 Introducing the minimal supersymmetric standard model. Here's the executive summary. It's a list of all the chiral super fields that we have. We had to add an additional Higgs doublet, okay? And the first thing I want to talk about today, begin with is, so we talked about the various soft supersymmetry breaking terms that we would need to add, okay? And so now we want to begin to talk about the phenomenology of this model. And I want to start, of course, with the Higgs. The Higgs is the main motivation for supersymmetry. The Higgs has been discovered. It's certainly the natural place to start. Now, we have a lot of scalar fields in the theory. We have scalar quarks, leptons. We have quarks, leptons, and so on. But we are going to assume that they have positive quadratic terms. In other words, positive mass squared terms in the potential so that they are, when we minimize the full potential of all scalars, they will have zero vebs. That means that we can discuss the Higgs separately. We can just set to zero the vebs of all the other scalars, okay? So we can discuss the Higgs in isolation. So let's first of all step back and think about what happens if we do not have supersymmetry breaking for just a second, okay? If we don't have any supersymmetry breaking, we look at the Higgs, then the only contributions to the Higgs potential come from the D term potential from the SU2 cross U1 gauge interactions and from this mu term, this term right here in the super potential, okay? And so this term right here will give us a positive mass squared for the scalar Higgs and the D term gives us a positive quartic and so we have a nice stable potential, that's good, but no electroweak symmetry breaking, okay? So, yeah, here's the F term. The F term is just a positive mass for H up and H down. The D term here looks like this. It's the square of the sum of generators. This form right here is not the most useful form. You can simplify it using this identity here for any SU1, for SU2 generators, okay? And we can find a more conventional looking form of the D term here. So these are the only contributions to the Higgs potential in the supersymmetric limit, okay? So but from this we already see sort of what we're dealing with, we're dealing with a model with two Higgs doublet with a specific sort of potential, okay? And as I said here, electroweak symmetry is unbroken in the supersymmetric limit. So now, let's add the soft Susie breaking terms and what are the terms that involve H up and H down? Well, we have again, mass, independent mass terms for H up and H down and now they can be different. The mu term gave the same mass, right, for H up and H down and we have this so-called B term right here, okay? The sort of holomorphic quadratic term. Okay, and now, and notice it's very important there are no additional quadratic terms, right? The quadratic terms, the dimensionless couplings we cannot change with soft Susie breaking. Now, these mass terms here are just, as far as we can say, they're free parameters, they're phenomena, we don't know what they are, we can choose them to be anything we like and in particular they can be negative. So we can have negative quadratic terms. So now we can have electroweak symmetry breaking. We can have the origin be unstable, okay? And I'm sorry, sometimes I left in these little H's, okay. But anyway, if you look back at the supersymmetric terms, you can see that everything there was manifestly real, all the couplings were manifestly real in the Higgs potential. B could in principle have a complex phase but we could rescale or refaze the fields, you should be capital H's here to make B real and positive without any loss of generality, okay? So summarizing everything so far about the Higgs, here is the tree level potential for the Higgs, the two Higgs fields in the MSSM, okay? And here I've written these combinations here, the quadratic terms I've written, I've combined them here. So the quadratic terms for the Higgs mass are a combination of the soft Suzy breaking parameters in the mu term, that'll be important later, okay? Okay, so now let's minimize this potential, okay? And when we have a two Higgs doublet model like this, for one of the Higgs doublets we can do what we normally do in the standard model, namely we can just choose a gauge to make H have this form. If H has some non-zero entry here, we can just make an Su2 transformation to rotate it down here. If there's a phase here, we can use a U1 hypercharge transformation to rotate the phase away, okay? But once we've done this, then we've used up our complete freedom of making, we can't make any further gauge transformations without messing this up. So the most general form of the down type Higgs VEB is like this, it's got arbitrary things in both positions, it's got arbitrary phases, okay? And all these phases are angles, theta, delta and eta, all these things are angles and all these angles have to be determined by minimizing the potential, okay? So you're just supposed to plug this in to the previous potential and do the minimization, all right? And when you do that, you find that all these angles are zero, they're just minimized at zero, okay? And so I can write this form right here, okay? And this is very nice because if you remember that H up and H down had opposite hypercharge, this is exactly the direction of the VEB that preserves electromagnetism. So when you have a two-higgs doublet model, you cannot take for granted that you have the right symmetry breaking pattern. It could happen that you break electromagnetism. So our first phenomenological success, if you like, of the MSSM is that in fact it does, it can break electro-week symmetry in the right pattern, okay? Now we have two VEBs, V up and V down. We have two non-zero VEBs in general. And so the sum of the squares of these things is what enters into the W and Z masses, okay? Because we have one kinetic term for this that gives us a W mass squared proportional to this squared and another term that gives us a contribution to the W mass squared proportional to that term. So it's the sum of the squares of these things that has to be equal to the V, the VEB and the standard model. Yes, okay. So the question is, I'm making it a little bit more brief, but I think you're saying that we are setting the other VEBs to zero by hand, okay? But that should be justified in some way, okay? Okay, so you're asking about somehow the renormalization conditions and their connection to the other scalar masses. I'm sorry if I'm not maybe quite getting your question, but I think that renormalization doesn't have, renormalization enters only in the following way. When we're talking about these parameters here, we're talking about parameters renormalized at the TEV scale, okay? And so at the TEV scale, we're going to need these parameters, for example, I don't have parameters up here, but the parameters in the Higgs potential, they have to be renormalized at the TEV scale and we will have phenomenological constraints on what they can be. Similarly, the squark and slepton, the other scalers that we have in the theory, the squarks and sleptons and so on, they all must have positive masses at the weak scale, okay? Now, they may not have, in principle, they could have negative masses at some renormalized at some much larger scale and we may have to worry about that if we're interested in the connection to physics at much larger scales. But here we're just working at the TEV scale. Yeah, the question is, are we setting the VEVs equal to zero by hand because we have positive mass squares at the weak scale? So I would say that, yes, we are assuming that the VEVs, sorry, we are assuming that the mass squares of the other scalers are positive at the weak scale, but that means that their potential is then minimized at zero, so it's not an extra assumption. Their potential, if we minimize it, it will just have a minimum at zero. That's right, but the potential is a sum of a million terms, not a million, but a lot of terms, right, for all the scalers and what I'm saying is that if you minimize the full potential, the squarks and sleptons will have their minimum at zero VEV, okay? Because we're assuming that the quadratic terms are positive, okay? Right? So the potential looks like some squark mass squared, squark squared plus some slepton mass squared, slepton squared and many, many other terms, but if all of these things are positive, right, then minimizing this potential just gives us that those VEVs are zero, okay? It is a constraint on the theory, but it's one that's easily satisfied, okay? Other questions? Okay? So it's conventional to take the ratio of these VEVs and call it tan beta, so we have a little triangle like this, okay, with V up and V down on the sides and V on the diagonal, and now these parameters, V up and V down, the VEVs of the Higgs fields are determined by minimizing this potential. We have two VEVs, V up and V down, so we have two minimization conditions, and they can be written like this, okay? So they give some relationship between the parameters that we want, namely the parameters, for example, MH up effective and so on, and the VEVs, okay? Now the VEVs are sort of hidden here in MZ and beta, okay? So beta is the ratio of the VEVs, and MZ is of course proportional to the VEV, okay? So this is, so here this is really, this is the traditional way of writing it, or one of the traditional ways people write it, you'll see these kinds of things, but it's really just a condition that determines the VEVs in terms of the fundamental parameters, okay? All right, now this way of writing things is helpful if you want to understand, you know, relationship between parameters and what we observe, but it's important to understand what's going on here, why is MZ, so the important point about these equations for us here is that it's really telling us that if we want to avoid any big cancellations here, we need all these parameters to be of order MZ, okay? So MZ is appearing because the quadratic term is proportional to this combination of gauge couplings, and if we want to avoid any cancellations here, all these mass parameters should be of order MZ, okay? And this is arising, stepping back a little bit, there's a lot of, there's two Higgs doublets, there's a lot of stuff going on here, but the basic point is just whenever we have a Higgs model, the VEV squared is the quadratic term divided by the quartic term, okay? And here the quadratic terms are these various soft terms, okay, and the quartic term is exactly G squared plus G prime squared, okay? And so, of course, MH squared comes out to be, should be of order MZ squared, so that's what's going on, barring any weird big cancellations, okay? So it's not a surprise, okay? So if we get this minimum that we want, then we want to know, we want to know actually about the 125 GV Higgs, and we also want that we've observed at the LHC, and we also want to know what about all these other Higgs's that we have, right, where are they? So let's just take our two doublets, and we expand around the VEVs, okay? So the excitations in the VEV directions are neutral, and the excitations perpendicular to the VEV directions are charged, as usual, and these various Higgs fields here split up into a pair of, oops, now there's, now there, yeah, no, no, it's right, it's right. They're too neutral, too CP even neutral, too CP odd neutral, and too charged scalars, right? But now, as we know from the standard model, we know that there are, among these, when we work out the mass matrices, there will be three massless Nambu Goldstone bosons that are going to become the longitudinal components of the W and Z, right? And the same thing has to happen here, out of these six fields, right, or however you count, out of these fields, three of them are going to become the longitudinal components of the W and Z, okay? So if we remove those from consideration, the physical scalars, right, not the ones that become the longitudinal W and Z, there remain two CP even guys, one CP odd and one charged Higgs scalar. And you can work out the second derivative of the potential and plug back in the conditions that for the VEVs that I wrote down before, okay? And this is done in many, in detail, the details of what I'm doing, for example, are extremely very nicely explained in the review by Steve Martin that I gave earlier, okay? And so you can find the masses for all of these guys. Notice that, you know, it's a two by two mass matrix, so you're finding the eigenvalues as a quadratic equation so we can write down simple solutions. Notice that the A mass, the CP even, sorry, this should be CP odd, yeah, okay, that guy, that slide guy again, I've talked to him, but he keeps screwing up. They, this should be odd. The A zero, the neutral CP odd guy has an extremely nice simple formula in terms of the fundamental parameters. So it's often done to use this as an input parameter, because it almost is an input parameter, okay? And so we've written the masses of the CP even, neutral guys in terms of A zero, right? Which is really just the sum right here, okay? All right, now of course, the burning question that you have is we're interested in the 125 GB Higgs, and so of course, oh, I should also just explain the notation, it's very conventional, so it'll help you read the literature. Usually we use the little h zero to be, yeah, to denote the light guy, okay? So whichever guy is the lightest is the little h zero, and that's a mixture of the other neutral guys, you can find all the formulas for this in the review by Martin, for example, okay? So we wanna identify the little h zero, the light CP even guy with the Higgs boson that's been observed at the LHC, right? Okay, so let's see if we can fit that, what that tells us from that formula, okay? And the answer is that it doesn't work, it fails. And the reason, if you go back to that formula, if you look at it a little more carefully, you can actually see that, so there's an A's here, MA here, MA here, which is really just these parameters, so this is really a function of MA, right? And you can actually see that this is a monotonically increasing function of MA, it's not hard to see that, okay? And then if you take MA goes to infinity, take this thing to be as big as you can, you get MZ cos two beta, that's bad, that's less than 91 GeV, okay? That's not only incompatible with the LHC discovery, it was already ruled out in the 1990s by LEP, okay? And it's not hard to understand what the root of the problem is, okay? Once again, if you step back from the complications of these formulas, the point is that the quartic, we have some Higgs potential, the quartic coupling is G squared plus G prime squared. That's a direct result of supersymmetry, right? Supersymmetry, this beautiful symmetry fixed, a lot of fixed, the renormalizable couplings of the theory, it fixed the quadratic coupling to be, the quartic coupling of the Higgs to be related to the gauge couplings. So this is a very intrinsic property of supersymmetry. And once the quartic coupling is fixed to be this, well, generically a Higgs mass would be the quartic coupling, Higgs mass squared is the quartic coupling times the VED squared and indeed that is MZ squared. And what this thing says is that if you put in all the details, you can't get anything bigger than MZ squared. So this also tells us that we need some additional, what we need to do to fix this problem. I'm sorry, yeah? So go ahead and shout it out, I'll repeat the question. To make this mod, to get as close as I can to 125 GeV. So I don't really have to take it infinity, it's enough if I take it to be 400 GeV, I might get instead of 91 GeV, I might get 90 GeV or something. I mean the point is just that I wanna take MA as large as I can. I don't literally take, I never take anything to infinity, right? But I need to take it large. So I'm being as optimistic as I can and trying to fit what I want, okay? Sorry, did I repeat the question? The question is what was the point of taking MA to infinity? Sorry, other questions? Okay, so we need, it's clear from this what it is that we need to solve the problem. The problem is just this quartic. This quartic is too small. The gauge couplings are what they are, we can't change them. They are too small for this purpose, okay? Now in fact, such contributions do exist within the MSSM itself, namely loops. We have so far just considered the tree level potential, okay, if you're a phenomenologist, you do tree level until you get desperate and then you look at loops, right? When they don't work, so okay. So let's look at loops, okay? Is there any reason to think that loop effects could be big in this case? Well, loop effects can be big precisely when something is small at tree level, right? When something is sort of anomalously small at tree level, loops can be important. This may be such a case because G squared plus G prime squared is a relatively small number. It comes with a one eighth, you know, in the potential in the quartic and so on. And so anyway, what we want are the biggest loop corrections obviously and the biggest loop corrections come from the particles with the biggest coupling to the higgs and those are the top and the stop, okay? So just as in the standard model, we have a left-handed and a right-handed top, okay? So here's the translation between our left-handed vial language and the more conventional left and right-handed language we use in the standard model. In the MSSM, we have left and right-handed stops, namely we have the scalar super partners of the left and right-handed top fields, okay? And so we actually have two scalar tops. We have stop left and stop right. And after electro-icymmetry breaking, these guys have the same quantum numbers and therefore they can mix, okay? In the standard model, we don't think of the top left and the top right as mixing. It's because they have different Lorentz quantum numbers. One is a left-handed fermion, one is a right-handed fermion. But these are scalars, right? The stop left and the stop right are scalars. They can certainly have the same Lorentz quantum numbers, so after electro-icymmetry breaking, they have the same gauge quantum numbers and so they can mix and they do, okay? And the whole, the complete, I'm not gonna write down the complete stop potential. One of the many things I have to choose, I chose to leave out in all of this. But I'm just gonna neglect the stop mixing, okay? It doesn't qualitatively change, it changes the story only by some details, okay? So I'm just going to assume that the stop left and the stop right have the same mass and they don't mix. I'll make a little comment about mixing at the end. So what are these loops? Well, these are actually, remember, we talked about some of the, we began this series of lectures by talking about top and stop loops, okay? So here they are again, but now we actually hopefully know a little bit more about supersymmetry than when we started. So we have this super potential coupling, right between the third generation left-handed quark, H-up, right-handed up-type guy. And now this thing here is the same thing that gives us the Yukawa couplings also gives us a contribution to the potential. If we take the derivative of this with respect to the squark fields here, we'll get a quartic term like this that involves H-up and the squark fields, okay? So we get a quartic coupling like this between H-up and stops, okay? So if we want to correct the quartic, we have to look at diagrams like this, okay? And if we include some of the effects that give us stop mixing, there are a few other contributions, but this is already good enough for the parametrics, okay? Now, you should always estimate things before you calculate them, right? And so without doing any calculations, we can see that each of these diagrams has four powers of y-top. This one it's obvious because this is the top loop, I have one, two, three, four powers of y-top. This one right here also has four powers of y-top because the quartic coupling is proportional to y-top squared, okay? So we have a y-top to the fourth here. We have a one over 16 pi squared because every loop always gives you a one over 16 pi squared, that's how it is. It comes from the d-fourth momentum integral. Vector of three for colors. And now we know that this thing has to be a dimensionless thing. It's a correction to a Higgs quartic, so therefore it's dimensionless. So we can't put any, the only dependence that we can have on the masses, m-top and m-stop, can be logarithmic, right? And furthermore, we know that these effects, these corrections have to go to zero if m-stop equals m-top. Because if m-stop equals m-top, then as far as these diagrams know, this is a completely supersymmetric theory, okay? So with all putting that all together, we know that it has to have this form, okay? Just without doing any calculation. Okay? All right? And then we can take this thing and we can plug it into the correction to the physical Higgs mass. It will be a correction proportional to the correction to the quartic times the VEB squared. Okay? So these squigglies mean I'm doing order, I'm doing simple estimates, putting in the factors that I know. Okay? So, sorry, thank you, yes? I'm sorry, you're gonna have to repeat that. Okay, you're asking did we already use supersymmetry to cancel the contributions to the Higgs mass? So my answer to that is stay tuned, wait just a few moments, we're gonna get to that, exactly that topic. But yes, we're talking about exactly, this is exactly the context in which we said at the very first lecture, we were canceling the loop contribution to the Higgs mass squared. That exactly happens when we consider the mass squared, which we will in just a few slides, okay? So if I did not, don't answer your question then, please ask it again, okay? So here, this was a very simple estimate, okay? How well does it actually do? It's essentially perfect, okay? Because if you go and look up the actual formula that you find, okay, and the actual formula with all the factors, here it is, okay? It's just changed by a factor of two from what I said, okay? So, all right? So, you know, you should never be afraid of doing these kinds of estimates, okay? Because it really is good, okay? There's one thing that comes in here that I've neglected, which is there's actually a mixing angle among the Higgs because the 125 GB Higgs is a mixture of the other states and there's a parameter involved there, but I've made the optimistic choice. I'm always making the optimistic choice here to make the model fit as well as possible, which is in this case to assume that the mass eigenstate is nearly the uptype guy, okay? So, the light Higgs we've seen is dominantly the guy that couples to the top quark, okay? That is the optimistic choice here, all right? And so, you can plug into this, you can now plug into this. We know what y-top is, we know what m-top is. The only thing in this formula we don't, oh, I'm sorry, it's very important. There should be a four here, okay? Sorry, there's a four here, you know, and that just came from the fact that, you know, these things, this is y-top to the fourth. So, I did actually get the number of powers of y-top, right? Okay, there should be a y-top to the fourth here. Okay, so we know y-top, we know m-top, the only thing in this formula we know v, the only thing in this formula we don't know is the stop mass, right? So, we can just figure out how big of a stop mass do we need to make this work, okay? And the answer is 880 GeV, if you plug in the numbers, okay? However, it turns out that estimating the stop mass is an extremely delicate thing. In other words, estimating the value of the stop mass in the MSSM that I need to get the right Higgs mass is a very delicate thing. And you can kind of see that because it appears in the logarithm, right? And so, if something appears in a logarithm, then this value is very insensitive to that. Conversely, this value is not very well determined, right? And in fact, higher loop corrections are enormously important. And just to give an example, one loop, an X-order loop calculation that's very important is just the QCD correction to the top mass itself. At tree level, I would say that m-top is y-top v over root two, right? It should actually be a v up, I guess, okay? All right? But at one loop, there's a correction, it's a QCD correction. And that correction is huge. So, this number, if I use, if I include the one loop correction, instead of getting 173 GeV for this combination, I would get 163, okay? And if I just put in this correction, just as an illustration, just put in only this correction and correct this formula, I would find that the stop mass needs to be 1.3 TeV instead, okay? But that's just one example, okay? There are actually many other QCD corrections, okay? And here's the full result of an official, this is a whole industry of people who do this, okay? Here's one of the most recent results on this, okay? So I need to explain what you're looking at. On the vertical axis, you have the stop mass. This is the stop mass you need to get 125 GeV Higgs, okay? That's just one number. What's plotted here on the other, the horizontal axis is this A parameter. This is this cubic scalar interaction. This gives rise to the mixing between the stops, okay? Which is the thing that I've neglected. So what I've been talking about is A equals zero right here. And we see that the answer is, it's about 10 TeV actually, okay? It's about 10 TeV. You actually need a 10 TeV stop if you don't have any mixing to make this work, okay? So one lesson here is that people who do calculate radius, sometimes it is very important to calculate radiative corrections very accurately, okay? Now if we have a stop that is that heavy, we really have to worry about the problem that we started with at the very beginning of the first lecture, which is to say that the whole point of supersymmetry, the whole reason we're talking about this is that we wanted to cancel the quadratic contribution to the Higgs mass parameter from heavy particles. But now we're being forced to introduce a heavy particle, namely the stop itself. So we should go back to the quadratic corrections to the Higgs mass. And here they are, okay? This is back from the first lecture, exactly the, I think it's supposed to be the same formula, okay? All right, all right? And now let's see how big it is. Well, how big is this log? This log comes from the fact that supersymmetry is broken at some scale lambda, which we've seen is generally much, much larger, okay? But again, let's be optimistic. We're trying to be as optimistic as possible. And so we'll just assume that this log is like one, okay? In other words, lambda is not very much bigger than the stop mass. Supersymmetry is broken around 10 TeV roughly, okay? And then what we find is that this correction to the Higgs mass squared parameter is about a thousand times bigger than what it needs to be to break electro-week symmetry. Okay? So if you remember that we had some effective MH up squared, which was the MH up squared in the mass parameter plus the mu term, right? Okay? And so what we're talking about here are that the stop corrects this thing. The stop loop corrects this thing right here, okay? But we can always choose this to cancel this. But now this is 10 to the three times too big, so we have to cancel these two things to one part and 10 to the three, okay? Now, if we allow the stop mixing, things get a lot more complicated, okay? But it helps a bit because you can see that if you have very large values of this stop mixing parameter atop, then you can live with a stop of around two TeV or maybe 1.5 TeV or something, okay? And then if you do the same thing, you have to worry about the fact that you do get an additional contribution to the Higgs mass squared, also proportional to this parameter, but you can still get, you can prove things a bit, you can make it instead of thousand times too big, it's only about a hundred times too big. And this is, again, being extremely optimistic. The problem is much worse if you have high scale supersymmetry breaking. What is the imposed constraint on this figure right here? Okay, what is, here in this figure right here, we have just looked at the Higgs potential, okay, including all radiative corrections and including this stop mixing parameter. And what we found is the allowed parameter space that reproduces 125 GeV Higgs boson, okay? These are extremely sophisticated calculation. They go to two loops. Some effects to higher loops, okay? So this should worry you. If you're worried, you're right. You're paying attention, okay? Because eliminating exactly this kind of quadratic sensitivity was precisely the motivation for supersymmetry, okay? Let's let that sink in. So this motivates actually, so how should we think about this? Well, before we get too depressed, we should realize that this was specific to the MSSM, the minimal supersymmetric standard model, okay? Supersymmetry is a much bigger idea, although you wouldn't know that by going to a lot of conferences, right? But it's true. Supersymmetry is a much grander idea than the MSSM. And so this actually strongly motivates models of supersymmetry that go beyond the MSSM, okay? Sometimes the minimal thing is not enough. If we had minimality, we would only have one generation, right? You and I are just made out of one generation. We don't need anything else, but we have it. We don't know why, right? So here, one example of this is if we go to the next to minimal supersymmetric standard model, that is take the MSSM, take the same wallet card that I've been showing you, add to it a single chiral superfield S. So we add one more thing to the list. It is a chiral, it is a gauge singlet. And now, when I have this additional superfield S, I can add additional terms to the Lagrangian. I can add to the super potential terms like this, lambda S H up H down and S cubed, okay? And now I can get an additional, if I look at the F term, there's a term which is proportional to DWS squared, which looks like this, and that has an additional quartic. And now I have a tree level quartic in my Lagrangian and lambda is just some new parameter in my model. I can choose it however I want, okay? And I can choose it big enough to get 125 GV Higgs mass. No problem. Now, of course, you would want to do the detailed phenomenology of this and maybe you will discover some issues and there are, but basically it's fine, okay? Questions? Yes, please. Sorry, can you just speak up? Because I can't hear you, I need to hear you. Yeah, I think the question is, when I showed the stop loops, was I calculating a contribution to the effective potential? The answer is yes, absolutely, that's what I'm doing. So, the technical thing is that I'm computing the Coleman-Weinberg potential, right? So, at tree level, we have a potential that we read off from the Lagrangian. At higher loop order, there are corrections to that, okay? If you want to read about those, I'm sorry, I'm an old guy, but the standard reference back in my day was a physics report by Mark Scher, S-H-E-R. But I think that's probably still a decent reference. Okay? Also, many quantum field theory textbooks talk about it a little bit. Okay, so that is what it is, okay? So, now, but we also have to talk about the rest of the model, okay, not just the Higgs. Okay, there's lots of things in the MSSM. That's why people, there are people who, as far as I can tell, they literally spend their entire lives working on the MSSM. So, anyway, what is the, we can ask, okay, we have many, many other particles in the MSSM. What do we expect? Okay, we have many parameters, so if we don't have any expectation, we're... Okay, so, in fact, we have an extremely strong hint, extremely strong hint, it's so strong that we would be foolish to ignore it, and that is that we have, remember when we wrote down the soft scalar masses, we have a lot of these masses because we have three generations of quarks and leptons, and so these mass squared things are really three by three matrices, and they're, right? They're Hermitian, but they're otherwise arbitrary a priori, okay, and in particular, there's no a priori reason why these three by three matrices should be diagonal in the same basis that diagonalizes the quark and lepton masses, okay, and if that's the case, then we have a new source of flavor violation, right, in the theory. In addition to the Yukawa couplings, we have these things, okay? So for example, and these are really dangerous because for example, one of the most sensitive or the most sensitive thing is KK bar mixing in the standard model because of the off diagonal Yukawa couplings, we get box diagrams like this, and they give us some for fermion interaction and notice it's suppressed by MW to the fourth, okay, and it's proportional to the charm course mass squared, that's a complicated thing that comes from the gym mechanism, okay, and there are other contributions as well, anyway, but this is complicated, but the point is, it's sort of a light, it's like this basically, okay? All right, in the MSSM, we have similar box diagrams where we replace various particles by their super partners, and actually one of the biggest one involves the exchange of a gluino, so an S quark can emit a gluino and turn into an S squark. The S squark can mix with a D squark and via a gluino turn back into a D and then the opposite thing down here, okay? So this kind of diagram can occur, and again, you should not be afraid to estimate these things, it's a one loop guy, it has one, two, three, four powers of the strong coupling, it needs two insertions of these off diagonal mass squares, and so the rest of the dimensions, since this is a dimension six operator, they have to be, there has to be some M Susie to the sixth, these are the heavy masses that are flowing around this loop, so they're of order M gluino, M squark, something like that, okay? And now we need this to be small about the same size as that, because this thing right here explains KK bar mixing, the standard model does a good job of explaining KK bar mixing, so this can't be any bigger than that, and that tells us that this off diagonal mixing, compared to this guy right here, has to be something like 10 to the minus three, if this should be M Susie here, if these Susie masses are near a TEV. To be honest, if you really plug in these numbers, you'll get a less dramatic effect here, this number really comes from the detailed calculation, but I've used the value from the detailed calculation, and it's not just this one entry that has to be small, there are many other entries that have to be small, and there's also BB bar and DD bar mixing, I mean the story repeats many times for many other off diagonal entries, okay? So this very strongly suggests that there should be some reason that we should not have, these cannot be arbitrary matrices, obviously, so there should be some principle underlying the flavor structure, and there's a very, very simple and natural ansatz, which is just that these three by three matrices are proportional to the identity matrix, okay? All right? And this ansatz, this invariance under this, this ansatz is invariant under the flavor rotations that diagonalize the quark and lepton masses, okay? Because basically these guys transform oppositely, okay? The two indices transform oppositely under those rotations, and similarly for the A terms, we can make an ansatz that they are proportional to Yukawa couplings, okay? And there are very good theoretical justifications for this I'm not gonna talk about right now, I'm not gonna get to talk about in this lecture, but you can also have very good theoretical justifications for this, it's not just in phenomenological ansatz, okay? So if we do that, if we make this simplification, we do have a vastly reduced parameter space, I think somebody counted there's like 105 parameters independent before I do this, now here's what I have left, I have one scalar mass for every type of quark and lepton, in addition I have the soft scalar masses for the two Higgs, I have Gagino masses for U1, SU2, and SU3, I have three A terms, and I have mu and B mu, okay? Now this is all ready to, this is still too much, okay? I don't know about you, but I can't think in a, whatever this is, 10 dimensional parameter space, nobody can think in 10 dimensions, okay? So we still need some guidance from theory or at least some or phenomenological explanations, we need some guidance from somewhere, okay, about where to look, I think a very natural place to look is to think about high scale supersymmetry breaking, that's what we talked about before, if we have some hidden sector that breaks supersymmetry at some very high scale, okay? Then I remind you, but I now say it in the context of the MSSM, that we just have some chiral field X from the hidden sector, its F term gets a VEB, and then we can write down all these various kinds of higher dimension operators involving powers of X over M, okay? And the nice thing about this is that if you work this out, this generates all the supersymmetry breaking parameters that you need, all the things on the previous slide, all of the same order of magnitude, FX over M, so if we assume that there's a single scale M that controls all of these terms, we automatically get all the soft breaking of the same size, and then if M is some very, very high scale, could be the plank scale, that's a choice that's often made, then you just have to choose F to be the right value to make F over M to be one TEV. So this is extremely compelling, and one of the things to note that in fact here, this particular term right here, generates actually the mu term. So I could write the supersymmetric part of the theory as having no dimensionful couplings, and this automatically generates the mu term with the same size as the soft Susie breaking masses. In most other frameworks of Susie breaking, it's much more complicated to explain that, okay? Here it comes out very naturally. So this is extremely compelling, okay? And indeed there's further support for this idea, here we're being very, very bold. We're, if we're talking about Susie breaking at say 10 to the 11 GEV, we're extrapolating this theory over many, many orders of magnitude, okay? But one of the reasons we dare to do that is because of one major success of extrapolating physics to very high energy scales, which is grand unification, okay? So grand unification works extremely well in the minimal supersymmetric standard model. So the particle content of the MSSM was of course motivated by minimality, the minimal extension of the standard model, and we got what we got, okay? And what it turns out is that if we include the current, if we don't include, if we just look at the standard model, we get these dashed lines and we see that the three couplings do not quite unify, okay? That's what these dashed lines are. But if we include exactly the content of the MSSM, we have a threshold here, so we follow sort of the standard model, but then we have a threshold at around one TEV, and then things unify very beautifully. They unify in a really precise way, right? And there are no adjustable parameters in this plot. Nothing, right? Three lines meeting at a point with no adjustable parameters. And these widths here are the error bars. And it's quite striking because actually it turns out that the fact that we added, remember we added an extra Higgs doublet and that seemed like kind of a, I don't know, that seemed like kind of a, maybe it would have been better if we didn't do it kind of thing. But here it turns out that the differential, that makes a very important contribution to the differential running of these things right here. If you added more Higgs doublets, for example, you would screw this up, okay? You'd get back something more like that, okay? So this is something that absolutely didn't have to work. It worked. Maybe there's a hint here. Maybe there's a strong hint here, okay? So if we take this seriously, let's try to sort of think about the simplest thing we could think about. If we take this seriously, then unification suggests that the three gauge groups of the standard model are unified. So we would expect that at the gut scale, these gauge-genome masses should be equal to each other, right? That's a very, it would be a natural thing to sort of guess, but in unification it's actually required, right? And then we can ask, okay, but that's a 10 to the 16 GEV at the unification scale. What happens at the TEV scale? We have to look at the renormalization group equations for the gauge-genome mass, and they have the property, oops, these should be squared, sorry. This should be, this should be squared down here, okay? All right, M1 over G1 squareds. The gauge-genome masses divided by the appropriate gauge coupling squared should all be equal to each other at all scales. It's a simple consequence of that renormalization group equation. And so at the weak scale, you expect these, man, this should be three, two, one, instead of one, three, three. I'm gonna fire that guy, man. Okay, so all right, so you can see from this that the heaviest guy is the guy with the biggest gauge coupling, right? So this should be M3, M2, M1 is five, two, one, roughly. So there should be some ratio like that. That's what we believe from this. And now what about the scanner masses? Unification doesn't tell us, the unification does tell us that a lot of these things should be equal, things that are in the same multiple should be equal to each other, right? So a lot of the scalar masses should also be equal to each other. Renormalization group equations though are not the same for them because, for example, the colored squarks are charged under color, the biggest coupling at low energies, okay? And so if we look at the renormalization group equation, there's a big term involving the gluino mass correcting the squark masses, okay? And this negative sign, remember, means that when you run down, you're gonna get a positive contribution because we're running to low energies. And so these things tend to make the squarks of order the gluino mass at the TEV scale. That's what these renormalization group equations tend to do. Similarly, one that's easy to forget about but very crucial is the parameter MH up, right? That's, we really care about that. That's the one that breaks electroic symmetry for us. And what you see is there's, the reason we care about this is there's a big contribution. Y-top is the other big coupling, right? The two biggest couplings in the standard model are the QCD gauge coupling and Y-top, right? So whenever you wanna look at RG effects, those are the first things you should look at. And this contribution is positive in the RG equation. That means that it makes MH up-squared run negative. That's good because we do want it to be negative. So that's kinda nice. That means if it starts off positive at the gut scale, it can run negative. And some people really like that, okay? But anyway, it's fine. But the point is it's proportional to a squark mass. Squark mass is one of the big guys, right? So just like this makes the squarks be of order of the gluino mass, this makes MH up-squared be of order M-stop. That's bad. That's very bad for naturalness, right? Remember? Because we actually, right? We would actually like this to be at least a loop factor smaller than that. But this makes it equal. So it makes the naturalness problem even worse than what I've estimated before. But that's what it gives us, so, okay? And just to show you that this is not just some squiggly stuff. Here's an example. This is a very common thing to do is to have a phenomenological parameter space where you assume that all of these guys right here are equal to, all the scalar masses are equal to a common value at the gut scale. All the gaugino masses are equal to a common value. And here you can see, here they all are. At 10 to the 16 GV, all the scalar masses have the same. But then as you run down, the squarks get, okay, so first of all, here's the gluino mass. The gluino mass goes up. The squark mass get dragged up with it. The sleptons stay down here. And here is the uptype Higgs mass. Okay, it's going as negative as it possibly can. It's trying to catch up with the, okay? This is a typical sort of thing that happens, okay? And of course, this is just one possibility, right? I've made many, many choices. You could say, well, why this, why that? There are many, many other possibilities. And here are just some example spectra, okay? And if you stare at these kinds of pictures too long, you will go blind. And in fact, many people do, okay? They spend their lives staring at these kinds of pictures. But these are just some theorists. You should really think of these things as just some examples, right? We need, it's useful to have worked examples to do phenomenology, but you shouldn't take any of these things too seriously. But this gives you a little bit of an idea of what is involved in a complete model. So I have here, here's the light Higgs. Here's some neutral fermions, which are the, I'll talk about these a little bit later. Anyway, you can see there's a lot of particles. And once I have one of these models, I do predict this complete spectroscopy. And I can now go and ask whether this is how, what kind of experimental constraints there are on these kinds of spectra, okay? But one of the things that almost all the spectra that come from any theory have in common is that the colored super partners, the gluino and the squorts are heavier than the non-colored ones. Yeah, that's in common with all of these. Okay, so another thing that I need to talk about before we talk about the phenomenology is our parity. So when we first wrote down the MSSM, we noticed that we could write down all these terms here. These terms that violate Baryon number and Lepton number, okay? And if we, these terms would therefore all have to be highly suppressed, and so it's very natural to consider the possibility that they're really absolutely suppressed or suppressed so small that we can completely ignore them, right? And this has an extremely striking consequence. If you forbid these terms, then if you look at the MSSM, then there's a Z2 symmetry, okay? Under which all the ordinary particles, namely the fermions that we've seen, the gauge bosons that we've seen, and the Higgs's among one of which we've seen, okay? All of these guys are even under this Z2 symmetry. All the particles that we have not seen, not all of them. You know, the scalar partners of all these particles here are odd, namely the squarks, the gageinos, and the Higgsinos, they're all odd under this symmetry. So this symmetry, the Z2 symmetry is called r-parity, okay? And yeah, okay? And it's usual to call these supersymmetric particles, although I don't know why its partner is any less supersymmetric than this particle, but that's the terminology, okay? All right? So this r-parity symmetry has an extremely important, several extremely important, well, has one really extremely important consequence that has many consequences itself, which is that the lightest particle that is odd under this Z2 symmetry is stable, because it would have to decay to something, some state that is Z-odd, that is r-parity odd. It could of course decay to another r-parity particle plus some other stuff, but it's the lightest guy, so it can't do that. It's the lightest r-parity odd guy, so it can't do that. So the lightest r-parity odd particle is absolutely stable, okay? If that particle is electrically neutral, then it's a candidate for a dark matter particle, right? A WIMP needs to be a neutral particle, massive around 100 GeV, with renormalizable interactions with the standard model. Check, check, check, right? So we have particles like that in the MSSM. We didn't ask for them, but we got them, and they're at least the two most natural candidates. One is the neutralino. Neutralinos are mixtures of neutral fermions, right? Any particles that have the same gauge and Lorentz quantum numbers, the same gauge quantum numbers after a lecture week symmetry breaking and the same Lorentz quantum numbers can mix. And so in the MSSM, we have the Bino, the gauge partner of the U1 hypercharge gauge boson, the diagonal component of the Wino, the neutral Wino, and the neutral Higgsinos. Those are all neutral fermions after a lecture week symmetry breaking, they can mix. And the lightest guy of this is a very natural candidate for the lightest supersymmetric particle, the LSP, okay? And that happens in many, many models. Most models are, most models, I would say, okay? But there's another candidate that is also extremely natural, which is don't forget supergravity. You might think gravity doesn't matter for particle physicists, but actually sometimes it does because the, remember the, or don't remember, but I'm telling you again, the gravitino mass is F over M plank, right? The gravitino mass is F over M plank, but the Susie breaking scale was F over M. What was M? M was something that had to do with the hidden sector dynamics. If M is M plank, then these are equal, and then M3 halves is of order at TEV, but if M is any smaller than that, if I have lower scale Susie breaking, then the gravitino is always light and is the natural candidate for the LSP, okay? So these are two extremely natural candidates for the LSP. Yeah, they're heavy, okay? So, well, they're only floating around if they're stable. So the question is what happened, sorry, the question is what happened to all those particles in the hidden sector, okay? It's a similar question to what happened to all those particles at the gut scale. If I have a grand unified theory, that theory has many, many more particles, right? What happened to all of them? Well, the first answer is that they have huge masses, right, so, and we're not bumping into them today because they're unstable, presumably. Now, of course, it's quite possible that some of them, it's possible that some of them could be stable and might have cosmological consequences, might even be dark matter or something, and we can think about that, but there's no immediate, there's no immediate problem, okay? Right? So yeah, there's a lot of room for fantasy there, okay? Okay, all right, okay. So now I have enough to tell you a little bit about how we look for supersymmetry, okay? And you can already see the answer from this slide, this is where's Waldo, where do you look? You have to look everywhere, right? There isn't any, it's not a strong reason to look in any particular corner of that picture, right? There's no, all right? So you should think about supersymmetry the same way, I would say, okay? So I should, well, I should warn you that there's no way to give a topic, I'm now gonna spend the rest of the lecture talking about how we should look at Susie's searches, and there's no way to talk about this without editorializing, so I'm not going to make any apologies for that. And I've been editorializing the whole time, but you know, but anyway. Okay, so what is a generic, what is a generic signal in supersymmetry, okay? So we're gonna assume r parity, okay? So because of r parity, these supersymmetric particles have to be produced in pairs, right? They can't be produced, so they're produced in pairs. And then each of these guys decays, if he's not already the lightest guy, he decays, right? If he's not the lightest LSPE decays. And what do they decay to? Well, they have to, at the end of the decay chain, there has to be the LSPE, right? Because everything has odds to be r parity odd along the way. Okay, that's the generic signal. And so what is this LSPE? Well, here I'm gonna assume that the LSPE is neutral. I'm sorry, I should have made that assumption explicit here. I'm gonna assume that the LSPE is neutral, electrically neutral. If it's electrically neutral, it's like a neutrino, right? A neutrino, if you produce a neutrino in a detector, in a particle detector, unfortunately, we don't have the money to make the detectors big enough to detect neutrinos, we would have to make them, I don't know, miles, I don't know, miles? Probably, I don't know how big they'd have to be. It's a good exercise to figure out how big of a detector would you need to do tracking for neutrinos. Anyway, we can't, we don't have the money. So they just escape the detector, they don't interact. And so we have, we have, we see them, quote unquote, by the fact that we don't see them. Namely, there's an imbalance in the momentum transverse to the beam axis, okay? We can't keep track of the momentum along the beam axis. It's just too messy, right? There's too much crap going down the beam, all right? So we see them as missing transverse, people say missing transverse energy, which is really the magnitude of the missing transverse momentum, okay? So a very typical example, a prototypical example is that we produce a gluino, we produce a pair of gluinos, right? Gluinos are good because they're strongly interacting. In fact, they're in the octet representation of color. They have a huge, huge cross-section. So we produce a pair of gluinos, and what does each gluino want to decay to? Well, he wants to decay through its biggest coupling, which is the strong coupling, to a squawk and a quark. So if a squawk is lighter than the gluino, then the gluino will decay to a squawk. But then the squawk can decay down to, if it's the lightest squawk, it can't decay anymore through the colored interaction. It has to decay through an electro-weak interaction to a quark and let's say a bino, which are a component of the not lightest neutrino, okay? So it's very typical, it depends on the spectrum, of course, what's lighter than what, but it's very typical that you produce one superpartner and it decays via a cascade, right? So at the end of the cascade is the lightest, the LSP, okay? So in this particular case, if we produced a pair of gluinos, we would get four jets because we get two gluinos and then we'd get missing energy from the two neutrino that just escaped the detector. And as I said, there are many, many possibilities because we can have many, many spectra, right? Okay, it's a bewildering number of spectra, okay? So what you have to do is you have to look everywhere and that seems like that's mathematically impossible, but if you look into this, you will see it's truly impressive, the number of different searches, the number of different final states and number of models that have been searched for in a dedicated way by both the Atlas and the CMS collaboration, okay? But you can actually understand the broad features of all of these searches in a very simple way, which is that of course, every experiment at a collider is ultimately just a counting experiment. You're counting events, right? And what you're looking for is an excess of events over what is expected in the standard model. And so the thing that controls your reach is just the event rate, okay? And typically you have to have between 10 and 100 signal events in order to see a signal, okay? It depends on the standard model background, of course. So this is an extremely broad brush summary. And here's a very beautiful broad brush sort of plot made by Nathaniel Craig, that basically, that here on the vertical axis is the cross section in Pico Barnes for various kinds of super partners that we're interested in looking for. Let's start with squawk, gluino, gluino, gluino, stop, stop, neutralino, chargeino, the chargeino is the mixture of the charged fermions, the guino, and the charged hexenos, okay? And sleptons. And you can see that the colored guys have the biggest cross section here. And this is the mass of those particles, okay? That's the mass of the particles. So the cross sections are very, very rapidly falling functions of the mass, of course, right? And now here is the cross section that you need to get between 10 and 100 events, okay? So this is, these regions here are where you, the bounds will be, okay? And this is, this plot is for the eight TEV LHC run, the one that we just finished, okay? You can make a similar plot for the 14 TEV run. I wish I had that plot, but I don't. I'll show some other things, okay? Okay? And when I said there were a lot of searches, you know, this is a very, very executive summary. Each one of these is a supersymmetric search. There's a little footnote here somewhere that says only a selection of the results are shown here. So this is a executive summary of the executive summary of the executive summary of all these searches. And, you know, the very, very rough message is each one of these searches is reduced to one bar, okay, which just tells you the reach and mass of whatever particle they were looking at. And the here is one TEV. And, you know, there's a sort of this green tide which is sweeping over to one TEV, okay? So this just gives the message that there are many, many searches. And of course, these are all limits. There have been no signal observed, not for want of trying, right? If there had been a signal observed, believe me, you would have heard about it. And then I would have talked about it in the first lecture, right? All right? So we have to ask, in the absence of any such signal, what should we think about supersymmetry? How should we assess the impact of all of these searches? Okay? Now there are many ways to do it. It's complicated. I'm gonna just talk about one way to think about it. It gives you some way of trying to assess the impact. And that is, we can ask how natural is supersymmetry in light of these various types of non-observation that we've, from the LHC? And so we can ask which supersymmetric particles are most relevant for solving the problem of the large, solving the problem of explaining the Higgs mass, okay? Remember that we may have a problem in the MSSM, but maybe in some extension we can solve it and the searches would apply in those models as well. So the parameters are the Higgs-Zeno mass because remember that the mass parameter that enters the potential for the uptype Higgs mass is a combination of the soft mass and mu, right? However, mu independently also directly is the Higgs-Zeno mass. It is the Higgs-Zeno mass, so it plays this dual role. And so this mu, namely the Higgs-Zeno mass, cannot be too heavy, otherwise I'm getting a big cancellation here, okay? And so if I roughly estimate the cancellation, I need a 20% tuning for a 200 GeV mu. And just to explain what I mean by tuning, small tuning is bad, right? 10 to the minus three tuning is worse than 20% tuning, okay? So this is just a measure of the amount of cancellation that I would need, okay? The other one is of course the top mass and if you want a 20% tuning, you should have a 600 GeV stop, roughly. And the other one that might not, might not immediately realize is the gluino mass. Because the loop corrections to the gluino mass to the stop are so big, then to get it, if you want 20% tuning, you can't have more than about a 900 GeV gluino mass. And of course, you shouldn't take these 200, 600 and 900 numbers, you shouldn't take them in any religious strong sense. But this just gives, I'm trying to give you some benchmark for where you should start worrying, perhaps, about naturalness, okay? Okay, so we want to use this, so if you take this point of view that this is what I care most about, you should look at searches for producing stop quarks, okay? And here is a plot from CMS. I have to say, I apologize, somehow all the plots, the next plots I found, the plots for CMS were better illustrated of what I wanted to say, but there are similar limits, equivalent limits from Atlas, okay? But anyway, here are the stop limits, here is the, these assume that the stop decays to a top plus a neutrino, in most cases, sometimes into a charge, you know? There is various kinds of stop decays. Here is the stop mass, here is the LSP mass, okay? And you have to live underneath these things, these are the excluded regions. And here, for comparison in the yellow, I've put these naturalness constraints. Okay, and you can see that, oh yeah, it looks pretty good. We're probing most of the quote unquote natural parameter space if we take these hints seriously, okay? The situation is much stronger for gluino searches, because the gluino, even though it can be heavier, it has a much bigger cross section, and its signals are much more striking, okay? It decays, therefore it decays with much more energy. And so here are the gluino limits, lots of gluino limits, and they're plotted in the plane of the gluino mass and the LSP mass, okay? And here's the natural region, the same natural region I estimated before, it is now obliterated, okay? But the life of an experimentalist is very hard, because it's very hard to satisfy theorists, because we went and told the experimentalists they should exclude this, and they did, but then we say, oh, oh, oh, but you got 1300 TV, well now the tuning is only 10%. But you can imagine how much work went into each one of these searches. But anyway, if we really believed in naturalness, we should have found it, right? Okay, here's, I wanna show one set of plots, I'm gonna go over by just about five minutes, okay, but not more than that, so I'm aware of the time. I want to talk about electro-weak enos, okay, electro-weak enos are the neutral and charged fermions, the Higgs enos, weenos and beenos, okay, which are generally lighter, and I just want, even though those don't have much directly to do with naturalness, it's striking that, I show this plot because there are actually quite strong exclusion limits, and notice that along this diagonal line right here, the heavier guy can just barely decay to the lighter guy, there's like zero phase space for that. There's essentially no missing energy in this signal because all the energy is going into the, there's no missing transverse energy along this line, essentially, and yet the bounds are extremely strong here. This plot also illustrates another point, which is that what are these different tiles here? Each one of these tiles right here is actually a different search, because if you have a mass that lives, say, in this little guy right here, you should do a different search because your signal looks different than if you're up here, and for each one of these tiles they have done the optimum search, okay? So I'm just trying to give you an idea of how hard people have been looking for supersymmetry. They have optimized this plot everywhere here, okay? Now, that was all these plots that I showed you are the present limits for the past 8TEV run of the LHC. Of course, we're now just starting a 13TEV run, which is going to have an enormous increase in both energy and luminosity, okay? So what are the prospects for these various searches? Here is the stop search at 14TEV. Even though we have a 13TEV run, these projections were made for 14TEV, it doesn't matter, okay? All right, it's not a big difference. Here are the present projected, the present stop searches roughly, sorry, yeah, okay. Yes, here are the present limits, okay? And here is the natural region, and these two things here are the projected limits at 14TEV. This one is made under more optimistic assumptions than this one, okay? The assumptions have to do with assumptions about errors, not theory. Theory, they don't know what the experiment errors will be until they do the experiment, and so that's the uncertainty here. But you can see it's a huge increase. We're improving much, much larger parameter space. This parameter space we're presently probing is almost negligible compared to what we're gonna get, okay? And for gluinos, it's even more impressive. Here was that natural region, here were the previous bounds, bloop. Here's what we're gonna get at 14TEV, okay? So there's a huge increase in reach here, okay? Here's electroweak enos, okay? And here again are the present and projected reaches, okay? Now here I wanna make a point, which is that we have these huge uncertainty in, you know, it would be much better to have this than have this. Are the experimentalists gonna be able to do this? I predict yes, okay? Because they always do. Experimentalists at Hadron machines always do better than they are supposed to do. Here's an example. This plot was actually made before they actually did the search at ATV. This was what the ATV bound was supposed to be. Here's what it is. I just superimposed the other plot on top, okay? I sort of stretched it to make it the same. Here's what they actually got, right? Remember underneath there was this little bloop thing. So I predict they're gonna do this or better, okay? They always do that. They're always conservative, okay? So anyway, it looks fantastic. They're gonna get a huge amount of reach. So that's really all I've got time for, okay? So I'm just gonna leave you with some final thoughts, okay? So already in the 1990s, a very distinguished Italian theoretical physicist, Guido Altarelli said the train of supersymmetry is late. It's very late now, right? We really expected it much sooner, okay? People have been working on supersymmetry for a long time. Many people, you know, we have to think about it should we keep going. I think it's fair to say that supersymmetry needs either fine tuning or non-minimal structure. Maybe both, probably both, I don't know. Okay? One thing I wanna leave you with though about the tuning, I wanna really emphasize is that it is one tuning. It's one tuning. Why is it one tuning? All the way back in the beginning of the first lecture, I emphasize that the standard model has one relevant parameter. If you tune that one relevant parameter, the standard model is perfect. And that has its reflection in any extension of the standard model. If I can tune to get the standard model, I'm really only tuning one combination of parameters. The combination of parameters that ends up being the Higgs mass, the 125 GB Higgs mass. So there's one combination of parameters in the MSSM that is tuned and it's basically MH up squared, okay? In the MSSM because of the large effect of the stop, that's what it is. And so a very wise man from the 13th century told us that, taught us that when we think about nature, we should not multiple, entities should not be multiplied beyond necessity, right? We should not introduce more things, more concepts, more particles than we have to, okay? And there's a lot of truth to that, right? Even all these centuries later. And especially if you look at the alternatives, I would say that the MSSM possibly with some 1% tuning or maybe even worse is still perhaps the most compelling framework for particle physics. Actually I would probably say the MSSM are some extension, okay? So supersymmetry with 1% tuning, okay? And so what are the implications of that? If we take this point of view seriously, we would say the most likely place to find supersymmetry is right around the corner. And it sounds like a joke, but it's not, it's true. If we take these ideas seriously, that's what it implies, okay? And so we should really keep looking for these supersymmetric signals we shouldn't give up if we take this kind of point of view, all right? But finally let me just say that in science we always have the privilege of standing on the shoulders of the people that came before us and certainly we all benefit from that and this lectures have been my attempt to try and pass the torch to you guys. You guys are the future of science. You guys have to figure out what to do with the situation that we've left you with and I wanna thank you very much for your attention. Thank you.