 Before starting anything. OK, let's start. OK, OK, good. Testing, Micazon, people at the back can hear me. OK, welcome back. So so far, we've talked about the range of possibilities for what dark matter could be, the constraints on the properties of dark matter, and on how you might produce dark matter in the early universe under these in two general categories of models. So what I want to move on to now for this lecture and the next lecture is possibilities for how we might search for dark matter and learn more about its properties beyond what we've already talked about. Now, so I hope you've absorbed from this point, most of what we know about dark matter to date comes from observing the positive statements as opposed to negative statements about what it doesn't do. Come from observing its distribution in the universe and looking at its gravitational effects on visible matter. But that is not enough to distinguish between even such very different dark matter models as 10 to the minus 20 EV scalars that form some kind of condensate and 10 TV supersymmetric partners of the standard model particles. So we've learned a lot. There's an enormous range of possibilities still to go. If you want to be able to tell the difference between different dark matter candidates, the obvious discriminator to look for is non-gravitational interactions with the standard model. So you heard yesterday, I believe, from Robert Lazenby about searches for axion dark matter that you could conduct looking for its interactions with photons and with the standard model. So today I'm going to continue that direction and talk about other terrestrial searches for dark matter. So there's a wide range of such searches. So you talked yesterday about axion and ALP searches relying on the interconversion, which typically rely on the interconversion between axions and photons, the fact that the presence of axions can distort electromagnetic fields. We can also look for searches for just particles scattering off targets. So once the dark matter mass is heavier than about the milli-EV scale, then its wavelength is shorter than the wavelength of your typical experiment in a lab, shorter than a few meters. So then it makes sense to think about dark matter as individual particles. So this is going to be the main focus of my lecture today. This is a classic long-standing way to search for dark matter. So what you're really looking for is the targets recoiling when the dark matter particle strikes it. This is often called stare detection. Then we also have accelerator-based searches of various varieties. These can either be searches for the dark matter itself, particularly at high-energy colliders such as with the Large Hadron Collider, obviously being the big current example. You can look for the production of dark matter particles in conjunction with other visible particles and look for the apparent absence of a lot of missing transverse energy in your colliders. You can also, in models where the dark matter interacts with other new particles, where perhaps the interactions with the standard model go through other new particles, you can search for those mediator particles directly. If the mediator's light, that's sometimes something that you can do at much lower luminosity machines. There's been a lot of work in recent years to build up non-traditional searches both on accelerators and on the direct detection front to look for very light and weakly coupled particles, very long-lived particles that can be produced and then decay at some subsequent point with characteristic signatures. I'm probably not gonna have time to go into a lot of detail on that, so I'll point you to two reviews on this. I'll say some briefings about accelerator-based searches if I have time at the end, but for classic LHC searches, there's a review which is written for theorists from a couple of years ago by Karl Hoeffer, Alex Karl Hoeffer. This is probably not a comprehensive review, but it has a lot of references and is a good overview. And there's also a pretty good review as a white paper report of a range of possibilities, including for new axion searches, but also for new mediator searches of accelerators and new way to do dark matter and direct detection that is a cosmic visions report from the US. I'll say what I can about this, but especially in this third area, there's a wide diversity of search channels. There are a lot of different things that you can do, so to the degree that I talk about it today, it'll mostly be about general principles. So where I wanna begin today is for this classic search strategy for if you can think of dark matter as a particle, not as a wave, then when it scatters off standard model particles, you can look for those particles recoiling against apparently nothing. So let's work out how that works and its advantages and limitations as a search strategy. Okay, so what you are looking for, visible target. In order, I mean, in principle, there's a gravitational scattering, but it's very weak. So what we are searching for here is a non-gravitational interaction between the particles. So our cartoon is we have a target fitting here. This is traditionally an atomic nucleus. It could, in principle, be an electron or something else. My dark matter particle lies in. So if first time progresses, our target recoils and the dark matter also gets deflected, goes off in its way. We can't see the dark matter. What we're looking for is this, is this momentum kick. So how do we observe that recoil? So you can see the effects of this recoil through looking for, for example, ionization in the target for phonons, sound in the target for scintillation, or excitation. So depending on what material you use as your target, that changes the available signatures, but they're all looking for basically the same thing, which is this momentum transfer from the dark matter to the target. For very light dark matter, you may be interested in looking on absorption onto the target rather than this kind of recoil. And that's sort of similar in principle, but a different sort of strategy. So of course, a difficulty with these experiments always is that there may be many other reasons for an atomic nucleus or an electron to jump other than being struck by a dark matter particle. So these experiments are typically underground to reduce backgrounds and also heavily shielded. We talked a few days ago about these experiments using lead from ancient shipwrecks in the Mediterranean to try to reduce their backgrounds from radioactivity and from cosmic rays coming in. So in principle, the signal that you're looking for, first it's directional because the earth is moving around the sun, the sun is orbiting around the galaxy, the dark matter halo is relatively stationary compared to that orbit. So we know which direction we're moving in through the galaxy. So in principle, all these DM particles are coming in from a well-defined direction that we know about. So in principle, the signal is directional. It's also time dependent because of the motion of the earth around the sun and the motion and actually the rotation of the earth as well, the direction that the dark matter should be coming from relative to your lab is changing. However, at present, the vast majority of detectors do not keep directional information. So what you're looking for is just an overall scattering rate potentially with a time modulation. Most direct detection experiments today take the approach of trying to reduce their backgrounds to zero and just searching for any signal at all above the expected backgrounds, which are typically less than one event over their cycle. But it is possible to exploit this kind of time information to look for what we call a modulation signal on top of hopefully non-modulating backgrounds. So such a modulation signal arises from the fact that because the earth is moving around the sun and the sun is moving around the galaxy, the flux is enhanced when the earth is moving directly into this effect of wind due to the sun's motion. So when the earth and the sun are effectively moving in the same direction relative to the galaxy, you get an enhanced flux compared to where the earth is moving counter to the sun's motion through the galaxy just due to the relative velocity compared to the dark matter halo. So it's just the V in n sigma V is changing. Have the galactic sun is going this way, then there's an enhancement when the earth is doing this and a suppression when the earth is doing this. So that time dependence is also a possible signal. And for a long time for something on the order of, well, since the late 1990s, so coming up on 20 years now, there's actually been a claim of a signal in the Dharma experiment, which uses this annual modulation technique. Of course, the danger of this annual modulation technique in experiments where you have not reduced your other backgrounds to zero, is that it's possible that some of the backgrounds may also be modulating on time scales of a year. It's not that hard to think of things that change with a time scale of roughly one year, although it is quite non-trivial to find an explanation for what Dharma's seeing that hits all the criteria. So the flagship experiments at the moment are the large xenon-based experiments. They use xenon as a target material and the current level of the constraints and the strategy. So I'm first gonna show you what they can do and then I'm gonna tell you how they do it. If we plot the cross-section in square centimeters against the mass of the dark matter, and I'll do it in GV here, then what you see is that the constraint curve has a very characteristic shape that looks something like this. This is a log-log plot. So this cutoff is somewhere around the, let me draw the line there. This corresponds to about 10 GV. So it masses below a GV. These constraints get quite a lot weaker. There are other experiments that start to kick in and continue the constraint down to about the GV scale. And there are new ideas for lots of experiments that can prove this sub-GV scale. The minimum of this cross-section constraint is at about 30 GV. Above this scale of about 30 GV, the limit on the cross-section just gets weaker essentially linearly with the mass. So what you can constrain as previously is the cross-section divided by the mass. But this cutoff at low masses is different. This cutoff at low masses is coming from the problem when the dark matter is too low mass and moving too slowly, this recoil is just very small and hard to see. So a lot of work on the light, dark matter end of the spectrum is a matter of trying to be able to detect tinier and tinier energy recoils. The minimum of this plot is a cross-section of about four times 10 to the minus 47 square centimeters. As we talked about earlier, this is an extremely tiny cross-section. If you take a, if you would ask if I just had 100 GV dark matter scattering through the Z-boson, it would now be many orders of magnitude above this constraint somewhere up here. So we are now starting to probe models where the dark matter into scatters off, the nuclei scatters off standard model particles through the Higgs boson and through one loop level diagrams. It's kind of at the status where we are. Okay, so how do we get, yeah, okay, so question. So the reason, yeah, so you will see this in detail in a moment, but parametrically, the reason for this cutoff is because the recoil energy is getting too small to see. Why is there, what sets the scale of the recoil energy? Turns out to be basically the mass of the target versus the mass of the dark matter. So if the, this recoil energy starts to decrease rapidly with dark matter mass, once the mass of the dark matter is substantially smaller than the mass of the nucleus on which you're scattering, if you think about what are the possible range of masses of atomic nuclei? Well, the very lightest one of course is hydrogen which is about one GV but we usually don't use hydrogen as a target material for other reasons. So the Xenon experiments, their target mass is about 100 GV. So once the dark matter gets sufficiently lighter than that. But I mean, this is a quantitative question about how big a recoil they can see but the basic scaling is that once your dark matter mass is very small compared to your nuclear mass, then it's, so this is basically just like billiard ball scattering kinematics. If I bounce a ping pong ball off a bowling ball, the bowling ball doesn't move very much. The recoil ends up scaling as the square of the dark matter mass and it drops off pretty rapidly. And for Xenon where that cutoff starts to become just, okay, we just can't see a signal anymore is down around the future GV scale. But yeah, so it's the comparison between how heavy is the dark matter and how heavy is the target. So, and that in turn means that I mean, you can play games by using different targets, germanium and silicon experiments which have lighter targets can go to somewhat lower energies than Xenon. The CDMS light experiment sets stronger constraints than Xenon and sort of the one to 10 GV range, maybe a little bit below one GV. But once your dark matter mass gets appreciably below one GV, it's really hard to see through this nuclear recoil strategy. I believe Robert is gonna tell you more about other strategies to detect very light dark matter. But okay, so let's see the actual calculation version of what I just said. So let's consider a dark matter mass which I'll label as M chi. We often use chi as a symbol for the dark matter field. And let's consider scattering off a target of mass MT, M target. So, and we'll work in the lab frame first. So let's say this target is at rest in the lab frame. This assumption that the target is at rest, it's usually an okay assumption for atomic nuclei. It can be a pretty bad approximation if you start looking at scattering off electrons, especially if they're deeply bound electrons which can be moving pretty fast. So okay, what I wanna calculate, so what's my observable in these experiments? What I wanna calculate is the rate, is the differential rate with energy of these recoils. So this is what I can measure, I can measure how many recoils occurred and what the recoil energy looked like. So I wanna calculate the differential rate with it as a function of the recoil energy. Okay, so this, everything is very non relativistic in this problem. And at the moment I'm just going to assume just an elastic scattering, okay? This is an assumption. If there's structure in the dark matter sector, you could potentially have a case where the dark matter scatters into a different dark matter like state in this process or alternatively, people have also considered, once you have very low energy scatterings and you're scattering off molecules, for example, you could excite other states of the molecules. So it is possible for an elastic scattering to happen in principle. But here is the first pass, I'm going to assume elastic scattering. Okay, so if I'm assuming elastic non relativistic scattering, this is back in your first year undergrad physics classes. So our kinematics are before the scattering, we have a target and we have a dark matter particle approaching with some velocity v and mass m chi. After in the lab frame, our target is going to recoil with some recoil angle for this theta and some recoil energy which we'll label by recoil. And our dark matter particle is going to recoil off with likewise some angle which I'll call theta prime and some velocity v prime. So now we can just write down our energy and momentum conservation equations. Our energy conservation says, again, this is non relativistic, we're going to assume the masses are conserved. So before our energy is just the kinetic energy of the dark matter particle, afterwards it's the sum of the recoil energy and the new kinetic energy of the dark matter. We can do our momentum conservation calculation. So before the collision in the lab frame, it's m chi v afterwards, it's just the sum of the momentum. So this is the x direction momentum, the net momentum of this object. We can write as square root two times m times the kinetic energy of the target and in the y direction, we can just write out this equation. So now this is a straightforward calculation, we just need to eliminate, so we want to eliminate v prime and cos theta prime and get our recoil energy as a function of the recoil angle because this is the part of the interaction, this is what we can actually observe. So then, so eliminating v prime and theta prime from these three equations, we end up with an equation for a recoil that just looks like, that just takes this form where here, so theta is the scattering angle, this mu here is the reduced mass, is the reduced mass of the system. So we can already see just from this before having computed any differential rates or anything, that the recoil energy runs between zero and between two mu squared v squared over mt, the maximum over mt. So this reduced mass, when one of the particles is much lighter than the other, the reduced mass just reduces to the mass of that particle, so if the dark matter mass is very light, this just becomes m chi, if the dark matter mass is very heavy, this just becomes m t. So we can see that this maximum recoil energy in the limit that the dark matter is much heavier than the target mass, the maximum recoil energy is just a constant that only depends on your target mass, but once the dark matter is much lighter than the target mass, then this recoil energy scales as the square of the dark matter mass. This is basically why we lose sensitivity for sufficiently low dark matter masses. Okay, so there's another thing that we can take from this, so what's the v in this? So in a realistic calculation, we're going to integrate whatever spectrum we get over the distribution of velocities of the dark matter particles. So in one way, to look at what we just derived is that suppose we're looking, so we don't know what any individual dark matter particles scattered by, we're going to see a spectrum that's the integration, that's the sum of the effects of scattering of dark matter particles at a range of different velocities. So at a given e-recoil, this also tells us that only particles with velocity higher than a certain threshold can contribute. So the thing that eventually limits the size of the, the thing that eventually limits the size of the recoil energy will just be when does this minimum velocity cut off run into the highest velocity of dark matter particles in the neighborhood of the earth, that's basically governed by the galactic escape velocity. The particle, the dark matter particles that are around the earth are bound to the Milky Way galaxy. If they're going too fast, they would escape from the galaxy and would not be around anymore. So we can do, so we can do a quick estimate for the, for WIMP-like particles. So for heavy WIMPs, let's take the target to be a nucleus. That says that mass of the target is about 10 to 100 GV. Now, as we just said, if the mass of the dark matter is much heavier than the mass of the target, this mu just reduces to the mass of the target. So consider M chi significantly greater, or well, really only has to be somewhat greater than the target mass. Then we have, then we have e-recoil is going to be of order. Typical e-recoil is going to be of order mu squared V squared over M target, which is going to be roughly just MTV squared. The typical velocity of dark matter in our halo is about 10 to the minus three times C, a few hundred kilometers per second. The escape velocity from the Milky Way at our radius is about 500 or 600 kilometers per second. So plugging in this estimate, we see that the typical recoil energy is going to be about 10 to the minus six times the target mass. So heavier targets mean that this cutoff, that this requirement of the dark matter mass be higher than the target is a more stringent requirement, but they also give you bigger energy recoils in this case. So for 10 to 100 GV targets, atomic nuclei, this is going to be about a 10 to 100 KV nuclear recoil. So these are the kinds of typical energies that we need to look at for scattering a very heavy dark matter. But if I were instead to take 100 GV target and a one GV dark matter particle, then this would be V squared times one GV times one over 100. So V squared times our 10 MEV. So now instead of looking at 10 to 100 KV recoils, I'm suddenly looking for 10 EV recoils. So I go down to, as I go down to the GV scale. So that's why it's hard to detect like dark matter with the strategy. Okay, so that tells you broadly what energy range, experiments like this would want to look in as a function of the dark matter mass. So what about the rate of these scatterings and what about the spectrum of the recoils? So now we start to get into something that's a little bit more complicated than just the billiard ball scattering. And what people do is make a lot of, so I'm gonna now tell you how to read plots like this. What people do to do this calculation is make a bunch of simplifications to get a sort of standard picture which allows experiments to be compared to each other. So we'll go through those now. I'll tell you instead of where they can break. So in principle, the spectrum and rate of these scatters, what ingredients do you need to work this out? First, you need the amplitude for scattering on individual nucleons. Now, this calculation has two components. This is a nuclear physics calculator. This has a particle physics component which is how does the DM couple check walks in blue arms? But because these energy transfers that you're looking at are way below the QCD scale, you're at maybe a best at about a hundred, maybe the highest recall you're looking at is about a hundred KV. We're not really scattering of quarks and gluons, we're scattering off the bound systems. So we can ask also like what's the effect of quark and gluon content of the nucleus? So the effect of matrix elements for how these low energy dark matter particles would feel the different quarks inside the nucleus is a calculation that you can do using lattice QCD. So we take these nuclear matrix elements as inputs into the broader calculation. So this, then you need to be able to go from this to what you actually care about with the amplitude for scattering on the whole nucleus. Now, in getting to this question, you want to ask, you need to ask the question of okay, what is the, how do these amplitudes from different nucleons interfere? Do they interfere constructively? Do they interfere destructively? And so that's one question. And also, what is the effective nuclear form factor here? This particle comes in with this relatively low recoil energy. How well can it, is it only gonna scatter off nucleons that are on the outside of the nucleus? How deeply can it probe in? So this, we parameterized by a nuclear form factor and the standard in, the standard in the literature is to just, is to often just use something simple, a simple prescription, which is just called the Helm form factor. Okay, so now this gets us to an amplitude for scattering on the nucleus, having made either these calculations or these assumptions about these intermediate steps in the process. Then from here, what we actually want to get at the end is the scattering rate. So that also includes questions about what is the density of dark matter? This is a number, it doesn't have particle physics uncertainties, it has significant astrophysical uncertainties. So the standard assumption, and this is not based on a model, this is just based on observations, is to take the local dark matter mass density to be somewhere between 0.3 and 0.4, PV per centimeter cubed. This is probably a pretty reasonable estimate provided that our solar system is actually, has the typical dark matter density for a location this far away from the galactic center. It's, yeah, so there are, we can't actually measure the dark matter density in the solar system directly. You have to measure in a larger region around us. If there is a dark matter clump or a dark matter stream passing through our solar system, this density might be higher. If we got unlucky and live in a place where there's not a lot of local dark matter density, it could in principle be lower. So and then the other question is, what is the velocity of dark matter? And for that, the standard approach, so the standard approach that's used to get plots like this is to assume some just Maxwellian distribution of velocities for the halo, sometimes called the standard halo model. Okay, so the usual assumption is that the velocity distribution of the dark matter in the galactic frame is essentially just this isotropic Maxwellian or Gaussian like distribution with an upper cutoff on the velocity of 500 or 600 kilometers per second. So a lot of model building to come up with, alternative to come up with models which might evade or which might evade direct detection constraints or might allow consistency between two different direct detection experiments basically consists of changing some of these assumptions. So the standard simplifications I've mentioned that are used in the calculation. I've mentioned some of these already. People often assume a fairly simple form for the form factor. They assume a local density value in this range. They assume a Maxwellian distribution for the velocities and a V naught of usually 150 or 200 kilometers per second. So the standard simplifications that we make on this side of the problem in the calculation of the amplitudes to say assume that we can just parameterize. We're just going to leave the couplings of dark matter to protons and neutrons separately as parameters in our analysis. And we're going to set them and typically for these calculations people set those amplitudes equal to each other. Given a particle physics model you can compute from the nuclear matrix elements and from the particle physics how the dark matter actually couples to protons and neutrons. And we typically assume that these numbers do not depend on the velocity of the dark matter or the momentum transfer between the dark matter. That's not generically true. You can come up with dark matter models where the scattering is momentum dependent. So and in the standard calculation it's often also assumed equals FP. So just as an example of dark matter model building that people have done in the past for example if you have one experiment that is excluding your favorite dark matter model and it's fine with everything else but for example xenon is rolling it out. It's above that line. One game you can play is what if my dark matter had couplings to neutrons and protons of opposite sign and the relative sizes of these couplings was contrived just so that it really hated to scatter on xenon. So the dominant isotope in xenon experiments has a specific number of neutrons and a specific number of protons. If you're willing to tune these couplings against each other you can get a very suppressed signal in xenon experiments. This is very fine tuned. I don't know of a particularly good explanation for it other than maybe the universe hates xenon experiments. And this is somewhat jokingly called xenophobic dark matter but xenophobic but so like this is just an example of these are the assumptions that go into the standard calculation. If you perturb these assumptions a little bit you can sometimes quite drastically change the signals appearing in a given dark matter experiment. Xenophobic dark matter isn't like perfect at killing the signal even in xenon because a typical xenon experiment has an admixture of several isotopes in it so you can't set it to zero but you can make it small. Question? Yes, that's right. Good, good question. Yeah, so these standard simplification so these standard simplifications are true for I mean they're true for some models and not for others. For example, people often think about if I've got an interaction through something that is effectively just like coupling to electric charge then for the proton it will, the coupling will be whatever it is for the neutron it will be zero. If I go through the Higgs boson it will depend on the, then your coupling will depend on the quark contents and the masses of the quarks in the proton and the neutron. I guess then it will probably be pretty, I mean it will, it should be pretty similar. But so if the only difference here is that your Fn and your Fp I mean have somewhat different, like are not exactly the same it's usually pretty easy to rescale the cross section constraints. Like it's, well as we'll see in a moment if you assume purely constructive interference between different nucleons then the number that actually enters into the cross section is just number of protons times Fp plus number of neutrons times Fn. You get a constraint on that quantity once then then once you tell me what Fp and Fn are it's easy enough to figure out what that constraint means like the overall, whether your model is dead or not. But sometimes, yeah. So this assumption that Fn equals Fp is like it's an assumption that's often made just so that we can say, okay, here's just so that we can settle just so that you can do something definite but it's pretty easy to recast too. I mean as we'll see under certain assumptions the only thing that matters is a particular linear combination of these quantities and so then the assumption that they equal is not really important. But things like changing the momentum dependence for example, that's much harder to recast. So the usual categorization is that if the scattering is spin independent then you just assume completely constructive interference and if it's been dependent then you assume that it's constructive or destructive depending on the spins of the nucleons that you're scattering off since. So basically then your final answer ends up depending on your number of unpaired nucleons and are they protons or neutrons? If Fp and Fn are not the same. So yeah. So this is why, incidentally, I was about to write this. This is why the constraint I just showed you was on spin independent scattering. The constraint on spin dependent scattering is orders of magnitude weaker. The reason for that is that in one case we have fully constructive interference between order of 100 nucleons and in the other case we don't. So okay, so this is one assumption. So another, so this is what I just said. We can consider, is there a question over there? We can see the two cases of spin dependent and spin independent scattering. So in the spin independent case the nucleon amplitudes are typically taken to add coherently and thus the overall rate is just proportional to the atomic. Well, if Fp and Fn are the same the overall rate is just proportional to the atomic mass squared. For spin dependent scattering on the other hand the rates from paired nucleons with opposite spins and the same identity cancel out, spin up, one is spin down, these cancel out and so the rate scales as the total spin squared. For this reason it can also be true that what's a good target for a spin independent experiment may not be the best target for a spin dependent experiment. So for spin, this is why Xenon is such a good target for spin independent scattering because it has a high atomic mass. So you get to take advantage of having a mass of, so the trade off here is basically high atomic mass gives you lots of nucleons to scatter on. You get to square that for coherent scattering so your rate becomes large but as we discussed over here, high atomic mass also, so as we discussed over here, high atomic mass also means that it's easier to be in a situation where the dark matter is lower than that mass and then your recoil energy is suppressed. So high atomic mass targets like Xenon tend to be the very, give the very best constraints on spin independent scattering at mass scales above about 10 GV. For spin independent, for spin dependent scattering or at lower mass scales, there's more competition. So yeah, so just to say again, what I said previously in a lot of exotic or non-standard dark matter models, basically all that's going on is that some of these assumptions are being changed. Like the assumption of elastic scattering is being changed, the assumption of momentum independence is being changed, the assumption of FN equals FP is being changed in a way that makes it hard to see a signal in particular experiments. The assumptions of the standard halo model are being changed. It would be like a dark disc present or a clump in our neighborhood and that can change both the velocity distribution and the local dark matter density. So to a first approximation, when comparing if you're trying to have a situation where a signal in one experiment is real and the signal in another experiment is not, the zero-thorough thing that model builders do is basically go through this list and say, okay, what could we change? We would violate the standard assumptions. Okay, so under all these standard assumptions, let's look at how the calculation is done. Okay, so let's, to work out the spectrum, it's easiest to do this calculation in the center of mass frame. So in the center of mass frame, the situation is prior to the collision, we have a dark matter coming in, a dark matter particle coming in and a target coming in with equal and opposite three momenta and after the collision in the COM frame, we'll have the scattering with output momenta of K, which I'll label K and minus K. Now, because this is an elastic scattering and we're working in the center of mass frame, energy conservation just requires that the magnitudes of these incoming and outgoing momenta are equal, you can show that they're given by the reduced mass of the system times the relative velocity of these particles. So the three momentum transfer, it's convenient to write this in terms of the momentum transfer. We can just write this in terms of the input parameters and the scattering angle theta. So I'm gonna take theta to just be the angle between K and P in this frame. So that we can, now this three momentum transfer, if we look at what the three momentum transfer is in the lab frame, we saw that the target went from zero momentum to having a momentum that is our square root two MT times the electron times the recoil energy. So we can write this expression as being equal to the square of that. So this recoil is the lab frame recoil energy. Okay, so we can write the recoil energy in this form, so this is similar to what we did previously, but now the theta here is the lab frame scattering, sorry, is the COM frame scattering angle, not the lab frame scattering angle. So what this tells us is that the, so this relationship between the scattering angle in the lab frame and the recoil, sorry, the scattering angle in the center of mass frame and the recoil energy in the lab frame allows us to relate the differential rate by recoil energy, which I said was my observable, to the differential rate by lab frames by center of mass frame scattering angle. So if I assume that there's no dependence on the azimuthal angle in this lab frame that I can rotate this around its axis and there's no change in the process, then I can also rewrite this. So this is the, okay, so this is a differential scattering rate, this in angle, center of mass frame angle, this is related to the cross section, the dark matter number density times the number of target particles, times the relative velocity, times the cross section with respect to the omega. So this is the, so the rate is proportional to both the number of dark, number density of dark matter particles and the number of total target nuclei in your volume. And so this is just the same sigma vn rate that we looked at earlier, but multiplied by Nt. So this object, d sigma by d omega, this is what you usually calculate from Feynman diagrams. So, okay, so this d sigma d omega and the center of mass frame can be written in terms of the matrix element, the amplitude for this scattering in this form, this is just the expression that you'd have for our two to two scattering from Pasco Nishota or your favorite QFT textbook. So if we, so this is where these assumptions that I mentioned earlier about coherent addition of contributions from the nuclei come in and that we can summarize the matrix element for proton and neutron scattering just by these numbers. So we now say, okay, this contribute, assuming so for spin independent scattering, this amplitude as an overall form factor tells you, so this is how we go, how we take into account that the momentum is pretty soft, it may not be able to prove the whole nucleus. So this completely summarizes the degree to which you cannot just treat the nucleus as a linear combination of this many nucleons times the number of protons times this FP parameter plus the number of neutrons, the atomic mass minus the atomic number times the neutron parameter. And what your particle physics model will actually give you is these numbers. So then we can rewrite this observable, this differential rate with respect to energy as so this N chi, the number density of dark matter we'll rewrite that as the rho chi, the mass density of dark matter, f of q is the nuclear form factor. So it's a function of the momentum transfer because that tells you essentially how deeply you can probe into the nucleus. So it tells you how much of the nucleus you can see. And yeah, in principle, this is like the Fourier transform of the proton and neutron density within the nucleus. And people usually make some simplified approximation for what it is. Okay, so putting this all together, I get a one over V rel squared from this pre-factor here, but a factor of V rel from DRD omega. So I end up with an overall one over V rel scaling. Get three factors of M chi from there, one over 32 pi. Okay, so this is the expression that we end up with for the overall scattering rate. And then what's actually plotted in those constraint plots is what's called the effective single nucleon cross section. The very last line, so this is just the form factor, which is, so the form factor, which is a function of q, which depends on recoil energy through this relationship. It's the square root of two MT times ER squared. And then this is just the matrix element, the z times Fp plus a minus z times Fn squared. So basically, so broadly speaking, this part of the equation tells you about your target material. This tells you about the astrophysics plus DM mass. This form factor is a statement about the physics of the nucleus. And this is what comes from your particle physics model. You can see I mean, so if I were to turn on some appropriate momentum dependence in these matrix elements, then basically Fp and Fn would also become functions of q squared, and hence of the recoil energy. So I would just have some more non-truville dependence on the recoil energy. In this case, the recoil energy dependence, we'll talk a little bit more about that in a sec. And also, this is the rate for one specific velocity distribution. So the real energy, so in reality, we're going to take this rate and integrate over the assumed distribution of the relative velocities between the dark matter and the target material. So what's actually plotted? Okay, so that's our observable. What's actually plotted on constraint plots is the effect of single nucleon cross-section. So the effect of single nucleon cross-section will define, okay, so this is the object that actually gets plotted on the constraint plots. What it is, is take the scattering cross-section on a target, so assume that d-sigma by d omega doesn't have any angular dependence, okay? So you assume that the matrix element is just a number that doesn't depend on the scattering angle. In that case, this is very true, this d omega is very trivial, we just multiply by four pi. That gets you your overall cross-section of the scattering of the dark matter on the target. So then you rescale this cross-section, so there's a mu squared dependence in the mu squared dependence here. This is the reduced mass between the dark matter and the actual target. You rescale that to sort of get the effect of cross-section that you would have. If instead of scattering off the whole nucleus, you were scattering off just a single nucleon, okay? My setting q equals zero here, we also, we just set the form factor, we just set the form factor to one. That's how it's conventionally normalized. So this is sort of like what you would expect the cross-section to be if you were just scattering off a single nucleon. So you stick in the reduced mass for the new, so this is one nucleon. So, and then you divide because this is spin-independent coherent scattering to get your effective single nucleon cross-section, you just divide by the atomic mass squared. So this is a definition. So this object, so we can relate that back to the observable using the equation over there as the single nucleon cross-section. So we can relate, so the single nucleon cross-section then becomes just an assumed to be independent of momentum transfer pre-factor. We put in the various factors. This, so this tells you about the single nucleon cross-section, the reduced mass of the dark matter nucleon system and the mass of the dark matter. So this is mostly particle physics. This is entirely particle physics. This piece tells you about your target. Tells you about the, so this signal depends on the total mass in your target multiplied by the atomic, multiplied by the atomic mass squared. We end up with this, we recover this one of VRL dependence and this rho chi dependence. So this is just counting the number of dark matter particles available and how fast they're going. So this is our astrophysics piece and then here we have our form factor which describes going between the single nucleon behavior and the nuclear behavior. So when they actually plot these objects, the constraint, they have taken the, they have, so once you know the dark matter mass, you know every factor in here instead of sigma chi n, you take the constraint that they give you on sigma chi n that tells you the size of this pre-factor. They know the mass of their, the experimentalists will know the mass of their target, the number of particles in their target and the atomic mass of their target. They've made a set of standard assumptions on the astrophysics here and the VRL, although again, so this is not the final answer. The final answer is an integral over the VRL distribution of this equation, but there's an assumption made on that distribution on the local density and there's an assumption made on the form factor. So they've compared this observable quantity to the data in order to set a constraint on this parameter which is actually what's being plotted. I mean we could plot this, we could plot this object as in terms of the scattering cross section on individual nuclei but that would make it a bit less, a bit more difficult to compare different experiments with different target masses to each other. So this is the convention in the field that everything's expressed in terms of this single nucleon cross section. So it's the single, it is this appropriately averaged cross section of the cross section on protons or the cross section on neutrons. This doesn't have to line up to the actual physical scattering cross section on any particular nucleon. Yeah? Yeah? Good. So the question was I have a big tank of xenon, what do I actually look for? So if I have a big tank of xenon, what I look for is when one of these recoils occurs, it produces a trail of ionization in the xenon and also a trail of scintillation line. So you then, you drift those electrons across the chamber, you think you see the, so you drift the electrons across the chamber and you basically count number of electrons produced, count number of photons produced. Then there are, so in xenon, you need to take into account what the, you need a calibration for how does recoil energy translate into number of photons and number of electrons thus produced. This ratio, the ratio of photons to electrons that you expect differs between different kind of recoils, like whether it was a nuclear recoil or an electron recoil, and they use this to discriminate, so to look primarily for nuclear recoils for WIMPs. But yeah, for xenon, so the question of if I cause a xenon nucleus to recoil, then how much scintillation light and how much ionization light and how many ionization electrons do I get is calibrated based on separate studies. Like using, I don't remember what kind of beam I use for like you can hit it, you can hit it with neutrons and see what happens. So, but at low energies in particular, those measurements are very difficult and it's been a cause of a lot of debate over the years because that sensitively controls how small or equal energy you can actually see. Yeah, so you can, so you can do, you can use, there's actually a lot of interest in using argon-based, and using argon-based detectors. And people talk about using time projection chambers to try to do directional detection because if you can really see like the track, if you can tell what direction the particle came in from, you can see a, if you can tell what direction the recoil is in, then you can, for a number of recoils, then you can get an inference about the direction that the particle came in on and then you could use it to, ideally with enough statistics, you could then see that rotating over the course of the day, which would be a very clean signal for something cosmological. I guess argon has a lower atomic mass than xenon and so if I'm remembering my periodic table correctly. And so, and I mean that, so you don't get, if that's right, you don't get as much benefit from the a squared factor. And yeah, there's also just a question of like what you can do in terms of reducing other backgrounds. The xenon experiments have done a lot of work to get their backgrounds down. Xenon is also like pretty effectively self-shielding. It's dense enough that a lot of the scatters happen in outer regions of the detector and the inner region then remains very clean and that's not true for all other materials. But yeah, I mean argon's not a bad material for this kind of thing either. Okay, so okay, so now we understand, so this is so far all been, as I said, for just a single value of VRL. So really the dark matter has a distribution of velocities and the physical spectrum that we see. Yeah, so we're just saying you measure the energy of an individual recoil by looking at how many photons or electrons or phonons or other similar signatures of the recoil are produced. Ideally, you would get a number of different events and you could then actually make a spectrum of the rate as a function of recoil energy. In practice these experiments haven't seen any signals yet so these are instead just used to set constraints. So in reality, this energy recoil rate is just going to be these same pre-factors. We have to integrate this one over VRL piece over a distribution where V, so this is my relative velocity distribution. I've dropped the REL sign because it's not super relevant here. But we need to think a little bit about the limits of integration on this. We know, as we discussed earlier, that at a given recoil energy, we only get contributions from relative velocities above some Vmin. So this interval starts at some minimum value which depends on the recoil energy. This is then just an integral over the distribution of velocities in a halo. So the example that is usually taken, the standard halo distribution, this f of b function looks like this and up to some. So that integral then becomes, so in that case that integral, in the limit that the Vescape is large enough that this exponential is pretty small by Vescape, this asymptotes to an expression that looks like this. So where the ER dependence here completely comes from the fact that the lower end of the integral Vmin is just controlled by the recoil energy. Once you fix the recoil energy, once you fix the reduced mass, then there's a direct relationship between this Vmin and ER. So in this case, what, sorry, V0 is the same as what? No, sorry, no, V0 is the velocity dispersion. So this is typically about 150 to 200 kilometers per second. Vescape is typically about 500 to 600 kilometers per second and the standard assumptions, okay? Vescape is the galactic escape velocity, V0 is just the typical velocity of a dark matter particle in the halo, so the different biofactor of you, but thank you for the question. Okay, so what we see, so we saw previously that for a fixed V rel, the recoil energy was determined by, went up to this maximum range of two mu squared V squared over MT. So we see that here too, but now that we have a range of possible velocities instead of just being a sharp cutoff, what we see instead of this exponential dependence on the recoil energy at low recoil energies, you get contributions from the entire velocity distribution of halo. Anything, no matter how fast it's going, can always give you a basically zero energy recoil if it's just a glance in collision, it doesn't scatter very much. But at high recoil energies, you're really only sampling the most extreme dark matter particles. So as a consequence, we get, so we see that this is an exponentially falling spectrum, which so even if you have a heavy target, even if you have relatively heavy dark matter, it's still always true that if you could go to very low recoil energies, you could see more of the signal. The spectrum always starts at zero in this case and then drops off exponentially. More generally, even if we don't make this assumption about this momentum distribution, we know that we can see as we increase recoil, we increase V min, and that always will monotonically decrease this integral because the integrand is positive. So more generally, we can say DR by DER is always monotonically decreasing. So as a consequence, just sensitivity to low energy recoil is really crucial for all these experiments, but especially so once you, as you go to low dark matter masses, where this scale in the exponential can be extremely small. Okay, that's more or less what I wanted to say about classic WIMP direct detection. It's now clear what's actually being plotted and what assumptions are being made to go between this object and the observable and what they're actually measuring in the experiment. Are there any questions? Yeah, okay, so the question is like how much can we trust these constraints then if I'm sitting, and as usual with this, the question is, so this sort of like, what are the sizes of the systematic uncertainties? Sorry, some of these uncertainties you can mitigate by looking at different, by looking at multiple different experiments. So I mean, another way to say this is if I have some theoretical model and it's nominally ruled out by some experiment, like how much should I trust that exclusion? So the first thing I'll say is that if you're right on the borderline of the exclusion, then yeah, you're probably not really excluded. Like if you can, if the uncertainty bar on the local dark matter density is, there are a number of estimates, some of which have very small error bars, but the scatter between the measurements is often larger than those error bars. So I mean, I would say that the uncertainty on those local measurements is realistically not less than about 20 or 25%. So if you're right on the boundary, fine, take 0.3 GB per cubic centimeter as the local dark matter density instead of 0.4, and you're probably fine. And that's under the assumption that our solar system is a representative point for places in our general neighborhood. And that may not be true. In scenarios where you have, a non-negligible fraction of the dark matter in streams of dark matter or discs of dark matter, then these numbers can change by a lot. The question of how probable it is that we're in such a stream and it has a significant contribution of the dark matter density, if the dark matter is standard cold dark matter, if that's what your particle physics model is, then there's probably not a high probability that we are in some dramatically over dense or under dense region. But we could be in a mildly over dense or under dense region. And if you do have say some sub-dominant dark matter, if your model includes some strong interactions that cause the dark matter to have a very different distribution than it does in the collisionless case, then this number may be pretty off. The bigger effect of in a purely cold dark, cold collisionless dark matter model, the bigger effect of taking into account carefully the detailed distribution of dark matter is more likely on this velocity distribution. I mean, just seeing that the energy recoil is pretty sensitively related to the velocity distribution. If you change the typical velocities by, if you change this distribution, then you're changing something that is falling exponentially. You don't need to shift it to the side very much to have a pretty big impact on what the constraints look like. My colleague, Marie-Angela Lysanti at Princeton and her group have done a lot of work on trying to sort of figure out what the velocity distribution of dark matter in our neighborhood should actually look like, taking into account streams and so on. This Maxwellian distribution is probably not a great approximation. The real approximation, the real distribution, my memory is that it's like, it's somewhat flatter. It doesn't fall off quite as fast at high velocities. And that can change these constraints by a factor of a few. So the cross-section. And then there's questions like, okay, what if I play around with say, FP versus FN? What if I change the momentum dependence? So people have played a lot of games along these lines, often like trying to explain the Dharma modulation signal while not being ruled out by other experiments. You can partly mitigate this by, so if you would have expected to see a signal in both CDMS and Xenon, you can tune one of them away by playing with the FP and FN couplings. But you can't tune both of them away. If you have, if your model naturally has, like it's spin independent cross-section vanishes and it's spin dependent cross-section is the dominant one, then it will be much less constrained by direct detection than a spin independent model in general. Similarly, if you have some like steep Q squared scaling, like some suppression at low momentum transfer in your model, then direct detection just generally will be a less powerful way of finding it compared to other probes. So, yeah, I mean, at zero I thought I would say if your model is sitting right on this limit, maybe you know, don't despair, you may still be okay. If you're trying to explain a particular signal while being consistent with all other experiments, then there are games that you can play, some of them just generically suppress all direct detection signals, others change the signal in one experiment relative to another, but your ability to do the latter is severely constrained by the fact that multiple experiments with different targets have not seen anything yet. Yeah, question. I think signal one tonne is pretty close to zero background. Like the number of background events expected per year is something in the single digits. I don't remember whether that is one or two or three, but it's a small number. These experiments have done a lot of work to make themselves effectively background free. If you were to get a signal, your signal essentially just has to be a decent number of Poisson sigma away from zero in order to be detectable. The background is very small. So, yeah, that's the good news, but the bad news is that we haven't seen is that these experiments are fantastic and amazing and they run for a year with a tonne of xenon and they see one event or two events. Okay, so it's basically time for the coffee break, but so in five minutes, let me just say something briefly about the other kind of earth-based searches which are searches at colliders. And I can say a bit more about this at the start of next time. We'll just say something very briefly about this. Okay, so the basic issue with dark matter at colliders is dark matter is stable. It's stable on a timescale of the age of the universe, so it's not going to decay within your collider if you produce it. If there are other particles accompanying the dark matter, you can possibly look for their decay products, but generally what you're looking for in collider searches, if you're looking for a production of the dark matter directly, is you're looking for missing transverse energy and momentum. So if the DM, so the dark matter is stable, it will escape. So what it should show up out, so we'll show up. So most classic dark matter searches at the LHC are what's called mono-X searches, which is to say what you're looking for is similar in way to direct detection is what you're looking for is a visible partner recoiling against an invisible dark matter that's produced in the collision. So for example, we could have a situation where we produce, suppose there's some mediator between the dark matter and the standard model that interacts with quarks. We could go through that mediator. Produce dark matter particles. This is like the inverse to the annihilation process that I was talking about earlier that we talked about as the production, but if this was all that happened, you would not see any final state signals, but we can radiate off some standard model particle from the initial quark state and then look for, we see an X, but we don't see any of the other particles that should accompany it. So this X could be a gamma, a photon or a jet or a W or a Z or a Higgs, et cetera. And then so we can look for the decay products of this X accompanied by missing energy. We can similarly, we could have, again, we can radiate off the mediator instead. If this is say a gauge boson, we can radiate off a Higgs boson. Sorry, I guess I shouldn't have written the Higgs here since I drew it as a vector. So we could look for a Higgs being radiate off some mediator. Again, we'd produce dark matter particles in the final state. We could also look for, maybe we could have some kind of situation where there's a vertex that directly produces the dark matter particles and also the X, there could be some particles in the dark sector or the dark matter itself could directly couple to standard model particles. Such that you effectively get radiation off the dark matter side of the interaction. So we could get, so we can get these extra particles either by radiation from the standard model side of the interaction can generate. So we can make something on the dark sector side some partner particle. So in Susie searches, for example, I might produce some heavier supersymmetric particles which when they decay will produce both standard model particles and the lighter supersymmetric particle which is our dark matter candidate. You can also have specific searches for other specific models. For example, we know in Higgs, you know dark matter would have partner particles that are charged and that are just slightly heavier than they are. So you could look for production of those charge particles. They would decay on length scales that can be seen within the collider. So you could potentially look for the, at least particles that are closely related to the dark matter that way. There's are essentially two broad approaches to categorizing these searches. So this is the very general idea of many collider searches but there are many, many variations on this. Unlike indirect detection, we can't really just say, okay, I've got some generic spectrum. I'm gonna look for it. The particle physics is all encoded in basically just a pre-factor. You often in collider searches need to specify what specific kind of model you're looking for. And there are essentially three approaches that I know of to that. Approach one is to say, right, I have some complete model. I have some supersymmetric model. For example, I have some minimal supersymmetrics down to model parameter space. I'm going to scan over that and I'm going to ask, how does this particular search constrain that UV complete full model? This is great provided that you think the answer lies within that UV complete full model. If you don't, it can be difficult to recast those constraints to other more simplified models and to figure out what physics you are really constraining. So then the sort of next less complete, maybe more generic option is to work instead with some kind of simplified model. Like to say, okay, let's suppose, for example, we're going to posit a dark sector that contains the dark matter plus some mediator particle. Like in this situation, let's allow the mediator to be a scalar or a pseudo scalar or a vector or a pseudo vector. Write down the kinds of, write down the kinds of diagrams that we can get, use our observations or non-observations of a signal in these channels to constrain, say, the coupling of the mediator to the dark matter, the coupling of the mediator, the standard model and the mass of the mediator. This, I like this strategy personally. It can be generalizable to a wide range of models. The potential danger with it is it's possible to write down simplified models that don't actually have a consistent UV completion and that can give you very strange results. For example, if you just posit some vector mediator with axial vector couplings and it has some mass, you actually get inconsistent non-physical results if you don't also include some mechanism to give that vector boson its mass, such as a Higgs field in the dark sector. So, simplified models can be very useful, but you have to be cautious that what you're writing down is actually a reasonably complete model and that you haven't left something out that could significantly alter the phenomenology. And then even further in this direction, you have effective field theory based strategies where you just write down the operators that would describe these various processes. This is in some ways the most model independent of all. You're just looking at operators. You're just assuming that the only relevant dynamics is the dark matter and everything else can be integrated out, just expressed as strength of some interaction between the dark matter and the standard model particles. The danger here is again, as in the simplified models case, you can write down operators that don't actually fit into a consistent UV completion. And also that in order for this to be valid, it has to be true that the particles you're integrating out are heavier than the energy scale of the interaction that you're looking at. And you can run into situations where, as a result, you can't rule out any parameter space at all because the only things that you can potentially probe are where the media does a lighter than the energy scale that you're dealing with. And thus the effective field theory just isn't valid. So that's my 10 minute introduction to dark matter searches at colliders. I will again point you to those reviews that I gave references to at the start for more detail. If people have specific searches that they want me to talk about or chat about, I'm happy to do that in the discussion session. But that's the sort of very broad picture of the kinds of searches that you can do with colliders. Thanks very much.