My paper on the SU(7,2) model has appeared on the arXiv today, so you can download it from https://arxiv.org/abs/2407.18279 if you want. I’m busy adding to it and correcting it, of course, so an update will appear in due course. The main theme is that the finite symmetries that appear in the Standard Model are not “just a phase” (which in technical language means “a subgroup of U(1)”), but are important in their own right.
To illustrate what I mean, take the Standard Model gauge group (SU(3) x SU(2) x U(1))/Z_6. What is that quotient by Z_6 doing there? What is it for? A quotient group is math-speak for forgetting something, or ignoring something. A typical example is SU(2)/Z_2: the original group SU(2) consists of 2×2 matrices, and for every matrix there is another matrix which is its negative. If you don’t care about the signs, you just take the quotient group SU(2)/Z_2, and the signs disappear, leaving everything else intact. Physicists do care about those signs, but they think they are “just a phase”, so what they do is add in a U(1), so they can talk about phases, and take out the Z_2, to give the group U(2) = (SU(2) x U(1))/Z_2. So while SU(2) x U(1) has got two signs in it (one in SU(2) and one in U(1)), the group U(2) has only one.
Clear so far? What should also be clear is that by this construction, we have thrown away some information. We have forgotten a sign, which we are therefore ignoring. We have lost a vital bit of information. We may not know what this bit signifies physically, but it may be something important like the sign of the electric charge. Actually, this is exactly what it signifies in the model I’m talking about here, but it doesn’t matter exactly. What matters is that it could be something important. That something important is not just a phase.
The Standard Model also throws out a Z_3, that acts as scalars in SU(3), by exactly the same process of extending the scalars to U(1), and then throws away one of the two copies of Z_3 in SU(3) x U(1) to give a group U(3) = (SU(3) x U(1))/Z_3. So again, there is a triplet symmetry that has been thrown away, forgotten and ignored. It could be something important, like the generation symmetry of electrons. Actually, this is exactly what it signifies in the model I’m talking about here, but it doesn’t matter exactly. What matters is that it could be something important, that is not just a phase.
As if that wasn’t bad enough, the Standard Model then identifies the two copies of U(1) with each other. This doesn’t just throw away a finite symmetry, it throws away an infinite symmetry. What could this infinite symmetry be? Of course, it is a “phase”, but it is a phase that is attached to a relationship between finite charge and generation symmetries. So it has meaning. It relates to electron/positron charge and to electron/positron generation, but it is continuous. So it must relate to neutrino/antineutrino generations, and it tells us that the neutrino generation symmetry is continuous, unlike the electron generation symmetry, which is discrete. In other words, by throwing away this copy of U(1), the Standard Model has thrown away neutrino oscillations. Bad mistake that.
So, anyway, if we take the full group SU(3) x SU(2) x U(1) x U(1), instead of the bowdlerised SM version (SU(3) x SU(2) x U(1))/Z_6, then we gain three things that the SM cannot understand:
- The sign of electric charge;
- The three generations of electrons;
- Neutrino oscillations.
In other words, we don’t have to add anything to the Standard Model to get explanations for these three experimental facts, that are apparently so hard to understand. We just have to not throw away what was there all the time. All that stuff they threw away, thinking “it’s just a phase”. It’s not just a phase, it’s physical reality. Get used to it.
In particular, the single copy of Z_2 x Z_3 that is in the SM is replaced by two copies, that is by Z_2 x Z_2 x Z_3 x Z_3 in the SU(7,2) model. Well, maybe that’s not quite accurate, since the SM does have another copy of Z_2, being the scalars in the Lorentz group SL(2,C). But what the SU(7,2) model has is a Z_4 in place of that second Z_2. So in any case, we’ve got an extra factor of 2 over and above what is in the SM. There is also another factor of 3, since one of the two copies of Z_3 turns out to be a quotient Z_9/Z_3, and if there is anything you have learnt from this post it should be, never take a quotient group if you can avoid it, because a quotient group throws away information that might come in handy one day. That is the first rule of hoarding: never throw anything away, because you never know when it might come in handy.
There is a picture in the paper that I think illustrates my point. It is a phase diagram, with a whole load of lines emanating from the centre (the origin of coordinates). The angles between these lines are what is important. They illustrate what happens if you make everything be “just a phase”. There is a copy of Z_3 for electron generation symmetry, and a copy of Z_4 which I haven’t labelled, but which is really a symmetry that switches weak hypercharge with the third component of weak isospin. There is a copy of Z_9 in the background, controlling what is going on. And then I’ve marked most of the mixing angles of the SM on the diagram.
For example, I’ve marked the mass axis, obtained from the experimental values of the masses of the three generations of electrons. From that one data point, I get two mixing angles for sure: the mixing between first and second generations of both leptons and quarks. I also get, though less surely, the two CP-violating phases, of 67 and 200 degrees. Mixing with the third generation is more complicated, and needs two more data points, arising from adding in the proton and neutron masses, as I explained in an earlier paper. From these two data points, I get three more mixing angles for sure, and possibly one more. That accounts for all except the Weinberg angle, which I already explained in the same paper with a similar diagram that related the Z_2 charge axis to the Z_4 weak hypercharge/isospin square.
So you see how useful these finite groups have become? You see why you shouldn’t throw them away, or forget them, or ignore them? You see how they explain most, if not all, of the 9 mixing angles in the Standard Model? You see how the Standard Model treats them as “just phases”, which means they have to be measured, when my model treats them as “not just phases”, but fundamentally geometrical in character, which means they can be calculated? Which model is better?