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Gauge bosons

The Standard Model of Particle Physics is based on the Yang-Mills paradigm, in which there are (unitary) “gauge groups”, whose adjoint representation contains “gauge bosons” that mediate the forces. In the first case, (quantum) electrodynamics, the gauge group is U(1), which is one-dimensional, and the gauge boson is the photon. The adjoint representation is the same as the “vector” representation and the same as the unitary representation and the same as the trivial representation, so it is not immediately obvious how to generalise. Should we generalise adjoint, vector or unitary?

The second case is SU(2), for the weak force. Here the adjoint representation and the vector representation are again the same, and are three-dimensional, but the unitary representation is two-dimensional complex or four-dimensional real. This reflects the fact that there are three “gauge bosons”, the W+, W- and Z bosons. But the situation is complicated by the “mixing” with electrodynamics and U(1). So the group becomes U(2), which is four-dimensional (real). The vector representation now becomes complex (three-dimensional), while the “spinor” (unitary) representation is two-dimensional complex or four-dimensional real. There is a distinct possibility that these two four-dimensional representations have been confused with each other, and that it is the unitary representation that describes the gauge bosons, not the adjoint representation.

Indeed, the formalism of electroweak mixing takes place in a four-dimensional representation, and involves taking weird linear combinations of the four dimensions. This makes no mathematical sense in the adjoint representation, which splits 1+3 without the possibility of mixing. But it does make sense in the unitary representation, which is irreducible, so you can mix things up to your heart’s content.

The third case is SU(3), where the adjoint representation and the unitary representation are different. The former is 8-dimensional real, and gives the 8 gluons of the Standard Model. The latter is 3-dimensional complex, or 6-dimensional real, and gives only 6 coloured gluons, and does not include the two colourless gluons of the SM. Is this a problem, or not?

Well, let’s consider what happens when we try to make Grand Unified Theories. The idea that Georgi and Glashow had in 1974 was to unify SU(2) and SU(3) into SU(5). All well and good, but what about the gauge bosons? If they live in the adjoint representation, there are now 24 of them, compared to the 1+3+8=12 in the Standard Model. What do the other 12 “gauge bosons” actually do? Well, apparently they cause protons to decay. But protons do not decay. So this is obviously wrong. But what exactly is wrong? Is it the whole idea of unification, or is it the assumption that the gauge bosons live in the adjoint representation?

If we use the unitary representation instead, then we get the 6 coloured gluons, plus the four electroweak mediators, and nothing else. No new forces, no proton decay, no new particles, no nothing. That’s a lot better, don’t you think? All we’ve lost is two measly colourless gluons, that nobody has ever seen anyway, so what’s the problem? There is no problem, except that a sacred cow has been slaughtered. Oops. Guilty as charged, m’lud.

If you’re worried about those colourless gluons, you can ask Pati and Salam for them. They decided it would be a good idea to unify the three colours of quarks with a fourth “colour” for leptons, and extend the gauge group from SU(3) to SU(4). Again, if you consider that the gauge bosons lie in the adjoint representation, then you get an extra 1+6 of them. Well, the 1 represents the group U(1), so it may be the photon or the Higgs boson or something like that, so that’s OK, but what about the 6? Where are these extra particles, what are they doing? Well, you can avoid this problem by just considering the unitary representation instead. Then you get the extra two gluons that you wanted, and nothing else.

Let’s go back to SU(5), and split it as SU(4) x U(1), so that we get 8 gluons and the photon in the unitary representation. But surely we need 10? Yes, of course, the photon comes in two polarisations, so that makes up the full set of 10. Now split SU(5) as SU(3) x SU(2) instead. What happens? The four “electroweak gauge bosons” now appear in massless form, as two colourless gluons and two (virtual) photons. Where do the masses of the W and Z bosons come from? Well, in E8 they come from mixing with General Relativity, which grabs two of the 10 dimensions, leaving only the gluons behind. But that’s another story, for another day.

How is that sacred cow doing? Is it thoroughly butchered by now? Is it ready for the cooking pot? Or would you prefer to bury it?


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