 Can everybody, yeah. So I decided to simplify the title because it's really mouthful as you noticed. And by the way, if we are running out of time, even emergency supersymmetry will become an optional topic. Let's see if we can make it. There are several slides. So this is the plan of the talk. I want to introduce you, well, you have seen it several times already during the conference to the local potential approximation, but in a slightly different way. Only mildly different. And then I want to show you how to derive through the epsilon expansion, unitary multi-critical models, and through a new expansion, non-unitary multi-critical models using the local potential approximation. And finally, if we have time, we're gonna talk about emergency supersymmetry. So this is the brief introduction to the LPA. As I said, it's slightly different. I want to start in a top-down approach, let's call it. So I'm suggesting to pick up our favorite renormalization group flow, for example, through the Wetterich equation. And I'm defining what it is known as the next to the order in the derivative expansion as a truncation of the space of all the functionals which are flowing through the RG, which contains the most general terms up to second order in the derivatives of a scalar field phi. Now, if you have your RG flow and you have your truncation in this, you can project the RG to your truncation and derive a flow for an effective potential and a field-dependent way function renormalization in this way. So this is kind of a top-down because through the second-order derivative expansion I want to define what I call as the local potential approximation. If you pick up your truncation in the next to the order, you can approximate it so that the way function renormalization is just one or it's just a scale-dependent constant in phi. And in this case, you obtain what we typically call as the local potential approximation or the local potential approximation prime. Now, the local potential approximation prime is the one that I'm interested on because if you further redefine your fields into dimensionless renormalized quantities like so, you can extract through the LPA a formula for the anomalous dimension of your scalar field. So the flow that we just considered hides some explicit cut-off dependence which appears explicitly through the cut-off and a modified version of the propagator that contains the cut-off. So as I said, I like the idea of starting from the next to the order of the derivative expansion because I'm able to, even if I'm taking the limit of the constant Z, which goes into the local potential approximation prime, I'm still able to retain kind of a flow for a field-dependent way of function renormalization and I will use it later. So if you move to dimensionless quantities, this is how the flow looks, for example, the flow of the potential in the specific limit has a scaling part, which is just canonical scaling. And then I'm writing it as a series, you will see why of powers of the second derivative of the renormalized potential. Now all the constants that you see here are cut-off dependent. So they depend on G, then depend on R and they're fixed once you fix the cut-off, but they don't want to fix them for now and likewise for the flow of the way function renormalization. Now my objective, as I said, is to obtain unitary critical models using the local potential approximation. Now to define what the unitary critical model I want to define, what it is, I want to do it by starting from there, let's call it the bare version of the potential. So a phi to the n critical model is characterized by an interaction of the type phi to the n and by a coupling. Now the coupling, the model has an upper critical dimension which I call dn and it's the dimensionality for which the coupling in front of the interaction is dimensionless. So for example, for n equal to one, which would be phi square, the upper critical dimension is infinity, so this is just a massive theory, non-interacting. n equal to two is phi to the four and we know it has upper critical dimension four, phi to the six has upper critical three and so on. Now to study these models in the epsilon expansion, what you typically do is you have these theories which are Gaussian at the upper critical dimension and you're going slightly below the upper critical dimension by expanding the dimensionality in epsilon from the critical dimension. And what you know is we know it because we know that these models have non-trivial phases, they all have non-trivial theories below the upper critical dimensions. Another condition is that if epsilon goes to zero, you are going back to the upper critical dimension and the theory should become Gaussian in this way. Now there is a small appropriate scaling which will turn out to be useful and it's a technical, I'm redefining the field in this way using a cut-off dependent constant as well as the dimensionality and the anomalous dimension. And this is the new rescaled field that I'm considering the variable X. It's the reason why I'm doing this scaling is that in this way, the flow of the potential is a second order, let's call it differential equation normalized to one half in front of the power one of the second derivative of the potential. And this will turn out in a second particularly useful. So here I'm using, by abuse of notation, I'm using the same names for the cut-off dependent constants even though they are actually rescaled through this mechanism. And likewise for the wave-functional normalization, field-dependent wave-functional normalization in this limit. So the fact that the second order, sorry, the fixed point equation for the flow of the potential is like this with the cut-off dependent part. But if you think about the fact that close to the upper critical dimension, the potential should be small because it has to interpolate with the Gaussian solution when epsilon goes to zero, it is legitimate to just in a first order approximation to just compute fixed point solutions by solving the linear equation here. And then eventually add orders coming out from the powers of the second derivative. So it's useful to define this operator. And it, as I said, in the limit epsilon equal to zero, the solution should be a solution of this linear equation. So it's just D acting on the potential equal to zero should be x here. And we know the solutions, the bounded solution, the power low bounded solutions of this second order differential equation are the Hermite polynomials. And they are characterized by the index n, which is the index of the critical theory. So the idea behind solving in the epsilon expansion DLPA is the idea behind giving this ansatz to the potential solution, which is proportional to the Hermite polynomial, the 2nth Hermite polynomial. Now, since the equation is linear, the ansatz is not determined by, does not determine the overall coefficient of the ansatz. We know it has to be somehow proportional to epsilon because the solution has to go to zero for epsilon goes going to zero. But then there is still an unknown coefficient cn. And how do we fix the cn? So the advantage of having defined the operator dn and using the Hermite polynomials is that the operator dn generates terms which are orthogonal to the Hermite polynomial h2n. So, and we have a norm, which is e to the minus x square. So we have orthogonality conditions, but we also know integrals of three Hermite polynomials which have, which are bounded for certain values of the coefficients. And so from the ansatz, we can and thanks to the properties of d and the Hermite polynomials, we can to any desired order in epsilon, we can construct the potential, the rescale potential as an expansion in the parameter epsilon and using the Hermite polynomial. So this is how the expansion in epsilon of the potential looks. And likewise, since I retain the ability of talking about z dot, I can project over Hermite polynomials and construct formulas for the anomalous dimension. Again, as an expansion on the powers of epsilon. So this is interesting because there is an exploit of this fact, I can determine eta, eta by projecting the flow of the wave function and normalization, even if it is field dependent in this limit and I'm projecting over the zero Hermite polynomial. And this is in practice a new way of computing eta in the local potential approximation which I've seen only in papers by Osborne and Twig but I've never seen exploited in the functional g literature. So this would be how the formula looks like and it's an integration using the norm. So it's basically an integration over the field. And so this is just a small detour from what I was saying, but this eta has some nice properties. For example, it's non-zero even when the third derivative of the potential is zero at zero. It correctly interpolates, well, almost correctly interpolates with the perturbation theory result that you will see. It's proportional to epsilon squared when epsilon is going to zero. So it's behaving like a perturbation theory. And since the measure case fast, you don't really need global solutions in principle. So if you have them, it's better, but you can compute this anomalous dimension by just having a solution which lives on a finite size if you didn't solve the full asymptotic behavior. And this formula admits a generalization to the dimensions. If you want to see the Polchinski's version, just look at this paper. So you can solve, as I said, iteratively, ordered by order in epsilon and obtain all the coefficients of the ansatz that I was showing you before as a functions of n, which is very interesting. And again, I suggest you to see Osborne's papers for the Polchinski's version. So we have, for example, up to order epsilon squared. It's relatively easy. You can have an analytic form of the solution which depends on n. So you can, by setting up n, you can choose the kind of critical model that you are interested on. So another nice excursus, the tour, would be that in principle there is a new criterion which you can use together with others for optimization which would be what I call epsilon squared matching of the anomalous dimension. So we know that the LPA that we consider does not include all operators which are generated at one loop or two loops or any order of loop that you so desire. So it's not, for example, two loop exact. But what you can do is you know the result from perturbation theory which would be this eta pt which is the left hand side of the formula. And you can try to match the perturbative result with the result which comes from the LPA which depends on the cutoff even though it behaves correctly in epsilon squared. It depends on the cutoff and so you can use, you can simplify this formula and try to constrain the cutoff dependence based using the fact that you have the cutoff freedom to try to match the perturbative result. Again, I'm not saying that the LPA, that the function Rg gives a wrong eta. I'm saying that the truncation to the local potential approximation gives a wrong eta. So this is nice because as I said, if you have a parametric dependence of the cutoff, you can try to fix at least one parameter using this formula. And I'm saying that Puczynski equation, so this goes for Wetterich equation. Puczynski equation does it automatically even though it's a little bit tricky. Puczynski equation in the LPA does not depend on any parameter so there wouldn't be any chance to actually display a wrong result in that case. But it would be a matter of debate. So to compute the spectrum of the solution, we just deform in a standard way the rescale potential and the linearized equation for the deformations also contains the operator D. So what you can do is you can compute the spectrum for the part which is proportional to the operator capital D, which is just a normal spectrum labeled by natural numbers and then use the quantum mechanics perturbation theory ordered by order in epsilon to compute the corrections. And the result is pretty nice. The spectrum up to order epsilon, which is this guy, is fully universal in the sense that it's exactly the same that you would obtain from perturbation theory and does not depend on any cutoff. So it agrees basically with the scaling dimensionalities of the operators phi 2k, each of them living in the 2n theory. This is a lucky coincidence, I would call it, because the mixing between some operators which would go beyond the LPA is relatively simple and ensures the fact that this formula is universal. So it's not a general property, I would say, of the local potential, but it is nice to have. So for example, this is the spectrum of the Ising class and this is the spectrum of the critical Ising, which goes for n equal to three. Of course you can check the spectrum, I mean, this is a super simple spectrum, but you can check for universalities. So this is just volume scaling and these two, sorry, this guy and this guy you can determine directly from the anomalous dimension if you're interested. So this is all nice, but what about beyond the epsilon expansion? Well, you can pick up your analytic solution toward the epsilon square and you can pick up your analytic solution and compare it with some numerical estimate based on any method. So for example, this is for the Ising class at epsilon equal to one over 10, so slightly very close to d equal to four. You see that the dashed one is the numerical one and the thick one is the analytic one, they basically coincide, but if you go all the way down to d equal to three, so for epsilon equal to one, the two solutions are quite different. Now we know that by comparing terms, we can estimate that the expansion is supposed to fail at around epsilon times x squared equal to one and we also know by the original scaling that it has to fail for d equal to two, so this is not a good method for d equal to two. Of course, you should be using full numerics or since you know what are the CFTs, you should be using CFT. The second part of the talk, and I guess I would be skipping the third one, because of time constraints, is to basically repeat this analysis for non-unitary theories and using this new thing, which I call epsilon to the one half expansion. So we are now interested to fight to the two n plus one critical models and I'm putting an imaginary unit in front because there is actually a more general symmetry different than a simple parity which is protected by the functional RG, by any functional RG equation, which would be under parity, the potential goes into its complex conjugate. And for example, if you split your potential into a symmetric and an anti-symmetric part, the anti-symmetric part thanks to this property has to have an imaginary unit in front. And so in principle, we can have so anti-symmetric solutions like five to the two n plus one. And, but we need an imaginary unit in front. So these are the potential, the bare potentials that I'm interested on and these are their critical dimensions. So six would be the critical upper critical dimensions of the Lee Young model, if you're familiar with that. So the new answer involves solving at lower store the differential operator that we defined above but with index two n plus one. And I found that a consistent expansion does not rely on an epsilon expansion but it relies on a square root of epsilon expansion. So you start with an anti-symmetric part which goes like square root of epsilon but then there is a symmetric part which goes like epsilon. And if you keep going, there will be epsilon to the three half anti-symmetric, epsilon to the two symmetric and so on. You know, as they mentioned differently from the previous model is proportional to epsilon which means it's non-zero at order of epsilon. And as all physical quantities, it's analytic in epsilon. So the potential does not have, the potential can expand into a square root of epsilon but the critical exponents and anomalous dimension will not, they will expand into integer powers of epsilon. And parity as we saw it implies that this coefficient has to be purely imaginary. And this is for example what you get for the Lee Young class. This is n equal to one at dimensionality 5.9, so slightly below the upper critical six. The solution has the anti-symmetric part which is the purely imaginary part and the symmetric part which is the dashed curve. It's like an inverse parabola at order epsilon. So this is the spectrum which you can compute. It also has some spectral relations among the entries. So I have a small conjecture and a candidate other model. So the conjecture would be that these solutions which we know from the upper critical dimensions should interpolate with some series of unitary, non-unitary CFTs which is M2 to M plus three. I know I'm right for n equal to one because M2 five is the Lee Young that you have seen before. But this would be an interesting thing to try to understand a little bit better. A corollary is that we know that M2 seven which is n equal to two is in the same universality class as the Bloom-Capelle model which is coming from the study of spin chains and it's a critical non-ermitian model at imaginary magnetic field. So it's new, it's interesting and it's not that explored. There are only a couple of papers by Fongelin and a nice observation for the Bloom-Capelle model would be that the upper critical dimension is likely above three and so the epsilon expansion is expected to work well. I hope in dimensionality three which is dimensionality interesting for natural reasons. So I hope there is more work to come on this topic and I will be skipping this part and go straight to the conclusions. So the conclusions for the first part is that several features of Wetterish flow and the local potential approach can be appreciated by solving it through the epsilon and the square root of epsilon expansions. At least this is my feeling. I've learned a lot by doing these expansions on how the solutions actually develop from the upper critical dimension and how they are structured. The approach suggests new ways to compute eta which are in principle useful also for numerical reasons and a criteria for optimization which I think can be explored because it might give better results for the anomalous dimensions. And what you have not seen is that you can use the local potential to, you can generalize the local potential for supersymmetric applications and study the emergence in the infrared of supersymmetry. For this part, I suggest you to find to be us and which has a nice poster on emergence supersymmetry and ask him a lot of questions. Thank you.