 Sorry, I was checking my internet connection. OK, maybe we can start already. So we are very happy to have Tom Yanlin, who will start today his first lecture on the direction of wider matter with condensed matter systems. So please, Tom Yanlin. OK, thank you. Yeah, good to see you all here. And as Mr. Joan already said, please feel free to interrupt with comments and questions. I'll be giving two lectures talking about direct detection of light dark matter, particularly how we can use condensed matter systems, solid state systems as will be the emphasis to detect KV to GV scale dark matter scattering. OK, so to start with, let me just tell you the plan for these two lectures. So today I'll try to keep a little bit more introductory and lighter, talking about dark matter direct detection currently, focusing on WIMPs and motivating the need for condensed matter systems as a powerful way to detect sub GV dark matter. Well, condensed matter systems and other novel direct detection systems and techniques. So it'll be broken into roughly two parts. First, we'll overview dark matter direct detection with WIMPs and get into a little bit more detail. And I'll draw some analogies with the condensed matter or other systems. And then in the second half, I'll motivate these solid state systems and other systems. And just give a general overview of the kinds of excitations we have in solid state materials. So kind of just a condensed matter review, a brief review, so we get the right language. And then in the second lecture, we'll talk more in detail about what the kind of interactions for dark matter, what kind of interactions we have for dark matter in these systems, how to calculate the rates, current status of experiments. So as I go through today, what I'll draw on is mostly my lecture notes from 2019. So also some additional material. And then if you are interested in more detail in direct detection of WIMPs, I've also put another recent review. All right, so to get started, I want to talk a bit about direct detection in general. And I know you saw a little bit of that already last week. But still, it's good to get a reminder. So the goal, of course, of direct detection is to search for the dark matter in our galaxy and look for its particle interactions. So let me draw the cartoon again of us on Earth. And as we're orbiting, say, the center of our galaxy, which I'll put here, where there's a black hole. And here's our sun. And so together, we are moving around the center of the galaxy right now at about 240 kilometers per second. On top of that, we have some small amounts. Actually, I forget the number, 20 or 30 kilometers per second of the Earth moving around the sun itself. Whereas we live inside of this larger dark matter halo with approximately isotropic velocity distribution, we think. At least that's the simplest model. And so because of that, we'll see due to our motion through this halo, the wind of dark matter coming from, as we say, Cygnus, velocity 103, or a few hundred kilometers per second. So it'll extend up to about 2 or 3 times 10 to minus 3. And my cartoon is not very accurate, of course, in that it's really going to be lots of random velocities. So we'll go like this, et cetera. But on average, that's the typical velocity we'll use. And so as you saw last week, this is an important probe because it's complementary to other kinds of probes of dark matter. It's a way of directly getting at the interactions of the long-lived particles in our galaxy. If we compare with collider searches, it's hard to tell when we produce something that's long-lived in a collider if it's really going to be dark matter or just long-lived on a dark matter, meaning long-lived on a cosmological time scale, or just long-lived on Earth, on Earth-length scales, let's say. And of course, depending on the model, it may show up in one experiment or another. That's the importance of having these complementary approaches. All right, so let's review then the searches for WIMPs and just do the quick estimate of the scattering cross-section. So you saw already that one of our leading ideas for how dark matter might be produced, the relic abundance, is through some thermal freeze-out, for instance, through a heavy particle. And I'll just put a Z here as an example, but it could be some other heavy mediator particle. This process annihilation could lead to thermal freeze-out and give us the right relic abundance. And that was true if dark matter masses are around 100 GeV to TeV for this, sometimes called WIMP miracle. And so because this also leads to a scattering cross-section, we can estimate the kinds of scattering rates and that will tell us something about the kinds of experiments we want. So if we estimate this scattering rate of this process, I've drawn the diagram with quarks and then we'll discuss a little bit shortly how this translates into interactions with nuclei. But in any case, if it couples to quarks, then dark matter can scatter off of nucleons, in particular nuclei. And the cross-section is going to scale with some alpha weak squared for C to the fourth coupling. And there's going to be a squared enhancement where A is the atomic, sorry, it's not the atomic number, it's the mass number of the nucleus. So overall, the typical dark matter nucleus cross-section can be estimated for the Z mediator. So to turn the cross-section, I just include the mass squared factor. And this turns out to be about 10 to the minus 30 centimeters squared if you pick a heavy nucleus. And if we estimate a rate, we can do that by taking the local dark matter density times sigma n times v. And through gravitational measurements, it's possible to get an estimate of what the local dark matter energy density is. This is rho chi, which is usually estimated to be around 0.3 to 0.4 gv per centimeter cubed. And then we have v being around 10 to the minus 3 speed of light. And from that, you would get 10 to the minus 25 per second. OK, so what does that tell us immediately? That tells us, even for this candidate, which has fairly large cross-sections compared to what we're now searching for, then heavy nuclear good targets that comes from the enhancement of the cross-section here with a squared. It also comes from the fact that, as we'll discuss, the kinematics are more ideal if we're talking about heavy nuclear, because the mass of the target is more similar to the mass of the dark matter around 100 gv. At the same time, we see that the rates are very small. So we're going to need a large target exposure. So waiting a long time with a large amount of nuclei. So we'll see kilograms to tons of our target material. OK, so let me introduce now the kinematics of dark matter scattering. And this will appear again and again throughout the talk. So I want to spend throughout the lecture. I want to spend quite some time making sure we're introducing this. So let's just start with the labeling, the incoming dark matter with momentum p, outgoing dark matter with momentum p prime. And the picture we'll have is that this nucleus is just at rest in a material, which is not entirely true, but it'll be a good approximation for the scales that we're talking about. And then when dark matter scatters off a nucleus, it'll be given some momentum q. The energy deposited, I'm typically going to label omega. So that's just given by the non-relativistic energies for the dark matter. OK, so then we end up with this relationship between the momentum transfer, the energy deposited, and the p, which is the initial momentum of the dark matter. So p over mx could also just be replaced with a velocity. OK, so if we draw this now in momentum transfer versus energy, the relationship below basically defines a parabola, where the parabola depends on the specific dark matter velocity I pick, of course. It depends on this angle. But for the moment, we can take this to be as large as possible. So the angles are exactly aligned. And the velocity is of around 10 to the minus 3. So OK, and so that defines this region here. And all dark matter scattering has to happen within this phase space. So this is an upper bound on where the dark matter scattering can happen. If you want to go outside that, you need higher dark matter velocity. And so certainly for some large enough velocity, we just don't think there's any dark matter particles outside of this region. But for a given dark matter velocity, you're bounded to be within this region. And so if you look at this parabola, basically just tells you that this scale is roughly mx v0. And v0 I'm going to use to mean basically the velocity of 10 to the minus 3. And the maximum here is 1 half mx v0 squared. OK, so this region of this phase space for dark matter scattering that I've drawn really is just, the only thing it depends on is the fact that we've assumed dark matter particles are non-relativistic and non-relativistic. And that they get a lot of the masses. Once we identify this, and we see the typical scales, the next question to ask is what kind of targets have available excitations that would land in this phase space? So as an example of what I mean for nuclear recoils, we are now required to have that the energy deposited is q squared over 2mn, which is assuming that these new fair can be treated as free particles in the target. And so if we draw what that looks like in this phase space, this is my tent. OK, add a q squared over 2mn formn that's similar to mchi. You'll get a dispersion that lands right in the middle of this allowed phase space, which is what we want. Whereas if we had the nucleus mass much larger than mchi, then that would be less ideal, because basically you're not at the point where you're getting a typical amount of kinetic energy similar to the dark matter kinetic energy getting a lot less. And as we'll see, that'll be the case basically for light dark matter. And of course, you could also do the other case where m is much less. OK, so to just do that a little bit more explicitly, I'll just plug in now. I'll just plug in now this purple equation. And if we solve it, we'll get that q squared over the reduced mass of the dark matter in the nucleus equal to pq cosine theta, which tells us that the typical momentum transfer will be around 2 nu chi n v. And that tells us the scale of the recoil energies. So the typical numbers for this, if the dark matter and nucleus mass are similar, this is going to be around 10 MeV, 100 MeV. If the dark matter, the nucleus are both around 100 GV. And this is going to be around 10 KeV, the other extra factor velocity. All right, so at the end of this, what we learned is that the typical scales of the process we're looking at are fairly high, well, high relative to the condensed matter system I'm talking about. So the momentum transfer is around 10 MeV, which is basically probing length scales that are much shorter than, say, the interparticle separation of the nuclei. So when we deal with an actual target material, of course, we don't usually want to have a gas of nuclei because we want to have so many target nuclei. Most experiments are dealing with liquid or solid. There are gaseous experiments. But if you're dealing with a liquid or solid, then you can ask, what's the interparticle spacing of all these nuclei? And for such a high amount of transfer, you're not going to be able to probe that. You're going to have the individual nuclei themselves. And similarly, if we look at the energy scales, 10 KeV, this is actually high enough that it is a good approximation to treat these nuclei as free targets. But it's also not so high that you need to worry about things like nuclear splittings. Those are generally higher scales of MeV. So it turns out to be a pretty good approximation to treat these as free particles and scattering off of single nuclei for limb experiments. And so I'll return to this diagram later when we talk about what kind of excitations are possible for condensed matter systems. So now I want to review briefly the calculation for the rate for nuclear recoils. I'm particularly drawing some parallels with the calculation we'll be doing in condensed matter systems. So rather than focus on the z-mediator, I'll just consider some operator, some heavy mediator, and consider the operator with vector currents. I just won't even write a mess. So this coefficient in front will be some 1 over mass squared constant. So for direct detection, where we want to really focus on the interactions with the nuclei, we first need to take some non-relativistic limits and consider the interaction with nucleons. So for this, it's just basically the charge, some vector charge. So the matching is not too complicated. For other cases, it'll be a little bit more complicated to go from the quark to the nuclear. And we want to further take the non-relativistic limit. And we want to decompose this into a few operators that are the leading operators that appear in the non-relativistic limit, which include one. So it's an operator. The momentum transfer, the relative velocity. So these are the leading invariant quantities under Galwayan, rather than the usual full Lorentz symmetry. So we have the momentum transfer and the relative velocity of the dark matter and the nucleus, as well as the spin of the dark matter and the spin of the nucleus, or sorry, of the nucleon. I apologize. These are all little ns, so these are the ones. So in dealing with various operators, there's a way to decompose these all into these leading quantities. And there's a nice paper that does this. I don't have the archive number, but it's not even going to get all the authors correct so I apologize. Patrick, X2, et cetera, that does this, called the EFT of dark matter direct detection. So for the vector operator in particular, this one, we can decompose it in the non-relativistic limit. So I'll break it apart into the time component and spatial components. And we can basically use the symmetries to determine which of these things will appear. Basically, for the first one, under Galilean boost, nothing will happen, so it'll just be one. And in particular, this corresponds to the nucleon number density as the scalar with respect to boosts. The second one is three vectors. So it'll be a vector with respect to boosts. And this one is going to look like if you just did a spinner decomposition, it would look like this. The momentum transfer would appear, which also makes sense because that's the only thing in the list of possibilities, which is a vector and depends only on the nucleon. So the relative velocity will depend on both the dark matter and the nucleon. Thank you for posting the link to that paper. And so this one is, as you can see, it's typically suppressed by q over mn, which is going to be 10 minus 2, roughly. And so it's possible to just neglect this operator. And so what we find, so leading coupling for the vector mediator is to nucleon number density in non-relativistic limit. And that's what we would call spin independent scattering as well, because this doesn't depend on the dark matter spin or the nucleus spin. Well, spin independent scattering refers to dependence on the nucleon spin. All right, so continuing our discussion of this rate, we have that the leading operator is going to be the nucleon number density, which I'm going to write as rho, so it may be more explicit. And you could write it like this. So i is summing over all the nucleons inside a nucleus, which you could also write as sum of i, u, v, i, q, i. And then when we go to calculate our rate, then what we have is a non-relativistic matrix element, which depends on a dark matter piece. And I'll just keep the zero component there as well. And this piece, which depends on the initial and final state of the system you're looking at in the expectation value of this number density operator. So I won't go through all the details of how you get the cross-section. Of course, then we would take the matrix element squared. We would integrate over the face base, et cetera. But I want to focus a bit on this part of the matrix element because that matrix element will appear very analogously when we turn then to condensed matter systems. So the point I want to make is that in dealing with this, this is something that depends on all the nucleons, protons and neutrons inside the nucleus. And technically, or what I and F are, are nucleus states. So for instance, if you took a nuclear shell model, it would be many body states in all the nucleons. So because we're dealing with elastic scattering, the final state is basically just the initial state but boosted with some velocity. And so the matrix element we care about can be reduced to just n, n where these are both the ground states of this nucleus. So what we're dealing with is we're coupling to a density operator in some constituent of the system. And then we want to evaluate that in the general many body states, in this case, a nucleus made of nucleons. And we can approximate it, in this case, with, say, just a calculation of the density inside. So if you have a model of the nucleon density inside, you can just calculate it directly. And you can do that with shell models or an example of a common one that's used is a Helm form factor. All right, so this form factor will appear in our rate calculation. Actually, here we go. In our rate calculation, I've shown the result for this Helm form factor as a function of here, the recoil energy, which is just related to q by q squared over 2 mn. And so at low momentum or low energy, so low recoil energy, which corresponds also to low momentum transfer, the Helm form factor just amounts to summing up all the nucleons inside this material. And you'll get this a squared, which is the enhancement I talked about near the beginning. It's coherent enhancement over all the nucleons. But at higher momentum transfer, then you'll see some, you're basically probing within this nucleus. And you'll see some interference and some suppression, for instance, for xenon here at higher recoil energies. So this form factor is basically some form factor for the target system telling us how we're able to probe it. And so there will be an analogous quantity later on called the dynamic structure factor for materials. And so I said I'll skip the details, but if we go now and use this matrix element square to calculate a rate, this is what the differential rates look like. So nt here is a number of nuclei per kilogram of material, or dark matter number density. This is the dark matter nucleon cross-section. And then finally, we have an integral over the dark matter velocity distribution on Earth. So that's what f of v is, along with some step function that tells us we're only integrating over velocities large enough to create this nuclear recoil. So this function encapsulates the astrophysics side, sometimes called eta v min, where v min is the minimum velocity for that recoil energy, which here is q over 2 mu chi n. So this gives a rate per unit time and per unit target mass and per unit recoil energy. And so usually, I'll talk about an exposure, which is the amount of time and the amount of target mass for a given experiment. For instance, we'll talk about experiments ranging from kilogram day to ton year, which is where we are now, the kinds of exposures for wimp detection. All right, and here's a plot then of that differential recoil rate shown in the bottom right for a xenon target and different dark matter masses. And you'll see as we lower the dark matter mass to well below the xenon mass, xenon is most similar to the mass of the 150 gv1. As we lower the dark matter mass, then our spectrum becomes much more, well, the maximum energy, of course, is going to decrease with the dark matter mass. And you also learn from this that the spectrum of the recoils is generally exponential in this recoil energy, because we're often most sensitive to what this dark matter velocity distribution is, which has an exponential tail at high velocities. OK, so in the last minutes of this first section, then I'll just review what the existing experimental salts look like and their techniques. So one important point about the searches for wimps at higher mass above 10 gv and dark matter mass is that they're typically trying to use at least two different handles on nuclear recoils. So for instance, in this diagram, I've taken the graphic from the xenon collaboration. So if dark matter comes in and recoils off a nucleus, then there will be two types of signals that can be observed. One is immediately after hitting the nucleus, there's S1 signal, which is scintillation light that's produced just from that initial collision. So there's an immediate pulse from the scintillation light. At the same time, a bunch of electrons are being produced. And there's an electric field being applied. The electrons are being drifted. And that's our S2 signal when sometime later, the electrons can also be detected once they reach the top of this. And they're converted to a light signal as well. So in that example, the experiment is detecting two channels of charge and light and able to use the ratio of the two to help distinguish dark matter events from other background events. Another point about these is that radioactive backgrounds are one of the big challenges. And for instance, if you look at the most recent xenon one time collaborations result, those are the leading backgrounds, leading irreducible backgrounds right now. So the situation will differ somewhat once we get to the low dark matter, light dark matter mass, light dark matter searches, where the backgrounds will change, in addition, we won't be able to detect necessarily multiple channels for each recoil. So this is a recent plot showing the exclusion limits. See, the strongest at around, strongest at around between 10 and 100 GV in mass, the reach here or the exclusion curve cuts off at around 10 GV due to threshold energies around KEV for detecting these nuclear recoils with two channels. So if you want to do this two channel detection, it's going to be much harder at lower energies. So often for these, there's a typical threshold of around KEV and that's setting the lower boundary of this exclusion. The upper boundary of the exclusion is set mainly by the dark matter number density. So this is going as 1 over mx. And that's basically coming from the fact that we have a dark matter number density of kai over mkai or pro-x over mx in our rate. And this is just a zoom in on that low mass region. So this starts at 10 GV. And you see the xenon-based experiments are dropping off many orders of magnitude here. And there are some experiments going to as low as 1 GV. But the sensitivity curves are much higher, of course, in cross-section. OK, so what we've learned is that the nuclear recoil approach is not going to be optimal for sub-GV dark matter. The kinematics is just off. We've seen the phase space for dark matter scattering. And if the new mass is much heavier, then it's not optimal. We're not able to get much recoil energy. And so in the rest of these lectures, I'll be talking about how we can extend this down. And we'll go down to roughly kV. And we see that in terms of the direct detection cross-section, it's really quite open. The allowed cross-sections are much larger. And there's all this open parameter space here. So what are the things that live in this parameter space? For MEV to GV masses, we will have additional thermal relic candidates, thermal, let's say, freeze-out candidates, actually, let me, more specific. That's usually cut off of around 1 to 10 MEV if we're talking about freeze-out from the standard model. Because otherwise, if the mass is below that, then that might affect BBN. That can affect BBN. But there are also interesting dark matter benchmarks below that. So there's secluded freeze-out. So you saw freeze-out, but you're not directly freezing out from the standard model, as well as freeze-in. These are possibilities down to order few keV. And this can be accomplished if there's mediators beyond the heavy mediators that I talked about. So sub-GV dark matter, at least thermal relics with sub-GV dark matter, let me write that, requires light mediators. And a common one will be the dark photon mediator. So this dark photon mediator, I'll discuss a little bit more tomorrow. But basically, it'll couple to the electromagnetic current with the additional factor in front of epsilon, which is the kinetic mixing. And also, couple to dark matter with coupling G chi. And usually, this mass will be below GV, as well. And so we'll consider interactions with light mediator exchange, which, if the mediator is very light, will have this kind of amplitude. And so this kind of signal might favor even lower recoil energy than before. And if we look at this coupling, it also implies that there's a coupling with electrons and not just nuclei that you might want to explore. So in summary, there's a range of dark matter models, thermal relics below this GV scale. And we'll see that they can give some interesting signals that are particularly peaked at low energy and that they also motivate searches beyond nuclear recoils with electrons, in particular. So that concludes the first part of today's lecture. And I guess we'll take a break now, right? Take any questions? Yeah, sounds good. Let's take five minutes. So we come back at 6 here. I'll show what time is there. See you in a bit. All right. Maybe we can resume the lecture. OK. So now we're on to part two of this lecture, giving an introduction to why condensed matter systems are interesting for this sub-GV direct detection. Excuse me, Don Gian, maybe you can take one question before we resume, because I see there's a raised hand. Probably it's from before. Couple. So, Maria, whenever you want. Hi. Thank you for the lecture, Don. I wanted to ask you a question about the previous blood. What does it mean, the yellow region and the green curves? Yeah, I didn't go over those at all. But let me, yeah, thank you. So the green curves are two projections for future nuclear recoil experiments going to lower threshold. So with two different targets, silicon and germanium. So they aim to get to threshold, I think, of around 50 EV, if I remember the number correctly. So they want to reduce their threshold. And that's a projection from 2016 paper that they're working towards, but it's a future reach. The yellow is a calculation of discovery limit due to coherent neutrinos scattering. So there's solar neutrinos that the flux below a certain energy becomes much larger. So there's a kind of cliff there at around 8 EV. And so if you have solar neutrinos, they give a signal that's almost exactly the same, except for the spectrum, as a nuclear recall from a dark matter. And so you can compare that and ask, at what point in the cross-section for dark matter do you get, say, a similar rate between coherent neutrinos scattering and from the dark matter? And that's roughly where the orange is. It's not exactly because you can try to distinguish neutrinos and dark matter from the recoil spectrum. OK, thank you. All right. My next question is by Dong. Hi, Professor. So I have two questions. The first one is for the nucleon scattering cross-section, did we calculate that by assuming the nucleon is at rest and take the non-relativistic limit for the dark matter? So we assume the nucleus is at rest, but the nucleons themselves, they're not at rest in the sense that the nucleus we can treat as a bound state of the nucleons. So we did assume the nucleons, or we took the non-relativistic operator of coupling with the nucleons, but we didn't assume the nucleons are at rest, just the nucleus. So how can we calculate that cross-section by using the center mass system? Right, that's where this form factor comes into play, because the thing that's actually, here we go. The thing, so one thing I said earlier was that if we look at the energy scale of the scattering, it's not high enough to excite nuclear states. So you can't excite internal nuclear states. So the only thing that's going to happen is the nucleus is going to be boosted, or it's going to move, right? It's going to recoil. And so the details of how the nucleons are distributed in the nucleus all appear in this form factor when we, basically, the density distribution inside the nucleus. So the picture is we're just coupling to the nucleon number density, but then there is a target nucleus specific quantity that you need to figure out, which is that distribution inside. And that's where the, you can think of as the wave function of the nucleons. That's where it comes into play. Oh, OK. And my second question is to building the larynion for the interaction between dark matter with the nucleon. If we consider like the vector current, we will multiply with the vector current, we will have this. But if we consider like a vector current multiplied with an axial current, so what will happen? Yeah, that's a good question. There's other form factors you would have to calculate then. So this one is just for the vector. But if you have other, like say an axial vector here, then you would have a spin dependent interaction. So you would have an SN operator appearing, for instance. And then you would have to calculate a spin matrix element. So you'd have to calculate SN of the nucleons inside the nucleus. So that's actually one of the things that's kind of done in this paper, is like using nuclear shell models to calculate those matrix elements. OK. OK, and sorry. And my third question is when you introduce a parameter that's called kinetic missing term. So you have the two dark matter colliding into a dark photon and then going now to standard model particle. But should we include like another two diorama, like the dark photon will missing and reduce a photon or reduce a z bosom and the particle will decay into standard model particle? Yeah, the mixing is already accounted for when I write it in this basis. So maybe I can discuss that more next time. But the kinetic mixing you can rotate into this interaction term. So it's already there. So in principle, there is also like a prime coupling with like, I guess you could call it z, something like this. But it tends to be very suppressed for low mass dark photons, for sub-GV dark photons. It's very suppressed. OK. Thank you, Rukes. There are no more questions, so you can proceed if you wish. OK, great. All right, so in the second half, we'll discuss the motivation for condensed matter systems. And this plot now is taking this dark photon model. So it's a particular limit of the dark photon model where you have freeze in as a potential benchmark. So that means on this line, you could produce all of dark matter through a process called freeze in. I'm not going to go too much into detail about that. That's just an example where, unlike freeze out, the dark matter is never in equilibrium. But you do get the right amount of dark matter. And up here, I'm showing a projection, for instance, for the next generation nuclear recoil experiments. And I'm also showing on here example projections for experiments looking for electron excitations in silicon and germanium. And that goes down to around MEV. And I've also shown experiments that could look for dark matter scattering off of a condensed matter system and creating phonons. And these are different, again, solid state materials. So one thing I want to draw your attention to is when we are looking below a GV, we are going to be looking at energy and momentum scales where we really do have to consider the fact that the particles we're scattering off of in a solid state or liquid, they're not free particles. We have to account for that fact. We can't treat them as some nucleus at rest. And the other thing to know is that if we talk about the exposures, the nuclear recoils here are going to be ton year or multi-ton year. And the lines I've drawn for the solid state ones, these are all kilogram year in the limit of no background, which is another caveat I'll get to. And what you see is that, roughly speaking, all of these are scaling along this line. There's a kind of best reach which goes along this line. And that's exactly this 1 over m chi sensitivity scaling with the dark matter number density. So we had a rate proportional to rho chi over m chi. And so if we look at the rate for sub GV dark matter, because of this enhancement factor, even with relatively small experiments, it is possible to get quite strong potential sensitivity in cross-section because of this. And so we'll see that the rates can be quite high. And that's very important because as we continue to lower the energy thresholds to get to this low mass dark matter, we're going to be potentially running into lots of low energy backgrounds. And it's going to be hard to build these with very large target volumes compared to xenon experiments. We can have tons. And you can do these look for KV nuclear recoils. It would be very difficult to have a ton of silicon and look for much lower energy excitations. But that's OK because we only need kilogram scale. We generally, well, a good target is kilogram scale experiments to get to interesting benchmarks. All right, so let me go back to this diagram I drew earlier. Remind you of the energy and momentum for dark matter scattering, momentum transfer. That was 1 1⁄2 MV squared. So let's say we were looking for dark matter of around an MeV in mass. And this momentum transfer will be KeV. And the energy will be around MeV. And as I mentioned earlier, we want to look for excitations that kind of live in this plane. And if we looked at nuclear recoils for those masses, it would be some parabola, which has an extremely small slope. So it would definitely not be optimal. And this is even probably too high of a slope here. So it lives somewhere down there, whereas what we really want is some kind of excitations that live right in the middle of this thing. So if we had free electrons at rest, that would give you exactly the same thing, Q squared over 2 Me. And that would actually potentially be right in the middle of this thing. So that would potentially be right in the middle of this. But we don't have those. Of course, it's possible to have small numbers of free electrons. But if we're talking about a typical material target, then the electrons are not free. They're in bound states. And so we can't exactly use this dispersion. But still, if we look at this, that kind of motivates already looking at electron recoils. That's in addition to the fact that if you have a dark photon mediator, then you will have electron scattering. So there's two reasons to consider those. So now I want to outline some of the possible states that could. We could excite at these energy scales, the light scales. So we do have available electrons to scatter off in atomic states. And so here, I want to look at the available energy to excitation. And that's going to be order alpha squared Me, which is around 14 EV or so, or it is around 14 EV. And it turns out that the favored momentum transfer will also be around alpha Me, which is 5 KeV. So that lives right in the middle of the space space as we want. So atomic states are a good, basically, atoms are a good potential target. And indeed, one of the first probes of dark matter electron scattering used xenon as an electron scattering experiment. So in this approach, it's kind of neglected the fact that we really have a liquid xenon. And we just treat the xenon as individual atoms, assuming that the atomic states are not modified too much. And already about a decade ago, they showed that they could see single electrons being excited. So in particular, a single xenon atom being ionized leads to a single electron being extracted. And they could see down to that single electron. But then it's down to that single electron. We could also consider molecular states rather than single atoms. If we consider molecular states, we'll get similar types of electron transitions. So once we have a molecule, we have to consider how the bound states of the electrons get modified. But the typical energy momentum scales are similar. With molecules, you can also consider vibrational modes. And these will have energy 0.1 to 1 EV. So instead of thinking about the dark matter hitting a nucleus and just the nucleus recoiling, let's say the dark matter hits one of the nuclei in this molecule. Or the dark matter comes and scatters off a nuclei in this molecule. And it'll excite a vibrational mode of order 100 mil EV or an EV. That could also be a good candidate. It could also just break the bond itself. And that will be a larger energy threshold, more like tens of EV. And then as we continue to increase the density of our potential target, we go into solid state systems. And for electron excitations, then we have a similar possible momentum transfer of KEV. But the available or the possible threshold for electron excitations is lower. It can be EV in semiconductors. Or it could be even smaller. Really, it can go to 0 in a metal. So you can consider a wide variety of solid state materials, which is interesting because you can then pick your threshold energies to match different dark matter masses. So that's for electrons. The vibrational excitations in solid state materials turn into phonons. And these will have lower energy below 100 mil EV. So if we're talking about dark matter scattering off a nucleus at order MEV mass or below, we actually can't really talk about that because the fundamental excitations will be phonons, which are the coupled excitations of all the nuclei. And you could also consider bond breaking in a solid state material as well. That'll be a similar amount of energy to basically displace a nucleus entirely from its lattice site. All right, so what does that mean in terms of the mass range of dark matter we're sensitive to? When the energy threshold is around 10 EV or so, so that's the case for this one, for this one, that corresponds to dark matter masses of around 10 MEV since the kinetic energy will be around 10 EV. Sorry, that also applies to molecular electron excitations like your bond breaking atomic states. If we go a little bit lower to about 0.1 to 1 EV, we could look at vibrational modes in molecules. We could look at solid state systems with EV gaps. Those, I would say, go down to about, they can probe dark matter masses above about MEV. And lastly, we could go even lower. And that means even lower thresholds in the electron excitation or very low energy phonons. And that would allow us to get to KV to MEV dark matter. So all of these different systems have been proposed in the literature in recent years. And there are still lots of ongoing work like exploring new systems. But I want to mainly focus in these lectures on the solid state systems, in part because you see, one, as you see, if we look at the solid state systems, there's really there's so many possibilities you can use them for somewhat heavier dark matter. If you're doing bond breaking, you can use them for MEV scale to 100 MEV scale dark matter. You can also use them for really light dark matter below an MEV scale. So they offer a really wide range of possible excitations. There's many others beyond what I've talked about. So this is electron excitations and phonons. Those are the ones I'll emphasize because they're the most common. But I'll just add, in an actual condensed matter system, there will be other potential excitations that you could try to exploit, like spin waves or other things. So it's both the kinematics, the flexibility in the kinematics for solid state systems in matching light dark matter, and the availability of lots of different types of excitations. So you can even think about how you can use that to combine different solid state systems to figure out what kind of dark matter model you're looking. You would see if you saw something. So a lot of this was already pointed out in the original paper on this, which should have been cited on the previous one, but it's cited now in this slide by Essie Martin and Volanski pointing out that dark matter electron scattering is interesting and the possibilities for using some of these systems. So while that's just kinematics, we should probably talk more practically about some other key considerations. We want to consider this as a real experiment. So let me start with a few of these. So one is just thresholds. As I mentioned, the xenon experiments have a nuclear recoil threshold of about a KEV. And as I said, that's really coming from the, or one of the key things it's coming from is that they're requiring two channels to see a nuclear recoil. But if they don't require two channels and they're just talking about a single electron being produced already in one decade ago, xenon 10, even though it's not what it was designed to do, they showed that they could do, they could look for single electrons being produced. And that was a threshold of around 12 EV. So already it was shown a decade ago that we could consider, start to consider, quite low thresholds. And so one EV is not too far off from that. And indeed, now their experiments with one EV threshold. Another question you might ask is, what about exposures? Well, I already argued that when we're looking at low mass dark matter, we don't need quite as much exposure to reach pretty interesting cosmological benchmarks and dark matter models. So just as a point of comparison, the xenon 10 had about 15 kilogram days. That's already producing pretty interesting limits. And as I argued earlier, these are small scales. So we need gram to kilogram size. So we don't have to worry about reaching tons size for now. That is, of course, assuming that backgrounds are small. And so the case is that known backgrounds are small. The xenon experiments in looking for KEV nuclear recoils are fighting against other kinds of events at KEV, such as coming from radioactive decays. If we're looking at EV scale energies, the radioactive decays don't matter as much. The rates in that energy range is not quite as important. But of course, a big challenge is that this is a new energy regime. So known backgrounds, meaning radioactive decays, should be small, but this is totally new. And there's certainly unexpected backgrounds, including ones that we're learning about now. OK, so that's a little hard to answer, but there's a lot of promise that these experiments are feasible going ahead. And the question for theorists is to now how do we calculate the scattering rate inside this condensed matter system? So I'm going to draw an analogy here to the matrix element that we wrote down for nuclear recoils and argue that for some of the common models that we'll consider, we'll still couple to the nucleon and number density, but we'll also couple to the electron number densities. And so the kind of matrix element squared we'll get looks quite similar. So I have an initial state and a final state. And I have the electron number density. So this is the electron number density. And normally I would have the nucleon number density, but I'm going to integrate that out. And this instead is the ion number density. So what that means is I'm going to treat this solid state system as a combination of valence electrons, which are the most weakly bound electrons, and ions, which are the nucleus. So this is nucleus plus tightly bound core electrons. So in the model of the dark photon, this is relatively more straightforward because I can just talk about the electromagnetic charge of the nucleus plus the tightly bound core electrons. So similar to before when we were calculating the form factor. Here we are calculating some sort of form factor. But the initial and final states are condensed matter states or many body states. So it's a condensed matter system. And now instead of just summing over, say, the nucleons in one nucleus, we're really summing over all the electrons in the target material and all the ions in the target material. And let me point you out that we are past half, half past six. So you can take five or ten more minutes, but then we would open the Q&A. How are you doing regarding the material that you wanted to cover? Well, I think I can just, yeah, I can finish up in a couple minutes, yeah. OK. Yeah, so I had some extra material introducing the excitations in the condensed matter system. But I won't go into it because it's OK. It's more background material. I think the things I'll just point out for today are what the excitations are in the condensed matter system. OK, so this is the kind of rate that I'll be focusing on calculating next time. The question that we have is, OK, I'm coupling to all, I'm coupling to the electron number density operator in this condensed matter system. What are these states I and F? How do I even start to write them down? And the answer is a bit complicated. But I'm going to start by just introducing possible states for I and F. And those are just single electron and single phonon excitations. So if I had the most general Hamiltonian for this system, I would have something very complicated with summing over many, many electrons and ions. So these are summing over all electron kinetic terms, coulomb interactions between potential between electrons and ions, as well as ion-ion interactions, electron-electron interactions. So that would be very complicated. And it can be simplified. And we can look at two basic types of excitations in a simplified picture. So the idea is that there is this complicated Hamiltonian. But there's an effective theory with low energy excitations. And there's two types that we'll consider. One is phonons, so for this toy model. And the phonons are the collective motions of the ions in this solid state. So I said our picture of the solid states will be lots of ions, periodic lattice of ions, as well as delocalized electron states on top of it. So instead of hitting a single nucleus and getting a nuclear recoil, we get phonon excitations. And the dispersion for the phonon excitations is shown in an example on the right. So this is now momentum, or this is momentum. And this is in units of pi over a. So this is kind of like keV at the edges, or a few keV. And what you'll see is you'll get acoustic phonons, which have a linear dispersion. And you might also get gapped optical phonons, which appear if you have, for instance, two different types of ions in your lattice. And so they can oscillate together, which gives these acoustic phonons. So if you can oscillate out of phase, then you get the optical ones. So you'll get phonons up to about, as I said, 100 milliEV. And I'll just skip this, too many slides. And you also get electron excitations, which gives a similar kind of band structure. So what I'm showing here, you can take to be a momentum. And this is also going to be order keV. And this band structure now shows the dispersion of available states for electron excitations. OK, so since we're out of time, I'll spend a little bit more time at the beginning of next one, making sure we're on the same page with reviewing electron and phonon excitations. But to conclude, motivated-wise solid state materials are interesting direction for sub-GV dark matter. And next time, we'll discuss more about how to think about the rate and how to calculate the rate in these materials, look at the proposed experiments and current experimental status. So since I'm out of time, I'll just stop there. Thank you. Thanks, Nanyan. All right, so now we can start the Q&A session, which will not be recorded. So don't be afraid to ask any question. First question is by Tom. Please go ahead. Thank you, Professor, for a wonderful lecture.