Generalized Gross-Neveu Universality Class with Non-abelian Symmetry
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Gracey, John
We use the large N expansion to compute d-dimensional critical exponents at O(1/N^3) for a generalization of the Gross-Neveu Yukawa universality class that includes a non-abelian symmetry. Specific groups correspond to certain phase transitions in condensed matter physics such as graphene. The effect of the non-abelian symmetry on the exponents is evidenced by the appearance of rank 4 Casimirs in the higher order 1/N analytic corrections. These in effect tag the light-by-light diagrams. The main benefit of the final expressions for the exponents is that one can recover previous results for a variety of universality classes in various limits.
 Okay, thanks Eric and thanks for asking me for this meeting. It's a pleasure to speak at Derek's celebratory meeting. It's a double celebration because it's 25 years this year since we first met in Payser in April at the egg cat meeting where David and Derek were exponentially developing the knot theory and subsequently wrote a paper together. And some of the work, some of the work I'm on, we touch on, we'll be mentioning later on. But all this sort of quite correctly, given a craze of Derek's support through their careers, and I've certainly benefited from that through this Mercator Fellowship that Derek got me several years ago. But that's just another aspect of Derek's support for the community at large. He also just organizes our meetings for us and gets us together and gets training together, I think it's just working together. And I think we should be very grateful for that. And it gets us to meetings like this where we share different techniques to develop the subject, particularly QFT. But I'm going to speak today on a piece of work that came out of work with colleagues in Germany, collaborators in Germany, Andreas Dynand Cologne. We're working in Kinesmatter theories. And this works based on an older paper, a more recent work that's being written up at the moment. But since we finished that project, I suddenly realized I could have done a more general calculation. And this is what this talks about, which is where the non-Abelian synergy comes in. Excuse me. So what I'm interested in, what we're all interested in, the structure of QFT, and the development of the precision calculations to be able to understand the structural aspects and also get numbers out for real-world phenomena. And that's mostly in this area, tends to be particle physics, pushing put out beta functions to high loop order, and trying to see beyond the standard model. But there's other areas of science that QFT underlies in one of the material science of condensed matter physics. And that particularly typically the connection with continuum field theory is in phase transitions and materials. And in the last 10, 15 years, with the development of this wonder material called graphing, people are looking at these materials now to try and understand new phenomena. The graphing is a sheet of carbon atoms joined together in a hexagonal lattice. It's one atom thick, which doesn't always mean it's flat. It can be slightly crumpled. But it's got a rather curious property in that it's incredibly strong for something so thin. But if you stretch it or contract it a bit, it undergoes transitions from what you'd expect carbon to be able to be inserting to more conducting phase. And if we can harness this as a material, it opens up a lot of opportunities to develop new materials and a macroscopic scale. So what I'm going to focus on is some of the theories that underlie these phase transitions in the quantum field theory. And they're not unrelated to models we have in particle physics and the standard model. The key thing I've been talking about today is this gross Navoia-Calva type of interaction, which is basically a scalar field with permeants and no structure in terms of large structure. However, because we're working with electrons, you can add in QED quantum electrodynamics to govern the dynamics of the electrons as they move around the sheet of carbon atoms. I'm not going to focus on that today. We've done working that before. One of the connections is that because the seminar is underlying models, which are like four primary interactions in low dimensions with two and three dimensions, we could get an idea of looking at spontaneous symmetry breaking in standard model by the mimic of the theory that undergoes, underlies the phase sensations. And the more complicated standard model is spontaneous symmetry breaking. But in a scenario where you don't have to dig a tunnel underneath Geneva, which is 25 kilometers round, and it's a bomb of a lot of energy into it, you can do it in the lab, that's the hope. Whether it's completely parallel is another question, but it offers an opportunity to look at phase sensations in simplest scenarios. But the main issues, the phase sensations are not only the Wilson-Fisher fixed points of quantum field theories that we look at. The Wilson-Fisher fixed points are something which we can do in the future. We in fact use in dimensional regularization in effect when we look at the basic renormalization before we put the dimensions to be back to be four dimensions or two dimensions or whatever. So the focus here will be on grossed-in-bore model. But the insight I've gotten recent years working with the condensed matter committee is that they endure these theories now with what is I regard as a non-nibelian symmetry, which for us who do Gaetz theories is second hat. It's basically another structure, group theoretical structure. We've seen talks this week on double copies and other aspects of the group symmetry restrictions on field theories that just becomes an extra layer which we can handle. But the other interesting development in recent months is a new universality class has come out of studies of spin models. So on the condensed matter side, people look at spin theories where they have a discrete lattice and they put polyimages at the corners and they describe the phase sensations with lattice type models. And these models are also in the same universality class of criticality. It allows you to look at phase sensations. But one of those spin models has given rise to a generalized grossed-in-bore model which is called the fractionalized one. And it's got a different group structure. And that's the focus that my collaborators have been looking at which is where I was brought in to do some calculations. So that's the model I'll be focusing on primarily today. So let me just go back and just do a wee bit of basics on phase sensations. I'm going to talk about continuum field theories. In other words, honest-to-goodness data-menstrual field theories with no discretization of space-time. And normally we focus on calculating beta functions. And those beta functions are delused to above a coupling constant to high energy scales or low energy scales in standard model or QCD, quantum chromodynamics. To be able to have some idea of what the behavior observables are, what are objects in nature, which we can calculate on theory, which are independent of Gibbs, parameter independent of the renormalization scale. But there are other what I call observables or renormalization group of right quantities. But in what's simpler to derive, and they come from phase sensations or fixed points in the renormalization group flow. So fixed points are defined by those coupling constants, which start at beta of G equals naught. So this equation here, if you can see my mouse, the solutions to this define the fixed points. Now, I've typically given a one coupling theory, but you have a vector of beta functions with a set of coupling constants. Every beta function by analysis and set of points, which all the beta functions by analysis define the fixed points. So it's true for multi-coupling theories as well. There's several types of fixed points. The trivial one at the origin is known as the Gaussian fixed point. And it's the one which is where we typically do perturbation theory around. Coupling constants are smaller in the neighborhood of the origin. And so we can have an abalance perturbative approximation in that region and get back to the free-field theory, which is what the Gaussian fixed point is. However, the more interesting one is that is the one that derives from the Wilson fixed point. And this is defined by the d-dimensional theory. So if you imagine that if you could calculate in d-dimensions, where d is not the critical dimension of the theory, whether the dimension slightly above or below are somewhere off into the complex plane away from the integer dimension that you're working in. Then the beta function develops a term at the start, which is order epsilon. Repsilons close to the critical dimension of the field theory. Okay. Now this is basically reflecting the dimensionality of the coupling constant. A and B are the one and two libs coefficients of the beta function and so on. Now I've typically written A and B here as epsilon independent. And that means I'm in the MS bar scheme. If you're in a scheme, which was not MS bar, in other words, with a finite renormalization, these coefficients would depend on d or epsilon effect. And that would be a different scheme. But let me just, what I'm saying today is independent of the scheme. So it doesn't quite matter what scheme I'm in. But the fixed point defined by the zero is the beta function. If I take the first two terms here, obviously I can get the zero from G equals not. But I also get a non-trivial zero, which is order epsilon from the origin. Now say epsilon is a complex variable. So the coupling could be complex. And we overlooked that in d dimensions because we're not in a reality situation. So issues about unitarity, breaking, etc. are not really relevant. It's only in the limit going back to four or two, whatever is relevant. And obviously the fixed point depends on the coupling of the coefficient of the beta function, which can include group structure and all the symmetry properties of the theory. And there's obviously corrections from the b term and so forth, all the way down the line. And since I've gone and deal with the growth of a model, this is an example of the one loop term. It's got n dependence where it's an SUN symmetry. I'm putting f on the group there because it's a flavor symmetry. Now, what's the point of the fixed point? Well, if it was a gaze theory, the gaze dependent object, but despite that, you can calculate gaze independent, scheme independent information from the fixed point. And it comes through what we call critical exponents. If I take the wave function minimization group function and evaluate it at the fixed point, I get something called eta, which is the wave function in almost dimension. I'm not going to be a pure number or going to depend on the group cascumers or any parameters in theory, but it's independent of scales as a physically measurable object. Of course, we truncate a series to certain orders and perturbation theory. And by that, we sort of have to terminate and so would be truncation errors. But it's a measurable quantity. And that's what the phase transitions are governed by these exponents. Obviously, you can calculate all the ones the critical slope is a correction to scaling exponent. And obviously, the beta function vanishes at the fixed point, but the slope doesn't. It's a renormalization. Again, this is measurable. And these define the properties of the theory in d dimensions. Obviously, I did the perturbation expansion up here with order epsilon. But in principle, these are d dependent and can exist beyond the critical dimension up to any dimension you like. So they're very universal properties of the theory in all space times. Okay, now, how do we work with this? Well, we have to do an approximation. There's various approximation multicolored those things numerically. There's the dimension conformity field three more recent years functional renormalization group where you can evolve operators across the spectrum using the Wilson renormalization group formalism. You can do prediabasic theory and then use maps to probably approximates between various dimensions and get a behavior. And then there's the large n expansion, which is what I worked on over the years. And whatever you do, you have to do something to get to the dimension of interest and then condense matter theory is three dimensions, which is always one away from two or four. So when we do epsilon expanses, we have to sum down from four or sum up from two. And that's not straightforward. Some of you are aware. Now, there's a secondary aim in this talk is to look at aspects of the large n formulas for another problem I'm interested in. I've decided to do this for a general lead group. The problem with that would become apparently putting in a non-Avelian symmetry is trivial in some sense, but the motivation is to try and understand how the higher order calcium is that will come in and perturbation to come in and then RITN expansion. And this is a toy model where you can do this. So that's the secondary issue because ultimately, we'd like to be able to understand Gaetz theories with an un-Avelian symmetry and similar methodology. So let me get down to Lagrangians. We know the four Fermi interacts in four dimensions and normally normalizable, but it's renormalizable in two. And this is the gross nebulin Lagrangian that was written down in 1974. And it has got several very interesting properties. It's renormalizable. It has the same background to Quartic theories with scalar fields in four dimensions, but it's asymptotically free in two. I've got an SUNF, I've got an SUNF-Fleber symmetry. But as with any Quartic interaction, you always linearize the interaction by introducing auxiliary field. And it's this Lagrangian which is the key thing to write all the calculus to do. Obviously you can put a mass and critically masses are irrelevant because they don't scale. The masses run zero. Fixed points of scale free, so the masses are irrelevant. The sigma field here doesn't propagate, but it defines the dimension of the sigma field through the interaction. No. One interest for models like this is that they got popular again in the early 90s because they could be models of composite Higgs fields, where the sigma would play a role of a, so the sigma would be the Higgs field and then so forth, so it would be the composite fields at the fields that make up the composite Higgs field. Except this is two dimensions. You have to have a four dimensional theory. Well, you can construct a four dimensional theory with that property. It's called the Gross-Nivouria-Cavan model and it's termed the ultralidal completeness. And what that means is this is not renormal, this is not, this is super normalizable forward and make it renormalizable. The sigma field has to become an order kinetic term, but if that becomes an order kinetic term, you have to record the interaction and that would be your Higgs potential. And this was shown by Zinjurston in the early 90s. So that would be the corresponding theory of four dimensions that's in the U.S. and the class of this theory in two because it's the same core, a cubic interaction, the same core fermi-i kinetic term. And this operator is not relevant in four. The quarter value vertex would be relevant in four. So the sigma field depends changes across dimensions. Now, the extension that people looked at for the condensed matter problem for the Mott transition years ago was a variation that goes into one with the Karel-Heisenberg one. And the only difference is to put a Paul-Eye spin matrix in here in Lander. On, you can linearize it, but you have to have a vector of scalar fields. And I've written down the other completeness of this theory. It's just by analogy. So the pi field would have to have an order kinetic term, the propagate, and obviously you have to have a quadratic term for randomizability. But the same core interaction here spawns the other interactions as you go through the dimensions. If you go down in dimensions, this becomes irrelevant. This becomes irrelevant, but the other operators become relevant, and that makes it re-organisable in two dimensions. But in between two and four, there's a three-dimensional theory that describes the phase transitions in the graphene sheet. Now, the fractionalized gross liberal model that I mentioned earlier on, the more recent one, is a variation in that where you replace the parli matrices by S-O3 matrices, and the S-O3 matrices satisfy this relation, which you can just encode into your files. And that's why I came in on the more recent work with my colleagues, is to do the S-O3 calculation. So I reworked all my S-O2 calculations with a different group theory structure. And then Dumbo realized afterwards, I should admit it for an already lead group, and start it from a more general case with the usual group casamiers. Then I could have taken the S-O2 and S-O3 results as a corollary of more general result. So it's a more general universality class with non-abelian symmetry that I want to focus on now, and go through the construction. And basically, the large N expansion technique, which I use, is to work completely now in the universal theory with facility as technique from the early 80s, and just apply the formalism from scratch. So I mentioned earlier on about the ultravalent completeness. Well, if you extract from what that, if you extract the lesson from that, this means the large N theory will work with this Lagrangian. We've got the core kinetic term and the core interaction with the non-abelian surgery. And some function of all the other scalar fields, and that varies as you go through dimensions, but it's irrelevant for the large N expansion. I call these spectators, and I've rescaled a coupling constant out here to have unit coupling, because it criticizes there is no coupling constant. It's just replaced by some number. So you just start with this version of the theory. This comes along for the right. Everything in this f of pi in the fixed dimensions is contained within this interaction as effective vertices. So life becomes a bit simpler. The other aspect of the formalism that we work completely in coordinate space, and I want to focus a lot more on coordinate space and having the past, because Oliver is giving a talk tomorrow on his graphical functions and the similarities, but differences. And I want to sort of add some of the ideas to that debate this week. So in coordinate space, the propagators all have a scaling form with potentially corrections to scaling, which I'll come back to later, but the key focus is on the leading term. And the dimensionality of the field in the Lagrangian at critical count, they determine the power of the propagator. And if you look at the dimensional analysis of the Lagrangian, the Fermion field has dimension, which is d over 2. So I'm setting d equals 2 mu here just for a shorthand. And the pi field has a propagator, which has got one, but there's corrections to both. And these are the anomalous pieces. The mu and the one are the canonical pieces. And eta is the anomalous dimension of the Fermion field as I had before. And chi-pi is the vertex anomalous dimension of the operator that we have in the theory as well. So both operators have anomalous dimensions, reflected by chi and eta. And just for the future, the discombination of exponents is d plus 1 minus chi. And chi and eta will have order 1 of n expansions, starting at 1 of n, with mu dependent or d dependent coefficients, which also depend on the group casemers. And the beauty of the facility of method is you just take these propagates and put them in swingard-eisen equations. And we'll hear more about swingard-eisen equations later. And you just self-consistently solve for the unknowns, which are the exponents and an amplitude for combination of a and c, a and c are just x independent objects, just to get a scale. That's a good number on the organization for the propagators. And that's, quote, all you do. And these are the leading order diagrams, and next to the leading order diagrams are the two point functions. Of course, you have to calculate the integrals. But in principle, this is all you have to look at. And you note that in the diagrams are written down, there's no dressings of the propagators. And that's because the anonymous dimensions automatically incorporate the self-energy corrections on each line. So you automatically sum up those diagrams by taking the generalized anonymous dimension. What you haven't done is got rid of vertex corrections. So there's two diagrams, one over n squared, the vertex corrections, these two, which are actually divergent, as you would sort of half expect in quantum field theory. But you can analytically regularize with the delta. And if you do that, it's possible to calculate them. We'll have simple poles in delta at this order and double poles in next order. Now, how do we calculate? We use what's known as conformal integration, or uniqueness. So the rule for uniqueness or conformal integration is that at each vertex, you have a sum of exponents. You integrate over the vertex point. So this is not like quantum space where you integrate over loops. You integrate over the points where the edges join. But the powers of the propagator joining that point are not one that there would be in perturbation theory. They're different. They're taken generic alphabet in gamma here. But the rule is this is not possible to integrate except in a restricted set of cases, the counter-blank number of cases. And this is the first case where you can integrate this vertex. It's when alpha plus beta plus gamma is 2m plus 1. So it's d plus 1. If that condition is satisfied, then you can meet the character at the integral. You get some overall normalization factor where it's just products of gamma function, which I'm not going to dwell on. These always come along for the right every time I do an integration. And you replace the vertex where you're integrating by just the product of three propagators. This is sometimes called the star triangle relation. But it's in coordinate space. There's a parallel of it in the momentum space, of course. There's a duality with it. If the line, the propagator opposite the point for you, the opposite the original propagator you integrate over is dual to it. So it's mu space time dimension of 2 minus the experiment. If the square amunces are adjusted for the x slash or the y slash, just plus 1. Now, seven aspects I need to point out. For the ghost number you can with Carol Heisenberg vertex, 2 alpha plus gamma is 2m plus 1 minus chi pi. So unfortunately, the vertex is not quite correct for uniqueness in the large n setup. However, with the regularization, you can calculate with subtraction methods to be able to get around that with still using uniqueness to the ways around it. And the second point is there's various ways of proving this formula. One of which is just do the Feynman integral. Either you have the Feynman integral associated with it by using the condition and you end up with hyper geometric function, which simplifies into a geometric function when the condition is satisfied and you get down to the right-hand side. That's brute force. The other way to do it is by conformal transformation and I'll mention those later on. And the rule for the conformal transformation is that when any line joins, so you define an origin for an external point and then take a conformal transformation. So the rule is that you're clear every line joining the origin by 2 mu minus the sum of the exponents joining that point, which here would be 2 mu plus 1 in the case of fermionic vertex. It would 2 mu plus 1 minus alpha minus beta minus gamma, which is not if the condition is satisfied. So then you just get an integral of a bubble or a chain integral, which is what this is effectively. And that's the rule. There's no rule for QCD because it's got a gamma matrix inside it. Now, just applying the formulas, we get these results, works order and next deleting order. And the reason I'm showing these is several points. You see the group customers coming in from before, CACF. And also this combination of side functions coming. I've written this side function so it's got a zeta expansion near two dimensions. And these are typical of what you get, but the leading on next deleting order is 3 will to get. The corrections of scaling that I had in which had the exponent lambda can also be incorporated in the formalism. And in the Kinesma line, this is the correlation scale exponent. And it's slightly more involved because you have to calculate the two point function scaling function before you start with Kresens. And involves a hard set of gamma functions incorporated in this notation. But there's a subtlety with the gross Neville case, which I want to point out to you now. It's to do with this function Q that comes in mu, I say lambda at leading orders, mu minus one dimension of space time over two minus one. And if you look at that formula with a specific example of the exponents we've got, it turns out this is singular. And that singularity means there's a reordering of diagrams in the Dyson-Swinger equations. So you need higher order diagrams to be able to go to one over n squared to get the second order correction to the exponent lambda. This means just more diagrams to calculate. And these are the diagrams where the dot indicates the insertion of the correction to scaling. So this is a vertex correction on the trulip self-energy, likewise this one with the insertions in line. This is a vertex correction on the external leg. And then you get the usual characters, the primitive norm planar. And these other diagrams, which are, well, norm planar as well, with these various decorations with the insertion. Now I've illustrated these graphically because those were two gates series automatically go light by light, which is not something out of Star Wars. It's basically this set here. If I cut down this line, I get four external wiggly lines, which you could write as photons. And that would be two goes to two photons scattering. And these are regarded as tougher diagrams. But there's two bolted together here. There's two bolted together here. And each one's got a group generator at the vertex. So there's a lot of groups that we associate it with this, which we'll have to take into account for the calculations. And the typical combination is written here. A and B are external group indices. And C, D and E are the internals. I've written them in sort of different orders here, because the way the pie field goes across varies from the left loop to the light loop. But it's proportional to delta AB. No. In the mid-70s, mid-80s, from Arsene von Littberg and Lauren calculated the polar better from SecuCity. And they had to take account of the group theory graphs, graphs for this type of group theory. And they have a very elegant package called color.h, which allows me to do the group theory for this problem. And it involves rank four fully symmetric tensor DABCD. The F here means the fundamental. It could be an adjoint, for example. There's a rank three symmetric tensor, which we all know about in league groups, which is DABC. And this calculation is absent. But I get the DABC tensor in the problem, which is going to be products of these two things when there's a shake down on the calculation. So it's possible to code this in automatically and run through the calculation. And this is the second order correction, which is significantly more complicated than the original one for the original Ising model, versus the case from the mid-90s, which was the only thing I did separately. But you see the rank four customer is coming in here at second order large end. And there's seven new functions here that come in this five comes in. There's a lot of decorations. And the rest of it's on the next sheet, unfortunately, the formula got a bit big. And there's this data here, which are the second, I said the derivative of the side function. And these incorporate status. So if you were to expand these objects out an epsilon expansion, they are four dimensions, they would correspond to renomous this group functions near four dimensions or equally two dimensions. If you did it near two dimensions, and this would show you that two orders in large end, the corresponding renomous this group function to all orders at this order in large end would only have a epsilon and status. It's not a proof. It's just an observation because obviously there's going to be non status further down the line. At higher order large end higher order per device. So it's somewhere to get them. But the key thing here is now I've got this general formula I can specify groups very simply and take limits. I'll talk about that towards the end. Now, the other aspect of the facility of work from his school, honking in and Pismac as well, was that he pushed the calculations to one of the grand cubes. This is a can stress how much this is a well ahead of its time. This was in the early 80s. The form was set in stone earlier on by Poliakoff in 1970 on Paraisie a few years later, where they were doing what's now known as the conformal birch strap method. In essence, the way I'm using what's now known as the large end conformal birch strap method, which is what I've had to rename it after conformal birch strap people come in and corner that market. But if you remember before, I didn't have dressings on the two-point functions, but I had vertex corrections. This method goes around that by getting ready ready vertex corrections in the self consistent serendice and formalism. So these are the diagrams that contribute to the calculation of eight of three. You calculate the vertex functions and through a formalism that I'm not going to go through. You gain the eight of three. These are all primitive diagrams because there's no vertex corrections, but the dot that's on each vertex is not accidental. It's deliberate. The dot represents what's called a Poliakoff conformal triangle. Now, if you remember that the vertex of the theory is set in such a way that it's not unique, but the Poliakoff conformal triangle is in some sense mimicking not the absence of uniqueness but allows you to calculate these diagrams exactly. So let me explain what a conformal triangle is. So if you're most an ordinary vertex with arbitrary exponents meeting at a point, Poliakoff replaced that point by a triangle with arbitrary a1, a2, a3 exponents internally. Obviously, there's some proportionality function here. Would you choose these internal exponents which are completely unseen and not relevant to the field for each vertex by making each vertex of the triangle unique? So these are the conditions here. So if the triangles are all unique, then there's lots of tricks you can use to evaluate the integral. And this is the conformal transformation that you use. You set x goes to one over x, but the perfect is x me equals x me over x squared. Or if you take what would happen in the product of in a propagator would be x slash minus x, y slash goes to x slash minus y slash y slash over x squared, y squared minus x. But if you build a lot of these together with a y slash minus z slice, there'll be a y slice here and another y slice for the next propagator. So one of the y slices, so the pair of y slices can't stop at y squared. So you retain the same string of gamma matrices around a diagram after conformal transformation as the original diagram. Clearly because there's no gamma matrix in the vertex and that allows a lot of tricks to be used. So with these vertex, we're clear as by a conformal triangle you can calculate the diagrams. But leading orders, as David reminds us many occasions, the hardest diagrams sometimes are calculated by the previous loop order to the higher order in the expansion parameter. And that's the case for the leading diagram, which is just the triangle. So this is the full setup for the conformal triangle definition of the original one loop triangle diagram. So each vertex is now replaced by its conformal triangle. These internal ones are chosen purely because of the alpha here, here and here and the external one that haven't drawn to the top. There's a regularization coming from the external lines because the equations as I said before are divergent, so you have to regularize each diagram. And there's two regularizations, the delta here and the delta prime here. And I've arbitrarily put the origin at the top. So how do we evaluate this? Well, every vertex now is unique. If I take a conformal transformation, I replace every line joining the origin by the 2 mu plus 1 minus the sum of the exponents at a vertex. If you do that, you get this. It's the same diagram, but the zero, this line's removed because the sum of the exponents at that line are 2 mu plus 1. Now, that means the only points to integrate are the ones on the hexagon that goes around here, x and y are external. So this immediately gets you down to a two-point function. And this is an integration point, and that's an integration point. So I'm going to immediately integrate these two diagrams. And I'm not carrying factors through this in the slides, but you imagine the state of paper per se where you carry the factors. And you get this diagram, which is two-point function, which you have four integrations on, and a lot of parameters. But this is where you have to think. And the facility of trick was to write this as a diagram that looks the same without the top line removed, but you multiply by a function which represents the discrepancy or the compensation for the absence of it. This can be evaluated because it's just, again, simple chain integrals. And you get a result, which is this. Sorry, this which is a result, but the compensation function can be written in this form. Or basically, it's because it's a function of this combination of parameters. So the orders were calculated, and then you can write down the expansion. It's in an exponential because it's easier formulation because we always write a gamma function x-logs, and it's got an easier expansion. And then the idea is you calculate the coefficient x1 to x4. Well, how do you do that? Well, the very simple thing is you took the original, took the original point here to be at the top for the origin. There's no very priori reason why I should pick that point. If you pick this point here as the origin, I'll do the conformal chance to make these two lines disappear. And this collapses the other way. But if you look at that, the common estimate is going to be different for the behavior of the compensation, which is going to be delta prime on delta pi. Likewise, if you pick this point here, the collapse will be here and here. And so the discrepancy function will just depend on delta pi. So we've got three different ways of evaluating this diagram, but they've all got to be equivalent. They'll all have a similar compensation function like this, but a different expansion problem because the argument would be delta prime minus delta tilde, as I said, and fc would be delta tilde. And so three different ways, plus all the factors that are neglected right down in the slides, brought together, they should all be the same, and that's sufficient to be able to solve this to two orders. That allows people to write down sufficient terms and expanse of leaving on a diagram to be able to evaluate for age of three. And this is just an example. I haven't written down all the terms, but you see that you get the corrections involving obviously the problems themselves, written out properly, but you get these combination of psi functions, which is where the vector we come from in the results of Schumann's so far. The higher order diagrams come in in the corrections, these are the two formulas to find the vertex function. The first one here, I represent the fact all the diagrams have to sum up to one. It's a very simple self-consistency equation, and that fixes one of the internal parameters. The second equation is in effect equivalent to the two-point function, calculated as we did earlier, but summed on the vertex functions. It's easy to calculate this. The higher order diagrams have been evaluated years ago, it's just a question of decorating with group theory, but the leading order diagram is a bit harder to calculate, which is what I've developed on it. So in all of his formulas, he sends off a point to an infinity for a three-point function, and that in some sense is similar to what I do with sending the conformal transformation in. I just make one point irrelevant and you're left with two-point functions. It's similar, it's probably not connected, and it's just I want to highlight that this morning in anticipation of what all of us are going to talk. I'm sorry, all of us put pressure on you. So what do you get? Well, you get a result which is not the similar to lambda 2 in the sense, structurally it's got the Casting-Age structures. Obviously, this is now tags for the light-by-light diagrams come in in the calculation, which is kind of nice because you can sort of look at the structure of these diagrams through perturbation theory if you just focus on the expanse of these objects in parts of epsilon. The other feature that comes in is that there's another new function, which I call psi. This is a known function from early times when facilities work on this, and it appears a non-Caroly symmetric theories at large animal around cube. I've never quite understand the Carole 70 aspect of this, but it's come up in two calculations, one of which has been written up separately, one of which I've known from the early 90s, mid-90s from the rest of the model in super symmetry. But it's a hardly complicated function to define. If you take this two-loop integral with decorations, well, obviously, epsilon depends, but d demands mu minus 1 and it regularizes the central line. It's the derivative of this integral, it's the derivative of the logarithm of this integral with delta equals naught, that defines psi. Now, we know if we've got a two-loop function, a two-loop two-point function, if we have epsilon ordered from one on the lines that join at external points, we can expand that in zetas. If there's any decoration of order epsilon on the center line, that's a problem, and that brings in the multiple zetas, basically, which is what Derek and David have been looking at in the mid-90s. This is the connection with where David and Derek and I got working together in these problems. In our paper, David showed that this integral can be unlifted expressed as a hypergeometric function in d dimensions. But the complication is because the difference is, with respect to the delta parameter here, you have to differentiate the arguments, the parametric arguments of the hypergeometric function, to really define the function, and that makes it harder to write it down as a closed form function of 1, as we know. However, what we did extract in that paper was the epsilon expansion of psi near two and four dimensions, and it has zeta-3, or zeta-3, 5, it has u6, 2, and no music. But we now know it's all orders in epsilon, and that's sufficient. David and Derek wrote down an integral in phi-4 theory in their early work with these multiple zetas structures, but it was never shown whether it was realized in the free theory. Oliver and others have now found those in the free theory in phi-4. This was another example, an early example of where the zeta-5, 3 can appear in the free theory, because we expand out in the parts of epsilon to find it in a certain order. It's probably the earliest order, but it comes in. It is possible to evaluate the integral exactly in three dimensions. It involves log 2s and zeta-3 over pi squared, and that's purely because in three dimensions you can exploit three dimensions on the uniqueness of the diagram exactly, right of time. I haven't talked about checks on the results, but those of us who look at non-abillion-gay theory, if we've got a group generally inside interaction, for say, young girls, we immediately get QED from those calculations, by just notionally taking T to be 1, just taking probably really mathematically unsigned limit of setting the generator to 1. Well, what that really means is you replace the cacomers by the corresponding object in the center limit. So the abillion limit would be taken, Cf to be 1, Tf to be 1, Tf to be 2, of Tatb. Obviously, the group cast me over a couple of one, but the abillion limit has to have FA, BC to be 0, so that CA goes to 0. I reproduce everything that was known before in the gross removal model, by taking that limit from the results I've presented today. The other results that have been known is the SU2 mot-cont's interditing phase structure, which is slightly different than SU2. So it's still not a million, so it's three quarters to half. CI goes to two, but the group cast is five to 64. That reputages what I had before. And the more recent one which we've done is SU3, which has got a slightly different structure. If you go and look it up on some table, the group cast was 20 over three. I'm taking those elements and reputating what I did with my colleagues. But also all these results agree with all that's known in perturbation theory. For most of these models, it's either two-dimensional or four-dimensional, but in this case, the fractionality one, we've only got four-dimensional results. We've calculated out three looks in that and got agreement with the large N, so they're both tally off, which is reassuring. Now, once you've got a generalized result, you can just look at different group structures. And so I've written down the three-dimensional results here just for the adjoint quotes for a bit of fun. So put the firm into the firm fundamental. The group cast mirrors obviously change, but the reason I've written this down is it shows you the structure more succinctly in three dimensions for the light by light diagrams having the facts. And it identifies this term as a light by light contribution over and above this contribution. But for Aida, for Aida, you go slightly differently. I kept this combination here because this tends to pop up a lot for the adjoint case. It doesn't quite do a couple here, but if you're interested, you put in the numerical values and see what it looks like. But that's the kind of more general aspects that one can do with this form. Since doing this, I've been looking at other university alty classes with a different underlying interaction. I'm playing around with that and applying the same tools to sort of look at all the problems that were interactable. It seems to be a really robust way of doing things, taking a general lead group approach. So just let me conclude. I've looked at a more general type of critical alty class, university alty class for this fundamentalizing growth type interaction. It's opened up a whole idea now of looking at more general structures. I can't emphasize enough on what the color park is in form. It's been makes life easy for doing the group theory. You don't have to sit there with a pen and piece of every minute group generally. It is around the diagram. It's easier to identify structures inside results and in principle, all orders are part of basically by integrating through the epsilon expanse. So I'll finish there. Thanks everybody. Thank you, John. Thanks for your nice talk. Please feel free to unmute yourselves and raise your hand or speak just up if you can. I have an immediate question to go just one slide back. The three-dimensional thing you showed. So since you have this result for an arbitrary Lie algebra, is there any special cancellations that you see for, I mean for example here, I could see this bracket that you mentioned in the third line from the bottom. So you can imagine that there's a special situation with these vanishes and you drop the zeta three term. Things like that. I did look at that off the top of my head. I think it doesn't happen. Okay. This is actually a quartic polynomial in the number of colors. It has to depend on if it's going to happen, it's going to happen for low values of colors and I feel it doesn't. I should say I haven't looked at the exceptional groups. Yeah, so I was just wondering since you have all this data, I'm just curious if there is something funny happening. Since you've raised that, I think of what I'll do is I'll look at it after, come across and I'll put it in the right up. I'll try to make some room to make some room to come in the right up. Just yeah. Yeah, thank you. We have a question from David. Yes, you've reminded me that in 1996 we did epsilon expansions about two and four dimensions to get towards three and there we encountered multiple zeta values. But just out of mathematical curiosity, I wanted to know what the epsilon expansion was about three dimensions. And if you remember, Toliakotikov and I actually did that because I knew that you would get alternating Euler sums, the same things you get in the magnetic moment of the electron. Is there any use of those epsilon expansions that we did about three dimensions? There could be, David, because I didn't put the slide up. I had it in an earlier version of the talk. I think you may have seen the slide before. What you can do now is you can plot the behavior of the exponents in d dimensions from two to four. This is important because not only can you do that in large n now, we're using a paddy approximate across the dimension. You also do it in protobase theory. You know in protobase theory in two dimensions, up to say four loops, and the four dimensions down from four dimensions to four loops, you can do what's known as a matched paddy approximate where you use the paddy information near two and near four to write it down and interpolate the behavior of the exponent across the dimensions. You can plot that. You also, in the function renormalization group, they can plot the behavior of the exponents across d dimensions. They all have a similar sort of, it's like a hump, like a bell, but it's offset sometimes about three dimensions. What we could do with the epsilon span around three is expand in the neighborhood of three to see how well the maximum matches the shape of the other ones. Now it's not something, until you mention it, it never occurred to me. I can't remember what order in epsilon you went to. It's obviously going to be breakdown, but what it could do, it could influence the perturbative expansion that people do from two to four with the epsilon expansion from a high order perturbation to try and do some sort of global fit based on the limited data we know from perturbation in large M. It's an exercise that could be worth doing, but it's not something I might have the tools do immediately. Thank you. I see some questions at the moment, so let's thank John again. Here.
 Okay, thanks Eric and thanks for asking me for this meeting. It's a pleasure to speak  at Derek's celebratory meeting. It's a double celebration because it's 25 years this year  since we first met in Payser in April at the egg cat meeting where David and Derek were  exponentially developing the knot theory and subsequently wrote a paper together.  And some of the work, some of the work I'm on, we touch on, we'll be mentioning later on.  But all this sort of quite correctly, given a craze of Derek's support through their careers,  and I've certainly benefited from that through this Mercator Fellowship that Derek got me  several years ago. But that's just another aspect of Derek's support for the community at large.  He also just organizes our meetings for us and gets us together and gets training together,  I think it's just working together. And I think we should be very grateful for that. And it gets us  to meetings like this where we share different techniques to develop the subject, particularly  QFT. But I'm going to speak today on a piece of work that came out of work with colleagues in  Germany, collaborators in Germany, Andreas Dynand Cologne. We're working in Kinesmatter theories.  And this works based on an older paper, a more recent work that's being written up at the moment.  But since we finished that project, I suddenly realized I could have done a more general  calculation. And this is what this talks about, which is where the non-Abelian synergy comes in.  Excuse me. So what I'm interested in, what we're all interested in, the structure of QFT,  and the development of the precision calculations to be able to understand  the structural aspects and also get numbers out for real-world phenomena. And that's mostly in  this area, tends to be particle physics, pushing put out beta functions to high loop order,  and trying to see beyond the standard model. But there's other areas of science  that QFT underlies in one of the material science of condensed matter physics.  And that particularly typically the connection with continuum field theory is in phase transitions  and materials. And in the last 10, 15 years, with the development of this wonder material called  graphing, people are looking at these materials now to try and understand new phenomena.  The graphing is a sheet of carbon atoms joined together in a hexagonal lattice.  It's one atom thick, which doesn't always mean it's flat. It can be slightly crumpled.  But it's got a rather curious property in that it's incredibly strong for something so thin.  But if you stretch it or contract it a bit, it undergoes transitions from what you'd expect  carbon to be able to be inserting to more conducting phase. And if we can harness this as a material,  it opens up a lot of opportunities to develop new materials and a macroscopic scale.  So what I'm going to focus on is some of the theories that underlie these phase transitions in the  quantum field theory. And they're not unrelated to models we have in particle physics and the  standard model. The key thing I've been talking about today is this gross Navoia-Calva type of  interaction, which is basically a scalar field with permeants and no structure in terms of  large structure. However, because we're working with electrons, you can add in QED quantum  electrodynamics to govern the dynamics of the electrons as they move around the sheet of  carbon atoms. I'm not going to focus on that today. We've done working that before.  One of the connections is that because the seminar is underlying models, which are like  four primary interactions in low dimensions with two and three dimensions, we could get an idea of  looking at spontaneous symmetry breaking in standard model by the mimic of the theory that  undergoes, underlies the phase sensations. And the more complicated standard model is  spontaneous symmetry breaking. But in a scenario where you don't have to dig a tunnel underneath  Geneva, which is 25 kilometers round, and it's a bomb of a lot of energy into it, you can do it  in the lab, that's the hope. Whether it's completely parallel is another question, but it  offers an opportunity to look at phase sensations in simplest scenarios. But the main issues,  the phase sensations are not only the Wilson-Fisher fixed points of quantum field theories that  we look at. The Wilson-Fisher fixed points are something which we can do in the future.  We in fact use in dimensional regularization in effect when we look at the basic renormalization  before we put the dimensions to be back to be four dimensions or two dimensions or whatever.  So the focus here will be on grossed-in-bore model. But the insight I've gotten recent years working  with the condensed matter committee is that they endure these theories now with what is  I regard as a non-nibelian symmetry, which for us who do Gaetz theories is second hat. It's  basically another structure, group theoretical structure. We've seen talks this week on double  copies and other aspects of the group symmetry restrictions on field theories that just becomes  an extra layer which we can handle. But the other interesting development in recent months is  a new universality class has come out of studies of spin models. So on the condensed matter side,  people look at spin theories where they have a discrete lattice and they put  polyimages at the corners and they describe the phase sensations with  lattice type models. And these models are also in the same universality class of criticality.  It allows you to look at phase sensations. But one of those spin models has given rise to a  generalized grossed-in-bore model which is called the fractionalized one. And it's got a different  group structure. And that's the focus that my collaborators have been looking at which is where  I was brought in to do some calculations. So that's the model I'll be focusing on primarily today.  So let me just go back and just do a wee bit of basics on phase sensations.  I'm going to talk about continuum field theories. In other words, honest-to-goodness  data-menstrual field theories with no discretization of space-time.  And normally we focus on calculating beta functions. And those beta functions are  delused to above a coupling constant to high energy scales or low energy scales  in standard model or QCD, quantum chromodynamics. To be able to have some idea of what  the behavior observables are, what are objects in nature, which we can calculate on theory,  which are independent of Gibbs, parameter independent of the renormalization scale.  But there are other what I call observables or renormalization group of right quantities.  But in what's simpler to derive, and they come from phase sensations or fixed points in the  renormalization group flow. So fixed points are defined by those coupling constants,  which start at beta of G equals naught. So this equation here, if you can see my mouse,  the solutions to this define the fixed points. Now, I've typically given a one coupling theory,  but you have a vector of beta functions with a set of coupling constants. Every beta function  by analysis and set of points, which all the beta functions by analysis define the fixed points.  So it's true for multi-coupling theories as well. There's several types of fixed points.  The trivial one at the origin is known as the Gaussian fixed point. And it's the one which is  where we typically do perturbation theory around. Coupling constants are smaller in the neighborhood  of the origin. And so we can have an abalance perturbative approximation in that region  and get back to the free-field theory, which is what the Gaussian fixed point is.  However, the more interesting one is that is the one that derives from the Wilson fixed point.  And this is defined by the d-dimensional theory. So if you imagine that if you could  calculate in d-dimensions, where d is not the critical dimension of the theory,  whether the dimension slightly above or below are somewhere off into the complex plane away  from the integer dimension that you're working in. Then the beta function develops a term at the  start, which is order epsilon. Repsilons close to the critical dimension of the field theory.  Okay. Now this is basically reflecting the dimensionality of the coupling constant. A and B  are the one and two libs coefficients of the beta function and so on. Now I've typically written A and  B here as epsilon independent. And that means I'm in the MS bar scheme. If you're in a scheme,  which was not MS bar, in other words, with a finite renormalization, these coefficients would  depend on d or epsilon effect. And that would be a different scheme. But let me just, what I'm saying  today is independent of the scheme. So it doesn't quite matter what scheme I'm in. But the fixed  point defined by the zero is the beta function. If I take the first two terms here, obviously I can  get the zero from G equals not. But I also get a non-trivial zero, which is order epsilon from the  origin. Now say epsilon is a complex variable. So the coupling could be complex. And we overlooked  that in d dimensions because we're not in a reality situation. So issues about unitarity,  breaking, etc. are not really relevant. It's only in the limit going back to four or two, whatever  is relevant. And obviously the fixed point depends on the coupling of the coefficient of the beta  function, which can include group structure and all the symmetry properties of the theory.  And there's obviously corrections from the b term and so forth, all the way down the line.  And since I've gone and deal with the growth of a model, this is an example of the one loop term.  It's got n dependence where it's an SUN symmetry. I'm putting f on the group there because it's a  flavor symmetry. Now, what's the point of the fixed point? Well, if it was a gaze theory,  the gaze dependent object, but despite that, you can calculate gaze independent, scheme independent  information from the fixed point. And it comes through what we call critical exponents. If I  take the wave function minimization group function and evaluate it at the fixed point,  I get something called eta, which is the wave function in almost dimension.  I'm not going to be a pure number or going to depend on the group cascumers or any parameters  in theory, but it's independent of scales as a physically measurable object. Of course, we truncate  a series to certain orders and perturbation theory. And by that, we sort of have to terminate and so  would be truncation errors. But it's a measurable quantity. And that's what the phase transitions  are governed by these exponents. Obviously, you can calculate all the ones the critical slope  is a correction to scaling exponent. And obviously, the beta function vanishes at the fixed point,  but the slope doesn't. It's a renormalization. Again, this is measurable. And these define  the properties of the theory in d dimensions. Obviously, I did the perturbation expansion  up here with order epsilon. But in principle, these are d dependent and can exist beyond the critical  dimension up to any dimension you like. So they're very universal properties of the theory in all  space times. Okay, now, how do we work with this? Well, we have to do an approximation. There's  various approximation multicolored those things numerically. There's the dimension conformity  field three more recent years functional renormalization group where you can evolve  operators across the spectrum using the Wilson renormalization group formalism. You can do  prediabasic theory and then use maps to probably approximates between various dimensions and get  a behavior. And then there's the large n expansion, which is what I worked on over the years.  And whatever you do, you have to do something to get to the dimension of interest and then  condense matter theory is three dimensions, which is always one away from two or four.  So when we do epsilon expanses, we have to sum down from four or sum up from two.  And that's not straightforward. Some of you are aware. Now, there's a secondary aim in this talk  is to look at aspects of the large n formulas for another problem I'm interested in.  I've decided to do this for a general lead group. The problem with that would become apparently  putting in a non-Avelian symmetry is trivial in some sense, but the motivation is to try and  understand how the higher order calcium is that will come in and perturbation to  come in and then RITN expansion. And this is a toy model where you can do this.  So that's the secondary issue because ultimately, we'd like to be able to understand Gaetz theories  with an un-Avelian symmetry and similar methodology. So let me get down to Lagrangians.  We know the four Fermi interacts in four dimensions and normally normalizable, but it's  renormalizable in two. And this is the gross nebulin Lagrangian that was written down in 1974.  And it has got several very interesting properties. It's renormalizable. It has the same  background to Quartic theories with scalar fields in four dimensions,  but it's asymptotically free in two. I've got an SUNF, I've got an SUNF-Fleber symmetry.  But as with any Quartic interaction, you always linearize the interaction by introducing auxiliary  field. And it's this Lagrangian which is the key thing to write all the calculus to do.  Obviously you can put a mass and critically masses are irrelevant because they don't scale.  The masses run zero. Fixed points of scale free, so the masses are irrelevant.  The sigma field here doesn't propagate, but it defines the dimension of the sigma field  through the interaction. No. One interest for models like this is that they got popular again  in the early 90s because they could be models of composite Higgs fields, where the sigma would  play a role of a, so the sigma would be the Higgs field and then so forth, so it would be the  composite fields at the fields that make up the composite Higgs field. Except this is two dimensions.  You have to have a four dimensional theory. Well, you can construct a four dimensional theory  with that property. It's called the Gross-Nivouria-Cavan model and it's termed the ultralidal  completeness. And what that means is this is not renormal, this is not, this is super  normalizable forward and make it renormalizable. The sigma field has to become an order kinetic term,  but if that becomes an order kinetic term, you have to record the interaction  and that would be your Higgs potential. And this was shown by Zinjurston in the early 90s.  So that would be the corresponding theory of four dimensions that's in the U.S.  and the class of this theory in two because it's the same core, a cubic interaction,  the same core fermi-i kinetic term. And this operator is not relevant in four. The quarter  value vertex would be relevant in four. So the sigma field depends changes across dimensions.  Now, the extension that people looked at for the condensed matter problem  for the Mott transition years ago was a variation that goes into one with the  Karel-Heisenberg one. And the only difference is to put a Paul-Eye spin matrix in here in Lander.  On, you can linearize it, but you have to have a vector of scalar fields.  And I've written down the other completeness of this theory. It's just by analogy. So the  pi field would have to have an order kinetic term, the propagate, and obviously you have to have a  quadratic term for randomizability. But the same core interaction here spawns the other interactions  as you go through the dimensions. If you go down in dimensions, this becomes irrelevant.  This becomes irrelevant, but the other operators become relevant, and that makes  it re-organisable in two dimensions. But in between two and four, there's a three-dimensional theory  that describes the phase transitions in the graphene sheet. Now, the fractionalized gross  liberal model that I mentioned earlier on, the more recent one, is a variation in that where  you replace the parli matrices by S-O3 matrices, and the S-O3 matrices satisfy this relation,  which you can just encode into your files. And that's why I came in on the more recent work with  my colleagues, is to do the S-O3 calculation. So I reworked all my S-O2 calculations with a  different group theory structure. And then Dumbo realized afterwards, I should admit it for an  already lead group, and start it from a more general case with the usual group casamiers.  Then I could have taken the S-O2 and S-O3 results as a corollary of more general result. So it's a  more general universality class with non-abelian symmetry that I want to focus on now, and go  through the construction. And basically, the large N expansion technique, which I use, is to work  completely now in the universal theory with facility as technique from the early 80s,  and just apply the formalism from scratch. So I mentioned earlier on about the ultravalent completeness.  Well, if you extract from what that, if you extract the lesson from that, this means the large N  theory will work with this Lagrangian. We've got the core kinetic term and the core interaction with  the non-abelian surgery. And some function of all the other scalar fields, and that varies as you  go through dimensions, but it's irrelevant for the large N expansion. I call these spectators,  and I've rescaled a coupling constant out here to have unit coupling, because it criticizes  there is no coupling constant. It's just replaced by some number. So you just start with this version  of the theory. This comes along for the right. Everything in this f of pi in the fixed dimensions  is contained within this interaction as effective vertices. So life becomes a bit simpler.  The other aspect of the formalism that we work completely in coordinate space, and I want to  focus a lot more on coordinate space and having the past, because Oliver is giving a talk tomorrow on  his graphical functions and the similarities, but differences. And I want to sort of add some  of the ideas to that debate this week. So in coordinate space, the propagators all have a  scaling form with potentially corrections to scaling, which I'll come back to later, but the  key focus is on the leading term. And the dimensionality of the field in the Lagrangian  at critical count, they determine the power of the propagator. And if you look at the dimensional  analysis of the Lagrangian, the Fermion field has dimension, which is d over 2. So I'm setting d  equals 2 mu here just for a shorthand. And the pi field has a propagator, which has got one,  but there's corrections to both. And these are the anomalous pieces. The mu and the one are the  canonical pieces. And eta is the anomalous dimension of the Fermion field as I had before.  And chi-pi is the vertex anomalous dimension of the operator that we have in the theory as well.  So both operators have anomalous dimensions, reflected by chi and eta. And just for the future,  the discombination of exponents is d plus 1 minus chi. And chi and eta will have order 1 of n  expansions, starting at 1 of n, with mu dependent or d dependent coefficients, which also depend on  the group casemers. And the beauty of the facility of method is you just take these propagates and  put them in swingard-eisen equations. And we'll hear more about swingard-eisen equations later.  And you just self-consistently solve for the unknowns, which are the exponents and an  amplitude for combination of a and c, a and c are just x independent objects, just to get a scale.  That's a good number on the organization for the propagators. And that's, quote, all you do.  And these are the leading order diagrams, and next to the leading order diagrams are the two  point functions. Of course, you have to calculate the integrals. But in principle, this is all you  have to look at. And you note that in the diagrams are written down, there's no dressings of the  propagators. And that's because the anonymous dimensions automatically incorporate the  self-energy corrections on each line. So you automatically sum up those diagrams by taking  the generalized anonymous dimension. What you haven't done is got rid of vertex corrections.  So there's two diagrams, one over n squared, the vertex corrections, these two, which are  actually divergent, as you would sort of half expect in quantum field theory. But you can  analytically regularize with the delta. And if you do that, it's possible to calculate them.  We'll have simple poles in delta at this order and double poles in next order.  Now, how do we calculate? We use what's known as conformal integration, or uniqueness. So the  rule for uniqueness or conformal integration is that at each vertex, you have a sum of exponents.  You integrate over the vertex point. So this is not like quantum space where you integrate  over loops. You integrate over the points where the edges join. But the powers of the propagator  joining that point are not one that there would be in perturbation theory. They're different.  They're taken generic alphabet in gamma here. But the rule is this is not possible to integrate  except in a restricted set of cases, the counter-blank number of cases.  And this is the first case where you can integrate this vertex. It's when alpha plus beta plus gamma  is 2m plus 1. So it's d plus 1. If that condition is satisfied, then you can meet the character  at the integral. You get some overall normalization factor where it's just products of gamma function,  which I'm not going to dwell on. These always come along for the right every time I do an  integration. And you replace the vertex where you're integrating by just the product of three  propagators. This is sometimes called the star triangle relation. But it's in coordinate space.  There's a parallel of it in the momentum space, of course. There's a duality with it. If the line,  the propagator opposite the point for you, the opposite the original propagator you integrate over  is dual to it. So it's mu space time dimension of 2 minus the experiment. If the square  amunces are adjusted for the x slash or the y slash, just plus 1. Now, seven aspects I need to point  out. For the ghost number you can with Carol Heisenberg vertex, 2 alpha plus gamma is 2m plus  1 minus chi pi. So unfortunately, the vertex is not quite correct for uniqueness in the large n  setup. However, with the regularization, you can calculate with subtraction methods to be able to  get around that with still using uniqueness to the ways around it. And the second point is there's  various ways of proving this formula. One of which is just do the Feynman integral. Either you have  the Feynman integral associated with it by using the condition and you end up with hyper geometric  function, which simplifies into a geometric function when the condition is satisfied and you get  down to the right-hand side. That's brute force. The other way to do it is by conformal transformation  and I'll mention those later on. And the rule for the conformal transformation is that when any  line joins, so you define an origin for an external point and then take a conformal transformation. So  the rule is that you're clear every line joining the origin by 2 mu minus the sum of the exponents  joining that point, which here would be 2 mu plus 1 in the case of fermionic vertex. It would 2 mu  plus 1 minus alpha minus beta minus gamma, which is not if the condition is satisfied. So then you  just get an integral of a bubble or a chain integral, which is what this is effectively.  And that's the rule. There's no rule for QCD because it's got a gamma matrix inside it.  Now, just applying the formulas, we get these results,  works order and next deleting order. And the reason I'm showing these is several points.  You see the group customers coming in from before, CACF. And also this combination of  side functions coming. I've written this side function so it's got a zeta expansion near two  dimensions. And these are typical of what you get, but the leading on next deleting order is 3  will to get. The corrections of scaling that I had in which had the exponent lambda  can also be incorporated in the formalism. And in the Kinesma line, this is the correlation  scale exponent. And it's slightly more involved because you have to calculate the  two point function scaling function before you start with Kresens. And involves a hard  set of gamma functions incorporated in this notation. But there's a subtlety with the  gross Neville case, which I want to point out to you now. It's to do with this function Q  that comes in mu, I say lambda at leading orders, mu minus one dimension of space time  over two minus one. And if you look at that formula with a specific example of the exponents we've  got, it turns out this is singular. And that singularity means there's a reordering of diagrams  in the Dyson-Swinger equations. So you need higher order diagrams to be able to go to one over n  squared to get the second order correction to the exponent lambda. This means just more  diagrams to calculate. And these are the diagrams where the dot indicates the insertion of the  correction to scaling. So this is a vertex correction on the trulip self-energy, likewise  this one with the insertions in line. This is a vertex correction on the external leg.  And then you get the usual characters, the primitive norm planar. And these other diagrams,  which are, well, norm planar as well, with these various decorations with the insertion.  Now I've illustrated these graphically because those were two gates series automatically go  light by light, which is not something out of Star Wars. It's basically this set here. If I cut  down this line, I get four external wiggly lines, which you could write as photons. And that would  be two goes to two photons scattering. And these are regarded as tougher diagrams. But there's two  bolted together here. There's two bolted together here. And each one's got a group generator at the  vertex. So there's a lot of groups that we associate it with this, which we'll have to take into account  for the calculations. And the typical combination is written here. A and B are external group  indices. And C, D and E are the internals. I've written them in sort of different orders here,  because the way the pie field goes across varies from the left loop to the light loop.  But it's proportional to delta AB. No. In the mid-70s, mid-80s, from Arsene von Littberg and  Lauren calculated the polar better from SecuCity. And they had to take account of the group theory  graphs, graphs for this type of group theory. And they have a very elegant package called color.h,  which allows me to do the group theory for this problem. And it involves rank four fully  symmetric tensor DABCD. The F here means the fundamental. It could be an adjoint, for example.  There's a rank three symmetric tensor, which we all know about in league groups, which is DABC.  And this calculation is absent. But I get the DABC tensor in the problem, which is  going to be products of these two things when there's a shake down on the calculation.  So it's possible to code this in automatically and run through the calculation. And this is the  second order correction, which is significantly more complicated than the original one for the  original Ising model, versus the case from the mid-90s, which was the only thing I did separately.  But you see the rank four customer is coming in here at second order large end. And there's  seven new functions here that come in this five comes in. There's a lot of decorations. And  the rest of it's on the next sheet, unfortunately, the formula got a bit big. And there's this data  here, which are the second, I said the derivative of the side function. And these incorporate  status. So if you were to expand these objects out an epsilon expansion, they are four dimensions,  they would correspond to renomous this group functions near four dimensions or equally two  dimensions. If you did it near two dimensions, and this would show you that two orders in large end,  the corresponding renomous this group function to all orders at this order in large end would only  have a epsilon and status. It's not a proof. It's just an observation because obviously there's going  to be non status further down the line. At higher order large end higher order per device. So it's  somewhere to get them. But the key thing here is now I've got this general formula I can specify  groups very simply and take limits. I'll talk about that towards the end. Now, the other aspect  of the facility of work from his school, honking in and Pismac as well, was that he pushed the  calculations to one of the grand cubes. This is a can stress how much this is a well ahead of its time.  This was in the early 80s. The form was set in stone earlier on by Poliakoff in 1970 on  Paraisie a few years later, where they were doing what's now known as the conformal birch  strap method. In essence, the way I'm using what's now known as the large end conformal birch  strap method, which is what I've had to rename it after conformal birch strap people come in  and corner that market. But if you remember before, I didn't have dressings on the two-point  functions, but I had vertex corrections. This method goes around that by getting ready  ready vertex corrections in the self consistent serendice and formalism. So these are the  diagrams that contribute to the calculation of eight of three. You calculate the vertex functions  and through a formalism that I'm not going to go through. You gain the eight of three.  These are all primitive diagrams because there's no vertex corrections, but the dot  that's on each vertex is not accidental. It's deliberate. The dot represents what's called  a Poliakoff conformal triangle. Now, if you remember that the vertex of the theory is set  in such a way that it's not unique, but the Poliakoff conformal triangle is in some sense  mimicking not the absence of uniqueness but allows you to calculate these diagrams exactly.  So let me explain what a conformal triangle is. So if you're most an ordinary vertex with arbitrary  exponents meeting at a point, Poliakoff replaced that point by a triangle with arbitrary  a1, a2, a3 exponents internally. Obviously, there's some proportionality function here.  Would you choose these internal exponents which are completely unseen and not relevant to the  field for each vertex by making each vertex of the triangle unique? So these are the conditions here.  So if the triangles are all unique, then there's lots of tricks you can use to evaluate the  integral. And this is the conformal transformation that you use. You set x goes to one over x,  but the perfect is x me equals x me over x squared. Or if you take what would happen in the product of  in a propagator would be x slash minus x, y slash goes to x slash minus y slash y  slash over x squared, y squared minus x. But if you build a lot of these together with a y  slash minus z slice, there'll be a y slice here and another y slice for the next propagator.  So one of the y slices, so the pair of y slices can't stop at y squared. So you retain the same  string of gamma matrices around a diagram after conformal transformation as the original diagram.  Clearly because there's no gamma matrix in the vertex and that allows a lot of tricks to be used.  So with these vertex, we're clear as by a conformal triangle you can calculate the diagrams.  But leading orders, as David reminds us many occasions, the hardest diagrams sometimes are  calculated by the previous loop order to the higher order in the expansion parameter.  And that's the case for the leading diagram, which is just the triangle. So this is the full setup for  the conformal triangle definition of the original one loop triangle diagram. So each vertex is now  replaced by its conformal triangle. These internal ones are chosen purely because of the alpha  here, here and here and the external one that haven't drawn to the top. There's a regularization  coming from the external lines because the equations as I said before are divergent,  so you have to regularize each diagram. And there's two regularizations, the delta here and the delta  prime here. And I've arbitrarily put the origin at the top. So how do we evaluate this? Well,  every vertex now is unique. If I take a conformal transformation, I replace every line joining the  origin by the 2 mu plus 1 minus the sum of the exponents at a vertex. If you do that, you get this.  It's the same diagram, but the zero, this line's removed because the sum of the exponents at that  line are 2 mu plus 1. Now, that means the only points to integrate are the ones on the hexagon  that goes around here, x and y are external. So this immediately gets you down to a two-point  function. And this is an integration point, and that's an integration point. So I'm going to  immediately integrate these two diagrams. And I'm not carrying factors through this in the slides,  but you imagine the state of paper per se where you carry the factors. And you get this diagram,  which is two-point function, which you have four integrations on, and a lot of parameters. But this  is where you have to think. And the facility of trick was to write this as a diagram that looks  the same without the top line removed, but you multiply by a function which represents the discrepancy  or the compensation for the absence of it. This can be evaluated because it's just, again, simple  chain integrals. And you get a result, which is this. Sorry, this which is a result, but the  compensation function can be written in this form. Or basically, it's because it's a function of this  combination of parameters. So the orders were calculated, and then you can write down the  expansion. It's in an exponential because it's easier formulation because we always write a  gamma function x-logs, and it's got an easier expansion. And then the idea is you calculate  the coefficient x1 to x4. Well, how do you do that? Well, the very simple thing is you took the  original, took the original point here to be at the top for the origin. There's no very priori  reason why I should pick that point. If you pick this point here as the origin, I'll do the conformal  chance to make these two lines disappear. And this collapses the other way. But if you look at that,  the common estimate is going to be different for the behavior of the compensation, which is going to  be delta prime on delta pi. Likewise, if you pick this point here, the collapse will be here and here.  And so the discrepancy function will just depend on delta pi. So we've got three different ways of  evaluating this diagram, but they've all got to be equivalent. They'll all have a similar compensation  function like this, but a different expansion problem because the argument would be delta  prime minus delta tilde, as I said, and fc would be delta tilde. And so three different ways,  plus all the factors that are neglected right down in the slides, brought together,  they should all be the same, and that's sufficient to be able to solve this to two orders.  That allows people to write down sufficient terms and expanse of leaving on a diagram  to be able to evaluate for age of three. And this is just an example. I haven't written down all the  terms, but you see that you get the corrections involving obviously the problems themselves,  written out properly, but you get these combination of psi functions, which is where the vector  we come from in the results of Schumann's so far. The higher order diagrams come in in the  corrections, these are the two formulas to find the vertex function. The first one here,  I represent the fact all the diagrams have to sum up to one. It's a very simple self-consistency  equation, and that fixes one of the internal parameters. The second equation is in effect  equivalent to the two-point function, calculated as we did earlier, but summed on the vertex functions.  It's easy to calculate this. The higher order diagrams have been evaluated years ago,  it's just a question of decorating with group theory, but the leading order diagram is a bit  harder to calculate, which is what I've developed on it. So in all of his formulas,  he sends off a point to an infinity for a three-point function, and that in some sense is  similar to what I do with sending the conformal transformation in. I just make one point irrelevant  and you're left with two-point functions. It's similar, it's probably not connected, and it's  just I want to highlight that this morning in anticipation of what all of us are going to talk.  I'm sorry, all of us put pressure on you. So what do you get? Well, you get a result which is not  the similar to lambda 2 in the sense, structurally it's got the Casting-Age structures. Obviously,  this is now tags for the light-by-light diagrams come in in the calculation,  which is kind of nice because you can sort of look at the structure of these diagrams  through perturbation theory if you just focus on the expanse of these objects in parts of epsilon.  The other feature that comes in is that there's another new function, which I call psi.  This is a known function from early times when facilities work on this,  and it appears a non-Caroly symmetric theories at large animal around cube. I've never quite  understand the Carole 70 aspect of this, but it's come up in two calculations, one of which has been  written up separately, one of which I've known from the early 90s, mid-90s from the rest of  the model in super symmetry. But it's a hardly complicated function to define. If you take this  two-loop integral with decorations, well, obviously, epsilon depends, but d demands  mu minus 1 and it regularizes the central line. It's the derivative of this integral,  it's the derivative of the logarithm of this integral with delta equals naught, that defines psi.  Now, we know if we've got a two-loop function, a two-loop two-point function,  if we have epsilon ordered from one on the lines that join at external points, we can expand that  in zetas. If there's any decoration of order epsilon on the center line, that's a problem,  and that brings in the multiple zetas, basically, which is what Derek and David have been  looking at in the mid-90s. This is the connection with where David and Derek and I got working  together in these problems. In our paper, David showed that this integral can be unlifted expressed  as a hypergeometric function in d dimensions. But the complication is because the  difference is, with respect to the delta parameter here, you have to differentiate the arguments,  the parametric arguments of the hypergeometric function, to really define the function,  and that makes it harder to write it down as a closed form function of 1, as we know.  However, what we did extract in that paper was the epsilon expansion of psi near two  and four dimensions, and it has zeta-3, or zeta-3, 5, it has u6, 2, and no music.  But we now know it's all orders in epsilon, and that's sufficient. David and Derek wrote down an  integral in phi-4 theory in their early work with these multiple zetas structures, but it was never  shown whether it was realized in the free theory. Oliver and others have now found those in the  free theory in phi-4. This was another example, an early example of where the zeta-5, 3 can appear  in the free theory, because we expand out in the parts of epsilon to find it in a certain order.  It's probably the earliest order, but it comes in. It is possible to evaluate the integral exactly in  three dimensions. It involves log 2s and zeta-3 over pi squared, and that's purely because in  three dimensions you can exploit three dimensions on the uniqueness of the diagram exactly,  right of time. I haven't talked about checks on the results, but those of us who look at  non-abillion-gay theory, if we've got a group generally inside interaction, for say, young  girls, we immediately get QED from those calculations, by just notionally taking T to be 1,  just taking probably really mathematically unsigned limit of setting the generator to 1.  Well, what that really means is you replace the cacomers by the corresponding object  in the center limit. So the abillion limit would be taken, Cf to be 1, Tf to be 1,  Tf to be 2, of Tatb. Obviously, the group cast me over a couple of one, but the abillion limit has  to have FA, BC to be 0, so that CA goes to 0. I reproduce everything that was known before  in the gross removal model, by taking that limit from the results I've presented today.  The other results that have been known is the SU2 mot-cont's interditing phase structure,  which is slightly different than SU2. So it's still not a million, so it's three quarters to half.  CI goes to two, but the group cast is five to 64. That reputages what I had before.  And the more recent one which we've done is SU3, which has got a slightly different structure.  If you go and look it up on some table, the group cast was 20 over three.  I'm taking those elements and reputating what I did with my colleagues.  But also all these results agree with all that's known in perturbation theory.  For most of these models, it's either two-dimensional or four-dimensional, but in this case,  the fractionality one, we've only got four-dimensional results. We've calculated  out three looks in that and got agreement with the large N, so they're both tally off,  which is reassuring. Now, once you've got a generalized result, you can just look at  different group structures. And so I've written down the three-dimensional results here just for  the adjoint quotes for a bit of fun. So put the firm into the firm fundamental. The group  cast mirrors obviously change, but the reason I've written this down is it shows you the structure  more succinctly in three dimensions for the light by light diagrams having the facts.  And it identifies this term as a light by light contribution over and above this contribution.  But for Aida, for Aida, you go slightly differently. I kept this combination here because this tends  to pop up a lot for the adjoint case. It doesn't quite do a couple here, but if you're interested,  you put in the numerical values and see what it looks like. But that's the kind of more general  aspects that one can do with this form. Since doing this, I've been looking at other university  alty classes with a different underlying interaction. I'm playing around with that and  applying the same tools to sort of look at all the problems that were interactable. It seems to be  a really robust way of doing things, taking a general lead group approach. So just let me conclude.  I've looked at a more general type of critical alty class, university alty class for this  fundamentalizing growth type interaction. It's opened up a whole idea now of looking at  more general structures. I can't emphasize enough on what the color park is in form. It's been  makes life easy for doing the group theory. You don't have to sit there with a pen and piece of  every minute group generally. It is around the diagram. It's easier to identify structures inside  results and in principle, all orders are part of basically by integrating through the epsilon  expanse. So I'll finish there. Thanks everybody. Thank you, John. Thanks for your nice talk.  Please feel free to unmute yourselves and raise your hand or speak just up if you can.  I have an immediate question to go just one slide back. The three-dimensional thing you showed.  So since you have this result for an arbitrary Lie algebra, is there any special cancellations  that you see for, I mean for example here, I could see this bracket that you mentioned  in the third line from the bottom. So you can imagine that there's a special situation with  these vanishes and you drop the zeta three term. Things like that. I did look at that off the top  of my head. I think it doesn't happen. Okay. This is actually a quartic polynomial in  the number of colors. It has to depend on if it's going to happen, it's going to happen for low  values of colors and I feel it doesn't. I should say I haven't looked at the exceptional groups.  Yeah, so I was just wondering since you have all this data, I'm just curious if there is  something funny happening. Since you've raised that, I think of what I'll do is I'll look at  it after, come across and I'll put it in the right up. I'll try to make some room to make  some room to come in the right up. Just yeah. Yeah, thank you. We have a question from David.  Yes, you've reminded me that in 1996 we did epsilon expansions about two and four dimensions  to get towards three and there we encountered multiple zeta values. But just out of mathematical  curiosity, I wanted to know what the epsilon expansion was about three dimensions. And if you  remember, Toliakotikov and I actually did that because I knew that you would get alternating  Euler sums, the same things you get in the magnetic moment of the electron. Is there any  use of those epsilon expansions that we did about three dimensions?  There could be, David, because I didn't put the slide up. I had it in an earlier version of the  talk. I think you may have seen the slide before. What you can do now is you can plot the behavior  of the exponents in d dimensions from two to four. This is important because not only can you do that  in large n now, we're using a paddy approximate across the dimension. You also do it in protobase  theory. You know in protobase theory in two dimensions, up to say four loops, and the four  dimensions down from four dimensions to four loops, you can do what's known as a matched  paddy approximate where you use the paddy information near two and near four to write it down and  interpolate the behavior of the exponent across the dimensions. You can plot that. You also,  in the function renormalization group, they can plot the behavior of the exponents across d  dimensions. They all have a similar sort of, it's like a hump, like a bell, but it's offset sometimes  about three dimensions. What we could do with the epsilon span around three is expand in the  neighborhood of three to see how well the maximum matches the shape of the other ones.  Now it's not something, until you mention it, it never occurred to me.  I can't remember what order in epsilon you went to. It's obviously going to be breakdown,  but what it could do, it could influence the perturbative expansion that people do from  two to four with the epsilon expansion from a high order perturbation to try and do some sort of  global fit based on the limited data we know from perturbation in large M.  It's an exercise that could be worth doing, but it's not something I might have the tools do  immediately. Thank you. I see some questions at the moment, so let's thank John again.  Here.
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