Twistor string & the integrability of N=4 Sym

0:00 Let me begin by motivating why it is reasonable to ask this question. There are, roughly speaking, two main research areas nowadays. One of them is the ABS-CFP correspondence and the other one is the twistor string. And the maximal expression that we can miss here in four dimensions plays a crucial role in both of them. It is the CFP part of the ADS-CFP correspondence, and it is also the main character of the Fisker string. It is therefore particularly natural to us whether it is true or not. In fact, we heard yesterday from Witten that he constructed the twistor string in an attempt to better understand the EACST correspondence. Now, by just staring at this diagram, one can learn basically nothing. However, one might go in a little bit more detail and attempt to match things that we know on the two sides. From the ADCFT correspondence as we've heard in the previous talk, there is an important piece of information which emerged, which is integrability. Integrability means various things. From the Geishari standpoint, it means that there is a spin chain which describes the calculation of anomalous dimensions. From the worksheet perspective of spins in ABS, it means that there exists an infinite number of conserved quantities. These quantities are conserved both classically, as Dana-Volginstein Lai showed, and quantum mechanically, as Nathan Berkovich showed quite recently. So, integrability. The question is really, what does it mean here, and I will not answer it, because I can't, the purpose of my talk will be to collect pieces of evidence which exist on, mostly on this side, and ask the question what do they mean on this other side, so the plan, first I will review the derivation of spin chains from Feynman diagrams.

2:30 Now, in this, the first thing that I will do is this, the format in which the spin chains emerge more clearly cannot, unfortunately, be translated easily to twistor space, the reason being that it requires too much offshore information, which is not available from twistor space. And after we've seen how these guys emerge, I will give you another method. Which uses entirely on-share information and which, at least in principle, can be translated into research in a rather straightforward way. And this will lead to somewhat sharper questions than the ones asked in the title. After doing this, I will not be able to read this and I will present a speculation, which, as with all speculations, might seem to be wrong. Well, anyway, I think it's an interesting population. In the gravity, spin chains in the context of the calculation of anomalous dimensions, spin chains emerge reasonably naturally. As we've heard in the previous talk, the gauge-invariant operators, a single-phase gauge-invariant operator, say something like trace-by-one, that is, by m. Where 5.1 and 5.n are some arbitrary fields and derivatives of fields appearing in any of those 4.n mils or perhaps even in another theory. So a single trace cage invariant operator can be putting up one-to-one correspondence with a purely one-dimensional lattice by identifying the first object in the trace on the first side, the second on the second side, and so on.

5:00 The periodicity of the trace becomes the periodicity of the lattice, therefore things are natural. If we want to look at collections of operators, we will, it means that inside the single trace operator, we can put, position various fields. This translates into the spin chain being the opportunity of putting various things at the size of the chain, at the size of this lattice. This means that it is like building a spin with the number of states equal to the number of fields which can appear at each building in the space. Therefore, we have a spin chain. Now, this is, in fact, only a map between the operators and the states of the chain. Now, what is the Hamiltonian of the chain? A natural equation in gauge theory is the calculation is that the randomization group evolution, and the randomization group evolution looks very much like one, like a , like you and me. The quantities gamma and d which govern the randomization group evolution are called anomalous dimensions. Now, from a chain perspective, one is interested really in a Schrodinger-type equation, d by dt of some state, because the Hamiltonian happened in this state. So, the goal is to identify the anomalous dimension matrix of operators with the Hamiltonian of the chain. So, all we have to do is compute anomalous dimensions. This is a well-oiled machinery in quantum field theory. They can be computed in fact in various ways. The way which leads most naturally to the appearance of the spin chain is the calculation from the z-factors, from the randomization factors.

7:30 One has an operator A denormalized, which was ZAB, which depends on the randomization scale, times the Bayer operator. There are very few derivatives of the Z-factor. Now, to compute these Z-factors, one computes Feynman diagrams. And the Feynman diagrams look like these. This is our alterator. It has a bunch of fields sticking out. And one computes essentially all possible Feynman diagrams with one insertion of such an alterator. And the other vertices coming from the Lagrangian of the point of view theory. The two Feynman diagrams contributing to the generalization of operators of this kind, phase yi1 and phase yin, where yi is any one of the field y1 or y2. Final diagrams let me also tell you what the results are. It turns out that gamma is in fact a sum, which looks like gamma i i plus 1, and gamma i i plus 1 is the identity of the other plus, after some relative numerical coefficients, the permutation of the other acting on the side pi and i plus 1.

10:00 One can make things more complicated. One can have rather incorrectness, go to higher loops. The structure remains more or less the same. At L-loops, one has a sum of gamma i up to gamma i plus L. The interactions will involve L adjacent sites. We compute the Z-factors if we can. We take the logarithmic derivative and we find the anomalous dimension matrix and as the string-chain homodominant. Now, this discussion applies to a large extent to any quantum theory. Scalar filtering can compute these things and find that there is a spin chain which governs the RG flow of operators. The thing which is special about the spin chains appearing in the context of the Nicos-Florian muons is that they are integrable. How to define integrability is a little unclear. For quantum theories, one might be tempted to say that one should be able to find the change of variables in which the theory becomes free. In this context, the definition I would take is that there are as many integrals of motion as eigenvalues of the Hamiltonian. Therefore, it should be possible to determine the eigenvalues by entirely algebraic methods. You must have meant as many integrals as to be free. Yes, definitely. Now, let me list very briefly the results which are known for any post-war or anything else. At one look it is known that the twist theory is wrong. At one look we know these matrices, as we've seen in the previous talks,

12:30 in the previous talk, they are given by the projector onto... The representation labeled by an integer j in the product of two singleton representations after a sitting of the size i and i plus 1. In the operators, a smaller set of fields than the set of fields of n plus 4, one knows more. In the operators of this kind, one actually has derived or conjectured results up to five loops. These are not computed by Feynman diagrams, but by purely algebraic methods, together with input from string theory. The idea is to use Feynman diagram information to make an answer for the types of terms which can appear in gamma ij, which is occasionally also called hij. And then use the type of algebra should be a quantum, should be a symmetry of the quantum each day and find the coefficient which I left free by the type of reaction we can compute final diagram as a reaction. Now, the variability again at one move means that one can diagonalize. Come on. And this is typically done by the use of some form of data answers, which I will not describe. At higher loops, the notion of integrability should be taken in a perturbative sense. At higher and finite number of loops, it should be taken in a perturbative sense. If gamma is a polynomial in the coupling constant, then we cannot trust its eigenvalues beyond the higher power of the coupling constant that we have to begin with. Now, the old frame is appearing up there, which formed the so-called H-U-2 sector, because there are groups that have been formed due to the major representation of H-U-2.

15:00 There exists a completion of the dual Hamiltonian, which turns out to be integrable to all orders in its, what in that case plays a role, the Huffman constant. It turns out that that integral of completion is not Hamiltonian appearing from an any-cause, from the equation perspective. However, that completion led to a conjecture. or the anomalous dimensions of those kinds of alternatives to all of these. This conjunction still remains to be proven, and also an interesting open question is how does it extend to the other sectors of the theory, or how does it extend to the blue theory, and it would certainly be an interesting and important result figuring out how this is done. Now, this is really all I was planning to say about how spin chains appear in the calculation of anomaly dimensions. Now, what about... I would be happy to go into details later on if anybody is interested. Now, the question is, what does this have to do with the tweezers? The conclusion is that it has very little to do with the twistor theory. The reason is that both in the formulation in terms of the string theory of the V model that Witten proposed as well as in the formulation in terms of an open string theory that Berkowitz proposed, one really deals with on-shell states and these formulations are geared towards computing scanning quantities of some sort. However... Well, scaling up, these are Feynman diagrams. They rely on computations of Feynman diagrams. However, their algebras are on shell, whereas here, their algebras are on shell because this vertex here, the insertion of algebras, makes things very bad. Those diagrams aren't actually part of, don't really come from any classical diagram here. So, there seems to be a problem.

17:30 A potential solution comes from the string theory of the word theory that Nathan proposed. The idea is that in that string theory there are a lot, essentially an infinite number of ordinary fields, which, roughly speaking, can have the potential of keeping the whole set of n equals 4 fields off-shell, Approach or calculation of such diagrams from that perspective. Unfortunately, the status of computations of glucose in that framework is not very clear at the moment, and there is the problem of contamination with components through gravity states. Therefore, I will not dwell on that for too long. Instead, I will go to the next possible solution, which is distressing the calculation of anomalous dimensions purely in terms of the calculation of function of skeleton gas. This is not something new. This has been largely known for the past 30 years or so, and it goes by the name of... The method involves computing the so-called Altari-Liparesi kernel, and then the anomalous dimensions of certain kinds of operators are related to the Manning transforms of these Altari-Liparesi kernels. Before I will describe this a little bit, I will not go into the technical details because I don't exactly know them, I will try only to give you the physical information which is necessary for the purpose of this talk, and perhaps it is your appetite to read about this. Before going into that, let me mention that this is quite a powerful method. The two-loop calculation of the alteric-barotic kernel in QCD has been completed by Mohr, Vogt, and Dermasso. And out of that, Voigtvogt, Vipatov, and Delejani extracted the three-loop anomalous dimensions of these two operators, N-u-s-v-o-r-i-a-m-u-s.

20:00 Computing this from an ordinary time-on-diamond perspective is particularly difficult. Not that the calculation of the Altairi-Poizzi kernel is very easy, but anyway, it suits our purpose of expressing everything in terms of Feynman diet, in terms of what is happening. So, there is no simple, there is no gentle way of saying this. The Altairi-Poizzi kernel is the integral kernel which governs the randomization and group revolution of part and distribution. Now, partner distribution functions are the functions which describe how partners behave in hand runs. There are many reasons to study these functions, at least in QCD. For starters, LHC will come along, and knowing what happens among scattered protons requires knowing the distribution functions of the protons. The most important feature of these functions is they are universal in the sense that they can be computed from some process where computations are easy and then they can be used in your other favorite process where they can't actually be computed directly. The only thing that one has to take into account is their idea evolution. Which is governed, as I was saying, by the Altairi-Farisi kernels through the DGLAT equation, which looks like this. And this is the famous Altairi-Farisi kernel that we are interested in.

22:30 We'll record in this. Besides being part of this equation, this guy has a physical meaning, which is in fact implied by this equation, if we write it in the following way. Let me ignore the indices for the moment. The idea will be clear just by considering a single kind of partners. So one can rewrite this in a simple way as a double integral, the same function is given there, some delta functions, and then some more delta functions. And the integrals are over y and z and u and v. Now, if you stare at this equation, it is clear what this bracket means. It means that this function represents the probability for a Python with a momentum fraction z to exist in some other Python and the emission of such an object. We'll change this function from its original form to this variation. Now, there are two ways of thinking about this. One of them is due to Groswilczek and Politzer, and it is in terms of operators. In that case, the evolution is governed by the anomaly dimensions of operators and training. The calculations in that case are very similar to the ones that I was outlining in the beginning, so that perspective is not very useful for us.

25:00 The other way of thinking about it is in terms of this equation, which is due to, well, in particular, this kernel, which is due to Autoregion Parisi, and the relation between the two implies a relation between PAB and the anomalous dimensions of operators appearing in the description of L. So, this is the equation that I am trying to suggest that can be used instead of this thing to identify the speech in Hamiltonian with quantities in the twistor space. Now, how does one compute this? There are, roughly speaking, two methods. And they both rely on the fact, well, the fact that there are two methods rely on the fact that scathing numbers of padrons factorize, and they factorize in the following way. Here are two padrons, and here is a part one exchanged between the two. Actually, they're called as a convolution of L, which describes the low-energy physics of atoms and the distribution function. Convolution of this partial distribution function with some high-energy process, which is in principle computable from perturbative QCD. This factorization happens on the scale of μf, which is the same scale which appears here.

27:30 The fact that there are two methods of computing these guys is that μf appears here as an ultraviolet couple, whereas on this other side it appears as an infrared couple. The calculations treating this guy as an ultraviolet couple are again in terms of anomalous dimensions, and we don't want to do that. The one in terms of an infrared problem is what I'm planning to emphasize. The idea is that from a high-energy perspective, when one confuses an altitude in a preserve at a specific point, also sub-divergences, those sub-divergences are normalized away in the usual way. Some infrared divergences are fine by taking care of the incoming and outgoing state, and some other divergences are not fine by themselves, but they have to be absorbed into the phyton distribution functions. The idea is, because this whole process has to be well defined, which means that the divergences which are absorbed do cancel the divergences that would appear here. When one computes this as a separate thing, therefore they do govern the renormalization of the revolution. Now, what do we expect to find when we compute these guys? So what are splitting amplitudes? Or rather, splitting amplitudes are the amplitudes which govern the collinear behavior of scattered amplitudes in ordinary QCD or in this case, you know, in the case that we are aiming for any phosphorylamines. They have a universal form and this is very good because we expect that this is quite universal.

30:00 There are no more names to say what is the amplitude of finding a partner with momentum k1 inside a partner with momentum k2. As I was mentioning earlier, this kernel roughly speaking says what is the probability of finding a partner inside another partner. Therefore, one expects a relation on this side. This expectation is, roughly speaking, correct. At some leading order, one finds corrections to it, but again, morally speaking, the integration is correct. As far as I know, now, let's go back to how one actually computes this. Computing these guys and expressing it in terms of speeding integers, as I was mentioning earlier, can be done using There are presumably preservative calculations in this kind of energy process. One can, in principle, just compute the amplitude and extract the divergence at the end. This is generally hard, though with the recent advances in loop calculations using twisted string-inspired methods, this might actually be doable. The idea being to simplify our life as much as possible to start with and identify, ultimately, what diagrams and what divergences one should compute, and compute just those, and not worry about the rest. Now, I'm not about to go into the details. Let me just thread down one for you. It has an illustration. I would say a decay rate, a renormalized decay rate, would look like this. I will say in a second what the letters mean.

32:30 H0, H1, and higher H's represent the bare decay rate. So one starts with this, one normalizes away everything one may need to take care of problems and the entire divergence left in here must be cancelled by This guy, which determines, you know, and he also determines in principle the higher p's by looking at higher powers of alpha. Now, under all this talk, let me put up a formula which expresses this. p is, after some coefficients which I will not worry about, a delta function with some coefficients, The absolute value of the field of splitting energy, which is how do we try to extract the two-part scale of S-22. And then there are some further things coming from the natural regularization. So this is all taken out of the paper, so it will also be there. Expanding this in the powers of epsilon and keeping the leading term reproduces the results of algebraic and parity.

35:00 So anyway, this is a formula of the kind that we want. We have this least which we needed in order to compute the anomalous dimensions. And we have it expressed in terms of an amplitude. Everything here, all the particles appearing here are on shell, so this would be some amplitude, and when K1 becomes part of the K2, then this amplitude factorizes into something with 0 lux, and this is the speed limit. So here everything is on shell. This in principle has an interpretation from twistor space. And one might attempt at least a base interpretation for integrability using such a formula. Now, all the words I said apply to QCD. In N equals 4, the operators which are known to be readable using such a method are the following. Some number of derivatives and another view on this term. Operator containing fermions, which are along the block to the normal radius e to the m long down. This is very schematic. And operators containing scalars, which look pretty much the same. The derivatives here should be symmetrized in their vector indices. There are two more kinds of operators which I won't bother with. So, in this context, one has a kernel with a 3D matrix, a 3x3 matrix, because there are two kinds of operators, and if one takes the results for the ones that are for the leading, if one takes the leading terms of the species and does the milling transfer, one finds the Hamiltonian that Luis Dolan showed us in the previous talk, the thing expressing terms of side functions.

37:30 The idea is that because this is supposed to give the leading term, one can just take this result and algebraically reconstruct, algebraically add the further fields, so one can essentially insert further fields inside these traces without paying any penalty as far as the complexity of the calculations. Let me sharpen a little bit this question. We have here an equation, so let me write out a minute, I'm committed, so we have an equation which relates the scattering entities with the grammar information. Or, from the original discussion with Vespinchen and Hamiltonian, we know that this has certain integrability properties. The question is, what does this relation imply for the Witten space structure of this amplitude? Perhaps to a T-level, this is a little too simple. One can, in principle, go beyond T-level. And since we expect that Young-Woods is integrable beyond one loop, this equation should imply something for the twistor space structure of Young-Woods appearing here. The question is why? Now, I was saying a second ago that the reading order one can add more fields in here without paying any penalty. The question is what happens at higher loops?

40:00 At higher loops, one can all do this, and one can separately do, say, operators with three kinds of fields in here, which are called higher-loop stroke, and that takes at least three operators. The question is, this time for the QCD people, is it possible to use a similar formula as for at least three operators, and if so, how does it work? The next question is what does this mean at the end of the research phase. The other different question is the following. Even though I didn't emphasize it when I described briefly the distinction between the components, Higher-look Hamiltonians involved in Penrose do not have manifest supersymmetry. The reason is, in fact, easy to see. The calculations are done in an opponent formula. In that formula, supersymmetry generators receive quantum corrections. Therefore, they will not be They will act in a somewhat nonlinear way on the fields, on the original fields of the theory, therefore the original supersymmetry cannot look at any quantum level like the original supersymmetry. This leads to specific components which do not have unified supergroup symmetry. On the other hand, if we believe that such a migration actually holds at higher loops, And we know the fact that the twistor space is one way of keeping manifest the supergon-formal symmetry. The question is, is it possible that such a formulation will produce for us better expressions for speech in Hamiltonians, expressions which will have manifest supergon-formal symmetry. Well, it doesn't matter if we don't know and know. One may still hope that, again, the new technologies for computing new computers will at least simplify our life in computing anomalous dimensions by providing for us better, easier, and simpler expressions for these computing amplitudes.

42:30 And in the remaining five minutes, I will describe the speculation, and the speculation involves the Yangian symmetry and the associated Katmody symmetry that we've heard about in the previous talk. In the context of one of the dimensions is in this way. In the context of ABS-CFT, there is another way, and that is by doing a conformal transformation and map R4 is 3 plus R, where this R is the time flown, is the RG time, is the RG time in here. On the side of the map, one maps anomalous dimensions for energies. So, another way of thinking of anomalous dimensions is in terms of energies of states. Now, for each state one can construct a solution of the equations of motion. In principle, one might attempt to construct a solution of the quantum equation of motion. Well, we are not quite there yet, so we will content ourselves with the classical equations of functions. So, given that there is a Casamutti symmetry on this side, one would expect a perhaps complicated Casamutti symmetry on this other side.

45:00 Now, it is not immediately clear how this works, but there is some good news. Not too long ago, Meisinger showed in the context of self-dual supersymmetric chemistry that it is possible to construct such a cosmology symmetry by starting from the superspace constraints of supersymmetric cellular units. Now, he also suggested that such a cosmology symmetry exists for Fourier units. And there are earlier results in this direction. A paper about 20 years ago by Defshank constructed such symmetries by starting again from the special constraints. I was planning to show you how this works, but, okay, I won't. Now, there are certain caveats that one might present against such a relation. The most obvious one is that, though I didn't mention, all the work that has been done on this Well, for the sake of the argument, let us assume for a second that the math with 3 plus r is not for the distance of these entries, we won't be able to just map it with conformal consternation, we'll prefer the conformal consternation than this math, but perhaps more serious caveat is that in their construction there is no mention anywhere of the fact that A possible way around this is that since all these guys are classical, at a classical level, all Feynman diagrams are Feynman, and solving equations of motion is like summing up Feynman diagrams with appropriate stasis.

47:30 Anyway, the speculation would be that the Cosmode symmetry appearing on this side, which is a symmetry of the supercase constants and therefore a symmetry of the classical equations of motion of the universe for e and s, is directly related to the Cosmode symmetry appearing in the calculation of the anomalous dimensions. Cosmode symmetry as we've heard about in the previous lecture. Assuming that this calculation turns out to be right, there are at least two questions one can ask. There are probably more, but I can't think of them right now. The question is how do we see this from a physical perspective and introduce the how-do-you-have-on-service with the gainer limit that one should take, which doesn't seem to appear in the calculators of these geometry generators. And last but not least, the question would be how does this help us with understanding strings in APS? Well, I guess I'm done with asking questions, so let me close by hoping that I managed to convince you that asking the question in the Bible is interesting, and that if the answers to the digital questions that I put up here is the most negative possible one, the twistor theory has the potential of Useful information for the calculation of homomorph dimensions of operatives in English and Spanish. Christian?

50:00 I'm not sure. If not doctor, I think Roger would be. The new development in clisters is that it has the wide application all the way from clister springs, as we've heard, to collider physics. So this will be a pedagogical talk where I'll explain about scattering amplitudes and particle physics and how twistors can help. So first a brief overview of particle physics for the mathematicians. If you're a particle physicist, you can capture that. Then I'll explain a little bit about particle colliders and scattering amplitudes, finding diagrams. The status of the standard model of scattering amplitude calculation, what we can calculate, basically what the frontiers of calculation are. And this has to do with twistor space, with twistors because of the recent understanding gauge theory amplitudes are much simpler than we expected. So I'll explain a little bit about the helicity which we've heard about and some basics of twistor space amplitudes within QCD amplitudes. What we learned from twistors and then finish up by talking about applications to electroweak theory. We have the quarks, up, down, charge, strength, top, bottom. So this is the nuclei of both protons and neutrons. We have the electrons, neutrinos, which is in the news.

52:30 And then here the electron muon and tau. The general structure is that magnetism and electricity are united in QED, quantum electrodynamics, which is then united with Fermi's weak force into the electroweak theory. And then another branch is quantum thermodynamics. We're sitting right around here in our experiment and the question is what's beyond what's beyond it. So the general structure is the theories behind the standard model of particle physics, our gauge theory, the quantum thermodynamics is based on the colour group Fe3, The blue on the fourth category that I mentioned, the fourth category with colored colors, the left-armed zone, and one of the characteristics, the key characteristics of this theory, the Nobel Prize in the report to Gross-Wilting and Pollitzer for asymptotic freedom, the statement that the forces become weak at high energies, and that allowed us to do perturbation theory, to do Feynman diagrams, in a meaningful way, even in a strongly quadratic theory, because of the... In the electroweak theory of the radioactive decay, this is described by mass-evector bosons.

55:00 In the case of mass-evector bosons, it's actually not completely obvious how you're supposed I'll explain how you can apply these ideas indirectly by looking at the case of electroweak theory coupled with thermodynamics, which is just a little bit about the key questions in the standard model. When I showed you a few transparency facts of particles in the standard model, I left one out. The question of the Higgs boson is something very important that we don't understand in the standard model. The question of spontaneous symmetry breaking is a key issue. There's a heat potential in the standard model of this double well, and the symmetry grades become more and more important in the minimum energy here, which is an asymmetric point. But there's no understanding in the standard model of what the origins of this potential is, which is put in by hand. And this is a key issue for the colliders, to explore the electroweak symmetry grading. I experiment to find the Higgs boson, find its properties. There are many other questions, but the standard question is the hierarchy of scales. You can look at the particles in the standard model. There's the electron, the electroweak scale, there's a factor 10 after electroweak. Then if you start thinking beyond the standard model, the place for grand unifications, that's where the electroweak and the...

57:30 Functions from the chromodynamics is unified into the 15D, the question of gravity, where do the scales come from. Clearly we need to get more knowledge about what's beyond the standard model before we can really pin down an answer here. There's many ideas for what's beyond the standard model. There are probably as many scenarios as there are particles. There's probably many more scenarios beyond the standard model. And accelerators will tell us very directly what's going on at 1CBD. Certainly it will shed light on, the accelerator will shed light on these very important questions of particle physics. These are important questions that are very difficult to commend this expenditure of resources. This is a certain site. These are the large hadron colliders under construction right now. So in that circle that we see is just the line that's in the middle? Yeah, it's just the line that's in the middle. It's all underground. It's all underground. This is so you can actually see something. This here is the photograph where someone drew something on it. Anyway, the scale, to get an idea of the scale that's involved, here is the airport. That's what you need when you need an airport runway. So that sets the scale for this main thing. Besides having a colossal slider, there are colossal detectors. This one here is called the CMS detector. The C stands for compact because it's very small. This is a person here. The reason why it's called compact is the other one is a lot larger. This detector for detecting the scattering of particles costs about $500 million. The basic idea is particles coming from both sides, there's an interaction in the middle here, and then particles that are created are scattered in various directions, and the detector tracks this.

1:00:00 Here's a photograph of some inner pieces. Again, this is not exactly chilled. This is an example of a detector. I picked this one because it's clean so you can actually see a few tracks. So this is looking straight down at the beam pipe. So the particles are coming straight in there and then they scatter off in various directions. The particles pass into the magnetic field. And this picture here shows you the scattering that's detected around the outside of the detector. So this takes the detector and just unrolls it. And then see what a particle is. So this is an example of two electrons and two muons. Then you can keep track of all the scattering that happens. A theorist sees something completely, detectors or events or anything like that. What we see are this general picture of protons coming in, so protons and antiprotons. You can look a little unhappy because it's black still partially. But the constituent, the partons, are described by fusion function. It tells you exactly what the constituent inside here, the quarks and gluons, you find in here. It's something that has to be measured experimentally and then fed into new experiments. Where the constituents strike each other through the, let's say, through the gluon chords here.

1:02:30 That's described by Feynman's diagram, which I'll tell you a little bit more about. This is, we'll call it the hard scatters. I try to call it hard because it's hard to calculate, even though it's really hard to calculate. The soft parts, that's the low energy parts. The hard is called hard because it's high energy. The soft is low energy. Those are also very hard to calculate in many ways. The only part is relevant to the twistors, at least so far, that's directly relevant, although in Radu's talk we clearly noted that there is some connection between fusion functions and the algebras, but at least this part here is very clear how the twistors apply. There are a number of different types of diagrams where the quarks and gluons have to recombine back into the hadrogs and mesons and the physical particles that we actually see. Once again, this is the only part that I'm going to focus on here, which is the part that can be described by the Feynman diagrams. The Feynman diagrams are very systematic way of doing things, completely textbook material. Let's say this would be a fork annihilating against an anti-fork, painting with a blue arm, and then reanimating that toward an anti-fork. There's a set of rules that you can use to build up your textbooks. Completely mechanical set of rules that you can follow. In fact, it's very easy to teach your computer to do this. In that sense, it's very mechanical. And there would be the leading order. Let's say this would be the leading order diagram for... There's an extra leading order diagram where you can have extra gluonic radiation or you can have loop correction. From the point of view of the standard model, it would be loop correction in this part, an extra leading order, much harder to calculate. But as a matter of principle, all timing diagrams tell you how to get any scattering effort.

1:05:00 Of course, as a matter of fact, it's not necessarily. And the reason why it's not true is because it's a technical question of difficulty of calculation. So there's a question for precision. When an experimenter comes to a theorist and says, I've done this very precise experiment, and this would be an example right here on the left, the measurement of the strong coupling constant, it's been done to this incredible precision. There's enormous work. Done by an experiment of colleagues who pushed down the theoretical uncertainty to a really remarkable level, less than 1%, and then you come to the theory, and then the theory says, oops, I can't calculate it, I haven't been able to do it, it's too hard. Even though enormous effort goes into this theoretical effort of trying to do these calculations, the dominant error here actually has to do with the missing higher order correction. If you have precision, of course, it makes it much easier to uncover deviations. You're always looking for new physics. What's beyond the standard model? Well, if you can't really have precise predictions for your standard model, then it's going to be that much harder to find new physics. I'll show you some examples of that. I should mention that because this coupling consciousness is actually rather large, even though we're doing perturbation theory, You can have a lot of gluonic radiation, a lot of production of quarks and anti-quarks, and so you also need computations of large numbers of legs, large numbers of particles, and this actually here is a very famously bantled, it's called a loop-air plot, but anyway, it's just a plot that, the idea is that using precision calculations... Even though we have not discovered the Higgs, we can still see the effect in experiments, and mathematics is really different from the Heisenberg uncertainty principle even though you can't really create it due to the short amount of time you need to do it. So that drives our desire to compute the higher orders, larger numbers of letters.

1:07:30 So this would be an example, this is actually a really good example to show you the problems of why you really need precision. So, this is the Tevatron Higgs search. So, this here is a plot of kinematic invariants in invariant masses of B quarks. It just turns out that the B quark is the one you want to be looking for for the Higgs decay. The Higgs couples very well to the B quark. So it's just the number of particles for this kinematic invariant. We have some measurable quantity, and there should be some plot, and if you have the Higgs, you're supposed to find the bump. So here's the bump. It's in the case of reduction of the Higgs in association with the W. It happens to be the very best carat, so this is the very best way to do it as a tevatron. And as you can see, this bump is not much of a bump. I wouldn't call it a bump, I'd call it a... There are other ways to produce W's and B quarks. Here's one of them, gluon, which is the B quarks, and then a W boson, and that's the background. If you add up all the backgrounds, that's the lower black curve, and then you add in the Higgs. Well, the kinematics of the Higgs events must be different. So if you use all the kinematic particles, it must be better than the Higgs. The problem is always background. You have to kill the background much better than you kill... It has to actually be much, much... You have to knock down background much, much better than you... In this case, the angular distribution between the b-partets must be quite different from the b-partets.

1:10:00 But if there's a lot of background, then you're, you know, this is what the problem is. If there's a lot of background, then, then, anyway, in this case, there's another problem which is much more serious, and that's luminosity. So this will only work, so the point here is that, this is an example where it's very difficult to find, so the point here is it's very difficult to find the signal unless you really understand the background. And you need these higher-order calculations in order to understand exactly what the background is doing. But unfortunately, the point is moot with the Tevich problem because it's a luminosity problem. So this is almost certainly not going to happen with the Tevich problem. Now, actually, look, why don't you hold on. Hold that thought. So there's a W boson here, right? Okay, hold that thought. So wait 20 minutes, I'll give you the answer to the W boson. The B port, if you actually ignore the masses, it's actually very small. It's a relatively small effect on the masses of the B port. So everything is massless here except for the W. The only thing, the much better finding the Higgs boson. In fact, you don't need theorists really to find Higgs bosons. You just need enough patience. And with enough cases, this is a few years of running, and this isn't a case of having a low mass case, so this is an opportunity to see if this is real data, but it's just simulated data. And what happens is, with enough running, collecting enough data, this is a few years of running, you find, but of course it's still the question of what exactly is this peak?

1:12:30 And finally, the peaks. Is this truly the Higgs of the standard model, or is it a slightly different Higgs, or is it something else? And for that, you still need your precision calculations. But the point is that the Higgs boson will almost certainly be discovered at the LHC, which is just a matter of time for it to happen. I can mention if it's a high mass, a higher mass Higgs, that it'll happen very quickly. You turn it on and the peaks will appear. And that has to do with the channels that open up. And there are actually many calculations. This list is maybe overblown and some are less important than others, but this is a list list. And there's many examples. In fact, you can see there's many of them with W bosons here. Again, it's masses. Now, what's the state of the art? What can we actually calculate? Well, for example, here. W plus two jets, J stands for jets, the jets come from these quarks or gluons, you get these crazy particles of jets that come out of the final state. So this would be W plus two jets, so two gluons come in, and then you get two quark jets, W bosons, W in this corner, with a two there instead of a five. And what we can calculate right now, this is it. We would have thought that by being able to define the diagrams, go ahead and compute whatever you want, this is the state of the art. Five points. Typical, typical thing to find in the literature with, uh, for example, your two-quarters is W or the H-boson, and six is so far known as the qubit. Six points. Now let's go on to higher loops. What's the, what's the situation of higher loops?

1:15:00 And there's actually many very impressive calculations. The anomalous magnetic moment is probably one of the most impressive around. This has to do with the perception of electrons and muons and magnetic fields. It's in very high order, incredible precision. It's really a triumph of quantum field theory. Another good example is the four-loop QCD beta function. The beta function controls the running of the coupling constant in function-terminal dynamics to four loops, the 50,000-Kleiman diagram, and I only had the patience to draw one of them, and if you look at the class of these calculations, they're all in the class One kinematic variable or zero kinematic variable? One kinematic variable is as good as zero because it scales out of the puzzle. So in the last few years, a really big advance has been the ability to calculate more than one kinematic variable. If you're outside the field, you'll see one more kinematic variable, but believe me, in our field... Collider physics. This is a big deal. The fact that we can do new classes of computations with a lot of new physicists and scientists. And, of course, the key to this progress is people. When I go into technical details, I'm kind of astonished with young people, these bright, energetic people that come into the field that really push forward on these higher-loop calculations. And the basic idea of how it works is that you've done it in a rather brute force way. The typical error would be a difficult approach of how you do it. You consider the set of integrals of the problem. And you consider these total derivatives via some arbitrary vector that you choose appropriately made out of either loop momenta or external momenta, like these Ki. In dimensional regularization, when you can analytically continue away from four dimensions to three dimensions with no surface terms, alternative issues about singularities or how things behave in proximity...

1:17:30 It all works very nicely so that you can construct these very simple total derivatives. If you construct enough of these total derivatives, you get a system of equations, a large system of equations. In typical problems, you can have 10 to the fifth equations, and you just systematically mark through them one by one with Gaussian eliminations until you have found the solution. In terms of a set of basic masters, called master integrals, all integrals are expressive entities. And then at the end you can construct a set of differential equations using this technology for the master integrals and then solve the master integrals. There's also, I should say, besides the issue of being able to do these good calculations, doing the integrals, there's important questions, difficult questions having to do with turning the calculations into physics, into an honest prediction, a prediction for a collider, but that's another story, subject to another lecture on this one. Although these techniques are brute force and they can take quite an effort to implement, there's really been a tremendous progress in the last few years in doing these higher loop calculations. So in amplitudes that are relevant for flyers, there's lots of new calculations. So these are calculations of amplitudes for E plus, E minus. So this is where E plus, E minus flyers are famous. The two-loop, two-to-two scattering, the QCD, all the processes that can work well, the photons, the gluons of the photons that make new backgrounds for Higgs, and so on and so forth, including cases of electroweak and the WMZ boson.

1:20:00 But there are some very nice examples of true next-to-next leading-order calculations. This involves two-loop calculation systems, Lansbixen and its collaborators. What this shows you is a plot of a measurable, this is called the rapidity here, Defining the terms of energy and momentum along the beam axis. Think of it just as an angle. I mean, roughly speaking, it's just the angle away from the beam axis, roughly speaking. It's a cross-section against this measurable quantity, this activity, and if each activity tells you the number of events. If you do a leading-order calculation, that's tree level, there's a tremendous theoretical uncertainty in your prediction of what particularly does. This uncertainty is just done by voodoo and black magic to get some kind of uncertainty when you compute it on the next leading order, one higher order. In fact, you can see that the answer seems to be outside of the uncertainty. But anyway, the point is that you now look at the next-to-next leading order, where there's this red line, and the red line, the thickness of that line, is the estimated theoretical uncertainty for that. It's basically not there at all. One percent predictions seem to be good for one percent. And there are various other issues that you have to be very careful about before you get to some percent, but anyway, to be completely clear, this example shows... So what goes into these types of calculations? So we start off with our quantum field theory, If we're doing QCD, quantum thermodynamics, or electroweak, or if we're doing n equals 4, whatever n equals 4 is, whatever your quantum field theory is, you need some set of clever ideas. And I'll explain what I mean by that set of clever ideas. By going through these computational rules, usually we have to resort to some computer algebra.

1:22:30 In many cases, final answers are so large that no human can possibly do it without making a mistake. So what's an example of a clever idea? Well, let's consider what should be a completely trivial calculation. It's really, in principle, something that should be very simple. This is five blue arms scattering, so two blue arms coming in, three coming out, in whatever your favorite theory is. It could be even n equals four superyang mils. You can do this by opening up your textbook, following the assignment rules, and if you get tired you can do it. And this is what you get. This is written in some microscopic font. It's actually only a small fraction of the answer. I believe it's about 1 25th of the answer. In here there's a memento that contains and there's a polarization vector that describes the state of the gluons. Now clearly if you look at this, you will decide there's nothing simple about this, that the scattering amplitudes and gain theory are just a mess. So a clever idea is the use of felicity. And felicity, we're getting closer to felicity. The colourization vector is blue on in terms of spinors. We use the gamma matrix which does the translation of the spinner representation and vector representation. These formulas are fancy ways of doing particular colourizations of light. The typical formula for these inner products of these angular fractions. So in QCD we'll see these angle brackets in the tops and twistors. We'll see it's the same thing, just written a little bit differently in terms of the spinors and the anti-symmetric hexagons.

1:25:00 So these are really one and the same here after some normalization. They're square roots of the rest of the products times the phases. And what happens is the same result that was on two parenchrances ago that was completely illegible and very hard to understand, the complete content of that formula is described here. If you have felicities with circular polarization one way with all pluses here, and if they're all pluses you just get zero. So now we can start to see there's some simplicity. This is the first example of the simplicity that you can really see in the speed theory. I can mention that the gauge group, there's no gauge group here, all the color factors are stripped away by this color ordering that's pretty similar to what's done in open string theory. There's a Feynman diagram which of course we can scramble with stuff in some way. So what about loop level? So what's a good trick for doing loop level calculations? Well let's consider a tree level calculation. So this is a tree diagram. And these particles here are physical. They satisfy the on-shell conditions, but they're massless, so that the 4-momentum-square, energy-square minus the 3-momentum-square, that should be zero, because of these massless particles. And now let's consider a loop diagram, a loop amplitude. Now wouldn't it be neat if you could just take this fellow at tree level and just turn it somehow into a loop calculation? Well, with Feynman diagrams, you can't do that, because... Right here in the intermediate step, the intermediate state, so these are off-shell. The L squared is not equal to zero. So this diagram here, it kind of looks like you take two of these and just glue them together to give you this or this finding diagram. But finding diagrams don't work that way.

1:27:30 So when you go to a loop of any calculation in Feynman diagrams, the Feynman diagrams, it's a new calculation in the sense that you have not used the information from the previous calculation. Like, first you do this one, and then you try to do that one, but you can't see this straight and clear. Well, what's clear about, this is a general rule, this is always going to come true. If you want a good way to calculate, you must have a way of taking... Previous calculations return them to new calculations. So, you know you're supposed to take bottles and cans and so on and recycle them, turn them into new bottles and cans. Well, you've got to do the same thing with your calculations. And there are ways of doing this with the recursive approach, which allows you to recycle old calculations into new ones in a very systematic way. This is due to variance of yield. And so Kossauer had a light-tone formulation and a little bit done on loop calculations using recursive approach and loop calculations. Now there's much more recent work that exists this year also on recursive approach. And again, the key is to recycle the information so that one calculation feeds into the next one, feeds into the next one, and so forth. And then this unitarity approach, a way to actually feed these tree calculations into loop calculations. And I'm not going to say that much about it, but let me just insert a few things from these talks. We did some cuss hour on the unitarity method. But the basic idea is that any amplitude in any massless theory So, n equals 4, qcd, pick whatever your favorite math is theory. It's slowly determined from the d-dimensional tree amplitude to all loop orders. You don't need off-shelf Feynman diagrams. All the information is already present as a tree amplitude. You just have to understand how to extract it. And then something getting better from the point of view of twistors is in certain theories, massless supersymmetric gauge theories, which satisfy a certain power counting criteria of one loop, you can use four-dimensional tree amplitudes. You don't need to be dimensional on them.

1:30:00 And four-dimensional means that you're allowed to use helicity and twistors. And I can mention that we may have many talks on, we've already had talks on an equal score and we will have many more on an equal score superhuman reveals. And this is an important theory from many points of view, including the fact that, well, the close relative of QCD. And we can learn a lot about the perturbation theory of QCD by looking at this maximal supersymmetric theory. And as I said, there will be, there were many talks and there will be. There's a nifty thing that you can do to generalize unit parity. I'll give you some more details. And the basic idea is, again, these blocks here represent on-shell tree amplitudes. And it's a way of turning these tree amplitudes into loop amplitudes in a very systematic way that Kossauer will describe. So let's get on to twistor space. We're here because of these really amazing developments in the twitching space scattering amplitudes and their connection to the topological string theory. So, as we heard from Ed and we'll hear in a later talk, which was also from Ed, we'll continue discussing these results on these. Now, I should mention that the link to the string theory is for any of the four superannual mills, but at tree level, there's almost no difference. If you're considering cases of pure glue, then the quarks or the scalars or the luinos, they don't play a part. Also, it turns out that the difference between quarks and luminos is very important. It's just a question of doing some color re-arranging. So, at tree level, if we're talking about tree level, Rangel's quark theory might as well be QCD. So, after the twistor transform, the remarkable observation of Ed is that the amplitudes live on certain curves in the twistor space. So this is in the twistor space, the points represent the external points.

1:32:30 So it can live on higher degree curves or degenerate higher degree curves with intersecting lines in the twistor space. And this structure here is very important in that it essentially points to a simplicity in the scattering applications of the n equals 4 and also QCD for that matter. When they understood and exploited this paper, they took the Park-Paylor Amplitude. This is an amplitude of QCD. This was already discussed in the Ayers talk. This is n-poined with two minus multiplicities and the rest plus and has a very simple structure and the idea is to use this as a vertex. The idea of how to do this, it comes straight from the Twister space picture, of the simplicity. The simplicity that's observed here in the twisted space had to be reflected in the scattering amplitudes back in the mental space, which led to this idea of constructing these amplitudes in this nice way using these MHD vertices. And the key point, this is really the key, is amplitudes are much simpler than anybody anticipated. We knew that the MHD amplitudes were simple. But we didn't know how simple all the other amplitudes are. Now we know how simple they are. And that's the key for unlocking a lot of things. And we'll hear many talks about computations that follow from this. So what about masses? What are you talking about with W bosons?

1:35:00 Well, W goes on to the very end.