 To je videla sega otvaru vegga. Vz solej sega kot sem da bo je teba deločne preddelavanje z k stri bi je izgleda. Spetra vsakazila o cosvenih modelov. Vozunje od vsi, da je tudi izgleda, neč nekaj, nekaj. Protočno je, da jih ima rejbo volj. Kam je jih ga za del nekaj. Kaj je tudi tas ni deleti components. Zato v zelo, ta tanto se veče vše, ko je obista v spetra z tudi. kako zelo našem je odličil. So, da smo vse z vse. Je toga, kot je to, da se začem, nekaj je, kot je to, ki se vse, način se pravamo, in nekaj, da se zelo našem vse. In zelo, da je to, ki se začem, nekaj, da se začem. To se začem, kaj je to, kaj je to, kaj je to, ki se začem. s4. Kaj? Zelo, kaj je 10 ruprezentacij, in je to naparete v s4 rupre, v kaj je 6. Od s6 je s vsega ruprezentacija, plus je še in ekstra večka, ki, zelo, pravite, od večka nolega in gorsom teore, je še in ekstra večka, in vem z tem, da je tjena tega, ko je vsega iz takrat, ali to, da je doljston, vzelo, tako doljston, vzelo sa svim svojom. Da, oče, vzelo, vzelo v svojom svojom kot doljstv. Zelo pristite, da sem taj zelo tudi iz vsega generatora. Ne, taj zelo pristite, ali sem taj zelo generatora, ko je v 10, in nej se v 6, začel, kaj je ta,の1 Soup panes I Don't yetago light. So A gain are six as a four generators since our broken ones with the notation so quote unquote unbroken. T'cote speak broken these four ones with the same terminology we used yesterday so basically you already see that from this breaking there are four real goldstone in se je zrčil drugi komponenti del 1.6 dublet. Zazim smo na različke trakje. Znači, da prišličen na naprej, ko ustravimo enakvom modelom, kako g nebo h, je, da h, kaj je sovr, svoj tudi, da je dobroje, da je tukaj však. Začal si, da je zelo, da je to pričočena? Pa, da ne ne videl, pa da vedel sem, da je tukaj však však, da je tukaj však s veči što je zelo, veči je boščena. To je, da je tukaj však. Pojede je in vsevem, nekaj sej sej sej vsevem, Kostodiasimetri, ki je tukaj Kustodiasimetri. Zato vzivim, kako je minima kompositivna modela izgleda na taj coset, in vsega, kako je izgleda na taj SO4 unbrokovan grub. Vsega je, da SO4 je tukaj algebra, tako zelo, da je vsega vsega, ovo je vsega vsega vsega vsega, do vsega vsega vsega vsega, the two left cross S-U2, S-U2 right. I will briefly show this, because this allows me also to introduce some notation that will be useful in what follows. So, that is the K input. So it is independent and unrelated S-U2 left and S-U2 right transformation. This piece S-U2 left commutes with this one, with S-U2 right in komutacije generatorjne, vzelo vzelo in japočke vse japoče. Vzelo vzelo in tudi se zelo vzelo, ko je tudi bilo vzelo vzelo, ki je zelo vzelo 2,2. Zelo se, da je tudi vzelo, da tudi vzelo vzelo iz Vzelo, su tudi vzelo iz Vzelo iz Vzelo, Or explicitly the sigma matrix transforms as sigma that goes to g left, which is the SU2 left transformation, times sigma times g right diagonal. Furthermore, this 2.2 is a pseudo real 2.2, so it is a matrix which is subject to an extra condition, which is that the conjugate of this matrix is equal to sigma 2 sigma sigma 2. And you can check by yourself that this is a consistent restriction, which means that if I start from a matrix that obeys this condition, if I act with the group I end up with a matrix that also obeys this condition. So that's a valid restriction that in total gives us to this 2 by 2, you see this matrix is 2 by 2, there is g left, which is in the 2 representation, g right in the other representation here. So this is a complex 2 by 2 matrix, which a priori would have 8 degrees of freedom, but after this condition it has only 4 real degrees of freedom. And so this sigma matrix can be written in terms of 4 real numbers, which will eventually correspond to the 4 real components of the 4 plate of SO4. So this is something like 1 over square root of 2, i sigma alpha, the 3 poly matrices, pi alpha. I'm calling this pi because they are going to be the Gorshton bosons in our theorem, and in particular they are going to be the components of the Higgs, these guys, plus the identity, the 2 by 2 identity, multiplied by the fourth, pi fourth, ok. And now we can check by using this specific representation that the 2 groups are locally isomorphic, because you can take the trace of sigma dagger sigma, you can compute it, and discover that it is the modulus of pi square, so it's the norm of pi. You can also check that this trace is invariant if you do this operation, because you get only g right and g right dagger on the left and on the right, the trace is cyclic, so keral transformation preserves this trace, which means it preserves the norm of pi, and you also know that SO4 is defined locally as the group of the most general rotations you can do on the fourth plate pi. So this tells us that any SU2 left cross SU2 right transformation must actually be an element of SO4, so it acts as an element of SO4 on pi, so SU2 left cross SU2 right is contained inside SO4, or it's equal, but then if you count the number of generators, you discover that SO4 has six generator, SU2 left cross SU2 right has also six, three from here plus three from there, and it can be shown that there cannot be two lig groups, which is one, the subgroup of the other, and they have the same number of generators. The only possibility is that they are equal, or better said, they are locally equal, in the sense that they are equal only for transformation, which are closely connected to the identity. This means in particular that the, so we can, for our purposes, continuously switch from the SO4 or SU2 left cross SU2 right notation, and so it also means that the generators of SO4, this t-literate here, so they actually obey the same algebra of SU2 left cross SU2 right, and so they can be split into two sets, a t-alpha left and a t-alpha right, with alpha going to one to three, and these two sets commute with each other, which correspond to the fact that this is a product of groups, and the algebra of these two subsets is just a one of SU2. So that's going to be our notation for the SO4 generators, written in these bases, these close SU2 bases. There is another couple of other things that are going to be useful in what follows. So this sigma field, so now you see SO4 is made of this SU2 right, so when we identify the standard model group inside here, the obvious thing to do is to identify SU2 left factor as the SU2 left of the standard model, and to go and look for the U1 hypercharge to be inside this SU2 right. So what we are going to do, we are going to identify the electroweak group as this SU2 left is equal to the SU2 left, and the U1 hypercharge is corresponds to a generator Y, which is the third right-ended generator of the SO4. And so now we can also read from here how the Higgs doublet will be made, I mean what the Higgs doublet will be made of. So one SU2 left transformation, acting on this sigma 2 by 2 matrix, it corresponds to acting with SU2 left on the left. While the hypercharge corresponds to a rotation of SU2 right, which is diagonal, because it is just the exponential of the third SU3 right generator, which is sigma 3 over 2, and so this operation corresponds to the following identification of sigma with the Higgs. So this sigma matrix we can write it as h conjugate h. So h is the column of doublets, so h is just the standard model X doublet h up, h down, hc is i sigma 2 h star, so is the conjugate X doublet, you encounter also in the standard model. hc transforms exactly like a doublet under the SU2 left, so you see that both the columns transform like doublets, and so this is consistent with the fact that you want SU2 left to act as matrix multiplication from the left. So line column, this gives that SU2 left acts on this thing exactly as it has to be, in the sense that trace the Higgs as a doublet. Then hc is the opposite hypercharge as h, and this is correct, because the hypercharge is going to be here the conjugate of g right, which is going to be the exponential of the, so e to the i t3 right, plus or minus 1 half above or below. So by this identification we identify an X doublet, which as in the standard model, lives in the two 1 half, so in the doublet with 1 half hypercharge. Now that we have the map between the complex X doublet notation and the pseudo-real 2 by 2 matrix notation, since we know what the pseudo-real matrix is in terms of the four-plat pi, we can finish the exercise and derive the map between the Higgs and the pi's. So these four real pi's will be equivalent to the Higgs and the equivalence goes like that h, so h up, h down is equal to 1 over the square root of 2 pi 2 plus i pi 1 pi 4 minus i pi 3. Ok. And we are always going to use independently all these different notations. Ok. Of course the four-plat notation is the one which is more useful to formulate the non-linear sigma model, but in the end we want the Higgs doublet notation. Ok. So that's the first, the first thought. Now I hope you are convinced that inside the SOF4 there is the standard model group that we need, but there is also custodial symmetry. I think peskin lecture discusses a bit custodial symmetry, so I will go fast. So custodial symmetry is the following observation. If the Higgs takes its web, which is the first component equal to 0 and the second component is real, V over the square root of 2, ok, then the sigma matrix which is written here just becomes proportional to the identity, so the web of the Higgs written in the sigma matrix notation is V over the square root of 2 times the identity. Ok. And so you see that it does not break completely these SU2 left cross SU2 right group. Remember this was acting as G left on the left and G right dagger on the right, so this configuration which is proportional to the identity preserves transformation in the vector subgroup. You can check. I mean obviously web of sigma under the transformation which has G left equal to G right equal to G vector. Well, it remains the same, ok. It remains the same, because you have GV dot sigma GV dagger, ok. Given that this is proportional to the identity I can commute and I get the same, ok. So after even after atroic symmetry breaking there is a symmetry that remains in this theory given that there was this original SU4 and this symmetry is SU2 vector also called SU3 which is the same SU2 custodial custodial symmetry. The reason why this symmetry is important is that the coupling and I think Peskin explained this to you, the coupling of the W left which occurs with this generator here, do not break this symmetry. So under custodial the W mu alpha fills of the standard model form a triplet of custodial and this offers a certain number of selection rules in particular in the mass of the W's that makes this more easily compatible with observations, ok. We will see an effect of custodial symmetry in a couple of minutes but just remember that to build a reasonable composite theory if you don't want to do an enormous fine tuning and enormous small psi you need to have a group which not only contains the electro weight group but it also contains this enlarged SU4 which is SU2 left cross SU2 right. Ok, so aside from this little group theory complication the minimal composite is Lagrangian is just the same form of the one we wrote in the abelian example so we are going to take a scalar field of course this scalar field now is a phi plate of SO5 is not a triplet of SO3 so transpose mu phi ok, we are going to add the potential minus G star squared over 8 phi transpose phi model squared minus F squared ok, just the same one you had before now the covariant derivative is obtained by gauging the SU2 cross U1 generators and we already decided how to do that in particular we are going to have that so the W mu alpha gauge SU2 left and the generators we call them T alpha left so minus I G prime D mu this is the hyper charge gauge boson that gauge is the hyper charge which we identified as T3 right applied, of course that is the one derivative of phi ok, so this is what we mean when we say that the electro weak interactions are introduced by gauging ok, are introduced by plugging into the covariant derivative if you want by coupling the gauge fields to the current that corresponds to the appropriate generators that we have chosen to represent the standard model ok, and all the derivations we did last time can be done in exactly the same way in particular you can go to a nonlinear basis which is going to be phi equal to the Gorshton matrix U of pi acting on F plus sigma sigma is going to be the resonance and once again it has the mass I remind you M sigma equal to M star equal to G star F and the pi's are going to be the massless Gulton boson now we have four of them as I told you and the Gorshton matrix just to be explicit the Gorshton matrix is e to the I square root of 2 over F so even the same normalization we had last time pi I t at I where now of course the sum runs over all four indices so we can work out everything like before and I'm going to give you the result for the Lagrangian for these Gorshton X in the unitary gauge so let's go to the unitary gauge where H is equal to 0B plus H over a square root of 2 capital B is the verb of the composite X this is the fluctuation and let's see what we get so we get the usual kinetic term for the Higgs that stays alone then we get and here is where we get F square over 8 sine square V plus H over F that multiplies so there is G square the master for the W so W1 square plus G square W2 square plus the following thing GW3 minus G prime B square and custodial symmetry is what is ensuring that in front of this third term in front of this plus here there is just a one and the reason is that when there is no hyper charge and only the SU2 left gauge bosons are gauged custodial symmetry is unbroken and the 3Ws form a triplet of this symmetry this master must be compatible with this symmetry and so the mass must be W1 square plus W2 square plus W3 square up to only one coefficient which is G square so that's essential that here we have one and that is enforced by custodial symmetry as it happens of course in the standard model and then as in the standard model what we are going to find here from this square from this square bracket is an expression that is G square times the charge W mass model square there is actually a factor of 2 coming from the square root plus 1 over 2 the cosine of the binary angle square z square ok, so here I just made the usual binary rotation to go from W3 and B to the z and I switched to the complex component complex notation for the W field and from here you see that well, the usual relation between the W and the z mass ok, that you have a tree level in the standard model is present at tree level also in this also in this theory ok, so the relation I am talking about is of course MW square equal to CW square MZ square so when this takes a path this gives a mass to these guys but it gives them with this relation ok, in this form so as you know in composites I can always recover the standard model in any situation so even if I had started with a model without custodial symmetry I would still have found this relation but with corrections with corrections that come from some complicated trigonomic function that can be different here than there and this would have been a big pharmacological problem because it would have required a very small xi a very small xi parameter ok, so here we see now electro-wiximally breaking so in particular how the W and z get the mass from this first term in particular we find that the W mass is one-half gf sine V over f and in analogy in the ordinary standard model where just the formula is mw equal to one-half gv let's call V all this part ok, so this has to be fixed at that 246 gv ok and it is the other mind by combination among the composites x-vebs capital V and s ok we also compute xi which is V square over f square and we find that it is given by sine square V over f again it's the sine square of the misalignment angle as we did as we argued on general grounds and we verified those specific except and then I can expand the sine square and find the x interactions one x to W to x to W three x several W's and similarly to again to what we did in the billion case and only that now we are talking about real, say, word Higgs something that could be really present and so let me rewrite the result which by the way is identical so we have a modification of the copying to W and to Z with the single Higgs that come from expanding this at the first order which is universal for the W and for the Z and this is again the consequence of the soldier's symmetry and so it's called Kappa V which is Higgs vector vector in, say, composite Higgs divided by G Higgs vector vector in the standard model which is the one we already see square root of one minus sign the question was how large, how bad would it be if I didn't have to study a symmetry in my theory in practice what you have to imagine that rather than finding sine square V, so this trigonometric function factorized out here you will find one trigonometric function of the W's and one different one for the neutral part so, of course, in the limit F going to infinity everything goes to the standard model because of the usual vagumizalignment mechanism but the deviations are over their psi, are over their B square over S square, sine square B square over F square basically you will have found in the case of no custodial a correction over the psi to this relation now this relation well this relation has been measured very precisely, the corrections from this identity come in the standard model from loop effect so loop effect of the standard model correct roughly speaking this relation by around one percent the measurement is at around one per mil and the one percent works with the standard model and so this means that you can tolerate deviations into this relation so you can tolerate extra deviations of the order of per mil at most which means psi equal to one per mil or tuning equal to thousand so if there are no more questions on this part I will move to the fermions so this is the leading or good question so you can view this in several ways so you can view this as the low energy effect the full low energy effect that you find in the theory including the change of every resonances so basically what I claim is that this result is the one that you have at leading order in a derivative expansion so it is subject to corrections important which typically are not very important this is why I stress so much this result is controlled by basically the W mass over the resonance mass square so they are typically negligible with respect to that they come in the full say non-linear sigma model analysis from higher dimensional operators while these ones are only the ones of dimension 2 of two derivative order so there are corrections but they are small it is also important to remark that even though I derived this coming modification in this model it is possible to show that as far as this calculation is concerned the dynamics of the goldstone boson is universal and these results are really universal at the leading order in the derivative expansion but to show this will require more techniques that I'm not going to introduce so now we can we can think so basically for gauge sector we made it with gauge fields and the X field electroissimally breaking giving mass to them and so on now we have to worry about fermions so we need to generate fermion yukavas and of course fermion masses after the X takes effect so here it is interesting to make a digression before to introduce the partial compositiveness framework because it is an interesting quantum field theory subject related with the alternative to partial compositiveness so I told you that we are going to do this by partial compositiveness but that's not the way in which fermions get their mass in the standard model and this is not the way in which the fermions get their mass in the yukavas in say old fashioned technical models so let me first let me do a first attempt to generate these yukavas in the following way that's the old fashioned technical way so what I need to do I need to write down some interaction of the fermions remember that the fermions were living in a different sector in the elementary sector while the X was in the composite sector and so I have to write some interaction among the elementary sector and the composite sector and remember I'm willing to believe that this theory is going to be generated at a very high scale which I call the lambda uv and then it flows until the confinement scale m star and the physics I'm studying is at m star but I would like my theory to be well defined starting from a uv scale which is far above m star so what I want to inject into a theory is an assumption on this interactional agrangian at the scale lambda uv and the first thing I can do to make the fermions interact is by writing bilinear couplings so I will have my composite sector that perhaps it delivers a scalar operator which I call OS that happens to have the quantum numbers of the Higgs field that's not difficult to imagine that and so I can write something like Q bar OS actually here I will need OSC which is the i sigma 2 OS star conjugate like for the Higgs T write so perhaps that's the leading so the lowest dimensional interaction I can write in my theory perhaps this is the leading interaction which obviously makes Q and T talk with the Higgs the Higgs takes a verb so makes Q and T get a mass at low scales this operator has a coefficient which I call lambda t and it has however it must have the correct energy dimension and we don't know what is the dimension of the scalar operator let's call it d and so here we have to put lambda uv to the d-1 that's the usual dimension analysis exercise we did the very first lecture so if you have that the scale so the physics scale where you are writing your interactions lambda uv then lambda uv waits all the dimension full interactions ok and then I could write the same thing a similar thing for the bottom ok this operator OS I said must be must have the quantum numbers of the Higgs must be in the 2 1 half of the electric wave group it's equal to its dimension in energy dimension of OS actually that's the way which the fermions communicate with the Higgs in the standard model and there is it's particularly simple what we have we just have one elementary operator OS which is the X doublet that has a dimension equal to 1 actually here when I speak about dimension I'm most properly speaking about scaling dimension which need not to be exactly integer and indeed they are not integer even in the standard model there is some amount of anomalous dimension so the dimension is close to 1 but not exactly 1 in the case of the standard model then there are other examples like technicolor in technicolor this operator was completely different was a fermion bilinear was 2 say 2 technichermions ok so new particles with the technicolor gauge group which made a bilinear so given that in technicolor the strong sector was almost a free theory at varied energies the dimension of this thing was more close to 3 ok because it's a fermion bilinear then to 1 there is a problem with this picture there is a structural problem with this picture that can't if so there is a problem if d is larger than 1 ok significantly larger than 1 and the problem is that if d is significantly larger than 1 this interaction operator which I have written is a high dimension operator it's an irrelevant operator as a dimension larger than 4 and so it's a fact that low energies are gonna be rather small ok what we can do is that we can define the effective strength of lambda t lambda t was the coupling I defined in the uv I can define an effective lambda t coupling strength at the scale m star that is at the scale of confinement where I want to define where I want to see the implication of my theory in the following way so the lambda t comes together with this 1 over lambda uv to the d-1 ok so better be here a lambda uv d-1 into the numerator and on the numerator I put the typical scale that corresponds to the processes which I'm interested in ok so which is m star so m star to the d-1 times lambda t in the uv so basically what I'm saying is that the coefficient of the operator which is lambda t over lambda uv to the d-1 will appear will affect processes but at low energies only in a dimensionless combination because I want a dimensionless thing which is m star to the d-1 over this one ok this can be done precisely by Kalansimanski running equation but doesn't matter much here the point is just that now the effect of this operator can be tremendously suppressed if d is larger than 1 because m star over lambda uv is very very small ok so in particular the first thing I would like to get out of this is the yukawa coupling and in particular the yukawa coupling of the top quark that's why here I started from the top yukawa terms because that's the one which is problematic and the yukawa of the top will be exactly proportional to this lambda t written here and so it will be let's write it once again m star over lambda uv everything to the d-1 lambda t ok and this can become a huge suppression if there is a big-scale separation between m star and d m star and lambda uv ok so the problem here is that we are not gonna be able to reproduce a sizable enough top quark yukawa if d is not close to 1 and if and if the hierarchy between m star and lambda uv is large enough after all we wanted all this thing to explain the size of the x-mas that will emerge after composing the tricks and everything from m star but the main point was here we want to explain a big hierarchy between lambda uv and m star and here we are not able to do that ok now one can play around with numbers and try to see which is the maximum lambda uv that you can get well this depends on how large can it be the lambda t coupling in the uv typically there is a bound of perturbativity so I could be more precise here but so imagine m star over lambda uv of the order of 10 to the 12 ok which corresponds to 10 to the 15 in uv physics scale here you can imagine that this is an enormous suppression ok of course this was for d larger than 1 for instance in the standard model there is not this problem because this is over the 1 however in the standard model there is another problem that is the fact that the x-mas term is dimension 2 so as we saw in the first lecture is the guy that causes the arising of the hierarchy problem so people have been considering the possibility of some hypothetical theory where the dimension like similar to the standard model is close to 1 but not exactly 1 is 1 plus epsilon ok such that after all this is not a big deal and you can eliminate this big suppression however some other people showed that if d is 1 minus plus epsilon so if the dimension of os is 1 plus epsilon ok in a general even strongly coupled near conformal fixed point theory well then the dimension of the square of the os operator is equal to 2 plus order epsilon correction in the standard model this is close to 1 and indeed the x-mas term which is os square as dimension 2 now the x-mas term or os square are dangerous operator because they are singlet operator they are operators that can be put inside the Lagrangian and so if you have one such operator of dimension 2 or close to 2 you do reintroduz the hierarchy problem you will need to have only operator of dimension larger than 4 not to have the hierarchy problem so this result implies that basically there is no way out we cannot expect any theory that gives a realistic top yukawa which has a big gap as we want to have and at the same time does not suffer of the hierarchy problem because it will always have in his own Lagrangian so it will always be something like the standard model and this is the main motivation for considering alternatives and for considering partial composiness and I stress this because partial composiness has a lot of implications physical implications so it is better to understand where it comes from partial composiness is this idea that the interaction Lagrangian now is not made of bilinears but is made of linear couplings so now you have something like q bar left and now you have some strong set operator which now is not a scalar anymore it is a fermionic operator which we call here OF t left k it mixes with the q left third family doublet and the strength of the interaction we call it lambda t left and now here the scale will be at the power d left minus 5 over 2 similarly I need to couple the top right to make in the end up and to arise so lambda top right lambda u b dimension d right which is the dimension of the corresponding operators minus 5 alpha OF t right so now what is the difference well now the difference is that well there is no much different from before only numeric changes so now we have that the effect of these couplings for instance lambda t left at the scale m star can still be suppressed by an amount m star lambda u b to the d left let's say minus 5 alps you see where these 5 alps come from right comes from the fact that sorry I forgot the top right comes from the fact that here you always have one elementary fermion which has dimension 3 alps and so here you want to get dimension 4 and you do this by this d left minus 5 alps so now there is this power here d left minus d left lambda d left so now what do you have fermionic operators to avoid the big suppression are fermionic operators which have a dimension close to 5 alps so partial composiness in order to generate the top mass requires the existence existence of fermionic operators with the ok so is this ok well clearly even if you are very naive and you consider the square of this operator you take the square of this fermionic operator and you assign to it a dimension which is around 2 the dimension of the operator itself so you consider something like OF bar OF you end up with the dimension of 5 and which is not dangerous from the hierarchy problem it's an relevant operator furthermore there are examples of theories that obey all the rules of conformal field theories which are the 5 dimensionographic theories operators of dimension 5 alps are actually easy to obtain what is somehow lacking in this picture is a microscopic realization of a theory with these one operators and with these specific dimensions I don't think there has been still enough work on this because it's a very interesting subject of course the simplest thing you can imagine is the following in dimension 5 alps even in a weekly interacting free theory where the dimension are just counted by energy dimensionality you may want to consider an operator which is made of a scalar times a fermion like you have a scalar constituents weekly couple to a fermion constituents of course this requires a scalar to be made and then it's better than in supersimmatic theories rather than in ordinary non supersimmatic theories or there are possibilities like that this dimension 5 alps is generated because of the running towards a because the theory in the UV is not weekly coupled and so it has a fixed point with particular properties and seems not unlikely that one operator of dimension 5 alps of scalar dimension 5 alps can exist so that's what we need to assume to do the top mass where there is an hierarchy between M star and lambda UV which is significant enough for us to be happy or satisfied a couple of implication of this partial compositiveness idea so there is a striking implication which marks a big difference of this with the technique color way so you see that now I'm writing before I was writing like Yukawa like objects and so the color index of the quarks was contracted implicitly between the quarks but here I'm writing a linear capping so I'm writing Q coupled to O so to make this comply with QCD gauge invariance it must be that O must carry QCD color so this implies that the composite sector must carry color so the constituents of the composite sector must carry color charge and the color group so SU3C must actually be a global symmetry of our theory so basically this immediately implies us to enlarge the global symmetry content of our theory from SU5 that contains the standard model was broken to SU3 color times SU5 and not only this this has important implication because if you have that the composite sector is colored it's clear that it will also deliver colored resonances so this makes colored resonances which is something you did not even think to have in technique color for instance where you had no reason for the composite sector technique color sector to be to be colored ok second we can try to explain a little bit this is called partial composiness which I still didn't do ok so this has to do with the fact that here I have a mass mixing, something like this I have Q mixed with the operator and I can view this more precisely in the following way so this elementary composite mix let's just write it as lambda Q bar O ok all the lambda UV scale powers inside the O now this operator O is supposed to be a gauge invariant operator gauge invariant with respect to the strong sector confining group so OF is an observable operator and as such we expect it to be associated with particles there is a sort of correspondence between operator and particles which is the usual thing that an operator interpolate from the vacuum for a field so even that I have OF I do expect to have particles with suitable quantum numbers such as to be excited from the vacuum or destroyed from the vacuum by the operator OF so there are going to be particles associated to OF associated means they have to have the same quantum numbers otherwise this one would not, would vanish so the particles I have to have for instance there is a particle Q which belongs to the same representation as this operator so let's pallid out all of it so this operator couples with the standard model left handed doublet so it is in the 2, 1, 6 of the standard model we said it also the quark is also QCD color and for this reason we say that this operator has to carry QCD color in particular it is in the triplet of color and so we do expect to have sorry it is a fermionic operator so we expect to have fermionic resonances with this one quantum numbers at least this resonance is going to be massive so what we do expect are massive spin one half particles associated with these operators that I am calling Q so out of this interaction term when I write down a phenomenological low energy Lagrangian that involves the quark and also the heavy particles this will result not surprising in a mixing that I can conveniently parameterize as lambda times F F is the course on boson X decay constant times Q bar Q so that's just a mass mixing between the Q left of course and this so then you also have a mass term for Q itself which is of the order of the M star scale clearly so M star Q bar Q and so you have a mass matrix involving Q left and Q right component capital Q and little Q so you diagonalize the mass matrix and you find that the physical massless eigenstate which emerge from these two terms is not exactly the Q but it's a combination of little Q and capital Q so the physical the physical state the physical Q well, before that we seem to be breaking so the massless Q left guy is actually going to be some cosine of theta little Q field so little Q state plus sine theta ok where this sine theta is the compositeness fraction in which you have to rotate the matrix to diagonalize this mass matrix and you can immediately find that this sine theta is of order lambda F over M star ok and since I told you several times that M star the resonance mass we call it also G star F ok with this is lambda over G star ok so this is called partial compositeness because the physical states are partially and by the amount which is sine theta made of composite there is a freedom and this partial compositeness so this holds of course for the Q but of course it also holds for the for the top right which was written there you have lambda T bar right of F T right so in this case you will have another state in the singlet with the appropriate quantum number that is called T tilde ok and then the thing works in exactly the same way so also the top right is partially compositeness by an amount which is a weather so in this case we have Q so the coupling was Q left so the amount of compositeness was lambda left ok in the other case you will have lambda right lambda top left or lambda top right ok so and this one is the thing that also makes in the end the Yukawa originate at least in a schematic in a schematic picture so in order to make the quark talk with the hicks what do we have to do the quarks in elementary sector so they cannot talk with the hicks before they go to the elementary sector and to go there they mix with the fermionic operators so basically all what makes the quarks interact with the hicks is mixing which then transforms the Q left and the top right in their corresponding Q and T partners and then the Q and T partners are resonance so they live in the composite sector so they can interact with the hicks as much as they like ok and they will do that with the strength again g star g star is and is gonna be the typical coupling strength of all the vertices of the composite sector and so this gives a diagram out of which you can generate the Yukawa of the quarks in particular the Yukawa of the top and you can estimate to be g star times the compositeness fraction so sine theta left sine theta right which is so lambda T left lambda T right over g star now the g star power for the moment doesn't matter much ok, that's the correct one here we just want to notice that the partial compositeness mechanism controls the generation of the Yukawa and so how much are partially compositeness the fermions that is how large or small I have to take this lambda T left or lambda T right depends on how much or small I want to be the Yukawa so for the case of the top I know that I want a large Yukawa and it's going to be a large compositeness fraction ok, while for the case of the other quarks which have a very small Yukawa well this lambda left and lambda T right better be rather small so the picture works with a large partial compositeness for the top ok and instead a limited or very small or negligible partial compositeness interaction for the other quarks ok, in particular this makes have a hard time to interact directly with the composite sector and this is helpful in certain respect with flavor phenomenology but not ok ok, the question was the difference between the partial compositeness and the technicolor bilinear way let me tell you this once again so the only difference is that in the bilinear way in the bilinear technicolor way I'm sure I need the scalar operator of dimension close to 1 and I'm sure that the scalar operator of dimension close to 1 reintroduces the hierarchy problem in my theory destabilizes the hierarchy between m star and lambda uv and I'm sure about that because I'm sure that the operator's scalar square it's a singlet operator it's something you can put in the Lagrangian it's like the x-master and so you have an operator of dimension less than 2 so around 2 let's say which would have a huge coefficient and this destabilizes the hierarchy while in the fermione case on one end we don't have such a theorem and we have good reasons to believe that such a theorem cannot exist simply because 5 halfs plus 5 halfs is 5 and because we have examples and so we think that leaving somehow close to 5 halfs is reasonable ok 5 halfs but 5 halfs is not a bad number ok so scalar operator of dimension 2 is a bad number sorry dimension 1 is a bad number because it implies having a scalar square operator of dimension close to 2 fermionic operator of dimension 5 halfs is compatible ok with can occur even in natural theories ok please ok the question so whether having introduced having discussed it here only one resonance is enough or not ok so first of all that one resonance is it could be heavy ok it could be 2TV and we have not yet reached that level at the LAC but of course we are searching for that resonance ok but it can be naturally heavy because it has a mass which is dictated by the composite sector confinement scale that indeed we wanted to take in the TV or multi TV region ok so this guy is heavy first part of the question second part of the answer so that is an illustrative thing because you have to imagine that on top of this Q there is going to be many other resonances in particular in strongly gavel sectors there are towers of particles replicas typically say some splitting in mass so you can think to this as the lightest but this is not the complete calculation this is a sketch other resonances are expected also in the fermionic sector if there are no other questions I can go to I can go indeed to a calculation I can really show you well first I will show you indeed how does the coupling of the Higgs to the resonance look like good question the coupling of the Higgs to the resonances which is this one here in principle you have to be careful they will be subject to certain selection rule dictated by the fact that this guy is in most of both so I am telling you that this one coupling is indeed present is indeed present and it originates intrinsically inside the composite sector so you have to imagine this I don't know to be in analogy with QCD that's large, that's G-star that's a big coupling interaction because it's a strong sector interaction so if it's not forbidden by some symmetry this is the naive sites we do expect naive sites common to all these couplings of the composite sector that I am calling G-star it was also in the Lagrangian this is the same G-star I put in the Lagrangian and indeed, but this I am going to go back on this tomorrow this G-star also in that Lagrangian is precisely the guy that controls the interaction for instance the 3-sigma interaction is G-star proportional to G-star and the 4-sigma interaction is G-star squared and this has a reason for that but ok, we will come back on G-star ok, let's build some sort of a model or at least let's specify better the quantum numbers of these fermionic operators so already told you that so, the fermionic operators for the top left and for the top right have, now let's neglect color ok, let's ignore color for the moment doesn't play in error these are in the 2, 1, 6 of the electro wave group ok, like the corresponding doublet and this is the 1, 2, 3 ok, this is what we would conclude from this formula because of course we don't want to break explicitly the standard model group however these operators ok, are the operators of the composite sector in the UV so, you should imagine if one day we will know this theory to write them in terms of this technical sequence of this theory and to write them in a phase of the theory which is far above the scale of explicit breaking of the Gorson symmetry so, I am writing this operator in terms of costituences for which the Gorson symmetry is still an exact symmetry ok, so, sitting in another way this fermionic operator will come as the product of constituents with their own global group G which will be a flavor group symmetry all this is to say that actually these fermionic operators we are not happy to know what are their standard model representation we really would like to know where they live in terms of G representation so, the OF feel complete multiplets of G of the full global group, not even only of H really of the full thing for instance, in technicolor psi bar psi is a 2,2 of SU2, let's call SU2 right in technicolor it's just chiral theory and it's a full group it's a full multiplet of the full of the full group of the full group, even before the break ok so, these OF are actually larger multiplets of G and the only requirement we have on them is that they have to contain these components these 2, 1, 6 and 2, 2, 3 component in order to be capable to talk with the course in the way we say ok so, let's make a couple of examples which are the 5 of SU5 and the spinorial 4 of SU5 ok, here where I'm talking about the minimal compositing model SU5 so let's see what happens to this 5 so, this 5 of SU5 let's first decompose it under SU4 gives a 4 plus a 1 the 4, remember is the representation where the Higgs lives so, I already told you that this 4 is actually equivalent to a 2,2 actually, these are fermionic operators so, this is a complex 2,2 it's not a pseudo real 2,2 ok, so it has more it has twice the degrees of freedom plus 1 ok so, we know that the standard model acts like SU2 left here and U1 right here so, now we can further decompose this under the electro week group of the standard model and so, we will get from the 2,2 a2 1,2 which correspond to take the doublet part here and the first of the doublet elements ok, which has 1,2 because the hypercharge is 1,2 the hypercharge is T3 right remember, the hypercharge is the third SU3 right generator so, given that this is a doublet of the SU2 right, it has eigenvalues which are of T3 right, which has plus 1,2 and minus 1,2 in the case of the Higgs so, actually this two representation were 1, the C conjugate of the other and so, there was not doublet plus the singlet, which is a singlet under everything so, it has 0 hypercharge so, clearly this one does not work does not contain a 2,1,6 and not even a 1,2,3 you can do it for the 4 the 4 the 4 is the spinodium of SU5 and it decomposes in the 2,1 plus the 1,2 which means basically that so, this is just a doublet of SU2 that does not transform under the right and this is vice versa and so, this 2,1 now is still a doublet of SU2 but now is a singlet of SU2 right so, it cannot have any hypercharge the hypercharge is equal to 0 while the other two have one are singlets and then they have the hypercharge so, there is 1,1,2 plus 1, minus 1,2 again, nothing which we can use to make up so, to have a component we can make talk with the quarts and you can show by if you are good in group theory you can show that you are not never gonna find quantum numbers that you like in SU5 but we don't care of these details and so, what we do is that we further extend the group G so actually, this is another natural thing to do and also well it's not particularly implosible that your group is actually not SU5 but SU5 cross a U1x like before we introduced the color group and now we introduce this U1x global group this is unbroken the U1x and the reason why we introduce this extra global charge is that because now we want to change our definition of the hypercharge because it was our definition of the hypercharge as t3 write that made these bad numbers appear and so now what I want to do is to change the definition of the hypercharge in ratio from y equal to t write 3 ok, I want to make it become t write 3 plus x where x is the U1x charge of the fermionic operator which I'm gonna choose so, what I'm gonna do is to, and if you don't mind I can write directly here I can have a five plate with U1x charge of two thirds and then if I go until the end of the decomposition I just have to add two thirds to the hypercharge and so I find a doublet with hypercharge 7 over 6 which is still not ok but then I have a doublet with hypercharge 1,6 and a singlet with hypercharge 2,3 ok so now I have the good way to couple the top right and a good way a good components to couple the q left what would happen if I had the extra would happen that the x would be neutral because the x starts from a goal stone in S05 so its charges are fixed and cannot be transformed under S05 so if I just give x equal to U1x the x would have no hypercharge I predict a question what we charges are yes, yes from this decomposition knowing that these operators are also associated with particles you can indeed read that is gonna be a doublet with 7,6 hypercharge which contains an exotic charge state in particular in this case in this other case which is the fourplet which is not very popular because it has troubles with ZBB bar so I'm not gonna talk about this but in this other option is different in this other option you have four ones so you have to give hypercharge 1,6 to the singlet and so you get a 2,1,6 plus plus 1,2,3 plus 1,1,3 so you see that this is particularly nice because you can feed not only the top right and the q left but also the bottom right in the same representation it's an aesthetic thing it's not important but here in this case you do not have exotic charges but you are in troubles with some properties related with ZBB bar so I will now take I hope 5 more minutes to finish the exercise so to start and finish the exercise of using this information which is written on this side of the blackboard to really compute the Higgs couplings modification so the modification of the Higgs couplings to fermions by using the formalism of nonlinearly realized symmetries and what I need to do that is to do a trick and then do a calculation so the trick is the following so let's write our elementary composite interaction in the following way the coupling lambda t left the operator the operator now we say this fermionic operator is actually a multiple of the full S of 5 in particular we are going to take the case of a 5 so we are going to take this case here so this is a 5-plat and so it has 5-plat indices i i goes to 1, 2, 3, 4 and 5 and the quarks are instead do not feel a complete multiplat the q left is only 2 and so this coupling is going to break the g symmetry and the way we can visualize this is to introduce here something like q bar left, q bar t left i not to be confused with the capital Q we had previously so this capital Q is just a 5-plat an incomplete 5-plat made of the two components of the q left doublet so basically this q t left explicitly is 1 over square root of 2 minus i b left minus b left minus i t left t left and zero why these numbers so this at the right numbers to select the components of this q left which now you see as 5 components so it's really a 5-plat this is just to select the components of this 5-plat that correspond to the doublet of chart 1-6 in such a way that when I multiply this with that I really end up with what I wanted so with the doublet mixed only to the components of the OF that have the right standard model 1-2 numbers and for the top right is even more simple I just define a capital T right which is an incomplete 5-plat of very trivial form top right down here so it's only the last component of the 5-plat that is corresponding to a singlet and so it's the guy which then is 1-2-3 here so this one I can write as lambda t right t bar right i OF t right i ok so this rewriting is sort of similar to the rewriting you do when you use spurions as Marcus explained to you here it's even a bit more simple so that is the so called method of sources that you may have heard already that is the following idea so suppose I want to compute the Yukawa couplings or better I want to compute all the couplings of fermions ff bar with the higgs ok and I'll be the right number of higgs but I know that well the composite sector is the strongest sector so it's the one that leads the most so the most relevant result is the one that comes from pure strong sector interaction basically I want to compute all the possible I imagine computing all the possible diagrams where there are only for instance this one a fermion mixing to another composite fermion like similar to one we saw before then missing again to nf with x's attaches or even I don't know loop contributions like nf that goes into fermion and some other composite boson with all the x's attached here so I want to keep all this contribution but I don't care of any diagram where the f elementary field is itself propagating so for instance this one I don't want to include I will be able to include it afterwards after having done the calculation that follows by doing one loop of the resulting effective Lagrangian so this one with f inside I don't want to put it ok so and this is a good approximation because this f results into elementary cabling, new elementary cabling insertion, new lambda t left insertion which are weak ok so if I want to do this for doing this the fact that this capital Q and capital T multiples were incomplete doesn't matter at all it doesn't matter because the only way you can say that this capital Q was not made of really five different fields is when you look at the kinetic term and the propagators because then if you open this you will discover that this be left as a propagator while this one is set to zero which means as if you want an infinite kinetic term ok so it's only when you open the box of the propagators that you realize that these sources so these fields here are actually being sent to this specific value and that's the method of sources so you regard all these Q and T as an external source again a non-propagating external source and you compute correlators by integrating out all the rest all the composite sector but not taking into account the propagation of the sources this you can do again afterwards and if you use this trick the things enormously simplify because QI now is a five-plat ok so as far as this calculation is concerned I can even if the symmetry is broken by the fact that this clear is not a five-plat is an incomplete five-plat for instance even if the symmetry is broken I can still use the predictive power of the symmetry so I'm gonna do quickly the following thing so I want to get the effective Lagrangian because I want the Yukawa Kapliks and all this for the capital Q for capital T and for the Higgs which emerges as a pseudo-Lambu-Gorston bosom and this I will do by imposing that this is S of 5 invariant so under S of 5 the Q and the T transform as a five-plat so let's say GQ and this goes to GT ok and how the Higgs transform well you know it's complicated it becomes simple only if you look at the Gorshton matrix U of pi or U of H ok, I'm continuously reaching notation Gorshton pi or Gorshton H is the same so I told you that the Gorshton matrix under S of 5 transforms into G U H to the minus 1 where G is the S of 5 transformation itself and H is a transformation inside S of 4 which I repeat you once again so H is a transformation inside the subgroup so like we saw for S of 3 broken to S of 2 is gonna have a block diagonal form there's gonna be here a 4 by 4 S of 4 transformation and here there's gonna be nothing ok so which kind of terms I can construct now I think it becomes a very simple question so take the Gorshton matrix to the minus 1 it will transform now with H on the left and G dagger or G minus 1 to the right and apply it to Q for instance then the way this thing transforms is just that this goes into H U to the minus 1 G to the minus 1 but then there is G from the Q transformation and this cancels and so you see that this guy here transform with H U to the minus 1 Q so if I take the the sources, I act with the Gorshton matrix I find something that actually does not really transform with S of 5 ok, it transforms only as it was made of it was in reducible representation of S of 4 because the matrix that acts on it is block diagonal so for instance the first 4 component of this guy they transform like a 4 part of S of 4 while the last component does not transform at all ok that is the essence of C C W Z in a simple example a way to write down all this all this invariant in a systematic way and so what one does given this observation is to take this U minus 1 Q and decompose it in the first 4 and in the last component ok, so for instance you may have something like acting on Q which is again 5-plat you can write it as a 4-plat this is typically called Dresed sources because they are Dresed by the U field by the Gorshton field and these now are nontrivial objects they contain not only the elementary fermions but also the Higgs that comes when I multiply with U so we typically have in this case you have a Dresed source that transforms in the 4-plat and the Dresed source that transforms in the singlet Dresed in the singlet ok ok, so then let me just write down the result if you I mean questions you can ask me next time I want to end and now I have to stop so I want to end with this before discussing the implication but already this will be enough so I will write down the most general zero derivative Lagrangian involving Q,T and the Higgs ok and this happens to be basically the master for the Dresed source so this happens to be to be proportional I don't have time now to discuss the proportionality factor to Q bar,T left Dresed singlet T, right Dresed singlet ok so tomorrow we will see what is the proportionality factor and we will work this out explicitly you see already that that's a non-trivial object I'll tell you once again this depends on the Gorsons in a non-trivial way so inside here we are indeed going to find a lot of interactions so we have two interactions with two elementary quarks so with the Q left and the top right and then here there will be the Higgs in many possible different powers coming from this complicated non-linear structure and in particular we will predict the deviations of the single Higgs coupling to two quarks