 Tako, da bo v alle, ki so vsoživati na ljudev, z kjer sem... najlednije vse odnočenje z kratiljstvom sputinu, z kratiljstvom sputinu za hljevstvčenje. I smo bojali s tem, na kratiljstvu, na kratiljstvu, na dnez. Zdaj, Ma želim, da se izgledo, in nekaj je što izgledo, začešno raz什 izgleda, ki pa drugi dom, prav freaking, začo, začo, začo write, začo. Pa je denovil, da je naštrak jesno upod zbvenici! Nel jaz, ki se vsi in su sločilo, več deločni hrhej. Kaj je hrhej in skupio, kaj je vsega hrhej v barjan sivljajstva, Kubar, left, dresned, Cinglet, top-right Cinglet dresned. Ko počustili v kardeticu vsi ccwz teklonov z vrhanja ki so početni nekaj, ki so prišli spremni in se bo tukaj. I na pomečnega, ki so prišli spremni in se bo prišli spremni in se bo prišli spremni. To je prišla in vsezni, ki je konečnje in vsezni, in to ki so prišli spremni in vsezni, in še je početno g-5 pečnje. Ne, tako ta je... Ta je vse sebe, ko je vse sebe. In tako in v eni nekaj zelo. Zato prejel sem, da bo što sebe vse začeli tudi. Vse začeli tudi, da je vsebe vsebe, da so vsebe vsebe vsebe in tudi x. Ko se neč negačno matriče, ki jo jaz jaz v tudi. Način je, da je vsebe vsebe vsebe, da je tok, da je vsebe vsebe vsebe, So after all the coefficient I'm gonna write in front of here doesn't matter much because it will be reabsorbed in the definition of the top mass. It will be fixed by reproducing the top mass. However this coefficient is, so it's parking parameterized like this, there is an order one number C-top, then there is a scale because this is like a master, so it has to have dimension one, so there is a star, the typical scale of the composite sector, and then there's gonna be the couplings of the source with the composite sector. The Q-less source was coming with lambda t-left, the t-right was coming with lambda t-right. And so whenever one of these things enter there should be together his own coupling, and finally there is in the denominator here something that I'm gonna explain you the origin of in few minutes, and it is the usual G-star, I will discuss it in more details. So G-star was this typical coupling of the composite sector that was somehow entering in our estimates, and here it comes to the second power. OK, you can rewrite this in the unitary gauge leading to the same prefactor times the following function, which is 1 over 2 square root of 2, the sign of 2v plus h over f t-left t-right. So v is the web, so h is the fluctuation, when h is equal to zero, the first thing this does is to give a mass to the top quark, which you can read from here. It depends still on this c-t coefficient, so m t is equal to, if you want it explicitly, c t lambda t-left, lambda t-right over G-star square, m star root square of xi, 1 minus xi, divided by square root of 2. OK, it doesn't matter, whatever it is it will be fixed by this c-t. Of course we would like c-t to be over the 1, so this combination, lambda t, g-star, et cetera, they should be rather, these lambdas in particular should be designed in a suitable way, such that this is the order of magnitude of being the top mass. But once I've done this, I can re-express the coefficient c-t in terms of the top mass and get definite predictions. So now I get something like minus m t over 2, 1 over square of xi, 1 minus xi, sign of 2v plus h over f, and then there is t-bar-left, t-right, or if you want the Dirac mass term for the top. OK, so of course in the standard model you just will just have found rather, as usual, rather than sign of 2v plus h over f, you would have find just v plus h over f, sorry v plus h, OK, so you would have just find v plus h rather than this complicated trigonometric function. So this will deliver, expanding this in powers of h, it will deliver coverings of the hicks to tops, but with the modified strength. And this I can immediately show by expanding this in h. So I will find the first term, which is the top mass m t, t-bar-t. Then there is minus a parameter I call k-t, and here I will encapsulate the differences with respect to the standard model. So m t over v, that's the linear term, t-bar-t. And then there are also higher powers of the hicks that come from expanding, which of course are not present in the standard model. And so there are new couplings like c2m of the top, divided by v square, h-square, t-bar-t, for example. And so k-t is the modification of the top core coupling, so it's the coupling hicks top-top in composite hicks, normalized to that of the standard model, which if you compute it, you find it 1 minus 2 psi over root square of 1 minus psi. So as usual is something that for psi goes to 0 goes to 1, but otherwise when psi is not dramatically small, there is not so much fine-tuning, it must deviate with respect to the standard model prediction, and this is something you can measure also at the LHC. Another thing that you can measure at the LHC, you will be able to measure at the LHC with some accuracy, is the possible presence of this c2 coupling, which is an interaction involving two top quarks and two hicks. And well, this will be perhaps measurable in double-x production, I mean some error bars, and the value in this case is c2 equal to minus 2 psi. So it's a direct sign of new physics as it involves an operator, which as you see here is dimension 5. So this was for, well, so right now we have an almost complete picture of the hicks couplings. The hicks couplings are measured mainly by x-production time-bright ratio, as you've seen in the Michael Peskin lectures, and so you know that they are measured together. So you measure together the coupling to the vectors, the coupling to the fermions, to the top and to the bottom. So we do already have a prediction for the deviation of the coupling to vectors, which we call kv, which is square root of 1 minus psi, which we determined at the very beginning of these lectures, and well. And this prediction is completely universal, so it only depends on the choice of the coset s of 5 over s of 4. Next we have this one, this one written here, which instead is more model dependent. So in particular the one I called here kt should actually be called kt5, so it's the prediction of the modified top coupling obtained in the 5 representation, and it's this 1 minus 2 psi over root square of 1 minus psi. But this could be changed in another representation. For instance we saw that the 4 representation, even though this fabric, I tell you this fabric, it also could work for generating the top mass, and this gives to kt4, which is slightly different, t square root of 1 minus psi. So when you move psi in the plane, if you want kt kv, this corresponds to a different trajectory, and in principle if we are able to measure these deviations well enough, we can distinguish between this option or this other option. To make actual prediction, you also need the coupling to the bottom pork, and these modifications. Now for the bottom, well, again there is some ambiguity, depending on which kind of representation of the composite operators you use to give mass to the bottom. However, for the most common examples, if you have the top in the 5, you also have the bottom in the 5, and so what happens is that in all cases that are commonly studied in the literature, there are counter examples, what happens is that kb is always equal to kt, even though it's not strictly needed. So basically the deviation of the bottom and top couplings are the same, and so they can be encapsulated in a unique correction kf, which is what our experimental colleagues like to do of course for simplicity, to have a fit to the x couplings involving only two parameters, which is kv and kf. So I can show you the result of this little exercise, overlaj, overlajd with the experimental, the correct experimental data, which is over there. So you see that these lines correspond to the, so this is the standard model in the plane kvf, and this is the minimal composities model with four representation, this is the one with the five representation, so you see they are a little bit tilted. You also see that the error bars are, this is one sigma for instance, are relatively large, still as you can expect, these have just been discovered. CMS is this blue curve here, and is somehow generous with composities, it allows a lot of values of size, atlas is more and more moved on the right, and so you see that at two sigma it's already capable to exclude from 0.1, from, let's say, 0.15. So that's one of the constraints we do have on composities as of today, even though it's probably not the strongest one. But I think it's the simplest one to get, and that's why I decided to derive all these couplings for you. Okay, is there any question? Sorry, sorry, sorry? Yes, atlas, good question. So the question was whether, so the fact that atlas is preferring larger couplings with respect to the standard model, and this is true, and whether the composities unavoidably give smaller as it does, you see. Now for any composities with a compact coset that is typically going to be the case, any composities, with a compact coset is going to be the same. There are examples of composities based on non-compact cosets, like the complexified version of SO5, which then have something like, I mean, one plus, okay? It is also true when you play this kind of games that you have to take into account that here we are assuming that there is only one X doublet, so we are doing the minimal composities model where there is only one X doublet. But actually it's sufficient to imagine a bit larger coset to have more X particles, okay? One doublet, one singlet, and then you will have a more complicated phenomenology because these extra hexes can mix with the other ones and so can backreact in that plot in an untrivial way. So that plot, if you want, is made on the assumption that the direct searches of other scalars will have no result, will have no result, so these other scalars are very heavy and so they do not mix because otherwise these results, if you want, these predictions will be contaminated by new effects which will sign and direction is a bit difficult to predict. Okay, so now I want to move to the last piece of information that you need. If you want to understand a paper on compositics, which is called the power counting, okay? So what is a power counting? A power counting is a systematic way to estimate the size of operators in effective Lagrangians, okay? And the power counting then reflects an assumption on the UV theory, okay? On the way it behaves, generating these low-energy effective operators. Power counting is needed for at least two reasons. If you look at the, for instance, the predictions we derived until now, right? You derived the top Yukava Lagrangian prediction. Similarly we derived the counting of the X2 vectors coming from the gauge field master and so on. And in that case we have been rather lucky because in both cases the pre-factor of this single operator was fixed. In this case it's fixed by the fact that you have to reproduce the top mass. So it's measured if you want the coefficient of effective operator to phenomenology. But you do have clearly corrections, okay? For instance, to this formula you may consider the effect on the X2 top covering modifications due to higher derivative operators. Operators involving not only, I mean, not only fields, but also derivative acting on the fields which will change the kinetic term and so backreact in different couplings. And in general there are several sources of corrections to these leading order results that you need to estimate. And in order to estimate them you need an estimate of the effect of the operators that you have neglected at this leading order level. And for this you need an estimate of their size which is power counting. And also another very important thing is that there is not only goals on bosons and standard model fermions in this theory. We saw that there are plenty of resonances of new particles whose dynamics cannot be just fixed by the nonlinear sigma model predictive power, okay? So the sigma model constrains these dynamics, but it really doesn't tell us specifically one Lagrangian with all its coefficient. And also to estimate the size of the couplings and the interaction strength of these resonances to be found at the LHC or to be searched for at the LHC you need this power counting rule, okay? So there is one specific power counting which is done in these theories, which is called one scale, one coupling power counting or it's also called Siege power counting from the paper called Strongly Interacting Light Hicks where this was introduced in this framework. And this power counting is as follows. And so the way this power counting works can be easily illustrated by, again, our first example of a compositing theory in term of a linear scalar theory, okay? So if you go back to your notes, you should discover that the Lagrangian that we have takes, well, the following four, okay? So there is the kinetic term for sigma, there is the mass term for sigma, and the mass term of sigma, even though it was not written in that way, at the beginning is like this, okay? So there is a mass for sigma, which we call M star, and it is G star times F. G star was that coupling parameter that was appearing in the original potential. Remember, it was G star squared over 8, no, 5 squared minus F squared, okay? So the Lagrangian was written in terms of G star and F, but now we can rewrite it in terms of M star, okay? And G star. And then if you do this, you discover that this is, so that's the mass term, that's the interaction term, which is G star over 2 times M star, sigma cube, okay? So a three-linear sigma interaction is weighted by the coupling G star, not surprisingly, it is dimensionful, so it has also M star. Then there is the quartic coupling, which is just G star squared over 8 sigma to the fourth, which has two power of the coupling, that this is for power of fields interactions. And then there was all the complicated stuff, including the Higgs, which we can see, the Higgs or the Gorsons, which you can rewrite as 1 plus G star sigma over M star. So here you add at the beginning sigma over F, but I just traded it by M star and G star squared, multiplied by that complicated expression, which, however, altogether was the pure pi on or Goldstone Lagrangian, okay? So let me call it, let me call all these L pi, okay? For the time being, let's focus on the gaugeless situation. So let's do everything when the gauge couplings G and G prime are equal to zero, okay? We will introduce them later. And so here in this Lagrangian there are just fields and ordinary derivatives, and there is no gauge fields. And this Lagrangian L pi, even without writing the complicated form it had, you can check that, I mean, schematically, it is of the following form. It is F square in front of it. Then there are two derivatives in all the terms that we have, okay? So D square, and then there is pi, okay, divided by F, okay? All these to an arbitrary power, actually it was even a trigonometric function, so you should imagine expanding it, okay? It has this structure. This structure, remember, came from the fact that the Gurson matrix depends only on pi over F, but in front of this you have an F square prefactor, okay, that's, you have two derivatives and this is what you get, okay? So all the terms in this thing have this form. And you can express these in another way, okay? Again, using F and G star and star relation, you can find that the full composite sector Lagrangian as written here takes the following form. So if you check term by term, so both the terms involving sigma and pi, they obey the following structure. So there is m star to the fourth over G star square, okay? Which I factorize in front of everything. And then there is a dimensionless polynomial of interaction, okay? So this is a functional, which contains derivative weighted by m star, okay? And the fields, okay? And the fields are weighted by G star pi over m star, okay, which just means pi over F, and G star sigma over m star, okay? Okay, so what we can say about this Lagrangian, about this structure, okay? This is gonna be our power counting formula, which now we are gonna justify in some details. First of all, we recognize that it's not surprising that pi and sigma enter into the interactions in the same way, because they are both composite particles, okay? And it's also, if you want, not surprising that the only scale that enters in the problem, and there is only one scale, okay, which is the strong sector scale m star, and the only thing which is surprising is the way in which I'm placing this G star couplings, which for the moment you may not see why it is so, okay? We can just verify that this form matches with the explicit Lagrangian written on the other side, which is then a trivial exercise. However, it's a bit trivial to check that this structure of the Lagrangian preserves, is maintained, okay, survives and plays a little bit around for this Lagrangian. Suppose, for instance, that you want to compute the low energy effective field theory only for the Gorshton fields, okay, obtained by integrating sigma out, okay? So I integrate out sigma, and I get another Lagrangian, and this Lagrangian happens to be of exactly the same form. I can verify this very easily. So to integrate out sigma, the first term which I find is just the term when I just set sigma to zero in all places. So the first term of this effective Lagrangian is the one we used already, okay? So it's just the Gorshton Lagrangian, and it's a two derivative term. So it's an L2, okay, which is just coincided with the old Gorshton Lagrangian, L pi. Okay, which, if you look at it and you compare with that, you see that it has exactly the right form to match our expectation. Then you have another term which starts with four derivative and the one at six derivatives, and so on. And the one at four derivative, let's look how it's obtained. So Lagrangian for derivative is like this. You integrate out sigma, and so the only interaction of sigma with the Gorshton comes from the square of this term. When you take the mixed product, you have something like just two g star over m star sigma multiplied by all these L pi Lagrangian. So diagrammatically, this means that the four derivative Lagrangian comes from this diagram here. There is a Nell pi blob here that couples with sigma and another L pi on the other side with the sigma field propagating in between. Here there is the sigma propagator, but I'm constructing a low energy effective field theory, so this will just be minus one over m star squared when I expand the propagator. And here I just have the strength of the interactions, which I read from that formula, from the mixed product, this star over m star squared, because there are two such vertices, then there is one over m star squared, then there's gonna be some numerical coefficient, and then there's gonna be L pi squared. See, now it's a four derivative Lagrangian because each L pi contains two derivatives. And if you properly count, and if you remember that L pi Lagrangian is itself of this form, so you have that all this is g star squared over m star to the fourth, so times L pi, right? L pi obejs this rule, so it is itself of order m star to the fourth that then goes to the eighth over g star squared that becomes to the fourth, and then it's the product of two polynomial that I will just write it as one, that of d over m star, g star pi over m star, g star sigma over m star. So if you simplify things, this eighth becomes four, and this four becomes two. And so you recover that even the higher derivative order Lagrangian has the form predicted as the form dictated given by that power counting over that. So the structure of the power counting persists if you integrate out one or some resonances out of this more complicated Lagrangian. If you like to do so, you can go on. For instance, you can look at the dimension 6 Lagrangian which comes from two contributions. The first one is when here I pick the first term in expansion of the propagator, which is p squared over m star squared down to p squared r derivative. So this is exactly two more derivatives and two more m star. So it still obeys this formula here. There is another term in the L6 Lagrangian which is also schematically like this. So you have a vertex with three sigmas and blobs attached. And so you can check that this is dimension 6 and this also fits into this formula. OK. So the next thing about this power counting is that it is actually justified in two completely different directions and only the simplest one will be actually explained here. But the second one is more interesting. OK. But let's go to the first one. So one way I can understand this formula is in term of dimensional analysis. OK. But not just the natural units dimensional analysis. OK. So what I'm going to do is for a minute is I'm going to forget each bar. So I'm going to take each bar different from C, different from 1. OK. And I'm going to do dimensional analysis in the old fashioned way where you do have three fundamental units which are length, mass and time. OK. That you can trade for, which is more convenient for us, length, time and energy, which, of course, is mass length squared over time squared. OK. In these units, the plant constant, of course, is not dimension. Dimension less the reduced H bar plant constant has the dimension of energy times time. And, well, in this dimension happens to be, of course, the same dimension of the quantum mechanical action. In the path integral you always have into the I action over H bar. And so this must be dimension less now, the exponent. And so we know that the dimension of the quantum mechanical action is the same as H bar of, that is, AET. OK. Furthermore, I know that, so from here I can read the dimension of the Lagrangian density. OK. So the action is the integral in d4x of the Lagrangian density. d4x is always defined, so x is always defined to be c t comma x mu, sorry, x vector. OK. So each of these x has actually a dimension of length. OK. So the dimension of this d4x is length to the fourth. And thus the dimension of the Lagrangian density is the dimension of s, which is the dimension of H bar, divided by length to the fourth. And furthermore, always because of this fact that x has dimension length, derivatives have dimension one over length. OK. So this is trivial, and then it's also trivial to see what are the dimension canonically normalized fields. For instance, a scalar field has a kinetic Lagrangian, which is d5 squared. And so its dimension must match with the dimension of a Lagrangian, must match with the dimension of H bar, divided by L to the fourth. And so the dimension of a scalar field is equal to the dimension of H bar to the one alpha over length. OK. We normally say that the field has dimension of mass, which is true in natural units. Indeed, here we find one over length, which is equal to mass in natural units. But if you are more precise, there is also this square root of the H bar dimension. By the way, the gauge fields have the same dimension, because the vector fields have always a Lagrangian with two derivatives like this. The fermions are a bit different, because for the fermions the kinetic term contains only one derivative, right? So you have psi bar, this less psi. And so for the fermions, you have that dimension of a fermionic field, OK, is H bar to the one alpha. You see, there is always H bar to the one alpha, because kinetic term is always quadratic in the fields. But the power of L is different, corresponding the energy in natural units is different, and it's L to the three alpha. So this suggests that, well, in the first place we should define a symbol for this H bar to the one alpha dimension that enters in all places. So actually we will define a coupling dimension C, which is the dimension of H bar to the minus one alpha, OK? And so we can start now reading the dimension of the various object that we have typically in our Lagrangian. For instance, the dimension of the electric charge, OK? The electric charge appears in the interaction of QED, like E, C bar, A slash psi, OK? And you can count this dimension, OK? And compare it with the dimension that Lagrangian has to have, OK? And so this one you impose to be equal to H bar over length to the fourth, which means one over coupling square, one over length to the fourth. And so you discover that the electric charge has dimension C, and that's why I call it a coupling dimension, OK? So the electric charge in non-natural units is not a dimension as object, it has a dimension, which are dC's, which are minus one over the dimension of the square root of H bar. And similarly, many other parameters of the standard model you are used to, and sometimes people regard them as numbers. They are not numbers and this makes a difference. I mean, when you look at the calculation, you can immediately sometimes conclude if it's wrong or not by this coupling dimension analysis. For instance, the Yukawa coupling of the fermions, let's say the Yukawa of the top is also a coupling, it's properly called a coupling. You can check by the same way. The dimension of something that is called a coupling is the quarter gigs coupling, lambda for H, OK? It's not a coupling, it's a coupling square, OK? The reason being it appears like multiplies phi to the fourth and not a phi cube interaction, OK? And that's why, for instance, supersymmetric three-level relations between the x-quartic and the gauge couplings, given that the gauge couplings as dimension C are never lambda equal to C, OK? It's always lambda equal to G, equal to G square, OK? Because of this. All masses have dimension of one over length in this thing. So, for instance, the mass of the W or the Z or whatever else has the dimension of one over length, OK? And I could go on with several examples. But here, let's see why this is useful. It's useful because I can recover that formula by dimension analysis. So, we call theories of the type which have one scale, one coupling. If they happen to have only two dimension full parameter, OK? Which are a scale and star, which has the dimension of one over length. And the coupling that is an object called G star, which has the dimension of C. And then you can see what the Lagrangian has to be just on the basis on dimension analysis, OK? For instance, the linear sigma model Lagrangian that we start the usual LC, right? That's clearly one scale, one coupling Lagrangian in a trivial way. I mean, it's one half d sigma d phi square potential, you see, no? The G square over 8 phi square minus f square which is m star square over G star square, OK? So, you clearly see that here there is only one dimension full guy G star and another dimension full guy m star, OK? So, if you want this trivial that we got even after our complicated field definition, right? Which was a bit complicated to perform, we got the Lagrangian that obeys that relation, OK? As we will see. It's just dimension analysis. There is a less trivial example which are the five-dimensional holographic models where in order to really get the couplings and to check that they obey that relations you will need to go through some complicated Kaluza-Klein reduction in warped space whatever it is but that's also very simple because in the five-dimensional models you have one scale which is the infrared if you want to cut off of the theory which is one over length and you have a coupling which is the five-dimensional coupling which is not dimension which is not dimensionless in length in five-dimensional gauge coupling as a dimension of C L to the one half you combine the two and you indeed end up into one scale one coupling paradigm, OK? So, you understand that this immediately simplifies our lives in the way we can parametriza understand how the results of even complicated theories behave, OK? So, but that still didn't get to the point to show you this correspondence the fact that if there is only one scale in one coupling the effective Lagrangian looks like this I will do it immediately with an important specification so I work in that hypothesis and I also work at the three-level order so I imagine having plenty of particles and stuff like for instance in the complicated five-dimensional graphic models I integrate them out and I get an effective Lagrangian for the Gorsons and perhaps some of the resonances when I do this at the three-level there is no dimension full parameter that appears because at three-level there is no parameter in particular there is no H bar, OK? that enters into the calculation and so the result must have the correct dimension just and these dimensions should be reproduced by just these elements that we have here and star and g star and so you have a pre-factor which is one over length to the fourth coupling square which is one over coupling square which is then star to the fourth over g star square and then all the rest must be dimensionless dimensionless in all possible senses in the sense it must have zero length dimension zero coupling dimension and so that's what we wrote there derivative come with m star fields given the dimension of the fields that we discovered over there they have to come with g star to compensate for the coupling dimension and then one over m star for the length dimension and this is for all the bosonic field so g star sigma over m star also g star pi over m star the only one which are different are the fermionic fields because the fermionic field has a little bit different dimension so for them the true power counting formula is g star psi fermionic if you want resonance field divided by m star to the three else so this is a three level result I've been careful in stress in stressing that because of course well perturbation theory so semi classical expansion introduces in the game another dimensionful parameter which is h bar itself ok so if you want to estimate the one loop effect on this Lagrangian suppose you have some operator in this effective Lagrangian that for some reason cannot come at three level there are examples in this case coefficient is a bit smaller and the thing goes like this the one loop contribution well it gets an h bar because we know how to count the powers of h bar that enter into loops there is one for each loop next there is a one over 16 pi square that well it comes from the loop integral in unavoidably and then there's gonna be something which must match the dimension and so you need the g star square remember that h bar has one over coupling square dimension so you need two couplings to compensate and then you put the g star square and then on top of this that's all the rest you needed before so then start to the fourth right over g star square times L hat so there is basically a suppression of loop effects with respect to three that is this part here which if you now go to natural in the end of the calculation h bar equal to one is g star over 4 pi square ok the usual loop factor loop factor suppression that if you want you can understand in this way also important is that this this formula the way I derived and it's obvious from here it really only holds for perturbatively or say semi perturbative theories because well here I'm using the concept of of this dimensions that have a different impact depending on the order in perturbation theory I go I mean that if there is no perturbative expansion which means that if g star over 4 pi is not smaller or possibly much smaller than one then it's clear that three and loop level contribution they all come together right and so there is no way in which you can maintain this coupling dimension if you want the coupling dimension gets obscured by h bar that enters in all possible ways ok so that's the justification of the one scale one coupling power counting for theories that have some weekly coupled regime ok at the resonance scale from weekly coupled theory or resonances that's what's gonna happen for completely strongly coupled theories it's rather interesting so that this could be true ok so there is a I mean I'm not gonna enter into this but so largenci gauge theories like QCD in the limit of a large number of colors they are genuinely strongly coupled theory in the sense that the UV description of this is a strongly coupled theory it goes through a point where the tough coupling where the coupling the intrinsic coupling of the theory it's 4 pi it's non perturbative but there are powerful theorems that tell us that the coupling among the resonances ok is suppressed at largenci ok and by these powerful theorems it's possible to show that in some particular relevant sector of these theories in particular in the mesonic sector the expected power counting rule for the interactions involving mesons in particular is exactly this one ok and this result I mean is one of the of the robust evidence of the possibility of duality between a weakly coupled theory and an underlying largenci strongly coupled theory so I will just write this for you so at largenci you can actually demonstrate that you get the same parametric formula with a coupling with an effective coupling right G star an effective resonance coupling of 4 pi over square root number of colors ok so it becomes weaker and weaker as the number of colors increase and so as the number of colors increase you can imagine so you are legitimate to imagine a strongly interacting composite sector based on the genuine say strongly coupled dynamics which however doesn't have maximal coupling maximal non perturbative coupling so there is some track of a weak coupling please can you speak loud? no the question was where it depends on top coupling it does not depend on top coupling because I mean it cannot depend on top coupling because I am in a regime where the top coupling is 4 pi is non perturbative so it is like it was not there but if you look for instance the Kolemann lectures or in the paper by Whitten on barions there is a very barions allergenci there is not the 4 pi because a string person do not want to see the 4 pi ok the top coupling the question is whether the top coupling if you want but it is not this that list the 4 pi ok some I can tell you later ok by this logic we can also very easily complete this power counting formula for the other interactions which we haven't considered yet so for instance well let's write down here we have introduced the gauge interactions ok and and then it's clear that the interaction the gauge sorry the gauge interactions are which we introduced in the covariant derivatives basically have a form l int equal to g a mu ok the corresponding gauge coupling to the corresponding gauge fields times the global current these these particles attach to g we said is a gauge coupling so it has dimension of c and when you see a new field in your Lagrangian well you have also to attach a g ok so the combination that has to enter into your power counting so it's g times a mu ok which is already dimensionless because this is c and this is c to the minus one from the coupling viewpoint and all what you need to make it dimensionless is one say length inverse length power so you have m star ok so when you have an elementary field insertion a campaign with its own coupling g then the say in the power counting here you have to consider like this ok you have to consider g a mu divided by m star rather than g star and that's not surprising at all it's clear that the composite sector has its own coupling g star the elementary sector has a completely different coupling g and so g replaces g star in this form similarly for the fermions now for the fermions is a bit more conventional if what I'm gonna tell you because clearly the fermionic operators we don't know how they are normalized right so we just wrote lambda f bar o let's say elementary composite operator the fermionic operator then can be normalized the way you like you can absorb powers of g star here and there so it's just conventional that we say that the elementary composite coupling lambda has a dimension of coupling ok otherwise you will find the appropriate powers of g star in places it would not be observable but the important thing is that it has one of these dimensions and so in this assumption you do have that a fermion like an elementary standard model fermion interacts weighted by its own elementary composite coupling so you have lambda lambda f over m star in this in this expression here in this diagram here ok the simplest application of all these is of course the estimate of the coefficient of the yukawa of the top that that I already anticipated so the yukawa of the top is a guy made of two elementary elementary top left and top right so let's say f bar left f right and then there is some complicated function of the let's call it f as well function of the hicks ok like this so the hicks must come is a gorshton so it must come with g star so g star h over m star which is of course h over f ok and the prefactor must be must be let's say m star to the fourth over g star square I'm just applying the formula and next you are gonna you're gonna have for each for each fermionic field you're gonna have 1 over m star to the three ales because of their dimension so you have 1 over m star cube and next you have the elementary composite couplings that is the lambda left and the lambda right ok all this simplifies and you find what we had at the beginning so you find lambda so and lambda left lambda right over g star m star ok that's very that's very simple ok I want to show you very simple application of this power counting rule to some phenomenology of of some kind of resonant particles to be searched for at the LAC you will see that it gives a rather non-generic prediction for this kind of particles so we already discussed in general the arising of resonances ok so we know that there are resonances that are associated to operators operators of the composite sector give drives to resonances with the same quantum numbers and here I want to briefly discuss vector resonances ok with vector I mean spin one resonances and to understand the spin one resonances you first need to find out which are the corresponding operators and the most appropriate operator for spin one resonances of course spin I mean is of course a vector Lorentz vector operators and so these are the particles associated in particular to the global current operators so for example the global currents we are talking about are those of the Composite X group S of 5 and and these currents of course are in the joint of the group which is the 10 representation ok and we do expect resonances for each of these 10 and these resonances the spectrum of these resonances given that we have the spontaneous breaking S of 5 to S of 4 is gonna be classified in terms of only S of 4 representation and so to read which particle I have I have to decompose the 10 representation into S of 4 and this gives me 3,1 plus 1,3 which is the 6 of S of 6 sorry of S of 4 you remember plus the 4 plate which is a 2,2 of resonances ok we also have other other global groups we saw you have the U1X for instance ok and this leads to one singlet resonance which of course is just a singlet of everything but it has some non-trivial phenomenology and also this has been studied at DLHC another thing which has been studied at DLHC in the vector resonance sector is the SU3 of color ok partial compositeness obliged us to introduce color as a global symmetry of the composite sector and so this obliged us to have octet of resonances of course SU3 is unbroken so this will just show up as octets of colors these last particles also have another name which are called caluzagline gluons because the first place they arise was from caluzagline reduction of 5 dimensional holographic like theories ok so what I want to study here is one piece of this SO5 say 10 multiplet and in particular these one representations here so the particles which are you see the decomposition now in the decomposition this one is the joint of SU2 left this one is the joint of SU2 right so I want to study these particles here which are those that correspond they are in the 3.1 so they are triplet under SU2 left with zero hypercharge so they are heavy vector heavy resonance when every triplet of resonances in the 3 zero of this standard model group ok these particles contain then 3 degrees of freedom which get arranged into a raw zero so into a neutral I call it raw the resonance the vector resonance and raw mu plus or minus and I want to briefly summarize what we can tell about the phenomenology of these particles on the basis of symmetries and power counting so as far as symmetries are concerned I I'm gonna only use for simplicity here they linearly realize the SO4 group ok so I'm just gonna write operators following not the full SO5 but only SO4 because it's easier and in this case the result wouldn't change SO4 acts in a simple way so raw mu which is this triplet here can be written in a matrix notation as if you want raw mu zero square root of 2 raw mu plus square root of 2 raw mu minus minus raw mu zero and and it transforms in the 3,1 of the of the SO4 group which means that raw goes to g left raw g left dagger ok so it doesn't transform on the right and it is in that joint of the left transformation while we also have the other important element here is the X field which I show you can be written as a 2 by 2 to the real matrix sigma that transforms as g left sigma g left dagger sorry, g right dagger this one transforms with both so these particles from the viewpoint of the composite sector will have a mass ok, which we call m raw of the order of of m star ok if you want trivial power counting analysis they have a mass of the order of the confinement scale they are and so by using symmetries and selection rules we can first of all understand how they couple to the Higgs ok so the coupling of these resonances with the Higgs will be entirely fixed by a unique parameter because there is a unique operator which can mediate raw h h interaction which is g which is a coupling which by power counting is of the order of g star over 2 i trace of raw mu and then here there is sigma d mu with the two arrows which means acting on the right and then minus the one acting on the left the usual notation sigma dagger so as of force symmetry tells us that there is a unique interaction of the raw with two Higgs and this already tells us a lot ok about the phenomenology of these guys Michael Peskin told you about the equivalence theorem right and so you already know that inside the Higgs doublet h equal to h up h down there are several interesting things there is not only the physical Higgs but there is also the longitudinal component of the W boson represented by a scalar W left plus which is just equal to h up and there is also the longitudinal component of the Z boson that comes from the imaginary part of the h down so you have something like v plus h over square of 2 which then comes together with i z left over square of 2 ok so and this is the longitudinal component of the Z boson so given that I can control all this interaction with the Higgs I can control all the interaction of the rho with the longitudinal vector bosons and of course also with the physical Higgs boson here and so I will be able to predict as I will show you now in a second all these all these interactions at the same time so here I have written the VEV but the Higgs VEV doesn't matter much here so I can neglect electro-hysimely breaking electro-hysimely breaking is effectively electro-hysimely is effectively broken ok so you can work it out you know what is sigma you know what is h and you will you will work out the couplings that you get out of out of here ok so for instance you are going to get something like g v rho mu 0 and you are going to get i w plus d mu w g 2 minus so these vertex minus the conjugate minus i w minus d mu plus g 2 d mu plus g 2 d mu plus g 2 d mu plus g 2 which we learned it needs not to be the maximum coupling, not to be four pi is a parameter of the theory. Of course, if these really needs to be a strong center we would like these to be a rather strong, in particular we would like this to be four five two three. Well there are other pieces that for instance there is minus h d mu Longitudinal Z plus Longitudinal Z dimuH, oče je izgleda, ki je zelo 0, pa x in z. Zelo samih kapaling, izgleda v samih operatora, kot nalegovati transišljenih, zelo vse igrojno. Zato dobro smo preddikovati. Zelo, kot zi roba plusi, komponentom roba, izgleda vzpe vzet, 2x in 2z. Kaj se priključanje in došliči, porom si se nakeče. Zgledajte samo vzora gamma z rosim. Wzido vzdo vzdo, plus wzido vzdo. Minus je izgleda gamma z rosim. 2zx is equal to gamma of rho plus 2wx is equal to gamma of rho plus 2wz. Now if I did the calculation right, this should be, I can also compute the final number which is g star square over 12 pi m star, or m rho, in the mass of this pi. So provided g star is not extremely large, after which this particle should start becoming too broad to be seen, all these final states are being searched for at the LAC. So sometimes people wonders how we can test composite takes, given that it's so big, it's full of things that we cannot compute. I think I showed you that not only in the x-carb modification, but here also at the level of resonance production, we do have extremely sharp predictions. So in particular, all these different vector boson channels can be studied at the LAC, and they happen actually to have even a comparable reach. And so having such a sharp prediction like that, the partial width, and so the partial branch, the relative branching ratios are the same, is an important constraint by which you can correlate different channels and make the search more effective. The other thing, which is of course very important for the phenomenology, is the production rate, and also the branching ratio to leptons, given that these particles are also searched for in their decay to leptons. And to derive this, we can use our power counting again. So I want to consider the couplings with the light standard model quarks and leptons, let's say light standard model fermions. Those are the ones, which are relevant of course for the production. And our power counting formula immediately tells us that a direct interaction between the raw and the quarks is extremely suppressed. And the reason is that in the formula you do have the partial compositing and the scapling associated with the light standard model fermions, not with the top, the one with the top are sizable, but the one with the light families. Now we will see what they are, star raw over m star. Here I am applying the power counting formula, lambda q, let's say the quark, but it would be the same for leptons, m star three-halves, lambda q m star three-halves. So this is lambda q, say square divided by g star raw q bar q. While I remind you that the yukawa of the quark q, like it was for the top, is fixed like lambda q left, lambda q right over g star. So that's basically the same that we find here. If there is not a huge hierarchy which you don't expect it to be between the q left and the q right composite couplings, that's basically you can estimate as the yukawa coupling. Let's say the yukawa coupling of the up or of the down of the quarks that are inside the LAC, so this is very, very small. So this term will not help us to produce the resonance at the LAC. Resonance at the LAC will be produced through another operator for which power counting estimate is also needed. And this is a mixing, so there is a mixing, in particular with the W boson. This mixing is just one allowed term in effective Lagrangian, so it's there. Of course it's controlled by an unknown order one coefficient. You can normalize it like this, c over 2, then power counting m star to the fourth over g star square. And mixing is made like we have the standard model gauge field strength. We have to use the field strength because of SU2 gauge invariance. So W mu nu alpha with its own coupling strength, this is the SU2 left coupling strength here, divided by m star, m star square, because I have one W mu field and also one derivative. And each of them cost m star, so there is one m star square. And then there is the rho. The rho comes with its own coupling g star, rho here I have to have two derivatives, so let me write this as rho mu nu alpha, d mu rho nu minus d nu rho mu, divided by m star square. So that's the thing, which is just c over 2 g over g star, W mu nu, rho mu nu. So you see that this mixing is present and it's going to mediate the interaction of eventually it's going to mediate the interaction with the fermions, but you see already that it is a little bit suppressed by the presence of this g over g star. G is the rectal wave coupling, this is the new strong sector effective coupling, so this thing is expected to be smaller than one. How small or large depends on g star, depends on this other coupling, on this parameter that we see before, the one that controls the interaction strength with the retrovik bosons. Now I will not bug you with the calculation of the vertex that comes from this, I mean the calculation is very simple, there is a kinetic term mixing, and so you have to eliminate it before doing anything, and the simple way to eliminate it is to redefine the W field as W plus g over g star zero. And when you do this operation, you make kinetic terms and mass terms, everything diagonal, very nice, but of course you introduce from the usual W boson coupling to the quarks, you spit out the term, which is proportional to rho. So in particular, like in the standard model, you have something like W mu a g q bar left, rho t mu t a q. And when you do this operation, you do get from here a vertex with the quarks, that it goes like g square, because there was a g also in front of the, so there are two g's, one is in front of the original vertex and comes from the shift, g squared over g star, rho mu a q bar, there is a c also, three parameters, got the one q bar left, gamma mu t a q. So also in this sector, the coupling of this vector to fermions is completely predicted, because it has exactly the same format as the SU2 standard model current, it is controlled by one second parameter, on top of g star there is also this order 1c, but aside of this, given that c is already one, that's a strictly well predicted. It's predicted and it's also predicted to be rather small. Typically, one typical example of new physics theories are z prime models, which have typically normalizable wicked coupled origin, and so you typically imagine that they do have a coupling to quarks, in order, let's say, of the rectroweak couplings of g. Here you get the result, which is g over g star smaller, that of course is very important, if you then go to compute the cross-section, because all this then enters still to the square when you compute the cross-section at the LAC. And so this makes that the production rate of this thing is rather smaller. It's... Well, if I have a second afterwards, I will show you... I think I will show you the limits. And sorry, the last thing is that by this vertex, so here I wrote it for the quarks, I could have written also for the leptons, and so in general for all the fermions, from this formula I can also compute the partial width of the, say, rho plus going to fermion-fermion prime, which you can completely predict together with the width of the rho zero going to fermion-antifermion, there is a factor of two, multiplicity between the two, and this is... Well, it's the number of colors, which is there or not depending on whether the fermion is a quark or a lepton, and then there is this suppression factor c over gd square m rho por m star over 48 pi. So, in comparison with the previous one, which is written here, which is... which has g star square rather than this g square over g star over these squares, it's clear that the branching ratio to fermions is typically small, so the gamma of the rho to fermions is much smaller than the one to vector boson, and so, in particular for large g star, and so the phenomenology is very simple, at large g star you do have a reduced cross-section, which makes the thing most difficult to see, and the first place where you should see it, at very large g star is in the diboson final state, because the branching ratio to diboson is much larger, while dileptons are much more difficult, so even if the search are rather accurate, are rather powerful, they are not yet there to see them, and, well, I can show you a couple of plots, if this thing works, and if you don't have some question in the meantime, that is not a Spurian analysis. Spurian analysis has to do with symmetries. In the end, the question was about Spurian analysis. I think you cannot give a charge to m star and g star in any meaningful way to obtain this. If you want, you don't have to give them a charge, you have to give them a scaling dimension, which is the coupling. So giving a dimension to things is like allowing them to scale one independently from the other. And so I think the coupling counting is exactly doing what you are saying. But Spurian analysis is a conceptually difference. We also do it always in composities, but it has to do with symmetries. No, the symmetry here does not control everything. You see, the symmetry does not... The symmetry tells me these relations. The symmetry tells me that this branching ratio are related, this branching ratio are related, but it doesn't tell me how to estimate the relative size of the, I don't know, the branching ratio to fermions, that's another... If you want, it's a dynamical information. And now, this is an example of how we are searching, how the LSE is searching for these resonances and what is doing at the level of mass reach. So the yellow thing here is the search in dileptons, two-leptons final state, which, as you see, are very effective when the... This gv here means g star. So when g star is one, is of the order of the retroic coupling, then there is not suppression, neither of the branching ratio to leptons nor of the production rate. And so the limit is very strong, is what our experimentaries colleague like to call the limit on vector resonances, which is 3TV. But as you see, if you actually have the preference for a large g star, given that you are dealing with strongly coupled sector, two things happen. First of all, the limit becomes much more weak because we have less cross-section and also this yellow curve loses predictive power and you basically have to rely on the blue curves, which are searches into diboson final states. Overall, a plausible limit for the current ATV searches, let's say if you stick at gv around 3, which seems plausible for a composite sector and it also complies with other considerations, the limit is of around 2TV, which is not yet very much in trouble with respect to naturalness. And so the LAC instead at 14TV will be able to say much more about this. And this is for instance shown here, where you have this small area here, which is the whole plot of before. This is the ATV LAC. That's the LAC. That's the 300 investment at 14TV. You see, a small coupling, they really exclude extremely large mass, but still, it's capable to test composites up to around 3TV if we take coupling equal to 3 as a reference. So that's gonna be... And also you can notice here that more luminosity, which you could obtain by iluminosity is not useless for composites. It's typically meant to be useful for a possible application, but in the case of composites it is not, because it doesn't do much in the same mass direction, on this direction, but it does a lot in the coupling direction. When you increase this coupling, as we saw there, the coupling to course gets reduced and the solar cross section gets reduced and the discovering signals with small cross section is what iluminosity adds you in doing. And if you look at the other plot, you can imagine having an 100TV collider and that's the impact it would have in the mass coupling plane of, say, this vector resonance and in general of compositic idea. Oh, there is a question. The question is whether these masses should really be at the TV scale. The answer is always the same. I find well, if you believe in fine tuning, well, I don't see. If you want to have a model where the Higgs mass is natural, which means that the Higgs mass is not only calculable in line of principle, but it's also calculable in concrete, then you cannot have all these large fine tuning. And so your masses should be in the TV range. Okay? But if you have other reasons to do compositics which I don't know which one they are, but there could be, I mean, there could be several. If you have other reasons, then you can go on for this also at FCC. Good question. So the question was about the retro-reposition test. I have here on the notes an estimate of retro-reposition test contribution precisely from this resonance, but I didn't have time to discuss that. The situation with retro-reposition test is the following one. Retro-reposition test parameters which are relevant for this discussion are the S parameter and the T parameter. Probably past in lectures you saw what these things are. And the vector resonance precisely by the covering we already wrote down gives a contribution to the S parameter. Not to the T parameter because of custodial, but it is noted here as this contribution, vector resonance contribution. Next, there is another effect which I mentioned quite a couple of times. That is, if you modify the X-Caplic, this back reacts into retro-reposition test by the relative connections from the standard model. And so this red thing here depending on side. So the thing is that this contribution goes in exactly in any direction with respect to how the ellipse is. You see it goes really standard model is just a little bit one sigma down here and they go immediately very much away. So I would say that this side is 0.25, which is depicted here is probably excluded. And then however there is a further model dependent contribution that depends on fermionic resonances. And this is denoted here as this line here so easily go back. So you see on this plane I'm not working out here so that I cannot hope to go back even with the resonance which is reasonably light. For instance this line here is the strict retro-reposition test bound from S on this resonance. This corresponds to a certain length if you want of this vector contribution here. So 2TV is marginally inside what is not still completely excluded by retro-reposition test. You see if you work enough in any model you can account for any signature. But it is clear that for example a vector resonance decaying to diboson like this one has really no reason to be there in super symmetry. While it has a reason to be there here at the same time it is clear that if I find a squork I don't think to composite x. I don't think that would be distinguished to such a different scenarios like super symmetry and composite x would not be difficult. Of course it would be much more difficult if super symmetry or composite x is discovered to characterize what exactly has been discovered the details of the model. That is going to be much more difficult and that would require under TV. If there is no access who knows, there is a little access in the diboson searches. If this was true it is compatible with resonance like this at 2TV, if that was true then at least for the first at the beginning it would be obvious that this is on the composite side not in the supersymmetic side. But all the models that have extra gauge bosons depends on the origin but if you give them a microscopic weakly coupled origin in term of a gauge theory then they do not show this nice thing of the suppression of the coupling two quarks and two leptons. So they typically badly excluded by dilepton searches which doesn't mean that you cannot make them survive, for instance the only couple of two quarks and not two leptons but then you have to start telling me why you want to play this game while the thing is like that. I think we have no, I will question so.