 OK, so greetings. This is the fourth and last lecture of this series. But I will be around tomorrow. If any of you have more questions, I will not be around on Saturday. So tomorrow is your last shot to find out anything that happens to be in my head that you'd like to know. OK, the last lecture in this course in weak interactions is about the Higgs boson. And the Higgs boson, of course, is the last piece of the SU2 cross U1 theory, a very necessary piece, as we've seen through the week. And what's very interesting, of course, is that in 2012, about three years ago, it seems that this particle was discovered. And so we can actually learn something directly about its properties from the results of the LHC experiments. And so what I'd like to do in this lecture is to discuss the Higgs from a purely standard model point of view. Namely, what does the standard model predict about the Higgs boson? What we'll see is that the standard model really makes a, first of all, very precise, and secondly, very rich story of the properties of the Higgs boson. And then we'll talk a little about, to what extent the predictions of the standard model are actually verified experimentally. So there is another side to the Higgs story, which is the subject beyond the standard model. Namely, why is the standard model a totally inadequate theory of the Higgs, something that I happen to believe very strongly? How do we make a better theory of the symmetry breaking of electoral weak interactions? And how do we test that theory in relation to the standard model theory, in particular, by studying in more detail the properties of the Higgs boson? And I think Marcus Ludi and Andrea Vultzer will touch on various aspects of this as they go through their lectures. But today it's the standard model, and we'll see what the standard model has to say about this particle. Actually, if you just start all the way over on this side of the blackboard, the standard model theory of the Higgs boson is really very simple. We start out with a field phi, pi plus 1 over the square root of 2, v plus h plus i pi 0, which is a general complex two-component doublet, a field that transforms under SU2 cross U1 as an isospin 1 half and hypercharge plus 1 half field. We then let this field develop the vacuum expectation value, v. We then go to unitarity gauge. So we gauge away the Goldstone bosons, which we are always free to do. And if I go to the unitarity gauge, then this expression becomes simply this expression, where v is the vacuum expectation value, and h is the last field that we cannot succeed in gauging away. And this remains a dynamical field, and its excitations would be dynamical particles, which are the Higgs bosons. This expression means that it's more or less trivial to look at the standard model Lagrangian and write down all the vertices of the Higgs field to the various particles, namely, wherever there's a v, you replace it by an h. And so if you have any fermion, the v appears in the fermion mass term, and the coupling of the fermion to that field is just mf over v times h. And it's as simple as that. For a gauge boson, you have a mass squared. So the Higgs comes in quadratically, and you get 2, for example. So this is a vector boson, mv squared over v times h. And actually, it works even for the Higgs field itself. The trilinear Higgs coupling is, I believe, it's minus 3 lambda m Higgs squared over v times h. So from these values, we see that the Higgs, in principle, couples to everything, although some couplings are larger than others, the Higgs, in principle, can access any particle of the standard model that gets mass through SU2 cross U1 breaking. OK. Actually, now we can ask, what is the phenomenological profile of the Higgs? If I make a Higgs boson, what does it decay to? And it's clear from these formulae that the Higgs should decay to the heaviest particles in the standard model. So from these formulae, you would say the dominant decays of the Higgs boson would be Higgs to tt bar and Higgs to w plus w minus and Higgs to zz, because those are the particles of the standard model with the largest mass. And so the largest coefficients in this expression. And the only problem is that it turns out that the mass of the particle we identify now as the Higgs boson is about 125 GeV, which means that none of these modes are accessible energetically. And so we then have to look for other ways that the Higgs can decay. Of course, the Higgs couples to everything, so in principle, it can decay to anything that's allowed energetically. And what's going to be interesting is that when we start enumerating the possibilities here, there are going to be a lot of them, because these things that are naively the major modes are all excluded just by energetics. And so let me make a little list over here, and then we'll go through these. First of all, the Higgs can decay to fermions. So the next heaviest fermion is BB bar, tau plus tau minus, CC bar. You can go down the list, mu plus, mu minus, SS bar, et cetera. Secondly, the Higgs can decay to, if it can't decay to on-shell W and Z bosons, it can at least decay to off-shell W and Z bosons. So there are decays to WW star and ZZ star. Thirdly, it'll turn out that the Higgs can decay through one-loop processes to glue glue and gamma gamma and also gamma Z, which I'm not going to talk about in this lecture. And in principle, all of these modes can be important by virtue of the fact that the modes that are listed over there are absent. And this is actually an amazing situation, because what we've got to do now is to try and make a theory that encompasses all of these various Higgs decays. And then we can also do experiments and see which of these decays are actually observed experimentally. And I remember, and the Higgs Discovery Seminar in July of 2012, Fabiola Gianotti in the Atlas presentation made this point. And she said, thank you, nature. Because it makes the Higgs a very rich object for experimentalists to study and play with. So the next thing we have to do in this lecture is to actually make that theory. That is, I want to go through these three classes of Higgs decay modes and try and work out the formulae that would tell us what is the Higgs partial width into each of those modes. So the easiest one to start with is the fermions. So let's do that. It's really simple. The matrix element is minus i mf over v, as I said, over there. U dagger right, v right, and similarly for left. If you now put in the relevant spinors, this is minus i mf over v. What you'll find when you actually compute this is you get a factor of the momentum of the spinors. It turns out that if you have a fermion anti fermion, anti fermion system, if you want to have a scalar particle, it has to be in the p wave. So the momentum comes in. And when you work out the details, you find the partial width looks like this. h to ff bar is mf squared mh over a pi. There'll be a 1 over v squared in it. There is, because of the p, a factor of 1 over mf squared over mh squared to the 3 halves, I believe. And then you can eliminate v in terms of the w mass and come to a formula that looks like this. mf squared mh mw squared in the denominator. There's an alpha w, there's an 8, and yes, 1 minus 4 mf squared over mh squared to the 3 halves. This factor is more or less totally irrelevant for all of the fermions that I wrote down here. It's only really important if you think about off-shell Higgs's decaying to top quarks. Well, now we're done. This formula is very simple. For the leptons, it receives very small radiative corrections. And you can just compute the partial widths. So, for example, directly, Higgs to tau plus tau minus is 260 keV. And the smallness of it comes from the factor here m tau squared over mw squared. So the Higgs is intrinsically a very narrow object. The decay of the Higgs to mu plus mu minus is even smaller. It's 9 keV. So we'll compute a few more of these partial widths, but they all turn out to be really small. And in fact, when we've got the whole picture assembled, what we're going to find is that the total width of the Higgs in the standard model is only about 4 meV. And so it's for 125 keV object. This is a tiny width, which is essentially indiscernible on any collider. You can think about special methods to get near this order of magnitude. But to measure this number precisely, I'm not going to guess the next decimal place, to measure this number precisely at any collider is really a big challenge. Okay, so much for leptons. Now, how about quarks? So for quarks, there are some other considerations that we have to worry about. First of all, there's a color factor of 3, which you have to put in. Secondly, there's a large radiative correction, which begins 1 plus 17 over 3 pi alpha s of m Higgs. And then there are higher order terms. When you add it all up, this factor is about 1.24. So it's quite a significant radiative correction. Incidentally, this correction is known to end of the fourth LO by some amazing work by the Karlsruhe group, Baikav, Chiturkin and Kuhn. And if anyone is curious about how you can possibly calculate that many orders of perturbation theory, you can talk to me offline. It's one of these, the really impressive perturbative QCD calculations. There's another problem, too, which is that we have to worry about the mass that goes here. So the factor of mf squared over mw squared, we should probably write as mf squared ms bar at the mass of the Higgs divided by mw squared. And this brings in some more complication, which is actually important if you want to quantitatively understand the relation of various Higgs decay modes. These are not the quark masses that you're used to. And I'd like to say just a word about that. So you folks all know, I hope, that quark masses in QCD evolve according to an expression that let's write them this way. So typically what you quote for a quark mass is ms bar mf of mf. However, these quark masses evolve when you go to high scales. And the evolution looks something like this. It has in the numerator an alpha s of m Higgs over alpha s of mf. So this is a smaller number than that. So this gets smaller as you go to high values. With the exponent 4 over b0, where b0 is the usual first beta function coefficient. So with a rather large exponent. And then there's a 1 plus order alpha s. Correction to this. What that means is that the values of the quark masses that go into this expression and eventually the evaluation of the partial width are different from the ones that you're familiar with. And let me just write a little summary here. So if I have mf of m H, and I'm interested in this for b, c, s, d and u, the values that come out of this expression are for the b it's more like 2800 instead of mev, which is what you're used to. So the b partial width is significantly smaller than what you thought. The trump quark mass is 700. So the charm is just a lot lighter than the tau, which hardly evolves at all. And in fact, it's sufficiently small that it doesn't even overcome the color factor of 3. Doesn't bring it above the rate to tau plus tau minus. These other masses are really small. They're like 63 and 1.5, which are significantly smaller than what you're used to. Again, the interesting thing about this number is that the partial width for Higgs to tau plus tau minus is greater than the partial width for Higgs to cc bar, even though that's probably not what you would have expected. So now we haven't yet got all of the final states. We've only looked at the ones so far in this column. Let me put a little zero when we've calculated, I've given you at least most of the formula for calculating that width. But we can already talk about the magnitude of the branching ratios. It turns out that the decay of the standard model Higgs, the decay of the Higgs to BB bar is predicted by the standard model to be the dominant decay. The branching ratio of Higgs to BB bar in the standard model is about 58%. And that being so, this theory allows us to determine all of the other branching ratios. And let me just write them down. So for tau, it's supposed to be about 6.3%. For charm, about 3%. For strange, I don't know how to measure this, but if one of you has a good idea, please let me know. 3 times 10 to the minus 4 and mu 2 times 10 to the minus 4. It's actually thought that this number is measurable at the LHC. Because the Higgs would decay to a muon pair, two isolated muons of opposite sign, whose mass sums very precisely to 125 GeV, over a background about 10 to the 5th larger of Drayyan muons. And so it's thought that with extremely high statistics, one can actually pick out that signal. And so that's very interesting. Obviously it's interesting that the ratio of these numbers is supposed to be exactly the ratio of the mass squared. Similarly, if you can measure the Higgs coupling to TT bar, the ratio of that to the coupling you would get from this is supposed to be the ratio of the mass squared. So anytime you can make a comparison like that, it's often a very sensitive test of whether the standard model is doing the right thing. And so this is really a very interesting goal for the experiments. As I say, they believe it's feasible, although when you look at the backgrounds, it's really pretty scary. Okay, now maybe we should go ahead and talk about the things in this column, W and Z off shell. So here again, some of the physics that we discussed in the previous lecture comes up. If I had W's on shell, this vertex would be, as we've said already, MW squared over V times G mu nu, I should have written, epsilon mu of W plus epsilon nu of W minus. So it's an example where we get a dot product of W polarization vectors of the kind that I was talking about yesterday. And so in fact, this thing will turn into M Higgs squared divided by 2M W squared for exactly the reason that I told you yesterday, sorry, for the longitudinal polarization. And for transverse polarizations, it's of order one. So this vertex prefers the longitudinally polarized Higgs and I think there's, it's probably pretty easy to understand why that occurs. This quantity here again transmutes what's here basically a gauge coupling. This ratio is the ratio of a Higgs coupling to a gauge coupling. And so it transmutes the gauge coupling here into a Higgs coupling as just, that would be the prediction of the Goldstone-Boson equivalence theorem. For a longitudinal longitudinal, this should exactly be the Higgs to two Goldstone bosons. And the coupling here should scale with the intrinsic, so this coupling here should be something to water lambda v. It should be scaled with the strength of the Higgs coupling. Actually, the Higgs coupling is also related to the Higgs mass at tree level. M H is the square root of 2 lambda times v. And so the Higgs mass being not so large, the coupling is not so strong. Nevertheless, this number is about two. And so even for the 125 GeV Higgs, the contribution of longitudinally polarized bosons is significantly enhanced over the contribution of transversely polarized bosons. And that's actually going to come up in a minute, so keep that in mind. Now, if we don't have enough energy to make on-shell states, we can nevertheless make off-shell states. That is, you can compute this Feynman diagram. Let me say zz for the moment. And then the z perhaps can decay to leptons or quarks or neutrinos for that matter. And then in this diagram, the final state particles are on-shell, but these two z's can be off-shell. Intuitively, you might think you want one of these z's to be as close to mass-shell as possible and the other significantly off-shell. And if you want to actually make a preliminary calculation of this branching ratio, that's a pretty good approximation. It's good to about 15% accuracy. I think I put on the next slide the actual mass distribution that you have. Please excuse me, it's an old slide, so this is 120 rather than 125 GeV Higgs. But you see basically what the story is. For z and w, one of the bosons is very close to the mass-shell, although it cheats a little off the mass-shell, and the other has a broad distribution at much lower masses. And maybe I should just say one of the experimental problems is that the guy which has a low mass decays to very soft leptons, and in the LHC everything drowns out soft leptons, so it's a major experimental problem to actually collect all the leptons from these events. Nevertheless, the LHC experiments succeed in doing that with good efficiency. Okay. The branching ratios that you find when you compute these diagrams are actually somewhat large. The intrinsic strength here is much larger than BB bar because it's got an mz squared or mw squared in the numerator rather than mb squared. You pay a price of alpha when you go off-shell, but it's still substantial. So when you actually do the calculation, ww star is about 23%. The branching ratio to zz star is about, I think, 2.7%. Let's see if I remembered those numbers correctly. Something that should be noted is because this particle is going off-shell, the sensitivity of the branching ratio to the mass of the Higgs is actually very strong. Delta gamma over gamma, and if I put delta m Higgs over m Higgs here, there's a 14, something about a 14 that goes here. So if you want to make an accurate theoretical prediction to 1% of these branching ratios, you would better know the Higgs mass to 100 MeV, and that's also something that's quite challenging, although that number, again, people think is within the reach of the LHC. Yes. So what I plotted is the mass of the dilapton pair. So the way I generated that is I wrote a Monte Carlo program that computes the, that generates events according to the square of this Feynman diagram, and then these guys here have a 1 over q squared minus mz squared. So that Feynman diagram will have big peaks when one of the invariant masses comes close to the z-mass. Sorry? No, it's the mass of the dilapton pair. Exactly what it says. So this is one mz, and this is the other mz, and I plotted them both up here. So you have this integral which has various peaks in it, and one of the peaks has this guy close to the mass shell, and the other peak has this guy close to the mass shell. Okay, now, one of the very fascinating things about these decays is the fact that there's a lot of information in a four-body final state. And so one can actually try to learn about the properties of the Higgs boson using the Higgs decay. So let me, here's my Higgs boson again. So now I have two z's which are both slightly off-shell as we've just seen, they decay to two leptons. And this is not a really common process. Remember that the branching ratio of z to e plus e minus or mu plus mu minus is only 6.7%. So when you square that, you get, I think, I guess 5 times 10 to the minus 3 or something. So it's only a very small fraction, and when you finally put everything together, you get down to 1 times 10 to the minus 4 of all z decays. Nevertheless, there are some. And so one can then look at these special configurations where you have pairs of leptons, the four momenta summing to 125 GeV. So it's actually giving the mass of the Higgs boson. And then there are lots of angles to measure in this process. So there's an opening angle here, which is the same thing as the angle of the lepton from the z decay in the z frame. And there's an angle phi, which you might think of as the angle between the two decay planes of the leptons. And both of those have information. This angle we already discussed in the previous lecture is the longitudinal versus transverse polarization analyzer for the z. So if I expect that longitudinal polarization dominates, I expect this angle and also this angle to peak toward 90 degrees. This angle has another significance, which is quite interesting. And let's just talk a little about that. For the interaction that I wrote over here, the polarization vectors of the z's would tend to line up. If the z's are linearly polarized, they would tend to line up. If I had a pseudo-scalar meson, then I would have my meson, let's not call it a Higgs, let's just call it phi. And then the polarization vectors, sorry, maybe I should just say the kind of interaction that would allow a Higgs to decay to two z's from such an interaction would be something like epsilon mu nu lambda sigma, the f mu nu for one z, and the f lambda sigma for another z. So this thing leads to a situation where the two polarization vectors tend to be orthogonal. You get some structure like p dot epsilon one cross epsilon two. And so for a pseudo-scalar object to came through an effective interaction like this, the two plane, the decay planes would tend to be orthogonal to each other. And then finally, there's another choice, which you can call the dilaton model of this resonance, where the vertex is not that, but instead it's f mu nu squared. So this is a zero plus object but this object here contains only transverse polarizations. And so then these angles will peak toward forward and backward rather than peaking in the center. And so it's very interesting to do a complete angular analysis of these decays with all of the decay angles which are shown in this slide. And I think CMS pioneered this. There's a fellow named Andre Gritzon at Johns Hopkins who really decided that this was important and wrote a code that allows people to do this, which has since been widely shared. So it's kind of interesting to see what the distributions of these angles are, although I must say we're at a very primitive state of the art. And for this I really do need the pointer. So here is the decay angle. Here's the data. There are about 25 of these four lepton events in the whole CMS sample. If you squint at it, you can see that the red curve, which is peaked toward cosine theta equals zero, is kind of a better representation of the data than the dotted red curve, which would correspond to purely transverse Z production. Similarly over here, this is the phi angle between the two decay planes. And again, if you squint at it, you would see that the red curve, which is the prediction of this interaction, is marginally preferred over the dotted curve, which would be the prediction for a pseudo scalar higgs. If you are totally persuaded by these plots, you just haven't looked at enough data. But at least there's a suggestion that it's true. In each case, it's about a one sigma preference. You can add more observables. These are only two. And here's the summary, which is now published by CMS. So this compares the hypothesis that the higgs interaction is exactly this to various other hypotheses. And these are the likelihood distributions. And the arrow indicates what the actual observation is. So, oh man, I can't even read this. The cases considered, let me come up to here, are this one here is zero plus versus zero minus. This one here is the contrast with this hypothesis, which as you see is actually excluded about the one and a half signal level. The others have to do with various spin two models. And all of this is some, I think one should really say, rather preliminary evidence that the quantum numbers of the higgs of the 125 GeV resonance are actually zero plus, and that the form of the vertex is actually of this form. Now, you wouldn't put that in a textbook, but I think if we come back in about three years and get the whole accumulated run two sample of VLHC, you'll have some amazingly separated distributions that you would put in a textbook. Of course, you might only put them in a textbook if it turns out to be the right answer, but we're going to find out. I should say probably one more thing, which is that none of these slides compare to spin one. And actually, spin one was known to be excluded from the very beginning by the fact that the higgs, as I'll discuss in a moment, is observed to decay to gamma-gamma. Yes, please. Okay, so there are six angles, and I showed you distributions for individual angles. And in those distributions, there was kind of a one sigma preference for the standard model higgs over other hypotheses. So now, if you include all six angles, and they're correlated, these are the likelihood distributions that you get. And so, of course, there'll be more discrimination. Although, as you see, it's never great. It's about three sigma now for the discrimination against the pseudo-scalar higgs, and only about maybe one and a half for the discrimination against this dilutan hypothesis. Okay, I'm sorry. It's only 25 events. There's not much you can do with it. If you observe a resonance decaying to gamma-gamma, you actually know that the resonance cannot be spin one. And this is something that's actually pretty easy to understand at the first level from the Helicity point of view. A photon is a massless boson. It's only transversely polarized. So there are only two cases. Either the polarizations are the same or they're opposite. And this leads to J3 equals zero, and this leads to J3 equals two, but you can never get J3 equals one. And this doesn't really complete the argument, but if you now take this observation and run with it, using the fact that the final state wave function also has to be both symmetric between the two photons, it actually excludes the hypothesis of J equals one. This is something called the Landau-Yang theorem. And so we really know that that resonance is not a spin one resonance. The evidence that it's not a spin two resonance is the discrimination on the right-hand side of this plot. Okay, well now we're making some progress. So now the last class of modes that we have to talk about are these loop-induced modes. And they have another interesting story that let me see if I can get it straight for you. So in the standard model, we have the following one-loop processes. And now what I've already shown you is that these guys here are all basically Mf squared over M Higgs squared suppressed. These guys here are order alpha-week suppressed. So that allows these processes to become order one significant on the same footing as the others. And so let's draw the relevant one-loop diagrams. So first of all, Higgs to glue-glue. The diagram would look like this. The Higgs couples to some fermion here, which then decays to two gluons. Maybe I should say there are two diagrams that we should consider. For Higgs to gamma-gamma, you can draw the same diagrams, but also there are diagrams that involve W bosons in the loop. And so now what I'd like to do is maybe not compute those diagrams, but at least give you some argument about what the values of those diagrams should be. So first of all, we can make a couple interesting observations about these diagrams. Let me start from the case of Higgs to glue-glue, for which we just have the fermion, so it's a little simpler. So this diagram plus its partner has the following structure. First of all, it's going to be something like the Eucala coupling that sits here, yf. There are two powers of g sub s. And then if we write the glue on Memento k1 and k2, this expression has to be gauge invariant. And so we have to find something like g mu nu, k1.k2 minus k1 nu, k2 mu. It is a three-point vertex, which has dimensions of mass to the one, so there's got to be some mass in the denominator to give this the correct dimensions. And now there's an interesting observation that if the mass of the fermion is greater than the mass of the Higgs boson, then yf over M is going to be dominated. The mass of the momentum running in the loop will be of order the heavy mass. And so this will be yf over M, which is actually v over the square root of 2. It's of order 1 when the mass becomes very heavy. So the diagram is relevant when the mass is heavy. On the other hand, if Mf is less than M Higgs, this yf over M is going to be dominated by the mass of the Higgs, Memento of order of the mass of the Higgs in the loop. So it'll be yf over M Higgs, which means that there's going to be an Mf over v suppression of the diagram. And so this is a very odd case in terms of thinking about loop diagrams in quantum field theory. Ordinarily, when you have a loop diagram in quantum field theory, the lighter is the thing in the loop, the more important it is. But here it's exactly the opposite. If the fermion in the loop is light, you get a mass suppression. So this suppression turns out to be about 10% for bottom quarks, and it's even less for all the other lighter quarks. On the other hand, if the thing running in the loop is heavy, the top quark is the example in the standard model, then you get something which is independent of the mass and nonzero, and which, by the way, is a lot larger than this. And so what happens is that in the standard model, the top quark gives the only significant contribution, with, as I said, about a 10% correction in the amplitude from the bottom quark. And similarly, in Higgs to Gamma Gamma, the only fermion that contributes significantly is the top quark. So it's just odd. The loop decays of the Higgs are dominated only by those fermions that the Higgs is not energetically permitted to decay to, and it's something to remember. By the way, if there were a fourth generation, there would be three quarks here, which would all give the same contribution by this estimate. And so the Higgs to glue-glu partial width would then be nine times what it is in the standard model. Now, in addition to running the reaction this way, where the Higgs decays to glue-glu, you can run the reaction this way where glue-glu forms a Higgs. We'll talk about that in just a moment. And so that means that the Higgs cross-section would be nine times what it is in the standard model. And that's just manifestly not correct given the experimental data. So this totally obliterates the idea that there is a fourth chiral generation. There could still, I say, be heavier quarks, which are vector-like coupled, which don't mainly get their mass from the Higgs boson. But the idea of a heavier generation of quarks that gets its mass completely from electro-reximetry breaking, that's just totally excluded now by the fact that this partial width turns out to be more or less right to give the observer value for the Higgs cross-section. Okay. So now, I guess the next step is to actually compute this diagram. So now we have to roll up our sleeves and compute the diagram. But fortunately, I'm going to bypass that in the following way. I'm not going to try and get the exact answer. I'm going to try and get the answer in the limit where M Higgs is much less than M top. And I'm going to try to do this by considering, let's say, a zero mass Higgs and thinking about what its couplings are. Yes. Oh, I'm sorry. So this is dimensionless. Oh, sorry. Please excuse me. Oh, sorry. It should be a lambda v squared, something like that. Sorry. Oh, here. Oh, yeah, this is not very good, is it? Oh, yes. Thank you. Okay, in any event, it's independent of the mass. That's the point I wanted to make. I'm sorry, I couldn't hear you. Yeah, here it doesn't care what's the mass of this heavy particle. Right? Yeah, that's right. It doesn't care. Here, why don't you give them the microphone? So yeah, I'm confused because in the, as I understand, of course, the, why this is the case once you calculate the diagram, but for the action, there's a, there's a way to calculate the same diagram that is just by rotating away the action and it shows up in the field strength, right? Oh, yes. For the axiom, there's a similar thing. Right. If you have heavy particles, right, it's independent of the mass. It's independent of the mass. So that's, I guess, that's exactly the same that what is happening here, right? That's right, that's exactly right. But in the argument in the axiom, there's no reference to what is the mass of the quark that is running in the loop that gives you the diagram. No, but it's also true for the axiom that if you have a light quark in the loop, it cancels out. Or it's mass suppressed like this. If you go through the argument, it becomes a heavy quark. Oh, it does? Yes. Okay, good thing. Okay, so now we've got to compute this diagram. I think obviously I'm not going to stand here in half an hour and compute that diagram for you exactly. So let me use the following trick. Let's remember that the top quark also contributes to the QCD vacuum polarization. And what it contributes to the QCD vacuum polarization is a term that looks like this. k squared g mu nu minus k mu k nu times the trace of TATB times alpha s, and then you have to actually do the computation and get the number 3 pi. And there's a logarithm, because it's a logarithmically divergent quantity, lambda over m top square. So aside from this number here, which you have to get from actually computing the vacuum polarization diagram, I hope everything else is obvious to you. It's got to be gauge invariant. The color structure is this, which by the way is a half delta AB. It's got to have an alpha s in it, because obviously, and it's logarithmically divergent, so it'll have this structure. Okay. Now, we're doing quite well, because, oh, and remember here, of course, that mT is equal to yT times V over 2. So now what I told you was, if you want to get the coupling of the Higgs to anything, what you want to do is to take H, or just take the derivative of the expression with respect to V. And so if I just take this formula and compute d by dV of this expression and associate that with the Higgs coupling, then what that gives is the following expression. I k squared g mu nu minus k mu k mu times delta AB times alpha s over 3 pi times 1 over V. And interestingly, so this computes the coupling of a Higgs with momentum zero. The Higgs isn't really at momentum zero, it's really at momentum mH, but in the limit mH is much less than mT, this is a pretty good approximation, and what you can see is that this structure here has exactly the same form as the structure we needed here, and we've now identified all the coefficients. So we're now really totally set. We have calculated the coefficient of this expression. It has to be, please excuse me, it has to be an alpha s here and a 3 pi, and this is V, and we're totally done. Now we have this amplitude, we could plug it into an expression and try and compute the partial width. Now maybe I should say, and okay, and when you do that, what you find is gamma H to glue-glue is equal to, we'll get an alpha s squared, we'll get an alpha w from the V, you get 72 pi squared, and you get mH cubed over mw squared. Now, if you were to do the half hour of work and actually compute the Feynman diagram, you would get this for any value of the mass. And let me just write the expression so that you see I'm not cheating you. Alpha w of s squared over 72 pi squared mH cubed over mw squared times 3 halves tau minus, sorry, times 1 minus the arc sign, I'm sorry, I'm getting ahead of myself, tau minus 1, the arc sign of 1 over the squared of tau squared squared, where tau is equal to 4 mH squared over mT squared, sorry, 4 mT squared over mH squared. Good, now we're all set. And it's a little exercise to plug this into here, take the limit mH goes to 0 and show that it reduces to that. But the main feature of it, the big coefficient outside, comes from a relatively simple argument here. By the way, this is a large partial width. It's about 8% of the total width of the higgs boson. It's actually possible to use the same strategy to get most of the terms in the higgs to gamma-gamma rate. And let me write down the answer and just explain where these terms come from. So the higgs to gamma-gamma amplitude is going to look something like this. I k squared g mu nu minus k mu k nu in the limit. So this is, again, will be k1 dot k2, k1 nu k2 mu. There'll be now an alpha over 4 pi. There will be a minus 22 thirds. And this is the SU2 beta function. There's a 4 thirds times color, times the charge of the top quark. And please notice with the opposite sign. And that comes from this diagram. And then if you compute this in Feynman gauge, Feynman Toft gauge, there's another set of diagrams which involve the charged higgs boson, the unphysical charged higgs boson that you've eliminated. And that gives another plus 1 third here. And so that's the expression that you get for the mass of the higgs much less than both the mass of the w, or rather twice the mass of the w. So this is what you get, again, higgs much less than 2mw and 2m top. And these are not, especially this one, it's not a very well-satisfied inequality. But actually it doesn't do so bad at getting you the right answer for this width. And so the final expression that comes from that is alpha w alpha squared over 144 pi squared. mh cubed over mw squared. And then you see it's 21 fourths minus 4 thirds or something. In any event, when the smoke clears and when you get all the mass factors correct, it turns out that this mode is an 8.6% branching ratio. This is a 0.2, or rather 0.23% branching ratio. This one is about 0.1%. You can calculate that in a similar way. Okay, so now we have the complete phenomenological profile of the higgs boson as predicted by the standard model. And we can ask, well, what does this look like? What does nature say about it? By the way, it used to be an interesting thing before people knew what the higgs mass was to compute these branching ratios as a function of the higgs mass. And I think it's still interesting because there's a lot of physics in this. The correct value of the higgs mass, at least interpreting the 125 GeV resonance as the higgs boson, is here. And so you see this is a logarithmic scale. The BB bar dominates. The next one is ww star, glue glue, tau plus, tau minus. It's a lot bigger than charm. This is the z, z star. And then these photon loop induced modes here, and mu plus mu minus is way down here near 10 to the minus 4. Please notice though that if the higgs were at 200 GeV, where it would be allowed to decay on shell to w, w and z, z, those would be completely dominant and everything else would be totally submerged into the background. And interestingly, as soon as the w goes on shell, even before the z goes on shell, it completely dominates the higgs decay modes. I think it is a good exercise if you are interested in getting into higgs physics to sit down and write yourself a code that generates this figure. You won't get it exactly right because all the higher order corrections are in this figure. There is information in this lecture to get it maybe right to 10 or 15%. And you can see all the qualitative dependence. It's really fun. It's an exercise you should do if you want to get into higgs boson physics. Okay, so now we just have a little time left. So let me tell you a little about the extent to which all this stuff is actually verified by H.C. To discuss that, we have to discuss the various higgs production modes in proton-proton collisions. Yes. The higgs-to-grammar-gammar branching ratio is about 0.23%. Yes. How come it was the mode that the higgs was found in the LAC if the branching ratio was small? I'm about to explain that. Just hang on a minute. 2.3%. ZZ star is, what did I say? 2.7%. But if I insist that it's ZZ star going to 4 leptons, then this is 10 to the minus 4. Okay. So the two modes in which, as you all know, the higgs was discovered were this one and this one. And those are just ridiculously small branching ratio modes. So in fact, the higgs cross-section at the LHC is such that you need to have a billion pp collisions for every higgs you make. But the discovery modes are such that you need 2 trillion for every higgs that you observe. And it's just really scary. It's amazing that these people could actually find it. But why that's the strategy we're about to discuss? Yes. Yes. We'll get there. Okay. So let's just a little more systematically. Let's think about how pp can create a higgs. Well, first of all, you could have the process which is the inverse of the one that I just talked about. Glue glue making a higgs. And this actually turns out to be the dominant cross-section. This is, I think, in the previous run about 30 picobarns, which is big for a, quote, new physics LHC cross-section. Another mode for higgs production is the process that I discussed yesterday, W fusion into a higgs boson. This is interesting because when the jets produce W bosons here or Z bosons, they scatter forward. So this has a higgs accompanied by forward jets in each direction. And you can actually use that as a tag that it is a higgs-like event. Although I should point out that there are backgrounds because WW fusion can also make a Z boson. And the Z boson, as we know, has many of these same decay modes, in particular BB bar. So this can possibly fake the signal of the higgs. The other way is QQ bar annihilation through a diagram like this. Let's say WW, radiating a higgs or ZZ, radiating a higgs. So in this process, the higgs is produced accompanying a W boson or a Z boson, which can then be used to tag the higgs. And so this dominates the cross-section. But these other processes are often useful because the higgs isn't produced in isolation. And you can try and use the other particles in these events to enhance the signal over background that the thing you're looking at is actually a higgs. So on the next slide, we just show a calculation of these cross-sections as a function of center of mass energy. Please notice that they all go up very strongly as a function of center of mass energy, which is actually great because a couple years ago we were there and now we're going to be here. So you're going to get from any of these processes a factor of two almost in the cross-section. The red line is the glue-glue. Now, you have to put the production together with the decay and try to visualize what an experimental search for the higgs boson would look like. You're looking for some particular kinds of events and what do they look like. So as you just heard, the obvious thing to do is to look for the most dominant process in terms of cross-section. Glue-glue to higgs goes to BB bar. And what this would look like is an LHC event with a BB bar pair that is two jets opposite, which are both B-tagged. And of course, there'll be some other stuff in the event as well. The problem with this is that glue-glue can just go to BB bar through an ordinary QCD diagram. And if you want to make a BB bar state where the two-jet mass is about 100 GeV, the cross-section for this process is a million times the cross-section for that process. So it's pretty much hopeless to find this signal over this overwhelming background. And so what the LHC experiments decided to do is to take reactions where the final state is extremely characteristic, where it's leptonic or QED, but definitely not strongly interacting, and look at modes where you can completely reconstruct the Higgs boson. And if you go through this table, okay, well, that's a hadron, that's a hadron, that's a hadron. This is incredibly small. Tau always decays with the neutrino, so it's not going to be clear. That's hadronic. This always decays with either hadrons or neutrinos. It's really only these two modes that are totally characteristic. But fortunately, it's actually there. So you've all, I'm sure, seen plots like this where you look at the spectrum of events with two photons, and you compute the invariant mass in the two photons, and lo and behold, there is a little bump on that spectrum. The dominant thing in the spectrum is QQ bar annihilation to two photons. And the little bump is a resonance which has by now very high significance, 10 or 12 sigma in both of the experiments. So that's the Higgs boson. Similarly, if you collect these four lepton events from this process, if you just count in this peak, that red peak has about 15 events in it. So that's a very small number, but it's a very large enhancement above background. It's a very narrow resonance. It's in exactly the same place as the resonance from the photon signal. And so this would be the corresponding to the Higgs disease star mode. Yes. How can we measure the light fermion and Eucala couplings? That's a very important question. At the LHC, there isn't really a strategy. So we'll talk a little about the evidence for Higgs to BB bar. That's already hard. Higgs to CC bar is extremely hard. What people are thinking about is you look for, please excuse me, a JSI and a photon, for example, and you try and reconstruct those into a Higgs boson. The theoretical error in that is large. For SS bar, there's no strategy. And so, yeah, this is a very important problem. If you could figure out a way to measure these Eucala couplings, you could write a paper that would attract a lot of attention. Okay. So I want to show you one of these events. Here's an event display from Atlas. As we're going to see in a moment, when you see event displays that people say are Higgs bosons, usually they have a rather low probability of actually being Higgs boson events. But this one is really probably a Higgs boson event because you see that the peak is very large in the background. You see in this figure two muons. Those are the things that have very long tracks that go through five meters of hadron calorimeter and are seen in the outer detector. You see two hits in the electromagnetic calorimeter. Maybe it's a little more obvious. A little more obvious over here. These are the muons. These are the electrons. The sum of the four four momenta is 124 GV. And so this is in that peak and probably is a Higgs boson event occurred at observed at the LH6. Okay. So now we need to talk about the other modes. So now this one is observed. This one is observed. What about these others? So the next one to actually be observed was W plus W minus. And here what you want to look at is what people did look at is glue-glue production of a Higgs which decayed to W W star for which each one decayed to lepton and neutrino. And it's, I hope, obvious to you that there's going to be a background from QQ bar annihilation to W plus W minus into the same final state. And so what you have to do is to understand this cross-section extremely well. There are some kinematic differences between these cross-sections. So some cuts are made to enhance the region where this is preferred over the region where this is preferred. But ultimately it comes down to being able to normalize this process extremely well and recognize that there are more dilepton pairs coming from the ones that this process can account for. And I think the next slide shows that, yes. So the dark blue one here is the contribution to these histograms from W plus W minus. There are other contributions. Actually top turns out to be a large contribution because if the top decays leptonically, it's decaying to be lepton neutrino, the bottom quark jets look like any other jets that are produced in the LHC environment. The top curves, let's see. These two curves I think are electrons and muons, yes. These are electrons and muons left and right. And then top and bottom, it's events with zero jets, r equals 0.4 greater than 20 GeVPT jets. Zero jets and one jet. By the time you get to two jets, it's all coming from top quarks. So it's only these samples that are relevant. And that little red slice at the top is the prediction of the standard model Higgs production. And as you see, it is actually statistically significant that that slice exists, although the precise measurement of it has large uncertainty. So to the extent that there's the red explains an excess of the data over what you have in the blue, you've observed the Higgs in this decay channel. For tau, the story is going to be similar. We're going to look for Higgs to tau plus tau minus in a similar kind of setting. So you could look for a glue-glu to Higgs to tau plus tau minus. But it turns out that a more effective way is to look for W fusion. So W fusion to Higgs to tau plus tau minus accompanied by tagging jets, two forward jets. And then the tau can decay in various modes, to E new new, mu new new, or a new and a pion or row and A1, some kind of hadron, a very low multiplicity hadron. A typical event that you might be looking for looks like this. So once again, I think I should lead you through it. There are two forward jets, which are the blue cones. All these tracks here are low momentum tracks. That's what you're supposed to always ignore when you look at LHC event displays. However, over here you see an electron track with electromagnetic energy deposition, which is very significant. And over here you see a muon that's penetrating the whole detector and is seen in the outer tracker. So that would be the signal where you have the two forward jets, an electron and a muon from the tau on the other side. Now, what are the backgrounds to this? It's very challenging because, as I said before, you can have W fusion to a z and the z can also decay to tau plus tau minus. And that turns out to be the dominant background and the only way you can distinguish them is to measure the mass of the tau plus tau minus pair. Unfortunately, there are missing neutrinos, so it's not easy to measure that mass, but it's not impossible. And the trick that it's used is one which is quite general at Hadron Collider, so it's worth talking about it. The tau coming from either the z or the higgs is highly boosted. So all the decay products, including the neutrinos, are highly collimated. So you have an observed particle and then there's a missing particle. However, you measure the total missing transverse momentum. So you don't know what the momenta of these neutrinos are, but you know the sum of their transverse momenta just by momentum balance in the event. And so there are two pieces of information to get those two neutrino masses and their two unknowns. So you can solve for the two unknowns, calculate the neutrino momenta, assuming they're collinear with the charged particle momenta, and then add everything up to get the mass of the parent. And this technique has an accuracy of about 15 MeV in the mass, 15 GeV in the mass, so it's enough to tell you roughly is it a higgs or is it a z. And so if you now look at the data, this is now from CMS, you see some distributions like this. So the yellow here is the contribution from the z. The little bit extra here, here, maybe there's a little up there, there's evidence definitely in this particular kinematic region for something extra, and that would be the contribution of W fusion going to tau. And so when you put all of these samples together, I think I put that plot on, no I didn't put that plot on here. There is actually significant evidence, and I think three sigma and CMS and four sigma and ATLAS, that this higgs production to tau plus tau minus is taking place. So the higgs is seen to decay, as I said, to W plus W minus to tau plus tau minus. Okay, we're making our way along this plot. The fact that the production cross-section for gamma gamma is as large as it is tells us that this is in the initial state. And so we're making some progress. The last really important mode that we want to find is BB bar. And this one, I've already told you why it's hard. And the evidence I must say is still a little ambiguous, but I'll show you what it is. The way it's done is to look at QQ bar going to a vector boson, which radiates a higgs, which then decays to BB bar. And so the vector boson here serves as a tag that this is a higgs-like event. Now, unfortunately, there's also a Z, the possibility of a Z which decays to BB bar. And you can also have just plain vector boson production in association with a gluon, which by QCD splitting splits to BB bar. And so there are lots of difficult backgrounds associated with this higgs process. So I have just, so here is the current final result from run one from Atlas and CMS. So this is background subtracted except for the contribution from the Z. So you see the Z, you see a little excess on the high mass edge of the Z, and there is roughly a couple sigma XS in each of these plots, and that's taken to be, it's not yet quote, observation, but it's some very preliminary evidence that this mode exists. Now, once you find this page of the paper, you can turn back and see what the previous page looks like. And it looks like this. So here are all the backgrounds which are enumerated. Once again, I have to read the legend here. These are some particular decay modes. But for example, here this is Z plus BB bar. This is top. Top infects all of these analyses. And the little tiny red thing at the top is the signal which after you subtract the background gives you the previous curve. So these are very, very difficult analyses. And the hardest part is normalizing all of the various components here which are not Higgs, which requires the whole art of perturbative QCD and the ability to take estimates from non-Higgs regions of parameter space to get those normalizations correct. So finally, at the end of the day, we come to a set of estimates which here's the collection from 1-1 of the particle data group. So what's presented here is the signal strength. Now what is a signal strength? The signal strength is a measure of the rate of a given process. So what we're looking at is pp going to some final state AA bar. And what this is is the cross-section for pp to make the Higgs boson. The branching ratio for the Higgs boson to decay to AA bar divided by the standard model expectation for that quantity. And so there's a lot of information in this and it's of different kinds. Let me just say that this, if the thing that's producing the Higgs is B, this quantity here is proportional to the Higgs to decay to BB bar. So this would be glue-glue, for example. And for a branching ratio, the total width goes into the denominator. So these are highly composite quantities. It has to do with the production mode, the decay mode, and whatever the total width is. But whatever's going on in there, you see that it's working out really pretty well at the 20-30% level. So zero is no Higgs. You see that for H to BB bar, there's still maybe the possibility that it isn't there. But for the others, the reactions are clearly established. One is the standard model. And most of those points are within 20 or 30% of the predicted standard model value. So really, you know, for the first round of data collection on the Higgs, the standard model is really holding up quite impressively as the model for what this resonance is. By the way, the complexity of mu is something maybe to take note of. So for example, for Higgs to gamma-gamma, you note that the atlas value is maybe 1.5 sigma-high. And if you wanted to explain that, there are various ways to do it. One way is to say that the Higgs to gamma-gamma branching ratio is just large that the Higgs to gamma-gamma partial width, rather, is just larger than what the standard model predicts because, let's say, there's some extra fermion circulating in the loop, or actually some extra boson circulating in the loop. The other is to say the Higgs production from glue-glue is larger because there's an extra colored fermion circulating in the loop. And the other is to say that the total width of the Z is smaller than the standard model would have predicted because there's some suppression of the Higgs decay rate to BB bar. And at the current moment, it's very difficult to sort out different hypotheses of those. You have to basically make a global fit to this whole set of data and try and see which hypothesis you think is the most plausible with respect to the whole dataset. To get a 20% enhancement of the accuracy of the whole of each data point is only 20%. It's pretty futile. But in the next few years, we double the Higgs cross-section at this higher energy. There's much more data that's going to be collected. This whole picture can come into sharper focus. Finally, let me make one more comment, which is that when you study the Higgs beyond the standard model, it turns out that usually the prediction is that the properties of the Higgs boson, the various partial widths that you find, are pretty close, maybe within 10%, even within 5% of the standard model values. The reason is that whereas we saw an example of the fourth generation where an extra particle in the loop can dramatically enhance the size of the Higgs partial widths, typically those enhancements are suppressed by m Higgs squared over m squared. So that's a number that's a few percent for masses that are close to a TEV, that is the things we haven't yet discovered at the LHC. So it's actually incredibly interesting not only to go to the 5% level in the properties of the Higgs, but even to go to the 1% level. And for that, probably, we have to go away from the LHC to a next generation electron-positron collider. This is something that's really interesting to me at the moment to look forward to that and try and make that happen. So eventually, through the LHC program, accumulating thousands of inverse femtobarns and making these analyses much more robust, maybe even building another accelerator that would be able to see the Higgs much more cleanly and probe these processes. We're going to learn a lot more about this particle and I think it's going to be very exciting, and I encourage you to start thinking about this. So thank you very much.