 Well, yesterday we did a lot of ancient history, which is to say things that happened before you were born. Today we're going to do some modern history, which is to say, how many of you were born before 1990? Oh, okay. That's more of you than I thought. In any event, things that happened when you were quite young, but which definitely should not be forgotten. So today we're going to talk about the precision study of electroweak interactions, and in particular the experimental evidence for the SU2 cross U1 structure of the weak interactions. And it's, as you see, a quite interesting subject, not only from a theoretical point of view, but also from the point of view of the experiments, which were very beautiful. So we'll get to that in a moment. The first thing I wanted to do today is to write some, oh, let's see, before I begin, I should point to some references. Someone asked me yesterday if there were lecture notes. And I didn't make up lecture notes specifically for this set of lectures. However, I have on my website some lecture notes, which I asked Ms. Van Buren to refer you to. So if you go to my web page and then this site, GGI 15, you'll find lecture notes for some lectures that I gave at the GGI earlier this year. And if you go to the analogous site called Physics 152, you'll find lecture notes for a particle physics class that I gave at Stanford. And I think if you mix and match the relevant topics, you'll find most of the information that's covered in these lectures. In particular, the Wednesday and Thursday lectures will be very similar to the lectures five and six of the GGI series. So hopefully if you want to read and see the derivation of the formulae at a slightly slower pace than I did it yesterday, that's probably a good place to look. If there are some things that are missed out, you can ask me about that. Are there any questions from yesterday before I proceed? If any of you ask me questions informally, that's fine. But anyone have any questions that are still unanswered that you'd like to air in public? Okay, well, as I say, I'll be around the whole week, so take advantage of that. Okay, the first topic for today is that I'd like to write explicitly the formulae for the masses and especially partial widths and branching ratios of the W and the Z. Now the W and the Z are some of the basic objects of LHC physics. And I think those of you who want to do phenomenology probably want to actually memorize the numbers that I'm going to calculate now and just keep them in your head for handy reference. It's just essential to have the concrete picture of this in your mind when you do almost anything that concerns LHC. So let's start with the W. The WDKs to Enum, Unu, Townu, UD bar and CS bar, of course, it's not massive enough to decay to TB bar. And there are also Kibibo suppressed modes that I'm not going to talk about. Once again, everything that involves a CKM angle is relegated to Ligeti's lectures. The partial widths for these DKs are very easy to calculate and maybe I should just do that. For example, W to E plus nu, the matrix element has this G over the square root of 2 that we derived yesterday, a U dagger for the neutrino, sigma, V for the electron and a polarization vector for the W boson. This quantity we can compute in a frame where the neutrino goes forward and the electron goes backward. It's, I hope, well known to you that this object is equal to the square root of 2 times the polarization vector for the relevant spin. So the neutrino is left-handed, the electron is right-handed. So the polarization vector that we need is, oh, sorry. I should have done this the other way around. Please excuse me. Let's put the positron going forward because that's the observable particle. The neutrino is left-handed. The polarization vector is this. It's in the final state so we complex conjugate it. We would dot it with a W polarization vector. And the value of that matrix element for the W polarization vector that overlaps with this is G over the square root of 2. Oh, and sorry. Please excuse me. And then 2E electron 2E nu, which is Mw. And then there's the square root of 2 here. OK. So that's pretty easy to find. The partial width is then given by 1 over 2Mw, 1 over 8 pi, a polarization average, and gMw squared. And I guess that all works out to alpha w over 12 pi. Sorry, over 12. The pi is in the alpha w times Mw. And this will be basically the same for every one of these decays. So we'll get a factor of 3 here. For these two, we'll get a factor of 2 and some QCD radiative correction. This factor is about 3.1. When you put everything together, you get a width for the W that's about 2.1 GeV. And that's an interesting number just to have in your head when you think about the detection of the W at the LHC. There's one more very interesting. Oh, that also gives you the branching ratios. So the branching ratios are about 11% for these three modes and 34% each for the hadronic modes. So very important. Most of the decays of the W are hadronic. At the LHC you can find these things when the W is boosted using some techniques. I don't know if Michelangelo is going to talk about boosted or not. But there are techniques to differentiate boosted Ws from jets. And so you have access to these. Typically the first round LHC analyses used only the leptonic modes and mainly just the in-mute. So 22% efficiency for branching into those modes. There's one more very important aspect of the W decay which is its angular or polarization structure. So this is something that I was emphasizing yesterday but now let's come back to it. You can take this vector here and dot it with W polarization vectors. So for the right-handed W the vector is this. For the left-handed W in the rest frame this. And for the longitudinal W this. And what comes out of that are angular distributions which would be if this is the W direction we bring the W to rest and then the lepton goes off here and theta is this angle. The angular distributions are characteristically if you just dot these vectors into this one 1 plus cosine squared theta, 2 sine squared theta and 1 minus cosine squared theta. So it's very interesting the direction of the lepton identifies the polarization of the W which is often very important. If the W for example came from a Higgs interaction it would typically be longitudinally polarized and its distribution would be central in cosine theta whereas more ordinary Ws just radiated from quarks would typically be transversely polarized and so one can distinguish them if one can reconstruct this angle. So that's often a very important strategy when you do LHC analysis. Maybe I should remind you that these angular distributions are the same ones that appear in electron-positron annihilation and just over here I'd like to record these. d sigma d cosine theta for electron-positron annihilation into let's say mu minus mu plus depends on the polarizations of all of the fermions that are involved. If I have left-handed electrons annihilating to left-handed muons this is a 1 plus cosine theta squared angular distribution the right-handed electrons annihilating to left-handed muons give you a 1 minus cosine theta squared distribution. Actually these distributions are the same distributions as the 1 and the 1 minus y squared that appeared yesterday when we talked about neutrino scattering it's just the same expressions in different kinematic variables. But this will be rather important to us later in the lecture. Please notice that if you have massless fermions annihilating you can't get to the longitudinal polarization state. You always make a transversely polarized virtual photon or Z boson which then has either this or this angular distribution when it decays. So keep these formulae in mind. If I start erasing them yell at me and we'll come back to that. Okay, so maybe this is a good point to just make some notes about some other parameters that maybe are things that are interesting to keep in mind and even commit to memory to do LHC phenomenology. It's very cool to me to write the dimensionless strength of all the interactions of the standard model. And so let me do that here. So alpha is always g squared over 4 pi of the appropriate couplings. I'm going to look at everything at a q squared of mz squared which is maybe a little low but at least much more appropriate for LHC than the standard place of lower energies. Alpha is 1 over 128. Alpha weak, which is to say g squared over 4 pi, is 1 over 29.6. Alpha prime, 1 over g prime squared over 4 pi is 1 over 98. And one should add to this alpha s is about 1 over 8.4. So it's interesting with those numbers in your head you have right in front of you the dimensionless strength of all the interactions that appear at the LHC. Well, there's one more really important one which is alpha t is y t squared over 4 pi which is 1 over 14. So the top quark is actually quite strongly coupled on the scale of the other interactions. It's not quite as strong as QCD but that's also something to keep in mind. In any case, we can use these numbers to evaluate, for example, the formula that I wrote here and all the formulae that will appear later in the lectures. Yes, I'm sorry? Oh, I'm sorry, this is g prime squared over 4 pi. So the relation between this and this is tan theta w. Okay. Okay, now with that in mind we can go on and let's see, just extract the other weak interaction parameters. G fermi over 4, well, let's see. Yesterday I wrote the relation g fermi over the squared of 2 is g squared over 8 and w squared. So, please excuse me, this is 1 over 2 v squared and from that we get directly the, at least at lowest order in Higgs interactions, the Higgs vacuum expectation value of v of 246 GeV. So that times these couplings sets the scale of everything in the standard model. Mw, as we saw yesterday, is GeV over 2. This works out to 80 GeV and mz is GeV over 2 in the standard scheme divided by cosine theta w. This works out to 91 GeV. So part of the Precision Electro-Week program would be to measure these two quantities extremely accurately and we'll talk about how people do that. Eventually this number is going to be very important to actually extract the value of sine squared theta w with maximum precision. Okay, so that's a quite straightforward exercise. The next thing that we have to do I think is to do the analogous thing for the z. And in fact that's very easy because the algebra of working out the matrix element is almost exactly the same. So when we look at the z matrix elements we just have to look at this expression here and see what has to be replaced. So basically this factor of the square root of 2 no longer appears. So for the z there's a times 2 and there's also for every chiral species this quantity qz squared where qz is the isospin minus the charge times sine squared theta w which we talked about yesterday. And so very quickly then you can get to the following formula for a z partial width. z to some chiral flavor is alpha over 6 times mz squared times qzf squared where, please excuse me, qzf is i3 minus q sine squared theta w. Now for the w these numbers were just one but for the z it's quite different because the various members of a generation have all kinds of values of i3 and very different values of q. And so to work with this formula it basically is interesting oh and please excuse me one more thing I left out there's an extra 1 over cosine squared theta w in the couple. Now it's the correct formula. Okay, so now we need some systematics of the qzf's and I like very much to express that in the following terms. Let's define sf to be the qz for the left-handed particle plus the qz for the right-handed particle squared. And I'm also going to define af to be the asymmetry that is the thing that's proportional to the rate of predicting the left-handed particle minus the thing that's proportional to the rate for predicting the right-handed particle over the sum of those quantities. So now it's illuminating to make a table. And so let me do that here. So I'm going to write the flavor here the, oh man I boxed myself into too small a corner. I'm going to write the flavor the qzl, the qzr the numerical value of sf for sine squared theta w equals 0.231 which is pretty close to the correct value and the numerical value of af. The flavors are neutrino, electron, u and d and then they're repeated for each generation. Here I'd have a half here minus a half plus sw squared here a half minus two thirds sw squared and here minus a half plus one third sw squared and then in this corner we just drop the isospin so these things may or may not exist but whatever they are they don't couple to the zebo's on sw squared two thirds sw squared and one third. Yes, okay. I'm sorry, I'll wake up sooner or later. I'm sorry. It's over here this is correct it's mw to the one. Good. Okay now the numerical values. So the numerical values are basically really different. They're somewhat different in this column for example sine squared theta w is close to a quarter so this is about a half this sorry so this is about a half this this is also a half and when you square it and add them up you get about half the strength for neutrino one eight five. So interestingly the neutrino the one particle you can't see turns out to be the particle that's most strongly coupled to the z I'm sorry about that but that's the way nature works and the asymmetries are also very interesting here the asymmetry is one of course very difficult to observe these two quantities are about minus a quarter and a quarter so they're almost equal this one's a little larger with a 15% asymmetry here the asymmetries are larger 0.67 and in this column one third of sine squared theta w is a tiny number and you square it so the asymmetry is almost maximal 0.94 and so as opposed to the w case where everything kind of comes out equal the z case is very different every flavor within a generation kind of has its own personality with respect to z decays and these numbers are if you look at them in detail all over the map the numbers are all generated by this formula which is the result of SU2 cross U1 gauge invariance and so it would be totally amazing if you could actually measure all the numbers in this table and see if they come out right and so this is basically what was done in the 1990s at CERN and at SLAC one built large colliders dedicated more or less specifically to this purpose and tried to do experiments that were as accurate as possible so let me just show you here we talked about this yesterday let me skip ahead a few slides I'm sorry, here we go so this is a slide that Michelangelo alluded to in his talk but it's interesting to see what the actual data looks like E plus C minus annihilation in QED as you know the cross section is like one over the center of mass energy squared so basically the cross section falls off as the energy gets large plus at 91 GeV there should be a Z resonance in this process E plus E minus annihilating to Z and you should see an enormous enhancement and indeed that's what the data says the cross section goes up by a factor of a thousand and if you build an accelerator specifically to operate at that energy you get an enormous bonus and a large event sample and that's where basically all Zs, on-shell Zs decaying at rest in the laboratory and so you have the ability to test the predictions that are in this table in the 1990s this was done with two dedicated accelerators the original version of LEP at CERN which accumulated among four experiments 12 million ZDKs an accelerator called SLC at Slack which accumulated a half million ZDKs but also with a high degree of electron beam polarization whose role we'll see as we discuss the experiments so now what we can do is to discuss how the numbers in this table lay out in terms of observables that you can actually measure in this environment electron-positron annihilation in defermion pairs and we'll try and get some idea of how this can be viewed directly experimentally so maybe the first thing to work out is what are the partial and total widths of the Z numerically this is not so hard to do from the formula that I wrote over there after all of your corrections, thank you very much so it's alpha WMZ over 6 cosine theta W and then the contributions are three neutrinos three charged leptons two up quarks three although I'm going to write 2.98 with some ulterior motive down quarks I'm getting a QCD correction and when you work it all out you get 2.49 GEV for the Z width of course this is a zeroth order with no radiative corrections but in fact what we'll see is that this number is already in very good agreement with the experimental value and the radiative corrections turn out also when you compute them to be relatively small now how do we test this prediction well probably the easiest way is to actually measure the line shape of the resonance which is the Z the Z appears in a propagator as 1 over S minus MZ squared MZ gamma Z so it's a traditional kind of bright vigner resonance it's relatively narrow, this number is much smaller than the Z mass of 90 GEV so this is a very good starting point for an analysis however there is an important complication which is true for the Z and for anything that we're going to measure precisely in the future Michelangelo emphasized yesterday that the de-glape or ultraliparizia evolution of the initial state is very important in proton-proton scattering but actually here it also can't be ignored in electron-positron scattering in particular when electron and positron annihilate to the Z you have the possibility of radiating almost collinear photons and these photons have logarithmically enhanced rates of production for the same reasons as in Michelangelo's talk yesterday initial state gluons have an enhanced production by logarithms in a high energy proton-proton environment so something that you have to do is to actually compute this radiative correction and apply it to the bright finger line shape sorry, I told you I was leaving something for the future in writing that oh, the question was why did I put a non-integer quantity here and the answer is I'm going to answer that question in about five minutes please excuse me, I just did that to tease you but I will give an answer to that question okay so what you need to do then is to fold the bright-vigner cross-section with some appropriate parton distributions for as it were photons which are partons of the electron generated by the same kind of processes that Michelangelo talked about yesterday the formula was actually very pretty it was first worked out by Fadin and Karayev the relevant parameter beta is oh, I didn't write it down, alpha over pi times the logarithm of s which is basically mz squared here divided by the mass of the electron squared minus one and the formula is the integral dz of beta times 1 minus z to the beta minus 1 there's some first-order correction to this there are higher-order corrections as well times sigma of s times 1 minus z sorry, good so a fraction z of the energy of either the electron or the positron is radiated in this way this is effectively the convolution of two parton distribution functions representing these emissions of course the electron is an elementary particle so we can also compute the initial condition in the sense that Michelangelo described yesterday the function quite precisely in fact the function is known to two-loop order I didn't write the whole expression that actually has a dramatic effect on the shape of the resonance first of all it basically causes the resonance to slosh out to large values and gives it a long tail and this height is about 75% of the nominal resonance height so you have to take it into account when you do the measurement but when you do the measurement so this now represents hundreds of man years both on the theory side and the experimental side you get a comparison of theory and experiment which looks like this I always like to say this is a totally amazing oh by the way, you see those little red points that's the data, some of you have sharp eyes will notice that there are error bars on those data points this is a totally amazing confirmation of the standard model because the detailed structure of this curve contains all three of the parts of the standard model the QED of course through the calculation that I've just described the strong interactions through the factors of 3.1 that appear to enlarge the width of the Z and the weak interactions because one also has to take into account radiative corrections to this vertex which involve W and Z you put it all together, it just works spectacularly well here is this is the result of the OPAL experiment, here's the official compilation of the four LEP experiments and actually this curve nicely shows the effect of the radiative correction which as you see is an order one effect the results are that the mass that is to say the nominal resonance position and the width of the resonance are extracted to fantastic accuracy MZ is 91.1875 MZ is 91.1875 with an error of 2 MeV the width 2.4952 with an error again of about 2 MeV and in exceptional agreement even with the simplest estimate that you would get from the standard model and maybe I should say in better agreement as you include the electro-weak radiative corrections in addition to maybe I should just say and I won't discuss this further but those of you who would like to discuss it offline who don't know this story it's very interesting to get this number with the accuracy that's quoted involves a lot of detailed accelerator physics you have to do the energy calibration of the LEP ring which is in what's now called the LHC tunnel it's a 27 kilometer tunnel so you have to understand the strength of magnets in a device that you built which is macroscopically large and some very interesting subtleties arise that I'm glad to talk to you about offline but it's just a tremendous achievement to get to this level of accuracy in this quantity ok in addition to the total width one can actually experimentally verify some particular partial widths the way you do that is basically you look at the events and I think I included in these slides I won't go through them in much detail but there are several events that were recorded by the SLD experiment at Slack those of you who know something about elementary particle detectors will immediately see that this is an electron positron event this event is tau production and this event is two jet production which is QQ bar production those of you who just can't look at it and say that your background needs to be improved a little but I'm not going to say more about that here for the moment by the way this is the angular distribution of the quark jets measured by olive at the z this is supposed to this function which is you notice cut off at the midpoint is supposed to be a 1 plus cosine squared theta and by I you can see that that again the zero thought result is just a beautiful fit to the data that's provided here with very high statistics now some of the quantities which you can extract from the theory and then try and measure are first of all the leptonic fraction that is the branching ratio to hadrons divided by the branching ratio to one lepton pair let's say muons it's predicted by this theory at lowest order to be 20.6 the experimental value is 20.7 six seven with an error in the one two three fourth decimal place and another quantity which is very interesting is Rb that is to say the branching ratio for z to bb bar divided by the branching ratio for z to hadrons so here we're going to look at the events which are jetty events so events where the z is decaying into jets and try and figure out what fraction of those in what fraction of those does the z decay to bottom quarks it's a somewhat interesting business and I wanted to start by looking in more detail at one of these event pictures this is an event identified as z to bb bar it's very hard to see from this particular event display view what you see is just z decaying into two jets however in the lep experiments and in particular in the SLC experiment there were precision silicon devices located a few centimeters from the interaction point and in this particular event the view of the tracks as measured by those devices looks like this the scale is centimeters and as you see there's a fraction of a centimeter displacement for this I guess I really do need the pointer there's a fraction of a centimeter displacement of this vertex and this vertex from the primary interaction you all know that the lifetime of the z of the b quark is one and a half picoseconds which is to say half a millimeter and then you get to multiply it by the Einstein time dilation and so something that's a fraction of a centimeter away from the interaction point which decays to a secondary vertex you can identify as a bottom quark this was done very precisely at lep and here's the distribution of that unfortunately this is a variable called tagging variable not centimeters but it's the same idea something which is displaced from the origin on this plot is something which is significantly where the vertex is significantly outside the position of the primary vertex something that's done in this analysis which is very interesting is to sign the vertices so here's the primary vertex it produces a jet which goes in this direction if there are tracks in the jet that extrapolate to a point in front of the vertex you assign that as positive if there are tracks in the jet that seem to form a vertex behind you give this a negative sign this can occur if a bottom meson actually moves from here to here and decays this can occur only by mis-measurement and so if you compare the negative side of the diagram to the positive side of the diagram you can actually measure the experimental error associated with the vertex finding the results are Monte Carlo for UDS and then the slightly hatched region is the Tron quark and then the bottom quark and the whole histogram is fitted to the data as you see extremely precisely so if I find the right piece of paper we then get a very precise value of this quantity Rb which turns out to be 0.2162966 this is a composite of the four lep experiments now what's the value of Rb in the standard model well one can just please excuse me ignoring this factor here just calling it 3 for the moment construct Rb from the quantities I've given here and the value is 0.220 which is in large disagreement with this number however it turns out that there are some important radiative corrections that can't be omitted namely the correction where the Z turns into a top quark or where the Z turns into a W pair which then by exchange of a top quark will turn into Bb bar the contribution of these diagrams is about 2% it's proportional to alpha W times m top squared over m W squared so it's enhanced with respect to the usual scale of one-loop electroweak corrections and it's negative so when you subtract it from this number you get a number very close to the one that I showed here theory and experiment are about one sigma apart for this quantity now this is something I think to raise your antennae on when you talk about beyond the standard model because in theories of new physics beyond the standard model it's also very straightforward to generate corrections to this vertex for example if there's some heavy top heavy vector like top quark as you would have in composite Higgs models you can typically make a diagram that looks like this with some kind of exchange here and get corrections to this vertex that are of the same percent level order with respect to the standard model and in fact in the mid-90s this number was actually measured incorrectly due to I think the confusion of charm quarks for bottom quarks and there was a large flurry of theoretical literature on how you could generate plus 3% and 4% corrections to this vertex very straightforwardly in models of composite Higgs, technicolor, etc so this is now a very important constraint on physics beyond the standard model and this is something I think that Marcus Lutti will probably come back to in his talk okay well now you're doing good now we've talked about the SFs now I'd like to talk a little about the AFs the AFs are very interesting because they have even more variation a lepton has a 15% left-right asymmetry a bottom quark has a essentially complete left-right asymmetry and that's certainly something you'd like to be able to measure experimentally so how can you do that so the lepton asymmetry you can attack by various methods all of which actually deserve some comment the first way is just to attack it directly which was done by the experiments at the SLC so these experiments achieved about 90% electron beam polarization and then you could literally just measure the cross-section for a left-handed electron to make a Z minus the cross-section for a right-handed electron to annihilate a positron and make a Z divided by the sum of those quantities and this is supposed to be the A lepton the actual experimental setup is very beautiful somewhere over here near very close to the coastal range hills there's the electron gun so there you're shining a laser on a strained gallium arsenide target and trying to eject electrons with specific polarization then you have the 3 kilometer long slack accelerator then some transfer lines that add another kilometer and a half and here's a detector you basically vary the polarization of the laser here in some random way and you look for a correlated pattern here of changes in the rate of producing Zs so very roughly almost every systematic that you could imagine for this experiment cancels and it's possible with the very small statistics of the slack experiment to get a very accurate result which please excuse me I did write down here somewhere so this gave an A e of 0.1513 with an error of 0.21 the other ways that you can do this are to look at various final state particles so one of them for example is the tau now the tau we're very lucky is a particle that decays through weak interactions in a way that we completely understand this basically goes back to the things about v-a that I told you yesterday and it's a kind of practical application of that theory for example if I have a left handed tau that's going to decay for example to a pion and the neutrino the neutrino is always left handed the pion is spinless so in the case where the tau is left handed the pion goes backwards in the tau frame which means in the lab frame it'll be slow if it's a right handed tau the pion goes forward just by angular momentum in the lab frame and it'll be in the tau frame and it'll be a fast pion in the lab frame and it's not so hard to just plug in the v-a theory and find that the sigma d cosine theta and the tau frame is one minus cosine theta and this leads to a one minus z sorry for a left handed tau and this leads to a one minus z distribution where z is the energy of the pion divided by the energy of the tau in the lab similarly if the tau decays leptonically we saw that at the end point when the lepton has its maximum value it recoils against neutrinos that balance their spin and so only the left handed electrons survive out here so what that means is that the electron is fast for a left handed tau and it's going the other direction so it'll be slow in the lab for a right handed tau and then you can work this out also for the other tau decay modes the major decay modes rho and a1 so all of these things were measured at left and actually it's really quite beautiful here is a characteristic tau event so at the LHC tau's are hard to find and quite subtle because there are thousands of quark and gluon interactions for every tau at an E plus E minus collider the production is much more democratic and the tau events are extremely clean in particular this kind of one against three track topology is an indication that you're producing tau pairs this event probably in the lower part has this decay tau to pion and the neutrino with a single pion track that you see there so in the experiments you separate out the various decay modes and you look at the energy distribution of visible products and here's what the data looks like so of course it's slightly modified for the detector acceptance the first thing is in red the one minus z and in black the one plus z distribution that you would expect for a right handed tau the red distribution has 15% more events and so you can measure the asymmetry directly in the right hand side you see the tau decay to a muon and again you see the muon spectrum is predicted to be harder for the left handed tau than the right handed tau the black is a fit to the sum of those distributions and as you see it's really extremely well fit by that and in the end what you come to is going all over the place a number Ae of 0.1465 with an error of 0.33 it's a little smaller than this one quite consistently finally you can try and measure this parameter from flavor dependent forward backward asymmetries so I wrote over here the angular distributions in E plus and E minus and this distribution has a forward backward asymmetry which is three quarters or 0.75 this one a forward backward asymmetry which is minus three quarters and so by adding appropriate components you can try to reconstruct the forward backward asymmetry on the Z pole for a particular process and in particular if you have unpolarized electrons going to unpolarized FF bar the forward backward asymmetry on the Z resonance is that factor of three quarters the polarization asymmetry for the electron and the polarization asymmetry for F a particular felicitous one because you can find these events readily is the AFB for B quarks in that case this number here is essentially maximal and so it's supposed to be a very good measure for the Ae the number that's found is actually quite discrepant from the numbers over there but it's still in the right ballpark for the standard model prediction 0.407 0.23 I should say not understood why this number is not in good agreement with this number and experimentalists arm wrestle about this actually it continues to be true eventually we'll have a higher energy or actually a higher luminosity E plus E minus collider at the Z and this will be resolved but in any event what you see about 14 or 15% really does confirm the fact that this number here in the table is rather low on the other hand one can go look at the bottom of the table where you get very large values of asymmetry there the very interesting experiment to do a direct confirmation again came from the SLC because if you can produce a left-handed electron beam the bottom quarks are supposed to be almost all left-handed and so this should be a forward distribution one plus cosine squared theta if you had a right-handed electron beam you should have a backward distribution you don't get exactly these distributions because it's not trivial experimentally to distinguish B from B bar but there are tags that do that quite powerfully and so what the data looks like coming out of the experiment is this and yeah it's really right if you have an electron a left-handed polarized electron beam the distribution is strongly forward if you have a right-handed polarized electron beam the distribution just turns around and now it's all backward and by the way the difference in the normalization of these two distributions is in effect another way of measuring AE and again reflects the 15% asymmetry that you talked about so the number for please excuse me the number for RB that comes out of this analysis is 0.923 with an error in this decimal place actually one can also at the SLC find the trump quarks and measure the R for the trump quarks it's much lower precision but again in good agreement with this number in the table and so it's really very interesting that one can try to really go through and measure every number in that table as well as you can and find agreement with the standard model let's just say as well beyond the accuracy of this table it's really at the accuracy that requires one-loop electroweak corrections which have all been calculated and these numbers with their full precision are in good agreement with the fully precise theoretical calculations now there's one relatively new addition to the precision electroweak literature that's worth talking about it's really now going to be much improved in the future of the LHC and that's the measurement of the W mass the W mass you would think the right way to measure that is to produce E plus E minus to W pairs and then try to directly measure the mass of the two jet system that originates from the W in that environment and that was done at LEP the second stage of LEP with about a I believe 40 MeV accuracy on the W mass the statistics of LEP was unfortunately very low there was some decision to dismantle LEP to build a proton collider of some kind and so one didn't really reach the full statistics that one might imagine for this experiment but on the other hand it turns out to be possible to measure the W mass extremely accurately at Hadron colliders and this has already been demonstrated by the Tevitron experiments CDF and D0 and in particular there's a guy at Duke named Ashtosh Kotwal who's literally put his life into this and has achieved some really amazing results he's a member of the CDF collaboration and so I'd just like to show you a few pieces of this analysis so ordinarily what you would think is that at a Hadron collider you make a W which then decays let's say to an electron and the neutrino the electron is precisely measured or a muon here the neutrino is of course invisible and so as I showed you yesterday you see events with only a lepton and basically nothing else in the event and then from that you have to reconstruct the W now you can do that by making a theory of the transverse momentum distribution of the W I think of the lepton I think here you see that theoretical prediction nominally in the theory as a function of Pt the cross-section has a at leading order a singularity at a point which is Mw over 2 and then various effects including QCD will round this into a shoulder and so that's what's reflected in this distribution however there's a trick which makes it easier to do a very precise measurement and that is to construct something called the transverse mass so for the transverse mass you work only in transverse quantities and so what you would do is to do the following you take the electron for which you've measured the full three momentum the neutrino you interpret the transverse part of the momentum is equal to the missing momentum when you've measured everything else in the event so then you can construct the PE plus emu vector the transverse part please excuse me minus the transverse parts separately of the two momentum vectors and one can prove that this is bounded above by Mw so what you get for this transverse mass distribution is a distribution which theoretically would have an end point at Mw and in fact it turns out that the events are rather bunched up toward the end point so in 0th order there's a sharp distribution this is smeared out a little by the W width and by some QCD effects and one then gets not a distribution like this on a scale from 0 to 100 but the distribution on the previous slide which please notice has now cut off at around 30 GeV so the end point is much sharper and you can use that theory to determine the W mass now unfortunately to get the W mass to high accuracy you have to do a very detailed in situ calibration of all these quantities the missing energy the electron and muon momenta and in the tevitron experiments there are all kinds of very interesting methods to use experimental data to use the analogous quantity for the Z of course one knows the Z mass very accurately as I told you from LEP to do those momentum calibrations and in the end what you get is just a spectacularly accurate measurement of the W mass so I think I put that here so this is now a summary of the current status of the tevitron experiments and please excuse me what you see here is per experiment about a 20 MeV error on these quantities when you put it together it's about a 15 MeV total error on the W mass that number you see here and I guess I quoted 40 it says 30 when you do the composite in any event this measurement is the equal in terms of a constraint on the standard model of any of the LEP experiments that I discussed to you for the structure of the Z0 and again it turns out to be a very beautiful confirmation of the precise value which is predicted in the standard model so now we have about 15 minutes left and I wanted to just say a few words about the use of electroweak data to constrain physics beyond the standard model in a kind of general analysis so here there's a notion which came up someone asked me about yesterday but I promised that today I would actually define it and discuss it which is the notion of oblique electroweak corrections and this is the following idea that I think is it's not precise but it's an idea that assists your thinking I believe in a very nice way so there are lots of reasons to study physics beyond the standard model but I think the most important one for me is the question of electroweak symmetry breaking we postulate this Higgs boson we'll talk a little more about that on Thursday is it an elementary particle we postulate that it has a potential of the symmetry breaking form basically in the standard model we just write down some function and give the coefficients the correct sign so that the potential has that form so we would like to know where that structure comes from is there a deeper theoretical explanation for why the Higgs has the potential it has and why the Higgs eventually does what it does condensing into the vacuum and influencing any particle that goes through the vacuum of space to do that there are lots of theories and those theories typically introduce new particles and those new particles have the property that they must couple to SU2 cross U1 because ultimately they have to do something about the Higgs boson but those particles needn't couple to light fermions and of course things that modify the Higgs sector typically must be very weakly coupled to the fermions that we typically do experiments with electrons, up quarks, down quarks, etc. so if you start thinking about radiative corrections to some process for example let's think about the radiative corrections to let's say a lepton pair production the Z width to go to a lepton pair there are radiative corrections that come from new particles that can come in the vertex for example something like this so we could have some kind of heavy lepton here that would be mediated by W however if the rationale for that lepton is to make modifications in the Higgs sector probably the coupling here would be something like M e over M l which would be an extremely small number in addition okay so you can think of various ways to inject radiative corrections into processes like this in standard model physics when you estimate the effect of those interactions typically you find suppressions like this and in the quark sector as Zoltan I think will explain to you one literally can't introduce large numbers here artificially because various constraints from flavor physics tell you that such events can't be present except below a very low level on the other hand that heavy object usually occur in a vacuum polarization because to occur in a vacuum polarization it only has to couple to SU2 cross U1 and this contribution can be an order 1 radiative correction on the same level as one loop radiative corrections let's say that come from W bosons and so it's interesting to try and make a theory of the effect of beyond the standard model such a weak that uses this idea we ignore the corrections to vertex diagrams and box diagrams and concentrate on the corrections to vacuum polarization now when you do that we can now introduce one more idea which now is maybe more motivated than ever by the LHC experiments which is that whatever new particles that we're introducing they're going to be rather heavy and so I'm going to assume that the mass of any new particle that we introduce is much larger than the Z mass many of you know that this is not actually required by the LHC data but it's required as it were pre-mephation you need some special circumstances in order for the LHC not to exclude new particles that are 100 GeV this way we can think about taking the various vacuum polarization diagrams and just Taylor expanding them in momenta because we're always going to evaluate the momenta much lower than the scale of the new particles that we introduce now there are four vacuum polarizations that appear namely the photon the photon Z mixing the Z vacuum polarization and the W vacuum polarization and as we do a Taylor expansion there's pi of zero and pi prime of zero and the first corrections will be here I'm going to ignore pi double prime of zero now QED the gauge symmetry is exact and that requires this vacuum polarization to vanish at Q squared equals zero so this is actually not present and this is not present and so if we Taylor expand to this level there are only six amplitudes that appear and so now we're definitely making some progress there are six quantities that in this view of oblique electroweak corrections can be modified by new physics and then we can try and put them together into some theory now there are two more things that you need to understand to make this theory more complete the first thing is to remember that at the tree level electroweak theory has these three fundamental parameters within the standard model G, G prime and V those quantities are not known a priori and they're all renormalizable quantities so we have to determine them from experiment and I think a particularly clear way of thinking about this is to say there are going to be three reference values very accurate experimental numbers that we're going to use in order to determine these three parameters and then everything else potentially is free at the time this theory was made those parameters were alpha of mz G fermi and mz so each of these quantities is known to about the 10 to the minus 5 level actually G fermi to the 10 to the minus 6 level alpha is known very precisely but what you need is the value of alpha at mz so you need to evolve that with the renormalization group from q squared equals 0 up to q squared of order mz which is actually the major source of error in this quantity today in view of this slide you might think about replacing this by mw and building a different theory but I'm going to just talk about the traditional one okay so then in terms of these numbers we can define a reference value of sine squared theta w sine squared theta w0 by the following formula is alpha of mz divided by the square root of 2G fermi mz squared and this number is again about 0.231 okay so now let's say that you have some precision electroweak observable for example the leptonic asymmetry so the leptonic asymmetry or any asymmetry can be used to define the value of sine squared theta w by using the tree level formula let's remember this is a half minus sine squared theta w plus minus sw squared divided by the sum this is when you ignore small terms basically 8 times a quarter minus sw and let me call this sw star in a similar way to the other asymmetries that you see here also these rates that go into the z-physics depend on this on-shell z-coupling s star so we can try and write formulae for the accurately measured standard model parameters m and s star and so to do that what you have to do is the following you have to first of all write down the diagrams that directly contribute to these quantities so for example if you have a z-photon interference here then that will affect the asymmetry of this vertex because now instead of being a chiral vertex it's a vector like vertex so it slightly shifts the asymmetry however if you compute only those diagrams you don't get a finite result because somehow you have to take into account the renormalization of these basic electroweak parameters so there are going to be subtractions in the result that are generated by some other diagrams and so in particular to analyze this quantity you have to compute these z-photon interference diagrams and you have to compute all the diagrams gamma gamma w w and zz that come into the calculation of these quantities so you can make the correct changes in the counter terms when you do that what you find is that the answers are given in terms of combinations of these six vacuum polarization diagrams and those combinations are ultraviolet finite in a renormalizable theory and they would affect then the arbitrary influence that new physics has and conventionally those combinations are given some names st and u so let me write them down here s is pi I need to introduce some notation first so let me consider the currents 3 which is the third component of isospin q which is the thing that the electric charge couples to and 1 which is the one component of isospin then the photon-photon vacuum polarization is I'm going to call pi qq of q squared the photon z vacuum polarization is e squared over cwsw pi 3q minus s squared pi qq so this is the I3 minus s squared q coupling of the z and similarly for z and the w I think it's just easiest to write it as pi 11 of q squared so that you can see now how the su2 custodial symmetry is going to come in in terms of those quantities you can then define s as pi 33 of mz squared minus pi 33 of 0 minus pi 3q of mz squared and this quantity turns out is finite in a randomizable theory and there's a quantity t which is defined in this way pi 11 of 0 minus pi 33 of 0 and there's a quantity u I'm going to describe in a moment now these quantities are very nice in terms of as we're distinguishing different origins of new physics t is explicitly custodial su2 violating it differentiates the 11 and the 33 components of the vector boson vacuum polarization so things that violate custodial symmetry will give contributions to here this is also proportional to the thing called the row parameter the coefficients by the way give these quantities a contribution of order one from one loop corrections this quantity respects custodial symmetry and it basically has to do with the first derivative of the vacuum polarization as you go away from 0 so it's the effect of explicitly new things at high mass and u is a combination that I believe has pi 11 of n w squared here analogously to this so this one has both the derivative with respect to q squared and the violation of custodial symmetry and it's typically very small compared to these two in modifications of the standard model of new particles well, what's nice about these expressions is that these three expressions are the unique combinations of these three six quantities that are free of ultraviolet divergences and they're basically orthogonal to the combinations to the calculation of these quantities from observable quantities and so one can then write an extremely simple formula which I was about to do here so m w over m w squared over m z squared at tree level that's cosine squared theta w so in leading order that will be equal to the cosine squared theta w that you would derive from this formula and then radiative corrections to generate a correction of the following kind similarly the value of sine squared theta w that governs the Z-A symmetries would be equal to this in leading order and it would be modified if there are corrections from new particles beyond the standard model and again you can work out the formula it looks something like this and now if you have the measurements of these quantities over here you can then compare them to the reference value and fit for S and T and get constraints on literally any kind of new physics that you want to postulate within the bounds of this general idea that the corrections are oblique that they're not coupling directly to light fermions but only coming from physics beyond this only coming from basically physics that affects the Higgs sector so here is the current status of the fit to those quantities now let me just talk a little about how various known parameters of the standard model affect these quantities one interesting observation is that top and Higgs basically obey rather strongly this oblique assumption and so they give corrections both to S and T for the top quark the effect is logarithmic in the top quark mass for the T parameter the effect is actually quadratic in the top quark mass so this is again a situation where we're seeing radiative corrections enhanced by a factor of mT squared over mW squared I'm going to talk a lot about that phenomenon tomorrow for the Higgs both effects are logarithmic over mZ squared and so if here is your standard model value in the ST plane increasing the top quark mass actually it turns out to be an enormous excursion in that direction and increasing the Higgs mass turns out to take you along a curve like that which is again an order one excursion in S and T up on this slide I have the current constraints on S and T from the current precision electroweak fit and actually the constraints are rather strong S is minus 0.05 this is the fit of the G-fitter group plus or minus 0.11 T is minus 0.09 again perfectly consistent with the standard model with an error of about 0.1 for comparison one heavy SU2 cross U1 doublet turns out to make a delta S of about 0.05 so it's just barely consistent with this and a heavy generation would be a delta S of 0.2 which is already excluded at 2 sigma bound on S so sorry I have it multiplied by 3 this would just be a lepton doublet so in general new physics will contribute at this kind of 10 to the minus 1 level to S and T and in fact if you have a composite Higgs model that typically has a structure in the Higgs field and some new vector like fermion partners you can easily get larger values so this kind of general constraint is probably the strongest constraint on building Randall Sundrum or composite Higgs models this again is something that Marcus Ludi will talk about in his lecture so well it's probably time I stop but that's the theory of precision-electro-week interactions and I think it's something that you need to carry in your head because these incredibly beautiful experiments are very powerful not only in telling us that SU2 cross U1 is really right in a sense maybe even stronger than what we would have expected but it also puts strong constraints on what can happen at the next level that you need to constantly keep in mind ok thank you very much