 to have today, Stefania, to deliver the third lecture on the standard model. Please. Okay, very good. Thanks for the introduction and welcome back everybody. So today, as I was mentioning yesterday, we talk about electric symmetry breaking. And then we'll discuss quite a little bit about today's phenomenology. So that's the plan of today. Let's write it down. Discuss. Let me abbreviate with electric symmetry breaking and Higgs phenomenology. Okay, that's the plan for today. So let me start with a short note about fermion masses. So during the first class, we brought down the Dirac Lagrangian for a Dirac spinor. We saw that in the Dirac Lagrangian, we were adding a master mod this type. The website again was a four dimensional Dirac spinor. And this, no, we saw that it was allowed by Lorentz invariance and everything was fine. And it was leading to the master in the Dirac equation. But what happens if I have a fermion psi that is charged under our electric symmetry, so under SU2 left times U1, then what happens to the master? Okay. So let's expand this master in terms of left-handed and right-handed components of the spinor. Because I can always write down psi as a sum of left and right in the sense that I can write down this simple identity where psi left is defined as the projector operator applied to psi. And then similarly, and then it's simple to see that indeed this is an inequality. And then for our mass term, we can write it down again as sums as a sum of left and right. And this leads only to two terms. So the others are equal to zero. Namely, we have the combination of psi left and psi right and the opposite one psi right and psi left. Or the other combinations are zero because of the fact that we have to remember that if we apply p left and p right, then we get zero, the combination on orthogonal directions. So it's easy to see that if we use the fermion representations of SU2 times U1 that we have seen yesterday, then all combinations are not gauging variant. So in the sense that if I take any psi left and any psi right, any left-handed or right-handed quirk or lepton that we introduced yesterday and we combine them in this way, then we don't get singlet or we don't get the singlet of SU2 times U1. So we cannot write down a gauging variant mass for the fermions that we introduced yesterday. So that's the conclusion. But then the question is, we know experimentally that the fermions of the standard model do have a mass. Some of them are indeed very massive. And then the question is, how do we write a fermion mass term? Also, to introduce a little bit of jargon, what we say is that the fermions of the standard model at Karel, because as we have seen yesterday, the left and right components couple in a different matter with the gauge bosons of the standard model. And because of this reason, we cannot write down a mass term, an explicit mass term. So let's see what happens. So what if we introduce a new field? Let me call it phi. So this is a scalar field that does transform as a doublet under SU2 and it has a hypercharge equal to 1 half. Okay, so this side again, the quantum numbers under SU2 left times U1 y. So if I have this new degree of freedom, actually I am able to show that I can write down mass terms for the fermions of the standard model. So in particular, I can add to the Lagrange and the following term as an example, I can write down, I'm using the notation of yesterday for the fields. So I can take the left-handed quark, the doublet of SU2, I put that together with a scalar field that I just introduced, and then also together with the right-handed down quarks that we introduced yesterday. And this is singlet actually under SU2 times U1. Just to give you a brief reminder of the quantum numbers. So this guy here we introduced yesterday, this was a 2 minus 1 6. This we just said that this is a 2 1 half. And this is a 1 minus 1 third. Now I'm taking the conjugate of Q left. So this is, yeah, okay, I already put the minus sign actually here. So this is for the conjugate of Q left. And then what you can see is that for the hypercharge, indeed, this sum gives us zero. This is telling us that we don't transform under the U1 of hypercharge. And then also we see that we are putting together two doublets. Let me write it down in a color. So we have two doublets, two times two. And actually we can show that if we combine two doublet representation, we obtain a singlet representation. That is what I'm interested to. And then actually also a triplet representation. But this to say that there is a way to combine two doublets to give a singlet. And this is the singlet combination that I want to put in my Lagrangian to have a term that is a gaging variant. So this term is okay. I can put it in the Lagrangian. It's also okay under, you know, the Lorentz symmetry. And if the scalar develops, sorry, in a trivial vacuum, namely that the minimum of, at the minimum energy of our system, we have phi, let me denote this, the vacuum that is of the scalar field. If this is different from zero, or I can write this condition in components. So since phi is a doublet of SU2, with hypercharge one plus one half, I can write it down as, so these are the SU2 left components. Notice that this is the electric charge, because we have seen that the electric charge is equal to T3 plus y. And then with the hypercharge, we introduce these electric charges. So at the vacuum, at the minimum, we have the following configuration. We have a zero here, and some value that is non-zero here. So let me call it V over square root of two, where this is the framesome zero. And so we can rewrite this term of the Lagrangian using this vacuum. Again, I can write down explicitly the SU2 components. Let me put here a coupling constant. This is a coupling constant that is actually dimensionless. We can check the dimensions. And so this is giving us your left, the left, then we have zero V over square root of two, the right. And this we see that is leading to the following term in the Lagrangian, namely V over square root of two, the right. And you can see here that this is a proper master for downcorks. Hi, Siphania. So when you're writing, phi develops a non-trivial vacuum, which is not equal to zero. So what does that mean? How it is not equal to zero? Yeah, so for now, I just stated it. I will get into that in a few minutes. So maybe hold that question, and then if it is still not clear, maybe ask again. But I agree with you. For now, I just stated it. I didn't explain how that happens. I just said, suppose that this is what happened, and then we get this master. Yes, that was just abstract, actually, here. So you see that using this additional field that I introduced, this scalar field, I'm able now to write down a master if this field has indeed a non-trivial vacuum configuration. So that's the conclusion of this little exercise. Of course, I can also generalize it. And then what I can write down is the Yuccava Lagrangian, that is a new piece of the standard model that Lagrangian that we have we didn't introduce yet. Namely, I have the term that we just introduced, this term here. But then I can also write down two additional terms that are gauging variant. Namely, I can write down this term here with a five-dagger, you're right. And then one term that will generate the mass of our charged leptons. So this is all I can write down based on gauging variants using the fermion fields that we introduced yesterday. And note that with the fermions that we know of, we cannot write a mass term for neutrinos. Because yeah, this term here is simply giving us a mass for charged leptons. But we don't have the corresponding one for neutrinos. So this is already telling us that this Mas Lagrangian, these Yuccava interactions miss something if we want to explain the observation of neutrino masses. And this, as I already anticipated, we'll discuss a little bit more on Friday. So having said this, let's go back to the question of, I think, Sonali is your name. So how to achieve a SU2 left breaking minimals? So I will explain what I mean with this. So we want to see how to achieve a vacuum configuration of the Higgs that is not trivial as I was introducing. So the first part of the exercise is understanding what type of Lagrangian I can write down for this scalar field phi. So let's see what I can write down based on Lorentz and gauging variants. So first, I can write down a kinetic term for this scalar field. So I have to use the covariant derivative. And this follows exactly the discussion that we have seen yesterday for the other fields of the standard model. So this will be a covariant derivative that contains terms with the gauge bosons of SU2 times U1, so namely what we call a mu and w mu a. And then using the tools that we discussed yesterday, we can show that this is indeed a gauging variant. But then I can also write down an additional term that is the potential for the scalar v of phi. That will be a scalar function of phi. So let's see what terms I can plug in. So I can write down a term like this, or this will be a phi dagger phi. In terms of dimensions, mu has mass dimension equal to 1. But this is actually not the full story because I can also add a quartic interaction of this type where this lambda is a dimensionless coupling. Now you can see that for a phi that is a doublet and that has some hypercharge, both these terms are gauging variant. I don't have to worry about having something that transforms on the SU2 times U1. So in principle I'm allowed to put both of these two terms. And actually we can check that if we want to write down terms that have a dimension up to 4, a mass dimension up to 4, these are the two only terms that I can write down in my Lagrangian, in my scalar potential. Now this is a scalar function and we can study it and see where this potential develops a minimum. So first let me write that this is a gauging variant. I already studied it in words. So what we can do is to minimize V of phi. You know, this is a relatively simple exercise and I can write down that phi has, since it is a complex SU2 doublet, it is composed by four real scalar fields. That I can write in this way, just in components. And what I can see, what I will find is that actually the minimum of the potential corresponds to my square over lambda if my square is bigger than zero and also lambda is bigger than zero. So of course I mean this is a potential that is multi-dimensional but at the end of the day, you know, if you have these two conditions that are satisfied, I'm really bad at drawing but you will have a potential like this. Well, and this is your Higgs field vacuum expectation value. And this is actually, this quantity here is what I defined earlier as V square over 2. So this is what I denote. So this is our vacuum expectation value of the Higgs boson. Okay. So obviously this is a very brief discussion about, you know, having this Higgs SU2 doublet developing a VEV but hopefully, you know, it gives a bit of an idea of how to achieve a theory with a minimum that is not trivial. Sonali, what do you think? Is that this a bit better now or mute it or maybe the hand is like the thumb up, I guess. I don't know. But yeah, there is also another question. Okay. So let's proceed with Vinat. Yeah. Sorry, I didn't hear you. Can you speak louder? Yeah. Yeah. What is the reason behind taking the potential, the formal potential like this? So this is again the usual reason that we want to write down something that is Lorentz invariant but for a scalar, this is a three-dimensional task. And then also we want something that is gauge invariant. So suppose that, so let's write here. So the conditions are Lorentz, gauge invariance. And also I want to have operators, so terms in the Lagrangian that have a mass dimension that is up to four. Okay. So if I want to have terms in the Lagrangian with the dimension up to four, in principle, I can already write down terms like this or square or, you know, I mean. This is trivially what I can write down and then combinations like this or this and so on, right? But then if you look at all the other extra terms that I didn't write down here, say, I don't know, take this term here. What you can show is that this term is not gauge invariant because this fly field is a substitute doublet and it has an hypercharge that is different from zero. So if you impose gauge invariance, you will see that these are the only two terms that you can write down in your scalar potential. And then, you know, in front of these two terms, I'm putting some coefficients that in principle I don't know. There can be anything at this point. I mean, at this point of the discussion, I just know what is the mass dimension of these coefficients in front. Does it clarify your question? Yes. Thank you. Sure. And then if the requirement is to have a vacuum that is not trivial, what we found is that we need to satisfy also these conditions. So the signs of these coefficients of these operators are fixed. Okay. Otherwise, you know, either you get the potential that goes to minus infinity at for large field configurations in the case of lambda t is smaller than zero, or you get a simple potential that has a minimum at zero. So these conditions here are necessary to have a non-trivial vacuum. All right. I think Sonali has a bigger question. So I'm muted here. Yeah. Hello. Hey. Yeah. So when you write phi not equals to zero, that means lambda is greater than zero and where the Higgs field is coming into the picture, right? Correct. Yes. Okay. Okay. Yeah. Understood. Thank you. Okay. Sure. Okay. Very good. So now that we have this, let's see the appearance. Let's discuss the appearance of the Higgs and W and Z particles. So far what we have is this scalar field phi. And then also you remember from yesterday we had the gauge. Actually, I called it a beam, sorry, different notation, but this was the hypercharge gauge boson. And then we had the gauge bosons of SU2. Hello. Thank you. I have a question regarding the symmetry breaking mechanism. Shouldn't we break to SU2 cross U1 because you wrote we broke, we break SU2? Yeah. Thank you. What did I write? What did I write that I broke SU2? But yes, I fully agree. It's a little bit sloppy. Yeah. Let's see. So you won why breaking. So there is a difference in breaking the SU2 or SU2 cross U1? Yeah. So in this case, so we'll see that indeed we are breaking a combination of this group of the SU2 times U1. And this we'll discuss right now. But we see that U1, the hypercharge is also broken because the gauge group that is left unbroken is a different U1, is the U1 of the electromagnetism that is a little bit different than the hypercharge, as you will see in a moment. Okay. Thank you. So in this sense, yeah, you are breaking both. Just that something will be left unbroken, as I said, as you will see in a second. All right. Okay. So these are the fields that we have and that we discussed already today, I guess, and today. But now let's see how we get to the Higgs particle W plus Magnus and C. Okay. We want to understand this step. So to do so, it's pretty convenient to write down the scalar field in, I would call them polar coordinates. In the sense that you see that we started and I brought it down above the field configuration like this. And now I can write down explicitly the minimum this way. And so this phi 124, a real scalar field, so new degrees of freedom that I'm adding to the Lagrangian understanding model. But actually, we can write down this field in a more convenient way, namely in the following manner. So I put here in front the vacuum expectation value. I have the two SU2 components. Let me put them here, 01. And, well, let me write, sorry, let me write first the exponent of E, sorry, i pi i of x v sigma i over 2. So let me write the full expression and then I will discuss it. 1 plus h x over v. So let's see what is this this beast. So these are our Pauli matrixes that we know of. So as you see here, I'm putting some fields. So these are real scalar fields. And I have three of them, one for each Pauli matrix. And then I know that my complex SU2 doublet scalar field has four degrees of freedom. Here I put three and then I add here one more. And this is another real scalar. So that's just a different way to write down my SU2 doublet scalar field. And then what we can do is to take the Lagrangian that we wrote above. So to take this Lagrangian here, I love phi. And then plug in this expression for the scalar field. So if I take the term, the kinetic term, and I replace phi with its valve, what I can show is that I get the following terms in the Lagrangian. So I will have a term like this. And then I have another term that looks like this, gw mu 3 minus. So as you see, so here I have the fields that we saw yesterday. Okay. And what you can recognize from here are simply mass terms, let me write in a different color, mass terms for the SU2 times C1 gauge bosons. Notice that you have these combinations here, right? So you see that you are mixing the gauge bosons corresponding to the third component of the SU2 isospine and the gauge bosons corresponding to the hypercharge. So this shows that we have a theory, a unified theory for electric interactions where basically your SU2 left and you wonder why I don't have the same footing because the third component of SU2 is just mixing with the hypercharge part. Okay. And this is, you know, we have, this is attribute, I mean, we have glascio, Weinberg, and salam. We introduced this unified way of looking at the electricity of the standard model. Okay. Now, of course, you know, we can, you know, diagonalize the system and find the gain values and the gain vectors and so on and so forth. And then what will come out are the physical gauge bosons that we know of, namely the W boson, W plus minus and the Z boson and the photon. So I can diagonalize the system. So I will have the appearance of the Z boson and of the photon. This is the photon. These are linear combinations of these W mu3 and B mu. So we'll have this is a angle that multiplies my W mu3 and then I have sine of theta W mu3 plus sine of theta B mu. Well, this theta is the Weinberg angle. So we can show that the sine of theta is given by a combination of gauge couplings of SU2 and U1Y. Namely, you have this combination here. So the gauge coupling of the hypercharger over the square root of the sum of the two. And then we can compute, you know, the masses of these two gauge bosons and we find that actually the mass of the photon is equal to zero. And the mass of the Z, let me write the square is equal to B square over four. We did expect and we did that this is proportional to the vacuum expectation value of the Higgs. And then you have this combination of couplings. So it's quite, you know, remarkable and is that we get a degree of freedom, namely the photon that remains massless. And this, you know, for these are a bit more for the experts. This is the field. So M mu is the field that corresponds to the unbroken generator. In the sense that we have, as we said already many times, we were starting from this gauge symmetry. We show that the Higgs field is breaking the symmetry. But then actually there is a remnant, so a gauge brew group that is not broken by the Higgs field. That is the U1 electromagnetism. And the generator, as we mentioned, I think yesterday, so the generator is given by the combination of T3, that is the third component of the isospine, plus Y, that is the hypercharge. Okay. So it's done this here. That's why basically, you know, we were, when we introduced for the first time the hypercharge, we connected to the electric charge in this way. But there is really a reason. So this type of breaking that is coming from the electromagnetism. Sorry, from the Higgs mechanism. Okay. And so this is for these two gauge bosons, the photon and the Z boson. And then I can also look at the other two. Let me put here an arrow. So we have this W1 and 2. You see that there are already mass eigenstates, but I want to build something that is also a eigenstate of the electric charge. That's why I define W mu plus minus, again, state of the hypercharge as the combination 1 minus plus I2 over square root of 2. And then we can show that the mass of this particle is equal to G square over 4 V square, again, proportion to the vacuum expectation value of the Higgs. Okay. So going back to our initial cartoon, so if we look here. So what happens is the following. So here in terms of number of degrees of freedom, we're starting with four degrees of freedom for the Higgs field. And then these were massless gauge bosons, and therefore they had two degrees of freedom each of them. Okay. Like a photon where you have two polarizations. We ended up after this Higgs mechanism with one degree of freedom that is this Higgs particle here that we introduced when we were writing down this scalar field. And then we had, well, I should obviously add the photon. We had these three fields here that we're acquiring plus one degree of freedom because they are getting a mass. And therefore they are getting one additional polarization. Okay. So you see that basically what I'm doing here is to reshuffle the degrees of freedom. So sort of, you know, the Higgs field is sort of losing three degrees of freedom in favor of giving the degree of freedom to the W and the Z bosons that are acquiring a mass. So what are the degrees of freedom that the Higgs field is losing? These are the pi fields that we write down here. And in fact, again, this is, you know, if we want to go a little bit more into the details of the Higgs mechanism, these are called Goldstone bosons. I don't have much space. But these bosons have the nice property that they disappear from the Lagrangian. In a particular gauge that is called Unitary Gauge. So basically, I don't plan to go into the great details of this statement. But what I want to say with this is that you can use gauge transformations, so this S2 times C1, to go to a gauge in which these pi fields do not appear anymore in the Lagrangian. And in fact, these are the degrees of freedom that we are losing because they will appear as a longitude in a polarization of this massive gauge fields after a little ximetry breaking. Okay. Is there some question? There's a question in the chat by Mitesh, which says, is there any geometrical interpretation of the Weinberg angle? I'm not sure what he means, geometrical interpretation. But so the Weinberg angle is just parametrizing how much this third component of the isospine and dipesh charge are gauge bosonal mixing. Yeah. Okay. Excellent. So the conclusion of this is indeed, I mean, we explained in very briefly the Higgs mechanism and the appearance of of massive gauge bosons. And I also mentioned a little bit what we mean with, you know, fields that sometimes you hear that some fields are getting eaten by the gauge bosons. And this is, you know, a sort of a colorful way of saying that we have some fields that are the Gorsum bosons that indeed can disappear from the Lagrangian in favor of longitude in a polarization of gauge bosons that get a mass through this mechanism. Also, the last note that I wanted to mention in this context is that if you look at these masses that have rolled down, so that the mass of the Z boson and the mass of the W boson, and also, actually, you put together with the information on the Weinberg angle, you find a nice relation, namely, you can define what we call the raw parameter. And this is given by the following combination of masses and Weinberg angle. And you can see, you know, using the equations that I gave you above that this is equal to one. And this is related to the custodial symmetry that maybe we'll have time to discuss today. So, related, and this is a symmetry that I mentioned at the very beginning of this course. And we'll discuss a little bit later today or tomorrow, just briefly. But this is, you know, to notice since I've rolled down the seven masses and Weinberg angle in this slide. Okay. Maybe it's a good moment to take a break, five minute breaks, if you are already here. Yeah, in two minutes. I have one more question. So, we have achieved all of these, you know, studying the kinetic term of the Higgs field. Okay. Once that we replace the Higgs with the vacuum. But then also, from this kinetic term, I can plug in, so use this expression for phi. And then we can extract the couplings of the Higgs particle with the gauge bosons. So, with the Z boson and the W plus manias. And this is, you know, a good exercise we can do that we can do at all. Can give you the answer because this is something that I will use in the second part of the class. Namely, what we will obtain are couplings of this type where you have two gauge bosons and the Higgs. And so if you put here the W boson, you get this coupling scale like the mass of the W boson. Similarly, you can do the same thing for the Z boson. And now you will have a scaling with the mass of the Z boson. And then you have also four point interactions, so between two gauge bosons and two Higgs. So you have W, W, Higgs, Higgs. And this is again proportion to the mass of the W. And then finally, Z, Z, Higgs, Higgs that is proportion to the mass of the Z. So the reason I'm showing you this is that, okay, this is a good exercise to do. And then also because we learned that the couplings are proportional to the mass or mass square, if you want. Okay. So we learn already from here that the Higgs couples more to more massive particles. Okay. So that's what I wanted to say to complete the discussion of this part of the Higgs Lagrangian. And yeah, we are ready for taking a break. Or I will absolutely take questions. I see that there are a couple of questions. I suggest that we take them after the break because maybe there will be many others. Sounds good. Okay. So let's stop for five minutes and then we'll take the questions and resume. Yeah. Very good. All right. Maybe we can resume taking some questions. So the first. Okay. So we have a question by Banjeles. Please go ahead. Yes. Yes. Thank you. I would like to ask about this custodial symmetry. What exactly is this? Oh, yeah. As I was mentioning, I plan to discuss it. Either at the very end today or at the beginning of tomorrow. Just very briefly to say approximate global symmetry of the standard model. And yeah, we chat more about it. I'm just delaying the question, but I'm planning to discuss it. I mean, it's possible that I would prefer tomorrow because later today. Okay. I'm going slow. So it's very probable. They like to record it anyways. Yeah, that's right. So the next question is by Abid. Yeah. Hi, Professor Gori. I have a couple of questions from the linear combination of JNU and ANU that you have written. So my question is first, I want to clear one thing from my understanding that out of those four degrees of freedom, two of them was eaten by W plus and minus and they acquire mass. And third one is eaten by G boson. Right. That's correct. Yeah. And the remaining one was the observed Higgs boson in experiment. Correct. Yeah. Exactly right. Yes. Yeah. So my question is that linear combination you have written for JNU and ANU, is it a fixed that linear combination that JNU is always written with the negative and AMU is positive or can't we flip it? Is it random? Yeah. No, it's written that way. It's simply when you diagonalize, you find the system and you find eigenvectors. You have that combination if you define the angle in this way with this charge of signs. And it's something that you can check it. But yeah, that's right. It's coming from here. You see, you know, you have this minus for the way that we have defined these interactions and then, yeah. Okay. And one other question. I'm expecting one mathematical step if I'm not wrong, that how is there any steps to derive that Mj square expression that you have written? Yeah. That's right. I mean, the only thing I'm doing here, I mean, I'm not showing the steps, but is to take this Lagrangian, this part of the Lagrangian here, and then I want to find the eigenvectors and eigenvalues of this Lagrangian. And then one of the eigenvalues is given by this, this mass square here that correspond to the Z mu eigenvector that I wrote down here. Yeah. Okay. Okay. And my last question is about the Weinberg angle that this unification of this theory is dependent on parameters. There are Weinberg angle G and G prime. Those G, one of the G is constant, that is, I think, electromagnetic coupling constant. So my question is how many unknown parameters were there at that time when this theory was developed? I mean, so all. Yeah. So as I think we mentioned yesterday, so there are only two, for the later week interactions, there are only two coupling constants that are what I called here G and G prime. And this is then fully defined, fully defining my gauge theory before the last week's symmetry breaking. So these are the only two parameters that are in the game. I apologize because as yesterday I was not present because of power load setting problem. Oh, sure. Yeah, of course. Yeah. So without measuring Weinberg angle and one coupling constant, we cannot predict the Z masks, right? At least we have to measure one parameter experimentally. Yeah. So in this time, so the mass is generated after that to be symmetry breaking. So you have also an additional parameter in the game that is this web. And this web is, you know, in the standard model, the standard model doesn't predict the web. So, so we need also a measurement of some mass to determine this web. So on top of these gauge couplings that I was mentioning, this G and G prime after the last week's symmetry breaking, you have also this web that is an unknown. So you need some additional measurement to determine it. Okay. Yeah, that's right. Because from this theory, we can predict the mass object by knowing the value of G and G prime. But what about V? Because that expectation value is the vacuum expectation value. Yeah, that's right. Yeah. Yeah. So also that is something that we don't know from the theory point of view. So as I was mentioning, so we need the measurement in such a way to determine this V. Okay, okay. Yeah. Okay. So we have to measure the V from known parameters. Okay. Yeah, that's right. That's right. Yeah. Yeah. Correct. Okay. So at the end of the day, I mean, we mentioned this a little bit, but so for example, at the lab, the E plus C minus collider running in the past, we had many, many measurements including, you know, the Z boson mass or the Z couplings to fermions. And then the idea is to do a global feat of all these measurements and then extract the information about the coupling constants. I mean, this G and G prime and the web that are, you know, unknown in the standard model in the sense again that the standard model is not telling us what is the value of these parameters. These are just three parameters for the standard model. Okay. My last question is from that equation QEM. Is that a Gellman, is it even a formula for electric and some kind of similar equation? So I have to say I don't, I don't remember that, that, that equation. So the Gellman, but it can be, I have to check, I don't remember the name. Yeah, here I presented it in terms of where was it? Yeah, this, this equation here that is, you know, determining the generator of the group that is not broken by the, by the Higgs vacuum expectation value. Yeah, because I think why is simply hyper charge, right? Correct. Yeah, yeah, yeah. Because of that I have a doubt because this equation looks similar to Gellman, but there it was, I think Y by 2 was there. Oh, so these factors are two, sometimes in books you find these two, I think, or that, yeah, it's a matter of normalization. Sometimes the hyper charge is defined with a factor of 2 compared to these notes, sometimes not. So long as everything is consistent is, it doesn't matter. Yeah, yeah, yeah. Okay, okay, thank you. Okay, next there is a question by Svi Ruba. Hello. Yes, can I ask you my question? Yes, please. Yeah, is it possible to estimate the relation between Higgs mass and its vacuum expectation value in standard model physics? Yes, we get to there in a, in a bit, I believe we say something about it. Let's, let's delay a little bit this question and then we chat again if there is something of care. Yeah. Thank you so much. I had another question and what I asked in the chat box and it was answered, but still I want to check it again that actually I was trying to calculate the row parameter in case of beyond standard model using two triplets Higgs scalars and one of them was having a complex VV. Now, someone in the chat box suggested that using that U1 symmetry, we can choose always the vacuum expectation value to be real. But in my case, I'm actually having a complex VV. I don't know if I need to use the real component of the VV in the row parameter calculation or I should use the modulus only. Okay, so it depends a bit on the question. So in the standard model, so what we are discussing in this course, indeed, we have freedom to choose. Give me a second. So we, so we have some gauge freedom in such a way to gauge and also refacing freedom in such a way to choose the vacuum expectation value of the Higgs of this form in the sense that it is on the lower component of the doublet and this real. So this is for the standard model. Now, if you go beyond the standard model, sometimes it's not true. So sometimes you might have like vacuum that are much more complex. I mean, that so sometimes you can have a vacuum in that are breaking different type of symmetries, including CP. So that one has to be a little bit careful. So I don't know the exact model that you're considering, but sometimes it can be that the vacuum is not as trivial as sorry, as in the case of the standard model. So particularly I'm working with two triplet Higgs scalars and one of them acquires vacuum expectation value complex. Yeah, yeah. Yeah, it can very well be that then you cannot get rid of of that contrary to the standard model. Yeah. Yeah. But yeah, maybe we can, I mean, we can chat a bit more later or soft line about this because this goes pretty much beyond what we're discussing here that is the standard model. But it's true that it can be much more involved than in the standard model case where the vacuum structure is pretty simple. Okay. Thank you so much. Okay. So if there are no other questions, so there are a couple of more things I wanted to mention before entering the discussion of the Higgs phenomenology, namely that so there are once now that we found finally, you know, the W and the Z bosons, we can compute all the couplings of, for example, the fermions with these gauge bosons. So we'll have a fermion W or Z couplings. And again, this is a good exercise to try to compute these couplings. And they all come from from covariant derivatives. So we saw that if you take a fermion again, psi i of the standard model, then you have terms in the Lagrangian like this. And these are indeed inducing fermion couplings to W and Z. And what you can show is that, again, I would suggest to compute this at home, is that the Lagrangian will contain terms like G over the cosine of the Weinberg angle, C mu. And then you have the couple, I mean, the third fermion, so the standard model. Then you have here T3 minus sin square of theta, the electric charge of the fermion, and then psi i. So this is the coupling of two fermions and the Z boson. Then you have couplings with the W bosons that look like this. This time, as we have learned several times, this is only for left-handed quarks and the leptons. That's why I'm putting here a left. And then we have also couplings with the photons, with a photon that is given by the electric charge, and then A mu, and then a vector coupling gamma mu psi i. With this electric charge, actually I didn't define it, but you can show that this E, that is the electric charge of the electron, is given by G, that is the SU2 coupling constant times sin of theta. As we said, everything can be expressed in terms of this G and G prime. So that's what we are doing is now writing down the full Lagrangian of the standard model after the electric symmetry breaking, because now we have really all the ingredients for doing. So we can generate all the couplings that then we can test experimentally. Yeah, I see there are some questions. Yeah, let's take them. So the first one by Sri Rupa, please go ahead. No more questions. I forgot to just do all my questions. Okay, okay, okay. So there are no more questions, I think. Okay, and then a last comment, maybe yeah, let's not go much into the details, but I wanted to mention this because it's a very important aspect of everything that we are discussing is that we have presented electric symmetry breaking that is happening spontaneously in the standard model. Actually, yeah, I should write down this somewhere. Maybe let me pass you a question, because it was missed before, by Shantanu, please go ahead. Hi, Professor Bowden. Hello. So I need to ask you a very simple question. I'm wondering, there's some coupling between G boson and Higgs. Two G bosons and Higgs coupled to each other and give us a baron detector. So we can calculate them, but two Higgs doesn't couple with G boson. What is the reason behind that? Oh, you can see it also based on the age invariance if you want, and then it comes out once that you expand this covariant derivative if you do it properly. Yeah, okay. So what I wanted to mention is the following. So suppose that we don't know anything about the Higgs mechanism, but then we do know from experiments that the W boson has a mass. So if no Higgs mechanism, but the W boson has a mass somehow coming from some unknown source or some unknown UV physics, we end up actually having a problem. So, namely what we can do in a theory like this. So the standard model of particle physics as we have learned until yesterday before electricity symmetry breaking, but with a massive gauge boson. Then since the W boson has a mass, what we can do is to compute the scattering of two longitude and highly polarized gauge bosons. So WLWL. So this is a longitudinal polarization. So we can compute the cross section for two longitude and highly polarized W boson to go to two of them. Okay. And so we can compute this process in the standard model without the Higgs. And we end up having a problem, namely that the matrix element that we find summing over, you know, all the diagrams contributing to this process scale, scales like some coupling constant square, S that is the center of mass energy over the mass of the W boson square. Okay. So this is a problem. You might wonder why this is a problem. The reason being that, you know, you can think about doing the experiment, so, you know, starting your scattering and going to energies that are bigger and bigger in principle infinity. Okay. And what you see is that indeed this cross section is blowing up. Okay. So that tells us that the theory is a certain point is not well defined anymore. So as we say, we have problems with perturbative unitarity. So meaning that if we go to a bit really high energies, so in the UV, your theories start to be, start to have, start to be in trouble with unitarity. Okay. And so this is showing that we can compute what is the bound from perturbative unitarity. And we learn, so the bound is at energies not too far away from the TV scale. Let me call it two TV, just to give you an idea. So this tells us that we need something. So we cannot write down really the standard model theory without the Higgs, without something, you know, and with masses of these gauge bosons. Okay. And once that we introduce the Higgs, so this is, let's see. So this is the result without the Higgs. So with the Higgs, so once that we take the Lagrangian of the standard model that we have discussed in these three classes, this process will receive additional contributions. As for example, you know, you will have diagrams like this where you have a Higgs boson in the middle. And now the cross section is well behaved in the sense that you don't have this problem of perturbative unitarity. Okay. So this was really a very, very, very strong argument to believe that the LHC, you know, before it started running to discover something, something that was unitarizing this WW scattering. And because these are the energies tested by the LHC. And indeed, the LHC, as we all know, discovered the Higgs particle in 2012. Okay. So I can show you a slide about this. I find it pretty cute. So this is a slide. You can see my screen, right? The slide. Okay, great. So this is a slide about the Higgs boson discovery. So okay, this is a, you know, a picture that was taken the day of the announcement at CERN for the 4th of July 2012. As you see written here, so this was done using really a small amount of luminosity. It's LHC. So only five inverse Fentobarno 7 TV date and five at 8 TV. And for comparison, now we have something like 140 at 13 TV. And these are the two plots. I mean, two of the plots that the two discovery papers by Atlas and CMS showed us. One for the Higgs, they came to two Z bosons and two four leptons in the final state. We'll discuss a little bit of phenomenology of the Higgs, so we'll understand better why the Higgs did case in this manner. So what you see in this plot are really data. So this is, you know, these are data points. And the invariant muscle, the four leptons should give us the muscle, the Higgs boson that was unknown before the Higgs boson discovery. And you see that around 125 GB, that is the mass that we have measured, we had this excess of events, this tiny bit, because you see that we had only a few events back then. And this was in blue, in dark blue, you see what was expected by the standard model. And you see these two data points that are giving us this small bump at around 125 GB. And then similarly, we had the study of the Higgs, they came to two photons. And again, you see this small bump above the predicted background. And this was obviously super exciting. So having said that, I can go back to my iPad. And what we want to do now is to study a little bit of the properties of the Higgs boson. And let's see. Okay. So back to the iPad. So this is our section 3.4 properties of the Higgs boson. So we have to study the properties, we have to understand how the Higgs couples interact with all the other particles of the standard model. We have already learned that the Higgs couples to the W and the Z in a way that is proportional to the mass square. Okay. So this is what we already mentioned. Then we might wonder about the Higgs coupling to fermions. So two quarks and leptons. So these couplings are coming from the Yuccavala grungian that we have already introduced. So let me write it down again here. So this was our phi scalar doublet, q left. And then what we can do is to replace this simply with the vacuum plus the Higgs particle. And then if we do this exercise, what we will see, for example, if you take, I don't know, this first term, you will have something like d left, then you have v plus h square root of 2 d right. And you see that, so this term here is giving us the mass of the, in this case, the right, the down quark. And here you have a coupling of the Higgs with down quarks. And this coupling is once more proportional to the mass. So this will be the mass of the down quark over v. Okay. And similarly to for all the other couplings of the Higgs with famines. So we always find that it couples proportional to the mass. Can I pass your question now by Abid? Please go ahead. Yeah. My question is that Higgs couples to W1 or Z with proportional to this. So it gains mass before coupling or after coupling? Because if it's proportional to mass, then it should first interact with Higgs field, then acquire mass. Yeah. So let's be a bit careful not to confuse the Higgs field with the Higgs particle. So the Higgs field is this by here. And then I'm writing all the couplings, and then I'm studying the vacuum and so on and so forth. And, and then, I mean, the masses are generated once I look at my theory at the minimum. And then I do a lot of recent breaking and so on. But then I have also this physical degree of freedom. That is the Higgs particle and we learned that the Higgs particle couples to the massive cage bosons. Hopefully, yeah, it answers to your question. Maybe. Yeah, that cup. Yeah, that matter couples to Higgs field, then it's a square mass. I think it is the right statement than matter field couples to Higgs particle to as well the mass. I think which one is the right that I want to clear. Can you repeat the last part of the last question? Sorry. Yeah, yeah, there are some confusion between the statement that it is the Higgs field, which couple or formulaic field which couples to Higgs field then acquire mass rather than Higgs boson which couples to the fermion to get the mass. So it's the Higgs field. So this by here. That is giving mass to the massive cage bosons. And as a consequence of the Higgs mechanism, you got that you have a new degree of freedom that is the Higgs particle that is coupled to the massive cage bosons. But this is the Higgs field. So this by here that I'm writing down here that gives mass to the massive cage bosons for the standard model. Okay, okay, okay. My another question is from that Higgs couples to fermion. Is it possible to couple Higgs field with any kind of that fermionic field which is acquiring more heavy mass than Higgs boson because we experimentally know the mass of Higgs boson. Yeah. So in principle, you can couple the Higgs to other type of fermions beyond the standard model. That's the possibility. Yeah. And those can also be quite heavy. Of course, I mean, if they are super heavy, then you have a problem of having a perturbative theory because your couplings need to be very large because we know the vacuum expectation value of the Higgs. Okay, yeah. I asked this question because already we have one massive fermionic field that is our top quad field. Yeah. And the mass is much heavier than Higgs boson. So that top quad field is acquired mass through the same mechanism. Right. Yeah. Okay. So it is possible to gain mass with Higgs through spontaneous breaking even more heavy than the field itself, field quanta itself. Yeah. In principle, yes. In practice, as I said, you cannot go too heavy because at a certain point your couplings are really huge. And also at the same time, large couplings mean that most probably, you know, in general, you have to agree with experiments, right? So if you have a Higgs couple, that couple is very strongly to some other particle, then you have to understand, you know, how you link this statement to experimental surgeries. So one has to be a bit careful not to be in tension with experimental analysis, but in principle, yes. Yeah. Okay. So I can assume it that top quark mass is not quite acceptable according to this mechanism. Is this still a contradiction or some controversy? No, I mean, the top quark mass is just it's just coming from the Higgs mechanism. And I mean, it's just this term of the Lagrangian, right? Once that you replace the Higgs with a web. And yeah, that's the there is no problem here. Yeah. Okay. Okay. Okay. Okay. Thank you. All right. Next question is by Sonali. Yeah. So this is a generalized question. I want to know that can we estimate how much amount of mass we are getting from Higgs? Like, for example, B quark mass is 4.65 GB. So can we estimate how much amount of mass is coming from Higgs? Obviously, 100% mass we are not getting from Higgs, right? For elementary fermions, that's the statement of the standard model that the mass of the elementary fermions is coming from this Higgs mechanism. So yes, it is 100%. Yeah. For elementary fermions. 100% of mass is coming from Higgs. Yeah. For, as I said, for elementary fermions, if you are thinking about, I don't know, the proton mass or whatever, then this is a different story. But if you are thinking about elementary fermions, yeah, this is the statement of the standard model. Yeah. Okay. So it can be different for protons. Yeah. That's right. Absolutely. Yeah. Yeah. For protons, it's about 5%. The rest is coming from color, color dynamics from confinement. Okay, okay. Thank you. We have another question, I think. Yeah, Abhi, do you still want to ask a question? Yeah. Yeah. My question is about the Higgs self mass. Is there an explanation how Higgs acts like a mass? Because other particles are very large. Yeah. So I'm getting there right now. So just, yeah, stay with me. Okay. Yeah. So indeed. So I was explaining the coupling of the Higgs with gauge bosons and fermions. And then we might wonder, are there other interactions of the Higgs? Yeah. So other interactions of the Higgs. And actually, there are other interactions that are coming from the scalar potential that we wrote down already. Here, write it down again for completeness. So what we can do is, as I said, we have this non-trivial minimum. So what we typically do is to trade these three parameters, so the standard model of the grand general, with more physical parameters, namely the vacuum excitation value, B, and the mass of the Higgs boson that we have measured. So this is a simple exercise that we can do at home. And we have already seen the relation actually between B and mu and lambda. We can also extract the mass of the Higgs. And actually, you know, we get this nice relation, namely that the mass of the Higgs square is also proportional to the VEV and square in this way. So we see that, you know, we would expect to have the mass of the Higgs that is at around the electric scale, roughly speaking, so at around the Higgs vacuum excitation value. So having said that, once that we change these variables, we can rewrite the scalar potential for the Higgs particle. And what we find are these three terms. So you have a mass term for the Higgs. Then we have a H to the third power type of interaction with this pre-factor. And then we have a H to the fourth that is multiplied by the mass square over 8 V square. So the reason I'm showing you this is that, so first, yeah, you, some of you were asking about the mass of the Higgs. And we see that this is a relation with the vacuum expectation value. And then also we learned that we have Higgs auto interactions. So interactions of the Higgs with itself. So we'll have couplings like this and this. Okay. And then also what we see is that this type of potential, so the scalar potential of the standard model is highly predictive because we know these coefficients as soon as we measure the mass of the Higgs and we measure the vacuum expectation value, at least indirectly through the measurement of the gauge boson masses. Okay. So the standard model is highly predictive. I mean, this is a general statement but we are discussing this in this context of the scalar potential. Okay. And then let's think, am I missing any other interaction of the Higgs with anything? Seems to be pretty much complete. But from these nodes, we see that we don't have any coupling of the Higgs with massless gauge bosons. And this is the last bullet point I want you to discuss in this context. So does the Higgs couple to photons or gluons? Okay. We might say no because the mass of the photon and of the gluons is equal to zero. And then we have learned that the Higgs couple is proportional to the mass. But then also we might say yes. So we have seen that the Higgs was discovered through its decays to gamma gamma, so to photons. I showed you 10 minutes ago the slide. So, you know, we have to understand how we have this coupling, how to generate this coupling. And in fact, you know, the right answer is yes, just that we need to go beyond the three-level in perturbation theory. And so at the three-level, indeed the couplings are equal to zero. But at one loop, we can write down on final diagrams like the following, where you have a loop of the corks, of the standard model. And here you attach two gluons. Here you have a Higgs. So you see that this is an effective Higgs-gluon-gluon type of interaction. And then similarly for the coupling with photons, I can do the same thing. So I can plug in here the corks of the standard model that are charged under electromagnetism. But then I also have another diagram with the W boson that is also, that has also an electric charge. Here I put my two photons and this is the Higgs. So these two diagrams are generating my effective Higgs-gamma-gamma coupling. That is the coupling that was responsible of the Higgs decay to the photons that we have discovered at the LHC. Now, let's see. So I got many questions. So what do you think? Can I continue a little bit of discussion for, or it would be a good... Sorry, you can take five more minutes, if you wish, and then we'll take questions or five, ten, whatever. Yeah. So let me stop sharing my iPad and then I want to show you a couple of slides again. But yeah, before doing so, I wanted to, so that we don't lose track of what we are doing. So in this page here, I wrote down all couplings of the Higgs with the other standard model particles. And you see that once that we specify the masses and we basically define all the couplings of the Higgs boson with any other particle of the standard model. And then we can start to make predictions for the LHC. And this is what I'm going to show you in these couple of slides. So what is the phenomenology of the Higgs at the LHC? So let's discuss a little bit the production and the decay of a Higgs particle at the LHC. So first I want to mention that at LHC that we're colliding protons, the production of the Higgs is a very, very rare event in the sense that we really need many, many, many proton-proton collisions in such a way to produce a Higgs boson. And in this histogram here, so what I'm showing is how many proton-proton collisions we need, so something like 10 to the 10, in order to produce a Higgs boson either in what we call gluon fusion that is represented here by this Feynman diagram, where basically, effectively, you are taking two gluons of the protons that you are colliding, or the protons that you are colliding to produce a Higgs through this Feynman diagram here. This is the Higgs-gluglu one-loop interaction that we just brought on my iPad. And then we have also some bleeding production modes of the Higgs boson that are even more rare. So this is the Wettel boson fusion, number two. We see the Feynman diagram here. So this is happening because of the Higgs coupled, the Higgs coupled to the W and the Z of the standard model. And then we have the Higgs produced in association with the W or Z. We have this Feynman diagram here called Higgs-Strahlung. And then finally, we have here a diagram for the Higgs produced in association with two tops. And we have this diagram here. So overall, you see that indeed we need incredibly many proton-proton collisions to produce a Higgs particle. So it's very low probability. But since we know that we have incredibly many proton-proton collisions at LHC, indeed we were able to produce quite a few of them. I think we are talking by now by a couple of millions or a bit more even. And so this is for the production of the Higgs at LHC. And then for the once that we produce the Higgs, the Higgs will decay. It's not a stable particle. And then we can make predictions using the standard model Lagrange that we have studied Monday, Tuesday, and today. We can make predictions for the several probabilities of the Higgs to decay to whatever particle in the standard model. So in this plot that you see here on the left, I'm showing what are the branch ratios, so the several probabilities for the Higgs to decay as a function of its mass. Obviously, you know, before the Higgs was on discovery, we didn't know the Higgs mass. And now we know, so we are here at 125 GB. And what is interesting is that being here, you see that you have several of these lines that correspond to a branch ratio that is not too incredibly tiny. So we know that having this mass, we will be able to measure many of these decay modes of the Higgs boson. So you see that the Higgs is mainly decaying to BB bar, so two bottom quarks. Then we have the Higgs decay to WW. We have the Higgs decay to tau tau, charm, z boson, gamma gamma, z gamma, mu mu. So we have all of these that can be measured by the LHC. And we have very precise predictions from the standard model Lagrangian. Okay, so this is what we see in this plot. In the plot on the right, what I'm showing is the total width of the Higgs boson. Again, if we are here at 125 GB, we see that the width is tiny. It's like a few amoeba. And this is telling us that the lifetime is, so actually the Higgs is decaying pretty quick. So as soon as you produce it, the Higgs also will decay. And here I leave a little exercise. So using the standard model Lagrangian that we have learned from Monday to today, we could try to compute the several widths of the Higgs into WW, ZZ, as well as fermions. So the fermion calculation is easier. So this is what I would start with. And then we can check that indeed we get roughly this type of decay. So this type of franchinations. So this is for the standard model predictions. And then, yeah, I think we should delay to tomorrow. We would like to understand how we can match and we can compare these standard model predictions with data. So what experimentalists are giving us and what we learn using data on the Higgs coupling to the other standard model particles that we have introduced today. So that's, yeah, I think the task for tomorrow, for the first part of tomorrow. So I think I can stop here and take questions. Perfect. Thank you. So now we can go to the Q&A session now. So, okay, that was I was expecting the recording to stop. So any questions? Okay. So we have a question from Stefano. Please go ahead. Hi. So you showed us how to have a decay of the Higgs in photons in your notes. And the second diagram involves the W, the boson and the photon. Well, the interaction between the W and the photon, does it come from the the kinetic term of the Higgs field? And no, it does come. So let me go back to the iPad. I think let's go back maybe to the notes of yesterday so that I can show you directly there. Let's see. If I find it otherwise, I will write it again. Okay, here it is. So yesterday, you remember that we introduced, so this is part of the notes of yesterday, we introduced the covariant, so the Lagrangian that has a SU2 local invariance. And so you remember that we said that if we want to write down this term, this F mu nu F mu nu for the corresponding SU2 gauge bosons, we need to add this term here in the definition of F mu nu. Okay. And this will lead to three-point interactions and four-point interactions. And this is basically the starting point of that. Of course, you have to work a little bit because then, you know, you have to redefine, you have to define the W plus minus and so on. So you have an SU2 times U1, but it comes from this type of terms. Yeah. And so the Higgs has nothing to do with that. So for this coupling here that you were mentioning, so this is the coupling that you were talking about, right? Yeah. Yeah. Okay. Thanks. All right. Next question by Fenwei. Please go ahead. Hi, Stephanie. So I have a question about group theory. So can you give me more details about two cross two equals to one plus three? Yeah. I'm not understanding. It's notation. Yeah. Let's see where did I write it once. Okay. Here it is. Yeah. So first let me give you a reference and then I will mention a few things. So there is a nice a nice book by Georgi. I don't remember exactly the title is something like the algebra in particle physics or something like this particle physics roughly. I am not sure about the title. Let's put a roughly this where this is a, you know, a very good reference if you want to learn more about the groups from a particle physics perspective, not, you know, not for mathematicians if you want. So going back to your question. So here the idea is so you have two doublets, you know, just generic doublets that I can write down in this way. And then you have to understand what type of products you can do between these two doublets. Okay. So one thing you can do to put together these two doublets is to build the singlet. So you can be the singlet simply doing the scalar product between the two doublets. So you will have let's see a terms like this. And what you can show, so we have introduced SU2 transformations. And what you can show is that if you put together the two SU2 doublets in this way, so with the scalar products, what you get out is a quantity and object that transforms as a singlet under SU2. Okay. But then this is not the only possibility because in principle you can put together the two doublets using the epsilon tensor. And if you do so, you can again demonstrate that you get the triplet representation of SU2. So that once that you do a SU2 transformation on this, on this object, you get something that is transforming that is not a singlet. Yeah. So this is what I meant with this two times two that is equal to one point one plus three. So that you have two possibilities of building a representation that is either a singlet or a triplet. But yeah, you can find many, many more details obviously in this book if you want to learn more about the Lie groups for particle physics. Yes. Thank you. So another question is about the diagonalize of the W boson. So when you calculate the eigenvector of the W boson, so can we just use W mu one and W mu two as the eigenstay for the W boson instead of the combination? Yeah. So indeed, these W mu one and W mu two, there are mass eigenstates as you can see from here and here. So from that point of view, it's perfectly fine. The problem, the reason that you don't want to use those is that they are not, they are not eigenstates of electromagnetism. And since electromagnetism, so this U1 of electromagnetism is unbroken, you would like to work with fields that are eigenstates of this quantum number. That's why we define these two combinations here and these two combinations are indeed eigenstates of the electric charge. That's why we do that. Not to diagonalize masses, but to have something that has good transformation properties under electromagnetism. Great. Other questions? Yeah. Next question by Abhishek. Please go ahead. Hello. I have a question related to the X field. In the X field, we actually have four real degrees of freedom, right? Right. And when the spontaneous symmetry breaking happens, three of them are eaten by the gauge bosons. And so when we are walking above the energy scale of electroveic, I mean, when the symmetry is not broken, do we explicitly see these extra three degrees of freedom in the detector or do we have any signal that says these? Yeah. Yeah. So at the energies that we are testing in our labs, we are working, I mean, we have a lot of symmetry breaking and then we have all the masses of the gauge bosons and so on. So the energies that we are testing are energies where the electric symmetry is broken. Yeah. So you can ask the same question from a cosmological perspective and at a certain point, you restore the electric symmetry in the early universe and then this is a different story. But for now, for the nature that we are studying today, then electric symmetry is broken. And then we don't have these goldstone bosons in the sense that we have the double and the Z that are massive with a longitudinal polarization. But currently we have experiments being conducted at 13 or 14 TV, which is above the electroveic scale. So do we not see this massless scale of fields? No, the electric symmetry is broken at those energies. Yeah. I mean, when you measure the, you know, you have your W and Z boson that are massive at those energies. So at which scale do we have gauge bosons to be massless then? Yeah. Yeah, I think, Johan, you know the answer to this. I was distracted by the question. The question was? My question is, when do we have these gauge bosons of electric symmetry? I mean, is it to become massless? I mean, what is the exact energy scale? Is it 150 GeV, which is the electric scale or? No, no. So they're always massive. However, at high energy, at very, very high energies, it could be a good approximation to to elect the masses. There is no more than that. Yeah, just that, I mean, I was linking that to a cosmological question that you have in the early university, where you don't have electric symmetry breaking, and you're at very, very high temperatures. I don't remember the exact, you know, phase transition, but you know, when it happens, but it's, you know, the phase transition. Yeah, that's right. So this is, yeah, this is something I was mentioning, because otherwise, I mean, for our purposes, as I said, for laboratory tests, you have always massive WNC bosons. Sometimes, yeah, as Johan was mentioning, sometimes for a matter of calculation, it's useful to maybe neglect the masses, because they are a little bit smaller than the energies that you are testing, as you were saying. But electric symmetry is still broken. So when the symmetry is restored, do we not see the scalar field then? So do we always have to assume that W bosons are massive, even at 10 to the power of 10 GeV? Is that what you're saying? Yes. Yeah, that's right. Okay. Yeah. So it's not just by electric symmetry breaking. Okay, I get it. I think I understand it. Thank you. Sure. All right. Next question is by Abid. Hello. Hello. Yeah, my question is about the left handed and right handed component of the fermionic field. When we are saying that fermions acquire mass through the Higgs coupling, so we are specifying, specifically, the left component of the fermionic field, right? Yeah. So the mass terms are always combinations of the left handed field and the right handed field. As you can see from here, let me highlight there. So you see that I'm taking a left handed field and the right handed field. I don't know if that answer your question. Yeah. Yeah. My question is when we are talking about the group symmetry in electric, we are talking about is Q2L, that is the left handed fermionic field. So yeah, that is my question. So what happened with the right handed fermionic field? Yeah, so the right handed fermions don't couple to the W bosons as we were mentioning yesterday. So that's the reason that the standard model theory is carol in the sense that the left handed fermions are different from right handed fermions. They couple differently. But then your color symmetry, so the fact that the left and right fermions are different is broken by the fact that you are introducing this mass term, this type of mass terms that are put in together left handed fields and right handed fields. Okay. So Higgs field is not coupled to right handed fermionic field? No, it is. You see this combination. So if you write it explicitly, let's see it here. I brought it down again later in the notes. Yeah, so if you take these three terms in the Lagrangian and then you replace the phi field, what you will end up having are couplings like this, where you have a D left and a D right. So basically this will generate a coupling like this, right? The right and the Higgs. So you see that the Higgs couples the combination of the left handed field and the right handed fermion and the right handed fermion all the times. Okay. My next question is that even in left handed field, noitinos are very exceptional compared to quarks. Like both quark and leptons, they are present in doublet, right? All the six generations of quarks and leptons. But noitinos are not acquiring mass through Higgs mechanism, although electron, muon, tauon acquire through the same mechanism. So my question is being left handed noitinos, why they are not acquiring mass? I think they have different mechanism for mass generation. Yeah, that's right. So we'll discuss more of this on Friday. But just one word is that if you add sterile noitinos, then you can write down couplings of the Higgs with noitinos as well. Just that you need to add these additional fields in such a way to be able to write down a yukawa like this for the Higgs field. But we'll see this better on Friday. Okay, okay. Thank you. Thank you. Anyway, it's a very nice presentation. Okay, very good. Okay, I think we'll have one last question that we'll take. And then I think we should leave now by Abishek, that's one. Sorry, I already asked the question. Okay, because your hand was still up. Okay, so then we are on time and we are done for today. Okay, great. So many thanks for the lecture and see you tomorrow. Yeah, see you all tomorrow. Thank you for the questions. Bye.