 we have the last lecture on a standard model, please. Okay, very good. So welcome back everybody. So today we want to finish briefly the discussion of the quark flower physics. We'll introduce a little bit about the Neutrino physics and then we'll talk about why beyond the standard model physics. So that's the plan of today. So I will start with connecting with what we have seen yesterday about the CKM matrix. So let me show you a couple of slides. Let's see. You can see my slides, right? Okay, very good. So yesterday we introduced the CKM matrix. And we have seen that it does parameterize the coupling of the several quarks with the W boson. We have also learned that this matrix is unitary. And what we can show actually is that, as you can see here in this slide, the CKM is fully described by four parameters that are, as we see here, these are three angles and one face. So the three angles you see here are these one, two, one, three, and two, three angles. So these are rotations on orthogonal axes, x, y, and z. And then we have this phase that I call delta here. And actually, you can show that this brings CP violation. So it will bring CP violating interactions in the standard model Lagrangian. And actually, this is interesting that this is the only source of CP violation of the standard model. I put here almost only because, you know, we have also a strong CP phase in the QCD settler of the standard model Lagrangian, but we didn't talk about this in this lecture. So in the electric interaction, so this is the only CP violating phase, including also the Higgs interactions. So this is what you see here is the standard parametrization of the CKM matrix. Also, as we're seeing in the next slide, actually, there are many, many measurements of the CKM matrix. And what we have learned is that you can think about this matrix as being equal to the identity plus corrections that are small. And the way that we can parametrize this structure is using what we call the Wohlfeldstein parametrization, namely using a small angle that is called the Khabib angle, this lambda parameter, and then expanding in power of this lambda. And then you can see this expansion up to lambda to the third power type of corrections. And you see the appearance here of these other parameters that in general and in the tau of the older one are roughly. And as expected, also here in this parametrization, we have four free parameters that are completely describing these metrics. So these are the two parametrizations that we usually use for the CKM matrix. And now let's connect these two measurements because these four parameters of the CKM are not predicted by the standard model. So we need to have at least four measurements, if not more, to determine them. So the way that we present experimental results is using what we call the unitary triangle. So let's see what this slide is telling us. So we learned and I mentioned already several times that the CKM matrix is unitary. Then we can write down the unitary relations between several elements of this matrix. And here is what I write, where A and B can be any number between 1 and 3. So there are six unitary unitary triangles. Because you can think about these several terms here, simply vectors in a 2D plane. And then if you have some these vectors, you have to obtain this triangle that is closing. I hope that that makes sense. And in particular, what we do is to interpret the measurements in terms of the specific triangle that you see here. So this combination here of the CKM elements. And what we do typically is to normalize one of the sides to one. So you see that the other sides are normalized to this combination of VCB, VCB star. And then you see that I can close the triangle at some vertex that I need to determine experimentally. And so this is a, this is theoretically what I get from just thinking about unitarity. Let's see what my measurements are telling me. So in this plane here, so this is the row bar at a bar plane where row bars, row bar and data bars are defined here. You have row bar that is basically the row that we have seen in the Wolfenstein parametrization. So this row here. Normalized by some combination of the KB boeingol. And then similarly for data bar, so this is what we see here on the X and Y axis. And then in this plane, I start to put all my measurements. Okay, so I want entering to the details. So you can ask me about these, what the several regions in color correspond to the measurements, to several measurements. And what we know for your unitarity, if unitarity works, is that all these measurements need to pass from one point that is identifying this vertex of the unitarity triangle. And then indeed, if you look into the details of this plot, you can see that all the measurements indeed pass from here. And actually, we can determine this vertex here in quite good precision. And if you're curious, if you go to the Sihem Peter webpage, you can see, you can find actually plots also of these feet in the past, using past measurements. And you see that in the past, this point was determined with much, much smaller accuracy. Here, I'm reporting a plot of 2004. It's a little bit small, but you can see that these bends here were much, much broader. So the uncertainty in all these measurements was much bigger. Okay. But then, so what we conclude from this plot is that indeed, the measurements that we do seem to agree with the prediction of the standard model, namely that the Sihem is a unitary matrix. Okay. I'm not sure if there are questions on this. Yeah. Otherwise, I can show you the, you know, some of the measurements that are entering in the determination of the Sihem elements. I'm listing them here. So this is not the full story. There are many additional measurements, but these are some of the measurements that are determining directly the seven elements of the Sihem. I don't know. We can take this entry here, VCD, and you will have dependence on VCD if you study the D-meson decaying to a pion, a electron and a neutrino, because in this transition you have, so in this decay you have a CD type of transition, a charge current. So here you will have your W boson, and then you have all these other, you know, processes for the other elements of the Sihem. Some of these processes are determined with a super, super good accuracy. So in blue, we see here the neutron decay that is determined at a level of 0.01%. Some other decays like k2pion, no, it's determined with an uncertainty of 0.1%, and some have, you know, an uncertainty that is a bit larger. Now, you put all together, and indeed you see that you have a Sihem, as you see here, that is approximately given by the identity plus corrections. So you see that these off-diagonal terms are non-zero, but they are relatively small. So the Wolfgang parametrization makes sense, so, you know, to have this expansion in power of lambdas of the Cabebo-Ingo. And then also what this structure is telling us is that indeed we expect to have a flower violation, namely, you know, the transition between different flavors, say between the first and second generation would pick this element, between the first and the third, I would pick this element, and we see that these transitions are suppressed by small CKM elements. And yeah, I would suggest you to keep in mind the structure, because then we'll compare the structure to the structure that we have in the Neutrino sector, as we'll discuss maybe in, I don't know, 20 minutes or so. In the meanwhile, there's a question, if you want, I guess. Yeah, hi. So my question is, so this measurement of the unit in terms of the CK matrix, this is also a measurement of the number of the generations, right? So if there were like four generations, then the CK matrix would not be unitary. So is this measurement best measurement which we have for the number of generations, because this looks like a very good measurement already, but are there other measurements which we probe this in a better way actually? Yeah, very good. So I feel agree with you that if you have a fourth generation, then what would happen is that you have a relation like this, but you have a fourth term. So if you just restrict to the three generations that we have discovered that you wouldn't have any more this unitary relation. So I agree with what you say. And this, as a consequence, set a bound on the existence of a fourth generation, and in particular on the mixing that you would have with the other generations of the standard model. Okay, so I agree with what you say. Now, this bound is relatively stringent. We have other bounds. I'm not sure there is a comprehensive comparison of these bounds, because also they depend a bit differently on parameters, but there is a very, very stringent bound on the fourth generation of quarks coming from Higgs physics, because basically what you learn is that, so if you have a fourth generation mixing with the other generations of the standard model, you would have an extra contribution to glonfusion Higgs production, because you will have an extra quark that is running in the loop to produce the Higgs. And this is very stringently constrained. So there are constraints also from Higgs physics. Okay, thank you. Mm-hmm. That's a great question. Another question while we're here. A digit? Yeah, I have a question from previous slide. Yes, this one here. Yeah, yeah. Here, the thing that you have mentioned, the relation, the unitary relation between CKM matrix element, that what experimentally we test to be about this relation. So what is the current update about this experimental test of the relation? Is it exactly zero or very close to zero, or that we found in experiment? Yeah, so of course there are experimental uncertainties. You cannot say that something is exactly zero. But then you can see, maybe I can zoom. You see if they zoomed the version of my slide. Yeah, yeah. So if you zoom here, the fact that we have a zero plus minus something comes here from the fact that all these curves cross in a point, but you have a little bit of uncertainty. So you have a little bit of freedom to put some non-unitarity here, but it's a tiny bit because you see that, I mean, basically every curve pass from the same point with a very small uncertainty. Okay, thank you. And then also these measurements are going to improve in the future. So in the future we'll have even a better determination of these vertex or equivalently a better determination of this relation here being equal to zero. Okay, thank you. Okay, excellent. So if there are no other questions, I can go back to my iPad. So this is what I wanted to show you for the determination of the CKM metrics. So let me share my iPad. Okay, here it is. So now that we have, so yesterday, if you remember, we discussed about electrolyte precision measurements. So yesterday, electrolyte precision measurements. Today we have just seen the determination of the CKM metrics. So precision, determination of CKM. And then the conclusion indeed is that, again, the standard model, so the predictions of the standard model are confirmed experimentally with some uncertainty. So standard model, let me write that the standard model works in the sense that this set of experiments are confirming these standard model predictions. And what is interesting to keep in mind is that, and this I mentioned a little bit here and there, but now we have everything that we can put together, is that the standard model give us a structure of interactions. But then we have always, you know, three parameters that we need to determine. And we mentioned that so for three parameters count. So in the gauge sector, we have only three parameters. And these are the gauge couplings. So we'll have the strong coupling. We didn't talk about that because we were discussing electric interactions in these lectures, but we don't have to forget about QCD. Then we have the SU2 gauge was on coupling. And then we have the average charge. So this is quite, you know, the gauge Lagrange on the standard model is very elegant, very well-determined, because we have only a small amount of three parameters. Then we take the Higgs part. So the Higgs Lagrange on the kinetic term and the potential. Here we have two three parameters. And these are, you know, depending on which one you choose, but can write the mass of the Higgs as well as the valve of the Higgs. So also here, pretty easy, if you want, only two three parameters. And then lastly, we discussed the flowery interactions. So the Yukawa part of the Lagrangian. So Yukawa. And here it comes to the sector where we will have much, much more freedom in the standard model. Because basically, you have 10 three parameters in the quark sector. So these are the six masses, three plus three in the up and down sector, plus the four parameters of the CKM that we just discussed, the three angles plus one phase. And at this level, then we have also additional parameters in the left on sector. So far, we didn't introduce Neotrinos. We didn't introduce Neotrinos. So effectively, so far we have only three, three, three parameters in the left on sector of the standard model that are the three charged left on masses. So this is a left on sector charged left on masses. Now, you can look at the values of all these three parameters. Again, coming from experiments. And what you see is that the values are very hierarchical in the sense that you have some of the quark masses are much, much larger than other quark masses. And some of the CKM elements are much larger than others. And so the parameters are hierarchical, you know, the typical example is that the mass of the top is something like five orders of money to the bigger than the mass of the first generation up quark. And since we don't know how to explain these hierarchies, this is actually what we call an open problem in particle physics. And this is what we call the standard model flavor puzzle. Namely that we would need an explanation as of why we have these super large hierarchies in parameters as measurements are telling us, but we don't have really a good mechanism to explain these hierarchies theoretically. Okay. So this is all I wanted to say about, you know, the standard model action. And a little bit of phenomenology. So we study phenomenology of the hex and a little bit of flower phenomenology. And now as I as I promised a couple of days ago, we want to study a little bit of Neutrino physics. Of course, this is not going to be a comprehensive study, obviously, but I wanted to give you some some highlights of what we know and also what we don't know. So this will be our section 4.4. So Neutrin. And what we'll do is to focus on Neutrino masses. And the conclusion as we see is that the generation of Neutrinos, generation of Neutrino masses is an open problem. So we don't yet know the exact mechanism for the generation of these masses, contrary to, you know, the generation of charged laptop masses as well as quarks, that is the Higgs mechanism that we have discussed together. So what we already mentioned is that in the standard model, as I defined during these lectures, the standard model Neutrinos are massless. So the Neutrinos that are coming from the lepton doublet that we defined as nu L and charge lepton L. Okay, so these guys here are the massless particles. At the same time, so this is the first bullet point, second bullet point is that experimentally. So we have discovered that Neutrinos oscillate. So what it means is that if you start say with an electron Neutrino in some experiment or coming from the sun or from your or from the atmosphere, a certain point this electron Neutrino can convert or we say can oscillate to a Neutrino as an example, or vice versa, you can have the the oscillation between a Neutrino and a Neutrino while it propagates and so on and so forth. So briefly, the way that it was discovered was measured in the sun, actually. So we had in the sixties, a short parenthesis, so in the sixties we had what we were calling the solar Neutrino problem. Namely, the problem understanding the amount of Neutrinos coming to us from the sun, because now we have the sun that is sending us continuously electron Neutrinos. The reason being that we have the nuclear fusion chain in the sun, so, you know, we start with protons that are giving us Neutrinos, but then we have also, you know, more energetic processes taking place in this chain of nuclear fusion and we produce also higher energy Neutrinos that are coming to us and in particular we have this reaction here that is giving us energetic Neutrinos that we see. Okay, but then you see that from this nuclear fusion chain we always get electron Neutrinos and this is coming from, as you see, from the case of nuclei. So we have actually a solar model for these nuclear fusions and we can predict the spectrum of these Neutrinos. We can predict the spectrum and in particular the number of electron Neutrinos that is coming to us, but what we saw was a smaller amount of Neutrinos, so we saw less Neutrinos than predicted and then, you know, people were working on changing the models for our sun or changing something else and what we found out is that the way to explain this observation, so the explanation was that electron Neutrinos, while propagating between the sun and us, were oscillating and becoming some of them Neutrinos and Neutrinos were not seen by our detectors. So this was in the propagation to us and then plus we only detect electron Neutrinos. You might wonder why for completeness I like to mention this and the reason was that electron Neutrinos were interacting with our detectors done of chlorine, so this was the experiment done for the first time at Homestake Mine in North Dakota. So we had the following reaction giving us argon plus an electron and this is what was observed and of course to observe the electron you needed to start with electron Neutrinos. So this was my small historical parenthesis, but I see that there is a question now. Yeah, hello. Yeah, my question is about Neutrinos masses. It is actually in standard model, leptons and their Neutrinos are in their double, presented in the doublet form, in the doublet. Right, so but it is very strange that while all other leptons except Neutrinos, they acquire mass through Higgs mechanism that you explained last time, but for me on this Neutrinos didn't get or acquire mass through the same mechanism. So why is like this, why is like this? Yeah, so the reason is that so you might wonder why they don't get the mass from the same Lagrangian term as the charged leptons. So we wrote, sorry, let me write it here. So we wrote down yesterday, I believe, and also the day before the lepton you cover the same in this way. You have this doublet here and then you have a right that is the right handed charged leptons of the standard model. And actually, you can see that once that you replace H with its vacuum expectation value, what you obtain is simply a mass for the charged leptons. So you will get you're risked to ER and you don't get anything for the neutral component, so for the Neutrinos. And the reason being that here you have a charged lepton and you need to conserve charge and the electric charge. So this you cover here doesn't lead to any masses of the neutral component of this leptended leptons of the standard model. We see that if you introduce there are Neutrinos, you can write down a Yocawa that is giving mass to the Neutrinos, but this we'll see in a second. But with the field content of the standard model that we have presented so far, you cannot write down Neutrino masses. Does it make sense? Okay. Yeah, my next question is from same topic that it is massless in standard model except all other fermions and bosons. But one professor during this discussion in one presentation, he told me about CISO mechanism. So my question is that there is one mechanism that Neutrinos can acquire mass. But my question is that like Higgs mechanism which is well established and experimentally verified. This is the I think one of the best theory that we can explain how particle acquire mass. But about CISO mechanism, is it a well established theory or it is a proposed way to provide? Yeah, it is not. But yeah, we'll get there in a second. So we'll discuss that. And then if you have some questions, please. But we'll get there. Okay. Okay. Okay. Thank you. Thank you. Sure. So we discussed that Neutrinos oscillate. And then a consequence of that is that actually Neutrinos do have a mass because you can do a nice exercise in quantum mechanics to study the probability, for example, for oscillating between starting from an electron Neutrino and going to a muon Neutrino. And after a certain length of propagation. And what you will learn is that if this electron Neutrino has some momentum P, this probability will be proportional to the sine square or the difference in masses of electron and muon Neutrinos over 4P times L, where L is the propagation length that we see here. And then you have also some dependence on the mixing, let me call it sine square of theta. So this will be at the end of the day mixing in the PMNS matrix. So the corresponded with the CKM matrix. But we'll see what that is. So to have this probability that is nonzero, you need to have at least, so for this at least one mass that is different from zero. And in general studying all this oscillation of oscillations of Neutrinos, we learned that at least two Neutrinos need to have a mass. Okay. So we need to introduce some BSM physics or something that is explaining this observation of the masses. And so what can we write down in our Lagrangian to take care of this problem? So we'll give you two possibilities. So first one, the first is discussing an effective theory, EFT. So what are the operators that can take care of Neutrinomas generations? So the operator that can write down is a dimension five operator that looks like this. So I take the left hand in the left hand of the standard model, this S2 doublet. I put it together with a Higgs doublet. And I know that I can form a singlet out of it and then I square it. Now if you count the dimension of this operator using the dimension of the fields that we discussed during the first class, you see that this is dimension five. So in order to have a Lagrangian term, you need to divide by some mass scale, lambda, say a new physics mass scale. And then in general, you can have some parameter dimensionless parameter in front. And this is a term that you can put in your effective Lagrangian. And what you can see from here is that once that the Higgs get a VEV, this will give you indeed a Neutrino mass, okay, a Neutrino mass term in your Lagrangian. And in particular, this will be C over lambda V squared, maybe some factor of two. But yeah, Max, you have a question. Yeah, sorry. Could you please again explain why we know that at least two new Neutrinos need to have a mass but not all three of them? Yeah, so just from this expression here, if you just focus on the oscillation between an electron and a mu, you see that this probability depends on this difference of masses. So in principle, you could put M1 equal to zero and M2 equal to zero, sorry, different from zero, and you could get a probability that is non-zero. So in this sense, from here, you only need one mass to be non-zero, not two. But then you have also many other measurements of oscillations. So you will have, yeah, oscillation with a tau or any combination, but they always depend on this difference in masses. So putting all data together, what you learn is that you'll only need two. So the third one can be still massless and everything would work, simply because you have always this type of combinations of differences. Yes, this is also one, nevertheless, all three Neutrinos would mix, even if one of them would be massless, right? That's right, that's right, yeah. Okay, very good. You want to, there's another question, you want to ask Julian? Yes, so what against the reason why we don't just introduce right-handed Neutrinos, even though we can't detect them? Sure, I will get there. So now I present the EFT and then I will introduce Teran Neutrinos. Just give me a moment, yeah. So from the EFT perspective, this is the operator that I can write down with the lowest dimension that can take care of my Neutrino masses and this operator has actually a name and this is what we call the Weinberg operator. Now, from experiments, we know that these masses are at most of the order of one EV, so they are super, super light. We know what is the vacuum expectation value of the Higgs. So if you want to have some sort of natural masses in the sense that these sea coefficients are of the order one, what you learn is that this new physics scale, so the new physics scale has to be of the order of 10 to the 13 GB. That is obviously a very high new physics scale, but this is what you learn just from a simple EFT analysis. So this is the first approach to Neutrinos masses. Just using, as I said, the field of the standard model and writing down your EFT. But then you can say, okay, I don't like EFTs. I would like to have some more UV-complete model such to generate Neutrino masses. And here it comes, my Teran Neutrinos. So I introduced new fields to the standard model Lagrangian. So these are the Teran Neutrinos that I call N. They are sterile because they don't interact with any gauge bosons of the standard model, so they are complete singlet states. And now if I have these new fields, I can write down new terms in my Lagrangian. So in particular, there are two terms that I can write down. So the first term is this term here, i, j, h, and j. Now this term here is allowed by gauge invariance. And also what I can write down is a term like this. So this is a Majorana mass for the sterile Neutrino N. And in fact, we see that the sterile Neutrinos are equal to their conugate. That's why we call them Majorana Neutrinos. And so a couple of minutes and then I will do a break. But so we can connect this UV Lagrangian to the EFT description that we saw above. And this is a note a little bit more for the experts, namely that using this Lagrangian, I can write down a Feynman diagram like this, where you have here a Neutrino, here my Higgs, sterile Neutrino, Neutrino, and the Higgs. And then what I can do is, as we discussed for the Fermi theory, is to integrate out this heavy sterile Neutrino, integrate out. And if I do so, I generate indeed the operator, that the Weinberg operator that we saw above. And with a coefficient that is given by this Yukawa coupling square over the mass of the Majorana mass of the sterile Neutrino. So the Weinberg operator I wrote down before is not really magic, but is something that comes out in more UV-complete models. So once that we integrate out these additional fields that I introduced. Yeah, I think this would be a good time for making a break. And then I will finish the discussion of Neutrino mass generation and we'll see what it means to have a Majorana or a Dirac Neutrino. Julian, you have a question. Yes, maybe to specify the question from before a little bit better. What I was wondering is why can't we just have a usual right-handed Neutrino and just the Yukawa term? Why does it always have to be a Majorana term as well? Yes, so we'll get to the second part of this discussion. Okay, good. So we'll delay it. Sorry. Be patient. Okay. Then we have Seishi Keshe. Hello, ma'am. So my doubt was related to why exactly the Weinberg operator looks different in both the cases. So like one case you have Y v square by M and the other one is lambda. Yes. So this lambda is just in a generic new physics scale and this is coming from dimensional analysis because I know that I need to write down a term in the Lagrangian that is dimension four. The operator is dimension five and then I need to divide by a mass. But I don't know what it is from the UFT perspective. Now, I can, you know, if this operator is coming from this model that I presented here, then this lambda is related to is the mass of the Starr Neutrino. But this is for this specific model. You can invent a different model where lambda will be, you know, the mass or some other heavy particle. It can be, you know, a heavy triplet scale or whatever. And so this to say that this lambda, this new physics scale, can be the mass of different particles depending on the theory that you are writing down. Yeah. Okay. Thank you, ma'am. So, Gautam as well. All right. So I have a question regarding the Weinberg operator and what it does with regards to the Starr Neutrino Lagrangian. So you get the Weinberg operator when you go towards more UV-complete models, right? You were cut, sorry. I missed a couple of seconds of what you said. No. So we get the Weinberg operator when we develop models which are more UV-complete and so we figure it out. But so that's the Weinberg, but the effect of field theory, the Weinberg operator, it still makes use of the Starr Neutrino Lagrangian. We're not actually adding any new terms. Right. So is the Starr Neutrino Lagrangian already not UV-complete or are the sectors that are not UV-complete? Yeah, that's right. So if you write the Starr Neutrino Lagrangian that we have discussed until yesterday, plus this Weinberg operator, at that point, the Lagrangian is not anymore UV-complete because you see that you are adding operators that have a dimension bigger than four. So you don't have even anymore a renormalizable theory. So in this sense, you know that you need something else. It's just that you sort of parametrize your ignorance about the UV theory using a dimension five operator. Right. So all we're doing is we're just introducing a new operator, but you're not actually using any new fields. Sort of. I mean, this will be a scale on new physics, right? But you're not introducing any new physics field. This is true. So that there are no particles associated with a new field like the Lagrangian. Okay. Correct, correct. Yeah. Okay. So, yeah, let's take a five minutes break and then we'll continue the Neutrino discussion. All right, Issa, ready to start? Yeah. Let's do it. Okay. So now that I have these Starr Neutrinos, I can introduce what we call the CISO mechanism. So the C. So that is simply, you know, the mechanism to generate masses of Neutrinos using the Lagrangian that we saw above. So this Lagrangian here, because, yeah, what you can do is to write down the mass metrics for everybody. Of course, you know, I have always to remember flavor indexes for simplicity. We can do just one generation to see how it works. And the mass metrics will look like this. For one generation, you have a zero here. Then the this term here that does depend on the on the Eucala of the Higgs with Neutrinos. Same here. And then here you have your Majorana mass. So this is what we call the Dirac mass. Reason that, you know, if we have only this mass term, the Neutrinos are Dirac Neutrinos. And then, no, we can diagonalize the system. And what we find is that we have light Neutrinos. With a mass that is given by minus. So let me call this Dirac mass M nu d for simplicity. So this combination of Yucca and VEV, then I have minus M nu d M to the minus one M nu d. Now you see that you have a natural suppression of these masses. So if M is large as we learned from the discussion of the Weinberg operator, then you get the sort of automatically light Neutrino masses because of this suppression. And this is what we call usually the, yeah, this is what we call the CISO mechanism thinking that, you know, you have a system like this. This is your big M and this is the light Neutrino masses, okay. Okay, now to go back to the question of Julian. So this is, you know, the Lagrangian that we see here contains all terms that I can possibly write that are consistent with, you know, Lorentz's engaging variants. That's why I brought them down. In principle, instead of having this Lagrangian here, I can have something with only one term, namely I can write down a Lagrangian simply with a Yucca interaction. Obviously this Lagrangian also gives you Neutrino masses, it's not in the same way as we get the quark and charge left on masses in the standard model. And so this is Neutrino masses after electric week symmetry breaking. Just that, you know, since we know that the Neutrino masses are at the level of EV or less than that, what we learn since we know the vacuum expectation value of the Higgs boson, we learn that this Yucca coupling, why know, cannot be larger say than 10 to the minus 11. That is, of course, a tiny number that introduces even a bigger standard model puzzle that we discussed before. So let's say even more standard model puzzle. But this is, of course, a possibility, namely to have Neutrinos on the same footing as the quarks and charge left on the standard model, simply a Dirac mass and that's it, you know. In fact, so let's compare these two Lagrangians, one and two. Okay, so one is here with the Majorana mass term and two is here. So how do these entangle between the two? So in one, again, I have my standard Neutrino Majorana mass. And this actually induces a breaking of the lepton number. So what is the lepton number? Let me indicate this with an L. So this is a global symmetry, that a U1 symmetry that is defined as, so if you have a lepton of the standard model can be both the left-handed or the right-handed, they have a charge plus one. Okay. If you have instead a anti-laptone, this will have a charge of minus one. And then for standard Neutrinos, it's the same. So you will have a standard Neutrino, you can assign the charge plus one to the standard Neutrinos. So this is the global symmetry that is actually preserved in the standard model, and this is an accidental symmetry. But then if you introduce a mass term like this, you can see that this is not anymore symmetric. So this is the prediction of the Lagrangian number one. For the Lagrangian number two instead, you don't have such a term, and therefore the lepton number is conserved. Okay. Now the question is how to test the breaking of the lepton number? Because if we start seeing a breaking of lepton number, then this can give us hints that we have the first Lagrangian instead of the second Lagrangian. So tests of lepton number violation. So there are several tests, and the most famous if you want is what we call the Neutrino less double beta decay. So what is this process? So we take an element with some mass number and atomic number A and Z, whatever that is in principle. And then we search for the decay to a different element that has two more protons. And then we need to conserve the electric charge so we have two electrons. But you see that here you don't have Neutrinos in the process. Obviously this process is breaking the lepton number because we start with a zero lepton number, we end up with plus two. And we can draw the Feynman diagram if you like. So this is, yeah, let's draw here. So you take a down here, you convert it to an up. So this is your Neutron that goes to a proton. Here you have a W. And then you do the same here. So you have a down that goes to an up. Here is a Neutron and here is a proton. And here you will have the two electrons. Okay. Now if we discover this process, it's telling us that indeed we have a breaking of the lepton number and this would tell us that maybe this is the right framework. Okay. There are many searches. So there are several experiments that are looking for this process and so far we had limits. So we don't have yet an evidence for this process to happen in nature. And in particular there are limits on the half-life of this process. These are pretty stringent. So we're talking about a half-life of the order of 10 to the 25 years. That is obviously huge. Yeah, there is a question. Yes. Professor, I have one question regarding this capital N. So capital N is sterile neutrino here, right? Yeah, capital N. So these pills here, yes. Yes. So I want to know that how sterile neutrino is different from the ordinary left-handed neutrino. Yeah. So the quantum numbers are different. So we introduced them here. You see that is a singlet under... So this would be SU3 times SU2 left times U1. Instead for the left-handed neutrinos, they belong to the doublet of SU2. So this will be a 1, 2, and let's see, minus one-half. But we call sterile. So from the point of view, when we call something sterile, we mean that it is not going to interact with any other particle or something. Yeah, that's right. In this context, it's sterile because it doesn't interact with gauge bosons, as you see here, because it is a complete singlet. Okay. Thank you. Okay. So this is all I wanted to say about the neutrino mass generation. And so the conclusion here is that a conclusion is that we don't yet know if neutrinos are Dirac. So if they behave exactly as quarks and leptons, charged leptons of the standard model or Majorana. Okay. Max, you have a question. Yes. So I have a question about the process of the new neutrino and star beta decay. So how sensitive is it on the Majorana mass? Yeah. So you can compute the rate for this process using the model, this model here. So the number one, and then you can see your dependence. In general, if you're wondering about the direct bound on this mass, I don't have the number on top of my head. But you can compute it because here what you would need is sort of a Majorana mass for your neutrinos. Otherwise, this process is not happening. I was more wondering in principle. So if this process is not observed, if this really is a good indication that the neutrino is not Majorana, or if it is also at least as good an indication that the Majorana mass is very, very large. So I was wondering about the relation as part of your case. Yeah. Yeah. I mean, you can specify obviously the model and compute it. So in general, I think that this process is more in one-to-one correspondence with lepton number violation. Then, I mean, this Majorana mass and this, you know, CISO that is actually type one, but I don't want to discuss about the other types. So this type of models is introducing a lepton number violation. And therefore, you can add processes like this, but this is not unique. But you can do your calculation in this model and then see what you get. Okay. Thank you. And Deep also would like to ask you. Yes. Hello. I have a question regarding this mass term for style neutrino. Is it gauge invariant or not? We are not looking into that for the time being? Yeah. Certainly. It is gauge invariant. And the reason that it's gauge invariant is precisely that n is a sterarn neutrino. So it's singlet under any gauge interaction. Yeah. So it is gauge invariant. Otherwise, I wouldn't write it in the Lagrangian. And Gautam? Yeah. In terms of differentiating between the direct marijuana, what type of tests or experiments do people, like, can you have model dependent collider experiments that tests give the neutrino to be derived from marijuana? Yeah. So, sorry, I got a little bit distracted. Can you repeat the question again? Sorry. So I was wondering because a large portion of the experiments are model independent, right? Is it possible to have model dependent tests that can or is there something that is going on at price to differentiate between a derived marijuana or is it simply just a measurement of masses? Yeah. So there are several ways. And so you can, of course, directly search for sterarn neutrinos at your colliders. The problem is that, so in the minimal models, you expect to have these sterarn neutrinos to be very, very heavy. So that's, I mean, forget about testing a scale of 10 to the 13 GB. But there are models in which these sterarn neutrinos are lighter, and then you can search for those. And then you find them. Yeah. So the issues sort of. Then there are also searches at LHC for lepton number violation. So there, you know, you search for processes that are explicitly breaking the lepton number. So there are several different ways of testing this idea of having marijuana versus sterak, especially focusing, as I said, on lepton number violating analysis, I mean, searches. Yeah. Thank you. Okay. So to conclude the discussion about the neutrinos masses, so I wanted to mention that, you know, since you have this interaction here, you can play the same game, you know, of diagonalizing the system and so on, in the same way as we have discussed for quarks. And then what you end up getting is what we call the PMMS matrix. And this is coming from Ponte Corvo, Maki, Naka, Gawa, and Sakata. So this is the unitary matrix that correspond to the CKM matrix in the quark sector. So again, same discussion just at now. And again, using oscillation of neutrinos, so data from oscillations. What you can see is that the structure of this matrix here is quite different than the CKM matrix. So let me draw this cartoon, hopefully it's roughly in scale. But yeah, you get something like this. This is bigger. This is small. And this is kind of smallish. So you have a structure like this that you see that is not identity plus corrections as for the CKM. Because indeed for the CKM, we saw that we had some structure like this where these are the big numbers. These are a bit smaller. These are still smaller and these are tiny. So you see that the structures are very different. And then you can ask the question of why the structures of mixing between neutrinos and quarks is so different. Okay. But this, you know, to complete this discussion of neutrinophysics. And of course, I mean, there are many measurements that will come in the future to test more these PMNS metrics because indeed the uncertainties here, the experimental uncertainties are larger than those in the CKM. Okay. So the CKM elements are determined with a much better uncertainty, experimental uncertainty. Okay. So this is, yeah. Can I read a question from the chat from Paulina, whose microphone isn't working. In Lagrangian 2, the mass of the sterile neutrinos is very tiny, like the one of the standalone neutrinos. And in Lagrangian 1 is much bigger. Can we say anything about the nature of the mass term of the neutrinos if you can measure somehow the mass of the sterile neutrinos? Yes. Yeah. That's the answer. But yeah, we need to find sterile neutrinos to learn more about this mechanism. Until then, it's complicated because, yeah. Obviously, as I said already, I mean, it's a kind of a complicated task because in a minimal model, you expect this sterile neutrinos to be really, really heavy as based on this dimensional analysis or the Weinberg operator, you would expect a sterile neutrinos at the level of, with the mass at the level of 10 to the 13 GB. Then there are other models in which you lower this scale and then you can have votes to discover them at colliders. But it's not a guarantee. Yeah. Okay. So that's all for neutrinophysics. And now for the remaining time, I wanted to tell you a little bit, a tiny bit about why to go beyond the standard model, because so far, these lectures were about the standard model, and I hope I convinced you that the standard model is a great theory that explains a huge amount of data. Still, many of us are working on beyond the standard model, including myself, and then why do we do so? So there are many open questions that the standard model is not able to answer. So open questions, particle physics that are not addressed by the standard model, some are more theoretical. We mentioned, I think, yes, there was a question about the hierarchy problem. And I told you that this is, again, very roughly speaking, the problem of understanding why the mass of the Higgs is so light, in the sense that once that you specify your UV model, you learn that the mass of the Higgs is sensitive to any UV scale. So it's not stable, but so we don't understand how to keep it stable at around the left-wing scale, so at 125 GeV. And this is a problem coming from the fact that the Higgs is a scale, or if it was a fermion, we wouldn't have this problem, because the mass is propagated by the carar symmetry. Also, theoretically, we mentioned that we have this standard model clever puzzle. We don't know the origin of all these hierarchies in between cork masses, left-wing masses, CKM elements, and so on. We don't know how to write down a unified theory for the gauge group of the standard model, so SU3 times SU2 times SU1 plus gravity. So in these lectures, we saw how to write down a unified model for SU2 times SU1, but we would like to have something even more unified, and we don't have it yet. And then there are problems that are more, if you want to, observational, I would say. Well, some are crucial, like we have what we call the baryon anti-baryon asymmetry problem, so we don't understand why they, so we don't have a theory framework, an accepted theory framework to explain why we have more matter than anti-matter in the universe as we see it now. And this might hint to connect to something that we have mentioned during these lectures. It might hint to new sources of CP violation beyond the phase that we have in the CKM metrics. Then, yeah, we have discussed a little bit about the neutrino masses to explain them, and we need to go beyond the standard model. So beyond the standard model field content. And we have the problem of dark matter, so no particle of the standard model can be a good dark matter candidate, and I can add to the list. So there are many open fronts in particle physics that cannot really be addressed in the framework of the standard model, and so we would like to understand what is a good theory beyond the standard model. Also here I would like to add that we have some anomalies in data. So the only data that I discussed with you during these lectures are data that indeed agree with the standard model, with some of the predictions, but we have several that seem to show a bit of discrepancy. We are not yet at the level of discovery, but still there are some discrepancies here and there that we would like to understand better in the future. And then of course, I mean, if you buy, you know, this story of having all these open fronts and that you need the BSM, then you need to follow some sort of rules in the sense that I hope that in these lectures I convinced you that you need to write down theories that sort of respect at least roughly some symmetries. So in BSM, since the standard model can explain so well data in general, we know that BSM theories cannot break, let's say too badly, the approximate symmetries of the standard model. Because otherwise, you know, already, you know, naively that you will be in trouble with data. And so usually in these lectures also to, you know, wrap up a little bit what the discussions that we had, we saw that we have this custodial symmetry that again cannot be too much violated. Also, we mentioned that we have the big flavor symmetry, approximate symmetry, that is the 3 to the fifth power and so on. So this gives us some guiding principles to write BSM theories. But of course, you know, we have a big range of possibilities. And what is very interesting is to try to connect these BSM theories to data and then to try to design new measurements to eventually discover this BSM theories. So Max, you have a question. Yes, I would like to ask you about one item on this list. So you said that one reason for the BSM is to have this unified theory of the standard model and gravity. But actually, so I have seen many BSM theories already. But so besides, of course, the spring theory, I've never seen an example of BSM theory, which really includes gravity. So is this just my impression that people are not really working so much on that? Or are there really BSM ideas, which which include really gravity as well? Yeah, people are working, but I agree that, I mean, so far, there is a string theory that is doing, I mean, that is going this direction. And obviously, there are people working in string theory. Beyond that, yeah, there is not much if you want to, yeah. Yeah. Thanks. Okay. So, so in the meantime, what I wanted to do, and then really wrap up, is to mention a couple of anomalies that we have seen in data. The reason I do so is because several reasons. So first is that I find this exciting. Of course, if this anomalies will be confirmed, then we have BSM. So large part of the community is quite excited or at least interested to this type of anomalies that I will present. And two is that I don't want that you go home and say, okay, the standard model is the full story, all data agree with the standard model we are done. So I want to show you that indeed, that there are, there is data that doesn't agree with the standard model that well. Okay. So this is our section 5.2. So this was a 5.1. So anomalies in data. So there are several and for this discussion, I decided to focus on two. So one is the G-2 or the Mion, and the other are, let me call them lepton, flavor, universality ratios or what we call in jargon RK and RK star. Now, I have the impression I cannot discuss in details both. So I, why don't we do the following. So I, let's see, if you can raise a thumb or a hand, whatever, if you want G-2, let's see how many G-2 are in the audience. Otherwise, I do RK and RK star. So up for G-2. And if not, I will do RK and RK star. But this is dangerous because maybe people are not paying attention and then... I know, they need to, there's also comments in the chat. So it's growing, so we need to give a maximum time to survive the 24. That's G-master, right? Now I lower, I lower all hands now. Yeah, and then we do RK and RK star. We see if we get more than 24 or not. So it's for RK and RK star. You cannot raise both. You have to make a decision. Well, I will post our selection notes on the other, even if we don't discuss now. But the chat doesn't count. I can't count. So yeah, let's do G-2 then. We have 24. The key are we. Yeah. And as I said, I will post some selection notes on RK star. And then you can send me an email if you want to learn more as well. And I can send you references. Okay. So for this, see lecture notes. And let's do this. Okay. So let's start very basic. So what we want to do is to understand what is the effective interaction of a lepton with a photon. And this can be whatever lepton you can have an electron or you can have a muon. And then we'll focus on muons because we want to study the G-2 of the muon. So we studied this when we study QED. For example, Pesky and Schroeder has a a nice discussion of this effective interaction. And so what you can write down for this effective interaction is something like this, where here you have your spinors of the leptons that have the momentum P and P prime. Here you can put two structures, two different Lorentz structures. One is with the gamma muon and the other one is with sigma mu nu gamma nu over 2mf2 of Q square. And then here you have the other spinor, U of P. Now, this is what you get here. Once that you impose Lorentz invariance and also you impose parity conservation as well as the word identities. And these are form factors, form factors. And they do depend on Q that is the momentum exchanged in the process. Now, at the three level in the standard model, at three level, what you get is that the first form factor is simply equal to one. And instead, the second one is equal to zero. And once that you go beyond the three level, this is not anymore the case. And these form factors actually can be connected to g-2. So in general, we know that we can write down the magnetic moment of the muon in terms of its spin. So the spin S. And then here we have my electric charge over two times the mass of the muon. And here I have g, let's write it in red, where this g is equal, again, linking to the form factors as of above, g is equal to two times f1 computed at zero plus f2 computed at zero. And so we have seen that this is equal to one at the three level. And so what you can define is actually the g-2 over 2 that, again, you can connect to these form factors. And this is a quantity that we are measuring. So this is measured and predicted by the standard model at high precision. Of course, I introduced this in the context of QED, but the level of calculations that we have now are much, much beyond QED. So we are adding weak interactions as well as QCD corrections here. So in particular, just to give you an idea of how well the standard model is understood in the prediction for this observable. So let me quote this paper here. This is the archive number that came out roughly a year ago. So this is a paper of the g-2 theory initiative. So this is a working group of theorists that came together with the goal of recommending a value for the standard model prediction for this observable emu. And what they quote in this paper is that the emu of the standard model is equal to this number. So let me write all the digits. 5, 9, 1, 8, 1, 0, 4, 3. So the uncertainty is in the last two digits. And you see that all of these digits here are significant. So we have a precision at the level of less than one part per million. So this is one of the observables that is most precisely predicted by the computed in the standard model framework. And so one can, of course, discuss a lot about the standard model prediction. I just wanted to mention for the people maybe who already heard something about this is that one of the largest theory uncertainties is coming from hydronic vacuum polarization contributions. So these are diagrams of this type, just schematically where you have your two millions. Here is your photon. And here you have loops of hydrons. And there are two ways of computing these contributions. One is directly using data from E plus and minus colliders mainly. And then the other way is using lattice calculations. And yeah, there is a lot of delay on the agreement of these two methods and so on, but of course this would lead to a long discussion about this. I just wanted to mention in case people heard about this discussion. But having said that, we see that the recommended predictions of the standard model is super, super, super accurate. And then we can compare this prediction with the experimental measurement. So I have a slide on this. So I'm going to stop sharing my iPad and I will show you the slide. Let's see. Here it is. Okay. So in this slide, I'm showing the latest results on g minus 2 at the muon. So this plot is taken from this paper here. This is the archive number. And this is a paper of the g minus 2 collaboration at Fermilab. So this is an experiment that started running a couple of years ago at Fermilab. And what this experiment wanted to do is a better measurement of g minus 2 at the muon if compared to the previous measurement that was done at Brookhaven National Lab already 15 years ago. So this is the archive number of the previous measurement. So if you just look at the previous measurement, so this data point here in blue, you see that you have some deviation. If you compare this measurement here with the standard model prediction, that is this green band here. And back then there was an anomaly at a level of 3.7 sigma. So this was the significance of the deviation between the standard model prediction and the experimental measurement. Now Fermilab came, so there was an announcement two months ago or so by the g minus 2 collaboration at Fermilab. So a new measurement. And you see where the data point is. Okay. So this data point is quite consistent with the Brookhaven measurement. The uncertainty is a tiny bit smaller, very comparable. And you see that actually both measurements are having uncertainty that is better than one part per million. So this measurement is amazing, right? It's super, super precise. Then they have done the combination between the Fermilab measurement and the Brookhaven one. And this is the purple band that you see here. And then combining the two, what you get is an anomaly with the significance at a level of 4.2 sigma. Okay. That is pretty large, taking a face value. And that's why people are quite excited about this. And then here I'm also quoting what deviation you have between the experimental measurement and the standard model. And you see that is at the level of say 2, 10 to the minus 9. That is a relatively sizable effect. And so from my perspective, it would be very interesting to see what more data will tell us and also what new investigations about the standard model prediction will tell us. And from the experimental point of view, we know that in the coming years, Fermilab is supposed to give us a more precise measurement. So the level of uncertainty that we expect to get from Fermilab is something like four times better than what we have now. So we'll learn much more in the coming years. Okay. And so this is just to summarize where we are from the standard model perspective and from the experimental perspective. And then as you have probably seen in the archive, there are many papers after this announcement. And in general, there have been many papers since 15 years about how to explain the G minus 2 anomaly. And it will be very interesting to see what comes out of it. So if you have questions, we can chat a little bit about possible BSM explanations. Otherwise, yeah, this is all I wanted to say about G minus 2. So I see that there is some question. Yes, let's see. There's Anna first. Hi, thank you so much.