 about, you know, what is CP violation? What do we know? What are we going to learn? And what is going on with long-based on neutrino oscillation experiments today? And, you know, what are some perhaps surprises there? And this is work done with Rebecca on the right and Yulia on the, sorry, Rebecca on the left and Yulia on the right. And I should say I'm speaking from occupied to talking to them. So before I get into kind of the latest state of the art, I'd just like to review a little bit on neutrino oscillations. I think this plot here shows very nicely the history and where we are today and what is kind of the phenomena of the six oscillation parameters. So in 1998, you know, Super K saw this thing at 5.1 sigma, which indicated the neutrino oscillations, which immediately added seven parameters to our model of particle physics, of which six of them can be probed by neutrino oscillation, which is a non-trivial statement. So since then, so theta two three is the mixing angle that Super K, you know, is sensitive to and really new mu disappearance experiments, which is what atmospheric neutrinos predominantly are are sensitive to. And over time, this has improved, honestly, not very much, right? It's improved a little bit, but many experiments are sensitive to this and it's still as tough to know. It is close to maximal, which is right in the middle here. There was a brief period where we thought it might not be maximal from a NOVA analysis that then turned out to be perhaps a little bit, that had a little bit of a problem there and they corrected that and that went away. But so this is the first thing that was measured historically, but it's something we still don't know very well. We don't know if this is maximal, if it's above maximal or below maximal. And that's called the octane question. We'd like to know that. At the same time, Super K measured a frequency, of course, that dictated when these oscillations are happening in, you know, L over E space distance and energy. I didn't measure it super well, but many, again, many experiments are sensitive to it and it has steadily improved. In particular, we got a very robust measurement in 2012 from Dibe and Reno. And this really nailed it down. Now there are two lines here for the best fit cases, if you look very closely. These are the normal ordering and the inverted ordering. And we don't know, you know, in some sense, which of these is correct. Put another way, if we could tell through the higher value or the lower value, we would know the sign of this, whether it's positive or negative. And I'll talk more about this later. Now, a few years later in around, you know, 2000, 2001, 2002, snow, a solar neutrino experiment, you know, confirmed the solution to the solar neutrino problem is due to the neutrino oscillations and the LMA solution. And that provided a decent measurement of theta one two, which with more data, they improved. And then, you know, 2007 and on a few years after that, Camland, a reactor experiment, had an independent measurement with comparable precision. So we see the air is kind of improved and then we kind of settled on basically the same value since then. But you'll notice that, you know, solar neutrinos do a terrible job of measuring delta M squared to one. There uncertainties are still today very large, but reactor experiments do a great job. And then they give us a nice measurement of this. Although to be clear, there's only really one measurement here, which is from Camland. So we just hope that this number is what we think it is. Now, as this story was developing with these four oscillation parameters, there was a sort of theoretical bias growing that perhaps theta one three should be zero or at least very, very small. Not everyone thought this, but many people did. And to that extent, to that end, you know, these experiments, Diane Bay and Reno, and also Double Show, went out to measure assuming that it was maybe a degree or less. Well, as we know, it's around nine degrees. So they measured it very easily at very high significance. And in fact, this first measurement, you know, came with only two months of data. And of course, they continued collecting data and we now know if theta one three very, very accurately, which is nice. This brings us to the final parameter in this story, which is, you know, as you can see, this here from zero to two pi is the entire physical range of delta, of course. And for most of us, we had no information on it. And that's not, you know, that's not an accident because until we know all the other oscillation parameters, this parameter actually isn't even physical. But unless we, you know, if theta one three ended up being zero, then delta is not a physical parameter, it's irrelevant. You can sort of gauge it away in a sense. But now that we know, you know, as of 2012, we knew this is non-zero. So, you know, now this can start taking values and for long baseline appearance measurements, which is the best way to prove it, we start to get some indication and we have some information that it's probably not between zero and pi, although that's not, you know, there's still some amount of values at three sigma, but in general, it's probably in this upper region here. And of course, really understanding what it is and nailing this number down is the goal of upcoming oscillation experiments. So what is CP violation? This is something we sort of take for granted now. I mean, but at one point in time, CP violation was very shocking. People really thought that the Lagrangian of particle physics was going to be invariant under CP. Actually, I thought it was gonna be invariant under C and P and I thought, oh, maybe it'll be invariant under CP. We now know that that's not true. But the story I think is a little bit more subtle than that. I don't think we can just say, you know, CP violation is broken, so we expect it to be broken everywhere, right? There are at least three places now that we know that CP violation could appear. There could also be minor on a phases, but I'm not gonna talk about those. But without talking about those, they're three places. And one isn't a strong interaction, right? This so-called GG dual term, which gives rise to theta bar. And, you know, if we rescale it, such that it's a number between zero and one, the data shows that it's less than 10 to the minus 11. So we have a, you know, I would say a very good measurement of this and we see no evidence for it, of course. And there's no explanation for this, right? This is, you know, this could be due to axions, but we have not seen any evidence of axions yet. Now in the quark sector, and this is where CP violation has been detected, if we parameterize in terms of the Yarl Skog divided by the maximum, so this is again a number between zero and one, at least we have measured it. We know what it is, it is certainly there, but it's kind of smallish. It's not, you know, it's not like it's smaller than the electron or anything, but it's, you know, it's not an order one number, certainly. So that's kind of interesting. And then of course there's the third place, which is in leptons. And we know that this can be physical, as I just mentioned, because neutrinos have mass, the masses are different and all three mixing angles are non-zero. But at the moment, our constraints are not very good. It could be almost one. They've ruled out some of the space, but not much. Okay, so I think in this context, understanding CP violation for neutrinos is a top priority. Also, people have questions, if I'm speaking, you know, if they're very simple questions or very detailed questions, please don't hesitate to put it in the chat or wherever and it'll be read out to me and please interrupt me whenever. So my talk is gonna be sort of split up into two movements. The first movement is just kind of some understanding of how does CP violation, how should we be writing down CP violation for neutrinos? And the second one is going to be looking at the latest T2K and NOVA data. What's going on? Is there anything funny going on? And if there is, how can we understand that? What does that really mean? All right, so let's take a step back and think about how do we parameterize the PM&S matrix? Well, this is a three by three matrix. We're only gonna focus on three generations here. And in general, we expect it to be complex for the reasons I mentioned above. So you would think that, so on paper, it appears there's 18 degrees of freedom. But in fact, of course, there are fewer than that. Uniterity, the fact that the probability should be conserved provides nine constraints. So that takes us down to nine degrees of freedom. We can reface the charged leptons. So that means each charged leptons, say electron, muon, and tau can get an arbitrary complex phase, which removes three degrees of freedom. You can do the same thing for neutrinos, which is three more degrees of freedom. But one of those is linearly dependent on the three from the charged leptons. So it's only two more degrees of freedom removed for four degrees of freedom in the final answer. Now, to be clear, if neutrinos are morona, then this neutrino free-facing cannot happen. However, in ultra-relativistic experiments, such as every neutrino oscillation experiment we've ever done or will ever do, that you can't tell the difference. And so if you can do it in one case, you can do it in both cases. So you can do it in the Dirac case, so you can do it in the moronic case as well. But just because we went from 18 parameters down to four parameters, that does not indicate how we should write it down. And there are different ways to do it, each of which are equally valid. And I'm gonna be really going into understanding what does that mean? So how have people done this in the past? Well, lots of ways, of course. The common way, of course, the way that is done in practice is three rotations in a complex space on one of the rotations. And you can do this in a bunch of different ways and I'm gonna talk in detail about that. You can even do the same rotation twice. You can do rotation across one axis, a different axis, and on the same axis again. And that will cover the space. You could instead parameterize a three by three matrix as a linear combination of certain Gilman matrices, or really any generators of SU3. I mean, you have to pick certain ones, yet you can't do it willy-nilly, but you can cover the space that way as well. And people have done that. And in fact, there was a time, I think, where people thought that might be the best way to do it. There's also the fact you can do four complex phases. There's no reason why there has to be only one number associated with a complex number. And in fact, Boris Kaiser, who was one of the authors here, told me that they went out to do this for the quark matrix. Of course, that's the same procedure for quarks and leptons. They wanted to show that you couldn't do it because he thought there was some reason why, only having complex numbers cannot be enough to cover the space that you need to. And then they found that it didn't work. And of course, they then realized that you can write this as four complex phases. So I think that's a very interesting case to keep in the back of your mind that there's no fundamental reason why we need to have exactly one complex phase. There's also a perturbative description, which Wolfenstein pointed out, which is very popular for the CKM matrix. Of course, that's not so good for the leptons, which has very large mixing, but that is perfectly valid parameterization as well. And clearly you could do any number of ways to turn those 18 parameters into four parameters in such a way that you cover the appropriate space. So we're gonna focus on the sequence of rotations since that's what everyone does. And we will show why, how you can get tripped up here, but also how this is in general a good procedure. So these are these rotations and I'm just gonna label them like this. You know, the one rotation is the one where the first row and the first column is the proportional to the identity and two and three and so on. I'm putting the complex phase on this beta one three rotation, that doesn't matter too much. We're not gonna get too caught up on that right now. When we multiply it all out, we get this in the usual form. This is what we're gonna call the PDG because this is what's in PDG. And in terms of notation in this notation, it's written as one, two, three. So that's easy to remember. But what if we did things in a different way, right? There are a number of different ways to write this. There's six different ways where there's no repeated rotations. And if we include repeated rotations, there are another six ways. So how does this sort of our understanding of these parameters change? Now, of course, the fundamental parameters change. So the values of them change, but the physics does not actually change. The physics is, of course, invariant under these transformations. And we want to be sure that our conclusions that we're drawing are not dependent on one particular choice of parameterization. But it turns out that a lot of conclusions that people discuss are dependent on the choice of parameterizations. And this is something just to be aware of that we can get very different conclusions if we do things differently. And that's what this plot here shows, which is that for a given value of delta in our parameterization, let me think of it, which is what I showed in the earlier, the first slide, if I use a different parameterization, of course, it'll be a different number. Now in these blue and orange parameterizations, as you can see, it's not very different. And there's a small shift, but it's not a big deal. But in these other ones, these red, green and purple, the shift is, the change is dramatic. And in fact, what you can see is that with no information on delta, that is if we ignore the appearance data from T to K and Nova, and we just say we've measured the three mixing angles. So this is where we were maybe 10 years ago or so. I can already tell you what the value of the complex phase is to within 150 degrees to 210 degrees. That's 30 to three precision at one sigma, just from measuring those three mixing angles. And of course, I've not magically learned anything. Our precision on the mixing angles in that parameterization would be worse. But this tells you that, this notion of how well we know delta is very dependent on the choice of parameterization for what we mean by what delta is. And to be clear, the uncertainties on these oscillation parameters has a very small effect. That's what these bands are, is the uncertainty on the oscillation parameters. So we'll see a lot in the literature, including like the Dune TDR and Snowman reports and all these things. But we want to constrain 50% of all possible values of delta. But that's clearly parameterization dependent thing because we've already ruled out in some parameterizations, the majority of delta space. So how does that actually work? How can you go from a case where, you put in no information about delta and you get out that you have information. But we decided to break it down into sine delta and cosine delta. And we see in cosine delta that in these parameterizations, the ones that are colored red, green and purple, which have certain combinations of the rotation matrices, the cosine delta is close to negative one. I mean, that indicates something. So we looked at the Pumines matrix and this is what it is in our parameterization with no information on delta. So there's delta appears in these four elements and there's some square root with something in it. What we see is that all these elements in general are fairly large in magnitude except for this one in the corner, which as we know, this is what we call theta one three, really it's sine theta one three and it's point one five. So it's a bit small. Now in these other parameterizations, well, what happens is the location of these complicated elements, the ones with square root of a sum or a difference appear in different locations in this matrix. And in particular in these ones, the ones that are red, green and purple on the previous slides appear in this spot here, have a complicated term like this up there. Now, what that means is that it's something of the form square root of a plus b cosine delta. And since a and b are some functions of in practice, these other elements, which must be sort of largeish in order to get them to be sort of smallish, cosine delta is gonna have to be kind of close to negative one to get a cancelage. So that's what's going on. We can see this in other ways. We can see that you can translate from one parameterization to another one, very simply, we worked out some nice expressions for this. And this is basically the fact that sine delta is, in one parameterization is proportional to sine delta in another parameterization with some pre-factor. And we calculated what these pre-factors are. And you can see here that these pre-factors are just some slopes and they're just the slopes of these lines, which you can see differ from one in these other parameterizations. That's just if you want to investigate this more, you can do this fairly easily without doing all the work. You can also ask about precision on delta and how that changes with different parameterizations. And that's shown here for different true values, what kind of precision do you have on delta? Well, let's suppose that we have 15 degree precision on delta in our parameterization. Again, in blue and orange, you'll have again around 15 degrees precision. But in these other cases, you'll have much more precision. In fact, in some cases, possibly as good as one degree precision. So again, this notion of 15 degrees, one degree, things can change just by changing the parameterization. And this matters in a real sense because we can look at the physics goals of the field. And this is from the snow mass process, which is a process done in the USA about designing our experiments and strategizing as the field. But of course, this affects the global picture of neutrino experiments and particle physics as well, in some extent. And they say in the use for motivation, that the CKM angle, which is 70 degrees with around four degrees precision. So maybe we want five degrees for our delta. But of course, this is a parameterization dependent statement. This is not a fundamental physics statement, okay? So I've been harping on delta. So this means I have to provide an alternative. I can't just say what's bad. I have to say what's good. And some of you probably can guess what that is. But what we can do is we can second, what is something that is physical that we measure that doesn't depend on the parameterization? And that's gonna be the Yaw-Scog invariance course. And it's a parameterization independent thing. And this is Cecilia Yaw-Scog's paper from 1985. She actually had to fight pretty hard to convince people it was right, even though it's fairly easy to identify. People kept writing papers saying she was wrong. And they were all wrong. And then of course that was focused on quarks. But the same is true for neutrinos. And in vacuum, at leading order, the difference between an appearance mode, so this is new mu to new E, and neutrinos and an anti-neutrinos is to a pretty good approximation in this quantity here. And so there's a pre-factor, eight pi. There's this Yaw-Scog invariance. And there's the ratio of delta M squared. So what is measured in some sense is really just the Yaw-Scog. And to get delta, you then have to put in values for every other mixing angle, okay? And in particular, this requires input from example from solar experiments, from reactor experiments. And then in the long baseline accelerator experiments that are measuring this, like T to K and NOVA and the next generation doing in hyper-K, they of course don't measure these solar parameters. They're gonna take them from other experiments. But instead they could just report the Yaw-Scog, which is what they actually do measure. And the problem of course, is that if they take different values, depending on if they use this global fit or that global fit, or the KAMLAN number or the Juno number, which presumably will exist at some point, or the best fit to solar data, you'll have different numbers here and you'll get a different value for sign delta, which can then really confuse the whole picture. But the Yaw-Scog is a better way. And there's a matter of fact, which plays a role for these experiments somewhat. It's, we've showed that this is easy to count for and these problems don't propagate very far. Okay, so I wanna start, I wanna get people to start thinking about the Yaw-Scog invariant as the way to parameterize how much CP violation there is. As I said, again, 50% or 75% or whatever of delta space is not a great thing. Precision on delta is not a great thing. And there's a third statement, which is also mentioned a lot, which is that if delta is pi over two or three pi over two, that that's maximal CP violation. But that's totally not true at all. And this figure here schematically shows that. And in fact, maximal CP violation is already ruled out at many, more than a hundred sigma, right? And because for maximal CP violation, you need the mixing angles to be just right in addition to delta, right? So you need theta one two to be 45 degrees, but we know it's around 32 degrees. Okay, that's ruled out by my estimate, roughly 15 sigma. We need theta one three to be this number, which is around 35 degrees, course from Di-Van Reno, which measured this to be around eight and a half or nine degrees. That's many, it's about a hundred sigma. I feel a little ridiculous writing a hundred sigma since I have no idea what a hundred sigma means, but we can write that down anyway. Now, theta two three is, as we know, close to 45 degrees. So maybe that is the maximal value. Well, these other two, theta one two and theta one three, that's already very much ruled out. So here's this again, showing us with numbers and we kind of read this from left to right. CP conserving is in the middle. This would be yours, looking very into zero. The top is plus and the bottom is minus. And the range, the maximum values you can get from uniterity. Okay, that's the whole range of this plot here. And so from uniterity can be anything, but from the measurements of snow and the conformation from camp land of theta one two, we cut down the space to 91% of the space. Then from Di-Van Reno, of course, eight and a half degrees is pretty far from 35 degrees. So that's 34% of the space. And then long baseline experiments while they're a bit confusing as we'll see in a minute, but that rules out against some of the values of the positive Yorovskov at some significance. So we've ruled out about three quarters of the space. We're still certainly consistent with CP conserving and the data slightly prefers negative values in whatever sense that makes sense. So we have these different parameterizations. We have to use some parameterizations. And the question is, which one is best? Well, of course, you can define best however you want, but let's make some reasonable definition here. We want to be able to write down two-flavor pictures where we can. Now, for long baseline appearance, which is what I've been talking about a little bit, there's no way to write this down in a two-flavor picture, but for other experiments you can. Solar experiments, it turns out, is basically just measuring Ue2. Similarly, long baseline reactor like camp land in Juneau is also predominantly measuring Ue2. So we want this thing, which is this factor here, which is going to be proportional to Ue2 squared. We want this to be simple and not complicated, right? And complicated means it's the sum or difference of trig functions. And the problem is that if you constrain that, you can't really constrain one parameter. Of course, our information doesn't change, our knowledge of physics doesn't change, but it's a little bit harder to represent your results in a clean way. So we want Ue2 to be simple, medium baseline reactor that's like diabetes in Reno, they're predominantly measuring Ue3, and then atmospheric and long baseline accelerator disappearance. So this is like Super K, T2K, Nova, Ice Cube, and Minos, they're basically measuring Ue3. And we want these three elements to be simple because it turns out that these are the ones that are easiest to measure. And okay, the parameterization we all use, one, two, three is the only one that has that criteria, which is of course not an accident. Now other priorities may have different, may prefer different things, but in general, this is a good choice. There's a reason why we do things the way we do. And I talked about the order of the sequence of rotations, what about the complex phase? In practice, it doesn't make much of a difference. The convention that we all use for a historical reason, which isn't really valid anymore, is to put it on the one-three rotation. There are some benefits to putting it on the two-three rotation in that handling a matter effect is a bit easier. This doesn't actually change the value of Delta. Other than maybe a minus sign depending on how you do it. It doesn't really make a difference. I'm just gonna say let's keep with the way everyone's been doing it. Now, before I wrap up this section, I just wanna make a brief comment and step into the world of quarks and say that the standard parameterization used for quarks is the same for neutrinos. And we get these three mixing angles. We've got the capoebo angle and then a small angle at two degrees and a very small angle at point two degrees. And this complex phase, which I mentioned earlier is around 70 degrees. It's in the last 10 years, it's shifted down about one degree. And this looks like large CP violation. I 70 degrees in this context is large. Sign of this is point nine three, which is almost maximal in some sense. But as I said earlier, the Yarskog is three times 10 to the minus four of the maximal value. So that's small. So is CP violation big or small? The answer is it's small, but this angle here will mislead you. And so this is a concrete example of this. And in fact, if I use a different parameterization, which let's just pick two, one, two as an example, then the complex phase is actually about one degree. So clearly, this notion of complex phases as an indication of the amount of CP violation is not a robust statement. I can in a realistic situation go from 69 degrees to one degree just by changing the parameterization. Now throughout all of this, I've been harping a lot on Delta and saying we should be using the Yarskog invariant and that's completely true, but there is a caveat, I would be remiss if I did not point out that we do have to discuss Delta a little bit. And basically the reason is because given the three mixing angles and the Yarskog, there's one piece of information that's missing about the PMS matrix or the CKM matrix. And that is the sign of cosine Delta. You have to say this out loud, I'm silly, but sine SIGN of cosine Delta, because if you just know the Yarskog, all you can get is sine Delta and then I cannot tell if cosine Delta is positive or negative and that sine is physical. For example, new mutinumia disappearance depends on this a tiny bit and you have to know if it's positive or negative. So what does this mean? Well, T to K and a hyper K, they're basically only have sensitivity to sine Delta. So they should report the Yarskog. They can tell you about CP violation, but they tell you about Delta, they cannot really provide a lot of information there. And Nova and Dune, because of their higher energies and their larger matter effect, they have some sensitivity to cosine Delta. So it makes sense to report, of course, the Yarskog, which tells you about CP violation and Delta, which tells you about the parameter in the mixing matrix. And why does cosine Delta matter? Why do we care about this? This affects models. So as I've said before, of course, if you only know sine Delta, you don't know if cosine Delta is positive or negative, but if you want to understand flavor models and why the mixing matrix is structured as it is, which is one of the biggest open questions in our understanding of the standard model, pretty much all of them predict cosine Delta. And you can see this in this review by my post-doc. Here's one example that I put together where you cared about cosine Delta. And here's another comparison of many models. And you can see that if I only knew is that the absolute value of cosine Delta was, say, 0.3, and I couldn't tell the difference between this model goal ratio and try by maximal, which have different predictions as well for the other parameters. So it really matters to know what this is. So I'm gonna briefly summarize here, which is to say that the complex phase in different parameterizations can be very different looking than Delta in our parameterization. And the narrative that we have built up and the intuition we have is completely, completely changes. There's no maximal CP violation, dive bay, renal, snow, they've all ruled this out at very high significance. And really, when you're talking about in particular CP violation, you should be talking about the Yorskog invariant, the Yorskog coefficient. And, you know, but then we have to, we do have to make some choice about the parameterization. And the one we use is good and we use it for a reason. All right, so I'm gonna shift gears a little bit and move to the second phase of my talk, which is about T2K and NOVA. And they presented this plot at the last neutrino conference. Of course, we're coming up on the next one now in a month or two in Korea, and I hope people tune in for that. But we see this and, you know, it looks kind of interesting, right? So T2K prefers the region shown in black, NOVA prefers the region shown in blue. And these are these fundamental parameters, Delta and theta two three from now on, I'm just fixing us to the usual parameterization and the usual definition of Delta. And, you know, T2K wants to be at around three pi over two and maybe an upper, the upper octant for theta two three. NOVA doesn't really care about Delta or the octant, except the one thing they are, they do seem to think they know is that it's not at Delta of three pi over two and the upper octant. Okay, now the significance is clearly not very large, things overlap a lot, but let's pretend that this is a real thing. Now, before I do that, I have to explain the mass ordering. This is the one parameter or the one thing in neutrino oscillations that I have not discussed yet, but this is actually really important, not just because we have to measure, we have to measure all the parameters in our model and that is our jobs as physicists, but it also has important implications for other measurements that people are trying to do. In particular cosmology, they're measuring the sum of the neutrino masses with very good precision and the story plays out differently in the normal and inverted orderings. Neutrinal still beta decay, it is easier to probe if the mass ordering is inverted than normal. It does affect tritium endpoint experiments such as catrin, although catrin is not sensitive enough to do this in all likelihood and the cosmic neutrino background, if we ever measure that, there's potentially a significant impact depending on the mass ordering. So what is the status of the mass ordering today? It's actually pretty complicated. So individually, NOVA and TDK, which both have some sensitivity at the maybe one and a half to two sigma level, they both prefer the normal mass ordering over the inverted order individually, but when combined, they prefer the inverted ordering over the normal order and this is a surprising thing and I had to do it the last slide. It's because they prefer different values of Delta that can be resolved by flipping the mass ordering. Now, super K has some sensitivity in their atmospheric data. It's a little bit complicated, but they do have some and they prefer the normal ordering over the inverted ordering. So when you combine points one, two and three, even though NOVA and TDK together prefer the inverted ordering, super K prefers the normal ordering more. So the combination prefers the normal ordering. And you get some information from Diabane Reno because they measure the same Delta M squared in a different way. And if you combine that, you can get some information at around the one signal level. And that also slightly tends towards the normal ordering, but that's not enough to really change this narrative. So all in all, the sensitivity is at the maybe two to three signal level, depending on who you ask. One can also ask the question about the mass ordering and is it robust? So let's imagine that we measured it very well and all of our oscillation experiments agreed. So we had TDK, NOVA agreed, super K atmospheric agreed. We had, you know, June, we had June and we had everybody all agreeing. The problem is that I can, if we care about new physics and that's what I'm gonna be talking about, you cannot determine the mass ordering in the presence of new physics. And what that looks like is this. So if you have, for example, the normal ordering measured with no new physics and epsilon here is some parameter that just parameterizes the size of this new physics, that is exactly equivalent, exactly equivalent in oscillations to the inverted ordering plus some amount of new physics for some parameter of negative two. It also works the other way around, okay? So it doesn't matter how good of an oscillation experiments you do, you cannot tell if the mass ordering is normal or inverted and that will affect things for cosmology. So perhaps if cosmology and oscillations disagree, maybe we'll have to take a look and see, can we probe, can we understand if epsilon EE is equal to negative two or not? Negative two, this is known as LMA dark. And this can be broken, right? You can determine if there's epsilon EE of negative two by looking at scattering experiments. So the same process that would affect them, the oscillation experiments would also affect scattering and this is a paper which actually just appeared. This is a paper which actually just appeared a few days ago. But basically the problem is that scattering is only sensitive to new physics heavier than a certain scale. So older experiments like Charmin-Newtab ruled out this epsilon EE of negative two for sufficiently heavy mediators more than around 10G EV, coherent, a recent experiment that measured the seventh process, coherent elastic neutrino nucleus scattering ruled this out for mediators heavier than around 50 MEV and then also cosmology we show rules this out for light mediators. So if you have things that are too light then that'll mess up BBN and the CMB but that leaves a small window there and we recently show that data from a reactor experiment rules this out for any mediator mass but there are still some ways in the flavor structure to rule it to get around this which I would say this is just an ongoing direction. So it is still possible that the mass ordering might not be robust but it requires increasingly precise values of new physics. All right, now I'm gonna go back to T2K and Nova as I said, I'm motivating this by the fact that they seem to disagree the significances are not very high but it still is, I think a very valuable exercise to address this for a couple of reasons. One is, if they continue to disagree what kinds of things are we looking at and can we resolve this with new physics? And in particular, if we do resolve it with new physics what other experiments can rule this out or confirm it? And this applies both to Nova and T2K as well as the next generation experiments Dune and HyperK. I'm gonna skip this slide. So what kind of new physics can do this? Well, there's a number of things that we talk about in neutrino oscillations and this is an incomplete list. I apologize if your favorite scenario is not listed here but one obvious thing is sterile neutrinos, neutrino decay, and neutrino decoherence if they're decohering faster than we expect that'll change the oscillation probabilities. What if neutrinos are interacting with the dark matter and that modifies the parameters in a non-trivial way? You can also look at Lorentz and Verne's violation or CPT violation. You can do a more agnostic approach than sterile neutrinos and look at unitary violation. And these have been, some of these have been looked at by a few people in the context of this tension it turns out none of those work in data. The one that does work and one that I'm going to be presenting today is not standard interactions with a complex CP violation phase. And I'll explain what that means but the basic story is that NSI is like a matter of fact but with more flavor information. And in particular, because T to K is at lower energies they're at about a half a GEV while Nova is at higher energies, one and a half to two GEV, this leads to different matter effects and because of that NSI will manifest differently. So that's something different in the different experiments so this will do that. It's been known theoretically from understanding a neutrino oscillation probabilities that new complex NSI phases are somewhat degenerate with the standard phase. So we're going to want a complex phase to move Delta around between the two experiments. And how you should think of this is that since T to K is closer to vacuum they basically are measuring the vacuum parameters and Nova is measuring some combination of the vacuum parameters and the NSI parameters. It's not exact but it's a good picture. So if you don't know about NSIs this is my one slide review of this. NSIs is an EFT framework. So we're right down to dimension six operator where we've got some neutrinos with some flavor structure interacting with some fermions. And we mainly are thinking about matter fermions. So electrons up course and down course. And then we parameterize it in this perhaps somewhat unusual way with these Epsilon's and these Epsilon's are proportional to G Fermi. So this is how much bigger or smaller things are compared to the standard model of matter potential. Now it's a little bit difficult to construct a viable model that gives rise to this operator but it can be done and there's many papers that do this and there's actually some more since then that I haven't included here. And the point is to relate some Lagrangian that will affect scattering to oscillations in a relatively model independent way or at least as model independent as you can get. And the point is that these Epsilon's can then be recast and put in the Hamiltonian for oscillations. So this is the usual Hamiltonian we've got the vacuum term here the mixing matrix, the mass squareds and then the mixing matrix again and then A is the matter potential and this one is the standard model term these Epsilon's are all the new physics terms. Now there are a lot of NSI parameters. Depending on how you break it down you can have different flavors, different couplings to matter fermions. You can have different tile structures. You generally can actually have hundreds of different NSI parameters. Obviously that's ridiculous. Our data is not very good. We can barely measure the three flavor oscillation parameters. There's no way to constrain 270 parameters. So we're gonna break it down to just a couple of parameters and we're gonna focus on off diagonal with complex phases and just do one of these three at a time. So two parameters at a time, the magnitude in the phase Epsilon e mu, Epsilon e tau and Epsilon e tau. I'm gonna skip this. So we can approximate the size of this effect before we look at the data. We're gonna make this on Zatz or these two on Zatz that if two experiments are measuring different things it's because of NSI and not because of something else. And we're gonna assume that NSI mainly modifies Delta. In practice it modifies everything but the impact of some parameters is bigger than others. And so we make this thing. We say, well, what's going on the left is the reality. There is some new physics and there's some true value Delta. But experiments of course they do things assuming a regular three flavor picture and they extract some measured value Delta and we're gonna say that these should be the same. And you do this for neutrino mode and anti-neutrino mode and you do this for multiple experiments, T, K and O. And you combine them and you see what you get. What we did is we showed that to a good approximation the magnitude of this new physics whether it's Epsilon e mu or Epsilon e tau is some pre-factor and W beta is just different octants whether it's e mu or e tau. And then this term here and this term really contains all the physics. The numerator is how what they measured is different. So sine Delta T to K is what T to K thinks they measured and sine Delta is what Nova thinks they measured. And then the denominator is the matter potential. So this is how the experiments are different, right? Nova is a higher matter potential than T to K. And so that indicates why would the measure is different. And if you plug in the numbers, you get around 0.2 and you can check this equation roughly makes sense. You can also estimate the phase and you find and the details are a bit subtle but basically that the true value of Delta is basically with T to K measures which is as we saw earlier around three pi over two and you can check and you'll find that the new complex phase of the new interactions is gonna end up being roughly the same thing as what we expect. I'll skip this as well and jump right to the numerical analysis. So this is the results, this is what we all wanna know. And to do this, we actually have to do a fit to a couple of experiments. So we use Nova and T to K, we use their appearance data and these are approximately single bin measurements. They don't really have a lot of spectral information and that's by design. And then we approximate them in this way where X, Y and Z are some numbers that encompass Z is the background rate, X is a combination of the flux, detector volume, efficiencies and cross sections and then Y is the wrong sign left on contribution for in neutrino mode. So in neutrino mode, there are some anti-neutrinos produced the far detectors cannot differentiate between neutrinos and anti-neutrinos so you have to include that. In neutrino mode, it's not that important but an anti-neutrino mode is quite important because the neutrino cross section is larger and also the intrinsic anti-neutrino rate is sorry the intrinsic neutrino rate in the anti-neutrino mode is larger than the other way around. So you have to include this. We can then parametrize the input from the disappearance which there's some details from this that the results don't matter on how this is done too much, which is good because the experiment to make it a bit difficult to extract this information but the information is there anyway. We then plug in the standard experimental parameter so NOVA is at close to 2G EV, slightly more density because it's going deeper into the crust and the crust density increases as you go down and the baseline is of course a factor of two and a half times longer, which is why the energy is two and a half times higher. So they're both at the oscillation maximum and TK is lower and shorter baseline. Now to do a fit with appearance data you need all six oscillation parameters. And of course there's information about Delta which is what we're really paying attention to as well as theta two three and Delta M squared three one but you also need theta one three, theta one two and Delta M squared two one. You have to be careful about where this information comes from. If you use global fits, you'll get theta one three that includes information from T to K. So you'll double count, can't do that. So we did was we used Diabay for theta one three they also took additional information of Delta M squared three two and this is in vacuum so NSIs won't affect this. And then for the solar parameters we use Cameland which gives us theta one two and Delta M squared two one. And again Cameland is also in vacuum or almost in vacuum. So NSIs doesn't affect this either. Now we also know that the problem is Cameland does not tell us what Delta M squared two one is positive or negative. We need solar data from that. And of course snow famously told us that Delta M squared two one is positive because the appearance probability sorry the disappearance probability decreases at a higher energies instead of increases. That's because they saw an electron neutrino rate of about a third and not two thirds. Now this result depends on non-standard interactions. So non-standard interactions would change this story. Turns out it doesn't cause that the parameters we're considering here doesn't cause a problem. So this is fine. But you have to check this kind of thing when you're doing these analyses. So before I get to the new physics of course I'll hold off the best for last. I'm going to say a few words about the standard oscillation parameters. And I'm presenting this in terms of the Y-scog invariant because of course I've emphasized that that's important. And on the X-axis we have the Y-scog invariant the middle is CP conserving anything away from the middle is CP violation and Y-axis is theta two three. The middle is maximal mixing the upper is the upper octant the bottom is the lower octant and T to K kind of as we saw before prefers and this is our fit which is quite consistent with the experiments fit prefers negative values of the Y-scog invariant and the upper octant Nova is pretty much okay with anything except for where T to K is and we see that the combined fit shown in orange is a bit bigger than the T to K region which is not surprising. The switch to the inverted ordering things do tend to agree better because Nova prefers negative values of the Y-scog as does T to K. So you can kind of get a handle here for why the combined fit prefers the inverted ordering because there's not this slight tension in the amount of CP violation. Of course the significance there is not very high. So we plug in the new physics and this is what we get. So these are our results. So the X-axis is the magnitude of this parameter. So the left side at zero is the standard model and on the Y-axis is the complex phase. So zero and pi and two pi are if the anise is real and this is for epsilon mu and epsilon eta. We're gonna focus on epsilon mu and we see we get a preferred region here at a complex phase. And something to note is if you only did an analysis with real anise, you would totally miss this, right? You would see a slight preference here ish but you would miss a better fit up there. The orange regions are preferred at increasing units of delta Chi squared and the dark gray regions are just favored at 90% confidence level. I've also drawn on here the ice cube region. So ice cube in their atmospheric between and also has sensitivity to the same anise parameters and they just favor anything to the right at 90% confidence level. So they just favor the best fit point but I mean in any joint fit which is including ice cube is a bit tricky but certainly possible to be done in the future. This is not exactly ruled out. And you would expect a preferred region actually just a little bit lower but in general, things would be relatively compatible. More importantly, we see that ice cube sensitivity is right at the same level here. So ice cube is a great way to understand this. We can recheck those, these are the actual numbers if you care about the details. We can recheck our analytic approximations which were around 0.2 and a complex, a new complex phase of three pi over two and the standard complex phase of three pi over two. And in these scenarios epsilon union epsilon e tau we find that the values are quite close what we predicted around 0.2 and a little closer to 0.3 complex phases of 1.5 and 1.6. So again very close. And again, the value of delta was very close. So before I ramp up, I just want to say where can we test this? We know in neutrino physics we have a sort of a history of perhaps finding anomalies that we then can struggle to confirm or refute. And this is not one of them. Now this anomaly is much less significant than others in the past but this is something we can understand. And the reason is because the ways to probe these epsilon union epsilon e tau frames are long baseline but also atmospheric. And as we saw, they're both fairly similar. This is ice cubes plot in the same thing. Again, this dash curve is the one that I showed before. And then they actually have a best fit value and a kind of a similar region if I'm to put on my tinfoil hat. I believe that this is a real thing. This is even less significant than the other not very significant things but they've got a lot more data in the bank which might be presented in the next few months. They'll see if they can complete that analysis in time but it'll be very interesting because they can rule this out or see it. And the nice thing is that the systematics between ice cube and TDK and NOVA are completely different. The fluxes are different, the cross sections are different and the oscillation physics is different. So it's very unlikely to have any kind of systematic problem. Super-K also does this. Their sensitivity is about the same and they're basically they're not gonna accumulate more data that's not gonna really change their answer. You can also probe this with scattering experience as I mentioned earlier, such as coherent and the constraints are the same order of magnitude but it's model dependent. So you can always change the scattering experience to get around this if you want to in a very reasonable way. So with that, I will summarize and get to the end and just review again what I said. The first half of my talk I mentioned parameterizations and you have to be a little careful about it and really we should experiment and theory should both be presenting and discussing things when we're talking about CP violation in terms of Cecilia Arles-Kogg's invariant. Then as for NOVA and TDK, the story is interesting and I think a little bit complicated and that we've got this NOVA and TDK tension and they both prefer the normal ordering but together they prefer the inverted ordering. This improves the goodness of fit a little bit but not much, but the goodness of fit can be very much improved and the tension is fully resolved in the presence of new physics. All those other new physics scenarios like steriles and TDK and so on don't explain the tension. You can approximate it analytically so you don't have to do a fit, you can just plug it in, see what kinds of new physics you would need and then go and check and see if it's ruled out by another experiment. And I think what's really cool about this is that if this ends up being right, which you know, who knows, that this new physics would introduce yet another source of CP violation and really would money the puzzle about what is going on in CP violation even further. Hasn't been ruled out by other things but it will be tested soon by them such as ice cube and coherent. So obviously we should all be paying attention and that's it. Thank you very much. Thank you very much Peter for the nice and super interesting webinar. I like it a lot. So let's start with some question from the, I mean, for the people that is following the light transmission in YouTube, you can write the question in the chat. So later we can refer the question to Peter but to give you time to do that, maybe we can start with some question from here from the people that it's attending the Zoom session. So I don't know if some of you guys have some question for Peter. I could ask a couple, thanks. So thank you for the talk. It was very nice. I was wondering about the first part of your talk where you looked at the different barometer sessions and I was wondering if you had a look at what happens if you take into account also the mass ordering, right? Because if you have the mass ordering, the normal ordering, right? Okay, it's normal order, no problem. But if you have inverted ordering, right? The M3 should be switched to the lightest value as in the quarks and the leptons, right? That the lightest one is number one and that will rearrange your PMS matrix. So I don't know if you had a look at that. Yeah, so for the PMS matrix, it doesn't care about the mass ordering, right? I mean, I can write down, so let's go, I wrote down the Hamiltonian for oscillation somewhere, right here, I mean, obviously let's ignore the NSI, but we've got some mixing matrix and some mass and some mixing matrix. And the elements in this M squared can be ordered in whatever way you want, right? You can order them in some, and this is something I didn't talk about and maybe I should have and this will hopefully encourage me to do a better job in the future. But this, you know, you can write this down in whatever order you want. Now, naively you would think from quarks that you should write them in increasing order. And actually for neutrinos today, that's not a good idea because we don't know that as you point out. We know the electron neutrino fraction and that's how I choose to define them, by decreasing electron neutrino fraction. If you want to reorder that, that's fine, of course you can do that. You can, let's say that the mass ordering is inverted and let's say we want to write them such that M1 is less than M2 is less than M3, which is a very reasonable thing to do if we knew the mass ordering with any confidence. And then that would cause a reorder. That however does not significantly change this parameterization discussion. The reason is because I don't really have a good plot for that, but basically all you're doing is just redefining which one is which. And you're just shuffling the columns around, right? And so nothing really changes here. You may want to choose a different parameterization that might be true. But if you choose to, if the mass order is inverted and you choose to order the mass as an increasing value, we may want to reevaluate how we do this, but that's just, I think that's just changing the order of the indices here. But yeah, but I think that's definitely a good point and that's something to think about if the mass ordering ends up being inverted as a field. I think separately from that we're gonna want to reevaluate how do we define M1, M2 and M3? And I'm sure I can guarantee you right now that if that happens, there'll be a big split in the field. And some people will say, we should order them an increasing masses because now we know what they are and that's what we should do. And some people will say we should keep it the same way because otherwise everyone will be confused and do things wrong. And then this will cause even more confusion and even more things if you've done wrong. Yeah, that's a very good point. Thanks, can I make another one, another question? Please. You showed that Yarskot parameter in the standard parameterization, in your other parameterizations, does it change its shape? Yes, it does, great question. I did not write it down. I was not very comprehensive. Yeah, so this is what it is in our parameterization. Okay, assuming we're just talking about three rotations in a complex phase, it's always gonna be sine delta, right? If delta is zero or pi, then the matrix is real and therefore there is no CP violation. So it has to be sine delta. You see, it is sine and cosine of every mixing angle except for theta one three, which there's an extra cosine. There's a couple of different ways to see this and this has been discussed a lot in the literature, but it's the middle rotation is what it is. So if we go to one of these other parameterizations, you know, like one, two, three is what we have. But if we do one, three, two, then the middle rotation is three, which is the one, two. So then what you'll have is of course, all the angles will be different, right? But also the expression will be different in that the one that has an extra cosine is not theta one three, it is theta one two. It's whatever's the middle rotation, depending on the order here, it's different. So that's the only thing that changes. Oh really, and is it still the sine of whatever phase you have? Yes, exactly. It's still sine of the complex, the new complex phase, sine, I mean, delta prime or whatever. And of course, all these angles are now new angles that don't, you know, the new theta two three is no longer 45 degrees or whatever it is. But yeah, so it's just this square changes, you know, this extra factor of C one three, you know, what might be C one two or whatever. Yeah, but that's it, you know. Okay. And so it's one. Thank you very much. Yes, when we're actually calculating what we do is we say that you're also comparing is the same in different parameterizations. We then calculate it one way and the other way and then we match them onto each other in addition to some other matchings. And that's how you do it numerically. Thank you very much. Okay, we have, I mean, if there is somebody here that wants to ask it, but we have three questions from the YouTube audience. So the first one is from Adrian Thompson. He asked, does the jargon remain robust if we add the fourth neutrino into the mix? That's a great question. The answer is no. In the presence of four neutrinos, right? So yeah, so let's go back to, this is a very good question. So everything was assuming relatively vanilla stuff here for a reason. So this is, you know, three by three, we ended up with four degrees of freedom. As you may know, with an extra neutrino, there are three new degrees of freedom. Okay, so now up to seven. And the way it's usually parametrized is instead of three rotations with one complex phase, it is five rotations and three complex phases. So we end up with two new, sorry, three new angles, six rotations, and three new angles and two new complex phases. So sorry, it is, yes, eight parameters. And you can check by uniterity and field invariance. You can check that that's correct. There are, people have shown that there is no longer a single, useful Yorva-Scalic invariant, there are multiple ones. And the reason can be seen in a couple of different ways. One is from looking at the PMNS matrix in uniterity. The other way is to look at this. So this, you know, I sucked a lot of things into the right here, but what's really going on is that in three flavors, there are the CP conserving terms, which look like sine square delta M squared L over four E. And there's three of those for each of the three delta M squared. And there's one term, which is a triple sine terms, the product of, you know, sine of delta M squared L over four E for the three different delta M squares, okay? And that's, and then you make some approximations and you get this expression here, which is very nice and simple. But if you have four neutrinos, that doesn't work anymore. The CP violating term in the vacuum appearance probability is much more complicated. And the pre-factors, so it turns out that the pre-factors for all three terms here end up being the same. And that's why it's the Yorva-Scalic invariant that's what's so elegant about this. In the other case, they're not the same. There are a couple of different Yorva-Scalic invariants. So in that case, I would say you definitely need to use the complex phases. And so it's usually parameterized like delta one three, delta one four and delta two four or something like that. And I think in that case, there's no way around the fact if you're doing a study with sterile neutrinos in an appearance experiment, you know, dune, hyper K or whatever, you really have to do, it's a much more complicated scenario. And the reason why this works out so elegantly, right? You can ask them, why is it so like simple and pretty in this case? And the reason of course is because three is the minimal number of generations to have CP violation, right? With two, as we know, you cannot have CP violation of any kind. Three generations, you can, you know, but there's only really one way to have it, whether it's just, whether you say that's one complex phase, which I would say is not a good way to say it because of course I can write the PMNS matrix or the CKM matrix for four complex phases. But a true statement is that there is one relevant CP violating invariant, which is the Yarl Scott invariant. Yeah, that's a great question. Thanks. Okay, thanks. Adrian can, he can write in the chat if he has further questions, but let's give another question from the chat. He's from Orlando Perez. He's asking, what happened if you have only NSI in production and detection? Can you explain the difference between T2K and NOVA? That's a very good question. I don't know. So Orlando's asking about charge current NSI, which modifies the productions. The production is from pie indicates predominantly and the detection is from charge current processes, which in principle are related, but not the same. And there's been some studies recently, really going into how those two are related and how you can't do anything willy-nilly, which is what people have done in the past. I would say there's a reason we didn't look at, well, I say there's a, well, so charge current NSI will also depend on energy. So you'll get things being different for NOVA and T2K, which is good. The problem, of course, that anyone who works with charge current NSI knows is that, I had this slide here where I alluded to the fact that building models with large NSIs is difficult, but here I was just focusing on neutral current NSI. I didn't say that, but it is doable. With charge current NSIs, it's extremely difficult to evade other constraints. And I would say that is the first thing to keep in mind. That said, if you don't care about those constraints, which I don't know if they can be evaded in any realistic model, if you don't care about them, which is fine, then you can probably explain this T2K NOVA thing like that. I don't know if anyone has looked at it though. Yeah, thanks. Okay, maybe Orlando may also one moment, if you can write in the chat, but in the meanwhile, I don't know if there are other questions from Walter or Nicolas has questions to address to Peter, otherwise there is still another one in YouTube. This is from Jorge Diaz. I'm gonna read all the, there are two questions, more or less. CP violation in neutrinos is brought up as a way to explain the matter-antimatter asymmetry. What value of delta is needed for this or any non-zero value will do? If so, how small is needed? I mean, if you can comment on that. Yeah, that's a great question to topic that comes up a lot. People, when they motivate CP violation in neutrinos, they often say, you know, very anti-symmetry of the universe, Sacherov conditions, therefore this. And I did not say that. And that was not an accident. So I very much appreciate getting the question. I presented CP violation and motivated it this way, which I hope is motive. I mean, of course, we can all care about whatever we wanna care about. I think this is a very interesting story. Very anti-symmetry of the universe is as interesting or more interesting story. And from Sacherov conditions, under relatively general, although not completely general scenarios, we need CP violation. This thing from quarks times 10 to the minus four is too small to do anything. This of course could have done something, but it's way too small. And so people pointed out, you can get, you know, leptonic symmetry from the leptonic sector. And then from the Svelon process, which converts basically particle number to particle number, you can convert that to baryons and people have shown that that can work. However, it turns out that there is nothing, and this is the really the full answer and I'll go into more detail in a second, but there is nothing that we can measure in any neutrino oscillation experiment that can tell us if leptogenesis either is or is not the cause of the baryon asymmetry of the universe, right? We can measure delta or the Arles-Gaud, the PMS matrix, if you prefer, to be zero with arbitrarily good precision or we can measure it to be one or negative one or a half or negative a half. And maybe neutrinos are explained in baryon asymmetry, maybe they don't, but it does not tell us one way or another. And I can give an example of this. There's a paper by Jessica Turner and collaborators a couple of years ago investigating this and they developed a model. So this is a model-specific statement where they have neutrinos causing leptogenesis and as perigenesis and they get the right baryon asymmetry. How it works, however, is via the seesaw. And basically the problem is that for the neutrino masses, the temperature at which some CP violating process would happen, it doesn't work, right? It has to, you can't, you don't need just CP violation. It has to be out of thermal equilibrium. So it has to happen during a first order phase transition for neutrinos and the masses that we know that they have around point one EV or so, it doesn't happen. But if you have a new sector, of course you can do things willy nilly and let's say you have some heavy neutrinos with some mass, well beyond the TV scale, you can do this and that's fine. You can then choose to relate those parameters to the parameters down here and say there's some symmetry that relates them. Fine, of course there's no way to test that since we cannot measure things well beyond the LHC. So we would never know if that symmetry is there. Moreover, in that relationship that they chose, so even if you believe their model, even if you know the scale of which it happens, the problem is that while there is a connection to Delta in that specific model, it depends on the scale of the new physics, it also depends on both amyron phases, right? So even in a very specific scenario, you can't get around the fact that there are three complex phases on the low energy side. We can measure one of them from oscillations, which we presumably will in upcoming experiments. There are two more associated with monorhonic phases. Now, if we measure neutrinos with double beta decay, we measure with arbitrary precision and we get the nuclear matrix elements down, which may be extremely challenging. But let's say we do that, that gives us one bit of information which tells us the absolute neutrino mass field. If we then measure cosmology arbitrarily precisely, which is extremely difficult and extremely unlikely in my opinion, we then get one more piece of information which tells us one more of the monorhonic phases, which tells us one of the two monorhonic phases. But the second one is impossible to probe in any experimental configuration. And even in a very specific model where everything was working out just right, you still cannot tell what both monorhonic phases are, which means that I can always sort of change the Delta that we're measuring and the other monorhonic phases and cancel them out. They're basically fully degenerate with each other even in a specific model. And of course, model could be any old thing and it could be, you know, and the scale of the new physics could be at, you know, 1,000 TeV, at a million TeV or whatever, in which case that changes the amount of CP violation. So the connection between neutrino oscillation experiments and baryonisim into the universe is extremely tenuous and not something to take very seriously. Hopefully that's not too harsh, but I think that's correct. No, it's okay. So I don't know if there are the last question because we are already on the time. So maybe, okay. So first of all, we thank Peter for the very nice webinar. I like it a lot. And for the people that is following the transmission, we meet more or less in two weeks from now for another Latin American webinar of physics. And of course, take care of all of you and see you in the next time and keep in touch. Thank you very much. And thank you to the law group for inviting me. Okay.