 So, where we finished off on Monday, I've gone through this set of tools and concepts from effective field theory that are useful for thinking about beyond the standard model physics. And we finished up by studying the pions, where we saw that the charged pions have a coupling to electromagnetism that breaks any possible shift symmetry on those charged pions, whereas the neutral pion does not. And then this leads us, even if we don't know what the UV completion is, this leads us to expect that there could be corrections to the charged pion mass proportional to this parameter, which is just the QED gauge coupling, which do not enter for the neutral pion. So we expect some sort of mass splitting if we do a dumb estimate, naive estimate, which shows that something should show up around the energies of 750 MeV or so, if you want to explain the charged neutral pion mass splitting from these corrections, unknown corrections from the UV physics, and indeed that's precisely what happens, and in the full UV story where you have the Rho mesons and so on, you can actually just go ahead and calculate this mass splitting, and that naive sort of estimate works out very well. When we then go to the Higgs, we can imagine that the Higgs is a scalar, perhaps like a pion or something like that, and there's some UV story that tells us why it has the mass scale that it does. It tells this UV story, it will tell us where the weak scale comes from. In analogy to the pions, where the pions are some light scalars, of course if we just discovered the pions, we could have just said, oh well, job done, we've discovered some particles and we leave it there, but as scientists that's not what we do. We ask where are these mass parameters coming from, where does the scale come from, how can we understand its interactions, how can we understand the structures and symmetries involved in pion physics, and we want to do exactly the same thing with the Higgs. We don't know where this particle comes from, we don't know where the weak scale comes from, so we want to apply the same sort of approach in order to understand what the full UV story that explains where the Higgs sector comes from, we want to understand what structures it may take, and we can undertake exactly the same sort of exercise, so we stare at the action for the Higgs, and already you see a number of things that go wrong, so just like the pions, if we want to imagine that the Higgs is maybe lighter than the scale of the UV completion, and we know that that's pretty much, that that is true from the LHC results essentially, so we want to imagine that the Higgs is lighter than the UV completion, so maybe there's some sort of approximate shift symmetry just like the pions had, but then we can do exactly the same thing, and we notice that we have parameters in the action that are telling us that this shift symmetry is broken, we have the two gauge couplings, we have the cordic interaction, and we have the yukawa couplings with fermions, so we can estimate, again now this does not mean that we're literally drawing loops of gauge bosons or loops of Higgs bosons or loops of top corks or something like that, but just the fact that these parameters exist and they break the shift symmetry can lead us to estimate that there should be, we would expect in a typical UV completion to have corrections to the Higgs mass squared, which are proportional to things like the gauge coupling squared, the cordic, or let's call this y actually, the yukawa coupling squared multiplied by the cutoff squared over something like 16 pi squared, this is exactly what we did for the pions and we see that this naive guesstimate works very well, but then if we do this we see that really, especially when we look at the top yukawa coupling, that if these corrections to the Higgs mass are not going to be larger than the actual measured value of the Higgs mass itself, this tells you that lambda should be sitting really just around the corner, similarly to what happened with the pions, it's telling us that lambda should probably kick in, by around there should be some new physics if things are fully in analogy to the pions, kicking in around 500 GeV, and again this is not, you know, this is not some magic or hocus pocus, we're doing exactly the same thing for the Higgs that we did for the pions, but the LHC has run and as yet we have not seen anything if the new particles of the UV completion were colored if they were charged under QCD, then for sure we would have expected them to see them if they had mass in the ballpark of 500 GeV, unless they were doing something particularly clever and elusive to hide, that is possible, you can hide QCD charged particles with that mass scale at the LHC, but it's not easy, or it could mean that whatever particles live in this UV theory maybe don't have much to do with QCD, maybe they're just electroweak charged or they're gauge neutral or something like that, we will come to an example like that at the end of this lecture, in which case they could still very well be hiding around this mass scale, but nonetheless this is a big puzzle, and it's a puzzle that was anticipated long before LHC running of course, because if we take the cut off to be say the Planck scale, if we imagine that that's around the ballpark of the Planck scale, then these corrections would have been enormous, so people have been thinking about the hierarchy problem for a long time, but really what the LHC has done is make this, is really focus that the hierarchy problem and make it much more acute, and various people have reacted in various ways to this from all the way from investing all of their time to working on the hierarchy problem and finding new ways to understand this puzzle on one end of the spectrum all the way to just totally burying their heads in the sand and thinking about something else, but as you saw this is sort of hand-wavy, this is a rough estimate, and it shows that the hierarchy problem is not something that, it is something that you can calculate in a given theory, in a given UV completion that tells you where the weak scale comes from, that really predicts the value of the weak scale, you can calculate it and you can make all of these things very quantitative with the correct coefficients out the front here and everything like that, but while we don't know what the UV completion is it's sort of, it's more qualitative than strictly quantitative, but you can see it's a form of a strategy, this is perhaps a hint, gives us a hint about what possible UV completions could look like, and this is why it's worth paying a lot of attention to, it may be the one piece of evidence that we have about the next layer, the structure of the next layer of physics, just like with the pions we could tell a little bit about what the UV completion would look like even if we hadn't measured it, and here this may be a hint as to what is going on, so it's worth taking seriously, so the rest of this lecture, not this afternoon's, but this lecture I want to talk about it, I'm going to talk about, my plan is to talk about approaches to the hierarchy problem that are not necessarily textbook material, if you want on Friday we can do things like extra dimensions and supersymmetry, in fact I'll let you decide what you prefer, we'll have a vote, but what I want to try and give you a taste of is sort of more up-to-date material, theories that are maybe not even widely, completely widely accepted or are just at the start of being developed, to give you a sense or a notion of the sort of ideas that people are exploring now, again my logic for doing this is that I think it's much more important that you see the various tools that are being used and the ideas that are coming out these days rather than a detailed lecture course on one specific approach, before that though I want to discuss three or four strategies for understanding what's going on and in very little detail at all, just to give you a sense of some things you may have heard of or some of the really more exotic possibilities that could be going on, so one approach that you've probably heard of I think is known as skill invariance, so how do people invoke skill invariance, so you will note in the standard model, if I just write down the standard model living on its own, the second or the fourth option we discussed yesterday when looking at those different possibilities for the mass parameters is that all of the dimensionful parameters are zero, you have a skill invariant theory, so you'll note that in the standard model if we just write it down on its own the only parameter that breaks skill invariance is the mass parameter here, it's like the Spurion for breaking skill invariance, so if there's no other parameter in the theory that breaks skill invariance then this is perfectly natural for this guy to be small because it doesn't get any large quantum corrections from anywhere else, it will just stay where it is, so you could argue that within the standard model there's no hierarchy problem whatsoever because the theory has an approximate skill invariance and I would tend to agree with that within the standard model alone that is essentially the case, of course it's not fully skill invariant when the RG evolved, the gauge couplings run and so on, but at least that occurs in a relatively controlled manner, the hypercharged gauge coupling at very, very high energies becomes very large and non-perturbative, so even within the standard model there is actually a breaking of skill invariance in the UV, but I'm happy to sweep that under the carpet for a moment and just talk about the standard model, but the problem you see with the hierarchy problem isn't the standard model, as I said yesterday, there are many ways that you can actually have a light skill within a given quantum field theory, the problem is that we don't believe that the cutoff of the theory is skill invariant and a way that the UV completion is skill invariant and a reason to see this is again to keep going back to this pion analogy, I could have said exactly the same thing about the charged pions, remember yesterday we saw that the charged pions coupled with a photon, the neutral pions not, and I could have taken exactly the same argument and said, well look, when I look at the pion action, there's the kinetic terms and the gauge couplings, they all respect skill invariance, so there should be zero mass corrections to the charged pions and the neutral pions, so who cares, they should still have the same mass to leading order, of course that would have been completely wrong and you would have actually got the physics entirely wrong because the breaking of the skill invariance lives entirely in the UV, the UV theory itself breaks skill invariance and that's what leads to a mass correction to the charged pion relative to the neutral pion, so it doesn't work for pions and I'm not totally convinced, of course it's a very very interesting avenue to consider, but I'm not totally convinced that it can work in the most simple setups and really what, if you want to flesh this out as an approach to the hierarchy problem, what you really need to do is fully demonstrate that you can have a skill invariance all the way up to infinity, all the way to the 4UV, yep, yes, pardon this one here, no, so I can't hear you, you need to speak up, I can't hear you, so it will, when RG evolves, it will, there'll be a one loop RG evolution which breaks skill invariance, just a simple exercise, so there's the skill invariance, the symmetry that I wrote down yesterday where you rescale the fields and the derivatives and spatial coordinates, you can do the same thing here, I can't, you really need to speak up, I can't hear you, okay, yes, yeah, in different dimensions, dimensions other than four lambda carries a mass dimension, so it does break skill invariance but not in 4D, okay, super, so yeah, so the hard work, the really interesting work which some people have been trying to do and I think it's super interesting is to really demonstrate that you can embed the standard model within gravity in such a way that the theory crosses no new mass thresholds, there's essentially no UV scale, it flows all the way to some sort of conformal field theory in the UV and it's not impossible but it looks difficult. Another approach which I think is extremely interesting but I've not seen any explicit example that really works is that this expectation really relies heavily on the standard effective field theory picture for how the effects of UV scales can come in and can feed into the infrared theory and this is known as sort of the basic, this basic picture of effective field theories in quantum field theory is known as sort of the Wilsonian picture and it is based, you know, the way I presented it because obviously I don't have time to go into the details is what it's based on is some very simple notions of quantum field theory that hark back to the early days of the operator product expansion and things like this where you're working with a Lorentz invariant unitary cosal quantum field theory but one interesting thing is that perhaps Q of t itself is starting to break at the weak scale or above. I don't know how that would be but if you can screw around with these notions of unitarity or perhaps causality, they're all tied together or maybe locality, I'm not sure, then you could hope to actually break this picture of the hierarchy problem in such a way that you could have what appears to be a separation of scales without any extra symmetries but nonetheless where these sorts of things will cancel and there's an example where you can see that this can happen that has been studied it's not I don't really think it's a really it's not a working example but it's an interesting approach and this is known as the lee wick type of theories and in these lee wick type of theories essentially in the UV as far as we can understand there being a UV because when you see what I'm about to say it's very hard to even understand what you mean by scales when you get to this theory so let's say at short distances what enters along with the normal particles are particles that have a propagator so the normal particles would have the normal propagator something like this but then there are additional particles that show what a mass scale big M who have a propagator that looks like this and they have the same couplings as all the standard models and standard model particles and so on and you see that when p squared becomes large these two propagators actually start to cancel but what's happened here is that you see the kinetic term the residue of the pole for this propagator is negative which means that you actually have a ghost in your theory which means you have negative norm states so you have states with you know I working in a Hilbert space which is not does not have a totally positive norm states and so far as you can define one Hilbert space like that and so this is an approach to the problem where it's been demonstrated that if you have these sort of ghost particles around then this cancellation means that you don't get the the usual story of the heavy scale feeding into the to the IRS the IRS scalar mass nonetheless there are enormous problems with ghosts for example your vacuum is unstable you can just arbitrarily decay to lower energy back lower energy vacuum energy from our perspective by producing these these ghost particles but I think it's very very interesting and maybe maybe this is a sort of approach like with people who sometimes label the sort of general approach as things like UV IR mixing where because of messing around with the usual tenets of quantum mechanics the the physics at high scales at small distances actually mixes with physics at low scales again screwing up that scale separation of effective field theories so it's very interesting but I'm not going to talk about it and another one is Anthropics which is also a very very interesting approach in the idea for Anthropics is imagine that there are many different vacuums out there where the standard model parameters take different values so there's some sort of we call the landscape where where say this point in the vacuum some scalar field that sets the background value of say a gauge coupling or the Higgs mass or whatever takes one value but in a different vacuum over here takes a different value and we're in one of these vacuums which thanks to inflation has grown very very big so all of these other vacuums exist and as far as you know they exist in and in a sort of academic sense they don't exist in the sense of being able to travel over there measure them or study them in any way they're outside our horizon and they'll always be outside our horizon and on the list these different vacuums exist they have different values of parameters and then we would only find ourselves in a vacuum in which the parameters took the sorts of values that could lead to life we would only find ourselves as life in a vacuum that can host life this isn't an empty argument it's a very powerful concept I think some people sort of have a visceral reaction to it I don't it's a perfectly plausible thing in my mind but what it requires if you wanted to use this to explain that small value of the weak scale it requires that are that the formation of life for needs the weak skill to be very small to be near where it is or another another way of another aspect of it is that it requires that even if all of the different parameters can take different values that nonetheless the weak skill want to be very small and there are there are papers that it sort of try to attack this problem none of them super you know super convincing in my opinion but this may be if someone could find that there's some you know really core property of nature that requires the weak skill to be small that then this could potentially explain not why the weak skill is small but why we're in it happened to be in a universe where the weak where the weak skill is small and the last one which I think is by far the most radical is to to essentially break calculability by by this I mean something very very definite so the the whole notion you know the whole reason we're puzzled about the hierarchy problem is that just like the pions we believe that this is an effective field theory and we're measuring its parameters and that the values of those parameters will have an explanation in the UV in some UV theory that we will be able to derive the value of the Higgs mass and the weak skill from the parameters of a UV theory again this is not an exotic notion this is precisely what happens with pions we could have we could have with pions we could have measured them and said okay done I've got some scalers they've got some masses I'm going to set the masses just by my experimental measurements choose my counter term to be thus and forget about it and not ask any more questions or build any more experiments we could have done that but of course that that would have been a very cowardly and lazy approach because it would have been saying well let's just stop asking questions about nature and wash our hands of it the reality is that the pion masses we can understand in terms of the the QCD gauge coupling at high scales and the Yukawa coupling at high scales and we can run everything down flow from the UV theory to the IR theory and we can calculate in terms of those microscopic parameters precisely where the pion mass comes from we can put it on the lattice and we can calculate it and we see what it is we're in the same situation with the Higgs we could say well look I don't care about quantum gravity I don't care about you know the land I pull for for for hypercharge I don't care about dark matter or anything else I have measured the Higgs mass I'm going to set a counter term to cancel of the quadratic corrections I'm going to assume that I can never calculate it or derive it from some high scale I just set a counter term wash my hands of it and say that I'm happy let's go and do some go and do something else of course that would be essentially giving up on on on the scientific hope of understanding where the weak skill comes from where where the the Higgs sector the standard model comes from so I think this is very very radical it's in my opinion it's very it's so radical that it's essentially throwing major scientific question out the window because you don't want to talk about it but nonetheless it you could do that you can just you could just say well maybe this is all of nature and and there is no microscopic origin for this parameter we just have to measure it and and that's the only the only understanding we'll have okay so now it's just some more more concrete ideas okay so one of the the main ones I think the one that's most useful and illustrative and sadly for for you guys means I will keep going on about pion physics is the notion that the the Higgs could really be a pion could really be a pseudo goldstone boson and I'm going to make a little make that the notion of pseudo goldstone bosons a little bit more concrete in a second and this is an idea that's been around for a long time it's actually I think you know was first studied sometime in the 80s but it became very very popular about a decade ago and it's really an alternative to supersymmetry you sure you've all heard of supersymmetry but this isn't something that's as well known I think to the general public but it's a totally reasonable possibility and and I think it's worth explaining and the other reason I think it's worth going into it is that there's some useful tools involving group theory and continuous symmetries that that are useful in other areas of physics as well so what's the idea so so so what is it go what is a pseudo goldstone boson so when you break a continuous global symmetry I'll call it the symmetry group G so this could be something like for the pions it was s u2 left cross s u2 right could be s u 3 s o 10 whatever you want it's just some continuous global symmetry and if it is spontaneously broken to some subgroup each so for example if this was s u3 this could be s u2 it's just some subgroup of the original continuous symmetry group and then we have broken and unbroken generators so essentially G is broken to each so for each the sort of the manifold of this continue continuous symmetry will be described locally by by group generators by moving around on group generators so for s u3 there would be it and s u2 there would be three and so on and there are unbroken generators which describe essentially the the group that is that is unbroken so this is still we say this is unbroken more specifically you'd say that it's linearly realized and then there are the set of generators that we denote by the coset which is G divided by hg mod h in some sense where this is now not you know just two numbers this is a group theory notion but these are all the broken generators more specifically the language I prefer is is nonlinearly realized the reason I prefer linearly realized versus non nonlinearly realized is that the notion of breaking the group spontaneously the word breaking or broken is actually a bit misleading the symmetry all of the symmetry is there the entire time it never went away but what has happened is that where we see the symmetry acting in the normal way for this guy here which would be so for example if it were u1 symmetry you take some some element that transforms under that u1 and you rotate it by a phase you multiply it by e to the i theta that's the normal way we see the symmetry being respected here the way you see this the sorry the symmetry being this the way the symmetry acts on the representations here the way you see it is actually much more subtle the way it arises is essentially as a shift symmetry of a massless boson and we call that boson a NGB Nambu Goldstone boson so it was those guys who showed a long long time ago that for this story for every spontaneously broken continuous symmetry the broken generators will all be accompanied by an exactly massless scalar field and this is a general non-perturbative proof of course if you have why is there a P if this symmetry is actually a little bit explicitly broken just like with the pions the cork masses broke the symmetry a little bit explicitly then you may have the remaining symmetry and the unbroken symmetry here but now these what would be massless goldstone bosons actually become massive goldstone bosons because it was never a full symmetry anyway but that mass can be small if the breaking of the symmetry is a small parameter just like we saw with the pions okay so is that is that relatively clear and I'm I appreciate that some people will not be familiar with group theory so I'm not going to try and avoid doing too much of it but are there any questions at this point excellent okay so how could this work for the Higgs and so could this work for the Higgs the the answer is yep pretty much but with some details so there's a simple recipe and actually there are many there was a period where there are many many papers written demonstrating various ways you could realize this recipe but the underlying physics is essentially always the same so the number you can count the number of guys here and the number of guys here and you see that the number of gold stones is equal to the dimension of the unbroken group minus the dimension sorry the dimension of the the the initial group minus the dimension of the unbroken group so that's just the dimension of this so the number of broken generators is given by this formula here however we can also add some extra an extra ingredient which is that we could gauge some subgroups of these symmetry so what happens when when we gauge this story we start with a instead of a global symmetry we start with the gauge symmetry or gauge redundancy I will say symmetry just because the language translates nicely in between gauge and global for this story so I will use gauge and global rather than gauge redundancy but you should keep in mind that they're really very different things so imagine we gauge this symmetry and it's spontaneously broken to this gauge symmetry then we would we no longer because of the Higgs mechanism we no longer get this number of gold stone bosons for the broken generators because each gold stone boson is eaten by the gauge particle of of the spontaneously broken gauge symmetry so in the standard model we start with the the Higgs doublet of it has four scalars in it but when we break s u2 cross u1 to u1 we've spontaneously broken three generators and those the the gauge fields for those generators eat three of those scalar degrees of freedom so this is known as the Higgs mechanism so had we gauged had we gauged this whole story we would get zero gold stone bosons in the spectrum we get no masters goldstone bosons at all because they get eaten to become the longitudinal component of the gauge fields so if I denote a gauged group with a tilde over the top then if we gauged some aspects of the of of some some some subgroup of G and gauge some subgroup of H then the number of total number of goldstone bosons remaining would be this guy here essentially because the gold stones that would have come from here that correspond to a gauge symmetry actually get eaten to become longitudinal components okay so if we want to get the Higgs doublet so the Higgs double I said had four scalar fields living it in it if we want to get the Higgs doublet out of this then this guy here for our recipe has to be greater than or equal to four we're going to accommodate the full the full standard model Higgs doublet so as I said there are many papers written essentially furnishing a whole load of different possibilities for this you can start with it's basically a counting exercise what group can I start with G broken to H that with maybe some subgroups of those two groups gauged that will give me four or even a number close to four and I'll worry about the other details later yep pardon yes so the tilde group is a gauged subgroup of G so and equivalently for H's so for example I could have an s u3 global symmetry and I will we will do this actually you can start with an s u3 global symmetry so you can imagine this as the the space of unitary 3 by 3 matrices and it acts in the normal way on say a three-plit but I can then actually gauge I could have gauged a full s u3 symmetry as well but I can also choose to gauge a subcomponent of that symmetry where for example and here I have the so I implicitly included the generators in here but say in here I would have the generators of a full s u3 but I only give gauge fields to say that the s u2 subgroup of that full s u3 so let's do an example so if we break for example s u3 cross u1 x to s u2 cross u1 x then we know we have eight generators in here and one in here and we have three generators in here and one in here and 8 minus 3 is 5 so we get five goldstone bosons left over five is pretty close to four so so why don't we just start with that we've clearly got one extra goldstone boson but let's forget about it for a little while and work with this theory so this is what one of the first sort of explicit examples of a of a pngb Higgs model and again I'm not specifying anything about the uv completion just like with the pions I'm not saying telling you what you know strongly coupled gauge group is living in the uv that has fermions that condenses and things like this I'm not specifying any details like that just from the group theory structure alone whatever does this spontaneous breaking will give you five goldstone bosons okay so how do we understand the effective field theory of goldstone bosons and this is something that was worked out a long time ago by typically referred to CCWZ or CCWZ and a number of papers also various other people were were involved so it isn't just CCWZ but that's these are the clearest papers explaining this construction and even though we don't know the uv completion we can parameterize right easily right down just based on symmetry structures alone the effective field theory for goldstone bosons and the way this is typically done is that you would take some some field that contains the the goldstone bosons to be expressed as the vacuum expectation value that breaks the symmetry so for example if I had something if I was breaking SU3 and I had something that transformed in a three-plit so let's see say that sigma goes to a unitary matrix times sigma that's the transformation and this is a three by three matrix then if this so this is a three by three vector sorry not three by three this is a three dimensional vector then if I get a vacuum expectation value for example for for the the last component I can actually up to up to the symmetry choose any component or any mixture of components but if I get a vacuum expectation value then the symmetry will be broken just like the Higgs you have the Higgs doublet and the neutral component gets a vacuum expectation value and that's spontaneously breaks SU2 symmetry I can do the same thing here so I have a three-plit and it gets a vacuum expectation value which I will call sigma zero so sigma zero would be zero zero f where f is the vacuum expectation value so I could do this for example in a very Higgs like manner I could have some scalar field that's living in the three and I could write down a Higgs like scalar potential with a quartic like this and and sorry that should have a minus sign a quartic like this and mass squared if the mass squared is negative like for the Higgs you get a Mexican hat potential you get a vacuum expectation value and you spontaneously break the symmetry or it could be broken by some non-perturbative dynamics there could be some strongly coupled gauge group in the UV which leads to a cork condensate that has the same quantum numbers it's still in a three-plit and that spontaneously breaks the symmetry either either is fair game but then we can parameterize the Goldstone bosons by including this this vacuum writing sigma the full sigma field which here would have six scalar degrees of freedom it's there's three complex fields so there's a real and imaginary field for each component here so the six degrees of freedom but I know that I only have five Goldstone bosons the other three scalar fields will typically become heavy there's nothing to keep them light because there's no symmetry they aren't Goldstone bosons so I can actually forget about those other three fields and just work with my Goldstone bosons and then you can parameterize how they live in the sigma field by this construction where we write sigma is the vacuum expectation value multiplied by this exponential that contains the broken generators so these are the generators of s u3 that's been broken so these are all of the generators and if you think of them in terms of three by three matrices that when they act on the vacuum expectation value do not vanish so for example the generators living up in here you know I've got the three generators of s u2 if I call this little corner s u2 and I act on this vacuum expectation value you see that it vanishes so those are the unbroken generators of the broken generators correspond to elements that live that are non-zero along these pieces here and there are five of them and I can parameterize I take those those generators and for each generator I know there will be a mass of Goldstone boson so I literally contract together a Goldstone boson for each generator and typically these might be dimensionless but often people work in terms of a dimensionful scalar fields in which case you have the vacuum expectation value living in here and then what you can do is to understand the physics of these Goldstone bosons you write everything down in an s u3 symmetric manner so you just use the usual the original symmetry you work with this this three-plate this fundamental representation of of the s u3 symmetry and you write down a theory that is completely s u3 symmetric in terms of of this guy this field here and it turns out that this will capture all of the interactions you write down a general theory with all of the allowed terms this will capture all of the possible interactions of the Goldstone bosons notice this parameterization of the fields may look a bit unfamiliar to you but it's it's really not that exotic it's just like if you had a scalar field a complex scalar field you could write it as as for example you know the real component plus i times the imaginary component that's one way of writing it but you could equivalently write it as some as the radial component in this is a field these are both fields and an angular component if you wanted you could parameterize it there's two degrees of freedom here there's two degrees of freedom here it's purely a matter of choice there's no physics contained in writing them different ways it's just how you may wish to parameterize it and here you could also find other more complicated ways of parameterizing all the all these Goldstone bosons but it turns out that this is very slick and convenient but it's not not exotic in and really in any way okay so what happens for the Higgs let's actually do this construction this it's called CCWZ let's do this construction for the Higgs and so we do this and as I said all the broken generators live along here the extra one that we don't want corresponds to one of the diagonal generators just for the sake of simplicity now I'm going to to ignore it this is a bit this is illegal but I'm going to do it nonetheless and so we're just going to work with four four generators so we have four scalar fields with four Goldstone bosons here and the generators I'm interested in are the ones that correspond to these entries here and when you do that we can write that the pi a summing over the a's of course is written as 0 0 0 0 and then we have Higgs 1 Higgs 2 Higgs 1 dagger Higgs 2 dagger 0 where these guys are complex so we have a complex field here and a complex field here and they're they're Hermitian conjugates so what this means is we have four degrees of freedom and you can see this is already making up the Higgs doublet the Higgs doublet the standard model Higgs doublet will just be Higgs 1 Higgs 2 so when we do this this big sigma field that we can use to to write down all of the interactions we just respect the the s u 3 symmetry and write down everything we're allowed to do and in terms of the the the big sigma field it looks like this this is going to be a bit of a mass and where we're at mod H is just equal to the the magnitude of the Higgs doublet okay so so this this describes our sigma field we can we can now parameterize all of its interactions so for example you could write down the kinetic terms for global symmetry they would look like this this is just a standard kinetic term for these for this scalar field sorry they should be dimensionally there should be an F squared out the front oh no no sorry and so these are the kinetic terms the scalar field if you want to to if you don't trust me and you want to explore this you can explicitly put this in so explicitly put this sigma field which is expressed here put in the derivatives and see what happens and what you'll see is that you get standard kinetic terms for the Higgs field plus some some higher order interactions involving for example four Higgs fields and two derivatives six Higgs fields and two derivatives and so on we can also choose this this procedure I mentioned here of gauging a subgroup this g tilde guy here we can choose to gauge a subgroup of the the global symmetry and what we do is very very simple this is a three-plit we can see that this is s u3 invariant so what we do is we just promote some part of that s u3 gauge symmetry to s u3 global symmetry to a gauge symmetry where now we do exactly what you would expect you just promote the the derivatives to the gauge covariant derivatives something like that and if we had spontaneously broken if the and then if the whole thing was a gauge symmetry then all of these goldstone bosons would have been eaten so this would actually give you just the normal gauge boson mass terms plus the the the kinetic mixing terms between the goldstone boson on the gauge fields but what we're going to want to do is gauge some subgroup because we know that the s u that the the Higgs doublet Higgs 1 Higgs 2 here and transforms under s u2 left cross u1 hypercharge yep and so so if you put this guy so that the sum over these guys enters here and if you put this guy in here then that's what you get so it's do the the the matrix exponentiation mathematical can do this for you now if you just put this in a mathematical this is what you get I think it's just pretty neat but you should of course do by hand yourself and I was gonna say yeah yes so so we know that the the Higgs doublet transforms under s u2 left cross u1 hypercharge which means that these gauge derivatives now would just be the usual standard model ones so d mu is d mu plus i g you know the sum over all of the gauge fields I wrote it down over there so I don't need to write it again okay so now we have gotten pretty far there's a little bit of quasi formal stuff not much though but what we have now is we have goldstone boson Higgs so we can understand why it's light and it has the observed gauge interactions however gauging a subgroup of a global symmetry breaks the global symmetry I can see that because now actually I should write this down what did I write down over there notation so d i g so this is the gauge covariate derivative now these guys are living in the generators for this guy and this guy all live in this space here so I have you know the z plus b and z minus b as z plus a and z minus a and things like this living in here so I w zero let's write it like that this is all schematic here I'm not putting in the actual specific coefficients but here I have not gauged anything so this gauge covariate derivative as it acts on this three-plit is like this now let's perform an i global s yep yeah so you gauge some you have an additional you want symmetry that this this is a good question that this sigma transforms under so in terms of group theory this would be like having a u3 rather than s u3 so there's some extra bit but it has a different a different gauge coupling that lives along the diagonal as well very good question okay so now let's if I perform an s u3 global symmetry transformation corresponding to the generators that lived the corresponding to the broken generators the one that lived along here and you can see they'll get all mixed up and tied up when I act on sigma they'll get mixed up and tied up with the gauge fields here so it turns out you can just do this exercise yourselves you can see that actually the fact that I've gauged a subgroup of the full global group explicitly breaks the global symmetry so now I have an explicit breaking of the global symmetry and it's just like with the pions if you remember that QED interacted with the charge pions but not the neutral pion and that is explicitly broken the global symmetry that was keeping them all the same which was like an s o3 symmetry so just like with the pions we expect a correction to the to the Higgs mass I'll go back over here we expect a correction to the Higgs mass which looks like for this class of models which would look exactly like the pion mass correction so delta m Higgs squared has the scale something like g squared over 16 pi squared lambda squared where lambda is whatever the scale of the cutoff is where the the sorry the UV not the cutoff the UV completion is ok so what this is telling us is that even if the Higgs is a pseudo goldstone boson that if it's going to be light and avoid getting very large corrections from this from from whatever the UV completion is as I said it may be some perturbative story where you just have some scalar field it gets a vacuum expectation value or it could be some non-perturbative thing involving a strongly coupled gauge group in the UV doesn't matter there should be new physics the cutoff physics shouldn't be too far away if these corrections aren't going to get much larger than the weak skill and in fact when we put in the the known gauge couplings this would probably not be more than say TV or a couple of TV and there are ways to get around this you can make this pre-factor even smaller this is known one class models is known as little Higgs models where you what you do is you enhance the group theory structure further you would say for little Higgs models you do two copies of this you do two s u threes and they're broken spontaneously broken to two s u twos but what you do is you only gauge group the s u to subgroup of one of those s u threes and then it turns it turns out that this correction is delayed to higher loop orders it would only come in at two loops so it's smaller and so on so the price you pay you add extra baggage which is more tech group theory technology which means more fields but you can suppress these corrections but at the first naive pass these corrections will always be there right no very good question so yes so if the Higgs was a PNG be it's not from this correction alone it's not guaranteed that you would expect to see it because this may only relate to the electoral week sector and the LHC is actually has a lot of blind spots for purely electoral week charge stuff you can get away with electoral week charge particles that are actually still at you know 200 gv or something like that yep good very good question no so so no that wouldn't change because still in fact I think ignoring it made it a little bit what actually makes a little bit worse because what I did was just throw away a bit of the global symmetry so I broke the global symmetry by ignoring it in practice if you do this example and you don't throw that goldstone away but he doesn't help with this because it was the gauging of the subgroup that actually broke the symmetry and that's the reason if I hadn't gauged the subgroup then I would have a full global symmetry it wouldn't have been broken in this correction wouldn't have been there but that additional scalar or these additional goldstones that can come along for the ride can be useful for other things so for example there are models where they have an extra Z2 symmetry which stabilizes them and they could be a dark matter candidate or they could play other roles and for example modifying how the electoral week phase transition occurs things like this so they're not useless in fact this model this s u3 model this is in the homework gives rise it's not something that is is seriously studied or and wasn't seriously studied it's a very good example but the reason it's not seriously studied is that if I use the s u3 symmetry alone there's nothing to forbid me writing an operator that looks like I don't think so so there's nothing to yeah just to forbid me writing an operator like this and one of the problem sets in homework is to explicitly put in the Higgs doublet so in terms of the Higgs doublet as I wrote it there you can see that if I forget the signs and the cosines if I take the small h limit then this looks like just Higgs dagger d mu Higgs squared and you have the form of d mu this is in standard standard model textbooks q of t textbooks and you can put this in and you see that this operator it's known as a higher dimension operator but as I said if we're following this prescription we don't know what the uv completion is we have to write down all of the operators that are allowed by the symmetries and we typically expect them to be there unless there's some extra symmetry reason why they shouldn't be and it turns out that this operator would be there and actually gives a large correction to this difference in mass between the w and the z mass and this is something that was very well constrained by the large electron positron collider by lep at CERN and so actually this class of theories is pretty much ruled out unless you do a lot of extra work to try and get rid of this but it turns out there is a symmetry why that could forbid this operator it's known as custodial symmetry and other classes of models have that symmetry so for example if you have if you played a similar game but you did s05 to s04 and things like this and then you have custodial symmetry and that forbids this operator which makes those models more attractive okay but that's that's one of the in one of the problem sets you literally just what you have to do is literally just plug in the Higgs doublet the VEV of the Higgs doublet so the VEV of the Higgs doublet is you know 0v and put in the gauge covariant derivatives and see what happens to the W and Z mass yes yes ah no exactly so so what have we gained with respect to what we already thought would happen for the Higgs in the first place even if it wasn't a goldstone boson and we will get to that for the the tops let me get to the tops and you'll see so let's do the tops yes absolutely so so exactly I mean that's that's sort of the message is that if you try to make the Higgs be like a pion for the gauge interactions you still you still don't get get out get very far but when I get on to another class of theories called the twin Higgs you'll see that you you do so this is the first pass this is like the first pass of our plane and now we we loop around and come back again and okay so what about the top quarks so as was just pointed out you know the first thing we wrote down this morning was that for the top quarks you expect to get corrections that go like the yukawa squared times the cutoff squared so what happens for a pngb Higgs so how do we write the top interaction so we embed everything has to be now s u3 symmetric so we embed the top cork doublet into a top cork triplet so we have for example the the left-handed doublet will go like top left bottom left and now we have to make it a full triplet so we there's an extra field to fill out the full triplet we'll call it big t and then you can write down an s u3 invariant interaction u dot sigma times t right and you see here that this is completely s u3 invariant so this respects the global symmetry so if it respects the global symmetry it cannot give rise to a mass for the goldstone boson because the goldstone boson is protected by the s u3 symmetry s u3 symmetry hasn't been broken so fantastic so we now have the top cork yukawa and we have not generated this term but we have a great big problem which is that you see when the Higgs gets a verb I will marry up the top right with for example the the top but then I'll have not just the bottom left hanging around but also this guy here massless so this this predicts a new extremely light colored fermion which we haven't seen it doesn't exist so so there's a problem so this can't work now one thing you could do is explicitly just do something sloppy like when we gauge the subgroup here do something sloppy and just literally delete this from the theory so pretend you know bury your head in the sun and pretend it doesn't exist set this to guide to zero and then we still have the top cork yukawa but again the then we've explicitly broken the symmetry by setting it to zero and we expect to get exactly the same correction that we saw earlier which goes something like three lambda top squared lambda squared where it land is the scale of the uv completion time 16 pi squared so we haven't then we still not gained anything we've seen that if we try to construct a theory where the the Higgs is a goldstone boson but has just the standard model field content and the standard model interactions that despite all of our extra work we've gained absolutely nothing relative to what you would naively expect the last escape route is to keep this extra guy in the theory big T and then we can introduce some explicit breaking of the symmetry so we write down an interaction that is not s u3 invariant but does give a mass to big T and it looks like this where we've added and added in by hand an extra fermion which has the the complex the conjugate quantum numbers of big T so we write this down and we can now anticipate what will happen so if we are at low energies then this theory looks very not s u3 symmetric but say we went to very very high energies way way above the mass of empty at very very high energies this operator is not not very relevant for for the physics empty is a small number say we say empty was 500 gv and we're working at five TV or something like this and so the only relevant interactions up at those scales are the you cows and they look s u3 symmetric so in this theory in the UV we have an s u3 symmetry which has been we say softly broken by mass parameter for the tops and it turns out that you can actually just go ahead and so what this means is that then we expect to get corrections like this but the only term we have again let's use the spurion argument the only term we have in the theory that's breaking the s u3 symmetry is this term empty using the symmetry arguments that I outlined yesterday so we would expect that the corrections to the Higgs mass squared must scale like something like three lambda top squared three just comes from the fact that there's QCD so there's three guys in here lambda top squared over sixteen pi squared there's only one spurion that's breaking the symmetry which is this MT guy so that should go like MT squared and in fact you can actually just go ahead and calculate in this theory you calculate the top loops with Higgs is and you get also from from this interaction from the cosine you notice the cosine bit of the end so the big T is living in the third component the cosines living in the third component so you get big T and then a cosine from here cosine looks like when you expand it out it looks like one plus Higgs squared so you get an interaction that looks like Higgs squared big big T squared and our Higgs squared big TT right when you integrate it out you get something like this this if anyone's interested this could have been an exercise for the problems I took it out but if anyone's interested you can do this explicit calculation and this is exactly the answer you get yep I can't sorry well I could yeah I can I mean it's just a matter of nomenclature or cut QC yeah I could cut call this guy TC or whatever okay so what does this tell us so this tells us that you can get away with having actually large top yukawa for a PNG be Higgs but the price you pay is that there must be a new colored fermion living somewhere around and the heavier you make him the more the the bigger the corrections to the Higgs mass become and if this guy is heavier than say four or five hundred GV these corrections start to become bigger than them the mass of the Higgs itself so there's you start to have to fine tune the theory essentially you're getting big quantum corrections so you need to add in a bear source of explicit breaking that would give mass to the Higgs which exactly cancels this to leave a leave a light Higgs goes on so the price you're paying for for winning PNG be Higgs is that there have to be new degrees of freedom new things you can search for the LHC and in this case they're colored and colored new fermions which which stick out like a sore thumb you should be able to see them okay and again all this is you know follows the naive ex the the naive expectations that I that I outlined from those sort of basic EFT tools so now I want to discuss a class of theories which has been studied more recently but it's definitely not textbook material as I said for textbook material you can look at textbooks but this is a class of theories that will became very popular about four years ago it was actually written down about twelve years ago I think now and it's known as the twin Higgs and it uses some of these tools but in a sort of a curious way not to avoid as you can see from these arguments it's pretty inevitable sorry yep right so yes you also have to have the bottom cork you call an epsilon in here be right so you have to explain all of the observed particle masses so you do have to have a bottom you call but you can't escape the top you call I'm not adding it to as a model building tool I'm adding it because it exists in nature okay well this is so this is the bottoms living in here so that you start with the standard model doublet so yeah I'm calling this T but that's just a name I could call him X for example it's not really that he's a top he actually has a different hypercharge and things like this yep very good yes MT turns out to be I'm sorry not empty there you get a mass oh no sorry so so you're saying you would expect to show up here it so it does at some point but it's sort of sublating so let me so so MT so this so MT is the only parameter that broke this symmetry so this can only be empty but that that does show up in the calculation so you you've spotted something something non-trivial here which is that when I expand out the cosine bit living in here I have something like lambda top each squared over F coupled to T and then T right that's the actual interaction but then you diagonalize to the mass basis because also you've given a coupling of order of size F to the that mixes T and T right so you diagonalize to the mass basis and then when you have diagonalize to the mass basics basis and you calculate this loop there's actually an F which cancels in the numerator in here cancels that F squared in here but things like you know MT to the 4 over F squared absolutely so they won't go it won't go exactly like this because well maybe it will with another H bar so at higher loops it could so so so this is a very good question so in general you could have something like this it won't go exactly like this because you see this has to have dimension of of mass if it's one loop of mass squared and remember I pointed out yesterday that F F has dimensions of field which actually has dimensions of mass divided by coupling basically yes exactly the way which is why that I wanted to show you that H bar stuff because you could if you just do normal dive naive dimensional analysis you don't know but I can already see that there has to be an extra coupling squared in here or an H bar so it's higher loop so it lets you count very quickly see where all of the loops and couplings have to come in but absolutely okay so this this class of models is known as the twin Higgs it was unpopular because pre-LHC there are essentially much more attractive looking models on the market and this twin Higgs model was sort of seen as a sort of an academic curiosity but not something to be taken too seriously yep this this guy here yeah so that's this guy here so Higgs squared so if you look at the I've got the third component in here contracted onto the third component of sigma and the third component of sigma is F cosine of of mod H over F cosine is one plus H squared cosine of H is one plus H squared over F squared and H squared is just this guy here squared so that's where it comes from yep right but we did that already that's what I did before the tops right okay so so yeah so that was the term we just did which was the this guy here delta M squared goes like G squared lambda squared but the point is that if MT is small the correction for the top mass from the top yukawa is small so and also and the top yukawa is the biggest coupling in the the game so that's what you want to worry about the most if I do this naive thing the top yukawa is going to give me the biggest corrections the gauge are always there but they're not too bad if the UV completions living around a few TV you're not in too much trouble but the top would kill you this trick lets you take the cutoff by introducing some small explicit breaking take the cut off a way up so then you would maybe saturate saturate the gauge corrections but at the price of having to introduce new degrees of freedom and in fact even for the gauge you can play tricks using this twin Higgs stuff to to sorry this little Higgs stuff to make these gauge as I mentioned the gauge correction smaller but the price there is that you always pay for pay the price of the requirement of additional great degrees of freedom every time you try to break this naive thing this naive expectation you when you achieve it you always find that you've got new degrees of freedom that are soaking up some some symmetries okay so so let's just over ten minutes so I want to talk about this so I'm only going to do ten minutes in this because it's it's not textbook material and it's not something that everyone works on but I think it's useful because it's something that people are working on at the minute or at least have been in recent years and it illustrates some how you can evade some of these naive expectations particularly that we've got new colored particles in sort of a cute way but again you'll see there's always a price to pay so the idea of the twin Higgs this is by Chaco and Harnick and Go is that you have a standard model just the normal standard model that we know and love and then you add a complete copy called the twin standard model and this is really for our purposes today it will be an exact copy so there will be top corks and the Higgs and a photon and everything in here but it's not the same top corks and Higgs and photon as in here it's an exact copy so these guys are not charged under our our own our own QED they're charged under their own QED so our particles our electrons do not scatter off those electrons via the photon they have their own photon so at this level it's completely decoupled sector you wouldn't know it exists except for maybe cosmologically and then the only interactions consistent with the gauge symmetries are not the only but the most important interactions consistent with the gauge symmetries I'll call this the twin Higgs this is the Higgs from the twin sector or for example we can have a mass term and then a quartic interaction and what we will impose on the theory is an exchange symmetry so this is a symmetry we call it a z2 symmetry where we literally lift every particle in here and we swap it with every particle in here so we go do that and if all of the gauge couplings and yukawa couplings are identical on both sides then that will leave the theory unchanged because I've taken a photon with you know the coupling of e to the electrons and I've entirely transplanted this guy but if this guy also had a photon twin photon with the twin electron with exactly the same gauge coupling then the theory the Lagrangian that you had looks exactly the same so we impose an exchange symmetry and we can see that this then means that if you respect the exchange symmetry then the mass squared for the Higgs and the twin Higgs looks exactly the same so I can write it like this this term here break respects that exchange symmetry there's also a term that I want you to ignore for a little while I'll call it lambda tilde which also respects the exchange symmetry which looks like this so what we see here is that we have if we ignore this term for them for the moment pretend it doesn't exist if we have a theory that spits out this this action that some UV skills say you know three TV or something like this then we see we have the exchange symmetry has enforced at the quadratic level that we actually have an enhanced symmetry and as far as the masses are concerned which is that we can write let's call H tilde as H and then each twin so this is a doublet and this is a doublet so this is a four-plot there are four complex scalar fields in here and we can see that this actually has a symmetry where I could write this whole potential as minus m squared Higgs tilde squared so that should be minus minus lambda over two Higgs tilde to the four and we see that because this is a four-plot this actually has an accidental SU four symmetry because it's a four-plot and I've written it in an SU four symmetric manner again we're forgetting about this guy so now I imagine that big Higgs tilde gets a vev so I'm just for the sake of not confusing you remember this is two doublets so this gets a vev for example in the fourth component we'll call it F then this breaks SU four to SU three this means that we get the number of goldstones we get equal to 15 that's the number of generators in SU four minus eight which is the number of generators in SU three which is equal to seven so we had seven goldstone bosons but we have full this is a twin standard model and this is the standard model so we actually have gauged this guy here is a standard model Higgs double so he is actually charged under the standard model gauge interactions under this SU two gauge group and this guy here is charged under the twin standard model gauge interaction so this is the twin SU two gauge group so we've actually gauged an SU two subcomponent of this SU four so when this guy gets a vev this doublet gets a vev you see here then just like for the Higgs in the standard model this this vev spontaneously breaks the twin SU two left across twin you want hypercharge and three the twin gauge bosons become massive so that formula I wrote down before with the g-tildes and the H Tildes for gauge subgroups apply so we actually have three of those goldstones have become eaten to become the longitudinal components of the twin gauge bosons so now we have minus three which is equal to four and Bob's your uncle we have exactly we what we need for these four guys here to be pseudo goldstone bosons so this is again a PNG be Higgs type approach okay so that's great but as we saw before we've now done things like gauged subgroups so we expect to get corrections in this story corrections to the to the scalar masses that look like g squared times the cutoff squared so let's calculate what they are okay so we know already from the standard model sector that the correction to the Higgs mass will look like so let's say delta v at one look we expected to go like g squared over 16 pi squared lambda squared Higgs squared so we expect to be in trouble but of course there's also the one loop corrections from the cutoff that go like g twin squared over 16 pi squared lambda twin squared Higgs twin squared and this is where you start to see a little bit of the magic of the twin Higgs and this is really the I think the reason that these guys wrote the paper so we have two different mass corrections from the gauge sector that we've seen now I don't know 20 times for scalars we don't know what's going on at the cutoff but we expect these corrections to be there and they look like this and like this and so in principle they look different but we have an exchange symmetry so this exchange symmetry is respected g twin is exactly the same as g so I can just write this lambda twin if the even the UV physics respects exchange symmetry lambda twin is just equal to lambda so I can simply write this as g squared over 16 pi squared lambda squared Higgs squared plus Higgs twin squared which I can write as g squared over 16 pi squared lambda squared Higgs tilde squared so this is now the four-plot and you see that these unknown corrections from the UV will actually respect the su4 symmetry that you started off with so they will not generate a mass for the Higgs for the goldstone because they respect the the initial symmetry in the first place and this is all because of the exchange symmetry that sets this bit equal to this bit which tells you then that at one loop you will not have quadratic sensitivity to the to the cut off from the gauge loops and in fact there will be gauge partners which are the turn out to be the gauge bosons that the the W and Z bosons of the twin sector actually cancel the quadratic sensitivity coming from the standard model gauge bosons and you get this term here and that you naively expect gets eliminated at one loop what about the tops that the story for the tops is similar this is that was the gauge and now we add in the yukawa's and we have exactly the same thing lambda top squared over 16 pi squared lambda squared Higgs squared blah blah blah and you see again because of the exchange symmetry this looks like lambda top squared over 16 pi squared lambda squared Higgs squared plus Higgs top squared if you want you can also you can just go and calculate these things diagram diagrammatically put in all of these matter content go to the diagonal mass base and integrate things out but this is the answer you will get and you see that again it respects the su the top corrections respect the su4 symmetry so one loop there are no there's no quadratic sensitivity of the cutoff to coming from the the fermionic couplings but what's really peculiar here is that we've we haven't added a lecture week charge fields and we haven't added a QCD charge field so in the standard pngb model you saw that to to add in the top yukawa in an su3 symmetric way we had to add in something in this multiplet but because it's in a multiplet if we respect the su3 symmetry this guy had to be colored because these two guys were colored what we've done here is pretty slick because what we've added is stuff that's completely uncolored it's completely gauge neutral impossible to see at the LHC and we've eliminated these these quadratic divergences so this makes the twin Higgs a very interesting model and this is you can see now y in retrospect yep they're both gauged so they're both because this VEV here so there's an su2 that acts on this guy part and an su2 that acts on this guy here so we've gauged them both they're both identical but this vacuum expectation value only breaks this su2 in other words when I give a verb to this that's the same as giving a verb to this guy so this guy only breaks the twin su2 but this guy didn't get a verb so the standard model su2 at this level remains unbroken so you can see in retrospect why this became popular after the first set of results of the LHC came out because this is a class of theories where you can understand that at least the little hierarchy problem the notion that there's a some UV completion sitting around the corner could actually be not so severe and because there's a symmetry a symmetry story under that gives an explanation why why there aren't large corrections to the Higgs mass but as I said you always pay the price of additional degrees of freedom additional particles you can search for and that is always the case and in this case you have those particles but it just so happens that they're completely gauged neutral and because they're completely gauged neutral and they're very hard to see at the LHC so they could be out there but we haven't seen them yet in some cases they're more easy to see from cosmological observations because if you truly have a twin copy here then you also have twin neutrinos twin photons and they can be relevant cosmologically the only interaction that you can use at colliders comes through when I square this guy I get a mixing term between the Higgs and this the twin Higgs so you get the scaler sector of the standard model gets modified it looks like you have a new singlet living there so it's very interesting phenomenologically and finally there's a skeleton in the closet which is that all of this magic was happening because of the exchange symmetry but this term the exchange symmetry was enforcing at the quadratic level an accidental su4 symmetry just because I have two doublets that have the same mass terms this term here respects the su4 symmetry and the su4 symmetry is the reason that we have a goldstone in the first first place but this term here does not so if you add in this term which you would expect naively to be there if you add in this term then this whole story breaks down because you will get a mass for the Higgs that is correction for the Higgs that's proportional to lambda tilde times f squared where f is the symmetry breaking scale in the twin sector so there's an additional I'm not going to go into it now but there's an additional model building requirement is that your UV completion should for some reason and spit out this term being very small at the cutoff scale doesn't have to be zero but you need it to be small yep no so exact so so these were the problems yeah exactly so so but the exchange that was that that's the point of the model so you have exactly these problems for both sectors that's exactly what you have so you have this guy for our sector and this guy for that sector but you see that if you have the exchange symmetry then these terms are identical to those terms so they actually respect the su4 symmetry the su4 symmetry is not being broken at the quadratic level thanks to the exchange symmetry you didn't have the exchange symmetry absolutely you would have that that problem but furthermore you you you so I was saying you need to have a theory that makes this small yep so square this you get Higgs to the 4 plus 2 Higgs squared Higgs top Higgs t squared plus plus Higgs t to the 4 whereas this modifies the cross term between them yeah yeah but it's a specific form this is su4 invariant and this is not very good no and the reason is that say I started at some high scale so this is what I was about to say actually say I started some high scale let's call it 4 TV then and if at that scale this guy vanishes then I have to go to lower energies and see what happens so I RG evolve and what happens is that I generate even just from the standard model interactions and the twins standard model interactions I generate terms corrections of the quartic that look like lambda top to the 4 must be over 16 pi squared Higgs to the 4 plus Higgs t to the 4 and then there'll be a log of some you know that's called m over f when I run down to the f scale and you can see that this is not su4 symmetric I can absorb this into an su4 symmetric term plus a non-zero lambda tilde and which is telling that even if you get the magic to work in their models that can do it if you get the magic to work such that at the cutoff scale this guy vanishes and it looks su4 symmetric the RG evolution alone will break that su4 symmetry and as I said if if you have this term it generates a mass for the for the Higgs a mass correction for the Higgs proportional to lambda tilde times f squared so if you want that to not be too big you need this term to be small and the only way you can get this term to be small is if the high scale is not too far away if this log happens to be small of this log were big you'd be totally doomed so that's one that's that's that's the main thorn in the side of these models so you so you cannot push these particles you cannot push f and these other particles arbitrarily heavy without paying a price the second thing is that even at two loops and this cancellation does not this these guys do not look necessarily su4 oh no sorry it does it goes to arbitrarily high loop works at arbitrary high loops for this for the quadratic piece but the quartic pieces are our problem yep so we'll go yes exactly that that is that's precisely the problem that's how it hits you a good question so yes so naively yes exactly so if I did this have I started with this that you would actually because of the exchange symmetry you'd have equal vev in the twin and the standard model sector there would be a aligned equally in that so what you need to do is you just like for this guy we added a little bit of su3 breaking that we could get away with you add a little bit of su4 breaking so a mass term that's not exchange symmetric one way of doing it is to do that which pushes all the vev to be mostly in the in the twin sector that answer your question okay so very good the second option is people something that people do a lot I really dislike it because the only reason this thing worked was that the physics of the cutoff which you don't necessarily know what it is I mean you can you can write down super symmetric models you can write down the whole thing that works all the way up to the point skill if you want but but the physics of the cutoff you don't necessarily know what it is now I imagine I take this twin center model and literally just throw a bunch of particles in the trash can say gone then that's a very hard breaking of the symmetry it doesn't show up in the in the infrared just through these effects you don't see any problems but what it means is that in the UV you have a very very hard breaking of that symmetry so it's very very dangerous so I don't like that option but people do it to in some sense just study the phenomenology of the lightest of the of the most relevant particles the alternative is to get rid of them in some way that's soft in the sense that it doesn't spoil the the UV story so you can do that for example you could have additional fields that pair up with the neutrinos to give them a mass the twin neutrinos to give them a mass that makes them not too dangerous you could have a field in the twin sector that gets a small vacuum expectation value and breaks twin electromagnetism so you have no light photon so you can play games but it so you can make the the the cosmology safe or an alternative is actually to keep all of this stuff and have find a way to have the cosmology such that you maybe have a low reheating temperature so that you never really access these interactions very efficiently and you only reheat into the standard model which would mean that you only get standard model particles being hot and making up our universe and there's a very small number density of these particles which then helps you evade cosmological bounds but this is a whole program of study that people you know try to find ways around around the cosmological issues yeah no no so so there can be kinetic mixing if you can calculate it some very high loop order it would be there but it could also be bigger and that can actually be interesting cosmologically people written papers with that but there's it's not it's not it's sort of like with super symmetry right so nature is not super symmetric and no one ever believed it was I hope but we know that that you can break super symmetry in such a way that you preserve the night UV story at the price of modifying the IR story so that's soft super symmetry breaking you add soft mass terms and so on and the game is exactly the same here you can preserve the nice UV story and while still in IR having breakings of the exchange symmetry that make it you know phenomenologically and cosmologically acceptable you may have an aesthetic aesthetic preference about you know you may dislike that and I think that's that's totally fair all I would say is that nature thus far doesn't really seem to care at all about our aesthetic preferences so we know we have to explore all possibilities finish there let's take