 Thank you. Good morning, and thanks to the organizers for the imitation and Hope you enjoyed the talk. I mean, I try to make it partially pedagogical because somehow Kind of notice that we many young participants and part Advanced so hopefully you can actually enjoy both sides but I Also make the plan from the beginning so you know also what you might be missing in the end and you can ask if you want to And the plan is like this. I will first define what Wilson Uses word fundamental because we often say we work on fundamental interactions But there's actually a rigorous way to define a fundamental interactions that we can use at least them mathematically Then I will describe completeness into the freedom and how you transit from completeness into the freedom to a complete us into the safety I will construct and I want to mean construct I really mean construct a controllable asymptotically safe theory for dimensions I will then describe the a theorem What is the essence of the a theorem in perturbation theory and how I can calculate it for asymptotically safe field theory? then I also know that people here works also on thermodynamics, so so will show the asymptotically safe thermodynamic of Sorry thermodynamics for asymptotically safe quantum field theory and also confronted to with the I will describe with the F theorem and show that the F theorem fails in the case So you can't really use the F theorem in general and all this is going to be precise So then I will speculate a little bit more with just a second Then I will discuss In a less controllable theory, which is QCD Different number of flavors in the multi-flavor case what I call the QCD windows 2 comma 0 So how do you actually go beyond the standard conformal window and how the QCD conformal window 2.0 might look like? For example by adding asymptotically safe window finally I will translate it to supersymmetry where you can make a number of precise results I will call them exact Beyond perturbation theory. I will lay down for you the No perturbative constraints that we can use to constraint conformal field theories in four dimensions And I finally conclude and I will show also how hard it is to make an and why it is hard to make an asymptotic safe field theory supersymmetry Very good. So but my motivation I mean, why am I actually interested in that is because if I look at the standard model I see gauge fields I see fermions and scalars in particular the Higgs The interaction of the standard model are SU3 cross SU2 cross U1 I would typically say the standard model is a gauge theory But in truth as soon as we add the Higgs there are new kind of accidental interactions Which are not Driven by gauge interactions and these are the Eucala sector of the standard model and the scalar self-interruptions The first is responsible for all the flavor program and the second is Responsible for whether or not the Higgs is what we think is the Higgs right now So that second part and it would be very hard for LHC at least for the future to Pin down the self-interrupting of the Higgs So this is the status of the standard model so far But it is a fact that the standard model so far works So we need to take that to the face value and it can be embodied very simply into a schematic Lagrangian like that Where you see indeed the gauge term the fermion interaction with a gauge a generic You have interactions the Higgs and his own self-interruptions. Of course, I'm sticking to specific schematics, but that what it is and And This is the summary of what I just said in words and it's a fact that the gauge Eucala theory Described just under model perhaps that matter But in three dimensions the similar Lagrangian can be used to mitigate the next matter face transition If I go in two dimension you have graphene if you go to higher dimensions So this is actually the effect if your theory for strength here So it is a factor we need to explore gauge Eucala theories and try to solve them If we can So it is a universal description of physical phenomena so far Now as I promised cannot define a fundamental field theory. Yes mathematically I can and I will go back to Wilson a fundamental field theory is a theory which is valid and I'm setting aside gravity for a second at arbitrary short and large distance scales This can be achieved if you have an ultraviolet fixed point. This can be And I'm it can be trivial or interacting for example in this theory I have an ultraviolet fixed point and There's a gosh a fixed point of the origin think of this Gaussian fixed point as a QED like theory And then the coupling gets strong at one point you will reach a fixed point this theory on this Line is well-defined both at very very low energies and a very very high energies the theory is Does not have any land of course These are other irrelevant directions so in the recipe you want short distance conformality That all permits to have a well-defined limit if you ever worked on lattice quantum field theory. This is a Fundamental requirement to be able to set in the lattice spacing to zero and I want that to be a complete you be fixed points For example in the gauge of cavatier I showed you before there are several couplings that one of them is either as Synthetically safe or asymptotically free while the others run wild doesn't make the theory fundamental So I want a theory which is complete in all the couplings the dimension of the critical surface at the interact in fixed points defined for you the Predictiveness of the theory right it tells you the surface of relevant directions in this case for example in these two Couplings it is one dimension as predictive as QCD and if you have some mass operators say for example the mass of the Higgs We often say the mass of the X is a relevant operator That is true from the infrared physics point of view not from the ultraviolet point of view Unless the anomalous potential of the Higgs runs wild and mass of the X is irrelevant from the ultraviolet physics Only cares about the low energy physics that drives you away from a potential low energy fixed point Or the low energy dynamics so the hierarchy problem if you still like to discuss about the hierarchy problem It's not a UV problem. It's an infrared problem Stabilization of the infrared scheme now according to Wilson the Sunder mold is not a fundamental field theory because Does not have a either interacting or elementary or a non-interacting fixed point in the ultraviolet one of the famous example of a theory which is well-defined is of course QCD Which we know is asymptotically free and that means that there is a trivial provided fixed point Means that this is the cartoon of the coupling at high energy the coupling Reaches a non-interacting fixed point and it reaches it to logarithmically however, and That's what I just say however in in in you can determine precisely in perturbation theory because It's actually non-interacting so it's enough one loop to dictate information around here The situation gets more interesting in this case in infrared because different phases can be in vision infrared And this has been a long program being still going on to what is the physics in infrared for example if you change the number of flavors But imagine a bit here you were first up I add those of scalars and the theory in the gauge coupling as a perturbative ultraviolet trivia fixed point Then I would require for example that all the margin of couplings vanish in the ultraviolet These are cool the complete asymptotically free conditions. It can be all of that use them a loop for all the couplings You can demonstrate that you do need the gauge coupling to drive complete asymptotic freedom And of course in infrared you can still have either for example kerasimetry breaking confinement and so forth so An example of why do you need the? Gauge coupling imagine if you have any you have interactions at one loop you have the this is the you have interaction See this is the second coefficients typically positive, and that's what you typically Say that this this theory is a land of all because it grows quadratically with alpha and this has a Because of the interaction with the gauge this one confuses typically negative. So by adjusting alpha H as Initial condition you can make the second-term win and this will drive you Alpha H to be asymptotically free I can be more specific and look at the face diagram here I have alpha H and alpha G you can see that for the values of alpha H below the above this line fixed point line you can see that The alpha H is not well-defined in ultraviolet mean my other such that The this goes to the toward infrared and if I try to go to ultraviolet alpha H explodes at risk perturbation here However, if I start in the region the parameter space below this green line You see that both alpha H and alpha G vanishes Close to the ultraviolet x1 and therefore this theory is a Wilsonian theory and it admits a continuum limit So the same gauge of cavatieri can describe either not well defined here in ultraviolet at least in perturbation theory And it completely will define here in ultraviolet We are only scratching the surface of these theories This is not a new idea That's why typically I did not present very much in the past because goes back to chain I can only and then Callaway wrote even a report about that And it was recently reprehended and in the paper with cloud you and Thomas We looked actually at the high-order corrections where you also see the infrared fixed point emerging in these theories Very good that all I'm going to say about complete asymptotically free field theories So in summary, they have a non-interacting ultraviolet fixed point and they have a logarithmic scale dependence In the high energy But there's another possibility that might actually happen in the ultraviolet and you have in fact an interrupting fixed point In this case you reach. Oh, sorry. It was to be too fast. In this case. Oops. Let's do it first It's it doesn't I mean you can see when it goes from logarithmic to power low. You feel it So so now you see that I am reaching ultraviolet fixed point an Interacting one and now I reached it with a power low Which means the critical exponents or the scaling is point will not vanish in this case That's a theory like this exist I know that this community has been working very hard for Gravity to try to demonstrate whether a gravitational theory where in fact asymptotically safe and There are examples in two dimensions three dimensions. So for so on actually when I mentioned depends on the theory But let's look in four dimensions. So Let me take a theory that looks very much like the one I introduced at the beginning This is a gauge Yukawa theory. So I have in fact the gauge interaction the fermions I have the Yukawa and I have the interaction of the Higgs now Take a this as Qcd plus a Higgs right in this case the Higgs does not carry colors So the Higgs is colorless, but it does carry some self-interaction. This is the group theoretical structure of this theory in terms of the field content and the exact Quantum global c-meters of the tier as you can see this theory looks very much like Qcd I only added it if you want an elementary mason to the tier and I want to I want to see whether this theory Can lose a synthetic freedom, but enact a synthetic safety Because I want to prove a result. I will actually I will go into a the large number of colors Large number of flavor limits such that the ratio is a continuous variable and it can use it as a control parameter for my tier So this is the way you prepare your coupling. This is the Yukawa. This is the Single trace and this is a double trace operator and this is the gauge Interactions, okay, and as I promised I'm taking the I'm taking the Veneziano limit Which means infinite number of flavors and infinite number of colors so that the ratio is a continuous parameter Our loop and this is the normalization time Depending on the coefficient B. I can either have a Synthetic freedom if B were positive and at least for the gauge coupling It wouldn't make automatically complete asymptotically free fifth year So but in the gauge could be asymptotically free or I would could actually by tuning the number of flavors I can lose a synthetic freedom exactly at the same value where it happens in Qcd and typically we never really wanted to use this Tier right I was told since it was a young younger That please stay away from these tears right if you want to do model building users into the freedom Because we don't know whether these tears will define ultra value because it has to land up all Now let me define for you a parameter a small parameter Which is going to be the distance from the loss of us into the freedom and let me choose this Parameter to be very very small so that I can have controllable perturbation theory in Epsino and let me ask whether for example, I ordered corrections that can professionally fix the Fixed one now I looked and I go and look at the second coefficient in the gauge Keep in mind. This is a bit more complicated than just a gauge of the you have and so first on and now what happened is that If C if B where my nice B's positive if C were negative and alpha G were small Then yes, I could in fact imagine a bentax type ultraviolet fixed point However, there was a theorem It is impossible to do this in gauge theories with fermions alone This was demonstrated by casual as a David Gross told me they were afraid The quote is something that that could happen so they wouldn't push they wouldn't have pushed very hard for a synthetic freedom But he came back empty-handed. He said look that's a beautiful paper if you have a gauge theory with fermions in arbitrary Representations the C coefficient will never be such that you can have a controllable ultraviolet fixed point So at least in perturbation theory that are not asymptotic I say 50 years and then it makes the typical mistake of a theoretical physicist He proves the theorem and then at the end he says I would have spread that the same happens with scalars But he didn't prove that Let's look at what happens when I add the as soon as I have scalars in this case I might have and like in this case you have interactions right now You see that I'm already took the large the Veneziano limit This is the essence of the casual theorem You see the this coefficient when the absence positive and I've already lost us into the freedom can never change sign So he was right. You cannot balance alpha g to the first coefficient But now when you have you come interactions you get the negative contribution and this come from the from the exchange with the Higgs Of course, you also have to look at the beta function of you cover and you see that again A negative conclusion that by the way, this is what you like from complete asymptotic freedom So it's actually happening a similar thing here and you see that these two now you can set them to zero You cannot always do but in this particular theory you can And you look for the zero and now you discover of course a Gaussian fix pond at the origin Which is QED luck. I will call this non-Abelian QED because I lost us into the freedom But I also get an interacting ultraviolet fixed point which I can professionally span in epsom at this point I have a Taylor response in epsom, of course, it's still an asymptotic series But if epsom is sufficiently small all the instant on correction will go like e to the minus one over epsom So they will be exponentially Vanishing and therefore I can trust the Benzakish fixed point Of course higher terms here will be corrected by next next leading order so far so on but not the leading orders But we can prove that and this is in fact You know the Delocation of the fixed point is a function of epsom, right? And you can see how the next lead in order next nice leave the order change slightly location when epsom becomes big But not in the ridge in a controllable regime of a small epsom as you should be Now it's fun when you have a theory to be that what you can calculate let's calculate everything that could be interesting So I can for example look at the scaling exponents which told me how fast I reached the fixed point Having two couplings at the moment. I have one Relevant directions and one irrelevant directions from the ultraviolet fixed point point of view Okay, you can see that the irrelevant directions is quadratic and this is the direction I can QCD drives you toward the infrared This is the relevant direction. This is the relevant and this is a true ultraviolet fixed point where it has is It has basically one dimensional critical surface Which is cases a line and is exactly this so which means that in the Deep ultraviolet along this if you go along this line you reach the ultraviolet fixed point to the gauge coupling to cover Cabin freeze as you go to a low energy you go to an uninterrupting filter Here you could put mass of quirks mass of scalars You can deform the Gaussian fixed point to make something like QCD actually I can reconstruct QCD If I want to and in the ultraviolet that you will not have the asymptotic so QCD So and as you know experiments have never improved QCD to be asymptotically free They always showed that it works around here. So in fact, I can in principle complete QCD In an asymptotically safe quantum field theory even with imperturbation here This is the irrelevant direction tells you basically that on the line of physics called all the separatrix I will actually have an exact asymptotically safe quantum field here The next next reading order will make this a bit brief But it's very interesting part and you know it would require much more than an hour to go through it But the point is that you have Next next thing order which means three loops in the gauge two loops in your cover one loop in the soft coupling the self-capping start running So I can possibly neglect them One is a single trace one a double trace is a one-on-factor that the double trace in the large and leave it becomes a spectator Capling which means you can solve it at the end once you solve for the first three and these are the back reactions on Three loops for the gauge and two loops on the ocava of the sex coupling You can see that because of the Higgs is neutral the only Gage can only coupling the fields it is in fact the you cover and it comes only from the as promised Alarm in the national unity from the single trace So I need to solve the problem of gauge you cover single trace and then plug it back in the double trace This is the find that the double traces, you know, it's typically Spectator now I Asked you to trust me on that but you should never trust anybody so And since it's a calculable field theory, I urge you to go and check right Just in the spirit of going to be fast, but you can demonstrate there are several fixed point That need to pass a number of consistency conditions and one survives So it's not obvious that you automatically the fat one you have a scalar Firmium field theory the value of a fixed point even in perturbation theory doesn't guarantee you that you have a fixed point You can have a column on my work mechanism. You can have several things. You need to actually controllably check that Once all these checks have been done you demonstrate that in fact there is still a fixed point in all the couplings So the theory is a complete asymptotically safe quantum field theory These are the critical a spot and the scaling exponent for all of that and again at the surface is one dimensional Which means that the gauge cabin drives the dynamics here a bit like in completeness totally free It is the gauge coupling that forces the theory. So gauge is it's a good thing It's not the best thing but The Higgs here ties the theory together because with other scalars the theory will have a land-out ball So for the first time you have a dynamical reason for having scalars into the tier if you're coming From it, you know the background I come from like technical or you don't like scalars There are nuisance In this case scalars are actually needed for the dynamics and not for supersymmetry now if this is actually a three-dimensional Face diagram where you also see the single trace the gauge coupling at you cover Reaching indeed the ultraviolet fixed point that this is the projection in the alpha y alpha G Now once you have to fix point you can always define a globally defined line that links to fix point This typically non perturbative statements But in this case you can solve it and It's a globally defined also known as separatrix because the separates different region parameter space here different region basics and If I project the beta function over this line for both the gauge and you cover Because of the dimension is one you can try to show that the asymptotically safe behavior the beta function indeed Even ultraviolet the fixed point the one you were actually hoping to get In a bit a GP the why but actually even in the scalar couplings, right? They all have the same behavior and in fact a more dramatic way to show that this I can just show the running Typically, I'm very worried where it to show running of couplings because couplings are not physical quantities But in Benz X regime you can actually control of we show them to It always show physical Quantities that for example critical exponents for so This shows that that is correct now at the you can also see that the double trace Coupling is negative and you could be worried about the column of member But we have calculated that and because the sum of two couplings positive the theory is well-defined even in ultraviolet Very good. So as I say before In my eyes was the first time where I saw the need for of scalars to make the theory fundamental I was also sure what can happen even without scalars at least that's a conjecture. It's no longer as precise as this Very good Now I would like to be able to calculate everything and then this is a bit more pedagogical I would like to know to calculate the a function and on this theory, which I can and how do I do that? I know that if I take a conformal field theory in this case can be thought as the trivial field theory, right? Without interactions then even the quantum corrections on a trivial field theory gives me a marginal operators a marginal deformations That I will consider as a going away from the conformal field theory and I will upgrade that just a technical trick the couplings to be a function of the space time and Make a conformal transformation, which is equivalent to a This case deletion of the metric tensor metric And I will redefine the couplings a function of the scale Under this variation I can Calculate at least formally the variation of the of the Generating function of the theory and this is the what you get in general Okay, so this is the essence of The fact that you have that the trace of the energy momentum tensor is proportional to the beta function of the theory In a trivial background, but in an interior background you you you get these extra corrections This is the Euler density the Einstein tensor and Sigma and all decays this symmetric metric they now become function of the gauge couplings Even though you have a background These functions are well defined even in flat space That's all the trick, okay So it's a trick to calculate this object By putting this theory on to your background and if you do two while Transformations, you know that because of the be a nature of the while anomaly to independent Consecuity transformation of while While transformations commute and that you can use it to relate couplings and have a so-called gradient flow in the end I can define over this Function the a function which was a coefficient the Euler term and the beta function which now becomes a vector of your space So already think about a bit of function of the theory are not fundamental quantity. They're a vector space Fundamental quantity will be skillar quantities like an a function, but you need to correct it by By minus W be a W is another one for if you differentiate the now this a till the function So the a theorem is as one of my students used to say it's the name is wrong in in many in many ways It was never a was an a till day was never a theorem so far, so But in perturbation theory we can prove it so the variation of it in the respect to the coupling Can be now its proportion of the beta function for a gauge you have a theory this term vanishes And you can then calculate also the derivative of the a till the respect to the end log of the scale and you Get the quadratic form Now you immediately see that if chi were positive this leads to a negative monotonically decreasing You can show this in perturbation theory, but beyond perturbation theory you need to perhaps use analyticity in In the optical theorem and this is a bit more delicate So I'll be using this in perturbation theory and since I was calculating a theory a fixed point in perturbation theory The a variation better be positive So I will do that for the theory then Now you see that I will calculate a in ultraviolet to minus a in infrared and this is the value get is directly proportional To this order to the scaling exponent of the theory to the relevant scaling exponent here up mean Minus a sign multiplied by minus one And this is a fact that we already checked to the next leading orders not automatically the scaling exponent So unfortunately that's not General and chi is a calculable quantity. So I Will say that We also calculated other things like the bootstrap for the composite operators of our song This theory is a well-defined conform of the theory in four dimensions, which is indeed asymptotically safe Now once again, let me just once you have a theory It's fun to check other things like let me calculate for some of the terbonomics of this theory and see how does it look Because you know different from QCD this theory is not as an interrupting ultraviolet fixed point So how is it the thermodynamics in ultraviolet and that out how that depends on the temperature? It's something that was worked together with you Krishka and This is the pressure to the next next leading order Again calculating now is a function of T and already in a menace no limit So this is the pressure you see the immediately that the next next leading order you get the well-known non-analytic terms and This is the idea gas limit And this is next leading order term and this is the next to next leading order term right in this theory There are firm and scalars and gauge interactions Now this is the for different epsilon. This is the pressure normalized to the to the Zero-thorough-order pressure this the and you can see that the radius of convergence of finite temperature Worse than or compared to the radius of convergence of the theory in absence of finite temperatures as well known Because the you have a non-analytic sponge Out of that I can of course calculate that immediately the entropy density of the theory and it's nice to plot s minus p because this is directly proportioned to the Beta function of the theory at zero-temperature So it's actually a way to measure the beta function if you were able for example in on the lattice to calculate independently the entropy density and the pressure But another interesting application is to check the f theorem I don't know many views a quantity with that to be something that all enough people discuss that two was never a theorem That is a conjecture and I would say on worse standing than the a because a at least for super similar theory is considered to work So this was before words on time and go by Tom upper quiz Scottish mouths and he tells you that if you count the degrees of freedom according to the Coefficient of the free energy, which is nothing better than the pressure divide by t to the fourth Times 90 over pi square this coefficient Intuitively it counts a number of degrees of freedom of your theory So the infrared should be less than the ultraviolet. Well, I have an asymptotically safe field here This is true for a number of asymptotically free field theories. So let's check if it's true for an asymptotically safe So we did that I spare for you the attic and we don't seem to see that is the case Please double check the results to I mean we feel confident about that So f is violating asymptotically safe Of course, you could say well, but it's an asymptotic safe field theory. Why do I care? Perhaps f doesn't apply there But the a theorem fun works, right? So you have a function that actually the difference is indeed positive even if we're not asymptotic I say field theory. So there is no reason to doubt that it should work in principle also for an asymptotically safe How am I doing? excellent, so so far I presented everything which was Under control now. It's always nice to think a little bit ahead and for example take QCD just QCD and Change a number of colors flavors and in particular, I could also change our colors. It's no no problem with that and Generally, you plot this diagram, which is There is certainly a a line above which you lose us into the freedom. This is you know An exercise for a master student. So when that's QCD loses us into the freedom This is the lemon over over to NC and then if you go a bit Grow a little bit older. You can also say that just below this line and in particular in the Venezuelan limit I can sorry in the in the in the middle of it. I can prove that there is an infrared Interactive fixed point and You can you expect this at one point to end And you make what is known as a conformal window. That's the traditional conformal window for which number of lattice Efforts has gone for the past ten years to investigate that So question is it over? Was that the only window we can imagine to add? Even if you were to do that, I mean, which is very hard in any case Well There are indications not as solid as in the case I mean certain not as proved as before where if you go to infinite number of flavors You might enact an ultra valid rati fixed point a fine number of colors now One thing I can say is that if I assume that there is a large NF QCD As an asymptotic I guess a fixed point Then because there is no Benzax ultra valid fixed point there must be a critical number of flavors, right? So I'm just Using the fact that since I know that in perturbation theory I cannot get by casual an ultra valid rati fixed point There must be a critical number of flavors about which that has to up So they might be from there on In asymptotically safe for you So the conformal window 2.0 might look more like this Because if I look about the conform of your theory that could be like that too, right? So we only plotted the lower one But they probably will be also an upper one. I mean where you the theory becomes safe and On general grounds without even doing a computation. I can say that I would expect a critical number of flavors About which the theory is asymptotically safe There must be an unsafe region where the theory cannot be used as a fundamental for theory It can be as a low energy and fatty for theory non-habilian QCD, sorry non-habilian QED And I don't know the order of the first transition here Could be continuous can be first order could be all orders. We'd studied with Pica with Claudio Pica and with Daniel Something up here where you can have some kind of control. I'm not completely confident about that Infine number of flavors with a function is to be a better understood, but nevertheless there will be the What I would consider the conformal window 2.0 Very good so this now we leave the realm of Not super symmetric field here. I try to step into super symmetric safety or unsafety Everything's clear so far. I hope I was going you know not too fast I tend to speak fast as Italian Okay, so I've used perturbation theory so far and when I didn't use perturbation theory I was very careful in telling you I was a conjecture So please don't go around say that was an exact result was not for the QCD case, but it is for the Gejo-Cava theory Of course, I know as soon as you have a theory for which even in perturbation theory you have a Preciser is how you can trust you will you typically expect that if you supersymmetrize you make it better Right how many times you heard well, of course in super sim is going to work better or I can go beyond perturbation theory So let's go and see So What can I use in supersymmetry that I cannot use? automatically in For example, I showed you before Like you CD a different number of flavors But certainly even if for those two that could in principle discuss some utility constraints But that will be trivia because in perturbation theory will never really violate that So I will want to go beyond perturbation theory So I know that operators that belongs to unitary representation of the super conformal group typically actually always have to Need to abide some lower bounds an interesting one is for the one of these pin zero operators So that the the dimension of the operator must be always larger equal to zero and you get equal to one only for a non-interacting filter Now again, this is a specific thing of supersymmetry car of primary operators. I mean car operators Have their dimensions and the our charge which is nothing but and then you an axial charge of this theory Which in the case of supersymmetry, there is a special you an axial which is in the same super multiplet of the conformal anomalies In this case you can calculate it by knowing just the our charge and that's if you want one of the big Strong points for supersymmetry, I cannot do this for Non-super symmetry field here is there is not any exact relation between the dimension of the operator and in an axial charge and it's very easy and What I'm going to be easily understood in supersymmetry because trace anomaly and axi anomalies get related in supersymmetry And because axi anomaly is an ABJ anomaly is one loop exact that Immediately, you know makes the trace anomaly richer in QCT or non-super symmetry quantity. That's not true This is to do so with a little more physicality of the theory For example, if you a Squirt I mean if you have a car super field, which is my generalized quirks right now besides the quirks also of the squirks you have also a relation that tells you that the The dimension of the operator so say to the air charge, which is automatically now fixing for you the Anomalous dimension of your field This I can forget again in non-super symmetry field This is one thing I'm going to use the other thing I'm going to use a central charges, which is central charges in principle You can use Please you can use any linear combination of these central charges, but this one's I'm going to show to you are interesting because they Abide the positivity conditions and they derive from the stress energy trace Anomaly one of that is the again the a function, which is not a supersymmetric version of that all of them depend For a conformal field theory on the R charge of the tier Or all the R charges of the tier now The a of R is a conform anomaly of a supercompromable theory and it's nothing but that you can show that the Anomaly is associated to to the tuft anomaly conditions for the u and r symmetry It is the way you derive it is because it's a coefficient proportion to the square of the dual the Riemann curvature This is his own expression right is trace you and cube minus trace you This is the short end notation for that There is also a more Specific expression in terms of the F charges then this is a C function and this is a proportion to the square of the vile tensor And that's the expression and again is very similar to the a but with different coefficients for the the anomalies This one is a linear commissure of the u1 cubed and u1 in typically u1. We also call the gravitational anomaly p of R is associated to the flavors of the theory right It's a proportion to the square of the flavor symmetry field strength and you can think about this is indeed one u1 A triangle with one u1 And two gauged flavor symmetries in the end and that actually is what gives you that The a theorem is an extra condition all this function must be positive of the conform of fixed point But a abides at one further condition that's b and c don't And this is due to cardy It's a delta a must be positive good So let's calculate delta a for a generic quantum field theory both in ultraviolet and infrared at a fixed point Now this is the generic expression you have Where r i is the dimension of the matter field for example this were quirks this will be a NC This is now you see that this coefficient the first coefficient is always positive the plus sign Is for an asymptotically safe quantum field theory the minus sign is for an asymptotically free So if you do in the cyberg phases and infrared you'll be using the minus sign because you have an Interacting fixed point in infrared and an ultraviolet free fixed point that tells you that there is a change In sign they need to require For the other term to be positive now you immediately see the the problem And now this is actually the the essence of why it's harder to make asymptotically safe field theory is In supersymmetry, okay? It is that it's stronger constraint for an asymptotically safe quantum field theory than it is Than the case an asymptotically free the reason being that at least one of the r charges needs to be very large Or the asymptotically safe why for asymptotically free our charge needs to be small But that's good because that's what you get already almost from perturbation here Makes sense That is the Issue good Now I go slowly to construct the uh beta functions of my theory And again I can show that the gauge coupling beta function are proportional to the abj anomaly Where now the anomalies of u1 are charged and two gauge and two gauge And two gluons You see that the whole related triangle anomalies as I was saying because in the supersymmetry they get related with the tris anomaly And this is the expression where two of g are the normalizations of the quadratic generator of the tris normalization of my generators for something qc this will be n And this is if there's a fundamental representation this will be one half If I have also yukawa couplings in this case you have yukawa like interactions Then you have another beta function which is actually proportional to the r charge of the operator that makes up your yukawa uh interactions minus two because this has to be always uh Normalized to two and the beta function when you manage it it constrains the r charge of the operator to be equal to two Makes sense I mean in in in supersymmetry I'm reducing most of the things to algebraic things right in the in the first case I had to calculate the beta function to show that that I can show in general that certain things cannot happen So let me take the most naive generalization of the work with daniel, which is a super qct with an elementary mason Now super mason you can see that the this this part of the table is the same But because you have a gluino you also have a u and r symmetry what makes the difference here Really is the gluino and and there of course the supersymmetry in the end Now you immediately know that uh, you can immediately check that you lose us into the freedom when a knife is larger than three c And I am allowed to add the super potential to this theory that allows you to have an x interaction And in this year, I cannot have self-coupling so in principle my life will be easier because I don't need to check Whether a self-coupling are asymptotically safe would be enough to prove it for the yukawa and the and the and the gauge coupling So of course, I first do the most next things which calculate the In perturbation theory the beta function and you find out that unfortunately in this case The yukawa interaction is not strong enough to do it and this we knew since a long time So this theory at least like qcd cannot have a perturbative intrastable Ultraviolet fixed part, but perhaps it could happen a perturbatively right. I just told you qcd could in principle do that Maybe this can happen In supersymmetry I can just declare that that happens and check whether the conformal field theory exists, which means Passes all the tests. I just showed it to you Let me let me do that So assume a new perturbative in fixed point emerges, okay Then I can calculate for example the dimension of the kara super field h, which is a gauge singlet And this object when nf is larger than 3c violates the unitarity bound. Oops So Maybe this theory doesn't have an ultraviolet fixed point But not so quick Because I might be lucky that at the moment where I hit the conformal fixed point Yukawa interactions have disappeared the Higgs decouples And then this is no longer constrained and there are cases like that So although violates the unitarity bound, there is a potential loophole H is free and the couples of the fixed point So I need to study the reforger theory without the Higgs and only if that theory without the Higgs Also does not have a conformal fixed point. Then I am home safe But this theory without the Higgs is super QCD now. I'm actually studying the cyber when you lost us into the freedom Cyber only showed it to you the conformal window for infrared fixed point, but he didn't tell you what happened when he lost us into the freedom So let's see the end of the story on the other side of the line super QCD So you can immediately show that the unitarity bound is not sufficient What does it mean that I can construct all the gauge singlet operators that defines my modular space And which are baryonic operators and mesonic operators and all and behold they all pass the unitarity test So in principle this theory could never conform a fixed point. That'll be very nice There is something that however it doesn't Speak in favor of that and that is the a function If you now calculate the variation of the a function you find out that it violates the bound So if you assume the ultraviolet fixed point exists Then delta e will be negative This wouldn't be a good asymptotically safe counterfeit here period So now I've proved that also the theory with a nice and a good and do it You can generalize that So we can show that another being a super QED with and without a higgs cannot be asymptotically safe So at best these are theories that you can consider. It's a low-energy effective theories Remember cyber use those but only as infrared fit here. It's not as ultraviolet It's effective Interaction so you can do that Like the Karela grande and he don't care that the Karela grande does not have any safety in itself But you can use the low energy Okay You can then generalize using a maximization because there are cases where the R charge cannot be calculated And therefore you need to have another principle to fix the R charge of the theory. This typically happens when you have for example a joint map So key points so far Gauge plus Fermi plus scalar theories can be fundamental there at any energy scale either because they can be complete asymptotically free Or because they could be complete asymptotically safe in either case Gauge interactions are essential to make the theory fundamental So perhaps it's not by chance that we are ruled by gauge interactions Simultaneously, however, if you lose asymptotic freedom and gauge interactions alone cannot do it That at least in perturbation theory you seem to want a scalar outside perturbation theory you can still do it without scalars But it's not a proof the results Both for complete asymptotic freedom and complete asymptotic safety presented so far are based Only on perturbation theory and when I couldn't do it. I used supersymmetry to check where that wasn't possible So we can say that we discovered the uv complete non-Abelian quantum QED like theory. So non-Abelian QED theories can be fine. There's absolutely no problem about that I mean ultraviolet The n equal to 1 Susie Causing theories are unsafe though and that I can prove it to non perturbative beyond For any NF and an NNC provide NF is larger than 3 NC Also show that the result for the asymptotically safe thermodynamics, which is an interesting stuff You can do many other things on that that shows for example that uv is this conjectured F tier So there is no reason to enact it I just conjectured for you a potential QCD conformal window version 2.0 because in fact I don't reason I don't see why I should always spot only one window There could be two one for asymptotic safety and one for asymptotic freedom And that this can do both. There's absolutely no problem Actually, it's not very hard to change a number of flavors on a code And I'm rephrasing it a bit my need of scalars because I show that we need the scalars in perturbation theory for Tears like QCD where it loses in 30 freedom, but you might actually not need them at all So in this case the Higgs is a bit like the shoelace right when he loses in 30 freedom Right, so he is needed to make the tie the tier the tie the theory together It's not a nuisance in the theory It's a nuisance For other kind of reasons. I did not solve the Iraqi pro But if you set that for aside for a second then actually you can need the Higgs for different reasons So at look I mean there are many many things to do and we only scratch the surface You can extend to other carriage theories and space time dimensions some of this is already happened others are on the way There are interesting things about the grand unified theories like Susie because some of these actually are not asymptotically free So what happens to those? We have the result, but he's for another talk I would love to start calculating with some new critical exponents and even new maximum reasonable mhb's maximally elicited by leading amplitudes for this theory There are many things that make this theory very close to n equal to 4 Because it has an ultraviolet fixed point but n equal to 4 that is a fixed point for all values of alpha You can go beyond perturbation theory if you don't have Susie you can use lattice dualities holography truncations This is up to you You can think of a new ways of unifying flavor because if you think about it, right? I had one old yukavas above the mass of the Higgs will actually unify in a single Interaction yukava scalar interactions and gauge coupling They all now dictated by the same scaling exponent of the theory As George I myself say that that's better than god unification because even in ordinary god unification You all unify gauge but not self interactions and yukavas This case you have that the condensed matter physics of that predicts a universal behavior for all the couplings whether this happens the reality is all another story You can play with now different models of dark matter inflation And of course it lends some hope also for gravity although it's a completely a completely different theory in practice. Thank you