 Then other people, they're sleeping and have all these nightmares. Yeah, I guess we should start. Okay, let's start. Let's start? What do you think? Okay. Thank you very much. I definitely appreciate you waking up so early just to come to my class. And I really appreciate all of you who forced me yesterday to get into my dancing stuff. It was really fun. I enjoyed so much. The only problem was that I was, I can hardly walk. So let's see. The worst case scenario, I had a plan B. I will sit here in the big chair and we will talk. But let's start and hopefully it will be okay. So what I want to keep going and talk today is about flavor physics. And what we did yesterday, we defined the standard model. Let me ask you a wake up question in the morning. So what is the symmetry group? Okay. I know it's an easy question. In the morning you have to ask easy question like, how are you? Okay. So you one, why? Okay. And then we have the fermions. They have a very cute name. You remember? They're all over there. It's called the Cudel. Cudel. And then we gave them some number and then there was one Higgs. And there's three of the Cudels. And then we basically, what we did, we follow our procedure. We write the Lagrangian, the most general one and we start expending and we looked into the coupling interaction that the standard model has. And we have the following interaction. We have the interaction of the photon and the gluon. The interaction are diagonal and universal and parity invariant. So they are proportional to either E for the photon or G strong for the gluon and the parity invariant. And then we talked about, we didn't talk about the coupling of the Higgs. I'm just telling you that the Higgs coupling is also universal and universal. Let me say, I should have said universal mean diagonal but with the same value of the coupling. Well, diagonal mean that things are, there's no off diagonal term. There's no like UC coupling. But the coupling is not the same. So that's the difference in universal and diagonal. So the Higgs coupling is proportional to the mass of the fermion and it's diagonal but not universal. It's proportional to the mass of the fermion. So the heavier fermion cup is stronger to the Higgs. And then we show the coupling to the Z. We kind of did it and we find that the coupling of the Z is proportional to T3 minus q sin square theta w. And it's p-violating and it's universal. That is for all the quarks that are, for all the down type left-ended quarks, you have the same value here. However, it's not the same for left-ended and right-ended. And we can see it through the T3 because left-ended field, the T3 is either plus or minus half. But for right-ended field, this T3 is zero. So we see that we have p-violation. And then we talked about the w which we spend most of the time yesterday and we find that the part of the w is proportional to something like qi, vij, dj such that this v is the CKM matrix. And then we learn how to count. And we find that the CKM has four parameters, three angles and one phase. And we also find that the w couple only two left-handed. Very good. And then we went on and discussed the structure of the CKM and we find that the structure of the CKM is very roughly one lambda cubed, lambda one, lambda squared, lambda cubed, lambda squared one. And one thing that I didn't actually discuss yesterday that I should have is I didn't tell you what are the masses of the quarks. So I make a big deal about the fact that the CKM doesn't look like a generic matrix. It looks like they have a structure. It's very close to the unit matrix. But I didn't talk about the mass of the quarks. And the mass of the quarks, the mass of the up-type quarks are roughly five MeV 144 GV and 174 GV. That's the U, U, C and T. And for the down-type quark, it is roughly 9 MeV, 100 MeV and 4.2 MeV. And that will be the d, s, and b. And one thing that we see here is that actually also the quarks, the masses of the quarks are not generic. They're not generic in the form that you see there's a very big hierarchy between them. And we don't know what to expect if you just, if I tell you give me six number, I expect you to give me six number that are distributed evenly. And somehow you see here that are very, very different. Here you have 5 MeV, 1 GV, almost a factor of 1,000 and almost 200 GV. So that's another factor of 100. And in the down-quark, there's a factor of 10 here and another factor of maybe 100 or something like this. So you see that we have this really non-generic structure for the masses of the quarks. OK? Yes? 4.2 GV. GV? Thank you. Anything else that I missed? But actually, before I go on, let's actually discuss a little bit. How do we even can talk about masses of the quarks? Does it make sense to talk about the mass of the quarks? OK. So you have three options. You have 2.7 GV. Does it make sense? You should say yes, no, or it depends. How many people vote for yes? Nobody. Nobody. Yes. OK. One person who thinks it makes sense. You should have said it makes sense because I write in them. So I wouldn't write it if it wouldn't make any sense, right? How many speakers say it's totally garbage? I learned it. I actually, when I was in the UK, in the flight to the UK, the flight attendant, she said, rubbish. You know the difference? And I said, oh, in America, they say garbage. And she said, garbage is rubbish. It's such a nice word. Anyway, so how many things that it depends? Good. OK. Very nice because usually that's the answer to everything. It depends. It's kind of a good answer. OK. So what does it really mean to talk about the mass of the quarks? It does make a lot of sense in the following sense that when we think about the mass of the electron, how do we measure the mass of the electron? We take the electron to infinity. We isolate it, and we measure the mass. And the idea is that we measure the mass because when it's got to infinity, there's no interaction. And actually, all the energy can be attributed to the mass. However, we cannot really measure the mass of the electron if the electron was part of a bound state because then you have to take into account the binding energy, and you cannot say, oh, that's binding energy, and that's the mass of the electron. And we know that the mass of the hydrogen atom, for example, is less than the mass of the proton plus the mass of electron by 13.6 electron volt, which is the ionization energy of the hydrogen atom. But for the hydrogen atom, we say yes. I know what to do. I take my hydrogen atom, take the electron to infinity, which is like 1 centimeter away, and take my proton away, and I measure the mass. But for quarks, I cannot do it. Why I cannot do it? Why I cannot take my quarks away? Because of confinement. So what do I really mean by measuring the mass of the quarks? Moreover, I even told you that at the infrared, there's no meaning to talking about quarks. Quarks only exist at the UV. So if something only exists at the UV, clearly I cannot take it to infinity and measure the mass. Therefore, quarks only exist when there are actually a lot of interaction around them. So why are I still talking about the mass of the quarks? So one way to go, which I like to think about when I think about quark masses, is that it just represents some result of some experiment. And that's the way we do physics. We do one experiment, and using the result of one experiment, we can make prediction in the other experiment. So there's just some parametrization of the result of one experiment, and using this parametrization and can use it in another experiment. OK? And that's for me the best way to think about it. Another way to think about it is I say yes. Masses run just like any coupling in nature. Like we talked about the fact that the coupling, the electromagnetic couplings run. Also the mass runs. The measurement of the mass depends on the scale that you are doing it. Now usually, when we measure the mass of the electron, we mean the scale is zero. We measure the mass of the electron when it's far away so there's no scale associated with it. When we measure quark masses, we measure them at a scale that is very high, because we want to measure them at a scale far away from the confinement. OK? So these masses were measured at 2GV, except the mass of the heavy quark that we measure at the mass. So the mass of the top was measured at the energy of the top. And the mass of the B was measured at the energy of the B. But for the UCDNS, it was measured at 2GV. So we want to understand that these masses are the mass at 2GV. And these masses at 2GV basically have nothing to do with the mass, the way we think about mass in the usual sense that, you know, you have particles with the mass and it's the react to gravity and all this. It's just a parameter in my well-aggrandian that we can measure. Yes? So I'm not going to get into too much details of actually how we do those measurements. And let me say one more thing that you will not be surprised that these masses are totally depend on your regularization scheme. So these are MS bar masses. And then you can define another regularization scheme, some HD, HD bar, pole masses, and you get different answers, okay? So they are totally unphysical, okay? And the way we measure them, so the mass of the B, there's so many ways to measure it. And it depends, you can take the Upsilon. The Upsilon is a BB bar, a bound state, and divide by two and define this as to be the mass, or you can actually look for some resonances. There's many ways. As I said, the important thing is that you understand how you take one measurement and apply it to another measurement, okay? But at the end of the day, what we learned, and that's what I wanted to say, is that we see here there's also very non-trivial structure, okay? And one way that kind of, if I tell you that I just randomly generated number and I look onto this number, I said, oh, if instead of generate the number, I generate the log of the number, then they maybe look a little bit distributed randomly, okay? So maybe the physics is not in the masses, but somehow the log of the masses where the physics is. So people write some model, they try to take the physics into the log, and the manual of those exist, but we don't know if they are, you know, how much physics they are in those. But one other thing to mention is that both the CKM and the masses are related to the Yukawa matrices, right? Because these are the, they are the eigenvalue of the Yukawa times the VEV, and these are, they are related to the mixing angles of the Yukawa, the mismatch between the two Yukawa's, right? And one thing that actually come naturally is that if you have a matrix and, let me back up a second, three by three matrices are so much more complicated than two by two, but two by two by that stage of your career, you should really have all the intuition of two by two. You look at the two by two matrix and immediately you kind of understand what's going on. So in a two by two matrices, okay, I want to ask you the following question. What is the relation between the, generally, for a generic matrix? If you have very different eigenvalue, what is the, in general, the mixing angle? And the answer is that in general, when I have a matrix of the following, something like, let's do it like zero, epsilon, epsilon one, okay, you look into this matrix. What is your intuition? Very quickly, what are the eigenvalues? Roughly, very roughly. That's what the wake-up call. You all have to know when you see this, okay, we talked about it yesterday, like during the dance, but because of the noise, nobody helped me. I wanted to explain that to first order, you look at this, to first order how it looks like, zero and one, because epsilon is small. So to zero's order, you say it's diagonal, so to leading order, the eigenverses are zero and one. And then you say, well, but it's not really zero because the determinant, what is the determinant of this? Is epsilon squared? And the determinant is a product of the two eigenvalues. So you say, since one of the eigenvalues is one and the determinant is epsilon squared, the eigenvalue should be epsilon squared and one, okay? Another way to look at it, you say the trace is one and the determinant is epsilon squared. So I have a sum of two numbers is one. The product is epsilon squared, so there should be one and epsilon squared, roughly speaking, okay? So I say that it's epsilon squared and one. Do you see it? It's all those things that we should just kind of get used to. And what is the mixing angle? What is the mixing angle that take to diagonalize this? I know it's early in the morning. I know, I wake up before you guys, I promise. I will prepare the lecture, okay? Since sunrise, the first sunrise come to my room, I wake up into FCN scenes, okay? So how do we, what is the mixing angle here? You should do it, you should do it, yes? Epsilon, very nice. You see that this is epsilon away from diagonal? Do you see it? Yes, so it's epsilon. And actually there's some, the formula tells us that tangent to theta is equal. Let's write it like this. A, let's leave it. For a generic matrix, A, B, C, C, okay? Tangent beta is 2C over A minus B, okay? So those are formulas that you just need to remember, okay? So you kind of get the intuition. And in particular where C is much smaller than A over B, you get that theta is equal to C over A minus B. So in this case, theta is equal to, theta is about epsilon, okay? So one thing I want to say is the following thing. When you have hierarchy, when you have hierarchy of eigenvalues, it's come very naturally with small mixing angle. That's a general statement. It's not a theorem. There are examples that violate this idea. But the generic feeling is that large hierarchy in the eigenvalues come together with very small mixing angles. That's the general statement. And that's what we see in the CKM and the Quark masses. We see hierarchy of masses come together with small mixing angle, okay? So somehow what we conclude is that somehow there might be some hierarchy in the Yukawa, okay? So the Yukawa is not generic. And the fact that the Yukawa is not generic give us this kind of structure. Okay. So that's basically just kind of, wow, wow. 15 minutes and I already reviewed what I was saying yesterday. So let me go on. And before I go on, I want to touch upon some, it's almost vocabulary. But when we start doing flavor physique, we start using meson and hadrons. And what I find out is many times just remembering the names of the hadron and what really they mean. It's a little hard so I want to very quickly review and remind ourselves what are those hadrons. And I know that many of you know it very well. But let me just remind you. So when we are talking about now, I want to kind of review what we have at low energy QCD. What's happened for the, yes. So actually I found all six to be out of place, okay? And that's really the important thing. There's nothing in the structure all together that tells you it's interesting. And one way to look into those and again, it's become same philosophical. I said I expect the Yukawa to be all the one and therefore there's only one that is in place, which is the top and the other five are not in place, okay? But then usually you say, well, you know, we kind of say the majority is what we call the use to it and then this looks like exception. So it's very, it's kind of philosophical. But the point is that you see some structure and it's interesting and people spend a lot of time thinking about why there is, okay? So I want to talk about the structure of low energy QCD and we talked about the fact that we have confinement and then when we have confinement, what emerge are set of hard runs and we distinguish between two kinds of hard runs, which we distinguish between what we call resonances, a resonance and a stable and a stable meson, okay? And in the PDG, if you don't know if something is a resonance or no, a resonance appear with a number in parentheses. For example, when you open the PDG, you see something like this, okay? Or you can see something like this. So what does it mean? What does it mean, row 770? Row 770 mean that I have a resonance with a mass of 770 MeV and this resonance, we call it resonance because it could decay via QCD interaction. It's decayed by emitting a pion. This is something that QCD allowed it to happen and the width of those resonances are very, very, very big, okay? The width of the row, anybody? I know it's some trivia, but maybe yesterday after the dance, you opened the PDG. Anybody after the dance opened the PDG? Nobody, yeah? Just me. I couldn't fall asleep, so I opened the PDG. So the width of the row is like 150 MeV. It's a very large width. 150 MeV is about a quarter of a quarter or a third than the mass, okay? That's the meaning of a resonance that it's very, very wide, okay? And what is a stable particle? A stable particle in the PDG appear without a number. For example, a K or a B or a D which are actually all the ones that we care about or a pi. And the width is much, much, much smaller than those, okay? So the width of the K is about 18 orders of magnitude smaller than the width of the row. And in the standard model, we understand the following. QCD tells us that those mesons are stable, why? Because they are made of the lightest, they are the ground state of QCD with this specific quark. And it's only the weak interaction that make this quark to decay, okay? Now, the name is a little bit misleading because we say it's a stable particle, but the B, for example, the B have a lifetime of a picosecond. So picosecond is not very stable in everyday life, right? And the K-on depend which K-on have a lifetime of order of 10 to the minus 8 second. It's not very stable. But it's stable compared to those ones, okay? It's much, much, much decay much slower than this one. So when we talk about the weak interaction and when I care about the weak interaction of the quarks, all I care about is the weak interaction of those mesons, okay? So those resonances to leading order when we do flavor physics, you don't care about them, okay? Of course, we care about them at some point, but to leading order, we care about those quarks, those mesons only, okay? So when I try actually to do measurements and understand how I actually measure the interaction of the quarks, all I have to do is to look into the equivalent of the mesons. And now comes the big question and that's the huge question. And the question is how do I know that I deal with this crazy QCD bound state? How I related this crazy QCD bound state to the quarks, okay? So the way we do it, we use what we call the quark model. And the quark model is just a model. It's not based on any truly fundamental theory, but it's still we have quite a lot going on with it. But we use the quark model and we say, the quark model tell us what quarks make of those mesons, okay? So for example, the cone is made out of someone. Someone tell me what the K is made out of. Let me do K plus. The K plus made out of, it's made out of an S bar and a U. You see that this charge of the S bar and the U add up to be a charge of plus one. And this one have a strange quantum number of minus one and that's what we, or plus one, that's what we have for the K plus, okay? So you say the K plus is made out of S bar and a U. It's a very complicated bound state. That doesn't make a lot of sense to talk about it as a bound state of two quarks. But what I care about is that it's have the quantum number of these things, okay? Under QCD, it's have one S quantum number and one non-S quantum number. So then this K on decay. And this K on decay, for example, not for example, most of the time this K on decay too, anybody, sorry? Two pions and two pions. Anybody, nobody opened the PDG. What the K on the K plus decay too? So the K short decay two pions. The K short decay into two pions. That's basically 100% of the time, okay? So the K short decay two pions, 100%. So the K plus decay to what? Yes, exactly which leptons? Very specifically. Muon and a neutrino. Wow, we really, we really need to go over those basic PDG stuff. Open the PDG please, okay? Today, you have some time, go to the beach, take your PDG with you and start looking on those decays. And so actually a K plus decay to Muon and a neutrino, 64% of the time. So about two-thirds of the time a K plus decay to Muon. Okay, it's a huge number. Anyway, the point is it's following, what I tried to make is it's following. How do I actually make the connection between the K on decay and the parameter in the Lagrangian? The parameter in the Lagrangian is the one that actually corresponds to this V us. Because if this diagram in the standard model, what's happened is that I have an S, the decay into a U, emits a W and the W go to the Muon and the neutrino. So this is the diagram that I have in the standard model for SDK. And the question, how this diagram of SDK is related to the decay of the K plus, okay? And you would say, well, I really know how to calculate decay of three particles. How do I know it? Because I study QFT, because people did it forever. And I know how to think a decay of three quark decay, okay? And when you study quantum filtering, not if you just, the very basic, even in this little, what I was telling you about quantum filtering, we say the way we are doing it, we think about a harmonic oscillator, the decay. But a harmonic oscillator just have, you know, there's just a kinetic term. So it's a free particle. So we really know how to, how free particles decay. But this S, it's bounded inside a kaon. And not only that it's bounded inside the kaon, I was telling you there's like no way to even think about it as an S quark. It's just some complicated ground, bound state of QCD. So how do I relate to this complicated bound state of QCD? That happened to have the same quantum number, the same kind of interaction as an S by an U, into the decay that I'm really after, which is this diagram, okay? So you see the question. The question is this following. When I do weak interaction of quarks, what decays are hadrons? And what I want to measure are decay of free quarks. But free quarks do not exist when I look at mesons. So the question is how I do relate the decay of the hadrons into the decay of the free quarks? This question is understood? Good. And so there's actually many ways to go. And one way is to do QCD parametrization, which is a very, very important method. This method is called parametrize our ignorance, okay? When you don't know something, what you do, you kind of look around and say, that's I know, that's I don't, that's I know, until you really, really know what you don't know, okay? So you parametrize what you don't know. So here I was just telling you a big deal. I said, how do I relate quarks to mesons, okay? But then you can actually do some algebra based on Lorentz invariance, based on some isospin, based on whatever other symmetries you have, until you come and say, that's exactly what I don't know. So for example, when I look at K plus going to mu nu, you can actually do those little tricks until you say there's only one number that I don't know. And this one number is called the K on decay constant, F sub K, okay? So that's very nice. You take something that you don't know and you walk on it until you say the output is something else that you don't know, okay? So you made a lot of progress, right? You take something that you don't know and there's something that you don't know. And then at the end of the day, there's one number that you don't know, for example, the K on decay constant. And then you spend a lot of time trying to actually calculate it. And for example, this specific number, the K on decay constant is now calculated on the lattice to amazingly good precision, okay? So sometimes you can actually use the lattice to get those numbers. Sometimes you can use some models to get those numbers. And sometimes you are smart enough to actually find some observable that this number cancel in some ratio and then you can still get some intuition about those kind of things, okay? And I want to emphasize the following. I think most of my physics career was developed to this one specific question, okay? I think more than half of my papers are related exactly to this question. And that's the question, how we overcome QCD in order to get sensitivity to the weak interaction, okay? And this is an extremely important question and we do not have a simple answer. We have a lot of little tricks but there's no one big idea. When we do quantum filter, we have one big idea, which is perturbation theory and Feynman diagram. So we do perturbation theory and Feynman diagram and that basically solves almost all that we need to do in QFT. When we come to this specific question of how we probe the quarks and we have QCD, this question is not very simple and there's a lot of little tricks that we are using, okay? And not only me, many people asking this question. However, I'm not going to talk much about it although I really like those kind of subjects. I want to go on. So to leading order, the answer is this following. To leading order, you look into some decay of mesons. You write them as decay of quarks you write the Feynman diagram of the quarks and you say there's also some QCD that someone can take care of and at this moment let's not worry about it. And that's a very good approach to don't think about QCD as of now. It will come back later but at the level of discourse, at the level of the introduction, don't worry and later on hopefully some of you are excited enough and I know some of you already work on such things and you ask the question of QCD, okay? So when we ask how do I measure VUS, I measure VUS by looking of this decay, okay? And if I ask you how would I measure VCD? VCB. How would I measure VCB? Can someone give me a decay that will be set? So, oh, I should have said. Everybody see that this is proportional to VUS, right? I have here VUS. So if I measure the rate of this process, I can measure VUS. I can extract VUS from this measurement. That's what I have in mind. So how would I get sensitivity to VCB? Yes? So which one is heavier? The C or the B? The B. So the B is heavier than the C. The C is 1.4 GV and the B is 4.2 and then I should actually look for a B decay and a B decay to what? To something that has a C and that's a D. The meson and then something like MUNU. So if you look into this, the diagram, the B, the B meson is made out of a D and a B bar and it can decay to a, let's do like this. I have a B plus. A B plus is, ah, it's a B bar and a U. That's a B plus and the B, the B bar is decayed to a C bar and a MUN and a neutrino, okay? And this one is proportional to VCB. So to lead in order to get sensitivity to VCB, I look for something like a B meson decay into a D meson decay and then I have to worry about QCD and a huge amount go into it until at the end we believe that we kind of understand everything that mostly what's going on and we get VCB. Good? We're good with our understanding of how we actually do those kind of things, okay? Yes? Good. So now, yes, I'm not going to discuss much but the question is experimentally, how do we actually know that I have a B meson and a D meson? So the point is that in order to actually see those, the way we know that we have a B meson is once we measure the mass of the B meson we can have reconstructed, okay? It depends if you are working in an E plus collider or in a Hadron collider. And when you look at the D, you decay the decay product of the D. For example, the D many times decay into a K pi. A K plus, say a K plus pi minus. And then you reconstruct the K on and the pi on and you look for the environmental mass of the K and the pi and you understand that you have a D and you also have a muon and you can reconstruct the energy of the muon in an E plus and minus, for example. You can reconstruct the energy of the neutrino. So in order to actually fully do this kind of measurement you have to look at the total part of the event, okay? So I'm not going to discuss too much but the point is that yes, it's not a trivial measurement and you have to look how those particles decay. And kind of to give you the feeling, actually, yeah. So the lifetime of the B meson is about 1.6 picosecond and the lifetime of the D depends which D is between 0.4 to 1 picosecond or the D zero and the D plus. So 1 picosecond even with a large boost is no more than few microns at most, okay? So they really decay at the very, very beginning of our detector. They're not going into the outside of the detector. What's really go outside of the detector is basically the cations and the pions and the cations have a lifetime of something like 10,000 more than the B. So you see just cations and pions going into your whatever calorimeter and everything. Okay, so what I want to do next I want to open the PDG and you think I was joking that I said that last time when I couldn't do anything I was opening the PDG. So I did. And I did the following thing. I, yes. How do I get the CKM element and the decay constant? No, no, no, no. So again, what you are really measuring at the end of the day you are measuring some combination of some QCD factors that could be a decay constant or in this case something else that called form factors. So you always measure CKM times the form factor, okay? And then you have to use all those smart ideas how to actually get the form factor and separate those two, okay? And then you use things like heavy quark symmetry, Lorentz invariance, lattice, a lot of things you put together until you kind of believe that you kind of understand well enough the form factors. And there's always this worry that you don't really understand what you are doing because it's QCD, okay? But as I said, it's a huge topic that I spend a lot of time in but I don't want to get into it. The point is that that's really that question that we are dealing with but I'm kind of going around it and I'm telling you some result without getting into the interesting part. So I open the PDG and I look into some decay. So I measure some decay rate and I want to do the following exercise. I want just based on the measurement I want you to stare at these measurements and tell me what kind of a structure do you see emerging, okay? So I'm doing, I was looking into the following one. We're going to D muon neutrino and the branching ratio is 2 times 2 times 10 to the minus 2. Then I did B plus going to pi muon neutrino and the branching ratio is 1.5 times 10 to the minus 4. And then I look for B plus going to K plus mu plus mu minus and the branching ratio is 4.4 times 10 to the minus 7. Then I start doing D meson decay. I was very effective and I was, it was K muon E and I found that this is 3.3 times 10 to the minus 2 and I did D going to K and going to pi muon neutrino, pi muon nu and that was 2.4 times 10 to the minus 3. And then I did K plus going to mu nu and that's 64 percent and then I did K plus going to, not K plus, K long, going to mu plus mu minus and that's 7 times 10 to the minus 9. And I forgot, I also did D going to pi mu plus mu minus and that's the bound that is less than 7 times 10 to the minus 8. Okay? So I just throw some number on the blackboard and I want you to stare at one thing I should say. I didn't actually talk about K long and K short. So the neutral K is actually mixed and we're going to talk about it I think tomorrow. The mix and the mass eigen set are called K long and K short. But to leading order you can forget about all these subtleties what I told you is that this one you can think about S decay and also here is the S d decay. Okay? I want you to stare at this on those numbers. Experimental measurements, usually effort going to doing all those measurements. And I'm just asking you what kind of pattern do you see? Do you see some specific pattern? And I want you to make some conclusion about this pattern. Stare, stare. Let's do the usual one that we required for a minute. So I'll be quiet for a minute, talk to your neighbor, talk to people and say, what do you see from those numbers? Okay? And if you don't see one number, let me know I will make it bigger. Okay? Try to see, talk to your neighbor and say, what do we see here? There's not too many numbers here. I put eight numbers on the board. Please talk to your neighbor, see what you can conclude about these numbers. Okay. So, can someone tell me like, yes. The one very important result is the following thing. It's the fact that when we see that we have a pair of charge left on in the final state, here, here and here, the decay rates, it's much, much smaller than the decay rate when I have a pair that is one charge and one neutral. So the new, new final state is much, much, much more likely to appear than a pair of muons. You see? For B physics, it's something 10 to the minus 2 to 10 to the minus 7. In D physics, it's from 10 to the minus 2 to less than 10 to the minus 8. We don't know how much. And in Keon physics, it's from 1 to 10 to the minus 8. So here's 8 order of magnitude. Here is 5 order of magnitude. Here is at least 6 order of magnitude. Okay? So just looking at the data, we make this following very interesting conclusion. That somehow in nature decay into a, when the left on final state is charge, the combination is charge 1, the decay rate is much, much bigger to decay where the combination of the left on is 0. The charge of the left on is 0. Okay? And I'm going to discuss it in a second. These decays are called flavor changing charge current and these are flavor changing neutral current. Let me, probably I should explain it right now. These are flavor changing neutral current because you see that the change, the flavor of the quarks is changed in a way that the charge of the flavor doesn't. Okay, so if I look for B plus, B plus that go to K plus, what's happened here? A B plus is a B bar U and a K plus is an S bar U. So you see that the B become an S. So the quarks change from a B to an S. The quarks change from one to another quark that have the same charge. That's called FCNC. FCNC, flavor changing neutral current. Very important acronyms. You really need to know it. FCNC, it's so important. And then we look for example for B plus decay into a D plus and a B plus decay into a D plus, it's B U and it's decay into a C. Decay into a C bar and a U. Okay? So this is the B plus going to a D, D0 bar. Okay? And what we see here, we see that we have a B going into a C. Here we have a B going into a S. That's flavor changing neutral current. This one is flavor changing charge current. This one is FCCC, flavor changing charge current. So see here the flavor of the quark change such that one quark went to another quark with a different charge and here the quarks, the flavor change such that the new quarks has the same charge as the decaying one. And here the decayed quarks have a charge that is different. And what we found, conclusion number one, conclusion number one is that FCCC is much, much larger than FCCC. Okay? And the fact that FCCC are so small is very important for our understanding of what's going on. That was conclusion number one. Conclusion number two, what other thing you can actually get just from staring at the number? BKM. So you're totally right. But what I like to do, I like to actually look at the number and just tell something about those and then we interpret them. I didn't actually interpret this result yet. And we're going to interpret in terms of... So can you interpret this result in terms of without saying the name CKM? What do you really see? Yes? You kind of know the answer from the standard model. I want you to just get it from the number. So what you are actually saying is the following. The transition from first, from three to two is much bigger than from three to one. Yes? B to D is much bigger than B to Pi. And here, D to K is much bigger than D to Pi. So what we see here, here we see that three to two is bigger than three to one. And here we see that two to two is much bigger than two to one. Okay? So what we conclude, that's the point number two that you can get from the data, is that three to two is much larger than three to one. And two to two is much larger than two to one. Okay? You see just that's based on the data. Yes? And I want to say this is a very generic result. It doesn't really very important the exact number. You just see this structure. Okay? Factors of hundreds and tens and millions. Okay? So these two conclusions are just, you just look at the data and you see these patterns. Okay? Are you with me? Yes? So I, yes. Oh, the generation, sorry. So the third generation quark into two-generated quark, the width of three-generation into two-generation is much bigger than the third-generation quark decay into a first-generation quark. And here a second-generation quark into a second-generation quark. So this would be C to S is much larger than C to, three to D. And here that's B to C is much larger than B to U. Okay? That's what I mean. So what I will ask you to do, I don't, we don't have any more like homework session, but try to do for homework is do the measurement that is open the PDG. And believe me, you know, we are not asking too much. This experimentalist works for years. The people spend all their life to make one measurement and all I ask you to do is to open the PDG and you're gonna do it. I mean, give them some respect, okay? They work so hard just for you to open and see the number, okay? So open, look at the number of the PDG and find actually more of those. Just look at the PDG, stare at the PDG and try to think about other decays where we could see the similar pattern and what you're gonna find out that no matter where you look at the PDG, these patterns always emerge. It's always the same thing, okay? These two properties are always there, okay? So, what I wanna say is the following. So far, I was just telling you some theoretical stuff and I kind of mentioned experimental data, but now I came to this thing that you just look at the data and you stare at the data and you see so much in the data and what I wanna do is like from this data actually say how good is the standard model picture of CKM that we were just talking about, okay? So I actually like to go the other way than before. So before I was just doing like telling you about the story and telling you that the CKM had some structure and I want to go the other way around. I actually look at the data and I say how these two things are explained in the standard model, okay? So you already know the standard model and I'm asking you what are the fundamental thing that happened inside the standard model that give us this kind of structure, okay? So can someone tell me that within the standard model how the standard model explain the fact that FCC is much larger than FC and C? How the standard model explain it, yes, yes. And so the point is as following, flavor changing charge current is mediated at three level in the standard model. Flavor changing neutral current is forbidden at three level in the standard model. You mentioned the Z, but actually in the beginning of the lecture I was careful enough to write all the other mesons, the gluon, the photon, the Higgs and the Z, all of them are diagonal. So to leading order to a three level, we have only this and we all know the three level is much bigger than one loop, okay? By how much? By how much a one loop is suppressed? Yes, it's one loop is one over 16 pi squared which is roughly a hundred, okay? So we're expected here to be a factor of a hundred. But actually the numbers are much bigger than a hundred, okay? So we kind of understood some, do you see what I'm saying? So the difference between this and this is five order of magnitude, okay? And we're expected to be a hundred, maybe if we square it, maybe a thousand, maybe a 10,000, but it's not such a big number. And here it's huge. Here we have eight order of magnitude and we're expected to be less. So what I'm going to actually discuss soon is why this suppression is much more than one loop. The suppression of flavor changing into our current in the standard model, you gave me the leading order result which is that it's three level versus one loop and soon I'm going to actually tell you that there's actually more suppression of the flavor changing into our current as we see from the data. And if you like to do a little more precise we see that the FCNC in the B meson is less suppressed than FCNC in the key on sector, the FCNC is most suppressed than the B. So also this we're going to understand when we actually do a little more digging into the standard model. But this is the result of FCNC much bigger than FCNC, okay? How the standard model explain a point number two which you already kind of tell me before I was actually making point number two is the CKM. So now let me ask you the following question. Which of those properties are there generic? Are there there in any standard model in S standard model? Or are there there only in this standard order? The one that we actually measure in nature? So I'm asking you about those two things. Are there there in a generic standard model or only in our standard model? Someone, yes? Which one is generic? So this one is generic because three level versus one loop is generic to the standard without measuring any parameter. This one is very specific to the structure of the CKM, okay? And what I'm going to show you next is actually the extra suppression that we see and I will tell, I just told you the extra suppression. This extra suppression is also unique to our standard model, okay? So this fact is actually part of it is generic and part of it is unique and this one is only in our standard model, okay? So one thing that we learn when we do flavor physics and like what we do when we do like Higgs physics and those kind of things, that there in flavor physics there's a lot that depend on the fact that our standard model is kind of different, okay? Very good. So what I want to do now is to talk a little more about flavor chain unital current and I really like to emphasize that the concept of flavor chain unital current is an extremely important concept in physics, okay? And you always have to kind of when you think about physics and models beyond the standard model you have to think about flavor chain unital current. It's a very big deal of what's going on, okay? So what I want to ask is the following question is what is really the reason or the deep reason that we don't have flavor chain unital current in the tree level and then we are going to do it at one loop and we're gonna see how come that we have this extra suppression, okay? So let's see what is the fundamental ingredient that doesn't make the guarantee that we don't have FCNC at tree level. Let me back up. I want you to understand that when people propose the standard model they know about this data and they actually build the standard model so that we have to make sure that FCNC is not there. So they built it in a way to make sure there's no FCNC at tree level and what I want you to understand is what are the tools or the model building tools that one can use in order to make sure there's no FCNC and you may use those tools when you do a model building beyond the standard model because whenever you do model building beyond the standard model you also want to make sure there's no FCNC because if there are large FCNC it's contradicting the data, okay? So there's four bosons. The gluon, the photon, the Higgs and the Z. And there's some different fundamental reasons of why we don't have FCNC in both of them, in all of them. So let me start with the gluon and the photon. So why the gluon and the photon do not have FCNC? Anybody have any intuition? Why the FC, why does... Let me ask the following question. Can you think that you can build a BSM model where you have photon FCNC? Based on the way I asked the question, what do you think the answer is? So can I build a new physics model with the photon couple to an SND? Is there a way for me to think about some BSM model that give me this vertex? No, very nice, perfect answer. Why? Yes, why? No, actually, it's not important, it's not the quality. Okay, let me kind of back up and come back to the big picture, okay? So the big picture is like this. When something is forbidden, why it is forbidden? Because of... So I asked you, can I do it? And you said no. So it is forbidden because of... Because of a symmetry. If something is forbidden, it's because of a symmetry. And I asked you, can we do it? And you say no, and I said that's correct. No, so there's a symmetry. So now I want you to be very precise and say, which symmetry forbid photon go to SND at tree level? So there's a tree level coupling. And there's a symmetry that guarantees that this one cannot happen at tree level. What's the name of this symmetry? Gate symmetry, which gate symmetry? Electromagnetism. The gate symmetry guarantee that the massless gauge boson mass-diagonal, mass-coupled in a diagonal and universal way. And you can actually prove it with some formal ways. But the intuition is that when you have a non-broken symmetry, the kinetic term is still just the demu of the unbroken generators. So this one, the coupling of the photon come together with the kinetic term, come together with the derivative. And the derivative is always diagonal and universal. It's always canonical. That's the meaning of being in the canonical basis. That's the meaning that we understand how things propagate in time. So gauge symmetry guarantee that we cannot have this. That's gauge symmetry. And that's very important because once in a while, you have someone, a friend, a student, someone who has some new physics model and they do calculation. And they said, wow, look, I have a very cool result. My photon coupled to an SND. And they come to me and say, look, it's very cool. And I said, you are wrong. I said, how do you know? And I said, because there's a symmetry. I said, but look, I did everything. I said, you were wrong. And then they go back and indeed they were wrong. Because there's a symmetry. So you have to remember that the symmetry that guarantee photon and gluon are diagonal. So here it's kind of easy. You don't need to work hard in order to make sure. The Higgs and the Z is a little different. So why the Higgs? What is the fundamental story of the Higgs? Why the Higgs coupled diagonally to the quarks? Let me ask it differently. Can I have a theory when I have a Higgs that's coupled to T and C at three levels? So this Higgs is the physical Higgs. It's not the field. It's not the field. It's the physical particle. So can I have a BSM model where I have some kind of a Higgs, maybe not our Higgs, but some relative of the Higgs or even our Higgs that's coupled at three levels at T and C? So it's a yes-no question. Can we find a new physics model where I have a Higgs coupled to Tc? Okay. There's some kind of a mixed reaction within the crowd. Okay. And the answer is yes. And if you like the answer is yes because there's no symmetry that guarantee it. Okay. Here there's a symmetry and here there's no symmetry. Okay. So let me explain what's really going on in the standard model. Why the standard model have the Higgs coupled diagonally? And then we understand how fragile is this condition? How easy it is to actually break this condition? Okay. So the reason that the Higgs coupled diagonally is the following thing. We write the Yukawa interaction. So let's take the Yukawa, say for the down type quark, yq d phi. Okay. And then I take this phi and I expand this phi. So instead of phi, I write, maybe I should write it up there. So instead of phi, I write v plus H where H is a field. Okay. And then I have yq, yq d, v plus H. And then what I do in order to move to the math basis, in order to move to the math basis, I diagonalize y. And when I diagonalize y, automatically also the coupling of the Higgs is diagonal because you see the proportion to the same thing. Okay. Do you see it? It's the fact that the reason that the Higgs have no FC and C coupling at three level is because it's the same matrix that give me the coupling and the masses. So when I diagonalize the Yukawa and I diagonalize the mass and I move to the math basis, I automatically diagonalize the coupling of the Higgs. You see this? So now can you think about an example where this is not the case? Can you think of an example where the coupling of the Higgs is not diagonal in the same basis that the mass is diagonal? Yes? So the model from yesterday? The model from yesterday, that's a very good example. And actually in my original question, this was part of it but I reduced it because I didn't want it to be too long of a question. What happened yesterday? You remember in the question yesterday, we have contribution of the mass from two sources. There was the bell master and the Yukawa. So the mass matrix in the model yesterday, the mass matrix there was some m plus Yukawa. That was the mass matrix. And what is the coupling of the Higgs? That's the mass. And the coupling of the Higgs was just y. The Higgs coupling was just y. So when m is equal to 0 and I diagonalize y, I diagonalize both. But in the model yesterday, when I also have some bell master, what's happening is that when I diagonalize the mass, I do not necessarily diagonalize y. What I diagonalize was m plus y. So when I diagonalize m plus y, it doesn't guarantee that y is diagonal, OK? So that's one example. One example is when I have bell master. Can you think about another example where I can break this interesting case that the Yukawa and the mass are diagonal in the same basis? If I have an extra Higgs, so if I have an extra Higgs, what would happen? What would be the mass? The mass would be something like v1, y1, plus v2, y2. Because I have two Higgs, so I have two vebs and two Yukawa. And what would be the coupling of the Higgs? Say the coupling of H1 would be something like y1, right? So in order to diagonalize the mass, I diagonalize this sum, where the coupling of the Higgs is only y. And in general, they are not diagonalizing the same basis, OK? So you see that actually the fact that the Higgs coupled in the diagonal ways to the quarks is far from a trivial statement. It's not like the photon and the glue under this guarantee by symmetry. It's very special. You have to make sure that all the masses for the quarks come from one source, which is the Higgs, OK? It's actually this requirement of minimality. And many times when people ask like, why you only put one Higgs in the sector? You say, it's minimal. I don't need more. If I put more, I don't need it, so I don't want it, OK? But now we learn that actually the fact that we have only one Higgs, it's not only minimality. It's actually an important consequences, OK? If I have more than one Higgs, I would have FCMC at three level. Good? Do we have to worry about Higgs FCMC? So let's say that I do have a model where the Higgs coupled in a non-diagonal way. Let's take the model from yesterday. In the model from yesterday, we actually have this Higgs FCMC because it's this case, OK? And in the model yesterday, I ask you exactly those two things. So how do you think Higgs FCMC will affect K long to Mu Mu? Is it important? Is it significant? Is it not? What is your feeling? Very nice. So it would be small because when I talk about K on physics, the Higgs coupling is somewhat propellant to the masses. And generically, we expect this to be suppressed by the yukawa of the S or something like this. So actually even if I have Higgs FCMC, it might be safe for the K on. But it might be more problematic for the B and it may be much more problematic for top decays, OK? However, in the standard model, we built it in a way that there's no Higgs FCMC and we understand what are the ingredients. The ingredient is that we have to guarantee that there's only one source of masses, which is the yukawa coupling. If I have two yukawa or if I have bare master, this is not there anymore and I can have FCMC. Good. So that was the Higgs. And now let's move on to... I don't want to erase it. I don't want to erase this, but I will erase it. On the last one I want to talk is the Z. And we actually kind of mentioned it yesterday when we discussed the homework. And the idea that the Z doesn't have FCMC is because of the following property. So I'm going to state the statement. The statement is as following that the Z and it's actually more general than the Z. If you do model building beyond the standard model, it's correct to any neutral gauge boson. In general, gauge boson beyond the standard model that are neutral are called Z primes, OK? So any Z or Z prime. Z prime is a generic name for neutral gauge bosons beyond the standard model that are heavy. So for any Z or Z prime, there's no FCMC if the following condition happens, OK? If all quarks with the same charge, with the same Q also have the same T3, OK? Basically, let me say it a little bit more generally. I do not have flavor chain neutral kind. If all the fermions in the unbroken representation belong, all the fermions that have the same representation in the unbroken group also have the same representation in the broken group, OK? That is, in the standard model, what will we find? We find that we have, say, look at the downtag quark. The downtag quark, I have the D, S and B. And all of them have charge of minus a third, OK? In principle, having charge of minus a third can come from very, very different Q's, because Q is T3 plus Y. So I could have many, many fermions that have the same Q, but they could come with different T3's, yes? I could have Y equal to 1 minus a third, and T3 is equal to 0. And I can have T3 is equal to a minus a half, and Y is equal to a third, and I get a third, or a sixth, and I get a third. So what I'm saying, you see that the same Q does not guarantee that it's coming from the same T3, OK? The way we built the standard model was that we built the standard model in exact copies. So in the standard model, all the quarks with the same electric charge also have the same T3, yes? What happened if we didn't? Let's say that I had quarks that do not. For example, in the homework yesterday, I had some quarks. The S actually had a T3 of 0, and the D has a T3 of minus a half, but they both have the same electric charge. And when you have two things that have different T3 and the same electric charge, pa, pa, pa, pa, pa, pa, you did it not, I won't say three people, right? You did it, but you really, I want you to do it. It's so much fun. It's like matrices, and you diagonalize them, and you know, it makes you very happy. When you said, one day that you really said, do it, okay? It's, you know, I say, ah, I remember you've said, and then you'll be happy, okay? So the point is that you do it, and you see that then in general, the ZFCNC is not diagonal, okay? You have coupling of the Z that is not diagonal, and we get FCNC. So what we conclude is that when we built the standard model, people were using this trick that we say, when I have generation, all the generation must have the same, totally the same structure, okay? And when you try to add stuff to the standard model, and many times people add stuff to the standard model, such I have the same Q, but the different T3, I do have a problem, okay? And in general, if I have it today, I have problem, and that's what you see in the lecture, in the homework yesterday, okay? So what do we conclude? We conclude that the photon and the gluon is kind of safe. We never have FCNC with them. The Higgs and the Z are far from trivial, and you have to actually, when you do model building, you have to be very clever and very careful about how to do it, so it's far from trivial, okay? So whenever you do model building, don't forget it, okay? You have to make sure that you do those kind of things to avoid big contribution, large contribution to FCNC. Any question on this? Nothing? Okay. Yes. So basically, the fact that the gluon and the, it's basically the fact that you look at the kinetic term. So let's look at the kinetic term. The kinetic term is something like, say, u bar d slash u, and the d slash must be diagonal in the canonical basics. Not only the diagonal, the coefficient is one. So in flavor basis, d d slash is, must be equivalent to the unit matrix, right? And the d slash contains both the coupling to the unbroken gauge boson and the kinetic and the d mu, so it must be diagonal because it's come from the same place where the d mu and the d mu is diagonal because that's how I define my propagating state, okay? So what I want to do next, yes. Yes, so the question is what's happening in two-higgs doublet model, and if I have two-higgs doublet model, there's no guarantee that actually the mass, the resulting coupling is equivalent to the mass. Maybe there's some really cool cancellation between the two-higgs doublet such that the coupling of each-higgs is very, very large, but the sum somehow is very, very small, okay? That could be the case. What I'm just saying is that it's not generic. I just say that in a generic case, you have to worry about it. But of course, you can actually do some tricks and people have some, you impose some symmetry. Two-higgs doublet model is very popular and there's a lot of kind of model building going into the case. I'm just saying that in general, you have to remember this kind of constraint. Okay, so what I want to do next, I want to move on to the standard model at one loop and the big picture is kind of a follow. What we did so far, I was telling you about the standard model and we built the standard model at three-level, okay? And the standard level kind of give us, roughly speaking, the right things, okay? So we talked about the row equal one. You remember the row equal one? And I told you, yes, you measure it and it's agree. We talked about flavor chain neutral kind. We talked about the structure of the CKM and all this and we did all these measurements and we see that this agree, okay? However, I agree only to leading order because in the standard, there's no FC and C at all at three-level, but we do see some FC and C very, very small, okay? And we kind of understand where it's coming from. It's come from one loop and actually also the row equal one relation, it's not exact when you make very, very precise measurement, you find that it's actually violated. It's more like 1.01, this row equal one relation. Depends actually how you define it, et cetera, et cetera, okay? So the point is as following and I want to emphasize how impressive this point is, okay? When you actually built a new model, when you think about a model of physics, at the very beginning, you can make only very rough prediction. You experiment results, give you prediction at the point of 10, 20, 30, 40%, okay? And you model a three-level, explain all of this and later on, you can start making more and more and more precise measurement until eventually you come to the point that you actually start being sensitive to deviation for three-level and that's really what you want. That's what physics is all about. You remember we talked about it several times. In physics, we want to actually go beyond our leading order. We want to go from three-level to one-loop. And what's happened in the standard model in the 70s and 80s, people were at the three-level level, okay? They were actually doing precision, the calculation was to get the standard model at three-level. And later on, roughly speaking, we started being precise enough that we can actually start seeing deviation from three-level prediction and we can actually start probing the standard model at one-loop, okay? And what we found out is very, very interesting thing. So historically, when we start looking into the standard model at one-loop, we start being sensitive to particles that we didn't know that they exist, okay? And historically, for example, the charm, the fact that we have charm was predicted exactly before because this one was not seen. So in the 70s, people didn't see K long to Mu Mu and they said, why it is? And they said, oh, there must be a charm and if there's a charm, I understand this question. And it was rather amazing because in 1970, people predicted the charm have a mass of 1.5 GeV just based on going beyond three-level, okay? And later on, people were able to actually predict the mass of the top. So that's come from K long, K long to Mu Mu, predict MC equal to 1.5 GeV and BB bar mixing. We are going to talk about it tomorrow about BB bar mixing. BB bar mixing when it came out, it's make the prediction that M top is equal to 150 GeV and kind of an interesting history. So people were sure that the mass of the top is less and they were using the following really cool argument. I'd like to present the argument to you and I hope that you will be impressed by this argument, okay? So the argument is this following. The mass of the B is roughly 4.5 GeV, okay? The mass of the charm, let's do the mesons. The mass of the B meson is about 5 GeV. The mass of the D meson is about 1.8 GeV, okay? The mass of the charm is about half a GeV, okay? I really like this argument and I messed it up. Let me do it nicely with the big letters. B, about 5. D, about 1.8. K, about 0.5. Pi, about 0.15, okay? So you see this is U, U and D. This is strange. This is charm. This is bottom. So do you see a pattern here? Yes? So going from here to here is about a factor of 3. Going from here to here is about a factor of 3 and going from here to here is about a factor of 3. Do you see? So what would be the mass of the top? 15 GeV, okay? So somewhere in the 80s, many people said, you know, it should be around 15 GeV. You always had the factor of 3. And people actually build colliders that try to look for the top at 30 or 60 GeV. It's called the Tristan Collider in Japan. And then, at 1980, people measure BB-1 mixing and using this one loop thing that I was telling you, the answer was 150, which is a factor of 10 more than what we were expecting. And many people wrote papers about why, you know, this BB-1 mixing result is not really correct. Maybe it's some new physics. Maybe we don't understand things. And it took people some time until they actually accept the fact that the top may be much, much heavier than what we expect from this very naive argument, okay? So we actually measure the mass of the top was first. We kind of indirectly measured the mass of the top from BB-1 mixing. And then we actually did also electroic precision measurement. And we did electroic precision measurement, which is one loop correction to the row equal one parameter. And from this, we got m top about 150 GeV, which is roughly correct, right? I mean, we measure it to be 170. So this is kind of okay. And from there, we also got that mh have to be less than 150, I think, between something like this. Less than 150 or 140, I think 140. And the direct search is back then was above 90. So using the direct search above 90 and the electroic precision, we knew that the Higgs have to be between 90 and 140. So when the LHT kind of start working, we knew that that's where the Higgs should be. Of course, we really were not sure because maybe we don't fully understand BSM physics. But within the standard model, we knew that the Higgs should be at this point, okay? So I hope you are impressed with this thing. What we are doing as a community that we are having precise enough, very, very precise measurement of observable that we are sensitive to one loop correction. And the one loop correction are those where we have parameters that appear from the heavy sector of the theory, like the top and the Higgs. And based on this, we are able to actually measure the parameters of the heavy sector. So we are able to measure these masses, okay? Good, so let me start with, we take a little detour away from flavor physics. I want to come back to the fontan symmetry breakage sector and talk a little bit about how we use the one loop correction to actually learn something from the electroic sector. And tomorrow we go back and study how we do it in flavor physics, okay? So I want to talk about the important row equal one relation. The row equal one relation. And it's actually, there's more to this. There's a full program and this program called electroic precision measurement, electroic precision measurement. And this program of electroic precision measurement is that you measure many, many observable that are sensitive to sinus state W and the masses of the W, the masses of the Z, all those coupling. You make many, many, many measurements. And in three level, all those whatever, 40 measurements, it's actually 20 something. 20 something measurements. Also atomic party evaluation, the one that we talked about yesterday. You make all those measurements and all of them at three level agree. But in one loop they are different, okay? So for example, I can do a one loop correction to the row equal one. So these are the one loop correction to row equal one. It will be something like this. That will be the W and here I have like T and a B and this will be the Z. And for the Z, I have here a TT or a BB, okay? So I have one loop correction to, so this is a correction to the two point function. This is correction to the mass of the W and this is correction to the mass of the Z, okay? I know if you didn't do quantum filter, this looks a little bit mysterious, but hopefully you understand when you take the propagator and you do some correction to the propagator, you get correction to the mass, okay? So correction to the W, you have a T and a B loop because it must have a different charge and correction to the Z, it's a TT and a BB, okay? And the result is different. The correction to the W and the mass is different depending on because the mass of the T and the B is not the same. This diagram is different than this diagram, okay? So you see that we have deviation from row equal one because I actually change a little bit the mass of the W and the Z. Do you see it, okay? How large do you expect this deviation to be? Roughly, order of magnitude, yeah? Times, times, times the loop factor, okay? Good, so the mass of the top, I have to normalize it to something, so it's a dimension full. So it's the mass of the top over the mass of the W, so it's kind of order one. So it should be order one times the loop factor, so we expect it to be 1%. So people were able to actually measure the mass of the W and the Z to precision that is better than 1%, so we can start being sensitive to this correction. And we find that the correction to row equal one is of order 1%. And you have to be a little more careful about how you actually do it, but roughly speaking, that's what we get. We get correction of order 1%, okay? And based on this, we could actually predict the mass of the top because the correction of order 1% proportional to the mass of the top, okay? And what we find out, I'm not doing the calculation, I'm just telling you that the result is such that within some very crude approximation, the result is such that row minus one is propelled to one over 16 pi squared times mt squared over mw squared, sometimes number of order one, the time not writing for you, okay? So this kind of give you the 1% and it's sensitive to the mass of the top. And based on this, we are able to say that the top is roughly 150 GeV, just based on the measurement of the deviation of row equal one. So we measure the deviation from row to one and we were able to get the mass of the top, okay? Does it look weird this result to you? Very nice. So why would you think that the first generation should have a bigger contribution than the heavy one? There's no log. I didn't write any log. But you say, oh, there should be a log. You cannot do a loop diagram without a log. There are actually some loop diagrams without a log. We're going to talk about them tomorrow. But there should be a log, right? If you did any loop diagram, you know there's a log, right? But actually, this is exactly the point that I was about to discuss. So you should be surprised that it's actually the top and not the U. But not because there's no log. There's some bigger, deeper reason that you should be surprised. Anybody is surprised why there's top and not U? Anybody can understand why they are surprised? Yes? Anybody? What should be our intuition? And it's based on our, there's so many ways to go around this intuition. Let me start with the intuition of second-order perturbation theory. In second-order perturbation theory, when I actually do some measurement of on-shell particles, what do I have to feel more? The light particle or the heavy ones? If I have a very, very heavy particle, second-order perturbation theory tells me that I don't care about it because the contribution is suppressed by one over delta E. Another way to think about it is the propagator. The heavy propagators are suppressed by one over the mass of the heavy propagators. Okay? Another way to say it, it's called the decoupling theorem. The decoupling theorem states that if you have a particle that is very heavy, it cannot affect your physics. It's very intuitive. I really hope you have this intuition. When you go to a very, very heavy masses, they cannot affect our light spectrum. Do you see this? This is called scale separation in physics. Scale separation tells me that I don't care what's going on at the UV when I do infrared calculation. It's a very deep and important thing. Can you imagine that it wasn't like this? That I couldn't do physics. I mean, to understand the physics of this chalk, I really don't care about the fact that these, actually, these quarks make the protons inside the atoms that make the molecules that combine into making the chalk. I really don't care. That's why it's so important, because could you imagine that we did care about the fact that quarks make the proton for this? Why I don't care? Because it's a very UV physics. When I do infrared physics, I don't care about the UV. Make sense? Do you see it from second order perturbation theory, the one over the delta E? Yes? When I have one over delta E, when the mass of the particle is very heavy, its energy is very large, so the one over delta E goes to zero, because I have one over the mass of the heavy particle. Look at the propagator. When I look at the propagator, I have one over the mass squared, so it's suppressed. Yes? And this is totally the opposite. Instead of being one over the mass of the particle, it is quadratic in the mass of the particle. Basically, if I take the mass of the top to infinity, this one diverge. Doesn't make any sense. Are you surprised now? A little bit? Surprise. Can you? Okay, one of you. I want one of you to just do this. Do it for me. Just one of those. Wow. How come it's empty squared? It's like, no way. I cannot believe you. There's no way it's empty squared. Like, why are you doing it to me again? Okay, so what is it? What is really going on? Anybody? Do you see the puzzle? The puzzle is that somehow, the sensitivity of my loop integral instead of being suppressed by the mass of the heavy particle is quadratic in the heavy particle. Anybody knows the answer? Yes? Thank you very much. So what she said, and let me rephrase it, but she's totally got the right answer. The intuition that things are decoupled goes under this very implicit assumption that I didn't tell you. The fact that when I take the mass of the particle to infinity, I keep the coupling the same. However, this theory, the mass of the top is not a fundamental particle. The mass of the top is derived from the Yukawa interaction. So when I take the mass of the top to infinity, what I'm really doing, I take the Yukawa interaction to infinity. And now there's a whole different intuition. When I have a coupling, I take the coupling to infinity, then you are not surprised at all that the effect becomes large. Because when the coupling grows, also the effect goes. Now, you don't really see that actually the Yukawa coupling enter into this diagram, but secretly it is because of the way the Higgs mechanism works, et cetera, et cetera, but I'm not going to get into this point. All I want you to understand is that because the mass of the top derives from coupling, making the mass of the top heavier, it's like making the coupling bigger, and therefore we have sensitivity to coupling, not to mass, and therefore actually we expect to see an effect that grows as the coupling grows. So we have this kind of two effects. We have growth of the coupling and growth of the mass, and we don't know how they play out, and they play out in a way that the growth of the coupling is more important than the growth of the mass, so we get an effect that actually grows with those things. So our intuition that we just developed a few minutes ago that it should actually disappear is not, it's correct only in cases where the coupling, where the mass is a fundamental mass that is not related to coupling. So let me just close with one last remark, yes. So eventually it's all related to the Yukawa in kind of a somewhat complicated way. And if you have two X-dablet models, it will still scale like empty squared. If you have a model with vector like quarks that have a bare mass term, there will be some combination here that will be a little different. So let me just conclude. Okay, go ahead, go ahead. Yeah, yeah, yeah. And I will be a short answer then. For z prime? Yeah, yeah, yeah, yeah, yeah. So there's actually here it's only g, and here there's also some g prime. Okay, so there's also both g and g prime, and therefore that's how you also get sinusoidal. I'm definitely not going to get into the details. So when g prime goes to zero, it will have the mass of the z and the w become actually identical. Actually, you increase the symmetry of the theory. There's much more things going on. I'm really not going to get into it. It's a whole topic of electric precision measurement. I just want you to hear about it and understand that it's there. And I want to conclude with the following thing. That now that we actually measure those and we know the mass of the top, we can use electric precision correction to put bounds on higher order interaction. In particular, we can put a bound on an operator of the form h d mu h squared. And this is a dimension six operator. Everybody see that this is dimension six operator? So using this kind of thing, we can put bounds on dimension six operator and since it's dimension six, I can write it like this. And assuming that all of my unknown is in this K lambda, the current status of those electric precision measurement is that roughly speaking, we are sensitive to lambda. We know that lambda is roughly larger than 10 TeV. Which is kind of what we somewhat expect because it's kind of one loop above the weak scale. So the weak scale is 100 GeV. Actually, I think I never mentioned this number. The mass of the W and the Z is roughly 100 MeV. And one loop above them is roughly 10 TeV. So we can, everything fits nicely. We see everything agree with the standard model. And the bounds that we get from the variation from the standard model to one loop is roughly at the scale naively. If I have new particles, this new particle has to be with a scale that is roughly bigger than 10 TeV. And I like to emphasize here that when you do model building and you want your particle to be detected at the LHC, you want the masses to be roughly one TeV. And since this one tells you that the masses have to be roughly above 10 TeV, you have to be a little smart and you have to worry and you have to do some model building tricks in order to allow your new particle to be at the TeV such that they will not actually mess up with the bounds from the row equal one relation. And of course, there are some model building tricks and I'm not going to talk much about them and I'd be happy to talk about them in private. So that's all I wanted to say for today and we have one more lecture tomorrow. I hope we will actually be able to wake up in time. I will be able to wake up in time and then I will go on on Flavor. Okay, thanks. Thank you very much.