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.

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.

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.

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.

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. Thanks!

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.