Unitarity & Twistors

0:00 I'm from Australia, also at UCLA, and Rondon Roybein is a candidate of Princeton. We've heard a lot about twisters and amplitudes in the gauge theories, and about n equals four supersymmetric gauge theories in particular, and Steve Byrne has already talked about some of the motivations for studying these amplitudes, so I won't. In fact, I was given two briefs by the organizers. The first was to mention the currents, the maturity, and twisters, so that's fine. And the other brief was to review the unitarity-based method, which will take the most of the remaining 50 minutes or so. Before I do that, however, I wanted to open with some comments and background about tree aptitudes, and I wanted to take the point of view of asking about the computational complexity in the sense that how many operations, multiplications, additions, and so on and so forth, does it take to evaluate an amplitude? If we take a textbook Feynman diagram approach, well, there's a factorial number of diagrams, so it's a factorial complexity. And that's a very, very hard problem. But fortunately, with color ordering, which a number of speakers have already mentioned, when that reduces to an exponential number of color order diagrams. So that reduces the problem to exponential complexity. For any given n, of course, there's order 2 to the n different helicity. So in some sense, we have 2 to the n different quantities to compute. It's at least an exponential complexity. But we can still ask, what about the complexity of each helicity amplitude? to. And to ask about that, it's nice to go back to the original recurrence relations, which you could do to Behrens and Kruger about 15 years ago. And you could think of them as a solution to Dyson-Schwinger-like equations for tree-level gauge theories. The idea is

2:30 basically you have one optional leg, you follow it into some arbitrary sum of diagrams, and it can meet either a three-point vertex or a four-point vertex, and there's a propagator attached to the other two legs, and what's at the other end is again another current of the same form, but with a smaller number of legs. And the key point here, and this is kind of a philosophical point about any calculations you want to reuse as much as possible, it's the same smaller current that appears at different stages of the calculation. So there's only a polynomial number of different currents that are needed. And when you work through the counting, you see that they're in fact needed most into the fourth operations for each helicity amplitude, for generic helicity So that's much better than the exponential you might have. We've seen that twisters give new representations for tree aptitudes. Peter Zwerchak talked about the Cachazzo-Zwerchak-Witten construction. We have simple vertices and rules. Mark Spradlin talked about the Reumann Spradlin-Volovich representation in terms of box coefficients, and we've just heard about the Pachazzo-Feng-Witten recurrence relation where you get a representation, a very interesting representation, in terms of smaller on-shell amplitudes, which is a very interesting idea. And these give, all these different representations, give you very nice analytic forms, and in special holistic cases, in fact, you do better in order to enter the fourth operations. And, to me, the main point I want to say here is that I think it's not going to be the last word. There's probably some more interesting ideas waiting to come out. One thing I wanted to mention is that, of course, the CSW construction can also be recast recursively. So, again, the starting point are the Park and Taylor amplitudes from the mid-1980s, of cyclic structure in the color-only form, and as Peter described, you turn these into vertices with off-shell continuations, and then you build all possible diagrams out of these vertices connected by scalar property.

5:00 So if you now think about how can you reorder this construction, then when you start by drawing down diagrams, The first thought you might have is to go the same way that Behrens and Fitta did, which is to go into the diagram and try following a single leg. But that's an unwieldy thing to do here, at least from a computational complexity point of view, because you have vertices with an arbitrary number of legs, and so you get arbitrary numbers of sums. So instead, if you want to break this up into smaller pieces, you can think of cutting one of the propagators. these diagrams into two sets by cutting some property. Of course, there's not a unique choice. There's many different ways you can do this. But let's just imagine cutting on one of them and you can simply group all the vertices, all the diagrams building up on one side of this slice into a new kind of vertex. So you lump all of these together and you can treat them as a new higher degree vertex. So when you do that, then on the right-hand side, you now get a degree 3 vertex, and again, if you do that on the left-hand side, you get a degree 5 vertex. And so that more complicated diagram now has a representation in terms of this particular simple diagram. And as Mark Spratlin mentioned, these higher degree vertices have a natural interpretation in the twist-to-string theory in terms of connected and disconnected instantons. If you start to look at connected instantons and semi-connected contributions, then you get exactly these sorts of higher-degree vertices. Of course, you can unwind this, and the one thing I haven't told you is how do I choose which propagator to cut? The neat thing about the CSW construction, which is not, of course, generically true in field theory, is that, in fact, any way of cutting them gives you the same combinatorics. So what you do is you simply sum over all of them in an average. So a degree D vertex, then, will be given by a sum of vertices where you split into one in degree D. Again, each of these things is a sum of diagrams as described on previous transparency,

7:30 and all the way down to the most nearly symmetric form. I hope we can maybe dim the lights a little bit. So I was told that the budget for the conference did not include a lot of the classes for you to be able to see and study the formula as given on the lower part of the transparency. But it's actually not important. The main point I want to get across is you just end up with a double sum here addressing this equation with all possible external legs, and then you have something you can then feed to a computer, a recursive form of the CSW construction. Another issue that, of course, the phenomenologists and experimenters care a lot about is going beyond your QCD. QCD is an important part of collider experiments, but it's important in conjunction with other gauge bosons, and also Higgs's, which is going to be the first experimental target of the LH2 plastics, and we'll talk about that in the following talk. And I just wanted to briefly mention how you can put in W's and Z's. Again, you're interested primarily in these as external particles, if you like, to the QCD amplitude, so you have some process where, let's say, gluons are fusing together, they're emitting some bunch of gluons and quarks, a W emitted, which in turn decays into a lepton neutrino. And the basic idea is to use a hybrid formalism, to use both CSW and recursive currents, and of course we factorize the process into, let's say, the W decaying into the lepton-neutrino pair, and then there's the W current. And what we need to do there is basically add two vertices, two new kinds of CSW vertices And the one that's sort of maybe a little bit peculiar looking is the gauge boson vertex with nothing attached to it. And that, again, can be rephrased in terms of the same kind of off-shelf continuation that was used by Brandhuber, Spence, and Travolini in their loop computation. And then beyond that, with these two new vertices, you just do a standard kind of CSW construction.

10:00 recursive form, and it turns out that the recursion equation I showed you in the previous transparency is exactly the same equation that you need in order to turn this into a general amplitude. that's as far as trees are concerned, but we want to spend most of the time here really talking about loop calculations. And so, again, it's good to think about what the textbook be. We have a Lagrangian, and we derive vertices and propagators, and we're supposed to sew these together, these vertices and propagators, into loop diagrams. So when he's doing some endpoint calculation, in general, we're going to end up with some integrals to do. They're going to be everywhere from bubble to endpoint integrals, and those integrals will have powers of the loop momentum inside them. In fact, they'll range anywhere from scalar integrals up to n-point tensor. And each of those are multiplied by coefficients, which are functions of the external momentum polarization. There are standard techniques, which in their earliest forms date back to Brown and Kleinman, I guess about half a century ago, of reducing those tensor integrals to sums of simpler integrals. And there are some cleverer methods that seems to be a Dutch specialty idea. And at one loop, it turns out that generically, both originally as was done in four dimensions and in four minus two epsilon dimensions, you can reduce these to bubbles, triangles, and boxes. So it's a technology that can be used and has been used very, very extensively over the last 50 years for doing calculations. You can apply the same ideas to color-ordered amplitudes. So you write down color-ordered Feynman rules, and you can use the spinner-holicity method at the end of the day to get simpler expressions, to get helicity amplitudes. But the basic problem here is that we know that in doing the tree computations, all these new representations, and one of the interesting questions, of course, will be to see if there's some additional symmetries that are relating all these different expressions that people have before in the last year, all those different representations involve a lot of cancellations. And if you're just going to do the loop amplitudes this way, you're losing the advantage of all those cancellations. You're having to redo them again

12:30 at loop level. So a lot of calculation effort would be wasted. So for example, if we were to use traditional methods in the n equals 4 supersymmetric one-loop seven-point amplitude, a calculation was done in collaboration with Lance Dixon, Victoria Belduca, and Steve Byrne in the middle of last year, then we'd have about 227,000 diagrams. Well, what does that mean? If we were to draw them each in a one-inch square box, then it would take three bound volumes of FISRAVD just to draw them. And if we hired a draftsman and he was sufficient, he was able to draw one diagram a minute, And it would take him 22 months full time just to draw them all. So that's clearly not the way that you want to do the calculation. The question is, can we take advantage of the original tree-level recurrence relations and of all these twister-based ideas for reducing the computational methods? And in an environmentally conscious era, the answer has to be that we should reuse and recycle. Let's do it. And the key to doing that is a very basic property of field theories. It's just conservation of probability. So for the scattering matrix, that just translates into this statement, S dagger S is equal to the identity matrix, or if you write it in terms of the transition matrix, you discover that, in a loose sense, the imaginary part of that matrix is given by its square. More precisely, if you put in the Feynman i-epsilons, then the discontinuities in various variables are again given by the square. Now, ultimately, of course, we're doing this in the context of a field theory. We're not interested anymore in the question of what happens if it's some general theory that cannot be expressed in terms of the logic. So it's useful to translate it into Feynman diagrams, and you can do that using the Katkowski rules. So you have some Feynman rule. Everything is going to be ultimately some sum of these objects in the numerator as well, and we want the discontinuity in the k-square channel. k is some sum of external momentum.

15:00 And so what Dick Katkusky tells us is that we should replace these two propagators by delta functions. We should put these propagators basically on shell, and this delta plus is just the positive energy on shell change. And when we do that, what we obtain is a phase space integral. So we have a loop integral we've now replaced to the propagators with delta functions. That object is exactly a Lorentz invariant phase space integral. Or, diagrammatically, we have a loop integral which we would now write as the product of two tree diagrams integrated over the high phase phase. In the bad old days of dispersion relations, I guess I can feel comfortable saying that since it's been said before in the questions in the previous lecture. So in those bad old days, we wouldn't know anything about field theory or Lagrangians, to recover the whole integral, the real part, basically just using analyticity, would be written in terms of the imaginary part, and there'd be a potential ambiguity. That ambiguity would vanish so long as the function we're integrating here had sufficiently good behavior as the invariant goes to infinity, that is to say, in the ultraviolet region. Now, if that condition isn't satisfied, Then there are so-called subtraction ambiguities, and these correspond to rational terms in the full amplitude that don't have discontinuities. As I said, of course, we want to use the maximum information that we have, and we know there's an underlying with Ranjan field theory. Everything is ultimately expressible in terms of finement. So what we want to do is to not do the dispersion relations, but take advantage of all the sophisticated techniques that one has for doing the finement. Identities, reduction techniques, differential equations, reduction to master intervals, and a whole host of other ideas. The other key idea is that we don't want to do this at the diagram level. We want to do this at the amplitude level. And that's how we can take advantage of the cancellations

17:30 at lower orders, because that cutting relation can also be applied to sums of diagrams. And so, if you're now looking at the cut in a given channel of the sum of all the diagrams for some process of interest, you're effectively going to be throwing away those diagrams that don't have a cut in that channel. So where one or both of those propagators are missing. And then you're going to get the sum of all the diagrams on each side of the cut. And each of those is an on-shell tree aptitude, so we can use any of the compact representations that have been put forth and plug them in to the computation we're doing. And that leads to the so-called unitarity-based method for doing higher-order calculations, and it's a tool that has been used extensively in the last 10 years by a limited, admittedly limited set of people for doing one- and two-loop calculations. There have been calculations both with fixed numbers of external legs and also a certain number of all-end calculations. It also turns out to be a very useful tool for doing tonal proofs. For example, the all-order structure of linear factorization is proving used in this technique. technique, and unlike many formal techniques at the same time that you can use it to prove, you also get a very compact representation for the factorization function, in this case the two-loop splitting amplitude, which this computation was finished last year. Of course, in the last year or so, there's been a number of other groups that have relied in some form another on this approach. So the way this works at one loop in an explicit computation is that we're going to compute some set of channels. Each of those channels we have to compute the tree amplitudes on either side. We're going to form the phase space integrals that come from the interference of the trees on either side of the cut. We're going to reconstruct the corresponding Feynman integrals. We're not going to do a dispersion over or reconstruct a Feynman integral. We're going to perform a set of reductions to some master integrals. And finally, we're going to assemble the answer.

20:00 So again, we have now a similar figure to what I drew earlier, except that the blob now tells us that we have the sum of all the diagrams going in, so we have an on-shell tree amplitude. We're going to multiply two of them together across the cut. internal momenta is going to be promoted to loop integral, so we're going to put in the propagators that are missing. And we have to figuratively sum over the different channels, over the different cuts. Now, in general, one has to do this in 4 minus 2 epsilon dimensions, and I'll explain in a little bit more detail why that's an important thing to do. This reflecting an observation of Van Nervin from about 20 years ago where he used this technique to compute some two-loop integrals, namely that in a certain sense, when you're working in dimensional regularization, the dispersion relations can be thought of as convergent, and so that eliminates the subtraction end of the universe. It turns out that at one loop, when you're working in supersymmetric theories, as we all know, supersymmetric theories have better ultraviolet behavior, and that's reflected in the fact that you can actually do all the algebra in four dimensions strictly, and it's only the loop integral in order to control infrared or perhaps ultraviolet divergences that you need to do in 4-minus-2f-sun dimensions. Now, I've written here a sum over all channels, but you're not supposed to blindly some, what you're supposed to do is merge channels. And the reason you have to worry about this is that some integrals actually have cuts in multiple channels. For those integrals, those coefficients, you don't actually need to compute both of the channels. You could and use it as a consistency check if you wanted to, but you're only supposed to pick it up once. And so essentially, this is a procedure for finding a function which has the correct cuts in all channels. And it's something which you can teach a computer mechanical way. Now, dimensional regularization, as particle physics know, is a very useful tool. It's proved a very powerful tool for doing, of course, ultraviolet computations. And also, as an infrared regulator, it's really played an essential role in a lot of the jet

22:30 physics computations. I think it would have really been, from a practical point, impossible possible without it. Here, in the unitarity-based method, it's playing a third role, and that's to give us a handle on the rational terms. And I like to sometimes express this in a very pedestrian way. There are different ways one can argue with this. In a massless theory, any amplitude in dimensional regularization is always going to have a factor of some negative an invariant to the minus epsilon. So if we now go to higher order in epsilon, then what started out as a rational term at order epsilon over 0 is multiplied by some factor of logarithm minus S. That factor has a discontinuity, and so it can be accessed without any subtraction and the UADs. As I mentioned, at one loop, one can show that, even in dimensional regularization, you go down to a basis consisting of box, integrals, triangles, and bubbles, and in the n equals 4 theory, again, because of the even better ultraviolet behavior of the theory, you can show that all you need are the box integrals. And the box integrals here have external legs whose arguments are sums of the original external momenta. So we started out with n different momenta, and we grouped them into four sets. So if you, of course, take a sum of masses of momenta in general, you get a mass of momenta. There's one mass, three mass, and four mass, of two mass boxes. If anyone is curious as to why one is called easy and hard, ask them in the question section. So again, if one is looking at the MHV amplitude at one loop, this is particularly simple. Let's place, for example, the two negative elicities on one side of the cut, and what's going to happen is that everything is positive elicity, All the external legs on the other side are positive helicity. That means in order to get a non-vanishing amplitude, we have to have these two gluons being negative helicity. That forces these to be positive. That means on both sides of the cut, we have an MHV amplitude.

25:00 And so we have a simple effect formula. It turns out that for all m, given the structure, you start out with a hexagon integral. There's some simple algebra that reduces it to a set of boxes. In fact, the only boxes that show up are the so-called easy-to-mass boxes with opposite-edge masses. And the form of the answer that you get is also amazingly simple. You get the tree amplitude out in front, and basically you get a sum over these boxes of the box times the dimension core denominator. And, again, this is something that you can really do on the back of an envelope using a unitarity-based method. Now, the knowledge of the basis, which is special to one loop, actually opens the door to new methods of computing the opportunities. Because the only thing you really need to compute are the coefficients. The integrals are all taken care of, and we've already heard from Ruth Frito of the algebraic approach that was proposed, a very elegant algebraic approach that was proposed by Freddy Cachazzo based on the holomorphic anomaly. Now it's always good to use knowledge of things like a basis if you have it, but I do want to emphasize that you don't need to have a basis, a predetermined basis, in order to apply the unitarity-based method. And in particular, the method also applies and has been used in higher loop calculations. Here, the structure is a little bit more complicated, but it's the same kind of idea. We have contributions now where on one side of the cut, you're going to have a one-loop amplitude, and it's still two particles going across the cut. Of course, it can appear on either side of the cut through independent contributions. And in addition, you have contributions where you're cutting more loops, let's say you're cutting two loops at once, so you now have, again, a product of two tree amplitudes, but let's say three particles crossing the cut. And the technical details are perhaps a little bit more complicated, but the basic approach You ultimately are finding, reconstructing an integrand or an integral that has the correct

27:30 cuts in all the given channels. And one thing that we realized over the years of doing this is that you see a loop amplitude here. Of course, the temptation is to replace it by its cuts as well. And that's, in fact, something that you can do. And because if you think about it from a Feynman diagram point of view, what a cut is really this phase-space integral is that you're acquiring two propagators. You're looking at all contributions that have a specific pair of propagators present corresponding to some massive channel. But there's no reason to stop it, too. You can go on. You can acquire more than two propagators to be present, and that's what gives you the idea of the generalized price. And the virtue, of course, is that you're breaking up the amplitude into smaller pieces because you're now looking at pieces that have more constraints on them. So you have smaller pieces and they're simpler pieces. And again, even in cases where, let's say, the original cut has non-MHV structure, the higher cuts, for example, the triple cuts, may now have MHV amplitudes in each of the cut pieces. And so again, that's something that you can use a simple, more straightforward expression in order And so, in fact, initially we used this in the computation of the z to q-q-bar glu-glu amplitude about a decade ago, and it's been a technique that's been used in higher loop computations, and most recently in the all-m next-to-mhb amplitude, which Lance is going to talk about in more detail in the next talk. But I do want to show you just some pictures, so the kind of triple cut that you would get that would, for example, show up in the all-in next image, the amplitude, again, you'd have three, you're breaking it up into three different components, there's no particular phase space integral interpretation, but it doesn't matter. And again, at higher loops, as suggested in the previous discussion, you can imagine cutting both loops, and then you have a product of tree amplitudes which you can readily compute. One of the interesting results that was obtained using the Tarity-Base method and some of these

30:00 other ideas is a very interesting relation for planar to the amplitudes. I should mention the original form in terms of integral functions, which was, at the time, were not known for the two-loop amplitude, was done by Zvi Perum in collaboration with Joel Rozofsky and Bryce Yan. And when we were finally able to compute the integrals that went into this function, Then we discovered that in a certain sense, if you take the ratio of the amplitude to the tree, you discover that the two-loop amplitude is in a certain sense the square of the one-loop amplitude. And I should mention this is done in collaboration with Baris Anastasio, who is currently a postdoc at the A.T.R. in Syria. And we certainly expect that this result will generalize, and we're certainly hopeful that that these twister-based methods will shed some additional light on it. Now, once you know the integral basis, then you can, of course, as mentioned by Ruth Virou, use that to give a general solution for the n equals 4, one of the coefficients. The thing that's the really interesting observation that Virou, Cachazzo, and Thang made is that you can actually do this even for three-point vertices. So you have something that for real momenta vanishes, but if you take complex momenta, they don't, and then you can apply the same kinds of ideas to those vertices. Now, when you do these calculations, this is an example of the kinds of results that you get at the 7 point you get a whole bunch of different kinds of singularities showing up in the coefficients so there's some of them that are physical singularities for example these two particle spin-off products are the collinear singularities that you expect the final amplitude to have there are multi-particle singularities the same nature. And you also get a set of unphysical singularities in each of the coefficients.

32:30 And those things cancel in the amplitude as a whole. And it's interesting to note that these you can think of as kind of generalized CSW continuations where the reference momentum is perhaps now being chosen to be one of the external legs instead of an arbitrary light And it's going to be interesting to see whether these kinds of objects are more general building blocks, even if they show up with these unphysical singularities. So let's talk a little bit about how the calculation of the all-N amplitude, all-N-NMHV amplitude, so now three negative velocity gluons in the N equals 4 supersymmetric theory of this time. So first of all, it's easy to see from the quadruple cuts that the four mass boxes are absent in this calculation. And the natural thing to do is to look at the triple cuts because then each of the box coefficients is going to decompose into three different vertices. And in fact, this is the way that you can calculate the three mass box coefficients. From those same triple cuts, you can also calculate the hard two-mass box coefficients. Again, you have to look at those boxes that have cuts, multiple cuts with, in our case, real external, real momentum. You can also use the unitarity-based method to calculate the easy two-mass coefficients, but it turns out that the infrared equations actually give you a more straightforward and a much more elegant form for the solution, the structure of which Lance will talk about. I want to talk a little bit about the background. See how much? So let's actually go through a calculation of one of the three-mass box coefficients. This is maybe not quite a back-of-the-envelope calculation, and maybe you need a large envelope.

35:00 But it's something that you can carry out in a straightforward way. So we're looking at an object that has a triple cut, and we're going to write down those three vertices that I showed in the diagram of P-transparency's back. You immediately recognize there's a certain factor that is independent of all the cut momentum. You can just pull it out. So in fact, independent of the number of legs you started out with, you see that you have a fixed number of denominator factors that, eight of them in this case, that depend on some of these cut or effectively depend on the loop limit. So the first thing we can do is we can take advantage of these numerator factors that essentially go from one massive leg to another. Those are not present in the original characteristics. And we can use something called a Staubman identity, again, the Dutch contribution, to do some partial fraction. The point of partial fractioning is that we're going to take this octagon integral and ultimately reduce it to integrals that have fewer of these denominator factors. Each of these denominator factors, when you complete the square, is effectively going to lead to a propagator. So instead of 8 propagators, we want to get down to a situation where we have only 4. So we partial fraction. And then it turns out that this is one of the insights that came from doing the seven-point calculation. The other, of course, insight from the seven-point calculation was the realization that there are simpler forms of the tree than have previously been written now. So we take the general form of these cubic denominators, and again, you can use that to partial fraction and reduce additional pairs of these denominators to forms where you have a single denominator involving the loop momenta and some other object which you know ultimately has to show up in the answer to the coefficient. Finally, our four propagators are going to come from the three-cut momenta. If we promote this back to a loop integral, we're going to replace those cuts with 1 over Li squared, so we have propagators there, and in addition one factor that's left over, one of these spin-off problems. And if you now complete the square here

37:30 to get the propagator and do a little bit more algebra, you find that you basically get a form with four of these cubic denominators and, of course, the overall factor, NHV-like factor in front. And finally, in a very natural way, you again get the box denominator to appear. So the natural kinds of functions, in a sense, are not exactly the box integrals They're the box integrals multiplied by their denominators. That's actually all I'm going to say about the structure of the all-N result. For a more complete discussion, you'll have to wait a few minutes for the last part. One other subject I wanted to touch on is that of the infrared consistency equations. The first thought, of course, is that we're thinking about n equals 4 supersymmetry, it's a finite theory, so why am I talking about divergences, why are there 1 over epsilon? It's an ultraviolet finite, the beta feature is zero, but there's still infrared divergences that are due to soft gluons. And these arise from regions of the loop integration where a gluan becomes, one of the internal gluons becomes soft. And those soft divergences are universal in negation. In general, there are also collinear divergences, which in any historic case at one loop are absent. And, in fact, the structure of the soft-guan divergences in the n equals 4 case is exactly the same as in QCD, because the guan perturbations are really the same. So it's another example of where studying the n equals 4 case also teaches you a lot about the real-world QCD. If we were building a physical cross-section, probably we wouldn't be building a physical because that's kind of hard to do experiments on. But if you're building a physical cross-section in QCD, then what would happen is that there are also going to be contributions with real emission, where that emitted gluon is also becoming soft. And that, again, is something you regulate in dimensional regularization.

40:00 It has a universal structure, and it will cancel the infrared divergences that show up here. That cancellation is, again, a reflection of unitarity, that goes by the name of the Kinoshter-Li-Malemberg theory. It's a very general statement about on a field theory. At one loop, these infrared divergences have a very simple structure. They're just given in terms of a double pole. There's a certain epsilonic power, and then multiplied by the triumphant. The fact that it's the triumphant theory that shows up here is really essential for this cancellation to work. There's also a two-loop version of this formula, which was written down a number of years ago by Stefano Gattani. In fact, the origin of this formula is in a, if you like, the real emission calculation that was done by Valtekina and Nigel Glover 13 years ago. So the basic idea is we want to take a one loop amplitude and we now want to examine the coefficients of the various logarithms divided by epsilon Again, I had a 1 over epsilon squared which has no logarithms If I expand that power, I now get logarithms over epsilon And if we look at the nearest neighbor invariance then we expect to get a function a multiple of the tree amplitude other invariants, then we should get zero. So we take the, in each box, there's a contribution to one of these logarithms. Typically the coefficient zero plus or minus one plus or minus one half. And so we take those appropriate linear combinations of the box coefficients and that gives us an equation relating them to either the tree amplitude or the zero. Turns out there's a quadratic number of equations. And at least for odd n, that's enough to solve for all the easy two-mass and one-mass coefficients in terms of the three-mass and the hard two-mass ones. Those have a simple expression that comes from the triple cuts. There's a redundancy for even n, but then by taking collinear limits, you can show that the solution also holds for even n. So that's one use you can make of it. The other use, if you turn this around, and know what the coefficients are, then you can solve for the tree in terms of the known

42:30 coefficients. And that was a particularly nice form of that was shown by Mark in his talk. And that inspires a new set of recurrence relations, which Ruth talked about in the There's another aspect of infradivergences that's worth pointing out. We saw in Gabella Travolini's talk yesterday that you can map the MHP diagrams into what's basically a unitarity-based calculation. And the basis, in a sense, the basis of one of the boxes obscures this fact a little bit, but a generic diagram actually doesn't have any infrared divergence. The cut, you can write down it's finite, and if you were to do the dispersion interval, you'd again find something that's finite. It's only those diagrams that have a four-point vertex where you have the infrared divergences. So that means that from the point of view of the twister string theory, you can separate the issue of understanding aspects of it from the question of infrared divergences. It also turns out to be possible, and this is work done in collaboration with Josef Bena and Radu Rojvan and Spi, of course, that you can define a twister space regulator for those objects. And this is, it's a formal construction, but it's a little bit reminiscent of the way that you can put dimensional regularization into the usual heterotic string theory, again, in order to get and prevent a regular. I'd like to close by emphasizing that unitarity is a very natural tool for doing calculations, loop calculations in a context where you're dealing with on-shell amplitudes. It's something that meshes very naturally with the twister methods. And there's a large body of explicit results

45:00 that have been done that are useful both for phenomenology and for twister investigations. I think that it's interesting to note that the investigation of Twister string theory has a little bit of an experimental flavor to it. That by looking at results of explicit computations, you get a little bit of insight into the Twister string theory, and that in turn leads to new ideas for doing calculations. And that's a very virtuous cycle, which I hope will continue. Thank you. Thank you very much. So, do these methods, do these methods sort of extension, which applies in that signature? Well, I'll give you my personal viewpoint, which is that some of the discussions that we heard in terms of, quote, two signature twisters, I think are more naturally thought of as involving continuations to complex momenta, a 3-1 kind of signature. And I at least don't understand what exactly would be meant by unitarity in a 2-2 kind of signature. So one way of defining amplitudes in that context would be to do the calculation in the 3-1 signature, and analytically continue the result at the end. And that would avoid the question of having to think about what do you mean by rick rotation, what do you mean by Feynman I-Epsilon, Z-2-2, all of which are questions which I don't really know the answer to. Can you summarize to the extent which these methods are useful for supergravity?

47:30 They are very useful, and there's a large body of work which has been done by Weyland, Stixson, Sviburn, Dave Dunbar, Joel Rizofsky, and Maxim Perlstein. Am I forgetting anyone? Okay. And, in fact, I wasn't really planning to talk about it, but, of course, the observation that n equals 8 supergravity is, in fact, no counterterm at three loops, is originally to do to that work so I think there's very interesting statements that have been made and I think in the future will continue to be made about higher behavior in any state on this based on the same kind of ideas So do you think the distribution, I think, could be generalized to normal supergravity, where the propagations are located, of course? What is the sense of a loop amplitude into formal supergravity? What? What's the question? What is the sense of a loop amplitude into formal supergravity? What is it describing? There may be a formal construction that lets you do it, but I'm not sure is the answer. So thank you very much for the chance to speak here. It's second in the tag team on the talk. So both the topics I'm going to cover have already been promised by David in his talk.

50:00 And the two topics are, one will be to just describe the kinds of functions or the full results for the next MHD one-loop amplitude and then I'd like to discuss their twister space structure and also some of their structure in terms of more conventional poles and momentum variables because I think that sort of has some use as well. And then I want to discuss the extent to which you can organize amplitudes Higgs boson into Quister's phase, or according to the NHG rules developed by Pachazzo, Spurcheck, and Whitney. And so that was, the first work was done in collaboration with Steve Arne and David Kotzauer, and the second with Nyberg Glover and Bell. So the, first I'll give some motivation for the first topic. I'll describe the complete results and then I'll analyze them according to trister space and factorization properties. And then we'll go on to the Higgs case and I'll sort of describe the way we've organized the Higgs computation and there's a very similar related structure that comes up in some other campuses, so I'll make sure we get there. So, a lot of the motivation has been given before, so I can actually be pretty brief here. Steve Byrne covered this very nicely. We're interested in these one-loop amplitudes with large numbers of external legs because they form, they enter into eventual numerical programs that will help us predict the production of jets of hadrons, originally arising from quarks and luons, in hadronic colliders like the Tevatron and the Large Hadron Collider. Now, to be honest, probably the most interesting things for the experimentalists are not processes purely with jets, because there's just too many of them from QCD. You kind of have to look for other signals to uncover new physics in many cases. That's not a firm and fast rule, but if you looked at the laundry list

52:30 that Zeke presented of processes, almost all of them had other objects in there. But because we understand the pure QCD case and even better the N equals 4 superior A. Mills case, we can use that to understand, perhaps help understand how to organize more complicated calculations or calculations in extensions of QCD. And also, these results can always be used to sort of benchmark numerical programs. At some point, QCD is still a fairly simple theory when all the quark masses are set to zero. The intrinsic presence of masses is probably going to make many processes still incapable of analytic computation. to construct these amplitudes completely numerically. But it would be nice to have these test beds in pretty complicated situations where you can understand how well the numerical programs are working. So that was sort of phenomenological motivation. Now, we've heard an awful lot of discussion of twister space and the close correspondence between these amplitudes and their structure in twister space. It's now becoming extremely well understood, at least empirically, at the tree level. We would like to add some more data at the loop level to see how amplitudes or pieces of amplitudes are localized. And, of course, if you have a large number of legs, then if there is some kind of localization onto fixed curves, it's easier to see the more legs you have. If you don't have enough legs, the situation might be degenerate. That was our original motivation, in fact, to work on the seven-point amplitude. And there's also just an intrinsic interest, at least for me, is to understand how these amplitudes fit together, how all the different pieces of them conspire to make it a consistent amplitude. So this equation has been given at least twice previously, but here it is again graphically. it provides our excuse when we have to talk to experimentalists and explain to them why we're in twister space instead of computing QCD cross-sections directly. And so the excuse we give is that, well, a QCD amplitude like this with the gluon going around

55:00 can be rearranged so that one of the components is N equals four super-nag mills. So we might as well go off and compute this one first, and we'll come back and get around to these later. So that works up to a point. But anyway, it's a pretty simple rearrangement. We just add and subtract four fermions in the adjoint representation, and then we can add six scalars, subtract eight of them, four times two, and add two back again. And sure enough, these guys cancel out, and we sum them vertically, and so a gluon is equal to a gluon. But it's also equal to the sum of this n equals 4 superiagnuels multiplet, and there's another supersymmetric contribution, the n equals 1 chiral. And so this is a very useful way to do the decomposition. It's not just making a mess out of this because, in fact, each line here has a separate analytic structure in all the examples that have been worked out so far. and basically because they have different spins or multiples running around in the loop. And the scalar one has usually been the most recalcitrant piece because it doesn't have the cut constructability properties that other speakers have mentioned that these supersymmetric ones have, and so it's usually the last of fall. But nevertheless, it has been worked out up to the five-point case, large pieces of it hasn't worked out recently beyond five months by the group from Queen Mary, Bedford, Brandhuber, Spence, and Crowley. So actually, I just want to display some of the information that's been presented by other speakers a little graphically, and I'll use the same kind of notation in the second part part of the talk, so for this part you can just indulge me, but I want to sort of display the different possible types of multiplicity amplitudes you can have, and we'll just consider the end gluon case, which has been, a lot of the attention is focused on that case. So if we're doing scattering amplitudes in Minkowski space, then we need to have at least four gluons, say glu-glue to glu-glue. And so on the vertical axis, I'll put the total number of felicities, that is the total number of gluons, n plus plus n minus. n minus is

57:30 the number of negative felicities. And then on the horizontal axis, I'll put the difference between the plus and negative felicities, the number of them. So this line here is where n- is equal to 0. We get 4, 5, 6, 7 Guilans, all with plus felicity. The next line in has exactly one negative felicity, 2 negative, 3 negative, and so on. And I could have shown the other half of this plot, but in the QCD case, the answers over here are exactly obtained by reflection across the vertical axis. That's just parity, which exchanges the number of plus the number of minus felicities. So the first 4-gluon scattering aptitudes were computed in a helicity basis in 1991. Results for cross-section summed over felicities were known a bit earlier. And they were first done in the pure glue case, and then within, I should state that But it was also understood at that time that these amplitudes would always be zero in the supersymmetric case, but they become non-zero, as indicated by the fact that there was a non-trivial computation here, in the non-supersymmetric case like TCD. And... Oops. Going back in time. Okay, then by 1994, that was a pretty good year for Nguyen computations, and by the end All the, and some of these were known even earlier, these were computed by Malin after Anzatz by some of us, and these were computed by Malin. These are interesting amplitudes. They vanish in any supersymmetric case, so these green dots just indicate, the filled in small dots indicate the scalar contribution in this decomposition that I described earlier. interesting functions in that they're purely polynomial. You can show on the back of a very tiny envelope that all the cuts vanish just by saturating, trying to saturate them with felicities. It's the same sort of counting, in fact, that will lead you to not, well, to be unclear about how to draw in any MHB diagrams.

1:00:00 So the first amplitudes with cuts and the first non-trivial super-symmetric amplitudes lie here. These are the configurations that at tree level are given by the Park-Taylor NHD amplitudes, which we've seen multiple times by now, but here I'm describing the one-loop amplitudes. and so the Taylor contribution was known up to the five-point level and the supersymmetric ones are now known, n equals one and n equals four, for the MHV and also the non-MHV six-point. So you can see a trend developing which is that theorists like to calculate in like this. Now experimentalists, they can't measure the publicity of fluons, so they like you to give results going up like this, because they want to have everything filled in at a given number of legs so that you could give them a prediction for a cross-section summed over all fluons. So sometimes you end up calculating an infinite amount of useful stuff to give them one useful if you want to look at it that way. But, obviously there are other ways to look at it. So, actually, very little happened in this plot, particular plot, over ten years. But then this year was another good year, or last year now, excuse me. So all the stuff in Brown was what was known before, and then the new things were, as we've heard already, as I'll talk about in more detail, the amplitudes of three negative velocities in N equals four super nine mils. And in addition to that, I put in cyan, which you can not see so well, partial results. So I mentioned earlier that the group between Mary has contributed the terms in the scalar amplitudes that have cuts. And so we're channelizingly close to having there's just some polynomial terms or rational function terms that are missing. And also, there was one of the three helicity amplitudes in the n equals 1 chiral case, and other pieces of other helicities, n equals 1 chiral, that's this blue circle,

1:02:30 by Stephen Bitter, Bjorn Bohr, Dave Dunbar, and other collaborators. So that's sort of the current status, but who knows, by next week it may all look different. Because, in fact, things are happening pretty quickly right now. So previous speakers have also described the fact that an equal score super-Yang-Mills we decompose any endpoint one-move amplitude into a linear combination of certain rational variables multiplied by integral functions. And these functions are scalar box integrals multiplied by a particular factor which was chosen to sort of separate out power law dependence on the spinners of Lamenta into the C alphas and leave only slow logarithmic dependence in the f's on the kinematic variables. For example, here's the easy two-mass box with momentum invariants, mantelstand invariants if you like, s and t, and these are clusters of legs coming from the original endpoint amplitude clustered into these two masses, and there's a formula for it. This is the old-fashioned formula with five dialogues, and Gabriela Cavallini showed you the newer, simpler version with four. And you can see that it's defined so that everything depends logarithmically or di-logarithmically. If you expand these out in epsilon, these terms will also be logarithms in each order in epsilon. And there's a factor here that was introduced to make that happen. Similarly, you can write down functions through this order in epsilon, epsilon to the 0, what you need for the regulated cross-sections of four dimensions, and they're known for all the other box integrals. Now, I want to describe a little bit later the analysis of the twister space structure and the prescription for how to do this in a simple fashion was described in Witten's paper.

1:05:00 But the vanishing relations on curves and twister space lead to differential equations the spinner or closer to the momentum space in which the amplitudes are traditionally written. And so the condition that three points, i, j, and k, lie on the same line is that the antosymmetric tensor and the twister coordinates contracted with those three points should annihilate the amplitude. And that leads to a first order differential equation involving derivatives with respect to the anti-holomorphic spinner variables. So these spinner variables enter the square bracket spinner products, and then you also get a pre-factor of the of the holomorphic variables. So, if an amplitude is entirely holomorphic, then all the points will lie on a single line. It doesn't have any anti-holomorphic spinner dependence. On the other hand, three points could be collinear. This could be I, J, and K. Three collinear. If they only appear holomorphically, but there's another important sufficient condition, if they all appear through a momentum sum, that may contain them and any other number of variables. Then when you apply this collinear operator to this sum, using the chain rule, you obtain an expression which turns out to be zero through the Chauvin identity. So that's another sufficient condition. And then similarly, we've also heard discussed in Ruth Biddo's talk that coplanarity can be established by acting with a particular secondary differential operator, which arises from the hole contraction with the anti-symmetric tensor. So we'll look at the behavior of those pieces of the amplitude under those operators. So, this is very easy to do in the MHB case, that first set of n equals 4 superanguels series. In that case, the box coefficients turn out to be either zero or exactly equal to the

1:07:30 tree amplitude through the Park Taylor formula. And these are well known, because they're all holomorphic, to lie on a line in twister space. One of the initial questions before we got involved with these computations about a year ago was what is the full structure of non-MHV coefficients for which very few were known at that time. We just had a few for the case of N4-6, and they took on a fairly random looking form. There are some things that you can understand without, I mean, to be understood from structures of integrals, for example. These denominators here are sort of longer versions of the spinner inner products where, which can also be thought of as inserting a sum of slash momenta, the assignment slash into, in between two spinners. So, and I guess David Kossauer already showed a number of other examples that we got for the seven-point coefficient and pointed out how each of these denominator factors sort of played a special role, either in physical singular regions or in unphysical regions. So, before the all-end results were known, Brito, Cachazzo, and Feng computed one of the non-MHV seven-point amplitudes, and we computed the complete set of four Felicity configurations. And as was mentioned in Brito's talk, there are 35 boxes in each case, And so you need each of those coefficients. Now actually three of the four cases have a flip symmetry, so there are really only 19 independent still. It adds up to 90 some odd sort of separate coefficients, which had some regularities to them, but not enough to be quite pleasing to the eye. So I'm not going to show them here, unfortunately for you. And there were also several different types of cuts that we used to obtain these coefficients.

1:10:00 Typically, they were obtained first as rather large but analytic expressions, but these expressions had the virtue that they could be evaluated numerically quite easily, and as many times as you wanted. They also, we knew that they were not all independent, so we could realize that we didn't have to sort of simplify them all because of these linear relations due to the infrared And so using that information, we could construct simple guesses in cases where we didn't have simple expressions to start with, and then we could probe the behavior numerically by going into all the different regions where these denominators would be singular and examine what was going on there. And in that way, we found that the coefficients collapsed to a small number of terms. But that sort of set the stage for understanding the patterns in the all-n case. So what we found is a simple unified description of all of these C alphas in the general, next to MHV, endpoint amplitude from studying these results, computing additional triple cuts, and so on. First of all, the building blocks, as David Kossauer mentioned, coefficients of the most complicated boxes, that is the ones with the most masses, once you understand that the four-mass box coefficients are identically zero. And here's a formula which is also in agreement with the work of Grito, Chazzo, and Feng. And the three-mass box coefficient has a dependence on the kinematic variables. So what are these variables? S is the single leg that comes in here, and then A, B, and C are the three masses going around the loop. And so A stands for a set of momenta, and you see the sum of all those momenta appears inside here, and similarly for the sum of the B and the C momenta. And then, A minus, let's see, B1 denotes the first number of B, B minus 1 the last number of B. So these are these factors that straddle triple cuts that you could draw through here. And David

1:12:30 showed you where those, how those arose in the triple cuts. And you see there's these four sets of denominators, and typical factors for cosine or singularities appear here, and I'll say more about this later. The only place that the velocity dependence enters is in this numerator factor, which is raised to the fourth power in a way that's reminiscent of the original Park Taylor formula, where it is a spinner product raised to the fourth power as well. So, that one depends on the cases where, which luons belong to which cluster. This one is where there is one from each, and this is a form of it which shows that it can be written in a form having almost very similar structure to these denominator factors. So the two-mass hard could be obtained from the triple cut as well as a typical two-mass hard integral. It can also be obtained by thinking about these soft limits as you take one of the legs s to become, have almost zero molenta. As you do that, you go from an n plus one point amplitude to an n point amplitude. You see, you start off with a three-mass coefficient, which you know, and another one over here, and you end up with the kinematics of a two-mass integral. And so if you collect the coefficients of the one two-mass integral in the endpoint amplitude would then require the known consistency, known form of the soft limits. That determines this coefficient depending on two singlet legs and two masses. In terms of three-mass coefficients where you have deleted, you have considered a two-mass-like configuration. So it's a set with a single number. And so this shows you the soft limits of where you take a Guon to become soft are independent of the felicities of the other remaining hard Guons.

1:15:00 And so this equation is felicity independent. So the other box coefficients are more complicated, like easy two-mass and one-mass. So they were deduced by solving the infrared equations, or finding a solution that agreed numerically to the results of the infrared equations. And this is what the solution looks like in the easy two-mass case. And I just show it schematically here. If you have an easy two-mass one, that's characterized by two singlet legs on opposite sides of the integral, and two masses also on the diagonally opposite side. And the answer is a double sum of these three mass coefficients where the configuration is all terms where this S2 leg gets buried in the first mass clockwise. And there's a double sum because you can break this here and here at two places. So there's a double sum respecting this constraint and certain other, these guys can't get, you have to have at least two legs here, and then there's another term which is sort of the same thing starting from the S2 side and working around this way. So I'll sort of explain where that comes from by looking at multi-particle factorization in a minute. Strangely enough, the same equation, well, it's not strange, it sort of has to The same equation holds also with the reverse candidness, but there's an identity involved in there which can be observed numerically, but hasn't been proven analytically. The one-mass case is only epsilon more complicated than the easy-two-mass case. You take the ED2 mass case, and you consider resetting one of the clusters to have a single leg, and then you add one more term. So the twister structure of these coefficients. Now, as was pointed out earlier, you can give homomorphic anomaly arguments to constrain the twister structure of box coefficients, Cachausa showed how to do that, and it's been exploited since then in various cases. In particular, as Rich Vitto mentioned, our group and their group showed that these NMHV

1:17:30 box coefficients are completely Co-Clinger, so they're annihilated by this K operator for any subset of the N legs in the amplitude. And so that's a nice general argument, and indeed there was nothing wrong with it, because now we have the set of all coefficients, and we can actually explicitly verify by acting with this operator analytically that it annihilates each of the C3Ms. And all the other ones, remember, are linear combinations of these C3M functions, and since it's a linear operator, we can verify it for all coefficients. So, in fact, though, the C3Ms have a more restrictive structure, which is, in some sense, completely independent of the helicity configuration. It only depends on the kinematics of the integral in question. And this is what it looks like. So, we know they all lie in a plane, so I'll use the plane of the screen. And then, in addition, the line, each cluster lies on a line, and the A and C lines happen to intersect in the point S. This is something we saw empirically already at the seven-point level, but we weren't completely sure about it because we only had seven points. In fact, you can see that ladder condition. There's a bit of algebra to show this explicit constraint, but the collinear conditions are very easy to see. Recall that the only issue in checking the collinearity is where the anti-holomorphic spheres are. So one place they show up is in factors in the free mass coefficient that have a ka slash and a kb slash inserted between these two. We wrote standard spinner products, then there's some holomorca pieces, but there's an anti-holomorca one that includes legs of A and legs of B. However, because the legs of A only appear as a sum, which can actually be extended to include S, because SS is zero, that means that S and all A belonging to A can be on the same line, as shown here. And it obviously

1:20:00 also prevents all KBs to be on another line. You have other factors with KCs which similarly allow S and C to be on the same line, hence the intersection in S. And then finally, the only other non-holomorphic factor is kb squared, which puts all the b's, or allows all b's in the yellow line. So let me just give a kind of rough correspondence between what you might get if you extended the computation of Rand-Uber, Spence, and Travolini, he described we talked about yesterday, to the NMHV case, you would draw that, you would presumably write down MHV vertex diagrams of this general sort, three in a ring, and you might put one off of here. This seems to correspond quite closely to the pictures of these coefficients, and in the sense that these legs should be associated with C, these legs with A, these legs with D, and we'll ask what we can worry about later. But this one here would naively give some picture like this where we join blue and green correspond to these two, and this could be coming out of the plane. Now we know from the holomorphic anomaly arguments that the structure can't be non-planar. And similarly, in the MHV loop amplitude, you take this wobble away, we know that these two lines really are the same line. The coefficient is the collinear Park-Taylor amplitude. But even then, if it collapses to a line, it still looks more degenerate than what we see here. So that's sort of an open question. why we don't see that part. By the way, you can extend these polomorphic anomaly arguments go on to say something about the structure at NNMHV and there you will have now have four mass coefficients and if you can show easily extending an example worked out and shown by

1:22:30 Grito also with Pachazzo and Feng, that you get this kind of non-planar structure, and you probably continue to get more ring-type, some kind of corrugated crown as you go on to hire even more holicities. So, a lot of people sang the praises of these amplitudes and how amazingly simple they are, but I want to be a little bit cantankerous and ask the question, why aren't they simpler? I mean, that is to say, how do you know when you've got the final, simplest form for an amplitude? And, I mean, one answer, sometimes simplicity is in the eye of the bigger, because there can be alternate forms that trade off different nice properties against each other. In fact, yeah, so the sort of minimal amount of complexity is required force the general analytic structure of amplitudes. We've heard a lot about the branch cut behavior in unitarity, and a little bit maybe about the collinear pole behavior. As two particle invariants get small, or two particles become collinear, you can understand various denominator factors. And here I just want to look a little bit at how the multi-particle pole behavior here works in these particular amplitudes. So multi-particle poles separate scattering amplitudes to different space-time regions, each containing a non-trivial scattering amplitude. So the general structure is, actually this was pointed out by Byrne and Chalmers a while ago. The one-loop amplitude factorizes into a one-loop times a tree on either side of the pole, plus there's a kind of non-local piece that's due to the existence of infrared divergences. Roughly speaking, soft luons can flow along here and tangle things up a little bit. And in the case we're interested in at the moment with three negative felicities when you saturate the intermediate state with velocities and you find that you get MHV multiplying MHV in each case. So another point that helps you understand the structure of the NMHV result is the fact that the MHV result

1:25:00 contains only a specific type of integral, the easy-to-mass boxes. This is just one of these f functions here, and the coefficient is the tree. So what this means is that every easy-to-mass box coefficient in this amplitude has to have a pull as every sub-cluster inside its mass goes on shelf. And that's because each of those factorizations will result in a different leg of this expression being identified as the leg P in this factorization. So you have to find a large number of poles in these easy two-mass ones. And in fact, most of them are identified explicitly in this three-mass formula. There are also some hidden poles that contribute a little bit to the picture. of them are found here. The residue of this pole is a little algebra exercise to see that this quantity always factorizes in the limit that kb squared goes to zero into the product of two tree amplitudes. And that has the right form that we want because we want a tree here and one residue and we want another tree from the other side of the, of p. So we want tree times 3, that in fact gives you one of the easy two-mass boxes where p is a singlet leg. More complicated ones are where p is buried in one of these clusters. So these blue legs here are supposed to be the legs in the factorization, and we're looking at this two-mass-easy coefficient, and we have this formula that relates it to two-mass coefficients. And so what you find is that the reason, in some sense, that you had the sum over all of these different double sums was that you could always choose the gap here. So these three legs were precisely the cluster that appears in this hole. So we have a match between the shape of this formula and each multi-particle pole. So that's, I'm not saying that's the simplest possible way, but it's typical that when you have it written in this way,

1:27:30 a single multi-particle pole goes with a single term. And it's already been mentioned that you can reconstruct the tree amplitude using the incorrect equations, and this is one example of the seven-point level. But here, let me just give a heuristic, twister-space picture of what's going on, because there are certainly many other such equations, and very efficient ones have been found now by Norban, Strabo, and Volovich, and exploited further by and then, in any case, there are definitely, some are better than others, but there are a large number of these equations, but they all have the generic form, because the generic twister space structure of these box coefficients are these planar diagrams. The sum here somehow gives you a sum here, which I'm just writing down, say, from the CSW rules directly for the tree amplitude. So there's some kind of relation like that in general. And again, this sort of recognition that you get these simple tree amplitudes has been exploited and extended. Okay, so I have just a few minutes left. Fortunately, I could be pretty brief I wanted to discuss the extension to amplitudes involving Higgs bosons. The Higgs boson plays a pretty fundamental role in the standard model, as Steve Verne pointed out. Well, some people decide whether the Higgs boson is part of the standard model, but if it is, it is certainly the only undetected particle of the standard model. And if it exists as an elementary scalar, it is the only scalar. Furthermore, it underlies electroweat symmetry breaking, which is responsible for the masses of the quarks, the leptons, the vector bosons. So we'd like to discover it and measure its properties as well as possible. And that involves computations at tree level in higher order, its interactions with the partons. In particular, the dominant way it's produced with the Large Hadron Collider is via a loop containing a top quark. The Higgs likes to couple to max, and the top is the heaviest quark, so it has a big

1:30:00 coupling here. But there are no tops inside the proton, the first approximation. So instead, we bring gluons in from protons and produce a virtual top. And so you're interested in this kind of process, plus then radiation of gluons off of all these lines here. So, in fact, the top cork is quite a bit heavier than the mass of the Higgs incurred from certain precision electrolyte data. And that means that it doesn't move very far in this loop, or that the space-time picture is that it shrinks to a point, and its effects can be summarized by an effective interaction, which is just the Higgs field multiplied by the standard QCD Lagrangian. So, we're interested in computing amplitudes like those in QCD, but where we have one more Higgs boson, because producing two is a quite small probability. So, the point with one is most important. And there's some experimental evidence or to go off of, which is that some of these were computed up through the four and five point cases by these groups, and there was one simple form observed that we initially thought would be a good thing to look at, where all the gluons are plus. But playing around with this, we could not get an MHV formalism to work. And we think it has something to do with the fact that this numerator sort of secretly has anti-holomorphic stuff in it. That is, the Higgs mass squared is a sum of momentum invariants that can be rewritten and contain anti-holomorphic as well as holomorphic objects. This is kind of reminiscent of one-loop amplitudes with all plus solicities and gauge theory and also features of gravity amplitudes. out that you can resolve the problem by splitting the amplitude, or actually splitting the Lagrangian in the self-dual and anti-self-dual parts. So we could think of the Higgs field as a scalar and put in a fictitious, actually in supersymmetry it's real, a pseudoscalar. I mean it's an actual particle, pseudoscalar in supersymmetry is true. But we can package them

1:32:30 into a scalar field, which is the lowest component of the superfield, and supersymmetry can be used to motivate this, but also the fact that when you use this field, what phi couples to is the self-dual part of the field strength, and what phi dagger couples to is the anti-self-dual part. And then what you find is that the phi amplitudes with lots of gluons are described by MHV rules. And one of the kind of amazing things when you first see it is that the phi-MHV vertex is identical to . . .