 Zdaj sem očustila, da je bilo včasno, da smo prišli v primeru, zelo, skupovati, karakterizacijo kompositivno scenari, v tem, da je tudi sekta, ko je kompositivno sekta in vse elementarje, in vse interakcije. izgledaj. Tako, da bomo so vse zelo zelo pridike, da mi nekaj, atukaj da se pravim proveri, kot sredanje korist, je, da veliko se konfajmenja srežina. H, ok. In tudi hiz, je tudi nambugorstvom bozon, tudi tudi brejting. Ok. Zato, skala, kaj je brejting, tudi je nekaj konek, kaj je tudi brejting, kaj je brejting, tudi skala, kaj je brejting, tudi brejting, tudi brejting, tudi brejting, tudi brejting, ok. Zato nekaj konek. Hvala, da se vzit, kaj je F, skala, srednje parametra in tudi brejting, nambugorstvom, tudi brejting. V zelo, nekaj se bozon, kaj je zelo, kaj je tudi brejting, kaj je konek, kaj je Fisix izgleda tudi mstar, kaj je več več več energija, je TV, ali je infrared, kaj je več več energija. Zdaj smo prišli všeč na teori. Zdaj sem vse predstavljeni Fisix, kaj je tudi tudi spontanusimelj, in znamenjo, kako bi možemo obtajati tudi vseh bozon in ampugolston bozon, in nekaj je zelo, da nekaj je vrkazib, in pa je zelo, ki je zelo, that the mechanism of vacuum misalignment. So, the thing is as follows, so there is a global group G, and it has of the composite sector, it has a sub group to which it will be broken, which is called H, da je vse elektroviko vsega vsega. Zato je to vsega, da je vsega. Kroče, je vsega iberčača. Zelo se zelo izgleda na vsega vsega. In da se vsega vsega, da je vsega, da je vsega.uen drive, kjer Skolješ, napravlj the set of generators of the Group-G and let's split them into two sets, as you typically do when you see a spontaneous-stim dassom breakdown. The first set is called the t-little A, and the second is the AT-A-AT. The first set is a number of kaksi announcement that put together. They are capable to generate the algebra of the subgroup H. And one we are going to call them the unbroken generators. While these other ones are the broken generators. The unbroken generators define the theatra of H, while the broken are allwhat is left. And we call them, Those that define the coset space z程u h As you understand the choice of the generator's basis in a theory, which is perfectly invariant, is completely arbitrary, it's like a choice of a reference system Ok? However we know that this g symmetry is eventually going to be broken by several effects. among which by the fact that we want to to, ki se vidjem, boleh je tako, tako g generator in hreč, da jim hrečenim elektrovikem. In posredi z državcij grečenih generatora, kaj igraš vzicega z večnega kriče, in zelo, ki hija neč ovo, lež이š taj ampelok, je naredilo, še naredilo, da imamo, ki je od nekaj težek, in neč dobro, leži nekaj frasil nekaj. Ispečen in staj željno, da elektrovarjne generatorje, in nekaj, da je dobrovršen, na način, je nekaj vzelačen, ali nekaj obršen, da je tačnja. In tako, da nekaj komponent nekaj, da smo vsega vzelačen. Zde vsega, elektrovarjne, the belongs is inside age. Of course, all these are reversed. Contains A and contains H. So we can also define, given these basis of the generators, what we will call a reference vacuum. So our fields will be... our theory will have plenty of operators, among which there will be scalar operators, composite or elementary doesn't matter, so those who can take a web, and this defines representation of the group, and inside this space of field we can take a special direction, f, which is defined such that ta applied to f is equal to zero, and ta at applied to f is different from zero. So the reference vacuum is defined in such a way that if the true web of the field phi had to be parallel, had to coincide with f, so if the true web is unlined with f, then what happens is that f does not break really what we call the unbroken generator, it breaks only the broken ones, and since the electro-wick generators are part of these ones, the electro-wick group is unbroken. So if the true web of the theory coincides with the reference vacuum, there is no electro-wick symmetry breaking. Of course it's not what's going to happen, and the amount by which the true vacuum is not parallel to f is exactly the vacuum misalignment we are talking about. And it induces electro-wick symmetry breaking. That's why here I said quote and quote unbroken and quote and quote broken because actually this way of splitting the generators does not correspond to what the vacuum will decide to be. It's just a reference system which is defined through the reference vacuum. So the reference vacuum is useful also for another thing. So we can use the reference vacuum to parametraze the fluctuation that corresponds to the massless-gulstombosons. So if I take the reference vacuum which is one of the life vacuum of the theory in the case in which there is no explicit breaking, they are all the same, and I act on it with an element of g along the broken generators, I end up with something like this into the i theta a hat t hat a hat where this theta written here are fields which depends on x and as you know from the general proof of the Golston theorem those are massless fields. They are massless fields because if I set them to constants the configuration I'm studying is actually equivalent to the vacuum so it has the same energy which means that constant thetas do not cost any energy which means that constant theta so that theta has no potential. It has no mass, it's a massless state and what we call the Golstombos. Similarly, so a priori in a theory with a completely exact g group this theta fields have no mass, they have no potential so they are flat direction, so their web spans all possible vodka which are obtained by f rotating but then we should imagine that actually we are introducing explicit breaking so there will be a potential for theta being generated and so from this potential of theta you will get actually a definite value for the vacuum expectation value of theta a and so you will be able to determine the true vacuum web of phi equal to e to the i web of theta a hat t hat a hat and now you start seeing what happens so if the web of the Golstons happen to be zero so the true vacuum is aligned with f and then there is no electroveximity breaking because the web is only breaking the generators which were orthogonal which were not containing the electrovec group the electrovec group was entirely inside these ones but if theta takes a web which is different from zero then you do break the electrovecimetry ok so theta defined in this way behaves exactly like the x field of the standard model if it takes a web it breaks the symmetry ok and so this parameterization is the simplest one to appreciate how a Golston boson theta field can actually behave like the x ok we can draw a picture of this so this vector is f ok and the norm of this vector is the order parameter for the g2h breaking what we called before little f and orthogonal to this there is the plane of h transformation so h is made of transformation which are orthogonal which lives invariant to this f and we say that inside h there is the electrovec group so you can imagine the electrovec generators as again acting as transformation in this plane and then so this is the reference vacuum is not the true vacuum the true vacuum is this one is a vector with the same norm f but with an angle ok with respect to f and this angle is set by web of theta so let me be sloppy here and call web of theta now the modulus theta has different components ok as one component for each broken generator let's say modulus of theta so this measures the angle between this and that so by trigonometric we can understand how much is broken the electrovec group so again if theta is 0 no breaking because I can freely rotate if theta is a little bit misalined then all what matters for the breaking is the projection of the true vacuum on this plane which makes an amount of breaking v v is the is what we would like to be 246 which means the electrovecimery breaking order parameter which is the other mind as v equal to f sine web of theta ok so that's a generic theory of vacuum misalignment you do find a theory where where the goldstone bosons behave like the hicks so they trigger their bad triggers electrovecimery breaking and the amount of electrovecimery breaking that you have ok is not fixed so it's related with the amount of breaking f of the full g2h times a free parameter which is the misalignment angle theta ok this misalignment angle is also called defined in terms of another parameter which is called xi so let me introduce it for future use xi is v2 over f square and so it is just the sine square of web theta ok and you see it measures the square of the hierarchy that we can have in the theory between v 246 and f ok so depending on what is xi depending on whether v over f is larger or smaller so is equal to 1 or is much smaller than 1 you get completely different say results and phenomenology so if xi is close to 1 so let's say it's just equal to 1 ok what this means it means that geometrically the true web is pointing in a random direction with respect to f so somehow it made no sense at all to define an f and to identify broken and unbroken in that way because the web is in a direction that breaks if you want at the same level both the a at the electric week generators which are only with ta so for xi equal to 1 the break the size of the breaking of the electric week symmetry is just comparable with the size of the breaking of the extra generators so somehow is like not having put at all the extra symmetries and indeed this xi equal to 1 limit is called the Technicolor limit because remember what was happening if you know Technicolor, in Technicolor you were having a theory which had a group g a global symmetry group g which contained the electric week group but h the subgroup was too small was only the u1 electromagnetic so the breaking g to h was directly breaking the electric week symmetry ok there was no intermediate h and here is happening just the same thing because the web is after all aligned in a random way 90 degree such that the web is breaking the electric week symmetry directly so the xi equal to 1 limit is not interesting in composites because it is just adding more complication to Technicolor theories and the case which is instead interesting is when xi is smaller than 1 or let's say much smaller than 1 so when there is a gap between b and f so you can consider this in in a limiting situation so v has to be fixed because v is 246 is the one we need for the realistic electric week symmetry breaking that you observed but still if xi is going to 0 so we can achieve xi going to 0 by f going to infinity so remind you, remember that f and m star are not the same thing m star was the confinement scale but they are related so basically now if I allow for xi to be very very small I can consider a limiting which I keep all the electric week physics at the same place while the confinement scale and f they just grow and go to infinity and so they eventually the couple so and indeed in the original paper by this was an example to show how you could get the microscopic origin of the electric week theory with xi let's say 10 to the minus 5 far from what could be observed at that time and even today by this decoupling limit by this decoupling limit and so given that there is this decoupling limit we do expect that in this decoupling limit all what is left in the theory is the Higgs boson with properties that must be similar to those of a normalizable Lagrangian of any Higgs boson because if you can generate a gap it means that there is a range of energies in which you can live with the theory in which there is only the Higgs plus of course the electric weak bosons and the fermions that we will introduce on top and there is nothing else up to very high energy scales and the only way for this big gap to be generated is if the theory that you have reduces to the standard model you know very well that if you start modifying, tilting a little bit the standard model couplings then you encounter say inconsistencies and the cutoff which can be at low scale so if you want to really reach this decoupling you know that in between these two scales that's gonna be the standard model now we are gonna verify this in an example in a while but please keep in mind that it is a generic expectation in all these scenarios that all the couplings all the couplings of the Higgs but not only also the couplings of the electric weak bosons so all the couplings in the electric weak and also in the fermion sector of composite Higgs particles so particles with composite Higgs origin are actually similar are actually equal to those of the standard model up to corrections of order psi and this is what makes the advantage of composite Higgs with respect to technicolor because in technicolor psi is 1 so these corrections are really always over the 1 which means that you can measure the coupling of the Higgs to vector boson and find that it is 2 in the units where the standard model is 1 so this cannot occur in composite Higgs because there is this limit that you can take in which the new physics decouples and you are left only with the standard model always gonna be the case in all models of composite Higgs and again now we are gonna do a very simple example to verify this please let me say this one second because it's important it's a bit complicated but it's important so when we speak about vacuum misalignment we have to understand misalignment with respect to what so one of the two things is the true value so that's the true value and the other thing must be well, must be a reference must be a reference system and the right reference system is this vacuum F this vacuum F is nothing it's simply the direction of the vacuum in which electro symmetry would not be broken yeah, question was about the reference vacuum the reference vacuum is again, it's like setting a reference system in the Lie algebra only that if you do it in a complicated way you get a very complicated and there are papers where it's done in a different way, which makes things completely messy, please vector space which we define the group or... which object? F like in the general theory of spontaneously broken symmetries imagine you have all your fields all your theory in particular you need the bosonic ones which can take a web in a Lorentz invariant way in a logical way only those which can take a web and this will form a big multiplet as big as you like irreducible representation doesn't matter all this is all algebra and this is one possible configuration in that field space thank you so I hope this is clear but I'm sure it will become even more clear with the example that follows but sorry before this before this so it's important you understand clearly what we are doing so we are we have found a unique parameter psi that is capable to control all the possible deviations of the predictions that will come in a composite x theory with respect to the standard model actually I should be more precise because you may define composite x theories which contain more than one x doublet for instance you can do the composite 2x doublet model or a doublet plus a scalar so in this case of course the coupling will not just go to the standard model but they will go to the normalizable corresponding normalizable weekly coupled theory of 2x doublet model or of extra singlets or whatever you want there is a question you are asking what I need the group G for if it was not for the presence of the group G there would not be any broken generators and so any gorshton bosons to talk about I break them by the web of the gorshton bosons which live in the cos g over h and also if you want so these generators here which is the fact that these guys are gorshtons is essential to make the Higgs light this is what I say in the first lecture if I don't have this part there is no protection there is no protection for the Higgs mass again the important thing the important thing was that it's true I have this parameter for a model builder used to zillion of constraints from all possible place that can invalidate your theory having one such such a parameter is great because you know that you can systematically recover the standard model on the other hand it's obvious that when you do this it's clear that you are hiding your extra physics so differently from the original say ideological very good paper by Georgian Kaplan here we don't want psi to be 10-3 or 10-5 because we believe even though there is not conclusive evidence that in the models we do have setting psi small is a fine tuning so in principle psi is defined by the sine square of a web of something this something can go on the whole circle so you imagine that this sine goes from 0 to 1 so in average it will be one half well one over whatever so if I want instead the psi of the order of 0.1 as of now is what we think we need 0.1 or 0.2 then I have to accept that something strange has happened by an amount which we can call fine tuning which has to occur in some place in which place in the X potential because psi is fixed by the web of theta which in turn is fixed by this potential that you will generate it will have three parameters which we will be able to adjust but this will cause a certain amount of fine tuning so and there are no models in which the fine tuning is less than one over psi so given psi you can quantify how much tuning you are doing in this theory on the other hand is true that model building is full of plenty of possibilities perhaps in model building that we have not yet explored and so if the X potential has some particular peculiar structure because of symmetry of selection rules and such that it has a mass term which is parametrically smaller than the quartic you may try to naturally accommodate a small psi but as far as I know there is no working example of this so for the time being when you say psi equal to 0.1 you have to imagine of making a tuning of at least 10 questions Does that mean we can never rule our technique color by experiment? It's not technical Sorry Composite Hicks No, in the same very same way you can never rule out to estimate So the psi doesn't have any implications for the only implication it has there is some fine tuning involved and if we accept there is fine tuning Composite Hicks will always survive You can go on your rule life Sorry I just don't understand how you define this field in terms of the vacuum but then the other term is just broken like the goeson boson belongs to the broken parameter I don't get how Let's say let me try Well the simplest example is when you have the theory we are going to show in a second is the theory of scalar fields A scalar multiplet in certain representation of this g group Now this has a family of vacua The vacuum of this is not only one because the g symmetry as long as the g symmetry is an exact symmetry you can always act on the vacuum with the rotation on the group and you will get the same energy so you will get another vacuum So If you now try to if you want in the lowest mass fluctuation of this field phi you can parameterize them in this way So you take one vector any of the possible vacua and you act on them with a similar transformation If you do a similar transformation on the H generators nothing changes because this F was invariant under the similar transformation If you act with the broken generators you do have a different field configuration which is the one that the gore stones populate when they are turned on So it is the answer if you want for the gore stones Yeah So I was kind of trying to draw an analogy with what happens in the standard model So in the standard model the vacuum expectation value can be always chosen to be in the real direction by the kind of residual symmetry I mean If you say that this G breaks to H and then this vacuum has this symmetry this phi goes to exponential something times F Can't we use that symmetry to always align the wave direction with F I mean in that case the gy will not be a physical parameter So exactly So if the symmetry is completely unbroken So if G is exact then two things happen When I say that H that the gore stones have no potential so there is no preference value for this wave of theta and second that this wave of theta is unphysical exactly because what you say because you can always absorb the wave of theta by the definition inside phi So the only reason why theta instead gets a potential and gets physical is because there is the explicit breaking If you want if I don't know where to project here so here I get electroic symmetry breaking because I know I am projecting on this plane and why exactly on this plane because this is the plane where there is the electroic group not in this other plane I understand that this B breaks the electric symmetry but my question is that can't we make always that angle 0 by using that symmetry If it is not an exact symmetry when you do this operation you will reintroduce theta somewhere else So you are introducing an explicit breaking term for the G in the potential The explicit breaking is the fact that we identify part of this generator with the electroic generators so in concrete the fact that we gauge the W in the Z boson I will show you when I build an example in one second In this example there will be three generators one will be gauged and this breaks the symmetry Can you still hear me? The microphone looks No Ok, now I tell you the example if it doesn't work So if you will still have question after this example you can please ask me, ok? So the example is the simplest possible Higgs model which is the abelian abelian abelian of course composite Higgs So I will show you now how to construct a model I mean similar to the usual U1 X model that you study in all textbooks, ok? So this is done by taking an SO3 global symmetry group and imagine having a system that breaks this spontaneously to SO2 SO2 is just U1 so this U1 is going to be the U1 gauge symmetry of the abelian X model So you can do this in the simplest possible way by a representative version of the composite sector it's not a good representative, ok? It's just one example theory that does like this the theory of a real triplet of fields which transform under SO3 as a triplet, so phi goes to g phi g belongs to SO3 Next you will write down the simplest possible Lagrangian for this field and you will regard this Lagrangian as a representative Lagrangian for how say the composite sector could look like Again it's very different from a strongly interacting complicated theory it's just an example So you will have a kinetic term d mu phi transpose d mu phi and then you will have minus some potential, g star square over 8 phi transpose phi minus f square ok? So, sorry and I will use it only once and by the way this is called g star this is called g star and and ok, let me tell you what g star means So the vacuum in these three space now you see what is the space defined off the space of the field phi was a general thing in the previous part and now it's just 3 real fields so the space is R3 and inside this space of field configuration there is the family of the vacua which is just the two sphere because the minimizing this potential leads to a family of vacua characterized by the web of phi modulus equal to f and see so g star square over 8 for normalization ok and here there is phi transpose phi minus f square I will now I have to choose the representative vacuum ok? Let me choose it in the following way so I work in a field basis where f is just 0 0 f ok? So, associated with this reference vacuum there is the choice of the generators which are like this so I will take one generator which is unbroken and it's called t and it's 1 over root square of 2 and it's block diagonal so it's sigma 2 0 0 0 ok? It's a 3 by 3 matrix and then there are two broken generators the first one is t at 1 which is 1 over root square of 2 so the rotation in the other direction this is the rotation in the 1 2 direction and this one for instance 0 0 minus I 0 0 0 I 0 0 is the rotation in the 1 3 direction and then similarly I will have t at 2 t at 2 which will look exactly the same just that minus I goes down here and I goes here ok? So, there are a number of broken generators which are these two I can find how many Gorshton bosons I have I have two Gorshton bosons two real Gorshton bosons ok? Which combined together will make one complex field and this one complex field is gonna be the complex 6 field of the Abelianics model so to study this theory and to see exactly where the Higgs comes from you have studied I mean you should know so to change field basis so rather than using phi 1, phi 2 and phi 3, the components of phi it's better to go to the nonlinear basis in which phi is described as follows so you you start from a reference configuration which is 0 0 f plus sigma of x ok? and this is a rotation so you apply to this e to the i square root of 2 over f pi i of x t at i ok? So, what is this? So, this is exactly what I wrote in the previous slide so if I set sigma to 0 here I just have the representative vacuum on which I act with a broken symmetry transformation and this is allows me to describe 2 degrees of freedom of the 3 that I had inside phi ok? which basically correspond in the 3 space this is the angle I form with respect to this vector 0 0 1 so I move in this way and then there is sigma which controls the norm of the vector it's the radial mold and that's not gonna be a goldstone bosom I already see that sigma is gonna be a massive a massive particle that given that I'm interested in describing something that looks like a composite sector I will call it a resonance ok? while these ones are the goldstone bosoms and they're gonna be massless so the non-bosom bosoms ok? so you can really work out this exponential explicitly it's just a bit complicated matrix I'm not gonna write it down instead I will write down so this is starting from that Lagrangian you can use this nonlinear basis and you can describe of course exactly the same physics in terms of the variables which are sigma and pi and the Lagrangian becomes like this I want to write it once to illustrate a couple of things so there's gonna be one half d sigma square minus g star f square over 2 sigma square so I'm gonna get a kinetic term and the master for sigma and take note of this formula that's gonna be useful in what follows the mass of the sigma which is a resonance so sigma has a mass which I expect to be of the order of the confinement scale so let me call sigma equal to m star this mass is given by g star f so it's different from zero can be arbitrarily heavy if f is large or if g star is very large gaping then there are some interactions like for instance g star square f I will use this interaction at a certain point so just take note for the moment sigma cube minus g star square over 8 sigma to the fourth and then I have a rather complicated Lagrangian which involves sigma in the Goldstones which is like this there is one half there is one plus sigma over f square and then we get again this structure here so f square over pi square pi is the norm of the pi vector there are 2 pi so pi is the square root of pi squared so this is pi squared sine squared pi over f d pi modulo square there is other complicated terms like f squared over 4 pi to the fourth pi squared over f squared minus sine squared pi over f d derivative of pi squared itself everything squared so let's not take a couple of simple things about this Lagrangian first of all if you expand it around pi equal to zero you find that it has no singularity as it has to have in spite of having one where pi is in all places so it just gives a kinetic third for pi it gives no master for pi obviously because we know in general that this is a number works on boson so it has no mass and it gives a number of interactions and also know this one thing that all these complicated polynomial and nonlinear functions of pi are actually always controlled by the ratio pi over f so the Gorsons appear in the Lagrangian basically as pi over f up to an overall multiplicative factor and this happens because the Gorsons appear through this matrix here which has been written as pi over f and the reason why I have written it as pi over f is because I want pi to be canonical in normalized in the previous example I was calling if you want all these exponents to be theta it was a dimensionless field and I wanted dimension one scalar field so I normalize it by f just one word of notation this is exponential so e to the i square root of 2 pi pi t at i is called the Gorson matrix and it is denoted as u of pi and you see yes please it depends only on pi over f there is an f square to pi to the fourth because you see there is an f square which comes from the pre facto so the Lagrangian is I'm not sure I understand what you're doing so you have an SO3 group and you have a field in it and will I be right in saying you have two directions of vacuum expectation value and something like fixed angle between them and these signals you see the vacuum is a sphere because it's controlled by it's a two sphere just the norm of pi is fixed so the space of the sphere has two coordinates which are the pi and then there is one radial mode ok the radial mode is equal so you have one direction where you have non-zero vacuum expectation value and then you express this in then I simply this is just a field redefinition this is a field redefinition it's a complicated field redefinition that is needed to illustrate a couple of things ok and this vacuum is alignment when you get this angle between one field the vacuum is alignment here you will have to do with the web of pi when pi gets a web so the web so web of pi equal to f is the reference vacuum and if web of pi is different from zero then the web doesn't point along f but it's a little bit tilt ok so web of pi is the misalignment angle so at the end you can break to s2 and then to u1 no s2 is u1 ok so at the end of everything I will have broken everything because exactly I want to build the u1 in this model in which the u1 is broken ok no here I am writing just 2 pi1 and pi2 2 real fields oh this is pi squared so this is the norm of pi so it's square root of pi12 plus pi2 squared yeah it's the model ok ok so so there is a couple of so let me so a couple of messages which come from this Lagrangian so one is that the Gorshton are massless the second is that the Gorshton interaction are all up to a proportionality factor of f2 controlled by pi over f there is pi over f in all places and the other the Gorshton Lagrangian appears very complicated nonlinear ok and it's clear that all these nonlinearities will eventually trigger new physics effect in the X-caplex and this is what we are going to compute in a second ok what is less trivial when you look at this Lagrangian is that this Lagrangian is complicated but it's extremely symmetric Lagrangian ok it has a number of symmetries and basically 3 symmetries that correspond to all to each of the 3 SO3 generators we started from because this Lagrangian as I told you is just a rewriting in a different coordinate for the fields of the original Lagrangian that add this symmetry ok with a poorly chosen name G for the transformation ok so all these G's will have something some effect also here and the reason why the Lagrangian is so complicated and it's difficult to identify all these symmetries is that the symmetries now act on pi in a complicated called nonlinear way so are nonlinear realized on pi it works like this so we wrote phi look at phi phi is a Gorshton matrix which is this exponential of matrices pi which depends on the Gorshtons applied to applied to f plus zero zero f plus sigma ok so suppose suppose operating on pi with the following transformation so take pi send it to of G G is an element of SO3 so this map defines the representation so the rule to change pi into something else depending on what G is so this is the representation of the SO3 group that is defined implicitly in the following way so it is such that U of pi G is equal to G but U this is matrix multiplication U is just a 3 by 3 matrix like G U dot pi times H to the minus 1 so this H to the minus 1 is another matrix it depends on pi and on G and the important property of this H matrix is that it is not a full SO3 matrix but it is just a matrix that lives in the SO2 ok so H belongs to SO2 or if you want to see this explicitly H is a block diagonal matrix made of here some liter H which is basically the exponential e to the i some psi sigma 2 sigma 2 is the rotation of the 2 by 2 and here is 0 0 0 0 1 it is possible to show that so it is not trivial so it is not trivial that I can find an action so I can find a rule to change pi into pi G such that the Gorsom matrix changes into this way but it is true for any pi G and H and by the way is the starting point for a very brilliant construction called Callen-Colem and Bessen-Zumino construction of non-linearized C matrix ok so this thing exists so if this operation exists then I can act on this I can act with this operation on phi and look what I find I find phi that goes into U of pi G phi which is just G U of pi times H to the minus 1 applied to 0 0 F plus sigma ok but this one has zeros in the first 2 component so it doesn't care of an SO2 rotation and so H is like it was never there and then the result of this transformation on pi while leaving sigma invariant is just means that phi goes to G dot phi so it is a symmetry transformation ok it is just one of the SO3 symmetry transformation so it exists this complicated rule that well that allow us to change pi leaving the Lagrangian invariant so it's a perfect symmetry of the Lagrangian ok so Lagrangian here is the most general possible two derivative Lagrangian here there are two derivatives here also here ok that you can write compatibly with this complicated non-linearized symmetries there are cases in which this transformation actually becomes very simple so if I consider a transformation an element G inside of taking a generic one in SO3 let me take G equal to GH to the SO2 subgroup ok so in this case you can show that actually the H matrix is just GH itself ok so in this case you can work this out explicitly and find that the transformation you need is only a rotation so pi that goes to pi as a doublet that goes like e to the i some say zeta pi this is the way in which the Gorsons transform under the Brogdon transformation the general rule is the following you take the adjoint of SO3 and you see that it decomposes into a1 under the SO2 group plus another representation which is the doublet of SO2 and if you look at this other representation is how the Gorsons transform ok so that's a general result from the Gorson theorem if you have the adjoint of G you decompose it into H you find necessarily the adjoint of H plus something else this something else is how the Gorsons transform ok and in this case they just form a doublet of SO2 and this you can verify immediately that the Lagrangian is invariant it's all function of d pi squared pi squared it's a bit more complicated the case in which the transformation do not belong to H I can work it out at infinitesimal level for you so if G is like the identity plus i so let's make a transformation along the Brogdon generators alpha i at t at oh so i t at i ok so these are two transformation parameters associated to the Brogdon generator 1 and to the Brogdon generator 2 I can write for you the full formula which is that pi this transformation pi to pi g becomes pi that goes to pi plus so that's the modulus of pi cotangent pi over f alpha plus f over pi minus cotangent pi over f then there is alpha scalar pi and then pi over pi ok so if you are bold enough you can take this operation do it on the Lagrangian and check that it's indeed invariant ok even if of course at the first time at first time you would not have said and this is this transformation that constrains the form of that Lagrangian to be like that, to be unique it's interesting also to look at this formula in in the small pi field limit ok because this becomes pi plus if you expand pi cotangent alpha you get f alpha plus higher orders in pi square so if you are at low pi you can view this transformation just as a shift of pi see yeah it's strange it's f over pi this node so wherever if you like ok so that's something you may have heard about this pi goes to pi plus something which is proportional to the transformation parameter that's the shift symmetry ok and in particular the shift symmetry is what guarantees that pi cannot have any mass and any potential and nothing else because a non derivative term will never survive becoming invariant under this transformation so that's the basic of the formalism of non linearly aligned symmetries that we are going to use in the next lecture now I want to complete the construction of the a million compositing theory so now we want to introduce the gauge fields so we need the u1 gauge field and so we have to choose one of these three generators to be our to be identified with our u1 u1 gauge symmetry given the way in which we parameterize the generators we have already decided actually that the generator of the u1 which is going to be made local is the unbroken one so it's t, the one that was rotating one into two ok and then I can change my Lagrangian into the final result by just by just introducing a gauge field and coupling it to this generator so I just promote d mu phi to a covariant derivative which becomes d mu phi equal d mu minus i square root of 2 some electric charge which will be the coupling of this gauge field which I am introducing now square root of 2 normalization amu t ok so now the system becomes a triplet coupled to a u1 gauge field that gauge is only one of these triestot regenerator so it's clear that this theory now doesn't have anymore an exact g symmetry so you could immediately guess how this Lagrangian becomes it will basically gain covariant derivatives in many places to make things even more familiar with respect to the usual abenianics model I can also use switch to complex notation in which a complex x doublet is defined by the two component of pi1 minus i pi2 over square root of 2 ok so pi1 and pi2 they rotate with each other which means that h gets a phase under the u1 transformation so it just transforms like you expect for the normal x doublet then I can complete the exercise by writing down the Lagrangian I get for this object so the Lagrangian for the hicks now I'm setting the sigma resonance to zero you could reintroduz it in a second given the previous formula so it's just the same as before only in the complex notation so f square over 2 modulus of h square sine square square root of 2 web of h over f d mu h modulus square ok so that's basically a more complicated version of the usual d mu h square that you find that you write down in a normalizable x model Lagrangian the main difference is that rather than having one here I have this sine square over h square which however as you see for h small just becomes one another term which is f square over 8 modulus of h to the fourth then there is 2 h modulus square over f square and then there is some other trigonometric function ok so like minus sine of 2 square root of 2 modulus of h over f d mu h modulus square square ok the covariant derivative here so actually this is not covariant derivative but square which is a singlet so you don't need to put any covariant derivative and that's just the ordinary d mu of the x d mu minus i e a mu so let's now to verify here a couple of things I told you so first of all trying to expand for f going to infinity to try to expand this Lagrangian for f going to infinity you immediately realize that you just end up with ordinary renormalizable abelianics model Lagrangian so dh square and the reason is obvious because I told you that the x can only enter up to a multiplicative factor as h over f so if I start sending f to infinity all the complicated non-linearities they just drop and I'm left only with the leading term which are those with less power of the x so are just those which correspond to d equal 4 operators I can write in this theory in variant is this one so I must end up here for f going to infinity and so at the leading order in f going to infinity which means psi going to exactly 0 that's just recovers a standard model automatically and try to do anything ok then the corrections that will induce ok if this was a true theory of electroic symmetry breaking they would induce for instance its coupling modifications are terms like h model square over f square dh square plus many others but all these are two derivatives two derivatives acting o h and then any power of h over f correction that's a class of operator which is under study at the LHC and for future collider now we will next time probably we will say more about this so first thing for f going to infinity f going to infinity ok I get the renormalizable renormalizable ok a billion hx model or if you want I get an elementary hx model the composites tends to become elementary it's like random coincides the other interesting thing which I can notice so the other important thing which I can have to do is to show you how the electroic symmetry is broken assuming of course that this h will have developed a potential by you will buy the fact that the symmetry is broken ok we will not discuss right now the potential generation but h as a potential so it will take a bad and then the first thing we can do is to go to the unitary gauge where the hx is taken on the real direction so b plus h over square root of 2 ok and here I mean somebody who asked before the question about whether they can redefine a way v he can do this so try to set to zero the gauge coupling ok which appears in the covariant derivatives ok and you will discover that indeed v has no effect ok why because the only way in which v has no effect is because there is an explicit breaking of the symmetry which is coming here which is coming from the gauging which is coming from the fact that I have gauged only one of the three generators and this has broken the global symmetry and so the hx is not anymore an x hat goes to both and you do see this also from the result let's just go to the unitary gauge and write the Lagrangian becomes incredibly simple one half dh square kinetic turn for the x plus e square f square over 2 sine square v plus h over f a mu a mu ok you see the only place where the v remains otherwise is just the free Lagrangian ok master's free Lagrangian is where there is the gauging ok and the formula is I think is remarkably simple and it's remarkably similar to the one that you will get in the usual abinian x model in the usual abinian x model just rather than f square sine sorry this is sine square obviously sine square v plus h over f you would have got instead of all this thing here just v plus h square that's the only difference you replace v plus h square with this little bit more complicated trigonometrical function so I want now to read the various pieces that appear here like the W boson master and the interaction ok so obviously I hope the notation was clear v here denotes the web of the Gorson boson hicks liter h is the physical fluctuation ok so if I start Taylor expanding for small h well in series of h I get no h interact term which is a master and I start getting interactions let's start from the master so the mass of this field a which I get after the compositing that's taken a web is e f sine v over f so you see the usual story it's like I anticipated so let's call this e times v in ordinary abelian compositing you define the web of the hicks it's just the m a over e ok so just to make just to make the analogy let's call v all what multiplies f and so v is f sine v over f so v over f is the misalignment angle because it was entering in the spawn of the Gorson matrix this f normalization that was just theta but you see that this is the same formula as before v equal to f sine of the misalignment angle so xi an important parameter before v square over f square is as it has to be is sine square so I have this parameter I can control, I can use to control v which up to a constant which is the covering constant gives the mass of a so I can keep for instance v finite ending f to infinity and go to this limit to the f to infinity limit and so I can systematically recover the standard model and this we can see also the elementary hicks model and this we can see also by computing explicitly out of this Lagrangian the couplings of the of the hicks that's just some trigonometry so I go here I expand the mass term in powers of h and I find one half m a square a square then there is one because we normalize everything to have the correct let's say mass for this gauge field we have solved the prefactor inside this then you expand you get sine square, you get sine cosine and then you have to use some trigonometric relation to re-express these signs in terms of xi so you have in h, you have 2 1 minus xi h over b you can check this and this is corresponds to an interaction involving 1x field and 2 gauge fields then there is 1 minus 2 xi h square over b square minus 4 third xi square root of 1 minus xi h cube over b cube and of course I can continue and all these terms correspond to interaction of 2 gauge fields with an increasing number of powers in number of x legs so the first 2 such interactions are present also in the elementary x case where that thing is just b plus h square so you get up to h square and but of course the coefficient is different so look at what we find so the composite x vertex of the x and 2 gauge 2 gauge fields so is given by this 2h over b which just reproduces the one you find for the ordinary the normalizable elementary x theory so I factor out 2h and there is only a multiplicative factor square root of 1 minus xi of course it goes to 1 when xi goes to 0 all the zillion of arguments normizability, f go to infinity in volume misalignment you can verify this explicitly here there is also the 2x 2 vector bosom vertices which is the elementary result multiplied by another factor so you can always 1 minus 2xi controlled by the same parameter there are also new interactions so this interaction for instance in the case of the physical x has been measured already at certain point I will show you how much correction we can tolerate this one has not been measured yet for the standard model and we probably not be measured at the LAC but ideally there are also all these correlations also the correlation between the change of this and that is predicted and testable in the model then there are also higher order interactions more hexes and two gauge fields which are absent in the elementary case and indeed you see that once you normalize them properly with the scale v that corresponds to the physics of the hex you see that the scale xi so they are proportional to xi and so they go to 0 they become weak when xi goes to 0 ok so I think I'm supposed to to stop here and tomorrow we will briefly illustrate the minimal potentially realistic model of composites with the full breaking of SU2 but there will be no need of any other calculation actually the results and the form of the Lagrangian that we will get are exactly already written at this blackboard so I can already anticipate you that the coupling of the x2 vector boson will be modified by this amount and next we will start working hard to arrive on time at the end of this lecture to compute not only the x coupling to vectors which cannot be measured alone at the LAC they come together with the coupling to fermions so we will need also a prediction of the coupling to fermions ok to make the thing more interesting ok there is a question over there the question was where is the master term of the Higgs the master term of the Higgs will come from the Higgs potential ok so it is an effect that as you see is not present in this Lagrangian ok where it comes from it comes at the loop level because when I gauge the u1 is true that I break the symmetry but I don't break it directly by introducing a master I break it so to use a master I have to do things like this so for instance this coupling ok of the Higgs with the gauge field ok will be capable to generate so this one for instance will be capable to generate a master only at loop level by taking loops of the gauge fields itself of the gauge field itself basically by e the loop will have to contain yes the potential and the master will be generated by loops of the gauge field and so they will be proportional to e e squared over 16 pi squared so how can you arrange this value to be well the value of today if the only free parameter for the master is e ah ok so the full story well, first of all it's not so far so if you we should do a more complete model introduce many other things because the estimate of the size of the Higgs depends on several parameters ok so I wish I would get there but given the speed I go to probably I will not but a priori there is no problem and actually the value of the Higgs mass which is predicted by Higgs loops so the Higgs potential is not fully predicted ok because you should know the full theory and this loop is not a loop that you can compute ok you have to specify the full theory but if you estimate it up to order one coefficient the value of the Higgs mass that you get from the gauge loops is actually not far from 125 ok what is a bit problematic is to take care of the loop of the top that also contributes to that and that tend to give a too heavy too heavy Higgs ok but if you get the exact correct mass for the Higgs it will be a very strong confirmation that the mix is in this I told you to get the exact correct mass I have to build a UV theory UV realization of all these with all the constituents then I have to go to a lattice friend and ask him to compute all possible correlators which I need so presumably this will be a strong confirmation but it's model dependent that's outside ok and this is also why I would not spend so much on the Higgs mass because instead the Higgs coupling modifications are not model dependent ok your lattice friend will find what I'm saying independently of what are the details ok of the composites so presumably the Higgs has been seen at the LHC and they can constrain this exire parameter what is the constraint as it stands and does that directly already constrain the scale of of the composites Higgs there are two kind of constraints there is the constraint coming from the coupling measurement that as you see here basically constraints on LHC and so the constraint F the parameter F and this constraint corresponds to XI which is of the order of 0.2 or 0.1 if you take atlas or CMS because atlas is more moved in the bad direction for composites the sensitivity the spectral sensitivity is 0.2 so more or less you can say 0.1 if you take the atlas stick ok the other constraints on this XI come from a retro-whip position testing in an indirect way I already mentioned this so these couplings of the Higgs are modified and so this will show up in the loops that do retro-whip position test physics that is sensitive to and then there are the constraints of the LHC instead of the rest of the scale on the partners on the resonances and just additionally so this XI is also to be related to some fun tuning measure as well this is regarded to be a measure of how much fine tuning the theory is but again if I'm not sure we have tried to play enough with the potential of the Higgs with the structure of the potential of the Higgs to be completely sure that there couldn't be some structural reason while XI is more for instance the detail X models in the end boil down to particular realization of composites in which you do have a reason to have say a quartic which is larger than the master in this effective potential for the Higgs and so you can naturally have and tend not to work or to have other problems but ideally it's not impossible sorry how do they triple and quartic Higgs couplings modify here in a vertex with three Higgs or four Higgs as in the standard model so this this depends on this is part of the potential obviously no and so there is not a unique prediction before you tell me the details of this potential so if you assume that the potential only comes from gauge fields I can predict the functional form of the potential like it will be sine square plus sine to the fourth with two parameters so if I set the bab and the X mass by these two parameters that your lattice guy has to compute then I can predict for instance the trilinear X coupling and the quartic and you know that most of the contribution to the X mass will most likely come from the top sector so if you specify the top sector some details of the top sector I will tell you tomorrow which details you can predict also that contribution so there are special cases in which the functional form of the X potential is so constrained that you can predict also the trilinear and the quartic but there are also more general models in which the functional form is more complicated you don't have only two parameters and so the trilinear is still a free parameter it's more dependent in general it's more dependent thank you so I don't understand these couplings that operator that you have there is the muh squared and then squared these couplings here I understand that but so in that in that first line there in your Lagrangian for the H you have the last operator that you have is the muh squared and then squared do the calculation so this is a motion for H and it modifies that at the same effective dimension I guess so d muh squared so the wallmaster only comes from here because here there are no gauge fields because this is the derivative of a singlet you can use equations for motion in that use whatever you want so to find that result you just have to take this piece then this one conspires with the other to give what your friend was asking so to give the fact that the kinetic term of the X doesn't depend on B because it doesn't depend on E so there is no breaking of the symmetry and so here it must be like this but this comes when you sum this and that for reading this result that result is just yeah it's only the first part but that's what I don't understand because you can always use waves of motion there and that will modify your couplings but I want to use the question of motion use the question of motion properly the couplings should not be modified because it depends what you do you can move of course you can move let's see what you can do of course you can move things around for instance you can K change the kinetic term if you do change the kinetic term then you can write the couplings in a different way but of course the X is not not anymore the trivial kinetic term so that's probably what you are doing but you can show me the changing result there is something I didn't get completely I don't know if I didn't understand your answer but you said that the Higgs mass is not an issue for you no, I didn't say that go on with the question well, you said at least that you can get the Higgs mass exactly right now it depends on the model so the Higgs mass the Higgs mass is not predicted in this model because it depends on order 1 parameter which in turn depends on many details of the composite vector you don't even know and even if you knew you would not be able to compute so in this sense the Higgs mass is not a prediction that can be directly falsified then you can ask yourself whether the size of the Higgs that you observe is reasonable or not so how large or small we do estimate to find the Higgs mass like in supersimary but there are boundaries and there is reasons of plausibility of this Higgs mass now the general message is that due to the top contribution and not to the gauge the gauge are fine, the top is a bit larger due to the top contribution in supersimary you would have liked to have the lighter sticks in spite of what people can tell you but in supersimary the Higgs should have been lighter not Marcus, but somebody else and in composite ticks you would expect it to be a little bit heavier so the top tends to give the contribution to the Higgs mass in this sense can be accounted for but it pushes down other particles which can give constraints so it's an issue it leads to some issues so you put it like in supersimary you don't have the right mass unless you make some fine turning or something a way out would be to do some fine tuning so the full story is like this you can have the Higgs mass light if you light compatible with data so without fine tuning if you lower certain part of the spectrum by some factor so let's say that the spectrum of the gauge resonance is at 3 TV and the other guys that are called top partners are at 1.5 TV or 1 TV or so but then you should find these particles and you have to look at the other bounds so somehow it's similar to the top stop situation in supersimary I'm sorry I didn't understand how did you arrive to the vacuum expectation value this is not I'm assuming that there is a potential there is a potential and in this theory I wrote there is actually a potential which you could compute you could not compute it in this particular theory this would be quite a divergent but it comes from loops of the gauge fields that break the symmetry and give a potential to the Higgs so there is this little potential and it has a minimum I call it V I haven't said anything on the origin of this potential I told I explained that how it could be there but I haven't told you I haven't told you how it depends on the Higgs field how it's formed, its estimate I haven't told you this so just taking this Lagrangian this Higgs vacuum expectation value comes only from loop no the potential comes only from loop effect it doesn't mean that the web is not loop suppressed we wish it was so there is a question sorry if the symmetry breaking potential for capital H comes from loop effects then why are we sure that it has the symmetry breaking form maybe it is very good sure and that's an interesting story it comes out that the gauge contribution so this depends on two things it depends on the functional form of this potential as a function of the ghoulstone of the Higgs and this you can predict in terms of ghoulstone symmetry sine square in some cases or sine cosine it's fixed and also depends on the overall sine of this whether the sine is positive or negative you can have now the gauge contribution gives a positive master so only the gauge contribution which I've written there would not work would give a positive one even thought there are cases in which a gauge can be a negative and this was actually an issue when Georgian couple were building their models because they were not finding out to achieve the negative web they could do by adding extra stuff there was an extra z prime gauge the elementary z prime because instead the fermions give a negative contribution the top in particular gives a negative mass contribution but at the time Georgian couple could not know that the top was so heavy and so that the yukawa of the top was so large so they thought the top yukawa contribution is negligible with the top yukawa we have now instead this contribution is actually a dominant one so it's typically natural to have the right sine for the X mass taking literally this model we have to add something extra to get symmetry breaking for potential first of all you don't have to take literally this model because you understand this is a linear scalar theory so this suffers of hierarchy problem just as a toy model don't take it this theory is very different from a composite sector first of all it suffers itself the problem as I said second it has only one resonance which is a scalar resonance while you have no reason to expect that you will expect most actually vector resonances rather than scalar resonances so I don't recommend you to take this model seriously anyhow if you take it as it is it also does not allow you to predict the X mass because the X potential is quadratically divergent in this theory in this simplified theory because it's part of a more complicated sector and then I can argue whether it's larger or smaller but you cannot take this as a representative model no more questions? let's think and do it again