 I'll tell you already. Yes, I am. All right, then welcome back everybody. And we're going to start with the second lecture from QCD and Colliders by Kacai. OK, thank you. Apologies, I still have the same problem of the iPod as yesterday, so let's try this way. It's not ideal, but at least it should allow us to get this to work. So we talked yesterday about colliders and detectors. And before moving on to QCD, I would actually like to, oh, sorry, I don't have my camera on, let me put it on. There. Before moving on to QCD, I would like to be able to give you a feeling of what we are actually looking at in colliders and how. So I want to more specifically, I want to talk about kinematics, kinematical variable, not colliders. So as you remember, things collide head on, and you have your detector inside what is essentially a cylinder. And then here you have your interaction. And then out of this interaction, of course, things come out and fly in all direction. And what you actually do as an experimentalist, you try to measure these things on your detector. So the detectors we saw yesterday are essentially concentrical cylinders of detectors, most of the times, and then typically you have end caps at the end on one side and on the other. So you have, so here, these are the end caps. And then this is the central part of your detector. So you detect particles at the surface of each of these concentric cylinders. And essentially, you cannot look down the beam pipe because, well, that's where the beam pipe is. So of course, you can't go exactly inside, even though you try, of course, to put your detector as close as possible in this region here. So you can get also particles coming in at a very shallow angle. So what kind of variables you actually look at in this kind of thing? So we typically set up, so you have your beams. So these are your proton and the other proton that comes from the other direction. Here you have your interaction. And so the first thing you do, you set up a polar angle theta from the beam pipe. That's one of the variables you look at, even though we will not actually, most of the times, use it with that particular variable, but this is the idea. You look at the polar angle. And the other thing you look at, with respect to the cylinder, you look at the azimuthal angle that gives you essentially the way to go around the cylinder that way. We then typically measure not so much overall energy and momentum, because as I said, you typically not have that much access to what happens in the beam direction, but you end up looking at things like transverse energy. So t, e sine theta, or transverse momentum, pt. And this is, of course, the component of the momentum that flies out perpendicular to the beam pipe. So as I was telling you yesterday, especially in a hard-run collider, it plus and minus colliders can be a little different, because, well, especially if you produce hard runs, you have electrons coming in. So you may look at a more of a spherical aspect of a detector. On the other hand, on hard-run colliders, you typically tend to use this cylindrical geometry and therefore concentrate on these kinds of transverse variables that I mentioned. To give you a better feeling for what you're actually looking at in a collider, this is an example of the kind of event that you see at the LHC. And you have three different representations of the same event. This one is the event as you would see it if you were looking down the beam pipe towards the detector. So you have your interaction at center, and then you have all the tracks of the particles, of course, roughly back to back coming out in the transverse direction. Remember, so the beam pipe is down here, and then this is the transverse direction. These are actually the transverse directions. And then you see here, represented as towers, the amount of energy that has been deposited. Typically, one color is the electromagnetic calorimeter, and the other color is the hydronic calorimeter. I cannot remember now which is which, but you get the idea. You get an event back to back, momentum conservation, of course, and then the energy flies out in a transverse direction, and this is where it is recorded in your detector, sorry. This is another view, which is the same thing, but now you are looking at it perpendicular from the beam. The beam is here, so this is one beam, and this is the other beam. The particles collide here. The results fly out again back to back. This would be the transverse component, of course, there will be an angle, and there will be a projection over the transverse component in both cases. That is, of course, also a longitudinal component of this momentum, but then otherwise, you still get the same deposits in the endicolorimeter. And finally, this is yet another representation. This is called a LIGO plot for obvious reasons, because it's like you are stacking LIGO blocks over a plane, and now the interesting thing that I wanna mention here is what kind of variables we are using to actually find a position on a plane. So on one direction, there's no surprise. You have the azimuthal angle phi, so this is the angle that goes around your detectors, essentially. It measures, for instance, on something like this. So this will be phi, or rather, sorry, this will be phi. On the other hand, on the other direction, you don't have the polar angle theta, but you have this other variable eta that I have not defined yet. I will define it in a moment. It's called the pseudo-rapidity, and we'll see in a moment where it comes out of and why it is a useful variable for Hadron colliders. Also, another thing you may wonder, what are these jets that you see here? Well, we'll talk about them in one of the coming lectures. They are a useful object that is constructed to analyze strong interaction events. For the moment, it will be enough to say that, as you can see, these jets are essentially related to these fairly collimated beams of particles that come out of the interaction. So this is a characteristic of strong interactions, which is strongly related by the way to the way strong interactions behave. So actually, the fact that you see jets and you don't simply see Hadrons coming out with a more spherical symmetry everywhere is strictly related to how QCD works, and we talk about it, but for the moment, a jet of Hadrons are these beams of Hadrons that you can even visually see in the detector. And on this plane here, they are simply represented as areas where you have a lot of energy. Now, let's write down some kinematics. So let's say that the two beams have momenta PA equals square root of S divided by two, one, zero, zero, one. So this is the beam, the proton beam that travels, let's say in the left to right direction, just to be definite. And PB, of course, is the other beam, root S divided by two, one, zero, zero, minus one. Minus one, of course. So these are the two beams. Root S is the center of mass energy and it's defined by PA plus PB squared, which is equal to two PA dot PB. And then this is defined as S. And so you can see that this is the center of mass energy, the total momentum of the colliding beams. Let me also anticipate, and this is something that we'll see later on again, that when you collide two protons, so head on, what collides are not actually, I mean, the hard collision is not given by the full collision of the two protons because typically what collides are smaller objects that we now know today are contained in the protons, or what quarks and gluons, of course. And each of these quarks and gluons will typically have a fraction of the proton momentum. So for instance, we may have some proton of momentum PA here, but then what actually collides is a quark with momentum PA equal X1 PA. And on the other side, you have something similar a momentum PB, some PB equal X2 PB. And then here's the collision. So you also have a center of mass of the hard collision, center of mass energy, which is denoted by typically S hat and given by the square root of X1 PA plus X2 PB squared. And then if you do the calculation, this is nothing but X1 X2 S. So when you say, oh, at the LHC, we are colliding protons at 14 TV, that's certainly true, but the energy available for the hard collision is more like X1 times X2 times, well, square root of X1 X2 times 14 TV. So it will definitely be generally smaller these fractions go from zero to one. So this is for the incoming momentum, for the outgoing momentum, let's say P mu, we denote it as, so some energy, then it's momentum sine theta cos phi, then there is P sine theta sine phi, oops, sorry, and then there is P cos theta. So you see that this is a longitudinal component called P parallel or actually also PZ and these two here give the momentum transverse to the beam. And of course, we are in a relativistic kinematics and therefore E squared is equal to PT squared plus P parallel squared plus M squared. This is the relation that Stefania Gorey wrote down yesterday. From these things, let us introduce two more very relevant variables for hard on colliders. The first one is the transverse mass and this is defined as MT equal to the square root of PT squared plus M squared. So you can see it's the same equation as the mass, but it's missing the longitudinal component. And then another quantity is the rapidity. Rapidity is a bit strange as a name and the definition is perhaps even stranger, but then we'll see what properties make it very useful. It's defined as Y equal one half log E plus PZ divided by E minus PZ. And as a first exercise, let's suggest that you try doing, show that this is also equal to log E plus PZ divided by MT without a one half. So this is the first exercise that you can try doing to familiarize yourself a little with these things. So transverse mass and rapidity. Now you can use rapidity and then we'll see what why it's useful to rewrite the expression that we have here for the momentum. We can rewrite it using rapidity in the following way. So P mu, the same P mu that we had before can be written as MT times hyperbolic cosine of rapidity for the, this is the energy component. PT cos phi, PT cos phi. Sine phi and then MT again. Sine hyperbolic sine of rapidity. It is perhaps a half a page calculation to show that this is indeed the case. And perhaps again, I would not do it in details, but you can easily check again using the definition and manipulating a little bit. You can try showing that this is indeed the case. So why we introduce such a weird variable? The reason is that it has very useful symmetries or rather transformation properties with respect to boosts along the beam axis. Why are we interested in boosts along the beam axis? Well, the reason is, as I was telling you, you collide protons and of course the machine is such that the two beams have the same energy. So in principle, the detector are in the center of mass frame of the proton-proton collision. But then, as we just said, what collides is not so much the protons, but it's a constituent of the proton. And these constituents that are extracted from the two protons are not necessarily extracted with the same fraction, meaning that typically one of the two constituents will have a larger momentum or energy than the other. And so you will have a collision where let's say, well, let me call it quark, since it will be a quark or a blue one, will have perhaps a bigger momentum and the quark that comes from the other proton will have a smaller momentum. This means that, of course, the center of mass of the collision will not be stationary in the lab. The center of mass of the collision will be boosted in one or the other direction along the beam axis. And then, therefore, all the momentum of the particles coming out will be boosted in the same way. So this means that it is useful to work with variables that have a, how can I say, a simple or transformation property with some nice properties with respect to Lorentz transformation. And rapidity is such a quantity because when you go from one frame, so let's say you go from one frame to another frame, a prime, the rapidity that you have in one frame actually transforms as y prime equals y plus some quantity. So it's an additive transformation of the rapidity. And this quantity, this is actually, I think it's, if I remember correctly, I think this is the inverse tangent of the velocity, inverse hyperbolic tangent of the velocity of one frame with respect to another. Please check this, I'm not 100% sure. So the two frames move, have a velocity, and then this is how rapidity transforms, which means that, of course, when you measure things related to rapidity differences, which is, of course, something that we would want to do in many cases. So this is y1 minus y2, this transforms simply as y minus y2 equals y1 minus y2. So there is no change in rapidity differences because rapidity transforms this way. So this is a Lorentz invariant and this is, therefore, something useful to characterize many characteristics of the collisions. And remember, this is particularly useful because since these constituents are hidden inside the proton, you typically do not know exactly the boost of each collision and you cannot reconstruct it easily very well because you don't observe every single particle produced in the final state. So to work with quantities that do not depend on your knowledge of the precise center of mass of the collision, of course, is something that is useful. Another way, of course, is looking at transverse properties that also do not depend on the boost of your collision. So this is why we want to introduce this. I was wondering whether I should show this transformation property or perhaps I can just hint how to do the proof, how to prove this thing? There is a question. Yes. Yeah, hello, can I ask you a question about one expression on the previous slide? So when you define this p as a small a equals, for instance, x1 and p in capital A, so these p's, they are for momentum, right? It is a for momentum, yes. But why is it so obvious that this spatial momentum scales in the same way and so with the same x1 as the energy of the quark in relation to the proton? So why is it the same x1? Because we are neglecting the masses. The energies are solar, I mean, one TV or seven TV for a proton of mass, one GV. You can almost, I mean, you can essentially consider the proton as massless and the quarks and the gluons, we do the same. So that's why you can scale the for momentum that way. Okay, okay, thank you. So let's quickly see how you can prove this. Perhaps I want to do all the calculations but I will just suggest how to do this thing. So just consider the Lorentz transform. So you have two beams again with relative velocity with each other, velocity beta. So you have, you know, of course, the Lorentz transform, PZ prime is equal to gamma minus beta gamma minus beta gamma gamma and E PZ and of course the PT prime is equal to PT. So this is what I was saying before. When you work with transverse quantities, you don't need to worry about the boost of the center of mass because these transverse quantities don't change. And you can rewrite the gamma factor, the boost factor as hyperbolic cosiness of omega and gamma beta is, so these are the well-known parametrization of these quantities that you use in Lorentz transformations. And then once you have this, it's easy to see, and I want to do the calculation explicitly, it's easy to see that E prime becomes eventually empty cos Y minus omega and PZ prime is empty. I need to change the page, sine minus omega. So these two things you can easily check, but this means that you can write the boosted momentum, the transformation of P mu to P mu prime is given by, you know, the transformation of P mu to P mu prime is given by, so empty cosiness Y minus omega PT, empty since Y minus omega. But this is, since we know how a four momentum was parametrized, remember, now I go back to the parametrization of the momentum that we had earlier, and it is this equation here. Where did I put it? Here, so you see that this is exactly the same form, but with now Y replaced by Y minus omega. And so you see that this corresponds to a new rapidity, which is Y minus omega, essentially, the different components of the momentum have not mixed as they would usually do in a normal Lorentz transformation. The only thing that has changed is that they now correspond to a different rapidity, which is not Y anymore, but it is simply this Y minus omega. And so this is what I was saying earlier that this transformation property of the rapidity is particularly useful. Another reason why rapidity is useful is that we can easily parametrize the phase space in a Lorentz invariant way. And so let me write the phase space D, let me write D4P delta plus P squared minus M squared. So this is one way of writing the phase space in a clearly Lorentz invariant way. This gives the well-known D3P divided by 2E. You must definitely be familiar with this equation if you've done any cross-section calculation in QD or in quantum field theory. Under this form, it is less clear that this phase space is Lorentz invariant, though, of course, it still is because it corresponds to the equation on the left. But what the interest says here is that this can be written in the following way. Oops, sorry. D P T squared, D Y, D Phi. So you, oops, D Phi. You see that I've now rewritten this in terms of the variables that was telling you we will typically use in heart on collision, the transverse momentum, the rapidity, and the azimuthal angle. So this is particularly useful in many calculations to be able to actually simply re-express the phase space in terms of this. And so it's clearly Lorentz invariant on the left and then what? Okay, perhaps the last thing I wanna say about rapidity is that one can show that the maximum value of the rapidity for the production of a particle in a given center of mass frame with center of mass energy, square root of S, it's given by log square root of S divided by M. And if you try plugging in numbers for, let's say, the LHC and the production of a typical Hadron, say one GV. So you should take a square root of S equal 14 TV and mass equal one GV. So production of one Hadron, fairly light Hadron indeed in 14 TV, assuming that all the energy, of course, goes into the collision, which is not then the case. So and then you get Y max of the order of mass. So this gives you an idea of how far in rapidity particles can go at the LHC. In practice, it will be less because there will be smaller energy because sometimes particles have larger masses and so on and so forth. But it gives you an idea when you see a number, well, how far it can go at the LHC. Again, in practice, the detectors will measure much more central areas like two, three, four. And I say central because of the reason I'm going to tell you now that this rapidity, which is a purely kinematical variable, so I thought that I could go back to what I, where is the definition, sorry, put it here. So if I go back to the definition, you see that this is something defined simply in terms of energy and momentum. It has no obvious geometrical meaning in terms of position on the detector cylinder. But it happens that for a massless particle, so let me introduce something called pseudo rapidity. This eta is that variable that I had shown you earlier to be on the Lego plot. Let me go back to that one, show you to get to it again. So here it is, this eta here, as you can see, it has an obvious geometrical meaning. It is something that describes the event in place of the polar angle theta, instead of placing the events at a given theta, I place them at a given pseudo rapidity. So this has a geometrical meaning that relates its value to a position on the detector, whereas rapidity is a kinematical variable related to the momentum of the given particles. Nonetheless, there is a relation between the two and this relation is particularly simple in the case of a massless particle, m equal to zero. So if m is equal to zero, the transverse mass, which was equal, let me remind you, m squared plus pt squared, of course will simply be equal to pt. And of course, also the energy will be simply equal to the value of the momentum. These are typical characteristics of the typical, I mean they are the characteristics of a massless particle. This means that the energy of the massless particle will simply be mt cos y. Remember it was mt here. This was mt, but now mt is equal to pt. So for a massless particle, remember that we are in this case. We are in this particular case. And pz is pt sin y. Now, this means that e plus pz is nothing but pt cos y plus sin hyperbolic y, which is nothing but pt e to the y. So, which is not something I need. Why do they write that? Why does it matter? So, now take again the definition of rapidity, y equal. Let's use the definition e plus pz divided by this time pt because again, we are in a massless particle. This is p plus p sin theta on the definition of the energy. This is simply the energy. And this is simply the longitudinal momentum divided by pt is simply p, sorry, I made a mistake here. This is cos e plus theta and this is sin e plus theta. And you can show fairly easily using standard manipulations that this is log of square root of one plus cos theta divided by square root of one minus cos theta. It just takes a few manipulations of trigonometric quantities. And using now the formulas for the half angle sinus and cosinus with respect to cosinus theta, you can rewrite this as minus log tangent of theta divided by two. And this is what is defined as being the pseudo rapidity. So you see when the massless, the mass for particle is zero, rapidity is equal to pseudo rapidity. And then pseudo rapidity has a clear geometric interpretation, description like this one. Pseudo rapidity is directly related to an angle to a position over the polar coordinate of the detector. To give you an idea of how this looks like, well, if you take a detector, then you will have a situation where eta equals zero corresponds to 90 degrees, of course, because the tangent of, if you go back to this equation here, for 90 degrees, the tangent of 45 degrees is equal to one and the log of one is equal to zero. So this is where you have, and then you have somewhere here, oops, sorry, you have somewhere here, eta equal, let's say 0.5, somewhere here, eta equal one and 45 degrees is here. This is 45 degrees. So it's eta equals 0.88, if my iPad had worked, I could have just pasted the pictures now and forced to draw them. Then you have somewhere here is eta equal to two, eta equal to three, et cetera, et cetera, et cetera. As you can see, it's not linear in the angle, of course. It is linear only in the central, it is linear only in this region here where eta is fairly small, but then there is due to the tangent, of course, that kind of behavior. So you will typically have events that happen around small values of eta are called central, small values of eta or also small values of y when the mass of the particle is sufficiently small with respect to its momentum that you can consider eta and y fairly close. At the LHC, where you have very high energies with respect to the mass of the particles, if you consider individual hydrons or individual leptons, essentially eta and y are almost identical. Things may differ quite significantly if you consider instead the production of a very massive boson or particle like a top quark or a W boson, they are masses of the order of 100 GV. If you measure them with momentum of the order of 100 GV or less or even just a little more, of course, this is not the case anymore. But for the individual lepton or the individual light quark, eta and y are very similar. And therefore this notion of central events related to small eta or small y and forward event here. These are forward events related to large values of eta stand. Then it matters of course, where you have your detector. If your detector is in a region like this one, if this is where you have your detector, you can only detect particle up to, well, eta equal one or two. Then it depends where your detector is. The end cap usually allow you to collect also particle at very large values of rapidity, pseudo rapidity. Then again, it depends how your detector is constructed. But this is how you describe where the particles are scattered after a collision. The last thing I can tell you about this rapidity, pseudo rapidity is that of course, since the two variables are not the same, if they become similar, it is useful to know that there is a Jacobian between the two. So if you look at the distribution in terms of eta, then there is this Jacobian, one minus m squared divided by m t squared cos squared y dn over dy. And this is another exercise that you can try doing, evaluate this Jacobian that relates distribution in the two different variables. And if you go and look at the distributions, so for instance, you have that the distribution or a typical pseudo, sorry, a typical rapidity distribution is something like this. There will be actually less, typically less, sorry, let me do it again. This is supposed to be symmetric. Okay, I'm a very bad drawer. So this is eta or y. So let's say this is minus two, two, minus four, four, et cetera, et cetera. These are typical numbers that you may find at a DLC. So this could be a typical distribution for dn over dy and the associated distribution for the pseudo rapidity could be something like this. This is the n over the eta. And you can see it's not monotonic anymore going towards the central region, there is this little dip that is essentially related to the presence of a massive particle. Of course, the two distributions will be identical in the limit of mass less particle. And you see that when the mass goes to zero, of course, this term in the Jacobian goes away and you're only left with the same distribution. So this difference here is related to the fact that the particle is a sufficiently massive with respect to the momentum that we are considering. And when you go and look at events at the LHC, you will see depending on whether they are massive or sufficiently mass less particles and depending on what the masses and the momentum are and what the experimentalists decide to plot, you will see either distributions of this kind or more distributions of that kind. Always symmetric left and right unless the collision is itself non-symmetric. It happens and some of the collisions recorded at the LHC are not symmetric, some high ion collisions for instance or proton heavy ion, then one of the two beams may be more energetic than the other. And therefore these distributions will be shifted by an amount. But as we saw earlier, this shift will really be a shift. When I mean a shift, I mean really a shift or something like this. It will be a rigid shift in one direction or in another by the amounts that the center of mass is shifted with respect to the center. But the shape of these distributions would be absolutely the same. And this is another, actually this is for, sorry. This is true for rapidity. And actually I have said something stupid. It is true for rapidity. It is also true for pseudo-rapidity if the mass is sufficiently small, but otherwise, no, it's not true for pseudo-rapidity. Pseudo-rapidity does not have this rigid transformation property under a boost. So let's say that what I said applies rather to the rapidity distribution. Okay, so this concludes the kinematics addition to yesterday's lecture about detector, what happens in a collider and what happens in a detector. I think this could be a good time to make a little pose before moving on to QCD. So far, we have described how we look at events, how the events are produced and how we look at events. And then starting from the next part, we start looking at the dynamics of how events actually are produced. Okay, thanks. So take a break, five minutes. Okay, let's start when you're ready. Let's start again. Okay, so as I was saying, as I was saying, we have seen, well, quickly, what we collide and how we, at least with what variables we observe the events in a detector at a collider. But as I said, your aim is to connect, of course, what you are observing to fundamental theories. And from that you have to calculate predictions, compare them to the result of the experiments and see if they agree or if they actually point to discrepancy or if you observe a certain particle that you were expecting to observe and so on and so forth. So of course, you need the whole set of theories of the standard model to do that. You will need not only strong interactions, QCD, but you will also need, of course, electrodynamics and you will need weak interactions. They all contribute to a collider event at the energy of the other of the LHC. Only gravity does not contribute to this kind of events of the four interactions that we know. So in a sense, electric physics for colliders could be a fully different set of lectures and a fully additional set of lectures with respect to this one that one should actually follow because you need everything in order to make real predictions. Here we concentrate on QCD, which in a sense you may consider as a background. QCD is the not interesting part of the collisions because typically what you want to study is beyond standard model physics at scales of order of several hundred GBs or one TV or possibly beyond. So in a sense, QCD and all the strong interaction processes are simply what obscures what you want to observe. But since you are at a hard rock collider, especially at the LHC, and since in any case, not only protons are strongly interacting particles, they are hard ones, but whatever is produced by protons or even by plus and minus collisions in some sense, all this eventually interacts strongly and the result of these strong interactions appear in our final state. It is almost unavoidable to actually control what these strong interactions do in order to understand better what you want. If anything, because sometimes backgrounds are produced by strong interaction. A typical example, for instance, is imagine you want to produce a hex boson in PP collisions that then decays to a BB bar pair, of course. This is a very nice process. And actually, since the hex couples proportionally to the mass to particles, this is actually a process that has a fairly large signal with respect to others, like the ones we mentioned yesterday, hex to gamma gamma and the hex to four leptons. So this would be produced more, but unfortunately there is a general PP2 BB bar background. So this is the signal and this is the background, which is also very, very large, actually even much larger than the signal. So at the very least you have to try to control the background well and to know it as well as you want in order to hope to be able to extract the signal. So being able to control intensity, this is a purely strongly interacting process. A process like this one is a key to then be able to observe what you want to observe, meaning some electroweak particle through its decay to more mundane, if you wish, particles like a pair of bottom cores. So this explains why starting QCD is necessary, even if your final aim is to discover BSM physics supersymmetry or whatever you want. So we have seen in the shells plot that I did try to draw yesterday where QCD lives and I can try redrawing this in fact, actually I had a better drawing to show to you, but since I cannot copy and paste things and oblige to draw it by hand. So again, you have your protons that collide. So these represent roughly the three quarks that are inside a proton. And then at some point what will happen is that it's not the full three quarks that collide, but it's some quarks and these may even emit other particles. And then these particles collide in the heart process. And in the meantime, the rest of the event will give what is called the underlying event. So anything that does not participate in the heart collision still does or something, produces a few low momentum particles that you still see in the detector and that still pollute your measurement of the event because don't forget that then around everything here, of course there is the detector that is observing. And so this heart process may produce, let's say, imagine we are producing a Higgs in this particular case. These Higgs may decay in some particles. Well, there could be tow leptons and that could be anything else. And then you also produce other quarks that will radiate some radiation. And then at some point they will hydronize and this atomization will produce the famous particles that you can observe in a detector as we mentioned yesterday. So this is the kind of process that you would have to deal with at the LHC. Again, around everything here is your detector. Your detector is going to pick up not only what you may want, meaning the Higgs decay into tow and tow plus and to minus, knowing that the tow, by the way, still decays in other particles. But you will also pick up everything here and then something will be produced by these two. And so you have to understand how this looks like, how this looks like in order to, and what happens in here, of course, in order to be able to measure this. And there are various stages here that you can perhaps, so this is some sort of initial state. This is something that happens at the protons before they collide, or at the part of the protons that does not participate in the heart collision. Then there is also something which is the heart interaction where the high energy physics is produced. High energy physics that in this case may be the production of a Higgs, or simply the production of a quark or a gluon at very high transverse momentum. Remember, the momentum here are of the order of one GV, scale of a proton, the proton size, 200 MeV, the proton mass, one GV. This is in a sense a low energy physics from the point of view of the LHC. What happens here is high energy physics. Here is tens or hundreds of GV, the mass of a Higgs, the mass of a W boson, the mass of a top quark in a sense also, 100, 170 GV. That's why I call it this hard. Hard means high energy. High energy at the LHC is tens or hundreds of GV. Soft, small energy is something like a few hundred MeV. And then again, there is a final state here, which is what happens after the hard interaction. The hard interaction produces a quark here, but we do not observe a quark in the detector. We observe the result of its hydrolyzation and fragmentation. By this I mean that this quark is not something we can observe because of a property of strong interactions we'll talk about it. We never observe naked quarks or gluons. We only observe the result of their transformation at long distance. Long distance again means beyond the radius of a proton. At distances longer than one femtometer, longer than one 10 to the minus 15, there are no free quarks and gluons. There are only hard runs. And these hard runs are formed in this process, which I cannot really calculate from first principles yet, but which I can at least model using QCD symmetries and seeing how things behave. So again, this is a non-perturbative process but that I have to account for and it is something that belongs to the final state. So QCD enters in all these interactions and we will actually try to talk about tools and methods to attack these three phases with the disclaimer that again, I will essentially only be dealing with the perturbative part of QCD. There is some perturbed QCD everywhere. There is some perturbed QCD here. There is some perturbed QCD here. There is some perturbed QCD here. On the other hand, specifically in the hedonization phase and in the underlying event phase, here there is non-perturbed physics. The part of QCD that I cannot calculate with perturbed techniques and I will actually not deal with it in these lectures. You should just need to know that here and here you need the non-perturbed models or methods in order to actually extract something. They are important. They do affect how you describe the event, but they are not the goal, the target of these lectures. I will instead talk about how quirks and gluons may emit other quirks and gluons on their way to the hard interaction. What happens at the level of the hard interaction? How do you calculate the hard interaction? And then finally, again, there is a similar process, how a quark and a gluon emits other quirks and gluons on its way to eventually hydronize and finally give you what is observed. Yes, okay, so I'll only talk about QCD and then you may say, oh, well, that's disappointing. You have a very complex process here. We have a theory, which is QCD and I cannot use it to calculate it. Yes, it is in a sense disappointing. It would definitely be better if we could calculate from here, let's say to hear everything every single step. In a sense, in principle, you can do it by using numerical simulation in practice. As I said yesterday, we do not have enough computing power to actually do it. So you still have to resort to approximations and everything so we can't do it. Still, you can predict a lot of things in QCD and to give you an idea of how actually, how can I say, not surprising, but how much of a progress this is, what we can do with QCD. I cannot show you, but I can read you a couple of quotes that were said in the 50s, 60s. So before the golden era of one point theory and the standard model as we know it today. So there is a quote that says the following from a very respected physicist like Lev Bando who said, I'm quoting, we are driven to the conclusion that the Hamiltonian method and by this he means QFT. So we are driven to the conclusion that the QFT for strong interactions is dead and must be buried, although, of course, with discerned honor. So Lev Bando in the 50s or early 60s had essentially given up on quantum field theory to describe strong interactions. He thought, and he was not alone, that it was impossible to do anything about strong interactions using quantum field theory methods. That was the time when people started going off doing regi theory and this kind of analyticity approaches to high energy physics instead of quantum field theory. And then of course people came back to quantum field theory and what you are doing today in phenomenology and high energy physics is mostly or perhaps even essentially or exclusively quantum field theory. But in the 60s and in the 60s, it was absolutely not obvious that you could attack the strong interactions problem using quantum field theory. And another quarter similar one by Freeman Dyson was the correct theory of strong interactions will not be found in the next 100 years. Well, he was proven wrong within 10 or 20 years because at the beginning of the 70s of the past century, QCD was discovered. So this should tell you how much of a in a sense of surprise was at the time that you could quickly go from, oh, it's hopeless. Forget it. Nope, let's do something else. Let's go fishing, but let's forget about doing calculations about strong interaction. So people had given up and then all of a sudden you could find QCD that has proven to be a viable theory for strong interactions. Note that I have not said it is the theory of strong interactions because there are some parts we have not yet proven, but all signs point to QCD being indeed the correct theory for strong interactions. And what was the big jump, of course, what allowed us to go from hopeless to actually, yes, it works, we can do it. Well, the big, of course, discovery was non-abelian gauge theory. So you know what a gauge theory is, I hope. I think that Stefania Gore will introduce them in a more pedagogic way, seeing what she has done yesterday. I think she's on that path. So in case you haven't seen them, she will talk about it. Gauge theories is, for instance, what gives you in a fairly straightforward way QED. QED is a gauge theory, but it's a fairly simple one. Perhaps it's the simplest gauge theory one can think of, an abelian gauge theory based on a simple phase rotation. So essentially you want a theory that is invariant under a transformation of its fields. And so abelian gauge theories like QCD had been thoroughly explored in the 50s and in the 60s and people had correctly reached the conclusion that they could not describe strong interactions using an abelian gauge theory. What they didn't know yet and what was discovered a little later is that instead, non-abelian gauge theory actually do allow you to have these particular characteristics that allow for a description of strong interactions. So what is QCD in terms of non-abelian gauge theory? QCD is again a non-abelian gauge theory based on the group SU3 color, not to be confused with the SU3 flavor that describes the mixing of the three lightest quarks up, down, and strange, and which is at the basis of Gelman's Eightfold Way and the Quark Modern. Both are SU3 groups. One is SU3 flavor, which is an approximate symmetry. This one is SU3 color, which is expected to be an exact symmetry of nature. So don't confuse them just because they are both based on SU3, SU3 is just the mathematical underpinning, but the two things are completely different. And then you have, with these, we have three quarks for each flavor. So you have three types of quarks. It's like having three different states of charge in a sense, instead of just one as for the electron. The electron has one electric charge, a quark has three color charge. So three quarks in the fundamental representation of the group, and eight blue ones in the adjoint representation. If you have not done these things, I'm afraid there's not much more I can do right now from a theoretical point of view. Again, I hope that Stefania will cover this in more detail, but we won't need the details. Let me just be said that the structure of the theory and the choice of what you have in one representation or in another fixes essentially everything in there. The beauty of gauge theories is that once you've made a few initial choices, everything else is fixed for you. So you don't have to think anymore. You just have to find the right group and you have to find the right representation of the group into which to put your particles. Everything else, then it's just turning the wheel and getting things out. So the transformations under which the Lagrangian is invariant, the following ones. So a quark, I'm writing it as Psi B, meaning a fermion, this is a direct particle Psi with color B transforms under the transformation as E theta C X T C AB Psi B. Now this is a generalization of the usual phase, local phase, I say local because this theta here depends on the spacetime point X. And you see that there is a, this is a generalization of what you find in QED because there is also this new term here which is called a generator of the SU3 group. And it's something that does not, I mean, in QED it's essentially unity. So it doesn't, it doesn't really. And then you have gauge fields. Again, these are the analog of the electromagnetic field except that again, there is an additional index C related to the color. And this transforms in the following way, A C mu T C plus one over G D mu theta C X T C again plus I a commutator. And this is the key point, theta C X A C mu. So the T A T C AB are the generators of SU3 and they have this key characteristic that they do not commute T A T B equal I F A B C T C. And the F A B C are the structural constants of the group. And they characterize completely the group. This is called the Li algebra of the group. If you know the F A B C, then all the group is set. And let me just point out that here these are indices in the fundamental representation. And this is an index. This is a capital C actually, it's not easy to see here. I will try to use capital indices for the joint representation and lower case indices for the fundamental one. So this is what you have. And A representation for these things are the eight Gelman matrices. Again, I cannot cut and paste, but you may have seen if you've seen the Quark model, you may have seen the eight matrices that are the generators of SU3 flavor. They are the same here, the eight Gelman matrices. They are three by three matrices. And they characterize completely this group. So what kind of Lagrangian can you write down in the, I thought I had something else but I can't find it here. No, it's later on, just a second. Ah, it's there. So what kind of Lagrangian can you write down then? You write down a Lagrangian that happens to be very, very similar to the one we would have written down for QD, but it has this characteristic of a non-Abelian group that completely changes its behavior. So let's write down this Lagrangian. Let's start with the QD one for reference. So LQD, this is something you probably know. So you have the Dirac Lagrangian, then you have the Maxwell equations, the Maxwell Lagrangian essentially. So these are freely propagating fermion, a freely propagating photon. And then you have the interaction term which is minus E, C bar, gamma mu, A mu C. So this is the interaction as you know probably very well. And then you also have that F mu nu is defined as the mu A nu minus D nu A mu. This is QD, an Abelian gauge theory based on a local phase rotation group U1. And as I was saying, I consider that this is known and that how you can calculate with this, how you can extract finite rules and so on and so forth. We won't need the details of the calculations, but of course it's useful if you already use this thing in calculation. Now let's write down the Lagrangian of QCD. As I said, it's going to be quite similar. This there will be a sum over flavors. We know that there is more than one kind of quark. So I'm summing over the different quarks. Okay, in the QED one, I could also have summed over all charged particles in the standard model, of course. I wrote down Lagrangian of QED for a single electron whereas here I am directly writing down the QCD Lagrangian for all quark flavors. So I'm not being totally consistent, but that's not the key point. So this is psi i bar i d slash minus mi psi. These are the different quark flavors. Then there is minus one fourth f mu nu and there is a C here. And then we come back to this C there. So this is the free propagating gauge fields, the gluons in this case. As you can see, the C tells you that there are more. And of course I'm always using Einstein notation for summation over repeated indices. So this is a sum actually of terms, each of where C runs from one to eight. And then there is the part with the interaction, which is minus again, the sum over flavors. G psi i bar gamma mu A C mu T C psi i. And again you can compare this interaction term here in QD with this interaction term here in QCD. Again, they are very similar. The difference being that the generators of the group enter here and you have not a single kind of photon like it's the case in QD, but you have many actually eight kinds of gluon. And the other difference is that the tensor f mu nu C is defined as the mu A nu C minus D nu A mu C. And then there is plus G f C A B A nu A A nu B. And again, the f A B C are the structure function, the structure constants of the group. Perhaps one thing I have not said but I should perhaps say it. Let me say it here. Let me open a bracket and say it. The generators T C, as I said before, they are actually matrices. They are proportional to the Gelman matrices and they act onto the three color components of a quirk field. So when I write something like this, what I actually mean in terms of indices is T C A B. This is, I'm sorry. This is a matrix A B acting on C B I where this is the color index and this is the flavor index. And if I had to write this in terms of explicit matrices, this would be a three by three matrices acting on a three vector. So this is what the action of one of the generators over a single quirk field would look like. It's this product here. So that's how you interpret the various indices I have been writing down. The C numbers, the eight matrices, C goes from one to eight and A and B both go from one to three. Okay. So I have another perhaps 10, 15 minutes, something like this. Okay. So let's see qualitatively, at least what are the differences between Q D and Q C D? Based on the way we've written down these Lagrangians. We'll see more detail perhaps tomorrow. So the first thing I wanna do is to expand the Lagrangian that I have written down. So L Q D, let's expand it. So I'm trying to write as quickly as possible because this is just a bit of a waste of time, but the writing I mean, it's not a waste of time. So I first write it down and then I will explain what the various terms given. And the last one, well, almost last. If you think the Q C D Lagrangian is complicated, well, the standard model one is much longer. Okay. So, and then let me just write, talk about it in a moment. So what do we have here? We have different terms and let us see what they look like in terms of a perturbative treatment. So now I've written down is, what I've written down here is just a classical Lagrangian. Normally what you would do is you take a quantum field theory, you quantize this Lagrangian, you do a perturbative interpretation and you write down Feynman rules that describe certain processes in perturbation theory. Of course, a Lagrangian is much more than perturbation theory. If you could solve it exactly, if you could solve its dynamics exactly, you would have an exact, a full solution. But most of the times we cannot solve this Lagrangian exactly, you can try put it in on the lattice. Yes, but at least the part I'm talking about is rather using perturbation theory to do calculations. And so after quantization, I'm now looking at a perturbative expansion. And in a perturbative expansion, what are the various terms here? So the first one, this one here is simply the Dirac Lagrangian. So I can see this simply as the propagation of a fermion of a quark. This one here is just the Maxwell equations. It's identical to what you have in QED essentially. And so you can interpret it as the free propagation of a particle which is quite similar in a sense to a photon, well, not quite because it has this additional degree of freedom which is color in the joint representation. But other than this, its Lorentz characteristics are quite similar to actually are the same as those of the photons. And so instead of drawing it as just a wiggle as a photon, we draw this as a coil and we call it a gluons. Then you have the second, the third term which is identical to what you have in QED except once again for this term here and the fact that the field here, the gauge field is colored, but nevertheless, you can still write it as something like this. It looks a lot like the photon electron interaction but it is actually a quark gluon interaction with the fact that this is actually a color gluon. So I write it as color C, so it has a particular and these two will also have some colors that will be changed by the interaction with the gluon. The other two terms are where things become interesting because these are terms that do not exist in QED. And you see that they are terms that contain respectively three powers of the gauge fields and four powers of the gauge fields. These terms do not exist in QED as I was saying. They are unknown. They are strictly related to the known abelian part of the theory because they are strictly related to the existence of this term here. And again, this term as you can see was not existing in QED. And this term is only there because the commutator between the generators is not zero and therefore the FCAB are not zero. If you set the FCABC to zero, this term goes away and the two generators commute so that the theory becomes an abelian theory and nothing happens anymore. So these are the result of non-abelianity of the theory and they can actually be interpreted after quantization and perturbation theory as the interaction between three gluons and the interaction between four gluons. These are new things and these are exactly the kind of interactions that make QCD special and we'll see how this QCD is special. This very non-abelianity also forces us and I tell you this mainly as a fact. I'm not going to use it in particular ways but when you quantize the theory you also have to include what are called ghosts actually called fadaev popov ghosts. They are additional term in the Lagrangian. They take this for their additional field. This is the covariant derivative that I have not defined but it doesn't matter since I'm not going to use it. Okay. So you have to add to the Lagrangian these terms there to eliminate unphysical degrees of freedom that would remain. You cannot quantize properly a non-abelianity theory unless you take care of these unphysical degrees of freedom and so one way of doing this is to include these additional terms. They are called ghosts because they do not participate in physical processes but they are needed while doing the calculation in order to get rid of unphysical degrees of freedom and again, they have their own final rules and when doing a calculation you have to use them and then they will take care of eliminating some terms in the calculation. Again, it's a difference with respect to QD. Okay. Perhaps... Yeah. I can perhaps keep a more detailed description of ghosts. There is a question. Yes. I can see the question in just a second. No, it's raised end. Okay. Hi. Thanks for the lecture. I just wanted to ask... Okay, I guess I'm kind of confused as to where... My question is... Okay, where the... What would be the underlying group structure for the flavor symmetry and how does it come into play? And second of all, when we do the spontaneous symmetry breaking for pion masses, are we breaking the SU3 flavor in that case and if that's the case, then are we breaking SU3, SU2? What is the group structure? Yeah. So flavor symmetry and color gauge symmetry are two very different animals. The flavor symmetry is a bit like iso-spin symmetry, or you can also say iso-spin comes from flavor symmetry. It's an approximate global symmetry. It's just the statement that in a strong interaction, you can replace like a new term for a proton. For instance, if you're doing SU2 iso-spin symmetry, or you can permutate up, down, and strange and do not change the strong interaction. So this is... But it is an approximate symmetry and it is approximate because in fact, Newton and proton are not exactly identical. They have different charges, they have different masses, and the same for the three quarks, up, down, and strange. Actually are very similar, but not identical. Again, for instance, they don't have the same mass and so on and so forth. And this is the reason actually why pushing the flavor symmetry up to four, five, and six flavors does not really work, because then the masses of the higher mass quarks start becoming really very much different. So you could try, but it doesn't really work that much. Instead, the color, the local gauge color symmetry is, as I was saying earlier, something that you postulate or you expect to be an exact symmetry of nature. And it's out of that one that you construct QCD, exactly the way you construct QED out of an exact U1 abelian gauge symmetry. I guess like the next question that I had is like, so I thought the pions are super relatively light because they're breaking the, well, the pions have masses, but they're light because they're breaking the approximate flavor symmetry. Is that wrong? Or are we breaking the color symmetry in that case? No, because I thought pions are like pseudo goldstone bosons because they start with this approximate. But they are related to breaking of the chiral symmetry. Another symmetry of QCD that I've not mentioned. Oh, so there are like three symmetries in play here. We have color, flavor, and chiral symmetry. Well, yeah. In a sense, flavor is not really related to QCD. I see. Flavor symmetry, I mean, even historically, flavor symmetry had been, well, flavor symmetry had been seen as isospin symmetry from the beginning on nuclear physics in the 30s or in the 40s. I don't know exactly when. And flavor symmetry had been written down by Gelman SSU3 in the beginning of the 60s. So that was well before the QCD Lagrangian. Yeah. Okay. You're right. That doesn't make sense for it to be related to pion masses. Then I guess I'm slightly, again confused as to like where pion masses are coming from and what symmetry are we exactly breaking in that case? That would, yeah. That would be a nice topic that I hadn't planned to cover indeed. I'll see if I can say more tomorrow, but I need to dig a little bit in order not to say stupid things. Thank you so much. All right. There are two questions. What do you want to do? There are hands or in the chat? Yeah, hands. Yeah, I can. I can stop here. And that's another question. Okay. So if you wish. Then add your battery to the Q&A session. Okay. So there's a question from Lalit Kumar Sahini. Hi, sir. My question is about the balloons. In quarks you said that there are three colors and we have experimental evidence for that. How do we know that there will be eight gluons and that will follow the adjunct orientation? Is there any proof or is it theoretically coming some? Let's say there are a number of, first of all, the number of gluons, the fact that there are eight of them is related to how you choose the representations of the group that you apply them to construct your Lagrangian. And there are a number of experimental measurements, especially from LEP in the 90s that depend precisely on the group structure that you have chosen to build the QCD Lagrangian. So essentially it's related to what I was saying earlier. You calculate predictions with your Lagrangian. You compare them to the experiment and then you deduce whether your Lagrangian was right or wrong in a sense. So there isn't, I'm not sure if there is something that really pinpoints to, yes, there are eight gluons, like there is for three quarks. Indeed, as you say, there is a very, very clear experimental measurement that tells you that there are three kind of colors for the quarks and it is, as you probably know, E plus and minus two hard rock cross sections. For the gluons, I'm not sure if there is an experimental measurement that really points to, oh yes, look, it's eight and not nine or seven. On the other hand, there are a number of measurements that fix the color structure of the group and therefore that fix the number of gluons. So it is experimentally checked that at least for what can be predicted perturbatively, the QCD Lagrangian built on top of SU3 color with the three quarks in the fundamental representation and the eight gluons in the general representation is the right choice. There is one question by Manuel. Hello, thank you very much for the lecture. I wanted to ask about the ghosts. So when I want to calculate the QCD cross section, I have to also include this kind of propagators and loops and something like this where the ghost is included. Do you see it right? Yes. Thank you. Yeah, indeed. If you forget the ghosts, you get the wrong result, which is not the case in QED because it's a billion and so you get away with it. Okay, thank you. You're welcome. Next question by I think Max. Yeah. So at one point you were saying that there are still a couple of things which remain to be proven to really be sure that QCD is the correct theory of the strong interaction. So I guess one of those things would be probably confinement, but so can you maybe say a couple of things or give a little list on what are still the open questions to be sure? I think, but definitely pretty much confinement. Indeed, it's actually something I was planning to cover later at this point or tomorrow, but indeed confinement has not been proved in QCD. There are hints that come both from perturbative calculations and from lattice calculation that indeed the theory may contain confinement, but nobody has yet been able to prove analytically that this is indeed the case. And I was saying yesterday there is a million-dollar prize for a proof of confinement. So it's definitely an open question. On the other hand, we already have plenty of numerical evidence that things turn out. I mean, all perturbative calculations work. The Hadron masses calculated with the lattice turn out to be quite precise and so on and so forth. So whenever we have been able to calculate things either using perturbational theory or using lattice approximations, we have found good agreement with experiment. So, but definitely, I mean, confinement is definitely what is missing. I agree. Next is Anna. Sorry. Can you move this? The hand again? Sorry.