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  Why is there no fundamental force following from the SU(4) symmetry?

+ 5 like - 0 dislike
2139 views

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?"


This post imported from StackExchange Physics at 2016-03-30 10:14 (UTC), posted by SE-user David Herrero Martí

asked Mar 28, 2016 in Phenomenology by David Herrero Martí (25 points) [ revision history ]
edited Mar 30, 2016 by Dilaton
Are you asking why nature works the way it does and not any other way?

This post imported from StackExchange Physics at 2016-03-30 10:14 (UTC), posted by SE-user Prahar
Well, I am asking for a physical or mathematical reason. I don't really know if there is one, that's why I asked.

This post imported from StackExchange Physics at 2016-03-30 10:14 (UTC), posted by SE-user David Herrero Martí
FWIW SU(4) has been proposed.

This post imported from StackExchange Physics at 2016-03-30 10:14 (UTC), posted by SE-user Qmechanic

1 Answer

+ 6 like - 0 dislike

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 finite group Z2). 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)×SU(2)×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.

This post imported from StackExchange Physics at 2016-03-30 10:14 (UTC), posted by SE-user asperanz
answered Mar 28, 2016 by asperanz (175 points) [ no revision ]
The SU(4) flavor is occasionally used to classify the baryons, as in this PDG reference. As you noted, SU(4) is not a very good symmetry, so it is only sometimes helpful.

This post imported from StackExchange Physics at 2016-03-30 10:14 (UTC), posted by SE-user Luke Pritchett

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