 OK, hello, everybody. My name is Joel Jones from the PUCP in Lima, Peru. And I'll be today's host of this webinar in the Latin American Webinars of Physics. Today we have a very nice talk about some rules for flavor parameters given by Martin Spingard. So before we begin, let me remind you that the questions can be asked via the YouTube live chat. And if you miss any live transmissions, you can watch it later on our YouTube channel. Let me also remind you that we have a WordPress page where we centralize all the information about the webinar. So anyway, regarding the speaker, we're having Martin Spingard, and he's currently a postdoc at the National Center for Theoretical Science in Taiwan. So he's giving this webinar like at 11 PM. So we really need to thank Martin for giving this talk. Martin got his PhD in Munich and has carried out postdocs at CISA and the kit in CalSchool. So the title of his talk is Some Rules for Flavor Parameters. So I'll leave you all with Martin. So they're all yours, Martin. Let me unmute you. Hello, everyone. OK. Yeah, thanks a lot for the invitation and the chance to present my work here and this webinar. It's my first time I give a webinar, and I hope it works. So I guess I'm supposed to share my screen now, right? That is correct. Let me try if it works. Can you see this? Yep, perfect. OK, great. So as Joel already said, I will talk about some rules for flavor parameters. And this was a work done in collaboration with a lot of people over the years, and you will find them below. So let's start with the outline of my talk. First, I give a short introduction about the motivation and the background of the some rules we present. Then I will first talk about electronic some rules, and then if time allows, I will also quickly mention some quarks, some rules, which are, one is much very well known, and the other one is very unknown, but they're both interesting. And then I will summarize and conclude. So let's just jump into the introduction. So the main motivation where all this comes from is coming from the flavor puzzle. In the standard model, actually most of the parameters in standard model are flavorful parameters. To be precise, it's something like 20 parameters for the months and the mixing angles. And here I show you a plot. It's a logarithmic plot of the permanent masses, which we know. There you see the quark masses and the electron masses. And they are vastly different, but on a horizontal line, they all look like being grouped, especially the bottom and the tower, the muon, and the strange, and the down and the electron are somewhat similar, which you can actually explain if you look at guts, but this I will not talk about today, basically. And in neutrinos, we don't know where they are, but they are somewhere at the bottom or actually slight or even below the slight. So this is the one part of the flavor puzzle, and the other part is the mixing angles, which is very weird because we know the quark sector for a long time, and the mixing angles are small, which the biggest one is only 13 degrees. And the electron sector, things are very different, where the smallest mixing angle is roughly of the same size as the biggest angle in the quark sector. So this made people question what's going on there, why are they so different? And they were looking for some explanations for these very different patterns and some deeper reason behind these peculiar things. But let me first summarize what is the essence of the flavor puzzle. You can categorize it in different questions. First of all, is why do we have three generations? We could very well live with having only two generations, or we could also have four generations in principle. There is no principle reason why it should be three and not any other number. Why has the model of Feynman master asked a different, why are they not much more similar? What is the origin of CP violation? And why are the quark's angles so small and the electric mixing angles so large? So these are the questions people are trying to understand for a long time now. And in the recent decade or so, one very popular approach was to use non-ambilance grid family symmetries to do so, where you start on the left-hand side with some family symmetry, which could be like A4, S4, T prime, delta 27, S03, S03, whatever you like. And then you break it to such that the subgroup, the main subgroup in the twin sector is different from the main subgroup in the charged electron sector. But because you have embedded in a larger group, you know the misalignment and from this misalignment of the two subgroups, the two surviving subgroups, you can predict the mixing pattern in the PMNS matrix, which find in bi-maximal, tri-bi-maximal, golden ratio, tri-maximal, and many, many more. So this was very popular until recently when they discovered or measured theta-1-3, where most of these patterns got somewhat disfavored because they all preferred usually a rather small theta-1-3. So then people added just a small perturbation to this picture to describe theta-1-3, and this results into mixing some rules. So you get a weaker prediction, so you do not get a precise prediction for the mixing anymore. What you will then find is the relation between the mixing angles and the direct speed variation phase in general. And here I'll show you one. And basically, this is a very common prediction in all the flavor models around. Not quite so common is another kind of sum rule in this flavor models. And these are the mass sum rules where the neutrino masses are related to the Majorana phases. So at the bottom the equation, actually complex equations of two equations. So these are the two kinds of sum rules in the electron sector I will discuss. And on the other hand, in the quark sector, there's the GST relation where most people quotes this paper by Gato Sattoria Antonin, which has nothing to do with this relation. I still don't know what the original reference is, but this is very old and known for a long time. So this is known for a long time. And there's also what we call the phase sum rule, which is not so well known, but it's very interesting. And I will tell you hopefully maybe later a bit, later more about this. So let's start with the atomic sum rules. To be precise, let's start with the mixing sum rules. So they are known for a long time. Many people worked on them. Actually, already when you start to look at the very early models, it was kind of always there. It took some time until people really understood what was going on. And Steve King was, which is people was one of the first to realize it fully and there was pushing for this, but all the many other people will turn this later on. And as I said before, basically, this is the relation between the mixing angles and deluxe the P phase, which emerges in different models. And this is very easy to understand if you just look at the PMS matrix because the PMS matrix is a combination of the charge-laptone sector mixing and the neutrino mixing. And usually what you will find in most models is that the new part, the neutrino mixing is a symmetric matrix where the one-three mixing angle is usually zero. But then what you can introduce is a one-two mixing in the charge-laptone sector, which is a small perturbation. And then you will find that you generate effectively a one-three mixing angle, which is a combination of the one-two mixing angles from the charge-laptone sector and the two-three mixing in the neutrino sector. But this kind of, if you do a diagnosis properly, you will find some relations emerging. And for example, for bimax mixing, you will then find that the science grade of theta one two, which is the physical mixing angle is one-half plus sine theta one-three cos delta. Yeah, theta one-three is again the physical mixing angle, the ones you measure. And delta is also the physical PM and S matrix, which you measure. For a bimax mixing, you will see what changes is the coefficients. So you will find one-third plus, and then there's a different coefficient in front of the sine theta one-three cos delta. But all these mixing patterns usually, they will give you one of a relation which looks very similar to this one. One thing you should be aware of is that most of these models are high-scale models. So you have a high-scale CISO, like 10 to the 14 GB, and this is the scale where this is realized, where this mixing sum was realized. So the physical mixing parameters that are written there are defined at the high-scale. So if you want to compare this to your measurement, it's low-scale, you have to do RG running. And this can be large, as you can see on this slide. In the typical size of subtractions, you would say usually it's 10 to the minus six, so it's small. But then you have time beta squared, which can be large. It can give you a factor of 100 easily. And then you have this m squared divided by delta m squared. And this is the neutrino mass squared divided by the mass splitting squared. And as it's well-known, the neutrino mass scale can be much, much larger than the splitix. And in this case, you can find easily very large corrections of 10% or so. But this is also known for a long time, but many people tend to forget about this one. And you find a similar running order for a Maran phase, and also for the Dirac phase, and the masses. Okay, for the masses usually, the running is not so large, but out of there, you can feel it. Important is that in the standard model, this will generically be small still because they don't find the time beta enhancement. But in the m system, this enhancement can be, can be quite sizable, yeah? So let's have a look what this means. So what we did then is a systematic study of the artifacts on this mixing samples. And here I take an example of try the maximum mixing, which is here on this slide, the lower one, the lower sum rule, which we try to test. So we define this to be defined as a high scale, and then we just let them run down to see what happens. And because we know theta one two, and we know theta one three, what you would get is a likelihood for delta, which is very narrow because theta one three, and theta one two are well known, which you can see here. So on the right hand side, I show you the plot. On the vertical axis is likelihood, and on the horizontal axis is delta, from zero to 360 degrees. The dotted line, this is coming from the global fit. So this is just data. There's no mixing sum rule so far. And if you switch on a mixing sum rule, you get, for example, this blue line, which is the standard model line, which means there's no running. This would really be if it just takes the sum rule to be realized face value. Then you see that the likelihood shrinks a lot. This is simply because you will find that the combined likelihood of theta one two, theta one three, and delta is never as optimal as it is just taking the global fit, because in the global fit, there's no influence on theta one two, theta one three. But there it enters, so the likelihood shrinks down. But then you see if I switch on RG effects in the MSSM, which is the green line and the red line, you see that the likelihood changes dramatically. In the normal ordering, not so much, but you see that it gets generally broader and also a bit more flat. While in inverted ordering, you can see that for the large mass scale at the bottom, the green line is almost as high as the dotted line. That means you really lose a lot of information. So the mixing sum rules can be heavily spoiled by RG corrections, but this depends a lot on the mass scale and also, yeah, if it's the MSSM with the large jambita. This is another sum rule I want to show you. This is if you allow one additional perturbation. So the sum rules get even peaker. And then you can see that on the lower plot, for example, with the large mass scales, the red line is basically the same as the line from the global fit, so there's really no information anymore. So the sum rules can be always fulfilled trivially just by adjusting the minor phases, which we don't know. And the direct phases are the usual. But the minor phases end, yeah. And then you can do it. So let's come now to the mass sum rules. Before I go into some more details, you give me some disclaimer. They're not related to any real symmetry. This was my latest paper. They're not specific to any mass mechanisms. They appear with the type one seesaw, type two seesaw, type three seesaw, the birth seesaw, and so on and so forth. And what the essence of this mass sum rules is that you start with the family symmetry. So you have no masses and no mixing. Then you break the family symmetry, but you break it in a minimal way, such that your low-energy neutrino masses will effectively depend only on two complex parameters. And because you want to describe three complex parameters by using only two complex parameters, you get two relations. And this is exactly a mass sum rule. Let me give you one example to show you what I mean. And this is from this SU5 times A5 cut flavor model, which we discussed in 2014. It is a type one seesaw, and it's this type one seesaw, which is the Eucava matrix, an neutrino Eucava matrix, is basically just a number times this matrix which I wrote down there. And the rights in the neutrino masses are given by this matrix. And you will see that there are only two complex parameters, which is the V2 and V3, which are symmetry, baking, flour, and beps. And I hope you're all familiar with the type one seesaw, but it's very easy to understand that also your light neutrino mass matrix will depend effectively only on the ratio of the Eucava coupling divided by this two complex parameters between V3 and V3. And effectively, you get then this mass sum rule at the bottom there, where you have the physical neutrino masses M1 and M2 M3. So they are positive numbers. And you have to do my runner places at one alpha two. Yeah. And this is one of the mass sum rules. And generally you can, any mass sum rule which is on the market, you can write in the form which I show here. So S is the sum rule. Then you have M1, M2, M3. They appear with the power of D, which can be plus one, minus one, one half, minus one half. And you can also have, and then the C1 and C2 are some numbers. They are fixed. There are some really numbers and not some three parameters, they're just numbers. And the data kites are also given by our underlying model. Also just some special numbers and not three parameters. For instance, the previous sum rule which I showed you before, and in this case, C1 and C2 would be one, D would be minus one, and the data kites will both be pi. Then you will get the sum rule I've shown you before. The funny thing is if you use this parameterization, you can categorize all sum rules which are on the market. And the only 12 known, the only 12 sum rules which are known in the literature. Which correspond to about 60 models. So some appear in several models and some only appear for rather special models. And here you see all the C's which appear, all the D's which can appear, and all the data kites which can appear. So when, well, if you have a model which just went into this, please let me know that we can edit. But these are all the sum rules which we could find in the literature. The only 12 of them, it's very easy to check them, much easier to check than 60 models. Before I go on, I also want to remind you, this is a sum rule, it's an equation, a complex equation. So you can provide it as an equation in a complex plane, which then forms a triangle. And because it gets zero, this is the triangle closest. And just keep in mind for later, that in this triangle, there's one angle which we call alpha. This alpha is minus d phi two plus data kites, right? Just keep this in mind, that there's one angle alpha. So what do they actually mean? These equations are the sum rules because these are some constraints on the marijuana phases. We have to look at some marijuana, marijuana physics dependent process and the most famous one is not really less than a meter decay. And here we see the typical plot which people show them. On the vertical axis, we have this MEE, which is the particle physics part of the rate of the tensile meter decay rate. And on the horizontal axis, we have the lightest neutrino mass, which can be either M1 or M3 depending on the ordering of the masses. And with the straight lines and the dashed lines, this would be the allowed regions to sustain that three light marijuana neutrino case without any mass sum rule. This would be the allowed regions, yeah? On the upper part, this is the inverse ordering case where M3 is smaller than M1 and M2. And the lower region is the one with normal ordering where M1 is smaller than M2 and M3. Now, if we have this sum rule number six, which I showed you before as well, we can really exclude a lot of parameter space. In such a model, you would only get for, if you use inverse ordering, you will only get the yellow region, which is there on the right, the small yellow band. And from the normal ordering, only the blue region would survive. So you see, this is like a very tough constraint because really a lot of the principle allowed region is basically already ruled out, just having this sum rule, yeah? Yeah. Again, this is, they put in a standard model there because again, there might be some RG corrections. But first, let me show you another sum rule, which is also interesting. This is sum rule number two. And this sum rule, you could not realize inverted ordering, which I will show you later, why this is not possible, but just keep in mind that some of the sum rules only go together with one of the two orderings. So if you would find inverted ordering, you might wonder if you can rule out this sum rule completely without looking into any other details. So this was the unperturbed case. What happens if you add corrections? Again, this most of the models are high-scale models. So you have to wonder what happens if you add RG corrections to the game. Do again, get all my predictions somewhat spoiled at least in certain parameter regions or what is a good prediction and what not? This is, RG corrections, you can never switch off. Something which is more model specific is in most models, you might have some high-dimension operators, and you might have some additional corrections from the charge-tran sector. There might be some corrections to your vacuum alignment and so on and so forth. So there are many things which might actually tell you that there is a small correction to your sum rule and you might wonder what it does to your predictions again. So those are two somewhat related questions but somewhat differently also as you will see. So the main questions which we want to discuss now is first of all, can we reconstitute forbidden orderings? So I'll show you one sum rule which forbids inverted ordering. Can we get this back if we add some small corrections? This is the first question. The second question is, how big is the impact on the distributed decay? Do you suddenly fill up all the allowed region from inverted ordering, normal ordering, or do we still have a rather strong constraint on the allowed parameter space? Let's have a look at this. Let's have its forbidden orderings. This is again sum rule number two. The one I showed you before. And as you can see here below, there's this course alpha three. This alpha is the triangle I mentioned before. So the cosine alpha should be between minus one and one. Otherwise, you would not get a triangle, simply speaking. Obviously. But this course alpha three, you can, using simple geometry, you can express it in terms of sodium masses and one and two and three. And because we want to get inverted ordering, we know that M3 is smaller than M2 and then the one. So you can make some estimates and you will find that the course alpha for inverted ordering would always be smaller than minus one. Independent of your mass scale, you would always find that it is smaller than minus one. And this is not good. And it tells us actually that we can never get inverted ordering on three level in this setup. But now the question is, so it's minus one, but what if RG corrections add up on alpha, start set at the low scale, you get the number which is larger than minus one and then things would be allowed again. So let's have a look at the RG corrections through course alpha, which I show here. And then we have this coefficient on front. It's a CY tau square, Y tau is the cover coupling of the tau C is just the number, a coefficient which is in the M2M, it's positive. Actually it's one. And then we have here this combination of masses which is positive as well. And then we have the log of the CISO scale divided by the Z scale, which is also positive. And that has a global minus sign in front. So this is actually bad because we start with the number which is more than minus one, and then by RG corrections we add also a negative number. So actually it makes things worse by adding RG corrections in the M2M. And this is a somewhat general statement. Basically all the forbidden orderings stay forbidden unless you go some very extreme regions of parameter space where you have like a time meter of 100 or an attrino masses of above electron volt or so, you have to go to really extreme regions which are basically a good out. So just by having RG corrections, you cannot reconcile forbidden ordering which is an important statement. For the second question, we have to do some numerical scans. These are just some details, so the scripts. And let's have a look at some results. So this is some rule number one which I've shown there below. And here on the left you see the standard model case which is again basically the same as if we would have no RG corrections. And the yellow region is the one from inverted ordering and the blue region is the one from normal ordering. And then I switch on RG corrections. In the middle I have the plot of star beta equals 30 and on the right I have the plot of star beta equals 50. And we see the star beta equals 30. Yellow band gets a bit broader. Also the blue band gets a bit wider, especially in the large mass region. This is where the RG corrections become also larger. But for time beta equals 50, we see that the yellow region actually becomes kind of broad. And also, yeah, but the blue region also becomes a bit larger. But in particular in the region which is all these favor by cosmology. So this tells us that quantitatively things change a bit. But we cannot fill out the complete region. In particular what I would like to highlight is that the point that there's always a lower bound on the mass scale, which we can see here because the blue band is constrained on the left hand side and yellow band is also not going to zero. And this number, the lower bound on the mass scale does not change by RG corrections a lot. I think in this case, it was most drastically changed by a factor of two or so. But generally speaking, this is rather robust in the RG corrections. So the forbidden orderings and the lower bound on the mass scale do not really change on the RG corrections. Let me show you another example which is somewhere number six, the one I mentioned before. On the left hand side, we have again a standard model case which is the same as we would have no corrections. And the right hand side, you see again that the allowed bands usually they widen a bit, but most notably they get broader in the region which is all these favor by cosmology. This one is also funny. I would just like to show you because there's the sum rule number eight. And there you will not only find a lower bound on the mass scale, but also an upper bound on the mass scale for normal ordering. This can also happen. So if you would find normal ordering with rather heavy neutrinos, this would also be put out to some. So let's come to the model specific corrections. They are a bit different. And here we chose to just parameterize them because this really depends on your model and what you do. And this is difficult to make some general statements. So we just said, we have some zero order masses MIT row which fulfills the sum rule. And then we have some corrections delta MI which can be arbitrary. So they do not have to fulfill the mass sum rule or they could maybe also fulfill another mass sum rule. We don't know. We didn't specify. Then we can, we take some rule and we expand in delta M. So we find also delta S. And because the S itself is a dimension for quantity because it's proportional to some power of some masses, we divide it by MN, which is a normalization scale. And we always choose it in such a way. Basically, we always choose it such that this is the largest mass scale. We divide by the largest mass scale which can also, depending if D is negative, it can also be the smallest one. This we did to avoid to artificially enhance any corrections because, yeah, if you think about this, if the mass scales become very light, there might be some enhancement which is artificial. So this is just some mass scale such that corrections remain small, so it should be small. And by assumption, this S zero advantages because to leading order, the MI zero's fulfill the mass sum rule. So, and then we again ask ourselves the question, can we get back forbidden orderings? And in this case, we have to say yes, and why can we do this here and not before? Here we can choose the sign of the corrections freely before the sign was fixed by, by the better function, basically, and the parameter values. But here we don't really know which sign the corrections have. So they can have the correct sign. But then the question is how large two corrections have to be to get the forbidden orderings? And this is what we answer in this table. And here we show the minimum correction which we would need. Let's say, again, sample number two, where forbidden ordering was inverted ordering. And if we have a mass scale, so if the light is not renowned, this case was 0.05 electron volts, you would need a 30% correction to get forbidden ordering. And this becomes larger. If you go to smaller mass scales, if the light is once 0.01 electron volt, it would go up to 100% correction. And then it doesn't make sense anymore to talk about a sample at all. And even with 30%, you might discuss it, if you really should still talk about a mass sample or not. But interestingly for the sample 10, 12, you might live with something like a 16% correction. So this is just an estimate. I'll also show you some plots later. Here's a plot. So here, again, it's the same plot as before. For sample number two, on the left-hand side, we have S hat, which is the properly normalized sample. Should be smaller than 10 to the minus four. So this is, again, but it is a case that there's no corrections at all. And this we find just normal ordering. But then we allow for a 10% correction in the plot in the middle. And then we see that in the region which is the favor by cosmology, the yellow region starts to form. We see a small yellow region. And then for a 30% correction, we see that there's, you really can notice that there's yellow band. So there's ordering which is allowed. But this is, again, this favor by cosmology and this is general statement. The forbidden orderings, which are suddenly allowed now by this arbitrary 30% corrections or so. They are all borderline excluded by cosmology. So if cosmology only gets better by a factor of five or so, this will all be ruled out. If your corrections say below 30%. This is just a map of the orderings which are forbidden allowed. So if you would find normal ordering, you could basically exclude some rules four and five. If you would find inverted ordering, you could basically exclude some rules two, three, 10 and 12. So this is a good prediction, the ordering. And another good prediction is as I said before, the lower bound of the mass scale which usually does not change a lot. So this could also help to rule out certain summaries. And so let me give you one powerful example. Just quickly, the model I mentioned before, and that's very funny. It's the golden rational kinemixing model. These are the parameters. We find there's a one, two correction coming from the charge depth on sector which is slated versus the carbibor angle because it's a gut model. So then we find theta one, three is built in nine degrees to a very good precision. This is related to the carbibor angle. And this is well known for some time now. We find a mixing sum rule and we find a mass sum rule. And from firming mass ratios because the gut model, we know that the time meter should be larger than 30. So basically this is one case where we can use everything which I discussed before. We have a mixing sum rule, we have a mass sum rule and we have large RG effects. So what happens if you put all of this together? At the high scale from the mixing sum rule, we can estimate that the theta one two at the high scale should be between 24 and 39 degrees. And from the mass sum rule, then we can estimate that M1 should be larger than 0.01 electron volts or M3. These are both times the lightest masses of new masses. Should be larger than 0.03 electron volt. And then you can just make some estimate for the RG running. And you will find that for normal ordering at the high scale, theta one two PMS, if I take the best fit value at low scales and just estimate how big it will be at high scale, it should be smaller than 33 degrees, which is fine because it's available well inside of the range. But for the inverted ordering, because the mass scale is higher, and the larger there will be much larger as well, theta one two suddenly drops down entire scale to a number smaller than six degrees. And this is not, this is not within the range which I was given in voting before. So this tells us that inverted ordering is not possible there. And this is, I mean, this is such as estimates, but this was all between our numerical scans, which solved the RG's numerically. And we didn't find any allowed parameter point and this is the reason why. So if you combine mass sum goals together with mixing sum rules, things get heavily constrained and can really rule out cases, which you wouldn't see if you only look at the one of the two. Now let me also quickly go through some things about quarks sum goals. So quarks are very similar, quarks are very different from latiners. Now latons, the quarks are very hierarchical, quarks which are very small. And this gives you a suggestion for the mass matrices. It's somewhat tempting to believe that they are also very hierarchical, that they are small after an element. And you can even say that once remixing in the up and down sector is basically negligible. And if you do this and use some parameterization of the mixing matrices, which you have this complex rotation and you assign a phase to the sign of the mixing angle, you can derive this approximations for the mixing angles. You will see that you create effectively once remixing because we know that the C cam once remixing is not zero, which is again related to the two stream mixing in the down sector and the one, now it's related to the one two mixing in the up sector and the two stream mixing in the up and down sector. If you meant to do one element of md very tiny, you will also find this relation between the one two mixing and the quark masses. So the theta one to d is roughly speaking square root of md over ms. And this leads to this famous GST relation where you find the relation between the CKM mixing angles and the CKM phase and the square root of md over ms. And the right hand side, I show you the plot where the blue line is coming from this sum rule on the left hand side. And with the dashed lines, this is the one sigma errors of the quark masses or the CKM phase. And you see that the error bars and the data, this is from 2010 is much, much, much too bad to tell us anything useful about this relation in the future. We could, at the moment, it's very well fulfilled, but the errors are much too large to say anything more interesting about this. That's why it's around for a long time and people are not so interested in this at the moment. A different story which is not so well known is in this problem solution, you will find that alpha, which in this case is the angle in this CKM unitary triangle. This is the plot from the utility collaboration. The alpha is the angle at the top there. And for this problem solution I showed you before, this is just the difference of delta one to D and delta one to U, which are the phases of the complex one to mixing. And this, from data we know it's very close to 90 degrees. So let's say it's 90 degrees. And if it's 90 degrees, there's one very simple pattern for the quark masses, quark mass matrices. And you can say, for example, that M U should be real. And in M D, you only make the two element imaginary and all the elements real. And then you will fulfill the sum rule and you can say that this might look like for 10 to 15 violation because it's a special value, in a sense. It's not an arbitrary phase, it's like a special, very special phase. And yeah, I will skip this one. Let me just mention, you can also try to use this kind of textures then to solve the strong repeat problem. You just say theta, so we say CP is fundamental. That means theta bar should vanish at the fundamental level. But the problem is then if you break CP, you have to break such that you get the right CKMCP violation and you do not generate an effective theta bar, which is too large. And so the argument of the determinant of the quark mass matrices should be zero. And this you can do by using the texture which we promoted before. You say that M U is the real matrix. So this will not give you any contribution to the theta bar parameter. And for M D, you use this texture, which I show you here. And if you look at the determinant, you see that it's just a product of the three stars. So it will be a real number. But still, because we know that there is this phase time rule, we know that the CKM phase is correctly reproduced. And yeah, there's a model implementation. So this is not just just talking and this is not just texture. You can really get this from the square symmetries. And this is basically going back to this paper, but I don't want to skip this now. But you can really write on a model which gives you this as a prediction and not just some hand-waving arguments. So let me summarize and conclude. On the one hand, we have to point some rules which people are more interested in the moment because they give you constraint on speed variation, especially on the decrease speed variation, which will be hopefully measured soon. It gives you a constraint on the mass scale and the ordering, which will also be tested very soon. Those are like three things which will be tested very soon and this is always interesting to see which models would still survive after the next big data release. The RGEs do not affect the qualitative behavior in particular of the mass sum rules because the orderings, the forbidden orderings are not really affected and also the lower one in the mass scale is not really affected. Of course, the real numbers change a bit, but the qualitative picture does not change. And one important statement is also, if you have both, if you have a mixing in the mass sum rule, things become very, very constrained and things are even more constrained than you would believe if you only have one to two. So as soon as we can boil down, they allow the mass sum goods to one or two, maybe we can have a look at the models again and see if they have some additional predictions which will tell us then that this is still good out. So hopefully there's only like a very few number of models which would survive the next big data releases. In the clock sector, things are a bit different. People are not so excited about it at the moment. We have this GST sum rule which is pretty much fulfilled and there's not so much improvement to be seen in the future. What I think is more interesting is actually the HILFOS-potential SP evaluation. As I mentioned, if you look at this, special values for the phases, so that's much like the potential SP evaluation. And the funny thing is that you can use this to solve the strong supply problem, which is one of the few remaining tough problems in particle physics. And it might give you a fresh look on this old problem and might give you a new way to solve it or like a new way to implement SP evaluation which people did not think about before so much. And if you don't find an axion, that might be really interesting. So yeah, thanks a lot for your attention. Okay, thank you very much, Martin. It was a very nice talk. Let's proceed with the questions. So let me remind our viewers, again, that you can ask questions through the YouTube live chat. But before we go into those questions, let's see if anybody on the audience has got a question. Yeah, I have a couple of questions if I can make them. Now, Martin, because I want to very nice your talk, and I'm very curious about the topic. In fact, I want to ask you how they change these some rules or these some rules is only in the assumption that you have for the case of neutrinos, treat light neutrinos or can be modified by you have a sterile neutrinos, the electron ball, KV, how can we spoil the some rules or all the some rules that you have, in fact? Well, for the master rules, I don't know. I mean, the models which I know, they always have three light neutrinos. They don't have a KV. I mean, there were some low-scale T source, but I think even they don't have a KV neutrinos. I would have to check, yeah? I'm not so sure. And in general, you might still find some rule, but I would have doubts because of course, as soon as you add a KV to this game, it's very difficult to fulfill it, right? Because this... Yeah, yeah, yeah. In fact, this is one of my questions because, yeah, for the some rules, you really need to have some hints of what is the size of the masses. But the funny thing is just one thing which might be interesting is, for this model I was mentioning here, what you don't see here is, but you will find also a sum rule for the right-ended neutrinos masses. So actually, if you have a low-scale CISO model, what you might find is that you find, effectively, a mass sum rule for the light neutrinos, but you would also find a mass sum rule, let's say, for your KV neutrinos. Because, and then if you look at the CISO, you will see that also for the mass sum rule below, I give, there would be some correction which is, yeah, which is electrific scale divided by CISO scale, you forget about type one CISO. But then, depending on your CISO implementation, you might find this that you have two sum rules, one for the light sector and one for the heavy sector. Sector, yeah. Ah, that's interesting. Yeah, another question, just I want to take advantage of the question, in your slide number 35, you show this blue and yellow bands, they just have a curiosity. So that means in this sum rule, you're assuming for any of these three bands that you have a kind of lower bound for the neutrinos mass, no? It's coming out from the sum rule. Yeah, so that's in some sense interesting generally it's hard to have a lower bound for the neutrinos mass, for the light, one of it, yeah. That's nice. But this is really a common feature of all the sum rules. All the sum rules will give you a lower bound on the mass scale, which is somewhat, it's easy to understand because if you just go smaller with the masses at some point, let's say M3 would be zero and then you could not fulfill the M1 plus 2M2 is zero anymore, right? Yeah. You can address the marana phases, of course, M1 is not two times M2, yeah, it cannot be. Yeah, and then you need maybe an extra source for it. Yeah, anyway, so and then the one that I was very curious because when you were mentioning the strong CP problem. Oh yeah. Because at the end, when you were with these hints of CP violation, because I didn't get it so well, how large is this hint in your case? Well, it's my model, so I say it's large. No, yeah, yeah, but in the sense of... No, I mean, of course, I mean, you can say it's just numerology, yeah? I mean, the point is in this parameterization, which I mean, basically it's this parameterization, which is a very naive guess of if I look at my quark masses, I would believe that this is a good parameterization. And then I will not find any, I would like very, very tiny ones remixing angles. And this is important, actually, if you have very, very tiny ones remixing angles in the down sector and in the quarks separately. And then if you go through this, it's simple algebra. It's really simple algebra. You will find that this alpha in the C-chemistry triangle is related to this two phases, delta one to D and delta one to U. And alpha, this is data, yeah? This is nothing I come up with and write so much. Data tells us that alpha is very close to 90 degrees. And, yeah, and then how do you get this, this phases delta one to D minus delta one to U close to 90 degrees? You just say delta one to D is 90 degrees and delta one to U is zero. And this you get most easily by having this very special phase structure of the quark masses. You say M-U is a completely real matrix and M-D is almost completely real a part of the 2-2 element, which is completely imaginary. So it's actually very surprising that with such a cheap choice of phases, you get the right amount of C-K-M-C-P relation. For me, this is, it's somewhat intriguing, but it's not, yeah, it could also be just an accident. So it's hard to tell. But the funny thing is, I mean, because we have a model which can do this. So if you could not cook up a model which does it, we would still say this is maybe just some numerical accident, but we can really get this, let me just show you because you're asking. How we did this, it's actually very simple. You say, basically you do your standard trick of your discrete family symmetry and then you have these flower ones, which are called your phi. And you have this alignments that say 0,0x and x, x, x, x. This you get for cheap in all these models. And the x in general, it's a complex parameter, you would say. But what do you do? You take basically your standard trick and then you add a set N, let's say first make it very simple, a set two. And then you have in your super potential an additional term, P is the driving field and you have phi squared plus minus M squared. And M is all the real because you have Cp is fundamentally conserved. So this is the real equation. You solve your F-term conditions, you will find that phi has to be either, so the vx in the end has to be either real or imaginary. This is all you have to do. You take your model, which you already have, you add a set two or set four, or what you like, yeah? You add these terms and then you can do this. Of course, usually you have to add more than one set two and things get a bit nasty. But in principle, this looks very cheap to do this, if you just look at this. So, I don't know, for me it's like, it's for sure the most explicit model of potential P-relation which solves the potential P-program I've ever seen. It's much more explicit than all these Nelson-Bar models I've seen, for sure. And we can also explain that the CKM phase is large with the old Nelson-Bar models, the problem switch because for them it was difficult to get this large P-relation. And so, in some way we can really make a progress in this direction with this approach, but if you want to believe in it or not, now it's hard to tell. Mm-hmm. No, yeah, it's interesting, in fact, because it's particular in this case of models, I mean. Yeah. Good, good. Yeah, I guess I don't have more questions for the moment. Yay! I don't know. Yeah, it's your turn. Sorry. Sorry, Robert. So, I don't know. I don't know if anybody else in the audience has got any questions. Okay, I have a couple, but maybe, okay, okay, awesome. So, all right, so, the first question I have is if there's any area of the m-beta-beta space versus m-space that would rule out a possible sum rule. I mean, besides the limit m goes to zero, right? Yeah. I mean, let's say here, for example, yeah, if you would be... So, if you go to 10 to the minus four, would... Yeah. Yeah, that's what I'm asking. Yeah, if you go to 10 to the minus four, I think all the sum rules are ruled out. I think the one which goes down the most is the sum rule number one. And you see, this is, if you have 10 to the minus four, for the largest mass, all the sum rules, you can forget about it. Right, so what's the lowest value of m-beta-beta? All right, I would have to check. I don't remember, but I think sum rule one was really one of the ones which went down most. There was not, the other ones did not, for sure they didn't go much further down, yeah? But yeah, well, this is for ordering for normal, I would have to see if I don't remember which one was the one which went down the most. But we have, in the second paper, with Alex and Julia. Yeah, because if you have inverted ordering, you can never really rule it out, right? I guess. Yeah. Not from an m-beta-beta measurement, right? Because you have no other way of measuring m, okay, maybe because more or less. Yeah, well, the master you can hope for some successor of Kotlin to find something, yeah? Oh, okay, I thought that was incompatible with Cosmology. Yeah, and this one I'm saying successor of Kotlin. All right, okay. The other question was about the likelihood for delta that you presented initially. There was some very strange structure that I didn't understand what came from it. Yeah, exactly, on the plot on the right, you see some very strong cuts, right? And there was another plot that you showed that at some point it would follow the normal likelihood of the global analysis. And then, yeah, exactly, right? So why are you having all of these very strong cuts? This is, yeah, we're all confused, but the point is the following, that if you look at the RGs, the RG corrections, the sign of the RG corrections for TITA-1-2 is basically fixed. So with RG corrections, you can only go in one direction. And this is in this case, because there's a modulus and the sign is fixed, so there's nothing you can do at leading order. This is why you will find this steep drop off, yeah? So this tells you this, because TITA-1-2 is of course very important in this business because you can only go in one direction and it's a high scale, there's a value which is fixed as well. So this gives you this cut off. But then you can make the RG corrections for TITA-1-2 very small and then try to adjust it by playing around with RG corrections for TITA-2-3 or so, and then you might, that's why you can then shift this cut to the right side. But this cut is usually coming from TITA-1-2, basically, that it falls off like this, yeah? I see. Okay, super. So right now, I don't know if there are any questions on the live chat. Oh, there are, but they're all from Roberto. Oh, that is a surprise. I'm kidding, Roberto. So I don't know if there's any. I write to them because do not forget, just that you can ask later. Yeah, I have my posted here. Okay, so if there are no more questions on the live chat, I don't know, no more questions on the audience. So I guess that's it. So thank you, Martin. We'll let you go to sleep. And thanks for all our viewers for being here. And we'll see you on the next webinar. So thank you very much. We'll see you around. Thank you.