 Yeah, it's it's it's wonderful to be back at ICTP. It's one of my favorite places to come and especially Lecturing here. I understand that there are 50 countries represented here in the audience, which is wow That's pretty cool. So I'm going to be talking about supersymmetry Here's a list of some of the topics. Can you see this thing? Okay here list of some of the topics I want to cover but I'm not going to spend time introducing it I'm just going to go right in. Let me just say that this is a vast subject There's an enormous literature on it. There's no way I can do justice to all of it and do it in all of it in detail and so I'm going to be doing different things at varying levels of detail. So in the beginning I'm going to be quite detailed hopefully to with the formalism to make sure that the formalism is Concrete and that you have something that you can really use to start Doing calculations on your own and then when we get into the more phenomenological aspects It's going to be more of a survey, okay? Frankly. I haven't really prepared. I'm only prepared sort of about a third of this Okay, so based on how things go today. I'm going to figure out how to do the rest of it So ask me questions and you can influence the rest of the lectures Because I want to start off with the motivation the main motivation for supersymmetry at least in a phenomenological context of course like this, which is the hierarchy problem so We it's a very common idea throughout all of physics that we don't we're not dealing with exact theories, right? We most theories that we have in physics are effective theories There are approximate descriptions of the physics that are valid in a limited dynamical range, right? So for example the never stokes equations that describe fluids are extremely successful in describing the detailed behavior of fluids But it's clear that they have to break down at atomic distances So for us the question in particle physics the question is what is the status of the standard model of? Particle physics is it an effective theory? Is it an exact theory if it's an effective theory? What scale does it break down? Okay, and it's worth Reminding ourselves that particle physics has been in a very privileged position for the last I don't know you know 80 90 years okay in the 1930s Fermi wrote down his theory of beta decay and He was made fun of by Oppenheimer and people like that because this theory had a cross-section That grew with energy and therefore broke down at an energy scale of a TEV that was in the 1930s Okay, then of course There were the proposal that the weak interaction instead of this silly model We had a much more sophisticated model with W's and Z's But if we only have the W's and Z's in a different process namely WW scattering also has a quadratic growth in energy In fact, it's parametrically exactly the same growth and it also breaks down at a TEV now. This is good Right, it's fantastic to have the theory that you the best tested theory that you know Break down at a finite scale and one that you can access experimentally so particle physics has been in this very very privileged situation and The LHC was built and finally in 2012 the standard model Higgs was indeed Discovered, okay, and that is exactly what's required to fix up this WWW scattering Okay, and so after having been in this situation where we knew there had to be new physics below the TEV scale We're now in a situation if we take the standard model with the Higgs We can extrapolate this all the way up to the Planck scale and there's no inconsistency. There's no guarantee of new physics Okay, and particle physicists are freaking out, right? A lot of people are freaking out But that's just the way it is. That's the normal situation in physics actually that we don't know that our theories necessarily break down, right? Classical mechanics broke down not because of any internal inconsistency of classical mechanics. It was just wrong, right? So we're in that situation now to the standard model Doesn't break down at high energy scales. Okay, so Maybe should we consider it to be an exact theory? Well, we can't do that Okay, because there are many many phenomena that cannot be explained by the standard model Okay, and I've listed some of them here and There some of these are just experimental facts for example neutrino masses in dark matter exist We need to explain them. They're not explained in the standard model. Okay? There are other things that are more theoretically motivated. So this is a list of these things But one of the things about this list is that there's none of these things There's only one of these here that really absolutely requires new physics at the TEV scale Which is the scale where we can do experiments right now. Okay, and that is the naturalness of the electroweak scale Okay, unfortunately, it's also probably the most theoretical and loosest of all of these things, but that's life So let me explain what this is. Okay, so First of all, let's remind ourselves of some very basic things about dimensional analysis. Suppose we have some Lagrangian Okay, and consider a coupling in that Lagrangian. It has some mass dimension Right reason units were h bar and c or 1 everything can be measured in mass And I'll use these little brackets to give the mass dimension So last mass dimension of n means that this coupling lambda is mass to some power. Okay, and so capital M is that mass scale Now imagine we treat this coupling as a perturbation. So we just do perturbation theory in that coupling Okay, we would get some lowest order result which is lambda to the zero plus terms of order lambda to the one and just by Dimensional analysis there has to be a power of some energy scale that appears there That'll be some energy scale associated with the process right some momentum transfer or the mass of an external state or something like that Okay, and what you can see is that dimensional analysis tells us where this perturbation is big and small right for n Positive positive mass dimension the perturbation theory breaks down at small e if n is negative It breaks down at large e and if n equals zero it works at all energies Okay, and so we call these things relevant irrelevant and marginal couplings That's the terminology and do introduced by Wilson Okay, and The thing to remember is that you have to remember there certainly an infinite number of irrelevant Couplings and that's because the fields in our Lagrangian always have positive mass dimension right scalars And gauge bosons have mass dimension one Fermions have mass dimension three halves and so if you can you can keep what can you do? You can keep adding more fields and more derivative operators and of course you get extremely You get couplings with extremely large negative mass directions. Okay, so how do we interpret this? How do we interpret the fact that we have all of these these possible couplings that are allowed by by the Symmetries of the problem. Well, the the sensible interpretation of this is that these terms arise from New physics at some high mass scale Okay, so if these terms if there's some new physics at some high mass scale And if you integrate out that physics and write an effective Lagrangian only for the light degrees of freedom You expect terms like this to be generated, but they'll be generated with a characteristic mass scale associated with that very high Scale physics, okay Sorry. Yes, please. I'm sorry Previous slide. Sure Yes At n equals zero all all the terms in the perturbation series will be one one one plus one one Oh, but I didn't I mean there's lambda itself So lambda would have to be lambda as a dimensionless number Okay, so I would have to I couldn't really use this formula because m to the zero you're right would be one Okay, so I would have to write it as as I would have to put a little epsilon in front of this or something Okay. Yeah, but so for the dimensionless case the parameter itself is dimensionless and it has to be a small number For the dimension full case You can see that the the perturbation whether the perturbation theory expansion works or not as controlled by whether the energy is large or small Okay. Yeah, thanks Yeah, sorry, do do please interrupt me and I'm sorry if I'm fine. I'm a little jet lagged I've just been up for 24 hours So if you might need to get my attention, but I really do I appreciate the questions very much Okay Okay, so so the idea here is that that if we if our theory is an effective theory If there are some new physics at some high-scale m then generically we expect an infinite number of higher dimension operators suppressed by powers of m but The converse of this up here is that there's always only a finite number of relevant or Marginal couplings there's always a finite number of those and those naturally dominate at low energies Okay, and those are the renormalizable couplings and this is the view of Ken Wilson this is the viewpoint of Ken Wilson about why renormalizable quantum field theories are good descriptions of nature as we know they are and the standard model is in a way the Perfect exemplar of this idea of this philosophy. Okay, it's a the standard model is a truly amazing Thing if you boil it down, okay Because you can define the standard model by saying that it's the most effective the most sorry the most general Lagrangian that you can write in terms of these degrees of freedom all of which we have experimentally observed now right, these are the quark fields the lepton fields and now that the Higgs the Higgs field and We assign them there the gauge quantum numbers that they must have that basically tells us how they interact with the spin one Particles that we know are there in the standard model We have three generations and then we simply write down the most general Lagrangian Involving with with with relevant and marginal couplings that we can that defines the standard model We don't leave anything out. We don't impose any extra symmetries. That's it So when I when I teach the standard model I always call this the standard model wallet card because if you carry this around with you you can in principle Reconstruct all the physics of the standard model from this card Okay, of course you have to figure out what the parameters are by comparing with experiment right and This is really a long way from the original wallet card that was published by the particle data group had a lot of more Experimental information in it right because there was really not much of a theory that described the particle interactions that were known at that time Now we can really summarize the theory in an equally in a very very succinct way Okay, and this effective theory where we just put in everything that is allowed and don't leave anything out put You know has an incredible number of successes some of which you're learning about here at the school I'm just going to list some of them. There are the weak decays of the Hadrons and leptons Quark mixing CP violation the intricate pattern of flavor changing effects Which can be summarized by saying that there are no flavor changing neutral currents Also the fact that there's a baryon and lepton number symmetry doesn't have to be put in it just comes out Okay, so this is a perfect exemplar of what we want from an effective theory We just look we write down the most general interactions. We can Compatible with a few basic facts about the low-energy world and voila, right? We agree with every experiment performed so far unfortunately Okay So There's just one little problem with this story Okay, and that is the fact that the standard model actually all the couplings in the standard model are in fact marginal They're dimensionless parameters except for one which is the quadratic term for the Higgs, right? The Higgs mass term so remember that that's negative It's not the physical mass of the Higgs the physical mass of the Higgs is is minus 2mh squared at At tree level that basically we'll just refer to this as the mass of the Higgs Okay, so that obviously has dimensions of mass. It's a relevant parameter Okay, and so if we believe in this philosophy that the standard model arises from integrating out some new physics at some very high-scale m You have to ask the question. Well this dimensional analysis that was so successful in allowing us to neglect all these higher dimension operators It suggests that mh squared should be awarded this capital M. It should be huge Okay So you can ask okay, but that's just dimensional analysis. Is this really a problem? Okay, because sometimes dimensional analysis fails. So this is a good way to do physics, right? You do dimensional analysis if it really fails then you found something interesting something something should something should be happening to fix this up Okay, now if we look at actual models rather than just doing dimensional analysis We start to see that this is not such an easy problem to get around Okay, so suppose you just throw in a new particle x, okay? And this particle couples to the Higgs it has some coupling to the Higgs and you can write down this one loop diagram Okay, now forget about quadratic divergences Divergence bedurvenge Okay, forget about it. All right, we normalize they're gone Okay, but this particle x has a physical mass that you could go out and measure Which I'm assuming is much larger than the TV scale. Okay, we may not be able to do the experiment But it's we could and so there's a finite correction, which is proportional to mx squared Okay, if you do this if you do this this kind of diagram for model after model after model after model after model Okay, and the reason that you always find this to be non-zero and proportional to mx squared can be traced to the fact that If you look at this operator right here this thing that we're correcting This thing is invariant under all the symmetries of the problem h dagger h is a completely a singlet You can't somehow forbid it by imposing some symmetry at least not very easily okay, and We do in fact expect particles like this We we we want particles like this to be there to address various problems that we have in physics for example The seesaw mechanism for neutrino masses. We might expect right-handed neutrinos at very high energy scales If we believe in grand unification, we must have particles just like this with masses of order 10 to the 16 g e v if we believe that That that that that that quantum gravity takes place at high energy scales We need some sorts of new particles like this and they definitely have to couple to the Higgs because gravity couples to everything Okay, and so at astronomically high scales. We believe we must have particles like this Okay, and whenever we look at particles like this They do give these kinds of contributions on the other hand experimentally. We need this parameter to be small Okay, so we seem to require some enormous cancellation. This is many orders of magnitude larger than the observed values We need some enormous unexplained cancellation Okay, that is not an explanation of why dimensional analysis fails. That's just a failure of predictivity Sorry, there's a question. Yes. Sorry. Thank you Can I go back to the slide before please? Yes Yeah, could it be that like the new particles the contributions of the new particles at this very large scale gut scale plunge scale Whatever could cancel each other directly like they could they they could but we would want to know that there's some principle that Underlies that right because we could already That is true We could we could assume that they're they cancel But then what would happen if you changed if you had to an X and a Y particle that were both at the gut scale They would have to cancel each other to fantastic accuracy. So if you changed the Y mass by point 00001 percent the cancel the Higgs mass would be something completely different Thank you. Yeah Go ahead. There's another one behind you then. Yeah, sorry. Is this scenario that the as a very energetic particles coupled to Higgs and in Coupling strings like one over energy unrealistic. So that this correction is always say order one over a small You're talking about these these these irrelevant operators from the from this previous slide or I mean the diagram You don't know the coupling. So if the coupling to the Higgs becomes very weak at large energies Okay, so here I have assumed that this coupling here is a dimension less coupling Okay, so the the the motivation for that and all of these kinds of models I need new particles that have a big effect on the Higgs Okay, for example in grand unification the Higgs and the Higgs and all the other particles need to complete themselves into larger Multiplets, okay, and so there have to be particles like this that have couplings that are related to the standard model Couplings, okay, so for example gauge X gauge bosons whose gauge couplings are in fact equal to those of the standard model So I'm I'm suppressing many many details here But in space so for example in in grand unified models There are very very Concrete reasons why these couplings have to be exist and even what their values are so that's maybe the cleanest case I understand but if you abandon on grand unification is there still a reason to assume that the coupling should be dimensional Well, if you I think you would if you you could try to abandon all of these things Okay, and that's actually one of the options. I'll talk about in the next slide You could just say well I don't I just don't want any new physics above the TEV scale at all Okay, and you can try to do that and people have thought about that So one of the our job in facing these very basic fundamental problems I think is to try everything that makes sense, okay So I'm gonna of course be focusing on supersymmetry in these lectures But there are many other things that have been tried that's that was my next slide actually Okay, so here I'm I'm just listing this is the basic what I've tried to describe here is the basic conundrum the hierarchy problem and Here I'm just listing. I'm not going to talk about these but I'm during the lectures. I'm happy to talk about anything during coffee breaks or in the discussion periods But some of the ideas that have been enumerated that in my opinion make sense, okay One which is the one that we will be focusing on is that in fact there is a symmetry that can forbid this term namely supersymmetry The Higgs compositeness is the idea that there's basically a cutoff on the Higgs the Higgs dissociates And so it doesn't make any sense to talk about those diagrams above the high energy scale and those of that will be discussed By Andrea Voltser in this course some that I don't think will be Discussed in this course you could just have this is the option that you were talking about There's just no physics of new new physics above the TV scale or the TV scale is the fundamental scale of physics That implies in particular you would need quantum gravity at a TV scale That's actually possible in the models with large extra dimensions. You could have anthropic Selections and I misspelled it and that's probably appropriate because I don't want to say anything about it And just to as a proof that you know I've given many many colloquial where I've said you know These are the only sensible ideas out there to the extent that they even are sensible but just You know not very long ago. There was a completely new idea proposed Which as far as I can tell makes sense and I'm not going to talk about it at all But it's a relaxation model. So that's a great inspiration for you for anyone who thinks that you know, there might be some Cool new ideas that nobody's thought of yet Okay, okay, but we're going to focus on the first one here that that in fact There is a symmetry that can that can explain why this thing is zero Okay, so to understand what the symmetry is let's notice first of all Let's notice that this the standard the standard model doesn't actually have any Fermion mass terms in it, right? It has Yukawa couplings, which become fermion masses when you have beds But let's suppose it did just suppose we add to it some parts some some fermions that have an explicit Master in the Lagrangian then we still even though that's a relevant interaction We would not have the same problem that we have with the the Higgs mass with the scalar mass And the reason what we would find instead if we if we if we had some x particle like this And we and we we looked at the corrections to the to the mass of some fermion We would find that the correction to the mass of the fermion is Proportional to the mass of the fermion itself. So instead of being proportional to mx. It's proportional to m psi So this is not a problem at all, right? Okay, and why is this the reason is because there's an extra symmetry in the limit where a fermion mass vanishes, right? There are extra chiral symmetries, and so the corrections have to be proportional to the parameter itself They can't be additive corrections Okay, on the other hand the scalar mass term doesn't have that character if we set the scalar mass term to zero There is not an additional symmetry that Actually that statement is actually not quite true if you if you there's no other no symmetry that's compatible with for example The quartic coupling and the yukawa couplings. Okay, because you could have a shift symmetry But anyway, there's no similar. There's nothing that forbids this that's compatible with the other Couplings in the standard model that we know have to be there Okay. All right So this term seems to be invariant under all symmetries But supersymmetry as we'll discuss finds a way of making this term not invariant even though it looks like it's invariant under Absolutely everything and the basic idea is that supersymmetry is a symmetry between bosons and fermions So in the in a supersymmetric theory We have the Higgs doublet and there's also a fermion doublet Something that has exactly the same quantum numbers as the the Higgs called a Higgs, you know And there's a symmetry that relates the Higgs and the Higgs you know properties and same for all the other Particles of the standard model Okay, and if you believe just for a second that there could be such a symmetry then the point is is that this Symmetry requires that the mass of the scalar be the same as the mass of the fermion, right? It makes sense that if the symmetry relates these two particles They must have the same mass, right and once they're Once they're forced to be equal by supersymmetry then this chiral symmetry means that it's natural to have the Higgs eno mass or the fermion mass be small and therefore it must be natural to have the skater mass be small So this is the basic idea behind supersymmetry and of course we're going to be making this a lot more precise okay So now of course this symmetry between bosons and fermions at minimum It would require that there should be one boson for every fermion and vice versa with the same quantum numbers And we don't observe that so the symmetry must be broken can't be exact in nature And so if we look at for example if we look at how this works if we we look at the top quark Which is the particle that has the largest coupling to the Higgs So this is sort of the biggest loop effect we could imagine that generates a Higgs squared mass If we just calculate in the standard model this top loop we would find that it's quadratically divergent Okay, so we're supposed to renormalize that away and then we find a term which is logarithmically divergent proportional to the top quark mass, okay now Everything that Ken Wilson has taught us Tells us that we should actually take this lambda dependence seriously and we see that in many many models and This quadratic dependence really does signal even though it can be renormalized away It signals the fact that you are sensitive to physics at very high scales as we saw before with these x particles, right? Now what you find in supersymmetries that you have a scalar and the scalar has Couplings that look like these quartet couplings, but this quartet coupling here is equal to y t squared So there's a factor of y t squared here And it's the same it's the same y t because of supersymmetry and then this completely different diagram has an opposite sign Quadratic divergence, which exactly cancels signaling the fact that the quadratic sensitivity to high scales is cancelled And you have some term left over Proportional to the mass of the particle going around this loop the the scalar partner of the top which is called a stop an S top right and so the final thing you get no quadratic divergence And you get a logarithmic Sensitivity to high energy scales that can be parameterized by the renormalization group And it's proportional to the difference of these two masses So if these two it's a stop mass we're equal to the top mass even this logarithmic sensitivity would vanish Okay, so this is the sort of paradigm for the way that supersymmetry is going to solve the hierarchy problem There's no more quadratic dependence on high energy scales, and we have a mild logarithmic dependence left Questions on this Yeah, if I just renormalized this lambda square away, why should I be still sensitive to it? Well that goes back to these these kinds of things right here, right? So in any concrete model, right? Well, actually, I don't have to go back here Okay, so let's say that you say who needs this stop I'm going to take the stop mass to be 10 to the 16 g e v Well, then you would have an enormous correction to the Higgs mass So the claim is that if that in these models right here in these models right well In these models as in any model if you have very heavy particles, right? They will contribute to the to the mass parameter of the Higgs Okay, but if the new particles the new heavy particles always come in pairs like supersymmetries Maybe valid at a very very high scale then I don't get these then these Could concentrate this is the super partner of the top not the super partner of something else So you could you could try to imagine a theory like that. Oh, yeah, okay? Yes Good, we're like all new particles which additional to these standard model particles come in those pairs That's right But unless there was some principle guaranteeing that supersymmetry cannot do that because supersymmetry has to relate the top to something Which must be the stop and Then the stop cannot be too heavy Okay, thanks, right, so it would have to be but there could be some other principle like that that would be very interesting Yeah, go ahead. I'm sorry The last sentence on the slide says that The mass stop should be smaller than TV ended couples to Higgs So shouldn't we have seen the sorry the stop of course not the top Shouldn't we have seen this particle and collisions then yes, and I'll talk more about that later Okay, I mean that's a question of the extent of which that's the problem is going to be a major subject later on So I'm not going to talk about it right now. Okay, but that's an extremely important point. Yes It's a colored particle. It's supposed to be below a TV scale. There's no excuse for not finding this, right? So let's we will talk about it. Okay Okay, so Now things are going to get get a little bit more technical There's going to be more equate they're going to be equations with actual equals signs and indices. Okay, so warning We're shifting modes here and There's I don't think there's any better way to start off a set of Lectures to graduate students than by invoking the great Sydney Coleman who gave such magnificent lectures in Arachae, which I never had the privilege to attend, which I certainly read many of them and Sidney was also famous for for for many witty things that he said And this is very true the career of a young theoretical physicist Consists of treating the simple harmonic oscillators and ever-increasing levels of abstraction and these lectures on supersymmetry will be no exception Okay, so let's just let's look at the very very simplest concrete example of supersymmetry Okay, so the very simplest quantum mechanical model that we can think of is a simple harmonic oscillator and I'm going to have so this is the Hamiltonian for a simple harmonic oscillators these bees are the standard creation and annihilation operators I call them bees instead of a's because B is boson. This is a normal bosonic Simple harmonic oscillator what makes it bosonic is that you can have any excitation number you want, right? And remember that when you go to quantum field theory a free quantum field theory is just one simple harmonic oscillator for every spatial Momentum and the fact that we have any number of levels is exactly the statement that you can have any number of particles with the same Momentum, so this really is a boson this really means a boson Okay, and we have the vacuum and the excited states and defined in the normal way. This is all should be very familiar Okay We can do the same thing what we can define a fermionic simple harmonic oscillator by defining Creation and annihilation operators f that satisfy anti commutation relations. Sorry. Yes. Oh I'm sorry because that should be zero. I'm sorry. It's just a typo. Yeah, that should be zero I'll try to remember to fix that before I post the slides Yes, thank you Okay, I'm glad somebody's awake All right Here I fixed it right so for the the fermionic creation and annihilation operators Satisfy anti commutation relations. Okay, and if this is not familiar This this is familiar to you whether you realize it or not because in free fermionic quantum field theory every Every moment spatial momentum mode is one of these kinds of harmonic oscillators Okay, so this really is a fermionic fermionic simple harmonic oscillator now because the creation operator squared vanishes You only have two states zero and one right and that makes sense in a in a free fermionic field theory You have two states for every spatial momentum. That's the Pauli exclusion principle can't put more than one particle in a state Okay, so that's it and now we're just going to combine them We're just going to put them both into the same Hamiltonian We just add the two Hamiltonians together the Bosonic guys commute with the fermionic guys Okay, so these two sets of operators don't talk to each other at all And now we just have states labeled by how many bosons we have and how many fermions we have right just by acting with the appropriate creation and annihilation operators Now the frequencies of these two harmonic oscillators are free parameters But the claim is that if we choose them to be equal then we have a Bose Fermi symmetry Okay, and the simplest way that we can see that is we just look at the spectrum Right because if we look at these states if they have the same frequency omega Then the energy of these states is just the sum is proportional to the sum of the Bosonic and fermionic occupation numbers Right and so we have a spectrum like this We have a ground state which is a unique state The first excited state could be either one boson and no fermions or zero bosons and one fermion The next excited state could be two bosons no fermions one boson one fermion and so on Okay, and so we have this exact degeneracy between we have the ground state is unique But every other state there's a there's there's there's a double degeneracy Okay, and as always in quantum mechanics these degeneracies are due to our due to symmetries, okay, and To see that it's actually a symmetry you can define the operator the sort of obvious operator that would take you between These two states that have the same energy namely you have something that denyelates a boson and creates a fermion Annihilates a boson and creates a fermion a boat annihilates a fermion creates a boson see I told you I was jet lag Okay, so the obvious thing and you can see that what this thing does is it well It does exactly what it's supposed to do and it's very easy to figure out that this thing commutes with the Hamiltonian Okay, you just do a little calculation to see that and then also an interesting fact is that if you square this operator Q you get the Hamiltonian up to a factor So this operator Q is the square root of the Hamiltonian in some sense, okay? So this extremely simple model right here. This is exactly What supersymmetry is going to be okay? It's going to relate particle fermionic and bosonic particle states exactly the way this Q does Any questions on this Yeah Yes, you can but for field theory That's not really a useful way to to do it because in field theory We really care about the particle basis, right? Even in the simple harmonic oscillator who needs p at Q when you can have a and a dagger, right? Well, P is I mean Q is basically a plus a a plus i a dagger P is a minus i a dagger or the other way around I forget, right? Or plus or minus a dagger whatever I don't even know I've forgotten but it's you know They're they're just some p and q's are just simple linear combinations of a and a dagger Okay I'd have to turn on my brain to figure out what they are and that's not happening Okay, I Had my brain on when I wrote these slides, so I'm outsourcing Any other any other questions? Okay, so this is just what I said This is literally true free field theory is literally just one harmonic oscillator for every spatial momentum So this is this is not just some silly toy model And so then you can immediately say well, wow, then I can easily imagine Theories where this works so if I just take a Dirac fermion because that's what I know And I have now a Dirac fermion has four particle states for each momentum each three momentum, right? So I'm going to need four Bosonic states right for every momentum So I just put in four real scalars just by hand and of course I have to have their masses the mass of the fermion and the mass of the scalar be the same right so that the Energy levels are the same and so this is a supersymmetric theory In fact, it has a non minimal supersymmetry, which is why we're not going to Study this theory Okay, so these so that this this really can be translated directly into to free field theory But we want to actually study the minimal supersymmetry and so we have to talk about Minimal fermions which are not Dirac fermions, and that's what I'm going to do next Okay, any questions before I do that? Okay, so I'm going to talk a little bit about vial fermions Okay, so the inevitable note on conventions I am actually trying to keep factors of I minus one two square root two and so on I'm using what I think are the most standard conventions that I can use this reference right here is a wonderful resource for all kinds of details and results and I highly recommend it and Like I said, I believe conventions should be conventional That also means the corollary of that is that if you don't like some of these conventions don't blame me, okay? I didn't make them up Okay, so what are vial fermions vial fermions are the minimal fermions in four dimensions Okay, so they are the basic building blocks for all other kinds of fermions including Dirac, Myrana everything Okay, so let's start with the Dirac representation the reason for starting with the Dirac representation is that it exists in any number of Dimensions and even any signature. It's completely universal So it's a good starting point in any dimension to find out what the minimal fermions are So let's go through that in the case of four dimensions So we have our gamma matrices and they have this famous anti commutation relation and then Dirac's Insight was that these if you take the commutator you get an Antisymmetric matrix, which is the generator of the Lorenz group, which is a generator of the Lorenz group Okay, so whenever you can find Dirac matrices you can find a new representation of the Lorenz group Okay, and so the way this works for infinitesimal Transformations Lorenz transformations we write the Lorenz transformation is the identity plus a small piece omega mu nu and The fermion then transforms as the same omega mu nu multiplied by these generators acting on the Dirac spinner Okay, so this part is hopefully Familiar or at least hopefully at least rings bells from your quantum field theory course So for finite transformations, we just exponentiate the Transformations and the Dirac bar transforms in the inverse representation Okay, this is also a classic part any any discussion of Dirac fermions will show this in some form Okay, now the way I want to look at this is I want to think about this in terms of an index notation Okay, so I want to think about Dirac fields as having an index a that goes from one to four It's a Dirac spinner index and these matrices s which are the exponentials of these Omega times generators These guys have an index structure like this and the inverse guy has an index structure like this This is just exactly what you do with Lorentz transformations when you have Lorentz transformations acting on the fundamental representation I'm using exactly the same kind of index notation here And just as it's very useful to have upper and lower Lorentz indices to keep track of Lorentz invariance It'll be useful to have upper and lower spinner indices to keep track of Invariance, okay, and the way remember that we keep track of Lorentz invariance is that we match the indices, right? Everything transforms according to its index structure, and we can do exactly the same thing for fermions So for example the gamma matrix has this index structure It has a new then it has two spinner indices, right because it's a matrix And now the claim is that this thing right here is it transforms according to its index structure Except that it doesn't transform. Namely, it's an invariant tensor. Okay, so to illustrate that concept Let me just remind you that this is something you actually know In the case of Lorentz transformations in the fundamental representation You know that the space-time metric 8mU nu is an invariant tensor because if I were to transform it I would transform it by this by this by these lambdas, but that's in fact equal to the original thing Okay, and the fact that omega mu nu is an invariant tensor like this is the basic reason why Contracting upper and lower indices gives you an invariant Okay, so exactly all of that transform Translates over to the spinner case this thing right here is an invariant tensor the gamma matrix direct gamma matrix And in this case it means a more complicated identity like this where the mu index transforms by a Lorentz transformation The upper A index transforms by an S and the lower B index transforms by an S inverse Okay, so these got these these kinds of expressions look complicated The whole Christmas tree of indices, but if you just the idea is very simple every index just transforms according to its It's its position Okay, and so in this way you can easily see that all the various Invariants that you know for direct matrices can be thought of as just arising in this way of contracting upper and lower indices Okay, any questions on that now The Dirac representation though as I said it's not the minimal Representation and the way we see that is there's a basis called the vial basis for the gamma matrices where they look like this Okay, and in that basis if I look at the commutator of two Gamma matrices I look at the Lorentz generators. They are block diagonal So whenever you see this two by two notation here for Dirac guys This is a two by two block structure and because these guys are block diagonal It means the upper two components and the lower two components transform under Lorentz transformations completely independently, right? They're they're their own representation individually. So the Dirac representation is reducible in these matrices here given by formulas like this Okay, so a lot of this notation is standard and I know that I'm going over it fast But you can these notes will be posted and there's there'll be a lot of reference material on as well Okay So as I said the upper two components and the lower two components transform Indic they transform among themselves So I have these left-handed vial spinners and these right-handed vial spinners that transform in this way And now just as for the Dirac fields. I want to have an index notation for these so I write the left-handed Guys with a lower alpha index and the right-handed guys with an upper dotted index and again like I said don't blame me for this Notation. This is a standard notation There is actually reasons for it, but anyway This is what it is. Okay, and you should just think you shouldn't think of dot as an operation on alpha You should just think of it as another alphabet. There's the dotted Greek alphabet because we're running low on alphabets, okay? Okay, and so here once again or here are these same equations here written out with the index structure Okay, so these are just the same equations and you can see upper and lower indices are nicely lined up The finite transformations the same way they exponentiate they have this index structure, okay? And now I can just define something with any number of upper dotted upper undotted Lower undotted lower dotted indices to transform in the way that they're supposed to namely If there's a lower lower Undotted index it transforms according to this s if there's an upper Undotted index it has to transform with an S inverse here. It is okay All right, so this is again just the way you would do it for Lorentz with different with an addition of a couple of different new kinds of alphabets All right, so I hope that's okay and Once again, you can show that all of these various objects that we introduce just like the gamma matrices They are invariant objects in the sense that they are equal to the thing that you get by transforming all of their indices Okay, and there's another invariant tensor, which is Which is this the anti-symmetric two by two matrix, okay? So this has two upper spinner to two of the same kind of spinner indices either upper or lower and it's anti-symmetric Yeah, okay, it's an anti it's anti symmetric and you can check that it also is an invariant tensor namely it satisfies this Okay, and the proofs of these things are basically just by checking They're two by two matrices, so you just check and they're also given in many places All right, but that's what I wanted to say about that Okay. All right, so here's the summary the executive summary We have these new kinds of spinners. They have two components So these indices alphas and alpha dots run from one to two and we have all these different kinds of invariant tensors Okay, and the rules are you can form Lorentz invariance if you just contract all the different kinds of indices that you have okay, and Here's what I'm saying here. You can check these identities by You it's smart to use identities like this for example this identity is very easy. Sorry. Where is it? Anyway, I don't know you can check here this identity here This is a smart identity to check first and you can use this to prove all the other identities involving the sigma matrices Okay One last thing is that charge conjugation actually relates the dotted in the undotted indices And the reason for that is that if you complex conjugate the sigma generator you get the sigma bar generator I didn't mention and again. This is notation is is conventional. So don't blame me bar Doesn't necessarily mean complex conjugation unless I tell you that it does so for example sigma mu bar has nothing to do With sigma mu. I mean at least not it's not the complex conjugate of it Okay, on the other hand, you can see that the sigma mu nu bar is closely related to the And there should be there should be a this should be an alpha dot. I apologize I'll fix this. I'll fix these typos before I put this online. This should be an alpha dot Okay, so it makes sense to write things like this if I take a lower Index a lower index spinner and I complex conjugated it turns it into a dotted index spinner Okay, because of this relation this transforms just like a dotted index And so the basic point is is that I can make all the different kinds of spinners out of the Vile spinners with a single lower dotted index because I can raise the index with epsilon alpha beta So I get an upper spinner index if I want a lower dotted index I just complex conjugated if I want to This should read. I'm trying to raise the index here. I'm trying to say Okay, I wanted to hear what I wanted to write is that if you wanted an upper dotted index You can raise the dotted index with epsilon alpha beta of the dagger thing. Okay, so I'll again I'll fix these things before I post them Okay Questions All right Finally now we can use this to write some Lagrangians. Okay, so let's write some invariant Lagrangians So let's write the most general quadratic Lagrangian for a single vile spinner psi alpha okay, and We can see that we can write this very Dirac-ish Kinetic term we just use psi dagger and psi We can track the indices out these these dotted and undotted indices with a sigma bar mu and we have a derivative here okay, and This looks just like the the Dirac kinetic term and then for a mass term We mix of psi with upside because we have the metric that we have involves to To spinner indices of the same type. Yes So I am not going to talk about parody and all of those things I have to leave there are many things I have to leave out for lack of time Okay, so when I talk about when I say Lorentz invariant I'm just going to mean the connected component of the Lorentz group namely boosts Okay, so, you know what what what Einstein the symmetry of space time is is clearly must include boosts and and translations and rotations right but it need not Contain parody and time reversal and those other things may or may not Well, that's what I've written here now psi is a left-handed Vile fermion So that's what I've written Yeah, yeah, so these this is a single. That's exactly what this is Okay, I've just left off the let the L Okay, so I didn't I didn't say that but I'm I just started leaving. Oh, these are all left-handed vile spinners I should have said left-handed here. Okay The ones with a single and lower undotted index of the left-handed guys okay, and Notice one thing is that if we look at this mass term right here This this mass term would be zero if psi was a commuting Classical field so if we think of psi as a classical field right normally we think of the grand gins is being classical And then we quantize them so if we think of this as a classical field this mass term would actually vanish Identically because epsilon alpha beta is anti symmetric right and psi alpha psi beta would be symmetric So this would vanish so just even having this mass term requires that these fields actually anti commute Okay, but that's actually what they're supposed to do in canonical quantization The fermion fields have to obey anti commutation relations So we can understand this from the fact that the h bar goes to zero limit of the operators should reduce to the classical fields okay This is this is usually sloughed over in most treatments of Quantum field theory, but it's true More conventionally What is usually not sloughed over is if you want to write a path integral for fermions It's a the path for me on path integral is over anti commuting fields Okay, but I claim that even if you think about canonical quantization carefully You will see that you have to have anti commuting classical the grunge in that you're canonically quantizing Okay all right Finally we need to check the that the Lagrangian is Hermitian so that the Hamiltonian is Hermitian Okay, and we need the definition we need to figure out what is the definition of complex conjugation for anti commuting Classical fields and we want that definition to agree again We want the the the classical fields to be the h bar goes to zero limit of the classical fields So we have to make the rules agree and one of the things about Complex conjugating operators is that it reverses the order of the operators regardless of any commutation or anti commutation relation they obey right so we have to actually define Complex conjugation of spinners to reverse the order of spinners even for the classical anti commuting fields Okay, and then we can actually check and I'm not going to go through these things because this is not a calculation You should do when you're jet lag you're guaranteed to get the wrong sign 50% of the time But you know you get it is instructive once or twice in your life to go through and do some of these Calculations and just see how many sign flips there are that you could get wrong Okay, and there are plenty of them. Okay, but the claim is that if you get them all right You will find that these objects would sort of anyway, they If things work, okay, so the Lagrangian is in fact Hermitian Now a lot of times You know these these expressions with all these spinner or indices will dazzle your eyes And so it's it's useful to sometimes omit the spinner indices you get much cleaner-looking expressions And it turns out that that that you always get the Dotted and undotted spinner indices contracted in these these directions. Okay, and you can see some examples of this here Okay So that was sort of a lightning introduction to To to index spinner ology I gave you a reference if you want to learn more This is a very useful exercise here if you want to understand how to quantize that theory of a single vial fermion This exercise like I said, I'll post these slides this exercise sort of walks you through the main steps Okay, and the bottom line is that there is a mode expansion for this guy in terms of these a and b daggers There's a single Annihilation and a single creation operators here. So this this massless vial fermion describes Two physical states Okay, and Yeah, you can You can you can have fun with this. Okay, but I'm not gonna go through it here Okay, so any questions on that? Okay So now I want to talk about Super symmetry actually super symmetry in quantum field theory. Okay, and now we can talk about the minimal We can talk about the minimal Theory that has super symmetry in four dimensions. We have the minimal fermion, right a left-handed vial fermion And so to match the number of degrees of freedom we need two real Scalers which we can assemble into a single complex scalar So now the number of degrees of freedom Matches, okay, and I'm going to take the mass terms to be equal namely zero I'm going to take them both to be massless. Okay, just take the very very simplest theory Okay, and I'm going to show that this in fact has a Super symmetry this this this theory has a super symmetry Now one thing to notice that will help us in guiding us and finding this is that this Lagrangian here actually has a u1 Symmetry under which the fermion and the scalar have the same charge So again if these guys are going to be related by some sort of a symmetry It sort of makes sense that they should have the same u1 charge Okay, so that will be a guiding principle that will help us Okay, so in fact what we're going to do is we're going to try to find a Super symmetry and in variance of this Lagrangian and it should have the property that the variation of the scalar should be like the Fermion the variation of the fermion should be like the scalar right it should exchange Fermions and bosons Obviously the Lorentz and the spinor indices have to match up It has to a scalar has to go to a scalar not some other you know beast and so on and We want everything to be compatible with this u1 in variance right here Okay, and if you do the exercise of writing the most general thing that you can Then for the scalar you find that you you get a term like this. Okay. It has to be some There there's whenever you have a transformation there has to be a parameter right of the transformation like for the u1 field There was the angle theta Now here this first. This is a scalar. This is a boson. This is a fermion So the spin the parameter has to be a spinner. It has to be an anti commuting field So this is a well-defined mathematical object, but you can't say how big it is you can't measure It's not a real number. It's an anti commuting thing and you see I can't have a Psy dagger term in here because of the u1 symmetry that I assume Okay, and if I look at this now I see that well the the scalar has dimension one the fermion is dimension three-half So this mysterious transformation parameter has mass dimension minus a half. Okay, so it's the square root of a length All right, and then I can write down the most general fermion Transformation that I can and if you think about it then this is the only way again It can you can only have a Phi here not a Phi dagger Okay, and this is the only way to hook up the Lorentz indices and I put an arbitrary factor C here I could have put a factor C here, but I can just absorb it into Psy, right Now I can compute now that I've written down the most general thing I can just compute the variation of the Lagrangian Okay, and I can see that the I can compute the the the variation of the kinetic term and For the scalar and for the fermion for the fermion you have to do a little bit of work Not very much, but you have to do some indexology. You have to integrate by parts Yeah, sorry, thank you. Thanks for making What do I mean by this right here? I'm sorry. Yeah, this is here. What I mean is just Is d mu is just shorthand for d by dx mu Is that was your question? Ah, okay. I'm sorry. Yeah, I could let me slow down a little bit here Thank you for slowing me down and waking me up. Okay, so One thing I notice is so first of all if I tried to let's try to write something without a derivative Okay, let's try to write something without a derivative. Okay, so then we would have to write del psi This has a spinner index. Okay, and now this has no indices at all Now you could try to write psi alpha like this Right, so you could try to write it without derivatives But now if you look at the dimensional analysis this guy has dimension minus this guy right here has dimension equal to one half This guy has dimension equal to one Okay, so I haven't actually thought this thing all the way through Maybe there's some better argument. Okay, but I would like the transformation Not to involve any dimension full parameter Okay, so the only other object that I can put in here is a derivative To make it dimensionally correct Yeah, it's also the thing that ultimately works, you know, so there there are actually I'm not going to discuss it here There are there are very fancy theorems that tell you that this is the only possible Sort of symmetry that you can have which we'll see at the end that this doesn't commute with space time symmetries And this is you can there there are theorems that tell you that that this is the only thing you can write down I wouldn't claim what I'm doing here is a proof But I'm at least I think I'm trying to motivate it strongly. Okay, that this form is the is the right form. Okay All right. So yeah, anyway, the once I have this derivative to I I can I can I can work out the The the change in the the fermion kinetic term and now to see after integration by parts I get something like this. Okay, so this started off with one derivative I integrated by parts to put both the derivatives on Phi now I have two Sigma matrices like this now notice that this is symmetric and mu and new So I can symmetrize this under mu and new and use this identity right here Okay, and then when I do that what I find is I this thing simplifies and I get something like this Okay, so it's You can see that it's you know, it's not It's not immediate But you do a few steps and you can actually see that the fermion kinetic term Changes into this and that now tells us what C needs to be Okay, you know for to be invariant C needs to be minus I as it turns out and so here is our transformation right here Okay All right Okay, and this is a supersymmetry transformation This is a transformation that this theory is invariant under When we have both the fermion the boson have zero mass now that we have a Should have put a box around this now that we have the symmetry we can work out what the neutral current is Okay, it's another tedious calculation, but you can it's a standard Calculation using canonical methods you can find that the neuter current looks like this So rather than deriving it Let's just check that this is in fact invariant that it's conserved when we When we impose the equations of motion if we take its divergence We get a term where the derivative hits the fermion and a term where the derivative hits the scalar So if we look at the term where the derivative hits the fermion If we look carefully we see that that vanishes on the equations of motion because we have a sigma bar alpha beta del mu psi Beta that's exactly the combination that vanishes according to the equation of motion for the scalar fermion for the for the boson piece we again use this trick we get these two two Derivatives on the scalar we can symmetrize these new new indices We get something proportional to box of the scalar which again is zero by the equations of motion Okay, so this is in fact conserved on classical solutions And that means now we can we can we can compute the charge associated with this We just take this spatial integral of the zero component of the charge, right? So notice that this charge has an extra spinner index here, right so There are many signs throughout all of this that That supersymmetry is really a space-time symmetry. It looks kind of abstract because it has these fermion these fermion Parameters, but here's one way in which it's a space-time symmetry the nerder current has an additional space-time index on it namely a spinner index, right so the The charge that you form from this also has an extra spinner index. It's a spinner charge, okay and it's given by the the integral spatial integral over the zero component and Classically this charge is is conserved. It's time-independent on classical solutions Okay, but we don't care about classical solutions. We care about the quantum theory So for the quantum theory what we do is we just expand these fields There's an alpha here And who wrote these slides Okay Yeah, obviously there's no alpha index here. So there's a it's copy and paste I guess so So anyway the the vial fermion field has a mode expansion, okay, and there are some spinner solutions here Which are solutions to the vial equation just like in the Dirac case You're used to having u's and v's which are solutions to the free Dirac equation here These guys are solutions to the vial equation So we have some mode expansion for these guys I call a's and b's the creation annihilation operators for the fermion and C and D are the creation and annihilation operators for the Scaler, okay, and now I can just compute I just take these expressions here without the index spurious index on the scalar and I just plug into the into this expression and Integrate, okay, and I do some manipulations and I find voila. I find q and q dagger are given by these expressions right here Okay, and notice that these have exactly the structure that we had in our very very simple Harmonic oscillator example, right? We had a both a single Boson and fermion thing We have a this is a Bosonic creation and a fermionic annihilation a fermionic creation Bosonic annihilation this is exactly a thing that takes away one Boson and gives you a fermion or takes away one fermion It gives you a boson, okay, and it nicely is weighted by this This this spinner solution here to give you take care of the indices, okay, and similarly for the dagger guy Okay, any questions? Yes That's right Yes, hold on to that thought sort of hold on to that thought Well, we'll say more about that later, okay, so so we'll see what What I mean there the natural the natural setting where all of this acts like Lorenz symmetries act on space time is called super space and that's something we'll talk about Okay, but now that we have these things we have completely explicit Expressions for these cues and cue daggers. They're made of these creation and annihilation operators these creation and annihilation operators obey commutation and anti commutation relations that we know, okay And so now we can let's see what am I saying here? Oh, yeah, let's sorry So before we do that before we okay, so let's let's look at what these cues do to states Okay, so how do we define a single fermion state? We act with an a dagger or a b dagger right that creates Respectively a fermion or an anti fermion Similarly, we get a scalar by acting with c dagger and an anti scalar by acting with a d dagger, right? This is the standard way we define Particle and antiparticle states for complex fields Okay, and now we can just work out exactly what these cues do and you can see what they do They take a fermion state and turns it into a scalar state the cue takes an anti fermion state and annihilates it But the cue dagger does the opposite it takes Annihilates a fermion state and acting on an anti fermion state gives you an anti Scalar this should be a phi here Lots of typos Okay, this should be a phi it's supposed to turn anti fermions into anti scalar Okay, and then similarly here if you act on a bosonic state the cue annihilates the Scalar, but the cue on an anti scalar gives you an anti fermion and then vice versa here Okay, so alright, so these cues do exactly what what they were advertised to do They create take one boson state turn into a one fermion state, etc Okay All right, so this is what supersymmetry is in a very very concrete form Okay So next thing I wanted to talk about the algebra, so I'm heading my way towards talking about this as symmetry, but any questions so I think I'm just going to I'm gonna Start this subject. I'll just because I want to stop in about three minutes, and then we'll just use the last 15 minutes for questions, okay, so As I was starting to say Before I got a little bit ahead of myself, you know, we can work out the The anti commutator of these two guys. I don't know why there's a prime there. Who wrote these slides? I have no idea. Okay. Anyway, there's not supposed to be a prime there What? Sorry say again. Oh, yeah. Oh, it's a comma. Ah Ah Thank you. Yes. Oh Very good. Yes From from where you're sitting. It looks more like a comment. Okay. Anyway So so you can work out this anti commutator and because we know the algebra of all the a's and c's This is a standard sort of an exercise and then what you find is You find this right here, okay? So you find that you can write it you can you find that you get a combination like this Which basically is the sum which is a sum of terms that counts the number of Each kind of particle and actually this is also not quite correct because this is what it is okay for the There there are two terms here involving fermions and for those terms They are multiplied by this and each of these no, no, this is correct No, this is correct You get you get each of these things involves one you and one you dagger because each of these terms involves at least one involves two firm non-zero terms involves one fermion and one when fermionic creation and one fermionic annihilation operator and That's actually not quite it anyway that comes out proportional to this and you get this okay, so I'm going to just Stop here and just say that we have remember for the very very simple Super symmetric harmonic oscillator we had that that q was the square root of the Hamiltonian We had q squared equals h well things can't be quite that simple in our Lorentz invariant theory right because these q's have spinner indices But if you look at this thing right here, p mu is the generator of space time translations so p zero is the Hamiltonian right and Q this anti-commutator is like a square So this is it is sort of a covariant Lorentz covariant version of saying that we have that these q's are square roots of translations Okay, space-time translations, and so that is yet another way to say that these are space-time Symmetries you clearly think that p mu is a space-time symmetry right but you can get a space-time Translation by anti-commuting two of these weird supersymmetry operators Okay, so I think that's a good place to start for now stop for now Place to start next time