 I'm really very much enjoying this conference because it brings together very different physics fields using very similar tools. So whenever there's progress in one field, we may be able to transfer it to a different field and so the talk I'm giving is inspired by particle physics, the Higgs sector of the standard model or related physics, but perhaps the ideas we had about this may be transferable to other different fields. I'd like to invite you to replace these words asymptotically free to any other property you would like to solve, any other problem you would like to solve in your specific system. So this is work done in collaboration, a very enjoyable collaboration with Luca Zambelli. There's a short paper out and there will be a longer one very soon, hopefully, and it goes actually back. The idea is to work also together with Michael Scherer, Stefan Rechenberger, Michael and Luca here and we'll give talks, but on different, very different, very nice subjects. So the inspiration came from the observation that whenever there's a problem in physics, you just put a scalar and you come up with a new model and usually it works very nicely. And this is a strategy that works in very many different cases, of course, most prominently in the case of getting gauge bosons massive, in general symmetry breaking, if you want to come from a gut-like, grand-unifat, like theory down to the standard model. In particle physics you have the CP problem where you also put a scalar like the axion and there are many other examples where you put scalars and in condensed metaphysics, of course, you can also play the same trick in many instances. And yeah, the showcase is certainly the standard model Higgs sector where the metaparticles and the interaction particles are kind of bound together by the Higgs and it has been discovered and that has been a major breakthrough for this, for this paradigm just for the scalar. But it comes with a price tag. There's a price to be paid. In particular the standard model. There's this problem called naturalness. Technically it appears in terms of quadratic divergencies of the Higgs mass and it's simply difficult to get the Planck scale separated from the low energy scales, Fermi scale, or QCT scale, if you put a scalar. But as LHC seems to tell us right now, nature is not natural, so we have to live with this problem. Probably so I won't say anything about it. I don't have anything to say about it. I can't solve it today. Also when you put a scalar, you typically have to put further couplings and if you put further couplings, that means you put further parameters. So you typically lose a little bit of predictivity and the challenge I'd like to face today now is and it fits very nicely with Francesco's topic is UV completion because typically when you put scalars there's a danger that the system becomes, well, has a land up hole in the ultraviolet becomes trivial in a mathematical sense. So going to very high energies and showing that the system can really be fundamental can be difficult or even impossible. So that's the topic I'd like to deal with today because if a system with gauge degrees of freedom and a scalar is asymptotically free, it can be fundamental as Francesco told us right now. So there's a currency that is accepted in the particle physics community to pay these prices, for instance super symmetry, technical or extra dimensions and then many many different proposals that in particular address the naturalness problem and some of them also address the UV completion problem. And of course also in our community we have ideas to offer for instance if you add gravity then some of these problems may go away or as just Francesco told us if you do it in a clever way with gauge-shakes your cover systems then you can have asymptotic safety and that leads to UV completion. So I'm after actually asymptotic freedom. I won't put any fermions even though I could but I just stay with the gauge sector and the scalar sector. So let me tell you what this triviality means in very simple terms, a perturbation theory. So this is the one loop equation that relates the bare coupling, 5-4 type coupling with the renormalized coupling and there's a log relation and if you fix the physics so if you fix the renormalized coupling at low energies and look up the bare coupling at higher energies then you observe this Landau-Poll singularity which inhibits that you can consider this theory as truly fundamental. Or you can reverse the logic you can insist on sending the UV cut off to infinity and this you can only do if the renormalized coupling goes to zero and then you end up with a non-interacting theory which is not what you want to have. And so for the pure scalar system, pure 5-4 theory, there's no mathematical proof but there is a strong, very strong lattice evidence accumulated over quite a number of years. So I think it's a safe statement to say that the 5-4 theory is trivial or not asymptotically free. Now if you think of the standard model there is a scalar sector, the 5-4 sector of the Higgs coupled to the SU2 left sector in a gauge version and of course also involving fermions but I just pick out the gauge interactions, say an SUN gauge theory coupled to a scalar field. So it has the mass parameter, it has a self-coupling of the scalar field lambda term lambda 5-4 and it has the coupling of the gauge field to the scalars in here. So we have three parameters and a two naive argument would be but sometimes here it that well of course the gauge theory is asymptotically free that means the gauge sector kind of switches off in the ultraviolet and then you're left with the firm, sorry with just the scalars and the scalar sector is trivial. So a naive argument is that whenever you write down such a theory it's likely to be trivial or not completable in the ultraviolet but this type of argument is actually too naive and this was already known in the very early days of asymptotic safety if you write down the beta functions. So this is the running of the gauge coupling, usual term on the right hand side, the Nobel Prize winning beta function of of course Wilczek Pulitzer with the famous minus sign and then you have a beta function for the scalar self-interaction, it shows a lambda squared term, lambda g squared g4 and it has these constants here which during my talk will only be numbers. So for SU2 these are pure numbers and this coupled system of differential equation can be solved exactly, again it was done in the 70s and this is the solution. You don't have to look at the detail, the only point I would like to make is there is a combination of these constants here which is called delta and the sign of this delta is decisive because in fact if this delta is positive, as I said a combination of all these beta function coefficients if this delta is positive then the solution tells you that the phi4 coupling is proportional to the gauge coupling. So kind of the scalar sector is locked to the gauge sector which means if the gauge sector goes to zero to asymptotic freedom towards the ultraviolet then also the scalar sector is asymptotically free and this is what you would like to have. Now the bad news is that for all SU engage theories this delta is negative. So in fact all SU and non-abili engage systems are not asymptotically free according to this line of reasoning and this you see if you just plot this coupling as a function of the gauge coupling and the gauge coupling goes to zero at some point you hit the lambda pole in the phi4 coupling. And in fact there are fewer lattice studies on this and they are kind of compatible with this picture even though some of them also point to the possible existence of a second order phase transition. And there are actually current studies going on to understand this system a bit better on the lattice. Okay so still my title was asymptotically free non-abili engage theory. So what do I have to talk about if the delta is always negative? Okay so let's assume that delta was positive. How can I do this? This was also known to the ancestors. You can add fermions and you can adjust the gauge group in such a way that this delta becomes positive. So then you have as I said the good news asymptotic freedom the gauge sector and the Higgs sector are locked and the really nice thing about it is that since there is this locking and these are related by a computable relation that means that the Higgs mass related to the phi4 coupling is proportional to the W mass related to the gauge coupling. So if you have this locking if you have such a system then you compared to the standard model lose one parameter. You can predict one parameter. That's very nice and that program has been tried to work out in more detail but the bad news is that typically if you start model building the residual symmetry after symmetry breaking from the Higgs phase it's too big to be to form a realistic model for the standard model. Another problem is that there's no really ordering principle. So there are many possibilities to write down these models by adding many fermions adjusting the gauge group and so on. So this is not what I have in mind today but let me just show you the properties of such models where this delta for some reason or another is positive. So let me show these properties. I emphasize what I show now is not the standard picture for perturbation theory in non-Abelian Higgs systems without fermions. It's a more complicated system which you need to have the delta positive but if the delta is positive then you get pictures like this. Here's the gauge coupling. Here's the phi4 coupling. Here's the Gaussian fixed point and if you start for instance with positive phi4 coupling positive gauge coupling you are attracted by the Gaussian fixed point towards higher energies and that's the manifestation of asymptotic freedom. And there are specific trajectories which attract or repel all the other trajectories so the green one here is attractive and you can really run into the Gaussian fixed point in this nice way towards the ultraviolet. So I will show stream plots where the arrows point towards the ultraviolet because I'm interested whether I hit the Gaussian fixed point or asymptotically free fixed point or not. Okay now I can rephrase the same statement the following way. So lambda the phi4 coupling is locked to the gauge coupling so I can form the ratio and that should go to a constant if the theory is asymptotically free. And the nice thing is if you write down the beta function for this psi here this ratio the gauge rescaled coupling then there's a factor of g squared and the polynomial in psi. It turns out that the green line I showed you before this one is the green line here and this is kind of the fixed point of this beta function but it's already a fixed point at finite values of the gauge coupling. So you can fix the gauge coupling to a small value and still you observe already a fixed point in this gauge rescaled coupling where this term here vanishes and that's exactly the green line. Or you can also think of it as well the Gaussian fixed point is of course a point but by this gauge rescaling I just kind of re-parameterize the point and it turns it's converted into a line. So each point on this line here is actually corresponds to the Gaussian fixed point. So hitting at some finite value still hits the Gaussian fixed point. I'd like to call these quasi-fixed points because they are already visible at finite values of the gauge coupling as zeros of certain beta functions. Okay, so things to ponder. Is there a general meaning to these quasi-fixed points meaning that I've set the gauge coupling to a finite value and I still observe beta functions which have zeros, interesting zeros. You can also view this rescaling in a different way. You can rewrite the lambda-44 coupling in terms of this rescaled coupling times a rescaled field and if the gauge coupling goes to zero asymptotic freedom then you can expect large fluctuations of the field. They are not penalized by a large action in this language and that of course happens for the 5.4 operator but if there are large fluctuations you may wonder what happens to higher order operators. Perhaps 5.6, 5.8 also become important. Okay, so the next five minutes or so of my talk I'd like to present to you an argument which is kind of on slippery ground. So if you presented this argument to me I would never believe you so you should do the same to me. But the nice thing is that in the present case it kind of works. Not completely, but it kind of works but it gives you a nice feel for what is going on here. So as I said I'd like to study higher dimensional operators so I put in addition to 5.4, 5.6 term and of course it's dimensionful so I have to put another scale. It's in the spirit of let's say effective field theory or a perturbative type of spirit. And of course you have to balance the additional dimensions by some scale. This you would do in effective field theory. Of course I'd like to go towards the functional RG so replace that scale by a sliding scale in the sense of the functional RG equations. And then of course the UV behavior should be visible for that scale going to infinity. And of course I don't want to stop at 5.6 but I can include 5.8 and so on. So I can include the full potential and infinitely many couplings here. And I may expand about zero field or I may expand about a non-benishing bath. It's kind of equivalent but one or the other formulation might be more convenient. Usually perturbation theory is done with this type of expansion because there are theorems that say well if you're interested in the ultraviolet then it suffices to look at the deep ultra, deep Euclidean region where you have asymptotic symmetry. But it's actually you can kind of, hold to this argument and it shouldn't take that too seriously as I will show you. Okay and then you can just derive by perturbative means in an effective field theory language you can derive the beta function of all these couplings here. They will depend on the gauge coupling and on many many couplings. So in the end you get an infinite tau of RG equations which are all coupled to each other. And the equations I showed you just before for the gauge coupling and the 5-4 term which you see the lambda 2 coupling you get back when you just delete the higher order equations and delete these other couplings. But now the point is even if you delete these higher order couplings even if you only keep the gauge coupling and the 5-4 coupling this beta function lambda 3 for lambda 3 is not zero. It still gets contributions from diagrams. So setting lambda 3 to zero is as good or as bad an approximation as setting it to any other value. You can set it to a constant you can set it to a function of the coupling whatever you want in that truncation in that approximation it's you can't tell as long as you don't include the full beta function. So let me take an agnostic viewpoint I just keep this higher order coupling in this 5-4 equation and it gives me an extra term and you can compute the coefficient and the point is now that you get back asymptotic freedom even without fermions so you get a positive delta now it's a delta prime if this lambda 3 this higher order term has a specific scaling with the gauge coupling and it involves a constant which are called zeta here. So if for some reason this higher order coupling would scale like this with a constant then I would be well off because I would find a family of asymptotically free trajectories in the system and it's one parameter family because here's this unfixed constant of course that's now a kind of a game which I can play but of course I would have to show that this really happens in the full system but let me just show you if you plot the flow towards the ultraviolet in the G-squared lambda 2 plane you again get these plots where trajectories are attracted by the Gaussian fixed point and again you can choose this gauge-free-scaled version and you have this new quasi-fixed point involving the new constant here the parameter which you can just put and in fact again the Gaussian fixed point has been smeared and you can find trajectories which hit the asymptotically free Gaussian fixed point now I don't have to stop at this low order I can go to any higher order and play the same game and I find the same picture as long as the highest order I don't include so if I go to order lambda n I don't include lambda n plus 1 if this higher order coupling skates like some power of the gauge coupling family of asymptotically free solutions and they are parametrized by one free-paramed okay? as I said this is slippery ground because you can easily fool yourself with arguments of this type and what you really have to check is the following so once you have the full series of all these once you have solved the full tower you can reconstruct the full potential and of course this potential has to be kind of physically meaningful so for instance it shouldn't have singularities it shouldn't be unbounded and so on so there are possibility checks which you first have to show but if this all works out then there is still this one parameter psi which remains and this of course has to be translated into a physically meaningful boundary condition for this potential and there are actually many further checks you have to do and it was I think Tim Morris who very carefully worked out all the arguments that typically rule out physically non-meaningful physically nonsensible potentials for instance you have to show that the perturbations about the fixed point potential are self-similar that there is a uniform convergence towards the fixed point and recently also the gravity context there has been this notion of singularity count which I will show you later so there are a lot of checks which you have to and this potential has to pass in order to be physically meaningful okay so now there is actually not only one parameter there is yet another one to make things more complicated or more interesting I motivated this rescaling of the couplings by rescaling of the fields and I rescaled it with a square root of g of course you can take any other power power p and do the same story as we did before and you end up with trajectories which are hitting the Gaussian fixed point which are asymptotically free for any value of p and they hit this Gaussian fixed point in a specific way depending on p so it turns out there are actually two parameters zeta and p that parameterize the boundary conditions for your effective potential or for your correlation functions in general okay so now I have to convince you that also in a bigger more comprehensible more meaningful framework and for this of course I use the functional rg I use the cost of the equation and of course from a mathematical viewpoint it's a functional differential equation and if you boil it down to a couple of sets of partial differential equations it's an initial value problem in the rg time or if you are interested in the fixed point equations then it typically boils down to second order ordinary differential equations couple of systems that require two initial boundary conditions to be solved so you expect already that the boundary conditions will be important for your class of solutions so I don't bore you with the details but what we specifically look at is the full potential for the scalar field and there is an equation for it we include enormous dimensions and we include the gauge the running of the gauge coupling purely perturbatively because we're interested in asymptotic freedoms so the perturbation theory should be fine at weak coupling so it's a second order pde you don't have to look at the details but what I want to like what I want to show you is that you can do this gauge rescaling with this general power p for the field and introduce a new field variable I call x and it's now a field variable that are of the potential which I call f now so f will be my scalar potential and x will be my field variable and it contains in an intricate manner gauge couplings so you see p pops up everywhere and so on it's a second order differential equation p is not fixed you choose a p and it should be bigger than zero but otherwise it's fine then you can solve this equation for p equals one in a weak coupling expansion so I'm at weak coupling I'm close to the Gaussian fix point and for p equals one here's the solution and in fact it involves one free constant theta and it involves this I like to call it Coleman Weinberg type potential so it's a typical phi for log phi term which pops up here and it's very generic it comes from the gauge loop the thing I'd like to emphasize is it contains a logarithmic singularity at the origin not so higher derivative I mean it's smooth and the first derivative is smooth but higher derivatives may contain divergencies at the origin but this is expected from the gauge loop but it's one parameter family depending on this coupling theta and also for the other powers of p there are solutions which you can work out in the weak coupling limit but this one looks particularly nice so let me show you how it looks like so again I switch on the gauge couplings to a specific value this green one here and then solve the quasi fix point equation and I get this Coleman Weinberg type solution and it looks like this here the green line it has a non trivial minimum and it goes like phi four log phi to infinity now if you go to smaller gauge couplings now you kind of switch on the RG flow towards the outer violet when the gauge coupling decreases and for smaller gauge couplings you again solve the quasi fix point condition and you get these other plots here and what you see is that the whole potential becomes flat so that's exactly what you expect from asymptotic safety that the interactions go away but yet another thing happens larger gauge coupling you have the minimum here and for smaller and smaller gauge coupling the minimum runs to infinity more quantitatively towards the outer violet G goes to zero the minimum runs to infinity in such a way that the W boson mass divided by the scale approaches the constant so that's a way how you can sidestep these typical arguments of deep euclidean region so there is in this system there is no asymptotic symmetry as always a kind of a vacuum or an expectation value of the field so let's try to understand so that's a scaling solution that runs towards the Gaussian fix point in a very non trivial manner and is asymptotically free so let's understand when you perturb the system what type of classification of relevant operators do we have it turns out it's very simple it's almost like a perturbation theory you have the relevant mass parameter on the higher order terms are typically irrelevant and you have a marginally relevant parameter in perturbation theory you would associate it only to the gauge coupling in the present system the marginally relevant perturbation is a combination of switching on the gauge coupling and switching on the potential in a non trivial way everywhere for every field but in a controlled and smooth manner so here for instance is the difference from the fix point solution I showed you before plotted for higher values of G of this marginally relevant perturbation and you see for larger gauge coupling it has this deviation for small and smaller gauge couplings it approaches very smoothly polynomially bounded the causal fix point potential I just showed you so this test is passed it really works now I'd like to give you a very important argument that tells you where the additional parameters and the additional solutions that you don't see in perturbation theory really come from so if you're familiar with say the easing model or n-type models RG flows, Wilson-Fischer type fix points what you typically do is to really fix the fix point potential is you do a singularity count or you should do it you can rewrite the flow equation which is a second order differential equation you can rewrite it in normal form so you put the second derivative on the left-hand side and all the other lower derivatives on the right-hand side and the typical structure that you find is 1 over the field squared and a numerator and a denominator so that tells you that in principle in order to find your solution you have to fix two boundary conditions or initial conditions but you have to be careful you have to avoid this fixed singularity at the origin and for this this numerator function has to have special properties so that fixes one of these boundary conditions and then you have a movable singularity when this denominator data here becomes zero and for this also you have to impose another condition and that kind of kills the two free parameters that you have in principle and you end up with zero parameters which means that those solutions are quantized so the Wilson-Fischer fixed point is really unique in three dimensions and this also leads to the quantization of multi-critical fixed points in dimensions lower than three so you end up with a zero parameter family the Wilson-Fischer fixed point is just a typical example now how does this work for our case first of all it looks very similar again for my potential function differential equation sorry ordinary differential equation I have the same type of fixed singularity at the origin and I have the same type of movable singularity so in principle you would say again you should end up with quantized solutions only one there shouldn't be any free parameter so where does it come from but actually if you remember in this Corsa fixed point solution I showed you there was this log type singularity and that means the origin actually shouldn't be as regular as it is in a scalar theory but the gauge loop induces log type singularities so it removes this first constraint here and you can permit for a larger class of solutions so the solution space is different from the one you know from scalar O and models for instance so you have to include these log type singularities so that means these are not I mean bad singularities the potential itself and derivatives stay regular at the origin but high derivatives can and should develop log like singularities and that effectively removes one constraint so that you end up with actually one parameter family of solutions and that's what we also see in full numerical solutions of this equation okay that's kind of the end of the story so I spoke too fast so I can speak very slowly now yeah so once you have convinced yourself or we have convinced at least ourselves that there are these asymptotically free trajectories in non-ambient Higgs models you want to use them for predicting something for the let's say standard model, you want to use them for the Higgs sector in the standard model and predict the infrared and of course that's in principle straightforward but since we are dealing with the full potential which you have to follow it's kind of technically complicated this last slide is under maintenance and the only result I'd like to present is so what is of interest is for instance the ratio of the Higgs mass by the W mass this you can compute and you can check which of these parameters which we have depends on or influences this ratio it just turns out that this ratio is linear and this zeta parameter and there's also p dependent coefficient so I forgot to tell you sorry I should tell you the p parameter still is in the game because it kind of defines which type of log singularity you have so for a different p you have a different type of singularity so you change your function space slightly with choosing a different p so even though it looks like a trivial rescaling that should drop out in the end it parameterizes the function space the Hilbert space of functions should use for finding the solutions so in this sense it is a physical parameter that belongs to the definition of your theory okay and so the good news is that you can have theories that have the correct Higgs to W boson mass ratio so there are trajectories that connect the Gaussian fixed point to the physical parameter space you're interested in and this is all you can expect and just to check so in the standard perturbative line of reasoning you have the Fermi scale the vacuum expectation value the Higgs mass and the W boson mass as physical observables and in this system here we have this relevant mass parameter which is kind of normal you have the marginally relevant direction which is kind of normal also from the Gage perspective and then these two further parameters are kind of exactly marginal you choose them this defines your theory of the game but there are two of them and with these you can just fix the theory you would like to have okay so conclusions the standard statement that non-Abelian Gage theories for any SUN are not asymptotically free I think is not complete the statement is only complete once you fix the boundary conditions for your effective action for your correlation functions and I claim that there are boundary conditions which are kind of natural including log-like singularities which allow for trajectories that are asymptotically free so in this sense even a non-Abelian Higgs model can be asymptotically free and UV complete as Francesco pointed out which is an important criterion so I just told you that this is that this holds for the non-Abelian Higgs case but you can also include fermions and play the same game and it seems that the same structures are at work and so we have found no criterion that rules all these log-like singularities they should be there but of course it might be that one of you has a good physical reason why one should exclude this type of solutions from the physical parameter space or physical function space and if so then that would reduce the type of solutions you have so my message to you working on very different fields gravity, condensed metaphysics is you should really take the boundary conditions for the flow equation seriously and really explore which boundary conditions are physically meaningful in many cases it's obvious but in other cases it might not be so obvious and as this example should tell you in some cases it makes a decisive difference for the properties of your theory so that's all