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I've recently been studying the symmetries that give rise to the bosons of the standard model. I have taught myself just enough group theory to kind of understand what $U(1)$, $SU(2)$, and $SU(3)$ mean, and how the generators of these groups are the identity matrix and the Pauli/Gell-Man matrices. However, I don't quite understand why electromagnetism/weak hypercharge and the $U(1)$ symmetry is not traceless, while the weak $SU(2)$ and strong $SU(3)$ are. I've seen some references that the existence of an element with a trace would give the strong force a self-interacting, color neutral, ninth gluon which would act like a photon and be long range. I can understand that. However, I don't understand why the strong and weak forces are $SU(n)$ and can't have one of these extra bosons, while electromagnetism is $U(1)$ and can. Is it simply that the strong and weak force aren't long range forces? I'd prefer an explanation that has some reference to a physical/measurable property rather than just being pure math, if possible.

Edit: I am not interested in the baryon/lepton number U(1) symmetries. This is not a duplicate of Why is the standard model gauge group $SU(3) \times SU(2) \times U(1)$ and not $U(3) \times U(2) \times U(1)$?. I am more interested in why a chargeless (e.g. colorless for strong) gauge boson doesn't exist for the weak and strong forces.

18th Shard
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  • Possible duplicates: https://physics.stackexchange.com/q/536742/2451 , https://physics.stackexchange.com/q/116831/2451 , https://physics.stackexchange.com/q/119190/2451 and links therein. – Qmechanic Jul 18 '23 at 16:21
  • My question is not answered by the linked question: the response there is in reference to the U(1) symmetries of baryon and lepton numbers as opposed to the more fundamental U(1) of weak hypercharge. It also does not address the physical interpretation why strong and weak symmetries don't produce long-range, uncharged bosons at all. – 18th Shard Jul 18 '23 at 16:47
  • Extra U(1)s would act like extra variant EM forces/fields, but nature chose to not realize them in practice. Nobody knows why. Nature does daffy things, and we strive to describe them as they are.... – Cosmas Zachos Jul 18 '23 at 19:03
  • So in other words, there is not any real reason for the lack of these bosons except that they don't seem to exist in the universe? – 18th Shard Jul 18 '23 at 19:16
  • Indeed, yes. In physics, we reason after the facts, not before them! That's what they mean by calling it an experimental science. Most Platonic overextensions have famously fallen flat on their face! – Cosmas Zachos Jul 18 '23 at 21:22

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