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I have already seen this question. It was answered that $U(3)$ can be decomposed into $SU(3) \times U(1)$, and $U(1)$ is already used for the EM interaction. Still, I wonder why the EM interaction influences the strong one. A linear combination of three gluon-antigluon states would be conceivable, as far as I can see. The matrix of gluon antigluon states has trace zero, but why should this depend on the fact that $U(1)$ has been used already for EM?

In other words, is the $U(1)$ transformation used in EM the cause that it's absent in the $SU(3)$ transformation used in the color force? Is this the reason $SU(3)$ was chosen and not $U(3)$, or was it the absence of colorless states of three gluon-antigluon states (would these be able to travel like real photons, like a kind of gluon balls?).

Qmechanic
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MatterGauge
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    I don't see anyone in the linked question saying that "U(1) is already used for EM". The point in the accepted answer (however correct it may be) is just that "a gauge theory with $\mathrm{U}(N)$" is similar to "a gauge theory with $\mathrm{SU}(N)$ and an $\mathrm{U}(1)$", so there's not really any point in talking about the former. – ACuriousMind Nov 18 '21 at 18:05
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    Do not forget that the groups are chosen so that they fit the experimental observations and data. Look how the observed symmetries led to the quark model https://en.wikipedia.org/wiki/Quark_model and the to color su(3) https://en.wikipedia.org/wiki/Color_charge – anna v Nov 18 '21 at 19:07
  • @ACuriousMind The second answer seems to imply that it's already used in EM. Thus my point is, could the color force be described by a U(3) theory, as the colorless combination of three gluon-antigluons is different from the photon. Of course all three color-anticolor gluon pairs are used but one combination is excluded because of tracelessness. Does the U(1) of EM already excludes this? – MatterGauge Nov 18 '21 at 19:25
  • @annav So it are indeed observations which led to it? If a colorlees gluon state had been observed it would be U(3)? – MatterGauge Nov 18 '21 at 19:30
  • The accepted answer to the question you link explicitly says: "The latter [the additional $\mathrm{U}(1)$] in principle corresponds to an additional gauge boson, but a theory of the strong interactions containing such a particle is inconsistent with experiment. " What exactly is unclear about that? – ACuriousMind Nov 18 '21 at 19:34
  • @MatterGauge or even something else, it is the consistency with data that picks the group structure. – anna v Nov 18 '21 at 19:41
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    @ACuriousMind The second answer confuses me. Not the accepted one. – MatterGauge Nov 18 '21 at 19:52

1 Answers1

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In short, no.

One of the answers to the other question does say that the strong force gauge group can't have an extra factor of $U(1)$ because "total phase rotations of the quark wave function are already part of the model", referring to the $U(1)$ factor in the Standard Model gauge group.

The $U(1)$ factor in that answer isn't electromagnetic $U(1)$, which is a different subgroup of $SU(2)\times U(1)$.

More importantly, though, the answer is just wrong. Identifying a $U(1)$ factor in the gauge group with complex phase is just a convention. You can add additional $U(1)$ forces; they aren't in the Standard Model simply because they aren't observed (as the other, better answer said).

benrg
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  • That's what I mean! Thanks! – MatterGauge Nov 18 '21 at 19:55
  • One more thing. When you say U(1) doesn't correspond to EM, do you mean it corresponds to the electroweak interaction? – MatterGauge Nov 18 '21 at 20:17
  • @MatterGauge I'm not sure I understand your question, but the $U(1)$ in the other question is the $U(1)$ factor in $SU(2)\times U(1)$, and $SU(2)\times U(1)$ is the electroweak gauge group. If that $U(1)$ was electromagnetism, then $SU(2)$ would be a separate force, and in particular there would be no electrically charged W bosons, which isn't what's observed. You could look at, e.g., this question. – benrg Nov 18 '21 at 21:42
  • Yes. I meant if U(1) is the factòr of the U(2)×U(1) weak interaction (or better, U(2)L×U(1)Y). – MatterGauge Nov 19 '21 at 02:12