 I mean, give them a minute or two, they're still wandering in. OK, ready to continue with the collider physics, please. OK, so last time, we talked about the basics of the Large Hadron Collider, how the accelerator works, some kind of rough parameters, the ideas of cross sections, and the coordinate system we use with rapidity and polar angle and azimuthal angle. Today, we're going to talk about the kinds of things that we can measure, observe at the LHC, and give you these sort of introductions of the kinds of things we would try to measure and calculate to find the standard model or beyond the standard model physics. So to begin, let's just think about the kinds of things we can measure. So first of all, if we have a particle, say it's a pion or a proton, the most straightforward things to measure are where it's going. So we can measure its angle is theta phi, which way it went. If we have the calorimetry information from where when the detector we can measure its energy, if we have its tracks, so this is from calorimeters, we can measure its three momentum from the tracker. So keep in mind, the tracker only measures charge particles. So if you really want to measure the momentum of particle, you can bend it in a field and see what happens to it. But if it's not charged, you don't have access to the actual information about the three momentum. We talk about the three momentum of a particle, say a neutral particle. We're reconstructing that three momentum from knowing where it went and knowing its energy. And then if you know its mass, you can determine its three momentum. But if you don't know its mass, then you still have one unknown. So the first approximation and everything at the LHC that we actually measure can be treated as massless. So we combine these things to produce the four momentum. But remember to construct the four momentum, if we don't actually have information about four degrees of freedom, we have to make some assumption about the mass, often assuming equal 0. Or if it's not 0, you can assume it's a pion mass, something like that. But these are the basic things that we have access to. And then from these, we try to construct other observables. So I lost my voice this morning. So another useful object is the missing transverse momentum. We sometimes write it like this as a four vector, which is defined as minus the sum over the visible particles of Pt. So this Pt is a two vector. So it's just in the x and y component. So we talk about the transverse momentum of a particle as the direction orthogonal to the beam. The reason we do this is because we can't measure all of the components of the momentum that go in the direction of the beam because there's a beam there. And most of the particles in the pros and cons collision will go down the beam. So we don't have access to the full momentum of the event. The only thing that's useful is the transverse momentum. And we can usually measure that pretty well. So the idea of missing transverse momentum is you measure everything you can see. And the missing transverse momentum is everything you don't see. So if there's a neutrino in the event, it would show up as the missing energy. So if you know exactly what you had, say it was the W boson that decayed to a muon, all you would see is the muon, and then you assume there's a neutrino. And from that neutrino, you can't get its full form momentum. But if you know the muon's transverse momentum, you could determine the muon's transverse momentum. And then the only thing you don't know is the z component of the muon's momentum. And it's energy. Anyway, this is a useful quantity. We talk about the missing energy, E t, miss, something like that, which is just defined to be the magnitude of missing E t. So it's the missing energy is really missing transverse energy, which is defined as the magnitude of the missing momentum. It's just terminology and people will talk about missing transverse momentum as a scalar, as a number, or as a vector interchangeably. Another observable is H t, which is the, it's defined in various different ways, but it's generally the missing transverse energy in some set of objects. So the total missing E t means everything in the event. You're trying to determine the total missing energy summing over everything. H t is a different kind of observable, which might be say if you had three jets in the event, plus some leptons, you might just sum over the transverse energy of the jets and ignoring the leptons. So it's something if you had some sub-process that you imagine might have missing energy, you could try to isolate it from the sub-process rather than the whole event. So for example, we might have H t is say, the sum over the jets of the P t. So it's the magnitude of that. Generally it's a scalar quantity and these quantities generally useful, but it depends the definition of H t is in universal. So when you read an analysis that talks about H t, you have to figure out what exactly are the objects you're summing over in calculating H t. Another useful object is the invariant mass, which is the sum over some object, I P i mu. So that's the square root of that. So this invariant mass, the idea is if you had say a Z boson decaying to a muon and anti-muon, you would sum over the muon and anti-muon and plot the distribution of that invariant mass and it should have a peak around the Z boson mass. And again, this depends on what things you're interested in. And it also depends on being able to reconstruct the formament of all the particles that you're interested in, which generally you can't do, but you approximate it by doing things like assuming particles are massless and using their energy and angle to reconstruct the momentum. So these are subtleties that often you forget about, but it's important to keep in mind when you're doing some precision calculation and you want to understand the uncertainties to know what actually goes into these measurements. So let's consider a particular process now. The first one we're going to talk about is Drillian. Drillian refers to the process protons collide and produces Z boson, which decays the leptons, usually electrons or muons. So the Feynman diagram for this is you would have, well, say quarks come in, it would be a quark and an anti-quark annihilated into a Z that would go to E plus or minus. So here my time is going this way. This is a Z boson. The cross-section for this we write as the sigma dS hat. So S hat is the invariant mass of the lepton pair. It will be written as dx1, dx2, f of x1, one f of x2 mu times the partonic cross-section for q, q bar t plus or minus through a z. So these objects here are parton distribution functions. The parton distribution functions tell you the probability of finding a parton, in this case a quark, inside a proton with a certain fraction x of the proton's momentum. So it's easier to see if we have a picture. So we have a proton momentum p mu and then inside it is a quark with momentum q mu. So we say its momentum is equal to x1 times p mu. So p mu is the momentum of the proton and we pull out some quark which has some fraction x1. So x1 is generally a very small number, like 10 to the minus three. So if the proton has momentum of six and a half TeV and the quark might have momentum energy, one second, of 100 TeV. So 100 TeV divided by six TeV is 10 to the minus two. So that's a typical number we would have for x. Yeah, some constraint. Well, x1 and x2 have to be less. Yeah, so we're integrating them from zero to one. That's right. So generally you would say, well, if x1 is very close to one, x2 can also be that close to one. But actually they're completely independent and this follows from factorization. So the x1, the amount of momentum going to one quark from one proton is completely independent of the amount of momentum picked out. So this is an approximation. It's not actually true, but it's true to an excellent approximation. So it's true up to corrections that go like lambda QCD divided by the center mass energy of the collision, which are generally very small. So for all practical purposes, we can assume this is exactly true. And it's never been proven this formula to hold in general processes, although actually for Drellian, it's proven to hold up to these power corrections. That's really the only process in which you can prove that this factorization holds. Yeah. Yeah, good. So this mu is some scale that you have to pick. So the scale associated with this process, the part and distribution functions are dependent on the energy at which you're pulling it out. So there's a kind of virtuality here of the quark associated with mu. Generally, you can calculate this for any mu and you can vary mu and see how the cross-section depends. If you pick mu very different from, say, s hat, so usually we would pick mu is something like root s hat, some scale associated with the hard collision. If you pick it to be very different, you can pick mu to fix to be 10 GeV. Then what you'll find is that the large, so this also depends on mu. And the mu depends of this, we'll cancel the mu dependence of that. So generally we choose mu to be something of order of the scale so that there aren't, so that perturbation theory works very well. Because if I try to calculate this cross-section, I might get something that this cross-section might look like one over mz squared times one plus alpha log mu over mz squared, something like that, dot dot dot. So if you don't wanna have a large logarithm here so if this log were really large, if there were 100, then this would be bigger than the leading order term and perturbation theory wouldn't work. So generally you choose this scale to be the same order as the other scale in your partonic process and because this scale is the same as that scale, you have to also choose this to be that scale. But the whole factorization formula is independent of mu, so you could pick mu to be anything you want, that would just mean that higher order corrections become important and you'd have to work harder to do it. So if you wanna save yourself work, you'd usually pick mu to be a scale associated with a partonic process and that determines what scale you pick for the PDFs. So as you see that the, yeah, so, okay, good. So we have a proton here and we have the other proton with p mu, I don't know, prime. And again, we'll pick out some other momentum q2 mu, which is x2, I guess I should call this p2, p2 mu and let me call this one. So we're picking out two, and then these guys scatter so that I have this quark come in and scatter and produce a z. So this is the, this part here is the partonic process I've indicated with a hat. And then the full cross-section proton is divided by the PDFs here. This is x2 mu, tells you the probability of finding a quark with this momentum fraction and then you scatter the quarks off of each other using Feynman diagrams. So this thing we calculate perturbatively and this is some non-perturbative object that has to be measured. So you measure, so it's been measured, people have been measuring them for 40 years. They measure them from a lot of different experiments. There's input to the PDFs from Tevatron from UA1, from deep and elastic scattering experiments at HERA and all of these are combined to global fits to PDFs. And if I have time, I'll talk more about what the partonic distribution functions are, but for now we'll just take them as observable measured functions that you can use to turn. But the point is that they're universal. So it doesn't matter what process I'm putting, I can calculate the probability of finding a quark inside the proton independent of how they're scattering. So we can't calculate these things, but we can measure them. We measure them once and for all and we can use them for any process, be it standard model or beyond the standard model. Yes. No, the s hat is calculated from the quark. So this s hat is the, so what is s hat? Here I'm defining s hat to be the momentum of the leptons. So P E minus plus P E plus squared. Well, right, so here I'm calculating in general, but you can rewrite this in a way that makes this more explicit. So let me do that. So to do that, it's helpful to define what's called the luminosity function, which is integral dx1 dx2 f of x1 mu, f of x2 mu times delta of x1 x2 s minus s hat. So this is now a function. So the point is that the PDS don't really appear separately. All that matters at a given value of s hat is that s hat is given by the product of momentum fractions times s. So to see that, notice that s hat here, which is q1 plus q2 squared is q1 squared plus q2 squared plus q1 dot q2. These are partons, so their masses are zero. So this goes to zero and then two q1 dot q2 is equal to two x1 times x2 p1 dot p2, which is x1 x2 s, right? Where s here is p1 plus p2 squared, which is two p1 dot p2. So the point is there's a relationship between the center mass energy of the full collision, which is 13 TeV. So s is 13 TeV squared and x1 x2 are the momentum fractions. And so if you know s, to find a certain s hat, you only need, it only depends on the product of x1 and x2. And this happens for a lot of processes of interest, because any two to two process only depends on this product x1 x2. So it's helpful to have these functions of s hat, which is useful for things like resonant production, which is a simpler object than the separate PDFs. So we don't need to know the separate PDFs. We just need to do a function that tells you how the luminosity depends on the center mass energy. So the idea is that this is some steeply falling function that looks like that as a function of s hat. And you can calculate it for different channels. So this would be the qq one, which is like this, and you would have glue glue, which looks a little different. And you might have the cork glue luminosity function. But it's a little easier to think about. It's a little more physical than the PDFs. It doesn't depend on some number between zero and one. It depends on the dimensionful scale. And it tells you how the probability decay is as it gets the higher and higher energy. I'll show you some plots of this in a second. So once you have this luminosity function, then this partonic cross-section becomes simply l of s hat d sigma d s hat. So the point is that the proton cross-section to produce the Drillian pair is related to the partonic cross-section to reduce the Drillian pair by a simple function. So if I wanted to know, say, s hat is the same as the Zebo's on mass, 90 GeV, then I would calculate this at 90 GeV. And I just multiply by this function at 90 GeV to translate from the partonic level to the proton level. So that makes it relatively easy. Yeah, P1 and P2 are the proton momentum. I just erased the picture. Well, they're massless compared to very high energy scales. So the proton is a GeV, which is a very small scale compared to 13 GeV. So generally, we neglect it. Otherwise, you can put in some 2mp squared here if you want a small correction. But generally, that's negligible. It depends on how precise you want to do things. But usually, we treat the proton as massless from this point of view. OK, so let's talk a little bit more about Drillian. So let's talk about this partonic cross-section. So d sigma d s hat, which is given by the Feynman diagram. Quarks come in. You have a photon, e plus, e minus, come out. So the coupling here is the electric. Let me write it as G. In general, for some resonance production, you'll always have couplings at each vertex. So overall, this scales like something like, I'm sorry, G squared to the fourth. And so the cross-section, let's be a little more precise about this. So it's roughly G squared over s hat minus mz squared from this propagator plus i gamma mz. So this gamma is the width of the z boson squared, which is G to the fourth over s hat minus mz squared squared plus gamma squared mz squared. So here I've added the width. So you can think about the width of a particle if an unstable particle has an imaginary part to the propagator. But when you square it, of course, the answer is real. So it gets a term that has to do with this resonant behavior. If gamma were 0, this would blow up at s hat is mz. But since gamma's not 0, you get some kind of broadening of it. This is called the Breitwigner distribution. It describes the shape of a cross-section near a resonance. Yes, the sigma hat, yes. Well, I mean, sigma hat from that context. Here I'm just drawing a Feynman diagram. Yeah, but to be consistent, I should go to the sigma hat. So just to kind of sketch what this looks like, just some kind of peak centered as a function of s hat. This is the sigma ds hat. It'll have some peak in order mz with the width of order gamma. So gamma is the width at half maximum of this Breitwigner curve. So that's the typical form for resonant behavior. Now, OK, but there's something funny about this, which is that we can think about the production of the z in two different ways. We can think of it as produces z, and then it decays to e plus, e minus, or we can think of it as producing the e plus, e minus pair through an intermediate z. But if you think about producing a z, what would be the cross-section for that? So if I ask what the cross-section is for, say, q, q to z, so that would just be from a Feynman diagram like this, where I have a g. So this would go like g squared. So if I look at the, so I would say, in the view where I produce a z and then it decays, the rate for producing a z goes like the square of the coupling. But here we say that the rate for producing a z and a k goes like the fourth power of the coupling. So they seem inconsistent. So you say, well, OK, but then there's some rate to decay, which must, you know, there's obviously a g over here, and there's another g over there. And so that's going to be the extra factor of g squared. But what's inconsistent about it is, once you produce a z, it's going to decay, right? So whatever it decays to, it's going to be proportional to g. But there's an order one probability that it decays, right? So it has to decay once I produce it. And therefore, there seems like there's an inconsistency that as I produce the rate to decay, and then I have some, I mean, suppose it only ever decayed to e plus or minus, right? Then it seemed like I would just produce it and go like g squared. And then it decays. It doesn't matter how small the rate is for it to decay. It eventually will decay, so I shouldn't have to pay that penalty. So does anyone know that the resolution of this paradox? I don't understand what I'm talking about. If you try to calculate the direct on-shell production of a z, it goes like g squared. But if I calculate the rate for production to k, it goes like g to the fourth. And that'll be no matter what it decays to, because it always couples with proportions of g. Yeah, but if I say suppose it only decays to this. If you do what? That's right. Yeah, so I guess it wasn't a great question, because the only way you know the answer is if you already know the answer. But that's right. So it has to do with the narrow width approximation, and that's what I want to talk about next. So what's going on here? When we talk about it being produced, so that means that you could think of it as an on-shell particle, right? We're effectively at less for a very long time before it decays, right? If I can talk about it as a particle itself, it's got to survive long enough to call it a particle. So it means survive long enough. It means that the coupling is getting very small, so the width is getting very small. So in the limit that this width is small, it becomes a very sharp resonance. And most particles that we think of as actual physical particles are long-lived compared to their mass. So in the limit that the width gamma is much less than mz, right? So if we could drop gamma altogether, this would blow up, this would be singular. So this kind of regulates that singularity, but you can see it's going to regulate it in such a way that this thing is integrable. So it's going to approach something that's very, very narrow, which looks like a delta function. If you work out all the factors, it goes like g to the fourth pi over gamma mz delta of as had mz squared. Again, it feels like gamma to the fourth, but the key here then is that this factor gamma, that was supposed to be narrow, in the limit that gamma is very small, well, why is gamma small? Because gamma is proportional to the raffer to the k, which is proportional to this small coupling, right? In fact, you can calculate gamma. Gamma is just this part of the process where z goes to whatever it goes to, which has a g squared, a g squared, gamma. So gamma goes like g squared. And so you see that in the narrow width approximation, we get a g to the fourth over gamma, which also scales like g squared. So what's the point of all this? That, generically, if I'm producing a particle, if I'm not right on the resonance, suppose I'm producing an s hat which is slightly below or slightly above mz, right? Suppose I look for an invariant mass left on pair or much larger than mz, or much smaller than mz. Then the rate is very small. It goes like g to the fourth. But you get an enhancement in the resonant region. If I look for the z to be produced on shell, and the rate only scales like g squared, which is consistent with this. So that's an important qualitative point, is that the cross section for s hat not equal to mz scales like g to the fourth, but the cross section for s hat approximately mz scales like g squared. This is called the narrow width approximation here. So the narrow width approximation is just taking the limit that the width goes, get small, where you approximate it. You treat the process as production and then decay of a particle. But the qualitatively of what it means is that you get a large enhancement, right? So in the limit that the width is small, this becomes a very, very large, right? So basically the ratio of this resonance to something off resonance, which is larger than gamma, is a factor of g squared, right? So the cross section goes down by a factor of g squared when you're away from the peak. And that's why we like to produce things on resonance. But that's the thing that we naturally produce on resonance. But also why if we want to produce a lot of a particle, we have to have a collider that's strong enough that has high enough energy to produce on resonance. Because if you're well below the resonance, you still produce things, but it's down by a factor of g squared. You know, on a g, so typically the typical scales for these couplings are 10 to the minus 2. So typically you're down by a factor of 10 to the minus 4 or something below resonance. OK, are there questions about this? Take a break and look at some figures. So starting from last time, this is the large-end collider, so you've all seen it. Just to give a sense of scale, it's Geneva on the background, Lake Geneva, you can see there. I mean, it's 27 kilometers, but it's helpful to see it overlaid with land to appreciate how big it is. How long have you been to a CERN? It's a good fraction about, no, about a third, maybe. So this is from last time. This is just an update. This was the luminosity recorded up to 2018. So again, these are these units of Hertz per nanobarne. So 21 was the peak luminosity. So again, you could think of it as 21 inverse nanobarne per second when it's running at peak luminosity. But you see all the downtime when it wasn't actually running at all. So if you integrate this, you get the total integrated luminosity, which has units of inverse femtobarne. So this was the run-to of the LHC in 2018. Got around 70 inverse femtobarne for CMS, and a similar amount for ATLAS. I want to show you this video from CERN. I don't know. These things are kind of fun. CERN produces some of these animations of what comes on CERN. But just to give you a little context of the LHC, it's fun to watch. So again, there's an overlay of the LHC on the Geneva region. It's just showing you where the hydrogen comes from. I guess it comes from a bottle. Because this is the cartoon of it. It's dripping off the protons and putting them into this RF injector. So the first stage of the LHC is a linear accelerator, which then it goes into the PS booster and then the SPS. So these were old accelerators at CERN, like UA1, involved this proton signature on in the early 80s. And then they upgraded it. They used the old accelerator from UA1 to inject into. So this SPS was the UA1 accelerator complex. And then they inject that into the large H-run colliders. So they build it up. And maybe for the upgrade, they'll use the LHC to inject it to a higher energy one. There's a little bit more to it than your mission. So these are the four detectors at four different points around the ring. This is some artist's picture of what a proton looks like. So that's an example of an event. This is a little bit about the acceleration. So they have these dipole magnets. So they have two beams, or side by side, in the big tunnel. Just showing how they cool it with liquid helium. What the field looks like. This is about a millimeter before it. Then they contract it from a millimeter down to about 10 microns. This is the way to see what the beam looks like. They have this wire scatter where they scatter some electrons. And then they use to figure out what the beam shape looks like. It's actually very difficult to measure what the actual luminosity is of the LHCs. So they have a lot of indirect to it. But this is a typical beam shape. I guess this is an artist's picture of what data looks like. Coming out. But they have these big data farms stored at CERN where they store and process all the data. This stops being interesting. Data reconstruction. I guess this is what I mentioned before. This is the tier one centers. Our scatter, they went through it pretty quick. But you can see there. Yeah, it's tier one and tier two centers. We're located around England. I guess they made this in England. And it's a fun video. There's a bunch more if you look at the CERN website. There's a lot of videos that you can look at. Here's an example of a trigger table that I mentioned yesterday. So these are the Atlas triggers from run two. You could see that this overall number here on the bottom, I don't know if you can read that, 1550 Hertz, which is the rate. So I talked about 200 Hertz. So 200 Hertz was the sort of early run frequency. But as technology has improved, they're able to upgrade that by an order of magnitude to almost 1.5 kilohertz. They talk about the level one triggers and the high level triggers. Level one triggers are hardware triggers, which gets to 85 kilohertz. And they have to reduce that now to the data recording rate of 1.5 kilohertz. But if you're interested in a particular process and you want to know if they're going to see it, you can look at your trigger table. And if it doesn't trigger, it's not saved. So you can study this if you want. Here's a picture of Atlas. Again, I mentioned before. Let me see, is there a pointer? Pointer, pointer. Again, for scale, there's people there. The middle region of the silicon, so that's all for tracking. So there's the pixel detector, the inner tracker, which is the highest resolution. And then the silicon tracker outside of that transition radiation tracker. So they combine this with cascading information, a different angular resolution depending on different kinds of particles looking for the inner detector. And then the electromagnetic calorimeter is this liquid argon here. That's to measure electromagnetic photons and electrons. Outside of that is a tile calorimeter, which is for hydronic calorimetry. And then the muon system is the outermost region. CMS is qualitatively similar. I'm not going to go into details between the two experiments. Let's see. So typically, this is how particles would show up. So photons don't leave tracks. They give all their energy in the e-cow. Electrons leave tracks and give all their energy in the e-cow. Muons just go through everything, and hopefully they'll bend here so you can figure out their momentum. Pions will give a little bit of radiation here, but mostly show up with hadrons. Neutrons don't give tracks and give all their energy in the hadrons. So these are basically the objects you see. And of course, neutrinos, which you don't see. But that's basically what the LHC produces. Here's a typical event display. So you have a collision in the middle, and they show different views of what the collision looks like. So the beam comes in here, and this is a view down the beam, and you can try to reconstruct in three dimensions. And this is what's called a LEGO plot, where you unroll the azimuthal angle and the pseudo rapidity, and you look at the height of these histograms and the energy deposit in different regions. So these are actually back to back, which is a little hard to see, but you can see that the azimuthal angle differs by about 180 degrees. So they're back to back in azimuthal angle. It's hard to see where they are in this picture. They're not back to back in pseudo rapidity, because it's boosted. So this thing, overall, most of the pseudo rapidity is in this direction. You can see here there's a bias towards this side of the detector. This event is overall boosted, but there's two jets that are still back to back in azimuthal angle. What do I want to say about this? Let me not talk about LHCB. Here's a plot of parton distribution functions. So the parton distribution functions are different for different species. The blue on one, so they all sort of vanish as X goes to one very, very steeply. So it's very, very unlikely that you'll find a parton with order one fraction of the proton's momentum. I mean, even at a tenth of the proton's momentum or even a hundredth, you'll find very little. But these, so up is most likely because the proton has two ups and a down. So the integral of the up is twice the integral of the down, but typically you'll find, well, you're most likely to find the momentum of a parton at the lowest possible momentum, and you're more likely to find ups than downs. And then gluons are dominant at everything with the highest momentum. These are the luminosity functions I mentioned, which is more, I think they're a little bit more intuitive. So again, these are things that die with energy. You can compare them at the LHC to the Tevatron. Remember, the Tevatron ran at two TV, and the LHC runs at 13 TV. So these luminosity functions, so for Drillian, you would look at the red one, which is QQ bar. And the main difference in LHC and Tevatron is also LHC is PP, and Tevatron is PP bar. So for Drillian, we would use this red curve. Suppose you want Drillian production at half a TV, you just look, okay, here's half a TV. And then you look up there to get 10 to the six. So this is in units of Pico bars. So that would tell you the cross-section is 10 to the six Pico bars, and then I'd have to multiply that by whatever we're calculating cartonically, which is this formula here, which would tell you the cross-section for producing a lepton pair with that energy. Yes, why what? Yeah, it's a peak of the third, right? Because the up quark, mostly you would expect, there's three valence partons in the proton. So if you didn't know any better, you'd say the up quark should have a third of the momenta, another up quark should have another third, and down quark should have a third. And so then you get two up quarks and down quarks, and this is twice that. So that's the origin of this peak. But you shouldn't interpret too strongly, because first of all, this is multiplied by X. So there is an enhancement near a third, but it's much larger than smaller X because it's multiplied by X. But that's the origin of what, if you didn't multiply by X, you would see a stronger bump. But that's where it comes from. So I want to show one more slide. So this is Drillian at the LHC. This is atlas data. This is early data, but it's five different femtovards, and here you see this resonant peak. So this is Drillian. This kind of shows you roughly the width of it, and there's this long tail of distributions of Drillian well beyond the Z pole. More Drillian. So this is the CMS plot of Drillian, where you all see the week, and it's a different scaling. We'll talk more about different kinds of observables. The questions about any of this stuff is Drillian, for example. Drillian is a nice standard candle process for any collider, because it tells you you can look for the peak of the Z, and since we know the Z mass, it's a good way to calibrate your energies. We know the Z mass very, very precisely, so it's a useful way to figure out that you know what you're measuring. It's very difficult to translate from the readouts of your electronics, which tell you something went there with some current to what the energy is in GEV. And so if you have something like this to calibrate it, it's very helpful, very high energy. I mean, in the lab, you can't really test. You can try your silicon by shedding some pions at it with a low energy accelerator, but to test it for Zs, for 50 GEV leptons, you need to have an LHZ. So you need to do these kind of in-situ calibrations and processes like Drillian are very useful for that. Drillian generally refers to lepton production, lepton pair production, you plus, you minus, you usually are muons. When you put us a W boson, you also produce two leptons, but W is charged, so you only get one charged lepton and the other one is a neutrino. And since you can't measure the neutrino, you can't make these kind of invariant mass plots. So W is much harder to study in Hadron colliders. So if you wanna find W and say measure the W mass, it actually is very challenging. And in fact, the best measurements of W mass do not come from the LHZ. Best measurements of the top mass don't come from the LHZ either, but for different reasons. That's what dependent factorization. Out in terms of PDS. Yeah, so that's, so factor, as I said, factorization has been proven for the Drillian process. That is, you can write the cross-section as this partonic thing convolved with PDFs up to correction suppressed by lambda QCD, right? But for any other process, we haven't proven factorization exists, but it seems to work pretty well. And you can sort of justify it semi-classically as these protons are independent and you can pick a quark out. But to actually make that precise, it's still an open question, so. But there's something different than factorization of your amplitudes, kind of limit or? Can you talk into the microphone? There's something different than factorization of your amplitudes in some kind of limit No, it's the same, but it's more complicated because there's issues that involve when you scatter protons that are harder to say when you calculate partons. So if you just have a Feynman diagram, it's easy. You can study properties of the scattering amplitude. You can study in perturbation theory. When you scatter protons, there's non-perturbative elements. In particular, there's what are called spectator quarks. So it's not just, you have to show that it's the only the proton matters. So when you scatter a proton, you would have a quark scatter like this, right? And that certainly happens, but you have these other guys, right? And you want to show that there aren't processes like this that contribute, that involve these spectator quarks. And that's what's very, very tricky to do, right? And sort of prove that this cancels. So this is sort of, the problem is there's a kind of potential between these. And you want to show that the potential doesn't affect the hard scattering. And this is the highly non-trivial thing that involve proton, Hadron scattering that you can't see in perturbative proton scattering. So there's proof of this is due to Collins-Oper and Sturman, but it's only for Trellian because it's very inclusive and it's only leptonic. So you can show that when you sum over everything, the things cancel that wouldn't otherwise. But if I had some Hadronic things in the final state, like a jet, then it's very hard to show that you can drop this compared to that. And that's what hasn't been shown. So yet, we have data and it seems to agree pretty well, but we're getting to the point where the precision is high enough that maybe we're sensitive to factorization violating effects. So it's an act of people are looking for observables that are sensitive to factorization violating and trying to understand better what factorization actually is. All right, thanks. Okay, so let's talk about W production. So here, we calculate the same kind of process. Quarks go in, a W, and then it would decay to say an electron and a neutrino. For a Z, it would be like an up and an up bar. For a W, it would be like an up and a down bar that produces a W. Calculate this protonic process. It's very simple, just using ordinary Feynman diagrams. It's a couple of line exercise. And you can write it as d sigma d cosine theta star is 3 eighths, one cosine squared theta star. So I've just normalized it so it integrates to one. So cosine theta star, so what is theta star? Theta star is the polar angle in W rest frame. Again, we have protons come in and a W is here and the electron would go out in a neutrino. So we're in the W frame and so things go back to back. We can pick any, I mean, the W is not generally produced at rest, it's produced with some boost, or it can even have transverse momentum. It's produced and associated with something, but if it's just quark quark to W, the W can in most have momentum in the Z direction. Since the observables we're interested in are generally gonna be boosting variant, things like pseudo rapidity or rapidity differences or transverse momentum, we can calculate those in any frames and the W rest frame is a particularly convenient frame. It's like the partonic center of mass frame. So what we wanna do is translate from theta star which isn't observable itself. Something that is observable like the transverse momentum of the W. So we wanna have the relationship between theta star and the transverse momentum. So to do that, let me draw a triangle here. So theta star is the scattering angle. Again, this is the same theta star here. Just drawing the triangle bigger. So this would be Pt of the electron is this direction. Pz of the electron is over here. And the magnitude of the electron momentum is here. But remember if the W is decaying to an electron with PE mu and an attrino with P new mu, since the W has energy Mw and it's decaying to basically two massless particles that go back to back, then each of those have to have energy Mw over two. So the momentum of the electron is Mw over two. Right, that is mass of the electron squared we're just saying is zero is the energy of the electron squared minus the momentum of the electron squared. And since the energy is Mw over two squared, we conclude that the magnitude of the momentum is Mw over two. Okay, so from this, we calculate that sine theta star is two Pt of the electron over Mw, right, it's this divided by that. The sine and so cosine theta star is square root of one minus two Pt I'm sorry, one minus four Pt squared over Mw squared. Okay, so now what I wanna do is change variables from d sigma d cosine theta to d sigma d Pt because what we can directly measure is the transverse momentum of the electron. So we get d sigma d Pt is three over Mw squared times, well one plus cosine squared theta is gonna be one minus two Pt squared over Mw squared, but there's also a factor times we need a Jacobian factor to go from Pt to cosine theta. So we need to take d cosine theta d Pt and that gives us another factor which looks like Pt divided by square root of one minus four Pt squared over Mw squared. So this is the, again, this is the Jacobian d cosine Pt d cosine theta star. So this distribution is just some quadratic function of cosine squared theta, whatever it looks like. Not particularly interesting. This one is much more interesting because it has this feature that it has a one minus Pt over Mw in the denominator. So what happens here? When I take Pt to go to Mw over two, then this thing becomes one and I go with square root of one minus one which is zero and it blows up. So this thing blows up Pt is Mw over two. What we'd expect is if this is Pt, this is d sigma d Pt, you would get something that gets singular at Mw over two. Of course, it's not actually gonna be singular because things are regulated. So if I just have a leading order, I get something that actually blows up. But in fact, if I put a higher order of facts, this won't of course be infinite infinite and I put the W with and things like that. But so instead of blowing it up, you get what's kind of an edge here. So we expect an edge near Pt is Mw over two and we get a peak and this peak is called a Jacobian peak. So a Jacobian peak comes from this Jacobian factor, right? The actual distribution in terms of the center of mass frame angle is not particularly interesting but when we're choosing to measure something in terms of Pt, we generate this feature in the distribution. But this feature is useful, right? Because remember, we can't reconstruct the mass of the W boson but suppose we didn't know what it was at all, right? What we can do is measure the lepton Pt and look at the endpoint of the lepton spectrum, right? And when we look at the maximum value of the lepton momentum, we can determine that this edge is Mw over two. So we just figure out where this is multiple by two and that gives us an estimate of the W mass, right? Well, if you just plotted cosine theta star, you wouldn't see anything interesting at all. So Jacobian peak is an artifact of a choice of variables but it's a useful way to represent the information that gives you information about the underlying a mass of the resonant particle plot here. So here's an example of what that Jacobian peak looks like on data. So this is Atlas data from ATV, 20 hundred spectro bars where you see, I mean, this is a real data version of what I was sketching. So let's see if the peak is in the right place. Let me know what the W mass is, 80. So you expect the edge of W over two, which is 40. So you can see that edge there in the plot. It's not so 40, so the sort of where it starts to turn over the location of the Jacobian peak is that. So it's not as sharp a peak as you would expect. But again, it's determined by higher order of facts. So the fact that it's controlled by higher order of facts makes it a problem because you have to calculate those higher order of facts if you're gonna do a fit. Usually you don't, so if you wanna extract the W mass from this, what you do is try to calculate this distribution as well you can and fit the whole curve. Not just look for the edge, but the fact that there is a feature there makes the fits somewhat easier to do. Notice here, however, we're only using the mention of the muon, right? But when at W decays, we have more information than just where the muon went. We don't know the complete momentum of the neutrino, but we know something about the neutrino, which is we can measure the transverse, the missing transverse energy in the event. And that gives us a handle of what the neutrino was. So you can imagine we can get an improved measurement of the W mass if we improve this observable to include information about the neutrino. And that's what motivates an observable called the transverse mass. I'll describe next. First, are there any questions about this? Yeah. Yeah, so here we're just looking at events with hard muons in them, right? So when a tau decays to a muon, well, you could often reconstruct the tau, right? So there's other things it decays to, and the muon is often soft, right? So, towels that go to muons are considered part of the W to muon branching ratio, right? So there's some small fraction of towels that go to muons, but generally these other things, part of the tau decay that you can use to distinguish towels from muons. But towels are harder, because they're essentially a hydraulic object. We'll talk about towels, probably not today, but the next time, tomorrow. So let me talk about transverse mass. I guess I'm just gonna talk about this briefly. Okay, so if we wanted to really construct the W boson mass, we would use the electron plus neutrino mass, energy minus the transverse momentum of the neutrino minus the z component of the electron. So if we knew everything, we could just plot this. So for Drillian, this is what we do, do we construct the z boson mass, but we don't know everything in this equation. So we know this guy. We know this guy. We know everything about the electrons. We know this guy. We know, well, as I said, we don't generally, well, maybe we can measure the electron energy. We can certainly put the electron on shell. So we assume M e squared is e mu e squared equals zero, then we know this. But we don't know this guy, and we don't know this guy. But we do have a constraint that the neutrino is massless. So a similar equation for the neutrino being massless relates this to this to this. So there's one unknown. So we can't construct the neutrino even using the missing, even using the transverse momentum of the neutrino, which comes from missing transverse momentum in the whole event. Nevertheless, if we're just plotting the electron, we're only using basically this single piece of information. The idea is we wanna try to incorporate, we wanna come up with an observable that also uses this information, which ends up being more stable to high-rotor effects because you, well, we'll see how it works. So this motivates the definition of the transverse mass. So M t squared is e e transverse plus e nu transverse squared minus p t nu. So it's the same formula as mass, except we don't include the z component of the momentum. So what is this transverse energy? So e e t squared is M e squared plus p t e squared. So all of these quantities, transverse energy and transverse mass are defined only in terms of transverse momentum, right? So they're gonna be boost invariant by construction. And this object involves a combination of the various transverse momentum of the different objects in a way that's, well, it's kind of like, so transverse mass is like mass without the z component. And energy is like energy without the z component. So let's think about what this transverse mass looks like. Okay, so we'll do, so we'll do an example. Suppose all transverse, right? So what does this mean? We have the proton comes in and we have some kind of scattering and all the momentum of the z and the electron are in a plane perpendicular to the beam. And so there's the electron and the neutrino. That is the pz and p of both the electron and the neutrino are zeros. This is one special case, but we can consider, so this is pz e is pz nu is zero. So what does that mean? That means that e, the transverse energy is the electron energy and the transverse energy of the neutrino is the neutrino energy. And therefore the transverse mass is just equal to the w mass. So in the case where there's no z component, the transverse mass is the w mass. So the transverse mass is kind of the projection of the mass onto the transverse plane. Let's consider another example. So the opposite case is where it's beam plane. So here we have the beam command and we have a single plane. So my electron and neutrino go off in this plane, right? The generic situation is that they're not planar at all. It's not perpendicular to the plane, but this plane will be rotated by some angle. But that case is a little bit more difficult to study. So let's just do this case to see what transverse mass looks like. Okay, so let's just work in the w rest frame since everything is Lorentz invariant. So in the w rest frame, draw these triangles here. Pt, pz energy, which is the magnitude of p. The invariant mass, so mw squared. If I have these two going back to back with energy e, it's just gonna be equal to two e squared. But the transverse mass squared is eet plus e nu t squared minus pt nu squared. Yeah, good. But these are back to back, so this is just zero. And so this thing is equal to two times pt squared. Which is four pt squared. While from this plot, we see that e squared is pt squared plus pz squared. So mw squared is four e squared, which is four pt squared plus four pz squared. So in this case, the transverse mass just has to do with the transverse momentum, but the w mass and the transverse momentum plus the z component. In other words, mw squared is mt squared plus four pz squared, which is greater than mt squared. And in general, you can show that mw is greater than or equal to, let's try it this way, mt is less than or equal to mw. So the transverse mass has an edge, has an endpoint at the w mass. So it's kind of a mass variable that is constructed so that it's equal to the w mass if the scattering is all transverse, but if it's all longitudinal, you have some correction proportional to the z component of the momentum of the different objects. But it's an object that's constructed that depends, so unlike the pt of the muon, it also involves the transverse energy, so it has more information, and that's really the fundamental reason why it's more useful. Let me just show you a plot of the transverse mass to be consistent with this. So this is transverse mass, again, these are atlas plots. So here you see there's an edge at mw, but it's really the location of the geography peak, so I didn't make that the formula, but you get a peak there at the same scale, but it ends up dying if you compare this to, if you compare it to the lepton, there's a much shallower tail, and so it ends up being a much more useful observable for measuring the w mass. It's useful for other kinds of observables, the generalizations of transverse mass that's useful for beyond the standard model searches, any particle that might decay to something with missing energy, there's generalizations of transverse mass that are useful for it. But so transverse mass and the pt of the lepton are the two observables useful for w mass determination at Hadron collider. Are there any questions about this? So the width of this plot is not determined by the width of the w mass. There's a lot that goes into it, so it mostly has to do with the broadening, comes from higher water corrections from the w is produced with some transverse momentum because it's produced in association with some other particle and that broadening. So if this were a plot of mw, it would be very narrow, it would look similar to these Drillian plots like that, right? If we actually reconstructed the w, the width would be much smaller. But the fact that we can't reconstruct the w is sort of constructing a different observable which is losing some information. And the fact that we're not reconstructing completely is the origin of this width. It's not determined by the intrinsic width of the w. Good question. Anyway, those are the two of the most important collider observables are transverse momentum and transverse mass. Let me continue then. So the next thing I want to talk about is jets. So what are jets? Jets are when you produce a quark, say a Z decays to a quark, a QQ bar pair, or you just directly produce quarks or gluons at a collider. Since quarks are not stable objects and they're essentially massless, what happens is you have a proton come in and you might produce some gluons scattered to gluons that these things tend to radiate as they move along and you end up with a collection of particles. So you might, these might all end up being pions that end up in the same, roughly the same direction. So this is a typical process of almost, I would say every single event of the LHC involves some types of jets. So this thing is called a jet. It's a collimated collection of essentially pions. So we think of jets as a representation of some hard quark or gluon produced in the collision. What we actually see are a collection of partons going in the same direction. So a jet isn't a precise thing. It's not, there's no, you get from first principles determine what the right way to define a jet is. But at the LHC and since the beginning of colliders, people have seen these collimated collections of particles and we've called them jets. So to make a jet precise, you need a jet algorithm, which is a way to define precisely what a jet is. As I should say, there's no unique definition of a jet. You use different definitions for different applications. So you've heard of things like the anti-KT algorithm or the Cambridge talking algorithm. These are different depth definitions. They're different conventions for defining what we mean by a jet and they all come with some scales. So you could ask for Cambridge-Occan jets of size R equals 0.4 or anti-KT jets of size R equals 0.7. These are different conventions and what you choose depends on what you're interested in. But a jet isn't some fundamental object with unique definition. So you have to think about how to adapt your definition of a jet for the kinds of things you're looking for. So an example of a jet algorithm would be a cone. So remember, we define radius as delta eta squared plus delta phi squared. So generally things are defined in terms of pseudo-rebitivity. Changes and differences in pseudo-rebitivity aren't exactly boost invariant, but they're pretty close. And if we treat it like the particle is massless than they are, but this is how we define distance. So it's one measuring distance. So this R would say one jet algorithm is take all particles with say delta R less than 0.7 radians. That's one way to define a jet, right? But of course, where do you center that jet? So you could choose things. You can say let me take the hardest particle in the whole event and center it around that. But choosing the hardest particle is not quite, it's a little tricky to choose the hardest particle because say there were two particles going in exactly the same direction with each of half the energy, right? Suppose I have one particle with 100 GV or I have two particles with 50 GV and suppose I have another particle going over here with 70. So from the point of view of the detector, I can't really tell the difference between one particle with 100 GV and two particles going in exactly the same direction that at 50 GV, right? I mean, suppose the pion decays right before I measure it, but I just measure the total energy. So again, if I'm just measuring energy and direction, I can't distinguish these two, right? If maybe there's a track and I can tell them apart, I can distinguish them, but it shouldn't really matter. So it's dangerous to try to pick the hardest particle as the seed at the center of your jet because it's not stable to these kind of distinctions, right? Yes, you can say that, right? So this leads us to the idea that we want to using a hardest particle may lead to strange results, right? That is to say there's nothing wrong with doing it. You can define an algorithm, take the hardest particle in the event and draw a cone around it of size 0.7. That is a jet algorithm and you can plot things with it and most things will look fine, but it'll have peculiar properties. If you try to calculate the distribution, you might find some divergences in perturbation theory, right, that have to be regulated. So it might be unusually sensitive to hydronization, right, because this kind of thing, you know, if I add 100 GV particle that admitted some other particle or fragmented into two particles in the same direction, whether I choose this or this depends on how the particle fragments. So it depends on kind of non-perturbative physics that we're not interested in. So this is a perfectly fine jet algorithm, but you're gonna get results that the distribution of the jets might be sensitive to physics that you don't really care about. So it might not be a good way to understand things you are interested in, which might be hard collision or some supersymmetric particle decay rates. So you can do this, but it ends up being sensitive to non-perturbative physics. Okay, so how do we make something not sensitive to non-perturbative physics? What you want is whatever your jet algorithm is, it shouldn't matter if I treat a particle of 100 GV as two particles with 50 GV. It also shouldn't matter if I take my event and I add some very low-energy particles, right? So let me do it over here. Avoid sensitivity to non-perturbative physics. We want algorithm care. If we take P1 goes to P1A plus, I don't know how to write this. If particle P were replaced by P1 alpha P and P2 is one minus alpha P, right? That is, I have, again, these are four momentum, right? So that's just what I was saying before. So we have P and you can replace it by alpha and one minus alpha P. So if you replace a particle with momentum P by two other particles going in the same direction, it shouldn't affect your any property of your jet. We also choose insensitive to adding low-energy particles. So this is formally in the limit that the energy goes to zero. And again, this is just a natural thing because the LHC is full of soft particles. So if you have something sensitive to all that soft crap, then your jet algorithm is gonna be very useful. So these two things together are known as infrared safety. So I think infrared and colonial safety, but I like to just call it infrared safety. So you want your jet algorithm to be infrared safe. There's a bunch of infrared safe algorithms on there. Most of these are kind of iterative clustering algorithms. Let me, we're running low on time. So let me just show you what an iterating clustering algorithm looks like. That, there's some jets, yeah. So a general clustering algorithm, basically the way it works is you take the particles in your event. So here I've drawn the particles as circles whose radius is portion of the energy. So you calculate the distance between them. So here you would have, I don't know, this is just a cartoon, but you see there's the, that represents the differences of distances. And then you have some different distance measures. So here I'm just taking angular distance. This is delta R, where I have some the pseudo-repetitive, maybe five. And then what you do is you merge the two closest particles. So I take those two that are separated by 0.3 and I put them on top of each other and into a single particle. And then now I have three and I just do it again. I repeat until, so now the closest ones are 0.7 and so on. I repeat until no particles are closer than some parameter R, which is the jet, the size of the jets I'm looking for. And what that means is I'll end up with particles. There's no clipping closer than one. So I say I have now two jets with R equals one. And well, so this is a generic form of an iterative jet algorithm where all you need to know is what the measure is, what is the distance measure between particles. Here, well, really here is a distance measure, which is just an angular distance. You could also take a distance measure that's weighted by energy to some power. And those are also common. So how do you choose what? I'll talk about that in a second. Let me just give you the examples of the other distance measures. So the pure angular distance is called the Cambridge-Occan distance measure. So Occan, so this distance measure Dij is square root of delta eta ij plus, well, it's really eta i minus eta j squared plus phi i minus phi j squared. The AT algorithm has Dij is minimum of Pti squared Ptj squared times delta r ij squared over r squared. And the anti-Kt algorithm as Dij is the minimum of one over Pti squared, one over Ptj squared. So the Kt algorithm chooses big momenta first, right, so the higher the momentum, the bigger the distance. And the anti-Kt clusters small momentum first. And Cambridge-Occan doesn't care about the momentum, it just uses distance. These three algorithms are all in common use. Anti-Kt seems the most unintuitive because it clusters soft particles first, but it ends up having jets with nice properties that end up being particularly local. So that's used most experimentally. This is sort of the standard algorithm used to find jets. Cambridge-Occan is the most physical because it's based on this distance measure, which is easiest to interpret. But you should be aware that there's different algorithms that are used in different processes. Right, so the question was how do you determine r? So there's trade-offs. So you can pick r and sometimes big r is useful and sometimes little r is useful. If I want some particles, suppose I have something decaying to two jets and I'm interested in its mass. So if I wanna get the mass right, I better make sure I have all the particles in the decay. So that would say you want a very big radius to make sure you don't lose anything. If I take r too small, I might lose some of the particles and I wouldn't reconstruct all the decay products of that resonance. So you wanna take r big to make sure you get everything in the resonance. On the other hand, if you take r very big, you also get other stuff. So there's a beam here and there's other particles that can go into the jet. So the bigger the radius of the jet, the more contamination I get. So the trade-off between making it bigger so that you get everything you want and making it smaller so you don't get stuff you don't want. And you have to choose a compromise that's appropriate for the process you're interested in and the particular pile-up situations and so on. I could show you some plots showing how this works. So first, let me show you this, which is, this represents what's called the catchment area of a jet. So this is KT, this is anti-KT, this is Kimberdaken, this is a seeded cone algorithm called SISCone. And so one advantage of anti-KT is that the jet, so what this means is, the way you compute this is you have some event and then you add basically zero energy particles and you see what region of the detector they got clustered from. And so anti-KT has this property that the jets are basically always circles. And that's really the thing that makes anti-KT fantastic. So you say, who cares about circles? But from an experimental point of view, it's very important to know where all the particles were in your jet because the detectors have very different systematics in different regions. So if you have a jet that involves particles from the central region and the forward region, the uncertainty on that jet has dominated by the worst parts of the detector, which is in the forward region. But if you have anti-KT jets, then if it's in the central region, all of the particles in the central region and generally have lower uncertainty on that jet. So really what's dominated the use of anti-KT is that they're easy to calibrate and they have lower uncertainties than other algorithms. Anti-KT was distance, you think they would be round but it turns out they're not. In terms of distance, so here's a showing a resonance, this is a, okay, let me read the plot here. So this is a 100 GeV resonance and we're trying to reconstruct it with R equals 0.3. And you see as we increase R, so this is what the jets look like. So what I'm gonna do is scroll through these plots to different values of R. And so we're looking for a peak at 100. And you see if I take R too small, I don't get all the particles in the jet. And if I take R too big, I get a lot of contamination. And the R that reconstructs the peak best is R around a 0.6, 0.7, right? And again, this depends on the resonance, depends on the scale you're looking for, but there's a kind of trade-off and you have to make that compromise. And there's different standards. So Atlas for a while was using R equals 0.4 and CMS was using R equals 0.7. And then Atlas tried using 0.6 and 0.8 and CMS was using 0.5 and 0.9. And so for some reason CMS was using odd-sized jets and Atlas even-sized jets. You couldn't actually directly compare them to each other which maybe they did on purpose. But where there's been some convergence and now 0.4 and 0.7 are kind of standard jet sizes. In some conditions you want these R equals one jets where you allow it to take everything including contamination and then afterwards you post-process it and try to remove the contamination. So there's been a lot of progress on understanding jets in the last, I don't know, 10 years or so and it's still very active in improving these jet algorithms. I mean, NTKT algorithm came up in 2008 which is not that long ago for me at least for you. It's probably you're in kindergarten. But it's recent history from my point of view and there's still a lot of active developments in the area of jet physics. So let me stop here for today and next time we'll talk about different particles and how to see them. Thank you very much.