There are two very different aspects of the symmetry-breaking of the weak nuclear force (the force responsible for radioactive decay). One is the breaking of charge symmetry, the other is the breaking of mass symmetry. Mathematically these behave differently, because the charge symmetry is discrete and the mass symmetry is continuous. It is therefore worthwhile considering them separately. But let us start with the way this symmetry-breaking is described in the Standard Model. There the group of symmetries is usually described as SU(2), that is Spin(3). The three coordinates of the space on which Spin(3) acts (as SO(3)) are called the three components of weak isospin.
But only the third component of weak isospin is actually used in the Standard Model. This effectively breaks the symmetry to Spin(2+1), and leaves the first and second components of weak isospin with nothing to do. Why are they there, if they just sit there and do nothing? Why does the theory not use them? I’ll come back to these questions in a moment. First, we must take into account the fact that the weak SU(2) is actually treated as a complex Lie group/algebra, rather than real, which means that in mathematicians’ language it is actually SL(2,C) = Spin(3,1) rather than SU(2) = Spin(3). In other words, we need to talk about four components of weak isospin, rather than three. The “fourth component” is, however, a scalar under real SU(2), and is called weak hypercharge.
At this point we can re-interpret the breaking of Spin(3) into Spin(2+1) as a conversion, within Spin(3,1), of Spin(3) into Spin(2,1). Or we can do both, and write Spin(2+1,1). The last two coordinates represent weak hypercharge and the third component of weak isospin, but it is no longer quite so clear which is which. Is weak hypercharge the scalar under unbroken Spin(3), or the scalar under broken Spin(2,1)? This is the kind of mathematical nicety that physicists generally have no patience with. It doesn’t matter to them, because they just stick in a square root of -1 to convert from one to the other, so that they can’t see the difference. But to a mathematician this is the difference between chalk and cheese, because one of them is right and the other one is wrong.
If we first consider the finite symmetries (charge, weak hypercharge), then all we have available are sign changes on the two scalars, and a rotation in Spin(2) = U(1). The latter acts on the the first two components of weak isospin, but not on the the third component, or on weak hypercharge. Therefore it does not appear in the standard formalism. It is the bit that is not being used. But it can be used for the generation symmetry (which means it is technically a trit rather than a bit).
You may consider this to be a radical or even nonsensical suggestion, but in fact it appears (somewhat disguised) in my published joint paper with Manogue and Dray on a proposed E8 model. In that model, it is the discrete generation symmetry that breaks the symmetry of the weak SU(2). In that paper, this symmetry-breaking arose from the structure of E8, which did not allow for a generation symmetry that is independent of the SU(2) weak symmetry. But here, it arises purely from consideration of the weak symmetry group itself, without any constraints coming from embedding the group in a larger model. It comes, in fact, from embedding the discrete symmetries of weak isospin into SU(2).
If you only consider the third component of weak isospin, then you don’t see any of this structure. But if you consider all three components, then what you get is tetrahedral symmetry of weak isospin. The important group is the binary tetrahedral group. The generation symmetry is then a subgroup of order 3, embedded in U(1), acting on the first and second components of weak isospin. I’ve described before how it works for electrons, but not for quarks or neutrinos. There is a recent paper on the arxiv, https://arxiv.org/abs/2409.15385 by Henrik Jansson, which extends this description of the quantum numbers to all the elementary particles (fermions). Where it differs from my description is that it does not deal with mass at all. But it deals with all the discrete symmetries, and that, I think, is very important.
So, let us now consider where mass arises in the model. In the Standard Model, it is the Dirac equation that defines mass, in terms of Dirac spinors. These spinors are physically quite distinct from the iso-spinors that I’ve been talking about so far, but they are mathematically very similar, which can lead to confusion. But what is important about the generation symmetry is that it changes the masses of the particles, and therefore it acts on spinors as well as isospinors. Otherwise, it could not change the solutions of the Dirac equation. So the question then becomes, how exactly does the generation symmetry act on the Dirac spinors? That is the question that my investigations in recent months are designed to answer.
I don’t know if I have the right answer yet, but there is a very curious aspect to the symmetry-breaking that comes out of the necessity for having a discrete symmetry of order 3 to act on both spin and weak isospin. That is that the conversion of unbroken SU(2) symmetry to broken SU(1,1) symmetry inside SL(2,C), that is the cornerstone of weak symmetry-breaking in the Standard Model, doesn’t contain any such symmetry of order 3. The reason appears to be that it does not contain a U(1) symmetry group for weak hypercharge. If we want to embed weak hypercharge correctly in the model, we seem to need SU(2,1) instead of SL(2,C). At this point, of course, I am effectively saying that the Standard Model description of weak symmetry-breaking using SL(2,C) is not just “incomplete”, but is actually “wrong”. So this is the point at which I earn my “crackpot” stripes.
But if you look carefully at what is happening in the Standard Model, you see that in the vertex factors for neutral current weak interactions there are factors of sin^2 of an angle. This indicates that there is a rotation embedded in the model somewhere. Essentially, it is a rotation from electric charge to weak hypercharge – in theory. But in the practical calculations, the balancing cos^2 term does not appear. That means it is a rotation from electric charge into nothing. Why is it a rotation into nothing? Because if you use the group SL(2,C) to do this mixing, then there is nothing there. But if you use the group SU(2,1) instead, then there is something there that you can actually rotate into. In other words, it is the absence of the cos^2 terms in the Feynman calculus that is the clue that there is something wrong at this precise point in the Standard Model.
That on its own does not prove that my model is right, or that the Standard Model is wrong. But my model has a mathematically rigorous description of electro-weak mixing, and the unification of weak hypercharge with all three components of weak isospin, whereas the Standard Model has nothing but a kludge.