 I hope everybody is all ready to go. I hope I am. So yesterday, we went out for a very nice dinner in the city. And we came back. It was almost 11, and I didn't finish put everything into the transparencies. So I will do a Blackboard talk today. Let's see how it goes. And of course, the good thing about Blackboard talks is that it's much simpler to actually get away from what you planned. And the bad thing about Blackboard talk is that it's very easy to get away from what you planned. So let's see how it will go. So what I want to start today, I want to start the flavor or physics part of these lectures. And basically, we keep doing the same as we did before. And we're going to actually fully define the standard model. And the difference from what we did yesterday where we defined the leptonic standard model in the leptonic standard model, we only studied the lepton sector. Today, we're going to start studying the weak interaction of the quarks. So we already talked very briefly on the QCD effect of quarks. And now we're going to talk about the weak interaction effect of quarks. And that's actually the topics of flavor physics. So if one asks you to define flavor physics, flavor physics is how the weak interaction actually play with the issue of quarks. So let's see what we have here. So I want to define the standard model. So we know how to define the model. I have to give you the symmetries, the fields, how they transform. And then we write the more general Lagrangian. We truncated it at some order, usually four. And we keep going. So I already started. So I gave you the symmetry. The symmetry is SU3 cross SU2 cross SU1. It's kind of nice. It's 1, 2, 3. And you know how to extend it. Then you have to be some four. But that's the symmetry. And then I tell you what the fields are. So I have three generations. So I have three copies of the following field. It's called the Q-del. Can you say Q-del? It sounds so cute, right? Q-del. So then you don't forget it. It's the Q-del field. And the Q-del fields are Q, U, D, L, and E. These three are called quarks. They are called quarks because they charge SU3 color. And these two are the leptons fields. They charge only under the SU2 cross U1. And these are the two that we talked about yesterday. And actually most of today I'm not going to mention them anymore. And I'm only going to concentrate on this one. And here are the representation of those fields. The Q, U, and D are triplet under SU3. So they charge under the strong interaction. And you see that I use this word charge under, although it's not like a number. It's some representation, right? So they are not singlet under SU3. Q, L is a doublet under SU2. And the right-handed fields are singlet under SU2. That's why we call it SU2 left, because it's only the left-handed field that are charged under it. And the right-handed field are singlet under the SU2. And then I gave you some number for the hypercharge. And there's one scalar field, phi, the same Higgs fields that we have for the leptonic standard model. Now it's good. Let's also make it easy on blackboard, right? On the transparency, you cannot do like this. So that's a singlet of SU3 and a doublet of SU2. Why it is kind of important that it is a singlet of SU3? What would happen if I have a Higgs that was not a singlet of SU3? What was the phenomenological problem of a model where the Higgs is not a singlet of SU3? I know it's early in the morning. The several problem, what is the biggest problem? The row parameter is also there, but that's one big problem. If the Higgs is charged under SU3, it will break SU3, very nice. So once the Higgs requires a wave, it breaks all the symmetry that is charged under. So if it was charged under SU3, it will break SU3 and will not have QCD as we know it. So that's why we want to make sure that this is what's going on. So I defined the model. So basically we done. So then we do this. We write the more general Lagrangian and then we have to start working with this Lagrangian. So let's start the usual thing. Which one of this is 0? Which one is 0? Very nice. So let's do like this, right, equal to 0. And the reason is that there's no, you see, there's no vector like pairs of quarks. There's no quarks that can acquire the wave and no leptons that can, sorry, that acquire mass because they are different representation. So therefore I don't have them, OK? And for the kinetic term, we basically, the kinetic term is the same as we have yesterday. The only little extra thing is that we also have the gluons. So the gluons we talked about when we talked about QCD, the kinetic term for the SU2 cross C1 were there when we talked about the leptonic standard model. So I'm not going to repeat it. It's basically just combining QCD and leptonic standard model here, OK? The Higgs potential is exactly the same as the leptonic standard model. So again, I'm not going to repeat it. We just have a Mexican head potential. And Higgs acquire the wave. Everything that we talked about yesterday is also going to be here. The interesting thing that are different from what we talked yesterday in the leptonic standard model have to do with the Yuccava interaction, this term. And it's this term that actually makes the big difference between the leptonic standard model and the full standard model. Or another way to say it is that the big difference between the weak interaction works on quarks versus weak interaction works on leptons has to do with the difference in the Yuccava, OK? So let me write the Yuccava explicitly. And let's see what we have. So before I'm going to do, before I'm going to write the Yuccava explicitly, I want to make one little definition. It's kind of technical, but I want you to see it is, if I have a scalar field phi, I can define another scalar field phi tilde. And phi tilde is defined by epsilon times phi. Now, many of you have seen it before. And I want to do a show of hand. In many books, instead of this epsilon, what you usually see is sigma 2, is the second Pauli matrix. So now I want to ask you, how many of you saw phi tilde with sigma 2? How many of you saw phi tilde with epsilon? Oh, nice. That's quite good, OK. So I want to emphasize that the sigma 2 is extremely confusing. I mean, it should be abandoned. There should be a law of not using sigma 2 on the Y. Because sigma 2 is a total basis choice, OK? And then for years, I say, like, why the hell sigma 2 is so special? What there is about the Y direction that is, like, so important? And of course, there's nothing there. Sigma 2 just happened to be the same as the epsilon tensile. The epsilon tensile is just 1 minus 1, OK? So that's kind of the definition that I love to use, because it's kind of basis independent. And this phi tilde is basically what it does for us. It's, ugh, nobody, it's too early in the morning. I was waiting for someone to correct me. OK, so it's actually the complex conjugate field. But on top of it, it also changed the up and down component, OK? So in phi, it's the down component that requires a verb. But for phi tilde, it's the up component that requires a verb. So you see, in the up component of phi is what we call phi die. So when you do this little trick of doing the tilde, not only does it put a complex conjugate, it also changed the up and down component of a doublet, OK? And the reason that it is important, because this is kind of the part of the field, this kind of field is what we are going to use for the yukawa. OK, so now that we have all the ingredients, let's actually go on and start writing the yukawa. So the yukawa, and I'm only care about the quarks. I'm not writing the leptons. So for the yukawa, the quarks, the thing is y uij q bar i uj phi tilde plus ydij q bar i dj phi, OK? And I didn't write the left, right? You can put back the left, right just to make life a little simpler. But what I did write, I did explicitly write the ij indices, OK? So the point is that, in general, this matrix is the yukawa has to do with two matrices. These matrix are called the yukawa matrix, OK? And this is a 3 by 3 matrix. Before we go on, what can we say about this matrix? Again, whenever you see a matrix, I know the first thing you want to do is diagonal. But maybe before you kind of just go straight forward and diagonalize the matrix, maybe you can do a little small talk with the matrix, OK? So you meet the matrix, what then usually you say to a matrix when we meet it. Nice to meet you. And then you start in, are you a mission? Why are you telling it to me? I mean, are you unitary? So when you meet this matrix, the yukawa matrix, we should do this role play, right? So what this matrix will tell you? So first of all, you ask the matrix, are you simple? Are you complex? And you say, yes, my life is very complicated, right? So it's a complex matrix, OK? And is it a mission? What do we know about it besides the fact that it is a 3 by 3 complex? Nothing. Very nice. So the quiet was because, but you could say nothing instead of just be quiet, OK? Make it awkward with the conversation with the matrix, right? You don't want to have that it happened. So this matrix actually is the most generalized 3 by 3 matrix, OK? And the reason that it's not a mission or anything is because those two fields are different. And we are so familiar that matrices in quantum mechanics are a mission. And the reason that many times matrices are a mission is if these two vectors were the same. But here I have a vector like these two vectors are different. This is q and this is u. OK, so we have the most general matrices. And after we get to know the matrix and after some conversation we say to the matrix, OK, it was nice meeting you, let's diagonalize in you, OK? So then you want to diagonalize the matrix. And the question is how do we diagonalize matrices, OK? And here some of you may have the intuition that we gain from quantum mechanics. And in quantum mechanics, when we have an operator, how we diagonalize any matrix? So I have some general matrix in quantum mechanics. And the way we diagonalize it is that I have something like this, right? So that's the way that we usually do things. We just multiply from both sides with v and v. And what do I know about this v? It is? It is unitary, very nice. So I multiply my general matrix in v. But when this formula is, so I said this one is equal to m diagonal. So when I can do this, I can do this only when m is Hermitian. So when I have an Hermitian matrix, Hermitian matrix can be diagonalized by v and v dagger. What happened when I have a generic matrix? When I have a generic matrix, the statement is that I can diagonalize it by using two different unitary matrices, OK? So I call this vL and I call this vR. Now note that the reason that I, so in general, vL is not equal to vR. If m is Hermitian, vL is equal to vR, OK? Now I want to emphasize that here vL at this point has nothing to do with the helicity of the field. This L is just because it's to the left of the matrix. And this R is because it's to the right of the matrix, OK? And interestingly enough, it will end up such that vL will also be relevant for the left-handed field and vR will be relevant for the right-handed field, OK? But here, they're not yet relevant. So how do I make diagonalization in practice, OK? And again, probably many of you have seen it before, but I would like to do it. So the way I do diagonalization in practice is that actually I look into this term that I have. And what I do, I insert unit matrices in the middle, and I happen to choose my unit matrix in a smart way, OK? So let me soon going to do this. So let's take, for example, just this one. And let's forget about this. We don't really need it. I just want to look for the flavor structure. So the flavor structure is something like this. I have here a qi. Then I have here the yij. And here I have the uj, OK? This is really the flavor structure of this, yes? And you see that I left here some space. Why? Because I want to multiply it by 1. So multiplying something by 1, but the unit matrix do nothing, right? So I did nothing by doing this, yes? And now, I don't know if anybody wrote a song. Like there's many ways to write 1, right? There's infinite way to write the unit matrix. So let's find a nice way to write the unit matrix, OK? So let's write here the unit matrix in the following form. Let's write it in the form of v left dagger v left. And here I have another i. And let's write this i as v right dagger v right, OK? Still I did nothing. I multiplied by 1. So this one become more interesting, right? And now, what I want to do, I said, oh, now this one, I choose, oops, the dagger should be inside. Not so easy when this one is a little wet. Anyway, so I choose this one to be diagonal, OK? So I choose what I'm saying. I'm choosing my v left and v right in such a way that this is diagonal. So this one is y diagonal, OK? And then this one and this one are my new fields, OK? And if eventually this one would be, say, if this was the energy, then this one I would call energy eigenstate, OK? And if this one is the mass matrix, then I would call this one the mass eigenstate. Good? OK, so this is really how we are to diagonalization. We always insert ones between things and we do this little thing. OK, so now let's go on and discuss how this is actually have to do with the masses of the fermions. So let's look into this. And soon we're going to diagonalize it. But before we diagonalize what I want to do, I want to ask what's happened after the Higgs requires a verb. So after the Higgs requires a verb, let's see what we have. So when we have spontaneous symmetry breaking, I can define q left is a doublet. So I can define q left is u left and d left, OK? And you may ask, why do I call this u and d? And someone should answer. Let's ask you. Why I would call this u and why I call this d? Yeah, I guess. Yeah, I was hoping for a simpler answer. I was hoping to the answer that this is the app. So I call it up and this is the down, so I call it down. And actually, that's what for years I was thinking that the reason. But then recently I found out that actually this is not the reason, and it's actually very interesting. So this is a doublet of SU2. This is only the left-handed field, OK? And the reason that we call the app up and the down down is not because they are doublet of SU2 left, but because they are a doublet of isospin, which has to do with the strong interaction, OK? So that's why we call it up and down. And then it just happened to be that there are also the up and down component of q, OK? So it's kind of really nice that the up and down are kind of accidental. And the left and right here are also not really related. But so I write my thing as up and down. And now let's see what's happened when I take this and apply it into the u-couple, OK? So what do I have? Let's do something like this. Let's look at only the SU2 index of the q bar. So I have q bar and a phi and a phi tilde, OK? From here, I have a q bar and a phi tilde. So let's write it explicitly. So explicitly, this will be u left, d left. And after the h equals ff, what do I do here? I have v and 0. So this one is equal to vul, OK? So when here in the up u-couple, after the v requires a v, I have here u left, u right. And what's happening in the down case? After the phi acquires a v, OK? So if instead of a phi tilde, I have here a phi. Or the different if I have a phi here, when I have a phi, then it is the down component that requires a v. And then I have here v times d left, OK? So after the h equals ff, this one is u left, u right. And this one is d left, d right. But let me emphasize that the d left and d right are fundamentally different fields at this point, OK? So where do I going to write it? Write it here. Because we all remember this by now, hopefully. Let me write the mass term. So after the h equals f, yes. There should be some bars floating around, yes. There's probably a few other things that I missed. But the point was that I just care about the up and down. But thank you. So let me write explicitly the mass metrics. So what I have after the things requires a v that I have something like this. yu times u left bar times u right times v, OK? That's what end up from the up you cover. And for the down you cover, I have yd, d left, d right. And now let's put the indices, the flavor indices here back. So here I have ij, ij, here I have ij, ij, OK? And now what I want to do, I want to do the diagonalization. How do I do the diagonalization? I insert all these unit matrices inside. And what we find out at the end of the day, that in the basis, this mass matrix is diagonal. So I should have said, now it's become a mass. And what is the mass matrix? The mass matrix is the v times the yu cover coupling. So you see that this is a mass matrix. Because I have just two fields left bar, so I'm right. And then I can define my mass basis. u left mass is then equal to v left up u left. So let me be explicit what I did here. I actually moved to a new basis, which I call the mass basis for the app. And the matrix here, it's vu because it works on the up-tile quack. And it's left because it's the one that works on the left. So how many matrices total do I have? The total number of matrices that I have is actually four. I have, for the up-tile, I have two, the v left and vr for the app. And then I have v left and vr for the down. Because I have to diagonalize also the down one. So just like this, I have four of those. I have then, u right m is v right up. u right and the same for the down. dm left is v left d. d left and d right m is v right d d right. So I have four, I define four different unitary matrices. Good, because I hope it's good. It's good? Good, yes, OK. So before I go on, I want to discuss what is physical here and what is unphysical. So what do I mean by something that is unphysical? Something is unphysical if I can choose a different basis and things are changing that's unphysical. If things, when I do a basis rotation and nothing change, I say it's physical, OK? Makes sense. So for example, when you talk about the magnetic field in the hydrogen atom, the magnitude of the magnetic field is physical, but the orientation is unphysical because it's up to me to choose in which way I define the z-axis. So here, we can actually use the following basis rotation. I can take this vector. This is three quarks. So three quarks create a vector. It is three in this space. So I can rotate between them. I can do a rotation on this vector. I know it's a little bit abstract because you think about it as field. But don't think about them as field now. Just think about them as just positions of three particles. And I can just make rotation in this space. So I can actually take this q and rotate it. When I rotate this q, this v is also become different. So another way to say it, the matrix that diagonalize my yukawa is as a freedom that has to do with the initial basis that I choose for my q's, u's, and d vectors. Makes sense? OK, yes. However, there's one combination that it is actually physical. And the one combination that is physical is the product between the u and the d. Why? Because q, this q, involves both the u left and the d left. So when I rotate this q, I rotate v left and d left with the same phase. That's clear? So what become physical? So if I rotate v left and v u and v d with the same phase, can you tell me some combination that is physical, that is invariant under the rotation of the q? So I tell you, v u left and v d left rotate with the same phase. So I tell you, v u left and v d left rotate with the same phase. Based on this, can you tell me something that is physical? Yes? At the dagger, very nice. Where? First, it doesn't matter? It doesn't matter? OK. You're right, it doesn't matter. And why did I choose it here? I choose it here because that's the standard convention. So this one is independent. It's physical. There's no freedom that is unphysical. And this one, it's going to be an extremely important matrix, probably one of the most important matrices in nature and probably definitely the most important matrix in my life. And that's called the CKM, also known as V. So many times we don't write the CKM. And CKM stands for Khabibu, Kobayashi, and Masukawa. Khabibu was the person who did it for two by two. Kobayashi and Masukawa moved to three by three. And therefore, they got the Nobel Prize and he didn't. Of course, the reason they got the Nobel Prize because they realized that when you go to three by three, it's something to do with CP violation. And that was the big deal. And soon we're going to discuss it. At this point, I think there's nothing about the CKM. All I'm telling you at this point is that this combination of matrices is physical. And all the other are not physical. All the others, there's some phase rotation are not physical. But there's one physical combination of all those Vs. And therefore, you will not be surprised that soon when I'm actually going to play a little bit more, you will find that this CKM matrix, which is physical, actually end up entering into the physics. And that's why we take care so much about this one. OK? Any questions on this? Yes. So actually, in order to define the fact that the CKM is physical, you don't really care about, you can do it actually without this one equalizing a VEV, OK? The point is that without this acquiring a VEV, you don't really, you say it's not the mass matrix that you work. You work saying the basis where the coupling of the Higgs is diagonal, OK? And only then you see that it's a matrix. So actually, we don't really need to give it a VEV before we do this calculation. But it's much simpler, and that's the way I like to do it. I mean, we kind of give it a VEV, and then we see it. But it's not really necessary, OK? This statement that the CKM is physical, it's a general statement. OK, so let's move on, and let's move to the interaction. So the interaction, where it's coming from in this Lagrangian, where the interaction coming from? From which term? The gauge interaction from which term? From the kinetic term, right? So what I care about now is the interaction of the kinetic stuff with my quarks, OK? And in general, we can think about how many bosons, how many types of bosons we can think about. So what I like to think about now, I want to actually talk about the interaction of the quarks with some bosons, OK? So I have some quark-quark with some boson line, OK? Whatever you want to do here, OK? So how many of those do I have? How many boson types do I have in this model? So we already talked about it, all of them, right? Yes? So here I have the Higgs. Here I have a gluon. And here, how many vector bosons do I have that are not gluons? Three, there will be the photon, the Z, and the W, OK? So actually what I like to do, I like to ask what are the interactions of the quarks with all of those, OK? Now out of these five, four of them are neutral, OK? This, this, this, and this have zero electric charge. And the W has a charge of plus and minus, OK? So I want to discuss the coupling of those things. So in particular, what I really care about is what quarks appear on these lines. I didn't put you some label, OK? So let's start with discussing the coupling of the Z, OK? So the coupling of the Z. And let me take a specific case. There's four cases, the up and down, up left, the left, et cetera. Let me take just the coupling into the U left, OK? So the general coupling of the Z is G over cosine theta W. Remember theta W from yesterday? Yes, good. Why the W? Why in a week? Let me make it, I stop. I try to write big, and then you see when the lecture goes on, it becomes too small. So when it goes below the cutoff, the UV cutoff, you tell me and I make it big again, OK? So the coupling of the Z is G cosine theta, and then you have here T3 minus sine square theta WQ. So for the case of the up type, for the left up type, what is T3? So now I'm caring about this. I said U bar I left time I Z time U left J, OK? So this is the general coupling that I have. So for the U left bar, what is T3? Plus 1 half, thank you very much. The reason that it is not 0 is because the U left is part of a doublet, and the upper component of a doublet is plus half, and the down component of the doublet is a minus half. So in this case, this is 1. And what is this Q? Oh, I should have said it. I forgot to say it. So Q is T3 plus Y. So what is the Q of the up type quag? The Q is 2 third, OK? So that's equal to 2 half plus a 6 equal to 2 third. So the electric charge is 2 third. Of course, everybody know that that's the one, but let me just kind of show it explicitly from the spontaneous symmetry breaking, OK? Now, in general, I put it into two indices, I and J. It's a little hard to see, so let's make them bigger. I and J, OK? However, before I do any rotation, this one must be diagonal. Why it must be diagonal? Because it's come from the kinetic term. And the kinetic term, we always work in the basis that the kinetic term is canonical. We call it canonical kinetic term, which is always diagonal and every coefficient 1, OK? So therefore, this I and J are actually the same, so I can say, actually, I multiply it by delta Ij, OK? So now, I know that in the flavor basis, my interaction is diagonal, OK? And I want to ask what happened when I moved to the mass basis, OK? So I want to see what's happened when I moved to the mass basis. So in order to do this, let me look only on the flavor structure. So the flavor structure that I start with is ui, ui, delta ij, uj. I forget about bars, left, all those. It's just a flavor structure. ui bar, i, delta ij, uj. Is it clear that that's our starting point? OK. And now, what I want to do? I want to move to the mass basis. How do I move to a mass basis? Instead of working with you, I work with a different basis, which is defined from here, OK, from this relation. So here in the relation, I said u at the mass basis is just v times u. So then I multiply by the dagger, and I know that u is equal to u times v dagger, OK? So let's plug it back here. And here I have a u bar. So then I say this one is equal to u bar m, which is in the mass basis, times v u left times v left dagger v left u, dagger times u m, OK? So what I did is the following. This one is u bar, and this one is u, OK? I just take u bar and write it like this, and u like this. OK, so now what's happened to the flavor structure? You see that the flavor structure is modified by the combination of these two matrices that appear here, OK? So now I started with delta ij, and now instead of delta ij, I have this combination. What is this combination? Someone have to like stand up and say, this is? This is? One, thank you. Yes, someone, you know, this is one. How do I know that this is one? Because it's v v dagger, v v dagger of the same matrix is one, OK? And the crucial point here is that this is the same matrix. Why did I write t? It's a u. That's what's confusing. Sorry, that's why, that's why. But you see, I modify what's happening is that I have a u and a u. So I do here rotation on the u, and I do here rotation on the u. So it's just u, u, dagger, which is one. So actually what we find out that doing this rotation from the flavor basis, the one that I started, into the math basis, do nothing for me, OK? There's nothing. And we kind of understand it because the coupling is universal. The coupling is proportional to delta ij in flavor basis. And when I do rotation on delta ij, nothing happened, OK? So what we find out, we find out this important result, that the coupling of the z to the uptype left quarks is diagonal. So the z can only come to u, u bar, c, c bar, and t, t bar, OK? And actually, you can generalize it and you find that for all the quarks is the same story. The z coupled in a diagonal way, not only diagonal, it's actually universal. The coupling is always the same number to all three generations, OK? And actually, you can do the same exercise also for the photon, gluon, and Higgs. And you find out that all the neutral bosons actually coupled diagonally in the math basis to the fermions. So all those, the photon, the z, the Higgs, and the gluon, always coupling in diagonal ways. They only coupled to u, u bar, c, c bar, t, t bar. There's no uc or ut or c, t coupling. Good? Very, very important, OK? Good. So now, we did four out of five. So four out of five were very, very easy. We find that things are diagonal and nothing really happened. And we almost lost hope for something interesting to happen. But you never should lose hope. If you did four out of five, there's over the fifth one. And maybe the fifth one will be more interesting. So what do you think it will? Yes. Of course, you know the answer. So now, let's do the same trick when I coupled the thing to the w, OK? So let's look for the coupling of the w. So the coupling of the w is something like this. It's minus g over square root of 2. I'm u left bar i w slash d left, OK? And in the flavor basis, when I start, these two are the same. I have di, di, and ui. They come from the same q. So I have q1. This will be u1 and d1. There are not much I can say it yet. I don't call them u, c, and t. So now what I need to do, I need to do the same trick and move to the mass basis. So how do I move to the mass basis? I actually, instead of this, I write here u bar m i times v u left. And instead of the d, that's instead of the u. And instead of the d, I write here v d left dagger. Why I change notation all the time? It's so bad. So the l and the d should be up and down. v u left v d left time d left mass, OK? So you see what I did. I basically did what I did. Before, instead of working in this interaction basis, I put the mass basis in, OK? I just do a basis rotation. And now what do I have in the middle? What is this? Ta-da, ta-da. Nice, huh? I really want to learn how to make the trumpet. So one day when I know how to do the trumpet, I will come. Ta-da, ta-da, ta-da, ta-da. Look, OK? I mean, what we found out is this really interesting. See, first, we kind of anticipate that the CKM will show up, right? Because it was physical and because we talk it in the beginning of the lecture, OK? And because you know it for, I don't know how many years and it should appear here. But you kind of see where it's coming from. And in general, in general, the CKM is non-diagonal. How do I know it? Because in general, this VLFQ and VLD should not be the same because this one is diagonal in one yukawa and this one diagonal is another yukawa. And in general, the yukawa should not be the same. And actually, we measure the masses of the quarks and they have different eigenvalues. So that's what we actually see in nature. We see in nature that the yukawas are not the same. And therefore, these are not the same. And while each of those matrices is unphysical, this combination is physical and it's appear here in the coupling of the W, OK? So we learn a very important lesson here. We learn that when we actually talk about quarks, we do not have each quarks can actually, we can have generation changing. And that's different from the leptons. In the leptons that we did yesterday, we have a U1 cubed accidental symmetry. The electron only coupled to the electron neutrino, the muon only to the muon neutrino, and the tau only to the tau neutrino. For the quarks, we find some signals. All the mass eigenstates can actually move to the other through the CKM matrix, OK? And the fundamental difference is the following. That for the leptons, I have only one yukawa matrix for the charged leptons. And then I diagonalize it and that's it. For the quarks, I have two yukawas and then I have two yukawas and they're actually this mismatch between the diagonalization of those two yukawas, OK? Yes. So the point is the following. All those are unphysical, right? And the yukawa are also unphysical. So what I'm telling is that if you give me some yukawa, which in principle is unphysical, you can calculate those. Now you can think about some UV model that make the yukawa actually physical and then also this one will be physical and you can do it. But within the standard model, I want to think about unphysical and therefore I try to concentrate on the physical ones, OK? So what I want to do now, I want to talk about how many parameters do I need to diagonalize the CKM? So everybody knows the answer. Not everybody, but almost everybody knows the answer. How many parameters I need for the CKM? Wow, no, not everybody knows the answer. Four, four parameters, why four? Because four is equal to three plus one, that's a very good answer. But actually, what I want to actually show you to you now is how we can actually calculate and see that the CKM matrix is parametrized by four parameter, OK? And it's a unitary matrix and what we're going to find out that this unitary matrix is parametrized by three mixing angle times one phase. And the way we like to think about unitary matrices, you can think about unitary matrix as an orthogonal matrix, orthonormal matrix, times some phases, OK? And it's sent out that the CKM, you can think about it as a three by C orthonormal matrix, parametrized by three angles, times phases. And in general, there could have been six phases. On a general unitary, it's three by three unitary matrix. But the CKM, it's sent out that five of these phases are unphysical and only one is physical, OK? Good, yes. So the flavor first are physical states. So let's first define what do I mean by physical. They are physical. The only thing is that they are not eigenvalue of the Hamiltonian, OK? And they are totally physical. The time evolution is not trivial. Only mass eigenstate, which are eigenvalue of the Hamiltonian, have a trivial mass evolution. And of course, we talked about the fact that this is an approximation. We have to neglect the case. But they are physical, OK? And then we talked about this little phase things. They don't affect the fact that the flavor eigenstate can be written as a superposition of mass eigenstate with the CKM, OK? OK, so now what I'm going to teach you, I'm going to teach you how to count, OK? And you may say, I learned how to count when I was very young. And I learned how to count in many languages. I was thinking how many languages I know how to count. And I think that up to four, I probably know like five languages, OK? But up to 10, I probably know only in three languages. So why do I care about teach you how to count? And I didn't mean to teach you in a new language. I really need to teach you how to count in physics, OK? And the answer is that many times it's kind of easy to see how to count. But sometimes actually learning how to count how many physical parameters you have is non-trivial. So that's what I want to teach you now is how to count. So let's do it here. So what I'm going to tell you is the following story. So I'm first going to state the result. And then we are going to actually explain the result. So let's think that I have some system. And this system, to leading order, have some accidental symmetry, OK? And then I add a perturbation. And when I add the perturbation, or going for higher order, some of these accidental symmetry are broken by adding the perturbation, OK? Then I ask how many physical parameters did I add? Because many times when I add the perturbation, the perturbation has many, many parameters. But some of these parameters are unphysical. And the answer is that when I add the perturbation, the number of physical parameters that I add are following the following formula, the number of physical parameters, the number of physical parameters is equal to the number of parameters that I add to start with minus the number of broken generators or broken symmetries, OK? So let me just explain what I just said. I said that the number of physical parameter that I add to the theory by adding the perturbation is equal to the total number of parameters that I add minus the number of broken symmetries, OK? So now let me give you an example, and then we actually, I hope that we will understand. And I'm going to come back to the example that we already talked about many, many times. And that's the example of the hydrogen atom in a magnetic field, OK? And in this example, everybody know what we do. We take the hydrogen atom, and we put the magnetic field in the z direction, OK? But let's assume that you have a student in the class and the student in the class come and say to you, hey, professor, I don't want to put it in the z axis. And I said, what do you mean you don't? I'm this side. I said, no, no, you don't decide anymore. Those days are gone. I decide. I say, yeah, you're right, OK? Which direction do you want to put the magnetic field? And the student said, I want to put it in some arbitrary direction. I want to have my b field in dx x hat plus by y hat plus bz z hat, OK? They said, now go and solve this question, OK? So I know that the answer should be independent on this stupid choice. Why you don't choose the z like everybody else in the world? Why you want to do this? But then how would I actually convince the student that I want to do it? So I said, well, one way to convince it, let's call this new direction the z direction, OK? And hopefully he will be convinced. But then he said, I don't see how you actually do it. Said, OK, I'll tell you how I do it, OK? I take my coordinate system, and I rotate it, and I do two rotations, OK? I first rotate around the x direction to get rid of this, OK? So I do one rotation to get rid of x, and then I do another rotation to get rid of y, and then I call it the z direction. Yes? Let me show it, OK? This is an arbitrary direction, OK? x, y, z, OK? Then I rotate, OK? The other rotation is a little harder, OK? And then I go to the z direction, right? So I actually did two rotations, yes? And now I tell the student the following thing, OK? You remember what was your symmetry before you put the magnetic field? The symmetry was three-dimensional rotation. That is, before you put the magnetic field, all these rotations was totally irrelevant. I could rotate as much as I want. I can get very dizzy, but it doesn't going to change the physics, right? But now when I put the magnetic field, making a rotation is actually do change the physics. Do change not the physics. Do change the value of these things, OK? And another way to think about it is that rotation around the x-axis that used to be a symmetry, it's not a symmetry anymore. And since it's not a symmetry anymore, I can use it to set bx equal to 0, OK? I can use the broken generator. I can use rotation around the x-axis in order to get rid of a parameter. Is that clear? So I have, in total, when I, in a three-dimension, I have three rotations around the x, y, and z direction, OK? And now when I put a magnetic field, when I choose a direction, two out of those three rotations are not symmetries anymore. Only the rotation in the plane perpendicular to my magnetic field is a symmetry. The other two rotations, when I do them, I can actually change the value of my system. And I can use them to set bx and by equal to 0, OK? So what we see here, we see that this formula is satisfied. What is the number of parameters that we have in the magnetic field? We have three parameters, right? Because I have bx, by, and bz. How many broken generators do I have? How many generators are broken when I put the magnetic field? I break two out of the three rotations, right? So here is minus 2, and that's equal to 1. And we know which is this one. This one is the one that we always do without thinking. We just call it the magnetic field in the z direction. Good? I always like this kind of things. I always think that, you know, if someone like, say, you know, like, at Cornell, I would think if the president of Cornell come into class, we see, you know, some advanced grad classes, and I say, you know, and do you see that 3 minus 2 equal to 1? And everybody, yeah, you know, it's so deep and everything. OK, so we learn how to count. Yes? You understand the intuition, so let's now actually apply it and see what we have in our cases, OK? So let's do it like this. In our case, I like to think about the following thing. I like to think about my original system of only a kinetic term. So my zero thing is only kinetic. And then I add a perturbation, which is the Yukawa, OK? And then using this trick, I want to ask how many physical parameters I have in my Yukawa. And then I'm going to see how many physical parameters I have in my system, OK? So let's look at the kinetic term. And I ask, what is the symmetry of the kinetic term? And I only look at the kinetic term of the quarks. So for the quarks, I have something like q bar d slash q plus u bar d slash u plus d bar d slash d. This is a different d, right? This is the Qawa and derivative. OK, so what is the symmetry of the kinetic term? So let's look only on this one. And I ask, what is the symmetry of this term, OK? So the symmetry of this term has to do, this is just a vector, it's a three-dimensional vector, because I have three generations. So it lives in a complex plane, in a complex space, OK? And the symmetry of rotation in a three, so it's totally symmetric under rotation in three-dimension. You see, it's nothing but the dot product of two vectors. Dot product of two vectors is just a number. So the numbers totally symmetric under all rotations. So if it was real space, what would be the symmetry? The symmetry would be S03. However, since it's a complex space, the symmetry is a symmetry that we call u3. And u3 is also Su3 times u1. And I'm just telling you that in a complex space, when I have three by three, I have nine generators. And other way, it's also related to the fact that the unitary matrix have nine parameters. It's the same counting. So this one have nine possibilities to make rotations, because it's three by three complex space, OK? So how many total symmetries do I have? I have here nine plus nine plus nine. So in this formula, let me erase it up here. In this formula, I'm not done yet. I have 27 parameters, OK? How many of those are broken when I add the yukawa? In order to do this, I have to ask what is the total symmetry that I left after adding the yukawa? So you remember yesterday when we did the leptonic standard model, we end up with u1 cubed. And in this case, because of the CKM, we can actually mix the generations. And the only one that survived is 1 u1. And this u1, we call bion number, OK? So we have u1 bion number as an accidental symmetry of the model. And this corresponds to the accidental symmetry of the rotation in the plane perpendicular to the magnetic field. So I have one that left. So out of this 27 that I start with, when I add the yukawa, the yukawa break 26 of those things, and only one survived. The one that survived is bion number, where all the quarks rotate with the same angle, OK? So how many broken symmetries do I have? 26. Now let's calculate how many parameters did I start with. When I add the yukawa, how many parameters did I add to the yukawa? So we talked about it that each yukawa matrix is a complex 3 by 3. 3 by 3 is 9. It's complex, so it's 18. And I have two of them, so it's 36, OK? Sorry about discounting, but therefore I have 10 physical parameters. Out of those 10 physical parameters, I have six masses. So this 10 is six masses plus three angles plus one phase. And I didn't actually tell you how I know if it's an angle or a phase, but you can actually do a little bit more sophisticated thing and see how many of those parameters are real and how many are complex. And you find out that out of these 10, nine are real and one is complex. So I know that I have nine real parameters and one phase, which eventually has to do with CP violation. But I'm not going to get into this discussion. But all I want you to see is that I actually know how to count and then I get this 10. And I get that the CKM have three angles and one phase. Any questions? So let me be quiet for a second. And let me ask you to do the following. I want you to actually calculate how many parameters I have in a 4 by 4 model. So let's assume that nature didn't have three generation by four generation. And I will ask you to calculate how many physical parameters we have in a 4 by 4 case. So I will be quiet for two minutes and please go on. Do the usual thing, calculate, talk to your neighbor. You understand the question, right? So please go. Please start. So I want to see that you really know how to count. OK, what did we get? Anybody? What's the answer? How much? 17. 17. How many masses? How many angles? How many phases? Is that 17? Yes? Yes, OK, so let's do the calculation together. How many broken generators do I have? So instead of U3, I have a U4. U4 have 16 parameters because it's a general 4 by 4. So I have 16. I have 16 times 3. 16 times 3 is 48. And out of 48, only one is surviving. So this one become 47. How many parameters did I start with? I start with 4 by 4 matrices. 4 by 4 matrices is 32 parameters. Time 2 is 64. And 64 minus 47 equals 17. And these kind of things I didn't actually told you how to do, but that I'm just telling you the answer. Good. OK, so I hope we learn how to count. And if you have any question how to count, ask me. But it's actually much more important than this. Because any time that you do any model building, you want to actually ask yourself the question, how many of my parameters are physical? And you want to make sure that you identify your physical parameters. And you don't work with things that aren't physical. Any questions here? Questions? Yes? OK, so let me kind of do the same. Let me kind of slow down and say it again. So I look at the kinetic term. So again, the picture is as following. I have my kinetic term. And I think that the Yukawa is in something that I add on. Just like I think about my hydrogen atom, and I add the magnetic field. So when I add the magnetic field, I give you three numbers. But then you say, oh, actually only one number is physical. So I want to do the same. It's just more complicated here. And the way I like to do it is I start with my kinetic term. And the kinetic term have a very large symmetry. And then I add the Yukawa. The Yukawa break a lot of those symmetries, just like the magnetic field breaks some of the symmetries. And those symmetries that I break, I actually can be used to eliminate parameters. So I want to count how many of those symmetries I break. Now, I start. I said, what is the symmetry before I do everything? So I'm just telling you that if I have a vector, if I have a n-dimensional complex space, the general symmetry of an n-dimensional complex space is what we call Un. And it's n squared symmetry axis. In a real space, it's n times n minus 1 over 2. In a complex space, it's n squared. And then I say, how many of those are broken? I said, all of them are broken by but one. And then I ask, how do I calculate this 47? I have a 4 by 4 space in the Q space, 4 by 4 in the U space, and 4 by 4 in the D space. So it's 16 plus 16 plus 16. That's how I get the 48. And only one is unbroken, so I have 47 broken one. And now I ask, how many parameters do I add when I add the two Yukawa matrices? I add 64. Because each matrix is a 4 by 4 complex. So it has 32 parameters. It's 16. 4 by 4 is 16. Each one is complex, so it's 32. And I have two matrices, so I have 64. So it's like the three parameters of the magnetic field that I add. So I add 64, and I ask, how many of those are physical? So the 64 is the total number physical plus unphysical. And then 47 generators are broken, so therefore I have 17. Is that clear now? Yes, good. OK. So what I want to do in the 27 minutes that I still have, I want to actually play a little more with the CKM. And there's so much to play with the CKM, because it's such an important matrix. But before that, I want to make two remarks, and then we start playing with the CKM. The first remark has to do with P and Cp violation. So the standard model clearly violates P. Why it violates P? Because it is a chiral theory. Chiral theory violates P, just from the start. Does it violate Cp? And the answer is yes, because it's have one phase. This one phase of the CKM implies Cp violation. And when you do the calculation for two generations, there's no phase. And for three generations, there is a phase. And that's the big deal about Kobash and Mascara, they understand. And I hope that you see that now that I teach you how to count, what Kobash and Mascara did is so trivial. But back then, people didn't understand that actually that's the way to do and calculate and all those kind of things. So we say, in general, the standard model violates P, and it could violate Cp. Why I said could violate Cp? Because I just told you that there is a phase, but I didn't tell you what the value of the phase is. What's happening if the value of the phase is 45 degree? Then Cp is violated. What's happening if the value of the phase is 0? Then Cp is conserved. That's why I said it could violate Cp. So you have to go and measure, and you measure, and you find that it's non-zero. And therefore, you know that it violates Cp. So I hope that you see the difference between P violation and Cp violation. P violation is there from the very start. I define the model, it violates P. For Cp, I said it could violate Cp. And then I have to go and measure it, and I measure it, and I found that this is the case. And let's lead me to a definition, and I want to make a distinguish between two things. I want to distinguish between a standard model, maybe I should do, a standard model, and the standard model. What is the difference between a standard model and this standard model? This standard model is a standard model that we have in nature. A standard model is just the model that I define. So when I say a standard model is a generic model that I don't know the value of the parameters. This standard model is a standard model where I know the value of the parameters. Does that make sense? Usually, we talk about this standard model. But many times, it's kind of very important to make the distinguish between what we have that is unique to the fact that we have a very specific value of parameters versus predictions that are there in any standard model that I have. So in particular, I would say that the fact that we have CP validation, it's generic, but you could have a case where we don't, okay? And what we're gonna take, talk about right now, we're gonna see that actually our standard model, this standard model, it's kind of generic when we talked about other sectors of it. But when we come to flavor, we found out that this standard model is not generic. It's have, the value of the parameters are kind of unique or kind of special. And many of the results that we're going to see in nature are actually specific to the fact that we have this standard model and no S standard model. So for a little bit today and most of the lecture tomorrow, I'm going to emphasize many times what predictions are there for any standard model and what predictions are there because our very specific this standard model. Is that clear? Good. So let me start talking about the CKM and some generic things. So first let's write the CKM. I don't know how many jokes did I break today. It's amazing. So I write the CKM in the following way. Here is the U, C and T. And here is D, S and B. So here is VUD, VUS, VUB. VCD, VCS, VCD, VTD, VTS, VTB, okay? To make things a little complicated, when you're going to study mixing in the neutrino sector, these indices are different. So for the CKM, the first index is the up type and the second is the down type. When you do the same for the neutrinos, the first index will be down and the other is up. It's totally arbitrary and since it's arbitrary and we only have two cases to study, we are smart enough to give equal probability for this. But just that you remember that it's actually the other way around, okay? And we went out and measured those and I hope to actually get into some discussion of how we measure those, either tomorrow or in the last lecture. And here I'm just going to give you the rough value of the results, okay? So this is the values of the CKM and the value of the CKM is something like this. I keep only two significant digits. 0.97, 0.22, 3.5 times 10 to the minus three, 0.22, 0.97, 0.04, 0.041, 8.5 times 10 to the minus three, 0.041 and one, okay? So that's what we measure and nothing, huh? I show you this and nothing, nothing. What do you think about such a matrix? So I think there's something that I do, when I teach I do it all the time and I really recommend it for you, okay? Whenever you see something like a new formula or a new matrix or something else, you stare at it, okay? And I always tell my student matrices or formula, it's actually totally fine to stare at them. They don't get offended, it's totally correct, okay? So stare at them, don't worry. Stare at this matrix, stare at this matrix and maybe it's not, it's large enough. The problem is this, it is 1.041, okay? So what do you feel when you see such a matrix? What come to mind? It is actually almost the unit matrix, okay? So let's do this, huh? Okay, so what we find out, I was telling you about the CKM, we do the general thing and generically, I just said the arbitrary unitary matrix, okay? And then I measure the value of the unitary parameter and you find out that it's very, very close to unit matrix and it should be an arbitrary unitary matrix, okay? And that's one important thing to remember and that's when I was, yes. Sorry? It should be symmetric or not? Yes or not? So actually no and the answer is that it's a general unitary matrix and a three by three unitary matrix should not be symmetric. And actually the surprising fact is to a very good approximation, this is equal to this and this is equal to this. And in a general unitary matrix, this is not the case. For a general unitary matrix, all those values are different, okay? For a generic unitary matrix. So the two by two is the same but on a three by three, they should not be symmetric, okay? And the surprising fact is that actually two out of the three are to a very good approximation symmetric, okay? So I want to emphasize the following point. The CK matrix that we have in this standard model, the standard we have in our nature is very different than what you naively expect, okay? Naively you expect it to be some order one unitary matrix and it looks instead of every entries, some order one number, it's very, very close to the unit matrix, okay? So I can write it something like this plus some order epsilon with some small perturbation, okay? So to a good approximation, actually the CK matrix is unitary, it's close to the unit matrix and therefore there's actually some sense to talk about generations. If the mixing maxing was order one, talking about generation would not make much sense because there's all of them mixed up but if this one is very close to the unit matrix, it makes sense to talk about first, second and third generation in the mass eigenstate, okay? And then we say, oh, all diagonal terms are very small so I still have some kind of a feeling that my generations are all live together and the drop into other generation is kind of small. Okay? So giving that, I wanna talk now about the parametrization of the CKM matrix and the point is the following. So while while the CKM is physical, I can still find many parametrizations for this CKM, okay? And the way to think about it is as following. So if I have a three by three unitary matrix, think about three by three orthogonal matrix. Three by three orthogonal matrix you can build out of three, multiplication of three two by two orthogonal matrix, okay? You're very much familiar with this from the Euler angles. You remember how you build the three by three Euler angles? You take two by two times two by two times two by two and you get a three by three, okay? And if you wanna build four by four autonormal matrix you can do it by multiply six two by two, okay? And the point is that you just, each of them is rotation around a different plane. And when you do it for unitary matrix you can actually do it the same with, some of them also have some phases involved, okay? So the way to think about the CKM, I can think about it as a product of three, I can multiply three two by two matrices. However, the order that I do it is unphysical, okay? The order is arbitrary. I can first define the one two and then one three and then two three. So there's actually many ways to actually define the angles that I want. So while the CKM is physical, the angles that I parameterize the CKM by is basically dependent, I can choose whatever I want. So the PDG choose for us what we call the standard parameterization and I'm not going to write it down but you can actually go to the PDG and find it. But this is an exact parameterization of the CKM. What I want to talk instead is about an approximate parameterization of the CKM, go under the name, the Wolfenstein parameterization. Wall, I'll write it wrong. I should have remembered this name. Wolfenstein, he passed away a few years ago, he did so many things. Wolfen, Wolfenstein parameterization. And what Wolfenstein did is a really cool thing and I hope you will appreciate how cool it is. So we have the exact parameterization of the CKM and the CKM is a unitary matrix and we have an exact parameterization. And what Wolfenstein did is said, oh, I don't want to be exact. I want to be approximate. Why? Because physics is the art of approximation. Yes, I do have an exact result but I don't want the exact result. I want the approximate result and he write it and you know, it's one of his biggest achievements. He's very, very famous for several things, okay? But he's also very famous for this really cool thing that you take an exact result and write it as an approximation, okay? And he said that the CKM is to a very good approximation given by one small parameter that I expand in times all the one parameters and the CKM is given by one minus lambda squared over two lambda A lambda cubed rho minus I eta. Here minus lambda one minus lambda squared over two A lambda squared. Here I have A lambda cubed one, one minus rho minus I eta. Here I have minus A lambda cubed and here I have one, okay? This matrix is not unitary. It's unitary only up to all the lambda cubed, okay? It's an approximation and actually you can expand it and you can keep higher and higher order term in lambda, okay? So why Wolfenstein did it? Because so many times an approximate result is better than exact result and I really hope you appreciate this fact and also in everyday life is the same, okay? Imagine that you say, oh, you know, that you have to remember exactly prices of things. Many times all you care about is the up, is the rough price of something, right? If you wanna and buy some milk and you say how much a milk would cost, do I really remember how much a gallon of milk is cost or a liter of milk depend on which country you are? Yes, gallon is only in the US, other countries? Okay, do I know, do I remember how much it is? I don't remember how much it is, but I know that it's roughly a few dollars. If I go and someone tell me, you know, this gallon of milk is actually $10,000, I should be surprised. But I don't really need to remember the exact how many cents I need to pay, right? So in our everyday life, we're so used to the fact that approximations are so used and that's also what we did in physics. So this is the way I like to think about the CKM matrix and this parameterization I do remember by heart while the standard one I do not remember by heart, okay? This one I remember. And you see that this one, actually in terms of the order of magnitude, it's kind of nice because we have something like this. It's all the one lambda, lambda cubed, lambda one, lambda squared, lambda cubed, lambda squared one. So what we learn from here is the following things. That to leading order, generations are conserved. The one, two generations, the transition probability between one and two is provided to lambda. The transition probability in two and three is lambda squared and the transition probability between one and three is lambda cubed, okay? And this other wolfage time parameterization, the A, rho and eta are supposed to be all the one. And the one small parameter which is lambda, that's already tell us a lot of physics, okay? Like if I ask you, what is more probable the transition between one and two or one and three? You say, well, of course one and two because one and two is lambda and one and three is lambda cubed. Is it a general statement or just a statement that we have in our standard model? It's our standard model. So all those kind of Wolfenstein hierarchy, it's something very specific to our standard model. It's not a generic property of the standard model, okay? Good, yes? No, no I didn't, thank you. Okay, so we talked about the Wolfenstein and the approximation of the CKM. And what I wanna do now, I wanna talk a little bit more formal stuff of the CKM and it has to exploit the unitarity of the CKM. Now since the CKM is a unitary matrix, I know that there's some relation between the column and the rows of this matrix. And in particular, if I take a unitary matrix, we know that the sum of two rows is equal to delta ij. So viq prime star is equal to delta qq prime, okay? So if I take two rows, if I take the same row and multiply it is one, if I take two different rows and dot them, I should get zero. That's the property of unitary matrices. So in particular, this relation, give me something like this, it's give me something like Vub, sorry, Vud, Vub star plus Vc, Vcd, Vcb star plus Vtd, Vtb star equal to zero. So this is a general statement of the unitarity of the CKM, okay? So just because the CKM is unitary, I know some relation between the CKM parameters, okay? In particular, what I see here, I have three complex numbers that add up to zero. And when I have three complex numbers that add up to zero, we can represent them as a triangle, okay? So think about three complex numbers that add up to zero. You can think about them as three vectors in two-dimensional space. So I can think about this as the u-side. The u-side, I just mean this term. And then I can think about this as the t-side. And this as the c-side, okay? So I can do this, plus this, plus this. And the fact that it's zero, tell me that I have a triangle, okay? This is a very important triangle. It's a very famous triangle. And since it's so famous it's have a special name, it's called the unitary triangle. It's the unitary triangle, the ut. Ut for unitary triangle, okay? So what is a unitary triangle? It sounds like a fancy name. A unitary triangle is a representation of the unitary relation of a unitary matrix as a sum, a three by three unitary matrix as a sum of three numbers equal to zero by a triangle, okay? And since this triangle is so important, only important triangle have names for their angles, okay? So if I, how many of you know the angles of the unitary triangle? Show of hands. How many people know what angles I wanna put here? You cannot believe it. Andrew, you should know it. Just like this, do like this, just Andrew. Only Andrew know the angles of one, two. What are the angles, okay, okay, okay, okay. Don't do like this, do like this. I wanna, yes, yes. Anybody actually remember what, okay, let's start following. If I just, if you were like stressed sometimes said, I have three angles, what names are you gonna give them? Okay, those everybody knows, okay, alpha, beta, gamma. How about phi one, phi two, phi three? Why not? Why, let's call them phi one, phi two, and phi three, okay? So now I have to actually put them into the triangle. So should I go in this direction or in this direction? Where should I start with alpha? On the top, on the top. Let's start alpha here on the top. Then which direction should I go? Let's go this direction. So call it beta and then we have here gamma. But then why don't we do it another way and call it phi one? So let's call beta phi one because I want to start with phi one, okay? And go that direction and I call this phi two and I gonna call this phi three, okay? Very nice. Now since this triangle is so famous, there was two set of names that were assigned to this triangle, okay? There was the alpha, beta, gamma convention and the phi one, phi two, phi three convention, okay? And it's become a big deal, it's become a huge deal because when people start measuring these angles, they were done at both in the US at Slack and in Japan at KK. And in the US, everybody used only the alpha, beta, gamma convention, okay? And in Japan, you only use the phi one, phi two, phi three convention, okay? And it's become such a big deal that since I was a postdoc at Slack, I used the alpha, beta, gamma conventions, right? And then I went to give a talk at Bell and they asked us to write the proceeding and in my proceeding it was alpha, beta, gamma and then the editor actually do an edit and whenever he actually changed all my convention into phi one, phi two, phi three because in Japan, you are not allowed to talk about alpha, beta, gamma, okay? So it was such a big deal back then. Now life is easy and now actually one big book and it's called The Legacy of the Bee Factories and in this big book, it's always appear like this, okay? So it's not only the jpeg side that people fight about the name, also this. So I hope you appreciate how important this triangle is if people really, really fight on the name of the angle of this triangle, okay? So we know the alpha, beta, gamma convention for this triangle and let me, I have one more minute, okay? Maybe I will need one more. I can argue that I started nine or three, okay? Let me just finish this, just this, okay? So we measure the, we define the unitary triangle and we can also do what we call the rescaled unitary triangle. So instead of this, we divide everything by the C side and when we, the C side, it sounds so nice, right? Some of you probably sit in a C side hotel. So we divide it by the C side and when we divide it by the C side, then this is zero, zero, this is one, zero and this apex is the row eta. So many times this unitary triangle, we say it's directly relate to the value of row eta and row eta in the Wolf-Fenschstein parameterization has to do with those two parts. Oh, I forgot to mention. When I look at the Wolf-Fenschstein parameter, I think we see that to leading order, a phase appear only in one, three and three one. So in order to see Cp version, you need to go to the one, three and three one. So I want you also to see this rescaled unitary triangle that have the apex up here. So let me close by saying that we have, there's actually some cool properties of the CKM is that the CKM have six unitary triangle, why six? Because I have three rows and three columns that I can multiply. And all those six unitary triangle, one can prove have, must have the same area, okay? And the area of the unitary triangle is related to some parameter that is called the Yarskog invariant. And Yarskog, she's a Swedish physicist. There is someone from Sweden here. Can you tell me this, how you said correctly? Yarskog, I'm good? Yarskog? Oh, so the Yarskog invariant, right? The Yarskog invariant, I'm so happy now. It's always cool fact, the Yarskog invariant. And it is, it's called J. I guess J is because Yarskog's card with J, right? Obviously, Yaw, all right? Okay, and one can show that this invariant is equal to the imaginary part of any kind of really cool combination of CKM. It's equal to V, I, J, V, K, L. V, I, L, star, V, K, J, star. Okay, that's how we define the J. And the area of all triangles, the area of all triangles is equal to the absolute value of J over two, okay? And just by the definition of this J, you can show the J absolute value have to be less or equal to one over six square root of three. Just some properties of three by three matrices that we get this kind of thing. And this is roughly one over 10, okay? So it's really open one, and then we measure it and the measured value is the J is equal to three times 10 to the minus five, okay? But that's another case that we see that our model is far away from a generic model. So a generic model tell you that J have to be less than 10%, less than 10 to the minus one, and we measure it to be 10 to the minus five, okay? So our standard model is not unique. This standard model is small, and that's also the statement that you might hear sometimes that people say in the standard model CP violation is small. So CP violation is small in the standard model is measured by this relation. By the fact that J that we measure, it's much, much smaller than what it could have been for in a generic model, okay? And that's of course related to the fact that the CKM matrix is very close to your unit matrix, and therefore this triangle, if it was a unit matrix, the triangle was squished, okay? Since it's very close to unit matrix, this triangle have a smaller area because they must be provided to the correction from the unit matrix. Okay, so it's still exactly an hour and a half since I started. Okay, so let me stop here. Thank you.