First impressions
arivero said:
The most common explanation I know . . .
Is this explanation satisfactory enough?
Asking "why" the laws of physics are what they are is always problematic.
When you say "common explanation" this rings two bells in my mind.
First, it isn't the only possible way that it could be explained.
Second, it is an "explanation" not a law of physics or mathematics that can be established from anything but the example it seeks to explain.
Some laws of physics are derived from other laws of physics and then explaining why they are so isn't circular. But this doesn't appear to be one of them.
If you want to explain why must massless particles travel at the speed of light and not experience the passage of time in their own reference frame, you can point to Special Relativity, which can be established from multiple theoretical arguments unrelated to this property of massless particles, and also from experimental evidence tending to confirm this physical law in various circumstances to the limits of observational precision.
In contrast, "anomaly cancelation implies the sum of electric charges of each particle must cancel generation" is an explanation of why electric charges must cancel out with the observation that this is what happens in real life as its sole observational support of example.
Within the theoretical framework of the Standard Model and its possible extensions, you are really just trying to use an axiom of the Standard Model to explain something that you assumed in the first place.
Why must there logically be anomaly cancelation at each generation?
One could imagine a universe in which, for example, anomalies must cancel globally and not at each generation, if they didn't cancel at each generation. That universe wouldn't be logically impossible, it would just be a universe with particles and laws of physics a bit different from our own. But we don't live there.
Thus, I agree with Vanadium that:
"I want my theory to be anomaly free" is not an argument I find compelling.
There are plenty of circumstances in physics where anomalies of some kind do arise, and we deal with that and use it as a tool to do physics. For example,
we calculate anomalous dimensions in effective QED, and the world doesn't come to an end.
Similarly, a pion is often called a "
pseudo-Goldstone boson", because it is almost a Goldstone boson, but rather than involving an exact symmetry, it involves a symmetry that real world physics closely approximates, but
anomalously, real world physics deviates slightly from.
Isn't assumption one circular?
As the OP notes, for this explanation to work one has to "assume the SM group and the SM particle content". But, if the outcome you are trying to explain is compelled by one of the axioms that you assume in the first place, then you're engaged in circular reasoning and the exercise is meaningless.
What does assumption three do? And isn't it false anyway?
I also don't see that it is important one way or the other to the explanation that "It needs to make sure that the nucleus is just protons and neutrons". This seems like a non sequitur and I don't see where you are going with it.
This also isn't an assumption that is really true. A nucleus doesn't just have protons and neutrons. It also has electromagnetically neutral mesons like pions and kaons that are carrying the residual strong force that binds the nucleus in atoms together between the protons and neutrons in that nucleus. And, of course, atomic hydrogen-1 doesn't have any neutrons in its nucleus. So you have things that aren't protons or neutrons in a nucleus, and you don't have to have both of them in a nucleus. This doesn't even begin to investigate whether the sea quarks of a proton or a neutron count or not, with the theory behind "not" being that they are part of particular protons and neutrons and not directly a part of the nucleus itself. And, the reason that there are no other stable hadrons in a nucleus is rather obviously because protons and bound neutrons are the only stable hadrons. The electron is a stable particle, but if any particle in a nucleus was a lepton, then the residual strong force wouldn't bind to it because leptons don't have color charge, so it wouldn't stay in the nucleus or be part of a structure bound by the residual strong force, and so, it wouldn't really be a part of the nucleus.
In sum, this question is unanswerable
Ultimately, the only real answer to "Why do proton and electron have equal and opposite electric charge?" is "because that's what we observe them to be to high precision," therefore a theory makes the logical leap of faith of assuming that it is true exactly, because this keeps the math clean and simple and seems to work.
While some other answer than "that's how it is" could hypothetically be possible for some laws of physics, in this particular example, I don't see how any elaboration or analysis that you could add to "that's how it is" would add any meaningful explanatory power or understanding to the question.
