 Yes, well, thank you very much for the opportunity to speak at this meeting. So the title of my talk has changed a tiny bit. And the reason for this is because I wanted, in fact, to highlight that we're in the process of really, truly, trying to understand what are the deep-down fingerprints and dynamical reasons for why something like asymptotic safety in a fundamental, say, four-dimensional quantum field theory might exist. There have been signs out recently that structural dynamical reasons for this might exist, but we've been looking into that, trying to nail it a bit deeper. And today, I will be reporting about the progress we have been made along those lines. Now, many of the things I'm going to talk about, many of the results have been achieved together with my students, Andrew Bond and Tuba, who are both of which are sitting here in the audience and are present. Good. So what is the motivation for looking into this? For me, the motivation is definitely the standard model of particle physics. Yes, it's a totally beautiful, local quantum field theory. It describes all those fundamental interactions we know, perhaps except gravity. OK, it has been verified with extraordinary precision. Yes, it's really a beautiful theory. It contains all the degrees of freedom we know up to now, including, very recently, the arrival of the Hicks. Now, from a theory perspective, what theoreticians do totally like with that type of theory is predictive. Turbatively renormalizable theory has predictive power, and we see this predictive power confirmed day by day now that the LHC is running. OK, now, of course, not everything is beautiful and shiny with the standard model. We also know that there are a large number of intriguing open challenges. So for once, we do know that the standard model ultimately can't be the final fundamental answer. It intrinsically has built in and highest energy scale beyond which it is no longer going to be valid. And even if this weren't the case, there are still many questions from a theory point of view, which we would like to have an answer for when it comes to standard model physics. For example, what is it that exactly determines the physics of the electroweak scale? How comes we have so huge hierarchies in the masses of particles? And ultimately, what are the missing pieces, which we haven't seen yet, but which hopefully would complement the standard model towards higher energies so that we would actually have predictivity up to ideally the Planck scale minimally or possibly beyond? So from that angle, there are many challenges, and many of you, including myself, have certainly also been interested in exploring this at the interface with gravity. So the big question is somehow, for a theoretician, what would be useful directions which we now can follow? So in model building, we know directions. So a lot of structural directions had been given to us, motivated by symmetry, super symmetry, for example. Okay, ideas to overcome challenges in explaining hierarchies within the standard model. But as we know by data from LHC thus far, this has not necessarily been totally successful. So it might be a good question to rethink the landscape of directions. Now, speaking of directions, we all know how important directions are, and when we arrived here at the conference, we've been given directions. Yes, namely these here. We all arrived, right, and we were invited to appreciate clear directions when we were looking to find the path to the registration. Yes, okay, so clearly directions are important, yes. Now, if we take this as an allegory for the standard model, of course, you might wonder what are the directions we could follow there, clearly there is one direction, a total light tower of the direction which has been given to us already back in 73, which has been this important discovery of asymptotic freedom. Asymptotic freedom is a deep down dynamical mechanism which can explain how fundamental theories remain predictive to highest energies. So it really is a very important cornerstone, not only in the understanding and construction of the standard model, but in providing ideas on how to perhaps go beyond. Now, I have to review this because it is so important, asymptotic freedom is so extraordinarily well established by experiment, for good reasons, Nobel prizes have been given for these discoveries. But not only this, I mean it is, as I said, an important building block in the understanding of constructing models beyond the standard model. And so what we do know, so the hard results we have are summarized here. The hard results about achieving predictivity up to highest energies means we do, by all means, we do necessitate non-Abelian gauge degrees of freedom. Without non-Abelian gauge fields, there is absolutely no way we can achieve predictivity to highest energies. And the difficult customers, of course, here are the Scala fields who by themselves would not be asymptotically free, but also you cover interactions, all those things we do know very well from standard model physics. It is only because of non-Abelian gauge fields that there is an avenue according to which such theories might become completely asymptotically free. Of course, there are some conditions under which need to be fulfilled and as I said, these conditions are not fulfilled even if we were to be taking the hypercharge out of the standard model. Okay, so what I'd like to do today is to open up perhaps a second avenue to offer directions on how we might think about completing the standard model beyond the physics we know for now. And this avenue I would like to discuss is related to the idea of asymptotic safety. Presumably some of you have come across this concept. Let me quickly mention what it is about. The central and new idea which makes it perhaps different from asymptotic freedom is to allow yourself to think that couplings at highest energies might not necessarily vanish. So this is what you normally get in asymptotic freedom but now we want to consider a scenario where this is not necessarily happening. Still, the important ingredient is going to be that we would need an interacting fixed point or else people will have difficulties believing that such a setting in fact is fundamental and predictive really up to highest energies. Now if this were to be the case, it would offer a set of opportunities to rethink ultraviolet completions. It may or may not be realized in nature. This is something which we still of course would then have to figure out in quantitative detail, yes, but at least it would offer an alternative. Now quite some of my motivation for working in this direction relates to some recent work which I've done with Francesco Zanino who reported about that two days ago. Okay, so a model where actually we managed to figure out in some example theory that this type of asymptotic safety in fact is realized in exactly solvable models. The big question here is was that just a curiosity of mathematics or is there more structure underneath actually opening up an entire window of opportunities for us to look into. I would like to do what I would like to do now is to show you that indeed there is an entire window of opportunity hidden underneath that result. Okay, so what I will do is for you today is I will discuss all, I will derive and discuss and analyze all weekly coupled fixed points in general gauge theories. You name the gauge group, you name the field content, yes, and I can tell you now, we can tell you the conditions which must be satisfied so that this theory either has or is not having interacting asymptotically safe UV fixed points. We will derive strict conditions including no-go theorems for asymptotic safety. Now this table is perhaps the asymptotic safety counterpart to the table I've shown to you a minute ago. Namely this is the summary table in fact of all results. So what I want to do is I will explain you in some depth how these results come about but I wanted to show you those results in the first place. Results look like this. There are a set of no-go theorems. So there are a set of very general gauge theories with simple or a billion gauge group no matter what the fermentic or meta content is. They will never be able to develop weekly coupled asymptotically safe fixed points. However, and this is the crucial point, as soon, and this is what we will show, as soon as you cover interactions are present, we can formulate certain conditions which need then to be met so that these totally general simple or a billion semi-simple or product gauge theories with a billion factors have ultraviolet asymptotically safe fixed points. So let's see how these results are coming about. Now, and to see that, I am going to start little by little. So we first start discussing just a simple gauge theory with a gauge coupling g. So we introduce little alpha and that is going to be the coupling we will be mostly interested in. Now we speak recoupling so we can do perturbation theory. That's the territory in which we are going to be operating. So in perturbation theory, we write down the beta function which up to two loop order is given here. Yes, and so we see the one loop coefficient which we tend to call minus b, and we see the two loop coefficient which we call c. Now in a general gauge simple gauge theory, b and c can be expressed in terms of the quadratic casimir of the gauge group in the adjoint and then it's going to depend on the fermionic and the scalar meta-content they will contribute according to their Dinkin index under the representation of the gauge group. And very similarly there's a second coefficient c, the two loop coefficient which also has fermion contribution, scalar contribution if the scalars are charged and a contribution coming from the gauge fields with the opposite sign. So in order to get a weakly coupled fixed point what we do need is that one loop and two loop cancel out and that this cancellations parametrically happens in a regime where the fixed point itself is still small so that we can trust perturbation theory. Now this will be our conditions. Now when you look into this expression you immediately see the following obvious things. Firstly both b and c can have either sign. It's going to depend on the specifics of your meta-content. If you have a lot of meta, the meta fields dominate and b is going to be negative. Now if b is negative the one loop coefficient is positive meaning you've just lost asymptotic freedom. Conversely if you have a little matter then the gauge fields dominate and asymptotic freedom persists. Perhaps similarly with the coefficient c you have a competition between meta which add up positively and the gauge fields. So one would have to throw more careful looking to this theory to understand what the hell is going on and what are its possible fixed points. Now what is already known for very very many years of course is that these theories can have infrared fixed points provided that b one loop and two loop coefficients are positive. If they are positive their ratio is positive we have a physical fixed point which is the famous casual bank sucks fixed point. Furthermore it's an infrared low energy fixed point how do we know that? Simple because b is positive so when b is positive it means the theories asymptotically free. The asymptotically free fixed point the Gaussian is the ultraviolet fixed point so the other one the new interacting one is there for an infrared fixed point. So the question is actually is it perhaps possible to have both b and c negative because if we have b and c negative then the corresponding fixed point would indeed be ultraviolet. Up to now there have been no examples of this type but the hard answer was missing. And now in order to progress with this we decided to rewrite the two loop coefficient in a slightly different form. So what we did here is we wrote the two loop coefficient by exploiting by inserting the one loop coefficient in it so we've been replacing the quadratic casimir by the one loop coefficient and reordered terms. So it's the very same expression but written in a slightly different way. So let's see what we can learn by this. So the first thing we learn is that if we lose asymptotic freedom meaning b becoming negative then this term here is going to be positive. b negative so its contribution is positive. So this is a term which is manifestly positive if asymptotic freedom is gone out of the window. Then we see there are two remaining terms coming either from the fermions or the scalars. So first of them you can just look at it it's manifestly positive definite. So no matter what your fermionic content is they are going to contribute positively to the two loop coefficient. So we already have a first result and actually this result has been discovered by Caswell back in 74. If we have a gauge theory coupled to fermions only then there is no way this theory is going to have a weakly interacting ultraviolet fixed point. Impossible! The right hand side is always going to be positive at the instant where b has become negative. Now let's look into the scalars. The scalar contribution as you can see from this expression is not manifestly positive definite. Now what does it mean? It means that if we manage to find a gauge group where the quadratic casimir of our scalars is smaller than 1 over 11 then you know the quadratic casimir in the adjoint then in fact the contribution by the scalars would have the negative sign. And in that case c, the two loop coefficient could potentially become negative in which case an interacting fixed point will rise and may become physical. Okay? So let's see how that goes. If we want to understand or exclude this case we must know whether this relation can be achieved in any gauge theory. So once we realize this and run myself with a scratch on our head that must be totally easy. Let's walk over to the library. Let's get out some dusty books about Lie Algebra and representation theory. Somewhere there the answer must be. Obviously all the things we know about Lie Algebra. But in fact we were digging for a little while and shying away from doing an actual computation by ourselves but then realized no. Somehow apparently people have not looked or didn't felt that this is an interesting or important question. We didn't manage to find the result. So we had to sit down and do this analysis by our own. So how do you do it? We used some... The idea which helped us in the end is realizing that we can always express of course the quadratic casemia in terms of a highest weight vector because these are one to one related to identify the irreducible representations under our gauge group. So this gives us some formal expression for the quadratic casemia in terms of these highest weight vectors. This is expressed as some scalar product formally speaking in this space of weight vectors for your representation. But the thing which is known about this scalar product is that this scalar product has some metric in it and this metric, these matrices these metrics are known explicitly for all Lie algebras. So in fact we can exploit this knowledge and analyze which representations lead to the smallest casemia and what that smallest casemia at the end of the day is. So the answer is this. We classified it for all simple algebras and what you get, you compute the smallest quadratic casemia and the quantity which ultimately is important is the ratio between the smallest quadratic casemia and the casemia in the adjoint. This ratio is what we call chi and it's given in over here. Now as you know quadratic casemies of course they can easily take very large values and in fact if you look into it there's not a lot of monotonicity in size when you run through your possible representations. But what came out for the smallest casemia in any Lie algebra is the following. It is always always related to the reducible representation with the smallest dimension and that turns out to be always one of the fundamental representations of your gauge group. So that's the result so then we can compute chi and we can also make a nice plot. So this is like good here you see the classic Lie algebras and their end dependence and the exceptional Lie algebras and no matter what you do however and that's perhaps the central result here the quadratic casemian units of the adjoint is bounded from below it's bounded from below by 3 over 8. So we now know that's the piece of information we got from representation theory and we can now come back to our main line of reasoning and realize that in fact we have no-go theorems. First no-go theorem is we should have had this in order to be able to generate an ultraviolet fixed point but it so turns out we have that. Now what are the implications? The implications are truly far-reaching because the implications tell you that if B the one-loop coefficient or minus B is the one-loop coefficient if B is negative so if we lose in any gauge theory if we lose asymptotic freedom then no matter what the two-loop coefficient is going to be positive proven. Okay? Therefore we have a no-go theorem meaning that if we only have gauge interactions there's no way of ever achieving an interacting UV fixed point end of stocking. Now let's see how this can be generalized. Okay? So you might think well let me try to get away of this no-go theorem by circumnavigating it through additional couplings. So the first thing you might think of is let me introduce more gauge couplings. So I write down yes maybe I think of a semi-simple gauge theory with product gauge groups okay I have more than one gauge coupling and this is how now my beta function would be looking like. Okay? So for each gauge group I have a one-loop coefficient I have the two-loop coefficients CAA the diagonal one as we discussed a minute ago and the new gimmick however which is now coming into play are the off-diagonal contributions CAB the mixing between the gauge groups. Okay? So if I had a theory like that what is the interacting fixed point? Well this theory has interacting fixed point provided that this simple linear equation has solutions for which alpha star remains positive for zero. Okay? This would then be the fixed points. Now let's see what do we know about these new mixing terms. Now first of all the non-trivial mixing terms we know how to write them down so they are related again to the quadratic Casimir and the Dinkin index of the representation of your particles but as you can see from the explicit expression it is manifestly positive. These contributions are positive. So what does what follows out of that? It follows that as soon as we have any gauge factor which is not asymptotically free. Okay? Meaning the factor BA one of them being negative then we know by what we showed a minute ago that all diagonal contributions so the CAA is positive and we see here that the off-diagonal are all positive also so we know that all these coefficients CAB are positive for all B. What does that mean? It's a no-go theory. Why? Think of finding a solution for this equation. We just said if BA is negative so if we've lost if only in one of those gauge groups, if we've lost asymptotic freedom, BA is negative but this equation must have a solution but we just established that all these CABs are positive and for a fixed point of view physical we need alpha star to be positive. No way to have a solution as soon as BA is negative it's impossible. So it's a no-go theorem. It's impossible to get that fixed point. Good. So this was part one. Three no-go theorems in all generality yes how not to get an asymptotically safe fixed point. So we've seen we can take gauge couplings we can take more gauge couplings it doesn't work. We've got two positive examples so something must be working so let's see what that is. First of all you might think well let me try to throw more meta couplings into it and the natural candidate would be scalar couplings if you have scalar degrees of freedom. However, scalar couplings will never be able to help you generate a fixed point in the gauge sector. How comes? They only start contributing for loop level depending on whether the scalars are charged or not. If they don't contribute at the two loop level where we have the wrong sign this then would mean that we need non-perturbatively strong effects coming from the scalars which is not compatible with our assumption of perturbativity for now. Not saying that this is excluded but at least it's not possible at weak coupling. The final candidate we may want to look into are yukawa couplings. Now yukawa couplings do contribute to the running of the gauge starting from the two loop level so indeed they will be there if we have yukavas so it's important to look into that. Now what do we know about yukawa interactions? So think of writing down yukawa interactions in some general form y index A is now the yukawa matrix of couplings. What is the contribution to the running of the gauge? Well, we've written it down here. It's this new term minus 2 y4 I'm adopting the notation which has been introduced by those who actually did that computation quite some time ago. And y4 is some expression which knows about the fermionic representation of your field and which knows about the yukawa matrices. The only thing which you need to know about y is that y is positive no matter what but see the minus sign here So the important message really is yukawa couplings always slow down the running of the gauge and even more if you're lucky the slowing down may be so strong that this yukawa contribution generates a zero so that the theory is having a fixed point. Now let's see how that could happen. So let's imagine that the yukavas get a fixed point by themselves. Then y4 would be some number of stars. So the way we can now look for a fixed point is to say, oh okay what this infect the effect of the yukavas quantitatively is that our one loop coefficient is shifted into a one loop coefficient B' which is now a different value because of the yukawa fixed point. Now the important point here no matter what you do B' is going to be bigger than B if this happens what is going to be the fixed point well you can clearly read it off the fixed point would then be not B over C but B' over C but as you can see something very important has happened here imagine you have a theory which is not asymptotically free say imagine you start with negative B and negative C is positive so this fixed point B over C would not be physical there is no such fixed point but now that you've switched on the yukavas the yukavas induce a shift and some negative B can be turned into a positive B' thanks to the yukava interactions once B' is positive we will have an interacting totally viable interacting natural violet fixed point that's the mechanism good so you can now dig a bit more deeply and look into what is really going on in the yukava sector so for the yukava sector we know it's beta function we postulated that that sector has a zero so let's see whether this is a good postulate we know zeros of the yukava beta function either mean the yukavas are zero that's the well-known gausson fixed point or the yukavas are proportional to the gauge yes with some numerical proportionality constants C A now if we exploit this piece of information we can actually compute explicitly y star and we will see that it is given by some number D just related to group theoretical factors and the structure of your yukava sector times the gauge coupling so in fact once you know that you can equally think of the yukava contribution in fact on its null line having modified factually the two loop coefficient C because this term as we just showed is proportional to alpha at those points where the beta function for the yukavas is vanishing so an alternative way to exactly look at the same thing is to realize that we have an induced shift of the two loop term but now the two loop term is being made smaller because D is always a positive number so there are two sides of one and the same metal we would find the reliable fixed point over C prime which is the same fixed point as we saw a minute ago except that now we have resolved this remaining implicit dependency so you see a first necessary condition for asymptotic safety of course you can play now this game for semi-simple gauge theories and you will see that the same type of logic in fact is going through we will find the generalized equations for the necessary condition for asymptotic safety once again takes a very similar form namely requiring that the shifted one loop coefficients once you have taken into account the effect of the yukavas are all positive good so I have explained to you how this table has come about I would like now to use the remaining few minutes I have to indicate a few further results which we found so one side result which we have is I have focused a little bit on fixed points which are asymptotically safe but at the same time our analysis gives you a complete overview of all possible weakly interacting fixed points engaged theories be it ultraviolet or infrared one and this is the table showing you that classification and so in the end of the day maybe the central new point to be highlighted here is that remember we started and realized meta content is very important to understand potential phase diagrams and phase structures of gauge theories so the one and two loop coefficients which do encode those things but as it now turned out there's the third quantity related to information coming from the Yuccava sector which for simple gauge theories is this quantity C prime and that quantity is playing an equally important role if you were to be imagining let's plot some phase diagram about what type of fixed points fundamentally really exist well then there are only four this is the one everybody has seen because this is the Gaussian fixed point of asymptotic freedom we have a gauge coupling down here we have Yuccava interactions we have trajectories emanating out of the Gaussian and actually entire families of asymptotically free trajectories getting out of it but no real interacting fixed point this happens if B is positive now if additionally C is positive then we also have the Benzaks fixed point sitting down here the Benzaks fixed point exists once the Yuccava are switched off it has an unstable direction so trajectories get out of the sky and would run into that plane towards the infrared ok these fixed points are very well known and well studied in the field now if additionally you also have that C prime is positive then that means that besides the fixed point B over C which is the Benzaks you have the fixed point B over C prime which is this one a fully interacting we call it a gauge Yuccava fixed point because both the gauge and the Yuccava have non-trivial fixed point values and in this setting this fixed point like an infrared sink if you have masses of course you still can run away from that fixed point so this fixed point factually corresponds to a second order quantum phase transition in this type of four-dimensional gauge Yuccava theory now finally the opposite case if you have lost asymptotic freedom so if B has become negative then the only way for acting fixed points is that well we know that C is positive anyway we've proven this but if then C prime is negative then B over C prime gives you a reliable physical fixed point which is the sky over here this is all there is because we've shown there's only one mechanism to generate that fixed point now if you generalize it to semi-simple gauge series what is going to happen is that your phase diagrams will in essence be direct products of the phase diagrams I've been showing you now before concluding let me mention two extensions and then I will show you my concluding slide first extension is we know that these interacting fixed points exist for simple gauge groups we have an example and the question is is the space of solutions populated even in the case where we have semi-simple gauge groups well Andrew Bond reported about these results so if you want to know more please speak to him but yes we have hard results showing that those guys do exist okay second extension you might wonder what is going to so I've been concerned not only with the couplings which have marginally canonical mass dimensions of course those which are perturbatively renormalizable but you might wonder and ask yourself what is going to happen if we include more complex higher dimensional operators higher powers in the fields operators which normally are being considered perturbatively non renormalizable well the important result which comes out here technology beyond what we did for now so we've done it with functional RG the results are shown by Tuba on her poster and we find that the fixed point does persist and even more so effective potentials remain stable as they must okay so here are some of her results which you can see in the poster so it's time to show my conclusions I try to show you how all weekly interacting fixed points of general gauge theories can be classified now in this set of fixed points there's a subset of candidate fixed points for ultraviolet completions in four dimensional gauge recovery theories and I've shown you necessary and sufficient conditions for these to exist okay in the sense of a finger print for where asymptotic safety is coming from the you cover interactions are the pivotal element they are the only one which in fact can negotiate a fixed point once asymptotic freedom is lost okay now this is offering maybe a window of opportunities to think about how particle physics models can be constructed even beyond the traditional paradigm of asymptotic freedom thank you very much