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STFC-RAL-CR03  R61: Right.

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STFC-RAL-CR03  R61: Okay, while we are getting started on this, welcome to the seminar. Let me introduce our speaker today.

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STFC-RAL-CR03  R61: Paul Harrison, Professor Paul Harrison worked,

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STFC-RAL-CR03  R61: on the WA78 beauty production experiment for his PhD at UCL, and on the C.P. Lear neutral Chem Experiment as a postdoctoral

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STFC-RAL-CR03  R61: research assistant at Liverpool. As an academic, that's

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STFC-RAL-CR03  R61: Queen Mary University of London, to work on paper, on the PIPAA content.

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STFC-RAL-CR03  R61: Papa, and became the first

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STFC-RAL-CR03  R61: physics coordinator in 1993. He also worked in layler phenomenology, writing a number of highly cited papers, particularly on tectonic mixing with Don Pegg Helkins and fields copy.

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STFC-RAL-CR03  R61: He moved to Warwick in 2004.

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STFC-RAL-CR03  R61: to farm the… Experimental Particle Physics Group.

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STFC-RAL-CR03  R61: And later joined the Atlas experiment, where he currently works on the CKM measurements using on-shell WDKs on top-air events.

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STFC-RAL-CR03  R61: In today's seminar, he will review his latest publication in phenomenology.

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STFC-RAL-CR03  R61: Welcome.

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STFC-RAL-CR03  R61: How'd you?

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STFC-RAL-CR03  R61: Okay, so, the outline of my talk,

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STFC-RAL-CR03  R61: The first two, sections, first three sections are completely standard, and I'm always quickly through them. But I do want to just remind them the mysteries of the corp, and

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STFC-RAL-CR03  R61: book, Masculine Mixing Spectra, and some of the… just a very brief mention of, actually, just one or two, historical efforts to explain them. And,

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STFC-RAL-CR03  R61: And then the mysteries of the Unitarian Triangle.

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STFC-RAL-CR03  R61: Under the new mass matrix picture.

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STFC-RAL-CR03  R61: This work was done with Bill Scott, and it's published, it came out, early last year in JHEP. Bill Scott, who, who, of this lab, retired, but he's still, still active, was still collaborating on, on various bits and pieces.

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STFC-RAL-CR03  R61: And, he apologized he couldn't be here today, his wife's sick. Then I'll talk about confronting the mass matrix texture with the data.

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STFC-RAL-CR03  R61: And, then talk about the symmetry to the texture, which to some extent are the most interesting part, but that's right at the end, and then a quick discussion.

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STFC-RAL-CR03  R61: Okay. So, the mystery of the quark mass spectra is, of course, that on a… on a log scale, the quark masses appear to be approximately

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STFC-RAL-CR03  R61: To form straight lines, for the charged two-thirds corks. Charged two-thirds corks in this cork are always red, and charged minus a third quarks are always, always blue. And, the slope of the red one is steeper than the slope of the blue one, and,

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STFC-RAL-CR03  R61: Actually, the… the graph…

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STFC-RAL-CR03  R61: somehow makes them look straighter than they are, because, because, the MC over MT is about 0.035, MU over MT is 0.002, and MS over MB is 0.02, and MD over MS is 0.05. But they're approximately, kind of, exponentially going, going with generation.

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STFC-RAL-CR03  R61: This, of course, is nothing new, I'm just reminding us, and it's been noted by many authors.

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STFC-RAL-CR03  R61: The masses are not predicted in the standard model. They come from the Higgs mechanism, they come from the coupling of the Higgs to the quarks, and they're free parameters in the standard model.

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STFC-RAL-CR03  R61: This hierarchy, as it's called, this, exponential increase in mass with family or generation is certainly not today within Sanball, and I'm not going to explain it.

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STFC-RAL-CR03  R61: At least not today, anyway.

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STFC-RAL-CR03  R61: There's a mechanism beyond the standard model, the Frogant-Nielsen mechanism, which works quite well to explain this, and there are lots of brain-inspired models and things which have some success in explaining these kinds of things. I'm not going to be talking about those either.

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STFC-RAL-CR03  R61: So, then on to the mystery of the quark mixing spectrum. As we know, the CK matrix couples that charge two-thirds quarks that charge minus a third quarks, and it's hierarchical in the sense that, expanding in the small

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STFC-RAL-CR03  R61: parameter lambda, the sign of the cable angle, this one here, which we just measured at Atlas, is proportional, roughly, to lambda points of order lambda squared, and this small one, the VUB, is of order lambda cubed.

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STFC-RAL-CR03  R61: And so this hierarchical structure also isn't explained, in the standard model at all.

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STFC-RAL-CR03  R61: So we've said that. But the masses, unexplained, and the mixings.

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STFC-RAL-CR03  R61: But both have these strong hierarchies. They both arise in the same place in the standard model, the Yukawa, or also called just the mass matrices, in the Yukawa couplings, essentially. And so it's tempting to think that these hierarchies might have a common, the reason for them might be common to the two.

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STFC-RAL-CR03  R61: It might be common, coming from the, essentially, the eucalyptic couplings of the Higgs to the quarks. So, very quickly, the weak interaction. In the gauge theory, we're dealing with gauge eigenstates, not mass eigenstates, and so we can just label them 1, 2, 3 for the families.

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STFC-RAL-CR03  R61: And, we have a universal coupling, so the CCM matrix is the identity, essentially, in this basis. And,

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STFC-RAL-CR03  R61: And so we can put this in a sort of matrix notation here, that the weak Lagrangian is given by us for the weak coupling, and then the W's coupled to the simple dot between the ups and the downs, with no matrix in between them, apart from the identity implicitly there.

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STFC-RAL-CR03  R61: So, the masses and mixings arise, as I've already mentioned, in the couplings of the Higgs, 2…

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STFC-RAL-CR03  R61: pairs of quarks of the same charge, and so for quark generations were 1 and 2, we call that Yukawa, with a subscript 1, 2.

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STFC-RAL-CR03  R61: remembering that these numbered ones, the gauge eigenstates. And after spontaneous symmetry breaking, the Higgs can be decomposed into the constant VEV and the physical Higgs. And so the diagram splits, you get the bit where

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STFC-RAL-CR03  R61: Phi is V over root 2, just gives you a V over root 2, Y12, and that's the mass term, which somehow couples the 1 left and the 2 right, which may or may not be the same.

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STFC-RAL-CR03  R61: And, then the residual, the physical Higgs also couples it. So this… this is how the masses are proportional to the Higgs couplings of the physical quartz.

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STFC-RAL-CR03  R61: Taking all the permutations.

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STFC-RAL-CR03  R61: you have a 3x3 matrix for charge plus 2 thirds quartz, and you have another 3x3 matrix for charge minus a third quartz, and they just get added in the Lagunogen. This is after SSB, and the mass matrices are just V over 2 times the respective,

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STFC-RAL-CR03  R61: Yukov matrices, and the important thing is these Yukov matrices are not diagonal.

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STFC-RAL-CR03  R61: So, it is possible to choose a basis now where they're commissioned without observable consequences, and that's what we'll do, but essentially, free rotations in the right-handed port fields, free redefinition of

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STFC-RAL-CR03  R61: Of the basis, on the right-hand fields.

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STFC-RAL-CR03  R61: Because they don't couple in the weak contractions, so you can always make these, Hermitian.

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STFC-RAL-CR03  R61: The physical particles, of course, are the mass eigenstates, as standard in particle physics. So usual unitary rotation, a different one for the ups than for the downs.

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STFC-RAL-CR03  R61: And then we get, the mass and the weak interaction Lagrangian can be written as a diagonal mass matrix for the U's, a diagonal mass matrix for the Ds, and,

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STFC-RAL-CR03  R61: The U mass eigenstate and the D mass eigenstate get coupled by a product of two unitary matrices.

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STFC-RAL-CR03  R61: So that's just the diagonalized mass matrices, and this is the sequence matrix. Always nice to remember that it's actually a product of two diagonalizing matrices.

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STFC-RAL-CR03  R61: So, this is the story of how the mass island values and the physical book mixings both originate from the original Yukawa matrices.

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STFC-RAL-CR03  R61: So they have a common origin.

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STFC-RAL-CR03  R61: So, how can we relate the mass ratios and the mixing

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STFC-RAL-CR03  R61: parameters, the mixing angles. One way is by enforcing something called texture zeros.

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STFC-RAL-CR03  R61: an idea by Holt Fridge from the 1970s, which has been immensely successful. There's thousands and thousands of phenomenology paints doing this kind of thing. They're needle excluded now, because the measurements of core classes and mixings are so…

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STFC-RAL-CR03  R61: so precise that it's difficult to find a model that still gets the data, and that's kind of what we've done, if you like. So the fridge…

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STFC-RAL-CR03  R61: I should be using the pointer more, I guess.

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STFC-RAL-CR03  R61: There we go.

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STFC-RAL-CR03  R61: The fridge matrix here has this form with these textures, and those are enforced by various symmetries that you can just sort of

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STFC-RAL-CR03  R61: choose which force these things to be zero. So the mass of the down-like, matrix, the 3x3 undiagonalized mass matrix here, proportioned to the power copies, is given by a normalization close to the B mass, and then this thing with these zeros, and these parameters A, D, and BD.

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STFC-RAL-CR03  R61: They can be complex, which introduces CP violation, and we find

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STFC-RAL-CR03  R61: once you diagonalize and solve for the masses, you can invert and find that AD is, a simple function of, of the mass ratios, and as is BD.

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STFC-RAL-CR03  R61: And they have these actual values.

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STFC-RAL-CR03  R61: This is diagonalized by the unitary matrix on the charge minus a third quartz, and it's got a sign of an angle, a sine of another angle here, and the product of the two signs. So if these angles are quite small, this will be extra small, and that's kind of…

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STFC-RAL-CR03  R61: kind of looking a bit like the CCAM matrix, so this looks like it might work, and of course, this is why this was a successful model.

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STFC-RAL-CR03  R61: And, we've got, that's one…

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STFC-RAL-CR03  R61: is given in terms of A and B, so it's just given by this, square root formula.

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STFC-RAL-CR03  R61: And is about 0.22, which is already very close to the commuto angle.

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STFC-RAL-CR03  R61: And, for this one here, the one… the 2-3 element, if you like, that's, smaller, but not… not a great deal smaller, it's the square root of ms over MB.

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STFC-RAL-CR03  R61: So it's encouraging, to say the least.

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STFC-RAL-CR03  R61: What does… which predict for VUS itself.

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STFC-RAL-CR03  R61: S2 and S3 were a bit too big, especially S, yeah, S2 and S3, actually. And,

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STFC-RAL-CR03  R61: what we haven't done is treated MU yet, so we want to treat MU symmetrically, so we… sorry, this is, interesting, but…

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STFC-RAL-CR03  R61: changer here. So we use the same form, the Fritsch matrix of three new parameters.

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STFC-RAL-CR03  R61: In red, to be… just to remind us that we're dealing with the two-thirds quartz. And the CKM matrix, then, is this product of unitary diagonizing matrices, remembering that the dagger is like an inverse. So it's like a rotation and a rotation back. So this is good for generating small mixing angles, you know, what's not to like there?

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STFC-RAL-CR03  R61: So, in the complex case, you get a phase that then becomes, observable.

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STFC-RAL-CR03  R61: And it gives CP violation, and together we get the prediction for VUS. This is the fridge prediction.

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STFC-RAL-CR03  R61: of, S, the sine of theta 1, and then the sine of theta 1 for the ups, with times the exponential of the X.

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STFC-RAL-CR03  R61: And, subbing in the mass ratios and everything, you get this plot here.

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STFC-RAL-CR03  R61: So you can… this is the experimental value of the sign of the Cabigo angle, 0.222, roughly, and here you've got the sinusoidal variation from the modulus of this complex number, essentially. And where they meet is where we're agreeing with the data.

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STFC-RAL-CR03  R61: And you can read off this delta tilde parameter, and that will tell you something about CP violation.

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STFC-RAL-CR03  R61: How about the predictions for VCB and VUB? Here, a second phase enters, this is the difference in the phases of the Bs. As in all these kinds of models, the absolute phases are unobservable, but relative phases are observable.

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STFC-RAL-CR03  R61: And one finds a similar formula with the

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STFC-RAL-CR03  R61: With the second, signs of the theta twos, if you like.

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STFC-RAL-CR03  R61: And again, there's a sinusoluble variation.

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STFC-RAL-CR03  R61: And this is the prediction, you know, which doesn't look too bad, and it gets you from the 0.14 we had before down to 0.07, roughly. But the experiment's very clear that it's 0.04, very close to 0.04.

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STFC-RAL-CR03  R61: The width of the band here is given by doing this fit either at, something like the weak scale, just above the weak scale, or at, a gut-type scale, and it doesn't help very much.

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STFC-RAL-CR03  R61: So these are too big, this angle is too big, and so fridge is now excluded. But it's got many features which are very attractive, which are worth preserving, is the point.

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STFC-RAL-CR03  R61: Also, the ratio of VUB over VCB, that's the smallest one, divided by the second smallest one, is predicted also to be lambda, which is unfortunately too small.

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STFC-RAL-CR03  R61: Well, can I ask a question? I don't know if you want to be interrupted or not. No, I don't mind. So, because I think you've sort of finished the bricks… I've finished the bricks, yeah, so…

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STFC-RAL-CR03  R61: The philosophy of it, or is that it's a kind of mathematical trick, where you make a prediction, and if it had agreed.

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STFC-RAL-CR03  R61: With the one data point we have, which is the universe that we seem to have, what the ratios are, then that tells you something about what the underlying

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STFC-RAL-CR03  R61: dynamics. Yeah, you might say, if it had fitted the data really well, especially with… now the data are very, you know, very accurately measured in many cases, then you would say, this is… this looks like maybe right.

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STFC-RAL-CR03  R61: And where does that get you? Do you see what I mean? Should we think of it as a… as a trick over mathematics, that it seems to be right, or does it then… does it relate to some sort of broader interpretation, then, of what's really going on? So, there are many models in which… which type zeros are natural.

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STFC-RAL-CR03  R61: So they're… they have additional discrete symmetries and things, and those just,

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STFC-RAL-CR03  R61: They constrain those zeros, either exactly or approximately, and so you might think those models which do that in a natural-looking way might be famous. I see, okay, thank you. So you're using it as a signpost to… to say, this…

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STFC-RAL-CR03  R61: this small collection of models, you know, which predict this kind of thing are… are likely to be right. One of them might be right, but there's, of course, some variations in the sort of,

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STFC-RAL-CR03  R61: In the space of models. Cool, thank you. Okay,

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STFC-RAL-CR03  R61: So, what have we got here? I'm just trying to remember which button is which. Yeah, okay, so, yeah, I'm a bit puzzled. I think this…

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STFC-RAL-CR03  R61: Anyway, it doesn't… it doesn't agree, as we've already said. I've taken these plots from this nice paper.

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STFC-RAL-CR03  R61: These, these… these people have… what they've tried to do as a recent… to… to… to…

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STFC-RAL-CR03  R61: to get… to find a… they're looking for a way to make Fritch agree, having been excluded about 20 years ago.

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STFC-RAL-CR03  R61: And, their approach was to use non-Hermitian mass matrices by some game which I must… didn't fully understand, but because their non-Hermitian may have more parameters, it's got 10 parameters, which is the same number of parameters you're hoping to fit, and they still can't fit the data. So, ultimately, their approach is also excluded, as far as I can tell.

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STFC-RAL-CR03  R61: Anyway.

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STFC-RAL-CR03  R61: So, that's sort of…

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STFC-RAL-CR03  R61: that's the end of the first half of the talk, which is the sort of the theoretical framework, if you like, and the historical attempts to explain this kind of thing. We employ some of Rich's stuff, but not all of it.

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STFC-RAL-CR03  R61: I want to talk, before I go on, about the Unitarity Triangle itself, because…

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STFC-RAL-CR03  R61: Some of the proofs we do have got very nice, geometric

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STFC-RAL-CR03  R61: justification for geometric proofs, which help understand around just a bunch of algebra, and it's nice for a talk. So, of course, the secant matrix is geometry. If you dot any pair of rows and columns, therefore, you get zero, the complex dot product.

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STFC-RAL-CR03  R61: And this is the sum of three complex numbers, which makes a triangular and complex plane, and its area is proportional to CP violation. It's the phase convention independent measure of CP violation, as

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STFC-RAL-CR03  R61: famously discovered by yourself.

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STFC-RAL-CR03  R61: you can either take the… the one that's exactly given by this, or you can divide through by one of these… these products, the model of one of these products, and get a base of unity. And there's 6 different triangles.

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STFC-RAL-CR03  R61: You can dot 3 rows in three different ways, you can dot 3 columns in three different ways. All the triangles are different shapes, and they all have the same area. That's the… that's a beautiful, beautiful result of the Alzhecog. And for the normalized one, it's just

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STFC-RAL-CR03  R61: portion half eta bar, where e to bar is the complex space, or the complex parameter in the CTM matrix, so it's clearly CP violated. If you want to shrink the triangle to zero, you get rid of CP violation, and…

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STFC-RAL-CR03  R61: see their matrix become real.

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STFC-RAL-CR03  R61: So…

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STFC-RAL-CR03  R61: What are the mysteries… there are new mysteries, which I haven't mentioned yet, the mysteries of the Unitarity Tribal.

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STFC-RAL-CR03  R61: So, it's been long realized, and I spent many years of my career measuring alpha and beta at Barbar, and we measured beta first, and it turned out to be, you know, close to 25 degrees or something, and then we measured alpha.

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STFC-RAL-CR03  R61: And, it's about 90 degrees. In fact, it's consistent with being exactly 90 degrees.

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STFC-RAL-CR03  R61: And that's been noted by many authors, and some people have built it into models.

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STFC-RAL-CR03  R61: And, beta, and it was actually, Tim Gershen, who donkeys years ago, pointed out, beta's compatible with pi by 8. Now, this is neurology, of course, but, you know, when you start to see a lot of numerical penisance, you can ask, well, is there some physics behind that, or is it just an accident?

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STFC-RAL-CR03  R61: So I took, I took Tim's, offhand remark quite to heart, and I thought about it for about 15 years before all this came out.

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STFC-RAL-CR03  R61: But anyway, so…

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STFC-RAL-CR03  R61: The angles, of course, once you've got two of them, they send them to 180 degrees, so you've got the third one is, you know, is 3 pi by 8 or something, and they're consistent with what we call these three special values, so it seems striking.

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STFC-RAL-CR03  R61: Is it a coincidence, or is it a smoking gun of some physics behind this? So, Bill and I decided to, try and cook up a scheme which incorporates this in a natural way.

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STFC-RAL-CR03  R61: You know, would this be a clue to what's lying behind?

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STFC-RAL-CR03  R61: these measurements. So, this is the Harrison Scott texture from our recent paper, and, both mass matrices should have this form, and I'll take you through it carefully. It's… it's normalized up to one in the biggest element here in the corner, and it's got, three, four real

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STFC-RAL-CR03  R61: parameters, the green ones, and the Q here is either up or down for 2 thirds or minus a third, so… so taken… these green ones are the same for the ups and the downs.

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STFC-RAL-CR03  R61: The lambdas are different types of bounds, that's why they're purple, because they are the way between red and blue, if you like. So Q is up and down. The lambda Q is a complex parameter.

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STFC-RAL-CR03  R61: It looks a bit like the Wolfenstein parameter, but it's complex. The argument of Q is unobservable, but the relative phase between Q up and Q down is observable.

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STFC-RAL-CR03  R61: In fact, the ratio of lambda U and lambda D

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STFC-RAL-CR03  R61: is, is observable, and it's a complex number, because they're complex, but in our approach, this is asserted to be minus i tan by tan pi by 8, which looks a bit random. This minus i is a phase of 90 degrees, and that's going to correspond to alpha.

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STFC-RAL-CR03  R61: At least in leading approximation. And this corresponds to the tangent of beta.

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STFC-RAL-CR03  R61: Quite amazingly, once you dynamize this matrix.

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STFC-RAL-CR03  R61: So, this lambda, this minus I, is the sole source of CP violation in this, in this texture, and as, as you can imagine, it's sort of maximal CP violation, modular, all the, all the other constraints.

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STFC-RAL-CR03  R61: And, that's sort of… that's sort of good, because the immutality triangle is kind of

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STFC-RAL-CR03  R61: Is the area as big as it can be when one of the angles is 90 degrees? I think you'll find it probably is.

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STFC-RAL-CR03  R61: So,

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STFC-RAL-CR03  R61: So this, as I've already alluded to, controls the… this ratio controls this real and imaginary parts control the ratio… control two of the angles of the entire triangle. There are already two free angles.

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STFC-RAL-CR03  R61: So I've said all this.

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STFC-RAL-CR03  R61: And the complex sum, lambda U plus lambda D, is a parameter to be fitted, and it turns out to be very close to Wolfenstein's lambda, so that's also neat. You know, it's taking the role of Wolfenstein, if you like, well, except that it's a mass matrix and not a mixing matrix, but there's a post-mapping between them.

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STFC-RAL-CR03  R61: And it's got 10 observables with same real parameters.

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STFC-RAL-CR03  R61: Fritz had…

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STFC-RAL-CR03  R61: 5 real parameters, so we're not as predictive as Fritch, but on the other hand, we're not excluded, so that's good.

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STFC-RAL-CR03  R61: So, the leading order solution, if you diagonalize, you just get a formula for the mass matrix, the diagonal mass matrix, in terms of some of the parameters.

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STFC-RAL-CR03  R61: And, it's hierarchical. As long as landlord is small, it's hierarchical.

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STFC-RAL-CR03  R61: We're not explaining why the number is small here, but we're using the fact that it's small to get the hierarchy here.

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STFC-RAL-CR03  R61: I talk about leading audits because you can solve these things with small rotations, and then you can, you can ignore.

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STFC-RAL-CR03  R61: Products are small rotations, then you're a leading order, or you can take… you can take everything into account, just, well, do the whole diagonalization numerically, then you've got the higher order, calculation.

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STFC-RAL-CR03  R61: So, there are three parameters to fit four mass ratios. So there's one constraint, and it's a constraint on the double mass ratio, which should be exactly equal to mod lambda U of lambda D squared, which is tan squared pi by 8.

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STFC-RAL-CR03  R61: which is, 0.172, and if you do the full calculation numerically, you get 0.176. And the experimental value. Mass ratios are now known to very high precision, to see Glattys QCD, and fit into the data, of course.

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STFC-RAL-CR03  R61: And, compared to 1.177 per sign, 0.002, so it's a rather, rather good fit. And,

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STFC-RAL-CR03  R61: there's enough parameters here to fit any MD or MD separately, so there's no prediction there.

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STFC-RAL-CR03  R61: We called the mixing.

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STFC-RAL-CR03  R61: one of the… either of the diagonalizing matrices takes this form, where I think the plus is for Q up and the minus is for Q down.

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STFC-RAL-CR03  R61: And, It looks at… it's of the same form, exactly, as the, as the…

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STFC-RAL-CR03  R61: Wolfenstein parameterization, so that's nice, mathematically. And when you take the UU times the dagger of UB, you get differences, but because of the signs, the differences become sums, and you'll remember I told you that lambda 0 is lambda U plus lambda d, so that comes from here.

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STFC-RAL-CR03  R61: And essentially, at the leading order, you know, you just sum that angle and sum that angle, and you get this. It's a bit more complicated in the corners.

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STFC-RAL-CR03  R61: But you get… so this is very… this is the actual prediction for the CKM ranges, or the calculation of the CKM ranges, and it's very Wolfenstein-like, with one lambda, two lanas, and three lanas, and an A. And of course, it's reverse engineered, it's built to work like this.

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STFC-RAL-CR03  R61: Because a lot of the formula you can just read off by inspection, even though you're diagonalizing 3x3 matrices, which is a bit of a…

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STFC-RAL-CR03  R61: a handful, normally.

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STFC-RAL-CR03  R61: So, there you go. So this also allows you, for example, to read off that lambda times rho minus i eta better be lambda U, and the other one, which isn't independent lambda times 1 minus rho minus i eta better be lambda T star.

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STFC-RAL-CR03  R61: So you read off lambda is lambda 0, A is A0, hence the notation, and rho plus i eta is the scaled value of lambda U star.

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STFC-RAL-CR03  R61: What about the unitarity triangles? I've already told you the answer. Now, this kind of essentially proves it.

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STFC-RAL-CR03  R61: This is the normalized unit out of the triangle, BUV star over A lambda Q, VTB over A lambda Q.

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STFC-RAL-CR03  R61: The blue one is lambda D star, and the red one is lambda U star.

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STFC-RAL-CR03  R61: So, the Lambos, in fact, turn out to be

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STFC-RAL-CR03  R61: The size of the unitary triangle, the lambdas that we have to pick in the mass matrices.

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STFC-RAL-CR03  R61: At the beginning, scaled by 1 power of lambda.

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STFC-RAL-CR03  R61: And… Jammoth?

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STFC-RAL-CR03  R61: Over here is the argument of lambda U star. Beta is the island of Lambda C.

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STFC-RAL-CR03  R61: And alpha, that's in a particular phase of invention, and alpha is the argument.

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STFC-RAL-CR03  R61: The ratio up to a minus sign.

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STFC-RAL-CR03  R61: So… At leading order, we've got alpha is pi by 2, 10 beta, is, is 105x8.

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STFC-RAL-CR03  R61: I'm sorry, I'm zooming ahead by mistake.

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STFC-RAL-CR03  R61: 10 beta… if once this is arrived, I get 10 beta is obviously that divided by that.

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STFC-RAL-CR03  R61: Which is, just lambda D overlander D.

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STFC-RAL-CR03  R61: Okay, yep.

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STFC-RAL-CR03  R61: So, those are… those predictions at leading order. The higher order corrections in rotations and things give very small, well, modest adjustments, and that actually helps the fit in the end.

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STFC-RAL-CR03  R61: So what is the fit? We do a precision fit to the data, we've got our 7 parameters, we've got the 10 inputs, we put them in, it's all done in Mathematica program, data, latest PDG, with some lattice, input as well.

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STFC-RAL-CR03  R61: We renormalized the common scale of, mass at the top, you know, the weak scale to start with, and we fit using full numerical diagonalization rather than any approximations.

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STFC-RAL-CR03  R61: I would get a terrible fit.

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STFC-RAL-CR03  R61: So, give up, it's all rubbish. Even though maths was pretty, 100 over 3… we got 3 degrees… 3 degrees of freedom, 10 rounds for 10 observables.

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STFC-RAL-CR03  R61: There's a tension between A, mass charmway around mass top, and mass stranger, mass beauty. There's a disaster. Or is it?

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STFC-RAL-CR03  R61: Because it's exactly these three quantities which run with renormalization scale.

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STFC-RAL-CR03  R61: Okay, so people get surprised when I tell them that the CK matrix runs with a renormalization scale, but it does. They're coupling constants, the new cars are coupling constants, they run like all other coupling constants.

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STFC-RAL-CR03  R61: It turns out that the Kabiba angle hardly runs at all. It's very, very, very slow running.

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STFC-RAL-CR03  R61: But, A, which is… essentially controls VCB, runs by about 13% from the weak to the gap scale, and these two ratios also run, actually, by this… essentially the same factor. This go… these two get bigger as you go from the weak to gap scale, and this gets smaller by about 13%.

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STFC-RAL-CR03  R61: While all the other parameters, lambda itself, it could be weighing on alpha, B to the angle, the shape of the unitarity triangle is not exactly precisely weighed, but to five decimal places is invariant under evolution in the standard model.

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STFC-RAL-CR03  R61: And these mass ratios are essentially invented.

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STFC-RAL-CR03  R61: So this suggests to see if we very knew the scale at which you're doing all these misfit might help the fit.

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STFC-RAL-CR03  R61: I'm finding it because it does.

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STFC-RAL-CR03  R61: We allow the renormalization scale that we use.

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STFC-RAL-CR03  R61: So you have… this involves evolving all the parameters, between… between the wheat scale and the gut scale, and redoing the fit, and making… making the scale variable and put it in your program. And we get a fix of 1 degree of… 1 over 2 degrees of freedom. We've lost the degree of freedom because we've added mu as a variable.

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STFC-RAL-CR03  R61: The best fit is about 10 to the 40V. The precision isn't fantastic, because these things are very logarithmically, so you don't get a great, like, all these kind of games.

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STFC-RAL-CR03  R61: You don't get a great precision on the scale at which… so this would be the scale at which

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STFC-RAL-CR03  R61: The textures are examined.

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STFC-RAL-CR03  R61: the assumption is that there's some new physics, like these additional symmetries, discrete symmetries, which would force some of these special parameters to take those values.

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STFC-RAL-CR03  R61: These are just the numbers for the record.

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STFC-RAL-CR03  R61: This curve is, well, these curves, these are the contributions chi-squared from A from MC over MT and MS over MB, and at the 10 to the 3, we were down here, which was responsible for the large chi-squared.

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STFC-RAL-CR03  R61: In fact, 10 to the 2, I said that we'd scale. This was our, you know, 100, basically 100 for the total price width. And as you go up in scale, they all… the fit gets better and better, and they all minimize at the same point.

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STFC-RAL-CR03  R61: Which is kind of lucky, I suppose, for us.

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STFC-RAL-CR03  R61: So, 10 to the 4TV, if this, ANZATs, these tech… if these textures have anything to do with new physics, then they apply at 10 to the 4TV. That might be the scale of new physics.

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STFC-RAL-CR03  R61: Just for the record.

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STFC-RAL-CR03  R61: These are the inputs we normalize to 10 to the 4TV. These are the fitted values. So of course, there are 7, seven floating parameters in this, plus the scale.

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STFC-RAL-CR03  R61: The predictions for the unitarity triangles, rather than being exactly 90 degrees, they vary a little bit by higher order,

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STFC-RAL-CR03  R61: Tons?

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STFC-RAL-CR03  R61: And so you get 91.3 compared to 91.6, 22.3 divided by 22.6. Our predictions have got very small uncertainties.

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STFC-RAL-CR03  R61: So experiments got a while to catch up, I suppose, but that's okay. And so these fitted values here are predictions.

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STFC-RAL-CR03  R61: Let's see.

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STFC-RAL-CR03  R61: There are two degrees of freedom.

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STFC-RAL-CR03  R61: Then we can, just to look at this, the pictorial representation I advertised at the beginning, we're going to need to define, these are the sides of the unit outer triangle, Z0 and the Z0 bar. It's just notation,

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STFC-RAL-CR03  R61: in fact, Z bar is 1 minus Z… Z0 bar is 1 minus Z0. They're not independent. And the sines and cosines here are all… are all,

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STFC-RAL-CR03  R61: sines and cosines of pi by 8. Quarter from the right angle. So…

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STFC-RAL-CR03  R61: That's… I call it the leading order unitarity triangle because, of course, this is the unitarity triangle at leading order in these small rotations.

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STFC-RAL-CR03  R61: And Z0 and Z0 bar are just proxies for the normalized lambda U and lambda D.

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STFC-RAL-CR03  R61: So the last section of the talk is the symmetries of the texture.

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STFC-RAL-CR03  R61: So…

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STFC-RAL-CR03  R61: the symmetries are properties of the paired system, NUMD, so somehow, the UCawa sphere have to be coupled, there has to be a reason for the MUs and the NDs to have the same form as matrices.

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STFC-RAL-CR03  R61: And for the parameters to be related to each other. So that's an unusual feature of this, this, this texture.

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STFC-RAL-CR03  R61: And the symmetries I'm going to discuss could be viewed as consequences of these forms.

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STFC-RAL-CR03  R61: Well, preferably, you might want to view them as ab initio symmetries. So, like a gauge symmetry, you start off and say, well, we've got these fields in our model, these are the gauge symmetries, and a Lagrangian is already defined.

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STFC-RAL-CR03  R61: So the idea here is… and the interactions are defined… the idea here would be that the symmetries are defined.

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STFC-RAL-CR03  R61: And that then defines the mass matrices.

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STFC-RAL-CR03  R61: obviously defines some constraints on the mass matrices. And the next slides outline what these symmetries are, and how they look.

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STFC-RAL-CR03  R61: So, in order to understand the pictorial representation of these symmetries, we just need a little bit of, unitarity triangle technology.

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STFC-RAL-CR03  R61: Under CP, all complex numbers in the mass matrices get complex conjugated. That's well known. And what that does to the unitarity triangle is it inverts it. So e to bar becomes minus eta bar. That's just a complex conjugation of the whole CTM matrix, and of course.

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STFC-RAL-CR03  R61: The triangles, the triangle gets reflected in the real axis, and that's a CP violation.

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STFC-RAL-CR03  R61: A rephasing of the parameters in… the mass matrices

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STFC-RAL-CR03  R61: can rephase rows and columns of the CK matrix. What that does is it rotates the…

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STFC-RAL-CR03  R61: unitarity triangle. And again, the oscill demonstrated that the observables of the unitarity triangle are its area, and the lengths of its sides.

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STFC-RAL-CR03  R61: the orientation is completely, unspecified. It's like a gauge transformation. You just rotate the… you rotate the triangle unobservably.

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STFC-RAL-CR03  R61: The physics is invariant. So there's a nice… well, I hope it's nice. This picture's from the paper. What is the symmetry that ensures that alpha is 90 degrees, effectively? And it's a symmetry of the mass matrices in terms of

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STFC-RAL-CR03  R61: these, parameters we defined, Z0 and Z0 are the sides of the standard unit character triangle. If you flip

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STFC-RAL-CR03  R61: Either Z0 or Z0 bar

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STFC-RAL-CR03  R61: give it a minus sign. So if you let Z… I want to construct now a unitarity triangle

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STFC-RAL-CR03  R61: The unitarity triangle from minus Z0 bar, and… sorry, minus said0, going down here, and…

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STFC-RAL-CR03  R61: plus said0 bar, then this green one is the resultant. This triangle here is a rotated form of this green one, so it's a rotated form of the CP-transformed unitarity triangle. By negating one of the sides, in other words, one of the lambdas in the mass matrix.

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STFC-RAL-CR03  R61: You flip…

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STFC-RAL-CR03  R61: sign a CP violation, and you rotate the triangle, but rotations are undoable, so it's just a CP transformation. That's the equivalent CP transformation. There's no other. The other example is to use

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STFC-RAL-CR03  R61: Oh, that is to use.

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STFC-RAL-CR03  R61: plus Z0 and minus Z0 finally get this triangle, which also, unsurprisingly, rotates into this triangle.

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STFC-RAL-CR03  R61: But it's only true when you've got 90 degrees at this angle.

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STFC-RAL-CR03  R61: If this angle were anything other than 90 degrees, a simple sort of just think about it.

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STFC-RAL-CR03  R61: Flipping this from plus to minus will then leave an angle here.

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STFC-RAL-CR03  R61: that's not 180 degrees, and then this triangle isn't the same shape as this triangle, and you've done more than flip the sign of CP violation, you've changed the magnitude of CP violation.

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STFC-RAL-CR03  R61: So you can change the magnitudes of some of the ZKM, sites.

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STFC-RAL-CR03  R61: Okay, so…

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STFC-RAL-CR03  R61: The combination of the flip in sign of either of these parameters and the CEP transformation is a symmetry, if and only if alpha is 90 degrees.

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STFC-RAL-CR03  R61: That's the… so, take it the other way, take it the other way around. Alpha is fixed 90 degrees by imposing this symmetry on the mass matrices.

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STFC-RAL-CR03  R61: Okay.

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STFC-RAL-CR03  R61: So symmetry is a bit taller. Yet, although the demonstration of this in pictorial terms is done with the leading organizing triangle, surprisingly.

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STFC-RAL-CR03  R61: If you do it algebraically, it's good to all orders.

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STFC-RAL-CR03  R61: So what symmetry requires that the annual beta is a quarter of a right angle?

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STFC-RAL-CR03  R61: Well… If we start off by assuming that this beta angle is not

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STFC-RAL-CR03  R61: Pi by 8, so let it be pi by 8 plus a bit more.

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STFC-RAL-CR03  R61: And then we try rotating, through minus 90 degrees. It's actually obvious that that… that this

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STFC-RAL-CR03  R61: That, I can't… This becomes something down here.

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STFC-RAL-CR03  R61: which is not a side of an angle of a triangle, that's a reflection of that. But if the angle beta tilde, which is just the generalization beta, is exactly pi by 8, then if you subtract,

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STFC-RAL-CR03  R61: High by 4 from it.

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STFC-RAL-CR03  R61: then that's equivalent to CP transformation.

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STFC-RAL-CR03  R61: So, it might… Some people think this is more pendulous, but… The symmetry is… that…

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STFC-RAL-CR03  R61: If you subtract pi by 4 from beta, and you get the CP transform triangle, then beta better be pi by 8.

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STFC-RAL-CR03  R61: And it's a special angle I work for, okay? So that's… that's… that's the second symmetry. And in fact, the two symmetries taken together, if you flip the sign of one side.

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STFC-RAL-CR03  R61: and subtract 5 by 4 from… from beta, you will get back to the beginning. That's the symmetry. That requires both angles to be 90 degrees

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STFC-RAL-CR03  R61: And a quarter of 90 degrees.

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STFC-RAL-CR03  R61: Okay, so that's the symmetry content.

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STFC-RAL-CR03  R61: of the mass matrices which predict the CKM angles and the masses which fit the data with 73 parameters.

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STFC-RAL-CR03  R61: Okay, so to conclude…

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STFC-RAL-CR03  R61: We've proposed a geometric hierarchical mass matrix texture. We don't explain the hierarchies. The idea was to try and explain these

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STFC-RAL-CR03  R61: Accidental, or coincidental, perhaps, particular angles in the inertility triangle, using symmetries.

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STFC-RAL-CR03  R61: The mass hierarchy and slopes are related

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STFC-RAL-CR03  R61: So this is interesting. The mass hierarchy, that's the slopes of the plots of the

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STFC-RAL-CR03  R61: From the very first slide, the ones that show the mass hierarchy, the blue and the red lines.

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STFC-RAL-CR03  R61: Those slopes are related to the sizes… the length of the unitary triangle sides. The symmetries constrain the forms.

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STFC-RAL-CR03  R61: Two discrete symmetries constrain the forms.

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STFC-RAL-CR03  R61: Alpha to be… approximately pi over 8 to be approximately pi over 8, and exactly that at leading order.

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STFC-RAL-CR03  R61: The hierarchy's not explained, but if we were to combine this by some model building with a Frank-Nielsen-type mechanism, we could also potentially, explain the hierarchies.

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STFC-RAL-CR03  R61: I've said all this before, 7 parameters to fit 10 observables with a chi-squared of 1 for 2 degrees of freedom. There's a single, precise prediction of the quark mass ratios, which fits the data spectacularly, and there are 3 predictions

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STFC-RAL-CR03  R61: For the entirety triangles, too.

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STFC-RAL-CR03  R61: You're talking triangle rectangles, two of which are independent.

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STFC-RAL-CR03  R61: Thank you.

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STFC-RAL-CR03  R61: Thank you, that was a very beautiful talk. Thank you.

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STFC-RAL-CR03  R61: Yes, let's move on to the question. Yeah, timeliness.

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STFC-RAL-CR03  R61: Thanks, Paul, that's very important. So on day 20, where you have your little diagram of,

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STFC-RAL-CR03  R61: Yeah, there we are. So, you pointed there to… A scale that's being

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STFC-RAL-CR03  R61: pointed out there by, we think, so I could… Can you relate that to… And the…

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STFC-RAL-CR03  R61: predictions in other contexts. If you just take that by itself and say, okay, this is the scale at which I expect new physics.

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STFC-RAL-CR03  R61: Can I then use that to…

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STFC-RAL-CR03  R61: Make predictions in other contexts for things that we might be able to observe.

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STFC-RAL-CR03  R61: Okay, that's a good question.

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STFC-RAL-CR03  R61: I mean… There are certainly… there's obviously lots of BSN models out there, that try and explain

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STFC-RAL-CR03  R61: Lots of things that are otherwise unexplained in the standard platform. And some of the neutrino models, which predict

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STFC-RAL-CR03  R61: Mixing matrices.

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STFC-RAL-CR03  R61: electrons that are close to what we see, approximately 5 by Maximal, if you like. Some of those models have scales

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STFC-RAL-CR03  R61: Put new physics at around this scale.

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STFC-RAL-CR03  R61: I mean, there are… there are models with new physics Occurring all kinds of scales.

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STFC-RAL-CR03  R61: Obviously the gut scale's a very special scale, for other reasons, but there are certainly

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STFC-RAL-CR03  R61: Should we say, Classes of models where

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STFC-RAL-CR03  R61: The rodent scale is about 10 to 4 TV.

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STFC-RAL-CR03  R61: I got messages from some of the genders to say, oh, my model already uses exactly that scale, that's great, you know.

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STFC-RAL-CR03  R61: I mean, certainly, what one could say is.

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STFC-RAL-CR03  R61: Related to that is, what do we say about electrons? And the answer is nothing at the moment.

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STFC-RAL-CR03  R61: We haven't attempted to include the electrons in this scheme.

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STFC-RAL-CR03  R61: Many years ago, we tried to do stuff with electrons, the driver matromix and stuff.

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STFC-RAL-CR03  R61: Marrying the two together, we haven't tried to do yet.

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STFC-RAL-CR03  R61: That is a question from Steven Burke on Zoom. Okay. Obvious question, what about electrons? Okay, so I think I've already pre-answered that. Yeah.

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STFC-RAL-CR03  R61: I'm not sure if he wants to make another comment.

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STFC-RAL-CR03  R61: Yes. So I've got a question about my understanding there. In that omission that you got, you explained correctly the Massey's run.

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STFC-RAL-CR03  R61: And… but yet there's also the explanation that the masses relate to the size of the unitary tube.

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STFC-RAL-CR03  R61: Triangle. Yes.

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STFC-RAL-CR03  R61: But the area of the unitarity triangle is invariant, so I don't…

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STFC-RAL-CR03  R61: There seemed to be some conflict, though. Okay, my perception…

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STFC-RAL-CR03  R61: So… the shape of the entire triangle is invariant, so the angle… so these…

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STFC-RAL-CR03  R61: The 90 degree angle… the angles in the corners don't run.

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STFC-RAL-CR03  R61: But the standard unitarity triangle, that's the… what I call the unnormalized one, the one you get if you just stop the rows of the CK matrix, and you don't normalize, say, the basis one.

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STFC-RAL-CR03  R61: that… the sides of that matrix are all scaled by VCB, and VCB does run with energy, so the unitarity triangle as a whole

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STFC-RAL-CR03  R61: does run with… energy scale. It's just the shape doesn't run.

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STFC-RAL-CR03  R61: So it evolves. It gets bigger and smaller as you… well, it gets…

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STFC-RAL-CR03  R61: Yeah, it gets bigger as you…

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STFC-RAL-CR03  R61: As you go towards the gut scale.

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STFC-RAL-CR03  R61: Does that implied CP violation, then? The CP viol… the scale… the amount of CP violation runs… Okay. So A, which is, you know, VCP is A, lambda squared, so A runs from 13% from the wheat to the gut scale.

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STFC-RAL-CR03  R61: And so, also, therefore, does the, unit temperature trend, which hopefully I find the right slide.

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STFC-RAL-CR03  R61: Archie was a bit of… Didn't be glad.

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STFC-RAL-CR03  R61: That works.

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STFC-RAL-CR03  R61: So this is the scaled unitarity triangle, but the normal unitarity triangle, it's just got VUB here and VTD here, times things that are essentially unity, and VUB is A lambda cubed times rho plus ieter or something. And so that's got A in it, and A scales…

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STFC-RAL-CR03  R61: A runs with energy, so… so both… both… so actually the area runs as A squared. Okay, so CP violation… So the QB violation should be, sort of, 26% more on that relative cut scale than it is at the weak scale, on the area of the triangle.

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STFC-RAL-CR03  R61: But it runs in such a way that it's designed for essentially invariant, I presume.

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STFC-RAL-CR03  R61: Actually, another one related to the one I asked you before called. So, I think if…

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STFC-RAL-CR03  R61: if I try and say what I think you said, which was that the fridge…

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STFC-RAL-CR03  R61: Was basically born of,

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STFC-RAL-CR03  R61: It being part of maybe some other theories that are, in some ways, natural, right?

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STFC-RAL-CR03  R61: Whereas your approach is born of wanting to make the numbers work. Yes, it's sort of built the other way up. Yes, yeah. We don't have a model for this, apart from the symmetries.

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STFC-RAL-CR03  R61: Okay, yeah, so… Okay, so there's not the opposite thing, which is to then say which of the theories that it fits into. You're not… you don't have that yet? Not yet, no, I mean… Do you have a feel for anything that might relate to it?

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STFC-RAL-CR03  R61: That's a nice question.

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STFC-RAL-CR03  R61: It's a very open question, I mean, I know, it's just so strange. Not off the top of my head, I mean, I think, you know, if I was going to start building a model, I would try and incorporate Frog at Nielsen. So Frog at Nielsen, they have some heavy, very super heavy bosons.

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STFC-RAL-CR03  R61: And then the fermions couple to these super heavy bosons, like the Higgs mechanism, or like with UCARA companies, and they can couple more than once. So you can have a fermion, and then the boson, then the fermion, then another boson. And the number of times they're coupled to that boson

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STFC-RAL-CR03  R61: It's proportional to the powers of these small parameters in the mass matrix, and it all works rather nicely.

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STFC-RAL-CR03  R61: And I talked to Steve, and I went to Southampton for the day, we had a nice day and chatted about this, but we couldn't immediately see a way to actually do it. So, so we're sort of a bit stuck at the moment.

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STFC-RAL-CR03  R61: But it works that way up, so… Yeah, yeah, you would use this to try and say, what kind of model has these… these discrete symmetries as well? The discrete symmetries…

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STFC-RAL-CR03  R61: of this, you know, negating the sign of the size of the notarity triangle, and that corresponding to the CP symmetry. That's a… it's a bit weird, it's a bit…

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STFC-RAL-CR03  R61: It is a bit unusual, and no one's come up with anything as yet, so maybe it's just that to them.

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STFC-RAL-CR03  R61: Well, and on that point, I mean, are there other places where it could…

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STFC-RAL-CR03  R61: be applicable. You mentioned electrons and that, so is it that… is there any indication that this approach might be something more universal in other surfaces? We've only got two examples of

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STFC-RAL-CR03  R61: permanent mixing, of dogs. We've got quarks and the leptons, and they're very different. I mean, completely outside our stuff, there's this idea of cork-lepton complementarity, where the deviations of the CKN matrix from just the identity, all being small.

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STFC-RAL-CR03  R61: correspond to the deviations of the lepton mixing from trivimaximal, and then tribomaximal would be somehow introduced to the leptons

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STFC-RAL-CR03  R61: by, by the seesaw mechanism.

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STFC-RAL-CR03  R61: And that would be the bimaxional bit. And then… and then a small perturbation of that.

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STFC-RAL-CR03  R61: would be both on the lacton side and the quartz side, and they'd be related to each other somehow. So you'd have approximate tri by Maximal and approximate identity for mixing. And there are models that do that, which are very, very interesting, very intriguing.

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STFC-RAL-CR03  R61: So, I mean, it would be nice to try and build something like that onto this, or vice versa, build this onto that.

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STFC-RAL-CR03  R61: Again, yeah, time's… time's finite.

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STFC-RAL-CR03  R61: Yeah, and I mean, it's only two sectors, only two-wheel sectors. The other sector that was in the background, etc. Of course, you don't have mixing in the same way, but just whether there are… Not in the same way. I mean, ultimately, it's all, you know, in the standard model, it's all… it's all your power captures. Exactly, yeah.

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STFC-RAL-CR03  R61: Plus, you know, the seesaw mechanism for doing those things we have.

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STFC-RAL-CR03  R61: We reasonably confident we expect.

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STFC-RAL-CR03  R61: So…

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STFC-RAL-CR03  R61: I'm not working with this at the moment.

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STFC-RAL-CR03  R61: just finished an Atlas paper, so… the BCE measurement. Yeah, it's excellent, yeah, basically.

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STFC-RAL-CR03  R61: Okay. Maybe a question from Sun. Is anything here on an explanation of why there are three generations? Or could it still work if they were four?

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STFC-RAL-CR03  R61: I would think the nice parts of this would almost certainly break down with four generations.

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STFC-RAL-CR03  R61: But I haven't thought about it in detail.

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STFC-RAL-CR03  R61: Three generations is very special, the CP violation. This thing that the CP… the triangle.

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STFC-RAL-CR03  R61: Area is the same for all kinds. That's no longer true with four generations.

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STFC-RAL-CR03  R61: So the whole Unitarity triangle idea breaks down. We end up with the unitarity… well, the unitarity quadrilateral with four generations, and…

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STFC-RAL-CR03  R61: And, lots of the really neat stuff. Three generations really is special in the world of CP violation.

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STFC-RAL-CR03  R61: And there are other… there are other things, aren't there, MT, where triangle anomalies cancel with free families or something? I don't know much about that. It's…

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STFC-RAL-CR03  R61: H3 stuff, but, I mean, I think four… four generations of…

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STFC-RAL-CR03  R61: There are lots of things that discourage us from believing that four generations are true, you know, the width of the Z and things like that. Of course, it's never… you can never say never.

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STFC-RAL-CR03  R61: But if there's a fourth generation, the extra quarts aren't like the other quarks.

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STFC-RAL-CR03  R61: They're not like the books we now have.

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STFC-RAL-CR03  R61: It would be very unnatural, I think, at this point.

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STFC-RAL-CR03  R61: Any more questions here?

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STFC-RAL-CR03  R61: Oh, and I just want to say, I love this team, it's always lovely to be here. It's nice to see so many old friends and faces that people haven't seen for a long time.

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STFC-RAL-CR03  R61: Yes. Yeah, that's a good transition to our lunch. So, if you would like to join, I hope you get some time. Yeah, I'll have some very, very quick. A very quick lunch with Paul.

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STFC-RAL-CR03  R61: Well, yes, you're welcome. Thank you.

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STFC-RAL-CR03  R61: Okay, yes.