 Yeah, I did I yesterday. Sorry. I went along Okay, let's begin BSM number three the strong city problem okay, look so I guess Everyone's getting a bit tired six hours six hours of lectures is one in one day is quite a lot But I hope I hope I can keep you all awake for the next hour and a half. So we're gonna change Change direction quite dramatically now In my first lecture on Monday, I pointed out that the two things I really wanted to talk about were the hierarchy problem and the weak scale and the strong CP problem So we're gonna delve into the strong CP problem today the reason For having this lecture today is that also, you know, Tracy and Robert both discussed the axion today So we're gonna come to the axion but from a more Theoretical angle and a less phenomenological angle. Maybe I might mention a very very briefly some phenomenology And then if we get time at the end We might be able to swing back towards the electro week hierarchy problem to discuss Some speculative ideas that sort of link the physics of the axion with the electro week hierarchy problem at the end These are by no means broadly accepted approaches to the hierarchy problem But an interesting set of ideas that's very sort of modern and relevant at the moment that people have been studying So hopefully we can swing back towards the hierarchy problem if we get time So the strong CP problem, I hope many of you have heard of it And it's all concerned with a particular Interaction in the standard model which you can write in various forms. I'm gonna write it this way So G is the QCD gauge coupling Theta is some constant term that we will discuss at length. So it's just some constant number And it as you'll see it takes angular values and then we have the the fully anti-symmetric tensor And and these are just the the field strength for QCD for the gluons So this term is really one of the most Remarkable and special terms that you can write down in the in the standard model It's I find it absolutely fascinating and it teaches us two very interesting lessons About Quantum field theory. So of course this has been studied at great length for many decades But similar terms that they're known as topological terms Have are very interesting and relevant in Wide area wide ranges of quantum field theories from condensed matter physics to string theory And all sorts of applications. So what we learn by studying the strong CP term is is Broadly applicable It's called strong CP because this term here you can see it violates CP This is a this combination here is CP odd. So if you have a non-zero value for this guy the You have CP violation in the strong sector and it teaches us two two important things The first is that sometimes Classical symmetry is not Quantum symmetry And this is really blew my mind when I when I first learned it. So we will come to that first and The second thing that you learn by this term is that total derivatives Matter and And what I mean by this is normally as I'm sure you're familiar with when we look when we calculate in some quantum field theory some Scattering amplitudes some perturbative scattering amplitudes some process for the LHC or or some other high-energy process There are many terms we could have in the action which are actually total derivatives And what we tend to do is we just throw them away Because we know when we evaluate the integral over a total derivative what we do is just evaluate the that term That's being differentiated on the boundaries of the surface and you know by by design when we set up a scattering process We're assuming that the fields vanish at infinity the values of the field fields vanish infinity So when you have total derivatives in the action, they don't contribute in any way to Proturbative scattering amplitudes. So normally we chuck them away forget about them who cares and This term here it turns out we will see it's actually a total derivative So you might think who cares and indeed in in any Proturbative process that you might consider scattering process involving gluons or anything like that That could potentially involve this guy You will see you would find if you calculated the scattering amplitudes that there'd be no dependence on it But what's really really interesting is that actually the physics? Does depend on this and hadronic physics that we will come to does depend on this which tells tells you that sometimes if you have non-trivial Field configurations the total derivatives can matter Okay, so the first bit quantum symmetries versus classical so consider For this for this application consider an s u3 gauge theory with some fermions in the fundamental So the the kinetic terms will look something like this. I'll work with Dirac spinners I prefer vile spinners, but I understand most textbooks work in terms of Dirac spinners, so we will do that and We have the kinetic terms which look like this Where this is the gauge covariant derivative involving the the QCD gauge fields That live in here the gluons so if we start this action we see that we can perform a rotation on This Dirac spinner which goes like e to the i I will call it alpha alpha is just a constant number gamma 5 So psi goes to e to the i alpha gamma 5 psi In terms of vile spinners this is like rotating the left and right-handed guys by separate phases that are equal and opposite and If you perform this rotation, this is a constant so the derivative doesn't care about it You can see calculate. It's a good exercise to do it. I think it's even in the exercises for for tomorrow You do this caro rotation and you will find that this term is is unchanged So what we say is that this that this is An axial symmetry you could also do just with the identity matrix in here and you would call out the vector symmetry But this is the axial symmetry So this is a classical symmetry However in a in a QFT it is not sufficient to have a symmetry It's not sufficient that the Lagrangian is unchanged or it's just changed up to total derivatives QFT is more than the Lagrangian. It's the full the full path integral So if we study the path integral it looks like this where we we actually Do the path integral over the the various configurations of the fermions And then we have the normal action. I'm setting h bar 2 to 1 So we see that when we when we do this transformation this term is unchanged So normally classically we would say that this is a symmetry of the theory, but that's not sufficient Indeed you have to convince yourself to really know that the theory the full quantum theory the path integral is invariant under this transformation then This term also has to be invariant Normally normally we don't worry about this so much. We just work with our classical action and forget about this term But in practice you can't always do that And so so how can you calculate determine whether or not this is invariant I'm not going to go through all of the details because it's quite long but very Enlightening calculation to do it But you will just remember if you if you were to discretize you can break this up into fields at different points and it looks like a very large multiple integral and we know how to deal with The change of the measure when we have a multiple integral We just calculate the Jacobian in this case It's a very large Jacobian But when we do multiple integrations say an integral integral over an area and we change change our coordinate definitions We always have to take into account the Jacobian and you can do that here and when you do that you see that the Path integral measure changes by an amount We'll call it the Jacobian minus 2 Under this change of variables which corresponds to the The rotation of these fermions And it turns out you can show the best old this which has been shown in a number of different ways But the the the most I think easily accessible method is used was really developed by Fuji kawa By studying the change of the the path integral measure And Fuji kawa showed that actually the value of this Jacobian Or jacobian to the minus 2 I guess All right, I'll just write it on this side The value of this jacobian term in the in the path integral measure is actually non-zero And it looks like the path integral over the fermions again So how does this come about? I'm where on earth did the the gauge fields come from so you Will remember that in here there's there's gauge fields living in the the covariant derivative So they look like as you'll remember from electromagnetism the gauge fields look sort of like a shift in the momentum of the fermion when the Derivative they enter in the same way that the derivatives enter and When you integrate over all of the the field configurations You just do a quadratic integral over the field configurations that shift in the momentum actually gets dragged along As you're integrating over all of the momentum of the the fermions and the path integral and And when you can consistently include it the whole time You find that you get terms that look like the the derivatives of the gauge fields when it's getting pulled along And you can rearrange them and you see that it takes this form here so what you see is that the the the Modified path integral so the path integral after I've done this this field this Carol rotation looks like the the initial path integral sorry about my My curly days are terrible. It looks like the initial path integral With an additional factor Which is not Equivalent to the original path integral that you started with this guy here So this is really remarkable. I think if you're not impressed or excited by this you can't be human because it's really It's really an incredible thing. Yep. Oh Sorry, yes, they shouldn't have been now Sorry, thank you. Thank you Pardon in the No, sorry, you're right. Sorry. I'm I must have No, you're absolutely right. I Thank you very much Absolutely, it's just that I move it across Thank you very much Okay So this is really remarkable. You start with a theory that has a classical symmetry And you can do all the classical physics you ever want to do with it and there will always be a conserved current and And You can calculate and know what their currents and things like this You can do all of the classical physics that you know and love but surely by Introducing quantum mechanics the quantum mechanics itself breaks the symmetry the fact that H bar isn't zero breaks the symmetry and which is really Totally amazing it's saying that this you want this this rotation, which you could call a you want axial symmetry Is not actually a symmetry of the path integral But you can only see it through quantum mechanical effects And it's physical it has it has real Consequences serious consequences because one thing I'm going to go back to pions again. I'm afraid I always go back to pions, but When we started with the global symmetries that were spontaneously broken by the court condensate in the first lecture I said that there was an s u2 left cross an s u2 right Global symmetry which is spontaneously broken. What's an approximate symmetry spontaneously broken to the the Diagonal s u2 So because the the dimension of s u2 cross s u2 is 6 and the dimension of s u2 is 3 We should have 3 goldstone bosons, which are the pions, but I I was being slightly dishonest at that point Because actually the the action has an s u2 left cross s u2 right cross u1 axial symmetry and classical symmetry So you should actually expect for goldstone pseudo goldstone bosons the three plant pions plus an extra guy. Yep Okay So you would expect to have four goldstone bosons and The fourth would have would be called the Eda prime or it's called the Eda prime But in fact, it's not light precisely because of this reason because you want axial symmetry is not actually a symmetry of the quantum theory Yep Good question So the question was if anyone didn't hear it could you have a symmetry that's a symmetry That's not a symmetry of the classical theory, but is a symmetry of the quantum theory So you have some quant compensating change in the path integral measure that that cancels it I don't know of any good examples my suspicion would probably be that maybe not because you would have to do something There's really There's really an h bar flying around and here I've said it to one So you'd have to do something I think in the classical action that would involve h bar, but I don't know It's a super question. Yep Sure, sure. Yeah. Yeah, you could add in effect terms in the effective theory that That would transform you could do something like that and was sort of yeah, absolutely Okay, so this so far was just for massless fermions. So now let's go to to massive fermions So when we add masses now our action is slightly changed We have the kinetic terms from before and I'm gonna put a Potential phase so I'm gonna write it. I'm gonna sandwich it between the two direct spinners As I said if you work with vial spinners, you could just see this as a relative phase between the left and right-handed guys So it doesn't have to be sandwiched between spinners, but the way I will write it It will be or we have e to the minus 2 I I'm gonna call it theta q gamma 5 plus the gauge pieces So this is just a generic phase that this Mass term could have so essentially if I wrote this is a mass for the Different mass for the left and right Carol spinners You could have Equal and opposites phases in the mass terms for those guys now So this is just a phase that could be there and in fact, you know in flavor physics We already know that we've seen an off diagonal relative phase. It's not not an axiom like this We've seen an off diagonal relative phase, which is The the strong CP I'm the strong CP phase. So I not the strong CP phase the CKM phase and So generically we expect there to be phases in the mass matrix There's no reason that a fermion mass matrix initially should be should be real quantity But typically what we do is we just rotate them away if they're unphysical We can can rotate them away and forget about them there after So we do a rotation to try and rotate this away and get rid of it So we do the rotation that Phi Psi, sorry goes to e to the I e to Q gamma Phi Psi and That removes the phase from the from the fermion mass entirely so the fermion mass now just becomes totally real But now we get a new because we've got this quantum anomaly This is the unknown as an anomaly whenever the classical symmetry is not a symmetry of the path integral we get a new term which Involves that phase We get a new term because that was not actually a symmetry We couldn't we know we can't actually get rid of it We get rid of it in the mass term and it pops up over here as the strong CP angle I'm Similarly, we could have started Right from the get go. There's no reason that that this term should initially have been zero in the first place So I'll just I would still call this theta and In which case that if this term was already there some term like that was already there We've got theta G squared over 32 Pi squared GG jewel and now in here we have a real mass term, but now we've got theta Q plus theta G squared over 32 Pi squared GG jewel and and so we've shifted what we originally defined as theta now includes the phase that was in the mass matrix and But we can't get rid of this combination I could similarly do a chiral rotation on these guys of minus i theta Q plus theta and Completely kill it from from this term, but it will show up again in the mass matrix You can never get rid of it Yep using what sorry? Yeah, yeah, so one way to say it you'll the famous triangle diagrams So you see it as an as a as a as a you can calculate exactly the same physics using a using the tri triangle diagrams and You'll get exactly the same result why I like the path integral one is that? It's absolutely equivalent so so so it's really fine to do it, but I like the path integral one for two reasons one is it's very Very physical in the sense that you have your action And you're seeing that a symmetry is spoiled But you're seeing really where it's coming from that this is sort of the classical bit of the of the theory And this is the quantum bit of the theory and you see that with Fujikawa's method You see that that's where it's coming up from and the next thing is that actually this is exact at one loop You see this is essentially a loop factor in here and the result is exact is to all orders so it's exact at one loop and That's another you can see this more directly from the path integral method than by drawing that drawing the triangle diagrams If you drew higher order diagrams, you would see that nothing shows up So I prefer the Fuji kawa way, but actually yes, there are two or three different ways of approaching this Okay, so yeah, so I could do a chiral rotation on the fermion fields and Eliminate this part entirely But I won't have eliminated the combination because it will show up again in the mass matrix now This would be theta q plus theta so we can't get rid of it no matter how it's like a whack-a-mole You hit it in one place and it'll pop up in another place and you keep doing this You'll never get rid of it if you convince yourself that you have gotten rid of it then Then you've made a mistake But why care? The reason I say why care is that if we study stare at this term we can see that it's actually total derivative and You can see that in a number of ways. I'm going to write down the explicit form of the total derivative So say we've got G squared over G squared theta over 32 pi squared GG joe I can equivalently write this as G squared theta over 32 pi squared D mu of J mu This J mu is a yep Right, right. So you're saying what if this is really QCD and these are the light corks And I can never measure a pole mass for a cork when the bare mass term is 4 Mev Yes, indeed a better way to see it is that is to avoid ever having to talk about a mass And you can frame it more simply which is that What has actually happened will come to this But what is actually happening is when you set this mass term to zero is that you're covering what's known as a petty quince symmetry a PQ Symmetry and because you have that PQ symmetry. I can you can perform a single PQ Redefinition without changing any of the physics which does eliminate this guy So a cleaner statement isn't to talk about massless or massive or poles or anything like that It's to talk about PQ symmetry and if you have a PQ symmetry then you can eliminate in This case from you have it if it's massless so you can do a rotation on the corks to eliminate this guy and That's one way of seeing it But For practical purposes, especially in the in QCD we'll really talk about the light corks and their masses Does that answer your question? Yes So what's the question so that I agree with that statement? I Okay, so you're really I mean again I would go back I would say that what you're really saying is that there's a PQ symmetry in the the standard model itself So when we add the axion we are promoted. We have a PQ symmetry Which is spontaneously broken but in the standard model itself You don't have if you write down all of the cork you cow was don't have a PQ symmetry in the standard model So that's why it doesn't work You set one of those corks to be massless you essentially regain a PQ symmetry So it's not that it's a mass term. It's really that it it's really the quantum numbers. It carries under the u1 axial Okay Okay, so so why? Why should I not care about this term at all? It's because it's a total derivative so we can write it as the total derivative of a current and this current is Known typically there are many names for it as the churned simons current just so that you actually believe me I'm gonna write it down. It's not pretty but Okay, so there it is So up to total derivatives if I if I If I act on this guy with with D mu I recover the gg duo term And as I said, we don't normally don't care about total derivatives because when we do some sort of perturbative scattering process They give no contributions to the perturbative matrix elements Essentially because the fields are vanishing at infinity and So that's why you shouldn't care about this term But then why you should care about it is that there are caveats to this logic which is that You can have field configurations Which do vanish at infinity, but still have non physical implications When you integrate over the boundary so a well-known example of this would be if you took take Magnetic field and you shield that magnetic field, but you still will have some electromagnetic vector potential Living outside it and if you take an electron and move it around that region that has a Non-trivial magnetic field even though it doesn't feel the magnetic field itself You can get a non non-trivial phase shift in the electron and you will know and love this as the Haranoff bomb effect Is that how you spell it? run off So this is something we we Have known for quite a while and it turns out that it's not completely analogous, but there's a similar there are similar field configurations in QCD Which even though they vanish on the boundary they can have non trivial implications for the physics when you integrate over all of the boundary And these are known as instantons There are various different types of these, but I'm going to talk about QCD instantons and essentially what they are if you now think of all of space as being like a sphere And you can have these objects living out on on this the three sphere We're living in here. These are field configurations that when you go out to Infinity the gauge fields actually do vanish but nonetheless the integral in the back the instanton background and Over these gauge fields will give a non non vanishing number and when there's a non vanishing theta angle One way you can one way to think about this is that if This is integrating over this term here Is that these the the gauge fields even though they're vanishing at infinity? They can essentially wind around the sphere in non-trivial ways so if you were to think about just as a simple example and if you're to live in 2d and Just something analogous to the electron you could have a u1 symmetry and You could have field configurations that that do vanish at infinity So the the radial components say of some scalar field. This is just a very sloppy analogy So to try and give you an idea of what the instantons look like, but if you had a u1 symmetry with a real scalar field You could have a field configuration where the scalar field let's let's call it the the radial mode and The phase they're both fields So this is a sorry a complex scalar field you know field configurations where even though row is going to zero out in by the infinity the Theta is non-trivial So theta as you go around could change by it to pi So it has to come back to to pi times where it started because it's an angular variable But you could have a field configuration of phi where you go around you go around the circle and when you come back to where you are The the phase of the field has actually changed by by 2 pi or and some integer times 2 pi Even though the actual magnitude of the field has gone to zero out of infinity Instantons are sort of like that Except they're much more much more complicated and if you were to you know if you write down the actual form of the instanton action and things like this Okay, so you can have instanton field configurations which are non-trivial But they are essentially winding around the sphere at infinity And just like the Haranoff-Bohm effect this can lead to physical effects in your theory But there are typically These effects are non-perturbative you can see that they're non-perturbative because just by the fact that this is a total derivative It's not going to give any perturbative contributions To your observable physics, but the non-perturbative effects when you actually calculate these the action for these instantons tend to scale In this following way they tend to be proportional to Factors like they always come out Looking like this each of the 8 pi squared over alpha Evaluated at some some rg scale So this is the the fine structure constant of the gauge coupling squared divided by 4 pi So this tells you that whenever your theory is perturbative If alpha is a small number then this exponent is a lot is minus a large number So these effects will typically be very very small. So that's why we never talk about them for the electroweak Interactions you have something like this for the electroweak interactions But because they're broken at the electroweak scale the relative the relevant scale in here is around the weak scale This is a small number. So it eats the minus a very large number is totally irrelevant But they're important for QCD because as we get down to the QCD scale This can start to become an order one number Which means that these instanton effects will typically be non non negligible and Indeed they lead to a physical we will see actually in a second where this Comes from but they lead to a physical implication For example, you could as I said before there are non-perturbative effects that give mass to the e to prime similarly if you have this non vanishing term for a non vanishing value for theta and this leads to Neutron electric dipole moment and this must have must have been mentioned earlier But you can see it in practice. How do you see it in practice? You can do and you can even you can calculate it You can do an axial transformation to move this guy back into the cork masses and you hold on to him We're going to do this a little bit later you hold on to it and then you just do the normal carol carol agron gene you calculate the Pion interactions and the pion masses and so on and you see that you get a phase that shows up in the in the neutron in neutron scattering and it shows up essentially in diagrams that involve pions So you can have something like Neutron a proton and a neutron and you can have charged pions interacting with the photon and there's a non vanishing phase shows up in here and And so there are diagrams like this You can go and look up the old papers from the 70s that do this and what it leads to is an electric dipole moment for the neutron which scales like the the strong CP angle so in fact in in in Quantitatively what it looks like is that it's proportional to minus theta plus theta q in our old notation multiplied by the up mass this is in in just a simple calculation, so and I've left out some some sort of QCD scale factors at the front Mu plus MD 2ms minus Mu minus MD There are more refined formulae for this So this this gives you a neutron electric dipole moment Which you can can search for you know, you can put a neutron in a background electric field and see if the energy level split proportional to the spin of the neutron and people have done this and Determined that this term here this pre factor theta plus theta q Or you could call us just theta bar has to be less than or equal to something like 10 to the minus 10 or 10 to the minus 11 Which is telling us that this overall phase that could have been living in and so there's one linear combination Which is physical living in the cork masses and the topological term Has to be extraordinarily small. I think it's actually more like 10 to the minus 11 in these units has to be extraordinarily small So this is known as the the strong CP problem Are there any questions at this stage? Of course, I went very quickly through the instant on stuff, but If there are any questions, yep Oh, sorry, yes, what's in the lecture notes online as well So don't worry too much about taking detailed notes, but M M up plus M down And this is 2m strange minus M up minus M down Okay, so this is a this is an enormous puzzle now It's not like the the hierarchy problem where we are wondering about some some UV completion Which we may be quad quadratically sensitive to something like that This is sort of different because this if you were to somehow set this to zero in the first place set theta plus Theta q this linear combination to zero in the first place It would stay pretty small and in fact the neutron edm. Yep This result is gauge independent and So if you set it small from the outset then then then it would stay small There are other contributions that to the to the neutron edm from for example from the CKM matrix and things like this But they're all small much smaller than this so we don't worry about them so much But the real puzzle is what boundary condition would set that somehow this term Knows so this term knows about this term We already know that there are phases in the quark mass matrix because we've measured it So why would the phases in the quark mass matrix know about the some phase in here such that they cancel their two values? Or just just so that they cancel down to some remarkable degree of precision You could say well there could be some theory at the UV which which sets it so that way and to which my answer would be show me Because we don't really have a convincing story for that So how can we understand how could we solve this problem is could we find some dynamical or some symmetry based Solution to this to the strong CP problem, and there are a number of approaches One has already been mentioned which is the master sub quark solution, so I won't I won't go into much detail, but you can see that if I Had zero mass here, so this term doesn't exist then I can do one rotation on one of the quarks of a value of theta q is equal to minus theta and exactly cancel this guy Without changing the rest of the action An even more dumb way to see that the mass massless up quark solution works or massless quark solution works Is that if I set MU to zero here this vanishes? Reflecting the same physics There's another one known as spontaneous CP violation and The notion here is that if you were to start with imagine it high energy So we're sort of it's like a boundary boundary value problem imagine it very high energies CP is a full symmetry of nature Because if CP were a full symmetry of nature it would forbid the existence of these CP violating terms They would not be they would not be allowed So that would go towards solving the strong CP problem and had we not measured CP violation in in the flavor sector through the CKM phase that would be a very believable solution because if if nature were CP invariant and Then it would be plausible that the UV physics is CP invariant and these guys vanish But the problem is that we have measured CP violation in nature nature isn't CP invariant So what you do so then this is known as as the Nelson Barre mechanism You can look it up The basic idea is that you have a CP symmetric theory at high energies And then to explain the the CP violation that we see at low energies at some point along the way you spontaneously break CP as a symmetry so it's a CP invariant in the UV and It's spontaneously broken at some scale and then it becomes a CP violating theory in the IR But you have to find a way to do that in such a particular way that When you break this CP symmetry spontaneously it doesn't show up in here But you do get a CP phase for the for the CKM the CKM matrix So it's it's hard and none of the models are particularly aesthetically appealing and Sometimes you need to to set some parameters to zero Without having a symmetry for them or or adfield a lot of field content But it is a possibility. I think it's one that people are starting to explore a little bit more these days Then it was sort of put to bed for a while and But it is interesting but difficult and then the the last solution which is by far the most popular is The axiom solution to the strong CP problem Okay, so how does the action work? So I'm going to sketch this again not in a huge amount of detail You can find more detail in the notes, but the idea is to start with PQ symmetry So this is one that lets us Rotate these corks You have to have it somewhere in nature as I said the standard model itself does not have a PQ symmetry so to have a to Realize a PQ symmetry you have to add some extra particles They might be heavy corks, so they might be an extra Higgs doublet and so on such that you can do a PQ transformation on some Coloured fermion somewhere in theory, so you have to add matter content when you do that You can have a PQ symmetry and then it you spontaneously break it No, sorry Peche Quinn, so it was it was Peche and Quinn that wrote this down and There's no the the definition of a PQ symmetry because you can have the very different There are many different models and it can involve The standard model corks or some heavy fields or whatever the definition of PQ symmetry is really one in which you can do Rotation that would have gotten rid of this term. That's the way the way I would define it So you spontaneously break this PQ symmetry at high energies and we already worked with this this The non-linearly realized symmetries this morning. So when we have the spontaneously broken symmetry We will have some non-linear realizations. We'll have some complex scalar field. I'm going to call phi That gets a vacuum expectation value, so this this could be a composite It doesn't have it's you don't needn't think of this is some truly fundamental field And but for for our purposes, let's let's think of it as just a simple scalar field It gets a vacuum expectation value we'll call F there's some radial mode row, which is like the Higgs, but that can be heavy so we can forget about them And we just do the the essentially the ccwz thing we discussed this morning But even more simply because it's just for a u1 symmetry and then we have The exponentiation and we have the axiom living in here, which is the goldstone boson of the spontaneously broken PQ symmetry And it turns out we'll go through the the details a little bit. It turns out that Of course, if this were PQ were an exact symmetry, which it is at the classical level this Goldstone boson would be exactly massless if there's no breaking But it turns out that it gets a mass And it gets a scalar potential which is just right just so that it will When it settles to the bottom of its scalar potential it will exactly cancel the the strong cp angle Okay, so how does that work? So I'm going to just going to show a toy model It's a model that works, but There are many other models on the market. This is just a very simple one. So imagine we start with an action Where we have some higher dimension We have some PQ symmetry such that the we have A complex scalar phi which has charged one under the PQ symmetry We're allowed a some higher dimension operator like this and then we have the normal terms or we have the higgs And and a pair of standard model quarks We can see that if we didn't have this guy, we couldn't perform the axial rotation on the quarks and leave this unchanged But now we can Essentially because when you pick up the Hermitian conjugate and the phase in here We'll mix with the phase in there But now we imagine at some scale this is this is spontaneously broken So this will pick up the the vev out the front. So we end up with just the standard quark mass terms for for the The standard quark yukawa which scale like f over big lambda So I'll just call that the yukawa coupling little lambda In here you have e to the ia over f And um when you take the Hermitian conjugate what this what this means and also when the higgs gets a vev So we'll set so we could say yeah, let's do this. We'll keep the higgs in for one line And then we have um In front of here, I could put in the sorry the the the phase The in the mass matrix as well But for now, let's imagine I've done a carol rotation just to push that all into here So we've got theta q plus theta. So then we've just got e to the i Gamma five a over f psi And then after a lecture week symmetry breaking this just becomes the quark mass something like that And if we wanted we could do a further U1 rotation to pull theta q plus theta that was living out here Back into the quark mass matrix. So then this becomes after that transformation becomes mq psi bar e to the i gamma five And then we've got a for f plus theta plus theta q okay Um, so this is now the the way the the axiom happens to be coupled to whatever quark it was that we chose It could be any of the quarks you like This is just sort of a toy model But below so let's let's pick for example the the the up and down quarks below the qcd scale Um, if these quarks are light they will condense We go back to pions again Below the qcd scale the quarks will condense we get a non-zero vacuum expectation value For this guy here Which we can just call f pi squared m pi squared And this is how you get You do this to get to to study pion physics. You do the ccwz Prescription in here. So we have the goldstones of the quark condensate, which are just the actual pions You do exactly the same procedure that we did this morning I'm going to skip it You should really include the pions in in the quark condensate to do the full axiom mass calculation But if you're interested in that you can read A very nice paper by Giovanni from a few years ago that does it in in a lot of detail But the long story short is that when we put this in and we take the Hermitian conjugate We get a term the Hermitian conjugate of of you know e to the i theta plus e to the minus i theta It's just a cosine. So what we actually get is a term that goes like f pi squared m pi squared a over f Plus theta Plus theta q Sorry cosine So we see when we minimize this potential as a function of the axiom background field we want the So this is this is the Lagrangian. So the the the scalar potential is minus that So when we minus this mind when we minimize the scalar potential the axiom field We'll want to take on a vacuum expectation value, which exactly cancels the strong cp angle And I've kept both of these terms together rather than calling it theta bar or something just to show you that it cancels the the independent terms that could have arisen in the in the first place And because this is exactly cancelled this it's taken a value such that the the the overall argument in here is zero You see that the effective theta term has has vanished. So you get no contribution to the the Neutron edm from this from this guy um So I've gone through it quite quickly Um But the there's a lot more detail in and references and things in the notes But just to recap The basic recipe which which works. I've shown you a toy a toy example of how this happens But the basic recipe is that you start with spontaneously broken u1 u1 petty Quinn You get a massless goldstone boson but because of I should say massless ish because of of the non-perturbative qcd Corrections which you can calculate just directly from pion physics So you might think well, I should start from from this term here Pull the axiom into this term here and calculate, but that would be much more difficult You get exactly the same answer But we can actually take the chiral Lagrangian for pions that we already had and that captures all of the non-perturbative qcd effects so we take the chiral Lagrangian with the cork condensate and Calculate what we get And it turns out that that this is no longer massless It becomes massive because of these non-perturbative effects, but you can very easily calculate the mass and the the the axiom potential Is such that when you minimize go to the bottom of the axiom potential You exactly cancel the the total strong cp angle now I showed this for for a particular model where I was discussing the The the light corks, but you could actually do this in many many different ways say you started with a theory Where you had this scalar field? Phi couple to some other corks without the higgs so not a higher dimension operator Then those other corks would get a mass proportional to f so f could be some very high High scale where the petty quince symmetry is spontaneously broken um And then you might say well how on earth could I calculate the axiom mass from pion physics when the axiom Is now a couple to some other very heavy corks in the theory vector like corks That may live at you know 10 to the 12 gb or something like this But it's actually quite straightforward because what you do is you do a unaxial transformation of those heavy corks And what that does is pulls the axiom all the way into this term here Then you do a unaxial transformation on the light corks And it pulls that the axiom right back into here in front of the light cork Back into the light cork mass terms And then you can just go ahead and calculate the axiom mass in exactly the same way So you use the freedom to shuffle this field around between cork masses and the topological term Um, and you can always pull it back into the cork mass matrix and calculate the axiom mass Okay, um, there's one thing to note which is very interesting Which is that as I said when we spontaneously break a symmetry You get if it's an exact symmetry you get a mass of goldstone boson And the way we see that symmetry being realized in the theory is as a shift symmetry a continuous shift symmetry of that massless goldstone boson Here you can see that there's no longer a shift symmetry The shift symmetry is gone, but what's interesting is you've actually, um You've not entirely gotten rid of it What you have left over is a discrete shift symmetry Which is that a over a goes to a plus 2 pi nf Where n is some integer If I shift the argument of the cosine by, uh, some integer times 2 pi then it all remains the same Okay, so those those are the the sort of the quick basics of um of, uh The strong cp problem and uh the axiom I think tracy showed earlier if you differentiate this twice you get the axiom mass So differentiating it twice you just get f pi squared m pi squared over Um f squared f could be any number. There's no there's no reason that this should um should be uh low energies and the the story goes that Um, you know, Weinberg and wilcheck tried to embed this in a weak skill model where f would be around the weak skill But the model is very very predictive And it predicts all sorts of when you put this in and you include the pions in the the carola grongian and It predicts all sorts of couplings of the axiom to the light mesons as a function of f And so it was very straightforward to go and search for the axiom right from the outset and it wasn't found And there are many many constraints. I can actually I think there's just about time to discuss one Constraint and where you see that f axiom has to be uh, uh very high scale indeed But it was originally Thought to be uh f was thought to be around the weak skill and the axiom would have been Quite light. You see it's f pi squared over f pi m pi over f So if f is around the weak skill this would be down at say an mev or something like that um But then there were two uh after that when it was realized that it couldn't be F couldn't be around the weak skill There were two models that were proposed and there are sort of classes of models known as the ksvz and the dfsv models Where you use heavy corks or heavy scalars and the spontaneous braking can essentially be at any scale you want So f can be can can be anywhere. It's essentially a free parameter um Okay, so I will do a little bit of axiom phenomenology um covering the bits I that um Robert and tracy are not covering as far as I'm aware So one interesting thing is that all of the axiom couplings because it's a goldstone boson then apart from The the non-perturbative qcd effects proportional to f pi and m pi all of its couplings um have to respect the shift symmetry So all of its couplings will be derivative So they will look something like the the interactions will look something like cj over f d mu a So this is this holds true for all of the the the sort of the classical Lagrangian because it's that guy didn't break the the The shift symmetry. It was only the non-perturbative effects proportional to m pi squared which broke the shift symmetry. So to a good approximation um This is what the couplings of the axiom looks like and then the it is interacting with some Some current I'll call it theta j and theta j could be all sorts of things. It could be A cork current an axial cork current it could be A leptone current Different models will will populate theory space to be all sort. There are all sorts of couplings you can have It could be the churn simons current for Electromagnetism I call it churn simons such that when I move the total derivative When I when I include move the total derivative just by integration by parts onto this side I would get ff dual for the photon and of course it can be the the the churn simons current for uh, what has to be the churn simons current for for Qcd as well and so on but the important lesson here and this is so so There's a shift symmetry and that dictates that this guy has to have derivative interactions that there should always be a basis After you do total derivatives in which you will see it has derivative interactions um And that will respect the shift symmetry But another tool that we learned from yesterday is that um, uh, we have a whole bunch of higher dimension operators You see that these are dimension dimension Three currents and this is Dimension two that's dimension one. So these are all higher dimension operators And what that tells us is that amplitudes matrix elements for for various scattering processes Will tend to be proportional to the energy Of the scattering process at hand whatever energy you're producing this axiom at um divided by f To some power and which depends on the process and the you know, how many axiom legs you have attached And uh, how high I mentioned the operator is Okay, so why am I telling you this? So as I said f can be a free parameter F could be a very large scale. It could be 10 to the 12 gv or something like this But the energies that we can access in the lab at most go up to say, uh, uh, a few tv So if this is 10 to the 10 or 10 to the 12 gv And this is 10 to the 3 gv If even if n is one that this this is a very very small number Which is telling you that axioms with large decay constants are very very weakly coupled So the only way to overcome this this very very weak coupling Um is to have some sort of very high luminosity experiment because the couplings are tiny and as i'm sure Uh, uh, robert will have mentioned, you know, if you have tiny couplings The only way to go is to to to discover something is to go to very high luminosity And it turns out that um, there are some very high luminosity experiments In our neighborhood Which are stars Stars are very very big Which means that if something happens in scattering within a star Because there's so many, uh Protons and and electrons in a star and photons in a star This means that because they're very very big Even if that scattering process is extremely rare, you see that this is a very small number Typical energies in a star around like the sun would be around a kv or something like this Even though this is a very very small number Um, you could hope to see observe the effects of axiom scattering Um, uh simply because there's so much stuff in a star that you could produce axioms Um where in such quantities that you have some large number here Or the sum over amplitude squared becomes a large enough number that this um is compensated And this indeed can happen. So in a star you can have processes like Gamma plus z e goes to a plus z e um And if this happens in a star, I'll just Uh call that z e This is just the the total number of some number of electrons Um You could produce an axiom which actually escapes the star so weakly coupled That you've produced it with some low energy say around a kv or something like that Um, but because the the the matrix element for re-scattering is so small it would typically escape the star and and flow out And this could actually can actually lead to a um a source for um cooling the star Which can be more efficient or comparably efficient to the standard model sources So for example, if you had a comparable diagram, there are lots of different diagrams that that can be important here But I'll just I'm just sketching the possibilities here Um processes involving for example photons scattering off of electrons that photon that that you've produced Compared to the the axiom that you produced will be reabsorbed inside the star So we can actually I'm going to do a super back of the envelope Calculation or estimate Which is that the rate for this process we know if if the axiom escapes the star Um, and you're working at some temperature the rate for this process must scale like If we just had n equals one so we're starting with an operator like this guy here for electrons So we have one derivative on the axiom divided by f So n equals one and the cross section goes like e over f squared and we're working at a temperature of around t So t is the sort of typical, um, uh, uh, energy scale within the star Then this process will be proportional to just on dimensional grounds t over f squared Typically multiplied by the the total amount of stuff in the star multiplied by n Similarly, if we do a If we estimate this process here all of the bulk stuff is not going to escape So what escapes can only be happening on the surface And then so so the the cooling rate from processes like this again doing this is extreme extremely sloppy Just to show you how you can use a star to probe axiom physics Um, this process will scale like here my coupling was e over f. That's my my guy with dimensions of coupling So for here we have two, uh, um electromagnetic couplings So I will write it as four pi Alpha just to be careful with my four pies. I didn't put a four pi in here. So I shouldn't put one on here Um scaling like the so not equal to proportional to Um scaling like the surface area of the star which will go something like and the two over three In a star like the sun and is something like 10 to the 57 So what you're seeing here is you have lower slower scaling with n for these sorts of processes Then you do for these sorts of processes and this can help you overcome this even if t over f is a small number So equating these two again just being very very sloppy I'm setting these two equal setting the energy to be around a few kv Our 10 kv if you do that, um, you know the electromagnetic coupling You know what n is you see that these guys are equal roughly equal um whenever F is around 10 to the 5 gv So this was very very sloppy if you're doing you have some project on axiom physics Do not do it like this open up cobalt and turner and uh and calculate everything properly Just on dimensional grounds and scaling with volume versus surface area You can see that you could hope with stars to probe Very large decay constants So you're probing couplings that have been generated at very high energies or sorry with very large decay constants So either low energy and small coupling or order one coupling and high energy and and this means that that uh by searching for axioms in this way There are even more sensitive processes involving white dwarves and and supernova you can hope to search for for axioms and in fact Um Using stellar processes like this you can exclude uh axiom decay constants all the way up to say 10 to the 9 gv or 10 to the 10 gv okay, um So i'm going to leave it there for the axiom for now and uh move on to something else. Are there any questions? Nope. Okay, super. So considering we have 15 minutes left I want to just like at the end of the last lecture I showed you this sort of much more speculative non mainstream theory, which was The the twin Higgs I'd like to do a similar thing here. So take this um With a health disclaimer. This is not You know widely you will not find this in in textbooks. It's not widely accepted but it is something that that um People are thinking about and have been thinking about Um in the model building community for the last uh few years Um, so I just want to sketch it and it involves axioms, but also involves the hierarchy problem. So it sort of connects the two main topics of these lectures Actually, I probably won't need more than that Okay, so the idea the setup is known as the relaxion And the idea is as follows this will again, um, utilize a bunch of the tools that we discussed in the the First lecture So the idea is to combine the strong cp problem and the axion solution with the hierarchy problem and try and find some cosmological explanation for why the Higgs mass might be far below the cutoff of the theory. So as I said before um, if you believe that there's some uv completion for the the standard model where the the um Electro week scale is calculable in terms of the the parameters of the uv completion Just like for the pion. It was calculable in terms of the quark masses and the uh, uh Qcd gauge coupling Um, if you believe that's the story for the standard model Then we have this puzzle because why um is the if that's the case And we've no symmetry in the standard model that can clearly protect the the the Higgs from getting Corrections to its mass just like there was no symmetry for the charged pion when it coupled to the photon Then why is it so light and this is a very uh interesting and uh super creative idea that Peter Graham, David Kaplan and Sergei Regendron put out. I guess it's four four years ago now and their main idea is that As we see in that so on the standard model, there's no symmetry Um, which makes which can justify why the Higgs could be much lighter than the the the scale of the uv completion where the electroweak scale is is generated um There's no symmetry reason but the point where Um Higgs squared is much much less than the cutoff squared Maybe a point of enhanced may not be a point of enhanced symmetry, but it may be special dynamically so How do they do this so the idea is that you will have some cosmological dynamics Which picks out this point in parameter space as a special one um and uh and and then that would the idea is that that would explain why the the Why you have a natural hierarchy So they take the standard model And treat it as an eft So then we do everything we would normally do. There's all the standard model terms And there's some large Expected correction to the the Higgs mass of order the uv scale so the uv scale will be big m So this might be large. It might be you know 10 tv or 100 tv or something like this So you take this this to be uh, uh The theory of nature and we're really taking seriously the notion that the standard model is an eft And then you add a scalar With a shift symmetry You'll see that the scalar sort of wants to be a goldstone boson So these will be the different steps um Couple this scalar to qcd So we have a the the usual term I think you should have a g squared in there 32 I squared f gg joe And you could see if we did one of those chiral rotations um In the the previous what we just what we just discussed that you could rotate this if you started off with some axion like coupling to the corks You do a chiral rotation you pull the the axion like coupling in here. So this is really how the axion couples to gg joe So this so far this story so far is just an axion model um And we saw earlier that perturbatively this this term here is not going to do anything to the axion because it's actually a total Um, this term is a total derivative. So I could move d mu j mu Move the d mu onto phi and just have j mu and you see that phi has a shift symmetry So this is not going to generate any scalar potential for for phi Apart from the non-perturbative uh effects that we already discussed that come out proportional to m pi squared f pi squared and then What they finally do is break the shift symmetry explicitly And they have some spurion. So this is again using all of the tools from the first lecture. They have some spurion g Which is the parameter that breaks the shift symmetry on phi. So you can have now couplings that look like g phi h squared You have couplings that go like g m squared phi Uh couplings that go like on half g squared phi squared Basically everywhere that phi enters with a coupling that breaks the shift symmetry. There should be a g coming along for the ride They chose to Define g as a parameter that has mass dimension one I still don't know why they did that. So i'm going to follow what they did so you can look up the paper But um, you could equally call this epsilon times big m if you wanted and it's all it all the physics is the same And you just write down you follow all the rules of aft you write down everything you're allowed to write down you just go for it and uh and uh See what you get So now let us think about what this what the potential for this scalar looks like um in terms of uh at zero temperature What i'm going to plot is the the Scalar potential as a function of this field phi This is the potential energy contained in the field So we see here from terms like this and we can add all of them suppressed by the the the cutoff higher dimension operators As far up as you want but what we see from this guy here is that we get some sort of smooth potential Doing something As i move as i change phi. This is just some boring polynomial and it will just do boring polynomial things However, as we're changing the the value of phi as i if i change the value of phi You see here that this term is actually changing the background value of the higgs mass squared So i could have my scalar potential without doing any fine tuning such that over here The higgs mass squared is positive But after i've changed so you see this has the opposite sign from the m squared. So as i'm changing phi if i increase phi the the um Higgs mass squared sorry it should be yeah as i increase Phi the higgs mass squared is becoming smaller and smaller and smaller and at some critical point Let's call it the background value of phi is just G phi is equal to minus m squared so Phi is just m squared over G big m squared over g at some critical point the Higgs mass squared the total higgs mass squared as a function of phi will pass through zero and start to become negative What that means is that the higgs mass that the the higgs will start to get a vacuum expectation value So what happens to the scalar potential for phi so that calculation we did earlier We found out that the the scalar potential for phi Goes like this It was just the x the the bit from from qcd was scaling like um f pi squared m pi squared cosine of phi Over f i've just absorbed the background qcd angle into a shift of phi. So i've just shifted this whole thing to set that to zero Which is fine So this goes like f pi squared m pi squared great but m pi squared depends on The the higgs veth as we pointed out yesterday Sorry on monday the the the Quark masses are like a spurion for explicit breaking of the su2 left cross su2 right Which means that the only thing controlling the pion masses has to be the quark mass terms If the quark mass terms went to zero the pion mass would be zero So this potential would actually vanish. So this is actually depending. It's proportional has to be proportional to something that scales like f pi cubed m quark cosine of phi f Which means that it's proportional to f pi cubed Times some quark yukawa's lambda up down whatever The veth of the higgs cosine of phi f Which tells you that to a very good approximation there's tiny corrections to this but to a very good approximation The axion potential Scales linearly with the the higgs veth But what I said was as as we go from left to right on this plot Phi is scanning the the the higgs mass squared which at some point crossed through zero So the higgs vacuum expectation value actually turns on around here And it starts to get bigger and bigger and bigger and as it's starting to get bigger and bigger and bigger The envelope of this cosine because this is just depending on the higgs veth is becoming bigger and bigger and bigger So you cross through this point And it doesn't actually look like this you're coming down There's some smooth function and then all of a sudden you get Wiggles that are growing and growing and growing So now you can see what the what the game will be the game here is to find a way such that you have some initial condition That isn't super fine tuned You just say that the the this field phi the relaxion starts somewhere around here And it will roll and as it's rolling the the it'll roll down its potential And as it's rolling the higgs mass squared will be changing it'll be Becoming smaller and smaller and smaller until the point that it goes through zero and becomes negative And then you want the axiom the relaxion field to get stuck somewhere Soon so stuck somewhere where the higgs mass squared has just gone through zero, but it may have some Some small value. That's the idea So how can this work? Well in order for it to get stuck This guy if we wanted to get stuck right here at the observed value for the the higgs veth Then this is telling you that the total derivative the if it's going to get stuck It means that has to be at a local minimum. What's a local minimum? It's just when the first derivative as a as of the the potential as a function of phi Vanishes, so it's telling you that the first derivative looks sort of like this There are other terms, but they're similar in magnitude. So dv by d phi I will be zero which is telling you that g m squared Is of order f pi squared m pi squared over f squared f sign of phi over f So that's where when it will get stuck Which means that if I choose g to be small Then I can ensure that it will get stuck. This will happen for some angle some phi over f I can ensure that if g is small that this will happen when this overall combination is small And thus it meaning it will happen when the higgs veth is small But the trick one of the tricks is that I can choose g in principle to be from an effective field theory perspective As small as I want because it's the only spurry on that perturbatively breaks the shift symmetry of phi So I can if in my uv completion, whatever it is the only shift symmetry breaking Is is proportional to g then that will be true and it will remain in the the infrared effective field theory as well So this is the game you roll down. There's a local minimum starts to appear if you satisfy this equation um g can be can f is going to have to be you know 10 to the 10 gv or something like this to satisfy all these stellar cooling bonds um But you can then choose g to be very very small such that uh such that you satisfy this Of course, you will have spotted that if this were just flat space the kinetic energy of phi would be conserved So as you roll down, there's no way you can get stuck in the local minimum If you're not doing cosmology and you just plot if you start with phi up here It's going to roll down indeed. There are little there are minima that show up here But um, it has enough energy to go the whole way down and back up the next slope and keep going But if you do this during inflation Do you have some background inflation occurring? Then this actually dumps the motion of phi the the equation motion for phi will be something like Uh phi double dot plus 3h phi dot Is equal to dv by d phi And you see that this is like a dumped harmonic oscillator So this adds a dumping term which actually slows the rolling of phi and it can reach just like for inflation a steady state Rolling where it's rolling at the same velocity the whole time Very smoothly in a controlled manner And then because that velocity can be slow it won't have enough energy to get up over the next the next hump And it will just settle down oscillate around its minimum and then then settle down there So they do this in an inflating background as well Now there are just two minutes a bunch of constraints you have to satisfy So for example, if you're going to scan be able to scan this is Enough to cancel off whatever the original Higgs mass squared was you have to travel through a field distance Which is of order? Delta phi has to be around bigger than m squared over g And the Hubble scale during inflation Has to be larger than m squared over m plank So Hubble just looks like the the square root of the scalar potential divided by m plank And you see that here if you plug in this field distance delta phi Back into this action you see that the total scalar potential height has changed by about m to the four They cut off to the fourth value which means that if This isn't going to be the infotone if there's some other infotone then there must be some contribution To the scalar potential which is roughly constant for the inflation to occur, which is larger than that You need to be able to form The the axiom potential in the first place Which tells you that the the Hubble scale during inflation has to be less than lambda qcd There are two two ways to see this one is that In Decider space you have effective effectively horizon of a given size which is related to the Hubble scale And for those instantons that I described to fit into that horizon And you need the strong coupling scale to be to be Larger than so that the size is the inverse of the strong coupling scale the strong coupling scale to be larger than The Hubble scale and it turns out that those instantons that dominate the contributions that Generate the axiom potential are in the ballpark of where the strong coupling is strong, which is around lambda qcd The other way to see this is that during In decider you have a sort of an effective temperature the Gibbons-Hulking temperature Which means that you can think of this as being like a finite temperature system And the axiom potential the instantons get washed out in the axiom potential if you're at high temperatures So another way of seeing this is that you can think of this as being like high temperature And if the temperature were too high you wouldn't generate the axiom potential at all Those are two ways of seeing it another constraint is that the Hubble scale during inflation Has to be less than g m squared to the third The reason for this is that you have fluctuations of the scalar field in decider Which are of order the Hubble scale so the scalar field You can think of it as fluctuating sort of in a quantum mechanical sense as it's rolling down You can't quite pin it down. It's fluctuating or you can even think of it as sort of being like finite temperature It's fluctuating around But if those fluctuations are in a given time period are larger than the distance It would have rolled during that time period classically down the slope Then you've completely lost predictivity because this thing is fluctuating more than it's rolling So you're you're just going to populate the all of the different Hubble patches that you get on the end of the day will have the scalar field five spread all over the show And there'll be some non zero but small probability that you find yourself in this point If you satisfy this classical versus quantum constraint Then this classical rolling is dominating it. So that will dominate where you expect to to land I'll come back to that in a second And and then you have to satisfy this that you can form a minimum in the first place for an order one angle so when you do this If you try and maximize the cutoff You get that the cutoff of this theory has to be less than about 10 to the seven gv's Times 10 to the nine gv over f to the one over six Um g has to be something in in dimensions in in gv Something like 10 to the minus. I think it was 26 gv So this isn't just a small number. This is an extremely small number You know if the cutoff is 10 to the seven Uh gv, then you're saying that if you are to write this as a dimensionless number epsilon then it would be something like 10 to the minus 33 Um the field range that you traverse is of order delta phi is of order 10 to the 40 gv Uh, which isn't just super plank in it's super duper plank in You know we worry about super plank in field excursions for scalar fields We gravity conjecture all of that sort of stuff This goes this blows all of that out out of the water. It's an enormous field excursion um and the number of E-foldings you require for this thing to have time to roll all the way down and do this is something like 10 to the 43 Now you may say who cares we see numbers like this in inflation all the time. This is the number of e-foldings This is in the exponent This is e to the 10 of the 43 is this growth of the scale factor Um There's some model building questions Raised by these parameters and these the the e-foldings the inflationary story Furthermore, um, so there are three three issues. So those are model building issues Some people don't care about it. Some people care about it a lot Um Another thing is that in this example that I showed it will stop It will only stop whenever theta over f isn't zero if this were zero It couldn't stop because this you would never balance this So it can only stop when theta over f is non zero, which means you have a non zero effective strong cp angle So this model actually predicts a large neutron edm this very simple model So you have to add extra ingredients to get around that And then the last thing um, which is more the last two things which are sort of more qualitative Or that uh, first of all if you've landed in this vacuum This also happens to coincide With the vacuum that has zero total energy or very close to zero total energy because we have a small observe a small cosmological constant now so in this scenario, um While for things like supersymmetry you may imagine there's some mechanism to relax the cosmological constant to zero at some late times This one, um, it changes it because it has to happen at very late times In the sense that this has to have known the some interplay now between the the cosmological constant And where you size the cosmological constant where you happen to land because had you landed here The weak skill wouldn't be much different But the total vacuum energy would be different by about lambda qcd to the fourth part Which is uh a gv essentially um So there's an extra puzzle with with um uh in terms of Um, uh the strong the strong tying this to the the cosmological constant problem and a final, uh issue which again some people are very worried about some people don't don't care about it at all is that if you embed this in an inflationary Epoch then basically all the things that can happen tend to happen So you can populate This this scalar field phi if inflation is happening for such a long time Um, it could have been populated um all over this this vacuum So then there's a question questions arise why why you didn't end up way way down here Or or or some other place about really how well you can calculate the prediction that you land here within some very long inflating background and within sort of a multiverse sort of Set up nonetheless I wanted to to tell you about this today in the last these last 15 minutes because I think this at least Illustrates a very interesting class of approaches that people are starting to develop now So this is one of now a handful of sort of models That try to use cosmology that mean that may be these symmetry based approaches that for the the electro weak hierarchy problem the the The origin of the the weak scale The symmetry based approaches that we've studied so far like the composite the pseudogolstone goes on Higgs approach And and other ones we can discuss tomorrow or on friday Maybe they're barking up the wrong tree. Maybe there's some very non trivial cosmological dynamics That actually picks out an interesting point Where you get a small weak scale? Because of some interplay between cosmological dynamics and the structure of the standard model. So I think this is a very interesting idea as I said, it's not Widely accepted as being a definite solution to the hierarchy problem that depends on people's own perspective But it certainly shows a very interesting class of of approaches that are now starting to get to be developed And I think I should leave it there Thank you very much