 Can you hear me? Yes. Okay. No, it doesn't matter. Yeah, okay. Now I am cohost. Thanks. Okay, whenever you are ready, we can start. I'm good, I think. All right. Then welcome everybody back. And we are ready to start with the last lecture on QCD and collider by Matteo, please. Yes, thank you. Good morning, everybody. So let's try to finish up what we were doing yesterday about calculability in simple processes like E plus E minus two to something. And then if time allows, I will also try to talk about part of distribution functions and deep in last is gathering and using atomic processes. Unfortunately, I'm a little bit so I'm afraid that I will have to compress things a little bit today ideally at least another lecture we needed to do things more properly but I picked up some delays during the throat. Okay, so we saw essentially in the past lecture that IRC safety means calculability in PQCD. So it means that you can expand order by order your observable in a power series in the company. Of course, this does not tell you anything about the reliability of this result. So the Perturbative series could still be very badly converging. And in fact, let me tell you a little secret. Perturbative series typically do not converge. They actually diverging in PQCD because we are neglecting the non perturbative part. This asymptotic divergence of the series typically signals that we are forgetting something, but eventually for phenomenology for simple processes for situations that are under control. Actually just calculating the first few terms of the series is enough to actually increase the accuracy. Before eventually the serious data. So let's not worry about this. This is the topic of renormalons. We touched very little about this yesterday or the before yesterday when somebody asked me how do you calculate the power of the power corrections and so on and so forth. So this is related to the eventual divergence of the perturbation theory. But again, let's keep that aside for a while and let's just consider that by calculating the first few higher orders of a power expansion, you are actually improving the accuracy of your of your prediction. So, as we said, once you define your observable properly, you can calculate in PQCD, even if you are not fully inclusive, you can work around the block north. Kinoshita Lee Nowenberg theorem by considering observables that are less inclusive, but still inclusive enough for calculability. That's what we discussed yesterday when we detailed out the conditions for infrared and collinear safety. So you see, if somebody tells you, oh, this is infrared and collinear safe, it means fine. You can go ahead, calculate higher orders in particular the QCD, and at least get a finite result, a finite number. You are ensured that the divergence is cancelled in virtual and real corrections. So what I want to do now is to actually say something about the, sorry, let me just get rid of this window. I won't say something about this, how can I say this structure in a sense of higher order calculations, which does not mean that I will calculate anything, but I will at least show how the various kinds of diagrams are arranged and what you consider to calculate what you need to calculate in order to calculate higher collage. So let's first set some nomenclature that we need in a moment. So let's call a process P, something where, well, something happens, I don't really know what, and eventually you have some final state that you observe. This final state could be a number of things. It could be a Higgs boson. It could be a W boson. So if you would have Pp2 Higgs, for instance, as a one possible process, you could have Pp2W boson. You could have Pp2W boson with nonzero transverse momentum, which means that the W boson recoils against some radiation. In that case, it becomes W boson plus one jet, typically, or you could have the two jets of the Sturman-Weinberg jet cross-section that we defined yesterday and so on and so forth. So by process P with some final state, I mean something that I can calculate in PQCD, because I have made sure that this particular process is infrared collinear safe. And then you may have this same process calculated to one loop that I can come into the question that I can depict like something like this, and the same process plus one leg that I can depict as something like this with the emission of the additional leg. So this is the additional emission. For the moment, I'm not saying anything about what I'm doing with the phase space and the integration over these things. This is at the amplitude level. I have one extra leg that gets out of the diagram. I can perhaps take the question. Yeah, it's very quick. Is the initial state included in the blob in the square? Yeah, in a sense, yes, it is included. So it's PP2, that's some final state that I wrote down. So I don't really care too much in this drawing here. I should note perhaps it's worth noting that P plus one leg or one emission is not the same thing as P plus one jet. P plus one leg is something that happens at the amplitude level. On the other hand, and it's not necessarily observable, it depends what you do with it at the level of the integration. P plus one jet, on the other hand, is a very specific observable process. As I was saying earlier, if I decide that I want to observe W plus one jet, this means that I am going to observe the production of a W boson that goes with some transverse momentum and the jet that typically recalls against it. And at least that I do have one visible jet in the event. So remember, a jet is an observable quantity, even if it is an artificial one. A jet requires a jet definition. So you can define what a jet is, like extermination and Weinberg did, and we'll see other jet definitions. But once you have defined it, a jet is an observable object. You reproduce the same definition at the theoretical level, at the experimental level, and you compare the two. But it's an observable. Instead, the emission of one leg is something that happens at the amplitude level and then it depends what you do with it. Let's see what, so, and then, yeah, exactly, sorry, I forgot to say that indeed this particular process, this P plus one leg, this is the follow. So what I had just drawn earlier, sorry, not like this, like this. This can contribute to two different processes. It can contribute to P plus one jet. And this will be this kind of thing, or it can contribute to simply P plus X, where this thing is integrated over. So it depends what you do with the integration of your phase space, whether a given amplitude contributes to one process or to, or to another. This is what you need to include as a Feynman diagram. And then, according to your choices, you may be calculating this or this. These are two different things that contain the same Feynman diagram. Okay, so let's see what the structure of this thing is. Suppose, so you have, now I have to try drawing things and as you've seen many times, I'm not very good at all at drawing things, but let us try. So, let's say that I have here my process P. And then let me draw a couple of, so I'm putting here, so I'm putting here additional loops over this axis, and I'm putting here additional QCD emissions, so legs. On the vertical axis, I'm putting additional alpha S powers. And so, for instance, I'm, now the process plus one loop, I can place it here, for instance, and the process one leg, I can place it here. So this is what you consider when you go one order higher. If you only can consider the, and then you can even go on, let me draw, perhaps draw also the other, you would have here, let me change color, perhaps P plus two loop, you would have here, this would be the process plus one leg at one loop. And you would have here, this is the process plus two legs. Okay, I think it's, well, I hope it's clear. Let me perhaps change the colors of these things if I can. It's consistent. And this one too, this one. All right, so I am essentially listing the kind of diagrams that that exist, starting from the one with the process I'm adding legs to the Feynman diagrams in this direction. So if this is, if this is P, then this is this plus one leg, and this will be this plus two legs. And on this side, this is this one loop. This is the same process, one loop, and this will be the process with one leg, and one loop, and so on and so forth. Okay, so now, in this particular configuration, suppose you are calculating exactly only this diagram here. At this point, the only process that you know exactly is this one, of course, the one without any other emission, you may call it the board, the board level, or the lowest possible level, in a sense. Suppose you go and you calculate also this one here. Okay, so now you have calculated this one. At this point, you have exact knowledge of the board level for the process. And you have exact knowledge of the process plus one leg. But this is of course not the full story for the process to next to leading order because as you know, you will also have to include the loops at that particular order. Once you calculate this one plus this one with a single emission, you are not yet done. And by the way, integrating over this leg will inevitably lead to divergences as we saw yesterday. So you will have to include something else and that something else will inevitably be the corresponding virtual corrections. Once you have included all these three things, so the born, the one loop and the one leg, and of course you have performed the usual sum. So you will have performed the usual, so it's sigma born plus sigma, sorry, not sigma, sorry, apologies. You would have done the matrix element for the born plus the matrix element for the virtual plus the matrix element. This is remember higher order correction for the real squared. This would have given you, of course, mb squared plus g squared m real squared plus, sorry, there is a square here already. So it's all another is a square element plus g squared twice real part of mb mv star. Okay, this would be your result and it is within these two things that you would have the cancellation of infrared and collinear singularities. So it is only once you've calculated everything that you have all the tools that you need. So that cancellation would take place essentially between these two diagrams here after they have been squared or properly multiplied with the interference. So you see, for instance, here, the virtual does not exist by itself. The virtual exists at a given order in perturbation theory after it has been calculated as an interference with with the with the ball and to draw this in terms of diagrams in order not to leave any ambiguities. This is something like this. Plus, this is something like this. And this is something like this. Okay. So this is what contributes to what so when I actually draw this what I mean is that there is an interference and interference can only take place along the lines where the same the final state is exactly the same. So you see along this line here, the final state is constituted only. Of that whatever that it is, it is the same the thing the same, the same one thing here, the same thing here, the same thing here and so on and so forth. So you these diagrams can all interfere with each other, because you can match the final state. On the other hand, this diagram could never interfere with this one, because here you have three particles in the final state, and here you have only two. So that won't won't happen. The final state must match for the interference take place on the other hand this one could interfere with that one. Because as you can see, they have the same final state in terms of particles. We don't need it now. Okay, so let's suppose you have calculated all this. And so what do you have out of this you have next to leading order correction for the process P. Or if you wish P plus X in the sense that you have integrated over the outgoing leg and so you don't observe it. It's a next to the go the correction to P. And what you have calculated is P plus X with X is the integration over everything. So now you have this particular triangle. This one here, which was meant to be a triangle and try again. Okay, so this triangle here is the next reading order triangle for the process P. You need to consider these three diagrams. If you want to calculate the next reading order corrections to the process P. And of course this diagram as I was saying earlier contains an interference among these two diagrams here. And it contains a cancellation of infrared and collinear divergences among these two diagrams here. If you wanted to calculate now instead, if you wanted to consider not so much the process P, but the process P plus one jet. Like for instance a W boson back to back with a jet. Where would you start? Well, you would start for instance from somewhere like this. So this process here would constitute the lowest order, the board approximation, if you wish, for P plus jet. So let me write it down. This would be of snow, not this color. This would be the lowest order for P plus jet. And this jet would have no substructure at all. This jet would be constituted only of that single emission that takes place. It's a very simple jet. You may imagine jet as bunches of particles collimated in one direction. It is certainly that in practice, but at the perturbative level, the simplest jet is simple is simply a single a single leg. Okay, so this is the lowest order for P plus jet. But now this will be the higher order correction with one further leg emission to that particular process. You see here I emit two legs. And this is the virtual correction to that particular process. So now you can play the same game and say that this you have another triangle that I now draw using a different color. This triangle here is another triangle, and this is the NLO triangle for P plus jet. If you want to calculate the next leading order corrections to the process P plus jet, you have to calculate all these three diagrams. And once again, of course, you will have interference between these two same final state. And you will have cancellation of infrared singularity between these two. Now, suppose that now you want to go back and you want to calculate not so much next to leading order correction to P plus one jet, but you want to calculate next to next to leading order corrections to the process P. So two orders above the leading order. So where would you do it? Well, how old would you need? Well, you would definitely need all these diagrams. You would definitely need zero emission, one emission, two emission. You need one loop. You need one loop to the P plus one leg emission. But you would also need what you have not yet calculated, which is this one here. You would also need to calculate the two loop correction to the basic process P. And so at the end of the day, you would have yet another triangle that I am, this thing is getting messy, apologies. So you would have yet another triangle here, which is an impossible color. Let me change this. What do I use? I use this one. Okay. Yet another triangle. And now this is the next to next to leading order triangle for P. If you want to calculate next to next to leading order corrections, and again, it is always infrared and collinear safety that ensures that you can actually calculate it. This is what you need to do. And you will have again, cancellation of infrared and collinear singularities along this line here at equal alpha s, cancellations take place at equal alpha s, of course, because you can so order by order in perturbation theory. And again, interferences can take place along all the lines where the final state is the same. So this diagram will interfere with this one. And these three diagrams will all interfere with each other and give various various orders. So this tells you quickly and easily this kind of scheme tells you what you need to calculate. Where is the difficulty here in doing this kind of things in practice? There are two main difficulties for today's technology. I mean, one thing is fairly easy. Calculating emission of legs is at the amplitude level, at least it's fairly easy. It's done even automatically in many, many different ways. Loop calculations. Well, you know how it works. You have to write down your final diagrams and calculate the loop integrals. It's difficult, especially when you have known equal masses or when you have many loops, but again, it is something that can be done fairly easily. Of course, it is still frontier research. I'm not saying you just sit down and you calculate the two look diagrams, but it is it can be done. On the other hand, and perhaps a bit surprisingly paradoxically, one of the things that is actually difficult is to implement the real and virtual cancellation for the infrared singularities, especially starting at next to next to leading. So next to leading, it's pretty standardized these days. There are a number of automated tools that actually implemented this automatically. The frontier of research today is to try to standardize and automate the cancellation of virtual and collinear sing virtual. Sorry, the cancellation of infrared and collinear singularities at the two loop, one loop, one emission and two emission level. So it is these cancellations that are the difficult thing to do because as you can imagine, as soon as you start having two particles in the final state, each of them could be either soft or collinear with any of the other particles in the face base. So there are many different combinations you have to take into account. And so this is actually a difficult technological problem. It's not a conceptual problem. Conceptually, it's easy. I've described it to you in five minutes. It is technologically difficult. This is the difficulty. And this is what people are trying today to automate in order to be able to do this kind of things. The more automatically that is actually actually possible. Okay. Questions about this. Yes. So, I was wondering now about the practical issues. Now, maybe you will explain this, but as far as I know, people tend to, well, if they do simulations, of course, through my graph and whatever to set, for example, a cut on the PT, let's say, or something else. And then, well, basically, then the problem is let's say solved. But yeah, my question would be then in this scheme when we set the PT cut, how would you then which diagrams would you look at? And how would you then sum up? Well, the PT cut could, at best, define your observer like imagine you want to observe the production of a Higgs boson in PP collision with a minimum PT cut. So you don't allow the Higgs boson to be just with zero PT. Okay. Of course, in order to have a PT cut, this Higgs boson will have to recoil against at least one jet. All right. So in this case, if you require your Higgs to have a minimum PT, it means that you are not looking at inclusive Higgs production, but you're looking at Higgs plus one jet. So the lowest order for your process will be this one here. You will have to, so let me, if you are calculating, so let's say that you are calculating Higgs production. So this is the simplest diagram for Higgs production, of course. This will give you a priority, zero PT Higgs, because the two gluons come head on from the beams. If you want these Higgs to have a PT, you have to meet at least another gluon. And this will give you something where you have one gluon, one gluon, you have the Higgs going out this way and you have the other gluon, oops, sorry, you have the other gluon going out that way. So this is PT larger than zero for the Higgs. Okay. So if you go back to my scheme here, this means that this is your leading order now. If you calculate this, this is your leading order, but still, if you want, of course, next to leading order correction to this process, you still have to calculate all these triangles. Of course, it means that you don't care about these things. So you can go to this order in alpha without worrying about these things. But still you have to calculate at least this triangle and you have to ensure cancellation of infrared and collinear singularities between these two guys here. All right. And so the choice, the choice of the process defines your starting point. Full Higgs cross-section, fully inclusive, you start here. Higgs plus one jet, you start here. And then it depends what kind of corrections you want to, you want to calculate. Of course, the loops will never contribute because if I go to my other, to my other diagram. So if I go here, loop corrections like this one will never contribute. This is PT equals zero always independently of how many loops you have. So a loop correction to this process will never contribute to a Higgs with a PT. So you don't need it. And indeed, you see that you don't need it because it is not contained in my triangle. The triangle always starts at the leading lower process and goes up with, goes up with sites that are parallel to this. So the, the triangle for Higgs plus one jet is this one. And it will never include this amplitude here. Okay. Thanks a lot for the clear answer. Max. Yes. Yeah. Hi. So my question is something very similar or maybe related to the one, to the previous question, because so I'm still confused about. So, so the really this tension between, between the process where you have so, so, so between, so between a jet, which only contains one, one leg or jet which contains many particles. So, for instance, let's say, I want to calculate this green triangle, but my leading only all the process is not a jet with just one particle, but jet with many particles. And then, and then I do this screen calculation. So I have to include one, one higher order. So the complexity of this calculation. So it's basically this is the same. So I can use the same computational techniques, or is it much more difficult to do it in the case with the many particle jet. I think I should think more about it, but I think that you don't get to have a jet with more than one leg as a leading process. Because, for instance, take, take, just, just a plus or minus to, let's say, qq bar, the leading process. So what this gives is back to back like that, and the leading process is two jet production. And each of the jets contain a single particle. If you want two particles in the jet, essentially you have to take a jet that is as big as the whole phase space. It's hardly a jet anymore, in a sense. And so if, if you, you can have more than one particle in the jet, I think it's not, it's not the leading situation anymore, because they take now the mission of a third balloon. Now, this could be a three jet event, like the one we saw yesterday. Or it could be that your balloon could be perhaps be here. But if the balloon is here, this is already a higher order correction to the two jet, but this would not, sorry, this would not be here, of course, in the case. So if the situation is this one, this is now a higher order correction to the two jet cross section. So I don't think, I mean, I should think better about it. But I don't think you can easily come up with a situation where the leading diagram for a jet contains already, sorry, where a leading, a leading, yeah, the leading diagram for a jet contains already two particles. But even if it did, I mean, I think the scheme would still hold. You would have to add more legs and more loops in order to go to higher orders. Okay, I see, but the thing is that at the time system, I'm missing something then, because so you were saying that already those and then so this next to next to the leading water calculations are already on the front here on for research, but then it would seem as if nobody is actually able to calculate really jet and jet processes with many particles in the jet. No, it's not that it's not that you can't. It's just difficult. I mean, people have calculated up to NNN. So three ends for fully inclusive his production, for instance, then the more particles you have in the jet. Sorry, the more particles you have in the process at leading order, the more difficult it becomes. So jet production, for instance, jet production. So his production has, let's do this, his production has as a leading diagram and then this can be contracted to a point. Just this. Okay, so it's a process. It's a two to one process and people have calculated next to next to next to leading order corrections. Jet production, on the other hand, has as a leading process, something like this, for instance, or if you wish. It said and there are many more diagrams and to this particular process only next to next to leading order corrections are known. And even then they have only been obtained three, four years ago, not much more than that. Because it is a much more difficult process, it has many more diagrams in the initial state and so many more diagrams for the correction. And then since it is a two to two process to begin with, next to next to leading order corrections have four particles in the final state. And then you have to deal with the phase space of four particles and all the cancellations of the singularity. The more complex the process is at poor level. And of course, the more difficult calculating corrections will be. But the scheme that I have already given you holds next to next to leading order correction will, for instance, need to a mission. We need one emission and one loop and we'll need. Let me also. And we need two loops besides everything else at next to leading order. You have to go back and construct all the diagrams that you need to find here. So we need two emissions, two loops, one emission and one loop, one emission, one loop. You need all these steps in order to complete the calculation. I see. I see. So that's the difference. So up to now, nobody has actually calculated processes within gen jets in the final states which contain many particles like 10 particles or whatever. Not also not at what is there. Well, you can calculate jet with 10 particles, but it would only be at three level. And still, no, still you can't. No, you can't. You can't because they would end up calling. No, you can't. No. So no, the answer is no, only up to up to two. Well, let's see three particles. So the most complex is three particles. And it's this one. It's where you get to have something like one jet here and the other jet there going back to back. So the most complex jet calculated fully with all the real and virtual is up to three particles. And the main reason for this are these infrared calling and intervention system. The main reason are that you need exactly all those corrections. Plus, plus the implementation of the consolidation. One more question. Can you hear me? Yes. Is there any solution for Calculate NL O process by using process with jet? Calculating NL O process using, sorry, by using a process with one jet. So you in higher order corrections to something plus one jet. Yes. Yes. Yes. Yes. Yes. Yes. One jet, Z plus one jet, they have all been calculated to at least next to leading, but it could also be next to next to leading. So the answer is yes, all these calculations to exist. Absolutely. They are a bit easier than these because they are processes where you have one week particle. And so that one does not radiate. It is massive. And the overall number of particles in the, sorry, the overall number of diagrams in the processes is smaller than for instance, processes like this. So those processes where you have one weekly interacting boson plus jet are typically a bit easier and we're done earlier. So yes. Thank you. Okay. So very quickly perhaps another few minutes before before the poison. Now that we have seen that there are classes of things that we can calculate and what the calculation looks like in terms of interferences and in terms of collinear singularity cancellation. Let's see very, very quickly. Now I won't go into detail because I really want to leave the time for after the pause to talk about the pattern distribution functions. So let's talk about a couple of observables that we can define and calculate any plus or minus collisions. And again, very quickly without any detail. So let me first talk about trust. So trust is what's called an event shape is essentially a function of all the moments. In the final state of an event that is sensitive to how these particles are distributed. And so for instance, if you define trust as the maximum with some arbitrary that you get an arbitrary direction and then you maximize the value of T over this and of this particular function. PI scalar and the oops, sorry, divided by the sum I PI with of course, as I said the unitary factor and so by maximizing you find a particular and direction that you call the thrust, the thrust axis. You can easily tell or fairly easily tell that this is IRC safe. Because well, you can easily see that. Remember the definition of infinite collinear safety. Soft particle emission and collinear splitings must not change the observable. So let's look at what happens here. If you emit something very, if you emit a P that is very, very soft. So if you meet the P that P goes to zero. So let's do it better. Soft emission P K goes to zero. But if you add to these sums, a zero P, of course, T is unchanged to a sum you add a term that is zero. Okay, so that's easy to see that T is unchanged. Now collinear emission collinear emission means that you take a PI. And you split it as one minus lambda PI plus lambda PI. This is the definition of collinear splitting. But now try for instance putting the numerator. This thing. So you have numerator becomes that PI the term PI and of course becomes one minus lambda PI. N plus lambda. Oh, sorry. PI N. But this is one minus lambda PI N plus lambda PI N. And then of course, this is simply PI dot N. And therefore once again, T is unchanged. It's actually quite interesting if you ever decide that you like physics archaeology to look at the paper that originally introduced like it is I mean it is always a good idea to look at old papers. I mentioned yesterday the Sturman Weinberg paper, I think it was a PRL 1977 that introduced the infrared and collinear safety with jets and the trust was introduced by fari. Also a PRL of 77 and it's actually interesting because well today we take everything for granted but at the time QCD was just in the very beginning so there are very interesting considerations on that actually why this particular variable is calculable in particular the QCD. There is reference about, there is a reference to the Sturman Weinberg paper. There is a reference to asymptotic freedom and so on and so forth. So if you ever want to look at that paper and since these are PRLs, they are short papers not like today where people write a hundred pages papers. I think all papers are very short. It means that sometimes they're a bit cryptic to read the language may not be the same that we use today so it takes some getting used to it, but some of the papers that survived and became historic are usually quite useful in terms of understanding how people were thinking at the time and what they were not taking for granted. So you can see all the considerations that we have essentially made asymptotic freedom IRC safety and so on and so forth, even though in a language that is not exactly the same, but it shows that already in 77 people were already calculating all these things and as you can see, they were already in a context of let's test QCD in E-minus colliders at the time that's what was there. There was a Petra at Daisy and this kind of E-minus machines that could be used to test QCD by checking the characteristics of the final state. So trust in particular is one such variable. Why? Well, because this is a variable that if the event is eventually distributed in a fairly isotropic way around some axis, trust axis T, the value becomes one of the order of one over two. Whereas if the event is more pencil-like, so with two back-to-back jets, trust is rather around one. And since you expect, of course, QCD events to be more pencil-like because of what we know of QCD as asymptotic freedom and the probability of emission in the soft and collinear region and so on and so forth, eventually this is what you can actually test and measure. Let me perhaps just quickly skip, yeah, we skip the calculation, I think, because we are not really learning that much. On the other hand, I can show you some results for trust that I have here, I think. Okay, so here you can look at these plots. So this is the distribution of the events as a function of trust. And you can see that most of the events as measured in E-minus collision are from a region where things tend to be back-to-back. And you see here trust tends to be one. And then there is a region where trust is smaller and that are more of this kind here. So most of the events, this is an history of the events. As you can see, most of the events are here, but not only that, you can actually predict the whole curve and see if your data follow the curve. So what happens here is that actually the data follow the curve well more in the region of three jets, but they do not follow the curve, the perturbative curve very well in the two-jet region. Why is this? The reason is what we saw yesterday, meaning that when you go in that region, you tend to have large perturbative corrections that you need to resum. And on top of it, you also need non-perturban corrections that you will never be able to calculate. And this is seen in this other plot here. So this is the fixed-order calculation to any variable. So now I have, sorry, what I have here at the bottom, you can consider it is similar to the plot at the top, but flipped. So the region where you have the two-jet area, the two-jet region is this one on the left. So here it is here. And here it is this one. So it's flipped. Okay. It is just another way of representing things. The two-jet region is on the right here and the two-jet region is on the left here. So sorry about this flipping, but otherwise, this is the same thing. So you see that a perturbative calculation eventually diverges in the small region here, which is the region again where trust goes towards one. You can resum to an order in perturbative QCD, these diagrams, and still you get this curve here. And this curve here starts being in better qualitative agreement with the behavior of the data. But still, if you go and look in detail, you see that the data that are here do not really agree very, very well with the perturbative prediction that is here. Why? Because there is a shift due to non-perturbative corrections. There is a power correction that for trust, for instance, is actually quite large because it is lambda divided by Q. It's linear. I told you yesterday, or the day before yesterday, that for the inclusivity plus and minus collision two-halves, it's lambda fourth over Q to the fourth. Here it is lambda over Q. So it is a much larger correction. And you can actually see it in the data, in the form of this shift here that you see. So you can actually see in the data that there is a shift from the perturbative prediction, there is some perturbative prediction to the data that is actually visible. And so it is a test in one sense of perturbative QCD, of the resumption of perturbative QCD, and of the particular parametric form that non-perturbative corrections can take. And you can perform zillions of this kind of tests and measurements in the plus and minus collisions, and they have actually been performed at LEP and in other machines. So another infrared and collinear safe. This is now becoming a bit late. Perhaps we should take the pause now because I will need more than two minutes to actually go on. So it may be a good time to actually ask questions if there are any. Take the pause and continue later. All right. So any questions? And now we take a pause. We resume at 58. Okay. In five minutes. Yes. All right. We can resume. Whenever you're ready. Let's start again. So we have seen trust, even though we haven't actually calculated anything, but just believe me that when we say that this is an event-shaped variable that is unobservable, something measurable, it is infrared and collinear safe. You can actually calculate it perturbatively to any order and compare it to data or a summit test for corrections and so on. So it's one way to study the structure of an event and check if this structure is consistent with the predictions of QCD. If you wish, it's a way to test QCD. Something which of course was much more fashionable 30 years ago when people were establishing QCD than today. Today we calculate these things not so much to test QCD, but rather to provide predictions for more complex processes where we produce other particles, especially electro weak particles or super symmetric particles on these kinds of things. But the techniques, of course, are the same because since QCD is everywhere, the problems related to QCD, the ones I mentioned, infrared collinear safety and the kind of stuff, you also find them when you calculate any other process. So another process, another observable that allows one to establish the structure of an event is something related to jets. So jets are more than what we just saw with Sturman Weinberg. Sturman Weinberg was a particularly simple situation because essentially you were pre-defining, I remember, Sturman Weinberg, you were essentially pre-defining some direction in any plus or minus collision and you were saying, oh, fine. I typically have two jets and then you were properly defining these two jets so as to make this cross-section infrared collinear safe and calculate higher order corrections, fine. But suppose you have an event like this. So imagine we are looking, this is supposed to be a circle. So suppose, in an event like this, imagine you are observing an event like this, it is quite easy to say, oh, fine, this is a two jet event. Easy. If you get something like, no, sorry, do it again. If you get something like this, it's also quite easy to say, oh, fine, this is obviously a three jet event. But suppose you get something like this, something like this. Okay, is this a three jet event? So one, two, and three, or is this a four jet event? One, two, three, four. Hard to tell. Hard to tell because, well, there will be borderline regions where the answer depends on exactly how you define the jets and what your resolution criteria is. If your jet is this big, then yes, this is a three jet event. If it is this big, on the other hand, this becomes a four jet event. So jets are more complicated objects than just what you get in Stem and Weinberg. And sometimes you cannot even know a priori how many jets you have in an event and where they are. Stem and Weinberg essentially decides beforehand where these jets are at a given angle and how many you have to. But in a real event, especially in hydraulic collisions that are more complex events, you typically will not know how many jets you may have and how they are best defined. And also this takes you to essentially the need for jet algorithm. A jet algorithm is a procedure that, or if you wish, sometimes it's called a jet finding algorithm, or in some cases a jet clustering algorithm. Because finding the jets is, in a sense, a process, a procedure that is set in place and that given an event like this one will eventually produce an answer. I am observing, according to the given definition, a three jet event or a four jet event with that particular energy or transverse momentum or whatever you're looking at. So jet algorithms have been used, especially in hydraulic collisions, because due to their complexity you don't know beforehand what the structure is and setting up things, alas, Stem and Weinberg, for more complex events, it's very complicated. Then there are two kinds of jet clustering of jet algorithm. They are the ones called the cone algorithm. Cone algorithm, essentially, you go around your event and try to find where the big fluxes of energy is. So a cone algorithm is a bit something like we do with our eyes. I look at this event and I say, oh, there is a big flux of energy here, there is a big flux of energy here, there is a big flux of energy here. So I'm essentially applying myself a cone algorithm to my event. And then there are other jet algorithms that are called sequential recombination algorithm. Let me just write it down, sequential. So instead of being a top-down algorithm where you look at the macroscopic event and try to identify where the big energy flow is, a macroscopic algorithm is a bottom-up algorithm. You combine particle two by two and you try to cluster them. This is why they're also called clustering algorithms. In mathematics and in other sciences, these are called agglomerative clustering, hierarchical agglomerative clustering algorithms. Because they take typically pair-wise particles and cluster them one by one in an iterative way until they build the overall cluster structure of the event. So for instance, in my previous drawing, a clustering algorithm would start taking particles that are nearby and so it would say, oh, I combine this one with that one. And then I may combine this one with these ones and I call this a cluster. And then I do the same for these three. And here same thing, I may combine this one with this one and this one. And then according to how the algorithm work, you would see whether you also combine this one with the other four or not and so on and so forth. So this is how they would work. I think that's a question. So my doubt is later clustering. So can we use some sort of machine learning unsupervised clustering algorithm where we generate a 2D image of where the particles strike the detectors and then use that for clustering? Absolutely. And it has been done quite a lot in the past few years, especially like everything else, machine learning has been applied. It's essentially an image recognition, image analysis and recognition. So yes, machine learning has been applied to these kind of things. Of course, any decent, I mean, if machine learning can recognize a cut in a picture, of course, it can try to decide whether this is a three or a four jet event. So no doubt you can do machine learning out of these things. The difficulty could be, is the procedure that machine learning is applying actually leading to something calculable? That's the difficulty because as a scientist or as somebody who's working with a predictive theory, you want to be able to use the theory to predict what you actually measure. And the real challenge with machine learning is actually to understand whether whatever it does can be replicated in terms of a calculation. So you can definitely define jets as a generic observable using machine learning. So in a way that you look at a picture and you say here there are three jets or here there are four jets. This is a machine learning. It's a neural network, your neural network in your brain is actually analyzing the picture and deciding whether it's three or four jet events or whatever. So definitely a machine learning can do this. On the other hand, typically you also want to be able to calculate and this is more challenging in fact. I think the jury is still out about whether one can set up machine learning techniques that at the same time also allow for sufficient calculability. But at least in principle finding the jets, it is certainly something that you can run through some machine learning application. So, as I was saying, and actually this dovetails well with what I wanted to say next. The thing about jet algorithms is that typically you want them to lead to calculable observables. Sturman Weinberg did that. Many of the cone algorithms that were used in other collisions in the 80s and 90s were not actually infrared and collinear safe, but people were still using them at the experimental level for lack of better alternative. On the other hand, in at least in the plus and minus collisions, people already in the 80s and in the 90s built infrared and collinear safe jet clustering algorithms and an example of such an algorithm is what was called Jade. It's an infrared and collinear safe jet clustering algorithm. And how does it work? Well, it works in the following way. I explained in detail how it works because it is a template for all the clustering algorithms that are used even today at the LHC. So let's say among calculate the invariant mass squared from all pairs of particles in an event. So if you have only two, well, it's easy. You have just one invariant mass. If you have three particles, you have M12, M23, M13 and so on and so forth. So M and Mij, let me just make sure that I define it properly. Of course, it is this and this is equal to actually two EI EJ. One minus cos theta IJ with obvious meaning of this. Then the second step is that you look for the smallest Mij among all the ones that you have calculated and you recombine I and J. By recombine, it typically means meaning that you sum somehow PI plus PJ. Then there are different recombination schemes that you may apply. There are different ways of recombining these things, but that's the idea. Out of two particles, you make a single object. And then you repeat this meaning you look for the net you re also with a new object. Of course, you have to update the list of all the invariant masses. So let me perhaps add this step here that I had forgotten to prime update list of Mij. Continue recombining the smallest and then three you repeat until this minimal Mij is larger than some parameter Y cut times the center of mass energy squared. So when the when the invariant mass of the two particles that you are trying to recombine is too large, that typically means that the jet is actually getting too broad, you stop and you say okay this is not a jet anymore. The idea is that you're still looking for something fairly collimated something fairly collimated tends to have smallish invariant masses. I say smallish because it still depends on the momentum but but that's the idea. So you stop once you would be recombining let's say two particles that go to hard particles that go at very large angle. These things have very large invariant mass. And it doesn't make sense to call two hard particles going that way a jet things that go like this are a jet but not things that go like that. So what's left are called jets. So as you can see, this is a jet finding algorithm, you construct a certain number of jets you don't know beforehand how many, but at the end once the algorithm has run, you have a certain number of objects that you call jets. This is trivially inferred and collinear safe. And you can see it easily because when you emit a soft particle of a massless particle, of course, the mass the mass remains zero and so it is immediately recombined remember the definition of inferred and collinear safety, you don't want to change the observable. The emission of a soft particle produces a new pair with zero invariant mass, which is therefore immediately recombined by the way the algorithm works. Remember you recombine the smallest invariant mass pair, and the same goes for the collinear emission, a collinear emission of a massless particle has a zero invariant mass. And so once again, this collinear branching is immediately reversed by the algorithm. And so the collinear branching does not change the observable. This is jade is inferred and collinear safe. Quite easy. And the advantage of these clustering algorithm is typically that it is quite easy to analyze their inferred and collinear safety because they are constructed as a step by step procedure. Yet, jet finding algorithms of the cone type are more like global minimization problems because you try to optimize the way the energy flows into cones, or you're trying to find the cones in the event where the energy flows and analyzing in terms of collinear safety, these global minimization problems is much, much more complex. And so it is difficult to even difficult to establish easily if a jet, if a cone jet algorithm is or not inferred and collinear safe. In this case, it is easy. So, I will not spend too much time doing the calculation, I will just say concerning jade perhaps. I want to consider a three jet rate with jade. So consider E plus E minus going to QQ bar G with of course some momentum Q equal to P1 plus P2 plus P3, the three momenta in the final state. And of course, let me write that in general, Mij squared be P1 plus P, sorry, Pi plus Pj squared, which is Q minus Pk squared, which is of course squared one minus two Pk dot Q and if you remember how we defined the X momenta in the QQ bar G emission, this is S1 minus Xk, where k is the third particle in the event. This is useful because so an event now I'm looking for the three jet fraction with jade. I want to calculate in this particular algorithm how many times I find the three jet in an event. Of course, it's an event that is composed only of three particles and so it's actually when the three particles are sufficiently far away from each other. So that's what you want to calculate. You may also try to calculate directly the next reading order correction to the two jet event but it is easier this way and then you subtract from the total cross-section because an event with three particles is only either a two jet event or a three jet event. So you always have sigma two jet plus sigma three jet equal sigma dot QQ bar plus G, of course, so it's easier to calculate the sigma three instead of calculating directly, but other than this. An event is three jet if one minus X i is larger than Y cut for all i. Why? Because if you look at this definition of invariant mass and if you remember the definition of jade that said that you stop when the mass is larger than Y cut S and that let me remind you it's this. Where is this here? You stop when M ij is larger than Y cut S. So if to begin with the three invariant masses of all the particles in the event, there are three possible combinations of invariant masses, of course, with three particles. If all three are already larger than the cut, well, there's nothing left to cluster. Nothing is clustered and the event is by definition of jade a three jet event. So let me write this means that all M ij are larger than cut or then stopping criteria if you wish. Okay. So this defines the phase space to calculate the three jet event in jade. So let me let me say one thing and then I will take the questions. I defined a jet algorithm that is a series of steps. Okay. Since it is defined as a series of steps that I execute in an iterative way, I can apply this part this algorithm to an event of any complexity. Meaning in particular that if I am looking at an experimental event, where in the final state I have not only a quirk and anti quirk and a gluon, but I have all sorts of hard runs that come from the fragmentation of these quarks and gluons, I can still apply the algorithm. Okay. So a jet algorithms should be applicable, both to theoretical results and to experimental measurements in order for you to be able to compare. So if you are an experimentalist who measures 20 pions and chaos and protons in the final state, well, you apply in an iterative way this algorithm to your event. On the other hand, if you are a theorist who has made a calculation like here, what you do is typically you translate the working of the algorithm in a phase space integration. Because as a theorist, you integrate over a phase space, especially because you can't deal with the singularities and virtual corrections and stuff like this through some iteration, at least not that easily. Perhaps one can come up with one way of doing it, but the typical way of doing a calculation in theory is to integrate particles over a phase space. So what I am doing here now, I am translating the iterative working of the algorithm in an integration region over the phase space. And I will come to that in a moment, but perhaps I can take the questions. Yeah, I was just wondering, how do you choose this Y cut? Because in the case where you draw a few slides ago, you draw a diagram that could be three jets or could be two jets. So you have particles close to each other. So how do you choose what's your Y cut? Excellent question. It's a free parameter. I'll show you a plot later on that explains. But yeah, the question is totally appropriate. Absolutely. It is, I mean, I can anticipate it's like a resolution parameter. It controls how many jets did you see in the event and we'll see a plot later on. Next is Monday. I was going to ask the same question. No, I will answer both of them at the same time in a minute. Okay. Okay. So let's keep continuing translating this. So this this this condition. So this gives eventually. So this is one minus X one larger than Y cut, which eventually means X one of course less than one minus Y cut. It's one minus X two larger than Y cut, which means X two less than one minus Y cut. Finally, it's one minus X three larger than Y cut, which means that one minus and remember X three in a two two. So in a QQG event is two minus X one minus X two, because of the relation X one plus X two plus X three equal to two. And so larger than Y cut. So this is X one plus X two larger than Y cut plus one, if I have not made a mistake. So now let's look at how this looks like on the face space. The face space is like this. So this is one. This is let's say it's X two. This is X one. How many these are 10 so like this X one and there is this. Okay, so this region here is the whole region available to X one and two and X three. The production of a QQ bar in the gluon lives in this in this region here where of course, let me write down a bit better things. This curve, this one is X one equal one and it's the maximum. This thing is X two equal one, which is of course the maximum. This one here is X three equal one, which means of course X one plus X two equal to one. And so where your radiation exists. I wanted to use your radiation exists in this triangle here. Now we start applying the cuts that we just derived. So this one's here. Now I want to actually write X one less than one minus Y cut X two less than one minus Y cut and X one plus X two larger than one plus Y cut. And so it turns out that the region where you can actually do this is the following one. So let me pick a color that is appropriate. So let's say this one here. Let's say that this is one minus Y cut. And remember X one can only live here or actually here because it is. This is the condition on X one a similar condition applies to X two actually it's a symmetric sorry. So this is also one minus Y cut and X two is forced to live below this thing. And finally, there is a third one, which is this one, which is here. This is one plus Y cut. So at the end of the day, the region that is available for the three jets cross section is that regional phase space. So and now you see how you can easily take your calculation that remember you're calculating you want to calculate F3 in Jade. And this is easily one over Sigma the integral over DX one the integral over DX two the. This Sigma qq bar G divided by DX one DX two the one that we calculated the other day or actually we gave the exact result. I only calculated the soft and clean up and here this phase space that well you can quite easily. So you integrate over a region that corresponds to this triangle here. And if you want to just write down the right numbers this is to my cat. This is one minus Y cat. This is one plus Y cat minus X one and this is one minus Y cat and you replace this of course. By what we calculated or we said one can calculate which is alpha s CF over two pi X one squared plus X two squared divided by one minus X one one minus X two. You will notice that we do not hit the soft and clean your singularity soft and clean your singularity leave here and leave here. And I think now yeah yeah leave here and here the soft and clean your singularity. As you can see integrating over this region here we never hit them why well because we have made the explicit decision to look for the three jet event a three jet event is something like this. All particles are sufficiently hard. I can see them. And none of these particles is collinear to each other. So of course I am staying away from the collinear region from the soft region and from the other collinear region. It's by construction. The phase space that I am cutting out by choosing a three jet. A three jet rate with a given Y cut is such that it stays away from the dangerous region so you don't need virtual corrections to calculate this three jet fraction, because the virtual correction only needed when you reach this area here because if you stay away from there, you can only calculate. You can only calculate at leading order a three jet cross section it's actually what you're doing here it's actually if I can't go back to my very messy drawing I'm actually calculating only this thing to jet plus one jet. I'm calculating at leading order the three jet cross section. And I don't need anything else because I'm only calculating the leading order. If I wanted to calculate F3J to next to leading order, then I would start including also the actual corrections but at leading order I don't need anything else. And let me write down the result because then I will take the question but let me write down the result because it is an interesting one. The result of this thing has the following form so alpha s CF over 2 pi. The fact that it leads with alpha s of course is clear you need one emission to make three to make three jets. You had only two at zero order back to back and you need one to make the third jet. Then this is looks looks like so three one minus to why cut let me just try to why see which makes my life a little easier log of why cut divided by one minus to why cut. Plus to log squared. Why cut divided by one minus why cut. Plus for the log. Why cut over one minus why cut. And then there are some constants by square divided by three plus six minus 12 why cut minus nine why cut squared divided by two. So what is interesting in this expression. What is interesting is that first of all it's calculable no surprise here we said we were away from the dangerous region so what you have here is the result of the integration of this over this. Okay, something perfectly calculable. What do you see is that you have terms like this log why see and this log squared why see that diverge when why see goes to zero. Why well it's clear why why see going to zero means moving this thing here towards the borders. It means setting a resolution for the algorithm, which is actually zero. And which means allowing two particles essentially collinear to each other to be called separate. Why cut is essentially the this the minimal distance that the particles must have in order to be called two different jets. If this distance goes to zero you are once again allowing particles to become collinear. And it's not a surprise since you are of course not including virtual corrections. It's not a surprise that your your cross section for a three jet rate blows up because it means that you are now integrating up to the borders space and so correctly this thing explodes. Now, the problem is not so much to stay away from zero. Of course, the problem is what happens if you are still in a region where why cut is a small and therefore these terms are large, even without exploding. So what happens if your perturbative correction is very large. So it's not so much a much a matter of divergence. It's a matter of poor convergence of the series. And that's where it's actually what you have in these situations like like here it's not too different from from what I had I had here. Where in this region here, as you can see, there is a big difference between the fixed order calculation and there is some calculation. So you typically in these cases you want to re-sum your calculation in order to improve it. What happens with Jade is actually that resumption does not work. I mean you cannot re-sum Jade the way it is defined. And that's why people eventually came up with another algorithm. I will talk about that algorithm in a moment, but first I want to show a plot and then I will take the question. The plot I want to show is this one. And this answer the question of what happens when you change why cut? Well, what happens? So first of all, this is what happens when you apply the algorithm to a real experimental event. So with many, many particles, the fact that you have many particles means of course that you can go beyond three jets. You see, if you have only three particles, at best you can have three jets. But if you have 15 particles in the event, if you have an event that is actually something like this, like this and like this, an event like this can have three jets or even like this can have four jets. Even if it has many, many particles, or you can have even more. So you see that as you reduce the value of why cut, the cross section for a larger number of jets increases because your resolution becomes finer and finer. So it's the equivalent of having an event like this, let's say. So you have an event like this. If your resolution is something like this, then this is a three jet event. But if your resolution is smaller, let me rewrite this thing. If your resolution is smaller and it is something like this, this becomes a four jet event. And this is what happens here. You apply to the same set of events, a Y cut that decreases. And you see that when Y cut is large, essentially all events are two jet events. We are here. These are two jet events. When Y cut is large, all events are two jet events. As Y cut decreases, the number of two jet events decreases and the number of three jet event increases. And then at some point, even the number of three jet events start decreasing, but the number of events that you recognize as four jet events increases, and so on and so forth. So again, of course, this is a resolution parameter. The smaller the resolution parameter is, the more details in the event you are seeing. And this should answer the question that was asked before, what happens if you change the value of Y cut? What happens is that the same event can be seen either as a two jet event or as a three jet event or as a four jet event and so on and so forth. Perhaps I can take the question now. Yes. Yeah, okay, coming back to the triangle plot you made. So you said the problem where the boundary on the left and on the upper boundary. So I guess the lower boundary it's fine because you don't have collinear or IR problem there. You mean this one, this one here. So this one is where the glue on the emitter glue on is hard because it is. So yes, I think yeah, I don't think. Oh wait, wait a second. No, wait a second. No, there's still momentum conservation if the glue on has the maximum allowed momentum. I mean that one of the quirks is infinitely soft. I can't remember. It could be that this means now wait a second. So here it is extra equal to one. No, no, so it means that the sum of x one plus x two is equal to one. But this could still mean that one of them is infinitely soft. I think there could still be there could still be soft singularities around I can't remember I good question I should should think about it. I can't remember. But wait a sec. Wait a sec. Potentially. Yeah, I think. I would like to think more about it. One more question from Monday. Hello. My question is regarding this fight cut like when we lower the value of why can we can have a better resolution but is there any like lower bound on the value of this fight cut. No, well in principle you can take whatever you want. But then the point is as you can see here that the smaller white cat is so the more exclusive. This is a typical feature of QCD. The more you're trying to look at details. The more difficult your life is, and vice versa the more inclusive you are, meaning the less details you look at the easier it is to make a calculation so this thing is made it explicit here. As you go to small white cat. These things end up increasing and at some point that will make the convergence of your perturbation theory very very bad. You can decrease it as much as you want. But at some point to your predictions won't actually make sense, unless you have properly resummed it. And even if you've resummed it you will still have larger number of corrections due to the fact that you're looking at very very tiny details. A general rule of thumb is, in order for things to make sense in quantum mechanics, you have to be sufficiently inclusive over something. And if you try to push too much into the details, something blows up. And because you don't have the full control of the theory. So let's say there is a difference between what you can do in theory, which is, you can set whatever value you want, but what in practice means to do it because you lose control of your productivity. Is there any like constraint comes on this biker from experimental side or something like that. On the value of white cat you say yes. Well, no, because again, you can decide that you process your, your data with whatever parameter you want. It's just that at some point it won't make much sense. Okay, thanks. White cat is not an observable. White cat is a parameter that you set to define the observable. Okay, thanks. Next is Ryan. Hey, so my question goes back a little bit concerns the graph that you had of the trust variable. Which one. Just a graph of the trust variable, and where you see the corrections to to resummation. Yes, this one. Yeah, exactly. So my question is, what is the difference between the pure resummed and the matched resummation. That's the it's you are combining the resummation in the region where the resummation matters which is this one here with the full calculation. So what you do essentially you you subtract from the resummed result and expansion of there is sorry, you subtract from an expert from the result of the equivalent of what is already provided by the fixed order calculation to avoid double counting. I would say you have an exponential you submit to to something that contains the first powers of the exponential already. So you take away those first powers that are already already contained in the exponential, and you sum the two things. Okay, so much that typically means I am combining a resummed result, a fixed order result and I avoided the double counting of the terms that are in both. Does this answer the question. Okay. Next is serene. Yes. Yeah, so my question was about. About sorry I can't hear you very well. Yes. Yes. So, when you mentioned, when it actually happens in a different experiment, you're talking about so this is often capitalization. I'm sorry, I really can't hear you very well. Hello, can you hear me now a bit better yes. No, it is the problem. So, what I was saying is that we were modeling jets based on the hard scattering, but we are clustering over the final space so we are going over skills here. So, does the sigma contain the hydronization part as well. No, so of course what you calculate here is a simply a perturbative prediction. And then indeed you compare this perturbative prediction, possibly modified by some by some number of the corrections to real data but the idea generally is that there is a fairly good much between what you predict with the patterns, and what you have with the other ones, unless you're probing to find details. It's called local pattern hard on duality, meaning that calculations made with patterns approximate quite well. What you actually get in real life with the other ones, unless you're probing the resolutions that are too small, and we are back once again to the fact that the more inclusive you are. The smaller number of corrections are. On the other hand, the more exclusive you go, the bigger the number of corrections are but still you have to try that fine line where you go exclusive enough in order to actually gather more information, but you're not breaking everything. So it's a bit of a.