MHV Rules in Gauge Theories

0:00 So the advertised title of my talk was How MHA Rodes in Rise and Gauge Theory, and I I have to tell you right away that I will not have the proof of the information that MHP awards are engaged here, but I hope I will be able to tell you something which is interesting, some interesting insights. So the talk will be in two parts. The first part, which is based on a published work with Nigel Glover, is about what the title is, how the figures can arise in gauge theory. And the second part, which will be probably the last third of the talk, would concern new applications of these very new decursion relations by Brito Kachazen and Frank. And I'm sure there will be more talks, so our briefs talk will tomorrow will be, will be, will be dedicated to this. I should mention that, I should mention that in the second part of the talk, I'll use, I'll, the discussion will be motivated by, by the work with Simon Badger, Lance Dixon, and Nigel Glover, and Simon is a student at Durham, but he's also present for this. So, the third part. Can MHV rules for general structures in Wheaton be derived directly from Geisner? And by MHV rules I mean the MHV rules which Peter explained so nicely in this talk. I am grateful to you for reviewing the subject and also for co-inventing it together

2:30 he's collaborated with this wonderful method which lets us explain. So, basically the idea... So, what we wanted to do was to derive the image with this novel diagrammatic image rules, directly in DHT. And we wanted to derive it from the Lagrangian off-shell formulation of the theory. And this will be one of the points where we will run into difficulties. So we did not want to derive these rules by looking at the amplitudes and analyzing their soft and collineal and multi-collineal similarities. The hope was to derive it directly from gauge theory, from Horschel rules, by some clever reorganization of perturbation theory. And if this could be achieved, right? So if we could achieve that, we would be able to understand how to use emission rules for cut non-constructible loops, that is for loops in non-supersymmetric theories, right? In non-supersenatic gauge theory, such as QCD, there are one-loop diagrams which have all positive helicities, or all positive and one negative helicities, and these diagrams do not punish, and these diagrams were so far impossible to calculate using MHV rules. If we can derive MHV rules or extend MHV rules from a reorganization of perturbation theory, we would be able to apply them for calculating such loops. Another related class of loops which cannot be calculated with MHV rules is another class of non-constructible loops where the states propagating on the loop are unphysical states. There are states not of plus or minus helicity in four dimensions, but there are some states that are helicity states in d dimensions. If d is not equal to four, you have more than two physical states. And then you could also address higher loops, and so far no one has attempted, or it's actually not known, whether the initially rules will work with higher loops. So, even though I will not be able to fully derive the MHV rules from the quadrants, I will be able to say something about how to apply MHV rules to calculations of these rules.

5:00 So, the continued list of motivation, I really want to understand, the continued list of motivation is this, we need a field theory derivation. The fact that MHV rules are so successful in deriding novel classes of tree-level amplitudes and deriding novel classes of loop amplitudes begs for fifth theory derivation. It does not exclude string theory derivation. In fact, the starting point, which I'm going to use in the next transparency, would be the self-dual Young-Neal's Lagrangian, precisely the object which appears in Witten's string theory, in Witten's topological B model. And there will be a certain overlap of what I'm going to say in one minute with these things mentioned in Witten's scope and also in Peter's scope. So it doesn't exclude string theory derivation. In fact, it can run in parallel. Secondly, if we can derive this rule from the first principles in field theory, we would not need to check the new results obtained with these elegant rules against the previously known expressions. Sometimes it is faster to derive a new amplitude which was known before. It's faster to derive it with new rules than to quote it, and then to quote the previously the two numeric, so maybe it's time to stop, if you have the generation, it would be nice to stop doing it, so there would be, would it be a better handle on loop amplitude, would be interesting applications for inclusion of massive particles in this approach, and actually the last part of my talk will deal explicitly with massive particles, and also Also it would clarify the role of twisted spells, self-duality, off-shell versus on-shell quantities. So let me start. So my starting point is going to be the anti-self-dual theory and middles, in the first order formalism. And this formalism appeared in the papers in the past, so the paper of Chalmers and Serial was dedicated to discussion of this formalism, which was also prominent in Whitten's paper in December 2003 and in Whitten's lectures.

7:30 So the action of the anti-self-dual Young-Neals is an integral D4X trace G-minu F-minu. is the usual field strength, so F-menu is made out of the gauge field A-menu, while G-menu is an elementary self-dual field. So in this approach, G-menu is not thought of as a field strength of anything. It's an elementary field, it's anti-symmetric in mu and nu, and it's self-dual. So here I'm in Niemkowski space, and the self-duality, I find that G is equal to G-star, and G-star is higher than epsilon-dual G. So, because gvenu is anti-self-dual, sorry, because gvenu is self-dual, excuse me, gvenu is self-dual, it selects only self-dual parts of fvenu. So, the same logarithm can be written as integral gvenu contracted with f-self-dual-venu, and f-self-dual is defined as fvenu plus the dual of fvenu. So the classical equations of motion, so if you differentiate with respect to g-menu you get a self-dual f-menu is equal to 0, and the equation for g is that the covariant derivative is respect to a, maximum g-menu is equal to 0. So because the mu-nu field is such that only self-dual component enters the action and the self-dual component by equation of motion has to be equal to zero, the H field mu is such that it can generate an arbitrary anti-self-dual mu. As such, a nu corresponds to a particle which has a positive unicity. And this is the connection between self-duality and the self-duality and the multiplicity basis. So a nu corresponds, a nu, which I will sometimes call an arrow plus, produces particles of positive unicity. g nu, which was self-dual by default, will be produced in particles of negative unicity. And the Feynman rules, the Feynman rules for this Lagrangian is that there is a propagator which connects G and A, and the errors on all these pictures below do not correspond to

10:00 a momentum flow, they correspond to a quillicity flow, so in order to draw pictures, next I will always be drawing a picture that in the propagated quillicity flows from minus to the incoming arrows correspond to positive felicities and I'm going to negative. So there's a propagating which connects G minus and A plus, and there's a single three-point vertex, which is plus, plus, minus, right? Because you have one power of G, and there are two powers of A in the commutator there. So what kind of diagrams can we draw? At tree level, and all the momenta here we need to think about the momentum will be considered in time so if we draw diagrams at tree level the most general tree level diagram will contain one g and that was supposed to be g minus it contains one negative helicity particle an arbitrary number of positive helicity particles and at one loop you can you can contract g with a like this using a ga propagator and you would have a single loop flowing out of the loop. And there are no high-loop diamonds. There are only trees, which are 1 minus all pluses, and only a single loop, which are all pluses. Nothing else. And this theory so far, so the theory we described so far, it's not the full Young Mills, of course. It's a part of Young Mills. It has the same spectrum as Young Mills, which is a positive and a negative curiosity state. It contains only one three-point vertex of a plus, plus, minus, star. It doesn't contain a minus, minus, plus. Three-point vertex, it doesn't contain a four-point vertex. But these things will appear soon. So at three levels, we have one minus, you know, plus, um, helicity amplitudes. These amplitudes, so if we calculate all these diagrams, we will discover that they vanish on the shell. So if we put all legs on the shell, they will vanish. off-shell. So all n-1 lags have p squared equal to zero and the n-1 lags have p squared not equal to zero. Then Bardeen showed a long time ago that in this case you will get the variance really at one off-shell current in QCT. So this subset, this kind of restricted anti-cell dual Young-Mills will give you a correct tree-level processes with one off-shell

12:30 with one off-shell lag, correct meaning that they would agree with the one-off-shell process calculated in blue QCD, in the universe. If you look at the one-loop interactions which appeared in the previous transparency, which are OI plus positive helicity amplitudes, this one-loop amplitude actually does exist on shell, so you can set all p-squares of external legs on shell, And it, again, you can calculate it in this soft dual theory, and you will find out that this result agrees with the same results in full QCD. This result was derived years ago by Bernkiesel-Possover and Malone, and Charles Sikin congeny, when other people argued, in fact, congeny did the calculation, which showed that this O plus one new pump, which is calculated in Sobdual-Yang-Nils, agrees with this efficiency difference. So now we need to... This appeared in the previous talks already before. We need to perform... We need to add... We need to add self-dual degrees of freedom to this anti-Sobdual-Yang-Nils-Lagrangian to turn it into full Yang-Nils-Lagrangian. And for gravity, I'll call this precision absolute deformation. And, well, I learned about this procedure from this stage. So, you take the G, U, F, M, U, self-dual Lagrangian, so this was the, this is what we had before, and you add a term which is basically epsilon, which is a parameter, G, U, G, U. So, in string theory, in topological B-model, this term arises from D and C. If you think about field theory, this is where it comes from. If you think about field theory, which is where we are now, you just add this term by hand. G appears most quadratically in this Lagrangian. You can integrate tau by using equations of motion. So, equation of motion for G is G, epsilon, and G is entering the result. We plug it back to the Lagrangian, and you discover that the Lagrangian goes like 1 over epsilon epsilon-dual-squared. And neglecting the total derivative term, which is never important in perturbation theory,

15:00 epsilon-dual-squared is the same as that we knew. So, the epsilon deformation of the purely anti-self-dual Lagrangian gives you the full young answer. nice, perturbation theory. We want to reorganize perturbation theory in Young-Mills as non-perturbation theory in terms of G squared Young-Mills, in terms of Young-Mills coupling constant, but in terms of this epsilon parameter. And see how it is related to this image we use. Since the epsilon will appear in the full Young-Mills action as one of epsilon, the meaning of epsilon, this is G squared Young-Mills. So the action of full Young-Mills, of course, goes But in identifying epsilon with G squared young mutes, you should remember that it corresponds to an asymmetric scaling of positive and negative helicity particles. So here epsilon was accompanying only the negative helicity particles, not the positive. So, this new Lagrangian has the following final perturbation theory, or final perturbation theory. So there's the same propagator, which connects a minus to a plus, there's the same vertex as before, which is plus plus minus vertex, and there is a new vertex, which is a two-point vertex, which is a GG vertex, so it connects minus to a minus. Since I want to treat this epsilon as a new vertex, I'm not treating it as a contribution to a propagator. It's a new two-point vertex. The idea, as I mentioned, is to reorganize the standard Gamil's perturbation theory in terms of epsilon powers and see whether we can repackage the whole perturbation theory in such a way that it will consist that the new elementary building blocks, in some way, would be MHD vertices. So it's order epsilon to be zero. So the idea is that we will work to a fixed order in epsilon and to all orders in this other vertex. So order epsilon to be zero when this vertex does not appear and when its determination is irrelevant, right? So we work to all orders in plus, plus, minus vertices, and we will get the same diagrams as we've had a couple of transparencies before. In three-level diagrams with one minus and all pluses, and one loop diagrams with all pluses. And to the epsilon to the one, and working to all orders in plus, plus, minus vertex,

17:30 we will actually get three-level amplitudes, which have now two negatives. And these are precisely the MHV three-level vertices which appear in Peter's story. Then if you work toward the epsilon squared, you will want to, in order to relate it to MHV rules, so toward the epsilon squared, you would have three negative vertices. You would want to repackage this amplitude with three negative vertices as made out of So we want to repackage the epsilon-squared diagrams in terms of two epsilon-tubicose diagrams. And you'll see in what sense we can succeed in this bar. There is an, just a side comment, there is an alternative way to think about this perturbation theory, an alternative and equivalent way, is to keep GA propagated the same as before, keep plus plus minus vertex the same as before but then instead of having this two-point vertex you can you can combine this two-point vertex with the plus plus minus vertex in such a way that you get minus minus plus vertex. It would be the first order in epsilon. So you can think equivalently on the defining rules for this epsilon-de-torn axis of Duell-Young-Mills is that it has one propagator which connects plus to minus and two types of three-point vertices plus plus And no four-point vertices necessary. So, addendum to the first order, you take a single minus-plus-plus-plus amplitude, which was in U and T-Soul-Dulian fields, and another one like this, so it has, on each side I have one minus, and then you connect by, you connect this two by the two point epsilon vertex. And so, the sum stands here, it's going to be summing over all possible minus plus plus plus amplitudes, I just drew some examples, but you have to sum over all possible bits of these forms, and they're connected in the middle by one epsilon. And this gives an MHP tree amplitude, so. I don't want to calculate MHP in this way, but I want to learn how to repackage everything in terms of this graph. And then you can, this is a tree diagram, and then you can have a one-loop diagram,

20:00 where you, for example, connect a plus to a minus with a propagator, so it becomes a one-loop amplitude, it has one minus, and all the pluses, and this, as I mentioned, are the amplitudes which exist in QCT. They do not exist in supersymmetric theories because they are ruled out by Susan Ward identities, but they exist in non-supersymmetric theories, and that's what we want to calculate. it. And as two loops, you can close another minus with a plus with a propagator, and this is a two loop for a plus amplitude, which also exists in QCD and has never been calculated as far as any number. And that's all. You can't have any more loops, because there are no minus lines left, which can be connected to your classes. So we discussed epsilon to the zero order, so now instead of drawing this tree with one minus and all classes, I just drew a blob with one minus and all classes. So there was a tree with one minus and no pluses, one loop amplitude with all pluses and no higher loops. At epsilon to the first order, there are MHV, there are MHV tree level amplitudes appearing. There are one loop and two loop contributions and no higher loops, now at epsilon squared order. This is where the non-tree will start, Mr. Post-Puskal, start. So the most general tree would be a tree with one minus, which I indicated as an arrow, of classes, which I didn't realize, so this is in the sense a skeleton weaver done. And there is another tree with one minus and no classes, which are not indicated, and the third tree with one minus and no classes, and they are now connected by two epsilon vertices, right? And that is the first principal object, which appears in the epsilon t4 and anti-cell dual Young-Mills, which is equivalent to four Young-Mills. So the question is, can you represent this epsilon square vertex in terms of two MHV vertices? So this was one MHV vertex, or MHV subamplitude, with one epsilon, right, which is connected to two trees, and another epsilon connected to two trees, and then you want to join these two trees by joining minus and plus.

22:30 And is this, the joining of two trees in axis of dual theory, is it again giving you a tree in axis of dual theory? So in other words, is the blob connected to this blob in this way, plus in the other way, gives you the same blob back? Naively, that's the case, right? If the blob corresponds to the most general tree, If one blob responds to a most general tree with one minus and all pluses, and you connect it to another most general tree with one minus and all pluses, in such a way that you eat one minus and all pluses, you have to end up with a blob which has one minus and all pluses, and if that's the case, you can repackage, you can repackage these amplitudes as made out of the mixed rules. Unfortunately, the story is a little more complicated. All the way and that way, but it can be not total k-efficient too much. So just briefly, and I'll explain it, so briefly, if we go to order epsilon to be p, this is not an explanation, this is what happens if you go to order epsilon to be p, you can have two-level diagrams with p-epsilon, so these diagrams can be arranged along the line, or they can have various bifurcations, and the question is, can you repartage all these diagrams by connecting in various ways MHP diagrams. And again, these are three-level diagrams, and there are also loop diagrams, but what's nice about loop diagrams, they just stop at maximum P plus one loops, and second, well, it's easy to discuss the principal content. So now, I'm going to do an example. How many minutes time? So remember Zee Bernstor when he said there is a simplest amplitude which you can calculate which is a kindergarten star, a minus minus plus plus plus five point amplitude and there was a page, there was a page of outputs with how you calculate this amplitude if you compute using Feynman rules. And this is the same amplitude, minus, minus, and, and, well, sorry, this is minus, minus, minus, and two pluses. So it's a, it's a, it's a, it's a simplest

25:00 NMHV amplitude, or it's a Googley MHV or MHV bar amplitude. It's the simplest object that exists, which has extremely simple form in terms of NMHV rules, or in terms of Park Taylor amplitude. And what I'm trying to do here, in order to see how it's made out of is to actually draw all defining diagrams in this deformed antisocial theory. So, there are black diagrams on the left. There are 12 of them. And they correspond, the sum of these things corresponds to a pull-ups. And the way it's organized is that, so there is, each amplitude has two crosses, which corresponds to two epsilon vertices, right? And there are three blobs. in the hollow-ordered perturbation theory, and I'm recapping the group theory part, as everybody says, and I'm looking only at the kinematic part, and it's cyclically ordered, so it's always 1, 2, 3, 4, 5. And now I'm looking at all possible arrangements of 1, 2, 3, 4, 5 in all the blocks, right? So, for example, this diagram corresponds to 1, 2, and 5 coming in the left block, 3 in the middle block, and 4 in the last block. correspond to all of these possible arrangements. And the outgoing arrows correspond to negative infinity blooms, which are 1, 3, and 4. 1, 3, and 4. So this diagram, now in blue I write down what stands between the crosses. If I now rewrite the middle block, which is the most important one, in terms of just final diagrams. So this is a very simple one. This just has one outgoing left, which is 3. So between the two, there is only 3. three-point vertex. But if I look at the diagram number five, which I'll investigate in detail in the next transparency, so the middle block has two, three, and five. So what can happen? Two, three come out this way, and five this way, or five comes out and down between two and three, or two and three go like this, or there's a deprecation between two and three and this. These are all possible choices of what you can have in terms of finding the of our theory. And so altogether there are 12 diagrams, and you should analyze all 12 of them, but you will analyze only the fifth one to see which words and what lessons are.

27:30 So that's the diagram drawn again. So this is a number 5 diagram with 2, 3, and 5 between and I wrote down all the arrangements of two three. Now, I want to, this is the first principle's result. So this is the, this is, well, if it's really final diagrams, this is what is the part, the number five part of the two one. So let's try to repackage it as coming from final rules, from, excuse me, from the initial rules. So, using the CSW MHV rules, there is a contribution from an MHV vertex, 3 minus, 4 minus, and 1 plus, connected to an MHV vertex with minus, and minus, 3, and 2 pluses. So, this is one of the new MHV contributions. There are others, right? I will call it, so this dashed line is only to guide the right. And this is where, it's not really a cut, but this is where I'm imagining I'm separating the left MHP from the right MHP. And I will call it, for convenience, S3,4 cut, which means simply that I'm separating 3,4 from the rest. Right? This is a part of the MHP distribution. And when I look at this diagram, now I want to cut, I want to separate, I want to choose, I would choose the integers which separate 3 and 4 from all the rest. So from this diagram, this is where the path lies, so this is 3, there was the 4 there, so it cuts 3 and 4 from all the rest. The second diagram gets a contribution with 3 and 4 cuts from all the rest, and the three remaining diagrams do not contribute to this emission rule, they will contribute somewhere else. They will contribute to some other arrangements of emission differences. Now, I look at another arrangement of MHT vertices, where there is one MHT vertex on the left, which has one minus and minus on the internal line and the plus, and the other MHT vertex has three minus and four minus. So I call it 2, 3, 4 cuts, because it has 2, 3, and 4 over them. And now, with the purple cuts, 2, 3, 4, I indicate how I would have to separate these diagrams in order to attribute them to this initial order.

30:00 And again, there are some diagrams which do not contribute, which are these three, and this diagram does contribute to this separation of universities, and this one does. Now, the key point is what happens to this diagram. Before I drew the lines, I had a single, final diagram. I had a single, final diagram coming from the epsilon-deformed G-80, or coming from this formation of antiseptic union. I had a single diagram, and now I have to count twice. And a single diagram, that's supposed to be the correct answer, but when I attribute it as coming from this MHV, from this MHV rules contribution, I count it once, because it has a black card there, and when I attribute it to be coming from this contribution, I have to count it again. This is the key essence of mismatching it back and forth. All the diagrams will be accounted for, and they all will be there, but they will not be, surprisingly, somehow, accounted for certain ones, and we will find a reason for that. So to summarize what you have so far, we are trying to represent one log with one minus in any number of classes between two epsilons, is coming from the MHV rules where we perform a cut between these two goals. And a single contribution, now, to some generic contribution between these two epsilons, it has a lot of, you know, it has a lot of lines between these replications, would in principle have as many blue cuts as are drawn there. So it could be attributed many, many times to different separation between dimension diagrams, so it would be over-counted. So what's the reason for this kind of unpleasant occurrence? So we have been willingly and on purpose operating these off-shell quantities in field periods. vertices and sub-diagons were honestly off-shell and this is how you do the calculations in quantum field theory, right? You calculate Green's functions on the off-shell and then you LZ amputate and then get on-shell amputated. The difference of this approach with the

32:30 with the MHV rules of CSW and also with the new approach of Brito-Pachazzi and Fang is that they operate with all-shell building blocks at all intermediate stages. And this is summation of implications which are very difficult to derive from the off-shell field theory. So, you can think about it this way. You could calculate an MHE, synchronous MHE diagram off-shell. Then you take it off-shell. And when you take it on-shell, a lot of individual diagrams will vanish. And I will find some particularly clear see how a lot of diagrams, in this case, will vanish. And after this diagram is vanished old shell, you then, following MHV rules, you then bring back the old shell building block of shell. So when you connect two MHV on-shell amplitudes together, by propagating you have to put one of the legs of shell, and when you put it of shell, the things which vanish are not going to reappear. So in the off-shell perturbation case, something we we are trying to do so far. There are too many diagrams. There are too many diagrams and we cover this step taking some intermediate building blocks on shell and then bringing them back of shell. This kind of magic I don't know how to derive in field theory from the Lagrangian methods, right, from the, from this, uh, from this matter. Um, maybe you say it's not surprising, I shouldn't have been trying to do it, but the hope was somehow to, the hope was that if you reorganize perturbation theory in a clever way, this, this step that you're combining on show quantities will somehow be not, not so crucial, but it took to be crucial. Anyway, Let's see where we can get with what we've got so far. So this is a general off-shell which I've shown before. It has one epsilon vertex and then it has all rather complicated, you know, topologically complicated one minus and all plus vertices. So we'll have to move about. However, and this is a fully off-shell result, and if I would turn some of all these diagrams and put it on a shell, since this is the right theory, I have to get a part and I would. But no sane person would do a calculation this way. There's only too many

35:00 diagrams. While if I write down an on-shell MHP diagram, and now, as you know in the CSW approach, MHP diagrams are not derived, they are taken for granted, they are taken from Park Taylor for me. But in the BCF new recursive approach, these on-shell diagrams are derived MHV diagons are derived from more elementary built-in blocks, which is plus plus minus and minus minus plus. So if I actually represent what follows from the BCF recursion direction, the MHV diagram would have an extremely simple form. So there would be no spider webs and branchings of these diagons, and there wouldn't be a sample, it would be a single diagram. with just a single diagram, which is pretty much a line, and it would have a one minus, and it would have another minus, and all the blocks are arranged in this way. If I have time, I will have it to experience you. We'll derive this expression using the approach. It's a straight line with no branching. If you don't see an epsilon there, you can And remember that minus minus cross vertices can be represented as, sorry, minus minus cross vertices can be represented as plus plus minus vertices with an axon on the edge. So, the axon is in the middle. So, it's a much simpler, on shell, this is a much simpler picture than on shell. I just showed you the picture, but I have to show where it rises from, and I will. So if I would draw, if I would combine now two MHV diagrams, represented in this DCF form, in order to get an MHV diagram, what's interesting, and so I would have to connect the picture looks like a connection between two straight lines, or if you think of a propagator as an intersection point, it looks very much like a twisted space picture of two intersecting straight lines. There haven't been any twisted space in this discussion, in my discussion so far, and these diagrams are drawn in momentum space in a particular on-shell fashion, with a particular choice of reference, but it's kind of an intriguing thing, that there are no biases there,

37:30 and the NMHV diagram really looks like an intersection of two lines, and an NMHV diagram would be three lines, et cetera, so maybe it's patients, maybe there is something different, maybe not. And if I come back to what I had before, which is a connect, which is a... So I take these lines now from the previous transparency and I advance them. So I'm now constructing... This was a famous conspiracy, where there were two MHV vertices attached to each other and they would form an N-MHV vertices. So now I would bend this line upwards and this line downwards, and this part would form a new straight line. This is exactly what is shown here. The purple thing is one MHV, the blue thing is the other MHV, and you actually see that there is only one place like Mr. Cut. So if my building blocks were old-style building blocks, as in the BCF, as what's emerging from the BCF recursion, I would not have any over-connect. which was the animation vertex would appear immediately with the addition to one but i would not be able to explain why of all the possible webs of diagrams i have to get the field theory only this one so far. Just a hypothetical question. Well, many times we ask the question during this workshop as well. How is it that physical scattering amplitude when you calculate the final rules? they really look like horrible answers, how does it happen that they turn out to be much simpler than anyone might have expected? This is because they are on-shell quantities. This is one of the answers, maybe there are other answers, maybe there are better answers, but what we always do in field theory, we operate with off-shell quantities, we calculate off-shell Green's functions, And somehow our, at least in the collider physics, at least in the particle physics, our ultimate game, our ultimate goal is to get on-shell scatter in amplitudes.

40:00 And we get on-shell scatter in amplitudes manipulated on-shell quantities. And that's the reason why, using standard weights, the amplitudes look horrible. But the two new methods, the CSW method and the BCS method, operate on shell quantities. And that is why, or this is one of the reasons why, they need much more elegant and straightforward answers. It's very difficult to derive from the official approach to the theory. So, we failed to fully derive the, we failed to derive NHV rules from the HDA that we can see where they would come from if I had this procedure of taking off-shell quantities on shuffle, if I had a proof of this procedure at intermediate stages. Are there any lessons we can learn for loops in QCD? So what lessons would we learn for loops in QCD if the approach went full? So, as I mentioned before, for the x1 to be zero, there is an all-plus amplitude, which is a cut. So, lessons for loops in PCD, I mean, can we get cut non-constructible amplitudes from MHP loops? Well, the simplest cut non-constructible amplitudes, all-plus, comes from epsilon to the zero limit, so it comes purely from the self-dual theorem, from before the MHP loops. No, MHB digress the third absolute to the first. This one I chose earlier. It's been calculated. It's been calculated in both theories. It decreased, as I mentioned, three times, and that's the right result. So it should be used as a new building block in MHB group rules in non-constructible theories. So let's use this vertex as a new building block, and let's go to order absolute to the first. So what's going to happen at order absolute to the first? As far as non-constructible loops, I can understand. I can take this loop with all classes, and I can connect it to an MHE tree. So this is an old building log, this is a new building log. In doing that, I get a one-loop amplitude, because this is still one loop, with one minus. And this is what Khashoggi's research in Ruten proposed in the second paper. This is a new building block, and this is an old building block.

42:30 And you combine the two, and you probably should get an amplitude, one of the amplitude is one minus in all classes. And they said that they tried it, and they couldn't get... couldn't find a non-shelf in... Well, they couldn't get, they couldn't finally derive the, the known answer for 1 minus and O plus 1 will accomplish it. And we tried to do and we couldn't derive it, and now we maybe know why. Because there is another diagram you can draw, which is an MHV tree, which is plus and minus, where plus and minus is contracted on itself. So no one in the MHV loop perturbation case so far was calculating the amplitudes where there is only one MHV contracted to itself. And if you change now, this is a new building rule, and this is a new building rule, the conjecture is that you will get all what you need at one loop. So this is a new building block which is all class and another new building block which was an old MHP, two-level MHP, self-contracted in the loop. And if you go, and nothing else at one loop. And if you go to two loops, you can self-contract MHP twice. And you can't do anything in everything. For example, to give an example of the loop calculation, so in addition to the Bedford-Brandt-Huber-Spenson-Toraglini type calculation, which Toraglini will review in his next talk, When they commuted loop amplitudes with two different MHV-2 level vertices were connected by a loop, one should add a loop. So here we are calculating MHV-1 loop amplitudes. So you can add MHV-3 two times to a loop, or you can add, basically, a next-to-MHV-3 to a loop, or you can have an MHV with a self-contracted line attached to another issue. And this conjecture would be, would give you the full answer for one loop, including cut-constructible parts, which is reproduced so far, and the cut-long-constructible parts,

45:00 which would vanish in super-semitic theories because super-partners will annihilate super-partners will cancel contributions in the loop but in the non-super-semitic theories this thing should definitely not vanish so now I'll move to the second part of the talk and I certainly don't want to review in great detail the recursion relation, the recursive approach of BCF, because we'll talk by Britain tomorrow, but I just need to, there's something which I want to say about some applications of this approach to massive and massive, of amplitudes of massive and massive spars. So I need to give some And the idea is that general tree level amplitude can be represented in terms of recursion relation which relates two amplitudes with fewer external legs. And the idea is that you mark two of the external lines, and you will treat a helicity, so everything is massless, everything is on strike, and you will treat a negative helicity spinner, which corresponds to this external leg as the reference spinner psi, and it will treat the positive helicity spinner, which corresponds to this leg, as a reference spinner sine tilde. And then, if this is your choice of the reference spinners, you can represent the amplitude in terms of lower point amplitude connected by a propagator. The key point in the approach is that all of the building blocks in this approach are entirely on start. They are entirely on shell, and this is achieved by shifting this momentum, and this momentum in this diagram, of course, is off shell, because it's the sum of all of this on shell momentum. It doesn't have to be off shell, it's off shell. But you don't use the V in the elementary amplitude, you use a shifted V with a cleverly chosen shift. And then in addition to it, you shift the external Mark's momentum with the same shift.

47:30 And the key point is that after the shifts are performed, this momentum becomes on shell. And this momentum, which we are on shell, is slightly modified by stay on shell as well. And I admit this is not a detailed review, but it will appear tomorrow, and if you run the sum all the way through, you can represent a large amplitude in terms of two simple building blocks, minus, plus, plus amplitude and plus, minus, minus, plus three-point amplitude. So if you look at the MHP altitudes in Newer Youngness theory, and you run this procedure which I will save time and will not do it. You break the amplitudes into smaller and smaller parts. You will end up with a single diagram which represents the whole MHV-3 amplitude, which has two minuses and all the rest are pluses. And the way it is calculated is that You choose this as a marked, as a marked, marked spinner and it has, so this is a negative helicity diagram spinner. And then this last will be a positive helicity spinner and you attach this three-point vertex. And then after you've calculated this, you attach this three-point vertex by treating still this is a negative helicity spinner and this is a positive one. And then you, you build it up like this all the way to this end and then you start here to his spinner and the reference spinner and then you build it this way until you get the columbage and a priori you might have expected that every time there will be a sum and there will be a lot of replications in different diagrams of theory but there's only one contribution which requires there's nothing else and in fact the distance between this minus and this minus So all this number of pluses is just how many pluses were separating the lines in the same way this way.

50:00 So this is this string representation, this line representation for the machine. And moreover, there's only one diagram. BCF were able to re-derive the Emparthi-Taylor expression, which is this very simple expression. And it's so simple because it kind of follows from the simplicity of this diagram, all this. Yeah, so basically every time you build it up, step by step, with attaching three-point vertex, you reproduce this function. It's a very, very beautiful way to see how simplifications occur on a shelf. So what has been achieved so far with this recursive version, which is very useful. So there is a particularly simple representation for MHD amplitudes, which I showed, and they look like straight lines, which is very nice. And there is an expression for these amplitudes derived instantly. Now, BCF also came out with new results for 8.3 amplitudes in a very compact form. This year we're working with gluons so far, but it's obvious that one can add quarks and gluinons and other massless particles in a straightforward way. This hasn't been done so far, but it's obvious that it can be done and will be done in the very near future. And there are going to be no problems there whatsoever. But also these rules, and I'm coming to a new part of the last part of the talk, is that massive particles can be included very efficiently in this recursive approach. And this is a work that we wrote this. How nearing them, nearing them out. So, for example, let's consider a simplest example where there is a massive and massless particles, of these Higgs, the massive Higgs, accompanied by, uh, finally immobilized mules. Right, and in the earlier work of, uh, with, with, uh, Lansdickson and Nigel Graber, which showed that there are MHV rules for construction as single Higgs to multi-glule amplitude, and these MHV rules, uh, incorporate, actually, two There are MHV and MHV bar building blocks, which are, needs to be, needs to be combined

52:30 into towers, and then you have to take an overlap of the two towers. So there's an interesting variation on the, on the standard MHV rules of how to get the amplitude of a single mass of cases gluons. And Lance, I know, will, will review this approaching part of his talk. But what's quite incredible about the recursion relations is that these two towers will be combined into a single contribution. And in fact, the natural building blocks from which all of the Higgs-Lysbluons amplitudes will arise would be an amplitude of... So, I need to decompose Higgs into a real scale of Higgs into a complex phi and a phi diagram. So, a real scale is decomposed into a sum of complex scale and a distribution conjugate. And then there is an amplitude of a phi with two negative helicity gluons. It's the smallest, a three-point vertex, and I'll call it three-point mHV. And another building block is a three-point mHV bar, where there are two positive helicity gluons with a phi diagram. And now from having just these two building blocks, plus the ordinary 3.mhp and 3.90 mhpp of blue vertices, the claim is that you can reproduce in a very efficient way all of the Higgs to multidimum amplitude. Here I calculated, it took me 15 minutes to calculate some of the amplitudes. And this is an interesting one, it shows that the Higgs can appear on the top as a pie, or it can appear on the bottom as a pie dagger, then this, in our old language, this would be a distribution from one tower, and this would be a distribution from the other tower, but now they're combined together naturally, and give this result. But it's not about reproducing the amplitudes which you knew before, even in a more efficient It's about doing something new, and you can do something new because in this recursion relations. These recursion relations are essentially not MHV rules, because you combine MHV and anti-MHV diagrams.

55:00 And if we are interested in the process which contains many Higgs, so there are a few massive particles and a few massless particles, which are quarks and gluons, until now we were not able to fully derive a multi-Higgs amplitude using MHVs, because if Higgs is made out of pi's and pi's diagrams, So if you write these two hexes, there will be a diagram where there is a phi, and there will be a diagram where there is a phi-diger, and in MHV rules, you don't combine MHV and MHV bar vertices, but in the recursion relation rules, you can do it. And all the indications are that this multi-hex diagram can be now very efficiently calculated in this recursion rule. This is the last slide. So we are now in the position that we can do calculations which are good for massive and massless legs. And the idea here would be the following. Who wants to calculate something which is important to particle physics, the immunology application, which has massive and massive particles? You can do the following. You first calculate this diagram, which has some external massive legs, but massive legs do not propagate. It's only massless legs which propagate. And we know now that recursion relation is probably the fastest presently, but we know what happens tomorrow, but this is presently the simplest way maybe to calculate this amplitude. And the massive legs are not propagating there, they're just external legs. And then you calculate another amplitude like this. And so a massive leg is just an off-shell leg. And then you connect it, it's already a shell because it has P2N which is non-zero, so you just connect these two by a massive propagator, and now you get, so each of these blocks is calculated using clever twist-inspired methods, and then they're connected by massive propagators, and you definitely can do multifixes, and we expect that we can do multi-Ws and multi-Zs using the vertex of Bern, Ford, Foster, and the collaborators for the massive W and Z. to do the same for the full report. Well, I want to apologize, because I've never done this, but it's been particularly dangerous in this case.

57:30 One of the two A-particle entities which are distributed to the DCF was actually, I presented it in my complex survey Yeah, I'm sorry, I should have mentioned, of course I'm aware, I'm aware of that. I was only referring to them in the sense of a recursion relation which provides, which provides a now, by now, proven formalism of how to do it. That is absolutely correct. Just examples in public opinion. Yes? Delia, can you show your graph again where you filtrate, trying to get to the 1 minus, all plus, and rest plus? 1 minus and 1 plus. Yeah, you've got a nutshell continuation of the all plus. The question I have is that term, if that loop just plugs back to itself, This looks like something where in the conventional regular day you use a picture up in the middle. And the second part is, don't you still have to specify a particular opt-out description? Yes, yes, you do. So, yeah. This is where computer-top would be. Yeah, so you say that... Right, so that this, this vertex, this contribution, somebody has to calculate it and find a clever way to do it. It's just, what was important for me is that, well basically in the, in the, in the epsilon-deformed self-doer theory, The way I was not incurred in general, is this vertex arises, this distribution arises. But it's just these are kinds of lonely diagrams which occur. So I call it a Machi tree, but now you can say, well, this is a long spider, a big fat spider made out of various tree level amplitudes, and then you, such that they have two minuses in any number of classes, and then you can pose it in itself. Well, I have not tried to calculate what me or somebody else will do, and if it vanishes the dimension of the literature. I don't know, maybe we can try something else.

1:00:00 But it seems to be the missing building, but we know that without these diagrams, we somehow could not reproduce the cool caps non-constructible lines of the one-manage. So I'm not, I don't say that it's a kind of working prescription of how to calculate it, but it would be very nice to get one. And let's say, uh, uh, the, the, the, the, the primary group, right, when they were using the MHV rules for combining two MHV diagrams in the loop, you can put it on your wish list that they, or somebody else, should try to, to, to use one MHV diagram for the next. Are there other questions? Let's thank you very much. And now we come to the last topic for today. Thank you. I would like to talk about some work which has been done in the past few months, together And the goal of our work was to apply the MHP diagram method, which has been described already in this morning several times by the computer rather now, in the case of new scientists. And I have to say, it was not entirely evident from the start that this might work, because it was known from the work that is written that this is in theory at new level does not So, there was no guarantee that this might work, but what we will see is that we, from

1:02:30 average we work in vertices, we are able to reproduce blue-pumped units in N1, N1, N1, and N1, and N1, and N1. This is a nice course which were given before mine, also in the review of some of the techniques we've heard about in the past 10 years, and especially the CSW rules were invented last last year, but a little more efficiently scattering amplitudes. So the main part of the program with the production will be the discussion of the construction of loop amplitudes for the energy vertices. And so we will see three cases. The first one with some detail. So the first case will be the calculation of the maximum capacity evaluating amplitudes, scattering amplitudes in N equals 4, from flowing energy vertices. And then we will see And then after one has some more confidence in the matter, one can try to compute new things. So the last slide will be of the calculation of the unconstructable part of the machinetic ability of learning amplitude in a plurality. And here I should stress that we will be able to obtain only one part of the amplitude, but it's not the rational terms that I mentioned a few years ago, but only the part which contains of course it could be nice to find out a way to calibrate anti-interactional functions in non-spersonal theories by some sort of things like a shadow of a shadow of a shadow of a shadow of a shadow of a shadow. that the usual way, the textbook way, to calculate scattering amplitudes is not efficient, and it is not efficient already when one has not so many external acts. So the conventional description for, for example, the mode scattering process is in terms of

1:05:00 what we need to be for each particle is a momentum, pi, for each particle, So, this turns out to be a very redundant and inefficient description. So, in general, when we try to compute a scattering amplitude in this way, then we get very complicated expressions and figure it also. So, yesterday, it's not very hard to try to do a lot of terms. But then, so, the point is that the final expression for these amplitudes sometimes turn out to be extremely easy, and so this invisibility is completely unexplained. And this is just a little table which has some paper, but already it must be solved. Which counts the number of diagrams if you want to go through some very easy three-level stuff. Processed with nuance, so already when you are two months going to seven months, you have this big number and more. I guess you don't want to know what happened, and you got ten. But nevertheless, we've seen already all the nice and simple and very short half-line expressions that one can get. So this is very clearly not the right way to . So in the past years, many new tools have been suggested in order to get a more efficient description. And the first one was the choral composition, so the introduction of super-quartilists, which are the second strict partial attributes, which do not depend on choral, so with a usual description, the choral is completely entangled with the remaining degrees of freedom, and in this way it is entangled. And the second important step is the introduction of the the speed of the energy formally and the last, that's what's happened, is the development of the energy dynamics. Tomorrow we will talk by Grito and Alan Tocs, which will discuss other ways of the dynamics. So, now I will review very, very briefly, after the period of stop, for that is the steps, and then we will apply this definition which was derived by the shadow of the energy commitment to the calculation of the dynamics. So, I will try to stress the last important points which will be relevant for the loop activation data. Okay, so, the goal, like I said, will be to do temperature vertices, move together,

1:07:30 and calculate loop lines. And, of course, we have to, since we don't know whether this might work, because there is still no proof of the CSW rules, we have to check against non-results. And the easiest result, which was known a long time ago, is the maximality of the amplitude in N4-4 civilian units, which was computed by using the half-constructed period approach by Bernhardt-Hitson-Talmar, and also were already in 1994. And then, and then one can talk. If things work, one can try to go down in square symmetry, and what is scattering of the amplitude and then once we're in a minute, so we can also do this calculation, and then we can write new results, like I said before, that they got constructed or part of the MHC stat in the outfield . And, of course, there are several reasons for doing this. Several of the reasons, but not the least, there is also the fact that it should be very important to know very precisely the background for the MHC. The first thing would be, for the composition, so here the idea is to separate out the color structure from the scattering amplitude. And, okay, let's start at three levels, so here, the remark is that the amputs are planar, which means that the scattering amplutes can be written as a single trace term times some other ampute, which does not contain color, and moreover, it's also color-olded. So this means that this quantity is simple in several respects, first of all, it does And then the second, there are no structures in here, because the environment we have here, in the environment we have here in the holes and cuts, will be only made by the sum of momentum from cyclically adjacent. So this is simpler to work. There will be only a subset of time and diagrams, which will be used. And if you go to the loop level, there will be more structures, multi-based structures, But we can see that in the larger unit, this will be subliminal. So we can still concentrate on the single-stress drop. This is what we will do in the following.

1:10:00 And now, . So even by simplifying the color, we are using a, if we describe a group in terms of a four vector, So, for momentum and polarization, we are using a very redundant description that there will be many, very many Lorentz invariants which can appear, and we don't want that. So, people found a more efficient description, which is, it starts from reviving the momentum in terms of a pi-spino by using the Tauly matrices, and then one noticed that for a particle, the determinant of this two-by-two matrix is zero, which means that the matrix has one and one, as we told us this morning, so it can be written in this way. For lambda A, a bosonic spinor is the plus one, say, and lambda-fiegel A, plus the opposite A is A. And here the important point is that not only the momenta can be written in this using these two spinors, but also the wave functions. So they would be both functions of lambda and lambda. So for example, for spin one-half, one can write this wave function, and here also the expression for the polarization vectors for nuance is going to express in terms of lambda and lambda and some additional reference. And, of course, we've seen many times in this conference, none of them are related to the coordinates in picture space by performing using the signature of the plus-plus-ninety-miners up to a platform, from the amount of people rider to the individual rider. So why are we doing this? Because this is the script from the scattering happening that we have such a very, very simple. So, for example, the partial ordered amplitude in O plus is zero, and so is the amplitude with only one minus and O plus is zero at three levels in any theory, and a two-level answers in any theory, which, like, they call us, for example, a non-vanishing in 30 minutes. And so the next, the first non-vanishing amplitude, here is the part of the amplitude, which is very difficult for.

1:12:30 And I'd say it's a polomorphic in the land of variables, written in this paper in a 10,000K. I'm pretty sure that there is a very nice dramatic result for this. So this amplitude in this space is localized on a line. So this is very small. And, uh, that's for the time of the film, slash, uh, describing video of the paper of 1986, where they wrote this, uh, amputee. Yeah. That's what, what, what is, uh, nice, the way they, uh, edit this example. So they still, of course, they comment about the simplicity of the amputee, but they also say, They found the student theories to continue with the experiment. So, they were ready to go. In 1996, I'm not surveying this case. Okay. Okay, so now this is the last point of the introduction. So, the next part, starting from this observation that are And while imagery in scattering amplitudes can be, partly speaking, considered as a normal interaction, Cachado-Savage-Cabry-Witton proposed to lift the scattering amplitudes to an effective vertex of some re-organized perturbation theory of Plan Bills. And this, of course, requires an off-shell description, because the vertex may have some literature of off-shell. And the other very nice thing is that these vertices will be connected in a seamless way by using scalar parameters. So when the power of this matter is that first of all enable people to reproduce non-amputals, but also to produce non-amputals with classic significations compared to the Feynman-Dibert approach. For example, this is the next one, which is 1 minus 2 minus 3 minus 1 plus, which is obtained by just some cool sets of diagrams, which is connected by a scale of the numerator, but also allowed people to have many in one in a much more efficient way. So here, the only function to use, because it will be important for the loop diagrams, is the off-shell description. So, when we have, for example, this three-level amplitude, and we use the MHP vertices,

1:15:00 we need to specify what these angular brackets 1L, 2L, etc. are, where L is its internal momentum. And, of course, L squared is not equal to 0, because L is the sum of 4 plus 5 plus 6 here. So we need to keep that, to give a prescription for this here. At some point, the written gave us this prescription. And this is a few other ways that I could just advise about their prescription. So, any non-null for vector, written as a non-vector, little l here, so little l a, little l, a dot, plus, and if this term was not present, then this would be null. But then we have another, z is a real number, types are reference non-vector, which we write as theta a to the little a dot. And if you want to go to compute what D dot L A is, you just multiply by D dot L A dot and you get these expressions. In this, you will recognize that this was an expression that Peter Savichard wrote this morning. And again, the nice part is that these denominators, so it's a field, it's not going to be relevant to the calculations done with M.H.U. Götis, so you can just... So now we know what to write for one L to L, and notice also that C here, this L parameter, is completely calculable in terms of the nominal vector and the represental vector eta, so find this format that you can adjust in the square of this format. So I need to look up for the following calculation here. If one uses a textbook approach to compute scattering amplitudes, one creates the problems of computing too many diagrams. And also, any recursive structure is not apparent, because if you compute an endpoint scattering and then you want to compute an n plus 1 scattering, you cannot recycle your old results we have to start a brand new calculation. This is somehow not nice. Whereas the CFW method allows to find also the

1:17:30 process structures which we're very far with. So clearly the next step is to try to apply this setup to the calculation of node diagrams. So this is what I will demonstrate now. So, just as you will see this feature many times, but just you need an idea, this is what you will do, you will join together two MHP vertices, obviously the vertices, joined by two state operators, and some of the only backups that you can draw this way. And I very recall at the beginning that some part was slightly surprising, perhaps, that is a big word, because the known is now, which is with theories, at the loop level, I'll go up to conform as the graph, so there are states which are around the states of where it is joining the loops, and this cannot be disentangled. But, so nevertheless, one can thus try, so try and join together the mHB vertices, we're cleaning, so v.q is the number of negative ABCDs that we want in our scouting activities, and in the number of loops, so we try to join the mHB vertices given by this formula, And we will recognize that this is the same format which describes the degree of the algebraic curve on which the scattering output is localized in the space, as we have told us, in the state of the paper. So, this is also the same problem that Peter Sarasak wrote with L equals 0. And, again, we will just do the same thing. We will join the two vertices by scale-up propagators, and we will use the as it was written in the previous class, and so on. We will use that particular decomposition for a non-null effect. And some of the diagrams that you can go in this way with a piece of cyclic ordering of the external laws. And so perhaps I can say that this is a slight difference with the cat-contracted video approach because here we create some of our diagrams. So we draw everything we can learn with some of them. Whereas with the cat-contracted video approach, if one sounds like a cat-contracted diagram, then we might have redundancies.

1:20:00 Okay, so, let's see what is the expression of the object we are going to compute. As I said, this was available to us in 1994 by using the abstracted linear approach. So the one loop I wish we had to be amplitude in N equals 4, so it means that it's very simple. It's expressed only in terms of this function. So this is called a two-mass easy-box function. And it's one of the diagrams I'm affecting by the theory, where two of the opposite legs, we have two are massless, and the remaining two opposite legs, capital P and capital U, are mass C. So this is a function of the whole immatural variance, p squared, q squared, which are the masses of these legs here, and of course, S and q, which are capital 2 plus little p squared and capital 2 plus little q squared. And the one that amplifies schematically is just the sum of our whole possible little between the Q of all these two-mast-easy-fos-fine. It's still a fos-fine. So this is our goal, which we want to use this in this object. So we have two types of diagrams that we have to consider. So again, so we have Q equal to 2. At one loop means two amitri vertices. So we have two amitri vertices joined by two scalar propagators. We have two cases, the case where the two negative vertices sit on one of the two hematric vertices, in this case. Hence, to have a configuration of an hematric vertex, we need two plus hematric here, and hence two minus hematric here. But also we have another possible diagram where one hematric is in two on one vertex, and the other one is attached to the other vertex. And in this case, we need another negative elicity in each of the Mhp vertices. So this can be either here, or so we have three possible elicity assignments for the diagram to sum over these two possible elicity assignments. And also, given the structure of the vertices, that is the Mhp scattering algorithm of shell, we can realize that in the first case, only the gluols can run in the loop, Whereas, in the second case, all the particles with an equal force or multiple can run, so we have more experiments and shared.

1:22:30 And notice that there are, we are only including physical particles with no ghosts, nothing like that. And after a loop of calculation, it turns out that diagram A and B are equal. And this would have been seen very easily by using an explicitly source-imagined version that is the naius provertec. So, one would compute only one value. So, schematically, again, our calculation will be solved. We have MHP vertex on the left, A left, one and then the other vertex on the right, and then we will have to integrate with some integration measure, which we will discuss later in the next part of the analysis, where we also include, for convenience, we include also the scale of the magnitude of 1 over L2 squared and 1 over L2 squared. So when the a left and a right are these MHP vertices, with the prescriptive of a shell, given like a shadow searching and written, we read it to the other. So there will be, so capital L1 and capital L2 are expressed in terms of null vectors, because L1 and Q2, and this will be, so this here is something here, So this will be those which will appear in the expression of the vertex, plus these extra curves. So, now let's start looking at the integration measure, because this is the, I mean, it's crucial to find a convenient way to express this integration measure to actually carry out the calculation. So the integration measure, of course, is as simple as that, with the length of action, left, and the left is done with some holding length of the left of the dynar. And then there is integration of L4L1, L4L2, and, as said, I'm going to include the propagators in the integration length here. So now let's see what happens in this moment. We will see that it can be written in a very suggestive form, which will actually allow us to do that.

1:25:00 Okay, so let's focus on a single dimension here. So the crucial point is that we want to convert the integration over the four components of the core vector of capital L into integration of the new variables which are little L and Z here. So I think that the most important thing is that P of L over L squared has a very simple form in terms of Z and the development of L. And so it's given by DZ over Z times something that we call the naive measure, because it was written out by naive first in his favor, L squared, which is given by this expression. That was an expression in terms of the spinos So I'm using it for the angular and square brackets. But the observation of Naevi is that this measure of the N is actually equal to this quantity for L to the plus of L squared, which is nothing but a non-examined phase-based measure for one particle. So, here we see that this measure is going to be decomposed into a phase-space measure, which will compute eventually the discontinuity of the diagram, times something that will become a dispersive measure. We will see this very much in detail in the extra classes, but the idea is that what we will compute is first a phase-space measure, sorry, a phase-space measure, which calculates the discontinuity of the diagrams, and then we will reconstruct the whole diagram by So, like in the old way, in the touch of the nanites, I see. But somehow the surprise is that there is personal interest that we have to calculate that's very easily. So, this is also how I describe it. Okay, one technical point, which is quite important, however, so when we rewrite, so we want to rewrite the answer to the data function here, in terms of this new quantity This is the realization of the formula in capital L, and when we do this, we see that the momentum which will appear in the curve measure will be this shifted momentum. So P there, no matter if it's the sum of all the momentum of the length of the diamond, but there will be a shift of period. So let's see what this moment there was in common.

1:27:30 So we had those two integrations that were before L1 and before L2. So we would have Pz1 over Z1, Pz4 over Z2 times this product of native metros. And since here there is only one combination of C which appears, C1 minus C0 is convenient for . And in terms of this variable z, the shift that we're making will be phi left z. So phi left minus z times the reference between our eta. And one of the things that we want to make sure at the end of the calculation is that the result is actually independent of the choice of eta. So, okay, the first thing we do is out z is z. And the second, so we will see the result in the next page. The first is that the product of these two phase-space measures, together with a vector function, precisely reconstructs a two-particle phase-space measure. So this is going to compute, really, the discontinuity of the diagram in a challenge. The challenge is, of course, determined by the momentum which appears here, and this is not p left, it's a shifted momentum. And this will be converted into a dispersion in the end. And also notice that this one of our computing contains, they do not contain total divergences, but they do contain total divergences. So we need to employ some sort of regularization. Of course, we just mentioned regularization, and this is the point where we start using it. So first, we convert all the spinon quantities into meta at this point. And we can dimensionally visualize this new method. And then . OK, so what we are trying to see after . The question is that the measure is, of course, in Z over Z times the phase phase measure, which, however, has this shifted momentum . And the result of this particular diagram, we have fixed M1 and M2, eventually some of all the choices of M1 and M2, will be eta over DZ over Z times the phase space of some expression.

1:30:00 So we'll see the next class. The next important thing for sure is that, so we want to see that this is the dispersion is equal, actually. So, if we write the expression of T left, T left V here and take the square, which is written here, and then just compute Vz over Z in terms of Vz squared, we find this relation. So now, it is entirely clear that this is going to be an expression, this is an expression. So the measure, our final result is that the integration measure we started with is a face-to-face measure to a cup. This is going to complete a cup interval with this momentum flowing in a V times a distribution interval. So now what's called in the next video is to perform the calculation. And just as you can see that it contains the part. This is integral, which is that it's going to be looking at a diagram across some particular cut, and then, as usual, the dispersion interval will start the whole diagram from the dispersion So, after some small calculation, one finds the following expression, so the one with the scattering amplitude is proportional to the three-level scattering amplitude times somewhere among the forces of the extreme legs of the vertex, times some quantity here. And by linear calculation, one finds that this is the 40 initial piers, and this is the integrand of a scalar box integrand. And so, to be completely a scalar box integrand, like L1 squared and L2 squared, they are present not here because they are precisely cut by this space-based micro piers. So, this is the tutorial representation of their computing. So we are cutting the leg L1 and the leg L2 here. So this red line here represents the cup. And the cup is precisely with momentum going in the cup, which is equal to P-leg to Z. So if we only write the momentum, which is here, we have P-leg. We have to perform this shift. So it's like what we're seeing if on this part of the diagram there is an extra leg,

1:32:30 implementing minus the z beta and the micro conservation on the other side is another extra-electric sphere with minus the plus z beta okay so now we will do this calculation just a reminder okay so the next thing one wants to draw for the that you can obtain in this way, and after you draw them and speckle them, you realize the problem. You realize that each state-of-box function which appears in the result, appears precisely once in each of these four cuts. So, the next thing you do is to collect these Cazzo, we're going to collect the coca...