 Good morning, everyone. It's glad to see you're still all here, still alive, fresh and kicking. So today we discuss, we're done more or less with a more formal, if you wish, part and today and tomorrow it will be mostly applications, phenomenology, an overview of the things that can be done with Hadron Coiliders, but with a focus on QCD. Of course, with the Hadron Coiliders, as I said at the beginning, we can look for new physics, we can do electroweak physics, we can do certainly flavor physics, but I will just focus on topics that are more directly related at least today to QCD. So this is the proton, those are the three valence quarks inside the proton and we already discussed this picture in which, if we start looking inside the proton at shorter and shorter distances, shorter and shorter timescales, of course, pretty much we can find everything, everything we want out of virtual gluons, if we again look with an even shorter timescale, we can get top quarks, we can get big quarks, we can get Z bosons and, you know, if we are even shorter, if we are taking an even smaller higher resolution out of a top quark pair, we can even create a Higgs boson. So pretty much everything happens in the virtual world inside the proton and if we want to materialize this phenomena taking place in the virtual world, we just have to add the proper energy to this intermediate virtual systems so that we can put on shell whatever particle we're interested in. So that we can do with coming in with our proton, we take a gluon from the proton and it just puts on shell the TT bar system and we create a TT bar pair. So here you recognize a glu glu goes to TT bar final state which we discussed shortly at the end of yesterday. If we take a quark or an antique quark from the proton colliding with this quark from the initial state, that provides the proper energy so that this Z can be put on shell and therefore we have QQ bar annihilation into a Z and of course if we connect a gluon to this gluon we're putting into the system potentially enough energy so that the Higgs boson can be produced. So what's nice about proton-proton collisions is that really we have access to pretty much every single phenomenon, every single process that you can imagine. In that sense we have a bit more variety of phenomena then we would have say E plus E minus because E plus E minus it's really driven mostly by electromagnetic processes. So before we get to several of these final states we have to go through higher order processes. So let me start with, you know, we started our lectures on Monday talking about PDFs. So let me go back to some interesting observables that can be used to learn more about the proton densities. And let me talk about one particular distribution which is the so-called W rapidity asymmetry and we start from discussing this in the context of proton-anti-proton collisions and then we do it for proton-proton. So if we want to create a W plus we take, we can create the W plus by taking a U quark and a D bar quark. Of course we could be taking the U quark from the proton or the U quark from the anti-proton and the D bar from the proton or vice versa. It's quite obvious that the most likely configuration is the one in which we take the up quark from the proton and the anti-D quark from the anti-proton because it's valence, valence and therefore we have more of that. Likewise if we want to produce a W minus we take a D and a U bar. Now it's intuitive and it's proven by data that the up quark distribution is larger than the down quark distribution inside the proton simply because in the proton there are up quarks and only one down quark. So in particular if we go to large x to quarks carrying a large fraction of the proton momentum it's more likely that we find the down. And that means that in this collision U D bar the W plus is more likely going in the forward direction, I mean in the direction of the proton rather than in the direction of the anti-proton. So if we look at the rapidity distribution, the longitudinal momentum distribution of the W plus, this is zero, this is rapidity and it will more likely be something with that shape. Of course PP bar is charge symmetric so if we do this for the W minus we get the symmetric distribution that will be something like something like that. So we see there is a slight asymmetry so if we look at the differential distribution versus rapidity this is proportional there will be the matrix element but then the PDF content will be the U quark content of the proton at x1 the D bar of the anti-proton plus of course there will be a contribution from the D bar in the proton and the up from the anti-proton and this of course is very small compared to that. We can do the same for the W minus, we have the U bar in the proton and the D in the anti-proton or the D in the proton and the U bar in the anti-proton which is the more likely situation. So we can take these, we can assume that we only have a balanced contribution so we neglect this and that which is a very good approximation and then we look at the difference divided by the sum that is precisely what we define as being in a symmetry and you can write out this the expression for the symmetry just by using these you can do it as an exercise and this we can rewrite in terms of the difference between r calculated at x2 minus r calculated at x1 where this r is the ratio between the down and the up quark distribution as a function of x. So what we learn from here is that the asymmetry is actually created not really by the fact that the up quark is bigger than the down quark but it's created by the fact that the ratio of the down quark divided by the U quark depends on x because you see if the ratio of the down quark distribution and the up quark distribution were constant let's say we're exactly a factor of two we have twice as many up as we have down so down divided by up could be equal to one half independently of x if that were the case then this ratio would be independent of x and then when we take this difference that would be exactly equal to zero okay so observing an asymmetry different from zero means that the ratio between the up and the down the quark distributions in x is not constant okay and this is what in fact is observed experimentally this is one example of the rapid distribution of a w boson measured by the cdf experiment you know the cdf experiment that is here experiments run at the Tevatron until about the beginning of the LHC until 2008, 2009, 2008 I guess starting from 1985 that was the very first collisions of the of the Tevatron so more than more than 20 years of operations so this is the rapidity of a w and this is the charge of symmetry which indeed is positive as we would expect because if we take the w plus minus the w minus you see it's a positive and symmetry it's more likely to produce w plus is in the forward region this red line is the theoretical prediction based on a specific set of PDFs and if one accounts for the fact that PDFs have some uncertainty we discussed we are certainty in the first lecture coming from the experimental data the fitting procedure etc you see there is this blue band and notice that these these of course is done at the time in which the measurement was taken so we're probably talking about the second half of the of the 2000s 2006 2007 perhaps and you see that this band is much broader than wider than the experimental uncertainty okay you see here this point the band is much larger than the experimental uncertainty and that means that if we were to take these experimental data as input as constrained for the PDFs forcing the PDFs to reproduce within say one sigma of the experimental data would clearly reduce significantly the uncertainty on the PDF would give rise to a much better set much more precise set of PDFs which then we can use in some other context of course we cannot use those PDFs to predict this quantity any longer because we are just using it for the fit right but once we have the ratio d over u that becomes relevant if you want to study for example the production of z boson in which a different combination of up and down appears so it's intuitively obvious that the w plus will go forward and the w minus will go backward and therefore there will be a positive asymmetry but of course we don't really see w bosons what we see is the decay products so latons typically and therefore what we measure is something that's based on the rapidity distribution of the latons there's something interesting happens here because there is a competition between the asymmetry driven by the PDFs and the asymmetry that's induced by parity violation by the fact that w decays the w interactions violate parity so let's look again in particular at the process let's consider the production of the w plus as we said this is dominated by a u quark scattering annihilating a anti down quark so weak interactions are parity violating it's only the left handed states that contribute so that means that the spin of the up quark has to point backward because this is the fermion so it's a left handed fermion and the right handed anti fermion so this is the configuration of the spins and that means that the w will have well polarization pointing backward as the w decays if we really want to minimize the amount of angular momentum that we have to put into the system in order to have conservation of angular momentum we would like to conserve the total angular momentum just by using the spins we can do that by making sure that you see now we have a decay so the particle going having spin pointing in that direction will be moving in that direction that has to be the anti fermion while the particle decaying in this direction we have a spin anti aligned and therefore we have to be the fermion if we have a w plus the fermion is the neutrino because the w plus cause e plus neutrino e plus is an anti fermion so it has to be right handed so in this decay the neutrino tends to go forward and the positron tends to go backward so you see now what the competition is because the w plus sorry and if we do for the w minus is exactly the opposite the w minus it's mostly produced by the quark coming from the proton so again we have exactly the same spin configuration for the final state and the fermion has to go in the direction of the quark another way of looking at this is that it's the fermion line that tends to continue here we don't have a flavor right because we go from quarks to leptons so what we have to follow we have to follow the fermionic line because gauge interactions conserve electricity and therefore if we had a given electricity state in the initial state the fermion that will be passed on to the final state fermion now in the case of the w minus it's the electron which is the fermion so it's the electron that goes forward and the anti neutrino that goes backward so the competition comes from the fact that as we just saw the w plus itself tends to go forward while the w minus tends to go backward but then when they decay the decay product tends to go in the opposite direction so the pdf pushes the w plus forward while weak interactions parity violation sends the positron so the final state lepton backward and that means that if we look at the asymmetry distribution not of the of the w which is always positive but of the lepton we get something but you know is will be a balance between these two competing effects and this is in fact the rapidity this is actually the pseudo rapidity but it's for leptons it's exactly the same the same quantity the pseudo rapidity of leptons this is the asymmetry okay in w decays and this is zero so you see that up to rapidity of the order of two indeed the asymmetry is positive so the pdf effects wins as we go more forward in rapidity it's actually the b minus a it's parity violation that wins and the lepton the positive lepton will tend to be less likely to go in the forward rapidity region than than the negative lepton and as an exercise you just you know think about it you just sit down and you try to understand why one when we go to really very very very large rapidity at some point it's unavoidable but parity violation should be the dominant effect so we will always have a asymmetry change sign when we go very very forward it's just that kinematical it's a kinematical effect okay now there are four plots because they correspond to presumably different ranges in in it it's different cuts and that shows that there is a kinematical a kinematical dependence but the message is all contained in just one of these one of these plots okay so this is a very cute example of an interplay between qcd and and weak interactions now in the case of a w the structure of the weak interaction is very is very simple it's very basic because it's strictly v minus a so there is no no parameter right there is nothing if these were the z boson decaying then of course there is a competition between v minus a and mv plus a between vector and axial that they don't come with a fixed coefficient minus one but there is the sine square of a weaker mixing angle in between okay so studying accurately the same asymmetry charge asymmetry between the positron and the electron in final states where a z is produced is an important probe of of sine square theta w and this is exactly what was shown in a peskin's lecture on yesterday of course it's done with great precision in any plus in minus that's where the best measurement of sine square theta w come from from from the asymmetries in z from the distributions in z decays but asymptotically with the great statistics that we have at the LHC we will be able to approach that that precision and hopefully one day match it and surpass it now in pp collisions things are slightly different first of all pp is not a charge symmetric uh uh initial state if we take the charge conjugate it becomes p bar p bar so things change and in fact in pp collision it's not even true that the w plus cross section is equal to the w minus cross section in pp bar the probability of producing a w plus or a w minus must be exactly the same right because just while in in pp that is not the case it's more likely the cross section to produce a w plus is larger than the cross section and producing a w minus the rapidity distribution has to be symmetric if we look at the individual w plus distribution there is no way there can be an asymmetry in rapidity because given that it's pp you know we cannot tell what's left what's left and what's right so it will certainly be symmetric with distribution here in this plot is shown in fact only the positive rapidity part so this is the w plus distribution but you should imagine that the negative part is exactly the reflection of this and the w minus distribution is given here and as you see it's lower just because the total the integral of that distribution is the total w minus production cross section which is smaller than the w plus this c s means the contribution coming from strange and char initial states so far we just focused in the simplified equations that we wrote before we focused on ud bar goes to w plus as a production channel but of course we could have also us bar this would be proportional to the sign of the cabiibo angle so it's suppressed because of the cabiibo angle and suppressed because there is less strange than there is a upcork and we can have of course cd bar which is doubly suppressed because there is the cabiibo angle which is small and because the charm there is a much charm and then there is c s bar as a possible contribution now these is not cabiibo suppressed so the cabiibo angle is almost is almost one here but we have a suppression because both the charm and the strange of course are smaller have a smaller pdf than up and down on the other hand at dlhc x starts becoming quite small for the production of the w because we have so much energy in the initial state and therefore there is enough room in q square for the evolution to allow both charm and s bar to grow and in fact the c s bar goes to w plus production rate is significant it's about 14 tv will be of the order of 30 percent of the total production rate and we see it from this picture so this is just the distribution it's less peaked forward than the w plus because this forward peak it is really due to the fact that to produce the w plus we have a upork against a d bar pork and now the d bar pork is c is not valence any longer as was the case in pp bar okay so there is really this significant significant a symmetry if you wish in the I mean shape in the distribution okay so we do have an asymmetry okay and this is really the absolute number of w plus is minus w minus is produced both in the forward in the backward region and this is the asymmetry that we get just by looking at the w's if we look at the decay products again we have the the effect of parity parity violation and again that will tilt and modify the asymmetry in fact the asymmetry driven just entirely from the decay spectrum leaving out the pdf effects as you see it goes into into this direction it's a it's positive now sorry this is the asymmetry for the lepton the negative lepton over the positive over the positive left there is a kinematical dependence as a function of a pt these are these are details and this is the experimental situation just as soon as vlhc started taking taking data which means we have a data and we compare them with the predictions and the pds that were available at the time the lhc started so pds but did not benefit in any way from the new lhc data these early measurements you see there are still rather large experimental uncertainties here it's mostly statistical and these bands the two plots are again because there are different sets of pds that are being used you see there is a very very broad range okay of predictions so there is a lot of room to improve significantly our knowledge of pds by improving reducing the uncertainties on the measurements and then tuning fitting the pds these as happened these are the latest now data from from run from run one five inverse pentobars as you see the experimental measurement is much more precise now look at the uncertainties the error bars are the uncertainty bars are very very small and this is now the comparison against the data there is one set here you see which is particularly outside particularly in disagreement with the data this is the so-called mstw 2008 and what happened here and these appears then in purple with a band that goes exactly you don't even see it because it's almost completely overlapping with the with the with the data points so it's brought inside and what has happened to these pdf fit in going from here to something that agrees was very simple was not making use of new data was not in particular using this data to improve the fit was simply to alter modify the parameterization of the pdf itself in other words we start from an assumption of a functional form for say the d-quark distribution would be one over x times one plus x square root of x plus log of x one makes an assumption of a functional form to introduce few parameters but then we'll go out and fit and of course once we freeze a functional form the best that we can do is to fit those parameters and but if that functional form doesn't really agree very well with the data there is no possible way we can we can improve it right if we try to fit a parabola with a straight line you know there is no way we can really do it very well so what they did they realized that there was an issue an issue that previous data had not exposed the previous data other observables were not sensitive to this specific aspect of the parameterization the move to what is called chebichev polynomial basis that's what cp stands for here cp deuteron deuteron is something else it's not really relevant so and by passing to these new basis of a complete expansion a complete set of polynomials that guarantees that you really can fit any function you want automatically that drove the fit so as to agree very well with the current data so these are just examples now one big question of course which which comes out of this is the following i use a set of data i use it to fix the pdfs what if there were some new physics hidden inside those data if the specific distribution here were driven by the fact that maybe there is some new particle but also decays leptonically it could be you know production for example of a neutrale in chargino in supersymmetry the chargino decays leptonically to a neutrale plus plus a lepton and the final state is lepton plus missing it so it looks exactly like the final state of a w and you know if it comes with a sufficient rate you can you can modify these these distributions so how do we know i mean is there a risk that by fitting pdfs we're really washing away possible new physics and we just parametrized new physics in the worst possible way which is absorbing it in the pdf and this is one of the big questions that everybody has to ask himself when when they go out and play these games so the always caution has to be exercised in using data in order to fit pdfs and and i will i would like to take an example from the past history because it's pertinent it's relevant to this to this discussion and has taught us a lot and this is the kind of example that we will have to to repeat in the future so it all starts from an old paper by professor pesking and and other colleagues who pointed out over three years ago that if quarks were not composite there would be very in or leptons were not composed sorry they were not fundamental if they were composite there were some underlying underlying structure then this could be exposed by looking of course at very short distances inside the quarks so this picture here is a picture of a scattering between a quark which is now made out of three prions three small components incidentally this is a concept that abu salam worked on for many many years and proposed for a long time the ideas that even quarks might be might be composite objects that one could have a prionic structure so it's relevant to discuss these in these in these whole so as we go to short distances we can expose directly the interaction between these components and one of the manifestations of these in the case of quarks would be a change in the rather for like distribution of of scattering and in particular if we go to very very high energy all of a sudden we would have an increase in the rate for example of of jets produced at large pt so this is the way the matrix the cross section the partonic cross section is modified for u u bar goes to u u bar scattering there is the qcd result and then there is an interference term and there is the square of the amplitude related to compositeness lambda is the scale at which quarks are composite so it's a fixed number and you see since it's in the denominator that means that the kinematical part will grow with with energy square so while the qcd part as i said yesterday it's dimensionless so it doesn't matter what the energy is of a collision here this part there is a one over lambda square and therefore it grows like u squared divided by t u squared divided by s and that means that if i go to higher and higher energy this term the interference between the compositeness and qcd grows relative to the qcd term itself okay so this stays constant as we go to high energy while this grows linearly with with the energy square for scattering at say 90 degrees and this is what happens so this line this is for a collider at 10 tv these are the days in which one was planning the the ssc for a collider at 10 tv this is the spectrum in transverse energy of jets that we get from qcd and if we have a compositeness k lambda of 5 tv 7.5 tv 10 tv etc this is the way the spectrum would be deformed and as you see we're talking about corrections which are not 10 20 percent but corrections that could be very very large factors of two or even 10 so we've been saying since 83 for many many years let's look at the high detail of the spectrum because if we see a discrepancy that's a sign that works are composites so 12 years later a paper comes out from from cdf at the teatron in which exactly this quantity has been measured these are the data points the solid line was the best theoretical prediction at the time based on next to the order qcd so already in the era of qcd precision in colliders and if we look more closely at this tail we see that there is exactly this departure from from the expected behavior of the departure that is foreseen by compositeness this different so this is the ratio between the the data and the theory right minus one so it should be exactly zero these different curves correspond to playing around with all of the pds that were available at the time so there is certainly some plus or minus 10 15 percent uncertainty but nevertheless nothing really gives rise to this this shape so if one had to stick to what had always been said this was the discovery of of compositeness but of course people started thinking a bit more carefully they said you know how well do we really know the pds right here we're talking about jets at 400 450 gv the teatron had beams with energy of 900 gv 900 over 900 was 1.8 tv so here we're talking about quarks or gluons carrying 50 percent of a proton energy x equal to 0.5 so it's a very large value of x in which experiment of course there were very very few data because it's very rare to find these quarks carrying all of these energies so the data are poor and therefore people started thinking about how do we convince ourselves that we understand pds of those values well one could have said okay fine we have to admit that this is an effect of pds we take this data and we fit into the pds but by doing that of course you pre-empt your possible discovery of compositeness so it's really a dilemma now the solution to this problem comes by looking a bit more closely at the kinematics of jet production and we we discussed yesterday what the kinematics is this is the beam line and I said we have three parameters one is the pt of the jet and then we have the rapidity the angle emission angle of one jet and the emission angle of the other jet so this is the longitudinal this is the proton and the anti-proton right so they don't have to be exactly they have to be back to back in the transfer plane so that pt is exactly the same but then they can have different angles because the initial state can can be boosted so what we're interested in x large and of course if we take a collision at 90 degrees at very very high energy we have indeed x large that's where those jet data were coming from if both jets are central both x's will be large but there is another way in which we can create we can generate configurations that probe very large x and that is to consider that not to take not both initial state quarks to have large x but just one so we have a very fast moving quark scattering against a very slow quark so this is x very large say of order 0.5 this is x say of order 0.01 so what we're creating in this way is a system that has an invariant mass which is relatively small because the total amount of energy here is small because there is a much energy coming from here so when we're looking at the collision with a small total energy means that we are probing a system with a mass of maybe 100 gb 200 gb and we know that 200 gb there is no new physics because we already explored the region of the 200 gb we know that at the scale of 200 gb quarks are not composite for example and that means that we are working in a situation in which the standard model has been fully tested and we can be guaranteed that there is no new physics so what we're looking at now is just the outcome of a collision of this initial state and that will be two jets of course of relatively of moderate pt but highly boosted in the forward region so if we go out and we do the measurement of two jets going in the forward region we know that we can rely on qcd and we will use that as a way of probing the the x distribution at large x in this slide there are a few a couple of equations which control somehow the kinematics and you can look at them and study them in this in this context so what people did they then just did the analysis of the pt spectrum not for centrally produced jets but for jets produced at larger and larger rapidities and this is what we we see here these are these zero jet data plotted as a function of rapidity and you see it goes down to rapidity between 2 and 2.5 that correspond to two few degrees so it's really going very very forward and in fact you see that the spectrum falls dramatically right and out here even though it's a pt of only say 200 gb we are really probing x in the range of 0.5 0.6 so we use this data now to do the feats of of the pds and when we redo the feet of the pds using this data and we plug it back into the data in the central region we get something which is now in in much better agreement with the data themselves and these are the final results from from the tevatron on the jet spectra and as you see again this is the ratio between data and theory so this should be equal to one yellow is the current uncertainties on so this is the the experimental uncertainty these red lines are the pdf uncertainties and you see that all of the data are perfectly consistent with the pdf and experimental uncertainty so there is no indication of the compositeness so this should really be the the example of how in the future we will have to use data to tune and fit pdfs now few selected results on jet physics from from vlhc here for you to to look at afterwards is a breakdown of all of the possible channels here we have quark quark when i say elastic scattering quark quark elastic scattering what i mean is literally uh gluon exchange in the t channel okay so it's elastic not in the sense of proton proton elastic collision but in the sense that there is a simple exchange of it's not an amiculation process it's really the exchange of a gluon uh in in the t channel and that is by far the dominant process when we're looking at very high it is right so this plot starts from about one tv and we see that it is already already dominant the now this dotted line is uh so all qq is quark gluon so you see the quark gluon the contribution with the gluon is always very important and it becomes the most important the solid line here is the total so down here up to about uh what is it one tv the most dominant process is quark gluon if we were to go to even uh to even smaller et's we would see glu goes to glue glue as being the dominant one it's this one that's shooting up if we go down to about 100 gb as i said yesterday that is the dominant process but as we go to very very high et it's a quark quark valence valence scattering that that dominates and this is a comparison between uh data these are relatively recent data it's an analysis uh completed uh uh last year of the data collected uh in 2011 as you see already now today from the experience of the tevatron data are always plotted as a function of rapidity so there are data in the central rapidity region that go up to about 2 tv transverse momentum and these are the most forward data from the region between 2.5 and 3 rapidity this is really very very short angle now these distributions are scaled so that one can look at them by different factors so you see the data multiplied by 10 to the minus 3 10 to the minus 6 10 to the minus 9 so that they get really separated once you correct for these and you look at the actual uh range in cross sections that is brought by these measurements you will find that there is 10 orders of magnitude okay so from the the the most frequent jets that are being measured to the rarest jets of the highest energy there is a factor of 10 billion in rate okay and the solid lines that go through and these points that we see correspond to the theoretical a theoretical calculation from first principles with uh convoluted with the pdf known at the time and as you see in a log scale there is perfect agreement here again is for different energies seven eight and 14 tv this is the fractional contribution from different channels so core glue on glue glue and qq and qq prime scattering and again as i said before above about 1500 2000 gv core core scattering becomes the dominant one one thing which is important is that now these measurements are becoming very accurate at the level of 10 percent the theory is accurate from the qcd perspective also at that level so we could start doing measurement feats of pdf's at the 10 percent 5 percent level and then that's a stage at which one has to stop and ask whether indeed all of the possible effects are being taken into account weak interactions typically play a minor role in qcd physics because weak coupling is is weak but nevertheless it's not so much weaker right after all alpha s is of order 10 percent alpha weak it's only a factor of a few smaller than smaller than that so at the level of few percent should be obvious that electro weak corrections play a role and this is a plot that shows the impact of electro weak corrections if you do the calculation of the jet rate including or not including electro weak corrections which means exchanges of w bosons for example instead of exchanging gluons what we see is that already at one tv we get about it's a five percent correction and at two tv it's a 12 percent correction two tv is the range where already data are exploring things once we go to 13 tv this year we will be going up to about four tv and as we see 10 15 percent this is the effect of of a core gluon contribution see core gluon at a couple tv is about 15 percent that means that if we want to hope to extract the gluon pdf by looking at at large x by looking at the by looking at the data there is no way we can do it without incorporating a little weak corrections as well because a little weak corrections are as large as the core gluon contribution right and if we want to control the core gluon we we have to include that these are other plots in which there are discussions comparisons between uh systematics coming from different experimental effects and theories so i can i can skip that okay so this is uh so far we were talking about typically die jet final states once we produce die jets as we saw yesterday in the case of the plus and minus plus and minus goes to qq bar 10 percent alpha s percent of the times we get extra hard radiation there is a third jet we saw that yesterday in one of the pictures and in pp collisions of course that's the case as well and in fact not only the third jet but we also have a fourth jet the fifth jet the sixth jet we have so much energy available that the probability of a meeting radiation is uh is very large and the probability of a meeting extra jets is also enhanced in fact it's enhanced by much more than just alpha s because the moment we have an extra particle produced we have so much energy that there is an immense amount of phase space that's available and the integration of this phase space contributes to additional additional log factors that enhance the rate so these are plots that document the analysis one of the early analysis from from atlas and what we see up here on the top left is the inclusive jet multiplicity that means that they look at final states with jets they reconstruct jets they count the number of jets and the plot the rate and these are jets with energy uh there is no threshold here i believe it's like 60 gv if i remember correctly the the pt uh the pt the minimum pt of a jet yeah it starts at 60 gv and we see that we have rate here all the way up to six uh six jets in fact it continues all the way up to 10 10 11 with the data we have today other this and and this is compared against against theoretical calculations the yellow band is the systematic uncertainty of the experiment and statistical uncertainty and you see in the scale is uh you see it right we're talking about an agreement at the level of plus or minus 20 so we can describe processes with six jets in the final state with an agreement between data and calculations at the level of 20 if you want to calculate at leading order forget about next leading order at leading order a six jet final state it means glue glue goes to six gluons and i cannot remember the exact number but the number of five man diagrams is in the range of a million so you have to sit down and calculate over a million very complicated five man diagrams because each five man diagrams is two gluons going to six gluons the number of terms if you expand it is immense and you multiply that by one million and that's just a matrix element you have to take the square with all of the interferences right so these things can be done nowadays in a almost routine way thanks to developments advances over the past 20 years and in fact they can be done even at next to reading order once you do the leading order exercise it's peanuts compared to what it means to promote everything to next to reading order and again the techniques allow that to be done so this is in principle a topic of discussion and of not just one but of a whole series of lectures something that I will not discuss at all but which is rather well documented in in the literature and it is one of the most interesting fields of development in collider physics nowadays so it's not just the rates that you know we we get right you can experimentally explore the the distributions the energy distribution of the leading jet of the second leading jet the third up to the fourth and you can go beyond at some point the statistics of course will hit you and even for the fourth leading jet you see it's not just the total rate but really the spectrum is in perfect agreement with the calculations okay now here is a few considerations because you know typically we say you meet an extra jet that costs alpha s so the probability to have an extra jet is more or less a 10 percent than having one jet less and the question is is that correct I mean to which extent this is a meaningful statement and to which extent we can use these rules of thumb to estimate rates for complicated multi jet final states so I have a couple of slides with some considerations here what is plotted here and sorry and the conclusion of this exercise will be to convince you that one has to be extremely carefully using these rules of thumb it's true that higher order corrections are of order alpha s but unless we define very carefully what is the observable we can be making serious mistakes in our in our projections okay so we have to start by defining clearly what is our observable here my observable is a final state with a given number of jets two three four five in which each jet has a minimum transverse energy 20 50 hundred okay so the final state is defined by the jet multiplicity jets above a given threshold and now I look at the rate and I do it for two jets three jets four jets up to five jets these are numbers that come from a theoretical calculation okay these are not data this is just uh one of those calculations that was mentioned in before and here are the ratios of three jets over two jets four over three five over four and as you see the ballpark of these rates is indeed say about 10 percent in the case of four over three it's about uh you know seven percent ten percent thirteen percent depending on the threshold the higher is the jet threshold the more it costs to emit an extra jet it's rather obvious right because we want to have a higher emission so this is reflected in these slopes and on average for a jet of say 50 gv adding an extra jet takes indeed something of the order of alpha s it's interesting that if we look at the ratio of three to two relative to four to three it seems that it costs much more to emit the third jet than to emit the fourth jet okay and you may think that it's a bit strange because if you have two jets you only emit one if you have three jets and you want to emit a fourth one god you know you already asked me to emit a third jet now you ask me to emit also a fourth one that should be even more expensive where it turns out that it's not because uh it's only three percent of the die jet events above 50 gv it's only three percent four percent that has the extra jet while it's ten percent of events with three jets that have a fourth jet and what is the explanation for this again it's not alpha s it's purely kinematics because if we take a two jet event with two jets above a given threshold say 50 gv we have a jet here and we have a jet there and that is 50 gv each now if we want to have a third jet it will also have to be above 50 gv it has to go some place so we have to add another 50 gv so the minimum amount of energy that this is costing us is at least increasing by 50 percent the energy required because even if there is no longitudinal momentum the two jets go back to back 50 50 that makes 100 gv if we want to have three jets they have to be at least 150 right and I can put them as a Mercedes like you know I can put them like this 50 50 50 in the transverse plane now this is the beam line okay that's the way to minimize the energy so the minimum energy configuration has 150 versus 100 so it's 50 percent more energy if I have three jets above 50 and I want to add the fourth jet I only add 50 more and now it's 50 divided by 150 that's what I'm adding so I'm only adding 30 percent of the energy already I had to put a lot of energy into creating three jets and now to add the fourth one it's only a 30 percent increase while for two jets going to three it's a 50 percent increase okay and that's why the more jets I have the less it will cost me to add the extra one I have to pay a lot to get to have many jets but once I have them I can add the extra one with a smaller cost up to the point where I'm really saturating phase space of course if I am already using all of the energy I have available in the proton-proton collision there is no more energy left for the extra one and there I start paying okay so these are the considerations behind this table and now we go to a completely different final state and now we define the final state not by the number of jets and their threshold but by the total amount by the invariant mass of the system so now we are looking at a multi-jet system with a total invariant mass in the partonic system above say 100 gv 500 gv 1000 gv so what I'm doing here I'm asking the question what's the distribution in jet multiplicity of a final state in which I have say a gluglu collision going to gluons and jets with the square root of s hat equal larger than some number 100 500 1000 and with this constraint I go and I calculate what's the rate of 2 3 4 etc jets so let's look and and now I have defined my jets my jets are above 20 gv but I'm not varying the jet ET threshold I'm just varying the total energy so this is these are the rates as a function of 2 3 4 of jet multiplicity you see that in the case of 100 gv threshold it's much more likely to have two jets than to have three four or five jets in the final state okay so indeed the two jet configuration is dominant if I go to 500 gv we see that now it's the free jet configuration which is dominant in other words if I put 500 gv into my system I'm more likely to come out with three jets than with two jets and in fact even three jets four jets are more likely than two jets and if I inject now one td in my system you see that two jets are peanut so I pretty much never find two jets I'm much more likely to find five or four or three okay so if we inject all of these energy you know the leading process is indeed the the two to two scattering if you wish but there is so much acceleration because I have so much energy that nothing prevents this quark from or gluons from radiating and the radiating of radiation with great probability and that radiation being emitted itself will form will form jets okay so as you see we get a completely different picture in the structure of a final state if we if we look at it using this observable okay so it's not true that two jets are more likely than three than four than five etc it depends on what you ask of the of the final state yes question so these are below why are they below they are below because I believe I put that line in the wrong place it's just it's this is an old version of keynote that I imported the slide from before and occasionally there are glitches like that so it is not physics okay in fact I was looking at it I said what the hell is going on but any more questions or deep doubts is this clear it's clear why it's important it's important right because often you know many branches of of high-energy physics there are things that one can do on a you know the classic back of the envelope right in plus and minus there are many things that can be done because at least the initial state is under control in you know some b physics in dk's one can you know do a matrix element one can calculate the kinematics things are much you know systems are much more constrained than it's there are many opportunities to get estimates as out of the back of the envelope calculation in Hadron collider physics everything is much harder right because everything you have to go through the pdf's but that you know you don't have in your head it's a cool everything is a computer code so but occasionally there are things that one can control analytically the example of a child pdf we did in the first lecture is one and then everything else one has to develop some sense of you know experience right it's like culinary art it's like cooking right you can read everything about how much salt you have to put and these and that how long you should cook but it's only by doing it for a long time that then you develop with this sense so it's important to look at the numerical results trying to put them in perspective and learn learn the skill of predicting what will happen in a given situation if i put a given set of cuts how will that impact that maybe you can get counter intuitive results such as three or four jets having higher rates than two jets for very solid reasons right i mean the reasons here are very simple but you have to think about it at least at least once okay now yes question you shouldn't repeat the question in the back from because i couldn't hear does the send the the energy determined whether your jets are collimated or is it only a function of the send of the kinematics of your system of course yes the question is whether it's just in very mass accounts or also the way we define the jets and if there is some minimal separation for instance everything counts of course this specific number here it's not written and you're right i should have written it there is an assumption on what is the cone for example that defines the jet and there is an assumption on how collinear the jets can be if i allowed the jets to be very very very close to the point that to me individual particles are jets then of course the jet multiplicity would be huge because it would be the multiplicity right here the jets are quite separated it's probably 0.7 in this delta r separation so that they are about say you know 40 degrees from from each other at least separation in in in space you know of course if one were to separate them more then needless to say these numbers would change but the picture is more or less qualitatively remains the same okay so these were global properties of the jets so we look at the jet as if it's a single entity oh i'm sorry sorry yeah in previous slides like two slides back another yeah in 50 gv why it is harder to produce fifth jet than fourth jet i mean there's 10 percent and there's 90 percent and without considerations it was easier well there are two reasons one is that at this level it could just be you know statistic you know the statistics of the calculation right so 0.9 and 0.9 and 0.1 plus or minus the uncertainty the calculation is the same but you see it's exactly the same here 0.13 0.13 0.71 0.67 it's slightly larger i think that the reason here is simply that we start hitting phase space because now we are asking five jets above you know 50 gv so at that point we really run into the fact that pdfs go down so we cannot arbitrarily add at some point this increases to stop because once we put so much energy in the system that there is no energy available we pay a lot of penalty to have the extra jet right so it's kind of a turnover and this is probably the crossover point but i think that this is really more like i should have maybe calculated the numbers more precisely so that this ratio were really that plus or minus one per me i think that it's like 10 percent with the uncertainty okay so as i said those were jets treated as single objects we just look at the energy of this cluster of particles what is very interesting is what happens inside the jet we discussed these in somehow yesterday as well we want to know how many particles there are inside what are these particles what is the momentum distribution when we talk about the energy of a jet we don't care whether that energy is carried mostly by a couple of particles or whether it's equally distributed among them and this has a crucial impact on two things on one side experimentally experimentally if we have one pion of 10 gv or 10 pions of one gv the experiment measures a different quantity okay because they interact with the detector in a way which is not exactly linear so the way we translate the measurement of the of the calorimeter into a total energy depends on whether it's 10 pions carrying one gv or one pion carrying 10 gv the other point is that the features of the way the jet develops depend on the nature of a jet if it's a gluon jet the gluon has more charge it will radiate more so there will tend to be more particles inside the jet than if it were a quark jet and a quark jet say a b jet has different features than a u an up quark jet because the b jet contains the b meson the b meson decays as a lifetime as we saw yesterday in mackerel peskin's lecture all of these features of what happens inside the jet are crucial to the give crucial information so the first thing to ask ourselves after we analyze the global properties inclusive properties of jets is whether we understand more exclusive properties of jets what's inside and these we do to start with by looking at what is called the jet fragmentation function the jet fragmentation function is the the distribution d n by d z where z where n is the number of tracks tracks inside the jet and z is the momentum parallel to the direction of the jet carried by a given track relative to the jet energy or transverse energy okay so we look at the various part we measure the jet energy we look at the various particles their momentum and the fraction of the jet energy carried by a particle is the variable z and we plot the n by d z it's a bit like the pdf's we're looking at the fraction the number of quarks carrying 10 percent of the protom momentum that's the pdf the number of particles carrying 10 percent or some percent of the jet energy is the fragmentation function and these are data compared against so this is done for jets of different energy higher energy jet of course are more accelerated so they radiate more radiating more that we have a higher multiplicity but having higher multiplicity means that individual particles will be carrying smaller amount of energy okay if I have a high energy jet instead of the multiplicity as a function of the of the jet the jet e t the multiplicity will go approximately logarithmically divided by some some scale that's the multiplicity like the average number of charged tracks in a jet it grows logarithmically because logarithmic is the probability of emitting of emitting a radiation of emitting gluons so as we go to higher e t there will be more particles more particles will have to share the given amount of energy so the average momentum fraction carried by each individual particle will be smaller that we have higher energy but z on average will be will be softer and and that's why we study this distribution as a function of a pt of a jet and if you were to look more carefully because this is a very compressed log scale you would indeed find the effect of scaling violation which is which is given by logarithmic dependence and this is the comparison between the data and the calculations these are calculated the different color points correspond to simulations obtained using different Monte Carlo, Spithia, Herwig, Sherpa and as you see again within perhaps 20 percent there is a very good agreement you can also say that within 20 percent there are discrepancies given that the data are very accurate and that means that this is a very useful observable to improve even more the Monte Carlo simulation so different Monte Carlo different shower part on shower Monte Carlo's give different predictions because there are slightly different ways to approach for example hadronization in the final stage and they reflect themselves in different predictions which at the level of plus or minus 20 percent can be improved the other thing that is of interest is to look at the at the mass of the jet itself the jet is born as a single as a single quark okay and the mass of a single quark or the gluon is equal to zero lambda qcd if you wish but it's a small number on the other hand as soon as the quark is produced and radiated by being accelerated it radiates radiates it evolves and at the end of the day we get all of our pions okay so the final state address if we look at the invariant mass now at the system so we do the sum of the momenta of all of the particles inside the jet square this is what we define as the mass of the jet square and this number is obviously greater than zero right so it's born as a quark at the end of the day it gets a mass and of course it's because somehow it's quantum mechanics right the final state the mass is not something that gets that gets preserved in the course of the of the evolution so it's of interest to because it's a probe of how well we describe the evolution of a jet to plot the invariant mass of a jet again as a function of a jet et and these are different distribution of the of a jet mass as a function of jet et is i believe again compared against compare against against the data and the agreement is is excellent the one place where this is of interest is when we go and we look at very high energy jets and we ask ourselves how do i know is there any way that i can use the information on the structure of this jet to learn about what was the origin of the jet because i drew this for a quark but how do i know if i just look at the final state particles but everything started from a quark or maybe started from a gluon or maybe it started from something even more exotic it could have been a top quark for example a top quark of course which is very slow the case to a w and a b so it's a very clearly reconstructable object as being a top but if this top is boosted to very very high momentum of course it decays to a w decays to a b which are almost collinear the w decays to a q q bar pair and at the end if it's very much boosted everything and up being ends up being inside the same the same jet so it's only by looking inside that i can learn something about the origin and one of the things that i need to know is really how the mass distribution of all of the particles inside the jet behaves and this is an example of a recent measurement you see 2014 from atlas in which they studied configurations of you know two jets or say a w recalling against jets relatively fat jets at high energy they go and very construct the mass of a jet and if it were only qcd jets there would be a given shape which is v's and you see there is a little bump here relative to what the expectation is and this bump corresponds to an invariant mass in the range of 80 plus or minus 20 gv and what that is is indeed a contamination coming from w's being produced at large pt w's with decay to qq bar so when we look at the jet and we just see addrons with a given energy you know we know nothing of where it came from if we look at the invariant mass of the system indeed in principle we can reconstruct some excess that corresponds to the production of a w so this is the observation of a w decaying to jets something that experimental is very hard to detect in a hadron collider by using these these tools and this is the boosted jets techniques that michael peskin was describing was mentioning yesterday okay one can use these observables to go back the jet spectra angular correlations and again ask the question do i see a possible substructure of quarks or are quarks still point like typically what one plots is distribution angular distributions expressed in expressed as a function of this variable chi it's a strange variable it's one plus the cosine of a scattering angle divided by one minus the cosine of a scattering angle as an exercise it's a trivial one just prove that this decay this distribution is nothing but d cos theta divided by sine to the fourth theta divided by two now why is sine to the fourth theta divided by two interesting what is it that goes like one divided by sine to the fourth theta divided by two what's the distribution what does one over sine to the fourth theta divided by two remind you of rather for very good exactly so rather for is the exchange of a photon of course but if we have the exchange of a gluon it's still exactly the same nothing changes because it's a spin one particle it's a gauge interaction so core core scattering is dominated by single gluon exchange so it has to obey the the rather for distribution and if we plot the n by the eta we should be getting something that if it's just rather fold would be entirely flat and these distributions done for digit final states of different digit invariant mass are pretty much flat on the scale that we see here there is slight deviations from being flat which are due to the acceptance of a detector the fact that it's not only t channel exchange there is also contamination from gluon fusion s channel u channel of course but when we compare the data against the expectation of the exact ucd calculation there is perfect agreement and the conclusion is that up to the scale of one divided by ten tv quarks still behave as elementary objects there is no evidence of compositeness so i have now a set of slides dedicated to the physics of a top quark let me just start i doubt i will be able to to finish it and then we can start tomorrow i have this is endless the lecture i have today is a gazillion slides and i don't have to go through all of it we just so top core production the domino production modes are glu glu fusion there is also a contribution coming from quark hantai quark in vs channel this is typically at the proton proton collider this is typically small and what we see here is for the lhc of the different lhc energies seven eight and fourteen tv this is the fraction of the tt bar production rate that comes from glu glu collisions as a function of the invariant mass of the tt bar pair so we're looking at tt bar final state we're looking at the production of threshold but also a very high invariant mass which is interesting because you know one is exploring maybe resonances exotic objects decay into tt bar and what is displayed here is the fact that up to very large tt bar invariant masses we're still dominated at the level of say 90 percent by glu glu initial state so even though as we go to very large x the glu one becomes smaller and smaller relative to the quarks still glu glu initial state dominates tt bar production all the way up to very very large x and the reason for that contrary to the jets right with jets we saw at 100 gv it's mostly glu glu goes to glu glu but as soon as we are above one tv or two tv it's all quark quark well the key difference is that here the alternative to glu glu production of tt bar the alternative is quark anti quark but the anti quark there is much less anti quark inside the proton than there is glu ones okay because the anti quark comes from very good on splitting so we have the anti quark is alpha s times the glu one okay so glu glu still is a winner over qq bar in addition there are other dynamical enhancements qq bar goes through the s channel so the cross section goes like one divided by the mass square of a tt bar pair while glu glu there is a t channel exchange so the cross section does not go like one over the mass of a tt bar pair square the cross section goes like one divided by the mass of the top quark squared and it's a fixed number okay the dimensions are given by the mass of the top so the cross sections the corrections due to being at large mass affect higher order terms so these are important ingredients these incidentally means just because the glu one is so important that we can use the tt bar cross section measured in a very precise calculation of the tt bar cross section in order to extract information on the glu one density okay again assuming that there is no new physics entering in tt bar production which has to be verified this gives you a picture of how accurately we can calculate the tt bar cross section this is the cross section at at leading order at next to leading order in qcd at next to next to leading order so you see the precision the rate grows slightly as we go as we add more terms but the precision increases dramatically these red and blue lines correspond to refinements of the fixed order calculation they come from very summation or soft blue ones it's theoretical refinements that as you see improve slightly the the accuracy so right now down here we are at the level of plus or minus three percent coming from the missing you know the not knowing higher order corrections beyond next to next to leading order there is a pdf uncertainty which is also the level of two to three percent and then there is some parametric uncertainties the value of alpha s the value of the top mass of course we don't know exactly exactly and that leads to some uncertainties but you know overall it means that theoretically we control the tt bar cross section to the level of perhaps four or five percent which is comparable to the experimental well experimentally they can do even better now one of the important things about the top is its mass in relation to the w mass that was also explained yesterday because they enter in all of these precise electrode precision tests of the electro week sector okay those oblique parameters that were discussed yesterday s tu you remember they were driven by the value of the top mass which appeared quadratically in several places and the value of the of the of the Higgs mass but through the electro week symmetry breaking the w mass is is a key player in that in that test as well now in fact the Higgs mass is known much much more accurately than than we can then we know for example the top quark and the leading uncertainties arise indeed from the w and the top and the top mass so to improve the measurement of the top mass is very important for what concerns the w mass i just want to point out one the measurement was was partly illustrated yesterday in in peskin's lecture one thing that it's worth noting is that say at the table throne but the same is true at the lhc the dominant source of the systematics today this is for example for cdf this is a list of all of the systematics and the single largest systematics is from part on distributions is the knowledge of the pds okay and the reason is that as we there is a the observable that is typically used is this transverse mass of the w that is this jacobian peak and the extraction on the information of the mass comes from a global feat of this distribution so it's not just the position of the peak that counts but it's the global shape and down here the region of low transverse mass is the one dominated by low pt low pt leptons low pt it's low pt because maybe the w has decayed with an angle so the very mass of the lepton plus neutrino system is always the w mass if the w decays at 90 degrees the lepton has pt equals to the w mass divided by two but if the decay of course is tilted then the transverse momentum will be smaller okay so the region of small transverse momentum correspond to the region of small pt that means forward emission and the experiments can only measure cannot measure all the way down to angle equals to zero they only measure in the central region so the details of the rapidity distribution of the lepton of a given pt are important in order to do the feat of this distribution but the rapidity distribution of the lepton as we saw the very beginning of the lecture is driven by the pds so information on how the w is boosted back and forth uh is crucial in order to do a proper feat of the of the w mass and that's why there is an important systematics that is left there and this systematics is pretty much the same at the level of plus or minus 10 mv at the at the lhc so i think i will just give you one final remark and then i stop here we discussed the precision of the top quark and i said the the mass of a top quark is an uncertainty in predicting the top cross section and therefore we can turn it around we can say if we had a very precise estimate of a top cross section we can use it to pull out the top mass now and this is an idea that that's being put forward by several people in these days i just want to to point out that the dependence of a cross section on the mass which is given here d sigma over sigma is about five times delta m over m or vice versa implies that if we want to have a one one percent measurement of the mass we need to control the cross section to a five percent level and five percent is pretty much the accuracy that we have today so if we have a five percent systematics we get the one percent determination of the top mass but one percent of 175 gv is about two gv this slide is old at the time when two gv was the direct measurement of the top mass now the mass of the top is known to about point seven gv say of the order of one gv so in order to improve on that in order to push the top mass precision below one gv we would need a measurement of the cross section with a total uncertainty including the experimental systematics and the theoretical systematics which is better than 2.5 percent and that's something that's very very very hard to imagine we will be able to to achieve we can possibly go down to say two percent but that again will only be one gv so if we really want to push the knowledge the measurement of the top mass to a precision of the order of few hundred mv is this is not really a good way of doing that and we do have to measure it directly something that can be done at any plus or minus collider operating at the t-bar threshold something that in principle can be done in hadron collisions it's very complicated for reasons that i can maybe illustrate tomorrow i can start tomorrow's lectures discussing this because it's a rather interesting topic so i stop it here