I've understood that the three fundamental interactions described by the Standard Model (the electromagnetic, the weak and the strong force) are thought to correspond (roughly) to gauge invariances under the $U(1)$, $SU(2)$ and $SU(3)$ group symmetries. Why isn't there a fourth fundamental force following from an (hypothetical) invariance under $SU(4)$ transformations?
Just to clarify, I'm asking for a possible argument relying on logic or theoretical reasons (say, there is perhaps some constraint which doesn't allow this correspondence to apply to $SU$(4)).
Edit:
Though I'll leave the original text unchanged, I'd like to add a possibly more precise way to reformulate this, as suggested by @Rococo:
"Can the Standard Model be extended in a straightfoward way to include an $SU$(4) gauge field?"
Best Answer
I think the crux of your question stems from the apparent pattern in the observed gauge groups appearing in the standard model. In particular, we see a $U(1)$, then $SU(2)$, then $SU(3)$, so if we follow the pattern we might guess this is just the beginning of an infinite series of gauge groups appearing, so the next would be $SU(4)$ (note this pattern isn't perfect, i.e. one would think we should use $SU(1)$, which is actually just the trivial finite group of one element). First I'll say that recognizing patterns and asking if there is an underlying explanation is absolutely essential to advancing physics from a theoretical perspective. And often the most profound breakthroughs come from seemingly trivial observations (the discovery of the different quarks seemed to follow a similar pattern: they had two, then it looked like 3 worked better, then they needed 4, and so on). So all that is just in support of the question, and also to refute the argument that the answer is "that is just the way nature is."
So once you have recognized a pattern, you should start asking whether the pattern solves existing problems with the your current understanding of the system. In the case of quarks, the two quark model did a good job explaining the pion particles that showed up at low energies. However, as more particles were discovered, it looked like they were arranging themselves into groups of $8$ or $10$ rather than groups of $3$. The explanation seemed to be that there was an underlying $SU(3)$ symmetry (not to be confused with the $SU(3)$ color gauge symmetry!), which required $3$ quarks, instead of the previous model based on $SU(2)$ symmetry with $2$ quarks. In fact, after thinking about how particles behaved under the electroweak interaction, they further realized a fourth quark was needed (although the corresponding $SU(4)$ symmetry you might guess is present is actually not, since the charm quark is too heavy to be considered on the same ground as the lighter three). Of course, now we know that there are $6$ quarks, and still people like to speculate whether there could be more.
So back to the original question of whether extending the pattern of the observed gauge groups solves any problems with the standard model. As far as I know, adding an additional $SU(4)$ symmetry doesn't do much other than add more particles that we haven't seen. So those prospects do not look good. However, a similar question related to the structure of gauge groups in the standard model is whether it arises from a grand unified theory (GUT), where the standard model gauge group appears as a subgroup of a larger gauge group. It turns out the smallest simple group that contains the standard model's $SU(3)\times SU(2)\times U(1)$ is $SU(5)$, and there are a number of interesting ways how the particles in the standard model arrange themselves into nice representations under $SU(5)$. This unification solves an interesting problem about how the gauge couplings in the standard model all seem to run to the same value at high energies, which would be an extraordinary coincidence in the absence of a GUT explanation. In this case, the simplest $SU(5)$ models don't seem compatible with data, but extensions involving $SO(10)$ or supersymmetry (as well as a host of other things) still look promising.
In fact, $SU(4)$ can show up as a subgroup of $SO(10)$, and so $SU(4)$ may play an important role in this GUT. I believe in this version of grand unification, lepton number plays the role of the fourth color. So for example, the three colors of up quarks and the neutrino arrange into a four color multiplet of $SU(4)$, and the three colors of down quarks combine with the electron to give another $SU(4)$ multiplet, which is kind of neat!
Anyway, I hope this gives you some intuition about how and why an $SU(4)$ gauge group could arise.