 Thank you very much Yeah, first of all, I would like to thank the organizers for allowing me to delay the lunch break for another about 40 minutes. I Will also talk about critical on models above four dimensions and In this case, I would like to add some of the ERG perspective to some of the things that were discussed by ego Klebanoff in the previous talk and Here I have the pleasure to represent The project that I've done earlier this year together with Astrid and and Lucas so As the outline of my talk I will start with a short introduction that basically repeats some of the things that were already introduced so I will Discuss a little bit the orn model below and above four dimensions and then Show how the work by Klebanoff and co-workers Put a little bit a different perspective to that But at the same time at least for somebody who is a practitioner of the ERG approach This this cubic model it's brought up basically three questions that we would like to address Within within the the FIG then of course I will not really thoroughly introduce the FIG approach on the last day of the conference once again But just highlight some of the features that I think are important and maybe advantages to to find out some To find some answers to these questions Yeah, and then I will set up the the calculations and compare and connect to these previous results and after all In the end or in the middle of my talk, of course, I will have to offer some answers to the questions I have raised So As for the introduction Yeah, we have seen during the conference a number of very nice sophisticated flow equations for all types of fields and all types of dimensions and I think even Yeah, this is nothing compared to the equation that I will show you in the following slide So please hold on to your seats Here's the beta function for the quartic coupling of an ON symmetric model In leading order of the epsilon expansion close to four dimensions, which is the upper critical dimension and of course everybody of you knows that the ON model below four dimensions has an interacting fixed point for a positive value of the quartic coupling and This fixed point is infrared attractive Basically meaning that we get in the infrared scaling solutions once we tune an additional relevant direction And we can basically explain Scaling behavior or critical behavior of statistical models when we now go to Dimensionality that is above four or in other words to an epsilon that is smaller than zero just in this very same calculation we still get a UV we get a non-trivial fixed point anyway, but as this basically is just a sign change here in this in this equation it means that we get the very same fixed point but this time with a negative quartic coupling and It is not infrared attractive, but UV attractive and At the same time so due to this negative quartic coupling an issue raises immediately That is the issue of stability that also was Discussed in the previous talk a little bit. So you would say from the first point of view This is a little bit discouraging. So one might not give up immediately, but there was this very nice work by Klibernoff and co-workers and they put a little bit of a new perspective into that question by basically How much start on a wish transforming the the quartic coupling in this theory and introducing a real scalar field Z which is basically here can be represented by an exchange diagram like that and Just to repeat very quickly what they found here so when you do this you have a cubic theory that has an upper critical dimension of of six and What they find is once they go to a quite large number of field components at leading order in the epsilon expansion that there's a real valued fixed point and As we have seen this is an example of a you you become can be interpreted as an example of a UV completion of the theory Or in other words It's something like an asymptotically safe model in that case so Then you can this this is really interesting. So you can start wondering What is the critical n in in higher orders of this loop expansion and you find that for example when you go to five dimensions? Which is naturally a dimension of interest you find 64 which is a really strong decrease as compared to the to the first order result and then you can go on to Next next order in the absolute expansion and apply resummation techniques and then it pops up again to to about 400 So there's no Well, I don't know what what the real value there or what the conclusion should be from this at the moment so and that brings me basically immediately to the To the to a number of questions with the first one being quite obvious. It's so what is the true number of the critical? the true critical number of field components in five dimensions and so I will Here add some ERG or FRG perspective to that to that question And it's motivated by the observation that there's obviously a strong dependence on the loop order in the perturbative RG approach The second question is that so when we introduced the model the cubic model starting from the pure or n model We were thinking of something that the the Model with the how about Satone which transformation is basically some sort of equivalent Model to the original or n models So at least on a classical level and now the question is whether this equivalence Basically holds if we go on and run the achieve law or in other words if we go to the quantum level of the theory so we can ask basically do these two theories really share the same set of critical exponents and the third question which I would like to ask and which is maybe Yeah, most obvious one also That is what is about the stability of the potential in this at the fixed point of this theory So a cubic theory after all is not bounded from below So somehow you should address this this question if you want to make sense out of this theory in five dimensions or About four dimensions in general so and I have to offer our first ERG perspective on these questions so The FRG approach Of course, we have all appreciated this equation of Christoph and So let me only highlight some of the some of the features that I think are interesting in this very context so what we use this equation to calculate beta functions for model couplings and to calculate interacting RG fixed points and Finally also critical behavior and critical exponents and One thing that I want to emphasize here To is that This equation allows you to connect in a controlled way to other to other approaches of course So for example, we can easily connect to the to the results from the leading order epsilon expansion and To make sure that we are really talking about the same fixed point at that point So that we're not just finding any fix an arbitrary fixed point in theory space And then we analyze it and finally we find out that this whole different story No, we can really in a controlled way connect to this fixed point and and look for the solutions Then there's a very Convenient feature that is the the flow equation also encodes non-perturbative information by this right-hand side where we have the This non-perturbative propagator and this basically also encodes higher order effects within the one group structure that is given here Then we can directly evaluate our better beta functions in five dimensions and And maybe this is the strongest point we can access information on the fixed point potential of the theory and study the stability Okay, so let me start we have this this cubic Theory with the how it's not on which field real value field z and we have to set up a truncation And what we do is basically we here for our for our first study We use this working horse truncation, which is basically a local potential approximation enhanced by these uniform wave function normalizations and Here the the central thing is basically the the effective potential which depends on the field invariant row Which is the squared of this field phi basically other yeah And then we have the z field of course, which is not bound by any symmetries so it can also appear in odd orders of the potential and So this is a two-field model and just to to mention this very quickly We have studied these similar to field and multi field models before and on this level of the truncation and have confirmed that For example for multi critical phenomena this this level of the truncation gives good estimates for critical exponents so We can put this ansatz into because of the equation and then Basically can find a closed form for the for the rescaled dimensionless effective potential Which is this quantity it depends of course on row and z again rescaled forms of that So in the first line, we basically have all the dimensional contributions and in the second and third line We get these threshold or these loop terms which also include the threshold connections from the non-trivial propagator So, I mean it's not about really looking at these equations It's just to show that we can really provide a closed form of these of these equations on that level Okay, so Yeah, now I have to to basically explain to you a little bit about the Expansion of the effective potential. So here we will we will make an additional approximation And we will do this in two different ansatzes. The first ansatz is basically that we we treat the Row part of the field so that the part of the potential that is formulated in the in the original field field fire only quadratic in this in this order and then we add basically a Z-dependent Potential which for example can be expanded in this tailor In this tailor expansion and this is so you can it's a severe approximation on the one hand But it's basically similar very similar to what we often do as a first estimate for fermionic models when we introduce For example the cross-nevere you cover theory and this has also shown to give some quite reasonable estimates for critical exponents in these in these theories then also this This approximation is basically sufficient to recover exactly the results from the epsilon expansion as we approach six dimensions and Very importantly, we can also study the large end limit for the full Separated Z-dependent part of the potential where no expansion in the z-field is actually needed So we can find an analytic analytical solution here for the for this part of the effect of potential and study stability and In a in a second ansatz, we will also Study a little bit the more comprehensive to field expansion and see what we can find out additionally on on that issue so Also, we have of course expressions for the anomalous dimensions which can be Represented by these diagrams, but as I said, we don't go very much into calculations for now It's just to show that these expressions are available So let's directly look at some of the results. So this is a comparison between the leading-order epsilon expansion results and different orders of our FIG ansatz for a number of field components of 2000 so we can we have to go To quite large value because in the epsilon expansion we know already that the fixed point only exists above about n equals 1000 and So what you see here is not a very advantageous plot because we are looking really at non-universal fixed point couplings So as soon as we approach to basically this perturbative limit close to six dimensions at some point They really merge then and they they converge nicely to each other But when we move away, we also see the onset of of non-perturbative effects basically And we can also compare for the with the anomalous dimensions here. There's a Limit of six dimensions it really nicely converges, but there are huge deviations as we approach five five dimensions So but what this shows is basically it allows us to really relate the FIG fixed point to the epsilon expansion Make sure we're looking at the same quantity here remit for six dimensions. It's quite nice And does not really depend on the particular choice of the number of field components that difference becomes sizable for When we approach five dimensions and what you also see in this plot is that the different orders of this LPA this is it's a finite finite expansion in the Taylor expansion. They are really lying on top of each other They are hard distinguishable. So if you look at the numbers, you see that there are still some differences But here it converges really very nicely Okay, so with this setup we can come to the FIG answer to the first to the first question namely what is the critical number of Field components in five dimensions and so just to remind you very quickly of what has been already introduced in the in the previous talk is that in the leading order epsilon expansion the critical number of field components is 1038 or 1039 and it does not depend on on the number of dimensions However, this is different when you go to higher Orders of the of the epsilon expansion there you get some dependence on the dimensionality actually you get a pretty sizable term in Epsilon squared and Also still sizable quite sizable term in epsilon to the three and this will also be the case in the FIG so We will find that the the critical number of field components is basically Strongly dependent of the dimension So here are some numbers to flash so when we are very close to to six dimensions We are of course also close to the leading order epsilon result But once we go to five point nine five point eight five point seven this number drops very quickly So this is already some some nice hint But if we look a little bit closer to how the fixed point values actually Evolve once we go from very large n to smaller n we see some also quite intriguing behavior So let's for example look at this curve Which is basically the purple the purple line which represents D equals five point nine the thing is that here at some point This is actually the 623 the fixed point values disappear into the complex plane Yeah, but when we try to to continue this curve We see that a real valued fixed point pops up again at smaller values of the field And so here you can basically add for yourself a guide Line that is a guide to the eye here in between the the fixed point values are actually complex But then it reappears again Yeah, and this can also be seen at the other fixed point coupling which is basically the C to the 3 coupling and When we decrease the number of dimensions these so these two separate islands they they approach each other and below Five point six five dimensions. They really merge and it seems like there is there could be a fixed point all the way down to to small to small n so I Think this this next slide basically summarizes a little bit more on what I've said And I have also added the results for the corresponding results for the anomalous dimensions here So we have two islands of on the n-axis where the fixed point exists for Dimensions that are sufficiently close to six dimensions and these islands merge at 5.65 so Yeah, I think I Would like to get to the next slide So this this behavior goes on once we we decrease the number of of dimensions. So this is basically starting at five point five point six five point five and so on until Five and you see that the behavior also becomes smoother and smoother. So what we also find here It's maybe interesting to look at this large n behavior So the the anomalous dimension of the five field always approaches zero and the one of the of the z field actually behaves like this it approaches the value which is six minus d which is also very nice with art actually and Yeah, this I can summarize these findings on the critical and a little bit in this in this plot here So basically in the hashed area, this is the region where the fixed point of this theory exists in The epsilon to the three expansion so three epsilon expansion and the whole gray areas actually the the FFG prediction now and here you see that there's a Rather small region where no fixed point exists in on the real axis But here this is basically the point where these so if you look at some particular dimension You get these two islands for large and for small and and they just merge at five point six five So that's the first Answer from the FFG perspective that I can offer. Let's go to the next next question that I have asked I know I had an additional remark So this is at the moment. It's more or less a numerical finding of the solution of our fixed point equation. So but We try to put some more thought into that to to to find reasons why we should why we could believe these These results why we think that they could be tentatively reasonable and the thing is That when you look at four plus epsilon Dimensions then you have an o n Wilson Fisher fixed point which I introduced in the beginning which is not stable But the the conjecture is that this fixed point is somehow connected to the six minus Epsilon dimensional case where we study the the cubic theory and here in the in the pure or n model above closely above four dimensions the the six this fixed point basically exists for all and so there's There's no limit to n as long as the epsilon Is small and so as the conjecture is that this fixed point in the cubic theory is basically some sort of analytic Continuation of the o n Wilson Fisher fixed point toward a four dimension We would expect for the cubic theory that there exists some non-interge critical dimension between four and six We are these islands merge and this is exactly what we basically observe We are the fixed point exists for all ends that are larger than zero and our result was the 5.65 basically which you which you see here So if you look just at the at the behavior of the epsilon to the three It also seems to to bend down and so they think that there would also be some room for for similar behavior that you could observe and Just as a short sight remark There's a very similar behavior in the three-dimensional a billion Hicks model. So There's also some sort of a fixed point that ceases to exist below a critical number of in this case complex scalars and Which is severely reduced at higher orders and there are also FIG fixed dimension Perturbative RG and also QMC results which suggests that the true number of Complex scalars where the fixed point ceases to exist is actually smaller than one So this is work by Bergaroff and I think Christopher was also also involved in that the fixed dimension is Herbert and Tisano which and There is a number of QMC results and I think I quoted here the the latest one and There's another hint that this small and solutions might be Sensible and that is even in the epsilon expansion when you do not a Decouple purely scalar mode, but you you make a tensorial decoupling of this Then you also find solutions for O2 and the O3 model basically which is work by Lucas was also involved in that and Igor Herbert earlier this year. So but this is pure epsilon expansion then Okay, so this brings me to the the second question. How am I doing in time? Okay, so I Will basically flash through it so we can calculate of course the universal University classes by looking at the stability matrix and extracting the critical exponents from the eigenvalues if you look at the So the the pure n-model in above four dimensions. There is In the large now I'm discussing now the large large n solutions you find two infrared relevant directions with analytically known Ritical exponents. So basically the correlation length exponent is 1 over d minus 2 just like you would expect it also from the situation below four dimensions and the second one is the sub leading exponent is 4 minus d and We can now Analyze our data on the cubic fixed point and what we basically find is so first of all that is something I already quoted we found that the anomalous dimension of the five field goes to zero and the one of the z field goes to 6 minus d and What we find additionally in terms of the critical exponents are three relevant directions now So the first one agrees basically with this one, which is just the inverse then the second one with this one And then there's an additional relevant direction clearly relevant direction with scaling dimension two So what we have in this situation is a three-dimensional critical hyper surface so What we can say from this finding is basically that the Universality class from the pure n-model seems to be embedded in the one of the of the cubic model because here we basically have to To place ourselves on a particular part of the hyper surface and then Basically the model as soon as we have restricted to a particular hyper surface a model basically behaves like like the pure One model in this large n-limit Okay, so now the final question is about the stability of the potential and Here I would like to to mention also some previous work that has been done in the in the FRG framework that were works by Roberto and John Paolo and also Peter Marti So they discussed the stability of the potential of the pure n-model above five dimensions so and therefore they they they Employed a mix of analytical and numerical Approaches that you can do within the FRG and what they basically find is that within the pure n-model without a Harvard-Sotonovic field no stable and physically admissible fixed point solution exists between four and six dimensions and Solutions what they find is that the solutions are either unbounded from below or they are singular and not globally defined So it's not acceptable However, so and this is where we would like to make our contribution This formulation within the pure or n-model Possibly misses some important non-perturbative information that is somehow encoded in the momentum dependence of the about Sotonovic field So we will basically try to analyze the same question again with this Extended model which is closer to the to the model that was formulated by Gliwanoff and co-workers Okay, so let's start again. I want to Show the the ansatz which we are making which it's a very much simplified ansatz which only is by linear in the five field and then has a general potential in the Z field and the thing is that if you look closely at the possible loop contributions that can can Go into the flow equation at that point you see that in the large end limit No higher orders in the invariant row appear at leading order in 1 over n so additional actions Interactions are basically only generated at sub leading order in 1 over n and also the feedback into these flow equations Of the of the effective potential is only sub leading. So basically this ansatz is a very well Very well suited ansatz or very well suited to truncation that captures basically all contributions That to the effective action at leading order in 1 over n so for large and it should work and just to be explicit in the following without even showing you Explicit expressions on the equations we will work in the in the sharp cut-off scheme because that showed to give us some nice computational simplifications Okay Now what's going on? So the flow of the effective potential in this large end limit basically can be written down in this in this form with only one Threshold functions the others are basically suppressed in this in leading order 1 over n and Then we have a separate flow equation for the mass of the five feet Which is just given by this very simple very simple Dimensional contribution we have already found out that the anomalous dimension of the five feet approach is zero And that means that the fixed point in the fire in the in the fire masses basically zero two So now we can write down the loop contributions to the zero coupling Which is the basically the Bosonic you cover coupling in this theory and it also comes basically only with a dimensional term That means the fixed point will be located at anomalous dimension equals six minus D also result that we have already found previously at the same time the expression for the anomalous dimension of the Z field reduces to This very simple equation so we can directly evaluate the fixed point value And that is very convenient because we have then the fixed point value for the G and for the M We can put it into the into the threshold function and directly solve the flow equation for the for the Z potential and That's what we do And it looks like this So it's not a very pleasant expression it is formulated in different For different values of the field in different ways, but it smoothly connects basically when you whoa That's now a disaster because it doesn't show my okay now Okay, I can I think I can make up for this so if you plot this if you plot this potential it basically looks like on the next slide the one that is properly displayed like the the gray curve and This is the result if we if we take the freedom that we have in the solution of this equation So the point is that you can always for this particular flow equation only and this will not hold for for other flow equations For example for the pure and model you can add the homogeneous solution with an arbitrary constant This will always remain a solution because the flow equation actually does not depend on 1 over 1 plus v derivatives of the of the potential itself, so you can do this and So neglecting this term basically gives you the potential that you see here in this slide and it's not bound from below so it has a It basically goes to minus infinity here, but if you add one of these non-alytic terms which you're allowed to do So the z to the five-half term then you see that Basically here the potential is not modified at least here close to this minimum, but you can you can get a Upward spending here on the side and you can basically save your potential, but at the same time So without the analytic non-analytic contribution those potential is globally unstable But if you add the non-analytic contribution the potential becomes globally stable But at the same time one has to say that in this case we don't have any more a Simple interpretation in terms of Feynman diagrams or a way to expand the theory basically so this is this is that finding and Well, I think I can come to my conclusions rather quickly. So As an add-on to this to this finding we also wanted to find out a little bit of the behavior at still large But a little bit smaller and So finite n and we therefore we have to go back to the two-field expansion because then the situation that the the second field Only appears quadratic in the potential is not true anymore. So we have to take into account these Can make it to two-dimensional Taylor expansion basically and if we just expanded if you just look at the z direction Then we see that this Taylor expansion Basically close to the minimum very nicely reproduces the result. So this is for n equals 10,000 still pretty large and The Taylor expansion even in pretty low orders approaches very nicely this analytic result and now if you look into the Into the phi direction. So this is the z direction. We have a second direction, which should also be stable What you find is basically that both both mass terms of both fields are positive So they basically produce a local minimum However in the feed I wrecked a phi direction all the higher order couplings have negative fixed-point values. So That's what you basically. I think the nicest blood is Basically here you see when you go to higher orders From blue to green to black Of the expansion in the five field then it gets deeper and steeper that you get an unstable potential. So I think the first thing to say or the most careful thing is that here We do not find an indication for stabilization for the potential Find more indication for destabilization of the potential Okay, I think that should bring me to my conclusions Time so Let me just summarize of what I've showed you. So What is the NC in D equals 5 and what we find is that below the critical dimension of 5.65? The real fixed point exists basically for all n Then what about the equivalence of the cubic theory and the original or n model? So we have two relevant Critical exponents that precisely agree at large n. However, the cubic model features an additional actually relevant direction. So What we can say is basically that the or n University class is embedded in that of the cubic model But the models seem to be not fully equivalent unless some of the couplings are placed on a critical hyper surface of the fixed Point so you can also put this in other terms You can really consider the the z field that has been introduced in how much on which kind of way as some some independent degree of freedom which really adds some additional information and Finally, what can we say about the stability of the fixed point potential at large and the Fixed point potential seems to be globally unstable unless non-analytic terms are added then at finite and the model exhibits a local minimum and the higher order terms are all negative and This suggests more like an instability at larger values of the five field and with this I'm finished and I thank you for your attention