 Thank you very much, and let me immediately tell you that In my talk, I would really would like to consider signboard on type models more precisely the FRG study of signboard on type models which seems to be a little orthogonal to the issue or topic of the Today afternoon I'm considering Higgs physics or whatever But I hope that I'm able to convince you that this is not the case So that's why I put these two words conformal field theory and Higgs physics Because what I would like to say what I would like to present you some application of signboard on type models models towards these Directions and let's see what we can see So why signboard on models first of all the definition a very brief definition of a signboard on model It contains the periodic self-interaction Phi is a real single component Scalar field so the simplest case whatever one can imagine the model contains two parameter the fury amplitude and beta And I put three issues here and these three issues is a kind of Commercial stuff. I would like to in the beginning of the talk show you that that the topic so this signboard on stuff can somehow related to the Some some possible talks of today. For example, let me just jump to the second one Higgs physics We have talks on Higgs physics in the afternoon. What do I mean relation of signboard on models to Higgs fixies? I will tell you BKT, this is the first one. I put it in because I consider it as probably the most important one This is the playground for signboard on model BKT means Berezynski, Kostelis, Thales phase transitions or topological phase transition and indeed signboard on type models can be used to attack BKT type phase transition and Pavel Jakubczyk will present at least for me a very exciting talk I hope so on the amplitude fluctuation O2 model xy model whatever and this signboard on representation Brings us a tool to attack the problem in a different way and there is this first issue Which is conformal field theory and more precisely? Zomologic of C function and C theorem. This is basically the the main Ingredients of my talk. So I put it in a kind of reverse order So the most I would like to speak about C function stuff a little Little introduction what I mean signboard on physics to vart Higgs potential and just mention a few words about BKT Of course limited time. I cannot really Put all the words what I had in my mind So if I wish to do this scenario go through this scenario Then certainly I need a method and the method is not surprisingly FRG Very very briefly in a nutshell So we know that the veterinary equation has a form which is the simplest in that case because we considering a simple single component scalar field theory You know the story are is the regulator function We also know that the regulator function should fulfill some requirements otherwise free to choose So this brings us some difficulties if we consider the exact FRG equation and solve it then the regulator dependence is not an issue But we we has to make truncations. So we know that the truncated flow depends on the regulator and we really need Approximation so I picked up to generally use approximation the first line is the so-called gradient or derivative expansion where the gamma the running gamma is Expanded in terms of the derivative of the fields. That's what I use very frequently and Leading order term Z the way function organization set to be zero That's the leading order, which is called local potential approximation So we will have results in LPA and also LPA prime. I will tell you the details further Approximation one can use for example a tailored expansion of the potential or the wave function organization And if we cut it then it's certain in approximation So our results and my results what I present today will depend on and cut will depend on How far I go to the derivative expansion? Okay, that's the general warning for you Now comes the sign Gordon stuff. This is the main issue. So let me first really go through how Different extension of the sign Gordon model can possibly be written. So This is as I told you a single component scalar field theory now. I defined the equal to Dimension I mean Euclidean dimension and indeed the model has two parameters you and beta And the model has two symmetries that to and periodicity reminds you that periodicity is a discrete symmetry So even in D equal to the model could have a non-trivial phase structure and indeed it has it undergoes the so-called topological or Btk phase transition, which is characterized with this critical value of the frequency beta square equivalent by There are some kind of issues where the sign Gordon model has been used mostly in statistical physics. I don't want to iterate it The next step the next idea to generalize this sign Gordon is just to add Explicit mass you may ask for what these are the reasons this model has been used again in super fluidity Superconductivity, etc But the idea what I would like to show you that probably this massive sign Gordon model can play some role in Higgs physics I will tell you what I mean, but it is clear if I add a master here Then it immediately breaks periodicity. So periodicity is gone and what remains is a Z2 symmetry So one expect a kind of easing like phase transition. That's one thing Another thing one can play with coupling of many sign Gordon's then you have a coupled system We can also use it for superconductors. I mean to describe vortex behavior, but at this point. I don't want to Tell you more about this kind of extension Instead I would like to play with the frequency beta now. That's the main issue of today Let me again record the usual sign Gordon and do the following trick in order to obtain the so-called cinch Gordon one has to do the following replace the real valued beta With an imaginary one so I beta then the cost becomes a cost and remind this is not a periodic function So one expect a Taylor expansion. I mean one expect that the Taylor expansion is a doable approximation for that model Go ahead. Then what we have is a kind of easing type fight to the 2n model We know that the fight to the 2n model has two phases even in the equal to now the question If you look up in a textbook, you will find that the cinch has a single phase. Why? Maybe the answer is clear for everyone, but I would like to tell you if you don't know What would be the solution of this kind of ambiguity or puzzle? Okay, so we would like to attack the problem by FIG and see what is then the phase structure of the cinch again and a kind of Generalization which is the so-called shine This is really the last one. I promise you the shine contains instead of an imaginary beta a complex one But please observe in order to be physically reliable Theory one have to take the real value which brings you a cost time a course and if beta one is zero then it is Cosh so a cinch and if beta two is zero then you come back to the sign golden So the shine somehow interpolates between the sign golden and the shine We will also consider the phase structure the C function of this model So I talk about C function. Let's be precise what I mean C function and Let's jump to the word of Conform of field theory or more precisely global and the local dilatation These are somehow related to each other and play a press special role in the equal to dimension Let me let it go step by step. So global dilaton symmetry means that changing let's say the lattice space by a Multiplicative factor lambda which do not it does not depend on the X. I'm in the spacetime If you go if you consider local dilatation, then of course lambda is X dependent Staying in the level of global symmetry one can immediately realize that this is nothing but scaling variance This is the cornerstone of our FRG We know that in the second order phase transition if the model under goes a second order phase transition It is self-similar. So it's scaling variant So one expected that if we really take seriously the FRG and look for the fixed points of FRG And that fixed that points the theories scale invariant. So one finds that this is a nice Situation for global Dilatation symmetry. This is a phase diagram of the sign Gordon showing you to the first time you see two phases The arrows indicate the direction of the flow. This is the separatrix somewhere here So this is one phase. This is the by the way the broken phase And that's the other phase and this is please observe that this is the line of attractive fixed point here They're line of repulsive fixed point and here you find this kind of IR convexity fixed point Okay, these are the fixed points. So one expect global Dilatation symmetry at these fixed points. Now, what happens if I switch to local one? I would like to really jump to be equal to be equal to special in this case The the the theory is not just simply conformal invariant, but the conformal group is infinite dimensional special situation because The conformal and the scaling variance are hand in hand So it means that what is trivially true that if the theory is conformally invariant then it is scary invariant But in the equal to the vice versa is okay. So once it is scale invariant, then it's conformal one So what one expect that the fixed points of the FRG? Also represents a conformally invariant field theory and one can identify The central charge of the corresponding conformal field theory So one expect that these fixed points can be dressed up with central charges and indeed this is true This is what we expect. So for example here a kind of Gaussian like is associated with C equal one the attractive region again We put C equal one and here is this an infrared we put C equals zero I mean we put how? By hand, but that would be really nice to see that FRG work and tell us that the favor that really Computably see that this are the central charges. How can we do it? In order to do it we should really add something as and then comes Zomologic of C theorem which tells us that if we are out of the fixed points We can still some do something we can define the so-called C function Which is a function of the couplings for the sine Gordon. It is the u tilde and the beta square. Okay, so that's a Function which is always a decreasing function from the u-way to the IR and we always take the value of the central charge It at the fixed at once can we construct such a kind of Function in the framework of FIG many papers one can iterate But now I would like to use Alessandro, Julio and Carlos result So taking the formula adopting the formula what they derived in a paper and Try to really use this in order to address such a questions Okay, so the goal is to see how it works for sine Gordon type models now in order to do so one has to make a little further step more precisely the Original sine Gordon contains the beta but in order to go beyond the LPA just for technical reasons It's good to make a kind of rescaling of the field So beta jumps here and becomes the wave function normalization still the model has two parameters Z and u tilde. Okay, that's just a technical trick But I also would like to remind you that this technical trick requires the careful treatment of the regulator So we are using power low type regulator Otherwise we can easily run into a trouble losing the scaling variance, but this is a technical issue So you see I always indicate power low type regulators equal one. So this is the mascot of here are the RG equations But consider this is still an approximation Because I use a single Fourier mode. I adopting the formula which is strictly speaking valid for LPA We discussed this issue with Alessandro Carlos, so you know what I mean So it's just a kind of approximation what we expect that's still viable still doable and let's apply it Okay, so three equations one for the C Function and one for the Z and one for the Fourier amplitude We are in D equal to two dimensions. So the flow equation always looks like this I'd switch to dimension less quantities and here are the results. What are the results? So what we wanted to do is the following Let's consider region one, okay What we want to do is to see that if we start from C equal one And if we go through an RG trajectory, it really runs to C equals zero This is the expected wishful thinking. Let me say and here here you find some results The colors correspond to various values of beta and the green one here is the green one is the best one Why because you see this is start from C equal one and if we decrease K It runs to C equals zero. So delta C is one This is what we expect because for San Gordon, we know that the Corresponding associated central charge is one and this trajectory is lying somewhere here But if we move away from that line and we start for somewhere here It is still produces us some reliable results. Of course under some numerical Uncertainties these are the uncertainties. So at least a 90% or 80% Accuracy we can we were able to reach that's the region one. So up to now This is this could be also considered as a kind of test of Alessandro's Julia and Carlos result and works fluently Now what happens in region two region three? I'm a bit confused telling the truth that we don't know how to associate a C here It's not a fixed point first of all and even if you extrapolate this Playing asymptotic safety or whatever then please observe this corresponds to large beta Large beta corresponds to small Z. So the kinetic term frozen out. What is then the C? Central charge of this theory. I don't know Nevertheless, if he really falls by hand to put C equal one in region three We have reliable results, but as you see numerically, this is not so stable So there is a question mark at this stage and I cannot give you the correct answer I'm open for discussion if you can tell me what you what do you think about what would be the theory in this case I would be very grateful. So that's all what I wanted to say about the same or no Let's switch to singe regarding singe This is a very fast way to show you how effective the effigy for sign or don't type of that Just think the linearized effigy dropping all the nonlinear terms Substitute it just this term take the Fourier expansion of both sides and one can easily reach To read of beta square equal 8 pi. It's so easy. So for sign Gordon the linearized RG works very well Let's try to adopt the same for the singe. Please observe. We have this I beta. So this Plus becomes a minus no beta square Critical value can be determined these signals that no big a taste transition appears in the singe good So what we have? No topological phase transition. This was a kind of trivial reason. Okay, but then what let's go and Blue brutally a kind of fully a Taylor expansion of the model go ahead It's a no it's a non periodic stuff one what one can see immediately that the coefficients are always positive So if you consider this as a fight to the 2n model, we are always in the symmetric phase That's why the model has a single phase. So We are somehow Confirmed that the textbooks are good even FRG tells us that the model has a single phase But let's see how the RG flows works This is what we did in a tricky way and these are the RG trajectories or lines Which really implies that the model has a single phase. So that's about the singe and if I wish to switch to the I wish to switch to the summary of the talk Then I would like to go through this and mention just the various the very last point Which means that what about the shine? What about other interpolating model if you are interested in there is a poster Precisely poster number two and where you can find this shine or whatever But the reason I would like to conserve three minutes for myself to say something about this kind of Hicks idea Now this is the very last thing but I would like to show just three slides, but this is the most questionable part Okay, so I'm very grateful if you can tell me that each time. This is a crazy idea. Forget it or go ahead I like this. So please really do Criticize what we what we plan to do together with my Italian friends. So the idea is the following Took the usual standard back mechanism I I go through this but probably you all know how it looks like take the fix sector Take the usual typical Hicks Lagrange Hicks potential which contains the parabolic and the quadratic term and Perform the steps which requires in order to dress mass for the Z and the W's This is the straightforward thing what we do always Just few things We use a certain parametrization V is of course the vacuum expectation value known from low to low energy experiment This is the Hicks miles measured by the LSE recently So lambda can be calculated using the formula. That's a known thing written in textbooks now What try to play with that and try to introduce other type of Hicks potential. Is it doable first of all? Yes, it is not forbidden. It's not a priori given by the standard model. So one can play with this Can one think about the higher order higher minima? Yes, it is true It's a it's a playground of today because we can play with stability We can play with the ideas in Inflaton dynamics relate the Hicks and the inflaton at the same time. It's a nice thing. I Put two papers here, but of course It is not the total number of papers related to the issues So I'm apologizing if I skip some important thing But I would like to show that it is doable But in every little paper when I when I go through the derivation I always found a kind of polynomial extension of the Hicks Why not try a periodic one seems to be a crazy idea But trust me this drastically changed the phase structure. Why because it has infinitely many minima infinitely many minima so no tailored expansion or no truncation of the tailored expansion It is simply forbidden Why we need this? First of all, let's go through the derivation or what I would like to say that the change Compared to the previous one. It's just this blue one a periodic stuff appear Okay, this is one thing one can fix beta in such a way that the vacuum expectation value of this periodic stuff to be equal to the Normal standard model Hicks potential then remains you you can be changed to fine-tuned to have the same Hicks mass Crazy idea. Am I right and if it becomes more crazy if you just consider a simple sine Gordon in D equal for if you Treat the model by FIG purely the sine Gordon model Be larger than two so in D equal to one finds a single phase now. It's wrong So it's not even doable. It's a single phase. We kind of do it's it's absolutely wrong But now comes the idea if we play with the So-called massive sine Gordon model which contains an explicit master which known to have two phases So it fulfills the requirement of having two phases, but it according to our previous examinations and studies the model Save in a sense that the parabola is always remains positive So we can have a chance to substitute it to the standard model Lagrangian Perform an RG and see whether the stability of this. I don't know how to say extended model remains the same or probably we can somehow escape the unstable Region so in another words, this is a kind of proposal. This is a book which is in progress But before we do any steps, I would be grateful for any comments But that's really the last one what I wanted to say. Thank you for your attention