 Thanks. Thanks to the organizers for allowing me this opportunity to speak. I'm going to be talking a little bit about this paper and some work in preparation with Daniel. Some of the topics I'm going to discuss will be covered in more detail in Daniel's talk on Thursday and probably much more coherently, so anything that doesn't make sense in my talk I'm sure will get cleared up then. Okay, so what is it that I'm trying to do? Well, I'm interested in four-dimensional gauge theories with semi-simple gauge groups. And the reason I'm interested in these theories is that we appear to live in a world of four dimensions and I'm happy to take that as a data point. It seems a reasonable class of theories to consider. And also we know that the standard model works very well for a wide range of phenomena and it is a gauge theory which has a semi-simple gauge group. And so again it seems reasonable that we might be interested in the behavior of these sorts of theories. We know very well that such theories can be asymptotically free and I'm interested to know if there are any other UV possibilities for these sorts of theories. And throughout this I'm going to be using perturbation theory and I'm going to be doing that because it will make life easier in a couple of ways and we'll be able to make some hard statements about such theories. So very quickly, we're going to be considering the renormalization group equations. So we've got some running couplings and their running is described by beta functions and that will be entirely determined by the field content and symmetries of our theory. And there's lots of ways that you can calculate these beta functions and as I've mentioned I'm going to be calculating them within perturbation theory. And in particular I'm going to be interested in fixed points. So this is points where the beta functions vanish. So depending on what happens to trajectories when they come into the vicinity of these fixed points we can classify them in terms of UV or IR theories. And I'm interested in ultraviolet fixed points and this will allow us to define QFTs up to highest energies. So that's the main thing. And in perturbation theory the way the feed functions are calculated is just through an expansion in terms of the couplings. So we just expand in the series and then these coefficients are determined by the particular theory of interest. And one of the reasons why I'm going to be using perturbation theory is that a lot of the heavy lifting has already been done and so in general renormalizable four-dimensional field theories these things have been calculated perturbatively and so we can make some strong statements based on the structure of these things. And also it's practically easier because it means calculations are a bit easier to do. And also it's a useful starting point because we're going to be using exactly perturbative theories. We can make very hard statements and this will be a good point to then if you want to look at things non-perturbatively it's a useful place to start from. So if we're restricting ourselves to perturbation theory I mean in general for a fixed point there's either two options either all your couplings are zero or all your couplings aren't zero or some of your couplings are non-zero at least. Now for in perturbation theory this is only going to be valid if we have small couplings and so what this means is that the couplings would be much much less than one so they can either be zero or they can be small. So I'm going to be bloating the cases where they're non-zero because we understand asymptotic freedom fairly well. And one of the other aspects of perturbation theory will mean that there will only be small corrections to the anomalous dimensions. So this means that the classical mass dimension of operators is still going to be what governs their relevance. And this means that we already know beforehand which operators are relevant, which operators are irrelevant and we only need to consider the marginal operators. So we don't have to worry about truncations and such. So I'm going to talk a little bit about what we need for this to occur. It's going to start with just a one loop beta function for the gauge coupling just in turn it only depends upon itself and the single coefficient B here determines exactly what happens to the gauge coupling in the UV. So the only fixed point we have is the Gaussian fixed point and depending on the sign of the B if it's positive then it's a UV fixed point and we have asymptotic freedom but it's negative then it's IR and the theory can't be UV completed at this sort of approximation. So this either means that we have a big problem or we need to worry about higher order corrections. And what determines B is purely in terms of the gauge group and the matter content. So we see that in general for a non-Abelian theory B can have either sign. We get contributions acting negatively from the gauge bosons and we have the other sign contributions from matter and so if we have no matter or a small amount of matter then we'll have an asymptotically free theory and if we have more matter then we lose asymptotic freedom. So what happens if we go to higher order? So if we have no other couplings in our theory then we get an additional term here, the C term which again will depend upon the matter content of the theory and we find that as well as the Gaussian fixed point we potentially have another fixed point given by this combination here. So if we want this to be physical we need alpha to be greater than zero so we need both of these guys to have the same sign and if we want it to be perturbative which it must be for our approximation to work then we need B to be much much smaller than C and so this is what the two-loop coefficient looks like. Again it's given purely in terms of the gauge group factors and what we find is that again the signs contribute oppositely from the gauge bosons and from the matter content and so again if we have not very much matter then C will be negative when B is positive and if we have lots of matter then the signs will flip and both of these cases we don't have an interacting fixed point. We also know that both of these signs can be positive this is the famous Banks-Ax infrared fixed point and there's lots of theories of which this is just one example. Now it turns out that it is not possible to have the other case where both B and C are less than zero so it's not possible to have a UV fixed point in this theory and I'm sure Danny will talk more about this on Thursday. So now Yigawa couplings these are the next thing to consider because they also contribute at two-loop to the running of the gauge coupling and they arise very naturally when we have theories which have fermions and scalars in them. So the term contributing comes like this in terms of the chasmier matrix and also these Yukawa coupling matrices and the important thing to notice here is that this contribution is always negative regardless of the details of our theory and so in theory this may be able to help us out because it may dampen the running of the gauge coupling sufficiently to allow us to develop an interacting fixed point. So to understand that we need to look at the running of the Yukawa coupling itself this is given in a general form like this where the E is a cubic in the Yukawa matrices and the F is linear and just from a dimensional analysis you see that the solution to this fixed point equation will have to be of this form where C is just some numerical matrix which depends upon the Yukawa structure of the theory. And so if we project the gauge beta function onto this solution we find that the net effect is that the coefficient the tulip coefficient C gets shifted to an effective tulip coefficient C' where this solution then determines exactly by how much it's shifted and now this plays the same role as C in the above case so we need B and C' to have the same sign here to develop an interacting fixed point and we see that because this Yukawa contribution is always going to be negative the C' is always going to be smaller than C so it may in fact be possible to have C' negative in which case we will get an ultraviolet fixed point if it doesn't have this sign then we will get an IR fixed point in the case where we have asymptotic freedom so that was all in general but here you can look at a specific example and this is the example that Francesco was talking about this morning so again all so far it's just been using a simple gauge group so we just have an SUK gauge group we have a matrix of uncharged scalars and we take the Venetian line limit where we let Nf and K go to infinity while keeping their ratio fixed and this allows us to have some small parameter which we have complete control over and that allows us to make sure that our couplings are exactly perturbative and we find that if we have a Yukawa term of this form which mixes it through a matrix then indeed this effective tulip term is negative which means that we end up with an ultraviolet fixed point and this will be a useful theory to consider when we are trying to build semi-simple theories as we have this example and we know it works it's a hard specific theory and for such theories the phase diagram is something like this so this is the gauge coupling here and this is the Yukawa coupling and we see that there's a fully interacting fixed point here and there's exactly two UV trajectories one which leads down to the Gaussian fixed point in the infrared and another one which flows off the strong coupling and we see that in the vicinity of the Gaussian fixed point we have lost asymptotic freedom and it's entirely infrared fixed point so what happens if we have a semi-simple gauge group? so all that happens is that the tulip feed function is almost the same the main difference being is that we now have two loop terms which mix the different gauge couplings together so I've written it here just with a single Yukawa but really this form generalizes if we have multiple Yukawa couplings so even before worrying about the Yukawa couplings we might think that maybe these off diagonal terms can help us maybe if these guys are negative we don't have to have Yukawa couplings maybe we can find a UV fixed point with these guys if they're contributing negatively but sadly they're not all these off diagonal terms are also positive and so you can see that this has the form of a matrix equation and the right hand side always has to be positive because all of these coefficients are positive and that means the left hand side has to be positive also so we have solutions to this equation and if the left hand side is positive that means we're sitting where we have asymptotic freedom so this doesn't really help us so again we add Yukawa interactions and it's very much the same story as before except now the Yukawa beta function will depend in general on all of the gauge group factors and so in general that means that all of these guys these tulip terms are going to be shifted by some amount may be shifted by zero but in general they can all be shifted and they'll all be shifted by some negative amount so we'll end up with some reduced matrix which is smaller numerically than the one we originally had and so now there's no constraint on these coefficients C so this doesn't have to be bigger than zero this should very much not be there so we can have the right hand side being negative which means that the left hand side can also be negative so it may be possible to have solutions where some of these factors are negative which means that we've lost asymptotic freedom and such a fixed point will have ultraviolet directions and this can be achieved either through the diagonal terms or we can do it through changing the off diagonal terms which is a mechanism that's not available in the simple gauge group case so as well as this fully interacting fixed point there'll be some partially interacting fixed points as well now that we have more than one gauge coupling we know that for each gauge coupling the free fixed point is always going to be there it's always going to renormalize in proportion to itself and so we have the possibility of setting one of the gauge group couplings to zero and finding an interacting fixed point in the other set and so in general there's going to be two to the number of groups that we have so for each gauge factor we can either have it on or sitting on its free fixed point and when we have these sorts of fixed points what we'll find is that generally there'll be marginal directions so when we try to linearize around the fixed point to understand the critical exponents there'll be some zeros there and to understand what happens in these directions then we'll need to look at the effective one loop term so instead of expanding the beta function around the Gaussian fixed point we effectively expand it around the fixed point it's partially interacting fixed point and we'll find that there's an effective one loop term which will receive contributions from the two loop gauge fixed point and also from the Yukawa couplings and there's no need for the sign of this b to be correlated with the sign of this b so just as a sort of little example to show with some slightly more concrete numbers if you have two groups, a single Yukawa coupling then we'll have four fixed points one of which is this partially interacting fixed point where alpha one is zero but alpha two sits on some interacting point and then to determine whether or not alpha one will have a UV direction we need to evaluate this one loop effective coefficient which will be helped by the Yukawa coupling and it will be hindered slightly by this alpha two fixed point and depending on the exact theory we'll find out whether or not this guy wins and whether or not then this becomes asymptotically free so what do we want if we're looking for a semi-simple example theory we want to, following the example of the simple case we want to have a controllable small parameter so we're going to take a generalized Veneziano limit we're going to take lots of, all of our flavor groups to be infinite and our gauge groups to be infinitely large while keeping the ratios of the various quantities fixed we'll need fields to have two of these large indices to make sure that they contribute appropriately and we're going to need something that speaks to both gauge groups so that they interact with each other and some free parameters so that we can play around a little bit so the first example is going to be based on the simple gauge group example we had so we just take two copies of it we have the same content just doubled one in one gauge group and one in the other gauge group and we're going to minimally couple them by adding just a single fermion that's in the fundamental representation of each of the gauge groups so that we have some of these cross terms appearing and if we do this it's fairly simple to calculate these pizza functions it's not very different from the simple gauge group case and then we find that we get an exact UV fixed point which is interacting in one gauge group and is free in the other one so it's something like asymptotic freedom across asymptotic safety and we do also have a fully interacting fixed point which in general is less relevant and it will be more of a crossover type so partially infrared then you may wonder is it possible to find a model where we have asymptotic safety where we've completely lost asymptotic freedom in the gauge theory so in that case we still had asymptotic freedom in one direction and the answer is yes so again we take the simple gauge group model that works we add a second SUM with additional fermions again in its fundamental we add some charged scalars now that we should charge under the second gauge group and then we add another Yukawa coupling which couples these new charged scalars to some of our original fermions and to this field which talks to both gauge groups which will allow us to change these off diagonal terms and again we can compute the beats functions fairly straightforwardly and we find that we can end up with three different exact fixed points two of which are only interacting in the first gauge group and change slightly in their Yukawa contributions and another one which is fully interacting in all four couplings we have a bunch of parameters that we can play around with in theory and depending on how we play around with them we see that in different regions different numbers of these fixed points are physical and non-physical so in these two cases A and C in fact only work if we set the second gauge group coupling to be exactly zero for all RG scales so in some sense these are just kind of generalized simple theories but for B, D and E we do have RG dynamics in both gauge groups and so we can kind of schematically look at what happens in the phase diagram for each so in this we've got the first fixed point which flows to the second and they both flow to the Gaussian and we have similar things going on in these two cases and here and here we have what looks more like the classic asymptotic freedom where we just have a single interacting fixed point flowing to the Gaussian and we can in certain cases take slices through coupling space to project the flow so this is in the case where we have a fully interacting fixed point and a partially interacting fixed point so I've projected it onto the Yukawa null line and so here what we see is in the vicinity of the Gaussian indeed we've lost asymptotic freedom in both gauge directions so it's fully infrared and we have a partially interacting gauge Yukawa fixed point which flows to the Gaussian and it also flows up to this fully interacting fixed point and so we have a two-dimensional UV critical surface here and this is in contrast to region E where we have only the fully interacting fixed point and the Gaussian and this is of the same form as in the simple gauge group case where we just have one trajectory running to the Gaussian fixed point and another one flowing off up to strong coupling so to conclude it is possible to have asymptotic safety in a theories with semi-simple gauge groups and exact perturbative fixed points we'll also have a range of partially interacting fixed points which in general will be more UV relevant and all of this can very easily be generalized to multiple gauge group factors and thank you for your attention