 We just quickly remind you of how we got here, and again what we're trying to do is to describe this phenomenon of neutrino oscillations, and the assumption we're making is that somehow we can produce a coherent superposition of different mass eigenstates. And because of that, the different mass eigenstates as they propagate, they acquire a different relative phase because of the propagation, and it's this relative phase during the propagation that allows your state to change from some initial state, which is usually a flavor state, into a different flavor state. So this is what we started talking about, and the next effect I wanted to discuss is the fact that this is related to the neutrino-dispersion relation. So let me remind you of that, it kind of goes like this. So we know that e squared is a p squared minus m squared. The neutrino mass is super small, so of course I can write that approximately this is a p minus plus m squared over 2p, so this statement here is correct because the neutrino mass is very, very small. So what we're saying, and this is completely hand wavy, but it's good for you just to get the general picture, the idea is this is the dispersion relation for one of the neutrino mass eigenstates, is this working, yes it's working, and for the other neutrino mass eigenstate it's a different mass, so the difference in the energies is proportional to the delta m squared over 2p. So this is comment number one, and that's why when we calculate this oscillation formula, we get that there's a sinusoidal behavior that's proportional to the energy difference between the two states, and we're saying that the energy difference is proportional to the difference of the masses squared divided by two times the energy. So what I started talking about yesterday is the fact that when the neutrino propagates in matter, even though the scattering probability is really tiny, it still feels the potential of the matter, and the impact of that is that you modify the neutrino dispersion relation. So this equation is no longer this equation, it turns into a different equation, so I said that you kind of add this thing called the matter potential. So because the neutrino dispersion relation changes, we have to go back and redo the neutrino propagation part of the discussion in order to take this matter potential into account. That's what's written up there. Okay, so in spirit, this is of course what you would do up to here if you only had one neutrino, and then all of this is completely irrelevant. What we really care about is the fact that if you have more than one neutrino and they have different dispersion relations, that allows your system to evolve in a non-trivial way. So the first term on the left-hand side is this piece over here, and if I'm writing my Hamiltonian in the flavor basis, so the flavor basis is the one where the thing that I call the electron neutrino is the one zero state, and the thing that I call the muon neutrino is the zero one state. So in this basis, that part of your Hamiltonian is not diagonal because the mass eigenstates are not the flavor eigenstates, they're linear superpositions of the flavor eigenstates. Okay, so again, the other thing I want to remind you of is that we can map our problem into a two-level system ordinary quantum mechanics problem, which is a problem that we should be very familiar with, so I'm going to be using that language a lot, so if you do have questions asked, otherwise this will be more confusing than it should be. So again, this is in hopes of making things simple. So again, that's the first part, it's basically saying that I have two different mass eigenstates, and the mass eigenstates are linear superpositions of the electron neutrino and the muon neutrino, which is why that piece of the Hamiltonian is not diagonal in the flavor basis. Then I have the other piece, that's the plus a zero zero zero piece. That's the piece that comes from this matter potential. The matter potential is interesting because there are actually different contributions to it, because as the neutrinos are propagating through matter, they feel the weak force due to neutrons, protons, and electrons. Now, most of the weak force is a neutral current force. That's whatever comes out of the Z boson. What's interesting about the neutral current force is that it doesn't care about the flavor. The electron neutrino, the muon neutrino, and the tau neutrino, they all see the same weak potential. So the weak part, or the neutral current part of the weak potential in this language would be some potential energy that's proportional to the identity matrix. If you remember your basic quantum mechanics, if you add another matrix to your Hamiltonian, which is proportional to the identity matrix, the eigenvectors of the Hamiltonian will not change, and the eigenvalues of the Hamiltonian will change by a constant, which is the same for all of the eigenvalues. When you calculate the propagation of your system, this shift in the eigenvalues of everybody is unphysical. It's kind of like saying, you know, I have a two level system, and then I take the energy and I move it up here, and that doesn't do anything. It just changes what the energy is. So the neutral current piece of the matter effect doesn't do anything. The charged current piece does, because the electron neutrino and the electron, they have a special relation that the muon neutrino and the tau neutrino don't have. And so the charged current piece, which is related with the neutrino electron scattering, is different for the electron neutrino relative to the muon neutrino and the tau neutrino. So that's the piece that I wrote there, and that's why that part of the Hamiltonian only has an EE element. Because it's diagonal in the flavor basis, and yeah, so that's the answer. So that's the idea. So it only affects electron neutrinos. It doesn't affect the muon neutrino. And then I've thrown out the piece that's proportional to the neutral current. Okay, so if we got this far, then the problem that we're dealt is I have picked the basis, and again, this is a two-level system, and the basis looks like this. And then I want to solve a quantum mechanics problem. And my quantum mechanics problem is very simple. I want to have some initial conditions like electron neutrino at t equals 0. This is my Hamiltonian that tells me what my wave function will look like, some distance L away from my source. And the reason I write it that way, because that's how the experiments look like. But hopefully, we are all super experts in ordinary quantum mechanics. So we can appeal to what we know about quantum mechanics to try to solve this problem, okay? So is everybody fine with that? So that's the proposition, and we're gonna appeal to what you know about quantum mechanics to write down what the solutions are. Or I will just write them in, the slides will write them down. And I mentioned later that this A term is proportional to the density of electrons in the medium. But that means that as you're moving along, you're always sensitive to the density of electrons in the medium. So if that density happens to be changing with position, that makes your problem very hard. Because it looks like a time-dependent Hamiltonian for a two-level system in quantum mechanics. It's not super hard, it's just kind of hard. But we know how to do it, because you learned it as an undergrad. If you had a very long quantum mechanics class, and you got to the time-dependent perturbation theory part of the class. If you didn't get there, you got very close to there, and you probably read the chapter on the train or something like that. There are cases where solving this problem is very easy. So one case, of course, is when this A parameter is position independent. So imagine that that's the case. Then we know how to solve the problem, and that's the first one we want to solve. So the idea is, of course, if the A parameters L independent, we just have a regular two-flavor system that we want to solve. Those are very easy to solve, right? It's a two by two Hamiltonian, and when you want to solve a quantum mechanics problem, the proposition is always the same. You're given a Hamiltonian, you want to diagonalize the Hamiltonian. You calculate the eigenvalues, you calculate the eigenvectors. You take your initial condition, which, for example, could be this one. You express that in the basis of the eigenstates of the Hamiltonian. You do the propagation, and then you can answer any question that you are asked. So, for example, that's what my Hamiltonian looks like. One thing that happens a lot in these metrenoflavor calculations is we often cheat by adding and subtracting something proportional to an identity matrix to make our systems look nice. So I am doing that along the way. So if you do that, and you calculate the eigenvalues and eigenvectors of the Hamiltonian, you end up with an oscillation formula, which is the one that's the first line in green there, which actually looks very simple. And it looks exactly like the formula that we had before, which is that the probability that something born as an electron neutrino will be measured as a muon neutrino goes like some amplitude, multiplied by sine squared of some phase. And the phase is proportional to the distance. What the matter effects will do in this case is that they will change the amplitude, and they will also change the phase, which is not a surprise, right? Because you don't have the, when the A parameter is zero, then you have oscillations in vacuum. Then you get the formula that we talked about before. When you turn on these matter effects, you get a phenomenon that looks the same, but the parameters change slightly. So that's the idea. There are lots of important things to appreciate. One is I write the expression for the modified frequency, and the modified frequency looks like that. And one thing that we want to remember is, if I were discussing anti-neutrino oscillations, then the anti-neutrino oscillations would have A modified into minus A. So the sine of the A parameter is different. So what this means is if you do neutrino oscillations through matter, and anti-neutrino oscillations through matter, then they oscillate differently. This is kind of a super cool result, because in principle, it tells you what neutrinos are relative to anti-neutrinos, which is kind of interesting. If you think about it, this is a CPT violation, because if I take the system and I apply a CPT transformation to it, I get a different answer. And this is not a surprise, because you're traveling in a background that's CPT breaking. Okay, why is that? Because you're traveling to a background that's made out of electron, and the CPT transform of that would be a background made out of positrons. And we don't know how to do that experiment. That would be a very, very hard experiment to come up with. But this is something that we want to keep in mind. And finally, the last thing which is very interesting, which I'll mention very, very quickly, is if you look at this delta sub m parameter, it is actually proportional to this object here. So delta, by the way, is this thing, delta m squared over 2e. And the one thing that you want to notice is the sign of the A parameter. Yes, that's a very good question. The answer doesn't change at all. And I'll talk more about that later if you want to, or maybe during the discussion if people have more questions. So for all practical purposes, whether or not neutrinos are a minor or a direct, it's totally irrelevant. Now, this is kind of confusing, because if the neutrinos were a minor, in principle, we're not allowed to say that there's such a thing as an anti-neutrino, so you can say, so what's this anti-neutrino business that you're talking about? And everything that I say for the neutrino and the anti-neutrino, if the neutrino were a minor or a fermion, and you really don't like to say the word anti-neutrino, you can still discuss the helicity eigenstate in the lab frame. So when you produce a neutrino, there can be a left-handed helicity state that's propagating, or there can be a right-handed helicity state that's propagating. For practical purposes, the left-handed helicity state is the thing that we ordinarily think about as the neutrino, and the right-handed helicity state is the thing that we think about as the anti-neutrino. So because we're lazy, even in the Majorana case, we keep using this language, even though we shouldn't be doing that. We should be calling all of them neutrinos, but one of them is a left-handed helicity. The other one is right-handed helicity, and it turns out that the left-handed helicity one always behaves like the thing that we ordinarily called the neutrino, and the right-handed helicity one always behaves like the thing that we call the anti-neutrino. What I mean by that, since I'm talking about it, I'll say something. So here's a neutrino coming in, it hits something, and a charged left-on comes out. What I mean by the neutrino is whatever it is that produces a negatively charged left-on, and if a positively charged left-on comes in, I usually call that the anti-neutrino. If the neutrino's a Majorana fermion, the same object can do both things. So they're both the neutrino, but it turns out, because the neutrino mass is so small, that if the helicity of this state is like this, it's left-handed, then almost always this will produce a negatively charged left-on. And if the helicity of this goes this way, then it always produces a positively charged left-on. And that's why we abuse language that way. So up to order, which is tiny. So that's why we can get away with that. Good, that's a very good question. Okay, so the thing I wanted to say is that the matter potential is proportional to a term that looks like this. And the one thing that's interesting is we know the plus or minus sign of this object here. So if it's for neutrinos, it's a positive number. That means that by looking at whether or not these two numbers add or subtract, I can measure the plus or minus sign of delta times cosine of 2 theta. Okay, and let me remind you that if I do oscillations in vacuum, this looks like this. And this is a very, very symmetric formula. Okay, it has a bunch of degeneracies in it. So for example, so here's my neutrino number one and my neutrino number two. So the delta m squared is defined like this mass subtracted off of that one. That's m2 squared minus m1 squared is my mass squared. And let me define my states in this way. So the number two state is heavier than the number one state, okay? And I can define that. So delta m squared is a positive definite number. You notice that this expression is proportional to sine squared of 2 theta. So it actually can't tell which one is bigger, cosine of theta or sine of theta. So if I do this experiment here, I can exchange sine and cosine. If you don't like cosine, you can do the squares. I can exchange sine squared and cosine squared, and I get the same answer. If I have oscillations in matter, if I change sine squared with cosine squared, cosine of 2 theta changes sine. And I apologize for the English language because sine and sine sound the same. But we'll get used to that. So the plus or minus sine of cosine of 2 theta will change. If you exchange sine theta with cosine of theta, so that means when you do experiments in matter, you can tell whether cosine squared theta is bigger than sine squared theta or vice versa. You can ask, why would I care about this? You care about this because there's physics here. So the neutrino number one state, its electron component is cosine theta. So if you have a neutrino number one, the probability that gets measured as an electron neutrino goes like cosine squared theta. So if cosine squared theta is bigger than a half, that means that the electron neutrino state is mostly in the one state, which is the lightest one. And if cosine squared theta is less than a half, that means that the probability of getting an electron neutrino is biggest for the heaviest state. So this is a physics question you can ask, but it's not one that you can answer by doing vacuum oscillations. But you can actually answer that with doing matter oscillations. And this will become very important in what I'll talk about later. So the point is, the physics of this is the following. The matter effect favors the electron neutrino. It makes the electron neutrino special, so it can tell whether the electron neutrino is mostly heavy or mostly light. That's what's happening here. So what the matter effects do is that, like I said, they change both the amplitude and the phase. And so the red line is the vacuum oscillation. If you add a matter effect, you either get the blue line or the black line and you notice that either your amplitude gets bigger and your wavelength gets bigger, or your amplitude gets smaller and your wavelength gets smaller. And the key thing is whatever happens for neutrinos, happens the other way around for anti-neutrinos. And if you change the electron state from being mostly heavy to mostly light, you also switch this behavior as well. It's very, very easy to understand why this curve looks like this. Okay, this I would have to plug it in. And the reason is, if you look at the behavior at very, very tiny L, the curves look the same, okay? And I'm not gonna tell you why, but if you stare at your Hamiltonian and you try to solve the problem in the limit where L is very small, the problem becomes very easy to solve. And then the matter effects don't matter. They don't do anything. They only start mattering at higher order, or if you have to propagate long enough for the matter effects to start impacting your oscillation. So that's a very nice and quick way of understanding that. So the next thing I wanna talk about, and I don't wanna spend too much time on it, is what happens when the A parameter depends on the distance? And the reason we wanna talk about this is because we wanna try to understand how neutrinos coming out of the sun oscillate. And the sun turns out to be super dense in the center of the sun. And it's not dense at all by the time you exit the sun, which is not a surprise, yeah. Yes, arbitrary units. So indeed it has no units, yeah. Not astronomical units, very good. Astronomical units is capital A, capital U, not little A, little U. Good, so other questions. So again, we wanna try to describe neutrinos coming out of the sun. And in order to do that, we really have to try to face this problem of a time dependent Hamiltonian. And again, this is a well known problem. It's a well known problem in atomic physics. There are lots of applications of this. One thing is, we know how to build two level systems in atomic physics. Adams do that, spins in a magnetic field do that. So imagine you have two spins in a magnetic field, and then you change the magnetic field over time. You kind of have your two spin systems, it splits up, and then you have a diode, and then you let your magnetic field change, so that your levels will kind of move back and forth like that. And then you can ask, what's the time evolution of my initial state? In order for us to do that, we're gonna discuss this in an approximation. And this approximation works really, really well. And the approximation will be that even though I'm changing the distance between my two energy levels or my two eigenstates of the Hamiltonian, I'm doing it slowly enough. And when I say slowly is that the change in the Hamiltonian is slower than the typical frequency of your system. When that happens, you can use what's called the adiabatic approximation. And if you remember basic quantum mechanics, and if you got to the time dependent perturbation theory discussion, then you've heard about this. If you haven't, then you just have to take my word for it. Okay, so that's the plan. But what we wanna do is, let's look at this Hamiltonian again. Again, it's a two by two matrix. And I know how to diagonalize this. That's good, so we know how to do that. So we wanna diagonalize this matrix. And in particular, we wanna calculate its eigenvalues and eigenvectors as a function of A, the A parameter. Okay? So if I take the A parameter to be zero, then diagonalizing this is very easy because we did it before. The eigenvalues are gonna be the mass eigenstates, neutrino number one and neutrino number two. And we even know what the eigenvectors are. So in this case, the eigenvalues would be m one squared over two E and m two squared over two E. And as I said, I can add and subtract something proportional to the identity. So let me make the zero eigenvalue have zero eigenvalue. So the lightest one to be zero. So that means that one eigenvalue is zero. The other one is a delta m squared over two E, which I'm calling capital delta. So I know how to diagonalize the matrix when A is zero. I also know how to diagonalize it when A is really large. So when A is super large, then if I look at the sum of those two matrices, I can forget about the first one. I only get the second one. And then my eigenvalues are gonna be capital A and zero. You can just stare at it and see. So the difference of the eigenvalues is capital A. And if I change A, this will change linearly with A. And of course, I can also get the eigenvectors in that case. And the eigenvectors are very easy because that matrix is diagonal. So the eigenvectors are gonna be the electron neutrino and the muon neutrino. Okay? So that everybody get that. So I can make a picture of that. It's here. So this is my eigenvalues as a function of A, also in arbitrary units. And so this is a plot of the different eigenvalues in that way. Of course, I always care about the difference. And I'm normalizing things in such a way that when A equals zero, the lightest eigenvalue is zero. So you can see the behavior that I alluded to before. So this is the heavy eigenvalue when A is very large. The lighter eigenvalue is something small and it doesn't change very much. When it's zero, this is the neutrino number two, and this is the neutrino number one, and the mass splitting is delta. And then of course, something has to happen in the middle. And that's what happens in the middle. And this is a very, very famous diagram. It's called the level crossing diagram. Because what you're doing is when A is super large, you have a small eigenvalue and a big one. You're making A small, so you're changing A and it kind of goes like this. And naively, you would expect your levels to cross like that. Of course, they don't cross, because once they cross, whatever's heavy becomes light and vice versa. So they kind of bounce off each other like that. And it goes like this, okay? So is everybody fine with that? So is this very, very clear? So now I want to solve the neutrino propagation, given that my eigenvalues are changing as a function of time. But I want to appreciate the fact that when the A parameter is very, very large, the electron neutrino is an eigenvector of the Hamiltonian. It's an eigenstate of the Hamiltonian, okay? So this means if I'm doing this problem in the limit where the A parameter is very strong, if I am born as an electron neutrino, I'm an eigenstate of the Hamiltonian. So my time evolution is very easy. Nothing happens, right? If I'm an eigenstate of the Hamiltonian, I stay an eigenstate of the Hamiltonian. This is where this adiabatic approximation comes in. The adiabatic approximation says that if your Hamiltonian is changing slowly enough and you're born as an eigenstate of the Hamiltonian, as the system evolves in time, you're going to stay an eigenstate of the Hamiltonian. That means that your state will change because the eigenstates of the Hamiltonian will change, but you will always be an eigenstate of the Hamiltonian. So does that sound vaguely familiar to some of you? Maybe? Have people heard about this before? Yes? Okay. So that's the idea. So what happens in this case is the electron neutrino is born as an eigenstate of the Hamiltonian. That means that as it propagates, that means as it's coming out of the sun, it stays an eigenstate of the Hamiltonian. So that means it's born here and it's whatever eigenstate corresponds to this eigenvalue. If it's over here, it's whatever eigenstate corresponds to this eigenvalue. If it's over here, it's whatever eigenstate corresponds to this eigenvalue. So when it gets all the way here, that's when you exit the sun, right? That's when the sun ends. You don't have any matter anymore. Your A-therm is zero. So that means that the electron neutrino over here is also an eigenstate of the Hamiltonian. That's the idea. This is called the MSW effect. I should mention that the ass in MSW effect is Alexey Smirnov, who worked here for a very long time before he retired. And he's in Germany now, enjoying the good weather. And W is of Ofenstein that you've already heard about from Yuval's lecture, who did lots of interesting things for physics. So the key thing is if this approximation holds, the electron neutrino is born as an eigenstate of the Hamiltonian, and it exits the sun as an eigenstate of the Hamiltonian, which means it exits the sun as a mass eigenstate, which means that now the electron neutrino has left the sun. It is now a mass eigenstate. Mass eigenstates don't do anything. They just start eigenstates of the Hamiltonian in vacuum. So it propagates all the way here, and then it gets to your detector. And what hits your detector is actually a mass eigenstate. It's the neutrino number 2. This is the MSW effect. And this is really interesting, because if I have a neutrino number 2 coming in and it hits something, the probability that it will produce an electron is proportional to sine theta squared. So the probability is proportional to sine squared theta, which is very interesting, because this number here can be anything between 0 and 1. And if you remember the sononutrino problem, if you remember the data, for high energy neutrino, I didn't emphasize this, but for high energy neutrinos, the survival probability of these electron neutrinos is about a third. So if I pick this number to be about a third, I can explain why the high energy neutrinos coming from the sun only behave like electron neutrinos a third of the time. The other cool thing is that as long as the matter potential is very large, compared with the delta M squared over 2E, this result is energy independent. So if I'm in the regime, well, delta M squared over 2E is much, much less than this matter potential in the center of the sun, then I expect my survival probability to be a constant equal to whatever number I like. And it doesn't depend on the energy. So is everybody fine with that? Now, of course, if the energy is super small, then this term eventually will become much bigger than that one. In that case, it's also easy to solve the problem, because if the delta M squared over 2E is much, much bigger than the matter potential, I can forget about the matter effects. They don't do anything, so I only get vacuum oscillations. And in that case, the oscillations are very, very fast. So in that case, I expect a survival probability that goes like 1 minus 1 half sine squared of 2 theta. So is everybody also happy with this? This is what averaged out oscillations look like. From the formula, from this vacuum formula, if I average out the oscillations, this is what I get. By the way, if I plug in a third, you can calculate what sine squared is. You can divide it by 1 half. This will be 0.6 or 0.7. So I can make a prediction for this behavior, which is when the energy is very small, my oscillation probability is about this 0.7. When the energy is very large, it's going to be about a third. And then something happens in the middle. That's when the calculation is hard. And if something happens, it kind of goes like this. So this is what the matter effects can do for me. And of course, I still haven't figured out what the delta m squared is. But I fixed that in such a way that I can choose where this feature here lives in my space. So is this clear? So this is how solar oscillations are going to work out. And you notice how the matter effects are very important. I've already talked about this. So again, this is a curve that I expect to get, something that kind of looks like this. Notice it's not 0.7, it's more like 0.6, but it doesn't matter. And these are the data, by the way. If I take the solar neutrino data and I interpret this in terms of an oscillation as a function of the energy, the different experiments give me these points. So the first thing you want to notice is that we don't have a whole wealth of data. We have a few points. But nonetheless, the points are really doing what we think it's doing. And the fit that we get is this sub blue curve. This is what this MSW effect is telling us for the right values of the parameters. So this is the idea. And that's how solar neutrinos work. There are a couple of things which still make solar neutrinos exciting. One is that if you look at our measurement of the solar neutrino spectrum, there is a gap in the measurements. So between about 1 and 1 half MEV and about 3 or 4 MEV, we have not made good measurements of the solar neutrino spectrum. So of course, if you're a theorist, you can fool around with how neutrinos oscillate in such a way that you screw up the oscillation in this region where we don't have any data. And of course, you can get away with that. And that's what these other curves in this plot here are supposed to mean. So one thing which is very interesting is that there's something called the boraxino experiment in Gran Sasso. And they are the ones that contribute to us this point. And they also worked super hard to come up with this red point as well. Oh, this actually works. OK, so to come up with this red point as well. But you notice that there's no data over here, which is kind of annoying. And this is the data that's the best data, but it only works at high energies. And again, the probability that we're talking about is this gray curve here. And it fits the data quite well. So this is how this works out. I want to skip this one. But the message that comes out from this, by the way, is, so again, we have a curve that looks like this. Our low energy neutrinos live here. The high energy solar neutrinos live here. So that tells us where this transition region has to happen. And this is governed by the value of the delta M squared. So the delta M squared that we need in the units that we like, it's about 10 to the minus 4 electron volts squared. A little bit less than that. OK, so the reason this is very, very important is that if somebody tells you the delta M squared, that means you can ask how to do experiments to probe for this. And again, the thing you want to keep in mind is if I tell you what the mass square difference is and you have an experiment and you know what the energy of your experiment is, you know how far away you have to go from your source to see this effect. So what people figured out is the following. If you take a nuclear reactor, the typical energies are about a few MEV, several MEV, 4567 MEV. So if you plug in 4567 MEV here and you know the delta M squared is about 10 to the minus 4 electron volts squared, you can ask how far away do you have to go in order to see this phenomenon turn on. So you have an oscillation length and you can calculate the oscillation length. And it turns out to be about 100 kilometers. So what you learn is the following. If I have a nuclear reactor here and I put a detector 100 kilometers away from my nuclear reactor, then I should see this effect. The other thing that you want to appreciate is the fact that I told you that the mixing angle squared is about a third. So this is a huge effect. We're not talking about a 1% effect or a 10% effect. This is a 50% effect. It's a gigantic effect. So people got excited about this. And I'm telling the story in the opposite order, but it doesn't matter. So they figured out, look, if I put a detector 100 kilometers away from a nuclear reactor, I can see this. So this is what people started working on at the end of the 1990s. And the problem, of course, is if you think about a nuclear reactor, it's emitting a lot of neutrinos. But it's not a neutrino beam. The neutrinos just go everywhere. So it's a very democratic source of neutrinos. You can put your detector anywhere and you get to see the neutrinos. However, the flux of neutrinos is also falling like 1 over the distance squared. So if you go 100 kilometers away, your flux of neutrinos is ridiculously small. So how do you fix that? The way that you get around this is that you need to build a really, really big detector. But 100 kilometers is very, very far away. So if you build a detector maybe the size of this auditorium, that would be a way to start. And there's an experiment that's being built right now that kind of looks like this. But if your detector is not so big as this room, there's another way of fixing this, which is imagine that you had more nuclear reactors. Imagine that there's one here. Maybe there's another one here. Maybe there's another one here. And then there's another one here. And then there's another one here. And then you have 10 of those. If they all happen to be about the same distance from your experiment, you can do this. And then you say, this is ridiculously expensive because I'm not going to build 10 nuclear reactors to do a neutrino experiment, which is true. Nuclear reactors are very expensive. If you think the LHC is expensive, look up the price tag for a nuclear reactor. But fortunately, people had prepared ahead of time. And it turns out that if you went to Japan, that's what it looked like. It had all of these very, very nice nuclear reactors that were all built very strategically 100 kilometers away from where the Super Communiconda experiment was. And I'm sure this was all planned. And of course, this is because Japan is an island, by the way. Being an island helps because nuclear reactors need water. And if you're an island, it's kind of like this. And you want water, so here's a lot of water. You put your nuclear reactors here. And then you have a mountain in the middle, and it works out. So that's how it works out. So this experiment is called the Kambland experiment. And the Kambland experiment had all the ingredients to see this oscillation of this order, this 10 to minus 4 electron volts squared. And this is the data from the Kambland experiment, which is over here. So again, it's a zero parameter fit because we already knew what the delta M squared was supposed to be. And you notice that the curve doesn't really look like a nice sinusoidal curve as a function of L over E. And this is because the nuclear reactors are placed at slightly different positions, so everything kind of averages out. So this is the main idea, and this is how you get a very, very good fit. And you notice that this is probably the first experiment to see a curve that oscillates like this, which is very exciting, because it really looks like an oscillation and doesn't just look like you're missing some neutrinos, but you actually get to see the survival probability move up and down, which is very, very exciting. And this experiment did not get the Nobel Prize, but it should have, according to me. Okay. So the last thing I want to talk about is there's another reactor neutrino experiment that people have done. So let me summarize where we are. So the idea is I told you about these atmospheric neutrinos, and I could explain that with neutrino oscillations between new mu and new tau with a delta M squared, which is about 10 to minus three electron volts squared. So we talked about that. And now I'm telling you that I can explain the solar data with the new E's disappearing at a delta M squared at around 10 to the minus four electron volts squared. And if I have three neutrinos, these are all the frequencies that I have. So one question you can ask is, this oscillation here is mostly new mu to new tau, but it's also possible to have, for example, I don't know, new mu to new E at this delta M squared. You know, you can oscillate into the other flavors. This statement here just means that the new mu's will predominantly oscillate into new tau, but there's no reason why the electron neutrinos don't participate in this fast oscillation as well, okay? So again, people tested for that, and they tested for that for doing the same thing that I told you about this Camblin experiment, except that you put your detector much, much closer to the nuclear reactor. Now you can ask, why didn't people see this before? And the reason they didn't see this before is because this effect turns out to be small, okay? So unlike the solar effect where the amplitude is really, really large, here you have a similar effect at a much smaller oscillation length, but the amplitude is very small. So nonetheless, this is a one parameter experiment because you know the delta M squared from the beginning, so if you place your detector only one kilometer away, and that's what they did in a couple of places, including in China, and you do your experiment in a clever way, so the way that you do that is your source is here, here's your one kilometer away, and then you wanna measure the flux of neutrinos there. Now you're looking for a small suppression of the total number of neutrinos that you get, and of course one challenge that you have is in order for you to establish that the number of neutrinos has changed, you have to know how many neutrinos you have to begin with, which is very hard, especially if you wanna do it at the 1% level. So we don't wanna have to do this, so one thing we do is, so this is my source, if I put another detector over here, this is a very short distance, so I assume that the oscillations are not started yet, so I measure my flux over here, and then I measure the relative change of the flux over here. This is the data in the bottom there, this is the number of events normalized to something as a function of the baseline of how far away you are from the detector, so you see a bunch of points in blue on the left-hand side, these are your near detectors, and they have a bunch of detectors, and you see that you get this many neutrinos in some units, and then you notice that if you go further away, you get relatively fewer neutrinos than you expect. And you get exact, and the red curve is the oscillation curve for the frequency that's already known, so you don't have any wiggle room in order to change the amplitude, the frequency of the curve, but you can change the amplitude, and if you do a fit to the amplitude, you get to measure another mixing angle, which is less than about 10% for sine squared of two theta. So this is a crash course on all of the oscillation phenomena that we have confirmed, and that we have observed, and that we have interpreted in terms of these oscillation parameters, and the situation is summarized here, and I also summarized it over here so we don't have to read it again, so I can also see electron neutrinos disappearing with the delta M squared, which is this one. So any questions about any of this? So the reason I'm talking about all of this is that we wanna ask what's the complete picture? So at the end of the day, all of these analyses here were done, assuming that there are only two neutrinos, which we know is wrong, so we better be able to take advantage of this, or otherwise we have to solve all the problems back again, assuming that there are three neutrinos. And the reason we don't have to do that is twofold. One is this number is way smaller than that number, and this probability here is very small, and I think this is what I said yesterday. It turns out that if you do a typical oscillation experiment, as long as you're not going to a super, super long baseline, you only get to see one oscillation frequency, so the two-flavor approximation works very well. And it turns out if you're doing experiments with electrons, but you're not super careful, or you're not super precise, the electron neutrinos don't like to participate in the short oscillation, so they only participate in the long oscillation length phenomenon, and that's why I think separate out in this very, very convenient way. Okay, so this is how you try to combine everything together. So this is what happens if you have three flavors. Instead of having just one mixing angle, you have a three-by-three matrix. You've all told you everything that there is to know about three-by-three matrices. As far as oscillations are concerned, these are three matrices, this three-by-three matrix is parameterized by three mixing angles in one complex phase. And what's written here is how we define the parameters of the matrix using mostly this canonical PDG parameterization. So for example, one phase is contained in the ratio of those two elements here. The other, sorry, one mixing angle is contained here. The other mixing angle is the ratio of those two elements here. The third mixing angle is the magnitude of this element, and then there's a complex parameter somewhere. And we like to put the complex parameter here, and then in these four elements here as well. Okay, so for doing three-flavor mixing, this is what we have to do. And the reason we don't have to reanalyze everything is because of this separation of scales that I told you about. And it turns out that the solar data is mostly explained by the mixing angle that lives here. The atmospheric data is mostly explained by the mixing angle that lives here. And this very rare oscillation of electron neutrinos at the short wavelength, that's described by the mixing angle that lives there. Now, we have to talk about the delta M squares because there are two of them. And we define them in different ways. But the definition is the following. And I'm gonna show pictures that look like this. So remember, I told you there's M1, M2, and M3. And I also told you that we don't know what those numbers are. And it turns out that just giving them names is not sufficient. You have to identify them in some way. Otherwise, this is just mathematics, okay? So we need to, I have to tell you who's one, who's two, and who's three. Or otherwise, we can always re-label things and everybody gets very confused. So the way that we do that is we define their order. And the way we define is we say that M1 squared is bigger, less than M2 squared. So here's M1, here's M2. By the way, it's kind of convenient to do this for the M squares because these are the numbers that we measure. And then there's neutrino number three. So neutrino number three is defined in such a way that it's either heavier than the other two, or it's lighter than the other two. In such a way that this difference here squared is bigger than that one in magnitude. So this is a definition of my states. It's a weird definition because it takes a long time to explain, but it turns out that these are two qualitatively different scenarios, so you should be able to tell them apart. And the reason we do this is because if we did it in any other way, our mixing matrix would be very strange. So we don't want to do that. Or the definitions of the mixing angles would change from one scheme to the other. This is referred as one neutrino mass ordering. This is the other neutrino mass ordering. This one here is called the normal ordering or the normal neutrino mass hierarchy. And this one here is called inverted. And you can ask me why if you don't figure out why this is called normal. And this is called not normal. And the reason is that in this case here, the neutrino masses seem to be ordered just like all the other masses that we know about. You know, the up quark masses or the down quark masses or the charged lepton masses. In this scheme here, it looks weird. So we think that this is weird and then this is normal. Okay, so that's the nomenclature. The reason we talk about this is because we don't know what the answer is. Okay, we don't know if the picture looks like this or if the picture looks like that. So if we take all of the data that we have, we can combine it into a very convenient table that looks like this. And this here contains the measurements of all of these mixing parameters if you take all of the data, except for some weird phenomena that I'm not gonna talk about that we don't understand yet. But if you take all of the data that we do understand, we can explain all of it. And we can actually measure almost all of these parameters ridiculously well for neutrino physics standards. So many of these parameters are measured at better than the 10% level. And some are approaching the few percent levels. Okay, which is kind of a very, very big deal. There are lots of things which are kind of entertaining to appreciate, which is the first mixing angle that we saw very, very clearly unambiguously was this one. And that turns out to be the one that we measure the worst, the error bars biggest. The last mixing angle that we saw, it's also the smallest one is this one. And it's also the one that we can measure the best. So it's one of those weird phenomena in physics. And here are the two oscillation frequencies that we can measure very well. And the key thing is of course, this table gets doubled up because I don't know if this delta M squared three two, which is the difference between this and that is a positive number or if it's a negative number because my experiments can't tell that either. Okay, so this is where we are with neutrino oscillations. There's still things that we haven't figured out. So one thing is people try to make a fit for the CP violating phase. And the reason that there are two pictures here is because we don't know what the mass ordering is. So if the mass ordering your analysis gives you one answer, if the mass ordering is the other one, you get a slightly different answer. But what we want to appreciate is the fact that what we seem to know about this delta parameter is that it's somewhere between 180 and 360 degrees at the few sigma level. But that's great improvement. We know it's not between zero and pi, which is something, but we still have a lot of work to do. And this is very, very recent data. And I do want to point out that the data are still in a state of flux. As new data comes in, the picture still changes sometimes relatively dramatically, especially when it comes to understanding the nature of the mass ordering and understanding the CP violation phenomenon. So one thing I wanted to mention is there was a preprint that came out yesterday from the NOVA experiment, which is this one. If you didn't notice it, you should have, it was the first one on happy EX. If you say, I don't care about happy EX, I say you should care about happy EX because it's the shortest one. So if you don't want to see too many papers, look at happy EX because it only has a few papers. And if you saw this one, this is most recent results from the NOVA experiment. The NOVA experiment is a long baseline experiment in the United States that's aimed at studying new mute to new reappearance at long baselines in order for you to study the small mixing angle, learn about the mass ordering, and maybe CCP violation. So to make a long story short, there's a lot of data. They're very excited about their new anti-nutritional results, blah, blah, blah. This is what we want to get to. This is what this experiment, which this is a paper from yesterday. On the left hand side, this is their measurement of the large delta M squared as a function of one of the mixing angles. And this is the mixing angle that we haven't measured very well, which is the atmospheric mixing angle. And you notice that their result is the black one, which is good, the regions are shrinking, but the error bar is still relatively big. I wanted to highlight the plot on the right hand side. This is their measurement of delta CP for the different mass orderings. And you notice that for the normal mass ordering, they can't measure delta CP at all. All values are completely allowed. As a function of this sine squared theta two, three, which is one of the mixing angles. And this is a result of that experiment. So the plot that I showed you just a minute ago, of course, does not contain these data because these data didn't exist before, but it does contain the collection of a lot of different data sets. So the way in which we're constructing this and constraining this is by combining all different kinds of data. And by the way, there's still the, this is a normal ordering. This is the inverted ordering. And they prefer this one here very slightly over this one at about the two sigma level. So the best fit points in quote are actually here, which is entertaining because here there's no CP violation. The CP violating phase is zero. So the message here is that we haven't really converged into a situation where the error bars are just shrinking, but the best fit points are still moving around, especially when it comes to CP violation. And the right answer will probably have to wait until better experiments come online. So we're gonna have a lot of fun with these parameters moving around, but it's gonna have to, we're gonna need a better experiment. Okay, so I wanna quickly summarize what it is we haven't measured yet in terms of oscillations. I've already talked about this. It's the CP violation. There's a fun question people like to ask, which is if you look at one of the mixing angles, you can ask if it's exactly equal to 45 degrees, a little bit less than 45 degrees or a little bit bigger than 45 degrees. Now I can tell you the physics of this if we stare at this picture here. So this is a picture of the masses squared. So that's the little cartoon I drew over there but with colors. What the colors mean, the colors represent the magnitudes of the mixing parameters squared. So for example, this is the number one state and the fraction of this bar that's colored red is a Ue one squared. So this is Ue one squared, Ue two squared, Ue three squared, which is tiny. And then if you look here, this is a U tau three squared and this is a U mu three squared. Now there's a mixing angle that tells you whether those two bars are the same size or not. And in particular, you can ask, is the neutrino number three more made out of mua neutrinos than tau neutrinos or is it the other way around? And we don't know what the answer is. And for some people, this might be an interesting question. For other people, it doesn't feel like a very interesting question. What sounds very interesting is imagine that the fraction of green and blue is exactly the same. That would be very weird. Whenever things look exactly the same, it usually smells like there's a good reason for it. So for people who like to build flavor models, this is something that they get excited about. Whenever there appears to be a symmetry that tells you that the amount of mua and tau neutrinos is exactly the same. So that's one thing that people get excited about. The other thing people get excited about is if you look at neutrino number two, which is this one, it seems to be very, very democratic in flavor. So it has the same amount of electron, the same amount of mua and the same amount of tau, roughly. People also get excited about that and maybe there's a symmetry reason for that. Of course, we always have to draw both pictures because we don't know what the right picture is. And finally, so these are the things we haven't resolved yet. But I do wanna step back and remind people of one thing which is, so we have this really great looking picture. But if you ask yourself, do we know that this is the right picture? The answer is we're not completely sure because we haven't done enough experiments yet. So when people talk about next generation neutrino oscillation experiments, they will often tell you that they wanna measure these numbers better. But what they're really trying to do is to test whether the formalism is correct or whether we're missing something, okay? So this is very important to keep in mind. So we're not just in the business of measuring oscillation parameters better and better as time goes on. We're in the business of figuring out if we get the physics right or not. And there are lots of ways of doing that but the idea is always kind of the same and I think that Yuval will tell you about this. When people were constructing the CKM matrix, they weren't sure that the CKM formalism was the right picture or not. So how do you check if your idea is correct? One way to check that is that your model has a finite number of parameters in it and we can measure those parameters. And the nice thing is once you've done that you can make predictions from more experiments. So another way of saying this is that if we measure the same parameter twice using the different experiments, if we get the same answer, that means that the model is right. If we get different answers, that means we're missing something. And oh, I don't have that plot yet. So I'll come back to this. I do wanna mention very quickly a couple of things about how we plan to measure things. One is we wanna figure out which picture is correct and how do we do that? There are lots of different ideas. The nicest and simplest ideas to use the matter effects that I was discussing earlier today. But remember what's different between those two pictures is where is the electron neutrino? So if you look at the picture on the left-hand side, the red, which is the electron part, that's mostly in the lightest states. On the other side, on the right-hand side, the red is mostly in the heaviest states. So when you do oscillations through matter, that should show up somehow. And in order to see this, we can go back to the, oh, okay. I wanna kind of rush through something because I wanna get to the end. So that's why I'm talking a little bit more quickly. You can ask all kinds of fun questions. Why don't we know the mass ordering yet? I think it's worthwhile to pause. And first of all, just to stare at this for a minute and appreciate the fact that those two pictures are very, very different, okay? They look kind of the same, but they're actually really different. So for example, in a world where the lightest mass is zero, in the left-hand side, one mass is very small, zero, the other mass is lighter, and then the other mass is heavier. For the picture on the right-hand side, even if the lightest one is zero, the other two masses are almost exactly the same. It doesn't look like it this way, but if you made a plus for the masses, if you want the differences of the masses squared to be very small for something that's heavy, those two heavy things have to be almost degenerate. So the physics can be very, very different. I mean, if you look around for all the masses that you've heard about, there are no two fundamental particle masses that are almost exactly the same, right? So is that, do we agree with that? You know, they're all very different. Now on the other hand, we know of many, many things that have masses that are almost the same. The proton and the neutron have almost exactly the same mass. The pions have almost exactly the same mass. The K plus and the K zero have almost exactly the same mass, and we're no longer surprised by any of that because we know why those masses are almost exactly the same. So imagine that I told you that the neutrino masses are almost exactly the same. We would be tempted to believe that that means something, probably important, okay? If it's something on the left-hand side, maybe it's not as important. So that's the idea. The reason we haven't measured this yet is because we have never done an experiment where you can see the two oscillation frequencies at the same time, nor have we done an experiment where the matter effects are important for the fast oscillations. So this is what we're planning on doing, and that's what the NOVA experiment is designed for. So the idea is, again, the same picture. If you have one mass ordering, your oscillations will look like the blue curve, for example. If you have the opposite mass ordering, the oscillations will look like the black curve, and then if you exchange neutrinos and anti-neutrinos, they're gonna behave like the opposite of those two pictures. Okay, the flipping is the other way around. Okay, so I don't wanna talk about that much more, but that's how we think we're gonna be able to do this, and in order to do this, you need an experiment with a lot of matter effects. And that's why the NOVA experiment is highlighted as interesting, because matter effects in NOVA experiment are about 30%. And the other computing experiment, which is in Japan, the matter effects are tiny, they're less than 10%. And in the next generation experiment in the US, they're hoping to have matter effects which are of order 100%. So it'll be qualitatively better. So I wanna say a few words about CP violation, which you've always gonna tell you about. What's cool about CP violation is that we have seen CP violation in the quark sector before in lots of different places. And all of it, all of those phenomena can be parameterized by a single parameter in neural Lagrangian, which is the phase in the CKM matrix. Now we know neutrinos have mass, so we know the leptons mix. That means that there's a new parameter that violates CP in the theory as well. So this is our first opportunity to see a new manifestation of CP violation coming out of the standard particle physics model. So it is a phenomena that we haven't seen yet, but we're very excited about that. And I have some comments about the theta parameter in QCD that Yuval has not told you about, but you can ask him. And that's something that we don't understand, but it's a very, very tiny parameter. So I wanna skip this part. And I wanna tell you a little bit how CP violation works in neutrino oscillations. So what we would like to do is to study new mu to new e oscillations. And then we wanna compare that with the CP conjugate channel, which is a new mu bar to new e bar. So in principle, if you measure these two oscillation probabilities, and CP is conserved, you better get the same answer. If they're different, then CP is violated. Now technically it's a little bit tougher than that because we know that the matter effects will do exactly the same thing. They will make the new e's and the new e bars oscillate slightly differently. So if we do our analysis and matter effects are important, we have to be able to separate out those two effects. But forgetting that, if you wanted to calculate your probability for new mu to new e, that probability would have an amplitude that looks like the thing on the top there. What I mean by amplitude is if you wanna calculate the probability, you have to square that number. If I do the same thing for antinutrinos, you get an expression that looks almost exactly the same, except that the couplings are complex conjugated. By the way, that's a good way of reminding yourself why you need complex couplings to see CP violation. The idea is very simple. When you calculate a diagram for some particle process, you have some couplings. When you calculate the CP conjugated diagram, the couplings all show up as complex conjugates of one another. So if the couplings were real, you have no hope of getting those two things to be different. If the couplings are complex, you have some hope. But that's not enough, you just have some hope, but no guarantee that you can see something. So the question we wanna ask ourselves is, can this complex number here squared not be the same from this complex number here squared? So is it possible, even in principle, for those two complex numbers squares to be different? And the answer is they can be different, but lots of things have to happen. And one thing that has to happen is if you stare at these expressions, they both look like the sums of two terms, right? So if one of those two terms is zero, then the squares of those two things are the same. So you don't get any CP violation. That's one comment. So this means a couple of things. One is if you look at those two terms, they both have an oscillatory thing that goes like e to the i delta minus one. And the delta here means delta m squared L over E, over two E. So what that means is if the deltas were very small, one of those two terms would be very small so you wouldn't see any CP violation. That's one constraint. The other one is if one of the mixing elements were zero, you also have nothing that you can see because one of those two contributions to the amplitude would vanish. And finally, you need the entries of your mixing matrix to be complex, otherwise you don't get the CCP violation either. So in terms of our experiments, it means lots of things have to happen. As far as setting up your experiment, it means that you want your baseline to be very long because you need both of your oscillation phases to be non-zero in order to CCP violation. That's kind of a requirement. And that's what we're doing as far as planning experiments. So we know what the experiments have to do. We can't control what the value of the physics parameter is, but we can control everything else. And that's what we've been working on. Okay, so I don't wanna talk about that. So I do wanna talk about a couple of things that are not oscillation related in the next 15 minutes or so because these are all interesting questions that we're spending a lot of time on. So one thing that I have said is we can measure the differences of the neutrino masses squared but we actually don't know what the neutrino masses themselves are because again, oscillations are not sensitive to that. So if you wanna answer this question as to what are the masses themselves, first of all, you can convert that question into a simpler question which is, what's the mass of the lightest neutrino? Because if you find that one and you know the differences of the masses squared, you can construct all the other ones. And I do wanna remind you also that there's a lot of physics there. If the mass of the lightest neutrino is super light, that could be interesting. If the mass of the lightest neutrino is relatively heavy, you might be in a situation where all of the neutrino masses are almost exactly the same and that again would be super interesting as well. So we wanna know what the answers are. And the cleanest way of doing this is to see a kinematical effect of a neutrino mass. That's what we would like to see. How do we do that? It's super hard because the masses are very small. Our best hope today is to study beta decay. And in particular, people like to study things like a tritium beta decay for lots of technical reasons that I'm not gonna tell you, but you can ask. And the idea is if you look at the spectrum of the electron that comes out of beta decay, that spectrum cares about the neutrino mass because it changes the kinematics of the electron. One thing that happens in particular is if you ask yourself out of that reaction there, what's the largest energy that the electron can have? That largest energy depends on what the neutrino mass is. If the neutrino mass is zero, the largest energy is something. If the neutrino mass is not zero, the largest energy is a little bit less than that something because you have to spend some energy to produce the neutrino mass, okay? So it's that very clear because this is very simple and should be very clear. I wanna show you a picture. So this is what the left hand side is a beta spectrum. That's what it looks like in some units. And if you zoom in on the very, very end of the spectrum, you have two spectra there. And one spectrum is what you get if the neutrino mass is zero. And the other spectrum is what you get if the neutrino mass is not zero. I have to pause and you can ask, what is that neutrino mass? The answer is that in practice, it's none of the neutrino masses, but it's a mixture of them. It's a linear combination of the masses that looks like that. So it's a weighted average of mass number one, mass number two and mass number three. And the weighting factors are the elements of the mixing matrix squared. And it's easy to understand why that happens because beta decay only has to do with the electron neutrino or with electrons. So you're always speaking to diagrams that look like this and you can either produce a neutrino number two or a neutrino number one or a neutrino number three and the weights of that are these elements squared. So again, the electron neutrino is not a particle, it doesn't have a mass, but I can define something that is often called the electron neutrino mass, which is a weighted sum of the neutrino masses like that. So again, if the mass looks like this, that's how the spectrum ends. If you've heard this story before, I wanna tell you how this measurement is done and how it's not done. So one feature you might notice is that the spectrum ends at a different point. So if you could measure where the spectrum ends, you would be able to measure the neutrino mass. And that's not how the experiment is done because we actually don't know where that point is very well anyway. At least not at this precision. So this is not a measurement of whether the endpoint is in the right place or not. This is a shape measurement. The thing you want to appreciate is that the shape of the spectrum in the red case and the blue case are different. So you can make a shape measurement to determine that. Okay, so that's how the measurement is done. Now you can ask yourself, why is that measurement very hard? And the reason is the curve on the left-hand side. So if you look at the region of the decay space where you actually care, which is where the neutrinos are doing something, that region, according to this plot, is less than 10 to the minus 12 or less than 10 to the 12 of all of the decays. But that means you need 10 to the 12 decays to get one electron that happens to be in that region there. So you need a lot of decays. And that makes for a very, very hard experiment, which I'm not gonna talk about, but you can ask me about it. There's something called the catrin experiment. This is a piece of the catrin experiment. It's a spectrometer. And basically the challenge that you have is the following. So I need a lot of decays, right? So here's my tritium source. It's spitting out electrons. Remember, I need to pick up all of these electrons. And I wanna measure their energy very well. And I really, really, really wanna make sure I don't lose any of them. So all of these electrons have to be accounted for in a way that I have a lot of control. And I have to measure their energies very, very, very, very, very well. So in order to do that, you use this capture solenoid to take your electrons, transport them into a region where you apply an electric field and this electric field makes the energy measurement for you. It's actually a barrier experiment. So what you do is you have your electron coming this way. You put an electric field going this way. Only energies above something will make it past the barrier and everything that's below that energy will not make it. And then if you can change the height of the barrier, you can actually do an integrated measurement of your spectrum for different values of this threshold. So basically what you can do with this curve is you can calculate an integral of this curve from some point on and then you can change the initial point and that gives you this measurement of the shape. And that's the Catherine experiment. You're gonna hear lectures about dark matter and people will talk about cosmology as well. So cosmology is a very good way of also learning things about neutrinos. In particular, we believe that there's a lot of leftover neutrinos from the Big Bang and those neutrinos have a huge impact on the time evolution of the universe. And that means that the universe cares if the neutrino mass is zero or not. So there is information out there mostly in the large scale structure of a matter. So if you look at how galaxies are distributed, the distribution of galaxies or the power spectrum of that cares about whether the neutrino mass is zero or not. I'm not gonna tell you that story because it's a long story, but the idea is by taking these cosmology measurements we can place a bound on the sum of the neutrino masses and the sum of those neutrino masses is known to be less than about an electron volt. All right. So I don't wanna talk about this but people can ask me about that later. So the last thing I wanna talk about is whether we understand the nature of the neutrino. And I think this will probably be the last thing that I'm gonna talk about because I'm running out of time but this is a big deal and it's something that we're not familiar with. So the question we're asking is is the neutrino what's called a Myrona fermion? And you've all heard about this before. There are different kinds of fermions. You can either be a Dirac fermion or you can be a Myrona fermion. What this question is really about it's about degrees of freedom. And one thing that we're very familiar with is that we're very, very familiar with the electron. So let's talk about the electron and I'm gonna allude to stuff that you've heard about before. We know that the electron particle is associated with two different fields. The left-handed electron field and the right-handed electron field. From a particle perspective, the left-handed electron field is responsible for left-handed electrons and right-handed positrons. So we're all very familiar with that. Of course, the right-handed electron field is associated with right-handed electrons and left-handed positrons. So if you count all together, there are four degrees of freedom. When it comes to the neutrinos, what you learn about from Yuval's lecture is that there's a left-handed neutrino field and the left-handed neutrino field is responsible for a left-handed thing, which we call the neutrino, and a right-handed thing that we call the anti-neutrino. This is all fine and good until you run into neutrino masses. And the same is, by the way, true about the electron as well. If the electron didn't have a mass, we could be talking about the left-handed electrons and the right-handed electrons as totally different particles and we would never get them confused with one another. But because the electron has mass, those two different particles get mingled together and we talk about them as one particle. So let me describe what the idea is in a little bit more detail. So the idea is imagine that you're an experimentalist and you're doing experiments and you discovered a new particle. It's the left-handed electron. So you're very excited, you discover a new particle and now just by staring at the left-handed electron, you can conclude lots of things. And the one thing you can conclude instantaneously is if you discover the left-handed electron, a right-handed positron has to exist as a particle. It cannot not exist, okay? And the reason for that is the CPT theorem. The CPT theorem says that if a left-handed electron exists, the right-handed positron has to exist as well. And from field theory perspective, it means that those two particles are associated to the same fundamental quantum field. Now if you keep doing experiments with your left-handed electron, you find out that there's also another one which is a right-handed electron, sorry, let me flip that. You keep doing experiments with this particle and then you discover that that particle has mass. So if you make that discovery, you also can instantaneously conclude that the right-handed electron also exists as a particle. You don't even have to do the experiment. The reason for this is very simple. It's this picture on the left and I think you've all already mentioned that. If your particle has mass, you can change the helicity and the helicity is reference frame dependent. Another way of making the same statement is you can go to the particle's rest frame and in the particle's rest frame, all that I'm saying is that if you see a particle with a spin up, you know that you can also see it with spin down and it would be very weird if all the particles only existed in spin up because you can always turn upside down and then the particle looks like it has spin down, okay? So because the electron has mass, all of those four particles have to exist because again, once you know that the right-handed electron exists, the CPT theorem says that the left-handed positron has to exist, okay? So is everybody fine with that argument? Let's do the same argument with neutrinos. We know that left-handed neutrinos exist. That means that right-handed anti-neutrinos also have to exist by the CPT theorem. Now I have discovered that neutrinos have mass. That means that for every left-handed neutrino, there has to be a right-handed friend that combines with it to give it a mass, okay? So that's the same argument that I made for the electron. What happens in this case is that the neutrino is special, is that there is a choice you can make, which is this choice here, which is I already have a left-handed thing and a right-handed thing. Why don't I make the right-handed thing that I already have be the mass partner of the left-handed thing that I already have? And that's a choice that I can make. And this was not a choice that I had with the case of the electron, okay? Because in the case of the electron, the CPT theorem told me that my left-handed electron was a partner of the right-handed positron, okay? And when I do a Lorentz transformation, my charge doesn't change. So there's no way I'm gonna confuse a right-handed positron with a right-handed electron. They have to be different things. The neutrino is special because it doesn't have any electric charge. So because the neutrino doesn't have any electric charge, it is possible that the Lorentz partner of the left-handed neutrino field is the same field that we had before. We don't have to invent the right-handed neutrino field necessarily, okay? We can, it's a choice that we make. And if we do invent the right-handed neutrino field, then the neutrino is just like everybody else. It's what's called the Dirac fermion. And if I don't invent the right-handed neutrino field, I only have two degrees of freedom. The thing that we normally call the left-handed neutrino and the thing that we normally call the right-handed neutrino are sufficient to fully describe all of the kinds of neutrinos that exist. So basically the Majorana versus Dirac question is basically asking the question, how many degrees of freedom does it take to describe a neutrino? Is the answer four or is the answer two? Now it's very peculiar that we don't know the answer because four and two are qualitatively different numbers. One is two times bigger than the other. But you can ask yourself, why don't we know the answer? And the answer is that the neutrino mass is very, very small, super small. And the point is, if the neutrino mass had been zero, exactly zero, then this Majorana or Dirac question doesn't exist. It's a non-question. For a massless fermion, you can't ask if it's Majorana or Dirac. Because if it is in quotes Dirac, it's Dirac partner doesn't couple to it. It's a totally different particle. It's kind of like asking, so I see an up quark, does the charm quark have to exist? And the answer is I don't care, it's up to you. One is not related to the other, okay? So this is the point, if the neutrino mass were zero, you can't ask the question if the neutrino's Majorana or Dirac. What this means is, if you can't ask the question, you better not be able to get the answer, right? You can't get the answer to a question you can't ask. Is that clear? So what this means is, if you can come up with an observable that can answer the question, that observable better fail when the neutrino mass is zero, which means that the probability for your observable is proportional to the neutrino mass in some way. But that means, again, if you invent a way of answering the question as a neutrino Majorana or Dirac, the answer has to be I don't know when the neutrino mass is zero, which means that the answer has to go to zero or it has to approach I don't know as the neutrino mass goes to zero, okay? So that's the key thing that you wanna remember. It's because the neutrino mass is so small and again, so small means a lot smaller than any energy scale of the physics problem that you have, of the physics observable that you have. So this is the answer. Now, what's the other way of getting the answer? The other way of getting the answer is that Majorana fermions have lots of interesting properties. Again, the way that you wanna think about them is that they're self-conjugate particles. The antiparticle and the particle are the same. And the reason I don't like to say this is because people get confused because it is true that the neutrino and the anti-neutrino are the same particle but there's still two different helicity states. And because the mass is so small, the two different helicity states usually don't mix at all and I normally call one of them the neutrino the other one the anti-neutrino. So even in the Majorana case, it kind of makes sense to use this nomenclature even though it's not correct. But the thing that you wanna keep in mind if the fermion is a Majorana fermion is that it is a self-conjugate state and what this means is it can't have any quantum numbers of any kind. Okay, so this is very important and it can't have any unbroken quantum numbers, by the way, so keep that in mind. And in particular, the neutrino seem to have an accidental quantum number which is called lepton number. And what happens is neutrinos produce charged leptons with negative charge, anti-neutrinos produce charged leptons with a positive charge. Experimentally, that's always true and that's associated to a conservation law called lepton number. So if the neutrino is a Majorana fermion, it better be that this conservation law is not exact. It has to be violated in some way. So if you give me just a couple of minutes, I wanna give you an example and then show you what the observable is that people like to talk about. And then I'll stop. So let me talk about a gedonkin neutrino experiment which is the following. So let's say I have a physics process where an electron hits something and produces a neutrino. I wanna describe that neutrino in the laboratory and I wanna describe it in terms of the helicity state of the neutrino that comes out. The state that comes out is mostly left-handed but if the neutrino has a mass, there's a tiny probability that you get a right-handed neutrino as well or a right-handed state as well. So hopefully this is clear for everybody. So imagine now that that object is gonna propagate and then it's gonna hit something. When it hits something, it has a probability of behaving like the left-handed state and that's the normal neutrino that we like to talk about and that object likes to produce electrons. So that's what happens most of the time that you produce an electron. Now imagine that the neutrino chooses to behave like the right-handed helicity state. So if the right-handed helicity state hits something and the neutrinos are the rack fermion, the right-handed helicity neutrino has no interactions virtually or to be specific, the right-handed chiral neutrino field has no quantum numbers so it doesn't do anything. So if the neutrino chooses to behave like the right-handed state and the neutrinos are the rack fermion, nothing happens. If the neutrinos are myerone fermion, this right-handed helicity state behaves like the thing that we normally call the anti-neutrino, which is of course right-handed. That means that if the neutrinos are myerone fermion, if it hits something and behaves like the right-handed helicity state, it will want to produce a positron. So if the neutrinos are myerone fermion, there's a probability that you start out with an electron in the initial state in the very beginning of the process and you end up with a positron. And that's how you check if the neutrinos myerone are the rack. Now the catch is that the probability for that to happen goes like the neutrino mass divided by the energy squared. So that's the problem. And that's why we don't do this experiment because it's a very, very hard experiment. What we do hope to do is a variant of that experiment, which is a nuclear physics experiment, which is here. So I think this is the last thing which I'll say, and I might say some things in the discussion as all the people ask. So we can look for a nuclear physics process that violates laptop number. And this is called a neutrino less double beta decay. It's a very, very rare, or it's related to a very rare nuclear physics process, which is the following. Imagine you have a nucleus, and somehow two neutrons decide at the same time to beta decay. So you have two neutrons, they convert into protons, they emit two electrons and two anti-nutrinos. This is called double beta decay. You can ask yourself, why would a nucleus choose to do that, which is super hard? The answer is because sometimes it doesn't have a choice. It's the only thing it knows how to do. So it will do that, and the lifetimes are ridiculous, like 10 to the 24 years or something like that. But people have measured that. They actually calculated that in the 50s. But if the neutrinos amyron affirmion, I have a different choice, which is, I can have my nucleus change to a different nucleus and emit only two electrons, but no neutrinos. That's kinematically allowed, and the only reason it doesn't happen is because of laptop number conservation. So laptop number forbids this from happening because your initial state has zero laptop number, your final state has two units of laptop number, and that breaks laptop number by two units. If the neutrino masses are the dominant contribution to this, the Feynman diagram that contributes to that is easy to understand. It's a WW scattering into two electrons via neutrino exchange. This clearly violates laptop number, and it only happens if the neutrinos amyron affirmion, and the last thing which I'll say is because we know what the neutrino masses are, if we postulate that the neutrinos are myron affirmions, we can calculate the rate for neutrino less double beta decay. So that means that by looking for this process, if we don't see a rate which is consistent with the neutrino masses, it would mean for us that the neutrinos have to be the rack fermions or that life is more complicated. And this is what's captured on the plot on the left-hand side. This is expectations of the observable to which neutrino less double beta decay experiments are proportional, and the experiments that we are performing right now, they are moving this vertical or this horizontal gray line downwards, and they're starting to get into the regime where we expect to see something if the neutrinos amyron affirmion, and for technical reasons, it's much easier to do that if the neutrino mass ordering is inverted. So there's a big campaign going on in the world to try to improve our sensitivity to neutrino less double beta decay, and we hope that in the next 10 years, the story might be qualitatively different, but it's gonna take 10 years. It's not gonna happen tomorrow. Okay, so I think I should probably stop here because we're gonna run out of lunch at some point. Thank you. Thank you very much.