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I've heard before that when looking for grand unified gauge groups, with a single gauge coupling, we want to use simple groups, in the group-theoretic sense of having no non-trivial normal subgroups. What is the reason for this? To me it doesn't seem as though there is a problem with having a GUT group (like $SU(3) \times SU(3)$ for instance) which is not simple, but just giving the group a single gauge coupling. Can anybody shed any light on this?

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  • $\begingroup$ Your proposal to "give the group a single gauge coupling" essentially means fine-tuning the three possible independent gauge couplings to all be equal. Physicists find theories in which this fine-tuning is forced by symmetry requirements (they usually don't call this "fine-tuning") to be more natural than theories in which it's simply postulated. $\endgroup$ – tparker Aug 31 '16 at 1:29
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If you don't impose some sort of simplicity restriction on the grand unified group, then the Standard Model gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ would already classify as a GUT group, making the search for a GUT pointless. There is no problem as such with non-simple gauge groups, but they just don't correspond to what we call "grand unified theories", and choosing "a single gauge coupling" isn't really what GUTs are about. One of the best motivations for GUTs is that we expect them to naturally solve the problem of the quantization of electric charge, in particular why the hadron/quark and the lepton balance each other so nicely. In order to solve this, you must forbid such things as the electromagnetic $\mathrm{U}(1)$ showing up as its own factor, so that its representation is induced by the stricter representation theory of the grand unified group, and not chosen as input as in the Standard Model.

This terminology is slightly at odds with saying that the electroweak $\mathrm{SU}(2)\times\mathrm{U}(1)$ "unifies" the weak $\mathrm{SU}(2)$ and the electromagnetic $\mathrm{U}(1)$, but there's not much we can do about that. Note, however, that at least the electromagnetic $\mathrm{U}(1)$ and the weak $\mathrm{SU}(2)$ are not the factors of the electroweak group - the electromagnetic $\mathrm{U}(1)$ embeds into the electroweak $\mathrm{SU}(2)\times\mathrm{U}(1)$ as a mixture of the isospin and the weak hypercharge.

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  • $\begingroup$ Thanks for the answer. If your objective in grand unification is to package particles into representations of a larger group to explain such things as quantisation of charge, then it seems to me that there is no problem with non-simple groups like $SU(3) \times SU(3)$, whose representations are labeled by discrete indices. – $\endgroup$ – gj255 Aug 31 '16 at 9:32
  • $\begingroup$ @gj255: In that particular instance, what is the difference between that as a GUT group and just having $\mathrm{SU}(3)$ as the electroweak group (maybe as a second stage of EW symmetry breaking)? I'm sure you can find something people would reasonably call a "grand unified theory" with a non-simple group, as I said, there's no problem with these theories as such. However, groups that are direct products $G_1\times G_2$ aren't viable candidates because there always remains the "fine-tuning" of why a particular field that transforms in one rep of $G_1$ transform in a certain other of $G_2$. $\endgroup$ – ACuriousMind Aug 31 '16 at 12:17
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    $\begingroup$ Well $SU(3) $ doesn't contain the SM gauge group --- when I say $SU(3) \times SU(3)$ (which was nothing more than an illustrative example) I'm supposing that one copy of $SU(3)$ is the QCD group, and the other contains the electroweak group. I've accepted your answer, since you raise some interesting points, but I will say that I don't feel your argument about quantisation of charge has any bearing on the question of simplicity of the GUT group. $\endgroup$ – gj255 Sep 13 '16 at 13:51

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