 So there's a message in the chat before we start. I don't know if it's up to you or yeah, it's up to you. All right. OK, so it is time. If you want, we can start. Shall we, Matteo? So let's begin with a resume of the lecture on QCD colliders, please. OK, thank you. And welcome back, everybody. Perhaps one thing I can do before beginning, I can try replying to at least some of the questions that were asked in the previous days. And sometimes I may have replied partially. Sometimes I said I need plot or something like this. So let me show, for instance, you this. OK, so one of the question was, how do we measure color factors? How do we make sure that the value of the CF and the CA that we predict in SUN are actually 4 over 3 and 3? And this is an example of measurements that have been performed at left. So be careful. In an E plus and minus machine, still, you are able to a lot of QCD. And so in the plot on the left, you see, well, it's a couple of different ways of showing things. And in the plot on the left, it's nice because you also see different groups, what kind of color factors different groups other than SU3 would give. And as you can see, the data are in good agreement with SU3 and the values that are predicted by QCD. And the plot on the right is somewhat similar. Here it's directly CA against CF that is measured through, of course, the measurement, experimental measurement of a number of observable that are sensitive to these color factors. And you see that once again, the values that come out experimentally are in good agreement with the values predicted by QCD, built as a theory of SU3 with the course in the fundamental representation and the gluons in the adjoint one. So again, it's an experimental confirmation that these are the right choices, knowing that, of course, you can build a theory out of any group and out of any representation, of course. Another question was, how do we actually make sure that the non-ability of the theory, in a sense, is seen that the gluon is indeed something that couples with multiple gluon vertices and something like this? So one example, one way of doing this, it's not necessarily the only one, but one way of doing this is to measure this angle here, the Bangson-Servas angle defined as you see here in the screen. This is an angle defined in terms of the production of four jets in E plus and minus collisions. P1, P2, P3, P4 are the moment of these jets or they're the decreasing value, I think. It doesn't really matter anyway. And you see in the plot that the experimental data do coincide with the QCD prediction and do not coincide instead with the prediction of a theory where the gluon is an abelian, in a sense, gauge-posing. And there are a number of plots like this. You can play this game under very many different forms. And every time, what you discover is that it is indeed the QCD prediction with the full SU3 that I showed yesterday that works. Another question was, how do we predict the power of non-perturban corrections? So I was hoping to be able to find something similar, something simple to give you. It turns out that, well, at least I haven't been able to find anything simple that would not have taken two hours of lecture to actually explain. The general idea remains what I said yesterday, meaning that you try to calculate as much as you can perturbation theory, resumption, and stuff. And from the ambiguity of the perturbative series, you try to deduce the power correction of the known perturbation one. But besides this very, very general argument, it's difficult to actually start writing equations. It would really take us too far. It would take too much time. On the other hand, I can point you to a couple of papers. One by pure accident came out yesterday on the archive. It's a nice review. And I just discovered completely by accident that this nice review, this one that I've given here, archive 2106-00276, it contains a nice introduction to renormalons. And renormalons are indeed this signal of non-perturbative corrections on top of perturbation theory. So you can definitely have a look at this. Another nice review is a review by Martin Benneke, again, titled renormalons. So here I don't have the exact reference, but it came out around a bit old, 1998, 1999. I'm not sure exactly the date. But if you Google it, of course, these days it's very easy to find anything on the web. So you don't need the exact reference with the name of the author and the title. And roughly when it came out, you can easily get it out. That's a huge review. It's multiple hundreds of pages, if I remember correctly, which is why I suggest that you first have a look at this archive paper here if you want a quicker introduction. But of course, Martin's review, even if it's from 20 years ago, it's probably one of the most complete piece of work on the topic. So I haven't answered the question, but I have at least a point of view to how to try to answer it yourself. OK, let me go back to our lecture. And let me get my notes so that we can continue. OK, so we were looking yesterday at E plus and minus. We just started, in fact, but we didn't do much. So we start again from this. So the idea is to study E plus and minus to QQ bar possibly plus other particles. We'll see that we'll eventually have a glue on here. And out of this, as we said, from the experimental point of view, you will get hard runs. But of course, what we calculate are rather the process for the particles. And this is a calculation that can already allows us to see some of the techniques and the tools that then can be used more generally in quantum colliders and also in general in collider physics. So let's start looking at the E plus, E minus, QQ bar. So let's say that we have a momentum L1 here in going, a momentum L2 here, a momentum Q, which is, of course, the sum of the first two. And then outgoing, you have a momentum P1 and a momentum P2. So that's a very well-defined process at a tree level. You have a center of mass energy, which is eventually equal to P1 plus P2. So oops, but not school, sorry, like this. And you write a cross-section for this process in the form ds is equal to 1 over flux. It's the usual expression. The matrix element summed and averaged. And then the phase space for two particles. So this is the standard thing. Let me also specify because I may use it later. P2, this is the d3 P1 over 2 pi cubed 2A1. d3 P2 to pi cubed 2A2. And then I also add in the delta of conservation of momentum. I want to explicitly the calculation, but you know how this works. You write down the Feynman rules for the leptonic current. You write down the Feynman rules in terms of spinors for the quark current. You write down the photon propagator. You sum, you contract, you saturate, you do diracology for the gamma traces and so on and so forth. And you integrate over the phase space, which is particularly simple in this particular case because it is a two-to-two process. And you end up with a result. And this result, as you certainly know, is something like this. So 4 pi divided by 3 alpha squared electromagnetic divided by s. And then there is a factor related to the electric charge of the quark and a color factor, which is related to the fact that there are three possible types of quark for each flavor in the final state. And this you probably know is one of the best experimental evidences for the existence of color and for the fact that that color is actually three. And as an example, I want to give you, let me see if I can go and grab a plot that I have here in my notes. Now I can't cut and paste, unfortunately, because of the technical issue that we have, but I should be able to at least show you this. OK, here it is. So here, this is the plot of the ratio of this cross-section in plus and minus two atoms divided by plus and minus two muons so that it eventually eliminates all the kinematical factors and all the constants. Eventually, you are left with your cross-section R. Sorry, your ratio R is equal to Nc, the sum of Eq squared. So this expression here is what you are measuring here. And you forget about the resonances. So don't look at where you have resonances because they are related to particular particles that we are not so much interested. Look at the areas where this thing is flat, which is what matters for us. And for instance, in this region here, so up to three quarks, this is exactly, sorry, this is up to two quarks, no, up to three quarks. Sorry, yes. This is exactly what you have. And you do the sum and you get that R should be equal to two. And this is exactly what you find with Charm. Sorry, with up, down, and strange. With three quarks, the prediction with this Nc is that R should be equal to two. If this Nc were not here, or if it were a different number, we would not find this value here, but we would find something different divided by a factor of three or something like this. So this is a clear evidence that we are on the right track and that this is the right expression. All the more so that when you start skipping and you go beyond this mass threshold, you go beyond the first three flavors and you go up to Charm and bottom, well, you find this step here and this step again is consistent with that number there. So fine, everything works nicely. And we do have another experimental evidence that indeed we have color in the right place. OK, now, having done this, this is nice, this is interesting. Let me, OK, I have the formula there, I think it doesn't matter. So now I want to consider higher order corrections. This was just the leading process. But what actually interests me and the reason why I'm choosing this process is that I can pretty easily calculate higher order corrections and actually we do it in some approximation. So let's consider these two processes, or actually this process and these two diagrams where I'm using again Q, Q bar. This is a gluon and I'm labeling this P1, I'm labeling this P2 and I'm labeling this K. And we'll actually look, it's actually simpler to simply a virtual photon going to my Q, Q bar and an N gluon. I don't need to take care of the electronic part at all. So let me write down the amplitude for this process. The amplitude for this process is the following one. So I have a usual U bar P1. This is IGS gamma mu TA. Then there is the propagator P slash 1 plus K. There is the electromagnetic vertex IEQ gamma mu. And then there is V2, VP2, the outgoing anti-quark. And then there is the polarization of the virtual, sorry, of the real photon. And actually this is, because it's not saturated. And then I have another term, which is the other process, U bar P1. Now IEQ gamma mu minus I, this is the propagator in the other diagram, plus K, IGS TA gamma mu VP2. And again, epsilon mu K. So this is the full matrix element, the full amplitude. I could, in principle, calculate it exactly. It's not a big, big deal. It's the work of half an hour if you know exactly what you're doing. It can be a few hours if you really have to try and find your way because you've never done this before. But it's not a big complication still. Since we don't need the full calculation, or at least we don't need to calculate the full calculation, let us just get an approximation. And the approximation that we want to make is the approximation of a soft gluon. So I want to consider this matrix element in the limit where K is much softer than P1 or 2. And my statement that I will not prove explicitly, but that can be fairly easily deduced, is the following one. In this particular approximation, we witness a factorization of the production of the QQ bar there on one side and the gluon emission on the other side. And the statement is the following. And in this approximation, M mu factorizes in the following way, U bar P1 IEQ gamma mu V2. And as you can see, this is nothing but the three-level amplitude. It is simply the amplitude for the emission of a QQ bar pair, a QQ bar pair. Sorry, I haven't yet said if it's a mass or not, from a virtual photon. So nothing special. And then there is another bit, which is the following one. Gs Ta P1 dot epsilon star divided by P1 dot K minus P2 dot epsilon divided by P2 dot K. And this is what is called the Iconal Current. And it describes, as I was saying, the emission of a soft gluon of the QQ bar pair. OK, so we have this factorized for. Why does this happen in physical terms? In physical terms, it happens because the hard distance, sorry, the hard collision, the creation of the QQ bar happens in a region of size 1 over square root of s. That's the hard interaction. It happens over a very small spatial distance, especially, of course, you assume s to be fairly large, several GBs or something like this. So this is a small distance. On the other hand, a soft gluon will have a very long wavelength. So a soft gluon will be something like this. The wavelength of the soft gluon will be much larger than the size of the region where the production of the QQ bar pair happens. And this explains in physical terms why you have this factorization. The reason you have this factorization, meaning the reason why the gluon emission is independent of the hard interaction, is that the gluon, since it has a long wavelength, cannot resolve the hard interaction. The gluon cannot know anything about the hard interaction. It's the same thing when you use an optical microscope and you try to look at the atoms. You cannot see the atoms because the atoms have a size which is smaller than the wavelength of the light that you are using to look at them. And this is exactly the same thing. The gluon is softer. It has a long wavelength. It cannot resolve what happens at a much smaller space distance where the hard interaction takes place. And so you have this factorization already at the amplitude level that allows you to essentially decouple the emission of a soft gluon from the hard interaction. And the reason why we are doing this is that now we can only worry about the part given by the gluon emission without carrying on making the calculation for the full process. Of course, all the considerations that we will be doing in the next few minutes could have been made using the full process. We are just simplifying our life. We are not going to discover anything much newer. So let's now proceed in the calculation and let's calculate the sum over the colors and the polarization of our matrix element. We have the amplitude written and factorized for. We want to do it squared and sum because we want eventually to go towards the cross-section. Because of the factorization, let me just quickly say this will be simply, of course, if the amplitude factorizes the matrix element squared also factorizes. And so we will have one term, which is simply what you would have found in the Born case. And then you have a term that you get from the Iconal current squared. And the term that you get from the Iconal current squared is cf gs squared to p1.p2 divided by p1.k p2.k. And again, this is factorized. The calculation is pretty simple. You get the usual cf. You get the sum over the polarization states of the gluon that you can do without costs because it's an abelian situation and so on and so forth. Now, let us include the three-particle phase space, which is also factorized. And so we have that we have the sum of m qq bar g times the three-particle phase space, which is equal to the matrix element squared times the two-particle phase space. And then the part that we have just calculated. So cf gs squared to p1.p2 divided by p1.k p2.k. And then the phase space, which is d3k divided by 2 pi cube to eg, where eg is the energy of the gluon in a given frame that we will choose later on as a matter for the moment. If you manipulate this a little, and in particular, if you choose an angle theta between the direction of p1 and the direction of k, we can rewrite this in the following way. So equal, again, m qq bar d phi 2. And then you have the part related to the radiation, which is eg d eg d cos theta d phi 2 pi 2 alpha scf over pi. And then the same, I haven't yet worked on this. I'll do this in a moment, p1.k p2.k. Now, let me remember that we are in the soft limit. So in the soft limit, we have that s, which is also q squared. If I write it in terms of the square of the virtual photon momentum, it's p1 plus p2 plus k squared, which we can approximate it to p1 plus p2, because k is much softer. And therefore, to p1 dot p2, we choose a frame in which we work such that p10, meaning the energy, is equal to the energy of the anti-quark equal s divided by root s divided by 2. This is the choice of the frame. And in this particular frame, we have that the term 2p1 dot p2, that was so far expressed in terms of invariance. So now we specify it to the frame that we have just mentioned. I will spare you the calculation, but it's really quite simple. And what you get is eg squared 1 minus cos squared theta. And remember, we have been using already a couple of times the soft approximation. So remember that all the results that I'm writing down are in that approximation. So now finally, q, q bar, g d phi 3 in this soft approximation is m, q, q bar squared. Oops, I had forgotten a few squares here in the apologies. d phi 2, and then there is the part related to the radiation. So 2 alpha s cf divided by pi, d eg divided by eg, d theta divided by sine theta, d phi divided by 2 pi. And this is a generic dp, the probability of gluon emission, of softer gluon emission, actually. The important thing, of course, in this context is that this term is universal. It has no reference whatsoever to what produced the gluon. It is just a universal probability of emission of a soft gluon. And we'll use the characteristic in a few contexts. Now, the first thing that, of course, you can see is that you can also rewrite it in the following way. So eventually, you can rewrite this dp k. So that's the probability of d energy, d theta ij. So the probability of a particle k splitting. Let me just, oops, let me rewrite it. You have a particle k that splits into a particle i and a particle j. It's a generalization of what we just saw. We saw the gluon being emitted off a quark. But in fact, you can generalize this quite quickly in QCD. And this is proportional to alpha s divided by the minimum of e i e j theta ij. And we'll see in a moment why this expression is useful. So the probability of emission in QCD is one over energy, one over angle. Note that this is a fairly classical result. If you have ever seen the expression for the Bremstra along the radiation in classical electrodynamics, you find exactly this spectrum. So we have performed a quantum mechanical calculation. Actually, a quantum field theory calculation. But eventually, we end up with a very classical result. So emission of soft radiation is a classical process, essentially. It remembers nothing of quantum, at least not in this context. OK. Oh, this is, of course, what happens with massless particles. I should make it clear. But massless particles are most of the time what we deal with. Because especially when you work at high energy, well, the first three quarks are fairly massless than it else. The gluon is also massless. But when you are at high energy, even the charm and the bottom and these kind of particles are typically fairly massless. So this is, by and large, the kind of situation that you work with, massless particles. So this is what you have. This is what we found in the calculation. Factorization of emission, probability of emission proportional to 1 over energy and 1 over angle. And this is quite a general result. Now, suppose I have now the matrix element squared. I have the whole expression. In fact, what I have here is essentially the full expression, even if only in the soft and collinear limit. Now, suppose I want to calculate a total cross-section. Total cross-section. I want to calculate a total cross-section for sigma e plus e minus 2 qq bar g. So how do I do this? Well, I integrate from 0 to whatever kinematical limit is appropriate, the matrix element that I found. So now let me explain that. Yes, it is true. I only found that matrix element squared in the soft approximation. It is true that I cannot integrate it into a region where the glue becomes hard because the approximation will not be valid. On the other hand, the characteristic that I'm going to point out is specific to the soft region. So I can afford only using the soft expression for pointing out what I'm going to point out. But again, of course, you won't get the exact result out of that equation. So the first thing integration you want to do is, of course, integrate the dEg over g term that you have here. You have it here. Of course. So you are integrating this from the emission of a very, very soft glue, so from 0 upwards. But if you integrate this thing, well, it's easy to see that this is logarithmically divergent. And this is what we call a soft divergence. This happens when I integrate down to 0 glue on energy. Of course, this is a divergence that happens because I'm integrating down to 0 glue on energy. And then there is another one, because when I try to integrate d theta divided by theta, well, I have the same problem. Or luckily, sorry, this would actually be d theta divided by sine theta. But near 0, of course, this is d theta divided by theta. And again, this is logarithmic, a little divergent. And this time, this is a collinear divergence. This happens when the emitted glue on is collinear to actually either of the quarts. Of course, these things would not be here if the particles were massive, but the particles are not massive. And so we can safely assume that these things are there. OK. So what do you do? Let me now, before telling you what we do, let me open a little bracket. And just tell you that you could write this in a different way, but that will also expose the singularity. So let's define the variables xi as 2pi dot q divided by q squared, where i equal 1, 2, and 3. And it corresponds to p1, p2, and actually k. So I can rewrite the cross-section using this variable, a-dimensional variable xi instead of the momenta. And the full result for the cross-section, so without the soft approximation, is a very simple expression. It's not so simple to derive, but it's a very simple expression when you calculate it. And it's the following, 1 over sigma 0, oops, sorry. So 1 over sigma 0 d sigma dx1 dx2 is given by cf alpha s over 2pi x1 squared plus x2 squared divided by 1 minus x1, 1 minus x2. And you see that since with, of course, 0 less than equal xi, less than equal of 1, and there is this relation x1 plus x2 plus x3 equal to 2. So you see that there are clearly two regions when x1 and x2 go to 1, where you get the same logarithmic singularity that you were observing in the integration in terms of theta and eg. So it's just a different way. Here, this 1 minus x1 becomes singular when x goes to 1. 1 minus x2 becomes singular when x2 goes to 1. And then you get the same kind of singularities that you were observing here. So this is a different way of expressing the cross-section. This is, by the way, the exact result. But other than this, of course, it still shows the same characteristics. It is a question. If you're in the chat, Zara asks, why you tell this gluon is soft? Which properties of gluons can show it is a soft? Zara, you are welcome to ask yourself and everybody. You are welcome to raise hand and ask yourself the question. Well, if the question is why the gluon is soft, the answer is I chose to look at the phase space region where the gluon is soft. Or perhaps I can let Zara. Yes, I allowed to unmute Zara if he or she wants. And in the meanwhile, there's another question. So I should put up first. So if she doesn't ask or explicitly, or he doesn't ask. Hello, can you hear me? Yes. I want to know how you can use this structure. And why we can say it's gluon soft. If we have other gluon, we don't count this structure. So in this particular process, I have decided to look at the region where the gluon is soft. Because it is there that I observe these characteristics that I'm telling you about. First, that there is a factorization of the emission. And second, that there are these divergences when I integrate over it. So I have decided to analyze this region because this is where I observe these things. If I do not want to look specifically at that region and I just do the full result, I actually get the same thing because I would get this thing here. And again, when I try integrating over the phase space that goes from 0 to 1, I would hit the same logarithmic singularities by integrating over those terms. So it's not so much that the gluon has to be soft. It is that I decided to study that region because it is in that phase space region that I observe these characteristics that are eventually important. Because by the way, these expressions here, forget for a moment the integration, but this expression here also tells you that QCD, or in general, any massless gauge theory, likes to emit soft particles at small angles. This expression here is quite universal, as I was saying. You also find it in classical electrodynamics. So it is emission of soft particles at small angles. This is how in these theories, particles are primarily emitted. All right. There's another question, another two questions. The first is from Daniel. Hi. Could you please go back to the slide after you draw the diagrams of this process when you were applying the Feynman rules? This one or this one? The next one, please. So I was wondering, why isn't the k slash not the one right after you draw the diagrams? This one? Oops. Yeah, that one. I was asking, ah, definitely. If that was the question, that was my mistake, so. Oh, sorry. Yeah, I was only quick checking, so thank you. OK, and then, sorry, Anna. Hi, thank you so much. This is a follow-up on Zafer's question. Is there any sort of practical situation when applying the soft blue on approximation is inappropriate? Well, sure. I mean, if you look at an observable that requires a hard glue on it, of course, you would not be giving a good example. The typical example, actually, there is a very, very simple example. Suppose you look at t plus and minus collision and you want to look for events where you have. So now I'm looking at a detector from the beam pipe. So I'm looking at, from the bottom of the cylinder. Suppose you want to observe an event with three hard particles coming out in the final space. So let me walk this back a moment, sorry. E plus and minus 2 q-q bar. So of course, the q-q bar had to have been back to back. So you have the collision here. There is a q-q bar. And then what will happen in QCD? We said that there is confinement. The q-q bar cannot live just by themselves. They must emit radiation. They must adornize. They do like to emit soft and collinear radiation. And so they will emit essentially radiation of this kind, mainly collinearly. And eventually, they will form what is commonly called two jets in the event. We'll come back to jets a little later. But for the moment, this is a typical two jet event. Now suppose instead you are looking at a process where what happens is that the q-q bar emits a gluing. If the gluing is soft or collinear, well, you don't see anything different from what you see here. Because the gluing will be emitted just very collinearly. It will simply be contained inside one of the two jets. So this process could very well be described by the approximation that I just made. On the other hand, if you plan to look for an event that's more like this and that then you have your additional radiation, whatever happens, and then you form the three jets. So this event here would be very badly described, of course, by the approximation of soft or collinear gluons. Because the approximation of soft and collinear gluons does not give a good description of the emission of a gluon at large angles with respect to the quark and the q-bar. So yes, I mean, the answer to the if there was the question, are there situations where you can't just stick to the soft and collinear approximation? Yes. The typical case is where you are looking at an observable that requires a hard gluon. So this is called a two jet event, for instance. And this is called a three jet event. Two jet events, the corrections to two jet events can quite well be calculated using the soft and collinear approximation, whereas the corrections to three jet events and the three jet event and its corrections will need hard gluons. Thank you so much. That's very helpful. OK. Then we have, in the meanwhile, Abhishek. Hello. Even in QED, we have a similar process where we emit soft photons. And those are, I mean, those diagrams are canceled with another one, which is a vertex diagram, right? Yeah, that was my next slide, in a sense. But yes. OK, same thing will happen even here. Sure. OK. Thank you. And finally, perhaps, Subham. Yeah. Hi. In this gluon emission process, does this momentum of the quarks could be substantially large? And then, sorry, can you hear me? Yes. And then there's this soft gluon. OK, so what actually sets the scale for the strong coupling here, the quark momentum, the quark energy scale, or the low energy? This is a very good question and also one that is very complicated to answer. The short answer is that the coupling in this process is still controlled by the hardness of the production of the Q-Q bar. So it will be the coupling at the scale set by the hard interaction, so the large quark and Q-bar momentum, and not the soft momentum of the gluon. But it's actually a complicated question with a complicated answer. OK, but is that an assumption or? No, it's a calculation. OK. But it depends on exactly what you're looking at. So it's actually, eventually you end up describing your observer in terms of a single coupling with a single scale. In practice, what happens is that you have to go looking for the dominant contribution to that particular process. Thanks. The chat is a question by Maruan. Jets means many particles, but I guess you'll answer that in the following. Yeah, jets will come. But in general, yes, jets roughly means many particles, most of the time, but that's especially because in practice, a quark or gluon is never just by itself. It's accompanied by all the shower and due to fragmentation and hydronization. So in experimental practical condition, yes, a jet is typically many particles. On the other hand, from the theoretical point of view, a jet may be described by a single particle. But we'll come back to that, indeed. OK, we can continue. And at some point, if you feel it appropriate, you can never break. So let's see a moment. Let me do a five minutes and then I'll do the break. So OK, so we have seen these divergences, but then we are trying to calculate a total cross-section that we would like to define it. So what's missing, indeed, what is missing is exactly what was just said. When you calculate corrects toward the alpha, there are some diagrams that we have not considered yet. And these diagrams are precisely the vertex correction and the self-energy corrections. These are diagrams that, via interference with the tree level, enter at the same order in alpha and therefore should contribute to the quantum mechanical calculation, the quantum theory calculation. But we have neglected them. And what happens is that, indeed, these diagrams are also divergent and do cancel in the infrared. So divergent in the infrared. Actually, they are also divergent and they will try to do it, but in the ultraviolet, either they cancel among themselves, like in this case in QED, or they will just be renormalized by ultraviolet renormalization. So what we are interested in here is the infrared divergence. So they are divergent in the infrared and they happen to cancel exactly the divergence of the real correction. Why does this cancellation happen? The reason why this cancellation happened is that you have to sum quantum mechanically all the degenerate states in order to calculate something. And when you have a gluon that is emitted very softly or that is emitted very collinearly, if you look here at this, suppose you're looking, I should draw an eye, but I'm not able to draw an eye. But suppose you look at that quark, you are not able to tell if the gluon is very soft or the gluon is very collinear, your eye or whatever instrument you are using will not be able to resolve the presence of that gluon. You will only see some energy coming into your detector. So this quark with that soft gluon, this quark with that collinear gluon will be identical for you to this quark here that is just a quark, to that quark there that is just a quark and to this quark there that is just a quark. In other terms, these two processes for you observer are identical to these ones. So there is no reason why you should consider some of them when calculating and not others. In order to get a meaningful prediction, you have to sum over all these processes. So in the infrared, you have to sum over all the degenerate process that contribute to your observer. This means that your cross-section will eventually be so you will have to regulate in practice what you do. You have to calculate your cross-section regulating the divergences, meaning finding a way to push the calculation until the end, but avoiding it becoming infinite. And so how you regulate this can vary. You can use a gluon mass or especially when you deal with non-abelian gauge theories, it's actually much better to use dimensional regularization. You use in particular, you work in d minus for epsilon dimensions with epsilon less than zero. And this thing regulates infrared divergences. Another option would be to use a gluon mass. As I was saying, it's perhaps simpler at this level, but it does not generalize easily to non-abelian gauge theories. So you do the calculation, you get a complicated expression. And I will write down for you the simplified expressions. So the simplified expression are the following for the real correction, meaning this one. Your result is the following, 2 alpha s divided by pi, some function of epsilon that will not matter. And then we have some terms that as you can see, go to infinity when epsilon goes to zero as they should. They should because we should find the divergence that we have already observed, for instance, in the soft and collinear approximation. Remember, epsilon going to zero is roughly the equivalent of whatever was preventing the gluon energy or the emission angle from going to zero. And when they go to zero, you were finding the divergences. So it's more than normal that this thing, even if it is regulated, then goes to infinity when the regulator goes to zero. But then you have also the virtual correction. Actually, you have, let me write here, the sum of the born qq bar plus the virtual corrections. So this one, we did not calculate. They are quite complicated to calculate. Let me just write down the result, which is the following 1 plus 2 alpha s divided by 3 pi, the same function h epsilon. And then there are minus 2 over epsilon squared, minus 3 over epsilon, minus 8 plus pi squared, plus order epsilon. And now the interesting thing, of course, is to take the limit of epsilon going to zero of sigma 0 plus sigma virtual plus sigma real. And you can quite easily observe that the singularities, the divergences disappear. And your result is finite. And it has of the form, the sum of the flavor, eq squared, of course. This is actually the result for, sorry, this is the result for R. OK. Anyway, sorry, let me do this then. It's easier. It's proportional to, because I don't have the right coefficients, it's proportional to 1 plus alpha s divided by pi. And then there will eventually be other corrections at higher orders in alpha that we have not calculated. So it is only after you sum both real and virtual corrections that you find a finite result. And the reason is that you have to sum over all the generate states. And if you consider only the real emission, that's not what you're doing when you go to zero energy or zero angle. OK, so I can perhaps answer the question and then it's a good time for the pose. OK, Manuel, you can ask the question. Thank you. I have a question regarding this sigma 0 and sigma y. What is it actually? What is it? What? The sigma 0 and the sigma y. So the sigma 0 is just the bond cross-section, the one that you would have calculated from squaring, sorry, just this. So this is just the bond. Actually, it is this. So this is sigma 0, the bond cross-section. And the virtual instead is what you get when you calculate the virtual diagram. So it is these diagrams here. And then they are actually the interference with the bond. So this is what gives the virtual. OK, thank you very much. And actually, it's actually this. OK. OK, there's more questions. So then, of course, you should need a few more minutes in the end. That's fine. Marwan? Yes, thank you. Professor, how to know that some corrections have important cross-sections, like gluon emissions? You calculate them. You calculate them because sometimes we have many diagrams. Yes, I mean, the only solid way of knowing is to calculate and then see what it gives. Of course, there are situations where you can guess if some corrections are more or less important. But at the end of the end, you can also predict some regions where some effects will be enhanced. For instance, as you can see here, we could predict that the emission of soft and collinear gluons is enhanced because this is the form of the emission probability. So there are some regions where you can actually expect some behavior. But in general, especially in more complicated processes, especially if you are far from clearly dominant regions, the only way of knowing is actually to calculate to do the job and calculate it. OK. Excuse me. Yeah. I have, I think, a couple of questions which are related to each other. The first one is, so this whole argument about why we have to sum this diagram with virtual gluons is because the gluon, it's not in the final state in the first place, right? The gluon is in the final state. I mean, not in the observable state. Well, it is integrated over. Let's put it this way. It is in the final state, but it is integrated over. I do not, I'm not looking at the gluon as if I could, but I'm not looking at the gluon. I'm not looking at the emission of a quirk, a non-a quirk, and the gluon. I'm looking at the emission of two or three particles in the final state, and I integrate over the phase space. Yeah, OK. So exactly, yes. That's probably what I meant. Yeah, I mean, it's an inclusive observable and not an exclusive one, if you wish. Yeah, OK. And the second one was in QED, these kind of diagrams with the photon would cancel between each other. So this argument wouldn't work in QED, right? Why? No, it's exactly the same behavior. And this is all the diagrams I've written down, as you've seen, are a billion. And on top of this, as you can see, this thing is proportional to CF, which is also a signal that everything I have here is actually equal to what you would have in QED. So this argument is actually identical to what you would have in QED. If you want to calculate the cross-section for E plus E minus 2 mu plus mu minus plus some photons, you would have exactly the same behavior. OK, thanks. And Max? Well, I mean, perhaps you have in mind the cancellation between vertex and self-energy in the ultraviolet. That's what happens. Oh, oh. OK, here we are in the infrared. Yeah, that's what I was. Yes, thanks. That brings it up. Thank you. So my question is also about the summation of the diagrams. So I understand why we have to sum all those diagrams. But so is it a priori kind of a logical necessity that those diagrams really cancel? Or is it more like in QCD or in QED? It is like a happy coincidence that this happens. But if it wouldn't happen, you would do something like in the UV divergencies to find a random normalization scheme and so on to take care of them. No, it's actually a theorem that they cancel. And I'll come back to it just after the post. So they must cancel. The reason they must cancel is that, in a sense, it's artificial, our separation of virtual corrections and real corrections. It's something we do because of the way we set up perturbation theory in quantum field theory. But it's not a physical separation. So we must consider them together. And they do must cancel. And I'll show it in a moment. OK, thank you. All right. OK, that's it. So we have a break. Five minutes, Matteo. Is it OK? OK. So we resume at 08. All right. All right. It's 08. So whenever you want to start again. Start. OK. Just wait for the recording to be restarted. All right. Wait. OK, so we just said that this cancellation is not an accident, but it is something that has to happen. As I said, it has to happen in part because, well, it was our fault that we were observing divergences because we were artificially splitting this situation in the way we set up perturbation theory. And in fact, there are a couple of theorems that tell you that this cancellation must happen. And these two theorems are called Bloch-Norzig and Kinoshita and Li-Nauenberg. So it is worth noting that, for instance, Bloch-Norzig is very old. It actually dates to the 30s. And this one is 1962 and 19, in the case of Kinoshita, 1964. So what do they say? Bloch-Norzig had actually been formulated, you guess it, for quantum electrodynamics. We didn't have it yet in the form of QED, but it was essentially the quantum version of electrodynamics. And it was formulated for, so Bloch-Norzig was for a billion theorists with massive fermions. This means that because the fermion is massless, you do not have a collinear singularity. Sorry, because the fermion is massive, you can't have a collinear singularity because you cannot split collinearly a massless particle of a massive one. It's a pure relativistic momentum conservation. It's something that you can't do. So you're only left with the potential soft singularity, soft photon. And what the theorem says is that infrared singularity cancel when something over all unobserved. And here we come back to the question that was asked, what kind of variable we are looking at? Yes, we are summing over all the photons, we are not observing them explicitly. So photons in the final stage. And this is a theorem that is valid to all order. So we have argued that this is what happens at first perturbative order. And actually we have not really tested it, but if we had calculated explicitly the virtual correction, we could have checked that indeed you do two calculations completely separately, you put them together, and then you can take the regulator to zero and it disappears and it leaves you with the final result. So even if we have not checked it, just believe me that if you wanted to check it, then you would indeed observe this cancellation. On the other hand, KLN was formulated more generally, also for massless theories. And what it says is that infrared and collinear divergences cancel when summing over all the generate states, oops, initial and final states. Okay, so one way or another, you see that you can cancel collinear and infrared singularities when you sum over everything. Now the point is that sometimes you either can't or don't want to do this. Well, perhaps you are just a bit tired of simply looking at full inclusive quantities. It's nice, but it's a number. E plus E minus two, all hard runs or all QQ bar gluons summed over, it's a number. You can do something with this number, but sometimes you can't do enough. There are analysis that you may want to perform experimental analysis that may require more differential observables. And so you may want to see if you can calculate predictions for observables that are less inclusive than the fully inclusive ones required by the application of Bloch-Norwich or Kinauch-Italy-Nauenberg theorems. Also, sometimes you cannot sum over everything. For instance, take a QCD process initiated by two protons, like what happens at the LHC. Two protons are an exclusive example of a hard run. They are not, you are not summing over everything. You are not summing over all hard runs because you do have a fixed adoronic state in the initial state of your collision. So when you do a calculation in QCD, especially if you have a collision of specific adoronic beams, you cannot apply KLN and have all collinear singularities disappear because you cannot go beyond the fact that you have a specific initial state. And so the idea of summing over all the generated initial and final states is not available to you. So this is to say that the Bloch-Norwich and KLN theorems do ensure cancellation of singularities under specific situations when you can easily sum over everything. But there are situations where you cannot apply them. And here the question is, what do you do if you cannot apply these things? And actually there are choices that QCD calculations make to work around this. This means that you set up your calculation in such a way to avoid the problems given by the fact that you cannot sum over everything. And these two strategies can essentially be summarized in factorization on one side and infrared and collinear safety on the other. So in essentially all the QCD calculations that people perform and that we may try to perform even though I suspect there will not be much time, we will try to work around the fact that we cannot use BN, KLN as they are by properly devising observables that have these properties. Factorization is essentially the fact that we will be dumping some singularities in an uncalculated non-perturban part of the proton. We'll hopefully see this tomorrow. Infrared and collinear safety on the other hand is a different thing and I want to describe it now. It is essentially related to the fact that KLN meaning full, so this particular characteristic here, so if you see safety, full inclusivity is a sufficient condition for cancellation, not a necessary one, which means that you can devise observables that have less stringent conditions that obey less stringent conditions but still guarantee the cancellation of singularities. Of course, the cancellation of singularities is necessary for perturbation theory. You want to do perturbation theory, you must ensure that you can cancel the singularities in the infrared region. If you are fully inclusive, that's fine. KLN tells you that you are guaranteed this cancellation but then since sometimes you don't want to be fully inclusive, well, there are ways to devise variables and observables so that you can still cancel without being fully inclusive but take some work and you have to be careful and a bit picky in the way you do things. So let's try to set up the conditions for IRC safety. IRC safety is essentially the statement that your observable is calculable, so this is essentially calculability in PQCD. Mind you, calculability does not mean good convergence of this series. You could still get very large corrections but at least they are finite. You can calculate them. And this means that you have to define properly your observable. So let's see what conditions unobservable must satisfy in order to be infrared and collinear safe and therefore in order to be calculable in PQCD. So let's define unobservable in the following way. So the usual one of our flux, then there is the sum over a number of particles at various orders in the final state, the integration over the phase space, the matrix element for production of N particle, V, oops, N particle phase space and the sum function of the momenta that eventually defines your observable. So each observable will be defined by a given SN. So if SN is equal to one, what do you get? What do you get when you just get rid of that? The total cross-section, okay? So the particular choice SN equal to one means that the observable is simply sigma thought. On the other hand, if I pick SN as something of the form X minus sum function again of the momenta, then lobservable becomes sum differential cross-section as a function of X, right? Okay, so for the total cross-section, we have seen the cancellation happening and we are unsure that the cancellation happens because of KLN. So for total cross-section, we have that the observable is proportional to, well, the integral of some mN squared d phi N plus the mN plus one squared d phi N plus one. And this is finite. This is the KLN block-norther cancellation. I'm summing the virtual correction to some final state with n particles. The real correction, you see that it's the plus one here and I'm getting a finite result. This is guaranteed by KLN. And we know that the cancellation happens in the soft collinear region. It is there that the real emission is divergent and it is there that it is canceled against the degenerate virtual diagrams that do not emit anything. So you have on one side, on this side here, you have no emission at all and on this side here, you have emission that is either very, very soft or I mean, either infinitesimally soft or absolutely collinear and the two cancel against each other. So this is a cancellation that happens in this region and that is explained at wall orders by the KLN theorem. So now, whatever I do, now imagine you go beyond the total cross-section and you start looking at something like this. So you start looking at something where you insert this and you ask yourself, am I spoiling this cancellation by replacing the variable, the function SN equal to one as I have here, here I have, so this year I have the equivalent of SN equal to one, of course, and here I have the equivalent of SN plus one also equal to one. So now I'm moving to a situation like this one where I am actually replacing SN by any particular function. And the question I'm asking is, am I spoiling the cancellation or rather, what should I do to avoid spoiling that cancellation? And the answer is that in order to avoid spoiling the cancellation, I must make sure that the soft collinear region is not changed. So to avoid spoiling the cancellation, I must leave the soft, well, not unchanged, perhaps what can I say? Well, let's say I must, I must ensure that in the soft collinear region, virtual and real maintain the same relation, same. Meaning, if they were canceling in the total cross section where the function S was absolutely the same in the two, I must ensure that this function chi remains the same in the two in that particular region. Otherwise, I will spoil something that was working. So I need, this means that I need that the function chi N plus one in the soft and collinear region must go to chi N because the cancellation was happening when SN plus one was equal to SN and therefore equal to one. So this relation of SN equal SN plus one plus one, I must maintain it at least in the soft and collinear region. And if I do, then the cancellation will happen. So in practice, what this means in practice, in practice, it means the following that either a soft emission or a collinear splitting must not change the observable. And I write it concretely XN plus one, which is a function of the various momenta. And then there is the splitting here. Let's say that PN has split into two collinear momenta like this, this must be equal to chi N P one, PN. So having split this has not changed, having split this has not changed the definition of the observable. And in the same way, chi N plus one, P one, PN, and then a soft emission that I describe by this zero here must be equal to chi N P one, PN. So if unobservable satisfies this condition, then infrared and collinear safety is a property that it has. And therefore I can be sure that real and virtual cancel, still cancel, let's say, even if the observable is not fully inclusive. Another way of saying this is that an IRC safe observable is such that, oh, let's say it must be invariant under a branching P i, P j plus P k whenever either P j is parallel to P k or either of them is soft. There's yet another way of writing of saying that this thing will be IRC safe, meaning I can calculate in perturbational theory if it has this particular property. Let's see some examples. What can be or cannot be infrared and collinear safe. Let's say gluon multiplicity. Suppose I want to calculate how many gluons they have in the final state. Definitely not an inclusive quantity. I'm not summing over all the gluons, I'm actually counting them. One, two, three, four, 27, 36, et cetera, et cetera. So this is obviously not IRC safe. For instance, a soft emission changes the number. You can emit a very soft gluon, expanding zero energy but it is yet an additional gluon. So you still conserve the energy but you have one more gluon and therefore this is not an infrared and collinear safe quantity. You cannot calculate in perturbation theory, this one. Another example is the hardest particle in an event. Suppose you want to calculate the momentum distribution for the hardest particle in an event. Is this a good idea? No, it's not. Why is it not a good idea? Well, because suppose your event looks like this. You have three particles. So one, two, three. Obviously, here it is the particle two that is hardest but then you can perform a collinear splitting and you find yourself in a situation like this where you have still the particle one, the particle two has been split in two and then you have the particle three. And immediately a collinear splitting has changed your observable because before the observable was it is the particle two that is hardest. And now after a collinear splitting it is the particle one that is hardest. And so this does not respect the conditions for IRC safety that tells you that in order for something to be IRC safe and therefore calculable, these operations must not change the. So what happens in practice is that since a virtual correction does never change the observable because imagine you have here a virtual correction. So the virtual correction will apply to the same observable but if at some point with this virtual correction you try to cancel something like this, you don't find it. The two don't match anymore. So the fact that a soft or a collinear emission change the observable makes it disappear the matching with the virtual that is needed to cancel the divergence. So only when soft and collinear do not change the observable, you can still match to the virtual and therefore cancel the singularity. Okay, so this is another example where it is not IRC safe. Instead, let's also see an example where this does work. Let's take a certain amount a certain amount of energy flowing inside a cone. Now, suppose you have a cone like this and you have some emission of particles inside then the same cone, let's do it more times. So if you have a little emission outside the cone, remember this is a finite amount, a very small amount of energy, you only want a certain amount of energy inside the cone. So this can still be allowed and not change, you can afford losing a little energy, it's integrated over, it's fine, or another way. If you have a particle that is split collinearly, these two are split collinearly inside the cone, of course nothing changes in the flow of energy inside the cone. So this is an observable that if properly defined, there are a few subtleties, this is potentially IRC safe and then you can calculate corrections to this kind of things. An example of this is actually, this is actually represents the first example of an infrared and collinear safe variable. It's the so-called Sturman Weinberg jets. And this, oops, that's a bit too old, written down in 1977 and it represents the first example where people understood the importance of defining properly infrared and collinear safe observables in order to calculate in QCD. So what they did, they were looking at the plus and minus collision and they were looking at the production of two jets, essentially, at some angle theta. And they said that a two jet cross-section, you get a two jet cross-section in this event if all got a fraction epsilon much less than one of the total E plus E minus energy, E is emitted within some pair of opposites. Some pair of opposites, I write it, directed cones of half angle delta. If you think about it, if you think how this is defined and if you think of the conditions of infrared and collinear safety that we gave earlier, you realize that this is an infrared and collinear safe quantity. And there is actually an analytical result for this. I will write it down because then I will comment about. So it's a sigma two jet. Remember, we are trying to have a non-inclusive observable. I'm looking at having two jets and not just anything. And this is equal to, well, sigma zero. And then it's proportional to one minus let's see, it's written in terms of G. So pi squared and then it's three log delta plus four log delta log two epsilon plus pi squared over three minus five over two. And here the important thing that I wanted to point out are these logs here. Why do you have these logs? Well, you have logs of the inclusiveness parameters that you are using. Here you have two inclusiveness parameters. You have the, so your observable is not fully inclusive but it is inclusive enough. And this inclusive enough is given by this epsilon here. Oops, I wanted to do this. This epsilon here and this delta here. You see that as epsilon and delta go to zero, you lose your inclusive enough because delta going to zero means that you squeeze the cones essentially down to nothing. And so you are not being a little bit inclusive around the linear region. And epsilon goes to zero means that you essentially allow no energy to escape. And so you are not inclusive enough around the soft region. Without the inclusive enough condition, you go back to a divergence. And the divergence is exactly the same that you had observed earlier. It is a double divergence in the collinear and in the soft log. So when delta and epsilon go to zero, you recover the double divergence that we had seen here before cancellation where was this here? This is the collinear divergence. Sorry, this is the soft divergence. This is the collinear divergence. They come from this thing here. And without an inclusive enough inclusiveness parameter, you make the divergence naked one more. You lose the cancellation with the collinear with the virtual. So these divergences is now finite, but you still find it in terms of potentially large logs. And then you may want, but this is a different topic and it's a different step. You may want to find a way to actually re-sum to order these things in order to improve the analytical, to improve the convergence of this series because this series is finite. But these logs could make the correction. This is a higher order correction. Could make it very, very large. Eventually, you may want to re-sum these things to all orders. Not all observables allow for resumption, but it is yet another characteristic that you may be interested in defining observables that are not only calculable, that's the minimum, but also re-summable so that you can improve. This result. Andrea, I was wondering, it's a 40, I may have, well, either I go on for another 10 minutes or I stop here and I take questions. But let's see if there are questions. And then we can, how about that? We stop, we see whether there are questions and eventually, so there is one question from Max already. Please. Yeah, I just want to make sure that I understand this properly, so this whole issue with this infrared emergency, so that this only appears because we are in perturbation theory, right? So if you were kind of infinitely powerful, it could really calculate QCD, then we could look at any of the observable we want and we assume that we wouldn't find any infrared divergencies, right? I suspect so, yeah, yes, I think so. It's essentially due to the use of perturbation theory and Feynman diagrams to set up the calculation. Yes. Zahra. How we can recognize that Godon is made by Quark's shoveling from the emission column. Sorry, can you say it again? How we recognize that? How we can recognize that the Godon is made by Quark's shoveling from the emission. No, you can't, you can't, you can't. I mean, that's, of course, when you talk about Quarks and Bluance, you are not talking about observable particles. So the only thing you can observe is the eventual products in terms of fragmentations, hard runs and something like this. So you have nothing that really pinpoints to a Bluance saying this come from a shower, this come from a hard emission or something like this. So you can't, you can just calculate what you need to calculate, but you cannot identify individual Bluance if this was, or their origin exactly if this was the question. Not to mention that, let's not forget. These are quantum mechanical processes. Eventually you only have probabilities. Even if we draw particles with nice little lines and we think we can recognize and see them one by one, that does not forget that what we calculate probabilities. And so typically a one-to-one identification with what you observe is not possible. I have another question. Imagine we have two points that they are near together. If an emission extract from the first pointed and the internal two second coin. I haven't called the question, I'm sorry. We have two cones and they are near together. Two, sorry. To what? Two cones. Two? Two cones. Two cones, yes, two cones, okay. In first, in first cone, we have emission extract from the cone and enter to another cone. The emission from one cone to another cone. Yes. That's not a problem. I don't really care where the emission comes from in here. The only thing I care is what I find eventually in the cone. So if a balloon is emitted from here and actually travels there, that's not a problem. I'm not looking at it. That's not an issue, not with this observer at least. All I worry about is how much energy could escape from the cones. In this particular observable. Not where the energy was coming from. And that's actually one of the key points. You look at the level of observable. What do you observe? What you observe is a flow of energy somewhere which may eventually be recorded by a calorimeter placed here and placed there. So this is something that you observe. On set of hand, your calorimeter will have a finite angular resolution. It cannot resolve two particles too close to each other. And it will also have a finite energy resolution. It will not be able to see very soft energy emitted around there. And that's why you need to be inclusive enough over the things that you cannot observe. Because these things will be done together with the virtual corrections. So for your calorimeter, so let me say perhaps, for your calorimeter that has a finite energy resolution, a process like this with a very soft gluon. And a process like this with a virtual correction are absolutely identical. Your calorimeter has no way of seeing this gluon. And therefore it has no way of distinguishing it from the virtual correction. And so you have to sum these two things together to ensure that everything cancels. So you have to be inclusive enough over everything that is degenerate. Remember, BNKLN theorems tell you that you have to sum over everything. Here, we are being a little less prescriptive. We don't say you have to sum over everything, but we say you have to sum over everything that is degenerate and you don't observe. So something under the threshold of observation is degenerate with the virtual and you must sum them together. And hence the definition that Sturman and Weinberg gave of the general section. But it is not a prescription on where things come from. It's rather a prescription on where things go and whether we observe them or not. Thank you. All right. Does Lalit also like to ask a question? Hi. My question is about like at parthenic level we have called it sigma m and sigma n plus one. Okay. There we're integrating for the extra glon from all the phase space. But here in this Sturman- Weinberg jet, we are only concerned with the particles that are inside the jet. So we are not integrating all everything. If I talk in terms of it's a parthenic process and going to the hydronic level and extrapolating up to that level. Sorry. In Sturman- Weinberg, we are at the perturbation theory level. So we are looking at quarks and gluons and then we will assume that the fragmentation will happen in a fairly collinear way. I'm not sure I got your question. I'm saying that initially when we were summing sigma n and sigma n plus one. Yes. You are summing over everything. So there are gluons that were hard also. So let's put it this way. What kind of, you have, you have, so let's say this is a QQ bar production at leading order. And obviously it contributes to a two jet cross section because it can only go to a situation like this. Then you can have a virtual correction which is also something like this with a virtual. And then this also can only contribute to a two jet cross section. Then you have the situation where you have a gluon emitted like that. And now the situation is a bit different because this could either contribute to a three jet cross section or it can still contribute to a two jet cross section. I'm sorry for bad drawing. So the real radiation can actually contribute to two different observables. It's the same diagram. It's the same Feynman diagram. It's the same amplitude. It's always QQ bar gluon, QQ bar gluon. But according to the angle of your gluon or to how large you decided to choose the cone, this same gluon could contribute to two different observables. And that's the choice of the observable and the integration of the phase space that does the thing. So there should be some... So go ahead. I think there should be some information that I'm cutting off the angle of the gluon up to that point. Yes. But if you are interested in the angle of the gluon, imagine that for some reason you are interested in... Where is this? For some reason you are interested in this angle here, let's say. Okay? If that's the case, then of course you do not look at the two jet cross section because in the two jet cross section, of course you don't see that particular angle, but you decide that you study your three jet cross section. But for the three jet cross section, be careful because the QQ bar G, let's say diagram is a leading order diagram. Because of course nothing with fewer than three particles can produce a three jet cross section. So the same diagram, the same QQG diagram, can contribute either leading order terms to the three jet cross section or next to leading order terms to the two jet cross section. And one more question. Okay. So now since you have written this QQ bar G as a leading diagram, so if I integrate over the phase space, this also will have divergence, how will this divergence cancel? It won't. Unless I put a code. This one won't. This one won't cancel because the three jet cross section, if you go to very small angle, you will encounter the, you will eventually encounter the divergence and there's nothing that can cancel it because the virtual correction belongs to the two jet cross section by definition. So you cannot actually push this one to zero. What you have to say is that you only look at three jets when theta is larger than a certain cutoff. And when it is smaller, well, you have to move to a three jet cross section if you want to study similar configurations. So it's been to say that for three jet events, I can only say a cross section in terms of the angle. I should have a cutoff at the angle. Yeah, yes. Sure. You have to keep the jets separate, absolutely. Okay, and in case of two jets, so since I'm taking a three jet event there, so there is that my drone is only inside the jet, not outside it. If you define, if you define, I mean, if you want to calculate a two jet cross section, then yes, with the Sturman-Weinberg definition, which is not the only jet cross section definition, but with the Sturman-Weinberg definition, what contributes to a two jet cross section is either a gluon sufficiently small, sufficiently close, or a gluon that is so soft that I don't care about. Okay, so it means it's an approximate solution. It's not approximate. It's a definition. You define what you call two jet in this way. It's not an approximation. The approximation is then the phi that you perform a particular calculation with truncated somewhere. But the definition of a jet cross section as given by Sturman-Weinberg, for instance, is a definition. Thank you. All right, thank you. Are there other questions? Still have a few minutes in case apparently not. So I guess we can close here, Matteo, and I'll resume tomorrow at this point. And so thank you again. Thank you, everybody. And we'll resume later on at five. Local time. See you tomorrow. Bye. Bye. Claps.