probably said lots of things that are wrong. The key thing which you wanna worry about is, okay, so you're producing a neutrino via some physics reaction, and that physics reaction conserves energy and momentum. So how can the same physics reaction that conserves energy and momentum produce two neutrinos that have different masses? But if you never thought about this, don't think about it, but it's too late now. That's the concern is how does this story actually make sense? I have lied in a few places, actually lots of places, but nonetheless in spirit the story is correct. So I want you to remember that. That's a very important thing to know. Even though it feels like there's a few things that are going on here that are wrong, it's not wrong enough to make the story wrong. But if you really sit down and think about this, the statement we're making is that there is a physics process, and that physics process can produce a coherent superposition of mass eigenstates. That's the assumption that I'm making. The key word here is coherent, that somehow the state that I produce has a non-zero probability of being different neutrinos with different masses. Now you can ask yourself, can that ever happen? And the answer is yes. And the answer is yes because of quantum mechanics. And as we know, quantum mechanics is very hard. So you have to think about it very carefully and people have. And this is one, I would say, interesting reference that tries to raise all of the problems of coherence. And when do you have to worry about the approximations you have made? And when you don't have to worry? And I have a summary of what you should be worried about there, and I don't want to read this, but I do want to say a very important thing that everybody should remember is that a lot of people have worried about this. I think what I said is that there's a very confused history of understanding neutrino coherence. It's also a very confusing history. There's a lot of discussions in the literature that are wrong. There are a lot of discussions in the literature that are not wrong, but they're very confusing and they don't add anything to the discussion. So don't go chasing these weird neutrino oscillation formulas wrong papers because they're not gonna help you in any way as far as I know. So keep that in mind. It's very important to remember that. Every once in a while, somebody will write a paper saying that the neutrino oscillation formula is wrong and that paper is wrong. Even the ones that haven't come out yet. So keep that in mind. But it's a very interesting quantum mechanics system that I described in an oversimplified way, but this oversimplified way of thinking about the problem is almost always correct. Okay, and when it's not correct, sadly enough, it doesn't matter. And I will try to mention that maybe a little bit later. Okay, so let's explain the data with this formula. So here are the, this is the data that I showed you. You will also get to do this if you do the homework. This is the formula for the survival probability of muonutrinos if you pretend that there are only two neutrinos. That's what we're pretending for now. It looks like the formula that I wrote down before. And again, the distance that the neutrino travels is something that I can measure. The neutrino energy is also something that I can measure. The stuff which I don't know is this a mixing parameter and this delta M squared. And the delta M squared is governing the oscillation length or the time scale for this process to start. So let's look at this plot, which is the one that I highlighted before. And let's try to explain this plot with this formula. It turns out it's very, very easy. So over here, when the distance is not very long, the survival probability of the muonutrino is one. It's 100%. So I can fix that by making the delta M squared smaller than something. If I make it smaller than something, and I tell you that if the neutrino travels only about 50 kilometers, nothing happens. It just means that if I pick this delta M squared to be super small, the oscillation doesn't have time to start. So that works out well. On the other hand, if I go all the way down here, you notice that this function here is kind of flat. So it doesn't really depend on the distance anymore. And the overall factor is about a factor of two. I miss about 50% of my neutrinos. How do I make this function here be kind of flat? The way I do that is by making the oscillation be super fast in such a way that when I integrate over these different distances here, my function goes up and down many times. It averages out to about a half. So over here, I expect my survival probability to be one minus a half times sine squared of two theta, which I also wrote down in the last slide. And if I pick sine squared of two theta to be one, one minus a half is a half. So I get this factor of a half here. So I can explain the behavior of the data for small distances and for super long distances. And then of course I need to transition from this behavior to that one. And I do that by picking this number delta M squared in such a way that I go from no oscillations to a lot of oscillations and that the middle is somewhere over here, where cosine of theta is a little bit less than zero, which is neutrinos that are coming from like over there. Like that. So that's how I explain the data. And that's the dotted line here. And you see that the dotted line fits the data ridiculously well. Now what's very exciting about this is the fact that I'm only talking about this panel and this panel is characterized by a range of neutrino energies. And to explain this panel, I have to fix the value of sine squared of two theta and I have to fix the value of delta M squared. And now I have no more parameters to fix. I am done fitting the data. I can then apply the same formula to all the other panels in the picture. And you notice that the dotted line fits the data very well everywhere. So it's a very, very non-trivial statement that this works out well, this works out well, this works out well, these data also work out well in such a way that this is a very non-trivial statement that I can explain the data with this oscillation formula. Now we have actually verified this experimentally in a very different way. So if this hypothesis is correct, that means that I have just determined what the delta M squared is. That means that if I set up an experiment in the laboratory where I know my neutrino energy, because it's mine energy, it's the one that I produced in the laboratory, I know how far away I have to put a detector in order to see this effect. So I can make a prediction if you wanna think about it that way. And that's what people have been doing for the last 20 years or so, is that they know where the detector should be so that you can see this effect. Because you know the delta M squared, say from the atmospheric neutrinos. So these are the older version of these experiments. There's a K to K experiment, was the first one to physically see this. There's the Minos experiment in the US. There's also an experiment at CERN, which is also done thinking data. And the thing which is important to remember if you're interested in neutrino experiments that are using accelerators, is that the distances that we need to do these experiments at are measured in hundreds of kilometers. And a hundred kilometers is a crazy distance for doing a particle physics experiment. Okay, so hopefully people appreciate this. You would never do this if you were not looking for something like that. Because you have nothing to gain at all by having an experiment that's a hundred kilometers away from your beam. Okay, this is very important. And the reason you wanna think about this is that if you produce a particle beam, even if you produce a great particle beam, that beam is not a perfect beam that's pointing that way. It has a little bit of a spread, right? You know, that's true of even a laser pointer. You know, the laser pointer gets bigger and bigger as you go further and further away. And the laser pointer is very, very well-colomated. So imagine that I had to point this laser pointer a hundred kilometers away. How big do you think the laser pointer would be? It would be very big. Okay, so we don't wanna do that calculation. Now imagine you're building a neutrino beam. A neutrino beam is a horrible beam. It barely qualifies as a beam, okay? And I won't have time to talk about this, so let me just say it now. The way that we produce a neutrino beam is by producing pions. That's how you do a neutrino beam. You produce a lot of pions. And the way that you do that is by shooting protons on a target. You shoot protons on a target. You know, the protons are going that way. You hit the target, you get a bunch of pions that are going everywhere. But mostly that way. Now the pions have charge, which is good. That means I can place them in a cleverly designed magnetic field in order to focus them. So I can take, say, all of my pi minuses and I can make them all go in the same direction. And I know how to do that. Now the problem is in order for me to do that in a way that I keep a lot of pions, I have a lot of pions that are all going in the same direction, but that beam is about this big. The pions live in a sort of a meters square kind of circle. And they're all going that way. And now the pions decay. And of course you will not be surprised to learn that when the pions decay, the neutrinos go mostly that way, but not exactly. They kind of spray out in some angle. So your neutrino beam in the best case scenario, as it's leaving the beam, it looks like that wall. That's your neutrino beam going that way. So if you have a detector 500 kilometers further down the road, which is probably gonna be in a different country, your neutrino beam is kilometers big. It's a gigantic thing. That has a good side, which means you don't have to aim very well. You're gonna hit your detector. On the downside, you're gonna miss all of these neutrinos that are going somewhere else. So going very far away is only a good thing because you have to see this phenomenon. On the flip side, it's also something to keep in mind. This also means that we got super lucky when we were discovering in quotes the standard model. Because imagine that this phenomenon occurred at a distance of like 10 meters instead of 100 kilometers. Imagine this a letterman experiment would have been very confusing because they were looking for this two neutrino flavor hypothesis by not seeing any electron neutrinos. But if the neutrinos had oscillated before they got to the detector, we would still be thinking about what's going on with neutrinos because they would be oscillating super fast. So because they don't oscillate super fast, you need these hundreds of kilometers, all of the experiments that are done inside of the laboratory don't get to see any of the oscillations. So the neutrinos behave like you think that they're supposed to behave. So that's the idea. So we did these other experiments and what this is supposed to tell you is that when you interpret the data using this oscillation hypothesis, the atmospheric neutrinos wanted to live here and these neutrinos observed in this other experiment want to live here. Again, if you look at a different experiment, the atmospheric neutrinos wanted to live here, this Samino's experiment is telling you that you want to live here in the parameter space. And the message is that the agreement is really, really good. That means that if the oscillation hypothesis is correct, you would expect these allowed regions to overlap with one another and they do overlap very, very well. Okay, so since I've talked very, very slowly, I want to try to explain solar neutrinos. And it turns out that even though the solar neutrinos are the oldest problem, if you try to use this formula here to explain the solar neutrinos, it doesn't work. It used to work before we got better data, so that's usually how life works. It's kind of funny that in order to explain the solar neutrino data with a formula that looks like this, you would need this neutrino oscillation length to be about the distance between the earth and the sun, which is possible. It's a long distance, but it is a logical possibility. And for a long time, this was a solution to the solar neutrino problem. But once the data got better, that was ruled out as a solution. And what turns out to be the problem is we have to remember that the neutrinos are propagating through some material, okay? And this is a subtle thing, so I'll spend these 10 minutes telling you about what's going on here and how you should think about this. The easy way of thinking about this is so the neutrinos are propagating through some material. As we know, the neutrinos don't really scatter, right? So there's no way that the neutrinos gonna hit something. That means it's not gonna hit anything and change its momentum or hit something and convert into something else because the probability for that is super small. On the other hand, as the neutrinos are propagating through material, they appreciate the fact that there's material around. And the way in which they do that is very, very similar to what happens to the propagation of light through a material, through a transparent material. So we know about transparent materials, like glass, and we know that the light is mostly happy to come through the material, but it doesn't do that with complete impunity. It actually knows that the material was there even though it doesn't scatter, right? You know, the phenomenon that we know about is refraction. You know, the light can refract in the material even though it's not scattering. And the thing you wanna remember about refraction is that when the light is propagating through the material, it still propagates just fine. It's just that its velocity changes, right? The index of refraction changes the velocity of light in that material. A different way of thinking about that, which is what we care about, is that the dispersion relation of light changes in the material. Instead of being, you know, E equals P, it's E equals P times something. I think. So there's dispersion relation changes, and that's exactly what happens with neutrinos propagating through a material as well. Even though they're not scattering, there is a potential that's generated by the material, and as the neutrinos are traveling through the material, there are dispersion relation changes. And because this neutrino oscillation phenomenon relies on the fact that the different mass eigenstates have different dispersion relations, if you change the dispersion relation, you're gonna change the way in which the oscillations happen, okay? So that's the story. So let me give you a little bit more detail. What happened to the detail? Oh yeah, here we go. So one is I can convert the neutrino oscillation formula into a quantum mechanics problem. Again, it's very important to first of all appreciate that this statement is true. It's a mathematical statement. I can take that formula and I can make it look like this, that the propagation of the neutrino in distance is given by some matrix multiplied by my state back again. And here I'm writing my state in such a way that the one zero state is the electron neutrino and the zero one state is the muon neutrino. And if I call this thing here my Hamiltonian, this looks exactly like an undergraduate quantum mechanics problem. We would call it a non-relativistic Hamiltonian propagation problem. And this statement is completely correct. It has nothing to do with Lorentz invariance. We didn't break Lorentz invariance. Nothing bad happened. It's just a mathematical fact that we can make the neutrino oscillation equation look like a two flavor system Hamiltonian problem in ordinary quantum mechanics. And this will be useful for us, okay? So this is what we call the Hamiltonian. And the Hamiltonian is proportional to the difference of neutrino masses squared divided by the energy. Okay, now let's look at the neutrino Lagrangian. And this is supposed to be the Lagrangian density. I apologize for that. So here's the neutrino kinetic energy and let's say that we care about the neutrinos with scattering off of electrons. So this is the interaction of say the charge-current interaction if I rearrange my fields in a way that it looks nicer that way, okay? So this is my Lagrangian. Now from this Lagrangian, I can talk about the one neutrino state and I can calculate the equation of motion for the neutrino, which is what we do all the time without thinking about it too much. But I wanna do a more complicated calculation which is I wanna calculate the equation of motion but I wanna pretend that I'm traveling through a medium that has a lot of electrons in it. So that means that when I calculate this equation of motion, in particular, this term here will give me another term that's also quadratic in the neutrino field and it will be multiplied by the expectation value of this operator here in this medium that has a lot of electrons in it, okay? So that will change my equation of motion. So is everybody mostly okay with this? Okay, so if I do this, by the way, we always do this except that always this term is zero. So we never propagate through any material but if we did, we could still go back and do the calculation anyway, okay? So this object here, E bar gamma mu E, if you write this out, this looks like E bar gamma mu E, first of all, this is a four vector. So this is some four vector that characterizes my electrons that I'm going through. I'm gonna make the assumption that the electrons are at rest in some reference frame. So if the electrons are at rest, there's no four vector for me to talk about. There's no, or let me say this again. There's no three vector in the picture, right? The electrons are all sitting there at rest. So this current here can't have any gamma I component because there's no three vector that I can talk about. If the electrons were all moving in one direction, I could talk about the velocity of those electrons but I can't talk about that. So that means that this only has the zero component. If I look at the zero component, the operator becomes E bar gamma zero E and E bar gamma zero E is equal to E dagger E. And again, if you remember the first class of quantum field theory, E dagger E is the number operator for your particle. So at the end of the day, this object here is proportional to the number of electrons per unit volume through which the neutrinos are propagating. And because I only care about the left-handed electrons here and because all of my electrons are at rest, I only get to see half of them because I only care about the left-handed part. Again, so for every electron that you see, they're kind of 50-50 left-handed and right-handed. So I get this factor of a half. What this means at the end of the day is that my Dirac equation for the neutrino propagating through this medium looks like this. So it's a very, very different Dirac equation. Here's the regular Dirac equation but then I have this extra piece which has to do with the fact that I'm traveling through matter. I can solve this equation and the solution is approximately the same. I get some e to the i p dot x except that my dispersion relation is not gonna be e equals p. It's gonna be a little bit different. The dispersion relation becomes that instead of the energy being the momentum, the energy is now the momentum plus something. And that something depends on the density of electron neutrinos in the medium, the number density of electron neutrinos in the medium. Finally, there's a plus and minus sign and the plus sign happens for neutrinos propagating in the medium and the minus sign happens for anti-neutrinos propagating in the medium. Now if you're very, very confused about all of this, you wanna go back to when you were talking about the Lagrangian for the electron in the presence of a classical electromagnetic field. And then you calculated the equation of motion for the electron or the equation for the electron and it had a propagating piece and then there was a piece with the a mu field in it. And then your Dirac equation is modified but you get the Dirac equation for an electron in the presence of a classical electromagnetic field. This helps out because it also explains to us why you get this plus or minus sign. The reason you get this plus or minus sign is because when you're calculating the propagation with the electromagnetic field and you calculate the Dirac equation for the electron, you have a certain charge and when you calculate the Dirac equation for the positron, the positron has the opposite charge and that shows up in your equation of motion automatically. You don't have to change the sign by hand to say that that's what's going on with the positron. So because this current here is a vector current, it looks like the electromagnetic field, at least in spirit and that's how you get this plus or minus sign. So this means, so again this is the message that I tried to send before, this means that the neutrino dispersion relation changes in the presence of matter. Which means that I have to go through the same exercise that led to this Schrodinger equation for me to calculate neutrino oscillations where now I have a different dispersion relation and just give me a couple of minutes. This term here has dimensions of energy. So it's called the matter potential because it looks like a potential energy. Looks like the neutrinos feel some potential energy and this shows up much better when you write this down again in this Schrodinger-like equation that governs neutrino oscillation. Here's the equation that we had before and now I add a new piece to my Hamiltonian in quotes which is proportional to this number, square root of two G Fermi times the number of electron, number of electrons per unit volume and you notice that if you believe that this is a Hamiltonian, this is also a contribution to your Hamiltonian. So from this perspective it also looks like a potential in the quantum mechanic sense. So that's why it's called the matter potential. And the key thing that we're gonna talk about the next time is we would like to be able to solve this equation here when this A parameter is not zero. And you notice that the oscillation equation will look differently. You won't get the same kind of oscillations anymore. And the other thing I wanna mention is this turns out to be a very difficult equation to solve. Even though it looks like a two state system in regular quantum mechanics. And the reason it could be a complicated equation to solve is because you're propagating through this material and the density of electrons in the medium might be different for different positions. So this is kind of like having a time-dependent Hamiltonian problem in ordinary quantum mechanics. So it's not just a regular two state system in quantum mechanics. It's a two state system with a time-dependent Hamiltonian. And by the way that's exactly what happens in the sun because in the sun the neutrinos are born in the center of the sun where the density is very high. They propagate out of the sun and the density falls off exponentially until they exit the sun and then they propagate through a vacuum. So it's this sharp behavior of the energy density, I'm sorry, of the matter density of the sun that actually explains how neutrinos oscillate when they're coming out of the sun. So I wanna talk about how this works because this is a useful thing to know. Okay, so let me stop here because we have to go to lunch. Thank you.