 Okay. I want to address a question that probably many of you have already thought about. When you teach critical phenomena, you always have in mind, am I sure that the critical exponents are the same on the two sides of the transition? This work has been done in collaboration with Frédéric Leunard, PhD student, and it's already published. A bit of history. The first thing you notice when you tackle this problem is that before the 70s in books or in reviews, people were very cautious about defining critical exponents that were not necessarily the same on the two sides of the transition. While starting from the 70s, the distinction started to disappear, and it's because the renormalization group arrived, and I should try to summarize the arguments that were given to explain why the critical exponent should be necessarily the same on the two sides of a second order of phase transition. I have some difficulty actually to explain this because I know that it's wrong, but okay, I'll do my best to explain what people are in mind. So the first thing is, I think a theorem that goes back to Karl Mann, who proves that if a theory is renormalizable in the symmetric phase, then it is also in the spontaneously broken phase. This was actually important for Gitch theories. Then the renormalization goes almost the same way in the two phases. Then you see the difficulty is that when there is a phase transition, let's think of the Ising model, for instance. So you have the temperature, you have the critical temperature here, and when you lower the temperature, you eat the critical temperature where there is a singularity. Of course, it's difficult to say anything when you cross a singularity. But of course, if you add a magnetic field in this direction, you can draw a continuous path that avoids the singularity. And along this path, everything is smooth and continuous. The next argument was that the scaling functions that you can write, okay, they are the same all along this path. So the renormalization goes the same way. The path avoids the singularity, everything looks continuous. And so the critical exponents are the same. You can even find a proof in the literature, look for instance at the Injustance book, that it's true for the O.N. model. The critical exponents are the same on the two sides of a phase transition. But actually, David Nelson, back in 1976, which means exactly 40 years ago, had an argument saying that this is wrong. And he claimed that when there are discrete symmetries, I'll actually explain in detail what it means in the following. Then, for instance, the susceptibility, the susceptibility is the response of the order parameter to a change of the external field. So think at the Ising model again. Actually, it will not, I shall not consider the Ising model for the following. But okay, for the definition of the susceptibility is the derivative of the magnetization with respect to the magnetic field. And then he said that the susceptibility in both phases will diverge at the transition, but not with the same exponents. And he made a calculation in the O2 model with discrete anisotropy where he showed this. Actually, when I started to work on this, what I wanted to prove was that David Nelson was wrong. But David Nelson is David Nelson. So it's right. Okay, since then, there has been a tremendous amount of work on this problem. And mostly what I shall consider in the following is XY system with O2 symmetry. So think at magnetic system, two components for the order parameter, a symmetry that in the absence of anisotropies would be O2. And you, for instance, consider what is called the cubic anisotropy, meaning that the spin has some preferred directions here. You can also consider, for instance, the hexagonal anisotropies. And these two anisotropies are indeed realized in real materials. Okay, where you have an hexagonal anisotropy, okay, and you have six direction in which the six preferred direction. So there has been a gigantic amount of work on this kind of system, either when you force the spin to point only in this four direction or six direction, and these systems go under the name of clock models, or simply the spin can move in all direction, but there are preferred directions. So you can think at a Mexican hat in which the valley is not flat, but it is modulated. So there has been a lot, a lot of work in two dimension and it's three dimension. The clock models, for instance, in two dimensions, there are many exact results about them. And in three dimension, they have also been enormously studied. And there has been a revival of interest for this system, these last 15 years. For instance, because of pyroclore, deconfine quantum critical points, and also the possibilities that maybe in these systems, there are two different phase transition, which actually has been rigorously proven in two dimension. And so people wondered whether it could also be the case in three dimension. And the reason was that numerically, people found that in the low temperature phase, there is not one correlation link, but two correlation links, two different correlation links that diverge with two different critical exponents. Okay. And I must say that actually there are three people who understood completely independently of David Nelson, or by the way, the paper by David Nelson, it has about 70 citations, I read the 70, and there is not a single one who just say anything about the fact that the critical exponent are different on the two side of the phase transition. So beside the paper for something else, that is called Dingerously Relievant Operators, I shall speak about this in a moment, but not a single one, just take care, pay attention to the fact, this very striking fact, said the critical exponents can be different. Okay. And actually, these three people, in 2000, they make a calculation in the case of cubic anisotropy, because they have made a six-loop calculation of some exponent, and they say, oh, okay, look, what we have done strongly suggests that the critical exponent gamma for the susceptibility can be different in the two phases. Unfortunately, precisely for this case, the difference is so small that it is unobservable, and I shall explain why in the following, but okay, they probably understood the whole thing. Let me present the argument under the form of a paradox. Let's imagine that you have an n component thing. What I shall have in mind most of the time is n equal to 2, but the argument is completely general. We could even think at different groups, not on ON, this is completely general, but for the sake of definiteness, let's consider the ON model to which you add a term, which here, as breaks explicitly the ON symmetry, because, for instance, either it has a term that is a cubic term, or by the way, this corresponds in the Hamiltonian, in this case, tau is nothing but the sum between i and n of phi i to the 4. Okay, this obviously breaks the ON symmetry. It is of degree 4 because the anisotropy is fourfold. This one would be over the 6. In this case, tau is phi 1 minus phi 2 square, phi 1 square plus 4 phi 1 phi 2 plus phi 2 square. It is of degree 6, and a q-fold anisotropy corresponds to tau of degree q in the field. And this tau has two properties. First of all, it is invariant under a discrete subgroup of ON, and second thing, it is irrelevant at the fixed point describing the phase transition. This is obvious for this term, which being over the 6 is not irrelevant already from the point of view of power counting. It is also true for this one. It's obvious because it is over the 4. We could think that it is that it is irrelevant, but actually it is not. For the n equal to 2k, this term is irrelevant. Not very irrelevant, but irrelevant. But okay, generally speaking, we shall consider tau as a near-relevant operator. So the argument goes in the following way. tau is irrelevant. Therefore, we can neglect it in the long-distance physics. Therefore, the attractive fixed point is ON symmetric. Therefore, the critical physics is identical to the usual ON critical physics, because the critical physics is given by the fixed point. Is there anyone who does not agree with this? Oh, there is one, but he already knows. So you are all wrong. This actually is right and wrong at the same time. Why is it wrong? Because what we think at the low temperature phase, the argument is right in the high temperature phase. It is wrong in the low temperature phase. In the low temperature phase, since we have discrete symmetries, we don't have any Galston bosons. Since we don't have any Galston bosons, the susceptibility, let's imagine that we are considering the x-to-y system with an anisotropy that is 4-fold, 6-fold, whatever anisotropy. In the pure model, which means without anisotropies, we can define two different susceptibilities, the transverse one and the longitudinal one. Here, if we consider this vacuum, then we have a transverse direction and we have a longitudinal direction here. We can define two different susceptibilities. In the pure model, of course, the transverse susceptibility diverges for all temperature below the critical temperature, but also the longitudinal susceptibility diverges, which is less obvious because at the mean field level, it does not diverge, but because of the fluctuation also the longitudinal susceptibility diverges. In the pure model without anisotropies, the two susceptibility are divergent. Here, since we do not have any Galston bosons, this means that the two susceptibility are finite in the low temperature phase and they diverge only when T goes to Tc, Tc minus. But this means that although irrelevant, tau matters for their behavior close to the critical temperature, at least in the low temperature phase. And since it matters, the susceptibility of the exponent for the susceptibility cannot be the same as in the high temperature phase, because in the high temperature phase, obviously tau is irrelevant and it plays no role. So the exponent gamma cannot be the same in the high and the low temperature phase. Tau is said to be a dangerously irrelevant operator for the susceptibilities. This name dangerously irrelevant operator goes back to Fisher in the 70s who realized this kind of thing and we all know a dangerously irrelevant operator. It is the fight to four operator of the usual ON model in dimension larger than four. This is also a dangerously irrelevant operator. This one, this tau is dangerously irrelevant for the susceptibilities. Okay, let us study this with our preferred tool, the NPRG. With a simple truncation, with a simple truncation, what we are going to do is the LPA. Here is the LPA prime. I have added a field renormalization. Actually, you can forget it. It will play a small quantitative role, but nothing spectacular. Okay. So for most of my seminar, I shall forget VZK. And we shall consider the local potential approximation. The potential here depends on two invariants, rho, which is the usual O2 invariant. I shall now consider the n equal to two k's on the six-fold anisotropy. So rho is the usual O2 invariant and tau is this guy here that is also written here. Okay. On top of the derivative expansion, we shall make a field expansion because we do not need to be functional, but surely since we want to compute something quantitative in three-dimension, the strong coupling regime of the theory in three-dimension has to be tackled with and so it will be very convenient to use a non-properative tool. Okay. So we turn the crank. Here what I've done is that I've taken the simplest field truncation where I retain only here kappa, which is the minimum of the potential. If I go back to my expansion here, you see that kappa is the minimum of the potential. I have chosen tau such that at the minimum of the potential it is equal to zero. So the expansion for tau is performed around zero. For rho, it is performed around the minimum. Actually, I have forgotten to write here that kappa depends on k. And I just keep kappa in my truncation, uk, and this coupling in front of tau and I just eliminate all the others. Okay. The flow equations, they look here not very pretty, not very ugly. And what we see is that we have two different masses, two different susceptibilities in this system. This is the inverse longitudinal susceptibility. This is the inverse transverse susceptibility, the masses of the problem. And we see that the transverse susceptibilities that should vanish if there were no anisotropies is proportional to land ethics. Of course, it is non-vanishing because there is this discrete anisotropy. And the longitudinal one is the usual one, the one that already exists in the automode. Okay. Here I have chosen the cutoff, the leading cutoff. You can forget this anomalous dimension that will not play much role in what follows. Okay. What is the flow? What does it look like? So here is the flow in the plain land ethics and u tilde. The tilde means dimensionless. I go to dimensionless quantities to see the fixed point structure of the theory. And what we get is three fixed point, the Gaussian fixed point, of course. The usual when landed six tilde is equal to zero, this is the usual O2 fixed point in three dimensions. And there is a third fixed point that is sometimes called the Nambo-Golston fixed point that also exists at lambda six is equal to zero. And this is the fixed point that drives the low temperature phase of the pure O2 model. Okay. So here we have the high temperature phase. Here we have a line that goes directly to fixed point. This is the critical line. And here we have two lines that goes in the low temperature phase. And for instance, if we consider this one, we see that there are five parts in the flow. The first part we start from an initial condition is very fast because lambda six decreases very fast. And we arrive close to the x-y fixed point. Then second part of the flow, the trajectory remains a long time around the x-y fixed point because it is a fixed point. So the flow is very slow here. Then it departs and it goes very fast close to this fixed point. So this is the first part of the flow, this transient regime here. The fourth part of the flow is that the flow will remain a long time around the Nambo-Golston fixed point. And then the fifth part of the flow is when the flow escape the Nambo-Golston fixed point. Okay. Here we retrieve the five different regimes for the flow. Here it is u tilde. U tilde, I remind you, is the coupling constant in front of the O2 invariant term, which means u tilde rho minus kappa square. And we find here the five regimes. The first one is a transient regime. We approach the x-y fixed point. Here there is a plateau, meaning that the flow is very close to the x-y fixed point. Then we escape here and we go close to the Nambo-Golston fixed point. The flow remains a long time, which means a plateau around the Nambo-Golston fixed point. And then finally, it departs from this fixed point. Okay. What about the susceptibilities? The susceptibilities where the transverse susceptibility, if there were no discrete anisotropy, would vanish all the time. So it does not vanish because there is a discrete anisotropy. Sorry, it's the yellow one, the yellow curve. And what we see is that it decreases extremely fast. Why does it so? Because here, the first part of the flow is very fast. Because the operator and the operator tau is irrelevant. So since it is irrelevant, the flow is very fast and goes very fast close to the x-y fixed point. We retrieve this feature here. The transverse susceptibility goes down very fast. Then the longitudinal susceptibility. The longitudinal susceptibility, even in the pure model, is not the inverse one, is not equal to zero. It goes to zero because of goldstone fluctuations. Okay. Already in the pure model. So what happens in the model with the anisotropy? So the beginning of the flow is exactly the same as in the pure model. And it's only here that it flattens. Here, you see this dashed line corresponds to the pure auto model. This is the real integration of the flow. Okay. So this dashed line corresponds to the pure auto model. And we see that the two flow departs precisely here. And here it flattens and it reaches a finite value. Clearly, we see on this flow that there are indeed two distinct length scales in the problem. One psi here that exists because the flow goes very close to the x-y fixed point. And here the anisotropy plays absolutely no role. So it will play a role at the end of the flow, but not when we are nearby the x-y fixed point. The physics is dominated by the x-y fixed point. And so here this correlation length is the correlation length in the low temperature phase of the pure auto model. But there is a second correlation length here. And this second correlation length corresponds to this last part of the flow when the system realizes that it is not the pure auto model and that actually it will not remain forever on the plateau corresponding to the number Galston fixed point that it escapes. And so this defined the second correlation length that is called psi prime here. Okay. So how is this psi prime defined? Well, this psi prime here I have... Of course, we can integrate the flow equations, but we can without any integration understand what is going on. When K is of order of the correlation length of the pure model, okay, we have integrated the non-trivial critical fluctuations. So it means that starting from this scale, the flow goes on evolve according to trivial dimension. So it means that the minimum of the dimension, full minimum of the potentials that correspond to the spontaneous magnetization does no longer change, which means that the dimension less magnetization, square magnetization, evolve according to its engineering dimension. Okay. So this comes only from the fact that starting from psi minus one, we have integrated all the non-trivial fluctuations. Okay. What is the value of kappa tilde when K is of order of psi minus one? Remember, the flow is close to the XY fixed point. So kappa here, the minimum of the potential is about its XY value. Okay. What about lambda six? Well, we think exactly the same way. Let's imagine that we have integrated the non-trivial critical fluctuation, the scale of the system is the inverse O2 correlation length psi, and then it evolves according to the engineering dimension of this guy. Now we can go a little bit further. What is the value of this lambda tilde six, lambda tilde six at the scale k equals psi minus one? Well, let's go back to the first thing here. So we started here, and at scale psi minus one, we are here, and we are ready to escape the XY fixed point. So what is the value of lambda six, lambda tilde six, when we are around here? Well, pretty simple to understand what is going on. This value is nothing but the initial value of the coupling, the Bayer one, and the flow has evolved very rapidly according to the eigenvalue of the XY fixed point in the direction of lambda six. Okay. Let's look at the picture here. There is clearly an eigenvalue of the flow in the direction of lambda six, and approximately lambda six has evolved in this direction according to this eigenvalue of the flows that I call lambda six. Okay. So put together, we obtain these two things. Okay. So what we see here, if we combine these two things, I recall here the definition of the transverse susceptibility, the dimension full transverse susceptibility after having integrated all the critical fluctuation, which means at scale psi minus one does no longer evolve. And so its dimensionless counterpart evolves according to its naive dimension. Okay. We could think the same way for the longitudinal susceptibility, and this is the result. So I go back to the flow of u tilde. I have written in red the terms that would be absent in the pure O2 model. So clearly this is zero, clearly this is zero. This one is not zero, but you can convince yourself that for k larger than psi minus one, it will be very small. Actually, we retrieve here that if we are in the pure model, this is equal to zero. This I three is equal to two. And we find the number Goldstone fixed point by balancing this term, which is negative in three dimension with this terms that is positive in three dimension. It's a square. Okay. But if we turn on the anisotropy, this term is not equal to zero. We have just seen in the in the last slide that it evolves according to its naive dimension starting from the scale psi minus one. And so this term start the m tilde that is start to increase. And the two flows will dive, the two flows between the pure O2 model and the one in the presence of the anisotropy will differ precisely when this term will be over the one. I should compare here the argument of the threshold function with ones as we see here. And so the definition of psi prime minus one, the scale at which the flow of the pure model on the anisotropic model start to differ, is given by this term start to play a role, a different role in the pure model on the anisotropic one. And this is when this is equal to one. So this is the definition of psi prime minus one. This psi prime, this scale, this new correlation length will diverge according to an exponent new prime. And the susceptibilities will also diverge according to two non-trivial critical exponents that I call gamma longitudinal and gamma transverse. Playing with the relation, I don't have time to show it, but playing with the relation I've just given before, which means this scaling for the minimum and for lambda 6, it is kind of trivial to prove how these new exponents are related to the old one, the old one, which means new exponent for the divergence of the correlation length psi, as well as gamma plus the divergence of the susceptibility in the high temperature phase. And we see that the difference comes precisely from the fact that we have these irrelevant operators, but dangerously irrelevant, because it is able to change the scaling of the model. So indeed the critical exponent are not the same. There is even a new one. This new prime does not have any counterpart. And now let's go to the numbers. So what we have done is to compute numbers. We've considered the LPA prime, which means we have a non-trivial field renormalization. We have pushed the expansion up to the 12. This is very easy to do. And we have completed this exponent corresponding to the irrelevant direction in the direction of the anisotropy for different kind of anisotropy, the cubic one up to the 12. What we find is that this exponent is tremendously large. It is extremely, it can be extremely, extremely large. And so you see our results are there in red. And some results in the literature are in black. So here is the result by the group of Lund Balance, Lu and Sunvik. And actually it works pretty well. But for this exponent, so actually there was a Japanese group that did not agree with the result by balance. So they redid the calculation of the new prime last year and they find these numbers that works pretty well with ours. Okay. So now I want to, so it works pretty well and I hope I have convinced you that there are, there can be different critical exponents with differences that are extremely, extremely large. It's not just the small numbers. And the philosophy is that the more irrelevant, the larger the difference between the critical exponent in the two phases. So now let me go to something that looks completely different, but that is completely the same. A problem that comes from the standard model. And this work is done with Nicolas Sheba. It is the problem of the hierarchy and the fine tuning in the standard model. Okay. For those of you who are not familiar with this question, let me remind you what it is. We consider the standard model and we imagine that the standard model is nothing but an effective low energy theory of a more fundamental theory. You can say, no, I don't want to hear about this because I know nothing about this super string or gut theory or whatever you want. Okay. But it's difficult to believe that just a renormalizable theory is nothing but an effective theory. The hierarchy problem is the following remark that in this case, there must be an ultra valid scale in the theory. And this ultra valid scale is certainly large. It's large because the precision test of the standard model does not see anything about this, this scale, which means that it's large. Of course, if you have in mind that it is the plan scale or the gut scale, then this ratio is not large. It is extremely large. Okay. And the problem of the hierarchy or the fine tuning problem is very much the same as in statistical mechanics where we know that we can produce a very large correlation length at the price of fine tuning a critical temperature close to the critical temperature, for instance. Okay. But this is very unnatural in statistical mechanics. Okay. We can turn the temperature and adjust as it will. But the universe is the universe. We cannot just change the parameters of the universe. There has been many proposals in the literature to avoid this problem. And the question is, can we produce naturally light scalars with something that is much simpler of this kind of scenario? This is what we have done following exactly the same idea. The idea is that irrelevant operators, the operators that in principle we just do not consider can exactly play this role. So we have devised a time model and it's only a time model up to now for the boson sector, for the scalars sector of the standard model. It is not realistic. I insist on this point. And what we need is to have three goldstone bosons that will be eaten by the gauge fields. And what we want is to have a naturally light particle. So three goldstone bosons, we break O4 down to O3. This will produce a goldstone boson. And we add here this kind of addysartra piece that I have considered here. Okay. So the action is similar to what we have considered above with the difference that instead of having two fields, we have two four vector fields. Okay. Because we are considering an O4 model, we have a problem with this time model. We break the custodial symmetry. That is an important symmetry. Anyway, we can run the flow and see what is going on. And this is my last slide. Here is the result. We consider two kinds of fine tuning. A fine tuning where we tune the temperature at the level of one person close to the critical temperature or the level of 10%. 10% is what not what I should call a fine tuning. Okay. And these are the curves. Let's consider the Z12 anysartra piece here or the Z12 discrete symmetry. So we start here and then we run the flow. And we find that the transverse model here acquire a mass that in 10 orders of magnitude, around 10 orders of magnitude smaller than the bare mass of the problem. And we have nothing that's for free. It's just because this mass, the mass of the transverse mode is driven by only a relive interpreter. Remember the exponent I got, the YQ I got is extremely, extremely large. So it means that the mass of the transverse mode goes down extremely rapidly. And this is exactly what we see here. Now, of course, the problem is to build something that is realistic from the point of view of the Sun and Marlore. And we have some ideas about this, this together with Nicholas Sheva. Thank you very much.