 Okie doke. So yes, this is the last BSM lecture and it's going to be slightly more disjointed than the other lectures, the reason being that we held this vote two days ago on what to have in the last lecture and there was large, almost no support for just supersymmetry, large support for supersymmetry in extra dimensions and reasonable support for extra dimensions, so I'm going to try and cover both of the topics, sort of half and half. Of course my job here is to partially decide or to make an informed decision about what should be in the course on your behalf and I should say that although I could tell from the vote that there was not much interest in supersymmetry and presumably that is because of the null results of the LHC, it's maybe seen the MSSM is maybe seen as a bit of an outdated topic to study. One should be careful when thinking about the reasons for that and I think there are sometimes sociological elements at play and for that reason that's one of the reasons I want to cover supersymmetry today and indeed the things I will cover for supersymmetry in extra dimensions as part of this goal I have of sort of equipping you with general tools, I want to go through some of the very basic and general features of these theories, of course I will cover the application that they have to the electro weak hierarchy problem but more importantly than that is that you get a feel for how they work structurally and the way you can do calculations within these frameworks and build models or whatever and that is because nonetheless although supersymmetry hasn't been discovered at the LHC it's still a very compelling candidate for a symmetry of nature at some point, at some energy there are many reasons why you might believe that that could be so and the other thing is that both in supersymmetry and extra dimensions there are tools that are useful in many many areas of theoretical physics without even thinking about the Higgs you know if you're working on if you end up working in slightly more formal areas of theoretical physics you could end up using moving between different numbers of dimensions fluidly all the time you know if you're working on ADS CFT or something like this and even when you're thinking about black hole physics it can sometimes be be useful to work in different numbers of dimensions if you're just working in some sub region or something like this so the tools of extra dimensions that I will discuss are broadly applicable not just to weak scale physics and similarly for supersymmetry you know supersymmetry is one of the last or is the last possible symmetry of space time that we could have and for that reason it comes up in all sorts of places that don't have any application to the Higgs for example when people are calculating scattering amplitudes or non-perturbative behavior in quantum field theories supersymmetry is an incredible useful tool because it can simplify calculations to the extent that you can actually answer a question and that can give you a hint about the behavior of more general quantum field theories or in its own right there are many interesting results in supersymmetric field theories that is purely academically very interesting and worthwhile knowing about even if they have nothing to do with with any immediate feature of nature okay so that's sort of my philosophy for these lectures so to try and equip you equip you more with a tool kit than with an in-depth study of some particular model and for for extra dimensions we will we will consider in the the the first 15 minutes or so the the extra dimensional Planck scale versus the four-dimensional Planck scale if we were to live in some say on some brain in some four-dimensional region confined to some four-dimensional region of a five-dimensional theory or if the dimensions are just so small that we can't I can't access them so many people have considered this question I'm sure you've heard of Kaluza Klein theories and so on and the one one reason to see that this the extra-dimensional theories can be interesting for the electroweak scale is to the the hierarchy between the electroweak scale and the Planck scale is to go back to one of the first things we did in the first lecture which is just dumb dimensional analysis when you put each bar back into the action so you had this as an exercise yesterday you can see that the Planck scale I the guy that lives in front of the coupling between the graviton and the stress energy tensor of the standard model is actually has dimensions of a mass scale divided by a coupling scale so this tells us that the thing so this is the theory where there this is sorry the the mass scale where the UV completion might kick in so where new resonances associated with with gravity might show up for example in string theory these would be the resonances of the strings and this is some coupling in the theory in a priori we don't know what this coupling is and but essentially that means that this is sort of from our perspective from as an effective field theory perspective we don't actually know where the UV completion of gravity might kick in if the coupling is order one then it means the UV completion of gravity is going to kick in around the Planck scale naively but if we have some some extremely small coupling this could kick in much earlier and in particular if the UV completion of gravity kicks in at the TV scale then this would require an extremely small coupling in the theory somewhere how might this resolve our puzzle about the hierarchy between the Planck scale and the weak scale well if the true UV completion of gravity kicks in at say TV or a few TV then we would expect the mass corrections to the Higgs from that UV completion to be around a TV or a few TV which are not huge much smaller than for example Planck scale squared so if we can find a theory with a very small coupling and we have a strategy that could explain the hierarchy between the weak scale and the Planck scale and indeed we can just use dimensional analysis imagine we're just doing something very dumb very back of the envelope not in any way precise and we say well what if there's an extra dimension in nature what is the only dimensionful say we're assuming we'll assume that this is a flat extra dimension what's the only dimensionful parameter associated with that extra dimension it's the length there's nothing else in the game that's the only other dimensionful unit we have to work with so let's construct something that has dimensions of coupling from the length and the only other thing we've measured which is the Planck scale so you do that and you see that the dimensions of m Planck are the minus one as the dimensions of coupling which is telling you that if you take are really really big you've got some effective coupling in the theory which is very very small and this is essentially one way we're going to see it from a much more orthodox perspective in a second but this is one way of seeing why the existence of a large extra dimension could provide you with precisely the small coupling you need to explain a large hierarchy and then the the basic idea from there is that maybe the fundamental Planck scale essentially there we will see that we'll call this m5 which will be the five dimensional Planck scale is not very far away at all so we would say that the you know the hierarchy between the Fermi scale and the fundamental Planck scale which is not the Planck scale that we measure I'll call it m fundamental and this could could potentially explain why there's why why these two things are actually don't have a hierarchy at all or at least they only have a small hierarchy and the only reason I say small is that LHC constraints have already pushed this up to be above a few TV okay so what are the how do we see this in in more detail so that's the sort of hand-wavy argument for why extra dimensions can can give you a strategy for understanding the Planck weak scale hierarchy but now we'll do this in a little bit more detail again I'm trying to give you a very qualitative handle on how things work so now we imagine we have some set of extra dimensions I'm not even going to limit us to one extra dimension just you know say it could be three or whatever and imagine at the moment that the extra dimensions are flat so there's no wacky geometry going on then if we have a particle in the extra dimension it must still satisfy you know p squared minus m squared equals e squared the usual equation we know from special relativity if this extra dimension is flat there's nothing special going on if we're living in the bulk of this extra dimension and we fire a massless particle it should satisfy the following equation where this is the this is you know it's like a lot p zero squared this is just you know e squared minus p squared equals zero it's massless so we have special we have Lawrence symmetry in the extra dimension so is everyone comfortable with this I'm seeing some puzzled faces so this is the three-dimensional this is our three-dimensional momentum and this is the extra-dimensional one so imagine we just lived in four dimension we recognize this equation right yes please say yes so now if we're in an extra dimension we just have the same old equation except it's extra dimensional so there's a the extra-dimensional component of the momentum in there as well but what do we see in 4d we in 4d we can't measure the momentum along the extra dimension we have no idea what it is what do we see in 4d well we can just rearrange this equation we see particles that satisfy an equation like this where this is the the 4d energy the 4d so the 3d momentum is equal to p e squared what we observe is particles who appear to have a mass what for our for from our perspective they do have a mass but that mass is just the momentum that's being carried in the extra dimension so in 4d this is just you know the you know p mu p mu equals m squared usual equation where now m squared is the the extra-dimensional momentum so you can see straight away straight off the bat if we have some extra-dimensional space I'll draw this one extra dimension but it could be n dimensions and we have we will have a massless mode in 4d and if we have a massless mode that mode has to carry zero extra-dimensional momentum and the operator that that measures the extra-dimensional is extra-dimensional momentum is the same as an operator that measures the 4d momentum which is just a derivative but in this case a derivative along the extra-dimension so we'll have a massless guy who satisfies you know if let's call this particle h for the graviton so dy it's called dm I will use capital letters capital you know m and n and l to describe extra-dimensional derivatives and then the usual mus and news and alphas and betas to describe 4d 4d derivatives so this guy is flat so this is telling us already that say we started off with the full-blown graviton in 5d and the graviton in 5d is massless in 5d just like the graviton in 4d is massless in 4d so the graviton in 5d satisfies this equation and that tells us that if we have a graviton in 5d if there's some some mode that has zero extra-dimensional derivatives then it will be massless in 4d and we can just recover a massless 4d graviton as usual but on top of that we will have additional modes that carry have non-zero momentum along the extra-dimension of non-zero momentum along the extra dimension and to our perspective from our perspective they will look like they just have mass furthermore in a flat extra-dimension we know what sort of wave functions live what sort of momentum you can have in an extra dimension we know from doing quantum mechanics as an undergrad particle in a box that if you live in a box your momentum is quantized in units of the radius of the box so we will have wave functions that look like this and so on and they have quantized momentum and which are integer units of the inverse radius of the box so the momenta the momenta will look like n over r where r is the radius of the box you can obviously generalize this to multiple dimensions which is telling us by again this dumb equation that from our 4d perspective we will have not only a massless zero mode for a graviton or anything else if you put a gauge field in the bulk of this extra dimension would be exactly the same we will also see massive quantized modes which from our perspective there will be an infinite tor of massive particles that correspond to the infinite tor of quantized momenta you can have in 5d so this is a very quick way of saying that you have a collusion client tor of states and it's inherently related to the fact that that you're working in an extra dimensional theory you can't it's really non-negotiable that all of these states will live there because they simply correspond to the allowed momentum states in the extra dimension of course if you have some weirdo geometry in the extra dimension then the states won't have the same collusion client spectrum as for a particle in a flat box but they will have the spectrum you expect for a particle in a curved box whatever the geometry of that box is okay is this reasonably clear yep super okay so now let's ramp up and go to a little bit more detail so we can study the metric and ask what the metric looks like so we can take a line element in 5d and it has to be it follows exactly the same form as it does in 4d in general relativity we have the the full d dimensional metric and then our coordinates dxm dxm so it's just a standard line element and we can now imagine that there's some background geometry of course we need to have Lorentz symmetry in 4d so there's no background geometry that depends on the 4d spatial space time components because we want to have a theory that reduces to a theory in 4d that has 4d Lorentz invariance but there could of course be non-trivial geometry going on in the in the extra dimension so in general we can have a metric that depends the background metric once you've solved Einstein's equations and done all the hard work a background metric that depends on the extra dimensions so we can always choose a frame in which this background metric i'm eliminating the extra dimensional fluctuations of the of the graviton at the moment we can choose a frame in which this extra dimensional metric has a form like this so this is some overall factor that depends on the geometry of the the extra dimensional space we will have a 4d metric i come g mu nu which is a function of just the 4d coordinates and the the extra dimensional part typically you can see you'll see you know if you study the literature scenarios where you have a factor just multiplying this or a factor just multiplying this or an overall factor you can always do well not always if there are singularities and things you can't do this but if it's a relatively smooth space you can typically do a coordinate transformation which pulls all the dependence out into some generic pre-factor in front of everything which is what we will work with and in here i've written g mu nu and not etym mu nu because i want to carry along with me the 4d massless graviton that we know will exist if we have a zero momentum eigenstate in the extra dimension so we're carrying along the the massless graviton but implicitly what i've done the full set of fluctuations in this case would look like you know e to the a and then you'd have a big matrix where here we have the the 4d components and then you do have vectors and scalars living along here and i've set them to zero so i'm not going to to going to discuss these extra extra dimensional fluctuations you can typically choose a gauge where most of them go to zero and one of them you can think of as this if for example if you just did 5d you can think of this as having a vector living in here a scalar living in here and a vector living in here and typically what happens one way of saying what's happening is that here we have the 4d graviton and it has two degrees of freedom and here we have a massless vector which is two degrees of freedom and a scalar which is two degree one degree of freedom and you can show it's quite a trivial exercise to show that a massive graviton in 4d has five degrees of freedom so one way you can interpret this is that each massive graviton we get in 4d so satisfying this equation so they have some extra dimensional momentum it's like having a for each one of them it's like having a massless graviton in 4d that is eaten like in the Higgs mechanism eaten a massless graviton and the massless scalar so just like with a vector a massive a massive vector in 4d we can think of that as being a massless vector with two degrees of freedom which eats a scalar to become a massive vector with three degrees of freedom that logic continues up to extra up to it for the for the graviton as well in extra dimensions where the the two you get five degrees of freedom for a massive graviton from two in here two in here and and one in here but we're going to forget about those fluctuations there's a lot of interesting things you can do with them we're going to forget about them and we're now just going to calculate the Einstein-Hilbert action for this metric so we calculate the the usual Einstein-Hilbert action and it's an integral over the entire space we now have the extra-dimensional plank scale so in four dimensions this would be m-plank and we have it's squared but in five dimensions it's cubed and so on m5 cubed and it's as always written like this you take the root of the determinant of the metric and the Ritchie scalar but now we can we can just separate this out again this is an interesting exercise there's a very simple procedure for taking the 5d I'll call this r5 the 5d Ritchie scalar for a metric of this form and writing it in terms of a 4d one plus some other terms that we're going to ignore because we're just interested in the 4d one because we're interested in what's going to happen to the 4d plank scale that's g4 so I've done I've done that procedure I've put plugged in this form of the metric in here and in here you get this over a factor that's n minus 2 over n this is all in the notes as usual by the way so if you don't get it all copied down that's fine and I've thrown away the extra terms that you get when you go from the 5d Ritchie scalar to the 4d Ritchie scalar so and the reason I've done this is those terms are there but what we're interested in is what in the end multiplies the 4d Einstein-Hilbert term the 4d Einstein-Hilbert term is just this guy here written in the usual way just like the 5d one is this guy here so we can calculate what the 4d one looks like and it has to be equal to well I'll write it what it is here so we will the 4d Einstein-Hilbert term should be equal to the usual guy d4x over m Planck squared root of minus g4 r4 of g4 comparing the 4d to this to the 4d part of the 5d we can then see that the 4d 4-dimensional Planck scale is given by this equation so we actually have a formula once we know what the extra-dimensional geometry is we know what this function is we have a formula for deriving the effective 4d Planck scale that we see for 4d gravity as a function of the 5d or n-dimensional geometry okay so now let's do some examples is that clear to everyone clear enough so let's look at the flat the flat case so now let's imagine we have n extra dimensions which are just in a box so they're all flat and we're just talking about some some extra-dimensional box then the 4d Planck scale is related to the n-dimensional Planck scale like this so it's the product over all of the extra dimensions essentially the volume and if you assume all the size of all of the extra dimensions to be the same just for simplicity then the the radius of the extra dimensions is given by a formula that looks like so this is just solving this equation here 2 times 10 to the 32 over n minus 4 minus 19 over 5 to the 2 over n minus 4 this is in the notes you can calculate it yourself as well 1tv over mn the n minus 2 over n minus 4 meters that's the size of the extra dimensions so we can plug in n here so for for n equals 5 so this is the total number so this is one extra dimension we plug this in and we get that the that r0 is of the order of the distance between the Sun and Jupiter which is of course ruled out this means that we would experience we are we are smaller than experience you know gravity on a distance scale smaller than this so we would it be experiencing five-dimensional gravity now and not four-dimensional gravity so this is completely ruled out you could also have ascertained this just by finding the r here that would give you a coupling small enough that this could live around the the tv scale however if we go to n equals 6 then this becomes something like a millimeter so already by going to to two extra dimensions you can generate a 4d Planck scale from a 5d theory with the Planck scale you know near the tv scale essentially then no hierarchy between the the the weak scale and the the underlying Planck scale for a theory which is on the boundary of of the sort of scales that can be probed gravitationally so this is a very interesting theory and this is motivated so the this framework was first put forward by our county hammad demopolis and valley and it really motivated many many developments but one of one of the interesting phenomenological ones is it is that it really motivated pushing hard on short distance probes of the the form of the gravitational interaction fifth force experiments and so on however what should be noted is that although we have brought the fundamental Planck scale all the way down to being near the weak scale there's another puzzle in both of these theories which is that in in units of the the electro weak scale you know we're talking you know 10 to the minus 17 meters or something like that would be the weak scale we actually have now we've lost one hierarchy which is the weak scale to the fundamental Planck scale but we've gained another one which is the weak scale to the radius the inverse radius is a very very low energy quantity here you're talking wavelengths of a millimeter in this case so there's actually another puzzle as to why you can have that hierarchy so in some sense you could argue that some people have argued that this doesn't resolve the hierarchy problem entirely it sort of shifts it into a different hierarchy which would be the hierarchy between the fundamental Planck scale and the actual size of the extra dimension and also typically i won't go through it but in in these theories the there is a a if you don't stabilize the the size of the extra dimension so if you don't do anything to fix the size the the the the distance between the two ends of the extra dimension then you'll have a massless scalar field which is known as the radion and it actually you can interpret it as living in in this component here of the metric and this radion is massless which means and the radion really measures the size of the extra dimension so it means it costs zero energy to change the size of the extra dimension so you have to stabilize it somehow you have to do some breaking of of 5D Lawrence invariance in such a way that the the radion is has a mass and a local minimum which fixes the size of its of its expectation value such that the the radius of the extra dimension is fixed but you can see then how this translates into another form of hierarchy problem because now we need to have a a a a a scalar with a very very small vev essentially just like we have with the Higgs despite the fact that there are other big skills floating around in the theory okay so so that is one class is a class of approaches and then following this I think you know a year or two later a different class of approaches was proposed which are known as Randall Sunderm models and here I'm going to use an analogy we don't have I'd like to get onto supersymmetry quite soon so we won't spend too much time on this but I'm going to use an analogy with with inflation so I'm sure you've all you're all familiar with the idea of inflation it solves hierarchy problems in cosmology so the flatness problem and the horizon problem can be interpreted as as being hierarchy or sort of they look like fine tune if you were to if you didn't have inflation they look like you have a set of initial conditions that are extremely fine tuned and how inflation solves that is that you have a a large cosmological constant in the past which gives you which when you solve Einstein's equations gives you the inflating solution um which looks like you know gmn is equal to my gmn um so this is just Einstein's equation it's the you know essentially the equations of motion for the metric um so you can let's take this analogy and and run with it a little bit so we look at the line element in uh in some geometry and in the inflating scenario we have it looks like this root 2 alpha over 3 t sum over the the extra dimensions or all of the dimensions this is when we have positive cosmological constant this is the usual inflating solution we see that there's um the the inflating scale factor here but if you flip the sign so if you have a positive sign when you solve Einstein's equations you can have a metric that looks like this um so now we have a sort of a an inflating scale factor when we flip the sign at the cosmological constant but this inflating scale factor um is now it doesn't act in this frame of course you can change frames to move things around here the the the scale factor was a function of time and multiplied all of the spatial coordinates here it's a function of one of the spatial coordinates and it multiplies all of the other space of the other space time coordinates in this way so we can use this sort of a quick analogy which is that that um if you imagine walking along uh um uh time essentially during inflation then scales are exponentially diluted as we move a long time so that's essentially what uh um the inflation is doing it's wiping out exponentially wiping out um uh scales in in in special uh in our special dimensions and with Randall Sundrom what you can hope to do is that as we move along some extra dimension so you sit at some point in an extra dimension and you move along and these are are these are the 4d space time components we see that the the 4d scales are going to be exponentially diluted as we move along and we can see that more specifically by actually considering the the Higgs mass in one extra dimension so we're going to put in put in a brain in the extra dimension and we're going to put a scalar we'll put the Higgs in living on that brain and we're going to move the position of the brain and yep um no you can you can choose you can choose so um so if we take for example the fermions the chiral matter this what I did here applies to to any particle particle living in the bulk so you could have them living on if you have them living on a brain and say the graviton living in the bulk then you have a fully four dimensional standard model and it's couplings it only feels the couplings to to the graviton and the kk mode to the graviton um that come from the wave functions evaluated at that position so that's the overlap between these wave functions and the brain but you can also um you could also put them in the bulk if you wanted and then the couplings come from from evaluating the wave function this is happens more in Randall's syndrome to be honest um evaluating the wave function integral of the the master's graviton overlapping with the the the uh master's Higgs or not master's light Higgs or master's fermions you can actually you can choose so and and actually with Randall's syndrome we'll get to it in a second but there's a game that you can play which is that well and with flat extra dimensions as well but it happens people use it more in Randall's syndrome you can even choose masses in the extra dimensions such that the profiles of the standard model fields are different you can have different wave functions and then when you integrate over those wave functions they can have different overlaps and this can for example explain the flavour hierarchy you could have essentially that that um you know the first generation is light because it's localized away from the Higgs whereas the third generation is heavy because it's localized towards the Higgs so there's a whole suite of model building games that you can play um to do with how the standard model fields are localized positioned within the extra dimension whether it's on you know strictly confined to a brain or or living in all of it or localized at one side of it oh exactly so so yes so so no indeed so yeah so for a flat extra dimension to explain the the weak scale indeed this would mean that you have a KK electron you know and the whole tower would be living there so yeah so you can't for this for this thing but um but what I mean for flat extra dimensions is that in fact if you choose a much smaller flat extra dimension so one that has um an energy uh uh that does not explain the small the the the electro weak hierarchy you can play these games in flat extra dimensions for flavour so you can have localization you can have the extra dimension become super symmetric so that it's partially explaining the hierarchy and then the the extra dimensional plank scale is at some intermediate scale there's a whole suite of things you can do but indeed that's the reason that um people tend to do this game of putting if you if you really want to explain the the the the um weak scale plank scale hierarchy when people put things in the bulk it's usually in Randall Sundrum because in Randall Sundrum the first KK mode is of order the curvature so of order uh uh related to the parameter the that's in here which is related to this parameter here of order the curvature so the curvature is high then um the first KK mode may be around a TEV or something like this so you could have all sorts of things living in the bulk but there's there was a huge industry in exploring all of these possibilities and there are many things you can do you can even for example have um uh uh you know the fermions confined to a brain and the gauge bosons in the bulk the higgs on a brain and the fermions and gauge in the bulk things like this um you don't want to break gauge symmetry so for example uh it's hard to have fermions in the bulk but not gauge in the bulk and things like this there are there are there are all sorts of things you can do okay so let's put in our put in the brain um in one extra mention and put the higgs in it so um and we're going to put it in Randall Sundrum so the 4d action on the brain looks like this so we have the 5d integral i'm going to call the extra mention y we have a delta function that confines it to the position y0 um to preserve diffeomorphism invariance you have the the determinant of the full 5d metric but you also have to divide by the um fifth component so i'll call it minus g55 so it should be minus in there oh no sorry plus g55 this absorbs essentially the variation here under a dif uh diffeomorphism so you have to have this factor in there and then we have the the usual kinetic terms g mu nu d mu higgs tiger d i'm not doing covariant derivatives but you can you know figure out how to insert them this lambda and then f squared i'll call it squared so this is the higgs potential you could call us v squared if you like but importantly in this in this frame here um if there's no hierarchy problem we expect this term here whatever the vev is or the mass term to be comparable to the other scales in the theory so we would expect this parameter on an eft basis to be you know around m5 or something like this not not too far from it but then let's insert the specific form of the metric um my root alpha is going to be uh become i'm going to call it k for this example so we insert the specific form of the metric and um integrate over y so we position ourselves at y0 and we have d4 of x e to the minus 4 k y0 e to the 2 k y0 e to mu nu d mu d nu minus lambda um but we see here that these two terms don't cancel so this higgs is not canonically quantized or is not uh does not have canonical kept kinetic terms so we need to give it canonical kinetic terms and to do that we just do a rescaling of the higgs such that we have the standard 4d kinetic term and we do that this is the the action that we would observe in our 4d world and you see that after doing this rescaling it's like a conformal rescaling the only parameter that has a leftover dependence on the position which is why not is the only dimensionful parameter leftover because the lambda times higgs squared is a quartic interaction which is scale invariant the only parameter that depends on it is this f squared here so we see that our what we measure to be the higgs mass the natural value of the higgs mass at a position y0 depends exponentially on the the the position so if we were to live at one end of the break one end of the extra dimension where y0 is zero we expect the natural value of the higgs mass to be around m5 but if we go um or comparable to that scale but if we go to um the other end of the extra dimension y is you know let's call it l or pi r which is whatever convention you want you see that you expect the natural value of the higgs mass that you measure to be exponentially smaller than the other typical scales in the theory and this is essentially the random sum approach but also um it has many many uh deep and interesting connections to um other formal aspects of of theoretical physics so for example using ads cft you can relate this ads 5d picture uh broadly speaking to a 4d cft picture where you have um an approximate conformal field theory just in 4d with large um uh with with with a lot of running uh going on where you get spat out uh at the scale at which this the conformal field theory is broken some light scalers so you can it can actually reinterpret um models of composite higgs theories where the higgs is spat out as a pngb as we discussed in the the second lecture um in a five-dimensional context so we have a tool this five-dimensional context gives us a tool for understanding structural aspects of of strongly coupled theories and strongly couple couple pictures of of the higgs bows on and so on so there are many interesting and deep interconnections these are just the absolute bare minimum basic tools for sort of playing around with extra dimensional theories but they are very very useful and they also gave um uh led to a sort of uh an explosion of a of um um technical tools that are for effective field theories that are unrelated to symmetries so at the start in the first lecture i really went on at length about how if you have um you know from an effective field theory perspective if you have parameters that appear to be uh smaller than you expect in your effective field theory that that um usually means there should be some sort of symmetry reason for that that there's some spurry on type field in the the ir and in the uv which um breaks the symmetry by a small amount but what this sort of unleashed on the theory community was a different approach which is locality essentially um imagine you have a field that has a a wave function that's strongly peaked over at this side maybe it's uh some scalar field or something like this and you have some other scalar field which has a wave function that strongly peaked over at this at this side then when you or maybe even you know very peak confined to this brain then when you integrate over the two wave functions to find out what the four d coupling of those particles looks like um it can be very small and the reason for its smallness isn't a symmetry it's locality in an extra dimension it's really um another term we use for this is sequestering and you can actually build this in it even without going to five dimensions because you can um do a thing called I don't have time to go through it but it's a lot of fun um dimensional deconstruction where you do what a lattice field theorist does what does a lattice field theorist do he takes a continuum theory and um at each point along an extra dimension there's an infinite number of degrees of freedom associated with each field that's that's living in that in a dimension and then you break that continuum theory up into a discrete theory so you break it up into a distinct set of lattice points so now you have a uh instead of a continuum of fields you have a a different field living at each point and it has connections with the next nearest neighbors which depend on the derivatives you know the derivative in space is just the value here minus the value here divided by the lattice spacing so this is just the the the definition of of of a derivative and it turns out you can do exactly the same thing for bsm physics so you can take your extra dimension turn it into a discrete set of points and you get a set of next nearest neighbor interactions and you can write this in a purely four-dimensional way and it allows you to use um this locality or or um uh uh sort of this locality in the extra dimension as a theory tool where it becomes essentially locality in theory space it becomes um uh dependent on how uh different sets of theories that neighbor each other in a in a lattice sense um interact so it's a very useful tool i i there's i'm not don't have time to go through it today but there's a lot on that uh in in the notes and it's a very useful and interesting tool um for constructing beyond the standard model theories or yes yes yes yes no it turns out um so there's a mechanism uh known as the gold burger wise mechanism um that makes this very easy in particular for um uh a random symptom model and here's how it works so you choose in your extra dimension you choose um you have a scalar with a bulk mass and when you solve so if you have a scalar with zero bulk mass then it will have um a mass spectrum and uh wave function dictated purely by the geometry so depending on what you choose for your boundary conditions um uh that sets it but now I imagine you put for the scalar let's call it phi you put a potential on this brain where phi is minimized um where with phi equals let's call f0 so it looks like a Higgs like potential just like on that just like there um on this brain and you put a separate Higgs like potential on this one with a different verb so it's telling you that the boundary conditions want you to have a verb that is at f f0 here and they want you to be have a verb which is f1 here but also the geometry dictates there's also a that that means when I expand about that verb there's a mass here and a mass there um so solving Einstein's equations across the brain that leads to a boundary condition on the derivatives of the field at that value so you solve the whole thing self consistently including in the bulk where you have to satisfy the equations of motion for the scalar field in the bulk and it will have some um uh uh equation some um solution which will have be some generic complicated function but you can solve it in some cases analytically and then what you see happens is that this guy here the curvature going how you curve from going from here to here um is dictated entirely by the geometry it's not negotiable once you've once you've solved the the boundary conditions with the masses then you have to have a certain gradient here and a certain gradient here and the only way you can get that certain gradient consistent with the bulk equations of motion is with some some guy like this now I imagine I move I try and move this brain I can know if I move the brain then because this has some gradient if I've moved it over here this keeps going down and the value of phi over here is no longer f1 so you see you've gone up the potential you've actually cost yourself some energy to shift the position of the brain which means that self consistently although this field phi we've just put in the bulk it doesn't look like it measures the size of the extra dimension but you can see here that that physically what's happening is you can if you have a potential for that field that's on the two brains that gives it wants it to have a one vev here and a different vev here then that will actually set self consistently when you minimize the energy set a radius for the extra dimension and the mass cost of changing that radius the fluctuation um the mass of the fluctuation of changing that radius is proportional to the mass on the on the brains of the scalar field so you can't you can't solve um the modulized stabilization problem in this way in Randall sunderm it turns out that works pretty naturally but in large extra dimensions you have to tune a parameter somewhere because like I said this has to be a very very large extra dimension which means that you need to have something that that has very um to stabilize it has is it going to involve very very small vevs and then you you come back up against this nationalist problem does that answer your question any more questions before yep ah no so so so i'm not advocating that you really do lattice physics i'm saying is that you use the tools of the lattice so you don't even need you don't need to put it on a computer so by the tools of the lattice i mean the formal tools the the tool of how you put a a quantum field theory on the lattice so how that works is imagine we have um uh and this will look very like condensed matter so imagine we have um an extra dimensional theory we'll put a scalar a massless scalar in 5d and i'm going to split up the the the derivatives so we have i'll call it d d mu phi d mu phi i'm going to work in mostly plus metric because i prefer it d y d y phi i'm probably going to make a mistake here so so um help me out if i do um so now let's turn this this extra dimension into a lattice so just the way a lattice field theorist would do it so now the integral over d y becomes a sum over extra dimensional points you know we're with this flat extra this uh flat extra dimension and now we're turning it into a set of points like this um a field living at a point y um now it becomes a set of fields living at each let's call it phi a living at it at each um lattice point so it's a if we're just turning one of the dimensions into a lattice then the the continuum now becomes a sum over a set of points where there's a 4d field living at each lattice point and finally what's a derivative well we know the derivative is d y of phi is the limit as a goes to zero of phi at i plus one a so it's i call it i would just call it phi i plus one um and x so let me just call it like this phi i plus one of x minus phi of i at x um where i plus one is labeling the position along the extra dimension um over a so it's that dumb uh so now you do this and our our new action becomes to go over d four x and then we have a sum of put the half out the front d mu phi i squared plus we now I have one over a squared phi i plus one minus phi i squared and let me put a big square bracket just so this is a dimensionally deconstructed scalar it looks like a sum of n where n is the number of lattice points and free scalars free massless scalars with next to nearest mass terms and you can diagonalize this this mass matrix and you will get a massless mode and well depending on the boundary conditions you will get a massless mode and a whole k k tower of states but they're no longer what you had before as being some um uh continuum of states in a flat extra dimension it would have been a k k tower like this going to infinity we've now limited because then there are infinite states because I said at each point in 5d you have a single um 4d field that's one way of thinking about it so you originally had a continuum so you have infinite numbers of 4d fields which is why you have to have an infinite number of k k modes but now we have a finite number so what we get is a band of states that terminates at some some value um and our our massless state here when we diagonalize this mass matrix and for example now imagine if we had had some field that interacts with phi at the end of the brain here in this scenario here our field that is just interacting with phi n at the end and when you diagonalize this mass matrix it will have some um overlap with the massless mode but also some overlap with uh the massive modes as well so yeah that's what I mean I don't mean really put it on the lattice but but use the tools of lattice lattice field theory yep it turns out in the rs case it's fine because all of the you're not putting in any hierarchically large or small parameters you just put in a bulk scalar you make the mass a little bit small the bulk mass a little bit small because you don't want to upset the geometry the this vev this profile for the field does actually change the geometry a bit but because you've not tuned anything if you calculate radiative corrections it stays not really tuned so it will change the parameters by a perturbative amount but it doesn't change the qualitative result okay yep yeah so the question was can you take in rs the the 5d cosmological constant to be afforded the fundamental skill and the answer is yes um which is why it's really a bona fide solution to the hierarchy problem of course we haven't seen the kk modes at the lhc so phenomenologically that's a different story but um we have to tune it a bit now but structurally there's really very little tuning you have to do anywhere and you could ask exactly the same question for large extra dimensions and say well why is my bulk cc so small because you essentially have one way of seeing it is that you have three in terms of cosmological constants you have three parameters you um have one on each brain so that's two and one in the bulk there's a cc for each of these terms there's a potential on all three in all three places now we know we're going to have to tune to get a small 4d cc so we know we can do one tuning to set this you know when you integrate over the all of the ccs to set the sum of those ccs to be small but in large extra dimensions there's an additional tuning which is that you in addition to that you have to set the bulk cc to be small as well as tuning them all to cancel because if you didn't set it to be small it would no longer be a flat extra dimension it would be rs so essentially rs is like taking a flat extra dimension and doing what you would expect from an eft perspective and just putting a m5 ish uh bulk cc in there okay any other questions nope okay so we will get through um a bit of supersymmetry probably not all of it but uh enough to give you an idea okay so again just like with with extra dimensions you know um there is merit in supersymmetric theories that goes well beyond you know the cm ssm or something like this and so you shouldn't dismiss supersymmetry as a sort of toolkit as as a weapon in your toolkit because you never know when you're going to want to use it you may be studying some property of whatever you end up doing some property of a strongly coupled qft or whatever and it may turn out that actually just the tool for the job for getting a good qualitative understanding of some part of physics is some part of relevant physics is to use a supersymmetric model as a first approximation so it's a very useful tool it shows up everywhere and also we should know I think any theorists should know about the possible spacetime symmetries of nature and supersymmetry is sort of the last the last port of call so supersymmetry is uh so we know that um in flat 40 space with the Poincare symmetry and supersymmetry is an extension of Poincare in the sense that um it extends the bosonic symmetries like translations rotations and boosts of Poincare symmetry to um a um fermionic symmetry and just like in any symmetry we have generators so remember for for our for uh standard sorry for the the global symmetries we talked about a few days ago we had generators which are just um the the for example for SU2 it's just the Poiley matrices for SU3 it's the Gilman matrices and they generate a transformation an infinitesimal chance transformation by shifting things say alpha i some infinitesimal parameter um proportional to the generator so if you have something in the fundamental representation you do a symmetry transformation and infinitesimal one by multiplying the fundamental you know vector by this exponentiated matrix we also know what generators look like for for spacetime translations for example you know x the field valued at x mu plus alpha mu where alpha is some small parameter is well approximated by x mu well it's by definition given by the full Taylor expansion five x mu plus you know half of plus all the other terms you know the proportional to alpha squared and so on and we can represent this transformation this translation infinitesimally as uh with a generator which is just the derivative acting on the field phi so p mu which is the derivative essentially is the is the generator for translations so this is something we all know about for SUZI the generator is uh the generator is called we'll call it q here and it is a fermionic variable so you can think of it as being a vile spinner it carries some uh some index in addition to the to the Poincare symmetry so every symmetry has an algebra you know the algebra for the normal groups that we yep oh sorry yes i'll try and write a bit bigger um there's an algebra for example for standard global symmetries which is given by essentially the commutation relations of the of the generators of that group um there's the point the algebra for the generators of the Poincare uh group um and in SUZI we we add to that an algebra so a symmetry should have the should have an algebra um which satisfies these commutation and anti commutation relations um as you can see these generators because they carry spin they essentially uh change the spin of of uh fields um and we can just like here we wrote an element of the group um as an infinitesimal parameter uh an exponentiation of an infinitesimal parameter multiplying the generators and we can do the same here if we have a an element of the the Poincare group we can write it as a function of now the full supersymmetric coordinate space so we've extended the 4d spacetime coordinates to include a fermionic coordinate which i'm calling theta so the full supersymmetric coordinate space um has group elements that look like this and when we we can act on so just like we can um uh uh generate additional elements of the group by multiple actions so we could act on this guy where i have alpha i replaced by beta i and this um is uh equivalent to a shift of alpha i plus beta i you can do the same thing here and when i act on um uh i'm not just going to write it out because of time but when i act on this guy with another supersymmetric transformation proportional to zeta then you get a group element that has this form except now x has been shifted so x prime of mu is x mu plus i theta sigma mu zeta bar minus i zeta sigma mu theta bar and where theta has shifted as a shift shifted proportionally to um zeta okay so these are the fundamentals of of the the group of supersymmetric transformations and you can see here i've replaced um what the derivative by the the derivative here by the generator which i'm calling p mu but you can remember its derivative and similarly you can um write uh an operator that generates supersymmetric translations um in the following way we will call it q so that a supersymmetric translation proportional to zeta is given just like here a translation proportional to x mu is given by um this operator here q okay so this is just a very very uh lightning review of the the properties of the symmetry but just like when when we were actually working with quantum field theories that have global symmetries we're interested in the actual representations we have fields for example su2 we have fields that live in a representation of su2 for the standard model we always work with the fundamental representations we just have vectors complex vectors with two components with qcd we have fields that live in the the fundamental representation of qcd so we have vectors fields that are a vector of three complex components so now we want to build up some sort of representation of supersymmetry so that we can study it at um in quantum field theories and what we have now is no longer fields but superfields so these are um fields that have some definite transformation under under supersymmetry and just like a field is a function of the spacetime coordinates a super field has to be a function of the superspace coordinates x theta and theta bar so we can just do the the most dumb thing you could imagine we say our super field is now called f for example of x theta and theta bar so now this is a generic function just like a field is a generic function this is a or up to some constraints like complex or real or whatever this is a generic complex function of um uh the superspace coordinates but what's interesting is that the thetas are spinners which means that they satisfy a grassman commutation relations which means that if the same component of a spinner encounters itself twice it vanishes this is something uh you can look up in in in wikipedia um so then the smallest thing so we can actually tailor expand so we tailor expanded here in terms of x mu but we can actually tailor expand in in terms of the superspace coordinates and we get a tailor expansion which actually terminates because once you have too many powers of theta because it's a grassman variable you get zeros so we can actually do this tailor expansion so we do f of x zero zero and so on and keep going and when we do this we find that this super field contains a a scalar field here uh a fermion here so theta if you remember is a spinner so this guy has to be a spinner another fermion another scalar theta squared is um uh the laurence contraction so there's an epsilon in there it's theta epsilon theta so you have if you were to think of this in two component spinners this gives you theta the top component multiplied by theta the bottom component if you had theta the top component twice it vanishes because it's grassman um and another uh a scalar and a vector and another fermion and finally a scalar okay so this um is not by definition this is just what happens if you expand tailor expand this in as a function of theta and theta bar and it terminates because we've absorbed as many we've got a in here we've got a theta theta one theta two and we have a theta bar one theta bar two if we put in any more theaters we'll have an at least a copy of um one of those components twice which will vanish okay and it has a definite this guy here has a definite super uh definite transformation so just like we know how the group acts for example in su2 how the the group elements act on the representation similarly we have the group elements here and they act on um uh this representation in a definite manner so a suz transformation proportional to zeta let me do this a bit lower a suz transformation proportional to zeta of this super field um is given by the contraction of zeta with the the generators furthermore so we we we now have our representation um for a a supersymmetric theory but furthermore because this this um uh expansion always terminates if i were to take the product of two different super fields call one f and call the other one h or something like this you take the product and the product has to have this form because of the termination of the expansion of the thetas it will always terminate so if i do f times h i will get a scalar here which is looks like f times h and i'll get scalars multiplied by fermions and so on it'll be a complicated mess but nonetheless it has itself has to be a super field so super field times super field is super field super field times any number of super fields is a super field um and that's only because this this uh it will always take this form because the the expansion um terminates okay so in some sense this is a reducible representation as we will see we can find smaller super fields that have definite transformations under supersymmetry and one of them is i'm not going to go through it because there isn't time but one of them the smallest super field essentially or one of the small super fields that is complex is known as a carol super field and it turns out that we can find a carol super field by taking a a general super field like this and eliminating some of the some of the components so we will eliminate um let's eliminate actually no let's not do it that way i'll just write it down there isn't uh isn't time so a carol super field looks like this so a of x is a complex scalar psi here is a fermion and f is a complex scalar and when i perform this supersymmetry transformation on this super field this carol super field the individual fields transform in this way sorry there's a lot of a lot of writing today okay so we can spot a few different things there's much more detail by the way in the notes the transformation of the scalar transforms it proportionally to zeta into a fermion the transformation of the fermion involves the bosons these two scalars and even more interestingly the the transformation of this component this scalar which is the theta squared component um is proportional to a total derivative you see this is zeta which is just some some constant spinner the poly matrix and then d mu of psi which is telling us that we've just discovered a susie invariant if we take we call this the f component usually if we take the f component of any some some super field doesn't matter what the super field is then under a supersymmetry transformation that f component will transform to a total derivative and as we've gone throughout length the total derivatives don't enter into any of the the perturbative um s matrix elements or anything like that so it's an invariant also um if we take the i'm not going to do it but if we take the theta bar squared theta squared uh component of any super field the same thing is true so we call this the d term the d term under a supersymmetry transformation transforms up to a total derivative and uh so does the f term of a chiral super field which is a a smaller representation you can see there's only two scalars and a fermion living in this field um and because of this we can actually write down generic um uh susie invariant theories so the last tool um you would come across is more of a of a slick way of writing these theories writing down actions for these theories this is all for n equals one supersymmetry by the way it changes a lot if you go to n equals two um so for n equals one supersymmetry we can write down now knowing that this guy transforms up to a total derivative and this guy transforms up to a total derivative means and combine that with the fact that a product of super fields is a super field means that we can take any polynomials of these super fields and their complex conjugates and extract d terms and f terms and we will have a susie invariant so the way we write that is uh is the following we write it as an integral over super space we're essentially the the definition of this integral uh d d squared theta is over theta squared is one but d squared theta over some number is zero this is sort of actually more of a definition than something you can derive i think um but for our purposes you can take it as a definition so then we can take the um for example this is called the the the Kepler potential so we're now taking general super fields that have thetas and theta bars in it and we extract the d term so we integrate over d squared theta and d squared theta bar to extract what we call a d term and this is known as the keeler potential just jargon busting you don't need to know what keeler potential means but this is how we refer to it and then um we can extract for chiral super fields which are just functions of uh of these uh complex scalars and uh fermions we can extract uh the f component which is the theta squared term from a an analogous integration over superface super space uh just over d squared theta and then a generic uh potential might have some uh linear terms i just call this call it f it's just my my choice of of uh notation some linear terms in the fields you might have some quadratic terms some cubic terms and so on and this is often referred to as the super potential okay so it turns out that when we extract we do this procedure and extract the theta squared theta bar squared and extract the theta squared terms from here we'd also usually add the Hermitian conjugate um we get a supersymmetric theory because we know we have constructed a theory which can under any symmetric transformation will transform up to a total derivative so i can always do integration by parts and kill what's being left over um okay uh i'll not go through gauge superfield so this is for chiral superfields but now you can see why we get um uh why we get these extra particles coming along for a ride people in supersymmetry theorists don't add super partner particles just for the hell of it or just because they um yep so phi is this super field so and phi i and phi j are just different super fields and the f component or d and d component of any super field is invariant up to a total derivative yes only the f component which is why we do this integral over super space so because of this definition this pulls out the uh this guy just pulls out the f component the bit proportional to theta squared from this whole product uh sorry i didn't explain that well and this guy pulls out the d component from this whole product here um okay super thanks um so we see that yeah theorists we're not just adding particles for the hell of it um it's because this the symmetry itself dictates it the fact that you actually have a represent representations of supersymmetry means that you have multiple fields packaged into this structure so if you have a scalar field like the Higgs you will have a fermionic field um uh which is the Higgs Eno and so on so don't uh don't believe the hype sometimes it's said that you know you know supersymmetric theories are totally ad hoc um and for sure they have lots of free parameters because there's a lot of flexibility in in performing this procedure but the symmetry itself the space time symmetry itself uh really dictates the structure um okay so when you do this you'll see here what you get here is the kinetic terms i won't go through it because but it's in the notes the kinetic terms for all of the different um uh uh fields here the scalar and for this fermion but what's interesting is you don't get a kinetic term for f yep pardon ah yes so so can you um very good question so the the whole story with the strong cp problem was total derivatives can be important um if you have uh interesting non-perturbative field uh configurations um it's a very very good question it goes well beyond my knowledge in the sense that the answer is no they they're not typically important for things like the mssm but i i'm imagine that on um uh in more supersymmetric theories or in different numbers of dimensions or or if they're put on different manifolds they could be important so um as in the respect of advocating susie is a general tool for playing with quantum field theories they could be important but for things like the weak scale story and the mssm they're they're not so we really have full supersymmetry but when i say could that doesn't mean that they are i just don't know um okay so we see that we get the kinetic terms we don't get a kinetic term for f and similarly here we don't get a kinetic term for the d's you can see immediately why that is because when i do phi dagger phi and multiply them together the theta squared theta bar squared component that involves f will be theta squared theta bar squared f f dagger without any derivatives whereas you can see here the theta squared theta bar component involving a will have d mu d mu a and a so that's just the standard kinetic term and the one involving the the fermion will have um uh did i write this correctly this should be the one involving the fermion sorry i think i must have written this incorrectly ah yes theta bar um will have say cross terms like a theta bar psi bar multiplying theta squared theta bar so when you pull out the theta squared theta bar squared guy you can see you get the kinetic term for the fermion from this derivative but f is not a propagating field so the number of propagated this because there's no kinetic term so the number of propagating fields um is just given by a and psi so we have the same number of degrees of freedom um similarly the masses of these fields um come from this term here and i've written m i j very suggestively because m i j is really the mass matrix for this theory so um when i extract the theta square how can i see this when i extract uh for example phi phi term so it's not phi phi dagger but just phi phi when i extract phi phi you see i get things like um uh theta squared i'm taking the theta squared component theta squared f a or i get um psi squared so the psi squared has a coefficient which is just m so you see it's a mass for the fermion for the scalar we get theta squared f and a so we get f a plus f a all dagger but we also have the term from the Kepler potential which is f f dagger so when we integrate that out solve the equations of motion for f we get m squared a squared so that's giving us the mass matrix and you see that because of this structure you see that the masses of the fermions is identical to the masses of the the the scalars there are all sorts of other nice properties you can play with so for example you can have other symmetries there's a particularly interesting one called an r symmetry where theta uh transforms under a u1 symmetry um theta goes to e to the i alpha theta but i won't go into it but um what's most interesting for and where we will finish up here for um the hierarchy problem and i actually i'll just do this in detail so you can see let's let's take a theory where we just have phi so there's no there's not different labels here there's just we have phi dagger phi and we have just a mass term m phi phi so it's extract um the full theory we'll get the kinetic terms for phi um and psi we'll get a term from the Kepler potential which is theta squared theta bar squared which will look just like uh f f dagger again as i said no no kinetic term for f and we'll get a mass term that goes like m psi squared plus psi squared dagger from the super potential and we will get cross terms as i said from this from the super potential that go like m scalar component f component and vice versa so we will get plus m f a is the scalar field plus formation conjugate f has no kinetic term so this is a kinetic term for for a and and psi f has no kinetic term so we can actually totally integrate it out we can do in the path integral we can do the Gaussian integral um straight off by hand because there's no kinetic term so we just get rid of them so when we when we integrate them out what's equivalent to doing that Gaussian integral is um essentially taking dl by df and setting it equal to zero uh minimizing this the potential essentially as a function of f and when we do that we see that f is equal to minus m uh a so we plug that back in here and what we see we get is the the same kinetic terms the same mass term for the fermions and uh mass term for the the scalars which goes like this so they have the same mass so supersymmetry has forced them to have exactly the same mass um for n equals this is a more general theorem that you can prove not in this way but um for uh for our purposes you can see that this just comes straight from this super field construction so they have the same mass so why is that interesting for quadratic divergences we can see here from this perspective m as it relates to the the fermion masses is the only parameter that's breaking the fermion chiral symmetry so we could have uh physics in the ultraviolet physics at high energies this is taking this EFT perspective physics way up in the ultraviolet and as long as it preserves the chiral symmetry it will never give any corrections to the to the fermion mass that are larger than m the fermion mass can't have you know one-loop corrections and so on but they they're always multiplicative rather than additive so if the uv pres preserves symmetry in that chiral symmetry supersymmetry in that chiral symmetry, the fermion mass is naturally light, but the supersymmetry also dictates that the scalar mass is tied to the fermion mass. They have to be equal. So if the theory is really supersymmetric, the scalar will always remain at the same mass scale as the fermion, but the mass of the fermion is protected by the chiral symmetry, which is telling you that if your theory is supersymmetric, you can comfortably have large-scale separations. You can comfortably have light fermions, but if the theory is supersymmetric, the scalar will always come along for the ride. Essentially, there's a very ad hoc way of seeing it, but it's borrowing the chiral symmetry of the fermion, which means that you can naturally have light scalars, which is really remarkable. You can do this in practice. You can write down a fully supersymmetric theory, where in a supersymmetric manner, these scalars and fermions interact with very heavy scalars and fermions, you know, ten orders of magnitude heavier, and you do the whole thing. You start at very high energies. You RG evolved down. You integrate out those heavy scalars and fermions. You do all of your EFT matching. You run all the way down to the light scale, and all that this has had is a small, perturbative, multiplicative correction to its mass. It's really miraculous in a supersymmetric theory. Scales involving scalars, fundamental scalars, can be arbitrarily separated without doing any damage to the scale separation with respect to the scalar fields. Okay, so there's much, much more than that in the notes, and I would urge you, if you're interested in Susie, to look into that. There's a lot more on the phenomenology and dark matter candidates and all sorts of things like that. But today, I really wanted to give you a taste of the symmetry that's underlying it, rather than the phenomenology, because I think the symmetry that's underlying it is always going to be relevant and interesting, and the phenomenology changes as collider probes evolve. But this is a symmetry which will always be of theoretical interest and is applicable to many areas of theoretical physics. So I'll leave it there.