 The first lecture of the day is from Matt McCullough from CERN, he's an expert of beyond the standard model physics, standard model physics, high energy, even low energy. Please enjoy. Okay, Rook, thanks Giovanni. So first of all I should say, looking at the audience, it's a real pleasure and honor to be able to give these lectures on BSM physics to such a diverse and interesting crowd of people. Before I start, I should say that the goal of these lectures is that you understand what I'm talking about. So if you don't understand what I'm talking about, that's my failure, not yours. And you should feel free and comfortable to ask questions and shout out. I'm a strong believer in the notion that there's no such thing as a stupid question, there's only such thing as a stupid answer. So if you ask a question, I don't answer it adequately, that's again my problem, not yours. So please feel free to shout out. Even if you think a question is very trivial, because you never know if every other person sitting here may be thinking exactly the same thing, because I've just not explained something well. So you never know who else is thinking the same thing, has the same question. So please shout out. Or otherwise corner me at coffee time, email me, whatever, it doesn't matter. But please feel free to get in contact, because I'm going to try and cover quite a lot of material in varying degrees of technicality. And if anyone, the goal is that no one should get left behind. Okay, so what is beyond the standard model physics? So why would, the question of why would you look beyond the standard model? You've been learning about the standard model in the last week. And it is clearly a tremendously successful theory. It's probably the most predictive and precise theoretical explanation for observations across a wide range of energies that we've ever had and may ever have, who knows, hopefully not, we'll do better. So why would we go beyond the standard model? And the reason is that there are many outstanding puzzles in fundamental physics. I will note some of them, but I'm not going to lecture on them. So the origin of the matter anti-matter asymmetry. Is my writing big enough for you all to read? I'll take that as a yes. The origin of neutrino masses, I think you had lectures on this last week. The nature of dark matter, you'll have lectures on this today. The origin, sort of more structural questions, the origin of the number of generators, generations, and particle, and the mass patterns. This is essentially flavor physics, is the mass patterns. The quantum description of gravity. The origin of the hierarchy in fundamental scales. Lack of CP violation in strong interactions. And the list could go on. So clearly, the standard model is extremely successful, but there are huge gaps in what it can tell us. And in fact, all of these are considered to be tremendously important puzzles. I won't be able to cover them all of them this week. So I'm going to pick a couple to focus on, which is these last two. But as theoretical particle physicists, we have a tremendous job ahead of us to try and take off answers to all of these questions. If, as a species, we don't ever reach a full understanding of all of these questions, then we have failed scientifically. We have a big job ahead of us and we have to keep pushing ahead to try to understand what's going on. Tracy will tell you about dark matter. It's probably one of the biggest concrete challenges we face in beyond the standard model physics at the moment. And it's sort of an embarrassment. There's a lot more dark matter than there is matter. So we have this exquisite understanding of a fraction of the total matter in the universe, but the rest of it, we can't even tell you what mass scale it lives up. Quantum gravity is a great big theoretical question as we know the gravitational interactions are not renormalizable. And this leads to lots of open questions as to what the quantum structure of gravity looks like. Origin of the number of generations in the mass patterns. You know, why are there three generations? Why not four? Why not two? And why do we have these very clear patterns in fermion masses that are calling out for an explanation? There's clear hierarchies from third to second to first generation. We have no good explanation of why that is. Origin of the neutrino masses is sort of related. We just take the standard model field content and write down all of the renormalizable terms we're allowed to write down. The neutrinos would have been completely massless. So in some sense, the neutrino masses must have an origin in physics beyond the standard model. And this one here, which is also an enormous puzzle sort of comparable in size to the dark matter puzzle, which is where did all the matter come from? If we just start the clock of the universe at very early times with an initial condition that has no baryon asymmetry or even if it has a baryon asymmetry after inflation would have been washed out. So we have no idea why we have an asymmetry in baryons today. But I'm going to discuss these two. First of all, because this one, I was asked to focus on the TEV scale. And that's sort of the area that I'm most comfortable with as well. So we will talk about the origin of the hierarchy between the weak scale and the Planck scale and why this is an enormous puzzle. And I also want to talk about the strong CP problem on Wednesday, which is why we observe a lack of CP violation in QCD. But there could be CP violation in QCD. And we would expect it to be there. So this is known as the strong CP problem. When discussing these puzzles, I'm going to try. There will be a lot of technical material. I'm going to try. There's a lot of jargon. And there's sort of a language gap when you're entering. I remember it when I started my PhD, that you come across a lot of jargon. And the only reason you don't understand it is they're not familiar with it. If someone explains it to you in one minute, you know exactly what you're talking about. So I'm going to try and introduce jargon. I'm going to try and explain what the jargon is every time the term that you may not be familiar with. But if I do start using jargon and I haven't explained what it is and you don't know what it is, then please ask me. OK. So what I want to do in this lecture is discuss a sort of general toolkit called effective field theory. And I will discuss it in the manner in which beyond the standard model physicists tend to use it. The reason I'm doing this is I think that to really fully understand the nature of these problems and their proposed solutions, one has to be a good effective field theorist. If you're not a good effective field theorist, you can sometimes confuse yourself about things like mass hierarchies and so on. And the other reason is that I'm sure you're all well aware that after a decade of LHC results, we haven't seen some of the more anticipated solutions to this problem here, the hierarchy problem. You'll be familiar with terms like supersymmetry or extra dimensions, composite Higgs models, and so on. As yet, nothing like that has shown up. And what that means is that my duty as your lecturer for these topics is not to give you a party line on any of these particular things. It's not to deliver six pairs of lectures and equip you with as many theoretical tools that can help you going forward to look for new solutions to these problems or to understand their phenomenology. So this will not be a monolithic lecture course. I will, of course, later on discuss supersymmetry, extra dimensional theories, composite Higgs models, and so on. But I think actually the most important aspect that you could take away from these lectures is that the broad set of tools contained within effective field theory that could help you go on to come up with your own better ideas that we have failed to come up with so far. So what is effective field theory? So it's something you're actually very, very familiar with. So if you think we can start in the UV. So I'll start with the jargon already. UV is very high energy, very short distances. And it's always relative to the topic at hand. So if I'm talking about physics at the TEV scale and I mentioned the UV, I mean far above or quite far above the TEV scale, if I'm talking about physics at the Pion mass scale, so around 100 MeV, what I mean by UV is actually probably will be a GV or where QCD becomes strongly coupled. That's where the UV completion comes in, which is QCD itself. In the UV, we may have some fundamental interactions. For example, between corks and gluons, something like this. This is corks and some gluons. And then when you start to zoom out, it starts to become idiotic to even discuss gluons. Gluons don't exist below a GV. There's no such thing as an asymptotic gluon. You can't scatter gluons at energies of, say, 100 MeV or something like this. They've become at those length, so by zooming out, I mean going to lower energy. And by the uncertainty principle, that just means going to longer distances. So you're essentially coarse-graining. You're zooming out from this description of physics where you have corks and gluons. And above a certain distance scale, you won't be able to resolve gluons or corks. It will not make any sense to discuss gluons and corks. But what you can resolve are still particles, but they're not fundamental. But things like pions and various baryons that eat the pions, the kaons, and so on. So when we zoom out, it makes sense not to talk about corks and gluons, but to talk about things like pions. And then, even zooming out even further, we might find that it's not even really useful to talk about pions anymore, but really nuclei. If we're zooming out at distance scales much longer than, say, 10 to the minus 15 meters, you will start to, it doesn't even make sense to talk about pions anymore. We'll talk about nuclei and maybe just neutrons and protons. And then as we go to the far IR, which means infrared, which is just sort of the longest distance scales relative to what I'm talking about. So for me, the IR in this case would be, for example, atomic physics, then it doesn't even really make sense to talk about nuclei and things like that anymore. You just talk about atoms and molecules and their interactions, things like dipole moments and so on. So if we were to try and do atomic physics in terms of QCD, it may be possible, but it would be an enormous waste of effort because there's a different effective theory that's appropriate at that distance scale than the QCD distance scale. Of course, we can recover all of these effective field theories by coarse-graining, by taking the theory we have and going to longer and longer wavelengths. We can always go in this direction if we have the fundamental theory in principle with an incredible amount of computing power, but it doesn't make much sense. The smart thing to do is to devise the appropriate effective field theory, which describes the relevant degrees of freedom at that scale. So if we're working here, we want to find the effective field theory of pions and their interactions with themselves and nucleons and so on. It'll be daft to talk about collions and quarks at that energy scale. And as I will go to show, there's a relatively well-defined procedure for constructing effective field theories and understanding how they're organized. But of course, one shouldn't be confused. When I'm talking about effective field theory, one should consider the field theory for the degrees of freedom that are relevant at that energy scale or at that distance, but that does not mean one shouldn't get confused about the energy scale at which the relevant physics may have some impact. So for example, in nuclear physics, we can have things like nuclear fusion in a star. And that is high-energy physics, which is related to very macroscopic processes. So we can understand those very macroscopic, very large objects and their physics by thinking about the appropriate effective field theory at the relevant energies of the interactions. Any questions at this point? No. Okay, so effective field theory, for some of you I'm sure is very familiar. For some of you, it may not be so familiar, but it is actually something that you should be familiar with. We have many examples in nature of effective field theories and their UV completions. So if I go in this direction, if I go up the way, I talk about a UV completion. So the microscopic theory that generates the effective field theory at the energy scale you're working at. I should also say that we don't know UV completions are not unique. If I start with an effective theory at some energy scale, there may be all sorts of different UV theories, microscopic theories, that give you the same effective field theory. So it's not a unique thing. You can't, I mean, to a certain extent, you can identify properties of the UV theory from your effective field theory involving, say, pions, but you can't unambiguously distinguish it. The only way to unambiguously distinguish it is to build an experiment and go to higher energies to go to smaller distance scales. The other direction is fine. We can always start pretty much with the UV theory, coarse grain it and find the effective field theory at least in principle. So there are examples of this where this play has shown up. As we know, if we take general relativity and work for many problems, not every problem, but for many problems, if we take general relativity and work at lowest order in one over m-plank, we recover the predictions of Newtonian theory plus some corrections that are small in a controlled expansion of, say, a length scale multiplied by m-plank. And in some sense, you can see how discoveries tend to happen in fundamental physics. We will often start by discovering the effective field theory with pions. We had the effective field theory for pions before we knew what the UV completion was. And this is actually a pattern that has shown up a number of times in fundamental physics. And I guess this is probably one of the main first examples where we had Newtonian gravity. It was working pretty well as an effective theory for describing the orbits of planets and so on, but it wasn't getting everything bang on right. And it had some theoretical puzzles, some mysteries embedded within it. And it was only when we understood general relativity that we could see how Newtonian gravity emerges as a sort of effective field theory of the more fundamental structure. Somewhere else that we've seen this sort of pattern is with Fermi theory. We started with the Fermi theory, which governs things like muon decays, for example. And we could look at the structure of the interactions and we see that they're non-renormalizable. So we know this is an effective theory and it must be UV-completed into some more fundamental microscopic theory. It won't, as we'll see later, it will not remain valid at all energy scales. And we had this before we had the full-blown SU2 cross U1 electroweak theory, which is the UV completion of the Fermi theory. If we take this theory and coarse-grain it, so average out all of the physics from short-distance scales, yep, yes. So the question was, is there a scale where you have to consider both halves of the theory, so the effective theory and the UV theory sort of at the same time? So you mentioned it in terms of, with respect to gravity, but this is a very general point, which is that absolutely yes. So the procedure, I will discuss this a little bit, but not much, is called matching, where you take your UV theory and at some, and you know it should match, so say we had QCD and the theory of pions and things like this. You know that the predictions at some energy scale, the predictions of those two theories should match. And you call that matching, so you take your UV theory, you derive its predictions at some energy scale, then you take the IR theory, the effective theory, and derive its predictions in terms of all of its free parameters, and they must absolutely match at some energy scale. And then you can continue to work at energy scales below that. And this is actually, this may sound a little bit unfamiliar, but it's actually something you're sort of doing already all the time when you do renormalization. If you think of how do we define the coupling constant for QED? It's just a free parameter in the QED Lagrangian, and we know it even receives infinite corrections at perturbation theory. So what do we do? We know that we don't have to worry about those large infinities if we match the predictions of that theory to an experimental measurement at some energy scale, then we define that to be the renormalized charge. That's actually a matching procedure in some sense, and you do exactly the same thing with effective field theories. Okay, another place that this is, so this is not just something in fundamental physics, but another place that has shown up is in the Ginsburg land-out, or land-out Ginsburg, whichever you prefer, theory of superconductivity. So you should Google this, because you will have seen in your standard model lectures probably the Higgs potential and how the electric week symmetry is broken. And it's really, the Higgs sector, the standard model is really like a relativistic non-Abelian version of this theory, which is the Ginsburg land-out model. Ginsburg land-out model is just a scalar field which has charge two, and it has a Mexican hat potential that you'd be familiar with, and at low temperatures it gets an expectation value that spontaneously breaks, essentially QED, and that's the reason you have superconductivity is essentially the photon has become massive. But this was just a dumb, not dumb, it was extremely clever, but this was an effective field theory that could not have been valid at all, energy scales, and it didn't give a microscopic description of why superconductivity was happening. It was sort of an empirical model that works very, very well, but does not tell you why superconductivity occurs. And actually, to understand the full story of superconductivity, we had to go to the UV theory, this is the effective theory, the UV theory, which is the full BCS theory of superconductivity, where you see that pairs of electrons have interactions with each other via phonons in the material, and that makes them actually become attractive to each other and then condense, and you can then interpret the scalar field in this theory as pairs of electrons in this theory as a condensate of pairs of electrons, much like in QCD, we have a scalar field like the pion, and we actually understand it in the more fundamental theory as being a condensate of pairs of fermions. So the story of effective field theories is ubiquitous, it is everywhere, and it's probably the most useful thing that you can, no, no, it's not. There are lots of useful things to learn, but it's a very useful thing to learn if you want to attack any problem in physics. Okay, so what can you read down here? Is this too low? So the last thing, so these are the UV completions, these are the effective theories in the infrared, and the thing that we want to know now, where we do not have a theoretical understanding, is what is the UV completion of the standard model? The standard model is an effective theory, it works extremely well at the energy scales that we've probed with high precision, like at the Large Hadron Collider, but it's crying out to be an effective field theory, for example, there are lots of free parameters, it sort of looks like an empirical model, there's a whole bunch of free parameters, the cork, the cowers, the gauge couplings, there's even a CP violating phase, mixing angles and so on, the Higgs mass and the Higgs cork, all of these free parameters that have no, that the standard model can never, ever tell you why they take the values they do, you just have to do an experiment, measure it just like with renormalization and QED, and match your observations to that experiment. So it's an effective field theory, but unlike these stories where eventually we understood the UV completion, we're still waiting to understand what the UV completion of the standard model is. And that's why the effective field theories are a great tool for understanding physics beyond the standard model, or at least organizing it. So what are the rules of effective field theory? I'm gonna give, this is gonna be very broad and basic, but there's a sort of an essential toolkit that is necessary for working with effective field theories, to help you organize interactions and organize structural aspects, things like symmetries. And we'll go through them step by step and try and build them all up. In a quantum field theory, you have to think about quantum fields, so you're thinking essentially about things like operators that create an annihilate fields, fields might be fermionic or bosonic, or they may have different spins. But the idea is to take every field and construct all the allowed operators. An operator is a product of fields, which makes an operator itself a field, multiplying fields just gives you a new field, it's a composite field that contains lots of fields, but it's still just a field. And these operators should be local. So this is what we will put in the end of the day into our action or our Lagrangian, from which we can then start evaluating predictions for things like scattering amplitudes or correlation functions. The operators have to be local, this relates back to essentially the requirement that you have cluster decomposition in your quantum field theory. So what happens here can be calculated without having to worry what's happening at the center of the galaxy. For example, if you have non-local interactions, then you have to know what's happening everywhere all of the time to even make a prediction. But if you have local interactions, which means that all of the interactions are evaluated at exactly the same spacetime point or they occur at the same spacetime point, then you can go ahead and calculate unambiguous predictions. Local interactions themselves doesn't mean that they're literally at the same infinitesimal spacetime point, because that spacetime point could be resolved into the UV completion. So for example, you might have pion interactions in QCD, and there are local interactions, so you write them in a Lagrangian, it's just some term, with four pions for example, all evaluated at the same spacetime point, but that doesn't mean as you go to infinitesimally small distance scales that will always appear as a point like interaction that can actually become resolved, and you see that you actually have quarks and gluons making up those pions and they're exchanging gluons between each other and so on. So it'll always be local in the sense of the energy scale that you're working at. Parameters may be dimensionful, so for example, there may be operators that have dimension greater than four, so the Lagrangian should have mass dimension four, and there may be operators that have mass dimension greater than four, and what you tend to do is soak up the coefficients of those operators with some dimensionful scale, I will typically in these lectures call it lambda, and we'll come back to what this scale means, what physical relevance this has for our effective field theory. One should also be careful about symmetries, so your effective field theory may have some symmetry or some gauge redundancy, and what this does is it constrains the form of your effective field theory, so if your effective field theory has some symmetry and there's some operator that you could have written down that violates that symmetry, then you are forbidden if that symmetry is present from writing it down, so this could be a hypothesized symmetry, or it could be a symmetry that you observe after the fact, so for example, there are in the theory of pions, there are some global symmetries that we observe, we observe that for example, the charged and neutral pions are very close in mass, and that's because of a fundamental symmetry, more fundamental symmetry that's actually present or approximate symmetry that's actually present in the UV completion. So we'll come back to this a lot, and so as a model builder and effective field theorist, you don't, there's sometimes noticed when discussing this, the aspect of symmetries, it's not that you're playing gauge, it's not that you get to decide what the symmetries are, but you may hypothesize one because you think it may explain some phenomena or you may empirically observe it, so when you're playing with an effective field theory, you can think, well, here's my general effective field theory, what if there was this symmetry then, or there was only a small amount of this, of breaking of some symmetry, then what physical implications would that have? Something that's very important, which comes back to this scale, lambda, is the relevance of operators. So for example, if we have some operator in our Lagrangian, which contains some interaction with a whole bunch of fields, I will come to this later, if we have some interactions, say for example, involving fermions, bosons, all sorts of things, I'll just call this O, and it's dimension larger than four, then I would typically write this guy here as being suppressed by some dimensionful scale lambda to the D of theta, theta's the operator here minus four, and I refer to this as sort of the cutoff, and there's also typically, and this is just some jargon busting again, and here there's a dimensionless number typically, which we often refer to as the Wilson coefficient, which is some unknown parameter that you may fix by observations or the UV theory may spit out for you. And what's important is that if I calculate some scattering amplitudes, so I have a bunch of states, say I have two states that I scatter at a collider to produce a whole bunch of other states involving this operator, then the scattering amplitude will tend to scale like the Wilson coefficient, and then unless there's some, if I'm working in this, if this is just containing fields, and this will tend to, and derivatives this will tend to scale like E over lambda to the D theta minus four. So what's this telling you? This is telling you that at energies well below lambda, that only the lowest dimension operators are gonna be important, because if I go to higher and higher, if this is a number that has to be less than one, otherwise I'm working, well, we'll come to that. So if this is a number that's less than one, the higher the dimension, then this is something less than one to a higher par. So this is telling you that the very high dimension operators, as I work at low energies, are much less important than the low dimension operators. And this is a very important concept. The second thing is if I work at energies approaching lambda, this becomes one, and all of the higher dimension operators start to become equally important, and in fact my amplitudes themselves may start to become larger than one, essentially. So I lose predictivity, which is telling you that the same thing that we've seen here and in these examples, if I go to higher and higher energies at some point, if it's truly an effective field theory, at some point the effective field theory will break down, and its predictions for me as an IR effective field theorist will become nonsense. This is telling you that there's some cutoff scale where some new description, the full UV completion has to kick in. So two things to remember, the higher dimension operators are not very important if you work at low energies, but as you start to approach lambda, the predictions of the theory will start to break down. There's another notion which I'll just put at the bottom here because it's less important for these lectures. Again, this is just to help with jargon. We sometimes talk about power counting. Apologies for my Belfast accent, that's power counting, P-E-O-W, I say power like sure. Power counting, which is that we may have theoretical ideas or prejudices as to what sort of patterns these things, these coefficients here may take, it's known as, as I said, the Wilson coefficient. So for example, if this Wilson coefficient is related to a small breaking of some symmetry, then it may actually be very, very small, or if the UV completion is a perturbative theory, then this may be, for example, a loop factor, some very small number. Or if it's a strongly coupled theory, it may be quite large, maybe you will see things like discussions of four pi counting and so on. So we essentially, when we're referring to power counting, we're essentially referring to the patterns of Wilson coefficients that can show up. Right, so it works sort of two ways. So empirically, starting from the bottom, you could look at the effective field theory and look at, you know, measure the patterns of Wilson coefficients, and that gives you an idea as to the power counting. So for example, if you have a strongly coupled theory in the UV, we have a power counting that's essentially an expansion in the UV coupling scale, often referred to as G star. And there's a thing, I strongly recommend you look it up, it's in my lecture notes, which I will post online, called naive dimensional analysis, which lets you count essentially derivatives versus fields in these operators and tell you how many times the G star will show up. And indeed, if you look at, for example, the interactions in QCD involving the Rho mesons and so on, it follows very well a power counting for that strongly coupled theory. The other side of the coin is that as a BSM theorist, you may have your effective theory, maybe it's a standard model, and you want it for whatever reason, you want to understand if it could be UV completed by a particular type of UV completion. So then you take that particular type, you impose the power counting that you expect for that particular theory, it may be a composite Higgs model or actually an extra dimensional model, things like this. And then that gives you a prediction for these higher dimension operators, for the Wilson coefficients of these higher dimension operators. Does that answer your question? Okay, super. Okay, so coming a little bit to this question of when these terms in the amplitude become large, we often refer to this as the cutoff. And there's a useful tool, a useful thing we should always remember that is that lambda, what you only measure this combination, say you measure this operator, whatever it is, in some scattering process at low energies, you measure this combination and it doesn't actually unambiguously tell you where the UV completion comes in. It tells you that by the time this starts to get big, something has to happen, you don't know what. But it may actually kick in before that. And you know this because you'll be familiar with this. If we just go to the Fermi interaction, I'm gonna be very sloppy and not even put in Lorentz structure here. But if you remember the Fermi interaction, looks something like g squared e bar e mu bar mu. There's gammas, one minus gamma five and things like this in here, the vector and axial pieces. Let's do it like this. I'm sitting in here, but let's not worry about them for the moment. Over mw squared. And what we see here is that g squared is essentially what I refer to as the Wilson coefficient and mw squared is the place, yep. You can write it a bit long. Pardon? You can write it a bit long. Ah, sure, yes. Sorry. Is that any better? No. No. No. Wow, really big. Okay. So with the Fermi interaction, we've integrated out. There's a g here and a g here. And what we've integrated out is we've done this in here is one over p squared minus mw squared, which is the propagator for the electric gauge boson. And in the effective field theory, which is the Fermi theory, we're coarse-graining. So what we're doing is we're saying, well, I'm gonna, we say integrate out, I'm gonna coarse-grain over all of the short-distance physics just to get the relevant predictions at large distances. And when you do that, you essentially shrink this guy to a point and write down the effective field theory that describes these interactions at low distances below the mass scale of the w. And what you get is an interaction that's called, known as the Fermi interaction, which scales like g squared over mw squared. Mw is the place. So this is what you measure. So this is like one over v squared, which is proportional to the Fermi constant. What you measure is the Fermi constant. It doesn't tell you, it only tells you this overall. It doesn't tell you what mw is. We know that mw is the mass scale at which this theory is uv completed. But g, as far as we're, from the low energy perspective, if g were something like four pi, you know, looking at the scaling of the amplitude here, if g were something like four pi, then we got the energies close to mw and we still hadn't seen anything. Then we know that the theory must be uv completed. But if g is small, your theory could be uv completed long before these amplitudes start to become one, because this e squared over mw squared is starting to become one. But g squared multiplied by e squared over mw squared is still small. So the cutoff, you shouldn't take it too seriously in the sense of the cutoff doesn't, if you measure just lambda or c times c over lambda to some part, the only information that's telling you is the point by which all hell has to break loose. You may actually access the uv completion before that point if it's a weakly coupled theory in the uv. Okay, and another important concept for our tour of effective field theory tools. I call it like widgets, you know, little bits of your sort of, you know, of your mechanic. You know, my dad has this great toolbox, a metal toolbox, you open it up and there's all these different layers of different types of tools for different things. That's what all of this is for the effective field theorist. And one very important concept is the concept of spurions, which is again, useful for understanding the structural aspects of the effective field theory that you're working with. So what is a spurion? So the idea is, imagine there's, in the uv or the IR theory, there's only one parameter that breaks some symmetry. Let's call this parameter epsilon. Then as epsilon goes to zero, you recover the symmetry. What this tells you then is if you can calculate to arbitrarily high orders in the, if this is respected, then you can calculate to arbitrarily high orders in perturbation theory, for example. And any observation that tells you about a violation of that symmetry will be proportional to this parameter epsilon. Because as you take epsilon to zero, the symmetry will be recovered. Is this, was that clear? Or is there, yep. No, absolutely yes. So it could be epsilon to the n or some polynomial of epsilon and so on. So this isn't something that you decide to impose on the theory, but it's something that you may discover empirically, or if you're working with some uv completion that you're interested in, which has an approximate symmetry, then this is a property that the IR theory will have. It's something that you're already familiar with. So pions are much lighter than the QCD scale. And that is related to precisely this fact that there's a global symmetry in the uv, chiral global symmetry, which lets you independently rotate all of the corks by different phases. And the only parameter breaking that global symmetry is actually very small. It's the Yukawa coupling of the up and down corks. That's a symmetry, an approximate symmetry, because the breaking parameter is very small compared to the other parameters in the theory. And so that's how you understand it in the uv theory and how that's manifested in the IR theory is that you see an approximate symmetry showing up that would be restored if some parameter went exactly to zero. So in the uv, that would be like sending the up and down cork masses to zero. In the IR, how you see this manifested is that the pions are very light. They have an approximate shift symmetry. They will come to this with their pseudo-goldstone bosons. They have an approximate shift symmetry, which is related to that approximate chiral symmetry in the uv theory, which is a very useful organizing concept. And one of the main tools we will use throughout. Because it's so important, I want to keep discussing it. So we will discuss it in the context of a specific example. So I'll write down a theory involving a scalar and a fermion. And we can sort of inspect what sort of symmetries this, so this will be our effective field theory. We can consider what sort of symmetries our effective field theory has, what sort of spurions we're working with to try and understand this better. So consider a theory where you have some complex, what would be complex? Yeah, complex scalar with a kinetic term and a vial fermion with a kinetic term. Sorry, is this too small still? Yeah, okay. We'll let them have some mass and we'll give Majorana mass to this fermion. And we'll also allow for, I'm not gonna specify them, but these two particles, so far this is just a free massive theory. Got a massive complex scalar and a massive vial fermion. Nothing interesting happening. But we will also allow for the possibility that there may be interactions between the scalar and the fermion. Okay, so what possible symmetries do we have here? There's actually a whole slew of different ones. I'll put them over here. So let's consider in the limit that m phi goes to zero. So at the moment, let's forget about the interactions and just inspect this part of the theory over here. If m phi goes to zero, can anyone tell me what symmetry we recover for phi? Who is this, sorry? Yes, precisely a shift symmetry. So in m phi goes to zero, the limit of m phi goes to zero, phi recovers a shift symmetry, which goes phi to phi plus some constant, superb. So in some sense, before we look at this, m phi is the spurion for shift symmetry breaking. If I take it to zero, I get a shift symmetry on phi. Then thinking about the interactions, what this means is if I have a shift symmetry on phi, say this is the only term that breaks the shift symmetry and I have a shift symmetry that's respected by all the interactions of phi, what that means is essentially that all the interactions of phi will involve a derivative acting on phi. If you're always differentiating it, then the derivative acting on the constant vanishes. So if I have, and this is really just like the pions, so I can have interactions of phi, interactions of the pions, which respect the shift symmetry, but I see that there's some small amount of shift symmetry breaking. But then what this helps me do is organize my structural understanding of the theory and organize, for example, what I expect for perturbative corrections in this theory, and even for non-perturbative corrections, because there should always be, if I'm considering m phi as a spurion, they should always be proportional to m phi, even if I can't calculate, even if I would need a supercomputer to actually calculate what's going on. The corrections, any corrections that break the shift symmetry will always be proportional to this parameter. Okay, what if I set m psi to zero? Can anyone tell me what symmetry we recover for the fermion? Yep, chiral symmetry, so this is a symmetry which would take the fermion and rotate it by some fixed angle, which is also a constant. And you see here you have psi dagger psi, so, and it's a constant, so the derivative doesn't do anything. So this guy remains unchanged. And again, same story, if these interactions respected the chiral symmetry, then what this is telling you is that it emphasized the only parameter in the game that's breaking the symmetry. So this is also something very familiar from the standard model. It's just like, for example, the electron eukawa. The electron eukawa in the standard model is the only term in the standard model that breaks the electron chiral symmetry. So then that means if I could go to arbitrarily high loop orders, or even if I have some non-perturbative physics above the standard model, I can understand that the electron always stays very light because there's a chiral symmetry protecting it, and any breaking of that chiral symmetry is proportional to the electron eukawa, which is a small number. Okay, what if m phi and m psi go to zero? Again, forget the interactions at the moment. Can anyone tell me what symmetry we recover? Who said, hands up, okay. Supersymmetry, that's next. So supersymmetry, but let's, okay, let's just, let's reverse the order a little bit. M phi equals m psi, then you recover supersymmetry. So we will come to supersymmetry probably at the very end of the lecture course. But when these two are equal, it turns out that there's this global symmetry in this action, which rotates phi via a fermionc symmetry into psi and vice versa, and it's known as supersymmetry. Was I gonna say more about that? No, we will come back to that. So what this means is something very deep. When m phi is equal to m psi, we have a symmetry, which is supersymmetry. So the difference in mass between those two guys is the spurion for supersymmetry breaking. Now, the important let, so now I imagine that all of these interactions happen to respect the supersymmetry. So the only parameters that are breaking the supersymmetry are when m phi is not equal to m psi. But I also told you that m psi is the spurion for the chiral symmetry of psi. So m psi can automatically be light and any breaking of that symmetry at higher orders or from some UV completion, if it respects these symmetries, will always be proportional to that guy. So that means that like I said for the electron, it's fine. We can easily understand why the electron, not why the yukawa is small, but how the electron is light, how the electron yukawa stays small for precise to that reason. But supersymmetry wants these two guys to be equal, which is what that's telling you that imagine you start off with these two guys equal and we have supersymmetry in all of the interactions. Then that tells you that whatever loop order and a perturbative calculation or whatever's going on with the UV completion, m phi will always stay equal to m psi, but m psi can happily be small. So they'll always be tied together. So essentially what supersymmetry is doing is it lets phi borrow the chiral symmetry of psi and it takes it along for the right, which is very useful when we come to think about the Higgs boson. Any questions about that? No? Okay, so this is the fourth one, back to this one. So what if m phi and m psi are identically zero and forget about the interactions for a moment? Can anyone tell me what symmetry we get? Yes, who is that, sorry? Superb, yes, scale invariance. So when we have scale invariance, this essentially lets us take the coordinate space and rescale it. So this is like this course screening. We're now sort of zooming. We're rescaling all of the space-time coordinates. But also if these two guys are zero and imagine L interaction is zero, then we can also rescale the fields. Sorry, this is maybe too small. Phi goes to phi over alpha. Psi goes to one of our alpha to the three halves. Psi. We can rescale the fields and the action will remain the same. So if the interactions also respect that scale symmetry, then this is a symmetry that we can use. And it tells us that if these are the only two terms that are breaking the scale symmetry, they're essentially the only two-dimensional terms in the theory. Then if the UV theory also respects the scale symmetry, then all of the scale-breaking terms that might be generated at higher loops or non-perturbative effects and so on will tend to come out proportional to the masses because they are the only terms that break the scale symmetry. This is something some of you may be, yep. Yes, so typically in 4D theories, yes. So the next thing I was gonna say, absolutely. So the question was, does scale invariance mean that you get conformal invariance? And the answer is typically yes. In every example that I've seen, I think it's essentially a theorem. Although I'm not sure if it's really totally accepted as that yet. Giovanni, is it at the perturbative level? Yeah, I guess. So yeah, so you recover full in this limit. If these guys go to zero and this guy respects the scale invariance, then you recover conformal symmetry. So places that you might, well, there are many examples of this. And this is a very important symmetry. And it's a very, very important symmetry in effective field theories. It's something that shows up all over the place because typically in, for example, condensed matter systems, but not only condensed matter systems, when you hit a phase transition, when you go through a phase transition, you start to get correlations on all distance scales, which essentially means you hit a conformal field theory. So conformal field theories are a very important end point of many effective field theory constructions as you're going down and you go to the deep IR. It's quite frequent that you will hit a conformal field theory at some point. Okay, is all of that reasonably clear? Yes, yes, absolutely, yeah. So yeah, alpha's here, just a real parameter. Yes, yes, it also hangs through. So say your UV theory say there's, okay, so another symmetry I didn't discuss because the mass parameter has nothing to do with it, but say there's a continuous symmetry acting on phi that goes like phi goes to e to the i theta phi, where this is a constant symmetry because this is, I chose phi to be a complex field here so I can rotate it by a complex phase. So here, this is just a global U1 symmetry and that I didn't, there's no terms here that I wrote that actually break it so that's why I didn't bring it up, but this is a very good example you bring up so we'll work through it. So imagine then this global symmetry is actually broken to some ZM sub, so it's like this guy has charged one. So say I've got, it's broken by some Spurion parameter that's called epsilon, which has a charge q equals, let's say minus n under this U1 symmetry and imagine this happens in the UV and I don't even need to know what happens in the UV. What are the implications for the IR theory? The implications for the IR theory is that I can now have, let's call it L epsilon, terms in my action which go like epsilon phi to the n because phi has charged n and epsilon has charged minus n, sorry phi has charged one, so one to the phi to the n has charged n, epsilon has charged minus n, so it has to enter in this way, something like this. And then this gives predictions, for example, a correction to the mass of phi and all sorts of things like this and predictions for how phi's can scatter off themselves and this is now a discrete symmetry so it leads to essentially another way of phrasing everything I'm saying is that there are selection rules in the effective field theory that tell you how the breaking of certain symmetries like this, or the breaking of continuous symmetries to discrete symmetries can show up. And you could do this, I could then break this Zn to a different Zn and have a similar story with a different parameter and so on. Yes, good question. So is this for the scale invariance? Yeah, yeah, exactly. So very good question, so this will come up when we discuss the hierarchy problem because if there's another mass scale at high energies, then that's already breaking the scale invariance. So if you, it can be, it's not so trivial to find an IR theory where all the scale breaking terms are small if the UV theory contains a dimensional full parameter, so if it explicitly breaks the scale invariance. Typically you can get them small if there are other symmetries flying around like the chiral symmetry or the shift symmetry and so on, but it's hard to have scale invariance on its own if it's badly broken in the UV, which is, we'll come back to this with regard to the Planck scale versus the weak scale. Yep, absolutely. This makes me very happy because then this is the goal of going through this argument, absolutely. So the problem with the Higgs, the hierarchy problem, isn't that it's a scalar. It's fine to have light scalars. I can sleep very comfortably when there's light scalars in nature. Precisely your point is the crux of it. It's that it has large interactions that break the shift symmetry. So indeed, if we were to take those interactions small, of all the U-cowas, we're really, really small. We wouldn't think that there's probably something around the corner because then we could understand that all of the corrections to its mass with regard to whatever's going on in the UV completion that we don't know have to be proportional to those parameters that break this shift symmetry. But you hit the nail on the head. What happens in the standard model, and we'll come back to this, is that actually the top U-cowa is pretty big. The gauge couplings aren't that small. So we observe big breaking of the shift symmetry. So then when we come to think about the UV, we expect big corrections to its mass that are proportional to those parameters. Does that answer your question? Anymore? Absolutely. Absolutely. So the question is with regard to the electron mass, as I just argued, and I can write down an equation for you, with regard to the electron mass, there's the U-cowa is the only parameter in the standard model that we see that breaks the chiral symmetry so we can understand that the electron mass is not small. But your question is, but why couldn't there be other parameters in the UV theory that are breaking it? So is this your question? So what's keeping it light? It doesn't have to be small, but we measure it to be small. So this is, again, this point, this going from the other direction empirically, we see that there's an approximate symmetry, which is the electron chiral symmetry. So that then tells us that if that is an approximate symmetry also in the UV theory, then it's automatic that the electron would be light because there's only one parameter that breaks the chiral symmetry, which is very, very small. It's 10 to the minus 7 or something like this. Is that, keep asking. Okay, right. Absolutely, so it is to do with technical naturalness, so we look at the interaction of the Higgs here, and there's a chiral symmetry, which, for example, rotates EC to E to the i theta EC, but that's not on its own, and that symmetry is, so we don't do anything to these fields. If I didn't have this term, that would be a symmetry of the theory, but I have this term which breaks it. But then what that means is that if the UV theory, if in the UV theory there's only this small parameter that breaks one small parameter that leads to this small parameter, that breaks that chiral symmetry, always any chiral symmetry breaking effects for the right-handed electron will always come out proportional to this guy. There'll never be larger corrections. Typically, not always, but typically in a quantum field theory, if you have a symmetry, a global symmetry, that global symmetry will be preserved by quantum effects. So it's not typically, we'll come to the strong CP problem where that changes a lot, but typically the symmetry will be respected by quantum effects, so any perturbative calculation you do, or even any non-perturbative calculation you do, will always maintain that symmetry. So now we break that symmetry a little bit. That tells you that then any breaking of that symmetry has to again come out proportional to that term that broke it. It's perfectly natural in the sense of technical naturalness. This is the essence of technical naturalness. It's perfectly natural for the electron to have a small mass. There's still a puzzle about why on earth this yukawa is so small, it's very, very small, and that is a valid and interesting question, but it's not that quantum corrections are gonna destabilize its smallness. Okay, so this first lecture, by the way I'm sure you can tell, is just, as I said, it's just a toolkit, and there's one last tool, which is related to this notion of power counting that I think is very useful, and it's not typically brought up, but it actually, I find as an organizing tool for model building and working with effective field theories can help keep you sane and stop making, stop you from making too many assumptions about the UV theory by feeling like you know more than you do, which is, this tool, which I find is useful, is to think carefully about masses and scales. So in particle theory, not even particle theory, particle physics 101, we're taught to set H bar and C equal to one because that makes our life a lot easier, but actually we lose a little bit of information when we do that, because we know how H bar enters the action in the path integral, it goes like S over H bar in the exponent, and that can actually help us carry our, H bar goes to zero is the classical limit, so holding on to H bar is actually not a totally stupid thing to do, because it can actually, it's like a tracer of, you know, it's like the line of breadcrumbs for quantum effects and how they enter. So if we put it back in, then of course it's definition dependent, but if you put H bar back into the action in a certain scheme, of course, because you can always rescale fields and coordinates and move things around, but in a certain scheme, H bar has units of energy times length, the Lagrangian has units of energy length to minus three, and so on, things like derivatives just have the same units as masses in my scheme, which is all in the notes that I put online, which is length to minus one, but importantly, couplings have, why will be a U-cow a coupling, have dimensions of inverse square root of H bar. Scalar-cordic interactions, which I will write like this, actually have dimensions of coupling squared, which I don't know how many of you have studied supersymmetry, but you know the D terms you get in a supersymmetric gauge theory, the scalar-cordics go like gauge coupling squared from the gauge theory, are you familiar with this? And it had to be, there's nothing else it could have been. You may have wondered, why does it always come in like coupling squared, but it's impossible for it to be anything else because a scalar-cordic has dimensions of coupling squared. Could never have been otherwise. Okay, so, and fields as well, scalars and fermions and bosons have dimensions and so on, but then if we look at an interaction in an effective field theory, it might take some form, which goes like one over something like lambda to the D minus four. There may be some derivatives, I'll call it, and the number of derivatives is n derivatives. This is all completely schematic, you know, you understand that I can contract Lorentz indices in different ways and so on. Some bosons, which I'll say, there's nb bosons. These may be vectors or they may be scalars, it doesn't matter. And some number of fermions, and the dimension of this operator is nd plus nb plus three over two nf, like this, and that's the engineering dimension in terms of mass units, but it also carries dimension in terms of coupling units because couplings have dimensions of inverse h-port to the half, and the full dimension of this parameter here, lambda, is m tilde, the dimensions of mass scale, which is the scale at which responsible particles will show up, for example, for the Fermi theory it was mw. But also it carries dimensions of coupling that go like c to the n minus two over D minus four, or it tells you that essentially that by measuring this guy, if it carries any coupling dimension, then that's not telling you unambiguously where the UV completion scale comes in because there might be some coupling, some small coupling or some large coupling that is an overall pre-factor in whatever is here that tells you that this guy here is not necessarily equal to m tilde. So we can use this again to do some simple exercises. So for the Fermi theory, I'm just gonna write this schematically, there's gammas, it's a four Fermi on interaction, over lambda squared, and the dimensions of lambda are mass over coupling, which is what we already saw. Lambda is, if one over lambda squared, it's mass over coupling, so the place at which it shows up, at which the full UV completion shows up will depend on what the size of the coupling is, and the Fermi theory, it wasn't so small, so the UV completion showed up at a scale similar to V, which is the same thing as lambda here, but if you have a small coupling, then it will show up much earlier than that, but in another place that may be less familiar to you is in gravity, so the graviton interacts with the stress energy tensor of the standard model, but m-plank actually carries units of mass divided by coupling, and so for example, in string theory, this is literally the strings, essentially the string scale divided by the string coupling, so we will go on to discuss the hierarchy problem, but the hierarchy problem isn't that there's the plank scale, because the plank scale isn't a mass, there's never gonna be, if you think about quantum gravity, there's never gonna be corrections from quantum gravity, even though we don't know what it is, that go like m-plank squared, multiplying Higgs squared, there's no Higgs mass corrections going like m-plank squared. There always has to be coupling in there to soak up this coupling dimension here, which again comes back to your question. The problem here is the standard model couplings with the Higgs, yep, and I think was, I think it was the total number of fields, I think it was nb plus nf, but I can't remember. It's in the notes, I don't have it here, and we can, I'll figure it out over coffee time. Okay, so this is a useful tool, so for example, and this is something that is actually, whether it's stated or not is used a lot in things like composite Higgs models where we believe that the next, just like with pions, we have a scalar and then we go to the UV completion, which is quarks and gluons. It may be the case that the Higgs is like a pion and that when we go to higher energies, we will see the full UV completion where there would be beyond the standard model quarks and gluons that are confining to give you light scalars, which would be the Higgs, and you have to use this tool to understand the nature of the power counting because if there's some strong coupling in there, it will enter in a certain way in all of these higher dimension operators. Okay, so now it's time to finally move on to the hierarchy problem and why so many theorists obsessed over the weak scale, but before we do that, I guess we will just have 10 minutes of that. Are there any questions about any of these sorts of EFT-ish tools or notions? Someone's complaining, but I don't think they're present. Okay, so now I want to discuss the hierarchy problem, which I think, you know, from the questions that are being asked, a lot of you are already anticipating. So, yeah, so we can understand the hierarchy problem. Again, we're going to lean heavily on this analogy with QCD and with pions. So, and we're going to use the Spurion arguments that we developed in the last sort of half hour. So, to understand what's going on. So, below the QCD strong coupling scale, so let's say this is lambda QCD, we see some sort of light-ish particles like the Rho mesons and things like this, the eta prime and so on. And then, hanging down well, below that we see the pions. And there are three pions. There's a pi zero, which has mass of 135 MeV. And there's the charged pions, which have mass of 140 MeV. So, working empirically, if you had never built a collider to go above the QCD scale, you would spot that there's something fishy going on here. Right, in some sense, you know, model builders and phenomenologists, theorists are detectives. And we can smell that there's some sort of circumstantial evidence that there's some structure because these guys are so close in mass, and why would that be so? And it turns out that it's related to an approximate symmetry in the UV. So, the pions are well understood as Goldstone boson, so I will come back to this in more detail later, but a Goldstone boson is a, an exact Goldstone boson is an exactly massless scalar. And the fact that it's massless is related essentially to a shift symmetry. And this shift symmetry is related to the fact that in the UV, what theory, whatever it is, there's a continuous global symmetry that's been spontaneously broken. And this notion can help us understand why there's such a big gap from all this QCD jump up here down to the pions because they are sort of like quasi-Goldstone bosons. They can happily be lighter because there's some approximate global symmetry, continuous global symmetry that's been spontaneously broken at this scale. If it were an exact symmetry, the pions, despite being made up of QCD stuff, would have been exactly massless. So, you should always remember that you shouldn't get confused that the pion mass is sort of close, for example, to the neutron or proton mass. And all QCD stuff must have mass around that scale. That's not necessarily the case. If there was an exact chiral symmetry in QCD, they would have been exactly massless. Okay, so we can package these pions into a representation of this global symmetry. So how we do that, I'll come back to this, but it's known as CCWZ construction. So we package these pions and my notation is pretty dreadful here because I'm calling them Pi I, multiplying by some of the Poiley matrices, which are the generators for the symmetry, which I will come to in a second. And we make the whole thing dimension, the whole exponent dimensionless. So this is a field, Pi I, there's three of them. This dimensionless, there's three Poiley generators. And these are the Goldstone bosons that arise when SU2 left cross SU2 right, sorry, SU2 right, is spontaneously broken to SU2 vector. In the middle. So you see that this group has three generators, three generators, this one only has three. So we've lost three, which means there are three generators for the continuous global symmetry, which have been spontaneously broken and that's why we get three pions. As I said, that's going a bit quick, but we will come back to it in more detail for the Higgs. So we know that if we have an explicit breaking of the diagonal piece of this already in the action, even though we don't know what the details of the UV completion are, we can write this down as a spurion, I'll call it M Pi, where you have the mass terms coming from something like a term like this and Pi squared is the spurion for breaking this symmetry, F Pi squared times the trace of this guy. So this guy explicitly breaks the axial part of these two global symmetries. And then we get a Pi on mass matrix that looks like this, on half M Pi squared Pi zero squared plus Pi one squared plus Pi two squared. And we can write this equivalently if we want to package them slightly differently. It's M Pi zero Pi squared Pi zero squared plus M Pi squared Pi plus squared. Just packaging these two guys now into a complex scalar field. Okay, so we can understand already a bunch of symmetries here. So if the axial part of this is only explicitly broken by a small parameter, our spurion for this will be M Pi. Then we can understand why they're naturally light and the only breaking of the shift symmetry is now this term M Pi. So if I took this term M Pi to zero, I would have a shift symmetry and they'd all be naturally light. So using my spurion arguments, I can see why these guys are naturally light compared to the UV scale. But there also, as you can see, this parameter didn't break essentially off diagonal pieces of this symmetry, which means that all of these Pions remained at exactly the same mass. This parameter is the same for all three of them. It doesn't discriminate between them. You can rotate them in amongst each other. This is sort of like an SO3 symmetry at this level and amongst each other, which is keeping them all light. So not only do we have an understanding in terms of the spurion arguments about why the Pions are light, but also why they all have the same mass. So we've done very well. This puzzle here of understanding this gap and why they're so close to each other is well resolved by spurion and symmetry arguments, even if we don't know what the UV theory is. But there's a great big elephant in the room which is that the charged Pions are charged. They interact with the photon. So when we look at the kinetic terms, the kinetic terms in our action look like a standard boring kinetic term for the pi zero. But for the charged Pion, we have the electromagnetic interaction. So this looks like a minor disturbance of the theory, but it's actually a very violent disturbance of our spurion arguments. Because while here we can understand that M pi is the only parameter in the game, maybe the only parameter in the game that's breaking the shift symmetry and the leftover symmetry which rotates all these Pions amongst each other and keeps them all at identical mass. We see that there's another parameter in this theory, E, which not only breaks the shift symmetry, can lead. Now, if I do a spurion argument, I should have corrections from the UV that go like E squared, then some big correction to the Pion mass squared. But it also breaks the symmetry. So it's telling you that the charged Pions don't even want to be light in the first place and their masses must be corrections to their masses proportional to this new spurion E. But also the photon is not interacting with the neutral Pion. So these mass corrections that break the shift symmetry from the UV theory should be different from for the neutral Pion. So what do we do as an effective field theorist? All we can do, we don't know what's going on in the UV, all we can do is write down all of the terms we expect to generate proportional to this guy here that might correct the mass of the neutral Pion. So it's called delta L mass. And we expect to have terms in the action that go like E squared, at least as sort of a loop factor, we don't know. And the cutoff squared, this is the cutoff, where all of the unknown physics is times pi plus minus squared. Of course we could also get correction proportional to m pi squared, but that's actually what we wrote down already when we implicitly included it in this term here. And this comes from, why you might imagine this, this is a very sloppy, yes? So you're saying that there's some unknown constant in front of this guy? Yes, exactly. But that requires, if you want that to be small, so this doesn't destabilize it, you're praying to the UV, you don't know. So you're asking for some sort of fine tuning or some sort of miracle for that to be small because you've used up all of your symmetry arguments. Ah, yes, yep. No, I'm saying, so if this were small, for example, say I take E to be really, really small, then this becomes small, but it's still always there. Does that answer your question? So we're nearly wrapped up, and I can discuss over coffee time if anyone has any questions. So a really dumb way of understanding how these might arise, but you should, I urge you not to think of it in this way, think of it in terms of spurions, but you can see that, for example, there are corrections to the mass of the charged pions from photons, which if you did something really dumb and just let the cutoff of the loop integral be lambda, where the new physics has to kick in, then these corrections go like that. But that's a sort of a dumb, perturbed way to think about it. More generally, symmetries will help. So now we sort of have a hierarchy problem and we've observed these masses, so we can actually use this as input, and if we don't want for m pi plus minus minus m pi zero to be less than five MeV, this actually requires that lambda should be less than or around the ballpark of 750 MeV. Otherwise, the mass splitting coming from this guy would be larger than we observe and the overall mass correction to the charged pion would be larger than we observe. So now we're using this hierarchy problem, this kind of spurions as a strategy to understand what's going on in the UV. So if you never built the colliders above this energy scale and you observed this, you would say something fishy is going on unless something kicks in at around 750 MeV that can explain why these guys are still so close in mass. Otherwise, these corrections would have been too big and what would have gotten rid of them. And of course, in practice, that's precisely what happens. In practice, there is something that shows up, it's the Rho meson and the rest of QCD and in the rest of QCD, you can actually calculate this correction and you get m pi plus minus squared minus m pi zero squared is, I'll have to cross over the boards here, three E squared over four pi squared m rho squared m A1 square, these are just meson masses in QCD, m rho squared plus m A1 squared times the log of m A1 squared over m rho squared. You can actually just calculate in QCD what this mass correction is and you get the right answer. What showed up was essentially the rest of the mesons in QCD. So you see that this strategy is extremely useful the Spurian argument is not, it is actually a quantitative tool. It's always a bit wishy washy, you don't know precisely as was pointed out, you don't know what the exact coefficient could be but it's telling you that something should kick in at around 750 MEV and that is precisely what happens. So that's the pions, you can see what's gonna happen with the Higgs but we'll do that in the next lecture. I'll leave it there. Questions?